%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM205^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n025.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 08:41:57 AM UTC 2026
% Result : Theorem 3.33s 1.97s
% Output : Refutation 3.33s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM205^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n025.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 04:16:57 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.48 ---- Original DTF file ---
% 0.31/0.48 thf(spec,logic,$$dhol).
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 % File : COM205^1 : TPTP v9.2.1. Released v7.0.0.
% 0.31/0.48 % Domain : Computing Theory
% 0.31/0.48 % Problem : Parallel extension to grammars and languages 84
% 0.31/0.48 % Version : [Bla16] axioms : Especial.
% 0.31/0.48 % English :
% 0.31/0.48
% 0.31/0.48 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.31/0.48 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.31/0.48 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.31/0.48 % Source : [Bla16]
% 0.31/0.48 % Names : parallel__84.p [Bla16]
% 0.31/0.48
% 0.31/0.48 % Status : Theorem
% 0.31/0.48 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.31/0.48 % Syntax : Number of formulae : 359 ( 127 unt; 67 typ; 0 def)
% 0.31/0.48 % Number of atoms : 618 ( 268 equ; 0 cnn)
% 0.31/0.48 % Maximal formula atoms : 7 ( 2 avg)
% 0.31/0.48 % Number of connectives : 4240 ( 68 ~; 7 |; 30 &;3873 @)
% 0.31/0.48 % ( 0 <=>; 262 =>; 0 <=; 0 <~>)
% 0.31/0.48 % Maximal formula depth : 28 ( 8 avg)
% 0.31/0.48 % Number of types : 4 ( 3 usr)
% 0.31/0.48 % Number of type conns : 287 ( 287 >; 0 *; 0 +; 0 <<)
% 0.31/0.48 % Number of symbols : 66 ( 64 usr; 5 con; 0-7 aty)
% 0.31/0.48 % Number of variables : 1207 ( 43 ^;1075 !; 15 ?;1207 :)
% 0.31/0.48 % ( 74 !>; 0 ?*; 0 @-; 0 @+)
% 0.31/0.48 % SPC : TH1_THM_EQU_NAR
% 0.31/0.48
% 0.31/0.48 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.31/0.48 % 2016-07-13 14:38:58.086
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 %----Could-be-implicit typings (9)
% 0.31/0.48 thf(ty_t_Product__Type_Oprod,type,
% 0.31/0.48 product_prod: $tType > $tType > $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_t_Sum__Type_Osum,type,
% 0.31/0.48 sum_sum: $tType > $tType > $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_t_Option_Ooption,type,
% 0.31/0.48 option: $tType > $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_t_DTree_Odtree,type,
% 0.31/0.48 dtree: $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_t_FSet_Ofset,type,
% 0.31/0.48 fset: $tType > $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_t_Set_Oset,type,
% 0.31/0.48 set: $tType > $tType ).
% 0.31/0.48
% 0.31/0.48 thf(ty_t_DTree_OT,type,
% 0.31/0.49 t: $tType ).
% 0.31/0.49
% 0.31/0.49 thf(ty_t_DTree_ON,type,
% 0.31/0.49 n: $tType ).
% 0.31/0.49
% 0.31/0.49 thf(ty_t_itself,type,
% 0.31/0.49 itself: $tType > $tType ).
% 0.31/0.49
% 0.31/0.49 %----Explicit typings (58)
% 0.31/0.49 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.31/0.49 bounded_lattice:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_cl_HOL_Otype,type,
% 0.31/0.49 type:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_cl_Lattices_Olattice,type,
% 0.31/0.49 lattice:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.31/0.49 finite_finite:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.31/0.49 semiring_char_0:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 0.31/0.49 distrib_lattice:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.31/0.49 semilattice_sup:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 0.31/0.49 bounded_lattice_bot:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 0.31/0.49 bounded_lattice_top:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 0.31/0.49 bounde1808546759up_bot:
% 0.31/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.31/0.49 bNF_Greatest_image2:
% 0.31/0.49 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_DTree_ONode,type,
% 0.31/0.49 node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_DTree_Ocont,type,
% 0.31/0.49 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_DTree_Ocorec,type,
% 0.31/0.49 corec:
% 0.31/0.49 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_DTree_Odtree_ONNode,type,
% 0.31/0.49 nNode: n > ( fset @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_DTree_Odtree_Ocase__dtree,type,
% 0.31/0.49 case_dtree:
% 0.31/0.49 !>[A: $tType] : ( ( n > ( fset @ ( sum_sum @ t @ dtree ) ) > A ) > dtree > A ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_DTree_Odtree_Occont,type,
% 0.31/0.49 ccont: dtree > ( fset @ ( sum_sum @ t @ dtree ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_DTree_Odtree_Ocorec__dtree,type,
% 0.31/0.49 corec_dtree:
% 0.31/0.49 !>[A: $tType] : ( ( A > n ) > ( A > ( fset @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_DTree_Odtree_Oroot,type,
% 0.31/0.49 root: dtree > n ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_DTree_Ounfold,type,
% 0.31/0.49 unfold:
% 0.31/0.49 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Finite__Set_Ofinite,type,
% 0.31/0.49 finite_finite2:
% 0.31/0.49 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.31/0.49 inf_inf:
% 0.31/0.49 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.31/0.49 sup_sup:
% 0.31/0.49 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.31/0.49 bot_bot:
% 0.31/0.49 !>[A: $tType] : A ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/0.49 ord_less_eq:
% 0.31/0.49 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.31/0.49 top_top:
% 0.31/0.49 !>[A: $tType] : A ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus,type,
% 0.31/0.49 parall1518086719_Nplus: n > n > n ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 0.31/0.49 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
% 0.31/0.49 parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
% 0.31/0.49 parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel,type,
% 0.31/0.49 parall556292031_r_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Product__Type_OPair,type,
% 0.31/0.49 product_Pair:
% 0.31/0.49 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Product__Type_Ocurry,type,
% 0.31/0.49 product_curry:
% 0.31/0.49 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.31/0.49 produc2004651681e_prod:
% 0.31/0.49 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Product__Type_Omap__prod,type,
% 0.31/0.49 product_map_prod:
% 0.31/0.49 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.31/0.49 product_rec_prod:
% 0.31/0.49 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.31/0.49 product_case_prod:
% 0.31/0.49 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.31/0.49 product_swap:
% 0.31/0.49 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Pure_Otype,type,
% 0.31/0.49 type2:
% 0.31/0.49 !>[A: $tType] : ( itself @ A ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Set_OCollect,type,
% 0.31/0.49 collect:
% 0.31/0.49 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Set_Oimage,type,
% 0.31/0.49 image:
% 0.31/0.49 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Set_Ovimage,type,
% 0.31/0.49 vimage:
% 0.31/0.49 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Sum__Type_OInl,type,
% 0.31/0.49 sum_Inl:
% 0.31/0.49 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Sum__Type_OInr,type,
% 0.31/0.49 sum_Inr:
% 0.31/0.49 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Sum__Type_OPlus,type,
% 0.31/0.49 sum_Plus:
% 0.31/0.49 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Sum__Type_OSuml,type,
% 0.31/0.49 sum_Suml:
% 0.31/0.49 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Sum__Type_OSumr,type,
% 0.31/0.49 sum_Sumr:
% 0.31/0.49 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
% 0.31/0.49 sum_rec_sum:
% 0.31/0.49 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Wellfounded_Oaccp,type,
% 0.31/0.49 accp:
% 0.31/0.49 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.31/0.49 lex_prod:
% 0.31/0.49 !>[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 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Wfrec_Oadm__wf,type,
% 0.31/0.49 adm_wf:
% 0.31/0.49 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Wfrec_Ocut,type,
% 0.31/0.49 cut:
% 0.31/0.49 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_Wfrec_Osame__fst,type,
% 0.31/0.49 same_fst:
% 0.31/0.49 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_c_member,type,
% 0.31/0.49 member:
% 0.31/0.49 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(sy_v_tr1a____,type,
% 0.31/0.49 tr1a: dtree ).
% 0.31/0.49
% 0.31/0.49 thf(sy_v_tr2a____,type,
% 0.31/0.49 tr2a: dtree ).
% 0.31/0.49
% 0.31/0.49 thf(sy_v_trA____,type,
% 0.31/0.49 trA: dtree ).
% 0.31/0.49
% 0.31/0.49 thf(sy_v_trB____,type,
% 0.31/0.49 trB: dtree ).
% 0.31/0.49
% 0.31/0.49 %----Relevant facts (256)
% 0.31/0.49 thf(fact_0_Nplus__comm,axiom,
% 0.31/0.49 ( parall1518086719_Nplus
% 0.31/0.49 = ( ^ [A2: n,B2: n] : ( parall1518086719_Nplus @ B2 @ A2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Nplus_comm
% 0.31/0.49 thf(fact_1_Nplus__assoc,axiom,
% 0.31/0.49 ! [A3: n,B3: n,C2: n] :
% 0.31/0.49 ( ( parall1518086719_Nplus @ ( parall1518086719_Nplus @ A3 @ B3 ) @ C2 )
% 0.31/0.49 = ( parall1518086719_Nplus @ A3 @ ( parall1518086719_Nplus @ B3 @ C2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Nplus_assoc
% 0.31/0.49 thf(fact_2_par__r_Oelims,axiom,
% 0.31/0.49 ! [X: product_prod @ dtree @ dtree,Y: n] :
% 0.31/0.49 ( ( ( parall1914194362_par_r @ X )
% 0.31/0.49 = Y )
% 0.31/0.49 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.31/0.49 ( ( X
% 0.31/0.49 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.31/0.49 => ( Y
% 0.31/0.49 != ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % par_r.elims
% 0.31/0.49 thf(fact_3_par__r_Osimps,axiom,
% 0.31/0.49 ! [Tr12: dtree,Tr22: dtree] :
% 0.31/0.49 ( ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 0.31/0.49 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % par_r.simps
% 0.31/0.49 thf(fact_4_root__Node,axiom,
% 0.31/0.49 ! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
% 0.31/0.49 ( ( root @ ( node @ N @ As ) )
% 0.31/0.49 = N ) ).
% 0.31/0.49
% 0.31/0.49 % root_Node
% 0.31/0.49 thf(fact_5_root__par,axiom,
% 0.31/0.49 ! [Tr12: dtree,Tr22: dtree] :
% 0.31/0.49 ( ( root @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) )
% 0.31/0.49 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % root_par
% 0.31/0.49 thf(fact_6_corec_I1_J,axiom,
% 0.31/0.49 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B3: A] :
% 0.31/0.49 ( ( root @ ( corec @ A @ Rt @ Ct @ B3 ) )
% 0.31/0.49 = ( Rt @ B3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % corec(1)
% 0.31/0.49 thf(fact_7_unfold_I1_J,axiom,
% 0.31/0.49 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B3: A] :
% 0.31/0.49 ( ( root @ ( unfold @ A @ Rt @ Ct @ B3 ) )
% 0.31/0.49 = ( Rt @ B3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % unfold(1)
% 0.31/0.49 thf(fact_8_dtree_Osel_I1_J,axiom,
% 0.31/0.49 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 0.31/0.49 ( ( root @ ( nNode @ X1 @ X2 ) )
% 0.31/0.49 = X1 ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.sel(1)
% 0.31/0.49 thf(fact_9_dtree_Ocorec__sel_I1_J,axiom,
% 0.31/0.49 ! [A: $tType,G1: A > n,G2: A > ( fset @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),A3: A] :
% 0.31/0.49 ( ( root @ ( corec_dtree @ A @ G1 @ G2 @ A3 ) )
% 0.31/0.49 = ( G1 @ A3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.corec_sel(1)
% 0.31/0.49 thf(fact_10_dtree_Oexpand,axiom,
% 0.31/0.49 ! [Dtree: dtree,Dtree2: dtree] :
% 0.31/0.49 ( ( ( ( root @ Dtree )
% 0.31/0.49 = ( root @ Dtree2 ) )
% 0.31/0.49 & ( ( ccont @ Dtree )
% 0.31/0.49 = ( ccont @ Dtree2 ) ) )
% 0.31/0.49 => ( Dtree = Dtree2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.expand
% 0.31/0.49 thf(fact_11_par__r_Opelims,axiom,
% 0.31/0.49 ! [X: product_prod @ dtree @ dtree,Y: n] :
% 0.31/0.49 ( ( ( parall1914194362_par_r @ X )
% 0.31/0.49 = Y )
% 0.31/0.49 => ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ X )
% 0.31/0.49 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.31/0.49 ( ( X
% 0.31/0.49 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.31/0.49 => ( ( Y
% 0.31/0.49 = ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) )
% 0.31/0.49 => ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % par_r.pelims
% 0.31/0.49 thf(fact_12_dtree__cong,axiom,
% 0.31/0.49 ! [Tr: dtree,Tr3: dtree] :
% 0.31/0.49 ( ( ( root @ Tr )
% 0.31/0.49 = ( root @ Tr3 ) )
% 0.31/0.49 => ( ( ( cont @ Tr )
% 0.31/0.49 = ( cont @ Tr3 ) )
% 0.31/0.49 => ( Tr = Tr3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree_cong
% 0.31/0.49 thf(fact_13_dtree_Ocollapse,axiom,
% 0.31/0.49 ! [Dtree: dtree] :
% 0.31/0.49 ( ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) )
% 0.31/0.49 = Dtree ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.collapse
% 0.31/0.49 thf(fact_14__092_060open_062_092_060exists_062tr1_Atr2_O_AtrA_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060and_062_AtrB_A_061_Atr2_A_092_060parallel_062_Atr1_092_060close_062,axiom,
% 0.31/0.49 ? [Tr1: dtree,Tr2: dtree] :
% 0.31/0.49 ( ( trA
% 0.31/0.49 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) )
% 0.31/0.49 & ( trB
% 0.31/0.49 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr2 @ Tr1 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % \<open>\<exists>tr1 tr2. trA = tr1 \<parallel> tr2 \<and> trB = tr2 \<parallel> tr1\<close>
% 0.31/0.49 thf(fact_15_dtree_Oinject,axiom,
% 0.31/0.49 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree ),Y1: n,Y2: fset @ ( sum_sum @ t @ dtree )] :
% 0.31/0.49 ( ( ( nNode @ X1 @ X2 )
% 0.31/0.49 = ( nNode @ Y1 @ Y2 ) )
% 0.31/0.49 = ( ( X1 = Y1 )
% 0.31/0.49 & ( X2 = Y2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.inject
% 0.31/0.49 thf(fact_16_Node__root__cont,axiom,
% 0.31/0.49 ! [Tr: dtree] :
% 0.31/0.49 ( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
% 0.31/0.49 = Tr ) ).
% 0.31/0.49
% 0.31/0.49 % Node_root_cont
% 0.31/0.49 thf(fact_17_dtree_Osel_I2_J,axiom,
% 0.31/0.49 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 0.31/0.49 ( ( ccont @ ( nNode @ X1 @ X2 ) )
% 0.31/0.49 = X2 ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.sel(2)
% 0.31/0.49 thf(fact_18_dtree_Oexhaust,axiom,
% 0.31/0.49 ! [Y: dtree] :
% 0.31/0.49 ~ ! [X12: n,X22: fset @ ( sum_sum @ t @ dtree )] :
% 0.31/0.49 ( Y
% 0.31/0.49 != ( nNode @ X12 @ X22 ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.exhaust
% 0.31/0.49 thf(fact_19_dtree_Ocorec__disc,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( corec_dtree @ A )
% 0.31/0.49 = ( corec_dtree @ A ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.corec_disc
% 0.31/0.49 thf(fact_20_par__r_Ocases,axiom,
% 0.31/0.49 ! [X: product_prod @ dtree @ dtree] :
% 0.31/0.49 ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.31/0.49 ( X
% 0.31/0.49 != ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % par_r.cases
% 0.31/0.49 thf(fact_21_par__r_Oinduct,axiom,
% 0.31/0.49 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 0.31/0.49 ( ! [Tr1: dtree,Tr2: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.31/0.49 => ( P @ A0 ) ) ).
% 0.31/0.49
% 0.31/0.49 % par_r.induct
% 0.31/0.49 thf(fact_22_dtree_Oexhaust__sel,axiom,
% 0.31/0.49 ! [Dtree: dtree] :
% 0.31/0.49 ( Dtree
% 0.31/0.49 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.exhaust_sel
% 0.31/0.49 thf(fact_23_prod_Oinject,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.31/0.49 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.31/0.49 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.31/0.49 = ( ( X1 = Y1 )
% 0.31/0.49 & ( X2 = Y2 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod.inject
% 0.31/0.49 thf(fact_24_old_Oprod_Oinject,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.31/0.49 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.31/0.49 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.31/0.49 = ( ( A3 = A4 )
% 0.31/0.49 & ( B3 = B4 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.prod.inject
% 0.31/0.49 thf(fact_25_dtree_Osplit__sel,axiom,
% 0.31/0.49 ! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
% 0.31/0.49 ( ( P @ ( case_dtree @ A @ F @ Dtree ) )
% 0.31/0.49 = ( ( Dtree
% 0.31/0.49 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
% 0.31/0.49 => ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.split_sel
% 0.31/0.49 thf(fact_26_dtree_Osplit__sel__asm,axiom,
% 0.31/0.49 ! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
% 0.31/0.49 ( ( P @ ( case_dtree @ A @ F @ Dtree ) )
% 0.31/0.49 = ( ~ ( ( Dtree
% 0.31/0.49 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
% 0.31/0.49 & ~ ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.split_sel_asm
% 0.31/0.49 thf(fact_27_par__def,axiom,
% 0.31/0.49 ( parall1899940088le_par
% 0.31/0.49 = ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% 0.31/0.49
% 0.31/0.49 % par_def
% 0.31/0.49 thf(fact_28_dtree_Ocase__eq__if,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( case_dtree @ A )
% 0.31/0.49 = ( ^ [F2: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree3: dtree] : ( F2 @ ( root @ Dtree3 ) @ ( ccont @ Dtree3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.case_eq_if
% 0.31/0.49 thf(fact_29_accp_Ocases,axiom,
% 0.31/0.49 ! [A: $tType,R: A > A > $o,A3: A] :
% 0.31/0.49 ( ( accp @ A @ R @ A3 )
% 0.31/0.49 => ! [Y3: A] :
% 0.31/0.49 ( ( R @ Y3 @ A3 )
% 0.31/0.49 => ( accp @ A @ R @ Y3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % accp.cases
% 0.31/0.49 thf(fact_30_accp_Osimps,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( accp @ A )
% 0.31/0.49 = ( ^ [R2: A > A > $o,A2: A] :
% 0.31/0.49 ? [X3: A] :
% 0.31/0.49 ( ( A2 = X3 )
% 0.31/0.49 & ! [Y4: A] :
% 0.31/0.49 ( ( R2 @ Y4 @ X3 )
% 0.31/0.49 => ( accp @ A @ R2 @ Y4 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % accp.simps
% 0.31/0.49 thf(fact_31_accp_Ointros,axiom,
% 0.31/0.49 ! [A: $tType,R: A > A > $o,X: A] :
% 0.31/0.49 ( ! [Y5: A] :
% 0.31/0.49 ( ( R @ Y5 @ X )
% 0.31/0.49 => ( accp @ A @ R @ Y5 ) )
% 0.31/0.49 => ( accp @ A @ R @ X ) ) ).
% 0.31/0.49
% 0.31/0.49 % accp.intros
% 0.31/0.49 thf(fact_32_accp__induct,axiom,
% 0.31/0.49 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 0.31/0.49 ( ( accp @ A @ R @ A3 )
% 0.31/0.49 => ( ! [X4: A] :
% 0.31/0.49 ( ( accp @ A @ R @ X4 )
% 0.31/0.49 => ( ! [Y3: A] :
% 0.31/0.49 ( ( R @ Y3 @ X4 )
% 0.31/0.49 => ( P @ Y3 ) )
% 0.31/0.49 => ( P @ X4 ) ) )
% 0.31/0.49 => ( P @ A3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % accp_induct
% 0.31/0.49 thf(fact_33_accp_Oinducts,axiom,
% 0.31/0.49 ! [A: $tType,R: A > A > $o,X: A,P: A > $o] :
% 0.31/0.49 ( ( accp @ A @ R @ X )
% 0.31/0.49 => ( ! [X4: A] :
% 0.31/0.49 ( ! [Y3: A] :
% 0.31/0.49 ( ( R @ Y3 @ X4 )
% 0.31/0.49 => ( accp @ A @ R @ Y3 ) )
% 0.31/0.49 => ( ! [Y3: A] :
% 0.31/0.49 ( ( R @ Y3 @ X4 )
% 0.31/0.49 => ( P @ Y3 ) )
% 0.31/0.49 => ( P @ X4 ) ) )
% 0.31/0.49 => ( P @ X ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % accp.inducts
% 0.31/0.49 thf(fact_34_dtree_Ocase,axiom,
% 0.31/0.49 ! [A: $tType,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 0.31/0.49 ( ( case_dtree @ A @ F @ ( nNode @ X1 @ X2 ) )
% 0.31/0.49 = ( F @ X1 @ X2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree.case
% 0.31/0.49 thf(fact_35_old_Oprod_Oinducts,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.31/0.49 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.31/0.49 => ( P @ Prod ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.prod.inducts
% 0.31/0.49 thf(fact_36_old_Oprod_Oexhaust,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.31/0.49 ~ ! [A5: A,B5: B] :
% 0.31/0.49 ( Y
% 0.31/0.49 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.prod.exhaust
% 0.31/0.49 thf(fact_37_prod__induct7,axiom,
% 0.31/0.49 ! [G: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
% 0.31/0.49 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F4: F3,G3: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G3 ) ) ) ) ) ) )
% 0.31/0.49 => ( P @ X ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_induct7
% 0.31/0.49 thf(fact_38_prod__induct6,axiom,
% 0.31/0.49 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.31/0.49 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 0.31/0.49 => ( P @ X ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_induct6
% 0.31/0.49 thf(fact_39_prod__induct5,axiom,
% 0.31/0.49 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.31/0.49 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.31/0.49 => ( P @ X ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_induct5
% 0.31/0.49 thf(fact_40_prod__induct4,axiom,
% 0.31/0.49 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.31/0.49 ( ! [A5: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.31/0.49 => ( P @ X ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_induct4
% 0.31/0.49 thf(fact_41_prod__induct3,axiom,
% 0.31/0.49 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.31/0.49 ( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 0.31/0.49 => ( P @ X ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_induct3
% 0.31/0.49 thf(fact_42_prod__cases7,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
% 0.31/0.49 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F4: F3,G3: G] :
% 0.31/0.49 ( Y
% 0.31/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G3 ) ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_cases7
% 0.31/0.49 thf(fact_43_prod__cases6,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.31/0.49 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F4: F3] :
% 0.31/0.49 ( Y
% 0.31/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_cases6
% 0.31/0.49 thf(fact_44_prod__cases5,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.31/0.49 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E] :
% 0.31/0.49 ( Y
% 0.31/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_cases5
% 0.31/0.49 thf(fact_45_mem__Collect__eq,axiom,
% 0.31/0.49 ! [A: $tType,A3: A,P: A > $o] :
% 0.31/0.49 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.31/0.49 = ( P @ A3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_Collect_eq
% 0.31/0.49 thf(fact_46_Collect__mem__eq,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A] :
% 0.31/0.49 ( ( collect @ A
% 0.31/0.49 @ ^ [X3: A] : ( member @ A @ X3 @ A6 ) )
% 0.31/0.49 = A6 ) ).
% 0.31/0.49
% 0.31/0.49 % Collect_mem_eq
% 0.31/0.49 thf(fact_47_Collect__cong,axiom,
% 0.31/0.49 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.49 ( ! [X4: A] :
% 0.31/0.49 ( ( P @ X4 )
% 0.31/0.49 = ( Q @ X4 ) )
% 0.31/0.49 => ( ( collect @ A @ P )
% 0.31/0.49 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Collect_cong
% 0.31/0.49 thf(fact_48_ext,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,F: A > B,G4: A > B] :
% 0.31/0.49 ( ! [X4: A] :
% 0.31/0.49 ( ( F @ X4 )
% 0.31/0.49 = ( G4 @ X4 ) )
% 0.31/0.49 => ( F = G4 ) ) ).
% 0.31/0.49
% 0.31/0.49 % ext
% 0.31/0.49 thf(fact_49_prod__cases4,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.31/0.49 ~ ! [A5: A,B5: B,C3: C,D2: D] :
% 0.31/0.49 ( Y
% 0.31/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_cases4
% 0.31/0.49 thf(fact_50_prod__cases3,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.31/0.49 ~ ! [A5: A,B5: B,C3: C] :
% 0.31/0.49 ( Y
% 0.31/0.49 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_cases3
% 0.31/0.49 thf(fact_51_Pair__inject,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.31/0.49 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.31/0.49 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.31/0.49 => ~ ( ( A3 = A4 )
% 0.31/0.49 => ( B3 != B4 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Pair_inject
% 0.31/0.49 thf(fact_52_prod__cases,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.31/0.49 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.31/0.49 => ( P @ P2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_cases
% 0.31/0.49 thf(fact_53_surj__pair,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.31/0.49 ? [X4: A,Y5: B] :
% 0.31/0.49 ( P2
% 0.31/0.49 = ( product_Pair @ A @ B @ X4 @ Y5 ) ) ).
% 0.31/0.49
% 0.31/0.49 % surj_pair
% 0.31/0.49 thf(fact_54_accp__induct__rule,axiom,
% 0.31/0.49 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 0.31/0.49 ( ( accp @ A @ R @ A3 )
% 0.31/0.49 => ( ! [X4: A] :
% 0.31/0.49 ( ( accp @ A @ R @ X4 )
% 0.31/0.49 => ( ! [Y3: A] :
% 0.31/0.49 ( ( R @ Y3 @ X4 )
% 0.31/0.49 => ( P @ Y3 ) )
% 0.31/0.49 => ( P @ X4 ) ) )
% 0.31/0.49 => ( P @ A3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % accp_induct_rule
% 0.31/0.49 thf(fact_55_not__accp__down,axiom,
% 0.31/0.49 ! [A: $tType,R3: A > A > $o,X: A] :
% 0.31/0.49 ( ~ ( accp @ A @ R3 @ X )
% 0.31/0.49 => ~ ! [Z: A] :
% 0.31/0.49 ( ( R3 @ Z @ X )
% 0.31/0.49 => ( accp @ A @ R3 @ Z ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % not_accp_down
% 0.31/0.49 thf(fact_56_accp__downward,axiom,
% 0.31/0.49 ! [A: $tType,R: A > A > $o,B3: A,A3: A] :
% 0.31/0.49 ( ( accp @ A @ R @ B3 )
% 0.31/0.49 => ( ( R @ A3 @ B3 )
% 0.31/0.49 => ( accp @ A @ R @ A3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % accp_downward
% 0.31/0.49 thf(fact_57_old_Oprod_Orec,axiom,
% 0.31/0.49 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 0.31/0.49 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.31/0.49 = ( F1 @ A3 @ B3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.prod.rec
% 0.31/0.49 thf(fact_58_in__lex__prod,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 0.31/0.49 ( ( 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 @ A3 @ B3 ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 0.31/0.49 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R )
% 0.31/0.49 | ( ( A3 = A4 )
% 0.31/0.49 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % in_lex_prod
% 0.31/0.49 thf(fact_59_internal__case__prod__conv,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
% 0.31/0.49 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.31/0.49 = ( C2 @ A3 @ B3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % internal_case_prod_conv
% 0.31/0.49 thf(fact_60_Inl__in__cont__par,axiom,
% 0.31/0.49 ! [T2: t,Tr12: dtree,Tr22: dtree] :
% 0.31/0.49 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 0.31/0.49 = ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr12 ) )
% 0.31/0.49 | ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr22 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Inl_in_cont_par
% 0.31/0.49 thf(fact_61_cont__Node,axiom,
% 0.31/0.49 ! [As: set @ ( sum_sum @ t @ dtree ),N: n] :
% 0.31/0.49 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 0.31/0.49 => ( ( cont @ ( node @ N @ As ) )
% 0.31/0.49 = As ) ) ).
% 0.31/0.49
% 0.31/0.49 % cont_Node
% 0.31/0.49 thf(fact_62_ssubst__Pair__rhs,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
% 0.31/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
% 0.31/0.49 => ( ( S2 = S )
% 0.31/0.49 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % ssubst_Pair_rhs
% 0.31/0.49 thf(fact_63_curry__conv,axiom,
% 0.31/0.49 ! [A: $tType,C: $tType,B: $tType] :
% 0.31/0.49 ( ( product_curry @ B @ C @ A )
% 0.31/0.49 = ( ^ [F2: ( product_prod @ B @ C ) > A,A2: B,B2: C] : ( F2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % curry_conv
% 0.31/0.49 thf(fact_64_finite__par__c,axiom,
% 0.31/0.49 ! [Tr12: dtree,Tr22: dtree] : ( finite_finite2 @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_par_c
% 0.31/0.49 thf(fact_65_curryI,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.31/0.49 ( ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.31/0.49 => ( product_curry @ A @ B @ $o @ F @ A3 @ B3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % curryI
% 0.31/0.49 thf(fact_66_dtree__simps,axiom,
% 0.31/0.49 ! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N: n,N2: n] :
% 0.31/0.49 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 0.31/0.49 => ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
% 0.31/0.49 => ( ( ( node @ N @ As )
% 0.31/0.49 = ( node @ N2 @ As2 ) )
% 0.31/0.49 = ( ( N = N2 )
% 0.31/0.49 & ( As = As2 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree_simps
% 0.31/0.49 thf(fact_67_finite__cont,axiom,
% 0.31/0.49 ! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_cont
% 0.31/0.49 thf(fact_68_dtree__cases,axiom,
% 0.31/0.49 ! [Tr: dtree] :
% 0.31/0.49 ~ ! [N3: n,As3: set @ ( sum_sum @ t @ dtree )] :
% 0.31/0.49 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
% 0.31/0.49 => ( Tr
% 0.31/0.49 != ( node @ N3 @ As3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % dtree_cases
% 0.31/0.49 thf(fact_69_curryD,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.31/0.49 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
% 0.31/0.49 => ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % curryD
% 0.31/0.49 thf(fact_70_curryE,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.31/0.49 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
% 0.31/0.49 => ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % curryE
% 0.31/0.49 thf(fact_71_sum_Oinject_I1_J,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 0.31/0.49 ( ( ( sum_Inl @ A @ B @ X1 )
% 0.31/0.49 = ( sum_Inl @ A @ B @ Y1 ) )
% 0.31/0.49 = ( X1 = Y1 ) ) ).
% 0.31/0.49
% 0.31/0.49 % sum.inject(1)
% 0.31/0.49 thf(fact_72_old_Osum_Oinject_I1_J,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,A3: A,A4: A] :
% 0.31/0.49 ( ( ( sum_Inl @ A @ B @ A3 )
% 0.31/0.49 = ( sum_Inl @ A @ B @ A4 ) )
% 0.31/0.49 = ( A3 = A4 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.sum.inject(1)
% 0.31/0.49 thf(fact_73_finite__code,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.31/0.49 => ( ( finite_finite2 @ A )
% 0.31/0.49 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_code
% 0.31/0.49 thf(fact_74_same__fstI,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,P: A > $o,X: A,Y6: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.31/0.49 ( ( P @ X )
% 0.31/0.49 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y ) @ ( R3 @ X ) )
% 0.31/0.49 => ( 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 @ X @ Y6 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % same_fstI
% 0.31/0.49 thf(fact_75_Inl__inject,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,X: A,Y: A] :
% 0.31/0.49 ( ( ( sum_Inl @ A @ B @ X )
% 0.31/0.49 = ( sum_Inl @ A @ B @ Y ) )
% 0.31/0.49 => ( X = Y ) ) ).
% 0.31/0.49
% 0.31/0.49 % Inl_inject
% 0.31/0.49 thf(fact_76_finite__set__choice,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,A6: set @ A,P: A > B > $o] :
% 0.31/0.49 ( ( finite_finite2 @ A @ A6 )
% 0.31/0.49 => ( ! [X4: A] :
% 0.31/0.49 ( ( member @ A @ X4 @ A6 )
% 0.31/0.49 => ? [X13: B] : ( P @ X4 @ X13 ) )
% 0.31/0.49 => ? [F4: A > B] :
% 0.31/0.49 ! [X5: A] :
% 0.31/0.49 ( ( member @ A @ X5 @ A6 )
% 0.31/0.49 => ( P @ X5 @ ( F4 @ X5 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_set_choice
% 0.31/0.49 thf(fact_77_finite,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A6: set @ A] : ( finite_finite2 @ A @ A6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite
% 0.31/0.49 thf(fact_78_old_Osum_Osimps_I7_J,axiom,
% 0.31/0.49 ! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A3: A] :
% 0.31/0.49 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A3 ) )
% 0.31/0.49 = ( F1 @ A3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.sum.simps(7)
% 0.31/0.49 thf(fact_79_Suml_Osimps,axiom,
% 0.31/0.49 ! [B: $tType,C: $tType,A: $tType,F: A > C,X: A] :
% 0.31/0.49 ( ( sum_Suml @ A @ C @ B @ F @ ( sum_Inl @ A @ B @ X ) )
% 0.31/0.49 = ( F @ X ) ) ).
% 0.31/0.49
% 0.31/0.49 % Suml.simps
% 0.31/0.49 thf(fact_80_finite__Plus__iff,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
% 0.31/0.49 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
% 0.31/0.49 = ( ( finite_finite2 @ A @ A6 )
% 0.31/0.49 & ( finite_finite2 @ B @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_Plus_iff
% 0.31/0.49 thf(fact_81_adm__wf__def,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType] :
% 0.31/0.49 ( ( adm_wf @ A @ B )
% 0.31/0.49 = ( ^ [R4: set @ ( product_prod @ A @ A ),F5: ( A > B ) > A > B] :
% 0.31/0.49 ! [F2: A > B,G5: A > B,X3: A] :
% 0.31/0.49 ( ! [Z2: A] :
% 0.31/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ X3 ) @ R4 )
% 0.31/0.49 => ( ( F2 @ Z2 )
% 0.31/0.49 = ( G5 @ Z2 ) ) )
% 0.31/0.49 => ( ( F5 @ F2 @ X3 )
% 0.31/0.49 = ( F5 @ G5 @ X3 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % adm_wf_def
% 0.31/0.49 thf(fact_82_Inl__cont__par,axiom,
% 0.31/0.49 ! [Tr12: dtree,Tr22: dtree] :
% 0.31/0.49 ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 0.31/0.49 = ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr12 ) @ ( cont @ Tr22 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Inl_cont_par
% 0.31/0.49 thf(fact_83_swap__simp,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.31/0.49 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.31/0.49 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.31/0.49
% 0.31/0.49 % swap_simp
% 0.31/0.49 thf(fact_84_swap__swap,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.31/0.49 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.31/0.49 = P2 ) ).
% 0.31/0.49
% 0.31/0.49 % swap_swap
% 0.31/0.49 thf(fact_85_finite__Un,axiom,
% 0.31/0.49 ! [A: $tType,F6: set @ A,G6: set @ A] :
% 0.31/0.49 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F6 @ G6 ) )
% 0.31/0.49 = ( ( finite_finite2 @ A @ F6 )
% 0.31/0.49 & ( finite_finite2 @ A @ G6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_Un
% 0.31/0.49 thf(fact_86_InlI,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A3: A,A6: set @ A,B6: set @ B] :
% 0.31/0.49 ( ( member @ A @ A3 @ A6 )
% 0.31/0.49 => ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A3 ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % InlI
% 0.31/0.49 thf(fact_87_Suml__inject,axiom,
% 0.31/0.49 ! [B: $tType,C: $tType,A: $tType,F: A > C,G4: A > C] :
% 0.31/0.49 ( ( ( sum_Suml @ A @ C @ B @ F )
% 0.31/0.49 = ( sum_Suml @ A @ C @ B @ G4 ) )
% 0.31/0.49 => ( F = G4 ) ) ).
% 0.31/0.49
% 0.31/0.49 % Suml_inject
% 0.31/0.49 thf(fact_88_infinite__Un,axiom,
% 0.31/0.49 ! [A: $tType,S3: set @ A,T3: set @ A] :
% 0.31/0.49 ( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T3 ) ) )
% 0.31/0.49 = ( ~ ( finite_finite2 @ A @ S3 )
% 0.31/0.49 | ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % infinite_Un
% 0.31/0.49 thf(fact_89_Un__infinite,axiom,
% 0.31/0.49 ! [A: $tType,S3: set @ A,T3: set @ A] :
% 0.31/0.49 ( ~ ( finite_finite2 @ A @ S3 )
% 0.31/0.49 => ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_infinite
% 0.31/0.49 thf(fact_90_finite__UnI,axiom,
% 0.31/0.49 ! [A: $tType,F6: set @ A,G6: set @ A] :
% 0.31/0.49 ( ( finite_finite2 @ A @ F6 )
% 0.31/0.49 => ( ( finite_finite2 @ A @ G6 )
% 0.31/0.49 => ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F6 @ G6 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_UnI
% 0.31/0.49 thf(fact_91_finite__PlusD_I2_J,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
% 0.31/0.49 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
% 0.31/0.49 => ( finite_finite2 @ B @ B6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_PlusD(2)
% 0.31/0.49 thf(fact_92_finite__PlusD_I1_J,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
% 0.31/0.49 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
% 0.31/0.49 => ( finite_finite2 @ A @ A6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_PlusD(1)
% 0.31/0.49 thf(fact_93_finite__Plus,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
% 0.31/0.49 ( ( finite_finite2 @ A @ A6 )
% 0.31/0.49 => ( ( finite_finite2 @ B @ B6 )
% 0.31/0.49 => ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_Plus
% 0.31/0.49 thf(fact_94_vimage__Un,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,F: A > B,A6: set @ B,B6: set @ B] :
% 0.31/0.49 ( ( vimage @ A @ B @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
% 0.31/0.49 = ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ ( vimage @ A @ B @ F @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimage_Un
% 0.31/0.49 thf(fact_95_vimageI,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,F: B > A,A3: B,B3: A,B6: set @ A] :
% 0.31/0.49 ( ( ( F @ A3 )
% 0.31/0.49 = B3 )
% 0.31/0.49 => ( ( member @ A @ B3 @ B6 )
% 0.31/0.49 => ( member @ B @ A3 @ ( vimage @ B @ A @ F @ B6 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimageI
% 0.31/0.49 thf(fact_96_vimage__eq,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A3: A,F: A > B,B6: set @ B] :
% 0.31/0.49 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F @ B6 ) )
% 0.31/0.49 = ( member @ B @ ( F @ A3 ) @ B6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimage_eq
% 0.31/0.49 thf(fact_97_UnCI,axiom,
% 0.31/0.49 ! [A: $tType,C2: A,B6: set @ A,A6: set @ A] :
% 0.31/0.49 ( ( ~ ( member @ A @ C2 @ B6 )
% 0.31/0.49 => ( member @ A @ C2 @ A6 ) )
% 0.31/0.49 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % UnCI
% 0.31/0.49 thf(fact_98_Un__iff,axiom,
% 0.31/0.49 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 0.31/0.49 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 0.31/0.49 = ( ( member @ A @ C2 @ A6 )
% 0.31/0.49 | ( member @ A @ C2 @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_iff
% 0.31/0.49 thf(fact_99_sup__apply,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.31/0.49 => ( ( sup_sup @ ( A > B ) )
% 0.31/0.49 = ( ^ [F2: A > B,G5: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G5 @ X3 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_apply
% 0.31/0.49 thf(fact_100_sup_Oright__idem,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A,B3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
% 0.31/0.49 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.right_idem
% 0.31/0.49 thf(fact_101_sup__left__idem,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.31/0.49 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_left_idem
% 0.31/0.49 thf(fact_102_sup_Oleft__idem,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A,B3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
% 0.31/0.49 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.left_idem
% 0.31/0.49 thf(fact_103_sup__idem,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X @ X )
% 0.31/0.49 = X ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_idem
% 0.31/0.49 thf(fact_104_sup_Oidem,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ A3 @ A3 )
% 0.31/0.49 = A3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.idem
% 0.31/0.49 thf(fact_105_sup__left__commute,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
% 0.31/0.49 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_left_commute
% 0.31/0.49 thf(fact_106_sup_Oleft__commute,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [B3: A,A3: A,C2: A] :
% 0.31/0.49 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C2 ) )
% 0.31/0.49 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.left_commute
% 0.31/0.49 thf(fact_107_sup__commute,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ( ( sup_sup @ A )
% 0.31/0.49 = ( ^ [X3: A,Y4: A] : ( sup_sup @ A @ Y4 @ X3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_commute
% 0.31/0.49 thf(fact_108_sup_Ocommute,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ( ( sup_sup @ A )
% 0.31/0.49 = ( ^ [A2: A,B2: A] : ( sup_sup @ A @ B2 @ A2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.commute
% 0.31/0.49 thf(fact_109_sup__assoc,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
% 0.31/0.49 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_assoc
% 0.31/0.49 thf(fact_110_sup_Oassoc,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A,B3: A,C2: A] :
% 0.31/0.49 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C2 )
% 0.31/0.49 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.assoc
% 0.31/0.49 thf(fact_111_sup__fun__def,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 0.31/0.49 => ( ( sup_sup @ ( A > B ) )
% 0.31/0.49 = ( ^ [F2: A > B,G5: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G5 @ X3 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_fun_def
% 0.31/0.49 thf(fact_112_inf__sup__aci_I5_J,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ( ( sup_sup @ A )
% 0.31/0.49 = ( ^ [X3: A,Y4: A] : ( sup_sup @ A @ Y4 @ X3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % inf_sup_aci(5)
% 0.31/0.49 thf(fact_113_inf__sup__aci_I6_J,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
% 0.31/0.49 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % inf_sup_aci(6)
% 0.31/0.49 thf(fact_114_inf__sup__aci_I7_J,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
% 0.31/0.49 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % inf_sup_aci(7)
% 0.31/0.49 thf(fact_115_inf__sup__aci_I8_J,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.31/0.49 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % inf_sup_aci(8)
% 0.31/0.49 thf(fact_116_Un__left__commute,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B6 @ C4 ) )
% 0.31/0.49 = ( sup_sup @ ( set @ A ) @ B6 @ ( sup_sup @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_left_commute
% 0.31/0.49 thf(fact_117_Un__left__absorb,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 0.31/0.49 = ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_left_absorb
% 0.31/0.49 thf(fact_118_Un__commute,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) )
% 0.31/0.49 = ( ^ [A7: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_commute
% 0.31/0.49 thf(fact_119_Un__absorb,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) @ A6 @ A6 )
% 0.31/0.49 = A6 ) ).
% 0.31/0.49
% 0.31/0.49 % Un_absorb
% 0.31/0.49 thf(fact_120_Un__assoc,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) @ C4 )
% 0.31/0.49 = ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B6 @ C4 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_assoc
% 0.31/0.49 thf(fact_121_ball__Un,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A,P: A > $o] :
% 0.31/0.49 ( ( ! [X3: A] :
% 0.31/0.49 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 0.31/0.49 => ( P @ X3 ) ) )
% 0.31/0.49 = ( ! [X3: A] :
% 0.31/0.49 ( ( member @ A @ X3 @ A6 )
% 0.31/0.49 => ( P @ X3 ) )
% 0.31/0.49 & ! [X3: A] :
% 0.31/0.49 ( ( member @ A @ X3 @ B6 )
% 0.31/0.49 => ( P @ X3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % ball_Un
% 0.31/0.49 thf(fact_122_bex__Un,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A,P: A > $o] :
% 0.31/0.49 ( ( ? [X3: A] :
% 0.31/0.49 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 0.31/0.49 & ( P @ X3 ) ) )
% 0.31/0.49 = ( ? [X3: A] :
% 0.31/0.49 ( ( member @ A @ X3 @ A6 )
% 0.31/0.49 & ( P @ X3 ) )
% 0.31/0.49 | ? [X3: A] :
% 0.31/0.49 ( ( member @ A @ X3 @ B6 )
% 0.31/0.49 & ( P @ X3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % bex_Un
% 0.31/0.49 thf(fact_123_UnI2,axiom,
% 0.31/0.49 ! [A: $tType,C2: A,B6: set @ A,A6: set @ A] :
% 0.31/0.49 ( ( member @ A @ C2 @ B6 )
% 0.31/0.49 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % UnI2
% 0.31/0.49 thf(fact_124_UnI1,axiom,
% 0.31/0.49 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 0.31/0.49 ( ( member @ A @ C2 @ A6 )
% 0.31/0.49 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % UnI1
% 0.31/0.49 thf(fact_125_UnE,axiom,
% 0.31/0.49 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 0.31/0.49 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 0.31/0.49 => ( ~ ( member @ A @ C2 @ A6 )
% 0.31/0.49 => ( member @ A @ C2 @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % UnE
% 0.31/0.49 thf(fact_126_vimage__Collect,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
% 0.31/0.49 ( ! [X4: A] :
% 0.31/0.49 ( ( P @ ( F @ X4 ) )
% 0.31/0.49 = ( Q @ X4 ) )
% 0.31/0.49 => ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 0.31/0.49 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimage_Collect
% 0.31/0.49 thf(fact_127_vimageI2,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,F: B > A,A3: B,A6: set @ A] :
% 0.31/0.49 ( ( member @ A @ ( F @ A3 ) @ A6 )
% 0.31/0.49 => ( member @ B @ A3 @ ( vimage @ B @ A @ F @ A6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimageI2
% 0.31/0.49 thf(fact_128_vimageE,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A3: A,F: A > B,B6: set @ B] :
% 0.31/0.49 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F @ B6 ) )
% 0.31/0.49 => ( member @ B @ ( F @ A3 ) @ B6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimageE
% 0.31/0.49 thf(fact_129_vimageD,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A3: A,F: A > B,A6: set @ B] :
% 0.31/0.49 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F @ A6 ) )
% 0.31/0.49 => ( member @ B @ ( F @ A3 ) @ A6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimageD
% 0.31/0.49 thf(fact_130_pair__in__swap__image,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,Y: A,X: B,A6: set @ ( product_prod @ B @ A )] :
% 0.31/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
% 0.31/0.49 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % pair_in_swap_image
% 0.31/0.49 thf(fact_131_internal__case__prod__def,axiom,
% 0.31/0.49 ! [C: $tType,B: $tType,A: $tType] :
% 0.31/0.49 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.31/0.49 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.31/0.49
% 0.31/0.49 % internal_case_prod_def
% 0.31/0.49 thf(fact_132_cut__apply,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,X: A,A3: A,R3: set @ ( product_prod @ A @ A ),F: A > B] :
% 0.31/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R3 )
% 0.31/0.49 => ( ( cut @ A @ B @ F @ R3 @ A3 @ X )
% 0.31/0.49 = ( F @ X ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % cut_apply
% 0.31/0.49 thf(fact_133_cuts__eq,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,F: A > B,R3: set @ ( product_prod @ A @ A ),X: A,G4: A > B] :
% 0.31/0.49 ( ( ( cut @ A @ B @ F @ R3 @ X )
% 0.31/0.49 = ( cut @ A @ B @ G4 @ R3 @ X ) )
% 0.31/0.49 = ( ! [Y4: A] :
% 0.31/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X ) @ R3 )
% 0.31/0.49 => ( ( F @ Y4 )
% 0.31/0.49 = ( G4 @ Y4 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % cuts_eq
% 0.31/0.49 thf(fact_134_old_Osum_Osimps_I8_J,axiom,
% 0.31/0.49 ! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B3: B] :
% 0.31/0.49 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B3 ) )
% 0.31/0.49 = ( F22 @ B3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.sum.simps(8)
% 0.31/0.49 thf(fact_135_old_Osum_Oinject_I2_J,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,B3: B,B4: B] :
% 0.31/0.49 ( ( ( sum_Inr @ B @ A @ B3 )
% 0.31/0.49 = ( sum_Inr @ B @ A @ B4 ) )
% 0.31/0.49 = ( B3 = B4 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.sum.inject(2)
% 0.31/0.49 thf(fact_136_sum_Oinject_I2_J,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,X2: B,Y2: B] :
% 0.31/0.49 ( ( ( sum_Inr @ B @ A @ X2 )
% 0.31/0.49 = ( sum_Inr @ B @ A @ Y2 ) )
% 0.31/0.49 = ( X2 = Y2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % sum.inject(2)
% 0.31/0.49 thf(fact_137_finite__imageI,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,F6: set @ A,H: A > B] :
% 0.31/0.49 ( ( finite_finite2 @ A @ F6 )
% 0.31/0.49 => ( finite_finite2 @ B @ ( image @ A @ B @ H @ F6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_imageI
% 0.31/0.49 thf(fact_138_InrI,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,B3: A,B6: set @ A,A6: set @ B] :
% 0.31/0.49 ( ( member @ A @ B3 @ B6 )
% 0.31/0.49 => ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B3 ) @ ( sum_Plus @ B @ A @ A6 @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % InrI
% 0.31/0.49 thf(fact_139_curry__case__prod,axiom,
% 0.31/0.49 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 0.31/0.49 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 0.31/0.49 = F ) ).
% 0.31/0.49
% 0.31/0.49 % curry_case_prod
% 0.31/0.49 thf(fact_140_case__prod__curry,axiom,
% 0.31/0.49 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.31/0.49 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 0.31/0.49 = F ) ).
% 0.31/0.49
% 0.31/0.49 % case_prod_curry
% 0.31/0.49 thf(fact_141_image__Un,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,F: B > A,A6: set @ B,B6: set @ B] :
% 0.31/0.49 ( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
% 0.31/0.49 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A6 ) @ ( image @ B @ A @ F @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % image_Un
% 0.31/0.49 thf(fact_142_sum_Odistinct_I1_J,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,X1: A,X2: B] :
% 0.31/0.49 ( ( sum_Inl @ A @ B @ X1 )
% 0.31/0.49 != ( sum_Inr @ B @ A @ X2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % sum.distinct(1)
% 0.31/0.49 thf(fact_143_old_Osum_Odistinct_I2_J,axiom,
% 0.31/0.49 ! [B8: $tType,A8: $tType,B9: B8,A9: A8] :
% 0.31/0.49 ( ( sum_Inr @ B8 @ A8 @ B9 )
% 0.31/0.49 != ( sum_Inl @ A8 @ B8 @ A9 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.sum.distinct(2)
% 0.31/0.49 thf(fact_144_old_Osum_Odistinct_I1_J,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A3: A,B4: B] :
% 0.31/0.49 ( ( sum_Inl @ A @ B @ A3 )
% 0.31/0.49 != ( sum_Inr @ B @ A @ B4 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.sum.distinct(1)
% 0.31/0.49 thf(fact_145_sumE,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 0.31/0.49 ( ! [X4: A] :
% 0.31/0.49 ( S
% 0.31/0.49 != ( sum_Inl @ A @ B @ X4 ) )
% 0.31/0.49 => ~ ! [Y5: B] :
% 0.31/0.49 ( S
% 0.31/0.49 != ( sum_Inr @ B @ A @ Y5 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sumE
% 0.31/0.49 thf(fact_146_Inr__not__Inl,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,B3: B,A3: A] :
% 0.31/0.49 ( ( sum_Inr @ B @ A @ B3 )
% 0.31/0.49 != ( sum_Inl @ A @ B @ A3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % Inr_not_Inl
% 0.31/0.49 thf(fact_147_split__sum__ex,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType] :
% 0.31/0.49 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.31/0.49 ? [X6: sum_sum @ A @ B] : ( P3 @ X6 ) )
% 0.31/0.49 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 0.31/0.49 ( ? [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
% 0.31/0.49 | ? [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % split_sum_ex
% 0.31/0.49 thf(fact_148_split__sum__all,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType] :
% 0.31/0.49 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.31/0.49 ! [X6: sum_sum @ A @ B] : ( P3 @ X6 ) )
% 0.31/0.49 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 0.31/0.49 ( ! [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
% 0.31/0.49 & ! [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % split_sum_all
% 0.31/0.49 thf(fact_149_old_Osum_Oexhaust,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
% 0.31/0.49 ( ! [A5: A] :
% 0.31/0.49 ( Y
% 0.31/0.49 != ( sum_Inl @ A @ B @ A5 ) )
% 0.31/0.49 => ~ ! [B5: B] :
% 0.31/0.49 ( Y
% 0.31/0.49 != ( sum_Inr @ B @ A @ B5 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.sum.exhaust
% 0.31/0.49 thf(fact_150_old_Osum_Oinducts,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 0.31/0.49 ( ! [A5: A] : ( P @ ( sum_Inl @ A @ B @ A5 ) )
% 0.31/0.49 => ( ! [B5: B] : ( P @ ( sum_Inr @ B @ A @ B5 ) )
% 0.31/0.49 => ( P @ Sum ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.sum.inducts
% 0.31/0.49 thf(fact_151_Inr__inject,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,X: B,Y: B] :
% 0.31/0.49 ( ( ( sum_Inr @ B @ A @ X )
% 0.31/0.49 = ( sum_Inr @ B @ A @ Y ) )
% 0.31/0.49 => ( X = Y ) ) ).
% 0.31/0.49
% 0.31/0.49 % Inr_inject
% 0.31/0.49 thf(fact_152_mem__case__prodE,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,C: $tType,Z3: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.31/0.49 ( ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
% 0.31/0.49 => ~ ! [X4: B,Y5: C] :
% 0.31/0.49 ( ( P2
% 0.31/0.49 = ( product_Pair @ B @ C @ X4 @ Y5 ) )
% 0.31/0.49 => ~ ( member @ A @ Z3 @ ( C2 @ X4 @ Y5 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % mem_case_prodE
% 0.31/0.49 thf(fact_153_old_Oprod_Ocase,axiom,
% 0.31/0.49 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X2: B] :
% 0.31/0.49 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.31/0.49 = ( F @ X1 @ X2 ) ) ).
% 0.31/0.49
% 0.31/0.49 % old.prod.case
% 0.31/0.49 thf(fact_154_PlusE,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,U: sum_sum @ A @ B,A6: set @ A,B6: set @ B] :
% 0.31/0.49 ( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
% 0.31/0.49 => ( ! [X4: A] :
% 0.31/0.49 ( ( member @ A @ X4 @ A6 )
% 0.31/0.49 => ( U
% 0.31/0.49 != ( sum_Inl @ A @ B @ X4 ) ) )
% 0.31/0.49 => ~ ! [Y5: B] :
% 0.31/0.49 ( ( member @ B @ Y5 @ B6 )
% 0.31/0.49 => ( U
% 0.31/0.49 != ( sum_Inr @ B @ A @ Y5 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % PlusE
% 0.31/0.49 thf(fact_155_Inl__Inr__False,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,X: A,Y: B] :
% 0.31/0.49 ( ( sum_Inl @ A @ B @ X )
% 0.31/0.49 != ( sum_Inr @ B @ A @ Y ) ) ).
% 0.31/0.49
% 0.31/0.49 % Inl_Inr_False
% 0.31/0.49 thf(fact_156_Inr__Inl__False,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,X: B,Y: A] :
% 0.31/0.49 ( ( sum_Inr @ B @ A @ X )
% 0.31/0.49 != ( sum_Inl @ A @ B @ Y ) ) ).
% 0.31/0.49
% 0.31/0.49 % Inr_Inl_False
% 0.31/0.49 thf(fact_157_Inl__Inr__image__cong,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,A6: set @ ( sum_sum @ A @ B ),B6: set @ ( sum_sum @ A @ B )] :
% 0.31/0.49 ( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A6 )
% 0.31/0.49 = ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B6 ) )
% 0.31/0.49 => ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A6 )
% 0.31/0.49 = ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B6 ) )
% 0.31/0.49 => ( A6 = B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Inl_Inr_image_cong
% 0.31/0.49 thf(fact_158_obj__sumE__f,axiom,
% 0.31/0.49 ! [A: $tType,C: $tType,B: $tType,S: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
% 0.31/0.49 ( ! [X4: A] :
% 0.31/0.49 ( ( S
% 0.31/0.49 = ( F @ ( sum_Inl @ A @ C @ X4 ) ) )
% 0.31/0.49 => P )
% 0.31/0.49 => ( ! [X4: C] :
% 0.31/0.49 ( ( S
% 0.31/0.49 = ( F @ ( sum_Inr @ C @ A @ X4 ) ) )
% 0.31/0.49 => P )
% 0.31/0.49 => ! [X5: sum_sum @ A @ C] :
% 0.31/0.49 ( ( S
% 0.31/0.49 = ( F @ X5 ) )
% 0.31/0.49 => P ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % obj_sumE_f
% 0.31/0.49 thf(fact_159_Plus__def,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType] :
% 0.31/0.49 ( ( sum_Plus @ A @ B )
% 0.31/0.49 = ( ^ [A7: set @ A,B7: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A7 ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B7 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Plus_def
% 0.31/0.49 thf(fact_160_obj__sumE,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 0.31/0.49 ( ! [X4: A] :
% 0.31/0.49 ( S
% 0.31/0.49 != ( sum_Inl @ A @ B @ X4 ) )
% 0.31/0.49 => ~ ! [X4: B] :
% 0.31/0.49 ( S
% 0.31/0.49 != ( sum_Inr @ B @ A @ X4 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % obj_sumE
% 0.31/0.49 thf(fact_161_split__cong,axiom,
% 0.31/0.49 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G4: A > B > C,P2: product_prod @ A @ B] :
% 0.31/0.49 ( ! [X4: A,Y5: B] :
% 0.31/0.49 ( ( ( product_Pair @ A @ B @ X4 @ Y5 )
% 0.31/0.49 = Q2 )
% 0.31/0.49 => ( ( F @ X4 @ Y5 )
% 0.31/0.49 = ( G4 @ X4 @ Y5 ) ) )
% 0.31/0.49 => ( ( P2 = Q2 )
% 0.31/0.49 => ( ( product_case_prod @ A @ B @ C @ F @ P2 )
% 0.31/0.49 = ( product_case_prod @ A @ B @ C @ G4 @ Q2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % split_cong
% 0.31/0.49 thf(fact_162_Sumr_Osimps,axiom,
% 0.31/0.49 ! [A: $tType,C: $tType,B: $tType,F: B > C,X: B] :
% 0.31/0.49 ( ( sum_Sumr @ B @ C @ A @ F @ ( sum_Inr @ B @ A @ X ) )
% 0.31/0.49 = ( F @ X ) ) ).
% 0.31/0.49
% 0.31/0.49 % Sumr.simps
% 0.31/0.49 thf(fact_163_Sumr__inject,axiom,
% 0.31/0.49 ! [A: $tType,C: $tType,B: $tType,F: B > C,G4: B > C] :
% 0.31/0.49 ( ( ( sum_Sumr @ B @ C @ A @ F )
% 0.31/0.49 = ( sum_Sumr @ B @ C @ A @ G4 ) )
% 0.31/0.49 => ( F = G4 ) ) ).
% 0.31/0.49
% 0.31/0.49 % Sumr_inject
% 0.31/0.49 thf(fact_164_image2__eqI,axiom,
% 0.31/0.49 ! [A: $tType,C: $tType,B: $tType,B3: A,F: B > A,X: B,C2: C,G4: B > C,A6: set @ B] :
% 0.31/0.49 ( ( B3
% 0.31/0.49 = ( F @ X ) )
% 0.31/0.49 => ( ( C2
% 0.31/0.49 = ( G4 @ X ) )
% 0.31/0.49 => ( ( member @ B @ X @ A6 )
% 0.31/0.49 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F @ G4 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % image2_eqI
% 0.31/0.49 thf(fact_165_surj__swap,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType] :
% 0.31/0.49 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.31/0.49 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % surj_swap
% 0.31/0.49 thf(fact_166_finite__vimageD,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,H: A > B,F6: set @ B] :
% 0.31/0.49 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H @ F6 ) )
% 0.31/0.49 => ( ( ( image @ A @ B @ H @ ( top_top @ ( set @ A ) ) )
% 0.31/0.49 = ( top_top @ ( set @ B ) ) )
% 0.31/0.49 => ( finite_finite2 @ B @ F6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_vimageD
% 0.31/0.49 thf(fact_167_sup__top__left,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A] :
% 0.31/0.49 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 0.31/0.49 = ( top_top @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_top_left
% 0.31/0.49 thf(fact_168_sup__top__right,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 0.31/0.49 = ( top_top @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_top_right
% 0.31/0.49 thf(fact_169_vimage__UNIV,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,F: A > B] :
% 0.31/0.49 ( ( vimage @ A @ B @ F @ ( top_top @ ( set @ B ) ) )
% 0.31/0.49 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimage_UNIV
% 0.31/0.49 thf(fact_170_UNIV__Plus__UNIV,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType] :
% 0.31/0.49 ( ( sum_Plus @ A @ B @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.31/0.49 = ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % UNIV_Plus_UNIV
% 0.31/0.49 thf(fact_171_finite__Plus__UNIV__iff,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType] :
% 0.31/0.49 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 0.31/0.49 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.31/0.49 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_Plus_UNIV_iff
% 0.31/0.49 thf(fact_172_Un__UNIV__left,axiom,
% 0.31/0.49 ! [A: $tType,B6: set @ A] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B6 )
% 0.31/0.49 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_UNIV_left
% 0.31/0.49 thf(fact_173_Un__UNIV__right,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
% 0.31/0.49 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_UNIV_right
% 0.31/0.49 thf(fact_174_finite__UNIV,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.31/0.49 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_UNIV
% 0.31/0.49 thf(fact_175_infinite__UNIV__char__0,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.31/0.49 => ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % infinite_UNIV_char_0
% 0.31/0.49 thf(fact_176_ex__new__if__finite,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A] :
% 0.31/0.49 ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.31/0.49 => ( ( finite_finite2 @ A @ A6 )
% 0.31/0.49 => ? [A5: A] :
% 0.31/0.49 ~ ( member @ A @ A5 @ A6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % ex_new_if_finite
% 0.31/0.49 thf(fact_177_Finite__Set_Ofinite__set,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 0.31/0.49 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Finite_Set.finite_set
% 0.31/0.49 thf(fact_178_finite__prod,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType] :
% 0.31/0.49 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.31/0.49 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.31/0.49 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_prod
% 0.31/0.49 thf(fact_179_finite__Prod__UNIV,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType] :
% 0.31/0.49 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.31/0.49 => ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
% 0.31/0.49 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_Prod_UNIV
% 0.31/0.49 thf(fact_180_finite__fun__UNIVD2,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType] :
% 0.31/0.49 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
% 0.31/0.49 => ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_fun_UNIVD2
% 0.31/0.49 thf(fact_181_UNIV__sum,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType] :
% 0.31/0.49 ( ( top_top @ ( set @ ( sum_sum @ A @ B ) ) )
% 0.31/0.49 = ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( top_top @ ( set @ A ) ) ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % UNIV_sum
% 0.31/0.49 thf(fact_182_finite__option__UNIV,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
% 0.31/0.49 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_option_UNIV
% 0.31/0.49 thf(fact_183_surj__image__vimage__eq,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
% 0.31/0.49 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.31/0.49 = ( top_top @ ( set @ A ) ) )
% 0.31/0.49 => ( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A6 ) )
% 0.31/0.49 = A6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % surj_image_vimage_eq
% 0.31/0.49 thf(fact_184_map__prod__surj,axiom,
% 0.31/0.49 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G4: C > D] :
% 0.31/0.49 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.31/0.49 = ( top_top @ ( set @ B ) ) )
% 0.31/0.49 => ( ( ( image @ C @ D @ G4 @ ( top_top @ ( set @ C ) ) )
% 0.31/0.49 = ( top_top @ ( set @ D ) ) )
% 0.31/0.49 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G4 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.31/0.49 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % map_prod_surj
% 0.31/0.49 thf(fact_185_map__prod__simp,axiom,
% 0.31/0.49 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G4: D > B,A3: C,B3: D] :
% 0.31/0.49 ( ( product_map_prod @ C @ A @ D @ B @ F @ G4 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 0.31/0.49 = ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G4 @ B3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % map_prod_simp
% 0.31/0.49 thf(fact_186_map__prod__imageI,axiom,
% 0.31/0.49 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B ),F: A > C,G4: B > D] :
% 0.31/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 0.31/0.49 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G4 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G4 ) @ R3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % map_prod_imageI
% 0.31/0.49 thf(fact_187_prod__fun__imageE,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G4: D > B,R3: set @ ( product_prod @ C @ D )] :
% 0.31/0.49 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G4 ) @ R3 ) )
% 0.31/0.49 => ~ ! [X4: C,Y5: D] :
% 0.31/0.49 ( ( C2
% 0.31/0.49 = ( product_Pair @ A @ B @ ( F @ X4 ) @ ( G4 @ Y5 ) ) )
% 0.31/0.49 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X4 @ Y5 ) @ R3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % prod_fun_imageE
% 0.31/0.49 thf(fact_188_finite__vimageD_H,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
% 0.31/0.49 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ F @ A6 ) )
% 0.31/0.49 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 0.31/0.49 => ( finite_finite2 @ B @ A6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_vimageD'
% 0.31/0.49 thf(fact_189_image__vimage__eq,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,F: B > A,A6: set @ A] :
% 0.31/0.49 ( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A6 ) )
% 0.31/0.49 = ( inf_inf @ ( set @ A ) @ A6 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % image_vimage_eq
% 0.31/0.49 thf(fact_190_surj__vimage__empty,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
% 0.31/0.49 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.31/0.49 = ( top_top @ ( set @ A ) ) )
% 0.31/0.49 => ( ( ( vimage @ B @ A @ F @ A6 )
% 0.31/0.49 = ( bot_bot @ ( set @ B ) ) )
% 0.31/0.49 = ( A6
% 0.31/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % surj_vimage_empty
% 0.31/0.49 thf(fact_191_le__sup__iff,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
% 0.31/0.49 = ( ( ord_less_eq @ A @ X @ Z3 )
% 0.31/0.49 & ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % le_sup_iff
% 0.31/0.49 thf(fact_192_sup_Obounded__iff,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [B3: A,C2: A,A3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 0.31/0.49 = ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.49 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.bounded_iff
% 0.31/0.49 thf(fact_193_bot__eq__sup__iff,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A] :
% 0.31/0.49 ( ( ( bot_bot @ A )
% 0.31/0.49 = ( sup_sup @ A @ X @ Y ) )
% 0.31/0.49 = ( ( X
% 0.31/0.49 = ( bot_bot @ A ) )
% 0.31/0.49 & ( Y
% 0.31/0.49 = ( bot_bot @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % bot_eq_sup_iff
% 0.31/0.49 thf(fact_194_sup__eq__bot__iff,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A] :
% 0.31/0.49 ( ( ( sup_sup @ A @ X @ Y )
% 0.31/0.49 = ( bot_bot @ A ) )
% 0.31/0.49 = ( ( X
% 0.31/0.49 = ( bot_bot @ A ) )
% 0.31/0.49 & ( Y
% 0.31/0.49 = ( bot_bot @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_eq_bot_iff
% 0.31/0.49 thf(fact_195_sup__bot_Oleft__neutral,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
% 0.31/0.49 = A3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_bot.left_neutral
% 0.31/0.49 thf(fact_196_sup__bot_Oright__neutral,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
% 0.31/0.49 = A3 ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_bot.right_neutral
% 0.31/0.49 thf(fact_197_inf__sup__absorb,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A] :
% 0.31/0.49 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 0.31/0.49 = X ) ) ).
% 0.31/0.49
% 0.31/0.49 % inf_sup_absorb
% 0.31/0.49 thf(fact_198_sup__inf__absorb,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 0.31/0.49 = X ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_inf_absorb
% 0.31/0.49 thf(fact_199_finite__Int,axiom,
% 0.31/0.49 ! [A: $tType,F6: set @ A,G6: set @ A] :
% 0.31/0.49 ( ( ( finite_finite2 @ A @ F6 )
% 0.31/0.49 | ( finite_finite2 @ A @ G6 ) )
% 0.31/0.49 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F6 @ G6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_Int
% 0.31/0.49 thf(fact_200_Un__empty,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.31/0.49 ( ( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
% 0.31/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.49 = ( ( A6
% 0.31/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.49 & ( B6
% 0.31/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_empty
% 0.31/0.49 thf(fact_201_Un__subset__iff,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 0.31/0.49 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) @ C4 )
% 0.31/0.49 = ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 0.31/0.49 & ( ord_less_eq @ ( set @ A ) @ B6 @ C4 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_subset_iff
% 0.31/0.49 thf(fact_202_vimage__empty,axiom,
% 0.31/0.49 ! [B: $tType,A: $tType,F: A > B] :
% 0.31/0.49 ( ( vimage @ A @ B @ F @ ( bot_bot @ ( set @ B ) ) )
% 0.31/0.49 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimage_empty
% 0.31/0.49 thf(fact_203_vimage__Int,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,F: A > B,A6: set @ B,B6: set @ B] :
% 0.31/0.49 ( ( vimage @ A @ B @ F @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) )
% 0.31/0.49 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ ( vimage @ A @ B @ F @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % vimage_Int
% 0.31/0.49 thf(fact_204_Plus__eq__empty__conv,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
% 0.31/0.49 ( ( ( sum_Plus @ A @ B @ A6 @ B6 )
% 0.31/0.49 = ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 0.31/0.49 = ( ( A6
% 0.31/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.49 & ( B6
% 0.31/0.49 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Plus_eq_empty_conv
% 0.31/0.49 thf(fact_205_finite_OemptyI,axiom,
% 0.31/0.49 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite.emptyI
% 0.31/0.49 thf(fact_206_infinite__imp__nonempty,axiom,
% 0.31/0.49 ! [A: $tType,S3: set @ A] :
% 0.31/0.49 ( ~ ( finite_finite2 @ A @ S3 )
% 0.31/0.49 => ( S3
% 0.31/0.49 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % infinite_imp_nonempty
% 0.31/0.49 thf(fact_207_finite__subset,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.31/0.49 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.31/0.49 => ( ( finite_finite2 @ A @ B6 )
% 0.31/0.49 => ( finite_finite2 @ A @ A6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_subset
% 0.31/0.49 thf(fact_208_infinite__super,axiom,
% 0.31/0.49 ! [A: $tType,S3: set @ A,T3: set @ A] :
% 0.31/0.49 ( ( ord_less_eq @ ( set @ A ) @ S3 @ T3 )
% 0.31/0.49 => ( ~ ( finite_finite2 @ A @ S3 )
% 0.31/0.49 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % infinite_super
% 0.31/0.49 thf(fact_209_rev__finite__subset,axiom,
% 0.31/0.49 ! [A: $tType,B6: set @ A,A6: set @ A] :
% 0.31/0.49 ( ( finite_finite2 @ A @ B6 )
% 0.31/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.31/0.49 => ( finite_finite2 @ A @ A6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % rev_finite_subset
% 0.31/0.49 thf(fact_210_finite__subset__image,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,B6: set @ A,F: B > A,A6: set @ B] :
% 0.31/0.49 ( ( finite_finite2 @ A @ B6 )
% 0.31/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F @ A6 ) )
% 0.31/0.49 => ? [C5: set @ B] :
% 0.31/0.49 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
% 0.31/0.49 & ( finite_finite2 @ B @ C5 )
% 0.31/0.49 & ( B6
% 0.31/0.49 = ( image @ B @ A @ F @ C5 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_subset_image
% 0.31/0.49 thf(fact_211_finite__surj,axiom,
% 0.31/0.49 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,F: A > B] :
% 0.31/0.49 ( ( finite_finite2 @ A @ A6 )
% 0.31/0.49 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F @ A6 ) )
% 0.31/0.49 => ( finite_finite2 @ B @ B6 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % finite_surj
% 0.31/0.49 thf(fact_212_Un__Int__assoc__eq,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 0.31/0.49 ( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ C4 )
% 0.31/0.49 = ( inf_inf @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B6 @ C4 ) ) )
% 0.31/0.49 = ( ord_less_eq @ ( set @ A ) @ C4 @ A6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_Int_assoc_eq
% 0.31/0.49 thf(fact_213_distrib__imp1,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.49 ( ! [X4: A,Y5: A,Z: A] :
% 0.31/0.49 ( ( inf_inf @ A @ X4 @ ( sup_sup @ A @ Y5 @ Z ) )
% 0.31/0.49 = ( sup_sup @ A @ ( inf_inf @ A @ X4 @ Y5 ) @ ( inf_inf @ A @ X4 @ Z ) ) )
% 0.31/0.49 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) )
% 0.31/0.49 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % distrib_imp1
% 0.31/0.49 thf(fact_214_distrib__imp2,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.49 ( ! [X4: A,Y5: A,Z: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X4 @ ( inf_inf @ A @ Y5 @ Z ) )
% 0.31/0.49 = ( inf_inf @ A @ ( sup_sup @ A @ X4 @ Y5 ) @ ( sup_sup @ A @ X4 @ Z ) ) )
% 0.31/0.49 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
% 0.31/0.49 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % distrib_imp2
% 0.31/0.49 thf(fact_215_distrib__inf__le,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z3 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % distrib_inf_le
% 0.31/0.49 thf(fact_216_distrib__sup__le,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % distrib_sup_le
% 0.31/0.49 thf(fact_217_inf__sup__distrib1,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.49 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
% 0.31/0.49 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % inf_sup_distrib1
% 0.31/0.49 thf(fact_218_inf__sup__distrib2,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [Y: A,Z3: A,X: A] :
% 0.31/0.49 ( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z3 ) @ X )
% 0.31/0.49 = ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z3 @ X ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % inf_sup_distrib2
% 0.31/0.49 thf(fact_219_sup__inf__distrib1,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A,Z3: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) )
% 0.31/0.49 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_inf_distrib1
% 0.31/0.49 thf(fact_220_sup__inf__distrib2,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [Y: A,Z3: A,X: A] :
% 0.31/0.49 ( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z3 ) @ X )
% 0.31/0.49 = ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z3 @ X ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_inf_distrib2
% 0.31/0.49 thf(fact_221_Un__empty__left,axiom,
% 0.31/0.49 ! [A: $tType,B6: set @ A] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B6 )
% 0.31/0.49 = B6 ) ).
% 0.31/0.49
% 0.31/0.49 % Un_empty_left
% 0.31/0.49 thf(fact_222_Un__empty__right,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
% 0.31/0.49 = A6 ) ).
% 0.31/0.49
% 0.31/0.49 % Un_empty_right
% 0.31/0.49 thf(fact_223_Un__mono,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,C4: set @ A,B6: set @ A,D3: set @ A] :
% 0.31/0.49 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 0.31/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ D3 )
% 0.31/0.49 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_mono
% 0.31/0.49 thf(fact_224_Un__least,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,C4: set @ A,B6: set @ A] :
% 0.31/0.49 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 0.31/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.31/0.49 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) @ C4 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_least
% 0.31/0.49 thf(fact_225_Un__upper1,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_upper1
% 0.31/0.49 thf(fact_226_Un__upper2,axiom,
% 0.31/0.49 ! [A: $tType,B6: set @ A,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_upper2
% 0.31/0.49 thf(fact_227_Un__absorb1,axiom,
% 0.31/0.49 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.31/0.49 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.31/0.49 => ( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
% 0.31/0.49 = B6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_absorb1
% 0.31/0.49 thf(fact_228_Un__absorb2,axiom,
% 0.31/0.49 ! [A: $tType,B6: set @ A,A6: set @ A] :
% 0.31/0.49 ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 0.31/0.49 => ( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
% 0.31/0.49 = A6 ) ) ).
% 0.31/0.49
% 0.31/0.49 % Un_absorb2
% 0.31/0.49 thf(fact_229_subset__Un__eq,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.49 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.31/0.49 ( ( sup_sup @ ( set @ A ) @ A7 @ B7 )
% 0.31/0.49 = B7 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % subset_Un_eq
% 0.31/0.49 thf(fact_230_inf__sup__ord_I4_J,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % inf_sup_ord(4)
% 0.31/0.49 thf(fact_231_inf__sup__ord_I3_J,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( lattice @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % inf_sup_ord(3)
% 0.31/0.49 thf(fact_232_le__supE,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A,B3: A,X: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X )
% 0.31/0.49 => ~ ( ( ord_less_eq @ A @ A3 @ X )
% 0.31/0.49 => ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % le_supE
% 0.31/0.49 thf(fact_233_le__supI,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A,X: A,B3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ A3 @ X )
% 0.31/0.49 => ( ( ord_less_eq @ A @ B3 @ X )
% 0.31/0.49 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % le_supI
% 0.31/0.49 thf(fact_234_sup__ge1,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_ge1
% 0.31/0.49 thf(fact_235_sup__ge2,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_ge2
% 0.31/0.49 thf(fact_236_le__supI1,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,A3: A,B3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ X @ A3 )
% 0.31/0.49 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % le_supI1
% 0.31/0.49 thf(fact_237_le__supI2,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,B3: A,A3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ X @ B3 )
% 0.31/0.49 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % le_supI2
% 0.31/0.49 thf(fact_238_sup_Omono,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [C2: A,A3: A,D4: A,B3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.31/0.49 => ( ( ord_less_eq @ A @ D4 @ B3 )
% 0.31/0.49 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D4 ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.mono
% 0.31/0.49 thf(fact_239_sup__mono,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A,C2: A,B3: A,D4: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.31/0.49 => ( ( ord_less_eq @ A @ B3 @ D4 )
% 0.31/0.49 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C2 @ D4 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_mono
% 0.31/0.49 thf(fact_240_sup__least,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [Y: A,X: A,Z3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ Y @ X )
% 0.31/0.49 => ( ( ord_less_eq @ A @ Z3 @ X )
% 0.31/0.49 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z3 ) @ X ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_least
% 0.31/0.49 thf(fact_241_le__iff__sup,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ( ( ord_less_eq @ A )
% 0.31/0.49 = ( ^ [X3: A,Y4: A] :
% 0.31/0.49 ( ( sup_sup @ A @ X3 @ Y4 )
% 0.31/0.49 = Y4 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % le_iff_sup
% 0.31/0.49 thf(fact_242_sup_OorderE,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [B3: A,A3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.49 => ( A3
% 0.31/0.49 = ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.orderE
% 0.31/0.49 thf(fact_243_sup_OorderI,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A,B3: A] :
% 0.31/0.49 ( ( A3
% 0.31/0.49 = ( sup_sup @ A @ A3 @ B3 ) )
% 0.31/0.49 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.orderI
% 0.31/0.49 thf(fact_244_sup__unique,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [F: A > A > A,X: A,Y: A] :
% 0.31/0.49 ( ! [X4: A,Y5: A] : ( ord_less_eq @ A @ X4 @ ( F @ X4 @ Y5 ) )
% 0.31/0.49 => ( ! [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ ( F @ X4 @ Y5 ) )
% 0.31/0.49 => ( ! [X4: A,Y5: A,Z: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ Y5 @ X4 )
% 0.31/0.49 => ( ( ord_less_eq @ A @ Z @ X4 )
% 0.31/0.49 => ( ord_less_eq @ A @ ( F @ Y5 @ Z ) @ X4 ) ) )
% 0.31/0.49 => ( ( sup_sup @ A @ X @ Y )
% 0.31/0.49 = ( F @ X @ Y ) ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_unique
% 0.31/0.49 thf(fact_245_sup_Oabsorb1,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [B3: A,A3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.49 => ( ( sup_sup @ A @ A3 @ B3 )
% 0.31/0.49 = A3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.absorb1
% 0.31/0.49 thf(fact_246_sup_Oabsorb2,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A,B3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.49 => ( ( sup_sup @ A @ A3 @ B3 )
% 0.31/0.49 = B3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.absorb2
% 0.31/0.49 thf(fact_247_sup__absorb1,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [Y: A,X: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ Y @ X )
% 0.31/0.49 => ( ( sup_sup @ A @ X @ Y )
% 0.31/0.49 = X ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_absorb1
% 0.31/0.49 thf(fact_248_sup__absorb2,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [X: A,Y: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ X @ Y )
% 0.31/0.49 => ( ( sup_sup @ A @ X @ Y )
% 0.31/0.49 = Y ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup_absorb2
% 0.31/0.49 thf(fact_249_sup_OboundedE,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [B3: A,C2: A,A3: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 0.31/0.49 => ~ ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.49 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.boundedE
% 0.31/0.49 thf(fact_250_sup_OboundedI,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [B3: A,A3: A,C2: A] :
% 0.31/0.49 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.49 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.31/0.49 => ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.boundedI
% 0.31/0.49 thf(fact_251_sup_Oorder__iff,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ( ( ord_less_eq @ A )
% 0.31/0.49 = ( ^ [B2: A,A2: A] :
% 0.31/0.49 ( A2
% 0.31/0.49 = ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.order_iff
% 0.31/0.49 thf(fact_252_sup_Ocobounded1,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.cobounded1
% 0.31/0.49 thf(fact_253_sup_Ocobounded2,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.cobounded2
% 0.31/0.49 thf(fact_254_sup_Oabsorb__iff1,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ( ( ord_less_eq @ A )
% 0.31/0.49 = ( ^ [B2: A,A2: A] :
% 0.31/0.49 ( ( sup_sup @ A @ A2 @ B2 )
% 0.31/0.49 = A2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.absorb_iff1
% 0.31/0.49 thf(fact_255_sup_Oabsorb__iff2,axiom,
% 0.31/0.49 ! [A: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.31/0.49 => ( ( ord_less_eq @ A )
% 0.31/0.49 = ( ^ [A2: A,B2: A] :
% 0.31/0.49 ( ( sup_sup @ A @ A2 @ B2 )
% 0.31/0.49 = B2 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 % sup.absorb_iff2
% 0.31/0.49
% 0.31/0.49 %----Type constructors (35)
% 0.31/0.49 thf(tcon_FSet_Ofset___Lattices_Obounded__lattice,axiom,
% 0.31/0.49 ! [A8: $tType] :
% 0.31/0.49 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.49 => ( bounded_lattice @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_1,axiom,
% 0.31/0.49 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Set_Oset___Lattices_Obounded__lattice_2,axiom,
% 0.31/0.49 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_fun___Lattices_Obounded__lattice_3,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 0.31/0.49 => ( bounded_lattice @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 0.31/0.49 => ( bounde1808546759up_bot @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 0.31/0.49 => ( bounded_lattice_top @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 0.31/0.49 => ( bounded_lattice_bot @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( semilattice_sup @ A10 @ ( type2 @ A10 ) )
% 0.31/0.49 => ( semilattice_sup @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( distrib_lattice @ A10 @ ( type2 @ A10 ) )
% 0.31/0.49 => ( distrib_lattice @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.49 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 0.31/0.49 => ( finite_finite @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( lattice @ A10 @ ( type2 @ A10 ) )
% 0.31/0.49 => ( lattice @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_4,axiom,
% 0.31/0.49 ! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_5,axiom,
% 0.31/0.49 ! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_6,axiom,
% 0.31/0.49 ! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_7,axiom,
% 0.31/0.49 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_8,axiom,
% 0.31/0.49 ! [A8: $tType] : ( distrib_lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Set_Oset___Finite__Set_Ofinite_9,axiom,
% 0.31/0.49 ! [A8: $tType] :
% 0.31/0.49 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.49 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Set_Oset___Lattices_Olattice_10,axiom,
% 0.31/0.49 ! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_11,axiom,
% 0.31/0.49 bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_12,axiom,
% 0.31/0.49 bounded_lattice_top @ $o @ ( type2 @ $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_13,axiom,
% 0.31/0.49 bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14,axiom,
% 0.31/0.49 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_15,axiom,
% 0.31/0.49 distrib_lattice @ $o @ ( type2 @ $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
% 0.31/0.49 finite_finite @ $o @ ( type2 @ $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_HOL_Obool___Lattices_Olattice_17,axiom,
% 0.31/0.49 lattice @ $o @ ( type2 @ $o ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_FSet_Ofset___Lattices_Obounded__semilattice__sup__bot_18,axiom,
% 0.31/0.49 ! [A8: $tType] : ( bounde1808546759up_bot @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_FSet_Ofset___Lattices_Obounded__lattice__top_19,axiom,
% 0.31/0.49 ! [A8: $tType] :
% 0.31/0.49 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.49 => ( bounded_lattice_top @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_FSet_Ofset___Lattices_Obounded__lattice__bot_20,axiom,
% 0.31/0.49 ! [A8: $tType] : ( bounded_lattice_bot @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_FSet_Ofset___Lattices_Osemilattice__sup_21,axiom,
% 0.31/0.49 ! [A8: $tType] : ( semilattice_sup @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_FSet_Ofset___Lattices_Odistrib__lattice_22,axiom,
% 0.31/0.49 ! [A8: $tType] : ( distrib_lattice @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_FSet_Ofset___Finite__Set_Ofinite_23,axiom,
% 0.31/0.49 ! [A8: $tType] :
% 0.31/0.49 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.49 => ( finite_finite @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_FSet_Ofset___Lattices_Olattice_24,axiom,
% 0.31/0.49 ! [A8: $tType] : ( lattice @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_25,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.49 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 0.31/0.49 => ( finite_finite @ ( sum_sum @ A8 @ A10 ) @ ( type2 @ ( sum_sum @ A8 @ A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Option_Ooption___Finite__Set_Ofinite_26,axiom,
% 0.31/0.49 ! [A8: $tType] :
% 0.31/0.49 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.49 => ( finite_finite @ ( option @ A8 ) @ ( type2 @ ( option @ A8 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27,axiom,
% 0.31/0.49 ! [A8: $tType,A10: $tType] :
% 0.31/0.49 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.31/0.49 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 0.31/0.49 => ( finite_finite @ ( product_prod @ A8 @ A10 ) @ ( type2 @ ( product_prod @ A8 @ A10 ) ) ) ) ).
% 0.31/0.49
% 0.31/0.49 %----Conjectures (1)
% 0.31/0.49 thf(conj_0,conjecture,
% 0.31/0.49 ( ( parall1518086719_Nplus @ ( root @ tr1a ) @ ( root @ tr2a ) )
% 0.31/0.49 = ( parall1518086719_Nplus @ ( root @ tr2a ) @ ( root @ tr1a ) ) ) ).
% 0.31/0.49
% 0.31/0.49 %------------------------------------------------------------------------------
% 0.31/0.49 ------------------------
% 3.08/1.78 ---- Embedded in THF ---
% 3.08/1.78 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.08/1.78 %%% Generated on Tue Feb 24 04:16:59 EST 2026
% 3.08/1.78 %%% using '$$dhol' embedding, version 1.3.0.
% 3.08/1.78 %%% Logic specification used:
% 3.08/1.78 %%% thf(spec, logic, $$dhol).
% 3.08/1.78
% 3.08/1.78 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.q1d5il6I29/DTF2DTF22976.p
% 3.08/1.78 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.08/1.78 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 3.08/1.78 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)).
% 3.08/1.78 thf(ty_t_DTree_Odtree, type, dtree: $tType).
% 3.08/1.78 thf(ty_t_FSet_Ofset, type, fset: ($tType > $tType)).
% 3.08/1.78 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.08/1.78 thf(ty_t_DTree_OT, type, t: $tType).
% 3.08/1.78 thf(ty_t_DTree_ON, type, n: $tType).
% 3.08/1.78 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 3.08/1.78 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_cl_Lattices_Odistrib__lattice, type, distrib_lattice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_cl_Lattices_Obounded__lattice__bot, type, bounded_lattice_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_cl_Lattices_Obounded__lattice__top, type, bounded_lattice_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde1808546759up_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.08/1.78 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 3.08/1.78 thf(sy_c_DTree_ONode, type, node: (n > ((set @ ((sum_sum @ t) @ dtree)) > dtree))).
% 3.08/1.78 thf(sy_c_DTree_Ocont, type, cont: (dtree > (set @ ((sum_sum @ t) @ dtree)))).
% 3.08/1.78 thf(sy_c_DTree_Ocorec, type, corec: (!> [A:$tType]: (((A > n) > ((A > (set @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))) > (A > dtree)))))).
% 3.08/1.78 thf(sy_c_DTree_Odtree_ONNode, type, nNode: (n > ((fset @ ((sum_sum @ t) @ dtree)) > dtree))).
% 3.08/1.78 thf(sy_c_DTree_Odtree_Ocase__dtree, type, case_dtree: (!> [A:$tType]: (((n > ((fset @ ((sum_sum @ t) @ dtree)) > A)) > (dtree > A))))).
% 3.08/1.78 thf(sy_c_DTree_Odtree_Occont, type, ccont: (dtree > (fset @ ((sum_sum @ t) @ dtree)))).
% 3.08/1.78 thf(sy_c_DTree_Odtree_Ocorec__dtree, type, corec_dtree: (!> [A:$tType]: (((A > n) > ((A > (fset @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))) > (A > dtree)))))).
% 3.08/1.78 thf(sy_c_DTree_Odtree_Oroot, type, root: (dtree > n)).
% 3.08/1.78 thf(sy_c_DTree_Ounfold, type, unfold: (!> [A:$tType]: (((A > n) > ((A > (set @ ((sum_sum @ t) @ A))) > (A > dtree)))))).
% 3.08/1.78 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.08/1.78 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 3.08/1.78 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 3.08/1.78 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 3.08/1.78 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.08/1.78 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 3.08/1.78 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus, type, parall1518086719_Nplus: (n > (n > n))).
% 3.08/1.78 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar, type, parall1899940088le_par: (((product_prod @ dtree) @ dtree) > dtree)).
% 3.08/1.78 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c, type, parall1914194347_par_c: (((product_prod @ dtree) @ dtree) > (set @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))))).
% 3.08/1.78 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r, type, parall1914194362_par_r: (((product_prod @ dtree) @ dtree) > n)).
% 3.08/1.78 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel, type, parall556292031_r_rel: (((product_prod @ dtree) @ dtree) > (((product_prod @ dtree) @ dtree) > $o))).
% 3.08/1.78 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.08/1.78 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 3.08/1.78 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.08/1.78 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 3.08/1.78 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 3.08/1.78 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.08/1.78 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 3.08/1.78 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 3.08/1.78 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.08/1.78 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 3.08/1.78 thf(sy_c_Set_Ovimage, type, vimage: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ B) > (set @ A)))))).
% 3.08/1.78 thf(sy_c_Sum__Type_OInl, type, sum_Inl: (!> [A:$tType,B:$tType]: ((A > ((sum_sum @ A) @ B))))).
% 3.08/1.78 thf(sy_c_Sum__Type_OInr, type, sum_Inr: (!> [B:$tType,A:$tType]: ((B > ((sum_sum @ A) @ B))))).
% 3.08/1.78 thf(sy_c_Sum__Type_OPlus, type, sum_Plus: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((sum_sum @ A) @ B))))))).
% 3.08/1.78 thf(sy_c_Sum__Type_OSuml, type, sum_Suml: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((sum_sum @ A) @ B) > C))))).
% 3.08/1.78 thf(sy_c_Sum__Type_OSumr, type, sum_Sumr: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((sum_sum @ A) @ B) > C))))).
% 3.08/1.78 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum, type, sum_rec_sum: (!> [A:$tType,T:$tType,B:$tType]: (((A > T) > ((B > T) > (((sum_sum @ A) @ B) > T)))))).
% 3.08/1.78 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 3.08/1.78 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.08/1.78 thf(sy_c_Wfrec_Oadm__wf, type, adm_wf: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > (((A > B) > (A > B)) > $o))))).
% 3.08/1.78 thf(sy_c_Wfrec_Ocut, type, cut: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ ((product_prod @ A) @ A)) > (A > (A > B))))))).
% 3.08/1.78 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.08/1.78 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.08/1.78 thf(sy_v_tr1a____, type, tr1a: dtree).
% 3.08/1.78 thf(sy_v_tr2a____, type, tr2a: dtree).
% 3.08/1.78 thf(sy_v_trA____, type, trA: dtree).
% 3.08/1.78 thf(sy_v_trB____, type, trB: dtree).
% 3.08/1.78 thf(fact_0_Nplus__comm, axiom, (parall1518086719_Nplus = (^ [A2:n,B2:n]: (((parall1518086719_Nplus @ B2) @ A2))))).
% 3.08/1.78 thf(fact_1_Nplus__assoc, axiom, (! [A3:n,B3:n,C2:n]: ((((parall1518086719_Nplus @ ((parall1518086719_Nplus @ A3) @ B3)) @ C2) = ((parall1518086719_Nplus @ A3) @ ((parall1518086719_Nplus @ B3) @ C2)))))).
% 3.08/1.78 thf(fact_2_par__r_Oelims, axiom, (! [X:((product_prod @ dtree) @ dtree),Y:n]: ((((parall1914194362_par_r @ X) = Y) => (~ ((! [Tr1:dtree,Tr2:dtree]: (((X = ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)) => (Y != ((parall1518086719_Nplus @ (root @ Tr1)) @ (root @ Tr2)))))))))))).
% 3.08/1.78 thf(fact_3_par__r_Osimps, axiom, (! [Tr12:dtree,Tr22:dtree]: (((parall1914194362_par_r @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)) = ((parall1518086719_Nplus @ (root @ Tr12)) @ (root @ Tr22)))))).
% 3.08/1.78 thf(fact_4_root__Node, axiom, (! [N:n,As:(set @ ((sum_sum @ t) @ dtree))]: (((root @ ((node @ N) @ As)) = N)))).
% 3.08/1.78 thf(fact_5_root__par, axiom, (! [Tr12:dtree,Tr22:dtree]: (((root @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22))) = ((parall1518086719_Nplus @ (root @ Tr12)) @ (root @ Tr22)))))).
% 3.08/1.78 thf(fact_6_corec_I1_J, axiom, (! [A:$tType,Rt:(A > n),Ct:(A > (set @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))),B3:A]: (((root @ ((((corec @ A) @ Rt) @ Ct) @ B3)) = (Rt @ B3))))).
% 3.08/1.78 thf(fact_7_unfold_I1_J, axiom, (! [A:$tType,Rt:(A > n),Ct:(A > (set @ ((sum_sum @ t) @ A))),B3:A]: (((root @ ((((unfold @ A) @ Rt) @ Ct) @ B3)) = (Rt @ B3))))).
% 3.08/1.78 thf(fact_8_dtree_Osel_I1_J, axiom, (! [X1:n,X2:(fset @ ((sum_sum @ t) @ dtree))]: (((root @ ((nNode @ X1) @ X2)) = X1)))).
% 3.08/1.78 thf(fact_9_dtree_Ocorec__sel_I1_J, axiom, (! [A:$tType,G1:(A > n),G2:(A > (fset @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))),A3:A]: (((root @ ((((corec_dtree @ A) @ G1) @ G2) @ A3)) = (G1 @ A3))))).
% 3.08/1.78 thf(fact_10_dtree_Oexpand, axiom, (! [Dtree:dtree,Dtree2:dtree]: (((((root @ Dtree) = (root @ Dtree2)) & ((ccont @ Dtree) = (ccont @ Dtree2))) => (Dtree = Dtree2))))).
% 3.08/1.78 thf(fact_11_par__r_Opelims, axiom, (! [X:((product_prod @ dtree) @ dtree),Y:n]: ((((parall1914194362_par_r @ X) = Y) => ((((accp @ ((product_prod @ dtree) @ dtree)) @ parall556292031_r_rel) @ X) => (~ ((! [Tr1:dtree,Tr2:dtree]: (((X = ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)) => ((Y = ((parall1518086719_Nplus @ (root @ Tr1)) @ (root @ Tr2))) => (~ ((((accp @ ((product_prod @ dtree) @ dtree)) @ parall556292031_r_rel) @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))))))))))))))).
% 3.08/1.78 thf(fact_12_dtree__cong, axiom, (! [Tr:dtree,Tr3:dtree]: ((((root @ Tr) = (root @ Tr3)) => (((cont @ Tr) = (cont @ Tr3)) => (Tr = Tr3)))))).
% 3.08/1.78 thf(fact_13_dtree_Ocollapse, axiom, (! [Dtree:dtree]: ((((nNode @ (root @ Dtree)) @ (ccont @ Dtree)) = Dtree)))).
% 3.08/1.78 thf(fact_14__092_060open_062_092_060exists_062tr1_Atr2_O_AtrA_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060and_062_AtrB_A_061_Atr2_A_092_060parallel_062_Atr1_092_060close_062, axiom, (? [Tr1:dtree,Tr2:dtree]: (((trA = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))) & (trB = (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr2) @ Tr1))))))).
% 3.08/1.78 thf(fact_15_dtree_Oinject, axiom, (! [X1:n,X2:(fset @ ((sum_sum @ t) @ dtree)),Y1:n,Y2:(fset @ ((sum_sum @ t) @ dtree))]: (((((nNode @ X1) @ X2) = ((nNode @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 3.08/1.78 thf(fact_16_Node__root__cont, axiom, (! [Tr:dtree]: ((((node @ (root @ Tr)) @ (cont @ Tr)) = Tr)))).
% 3.08/1.78 thf(fact_17_dtree_Osel_I2_J, axiom, (! [X1:n,X2:(fset @ ((sum_sum @ t) @ dtree))]: (((ccont @ ((nNode @ X1) @ X2)) = X2)))).
% 3.08/1.78 thf(fact_18_dtree_Oexhaust, axiom, (! [Y:dtree]: ((~ ((! [X12:n,X22:(fset @ ((sum_sum @ t) @ dtree))]: ((Y != ((nNode @ X12) @ X22))))))))).
% 3.08/1.78 thf(fact_19_dtree_Ocorec__disc, axiom, (! [A:$tType]: (((corec_dtree @ A) = (corec_dtree @ A))))).
% 3.08/1.78 thf(fact_20_par__r_Ocases, axiom, (! [X:((product_prod @ dtree) @ dtree)]: ((~ ((! [Tr1:dtree,Tr2:dtree]: ((X != ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))))))))).
% 3.08/1.78 thf(fact_21_par__r_Oinduct, axiom, (! [P:(((product_prod @ dtree) @ dtree) > $o),A0:((product_prod @ dtree) @ dtree)]: (((! [Tr1:dtree,Tr2:dtree]: ((P @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)))) => (P @ A0))))).
% 3.08/1.78 thf(fact_22_dtree_Oexhaust__sel, axiom, (! [Dtree:dtree]: ((Dtree = ((nNode @ (root @ Dtree)) @ (ccont @ Dtree)))))).
% 3.08/1.78 thf(fact_23_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 3.08/1.78 thf(fact_24_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) = ((A3 = A4) & (B3 = B4)))))).
% 3.08/1.78 thf(fact_25_dtree_Osplit__sel, axiom, (! [A:$tType,P:(A > $o),F:(n > ((fset @ ((sum_sum @ t) @ dtree)) > A)),Dtree:dtree]: (((P @ (((case_dtree @ A) @ F) @ Dtree)) = ((Dtree = ((nNode @ (root @ Dtree)) @ (ccont @ Dtree))) => (P @ ((F @ (root @ Dtree)) @ (ccont @ Dtree)))))))).
% 3.08/1.78 thf(fact_26_dtree_Osplit__sel__asm, axiom, (! [A:$tType,P:(A > $o),F:(n > ((fset @ ((sum_sum @ t) @ dtree)) > A)),Dtree:dtree]: (((P @ (((case_dtree @ A) @ F) @ Dtree)) = (~ (((Dtree = ((nNode @ (root @ Dtree)) @ (ccont @ Dtree))) & (~ ((P @ ((F @ (root @ Dtree)) @ (ccont @ Dtree)))))))))))).
% 3.08/1.78 thf(fact_27_par__def, axiom, (parall1899940088le_par = (((unfold @ ((product_prod @ dtree) @ dtree)) @ parall1914194362_par_r) @ parall1914194347_par_c))).
% 3.08/1.78 thf(fact_28_dtree_Ocase__eq__if, axiom, (! [A:$tType]: (((case_dtree @ A) = (^ [F2:(n > ((fset @ ((sum_sum @ t) @ dtree)) > A)),Dtree3:dtree]: (((F2 @ (root @ Dtree3)) @ (ccont @ Dtree3)))))))).
% 3.08/1.78 thf(fact_29_accp_Ocases, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A]: (((((accp @ A) @ R) @ A3) => (! [Y3:A]: ((((R @ Y3) @ A3) => (((accp @ A) @ R) @ Y3)))))))).
% 3.08/1.78 thf(fact_30_accp_Osimps, axiom, (! [A:$tType]: (((accp @ A) = (^ [R2:(A > (A > $o)),A2:A]: ((? [X3:A]: (((A2 = X3) & (! [Y4:A]: ((((R2 @ Y4) @ X3) => (((accp @ A) @ R2) @ Y4))))))))))))).
% 3.08/1.78 thf(fact_31_accp_Ointros, axiom, (! [A:$tType,R:(A > (A > $o)),X:A]: (((! [Y5:A]: ((((R @ Y5) @ X) => (((accp @ A) @ R) @ Y5)))) => (((accp @ A) @ R) @ X))))).
% 3.08/1.78 thf(fact_32_accp__induct, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A,P:(A > $o)]: (((((accp @ A) @ R) @ A3) => ((! [X4:A]: (((((accp @ A) @ R) @ X4) => ((! [Y3:A]: ((((R @ Y3) @ X4) => (P @ Y3)))) => (P @ X4))))) => (P @ A3)))))).
% 3.08/1.78 thf(fact_33_accp_Oinducts, axiom, (! [A:$tType,R:(A > (A > $o)),X:A,P:(A > $o)]: (((((accp @ A) @ R) @ X) => ((! [X4:A]: (((! [Y3:A]: ((((R @ Y3) @ X4) => (((accp @ A) @ R) @ Y3)))) => ((! [Y3:A]: ((((R @ Y3) @ X4) => (P @ Y3)))) => (P @ X4))))) => (P @ X)))))).
% 3.08/1.78 thf(fact_34_dtree_Ocase, axiom, (! [A:$tType,F:(n > ((fset @ ((sum_sum @ t) @ dtree)) > A)),X1:n,X2:(fset @ ((sum_sum @ t) @ dtree))]: (((((case_dtree @ A) @ F) @ ((nNode @ X1) @ X2)) = ((F @ X1) @ X2))))).
% 3.08/1.78 thf(fact_35_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ Prod))))).
% 3.08/1.78 thf(fact_36_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B5))))))))).
% 3.08/1.78 thf(fact_37_prod__induct7, axiom, (! [G:$tType,F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))))))]: (((! [A5:A,B5:B,C3:C,D2:D,E2:E,F4:F3,G3:G]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G)) @ E2) @ ((((product_Pair @ F3) @ G) @ F4) @ G3))))))))) => (P @ X))))).
% 3.08/1.78 thf(fact_38_prod__induct6, axiom, (! [F3:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: (((! [A5:A,B5:B,C3:C,D2:D,E2:E,F4:F3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F4)))))))) => (P @ X))))).
% 3.08/1.78 thf(fact_39_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B5:B,C3:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 3.08/1.78 thf(fact_40_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B5:B,C3:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2)))))) => (P @ X))))).
% 3.08/1.78 thf(fact_41_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B5:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C3))))) => (P @ X))))).
% 3.08/1.78 thf(fact_42_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,G:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))))))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D,E2:E,F4:F3,G3:G]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F3) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G)) @ E2) @ ((((product_Pair @ F3) @ G) @ F4) @ G3)))))))))))))).
% 3.08/1.78 thf(fact_43_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D,E2:E,F4:F3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F3))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F3)) @ D2) @ ((((product_Pair @ E) @ F3) @ E2) @ F4))))))))))))).
% 3.08/1.78 thf(fact_44_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 3.08/1.78 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 3.08/1.78 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A6:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A6)))) = A6)))).
% 3.08/1.78 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P @ X4) = (Q @ X4)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.08/1.78 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G4:(A > B)]: (((! [X4:A]: (((F @ X4) = (G4 @ X4)))) => (F = G4))))).
% 3.08/1.78 thf(fact_49_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2))))))))))).
% 3.08/1.78 thf(fact_50_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B5:B,C3:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C3)))))))))).
% 3.08/1.78 thf(fact_51_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) => (~ (((A3 = A4) => (B3 != B4)))))))).
% 3.08/1.78 thf(fact_52_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ P2))))).
% 3.08/1.78 thf(fact_53_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X4:A,Y5:B]: ((P2 = ((((product_Pair @ A) @ B) @ X4) @ Y5))))))).
% 3.08/1.78 thf(fact_54_accp__induct__rule, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A,P:(A > $o)]: (((((accp @ A) @ R) @ A3) => ((! [X4:A]: (((((accp @ A) @ R) @ X4) => ((! [Y3:A]: ((((R @ Y3) @ X4) => (P @ Y3)))) => (P @ X4))))) => (P @ A3)))))).
% 3.08/1.78 thf(fact_55_not__accp__down, axiom, (! [A:$tType,R3:(A > (A > $o)),X:A]: (((~ ((((accp @ A) @ R3) @ X))) => (~ ((! [Z:A]: ((((R3 @ Z) @ X) => (((accp @ A) @ R3) @ Z)))))))))).
% 3.08/1.78 thf(fact_56_accp__downward, axiom, (! [A:$tType,R:(A > (A > $o)),B3:A,A3:A]: (((((accp @ A) @ R) @ B3) => (((R @ A3) @ B3) => (((accp @ A) @ R) @ A3)))))).
% 3.08/1.78 thf(fact_57_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A3:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((F1 @ A3) @ B3))))).
% 3.08/1.78 thf(fact_58_in__lex__prod, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B,R:(set @ ((product_prod @ A) @ A)),S:(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) @ A3) @ B3)) @ ((((product_Pair @ A) @ B) @ A4) @ B4))) @ ((((lex_prod @ A) @ B) @ R) @ S)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A4)) @ R) | ((A3 = A4) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B3) @ B4)) @ S))))))).
% 3.08/1.78 thf(fact_59_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A3:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((C2 @ A3) @ B3))))).
% 3.08/1.78 thf(fact_60_Inl__in__cont__par, axiom, (! [T2:t,Tr12:dtree,Tr22:dtree]: (((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)))) = ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ Tr12)) | (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ Tr22))))))).
% 3.08/1.78 thf(fact_61_cont__Node, axiom, (! [As:(set @ ((sum_sum @ t) @ dtree)),N:n]: ((((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As) => ((cont @ ((node @ N) @ As)) = As))))).
% 3.08/1.78 thf(fact_62_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S:B,R3:(set @ ((product_prod @ A) @ B)),S2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S)) @ R3) => ((S2 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S2)) @ R3)))))).
% 3.08/1.78 thf(fact_63_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F2:(((product_prod @ B) @ C) > A),A2:B,B2:C]: ((F2 @ ((((product_Pair @ B) @ C) @ A2) @ B2)))))))).
% 3.08/1.78 thf(fact_64_finite__par__c, axiom, (! [Tr12:dtree,Tr22:dtree]: (((finite_finite2 @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))) @ (parall1914194347_par_c @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)))))).
% 3.08/1.78 thf(fact_65_curryI, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: (((F @ ((((product_Pair @ A) @ B) @ A3) @ B3)) => ((((((product_curry @ A) @ B) @ $o) @ F) @ A3) @ B3))))).
% 3.08/1.78 thf(fact_66_dtree__simps, axiom, (! [As:(set @ ((sum_sum @ t) @ dtree)),As2:(set @ ((sum_sum @ t) @ dtree)),N:n,N2:n]: ((((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As) => (((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As2) => ((((node @ N) @ As) = ((node @ N2) @ As2)) = ((N = N2) & (As = As2)))))))).
% 3.08/1.78 thf(fact_67_finite__cont, axiom, (! [Tr:dtree]: (((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ (cont @ Tr))))).
% 3.08/1.78 thf(fact_68_dtree__cases, axiom, (! [Tr:dtree]: ((~ ((! [N3:n,As3:(set @ ((sum_sum @ t) @ dtree))]: ((((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As3) => (Tr != ((node @ N3) @ As3)))))))))).
% 3.08/1.78 thf(fact_69_curryD, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A3) @ B3) => (F @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 3.08/1.78 thf(fact_70_curryE, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A3) @ B3) => (F @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 3.08/1.78 thf(fact_71_sum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,X1:A,Y1:A]: ((((((sum_Inl @ A) @ B) @ X1) = (((sum_Inl @ A) @ B) @ Y1)) = (X1 = Y1))))).
% 3.08/1.78 thf(fact_72_old_Osum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,A3:A,A4:A]: ((((((sum_Inl @ A) @ B) @ A3) = (((sum_Inl @ A) @ B) @ A4)) = (A3 = A4))))).
% 3.08/1.78 thf(fact_73_finite__code, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) = (^ [A7:(set @ A)]: ($true))))))).
% 3.08/1.78 thf(fact_74_same__fstI, axiom, (! [B:$tType,A:$tType,P:(A > $o),X:A,Y6:B,Y:B,R3:(A > (set @ ((product_prod @ B) @ B)))]: (((P @ X) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y6) @ Y)) @ (R3 @ X)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y6)) @ ((((product_Pair @ A) @ B) @ X) @ Y))) @ ((((same_fst @ A) @ B) @ P) @ R3))))))).
% 3.08/1.78 thf(fact_75_Inl__inject, axiom, (! [B:$tType,A:$tType,X:A,Y:A]: ((((((sum_Inl @ A) @ B) @ X) = (((sum_Inl @ A) @ B) @ Y)) => (X = Y))))).
% 3.08/1.78 thf(fact_76_finite__set__choice, axiom, (! [B:$tType,A:$tType,A6:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A6) => ((! [X4:A]: (((((member @ A) @ X4) @ A6) => (? [X13:B]: (((P @ X4) @ X13)))))) => (? [F4:(A > B)]: ((! [X5:A]: (((((member @ A) @ X5) @ A6) => ((P @ X5) @ (F4 @ X5)))))))))))).
% 3.08/1.78 thf(fact_77_finite, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => (! [A6:(set @ A)]: (((finite_finite2 @ A) @ A6))))))).
% 3.08/1.78 thf(fact_78_old_Osum_Osimps_I7_J, axiom, (! [B:$tType,T:$tType,A:$tType,F1:(A > T),F22:(B > T),A3:A]: ((((((((sum_rec_sum @ A) @ T) @ B) @ F1) @ F22) @ (((sum_Inl @ A) @ B) @ A3)) = (F1 @ A3))))).
% 3.08/1.78 thf(fact_79_Suml_Osimps, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C),X:A]: (((((((sum_Suml @ A) @ C) @ B) @ F) @ (((sum_Inl @ A) @ B) @ X)) = (F @ X))))).
% 3.08/1.78 thf(fact_80_finite__Plus__iff, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B6:(set @ B)]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A6) @ B6)) = (((finite_finite2 @ A) @ A6) & ((finite_finite2 @ B) @ B6)))))).
% 3.08/1.78 thf(fact_81_adm__wf__def, axiom, (! [B:$tType,A:$tType]: ((((adm_wf @ A) @ B) = (^ [R4:(set @ ((product_prod @ A) @ A)),F5:((A > B) > (A > B))]: ((! [F2:(A > B),G5:(A > B),X3:A]: (((! [Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z2) @ X3)) @ R4) => ((F2 @ Z2) = (G5 @ Z2))))) => (((F5 @ F2) @ X3) = ((F5 @ G5) @ X3))))))))))).
% 3.08/1.78 thf(fact_82_Inl__cont__par, axiom, (! [Tr12:dtree,Tr22:dtree]: ((((((vimage @ t) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inl @ t) @ dtree)) @ (cont @ (parall1899940088le_par @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)))) = ((((vimage @ t) @ ((sum_sum @ t) @ dtree)) @ ((sum_Inl @ t) @ dtree)) @ (((sup_sup @ (set @ ((sum_sum @ t) @ dtree))) @ (cont @ Tr12)) @ (cont @ Tr22))))))).
% 3.08/1.78 thf(fact_83_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 3.08/1.78 thf(fact_84_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 3.08/1.78 thf(fact_85_finite__Un, axiom, (! [A:$tType,F6:(set @ A),G6:(set @ A)]: ((((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ F6) @ G6)) = (((finite_finite2 @ A) @ F6) & ((finite_finite2 @ A) @ G6)))))).
% 3.08/1.78 thf(fact_86_InlI, axiom, (! [A:$tType,B:$tType,A3:A,A6:(set @ A),B6:(set @ B)]: (((((member @ A) @ A3) @ A6) => (((member @ ((sum_sum @ A) @ B)) @ (((sum_Inl @ A) @ B) @ A3)) @ ((((sum_Plus @ A) @ B) @ A6) @ B6)))))).
% 3.08/1.78 thf(fact_87_Suml__inject, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C),G4:(A > C)]: (((((((sum_Suml @ A) @ C) @ B) @ F) = ((((sum_Suml @ A) @ C) @ B) @ G4)) => (F = G4))))).
% 3.08/1.78 thf(fact_88_infinite__Un, axiom, (! [A:$tType,S3:(set @ A),T3:(set @ A)]: (((~ (((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ S3) @ T3)))) = ((~ (((finite_finite2 @ A) @ S3))) | (~ (((finite_finite2 @ A) @ T3)))))))).
% 3.08/1.78 thf(fact_89_Un__infinite, axiom, (! [A:$tType,S3:(set @ A),T3:(set @ A)]: (((~ (((finite_finite2 @ A) @ S3))) => (~ (((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ S3) @ T3)))))))).
% 3.08/1.78 thf(fact_90_finite__UnI, axiom, (! [A:$tType,F6:(set @ A),G6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => (((finite_finite2 @ A) @ G6) => ((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ F6) @ G6))))))).
% 3.08/1.78 thf(fact_91_finite__PlusD_I2_J, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B6:(set @ B)]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A6) @ B6)) => ((finite_finite2 @ B) @ B6))))).
% 3.08/1.78 thf(fact_92_finite__PlusD_I1_J, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B6:(set @ B)]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A6) @ B6)) => ((finite_finite2 @ A) @ A6))))).
% 3.08/1.78 thf(fact_93_finite__Plus, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B6:(set @ B)]: ((((finite_finite2 @ A) @ A6) => (((finite_finite2 @ B) @ B6) => ((finite_finite2 @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A6) @ B6))))))).
% 3.08/1.78 thf(fact_94_vimage__Un, axiom, (! [A:$tType,B:$tType,F:(A > B),A6:(set @ B),B6:(set @ B)]: ((((((vimage @ A) @ B) @ F) @ (((sup_sup @ (set @ B)) @ A6) @ B6)) = (((sup_sup @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ A6)) @ ((((vimage @ A) @ B) @ F) @ B6)))))).
% 3.08/1.78 thf(fact_95_vimageI, axiom, (! [B:$tType,A:$tType,F:(B > A),A3:B,B3:A,B6:(set @ A)]: ((((F @ A3) = B3) => ((((member @ A) @ B3) @ B6) => (((member @ B) @ A3) @ ((((vimage @ B) @ A) @ F) @ B6))))))).
% 3.08/1.78 thf(fact_96_vimage__eq, axiom, (! [A:$tType,B:$tType,A3:A,F:(A > B),B6:(set @ B)]: (((((member @ A) @ A3) @ ((((vimage @ A) @ B) @ F) @ B6)) = (((member @ B) @ (F @ A3)) @ B6))))).
% 3.08/1.78 thf(fact_97_UnCI, axiom, (! [A:$tType,C2:A,B6:(set @ A),A6:(set @ A)]: ((((~ ((((member @ A) @ C2) @ B6))) => (((member @ A) @ C2) @ A6)) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A6) @ B6)))))).
% 3.08/1.78 thf(fact_98_Un__iff, axiom, (! [A:$tType,C2:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A6) @ B6)) = ((((member @ A) @ C2) @ A6) | (((member @ A) @ C2) @ B6)))))).
% 3.08/1.78 thf(fact_99_sup__apply, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G5:(A > B),X3:A]: ((((sup_sup @ B) @ (F2 @ X3)) @ (G5 @ X3))))))))).
% 3.08/1.78 thf(fact_100_sup_Oright__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ B3) = (((sup_sup @ A) @ A3) @ B3)))))))).
% 3.08/1.78 thf(fact_101_sup__left__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 3.08/1.78 thf(fact_102_sup_Oleft__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B3)) = (((sup_sup @ A) @ A3) @ B3)))))))).
% 3.08/1.78 thf(fact_103_sup__idem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ X) = X))))))).
% 3.08/1.78 thf(fact_104_sup_Oidem, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ A3) = A3))))))).
% 3.08/1.78 thf(fact_105_sup__left__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z3)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z3))))))))).
% 3.08/1.78 thf(fact_106_sup_Oleft__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C2:A]: (((((sup_sup @ A) @ B3) @ (((sup_sup @ A) @ A3) @ C2)) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B3) @ C2))))))))).
% 3.08/1.78 thf(fact_107_sup__commute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X3:A,Y4:A]: ((((sup_sup @ A) @ Y4) @ X3)))))))).
% 3.08/1.78 thf(fact_108_sup_Ocommute, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [A2:A,B2:A]: ((((sup_sup @ A) @ B2) @ A2)))))))).
% 3.08/1.78 thf(fact_109_sup__assoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z3) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z3))))))))).
% 3.08/1.78 thf(fact_110_sup_Oassoc, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A,C2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ C2) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B3) @ C2))))))))).
% 3.08/1.78 thf(fact_111_sup__fun__def, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ B) @ (type2 @ B)) => ((sup_sup @ (A > B)) = (^ [F2:(A > B),G5:(A > B),X3:A]: ((((sup_sup @ B) @ (F2 @ X3)) @ (G5 @ X3))))))))).
% 3.08/1.78 thf(fact_112_inf__sup__aci_I5_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => ((sup_sup @ A) = (^ [X3:A,Y4:A]: ((((sup_sup @ A) @ Y4) @ X3)))))))).
% 3.08/1.78 thf(fact_113_inf__sup__aci_I6_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z3) = (((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z3))))))))).
% 3.08/1.78 thf(fact_114_inf__sup__aci_I7_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z3)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X) @ Z3))))))))).
% 3.08/1.78 thf(fact_115_inf__sup__aci_I8_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = (((sup_sup @ A) @ X) @ Y)))))))).
% 3.08/1.78 thf(fact_116_Un__left__commute, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ (((sup_sup @ (set @ A)) @ B6) @ C4)) = (((sup_sup @ (set @ A)) @ B6) @ (((sup_sup @ (set @ A)) @ A6) @ C4)))))).
% 3.08/1.78 thf(fact_117_Un__left__absorb, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ (((sup_sup @ (set @ A)) @ A6) @ B6)) = (((sup_sup @ (set @ A)) @ A6) @ B6))))).
% 3.08/1.78 thf(fact_118_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((((sup_sup @ (set @ A)) @ B7) @ A7))))))).
% 3.08/1.78 thf(fact_119_Un__absorb, axiom, (! [A:$tType,A6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ A6) = A6)))).
% 3.08/1.78 thf(fact_120_Un__assoc, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A6) @ B6)) @ C4) = (((sup_sup @ (set @ A)) @ A6) @ (((sup_sup @ (set @ A)) @ B6) @ C4)))))).
% 3.08/1.78 thf(fact_121_ball__Un, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ (((sup_sup @ (set @ A)) @ A6) @ B6)) => (P @ X3)))) = ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (P @ X3)))) & (! [X3:A]: (((((member @ A) @ X3) @ B6) => (P @ X3))))))))).
% 3.08/1.78 thf(fact_122_bex__Un, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),P:(A > $o)]: (((? [X3:A]: (((((member @ A) @ X3) @ (((sup_sup @ (set @ A)) @ A6) @ B6)) & (P @ X3)))) = ((? [X3:A]: (((((member @ A) @ X3) @ A6) & (P @ X3)))) | (? [X3:A]: (((((member @ A) @ X3) @ B6) & (P @ X3))))))))).
% 3.08/1.78 thf(fact_123_UnI2, axiom, (! [A:$tType,C2:A,B6:(set @ A),A6:(set @ A)]: (((((member @ A) @ C2) @ B6) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A6) @ B6)))))).
% 3.08/1.78 thf(fact_124_UnI1, axiom, (! [A:$tType,C2:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C2) @ A6) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A6) @ B6)))))).
% 3.08/1.78 thf(fact_125_UnE, axiom, (! [A:$tType,C2:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A6) @ B6)) => ((~ ((((member @ A) @ C2) @ A6))) => (((member @ A) @ C2) @ B6)))))).
% 3.08/1.78 thf(fact_126_vimage__Collect, axiom, (! [B:$tType,A:$tType,P:(B > $o),F:(A > B),Q:(A > $o)]: (((! [X4:A]: (((P @ (F @ X4)) = (Q @ X4)))) => (((((vimage @ A) @ B) @ F) @ ((collect @ B) @ P)) = ((collect @ A) @ Q)))))).
% 3.08/1.78 thf(fact_127_vimageI2, axiom, (! [B:$tType,A:$tType,F:(B > A),A3:B,A6:(set @ A)]: (((((member @ A) @ (F @ A3)) @ A6) => (((member @ B) @ A3) @ ((((vimage @ B) @ A) @ F) @ A6)))))).
% 3.08/1.78 thf(fact_128_vimageE, axiom, (! [A:$tType,B:$tType,A3:A,F:(A > B),B6:(set @ B)]: (((((member @ A) @ A3) @ ((((vimage @ A) @ B) @ F) @ B6)) => (((member @ B) @ (F @ A3)) @ B6))))).
% 3.08/1.78 thf(fact_129_vimageD, axiom, (! [A:$tType,B:$tType,A3:A,F:(A > B),A6:(set @ B)]: (((((member @ A) @ A3) @ ((((vimage @ A) @ B) @ F) @ A6)) => (((member @ B) @ (F @ A3)) @ A6))))).
% 3.08/1.78 thf(fact_130_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X:B,A6:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A6)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A6))))).
% 3.08/1.78 thf(fact_131_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 3.08/1.78 thf(fact_132_cut__apply, axiom, (! [B:$tType,A:$tType,X:A,A3:A,R3:(set @ ((product_prod @ A) @ A)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ A3)) @ R3) => (((((((cut @ A) @ B) @ F) @ R3) @ A3) @ X) = (F @ X)))))).
% 3.08/1.78 thf(fact_133_cuts__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),R3:(set @ ((product_prod @ A) @ A)),X:A,G4:(A > B)]: ((((((((cut @ A) @ B) @ F) @ R3) @ X) = (((((cut @ A) @ B) @ G4) @ R3) @ X)) = (! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X)) @ R3) => ((F @ Y4) = (G4 @ Y4))))))))).
% 3.08/1.78 thf(fact_134_old_Osum_Osimps_I8_J, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > T),F22:(B > T),B3:B]: ((((((((sum_rec_sum @ A) @ T) @ B) @ F1) @ F22) @ (((sum_Inr @ B) @ A) @ B3)) = (F22 @ B3))))).
% 3.08/1.78 thf(fact_135_old_Osum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,B3:B,B4:B]: ((((((sum_Inr @ B) @ A) @ B3) = (((sum_Inr @ B) @ A) @ B4)) = (B3 = B4))))).
% 3.08/1.78 thf(fact_136_sum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,X2:B,Y2:B]: ((((((sum_Inr @ B) @ A) @ X2) = (((sum_Inr @ B) @ A) @ Y2)) = (X2 = Y2))))).
% 3.08/1.78 thf(fact_137_finite__imageI, axiom, (! [B:$tType,A:$tType,F6:(set @ A),H:(A > B)]: ((((finite_finite2 @ A) @ F6) => ((finite_finite2 @ B) @ ((((image @ A) @ B) @ H) @ F6)))))).
% 3.08/1.78 thf(fact_138_InrI, axiom, (! [B:$tType,A:$tType,B3:A,B6:(set @ A),A6:(set @ B)]: (((((member @ A) @ B3) @ B6) => (((member @ ((sum_sum @ B) @ A)) @ (((sum_Inr @ A) @ B) @ B3)) @ ((((sum_Plus @ B) @ A) @ A6) @ B6)))))).
% 3.08/1.78 thf(fact_139_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) = F)))).
% 3.08/1.78 thf(fact_140_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F)) = F)))).
% 3.08/1.78 thf(fact_141_image__Un, axiom, (! [A:$tType,B:$tType,F:(B > A),A6:(set @ B),B6:(set @ B)]: ((((((image @ B) @ A) @ F) @ (((sup_sup @ (set @ B)) @ A6) @ B6)) = (((sup_sup @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A6)) @ ((((image @ B) @ A) @ F) @ B6)))))).
% 3.08/1.78 thf(fact_142_sum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,X1:A,X2:B]: (((((sum_Inl @ A) @ B) @ X1) != (((sum_Inr @ B) @ A) @ X2))))).
% 3.08/1.78 thf(fact_143_old_Osum_Odistinct_I2_J, axiom, (! [B8:$tType,A8:$tType,B9:B8,A9:A8]: (((((sum_Inr @ B8) @ A8) @ B9) != (((sum_Inl @ A8) @ B8) @ A9))))).
% 3.08/1.78 thf(fact_144_old_Osum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,A3:A,B4:B]: (((((sum_Inl @ A) @ B) @ A3) != (((sum_Inr @ B) @ A) @ B4))))).
% 3.08/1.78 thf(fact_145_sumE, axiom, (! [A:$tType,B:$tType,S:((sum_sum @ A) @ B)]: (((! [X4:A]: ((S != (((sum_Inl @ A) @ B) @ X4)))) => (~ ((! [Y5:B]: ((S != (((sum_Inr @ B) @ A) @ Y5)))))))))).
% 3.08/1.78 thf(fact_146_Inr__not__Inl, axiom, (! [B:$tType,A:$tType,B3:B,A3:A]: (((((sum_Inr @ B) @ A) @ B3) != (((sum_Inl @ A) @ B) @ A3))))).
% 3.08/1.78 thf(fact_147_split__sum__ex, axiom, (! [B:$tType,A:$tType]: (((^ [P3:(((sum_sum @ A) @ B) > $o)]: ((? [X6:((sum_sum @ A) @ B)]: ((P3 @ X6))))) = (^ [P4:(((sum_sum @ A) @ B) > $o)]: (((? [X3:A]: ((P4 @ (((sum_Inl @ A) @ B) @ X3)))) | (? [X3:B]: ((P4 @ (((sum_Inr @ B) @ A) @ X3))))))))))).
% 3.08/1.78 thf(fact_148_split__sum__all, axiom, (! [B:$tType,A:$tType]: (((^ [P3:(((sum_sum @ A) @ B) > $o)]: ((! [X6:((sum_sum @ A) @ B)]: ((P3 @ X6))))) = (^ [P4:(((sum_sum @ A) @ B) > $o)]: (((! [X3:A]: ((P4 @ (((sum_Inl @ A) @ B) @ X3)))) & (! [X3:B]: ((P4 @ (((sum_Inr @ B) @ A) @ X3))))))))))).
% 3.08/1.78 thf(fact_149_old_Osum_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((sum_sum @ A) @ B)]: (((! [A5:A]: ((Y != (((sum_Inl @ A) @ B) @ A5)))) => (~ ((! [B5:B]: ((Y != (((sum_Inr @ B) @ A) @ B5)))))))))).
% 3.08/1.78 thf(fact_150_old_Osum_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((sum_sum @ A) @ B) > $o),Sum:((sum_sum @ A) @ B)]: (((! [A5:A]: ((P @ (((sum_Inl @ A) @ B) @ A5)))) => ((! [B5:B]: ((P @ (((sum_Inr @ B) @ A) @ B5)))) => (P @ Sum)))))).
% 3.08/1.78 thf(fact_151_Inr__inject, axiom, (! [A:$tType,B:$tType,X:B,Y:B]: ((((((sum_Inr @ B) @ A) @ X) = (((sum_Inr @ B) @ A) @ Y)) => (X = Y))))).
% 3.08/1.78 thf(fact_152_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z3:A,C2:(B > (C > (set @ A))),P2:((product_prod @ B) @ C)]: (((((member @ A) @ Z3) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ P2)) => (~ ((! [X4:B,Y5:C]: (((P2 = ((((product_Pair @ B) @ C) @ X4) @ Y5)) => (~ ((((member @ A) @ Z3) @ ((C2 @ X4) @ Y5))))))))))))).
% 3.08/1.78 thf(fact_153_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),X1:A,X2:B]: (((((((product_case_prod @ A) @ B) @ C) @ F) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = ((F @ X1) @ X2))))).
% 3.08/1.78 thf(fact_154_PlusE, axiom, (! [A:$tType,B:$tType,U:((sum_sum @ A) @ B),A6:(set @ A),B6:(set @ B)]: (((((member @ ((sum_sum @ A) @ B)) @ U) @ ((((sum_Plus @ A) @ B) @ A6) @ B6)) => ((! [X4:A]: (((((member @ A) @ X4) @ A6) => (U != (((sum_Inl @ A) @ B) @ X4))))) => (~ ((! [Y5:B]: (((((member @ B) @ Y5) @ B6) => (U != (((sum_Inr @ B) @ A) @ Y5)))))))))))).
% 3.08/1.78 thf(fact_155_Inl__Inr__False, axiom, (! [A:$tType,B:$tType,X:A,Y:B]: (((((sum_Inl @ A) @ B) @ X) != (((sum_Inr @ B) @ A) @ Y))))).
% 3.08/1.78 thf(fact_156_Inr__Inl__False, axiom, (! [B:$tType,A:$tType,X:B,Y:A]: (((((sum_Inr @ B) @ A) @ X) != (((sum_Inl @ A) @ B) @ Y))))).
% 3.08/1.78 thf(fact_157_Inl__Inr__image__cong, axiom, (! [B:$tType,A:$tType,A6:(set @ ((sum_sum @ A) @ B)),B6:(set @ ((sum_sum @ A) @ B))]: (((((((vimage @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ A6) = ((((vimage @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ B6)) => ((((((vimage @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ A6) = ((((vimage @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ B6)) => (A6 = B6)))))).
% 3.08/1.78 thf(fact_158_obj__sumE__f, axiom, (! [A:$tType,C:$tType,B:$tType,S:B,F:(((sum_sum @ A) @ C) > B),P:$o]: (((! [X4:A]: (((S = (F @ (((sum_Inl @ A) @ C) @ X4))) => P))) => ((! [X4:C]: (((S = (F @ (((sum_Inr @ C) @ A) @ X4))) => P))) => (! [X5:((sum_sum @ A) @ C)]: (((S = (F @ X5)) => P)))))))).
% 3.08/1.78 thf(fact_159_Plus__def, axiom, (! [B:$tType,A:$tType]: ((((sum_Plus @ A) @ B) = (^ [A7:(set @ A),B7:(set @ B)]: ((((sup_sup @ (set @ ((sum_sum @ A) @ B))) @ ((((image @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ A7)) @ ((((image @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ B7)))))))).
% 3.08/1.78 thf(fact_160_obj__sumE, axiom, (! [A:$tType,B:$tType,S:((sum_sum @ A) @ B)]: (((! [X4:A]: ((S != (((sum_Inl @ A) @ B) @ X4)))) => (~ ((! [X4:B]: ((S != (((sum_Inr @ B) @ A) @ X4)))))))))).
% 3.08/1.78 thf(fact_161_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F:(A > (B > C)),G4:(A > (B > C)),P2:((product_prod @ A) @ B)]: (((! [X4:A,Y5:B]: (((((((product_Pair @ A) @ B) @ X4) @ Y5) = Q2) => (((F @ X4) @ Y5) = ((G4 @ X4) @ Y5))))) => ((P2 = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F) @ P2) = (((((product_case_prod @ A) @ B) @ C) @ G4) @ Q2))))))).
% 3.08/1.78 thf(fact_162_Sumr_Osimps, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C),X:B]: (((((((sum_Sumr @ B) @ C) @ A) @ F) @ (((sum_Inr @ B) @ A) @ X)) = (F @ X))))).
% 3.08/1.78 thf(fact_163_Sumr__inject, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C),G4:(B > C)]: (((((((sum_Sumr @ B) @ C) @ A) @ F) = ((((sum_Sumr @ B) @ C) @ A) @ G4)) => (F = G4))))).
% 3.08/1.78 thf(fact_164_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B3:A,F:(B > A),X:B,C2:C,G4:(B > C),A6:(set @ B)]: (((B3 = (F @ X)) => ((C2 = (G4 @ X)) => ((((member @ B) @ X) @ A6) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B3) @ C2)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A6) @ F) @ G4)))))))).
% 3.08/1.78 thf(fact_165_surj__swap, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 3.08/1.78 thf(fact_166_finite__vimageD, axiom, (! [A:$tType,B:$tType,H:(A > B),F6:(set @ B)]: ((((finite_finite2 @ A) @ ((((vimage @ A) @ B) @ H) @ F6)) => ((((((image @ A) @ B) @ H) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((finite_finite2 @ B) @ F6)))))).
% 3.08/1.78 thf(fact_167_sup__top__left, axiom, (! [A:$tType]: ((((bounded_lattice_top @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ (top_top @ A)) @ X) = (top_top @ A)))))))).
% 3.08/1.78 thf(fact_168_sup__top__right, axiom, (! [A:$tType]: ((((bounded_lattice_top @ A) @ (type2 @ A)) => (! [X:A]: (((((sup_sup @ A) @ X) @ (top_top @ A)) = (top_top @ A)))))))).
% 3.08/1.78 thf(fact_169_vimage__UNIV, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((vimage @ A) @ B) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A)))))).
% 3.08/1.78 thf(fact_170_UNIV__Plus__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((sum_Plus @ A) @ B) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) = (top_top @ (set @ ((sum_sum @ A) @ B))))))).
% 3.08/1.79 thf(fact_171_finite__Plus__UNIV__iff, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ (top_top @ (set @ ((sum_sum @ A) @ B)))) = (((finite_finite2 @ A) @ (top_top @ (set @ A))) & ((finite_finite2 @ B) @ (top_top @ (set @ B)))))))).
% 3.08/1.79 thf(fact_172_Un__UNIV__left, axiom, (! [A:$tType,B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ (top_top @ (set @ A))) @ B6) = (top_top @ (set @ A)))))).
% 3.08/1.79 thf(fact_173_Un__UNIV__right, axiom, (! [A:$tType,A6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 3.08/1.79 thf(fact_174_finite__UNIV, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 3.08/1.79 thf(fact_175_infinite__UNIV__char__0, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))))))).
% 3.08/1.79 thf(fact_176_ex__new__if__finite, axiom, (! [A:$tType,A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))) => (((finite_finite2 @ A) @ A6) => (? [A5:A]: ((~ ((((member @ A) @ A5) @ A6)))))))))).
% 3.08/1.79 thf(fact_177_Finite__Set_Ofinite__set, axiom, (! [A:$tType]: ((((finite_finite2 @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 3.08/1.79 thf(fact_178_finite__prod, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (((finite_finite2 @ A) @ (top_top @ (set @ A))) & ((finite_finite2 @ B) @ (top_top @ (set @ B)))))))).
% 3.08/1.79 thf(fact_179_finite__Prod__UNIV, axiom, (! [B:$tType,A:$tType]: ((((finite_finite2 @ A) @ (top_top @ (set @ A))) => (((finite_finite2 @ B) @ (top_top @ (set @ B))) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B))))))))).
% 3.08/1.79 thf(fact_180_finite__fun__UNIVD2, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ (A > B)) @ (top_top @ (set @ (A > B)))) => ((finite_finite2 @ B) @ (top_top @ (set @ B))))))).
% 3.08/1.79 thf(fact_181_UNIV__sum, axiom, (! [A:$tType,B:$tType]: (((top_top @ (set @ ((sum_sum @ A) @ B))) = (((sup_sup @ (set @ ((sum_sum @ A) @ B))) @ ((((image @ A) @ ((sum_sum @ A) @ B)) @ ((sum_Inl @ A) @ B)) @ (top_top @ (set @ A)))) @ ((((image @ B) @ ((sum_sum @ A) @ B)) @ ((sum_Inr @ B) @ A)) @ (top_top @ (set @ B)))))))).
% 3.08/1.79 thf(fact_182_finite__option__UNIV, axiom, (! [A:$tType]: ((((finite_finite2 @ (option @ A)) @ (top_top @ (set @ (option @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 3.08/1.79 thf(fact_183_surj__image__vimage__eq, axiom, (! [B:$tType,A:$tType,F:(B > A),A6:(set @ A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ F) @ ((((vimage @ B) @ A) @ F) @ A6)) = A6))))).
% 3.08/1.79 thf(fact_184_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(A > B),G4:(C > D)]: (((((((image @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G4) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F) @ G4)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 3.08/1.79 thf(fact_185_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G4:(D > B),A3:C,B3:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G4) @ ((((product_Pair @ C) @ D) @ A3) @ B3)) = ((((product_Pair @ A) @ B) @ (F @ A3)) @ (G4 @ B3)))))).
% 3.08/1.79 thf(fact_186_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A3:A,B3:B,R3:(set @ ((product_prod @ A) @ B)),F:(A > C),G4:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R3) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F @ A3)) @ (G4 @ B3))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G4)) @ R3)))))).
% 3.08/1.79 thf(fact_187_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C2:((product_prod @ A) @ B),F:(C > A),G4:(D > B),R3:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G4)) @ R3)) => (~ ((! [X4:C,Y5:D]: (((C2 = ((((product_Pair @ A) @ B) @ (F @ X4)) @ (G4 @ Y5))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X4) @ Y5)) @ R3)))))))))))).
% 3.08/1.79 thf(fact_188_finite__vimageD_H, axiom, (! [A:$tType,B:$tType,F:(A > B),A6:(set @ B)]: ((((finite_finite2 @ A) @ ((((vimage @ A) @ B) @ F) @ A6)) => ((((ord_less_eq @ (set @ B)) @ A6) @ ((((image @ A) @ B) @ F) @ (top_top @ (set @ A)))) => ((finite_finite2 @ B) @ A6)))))).
% 3.08/1.79 thf(fact_189_image__vimage__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),A6:(set @ A)]: ((((((image @ B) @ A) @ F) @ ((((vimage @ B) @ A) @ F) @ A6)) = (((inf_inf @ (set @ A)) @ A6) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 3.08/1.79 thf(fact_190_surj__vimage__empty, axiom, (! [B:$tType,A:$tType,F:(B > A),A6:(set @ A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((((vimage @ B) @ A) @ F) @ A6) = (bot_bot @ (set @ B))) = (A6 = (bot_bot @ (set @ A)))))))).
% 3.08/1.79 thf(fact_191_le__sup__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ Y)) @ Z3) = ((((ord_less_eq @ A) @ X) @ Z3) & (((ord_less_eq @ A) @ Y) @ Z3))))))))).
% 3.08/1.79 thf(fact_192_sup_Obounded__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A3) = ((((ord_less_eq @ A) @ B3) @ A3) & (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 3.08/1.79 thf(fact_193_bot__eq__sup__iff, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X) @ Y)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 3.08/1.79 thf(fact_194_sup__eq__bot__iff, axiom, (! [A:$tType]: ((((bounded_lattice_bot @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((((sup_sup @ A) @ X) @ Y) = (bot_bot @ A)) = ((X = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))).
% 3.08/1.79 thf(fact_195_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A3) = A3))))))).
% 3.08/1.79 thf(fact_196_sup__bot_Oright__neutral, axiom, (! [A:$tType]: ((((bounde1808546759up_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ (bot_bot @ A)) = A3))))))).
% 3.08/1.79 thf(fact_197_inf__sup__absorb, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)) = X))))))).
% 3.08/1.79 thf(fact_198_sup__inf__absorb, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ X) @ Y)) = X))))))).
% 3.08/1.79 thf(fact_199_finite__Int, axiom, (! [A:$tType,F6:(set @ A),G6:(set @ A)]: (((((finite_finite2 @ A) @ F6) | ((finite_finite2 @ A) @ G6)) => ((finite_finite2 @ A) @ (((inf_inf @ (set @ A)) @ F6) @ G6)))))).
% 3.08/1.79 thf(fact_200_Un__empty, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A6) @ B6) = (bot_bot @ (set @ A))) = ((A6 = (bot_bot @ (set @ A))) & (B6 = (bot_bot @ (set @ A)))))))).
% 3.08/1.79 thf(fact_201_Un__subset__iff, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A6) @ B6)) @ C4) = ((((ord_less_eq @ (set @ A)) @ A6) @ C4) & (((ord_less_eq @ (set @ A)) @ B6) @ C4)))))).
% 3.08/1.79 thf(fact_202_vimage__empty, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((vimage @ A) @ B) @ F) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))).
% 3.08/1.79 thf(fact_203_vimage__Int, axiom, (! [A:$tType,B:$tType,F:(A > B),A6:(set @ B),B6:(set @ B)]: ((((((vimage @ A) @ B) @ F) @ (((inf_inf @ (set @ B)) @ A6) @ B6)) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ A6)) @ ((((vimage @ A) @ B) @ F) @ B6)))))).
% 3.08/1.79 thf(fact_204_Plus__eq__empty__conv, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B6:(set @ B)]: (((((((sum_Plus @ A) @ B) @ A6) @ B6) = (bot_bot @ (set @ ((sum_sum @ A) @ B)))) = ((A6 = (bot_bot @ (set @ A))) & (B6 = (bot_bot @ (set @ B)))))))).
% 3.08/1.79 thf(fact_205_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))).
% 3.08/1.79 thf(fact_206_infinite__imp__nonempty, axiom, (! [A:$tType,S3:(set @ A)]: (((~ (((finite_finite2 @ A) @ S3))) => (S3 != (bot_bot @ (set @ A))))))).
% 3.08/1.79 thf(fact_207_finite__subset, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((finite_finite2 @ A) @ B6) => ((finite_finite2 @ A) @ A6)))))).
% 3.08/1.79 thf(fact_208_infinite__super, axiom, (! [A:$tType,S3:(set @ A),T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S3) @ T3) => ((~ (((finite_finite2 @ A) @ S3))) => (~ (((finite_finite2 @ A) @ T3)))))))).
% 3.08/1.79 thf(fact_209_rev__finite__subset, axiom, (! [A:$tType,B6:(set @ A),A6:(set @ A)]: ((((finite_finite2 @ A) @ B6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((finite_finite2 @ A) @ A6)))))).
% 3.08/1.79 thf(fact_210_finite__subset__image, axiom, (! [A:$tType,B:$tType,B6:(set @ A),F:(B > A),A6:(set @ B)]: ((((finite_finite2 @ A) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F) @ A6)) => (? [C5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C5) @ A6) & (((finite_finite2 @ B) @ C5) & (B6 = ((((image @ B) @ A) @ F) @ C5))))))))))).
% 3.08/1.79 thf(fact_211_finite__surj, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B6:(set @ B),F:(A > B)]: ((((finite_finite2 @ A) @ A6) => ((((ord_less_eq @ (set @ B)) @ B6) @ ((((image @ A) @ B) @ F) @ A6)) => ((finite_finite2 @ B) @ B6)))))).
% 3.08/1.79 thf(fact_212_Un__Int__assoc__eq, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C4:(set @ A)]: ((((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) @ C4) = (((inf_inf @ (set @ A)) @ A6) @ (((sup_sup @ (set @ A)) @ B6) @ C4))) = (((ord_less_eq @ (set @ A)) @ C4) @ A6))))).
% 3.08/1.79 thf(fact_213_distrib__imp1, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((! [X4:A,Y5:A,Z:A]: (((((inf_inf @ A) @ X4) @ (((sup_sup @ A) @ Y5) @ Z)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X4) @ Y5)) @ (((inf_inf @ A) @ X4) @ Z))))) => ((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z3)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z3)))))))))).
% 3.08/1.79 thf(fact_214_distrib__imp2, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((! [X4:A,Y5:A,Z:A]: (((((sup_sup @ A) @ X4) @ (((inf_inf @ A) @ Y5) @ Z)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X4) @ Y5)) @ (((sup_sup @ A) @ X4) @ Z))))) => ((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z3)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z3)))))))))).
% 3.08/1.79 thf(fact_215_distrib__inf__le, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z3))) @ (((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z3))))))))).
% 3.08/1.79 thf(fact_216_distrib__sup__le, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z3))) @ (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z3))))))))).
% 3.08/1.79 thf(fact_217_inf__sup__distrib1, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((inf_inf @ A) @ X) @ (((sup_sup @ A) @ Y) @ Z3)) = (((sup_sup @ A) @ (((inf_inf @ A) @ X) @ Y)) @ (((inf_inf @ A) @ X) @ Z3))))))))).
% 3.08/1.79 thf(fact_218_inf__sup__distrib2, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [Y:A,Z3:A,X:A]: (((((inf_inf @ A) @ (((sup_sup @ A) @ Y) @ Z3)) @ X) = (((sup_sup @ A) @ (((inf_inf @ A) @ Y) @ X)) @ (((inf_inf @ A) @ Z3) @ X))))))))).
% 3.08/1.79 thf(fact_219_sup__inf__distrib1, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((sup_sup @ A) @ X) @ (((inf_inf @ A) @ Y) @ Z3)) = (((inf_inf @ A) @ (((sup_sup @ A) @ X) @ Y)) @ (((sup_sup @ A) @ X) @ Z3))))))))).
% 3.08/1.79 thf(fact_220_sup__inf__distrib2, axiom, (! [A:$tType]: ((((distrib_lattice @ A) @ (type2 @ A)) => (! [Y:A,Z3:A,X:A]: (((((sup_sup @ A) @ (((inf_inf @ A) @ Y) @ Z3)) @ X) = (((inf_inf @ A) @ (((sup_sup @ A) @ Y) @ X)) @ (((sup_sup @ A) @ Z3) @ X))))))))).
% 3.08/1.79 thf(fact_221_Un__empty__left, axiom, (! [A:$tType,B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ (bot_bot @ (set @ A))) @ B6) = B6)))).
% 3.08/1.79 thf(fact_222_Un__empty__right, axiom, (! [A:$tType,A6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ (bot_bot @ (set @ A))) = A6)))).
% 3.08/1.79 thf(fact_223_Un__mono, axiom, (! [A:$tType,A6:(set @ A),C4:(set @ A),B6:(set @ A),D3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ C4) => ((((ord_less_eq @ (set @ A)) @ B6) @ D3) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A6) @ B6)) @ (((sup_sup @ (set @ A)) @ C4) @ D3))))))).
% 3.08/1.79 thf(fact_224_Un__least, axiom, (! [A:$tType,A6:(set @ A),C4:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ C4) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A6) @ B6)) @ C4)))))).
% 3.08/1.79 thf(fact_225_Un__upper1, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A6) @ (((sup_sup @ (set @ A)) @ A6) @ B6))))).
% 3.08/1.79 thf(fact_226_Un__upper2, axiom, (! [A:$tType,B6:(set @ A),A6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B6) @ (((sup_sup @ (set @ A)) @ A6) @ B6))))).
% 3.08/1.79 thf(fact_227_Un__absorb1, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((sup_sup @ (set @ A)) @ A6) @ B6) = B6))))).
% 3.08/1.79 thf(fact_228_Un__absorb2, axiom, (! [A:$tType,B6:(set @ A),A6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B6) @ A6) => ((((sup_sup @ (set @ A)) @ A6) @ B6) = A6))))).
% 3.08/1.79 thf(fact_229_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: (((((sup_sup @ (set @ A)) @ A7) @ B7) = B7))))))).
% 3.08/1.79 thf(fact_230_inf__sup__ord_I4_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 3.08/1.79 thf(fact_231_inf__sup__ord_I3_J, axiom, (! [A:$tType]: ((((lattice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 3.08/1.79 thf(fact_232_le__supE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A,X:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ X) => (~ (((((ord_less_eq @ A) @ A3) @ X) => (~ ((((ord_less_eq @ A) @ B3) @ X))))))))))))).
% 3.08/1.79 thf(fact_233_le__supI, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,X:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ X) => ((((ord_less_eq @ A) @ B3) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ X))))))))).
% 3.08/1.79 thf(fact_234_sup__ge1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 3.08/1.79 thf(fact_235_sup__ge2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [Y:A,X:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X) @ Y)))))))).
% 3.08/1.79 thf(fact_236_le__supI1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,A3:A,B3:A]: (((((ord_less_eq @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 3.08/1.79 thf(fact_237_le__supI2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,B3:A,A3:A]: (((((ord_less_eq @ A) @ X) @ B3) => (((ord_less_eq @ A) @ X) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 3.08/1.79 thf(fact_238_sup_Omono, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [C2:A,A3:A,D4:A,B3:A]: (((((ord_less_eq @ A) @ C2) @ A3) => ((((ord_less_eq @ A) @ D4) @ B3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C2) @ D4)) @ (((sup_sup @ A) @ A3) @ B3)))))))))).
% 3.08/1.79 thf(fact_239_sup__mono, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,C2:A,B3:A,D4:A]: (((((ord_less_eq @ A) @ A3) @ C2) => ((((ord_less_eq @ A) @ B3) @ D4) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ (((sup_sup @ A) @ C2) @ D4)))))))))).
% 3.08/1.79 thf(fact_240_sup__least, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [Y:A,X:A,Z3:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ Z3) @ X) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y) @ Z3)) @ X))))))))).
% 3.08/1.79 thf(fact_241_le__iff__sup, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X3:A,Y4:A]: (((((sup_sup @ A) @ X3) @ Y4) = Y4)))))))).
% 3.08/1.79 thf(fact_242_sup_OorderE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => (A3 = (((sup_sup @ A) @ A3) @ B3))))))))).
% 3.08/1.79 thf(fact_243_sup_OorderI, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((A3 = (((sup_sup @ A) @ A3) @ B3)) => (((ord_less_eq @ A) @ B3) @ A3)))))))).
% 3.08/1.79 thf(fact_244_sup__unique, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [F:(A > (A > A)),X:A,Y:A]: (((! [X4:A,Y5:A]: ((((ord_less_eq @ A) @ X4) @ ((F @ X4) @ Y5)))) => ((! [X4:A,Y5:A]: ((((ord_less_eq @ A) @ Y5) @ ((F @ X4) @ Y5)))) => ((! [X4:A,Y5:A,Z:A]: (((((ord_less_eq @ A) @ Y5) @ X4) => ((((ord_less_eq @ A) @ Z) @ X4) => (((ord_less_eq @ A) @ ((F @ Y5) @ Z)) @ X4))))) => ((((sup_sup @ A) @ X) @ Y) = ((F @ X) @ Y))))))))))).
% 3.08/1.79 thf(fact_245_sup_Oabsorb1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((sup_sup @ A) @ A3) @ B3) = A3)))))))).
% 3.08/1.79 thf(fact_246_sup_Oabsorb2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((sup_sup @ A) @ A3) @ B3) = B3)))))))).
% 3.08/1.79 thf(fact_247_sup__absorb1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((sup_sup @ A) @ X) @ Y) = X)))))))).
% 3.08/1.79 thf(fact_248_sup__absorb2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((sup_sup @ A) @ X) @ Y) = Y)))))))).
% 3.08/1.79 thf(fact_249_sup_OboundedE, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A3) => (~ (((((ord_less_eq @ A) @ B3) @ A3) => (~ ((((ord_less_eq @ A) @ C2) @ A3))))))))))))).
% 3.08/1.79 thf(fact_250_sup_OboundedI, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C2) @ A3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A3))))))))).
% 3.08/1.79 thf(fact_251_sup_Oorder__iff, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B2:A,A2:A]: ((A2 = (((sup_sup @ A) @ A2) @ B2))))))))).
% 3.08/1.79 thf(fact_252_sup_Ocobounded1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A3:A,B3:A]: ((((ord_less_eq @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B3)))))))).
% 3.08/1.79 thf(fact_253_sup_Ocobounded2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [B3:A,A3:A]: ((((ord_less_eq @ A) @ B3) @ (((sup_sup @ A) @ A3) @ B3)))))))).
% 3.08/1.79 thf(fact_254_sup_Oabsorb__iff1, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B2:A,A2:A]: (((((sup_sup @ A) @ A2) @ B2) = A2)))))))).
% 3.08/1.79 thf(fact_255_sup_Oabsorb__iff2, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A2:A,B2:A]: (((((sup_sup @ A) @ A2) @ B2) = B2)))))))).
% 3.08/1.79 thf(tcon_FSet_Ofset___Lattices_Obounded__lattice, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((bounded_lattice @ (fset @ A8)) @ (type2 @ (fset @ A8))))))).
% 3.08/1.79 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_1, axiom, ((bounded_lattice @ $o) @ (type2 @ $o))).
% 3.08/1.79 thf(tcon_Set_Oset___Lattices_Obounded__lattice_2, axiom, (! [A8:$tType]: (((bounded_lattice @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.08/1.79 thf(tcon_fun___Lattices_Obounded__lattice_3, axiom, (! [A8:$tType,A10:$tType]: ((((bounded_lattice @ A10) @ (type2 @ A10)) => ((bounded_lattice @ (A8 > A10)) @ (type2 @ (A8 > A10))))))).
% 3.08/1.79 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A8:$tType,A10:$tType]: ((((bounded_lattice @ A10) @ (type2 @ A10)) => ((bounde1808546759up_bot @ (A8 > A10)) @ (type2 @ (A8 > A10))))))).
% 3.08/1.79 thf(tcon_fun___Lattices_Obounded__lattice__top, axiom, (! [A8:$tType,A10:$tType]: ((((bounded_lattice @ A10) @ (type2 @ A10)) => ((bounded_lattice_top @ (A8 > A10)) @ (type2 @ (A8 > A10))))))).
% 3.08/1.79 thf(tcon_fun___Lattices_Obounded__lattice__bot, axiom, (! [A8:$tType,A10:$tType]: ((((bounded_lattice @ A10) @ (type2 @ A10)) => ((bounded_lattice_bot @ (A8 > A10)) @ (type2 @ (A8 > A10))))))).
% 3.08/1.79 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A8:$tType,A10:$tType]: ((((semilattice_sup @ A10) @ (type2 @ A10)) => ((semilattice_sup @ (A8 > A10)) @ (type2 @ (A8 > A10))))))).
% 3.08/1.79 thf(tcon_fun___Lattices_Odistrib__lattice, axiom, (! [A8:$tType,A10:$tType]: ((((distrib_lattice @ A10) @ (type2 @ A10)) => ((distrib_lattice @ (A8 > A10)) @ (type2 @ (A8 > A10))))))).
% 3.08/1.79 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A10:$tType]: (((((finite_finite @ A8) @ (type2 @ A8)) & ((finite_finite @ A10) @ (type2 @ A10))) => ((finite_finite @ (A8 > A10)) @ (type2 @ (A8 > A10))))))).
% 3.08/1.79 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A10:$tType]: ((((lattice @ A10) @ (type2 @ A10)) => ((lattice @ (A8 > A10)) @ (type2 @ (A8 > A10))))))).
% 3.08/1.79 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_4, axiom, (! [A8:$tType]: (((bounde1808546759up_bot @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.08/1.79 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_5, axiom, (! [A8:$tType]: (((bounded_lattice_top @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.08/1.79 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_6, axiom, (! [A8:$tType]: (((bounded_lattice_bot @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.08/1.79 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_7, axiom, (! [A8:$tType]: (((semilattice_sup @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.08/1.79 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_8, axiom, (! [A8:$tType]: (((distrib_lattice @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.08/1.79 thf(tcon_Set_Oset___Finite__Set_Ofinite_9, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((finite_finite @ (set @ A8)) @ (type2 @ (set @ A8))))))).
% 3.08/1.79 thf(tcon_Set_Oset___Lattices_Olattice_10, axiom, (! [A8:$tType]: (((lattice @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.08/1.79 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_11, axiom, ((bounde1808546759up_bot @ $o) @ (type2 @ $o))).
% 3.08/1.79 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_12, axiom, ((bounded_lattice_top @ $o) @ (type2 @ $o))).
% 3.08/1.79 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_13, axiom, ((bounded_lattice_bot @ $o) @ (type2 @ $o))).
% 3.08/1.79 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14, axiom, ((semilattice_sup @ $o) @ (type2 @ $o))).
% 3.08/1.79 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_15, axiom, ((distrib_lattice @ $o) @ (type2 @ $o))).
% 3.08/1.79 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16, axiom, ((finite_finite @ $o) @ (type2 @ $o))).
% 3.08/1.79 thf(tcon_HOL_Obool___Lattices_Olattice_17, axiom, ((lattice @ $o) @ (type2 @ $o))).
% 3.08/1.79 thf(tcon_FSet_Ofset___Lattices_Obounded__semilattice__sup__bot_18, axiom, (! [A8:$tType]: (((bounde1808546759up_bot @ (fset @ A8)) @ (type2 @ (fset @ A8)))))).
% 3.08/1.79 thf(tcon_FSet_Ofset___Lattices_Obounded__lattice__top_19, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((bounded_lattice_top @ (fset @ A8)) @ (type2 @ (fset @ A8))))))).
% 3.08/1.79 thf(tcon_FSet_Ofset___Lattices_Obounded__lattice__bot_20, axiom, (! [A8:$tType]: (((bounded_lattice_bot @ (fset @ A8)) @ (type2 @ (fset @ A8)))))).
% 3.08/1.79 thf(tcon_FSet_Ofset___Lattices_Osemilattice__sup_21, axiom, (! [A8:$tType]: (((semilattice_sup @ (fset @ A8)) @ (type2 @ (fset @ A8)))))).
% 3.08/1.79 thf(tcon_FSet_Ofset___Lattices_Odistrib__lattice_22, axiom, (! [A8:$tType]: (((distrib_lattice @ (fset @ A8)) @ (type2 @ (fset @ A8)))))).
% 3.08/1.79 thf(tcon_FSet_Ofset___Finite__Set_Ofinite_23, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((finite_finite @ (fset @ A8)) @ (type2 @ (fset @ A8))))))).
% 3.08/1.79 thf(tcon_FSet_Ofset___Lattices_Olattice_24, axiom, (! [A8:$tType]: (((lattice @ (fset @ A8)) @ (type2 @ (fset @ A8)))))).
% 3.08/1.79 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_25, axiom, (! [A8:$tType,A10:$tType]: (((((finite_finite @ A8) @ (type2 @ A8)) & ((finite_finite @ A10) @ (type2 @ A10))) => ((finite_finite @ ((sum_sum @ A8) @ A10)) @ (type2 @ ((sum_sum @ A8) @ A10))))))).
% 3.08/1.79 thf(tcon_Option_Ooption___Finite__Set_Ofinite_26, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((finite_finite @ (option @ A8)) @ (type2 @ (option @ A8))))))).
% 3.08/1.79 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27, axiom, (! [A8:$tType,A10:$tType]: (((((finite_finite @ A8) @ (type2 @ A8)) & ((finite_finite @ A10) @ (type2 @ A10))) => ((finite_finite @ ((product_prod @ A8) @ A10)) @ (type2 @ ((product_prod @ A8) @ A10))))))).
% 3.08/1.79 thf(conj_0, conjecture, (((parall1518086719_Nplus @ (root @ tr1a)) @ (root @ tr2a)) = ((parall1518086719_Nplus @ (root @ tr2a)) @ (root @ tr1a)))).
% 3.08/1.79 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.q1d5il6I29/DTF2DTF22976.p
% 3.08/1.79 ------------------------
% 3.27/1.84 ---- Cleaned THF ---
% 3.27/1.84 thf(ty_t_Product__Type_Oprod,type,
% 3.27/1.84 product_prod: $tType > $tType > $tType ).
% 3.27/1.84
% 3.27/1.84 thf(ty_t_Sum__Type_Osum,type,
% 3.27/1.84 sum_sum: $tType > $tType > $tType ).
% 3.27/1.84
% 3.27/1.84 thf(ty_t_Option_Ooption,type,
% 3.27/1.84 option: $tType > $tType ).
% 3.27/1.84
% 3.27/1.84 thf(ty_t_DTree_Odtree,type,
% 3.27/1.84 dtree: $tType ).
% 3.27/1.84
% 3.27/1.84 thf(ty_t_FSet_Ofset,type,
% 3.27/1.84 fset: $tType > $tType ).
% 3.27/1.84
% 3.27/1.84 thf(ty_t_Set_Oset,type,
% 3.27/1.84 set: $tType > $tType ).
% 3.27/1.84
% 3.27/1.84 thf(ty_t_DTree_OT,type,
% 3.27/1.84 t: $tType ).
% 3.27/1.84
% 3.27/1.84 thf(ty_t_DTree_ON,type,
% 3.27/1.84 n: $tType ).
% 3.27/1.84
% 3.27/1.84 thf(ty_t_itself,type,
% 3.27/1.84 itself: $tType > $tType ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_Lattices_Obounded__lattice,type,
% 3.27/1.84 bounded_lattice:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_HOL_Otype,type,
% 3.27/1.84 type:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_Lattices_Olattice,type,
% 3.27/1.84 lattice:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_Finite__Set_Ofinite,type,
% 3.27/1.84 finite_finite:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_Nat_Osemiring__char__0,type,
% 3.27/1.84 semiring_char_0:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_Lattices_Odistrib__lattice,type,
% 3.27/1.84 distrib_lattice:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 3.27/1.84 semilattice_sup:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 3.27/1.84 bounded_lattice_bot:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 3.27/1.84 bounded_lattice_top:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
% 3.27/1.84 bounde1808546759up_bot:
% 3.27/1.84 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 3.27/1.84 bNF_Greatest_image2:
% 3.27/1.84 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_DTree_ONode,type,
% 3.27/1.84 node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_DTree_Ocont,type,
% 3.27/1.84 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_DTree_Ocorec,type,
% 3.27/1.84 corec:
% 3.27/1.84 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_DTree_Odtree_ONNode,type,
% 3.27/1.84 nNode: n > ( fset @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_DTree_Odtree_Ocase__dtree,type,
% 3.27/1.84 case_dtree:
% 3.27/1.84 !>[A: $tType] : ( ( n > ( fset @ ( sum_sum @ t @ dtree ) ) > A ) > dtree > A ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_DTree_Odtree_Occont,type,
% 3.27/1.84 ccont: dtree > ( fset @ ( sum_sum @ t @ dtree ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_DTree_Odtree_Ocorec__dtree,type,
% 3.27/1.84 corec_dtree:
% 3.27/1.84 !>[A: $tType] : ( ( A > n ) > ( A > ( fset @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_DTree_Odtree_Oroot,type,
% 3.27/1.84 root: dtree > n ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_DTree_Ounfold,type,
% 3.27/1.84 unfold:
% 3.27/1.84 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Finite__Set_Ofinite,type,
% 3.27/1.84 finite_finite2:
% 3.27/1.84 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 3.27/1.84 inf_inf:
% 3.27/1.84 !>[A: $tType] : ( A > A > A ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Lattices_Osup__class_Osup,type,
% 3.27/1.84 sup_sup:
% 3.27/1.84 !>[A: $tType] : ( A > A > A ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Orderings_Obot__class_Obot,type,
% 3.27/1.84 bot_bot:
% 3.27/1.84 !>[A: $tType] : A ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.27/1.84 ord_less_eq:
% 3.27/1.84 !>[A: $tType] : ( A > A > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Orderings_Otop__class_Otop,type,
% 3.27/1.84 top_top:
% 3.27/1.84 !>[A: $tType] : A ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus,type,
% 3.27/1.84 parall1518086719_Nplus: n > n > n ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 3.27/1.84 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
% 3.27/1.84 parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
% 3.27/1.84 parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel,type,
% 3.27/1.84 parall556292031_r_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Product__Type_OPair,type,
% 3.27/1.84 product_Pair:
% 3.27/1.84 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Product__Type_Ocurry,type,
% 3.27/1.84 product_curry:
% 3.27/1.84 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.27/1.84 produc2004651681e_prod:
% 3.27/1.84 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Product__Type_Omap__prod,type,
% 3.27/1.84 product_map_prod:
% 3.27/1.84 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.27/1.84 product_rec_prod:
% 3.27/1.84 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 3.27/1.84 product_case_prod:
% 3.27/1.84 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 3.27/1.84 product_swap:
% 3.27/1.84 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Pure_Otype,type,
% 3.27/1.84 type2:
% 3.27/1.84 !>[A: $tType] : ( itself @ A ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Set_OCollect,type,
% 3.27/1.84 collect:
% 3.27/1.84 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Set_Oimage,type,
% 3.27/1.84 image:
% 3.27/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Set_Ovimage,type,
% 3.27/1.84 vimage:
% 3.27/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Sum__Type_OInl,type,
% 3.27/1.84 sum_Inl:
% 3.27/1.84 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Sum__Type_OInr,type,
% 3.27/1.84 sum_Inr:
% 3.27/1.84 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Sum__Type_OPlus,type,
% 3.27/1.84 sum_Plus:
% 3.27/1.84 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Sum__Type_OSuml,type,
% 3.27/1.84 sum_Suml:
% 3.27/1.84 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Sum__Type_OSumr,type,
% 3.27/1.84 sum_Sumr:
% 3.27/1.84 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
% 3.27/1.84 sum_rec_sum:
% 3.27/1.84 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Wellfounded_Oaccp,type,
% 3.27/1.84 accp:
% 3.27/1.84 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Wellfounded_Olex__prod,type,
% 3.27/1.84 lex_prod:
% 3.27/1.84 !>[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 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Wfrec_Oadm__wf,type,
% 3.27/1.84 adm_wf:
% 3.27/1.84 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Wfrec_Ocut,type,
% 3.27/1.84 cut:
% 3.27/1.84 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_Wfrec_Osame__fst,type,
% 3.27/1.84 same_fst:
% 3.27/1.84 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_c_member,type,
% 3.27/1.84 member:
% 3.27/1.84 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.27/1.84
% 3.27/1.84 thf(sy_v_tr1a____,type,
% 3.27/1.84 tr1a: dtree ).
% 3.27/1.84
% 3.27/1.84 thf(sy_v_tr2a____,type,
% 3.27/1.84 tr2a: dtree ).
% 3.27/1.84
% 3.27/1.84 thf(sy_v_trA____,type,
% 3.27/1.84 trA: dtree ).
% 3.27/1.84
% 3.27/1.84 thf(sy_v_trB____,type,
% 3.27/1.84 trB: dtree ).
% 3.27/1.84
% 3.27/1.84 thf(fact_0_Nplus__comm,axiom,
% 3.27/1.84 ( parall1518086719_Nplus
% 3.27/1.84 = ( ^ [A2: n,B2: n] : ( parall1518086719_Nplus @ B2 @ A2 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_1_Nplus__assoc,axiom,
% 3.27/1.84 ! [A3: n,B3: n,C2: n] :
% 3.27/1.84 ( ( parall1518086719_Nplus @ ( parall1518086719_Nplus @ A3 @ B3 ) @ C2 )
% 3.27/1.84 = ( parall1518086719_Nplus @ A3 @ ( parall1518086719_Nplus @ B3 @ C2 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_2_par__r_Oelims,axiom,
% 3.27/1.84 ! [X: product_prod @ dtree @ dtree,Y: n] :
% 3.27/1.84 ( ( ( parall1914194362_par_r @ X )
% 3.27/1.84 = Y )
% 3.27/1.84 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 3.27/1.84 ( ( X
% 3.27/1.84 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 3.27/1.84 => ( Y
% 3.27/1.84 != ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_3_par__r_Osimps,axiom,
% 3.27/1.84 ! [Tr12: dtree,Tr22: dtree] :
% 3.27/1.84 ( ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 3.27/1.84 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_4_root__Node,axiom,
% 3.27/1.84 ! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
% 3.27/1.84 ( ( root @ ( node @ N @ As ) )
% 3.27/1.84 = N ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_5_root__par,axiom,
% 3.27/1.84 ! [Tr12: dtree,Tr22: dtree] :
% 3.27/1.84 ( ( root @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) )
% 3.27/1.84 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_6_corec_I1_J,axiom,
% 3.27/1.84 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B3: A] :
% 3.27/1.84 ( ( root @ ( corec @ A @ Rt @ Ct @ B3 ) )
% 3.27/1.84 = ( Rt @ B3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_7_unfold_I1_J,axiom,
% 3.27/1.84 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B3: A] :
% 3.27/1.84 ( ( root @ ( unfold @ A @ Rt @ Ct @ B3 ) )
% 3.27/1.84 = ( Rt @ B3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_8_dtree_Osel_I1_J,axiom,
% 3.27/1.84 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 3.27/1.84 ( ( root @ ( nNode @ X1 @ X2 ) )
% 3.27/1.84 = X1 ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_9_dtree_Ocorec__sel_I1_J,axiom,
% 3.27/1.84 ! [A: $tType,G1: A > n,G2: A > ( fset @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),A3: A] :
% 3.27/1.84 ( ( root @ ( corec_dtree @ A @ G1 @ G2 @ A3 ) )
% 3.27/1.84 = ( G1 @ A3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_10_dtree_Oexpand,axiom,
% 3.27/1.84 ! [Dtree: dtree,Dtree2: dtree] :
% 3.27/1.84 ( ( ( ( root @ Dtree )
% 3.27/1.84 = ( root @ Dtree2 ) )
% 3.27/1.84 & ( ( ccont @ Dtree )
% 3.27/1.84 = ( ccont @ Dtree2 ) ) )
% 3.27/1.84 => ( Dtree = Dtree2 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_11_par__r_Opelims,axiom,
% 3.27/1.84 ! [X: product_prod @ dtree @ dtree,Y: n] :
% 3.27/1.84 ( ( ( parall1914194362_par_r @ X )
% 3.27/1.84 = Y )
% 3.27/1.84 => ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ X )
% 3.27/1.84 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 3.27/1.84 ( ( X
% 3.27/1.84 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 3.27/1.84 => ( ( Y
% 3.27/1.84 = ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) )
% 3.27/1.84 => ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_12_dtree__cong,axiom,
% 3.27/1.84 ! [Tr: dtree,Tr3: dtree] :
% 3.27/1.84 ( ( ( root @ Tr )
% 3.27/1.84 = ( root @ Tr3 ) )
% 3.27/1.84 => ( ( ( cont @ Tr )
% 3.27/1.84 = ( cont @ Tr3 ) )
% 3.27/1.84 => ( Tr = Tr3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_13_dtree_Ocollapse,axiom,
% 3.27/1.84 ! [Dtree: dtree] :
% 3.27/1.84 ( ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) )
% 3.27/1.84 = Dtree ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_14__092_060open_062_092_060exists_062tr1_Atr2_O_AtrA_A_061_Atr1_A_092_060parallel_062_Atr2_A_092_060and_062_AtrB_A_061_Atr2_A_092_060parallel_062_Atr1_092_060close_062,axiom,
% 3.27/1.84 ? [Tr1: dtree,Tr2: dtree] :
% 3.27/1.84 ( ( trA
% 3.27/1.84 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) )
% 3.27/1.84 & ( trB
% 3.27/1.84 = ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr2 @ Tr1 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_15_dtree_Oinject,axiom,
% 3.27/1.84 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree ),Y1: n,Y2: fset @ ( sum_sum @ t @ dtree )] :
% 3.27/1.84 ( ( ( nNode @ X1 @ X2 )
% 3.27/1.84 = ( nNode @ Y1 @ Y2 ) )
% 3.27/1.84 = ( ( X1 = Y1 )
% 3.27/1.84 & ( X2 = Y2 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_16_Node__root__cont,axiom,
% 3.27/1.84 ! [Tr: dtree] :
% 3.27/1.84 ( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
% 3.27/1.84 = Tr ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_17_dtree_Osel_I2_J,axiom,
% 3.27/1.84 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 3.27/1.84 ( ( ccont @ ( nNode @ X1 @ X2 ) )
% 3.27/1.84 = X2 ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_18_dtree_Oexhaust,axiom,
% 3.27/1.84 ! [Y: dtree] :
% 3.27/1.84 ~ ! [X12: n,X22: fset @ ( sum_sum @ t @ dtree )] :
% 3.27/1.84 ( Y
% 3.27/1.84 != ( nNode @ X12 @ X22 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_19_dtree_Ocorec__disc,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( corec_dtree @ A )
% 3.27/1.84 = ( corec_dtree @ A ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_20_par__r_Ocases,axiom,
% 3.27/1.84 ! [X: product_prod @ dtree @ dtree] :
% 3.27/1.84 ~ ! [Tr1: dtree,Tr2: dtree] :
% 3.27/1.84 ( X
% 3.27/1.84 != ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_21_par__r_Oinduct,axiom,
% 3.27/1.84 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 3.27/1.84 ( ! [Tr1: dtree,Tr2: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 3.27/1.84 => ( P @ A0 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_22_dtree_Oexhaust__sel,axiom,
% 3.27/1.84 ! [Dtree: dtree] :
% 3.27/1.84 ( Dtree
% 3.27/1.84 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_23_prod_Oinject,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 3.27/1.84 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 3.27/1.84 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 3.27/1.84 = ( ( X1 = Y1 )
% 3.27/1.84 & ( X2 = Y2 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_24_old_Oprod_Oinject,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 3.27/1.84 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 3.27/1.84 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.27/1.84 = ( ( A3 = A4 )
% 3.27/1.84 & ( B3 = B4 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_25_dtree_Osplit__sel,axiom,
% 3.27/1.84 ! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
% 3.27/1.84 ( ( P @ ( case_dtree @ A @ F @ Dtree ) )
% 3.27/1.84 = ( ( Dtree
% 3.27/1.84 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
% 3.27/1.84 => ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_26_dtree_Osplit__sel__asm,axiom,
% 3.27/1.84 ! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
% 3.27/1.84 ( ( P @ ( case_dtree @ A @ F @ Dtree ) )
% 3.27/1.84 = ( ~ ( ( Dtree
% 3.27/1.84 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
% 3.27/1.84 & ~ ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_27_par__def,axiom,
% 3.27/1.84 ( parall1899940088le_par
% 3.27/1.84 = ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_28_dtree_Ocase__eq__if,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( case_dtree @ A )
% 3.27/1.84 = ( ^ [F2: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree3: dtree] : ( F2 @ ( root @ Dtree3 ) @ ( ccont @ Dtree3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_29_accp_Ocases,axiom,
% 3.27/1.84 ! [A: $tType,R: A > A > $o,A3: A] :
% 3.27/1.84 ( ( accp @ A @ R @ A3 )
% 3.27/1.84 => ! [Y3: A] :
% 3.27/1.84 ( ( R @ Y3 @ A3 )
% 3.27/1.84 => ( accp @ A @ R @ Y3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_30_accp_Osimps,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( accp @ A )
% 3.27/1.84 = ( ^ [R2: A > A > $o,A2: A] :
% 3.27/1.84 ? [X3: A] :
% 3.27/1.84 ( ( A2 = X3 )
% 3.27/1.84 & ! [Y4: A] :
% 3.27/1.84 ( ( R2 @ Y4 @ X3 )
% 3.27/1.84 => ( accp @ A @ R2 @ Y4 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_31_accp_Ointros,axiom,
% 3.27/1.84 ! [A: $tType,R: A > A > $o,X: A] :
% 3.27/1.84 ( ! [Y5: A] :
% 3.27/1.84 ( ( R @ Y5 @ X )
% 3.27/1.84 => ( accp @ A @ R @ Y5 ) )
% 3.27/1.84 => ( accp @ A @ R @ X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_32_accp__induct,axiom,
% 3.27/1.84 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 3.27/1.84 ( ( accp @ A @ R @ A3 )
% 3.27/1.84 => ( ! [X4: A] :
% 3.27/1.84 ( ( accp @ A @ R @ X4 )
% 3.27/1.84 => ( ! [Y3: A] :
% 3.27/1.84 ( ( R @ Y3 @ X4 )
% 3.27/1.84 => ( P @ Y3 ) )
% 3.27/1.84 => ( P @ X4 ) ) )
% 3.27/1.84 => ( P @ A3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_33_accp_Oinducts,axiom,
% 3.27/1.84 ! [A: $tType,R: A > A > $o,X: A,P: A > $o] :
% 3.27/1.84 ( ( accp @ A @ R @ X )
% 3.27/1.84 => ( ! [X4: A] :
% 3.27/1.84 ( ! [Y3: A] :
% 3.27/1.84 ( ( R @ Y3 @ X4 )
% 3.27/1.84 => ( accp @ A @ R @ Y3 ) )
% 3.27/1.84 => ( ! [Y3: A] :
% 3.27/1.84 ( ( R @ Y3 @ X4 )
% 3.27/1.84 => ( P @ Y3 ) )
% 3.27/1.84 => ( P @ X4 ) ) )
% 3.27/1.84 => ( P @ X ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_34_dtree_Ocase,axiom,
% 3.27/1.84 ! [A: $tType,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 3.27/1.84 ( ( case_dtree @ A @ F @ ( nNode @ X1 @ X2 ) )
% 3.27/1.84 = ( F @ X1 @ X2 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_35_old_Oprod_Oinducts,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.27/1.84 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 3.27/1.84 => ( P @ Prod ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_36_old_Oprod_Oexhaust,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 3.27/1.84 ~ ! [A5: A,B5: B] :
% 3.27/1.84 ( Y
% 3.27/1.84 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_37_prod__induct7,axiom,
% 3.27/1.84 ! [G: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
% 3.27/1.84 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F4: F3,G3: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G3 ) ) ) ) ) ) )
% 3.27/1.84 => ( P @ X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_38_prod__induct6,axiom,
% 3.27/1.84 ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 3.27/1.84 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 3.27/1.84 => ( P @ X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_39_prod__induct5,axiom,
% 3.27/1.84 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.27/1.84 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 3.27/1.84 => ( P @ X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_40_prod__induct4,axiom,
% 3.27/1.84 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.27/1.84 ( ! [A5: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 3.27/1.84 => ( P @ X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_41_prod__induct3,axiom,
% 3.27/1.84 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.27/1.84 ( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 3.27/1.84 => ( P @ X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_42_prod__cases7,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) )] :
% 3.27/1.84 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F4: F3,G3: G] :
% 3.27/1.84 ( Y
% 3.27/1.84 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G ) @ E2 @ ( product_Pair @ F3 @ G @ F4 @ G3 ) ) ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_43_prod__cases6,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
% 3.27/1.84 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F4: F3] :
% 3.27/1.84 ( Y
% 3.27/1.84 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_44_prod__cases5,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.27/1.84 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E] :
% 3.27/1.84 ( Y
% 3.27/1.84 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_45_mem__Collect__eq,axiom,
% 3.27/1.84 ! [A: $tType,A3: A,P: A > $o] :
% 3.27/1.84 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 3.27/1.84 = ( P @ A3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_46_Collect__mem__eq,axiom,
% 3.27/1.84 ! [A: $tType,A6: set @ A] :
% 3.27/1.84 ( ( collect @ A
% 3.27/1.84 @ ^ [X3: A] : ( member @ A @ X3 @ A6 ) )
% 3.27/1.84 = A6 ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_47_Collect__cong,axiom,
% 3.27/1.84 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.27/1.84 ( ! [X4: A] :
% 3.27/1.84 ( ( P @ X4 )
% 3.27/1.84 = ( Q @ X4 ) )
% 3.27/1.84 => ( ( collect @ A @ P )
% 3.27/1.84 = ( collect @ A @ Q ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_48_ext,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,F: A > B,G4: A > B] :
% 3.27/1.84 ( ! [X4: A] :
% 3.27/1.84 ( ( F @ X4 )
% 3.27/1.84 = ( G4 @ X4 ) )
% 3.27/1.84 => ( F = G4 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_49_prod__cases4,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.27/1.84 ~ ! [A5: A,B5: B,C3: C,D2: D] :
% 3.27/1.84 ( Y
% 3.27/1.84 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_50_prod__cases3,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.27/1.84 ~ ! [A5: A,B5: B,C3: C] :
% 3.27/1.84 ( Y
% 3.27/1.84 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_51_Pair__inject,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 3.27/1.84 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 3.27/1.84 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.27/1.84 => ~ ( ( A3 = A4 )
% 3.27/1.84 => ( B3 != B4 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_52_prod__cases,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 3.27/1.84 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 3.27/1.84 => ( P @ P2 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_53_surj__pair,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 3.27/1.84 ? [X4: A,Y5: B] :
% 3.27/1.84 ( P2
% 3.27/1.84 = ( product_Pair @ A @ B @ X4 @ Y5 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_54_accp__induct__rule,axiom,
% 3.27/1.84 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 3.27/1.84 ( ( accp @ A @ R @ A3 )
% 3.27/1.84 => ( ! [X4: A] :
% 3.27/1.84 ( ( accp @ A @ R @ X4 )
% 3.27/1.84 => ( ! [Y3: A] :
% 3.27/1.84 ( ( R @ Y3 @ X4 )
% 3.27/1.84 => ( P @ Y3 ) )
% 3.27/1.84 => ( P @ X4 ) ) )
% 3.27/1.84 => ( P @ A3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_55_not__accp__down,axiom,
% 3.27/1.84 ! [A: $tType,R3: A > A > $o,X: A] :
% 3.27/1.84 ( ~ ( accp @ A @ R3 @ X )
% 3.27/1.84 => ~ ! [Z: A] :
% 3.27/1.84 ( ( R3 @ Z @ X )
% 3.27/1.84 => ( accp @ A @ R3 @ Z ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_56_accp__downward,axiom,
% 3.27/1.84 ! [A: $tType,R: A > A > $o,B3: A,A3: A] :
% 3.27/1.84 ( ( accp @ A @ R @ B3 )
% 3.27/1.84 => ( ( R @ A3 @ B3 )
% 3.27/1.84 => ( accp @ A @ R @ A3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_57_old_Oprod_Orec,axiom,
% 3.27/1.84 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 3.27/1.84 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.27/1.84 = ( F1 @ A3 @ B3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_58_in__lex__prod,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 3.27/1.84 ( ( 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 @ A3 @ B3 ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 3.27/1.84 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R )
% 3.27/1.84 | ( ( A3 = A4 )
% 3.27/1.84 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_59_internal__case__prod__conv,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
% 3.27/1.84 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 3.27/1.84 = ( C2 @ A3 @ B3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_60_Inl__in__cont__par,axiom,
% 3.27/1.84 ! [T2: t,Tr12: dtree,Tr22: dtree] :
% 3.27/1.84 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 3.27/1.84 = ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr12 ) )
% 3.27/1.84 | ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr22 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_61_cont__Node,axiom,
% 3.27/1.84 ! [As: set @ ( sum_sum @ t @ dtree ),N: n] :
% 3.27/1.84 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 3.27/1.84 => ( ( cont @ ( node @ N @ As ) )
% 3.27/1.84 = As ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_62_ssubst__Pair__rhs,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
% 3.27/1.84 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
% 3.27/1.84 => ( ( S2 = S )
% 3.27/1.84 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_63_curry__conv,axiom,
% 3.27/1.84 ! [A: $tType,C: $tType,B: $tType] :
% 3.27/1.84 ( ( product_curry @ B @ C @ A )
% 3.27/1.84 = ( ^ [F2: ( product_prod @ B @ C ) > A,A2: B,B2: C] : ( F2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_64_finite__par__c,axiom,
% 3.27/1.84 ! [Tr12: dtree,Tr22: dtree] : ( finite_finite2 @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_65_curryI,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 3.27/1.84 ( ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.27/1.84 => ( product_curry @ A @ B @ $o @ F @ A3 @ B3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_66_dtree__simps,axiom,
% 3.27/1.84 ! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N: n,N2: n] :
% 3.27/1.84 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 3.27/1.84 => ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
% 3.27/1.84 => ( ( ( node @ N @ As )
% 3.27/1.84 = ( node @ N2 @ As2 ) )
% 3.27/1.84 = ( ( N = N2 )
% 3.27/1.84 & ( As = As2 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_67_finite__cont,axiom,
% 3.27/1.84 ! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_68_dtree__cases,axiom,
% 3.27/1.84 ! [Tr: dtree] :
% 3.27/1.84 ~ ! [N3: n,As3: set @ ( sum_sum @ t @ dtree )] :
% 3.27/1.84 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
% 3.27/1.84 => ( Tr
% 3.27/1.84 != ( node @ N3 @ As3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_69_curryD,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 3.27/1.84 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
% 3.27/1.84 => ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_70_curryE,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 3.27/1.84 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B3 )
% 3.27/1.84 => ( F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_71_sum_Oinject_I1_J,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 3.27/1.84 ( ( ( sum_Inl @ A @ B @ X1 )
% 3.27/1.84 = ( sum_Inl @ A @ B @ Y1 ) )
% 3.27/1.84 = ( X1 = Y1 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_72_old_Osum_Oinject_I1_J,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,A3: A,A4: A] :
% 3.27/1.84 ( ( ( sum_Inl @ A @ B @ A3 )
% 3.27/1.84 = ( sum_Inl @ A @ B @ A4 ) )
% 3.27/1.84 = ( A3 = A4 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_73_finite__code,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( finite_finite @ A @ ( type2 @ A ) )
% 3.27/1.84 => ( ( finite_finite2 @ A )
% 3.27/1.84 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_74_same__fstI,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,P: A > $o,X: A,Y6: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
% 3.27/1.84 ( ( P @ X )
% 3.27/1.84 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y ) @ ( R3 @ X ) )
% 3.27/1.84 => ( 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 @ X @ Y6 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_75_Inl__inject,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,X: A,Y: A] :
% 3.27/1.84 ( ( ( sum_Inl @ A @ B @ X )
% 3.27/1.84 = ( sum_Inl @ A @ B @ Y ) )
% 3.27/1.84 => ( X = Y ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_76_finite__set__choice,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,A6: set @ A,P: A > B > $o] :
% 3.27/1.84 ( ( finite_finite2 @ A @ A6 )
% 3.27/1.84 => ( ! [X4: A] :
% 3.27/1.84 ( ( member @ A @ X4 @ A6 )
% 3.27/1.84 => ? [X13: B] : ( P @ X4 @ X13 ) )
% 3.27/1.84 => ? [F4: A > B] :
% 3.27/1.84 ! [X5: A] :
% 3.27/1.84 ( ( member @ A @ X5 @ A6 )
% 3.27/1.84 => ( P @ X5 @ ( F4 @ X5 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_77_finite,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( finite_finite @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [A6: set @ A] : ( finite_finite2 @ A @ A6 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_78_old_Osum_Osimps_I7_J,axiom,
% 3.27/1.84 ! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A3: A] :
% 3.27/1.84 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A3 ) )
% 3.27/1.84 = ( F1 @ A3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_79_Suml_Osimps,axiom,
% 3.27/1.84 ! [B: $tType,C: $tType,A: $tType,F: A > C,X: A] :
% 3.27/1.84 ( ( sum_Suml @ A @ C @ B @ F @ ( sum_Inl @ A @ B @ X ) )
% 3.27/1.84 = ( F @ X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_80_finite__Plus__iff,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
% 3.27/1.84 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
% 3.27/1.84 = ( ( finite_finite2 @ A @ A6 )
% 3.27/1.84 & ( finite_finite2 @ B @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_81_adm__wf__def,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType] :
% 3.27/1.84 ( ( adm_wf @ A @ B )
% 3.27/1.84 = ( ^ [R4: set @ ( product_prod @ A @ A ),F5: ( A > B ) > A > B] :
% 3.27/1.84 ! [F2: A > B,G5: A > B,X3: A] :
% 3.27/1.84 ( ! [Z2: A] :
% 3.27/1.84 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ X3 ) @ R4 )
% 3.27/1.84 => ( ( F2 @ Z2 )
% 3.27/1.84 = ( G5 @ Z2 ) ) )
% 3.27/1.84 => ( ( F5 @ F2 @ X3 )
% 3.27/1.84 = ( F5 @ G5 @ X3 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_82_Inl__cont__par,axiom,
% 3.27/1.84 ! [Tr12: dtree,Tr22: dtree] :
% 3.27/1.84 ( ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( cont @ ( parall1899940088le_par @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) )
% 3.27/1.84 = ( vimage @ t @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree ) @ ( sup_sup @ ( set @ ( sum_sum @ t @ dtree ) ) @ ( cont @ Tr12 ) @ ( cont @ Tr22 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_83_swap__simp,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,X: B,Y: A] :
% 3.27/1.84 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 3.27/1.84 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_84_swap__swap,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 3.27/1.84 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 3.27/1.84 = P2 ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_85_finite__Un,axiom,
% 3.27/1.84 ! [A: $tType,F6: set @ A,G6: set @ A] :
% 3.27/1.84 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F6 @ G6 ) )
% 3.27/1.84 = ( ( finite_finite2 @ A @ F6 )
% 3.27/1.84 & ( finite_finite2 @ A @ G6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_86_InlI,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A3: A,A6: set @ A,B6: set @ B] :
% 3.27/1.84 ( ( member @ A @ A3 @ A6 )
% 3.27/1.84 => ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A3 ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_87_Suml__inject,axiom,
% 3.27/1.84 ! [B: $tType,C: $tType,A: $tType,F: A > C,G4: A > C] :
% 3.27/1.84 ( ( ( sum_Suml @ A @ C @ B @ F )
% 3.27/1.84 = ( sum_Suml @ A @ C @ B @ G4 ) )
% 3.27/1.84 => ( F = G4 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_88_infinite__Un,axiom,
% 3.27/1.84 ! [A: $tType,S3: set @ A,T3: set @ A] :
% 3.27/1.84 ( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T3 ) ) )
% 3.27/1.84 = ( ~ ( finite_finite2 @ A @ S3 )
% 3.27/1.84 | ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_89_Un__infinite,axiom,
% 3.27/1.84 ! [A: $tType,S3: set @ A,T3: set @ A] :
% 3.27/1.84 ( ~ ( finite_finite2 @ A @ S3 )
% 3.27/1.84 => ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_90_finite__UnI,axiom,
% 3.27/1.84 ! [A: $tType,F6: set @ A,G6: set @ A] :
% 3.27/1.84 ( ( finite_finite2 @ A @ F6 )
% 3.27/1.84 => ( ( finite_finite2 @ A @ G6 )
% 3.27/1.84 => ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F6 @ G6 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_91_finite__PlusD_I2_J,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
% 3.27/1.84 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
% 3.27/1.84 => ( finite_finite2 @ B @ B6 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_92_finite__PlusD_I1_J,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
% 3.27/1.84 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
% 3.27/1.84 => ( finite_finite2 @ A @ A6 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_93_finite__Plus,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
% 3.27/1.84 ( ( finite_finite2 @ A @ A6 )
% 3.27/1.84 => ( ( finite_finite2 @ B @ B6 )
% 3.27/1.84 => ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_94_vimage__Un,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,F: A > B,A6: set @ B,B6: set @ B] :
% 3.27/1.84 ( ( vimage @ A @ B @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
% 3.27/1.84 = ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ ( vimage @ A @ B @ F @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_95_vimageI,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,F: B > A,A3: B,B3: A,B6: set @ A] :
% 3.27/1.84 ( ( ( F @ A3 )
% 3.27/1.84 = B3 )
% 3.27/1.84 => ( ( member @ A @ B3 @ B6 )
% 3.27/1.84 => ( member @ B @ A3 @ ( vimage @ B @ A @ F @ B6 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_96_vimage__eq,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A3: A,F: A > B,B6: set @ B] :
% 3.27/1.84 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F @ B6 ) )
% 3.27/1.84 = ( member @ B @ ( F @ A3 ) @ B6 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_97_UnCI,axiom,
% 3.27/1.84 ! [A: $tType,C2: A,B6: set @ A,A6: set @ A] :
% 3.27/1.84 ( ( ~ ( member @ A @ C2 @ B6 )
% 3.27/1.84 => ( member @ A @ C2 @ A6 ) )
% 3.27/1.84 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_98_Un__iff,axiom,
% 3.27/1.84 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 3.27/1.84 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 3.27/1.84 = ( ( member @ A @ C2 @ A6 )
% 3.27/1.84 | ( member @ A @ C2 @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_99_sup__apply,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 3.27/1.84 => ( ( sup_sup @ ( A > B ) )
% 3.27/1.84 = ( ^ [F2: A > B,G5: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G5 @ X3 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_100_sup_Oright__idem,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [A3: A,B3: A] :
% 3.27/1.84 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
% 3.27/1.84 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_101_sup__left__idem,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A,Y: A] :
% 3.27/1.84 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 3.27/1.84 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_102_sup_Oleft__idem,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [A3: A,B3: A] :
% 3.27/1.84 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
% 3.27/1.84 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_103_sup__idem,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A] :
% 3.27/1.84 ( ( sup_sup @ A @ X @ X )
% 3.27/1.84 = X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_104_sup_Oidem,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [A3: A] :
% 3.27/1.84 ( ( sup_sup @ A @ A3 @ A3 )
% 3.27/1.84 = A3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_105_sup__left__commute,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A,Y: A,Z3: A] :
% 3.27/1.84 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
% 3.27/1.84 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_106_sup_Oleft__commute,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [B3: A,A3: A,C2: A] :
% 3.27/1.84 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C2 ) )
% 3.27/1.84 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_107_sup__commute,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ( ( sup_sup @ A )
% 3.27/1.84 = ( ^ [X3: A,Y4: A] : ( sup_sup @ A @ Y4 @ X3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_108_sup_Ocommute,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ( ( sup_sup @ A )
% 3.27/1.84 = ( ^ [A2: A,B2: A] : ( sup_sup @ A @ B2 @ A2 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_109_sup__assoc,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A,Y: A,Z3: A] :
% 3.27/1.84 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
% 3.27/1.84 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_110_sup_Oassoc,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [A3: A,B3: A,C2: A] :
% 3.27/1.84 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C2 )
% 3.27/1.84 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_111_sup__fun__def,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ B @ ( type2 @ B ) )
% 3.27/1.84 => ( ( sup_sup @ ( A > B ) )
% 3.27/1.84 = ( ^ [F2: A > B,G5: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G5 @ X3 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_112_inf__sup__aci_I5_J,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.84 => ( ( sup_sup @ A )
% 3.27/1.84 = ( ^ [X3: A,Y4: A] : ( sup_sup @ A @ Y4 @ X3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_113_inf__sup__aci_I6_J,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A,Y: A,Z3: A] :
% 3.27/1.84 ( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
% 3.27/1.84 = ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_114_inf__sup__aci_I7_J,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A,Y: A,Z3: A] :
% 3.27/1.84 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
% 3.27/1.84 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_115_inf__sup__aci_I8_J,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A,Y: A] :
% 3.27/1.84 ( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 3.27/1.84 = ( sup_sup @ A @ X @ Y ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_116_Un__left__commute,axiom,
% 3.27/1.84 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 3.27/1.84 ( ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B6 @ C4 ) )
% 3.27/1.84 = ( sup_sup @ ( set @ A ) @ B6 @ ( sup_sup @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_117_Un__left__absorb,axiom,
% 3.27/1.84 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.27/1.84 ( ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 3.27/1.84 = ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_118_Un__commute,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( sup_sup @ ( set @ A ) )
% 3.27/1.84 = ( ^ [A7: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_119_Un__absorb,axiom,
% 3.27/1.84 ! [A: $tType,A6: set @ A] :
% 3.27/1.84 ( ( sup_sup @ ( set @ A ) @ A6 @ A6 )
% 3.27/1.84 = A6 ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_120_Un__assoc,axiom,
% 3.27/1.84 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 3.27/1.84 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) @ C4 )
% 3.27/1.84 = ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B6 @ C4 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_121_ball__Un,axiom,
% 3.27/1.84 ! [A: $tType,A6: set @ A,B6: set @ A,P: A > $o] :
% 3.27/1.84 ( ( ! [X3: A] :
% 3.27/1.84 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 3.27/1.84 => ( P @ X3 ) ) )
% 3.27/1.84 = ( ! [X3: A] :
% 3.27/1.84 ( ( member @ A @ X3 @ A6 )
% 3.27/1.84 => ( P @ X3 ) )
% 3.27/1.84 & ! [X3: A] :
% 3.27/1.84 ( ( member @ A @ X3 @ B6 )
% 3.27/1.84 => ( P @ X3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_122_bex__Un,axiom,
% 3.27/1.84 ! [A: $tType,A6: set @ A,B6: set @ A,P: A > $o] :
% 3.27/1.84 ( ( ? [X3: A] :
% 3.27/1.84 ( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 3.27/1.84 & ( P @ X3 ) ) )
% 3.27/1.84 = ( ? [X3: A] :
% 3.27/1.84 ( ( member @ A @ X3 @ A6 )
% 3.27/1.84 & ( P @ X3 ) )
% 3.27/1.84 | ? [X3: A] :
% 3.27/1.84 ( ( member @ A @ X3 @ B6 )
% 3.27/1.84 & ( P @ X3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_123_UnI2,axiom,
% 3.27/1.84 ! [A: $tType,C2: A,B6: set @ A,A6: set @ A] :
% 3.27/1.84 ( ( member @ A @ C2 @ B6 )
% 3.27/1.84 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_124_UnI1,axiom,
% 3.27/1.84 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 3.27/1.84 ( ( member @ A @ C2 @ A6 )
% 3.27/1.84 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_125_UnE,axiom,
% 3.27/1.84 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 3.27/1.84 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
% 3.27/1.84 => ( ~ ( member @ A @ C2 @ A6 )
% 3.27/1.84 => ( member @ A @ C2 @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_126_vimage__Collect,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
% 3.27/1.84 ( ! [X4: A] :
% 3.27/1.84 ( ( P @ ( F @ X4 ) )
% 3.27/1.84 = ( Q @ X4 ) )
% 3.27/1.84 => ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
% 3.27/1.84 = ( collect @ A @ Q ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_127_vimageI2,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,F: B > A,A3: B,A6: set @ A] :
% 3.27/1.84 ( ( member @ A @ ( F @ A3 ) @ A6 )
% 3.27/1.84 => ( member @ B @ A3 @ ( vimage @ B @ A @ F @ A6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_128_vimageE,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A3: A,F: A > B,B6: set @ B] :
% 3.27/1.84 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F @ B6 ) )
% 3.27/1.84 => ( member @ B @ ( F @ A3 ) @ B6 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_129_vimageD,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A3: A,F: A > B,A6: set @ B] :
% 3.27/1.84 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F @ A6 ) )
% 3.27/1.84 => ( member @ B @ ( F @ A3 ) @ A6 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_130_pair__in__swap__image,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,Y: A,X: B,A6: set @ ( product_prod @ B @ A )] :
% 3.27/1.84 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
% 3.27/1.84 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_131_internal__case__prod__def,axiom,
% 3.27/1.84 ! [C: $tType,B: $tType,A: $tType] :
% 3.27/1.84 ( ( produc2004651681e_prod @ A @ B @ C )
% 3.27/1.84 = ( product_case_prod @ A @ B @ C ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_132_cut__apply,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,X: A,A3: A,R3: set @ ( product_prod @ A @ A ),F: A > B] :
% 3.27/1.84 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R3 )
% 3.27/1.84 => ( ( cut @ A @ B @ F @ R3 @ A3 @ X )
% 3.27/1.84 = ( F @ X ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_133_cuts__eq,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,F: A > B,R3: set @ ( product_prod @ A @ A ),X: A,G4: A > B] :
% 3.27/1.84 ( ( ( cut @ A @ B @ F @ R3 @ X )
% 3.27/1.84 = ( cut @ A @ B @ G4 @ R3 @ X ) )
% 3.27/1.84 = ( ! [Y4: A] :
% 3.27/1.84 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X ) @ R3 )
% 3.27/1.84 => ( ( F @ Y4 )
% 3.27/1.84 = ( G4 @ Y4 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_134_old_Osum_Osimps_I8_J,axiom,
% 3.27/1.84 ! [A: $tType,T: $tType,B: $tType,F1: A > T,F22: B > T,B3: B] :
% 3.27/1.84 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inr @ B @ A @ B3 ) )
% 3.27/1.84 = ( F22 @ B3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_135_old_Osum_Oinject_I2_J,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,B3: B,B4: B] :
% 3.27/1.84 ( ( ( sum_Inr @ B @ A @ B3 )
% 3.27/1.84 = ( sum_Inr @ B @ A @ B4 ) )
% 3.27/1.84 = ( B3 = B4 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_136_sum_Oinject_I2_J,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,X2: B,Y2: B] :
% 3.27/1.84 ( ( ( sum_Inr @ B @ A @ X2 )
% 3.27/1.84 = ( sum_Inr @ B @ A @ Y2 ) )
% 3.27/1.84 = ( X2 = Y2 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_137_finite__imageI,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,F6: set @ A,H: A > B] :
% 3.27/1.84 ( ( finite_finite2 @ A @ F6 )
% 3.27/1.84 => ( finite_finite2 @ B @ ( image @ A @ B @ H @ F6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_138_InrI,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,B3: A,B6: set @ A,A6: set @ B] :
% 3.27/1.84 ( ( member @ A @ B3 @ B6 )
% 3.27/1.84 => ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B3 ) @ ( sum_Plus @ B @ A @ A6 @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_139_curry__case__prod,axiom,
% 3.27/1.84 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 3.27/1.84 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 3.27/1.84 = F ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_140_case__prod__curry,axiom,
% 3.27/1.84 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 3.27/1.84 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 3.27/1.84 = F ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_141_image__Un,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,F: B > A,A6: set @ B,B6: set @ B] :
% 3.27/1.84 ( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
% 3.27/1.84 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A6 ) @ ( image @ B @ A @ F @ B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_142_sum_Odistinct_I1_J,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,X1: A,X2: B] :
% 3.27/1.84 ( ( sum_Inl @ A @ B @ X1 )
% 3.27/1.84 != ( sum_Inr @ B @ A @ X2 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_143_old_Osum_Odistinct_I2_J,axiom,
% 3.27/1.84 ! [B8: $tType,A8: $tType,B9: B8,A9: A8] :
% 3.27/1.84 ( ( sum_Inr @ B8 @ A8 @ B9 )
% 3.27/1.84 != ( sum_Inl @ A8 @ B8 @ A9 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_144_old_Osum_Odistinct_I1_J,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,A3: A,B4: B] :
% 3.27/1.84 ( ( sum_Inl @ A @ B @ A3 )
% 3.27/1.84 != ( sum_Inr @ B @ A @ B4 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_145_sumE,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 3.27/1.84 ( ! [X4: A] :
% 3.27/1.84 ( S
% 3.27/1.84 != ( sum_Inl @ A @ B @ X4 ) )
% 3.27/1.84 => ~ ! [Y5: B] :
% 3.27/1.84 ( S
% 3.27/1.84 != ( sum_Inr @ B @ A @ Y5 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_146_Inr__not__Inl,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,B3: B,A3: A] :
% 3.27/1.84 ( ( sum_Inr @ B @ A @ B3 )
% 3.27/1.84 != ( sum_Inl @ A @ B @ A3 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_147_split__sum__ex,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType] :
% 3.27/1.84 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 3.27/1.84 ? [X6: sum_sum @ A @ B] : ( P3 @ X6 ) )
% 3.27/1.84 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 3.27/1.84 ( ? [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
% 3.27/1.84 | ? [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_148_split__sum__all,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType] :
% 3.27/1.84 ( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 3.27/1.84 ! [X6: sum_sum @ A @ B] : ( P3 @ X6 ) )
% 3.27/1.84 = ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
% 3.27/1.84 ( ! [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
% 3.27/1.84 & ! [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_149_old_Osum_Oexhaust,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
% 3.27/1.84 ( ! [A5: A] :
% 3.27/1.84 ( Y
% 3.27/1.84 != ( sum_Inl @ A @ B @ A5 ) )
% 3.27/1.84 => ~ ! [B5: B] :
% 3.27/1.84 ( Y
% 3.27/1.84 != ( sum_Inr @ B @ A @ B5 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_150_old_Osum_Oinducts,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 3.27/1.84 ( ! [A5: A] : ( P @ ( sum_Inl @ A @ B @ A5 ) )
% 3.27/1.84 => ( ! [B5: B] : ( P @ ( sum_Inr @ B @ A @ B5 ) )
% 3.27/1.84 => ( P @ Sum ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_151_Inr__inject,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,X: B,Y: B] :
% 3.27/1.84 ( ( ( sum_Inr @ B @ A @ X )
% 3.27/1.84 = ( sum_Inr @ B @ A @ Y ) )
% 3.27/1.84 => ( X = Y ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_152_mem__case__prodE,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,C: $tType,Z3: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 3.27/1.84 ( ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
% 3.27/1.84 => ~ ! [X4: B,Y5: C] :
% 3.27/1.84 ( ( P2
% 3.27/1.84 = ( product_Pair @ B @ C @ X4 @ Y5 ) )
% 3.27/1.84 => ~ ( member @ A @ Z3 @ ( C2 @ X4 @ Y5 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_153_old_Oprod_Ocase,axiom,
% 3.27/1.84 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X2: B] :
% 3.27/1.84 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 3.27/1.84 = ( F @ X1 @ X2 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_154_PlusE,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,U: sum_sum @ A @ B,A6: set @ A,B6: set @ B] :
% 3.27/1.84 ( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
% 3.27/1.84 => ( ! [X4: A] :
% 3.27/1.84 ( ( member @ A @ X4 @ A6 )
% 3.27/1.84 => ( U
% 3.27/1.84 != ( sum_Inl @ A @ B @ X4 ) ) )
% 3.27/1.84 => ~ ! [Y5: B] :
% 3.27/1.84 ( ( member @ B @ Y5 @ B6 )
% 3.27/1.84 => ( U
% 3.27/1.84 != ( sum_Inr @ B @ A @ Y5 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_155_Inl__Inr__False,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,X: A,Y: B] :
% 3.27/1.84 ( ( sum_Inl @ A @ B @ X )
% 3.27/1.84 != ( sum_Inr @ B @ A @ Y ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_156_Inr__Inl__False,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,X: B,Y: A] :
% 3.27/1.84 ( ( sum_Inr @ B @ A @ X )
% 3.27/1.84 != ( sum_Inl @ A @ B @ Y ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_157_Inl__Inr__image__cong,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,A6: set @ ( sum_sum @ A @ B ),B6: set @ ( sum_sum @ A @ B )] :
% 3.27/1.84 ( ( ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A6 )
% 3.27/1.84 = ( vimage @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ B6 ) )
% 3.27/1.84 => ( ( ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ A6 )
% 3.27/1.84 = ( vimage @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B6 ) )
% 3.27/1.84 => ( A6 = B6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_158_obj__sumE__f,axiom,
% 3.27/1.84 ! [A: $tType,C: $tType,B: $tType,S: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
% 3.27/1.84 ( ! [X4: A] :
% 3.27/1.84 ( ( S
% 3.27/1.84 = ( F @ ( sum_Inl @ A @ C @ X4 ) ) )
% 3.27/1.84 => P )
% 3.27/1.84 => ( ! [X4: C] :
% 3.27/1.84 ( ( S
% 3.27/1.84 = ( F @ ( sum_Inr @ C @ A @ X4 ) ) )
% 3.27/1.84 => P )
% 3.27/1.84 => ! [X5: sum_sum @ A @ C] :
% 3.27/1.84 ( ( S
% 3.27/1.84 = ( F @ X5 ) )
% 3.27/1.84 => P ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_159_Plus__def,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType] :
% 3.27/1.84 ( ( sum_Plus @ A @ B )
% 3.27/1.84 = ( ^ [A7: set @ A,B7: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A7 ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B7 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_160_obj__sumE,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 3.27/1.84 ( ! [X4: A] :
% 3.27/1.84 ( S
% 3.27/1.84 != ( sum_Inl @ A @ B @ X4 ) )
% 3.27/1.84 => ~ ! [X4: B] :
% 3.27/1.84 ( S
% 3.27/1.84 != ( sum_Inr @ B @ A @ X4 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_161_split__cong,axiom,
% 3.27/1.84 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G4: A > B > C,P2: product_prod @ A @ B] :
% 3.27/1.84 ( ! [X4: A,Y5: B] :
% 3.27/1.84 ( ( ( product_Pair @ A @ B @ X4 @ Y5 )
% 3.27/1.84 = Q2 )
% 3.27/1.84 => ( ( F @ X4 @ Y5 )
% 3.27/1.84 = ( G4 @ X4 @ Y5 ) ) )
% 3.27/1.84 => ( ( P2 = Q2 )
% 3.27/1.84 => ( ( product_case_prod @ A @ B @ C @ F @ P2 )
% 3.27/1.84 = ( product_case_prod @ A @ B @ C @ G4 @ Q2 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_162_Sumr_Osimps,axiom,
% 3.27/1.84 ! [A: $tType,C: $tType,B: $tType,F: B > C,X: B] :
% 3.27/1.84 ( ( sum_Sumr @ B @ C @ A @ F @ ( sum_Inr @ B @ A @ X ) )
% 3.27/1.84 = ( F @ X ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_163_Sumr__inject,axiom,
% 3.27/1.84 ! [A: $tType,C: $tType,B: $tType,F: B > C,G4: B > C] :
% 3.27/1.84 ( ( ( sum_Sumr @ B @ C @ A @ F )
% 3.27/1.84 = ( sum_Sumr @ B @ C @ A @ G4 ) )
% 3.27/1.84 => ( F = G4 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_164_image2__eqI,axiom,
% 3.27/1.84 ! [A: $tType,C: $tType,B: $tType,B3: A,F: B > A,X: B,C2: C,G4: B > C,A6: set @ B] :
% 3.27/1.84 ( ( B3
% 3.27/1.84 = ( F @ X ) )
% 3.27/1.84 => ( ( C2
% 3.27/1.84 = ( G4 @ X ) )
% 3.27/1.84 => ( ( member @ B @ X @ A6 )
% 3.27/1.84 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F @ G4 ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_165_surj__swap,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType] :
% 3.27/1.84 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 3.27/1.84 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_166_finite__vimageD,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,H: A > B,F6: set @ B] :
% 3.27/1.84 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H @ F6 ) )
% 3.27/1.84 => ( ( ( image @ A @ B @ H @ ( top_top @ ( set @ A ) ) )
% 3.27/1.84 = ( top_top @ ( set @ B ) ) )
% 3.27/1.84 => ( finite_finite2 @ B @ F6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_167_sup__top__left,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A] :
% 3.27/1.84 ( ( sup_sup @ A @ ( top_top @ A ) @ X )
% 3.27/1.84 = ( top_top @ A ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_168_sup__top__right,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( bounded_lattice_top @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A] :
% 3.27/1.84 ( ( sup_sup @ A @ X @ ( top_top @ A ) )
% 3.27/1.84 = ( top_top @ A ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_169_vimage__UNIV,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,F: A > B] :
% 3.27/1.84 ( ( vimage @ A @ B @ F @ ( top_top @ ( set @ B ) ) )
% 3.27/1.84 = ( top_top @ ( set @ A ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_170_UNIV__Plus__UNIV,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType] :
% 3.27/1.84 ( ( sum_Plus @ A @ B @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 3.27/1.84 = ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_171_finite__Plus__UNIV__iff,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType] :
% 3.27/1.84 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 3.27/1.84 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 3.27/1.84 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_172_Un__UNIV__left,axiom,
% 3.27/1.84 ! [A: $tType,B6: set @ A] :
% 3.27/1.84 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B6 )
% 3.27/1.84 = ( top_top @ ( set @ A ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_173_Un__UNIV__right,axiom,
% 3.27/1.84 ! [A: $tType,A6: set @ A] :
% 3.27/1.84 ( ( sup_sup @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
% 3.27/1.84 = ( top_top @ ( set @ A ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_174_finite__UNIV,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( finite_finite @ A @ ( type2 @ A ) )
% 3.27/1.84 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_175_infinite__UNIV__char__0,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 3.27/1.84 => ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_176_ex__new__if__finite,axiom,
% 3.27/1.84 ! [A: $tType,A6: set @ A] :
% 3.27/1.84 ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 3.27/1.84 => ( ( finite_finite2 @ A @ A6 )
% 3.27/1.84 => ? [A5: A] :
% 3.27/1.84 ~ ( member @ A @ A5 @ A6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_177_Finite__Set_Ofinite__set,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 3.27/1.84 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_178_finite__prod,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType] :
% 3.27/1.84 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 3.27/1.84 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 3.27/1.84 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_179_finite__Prod__UNIV,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType] :
% 3.27/1.84 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 3.27/1.84 => ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
% 3.27/1.84 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_180_finite__fun__UNIVD2,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType] :
% 3.27/1.84 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
% 3.27/1.84 => ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_181_UNIV__sum,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType] :
% 3.27/1.84 ( ( top_top @ ( set @ ( sum_sum @ A @ B ) ) )
% 3.27/1.84 = ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( top_top @ ( set @ A ) ) ) @ ( image @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_182_finite__option__UNIV,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
% 3.27/1.84 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_183_surj__image__vimage__eq,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
% 3.27/1.84 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.27/1.84 = ( top_top @ ( set @ A ) ) )
% 3.27/1.84 => ( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A6 ) )
% 3.27/1.84 = A6 ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_184_map__prod__surj,axiom,
% 3.27/1.84 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G4: C > D] :
% 3.27/1.84 ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 3.27/1.84 = ( top_top @ ( set @ B ) ) )
% 3.27/1.84 => ( ( ( image @ C @ D @ G4 @ ( top_top @ ( set @ C ) ) )
% 3.27/1.84 = ( top_top @ ( set @ D ) ) )
% 3.27/1.84 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G4 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 3.27/1.84 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_185_map__prod__simp,axiom,
% 3.27/1.84 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G4: D > B,A3: C,B3: D] :
% 3.27/1.84 ( ( product_map_prod @ C @ A @ D @ B @ F @ G4 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 3.27/1.84 = ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G4 @ B3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_186_map__prod__imageI,axiom,
% 3.27/1.84 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B ),F: A > C,G4: B > D] :
% 3.27/1.84 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 3.27/1.84 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G4 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G4 ) @ R3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_187_prod__fun__imageE,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G4: D > B,R3: set @ ( product_prod @ C @ D )] :
% 3.27/1.84 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G4 ) @ R3 ) )
% 3.27/1.84 => ~ ! [X4: C,Y5: D] :
% 3.27/1.84 ( ( C2
% 3.27/1.84 = ( product_Pair @ A @ B @ ( F @ X4 ) @ ( G4 @ Y5 ) ) )
% 3.27/1.84 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X4 @ Y5 ) @ R3 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_188_finite__vimageD_H,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
% 3.27/1.84 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ F @ A6 ) )
% 3.27/1.84 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 3.27/1.84 => ( finite_finite2 @ B @ A6 ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_189_image__vimage__eq,axiom,
% 3.27/1.84 ! [A: $tType,B: $tType,F: B > A,A6: set @ A] :
% 3.27/1.84 ( ( image @ B @ A @ F @ ( vimage @ B @ A @ F @ A6 ) )
% 3.27/1.84 = ( inf_inf @ ( set @ A ) @ A6 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_190_surj__vimage__empty,axiom,
% 3.27/1.84 ! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
% 3.27/1.84 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 3.27/1.84 = ( top_top @ ( set @ A ) ) )
% 3.27/1.84 => ( ( ( vimage @ B @ A @ F @ A6 )
% 3.27/1.84 = ( bot_bot @ ( set @ B ) ) )
% 3.27/1.84 = ( A6
% 3.27/1.84 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_191_le__sup__iff,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.84 => ! [X: A,Y: A,Z3: A] :
% 3.27/1.84 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
% 3.27/1.84 = ( ( ord_less_eq @ A @ X @ Z3 )
% 3.27/1.84 & ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 3.27/1.84
% 3.27/1.84 thf(fact_192_sup_Obounded__iff,axiom,
% 3.27/1.84 ! [A: $tType] :
% 3.27/1.84 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [B3: A,C2: A,A3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 3.27/1.85 = ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.27/1.85 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_193_bot__eq__sup__iff,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A] :
% 3.27/1.85 ( ( ( bot_bot @ A )
% 3.27/1.85 = ( sup_sup @ A @ X @ Y ) )
% 3.27/1.85 = ( ( X
% 3.27/1.85 = ( bot_bot @ A ) )
% 3.27/1.85 & ( Y
% 3.27/1.85 = ( bot_bot @ A ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_194_sup__eq__bot__iff,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A] :
% 3.27/1.85 ( ( ( sup_sup @ A @ X @ Y )
% 3.27/1.85 = ( bot_bot @ A ) )
% 3.27/1.85 = ( ( X
% 3.27/1.85 = ( bot_bot @ A ) )
% 3.27/1.85 & ( Y
% 3.27/1.85 = ( bot_bot @ A ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_195_sup__bot_Oleft__neutral,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [A3: A] :
% 3.27/1.85 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
% 3.27/1.85 = A3 ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_196_sup__bot_Oright__neutral,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [A3: A] :
% 3.27/1.85 ( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
% 3.27/1.85 = A3 ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_197_inf__sup__absorb,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A] :
% 3.27/1.85 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
% 3.27/1.85 = X ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_198_sup__inf__absorb,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A] :
% 3.27/1.85 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
% 3.27/1.85 = X ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_199_finite__Int,axiom,
% 3.27/1.85 ! [A: $tType,F6: set @ A,G6: set @ A] :
% 3.27/1.85 ( ( ( finite_finite2 @ A @ F6 )
% 3.27/1.85 | ( finite_finite2 @ A @ G6 ) )
% 3.27/1.85 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F6 @ G6 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_200_Un__empty,axiom,
% 3.27/1.85 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.27/1.85 ( ( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
% 3.27/1.85 = ( bot_bot @ ( set @ A ) ) )
% 3.27/1.85 = ( ( A6
% 3.27/1.85 = ( bot_bot @ ( set @ A ) ) )
% 3.27/1.85 & ( B6
% 3.27/1.85 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_201_Un__subset__iff,axiom,
% 3.27/1.85 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 3.27/1.85 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) @ C4 )
% 3.27/1.85 = ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 3.27/1.85 & ( ord_less_eq @ ( set @ A ) @ B6 @ C4 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_202_vimage__empty,axiom,
% 3.27/1.85 ! [B: $tType,A: $tType,F: A > B] :
% 3.27/1.85 ( ( vimage @ A @ B @ F @ ( bot_bot @ ( set @ B ) ) )
% 3.27/1.85 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_203_vimage__Int,axiom,
% 3.27/1.85 ! [A: $tType,B: $tType,F: A > B,A6: set @ B,B6: set @ B] :
% 3.27/1.85 ( ( vimage @ A @ B @ F @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) )
% 3.27/1.85 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ ( vimage @ A @ B @ F @ B6 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_204_Plus__eq__empty__conv,axiom,
% 3.27/1.85 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
% 3.27/1.85 ( ( ( sum_Plus @ A @ B @ A6 @ B6 )
% 3.27/1.85 = ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 3.27/1.85 = ( ( A6
% 3.27/1.85 = ( bot_bot @ ( set @ A ) ) )
% 3.27/1.85 & ( B6
% 3.27/1.85 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_205_finite_OemptyI,axiom,
% 3.27/1.85 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_206_infinite__imp__nonempty,axiom,
% 3.27/1.85 ! [A: $tType,S3: set @ A] :
% 3.27/1.85 ( ~ ( finite_finite2 @ A @ S3 )
% 3.27/1.85 => ( S3
% 3.27/1.85 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_207_finite__subset,axiom,
% 3.27/1.85 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.27/1.85 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.27/1.85 => ( ( finite_finite2 @ A @ B6 )
% 3.27/1.85 => ( finite_finite2 @ A @ A6 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_208_infinite__super,axiom,
% 3.27/1.85 ! [A: $tType,S3: set @ A,T3: set @ A] :
% 3.27/1.85 ( ( ord_less_eq @ ( set @ A ) @ S3 @ T3 )
% 3.27/1.85 => ( ~ ( finite_finite2 @ A @ S3 )
% 3.27/1.85 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_209_rev__finite__subset,axiom,
% 3.27/1.85 ! [A: $tType,B6: set @ A,A6: set @ A] :
% 3.27/1.85 ( ( finite_finite2 @ A @ B6 )
% 3.27/1.85 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.27/1.85 => ( finite_finite2 @ A @ A6 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_210_finite__subset__image,axiom,
% 3.27/1.85 ! [A: $tType,B: $tType,B6: set @ A,F: B > A,A6: set @ B] :
% 3.27/1.85 ( ( finite_finite2 @ A @ B6 )
% 3.27/1.85 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F @ A6 ) )
% 3.27/1.85 => ? [C5: set @ B] :
% 3.27/1.85 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
% 3.27/1.85 & ( finite_finite2 @ B @ C5 )
% 3.27/1.85 & ( B6
% 3.27/1.85 = ( image @ B @ A @ F @ C5 ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_211_finite__surj,axiom,
% 3.27/1.85 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,F: A > B] :
% 3.27/1.85 ( ( finite_finite2 @ A @ A6 )
% 3.27/1.85 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F @ A6 ) )
% 3.27/1.85 => ( finite_finite2 @ B @ B6 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_212_Un__Int__assoc__eq,axiom,
% 3.27/1.85 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 3.27/1.85 ( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ C4 )
% 3.27/1.85 = ( inf_inf @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B6 @ C4 ) ) )
% 3.27/1.85 = ( ord_less_eq @ ( set @ A ) @ C4 @ A6 ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_213_distrib__imp1,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A,Z3: A] :
% 3.27/1.85 ( ! [X4: A,Y5: A,Z: A] :
% 3.27/1.85 ( ( inf_inf @ A @ X4 @ ( sup_sup @ A @ Y5 @ Z ) )
% 3.27/1.85 = ( sup_sup @ A @ ( inf_inf @ A @ X4 @ Y5 ) @ ( inf_inf @ A @ X4 @ Z ) ) )
% 3.27/1.85 => ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) )
% 3.27/1.85 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_214_distrib__imp2,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A,Z3: A] :
% 3.27/1.85 ( ! [X4: A,Y5: A,Z: A] :
% 3.27/1.85 ( ( sup_sup @ A @ X4 @ ( inf_inf @ A @ Y5 @ Z ) )
% 3.27/1.85 = ( inf_inf @ A @ ( sup_sup @ A @ X4 @ Y5 ) @ ( sup_sup @ A @ X4 @ Z ) ) )
% 3.27/1.85 => ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
% 3.27/1.85 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_215_distrib__inf__le,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z3 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_216_distrib__sup__le,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_217_inf__sup__distrib1,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A,Z3: A] :
% 3.27/1.85 ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
% 3.27/1.85 = ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_218_inf__sup__distrib2,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [Y: A,Z3: A,X: A] :
% 3.27/1.85 ( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z3 ) @ X )
% 3.27/1.85 = ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z3 @ X ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_219_sup__inf__distrib1,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A,Z3: A] :
% 3.27/1.85 ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) )
% 3.27/1.85 = ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_220_sup__inf__distrib2,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( distrib_lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [Y: A,Z3: A,X: A] :
% 3.27/1.85 ( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z3 ) @ X )
% 3.27/1.85 = ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z3 @ X ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_221_Un__empty__left,axiom,
% 3.27/1.85 ! [A: $tType,B6: set @ A] :
% 3.27/1.85 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B6 )
% 3.27/1.85 = B6 ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_222_Un__empty__right,axiom,
% 3.27/1.85 ! [A: $tType,A6: set @ A] :
% 3.27/1.85 ( ( sup_sup @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
% 3.27/1.85 = A6 ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_223_Un__mono,axiom,
% 3.27/1.85 ! [A: $tType,A6: set @ A,C4: set @ A,B6: set @ A,D3: set @ A] :
% 3.27/1.85 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 3.27/1.85 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ D3 )
% 3.27/1.85 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_224_Un__least,axiom,
% 3.27/1.85 ! [A: $tType,A6: set @ A,C4: set @ A,B6: set @ A] :
% 3.27/1.85 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 3.27/1.85 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 3.27/1.85 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) @ C4 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_225_Un__upper1,axiom,
% 3.27/1.85 ! [A: $tType,A6: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_226_Un__upper2,axiom,
% 3.27/1.85 ! [A: $tType,B6: set @ A,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_227_Un__absorb1,axiom,
% 3.27/1.85 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.27/1.85 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.27/1.85 => ( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
% 3.27/1.85 = B6 ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_228_Un__absorb2,axiom,
% 3.27/1.85 ! [A: $tType,B6: set @ A,A6: set @ A] :
% 3.27/1.85 ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 3.27/1.85 => ( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
% 3.27/1.85 = A6 ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_229_subset__Un__eq,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( ord_less_eq @ ( set @ A ) )
% 3.27/1.85 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.27/1.85 ( ( sup_sup @ ( set @ A ) @ A7 @ B7 )
% 3.27/1.85 = B7 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_230_inf__sup__ord_I4_J,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_231_inf__sup__ord_I3_J,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( lattice @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_232_le__supE,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [A3: A,B3: A,X: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X )
% 3.27/1.85 => ~ ( ( ord_less_eq @ A @ A3 @ X )
% 3.27/1.85 => ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_233_le__supI,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [A3: A,X: A,B3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ A3 @ X )
% 3.27/1.85 => ( ( ord_less_eq @ A @ B3 @ X )
% 3.27/1.85 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_234_sup__ge1,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_235_sup__ge2,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_236_le__supI1,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,A3: A,B3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ X @ A3 )
% 3.27/1.85 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_237_le__supI2,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,B3: A,A3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ X @ B3 )
% 3.27/1.85 => ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_238_sup_Omono,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [C2: A,A3: A,D4: A,B3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ C2 @ A3 )
% 3.27/1.85 => ( ( ord_less_eq @ A @ D4 @ B3 )
% 3.27/1.85 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D4 ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_239_sup__mono,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [A3: A,C2: A,B3: A,D4: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ A3 @ C2 )
% 3.27/1.85 => ( ( ord_less_eq @ A @ B3 @ D4 )
% 3.27/1.85 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C2 @ D4 ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_240_sup__least,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [Y: A,X: A,Z3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ Y @ X )
% 3.27/1.85 => ( ( ord_less_eq @ A @ Z3 @ X )
% 3.27/1.85 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z3 ) @ X ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_241_le__iff__sup,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ( ( ord_less_eq @ A )
% 3.27/1.85 = ( ^ [X3: A,Y4: A] :
% 3.27/1.85 ( ( sup_sup @ A @ X3 @ Y4 )
% 3.27/1.85 = Y4 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_242_sup_OorderE,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [B3: A,A3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.27/1.85 => ( A3
% 3.27/1.85 = ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_243_sup_OorderI,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [A3: A,B3: A] :
% 3.27/1.85 ( ( A3
% 3.27/1.85 = ( sup_sup @ A @ A3 @ B3 ) )
% 3.27/1.85 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_244_sup__unique,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [F: A > A > A,X: A,Y: A] :
% 3.27/1.85 ( ! [X4: A,Y5: A] : ( ord_less_eq @ A @ X4 @ ( F @ X4 @ Y5 ) )
% 3.27/1.85 => ( ! [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ ( F @ X4 @ Y5 ) )
% 3.27/1.85 => ( ! [X4: A,Y5: A,Z: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ Y5 @ X4 )
% 3.27/1.85 => ( ( ord_less_eq @ A @ Z @ X4 )
% 3.27/1.85 => ( ord_less_eq @ A @ ( F @ Y5 @ Z ) @ X4 ) ) )
% 3.27/1.85 => ( ( sup_sup @ A @ X @ Y )
% 3.27/1.85 = ( F @ X @ Y ) ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_245_sup_Oabsorb1,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [B3: A,A3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.27/1.85 => ( ( sup_sup @ A @ A3 @ B3 )
% 3.27/1.85 = A3 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_246_sup_Oabsorb2,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [A3: A,B3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.27/1.85 => ( ( sup_sup @ A @ A3 @ B3 )
% 3.27/1.85 = B3 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_247_sup__absorb1,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [Y: A,X: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ Y @ X )
% 3.27/1.85 => ( ( sup_sup @ A @ X @ Y )
% 3.27/1.85 = X ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_248_sup__absorb2,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [X: A,Y: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ X @ Y )
% 3.27/1.85 => ( ( sup_sup @ A @ X @ Y )
% 3.27/1.85 = Y ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_249_sup_OboundedE,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [B3: A,C2: A,A3: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 3.27/1.85 => ~ ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.27/1.85 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_250_sup_OboundedI,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [B3: A,A3: A,C2: A] :
% 3.27/1.85 ( ( ord_less_eq @ A @ B3 @ A3 )
% 3.27/1.85 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 3.27/1.85 => ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_251_sup_Oorder__iff,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ( ( ord_less_eq @ A )
% 3.27/1.85 = ( ^ [B2: A,A2: A] :
% 3.27/1.85 ( A2
% 3.27/1.85 = ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_252_sup_Ocobounded1,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_253_sup_Ocobounded2,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_254_sup_Oabsorb__iff1,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ( ( ord_less_eq @ A )
% 3.27/1.85 = ( ^ [B2: A,A2: A] :
% 3.27/1.85 ( ( sup_sup @ A @ A2 @ B2 )
% 3.27/1.85 = A2 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(fact_255_sup_Oabsorb__iff2,axiom,
% 3.27/1.85 ! [A: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 3.27/1.85 => ( ( ord_less_eq @ A )
% 3.27/1.85 = ( ^ [A2: A,B2: A] :
% 3.27/1.85 ( ( sup_sup @ A @ A2 @ B2 )
% 3.27/1.85 = B2 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_FSet_Ofset___Lattices_Obounded__lattice,axiom,
% 3.27/1.85 ! [A8: $tType] :
% 3.27/1.85 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.27/1.85 => ( bounded_lattice @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_1,axiom,
% 3.27/1.85 bounded_lattice @ $o @ ( type2 @ $o ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Set_Oset___Lattices_Obounded__lattice_2,axiom,
% 3.27/1.85 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_fun___Lattices_Obounded__lattice_3,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 3.27/1.85 => ( bounded_lattice @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 3.27/1.85 => ( bounde1808546759up_bot @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 3.27/1.85 => ( bounded_lattice_top @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( bounded_lattice @ A10 @ ( type2 @ A10 ) )
% 3.27/1.85 => ( bounded_lattice_bot @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( semilattice_sup @ A10 @ ( type2 @ A10 ) )
% 3.27/1.85 => ( semilattice_sup @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( distrib_lattice @ A10 @ ( type2 @ A10 ) )
% 3.27/1.85 => ( distrib_lattice @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.27/1.85 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 3.27/1.85 => ( finite_finite @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_fun___Lattices_Olattice,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( lattice @ A10 @ ( type2 @ A10 ) )
% 3.27/1.85 => ( lattice @ ( A8 > A10 ) @ ( type2 @ ( A8 > A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_4,axiom,
% 3.27/1.85 ! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_5,axiom,
% 3.27/1.85 ! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_6,axiom,
% 3.27/1.85 ! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_7,axiom,
% 3.27/1.85 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_8,axiom,
% 3.27/1.85 ! [A8: $tType] : ( distrib_lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Set_Oset___Finite__Set_Ofinite_9,axiom,
% 3.27/1.85 ! [A8: $tType] :
% 3.27/1.85 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.27/1.85 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Set_Oset___Lattices_Olattice_10,axiom,
% 3.27/1.85 ! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_11,axiom,
% 3.27/1.85 bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_12,axiom,
% 3.27/1.85 bounded_lattice_top @ $o @ ( type2 @ $o ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_13,axiom,
% 3.27/1.85 bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14,axiom,
% 3.27/1.85 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_15,axiom,
% 3.27/1.85 distrib_lattice @ $o @ ( type2 @ $o ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
% 3.27/1.85 finite_finite @ $o @ ( type2 @ $o ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_HOL_Obool___Lattices_Olattice_17,axiom,
% 3.27/1.85 lattice @ $o @ ( type2 @ $o ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_FSet_Ofset___Lattices_Obounded__semilattice__sup__bot_18,axiom,
% 3.27/1.85 ! [A8: $tType] : ( bounde1808546759up_bot @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_FSet_Ofset___Lattices_Obounded__lattice__top_19,axiom,
% 3.27/1.85 ! [A8: $tType] :
% 3.27/1.85 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.27/1.85 => ( bounded_lattice_top @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_FSet_Ofset___Lattices_Obounded__lattice__bot_20,axiom,
% 3.27/1.85 ! [A8: $tType] : ( bounded_lattice_bot @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_FSet_Ofset___Lattices_Osemilattice__sup_21,axiom,
% 3.27/1.85 ! [A8: $tType] : ( semilattice_sup @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_FSet_Ofset___Lattices_Odistrib__lattice_22,axiom,
% 3.27/1.85 ! [A8: $tType] : ( distrib_lattice @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_FSet_Ofset___Finite__Set_Ofinite_23,axiom,
% 3.27/1.85 ! [A8: $tType] :
% 3.27/1.85 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.27/1.85 => ( finite_finite @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_FSet_Ofset___Lattices_Olattice_24,axiom,
% 3.27/1.85 ! [A8: $tType] : ( lattice @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_25,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.27/1.85 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 3.27/1.85 => ( finite_finite @ ( sum_sum @ A8 @ A10 ) @ ( type2 @ ( sum_sum @ A8 @ A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Option_Ooption___Finite__Set_Ofinite_26,axiom,
% 3.27/1.85 ! [A8: $tType] :
% 3.27/1.85 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.27/1.85 => ( finite_finite @ ( option @ A8 ) @ ( type2 @ ( option @ A8 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27,axiom,
% 3.27/1.85 ! [A8: $tType,A10: $tType] :
% 3.27/1.85 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.27/1.85 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 3.27/1.85 => ( finite_finite @ ( product_prod @ A8 @ A10 ) @ ( type2 @ ( product_prod @ A8 @ A10 ) ) ) ) ).
% 3.27/1.85
% 3.27/1.85 thf(conj_0,conjecture,
% 3.27/1.85 ( ( parall1518086719_Nplus @ ( root @ tr1a ) @ ( root @ tr2a ) )
% 3.27/1.85 = ( parall1518086719_Nplus @ ( root @ tr2a ) @ ( root @ tr1a ) ) ) ).
% 3.27/1.85 ------------------------
% 3.27/1.88 % (23085)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_22976 for (2999ds/27Mi)
% 3.27/1.88 % (23089)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/18Mi)
% 3.27/1.88 % (23083)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/183Mi)
% 3.27/1.88 % (23086)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/2Mi)
% 3.27/1.88 % (23087)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_22976 for (2999ds/2Mi)
% 3.27/1.88 % (23088)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_22976 for (2999ds/275Mi)
% 3.27/1.88 % (23084)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_22976 for (2999ds/4Mi)
% 3.27/1.88 % (23087)Instruction limit reached!
% 3.27/1.88 % (23087)------------------------------
% 3.27/1.88 % (23087)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.88 % (23087)Termination reason: Unknown
% 3.27/1.88 % (23087)Termination phase: shuffling
% 3.27/1.88
% 3.27/1.88 % (23087)Memory used [KB]: 1407
% 3.27/1.88 % (23087)Time elapsed: 0.003 s
% 3.27/1.88 % (23087)Instructions burned: 2 (million)
% 3.27/1.88 % (23087)------------------------------
% 3.27/1.88 % (23087)------------------------------
% 3.27/1.88 % (23086)Instruction limit reached!
% 3.27/1.88 % (23086)------------------------------
% 3.27/1.88 % (23086)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.88 % (23086)Termination reason: Unknown
% 3.27/1.88 % (23086)Termination phase: shuffling
% 3.27/1.88
% 3.27/1.88 % (23086)Memory used [KB]: 1407
% 3.27/1.88 % (23086)Time elapsed: 0.004 s
% 3.27/1.88 % (23086)Instructions burned: 3 (million)
% 3.27/1.88 % (23086)------------------------------
% 3.27/1.88 % (23086)------------------------------
% 3.27/1.88 % (23084)Instruction limit reached!
% 3.27/1.88 % (23084)------------------------------
% 3.27/1.88 % (23084)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.88 % (23084)Termination reason: Unknown
% 3.27/1.88 % (23084)Termination phase: shuffling
% 3.27/1.88
% 3.27/1.88 % (23084)Memory used [KB]: 1407
% 3.27/1.88 % (23084)Time elapsed: 0.004 s
% 3.27/1.88 % (23084)Instructions burned: 5 (million)
% 3.27/1.88 % (23084)------------------------------
% 3.27/1.88 % (23084)------------------------------
% 3.27/1.89 % (23089)Instruction limit reached!
% 3.27/1.89 % (23089)------------------------------
% 3.27/1.89 % (23089)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.89 % (23089)Termination reason: Unknown
% 3.27/1.89 % (23089)Termination phase: shuffling
% 3.27/1.89
% 3.27/1.89 % (23089)Memory used [KB]: 1918
% 3.27/1.89 % (23089)Time elapsed: 0.012 s
% 3.27/1.89 % (23089)Instructions burned: 19 (million)
% 3.27/1.89 % (23089)------------------------------
% 3.27/1.89 % (23089)------------------------------
% 3.27/1.89 % (23085)Instruction limit reached!
% 3.27/1.89 % (23085)------------------------------
% 3.27/1.89 % (23085)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.89 % (23085)Termination reason: Unknown
% 3.27/1.89 % (23085)Termination phase: Property scanning
% 3.27/1.89
% 3.27/1.89 % (23085)Memory used [KB]: 1791
% 3.27/1.89 % (23085)Time elapsed: 0.016 s
% 3.27/1.89 % (23085)Instructions burned: 27 (million)
% 3.27/1.89 % (23085)------------------------------
% 3.27/1.89 % (23085)------------------------------
% 3.33/1.90 % (23090)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/3Mi)
% 3.33/1.90 % (23091)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_22976 for (2999ds/37Mi)
% 3.33/1.90 % (23092)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_22976 for (2999ds/15Mi)
% 3.33/1.90 % (23090)Instruction limit reached!
% 3.33/1.90 % (23090)------------------------------
% 3.33/1.90 % (23090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.90 % (23090)Termination reason: Unknown
% 3.33/1.90 % (23090)Termination phase: shuffling
% 3.33/1.90
% 3.33/1.90 % (23090)Memory used [KB]: 1407
% 3.33/1.90 % (23090)Time elapsed: 0.004 s
% 3.33/1.90 % (23090)Instructions burned: 4 (million)
% 3.33/1.90 % (23090)------------------------------
% 3.33/1.90 % (23090)------------------------------
% 3.33/1.91 % (23088)Refutation not found, incomplete strategy
% 3.33/1.91 % (23088)------------------------------
% 3.33/1.91 % (23088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.91 % (23088)Termination reason: Refutation not found, incomplete strategy
% 3.33/1.91
% 3.33/1.91
% 3.33/1.91 % (23088)Memory used [KB]: 6524
% 3.33/1.91 % (23088)Time elapsed: 0.026 s
% 3.33/1.91 % (23088)Instructions burned: 47 (million)
% 3.33/1.91 % (23088)------------------------------
% 3.33/1.91 % (23088)------------------------------
% 3.33/1.91 % (23092)Instruction limit reached!
% 3.33/1.91 % (23092)------------------------------
% 3.33/1.91 % (23092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.91 % (23092)Termination reason: Unknown
% 3.33/1.91 % (23092)Termination phase: shuffling
% 3.33/1.91
% 3.33/1.91 % (23092)Memory used [KB]: 1791
% 3.33/1.91 % (23092)Time elapsed: 0.009 s
% 3.33/1.91 % (23092)Instructions burned: 15 (million)
% 3.33/1.91 % (23092)------------------------------
% 3.33/1.91 % (23092)------------------------------
% 3.33/1.91 % (23093)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_22976 for (2999ds/3Mi)
% 3.33/1.91 % (23093)Instruction limit reached!
% 3.33/1.91 % (23093)------------------------------
% 3.33/1.91 % (23093)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.91 % (23093)Termination reason: Unknown
% 3.33/1.91 % (23093)Termination phase: shuffling
% 3.33/1.91
% 3.33/1.91 % (23093)Memory used [KB]: 1407
% 3.33/1.91 % (23093)Time elapsed: 0.004 s
% 3.33/1.91 % (23093)Instructions burned: 4 (million)
% 3.33/1.91 % (23093)------------------------------
% 3.33/1.91 % (23093)------------------------------
% 3.33/1.91 % (23094)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_22976 for (2999ds/1041Mi)
% 3.33/1.92 % (23091)Instruction limit reached!
% 3.33/1.92 % (23091)------------------------------
% 3.33/1.92 % (23091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.92 % (23091)Termination reason: Unknown
% 3.33/1.92 % (23091)Termination phase: shuffling
% 3.33/1.92
% 3.33/1.92 % (23091)Memory used [KB]: 2174
% 3.33/1.92 % (23091)Time elapsed: 0.019 s
% 3.33/1.92 % (23091)Instructions burned: 38 (million)
% 3.33/1.92 % (23091)------------------------------
% 3.33/1.92 % (23091)------------------------------
% 3.33/1.92 % (23095)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/7Mi)
% 3.33/1.92 % (23095)Instruction limit reached!
% 3.33/1.92 % (23095)------------------------------
% 3.33/1.92 % (23095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.92 % (23095)Termination reason: Unknown
% 3.33/1.92 % (23095)Termination phase: shuffling
% 3.33/1.92
% 3.33/1.92 % (23095)Memory used [KB]: 1535
% 3.33/1.92 % (23095)Time elapsed: 0.005 s
% 3.33/1.92 % (23095)Instructions burned: 7 (million)
% 3.33/1.92 % (23095)------------------------------
% 3.33/1.92 % (23095)------------------------------
% 3.33/1.92 % (23096)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/16Mi)
% 3.33/1.92 % (23097)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_22976 for (2999ds/3Mi)
% 3.33/1.92 % (23097)Instruction limit reached!
% 3.33/1.92 % (23097)------------------------------
% 3.33/1.92 % (23097)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.92 % (23097)Termination reason: Unknown
% 3.33/1.92 % (23097)Termination phase: shuffling
% 3.33/1.92
% 3.33/1.92 % (23097)Memory used [KB]: 1407
% 3.33/1.92 % (23097)Time elapsed: 0.003 s
% 3.33/1.92 % (23097)Instructions burned: 3 (million)
% 3.33/1.92 % (23097)------------------------------
% 3.33/1.92 % (23097)------------------------------
% 3.33/1.93 % (23098)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_22976 for (2999ds/3Mi)
% 3.33/1.93 % (23098)Instruction limit reached!
% 3.33/1.93 % (23098)------------------------------
% 3.33/1.93 % (23098)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.93 % (23098)Termination reason: Unknown
% 3.33/1.93 % (23098)Termination phase: shuffling
% 3.33/1.93
% 3.33/1.93 % (23098)Memory used [KB]: 1407
% 3.33/1.93 % (23098)Time elapsed: 0.003 s
% 3.33/1.93 % (23098)Instructions burned: 3 (million)
% 3.33/1.93 % (23098)------------------------------
% 3.33/1.93 % (23098)------------------------------
% 3.33/1.93 % (23096)Instruction limit reached!
% 3.33/1.93 % (23096)------------------------------
% 3.33/1.93 % (23096)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.93 % (23096)Termination reason: Unknown
% 3.33/1.93 % (23096)Termination phase: shuffling
% 3.33/1.93
% 3.33/1.93 % (23096)Memory used [KB]: 1791
% 3.33/1.93 % (23096)Time elapsed: 0.010 s
% 3.33/1.93 % (23096)Instructions burned: 16 (million)
% 3.33/1.93 % (23096)------------------------------
% 3.33/1.93 % (23096)------------------------------
% 3.33/1.93 % (23099)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/7Mi)
% 3.33/1.94 % (23099)Instruction limit reached!
% 3.33/1.94 % (23099)------------------------------
% 3.33/1.94 % (23099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.94 % (23099)Termination reason: Unknown
% 3.33/1.94 % (23099)Termination phase: shuffling
% 3.33/1.94
% 3.33/1.94 % (23099)Memory used [KB]: 1535
% 3.33/1.94 % (23099)Time elapsed: 0.005 s
% 3.33/1.94 % (23099)Instructions burned: 7 (million)
% 3.33/1.94 % (23099)------------------------------
% 3.33/1.94 % (23099)------------------------------
% 3.33/1.94 % (23100)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/3Mi)
% 3.33/1.94 % (23100)Instruction limit reached!
% 3.33/1.94 % (23100)------------------------------
% 3.33/1.94 % (23100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.94 % (23100)Termination reason: Unknown
% 3.33/1.94 % (23100)Termination phase: shuffling
% 3.33/1.94
% 3.33/1.94 % (23100)Memory used [KB]: 1407
% 3.33/1.94 % (23100)Time elapsed: 0.003 s
% 3.33/1.94 % (23100)Instructions burned: 3 (million)
% 3.33/1.94 % (23100)------------------------------
% 3.33/1.94 % (23100)------------------------------
% 3.33/1.94 % (23101)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/4Mi)
% 3.33/1.94 % (23101)Instruction limit reached!
% 3.33/1.94 % (23101)------------------------------
% 3.33/1.94 % (23101)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.94 % (23101)Termination reason: Unknown
% 3.33/1.94 % (23101)Termination phase: shuffling
% 3.33/1.94
% 3.33/1.94 % (23101)Memory used [KB]: 1407
% 3.33/1.94 % (23101)Time elapsed: 0.004 s
% 3.33/1.94 % (23101)Instructions burned: 5 (million)
% 3.33/1.94 % (23101)------------------------------
% 3.33/1.94 % (23101)------------------------------
% 3.33/1.94 % (23102)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/18Mi)
% 3.33/1.95 % (23103)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/710Mi)
% 3.33/1.95 % (23104)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/6Mi)
% 3.33/1.95 % (23102)Instruction limit reached!
% 3.33/1.95 % (23102)------------------------------
% 3.33/1.95 % (23102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.95 % (23102)Termination reason: Unknown
% 3.33/1.95 % (23102)Termination phase: shuffling
% 3.33/1.95
% 3.33/1.95 % (23102)Memory used [KB]: 1791
% 3.33/1.95 % (23102)Time elapsed: 0.011 s
% 3.33/1.95 % (23102)Instructions burned: 19 (million)
% 3.33/1.95 % (23102)------------------------------
% 3.33/1.95 % (23102)------------------------------
% 3.33/1.95 % (23105)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_22976 for (2999ds/902Mi)
% 3.33/1.96 % (23104)Instruction limit reached!
% 3.33/1.96 % (23104)------------------------------
% 3.33/1.96 % (23104)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.96 % (23104)Termination reason: Unknown
% 3.33/1.96 % (23104)Termination phase: shuffling
% 3.33/1.96
% 3.33/1.96 % (23104)Memory used [KB]: 1535
% 3.33/1.96 % (23104)Time elapsed: 0.005 s
% 3.33/1.96 % (23104)Instructions burned: 7 (million)
% 3.33/1.96 % (23104)------------------------------
% 3.33/1.96 % (23104)------------------------------
% 3.33/1.96 % (23106)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_22976 for (2999ds/21Mi)
% 3.33/1.97 % (23103)First to succeed.
% 3.33/1.97 % (23107)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22976 for (2999ds/5Mi)
% 3.33/1.97 % (23083)Instruction limit reached!
% 3.33/1.97 % (23083)------------------------------
% 3.33/1.97 % (23083)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.97 % (23083)Termination reason: Unknown
% 3.33/1.97 % (23083)Termination phase: Saturation
% 3.33/1.97
% 3.33/1.97 % (23083)Memory used [KB]: 7291
% 3.33/1.97 % (23083)Time elapsed: 0.090 s
% 3.33/1.97 % (23083)Instructions burned: 183 (million)
% 3.33/1.97 % (23083)------------------------------
% 3.33/1.97 % (23083)------------------------------
% 3.33/1.97 % (23103)Refutation found. Thanks to Tanya!
% 3.33/1.97 % SZS status Theorem for DTF2THF_22976
% 3.33/1.97 % SZS output start Proof for DTF2THF_22976
% 3.33/1.97 thf(type_def_5, type, itself: ($tType) > $tType).
% 3.33/1.97 thf(type_def_7, type, set: ($tType) > $tType).
% 3.33/1.97 thf(type_def_8, type, product_prod: ($tType * $tType) > $tType).
% 3.33/1.97 thf(type_def_9, type, n: $tType).
% 3.33/1.97 thf(type_def_10, type, t: $tType).
% 3.33/1.97 thf(type_def_11, type, dtree: $tType).
% 3.33/1.97 thf(type_def_12, type, sum_sum: ($tType * $tType) > $tType).
% 3.33/1.97 thf(type_def_13, type, fset: ($tType) > $tType).
% 3.33/1.97 thf(type_def_14, type, option: ($tType) > $tType).
% 3.33/1.97 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 3.33/1.97 thf(func_def_1, type, sum_sum: ($tType * $tType) > $tType).
% 3.33/1.97 thf(func_def_2, type, option: ($tType) > $tType).
% 3.33/1.97 thf(func_def_3, type, dtree: $tType).
% 3.33/1.97 thf(func_def_4, type, fset: ($tType) > $tType).
% 3.33/1.97 thf(func_def_5, type, set: ($tType) > $tType).
% 3.33/1.97 thf(func_def_6, type, t: $tType).
% 3.33/1.97 thf(func_def_7, type, n: $tType).
% 3.33/1.97 thf(func_def_8, type, itself: ($tType) > $tType).
% 3.33/1.97 thf(func_def_9, type, bounded_lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_10, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_11, type, lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_12, type, finite_finite: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_13, type, semiring_char_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_14, type, distrib_lattice: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_15, type, semilattice_sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_16, type, bounded_lattice_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_17, type, bounded_lattice_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_18, type, bounde1808546759up_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.33/1.97 thf(func_def_19, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 3.33/1.97 thf(func_def_20, type, node: n > set @ sum_sum @ t @ dtree > dtree).
% 3.33/1.97 thf(func_def_21, type, cont: dtree > set @ sum_sum @ t @ dtree).
% 3.33/1.97 thf(func_def_22, type, corec: !>[X0: $tType]:((X0 > n) > (X0 > set @ sum_sum @ t @ sum_sum @ dtree @ X0) > X0 > dtree)).
% 3.33/1.97 thf(func_def_23, type, nNode: n > fset @ sum_sum @ t @ dtree > dtree).
% 3.33/1.97 thf(func_def_24, type, case_dtree: !>[X0: $tType]:((n > fset @ sum_sum @ t @ dtree > X0) > dtree > X0)).
% 3.33/1.97 thf(func_def_25, type, ccont: dtree > fset @ sum_sum @ t @ dtree).
% 3.33/1.97 thf(func_def_26, type, corec_dtree: !>[X0: $tType]:((X0 > n) > (X0 > fset @ sum_sum @ t @ sum_sum @ dtree @ X0) > X0 > dtree)).
% 3.33/1.97 thf(func_def_27, type, root: dtree > n).
% 3.33/1.97 thf(func_def_28, type, unfold: !>[X0: $tType]:((X0 > n) > (X0 > set @ sum_sum @ t @ X0) > X0 > dtree)).
% 3.33/1.97 thf(func_def_29, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 3.33/1.97 thf(func_def_30, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.33/1.97 thf(func_def_31, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.33/1.97 thf(func_def_32, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.33/1.97 thf(func_def_33, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.33/1.97 thf(func_def_34, type, top_top: !>[X0: $tType]:(X0)).
% 3.33/1.97 thf(func_def_35, type, parall1518086719_Nplus: n > n > n).
% 3.33/1.97 thf(func_def_36, type, parall1899940088le_par: product_prod @ dtree @ dtree > dtree).
% 3.33/1.97 thf(func_def_37, type, parall1914194347_par_c: product_prod @ dtree @ dtree > set @ sum_sum @ t @ product_prod @ dtree @ dtree).
% 3.33/1.97 thf(func_def_38, type, parall1914194362_par_r: product_prod @ dtree @ dtree > n).
% 3.33/1.97 thf(func_def_39, type, parall556292031_r_rel: product_prod @ dtree @ dtree > product_prod @ dtree @ dtree > $o).
% 3.33/1.97 thf(func_def_40, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.33/1.97 thf(func_def_41, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 3.33/1.97 thf(func_def_42, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.33/1.97 thf(func_def_43, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 3.33/1.97 thf(func_def_44, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.33/1.97 thf(func_def_45, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.33/1.97 thf(func_def_46, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 3.33/1.97 thf(func_def_47, type, type2: !>[X0: $tType]:(itself @ X0)).
% 3.33/1.97 thf(func_def_48, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.33/1.97 thf(func_def_49, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.33/1.97 thf(func_def_50, type, vimage: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X1 > set @ X0)).
% 3.33/1.97 thf(func_def_51, type, sum_Inl: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X0 @ X1)).
% 3.33/1.97 thf(func_def_52, type, sum_Inr: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X1 @ X0)).
% 3.33/1.97 thf(func_def_53, type, sum_Plus: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ sum_sum @ X0 @ X1)).
% 3.33/1.97 thf(func_def_54, type, sum_Suml: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > sum_sum @ X0 @ X2 > X1)).
% 3.33/1.97 thf(func_def_55, type, sum_Sumr: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > sum_sum @ X2 @ X0 > X1)).
% 3.33/1.97 thf(func_def_56, type, sum_rec_sum: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > sum_sum @ X0 @ X2 > X1)).
% 3.33/1.97 thf(func_def_57, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 3.33/1.97 thf(func_def_58, 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)).
% 3.33/1.97 thf(func_def_59, type, adm_wf: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > ((X0 > X1) > X0 > X1) > $o)).
% 3.33/1.97 thf(func_def_60, type, cut: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ product_prod @ X0 @ X0 > X0 > X0 > X1)).
% 3.33/1.97 thf(func_def_61, 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)).
% 3.33/1.97 thf(func_def_62, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.33/1.97 thf(func_def_63, type, tr1a: dtree).
% 3.33/1.97 thf(func_def_64, type, tr2a: dtree).
% 3.33/1.97 thf(func_def_65, type, trA: dtree).
% 3.33/1.97 thf(func_def_66, type, trB: dtree).
% 3.33/1.97 thf(func_def_82, type, sK0: n > product_prod @ dtree @ dtree > dtree).
% 3.33/1.97 thf(func_def_83, type, sK1: n > product_prod @ dtree @ dtree > dtree).
% 3.33/1.97 thf(func_def_84, type, sK2: (product_prod @ dtree @ dtree > $o) > dtree).
% 3.33/1.97 thf(func_def_85, type, sK3: (product_prod @ dtree @ dtree > $o) > dtree).
% 3.33/1.97 thf(func_def_86, type, sK4: product_prod @ dtree @ dtree > dtree).
% 3.33/1.97 thf(func_def_87, type, sK5: product_prod @ dtree @ dtree > dtree).
% 3.33/1.97 thf(func_def_88, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.33/1.97 thf(func_def_89, type, sK7: n > product_prod @ dtree @ dtree > dtree).
% 3.33/1.97 thf(func_def_90, type, sK8: n > product_prod @ dtree @ dtree > dtree).
% 3.33/1.97 thf(func_def_92, type, ph10: !>[X0: $tType]:(X0)).
% 3.33/1.97 thf(f756,plain,(
% 3.33/1.97 $false),
% 3.33/1.97 inference(trivial_inequality_removal,[status(thm)],[f754])).
% 3.33/1.97 thf(f754,plain,(
% 3.33/1.97 ((parall1518086719_Nplus @ (root @ tr1a) @ (root @ tr2a)) != (parall1518086719_Nplus @ (root @ tr1a) @ (root @ tr2a)))),
% 3.33/1.97 inference(superposition,[status(thm)],[f741,f753])).
% 3.33/1.97 thf(f753,plain,(
% 3.33/1.97 ( ! [X2 : n,X1 : n] : (((parall1518086719_Nplus @ X1 @ X2) = (parall1518086719_Nplus @ X2 @ X1))) )),
% 3.33/1.97 inference(beta_eta_normalization,[status(thm)],[f751])).
% 3.33/1.97 thf(f751,plain,(
% 3.33/1.97 ( ! [X2 : n,X1 : n] : (((parall1518086719_Nplus @ X1 @ X2) = ((^[Y0 : n]: (parall1518086719_Nplus @ Y0 @ X1)) @ X2))) )),
% 3.33/1.97 inference(argument_congruence,[status(thm)],[f750])).
% 3.33/1.97 thf(f750,plain,(
% 3.33/1.97 ( ! [X1 : n] : (((^[Y0 : n]: (parall1518086719_Nplus @ Y0 @ X1)) = (parall1518086719_Nplus @ X1))) )),
% 3.33/1.97 inference(beta_eta_normalization,[status(thm)],[f749])).
% 3.33/1.97 thf(f749,plain,(
% 3.33/1.97 ( ! [X1 : n] : ((((^[Y0 : n]: ((^[Y1 : n]: (parall1518086719_Nplus @ Y1 @ Y0)))) @ X1) = (parall1518086719_Nplus @ X1))) )),
% 3.33/1.97 inference(argument_congruence,[status(thm)],[f737])).
% 3.33/1.97 thf(f737,plain,(
% 3.33/1.97 (parall1518086719_Nplus = (^[Y0 : n]: ((^[Y1 : n]: (parall1518086719_Nplus @ Y1 @ Y0)))))),
% 3.33/1.97 inference(cnf_transformation,[status(thm)],[f572])).
% 3.33/1.97 thf(f572,plain,(
% 3.33/1.97 (parall1518086719_Nplus = (^[Y0 : n]: ((^[Y1 : n]: (parall1518086719_Nplus @ Y1 @ Y0)))))),
% 3.33/1.97 inference(fool_elimination,[status(thm)],[f571])).
% 3.33/1.97 thf(f571,plain,(
% 3.33/1.97 ((^[X0 : n, X1 : n] : (parall1518086719_Nplus @ X1 @ X0)) = parall1518086719_Nplus)),
% 3.33/1.97 inference(rectify,[status(thm)],[f1])).
% 3.33/1.97 thf(f1,axiom,(
% 3.33/1.97 ((^[X5 : n, X6 : n] : (parall1518086719_Nplus @ X6 @ X5)) = parall1518086719_Nplus)),
% 3.33/1.97 file('/export/starexec/sandbox/tmp/tmp.q1d5il6I29/DTF2THF_22976.p',fact_0_Nplus__comm)).
% 3.33/1.97 thf(f741,plain,(
% 3.33/1.97 ((parall1518086719_Nplus @ (root @ tr1a) @ (root @ tr2a)) != (parall1518086719_Nplus @ (root @ tr2a) @ (root @ tr1a)))),
% 3.33/1.97 inference(cnf_transformation,[status(thm)],[f707])).
% 3.33/1.97 thf(f707,plain,(
% 3.33/1.97 ((parall1518086719_Nplus @ (root @ tr1a) @ (root @ tr2a)) != (parall1518086719_Nplus @ (root @ tr2a) @ (root @ tr1a)))),
% 3.33/1.97 inference(flattening,[status(thm)],[f293])).
% 3.33/1.97 thf(f293,negated_conjecture,(
% 3.33/1.97 ~((parall1518086719_Nplus @ (root @ tr1a) @ (root @ tr2a)) = (parall1518086719_Nplus @ (root @ tr2a) @ (root @ tr1a)))),
% 3.33/1.97 inference(negated_conjecture,[status(cth)],[f292])).
% 3.33/1.97 thf(f292,conjecture,(
% 3.33/1.97 ((parall1518086719_Nplus @ (root @ tr1a) @ (root @ tr2a)) = (parall1518086719_Nplus @ (root @ tr2a) @ (root @ tr1a)))),
% 3.33/1.97 file('/export/starexec/sandbox/tmp/tmp.q1d5il6I29/DTF2THF_22976.p',conj_0)).
% 3.33/1.97 % SZS output end Proof for DTF2THF_22976
% 3.33/1.97 % (23103)------------------------------
% 3.33/1.97 % (23103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.33/1.97 % (23103)Termination reason: Refutation
% 3.33/1.97
% 3.33/1.97 % (23103)Memory used [KB]: 6396
% 3.33/1.97 % (23103)Time elapsed: 0.023 s
% 3.33/1.97 % (23103)Instructions burned: 40 (million)
% 3.33/1.97 % (23103)------------------------------
% 3.33/1.97 % (23103)------------------------------
% 3.33/1.97 % (23082)Success in time 0.111 s
%------------------------------------------------------------------------------