%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP141^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:53 AM UTC 2026
% Result : Theorem 17.81s 3.97s
% Output : Refutation 17.81s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP141^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n005.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:27:29 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.48 ---- Original DTF file ---
% 0.30/0.48 thf(spec,logic,$$dhol).
% 0.30/0.48 %------------------------------------------------------------------------------
% 0.30/0.48 % File : ITP141^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.48 % Domain : Interactive Theorem Proving
% 0.30/0.48 % Problem : Sledgehammer PHoareTotal problem prob_135__3259094_1
% 0.30/0.48 % Version : Especial.
% 0.30/0.48 % English :
% 0.30/0.48
% 0.30/0.48 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.48 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.48 % Source : [Des21]
% 0.30/0.48 % Names : PHoareTotal/prob_135__3259094_1 [Des21]
% 0.30/0.48
% 0.30/0.48 % Status : Theorem
% 0.30/0.48 % Rating : 0.67 v9.1.0, 0.33 v9.0.0, 0.67 v8.2.0, 0.33 v8.1.0, 0.50 v7.5.0
% 0.30/0.48 % Syntax : Number of formulae : 343 ( 160 unt; 76 typ; 0 def)
% 0.30/0.48 % Number of atoms : 633 ( 253 equ; 0 cnn)
% 0.30/0.48 % Maximal formula atoms : 21 ( 2 avg)
% 0.30/0.48 % Number of connectives : 4580 ( 104 ~; 14 |; 72 &;4109 @)
% 0.30/0.48 % ( 0 <=>; 281 =>; 0 <=; 0 <~>)
% 0.30/0.48 % Maximal formula depth : 28 ( 8 avg)
% 0.30/0.48 % Number of types : 6 ( 5 usr)
% 0.30/0.48 % Number of type conns : 655 ( 655 >; 0 *; 0 +; 0 <<)
% 0.30/0.48 % Number of symbols : 74 ( 71 usr; 8 con; 0-6 aty)
% 0.30/0.48 % Number of variables : 1348 ( 173 ^;1074 !; 28 ?;1348 :)
% 0.30/0.48 % ( 73 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.48 % SPC : TH1_THM_EQU_NAR
% 0.30/0.48
% 0.30/0.48 % Comments : This file was generated by Sledgehammer 2021-02-23 16:17:57.953
% 0.30/0.48 %------------------------------------------------------------------------------
% 0.30/0.48 % Could-be-implicit typings (7)
% 0.30/0.48 thf(ty_t_Product__Type_Ounit,type,
% 0.30/0.48 product_unit: $tType ).
% 0.30/0.48
% 0.30/0.48 thf(ty_t_Product__Type_Oprod,type,
% 0.30/0.48 product_prod: $tType > $tType > $tType ).
% 0.30/0.48
% 0.30/0.48 thf(ty_t_PLang_Ostate,type,
% 0.30/0.48 state: $tType ).
% 0.30/0.48
% 0.30/0.48 thf(ty_t_PLang_Ocom,type,
% 0.30/0.48 com: $tType ).
% 0.30/0.48
% 0.30/0.48 thf(ty_t_Set_Oset,type,
% 0.30/0.48 set: $tType > $tType ).
% 0.30/0.48
% 0.30/0.48 thf(ty_t_Nat_Onat,type,
% 0.30/0.48 nat: $tType ).
% 0.30/0.48
% 0.30/0.48 thf(ty_tf_a,type,
% 0.30/0.48 a: $tType ).
% 0.30/0.48
% 0.30/0.48 % Explicit typings (69)
% 0.30/0.48 thf(sy_cl_HOL_Otype,type,
% 0.30/0.48 type:
% 0.30/0.48 !>[A: $tType] : $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Obot,type,
% 0.30/0.48 bot:
% 0.30/0.48 !>[A: $tType] : $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Basic__BNFs_Ofsts,type,
% 0.30/0.48 basic_fsts:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Basic__BNFs_Osnds,type,
% 0.30/0.48 basic_snds:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 0.30/0.48 fun_reduction_pair:
% 0.30/0.48 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 0.30/0.48 fun_rp_inv_image:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_HOL_OThe,type,
% 0.30/0.48 the:
% 0.30/0.48 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_If,type,
% 0.30/0.48 if:
% 0.30/0.48 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_OSuc,type,
% 0.30/0.48 suc: nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Order__Relation_Oabove,type,
% 0.30/0.48 order_above:
% 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Order__Relation_OaboveS,type,
% 0.30/0.48 order_aboveS:
% 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 0.30/0.48 order_1409979114der_on:
% 0.30/0.48 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Order__Relation_Opreorder__on,type,
% 0.30/0.48 order_preorder_on:
% 0.30/0.48 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.48 bot_bot:
% 0.30/0.48 !>[A: $tType] : A ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.48 ord_less:
% 0.30/0.48 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Octvalid,type,
% 0.30/0.48 pHoare304044376tvalid:
% 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Othoare,type,
% 0.30/0.48 pHoare166426476thoare:
% 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Otvalid,type,
% 0.30/0.48 pHoare987424603tvalid:
% 0.30/0.48 !>[A: $tType] : ( ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoare_Ocnvalid,type,
% 0.30/0.48 cnvalid:
% 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > nat > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoare_Ocvalid,type,
% 0.30/0.48 cvalid:
% 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoare_Ohoare,type,
% 0.30/0.48 hoare:
% 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoare_Onvalid,type,
% 0.30/0.48 nvalid:
% 0.30/0.48 !>[A: $tType] : ( nat > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoare_Onvalids,type,
% 0.30/0.48 nvalids:
% 0.30/0.48 !>[A: $tType] : ( nat > ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoare_Ovalid,type,
% 0.30/0.48 valid:
% 0.30/0.48 !>[A: $tType] : ( ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PHoare_Ovalids,type,
% 0.30/0.48 valids:
% 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PLang_Obody,type,
% 0.30/0.48 body: com ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PLang_Ocom_OCALL,type,
% 0.30/0.48 call: com ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PLang_Ocom_OSemi,type,
% 0.30/0.48 semi: com > com > com ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PLang_Ocom_OWhile,type,
% 0.30/0.48 while: ( state > $o ) > com > com ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PLang_Oexec,type,
% 0.30/0.48 exec: state > com > state > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PLang_Oexecn,type,
% 0.30/0.48 execn: state > com > nat > state > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_PTermi_Otermi,type,
% 0.30/0.48 termi: com > state > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Product__Type_OPair,type,
% 0.30/0.48 product_Pair:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.30/0.48 produc2004651681e_prod:
% 0.30/0.48 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.30/0.48 product_rec_prod:
% 0.30/0.48 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 0.30/0.48 product_rec_set_prod:
% 0.30/0.48 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 0.30/0.48 product_rec_set_unit:
% 0.30/0.48 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 0.30/0.48 product_rec_unit:
% 0.30/0.48 !>[T: $tType] : ( T > product_unit > T ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.30/0.48 product_case_prod:
% 0.30/0.48 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Relation_ODomain,type,
% 0.30/0.48 domain:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Relation_OId__on,type,
% 0.30/0.48 id_on:
% 0.30/0.48 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Relation_OPowp,type,
% 0.30/0.48 powp:
% 0.30/0.48 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Relation_ORange,type,
% 0.30/0.48 range:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Relation_ORangep,type,
% 0.30/0.48 rangep:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Relation_Oinv__image,type,
% 0.30/0.48 inv_image:
% 0.30/0.48 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Relation_Oirrefl,type,
% 0.30/0.48 irrefl:
% 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Relation_Oirreflp,type,
% 0.30/0.49 irreflp:
% 0.30/0.49 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Relation_Orefl__on,type,
% 0.30/0.49 refl_on:
% 0.30/0.49 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Relation_Ototal__on,type,
% 0.30/0.49 total_on:
% 0.30/0.49 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Relation_Otrans,type,
% 0.30/0.49 trans:
% 0.30/0.49 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Set_OBall,type,
% 0.30/0.49 ball:
% 0.30/0.49 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Set_OCollect,type,
% 0.30/0.49 collect:
% 0.30/0.49 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Set_Oinsert,type,
% 0.30/0.49 insert:
% 0.30/0.49 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Set_Ois__empty,type,
% 0.30/0.49 is_empty:
% 0.30/0.49 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Set_Ois__singleton,type,
% 0.30/0.49 is_singleton:
% 0.30/0.49 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Set_Othe__elem,type,
% 0.30/0.49 the_elem:
% 0.30/0.49 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Wellfounded_Oless__than,type,
% 0.30/0.49 less_than: set @ ( product_prod @ nat @ nat ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.30/0.49 lex_prod:
% 0.30/0.49 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Wellfounded_Omeasure,type,
% 0.30/0.49 measure:
% 0.30/0.49 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Wellfounded_Omlex__prod,type,
% 0.30/0.49 mlex_prod:
% 0.30/0.49 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Wellfounded_Opred__nat,type,
% 0.30/0.49 pred_nat: set @ ( product_prod @ nat @ nat ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Wellfounded_Owf,type,
% 0.30/0.49 wf:
% 0.30/0.49 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Wfrec_Osame__fst,type,
% 0.30/0.49 same_fst:
% 0.30/0.49 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_Zorn_OChains,type,
% 0.30/0.49 chains:
% 0.30/0.49 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_c_member,type,
% 0.30/0.49 member:
% 0.30/0.49 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_v_Pa,type,
% 0.30/0.49 pa: a > state > $o ).
% 0.30/0.49
% 0.30/0.49 thf(sy_v_Qa,type,
% 0.30/0.49 qa: a > state > $o ).
% 0.30/0.49
% 0.30/0.49 thf(sy_v_r,type,
% 0.30/0.49 r: set @ ( product_prod @ state @ state ) ).
% 0.30/0.49
% 0.30/0.49 thf(sy_v_x,type,
% 0.30/0.49 x: state ).
% 0.30/0.49
% 0.30/0.49 % Relevant facts (256)
% 0.30/0.49 thf(fact_0_thoare_OAsm,axiom,
% 0.30/0.49 ! [A: $tType,P: A > state > $o,Q: A > state > $o] : ( pHoare166426476thoare @ A @ ( insert @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ ( product_Pair @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ P @ ( product_Pair @ com @ ( A > state > $o ) @ call @ Q ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) ) @ P @ call @ Q ) ).
% 0.30/0.49
% 0.30/0.49 % thoare.Asm
% 0.30/0.49 thf(fact_1_weaken__post,axiom,
% 0.30/0.49 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,C3: com,Q: A > state > $o,Q2: A > state > $o] :
% 0.30/0.49 ( ( pHoare166426476thoare @ A @ C2 @ P @ C3 @ Q )
% 0.30/0.49 => ( ! [Z: A,S: state] :
% 0.30/0.49 ( ( Q @ Z @ S )
% 0.30/0.49 => ( Q2 @ Z @ S ) )
% 0.30/0.49 => ( pHoare166426476thoare @ A @ C2 @ P @ C3 @ Q2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % weaken_post
% 0.30/0.49 thf(fact_2_thoare_OConseq,axiom,
% 0.30/0.49 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P2: A > state > $o,C3: com,Q2: A > state > $o,P: A > state > $o,Q: A > state > $o] :
% 0.30/0.49 ( ( pHoare166426476thoare @ A @ C2 @ P2 @ C3 @ Q2 )
% 0.30/0.49 => ( ( ! [S: state,T2: state] :
% 0.30/0.49 ( ! [Z2: A] :
% 0.30/0.49 ( ( P2 @ Z2 @ S )
% 0.30/0.49 => ( Q2 @ Z2 @ T2 ) )
% 0.30/0.49 => ! [Z: A] :
% 0.30/0.49 ( ( P @ Z @ S )
% 0.30/0.49 => ( Q @ Z @ T2 ) ) )
% 0.30/0.49 & ! [S: state] :
% 0.30/0.49 ( ? [Z: A] : ( P @ Z @ S )
% 0.30/0.49 => ? [Z2: A] : ( P2 @ Z2 @ S ) ) )
% 0.30/0.49 => ( pHoare166426476thoare @ A @ C2 @ P @ C3 @ Q ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % thoare.Conseq
% 0.30/0.49 thf(fact_3_PHoareTotal__Mirabelle__nfzgdgthbw_Ostrengthen__pre,axiom,
% 0.30/0.49 ! [A: $tType,P2: A > state > $o,P: A > state > $o,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),C3: com,Q: A > state > $o] :
% 0.30/0.49 ( ! [Z: A,S: state] :
% 0.30/0.49 ( ( P2 @ Z @ S )
% 0.30/0.49 => ( P @ Z @ S ) )
% 0.30/0.49 => ( ( pHoare166426476thoare @ A @ C2 @ P @ C3 @ Q )
% 0.30/0.49 => ( pHoare166426476thoare @ A @ C2 @ P2 @ C3 @ Q ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % PHoareTotal_Mirabelle_nfzgdgthbw.strengthen_pre
% 0.30/0.49 thf(fact_4_singleton__conv,axiom,
% 0.30/0.49 ! [A: $tType,A2: A] :
% 0.30/0.49 ( ( collect @ A
% 0.30/0.49 @ ^ [X: A] : ( X = A2 ) )
% 0.30/0.49 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % singleton_conv
% 0.30/0.49 thf(fact_5_singleton__conv2,axiom,
% 0.30/0.49 ! [A: $tType,A2: A] :
% 0.30/0.49 ( ( collect @ A
% 0.30/0.49 @ ( ^ [Y: A,Z3: A] : ( Y = Z3 )
% 0.30/0.49 @ A2 ) )
% 0.30/0.49 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % singleton_conv2
% 0.30/0.49 thf(fact_6_singletonI,axiom,
% 0.30/0.49 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % singletonI
% 0.30/0.49 thf(fact_7_termi_Ointros_I7_J,axiom,
% 0.30/0.49 ! [S2: state] :
% 0.30/0.49 ( ( termi @ body @ S2 )
% 0.30/0.49 => ( termi @ call @ S2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % termi.intros(7)
% 0.30/0.49 thf(fact_8_exec_Ointros_I7_J,axiom,
% 0.30/0.49 ! [S2: state,T3: state] :
% 0.30/0.49 ( ( exec @ S2 @ body @ T3 )
% 0.30/0.49 => ( exec @ S2 @ call @ T3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % exec.intros(7)
% 0.30/0.49 thf(fact_9_thoare_OCall,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ state @ state ),P: A > state > $o,Q: A > state > $o] :
% 0.30/0.49 ( ( wf @ state @ R )
% 0.30/0.49 => ( ! [S3: state] :
% 0.30/0.49 ( pHoare166426476thoare @ A
% 0.30/0.49 @ ( insert @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) )
% 0.30/0.49 @ ( product_Pair @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) )
% 0.30/0.49 @ ^ [Z4: A,S4: state] :
% 0.30/0.49 ( ( P @ Z4 @ S4 )
% 0.30/0.49 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S4 @ S3 ) @ R ) )
% 0.30/0.49 @ ( product_Pair @ com @ ( A > state > $o ) @ call @ Q ) )
% 0.30/0.49 @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) )
% 0.30/0.49 @ ^ [Z4: A,S4: state] :
% 0.30/0.49 ( ( P @ Z4 @ S4 )
% 0.30/0.49 & ( S4 = S3 ) )
% 0.30/0.49 @ body
% 0.30/0.49 @ Q )
% 0.30/0.49 => ( pHoare166426476thoare @ A @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) @ P @ call @ Q ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % thoare.Call
% 0.30/0.49 thf(fact_10_insertCI,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,B2: set @ A,B3: A] :
% 0.30/0.49 ( ( ~ ( member @ A @ A2 @ B2 )
% 0.30/0.49 => ( A2 = B3 ) )
% 0.30/0.49 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insertCI
% 0.30/0.49 thf(fact_11_insert__iff,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 0.30/0.49 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
% 0.30/0.49 = ( ( A2 = B3 )
% 0.30/0.49 | ( member @ A @ A2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_iff
% 0.30/0.49 thf(fact_12_insert__absorb2,axiom,
% 0.30/0.49 ! [A: $tType,X2: A,A3: set @ A] :
% 0.30/0.49 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A3 ) )
% 0.30/0.49 = ( insert @ A @ X2 @ A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_absorb2
% 0.30/0.49 thf(fact_13_hoare_OCall,axiom,
% 0.30/0.49 ! [A: $tType,P: A > state > $o,Q: A > state > $o] :
% 0.30/0.49 ( ( hoare @ A @ ( insert @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ ( product_Pair @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ P @ ( product_Pair @ com @ ( A > state > $o ) @ call @ Q ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) ) @ P @ body @ Q )
% 0.30/0.49 => ( hoare @ A @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) @ P @ call @ Q ) ) ).
% 0.30/0.49
% 0.30/0.49 % hoare.Call
% 0.30/0.49 thf(fact_14_empty__iff,axiom,
% 0.30/0.49 ! [A: $tType,C3: A] :
% 0.30/0.49 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % empty_iff
% 0.30/0.49 thf(fact_15_all__not__in__conv,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] :
% 0.30/0.49 ( ( ! [X: A] :
% 0.30/0.49 ~ ( member @ A @ X @ A3 ) )
% 0.30/0.49 = ( A3
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % all_not_in_conv
% 0.30/0.49 thf(fact_16_empty__Collect__eq,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o] :
% 0.30/0.49 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.49 = ( collect @ A @ P ) )
% 0.30/0.49 = ( ! [X: A] :
% 0.30/0.49 ~ ( P @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % empty_Collect_eq
% 0.30/0.49 thf(fact_17_Collect__empty__eq,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o] :
% 0.30/0.49 ( ( ( collect @ A @ P )
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 = ( ! [X: A] :
% 0.30/0.49 ~ ( P @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Collect_empty_eq
% 0.30/0.49 thf(fact_18_hoare_OConseq,axiom,
% 0.30/0.49 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P2: A > state > $o,C3: com,Q2: A > state > $o,P: A > state > $o,Q: A > state > $o] :
% 0.30/0.49 ( ( hoare @ A @ C2 @ P2 @ C3 @ Q2 )
% 0.30/0.49 => ( ! [S: state,T2: state] :
% 0.30/0.49 ( ! [Z2: A] :
% 0.30/0.49 ( ( P2 @ Z2 @ S )
% 0.30/0.49 => ( Q2 @ Z2 @ T2 ) )
% 0.30/0.49 => ! [Z: A] :
% 0.30/0.49 ( ( P @ Z @ S )
% 0.30/0.49 => ( Q @ Z @ T2 ) ) )
% 0.30/0.49 => ( hoare @ A @ C2 @ P @ C3 @ Q ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % hoare.Conseq
% 0.30/0.49 thf(fact_19_PHoare_Ostrengthen__pre,axiom,
% 0.30/0.49 ! [A: $tType,P2: A > state > $o,P: A > state > $o,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),C3: com,Q: A > state > $o] :
% 0.30/0.49 ( ! [Z: A,S: state] :
% 0.30/0.49 ( ( P2 @ Z @ S )
% 0.30/0.49 => ( P @ Z @ S ) )
% 0.30/0.49 => ( ( hoare @ A @ C2 @ P @ C3 @ Q )
% 0.30/0.49 => ( hoare @ A @ C2 @ P2 @ C3 @ Q ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % PHoare.strengthen_pre
% 0.30/0.49 thf(fact_20_ex__in__conv,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] :
% 0.30/0.49 ( ( ? [X: A] : ( member @ A @ X @ A3 ) )
% 0.30/0.49 = ( A3
% 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ex_in_conv
% 0.30/0.49 thf(fact_21_equals0I,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] :
% 0.30/0.49 ( ! [Y2: A] :
% 0.30/0.49 ~ ( member @ A @ Y2 @ A3 )
% 0.30/0.49 => ( A3
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % equals0I
% 0.30/0.49 thf(fact_22_equals0D,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A,A2: A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % equals0D
% 0.30/0.49 thf(fact_23_emptyE,axiom,
% 0.30/0.49 ! [A: $tType,A2: A] :
% 0.30/0.49 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % emptyE
% 0.30/0.49 thf(fact_24_mk__disjoint__insert,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,A3: set @ A] :
% 0.30/0.49 ( ( member @ A @ A2 @ A3 )
% 0.30/0.49 => ? [B4: set @ A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 = ( insert @ A @ A2 @ B4 ) )
% 0.30/0.49 & ~ ( member @ A @ A2 @ B4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mk_disjoint_insert
% 0.30/0.49 thf(fact_25_insert__commute,axiom,
% 0.30/0.49 ! [A: $tType,X2: A,Y3: A,A3: set @ A] :
% 0.30/0.49 ( ( insert @ A @ X2 @ ( insert @ A @ Y3 @ A3 ) )
% 0.30/0.49 = ( insert @ A @ Y3 @ ( insert @ A @ X2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_commute
% 0.30/0.49 thf(fact_26_insert__eq__iff,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,A3: set @ A,B3: A,B2: set @ A] :
% 0.30/0.49 ( ~ ( member @ A @ A2 @ A3 )
% 0.30/0.49 => ( ~ ( member @ A @ B3 @ B2 )
% 0.30/0.49 => ( ( ( insert @ A @ A2 @ A3 )
% 0.30/0.49 = ( insert @ A @ B3 @ B2 ) )
% 0.30/0.49 = ( ( ( A2 = B3 )
% 0.30/0.49 => ( A3 = B2 ) )
% 0.30/0.49 & ( ( A2 != B3 )
% 0.30/0.49 => ? [C4: set @ A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 = ( insert @ A @ B3 @ C4 ) )
% 0.30/0.49 & ~ ( member @ A @ B3 @ C4 )
% 0.30/0.49 & ( B2
% 0.30/0.49 = ( insert @ A @ A2 @ C4 ) )
% 0.30/0.49 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_eq_iff
% 0.30/0.49 thf(fact_27_insert__absorb,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,A3: set @ A] :
% 0.30/0.49 ( ( member @ A @ A2 @ A3 )
% 0.30/0.49 => ( ( insert @ A @ A2 @ A3 )
% 0.30/0.49 = A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_absorb
% 0.30/0.49 thf(fact_28_insert__ident,axiom,
% 0.30/0.49 ! [A: $tType,X2: A,A3: set @ A,B2: set @ A] :
% 0.30/0.49 ( ~ ( member @ A @ X2 @ A3 )
% 0.30/0.49 => ( ~ ( member @ A @ X2 @ B2 )
% 0.30/0.49 => ( ( ( insert @ A @ X2 @ A3 )
% 0.30/0.49 = ( insert @ A @ X2 @ B2 ) )
% 0.30/0.49 = ( A3 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_ident
% 0.30/0.49 thf(fact_29_Set_Oset__insert,axiom,
% 0.30/0.49 ! [A: $tType,X2: A,A3: set @ A] :
% 0.30/0.49 ( ( member @ A @ X2 @ A3 )
% 0.30/0.49 => ~ ! [B4: set @ A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 = ( insert @ A @ X2 @ B4 ) )
% 0.30/0.49 => ( member @ A @ X2 @ B4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Set.set_insert
% 0.30/0.49 thf(fact_30_insertI2,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,B2: set @ A,B3: A] :
% 0.30/0.49 ( ( member @ A @ A2 @ B2 )
% 0.30/0.49 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insertI2
% 0.30/0.49 thf(fact_31_insertI1,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,B2: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % insertI1
% 0.30/0.49 thf(fact_32_insertE,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 0.30/0.49 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
% 0.30/0.49 => ( ( A2 != B3 )
% 0.30/0.49 => ( member @ A @ A2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insertE
% 0.30/0.49 thf(fact_33_empty__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( bot_bot @ ( set @ A ) )
% 0.30/0.49 = ( collect @ A
% 0.30/0.49 @ ^ [X: A] : $false ) ) ).
% 0.30/0.49
% 0.30/0.49 % empty_def
% 0.30/0.49 thf(fact_34_insert__Collect,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,P: A > $o] :
% 0.30/0.49 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 0.30/0.49 = ( collect @ A
% 0.30/0.49 @ ^ [U: A] :
% 0.30/0.49 ( ( U != A2 )
% 0.30/0.49 => ( P @ U ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_Collect
% 0.30/0.49 thf(fact_35_insert__compr,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( insert @ A )
% 0.30/0.49 = ( ^ [A4: A,B5: set @ A] :
% 0.30/0.49 ( collect @ A
% 0.30/0.49 @ ^ [X: A] :
% 0.30/0.49 ( ( X = A4 )
% 0.30/0.49 | ( member @ A @ X @ B5 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_compr
% 0.30/0.49 thf(fact_36_singleton__inject,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,B3: A] :
% 0.30/0.49 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.49 => ( A2 = B3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % singleton_inject
% 0.30/0.49 thf(fact_37_insert__not__empty,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,A3: set @ A] :
% 0.30/0.49 ( ( insert @ A @ A2 @ A3 )
% 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_not_empty
% 0.30/0.49 thf(fact_38_doubleton__eq__iff,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,B3: A,C3: A,D: A] :
% 0.30/0.49 ( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.49 = ( insert @ A @ C3 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.49 = ( ( ( A2 = C3 )
% 0.30/0.49 & ( B3 = D ) )
% 0.30/0.49 | ( ( A2 = D )
% 0.30/0.49 & ( B3 = C3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % doubleton_eq_iff
% 0.30/0.49 thf(fact_39_singleton__iff,axiom,
% 0.30/0.49 ! [A: $tType,B3: A,A2: A] :
% 0.30/0.49 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.49 = ( B3 = A2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % singleton_iff
% 0.30/0.49 thf(fact_40_singletonD,axiom,
% 0.30/0.49 ! [A: $tType,B3: A,A2: A] :
% 0.30/0.49 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.49 => ( B3 = A2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % singletonD
% 0.30/0.49 thf(fact_41_hoare_OAsm,axiom,
% 0.30/0.49 ! [A: $tType,P: A > state > $o,Q: A > state > $o] : ( hoare @ A @ ( insert @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ ( product_Pair @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ P @ ( product_Pair @ com @ ( A > state > $o ) @ call @ Q ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) ) @ P @ call @ Q ) ).
% 0.30/0.49
% 0.30/0.49 % hoare.Asm
% 0.30/0.49 thf(fact_42_Collect__conv__if2,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,A2: A] :
% 0.30/0.49 ( ( ( P @ A2 )
% 0.30/0.49 => ( ( collect @ A
% 0.30/0.49 @ ^ [X: A] :
% 0.30/0.49 ( ( A2 = X )
% 0.30/0.49 & ( P @ X ) ) )
% 0.30/0.49 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.49 & ( ~ ( P @ A2 )
% 0.30/0.49 => ( ( collect @ A
% 0.30/0.49 @ ^ [X: A] :
% 0.30/0.49 ( ( A2 = X )
% 0.30/0.49 & ( P @ X ) ) )
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Collect_conv_if2
% 0.30/0.49 thf(fact_43_Collect__conv__if,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,A2: A] :
% 0.30/0.49 ( ( ( P @ A2 )
% 0.30/0.49 => ( ( collect @ A
% 0.30/0.49 @ ^ [X: A] :
% 0.30/0.49 ( ( X = A2 )
% 0.30/0.49 & ( P @ X ) ) )
% 0.30/0.49 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.49 & ( ~ ( P @ A2 )
% 0.30/0.49 => ( ( collect @ A
% 0.30/0.49 @ ^ [X: A] :
% 0.30/0.49 ( ( X = A2 )
% 0.30/0.49 & ( P @ X ) ) )
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Collect_conv_if
% 0.30/0.49 thf(fact_44_wf__empty,axiom,
% 0.30/0.49 ! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_empty
% 0.30/0.49 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,P: A > $o] :
% 0.30/0.49 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.30/0.49 = ( P @ A2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % mem_Collect_eq
% 0.30/0.49 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] :
% 0.30/0.49 ( ( collect @ A
% 0.30/0.49 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 0.30/0.49 = A3 ) ).
% 0.30/0.49
% 0.30/0.49 % Collect_mem_eq
% 0.30/0.49 thf(fact_47_Collect__cong,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.49 ( ! [X3: A] :
% 0.30/0.49 ( ( P @ X3 )
% 0.30/0.49 = ( Q @ X3 ) )
% 0.30/0.49 => ( ( collect @ A @ P )
% 0.30/0.49 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Collect_cong
% 0.30/0.49 thf(fact_48_ext,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.49 ( ! [X3: A] :
% 0.30/0.49 ( ( F @ X3 )
% 0.30/0.49 = ( G @ X3 ) )
% 0.30/0.49 => ( F = G ) ) ).
% 0.30/0.49
% 0.30/0.49 % ext
% 0.30/0.49 thf(fact_49_wfE__min_H,axiom,
% 0.30/0.49 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),Q: set @ A] :
% 0.30/0.49 ( ( wf @ A @ R2 )
% 0.30/0.49 => ( ( Q
% 0.30/0.49 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 => ~ ! [Z: A] :
% 0.30/0.49 ( ( member @ A @ Z @ Q )
% 0.30/0.49 => ~ ! [Y4: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 0.30/0.49 => ~ ( member @ A @ Y4 @ Q ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wfE_min'
% 0.30/0.49 thf(fact_50_the__elem__eq,axiom,
% 0.30/0.49 ! [A: $tType,X2: A] :
% 0.30/0.49 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.49 = X2 ) ).
% 0.30/0.49
% 0.30/0.49 % the_elem_eq
% 0.30/0.49 thf(fact_51_bot__apply,axiom,
% 0.30/0.49 ! [C: $tType,D2: $tType] :
% 0.30/0.49 ( ( bot @ C )
% 0.30/0.49 => ( ( bot_bot @ ( D2 > C ) )
% 0.30/0.49 = ( ^ [X: D2] : ( bot_bot @ C ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % bot_apply
% 0.30/0.49 thf(fact_52_old_Oprod_Oinject,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,A2: A,B3: B,A5: A,B6: B] :
% 0.30/0.49 ( ( ( product_Pair @ A @ B @ A2 @ B3 )
% 0.30/0.49 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.30/0.49 = ( ( A2 = A5 )
% 0.30/0.49 & ( B3 = B6 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % old.prod.inject
% 0.30/0.49 thf(fact_53_prod_Oinject,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.30/0.49 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.30/0.49 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.30/0.49 = ( ( X1 = Y1 )
% 0.30/0.49 & ( X22 = Y22 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod.inject
% 0.30/0.49 thf(fact_54_is__singletonI,axiom,
% 0.30/0.49 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_singletonI
% 0.30/0.49 thf(fact_55_prod__induct7,axiom,
% 0.30/0.49 ! [G2: $tType,F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.30/0.49 ( ! [A6: A,B7: B,C5: C,D3: D2,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C5 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 0.30/0.49 => ( P @ X2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_induct7
% 0.30/0.49 thf(fact_56_prod__induct6,axiom,
% 0.30/0.49 ! [F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.30/0.49 ( ! [A6: A,B7: B,C5: C,D3: D2,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C5 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 0.30/0.49 => ( P @ X2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_induct6
% 0.30/0.49 thf(fact_57_prod__induct5,axiom,
% 0.30/0.49 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.30/0.49 ( ! [A6: A,B7: B,C5: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C5 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 0.30/0.49 => ( P @ X2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_induct5
% 0.30/0.49 thf(fact_58_bot__set__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( bot_bot @ ( set @ A ) )
% 0.30/0.49 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % bot_set_def
% 0.30/0.49 thf(fact_59_is__singleton__the__elem,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( is_singleton @ A )
% 0.30/0.49 = ( ^ [A7: set @ A] :
% 0.30/0.49 ( A7
% 0.30/0.49 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_singleton_the_elem
% 0.30/0.49 thf(fact_60_is__singletonI_H,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] :
% 0.30/0.49 ( ( A3
% 0.30/0.49 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 => ( ! [X3: A,Y2: A] :
% 0.30/0.49 ( ( member @ A @ X3 @ A3 )
% 0.30/0.49 => ( ( member @ A @ Y2 @ A3 )
% 0.30/0.49 => ( X3 = Y2 ) ) )
% 0.30/0.49 => ( is_singleton @ A @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_singletonI'
% 0.30/0.49 thf(fact_61_surj__pair,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
% 0.30/0.49 ? [X3: A,Y2: B] :
% 0.30/0.49 ( P3
% 0.30/0.49 = ( product_Pair @ A @ B @ X3 @ Y2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % surj_pair
% 0.30/0.49 thf(fact_62_prod__cases,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
% 0.30/0.49 ( ! [A6: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B7 ) )
% 0.30/0.49 => ( P @ P3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_cases
% 0.30/0.49 thf(fact_63_Pair__inject,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,A2: A,B3: B,A5: A,B6: B] :
% 0.30/0.49 ( ( ( product_Pair @ A @ B @ A2 @ B3 )
% 0.30/0.49 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 0.30/0.49 => ~ ( ( A2 = A5 )
% 0.30/0.49 => ( B3 != B6 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Pair_inject
% 0.30/0.49 thf(fact_64_old_Oprod_Oexhaust,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.30/0.49 ~ ! [A6: A,B7: B] :
% 0.30/0.49 ( Y3
% 0.30/0.49 != ( product_Pair @ A @ B @ A6 @ B7 ) ) ).
% 0.30/0.49
% 0.30/0.49 % old.prod.exhaust
% 0.30/0.49 thf(fact_65_old_Oprod_Oinducts,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/0.49 ( ! [A6: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B7 ) )
% 0.30/0.49 => ( P @ Prod ) ) ).
% 0.30/0.49
% 0.30/0.49 % old.prod.inducts
% 0.30/0.49 thf(fact_66_bot__fun__def,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( bot @ B )
% 0.30/0.49 => ( ( bot_bot @ ( A > B ) )
% 0.30/0.49 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % bot_fun_def
% 0.30/0.49 thf(fact_67_is__singletonE,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] :
% 0.30/0.49 ( ( is_singleton @ A @ A3 )
% 0.30/0.49 => ~ ! [X3: A] :
% 0.30/0.49 ( A3
% 0.30/0.49 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_singletonE
% 0.30/0.49 thf(fact_68_is__singleton__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( is_singleton @ A )
% 0.30/0.49 = ( ^ [A7: set @ A] :
% 0.30/0.49 ? [X: A] :
% 0.30/0.49 ( A7
% 0.30/0.49 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_singleton_def
% 0.30/0.49 thf(fact_69_wf__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( wf @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ! [P4: A > $o] :
% 0.30/0.49 ( ! [X: A] :
% 0.30/0.49 ( ! [Y5: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R3 )
% 0.30/0.49 => ( P4 @ Y5 ) )
% 0.30/0.49 => ( P4 @ X ) )
% 0.30/0.49 => ! [X4: A] : ( P4 @ X4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_def
% 0.30/0.49 thf(fact_70_wfE__min,axiom,
% 0.30/0.49 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X2: A,Q: set @ A] :
% 0.30/0.49 ( ( wf @ A @ R2 )
% 0.30/0.49 => ( ( member @ A @ X2 @ Q )
% 0.30/0.49 => ~ ! [Z: A] :
% 0.30/0.49 ( ( member @ A @ Z @ Q )
% 0.30/0.49 => ~ ! [Y4: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 0.30/0.49 => ~ ( member @ A @ Y4 @ Q ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wfE_min
% 0.30/0.49 thf(fact_71_wfI__min,axiom,
% 0.30/0.49 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ( ! [X3: A,Q3: set @ A] :
% 0.30/0.49 ( ( member @ A @ X3 @ Q3 )
% 0.30/0.49 => ? [Xa: A] :
% 0.30/0.49 ( ( member @ A @ Xa @ Q3 )
% 0.30/0.49 & ! [Y2: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa ) @ R2 )
% 0.30/0.49 => ~ ( member @ A @ Y2 @ Q3 ) ) ) )
% 0.30/0.49 => ( wf @ A @ R2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % wfI_min
% 0.30/0.49 thf(fact_72_wfUNIVI,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ( ! [P5: A > $o,X3: A] :
% 0.30/0.49 ( ! [Xa: A] :
% 0.30/0.49 ( ! [Y2: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa ) @ R )
% 0.30/0.49 => ( P5 @ Y2 ) )
% 0.30/0.49 => ( P5 @ Xa ) )
% 0.30/0.49 => ( P5 @ X3 ) )
% 0.30/0.49 => ( wf @ A @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % wfUNIVI
% 0.30/0.49 thf(fact_73_wf__asym,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X2: A] :
% 0.30/0.49 ( ( wf @ A @ R )
% 0.30/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X2 ) @ R )
% 0.30/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A2 ) @ R ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_asym
% 0.30/0.49 thf(fact_74_wf__induct,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A2: A] :
% 0.30/0.49 ( ( wf @ A @ R )
% 0.30/0.49 => ( ! [X3: A] :
% 0.30/0.49 ( ! [Y4: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R )
% 0.30/0.49 => ( P @ Y4 ) )
% 0.30/0.49 => ( P @ X3 ) )
% 0.30/0.49 => ( P @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_induct
% 0.30/0.49 thf(fact_75_wf__irrefl,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.30/0.49 ( ( wf @ A @ R )
% 0.30/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_irrefl
% 0.30/0.49 thf(fact_76_wf__not__sym,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X2: A] :
% 0.30/0.49 ( ( wf @ A @ R )
% 0.30/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X2 ) @ R )
% 0.30/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A2 ) @ R ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_not_sym
% 0.30/0.49 thf(fact_77_wf__not__refl,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.30/0.49 ( ( wf @ A @ R )
% 0.30/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_not_refl
% 0.30/0.49 thf(fact_78_wf__eq__minimal,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( wf @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ! [Q4: set @ A] :
% 0.30/0.49 ( ? [X: A] : ( member @ A @ X @ Q4 )
% 0.30/0.49 => ? [X: A] :
% 0.30/0.49 ( ( member @ A @ X @ Q4 )
% 0.30/0.49 & ! [Y5: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R3 )
% 0.30/0.49 => ~ ( member @ A @ Y5 @ Q4 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_eq_minimal
% 0.30/0.49 thf(fact_79_wf__induct__rule,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A2: A] :
% 0.30/0.49 ( ( wf @ A @ R )
% 0.30/0.49 => ( ! [X3: A] :
% 0.30/0.49 ( ! [Y4: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R )
% 0.30/0.49 => ( P @ Y4 ) )
% 0.30/0.49 => ( P @ X3 ) )
% 0.30/0.49 => ( P @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_induct_rule
% 0.30/0.49 thf(fact_80_prod__cases3,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.49 ~ ! [A6: A,B7: B,C5: C] :
% 0.30/0.49 ( Y3
% 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B7 @ C5 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_cases3
% 0.30/0.49 thf(fact_81_prod__cases4,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/0.49 ~ ! [A6: A,B7: B,C5: C,D3: D2] :
% 0.30/0.49 ( Y3
% 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C5 @ D3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_cases4
% 0.30/0.49 thf(fact_82_prod__cases5,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 0.30/0.49 ~ ! [A6: A,B7: B,C5: C,D3: D2,E2: E] :
% 0.30/0.49 ( Y3
% 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C5 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_cases5
% 0.30/0.49 thf(fact_83_prod__cases6,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.30/0.49 ~ ! [A6: A,B7: B,C5: C,D3: D2,E2: E,F3: F2] :
% 0.30/0.49 ( Y3
% 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C5 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_cases6
% 0.30/0.49 thf(fact_84_prod__cases7,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.30/0.49 ~ ! [A6: A,B7: B,C5: C,D3: D2,E2: E,F3: F2,G3: G2] :
% 0.30/0.49 ( Y3
% 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C5 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_cases7
% 0.30/0.49 thf(fact_85_prod__induct3,axiom,
% 0.30/0.49 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.49 ( ! [A6: A,B7: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B7 @ C5 ) ) )
% 0.30/0.49 => ( P @ X2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_induct3
% 0.30/0.49 thf(fact_86_prod__induct4,axiom,
% 0.30/0.49 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 0.30/0.49 ( ! [A6: A,B7: B,C5: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C5 @ D3 ) ) ) )
% 0.30/0.49 => ( P @ X2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_induct4
% 0.30/0.49 thf(fact_87_old_Oprod_Orec,axiom,
% 0.30/0.49 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B3: B] :
% 0.30/0.49 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B3 ) )
% 0.30/0.49 = ( F1 @ A2 @ B3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % old.prod.rec
% 0.30/0.49 thf(fact_88_bot__empty__eq2,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( bot_bot @ ( A > B > $o ) )
% 0.30/0.49 = ( ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % bot_empty_eq2
% 0.30/0.49 thf(fact_89_wf__measure,axiom,
% 0.30/0.49 ! [A: $tType,F: A > nat] : ( wf @ A @ ( measure @ A @ F ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_measure
% 0.30/0.49 thf(fact_90_dependent__wf__choice,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
% 0.30/0.49 ( ( wf @ A @ R2 )
% 0.30/0.49 => ( ! [F3: A > B,G3: A > B,X3: A,R4: B] :
% 0.30/0.49 ( ! [Z2: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ X3 ) @ R2 )
% 0.30/0.49 => ( ( F3 @ Z2 )
% 0.30/0.49 = ( G3 @ Z2 ) ) )
% 0.30/0.49 => ( ( P @ F3 @ X3 @ R4 )
% 0.30/0.49 = ( P @ G3 @ X3 @ R4 ) ) )
% 0.30/0.49 => ( ! [X3: A,F3: A > B] :
% 0.30/0.49 ( ! [Y4: A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R2 )
% 0.30/0.49 => ( P @ F3 @ Y4 @ ( F3 @ Y4 ) ) )
% 0.30/0.49 => ? [X_1: B] : ( P @ F3 @ X3 @ X_1 ) )
% 0.30/0.49 => ? [F3: A > B] :
% 0.30/0.49 ! [X5: A] : ( P @ F3 @ X5 @ ( F3 @ X5 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dependent_wf_choice
% 0.30/0.49 thf(fact_91_thoare_OWhile,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ state @ state ),C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,B3: state > $o,C3: com] :
% 0.30/0.49 ( ( wf @ state @ R )
% 0.30/0.49 => ( ! [S3: state] :
% 0.30/0.49 ( pHoare166426476thoare @ A @ C2
% 0.30/0.49 @ ^ [Z4: A,S4: state] :
% 0.30/0.49 ( ( P @ Z4 @ S4 )
% 0.30/0.49 & ( B3 @ S4 )
% 0.30/0.49 & ( S3 = S4 ) )
% 0.30/0.49 @ C3
% 0.30/0.49 @ ^ [Z4: A,S4: state] :
% 0.30/0.49 ( ( P @ Z4 @ S4 )
% 0.30/0.49 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S4 @ S3 ) @ R ) ) )
% 0.30/0.49 => ( pHoare166426476thoare @ A @ C2 @ P @ ( while @ B3 @ C3 )
% 0.30/0.49 @ ^ [Z4: A,S4: state] :
% 0.30/0.49 ( ( P @ Z4 @ S4 )
% 0.30/0.49 & ~ ( B3 @ S4 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % thoare.While
% 0.30/0.49 thf(fact_92_hoare__sound,axiom,
% 0.30/0.49 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,C3: com,Q: A > state > $o] :
% 0.30/0.49 ( ( hoare @ A @ C2 @ P @ C3 @ Q )
% 0.30/0.49 => ( cvalid @ A @ C2 @ P @ C3 @ Q ) ) ).
% 0.30/0.49
% 0.30/0.49 % hoare_sound
% 0.30/0.49 thf(fact_93_Set_Ois__empty__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( is_empty @ A )
% 0.30/0.49 = ( ^ [A7: set @ A] :
% 0.30/0.49 ( A7
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Set.is_empty_def
% 0.30/0.49 thf(fact_94_com_Oinject_I4_J,axiom,
% 0.30/0.49 ! [X41: state > $o,X42: com,Y41: state > $o,Y42: com] :
% 0.30/0.49 ( ( ( while @ X41 @ X42 )
% 0.30/0.49 = ( while @ Y41 @ Y42 ) )
% 0.30/0.49 = ( ( X41 = Y41 )
% 0.30/0.49 & ( X42 = Y42 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % com.inject(4)
% 0.30/0.49 thf(fact_95_bot2E,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,X2: A,Y3: B] :
% 0.30/0.49 ~ ( bot_bot @ ( A > B > $o ) @ X2 @ Y3 ) ).
% 0.30/0.49
% 0.30/0.49 % bot2E
% 0.30/0.49 thf(fact_96_com_Odistinct_I25_J,axiom,
% 0.30/0.49 ! [X41: state > $o,X42: com] :
% 0.30/0.49 ( ( while @ X41 @ X42 )
% 0.30/0.49 != call ) ).
% 0.30/0.49
% 0.30/0.49 % com.distinct(25)
% 0.30/0.49 thf(fact_97_exec_OWhileFalse,axiom,
% 0.30/0.49 ! [B3: state > $o,S2: state,C3: com] :
% 0.30/0.49 ( ~ ( B3 @ S2 )
% 0.30/0.49 => ( exec @ S2 @ ( while @ B3 @ C3 ) @ S2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % exec.WhileFalse
% 0.30/0.49 thf(fact_98_exec_OWhileTrue,axiom,
% 0.30/0.49 ! [B3: state > $o,S2: state,C3: com,T3: state,U2: state] :
% 0.30/0.49 ( ( B3 @ S2 )
% 0.30/0.49 => ( ( exec @ S2 @ C3 @ T3 )
% 0.30/0.49 => ( ( exec @ T3 @ ( while @ B3 @ C3 ) @ U2 )
% 0.30/0.49 => ( exec @ S2 @ ( while @ B3 @ C3 ) @ U2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % exec.WhileTrue
% 0.30/0.49 thf(fact_99_termi_OWhileFalse,axiom,
% 0.30/0.49 ! [B3: state > $o,S2: state,C3: com] :
% 0.30/0.49 ( ~ ( B3 @ S2 )
% 0.30/0.49 => ( termi @ ( while @ B3 @ C3 ) @ S2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % termi.WhileFalse
% 0.30/0.49 thf(fact_100_hoare_OWhile,axiom,
% 0.30/0.49 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,B3: state > $o,C3: com] :
% 0.30/0.49 ( ( hoare @ A @ C2
% 0.30/0.49 @ ^ [Z4: A,S4: state] :
% 0.30/0.49 ( ( P @ Z4 @ S4 )
% 0.30/0.49 & ( B3 @ S4 ) )
% 0.30/0.49 @ C3
% 0.30/0.49 @ P )
% 0.30/0.49 => ( hoare @ A @ C2 @ P @ ( while @ B3 @ C3 )
% 0.30/0.49 @ ^ [Z4: A,S4: state] :
% 0.30/0.49 ( ( P @ Z4 @ S4 )
% 0.30/0.49 & ~ ( B3 @ S4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % hoare.While
% 0.30/0.49 thf(fact_101_termi_OWhileTrue,axiom,
% 0.30/0.49 ! [B3: state > $o,S2: state,C3: com] :
% 0.30/0.49 ( ( B3 @ S2 )
% 0.30/0.49 => ( ( termi @ C3 @ S2 )
% 0.30/0.49 => ( ! [T2: state] :
% 0.30/0.49 ( ( exec @ S2 @ C3 @ T2 )
% 0.30/0.49 => ( termi @ ( while @ B3 @ C3 ) @ T2 ) )
% 0.30/0.49 => ( termi @ ( while @ B3 @ C3 ) @ S2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % termi.WhileTrue
% 0.30/0.49 thf(fact_102_bot__empty__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( bot_bot @ ( A > $o ) )
% 0.30/0.49 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % bot_empty_eq
% 0.30/0.49 thf(fact_103_pred__equals__eq2,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( ( ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ R2 ) )
% 0.30/0.49 = ( ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ S5 ) ) )
% 0.30/0.49 = ( R2 = S5 ) ) ).
% 0.30/0.49
% 0.30/0.49 % pred_equals_eq2
% 0.30/0.49 thf(fact_104_Collect__empty__eq__bot,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o] :
% 0.30/0.49 ( ( ( collect @ A @ P )
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 = ( P
% 0.30/0.49 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Collect_empty_eq_bot
% 0.30/0.49 thf(fact_105_total__on__singleton,axiom,
% 0.30/0.49 ! [A: $tType,X2: A] : ( total_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % total_on_singleton
% 0.30/0.49 thf(fact_106_refl__on__singleton,axiom,
% 0.30/0.49 ! [A: $tType,X2: A] : ( refl_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % refl_on_singleton
% 0.30/0.49 thf(fact_107_wf__termi,axiom,
% 0.30/0.49 ! [B3: state > $o,C3: com] :
% 0.30/0.49 ( wf @ state
% 0.30/0.49 @ ( collect @ ( product_prod @ state @ state )
% 0.30/0.49 @ ( product_case_prod @ state @ state @ $o
% 0.30/0.49 @ ^ [T4: state,S4: state] :
% 0.30/0.49 ( ( termi @ ( while @ B3 @ C3 ) @ S4 )
% 0.30/0.49 & ( B3 @ S4 )
% 0.30/0.49 & ( exec @ S4 @ C3 @ T4 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_termi
% 0.30/0.49 thf(fact_108_linear__order__on__singleton,axiom,
% 0.30/0.49 ! [A: $tType,X2: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % linear_order_on_singleton
% 0.30/0.49 thf(fact_109_split__part,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 0.30/0.49 ( ( product_case_prod @ A @ B @ $o
% 0.30/0.49 @ ^ [A4: A,B8: B] :
% 0.30/0.49 ( P
% 0.30/0.49 & ( Q @ A4 @ B8 ) ) )
% 0.30/0.49 = ( ^ [Ab: product_prod @ A @ B] :
% 0.30/0.49 ( P
% 0.30/0.49 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_part
% 0.30/0.49 thf(fact_110_case__prod__conv,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B3: C] :
% 0.30/0.49 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B3 ) )
% 0.30/0.49 = ( F @ A2 @ B3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prod_conv
% 0.30/0.49 thf(fact_111_case__prodI,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B3: B] :
% 0.30/0.49 ( ( F @ A2 @ B3 )
% 0.30/0.49 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prodI
% 0.30/0.49 thf(fact_112_case__prodI2,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,P3: product_prod @ A @ B,C3: A > B > $o] :
% 0.30/0.49 ( ! [A6: A,B7: B] :
% 0.30/0.49 ( ( P3
% 0.30/0.49 = ( product_Pair @ A @ B @ A6 @ B7 ) )
% 0.30/0.49 => ( C3 @ A6 @ B7 ) )
% 0.30/0.49 => ( product_case_prod @ A @ B @ $o @ C3 @ P3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prodI2
% 0.30/0.49 thf(fact_113_refl__on__domain,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A,B3: A] :
% 0.30/0.49 ( ( refl_on @ A @ A3 @ R )
% 0.30/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B3 ) @ R )
% 0.30/0.49 => ( ( member @ A @ A2 @ A3 )
% 0.30/0.49 & ( member @ A @ B3 @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % refl_on_domain
% 0.30/0.49 thf(fact_114_prod_Ocase__distrib,axiom,
% 0.30/0.49 ! [C: $tType,D2: $tType,B: $tType,A: $tType,H: C > D2,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.30/0.49 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.30/0.49 = ( product_case_prod @ A @ B @ D2
% 0.30/0.49 @ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
% 0.30/0.49 @ Prod ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod.case_distrib
% 0.30/0.49 thf(fact_115_prod_Odisc__eq__case,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.30/0.49 ( product_case_prod @ A @ B @ $o
% 0.30/0.49 @ ^ [Uu: A,Uv: B] : $true
% 0.30/0.49 @ Prod ) ).
% 0.30/0.49
% 0.30/0.49 % prod.disc_eq_case
% 0.30/0.49 thf(fact_116_old_Oprod_Ocase,axiom,
% 0.30/0.49 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 0.30/0.49 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.30/0.49 = ( F @ X1 @ X22 ) ) ).
% 0.30/0.49
% 0.30/0.49 % old.prod.case
% 0.30/0.49 thf(fact_117_refl__onD2,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
% 0.30/0.49 ( ( refl_on @ A @ A3 @ R )
% 0.30/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
% 0.30/0.49 => ( member @ A @ Y3 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % refl_onD2
% 0.30/0.49 thf(fact_118_refl__onD1,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
% 0.30/0.49 ( ( refl_on @ A @ A3 @ R )
% 0.30/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
% 0.30/0.49 => ( member @ A @ X2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % refl_onD1
% 0.30/0.49 thf(fact_119_refl__onD,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A] :
% 0.30/0.49 ( ( refl_on @ A @ A3 @ R )
% 0.30/0.49 => ( ( member @ A @ A2 @ A3 )
% 0.30/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % refl_onD
% 0.30/0.49 thf(fact_120_case__prodD,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B3: B] :
% 0.30/0.49 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B3 ) )
% 0.30/0.49 => ( F @ A2 @ B3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prodD
% 0.30/0.49 thf(fact_121_case__prodE,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,C3: A > B > $o,P3: product_prod @ A @ B] :
% 0.30/0.49 ( ( product_case_prod @ A @ B @ $o @ C3 @ P3 )
% 0.30/0.49 => ~ ! [X3: A,Y2: B] :
% 0.30/0.49 ( ( P3
% 0.30/0.49 = ( product_Pair @ A @ B @ X3 @ Y2 ) )
% 0.30/0.49 => ~ ( C3 @ X3 @ Y2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prodE
% 0.30/0.49 thf(fact_122_total__on__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( total_on @ A )
% 0.30/0.49 = ( ^ [A7: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ! [X: A] :
% 0.30/0.49 ( ( member @ A @ X @ A7 )
% 0.30/0.49 => ! [Y5: A] :
% 0.30/0.49 ( ( member @ A @ Y5 @ A7 )
% 0.30/0.49 => ( ( X != Y5 )
% 0.30/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R3 )
% 0.30/0.49 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R3 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % total_on_def
% 0.30/0.49 thf(fact_123_total__onI,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ( ! [X3: A,Y2: A] :
% 0.30/0.49 ( ( member @ A @ X3 @ A3 )
% 0.30/0.49 => ( ( member @ A @ Y2 @ A3 )
% 0.30/0.49 => ( ( X3 != Y2 )
% 0.30/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
% 0.30/0.49 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R ) ) ) ) )
% 0.30/0.49 => ( total_on @ A @ A3 @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % total_onI
% 0.30/0.49 thf(fact_124_total__on__empty,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% 0.30/0.49
% 0.30/0.49 % total_on_empty
% 0.30/0.49 thf(fact_125_case__prodE2,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z5: product_prod @ B @ C] :
% 0.30/0.49 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z5 ) )
% 0.30/0.49 => ~ ! [X3: B,Y2: C] :
% 0.30/0.49 ( ( Z5
% 0.30/0.49 = ( product_Pair @ B @ C @ X3 @ Y2 ) )
% 0.30/0.49 => ~ ( Q @ ( P @ X3 @ Y2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prodE2
% 0.30/0.49 thf(fact_126_case__prod__eta,axiom,
% 0.30/0.49 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.30/0.49 ( ( product_case_prod @ A @ B @ C
% 0.30/0.49 @ ^ [X: A,Y5: B] : ( F @ ( product_Pair @ A @ B @ X @ Y5 ) ) )
% 0.30/0.49 = F ) ).
% 0.30/0.49
% 0.30/0.49 % case_prod_eta
% 0.30/0.49 thf(fact_127_cond__case__prod__eta,axiom,
% 0.30/0.49 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 0.30/0.49 ( ! [X3: A,Y2: B] :
% 0.30/0.49 ( ( F @ X3 @ Y2 )
% 0.30/0.49 = ( G @ ( product_Pair @ A @ B @ X3 @ Y2 ) ) )
% 0.30/0.49 => ( ( product_case_prod @ A @ B @ C @ F )
% 0.30/0.49 = G ) ) ).
% 0.30/0.49
% 0.30/0.49 % cond_case_prod_eta
% 0.30/0.49 thf(fact_128_lnear__order__on__empty,axiom,
% 0.30/0.49 ! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lnear_order_on_empty
% 0.30/0.49 thf(fact_129_refl__on__empty,axiom,
% 0.30/0.49 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % refl_on_empty
% 0.30/0.49 thf(fact_130_split__cong,axiom,
% 0.30/0.49 ! [C: $tType,B: $tType,A: $tType,Q5: product_prod @ A @ B,F: A > B > C,G: A > B > C,P3: product_prod @ A @ B] :
% 0.30/0.49 ( ! [X3: A,Y2: B] :
% 0.30/0.49 ( ( ( product_Pair @ A @ B @ X3 @ Y2 )
% 0.30/0.49 = Q5 )
% 0.30/0.49 => ( ( F @ X3 @ Y2 )
% 0.30/0.49 = ( G @ X3 @ Y2 ) ) )
% 0.30/0.49 => ( ( P3 = Q5 )
% 0.30/0.49 => ( ( product_case_prod @ A @ B @ C @ F @ P3 )
% 0.30/0.49 = ( product_case_prod @ A @ B @ C @ G @ Q5 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_cong
% 0.30/0.49 thf(fact_131_aboveS__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( order_aboveS @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ A ),A4: A] :
% 0.30/0.49 ( collect @ A
% 0.30/0.49 @ ^ [B8: A] :
% 0.30/0.49 ( ( B8 != A4 )
% 0.30/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B8 ) @ R3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % aboveS_def
% 0.30/0.49 thf(fact_132_case__prod__app,axiom,
% 0.30/0.49 ! [A: $tType,D2: $tType,C: $tType,B: $tType] :
% 0.30/0.49 ( ( product_case_prod @ B @ C @ ( D2 > A ) )
% 0.30/0.49 = ( ^ [F4: B > C > D2 > A,X: product_prod @ B @ C,Y5: D2] :
% 0.30/0.49 ( product_case_prod @ B @ C @ A
% 0.30/0.49 @ ^ [L: B,R3: C] : ( F4 @ L @ R3 @ Y5 )
% 0.30/0.49 @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prod_app
% 0.30/0.49 thf(fact_133_internal__case__prod__def,axiom,
% 0.30/0.49 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.49 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.30/0.49 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.30/0.49
% 0.30/0.49 % internal_case_prod_def
% 0.30/0.49 thf(fact_134_internal__case__prod__conv,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B3: C] :
% 0.30/0.49 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B3 ) )
% 0.30/0.49 = ( C3 @ A2 @ B3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % internal_case_prod_conv
% 0.30/0.49 thf(fact_135_mem__case__prodI2,axiom,
% 0.30/0.49 ! [C: $tType,B: $tType,A: $tType,P3: product_prod @ A @ B,Z5: C,C3: A > B > ( set @ C )] :
% 0.30/0.49 ( ! [A6: A,B7: B] :
% 0.30/0.49 ( ( P3
% 0.30/0.49 = ( product_Pair @ A @ B @ A6 @ B7 ) )
% 0.30/0.49 => ( member @ C @ Z5 @ ( C3 @ A6 @ B7 ) ) )
% 0.30/0.49 => ( member @ C @ Z5 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mem_case_prodI2
% 0.30/0.49 thf(fact_136_mem__case__prodI,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,Z5: A,C3: B > C > ( set @ A ),A2: B,B3: C] :
% 0.30/0.49 ( ( member @ A @ Z5 @ ( C3 @ A2 @ B3 ) )
% 0.30/0.49 => ( member @ A @ Z5 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A2 @ B3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mem_case_prodI
% 0.30/0.49 thf(fact_137_case__prodI2_H,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,P3: product_prod @ A @ B,C3: A > B > C > $o,X2: C] :
% 0.30/0.49 ( ! [A6: A,B7: B] :
% 0.30/0.49 ( ( ( product_Pair @ A @ B @ A6 @ B7 )
% 0.30/0.49 = P3 )
% 0.30/0.49 => ( C3 @ A6 @ B7 @ X2 ) )
% 0.30/0.49 => ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P3 @ X2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prodI2'
% 0.30/0.49 thf(fact_138_case__prod__Pair__iden,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
% 0.30/0.49 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P3 )
% 0.30/0.49 = P3 ) ).
% 0.30/0.49
% 0.30/0.49 % case_prod_Pair_iden
% 0.30/0.49 thf(fact_139_mem__case__prodE,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,C: $tType,Z5: A,C3: B > C > ( set @ A ),P3: product_prod @ B @ C] :
% 0.30/0.49 ( ( member @ A @ Z5 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P3 ) )
% 0.30/0.49 => ~ ! [X3: B,Y2: C] :
% 0.30/0.49 ( ( P3
% 0.30/0.49 = ( product_Pair @ B @ C @ X3 @ Y2 ) )
% 0.30/0.49 => ~ ( member @ A @ Z5 @ ( C3 @ X3 @ Y2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mem_case_prodE
% 0.30/0.49 thf(fact_140_case__prodE_H,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P3: product_prod @ A @ B,Z5: C] :
% 0.30/0.49 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P3 @ Z5 )
% 0.30/0.49 => ~ ! [X3: A,Y2: B] :
% 0.30/0.49 ( ( P3
% 0.30/0.49 = ( product_Pair @ A @ B @ X3 @ Y2 ) )
% 0.30/0.49 => ~ ( C3 @ X3 @ Y2 @ Z5 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prodE'
% 0.30/0.49 thf(fact_141_case__prodD_H,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,C: $tType,R2: A > B > C > $o,A2: A,B3: B,C3: C] :
% 0.30/0.49 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R2 @ ( product_Pair @ A @ B @ A2 @ B3 ) @ C3 )
% 0.30/0.49 => ( R2 @ A2 @ B3 @ C3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % case_prodD'
% 0.30/0.49 thf(fact_142_lex__prod__def,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( lex_prod @ A @ B )
% 0.30/0.49 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 0.30/0.49 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.30/0.49 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.30/0.49 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.30/0.49 @ ^ [A4: A,B8: B] :
% 0.30/0.49 ( product_case_prod @ A @ B @ $o
% 0.30/0.49 @ ^ [A8: A,B9: B] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A8 ) @ Ra )
% 0.30/0.49 | ( ( A4 = A8 )
% 0.30/0.49 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B8 @ B9 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lex_prod_def
% 0.30/0.49 thf(fact_143_prod__set__simps_I1_J,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,X2: A,Y3: B] :
% 0.30/0.49 ( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y3 ) )
% 0.30/0.49 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_set_simps(1)
% 0.30/0.49 thf(fact_144_prod__set__simps_I2_J,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,X2: A,Y3: B] :
% 0.30/0.49 ( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y3 ) )
% 0.30/0.49 = ( insert @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % prod_set_simps(2)
% 0.30/0.49 thf(fact_145_the__elem__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( the_elem @ A )
% 0.30/0.49 = ( ^ [X6: set @ A] :
% 0.30/0.49 ( the @ A
% 0.30/0.49 @ ^ [X: A] :
% 0.30/0.49 ( X6
% 0.30/0.49 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % the_elem_def
% 0.30/0.49 thf(fact_146_in__lex__prod,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,A2: A,B3: B,A5: A,B6: B,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
% 0.30/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ A5 @ B6 ) ) @ ( lex_prod @ A @ B @ R @ S2 ) )
% 0.30/0.49 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A5 ) @ R )
% 0.30/0.49 | ( ( A2 = A5 )
% 0.30/0.49 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B6 ) @ S2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_lex_prod
% 0.30/0.49 thf(fact_147_The__split__eq,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,X2: A,Y3: B] :
% 0.30/0.49 ( ( the @ ( product_prod @ A @ B )
% 0.30/0.49 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.49 @ ^ [X7: A,Y6: B] :
% 0.30/0.49 ( ( X2 = X7 )
% 0.30/0.49 & ( Y3 = Y6 ) ) ) )
% 0.30/0.49 = ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % The_split_eq
% 0.30/0.49 thf(fact_148_wf__lex__prod,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,Ra2: set @ ( product_prod @ A @ A ),Rb2: set @ ( product_prod @ B @ B )] :
% 0.30/0.49 ( ( wf @ A @ Ra2 )
% 0.30/0.49 => ( ( wf @ B @ Rb2 )
% 0.30/0.49 => ( wf @ ( product_prod @ A @ B ) @ ( lex_prod @ A @ B @ Ra2 @ Rb2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_lex_prod
% 0.30/0.49 thf(fact_149_old_Orec__prod__def,axiom,
% 0.30/0.49 ! [T: $tType,B: $tType,A: $tType] :
% 0.30/0.49 ( ( product_rec_prod @ A @ B @ T )
% 0.30/0.49 = ( ^ [F12: A > B > T,X: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % old.rec_prod_def
% 0.30/0.49 thf(fact_150_the__sym__eq__trivial,axiom,
% 0.30/0.49 ! [A: $tType,X2: A] :
% 0.30/0.49 ( ( the @ A
% 0.30/0.49 @ ( ^ [Y: A,Z3: A] : ( Y = Z3 )
% 0.30/0.49 @ X2 ) )
% 0.30/0.49 = X2 ) ).
% 0.30/0.49
% 0.30/0.49 % the_sym_eq_trivial
% 0.30/0.49 thf(fact_151_the__eq__trivial,axiom,
% 0.30/0.49 ! [A: $tType,A2: A] :
% 0.30/0.49 ( ( the @ A
% 0.30/0.49 @ ^ [X: A] : ( X = A2 ) )
% 0.30/0.49 = A2 ) ).
% 0.30/0.49
% 0.30/0.49 % the_eq_trivial
% 0.30/0.49 thf(fact_152_the__equality,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,A2: A] :
% 0.30/0.49 ( ( P @ A2 )
% 0.30/0.49 => ( ! [X3: A] :
% 0.30/0.49 ( ( P @ X3 )
% 0.30/0.49 => ( X3 = A2 ) )
% 0.30/0.49 => ( ( the @ A @ P )
% 0.30/0.49 = A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % the_equality
% 0.30/0.49 thf(fact_153_theI,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,A2: A] :
% 0.30/0.49 ( ( P @ A2 )
% 0.30/0.49 => ( ! [X3: A] :
% 0.30/0.49 ( ( P @ X3 )
% 0.30/0.49 => ( X3 = A2 ) )
% 0.30/0.49 => ( P @ ( the @ A @ P ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % theI
% 0.30/0.49 thf(fact_154_theI_H,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o] :
% 0.30/0.49 ( ? [X5: A] :
% 0.30/0.49 ( ( P @ X5 )
% 0.30/0.49 & ! [Y2: A] :
% 0.30/0.49 ( ( P @ Y2 )
% 0.30/0.49 => ( Y2 = X5 ) ) )
% 0.30/0.49 => ( P @ ( the @ A @ P ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % theI'
% 0.30/0.49 thf(fact_155_theI2,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 0.30/0.49 ( ( P @ A2 )
% 0.30/0.49 => ( ! [X3: A] :
% 0.30/0.49 ( ( P @ X3 )
% 0.30/0.49 => ( X3 = A2 ) )
% 0.30/0.49 => ( ! [X3: A] :
% 0.30/0.49 ( ( P @ X3 )
% 0.30/0.49 => ( Q @ X3 ) )
% 0.30/0.49 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % theI2
% 0.30/0.49 thf(fact_156_If__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( if @ A )
% 0.30/0.49 = ( ^ [P4: $o,X: A,Y5: A] :
% 0.30/0.49 ( the @ A
% 0.30/0.49 @ ^ [Z4: A] :
% 0.30/0.49 ( ( P4
% 0.30/0.49 => ( Z4 = X ) )
% 0.30/0.49 & ( ~ P4
% 0.30/0.49 => ( Z4 = Y5 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % If_def
% 0.30/0.49 thf(fact_157_the1I2,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.49 ( ? [X5: A] :
% 0.30/0.49 ( ( P @ X5 )
% 0.30/0.49 & ! [Y2: A] :
% 0.30/0.49 ( ( P @ Y2 )
% 0.30/0.49 => ( Y2 = X5 ) ) )
% 0.30/0.49 => ( ! [X3: A] :
% 0.30/0.49 ( ( P @ X3 )
% 0.30/0.49 => ( Q @ X3 ) )
% 0.30/0.49 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % the1I2
% 0.30/0.49 thf(fact_158_the1__equality,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,A2: A] :
% 0.30/0.49 ( ? [X5: A] :
% 0.30/0.49 ( ( P @ X5 )
% 0.30/0.49 & ! [Y2: A] :
% 0.30/0.49 ( ( P @ Y2 )
% 0.30/0.49 => ( Y2 = X5 ) ) )
% 0.30/0.49 => ( ( P @ A2 )
% 0.30/0.49 => ( ( the @ A @ P )
% 0.30/0.49 = A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % the1_equality
% 0.30/0.49 thf(fact_159_same__fst__def,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( same_fst @ A @ B )
% 0.30/0.49 = ( ^ [P4: A > $o,R5: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.30/0.49 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.30/0.49 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.30/0.49 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.30/0.49 @ ^ [X7: A,Y6: B] :
% 0.30/0.49 ( product_case_prod @ A @ B @ $o
% 0.30/0.49 @ ^ [X: A,Y5: B] :
% 0.30/0.49 ( ( X7 = X )
% 0.30/0.49 & ( P4 @ X )
% 0.30/0.49 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y5 ) @ ( R5 @ X ) ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % same_fst_def
% 0.30/0.49 thf(fact_160_Range__insert,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,A2: B,B3: A,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.49 ( ( range @ B @ A @ ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B3 ) @ R ) )
% 0.30/0.49 = ( insert @ A @ B3 @ ( range @ B @ A @ R ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Range_insert
% 0.30/0.49 thf(fact_161_same__fstI,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,P: A > $o,X2: A,Y7: B,Y3: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.30/0.49 ( ( P @ X2 )
% 0.30/0.49 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y3 ) @ ( R2 @ X2 ) )
% 0.30/0.49 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y7 ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) @ ( same_fst @ A @ B @ P @ R2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % same_fstI
% 0.30/0.49 thf(fact_162_Range__empty,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( range @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Range_empty
% 0.30/0.49 thf(fact_163_Range_Oinducts,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,X2: B,R: set @ ( product_prod @ A @ B ),P: B > $o] :
% 0.30/0.49 ( ( member @ B @ X2 @ ( range @ A @ B @ R ) )
% 0.30/0.49 => ( ! [A6: A,B7: B] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B7 ) @ R )
% 0.30/0.49 => ( P @ B7 ) )
% 0.30/0.49 => ( P @ X2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Range.inducts
% 0.30/0.49 thf(fact_164_Range_Ointros,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,A2: A,B3: B,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ R )
% 0.30/0.49 => ( member @ B @ B3 @ ( range @ A @ B @ R ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Range.intros
% 0.30/0.49 thf(fact_165_Range_Osimps,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
% 0.30/0.49 = ( ? [A4: A,B8: B] :
% 0.30/0.49 ( ( A2 = B8 )
% 0.30/0.49 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B8 ) @ R ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Range.simps
% 0.30/0.49 thf(fact_166_Range_Ocases,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
% 0.30/0.49 => ~ ! [A6: A] :
% 0.30/0.49 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A2 ) @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % Range.cases
% 0.30/0.49 thf(fact_167_Range__iff,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,A2: A,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.49 ( ( member @ A @ A2 @ ( range @ B @ A @ R ) )
% 0.30/0.49 = ( ? [Y5: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y5 @ A2 ) @ R ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Range_iff
% 0.30/0.49 thf(fact_168_RangeE,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,B3: A,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.49 ( ( member @ A @ B3 @ ( range @ B @ A @ R ) )
% 0.30/0.49 => ~ ! [A6: B] :
% 0.30/0.49 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B3 ) @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % RangeE
% 0.30/0.49 thf(fact_169_Range__empty__iff,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A )] :
% 0.30/0.49 ( ( ( range @ B @ A @ R )
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 = ( R
% 0.30/0.49 = ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Range_empty_iff
% 0.30/0.49 thf(fact_170_wf__same__fst,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,P: A > $o,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.30/0.49 ( ! [X3: A] :
% 0.30/0.49 ( ( P @ X3 )
% 0.30/0.49 => ( wf @ B @ ( R2 @ X3 ) ) )
% 0.30/0.49 => ( wf @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_same_fst
% 0.30/0.49 thf(fact_171_Rangep__Range__eq,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( rangep @ A @ B
% 0.30/0.49 @ ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ R ) )
% 0.30/0.49 = ( ^ [X: B] : ( member @ B @ X @ ( range @ A @ B @ R ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Rangep_Range_eq
% 0.30/0.49 thf(fact_172_Range__def,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( range @ A @ B )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( collect @ B
% 0.30/0.49 @ ( rangep @ A @ B
% 0.30/0.49 @ ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ R3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Range_def
% 0.30/0.49 thf(fact_173_Domain__insert,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,A2: A,B3: B,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ R ) )
% 0.30/0.49 = ( insert @ A @ A2 @ ( domain @ A @ B @ R ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Domain_insert
% 0.30/0.49 thf(fact_174_Domain__empty,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Domain_empty
% 0.30/0.49 thf(fact_175_Rangep_Oinducts,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,R: A > B > $o,X2: B,P: B > $o] :
% 0.30/0.49 ( ( rangep @ A @ B @ R @ X2 )
% 0.30/0.49 => ( ! [A6: A,B7: B] :
% 0.30/0.49 ( ( R @ A6 @ B7 )
% 0.30/0.49 => ( P @ B7 ) )
% 0.30/0.49 => ( P @ X2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Rangep.inducts
% 0.30/0.49 thf(fact_176_Rangep_Ointros,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,R: A > B > $o,A2: A,B3: B] :
% 0.30/0.49 ( ( R @ A2 @ B3 )
% 0.30/0.49 => ( rangep @ A @ B @ R @ B3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % Rangep.intros
% 0.30/0.49 thf(fact_177_Rangep_Osimps,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( rangep @ A @ B )
% 0.30/0.49 = ( ^ [R3: A > B > $o,A4: B] :
% 0.30/0.49 ? [B8: A,C6: B] :
% 0.30/0.49 ( ( A4 = C6 )
% 0.30/0.49 & ( R3 @ B8 @ C6 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Rangep.simps
% 0.30/0.49 thf(fact_178_Rangep_Ocases,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,R: A > B > $o,A2: B] :
% 0.30/0.49 ( ( rangep @ A @ B @ R @ A2 )
% 0.30/0.49 => ~ ! [A6: A] :
% 0.30/0.49 ~ ( R @ A6 @ A2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % Rangep.cases
% 0.30/0.49 thf(fact_179_RangepE,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,R: A > B > $o,B3: B] :
% 0.30/0.49 ( ( rangep @ A @ B @ R @ B3 )
% 0.30/0.49 => ~ ! [A6: A] :
% 0.30/0.49 ~ ( R @ A6 @ B3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % RangepE
% 0.30/0.49 thf(fact_180_DomainE,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,A2: A,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( member @ A @ A2 @ ( domain @ A @ B @ R ) )
% 0.30/0.49 => ~ ! [B7: B] :
% 0.30/0.49 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B7 ) @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % DomainE
% 0.30/0.49 thf(fact_181_Domain__iff,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,A2: A,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( member @ A @ A2 @ ( domain @ A @ B @ R ) )
% 0.30/0.49 = ( ? [Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ Y5 ) @ R ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Domain_iff
% 0.30/0.49 thf(fact_182_Domain_Ocases,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,A2: A,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( member @ A @ A2 @ ( domain @ A @ B @ R ) )
% 0.30/0.49 => ~ ! [B7: B] :
% 0.30/0.49 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B7 ) @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % Domain.cases
% 0.30/0.49 thf(fact_183_Domain_Osimps,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,A2: A,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( member @ A @ A2 @ ( domain @ A @ B @ R ) )
% 0.30/0.49 = ( ? [A4: A,B8: B] :
% 0.30/0.49 ( ( A2 = A4 )
% 0.30/0.49 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B8 ) @ R ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Domain.simps
% 0.30/0.49 thf(fact_184_Domain_ODomainI,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,A2: A,B3: B,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ R )
% 0.30/0.49 => ( member @ A @ A2 @ ( domain @ A @ B @ R ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Domain.DomainI
% 0.30/0.49 thf(fact_185_Domain_Oinducts,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,X2: A,R: set @ ( product_prod @ A @ B ),P: A > $o] :
% 0.30/0.49 ( ( member @ A @ X2 @ ( domain @ A @ B @ R ) )
% 0.30/0.49 => ( ! [A6: A,B7: B] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B7 ) @ R )
% 0.30/0.49 => ( P @ A6 ) )
% 0.30/0.49 => ( P @ X2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Domain.inducts
% 0.30/0.49 thf(fact_186_Domain__empty__iff,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.30/0.49 ( ( ( domain @ A @ B @ R )
% 0.30/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 = ( R
% 0.30/0.49 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Domain_empty_iff
% 0.30/0.49 thf(fact_187_old_Orec__unit__def,axiom,
% 0.30/0.49 ! [T: $tType] :
% 0.30/0.49 ( ( product_rec_unit @ T )
% 0.30/0.49 = ( ^ [F12: T,X: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % old.rec_unit_def
% 0.30/0.49 thf(fact_188_above__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( order_above @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ A ),A4: A] :
% 0.30/0.49 ( collect @ A
% 0.30/0.49 @ ^ [B8: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B8 ) @ R3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % above_def
% 0.30/0.49 thf(fact_189_inv__image__def,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType] :
% 0.30/0.49 ( ( inv_image @ B @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ B @ B ),F4: A > B] :
% 0.30/0.49 ( collect @ ( product_prod @ A @ A )
% 0.30/0.49 @ ( product_case_prod @ A @ A @ $o
% 0.30/0.49 @ ^ [X: A,Y5: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X ) @ ( F4 @ Y5 ) ) @ R3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inv_image_def
% 0.30/0.49 thf(fact_190_in__inv__image,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,X2: A,Y3: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( inv_image @ B @ A @ R @ F ) )
% 0.30/0.49 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_inv_image
% 0.30/0.49 thf(fact_191_wf__inv__image,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.30/0.49 ( ( wf @ B @ R )
% 0.30/0.49 => ( wf @ A @ ( inv_image @ B @ A @ R @ F ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_inv_image
% 0.30/0.49 thf(fact_192_rp__inv__image__def,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( fun_rp_inv_image @ A @ B )
% 0.30/0.49 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.30/0.49 @ ^ [R5: set @ ( product_prod @ A @ A ),S6: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R5 @ F4 ) @ ( inv_image @ A @ B @ S6 @ F4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % rp_inv_image_def
% 0.30/0.49 thf(fact_193_Id__on__empty,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Id_on_empty
% 0.30/0.49 thf(fact_194_Id__onI,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,A3: set @ A] :
% 0.30/0.49 ( ( member @ A @ A2 @ A3 )
% 0.30/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Id_onI
% 0.30/0.49 thf(fact_195_Domain__Id__on,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] :
% 0.30/0.49 ( ( domain @ A @ A @ ( id_on @ A @ A3 ) )
% 0.30/0.49 = A3 ) ).
% 0.30/0.49
% 0.30/0.49 % Domain_Id_on
% 0.30/0.49 thf(fact_196_Range__Id__on,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] :
% 0.30/0.49 ( ( range @ A @ A @ ( id_on @ A @ A3 ) )
% 0.30/0.49 = A3 ) ).
% 0.30/0.49
% 0.30/0.49 % Range_Id_on
% 0.30/0.49 thf(fact_197_Id__on__def_H,axiom,
% 0.30/0.49 ! [A: $tType,A3: A > $o] :
% 0.30/0.49 ( ( id_on @ A @ ( collect @ A @ A3 ) )
% 0.30/0.49 = ( collect @ ( product_prod @ A @ A )
% 0.30/0.49 @ ( product_case_prod @ A @ A @ $o
% 0.30/0.49 @ ^ [X: A,Y5: A] :
% 0.30/0.49 ( ( X = Y5 )
% 0.30/0.49 & ( A3 @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Id_on_def'
% 0.30/0.49 thf(fact_198_refl__on__Id__on,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] : ( refl_on @ A @ A3 @ ( id_on @ A @ A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % refl_on_Id_on
% 0.30/0.49 thf(fact_199_Id__onE,axiom,
% 0.30/0.49 ! [A: $tType,C3: product_prod @ A @ A,A3: set @ A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A3 ) )
% 0.30/0.49 => ~ ! [X3: A] :
% 0.30/0.49 ( ( member @ A @ X3 @ A3 )
% 0.30/0.49 => ( C3
% 0.30/0.49 != ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Id_onE
% 0.30/0.49 thf(fact_200_Id__on__eqI,axiom,
% 0.30/0.49 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 0.30/0.49 ( ( A2 = B3 )
% 0.30/0.49 => ( ( member @ A @ A2 @ A3 )
% 0.30/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B3 ) @ ( id_on @ A @ A3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Id_on_eqI
% 0.30/0.49 thf(fact_201_Id__on__iff,axiom,
% 0.30/0.49 ! [A: $tType,X2: A,Y3: A,A3: set @ A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( id_on @ A @ A3 ) )
% 0.30/0.49 = ( ( X2 = Y3 )
% 0.30/0.49 & ( member @ A @ X2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Id_on_iff
% 0.30/0.49 thf(fact_202_rp__inv__image__rp,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 0.30/0.49 ( ( fun_reduction_pair @ A @ P )
% 0.30/0.49 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P @ F ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % rp_inv_image_rp
% 0.30/0.49 thf(fact_203_preorder__on__empty,axiom,
% 0.30/0.49 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % preorder_on_empty
% 0.30/0.49 thf(fact_204_termi__while,axiom,
% 0.30/0.49 ! [B3: state > $o,C3: com,F: nat > state,K: nat] :
% 0.30/0.49 ( ( termi @ ( while @ B3 @ C3 ) @ ( F @ K ) )
% 0.30/0.49 => ( ! [I: nat] : ( exec @ ( F @ I ) @ C3 @ ( F @ ( suc @ I ) ) )
% 0.30/0.49 => ? [I: nat] :
% 0.30/0.49 ~ ( B3 @ ( F @ I ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % termi_while
% 0.30/0.49 thf(fact_205_termi__while__lemma,axiom,
% 0.30/0.49 ! [W: com,Fk: state,F: nat > state,K: nat,B3: state > $o,C3: com] :
% 0.30/0.49 ( ( termi @ W @ Fk )
% 0.30/0.49 => ( ( ( Fk
% 0.30/0.49 = ( F @ K ) )
% 0.30/0.49 & ( W
% 0.30/0.49 = ( while @ B3 @ C3 ) )
% 0.30/0.49 & ! [I: nat] : ( exec @ ( F @ I ) @ C3 @ ( F @ ( suc @ I ) ) ) )
% 0.30/0.49 => ? [I: nat] :
% 0.30/0.49 ~ ( B3 @ ( F @ I ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % termi_while_lemma
% 0.30/0.49 thf(fact_206_wf__iff__no__infinite__down__chain,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( wf @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ~ ? [F4: nat > A] :
% 0.30/0.49 ! [I2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ ( suc @ I2 ) ) @ ( F4 @ I2 ) ) @ R3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_iff_no_infinite_down_chain
% 0.30/0.49 thf(fact_207_wf__no__infinite__down__chainE,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
% 0.30/0.49 ( ( wf @ A @ R )
% 0.30/0.49 => ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K2 ) ) @ ( F @ K2 ) ) @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_no_infinite_down_chainE
% 0.30/0.49 thf(fact_208_execn_Ointros_I7_J,axiom,
% 0.30/0.49 ! [S2: state,N: nat,T3: state] :
% 0.30/0.49 ( ( execn @ S2 @ body @ N @ T3 )
% 0.30/0.49 => ( execn @ S2 @ call @ ( suc @ N ) @ T3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % execn.intros(7)
% 0.30/0.49 thf(fact_209_pred__nat__def,axiom,
% 0.30/0.49 ( pred_nat
% 0.30/0.49 = ( collect @ ( product_prod @ nat @ nat )
% 0.30/0.49 @ ( product_case_prod @ nat @ nat @ $o
% 0.30/0.49 @ ^ [M: nat,N2: nat] :
% 0.30/0.49 ( N2
% 0.30/0.49 = ( suc @ M ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pred_nat_def
% 0.30/0.49 thf(fact_210_while__rule,axiom,
% 0.30/0.49 ! [S2: state,B3: state > $o,C3: com,N: nat,T3: state,P: state > $o] :
% 0.30/0.49 ( ( execn @ S2 @ ( while @ B3 @ C3 ) @ N @ T3 )
% 0.30/0.49 => ( ( P @ S2 )
% 0.30/0.49 => ( ! [S: state,S3: state] :
% 0.30/0.49 ( ( P @ S )
% 0.30/0.49 => ( ( B3 @ S )
% 0.30/0.49 => ( ( execn @ S @ C3 @ N @ S3 )
% 0.30/0.49 => ( P @ S3 ) ) ) )
% 0.30/0.49 => ( ( P @ T3 )
% 0.30/0.49 & ~ ( B3 @ T3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % while_rule
% 0.30/0.49 thf(fact_211_while__lemma,axiom,
% 0.30/0.49 ! [S2: state,W: com,N: nat,T3: state,B3: state > $o,C3: com,P: state > $o] :
% 0.30/0.49 ( ( execn @ S2 @ W @ N @ T3 )
% 0.30/0.49 => ( ( ( W
% 0.30/0.49 = ( while @ B3 @ C3 ) )
% 0.30/0.49 & ( P @ S2 )
% 0.30/0.49 & ! [S: state,S3: state] :
% 0.30/0.49 ( ( ( P @ S )
% 0.30/0.49 & ( B3 @ S )
% 0.30/0.49 & ( execn @ S @ C3 @ N @ S3 ) )
% 0.30/0.49 => ( P @ S3 ) ) )
% 0.30/0.49 => ( ( P @ T3 )
% 0.30/0.49 & ~ ( B3 @ T3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % while_lemma
% 0.30/0.49 thf(fact_212_execn_OWhileTrue,axiom,
% 0.30/0.49 ! [B3: state > $o,S2: state,C3: com,N: nat,T3: state,U2: state] :
% 0.30/0.49 ( ( B3 @ S2 )
% 0.30/0.49 => ( ( execn @ S2 @ C3 @ N @ T3 )
% 0.30/0.49 => ( ( execn @ T3 @ ( while @ B3 @ C3 ) @ N @ U2 )
% 0.30/0.49 => ( execn @ S2 @ ( while @ B3 @ C3 ) @ N @ U2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % execn.WhileTrue
% 0.30/0.49 thf(fact_213_execn_OWhileFalse,axiom,
% 0.30/0.49 ! [B3: state > $o,S2: state,C3: com,N: nat] :
% 0.30/0.49 ( ~ ( B3 @ S2 )
% 0.30/0.49 => ( execn @ S2 @ ( while @ B3 @ C3 ) @ N @ S2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % execn.WhileFalse
% 0.30/0.49 thf(fact_214_exec__iff__execn,axiom,
% 0.30/0.49 ( exec
% 0.30/0.49 = ( ^ [S4: state,C6: com,T4: state] :
% 0.30/0.49 ? [N2: nat] : ( execn @ S4 @ C6 @ N2 @ T4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % exec_iff_execn
% 0.30/0.49 thf(fact_215_wf__pred__nat,axiom,
% 0.30/0.49 wf @ nat @ pred_nat ).
% 0.30/0.49
% 0.30/0.49 % wf_pred_nat
% 0.30/0.49 thf(fact_216_nvalid__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( nvalid @ A )
% 0.30/0.49 = ( ^ [N2: nat,P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 0.30/0.49 ! [S4: state,T4: state] :
% 0.30/0.49 ( ( execn @ S4 @ C6 @ N2 @ T4 )
% 0.30/0.49 => ! [Z4: A] :
% 0.30/0.49 ( ( P4 @ Z4 @ S4 )
% 0.30/0.49 => ( Q4 @ Z4 @ T4 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nvalid_def
% 0.30/0.49 thf(fact_217_termi_OSemi,axiom,
% 0.30/0.49 ! [C1: com,S0: state,C22: com] :
% 0.30/0.49 ( ( termi @ C1 @ S0 )
% 0.30/0.49 => ( ! [S1: state] :
% 0.30/0.49 ( ( exec @ S0 @ C1 @ S1 )
% 0.30/0.49 => ( termi @ C22 @ S1 ) )
% 0.30/0.49 => ( termi @ ( semi @ C1 @ C22 ) @ S0 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % termi.Semi
% 0.30/0.49 thf(fact_218_com_Oinject_I2_J,axiom,
% 0.30/0.49 ! [X21: com,X222: com,Y21: com,Y222: com] :
% 0.30/0.49 ( ( ( semi @ X21 @ X222 )
% 0.30/0.49 = ( semi @ Y21 @ Y222 ) )
% 0.30/0.49 = ( ( X21 = Y21 )
% 0.30/0.49 & ( X222 = Y222 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % com.inject(2)
% 0.30/0.49 thf(fact_219_execn_Ointros_I2_J,axiom,
% 0.30/0.49 ! [S0: state,C1: com,N: nat,S12: state,C22: com,S22: state] :
% 0.30/0.49 ( ( execn @ S0 @ C1 @ N @ S12 )
% 0.30/0.49 => ( ( execn @ S12 @ C22 @ N @ S22 )
% 0.30/0.49 => ( execn @ S0 @ ( semi @ C1 @ C22 ) @ N @ S22 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % execn.intros(2)
% 0.30/0.49 thf(fact_220_com_Odistinct_I13_J,axiom,
% 0.30/0.49 ! [X21: com,X222: com,X41: state > $o,X42: com] :
% 0.30/0.49 ( ( semi @ X21 @ X222 )
% 0.30/0.49 != ( while @ X41 @ X42 ) ) ).
% 0.30/0.49
% 0.30/0.49 % com.distinct(13)
% 0.30/0.49 thf(fact_221_com_Odistinct_I15_J,axiom,
% 0.30/0.49 ! [X21: com,X222: com] :
% 0.30/0.49 ( ( semi @ X21 @ X222 )
% 0.30/0.49 != call ) ).
% 0.30/0.49
% 0.30/0.49 % com.distinct(15)
% 0.30/0.49 thf(fact_222_exec_OSemi,axiom,
% 0.30/0.49 ! [S0: state,C1: com,S12: state,C22: com,S22: state] :
% 0.30/0.49 ( ( exec @ S0 @ C1 @ S12 )
% 0.30/0.49 => ( ( exec @ S12 @ C22 @ S22 )
% 0.30/0.49 => ( exec @ S0 @ ( semi @ C1 @ C22 ) @ S22 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % exec.Semi
% 0.30/0.49 thf(fact_223_hoare_OSemi,axiom,
% 0.30/0.49 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,C1: com,Q: A > state > $o,C22: com,R2: A > state > $o] :
% 0.30/0.49 ( ( hoare @ A @ C2 @ P @ C1 @ Q )
% 0.30/0.49 => ( ( hoare @ A @ C2 @ Q @ C22 @ R2 )
% 0.30/0.49 => ( hoare @ A @ C2 @ P @ ( semi @ C1 @ C22 ) @ R2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % hoare.Semi
% 0.30/0.49 thf(fact_224_thoare_OSemi,axiom,
% 0.30/0.49 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,C1: com,Q: A > state > $o,C22: com,R2: A > state > $o] :
% 0.30/0.49 ( ( pHoare166426476thoare @ A @ C2 @ P @ C1 @ Q )
% 0.30/0.49 => ( ( pHoare166426476thoare @ A @ C2 @ Q @ C22 @ R2 )
% 0.30/0.49 => ( pHoare166426476thoare @ A @ C2 @ P @ ( semi @ C1 @ C22 ) @ R2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % thoare.Semi
% 0.30/0.49 thf(fact_225_cnvalid__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( cnvalid @ A )
% 0.30/0.49 = ( ^ [C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),N2: nat,P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 0.30/0.49 ( ( nvalids @ A @ N2 @ C4 )
% 0.30/0.49 => ( nvalid @ A @ N2 @ P4 @ C6 @ Q4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % cnvalid_def
% 0.30/0.49 thf(fact_226_wf__less__than,axiom,
% 0.30/0.49 wf @ nat @ less_than ).
% 0.30/0.49
% 0.30/0.49 % wf_less_than
% 0.30/0.49 thf(fact_227_total__on__less__than,axiom,
% 0.30/0.49 ! [A3: set @ nat] : ( total_on @ nat @ A3 @ less_than ) ).
% 0.30/0.49
% 0.30/0.49 % total_on_less_than
% 0.30/0.49 thf(fact_228_measure__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( measure @ A )
% 0.30/0.49 = ( inv_image @ nat @ A @ less_than ) ) ).
% 0.30/0.49
% 0.30/0.49 % measure_def
% 0.30/0.49 thf(fact_229_nvalids__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( nvalids @ A )
% 0.30/0.49 = ( ^ [N2: nat,C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) )] :
% 0.30/0.49 ! [X: product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) )] :
% 0.30/0.49 ( ( member @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ X @ C4 )
% 0.30/0.49 => ( product_case_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ $o
% 0.30/0.49 @ ^ [P4: A > state > $o] : ( product_case_prod @ com @ ( A > state > $o ) @ $o @ ( nvalid @ A @ N2 @ P4 ) )
% 0.30/0.49 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nvalids_def
% 0.30/0.49 thf(fact_230_mlex__prod__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( mlex_prod @ A )
% 0.30/0.49 = ( ^ [F4: A > nat,R5: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R5 )
% 0.30/0.49 @ ^ [X: A] : ( product_Pair @ nat @ A @ ( F4 @ X ) @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mlex_prod_def
% 0.30/0.49 thf(fact_231_ball__empty,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,X5: A] :
% 0.30/0.49 ( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.49 => ( P @ X5 ) ) ).
% 0.30/0.49
% 0.30/0.49 % ball_empty
% 0.30/0.49 thf(fact_232_Ball__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ball @ A )
% 0.30/0.49 = ( ^ [A7: set @ A,P4: A > $o] :
% 0.30/0.49 ! [X: A] :
% 0.30/0.49 ( ( member @ A @ X @ A7 )
% 0.30/0.49 => ( P4 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Ball_def
% 0.30/0.49 thf(fact_233_wf__mlex,axiom,
% 0.30/0.49 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),F: A > nat] :
% 0.30/0.49 ( ( wf @ A @ R2 )
% 0.30/0.49 => ( wf @ A @ ( mlex_prod @ A @ F @ R2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_mlex
% 0.30/0.49 thf(fact_234_refl__on__def_H,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( refl_on @ A )
% 0.30/0.49 = ( ^ [A7: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ( ! [X: product_prod @ A @ A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ X @ R3 )
% 0.30/0.49 => ( product_case_prod @ A @ A @ $o
% 0.30/0.49 @ ^ [Y5: A,Z4: A] :
% 0.30/0.49 ( ( member @ A @ Y5 @ A7 )
% 0.30/0.49 & ( member @ A @ Z4 @ A7 ) )
% 0.30/0.49 @ X ) )
% 0.30/0.49 & ! [X: A] :
% 0.30/0.49 ( ( member @ A @ X @ A7 )
% 0.30/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % refl_on_def'
% 0.30/0.49 thf(fact_235_ctvalid__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( pHoare304044376tvalid @ A )
% 0.30/0.49 = ( ^ [C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 0.30/0.49 ( ! [X: product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) )] :
% 0.30/0.49 ( ( member @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ X @ C4 )
% 0.30/0.49 => ( product_case_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ $o
% 0.30/0.49 @ ^ [P6: A > state > $o] : ( product_case_prod @ com @ ( A > state > $o ) @ $o @ ( pHoare987424603tvalid @ A @ P6 ) )
% 0.30/0.49 @ X ) )
% 0.30/0.49 => ( pHoare987424603tvalid @ A @ P4 @ C6 @ Q4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ctvalid_def
% 0.30/0.49 thf(fact_236_valids__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( valids @ A )
% 0.30/0.49 = ( ^ [C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) )] :
% 0.30/0.49 ! [X: product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) )] :
% 0.30/0.49 ( ( member @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ X @ C4 )
% 0.30/0.49 => ( product_case_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ $o
% 0.30/0.49 @ ^ [P4: A > state > $o] : ( product_case_prod @ com @ ( A > state > $o ) @ $o @ ( valid @ A @ P4 ) )
% 0.30/0.49 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % valids_def
% 0.30/0.49 thf(fact_237_tvalid__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( pHoare987424603tvalid @ A )
% 0.30/0.49 = ( ^ [P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 0.30/0.49 ( ( valid @ A @ P4 @ C6 @ Q4 )
% 0.30/0.49 & ! [Z4: A,S4: state] :
% 0.30/0.49 ( ( P4 @ Z4 @ S4 )
% 0.30/0.49 => ( termi @ C6 @ S4 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % tvalid_def
% 0.30/0.49 thf(fact_238_cvalid__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( cvalid @ A )
% 0.30/0.49 = ( ^ [C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 0.30/0.49 ( ( valids @ A @ C4 )
% 0.30/0.49 => ( valid @ A @ P4 @ C6 @ Q4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % cvalid_def
% 0.30/0.49 thf(fact_239_valid__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( valid @ A )
% 0.30/0.49 = ( ^ [P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 0.30/0.49 ! [S4: state,T4: state] :
% 0.30/0.49 ( ( exec @ S4 @ C6 @ T4 )
% 0.30/0.49 => ! [Z4: A] :
% 0.30/0.49 ( ( P4 @ Z4 @ S4 )
% 0.30/0.49 => ( Q4 @ Z4 @ T4 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % valid_def
% 0.30/0.49 thf(fact_240_Chains__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( chains @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ( collect @ ( set @ A )
% 0.30/0.49 @ ^ [C4: set @ A] :
% 0.30/0.49 ! [X: A] :
% 0.30/0.49 ( ( member @ A @ X @ C4 )
% 0.30/0.49 => ! [Y5: A] :
% 0.30/0.49 ( ( member @ A @ Y5 @ C4 )
% 0.30/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R3 )
% 0.30/0.49 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R3 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Chains_def
% 0.30/0.49 thf(fact_241_irrefl__distinct,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( irrefl @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ! [X: product_prod @ A @ A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ X @ R3 )
% 0.30/0.49 => ( product_case_prod @ A @ A @ $o
% 0.30/0.49 @ ^ [A4: A,B8: A] : ( A4 != B8 )
% 0.30/0.49 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % irrefl_distinct
% 0.30/0.49 thf(fact_242_irrefl__less__than,axiom,
% 0.30/0.49 irrefl @ nat @ less_than ).
% 0.30/0.49
% 0.30/0.49 % irrefl_less_than
% 0.30/0.49 thf(fact_243_wf__imp__irrefl,axiom,
% 0.30/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ( ( wf @ A @ R )
% 0.30/0.49 => ( irrefl @ A @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % wf_imp_irrefl
% 0.30/0.49 thf(fact_244_irrefl__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( irrefl @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ! [A4: A] :
% 0.30/0.49 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % irrefl_def
% 0.30/0.49 thf(fact_245_irreflI,axiom,
% 0.30/0.49 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ( ! [A6: A] :
% 0.30/0.49 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R2 )
% 0.30/0.49 => ( irrefl @ A @ R2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % irreflI
% 0.30/0.49 thf(fact_246_irreflp__irrefl__eq,axiom,
% 0.30/0.49 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ( ( irreflp @ A
% 0.30/0.49 @ ^ [A4: A,B8: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B8 ) @ R2 ) )
% 0.30/0.49 = ( irrefl @ A @ R2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % irreflp_irrefl_eq
% 0.30/0.49 thf(fact_247_Powp__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( powp @ A )
% 0.30/0.49 = ( ^ [A7: A > $o,B5: set @ A] :
% 0.30/0.49 ! [X: A] :
% 0.30/0.49 ( ( member @ A @ X @ B5 )
% 0.30/0.49 => ( A7 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Powp_def
% 0.30/0.49 thf(fact_248_irreflp__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( irreflp @ A )
% 0.30/0.49 = ( ^ [R5: A > A > $o] :
% 0.30/0.49 ! [A4: A] :
% 0.30/0.49 ~ ( R5 @ A4 @ A4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % irreflp_def
% 0.30/0.49 thf(fact_249_irreflpI,axiom,
% 0.30/0.49 ! [A: $tType,R2: A > A > $o] :
% 0.30/0.49 ( ! [A6: A] :
% 0.30/0.49 ~ ( R2 @ A6 @ A6 )
% 0.30/0.49 => ( irreflp @ A @ R2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % irreflpI
% 0.30/0.49 thf(fact_250_trans__join,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( trans @ A )
% 0.30/0.49 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.30/0.49 ! [X: product_prod @ A @ A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ X @ R3 )
% 0.30/0.49 => ( product_case_prod @ A @ A @ $o
% 0.30/0.49 @ ^ [Y5: A,Y12: A] :
% 0.30/0.49 ! [Z4: product_prod @ A @ A] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ Z4 @ R3 )
% 0.30/0.49 => ( product_case_prod @ A @ A @ $o
% 0.30/0.49 @ ^ [Y23: A,Aa: A] :
% 0.30/0.49 ( ( Y12 = Y23 )
% 0.30/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Aa ) @ R3 ) )
% 0.30/0.49 @ Z4 ) )
% 0.30/0.49 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % trans_join
% 0.30/0.49 thf(fact_251_in__measure,axiom,
% 0.30/0.49 ! [A: $tType,X2: A,Y3: A,F: A > nat] :
% 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measure @ A @ F ) )
% 0.30/0.49 = ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_measure
% 0.30/0.49 thf(fact_252_trans__less__than,axiom,
% 0.30/0.49 trans @ nat @ less_than ).
% 0.30/0.49
% 0.30/0.49 % trans_less_than
% 0.30/0.49 thf(fact_253_less__than__iff,axiom,
% 0.30/0.49 ! [X2: nat,Y3: nat] :
% 0.30/0.49 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ Y3 ) @ less_than )
% 0.30/0.49 = ( ord_less @ nat @ X2 @ Y3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_than_iff
% 0.30/0.49 thf(fact_254_trans__lex__prod,axiom,
% 0.30/0.49 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 0.30/0.49 ( ( trans @ A @ R1 )
% 0.30/0.49 => ( ( trans @ B @ R22 )
% 0.30/0.49 => ( trans @ ( product_prod @ A @ B ) @ ( lex_prod @ A @ B @ R1 @ R22 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % trans_lex_prod
% 0.30/0.49 thf(fact_255_trans__Id__on,axiom,
% 0.30/0.49 ! [A: $tType,A3: set @ A] : ( trans @ A @ ( id_on @ A @ A3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % trans_Id_on
% 0.30/0.49
% 0.30/0.49 % Type constructors (5)
% 0.30/0.49 thf(tcon_fun___Orderings_Obot,axiom,
% 0.30/0.49 ! [A9: $tType,A10: $tType] :
% 0.30/0.49 ( ( bot @ A10 )
% 0.30/0.49 => ( bot @ ( A9 > A10 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Nat_Onat___Orderings_Obot_1,axiom,
% 0.30/0.49 bot @ nat ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Set_Oset___Orderings_Obot_2,axiom,
% 0.30/0.49 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_HOL_Obool___Orderings_Obot_3,axiom,
% 0.30/0.49 bot @ $o ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Product__Type_Ounit___Orderings_Obot_4,axiom,
% 0.30/0.49 bot @ product_unit ).
% 0.30/0.49
% 0.30/0.49 % Helper facts (3)
% 0.30/0.49 thf(help_If_3_1_T,axiom,
% 0.30/0.49 ! [P: $o] :
% 0.30/0.49 ( ( P = $true )
% 0.30/0.49 | ( P = $false ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(help_If_2_1_T,axiom,
% 0.30/0.49 ! [A: $tType,X2: A,Y3: A] :
% 0.30/0.49 ( ( if @ A @ $false @ X2 @ Y3 )
% 0.30/0.49 = Y3 ) ).
% 0.30/0.49
% 0.30/0.49 thf(help_If_1_1_T,axiom,
% 0.30/0.49 ! [A: $tType,X2: A,Y3: A] :
% 0.30/0.49 ( ( if @ A @ $true @ X2 @ Y3 )
% 0.30/0.49 = X2 ) ).
% 0.30/0.49
% 0.30/0.49 % Conjectures (3)
% 0.30/0.49 thf(conj_0,hypothesis,
% 0.30/0.49 ! [S7: state] :
% 0.30/0.49 ( ( pHoare166426476thoare @ a
% 0.30/0.49 @ ( insert @ ( product_prod @ ( a > state > $o ) @ ( product_prod @ com @ ( a > state > $o ) ) )
% 0.30/0.49 @ ( product_Pair @ ( a > state > $o ) @ ( product_prod @ com @ ( a > state > $o ) )
% 0.30/0.49 @ ^ [Z4: a,S4: state] :
% 0.30/0.49 ( ( pa @ Z4 @ S4 )
% 0.30/0.49 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S4 @ S7 ) @ r ) )
% 0.30/0.49 @ ( product_Pair @ com @ ( a > state > $o ) @ call @ qa ) )
% 0.30/0.49 @ ( bot_bot @ ( set @ ( product_prod @ ( a > state > $o ) @ ( product_prod @ com @ ( a > state > $o ) ) ) ) ) )
% 0.30/0.49 @ ^ [Z4: a,S4: state] :
% 0.30/0.49 ( ( pa @ Z4 @ S4 )
% 0.30/0.49 & ( S4 = S7 ) )
% 0.30/0.49 @ body
% 0.30/0.49 @ qa )
% 0.30/0.49 & ( ( ! [S: state,T2: state] :
% 0.30/0.49 ( ( exec @ S @ body @ T2 )
% 0.30/0.49 => ! [Z: a] :
% 0.30/0.49 ( ( ( pa @ Z @ S )
% 0.30/0.49 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S @ S7 ) @ r ) )
% 0.30/0.49 => ( qa @ Z @ T2 ) ) )
% 0.30/0.49 & ! [Z: a,S: state] :
% 0.30/0.49 ( ( ( pa @ Z @ S )
% 0.30/0.49 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S @ S7 ) @ r ) )
% 0.30/0.49 => ( termi @ body @ S ) ) )
% 0.30/0.49 => ( ! [S8: state,T5: state] :
% 0.30/0.49 ( ( exec @ S8 @ body @ T5 )
% 0.30/0.49 => ! [Z2: a] :
% 0.30/0.49 ( ( ( pa @ Z2 @ S8 )
% 0.30/0.49 & ( S8 = S7 ) )
% 0.30/0.49 => ( qa @ Z2 @ T5 ) ) )
% 0.30/0.49 & ! [Z2: a,S8: state] :
% 0.30/0.49 ( ( ( pa @ Z2 @ S8 )
% 0.30/0.49 & ( S8 = S7 ) )
% 0.30/0.49 => ( termi @ body @ S8 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_1,hypothesis,
% 0.30/0.49 ! [Y4: state] :
% 0.30/0.49 ( ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ Y4 @ x ) @ r )
% 0.30/0.49 => ( ! [S8: state,T5: state] :
% 0.30/0.49 ( ( exec @ S8 @ body @ T5 )
% 0.30/0.49 => ! [Z2: a] :
% 0.30/0.49 ( ( ( pa @ Z2 @ S8 )
% 0.30/0.49 & ( S8 = Y4 ) )
% 0.30/0.49 => ( qa @ Z2 @ T5 ) ) )
% 0.30/0.49 & ! [Z2: a,S8: state] :
% 0.30/0.49 ( ( ( pa @ Z2 @ S8 )
% 0.30/0.49 & ( S8 = Y4 ) )
% 0.30/0.49 => ( termi @ body @ S8 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(conj_2,conjecture,
% 0.30/0.49 ( ! [S: state,T2: state] :
% 0.30/0.49 ( ~ ( exec @ S @ body @ T2 )
% 0.30/0.49 | ! [Z: a] :
% 0.30/0.49 ( ~ ( pa @ Z @ S )
% 0.30/0.49 | ( S != x )
% 0.30/0.49 | ( qa @ Z @ T2 ) ) )
% 0.30/0.49 & ! [Z: a,S: state] :
% 0.30/0.49 ( ~ ( pa @ Z @ S )
% 0.30/0.49 | ( S != x )
% 0.30/0.49 | ( termi @ body @ S ) ) ) ).
% 0.30/0.49
% 0.30/0.49 %------------------------------------------------------------------------------
% 0.30/0.49 ------------------------
% 3.15/1.63 ---- Embedded in THF ---
% 3.15/1.63 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.15/1.63 %%% Generated on Tue Feb 24 08:27:31 EST 2026
% 3.15/1.63 %%% using '$$dhol' embedding, version 1.3.0.
% 3.15/1.63 %%% Logic specification used:
% 3.15/1.63 %%% thf(spec, logic, $$dhol).
% 3.15/1.63
% 3.15/1.63 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.UIk1xi5Kzw/DTF2DTF13980.p
% 3.15/1.63 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 3.15/1.63 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.15/1.63 thf(ty_t_PLang_Ostate, type, state: $tType).
% 3.15/1.63 thf(ty_t_PLang_Ocom, type, com: $tType).
% 3.15/1.63 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.15/1.63 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.15/1.63 thf(ty_tf_a, type, a: $tType).
% 3.15/1.63 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.15/1.63 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 3.15/1.63 thf(sy_c_Basic__BNFs_Ofsts, type, basic_fsts: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > (set @ A))))).
% 3.15/1.63 thf(sy_c_Basic__BNFs_Osnds, type, basic_snds: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > (set @ B))))).
% 3.15/1.63 thf(sy_c_Fun__Def_Oreduction__pair, type, fun_reduction_pair: (!> [A:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > $o)))).
% 3.15/1.63 thf(sy_c_Fun__Def_Orp__inv__image, type, fun_rp_inv_image: (!> [A:$tType,B:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))))))).
% 3.15/1.63 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 3.15/1.63 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.15/1.63 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 3.15/1.63 thf(sy_c_Order__Relation_Oabove, type, order_above: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 3.15/1.63 thf(sy_c_Order__Relation_OaboveS, type, order_aboveS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 3.15/1.63 thf(sy_c_Order__Relation_Olinear__order__on, type, order_1409979114der_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 3.15/1.63 thf(sy_c_Order__Relation_Opreorder__on, type, order_preorder_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 3.15/1.63 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 3.15/1.63 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.15/1.63 thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Octvalid, type, pHoare304044376tvalid: (!> [A:$tType]: (((set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) > ((A > (state > $o)) > (com > ((A > (state > $o)) > $o))))))).
% 3.15/1.63 thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Othoare, type, pHoare166426476thoare: (!> [A:$tType]: (((set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) > ((A > (state > $o)) > (com > ((A > (state > $o)) > $o))))))).
% 3.15/1.63 thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Otvalid, type, pHoare987424603tvalid: (!> [A:$tType]: (((A > (state > $o)) > (com > ((A > (state > $o)) > $o)))))).
% 3.15/1.63 thf(sy_c_PHoare_Ocnvalid, type, cnvalid: (!> [A:$tType]: (((set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) > (nat > ((A > (state > $o)) > (com > ((A > (state > $o)) > $o)))))))).
% 3.15/1.63 thf(sy_c_PHoare_Ocvalid, type, cvalid: (!> [A:$tType]: (((set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) > ((A > (state > $o)) > (com > ((A > (state > $o)) > $o))))))).
% 3.15/1.63 thf(sy_c_PHoare_Ohoare, type, hoare: (!> [A:$tType]: (((set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) > ((A > (state > $o)) > (com > ((A > (state > $o)) > $o))))))).
% 3.15/1.63 thf(sy_c_PHoare_Onvalid, type, nvalid: (!> [A:$tType]: ((nat > ((A > (state > $o)) > (com > ((A > (state > $o)) > $o))))))).
% 3.15/1.63 thf(sy_c_PHoare_Onvalids, type, nvalids: (!> [A:$tType]: ((nat > ((set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) > $o))))).
% 3.15/1.63 thf(sy_c_PHoare_Ovalid, type, valid: (!> [A:$tType]: (((A > (state > $o)) > (com > ((A > (state > $o)) > $o)))))).
% 3.15/1.63 thf(sy_c_PHoare_Ovalids, type, valids: (!> [A:$tType]: (((set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) > $o)))).
% 3.15/1.63 thf(sy_c_PLang_Obody, type, body: com).
% 3.15/1.63 thf(sy_c_PLang_Ocom_OCALL, type, call: com).
% 3.15/1.63 thf(sy_c_PLang_Ocom_OSemi, type, semi: (com > (com > com))).
% 3.15/1.63 thf(sy_c_PLang_Ocom_OWhile, type, while: ((state > $o) > (com > com))).
% 3.15/1.63 thf(sy_c_PLang_Oexec, type, exec: (state > (com > (state > $o)))).
% 3.15/1.63 thf(sy_c_PLang_Oexecn, type, execn: (state > (com > (nat > (state > $o))))).
% 3.15/1.63 thf(sy_c_PTermi_Otermi, type, termi: (com > (state > $o))).
% 3.15/1.63 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.15/1.63 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.15/1.63 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 3.15/1.63 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod, type, product_rec_set_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > (T > $o)))))).
% 3.15/1.63 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit, type, product_rec_set_unit: (!> [T:$tType]: ((T > (product_unit > (T > $o)))))).
% 3.15/1.63 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit, type, product_rec_unit: (!> [T:$tType]: ((T > (product_unit > T))))).
% 3.15/1.63 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.15/1.63 thf(sy_c_Relation_ODomain, type, domain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ A))))).
% 3.15/1.63 thf(sy_c_Relation_OId__on, type, id_on: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))).
% 3.15/1.63 thf(sy_c_Relation_OPowp, type, powp: (!> [A:$tType]: (((A > $o) > ((set @ A) > $o))))).
% 3.15/1.63 thf(sy_c_Relation_ORange, type, range: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ B))))).
% 3.15/1.63 thf(sy_c_Relation_ORangep, type, rangep: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (B > $o))))).
% 3.15/1.63 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 3.15/1.63 thf(sy_c_Relation_Oirrefl, type, irrefl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 3.15/1.63 thf(sy_c_Relation_Oirreflp, type, irreflp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 3.15/1.63 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 3.15/1.63 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 3.15/1.63 thf(sy_c_Relation_Otrans, type, trans: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 3.15/1.63 thf(sy_c_Set_OBall, type, ball: (!> [A:$tType]: (((set @ A) > ((A > $o) > $o))))).
% 3.15/1.63 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.15/1.63 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 3.15/1.63 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.15/1.63 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.15/1.63 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 3.15/1.63 thf(sy_c_Wellfounded_Oless__than, type, less_than: (set @ ((product_prod @ nat) @ nat))).
% 3.15/1.63 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.15/1.63 thf(sy_c_Wellfounded_Omeasure, type, measure: (!> [A:$tType]: (((A > nat) > (set @ ((product_prod @ A) @ A)))))).
% 3.15/1.63 thf(sy_c_Wellfounded_Omlex__prod, type, mlex_prod: (!> [A:$tType]: (((A > nat) > ((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A))))))).
% 3.15/1.63 thf(sy_c_Wellfounded_Opred__nat, type, pred_nat: (set @ ((product_prod @ nat) @ nat))).
% 3.15/1.63 thf(sy_c_Wellfounded_Owf, type, wf: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 3.15/1.63 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.15/1.63 thf(sy_c_Zorn_OChains, type, chains: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ (set @ A)))))).
% 3.15/1.63 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.15/1.63 thf(sy_v_Pa, type, pa: (a > (state > $o))).
% 3.15/1.63 thf(sy_v_Qa, type, qa: (a > (state > $o))).
% 3.15/1.63 thf(sy_v_r, type, r: (set @ ((product_prod @ state) @ state))).
% 3.15/1.63 thf(sy_v_x, type, x: state).
% 3.15/1.63 thf(fact_0_thoare_OAsm, axiom, (! [A:$tType,P:(A > (state > $o)),Q:(A > (state > $o))]: ((((((pHoare166426476thoare @ A) @ (((insert @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) @ ((((product_Pair @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))) @ P) @ ((((product_Pair @ com) @ (A > (state > $o))) @ call) @ Q))) @ (bot_bot @ (set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))))))) @ P) @ call) @ Q)))).
% 3.15/1.63 thf(fact_1_weaken__post, axiom, (! [A:$tType,C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P:(A > (state > $o)),C3:com,Q:(A > (state > $o)),Q2:(A > (state > $o))]: (((((((pHoare166426476thoare @ A) @ C2) @ P) @ C3) @ Q) => ((! [Z:A,S:state]: ((((Q @ Z) @ S) => ((Q2 @ Z) @ S)))) => (((((pHoare166426476thoare @ A) @ C2) @ P) @ C3) @ Q2)))))).
% 3.15/1.63 thf(fact_2_thoare_OConseq, axiom, (! [A:$tType,C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P2:(A > (state > $o)),C3:com,Q2:(A > (state > $o)),P:(A > (state > $o)),Q:(A > (state > $o))]: (((((((pHoare166426476thoare @ A) @ C2) @ P2) @ C3) @ Q2) => (((! [S:state,T2:state]: (((! [Z2:A]: ((((P2 @ Z2) @ S) => ((Q2 @ Z2) @ T2)))) => (! [Z:A]: ((((P @ Z) @ S) => ((Q @ Z) @ T2))))))) & (! [S:state]: (((? [Z:A]: (((P @ Z) @ S))) => (? [Z2:A]: (((P2 @ Z2) @ S))))))) => (((((pHoare166426476thoare @ A) @ C2) @ P) @ C3) @ Q)))))).
% 3.15/1.63 thf(fact_3_PHoareTotal__Mirabelle__nfzgdgthbw_Ostrengthen__pre, axiom, (! [A:$tType,P2:(A > (state > $o)),P:(A > (state > $o)),C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),C3:com,Q:(A > (state > $o))]: (((! [Z:A,S:state]: ((((P2 @ Z) @ S) => ((P @ Z) @ S)))) => ((((((pHoare166426476thoare @ A) @ C2) @ P) @ C3) @ Q) => (((((pHoare166426476thoare @ A) @ C2) @ P2) @ C3) @ Q)))))).
% 3.15/1.63 thf(fact_4_singleton__conv, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ (^ [X:A]: ((X = A2)))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 3.15/1.63 thf(fact_5_singleton__conv2, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ ((^ [Y:A,Z3:A]: ((Y = Z3))) @ A2)) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 3.15/1.63 thf(fact_6_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 3.15/1.63 thf(fact_7_termi_Ointros_I7_J, axiom, (! [S2:state]: ((((termi @ body) @ S2) => ((termi @ call) @ S2))))).
% 3.15/1.63 thf(fact_8_exec_Ointros_I7_J, axiom, (! [S2:state,T3:state]: (((((exec @ S2) @ body) @ T3) => (((exec @ S2) @ call) @ T3))))).
% 3.15/1.63 thf(fact_9_thoare_OCall, axiom, (! [A:$tType,R:(set @ ((product_prod @ state) @ state)),P:(A > (state > $o)),Q:(A > (state > $o))]: ((((wf @ state) @ R) => ((! [S3:state]: ((((((pHoare166426476thoare @ A) @ (((insert @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) @ ((((product_Pair @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))) @ (^ [Z4:A,S4:state]: ((((P @ Z4) @ S4) & (((member @ ((product_prod @ state) @ state)) @ ((((product_Pair @ state) @ state) @ S4) @ S3)) @ R))))) @ ((((product_Pair @ com) @ (A > (state > $o))) @ call) @ Q))) @ (bot_bot @ (set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))))))) @ (^ [Z4:A,S4:state]: ((((P @ Z4) @ S4) & (S4 = S3))))) @ body) @ Q))) => (((((pHoare166426476thoare @ A) @ (bot_bot @ (set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))))) @ P) @ call) @ Q)))))).
% 3.15/1.63 thf(fact_10_insertCI, axiom, (! [A:$tType,A2:A,B2:(set @ A),B3:A]: ((((~ ((((member @ A) @ A2) @ B2))) => (A2 = B3)) => (((member @ A) @ A2) @ (((insert @ A) @ B3) @ B2)))))).
% 3.15/1.63 thf(fact_11_insert__iff, axiom, (! [A:$tType,A2:A,B3:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B3) @ A3)) = ((A2 = B3) | (((member @ A) @ A2) @ A3)))))).
% 3.15/1.63 thf(fact_12_insert__absorb2, axiom, (! [A:$tType,X2:A,A3:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ X2) @ A3)) = (((insert @ A) @ X2) @ A3))))).
% 3.15/1.63 thf(fact_13_hoare_OCall, axiom, (! [A:$tType,P:(A > (state > $o)),Q:(A > (state > $o))]: (((((((hoare @ A) @ (((insert @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) @ ((((product_Pair @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))) @ P) @ ((((product_Pair @ com) @ (A > (state > $o))) @ call) @ Q))) @ (bot_bot @ (set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))))))) @ P) @ body) @ Q) => (((((hoare @ A) @ (bot_bot @ (set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))))) @ P) @ call) @ Q))))).
% 3.15/1.63 thf(fact_14_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 3.15/1.63 thf(fact_15_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X:A]: ((~ ((((member @ A) @ X) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 3.15/1.63 thf(fact_16_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X:A]: ((~ ((P @ X))))))))).
% 3.15/1.63 thf(fact_17_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X:A]: ((~ ((P @ X))))))))).
% 3.15/1.63 thf(fact_18_hoare_OConseq, axiom, (! [A:$tType,C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P2:(A > (state > $o)),C3:com,Q2:(A > (state > $o)),P:(A > (state > $o)),Q:(A > (state > $o))]: (((((((hoare @ A) @ C2) @ P2) @ C3) @ Q2) => ((! [S:state,T2:state]: (((! [Z2:A]: ((((P2 @ Z2) @ S) => ((Q2 @ Z2) @ T2)))) => (! [Z:A]: ((((P @ Z) @ S) => ((Q @ Z) @ T2))))))) => (((((hoare @ A) @ C2) @ P) @ C3) @ Q)))))).
% 3.15/1.63 thf(fact_19_PHoare_Ostrengthen__pre, axiom, (! [A:$tType,P2:(A > (state > $o)),P:(A > (state > $o)),C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),C3:com,Q:(A > (state > $o))]: (((! [Z:A,S:state]: ((((P2 @ Z) @ S) => ((P @ Z) @ S)))) => ((((((hoare @ A) @ C2) @ P) @ C3) @ Q) => (((((hoare @ A) @ C2) @ P2) @ C3) @ Q)))))).
% 3.15/1.63 thf(fact_20_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X:A]: ((((member @ A) @ X) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 3.15/1.63 thf(fact_21_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 3.15/1.63 thf(fact_22_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 3.15/1.63 thf(fact_23_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 3.15/1.63 thf(fact_24_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => (? [B4:(set @ A)]: (((A3 = (((insert @ A) @ A2) @ B4)) & (~ ((((member @ A) @ A2) @ B4)))))))))).
% 3.15/1.63 thf(fact_25_insert__commute, axiom, (! [A:$tType,X2:A,Y3:A,A3:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ Y3) @ A3)) = (((insert @ A) @ Y3) @ (((insert @ A) @ X2) @ A3)))))).
% 3.15/1.63 thf(fact_26_insert__eq__iff, axiom, (! [A:$tType,A2:A,A3:(set @ A),B3:A,B2:(set @ A)]: (((~ ((((member @ A) @ A2) @ A3))) => ((~ ((((member @ A) @ B3) @ B2))) => (((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B3) @ B2)) = (((A2 = B3) => (A3 = B2)) & ((A2 != B3) => (? [C4:(set @ A)]: (((A3 = (((insert @ A) @ B3) @ C4)) & ((~ ((((member @ A) @ B3) @ C4))) & ((B2 = (((insert @ A) @ A2) @ C4)) & (~ ((((member @ A) @ A2) @ C4)))))))))))))))).
% 3.15/1.63 thf(fact_27_insert__absorb, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ A3) = A3))))).
% 3.15/1.63 thf(fact_28_insert__ident, axiom, (! [A:$tType,X2:A,A3:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X2) @ A3))) => ((~ ((((member @ A) @ X2) @ B2))) => (((((insert @ A) @ X2) @ A3) = (((insert @ A) @ X2) @ B2)) = (A3 = B2))))))).
% 3.15/1.63 thf(fact_29_Set_Oset__insert, axiom, (! [A:$tType,X2:A,A3:(set @ A)]: (((((member @ A) @ X2) @ A3) => (~ ((! [B4:(set @ A)]: (((A3 = (((insert @ A) @ X2) @ B4)) => (((member @ A) @ X2) @ B4)))))))))).
% 3.15/1.63 thf(fact_30_insertI2, axiom, (! [A:$tType,A2:A,B2:(set @ A),B3:A]: (((((member @ A) @ A2) @ B2) => (((member @ A) @ A2) @ (((insert @ A) @ B3) @ B2)))))).
% 3.15/1.63 thf(fact_31_insertI1, axiom, (! [A:$tType,A2:A,B2:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B2))))).
% 3.15/1.63 thf(fact_32_insertE, axiom, (! [A:$tType,A2:A,B3:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B3) @ A3)) => ((A2 != B3) => (((member @ A) @ A2) @ A3)))))).
% 3.15/1.63 thf(fact_33_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X:A]: ($false))))))).
% 3.15/1.63 thf(fact_34_insert__Collect, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((insert @ A) @ A2) @ ((collect @ A) @ P)) = ((collect @ A) @ (^ [U:A]: (((U != A2) => (P @ U))))))))).
% 3.15/1.63 thf(fact_35_insert__compr, axiom, (! [A:$tType]: (((insert @ A) = (^ [A4:A,B5:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((X = A4) | (((member @ A) @ X) @ B5))))))))))).
% 3.15/1.63 thf(fact_36_singleton__inject, axiom, (! [A:$tType,A2:A,B3:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) => (A2 = B3))))).
% 3.15/1.63 thf(fact_37_insert__not__empty, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ A3) != (bot_bot @ (set @ A)))))).
% 3.15/1.63 thf(fact_38_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B3:A,C3:A,D:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C3) @ (((insert @ A) @ D) @ (bot_bot @ (set @ A))))) = (((A2 = C3) & (B3 = D)) | ((A2 = D) & (B3 = C3))))))).
% 3.15/1.63 thf(fact_39_singleton__iff, axiom, (! [A:$tType,B3:A,A2:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B3 = A2))))).
% 3.15/1.63 thf(fact_40_singletonD, axiom, (! [A:$tType,B3:A,A2:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B3 = A2))))).
% 3.15/1.63 thf(fact_41_hoare_OAsm, axiom, (! [A:$tType,P:(A > (state > $o)),Q:(A > (state > $o))]: ((((((hoare @ A) @ (((insert @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) @ ((((product_Pair @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))) @ P) @ ((((product_Pair @ com) @ (A > (state > $o))) @ call) @ Q))) @ (bot_bot @ (set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))))))) @ P) @ call) @ Q)))).
% 3.15/1.63 thf(fact_42_Collect__conv__if2, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X:A]: (((A2 = X) & (P @ X))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X:A]: (((A2 = X) & (P @ X))))) = (bot_bot @ (set @ A)))))))).
% 3.15/1.63 thf(fact_43_Collect__conv__if, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X:A]: (((X = A2) & (P @ X))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X:A]: (((X = A2) & (P @ X))))) = (bot_bot @ (set @ A)))))))).
% 3.15/1.63 thf(fact_44_wf__empty, axiom, (! [A:$tType]: (((wf @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.15/1.63 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 3.15/1.63 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) = A3)))).
% 3.15/1.63 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.15/1.63 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 3.15/1.63 thf(fact_49_wfE__min_H, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),Q:(set @ A)]: ((((wf @ A) @ R2) => ((Q != (bot_bot @ (set @ A))) => (~ ((! [Z:A]: (((((member @ A) @ Z) @ Q) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z)) @ R2) => (~ ((((member @ A) @ Y4) @ Q)))))))))))))))))).
% 3.15/1.63 thf(fact_50_the__elem__eq, axiom, (! [A:$tType,X2:A]: ((((the_elem @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = X2)))).
% 3.15/1.63 thf(fact_51_bot__apply, axiom, (! [C:$tType,D2:$tType]: (((bot @ C) => ((bot_bot @ (D2 > C)) = (^ [X:D2]: ((bot_bot @ C)))))))).
% 3.15/1.63 thf(fact_52_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B3:B,A5:A,B6:B]: (((((((product_Pair @ A) @ B) @ A2) @ B3) = ((((product_Pair @ A) @ B) @ A5) @ B6)) = ((A2 = A5) & (B3 = B6)))))).
% 3.15/1.63 thf(fact_53_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 3.15/1.63 thf(fact_54_is__singletonI, axiom, (! [A:$tType,X2:A]: (((is_singleton @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))).
% 3.15/1.63 thf(fact_55_prod__induct7, axiom, (! [G2:$tType,F2:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: (((! [A6:A,B7:B,C5:C,D3:D2,E2:E,F3:F2,G3:G2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C5) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3))))))))) => (P @ X2))))).
% 3.15/1.63 thf(fact_56_prod__induct6, axiom, (! [F2:$tType,E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))))]: (((! [A6:A,B7:B,C5:C,D3:D2,E2:E,F3:F2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))) @ C5) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F2)) @ D3) @ ((((product_Pair @ E) @ F2) @ E2) @ F3)))))))) => (P @ X2))))).
% 3.15/1.63 thf(fact_57_prod__induct5, axiom, (! [E:$tType,D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: (((! [A6:A,B7:B,C5:C,D3:D2,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C5) @ ((((product_Pair @ D2) @ E) @ D3) @ E2))))))) => (P @ X2))))).
% 3.15/1.63 thf(fact_58_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 3.15/1.63 thf(fact_59_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((A7 = (((insert @ A) @ ((the_elem @ A) @ A7)) @ (bot_bot @ (set @ A)))))))))).
% 3.15/1.63 thf(fact_60_is__singletonI_H, axiom, (! [A:$tType,A3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => ((! [X3:A,Y2:A]: (((((member @ A) @ X3) @ A3) => ((((member @ A) @ Y2) @ A3) => (X3 = Y2))))) => ((is_singleton @ A) @ A3)))))).
% 3.15/1.63 thf(fact_61_surj__pair, axiom, (! [A:$tType,B:$tType,P3:((product_prod @ A) @ B)]: ((? [X3:A,Y2:B]: ((P3 = ((((product_Pair @ A) @ B) @ X3) @ Y2))))))).
% 3.15/1.63 thf(fact_62_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P3:((product_prod @ A) @ B)]: (((! [A6:A,B7:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B7)))) => (P @ P3))))).
% 3.15/1.63 thf(fact_63_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B3:B,A5:A,B6:B]: (((((((product_Pair @ A) @ B) @ A2) @ B3) = ((((product_Pair @ A) @ B) @ A5) @ B6)) => (~ (((A2 = A5) => (B3 != B6)))))))).
% 3.15/1.63 thf(fact_64_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A6:A,B7:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A6) @ B7))))))))).
% 3.15/1.63 thf(fact_65_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A6:A,B7:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B7)))) => (P @ Prod))))).
% 3.15/1.63 thf(fact_66_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X:A]: ((bot_bot @ B)))))))).
% 3.15/1.63 thf(fact_67_is__singletonE, axiom, (! [A:$tType,A3:(set @ A)]: ((((is_singleton @ A) @ A3) => (~ ((! [X3:A]: ((A3 != (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))).
% 3.15/1.63 thf(fact_68_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((? [X:A]: ((A7 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))).
% 3.15/1.63 thf(fact_69_wf__def, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [P4:(A > $o)]: (((! [X:A]: (((! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X)) @ R3) => (P4 @ Y5)))) => (P4 @ X)))) => (! [X4:A]: ((P4 @ X4)))))))))))).
% 3.15/1.63 thf(fact_70_wfE__min, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X2:A,Q:(set @ A)]: ((((wf @ A) @ R2) => ((((member @ A) @ X2) @ Q) => (~ ((! [Z:A]: (((((member @ A) @ Z) @ Q) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z)) @ R2) => (~ ((((member @ A) @ Y4) @ Q)))))))))))))))))).
% 3.15/1.63 thf(fact_71_wfI__min, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((! [X3:A,Q3:(set @ A)]: (((((member @ A) @ X3) @ Q3) => (? [Xa:A]: (((((member @ A) @ Xa) @ Q3) & (! [Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Xa)) @ R2) => (~ ((((member @ A) @ Y2) @ Q3)))))))))))) => ((wf @ A) @ R2))))).
% 3.15/1.63 thf(fact_72_wfUNIVI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [P5:(A > $o),X3:A]: (((! [Xa:A]: (((! [Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Xa)) @ R) => (P5 @ Y2)))) => (P5 @ Xa)))) => (P5 @ X3)))) => ((wf @ A) @ R))))).
% 3.15/1.63 thf(fact_73_wf__asym, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A,X2:A]: ((((wf @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ X2)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ A2)) @ R)))))))).
% 3.15/1.63 thf(fact_74_wf__induct, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P:(A > $o),A2:A]: ((((wf @ A) @ R) => ((! [X3:A]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X3)) @ R) => (P @ Y4)))) => (P @ X3)))) => (P @ A2)))))).
% 3.15/1.63 thf(fact_75_wf__irrefl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A]: ((((wf @ A) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R))))))).
% 3.15/1.63 thf(fact_76_wf__not__sym, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A,X2:A]: ((((wf @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ X2)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ A2)) @ R)))))))).
% 3.15/1.63 thf(fact_77_wf__not__refl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A2:A]: ((((wf @ A) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R))))))).
% 3.15/1.63 thf(fact_78_wf__eq__minimal, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [Q4:(set @ A)]: (((? [X:A]: ((((member @ A) @ X) @ Q4))) => (? [X:A]: (((((member @ A) @ X) @ Q4) & (! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X)) @ R3) => (~ ((((member @ A) @ Y5) @ Q4)))))))))))))))))).
% 3.15/1.63 thf(fact_79_wf__induct__rule, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P:(A > $o),A2:A]: ((((wf @ A) @ R) => ((! [X3:A]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X3)) @ R) => (P @ Y4)))) => (P @ X3)))) => (P @ A2)))))).
% 3.15/1.63 thf(fact_80_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A6:A,B7:B,C5:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B7) @ C5)))))))))).
% 3.15/1.63 thf(fact_81_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: ((~ ((! [A6:A,B7:B,C5:C,D3:D2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B7) @ ((((product_Pair @ C) @ D2) @ C5) @ D3))))))))))).
% 3.15/1.63 thf(fact_82_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))))]: ((~ ((! [A6:A,B7:B,C5:C,D3:D2,E2:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ E))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ E)) @ C5) @ ((((product_Pair @ D2) @ E) @ D3) @ E2)))))))))))).
% 3.15/1.63 thf(fact_83_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))))]: ((~ ((! [A6:A,B7:B,C5:C,D3:D2,E2:E,F3:F2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2)))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ F2))) @ C5) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ F2)) @ D3) @ ((((product_Pair @ E) @ F2) @ E2) @ F3))))))))))))).
% 3.15/1.63 thf(fact_84_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D2:$tType,E:$tType,F2:$tType,G2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: ((~ ((! [A6:A,B7:B,C5:C,D3:D2,E2:E,F3:F2,G3:G2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C5) @ ((((product_Pair @ D2) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D3) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3)))))))))))))).
% 3.15/1.63 thf(fact_85_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A6:A,B7:B,C5:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B7) @ C5))))) => (P @ X2))))).
% 3.15/1.63 thf(fact_86_prod__induct4, axiom, (! [D2:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2)))]: (((! [A6:A,B7:B,C5:C,D3:D2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D2))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D2)) @ B7) @ ((((product_Pair @ C) @ D2) @ C5) @ D3)))))) => (P @ X2))))).
% 3.15/1.63 thf(fact_87_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) = ((F1 @ A2) @ B3))))).
% 3.15/1.63 thf(fact_88_bot__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((bot_bot @ (A > (B > $o))) = (^ [X:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y5)) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))))))))).
% 3.15/1.63 thf(fact_89_wf__measure, axiom, (! [A:$tType,F:(A > nat)]: (((wf @ A) @ ((measure @ A) @ F))))).
% 3.15/1.63 thf(fact_90_dependent__wf__choice, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ A)),P:((A > B) > (A > (B > $o)))]: ((((wf @ A) @ R2) => ((! [F3:(A > B),G3:(A > B),X3:A,R4:B]: (((! [Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z2) @ X3)) @ R2) => ((F3 @ Z2) = (G3 @ Z2))))) => ((((P @ F3) @ X3) @ R4) = (((P @ G3) @ X3) @ R4))))) => ((! [X3:A,F3:(A > B)]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X3)) @ R2) => (((P @ F3) @ Y4) @ (F3 @ Y4))))) => (? [X_1:B]: ((((P @ F3) @ X3) @ X_1)))))) => (? [F3:(A > B)]: ((! [X5:A]: ((((P @ F3) @ X5) @ (F3 @ X5)))))))))))).
% 3.15/1.63 thf(fact_91_thoare_OWhile, axiom, (! [A:$tType,R:(set @ ((product_prod @ state) @ state)),C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P:(A > (state > $o)),B3:(state > $o),C3:com]: ((((wf @ state) @ R) => ((! [S3:state]: ((((((pHoare166426476thoare @ A) @ C2) @ (^ [Z4:A,S4:state]: ((((P @ Z4) @ S4) & ((B3 @ S4) & (S3 = S4)))))) @ C3) @ (^ [Z4:A,S4:state]: ((((P @ Z4) @ S4) & (((member @ ((product_prod @ state) @ state)) @ ((((product_Pair @ state) @ state) @ S4) @ S3)) @ R))))))) => (((((pHoare166426476thoare @ A) @ C2) @ P) @ ((while @ B3) @ C3)) @ (^ [Z4:A,S4:state]: ((((P @ Z4) @ S4) & (~ ((B3 @ S4)))))))))))).
% 3.15/1.63 thf(fact_92_hoare__sound, axiom, (! [A:$tType,C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P:(A > (state > $o)),C3:com,Q:(A > (state > $o))]: (((((((hoare @ A) @ C2) @ P) @ C3) @ Q) => (((((cvalid @ A) @ C2) @ P) @ C3) @ Q))))).
% 3.15/1.63 thf(fact_93_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A7:(set @ A)]: ((A7 = (bot_bot @ (set @ A))))))))).
% 3.15/1.63 thf(fact_94_com_Oinject_I4_J, axiom, (! [X41:(state > $o),X42:com,Y41:(state > $o),Y42:com]: (((((while @ X41) @ X42) = ((while @ Y41) @ Y42)) = ((X41 = Y41) & (X42 = Y42)))))).
% 3.15/1.63 thf(fact_95_bot2E, axiom, (! [A:$tType,B:$tType,X2:A,Y3:B]: ((~ ((((bot_bot @ (A > (B > $o))) @ X2) @ Y3)))))).
% 3.15/1.63 thf(fact_96_com_Odistinct_I25_J, axiom, (! [X41:(state > $o),X42:com]: ((((while @ X41) @ X42) != call)))).
% 3.15/1.63 thf(fact_97_exec_OWhileFalse, axiom, (! [B3:(state > $o),S2:state,C3:com]: (((~ ((B3 @ S2))) => (((exec @ S2) @ ((while @ B3) @ C3)) @ S2))))).
% 3.15/1.63 thf(fact_98_exec_OWhileTrue, axiom, (! [B3:(state > $o),S2:state,C3:com,T3:state,U2:state]: (((B3 @ S2) => ((((exec @ S2) @ C3) @ T3) => ((((exec @ T3) @ ((while @ B3) @ C3)) @ U2) => (((exec @ S2) @ ((while @ B3) @ C3)) @ U2))))))).
% 3.15/1.63 thf(fact_99_termi_OWhileFalse, axiom, (! [B3:(state > $o),S2:state,C3:com]: (((~ ((B3 @ S2))) => ((termi @ ((while @ B3) @ C3)) @ S2))))).
% 3.15/1.63 thf(fact_100_hoare_OWhile, axiom, (! [A:$tType,C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P:(A > (state > $o)),B3:(state > $o),C3:com]: (((((((hoare @ A) @ C2) @ (^ [Z4:A,S4:state]: ((((P @ Z4) @ S4) & (B3 @ S4))))) @ C3) @ P) => (((((hoare @ A) @ C2) @ P) @ ((while @ B3) @ C3)) @ (^ [Z4:A,S4:state]: ((((P @ Z4) @ S4) & (~ ((B3 @ S4))))))))))).
% 3.15/1.63 thf(fact_101_termi_OWhileTrue, axiom, (! [B3:(state > $o),S2:state,C3:com]: (((B3 @ S2) => (((termi @ C3) @ S2) => ((! [T2:state]: (((((exec @ S2) @ C3) @ T2) => ((termi @ ((while @ B3) @ C3)) @ T2)))) => ((termi @ ((while @ B3) @ C3)) @ S2))))))).
% 3.15/1.63 thf(fact_102_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 3.15/1.63 thf(fact_103_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S5:(set @ ((product_prod @ A) @ B))]: ((((^ [X:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y5)) @ R2))) = (^ [X:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y5)) @ S5)))) = (R2 = S5))))).
% 3.15/1.63 thf(fact_104_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 3.15/1.63 thf(fact_105_total__on__singleton, axiom, (! [A:$tType,X2:A]: ((((total_on @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 3.15/1.63 thf(fact_106_refl__on__singleton, axiom, (! [A:$tType,X2:A]: ((((refl_on @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 3.15/1.63 thf(fact_107_wf__termi, axiom, (! [B3:(state > $o),C3:com]: (((wf @ state) @ ((collect @ ((product_prod @ state) @ state)) @ ((((product_case_prod @ state) @ state) @ $o) @ (^ [T4:state,S4:state]: ((((termi @ ((while @ B3) @ C3)) @ S4) & ((B3 @ S4) & (((exec @ S4) @ C3) @ T4))))))))))).
% 3.15/1.63 thf(fact_108_linear__order__on__singleton, axiom, (! [A:$tType,X2:A]: ((((order_1409979114der_on @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 3.15/1.63 thf(fact_109_split__part, axiom, (! [B:$tType,A:$tType,P:$o,Q:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A4:A,B8:B]: ((P & ((Q @ A4) @ B8))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P & (((((product_case_prod @ A) @ B) @ $o) @ Q) @ Ab)))))))).
% 3.15/1.63 thf(fact_110_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A2:B,B3:C]: (((((((product_case_prod @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A2) @ B3)) = ((F @ A2) @ B3))))).
% 3.15/1.63 thf(fact_111_case__prodI, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A2:A,B3:B]: ((((F @ A2) @ B3) => (((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A2) @ B3)))))).
% 3.15/1.63 thf(fact_112_case__prodI2, axiom, (! [B:$tType,A:$tType,P3:((product_prod @ A) @ B),C3:(A > (B > $o))]: (((! [A6:A,B7:B]: (((P3 = ((((product_Pair @ A) @ B) @ A6) @ B7)) => ((C3 @ A6) @ B7)))) => (((((product_case_prod @ A) @ B) @ $o) @ C3) @ P3))))).
% 3.15/1.63 thf(fact_113_refl__on__domain, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A)),A2:A,B3:A]: (((((refl_on @ A) @ A3) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B3)) @ R) => ((((member @ A) @ A2) @ A3) & (((member @ A) @ B3) @ A3))))))).
% 3.15/1.63 thf(fact_114_prod_Ocase__distrib, axiom, (! [C:$tType,D2:$tType,B:$tType,A:$tType,H:(C > D2),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (((((product_case_prod @ A) @ B) @ D2) @ (^ [X12:A,X23:B]: ((H @ ((F @ X12) @ X23))))) @ Prod))))).
% 3.15/1.63 thf(fact_115_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu:A,Uv:B]: ($true))) @ Prod)))).
% 3.15/1.63 thf(fact_116_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),X1:A,X22:B]: (((((((product_case_prod @ A) @ B) @ C) @ F) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = ((F @ X1) @ X22))))).
% 3.15/1.63 thf(fact_117_refl__onD2, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A)),X2:A,Y3:A]: (((((refl_on @ A) @ A3) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y3)) @ R) => (((member @ A) @ Y3) @ A3)))))).
% 3.15/1.63 thf(fact_118_refl__onD1, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A)),X2:A,Y3:A]: (((((refl_on @ A) @ A3) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y3)) @ R) => (((member @ A) @ X2) @ A3)))))).
% 3.15/1.63 thf(fact_119_refl__onD, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A)),A2:A]: (((((refl_on @ A) @ A3) @ R) => ((((member @ A) @ A2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R)))))).
% 3.15/1.63 thf(fact_120_case__prodD, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A2:A,B3:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) => ((F @ A2) @ B3))))).
% 3.15/1.63 thf(fact_121_case__prodE, axiom, (! [A:$tType,B:$tType,C3:(A > (B > $o)),P3:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C3) @ P3) => (~ ((! [X3:A,Y2:B]: (((P3 = ((((product_Pair @ A) @ B) @ X3) @ Y2)) => (~ (((C3 @ X3) @ Y2)))))))))))).
% 3.15/1.63 thf(fact_122_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A7:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: ((! [X:A]: (((((member @ A) @ X) @ A7) => (! [Y5:A]: (((((member @ A) @ Y5) @ A7) => ((X != Y5) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y5)) @ R3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X)) @ R3))))))))))))))).
% 3.15/1.63 thf(fact_123_total__onI, axiom, (! [A:$tType,A3:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((! [X3:A,Y2:A]: (((((member @ A) @ X3) @ A3) => ((((member @ A) @ Y2) @ A3) => ((X3 != Y2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y2)) @ R) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X3)) @ R))))))) => (((total_on @ A) @ A3) @ R))))).
% 3.15/1.63 thf(fact_124_total__on__empty, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((total_on @ A) @ (bot_bot @ (set @ A))) @ R)))).
% 3.15/1.63 thf(fact_125_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q:(A > $o),P:(B > (C > A)),Z5:((product_prod @ B) @ C)]: (((Q @ (((((product_case_prod @ B) @ C) @ A) @ P) @ Z5)) => (~ ((! [X3:B,Y2:C]: (((Z5 = ((((product_Pair @ B) @ C) @ X3) @ Y2)) => (~ ((Q @ ((P @ X3) @ Y2))))))))))))).
% 3.15/1.63 thf(fact_126_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X:A,Y5:B]: ((F @ ((((product_Pair @ A) @ B) @ X) @ Y5))))) = F)))).
% 3.15/1.63 thf(fact_127_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C)),G:(((product_prod @ A) @ B) > C)]: (((! [X3:A,Y2:B]: ((((F @ X3) @ Y2) = (G @ ((((product_Pair @ A) @ B) @ X3) @ Y2))))) => (((((product_case_prod @ A) @ B) @ C) @ F) = G))))).
% 3.15/1.63 thf(fact_128_lnear__order__on__empty, axiom, (! [A:$tType]: ((((order_1409979114der_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.15/1.63 thf(fact_129_refl__on__empty, axiom, (! [A:$tType]: ((((refl_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.15/1.63 thf(fact_130_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q5:((product_prod @ A) @ B),F:(A > (B > C)),G:(A > (B > C)),P3:((product_prod @ A) @ B)]: (((! [X3:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X3) @ Y2) = Q5) => (((F @ X3) @ Y2) = ((G @ X3) @ Y2))))) => ((P3 = Q5) => ((((((product_case_prod @ A) @ B) @ C) @ F) @ P3) = (((((product_case_prod @ A) @ B) @ C) @ G) @ Q5))))))).
% 3.15/1.63 thf(fact_131_aboveS__def, axiom, (! [A:$tType]: (((order_aboveS @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A4:A]: (((collect @ A) @ (^ [B8:A]: (((B8 != A4) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B8)) @ R3))))))))))).
% 3.15/1.63 thf(fact_132_case__prod__app, axiom, (! [A:$tType,D2:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ (D2 > A)) = (^ [F4:(B > (C > (D2 > A))),X:((product_prod @ B) @ C),Y5:D2]: ((((((product_case_prod @ B) @ C) @ A) @ (^ [L:B,R3:C]: ((((F4 @ L) @ R3) @ Y5)))) @ X))))))).
% 3.15/1.63 thf(fact_133_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 3.15/1.63 thf(fact_134_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A2:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B3)) = ((C3 @ A2) @ B3))))).
% 3.15/1.63 thf(fact_135_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P3:((product_prod @ A) @ B),Z5:C,C3:(A > (B > (set @ C)))]: (((! [A6:A,B7:B]: (((P3 = ((((product_Pair @ A) @ B) @ A6) @ B7)) => (((member @ C) @ Z5) @ ((C3 @ A6) @ B7))))) => (((member @ C) @ Z5) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C3) @ P3)))))).
% 3.15/1.63 thf(fact_136_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z5:A,C3:(B > (C > (set @ A))),A2:B,B3:C]: (((((member @ A) @ Z5) @ ((C3 @ A2) @ B3)) => (((member @ A) @ Z5) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ ((((product_Pair @ B) @ C) @ A2) @ B3))))))).
% 3.15/1.63 thf(fact_137_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P3:((product_prod @ A) @ B),C3:(A > (B > (C > $o))),X2:C]: (((! [A6:A,B7:B]: (((((((product_Pair @ A) @ B) @ A6) @ B7) = P3) => (((C3 @ A6) @ B7) @ X2)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C3) @ P3) @ X2))))).
% 3.15/1.63 thf(fact_138_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P3:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P3) = P3)))).
% 3.15/1.63 thf(fact_139_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z5:A,C3:(B > (C > (set @ A))),P3:((product_prod @ B) @ C)]: (((((member @ A) @ Z5) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ P3)) => (~ ((! [X3:B,Y2:C]: (((P3 = ((((product_Pair @ B) @ C) @ X3) @ Y2)) => (~ ((((member @ A) @ Z5) @ ((C3 @ X3) @ Y2))))))))))))).
% 3.15/1.63 thf(fact_140_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(A > (B > (C > $o))),P3:((product_prod @ A) @ B),Z5:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C3) @ P3) @ Z5) => (~ ((! [X3:A,Y2:B]: (((P3 = ((((product_Pair @ A) @ B) @ X3) @ Y2)) => (~ ((((C3 @ X3) @ Y2) @ Z5)))))))))))).
% 3.15/1.63 thf(fact_141_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R2:(A > (B > (C > $o))),A2:A,B3:B,C3:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R2) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ C3) => (((R2 @ A2) @ B3) @ C3))))).
% 3.15/1.63 thf(fact_142_lex__prod__def, axiom, (! [B:$tType,A:$tType]: ((((lex_prod @ A) @ B) = (^ [Ra:(set @ ((product_prod @ A) @ A)),Rb:(set @ ((product_prod @ B) @ B))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [A4:A,B8:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [A8:A,B9:B]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ A8)) @ Ra) | ((A4 = A8) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B8) @ B9)) @ Rb))))))))))))))))).
% 3.15/1.63 thf(fact_143_prod__set__simps_I1_J, axiom, (! [B:$tType,A:$tType,X2:A,Y3:B]: (((((basic_fsts @ A) @ B) @ ((((product_Pair @ A) @ B) @ X2) @ Y3)) = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))).
% 3.15/1.63 thf(fact_144_prod__set__simps_I2_J, axiom, (! [A:$tType,B:$tType,X2:A,Y3:B]: (((((basic_snds @ A) @ B) @ ((((product_Pair @ A) @ B) @ X2) @ Y3)) = (((insert @ B) @ Y3) @ (bot_bot @ (set @ B))))))).
% 3.15/1.63 thf(fact_145_the__elem__def, axiom, (! [A:$tType]: (((the_elem @ A) = (^ [X6:(set @ A)]: (((the @ A) @ (^ [X:A]: ((X6 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))))).
% 3.15/1.63 thf(fact_146_in__lex__prod, axiom, (! [A:$tType,B:$tType,A2:A,B3:B,A5:A,B6:B,R:(set @ ((product_prod @ A) @ A)),S2:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ ((((product_Pair @ A) @ B) @ A5) @ B6))) @ ((((lex_prod @ A) @ B) @ R) @ S2)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A5)) @ R) | ((A2 = A5) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B3) @ B6)) @ S2))))))).
% 3.15/1.63 thf(fact_147_The__split__eq, axiom, (! [A:$tType,B:$tType,X2:A,Y3:B]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X7:A,Y6:B]: (((X2 = X7) & (Y3 = Y6)))))) = ((((product_Pair @ A) @ B) @ X2) @ Y3))))).
% 3.15/1.63 thf(fact_148_wf__lex__prod, axiom, (! [A:$tType,B:$tType,Ra2:(set @ ((product_prod @ A) @ A)),Rb2:(set @ ((product_prod @ B) @ B))]: ((((wf @ A) @ Ra2) => (((wf @ B) @ Rb2) => ((wf @ ((product_prod @ A) @ B)) @ ((((lex_prod @ A) @ B) @ Ra2) @ Rb2))))))).
% 3.15/1.63 thf(fact_149_old_Orec__prod__def, axiom, (! [T:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ T) = (^ [F12:(A > (B > T)),X:((product_prod @ A) @ B)]: (((the @ T) @ (((((product_rec_set_prod @ A) @ B) @ T) @ F12) @ X)))))))).
% 3.15/1.63 thf(fact_150_the__sym__eq__trivial, axiom, (! [A:$tType,X2:A]: ((((the @ A) @ ((^ [Y:A,Z3:A]: ((Y = Z3))) @ X2)) = X2)))).
% 3.15/1.63 thf(fact_151_the__eq__trivial, axiom, (! [A:$tType,A2:A]: ((((the @ A) @ (^ [X:A]: ((X = A2)))) = A2)))).
% 3.15/1.63 thf(fact_152_the__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X3:A]: (((P @ X3) => (X3 = A2)))) => (((the @ A) @ P) = A2)))))).
% 3.15/1.63 thf(fact_153_theI, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((P @ A2) => ((! [X3:A]: (((P @ X3) => (X3 = A2)))) => (P @ ((the @ A) @ P))))))).
% 3.15/1.63 thf(fact_154_theI_H, axiom, (! [A:$tType,P:(A > $o)]: (((? [X5:A]: (((P @ X5) & (! [Y2:A]: (((P @ Y2) => (Y2 = X5))))))) => (P @ ((the @ A) @ P)))))).
% 3.15/1.63 thf(fact_155_theI2, axiom, (! [A:$tType,P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [X3:A]: (((P @ X3) => (X3 = A2)))) => ((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (Q @ ((the @ A) @ P)))))))).
% 3.15/1.63 thf(fact_156_If__def, axiom, (! [A:$tType]: (((if @ A) = (^ [P4:$o,X:A,Y5:A]: (((the @ A) @ (^ [Z4:A]: (((P4 => (Z4 = X)) & ((~ (P4)) => (Z4 = Y5)))))))))))).
% 3.15/1.63 thf(fact_157_the1I2, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((? [X5:A]: (((P @ X5) & (! [Y2:A]: (((P @ Y2) => (Y2 = X5))))))) => ((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (Q @ ((the @ A) @ P))))))).
% 3.15/1.63 thf(fact_158_the1__equality, axiom, (! [A:$tType,P:(A > $o),A2:A]: (((? [X5:A]: (((P @ X5) & (! [Y2:A]: (((P @ Y2) => (Y2 = X5))))))) => ((P @ A2) => (((the @ A) @ P) = A2)))))).
% 3.15/1.63 thf(fact_159_same__fst__def, axiom, (! [B:$tType,A:$tType]: ((((same_fst @ A) @ B) = (^ [P4:(A > $o),R5:(A > (set @ ((product_prod @ B) @ B)))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [X7:A,Y6:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [X:A,Y5:B]: (((X7 = X) & ((P4 @ X) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y6) @ Y5)) @ (R5 @ X)))))))))))))))))).
% 3.15/1.63 thf(fact_160_Range__insert, axiom, (! [A:$tType,B:$tType,A2:B,B3:A,R:(set @ ((product_prod @ B) @ A))]: (((((range @ B) @ A) @ (((insert @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A2) @ B3)) @ R)) = (((insert @ A) @ B3) @ (((range @ B) @ A) @ R)))))).
% 3.15/1.63 thf(fact_161_same__fstI, axiom, (! [B:$tType,A:$tType,P:(A > $o),X2:A,Y7:B,Y3:B,R2:(A > (set @ ((product_prod @ B) @ B)))]: (((P @ X2) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y7) @ Y3)) @ (R2 @ X2)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y7)) @ ((((product_Pair @ A) @ B) @ X2) @ Y3))) @ ((((same_fst @ A) @ B) @ P) @ R2))))))).
% 3.15/1.63 thf(fact_162_Range__empty, axiom, (! [B:$tType,A:$tType]: (((((range @ B) @ A) @ (bot_bot @ (set @ ((product_prod @ B) @ A)))) = (bot_bot @ (set @ A)))))).
% 3.15/1.63 thf(fact_163_Range_Oinducts, axiom, (! [A:$tType,B:$tType,X2:B,R:(set @ ((product_prod @ A) @ B)),P:(B > $o)]: (((((member @ B) @ X2) @ (((range @ A) @ B) @ R)) => ((! [A6:A,B7:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B7)) @ R) => (P @ B7)))) => (P @ X2)))))).
% 3.15/1.63 thf(fact_164_Range_Ointros, axiom, (! [B:$tType,A:$tType,A2:A,B3:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ R) => (((member @ B) @ B3) @ (((range @ A) @ B) @ R)))))).
% 3.15/1.63 thf(fact_165_Range_Osimps, axiom, (! [B:$tType,A:$tType,A2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A2) @ (((range @ A) @ B) @ R)) = (? [A4:A,B8:B]: (((A2 = B8) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B8)) @ R)))))))).
% 3.15/1.63 thf(fact_166_Range_Ocases, axiom, (! [B:$tType,A:$tType,A2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A2) @ (((range @ A) @ B) @ R)) => (~ ((! [A6:A]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ A2)) @ R))))))))))).
% 3.15/1.63 thf(fact_167_Range__iff, axiom, (! [A:$tType,B:$tType,A2:A,R:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ A2) @ (((range @ B) @ A) @ R)) = (? [Y5:B]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ Y5) @ A2)) @ R))))))).
% 3.15/1.63 thf(fact_168_RangeE, axiom, (! [A:$tType,B:$tType,B3:A,R:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ B3) @ (((range @ B) @ A) @ R)) => (~ ((! [A6:B]: ((~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A6) @ B3)) @ R))))))))))).
% 3.15/1.63 thf(fact_169_Range__empty__iff, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A))]: ((((((range @ B) @ A) @ R) = (bot_bot @ (set @ A))) = (R = (bot_bot @ (set @ ((product_prod @ B) @ A)))))))).
% 3.15/1.63 thf(fact_170_wf__same__fst, axiom, (! [B:$tType,A:$tType,P:(A > $o),R2:(A > (set @ ((product_prod @ B) @ B)))]: (((! [X3:A]: (((P @ X3) => ((wf @ B) @ (R2 @ X3))))) => ((wf @ ((product_prod @ A) @ B)) @ ((((same_fst @ A) @ B) @ P) @ R2)))))).
% 3.15/1.63 thf(fact_171_Rangep__Range__eq, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B))]: (((((rangep @ A) @ B) @ (^ [X:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y5)) @ R)))) = (^ [X:B]: ((((member @ B) @ X) @ (((range @ A) @ B) @ R)))))))).
% 3.15/1.63 thf(fact_172_Range__def, axiom, (! [B:$tType,A:$tType]: ((((range @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ B))]: (((collect @ B) @ (((rangep @ A) @ B) @ (^ [X:A,Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y5)) @ R3))))))))))).
% 3.15/1.63 thf(fact_173_Domain__insert, axiom, (! [B:$tType,A:$tType,A2:A,B3:B,R:(set @ ((product_prod @ A) @ B))]: (((((domain @ A) @ B) @ (((insert @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ R)) = (((insert @ A) @ A2) @ (((domain @ A) @ B) @ R)))))).
% 3.15/1.63 thf(fact_174_Domain__empty, axiom, (! [B:$tType,A:$tType]: (((((domain @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (bot_bot @ (set @ A)))))).
% 3.15/1.63 thf(fact_175_Rangep_Oinducts, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),X2:B,P:(B > $o)]: ((((((rangep @ A) @ B) @ R) @ X2) => ((! [A6:A,B7:B]: ((((R @ A6) @ B7) => (P @ B7)))) => (P @ X2)))))).
% 3.15/1.63 thf(fact_176_Rangep_Ointros, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),A2:A,B3:B]: ((((R @ A2) @ B3) => ((((rangep @ A) @ B) @ R) @ B3))))).
% 3.15/1.63 thf(fact_177_Rangep_Osimps, axiom, (! [B:$tType,A:$tType]: ((((rangep @ A) @ B) = (^ [R3:(A > (B > $o)),A4:B]: ((? [B8:A,C6:B]: (((A4 = C6) & ((R3 @ B8) @ C6)))))))))).
% 3.15/1.63 thf(fact_178_Rangep_Ocases, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),A2:B]: ((((((rangep @ A) @ B) @ R) @ A2) => (~ ((! [A6:A]: ((~ (((R @ A6) @ A2))))))))))).
% 3.15/1.63 thf(fact_179_RangepE, axiom, (! [A:$tType,B:$tType,R:(A > (B > $o)),B3:B]: ((((((rangep @ A) @ B) @ R) @ B3) => (~ ((! [A6:A]: ((~ (((R @ A6) @ B3))))))))))).
% 3.15/1.63 thf(fact_180_DomainE, axiom, (! [B:$tType,A:$tType,A2:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R)) => (~ ((! [B7:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B7)) @ R))))))))))).
% 3.15/1.63 thf(fact_181_Domain__iff, axiom, (! [A:$tType,B:$tType,A2:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R)) = (? [Y5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ Y5)) @ R))))))).
% 3.15/1.63 thf(fact_182_Domain_Ocases, axiom, (! [B:$tType,A:$tType,A2:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R)) => (~ ((! [B7:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B7)) @ R))))))))))).
% 3.15/1.63 thf(fact_183_Domain_Osimps, axiom, (! [B:$tType,A:$tType,A2:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A2) @ (((domain @ A) @ B) @ R)) = (? [A4:A,B8:B]: (((A2 = A4) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B8)) @ R)))))))).
% 3.15/1.63 thf(fact_184_Domain_ODomainI, axiom, (! [B:$tType,A:$tType,A2:A,B3:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B3)) @ R) => (((member @ A) @ A2) @ (((domain @ A) @ B) @ R)))))).
% 3.15/1.63 thf(fact_185_Domain_Oinducts, axiom, (! [B:$tType,A:$tType,X2:A,R:(set @ ((product_prod @ A) @ B)),P:(A > $o)]: (((((member @ A) @ X2) @ (((domain @ A) @ B) @ R)) => ((! [A6:A,B7:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B7)) @ R) => (P @ A6)))) => (P @ X2)))))).
% 3.15/1.63 thf(fact_186_Domain__empty__iff, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: ((((((domain @ A) @ B) @ R) = (bot_bot @ (set @ A))) = (R = (bot_bot @ (set @ ((product_prod @ A) @ B)))))))).
% 3.15/1.63 thf(fact_187_old_Orec__unit__def, axiom, (! [T:$tType]: (((product_rec_unit @ T) = (^ [F12:T,X:product_unit]: (((the @ T) @ (((product_rec_set_unit @ T) @ F12) @ X)))))))).
% 3.15/1.63 thf(fact_188_above__def, axiom, (! [A:$tType]: (((order_above @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A4:A]: (((collect @ A) @ (^ [B8:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B8)) @ R3)))))))))).
% 3.15/1.63 thf(fact_189_inv__image__def, axiom, (! [A:$tType,B:$tType]: ((((inv_image @ B) @ A) = (^ [R3:(set @ ((product_prod @ B) @ B)),F4:(A > B)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X:A,Y5:A]: ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F4 @ X)) @ (F4 @ Y5))) @ R3))))))))))).
% 3.15/1.63 thf(fact_190_in__inv__image, axiom, (! [A:$tType,B:$tType,X2:A,Y3:A,R:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y3)) @ ((((inv_image @ B) @ A) @ R) @ F)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X2)) @ (F @ Y3))) @ R))))).
% 3.15/1.63 thf(fact_191_wf__inv__image, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ B)),F:(A > B)]: ((((wf @ B) @ R) => ((wf @ A) @ ((((inv_image @ B) @ A) @ R) @ F)))))).
% 3.15/1.63 thf(fact_192_rp__inv__image__def, axiom, (! [B:$tType,A:$tType]: ((((fun_rp_inv_image @ A) @ B) = ((((product_case_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))))) @ (^ [R5:(set @ ((product_prod @ A) @ A)),S6:(set @ ((product_prod @ A) @ A)),F4:(B > A)]: (((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((((inv_image @ A) @ B) @ R5) @ F4)) @ ((((inv_image @ A) @ B) @ S6) @ F4))))))))).
% 3.15/1.63 thf(fact_193_Id__on__empty, axiom, (! [A:$tType]: ((((id_on @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.15/1.63 thf(fact_194_Id__onI, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ ((id_on @ A) @ A3)))))).
% 3.15/1.63 thf(fact_195_Domain__Id__on, axiom, (! [A:$tType,A3:(set @ A)]: (((((domain @ A) @ A) @ ((id_on @ A) @ A3)) = A3)))).
% 3.15/1.63 thf(fact_196_Range__Id__on, axiom, (! [A:$tType,A3:(set @ A)]: (((((range @ A) @ A) @ ((id_on @ A) @ A3)) = A3)))).
% 3.15/1.63 thf(fact_197_Id__on__def_H, axiom, (! [A:$tType,A3:(A > $o)]: ((((id_on @ A) @ ((collect @ A) @ A3)) = ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X:A,Y5:A]: (((X = Y5) & (A3 @ X)))))))))).
% 3.15/1.63 thf(fact_198_refl__on__Id__on, axiom, (! [A:$tType,A3:(set @ A)]: ((((refl_on @ A) @ A3) @ ((id_on @ A) @ A3))))).
% 3.15/1.63 thf(fact_199_Id__onE, axiom, (! [A:$tType,C3:((product_prod @ A) @ A),A3:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ C3) @ ((id_on @ A) @ A3)) => (~ ((! [X3:A]: (((((member @ A) @ X3) @ A3) => (C3 != ((((product_Pair @ A) @ A) @ X3) @ X3))))))))))).
% 3.15/1.63 thf(fact_200_Id__on__eqI, axiom, (! [A:$tType,A2:A,B3:A,A3:(set @ A)]: (((A2 = B3) => ((((member @ A) @ A2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B3)) @ ((id_on @ A) @ A3))))))).
% 3.15/1.63 thf(fact_201_Id__on__iff, axiom, (! [A:$tType,X2:A,Y3:A,A3:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y3)) @ ((id_on @ A) @ A3)) = ((X2 = Y3) & (((member @ A) @ X2) @ A3)))))).
% 3.15/1.63 thf(fact_202_rp__inv__image__rp, axiom, (! [A:$tType,B:$tType,P:((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))),F:(B > A)]: ((((fun_reduction_pair @ A) @ P) => ((fun_reduction_pair @ B) @ ((((fun_rp_inv_image @ A) @ B) @ P) @ F)))))).
% 3.15/1.63 thf(fact_203_preorder__on__empty, axiom, (! [A:$tType]: ((((order_preorder_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 3.15/1.63 thf(fact_204_termi__while, axiom, (! [B3:(state > $o),C3:com,F:(nat > state),K:nat]: ((((termi @ ((while @ B3) @ C3)) @ (F @ K)) => ((! [I:nat]: ((((exec @ (F @ I)) @ C3) @ (F @ (suc @ I))))) => (? [I:nat]: ((~ ((B3 @ (F @ I))))))))))).
% 3.15/1.63 thf(fact_205_termi__while__lemma, axiom, (! [W:com,Fk:state,F:(nat > state),K:nat,B3:(state > $o),C3:com]: ((((termi @ W) @ Fk) => (((Fk = (F @ K)) & ((W = ((while @ B3) @ C3)) & (! [I:nat]: ((((exec @ (F @ I)) @ C3) @ (F @ (suc @ I))))))) => (? [I:nat]: ((~ ((B3 @ (F @ I))))))))))).
% 3.15/1.63 thf(fact_206_wf__iff__no__infinite__down__chain, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((~ ((? [F4:(nat > A)]: ((! [I2:nat]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F4 @ (suc @ I2))) @ (F4 @ I2))) @ R3))))))))))))).
% 3.15/1.63 thf(fact_207_wf__no__infinite__down__chainE, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),F:(nat > A)]: ((((wf @ A) @ R) => (~ ((! [K2:nat]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F @ (suc @ K2))) @ (F @ K2))) @ R))))))))).
% 3.15/1.63 thf(fact_208_execn_Ointros_I7_J, axiom, (! [S2:state,N:nat,T3:state]: ((((((execn @ S2) @ body) @ N) @ T3) => ((((execn @ S2) @ call) @ (suc @ N)) @ T3))))).
% 3.15/1.63 thf(fact_209_pred__nat__def, axiom, (pred_nat = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [M:nat,N2:nat]: ((N2 = (suc @ M)))))))).
% 3.15/1.63 thf(fact_210_while__rule, axiom, (! [S2:state,B3:(state > $o),C3:com,N:nat,T3:state,P:(state > $o)]: ((((((execn @ S2) @ ((while @ B3) @ C3)) @ N) @ T3) => ((P @ S2) => ((! [S:state,S3:state]: (((P @ S) => ((B3 @ S) => (((((execn @ S) @ C3) @ N) @ S3) => (P @ S3)))))) => ((P @ T3) & (~ ((B3 @ T3)))))))))).
% 3.15/1.63 thf(fact_211_while__lemma, axiom, (! [S2:state,W:com,N:nat,T3:state,B3:(state > $o),C3:com,P:(state > $o)]: ((((((execn @ S2) @ W) @ N) @ T3) => (((W = ((while @ B3) @ C3)) & ((P @ S2) & (! [S:state,S3:state]: ((((P @ S) & ((B3 @ S) & ((((execn @ S) @ C3) @ N) @ S3))) => (P @ S3)))))) => ((P @ T3) & (~ ((B3 @ T3))))))))).
% 3.15/1.63 thf(fact_212_execn_OWhileTrue, axiom, (! [B3:(state > $o),S2:state,C3:com,N:nat,T3:state,U2:state]: (((B3 @ S2) => (((((execn @ S2) @ C3) @ N) @ T3) => (((((execn @ T3) @ ((while @ B3) @ C3)) @ N) @ U2) => ((((execn @ S2) @ ((while @ B3) @ C3)) @ N) @ U2))))))).
% 3.15/1.63 thf(fact_213_execn_OWhileFalse, axiom, (! [B3:(state > $o),S2:state,C3:com,N:nat]: (((~ ((B3 @ S2))) => ((((execn @ S2) @ ((while @ B3) @ C3)) @ N) @ S2))))).
% 3.15/1.63 thf(fact_214_exec__iff__execn, axiom, (exec = (^ [S4:state,C6:com,T4:state]: ((? [N2:nat]: (((((execn @ S4) @ C6) @ N2) @ T4))))))).
% 3.15/1.63 thf(fact_215_wf__pred__nat, axiom, ((wf @ nat) @ pred_nat)).
% 3.15/1.63 thf(fact_216_nvalid__def, axiom, (! [A:$tType]: (((nvalid @ A) = (^ [N2:nat,P4:(A > (state > $o)),C6:com,Q4:(A > (state > $o))]: ((! [S4:state,T4:state]: ((((((execn @ S4) @ C6) @ N2) @ T4) => (! [Z4:A]: ((((P4 @ Z4) @ S4) => ((Q4 @ Z4) @ T4))))))))))))).
% 3.15/1.63 thf(fact_217_termi_OSemi, axiom, (! [C1:com,S0:state,C22:com]: ((((termi @ C1) @ S0) => ((! [S1:state]: (((((exec @ S0) @ C1) @ S1) => ((termi @ C22) @ S1)))) => ((termi @ ((semi @ C1) @ C22)) @ S0)))))).
% 3.15/1.63 thf(fact_218_com_Oinject_I2_J, axiom, (! [X21:com,X222:com,Y21:com,Y222:com]: (((((semi @ X21) @ X222) = ((semi @ Y21) @ Y222)) = ((X21 = Y21) & (X222 = Y222)))))).
% 3.15/1.63 thf(fact_219_execn_Ointros_I2_J, axiom, (! [S0:state,C1:com,N:nat,S12:state,C22:com,S22:state]: ((((((execn @ S0) @ C1) @ N) @ S12) => (((((execn @ S12) @ C22) @ N) @ S22) => ((((execn @ S0) @ ((semi @ C1) @ C22)) @ N) @ S22)))))).
% 3.15/1.63 thf(fact_220_com_Odistinct_I13_J, axiom, (! [X21:com,X222:com,X41:(state > $o),X42:com]: ((((semi @ X21) @ X222) != ((while @ X41) @ X42))))).
% 3.15/1.63 thf(fact_221_com_Odistinct_I15_J, axiom, (! [X21:com,X222:com]: ((((semi @ X21) @ X222) != call)))).
% 3.15/1.63 thf(fact_222_exec_OSemi, axiom, (! [S0:state,C1:com,S12:state,C22:com,S22:state]: (((((exec @ S0) @ C1) @ S12) => ((((exec @ S12) @ C22) @ S22) => (((exec @ S0) @ ((semi @ C1) @ C22)) @ S22)))))).
% 3.15/1.63 thf(fact_223_hoare_OSemi, axiom, (! [A:$tType,C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P:(A > (state > $o)),C1:com,Q:(A > (state > $o)),C22:com,R2:(A > (state > $o))]: (((((((hoare @ A) @ C2) @ P) @ C1) @ Q) => ((((((hoare @ A) @ C2) @ Q) @ C22) @ R2) => (((((hoare @ A) @ C2) @ P) @ ((semi @ C1) @ C22)) @ R2)))))).
% 3.15/1.63 thf(fact_224_thoare_OSemi, axiom, (! [A:$tType,C2:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P:(A > (state > $o)),C1:com,Q:(A > (state > $o)),C22:com,R2:(A > (state > $o))]: (((((((pHoare166426476thoare @ A) @ C2) @ P) @ C1) @ Q) => ((((((pHoare166426476thoare @ A) @ C2) @ Q) @ C22) @ R2) => (((((pHoare166426476thoare @ A) @ C2) @ P) @ ((semi @ C1) @ C22)) @ R2)))))).
% 3.15/1.63 thf(fact_225_cnvalid__def, axiom, (! [A:$tType]: (((cnvalid @ A) = (^ [C4:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),N2:nat,P4:(A > (state > $o)),C6:com,Q4:(A > (state > $o))]: (((((nvalids @ A) @ N2) @ C4) => (((((nvalid @ A) @ N2) @ P4) @ C6) @ Q4)))))))).
% 3.15/1.63 thf(fact_226_wf__less__than, axiom, ((wf @ nat) @ less_than)).
% 3.15/1.63 thf(fact_227_total__on__less__than, axiom, (! [A3:(set @ nat)]: ((((total_on @ nat) @ A3) @ less_than)))).
% 3.15/1.63 thf(fact_228_measure__def, axiom, (! [A:$tType]: (((measure @ A) = (((inv_image @ nat) @ A) @ less_than))))).
% 3.15/1.63 thf(fact_229_nvalids__def, axiom, (! [A:$tType]: (((nvalids @ A) = (^ [N2:nat,C4:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))))]: ((! [X:((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))]: (((((member @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) @ X) @ C4) => (((((product_case_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))) @ $o) @ (^ [P4:(A > (state > $o))]: (((((product_case_prod @ com) @ (A > (state > $o))) @ $o) @ (((nvalid @ A) @ N2) @ P4))))) @ X)))))))))).
% 3.15/1.63 thf(fact_230_mlex__prod__def, axiom, (! [A:$tType]: (((mlex_prod @ A) = (^ [F4:(A > nat),R5:(set @ ((product_prod @ A) @ A))]: (((((inv_image @ ((product_prod @ nat) @ A)) @ A) @ ((((lex_prod @ nat) @ A) @ less_than) @ R5)) @ (^ [X:A]: (((((product_Pair @ nat) @ A) @ (F4 @ X)) @ X)))))))))).
% 3.15/1.63 thf(fact_231_ball__empty, axiom, (! [A:$tType,P:(A > $o),X5:A]: (((((member @ A) @ X5) @ (bot_bot @ (set @ A))) => (P @ X5))))).
% 3.15/1.63 thf(fact_232_Ball__def, axiom, (! [A:$tType]: (((ball @ A) = (^ [A7:(set @ A),P4:(A > $o)]: ((! [X:A]: (((((member @ A) @ X) @ A7) => (P4 @ X)))))))))).
% 3.15/1.63 thf(fact_233_wf__mlex, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),F:(A > nat)]: ((((wf @ A) @ R2) => ((wf @ A) @ (((mlex_prod @ A) @ F) @ R2)))))).
% 3.15/1.63 thf(fact_234_refl__on__def_H, axiom, (! [A:$tType]: (((refl_on @ A) = (^ [A7:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: (((! [X:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ X) @ R3) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y5:A,Z4:A]: (((((member @ A) @ Y5) @ A7) & (((member @ A) @ Z4) @ A7))))) @ X)))) & (! [X:A]: (((((member @ A) @ X) @ A7) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ R3))))))))))).
% 3.15/1.63 thf(fact_235_ctvalid__def, axiom, (! [A:$tType]: (((pHoare304044376tvalid @ A) = (^ [C4:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P4:(A > (state > $o)),C6:com,Q4:(A > (state > $o))]: (((! [X:((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))]: (((((member @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) @ X) @ C4) => (((((product_case_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))) @ $o) @ (^ [P6:(A > (state > $o))]: (((((product_case_prod @ com) @ (A > (state > $o))) @ $o) @ ((pHoare987424603tvalid @ A) @ P6))))) @ X)))) => ((((pHoare987424603tvalid @ A) @ P4) @ C6) @ Q4)))))))).
% 3.15/1.63 thf(fact_236_valids__def, axiom, (! [A:$tType]: (((valids @ A) = (^ [C4:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))))]: ((! [X:((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))]: (((((member @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))) @ X) @ C4) => (((((product_case_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o)))) @ $o) @ (^ [P4:(A > (state > $o))]: (((((product_case_prod @ com) @ (A > (state > $o))) @ $o) @ ((valid @ A) @ P4))))) @ X)))))))))).
% 3.15/1.63 thf(fact_237_tvalid__def, axiom, (! [A:$tType]: (((pHoare987424603tvalid @ A) = (^ [P4:(A > (state > $o)),C6:com,Q4:(A > (state > $o))]: ((((((valid @ A) @ P4) @ C6) @ Q4) & (! [Z4:A,S4:state]: ((((P4 @ Z4) @ S4) => ((termi @ C6) @ S4))))))))))).
% 3.15/1.63 thf(fact_238_cvalid__def, axiom, (! [A:$tType]: (((cvalid @ A) = (^ [C4:(set @ ((product_prod @ (A > (state > $o))) @ ((product_prod @ com) @ (A > (state > $o))))),P4:(A > (state > $o)),C6:com,Q4:(A > (state > $o))]: ((((valids @ A) @ C4) => ((((valid @ A) @ P4) @ C6) @ Q4)))))))).
% 3.15/1.63 thf(fact_239_valid__def, axiom, (! [A:$tType]: (((valid @ A) = (^ [P4:(A > (state > $o)),C6:com,Q4:(A > (state > $o))]: ((! [S4:state,T4:state]: (((((exec @ S4) @ C6) @ T4) => (! [Z4:A]: ((((P4 @ Z4) @ S4) => ((Q4 @ Z4) @ T4))))))))))))).
% 3.15/1.63 thf(fact_240_Chains__def, axiom, (! [A:$tType]: (((chains @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ (set @ A)) @ (^ [C4:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ C4) => (! [Y5:A]: (((((member @ A) @ Y5) @ C4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y5)) @ R3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X)) @ R3))))))))))))))))).
% 3.15/1.63 thf(fact_241_irrefl__distinct, axiom, (! [A:$tType]: (((irrefl @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [X:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ X) @ R3) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [A4:A,B8:A]: ((A4 != B8)))) @ X)))))))))).
% 3.15/1.63 thf(fact_242_irrefl__less__than, axiom, ((irrefl @ nat) @ less_than)).
% 3.15/1.63 thf(fact_243_wf__imp__irrefl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ R) => ((irrefl @ A) @ R))))).
% 3.15/1.63 thf(fact_244_irrefl__def, axiom, (! [A:$tType]: (((irrefl @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [A4:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ A4)) @ R3))))))))))).
% 3.15/1.63 thf(fact_245_irreflI, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((! [A6:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A6) @ A6)) @ R2))))) => ((irrefl @ A) @ R2))))).
% 3.15/1.63 thf(fact_246_irreflp__irrefl__eq, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((irreflp @ A) @ (^ [A4:A,B8:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B8)) @ R2)))) = ((irrefl @ A) @ R2))))).
% 3.15/1.63 thf(fact_247_Powp__def, axiom, (! [A:$tType]: (((powp @ A) = (^ [A7:(A > $o),B5:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ B5) => (A7 @ X)))))))))).
% 3.15/1.63 thf(fact_248_irreflp__def, axiom, (! [A:$tType]: (((irreflp @ A) = (^ [R5:(A > (A > $o))]: ((! [A4:A]: ((~ (((R5 @ A4) @ A4))))))))))).
% 3.15/1.63 thf(fact_249_irreflpI, axiom, (! [A:$tType,R2:(A > (A > $o))]: (((! [A6:A]: ((~ (((R2 @ A6) @ A6))))) => ((irreflp @ A) @ R2))))).
% 3.15/1.63 thf(fact_250_trans__join, axiom, (! [A:$tType]: (((trans @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [X:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ X) @ R3) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y5:A,Y12:A]: ((! [Z4:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ Z4) @ R3) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y23:A,Aa:A]: (((Y12 = Y23) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Aa)) @ R3))))) @ Z4))))))) @ X)))))))))).
% 3.15/1.63 thf(fact_251_in__measure, axiom, (! [A:$tType,X2:A,Y3:A,F:(A > nat)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y3)) @ ((measure @ A) @ F)) = (((ord_less @ nat) @ (F @ X2)) @ (F @ Y3)))))).
% 3.15/1.63 thf(fact_252_trans__less__than, axiom, ((trans @ nat) @ less_than)).
% 3.15/1.63 thf(fact_253_less__than__iff, axiom, (! [X2:nat,Y3:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ X2) @ Y3)) @ less_than) = (((ord_less @ nat) @ X2) @ Y3))))).
% 3.15/1.63 thf(fact_254_trans__lex__prod, axiom, (! [A:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B))]: ((((trans @ A) @ R1) => (((trans @ B) @ R22) => ((trans @ ((product_prod @ A) @ B)) @ ((((lex_prod @ A) @ B) @ R1) @ R22))))))).
% 3.15/1.63 thf(fact_255_trans__Id__on, axiom, (! [A:$tType,A3:(set @ A)]: (((trans @ A) @ ((id_on @ A) @ A3))))).
% 3.15/1.63 thf(tcon_fun___Orderings_Obot, axiom, (! [A9:$tType,A10:$tType]: (((bot @ A10) => (bot @ (A9 > A10)))))).
% 3.15/1.63 thf(tcon_Nat_Onat___Orderings_Obot_1, axiom, (bot @ nat)).
% 3.15/1.63 thf(tcon_Set_Oset___Orderings_Obot_2, axiom, (! [A9:$tType]: ((bot @ (set @ A9))))).
% 3.15/1.63 thf(tcon_HOL_Obool___Orderings_Obot_3, axiom, (bot @ $o)).
% 3.15/1.63 thf(tcon_Product__Type_Ounit___Orderings_Obot_4, axiom, (bot @ product_unit)).
% 3.15/1.63 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.15/1.63 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y3:A]: ((((((if @ A) @ $false) @ X2) @ Y3) = Y3)))).
% 3.15/1.63 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y3:A]: ((((((if @ A) @ $true) @ X2) @ Y3) = X2)))).
% 3.15/1.63 thf(conj_0, hypothesis, (! [S7:state]: (((((((pHoare166426476thoare @ a) @ (((insert @ ((product_prod @ (a > (state > $o))) @ ((product_prod @ com) @ (a > (state > $o))))) @ ((((product_Pair @ (a > (state > $o))) @ ((product_prod @ com) @ (a > (state > $o)))) @ (^ [Z4:a,S4:state]: ((((pa @ Z4) @ S4) & (((member @ ((product_prod @ state) @ state)) @ ((((product_Pair @ state) @ state) @ S4) @ S7)) @ r))))) @ ((((product_Pair @ com) @ (a > (state > $o))) @ call) @ qa))) @ (bot_bot @ (set @ ((product_prod @ (a > (state > $o))) @ ((product_prod @ com) @ (a > (state > $o)))))))) @ (^ [Z4:a,S4:state]: ((((pa @ Z4) @ S4) & (S4 = S7))))) @ body) @ qa) & (((! [S:state,T2:state]: (((((exec @ S) @ body) @ T2) => (! [Z:a]: (((((pa @ Z) @ S) & (((member @ ((product_prod @ state) @ state)) @ ((((product_Pair @ state) @ state) @ S) @ S7)) @ r)) => ((qa @ Z) @ T2))))))) & (! [Z:a,S:state]: (((((pa @ Z) @ S) & (((member @ ((product_prod @ state) @ state)) @ ((((product_Pair @ state) @ state) @ S) @ S7)) @ r)) => ((termi @ body) @ S))))) => ((! [S8:state,T5:state]: (((((exec @ S8) @ body) @ T5) => (! [Z2:a]: (((((pa @ Z2) @ S8) & (S8 = S7)) => ((qa @ Z2) @ T5))))))) & (! [Z2:a,S8:state]: (((((pa @ Z2) @ S8) & (S8 = S7)) => ((termi @ body) @ S8)))))))))).
% 3.15/1.63 thf(conj_1, hypothesis, (! [Y4:state]: (((((member @ ((product_prod @ state) @ state)) @ ((((product_Pair @ state) @ state) @ Y4) @ x)) @ r) => ((! [S8:state,T5:state]: (((((exec @ S8) @ body) @ T5) => (! [Z2:a]: (((((pa @ Z2) @ S8) & (S8 = Y4)) => ((qa @ Z2) @ T5))))))) & (! [Z2:a,S8:state]: (((((pa @ Z2) @ S8) & (S8 = Y4)) => ((termi @ body) @ S8))))))))).
% 3.15/1.63 thf(conj_2, conjecture, ((! [S:state,T2:state]: (((~ ((((exec @ S) @ body) @ T2))) | (! [Z:a]: (((~ (((pa @ Z) @ S))) | ((S != x) | ((qa @ Z) @ T2)))))))) & (! [Z:a,S:state]: (((~ (((pa @ Z) @ S))) | ((S != x) | ((termi @ body) @ S))))))).
% 3.15/1.63 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.UIk1xi5Kzw/DTF2DTF13980.p
% 3.15/1.63 ------------------------
% 3.15/1.66 ---- Cleaned THF ---
% 3.15/1.66 thf(ty_t_Product__Type_Ounit,type,
% 3.15/1.66 product_unit: $tType ).
% 3.15/1.66
% 3.15/1.66 thf(ty_t_Product__Type_Oprod,type,
% 3.15/1.66 product_prod: $tType > $tType > $tType ).
% 3.15/1.66
% 3.15/1.66 thf(ty_t_PLang_Ostate,type,
% 3.15/1.66 state: $tType ).
% 3.15/1.66
% 3.15/1.66 thf(ty_t_PLang_Ocom,type,
% 3.15/1.66 com: $tType ).
% 3.15/1.66
% 3.15/1.66 thf(ty_t_Set_Oset,type,
% 3.15/1.66 set: $tType > $tType ).
% 3.15/1.66
% 3.15/1.66 thf(ty_t_Nat_Onat,type,
% 3.15/1.66 nat: $tType ).
% 3.15/1.66
% 3.15/1.66 thf(ty_tf_a,type,
% 3.15/1.66 a: $tType ).
% 3.15/1.66
% 3.15/1.66 thf(sy_cl_HOL_Otype,type,
% 3.15/1.66 type:
% 3.15/1.66 !>[A: $tType] : $o ).
% 3.15/1.66
% 3.15/1.66 thf(sy_cl_Orderings_Obot,type,
% 3.15/1.66 bot:
% 3.15/1.66 !>[A: $tType] : $o ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Basic__BNFs_Ofsts,type,
% 3.15/1.66 basic_fsts:
% 3.15/1.66 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Basic__BNFs_Osnds,type,
% 3.15/1.66 basic_snds:
% 3.15/1.66 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 3.15/1.66 fun_reduction_pair:
% 3.15/1.66 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 3.15/1.66 fun_rp_inv_image:
% 3.15/1.66 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_HOL_OThe,type,
% 3.15/1.66 the:
% 3.15/1.66 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_If,type,
% 3.15/1.66 if:
% 3.15/1.66 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Nat_OSuc,type,
% 3.15/1.66 suc: nat > nat ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Order__Relation_Oabove,type,
% 3.15/1.66 order_above:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Order__Relation_OaboveS,type,
% 3.15/1.66 order_aboveS:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 3.15/1.66 order_1409979114der_on:
% 3.15/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Order__Relation_Opreorder__on,type,
% 3.15/1.66 order_preorder_on:
% 3.15/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Orderings_Obot__class_Obot,type,
% 3.15/1.66 bot_bot:
% 3.15/1.66 !>[A: $tType] : A ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.15/1.66 ord_less:
% 3.15/1.66 !>[A: $tType] : ( A > A > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Octvalid,type,
% 3.15/1.66 pHoare304044376tvalid:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Othoare,type,
% 3.15/1.66 pHoare166426476thoare:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoareTotal__Mirabelle__nfzgdgthbw_Otvalid,type,
% 3.15/1.66 pHoare987424603tvalid:
% 3.15/1.66 !>[A: $tType] : ( ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoare_Ocnvalid,type,
% 3.15/1.66 cnvalid:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > nat > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoare_Ocvalid,type,
% 3.15/1.66 cvalid:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoare_Ohoare,type,
% 3.15/1.66 hoare:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoare_Onvalid,type,
% 3.15/1.66 nvalid:
% 3.15/1.66 !>[A: $tType] : ( nat > ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoare_Onvalids,type,
% 3.15/1.66 nvalids:
% 3.15/1.66 !>[A: $tType] : ( nat > ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoare_Ovalid,type,
% 3.15/1.66 valid:
% 3.15/1.66 !>[A: $tType] : ( ( A > state > $o ) > com > ( A > state > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PHoare_Ovalids,type,
% 3.15/1.66 valids:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PLang_Obody,type,
% 3.15/1.66 body: com ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PLang_Ocom_OCALL,type,
% 3.15/1.66 call: com ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PLang_Ocom_OSemi,type,
% 3.15/1.66 semi: com > com > com ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PLang_Ocom_OWhile,type,
% 3.15/1.66 while: ( state > $o ) > com > com ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PLang_Oexec,type,
% 3.15/1.66 exec: state > com > state > $o ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PLang_Oexecn,type,
% 3.15/1.66 execn: state > com > nat > state > $o ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_PTermi_Otermi,type,
% 3.15/1.66 termi: com > state > $o ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Product__Type_OPair,type,
% 3.15/1.66 product_Pair:
% 3.15/1.66 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.15/1.66 produc2004651681e_prod:
% 3.15/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.15/1.66 product_rec_prod:
% 3.15/1.66 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 3.15/1.66 product_rec_set_prod:
% 3.15/1.66 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 3.15/1.66 product_rec_set_unit:
% 3.15/1.66 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 3.15/1.66 product_rec_unit:
% 3.15/1.66 !>[T: $tType] : ( T > product_unit > T ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 3.15/1.66 product_case_prod:
% 3.15/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_ODomain,type,
% 3.15/1.66 domain:
% 3.15/1.66 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_OId__on,type,
% 3.15/1.66 id_on:
% 3.15/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_OPowp,type,
% 3.15/1.66 powp:
% 3.15/1.66 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_ORange,type,
% 3.15/1.66 range:
% 3.15/1.66 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_ORangep,type,
% 3.15/1.66 rangep:
% 3.15/1.66 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_Oinv__image,type,
% 3.15/1.66 inv_image:
% 3.15/1.66 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_Oirrefl,type,
% 3.15/1.66 irrefl:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_Oirreflp,type,
% 3.15/1.66 irreflp:
% 3.15/1.66 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_Orefl__on,type,
% 3.15/1.66 refl_on:
% 3.15/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_Ototal__on,type,
% 3.15/1.66 total_on:
% 3.15/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Relation_Otrans,type,
% 3.15/1.66 trans:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Set_OBall,type,
% 3.15/1.66 ball:
% 3.15/1.66 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Set_OCollect,type,
% 3.15/1.66 collect:
% 3.15/1.66 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Set_Oinsert,type,
% 3.15/1.66 insert:
% 3.15/1.66 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Set_Ois__empty,type,
% 3.15/1.66 is_empty:
% 3.15/1.66 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Set_Ois__singleton,type,
% 3.15/1.66 is_singleton:
% 3.15/1.66 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Set_Othe__elem,type,
% 3.15/1.66 the_elem:
% 3.15/1.66 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Wellfounded_Oless__than,type,
% 3.15/1.66 less_than: set @ ( product_prod @ nat @ nat ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Wellfounded_Olex__prod,type,
% 3.15/1.66 lex_prod:
% 3.15/1.66 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Wellfounded_Omeasure,type,
% 3.15/1.66 measure:
% 3.15/1.66 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Wellfounded_Omlex__prod,type,
% 3.15/1.66 mlex_prod:
% 3.15/1.66 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Wellfounded_Opred__nat,type,
% 3.15/1.66 pred_nat: set @ ( product_prod @ nat @ nat ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Wellfounded_Owf,type,
% 3.15/1.66 wf:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Wfrec_Osame__fst,type,
% 3.15/1.66 same_fst:
% 3.15/1.66 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_Zorn_OChains,type,
% 3.15/1.66 chains:
% 3.15/1.66 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_c_member,type,
% 3.15/1.66 member:
% 3.15/1.66 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_v_Pa,type,
% 3.15/1.66 pa: a > state > $o ).
% 3.15/1.66
% 3.15/1.66 thf(sy_v_Qa,type,
% 3.15/1.66 qa: a > state > $o ).
% 3.15/1.66
% 3.15/1.66 thf(sy_v_r,type,
% 3.15/1.66 r: set @ ( product_prod @ state @ state ) ).
% 3.15/1.66
% 3.15/1.66 thf(sy_v_x,type,
% 3.15/1.66 x: state ).
% 3.15/1.66
% 3.15/1.66 thf(fact_0_thoare_OAsm,axiom,
% 3.15/1.66 ! [A: $tType,P: A > state > $o,Q: A > state > $o] : ( pHoare166426476thoare @ A @ ( insert @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ ( product_Pair @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ P @ ( product_Pair @ com @ ( A > state > $o ) @ call @ Q ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) ) @ P @ call @ Q ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_1_weaken__post,axiom,
% 3.15/1.66 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,C3: com,Q: A > state > $o,Q2: A > state > $o] :
% 3.15/1.66 ( ( pHoare166426476thoare @ A @ C2 @ P @ C3 @ Q )
% 3.15/1.66 => ( ! [Z: A,S: state] :
% 3.15/1.66 ( ( Q @ Z @ S )
% 3.15/1.66 => ( Q2 @ Z @ S ) )
% 3.15/1.66 => ( pHoare166426476thoare @ A @ C2 @ P @ C3 @ Q2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_2_thoare_OConseq,axiom,
% 3.15/1.66 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P2: A > state > $o,C3: com,Q2: A > state > $o,P: A > state > $o,Q: A > state > $o] :
% 3.15/1.66 ( ( pHoare166426476thoare @ A @ C2 @ P2 @ C3 @ Q2 )
% 3.15/1.66 => ( ( ! [S: state,T2: state] :
% 3.15/1.66 ( ! [Z2: A] :
% 3.15/1.66 ( ( P2 @ Z2 @ S )
% 3.15/1.66 => ( Q2 @ Z2 @ T2 ) )
% 3.15/1.66 => ! [Z: A] :
% 3.15/1.66 ( ( P @ Z @ S )
% 3.15/1.66 => ( Q @ Z @ T2 ) ) )
% 3.15/1.66 & ! [S: state] :
% 3.15/1.66 ( ? [Z: A] : ( P @ Z @ S )
% 3.15/1.66 => ? [Z2: A] : ( P2 @ Z2 @ S ) ) )
% 3.15/1.66 => ( pHoare166426476thoare @ A @ C2 @ P @ C3 @ Q ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_3_PHoareTotal__Mirabelle__nfzgdgthbw_Ostrengthen__pre,axiom,
% 3.15/1.66 ! [A: $tType,P2: A > state > $o,P: A > state > $o,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),C3: com,Q: A > state > $o] :
% 3.15/1.66 ( ! [Z: A,S: state] :
% 3.15/1.66 ( ( P2 @ Z @ S )
% 3.15/1.66 => ( P @ Z @ S ) )
% 3.15/1.66 => ( ( pHoare166426476thoare @ A @ C2 @ P @ C3 @ Q )
% 3.15/1.66 => ( pHoare166426476thoare @ A @ C2 @ P2 @ C3 @ Q ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_4_singleton__conv,axiom,
% 3.15/1.66 ! [A: $tType,A2: A] :
% 3.15/1.66 ( ( collect @ A
% 3.15/1.66 @ ^ [X: A] : ( X = A2 ) )
% 3.15/1.66 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_5_singleton__conv2,axiom,
% 3.15/1.66 ! [A: $tType,A2: A] :
% 3.15/1.66 ( ( collect @ A
% 3.15/1.66 @ ( ^ [Y: A,Z3: A] : ( Y = Z3 )
% 3.15/1.66 @ A2 ) )
% 3.15/1.66 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_6_singletonI,axiom,
% 3.15/1.66 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_7_termi_Ointros_I7_J,axiom,
% 3.15/1.66 ! [S2: state] :
% 3.15/1.66 ( ( termi @ body @ S2 )
% 3.15/1.66 => ( termi @ call @ S2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_8_exec_Ointros_I7_J,axiom,
% 3.15/1.66 ! [S2: state,T3: state] :
% 3.15/1.66 ( ( exec @ S2 @ body @ T3 )
% 3.15/1.66 => ( exec @ S2 @ call @ T3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_9_thoare_OCall,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ state @ state ),P: A > state > $o,Q: A > state > $o] :
% 3.15/1.66 ( ( wf @ state @ R )
% 3.15/1.66 => ( ! [S3: state] :
% 3.15/1.66 ( pHoare166426476thoare @ A
% 3.15/1.66 @ ( insert @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) )
% 3.15/1.66 @ ( product_Pair @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) )
% 3.15/1.66 @ ^ [Z4: A,S4: state] :
% 3.15/1.66 ( ( P @ Z4 @ S4 )
% 3.15/1.66 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S4 @ S3 ) @ R ) )
% 3.15/1.66 @ ( product_Pair @ com @ ( A > state > $o ) @ call @ Q ) )
% 3.15/1.66 @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) )
% 3.15/1.66 @ ^ [Z4: A,S4: state] :
% 3.15/1.66 ( ( P @ Z4 @ S4 )
% 3.15/1.66 & ( S4 = S3 ) )
% 3.15/1.66 @ body
% 3.15/1.66 @ Q )
% 3.15/1.66 => ( pHoare166426476thoare @ A @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) @ P @ call @ Q ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_10_insertCI,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,B2: set @ A,B3: A] :
% 3.15/1.66 ( ( ~ ( member @ A @ A2 @ B2 )
% 3.15/1.66 => ( A2 = B3 ) )
% 3.15/1.66 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_11_insert__iff,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 3.15/1.66 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
% 3.15/1.66 = ( ( A2 = B3 )
% 3.15/1.66 | ( member @ A @ A2 @ A3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_12_insert__absorb2,axiom,
% 3.15/1.66 ! [A: $tType,X2: A,A3: set @ A] :
% 3.15/1.66 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A3 ) )
% 3.15/1.66 = ( insert @ A @ X2 @ A3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_13_hoare_OCall,axiom,
% 3.15/1.66 ! [A: $tType,P: A > state > $o,Q: A > state > $o] :
% 3.15/1.66 ( ( hoare @ A @ ( insert @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ ( product_Pair @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ P @ ( product_Pair @ com @ ( A > state > $o ) @ call @ Q ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) ) @ P @ body @ Q )
% 3.15/1.66 => ( hoare @ A @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) @ P @ call @ Q ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_14_empty__iff,axiom,
% 3.15/1.66 ! [A: $tType,C3: A] :
% 3.15/1.66 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_15_all__not__in__conv,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A] :
% 3.15/1.66 ( ( ! [X: A] :
% 3.15/1.66 ~ ( member @ A @ X @ A3 ) )
% 3.15/1.66 = ( A3
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_16_empty__Collect__eq,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o] :
% 3.15/1.66 ( ( ( bot_bot @ ( set @ A ) )
% 3.15/1.66 = ( collect @ A @ P ) )
% 3.15/1.66 = ( ! [X: A] :
% 3.15/1.66 ~ ( P @ X ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_17_Collect__empty__eq,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o] :
% 3.15/1.66 ( ( ( collect @ A @ P )
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.15/1.66 = ( ! [X: A] :
% 3.15/1.66 ~ ( P @ X ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_18_hoare_OConseq,axiom,
% 3.15/1.66 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P2: A > state > $o,C3: com,Q2: A > state > $o,P: A > state > $o,Q: A > state > $o] :
% 3.15/1.66 ( ( hoare @ A @ C2 @ P2 @ C3 @ Q2 )
% 3.15/1.66 => ( ! [S: state,T2: state] :
% 3.15/1.66 ( ! [Z2: A] :
% 3.15/1.66 ( ( P2 @ Z2 @ S )
% 3.15/1.66 => ( Q2 @ Z2 @ T2 ) )
% 3.15/1.66 => ! [Z: A] :
% 3.15/1.66 ( ( P @ Z @ S )
% 3.15/1.66 => ( Q @ Z @ T2 ) ) )
% 3.15/1.66 => ( hoare @ A @ C2 @ P @ C3 @ Q ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_19_PHoare_Ostrengthen__pre,axiom,
% 3.15/1.66 ! [A: $tType,P2: A > state > $o,P: A > state > $o,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),C3: com,Q: A > state > $o] :
% 3.15/1.66 ( ! [Z: A,S: state] :
% 3.15/1.66 ( ( P2 @ Z @ S )
% 3.15/1.66 => ( P @ Z @ S ) )
% 3.15/1.66 => ( ( hoare @ A @ C2 @ P @ C3 @ Q )
% 3.15/1.66 => ( hoare @ A @ C2 @ P2 @ C3 @ Q ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_20_ex__in__conv,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A] :
% 3.15/1.66 ( ( ? [X: A] : ( member @ A @ X @ A3 ) )
% 3.15/1.66 = ( A3
% 3.15/1.66 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_21_equals0I,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A] :
% 3.15/1.66 ( ! [Y2: A] :
% 3.15/1.66 ~ ( member @ A @ Y2 @ A3 )
% 3.15/1.66 => ( A3
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_22_equals0D,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A,A2: A] :
% 3.15/1.66 ( ( A3
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.15/1.66 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_23_emptyE,axiom,
% 3.15/1.66 ! [A: $tType,A2: A] :
% 3.15/1.66 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_24_mk__disjoint__insert,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,A3: set @ A] :
% 3.15/1.66 ( ( member @ A @ A2 @ A3 )
% 3.15/1.66 => ? [B4: set @ A] :
% 3.15/1.66 ( ( A3
% 3.15/1.66 = ( insert @ A @ A2 @ B4 ) )
% 3.15/1.66 & ~ ( member @ A @ A2 @ B4 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_25_insert__commute,axiom,
% 3.15/1.66 ! [A: $tType,X2: A,Y3: A,A3: set @ A] :
% 3.15/1.66 ( ( insert @ A @ X2 @ ( insert @ A @ Y3 @ A3 ) )
% 3.15/1.66 = ( insert @ A @ Y3 @ ( insert @ A @ X2 @ A3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_26_insert__eq__iff,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,A3: set @ A,B3: A,B2: set @ A] :
% 3.15/1.66 ( ~ ( member @ A @ A2 @ A3 )
% 3.15/1.66 => ( ~ ( member @ A @ B3 @ B2 )
% 3.15/1.66 => ( ( ( insert @ A @ A2 @ A3 )
% 3.15/1.66 = ( insert @ A @ B3 @ B2 ) )
% 3.15/1.66 = ( ( ( A2 = B3 )
% 3.15/1.66 => ( A3 = B2 ) )
% 3.15/1.66 & ( ( A2 != B3 )
% 3.15/1.66 => ? [C4: set @ A] :
% 3.15/1.66 ( ( A3
% 3.15/1.66 = ( insert @ A @ B3 @ C4 ) )
% 3.15/1.66 & ~ ( member @ A @ B3 @ C4 )
% 3.15/1.66 & ( B2
% 3.15/1.66 = ( insert @ A @ A2 @ C4 ) )
% 3.15/1.66 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_27_insert__absorb,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,A3: set @ A] :
% 3.15/1.66 ( ( member @ A @ A2 @ A3 )
% 3.15/1.66 => ( ( insert @ A @ A2 @ A3 )
% 3.15/1.66 = A3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_28_insert__ident,axiom,
% 3.15/1.66 ! [A: $tType,X2: A,A3: set @ A,B2: set @ A] :
% 3.15/1.66 ( ~ ( member @ A @ X2 @ A3 )
% 3.15/1.66 => ( ~ ( member @ A @ X2 @ B2 )
% 3.15/1.66 => ( ( ( insert @ A @ X2 @ A3 )
% 3.15/1.66 = ( insert @ A @ X2 @ B2 ) )
% 3.15/1.66 = ( A3 = B2 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_29_Set_Oset__insert,axiom,
% 3.15/1.66 ! [A: $tType,X2: A,A3: set @ A] :
% 3.15/1.66 ( ( member @ A @ X2 @ A3 )
% 3.15/1.66 => ~ ! [B4: set @ A] :
% 3.15/1.66 ( ( A3
% 3.15/1.66 = ( insert @ A @ X2 @ B4 ) )
% 3.15/1.66 => ( member @ A @ X2 @ B4 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_30_insertI2,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,B2: set @ A,B3: A] :
% 3.15/1.66 ( ( member @ A @ A2 @ B2 )
% 3.15/1.66 => ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_31_insertI1,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,B2: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_32_insertE,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 3.15/1.66 ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
% 3.15/1.66 => ( ( A2 != B3 )
% 3.15/1.66 => ( member @ A @ A2 @ A3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_33_empty__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( bot_bot @ ( set @ A ) )
% 3.15/1.66 = ( collect @ A
% 3.15/1.66 @ ^ [X: A] : $false ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_34_insert__Collect,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,P: A > $o] :
% 3.15/1.66 ( ( insert @ A @ A2 @ ( collect @ A @ P ) )
% 3.15/1.66 = ( collect @ A
% 3.15/1.66 @ ^ [U: A] :
% 3.15/1.66 ( ( U != A2 )
% 3.15/1.66 => ( P @ U ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_35_insert__compr,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( insert @ A )
% 3.15/1.66 = ( ^ [A4: A,B5: set @ A] :
% 3.15/1.66 ( collect @ A
% 3.15/1.66 @ ^ [X: A] :
% 3.15/1.66 ( ( X = A4 )
% 3.15/1.66 | ( member @ A @ X @ B5 ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_36_singleton__inject,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,B3: A] :
% 3.15/1.66 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 3.15/1.66 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.15/1.66 => ( A2 = B3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_37_insert__not__empty,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,A3: set @ A] :
% 3.15/1.66 ( ( insert @ A @ A2 @ A3 )
% 3.15/1.66 != ( bot_bot @ ( set @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_38_doubleton__eq__iff,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,B3: A,C3: A,D: A] :
% 3.15/1.66 ( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.15/1.66 = ( insert @ A @ C3 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.15/1.66 = ( ( ( A2 = C3 )
% 3.15/1.66 & ( B3 = D ) )
% 3.15/1.66 | ( ( A2 = D )
% 3.15/1.66 & ( B3 = C3 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_39_singleton__iff,axiom,
% 3.15/1.66 ! [A: $tType,B3: A,A2: A] :
% 3.15/1.66 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.15/1.66 = ( B3 = A2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_40_singletonD,axiom,
% 3.15/1.66 ! [A: $tType,B3: A,A2: A] :
% 3.15/1.66 ( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.15/1.66 => ( B3 = A2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_41_hoare_OAsm,axiom,
% 3.15/1.66 ! [A: $tType,P: A > state > $o,Q: A > state > $o] : ( hoare @ A @ ( insert @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ ( product_Pair @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ P @ ( product_Pair @ com @ ( A > state > $o ) @ call @ Q ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) ) ) ) @ P @ call @ Q ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_42_Collect__conv__if2,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o,A2: A] :
% 3.15/1.66 ( ( ( P @ A2 )
% 3.15/1.66 => ( ( collect @ A
% 3.15/1.66 @ ^ [X: A] :
% 3.15/1.66 ( ( A2 = X )
% 3.15/1.66 & ( P @ X ) ) )
% 3.15/1.66 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.15/1.66 & ( ~ ( P @ A2 )
% 3.15/1.66 => ( ( collect @ A
% 3.15/1.66 @ ^ [X: A] :
% 3.15/1.66 ( ( A2 = X )
% 3.15/1.66 & ( P @ X ) ) )
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_43_Collect__conv__if,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o,A2: A] :
% 3.15/1.66 ( ( ( P @ A2 )
% 3.15/1.66 => ( ( collect @ A
% 3.15/1.66 @ ^ [X: A] :
% 3.15/1.66 ( ( X = A2 )
% 3.15/1.66 & ( P @ X ) ) )
% 3.15/1.66 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.15/1.66 & ( ~ ( P @ A2 )
% 3.15/1.66 => ( ( collect @ A
% 3.15/1.66 @ ^ [X: A] :
% 3.15/1.66 ( ( X = A2 )
% 3.15/1.66 & ( P @ X ) ) )
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_44_wf__empty,axiom,
% 3.15/1.66 ! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_45_mem__Collect__eq,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,P: A > $o] :
% 3.15/1.66 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.15/1.66 = ( P @ A2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_46_Collect__mem__eq,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A] :
% 3.15/1.66 ( ( collect @ A
% 3.15/1.66 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 3.15/1.66 = A3 ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_47_Collect__cong,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.15/1.66 ( ! [X3: A] :
% 3.15/1.66 ( ( P @ X3 )
% 3.15/1.66 = ( Q @ X3 ) )
% 3.15/1.66 => ( ( collect @ A @ P )
% 3.15/1.66 = ( collect @ A @ Q ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_48_ext,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.15/1.66 ( ! [X3: A] :
% 3.15/1.66 ( ( F @ X3 )
% 3.15/1.66 = ( G @ X3 ) )
% 3.15/1.66 => ( F = G ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_49_wfE__min_H,axiom,
% 3.15/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),Q: set @ A] :
% 3.15/1.66 ( ( wf @ A @ R2 )
% 3.15/1.66 => ( ( Q
% 3.15/1.66 != ( bot_bot @ ( set @ A ) ) )
% 3.15/1.66 => ~ ! [Z: A] :
% 3.15/1.66 ( ( member @ A @ Z @ Q )
% 3.15/1.66 => ~ ! [Y4: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 3.15/1.66 => ~ ( member @ A @ Y4 @ Q ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_50_the__elem__eq,axiom,
% 3.15/1.66 ! [A: $tType,X2: A] :
% 3.15/1.66 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.15/1.66 = X2 ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_51_bot__apply,axiom,
% 3.15/1.66 ! [C: $tType,D2: $tType] :
% 3.15/1.66 ( ( bot @ C )
% 3.15/1.66 => ( ( bot_bot @ ( D2 > C ) )
% 3.15/1.66 = ( ^ [X: D2] : ( bot_bot @ C ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_52_old_Oprod_Oinject,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,A2: A,B3: B,A5: A,B6: B] :
% 3.15/1.66 ( ( ( product_Pair @ A @ B @ A2 @ B3 )
% 3.15/1.66 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 3.15/1.66 = ( ( A2 = A5 )
% 3.15/1.66 & ( B3 = B6 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_53_prod_Oinject,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 3.15/1.66 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 3.15/1.66 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 3.15/1.66 = ( ( X1 = Y1 )
% 3.15/1.66 & ( X22 = Y22 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_54_is__singletonI,axiom,
% 3.15/1.66 ! [A: $tType,X2: A] : ( is_singleton @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_55_prod__induct7,axiom,
% 3.15/1.66 ! [G2: $tType,F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 3.15/1.66 ( ! [A6: A,B7: B,C5: C,D3: D2,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C5 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 3.15/1.66 => ( P @ X2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_56_prod__induct6,axiom,
% 3.15/1.66 ! [F2: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 3.15/1.66 ( ! [A6: A,B7: B,C5: C,D3: D2,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C5 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 3.15/1.66 => ( P @ X2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_57_prod__induct5,axiom,
% 3.15/1.66 ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 3.15/1.66 ( ! [A6: A,B7: B,C5: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C5 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
% 3.15/1.66 => ( P @ X2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_58_bot__set__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( bot_bot @ ( set @ A ) )
% 3.15/1.66 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_59_is__singleton__the__elem,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( is_singleton @ A )
% 3.15/1.66 = ( ^ [A7: set @ A] :
% 3.15/1.66 ( A7
% 3.15/1.66 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_60_is__singletonI_H,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A] :
% 3.15/1.66 ( ( A3
% 3.15/1.66 != ( bot_bot @ ( set @ A ) ) )
% 3.15/1.66 => ( ! [X3: A,Y2: A] :
% 3.15/1.66 ( ( member @ A @ X3 @ A3 )
% 3.15/1.66 => ( ( member @ A @ Y2 @ A3 )
% 3.15/1.66 => ( X3 = Y2 ) ) )
% 3.15/1.66 => ( is_singleton @ A @ A3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_61_surj__pair,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
% 3.15/1.66 ? [X3: A,Y2: B] :
% 3.15/1.66 ( P3
% 3.15/1.66 = ( product_Pair @ A @ B @ X3 @ Y2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_62_prod__cases,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
% 3.15/1.66 ( ! [A6: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B7 ) )
% 3.15/1.66 => ( P @ P3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_63_Pair__inject,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,A2: A,B3: B,A5: A,B6: B] :
% 3.15/1.66 ( ( ( product_Pair @ A @ B @ A2 @ B3 )
% 3.15/1.66 = ( product_Pair @ A @ B @ A5 @ B6 ) )
% 3.15/1.66 => ~ ( ( A2 = A5 )
% 3.15/1.66 => ( B3 != B6 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_64_old_Oprod_Oexhaust,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 3.15/1.66 ~ ! [A6: A,B7: B] :
% 3.15/1.66 ( Y3
% 3.15/1.66 != ( product_Pair @ A @ B @ A6 @ B7 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_65_old_Oprod_Oinducts,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.15/1.66 ( ! [A6: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B7 ) )
% 3.15/1.66 => ( P @ Prod ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_66_bot__fun__def,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType] :
% 3.15/1.66 ( ( bot @ B )
% 3.15/1.66 => ( ( bot_bot @ ( A > B ) )
% 3.15/1.66 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_67_is__singletonE,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A] :
% 3.15/1.66 ( ( is_singleton @ A @ A3 )
% 3.15/1.66 => ~ ! [X3: A] :
% 3.15/1.66 ( A3
% 3.15/1.66 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_68_is__singleton__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( is_singleton @ A )
% 3.15/1.66 = ( ^ [A7: set @ A] :
% 3.15/1.66 ? [X: A] :
% 3.15/1.66 ( A7
% 3.15/1.66 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_69_wf__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( wf @ A )
% 3.15/1.66 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.15/1.66 ! [P4: A > $o] :
% 3.15/1.66 ( ! [X: A] :
% 3.15/1.66 ( ! [Y5: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R3 )
% 3.15/1.66 => ( P4 @ Y5 ) )
% 3.15/1.66 => ( P4 @ X ) )
% 3.15/1.66 => ! [X4: A] : ( P4 @ X4 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_70_wfE__min,axiom,
% 3.15/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X2: A,Q: set @ A] :
% 3.15/1.66 ( ( wf @ A @ R2 )
% 3.15/1.66 => ( ( member @ A @ X2 @ Q )
% 3.15/1.66 => ~ ! [Z: A] :
% 3.15/1.66 ( ( member @ A @ Z @ Q )
% 3.15/1.66 => ~ ! [Y4: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R2 )
% 3.15/1.66 => ~ ( member @ A @ Y4 @ Q ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_71_wfI__min,axiom,
% 3.15/1.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.15/1.66 ( ! [X3: A,Q3: set @ A] :
% 3.15/1.66 ( ( member @ A @ X3 @ Q3 )
% 3.15/1.66 => ? [Xa: A] :
% 3.15/1.66 ( ( member @ A @ Xa @ Q3 )
% 3.15/1.66 & ! [Y2: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa ) @ R2 )
% 3.15/1.66 => ~ ( member @ A @ Y2 @ Q3 ) ) ) )
% 3.15/1.66 => ( wf @ A @ R2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_72_wfUNIVI,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 3.15/1.66 ( ! [P5: A > $o,X3: A] :
% 3.15/1.66 ( ! [Xa: A] :
% 3.15/1.66 ( ! [Y2: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa ) @ R )
% 3.15/1.66 => ( P5 @ Y2 ) )
% 3.15/1.66 => ( P5 @ Xa ) )
% 3.15/1.66 => ( P5 @ X3 ) )
% 3.15/1.66 => ( wf @ A @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_73_wf__asym,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X2: A] :
% 3.15/1.66 ( ( wf @ A @ R )
% 3.15/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X2 ) @ R )
% 3.15/1.66 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A2 ) @ R ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_74_wf__induct,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A2: A] :
% 3.15/1.66 ( ( wf @ A @ R )
% 3.15/1.66 => ( ! [X3: A] :
% 3.15/1.66 ( ! [Y4: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R )
% 3.15/1.66 => ( P @ Y4 ) )
% 3.15/1.66 => ( P @ X3 ) )
% 3.15/1.66 => ( P @ A2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_75_wf__irrefl,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 3.15/1.66 ( ( wf @ A @ R )
% 3.15/1.66 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_76_wf__not__sym,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A,X2: A] :
% 3.15/1.66 ( ( wf @ A @ R )
% 3.15/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X2 ) @ R )
% 3.15/1.66 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A2 ) @ R ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_77_wf__not__refl,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: A] :
% 3.15/1.66 ( ( wf @ A @ R )
% 3.15/1.66 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_78_wf__eq__minimal,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( wf @ A )
% 3.15/1.66 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.15/1.66 ! [Q4: set @ A] :
% 3.15/1.66 ( ? [X: A] : ( member @ A @ X @ Q4 )
% 3.15/1.66 => ? [X: A] :
% 3.15/1.66 ( ( member @ A @ X @ Q4 )
% 3.15/1.66 & ! [Y5: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R3 )
% 3.15/1.66 => ~ ( member @ A @ Y5 @ Q4 ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_79_wf__induct__rule,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A2: A] :
% 3.15/1.66 ( ( wf @ A @ R )
% 3.15/1.66 => ( ! [X3: A] :
% 3.15/1.66 ( ! [Y4: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R )
% 3.15/1.66 => ( P @ Y4 ) )
% 3.15/1.66 => ( P @ X3 ) )
% 3.15/1.66 => ( P @ A2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_80_prod__cases3,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.15/1.66 ~ ! [A6: A,B7: B,C5: C] :
% 3.15/1.66 ( Y3
% 3.15/1.66 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B7 @ C5 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_81_prod__cases4,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 3.15/1.66 ~ ! [A6: A,B7: B,C5: C,D3: D2] :
% 3.15/1.66 ( Y3
% 3.15/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C5 @ D3 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_82_prod__cases5,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
% 3.15/1.66 ~ ! [A6: A,B7: B,C5: C,D3: D2,E2: E] :
% 3.15/1.66 ( Y3
% 3.15/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C5 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_83_prod__cases6,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) )] :
% 3.15/1.66 ~ ! [A6: A,B7: B,C5: C,D3: D2,E2: E,F3: F2] :
% 3.15/1.66 ( Y3
% 3.15/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F2 ) ) @ C5 @ ( product_Pair @ D2 @ ( product_prod @ E @ F2 ) @ D3 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_84_prod__cases7,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F2: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 3.15/1.66 ~ ! [A6: A,B7: B,C5: C,D3: D2,E2: E,F3: F2,G3: G2] :
% 3.15/1.66 ( Y3
% 3.15/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C5 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_85_prod__induct3,axiom,
% 3.15/1.66 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.15/1.66 ( ! [A6: A,B7: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B7 @ C5 ) ) )
% 3.15/1.66 => ( P @ X2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_86_prod__induct4,axiom,
% 3.15/1.66 ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
% 3.15/1.66 ( ! [A6: A,B7: B,C5: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B7 @ ( product_Pair @ C @ D2 @ C5 @ D3 ) ) ) )
% 3.15/1.66 => ( P @ X2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_87_old_Oprod_Orec,axiom,
% 3.15/1.66 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B3: B] :
% 3.15/1.66 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B3 ) )
% 3.15/1.66 = ( F1 @ A2 @ B3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_88_bot__empty__eq2,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType] :
% 3.15/1.66 ( ( bot_bot @ ( A > B > $o ) )
% 3.15/1.66 = ( ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_89_wf__measure,axiom,
% 3.15/1.66 ! [A: $tType,F: A > nat] : ( wf @ A @ ( measure @ A @ F ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_90_dependent__wf__choice,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
% 3.15/1.66 ( ( wf @ A @ R2 )
% 3.15/1.66 => ( ! [F3: A > B,G3: A > B,X3: A,R4: B] :
% 3.15/1.66 ( ! [Z2: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ X3 ) @ R2 )
% 3.15/1.66 => ( ( F3 @ Z2 )
% 3.15/1.66 = ( G3 @ Z2 ) ) )
% 3.15/1.66 => ( ( P @ F3 @ X3 @ R4 )
% 3.15/1.66 = ( P @ G3 @ X3 @ R4 ) ) )
% 3.15/1.66 => ( ! [X3: A,F3: A > B] :
% 3.15/1.66 ( ! [Y4: A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R2 )
% 3.15/1.66 => ( P @ F3 @ Y4 @ ( F3 @ Y4 ) ) )
% 3.15/1.66 => ? [X_1: B] : ( P @ F3 @ X3 @ X_1 ) )
% 3.15/1.66 => ? [F3: A > B] :
% 3.15/1.66 ! [X5: A] : ( P @ F3 @ X5 @ ( F3 @ X5 ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_91_thoare_OWhile,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ state @ state ),C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,B3: state > $o,C3: com] :
% 3.15/1.66 ( ( wf @ state @ R )
% 3.15/1.66 => ( ! [S3: state] :
% 3.15/1.66 ( pHoare166426476thoare @ A @ C2
% 3.15/1.66 @ ^ [Z4: A,S4: state] :
% 3.15/1.66 ( ( P @ Z4 @ S4 )
% 3.15/1.66 & ( B3 @ S4 )
% 3.15/1.66 & ( S3 = S4 ) )
% 3.15/1.66 @ C3
% 3.15/1.66 @ ^ [Z4: A,S4: state] :
% 3.15/1.66 ( ( P @ Z4 @ S4 )
% 3.15/1.66 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S4 @ S3 ) @ R ) ) )
% 3.15/1.66 => ( pHoare166426476thoare @ A @ C2 @ P @ ( while @ B3 @ C3 )
% 3.15/1.66 @ ^ [Z4: A,S4: state] :
% 3.15/1.66 ( ( P @ Z4 @ S4 )
% 3.15/1.66 & ~ ( B3 @ S4 ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_92_hoare__sound,axiom,
% 3.15/1.66 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,C3: com,Q: A > state > $o] :
% 3.15/1.66 ( ( hoare @ A @ C2 @ P @ C3 @ Q )
% 3.15/1.66 => ( cvalid @ A @ C2 @ P @ C3 @ Q ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_93_Set_Ois__empty__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( is_empty @ A )
% 3.15/1.66 = ( ^ [A7: set @ A] :
% 3.15/1.66 ( A7
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_94_com_Oinject_I4_J,axiom,
% 3.15/1.66 ! [X41: state > $o,X42: com,Y41: state > $o,Y42: com] :
% 3.15/1.66 ( ( ( while @ X41 @ X42 )
% 3.15/1.66 = ( while @ Y41 @ Y42 ) )
% 3.15/1.66 = ( ( X41 = Y41 )
% 3.15/1.66 & ( X42 = Y42 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_95_bot2E,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,X2: A,Y3: B] :
% 3.15/1.66 ~ ( bot_bot @ ( A > B > $o ) @ X2 @ Y3 ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_96_com_Odistinct_I25_J,axiom,
% 3.15/1.66 ! [X41: state > $o,X42: com] :
% 3.15/1.66 ( ( while @ X41 @ X42 )
% 3.15/1.66 != call ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_97_exec_OWhileFalse,axiom,
% 3.15/1.66 ! [B3: state > $o,S2: state,C3: com] :
% 3.15/1.66 ( ~ ( B3 @ S2 )
% 3.15/1.66 => ( exec @ S2 @ ( while @ B3 @ C3 ) @ S2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_98_exec_OWhileTrue,axiom,
% 3.15/1.66 ! [B3: state > $o,S2: state,C3: com,T3: state,U2: state] :
% 3.15/1.66 ( ( B3 @ S2 )
% 3.15/1.66 => ( ( exec @ S2 @ C3 @ T3 )
% 3.15/1.66 => ( ( exec @ T3 @ ( while @ B3 @ C3 ) @ U2 )
% 3.15/1.66 => ( exec @ S2 @ ( while @ B3 @ C3 ) @ U2 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_99_termi_OWhileFalse,axiom,
% 3.15/1.66 ! [B3: state > $o,S2: state,C3: com] :
% 3.15/1.66 ( ~ ( B3 @ S2 )
% 3.15/1.66 => ( termi @ ( while @ B3 @ C3 ) @ S2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_100_hoare_OWhile,axiom,
% 3.15/1.66 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,B3: state > $o,C3: com] :
% 3.15/1.66 ( ( hoare @ A @ C2
% 3.15/1.66 @ ^ [Z4: A,S4: state] :
% 3.15/1.66 ( ( P @ Z4 @ S4 )
% 3.15/1.66 & ( B3 @ S4 ) )
% 3.15/1.66 @ C3
% 3.15/1.66 @ P )
% 3.15/1.66 => ( hoare @ A @ C2 @ P @ ( while @ B3 @ C3 )
% 3.15/1.66 @ ^ [Z4: A,S4: state] :
% 3.15/1.66 ( ( P @ Z4 @ S4 )
% 3.15/1.66 & ~ ( B3 @ S4 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_101_termi_OWhileTrue,axiom,
% 3.15/1.66 ! [B3: state > $o,S2: state,C3: com] :
% 3.15/1.66 ( ( B3 @ S2 )
% 3.15/1.66 => ( ( termi @ C3 @ S2 )
% 3.15/1.66 => ( ! [T2: state] :
% 3.15/1.66 ( ( exec @ S2 @ C3 @ T2 )
% 3.15/1.66 => ( termi @ ( while @ B3 @ C3 ) @ T2 ) )
% 3.15/1.66 => ( termi @ ( while @ B3 @ C3 ) @ S2 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_102_bot__empty__eq,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( bot_bot @ ( A > $o ) )
% 3.15/1.66 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_103_pred__equals__eq2,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( ( ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ R2 ) )
% 3.15/1.66 = ( ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ S5 ) ) )
% 3.15/1.66 = ( R2 = S5 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_104_Collect__empty__eq__bot,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o] :
% 3.15/1.66 ( ( ( collect @ A @ P )
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.15/1.66 = ( P
% 3.15/1.66 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_105_total__on__singleton,axiom,
% 3.15/1.66 ! [A: $tType,X2: A] : ( total_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_106_refl__on__singleton,axiom,
% 3.15/1.66 ! [A: $tType,X2: A] : ( refl_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_107_wf__termi,axiom,
% 3.15/1.66 ! [B3: state > $o,C3: com] :
% 3.15/1.66 ( wf @ state
% 3.15/1.66 @ ( collect @ ( product_prod @ state @ state )
% 3.15/1.66 @ ( product_case_prod @ state @ state @ $o
% 3.15/1.66 @ ^ [T4: state,S4: state] :
% 3.15/1.66 ( ( termi @ ( while @ B3 @ C3 ) @ S4 )
% 3.15/1.66 & ( B3 @ S4 )
% 3.15/1.66 & ( exec @ S4 @ C3 @ T4 ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_108_linear__order__on__singleton,axiom,
% 3.15/1.66 ! [A: $tType,X2: A] : ( order_1409979114der_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_109_split__part,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 3.15/1.66 ( ( product_case_prod @ A @ B @ $o
% 3.15/1.66 @ ^ [A4: A,B8: B] :
% 3.15/1.66 ( P
% 3.15/1.66 & ( Q @ A4 @ B8 ) ) )
% 3.15/1.66 = ( ^ [Ab: product_prod @ A @ B] :
% 3.15/1.66 ( P
% 3.15/1.66 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_110_case__prod__conv,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B3: C] :
% 3.15/1.66 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B3 ) )
% 3.15/1.66 = ( F @ A2 @ B3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_111_case__prodI,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B3: B] :
% 3.15/1.66 ( ( F @ A2 @ B3 )
% 3.15/1.66 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_112_case__prodI2,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,P3: product_prod @ A @ B,C3: A > B > $o] :
% 3.15/1.66 ( ! [A6: A,B7: B] :
% 3.15/1.66 ( ( P3
% 3.15/1.66 = ( product_Pair @ A @ B @ A6 @ B7 ) )
% 3.15/1.66 => ( C3 @ A6 @ B7 ) )
% 3.15/1.66 => ( product_case_prod @ A @ B @ $o @ C3 @ P3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_113_refl__on__domain,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A,B3: A] :
% 3.15/1.66 ( ( refl_on @ A @ A3 @ R )
% 3.15/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B3 ) @ R )
% 3.15/1.66 => ( ( member @ A @ A2 @ A3 )
% 3.15/1.66 & ( member @ A @ B3 @ A3 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_114_prod_Ocase__distrib,axiom,
% 3.15/1.66 ! [C: $tType,D2: $tType,B: $tType,A: $tType,H: C > D2,F: A > B > C,Prod: product_prod @ A @ B] :
% 3.15/1.66 ( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 3.15/1.66 = ( product_case_prod @ A @ B @ D2
% 3.15/1.66 @ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
% 3.15/1.66 @ Prod ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_115_prod_Odisc__eq__case,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.15/1.66 ( product_case_prod @ A @ B @ $o
% 3.15/1.66 @ ^ [Uu: A,Uv: B] : $true
% 3.15/1.66 @ Prod ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_116_old_Oprod_Ocase,axiom,
% 3.15/1.66 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 3.15/1.66 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 3.15/1.66 = ( F @ X1 @ X22 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_117_refl__onD2,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
% 3.15/1.66 ( ( refl_on @ A @ A3 @ R )
% 3.15/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
% 3.15/1.66 => ( member @ A @ Y3 @ A3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_118_refl__onD1,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
% 3.15/1.66 ( ( refl_on @ A @ A3 @ R )
% 3.15/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
% 3.15/1.66 => ( member @ A @ X2 @ A3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_119_refl__onD,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A ),A2: A] :
% 3.15/1.66 ( ( refl_on @ A @ A3 @ R )
% 3.15/1.66 => ( ( member @ A @ A2 @ A3 )
% 3.15/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_120_case__prodD,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,F: A > B > $o,A2: A,B3: B] :
% 3.15/1.66 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B3 ) )
% 3.15/1.66 => ( F @ A2 @ B3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_121_case__prodE,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,C3: A > B > $o,P3: product_prod @ A @ B] :
% 3.15/1.66 ( ( product_case_prod @ A @ B @ $o @ C3 @ P3 )
% 3.15/1.66 => ~ ! [X3: A,Y2: B] :
% 3.15/1.66 ( ( P3
% 3.15/1.66 = ( product_Pair @ A @ B @ X3 @ Y2 ) )
% 3.15/1.66 => ~ ( C3 @ X3 @ Y2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_122_total__on__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( total_on @ A )
% 3.15/1.66 = ( ^ [A7: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 3.15/1.66 ! [X: A] :
% 3.15/1.66 ( ( member @ A @ X @ A7 )
% 3.15/1.66 => ! [Y5: A] :
% 3.15/1.66 ( ( member @ A @ Y5 @ A7 )
% 3.15/1.66 => ( ( X != Y5 )
% 3.15/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R3 )
% 3.15/1.66 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R3 ) ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_123_total__onI,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
% 3.15/1.66 ( ! [X3: A,Y2: A] :
% 3.15/1.66 ( ( member @ A @ X3 @ A3 )
% 3.15/1.66 => ( ( member @ A @ Y2 @ A3 )
% 3.15/1.66 => ( ( X3 != Y2 )
% 3.15/1.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
% 3.15/1.66 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ R ) ) ) ) )
% 3.15/1.66 => ( total_on @ A @ A3 @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_124_total__on__empty,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_125_case__prodE2,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z5: product_prod @ B @ C] :
% 3.15/1.66 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z5 ) )
% 3.15/1.66 => ~ ! [X3: B,Y2: C] :
% 3.15/1.66 ( ( Z5
% 3.15/1.66 = ( product_Pair @ B @ C @ X3 @ Y2 ) )
% 3.15/1.66 => ~ ( Q @ ( P @ X3 @ Y2 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_126_case__prod__eta,axiom,
% 3.15/1.66 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 3.15/1.66 ( ( product_case_prod @ A @ B @ C
% 3.15/1.66 @ ^ [X: A,Y5: B] : ( F @ ( product_Pair @ A @ B @ X @ Y5 ) ) )
% 3.15/1.66 = F ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_127_cond__case__prod__eta,axiom,
% 3.15/1.66 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 3.15/1.66 ( ! [X3: A,Y2: B] :
% 3.15/1.66 ( ( F @ X3 @ Y2 )
% 3.15/1.66 = ( G @ ( product_Pair @ A @ B @ X3 @ Y2 ) ) )
% 3.15/1.66 => ( ( product_case_prod @ A @ B @ C @ F )
% 3.15/1.66 = G ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_128_lnear__order__on__empty,axiom,
% 3.15/1.66 ! [A: $tType] : ( order_1409979114der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_129_refl__on__empty,axiom,
% 3.15/1.66 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_130_split__cong,axiom,
% 3.15/1.66 ! [C: $tType,B: $tType,A: $tType,Q5: product_prod @ A @ B,F: A > B > C,G: A > B > C,P3: product_prod @ A @ B] :
% 3.15/1.66 ( ! [X3: A,Y2: B] :
% 3.15/1.66 ( ( ( product_Pair @ A @ B @ X3 @ Y2 )
% 3.15/1.66 = Q5 )
% 3.15/1.66 => ( ( F @ X3 @ Y2 )
% 3.15/1.66 = ( G @ X3 @ Y2 ) ) )
% 3.15/1.66 => ( ( P3 = Q5 )
% 3.15/1.66 => ( ( product_case_prod @ A @ B @ C @ F @ P3 )
% 3.15/1.66 = ( product_case_prod @ A @ B @ C @ G @ Q5 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_131_aboveS__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( order_aboveS @ A )
% 3.15/1.66 = ( ^ [R3: set @ ( product_prod @ A @ A ),A4: A] :
% 3.15/1.66 ( collect @ A
% 3.15/1.66 @ ^ [B8: A] :
% 3.15/1.66 ( ( B8 != A4 )
% 3.15/1.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B8 ) @ R3 ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_132_case__prod__app,axiom,
% 3.15/1.66 ! [A: $tType,D2: $tType,C: $tType,B: $tType] :
% 3.15/1.66 ( ( product_case_prod @ B @ C @ ( D2 > A ) )
% 3.15/1.66 = ( ^ [F4: B > C > D2 > A,X: product_prod @ B @ C,Y5: D2] :
% 3.15/1.66 ( product_case_prod @ B @ C @ A
% 3.15/1.66 @ ^ [L: B,R3: C] : ( F4 @ L @ R3 @ Y5 )
% 3.15/1.66 @ X ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_133_internal__case__prod__def,axiom,
% 3.15/1.66 ! [C: $tType,B: $tType,A: $tType] :
% 3.15/1.66 ( ( produc2004651681e_prod @ A @ B @ C )
% 3.15/1.66 = ( product_case_prod @ A @ B @ C ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_134_internal__case__prod__conv,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B3: C] :
% 3.15/1.66 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B3 ) )
% 3.15/1.66 = ( C3 @ A2 @ B3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_135_mem__case__prodI2,axiom,
% 3.15/1.66 ! [C: $tType,B: $tType,A: $tType,P3: product_prod @ A @ B,Z5: C,C3: A > B > ( set @ C )] :
% 3.15/1.66 ( ! [A6: A,B7: B] :
% 3.15/1.66 ( ( P3
% 3.15/1.66 = ( product_Pair @ A @ B @ A6 @ B7 ) )
% 3.15/1.66 => ( member @ C @ Z5 @ ( C3 @ A6 @ B7 ) ) )
% 3.15/1.66 => ( member @ C @ Z5 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_136_mem__case__prodI,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,C: $tType,Z5: A,C3: B > C > ( set @ A ),A2: B,B3: C] :
% 3.15/1.66 ( ( member @ A @ Z5 @ ( C3 @ A2 @ B3 ) )
% 3.15/1.66 => ( member @ A @ Z5 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A2 @ B3 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_137_case__prodI2_H,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,C: $tType,P3: product_prod @ A @ B,C3: A > B > C > $o,X2: C] :
% 3.15/1.66 ( ! [A6: A,B7: B] :
% 3.15/1.66 ( ( ( product_Pair @ A @ B @ A6 @ B7 )
% 3.15/1.66 = P3 )
% 3.15/1.66 => ( C3 @ A6 @ B7 @ X2 ) )
% 3.15/1.66 => ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P3 @ X2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_138_case__prod__Pair__iden,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
% 3.15/1.66 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P3 )
% 3.15/1.66 = P3 ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_139_mem__case__prodE,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,C: $tType,Z5: A,C3: B > C > ( set @ A ),P3: product_prod @ B @ C] :
% 3.15/1.66 ( ( member @ A @ Z5 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P3 ) )
% 3.15/1.66 => ~ ! [X3: B,Y2: C] :
% 3.15/1.66 ( ( P3
% 3.15/1.66 = ( product_Pair @ B @ C @ X3 @ Y2 ) )
% 3.15/1.66 => ~ ( member @ A @ Z5 @ ( C3 @ X3 @ Y2 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_140_case__prodE_H,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P3: product_prod @ A @ B,Z5: C] :
% 3.15/1.66 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P3 @ Z5 )
% 3.15/1.66 => ~ ! [X3: A,Y2: B] :
% 3.15/1.66 ( ( P3
% 3.15/1.66 = ( product_Pair @ A @ B @ X3 @ Y2 ) )
% 3.15/1.66 => ~ ( C3 @ X3 @ Y2 @ Z5 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_141_case__prodD_H,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,C: $tType,R2: A > B > C > $o,A2: A,B3: B,C3: C] :
% 3.15/1.66 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R2 @ ( product_Pair @ A @ B @ A2 @ B3 ) @ C3 )
% 3.15/1.66 => ( R2 @ A2 @ B3 @ C3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_142_lex__prod__def,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType] :
% 3.15/1.66 ( ( lex_prod @ A @ B )
% 3.15/1.66 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 3.15/1.66 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 3.15/1.66 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 3.15/1.66 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 3.15/1.66 @ ^ [A4: A,B8: B] :
% 3.15/1.66 ( product_case_prod @ A @ B @ $o
% 3.15/1.66 @ ^ [A8: A,B9: B] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A8 ) @ Ra )
% 3.15/1.66 | ( ( A4 = A8 )
% 3.15/1.66 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B8 @ B9 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_143_prod__set__simps_I1_J,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,X2: A,Y3: B] :
% 3.15/1.66 ( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y3 ) )
% 3.15/1.66 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_144_prod__set__simps_I2_J,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,X2: A,Y3: B] :
% 3.15/1.66 ( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y3 ) )
% 3.15/1.66 = ( insert @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_145_the__elem__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( the_elem @ A )
% 3.15/1.66 = ( ^ [X6: set @ A] :
% 3.15/1.66 ( the @ A
% 3.15/1.66 @ ^ [X: A] :
% 3.15/1.66 ( X6
% 3.15/1.66 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_146_in__lex__prod,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,A2: A,B3: B,A5: A,B6: B,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
% 3.15/1.66 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ A5 @ B6 ) ) @ ( lex_prod @ A @ B @ R @ S2 ) )
% 3.15/1.66 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A5 ) @ R )
% 3.15/1.66 | ( ( A2 = A5 )
% 3.15/1.66 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B6 ) @ S2 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_147_The__split__eq,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,X2: A,Y3: B] :
% 3.15/1.66 ( ( the @ ( product_prod @ A @ B )
% 3.15/1.66 @ ( product_case_prod @ A @ B @ $o
% 3.15/1.66 @ ^ [X7: A,Y6: B] :
% 3.15/1.66 ( ( X2 = X7 )
% 3.15/1.66 & ( Y3 = Y6 ) ) ) )
% 3.15/1.66 = ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_148_wf__lex__prod,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,Ra2: set @ ( product_prod @ A @ A ),Rb2: set @ ( product_prod @ B @ B )] :
% 3.15/1.66 ( ( wf @ A @ Ra2 )
% 3.15/1.66 => ( ( wf @ B @ Rb2 )
% 3.15/1.66 => ( wf @ ( product_prod @ A @ B ) @ ( lex_prod @ A @ B @ Ra2 @ Rb2 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_149_old_Orec__prod__def,axiom,
% 3.15/1.66 ! [T: $tType,B: $tType,A: $tType] :
% 3.15/1.66 ( ( product_rec_prod @ A @ B @ T )
% 3.15/1.66 = ( ^ [F12: A > B > T,X: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_150_the__sym__eq__trivial,axiom,
% 3.15/1.66 ! [A: $tType,X2: A] :
% 3.15/1.66 ( ( the @ A
% 3.15/1.66 @ ( ^ [Y: A,Z3: A] : ( Y = Z3 )
% 3.15/1.66 @ X2 ) )
% 3.15/1.66 = X2 ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_151_the__eq__trivial,axiom,
% 3.15/1.66 ! [A: $tType,A2: A] :
% 3.15/1.66 ( ( the @ A
% 3.15/1.66 @ ^ [X: A] : ( X = A2 ) )
% 3.15/1.66 = A2 ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_152_the__equality,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o,A2: A] :
% 3.15/1.66 ( ( P @ A2 )
% 3.15/1.66 => ( ! [X3: A] :
% 3.15/1.66 ( ( P @ X3 )
% 3.15/1.66 => ( X3 = A2 ) )
% 3.15/1.66 => ( ( the @ A @ P )
% 3.15/1.66 = A2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_153_theI,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o,A2: A] :
% 3.15/1.66 ( ( P @ A2 )
% 3.15/1.66 => ( ! [X3: A] :
% 3.15/1.66 ( ( P @ X3 )
% 3.15/1.66 => ( X3 = A2 ) )
% 3.15/1.66 => ( P @ ( the @ A @ P ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_154_theI_H,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o] :
% 3.15/1.66 ( ? [X5: A] :
% 3.15/1.66 ( ( P @ X5 )
% 3.15/1.66 & ! [Y2: A] :
% 3.15/1.66 ( ( P @ Y2 )
% 3.15/1.66 => ( Y2 = X5 ) ) )
% 3.15/1.66 => ( P @ ( the @ A @ P ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_155_theI2,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
% 3.15/1.66 ( ( P @ A2 )
% 3.15/1.66 => ( ! [X3: A] :
% 3.15/1.66 ( ( P @ X3 )
% 3.15/1.66 => ( X3 = A2 ) )
% 3.15/1.66 => ( ! [X3: A] :
% 3.15/1.66 ( ( P @ X3 )
% 3.15/1.66 => ( Q @ X3 ) )
% 3.15/1.66 => ( Q @ ( the @ A @ P ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_156_If__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( if @ A )
% 3.15/1.66 = ( ^ [P4: $o,X: A,Y5: A] :
% 3.15/1.66 ( the @ A
% 3.15/1.66 @ ^ [Z4: A] :
% 3.15/1.66 ( ( P4
% 3.15/1.66 => ( Z4 = X ) )
% 3.15/1.66 & ( ~ P4
% 3.15/1.66 => ( Z4 = Y5 ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_157_the1I2,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.15/1.66 ( ? [X5: A] :
% 3.15/1.66 ( ( P @ X5 )
% 3.15/1.66 & ! [Y2: A] :
% 3.15/1.66 ( ( P @ Y2 )
% 3.15/1.66 => ( Y2 = X5 ) ) )
% 3.15/1.66 => ( ! [X3: A] :
% 3.15/1.66 ( ( P @ X3 )
% 3.15/1.66 => ( Q @ X3 ) )
% 3.15/1.66 => ( Q @ ( the @ A @ P ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_158_the1__equality,axiom,
% 3.15/1.66 ! [A: $tType,P: A > $o,A2: A] :
% 3.15/1.66 ( ? [X5: A] :
% 3.15/1.66 ( ( P @ X5 )
% 3.15/1.66 & ! [Y2: A] :
% 3.15/1.66 ( ( P @ Y2 )
% 3.15/1.66 => ( Y2 = X5 ) ) )
% 3.15/1.66 => ( ( P @ A2 )
% 3.15/1.66 => ( ( the @ A @ P )
% 3.15/1.66 = A2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_159_same__fst__def,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType] :
% 3.15/1.66 ( ( same_fst @ A @ B )
% 3.15/1.66 = ( ^ [P4: A > $o,R5: A > ( set @ ( product_prod @ B @ B ) )] :
% 3.15/1.66 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 3.15/1.66 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 3.15/1.66 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 3.15/1.66 @ ^ [X7: A,Y6: B] :
% 3.15/1.66 ( product_case_prod @ A @ B @ $o
% 3.15/1.66 @ ^ [X: A,Y5: B] :
% 3.15/1.66 ( ( X7 = X )
% 3.15/1.66 & ( P4 @ X )
% 3.15/1.66 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y5 ) @ ( R5 @ X ) ) ) ) ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_160_Range__insert,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,A2: B,B3: A,R: set @ ( product_prod @ B @ A )] :
% 3.15/1.66 ( ( range @ B @ A @ ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B3 ) @ R ) )
% 3.15/1.66 = ( insert @ A @ B3 @ ( range @ B @ A @ R ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_161_same__fstI,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,P: A > $o,X2: A,Y7: B,Y3: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 3.15/1.66 ( ( P @ X2 )
% 3.15/1.66 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y3 ) @ ( R2 @ X2 ) )
% 3.15/1.66 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y7 ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) @ ( same_fst @ A @ B @ P @ R2 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_162_Range__empty,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType] :
% 3.15/1.66 ( ( range @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_163_Range_Oinducts,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,X2: B,R: set @ ( product_prod @ A @ B ),P: B > $o] :
% 3.15/1.66 ( ( member @ B @ X2 @ ( range @ A @ B @ R ) )
% 3.15/1.66 => ( ! [A6: A,B7: B] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B7 ) @ R )
% 3.15/1.66 => ( P @ B7 ) )
% 3.15/1.66 => ( P @ X2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_164_Range_Ointros,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,A2: A,B3: B,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ R )
% 3.15/1.66 => ( member @ B @ B3 @ ( range @ A @ B @ R ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_165_Range_Osimps,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
% 3.15/1.66 = ( ? [A4: A,B8: B] :
% 3.15/1.66 ( ( A2 = B8 )
% 3.15/1.66 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B8 ) @ R ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_166_Range_Ocases,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,A2: B,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( member @ B @ A2 @ ( range @ A @ B @ R ) )
% 3.15/1.66 => ~ ! [A6: A] :
% 3.15/1.66 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A2 ) @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_167_Range__iff,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,A2: A,R: set @ ( product_prod @ B @ A )] :
% 3.15/1.66 ( ( member @ A @ A2 @ ( range @ B @ A @ R ) )
% 3.15/1.66 = ( ? [Y5: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y5 @ A2 ) @ R ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_168_RangeE,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,B3: A,R: set @ ( product_prod @ B @ A )] :
% 3.15/1.66 ( ( member @ A @ B3 @ ( range @ B @ A @ R ) )
% 3.15/1.66 => ~ ! [A6: B] :
% 3.15/1.66 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B3 ) @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_169_Range__empty__iff,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A )] :
% 3.15/1.66 ( ( ( range @ B @ A @ R )
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.15/1.66 = ( R
% 3.15/1.66 = ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_170_wf__same__fst,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,P: A > $o,R2: A > ( set @ ( product_prod @ B @ B ) )] :
% 3.15/1.66 ( ! [X3: A] :
% 3.15/1.66 ( ( P @ X3 )
% 3.15/1.66 => ( wf @ B @ ( R2 @ X3 ) ) )
% 3.15/1.66 => ( wf @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_171_Rangep__Range__eq,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( rangep @ A @ B
% 3.15/1.66 @ ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ R ) )
% 3.15/1.66 = ( ^ [X: B] : ( member @ B @ X @ ( range @ A @ B @ R ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_172_Range__def,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType] :
% 3.15/1.66 ( ( range @ A @ B )
% 3.15/1.66 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( collect @ B
% 3.15/1.66 @ ( rangep @ A @ B
% 3.15/1.66 @ ^ [X: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y5 ) @ R3 ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_173_Domain__insert,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,A2: A,B3: B,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ R ) )
% 3.15/1.66 = ( insert @ A @ A2 @ ( domain @ A @ B @ R ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_174_Domain__empty,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType] :
% 3.15/1.66 ( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_175_Rangep_Oinducts,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,R: A > B > $o,X2: B,P: B > $o] :
% 3.15/1.66 ( ( rangep @ A @ B @ R @ X2 )
% 3.15/1.66 => ( ! [A6: A,B7: B] :
% 3.15/1.66 ( ( R @ A6 @ B7 )
% 3.15/1.66 => ( P @ B7 ) )
% 3.15/1.66 => ( P @ X2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_176_Rangep_Ointros,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,R: A > B > $o,A2: A,B3: B] :
% 3.15/1.66 ( ( R @ A2 @ B3 )
% 3.15/1.66 => ( rangep @ A @ B @ R @ B3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_177_Rangep_Osimps,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType] :
% 3.15/1.66 ( ( rangep @ A @ B )
% 3.15/1.66 = ( ^ [R3: A > B > $o,A4: B] :
% 3.15/1.66 ? [B8: A,C6: B] :
% 3.15/1.66 ( ( A4 = C6 )
% 3.15/1.66 & ( R3 @ B8 @ C6 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_178_Rangep_Ocases,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,R: A > B > $o,A2: B] :
% 3.15/1.66 ( ( rangep @ A @ B @ R @ A2 )
% 3.15/1.66 => ~ ! [A6: A] :
% 3.15/1.66 ~ ( R @ A6 @ A2 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_179_RangepE,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,R: A > B > $o,B3: B] :
% 3.15/1.66 ( ( rangep @ A @ B @ R @ B3 )
% 3.15/1.66 => ~ ! [A6: A] :
% 3.15/1.66 ~ ( R @ A6 @ B3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_180_DomainE,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,A2: A,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( member @ A @ A2 @ ( domain @ A @ B @ R ) )
% 3.15/1.66 => ~ ! [B7: B] :
% 3.15/1.66 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B7 ) @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_181_Domain__iff,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,A2: A,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( member @ A @ A2 @ ( domain @ A @ B @ R ) )
% 3.15/1.66 = ( ? [Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ Y5 ) @ R ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_182_Domain_Ocases,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,A2: A,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( member @ A @ A2 @ ( domain @ A @ B @ R ) )
% 3.15/1.66 => ~ ! [B7: B] :
% 3.15/1.66 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B7 ) @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_183_Domain_Osimps,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,A2: A,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( member @ A @ A2 @ ( domain @ A @ B @ R ) )
% 3.15/1.66 = ( ? [A4: A,B8: B] :
% 3.15/1.66 ( ( A2 = A4 )
% 3.15/1.66 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B8 ) @ R ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_184_Domain_ODomainI,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,A2: A,B3: B,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ R )
% 3.15/1.66 => ( member @ A @ A2 @ ( domain @ A @ B @ R ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_185_Domain_Oinducts,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,X2: A,R: set @ ( product_prod @ A @ B ),P: A > $o] :
% 3.15/1.66 ( ( member @ A @ X2 @ ( domain @ A @ B @ R ) )
% 3.15/1.66 => ( ! [A6: A,B7: B] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B7 ) @ R )
% 3.15/1.66 => ( P @ A6 ) )
% 3.15/1.66 => ( P @ X2 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_186_Domain__empty__iff,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 3.15/1.66 ( ( ( domain @ A @ B @ R )
% 3.15/1.66 = ( bot_bot @ ( set @ A ) ) )
% 3.15/1.66 = ( R
% 3.15/1.66 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_187_old_Orec__unit__def,axiom,
% 3.15/1.66 ! [T: $tType] :
% 3.15/1.66 ( ( product_rec_unit @ T )
% 3.15/1.66 = ( ^ [F12: T,X: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_188_above__def,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( order_above @ A )
% 3.15/1.66 = ( ^ [R3: set @ ( product_prod @ A @ A ),A4: A] :
% 3.15/1.66 ( collect @ A
% 3.15/1.66 @ ^ [B8: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B8 ) @ R3 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_189_inv__image__def,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType] :
% 3.15/1.66 ( ( inv_image @ B @ A )
% 3.15/1.66 = ( ^ [R3: set @ ( product_prod @ B @ B ),F4: A > B] :
% 3.15/1.66 ( collect @ ( product_prod @ A @ A )
% 3.15/1.66 @ ( product_case_prod @ A @ A @ $o
% 3.15/1.66 @ ^ [X: A,Y5: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X ) @ ( F4 @ Y5 ) ) @ R3 ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_190_in__inv__image,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,X2: A,Y3: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( inv_image @ B @ A @ R @ F ) )
% 3.15/1.66 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_191_wf__inv__image,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 3.15/1.66 ( ( wf @ B @ R )
% 3.15/1.66 => ( wf @ A @ ( inv_image @ B @ A @ R @ F ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_192_rp__inv__image__def,axiom,
% 3.15/1.66 ! [B: $tType,A: $tType] :
% 3.15/1.66 ( ( fun_rp_inv_image @ A @ B )
% 3.15/1.66 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 3.15/1.66 @ ^ [R5: set @ ( product_prod @ A @ A ),S6: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R5 @ F4 ) @ ( inv_image @ A @ B @ S6 @ F4 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_193_Id__on__empty,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
% 3.15/1.66 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_194_Id__onI,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,A3: set @ A] :
% 3.15/1.66 ( ( member @ A @ A2 @ A3 )
% 3.15/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_195_Domain__Id__on,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A] :
% 3.15/1.66 ( ( domain @ A @ A @ ( id_on @ A @ A3 ) )
% 3.15/1.66 = A3 ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_196_Range__Id__on,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A] :
% 3.15/1.66 ( ( range @ A @ A @ ( id_on @ A @ A3 ) )
% 3.15/1.66 = A3 ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_197_Id__on__def_H,axiom,
% 3.15/1.66 ! [A: $tType,A3: A > $o] :
% 3.15/1.66 ( ( id_on @ A @ ( collect @ A @ A3 ) )
% 3.15/1.66 = ( collect @ ( product_prod @ A @ A )
% 3.15/1.66 @ ( product_case_prod @ A @ A @ $o
% 3.15/1.66 @ ^ [X: A,Y5: A] :
% 3.15/1.66 ( ( X = Y5 )
% 3.15/1.66 & ( A3 @ X ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_198_refl__on__Id__on,axiom,
% 3.15/1.66 ! [A: $tType,A3: set @ A] : ( refl_on @ A @ A3 @ ( id_on @ A @ A3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_199_Id__onE,axiom,
% 3.15/1.66 ! [A: $tType,C3: product_prod @ A @ A,A3: set @ A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A3 ) )
% 3.15/1.66 => ~ ! [X3: A] :
% 3.15/1.66 ( ( member @ A @ X3 @ A3 )
% 3.15/1.66 => ( C3
% 3.15/1.66 != ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_200_Id__on__eqI,axiom,
% 3.15/1.66 ! [A: $tType,A2: A,B3: A,A3: set @ A] :
% 3.15/1.66 ( ( A2 = B3 )
% 3.15/1.66 => ( ( member @ A @ A2 @ A3 )
% 3.15/1.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B3 ) @ ( id_on @ A @ A3 ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_201_Id__on__iff,axiom,
% 3.15/1.66 ! [A: $tType,X2: A,Y3: A,A3: set @ A] :
% 3.15/1.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( id_on @ A @ A3 ) )
% 3.15/1.66 = ( ( X2 = Y3 )
% 3.15/1.66 & ( member @ A @ X2 @ A3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_202_rp__inv__image__rp,axiom,
% 3.15/1.66 ! [A: $tType,B: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
% 3.15/1.66 ( ( fun_reduction_pair @ A @ P )
% 3.15/1.66 => ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P @ F ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_203_preorder__on__empty,axiom,
% 3.15/1.66 ! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_204_termi__while,axiom,
% 3.15/1.66 ! [B3: state > $o,C3: com,F: nat > state,K: nat] :
% 3.15/1.66 ( ( termi @ ( while @ B3 @ C3 ) @ ( F @ K ) )
% 3.15/1.66 => ( ! [I: nat] : ( exec @ ( F @ I ) @ C3 @ ( F @ ( suc @ I ) ) )
% 3.15/1.66 => ? [I: nat] :
% 3.15/1.66 ~ ( B3 @ ( F @ I ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_205_termi__while__lemma,axiom,
% 3.15/1.66 ! [W: com,Fk: state,F: nat > state,K: nat,B3: state > $o,C3: com] :
% 3.15/1.66 ( ( termi @ W @ Fk )
% 3.15/1.66 => ( ( ( Fk
% 3.15/1.66 = ( F @ K ) )
% 3.15/1.66 & ( W
% 3.15/1.66 = ( while @ B3 @ C3 ) )
% 3.15/1.66 & ! [I: nat] : ( exec @ ( F @ I ) @ C3 @ ( F @ ( suc @ I ) ) ) )
% 3.15/1.66 => ? [I: nat] :
% 3.15/1.66 ~ ( B3 @ ( F @ I ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_206_wf__iff__no__infinite__down__chain,axiom,
% 3.15/1.66 ! [A: $tType] :
% 3.15/1.66 ( ( wf @ A )
% 3.15/1.66 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.15/1.66 ~ ? [F4: nat > A] :
% 3.15/1.66 ! [I2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ ( suc @ I2 ) ) @ ( F4 @ I2 ) ) @ R3 ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_207_wf__no__infinite__down__chainE,axiom,
% 3.15/1.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
% 3.15/1.66 ( ( wf @ A @ R )
% 3.15/1.66 => ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K2 ) ) @ ( F @ K2 ) ) @ R ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_208_execn_Ointros_I7_J,axiom,
% 3.15/1.66 ! [S2: state,N: nat,T3: state] :
% 3.15/1.66 ( ( execn @ S2 @ body @ N @ T3 )
% 3.15/1.66 => ( execn @ S2 @ call @ ( suc @ N ) @ T3 ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_209_pred__nat__def,axiom,
% 3.15/1.66 ( pred_nat
% 3.15/1.66 = ( collect @ ( product_prod @ nat @ nat )
% 3.15/1.66 @ ( product_case_prod @ nat @ nat @ $o
% 3.15/1.66 @ ^ [M: nat,N2: nat] :
% 3.15/1.66 ( N2
% 3.15/1.66 = ( suc @ M ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_210_while__rule,axiom,
% 3.15/1.66 ! [S2: state,B3: state > $o,C3: com,N: nat,T3: state,P: state > $o] :
% 3.15/1.66 ( ( execn @ S2 @ ( while @ B3 @ C3 ) @ N @ T3 )
% 3.15/1.66 => ( ( P @ S2 )
% 3.15/1.66 => ( ! [S: state,S3: state] :
% 3.15/1.66 ( ( P @ S )
% 3.15/1.66 => ( ( B3 @ S )
% 3.15/1.66 => ( ( execn @ S @ C3 @ N @ S3 )
% 3.15/1.66 => ( P @ S3 ) ) ) )
% 3.15/1.66 => ( ( P @ T3 )
% 3.15/1.66 & ~ ( B3 @ T3 ) ) ) ) ) ).
% 3.15/1.66
% 3.15/1.66 thf(fact_211_while__lemma,axiom,
% 3.15/1.66 ! [S2: state,W: com,N: nat,T3: state,B3: state > $o,C3: com,P: state > $o] :
% 3.15/1.67 ( ( execn @ S2 @ W @ N @ T3 )
% 3.15/1.67 => ( ( ( W
% 3.15/1.67 = ( while @ B3 @ C3 ) )
% 3.15/1.67 & ( P @ S2 )
% 3.15/1.67 & ! [S: state,S3: state] :
% 3.15/1.67 ( ( ( P @ S )
% 3.15/1.67 & ( B3 @ S )
% 3.15/1.67 & ( execn @ S @ C3 @ N @ S3 ) )
% 3.15/1.67 => ( P @ S3 ) ) )
% 3.15/1.67 => ( ( P @ T3 )
% 3.15/1.67 & ~ ( B3 @ T3 ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_212_execn_OWhileTrue,axiom,
% 3.15/1.67 ! [B3: state > $o,S2: state,C3: com,N: nat,T3: state,U2: state] :
% 3.15/1.67 ( ( B3 @ S2 )
% 3.15/1.67 => ( ( execn @ S2 @ C3 @ N @ T3 )
% 3.15/1.67 => ( ( execn @ T3 @ ( while @ B3 @ C3 ) @ N @ U2 )
% 3.15/1.67 => ( execn @ S2 @ ( while @ B3 @ C3 ) @ N @ U2 ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_213_execn_OWhileFalse,axiom,
% 3.15/1.67 ! [B3: state > $o,S2: state,C3: com,N: nat] :
% 3.15/1.67 ( ~ ( B3 @ S2 )
% 3.15/1.67 => ( execn @ S2 @ ( while @ B3 @ C3 ) @ N @ S2 ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_214_exec__iff__execn,axiom,
% 3.15/1.67 ( exec
% 3.15/1.67 = ( ^ [S4: state,C6: com,T4: state] :
% 3.15/1.67 ? [N2: nat] : ( execn @ S4 @ C6 @ N2 @ T4 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_215_wf__pred__nat,axiom,
% 3.15/1.67 wf @ nat @ pred_nat ).
% 3.15/1.67
% 3.15/1.67 thf(fact_216_nvalid__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( nvalid @ A )
% 3.15/1.67 = ( ^ [N2: nat,P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 3.15/1.67 ! [S4: state,T4: state] :
% 3.15/1.67 ( ( execn @ S4 @ C6 @ N2 @ T4 )
% 3.15/1.67 => ! [Z4: A] :
% 3.15/1.67 ( ( P4 @ Z4 @ S4 )
% 3.15/1.67 => ( Q4 @ Z4 @ T4 ) ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_217_termi_OSemi,axiom,
% 3.15/1.67 ! [C1: com,S0: state,C22: com] :
% 3.15/1.67 ( ( termi @ C1 @ S0 )
% 3.15/1.67 => ( ! [S1: state] :
% 3.15/1.67 ( ( exec @ S0 @ C1 @ S1 )
% 3.15/1.67 => ( termi @ C22 @ S1 ) )
% 3.15/1.67 => ( termi @ ( semi @ C1 @ C22 ) @ S0 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_218_com_Oinject_I2_J,axiom,
% 3.15/1.67 ! [X21: com,X222: com,Y21: com,Y222: com] :
% 3.15/1.67 ( ( ( semi @ X21 @ X222 )
% 3.15/1.67 = ( semi @ Y21 @ Y222 ) )
% 3.15/1.67 = ( ( X21 = Y21 )
% 3.15/1.67 & ( X222 = Y222 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_219_execn_Ointros_I2_J,axiom,
% 3.15/1.67 ! [S0: state,C1: com,N: nat,S12: state,C22: com,S22: state] :
% 3.15/1.67 ( ( execn @ S0 @ C1 @ N @ S12 )
% 3.15/1.67 => ( ( execn @ S12 @ C22 @ N @ S22 )
% 3.15/1.67 => ( execn @ S0 @ ( semi @ C1 @ C22 ) @ N @ S22 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_220_com_Odistinct_I13_J,axiom,
% 3.15/1.67 ! [X21: com,X222: com,X41: state > $o,X42: com] :
% 3.15/1.67 ( ( semi @ X21 @ X222 )
% 3.15/1.67 != ( while @ X41 @ X42 ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_221_com_Odistinct_I15_J,axiom,
% 3.15/1.67 ! [X21: com,X222: com] :
% 3.15/1.67 ( ( semi @ X21 @ X222 )
% 3.15/1.67 != call ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_222_exec_OSemi,axiom,
% 3.15/1.67 ! [S0: state,C1: com,S12: state,C22: com,S22: state] :
% 3.15/1.67 ( ( exec @ S0 @ C1 @ S12 )
% 3.15/1.67 => ( ( exec @ S12 @ C22 @ S22 )
% 3.15/1.67 => ( exec @ S0 @ ( semi @ C1 @ C22 ) @ S22 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_223_hoare_OSemi,axiom,
% 3.15/1.67 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,C1: com,Q: A > state > $o,C22: com,R2: A > state > $o] :
% 3.15/1.67 ( ( hoare @ A @ C2 @ P @ C1 @ Q )
% 3.15/1.67 => ( ( hoare @ A @ C2 @ Q @ C22 @ R2 )
% 3.15/1.67 => ( hoare @ A @ C2 @ P @ ( semi @ C1 @ C22 ) @ R2 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_224_thoare_OSemi,axiom,
% 3.15/1.67 ! [A: $tType,C2: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P: A > state > $o,C1: com,Q: A > state > $o,C22: com,R2: A > state > $o] :
% 3.15/1.67 ( ( pHoare166426476thoare @ A @ C2 @ P @ C1 @ Q )
% 3.15/1.67 => ( ( pHoare166426476thoare @ A @ C2 @ Q @ C22 @ R2 )
% 3.15/1.67 => ( pHoare166426476thoare @ A @ C2 @ P @ ( semi @ C1 @ C22 ) @ R2 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_225_cnvalid__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( cnvalid @ A )
% 3.15/1.67 = ( ^ [C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),N2: nat,P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 3.15/1.67 ( ( nvalids @ A @ N2 @ C4 )
% 3.15/1.67 => ( nvalid @ A @ N2 @ P4 @ C6 @ Q4 ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_226_wf__less__than,axiom,
% 3.15/1.67 wf @ nat @ less_than ).
% 3.15/1.67
% 3.15/1.67 thf(fact_227_total__on__less__than,axiom,
% 3.15/1.67 ! [A3: set @ nat] : ( total_on @ nat @ A3 @ less_than ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_228_measure__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( measure @ A )
% 3.15/1.67 = ( inv_image @ nat @ A @ less_than ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_229_nvalids__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( nvalids @ A )
% 3.15/1.67 = ( ^ [N2: nat,C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) )] :
% 3.15/1.67 ! [X: product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) )] :
% 3.15/1.67 ( ( member @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ X @ C4 )
% 3.15/1.67 => ( product_case_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ $o
% 3.15/1.67 @ ^ [P4: A > state > $o] : ( product_case_prod @ com @ ( A > state > $o ) @ $o @ ( nvalid @ A @ N2 @ P4 ) )
% 3.15/1.67 @ X ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_230_mlex__prod__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( mlex_prod @ A )
% 3.15/1.67 = ( ^ [F4: A > nat,R5: set @ ( product_prod @ A @ A )] :
% 3.15/1.67 ( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R5 )
% 3.15/1.67 @ ^ [X: A] : ( product_Pair @ nat @ A @ ( F4 @ X ) @ X ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_231_ball__empty,axiom,
% 3.15/1.67 ! [A: $tType,P: A > $o,X5: A] :
% 3.15/1.67 ( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
% 3.15/1.67 => ( P @ X5 ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_232_Ball__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( ball @ A )
% 3.15/1.67 = ( ^ [A7: set @ A,P4: A > $o] :
% 3.15/1.67 ! [X: A] :
% 3.15/1.67 ( ( member @ A @ X @ A7 )
% 3.15/1.67 => ( P4 @ X ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_233_wf__mlex,axiom,
% 3.15/1.67 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),F: A > nat] :
% 3.15/1.67 ( ( wf @ A @ R2 )
% 3.15/1.67 => ( wf @ A @ ( mlex_prod @ A @ F @ R2 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_234_refl__on__def_H,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( refl_on @ A )
% 3.15/1.67 = ( ^ [A7: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 3.15/1.67 ( ! [X: product_prod @ A @ A] :
% 3.15/1.67 ( ( member @ ( product_prod @ A @ A ) @ X @ R3 )
% 3.15/1.67 => ( product_case_prod @ A @ A @ $o
% 3.15/1.67 @ ^ [Y5: A,Z4: A] :
% 3.15/1.67 ( ( member @ A @ Y5 @ A7 )
% 3.15/1.67 & ( member @ A @ Z4 @ A7 ) )
% 3.15/1.67 @ X ) )
% 3.15/1.67 & ! [X: A] :
% 3.15/1.67 ( ( member @ A @ X @ A7 )
% 3.15/1.67 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R3 ) ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_235_ctvalid__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( pHoare304044376tvalid @ A )
% 3.15/1.67 = ( ^ [C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 3.15/1.67 ( ! [X: product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) )] :
% 3.15/1.67 ( ( member @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ X @ C4 )
% 3.15/1.67 => ( product_case_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ $o
% 3.15/1.67 @ ^ [P6: A > state > $o] : ( product_case_prod @ com @ ( A > state > $o ) @ $o @ ( pHoare987424603tvalid @ A @ P6 ) )
% 3.15/1.67 @ X ) )
% 3.15/1.67 => ( pHoare987424603tvalid @ A @ P4 @ C6 @ Q4 ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_236_valids__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( valids @ A )
% 3.15/1.67 = ( ^ [C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) )] :
% 3.15/1.67 ! [X: product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) )] :
% 3.15/1.67 ( ( member @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ) @ X @ C4 )
% 3.15/1.67 => ( product_case_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) @ $o
% 3.15/1.67 @ ^ [P4: A > state > $o] : ( product_case_prod @ com @ ( A > state > $o ) @ $o @ ( valid @ A @ P4 ) )
% 3.15/1.67 @ X ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_237_tvalid__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( pHoare987424603tvalid @ A )
% 3.15/1.67 = ( ^ [P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 3.15/1.67 ( ( valid @ A @ P4 @ C6 @ Q4 )
% 3.15/1.67 & ! [Z4: A,S4: state] :
% 3.15/1.67 ( ( P4 @ Z4 @ S4 )
% 3.15/1.67 => ( termi @ C6 @ S4 ) ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_238_cvalid__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( cvalid @ A )
% 3.15/1.67 = ( ^ [C4: set @ ( product_prod @ ( A > state > $o ) @ ( product_prod @ com @ ( A > state > $o ) ) ),P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 3.15/1.67 ( ( valids @ A @ C4 )
% 3.15/1.67 => ( valid @ A @ P4 @ C6 @ Q4 ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_239_valid__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( valid @ A )
% 3.15/1.67 = ( ^ [P4: A > state > $o,C6: com,Q4: A > state > $o] :
% 3.15/1.67 ! [S4: state,T4: state] :
% 3.15/1.67 ( ( exec @ S4 @ C6 @ T4 )
% 3.15/1.67 => ! [Z4: A] :
% 3.15/1.67 ( ( P4 @ Z4 @ S4 )
% 3.15/1.67 => ( Q4 @ Z4 @ T4 ) ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_240_Chains__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( chains @ A )
% 3.15/1.67 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.15/1.67 ( collect @ ( set @ A )
% 3.15/1.67 @ ^ [C4: set @ A] :
% 3.15/1.67 ! [X: A] :
% 3.15/1.67 ( ( member @ A @ X @ C4 )
% 3.15/1.67 => ! [Y5: A] :
% 3.15/1.67 ( ( member @ A @ Y5 @ C4 )
% 3.15/1.67 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R3 )
% 3.15/1.67 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R3 ) ) ) ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_241_irrefl__distinct,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( irrefl @ A )
% 3.15/1.67 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.15/1.67 ! [X: product_prod @ A @ A] :
% 3.15/1.67 ( ( member @ ( product_prod @ A @ A ) @ X @ R3 )
% 3.15/1.67 => ( product_case_prod @ A @ A @ $o
% 3.15/1.67 @ ^ [A4: A,B8: A] : ( A4 != B8 )
% 3.15/1.67 @ X ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_242_irrefl__less__than,axiom,
% 3.15/1.67 irrefl @ nat @ less_than ).
% 3.15/1.67
% 3.15/1.67 thf(fact_243_wf__imp__irrefl,axiom,
% 3.15/1.67 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 3.15/1.67 ( ( wf @ A @ R )
% 3.15/1.67 => ( irrefl @ A @ R ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_244_irrefl__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( irrefl @ A )
% 3.15/1.67 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.15/1.67 ! [A4: A] :
% 3.15/1.67 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R3 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_245_irreflI,axiom,
% 3.15/1.67 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.15/1.67 ( ! [A6: A] :
% 3.15/1.67 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R2 )
% 3.15/1.67 => ( irrefl @ A @ R2 ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_246_irreflp__irrefl__eq,axiom,
% 3.15/1.67 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
% 3.15/1.67 ( ( irreflp @ A
% 3.15/1.67 @ ^ [A4: A,B8: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B8 ) @ R2 ) )
% 3.15/1.67 = ( irrefl @ A @ R2 ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_247_Powp__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( powp @ A )
% 3.15/1.67 = ( ^ [A7: A > $o,B5: set @ A] :
% 3.15/1.67 ! [X: A] :
% 3.15/1.67 ( ( member @ A @ X @ B5 )
% 3.15/1.67 => ( A7 @ X ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_248_irreflp__def,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( irreflp @ A )
% 3.15/1.67 = ( ^ [R5: A > A > $o] :
% 3.15/1.67 ! [A4: A] :
% 3.15/1.67 ~ ( R5 @ A4 @ A4 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_249_irreflpI,axiom,
% 3.15/1.67 ! [A: $tType,R2: A > A > $o] :
% 3.15/1.67 ( ! [A6: A] :
% 3.15/1.67 ~ ( R2 @ A6 @ A6 )
% 3.15/1.67 => ( irreflp @ A @ R2 ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_250_trans__join,axiom,
% 3.15/1.67 ! [A: $tType] :
% 3.15/1.67 ( ( trans @ A )
% 3.15/1.67 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 3.15/1.67 ! [X: product_prod @ A @ A] :
% 3.15/1.67 ( ( member @ ( product_prod @ A @ A ) @ X @ R3 )
% 3.15/1.67 => ( product_case_prod @ A @ A @ $o
% 3.15/1.67 @ ^ [Y5: A,Y12: A] :
% 3.15/1.67 ! [Z4: product_prod @ A @ A] :
% 3.15/1.67 ( ( member @ ( product_prod @ A @ A ) @ Z4 @ R3 )
% 3.15/1.67 => ( product_case_prod @ A @ A @ $o
% 3.15/1.67 @ ^ [Y23: A,Aa: A] :
% 3.15/1.67 ( ( Y12 = Y23 )
% 3.15/1.67 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Aa ) @ R3 ) )
% 3.15/1.67 @ Z4 ) )
% 3.15/1.67 @ X ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_251_in__measure,axiom,
% 3.15/1.67 ! [A: $tType,X2: A,Y3: A,F: A > nat] :
% 3.15/1.67 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measure @ A @ F ) )
% 3.15/1.67 = ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_252_trans__less__than,axiom,
% 3.15/1.67 trans @ nat @ less_than ).
% 3.15/1.67
% 3.15/1.67 thf(fact_253_less__than__iff,axiom,
% 3.15/1.67 ! [X2: nat,Y3: nat] :
% 3.15/1.67 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X2 @ Y3 ) @ less_than )
% 3.15/1.67 = ( ord_less @ nat @ X2 @ Y3 ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_254_trans__lex__prod,axiom,
% 3.15/1.67 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 3.15/1.67 ( ( trans @ A @ R1 )
% 3.15/1.67 => ( ( trans @ B @ R22 )
% 3.15/1.67 => ( trans @ ( product_prod @ A @ B ) @ ( lex_prod @ A @ B @ R1 @ R22 ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(fact_255_trans__Id__on,axiom,
% 3.15/1.67 ! [A: $tType,A3: set @ A] : ( trans @ A @ ( id_on @ A @ A3 ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(tcon_fun___Orderings_Obot,axiom,
% 3.15/1.67 ! [A9: $tType,A10: $tType] :
% 3.15/1.67 ( ( bot @ A10 )
% 3.15/1.67 => ( bot @ ( A9 > A10 ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(tcon_Nat_Onat___Orderings_Obot_1,axiom,
% 3.15/1.67 bot @ nat ).
% 3.15/1.67
% 3.15/1.67 thf(tcon_Set_Oset___Orderings_Obot_2,axiom,
% 3.15/1.67 ! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(tcon_HOL_Obool___Orderings_Obot_3,axiom,
% 3.15/1.67 bot @ $o ).
% 3.15/1.67
% 3.15/1.67 thf(tcon_Product__Type_Ounit___Orderings_Obot_4,axiom,
% 3.15/1.67 bot @ product_unit ).
% 3.15/1.67
% 3.15/1.67 thf(help_If_3_1_T,axiom,
% 3.15/1.67 ! [P: $o] :
% 3.15/1.67 ( ( P = $true )
% 3.15/1.67 | ( P = $false ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(help_If_2_1_T,axiom,
% 3.15/1.67 ! [A: $tType,X2: A,Y3: A] :
% 3.15/1.67 ( ( if @ A @ $false @ X2 @ Y3 )
% 3.15/1.67 = Y3 ) ).
% 3.15/1.67
% 3.15/1.67 thf(help_If_1_1_T,axiom,
% 3.15/1.67 ! [A: $tType,X2: A,Y3: A] :
% 3.15/1.67 ( ( if @ A @ $true @ X2 @ Y3 )
% 3.15/1.67 = X2 ) ).
% 3.15/1.67
% 3.15/1.67 thf(conj_0,hypothesis,
% 3.15/1.67 ! [S7: state] :
% 3.15/1.67 ( ( pHoare166426476thoare @ a
% 3.15/1.67 @ ( insert @ ( product_prod @ ( a > state > $o ) @ ( product_prod @ com @ ( a > state > $o ) ) )
% 3.15/1.67 @ ( product_Pair @ ( a > state > $o ) @ ( product_prod @ com @ ( a > state > $o ) )
% 3.15/1.67 @ ^ [Z4: a,S4: state] :
% 3.15/1.67 ( ( pa @ Z4 @ S4 )
% 3.15/1.67 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S4 @ S7 ) @ r ) )
% 3.15/1.67 @ ( product_Pair @ com @ ( a > state > $o ) @ call @ qa ) )
% 3.15/1.67 @ ( bot_bot @ ( set @ ( product_prod @ ( a > state > $o ) @ ( product_prod @ com @ ( a > state > $o ) ) ) ) ) )
% 3.15/1.67 @ ^ [Z4: a,S4: state] :
% 3.15/1.67 ( ( pa @ Z4 @ S4 )
% 3.15/1.67 & ( S4 = S7 ) )
% 3.15/1.67 @ body
% 3.15/1.67 @ qa )
% 3.15/1.67 & ( ( ! [S: state,T2: state] :
% 3.15/1.67 ( ( exec @ S @ body @ T2 )
% 3.15/1.67 => ! [Z: a] :
% 3.15/1.67 ( ( ( pa @ Z @ S )
% 3.15/1.67 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S @ S7 ) @ r ) )
% 3.15/1.67 => ( qa @ Z @ T2 ) ) )
% 3.15/1.67 & ! [Z: a,S: state] :
% 3.15/1.67 ( ( ( pa @ Z @ S )
% 3.15/1.67 & ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ S @ S7 ) @ r ) )
% 3.15/1.67 => ( termi @ body @ S ) ) )
% 3.15/1.67 => ( ! [S8: state,T5: state] :
% 3.15/1.67 ( ( exec @ S8 @ body @ T5 )
% 3.15/1.67 => ! [Z2: a] :
% 3.15/1.67 ( ( ( pa @ Z2 @ S8 )
% 3.15/1.67 & ( S8 = S7 ) )
% 3.15/1.67 => ( qa @ Z2 @ T5 ) ) )
% 3.15/1.67 & ! [Z2: a,S8: state] :
% 3.15/1.67 ( ( ( pa @ Z2 @ S8 )
% 3.15/1.67 & ( S8 = S7 ) )
% 3.15/1.67 => ( termi @ body @ S8 ) ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(conj_1,hypothesis,
% 3.15/1.67 ! [Y4: state] :
% 3.15/1.67 ( ( member @ ( product_prod @ state @ state ) @ ( product_Pair @ state @ state @ Y4 @ x ) @ r )
% 3.15/1.67 => ( ! [S8: state,T5: state] :
% 3.15/1.67 ( ( exec @ S8 @ body @ T5 )
% 3.15/1.67 => ! [Z2: a] :
% 3.15/1.67 ( ( ( pa @ Z2 @ S8 )
% 3.15/1.67 & ( S8 = Y4 ) )
% 3.15/1.67 => ( qa @ Z2 @ T5 ) ) )
% 3.15/1.67 & ! [Z2: a,S8: state] :
% 3.15/1.67 ( ( ( pa @ Z2 @ S8 )
% 3.15/1.67 & ( S8 = Y4 ) )
% 3.15/1.67 => ( termi @ body @ S8 ) ) ) ) ).
% 3.15/1.67
% 3.15/1.67 thf(conj_2,conjecture,
% 3.15/1.67 ( ! [S: state,T2: state] :
% 3.15/1.67 ( ~ ( exec @ S @ body @ T2 )
% 3.15/1.67 | ! [Z: a] :
% 3.15/1.67 ( ~ ( pa @ Z @ S )
% 3.15/1.67 | ( S != x )
% 3.15/1.67 | ( qa @ Z @ T2 ) ) )
% 3.15/1.67 & ! [Z: a,S: state] :
% 3.15/1.67 ( ~ ( pa @ Z @ S )
% 3.15/1.67 | ( S != x )
% 3.15/1.67 | ( termi @ body @ S ) ) ) ).
% 3.15/1.67 ------------------------
% 3.15/1.69 % (14165)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/183Mi)
% 3.15/1.69 % (14167)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/27Mi)
% 3.15/1.69 % (14166)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/4Mi)
% 3.15/1.69 % (14168)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/2Mi)
% 3.15/1.69 % (14169)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/2Mi)
% 3.15/1.69 % (14170)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/275Mi)
% 3.15/1.69 % (14171)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/18Mi)
% 3.15/1.69 % (14168)Instruction limit reached!
% 3.15/1.69 % (14168)------------------------------
% 3.15/1.69 % (14168)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.69 % (14168)Termination reason: Unknown
% 3.15/1.69 % (14169)Instruction limit reached!
% 3.15/1.69 % (14169)------------------------------
% 3.15/1.69 % (14169)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.69 % (14168)Termination phase: shuffling
% 3.15/1.69
% 3.15/1.69 % (14168)Memory used [KB]: 1407
% 3.15/1.69 % (14168)Time elapsed: 0.003 s
% 3.15/1.69 % (14168)Instructions burned: 3 (million)
% 3.15/1.69 % (14168)------------------------------
% 3.15/1.69 % (14168)------------------------------
% 3.15/1.69 % (14169)Termination reason: Unknown
% 3.15/1.69 % (14169)Termination phase: shuffling
% 3.15/1.69
% 3.15/1.69 % (14169)Memory used [KB]: 1407
% 3.15/1.69 % (14169)Time elapsed: 0.003 s
% 3.15/1.69 % (14169)Instructions burned: 3 (million)
% 3.15/1.69 % (14169)------------------------------
% 3.15/1.69 % (14169)------------------------------
% 3.15/1.69 % (14166)Instruction limit reached!
% 3.15/1.69 % (14166)------------------------------
% 3.15/1.69 % (14166)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.69 % (14166)Termination reason: Unknown
% 3.15/1.69 % (14166)Termination phase: shuffling
% 3.15/1.69
% 3.15/1.69 % (14166)Memory used [KB]: 1535
% 3.15/1.69 % (14166)Time elapsed: 0.004 s
% 3.15/1.69 % (14166)Instructions burned: 5 (million)
% 3.15/1.69 % (14166)------------------------------
% 3.15/1.69 % (14166)------------------------------
% 3.23/1.70 % (14171)Instruction limit reached!
% 3.23/1.70 % (14171)------------------------------
% 3.23/1.70 % (14171)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.70 % (14171)Termination reason: Unknown
% 3.23/1.70 % (14171)Termination phase: shuffling
% 3.23/1.70
% 3.23/1.70 % (14171)Memory used [KB]: 1918
% 3.23/1.70 % (14171)Time elapsed: 0.011 s
% 3.23/1.70 % (14171)Instructions burned: 19 (million)
% 3.23/1.70 % (14171)------------------------------
% 3.23/1.70 % (14171)------------------------------
% 3.23/1.71 % (14167)Instruction limit reached!
% 3.23/1.71 % (14167)------------------------------
% 3.23/1.71 % (14167)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.71 % (14167)Termination reason: Unknown
% 3.23/1.71 % (14167)Termination phase: shuffling
% 3.23/1.71
% 3.23/1.71 % (14167)Memory used [KB]: 2046
% 3.23/1.71 % (14167)Time elapsed: 0.016 s
% 3.23/1.71 % (14167)Instructions burned: 28 (million)
% 3.23/1.71 % (14167)------------------------------
% 3.23/1.71 % (14167)------------------------------
% 3.23/1.71 % (14175)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/37Mi)
% 3.23/1.71 % (14174)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/3Mi)
% 3.23/1.71 % (14176)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/15Mi)
% 3.23/1.71 % (14174)Instruction limit reached!
% 3.23/1.71 % (14174)------------------------------
% 3.23/1.71 % (14174)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.71 % (14174)Termination reason: Unknown
% 3.23/1.71 % (14174)Termination phase: shuffling
% 3.23/1.71
% 3.23/1.71 % (14174)Memory used [KB]: 1407
% 3.23/1.71 % (14174)Time elapsed: 0.003 s
% 3.23/1.71 % (14174)Instructions burned: 3 (million)
% 3.23/1.71 % (14174)------------------------------
% 3.23/1.71 % (14174)------------------------------
% 3.23/1.72 % (14177)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/3Mi)
% 3.23/1.72 % (14176)Instruction limit reached!
% 3.23/1.72 % (14176)------------------------------
% 3.23/1.72 % (14176)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.72 % (14176)Termination reason: Unknown
% 3.23/1.72 % (14176)Termination phase: shuffling
% 3.23/1.72
% 3.23/1.72 % (14176)Memory used [KB]: 1791
% 3.23/1.72 % (14176)Time elapsed: 0.009 s
% 3.23/1.72 % (14176)Instructions burned: 15 (million)
% 3.23/1.72 % (14176)------------------------------
% 3.23/1.72 % (14176)------------------------------
% 3.23/1.72 % (14175)Instruction limit reached!
% 3.23/1.72 % (14175)------------------------------
% 3.23/1.72 % (14175)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.72 % (14175)Termination reason: Unknown
% 3.23/1.72 % (14175)Termination phase: shuffling
% 3.23/1.72
% 3.23/1.72 % (14175)Memory used [KB]: 2302
% 3.23/1.72 % (14175)Time elapsed: 0.012 s
% 3.23/1.72 % (14175)Instructions burned: 38 (million)
% 3.23/1.72 % (14175)------------------------------
% 3.23/1.72 % (14175)------------------------------
% 3.23/1.72 % (14177)Instruction limit reached!
% 3.23/1.72 % (14177)------------------------------
% 3.23/1.72 % (14177)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.72 % (14177)Termination reason: Unknown
% 3.23/1.72 % (14177)Termination phase: shuffling
% 3.23/1.72
% 3.23/1.72 % (14177)Memory used [KB]: 1535
% 3.23/1.72 % (14177)Time elapsed: 0.003 s
% 3.23/1.72 % (14177)Instructions burned: 5 (million)
% 3.23/1.72 % (14177)------------------------------
% 3.23/1.72 % (14177)------------------------------
% 3.23/1.72 % (14179)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/1041Mi)
% 3.23/1.73 % (14180)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/7Mi)
% 3.23/1.73 % (14180)Instruction limit reached!
% 3.23/1.73 % (14180)------------------------------
% 3.23/1.73 % (14180)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.73 % (14180)Termination reason: Unknown
% 3.23/1.73 % (14180)Termination phase: shuffling
% 3.23/1.73
% 3.23/1.73 % (14180)Memory used [KB]: 1535
% 3.23/1.73 % (14180)Time elapsed: 0.004 s
% 3.23/1.73 % (14180)Instructions burned: 9 (million)
% 3.23/1.73 % (14180)------------------------------
% 3.23/1.73 % (14180)------------------------------
% 3.23/1.73 % (14182)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/16Mi)
% 3.23/1.73 % (14183)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/3Mi)
% 3.23/1.74 % (14183)Instruction limit reached!
% 3.23/1.74 % (14183)------------------------------
% 3.23/1.74 % (14183)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.74 % (14183)Termination reason: Unknown
% 3.23/1.74 % (14183)Termination phase: shuffling
% 3.23/1.74
% 3.23/1.74 % (14183)Memory used [KB]: 1407
% 3.23/1.74 % (14183)Time elapsed: 0.003 s
% 3.23/1.74 % (14183)Instructions burned: 3 (million)
% 3.23/1.74 % (14183)------------------------------
% 3.23/1.74 % (14183)------------------------------
% 3.23/1.74 % (14184)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/3Mi)
% 3.23/1.74 % (14184)Instruction limit reached!
% 3.23/1.74 % (14184)------------------------------
% 3.23/1.74 % (14184)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.74 % (14184)Termination reason: Unknown
% 3.23/1.74 % (14184)Termination phase: shuffling
% 3.23/1.74
% 3.23/1.74 % (14184)Memory used [KB]: 1407
% 3.23/1.74 % (14184)Time elapsed: 0.003 s
% 3.23/1.74 % (14184)Instructions burned: 3 (million)
% 3.23/1.74 % (14184)------------------------------
% 3.23/1.74 % (14184)------------------------------
% 3.23/1.74 % (14185)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/7Mi)
% 3.23/1.74 % (14182)Instruction limit reached!
% 3.23/1.74 % (14182)------------------------------
% 3.23/1.74 % (14182)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.74 % (14182)Termination reason: Unknown
% 3.23/1.74 % (14182)Termination phase: shuffling
% 3.23/1.74
% 3.23/1.74 % (14182)Memory used [KB]: 1791
% 3.23/1.74 % (14182)Time elapsed: 0.010 s
% 3.23/1.74 % (14182)Instructions burned: 17 (million)
% 3.23/1.74 % (14182)------------------------------
% 3.23/1.74 % (14182)------------------------------
% 3.23/1.74 % (14185)Instruction limit reached!
% 3.23/1.74 % (14185)------------------------------
% 3.23/1.74 % (14185)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.74 % (14185)Termination reason: Unknown
% 3.23/1.74 % (14185)Termination phase: shuffling
% 3.23/1.74
% 3.23/1.74 % (14185)Memory used [KB]: 1535
% 3.23/1.74 % (14185)Time elapsed: 0.004 s
% 3.23/1.74 % (14185)Instructions burned: 9 (million)
% 3.23/1.74 % (14185)------------------------------
% 3.23/1.74 % (14185)------------------------------
% 3.23/1.75 % (14186)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/3Mi)
% 3.23/1.75 % (14189)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/710Mi)
% 3.23/1.75 % (14186)Instruction limit reached!
% 3.23/1.75 % (14186)------------------------------
% 3.23/1.75 % (14186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.75 % (14186)Termination reason: Unknown
% 3.23/1.75 % (14186)Termination phase: shuffling
% 3.23/1.75
% 3.23/1.75 % (14186)Memory used [KB]: 1407
% 3.23/1.75 % (14186)Time elapsed: 0.003 s
% 3.23/1.75 % (14186)Instructions burned: 3 (million)
% 3.23/1.75 % (14186)------------------------------
% 3.23/1.75 % (14186)------------------------------
% 3.23/1.75 % (14187)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/4Mi)
% 3.23/1.76 % (14187)Instruction limit reached!
% 3.23/1.76 % (14187)------------------------------
% 3.23/1.76 % (14187)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.76 % (14187)Termination reason: Unknown
% 3.23/1.76 % (14187)Termination phase: shuffling
% 3.23/1.76
% 3.23/1.76 % (14187)Memory used [KB]: 1535
% 3.23/1.76 % (14187)Time elapsed: 0.004 s
% 3.23/1.76 % (14187)Instructions burned: 5 (million)
% 3.23/1.76 % (14187)------------------------------
% 3.23/1.76 % (14187)------------------------------
% 3.23/1.76 % (14188)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/18Mi)
% 3.23/1.76 % (14165)Instruction limit reached!
% 3.23/1.76 % (14165)------------------------------
% 3.23/1.76 % (14165)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.76 % (14165)Termination reason: Unknown
% 3.23/1.76 % (14165)Termination phase: Saturation
% 3.23/1.76
% 3.23/1.76 % (14165)Memory used [KB]: 7675
% 3.23/1.76 % (14165)Time elapsed: 0.075 s
% 3.23/1.76 % (14165)Instructions burned: 185 (million)
% 3.23/1.76 % (14165)------------------------------
% 3.23/1.76 % (14165)------------------------------
% 3.23/1.77 % (14188)Instruction limit reached!
% 3.23/1.77 % (14188)------------------------------
% 3.23/1.77 % (14188)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.77 % (14188)Termination reason: Unknown
% 3.23/1.77 % (14188)Termination phase: shuffling
% 3.23/1.77
% 3.23/1.77 % (14188)Memory used [KB]: 1791
% 3.23/1.77 % (14188)Time elapsed: 0.010 s
% 3.23/1.77 % (14188)Instructions burned: 18 (million)
% 3.23/1.77 % (14188)------------------------------
% 3.23/1.77 % (14188)------------------------------
% 3.23/1.77 % (14190)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/6Mi)
% 3.23/1.77 % (14192)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/21Mi)
% 3.23/1.77 % (14191)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/902Mi)
% 3.23/1.77 % (14190)Instruction limit reached!
% 3.23/1.77 % (14190)------------------------------
% 3.23/1.77 % (14190)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.77 % (14190)Termination reason: Unknown
% 3.23/1.77 % (14190)Termination phase: shuffling
% 3.23/1.77
% 3.23/1.77 % (14190)Memory used [KB]: 1535
% 3.23/1.77 % (14190)Time elapsed: 0.005 s
% 3.23/1.77 % (14190)Instructions burned: 7 (million)
% 3.23/1.77 % (14190)------------------------------
% 3.23/1.77 % (14190)------------------------------
% 3.23/1.78 % (14194)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/5Mi)
% 3.23/1.78 % (14192)Instruction limit reached!
% 3.23/1.78 % (14192)------------------------------
% 3.23/1.78 % (14192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.78 % (14192)Termination reason: Unknown
% 3.23/1.78 % (14192)Termination phase: shuffling
% 3.23/1.78
% 3.23/1.78 % (14192)Memory used [KB]: 1918
% 3.23/1.78 % (14192)Time elapsed: 0.012 s
% 3.23/1.78 % (14192)Instructions burned: 21 (million)
% 3.23/1.78 % (14192)------------------------------
% 3.23/1.78 % (14192)------------------------------
% 3.23/1.78 % (14194)Instruction limit reached!
% 3.23/1.78 % (14194)------------------------------
% 3.23/1.78 % (14194)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.78 % (14194)Termination reason: Unknown
% 3.23/1.78 % (14194)Termination phase: shuffling
% 3.23/1.78
% 3.23/1.78 % (14194)Memory used [KB]: 1535
% 3.23/1.78 % (14194)Time elapsed: 0.004 s
% 3.23/1.78 % (14194)Instructions burned: 5 (million)
% 3.23/1.78 % (14194)------------------------------
% 3.23/1.78 % (14194)------------------------------
% 3.23/1.79 % (14195)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13980 for (2999ds/6Mi)
% 3.23/1.79 % (14195)Instruction limit reached!
% 3.23/1.79 % (14195)------------------------------
% 3.23/1.79 % (14195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.79 % (14195)Termination reason: Unknown
% 3.23/1.79 % (14195)Termination phase: shuffling
% 3.23/1.79
% 3.23/1.79 % (14195)Memory used [KB]: 1663
% 3.23/1.79 % (14195)Time elapsed: 0.005 s
% 3.23/1.79 % (14195)Instructions burned: 7 (million)
% 3.23/1.79 % (14195)------------------------------
% 3.23/1.79 % (14195)------------------------------
% 3.23/1.80 % (14197)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_13980 for (2998ds/779Mi)
% 3.23/1.80 % (14196)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13980 for (2998ds/377Mi)
% 3.23/1.81 % (14199)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_13980 for (2998ds/19Mi)
% 3.23/1.82 % (14199)Instruction limit reached!
% 3.23/1.82 % (14199)------------------------------
% 3.23/1.82 % (14199)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.82 % (14199)Termination reason: Unknown
% 3.23/1.82 % (14199)Termination phase: shuffling
% 3.23/1.82
% 3.23/1.82 % (14199)Memory used [KB]: 1791
% 3.23/1.82 % (14199)Time elapsed: 0.011 s
% 3.23/1.82 % (14199)Instructions burned: 20 (million)
% 3.23/1.82 % (14199)------------------------------
% 3.23/1.82 % (14199)------------------------------
% 3.23/1.83 % (14170)Instruction limit reached!
% 3.23/1.83 % (14170)------------------------------
% 3.23/1.83 % (14170)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.83 % (14170)Termination reason: Unknown
% 3.23/1.83 % (14170)Termination phase: Saturation
% 3.23/1.83
% 3.23/1.83 % (14170)Memory used [KB]: 7803
% 3.23/1.83 % (14170)Time elapsed: 0.135 s
% 3.23/1.83 % (14170)Instructions burned: 275 (million)
% 3.23/1.83 % (14170)------------------------------
% 3.23/1.83 % (14170)------------------------------
% 3.23/1.83 % (14203)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13980 for (2998ds/879Mi)
% 3.23/1.84 % (14204)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_13980 for (2998ds/17Mi)
% 3.23/1.85 % (14204)Instruction limit reached!
% 3.23/1.85 % (14204)------------------------------
% 3.23/1.85 % (14204)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.85 % (14204)Termination reason: Unknown
% 3.23/1.85 % (14204)Termination phase: shuffling
% 3.23/1.85
% 3.23/1.85 % (14204)Memory used [KB]: 1791
% 3.23/1.85 % (14204)Time elapsed: 0.007 s
% 3.23/1.85 % (14204)Instructions burned: 18 (million)
% 3.23/1.85 % (14204)------------------------------
% 3.23/1.85 % (14204)------------------------------
% 3.23/1.86 % (14206)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13980 for (2998ds/3Mi)
% 3.23/1.86 % (14206)Instruction limit reached!
% 3.23/1.86 % (14206)------------------------------
% 3.23/1.86 % (14206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.86 % (14206)Termination reason: Unknown
% 3.23/1.86 % (14206)Termination phase: shuffling
% 3.23/1.86
% 3.23/1.86 % (14206)Memory used [KB]: 1407
% 3.23/1.86 % (14206)Time elapsed: 0.002 s
% 3.23/1.86 % (14206)Instructions burned: 3 (million)
% 3.23/1.86 % (14206)------------------------------
% 3.23/1.86 % (14206)------------------------------
% 3.23/1.87 % (14207)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_13980 for (2998ds/30Mi)
% 3.23/1.88 % (14207)Instruction limit reached!
% 3.23/1.88 % (14207)------------------------------
% 3.23/1.88 % (14207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.88 % (14207)Termination reason: Unknown
% 3.23/1.88 % (14207)Termination phase: shuffling
% 3.23/1.88
% 3.23/1.88 % (14207)Memory used [KB]: 2046
% 3.23/1.88 % (14207)Time elapsed: 0.011 s
% 3.23/1.88 % (14207)Instructions burned: 30 (million)
% 3.23/1.88 % (14207)------------------------------
% 3.23/1.88 % (14207)------------------------------
% 3.23/1.89 % (14209)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_13980 for (2997ds/127Mi)
% 3.23/1.95 % (14209)Instruction limit reached!
% 3.23/1.95 % (14209)------------------------------
% 3.23/1.95 % (14209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.95 % (14209)Termination reason: Unknown
% 3.23/1.95 % (14209)Termination phase: Property scanning
% 3.23/1.95
% 3.23/1.95 % (14209)Memory used [KB]: 2686
% 3.23/1.95 % (14209)Time elapsed: 0.061 s
% 3.23/1.95 % (14209)Instructions burned: 127 (million)
% 3.23/1.95 % (14209)------------------------------
% 3.23/1.95 % (14209)------------------------------
% 3.23/1.96 % (14216)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_13980 for (2997ds/100Mi)
% 3.23/1.97 % (14196)Instruction limit reached!
% 3.23/1.97 % (14196)------------------------------
% 3.23/1.97 % (14196)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.97 % (14196)Termination reason: Unknown
% 3.23/1.97 % (14196)Termination phase: Saturation
% 3.23/1.97
% 3.23/1.97 % (14196)Memory used [KB]: 8571
% 3.23/1.97 % (14196)Time elapsed: 0.173 s
% 3.23/1.97 % (14196)Instructions burned: 378 (million)
% 3.23/1.97 % (14196)------------------------------
% 3.23/1.97 % (14196)------------------------------
% 3.23/1.99 % (14219)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_13980 for (2997ds/3Mi)
% 3.23/1.99 % (14219)Instruction limit reached!
% 3.23/1.99 % (14219)------------------------------
% 3.23/1.99 % (14219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.99 % (14219)Termination reason: Unknown
% 3.23/1.99 % (14219)Termination phase: shuffling
% 3.23/1.99
% 3.23/1.99 % (14219)Memory used [KB]: 1407
% 3.23/1.99 % (14219)Time elapsed: 0.004 s
% 3.23/1.99 % (14219)Instructions burned: 4 (million)
% 3.23/1.99 % (14219)------------------------------
% 3.23/1.99 % (14219)------------------------------
% 3.23/2.00 % (14216)Instruction limit reached!
% 3.23/2.00 % (14216)------------------------------
% 3.23/2.00 % (14216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/2.00 % (14216)Termination reason: Unknown
% 3.23/2.00 % (14216)Termination phase: Property scanning
% 3.23/2.00
% 3.23/2.00 % (14216)Memory used [KB]: 2686
% 3.23/2.00 % (14216)Time elapsed: 0.036 s
% 3.23/2.00 % (14216)Instructions burned: 103 (million)
% 3.23/2.00 % (14216)------------------------------
% 3.23/2.00 % (14216)------------------------------
% 3.23/2.01 % (14222)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_13980 for (2996ds/20Mi)
% 3.23/2.01 % (14223)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_13980 for (2996ds/86Mi)
% 3.23/2.01 % (14222)Instruction limit reached!
% 3.23/2.01 % (14222)------------------------------
% 3.23/2.01 % (14222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/2.01 % (14222)Termination reason: Unknown
% 3.23/2.01 % (14222)Termination phase: shuffling
% 3.23/2.01
% 3.23/2.01 % (14222)Memory used [KB]: 1918
% 3.23/2.01 % (14222)Time elapsed: 0.008 s
% 3.23/2.01 % (14222)Instructions burned: 20 (million)
% 3.23/2.01 % (14222)------------------------------
% 3.23/2.01 % (14222)------------------------------
% 3.23/2.02 % (14225)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_13980 for (2996ds/107Mi)
% 5.54/2.05 % (14223)Instruction limit reached!
% 5.54/2.05 % (14223)------------------------------
% 5.54/2.05 % (14223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.05 % (14223)Termination reason: Unknown
% 5.54/2.05 % (14223)Termination phase: Property scanning
% 5.54/2.05
% 5.54/2.05 % (14223)Memory used [KB]: 2558
% 5.54/2.05 % (14223)Time elapsed: 0.042 s
% 5.54/2.05 % (14223)Instructions burned: 86 (million)
% 5.54/2.05 % (14223)------------------------------
% 5.54/2.05 % (14223)------------------------------
% 5.54/2.07 % (14231)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_13980 for (2996ds/39Mi)
% 5.54/2.07 % (14189)Instruction limit reached!
% 5.54/2.07 % (14189)------------------------------
% 5.54/2.07 % (14189)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.07 % (14189)Termination reason: Unknown
% 5.54/2.07 % (14189)Termination phase: Saturation
% 5.54/2.07
% 5.54/2.07 % (14189)Memory used [KB]: 8443
% 5.54/2.07 % (14189)Time elapsed: 0.321 s
% 5.54/2.07 % (14189)Instructions burned: 710 (million)
% 5.54/2.07 % (14189)------------------------------
% 5.54/2.07 % (14189)------------------------------
% 5.54/2.07 % (14225)Instruction limit reached!
% 5.54/2.07 % (14225)------------------------------
% 5.54/2.07 % (14225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.07 % (14225)Termination reason: Unknown
% 5.54/2.07 % (14225)Termination phase: Property scanning
% 5.54/2.07
% 5.54/2.07 % (14225)Memory used [KB]: 2686
% 5.54/2.07 % (14225)Time elapsed: 0.051 s
% 5.54/2.07 % (14225)Instructions burned: 108 (million)
% 5.54/2.07 % (14225)------------------------------
% 5.54/2.07 % (14225)------------------------------
% 5.54/2.09 % (14234)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_13980 for (2996ds/448Mi)
% 5.54/2.09 % (14231)Instruction limit reached!
% 5.54/2.09 % (14231)------------------------------
% 5.54/2.09 % (14231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.09 % (14231)Termination reason: Unknown
% 5.54/2.09 % (14231)Termination phase: shuffling
% 5.54/2.09
% 5.54/2.09 % (14231)Memory used [KB]: 2302
% 5.54/2.09 % (14231)Time elapsed: 0.021 s
% 5.54/2.09 % (14231)Instructions burned: 41 (million)
% 5.54/2.09 % (14231)------------------------------
% 5.54/2.09 % (14231)------------------------------
% 5.54/2.09 % (14235)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_13980 for (2996ds/46Mi)
% 5.54/2.11 % (14235)Instruction limit reached!
% 5.54/2.11 % (14235)------------------------------
% 5.54/2.11 % (14235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.11 % (14235)Termination reason: Unknown
% 5.54/2.11 % (14235)Termination phase: Property scanning
% 5.54/2.11
% 5.54/2.11 % (14235)Memory used [KB]: 2046
% 5.54/2.11 % (14235)Time elapsed: 0.024 s
% 5.54/2.11 % (14235)Instructions burned: 46 (million)
% 5.54/2.11 % (14235)------------------------------
% 5.54/2.11 % (14235)------------------------------
% 5.54/2.12 % (14197)Instruction limit reached!
% 5.54/2.12 % (14197)------------------------------
% 5.54/2.12 % (14197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.54/2.12 % (14197)Termination reason: Unknown
% 5.54/2.12 % (14197)Termination phase: Saturation
% 5.54/2.12
% 5.54/2.12 % (14197)Memory used [KB]: 9594
% 5.54/2.12 % (14197)Time elapsed: 0.321 s
% 5.54/2.12 % (14197)Instructions burned: 781 (million)
% 5.54/2.12 % (14197)------------------------------
% 5.54/2.12 % (14197)------------------------------
% 5.54/2.12 % (14236)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_13980 for (2995ds/98Mi)
% 6.21/2.13 % (14237)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_13980 for (2995ds/507Mi)
% 6.21/2.13 % (14238)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_13980 for (2995ds/149Mi)
% 6.21/2.15 % (14236)Instruction limit reached!
% 6.21/2.15 % (14236)------------------------------
% 6.21/2.15 % (14236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.15 % (14236)Termination reason: Unknown
% 6.21/2.15 % (14236)Termination phase: Property scanning
% 6.21/2.15
% 6.21/2.15 % (14236)Memory used [KB]: 2686
% 6.21/2.15 % (14236)Time elapsed: 0.030 s
% 6.21/2.15 % (14236)Instructions burned: 101 (million)
% 6.21/2.15 % (14236)------------------------------
% 6.21/2.15 % (14236)------------------------------
% 6.21/2.15 % (14241)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_13980 for (2995ds/39Mi)
% 6.21/2.17 % (14241)Instruction limit reached!
% 6.21/2.17 % (14241)------------------------------
% 6.21/2.17 % (14241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.17 % (14241)Termination reason: Unknown
% 6.21/2.17 % (14241)Termination phase: Property scanning
% 6.21/2.17
% 6.21/2.17 % (14241)Memory used [KB]: 2046
% 6.21/2.17 % (14241)Time elapsed: 0.021 s
% 6.21/2.17 % (14241)Instructions burned: 40 (million)
% 6.21/2.17 % (14241)------------------------------
% 6.21/2.17 % (14241)------------------------------
% 6.21/2.18 % (14191)Instruction limit reached!
% 6.21/2.18 % (14191)------------------------------
% 6.21/2.18 % (14191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.18 % (14191)Termination reason: Unknown
% 6.21/2.18 % (14191)Termination phase: Saturation
% 6.21/2.18
% 6.21/2.18 % (14191)Memory used [KB]: 8699
% 6.21/2.18 % (14191)Time elapsed: 0.412 s
% 6.21/2.18 % (14191)Instructions burned: 905 (million)
% 6.21/2.18 % (14191)------------------------------
% 6.21/2.18 % (14191)------------------------------
% 6.21/2.19 % (14244)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13980 for (2995ds/6Mi)
% 6.21/2.19 % (14179)Instruction limit reached!
% 6.21/2.19 % (14179)------------------------------
% 6.21/2.19 % (14179)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.19 % (14179)Termination reason: Unknown
% 6.21/2.19 % (14179)Termination phase: Saturation
% 6.21/2.19
% 6.21/2.19 % (14179)Memory used [KB]: 10618
% 6.21/2.19 % (14179)Time elapsed: 0.470 s
% 6.21/2.19 % (14179)Instructions burned: 1042 (million)
% 6.21/2.19 % (14179)------------------------------
% 6.21/2.19 % (14179)------------------------------
% 6.21/2.19 % (14245)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_13980 for (2994ds/1087Mi)
% 6.21/2.19 % (14244)Instruction limit reached!
% 6.21/2.19 % (14244)------------------------------
% 6.21/2.19 % (14244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.21/2.19 % (14244)Termination reason: Unknown
% 6.21/2.19 % (14244)Termination phase: shuffling
% 6.21/2.19
% 6.21/2.19 % (14244)Memory used [KB]: 1535
% 6.21/2.19 % (14244)Time elapsed: 0.005 s
% 6.21/2.19 % (14244)Instructions burned: 7 (million)
% 6.21/2.19 % (14244)------------------------------
% 6.21/2.19 % (14244)------------------------------
% 6.51/2.20 % (14238)Instruction limit reached!
% 6.51/2.20 % (14238)------------------------------
% 6.51/2.20 % (14238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.20 % (14238)Termination reason: Unknown
% 6.51/2.20 % (14238)Termination phase: Saturation
% 6.51/2.20
% 6.51/2.20 % (14238)Memory used [KB]: 7291
% 6.51/2.20 % (14238)Time elapsed: 0.072 s
% 6.51/2.20 % (14238)Instructions burned: 150 (million)
% 6.51/2.20 % (14238)------------------------------
% 6.51/2.20 % (14238)------------------------------
% 6.51/2.21 % (14247)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_13980 for (2994ds/44Mi)
% 6.51/2.21 % (14248)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_13980 for (2994ds/432Mi)
% 6.51/2.22 % (14246)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_13980 for (2994ds/47Mi)
% 6.51/2.22 % (14247)Instruction limit reached!
% 6.51/2.22 % (14247)------------------------------
% 6.51/2.22 % (14247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.22 % (14247)Termination reason: Unknown
% 6.51/2.22 % (14247)Termination phase: shuffling
% 6.51/2.22
% 6.51/2.22 % (14247)Memory used [KB]: 2302
% 6.51/2.22 % (14247)Time elapsed: 0.015 s
% 6.51/2.22 % (14247)Instructions burned: 45 (million)
% 6.51/2.22 % (14247)------------------------------
% 6.51/2.22 % (14247)------------------------------
% 6.51/2.23 % (14246)Instruction limit reached!
% 6.51/2.23 % (14246)------------------------------
% 6.51/2.23 % (14246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.23 % (14246)Termination reason: Unknown
% 6.51/2.23 % (14246)Termination phase: Preprocessing 1
% 6.51/2.23
% 6.51/2.23 % (14246)Memory used [KB]: 2046
% 6.51/2.23 % (14246)Time elapsed: 0.016 s
% 6.51/2.23 % (14246)Instructions burned: 47 (million)
% 6.51/2.23 % (14246)------------------------------
% 6.51/2.23 % (14246)------------------------------
% 6.51/2.23 % (14251)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_13980 for (2994ds/62Mi)
% 6.51/2.24 % (14252)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_13980 for (2994ds/31Mi)
% 6.51/2.24 % (14203)Instruction limit reached!
% 6.51/2.24 % (14203)------------------------------
% 6.51/2.24 % (14203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.24 % (14203)Termination reason: Unknown
% 6.51/2.24 % (14203)Termination phase: Saturation
% 6.51/2.24
% 6.51/2.24 % (14203)Memory used [KB]: 9210
% 6.51/2.24 % (14203)Time elapsed: 0.412 s
% 6.51/2.24 % (14203)Instructions burned: 879 (million)
% 6.51/2.24 % (14203)------------------------------
% 6.51/2.24 % (14203)------------------------------
% 6.51/2.25 % (14251)Instruction limit reached!
% 6.51/2.25 % (14251)------------------------------
% 6.51/2.25 % (14251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.25 % (14251)Termination reason: Unknown
% 6.51/2.25 % (14251)Termination phase: Property scanning
% 6.51/2.25
% 6.51/2.25 % (14251)Memory used [KB]: 2430
% 6.51/2.25 % (14251)Time elapsed: 0.019 s
% 6.51/2.25 % (14251)Instructions burned: 64 (million)
% 6.51/2.25 % (14251)------------------------------
% 6.51/2.25 % (14251)------------------------------
% 6.51/2.25 % (14252)Instruction limit reached!
% 6.51/2.25 % (14252)------------------------------
% 6.51/2.25 % (14252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.25 % (14252)Termination reason: Unknown
% 6.51/2.25 % (14252)Termination phase: shuffling
% 6.51/2.25
% 6.51/2.25 % (14252)Memory used [KB]: 2046
% 6.51/2.25 % (14252)Time elapsed: 0.012 s
% 6.51/2.25 % (14252)Instructions burned: 32 (million)
% 6.51/2.25 % (14252)------------------------------
% 6.51/2.25 % (14252)------------------------------
% 6.51/2.25 % (14253)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_13980 for (2994ds/96Mi)
% 6.51/2.26 % (14254)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_13980 for (2994ds/453Mi)
% 6.51/2.26 % (14255)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_13980 for (2994ds/71Mi)
% 6.51/2.27 % (14234)Instruction limit reached!
% 6.51/2.27 % (14234)------------------------------
% 6.51/2.27 % (14234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.27 % (14234)Termination reason: Unknown
% 6.51/2.27 % (14234)Termination phase: Saturation
% 6.51/2.27
% 6.51/2.27 % (14234)Memory used [KB]: 9338
% 6.51/2.27 % (14234)Time elapsed: 0.181 s
% 6.51/2.27 % (14234)Instructions burned: 450 (million)
% 6.51/2.27 % (14234)------------------------------
% 6.51/2.27 % (14234)------------------------------
% 6.51/2.28 % (14256)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/83Mi)
% 6.51/2.28 % (14253)Instruction limit reached!
% 6.51/2.28 % (14253)------------------------------
% 6.51/2.28 % (14253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.28 % (14253)Termination reason: Unknown
% 6.51/2.28 % (14253)Termination phase: Function definition elimination
% 6.51/2.28
% 6.51/2.28 % (14253)Memory used [KB]: 2686
% 6.51/2.28 % (14253)Time elapsed: 0.030 s
% 6.51/2.28 % (14253)Instructions burned: 97 (million)
% 6.51/2.28 % (14253)------------------------------
% 6.51/2.28 % (14253)------------------------------
% 6.51/2.28 % (14255)Instruction limit reached!
% 6.51/2.28 % (14255)------------------------------
% 6.51/2.28 % (14255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.28 % (14255)Termination reason: Unknown
% 6.51/2.28 % (14255)Termination phase: Property scanning
% 6.51/2.28
% 6.51/2.28 % (14255)Memory used [KB]: 2686
% 6.51/2.28 % (14255)Time elapsed: 0.024 s
% 6.51/2.28 % (14255)Instructions burned: 74 (million)
% 6.51/2.28 % (14255)------------------------------
% 6.51/2.28 % (14255)------------------------------
% 6.51/2.29 % (14257)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/11Mi)
% 6.51/2.29 % (14258)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/41Mi)
% 6.51/2.30 % (14257)Instruction limit reached!
% 6.51/2.30 % (14257)------------------------------
% 6.51/2.30 % (14257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.51/2.30 % (14257)Termination reason: Unknown
% 6.51/2.30 % (14257)Termination phase: shuffling
% 6.51/2.30
% 6.51/2.30 % (14257)Memory used [KB]: 1663
% 6.51/2.30 % (14257)Time elapsed: 0.004 s
% 6.51/2.30 % (14257)Instructions burned: 11 (million)
% 6.51/2.30 % (14257)------------------------------
% 6.51/2.30 % (14257)------------------------------
% 7.05/2.30 % (14256)Instruction limit reached!
% 7.05/2.30 % (14256)------------------------------
% 7.05/2.30 % (14256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.05/2.30 % (14256)Termination reason: Unknown
% 7.05/2.30 % (14256)Termination phase: Function definition elimination
% 7.05/2.30
% 7.05/2.30 % (14256)Memory used [KB]: 2558
% 7.05/2.30 % (14256)Time elapsed: 0.027 s
% 7.05/2.30 % (14256)Instructions burned: 85 (million)
% 7.05/2.30 % (14256)------------------------------
% 7.05/2.30 % (14256)------------------------------
% 7.05/2.30 % (14259)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/6Mi)
% 7.05/2.31 % (14258)Instruction limit reached!
% 7.05/2.31 % (14258)------------------------------
% 7.05/2.31 % (14258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.05/2.31 % (14258)Termination reason: Unknown
% 7.05/2.31 % (14258)Termination phase: shuffling
% 7.05/2.31
% 7.05/2.31 % (14258)Memory used [KB]: 2302
% 7.05/2.31 % (14258)Time elapsed: 0.013 s
% 7.05/2.31 % (14258)Instructions burned: 41 (million)
% 7.05/2.31 % (14258)------------------------------
% 7.05/2.31 % (14258)------------------------------
% 7.05/2.31 % (14259)Instruction limit reached!
% 7.05/2.31 % (14259)------------------------------
% 7.05/2.31 % (14259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.05/2.31 % (14259)Termination reason: Unknown
% 7.05/2.31 % (14259)Termination phase: shuffling
% 7.05/2.31
% 7.05/2.31 % (14259)Memory used [KB]: 1663
% 7.05/2.31 % (14259)Time elapsed: 0.004 s
% 7.05/2.31 % (14259)Instructions burned: 8 (million)
% 7.05/2.31 % (14259)------------------------------
% 7.05/2.31 % (14259)------------------------------
% 7.05/2.31 % (14260)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/150Mi)
% 7.05/2.32 % (14237)Instruction limit reached!
% 7.05/2.32 % (14237)------------------------------
% 7.05/2.32 % (14237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.05/2.32 % (14237)Termination reason: Unknown
% 7.05/2.32 % (14237)Termination phase: Saturation
% 7.05/2.32
% 7.05/2.32 % (14237)Memory used [KB]: 8699
% 7.05/2.32 % (14237)Time elapsed: 0.208 s
% 7.05/2.32 % (14237)Instructions burned: 508 (million)
% 7.05/2.32 % (14237)------------------------------
% 7.05/2.32 % (14237)------------------------------
% 7.05/2.32 % (14261)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/343Mi)
% 7.05/2.32 % (14262)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/109Mi)
% 7.05/2.33 % (14263)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/253Mi)
% 7.05/2.35 % (14248)Instruction limit reached!
% 7.05/2.35 % (14248)------------------------------
% 7.05/2.35 % (14248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.05/2.35 % (14248)Termination reason: Unknown
% 7.05/2.35 % (14248)Termination phase: Saturation
% 7.05/2.35
% 7.05/2.35 % (14248)Memory used [KB]: 4093
% 7.05/2.35 % (14248)Time elapsed: 0.134 s
% 7.05/2.35 % (14248)Instructions burned: 433 (million)
% 7.05/2.35 % (14248)------------------------------
% 7.05/2.35 % (14248)------------------------------
% 7.05/2.35 % (14262)Instruction limit reached!
% 7.05/2.35 % (14262)------------------------------
% 7.05/2.35 % (14262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.05/2.35 % (14262)Termination reason: Unknown
% 7.05/2.35 % (14262)Termination phase: Property scanning
% 7.05/2.35
% 7.05/2.35 % (14262)Memory used [KB]: 2686
% 7.05/2.35 % (14262)Time elapsed: 0.034 s
% 7.05/2.35 % (14262)Instructions burned: 110 (million)
% 7.05/2.35 % (14262)------------------------------
% 7.05/2.35 % (14262)------------------------------
% 7.05/2.36 % (14264)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/140Mi)
% 7.05/2.36 % (14260)Instruction limit reached!
% 7.05/2.36 % (14260)------------------------------
% 7.05/2.36 % (14260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.05/2.36 % (14260)Termination reason: Unknown
% 7.05/2.36 % (14260)Termination phase: Saturation
% 7.05/2.36
% 7.05/2.36 % (14260)Memory used [KB]: 7291
% 7.05/2.36 % (14260)Time elapsed: 0.045 s
% 7.05/2.36 % (14260)Instructions burned: 153 (million)
% 7.05/2.36 % (14260)------------------------------
% 7.05/2.36 % (14260)------------------------------
% 7.05/2.36 % (14265)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/614Mi)
% 7.05/2.37 % (14266)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_13980 for (2993ds/202Mi)
% 7.56/2.40 % (14264)Instruction limit reached!
% 7.56/2.40 % (14264)------------------------------
% 7.56/2.40 % (14264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.40 % (14264)Termination reason: Unknown
% 7.56/2.40 % (14264)Termination phase: Saturation
% 7.56/2.40
% 7.56/2.40 % (14264)Memory used [KB]: 7547
% 7.56/2.40 % (14264)Time elapsed: 0.044 s
% 7.56/2.40 % (14264)Instructions burned: 141 (million)
% 7.56/2.40 % (14264)------------------------------
% 7.56/2.40 % (14264)------------------------------
% 7.56/2.40 % (14263)Instruction limit reached!
% 7.56/2.40 % (14263)------------------------------
% 7.56/2.40 % (14263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.40 % (14263)Termination reason: Unknown
% 7.56/2.40 % (14263)Termination phase: Saturation
% 7.56/2.40
% 7.56/2.40 % (14263)Memory used [KB]: 8315
% 7.56/2.40 % (14263)Time elapsed: 0.079 s
% 7.56/2.40 % (14263)Instructions burned: 254 (million)
% 7.56/2.40 % (14263)------------------------------
% 7.56/2.40 % (14263)------------------------------
% 7.56/2.41 % (14275)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_13980 for (2992ds/339Mi)
% 7.56/2.41 % (14278)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_13980 for (2992ds/283Mi)
% 7.56/2.42 % (14254)Instruction limit reached!
% 7.56/2.42 % (14254)------------------------------
% 7.56/2.42 % (14254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.42 % (14254)Termination reason: Unknown
% 7.56/2.42 % (14254)Termination phase: Saturation
% 7.56/2.42
% 7.56/2.42 % (14254)Memory used [KB]: 8827
% 7.56/2.42 % (14254)Time elapsed: 0.156 s
% 7.56/2.42 % (14254)Instructions burned: 454 (million)
% 7.56/2.42 % (14254)------------------------------
% 7.56/2.42 % (14254)------------------------------
% 7.56/2.42 % (14261)Instruction limit reached!
% 7.56/2.42 % (14261)------------------------------
% 7.56/2.42 % (14261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.42 % (14261)Termination reason: Unknown
% 7.56/2.42 % (14261)Termination phase: Saturation
% 7.56/2.42
% 7.56/2.42 % (14261)Memory used [KB]: 8315
% 7.56/2.42 % (14261)Time elapsed: 0.107 s
% 7.56/2.42 % (14261)Instructions burned: 343 (million)
% 7.56/2.42 % (14261)------------------------------
% 7.56/2.42 % (14261)------------------------------
% 7.56/2.43 % (14284)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_13980 for (2992ds/134Mi)
% 7.56/2.43 % (14266)Instruction limit reached!
% 7.56/2.43 % (14266)------------------------------
% 7.56/2.43 % (14266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.43 % (14266)Termination reason: Unknown
% 7.56/2.43 % (14266)Termination phase: Saturation
% 7.56/2.43
% 7.56/2.43 % (14266)Memory used [KB]: 8187
% 7.56/2.43 % (14266)Time elapsed: 0.064 s
% 7.56/2.43 % (14266)Instructions burned: 204 (million)
% 7.56/2.43 % (14266)------------------------------
% 7.56/2.43 % (14266)------------------------------
% 7.56/2.43 % (14287)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_13980 for (2992ds/298Mi)
% 7.56/2.44 % (14291)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_13980 for (2992ds/192Mi)
% 7.56/2.46 % (14284)Instruction limit reached!
% 7.56/2.46 % (14284)------------------------------
% 7.56/2.46 % (14284)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.46 % (14284)Termination reason: Unknown
% 7.56/2.46 % (14284)Termination phase: Saturation
% 7.56/2.46
% 7.56/2.46 % (14284)Memory used [KB]: 7164
% 7.56/2.46 % (14284)Time elapsed: 0.040 s
% 7.56/2.46 % (14284)Instructions burned: 134 (million)
% 7.56/2.46 % (14284)------------------------------
% 7.56/2.46 % (14284)------------------------------
% 7.56/2.47 % (14308)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_13980 for (2992ds/258Mi)
% 7.56/2.50 % (14291)Instruction limit reached!
% 7.56/2.50 % (14291)------------------------------
% 7.56/2.50 % (14291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.56/2.50 % (14291)Termination reason: Unknown
% 7.56/2.50 % (14291)Termination phase: Saturation
% 7.56/2.50
% 7.56/2.50 % (14291)Memory used [KB]: 3709
% 7.56/2.50 % (14291)Time elapsed: 0.058 s
% 7.56/2.50 % (14291)Instructions burned: 193 (million)
% 7.56/2.50 % (14291)------------------------------
% 7.56/2.50 % (14291)------------------------------
% 8.10/2.50 % (14278)Instruction limit reached!
% 8.10/2.50 % (14278)------------------------------
% 8.10/2.50 % (14278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.50 % (14278)Termination reason: Unknown
% 8.10/2.50 % (14278)Termination phase: Saturation
% 8.10/2.50
% 8.10/2.50 % (14278)Memory used [KB]: 7803
% 8.10/2.50 % (14278)Time elapsed: 0.085 s
% 8.10/2.50 % (14278)Instructions burned: 284 (million)
% 8.10/2.50 % (14278)------------------------------
% 8.10/2.50 % (14278)------------------------------
% 8.10/2.51 % (14320)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/90Mi)
% 8.10/2.51 % (14321)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/231Mi)
% 8.10/2.51 % (14275)Instruction limit reached!
% 8.10/2.51 % (14275)------------------------------
% 8.10/2.51 % (14275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.51 % (14275)Termination reason: Unknown
% 8.10/2.51 % (14275)Termination phase: Saturation
% 8.10/2.51
% 8.10/2.51 % (14275)Memory used [KB]: 7803
% 8.10/2.51 % (14275)Time elapsed: 0.100 s
% 8.10/2.51 % (14275)Instructions burned: 342 (million)
% 8.10/2.51 % (14275)------------------------------
% 8.10/2.51 % (14275)------------------------------
% 8.10/2.52 % (14287)Instruction limit reached!
% 8.10/2.52 % (14287)------------------------------
% 8.10/2.52 % (14287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.52 % (14287)Termination reason: Unknown
% 8.10/2.52 % (14287)Termination phase: Saturation
% 8.10/2.52
% 8.10/2.52 % (14287)Memory used [KB]: 8699
% 8.10/2.52 % (14287)Time elapsed: 0.086 s
% 8.10/2.52 % (14287)Instructions burned: 300 (million)
% 8.10/2.52 % (14287)------------------------------
% 8.10/2.52 % (14287)------------------------------
% 8.10/2.52 % (14327)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/458Mi)
% 8.10/2.53 % (14331)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/31Mi)
% 8.10/2.53 % (14320)Instruction limit reached!
% 8.10/2.53 % (14320)------------------------------
% 8.10/2.53 % (14320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.53 % (14320)Termination reason: Unknown
% 8.10/2.53 % (14320)Termination phase: Function definition elimination
% 8.10/2.53
% 8.10/2.53 % (14320)Memory used [KB]: 2558
% 8.10/2.53 % (14320)Time elapsed: 0.029 s
% 8.10/2.53 % (14320)Instructions burned: 92 (million)
% 8.10/2.53 % (14320)------------------------------
% 8.10/2.53 % (14320)------------------------------
% 8.10/2.54 % (14331)Instruction limit reached!
% 8.10/2.54 % (14331)------------------------------
% 8.10/2.54 % (14331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.54 % (14331)Termination reason: Unknown
% 8.10/2.54 % (14331)Termination phase: Property scanning
% 8.10/2.54
% 8.10/2.54 % (14331)Memory used [KB]: 2046
% 8.10/2.54 % (14331)Time elapsed: 0.012 s
% 8.10/2.54 % (14331)Instructions burned: 33 (million)
% 8.10/2.54 % (14331)------------------------------
% 8.10/2.54 % (14331)------------------------------
% 8.10/2.54 % (14265)Instruction limit reached!
% 8.10/2.54 % (14265)------------------------------
% 8.10/2.54 % (14265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.54 % (14265)Termination reason: Unknown
% 8.10/2.54 % (14265)Termination phase: Saturation
% 8.10/2.54
% 8.10/2.54 % (14265)Memory used [KB]: 8955
% 8.10/2.54 % (14265)Time elapsed: 0.183 s
% 8.10/2.54 % (14265)Instructions burned: 614 (million)
% 8.10/2.54 % (14265)------------------------------
% 8.10/2.54 % (14265)------------------------------
% 8.10/2.54 % (14245)Instruction limit reached!
% 8.10/2.54 % (14245)------------------------------
% 8.10/2.54 % (14245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.54 % (14245)Termination reason: Unknown
% 8.10/2.54 % (14245)Termination phase: Saturation
% 8.10/2.54
% 8.10/2.54 % (14245)Memory used [KB]: 8443
% 8.10/2.54 % (14245)Time elapsed: 0.351 s
% 8.10/2.54 % (14245)Instructions burned: 1089 (million)
% 8.10/2.54 % (14245)------------------------------
% 8.10/2.54 % (14245)------------------------------
% 8.10/2.54 % (14335)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/297Mi)
% 8.10/2.55 % (14338)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/987Mi)
% 8.10/2.55 % (14343)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/182Mi)
% 8.10/2.55 % (14344)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/182Mi)
% 8.10/2.55 % (14308)Instruction limit reached!
% 8.10/2.55 % (14308)------------------------------
% 8.10/2.55 % (14308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.55 % (14308)Termination reason: Unknown
% 8.10/2.55 % (14308)Termination phase: Saturation
% 8.10/2.55
% 8.10/2.55 % (14308)Memory used [KB]: 8315
% 8.10/2.55 % (14308)Time elapsed: 0.080 s
% 8.10/2.55 % (14308)Instructions burned: 260 (million)
% 8.10/2.55 % (14308)------------------------------
% 8.10/2.55 % (14308)------------------------------
% 8.10/2.56 % (14347)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/154Mi)
% 8.10/2.57 % (14321)Instruction limit reached!
% 8.10/2.57 % (14321)------------------------------
% 8.10/2.57 % (14321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.10/2.57 % (14321)Termination reason: Unknown
% 8.10/2.57 % (14321)Termination phase: Saturation
% 8.10/2.57
% 8.10/2.57 % (14321)Memory used [KB]: 7547
% 8.10/2.57 % (14321)Time elapsed: 0.069 s
% 8.10/2.57 % (14321)Instructions burned: 232 (million)
% 8.10/2.57 % (14321)------------------------------
% 8.10/2.57 % (14321)------------------------------
% 8.10/2.58 % (14359)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_13980 for (2991ds/79Mi)
% 9.25/2.60 % (14343)Instruction limit reached!
% 9.25/2.60 % (14343)------------------------------
% 9.25/2.60 % (14343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.60 % (14343)Termination reason: Unknown
% 9.25/2.60 % (14343)Termination phase: Saturation
% 9.25/2.60
% 9.25/2.60 % (14343)Memory used [KB]: 7675
% 9.25/2.60 % (14343)Time elapsed: 0.054 s
% 9.25/2.60 % (14343)Instructions burned: 184 (million)
% 9.25/2.60 % (14343)------------------------------
% 9.25/2.60 % (14343)------------------------------
% 9.25/2.61 % (14347)Instruction limit reached!
% 9.25/2.61 % (14347)------------------------------
% 9.25/2.61 % (14347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.61 % (14347)Termination reason: Unknown
% 9.25/2.61 % (14347)Termination phase: Saturation
% 9.25/2.61
% 9.25/2.61 % (14347)Memory used [KB]: 2814
% 9.25/2.61 % (14347)Time elapsed: 0.044 s
% 9.25/2.61 % (14347)Instructions burned: 156 (million)
% 9.25/2.61 % (14347)------------------------------
% 9.25/2.61 % (14347)------------------------------
% 9.25/2.61 % (14359)Instruction limit reached!
% 9.25/2.61 % (14359)------------------------------
% 9.25/2.61 % (14359)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.61 % (14359)Termination reason: Unknown
% 9.25/2.61 % (14359)Termination phase: Property scanning
% 9.25/2.61
% 9.25/2.61 % (14359)Memory used [KB]: 2558
% 9.25/2.61 % (14359)Time elapsed: 0.025 s
% 9.25/2.61 % (14359)Instructions burned: 79 (million)
% 9.25/2.61 % (14359)------------------------------
% 9.25/2.61 % (14359)------------------------------
% 9.25/2.61 % (14344)Instruction limit reached!
% 9.25/2.61 % (14344)------------------------------
% 9.25/2.61 % (14344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.61 % (14344)Termination reason: Unknown
% 9.25/2.61 % (14344)Termination phase: Saturation
% 9.25/2.61
% 9.25/2.61 % (14344)Memory used [KB]: 7547
% 9.25/2.61 % (14344)Time elapsed: 0.079 s
% 9.25/2.61 % (14344)Instructions burned: 183 (million)
% 9.25/2.61 % (14344)------------------------------
% 9.25/2.61 % (14344)------------------------------
% 9.25/2.61 % (14369)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_13980 for (2990ds/216Mi)
% 9.25/2.62 % (14370)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_13980 for (2990ds/114Mi)
% 9.25/2.62 % (14371)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_13980 for (2990ds/922Mi)
% 9.25/2.62 % (14373)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_13980 for (2990ds/182Mi)
% 9.25/2.63 % (14335)Instruction limit reached!
% 9.25/2.63 % (14335)------------------------------
% 9.25/2.63 % (14335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.63 % (14335)Termination reason: Unknown
% 9.25/2.63 % (14335)Termination phase: Saturation
% 9.25/2.63
% 9.25/2.63 % (14335)Memory used [KB]: 7675
% 9.25/2.63 % (14335)Time elapsed: 0.085 s
% 9.25/2.63 % (14335)Instructions burned: 298 (million)
% 9.25/2.63 % (14335)------------------------------
% 9.25/2.63 % (14335)------------------------------
% 9.25/2.64 % (14389)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_13980 for (2990ds/164Mi)
% 9.25/2.65 % (14370)Instruction limit reached!
% 9.25/2.65 % (14370)------------------------------
% 9.25/2.65 % (14370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.65 % (14370)Termination reason: Unknown
% 9.25/2.65 % (14370)Termination phase: Property scanning
% 9.25/2.65
% 9.25/2.65 % (14370)Memory used [KB]: 2686
% 9.25/2.65 % (14370)Time elapsed: 0.036 s
% 9.25/2.65 % (14370)Instructions burned: 116 (million)
% 9.25/2.65 % (14370)------------------------------
% 9.25/2.65 % (14370)------------------------------
% 9.25/2.65 % (14327)Instruction limit reached!
% 9.25/2.65 % (14327)------------------------------
% 9.25/2.65 % (14327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.65 % (14327)Termination reason: Unknown
% 9.25/2.65 % (14327)Termination phase: Saturation
% 9.25/2.65
% 9.25/2.65 % (14327)Memory used [KB]: 8827
% 9.25/2.65 % (14327)Time elapsed: 0.136 s
% 9.25/2.65 % (14327)Instructions burned: 459 (million)
% 9.25/2.65 % (14327)------------------------------
% 9.25/2.65 % (14327)------------------------------
% 9.25/2.66 % (14398)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_13980 for (2990ds/50Mi)
% 9.25/2.66 % (14399)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_13980 for (2990ds/368Mi)
% 9.25/2.67 % (14373)Instruction limit reached!
% 9.25/2.67 % (14373)------------------------------
% 9.25/2.67 % (14373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.67 % (14373)Termination reason: Unknown
% 9.25/2.67 % (14373)Termination phase: Saturation
% 9.25/2.67
% 9.25/2.67 % (14373)Memory used [KB]: 7291
% 9.25/2.67 % (14373)Time elapsed: 0.052 s
% 9.25/2.67 % (14373)Instructions burned: 183 (million)
% 9.25/2.67 % (14373)------------------------------
% 9.25/2.67 % (14373)------------------------------
% 9.25/2.67 % (14398)Instruction limit reached!
% 9.25/2.67 % (14398)------------------------------
% 9.25/2.67 % (14398)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.67 % (14398)Termination reason: Unknown
% 9.25/2.67 % (14398)Termination phase: Property scanning
% 9.25/2.67
% 9.25/2.67 % (14398)Memory used [KB]: 2430
% 9.25/2.67 % (14398)Time elapsed: 0.016 s
% 9.25/2.67 % (14398)Instructions burned: 53 (million)
% 9.25/2.67 % (14398)------------------------------
% 9.25/2.67 % (14398)------------------------------
% 9.25/2.67 % (14369)Instruction limit reached!
% 9.25/2.67 % (14369)------------------------------
% 9.25/2.67 % (14369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.67 % (14369)Termination reason: Unknown
% 9.25/2.67 % (14369)Termination phase: Saturation
% 9.25/2.67
% 9.25/2.67 % (14369)Memory used [KB]: 7547
% 9.25/2.67 % (14369)Time elapsed: 0.062 s
% 9.25/2.67 % (14369)Instructions burned: 216 (million)
% 9.25/2.67 % (14369)------------------------------
% 9.25/2.67 % (14369)------------------------------
% 9.25/2.68 % (14411)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_13980 for (2989ds/1021Mi)
% 9.25/2.68 % (14389)Instruction limit reached!
% 9.25/2.68 % (14389)------------------------------
% 9.25/2.68 % (14389)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.25/2.68 % (14389)Termination reason: Unknown
% 9.25/2.68 % (14389)Termination phase: Saturation
% 9.25/2.68
% 9.25/2.68 % (14389)Memory used [KB]: 2814
% 9.25/2.68 % (14389)Time elapsed: 0.048 s
% 9.25/2.68 % (14389)Instructions burned: 165 (million)
% 9.25/2.68 % (14389)------------------------------
% 9.25/2.68 % (14389)------------------------------
% 9.25/2.68 % (14416)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_13980 for (2989ds/267Mi)
% 9.25/2.68 % (14417)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_13980 for (2989ds/204Mi)
% 9.25/2.69 % (14418)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_13980 for (2989ds/26Mi)
% 9.95/2.70 % (14418)Instruction limit reached!
% 9.95/2.70 % (14418)------------------------------
% 9.95/2.70 % (14418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.95/2.70 % (14418)Termination reason: Unknown
% 9.95/2.70 % (14418)Termination phase: shuffling
% 9.95/2.70
% 9.95/2.70 % (14418)Memory used [KB]: 2046
% 9.95/2.70 % (14418)Time elapsed: 0.009 s
% 9.95/2.70 % (14418)Instructions burned: 28 (million)
% 9.95/2.70 % (14418)------------------------------
% 9.95/2.70 % (14418)------------------------------
% 9.95/2.71 % (14427)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_13980 for (2989ds/53Mi)
% 9.95/2.73 % (14427)Instruction limit reached!
% 9.95/2.73 % (14427)------------------------------
% 9.95/2.73 % (14427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.95/2.73 % (14427)Termination reason: Unknown
% 9.95/2.73 % (14427)Termination phase: Property scanning
% 9.95/2.73
% 9.95/2.73 % (14427)Memory used [KB]: 2430
% 9.95/2.73 % (14427)Time elapsed: 0.016 s
% 9.95/2.73 % (14427)Instructions burned: 53 (million)
% 9.95/2.73 % (14427)------------------------------
% 9.95/2.73 % (14427)------------------------------
% 9.95/2.73 % (14436)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_13980 for (2989ds/4436Mi)
% 9.95/2.74 % (14417)Instruction limit reached!
% 9.95/2.74 % (14417)------------------------------
% 9.95/2.74 % (14417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.95/2.74 % (14417)Termination reason: Unknown
% 9.95/2.74 % (14417)Termination phase: Saturation
% 9.95/2.74
% 9.95/2.74 % (14417)Memory used [KB]: 7675
% 9.95/2.74 % (14417)Time elapsed: 0.061 s
% 9.95/2.74 % (14417)Instructions burned: 207 (million)
% 9.95/2.74 % (14417)------------------------------
% 9.95/2.74 % (14417)------------------------------
% 9.95/2.75 % (14445)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_13980 for (2989ds/1539Mi)
% 9.95/2.76 % (14416)Instruction limit reached!
% 9.95/2.76 % (14416)------------------------------
% 9.95/2.76 % (14416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.95/2.76 % (14416)Termination reason: Unknown
% 9.95/2.76 % (14416)Termination phase: Saturation
% 9.95/2.76
% 9.95/2.76 % (14416)Memory used [KB]: 7803
% 9.95/2.76 % (14416)Time elapsed: 0.082 s
% 9.95/2.76 % (14416)Instructions burned: 267 (million)
% 9.95/2.76 % (14416)------------------------------
% 9.95/2.76 % (14416)------------------------------
% 9.95/2.77 % (14399)Instruction limit reached!
% 9.95/2.77 % (14399)------------------------------
% 9.95/2.77 % (14399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.95/2.77 % (14399)Termination reason: Unknown
% 9.95/2.77 % (14399)Termination phase: Saturation
% 9.95/2.77
% 9.95/2.77 % (14399)Memory used [KB]: 8443
% 9.95/2.77 % (14399)Time elapsed: 0.110 s
% 9.95/2.77 % (14399)Instructions burned: 370 (million)
% 9.95/2.77 % (14399)------------------------------
% 9.95/2.77 % (14399)------------------------------
% 9.95/2.77 % (14452)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_13980 for (2989ds/408Mi)
% 9.95/2.78 % (14455)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_13980 for (2988ds/5205Mi)
% 10.27/2.86 % (14338)Instruction limit reached!
% 10.27/2.86 % (14338)------------------------------
% 10.27/2.86 % (14338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.27/2.86 % (14338)Termination reason: Unknown
% 10.27/2.86 % (14338)Termination phase: Saturation
% 10.27/2.86
% 10.27/2.86 % (14338)Memory used [KB]: 9722
% 10.27/2.86 % (14338)Time elapsed: 0.315 s
% 10.27/2.86 % (14338)Instructions burned: 989 (million)
% 10.27/2.86 % (14338)------------------------------
% 10.27/2.86 % (14338)------------------------------
% 10.27/2.87 % (14483)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_13980 for (2988ds/1913Mi)
% 10.59/2.90 % (14452)Instruction limit reached!
% 10.59/2.90 % (14452)------------------------------
% 10.59/2.90 % (14452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.59/2.90 % (14452)Termination reason: Unknown
% 10.59/2.90 % (14452)Termination phase: Saturation
% 10.59/2.90
% 10.59/2.90 % (14452)Memory used [KB]: 8827
% 10.59/2.90 % (14452)Time elapsed: 0.126 s
% 10.59/2.90 % (14452)Instructions burned: 408 (million)
% 10.59/2.90 % (14452)------------------------------
% 10.59/2.90 % (14452)------------------------------
% 10.59/2.91 % (14371)Instruction limit reached!
% 10.59/2.91 % (14371)------------------------------
% 10.59/2.91 % (14371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.59/2.91 % (14371)Termination reason: Unknown
% 10.59/2.91 % (14371)Termination phase: Saturation
% 10.59/2.91
% 10.59/2.91 % (14371)Memory used [KB]: 9722
% 10.59/2.91 % (14371)Time elapsed: 0.310 s
% 10.59/2.91 % (14371)Instructions burned: 924 (million)
% 10.59/2.91 % (14371)------------------------------
% 10.59/2.91 % (14371)------------------------------
% 10.59/2.91 % (14494)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_13980 for (2987ds/4350Mi)
% 10.59/2.92 % (14499)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_13980 for (2987ds/68Mi)
% 10.59/2.94 % (14499)Instruction limit reached!
% 10.59/2.94 % (14499)------------------------------
% 10.59/2.94 % (14499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.59/2.94 % (14499)Termination reason: Unknown
% 10.59/2.94 % (14499)Termination phase: Preprocessing 3
% 10.59/2.94
% 10.59/2.94 % (14499)Memory used [KB]: 2430
% 10.59/2.94 % (14499)Time elapsed: 0.022 s
% 10.59/2.94 % (14499)Instructions burned: 68 (million)
% 10.59/2.94 % (14499)------------------------------
% 10.59/2.94 % (14499)------------------------------
% 10.59/2.95 % (14517)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_13980 for (2987ds/358Mi)
% 10.59/3.00 % (14411)Instruction limit reached!
% 10.59/3.00 % (14411)------------------------------
% 10.59/3.00 % (14411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.59/3.00 % (14411)Termination reason: Unknown
% 10.59/3.00 % (14411)Termination phase: Saturation
% 10.59/3.00
% 10.59/3.00 % (14411)Memory used [KB]: 10106
% 10.59/3.00 % (14411)Time elapsed: 0.317 s
% 10.59/3.00 % (14411)Instructions burned: 1024 (million)
% 10.59/3.00 % (14411)------------------------------
% 10.59/3.00 % (14411)------------------------------
% 10.59/3.01 % (14536)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_13980 for (2986ds/31Mi)
% 12.45/3.02 % (14536)Instruction limit reached!
% 12.45/3.02 % (14536)------------------------------
% 12.45/3.02 % (14536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.45/3.02 % (14536)Termination reason: Unknown
% 12.45/3.02 % (14536)Termination phase: shuffling
% 12.45/3.02
% 12.45/3.02 % (14536)Memory used [KB]: 2046
% 12.45/3.02 % (14536)Time elapsed: 0.011 s
% 12.45/3.02 % (14536)Instructions burned: 31 (million)
% 12.45/3.02 % (14536)------------------------------
% 12.45/3.02 % (14536)------------------------------
% 12.45/3.02 % (14537)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_13980 for (2986ds/279Mi)
% 12.45/3.06 % (14517)Instruction limit reached!
% 12.45/3.06 % (14517)------------------------------
% 12.45/3.06 % (14517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.45/3.06 % (14517)Termination reason: Unknown
% 12.45/3.06 % (14517)Termination phase: Saturation
% 12.45/3.06
% 12.45/3.06 % (14517)Memory used [KB]: 8059
% 12.45/3.06 % (14517)Time elapsed: 0.112 s
% 12.45/3.06 % (14517)Instructions burned: 358 (million)
% 12.45/3.06 % (14517)------------------------------
% 12.45/3.06 % (14517)------------------------------
% 12.45/3.07 % (14538)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_13980 for (2986ds/258Mi)
% 12.45/3.11 % (14537)Instruction limit reached!
% 12.45/3.11 % (14537)------------------------------
% 12.45/3.11 % (14537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.45/3.11 % (14537)Termination reason: Unknown
% 12.45/3.11 % (14537)Termination phase: Saturation
% 12.45/3.11
% 12.45/3.11 % (14537)Memory used [KB]: 7675
% 12.45/3.11 % (14537)Time elapsed: 0.082 s
% 12.45/3.11 % (14537)Instructions burned: 279 (million)
% 12.45/3.11 % (14537)------------------------------
% 12.45/3.11 % (14537)------------------------------
% 13.13/3.12 % (14539)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_13980 for (2985ds/26Mi)
% 13.13/3.12 % (14539)Instruction limit reached!
% 13.13/3.12 % (14539)------------------------------
% 13.13/3.12 % (14539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.13/3.12 % (14539)Termination reason: Unknown
% 13.13/3.12 % (14539)Termination phase: shuffling
% 13.13/3.12
% 13.13/3.12 % (14539)Memory used [KB]: 2046
% 13.13/3.12 % (14539)Time elapsed: 0.009 s
% 13.13/3.12 % (14539)Instructions burned: 28 (million)
% 13.13/3.12 % (14539)------------------------------
% 13.13/3.12 % (14539)------------------------------
% 13.13/3.13 % (14540)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_13980 for (2985ds/995Mi)
% 13.13/3.15 % (14538)Instruction limit reached!
% 13.13/3.15 % (14538)------------------------------
% 13.13/3.15 % (14538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.13/3.15 % (14538)Termination reason: Unknown
% 13.13/3.15 % (14538)Termination phase: Saturation
% 13.13/3.15
% 13.13/3.15 % (14538)Memory used [KB]: 8059
% 13.13/3.15 % (14538)Time elapsed: 0.080 s
% 13.13/3.15 % (14538)Instructions burned: 258 (million)
% 13.13/3.15 % (14538)------------------------------
% 13.13/3.15 % (14538)------------------------------
% 13.13/3.16 % (14541)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_13980 for (2985ds/373Mi)
% 13.66/3.23 % (14445)Instruction limit reached!
% 13.66/3.23 % (14445)------------------------------
% 13.66/3.23 % (14445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.66/3.23 % (14445)Termination reason: Unknown
% 13.66/3.23 % (14445)Termination phase: Saturation
% 13.66/3.23
% 13.66/3.23 % (14445)Memory used [KB]: 11129
% 13.66/3.23 % (14445)Time elapsed: 0.482 s
% 13.66/3.23 % (14445)Instructions burned: 1541 (million)
% 13.66/3.23 % (14445)------------------------------
% 13.66/3.23 % (14445)------------------------------
% 13.66/3.25 % (14542)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_13980 for (2984ds/253Mi)
% 13.66/3.27 % (14541)Instruction limit reached!
% 13.66/3.27 % (14541)------------------------------
% 13.66/3.27 % (14541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.66/3.27 % (14541)Termination reason: Unknown
% 13.66/3.27 % (14541)Termination phase: Saturation
% 13.66/3.27
% 13.66/3.27 % (14541)Memory used [KB]: 8571
% 13.66/3.27 % (14541)Time elapsed: 0.116 s
% 13.66/3.27 % (14541)Instructions burned: 375 (million)
% 13.66/3.27 % (14541)------------------------------
% 13.66/3.27 % (14541)------------------------------
% 13.66/3.28 % (14543)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_13980 for (2984ds/832Mi)
% 14.24/3.32 % (14542)Instruction limit reached!
% 14.24/3.32 % (14542)------------------------------
% 14.24/3.32 % (14542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.24/3.32 % (14542)Termination reason: Unknown
% 14.24/3.32 % (14542)Termination phase: Saturation
% 14.24/3.32
% 14.24/3.32 % (14542)Memory used [KB]: 7803
% 14.24/3.32 % (14542)Time elapsed: 0.074 s
% 14.24/3.32 % (14542)Instructions burned: 255 (million)
% 14.24/3.32 % (14542)------------------------------
% 14.24/3.32 % (14542)------------------------------
% 14.24/3.33 % (14544)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_13980 for (2983ds/668Mi)
% 14.71/3.43 % (14540)Instruction limit reached!
% 14.71/3.43 % (14540)------------------------------
% 14.71/3.43 % (14540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.71/3.43 % (14540)Termination reason: Unknown
% 14.71/3.43 % (14540)Termination phase: Saturation
% 14.71/3.43
% 14.71/3.43 % (14540)Memory used [KB]: 10234
% 14.71/3.43 % (14540)Time elapsed: 0.297 s
% 14.71/3.43 % (14540)Instructions burned: 995 (million)
% 14.71/3.43 % (14540)------------------------------
% 14.71/3.43 % (14540)------------------------------
% 14.71/3.44 % (14545)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=40:si=on:rtra=on_0 on DTF2THF_13980 for (2982ds/40Mi)
% 14.71/3.45 % (14545)Instruction limit reached!
% 14.71/3.45 % (14545)------------------------------
% 14.71/3.45 % (14545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.71/3.45 % (14545)Termination reason: Unknown
% 14.71/3.45 % (14545)Termination phase: shuffling
% 14.71/3.45
% 14.71/3.45 % (14545)Memory used [KB]: 2302
% 14.71/3.45 % (14545)Time elapsed: 0.013 s
% 14.71/3.45 % (14545)Instructions burned: 41 (million)
% 14.71/3.45 % (14545)------------------------------
% 14.71/3.45 % (14545)------------------------------
% 14.71/3.46 % (14546)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_13980 for (2982ds/294Mi)
% 14.71/3.48 % (14483)Instruction limit reached!
% 14.71/3.48 % (14483)------------------------------
% 14.71/3.48 % (14483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.71/3.48 % (14483)Termination reason: Unknown
% 14.71/3.48 % (14483)Termination phase: Saturation
% 14.71/3.48
% 14.71/3.48 % (14483)Memory used [KB]: 5884
% 14.71/3.48 % (14483)Time elapsed: 0.608 s
% 14.71/3.48 % (14483)Instructions burned: 1914 (million)
% 14.71/3.48 % (14483)------------------------------
% 14.71/3.48 % (14483)------------------------------
% 14.71/3.49 % (14547)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_13980 for (2982ds/187Mi)
% 15.43/3.52 % (14544)Instruction limit reached!
% 15.43/3.52 % (14544)------------------------------
% 15.43/3.52 % (14544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.43/3.52 % (14544)Termination reason: Unknown
% 15.43/3.52 % (14544)Termination phase: Saturation
% 15.43/3.52
% 15.43/3.52 % (14544)Memory used [KB]: 14072
% 15.43/3.52 % (14544)Time elapsed: 0.196 s
% 15.43/3.52 % (14544)Instructions burned: 671 (million)
% 15.43/3.52 % (14544)------------------------------
% 15.43/3.52 % (14544)------------------------------
% 15.43/3.53 % (14548)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_13980 for (2981ds/371Mi)
% 15.43/3.53 % (14543)Instruction limit reached!
% 15.43/3.53 % (14543)------------------------------
% 15.43/3.53 % (14543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.43/3.53 % (14543)Termination reason: Unknown
% 15.43/3.53 % (14543)Termination phase: Saturation
% 15.43/3.53
% 15.43/3.53 % (14543)Memory used [KB]: 8955
% 15.43/3.53 % (14543)Time elapsed: 0.256 s
% 15.43/3.53 % (14543)Instructions burned: 834 (million)
% 15.43/3.53 % (14543)------------------------------
% 15.43/3.53 % (14543)------------------------------
% 15.43/3.54 % (14547)Instruction limit reached!
% 15.43/3.54 % (14547)------------------------------
% 15.43/3.54 % (14547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.43/3.54 % (14547)Termination reason: Unknown
% 15.43/3.54 % (14547)Termination phase: Saturation
% 15.43/3.54
% 15.43/3.54 % (14547)Memory used [KB]: 7419
% 15.43/3.54 % (14547)Time elapsed: 0.055 s
% 15.43/3.54 % (14547)Instructions burned: 189 (million)
% 15.43/3.54 % (14547)------------------------------
% 15.43/3.54 % (14547)------------------------------
% 15.43/3.54 % (14546)Instruction limit reached!
% 15.43/3.54 % (14546)------------------------------
% 15.43/3.54 % (14546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.43/3.54 % (14546)Termination reason: Unknown
% 15.43/3.54 % (14546)Termination phase: Saturation
% 15.43/3.54
% 15.43/3.54 % (14546)Memory used [KB]: 7547
% 15.43/3.54 % (14546)Time elapsed: 0.084 s
% 15.43/3.54 % (14546)Instructions burned: 296 (million)
% 15.43/3.54 % (14546)------------------------------
% 15.43/3.54 % (14546)------------------------------
% 15.43/3.54 % (14549)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_13980 for (2981ds/1827Mi)
% 15.43/3.55 % (14550)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=70:si=on:rtra=on_0 on DTF2THF_13980 for (2981ds/70Mi)
% 15.43/3.55 % (14551)lrs+10_1:20_bd=off:cnfonf=lazy_not_be_gen:fe=axiom:hud=10:ins=1:nm=20:prag=on:sos=on:i=316:si=on:rtra=on_0 on DTF2THF_13980 for (2981ds/316Mi)
% 15.43/3.57 % (14550)Instruction limit reached!
% 15.43/3.57 % (14550)------------------------------
% 15.43/3.57 % (14550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.43/3.57 % (14550)Termination reason: Unknown
% 15.43/3.57 % (14550)Termination phase: Property scanning
% 15.43/3.57
% 15.43/3.57 % (14550)Memory used [KB]: 2430
% 15.43/3.57 % (14550)Time elapsed: 0.021 s
% 15.43/3.57 % (14550)Instructions burned: 72 (million)
% 15.43/3.57 % (14550)------------------------------
% 15.43/3.57 % (14550)------------------------------
% 15.43/3.58 % (14552)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=932:si=on:rtra=on_0 on DTF2THF_13980 for (2981ds/932Mi)
% 16.39/3.64 % (14551)Instruction limit reached!
% 16.39/3.64 % (14551)------------------------------
% 16.39/3.64 % (14551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.39/3.64 % (14551)Termination reason: Unknown
% 16.39/3.64 % (14551)Termination phase: Saturation
% 16.39/3.64
% 16.39/3.64 % (14551)Memory used [KB]: 8059
% 16.39/3.64 % (14551)Time elapsed: 0.091 s
% 16.39/3.64 % (14551)Instructions burned: 316 (million)
% 16.39/3.64 % (14551)------------------------------
% 16.39/3.64 % (14551)------------------------------
% 16.39/3.65 % (14548)Instruction limit reached!
% 16.39/3.65 % (14548)------------------------------
% 16.39/3.65 % (14548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.39/3.65 % (14548)Termination reason: Unknown
% 16.39/3.65 % (14548)Termination phase: Saturation
% 16.39/3.65
% 16.39/3.65 % (14548)Memory used [KB]: 8699
% 16.39/3.65 % (14548)Time elapsed: 0.114 s
% 16.39/3.65 % (14548)Instructions burned: 374 (million)
% 16.39/3.65 % (14548)------------------------------
% 16.39/3.65 % (14548)------------------------------
% 16.39/3.65 % (14553)lrs+1010_1:128_au=on:bd=preordered:cnfonf=lazy_pi_sigma_gen:plsq=on:plsqc=1:plsqr=32,1:sos=all:sp=occurrence:i=4714:si=on:rtra=on_0 on DTF2THF_13980 for (2980ds/4714Mi)
% 16.39/3.66 % (14554)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=8250:si=on:rtra=on_0 on DTF2THF_13980 for (2980ds/8250Mi)
% 17.21/3.86 % (14552)Instruction limit reached!
% 17.21/3.86 % (14552)------------------------------
% 17.21/3.86 % (14552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.21/3.86 % (14552)Termination reason: Unknown
% 17.21/3.86 % (14552)Termination phase: Saturation
% 17.21/3.86
% 17.21/3.86 % (14552)Memory used [KB]: 11769
% 17.21/3.86 % (14552)Time elapsed: 0.282 s
% 17.21/3.86 % (14552)Instructions burned: 932 (million)
% 17.21/3.86 % (14552)------------------------------
% 17.21/3.86 % (14552)------------------------------
% 17.21/3.87 % (14555)lrs+1002_1:1_apa=on:cnfonf=off:fd=off:flr=on:hud=1:lcm=reverse:prag=on:sd=1:sos=on:ss=axioms:i=770:si=on:rtra=on_0 on DTF2THF_13980 for (2978ds/770Mi)
% 17.81/3.97 % (14555)First to succeed.
% 17.81/3.97 % (14555)Refutation found. Thanks to Tanya!
% 17.81/3.97 % SZS status Theorem for DTF2THF_13980
% 17.81/3.97 % SZS output start Proof for DTF2THF_13980
% 17.81/3.97 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 17.81/3.97 thf(type_def_6, type, set: ($tType) > $tType).
% 17.81/3.97 thf(type_def_8, type, nat: $tType).
% 17.81/3.97 thf(type_def_9, type, state: $tType).
% 17.81/3.97 thf(type_def_10, type, com: $tType).
% 17.81/3.97 thf(type_def_11, type, product_unit: $tType).
% 17.81/3.97 thf(type_def_12, type, a: $tType).
% 17.81/3.97 thf(func_def_0, type, product_unit: $tType).
% 17.81/3.97 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 17.81/3.97 thf(func_def_2, type, state: $tType).
% 17.81/3.97 thf(func_def_3, type, com: $tType).
% 17.81/3.97 thf(func_def_4, type, set: ($tType) > $tType).
% 17.81/3.97 thf(func_def_5, type, nat: $tType).
% 17.81/3.97 thf(func_def_6, type, a: $tType).
% 17.81/3.97 thf(func_def_7, type, type: !>[X0: $tType]:($o)).
% 17.81/3.97 thf(func_def_8, type, bot: !>[X0: $tType]:($o)).
% 17.81/3.97 thf(func_def_9, type, basic_fsts: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > set @ X0)).
% 17.81/3.97 thf(func_def_10, type, basic_snds: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > set @ X1)).
% 17.81/3.97 thf(func_def_11, type, fun_reduction_pair: !>[X0: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > $o)).
% 17.81/3.97 thf(func_def_12, type, fun_rp_inv_image: !>[X0: $tType, X1: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > (X1 > X0) > product_prod @ set @ product_prod @ X1 @ X1 @ set @ product_prod @ X1 @ X1)).
% 17.81/3.97 thf(func_def_13, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_14, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 17.81/3.97 thf(func_def_15, type, suc: nat > nat).
% 17.81/3.97 thf(func_def_16, type, order_above: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 17.81/3.97 thf(func_def_17, type, order_aboveS: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 17.81/3.97 thf(func_def_18, type, order_1409979114der_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 17.81/3.97 thf(func_def_19, type, order_preorder_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 17.81/3.97 thf(func_def_20, type, bot_bot: !>[X0: $tType]:(X0)).
% 17.81/3.97 thf(func_def_21, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 17.81/3.97 thf(func_def_22, type, pHoare304044376tvalid: !>[X0: $tType]:(set @ product_prod @ (X0 > state > $o) @ product_prod @ com @ (X0 > state > $o) > (X0 > state > $o) > com > (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_23, type, pHoare166426476thoare: !>[X0: $tType]:(set @ product_prod @ (X0 > state > $o) @ product_prod @ com @ (X0 > state > $o) > (X0 > state > $o) > com > (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_24, type, pHoare987424603tvalid: !>[X0: $tType]:((X0 > state > $o) > com > (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_25, type, cnvalid: !>[X0: $tType]:(set @ product_prod @ (X0 > state > $o) @ product_prod @ com @ (X0 > state > $o) > nat > (X0 > state > $o) > com > (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_26, type, cvalid: !>[X0: $tType]:(set @ product_prod @ (X0 > state > $o) @ product_prod @ com @ (X0 > state > $o) > (X0 > state > $o) > com > (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_27, type, hoare: !>[X0: $tType]:(set @ product_prod @ (X0 > state > $o) @ product_prod @ com @ (X0 > state > $o) > (X0 > state > $o) > com > (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_28, type, nvalid: !>[X0: $tType]:(nat > (X0 > state > $o) > com > (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_29, type, nvalids: !>[X0: $tType]:(nat > set @ product_prod @ (X0 > state > $o) @ product_prod @ com @ (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_30, type, valid: !>[X0: $tType]:((X0 > state > $o) > com > (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_31, type, valids: !>[X0: $tType]:(set @ product_prod @ (X0 > state > $o) @ product_prod @ com @ (X0 > state > $o) > $o)).
% 17.81/3.97 thf(func_def_32, type, body: com).
% 17.81/3.97 thf(func_def_33, type, call: com).
% 17.81/3.97 thf(func_def_34, type, semi: com > com > com).
% 17.81/3.97 thf(func_def_35, type, while: (state > $o) > com > com).
% 17.81/3.97 thf(func_def_36, type, exec: state > com > state > $o).
% 17.81/3.97 thf(func_def_37, type, execn: state > com > nat > state > $o).
% 17.81/3.97 thf(func_def_38, type, termi: com > state > $o).
% 17.81/3.97 thf(func_def_39, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 17.81/3.97 thf(func_def_40, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 17.81/3.97 thf(func_def_41, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 17.81/3.97 thf(func_def_42, type, product_rec_set_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2 > $o)).
% 17.81/3.97 thf(func_def_43, type, product_rec_set_unit: !>[X0: $tType]:(X0 > product_unit > X0 > $o)).
% 17.81/3.97 thf(func_def_44, type, product_rec_unit: !>[X0: $tType]:(X0 > product_unit > X0)).
% 17.81/3.97 thf(func_def_45, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 17.81/3.97 thf(func_def_46, type, domain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0)).
% 17.81/3.97 thf(func_def_47, type, id_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0)).
% 17.81/3.97 thf(func_def_48, type, powp: !>[X0: $tType]:((X0 > $o) > set @ X0 > $o)).
% 17.81/3.97 thf(func_def_49, type, range: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X1)).
% 17.81/3.97 thf(func_def_50, type, rangep: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X1 > $o)).
% 17.81/3.97 thf(func_def_51, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 17.81/3.97 thf(func_def_52, type, irrefl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 17.81/3.97 thf(func_def_53, type, irreflp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 17.81/3.97 thf(func_def_54, type, refl_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 17.81/3.97 thf(func_def_55, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 17.81/3.97 thf(func_def_56, type, trans: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 17.81/3.97 thf(func_def_57, type, ball: !>[X0: $tType]:(set @ X0 > (X0 > $o) > $o)).
% 17.81/3.97 thf(func_def_58, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 17.81/3.97 thf(func_def_59, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 17.81/3.97 thf(func_def_60, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 17.81/3.97 thf(func_def_61, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 17.81/3.97 thf(func_def_62, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 17.81/3.97 thf(func_def_63, type, less_than: set @ product_prod @ nat @ nat).
% 17.81/3.97 thf(func_def_64, type, lex_prod: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 17.81/3.97 thf(func_def_65, type, measure: !>[X0: $tType]:((X0 > nat) > set @ product_prod @ X0 @ X0)).
% 17.81/3.97 thf(func_def_66, type, mlex_prod: !>[X0: $tType]:((X0 > nat) > set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 17.81/3.97 thf(func_def_67, type, pred_nat: set @ product_prod @ nat @ nat).
% 17.81/3.97 thf(func_def_68, type, wf: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 17.81/3.97 thf(func_def_69, type, same_fst: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 17.81/3.97 thf(func_def_70, type, chains: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ set @ X0)).
% 17.81/3.97 thf(func_def_71, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 17.81/3.97 thf(func_def_72, type, pa: a > state > $o).
% 17.81/3.97 thf(func_def_73, type, qa: a > state > $o).
% 17.81/3.97 thf(func_def_74, type, r: set @ product_prod @ state @ state).
% 17.81/3.97 thf(func_def_75, type, x: state).
% 17.81/3.97 thf(func_def_94, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_95, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_97, type, sP3: state > $o).
% 17.81/3.97 thf(func_def_98, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 17.81/3.97 thf(func_def_99, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 17.81/3.97 thf(func_def_100, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 17.81/3.97 thf(func_def_101, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 17.81/3.97 thf(func_def_102, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 17.81/3.97 thf(func_def_103, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_104, type, sK10: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 17.81/3.97 thf(func_def_105, type, sK11: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_106, type, sK12: state).
% 17.81/3.97 thf(func_def_107, type, sK13: state).
% 17.81/3.97 thf(func_def_108, type, sK14: a).
% 17.81/3.97 thf(func_def_109, type, sK15: a).
% 17.81/3.97 thf(func_def_110, type, sK16: state).
% 17.81/3.97 thf(func_def_111, type, sK17: !>[X0: $tType]:(set @ X0 > X0)).
% 17.81/3.97 thf(func_def_112, type, sK18: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > set @ product_prod @ state @ state > state)).
% 17.81/3.97 thf(func_def_113, type, sK19: state > state).
% 17.81/3.97 thf(func_def_114, type, sK20: state > state).
% 17.81/3.97 thf(func_def_115, type, sK21: state > a).
% 17.81/3.97 thf(func_def_116, type, sK22: state > a).
% 17.81/3.97 thf(func_def_117, type, sK23: state > state).
% 17.81/3.97 thf(func_def_118, type, sK24: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > (X0 > state > $o) > (X0 > state > $o) > state)).
% 17.81/3.97 thf(func_def_119, type, sK25: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > (X0 > state > $o) > (X0 > state > $o) > state)).
% 17.81/3.97 thf(func_def_120, type, sK26: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > (X0 > state > $o) > (X0 > state > $o) > X0)).
% 17.81/3.97 thf(func_def_121, type, sK27: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > state)).
% 17.81/3.97 thf(func_def_122, type, sK28: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > X0)).
% 17.81/3.97 thf(func_def_123, type, sK29: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 17.81/3.97 thf(func_def_124, type, sK30: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 17.81/3.97 thf(func_def_125, type, sK31: com > state > (state > $o) > state).
% 17.81/3.97 thf(func_def_126, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 17.81/3.97 thf(func_def_127, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 17.81/3.97 thf(func_def_128, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 17.81/3.97 thf(func_def_129, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 17.81/3.97 thf(func_def_130, type, sK36: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > X0)).
% 17.81/3.97 thf(func_def_131, type, sK37: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > state)).
% 17.81/3.97 thf(func_def_132, type, sK38: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 17.81/3.97 thf(func_def_133, type, sK39: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_134, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 17.81/3.97 thf(func_def_135, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 17.81/3.97 thf(func_def_136, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 17.81/3.97 thf(func_def_137, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 17.81/3.97 thf(func_def_138, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 17.81/3.97 thf(func_def_139, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 17.81/3.97 thf(func_def_140, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 17.81/3.97 thf(func_def_141, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 17.81/3.97 thf(func_def_142, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 17.81/3.97 thf(func_def_143, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 17.81/3.97 thf(func_def_144, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 17.81/3.97 thf(func_def_145, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_146, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 17.81/3.97 thf(func_def_147, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 17.81/3.97 thf(func_def_148, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_149, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 17.81/3.97 thf(func_def_150, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 17.81/3.97 thf(func_def_151, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 17.81/3.97 thf(func_def_152, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_153, type, sK59: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 17.81/3.97 thf(func_def_154, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 17.81/3.97 thf(func_def_155, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 17.81/3.97 thf(func_def_156, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 17.81/3.97 thf(func_def_157, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 17.81/3.97 thf(func_def_158, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 17.81/3.97 thf(func_def_159, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 17.81/3.97 thf(func_def_160, type, sK66: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 17.81/3.97 thf(func_def_161, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 17.81/3.97 thf(func_def_162, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 17.81/3.97 thf(func_def_163, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 17.81/3.97 thf(func_def_164, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 17.81/3.97 thf(func_def_165, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 17.81/3.97 thf(func_def_166, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 17.81/3.97 thf(func_def_167, type, sK73: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 17.81/3.97 thf(func_def_168, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 17.81/3.97 thf(func_def_169, type, sK75: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 17.81/3.97 thf(func_def_170, type, sK76: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 17.81/3.97 thf(func_def_171, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 17.81/3.97 thf(func_def_172, type, sK78: !>[X0: $tType]:(com > (state > $o) > (X0 > state > $o) > set @ product_prod @ (X0 > state > $o) @ product_prod @ com @ (X0 > state > $o) > set @ product_prod @ state @ state > state)).
% 17.81/3.97 thf(func_def_173, type, sK79: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 17.81/3.97 thf(func_def_174, type, sK80: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 17.81/3.97 thf(func_def_175, type, sK81: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > X0)).
% 17.81/3.97 thf(func_def_176, type, sK82: !>[X0: $tType]:((X0 > state > $o) > (X0 > state > $o) > state)).
% 17.81/3.97 thf(func_def_177, type, sK83: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 17.81/3.97 thf(func_def_178, type, sK84: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 17.81/3.97 thf(func_def_179, type, sK85: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 17.81/3.97 thf(func_def_180, type, sK86: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 17.81/3.97 thf(func_def_181, type, sK87: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 17.81/3.97 thf(func_def_183, type, ph89: !>[X0: $tType]:(X0)).
% 17.81/3.97 thf(f1339,plain,(
% 17.81/3.97 $false),
% 17.81/3.97 inference(avatar_sat_refutation,[status(thm)],[f1063,f1068,f1073,f1151,f1180,f1231,f1245,f1248,f1255,f1257,f1262,f1267,f1272,f1315,f1316,f1318,f1338])).
% 17.81/3.97 thf(f1338,plain,(
% 17.81/3.97 ~spl88_5),
% 17.81/3.97 inference(avatar_contradiction_clause,[status(thm)],[f1337])).
% 17.81/3.97 thf(f1337,plain,(
% 17.81/3.97 $false | ~spl88_5),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1336,f1113])).
% 17.81/3.97 thf(f1113,plain,(
% 17.81/3.97 ((sP3 @ x) = $true) | ~spl88_5),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1112])).
% 17.81/3.97 thf(f1112,definition,(
% 17.81/3.97 spl88_5 <=> ((sP3 @ x) = $true)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_5])],[avatar_definition])).
% 17.81/3.97 thf(f1336,plain,(
% 17.81/3.97 ((sP3 @ x) != $true) | ~spl88_5),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1335])).
% 17.81/3.97 thf(f1335,plain,(
% 17.81/3.97 ((sP3 @ x) != $true) | ($true != $true) | ~spl88_5),
% 17.81/3.97 inference(superposition,[status(thm)],[f1334,f980])).
% 17.81/3.97 thf(f980,plain,(
% 17.81/3.97 ( ! [X0 : state] : (((pa @ (sK21 @ X0) @ (sK19 @ X0)) = $true) | ((sP3 @ X0) != $true)) )),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f906])).
% 17.81/3.97 thf(f906,plain,(
% 17.81/3.97 ! [X0 : state] : (((((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK19 @ X0) @ X0) @ r) = $true) & ((pa @ (sK21 @ X0) @ (sK19 @ X0)) = $true) & ((qa @ (sK21 @ X0) @ (sK20 @ X0)) != $true)) & ((exec @ (sK19 @ X0) @ body @ (sK20 @ X0)) = $true)) | ((sP3 @ X0) != $true))),
% 17.81/3.97 inference(skolemisation,[status(esa),new_symbols(skolem,[sK19,sK20,sK21])],[f903,f905,f904])).
% 17.81/3.97 thf(f904,plain,(
% 17.81/3.97 ! [X0 : state] : (? [X1 : state,X2 : state] : (? [X3 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X1 @ X0) @ r) = $true) & ((pa @ X3 @ X1) = $true) & ((qa @ X3 @ X2) != $true)) & ((exec @ X1 @ body @ X2) = $true)) => (? [X3 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK19 @ X0) @ X0) @ r) = $true) & ($true = (pa @ X3 @ (sK19 @ X0))) & ((qa @ X3 @ (sK20 @ X0)) != $true)) & ((exec @ (sK19 @ X0) @ body @ (sK20 @ X0)) = $true)))),
% 17.81/3.97 introduced(definition,[],[choice_axiom])).
% 17.81/3.97 thf(f905,plain,(
% 17.81/3.97 ! [X0 : state] : (? [X3 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK19 @ X0) @ X0) @ r) = $true) & ($true = (pa @ X3 @ (sK19 @ X0))) & ((qa @ X3 @ (sK20 @ X0)) != $true)) => (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK19 @ X0) @ X0) @ r) = $true) & ((pa @ (sK21 @ X0) @ (sK19 @ X0)) = $true) & ((qa @ (sK21 @ X0) @ (sK20 @ X0)) != $true)))),
% 17.81/3.97 introduced(definition,[],[choice_axiom])).
% 17.81/3.97 thf(f903,plain,(
% 17.81/3.97 ! [X0 : state] : (? [X1 : state,X2 : state] : (? [X3 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X1 @ X0) @ r) = $true) & ((pa @ X3 @ X1) = $true) & ((qa @ X3 @ X2) != $true)) & ((exec @ X1 @ body @ X2) = $true)) | ((sP3 @ X0) != $true))),
% 17.81/3.97 inference(nnf_transformation,[status(thm)],[f885])).
% 17.81/3.97 thf(f885,plain,(
% 17.81/3.97 ! [X0 : state] : (? [X1 : state,X2 : state] : (? [X3 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X1 @ X0) @ r) = $true) & ((pa @ X3 @ X1) = $true) & ((qa @ X3 @ X2) != $true)) & ((exec @ X1 @ body @ X2) = $true)) | ~((sP3 @ X0) = $true))),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 17.81/3.97 thf(f1334,plain,(
% 17.81/3.97 ($true != (pa @ (sK21 @ x) @ (sK19 @ x))) | ~spl88_5),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1333,f1113])).
% 17.81/3.97 thf(f1333,plain,(
% 17.81/3.97 ($true != (pa @ (sK21 @ x) @ (sK19 @ x))) | ((sP3 @ x) != $true) | ~spl88_5),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1332])).
% 17.81/3.97 thf(f1332,plain,(
% 17.81/3.97 ($true != (pa @ (sK21 @ x) @ (sK19 @ x))) | ($true != $true) | ((sP3 @ x) != $true) | ~spl88_5),
% 17.81/3.97 inference(superposition,[status(thm)],[f979,f1331])).
% 17.81/3.97 thf(f1331,plain,(
% 17.81/3.97 ( ! [X0 : a] : (((qa @ X0 @ (sK20 @ x)) = $true) | ((pa @ X0 @ (sK19 @ x)) != $true)) ) | ~spl88_5),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1327,f1113])).
% 17.81/3.97 thf(f1327,plain,(
% 17.81/3.97 ( ! [X0 : a] : (((qa @ X0 @ (sK20 @ x)) = $true) | ((sP3 @ x) != $true) | ((pa @ X0 @ (sK19 @ x)) != $true)) ) | ~spl88_5),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1325])).
% 17.81/3.97 thf(f1325,plain,(
% 17.81/3.97 ( ! [X0 : a] : (($true != $true) | ((qa @ X0 @ (sK20 @ x)) = $true) | ((pa @ X0 @ (sK19 @ x)) != $true) | ((sP3 @ x) != $true)) ) | ~spl88_5),
% 17.81/3.97 inference(superposition,[status(thm)],[f1321,f978])).
% 17.81/3.97 thf(f978,plain,(
% 17.81/3.97 ( ! [X0 : state] : (((exec @ (sK19 @ X0) @ body @ (sK20 @ X0)) = $true) | ((sP3 @ X0) != $true)) )),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f906])).
% 17.81/3.97 thf(f1321,plain,(
% 17.81/3.97 ( ! [X0 : a,X1 : state] : (((exec @ (sK19 @ x) @ body @ X1) != $true) | ($true = (qa @ X0 @ X1)) | ((pa @ X0 @ (sK19 @ x)) != $true)) ) | ~spl88_5),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1173,f1113])).
% 17.81/3.97 thf(f1173,plain,(
% 17.81/3.97 ( ! [X0 : a,X1 : state] : (((pa @ X0 @ (sK19 @ x)) != $true) | ((sP3 @ x) != $true) | ((exec @ (sK19 @ x) @ body @ X1) != $true) | ($true = (qa @ X0 @ X1))) )),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1170])).
% 17.81/3.97 thf(f1170,plain,(
% 17.81/3.97 ( ! [X0 : a,X1 : state] : (((sP3 @ x) != $true) | ((exec @ (sK19 @ x) @ body @ X1) != $true) | ($true = (qa @ X0 @ X1)) | ($true != $true) | ((pa @ X0 @ (sK19 @ x)) != $true)) )),
% 17.81/3.97 inference(superposition,[status(thm)],[f1046,f981])).
% 17.81/3.97 thf(f981,plain,(
% 17.81/3.97 ( ! [X0 : state] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK19 @ X0) @ X0) @ r) = $true) | ((sP3 @ X0) != $true)) )),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f906])).
% 17.81/3.97 thf(f1046,plain,(
% 17.81/3.97 ( ! [X3 : state,X4 : state,X5 : a] : (($true != (member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X3 @ x) @ r)) | ((qa @ X5 @ X4) = $true) | ($true != (pa @ X5 @ X3)) | ((exec @ X3 @ body @ X4) != $true)) )),
% 17.81/3.97 inference(equality_resolution,[status(thm)],[f962])).
% 17.81/3.97 thf(f962,plain,(
% 17.81/3.97 ( ! [X3 : state,X0 : state,X4 : state,X5 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X0 @ x) @ r) != $true) | ((exec @ X3 @ body @ X4) != $true) | (X0 != X3) | ($true != (pa @ X5 @ X3)) | ((qa @ X5 @ X4) = $true)) )),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f889])).
% 17.81/3.97 thf(f889,plain,(
% 17.81/3.97 ! [X0 : state] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X0 @ x) @ r) != $true) | (! [X1 : a,X2 : state] : ((X0 != X2) | ((termi @ body @ X2) = $true) | ((pa @ X1 @ X2) != $true)) & ! [X3 : state,X4 : state] : (((exec @ X3 @ body @ X4) != $true) | ! [X5 : a] : ((X0 != X3) | ($true != (pa @ X5 @ X3)) | ((qa @ X5 @ X4) = $true)))))),
% 17.81/3.97 inference(rectify,[status(thm)],[f863])).
% 17.81/3.97 thf(f863,plain,(
% 17.81/3.97 ! [X0 : state] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X0 @ x) @ r) != $true) | (! [X4 : a,X5 : state] : ((X0 != X5) | ((termi @ body @ X5) = $true) | ((pa @ X4 @ X5) != $true)) & ! [X1 : state,X2 : state] : (((exec @ X1 @ body @ X2) != $true) | ! [X3 : a] : ((X0 != X1) | ((pa @ X3 @ X1) != $true) | ((qa @ X3 @ X2) = $true)))))),
% 17.81/3.97 inference(flattening,[status(thm)],[f862])).
% 17.81/3.97 thf(f862,plain,(
% 17.81/3.97 ! [X0 : state] : ((! [X1 : state,X2 : state] : (! [X3 : a] : (((qa @ X3 @ X2) = $true) | (((pa @ X3 @ X1) != $true) | (X0 != X1))) | ((exec @ X1 @ body @ X2) != $true)) & ! [X4 : a,X5 : state] : (((termi @ body @ X5) = $true) | ((X0 != X5) | ((pa @ X4 @ X5) != $true)))) | ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X0 @ x) @ r) != $true))),
% 17.81/3.97 inference(ennf_transformation,[status(thm)],[f343])).
% 17.81/3.97 thf(f343,plain,(
% 17.81/3.97 ! [X0 : state] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X0 @ x) @ r) = $true) => (! [X1 : state,X2 : state] : (((exec @ X1 @ body @ X2) = $true) => ! [X3 : a] : ((((pa @ X3 @ X1) = $true) & (X0 = X1)) => ((qa @ X3 @ X2) = $true))) & ! [X4 : a,X5 : state] : (((X0 = X5) & ((pa @ X4 @ X5) = $true)) => ((termi @ body @ X5) = $true))))),
% 17.81/3.97 inference(fool_elimination,[status(thm)],[f342])).
% 17.81/3.97 thf(f342,plain,(
% 17.81/3.97 ! [X0 : state] : ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X0 @ x) @ r) => (! [X1 : state,X2 : state] : ((exec @ X1 @ body @ X2) => ! [X3 : a] : (((X0 = X1) & (pa @ X3 @ X1)) => (qa @ X3 @ X2))) & ! [X4 : a,X5 : state] : (((pa @ X4 @ X5) & (X0 = X5)) => (termi @ body @ X5))))),
% 17.81/3.97 inference(rectify,[status(thm)],[f266])).
% 17.81/3.97 thf(f266,axiom,(
% 17.81/3.97 ! [X40 : state] : ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X40 @ x) @ r) => (! [X133 : state,X134 : state] : ((exec @ X133 @ body @ X134) => ! [X13 : a] : (((X40 = X133) & (pa @ X13 @ X133)) => (qa @ X13 @ X134))) & ! [X13 : a,X133 : state] : (((pa @ X13 @ X133) & (X40 = X133)) => (termi @ body @ X133))))),
% 17.81/3.97 file('/export/starexec/sandbox2/tmp/tmp.UIk1xi5Kzw/DTF2THF_13980.p',conj_1)).
% 17.81/3.97 thf(f979,plain,(
% 17.81/3.97 ( ! [X0 : state] : (((qa @ (sK21 @ X0) @ (sK20 @ X0)) != $true) | ((sP3 @ X0) != $true)) )),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f906])).
% 17.81/3.97 thf(f1318,plain,(
% 17.81/3.97 spl88_5 | spl88_13 | spl88_17 | spl88_11),
% 17.81/3.97 inference(avatar_split_clause,[status(thm)],[f1317,f1145,f1194,f1160,f1112])).
% 17.81/3.97 thf(f1160,definition,(
% 17.81/3.97 spl88_13 <=> ! [X0 : a] : ((pa @ X0 @ (sK23 @ x)) != $true)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_13])],[avatar_definition])).
% 17.81/3.97 thf(f1194,definition,(
% 17.81/3.97 spl88_17 <=> ! [X2 : a,X3 : state] : (((exec @ x @ body @ X3) != $true) | ((pa @ X2 @ x) != $true) | ($true = (qa @ X2 @ X3)))),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_17])],[avatar_definition])).
% 17.81/3.97 thf(f1145,definition,(
% 17.81/3.97 spl88_11 <=> ((termi @ body @ (sK23 @ x)) = $true)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_11])],[avatar_definition])).
% 17.81/3.97 thf(f1317,plain,(
% 17.81/3.97 ( ! [X2 : state,X0 : a,X1 : a] : (($true = (qa @ X1 @ X2)) | ($true != (exec @ x @ body @ X2)) | ($true != (pa @ X1 @ x)) | ((pa @ X0 @ (sK23 @ x)) != $true) | ((sP3 @ x) = $true)) ) | spl88_11),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1191,f1147])).
% 17.81/3.97 thf(f1147,plain,(
% 17.81/3.97 ((termi @ body @ (sK23 @ x)) != $true) | spl88_11),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1145])).
% 17.81/3.97 thf(f1191,plain,(
% 17.81/3.97 ( ! [X2 : state,X0 : a,X1 : a] : (((pa @ X0 @ (sK23 @ x)) != $true) | ($true = (qa @ X1 @ X2)) | ((sP3 @ x) = $true) | ($true != (exec @ x @ body @ X2)) | ((termi @ body @ (sK23 @ x)) = $true) | ($true != (pa @ X1 @ x))) )),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1189])).
% 17.81/3.97 thf(f1189,plain,(
% 17.81/3.97 ( ! [X2 : state,X0 : a,X1 : a] : (($true != (exec @ x @ body @ X2)) | ((pa @ X0 @ (sK23 @ x)) != $true) | ((sP3 @ x) = $true) | ($true != (pa @ X1 @ x)) | ($true != $true) | ($true = (qa @ X1 @ X2)) | ((termi @ body @ (sK23 @ x)) = $true)) )),
% 17.81/3.97 inference(superposition,[status(thm)],[f1045,f1049])).
% 17.81/3.97 thf(f1049,plain,(
% 17.81/3.97 ( ! [X2 : state,X3 : a,X1 : state] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK23 @ X1) @ X1) @ r) = $true) | ((pa @ X3 @ X1) != $true) | ((sP3 @ X1) = $true) | ((exec @ X1 @ body @ X2) != $true) | ((qa @ X3 @ X2) = $true)) )),
% 17.81/3.97 inference(equality_resolution,[status(thm)],[f986])).
% 17.81/3.97 thf(f986,plain,(
% 17.81/3.97 ( ! [X2 : state,X3 : a,X0 : state,X1 : state] : (((sP3 @ X0) = $true) | ((exec @ X1 @ body @ X2) != $true) | ((qa @ X3 @ X2) = $true) | ((pa @ X3 @ X1) != $true) | (X0 != X1) | ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK23 @ X0) @ X0) @ r) = $true)) )),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f909])).
% 17.81/3.97 thf(f909,plain,(
% 17.81/3.97 ! [X0 : state] : (($true = (pHoare166426476thoare @ a @ (insert @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (product_Pair @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (^[Y0 : a]: ((^[Y1 : state]: ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ Y1 @ X0) @ r) & (pa @ Y0 @ Y1))))) @ (product_Pair @ com @ (a > state > $o) @ call @ qa)) @ bot_bot @ set @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o)) @ (^[Y0 : a]: ((^[Y1 : state]: ((X0 = Y1) & (pa @ Y0 @ Y1))))) @ body @ qa)) & (((sP3 @ X0) = $true) | (! [X1 : state,X2 : state] : (((exec @ X1 @ body @ X2) != $true) | ! [X3 : a] : (((qa @ X3 @ X2) = $true) | ((pa @ X3 @ X1) != $true) | (X0 != X1))) & ! [X4 : a,X5 : state] : (((termi @ body @ X5) = $true) | ((pa @ X4 @ X5) != $true) | (X0 != X5))) | (((pa @ (sK22 @ X0) @ (sK23 @ X0)) = $true) & ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK23 @ X0) @ X0) @ r) = $true) & ($true != (termi @ body @ (sK23 @ X0))))))),
% 17.81/3.97 inference(skolemisation,[status(esa),new_symbols(skolem,[sK22,sK23])],[f907,f908])).
% 17.81/3.97 thf(f908,plain,(
% 17.81/3.97 ! [X0 : state] : (? [X6 : a,X7 : state] : (((pa @ X6 @ X7) = $true) & ($true = (member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X7 @ X0) @ r)) & ((termi @ body @ X7) != $true)) => (((pa @ (sK22 @ X0) @ (sK23 @ X0)) = $true) & ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK23 @ X0) @ X0) @ r) = $true) & ($true != (termi @ body @ (sK23 @ X0)))))),
% 17.81/3.97 introduced(definition,[],[choice_axiom])).
% 17.81/3.97 thf(f907,plain,(
% 17.81/3.97 ! [X0 : state] : (($true = (pHoare166426476thoare @ a @ (insert @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (product_Pair @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (^[Y0 : a]: ((^[Y1 : state]: ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ Y1 @ X0) @ r) & (pa @ Y0 @ Y1))))) @ (product_Pair @ com @ (a > state > $o) @ call @ qa)) @ bot_bot @ set @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o)) @ (^[Y0 : a]: ((^[Y1 : state]: ((X0 = Y1) & (pa @ Y0 @ Y1))))) @ body @ qa)) & (((sP3 @ X0) = $true) | (! [X1 : state,X2 : state] : (((exec @ X1 @ body @ X2) != $true) | ! [X3 : a] : (((qa @ X3 @ X2) = $true) | ((pa @ X3 @ X1) != $true) | (X0 != X1))) & ! [X4 : a,X5 : state] : (((termi @ body @ X5) = $true) | ((pa @ X4 @ X5) != $true) | (X0 != X5))) | ? [X6 : a,X7 : state] : (((pa @ X6 @ X7) = $true) & ($true = (member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X7 @ X0) @ r)) & ((termi @ body @ X7) != $true))))),
% 17.81/3.97 inference(rectify,[status(thm)],[f886])).
% 17.81/3.97 thf(f886,plain,(
% 17.81/3.97 ! [X0 : state] : (($true = (pHoare166426476thoare @ a @ (insert @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (product_Pair @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (^[Y0 : a]: ((^[Y1 : state]: ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ Y1 @ X0) @ r) & (pa @ Y0 @ Y1))))) @ (product_Pair @ com @ (a > state > $o) @ call @ qa)) @ bot_bot @ set @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o)) @ (^[Y0 : a]: ((^[Y1 : state]: ((X0 = Y1) & (pa @ Y0 @ Y1))))) @ body @ qa)) & (((sP3 @ X0) = $true) | (! [X8 : state,X9 : state] : (($true != (exec @ X8 @ body @ X9)) | ! [X10 : a] : (($true = (qa @ X10 @ X9)) | ((pa @ X10 @ X8) != $true) | (X0 != X8))) & ! [X6 : a,X7 : state] : (((termi @ body @ X7) = $true) | ((pa @ X6 @ X7) != $true) | (X0 != X7))) | ? [X4 : a,X5 : state] : (((pa @ X4 @ X5) = $true) & ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X5 @ X0) @ r) = $true) & ((termi @ body @ X5) != $true))))),
% 17.81/3.97 inference(definition_folding,[status(thm)],[f853,f885])).
% 17.81/3.97 thf(f853,plain,(
% 17.81/3.97 ! [X0 : state] : (($true = (pHoare166426476thoare @ a @ (insert @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (product_Pair @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (^[Y0 : a]: ((^[Y1 : state]: ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ Y1 @ X0) @ r) & (pa @ Y0 @ Y1))))) @ (product_Pair @ com @ (a > state > $o) @ call @ qa)) @ bot_bot @ set @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o)) @ (^[Y0 : a]: ((^[Y1 : state]: ((X0 = Y1) & (pa @ Y0 @ Y1))))) @ body @ qa)) & (? [X1 : state,X2 : state] : (? [X3 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X1 @ X0) @ r) = $true) & ((pa @ X3 @ X1) = $true) & ((qa @ X3 @ X2) != $true)) & ((exec @ X1 @ body @ X2) = $true)) | (! [X8 : state,X9 : state] : (($true != (exec @ X8 @ body @ X9)) | ! [X10 : a] : (($true = (qa @ X10 @ X9)) | ((pa @ X10 @ X8) != $true) | (X0 != X8))) & ! [X6 : a,X7 : state] : (((termi @ body @ X7) = $true) | ((pa @ X6 @ X7) != $true) | (X0 != X7))) | ? [X4 : a,X5 : state] : (((pa @ X4 @ X5) = $true) & ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X5 @ X0) @ r) = $true) & ((termi @ body @ X5) != $true))))),
% 17.81/3.97 inference(flattening,[status(thm)],[f852])).
% 17.81/3.97 thf(f852,plain,(
% 17.81/3.97 ! [X0 : state] : (($true = (pHoare166426476thoare @ a @ (insert @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (product_Pair @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (^[Y0 : a]: ((^[Y1 : state]: ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ Y1 @ X0) @ r) & (pa @ Y0 @ Y1))))) @ (product_Pair @ com @ (a > state > $o) @ call @ qa)) @ bot_bot @ set @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o)) @ (^[Y0 : a]: ((^[Y1 : state]: ((X0 = Y1) & (pa @ Y0 @ Y1))))) @ body @ qa)) & ((! [X6 : a,X7 : state] : (((termi @ body @ X7) = $true) | (((pa @ X6 @ X7) != $true) | (X0 != X7))) & ! [X8 : state,X9 : state] : (! [X10 : a] : (($true = (qa @ X10 @ X9)) | (((pa @ X10 @ X8) != $true) | (X0 != X8))) | ($true != (exec @ X8 @ body @ X9)))) | (? [X1 : state,X2 : state] : (? [X3 : a] : (((qa @ X3 @ X2) != $true) & (((pa @ X3 @ X1) = $true) & ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X1 @ X0) @ r) = $true))) & ((exec @ X1 @ body @ X2) = $true)) | ? [X4 : a,X5 : state] : (((termi @ body @ X5) != $true) & (((pa @ X4 @ X5) = $true) & ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X5 @ X0) @ r) = $true))))))),
% 17.81/3.97 inference(ennf_transformation,[status(thm)],[f660])).
% 17.81/3.97 thf(f660,plain,(
% 17.81/3.97 ! [X0 : state] : (($true = (pHoare166426476thoare @ a @ (insert @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (product_Pair @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (^[Y0 : a]: ((^[Y1 : state]: ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ Y1 @ X0) @ r) & (pa @ Y0 @ Y1))))) @ (product_Pair @ com @ (a > state > $o) @ call @ qa)) @ bot_bot @ set @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o)) @ (^[Y0 : a]: ((^[Y1 : state]: ((X0 = Y1) & (pa @ Y0 @ Y1))))) @ body @ qa)) & ((! [X1 : state,X2 : state] : (((exec @ X1 @ body @ X2) = $true) => ! [X3 : a] : ((((pa @ X3 @ X1) = $true) & ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X1 @ X0) @ r) = $true)) => ((qa @ X3 @ X2) = $true))) & ! [X4 : a,X5 : state] : ((((pa @ X4 @ X5) = $true) & ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X5 @ X0) @ r) = $true)) => ((termi @ body @ X5) = $true))) => (! [X6 : a,X7 : state] : ((((pa @ X6 @ X7) = $true) & (X0 = X7)) => ((termi @ body @ X7) = $true)) & ! [X8 : state,X9 : state] : (($true = (exec @ X8 @ body @ X9)) => ! [X10 : a] : ((((pa @ X10 @ X8) = $true) & (X0 = X8)) => ($true = (qa @ X10 @ X9)))))))),
% 17.81/3.97 inference(fool_elimination,[status(thm)],[f659])).
% 17.81/3.97 thf(f659,plain,(
% 17.81/3.97 ! [X0 : state] : (((! [X1 : state,X2 : state] : ((exec @ X1 @ body @ X2) => ! [X3 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X1 @ X0) @ r) & (pa @ X3 @ X1)) => (qa @ X3 @ X2))) & ! [X4 : a,X5 : state] : (((pa @ X4 @ X5) & (member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X5 @ X0) @ r)) => (termi @ body @ X5))) => (! [X6 : a,X7 : state] : (((pa @ X6 @ X7) & (X0 = X7)) => (termi @ body @ X7)) & ! [X8 : state,X9 : state] : ((exec @ X8 @ body @ X9) => ! [X10 : a] : (((X0 = X8) & (pa @ X10 @ X8)) => (qa @ X10 @ X9))))) & (pHoare166426476thoare @ a @ (insert @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (product_Pair @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (^[X11 : a, X12 : state] : ((pa @ X11 @ X12) & (member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X12 @ X0) @ r))) @ (product_Pair @ com @ (a > state > $o) @ call @ qa)) @ bot_bot @ set @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o)) @ (^[X13 : a, X14 : state] : ((pa @ X13 @ X14) & (X0 = X14))) @ body @ qa))),
% 17.81/3.97 inference(rectify,[status(thm)],[f265])).
% 17.81/3.97 thf(f265,axiom,(
% 17.81/3.97 ! [X132 : state] : (((! [X10 : state,X12 : state] : ((exec @ X10 @ body @ X12) => ! [X9 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X10 @ X132) @ r) & (pa @ X9 @ X10)) => (qa @ X9 @ X12))) & ! [X9 : a,X10 : state] : (((pa @ X9 @ X10) & (member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X10 @ X132) @ r)) => (termi @ body @ X10))) => (! [X13 : a,X133 : state] : (((pa @ X13 @ X133) & (X132 = X133)) => (termi @ body @ X133)) & ! [X133 : state,X134 : state] : ((exec @ X133 @ body @ X134) => ! [X13 : a] : (((X132 = X133) & (pa @ X13 @ X133)) => (qa @ X13 @ X134))))) & (pHoare166426476thoare @ a @ (insert @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (product_Pair @ (a > state > $o) @ product_prod @ com @ (a > state > $o) @ (^[X22 : a, X23 : state] : ((pa @ X22 @ X23) & (member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X23 @ X132) @ r))) @ (product_Pair @ com @ (a > state > $o) @ call @ qa)) @ bot_bot @ set @ product_prod @ (a > state > $o) @ product_prod @ com @ (a > state > $o)) @ (^[X22 : a, X23 : state] : ((pa @ X22 @ X23) & (X23 = X132))) @ body @ qa))),
% 17.81/3.97 file('/export/starexec/sandbox2/tmp/tmp.UIk1xi5Kzw/DTF2THF_13980.p',conj_0)).
% 17.81/3.97 thf(f1045,plain,(
% 17.81/3.97 ( ! [X2 : state,X1 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X2 @ x) @ r) != $true) | ((termi @ body @ X2) = $true) | ((pa @ X1 @ X2) != $true)) )),
% 17.81/3.97 inference(equality_resolution,[status(thm)],[f963])).
% 17.81/3.97 thf(f963,plain,(
% 17.81/3.97 ( ! [X2 : state,X0 : state,X1 : a] : (((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ X0 @ x) @ r) != $true) | (X0 != X2) | ((termi @ body @ X2) = $true) | ((pa @ X1 @ X2) != $true)) )),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f889])).
% 17.81/3.97 thf(f1316,plain,(
% 17.81/3.97 spl88_12 | ~spl88_17 | ~spl88_20 | ~spl88_23),
% 17.81/3.97 inference(avatar_split_clause,[status(thm)],[f1308,f1269,f1228,f1194,f1149])).
% 17.81/3.97 thf(f1149,definition,(
% 17.81/3.97 spl88_12 <=> ! [X0 : a] : (((qa @ X0 @ sK13) = $true) | ($true != (pa @ X0 @ x)))),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_12])],[avatar_definition])).
% 17.81/3.97 thf(f1228,definition,(
% 17.81/3.97 spl88_20 <=> ($true = (exec @ sK12 @ body @ sK13))),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_20])],[avatar_definition])).
% 17.81/3.97 thf(f1269,definition,(
% 17.81/3.97 spl88_23 <=> (x = sK12)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_23])],[avatar_definition])).
% 17.81/3.97 thf(f1308,plain,(
% 17.81/3.97 ( ! [X0 : a] : (((qa @ X0 @ sK13) = $true) | ($true != (pa @ X0 @ x))) ) | (~spl88_17 | ~spl88_20 | ~spl88_23)),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1306])).
% 17.81/3.97 thf(f1306,plain,(
% 17.81/3.97 ( ! [X0 : a] : (((qa @ X0 @ sK13) = $true) | ($true != (pa @ X0 @ x)) | ($true != $true)) ) | (~spl88_17 | ~spl88_20 | ~spl88_23)),
% 17.81/3.97 inference(superposition,[status(thm)],[f1195,f1274])).
% 17.81/3.97 thf(f1274,plain,(
% 17.81/3.97 ((exec @ x @ body @ sK13) = $true) | (~spl88_20 | ~spl88_23)),
% 17.81/3.97 inference(backward_demodulation,[status(thm)],[f1230,f1271])).
% 17.81/3.97 thf(f1271,plain,(
% 17.81/3.97 (x = sK12) | ~spl88_23),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1269])).
% 17.81/3.97 thf(f1230,plain,(
% 17.81/3.97 ($true = (exec @ sK12 @ body @ sK13)) | ~spl88_20),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1228])).
% 17.81/3.97 thf(f1195,plain,(
% 17.81/3.97 ( ! [X2 : a,X3 : state] : (((exec @ x @ body @ X3) != $true) | ($true = (qa @ X2 @ X3)) | ((pa @ X2 @ x) != $true)) ) | ~spl88_17),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1194])).
% 17.81/3.97 thf(f1315,plain,(
% 17.81/3.97 ~spl88_12 | spl88_21 | ~spl88_22 | ~spl88_23),
% 17.81/3.97 inference(avatar_contradiction_clause,[status(thm)],[f1314])).
% 17.81/3.97 thf(f1314,plain,(
% 17.81/3.97 $false | (~spl88_12 | spl88_21 | ~spl88_22 | ~spl88_23)),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1313,f1273])).
% 17.81/3.97 thf(f1273,plain,(
% 17.81/3.97 ($true = (pa @ sK14 @ x)) | (~spl88_22 | ~spl88_23)),
% 17.81/3.97 inference(backward_demodulation,[status(thm)],[f1266,f1271])).
% 17.81/3.97 thf(f1266,plain,(
% 17.81/3.97 ((pa @ sK14 @ sK12) = $true) | ~spl88_22),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1264])).
% 17.81/3.97 thf(f1264,definition,(
% 17.81/3.97 spl88_22 <=> ((pa @ sK14 @ sK12) = $true)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_22])],[avatar_definition])).
% 17.81/3.97 thf(f1313,plain,(
% 17.81/3.97 ($true != (pa @ sK14 @ x)) | (~spl88_12 | spl88_21)),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1312])).
% 17.81/3.97 thf(f1312,plain,(
% 17.81/3.97 ($true != $true) | ($true != (pa @ sK14 @ x)) | (~spl88_12 | spl88_21)),
% 17.81/3.97 inference(superposition,[status(thm)],[f1261,f1150])).
% 17.81/3.97 thf(f1150,plain,(
% 17.81/3.97 ( ! [X0 : a] : (((qa @ X0 @ sK13) = $true) | ($true != (pa @ X0 @ x))) ) | ~spl88_12),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1149])).
% 17.81/3.97 thf(f1261,plain,(
% 17.81/3.97 ((qa @ sK14 @ sK13) != $true) | spl88_21),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1259])).
% 17.81/3.97 thf(f1259,definition,(
% 17.81/3.97 spl88_21 <=> ((qa @ sK14 @ sK13) = $true)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_21])],[avatar_definition])).
% 17.81/3.97 thf(f1272,plain,(
% 17.81/3.97 spl88_23 | ~spl88_2),
% 17.81/3.97 inference(avatar_split_clause,[status(thm)],[f967,f1060,f1269])).
% 17.81/3.97 thf(f1060,definition,(
% 17.81/3.97 spl88_2 <=> (sP2 = $true)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_2])],[avatar_definition])).
% 17.81/3.97 thf(f967,plain,(
% 17.81/3.97 (x = sK12) | (sP2 != $true)),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f896])).
% 17.81/3.97 thf(f896,plain,(
% 17.81/3.97 ((((qa @ sK14 @ sK13) != $true) & (x = sK12) & ((pa @ sK14 @ sK12) = $true)) & ($true = (exec @ sK12 @ body @ sK13))) | (sP2 != $true)),
% 17.81/3.97 inference(skolemisation,[status(esa),new_symbols(skolem,[sK12,sK13,sK14])],[f893,f895,f894])).
% 17.81/3.97 thf(f894,plain,(
% 17.81/3.97 ? [X0 : state,X1 : state] : (? [X2 : a] : (((qa @ X2 @ X1) != $true) & (x = X0) & ($true = (pa @ X2 @ X0))) & ((exec @ X0 @ body @ X1) = $true)) => (? [X2 : a] : (($true != (qa @ X2 @ sK13)) & (x = sK12) & ((pa @ X2 @ sK12) = $true)) & ($true = (exec @ sK12 @ body @ sK13)))),
% 17.81/3.97 introduced(definition,[],[choice_axiom])).
% 17.81/3.97 thf(f895,plain,(
% 17.81/3.97 ? [X2 : a] : (($true != (qa @ X2 @ sK13)) & (x = sK12) & ((pa @ X2 @ sK12) = $true)) => (((qa @ sK14 @ sK13) != $true) & (x = sK12) & ((pa @ sK14 @ sK12) = $true))),
% 17.81/3.97 introduced(definition,[],[choice_axiom])).
% 17.81/3.97 thf(f893,plain,(
% 17.81/3.97 ? [X0 : state,X1 : state] : (? [X2 : a] : (((qa @ X2 @ X1) != $true) & (x = X0) & ($true = (pa @ X2 @ X0))) & ((exec @ X0 @ body @ X1) = $true)) | (sP2 != $true)),
% 17.81/3.97 inference(rectify,[status(thm)],[f892])).
% 17.81/3.97 thf(f892,plain,(
% 17.81/3.97 ? [X2 : state,X3 : state] : (? [X4 : a] : (((qa @ X4 @ X3) != $true) & (x = X2) & ($true = (pa @ X4 @ X2))) & ((exec @ X2 @ body @ X3) = $true)) | (sP2 != $true)),
% 17.81/3.97 inference(nnf_transformation,[status(thm)],[f883])).
% 17.81/3.97 thf(f883,plain,(
% 17.81/3.97 ? [X2 : state,X3 : state] : (? [X4 : a] : (((qa @ X4 @ X3) != $true) & (x = X2) & ($true = (pa @ X4 @ X2))) & ((exec @ X2 @ body @ X3) = $true)) | ~(sP2 = $true)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[=])],[predicate_definition_introduction])).
% 17.81/3.97 thf(f1267,plain,(
% 17.81/3.97 ~spl88_2 | spl88_22),
% 17.81/3.97 inference(avatar_split_clause,[status(thm)],[f966,f1264,f1060])).
% 17.81/3.97 thf(f966,plain,(
% 17.81/3.97 ((pa @ sK14 @ sK12) = $true) | (sP2 != $true)),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f896])).
% 17.81/3.97 thf(f1262,plain,(
% 17.81/3.97 ~spl88_2 | ~spl88_21),
% 17.81/3.97 inference(avatar_split_clause,[status(thm)],[f968,f1259,f1060])).
% 17.81/3.97 thf(f968,plain,(
% 17.81/3.97 ((qa @ sK14 @ sK13) != $true) | (sP2 != $true)),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f896])).
% 17.81/3.97 thf(f1257,plain,(
% 17.81/3.97 spl88_17 | spl88_5 | ~spl88_13),
% 17.81/3.97 inference(avatar_split_clause,[status(thm)],[f1256,f1160,f1112,f1194])).
% 17.81/3.97 thf(f1256,plain,(
% 17.81/3.97 ( ! [X0 : state,X1 : a] : (($true = (qa @ X1 @ X0)) | ($true != (pa @ X1 @ x)) | ((exec @ x @ body @ X0) != $true)) ) | (spl88_5 | ~spl88_13)),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1251,f1114])).
% 17.81/3.97 thf(f1114,plain,(
% 17.81/3.97 ((sP3 @ x) != $true) | spl88_5),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1112])).
% 17.81/3.97 thf(f1251,plain,(
% 17.81/3.97 ( ! [X0 : state,X1 : a] : (((sP3 @ x) = $true) | ((exec @ x @ body @ X0) != $true) | ($true = (qa @ X1 @ X0)) | ($true != (pa @ X1 @ x))) ) | ~spl88_13),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1249])).
% 17.81/3.97 thf(f1249,plain,(
% 17.81/3.97 ( ! [X0 : state,X1 : a] : (($true != $true) | ((sP3 @ x) = $true) | ($true = (qa @ X1 @ X0)) | ((exec @ x @ body @ X0) != $true) | ($true != (pa @ X1 @ x))) ) | ~spl88_13),
% 17.81/3.97 inference(superposition,[status(thm)],[f1161,f1048])).
% 17.81/3.97 thf(f1048,plain,(
% 17.81/3.97 ( ! [X2 : state,X3 : a,X1 : state] : (((pa @ (sK22 @ X1) @ (sK23 @ X1)) = $true) | ((exec @ X1 @ body @ X2) != $true) | ((pa @ X3 @ X1) != $true) | ((qa @ X3 @ X2) = $true) | ((sP3 @ X1) = $true)) )),
% 17.81/3.97 inference(equality_resolution,[status(thm)],[f987])).
% 17.81/3.97 thf(f987,plain,(
% 17.81/3.97 ( ! [X2 : state,X3 : a,X0 : state,X1 : state] : (((sP3 @ X0) = $true) | ((exec @ X1 @ body @ X2) != $true) | ((qa @ X3 @ X2) = $true) | ((pa @ X3 @ X1) != $true) | (X0 != X1) | ((pa @ (sK22 @ X0) @ (sK23 @ X0)) = $true)) )),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f909])).
% 17.81/3.97 thf(f1161,plain,(
% 17.81/3.97 ( ! [X0 : a] : (((pa @ X0 @ (sK23 @ x)) != $true)) ) | ~spl88_13),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1160])).
% 17.81/3.97 thf(f1255,plain,(
% 17.81/3.97 spl88_15 | spl88_5 | ~spl88_13 | spl88_14),
% 17.81/3.97 inference(avatar_split_clause,[status(thm)],[f1254,f1163,f1160,f1112,f1167])).
% 17.81/3.97 thf(f1167,definition,(
% 17.81/3.97 spl88_15 <=> ! [X1 : a] : ($true != (pa @ X1 @ x))),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_15])],[avatar_definition])).
% 17.81/3.97 thf(f1163,definition,(
% 17.81/3.97 spl88_14 <=> ((termi @ body @ x) = $true)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_14])],[avatar_definition])).
% 17.81/3.97 thf(f1254,plain,(
% 17.81/3.97 ( ! [X0 : a] : (($true != (pa @ X0 @ x))) ) | (spl88_5 | ~spl88_13 | spl88_14)),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1253,f1164])).
% 17.81/3.97 thf(f1164,plain,(
% 17.81/3.97 ((termi @ body @ x) != $true) | spl88_14),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1163])).
% 17.81/3.97 thf(f1253,plain,(
% 17.81/3.97 ( ! [X0 : a] : (($true != (pa @ X0 @ x)) | ((termi @ body @ x) = $true)) ) | (spl88_5 | ~spl88_13)),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1252,f1114])).
% 17.81/3.97 thf(f1252,plain,(
% 17.81/3.97 ( ! [X0 : a] : (((sP3 @ x) = $true) | ($true != (pa @ X0 @ x)) | ((termi @ body @ x) = $true)) ) | ~spl88_13),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1250])).
% 17.81/3.97 thf(f1250,plain,(
% 17.81/3.97 ( ! [X0 : a] : (($true != (pa @ X0 @ x)) | ((termi @ body @ x) = $true) | ((sP3 @ x) = $true) | ($true != $true)) ) | ~spl88_13),
% 17.81/3.97 inference(superposition,[status(thm)],[f1161,f1051])).
% 17.81/3.97 thf(f1051,plain,(
% 17.81/3.97 ( ! [X4 : a,X5 : state] : (($true = (pa @ (sK22 @ X5) @ (sK23 @ X5))) | ((termi @ body @ X5) = $true) | ((sP3 @ X5) = $true) | ((pa @ X4 @ X5) != $true)) )),
% 17.81/3.97 inference(equality_resolution,[status(thm)],[f984])).
% 17.81/3.97 thf(f984,plain,(
% 17.81/3.97 ( ! [X0 : state,X4 : a,X5 : state] : (((sP3 @ X0) = $true) | ((termi @ body @ X5) = $true) | ((pa @ X4 @ X5) != $true) | (X0 != X5) | ((pa @ (sK22 @ X0) @ (sK23 @ X0)) = $true)) )),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f909])).
% 17.81/3.97 thf(f1248,plain,(
% 17.81/3.97 ~spl88_1 | ~spl88_4 | ~spl88_15),
% 17.81/3.97 inference(avatar_contradiction_clause,[status(thm)],[f1247])).
% 17.81/3.97 thf(f1247,plain,(
% 17.81/3.97 $false | (~spl88_1 | ~spl88_4 | ~spl88_15)),
% 17.81/3.97 inference(trivial_inequality_removal,[status(thm)],[f1246])).
% 17.81/3.97 thf(f1246,plain,(
% 17.81/3.97 ($true != $true) | (~spl88_1 | ~spl88_4 | ~spl88_15)),
% 17.81/3.97 inference(superposition,[status(thm)],[f1168,f1242])).
% 17.81/3.97 thf(f1242,plain,(
% 17.81/3.97 ((pa @ sK15 @ x) = $true) | (~spl88_1 | ~spl88_4)),
% 17.81/3.97 inference(superposition,[status(thm)],[f1058,f1072])).
% 17.81/3.97 thf(f1072,plain,(
% 17.81/3.97 (x = sK16) | ~spl88_4),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1070])).
% 17.81/3.97 thf(f1070,definition,(
% 17.81/3.97 spl88_4 <=> (x = sK16)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_4])],[avatar_definition])).
% 17.81/3.97 thf(f1058,plain,(
% 17.81/3.97 ((pa @ sK15 @ sK16) = $true) | ~spl88_1),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1056])).
% 17.81/3.97 thf(f1056,definition,(
% 17.81/3.97 spl88_1 <=> ((pa @ sK15 @ sK16) = $true)),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_1])],[avatar_definition])).
% 17.81/3.97 thf(f1168,plain,(
% 17.81/3.97 ( ! [X1 : a] : (($true != (pa @ X1 @ x))) ) | ~spl88_15),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1167])).
% 17.81/3.97 thf(f1245,plain,(
% 17.81/3.97 spl88_3 | ~spl88_4 | ~spl88_14),
% 17.81/3.97 inference(avatar_contradiction_clause,[status(thm)],[f1244])).
% 17.81/3.97 thf(f1244,plain,(
% 17.81/3.97 $false | (spl88_3 | ~spl88_4 | ~spl88_14)),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1243,f1165])).
% 17.81/3.97 thf(f1165,plain,(
% 17.81/3.97 ((termi @ body @ x) = $true) | ~spl88_14),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1163])).
% 17.81/3.97 thf(f1243,plain,(
% 17.81/3.97 ((termi @ body @ x) != $true) | (spl88_3 | ~spl88_4)),
% 17.81/3.97 inference(superposition,[status(thm)],[f1067,f1072])).
% 17.81/3.97 thf(f1067,plain,(
% 17.81/3.97 ($true != (termi @ body @ sK16)) | spl88_3),
% 17.81/3.97 inference(avatar_component_clause,[status(thm)],[f1065])).
% 17.81/3.97 thf(f1065,definition,(
% 17.81/3.97 spl88_3 <=> ($true = (termi @ body @ sK16))),
% 17.81/3.97 introduced(definition,[new_symbols(naming,[spl88_3])],[avatar_definition])).
% 17.81/3.97 thf(f1231,plain,(
% 17.81/3.97 ~spl88_2 | spl88_20),
% 17.81/3.97 inference(avatar_split_clause,[status(thm)],[f965,f1228,f1060])).
% 17.81/3.97 thf(f965,plain,(
% 17.81/3.97 ($true = (exec @ sK12 @ body @ sK13)) | (sP2 != $true)),
% 17.81/3.97 inference(cnf_transformation,[status(thm)],[f896])).
% 17.81/3.97 thf(f1180,plain,(
% 17.81/3.97 spl88_13 | spl88_15 | spl88_5 | spl88_14),
% 17.81/3.97 inference(avatar_split_clause,[status(thm)],[f1179,f1163,f1112,f1167,f1160])).
% 17.81/3.97 thf(f1179,plain,(
% 17.81/3.97 ( ! [X0 : a,X1 : a] : (((pa @ X0 @ (sK23 @ x)) != $true) | ($true != (pa @ X1 @ x)) | ((sP3 @ x) = $true) | ((termi @ body @ x) = $true)) )),
% 17.81/3.97 inference(subsumption_resolution,[status(thm)],[f1156,f1053])).
% 17.81/3.97 thf(f1053,plain,(
% 17.81/3.97 ( ! [X4 : a,X5 : state] : (((termi @ body @ (sK23 @ X5)) != $true) | ((pa @ X4 @ X5) != $true) | ((sP3 @ X5) = $true) | ((termi @ body @ X5) = $true)) )),
% 17.81/3.98 inference(equality_resolution,[status(thm)],[f982])).
% 17.81/3.98 thf(f982,plain,(
% 17.81/3.98 ( ! [X0 : state,X4 : a,X5 : state] : (((sP3 @ X0) = $true) | ((termi @ body @ X5) = $true) | ((pa @ X4 @ X5) != $true) | (X0 != X5) | ($true != (termi @ body @ (sK23 @ X0)))) )),
% 17.81/3.98 inference(cnf_transformation,[status(thm)],[f909])).
% 17.81/3.98 thf(f1156,plain,(
% 17.81/3.98 ( ! [X0 : a,X1 : a] : (($true != (pa @ X1 @ x)) | ((termi @ body @ x) = $true) | ((termi @ body @ (sK23 @ x)) = $true) | ((pa @ X0 @ (sK23 @ x)) != $true) | ((sP3 @ x) = $true)) )),
% 17.81/3.98 inference(trivial_inequality_removal,[status(thm)],[f1155])).
% 17.81/3.98 thf(f1155,plain,(
% 17.81/3.98 ( ! [X0 : a,X1 : a] : (((termi @ body @ x) = $true) | ((termi @ body @ (sK23 @ x)) = $true) | ((pa @ X0 @ (sK23 @ x)) != $true) | ($true != $true) | ($true != (pa @ X1 @ x)) | ((sP3 @ x) = $true)) )),
% 17.81/3.98 inference(superposition,[status(thm)],[f1045,f1052])).
% 17.81/3.98 thf(f1052,plain,(
% 17.81/3.98 ( ! [X4 : a,X5 : state] : (($true = (member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK23 @ X5) @ X5) @ r)) | ((termi @ body @ X5) = $true) | ((pa @ X4 @ X5) != $true) | ((sP3 @ X5) = $true)) )),
% 17.81/3.98 inference(equality_resolution,[status(thm)],[f983])).
% 17.81/3.98 thf(f983,plain,(
% 17.81/3.98 ( ! [X0 : state,X4 : a,X5 : state] : (((sP3 @ X0) = $true) | ((termi @ body @ X5) = $true) | ((pa @ X4 @ X5) != $true) | (X0 != X5) | ((member @ product_prod @ state @ state @ (product_Pair @ state @ state @ (sK23 @ X0) @ X0) @ r) = $true)) )),
% 17.81/3.98 inference(cnf_transformation,[status(thm)],[f909])).
% 17.81/3.98 thf(f1151,plain,(
% 17.81/3.98 ~spl88_11 | spl88_12 | ~spl88_2 | spl88_5),
% 17.81/3.98 inference(avatar_split_clause,[status(thm)],[f1143,f1112,f1060,f1149,f1145])).
% 17.81/3.98 thf(f1143,plain,(
% 17.81/3.98 ( ! [X0 : a] : (((termi @ body @ (sK23 @ x)) != $true) | ((qa @ X0 @ sK13) = $true) | ($true != (pa @ X0 @ x))) ) | (~spl88_2 | spl88_5)),
% 17.81/3.98 inference(subsumption_resolution,[status(thm)],[f1128,f1114])).
% 17.81/3.98 thf(f1128,plain,(
% 17.81/3.98 ( ! [X0 : a] : (($true != (pa @ X0 @ x)) | ((sP3 @ x) = $true) | ((termi @ body @ (sK23 @ x)) != $true) | ((qa @ X0 @ sK13) = $true)) ) | ~spl88_2),
% 17.81/3.98 inference(trivial_inequality_removal,[status(thm)],[f1125])).
% 17.81/3.98 thf(f1125,plain,(
% 17.81/3.98 ( ! [X0 : a] : (($true != $true) | ((sP3 @ x) = $true) | ($true != (pa @ X0 @ x)) | ((qa @ X0 @ sK13) = $true) | ((termi @ body @ (sK23 @ x)) != $true)) ) | ~spl88_2),
% 17.81/3.98 inference(superposition,[status(thm)],[f1050,f1108])).
% 17.81/3.98 thf(f1108,plain,(
% 17.81/3.98 ((exec @ x @ body @ sK13) = $true) | ~spl88_2),
% 17.81/3.98 inference(superposition,[status(thm)],[f1079,f1080])).
% 17.81/3.98 thf(f1080,plain,(
% 17.81/3.98 (x = sK12) | ~spl88_2),
% 17.81/3.98 inference(trivial_inequality_removal,[status(thm)],[f1074])).
% 17.81/3.98 thf(f1074,plain,(
% 17.81/3.98 ($true != $true) | (x = sK12) | ~spl88_2),
% 17.81/3.98 inference(backward_demodulation,[status(thm)],[f967,f1062])).
% 17.81/3.98 thf(f1062,plain,(
% 17.81/3.98 (sP2 = $true) | ~spl88_2),
% 17.81/3.98 inference(avatar_component_clause,[status(thm)],[f1060])).
% 17.81/3.98 thf(f1079,plain,(
% 17.81/3.98 ($true = (exec @ sK12 @ body @ sK13)) | ~spl88_2),
% 17.81/3.98 inference(trivial_inequality_removal,[status(thm)],[f1076])).
% 17.81/3.98 thf(f1076,plain,(
% 17.81/3.98 ($true = (exec @ sK12 @ body @ sK13)) | ($true != $true) | ~spl88_2),
% 17.81/3.98 inference(backward_demodulation,[status(thm)],[f965,f1062])).
% 17.81/3.98 thf(f1050,plain,(
% 17.81/3.98 ( ! [X2 : state,X3 : a,X1 : state] : (((exec @ X1 @ body @ X2) != $true) | ((termi @ body @ (sK23 @ X1)) != $true) | ((qa @ X3 @ X2) = $true) | ((sP3 @ X1) = $true) | ((pa @ X3 @ X1) != $true)) )),
% 17.81/3.98 inference(equality_resolution,[status(thm)],[f985])).
% 17.81/3.98 thf(f985,plain,(
% 17.81/3.98 ( ! [X2 : state,X3 : a,X0 : state,X1 : state] : (((sP3 @ X0) = $true) | ((exec @ X1 @ body @ X2) != $true) | ((qa @ X3 @ X2) = $true) | ((pa @ X3 @ X1) != $true) | (X0 != X1) | ($true != (termi @ body @ (sK23 @ X0)))) )),
% 17.81/3.98 inference(cnf_transformation,[status(thm)],[f909])).
% 17.81/3.98 thf(f1073,plain,(
% 17.81/3.98 spl88_2 | spl88_4),
% 17.81/3.98 inference(avatar_split_clause,[status(thm)],[f970,f1070,f1060])).
% 17.81/3.98 thf(f970,plain,(
% 17.81/3.98 (x = sK16) | (sP2 = $true)),
% 17.81/3.98 inference(cnf_transformation,[status(thm)],[f898])).
% 17.81/3.98 thf(f898,plain,(
% 17.81/3.98 (($true != (termi @ body @ sK16)) & (x = sK16) & ((pa @ sK15 @ sK16) = $true)) | (sP2 = $true)),
% 17.81/3.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK15,sK16])],[f884,f897])).
% 17.81/3.98 thf(f897,plain,(
% 17.81/3.98 ? [X0 : a,X1 : state] : (($true != (termi @ body @ X1)) & (x = X1) & ($true = (pa @ X0 @ X1))) => (($true != (termi @ body @ sK16)) & (x = sK16) & ((pa @ sK15 @ sK16) = $true))),
% 17.81/3.98 introduced(definition,[],[choice_axiom])).
% 17.81/3.98 thf(f884,plain,(
% 17.81/3.98 ? [X0 : a,X1 : state] : (($true != (termi @ body @ X1)) & (x = X1) & ($true = (pa @ X0 @ X1))) | (sP2 = $true)),
% 17.81/3.98 inference(definition_folding,[status(thm)],[f835,f883])).
% 17.81/3.98 thf(f835,plain,(
% 17.81/3.98 ? [X0 : a,X1 : state] : (($true != (termi @ body @ X1)) & (x = X1) & ($true = (pa @ X0 @ X1))) | ? [X2 : state,X3 : state] : (? [X4 : a] : (((qa @ X4 @ X3) != $true) & (x = X2) & ($true = (pa @ X4 @ X2))) & ((exec @ X2 @ body @ X3) = $true))),
% 17.81/3.98 inference(ennf_transformation,[status(thm)],[f818])).
% 17.81/3.98 thf(f818,plain,(
% 17.81/3.98 ~(! [X0 : a,X1 : state] : ((x != X1) | ($true = (termi @ body @ X1)) | ($true != (pa @ X0 @ X1))) & ! [X2 : state,X3 : state] : (! [X4 : a] : ((x != X2) | ((qa @ X4 @ X3) = $true) | ($true != (pa @ X4 @ X2))) | ((exec @ X2 @ body @ X3) != $true)))),
% 17.81/3.98 inference(flattening,[status(thm)],[f369])).
% 17.81/3.98 thf(f369,plain,(
% 17.81/3.98 ~(! [X0 : a,X1 : state] : (~($true = (pa @ X0 @ X1)) | (x != X1) | ($true = (termi @ body @ X1))) & ! [X2 : state,X3 : state] : (~((exec @ X2 @ body @ X3) = $true) | ! [X4 : a] : ((x != X2) | ((qa @ X4 @ X3) = $true) | ~($true = (pa @ X4 @ X2)))))),
% 17.81/3.98 inference(fool_elimination,[status(thm)],[f368])).
% 17.81/3.98 thf(f368,plain,(
% 17.81/3.98 ~(! [X0 : a,X1 : state] : (~(pa @ X0 @ X1) | (x != X1) | (termi @ body @ X1)) & ! [X2 : state,X3 : state] : (~(exec @ X2 @ body @ X3) | ! [X4 : a] : ((x != X2) | (qa @ X4 @ X3) | ~(pa @ X4 @ X2))))),
% 17.81/3.98 inference(rectify,[status(thm)],[f268])).
% 17.81/3.98 thf(f268,negated_conjecture,(
% 17.81/3.98 ~(! [X9 : a,X10 : state] : (~(pa @ X9 @ X10) | (x != X10) | (termi @ body @ X10)) & ! [X10 : state,X12 : state] : (~(exec @ X10 @ body @ X12) | ! [X9 : a] : ((x != X10) | (qa @ X9 @ X12) | ~(pa @ X9 @ X10))))),
% 17.81/3.98 inference(negated_conjecture,[status(cth)],[f267])).
% 17.81/3.98 thf(f267,conjecture,(
% 17.81/3.98 ! [X9 : a,X10 : state] : (~(pa @ X9 @ X10) | (x != X10) | (termi @ body @ X10)) & ! [X10 : state,X12 : state] : (~(exec @ X10 @ body @ X12) | ! [X9 : a] : ((x != X10) | (qa @ X9 @ X12) | ~(pa @ X9 @ X10)))),
% 17.81/3.98 file('/export/starexec/sandbox2/tmp/tmp.UIk1xi5Kzw/DTF2THF_13980.p',conj_2)).
% 17.81/3.98 thf(f1068,plain,(
% 17.81/3.98 ~spl88_3 | spl88_2),
% 17.81/3.98 inference(avatar_split_clause,[status(thm)],[f971,f1060,f1065])).
% 17.81/3.98 thf(f971,plain,(
% 17.81/3.98 ($true != (termi @ body @ sK16)) | (sP2 = $true)),
% 17.81/3.98 inference(cnf_transformation,[status(thm)],[f898])).
% 17.81/3.98 thf(f1063,plain,(
% 17.81/3.98 spl88_1 | spl88_2),
% 17.81/3.98 inference(avatar_split_clause,[status(thm)],[f969,f1060,f1056])).
% 17.81/3.98 thf(f969,plain,(
% 17.81/3.98 ((pa @ sK15 @ sK16) = $true) | (sP2 = $true)),
% 17.81/3.98 inference(cnf_transformation,[status(thm)],[f898])).
% 17.81/3.98 % SZS output end Proof for DTF2THF_13980
% 17.81/3.98 % (14555)------------------------------
% 17.81/3.98 % (14555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.81/3.98 % (14555)Termination reason: Refutation
% 17.81/3.98
% 17.81/3.98 % (14555)Memory used [KB]: 8059
% 17.81/3.98 % (14555)Time elapsed: 0.123 s
% 17.81/3.98 % (14555)Instructions burned: 320 (million)
% 17.81/3.98 % (14555)------------------------------
% 17.81/3.98 % (14555)------------------------------
% 17.81/3.98 % (14164)Success in time 2.291 s
%------------------------------------------------------------------------------