%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP127^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:46 AM UTC 2026
% Result : Theorem 3.50s 1.94s
% Output : Refutation 3.50s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP127^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n008.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:27:24 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
% 0.18/0.46 ---- Original DTF file ---
% 0.18/0.46 thf(spec,logic,$$dhol).
% 0.18/0.46 %------------------------------------------------------------------------------
% 0.18/0.46 % File : ITP127^2 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.46 % Domain : Interactive Theorem Proving
% 0.18/0.46 % Problem : Sledgehammer Monomorphic_Monad problem prob_2789__7122292_1
% 0.18/0.46 % Version : Especial.
% 0.18/0.46 % English :
% 0.18/0.46
% 0.18/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.46 % Source : [Des21]
% 0.18/0.46 % Names : Monomorphic_Monad/prob_2789__7122292_1 [Des21]
% 0.18/0.46
% 0.18/0.46 % Status : Theorem
% 0.18/0.46 % Rating : 0.00 v8.1.0, 0.25 v7.5.0
% 0.18/0.46 % Syntax : Number of formulae : 326 ( 78 unt; 47 typ; 0 def)
% 0.18/0.46 % Number of atoms : 843 ( 228 equ; 0 cnn)
% 0.18/0.46 % Maximal formula atoms : 13 ( 3 avg)
% 0.18/0.46 % Number of connectives : 3576 ( 87 ~; 17 |; 40 &;3009 @)
% 0.18/0.46 % ( 0 <=>; 423 =>; 0 <=; 0 <~>)
% 0.18/0.46 % Maximal formula depth : 22 ( 9 avg)
% 0.18/0.46 % Number of types : 4 ( 3 usr)
% 0.18/0.46 % Number of type conns : 427 ( 427 >; 0 *; 0 +; 0 <<)
% 0.18/0.46 % Number of symbols : 47 ( 44 usr; 3 con; 0-9 aty)
% 0.18/0.46 % Number of variables : 1233 ( 74 ^;1078 !; 13 ?;1233 :)
% 0.18/0.46 % ( 68 !>; 0 ?*; 0 @-; 0 @+)
% 0.18/0.46 % SPC : TH1_THM_EQU_NAR
% 0.18/0.46
% 0.18/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:22:28.107
% 0.18/0.46 %------------------------------------------------------------------------------
% 0.18/0.46 % Could-be-implicit typings (5)
% 0.18/0.46 thf(ty_t_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT,type,
% 0.18/0.46 monomo1453983439ondetT: $tType > $tType > $tType ).
% 0.18/0.46
% 0.18/0.46 thf(ty_t_Countable__Set__Type_Ocset,type,
% 0.18/0.46 countable_Set_cset: $tType > $tType ).
% 0.18/0.46
% 0.18/0.46 thf(ty_tf_m,type,
% 0.18/0.46 m: $tType ).
% 0.18/0.46
% 0.18/0.46 thf(ty_tf_c,type,
% 0.18/0.46 c: $tType ).
% 0.18/0.46
% 0.18/0.46 thf(ty_tf_a,type,
% 0.18/0.46 a: $tType ).
% 0.18/0.46
% 0.18/0.46 % Explicit typings (42)
% 0.18/0.46 thf(sy_cl_HOL_Otype,type,
% 0.18/0.46 type:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Groups_Ominus,type,
% 0.18/0.46 minus:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.18/0.46 bot:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.18/0.46 ord:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.18/0.46 order:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.18/0.46 group_add:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.18/0.46 no_bot:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.18/0.46 no_top:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.18/0.46 linorder:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.18/0.46 preorder:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.18/0.46 order_bot:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.18/0.46 wellorder:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Odense__order,type,
% 0.18/0.46 dense_order:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.18/0.46 dense_linorder:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.18/0.46 ordered_ab_group_add:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.18/0.46 cancel146912293up_add:
% 0.18/0.46 !>[A: $tType] : $o ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Countable__Set__Type_Ocin,type,
% 0.18/0.46 countable_Set_cin:
% 0.18/0.46 !>[A: $tType] : ( A > ( countable_Set_cset @ A ) > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Countable__Set__Type_Ocinsert,type,
% 0.18/0.46 counta2111716221insert:
% 0.18/0.46 !>[A: $tType] : ( A > ( countable_Set_cset @ A ) > ( countable_Set_cset @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Fun_Ocomp,type,
% 0.18/0.46 comp:
% 0.18/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.18/0.46 minus_minus:
% 0.18/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_If,type,
% 0.18/0.46 if:
% 0.18/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM,type,
% 0.18/0.46 monomo2011693592ondetM:
% 0.18/0.46 !>[A: $tType,M: $tType,C: $tType] : ( ( ( countable_Set_cset @ A ) > M ) > ( M > ( ( countable_Set_cset @ A ) > M ) > M ) > ( ( countable_Set_cset @ A ) > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM__axioms,type,
% 0.18/0.46 monomo759963067axioms:
% 0.18/0.46 !>[A: $tType,M: $tType,C: $tType] : ( ( ( countable_Set_cset @ A ) > M ) > ( M > ( ( countable_Set_cset @ A ) > M ) > M ) > ( ( countable_Set_cset @ A ) > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM__base_Oaltc__nondet,type,
% 0.18/0.46 monomo955616121nondet:
% 0.18/0.46 !>[C: $tType,M: $tType,A: $tType] : ( ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > ( countable_Set_cset @ C ) > ( C > ( monomo1453983439ondetT @ A @ M ) ) > ( monomo1453983439ondetT @ A @ M ) ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc,type,
% 0.18/0.46 monomo439771545d_altc:
% 0.18/0.46 !>[A: $tType,M: $tType,C: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc3,type,
% 0.18/0.46 monomo1425736922_altc3:
% 0.18/0.46 !>[A: $tType,M: $tType,C: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc__axioms,type,
% 0.18/0.46 monomo555039548axioms:
% 0.18/0.46 !>[M: $tType,A: $tType,C: $tType] : ( ( M > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state,type,
% 0.18/0.46 monomo109450930_state:
% 0.18/0.46 !>[A: $tType,M: $tType,S: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( S > M ) > M ) > ( S > M > M ) > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state__altc,type,
% 0.18/0.46 monomo1036387116e_altc:
% 0.18/0.46 !>[A: $tType,M: $tType,S: $tType,C: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( S > M ) > M ) > ( S > M > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state__altc__axioms,type,
% 0.18/0.46 monomo1324021455axioms:
% 0.18/0.46 !>[S: $tType,M: $tType,C: $tType] : ( ( ( S > M ) > M ) > ( S > M > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetM__base_Oput__nondet,type,
% 0.18/0.46 monomo410456285nondet:
% 0.18/0.46 !>[State: $tType,M: $tType,A: $tType] : ( ( State > M > M ) > State > ( monomo1453983439ondetT @ A @ M ) > ( monomo1453983439ondetT @ A @ M ) ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_ONondetT,type,
% 0.18/0.46 monomo1975189248ondetT:
% 0.18/0.46 !>[M: $tType,A: $tType] : ( M > ( monomo1453983439ondetT @ A @ M ) ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Ocase__nondetT,type,
% 0.18/0.46 monomo1564381680ondetT:
% 0.18/0.46 !>[M: $tType,B: $tType,A: $tType] : ( ( M > B ) > ( monomo1453983439ondetT @ A @ M ) > B ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Orec__nondetT,type,
% 0.18/0.46 monomo1291740684ondetT:
% 0.18/0.46 !>[M: $tType,D: $tType,A: $tType] : ( ( M > D ) > ( monomo1453983439ondetT @ A @ M ) > D ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Orun__nondet,type,
% 0.18/0.46 monomo603716163nondet:
% 0.18/0.46 !>[A: $tType,M: $tType] : ( ( monomo1453983439ondetT @ A @ M ) > M ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.18/0.46 bot_bot:
% 0.18/0.46 !>[A: $tType] : A ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.18/0.46 ord_less:
% 0.18/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.18/0.46 ord_less_eq:
% 0.18/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.18/0.46 order_Greatest:
% 0.18/0.46 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_v_f____,type,
% 0.18/0.46 f: c > ( monomo1453983439ondetT @ a @ m ) ).
% 0.18/0.46
% 0.18/0.46 thf(sy_v_mergec,type,
% 0.18/0.46 mergec: ( countable_Set_cset @ c ) > ( c > m ) > m ).
% 0.18/0.46
% 0.18/0.46 thf(sy_v_x____,type,
% 0.18/0.46 x: c ).
% 0.18/0.46
% 0.18/0.46 % Relevant facts (256)
% 0.18/0.46 thf(fact_0_mergec__single,axiom,
% 0.18/0.46 ! [X: c,F: c > m] :
% 0.18/0.46 ( ( mergec @ ( counta2111716221insert @ c @ X @ ( bot_bot @ ( countable_Set_cset @ c ) ) ) @ F )
% 0.18/0.46 = ( F @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % mergec_single
% 0.18/0.46 thf(fact_1_cset__nondetM__base_Oaltc__nondet_Ocong,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,C: $tType] :
% 0.18/0.46 ( ( monomo955616121nondet @ C @ M @ A )
% 0.18/0.46 = ( monomo955616121nondet @ C @ M @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % cset_nondetM_base.altc_nondet.cong
% 0.18/0.46 thf(fact_2_cinsert__absorb2,axiom,
% 0.18/0.46 ! [A: $tType,X: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( counta2111716221insert @ A @ X @ ( counta2111716221insert @ A @ X @ A2 ) )
% 0.18/0.46 = ( counta2111716221insert @ A @ X @ A2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_absorb2
% 0.18/0.46 thf(fact_3_bot__apply,axiom,
% 0.18/0.46 ! [C: $tType,D: $tType] :
% 0.18/0.46 ( ( bot @ C )
% 0.18/0.46 => ( ( bot_bot @ ( D > C ) )
% 0.18/0.46 = ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % bot_apply
% 0.18/0.46 thf(fact_4_cdoubleton__eq__iff,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,B2: A,C2: A,D2: A] :
% 0.18/0.46 ( ( ( counta2111716221insert @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 0.18/0.46 = ( counta2111716221insert @ A @ C2 @ ( counta2111716221insert @ A @ D2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) )
% 0.18/0.46 = ( ( ( A3 = C2 )
% 0.18/0.46 & ( B2 = D2 ) )
% 0.18/0.46 | ( ( A3 = D2 )
% 0.18/0.46 & ( B2 = C2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cdoubleton_eq_iff
% 0.18/0.46 thf(fact_5_csingleton__inject,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,B2: A] :
% 0.18/0.46 ( ( ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 0.18/0.46 = ( counta2111716221insert @ A @ B2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 0.18/0.46 => ( A3 = B2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % csingleton_inject
% 0.18/0.46 thf(fact_6_cinsert__not__cempty,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( counta2111716221insert @ A @ A3 @ A2 )
% 0.18/0.46 != ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_not_cempty
% 0.18/0.46 thf(fact_7_cinsert__commute,axiom,
% 0.18/0.46 ! [A: $tType,X: A,Y: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( counta2111716221insert @ A @ X @ ( counta2111716221insert @ A @ Y @ A2 ) )
% 0.18/0.46 = ( counta2111716221insert @ A @ Y @ ( counta2111716221insert @ A @ X @ A2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_commute
% 0.18/0.46 thf(fact_8_cset__nondetM_Omerge__single,axiom,
% 0.18/0.46 ! [C: $tType,M: $tType,A: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,X: A,F: A > M] :
% 0.18/0.46 ( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
% 0.18/0.46 => ( ( Merge @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) @ F )
% 0.18/0.46 = ( F @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cset_nondetM.merge_single
% 0.18/0.46 thf(fact_9_cset__nondetM_Omergec__single,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,C: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,X: C,F: C > M] :
% 0.18/0.46 ( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
% 0.18/0.46 => ( ( Mergec @ ( counta2111716221insert @ C @ X @ ( bot_bot @ ( countable_Set_cset @ C ) ) ) @ F )
% 0.18/0.46 = ( F @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cset_nondetM.mergec_single
% 0.18/0.46 thf(fact_10_bot__fun__def,axiom,
% 0.18/0.46 ! [B: $tType,A: $tType] :
% 0.18/0.46 ( ( bot @ B )
% 0.18/0.46 => ( ( bot_bot @ ( A > B ) )
% 0.18/0.46 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % bot_fun_def
% 0.18/0.46 thf(fact_11_monad__altc_Oaltc__single,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M,X: C,F: C > M] :
% 0.18/0.46 ( ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc )
% 0.18/0.46 => ( ( Altc @ ( counta2111716221insert @ C @ X @ ( bot_bot @ ( countable_Set_cset @ C ) ) ) @ F )
% 0.18/0.46 = ( F @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % monad_altc.altc_single
% 0.18/0.46 thf(fact_12_run__altc__nondet,axiom,
% 0.18/0.46 ! [A2: countable_Set_cset @ c,F: c > ( monomo1453983439ondetT @ a @ m )] :
% 0.18/0.46 ( ( monomo603716163nondet @ a @ m @ ( monomo955616121nondet @ c @ m @ a @ mergec @ A2 @ F ) )
% 0.18/0.46 = ( mergec @ A2 @ ( comp @ ( monomo1453983439ondetT @ a @ m ) @ m @ c @ ( monomo603716163nondet @ a @ m ) @ F ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % run_altc_nondet
% 0.18/0.46 thf(fact_13_cset__nondetM_Omerge__empty,axiom,
% 0.18/0.46 ! [C: $tType,M: $tType,A: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,F: A > M] :
% 0.18/0.46 ( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
% 0.18/0.46 => ( ( Merge @ ( bot_bot @ ( countable_Set_cset @ A ) ) @ F )
% 0.18/0.46 = ( Return @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cset_nondetM.merge_empty
% 0.18/0.46 thf(fact_14_nondetT_Oexpand,axiom,
% 0.18/0.46 ! [M: $tType,A: $tType,NondetT: monomo1453983439ondetT @ A @ M,NondetT2: monomo1453983439ondetT @ A @ M] :
% 0.18/0.46 ( ( ( monomo603716163nondet @ A @ M @ NondetT )
% 0.18/0.46 = ( monomo603716163nondet @ A @ M @ NondetT2 ) )
% 0.18/0.46 => ( NondetT = NondetT2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.expand
% 0.18/0.46 thf(fact_15_cset__nondetM__base_Orun__altc__nondet,axiom,
% 0.18/0.46 ! [M: $tType,A: $tType,C: $tType,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,A2: countable_Set_cset @ C,F: C > ( monomo1453983439ondetT @ A @ M )] :
% 0.18/0.46 ( ( monomo603716163nondet @ A @ M @ ( monomo955616121nondet @ C @ M @ A @ Mergec @ A2 @ F ) )
% 0.18/0.46 = ( Mergec @ A2 @ ( comp @ ( monomo1453983439ondetT @ A @ M ) @ M @ C @ ( monomo603716163nondet @ A @ M ) @ F ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cset_nondetM_base.run_altc_nondet
% 0.18/0.46 thf(fact_16_altc__nondet__def,axiom,
% 0.18/0.46 ! [A2: countable_Set_cset @ c,F: c > ( monomo1453983439ondetT @ a @ m )] :
% 0.18/0.46 ( ( monomo955616121nondet @ c @ m @ a @ mergec @ A2 @ F )
% 0.18/0.46 = ( monomo1975189248ondetT @ m @ a @ ( mergec @ A2 @ ( comp @ ( monomo1453983439ondetT @ a @ m ) @ m @ c @ ( monomo603716163nondet @ a @ m ) @ F ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % altc_nondet_def
% 0.18/0.46 thf(fact_17_comp__apply,axiom,
% 0.18/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.18/0.46 ( ( comp @ B @ A @ C )
% 0.18/0.46 = ( ^ [F2: B > A,G: C > B,X2: C] : ( F2 @ ( G @ X2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % comp_apply
% 0.18/0.46 thf(fact_18_cset__nondetM__base_Oaltc__nondet__def,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,C: $tType] :
% 0.18/0.46 ( ( monomo955616121nondet @ C @ M @ A )
% 0.18/0.46 = ( ^ [Mergec2: ( countable_Set_cset @ C ) > ( C > M ) > M,A4: countable_Set_cset @ C,F2: C > ( monomo1453983439ondetT @ A @ M )] : ( monomo1975189248ondetT @ M @ A @ ( Mergec2 @ A4 @ ( comp @ ( monomo1453983439ondetT @ A @ M ) @ M @ C @ ( monomo603716163nondet @ A @ M ) @ F2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cset_nondetM_base.altc_nondet_def
% 0.18/0.46 thf(fact_19_cset__nondetM_Oaxioms_I3_J,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,C: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 0.18/0.46 ( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
% 0.18/0.46 => ( monomo759963067axioms @ A @ M @ C @ Return @ Bind @ Merge @ Mergec ) ) ).
% 0.18/0.46
% 0.18/0.46 % cset_nondetM.axioms(3)
% 0.18/0.46 thf(fact_20_comp__def,axiom,
% 0.18/0.46 ! [A: $tType,C: $tType,B: $tType] :
% 0.18/0.46 ( ( comp @ B @ C @ A )
% 0.18/0.46 = ( ^ [F2: B > C,G: A > B,X2: A] : ( F2 @ ( G @ X2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % comp_def
% 0.18/0.46 thf(fact_21_comp__assoc,axiom,
% 0.18/0.46 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G2: C > D,H: A > C] :
% 0.18/0.46 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G2 ) @ H )
% 0.18/0.46 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % comp_assoc
% 0.18/0.46 thf(fact_22_comp__eq__dest,axiom,
% 0.18/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
% 0.18/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 0.18/0.46 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.18/0.46 => ( ( A3 @ ( B2 @ V ) )
% 0.18/0.46 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % comp_eq_dest
% 0.18/0.46 thf(fact_23_comp__eq__elim,axiom,
% 0.18/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D] :
% 0.18/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 0.18/0.46 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.18/0.46 => ! [V2: A] :
% 0.18/0.46 ( ( A3 @ ( B2 @ V2 ) )
% 0.18/0.46 = ( C2 @ ( D2 @ V2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % comp_eq_elim
% 0.18/0.46 thf(fact_24_comp__cong,axiom,
% 0.18/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G2: C > B,X: C,F3: D > A,G3: E > D,X3: E] :
% 0.18/0.46 ( ( ( F @ ( G2 @ X ) )
% 0.18/0.46 = ( F3 @ ( G3 @ X3 ) ) )
% 0.18/0.46 => ( ( comp @ B @ A @ C @ F @ G2 @ X )
% 0.18/0.46 = ( comp @ D @ A @ E @ F3 @ G3 @ X3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % comp_cong
% 0.18/0.46 thf(fact_25_comp__eq__dest__lhs,axiom,
% 0.18/0.46 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
% 0.18/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 0.18/0.46 = C2 )
% 0.18/0.46 => ( ( A3 @ ( B2 @ V ) )
% 0.18/0.46 = ( C2 @ V ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % comp_eq_dest_lhs
% 0.18/0.46 thf(fact_26_comp__apply__eq,axiom,
% 0.18/0.46 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G2: C > B,X: C,H: D > A,K: C > D] :
% 0.18/0.46 ( ( ( F @ ( G2 @ X ) )
% 0.18/0.46 = ( H @ ( K @ X ) ) )
% 0.18/0.46 => ( ( comp @ B @ A @ C @ F @ G2 @ X )
% 0.18/0.46 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % comp_apply_eq
% 0.18/0.46 thf(fact_27_nondetT_Oinject,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,X: M,Ya: M] :
% 0.18/0.46 ( ( ( monomo1975189248ondetT @ M @ A @ X )
% 0.18/0.46 = ( monomo1975189248ondetT @ M @ A @ Ya ) )
% 0.18/0.46 = ( X = Ya ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.inject
% 0.18/0.46 thf(fact_28_nondetT_Ocollapse,axiom,
% 0.18/0.46 ! [M: $tType,A: $tType,NondetT: monomo1453983439ondetT @ A @ M] :
% 0.18/0.46 ( ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) )
% 0.18/0.46 = NondetT ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.collapse
% 0.18/0.46 thf(fact_29_nondetT_Oexhaust,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,Y: monomo1453983439ondetT @ A @ M] :
% 0.18/0.46 ~ ! [X4: M] :
% 0.18/0.46 ( Y
% 0.18/0.46 != ( monomo1975189248ondetT @ M @ A @ X4 ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.exhaust
% 0.18/0.46 thf(fact_30_nondetT_Oinduct,axiom,
% 0.18/0.46 ! [M: $tType,A: $tType,P: ( monomo1453983439ondetT @ A @ M ) > $o,NondetT: monomo1453983439ondetT @ A @ M] :
% 0.18/0.46 ( ! [Xa: M] : ( P @ ( monomo1975189248ondetT @ M @ A @ Xa ) )
% 0.18/0.46 => ( P @ NondetT ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.induct
% 0.18/0.46 thf(fact_31_nondetT_Oexhaust__sel,axiom,
% 0.18/0.46 ! [M: $tType,A: $tType,NondetT: monomo1453983439ondetT @ A @ M] :
% 0.18/0.46 ( NondetT
% 0.18/0.46 = ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.exhaust_sel
% 0.18/0.46 thf(fact_32_nondetT_Osel,axiom,
% 0.18/0.46 ! [Aa: $tType,A: $tType,X: A] :
% 0.18/0.46 ( ( monomo603716163nondet @ Aa @ A @ ( monomo1975189248ondetT @ A @ Aa @ X ) )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.sel
% 0.18/0.46 thf(fact_33_put__nondet__def,axiom,
% 0.18/0.46 ! [State: $tType] :
% 0.18/0.46 ( ( monomo410456285nondet @ State @ m @ a )
% 0.18/0.46 = ( ^ [Put: State > m > m,S2: State,M2: monomo1453983439ondetT @ a @ m] : ( monomo1975189248ondetT @ m @ a @ ( Put @ S2 @ ( monomo603716163nondet @ a @ m @ M2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % put_nondet_def
% 0.18/0.46 thf(fact_34_nondetM__base_Oput__nondet__def,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,State: $tType] :
% 0.18/0.46 ( ( monomo410456285nondet @ State @ M @ A )
% 0.18/0.46 = ( ^ [Put: State > M > M,S2: State,M2: monomo1453983439ondetT @ A @ M] : ( monomo1975189248ondetT @ M @ A @ ( Put @ S2 @ ( monomo603716163nondet @ A @ M @ M2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetM_base.put_nondet_def
% 0.18/0.46 thf(fact_35_nondetT_Osplit__sel__asm,axiom,
% 0.18/0.46 ! [B: $tType,M: $tType,A: $tType,P: B > $o,F: M > B,NondetT: monomo1453983439ondetT @ A @ M] :
% 0.18/0.46 ( ( P @ ( monomo1564381680ondetT @ M @ B @ A @ F @ NondetT ) )
% 0.18/0.46 = ( ~ ( ( NondetT
% 0.18/0.46 = ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) ) )
% 0.18/0.46 & ~ ( P @ ( F @ ( monomo603716163nondet @ A @ M @ NondetT ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.split_sel_asm
% 0.18/0.46 thf(fact_36_nondetT_Osplit__sel,axiom,
% 0.18/0.46 ! [B: $tType,M: $tType,A: $tType,P: B > $o,F: M > B,NondetT: monomo1453983439ondetT @ A @ M] :
% 0.18/0.46 ( ( P @ ( monomo1564381680ondetT @ M @ B @ A @ F @ NondetT ) )
% 0.18/0.46 = ( ( NondetT
% 0.18/0.46 = ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) ) )
% 0.18/0.46 => ( P @ ( F @ ( monomo603716163nondet @ A @ M @ NondetT ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.split_sel
% 0.18/0.46 thf(fact_37_rewriteR__comp__comp2,axiom,
% 0.18/0.46 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H: A > C,R1: D > B,R2: A > D,F: B > E,L: D > E] :
% 0.18/0.46 ( ( ( comp @ C @ B @ A @ G2 @ H )
% 0.18/0.46 = ( comp @ D @ B @ A @ R1 @ R2 ) )
% 0.18/0.46 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 0.18/0.46 = L )
% 0.18/0.46 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G2 ) @ H )
% 0.18/0.46 = ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rewriteR_comp_comp2
% 0.18/0.46 thf(fact_38_rewriteL__comp__comp2,axiom,
% 0.18/0.46 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G2: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 0.18/0.46 ( ( ( comp @ C @ B @ A @ F @ G2 )
% 0.18/0.46 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 0.18/0.46 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 0.18/0.46 = R )
% 0.18/0.46 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G2 @ H ) )
% 0.18/0.46 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rewriteL_comp_comp2
% 0.18/0.46 thf(fact_39_rewriteR__comp__comp,axiom,
% 0.18/0.46 ! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H: A > C,R: A > B,F: B > D] :
% 0.18/0.46 ( ( ( comp @ C @ B @ A @ G2 @ H )
% 0.18/0.46 = R )
% 0.18/0.46 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G2 ) @ H )
% 0.18/0.46 = ( comp @ B @ D @ A @ F @ R ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rewriteR_comp_comp
% 0.18/0.46 thf(fact_40_rewriteL__comp__comp,axiom,
% 0.18/0.46 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G2: A > C,L: A > B,H: D > A] :
% 0.18/0.46 ( ( ( comp @ C @ B @ A @ F @ G2 )
% 0.18/0.46 = L )
% 0.18/0.46 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G2 @ H ) )
% 0.18/0.46 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % rewriteL_comp_comp
% 0.18/0.46 thf(fact_41_type__copy__map__cong0,axiom,
% 0.18/0.46 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M3: B > A,G2: C > B,X: C,N: D > A,H: C > D,F: A > E] :
% 0.18/0.46 ( ( ( M3 @ ( G2 @ X ) )
% 0.18/0.46 = ( N @ ( H @ X ) ) )
% 0.18/0.46 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M3 ) @ G2 @ X )
% 0.18/0.46 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % type_copy_map_cong0
% 0.18/0.46 thf(fact_42_run__put__nondet,axiom,
% 0.18/0.46 ! [B: $tType,Put2: B > m > m,S3: B,M4: monomo1453983439ondetT @ a @ m] :
% 0.18/0.46 ( ( monomo603716163nondet @ a @ m @ ( monomo410456285nondet @ B @ m @ a @ Put2 @ S3 @ M4 ) )
% 0.18/0.46 = ( Put2 @ S3 @ ( monomo603716163nondet @ a @ m @ M4 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % run_put_nondet
% 0.18/0.46 thf(fact_43_nondetT_Ocase__eq__if,axiom,
% 0.18/0.46 ! [A: $tType,B: $tType,M: $tType] :
% 0.18/0.46 ( ( monomo1564381680ondetT @ M @ B @ A )
% 0.18/0.46 = ( ^ [F2: M > B,NondetT3: monomo1453983439ondetT @ A @ M] : ( F2 @ ( monomo603716163nondet @ A @ M @ NondetT3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.case_eq_if
% 0.18/0.46 thf(fact_44_nondetT_Ocase,axiom,
% 0.18/0.46 ! [A: $tType,B: $tType,M: $tType,F: M > B,X: M] :
% 0.18/0.46 ( ( monomo1564381680ondetT @ M @ B @ A @ F @ ( monomo1975189248ondetT @ M @ A @ X ) )
% 0.18/0.46 = ( F @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.case
% 0.18/0.46 thf(fact_45_ext,axiom,
% 0.18/0.46 ! [B: $tType,A: $tType,F: A > B,G2: A > B] :
% 0.18/0.46 ( ! [X4: A] :
% 0.18/0.46 ( ( F @ X4 )
% 0.18/0.46 = ( G2 @ X4 ) )
% 0.18/0.46 => ( F = G2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % ext
% 0.18/0.46 thf(fact_46_nondetM__base_Orun__put__nondet,axiom,
% 0.18/0.46 ! [B: $tType,M: $tType,A: $tType,Put2: B > M > M,S3: B,M4: monomo1453983439ondetT @ A @ M] :
% 0.18/0.46 ( ( monomo603716163nondet @ A @ M @ ( monomo410456285nondet @ B @ M @ A @ Put2 @ S3 @ M4 ) )
% 0.18/0.46 = ( Put2 @ S3 @ ( monomo603716163nondet @ A @ M @ M4 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetM_base.run_put_nondet
% 0.18/0.46 thf(fact_47_fun_Omap__comp,axiom,
% 0.18/0.46 ! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F: A > B,V: D > A] :
% 0.18/0.46 ( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F @ V ) )
% 0.18/0.46 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F ) @ V ) ) ).
% 0.18/0.46
% 0.18/0.46 % fun.map_comp
% 0.18/0.46 thf(fact_48_function__factors__left,axiom,
% 0.18/0.46 ! [A: $tType,C: $tType,B: $tType,G2: A > B,F: A > C] :
% 0.18/0.46 ( ( ! [X2: A,Y2: A] :
% 0.18/0.46 ( ( ( G2 @ X2 )
% 0.18/0.46 = ( G2 @ Y2 ) )
% 0.18/0.46 => ( ( F @ X2 )
% 0.18/0.46 = ( F @ Y2 ) ) ) )
% 0.18/0.46 = ( ? [H2: B > C] :
% 0.18/0.46 ( F
% 0.18/0.46 = ( comp @ B @ C @ A @ H2 @ G2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % function_factors_left
% 0.18/0.46 thf(fact_49_function__factors__right,axiom,
% 0.18/0.46 ! [C: $tType,B: $tType,A: $tType,G2: B > C,F: A > C] :
% 0.18/0.46 ( ( ! [X2: A] :
% 0.18/0.46 ? [Y2: B] :
% 0.18/0.46 ( ( G2 @ Y2 )
% 0.18/0.46 = ( F @ X2 ) ) )
% 0.18/0.46 = ( ? [H2: A > B] :
% 0.18/0.46 ( F
% 0.18/0.46 = ( comp @ B @ C @ A @ G2 @ H2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % function_factors_right
% 0.18/0.46 thf(fact_50_monad__altc_Oaxioms_I2_J,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 0.18/0.46 ( ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc )
% 0.18/0.46 => ( monomo555039548axioms @ M @ A @ C @ Bind @ Altc ) ) ).
% 0.18/0.46
% 0.18/0.46 % monad_altc.axioms(2)
% 0.18/0.46 thf(fact_51_cinsert__cDiff__single,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( counta2111716221insert @ A @ A3 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) )
% 0.18/0.46 = ( counta2111716221insert @ A @ A3 @ A2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_cDiff_single
% 0.18/0.46 thf(fact_52_monad__altc3_Oaxioms_I1_J,axiom,
% 0.18/0.46 ! [A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 0.18/0.46 ( ( monomo1425736922_altc3 @ A @ M @ C @ Return @ Bind @ Altc )
% 0.18/0.46 => ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc ) ) ).
% 0.18/0.46
% 0.18/0.46 % monad_altc3.axioms(1)
% 0.18/0.46 thf(fact_53_nondetT_Orec,axiom,
% 0.18/0.46 ! [A: $tType,D: $tType,M: $tType,F: M > D,X: M] :
% 0.18/0.46 ( ( monomo1291740684ondetT @ M @ D @ A @ F @ ( monomo1975189248ondetT @ M @ A @ X ) )
% 0.18/0.46 = ( F @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % nondetT.rec
% 0.18/0.46 thf(fact_54_monad__state__altc_Oaxioms_I2_J,axiom,
% 0.18/0.46 ! [S: $tType,A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 0.18/0.46 ( ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc )
% 0.18/0.46 => ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc ) ) ).
% 0.18/0.46
% 0.18/0.46 % monad_state_altc.axioms(2)
% 0.18/0.46 thf(fact_55_cDiff__idemp,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ B3 )
% 0.18/0.46 = ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiff_idemp
% 0.18/0.46 thf(fact_56_cDiff__cancel,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ A2 )
% 0.18/0.46 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiff_cancel
% 0.18/0.46 thf(fact_57_cDiff__cempty,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 0.18/0.46 = A2 ) ).
% 0.18/0.46
% 0.18/0.46 % cDiff_cempty
% 0.18/0.46 thf(fact_58_cempty__cDiff,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( bot_bot @ ( countable_Set_cset @ A ) ) @ A2 )
% 0.18/0.46 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cempty_cDiff
% 0.18/0.46 thf(fact_59_cDiff__cinsert,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,A3: A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ B3 ) )
% 0.18/0.46 = ( minus_minus @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiff_cinsert
% 0.18/0.46 thf(fact_60_cDiff__cinsert2,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,A3: A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ B3 ) )
% 0.18/0.46 = ( minus_minus @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiff_cinsert2
% 0.18/0.46 thf(fact_61_minus__apply,axiom,
% 0.18/0.46 ! [B: $tType,A: $tType] :
% 0.18/0.46 ( ( minus @ B )
% 0.18/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.18/0.46 = ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % minus_apply
% 0.18/0.46 thf(fact_62_cDiff__single__cinsert,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,X: A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiff_single_cinsert
% 0.18/0.46 thf(fact_63_monad__state__altc_Oaxioms_I3_J,axiom,
% 0.18/0.46 ! [A: $tType,S: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 0.18/0.46 ( ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc )
% 0.18/0.46 => ( monomo1324021455axioms @ S @ M @ C @ Get @ Put2 @ Altc ) ) ).
% 0.18/0.46
% 0.18/0.46 % monad_state_altc.axioms(3)
% 0.18/0.46 thf(fact_64_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( cancel146912293up_add @ A )
% 0.18/0.46 => ! [A3: A,C2: A,B2: A] :
% 0.18/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
% 0.18/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.18/0.46 thf(fact_65_diff__eq__diff__eq,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( group_add @ A )
% 0.18/0.46 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.18/0.46 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.18/0.46 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.18/0.46 => ( ( A3 = B2 )
% 0.18/0.46 = ( C2 = D2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_eq
% 0.18/0.46 thf(fact_66_fun__diff__def,axiom,
% 0.18/0.46 ! [B: $tType,A: $tType] :
% 0.18/0.46 ( ( minus @ B )
% 0.18/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.18/0.46 = ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % fun_diff_def
% 0.18/0.46 thf(fact_67_cinsert__cDiff,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ A3 @ A2 )
% 0.18/0.46 => ( ( counta2111716221insert @ A @ A3 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) )
% 0.18/0.46 = A2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_cDiff
% 0.18/0.46 thf(fact_68_order__refl,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( preorder @ A )
% 0.18/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_refl
% 0.18/0.46 thf(fact_69_csubset__antisym,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.46 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ A2 )
% 0.18/0.46 => ( A2 = B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubset_antisym
% 0.18/0.46 thf(fact_70_all__not__cin__conv,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ! [X2: A] :
% 0.18/0.46 ~ ( countable_Set_cin @ A @ X2 @ A2 ) )
% 0.18/0.46 = ( A2
% 0.18/0.46 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % all_not_cin_conv
% 0.18/0.46 thf(fact_71_cempty__iff,axiom,
% 0.18/0.46 ! [A: $tType,C2: A] :
% 0.18/0.46 ~ ( countable_Set_cin @ A @ C2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cempty_iff
% 0.18/0.46 thf(fact_72_cinsert__iff,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,B2: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ A2 ) )
% 0.18/0.46 = ( ( A3 = B2 )
% 0.18/0.46 | ( countable_Set_cin @ A @ A3 @ A2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_iff
% 0.18/0.46 thf(fact_73_cinsertCI,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,B3: countable_Set_cset @ A,B2: A] :
% 0.18/0.46 ( ( ~ ( countable_Set_cin @ A @ A3 @ B3 )
% 0.18/0.46 => ( A3 = B2 ) )
% 0.18/0.46 => ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsertCI
% 0.18/0.46 thf(fact_74_csubsetI,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ! [X4: A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ X4 @ A2 )
% 0.18/0.46 => ( countable_Set_cin @ A @ X4 @ B3 ) )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubsetI
% 0.18/0.46 thf(fact_75_cempty__fsubsetI,axiom,
% 0.18/0.46 ! [A: $tType,X: countable_Set_cset @ A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( bot_bot @ ( countable_Set_cset @ A ) ) @ X ) ).
% 0.18/0.46
% 0.18/0.46 % cempty_fsubsetI
% 0.18/0.46 thf(fact_76_csubset__cempty,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 0.18/0.46 = ( A2
% 0.18/0.46 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubset_cempty
% 0.18/0.46 thf(fact_77_cDiff__iff,axiom,
% 0.18/0.46 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 0.18/0.46 = ( ( countable_Set_cin @ A @ C2 @ A2 )
% 0.18/0.46 & ~ ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiff_iff
% 0.18/0.46 thf(fact_78_cDiffI,axiom,
% 0.18/0.46 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ C2 @ A2 )
% 0.18/0.46 => ( ~ ( countable_Set_cin @ A @ C2 @ B3 )
% 0.18/0.46 => ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiffI
% 0.18/0.46 thf(fact_79_cinsert__csubset,axiom,
% 0.18/0.46 ! [A: $tType,X: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
% 0.18/0.46 = ( ( countable_Set_cin @ A @ X @ B3 )
% 0.18/0.46 & ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_csubset
% 0.18/0.46 thf(fact_80_cinsert__cDiff1,axiom,
% 0.18/0.46 ! [A: $tType,X: A,B3: countable_Set_cset @ A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ X @ B3 )
% 0.18/0.46 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
% 0.18/0.46 = ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_cDiff1
% 0.18/0.46 thf(fact_81_diff__eq__diff__less__eq,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ordered_ab_group_add @ A )
% 0.18/0.46 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.18/0.46 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.18/0.46 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.18/0.46 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_less_eq
% 0.18/0.46 thf(fact_82_diff__right__mono,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ordered_ab_group_add @ A )
% 0.18/0.46 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_right_mono
% 0.18/0.46 thf(fact_83_diff__left__mono,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ordered_ab_group_add @ A )
% 0.18/0.46 => ! [B2: A,A3: A,C2: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.18/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_left_mono
% 0.18/0.46 thf(fact_84_diff__mono,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ordered_ab_group_add @ A )
% 0.18/0.46 => ! [A3: A,B2: A,D2: A,C2: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 => ( ( ord_less_eq @ A @ D2 @ C2 )
% 0.18/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_mono
% 0.18/0.46 thf(fact_85_csubset__cinsert,axiom,
% 0.18/0.46 ! [A: $tType,X: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) )
% 0.18/0.46 = ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubset_cinsert
% 0.18/0.46 thf(fact_86_dual__order_Oantisym,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [B2: A,A3: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.18/0.46 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 => ( A3 = B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.antisym
% 0.18/0.46 thf(fact_87_dual__order_Oeq__iff,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.18/0.46 = ( ^ [A5: A,B5: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.18/0.46 & ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.eq_iff
% 0.18/0.46 thf(fact_88_dual__order_Otrans,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [B2: A,A3: A,C2: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.18/0.46 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.18/0.46 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.trans
% 0.18/0.46 thf(fact_89_linorder__wlog,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( linorder @ A )
% 0.18/0.46 => ! [P: A > A > $o,A3: A,B2: A] :
% 0.18/0.46 ( ! [A6: A,B6: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.18/0.46 => ( P @ A6 @ B6 ) )
% 0.18/0.46 => ( ! [A6: A,B6: A] :
% 0.18/0.46 ( ( P @ B6 @ A6 )
% 0.18/0.46 => ( P @ A6 @ B6 ) )
% 0.18/0.46 => ( P @ A3 @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % linorder_wlog
% 0.18/0.46 thf(fact_90_dual__order_Orefl,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.refl
% 0.18/0.46 thf(fact_91_order__trans,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( preorder @ A )
% 0.18/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.46 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.18/0.46 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_trans
% 0.18/0.46 thf(fact_92_order__class_Oorder_Oantisym,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [A3: A,B2: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.18/0.46 => ( A3 = B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.antisym
% 0.18/0.46 thf(fact_93_ord__le__eq__trans,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ord @ A )
% 0.18/0.46 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 => ( ( B2 = C2 )
% 0.18/0.46 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_trans
% 0.18/0.46 thf(fact_94_ord__eq__le__trans,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ord @ A )
% 0.18/0.46 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.46 ( ( A3 = B2 )
% 0.18/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.18/0.46 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_trans
% 0.18/0.46 thf(fact_95_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.18/0.46 = ( ^ [A5: A,B5: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.18/0.46 & ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.eq_iff
% 0.18/0.46 thf(fact_96_antisym__conv,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [Y: A,X: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.18/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.46 = ( X = Y ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym_conv
% 0.18/0.46 thf(fact_97_le__cases3,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( linorder @ A )
% 0.18/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.18/0.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.46 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.18/0.46 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.18/0.46 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.18/0.46 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.18/0.46 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.18/0.46 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.18/0.46 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.18/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.18/0.46 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.18/0.46 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.18/0.46 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_cases3
% 0.18/0.46 thf(fact_98_eqcelem__imp__iff,axiom,
% 0.18/0.46 ! [A: $tType,X: A,Y: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( X = Y )
% 0.18/0.46 => ( ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 = ( countable_Set_cin @ A @ Y @ A2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eqcelem_imp_iff
% 0.18/0.46 thf(fact_99_cset__eq__csubset,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ^ [Y3: countable_Set_cset @ A,Z: countable_Set_cset @ A] : ( Y3 = Z ) )
% 0.18/0.46 = ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A4 @ B4 )
% 0.18/0.46 & ( ord_less_eq @ ( countable_Set_cset @ A ) @ B4 @ A4 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cset_eq_csubset
% 0.18/0.46 thf(fact_100_order_Otrans,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.18/0.46 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order.trans
% 0.18/0.46 thf(fact_101_le__cases,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( linorder @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.18/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_cases
% 0.18/0.46 thf(fact_102_eqcset__imp__iff,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,X: A] :
% 0.18/0.46 ( ( A2 = B3 )
% 0.18/0.46 => ( ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 = ( countable_Set_cin @ A @ X @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eqcset_imp_iff
% 0.18/0.46 thf(fact_103_eq__refl,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( preorder @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( X = Y )
% 0.18/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_refl
% 0.18/0.46 thf(fact_104_if__split__cin2,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,Q: $o,X: countable_Set_cset @ A,Y: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ A3 @ ( if @ ( countable_Set_cset @ A ) @ Q @ X @ Y ) )
% 0.18/0.46 = ( ( Q
% 0.18/0.46 => ( countable_Set_cin @ A @ A3 @ X ) )
% 0.18/0.46 & ( ~ Q
% 0.18/0.46 => ( countable_Set_cin @ A @ A3 @ Y ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % if_split_cin2
% 0.18/0.46 thf(fact_105_if__split__cin1,axiom,
% 0.18/0.46 ! [A: $tType,Q: $o,X: A,Y: A,B2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ ( if @ A @ Q @ X @ Y ) @ B2 )
% 0.18/0.46 = ( ( Q
% 0.18/0.46 => ( countable_Set_cin @ A @ X @ B2 ) )
% 0.18/0.46 & ( ~ Q
% 0.18/0.46 => ( countable_Set_cin @ A @ Y @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % if_split_cin1
% 0.18/0.46 thf(fact_106_eq__cmem__trans,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,B2: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( A3 = B2 )
% 0.18/0.46 => ( ( countable_Set_cin @ A @ B2 @ A2 )
% 0.18/0.46 => ( countable_Set_cin @ A @ A3 @ A2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_cmem_trans
% 0.18/0.46 thf(fact_107_csubset__trans,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.46 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubset_trans
% 0.18/0.46 thf(fact_108_linear,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( linorder @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.46 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % linear
% 0.18/0.46 thf(fact_109_csubset__refl,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ A2 ) ).
% 0.18/0.46
% 0.18/0.46 % csubset_refl
% 0.18/0.46 thf(fact_110_cequalityD2,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( A2 = B3 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ A2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cequalityD2
% 0.18/0.46 thf(fact_111_cequalityD1,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( A2 = B3 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cequalityD1
% 0.18/0.46 thf(fact_112_cequalityCE,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C2: A] :
% 0.18/0.46 ( ( A2 = B3 )
% 0.18/0.46 => ( ( ( countable_Set_cin @ A @ C2 @ A2 )
% 0.18/0.46 => ~ ( countable_Set_cin @ A @ C2 @ B3 ) )
% 0.18/0.46 => ~ ( ~ ( countable_Set_cin @ A @ C2 @ A2 )
% 0.18/0.46 => ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cequalityCE
% 0.18/0.46 thf(fact_113_antisym,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.46 => ( ( ord_less_eq @ A @ Y @ X )
% 0.18/0.46 => ( X = Y ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym
% 0.18/0.46 thf(fact_114_cequalityE,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( A2 = B3 )
% 0.18/0.46 => ~ ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.46 => ~ ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ A2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cequalityE
% 0.18/0.46 thf(fact_115_eq__iff,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.18/0.46 = ( ^ [X2: A,Y2: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.18/0.46 & ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_iff
% 0.18/0.46 thf(fact_116_csubsetD,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C2: A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.46 => ( ( countable_Set_cin @ A @ C2 @ A2 )
% 0.18/0.46 => ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubsetD
% 0.18/0.46 thf(fact_117_cset__eqI,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ! [X4: A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ X4 @ A2 )
% 0.18/0.46 = ( countable_Set_cin @ A @ X4 @ B3 ) )
% 0.18/0.46 => ( A2 = B3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cset_eqI
% 0.18/0.46 thf(fact_118_cin__mono,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,X: A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.46 => ( ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 => ( countable_Set_cin @ A @ X @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cin_mono
% 0.18/0.46 thf(fact_119_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A: $tType,B: $tType] :
% 0.18/0.46 ( ( ( ord @ B )
% 0.18/0.46 & ( ord @ A ) )
% 0.18/0.46 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 => ( ( ( F @ B2 )
% 0.18/0.46 = C2 )
% 0.18/0.46 => ( ! [X4: A,Y4: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.18/0.46 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_120_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A: $tType,B: $tType] :
% 0.18/0.46 ( ( ( ord @ B )
% 0.18/0.46 & ( ord @ A ) )
% 0.18/0.46 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B2 ) )
% 0.18/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.18/0.46 => ( ! [X4: B,Y4: B] :
% 0.18/0.46 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.18/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_121_order__subst2,axiom,
% 0.18/0.46 ! [A: $tType,C: $tType] :
% 0.18/0.46 ( ( ( order @ C )
% 0.18/0.46 & ( order @ A ) )
% 0.18/0.46 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.18/0.46 => ( ! [X4: A,Y4: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.18/0.46 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst2
% 0.18/0.46 thf(fact_122_order__subst1,axiom,
% 0.18/0.46 ! [A: $tType,B: $tType] :
% 0.18/0.46 ( ( ( order @ B )
% 0.18/0.46 & ( order @ A ) )
% 0.18/0.46 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 0.18/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.18/0.46 => ( ! [X4: B,Y4: B] :
% 0.18/0.46 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.18/0.46 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_subst1
% 0.18/0.46 thf(fact_123_le__fun__def,axiom,
% 0.18/0.46 ! [B: $tType,A: $tType] :
% 0.18/0.46 ( ( ord @ B )
% 0.18/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.18/0.46 = ( ^ [F2: A > B,G: A > B] :
% 0.18/0.46 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_fun_def
% 0.18/0.46 thf(fact_124_le__funI,axiom,
% 0.18/0.46 ! [B: $tType,A: $tType] :
% 0.18/0.46 ( ( ord @ B )
% 0.18/0.46 => ! [F: A > B,G2: A > B] :
% 0.18/0.46 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) )
% 0.18/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_funI
% 0.18/0.46 thf(fact_125_le__funE,axiom,
% 0.18/0.46 ! [B: $tType,A: $tType] :
% 0.18/0.46 ( ( ord @ B )
% 0.18/0.46 => ! [F: A > B,G2: A > B,X: A] :
% 0.18/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.18/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_funE
% 0.18/0.46 thf(fact_126_le__funD,axiom,
% 0.18/0.46 ! [B: $tType,A: $tType] :
% 0.18/0.46 ( ( ord @ B )
% 0.18/0.46 => ! [F: A > B,G2: A > B,X: A] :
% 0.18/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 0.18/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_funD
% 0.18/0.46 thf(fact_127_equals__cemptyI,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ! [Y4: A] :
% 0.18/0.46 ~ ( countable_Set_cin @ A @ Y4 @ A2 )
% 0.18/0.46 => ( A2
% 0.18/0.46 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % equals_cemptyI
% 0.18/0.46 thf(fact_128_equals__cemptyD,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,A3: A] :
% 0.18/0.46 ( ( A2
% 0.18/0.46 = ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 0.18/0.46 => ~ ( countable_Set_cin @ A @ A3 @ A2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % equals_cemptyD
% 0.18/0.46 thf(fact_129_ex__cin__conv,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ? [X2: A] : ( countable_Set_cin @ A @ X2 @ A2 ) )
% 0.18/0.46 = ( A2
% 0.18/0.46 != ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ex_cin_conv
% 0.18/0.46 thf(fact_130_cemptyE,axiom,
% 0.18/0.46 ! [A: $tType,A3: A] :
% 0.18/0.46 ~ ( countable_Set_cin @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cemptyE
% 0.18/0.46 thf(fact_131_mk__disjoint__cinsert,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ A3 @ A2 )
% 0.18/0.46 => ? [B7: countable_Set_cset @ A] :
% 0.18/0.46 ( ( A2
% 0.18/0.46 = ( counta2111716221insert @ A @ A3 @ B7 ) )
% 0.18/0.46 & ~ ( countable_Set_cin @ A @ A3 @ B7 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % mk_disjoint_cinsert
% 0.18/0.46 thf(fact_132_cinsert__absorb,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ A3 @ A2 )
% 0.18/0.46 => ( ( counta2111716221insert @ A @ A3 @ A2 )
% 0.18/0.46 = A2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_absorb
% 0.18/0.46 thf(fact_133_cinsert__ident,axiom,
% 0.18/0.46 ! [A: $tType,X: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 => ( ~ ( countable_Set_cin @ A @ X @ B3 )
% 0.18/0.46 => ( ( ( counta2111716221insert @ A @ X @ A2 )
% 0.18/0.46 = ( counta2111716221insert @ A @ X @ B3 ) )
% 0.18/0.46 = ( A2 = B3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_ident
% 0.18/0.46 thf(fact_134_set__cinsert,axiom,
% 0.18/0.46 ! [A: $tType,X: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 => ~ ! [B7: countable_Set_cset @ A] :
% 0.18/0.46 ( ( A2
% 0.18/0.46 = ( counta2111716221insert @ A @ X @ B7 ) )
% 0.18/0.46 => ( countable_Set_cin @ A @ X @ B7 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % set_cinsert
% 0.18/0.46 thf(fact_135_cinsertI2,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,B3: countable_Set_cset @ A,B2: A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ A3 @ B3 )
% 0.18/0.46 => ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsertI2
% 0.18/0.46 thf(fact_136_cinsertI1,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,B3: countable_Set_cset @ A] : ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ A3 @ B3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsertI1
% 0.18/0.46 thf(fact_137_cinsertE,axiom,
% 0.18/0.46 ! [A: $tType,A3: A,B2: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ A2 ) )
% 0.18/0.46 => ( ( A3 != B2 )
% 0.18/0.46 => ( countable_Set_cin @ A @ A3 @ A2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsertE
% 0.18/0.46 thf(fact_138_bot_Oextremum__uniqueI,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order_bot @ A )
% 0.18/0.46 => ! [A3: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.18/0.46 => ( A3
% 0.18/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % bot.extremum_uniqueI
% 0.18/0.46 thf(fact_139_bot_Oextremum__unique,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order_bot @ A )
% 0.18/0.46 => ! [A3: A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.18/0.46 = ( A3
% 0.18/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % bot.extremum_unique
% 0.18/0.46 thf(fact_140_bot_Oextremum,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order_bot @ A )
% 0.18/0.46 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % bot.extremum
% 0.18/0.46 thf(fact_141_cDiffD2,axiom,
% 0.18/0.46 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 0.18/0.46 => ~ ( countable_Set_cin @ A @ C2 @ B3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiffD2
% 0.18/0.46 thf(fact_142_cDiffD1,axiom,
% 0.18/0.46 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 0.18/0.46 => ( countable_Set_cin @ A @ C2 @ A2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiffD1
% 0.18/0.46 thf(fact_143_cDiffE,axiom,
% 0.18/0.46 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 0.18/0.46 => ~ ( ( countable_Set_cin @ A @ C2 @ A2 )
% 0.18/0.46 => ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiffE
% 0.18/0.46 thf(fact_144_csubset__cinsertI2,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,B2: A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ B2 @ B3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubset_cinsertI2
% 0.18/0.46 thf(fact_145_csubset__cinsertI,axiom,
% 0.18/0.46 ! [A: $tType,B3: countable_Set_cset @ A,A3: A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ ( counta2111716221insert @ A @ A3 @ B3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubset_cinsertI
% 0.18/0.46 thf(fact_146_cinsert__mono,axiom,
% 0.18/0.46 ! [A: $tType,C3: countable_Set_cset @ A,D3: countable_Set_cset @ A,A3: A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ C3 @ D3 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ A3 @ C3 ) @ ( counta2111716221insert @ A @ A3 @ D3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_mono
% 0.18/0.46 thf(fact_147_cDiff__csubset,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ A2 ) ).
% 0.18/0.46
% 0.18/0.46 % cDiff_csubset
% 0.18/0.46 thf(fact_148_double__cDiff,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.46 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 0.18/0.46 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ B3 @ ( minus_minus @ ( countable_Set_cset @ A ) @ C3 @ A2 ) )
% 0.18/0.46 = A2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % double_cDiff
% 0.18/0.46 thf(fact_149_cminus__mono,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,C3: countable_Set_cset @ A,D3: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ C3 )
% 0.18/0.46 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ D3 @ B3 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ C3 @ D3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cminus_mono
% 0.18/0.46 thf(fact_150_csubset__cinsert__iff,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,X: A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) )
% 0.18/0.46 = ( ( ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 ) )
% 0.18/0.46 & ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubset_cinsert_iff
% 0.18/0.46 thf(fact_151_csingleton__iff,axiom,
% 0.18/0.46 ! [A: $tType,B2: A,A3: A] :
% 0.18/0.46 ( ( countable_Set_cin @ A @ B2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 0.18/0.46 = ( B2 = A3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % csingleton_iff
% 0.18/0.46 thf(fact_152_cinsert__cDiff__if,axiom,
% 0.18/0.46 ! [A: $tType,X: A,B3: countable_Set_cset @ A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ( countable_Set_cin @ A @ X @ B3 )
% 0.18/0.46 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
% 0.18/0.46 = ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) )
% 0.18/0.46 & ( ~ ( countable_Set_cin @ A @ X @ B3 )
% 0.18/0.46 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
% 0.18/0.46 = ( counta2111716221insert @ A @ X @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cinsert_cDiff_if
% 0.18/0.46 thf(fact_153_csubset__csingletonD,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,X: A] :
% 0.18/0.46 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 0.18/0.46 => ( ( A2
% 0.18/0.46 = ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 0.18/0.46 | ( A2
% 0.18/0.46 = ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % csubset_csingletonD
% 0.18/0.46 thf(fact_154_cDiff__cinsert__absorb,axiom,
% 0.18/0.46 ! [A: $tType,X: A,A2: countable_Set_cset @ A] :
% 0.18/0.46 ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 0.18/0.46 = A2 ) ) ).
% 0.18/0.46
% 0.18/0.46 % cDiff_cinsert_absorb
% 0.18/0.46 thf(fact_155_cpsubset__finsert__iff,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,X: A,B3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) )
% 0.18/0.46 = ( ( ( countable_Set_cin @ A @ X @ B3 )
% 0.18/0.46 => ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 0.18/0.46 & ( ~ ( countable_Set_cin @ A @ X @ B3 )
% 0.18/0.46 => ( ( ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 => ( ord_less @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 ) )
% 0.18/0.46 & ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 0.18/0.46 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cpsubset_finsert_iff
% 0.18/0.46 thf(fact_156_monad__state__altc_Ointro,axiom,
% 0.18/0.46 ! [S: $tType,A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 0.18/0.46 ( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
% 0.18/0.46 => ( ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc )
% 0.18/0.46 => ( ( monomo1324021455axioms @ S @ M @ C @ Get @ Put2 @ Altc )
% 0.18/0.46 => ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % monad_state_altc.intro
% 0.18/0.46 thf(fact_157_monad__state__altc__def,axiom,
% 0.18/0.46 ! [C: $tType,S: $tType,M: $tType,A: $tType] :
% 0.18/0.46 ( ( monomo1036387116e_altc @ A @ M @ S @ C )
% 0.18/0.46 = ( ^ [Return2: A > M,Bind2: M > ( A > M ) > M,Get2: ( S > M ) > M,Put: S > M > M,Altc2: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 0.18/0.46 ( ( monomo109450930_state @ A @ M @ S @ Return2 @ Bind2 @ Get2 @ Put )
% 0.18/0.46 & ( monomo439771545d_altc @ A @ M @ C @ Return2 @ Bind2 @ Altc2 )
% 0.18/0.46 & ( monomo1324021455axioms @ S @ M @ C @ Get2 @ Put @ Altc2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % monad_state_altc_def
% 0.18/0.46 thf(fact_158_Greatest__equality,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [P: A > $o,X: A] :
% 0.18/0.46 ( ( P @ X )
% 0.18/0.46 => ( ! [Y4: A] :
% 0.18/0.46 ( ( P @ Y4 )
% 0.18/0.46 => ( ord_less_eq @ A @ Y4 @ X ) )
% 0.18/0.46 => ( ( order_Greatest @ A @ P )
% 0.18/0.46 = X ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Greatest_equality
% 0.18/0.46 thf(fact_159_GreatestI2__order,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [P: A > $o,X: A,Q: A > $o] :
% 0.18/0.46 ( ( P @ X )
% 0.18/0.46 => ( ! [Y4: A] :
% 0.18/0.46 ( ( P @ Y4 )
% 0.18/0.46 => ( ord_less_eq @ A @ Y4 @ X ) )
% 0.18/0.46 => ( ! [X4: A] :
% 0.18/0.46 ( ( P @ X4 )
% 0.18/0.46 => ( ! [Y5: A] :
% 0.18/0.46 ( ( P @ Y5 )
% 0.18/0.46 => ( ord_less_eq @ A @ Y5 @ X4 ) )
% 0.18/0.46 => ( Q @ X4 ) ) )
% 0.18/0.46 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % GreatestI2_order
% 0.18/0.46 thf(fact_160_verit__la__disequality,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( linorder @ A )
% 0.18/0.46 => ! [A3: A,B2: A] :
% 0.18/0.46 ( ( A3 = B2 )
% 0.18/0.46 | ~ ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.46 | ~ ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % verit_la_disequality
% 0.18/0.46 thf(fact_161_le__rel__bool__arg__iff,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ord @ A )
% 0.18/0.46 => ( ( ord_less_eq @ ( $o > A ) )
% 0.18/0.46 = ( ^ [X5: $o > A,Y6: $o > A] :
% 0.18/0.46 ( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 0.18/0.46 & ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_rel_bool_arg_iff
% 0.18/0.46 thf(fact_162_verit__comp__simplify1_I3_J,axiom,
% 0.18/0.46 ! [B: $tType] :
% 0.18/0.46 ( ( linorder @ B )
% 0.18/0.46 => ! [B8: B,A7: B] :
% 0.18/0.46 ( ( ~ ( ord_less_eq @ B @ B8 @ A7 ) )
% 0.18/0.46 = ( ord_less @ B @ A7 @ B8 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % verit_comp_simplify1(3)
% 0.18/0.46 thf(fact_163_ord__eq__less__subst,axiom,
% 0.18/0.46 ! [A: $tType,B: $tType] :
% 0.18/0.46 ( ( ( ord @ B )
% 0.18/0.46 & ( ord @ A ) )
% 0.18/0.46 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.18/0.46 ( ( A3
% 0.18/0.46 = ( F @ B2 ) )
% 0.18/0.46 => ( ( ord_less @ B @ B2 @ C2 )
% 0.18/0.46 => ( ! [X4: B,Y4: B] :
% 0.18/0.46 ( ( ord_less @ B @ X4 @ Y4 )
% 0.18/0.46 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_less_subst
% 0.18/0.46 thf(fact_164_ord__less__eq__subst,axiom,
% 0.18/0.46 ! [A: $tType,B: $tType] :
% 0.18/0.46 ( ( ( ord @ B )
% 0.18/0.46 & ( ord @ A ) )
% 0.18/0.46 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 0.18/0.46 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.46 => ( ( ( F @ B2 )
% 0.18/0.46 = C2 )
% 0.18/0.46 => ( ! [X4: A,Y4: A] :
% 0.18/0.46 ( ( ord_less @ A @ X4 @ Y4 )
% 0.18/0.46 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_less_eq_subst
% 0.18/0.46 thf(fact_165_order__less__subst1,axiom,
% 0.18/0.46 ! [A: $tType,B: $tType] :
% 0.18/0.46 ( ( ( order @ B )
% 0.18/0.46 & ( order @ A ) )
% 0.18/0.46 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.18/0.46 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 0.18/0.46 => ( ( ord_less @ B @ B2 @ C2 )
% 0.18/0.46 => ( ! [X4: B,Y4: B] :
% 0.18/0.46 ( ( ord_less @ B @ X4 @ Y4 )
% 0.18/0.46 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_less_subst1
% 0.18/0.46 thf(fact_166_order__less__subst2,axiom,
% 0.18/0.46 ! [A: $tType,C: $tType] :
% 0.18/0.46 ( ( ( order @ C )
% 0.18/0.46 & ( order @ A ) )
% 0.18/0.46 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 0.18/0.46 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.46 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 0.18/0.46 => ( ! [X4: A,Y4: A] :
% 0.18/0.46 ( ( ord_less @ A @ X4 @ Y4 )
% 0.18/0.46 => ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_less_subst2
% 0.18/0.46 thf(fact_167_lt__ex,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( no_bot @ A )
% 0.18/0.46 => ! [X: A] :
% 0.18/0.46 ? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % lt_ex
% 0.18/0.46 thf(fact_168_gt__ex,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( no_top @ A )
% 0.18/0.46 => ! [X: A] :
% 0.18/0.46 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 0.18/0.46
% 0.18/0.46 % gt_ex
% 0.18/0.46 thf(fact_169_neqE,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( linorder @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( X != Y )
% 0.18/0.46 => ( ~ ( ord_less @ A @ X @ Y )
% 0.18/0.46 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % neqE
% 0.18/0.46 thf(fact_170_neq__iff,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( linorder @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( X != Y )
% 0.18/0.46 = ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % neq_iff
% 0.18/0.46 thf(fact_171_order_Oasym,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [A3: A,B2: A] :
% 0.18/0.46 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.46 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order.asym
% 0.18/0.46 thf(fact_172_cpsubset__trans,axiom,
% 0.18/0.46 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 0.18/0.46 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.46 => ( ( ord_less @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 0.18/0.46 => ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cpsubset_trans
% 0.18/0.46 thf(fact_173_dense,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( dense_order @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 => ? [Z3: A] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Z3 )
% 0.18/0.46 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dense
% 0.18/0.46 thf(fact_174_less__imp__neq,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 => ( X != Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_imp_neq
% 0.18/0.46 thf(fact_175_less__asym,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( preorder @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_asym
% 0.18/0.46 thf(fact_176_less__asym_H,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( preorder @ A )
% 0.18/0.46 => ! [A3: A,B2: A] :
% 0.18/0.46 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.46 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_asym'
% 0.18/0.46 thf(fact_177_less__trans,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( preorder @ A )
% 0.18/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 => ( ( ord_less @ A @ Y @ Z2 )
% 0.18/0.46 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_trans
% 0.18/0.46 thf(fact_178_less__linear,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( linorder @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 | ( X = Y )
% 0.18/0.46 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_linear
% 0.18/0.46 thf(fact_179_less__irrefl,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( preorder @ A )
% 0.18/0.46 => ! [X: A] :
% 0.18/0.46 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_irrefl
% 0.18/0.46 thf(fact_180_ord__eq__less__trans,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ord @ A )
% 0.18/0.46 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.46 ( ( A3 = B2 )
% 0.18/0.46 => ( ( ord_less @ A @ B2 @ C2 )
% 0.18/0.46 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_less_trans
% 0.18/0.46 thf(fact_181_ord__less__eq__trans,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( ord @ A )
% 0.18/0.46 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.46 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.46 => ( ( B2 = C2 )
% 0.18/0.46 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_less_eq_trans
% 0.18/0.46 thf(fact_182_dual__order_Oasym,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [B2: A,A3: A] :
% 0.18/0.46 ( ( ord_less @ A @ B2 @ A3 )
% 0.18/0.46 => ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.asym
% 0.18/0.46 thf(fact_183_less__imp__not__eq,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 => ( X != Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_imp_not_eq
% 0.18/0.46 thf(fact_184_less__not__sym,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( preorder @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_not_sym
% 0.18/0.46 thf(fact_185_less__induct,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( wellorder @ A )
% 0.18/0.46 => ! [P: A > $o,A3: A] :
% 0.18/0.46 ( ! [X4: A] :
% 0.18/0.46 ( ! [Y5: A] :
% 0.18/0.46 ( ( ord_less @ A @ Y5 @ X4 )
% 0.18/0.46 => ( P @ Y5 ) )
% 0.18/0.46 => ( P @ X4 ) )
% 0.18/0.46 => ( P @ A3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_induct
% 0.18/0.46 thf(fact_186_antisym__conv3,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( linorder @ A )
% 0.18/0.46 => ! [Y: A,X: A] :
% 0.18/0.46 ( ~ ( ord_less @ A @ Y @ X )
% 0.18/0.46 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.18/0.46 = ( X = Y ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym_conv3
% 0.18/0.46 thf(fact_187_less__imp__not__eq2,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( order @ A )
% 0.18/0.46 => ! [X: A,Y: A] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 => ( Y != X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_imp_not_eq2
% 0.18/0.46 thf(fact_188_less__imp__triv,axiom,
% 0.18/0.46 ! [A: $tType] :
% 0.18/0.46 ( ( preorder @ A )
% 0.18/0.46 => ! [X: A,Y: A,P: $o] :
% 0.18/0.46 ( ( ord_less @ A @ X @ Y )
% 0.18/0.46 => ( ( ord_less @ A @ Y @ X )
% 0.18/0.46 => P ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % less_imp_triv
% 0.18/0.47 thf(fact_189_linorder__cases,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.18/0.47 => ( ( X != Y )
% 0.18/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % linorder_cases
% 0.18/0.47 thf(fact_190_dual__order_Oirrefl,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [A3: A] :
% 0.18/0.47 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 0.18/0.47
% 0.18/0.47 % dual_order.irrefl
% 0.18/0.47 thf(fact_191_order_Ostrict__trans,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.47 => ( ( ord_less @ A @ B2 @ C2 )
% 0.18/0.47 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order.strict_trans
% 0.18/0.47 thf(fact_192_less__imp__not__less,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( preorder @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ( ord_less @ A @ X @ Y )
% 0.18/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % less_imp_not_less
% 0.18/0.47 thf(fact_193_exists__least__iff,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( wellorder @ A )
% 0.18/0.47 => ( ( ^ [P2: A > $o] :
% 0.18/0.47 ? [X6: A] : ( P2 @ X6 ) )
% 0.18/0.47 = ( ^ [P3: A > $o] :
% 0.18/0.47 ? [N2: A] :
% 0.18/0.47 ( ( P3 @ N2 )
% 0.18/0.47 & ! [M2: A] :
% 0.18/0.47 ( ( ord_less @ A @ M2 @ N2 )
% 0.18/0.47 => ~ ( P3 @ M2 ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % exists_least_iff
% 0.18/0.47 thf(fact_194_linorder__less__wlog,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [P: A > A > $o,A3: A,B2: A] :
% 0.18/0.47 ( ! [A6: A,B6: A] :
% 0.18/0.47 ( ( ord_less @ A @ A6 @ B6 )
% 0.18/0.47 => ( P @ A6 @ B6 ) )
% 0.18/0.47 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 0.18/0.47 => ( ! [A6: A,B6: A] :
% 0.18/0.47 ( ( P @ B6 @ A6 )
% 0.18/0.47 => ( P @ A6 @ B6 ) )
% 0.18/0.47 => ( P @ A3 @ B2 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % linorder_less_wlog
% 0.18/0.47 thf(fact_195_dual__order_Ostrict__trans,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [B2: A,A3: A,C2: A] :
% 0.18/0.47 ( ( ord_less @ A @ B2 @ A3 )
% 0.18/0.47 => ( ( ord_less @ A @ C2 @ B2 )
% 0.18/0.47 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dual_order.strict_trans
% 0.18/0.47 thf(fact_196_not__less__iff__gr__or__eq,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.18/0.47 = ( ( ord_less @ A @ Y @ X )
% 0.18/0.47 | ( X = Y ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % not_less_iff_gr_or_eq
% 0.18/0.47 thf(fact_197_order_Ostrict__implies__not__eq,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [A3: A,B2: A] :
% 0.18/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.47 => ( A3 != B2 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order.strict_implies_not_eq
% 0.18/0.47 thf(fact_198_dual__order_Ostrict__implies__not__eq,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [B2: A,A3: A] :
% 0.18/0.47 ( ( ord_less @ A @ B2 @ A3 )
% 0.18/0.47 => ( A3 != B2 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dual_order.strict_implies_not_eq
% 0.18/0.47 thf(fact_199_monad__state_Obind__put,axiom,
% 0.18/0.47 ! [S: $tType,M: $tType,A: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,S3: S,M4: M,F: A > M] :
% 0.18/0.47 ( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
% 0.18/0.47 => ( ( Bind @ ( Put2 @ S3 @ M4 ) @ F )
% 0.18/0.47 = ( Put2 @ S3 @ ( Bind @ M4 @ F ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % monad_state.bind_put
% 0.18/0.47 thf(fact_200_monad__state_Oput__put,axiom,
% 0.18/0.47 ! [A: $tType,S: $tType,M: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,S3: S,S4: S,M4: M] :
% 0.18/0.47 ( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
% 0.18/0.47 => ( ( Put2 @ S3 @ ( Put2 @ S4 @ M4 ) )
% 0.18/0.47 = ( Put2 @ S4 @ M4 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % monad_state.put_put
% 0.18/0.47 thf(fact_201_monad__state_Oput__get,axiom,
% 0.18/0.47 ! [A: $tType,M: $tType,S: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,S3: S,F: S > M] :
% 0.18/0.47 ( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
% 0.18/0.47 => ( ( Put2 @ S3 @ ( Get @ F ) )
% 0.18/0.47 = ( Put2 @ S3 @ ( F @ S3 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % monad_state.put_get
% 0.18/0.47 thf(fact_202_leD,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [Y: A,X: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.18/0.47 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % leD
% 0.18/0.47 thf(fact_203_leI,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.18/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % leI
% 0.18/0.47 thf(fact_204_le__less,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ( ( ord_less_eq @ A )
% 0.18/0.47 = ( ^ [X2: A,Y2: A] :
% 0.18/0.47 ( ( ord_less @ A @ X2 @ Y2 )
% 0.18/0.47 | ( X2 = Y2 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % le_less
% 0.18/0.47 thf(fact_205_less__le,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ( ( ord_less @ A )
% 0.18/0.47 = ( ^ [X2: A,Y2: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.18/0.47 & ( X2 != Y2 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % less_le
% 0.18/0.47 thf(fact_206_order__le__less__subst1,axiom,
% 0.18/0.47 ! [A: $tType,B: $tType] :
% 0.18/0.47 ( ( ( order @ B )
% 0.18/0.47 & ( order @ A ) )
% 0.18/0.47 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.18/0.47 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 0.18/0.47 => ( ( ord_less @ B @ B2 @ C2 )
% 0.18/0.47 => ( ! [X4: B,Y4: B] :
% 0.18/0.47 ( ( ord_less @ B @ X4 @ Y4 )
% 0.18/0.47 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.47 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order_le_less_subst1
% 0.18/0.47 thf(fact_207_order__le__less__subst2,axiom,
% 0.18/0.47 ! [A: $tType,C: $tType] :
% 0.18/0.47 ( ( ( order @ C )
% 0.18/0.47 & ( order @ A ) )
% 0.18/0.47 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 0.18/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.47 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 0.18/0.47 => ( ! [X4: A,Y4: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.18/0.47 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.47 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order_le_less_subst2
% 0.18/0.47 thf(fact_208_order__less__le__subst1,axiom,
% 0.18/0.47 ! [A: $tType,B: $tType] :
% 0.18/0.47 ( ( ( order @ B )
% 0.18/0.47 & ( order @ A ) )
% 0.18/0.47 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 0.18/0.47 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 0.18/0.47 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.18/0.47 => ( ! [X4: B,Y4: B] :
% 0.18/0.47 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 0.18/0.47 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.47 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order_less_le_subst1
% 0.18/0.47 thf(fact_209_order__less__le__subst2,axiom,
% 0.18/0.47 ! [A: $tType,C: $tType] :
% 0.18/0.47 ( ( ( order @ C )
% 0.18/0.47 & ( order @ A ) )
% 0.18/0.47 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 0.18/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.47 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.18/0.47 => ( ! [X4: A,Y4: A] :
% 0.18/0.47 ( ( ord_less @ A @ X4 @ Y4 )
% 0.18/0.47 => ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.18/0.47 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order_less_le_subst2
% 0.18/0.47 thf(fact_210_not__le,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.18/0.47 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % not_le
% 0.18/0.47 thf(fact_211_not__less,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.18/0.47 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % not_less
% 0.18/0.47 thf(fact_212_le__neq__trans,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [A3: A,B2: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.47 => ( ( A3 != B2 )
% 0.18/0.47 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % le_neq_trans
% 0.18/0.47 thf(fact_213_antisym__conv1,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.18/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.47 = ( X = Y ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % antisym_conv1
% 0.18/0.47 thf(fact_214_antisym__conv2,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.47 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.18/0.47 = ( X = Y ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % antisym_conv2
% 0.18/0.47 thf(fact_215_less__imp__le,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( preorder @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ( ord_less @ A @ X @ Y )
% 0.18/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % less_imp_le
% 0.18/0.47 thf(fact_216_le__less__trans,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( preorder @ A )
% 0.18/0.47 => ! [X: A,Y: A,Z2: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.47 => ( ( ord_less @ A @ Y @ Z2 )
% 0.18/0.47 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % le_less_trans
% 0.18/0.47 thf(fact_217_less__le__trans,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( preorder @ A )
% 0.18/0.47 => ! [X: A,Y: A,Z2: A] :
% 0.18/0.47 ( ( ord_less @ A @ X @ Y )
% 0.18/0.47 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.18/0.47 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % less_le_trans
% 0.18/0.47 thf(fact_218_dense__ge,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( dense_linorder @ A )
% 0.18/0.47 => ! [Z2: A,Y: A] :
% 0.18/0.47 ( ! [X4: A] :
% 0.18/0.47 ( ( ord_less @ A @ Z2 @ X4 )
% 0.18/0.47 => ( ord_less_eq @ A @ Y @ X4 ) )
% 0.18/0.47 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dense_ge
% 0.18/0.47 thf(fact_219_dense__le,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( dense_linorder @ A )
% 0.18/0.47 => ! [Y: A,Z2: A] :
% 0.18/0.47 ( ! [X4: A] :
% 0.18/0.47 ( ( ord_less @ A @ X4 @ Y )
% 0.18/0.47 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 0.18/0.47 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dense_le
% 0.18/0.47 thf(fact_220_le__less__linear,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.47 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % le_less_linear
% 0.18/0.47 thf(fact_221_le__imp__less__or__eq,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [X: A,Y: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.18/0.47 => ( ( ord_less @ A @ X @ Y )
% 0.18/0.47 | ( X = Y ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % le_imp_less_or_eq
% 0.18/0.47 thf(fact_222_less__le__not__le,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( preorder @ A )
% 0.18/0.47 => ( ( ord_less @ A )
% 0.18/0.47 = ( ^ [X2: A,Y2: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.18/0.47 & ~ ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % less_le_not_le
% 0.18/0.47 thf(fact_223_not__le__imp__less,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [Y: A,X: A] :
% 0.18/0.47 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.18/0.47 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % not_le_imp_less
% 0.18/0.47 thf(fact_224_order_Ostrict__trans1,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.47 => ( ( ord_less @ A @ B2 @ C2 )
% 0.18/0.47 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order.strict_trans1
% 0.18/0.47 thf(fact_225_order_Ostrict__trans2,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.47 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.18/0.47 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order.strict_trans2
% 0.18/0.47 thf(fact_226_order_Oorder__iff__strict,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ( ( ord_less_eq @ A )
% 0.18/0.47 = ( ^ [A5: A,B5: A] :
% 0.18/0.47 ( ( ord_less @ A @ A5 @ B5 )
% 0.18/0.47 | ( A5 = B5 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order.order_iff_strict
% 0.18/0.47 thf(fact_227_order_Ostrict__iff__order,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ( ( ord_less @ A )
% 0.18/0.47 = ( ^ [A5: A,B5: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.18/0.47 & ( A5 != B5 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order.strict_iff_order
% 0.18/0.47 thf(fact_228_dual__order_Ostrict__trans1,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [B2: A,A3: A,C2: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.18/0.47 => ( ( ord_less @ A @ C2 @ B2 )
% 0.18/0.47 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dual_order.strict_trans1
% 0.18/0.47 thf(fact_229_dual__order_Ostrict__trans2,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [B2: A,A3: A,C2: A] :
% 0.18/0.47 ( ( ord_less @ A @ B2 @ A3 )
% 0.18/0.47 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.18/0.47 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dual_order.strict_trans2
% 0.18/0.47 thf(fact_230_dense__ge__bounded,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( dense_linorder @ A )
% 0.18/0.47 => ! [Z2: A,X: A,Y: A] :
% 0.18/0.47 ( ( ord_less @ A @ Z2 @ X )
% 0.18/0.47 => ( ! [W: A] :
% 0.18/0.47 ( ( ord_less @ A @ Z2 @ W )
% 0.18/0.47 => ( ( ord_less @ A @ W @ X )
% 0.18/0.47 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.18/0.47 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dense_ge_bounded
% 0.18/0.47 thf(fact_231_dense__le__bounded,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( dense_linorder @ A )
% 0.18/0.47 => ! [X: A,Y: A,Z2: A] :
% 0.18/0.47 ( ( ord_less @ A @ X @ Y )
% 0.18/0.47 => ( ! [W: A] :
% 0.18/0.47 ( ( ord_less @ A @ X @ W )
% 0.18/0.47 => ( ( ord_less @ A @ W @ Y )
% 0.18/0.47 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.18/0.47 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dense_le_bounded
% 0.18/0.47 thf(fact_232_order_Ostrict__implies__order,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [A3: A,B2: A] :
% 0.18/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.47 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order.strict_implies_order
% 0.18/0.47 thf(fact_233_dual__order_Oorder__iff__strict,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ( ( ord_less_eq @ A )
% 0.18/0.47 = ( ^ [B5: A,A5: A] :
% 0.18/0.47 ( ( ord_less @ A @ B5 @ A5 )
% 0.18/0.47 | ( A5 = B5 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dual_order.order_iff_strict
% 0.18/0.47 thf(fact_234_dual__order_Ostrict__iff__order,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ( ( ord_less @ A )
% 0.18/0.47 = ( ^ [B5: A,A5: A] :
% 0.18/0.47 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.18/0.47 & ( A5 != B5 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dual_order.strict_iff_order
% 0.18/0.47 thf(fact_235_dual__order_Ostrict__implies__order,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [B2: A,A3: A] :
% 0.18/0.47 ( ( ord_less @ A @ B2 @ A3 )
% 0.18/0.47 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % dual_order.strict_implies_order
% 0.18/0.47 thf(fact_236_order_Onot__eq__order__implies__strict,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order @ A )
% 0.18/0.47 => ! [A3: A,B2: A] :
% 0.18/0.47 ( ( A3 != B2 )
% 0.18/0.47 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.18/0.47 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % order.not_eq_order_implies_strict
% 0.18/0.47 thf(fact_237_bot_Oextremum__strict,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order_bot @ A )
% 0.18/0.47 => ! [A3: A] :
% 0.18/0.47 ~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % bot.extremum_strict
% 0.18/0.47 thf(fact_238_bot_Onot__eq__extremum,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( order_bot @ A )
% 0.18/0.47 => ! [A3: A] :
% 0.18/0.47 ( ( A3
% 0.18/0.47 != ( bot_bot @ A ) )
% 0.18/0.47 = ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % bot.not_eq_extremum
% 0.18/0.47 thf(fact_239_diff__strict__mono,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( ordered_ab_group_add @ A )
% 0.18/0.47 => ! [A3: A,B2: A,D2: A,C2: A] :
% 0.18/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.47 => ( ( ord_less @ A @ D2 @ C2 )
% 0.18/0.47 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % diff_strict_mono
% 0.18/0.47 thf(fact_240_diff__eq__diff__less,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( ordered_ab_group_add @ A )
% 0.18/0.47 => ! [A3: A,B2: A,C2: A,D2: A] :
% 0.18/0.47 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.18/0.47 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.18/0.47 => ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.47 = ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % diff_eq_diff_less
% 0.18/0.47 thf(fact_241_diff__strict__left__mono,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( ordered_ab_group_add @ A )
% 0.18/0.47 => ! [B2: A,A3: A,C2: A] :
% 0.18/0.47 ( ( ord_less @ A @ B2 @ A3 )
% 0.18/0.47 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % diff_strict_left_mono
% 0.18/0.47 thf(fact_242_diff__strict__right__mono,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( ordered_ab_group_add @ A )
% 0.18/0.47 => ! [A3: A,B2: A,C2: A] :
% 0.18/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.18/0.47 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % diff_strict_right_mono
% 0.18/0.47 thf(fact_243_cpsubsetD,axiom,
% 0.18/0.47 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C2: A] :
% 0.18/0.47 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.47 => ( ( countable_Set_cin @ A @ C2 @ A2 )
% 0.18/0.47 => ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % cpsubsetD
% 0.18/0.47 thf(fact_244_not__cpsubset__cempty,axiom,
% 0.18/0.47 ! [A: $tType,A2: countable_Set_cset @ A] :
% 0.18/0.47 ~ ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % not_cpsubset_cempty
% 0.18/0.47 thf(fact_245_cpsubset__eq,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( ord_less @ ( countable_Set_cset @ A ) )
% 0.18/0.47 = ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
% 0.18/0.47 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A4 @ B4 )
% 0.18/0.47 & ( A4 != B4 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % cpsubset_eq
% 0.18/0.47 thf(fact_246_less__cset__def,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( ord_less @ ( countable_Set_cset @ A ) )
% 0.18/0.47 = ( ^ [Xs: countable_Set_cset @ A,Ys: countable_Set_cset @ A] :
% 0.18/0.47 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ Xs @ Ys )
% 0.18/0.47 & ( Xs != Ys ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % less_cset_def
% 0.18/0.47 thf(fact_247_cpsubset__imp__fsubset,axiom,
% 0.18/0.47 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.47 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.47 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% 0.18/0.47
% 0.18/0.47 % cpsubset_imp_fsubset
% 0.18/0.47 thf(fact_248_cpsubset__csubset__trans,axiom,
% 0.18/0.47 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 0.18/0.47 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.47 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 0.18/0.47 => ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % cpsubset_csubset_trans
% 0.18/0.47 thf(fact_249_csubset__cpsubset__trans,axiom,
% 0.18/0.47 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 0.18/0.47 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.47 => ( ( ord_less @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 0.18/0.47 => ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % csubset_cpsubset_trans
% 0.18/0.47 thf(fact_250_csubset__not__fsubset__eq,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( ord_less @ ( countable_Set_cset @ A ) )
% 0.18/0.47 = ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
% 0.18/0.47 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A4 @ B4 )
% 0.18/0.47 & ~ ( ord_less_eq @ ( countable_Set_cset @ A ) @ B4 @ A4 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % csubset_not_fsubset_eq
% 0.18/0.47 thf(fact_251_csubset__iff__pfsubset__eq,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( ord_less_eq @ ( countable_Set_cset @ A ) )
% 0.18/0.47 = ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
% 0.18/0.47 ( ( ord_less @ ( countable_Set_cset @ A ) @ A4 @ B4 )
% 0.18/0.47 | ( A4 = B4 ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % csubset_iff_pfsubset_eq
% 0.18/0.47 thf(fact_252_monad__state__altc_Oaxioms_I1_J,axiom,
% 0.18/0.47 ! [C: $tType,A: $tType,M: $tType,S: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 0.18/0.47 ( ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc )
% 0.18/0.47 => ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 ) ) ).
% 0.18/0.47
% 0.18/0.47 % monad_state_altc.axioms(1)
% 0.18/0.47 thf(fact_253_cpsubset__imp__ex__fmem,axiom,
% 0.18/0.47 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 0.18/0.47 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 0.18/0.47 => ? [B6: A] : ( countable_Set_cin @ A @ B6 @ ( minus_minus @ ( countable_Set_cset @ A ) @ B3 @ A2 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % cpsubset_imp_ex_fmem
% 0.18/0.47 thf(fact_254_minf_I8_J,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [T: A] :
% 0.18/0.47 ? [Z3: A] :
% 0.18/0.47 ! [X7: A] :
% 0.18/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.18/0.47 => ~ ( ord_less_eq @ A @ T @ X7 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % minf(8)
% 0.18/0.47 thf(fact_255_minf_I6_J,axiom,
% 0.18/0.47 ! [A: $tType] :
% 0.18/0.47 ( ( linorder @ A )
% 0.18/0.47 => ! [T: A] :
% 0.18/0.47 ? [Z3: A] :
% 0.18/0.47 ! [X7: A] :
% 0.18/0.47 ( ( ord_less @ A @ X7 @ Z3 )
% 0.18/0.47 => ( ord_less_eq @ A @ X7 @ T ) ) ) ).
% 0.18/0.47
% 0.18/0.47 % minf(6)
% 0.18/0.47
% 0.18/0.47 % Type constructors (19)
% 0.18/0.47 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.18/0.47 ! [A8: $tType,A9: $tType] :
% 0.18/0.47 ( ( order_bot @ A9 )
% 0.18/0.47 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.18/0.47 ! [A8: $tType,A9: $tType] :
% 0.18/0.47 ( ( preorder @ A9 )
% 0.18/0.47 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.18/0.47 ! [A8: $tType,A9: $tType] :
% 0.18/0.47 ( ( order @ A9 )
% 0.18/0.47 => ( order @ ( A8 > A9 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.18/0.47 ! [A8: $tType,A9: $tType] :
% 0.18/0.47 ( ( ord @ A9 )
% 0.18/0.47 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.18/0.47 ! [A8: $tType,A9: $tType] :
% 0.18/0.47 ( ( bot @ A9 )
% 0.18/0.47 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_fun___Groups_Ominus,axiom,
% 0.18/0.47 ! [A8: $tType,A9: $tType] :
% 0.18/0.47 ( ( minus @ A9 )
% 0.18/0.47 => ( minus @ ( A8 > A9 ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__bot_1,axiom,
% 0.18/0.47 order_bot @ $o ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_2,axiom,
% 0.18/0.47 preorder @ $o ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.18/0.47 linorder @ $o ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
% 0.18/0.47 order @ $o ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
% 0.18/0.47 ord @ $o ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_HOL_Obool___Orderings_Obot_5,axiom,
% 0.18/0.47 bot @ $o ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_HOL_Obool___Groups_Ominus_6,axiom,
% 0.18/0.47 minus @ $o ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_Countable__Set__Type_Ocset___Orderings_Oorder__bot_7,axiom,
% 0.18/0.47 ! [A8: $tType] : ( order_bot @ ( countable_Set_cset @ A8 ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_Countable__Set__Type_Ocset___Orderings_Opreorder_8,axiom,
% 0.18/0.47 ! [A8: $tType] : ( preorder @ ( countable_Set_cset @ A8 ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_Countable__Set__Type_Ocset___Orderings_Oorder_9,axiom,
% 0.18/0.47 ! [A8: $tType] : ( order @ ( countable_Set_cset @ A8 ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_Countable__Set__Type_Ocset___Orderings_Oord_10,axiom,
% 0.18/0.47 ! [A8: $tType] : ( ord @ ( countable_Set_cset @ A8 ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_Countable__Set__Type_Ocset___Orderings_Obot_11,axiom,
% 0.18/0.47 ! [A8: $tType] : ( bot @ ( countable_Set_cset @ A8 ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(tcon_Countable__Set__Type_Ocset___Groups_Ominus_12,axiom,
% 0.18/0.47 ! [A8: $tType] : ( minus @ ( countable_Set_cset @ A8 ) ) ).
% 0.18/0.47
% 0.18/0.47 % Helper facts (3)
% 0.18/0.47 thf(help_If_3_1_T,axiom,
% 0.18/0.47 ! [P: $o] :
% 0.18/0.47 ( ( P = $true )
% 0.18/0.47 | ( P = $false ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(help_If_2_1_T,axiom,
% 0.18/0.47 ! [A: $tType,X: A,Y: A] :
% 0.18/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.18/0.47 = Y ) ).
% 0.18/0.47
% 0.18/0.47 thf(help_If_1_1_T,axiom,
% 0.18/0.47 ! [A: $tType,X: A,Y: A] :
% 0.18/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.18/0.47 = X ) ).
% 0.18/0.47
% 0.18/0.47 % Conjectures (1)
% 0.18/0.47 thf(conj_0,conjecture,
% 0.18/0.47 ( ( monomo955616121nondet @ c @ m @ a @ mergec @ ( counta2111716221insert @ c @ x @ ( bot_bot @ ( countable_Set_cset @ c ) ) ) @ f )
% 0.18/0.47 = ( f @ x ) ) ).
% 0.18/0.47
% 0.18/0.47 %------------------------------------------------------------------------------
% 0.18/0.47 ------------------------
% 3.41/1.82 ---- Embedded in THF ---
% 3.41/1.82 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.41/1.82 %%% Generated on Tue Feb 24 08:27:26 EST 2026
% 3.41/1.82 %%% using '$$dhol' embedding, version 1.3.0.
% 3.41/1.82 %%% Logic specification used:
% 3.41/1.82 %%% thf(spec, logic, $$dhol).
% 3.41/1.82
% 3.41/1.82 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.fjdGAlfEYi/DTF2DTF6822.p
% 3.41/1.82 thf(ty_t_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT, type, monomo1453983439ondetT: ($tType > ($tType > $tType))).
% 3.41/1.82 thf(ty_t_Countable__Set__Type_Ocset, type, countable_Set_cset: ($tType > $tType)).
% 3.41/1.82 thf(ty_tf_m, type, m: $tType).
% 3.41/1.82 thf(ty_tf_c, type, c: $tType).
% 3.41/1.82 thf(ty_tf_a, type, a: $tType).
% 3.41/1.82 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: ($o))).
% 3.41/1.82 thf(sy_c_Countable__Set__Type_Ocin, type, countable_Set_cin: (!> [A:$tType]: ((A > ((countable_Set_cset @ A) > $o))))).
% 3.41/1.82 thf(sy_c_Countable__Set__Type_Ocinsert, type, counta2111716221insert: (!> [A:$tType]: ((A > ((countable_Set_cset @ A) > (countable_Set_cset @ A)))))).
% 3.41/1.82 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 3.41/1.82 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 3.41/1.82 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM, type, monomo2011693592ondetM: (!> [A:$tType,M:$tType,C:$tType]: ((((countable_Set_cset @ A) > M) > ((M > (((countable_Set_cset @ A) > M) > M)) > (((countable_Set_cset @ A) > ((A > M) > M)) > (((countable_Set_cset @ C) > ((C > M) > M)) > $o))))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM__axioms, type, monomo759963067axioms: (!> [A:$tType,M:$tType,C:$tType]: ((((countable_Set_cset @ A) > M) > ((M > (((countable_Set_cset @ A) > M) > M)) > (((countable_Set_cset @ A) > ((A > M) > M)) > (((countable_Set_cset @ C) > ((C > M) > M)) > $o))))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM__base_Oaltc__nondet, type, monomo955616121nondet: (!> [C:$tType,M:$tType,A:$tType]: ((((countable_Set_cset @ C) > ((C > M) > M)) > ((countable_Set_cset @ C) > ((C > ((monomo1453983439ondetT @ A) @ M)) > ((monomo1453983439ondetT @ A) @ M))))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc, type, monomo439771545d_altc: (!> [A:$tType,M:$tType,C:$tType]: (((A > M) > ((M > ((A > M) > M)) > (((countable_Set_cset @ C) > ((C > M) > M)) > $o)))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc3, type, monomo1425736922_altc3: (!> [A:$tType,M:$tType,C:$tType]: (((A > M) > ((M > ((A > M) > M)) > (((countable_Set_cset @ C) > ((C > M) > M)) > $o)))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc__axioms, type, monomo555039548axioms: (!> [M:$tType,A:$tType,C:$tType]: (((M > ((A > M) > M)) > (((countable_Set_cset @ C) > ((C > M) > M)) > $o))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state, type, monomo109450930_state: (!> [A:$tType,M:$tType,S:$tType]: (((A > M) > ((M > ((A > M) > M)) > (((S > M) > M) > ((S > (M > M)) > $o))))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state__altc, type, monomo1036387116e_altc: (!> [A:$tType,M:$tType,S:$tType,C:$tType]: (((A > M) > ((M > ((A > M) > M)) > (((S > M) > M) > ((S > (M > M)) > (((countable_Set_cset @ C) > ((C > M) > M)) > $o)))))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state__altc__axioms, type, monomo1324021455axioms: (!> [S:$tType,M:$tType,C:$tType]: ((((S > M) > M) > ((S > (M > M)) > (((countable_Set_cset @ C) > ((C > M) > M)) > $o)))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetM__base_Oput__nondet, type, monomo410456285nondet: (!> [State:$tType,M:$tType,A:$tType]: (((State > (M > M)) > (State > (((monomo1453983439ondetT @ A) @ M) > ((monomo1453983439ondetT @ A) @ M))))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_ONondetT, type, monomo1975189248ondetT: (!> [M:$tType,A:$tType]: ((M > ((monomo1453983439ondetT @ A) @ M))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Ocase__nondetT, type, monomo1564381680ondetT: (!> [M:$tType,B:$tType,A:$tType]: (((M > B) > (((monomo1453983439ondetT @ A) @ M) > B))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Orec__nondetT, type, monomo1291740684ondetT: (!> [M:$tType,D:$tType,A:$tType]: (((M > D) > (((monomo1453983439ondetT @ A) @ M) > D))))).
% 3.41/1.82 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Orun__nondet, type, monomo603716163nondet: (!> [A:$tType,M:$tType]: ((((monomo1453983439ondetT @ A) @ M) > M)))).
% 3.41/1.82 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 3.41/1.82 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.41/1.82 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.41/1.82 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 3.41/1.82 thf(sy_v_f____, type, f: (c > ((monomo1453983439ondetT @ a) @ m))).
% 3.41/1.82 thf(sy_v_mergec, type, mergec: ((countable_Set_cset @ c) > ((c > m) > m))).
% 3.41/1.82 thf(sy_v_x____, type, x: c).
% 3.41/1.82 thf(fact_0_mergec__single, axiom, (! [X:c,F:(c > m)]: ((((mergec @ (((counta2111716221insert @ c) @ X) @ (bot_bot @ (countable_Set_cset @ c)))) @ F) = (F @ X))))).
% 3.41/1.82 thf(fact_1_cset__nondetM__base_Oaltc__nondet_Ocong, axiom, (! [A:$tType,M:$tType,C:$tType]: (((((monomo955616121nondet @ C) @ M) @ A) = (((monomo955616121nondet @ C) @ M) @ A))))).
% 3.41/1.82 thf(fact_2_cinsert__absorb2, axiom, (! [A:$tType,X:A,A2:(countable_Set_cset @ A)]: (((((counta2111716221insert @ A) @ X) @ (((counta2111716221insert @ A) @ X) @ A2)) = (((counta2111716221insert @ A) @ X) @ A2))))).
% 3.41/1.82 thf(fact_3_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X2:D]: ((bot_bot @ C)))))))).
% 3.41/1.82 thf(fact_4_cdoubleton__eq__iff, axiom, (! [A:$tType,A3:A,B2:A,C2:A,D2:A]: ((((((counta2111716221insert @ A) @ A3) @ (((counta2111716221insert @ A) @ B2) @ (bot_bot @ (countable_Set_cset @ A)))) = (((counta2111716221insert @ A) @ C2) @ (((counta2111716221insert @ A) @ D2) @ (bot_bot @ (countable_Set_cset @ A))))) = (((A3 = C2) & (B2 = D2)) | ((A3 = D2) & (B2 = C2))))))).
% 3.41/1.82 thf(fact_5_csingleton__inject, axiom, (! [A:$tType,A3:A,B2:A]: ((((((counta2111716221insert @ A) @ A3) @ (bot_bot @ (countable_Set_cset @ A))) = (((counta2111716221insert @ A) @ B2) @ (bot_bot @ (countable_Set_cset @ A)))) => (A3 = B2))))).
% 3.41/1.82 thf(fact_6_cinsert__not__cempty, axiom, (! [A:$tType,A3:A,A2:(countable_Set_cset @ A)]: (((((counta2111716221insert @ A) @ A3) @ A2) != (bot_bot @ (countable_Set_cset @ A)))))).
% 3.41/1.82 thf(fact_7_cinsert__commute, axiom, (! [A:$tType,X:A,Y:A,A2:(countable_Set_cset @ A)]: (((((counta2111716221insert @ A) @ X) @ (((counta2111716221insert @ A) @ Y) @ A2)) = (((counta2111716221insert @ A) @ Y) @ (((counta2111716221insert @ A) @ X) @ A2)))))).
% 3.41/1.82 thf(fact_8_cset__nondetM_Omerge__single, axiom, (! [C:$tType,M:$tType,A:$tType,Return:((countable_Set_cset @ A) > M),Bind:(M > (((countable_Set_cset @ A) > M) > M)),Merge:((countable_Set_cset @ A) > ((A > M) > M)),Mergec:((countable_Set_cset @ C) > ((C > M) > M)),X:A,F:(A > M)]: (((((((((monomo2011693592ondetM @ A) @ M) @ C) @ Return) @ Bind) @ Merge) @ Mergec) => (((Merge @ (((counta2111716221insert @ A) @ X) @ (bot_bot @ (countable_Set_cset @ A)))) @ F) = (F @ X)))))).
% 3.41/1.82 thf(fact_9_cset__nondetM_Omergec__single, axiom, (! [A:$tType,M:$tType,C:$tType,Return:((countable_Set_cset @ A) > M),Bind:(M > (((countable_Set_cset @ A) > M) > M)),Merge:((countable_Set_cset @ A) > ((A > M) > M)),Mergec:((countable_Set_cset @ C) > ((C > M) > M)),X:C,F:(C > M)]: (((((((((monomo2011693592ondetM @ A) @ M) @ C) @ Return) @ Bind) @ Merge) @ Mergec) => (((Mergec @ (((counta2111716221insert @ C) @ X) @ (bot_bot @ (countable_Set_cset @ C)))) @ F) = (F @ X)))))).
% 3.41/1.82 thf(fact_10_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X2:A]: ((bot_bot @ B)))))))).
% 3.41/1.82 thf(fact_11_monad__altc_Oaltc__single, axiom, (! [A:$tType,M:$tType,C:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Altc:((countable_Set_cset @ C) > ((C > M) > M)),X:C,F:(C > M)]: ((((((((monomo439771545d_altc @ A) @ M) @ C) @ Return) @ Bind) @ Altc) => (((Altc @ (((counta2111716221insert @ C) @ X) @ (bot_bot @ (countable_Set_cset @ C)))) @ F) = (F @ X)))))).
% 3.41/1.82 thf(fact_12_run__altc__nondet, axiom, (! [A2:(countable_Set_cset @ c),F:(c > ((monomo1453983439ondetT @ a) @ m))]: (((((monomo603716163nondet @ a) @ m) @ ((((((monomo955616121nondet @ c) @ m) @ a) @ mergec) @ A2) @ F)) = ((mergec @ A2) @ (((((comp @ ((monomo1453983439ondetT @ a) @ m)) @ m) @ c) @ ((monomo603716163nondet @ a) @ m)) @ F)))))).
% 3.41/1.82 thf(fact_13_cset__nondetM_Omerge__empty, axiom, (! [C:$tType,M:$tType,A:$tType,Return:((countable_Set_cset @ A) > M),Bind:(M > (((countable_Set_cset @ A) > M) > M)),Merge:((countable_Set_cset @ A) > ((A > M) > M)),Mergec:((countable_Set_cset @ C) > ((C > M) > M)),F:(A > M)]: (((((((((monomo2011693592ondetM @ A) @ M) @ C) @ Return) @ Bind) @ Merge) @ Mergec) => (((Merge @ (bot_bot @ (countable_Set_cset @ A))) @ F) = (Return @ (bot_bot @ (countable_Set_cset @ A)))))))).
% 3.41/1.82 thf(fact_14_nondetT_Oexpand, axiom, (! [M:$tType,A:$tType,NondetT:((monomo1453983439ondetT @ A) @ M),NondetT2:((monomo1453983439ondetT @ A) @ M)]: ((((((monomo603716163nondet @ A) @ M) @ NondetT) = (((monomo603716163nondet @ A) @ M) @ NondetT2)) => (NondetT = NondetT2))))).
% 3.41/1.82 thf(fact_15_cset__nondetM__base_Orun__altc__nondet, axiom, (! [M:$tType,A:$tType,C:$tType,Mergec:((countable_Set_cset @ C) > ((C > M) > M)),A2:(countable_Set_cset @ C),F:(C > ((monomo1453983439ondetT @ A) @ M))]: (((((monomo603716163nondet @ A) @ M) @ ((((((monomo955616121nondet @ C) @ M) @ A) @ Mergec) @ A2) @ F)) = ((Mergec @ A2) @ (((((comp @ ((monomo1453983439ondetT @ A) @ M)) @ M) @ C) @ ((monomo603716163nondet @ A) @ M)) @ F)))))).
% 3.41/1.82 thf(fact_16_altc__nondet__def, axiom, (! [A2:(countable_Set_cset @ c),F:(c > ((monomo1453983439ondetT @ a) @ m))]: ((((((((monomo955616121nondet @ c) @ m) @ a) @ mergec) @ A2) @ F) = (((monomo1975189248ondetT @ m) @ a) @ ((mergec @ A2) @ (((((comp @ ((monomo1453983439ondetT @ a) @ m)) @ m) @ c) @ ((monomo603716163nondet @ a) @ m)) @ F))))))).
% 3.41/1.82 thf(fact_17_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F2:(B > A),G:(C > B),X2:C]: ((F2 @ (G @ X2)))))))).
% 3.41/1.82 thf(fact_18_cset__nondetM__base_Oaltc__nondet__def, axiom, (! [A:$tType,M:$tType,C:$tType]: (((((monomo955616121nondet @ C) @ M) @ A) = (^ [Mergec2:((countable_Set_cset @ C) > ((C > M) > M)),A4:(countable_Set_cset @ C),F2:(C > ((monomo1453983439ondetT @ A) @ M))]: ((((monomo1975189248ondetT @ M) @ A) @ ((Mergec2 @ A4) @ (((((comp @ ((monomo1453983439ondetT @ A) @ M)) @ M) @ C) @ ((monomo603716163nondet @ A) @ M)) @ F2))))))))).
% 3.41/1.82 thf(fact_19_cset__nondetM_Oaxioms_I3_J, axiom, (! [A:$tType,M:$tType,C:$tType,Return:((countable_Set_cset @ A) > M),Bind:(M > (((countable_Set_cset @ A) > M) > M)),Merge:((countable_Set_cset @ A) > ((A > M) > M)),Mergec:((countable_Set_cset @ C) > ((C > M) > M))]: (((((((((monomo2011693592ondetM @ A) @ M) @ C) @ Return) @ Bind) @ Merge) @ Mergec) => (((((((monomo759963067axioms @ A) @ M) @ C) @ Return) @ Bind) @ Merge) @ Mergec))))).
% 3.41/1.82 thf(fact_20_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F2:(B > C),G:(A > B),X2:A]: ((F2 @ (G @ X2)))))))).
% 3.41/1.82 thf(fact_21_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(D > B),G2:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F) @ G2)) @ H) = (((((comp @ D) @ B) @ A) @ F) @ (((((comp @ C) @ D) @ A) @ G2) @ H)))))).
% 3.41/1.82 thf(fact_22_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A3:(C > B),B2:(A > C),C2:(D > B),D2:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => ((A3 @ (B2 @ V)) = (C2 @ (D2 @ V))))))).
% 3.41/1.82 thf(fact_23_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A3:(C > B),B2:(A > C),C2:(D > B),D2:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A3) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => (! [V2:A]: (((A3 @ (B2 @ V2)) = (C2 @ (D2 @ V2))))))))).
% 3.41/1.82 thf(fact_24_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E:$tType,F:(B > A),G2:(C > B),X:C,F3:(D > A),G3:(E > D),X3:E]: ((((F @ (G2 @ X)) = (F3 @ (G3 @ X3))) => (((((((comp @ B) @ A) @ C) @ F) @ G2) @ X) = ((((((comp @ D) @ A) @ E) @ F3) @ G3) @ X3)))))).
% 3.41/1.82 thf(fact_25_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A3:(C > B),B2:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B2) = C2) => ((A3 @ (B2 @ V)) = (C2 @ V)))))).
% 3.41/1.82 thf(fact_26_comp__apply__eq, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,F:(B > A),G2:(C > B),X:C,H:(D > A),K:(C > D)]: ((((F @ (G2 @ X)) = (H @ (K @ X))) => (((((((comp @ B) @ A) @ C) @ F) @ G2) @ X) = ((((((comp @ D) @ A) @ C) @ H) @ K) @ X)))))).
% 3.41/1.82 thf(fact_27_nondetT_Oinject, axiom, (! [A:$tType,M:$tType,X:M,Ya:M]: ((((((monomo1975189248ondetT @ M) @ A) @ X) = (((monomo1975189248ondetT @ M) @ A) @ Ya)) = (X = Ya))))).
% 3.41/1.82 thf(fact_28_nondetT_Ocollapse, axiom, (! [M:$tType,A:$tType,NondetT:((monomo1453983439ondetT @ A) @ M)]: (((((monomo1975189248ondetT @ M) @ A) @ (((monomo603716163nondet @ A) @ M) @ NondetT)) = NondetT)))).
% 3.41/1.82 thf(fact_29_nondetT_Oexhaust, axiom, (! [A:$tType,M:$tType,Y:((monomo1453983439ondetT @ A) @ M)]: ((~ ((! [X4:M]: ((Y != (((monomo1975189248ondetT @ M) @ A) @ X4))))))))).
% 3.41/1.82 thf(fact_30_nondetT_Oinduct, axiom, (! [M:$tType,A:$tType,P:(((monomo1453983439ondetT @ A) @ M) > $o),NondetT:((monomo1453983439ondetT @ A) @ M)]: (((! [Xa:M]: ((P @ (((monomo1975189248ondetT @ M) @ A) @ Xa)))) => (P @ NondetT))))).
% 3.41/1.82 thf(fact_31_nondetT_Oexhaust__sel, axiom, (! [M:$tType,A:$tType,NondetT:((monomo1453983439ondetT @ A) @ M)]: ((NondetT = (((monomo1975189248ondetT @ M) @ A) @ (((monomo603716163nondet @ A) @ M) @ NondetT)))))).
% 3.41/1.82 thf(fact_32_nondetT_Osel, axiom, (! [Aa:$tType,A:$tType,X:A]: (((((monomo603716163nondet @ Aa) @ A) @ (((monomo1975189248ondetT @ A) @ Aa) @ X)) = X)))).
% 3.41/1.82 thf(fact_33_put__nondet__def, axiom, (! [State:$tType]: (((((monomo410456285nondet @ State) @ m) @ a) = (^ [Put:(State > (m > m)),S2:State,M2:((monomo1453983439ondetT @ a) @ m)]: ((((monomo1975189248ondetT @ m) @ a) @ ((Put @ S2) @ (((monomo603716163nondet @ a) @ m) @ M2))))))))).
% 3.41/1.82 thf(fact_34_nondetM__base_Oput__nondet__def, axiom, (! [A:$tType,M:$tType,State:$tType]: (((((monomo410456285nondet @ State) @ M) @ A) = (^ [Put:(State > (M > M)),S2:State,M2:((monomo1453983439ondetT @ A) @ M)]: ((((monomo1975189248ondetT @ M) @ A) @ ((Put @ S2) @ (((monomo603716163nondet @ A) @ M) @ M2))))))))).
% 3.41/1.82 thf(fact_35_nondetT_Osplit__sel__asm, axiom, (! [B:$tType,M:$tType,A:$tType,P:(B > $o),F:(M > B),NondetT:((monomo1453983439ondetT @ A) @ M)]: (((P @ (((((monomo1564381680ondetT @ M) @ B) @ A) @ F) @ NondetT)) = (~ (((NondetT = (((monomo1975189248ondetT @ M) @ A) @ (((monomo603716163nondet @ A) @ M) @ NondetT))) & (~ ((P @ (F @ (((monomo603716163nondet @ A) @ M) @ NondetT)))))))))))).
% 3.41/1.82 thf(fact_36_nondetT_Osplit__sel, axiom, (! [B:$tType,M:$tType,A:$tType,P:(B > $o),F:(M > B),NondetT:((monomo1453983439ondetT @ A) @ M)]: (((P @ (((((monomo1564381680ondetT @ M) @ B) @ A) @ F) @ NondetT)) = ((NondetT = (((monomo1975189248ondetT @ M) @ A) @ (((monomo603716163nondet @ A) @ M) @ NondetT))) => (P @ (F @ (((monomo603716163nondet @ A) @ M) @ NondetT)))))))).
% 3.41/1.82 thf(fact_37_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E:$tType,D:$tType,A:$tType,G2:(C > B),H:(A > C),R1:(D > B),R2:(A > D),F:(B > E),L:(D > E)]: ((((((((comp @ C) @ B) @ A) @ G2) @ H) = (((((comp @ D) @ B) @ A) @ R1) @ R2)) => (((((((comp @ B) @ E) @ D) @ F) @ R1) = L) => ((((((comp @ C) @ E) @ A) @ (((((comp @ B) @ E) @ C) @ F) @ G2)) @ H) = (((((comp @ D) @ E) @ A) @ L) @ R2))))))).
% 3.41/1.82 thf(fact_38_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,E:$tType,F:(C > B),G2:(A > C),L1:(D > B),L2:(A > D),H:(E > A),R:(E > D)]: ((((((((comp @ C) @ B) @ A) @ F) @ G2) = (((((comp @ D) @ B) @ A) @ L1) @ L2)) => (((((((comp @ A) @ D) @ E) @ L2) @ H) = R) => ((((((comp @ C) @ B) @ E) @ F) @ (((((comp @ A) @ C) @ E) @ G2) @ H)) = (((((comp @ D) @ B) @ E) @ L1) @ R))))))).
% 3.41/1.82 thf(fact_39_rewriteR__comp__comp, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,G2:(C > B),H:(A > C),R:(A > B),F:(B > D)]: ((((((((comp @ C) @ B) @ A) @ G2) @ H) = R) => ((((((comp @ C) @ D) @ A) @ (((((comp @ B) @ D) @ C) @ F) @ G2)) @ H) = (((((comp @ B) @ D) @ A) @ F) @ R)))))).
% 3.41/1.82 thf(fact_40_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F:(C > B),G2:(A > C),L:(A > B),H:(D > A)]: ((((((((comp @ C) @ B) @ A) @ F) @ G2) = L) => ((((((comp @ C) @ B) @ D) @ F) @ (((((comp @ A) @ C) @ D) @ G2) @ H)) = (((((comp @ A) @ B) @ D) @ L) @ H)))))).
% 3.41/1.82 thf(fact_41_type__copy__map__cong0, axiom, (! [B:$tType,D:$tType,E:$tType,A:$tType,C:$tType,M3:(B > A),G2:(C > B),X:C,N:(D > A),H:(C > D),F:(A > E)]: ((((M3 @ (G2 @ X)) = (N @ (H @ X))) => (((((((comp @ B) @ E) @ C) @ (((((comp @ A) @ E) @ B) @ F) @ M3)) @ G2) @ X) = ((((((comp @ D) @ E) @ C) @ (((((comp @ A) @ E) @ D) @ F) @ N)) @ H) @ X)))))).
% 3.41/1.82 thf(fact_42_run__put__nondet, axiom, (! [B:$tType,Put2:(B > (m > m)),S3:B,M4:((monomo1453983439ondetT @ a) @ m)]: (((((monomo603716163nondet @ a) @ m) @ ((((((monomo410456285nondet @ B) @ m) @ a) @ Put2) @ S3) @ M4)) = ((Put2 @ S3) @ (((monomo603716163nondet @ a) @ m) @ M4)))))).
% 3.41/1.82 thf(fact_43_nondetT_Ocase__eq__if, axiom, (! [A:$tType,B:$tType,M:$tType]: (((((monomo1564381680ondetT @ M) @ B) @ A) = (^ [F2:(M > B),NondetT3:((monomo1453983439ondetT @ A) @ M)]: ((F2 @ (((monomo603716163nondet @ A) @ M) @ NondetT3)))))))).
% 3.41/1.82 thf(fact_44_nondetT_Ocase, axiom, (! [A:$tType,B:$tType,M:$tType,F:(M > B),X:M]: (((((((monomo1564381680ondetT @ M) @ B) @ A) @ F) @ (((monomo1975189248ondetT @ M) @ A) @ X)) = (F @ X))))).
% 3.41/1.82 thf(fact_45_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G2:(A > B)]: (((! [X4:A]: (((F @ X4) = (G2 @ X4)))) => (F = G2))))).
% 3.41/1.82 thf(fact_46_nondetM__base_Orun__put__nondet, axiom, (! [B:$tType,M:$tType,A:$tType,Put2:(B > (M > M)),S3:B,M4:((monomo1453983439ondetT @ A) @ M)]: (((((monomo603716163nondet @ A) @ M) @ ((((((monomo410456285nondet @ B) @ M) @ A) @ Put2) @ S3) @ M4)) = ((Put2 @ S3) @ (((monomo603716163nondet @ A) @ M) @ M4)))))).
% 3.41/1.82 thf(fact_47_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G2:(B > C),F:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G2) @ (((((comp @ A) @ B) @ D) @ F) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G2) @ F)) @ V))))).
% 3.41/1.82 thf(fact_48_function__factors__left, axiom, (! [A:$tType,C:$tType,B:$tType,G2:(A > B),F:(A > C)]: (((! [X2:A,Y2:A]: ((((G2 @ X2) = (G2 @ Y2)) => ((F @ X2) = (F @ Y2))))) = (? [H2:(B > C)]: ((F = (((((comp @ B) @ C) @ A) @ H2) @ G2)))))))).
% 3.41/1.82 thf(fact_49_function__factors__right, axiom, (! [C:$tType,B:$tType,A:$tType,G2:(B > C),F:(A > C)]: (((! [X2:A]: ((? [Y2:B]: (((G2 @ Y2) = (F @ X2)))))) = (? [H2:(A > B)]: ((F = (((((comp @ B) @ C) @ A) @ G2) @ H2)))))))).
% 3.41/1.82 thf(fact_50_monad__altc_Oaxioms_I2_J, axiom, (! [A:$tType,M:$tType,C:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Altc:((countable_Set_cset @ C) > ((C > M) > M))]: ((((((((monomo439771545d_altc @ A) @ M) @ C) @ Return) @ Bind) @ Altc) => (((((monomo555039548axioms @ M) @ A) @ C) @ Bind) @ Altc))))).
% 3.41/1.82 thf(fact_51_cinsert__cDiff__single, axiom, (! [A:$tType,A3:A,A2:(countable_Set_cset @ A)]: (((((counta2111716221insert @ A) @ A3) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ A3) @ (bot_bot @ (countable_Set_cset @ A))))) = (((counta2111716221insert @ A) @ A3) @ A2))))).
% 3.41/1.82 thf(fact_52_monad__altc3_Oaxioms_I1_J, axiom, (! [A:$tType,M:$tType,C:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Altc:((countable_Set_cset @ C) > ((C > M) > M))]: ((((((((monomo1425736922_altc3 @ A) @ M) @ C) @ Return) @ Bind) @ Altc) => ((((((monomo439771545d_altc @ A) @ M) @ C) @ Return) @ Bind) @ Altc))))).
% 3.41/1.82 thf(fact_53_nondetT_Orec, axiom, (! [A:$tType,D:$tType,M:$tType,F:(M > D),X:M]: (((((((monomo1291740684ondetT @ M) @ D) @ A) @ F) @ (((monomo1975189248ondetT @ M) @ A) @ X)) = (F @ X))))).
% 3.41/1.82 thf(fact_54_monad__state__altc_Oaxioms_I2_J, axiom, (! [S:$tType,A:$tType,M:$tType,C:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Get:((S > M) > M),Put2:(S > (M > M)),Altc:((countable_Set_cset @ C) > ((C > M) > M))]: (((((((((((monomo1036387116e_altc @ A) @ M) @ S) @ C) @ Return) @ Bind) @ Get) @ Put2) @ Altc) => ((((((monomo439771545d_altc @ A) @ M) @ C) @ Return) @ Bind) @ Altc))))).
% 3.41/1.82 thf(fact_55_cDiff__idemp, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((minus_minus @ (countable_Set_cset @ A)) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)) @ B3) = (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3))))).
% 3.41/1.82 thf(fact_56_cDiff__cancel, axiom, (! [A:$tType,A2:(countable_Set_cset @ A)]: (((((minus_minus @ (countable_Set_cset @ A)) @ A2) @ A2) = (bot_bot @ (countable_Set_cset @ A)))))).
% 3.41/1.82 thf(fact_57_cDiff__cempty, axiom, (! [A:$tType,A2:(countable_Set_cset @ A)]: (((((minus_minus @ (countable_Set_cset @ A)) @ A2) @ (bot_bot @ (countable_Set_cset @ A))) = A2)))).
% 3.41/1.82 thf(fact_58_cempty__cDiff, axiom, (! [A:$tType,A2:(countable_Set_cset @ A)]: (((((minus_minus @ (countable_Set_cset @ A)) @ (bot_bot @ (countable_Set_cset @ A))) @ A2) = (bot_bot @ (countable_Set_cset @ A)))))).
% 3.41/1.82 thf(fact_59_cDiff__cinsert, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),A3:A,B3:(countable_Set_cset @ A)]: (((((minus_minus @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ A3) @ B3)) = (((minus_minus @ (countable_Set_cset @ A)) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)) @ (((counta2111716221insert @ A) @ A3) @ (bot_bot @ (countable_Set_cset @ A)))))))).
% 3.41/1.82 thf(fact_60_cDiff__cinsert2, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),A3:A,B3:(countable_Set_cset @ A)]: (((((minus_minus @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ A3) @ B3)) = (((minus_minus @ (countable_Set_cset @ A)) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ A3) @ (bot_bot @ (countable_Set_cset @ A))))) @ B3))))).
% 3.41/1.82 thf(fact_61_minus__apply, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A4:(A > B),B4:(A > B),X2:A]: ((((minus_minus @ B) @ (A4 @ X2)) @ (B4 @ X2))))))))).
% 3.41/1.82 thf(fact_62_cDiff__single__cinsert, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),X:A,B3:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ X) @ (bot_bot @ (countable_Set_cset @ A))))) @ B3) => (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ X) @ B3)))))).
% 3.41/1.82 thf(fact_63_monad__state__altc_Oaxioms_I3_J, axiom, (! [A:$tType,S:$tType,M:$tType,C:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Get:((S > M) > M),Put2:(S > (M > M)),Altc:((countable_Set_cset @ C) > ((C > M) > M))]: (((((((((((monomo1036387116e_altc @ A) @ M) @ S) @ C) @ Return) @ Bind) @ Get) @ Put2) @ Altc) => ((((((monomo1324021455axioms @ S) @ M) @ C) @ Get) @ Put2) @ Altc))))).
% 3.41/1.82 thf(fact_64_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom, (! [A:$tType]: (((cancel146912293up_add @ A) => (! [A3:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2)))))))).
% 3.41/1.82 thf(fact_65_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B2) = (((minus_minus @ A) @ C2) @ D2)) => ((A3 = B2) = (C2 = D2))))))))).
% 3.41/1.82 thf(fact_66_fun__diff__def, axiom, (! [B:$tType,A:$tType]: (((minus @ B) => ((minus_minus @ (A > B)) = (^ [A4:(A > B),B4:(A > B),X2:A]: ((((minus_minus @ B) @ (A4 @ X2)) @ (B4 @ X2))))))))).
% 3.41/1.82 thf(fact_67_cinsert__cDiff, axiom, (! [A:$tType,A3:A,A2:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ A3) @ A2) => ((((counta2111716221insert @ A) @ A3) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ A3) @ (bot_bot @ (countable_Set_cset @ A))))) = A2))))).
% 3.41/1.82 thf(fact_68_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 3.41/1.82 thf(fact_69_csubset__antisym, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3) => ((((ord_less_eq @ (countable_Set_cset @ A)) @ B3) @ A2) => (A2 = B3)))))).
% 3.41/1.82 thf(fact_70_all__not__cin__conv, axiom, (! [A:$tType,A2:(countable_Set_cset @ A)]: (((! [X2:A]: ((~ ((((countable_Set_cin @ A) @ X2) @ A2))))) = (A2 = (bot_bot @ (countable_Set_cset @ A))))))).
% 3.41/1.82 thf(fact_71_cempty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((countable_Set_cin @ A) @ C2) @ (bot_bot @ (countable_Set_cset @ A)))))))).
% 3.41/1.82 thf(fact_72_cinsert__iff, axiom, (! [A:$tType,A3:A,B2:A,A2:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ A3) @ (((counta2111716221insert @ A) @ B2) @ A2)) = ((A3 = B2) | (((countable_Set_cin @ A) @ A3) @ A2)))))).
% 3.41/1.82 thf(fact_73_cinsertCI, axiom, (! [A:$tType,A3:A,B3:(countable_Set_cset @ A),B2:A]: ((((~ ((((countable_Set_cin @ A) @ A3) @ B3))) => (A3 = B2)) => (((countable_Set_cin @ A) @ A3) @ (((counta2111716221insert @ A) @ B2) @ B3)))))).
% 3.41/1.82 thf(fact_74_csubsetI, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((! [X4:A]: (((((countable_Set_cin @ A) @ X4) @ A2) => (((countable_Set_cin @ A) @ X4) @ B3)))) => (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3))))).
% 3.41/1.82 thf(fact_75_cempty__fsubsetI, axiom, (! [A:$tType,X:(countable_Set_cset @ A)]: ((((ord_less_eq @ (countable_Set_cset @ A)) @ (bot_bot @ (countable_Set_cset @ A))) @ X)))).
% 3.41/1.82 thf(fact_76_csubset__cempty, axiom, (! [A:$tType,A2:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ (bot_bot @ (countable_Set_cset @ A))) = (A2 = (bot_bot @ (countable_Set_cset @ A))))))).
% 3.41/1.82 thf(fact_77_cDiff__iff, axiom, (! [A:$tType,C2:A,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ C2) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)) = ((((countable_Set_cin @ A) @ C2) @ A2) & (~ ((((countable_Set_cin @ A) @ C2) @ B3)))))))).
% 3.41/1.82 thf(fact_78_cDiffI, axiom, (! [A:$tType,C2:A,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ C2) @ A2) => ((~ ((((countable_Set_cin @ A) @ C2) @ B3))) => (((countable_Set_cin @ A) @ C2) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3))))))).
% 3.41/1.82 thf(fact_79_cinsert__csubset, axiom, (! [A:$tType,X:A,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ (((counta2111716221insert @ A) @ X) @ A2)) @ B3) = ((((countable_Set_cin @ A) @ X) @ B3) & (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3)))))).
% 3.41/1.82 thf(fact_80_cinsert__cDiff1, axiom, (! [A:$tType,X:A,B3:(countable_Set_cset @ A),A2:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ X) @ B3) => ((((minus_minus @ (countable_Set_cset @ A)) @ (((counta2111716221insert @ A) @ X) @ A2)) @ B3) = (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)))))).
% 3.41/1.82 thf(fact_81_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B2) = (((minus_minus @ A) @ C2) @ D2)) => ((((ord_less_eq @ A) @ A3) @ B2) = (((ord_less_eq @ A) @ C2) @ D2))))))))).
% 3.41/1.82 thf(fact_82_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ C2))))))))).
% 3.41/1.82 thf(fact_83_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C2) @ A3)) @ (((minus_minus @ A) @ C2) @ B2))))))))).
% 3.41/1.82 thf(fact_84_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,D2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ D2) @ C2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ D2)))))))))).
% 3.41/1.82 thf(fact_85_csubset__cinsert, axiom, (! [A:$tType,X:A,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((~ ((((countable_Set_cin @ A) @ X) @ A2))) => ((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ X) @ B3)) = (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3)))))).
% 3.41/1.82 thf(fact_86_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ A3) @ B2) => (A3 = B2))))))))).
% 3.41/1.82 thf(fact_87_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A5) & (((ord_less_eq @ A) @ A5) @ B5))))))))).
% 3.41/1.82 thf(fact_88_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 3.41/1.82 thf(fact_89_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B2:A]: (((! [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) => ((P @ A6) @ B6)))) => ((! [A6:A,B6:A]: ((((P @ B6) @ A6) => ((P @ A6) @ B6)))) => ((P @ A3) @ B2))))))))).
% 3.41/1.82 thf(fact_90_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 3.41/1.82 thf(fact_91_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 3.41/1.82 thf(fact_92_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ A3) => (A3 = B2))))))))).
% 3.41/1.82 thf(fact_93_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 3.41/1.82 thf(fact_94_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((A3 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 3.41/1.82 thf(fact_95_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) & (((ord_less_eq @ A) @ B5) @ A5))))))))).
% 3.41/1.82 thf(fact_96_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 3.41/1.82 thf(fact_97_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 3.41/1.82 thf(fact_98_eqcelem__imp__iff, axiom, (! [A:$tType,X:A,Y:A,A2:(countable_Set_cset @ A)]: (((X = Y) => ((((countable_Set_cin @ A) @ X) @ A2) = (((countable_Set_cin @ A) @ Y) @ A2)))))).
% 3.41/1.82 thf(fact_99_cset__eq__csubset, axiom, (! [A:$tType]: (((^ [Y3:(countable_Set_cset @ A),Z:(countable_Set_cset @ A)]: ((Y3 = Z))) = (^ [A4:(countable_Set_cset @ A),B4:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A4) @ B4) & (((ord_less_eq @ (countable_Set_cset @ A)) @ B4) @ A4)))))))).
% 3.41/1.82 thf(fact_100_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 3.41/1.82 thf(fact_101_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.41/1.82 thf(fact_102_eqcset__imp__iff, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),X:A]: (((A2 = B3) => ((((countable_Set_cin @ A) @ X) @ A2) = (((countable_Set_cin @ A) @ X) @ B3)))))).
% 3.41/1.82 thf(fact_103_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 3.41/1.82 thf(fact_104_if__split__cin2, axiom, (! [A:$tType,A3:A,Q:$o,X:(countable_Set_cset @ A),Y:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ A3) @ ((((if @ (countable_Set_cset @ A)) @ Q) @ X) @ Y)) = ((Q => (((countable_Set_cin @ A) @ A3) @ X)) & ((~ (Q)) => (((countable_Set_cin @ A) @ A3) @ Y))))))).
% 3.41/1.82 thf(fact_105_if__split__cin1, axiom, (! [A:$tType,Q:$o,X:A,Y:A,B2:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ ((((if @ A) @ Q) @ X) @ Y)) @ B2) = ((Q => (((countable_Set_cin @ A) @ X) @ B2)) & ((~ (Q)) => (((countable_Set_cin @ A) @ Y) @ B2))))))).
% 3.41/1.82 thf(fact_106_eq__cmem__trans, axiom, (! [A:$tType,A3:A,B2:A,A2:(countable_Set_cset @ A)]: (((A3 = B2) => ((((countable_Set_cin @ A) @ B2) @ A2) => (((countable_Set_cin @ A) @ A3) @ A2)))))).
% 3.41/1.82 thf(fact_107_csubset__trans, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),C3:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3) => ((((ord_less_eq @ (countable_Set_cset @ A)) @ B3) @ C3) => (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ C3)))))).
% 3.41/1.82 thf(fact_108_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.41/1.82 thf(fact_109_csubset__refl, axiom, (! [A:$tType,A2:(countable_Set_cset @ A)]: ((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ A2)))).
% 3.41/1.82 thf(fact_110_cequalityD2, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((A2 = B3) => (((ord_less_eq @ (countable_Set_cset @ A)) @ B3) @ A2))))).
% 3.41/1.82 thf(fact_111_cequalityD1, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((A2 = B3) => (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3))))).
% 3.41/1.82 thf(fact_112_cequalityCE, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),C2:A]: (((A2 = B3) => (((((countable_Set_cin @ A) @ C2) @ A2) => (~ ((((countable_Set_cin @ A) @ C2) @ B3)))) => (~ (((~ ((((countable_Set_cin @ A) @ C2) @ A2))) => (((countable_Set_cin @ A) @ C2) @ B3))))))))).
% 3.41/1.82 thf(fact_113_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 3.41/1.82 thf(fact_114_cequalityE, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((A2 = B3) => (~ (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3) => (~ ((((ord_less_eq @ (countable_Set_cset @ A)) @ B3) @ A2)))))))))).
% 3.41/1.82 thf(fact_115_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) & (((ord_less_eq @ A) @ Y2) @ X2))))))))).
% 3.41/1.82 thf(fact_116_csubsetD, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),C2:A]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3) => ((((countable_Set_cin @ A) @ C2) @ A2) => (((countable_Set_cin @ A) @ C2) @ B3)))))).
% 3.41/1.82 thf(fact_117_cset__eqI, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((! [X4:A]: (((((countable_Set_cin @ A) @ X4) @ A2) = (((countable_Set_cin @ A) @ X4) @ B3)))) => (A2 = B3))))).
% 3.41/1.82 thf(fact_118_cin__mono, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),X:A]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3) => ((((countable_Set_cin @ A) @ X) @ A2) => (((countable_Set_cin @ A) @ X) @ B3)))))).
% 3.41/1.82 thf(fact_119_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A3) @ B2) => (((F @ B2) = C2) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ B) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C2)))))))))).
% 3.41/1.82 thf(fact_120_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((A3 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 3.41/1.82 thf(fact_121_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ C) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C2)))))))))).
% 3.41/1.82 thf(fact_122_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 3.41/1.82 thf(fact_123_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F2 @ X2)) @ (G @ X2))))))))))).
% 3.41/1.82 thf(fact_124_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G2:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F @ X4)) @ (G2 @ X4)))) => (((ord_less_eq @ (A > B)) @ F) @ G2)))))))).
% 3.41/1.82 thf(fact_125_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X)) @ (G2 @ X))))))))).
% 3.41/1.82 thf(fact_126_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G2:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G2) => (((ord_less_eq @ B) @ (F @ X)) @ (G2 @ X))))))))).
% 3.41/1.82 thf(fact_127_equals__cemptyI, axiom, (! [A:$tType,A2:(countable_Set_cset @ A)]: (((! [Y4:A]: ((~ ((((countable_Set_cin @ A) @ Y4) @ A2))))) => (A2 = (bot_bot @ (countable_Set_cset @ A))))))).
% 3.41/1.82 thf(fact_128_equals__cemptyD, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),A3:A]: (((A2 = (bot_bot @ (countable_Set_cset @ A))) => (~ ((((countable_Set_cin @ A) @ A3) @ A2))))))).
% 3.41/1.82 thf(fact_129_ex__cin__conv, axiom, (! [A:$tType,A2:(countable_Set_cset @ A)]: (((? [X2:A]: ((((countable_Set_cin @ A) @ X2) @ A2))) = (A2 != (bot_bot @ (countable_Set_cset @ A))))))).
% 3.41/1.82 thf(fact_130_cemptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((countable_Set_cin @ A) @ A3) @ (bot_bot @ (countable_Set_cset @ A)))))))).
% 3.41/1.82 thf(fact_131_mk__disjoint__cinsert, axiom, (! [A:$tType,A3:A,A2:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ A3) @ A2) => (? [B7:(countable_Set_cset @ A)]: (((A2 = (((counta2111716221insert @ A) @ A3) @ B7)) & (~ ((((countable_Set_cin @ A) @ A3) @ B7)))))))))).
% 3.41/1.82 thf(fact_132_cinsert__absorb, axiom, (! [A:$tType,A3:A,A2:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ A3) @ A2) => ((((counta2111716221insert @ A) @ A3) @ A2) = A2))))).
% 3.41/1.82 thf(fact_133_cinsert__ident, axiom, (! [A:$tType,X:A,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((~ ((((countable_Set_cin @ A) @ X) @ A2))) => ((~ ((((countable_Set_cin @ A) @ X) @ B3))) => (((((counta2111716221insert @ A) @ X) @ A2) = (((counta2111716221insert @ A) @ X) @ B3)) = (A2 = B3))))))).
% 3.41/1.82 thf(fact_134_set__cinsert, axiom, (! [A:$tType,X:A,A2:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ X) @ A2) => (~ ((! [B7:(countable_Set_cset @ A)]: (((A2 = (((counta2111716221insert @ A) @ X) @ B7)) => (((countable_Set_cin @ A) @ X) @ B7)))))))))).
% 3.41/1.82 thf(fact_135_cinsertI2, axiom, (! [A:$tType,A3:A,B3:(countable_Set_cset @ A),B2:A]: (((((countable_Set_cin @ A) @ A3) @ B3) => (((countable_Set_cin @ A) @ A3) @ (((counta2111716221insert @ A) @ B2) @ B3)))))).
% 3.41/1.82 thf(fact_136_cinsertI1, axiom, (! [A:$tType,A3:A,B3:(countable_Set_cset @ A)]: ((((countable_Set_cin @ A) @ A3) @ (((counta2111716221insert @ A) @ A3) @ B3))))).
% 3.41/1.82 thf(fact_137_cinsertE, axiom, (! [A:$tType,A3:A,B2:A,A2:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ A3) @ (((counta2111716221insert @ A) @ B2) @ A2)) => ((A3 != B2) => (((countable_Set_cin @ A) @ A3) @ A2)))))).
% 3.41/1.82 thf(fact_138_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 3.41/1.82 thf(fact_139_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 3.41/1.82 thf(fact_140_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 3.41/1.82 thf(fact_141_cDiffD2, axiom, (! [A:$tType,C2:A,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ C2) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)) => (~ ((((countable_Set_cin @ A) @ C2) @ B3))))))).
% 3.41/1.82 thf(fact_142_cDiffD1, axiom, (! [A:$tType,C2:A,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ C2) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)) => (((countable_Set_cin @ A) @ C2) @ A2))))).
% 3.41/1.82 thf(fact_143_cDiffE, axiom, (! [A:$tType,C2:A,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((countable_Set_cin @ A) @ C2) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)) => (~ (((((countable_Set_cin @ A) @ C2) @ A2) => (((countable_Set_cin @ A) @ C2) @ B3)))))))).
% 3.41/1.82 thf(fact_144_csubset__cinsertI2, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),B2:A]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3) => (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ B2) @ B3)))))).
% 3.41/1.82 thf(fact_145_csubset__cinsertI, axiom, (! [A:$tType,B3:(countable_Set_cset @ A),A3:A]: ((((ord_less_eq @ (countable_Set_cset @ A)) @ B3) @ (((counta2111716221insert @ A) @ A3) @ B3))))).
% 3.41/1.82 thf(fact_146_cinsert__mono, axiom, (! [A:$tType,C3:(countable_Set_cset @ A),D3:(countable_Set_cset @ A),A3:A]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ C3) @ D3) => (((ord_less_eq @ (countable_Set_cset @ A)) @ (((counta2111716221insert @ A) @ A3) @ C3)) @ (((counta2111716221insert @ A) @ A3) @ D3)))))).
% 3.41/1.82 thf(fact_147_cDiff__csubset, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: ((((ord_less_eq @ (countable_Set_cset @ A)) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)) @ A2)))).
% 3.41/1.82 thf(fact_148_double__cDiff, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),C3:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3) => ((((ord_less_eq @ (countable_Set_cset @ A)) @ B3) @ C3) => ((((minus_minus @ (countable_Set_cset @ A)) @ B3) @ (((minus_minus @ (countable_Set_cset @ A)) @ C3) @ A2)) = A2)))))).
% 3.41/1.82 thf(fact_149_cminus__mono, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),C3:(countable_Set_cset @ A),D3:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ C3) => ((((ord_less_eq @ (countable_Set_cset @ A)) @ D3) @ B3) => (((ord_less_eq @ (countable_Set_cset @ A)) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)) @ (((minus_minus @ (countable_Set_cset @ A)) @ C3) @ D3))))))).
% 3.41/1.82 thf(fact_150_csubset__cinsert__iff, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),X:A,B3:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ X) @ B3)) = (((((countable_Set_cin @ A) @ X) @ A2) => (((ord_less_eq @ (countable_Set_cset @ A)) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ X) @ (bot_bot @ (countable_Set_cset @ A))))) @ B3)) & ((~ ((((countable_Set_cin @ A) @ X) @ A2))) => (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3))))))).
% 3.41/1.82 thf(fact_151_csingleton__iff, axiom, (! [A:$tType,B2:A,A3:A]: (((((countable_Set_cin @ A) @ B2) @ (((counta2111716221insert @ A) @ A3) @ (bot_bot @ (countable_Set_cset @ A)))) = (B2 = A3))))).
% 3.41/1.82 thf(fact_152_cinsert__cDiff__if, axiom, (! [A:$tType,X:A,B3:(countable_Set_cset @ A),A2:(countable_Set_cset @ A)]: ((((((countable_Set_cin @ A) @ X) @ B3) => ((((minus_minus @ (countable_Set_cset @ A)) @ (((counta2111716221insert @ A) @ X) @ A2)) @ B3) = (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3))) & ((~ ((((countable_Set_cin @ A) @ X) @ B3))) => ((((minus_minus @ (countable_Set_cset @ A)) @ (((counta2111716221insert @ A) @ X) @ A2)) @ B3) = (((counta2111716221insert @ A) @ X) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ B3)))))))).
% 3.41/1.82 thf(fact_153_csubset__csingletonD, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),X:A]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ X) @ (bot_bot @ (countable_Set_cset @ A)))) => ((A2 = (bot_bot @ (countable_Set_cset @ A))) | (A2 = (((counta2111716221insert @ A) @ X) @ (bot_bot @ (countable_Set_cset @ A))))))))).
% 3.41/1.82 thf(fact_154_cDiff__cinsert__absorb, axiom, (! [A:$tType,X:A,A2:(countable_Set_cset @ A)]: (((~ ((((countable_Set_cin @ A) @ X) @ A2))) => ((((minus_minus @ (countable_Set_cset @ A)) @ (((counta2111716221insert @ A) @ X) @ A2)) @ (((counta2111716221insert @ A) @ X) @ (bot_bot @ (countable_Set_cset @ A)))) = A2))))).
% 3.41/1.82 thf(fact_155_cpsubset__finsert__iff, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),X:A,B3:(countable_Set_cset @ A)]: (((((ord_less @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ X) @ B3)) = (((((countable_Set_cin @ A) @ X) @ B3) => (((ord_less @ (countable_Set_cset @ A)) @ A2) @ B3)) & ((~ ((((countable_Set_cin @ A) @ X) @ B3))) => (((((countable_Set_cin @ A) @ X) @ A2) => (((ord_less @ (countable_Set_cset @ A)) @ (((minus_minus @ (countable_Set_cset @ A)) @ A2) @ (((counta2111716221insert @ A) @ X) @ (bot_bot @ (countable_Set_cset @ A))))) @ B3)) & ((~ ((((countable_Set_cin @ A) @ X) @ A2))) => (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3))))))))).
% 3.41/1.82 thf(fact_156_monad__state__altc_Ointro, axiom, (! [S:$tType,A:$tType,M:$tType,C:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Get:((S > M) > M),Put2:(S > (M > M)),Altc:((countable_Set_cset @ C) > ((C > M) > M))]: (((((((((monomo109450930_state @ A) @ M) @ S) @ Return) @ Bind) @ Get) @ Put2) => (((((((monomo439771545d_altc @ A) @ M) @ C) @ Return) @ Bind) @ Altc) => (((((((monomo1324021455axioms @ S) @ M) @ C) @ Get) @ Put2) @ Altc) => (((((((((monomo1036387116e_altc @ A) @ M) @ S) @ C) @ Return) @ Bind) @ Get) @ Put2) @ Altc))))))).
% 3.41/1.82 thf(fact_157_monad__state__altc__def, axiom, (! [C:$tType,S:$tType,M:$tType,A:$tType]: ((((((monomo1036387116e_altc @ A) @ M) @ S) @ C) = (^ [Return2:(A > M),Bind2:(M > ((A > M) > M)),Get2:((S > M) > M),Put:(S > (M > M)),Altc2:((countable_Set_cset @ C) > ((C > M) > M))]: (((((((((monomo109450930_state @ A) @ M) @ S) @ Return2) @ Bind2) @ Get2) @ Put) & (((((((monomo439771545d_altc @ A) @ M) @ C) @ Return2) @ Bind2) @ Altc2) & ((((((monomo1324021455axioms @ S) @ M) @ C) @ Get2) @ Put) @ Altc2))))))))).
% 3.41/1.82 thf(fact_158_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A]: (((P @ X) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X)))) => (((order_Greatest @ A) @ P) = X))))))))).
% 3.41/1.82 thf(fact_159_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X)))) => ((! [X4:A]: (((P @ X4) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X4)))) => (Q @ X4))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 3.41/1.82 thf(fact_160_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((A3 = B2) | ((~ ((((ord_less_eq @ A) @ A3) @ B2))) | (~ ((((ord_less_eq @ A) @ B2) @ A3))))))))))).
% 3.41/1.82 thf(fact_161_le__rel__bool__arg__iff, axiom, (! [A:$tType]: (((ord @ A) => ((ord_less_eq @ ($o > A)) = (^ [X5:($o > A),Y6:($o > A)]: (((((ord_less_eq @ A) @ (X5 @ $false)) @ (Y6 @ $false)) & (((ord_less_eq @ A) @ (X5 @ $true)) @ (Y6 @ $true)))))))))).
% 3.41/1.82 thf(fact_162_verit__comp__simplify1_I3_J, axiom, (! [B:$tType]: (((linorder @ B) => (! [B8:B,A7:B]: (((~ ((((ord_less_eq @ B) @ B8) @ A7))) = (((ord_less @ B) @ A7) @ B8)))))))).
% 3.41/1.82 thf(fact_163_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((A3 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X4:B,Y4:B]: (((((ord_less @ B) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 3.41/1.82 thf(fact_164_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B2:A,F:(A > B),C2:B]: (((((ord_less @ A) @ A3) @ B2) => (((F @ B2) = C2) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ B) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ B) @ (F @ A3)) @ C2)))))))))).
% 3.41/1.82 thf(fact_165_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((((ord_less @ A) @ A3) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X4:B,Y4:B]: (((((ord_less @ B) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 3.41/1.82 thf(fact_166_order__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B2:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ C) @ (F @ B2)) @ C2) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ C) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C) @ (F @ A3)) @ C2)))))))))).
% 3.41/1.82 thf(fact_167_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X))))))))).
% 3.41/1.82 thf(fact_168_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X:A]: ((? [X_1:A]: ((((ord_less @ A) @ X) @ X_1))))))))).
% 3.41/1.82 thf(fact_169_neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 3.41/1.82 thf(fact_170_neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 3.41/1.82 thf(fact_171_order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ ((((ord_less @ A) @ B2) @ A3)))))))))).
% 3.41/1.82 thf(fact_172_cpsubset__trans, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),C3:(countable_Set_cset @ A)]: (((((ord_less @ (countable_Set_cset @ A)) @ A2) @ B3) => ((((ord_less @ (countable_Set_cset @ A)) @ B3) @ C3) => (((ord_less @ (countable_Set_cset @ A)) @ A2) @ C3)))))).
% 3.41/1.82 thf(fact_173_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z3:A]: (((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Z3) @ Y))))))))))).
% 3.41/1.82 thf(fact_174_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 3.41/1.82 thf(fact_175_less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 3.41/1.82 thf(fact_176_less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ ((((ord_less @ A) @ B2) @ A3)))))))))).
% 3.41/1.82 thf(fact_177_less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 3.41/1.82 thf(fact_178_less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 3.41/1.82 thf(fact_179_less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 3.41/1.82 thf(fact_180_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((A3 = B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 3.41/1.82 thf(fact_181_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((B2 = C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 3.41/1.82 thf(fact_182_dual__order_Oasym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (~ ((((ord_less @ A) @ A3) @ B2)))))))))).
% 3.41/1.82 thf(fact_183_less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 3.41/1.82 thf(fact_184_less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 3.41/1.82 thf(fact_185_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A3:A]: (((! [X4:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X4) => (P @ Y5)))) => (P @ X4)))) => (P @ A3)))))))).
% 3.41/1.82 thf(fact_186_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 3.41/1.82 thf(fact_187_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 3.41/1.82 thf(fact_188_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 3.41/1.82 thf(fact_189_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 3.41/1.82 thf(fact_190_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ A3))))))))).
% 3.41/1.82 thf(fact_191_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 3.41/1.82 thf(fact_192_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 3.41/1.82 thf(fact_193_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X6:A]: ((P2 @ X6))))) = (^ [P3:(A > $o)]: ((? [N2:A]: (((P3 @ N2) & (! [M2:A]: (((((ord_less @ A) @ M2) @ N2) => (~ ((P3 @ M2)))))))))))))))).
% 3.41/1.82 thf(fact_194_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B2:A]: (((! [A6:A,B6:A]: (((((ord_less @ A) @ A6) @ B6) => ((P @ A6) @ B6)))) => ((! [A6:A]: (((P @ A6) @ A6))) => ((! [A6:A,B6:A]: ((((P @ B6) @ A6) => ((P @ A6) @ B6)))) => ((P @ A3) @ B2)))))))))).
% 3.41/1.82 thf(fact_195_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A3))))))))).
% 3.41/1.82 thf(fact_196_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 3.41/1.82 thf(fact_197_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (A3 != B2)))))))).
% 3.41/1.82 thf(fact_198_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (A3 != B2)))))))).
% 3.41/1.82 thf(fact_199_monad__state_Obind__put, axiom, (! [S:$tType,M:$tType,A:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Get:((S > M) > M),Put2:(S > (M > M)),S3:S,M4:M,F:(A > M)]: (((((((((monomo109450930_state @ A) @ M) @ S) @ Return) @ Bind) @ Get) @ Put2) => (((Bind @ ((Put2 @ S3) @ M4)) @ F) = ((Put2 @ S3) @ ((Bind @ M4) @ F))))))).
% 3.41/1.82 thf(fact_200_monad__state_Oput__put, axiom, (! [A:$tType,S:$tType,M:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Get:((S > M) > M),Put2:(S > (M > M)),S3:S,S4:S,M4:M]: (((((((((monomo109450930_state @ A) @ M) @ S) @ Return) @ Bind) @ Get) @ Put2) => (((Put2 @ S3) @ ((Put2 @ S4) @ M4)) = ((Put2 @ S4) @ M4)))))).
% 3.41/1.82 thf(fact_201_monad__state_Oput__get, axiom, (! [A:$tType,M:$tType,S:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Get:((S > M) > M),Put2:(S > (M > M)),S3:S,F:(S > M)]: (((((((((monomo109450930_state @ A) @ M) @ S) @ Return) @ Bind) @ Get) @ Put2) => (((Put2 @ S3) @ (Get @ F)) = ((Put2 @ S3) @ (F @ S3))))))).
% 3.41/1.82 thf(fact_202_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less @ A) @ X) @ Y)))))))))).
% 3.41/1.82 thf(fact_203_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.41/1.82 thf(fact_204_le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) | (X2 = Y2))))))))).
% 3.41/1.82 thf(fact_205_less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) & (X2 != Y2))))))))).
% 3.41/1.82 thf(fact_206_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X4:B,Y4:B]: (((((ord_less @ B) @ X4) @ Y4) => (((ord_less @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 3.41/1.82 thf(fact_207_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ C) @ (F @ B2)) @ C2) => ((! [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) => (((ord_less_eq @ C) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C) @ (F @ A3)) @ C2)))))))))).
% 3.41/1.82 thf(fact_208_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B2:B,C2:B]: (((((ord_less @ A) @ A3) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X4:B,Y4:B]: (((((ord_less_eq @ B) @ X4) @ Y4) => (((ord_less_eq @ A) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ A) @ A3) @ (F @ C2))))))))))).
% 3.41/1.82 thf(fact_209_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B2:A,F:(A > C),C2:C]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X4:A,Y4:A]: (((((ord_less @ A) @ X4) @ Y4) => (((ord_less @ C) @ (F @ X4)) @ (F @ Y4))))) => (((ord_less @ C) @ (F @ A3)) @ C2)))))))))).
% 3.41/1.82 thf(fact_210_not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) = (((ord_less @ A) @ Y) @ X)))))))).
% 3.41/1.82 thf(fact_211_not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.41/1.82 thf(fact_212_le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((A3 != B2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 3.41/1.82 thf(fact_213_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 3.41/1.82 thf(fact_214_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 3.41/1.82 thf(fact_215_less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 3.41/1.82 thf(fact_216_le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 3.41/1.82 thf(fact_217_less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 3.41/1.82 thf(fact_218_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,Y:A]: (((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (((ord_less_eq @ A) @ Y) @ X4)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 3.41/1.82 thf(fact_219_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y:A,Z2:A]: (((! [X4:A]: (((((ord_less @ A) @ X4) @ Y) => (((ord_less_eq @ A) @ X4) @ Z2)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 3.41/1.82 thf(fact_220_le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X)))))))).
% 3.41/1.82 thf(fact_221_le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 3.41/1.82 thf(fact_222_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) & (~ ((((ord_less_eq @ A) @ Y2) @ X2))))))))))).
% 3.41/1.82 thf(fact_223_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X))) => (((ord_less @ A) @ X) @ Y)))))))).
% 3.41/1.82 thf(fact_224_order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 3.41/1.82 thf(fact_225_order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less @ A) @ A3) @ C2))))))))).
% 3.41/1.82 thf(fact_226_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A5:A,B5:A]: (((((ord_less @ A) @ A5) @ B5) | (A5 = B5))))))))).
% 3.41/1.82 thf(fact_227_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) & (A5 != B5))))))))).
% 3.41/1.82 thf(fact_228_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A3))))))))).
% 3.41/1.82 thf(fact_229_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A3))))))))).
% 3.41/1.82 thf(fact_230_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,X:A,Y:A]: (((((ord_less @ A) @ Z2) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 3.41/1.82 thf(fact_231_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X:A,Y:A,Z2:A]: (((((ord_less @ A) @ X) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 3.41/1.82 thf(fact_232_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 3.41/1.82 thf(fact_233_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: (((((ord_less @ A) @ B5) @ A5) | (A5 = B5))))))))).
% 3.41/1.82 thf(fact_234_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B5:A,A5:A]: (((((ord_less_eq @ A) @ B5) @ A5) & (A5 != B5))))))))).
% 3.41/1.82 thf(fact_235_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 3.41/1.82 thf(fact_236_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((A3 != B2) => ((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 3.41/1.82 thf(fact_237_bot_Oextremum__strict, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ A3) @ (bot_bot @ A)))))))))).
% 3.41/1.82 thf(fact_238_bot_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((A3 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A3)))))))).
% 3.41/1.82 thf(fact_239_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,D2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ D2) @ C2) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ D2)))))))))).
% 3.41/1.82 thf(fact_240_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A3) @ B2) = (((minus_minus @ A) @ C2) @ D2)) => ((((ord_less @ A) @ A3) @ B2) = (((ord_less @ A) @ C2) @ D2))))))))).
% 3.41/1.82 thf(fact_241_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => (((ord_less @ A) @ (((minus_minus @ A) @ C2) @ A3)) @ (((minus_minus @ A) @ C2) @ B2))))))))).
% 3.41/1.82 thf(fact_242_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ C2))))))))).
% 3.41/1.82 thf(fact_243_cpsubsetD, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),C2:A]: (((((ord_less @ (countable_Set_cset @ A)) @ A2) @ B3) => ((((countable_Set_cin @ A) @ C2) @ A2) => (((countable_Set_cin @ A) @ C2) @ B3)))))).
% 3.41/1.82 thf(fact_244_not__cpsubset__cempty, axiom, (! [A:$tType,A2:(countable_Set_cset @ A)]: ((~ ((((ord_less @ (countable_Set_cset @ A)) @ A2) @ (bot_bot @ (countable_Set_cset @ A)))))))).
% 3.41/1.82 thf(fact_245_cpsubset__eq, axiom, (! [A:$tType]: (((ord_less @ (countable_Set_cset @ A)) = (^ [A4:(countable_Set_cset @ A),B4:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A4) @ B4) & (A4 != B4)))))))).
% 3.41/1.82 thf(fact_246_less__cset__def, axiom, (! [A:$tType]: (((ord_less @ (countable_Set_cset @ A)) = (^ [Xs:(countable_Set_cset @ A),Ys:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ Xs) @ Ys) & (Xs != Ys)))))))).
% 3.41/1.82 thf(fact_247_cpsubset__imp__fsubset, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((ord_less @ (countable_Set_cset @ A)) @ A2) @ B3) => (((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3))))).
% 3.41/1.82 thf(fact_248_cpsubset__csubset__trans, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),C3:(countable_Set_cset @ A)]: (((((ord_less @ (countable_Set_cset @ A)) @ A2) @ B3) => ((((ord_less_eq @ (countable_Set_cset @ A)) @ B3) @ C3) => (((ord_less @ (countable_Set_cset @ A)) @ A2) @ C3)))))).
% 3.41/1.82 thf(fact_249_csubset__cpsubset__trans, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A),C3:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A2) @ B3) => ((((ord_less @ (countable_Set_cset @ A)) @ B3) @ C3) => (((ord_less @ (countable_Set_cset @ A)) @ A2) @ C3)))))).
% 3.41/1.82 thf(fact_250_csubset__not__fsubset__eq, axiom, (! [A:$tType]: (((ord_less @ (countable_Set_cset @ A)) = (^ [A4:(countable_Set_cset @ A),B4:(countable_Set_cset @ A)]: (((((ord_less_eq @ (countable_Set_cset @ A)) @ A4) @ B4) & (~ ((((ord_less_eq @ (countable_Set_cset @ A)) @ B4) @ A4)))))))))).
% 3.41/1.82 thf(fact_251_csubset__iff__pfsubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (countable_Set_cset @ A)) = (^ [A4:(countable_Set_cset @ A),B4:(countable_Set_cset @ A)]: (((((ord_less @ (countable_Set_cset @ A)) @ A4) @ B4) | (A4 = B4)))))))).
% 3.41/1.82 thf(fact_252_monad__state__altc_Oaxioms_I1_J, axiom, (! [C:$tType,A:$tType,M:$tType,S:$tType,Return:(A > M),Bind:(M > ((A > M) > M)),Get:((S > M) > M),Put2:(S > (M > M)),Altc:((countable_Set_cset @ C) > ((C > M) > M))]: (((((((((((monomo1036387116e_altc @ A) @ M) @ S) @ C) @ Return) @ Bind) @ Get) @ Put2) @ Altc) => (((((((monomo109450930_state @ A) @ M) @ S) @ Return) @ Bind) @ Get) @ Put2))))).
% 3.41/1.82 thf(fact_253_cpsubset__imp__ex__fmem, axiom, (! [A:$tType,A2:(countable_Set_cset @ A),B3:(countable_Set_cset @ A)]: (((((ord_less @ (countable_Set_cset @ A)) @ A2) @ B3) => (? [B6:A]: ((((countable_Set_cin @ A) @ B6) @ (((minus_minus @ (countable_Set_cset @ A)) @ B3) @ A2)))))))).
% 3.41/1.82 thf(fact_254_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X7)))))))))))))).
% 3.41/1.82 thf(fact_255_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T:A]: ((? [Z3:A]: ((! [X7:A]: (((((ord_less @ A) @ X7) @ Z3) => (((ord_less_eq @ A) @ X7) @ T)))))))))))).
% 3.41/1.82 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A8:$tType,A9:$tType]: (((order_bot @ A9) => (order_bot @ (A8 > A9)))))).
% 3.41/1.82 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 3.41/1.82 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 3.41/1.82 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 3.41/1.82 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A8 > A9)))))).
% 3.41/1.82 thf(tcon_fun___Groups_Ominus, axiom, (! [A8:$tType,A9:$tType]: (((minus @ A9) => (minus @ (A8 > A9)))))).
% 3.41/1.82 thf(tcon_HOL_Obool___Orderings_Oorder__bot_1, axiom, (order_bot @ $o)).
% 3.41/1.82 thf(tcon_HOL_Obool___Orderings_Opreorder_2, axiom, (preorder @ $o)).
% 3.41/1.82 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 3.41/1.82 thf(tcon_HOL_Obool___Orderings_Oorder_3, axiom, (order @ $o)).
% 3.41/1.82 thf(tcon_HOL_Obool___Orderings_Oord_4, axiom, (ord @ $o)).
% 3.41/1.82 thf(tcon_HOL_Obool___Orderings_Obot_5, axiom, (bot @ $o)).
% 3.41/1.82 thf(tcon_HOL_Obool___Groups_Ominus_6, axiom, (minus @ $o)).
% 3.41/1.82 thf(tcon_Countable__Set__Type_Ocset___Orderings_Oorder__bot_7, axiom, (! [A8:$tType]: ((order_bot @ (countable_Set_cset @ A8))))).
% 3.41/1.82 thf(tcon_Countable__Set__Type_Ocset___Orderings_Opreorder_8, axiom, (! [A8:$tType]: ((preorder @ (countable_Set_cset @ A8))))).
% 3.41/1.82 thf(tcon_Countable__Set__Type_Ocset___Orderings_Oorder_9, axiom, (! [A8:$tType]: ((order @ (countable_Set_cset @ A8))))).
% 3.41/1.82 thf(tcon_Countable__Set__Type_Ocset___Orderings_Oord_10, axiom, (! [A8:$tType]: ((ord @ (countable_Set_cset @ A8))))).
% 3.41/1.82 thf(tcon_Countable__Set__Type_Ocset___Orderings_Obot_11, axiom, (! [A8:$tType]: ((bot @ (countable_Set_cset @ A8))))).
% 3.41/1.82 thf(tcon_Countable__Set__Type_Ocset___Groups_Ominus_12, axiom, (! [A8:$tType]: ((minus @ (countable_Set_cset @ A8))))).
% 3.41/1.82 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 3.41/1.82 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 3.41/1.82 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 3.41/1.82 thf(conj_0, conjecture, (((((((monomo955616121nondet @ c) @ m) @ a) @ mergec) @ (((counta2111716221insert @ c) @ x) @ (bot_bot @ (countable_Set_cset @ c)))) @ f) = (f @ x))).
% 3.41/1.82 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.fjdGAlfEYi/DTF2DTF6822.p
% 3.41/1.82 ------------------------
% 3.41/1.86 ---- Cleaned THF ---
% 3.41/1.87 thf(ty_t_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT,type,
% 3.41/1.87 monomo1453983439ondetT: $tType > $tType > $tType ).
% 3.41/1.87
% 3.41/1.87 thf(ty_t_Countable__Set__Type_Ocset,type,
% 3.41/1.87 countable_Set_cset: $tType > $tType ).
% 3.41/1.87
% 3.41/1.87 thf(ty_tf_m,type,
% 3.41/1.87 m: $tType ).
% 3.41/1.87
% 3.41/1.87 thf(ty_tf_c,type,
% 3.41/1.87 c: $tType ).
% 3.41/1.87
% 3.41/1.87 thf(ty_tf_a,type,
% 3.41/1.87 a: $tType ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_HOL_Otype,type,
% 3.41/1.87 type:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Groups_Ominus,type,
% 3.41/1.87 minus:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Obot,type,
% 3.41/1.87 bot:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Oord,type,
% 3.41/1.87 ord:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Oorder,type,
% 3.41/1.87 order:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Groups_Ogroup__add,type,
% 3.41/1.87 group_add:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Ono__bot,type,
% 3.41/1.87 no_bot:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Ono__top,type,
% 3.41/1.87 no_top:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Olinorder,type,
% 3.41/1.87 linorder:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Opreorder,type,
% 3.41/1.87 preorder:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Oorder__bot,type,
% 3.41/1.87 order_bot:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Owellorder,type,
% 3.41/1.87 wellorder:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Odense__order,type,
% 3.41/1.87 dense_order:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Orderings_Odense__linorder,type,
% 3.41/1.87 dense_linorder:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 3.41/1.87 ordered_ab_group_add:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 3.41/1.87 cancel146912293up_add:
% 3.41/1.87 !>[A: $tType] : $o ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Countable__Set__Type_Ocin,type,
% 3.41/1.87 countable_Set_cin:
% 3.41/1.87 !>[A: $tType] : ( A > ( countable_Set_cset @ A ) > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Countable__Set__Type_Ocinsert,type,
% 3.41/1.87 counta2111716221insert:
% 3.41/1.87 !>[A: $tType] : ( A > ( countable_Set_cset @ A ) > ( countable_Set_cset @ A ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Fun_Ocomp,type,
% 3.41/1.87 comp:
% 3.41/1.87 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 3.41/1.87 minus_minus:
% 3.41/1.87 !>[A: $tType] : ( A > A > A ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_If,type,
% 3.41/1.87 if:
% 3.41/1.87 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM,type,
% 3.41/1.87 monomo2011693592ondetM:
% 3.41/1.87 !>[A: $tType,M: $tType,C: $tType] : ( ( ( countable_Set_cset @ A ) > M ) > ( M > ( ( countable_Set_cset @ A ) > M ) > M ) > ( ( countable_Set_cset @ A ) > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM__axioms,type,
% 3.41/1.87 monomo759963067axioms:
% 3.41/1.87 !>[A: $tType,M: $tType,C: $tType] : ( ( ( countable_Set_cset @ A ) > M ) > ( M > ( ( countable_Set_cset @ A ) > M ) > M ) > ( ( countable_Set_cset @ A ) > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM__base_Oaltc__nondet,type,
% 3.41/1.87 monomo955616121nondet:
% 3.41/1.87 !>[C: $tType,M: $tType,A: $tType] : ( ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > ( countable_Set_cset @ C ) > ( C > ( monomo1453983439ondetT @ A @ M ) ) > ( monomo1453983439ondetT @ A @ M ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc,type,
% 3.41/1.87 monomo439771545d_altc:
% 3.41/1.87 !>[A: $tType,M: $tType,C: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc3,type,
% 3.41/1.87 monomo1425736922_altc3:
% 3.41/1.87 !>[A: $tType,M: $tType,C: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc__axioms,type,
% 3.41/1.87 monomo555039548axioms:
% 3.41/1.87 !>[M: $tType,A: $tType,C: $tType] : ( ( M > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state,type,
% 3.41/1.87 monomo109450930_state:
% 3.41/1.87 !>[A: $tType,M: $tType,S: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( S > M ) > M ) > ( S > M > M ) > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state__altc,type,
% 3.41/1.87 monomo1036387116e_altc:
% 3.41/1.87 !>[A: $tType,M: $tType,S: $tType,C: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( S > M ) > M ) > ( S > M > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state__altc__axioms,type,
% 3.41/1.87 monomo1324021455axioms:
% 3.41/1.87 !>[S: $tType,M: $tType,C: $tType] : ( ( ( S > M ) > M ) > ( S > M > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetM__base_Oput__nondet,type,
% 3.41/1.87 monomo410456285nondet:
% 3.41/1.87 !>[State: $tType,M: $tType,A: $tType] : ( ( State > M > M ) > State > ( monomo1453983439ondetT @ A @ M ) > ( monomo1453983439ondetT @ A @ M ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_ONondetT,type,
% 3.41/1.87 monomo1975189248ondetT:
% 3.41/1.87 !>[M: $tType,A: $tType] : ( M > ( monomo1453983439ondetT @ A @ M ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Ocase__nondetT,type,
% 3.41/1.87 monomo1564381680ondetT:
% 3.41/1.87 !>[M: $tType,B: $tType,A: $tType] : ( ( M > B ) > ( monomo1453983439ondetT @ A @ M ) > B ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Orec__nondetT,type,
% 3.41/1.87 monomo1291740684ondetT:
% 3.41/1.87 !>[M: $tType,D: $tType,A: $tType] : ( ( M > D ) > ( monomo1453983439ondetT @ A @ M ) > D ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Orun__nondet,type,
% 3.41/1.87 monomo603716163nondet:
% 3.41/1.87 !>[A: $tType,M: $tType] : ( ( monomo1453983439ondetT @ A @ M ) > M ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Orderings_Obot__class_Obot,type,
% 3.41/1.87 bot_bot:
% 3.41/1.87 !>[A: $tType] : A ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.41/1.87 ord_less:
% 3.41/1.87 !>[A: $tType] : ( A > A > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.41/1.87 ord_less_eq:
% 3.41/1.87 !>[A: $tType] : ( A > A > $o ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 3.41/1.87 order_Greatest:
% 3.41/1.87 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_v_f____,type,
% 3.41/1.87 f: c > ( monomo1453983439ondetT @ a @ m ) ).
% 3.41/1.87
% 3.41/1.87 thf(sy_v_mergec,type,
% 3.41/1.87 mergec: ( countable_Set_cset @ c ) > ( c > m ) > m ).
% 3.41/1.87
% 3.41/1.87 thf(sy_v_x____,type,
% 3.41/1.87 x: c ).
% 3.41/1.87
% 3.41/1.87 thf(fact_0_mergec__single,axiom,
% 3.41/1.87 ! [X: c,F: c > m] :
% 3.41/1.87 ( ( mergec @ ( counta2111716221insert @ c @ X @ ( bot_bot @ ( countable_Set_cset @ c ) ) ) @ F )
% 3.41/1.87 = ( F @ X ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_1_cset__nondetM__base_Oaltc__nondet_Ocong,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,C: $tType] :
% 3.41/1.87 ( ( monomo955616121nondet @ C @ M @ A )
% 3.41/1.87 = ( monomo955616121nondet @ C @ M @ A ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_2_cinsert__absorb2,axiom,
% 3.41/1.87 ! [A: $tType,X: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( counta2111716221insert @ A @ X @ ( counta2111716221insert @ A @ X @ A2 ) )
% 3.41/1.87 = ( counta2111716221insert @ A @ X @ A2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_3_bot__apply,axiom,
% 3.41/1.87 ! [C: $tType,D: $tType] :
% 3.41/1.87 ( ( bot @ C )
% 3.41/1.87 => ( ( bot_bot @ ( D > C ) )
% 3.41/1.87 = ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_4_cdoubleton__eq__iff,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,B2: A,C2: A,D2: A] :
% 3.41/1.87 ( ( ( counta2111716221insert @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 3.41/1.87 = ( counta2111716221insert @ A @ C2 @ ( counta2111716221insert @ A @ D2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) )
% 3.41/1.87 = ( ( ( A3 = C2 )
% 3.41/1.87 & ( B2 = D2 ) )
% 3.41/1.87 | ( ( A3 = D2 )
% 3.41/1.87 & ( B2 = C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_5_csingleton__inject,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,B2: A] :
% 3.41/1.87 ( ( ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 3.41/1.87 = ( counta2111716221insert @ A @ B2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 3.41/1.87 => ( A3 = B2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_6_cinsert__not__cempty,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( counta2111716221insert @ A @ A3 @ A2 )
% 3.41/1.87 != ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_7_cinsert__commute,axiom,
% 3.41/1.87 ! [A: $tType,X: A,Y: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( counta2111716221insert @ A @ X @ ( counta2111716221insert @ A @ Y @ A2 ) )
% 3.41/1.87 = ( counta2111716221insert @ A @ Y @ ( counta2111716221insert @ A @ X @ A2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_8_cset__nondetM_Omerge__single,axiom,
% 3.41/1.87 ! [C: $tType,M: $tType,A: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,X: A,F: A > M] :
% 3.41/1.87 ( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
% 3.41/1.87 => ( ( Merge @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) @ F )
% 3.41/1.87 = ( F @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_9_cset__nondetM_Omergec__single,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,C: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,X: C,F: C > M] :
% 3.41/1.87 ( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
% 3.41/1.87 => ( ( Mergec @ ( counta2111716221insert @ C @ X @ ( bot_bot @ ( countable_Set_cset @ C ) ) ) @ F )
% 3.41/1.87 = ( F @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_10_bot__fun__def,axiom,
% 3.41/1.87 ! [B: $tType,A: $tType] :
% 3.41/1.87 ( ( bot @ B )
% 3.41/1.87 => ( ( bot_bot @ ( A > B ) )
% 3.41/1.87 = ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_11_monad__altc_Oaltc__single,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M,X: C,F: C > M] :
% 3.41/1.87 ( ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc )
% 3.41/1.87 => ( ( Altc @ ( counta2111716221insert @ C @ X @ ( bot_bot @ ( countable_Set_cset @ C ) ) ) @ F )
% 3.41/1.87 = ( F @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_12_run__altc__nondet,axiom,
% 3.41/1.87 ! [A2: countable_Set_cset @ c,F: c > ( monomo1453983439ondetT @ a @ m )] :
% 3.41/1.87 ( ( monomo603716163nondet @ a @ m @ ( monomo955616121nondet @ c @ m @ a @ mergec @ A2 @ F ) )
% 3.41/1.87 = ( mergec @ A2 @ ( comp @ ( monomo1453983439ondetT @ a @ m ) @ m @ c @ ( monomo603716163nondet @ a @ m ) @ F ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_13_cset__nondetM_Omerge__empty,axiom,
% 3.41/1.87 ! [C: $tType,M: $tType,A: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,F: A > M] :
% 3.41/1.87 ( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
% 3.41/1.87 => ( ( Merge @ ( bot_bot @ ( countable_Set_cset @ A ) ) @ F )
% 3.41/1.87 = ( Return @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_14_nondetT_Oexpand,axiom,
% 3.41/1.87 ! [M: $tType,A: $tType,NondetT: monomo1453983439ondetT @ A @ M,NondetT2: monomo1453983439ondetT @ A @ M] :
% 3.41/1.87 ( ( ( monomo603716163nondet @ A @ M @ NondetT )
% 3.41/1.87 = ( monomo603716163nondet @ A @ M @ NondetT2 ) )
% 3.41/1.87 => ( NondetT = NondetT2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_15_cset__nondetM__base_Orun__altc__nondet,axiom,
% 3.41/1.87 ! [M: $tType,A: $tType,C: $tType,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,A2: countable_Set_cset @ C,F: C > ( monomo1453983439ondetT @ A @ M )] :
% 3.41/1.87 ( ( monomo603716163nondet @ A @ M @ ( monomo955616121nondet @ C @ M @ A @ Mergec @ A2 @ F ) )
% 3.41/1.87 = ( Mergec @ A2 @ ( comp @ ( monomo1453983439ondetT @ A @ M ) @ M @ C @ ( monomo603716163nondet @ A @ M ) @ F ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_16_altc__nondet__def,axiom,
% 3.41/1.87 ! [A2: countable_Set_cset @ c,F: c > ( monomo1453983439ondetT @ a @ m )] :
% 3.41/1.87 ( ( monomo955616121nondet @ c @ m @ a @ mergec @ A2 @ F )
% 3.41/1.87 = ( monomo1975189248ondetT @ m @ a @ ( mergec @ A2 @ ( comp @ ( monomo1453983439ondetT @ a @ m ) @ m @ c @ ( monomo603716163nondet @ a @ m ) @ F ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_17_comp__apply,axiom,
% 3.41/1.87 ! [C: $tType,A: $tType,B: $tType] :
% 3.41/1.87 ( ( comp @ B @ A @ C )
% 3.41/1.87 = ( ^ [F2: B > A,G: C > B,X2: C] : ( F2 @ ( G @ X2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_18_cset__nondetM__base_Oaltc__nondet__def,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,C: $tType] :
% 3.41/1.87 ( ( monomo955616121nondet @ C @ M @ A )
% 3.41/1.87 = ( ^ [Mergec2: ( countable_Set_cset @ C ) > ( C > M ) > M,A4: countable_Set_cset @ C,F2: C > ( monomo1453983439ondetT @ A @ M )] : ( monomo1975189248ondetT @ M @ A @ ( Mergec2 @ A4 @ ( comp @ ( monomo1453983439ondetT @ A @ M ) @ M @ C @ ( monomo603716163nondet @ A @ M ) @ F2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_19_cset__nondetM_Oaxioms_I3_J,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,C: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 3.41/1.87 ( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
% 3.41/1.87 => ( monomo759963067axioms @ A @ M @ C @ Return @ Bind @ Merge @ Mergec ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_20_comp__def,axiom,
% 3.41/1.87 ! [A: $tType,C: $tType,B: $tType] :
% 3.41/1.87 ( ( comp @ B @ C @ A )
% 3.41/1.87 = ( ^ [F2: B > C,G: A > B,X2: A] : ( F2 @ ( G @ X2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_21_comp__assoc,axiom,
% 3.41/1.87 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G2: C > D,H: A > C] :
% 3.41/1.87 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G2 ) @ H )
% 3.41/1.87 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_22_comp__eq__dest,axiom,
% 3.41/1.87 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
% 3.41/1.87 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 3.41/1.87 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 3.41/1.87 => ( ( A3 @ ( B2 @ V ) )
% 3.41/1.87 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_23_comp__eq__elim,axiom,
% 3.41/1.87 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D] :
% 3.41/1.87 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 3.41/1.87 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 3.41/1.87 => ! [V2: A] :
% 3.41/1.87 ( ( A3 @ ( B2 @ V2 ) )
% 3.41/1.87 = ( C2 @ ( D2 @ V2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_24_comp__cong,axiom,
% 3.41/1.87 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G2: C > B,X: C,F3: D > A,G3: E > D,X3: E] :
% 3.41/1.87 ( ( ( F @ ( G2 @ X ) )
% 3.41/1.87 = ( F3 @ ( G3 @ X3 ) ) )
% 3.41/1.87 => ( ( comp @ B @ A @ C @ F @ G2 @ X )
% 3.41/1.87 = ( comp @ D @ A @ E @ F3 @ G3 @ X3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_25_comp__eq__dest__lhs,axiom,
% 3.41/1.87 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
% 3.41/1.87 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 3.41/1.87 = C2 )
% 3.41/1.87 => ( ( A3 @ ( B2 @ V ) )
% 3.41/1.87 = ( C2 @ V ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_26_comp__apply__eq,axiom,
% 3.41/1.87 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G2: C > B,X: C,H: D > A,K: C > D] :
% 3.41/1.87 ( ( ( F @ ( G2 @ X ) )
% 3.41/1.87 = ( H @ ( K @ X ) ) )
% 3.41/1.87 => ( ( comp @ B @ A @ C @ F @ G2 @ X )
% 3.41/1.87 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_27_nondetT_Oinject,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,X: M,Ya: M] :
% 3.41/1.87 ( ( ( monomo1975189248ondetT @ M @ A @ X )
% 3.41/1.87 = ( monomo1975189248ondetT @ M @ A @ Ya ) )
% 3.41/1.87 = ( X = Ya ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_28_nondetT_Ocollapse,axiom,
% 3.41/1.87 ! [M: $tType,A: $tType,NondetT: monomo1453983439ondetT @ A @ M] :
% 3.41/1.87 ( ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) )
% 3.41/1.87 = NondetT ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_29_nondetT_Oexhaust,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,Y: monomo1453983439ondetT @ A @ M] :
% 3.41/1.87 ~ ! [X4: M] :
% 3.41/1.87 ( Y
% 3.41/1.87 != ( monomo1975189248ondetT @ M @ A @ X4 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_30_nondetT_Oinduct,axiom,
% 3.41/1.87 ! [M: $tType,A: $tType,P: ( monomo1453983439ondetT @ A @ M ) > $o,NondetT: monomo1453983439ondetT @ A @ M] :
% 3.41/1.87 ( ! [Xa: M] : ( P @ ( monomo1975189248ondetT @ M @ A @ Xa ) )
% 3.41/1.87 => ( P @ NondetT ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_31_nondetT_Oexhaust__sel,axiom,
% 3.41/1.87 ! [M: $tType,A: $tType,NondetT: monomo1453983439ondetT @ A @ M] :
% 3.41/1.87 ( NondetT
% 3.41/1.87 = ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_32_nondetT_Osel,axiom,
% 3.41/1.87 ! [Aa: $tType,A: $tType,X: A] :
% 3.41/1.87 ( ( monomo603716163nondet @ Aa @ A @ ( monomo1975189248ondetT @ A @ Aa @ X ) )
% 3.41/1.87 = X ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_33_put__nondet__def,axiom,
% 3.41/1.87 ! [State: $tType] :
% 3.41/1.87 ( ( monomo410456285nondet @ State @ m @ a )
% 3.41/1.87 = ( ^ [Put: State > m > m,S2: State,M2: monomo1453983439ondetT @ a @ m] : ( monomo1975189248ondetT @ m @ a @ ( Put @ S2 @ ( monomo603716163nondet @ a @ m @ M2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_34_nondetM__base_Oput__nondet__def,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,State: $tType] :
% 3.41/1.87 ( ( monomo410456285nondet @ State @ M @ A )
% 3.41/1.87 = ( ^ [Put: State > M > M,S2: State,M2: monomo1453983439ondetT @ A @ M] : ( monomo1975189248ondetT @ M @ A @ ( Put @ S2 @ ( monomo603716163nondet @ A @ M @ M2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_35_nondetT_Osplit__sel__asm,axiom,
% 3.41/1.87 ! [B: $tType,M: $tType,A: $tType,P: B > $o,F: M > B,NondetT: monomo1453983439ondetT @ A @ M] :
% 3.41/1.87 ( ( P @ ( monomo1564381680ondetT @ M @ B @ A @ F @ NondetT ) )
% 3.41/1.87 = ( ~ ( ( NondetT
% 3.41/1.87 = ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) ) )
% 3.41/1.87 & ~ ( P @ ( F @ ( monomo603716163nondet @ A @ M @ NondetT ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_36_nondetT_Osplit__sel,axiom,
% 3.41/1.87 ! [B: $tType,M: $tType,A: $tType,P: B > $o,F: M > B,NondetT: monomo1453983439ondetT @ A @ M] :
% 3.41/1.87 ( ( P @ ( monomo1564381680ondetT @ M @ B @ A @ F @ NondetT ) )
% 3.41/1.87 = ( ( NondetT
% 3.41/1.87 = ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) ) )
% 3.41/1.87 => ( P @ ( F @ ( monomo603716163nondet @ A @ M @ NondetT ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_37_rewriteR__comp__comp2,axiom,
% 3.41/1.87 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H: A > C,R1: D > B,R2: A > D,F: B > E,L: D > E] :
% 3.41/1.87 ( ( ( comp @ C @ B @ A @ G2 @ H )
% 3.41/1.87 = ( comp @ D @ B @ A @ R1 @ R2 ) )
% 3.41/1.87 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 3.41/1.87 = L )
% 3.41/1.87 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G2 ) @ H )
% 3.41/1.87 = ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_38_rewriteL__comp__comp2,axiom,
% 3.41/1.87 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G2: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 3.41/1.87 ( ( ( comp @ C @ B @ A @ F @ G2 )
% 3.41/1.87 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 3.41/1.87 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 3.41/1.87 = R )
% 3.41/1.87 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G2 @ H ) )
% 3.41/1.87 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_39_rewriteR__comp__comp,axiom,
% 3.41/1.87 ! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H: A > C,R: A > B,F: B > D] :
% 3.41/1.87 ( ( ( comp @ C @ B @ A @ G2 @ H )
% 3.41/1.87 = R )
% 3.41/1.87 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G2 ) @ H )
% 3.41/1.87 = ( comp @ B @ D @ A @ F @ R ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_40_rewriteL__comp__comp,axiom,
% 3.41/1.87 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G2: A > C,L: A > B,H: D > A] :
% 3.41/1.87 ( ( ( comp @ C @ B @ A @ F @ G2 )
% 3.41/1.87 = L )
% 3.41/1.87 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G2 @ H ) )
% 3.41/1.87 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_41_type__copy__map__cong0,axiom,
% 3.41/1.87 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M3: B > A,G2: C > B,X: C,N: D > A,H: C > D,F: A > E] :
% 3.41/1.87 ( ( ( M3 @ ( G2 @ X ) )
% 3.41/1.87 = ( N @ ( H @ X ) ) )
% 3.41/1.87 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M3 ) @ G2 @ X )
% 3.41/1.87 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_42_run__put__nondet,axiom,
% 3.41/1.87 ! [B: $tType,Put2: B > m > m,S3: B,M4: monomo1453983439ondetT @ a @ m] :
% 3.41/1.87 ( ( monomo603716163nondet @ a @ m @ ( monomo410456285nondet @ B @ m @ a @ Put2 @ S3 @ M4 ) )
% 3.41/1.87 = ( Put2 @ S3 @ ( monomo603716163nondet @ a @ m @ M4 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_43_nondetT_Ocase__eq__if,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType,M: $tType] :
% 3.41/1.87 ( ( monomo1564381680ondetT @ M @ B @ A )
% 3.41/1.87 = ( ^ [F2: M > B,NondetT3: monomo1453983439ondetT @ A @ M] : ( F2 @ ( monomo603716163nondet @ A @ M @ NondetT3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_44_nondetT_Ocase,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType,M: $tType,F: M > B,X: M] :
% 3.41/1.87 ( ( monomo1564381680ondetT @ M @ B @ A @ F @ ( monomo1975189248ondetT @ M @ A @ X ) )
% 3.41/1.87 = ( F @ X ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_45_ext,axiom,
% 3.41/1.87 ! [B: $tType,A: $tType,F: A > B,G2: A > B] :
% 3.41/1.87 ( ! [X4: A] :
% 3.41/1.87 ( ( F @ X4 )
% 3.41/1.87 = ( G2 @ X4 ) )
% 3.41/1.87 => ( F = G2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_46_nondetM__base_Orun__put__nondet,axiom,
% 3.41/1.87 ! [B: $tType,M: $tType,A: $tType,Put2: B > M > M,S3: B,M4: monomo1453983439ondetT @ A @ M] :
% 3.41/1.87 ( ( monomo603716163nondet @ A @ M @ ( monomo410456285nondet @ B @ M @ A @ Put2 @ S3 @ M4 ) )
% 3.41/1.87 = ( Put2 @ S3 @ ( monomo603716163nondet @ A @ M @ M4 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_47_fun_Omap__comp,axiom,
% 3.41/1.87 ! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F: A > B,V: D > A] :
% 3.41/1.87 ( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F @ V ) )
% 3.41/1.87 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F ) @ V ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_48_function__factors__left,axiom,
% 3.41/1.87 ! [A: $tType,C: $tType,B: $tType,G2: A > B,F: A > C] :
% 3.41/1.87 ( ( ! [X2: A,Y2: A] :
% 3.41/1.87 ( ( ( G2 @ X2 )
% 3.41/1.87 = ( G2 @ Y2 ) )
% 3.41/1.87 => ( ( F @ X2 )
% 3.41/1.87 = ( F @ Y2 ) ) ) )
% 3.41/1.87 = ( ? [H2: B > C] :
% 3.41/1.87 ( F
% 3.41/1.87 = ( comp @ B @ C @ A @ H2 @ G2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_49_function__factors__right,axiom,
% 3.41/1.87 ! [C: $tType,B: $tType,A: $tType,G2: B > C,F: A > C] :
% 3.41/1.87 ( ( ! [X2: A] :
% 3.41/1.87 ? [Y2: B] :
% 3.41/1.87 ( ( G2 @ Y2 )
% 3.41/1.87 = ( F @ X2 ) ) )
% 3.41/1.87 = ( ? [H2: A > B] :
% 3.41/1.87 ( F
% 3.41/1.87 = ( comp @ B @ C @ A @ G2 @ H2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_50_monad__altc_Oaxioms_I2_J,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 3.41/1.87 ( ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc )
% 3.41/1.87 => ( monomo555039548axioms @ M @ A @ C @ Bind @ Altc ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_51_cinsert__cDiff__single,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( counta2111716221insert @ A @ A3 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) )
% 3.41/1.87 = ( counta2111716221insert @ A @ A3 @ A2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_52_monad__altc3_Oaxioms_I1_J,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 3.41/1.87 ( ( monomo1425736922_altc3 @ A @ M @ C @ Return @ Bind @ Altc )
% 3.41/1.87 => ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_53_nondetT_Orec,axiom,
% 3.41/1.87 ! [A: $tType,D: $tType,M: $tType,F: M > D,X: M] :
% 3.41/1.87 ( ( monomo1291740684ondetT @ M @ D @ A @ F @ ( monomo1975189248ondetT @ M @ A @ X ) )
% 3.41/1.87 = ( F @ X ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_54_monad__state__altc_Oaxioms_I2_J,axiom,
% 3.41/1.87 ! [S: $tType,A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 3.41/1.87 ( ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc )
% 3.41/1.87 => ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_55_cDiff__idemp,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ B3 )
% 3.41/1.87 = ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_56_cDiff__cancel,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ A2 )
% 3.41/1.87 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_57_cDiff__cempty,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 3.41/1.87 = A2 ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_58_cempty__cDiff,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( bot_bot @ ( countable_Set_cset @ A ) ) @ A2 )
% 3.41/1.87 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_59_cDiff__cinsert,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,A3: A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ B3 ) )
% 3.41/1.87 = ( minus_minus @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_60_cDiff__cinsert2,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,A3: A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ B3 ) )
% 3.41/1.87 = ( minus_minus @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_61_minus__apply,axiom,
% 3.41/1.87 ! [B: $tType,A: $tType] :
% 3.41/1.87 ( ( minus @ B )
% 3.41/1.87 => ( ( minus_minus @ ( A > B ) )
% 3.41/1.87 = ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_62_cDiff__single__cinsert,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,X: A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_63_monad__state__altc_Oaxioms_I3_J,axiom,
% 3.41/1.87 ! [A: $tType,S: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 3.41/1.87 ( ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc )
% 3.41/1.87 => ( monomo1324021455axioms @ S @ M @ C @ Get @ Put2 @ Altc ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_64_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( cancel146912293up_add @ A )
% 3.41/1.87 => ! [A3: A,C2: A,B2: A] :
% 3.41/1.87 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
% 3.41/1.87 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_65_diff__eq__diff__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( group_add @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A,D2: A] :
% 3.41/1.87 ( ( ( minus_minus @ A @ A3 @ B2 )
% 3.41/1.87 = ( minus_minus @ A @ C2 @ D2 ) )
% 3.41/1.87 => ( ( A3 = B2 )
% 3.41/1.87 = ( C2 = D2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_66_fun__diff__def,axiom,
% 3.41/1.87 ! [B: $tType,A: $tType] :
% 3.41/1.87 ( ( minus @ B )
% 3.41/1.87 => ( ( minus_minus @ ( A > B ) )
% 3.41/1.87 = ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_67_cinsert__cDiff,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ A3 @ A2 )
% 3.41/1.87 => ( ( counta2111716221insert @ A @ A3 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) )
% 3.41/1.87 = A2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_68_order__refl,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_69_csubset__antisym,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ A2 )
% 3.41/1.87 => ( A2 = B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_70_all__not__cin__conv,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ! [X2: A] :
% 3.41/1.87 ~ ( countable_Set_cin @ A @ X2 @ A2 ) )
% 3.41/1.87 = ( A2
% 3.41/1.87 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_71_cempty__iff,axiom,
% 3.41/1.87 ! [A: $tType,C2: A] :
% 3.41/1.87 ~ ( countable_Set_cin @ A @ C2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_72_cinsert__iff,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,B2: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ A2 ) )
% 3.41/1.87 = ( ( A3 = B2 )
% 3.41/1.87 | ( countable_Set_cin @ A @ A3 @ A2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_73_cinsertCI,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,B3: countable_Set_cset @ A,B2: A] :
% 3.41/1.87 ( ( ~ ( countable_Set_cin @ A @ A3 @ B3 )
% 3.41/1.87 => ( A3 = B2 ) )
% 3.41/1.87 => ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_74_csubsetI,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ! [X4: A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ X4 @ A2 )
% 3.41/1.87 => ( countable_Set_cin @ A @ X4 @ B3 ) )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_75_cempty__fsubsetI,axiom,
% 3.41/1.87 ! [A: $tType,X: countable_Set_cset @ A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( bot_bot @ ( countable_Set_cset @ A ) ) @ X ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_76_csubset__cempty,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 3.41/1.87 = ( A2
% 3.41/1.87 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_77_cDiff__iff,axiom,
% 3.41/1.87 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 3.41/1.87 = ( ( countable_Set_cin @ A @ C2 @ A2 )
% 3.41/1.87 & ~ ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_78_cDiffI,axiom,
% 3.41/1.87 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ C2 @ A2 )
% 3.41/1.87 => ( ~ ( countable_Set_cin @ A @ C2 @ B3 )
% 3.41/1.87 => ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_79_cinsert__csubset,axiom,
% 3.41/1.87 ! [A: $tType,X: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
% 3.41/1.87 = ( ( countable_Set_cin @ A @ X @ B3 )
% 3.41/1.87 & ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_80_cinsert__cDiff1,axiom,
% 3.41/1.87 ! [A: $tType,X: A,B3: countable_Set_cset @ A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ X @ B3 )
% 3.41/1.87 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
% 3.41/1.87 = ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_81_diff__eq__diff__less__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ordered_ab_group_add @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A,D2: A] :
% 3.41/1.87 ( ( ( minus_minus @ A @ A3 @ B2 )
% 3.41/1.87 = ( minus_minus @ A @ C2 @ D2 ) )
% 3.41/1.87 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_82_diff__right__mono,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ordered_ab_group_add @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_83_diff__left__mono,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ordered_ab_group_add @ A )
% 3.41/1.87 => ! [B2: A,A3: A,C2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.41/1.87 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_84_diff__mono,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ordered_ab_group_add @ A )
% 3.41/1.87 => ! [A3: A,B2: A,D2: A,C2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less_eq @ A @ D2 @ C2 )
% 3.41/1.87 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_85_csubset__cinsert,axiom,
% 3.41/1.87 ! [A: $tType,X: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) )
% 3.41/1.87 = ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_86_dual__order_Oantisym,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [B2: A,A3: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.41/1.87 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( A3 = B2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_87_dual__order_Oeq__iff,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 3.41/1.87 = ( ^ [A5: A,B5: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ B5 @ A5 )
% 3.41/1.87 & ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_88_dual__order_Otrans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [B2: A,A3: A,C2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.41/1.87 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 3.41/1.87 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_89_linorder__wlog,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [P: A > A > $o,A3: A,B2: A] :
% 3.41/1.87 ( ! [A6: A,B6: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A6 @ B6 )
% 3.41/1.87 => ( P @ A6 @ B6 ) )
% 3.41/1.87 => ( ! [A6: A,B6: A] :
% 3.41/1.87 ( ( P @ B6 @ A6 )
% 3.41/1.87 => ( P @ A6 @ B6 ) )
% 3.41/1.87 => ( P @ A3 @ B2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_90_dual__order_Orefl,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_91_order__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A,Z2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.41/1.87 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_92_order__class_Oorder_Oantisym,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.41/1.87 => ( A3 = B2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_93_ord__le__eq__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ord @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( B2 = C2 )
% 3.41/1.87 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_94_ord__eq__le__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ord @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( A3 = B2 )
% 3.41/1.87 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.41/1.87 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_95_order__class_Oorder_Oeq__iff,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 3.41/1.87 = ( ^ [A5: A,B5: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A5 @ B5 )
% 3.41/1.87 & ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_96_antisym__conv,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [Y: A,X: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ Y @ X )
% 3.41/1.87 => ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 = ( X = Y ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_97_le__cases3,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A,Z2: A] :
% 3.41/1.87 ( ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 3.41/1.87 => ( ( ( ord_less_eq @ A @ Y @ X )
% 3.41/1.87 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 3.41/1.87 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 3.41/1.87 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 3.41/1.87 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 3.41/1.87 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 3.41/1.87 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.41/1.87 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 3.41/1.87 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 3.41/1.87 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_98_eqcelem__imp__iff,axiom,
% 3.41/1.87 ! [A: $tType,X: A,Y: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( X = Y )
% 3.41/1.87 => ( ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 = ( countable_Set_cin @ A @ Y @ A2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_99_cset__eq__csubset,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ^ [Y3: countable_Set_cset @ A,Z: countable_Set_cset @ A] : ( Y3 = Z ) )
% 3.41/1.87 = ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A4 @ B4 )
% 3.41/1.87 & ( ord_less_eq @ ( countable_Set_cset @ A ) @ B4 @ A4 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_100_order_Otrans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.41/1.87 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_101_le__cases,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ~ ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_102_eqcset__imp__iff,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,X: A] :
% 3.41/1.87 ( ( A2 = B3 )
% 3.41/1.87 => ( ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 = ( countable_Set_cin @ A @ X @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_103_eq__refl,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( X = Y )
% 3.41/1.87 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_104_if__split__cin2,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,Q: $o,X: countable_Set_cset @ A,Y: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ A3 @ ( if @ ( countable_Set_cset @ A ) @ Q @ X @ Y ) )
% 3.41/1.87 = ( ( Q
% 3.41/1.87 => ( countable_Set_cin @ A @ A3 @ X ) )
% 3.41/1.87 & ( ~ Q
% 3.41/1.87 => ( countable_Set_cin @ A @ A3 @ Y ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_105_if__split__cin1,axiom,
% 3.41/1.87 ! [A: $tType,Q: $o,X: A,Y: A,B2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ ( if @ A @ Q @ X @ Y ) @ B2 )
% 3.41/1.87 = ( ( Q
% 3.41/1.87 => ( countable_Set_cin @ A @ X @ B2 ) )
% 3.41/1.87 & ( ~ Q
% 3.41/1.87 => ( countable_Set_cin @ A @ Y @ B2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_106_eq__cmem__trans,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,B2: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( A3 = B2 )
% 3.41/1.87 => ( ( countable_Set_cin @ A @ B2 @ A2 )
% 3.41/1.87 => ( countable_Set_cin @ A @ A3 @ A2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_107_csubset__trans,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_108_linear,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_109_csubset__refl,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ A2 ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_110_cequalityD2,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( A2 = B3 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ A2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_111_cequalityD1,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( A2 = B3 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_112_cequalityCE,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C2: A] :
% 3.41/1.87 ( ( A2 = B3 )
% 3.41/1.87 => ( ( ( countable_Set_cin @ A @ C2 @ A2 )
% 3.41/1.87 => ~ ( countable_Set_cin @ A @ C2 @ B3 ) )
% 3.41/1.87 => ~ ( ~ ( countable_Set_cin @ A @ C2 @ A2 )
% 3.41/1.87 => ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_113_antisym,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 => ( ( ord_less_eq @ A @ Y @ X )
% 3.41/1.87 => ( X = Y ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_114_cequalityE,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( A2 = B3 )
% 3.41/1.87 => ~ ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ~ ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ A2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_115_eq__iff,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 3.41/1.87 = ( ^ [X2: A,Y2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.41/1.87 & ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_116_csubsetD,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C2: A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ( countable_Set_cin @ A @ C2 @ A2 )
% 3.41/1.87 => ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_117_cset__eqI,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ! [X4: A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ X4 @ A2 )
% 3.41/1.87 = ( countable_Set_cin @ A @ X4 @ B3 ) )
% 3.41/1.87 => ( A2 = B3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_118_cin__mono,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,X: A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 => ( countable_Set_cin @ A @ X @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_119_ord__le__eq__subst,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType] :
% 3.41/1.87 ( ( ( ord @ B )
% 3.41/1.87 & ( ord @ A ) )
% 3.41/1.87 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ( F @ B2 )
% 3.41/1.87 = C2 )
% 3.41/1.87 => ( ! [X4: A,Y4: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_120_ord__eq__le__subst,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType] :
% 3.41/1.87 ( ( ( ord @ B )
% 3.41/1.87 & ( ord @ A ) )
% 3.41/1.87 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 3.41/1.87 ( ( A3
% 3.41/1.87 = ( F @ B2 ) )
% 3.41/1.87 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 3.41/1.87 => ( ! [X4: B,Y4: B] :
% 3.41/1.87 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_121_order__subst2,axiom,
% 3.41/1.87 ! [A: $tType,C: $tType] :
% 3.41/1.87 ( ( ( order @ C )
% 3.41/1.87 & ( order @ A ) )
% 3.41/1.87 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 3.41/1.87 => ( ! [X4: A,Y4: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_122_order__subst1,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType] :
% 3.41/1.87 ( ( ( order @ B )
% 3.41/1.87 & ( order @ A ) )
% 3.41/1.87 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 3.41/1.87 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 3.41/1.87 => ( ! [X4: B,Y4: B] :
% 3.41/1.87 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_123_le__fun__def,axiom,
% 3.41/1.87 ! [B: $tType,A: $tType] :
% 3.41/1.87 ( ( ord @ B )
% 3.41/1.87 => ( ( ord_less_eq @ ( A > B ) )
% 3.41/1.87 = ( ^ [F2: A > B,G: A > B] :
% 3.41/1.87 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_124_le__funI,axiom,
% 3.41/1.87 ! [B: $tType,A: $tType] :
% 3.41/1.87 ( ( ord @ B )
% 3.41/1.87 => ! [F: A > B,G2: A > B] :
% 3.41/1.87 ( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) )
% 3.41/1.87 => ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_125_le__funE,axiom,
% 3.41/1.87 ! [B: $tType,A: $tType] :
% 3.41/1.87 ( ( ord @ B )
% 3.41/1.87 => ! [F: A > B,G2: A > B,X: A] :
% 3.41/1.87 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 3.41/1.87 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_126_le__funD,axiom,
% 3.41/1.87 ! [B: $tType,A: $tType] :
% 3.41/1.87 ( ( ord @ B )
% 3.41/1.87 => ! [F: A > B,G2: A > B,X: A] :
% 3.41/1.87 ( ( ord_less_eq @ ( A > B ) @ F @ G2 )
% 3.41/1.87 => ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_127_equals__cemptyI,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ! [Y4: A] :
% 3.41/1.87 ~ ( countable_Set_cin @ A @ Y4 @ A2 )
% 3.41/1.87 => ( A2
% 3.41/1.87 = ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_128_equals__cemptyD,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,A3: A] :
% 3.41/1.87 ( ( A2
% 3.41/1.87 = ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 3.41/1.87 => ~ ( countable_Set_cin @ A @ A3 @ A2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_129_ex__cin__conv,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ? [X2: A] : ( countable_Set_cin @ A @ X2 @ A2 ) )
% 3.41/1.87 = ( A2
% 3.41/1.87 != ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_130_cemptyE,axiom,
% 3.41/1.87 ! [A: $tType,A3: A] :
% 3.41/1.87 ~ ( countable_Set_cin @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_131_mk__disjoint__cinsert,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ A3 @ A2 )
% 3.41/1.87 => ? [B7: countable_Set_cset @ A] :
% 3.41/1.87 ( ( A2
% 3.41/1.87 = ( counta2111716221insert @ A @ A3 @ B7 ) )
% 3.41/1.87 & ~ ( countable_Set_cin @ A @ A3 @ B7 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_132_cinsert__absorb,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ A3 @ A2 )
% 3.41/1.87 => ( ( counta2111716221insert @ A @ A3 @ A2 )
% 3.41/1.87 = A2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_133_cinsert__ident,axiom,
% 3.41/1.87 ! [A: $tType,X: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 => ( ~ ( countable_Set_cin @ A @ X @ B3 )
% 3.41/1.87 => ( ( ( counta2111716221insert @ A @ X @ A2 )
% 3.41/1.87 = ( counta2111716221insert @ A @ X @ B3 ) )
% 3.41/1.87 = ( A2 = B3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_134_set__cinsert,axiom,
% 3.41/1.87 ! [A: $tType,X: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 => ~ ! [B7: countable_Set_cset @ A] :
% 3.41/1.87 ( ( A2
% 3.41/1.87 = ( counta2111716221insert @ A @ X @ B7 ) )
% 3.41/1.87 => ( countable_Set_cin @ A @ X @ B7 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_135_cinsertI2,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,B3: countable_Set_cset @ A,B2: A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ A3 @ B3 )
% 3.41/1.87 => ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_136_cinsertI1,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,B3: countable_Set_cset @ A] : ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ A3 @ B3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_137_cinsertE,axiom,
% 3.41/1.87 ! [A: $tType,A3: A,B2: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ A2 ) )
% 3.41/1.87 => ( ( A3 != B2 )
% 3.41/1.87 => ( countable_Set_cin @ A @ A3 @ A2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_138_bot_Oextremum__uniqueI,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order_bot @ A )
% 3.41/1.87 => ! [A3: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 3.41/1.87 => ( A3
% 3.41/1.87 = ( bot_bot @ A ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_139_bot_Oextremum__unique,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order_bot @ A )
% 3.41/1.87 => ! [A3: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 3.41/1.87 = ( A3
% 3.41/1.87 = ( bot_bot @ A ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_140_bot_Oextremum,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order_bot @ A )
% 3.41/1.87 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_141_cDiffD2,axiom,
% 3.41/1.87 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 3.41/1.87 => ~ ( countable_Set_cin @ A @ C2 @ B3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_142_cDiffD1,axiom,
% 3.41/1.87 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 3.41/1.87 => ( countable_Set_cin @ A @ C2 @ A2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_143_cDiffE,axiom,
% 3.41/1.87 ! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 3.41/1.87 => ~ ( ( countable_Set_cin @ A @ C2 @ A2 )
% 3.41/1.87 => ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_144_csubset__cinsertI2,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,B2: A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ B2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_145_csubset__cinsertI,axiom,
% 3.41/1.87 ! [A: $tType,B3: countable_Set_cset @ A,A3: A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ ( counta2111716221insert @ A @ A3 @ B3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_146_cinsert__mono,axiom,
% 3.41/1.87 ! [A: $tType,C3: countable_Set_cset @ A,D3: countable_Set_cset @ A,A3: A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ C3 @ D3 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ A3 @ C3 ) @ ( counta2111716221insert @ A @ A3 @ D3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_147_cDiff__csubset,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ A2 ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_148_double__cDiff,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 3.41/1.87 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ B3 @ ( minus_minus @ ( countable_Set_cset @ A ) @ C3 @ A2 ) )
% 3.41/1.87 = A2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_149_cminus__mono,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,C3: countable_Set_cset @ A,D3: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ C3 )
% 3.41/1.87 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ D3 @ B3 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ C3 @ D3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_150_csubset__cinsert__iff,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,X: A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) )
% 3.41/1.87 = ( ( ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 ) )
% 3.41/1.87 & ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_151_csingleton__iff,axiom,
% 3.41/1.87 ! [A: $tType,B2: A,A3: A] :
% 3.41/1.87 ( ( countable_Set_cin @ A @ B2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 3.41/1.87 = ( B2 = A3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_152_cinsert__cDiff__if,axiom,
% 3.41/1.87 ! [A: $tType,X: A,B3: countable_Set_cset @ A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ( countable_Set_cin @ A @ X @ B3 )
% 3.41/1.87 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
% 3.41/1.87 = ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) )
% 3.41/1.87 & ( ~ ( countable_Set_cin @ A @ X @ B3 )
% 3.41/1.87 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
% 3.41/1.87 = ( counta2111716221insert @ A @ X @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_153_csubset__csingletonD,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,X: A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 3.41/1.87 => ( ( A2
% 3.41/1.87 = ( bot_bot @ ( countable_Set_cset @ A ) ) )
% 3.41/1.87 | ( A2
% 3.41/1.87 = ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_154_cDiff__cinsert__absorb,axiom,
% 3.41/1.87 ! [A: $tType,X: A,A2: countable_Set_cset @ A] :
% 3.41/1.87 ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 => ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
% 3.41/1.87 = A2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_155_cpsubset__finsert__iff,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,X: A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) )
% 3.41/1.87 = ( ( ( countable_Set_cin @ A @ X @ B3 )
% 3.41/1.87 => ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
% 3.41/1.87 & ( ~ ( countable_Set_cin @ A @ X @ B3 )
% 3.41/1.87 => ( ( ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 => ( ord_less @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 ) )
% 3.41/1.87 & ( ~ ( countable_Set_cin @ A @ X @ A2 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_156_monad__state__altc_Ointro,axiom,
% 3.41/1.87 ! [S: $tType,A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 3.41/1.87 ( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
% 3.41/1.87 => ( ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc )
% 3.41/1.87 => ( ( monomo1324021455axioms @ S @ M @ C @ Get @ Put2 @ Altc )
% 3.41/1.87 => ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_157_monad__state__altc__def,axiom,
% 3.41/1.87 ! [C: $tType,S: $tType,M: $tType,A: $tType] :
% 3.41/1.87 ( ( monomo1036387116e_altc @ A @ M @ S @ C )
% 3.41/1.87 = ( ^ [Return2: A > M,Bind2: M > ( A > M ) > M,Get2: ( S > M ) > M,Put: S > M > M,Altc2: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 3.41/1.87 ( ( monomo109450930_state @ A @ M @ S @ Return2 @ Bind2 @ Get2 @ Put )
% 3.41/1.87 & ( monomo439771545d_altc @ A @ M @ C @ Return2 @ Bind2 @ Altc2 )
% 3.41/1.87 & ( monomo1324021455axioms @ S @ M @ C @ Get2 @ Put @ Altc2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_158_Greatest__equality,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [P: A > $o,X: A] :
% 3.41/1.87 ( ( P @ X )
% 3.41/1.87 => ( ! [Y4: A] :
% 3.41/1.87 ( ( P @ Y4 )
% 3.41/1.87 => ( ord_less_eq @ A @ Y4 @ X ) )
% 3.41/1.87 => ( ( order_Greatest @ A @ P )
% 3.41/1.87 = X ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_159_GreatestI2__order,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [P: A > $o,X: A,Q: A > $o] :
% 3.41/1.87 ( ( P @ X )
% 3.41/1.87 => ( ! [Y4: A] :
% 3.41/1.87 ( ( P @ Y4 )
% 3.41/1.87 => ( ord_less_eq @ A @ Y4 @ X ) )
% 3.41/1.87 => ( ! [X4: A] :
% 3.41/1.87 ( ( P @ X4 )
% 3.41/1.87 => ( ! [Y5: A] :
% 3.41/1.87 ( ( P @ Y5 )
% 3.41/1.87 => ( ord_less_eq @ A @ Y5 @ X4 ) )
% 3.41/1.87 => ( Q @ X4 ) ) )
% 3.41/1.87 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_160_verit__la__disequality,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [A3: A,B2: A] :
% 3.41/1.87 ( ( A3 = B2 )
% 3.41/1.87 | ~ ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 | ~ ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_161_le__rel__bool__arg__iff,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ord @ A )
% 3.41/1.87 => ( ( ord_less_eq @ ( $o > A ) )
% 3.41/1.87 = ( ^ [X5: $o > A,Y6: $o > A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 3.41/1.87 & ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_162_verit__comp__simplify1_I3_J,axiom,
% 3.41/1.87 ! [B: $tType] :
% 3.41/1.87 ( ( linorder @ B )
% 3.41/1.87 => ! [B8: B,A7: B] :
% 3.41/1.87 ( ( ~ ( ord_less_eq @ B @ B8 @ A7 ) )
% 3.41/1.87 = ( ord_less @ B @ A7 @ B8 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_163_ord__eq__less__subst,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType] :
% 3.41/1.87 ( ( ( ord @ B )
% 3.41/1.87 & ( ord @ A ) )
% 3.41/1.87 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 3.41/1.87 ( ( A3
% 3.41/1.87 = ( F @ B2 ) )
% 3.41/1.87 => ( ( ord_less @ B @ B2 @ C2 )
% 3.41/1.87 => ( ! [X4: B,Y4: B] :
% 3.41/1.87 ( ( ord_less @ B @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_164_ord__less__eq__subst,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType] :
% 3.41/1.87 ( ( ( ord @ B )
% 3.41/1.87 & ( ord @ A ) )
% 3.41/1.87 => ! [A3: A,B2: A,F: A > B,C2: B] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ( F @ B2 )
% 3.41/1.87 = C2 )
% 3.41/1.87 => ( ! [X4: A,Y4: A] :
% 3.41/1.87 ( ( ord_less @ A @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_165_order__less__subst1,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType] :
% 3.41/1.87 ( ( ( order @ B )
% 3.41/1.87 & ( order @ A ) )
% 3.41/1.87 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 3.41/1.87 => ( ( ord_less @ B @ B2 @ C2 )
% 3.41/1.87 => ( ! [X4: B,Y4: B] :
% 3.41/1.87 ( ( ord_less @ B @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_166_order__less__subst2,axiom,
% 3.41/1.87 ! [A: $tType,C: $tType] :
% 3.41/1.87 ( ( ( order @ C )
% 3.41/1.87 & ( order @ A ) )
% 3.41/1.87 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 3.41/1.87 => ( ! [X4: A,Y4: A] :
% 3.41/1.87 ( ( ord_less @ A @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_167_lt__ex,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( no_bot @ A )
% 3.41/1.87 => ! [X: A] :
% 3.41/1.87 ? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_168_gt__ex,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( no_top @ A )
% 3.41/1.87 => ! [X: A] :
% 3.41/1.87 ? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_169_neqE,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( X != Y )
% 3.41/1.87 => ( ~ ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_170_neq__iff,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( X != Y )
% 3.41/1.87 = ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_171_order_Oasym,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_172_cpsubset__trans,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ( ord_less @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 3.41/1.87 => ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_173_dense,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( dense_order @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ? [Z3: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Z3 )
% 3.41/1.87 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_174_less__imp__neq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( X != Y ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_175_less__asym,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_176_less__asym_H,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [A3: A,B2: A] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_177_less__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A,Z2: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( ( ord_less @ A @ Y @ Z2 )
% 3.41/1.87 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_178_less__linear,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 | ( X = Y )
% 3.41/1.87 | ( ord_less @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_179_less__irrefl,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A] :
% 3.41/1.87 ~ ( ord_less @ A @ X @ X ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_180_ord__eq__less__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ord @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( A3 = B2 )
% 3.41/1.87 => ( ( ord_less @ A @ B2 @ C2 )
% 3.41/1.87 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_181_ord__less__eq__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ord @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( B2 = C2 )
% 3.41/1.87 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_182_dual__order_Oasym,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [B2: A,A3: A] :
% 3.41/1.87 ( ( ord_less @ A @ B2 @ A3 )
% 3.41/1.87 => ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_183_less__imp__not__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( X != Y ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_184_less__not__sym,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_185_less__induct,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( wellorder @ A )
% 3.41/1.87 => ! [P: A > $o,A3: A] :
% 3.41/1.87 ( ! [X4: A] :
% 3.41/1.87 ( ! [Y5: A] :
% 3.41/1.87 ( ( ord_less @ A @ Y5 @ X4 )
% 3.41/1.87 => ( P @ Y5 ) )
% 3.41/1.87 => ( P @ X4 ) )
% 3.41/1.87 => ( P @ A3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_186_antisym__conv3,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [Y: A,X: A] :
% 3.41/1.87 ( ~ ( ord_less @ A @ Y @ X )
% 3.41/1.87 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.41/1.87 = ( X = Y ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_187_less__imp__not__eq2,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( Y != X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_188_less__imp__triv,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A,P: $o] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( ( ord_less @ A @ Y @ X )
% 3.41/1.87 => P ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_189_linorder__cases,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ~ ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( ( X != Y )
% 3.41/1.87 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_190_dual__order_Oirrefl,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A] :
% 3.41/1.87 ~ ( ord_less @ A @ A3 @ A3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_191_order_Ostrict__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less @ A @ B2 @ C2 )
% 3.41/1.87 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_192_less__imp__not__less,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_193_exists__least__iff,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( wellorder @ A )
% 3.41/1.87 => ( ( ^ [P2: A > $o] :
% 3.41/1.87 ? [X6: A] : ( P2 @ X6 ) )
% 3.41/1.87 = ( ^ [P3: A > $o] :
% 3.41/1.87 ? [N2: A] :
% 3.41/1.87 ( ( P3 @ N2 )
% 3.41/1.87 & ! [M2: A] :
% 3.41/1.87 ( ( ord_less @ A @ M2 @ N2 )
% 3.41/1.87 => ~ ( P3 @ M2 ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_194_linorder__less__wlog,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [P: A > A > $o,A3: A,B2: A] :
% 3.41/1.87 ( ! [A6: A,B6: A] :
% 3.41/1.87 ( ( ord_less @ A @ A6 @ B6 )
% 3.41/1.87 => ( P @ A6 @ B6 ) )
% 3.41/1.87 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 3.41/1.87 => ( ! [A6: A,B6: A] :
% 3.41/1.87 ( ( P @ B6 @ A6 )
% 3.41/1.87 => ( P @ A6 @ B6 ) )
% 3.41/1.87 => ( P @ A3 @ B2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_195_dual__order_Ostrict__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [B2: A,A3: A,C2: A] :
% 3.41/1.87 ( ( ord_less @ A @ B2 @ A3 )
% 3.41/1.87 => ( ( ord_less @ A @ C2 @ B2 )
% 3.41/1.87 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_196_not__less__iff__gr__or__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.41/1.87 = ( ( ord_less @ A @ Y @ X )
% 3.41/1.87 | ( X = Y ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_197_order_Ostrict__implies__not__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( A3 != B2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_198_dual__order_Ostrict__implies__not__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [B2: A,A3: A] :
% 3.41/1.87 ( ( ord_less @ A @ B2 @ A3 )
% 3.41/1.87 => ( A3 != B2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_199_monad__state_Obind__put,axiom,
% 3.41/1.87 ! [S: $tType,M: $tType,A: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,S3: S,M4: M,F: A > M] :
% 3.41/1.87 ( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
% 3.41/1.87 => ( ( Bind @ ( Put2 @ S3 @ M4 ) @ F )
% 3.41/1.87 = ( Put2 @ S3 @ ( Bind @ M4 @ F ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_200_monad__state_Oput__put,axiom,
% 3.41/1.87 ! [A: $tType,S: $tType,M: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,S3: S,S4: S,M4: M] :
% 3.41/1.87 ( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
% 3.41/1.87 => ( ( Put2 @ S3 @ ( Put2 @ S4 @ M4 ) )
% 3.41/1.87 = ( Put2 @ S4 @ M4 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_201_monad__state_Oput__get,axiom,
% 3.41/1.87 ! [A: $tType,M: $tType,S: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,S3: S,F: S > M] :
% 3.41/1.87 ( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
% 3.41/1.87 => ( ( Put2 @ S3 @ ( Get @ F ) )
% 3.41/1.87 = ( Put2 @ S3 @ ( F @ S3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_202_leD,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [Y: A,X: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ Y @ X )
% 3.41/1.87 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_203_leI,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ~ ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_204_le__less,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ( ( ord_less_eq @ A )
% 3.41/1.87 = ( ^ [X2: A,Y2: A] :
% 3.41/1.87 ( ( ord_less @ A @ X2 @ Y2 )
% 3.41/1.87 | ( X2 = Y2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_205_less__le,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ( ( ord_less @ A )
% 3.41/1.87 = ( ^ [X2: A,Y2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.41/1.87 & ( X2 != Y2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_206_order__le__less__subst1,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType] :
% 3.41/1.87 ( ( ( order @ B )
% 3.41/1.87 & ( order @ A ) )
% 3.41/1.87 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 3.41/1.87 => ( ( ord_less @ B @ B2 @ C2 )
% 3.41/1.87 => ( ! [X4: B,Y4: B] :
% 3.41/1.87 ( ( ord_less @ B @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_207_order__le__less__subst2,axiom,
% 3.41/1.87 ! [A: $tType,C: $tType] :
% 3.41/1.87 ( ( ( order @ C )
% 3.41/1.87 & ( order @ A ) )
% 3.41/1.87 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
% 3.41/1.87 => ( ! [X4: A,Y4: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_208_order__less__le__subst1,axiom,
% 3.41/1.87 ! [A: $tType,B: $tType] :
% 3.41/1.87 ( ( ( order @ B )
% 3.41/1.87 & ( order @ A ) )
% 3.41/1.87 => ! [A3: A,F: B > A,B2: B,C2: B] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 3.41/1.87 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 3.41/1.87 => ( ! [X4: B,Y4: B] :
% 3.41/1.87 ( ( ord_less_eq @ B @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_209_order__less__le__subst2,axiom,
% 3.41/1.87 ! [A: $tType,C: $tType] :
% 3.41/1.87 ( ( ( order @ C )
% 3.41/1.87 & ( order @ A ) )
% 3.41/1.87 => ! [A3: A,B2: A,F: A > C,C2: C] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 3.41/1.87 => ( ! [X4: A,Y4: A] :
% 3.41/1.87 ( ( ord_less @ A @ X4 @ Y4 )
% 3.41/1.87 => ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 3.41/1.87 => ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_210_not__le,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 3.41/1.87 = ( ord_less @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_211_not__less,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.41/1.87 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_212_le__neq__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( A3 != B2 )
% 3.41/1.87 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_213_antisym__conv1,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ~ ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 = ( X = Y ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_214_antisym__conv2,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.41/1.87 = ( X = Y ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_215_less__imp__le,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_216_le__less__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A,Z2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 => ( ( ord_less @ A @ Y @ Z2 )
% 3.41/1.87 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_217_less__le__trans,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ! [X: A,Y: A,Z2: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.41/1.87 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_218_dense__ge,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( dense_linorder @ A )
% 3.41/1.87 => ! [Z2: A,Y: A] :
% 3.41/1.87 ( ! [X4: A] :
% 3.41/1.87 ( ( ord_less @ A @ Z2 @ X4 )
% 3.41/1.87 => ( ord_less_eq @ A @ Y @ X4 ) )
% 3.41/1.87 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_219_dense__le,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( dense_linorder @ A )
% 3.41/1.87 => ! [Y: A,Z2: A] :
% 3.41/1.87 ( ! [X4: A] :
% 3.41/1.87 ( ( ord_less @ A @ X4 @ Y )
% 3.41/1.87 => ( ord_less_eq @ A @ X4 @ Z2 ) )
% 3.41/1.87 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_220_le__less__linear,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 | ( ord_less @ A @ Y @ X ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_221_le__imp__less__or__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [X: A,Y: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X @ Y )
% 3.41/1.87 => ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 | ( X = Y ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_222_less__le__not__le,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( preorder @ A )
% 3.41/1.87 => ( ( ord_less @ A )
% 3.41/1.87 = ( ^ [X2: A,Y2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.41/1.87 & ~ ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_223_not__le__imp__less,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [Y: A,X: A] :
% 3.41/1.87 ( ~ ( ord_less_eq @ A @ Y @ X )
% 3.41/1.87 => ( ord_less @ A @ X @ Y ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_224_order_Ostrict__trans1,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less @ A @ B2 @ C2 )
% 3.41/1.87 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_225_order_Ostrict__trans2,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.41/1.87 => ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_226_order_Oorder__iff__strict,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ( ( ord_less_eq @ A )
% 3.41/1.87 = ( ^ [A5: A,B5: A] :
% 3.41/1.87 ( ( ord_less @ A @ A5 @ B5 )
% 3.41/1.87 | ( A5 = B5 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_227_order_Ostrict__iff__order,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ( ( ord_less @ A )
% 3.41/1.87 = ( ^ [A5: A,B5: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ A5 @ B5 )
% 3.41/1.87 & ( A5 != B5 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_228_dual__order_Ostrict__trans1,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [B2: A,A3: A,C2: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.41/1.87 => ( ( ord_less @ A @ C2 @ B2 )
% 3.41/1.87 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_229_dual__order_Ostrict__trans2,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [B2: A,A3: A,C2: A] :
% 3.41/1.87 ( ( ord_less @ A @ B2 @ A3 )
% 3.41/1.87 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 3.41/1.87 => ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_230_dense__ge__bounded,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( dense_linorder @ A )
% 3.41/1.87 => ! [Z2: A,X: A,Y: A] :
% 3.41/1.87 ( ( ord_less @ A @ Z2 @ X )
% 3.41/1.87 => ( ! [W: A] :
% 3.41/1.87 ( ( ord_less @ A @ Z2 @ W )
% 3.41/1.87 => ( ( ord_less @ A @ W @ X )
% 3.41/1.87 => ( ord_less_eq @ A @ Y @ W ) ) )
% 3.41/1.87 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_231_dense__le__bounded,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( dense_linorder @ A )
% 3.41/1.87 => ! [X: A,Y: A,Z2: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ Y )
% 3.41/1.87 => ( ! [W: A] :
% 3.41/1.87 ( ( ord_less @ A @ X @ W )
% 3.41/1.87 => ( ( ord_less @ A @ W @ Y )
% 3.41/1.87 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 3.41/1.87 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_232_order_Ostrict__implies__order,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_233_dual__order_Oorder__iff__strict,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ( ( ord_less_eq @ A )
% 3.41/1.87 = ( ^ [B5: A,A5: A] :
% 3.41/1.87 ( ( ord_less @ A @ B5 @ A5 )
% 3.41/1.87 | ( A5 = B5 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_234_dual__order_Ostrict__iff__order,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ( ( ord_less @ A )
% 3.41/1.87 = ( ^ [B5: A,A5: A] :
% 3.41/1.87 ( ( ord_less_eq @ A @ B5 @ A5 )
% 3.41/1.87 & ( A5 != B5 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_235_dual__order_Ostrict__implies__order,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [B2: A,A3: A] :
% 3.41/1.87 ( ( ord_less @ A @ B2 @ A3 )
% 3.41/1.87 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_236_order_Onot__eq__order__implies__strict,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order @ A )
% 3.41/1.87 => ! [A3: A,B2: A] :
% 3.41/1.87 ( ( A3 != B2 )
% 3.41/1.87 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.41/1.87 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_237_bot_Oextremum__strict,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order_bot @ A )
% 3.41/1.87 => ! [A3: A] :
% 3.41/1.87 ~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_238_bot_Onot__eq__extremum,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( order_bot @ A )
% 3.41/1.87 => ! [A3: A] :
% 3.41/1.87 ( ( A3
% 3.41/1.87 != ( bot_bot @ A ) )
% 3.41/1.87 = ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_239_diff__strict__mono,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ordered_ab_group_add @ A )
% 3.41/1.87 => ! [A3: A,B2: A,D2: A,C2: A] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( ( ord_less @ A @ D2 @ C2 )
% 3.41/1.87 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_240_diff__eq__diff__less,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ordered_ab_group_add @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A,D2: A] :
% 3.41/1.87 ( ( ( minus_minus @ A @ A3 @ B2 )
% 3.41/1.87 = ( minus_minus @ A @ C2 @ D2 ) )
% 3.41/1.87 => ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 = ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_241_diff__strict__left__mono,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ordered_ab_group_add @ A )
% 3.41/1.87 => ! [B2: A,A3: A,C2: A] :
% 3.41/1.87 ( ( ord_less @ A @ B2 @ A3 )
% 3.41/1.87 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_242_diff__strict__right__mono,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ordered_ab_group_add @ A )
% 3.41/1.87 => ! [A3: A,B2: A,C2: A] :
% 3.41/1.87 ( ( ord_less @ A @ A3 @ B2 )
% 3.41/1.87 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_243_cpsubsetD,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C2: A] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ( countable_Set_cin @ A @ C2 @ A2 )
% 3.41/1.87 => ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_244_not__cpsubset__cempty,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A] :
% 3.41/1.87 ~ ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_245_cpsubset__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) )
% 3.41/1.87 = ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A4 @ B4 )
% 3.41/1.87 & ( A4 != B4 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_246_less__cset__def,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) )
% 3.41/1.87 = ( ^ [Xs: countable_Set_cset @ A,Ys: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ Xs @ Ys )
% 3.41/1.87 & ( Xs != Ys ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_247_cpsubset__imp__fsubset,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_248_cpsubset__csubset__trans,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 3.41/1.87 => ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_249_csubset__cpsubset__trans,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ( ( ord_less @ ( countable_Set_cset @ A ) @ B3 @ C3 )
% 3.41/1.87 => ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_250_csubset__not__fsubset__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) )
% 3.41/1.87 = ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A4 @ B4 )
% 3.41/1.87 & ~ ( ord_less_eq @ ( countable_Set_cset @ A ) @ B4 @ A4 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_251_csubset__iff__pfsubset__eq,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( ord_less_eq @ ( countable_Set_cset @ A ) )
% 3.41/1.87 = ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) @ A4 @ B4 )
% 3.41/1.87 | ( A4 = B4 ) ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_252_monad__state__altc_Oaxioms_I1_J,axiom,
% 3.41/1.87 ! [C: $tType,A: $tType,M: $tType,S: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
% 3.41/1.87 ( ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc )
% 3.41/1.87 => ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_253_cpsubset__imp__ex__fmem,axiom,
% 3.41/1.87 ! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
% 3.41/1.87 ( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
% 3.41/1.87 => ? [B6: A] : ( countable_Set_cin @ A @ B6 @ ( minus_minus @ ( countable_Set_cset @ A ) @ B3 @ A2 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_254_minf_I8_J,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [T: A] :
% 3.41/1.87 ? [Z3: A] :
% 3.41/1.87 ! [X7: A] :
% 3.41/1.87 ( ( ord_less @ A @ X7 @ Z3 )
% 3.41/1.87 => ~ ( ord_less_eq @ A @ T @ X7 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(fact_255_minf_I6_J,axiom,
% 3.41/1.87 ! [A: $tType] :
% 3.41/1.87 ( ( linorder @ A )
% 3.41/1.87 => ! [T: A] :
% 3.41/1.87 ? [Z3: A] :
% 3.41/1.87 ! [X7: A] :
% 3.41/1.87 ( ( ord_less @ A @ X7 @ Z3 )
% 3.41/1.87 => ( ord_less_eq @ A @ X7 @ T ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 3.41/1.87 ! [A8: $tType,A9: $tType] :
% 3.41/1.87 ( ( order_bot @ A9 )
% 3.41/1.87 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.41/1.87 ! [A8: $tType,A9: $tType] :
% 3.41/1.87 ( ( preorder @ A9 )
% 3.41/1.87 => ( preorder @ ( A8 > A9 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.41/1.87 ! [A8: $tType,A9: $tType] :
% 3.41/1.87 ( ( order @ A9 )
% 3.41/1.87 => ( order @ ( A8 > A9 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_fun___Orderings_Oord,axiom,
% 3.41/1.87 ! [A8: $tType,A9: $tType] :
% 3.41/1.87 ( ( ord @ A9 )
% 3.41/1.87 => ( ord @ ( A8 > A9 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_fun___Orderings_Obot,axiom,
% 3.41/1.87 ! [A8: $tType,A9: $tType] :
% 3.41/1.87 ( ( bot @ A9 )
% 3.41/1.87 => ( bot @ ( A8 > A9 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_fun___Groups_Ominus,axiom,
% 3.41/1.87 ! [A8: $tType,A9: $tType] :
% 3.41/1.87 ( ( minus @ A9 )
% 3.41/1.87 => ( minus @ ( A8 > A9 ) ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_HOL_Obool___Orderings_Oorder__bot_1,axiom,
% 3.41/1.87 order_bot @ $o ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_HOL_Obool___Orderings_Opreorder_2,axiom,
% 3.41/1.87 preorder @ $o ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 3.41/1.87 linorder @ $o ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
% 3.41/1.87 order @ $o ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
% 3.41/1.87 ord @ $o ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_HOL_Obool___Orderings_Obot_5,axiom,
% 3.41/1.87 bot @ $o ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_HOL_Obool___Groups_Ominus_6,axiom,
% 3.41/1.87 minus @ $o ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_Countable__Set__Type_Ocset___Orderings_Oorder__bot_7,axiom,
% 3.41/1.87 ! [A8: $tType] : ( order_bot @ ( countable_Set_cset @ A8 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_Countable__Set__Type_Ocset___Orderings_Opreorder_8,axiom,
% 3.41/1.87 ! [A8: $tType] : ( preorder @ ( countable_Set_cset @ A8 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_Countable__Set__Type_Ocset___Orderings_Oorder_9,axiom,
% 3.41/1.87 ! [A8: $tType] : ( order @ ( countable_Set_cset @ A8 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_Countable__Set__Type_Ocset___Orderings_Oord_10,axiom,
% 3.41/1.87 ! [A8: $tType] : ( ord @ ( countable_Set_cset @ A8 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_Countable__Set__Type_Ocset___Orderings_Obot_11,axiom,
% 3.41/1.87 ! [A8: $tType] : ( bot @ ( countable_Set_cset @ A8 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(tcon_Countable__Set__Type_Ocset___Groups_Ominus_12,axiom,
% 3.41/1.87 ! [A8: $tType] : ( minus @ ( countable_Set_cset @ A8 ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(help_If_3_1_T,axiom,
% 3.41/1.87 ! [P: $o] :
% 3.41/1.87 ( ( P = $true )
% 3.41/1.87 | ( P = $false ) ) ).
% 3.41/1.87
% 3.41/1.87 thf(help_If_2_1_T,axiom,
% 3.41/1.87 ! [A: $tType,X: A,Y: A] :
% 3.41/1.87 ( ( if @ A @ $false @ X @ Y )
% 3.41/1.87 = Y ) ).
% 3.41/1.87
% 3.41/1.87 thf(help_If_1_1_T,axiom,
% 3.41/1.87 ! [A: $tType,X: A,Y: A] :
% 3.41/1.87 ( ( if @ A @ $true @ X @ Y )
% 3.41/1.87 = X ) ).
% 3.41/1.87
% 3.41/1.87 thf(conj_0,conjecture,
% 3.41/1.87 ( ( monomo955616121nondet @ c @ m @ a @ mergec @ ( counta2111716221insert @ c @ x @ ( bot_bot @ ( countable_Set_cset @ c ) ) ) @ f )
% 3.41/1.87 = ( f @ x ) ) ).
% 3.41/1.87 ------------------------
% 3.50/1.90 % (6937)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_6822 for (2999ds/2Mi)
% 3.50/1.90 % (6939)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_6822 for (2999ds/275Mi)
% 3.50/1.90 % (6936)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_6822 for (2999ds/27Mi)
% 3.50/1.90 % (6940)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_6822 for (2999ds/18Mi)
% 3.50/1.90 % (6934)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_6822 for (2999ds/183Mi)
% 3.50/1.90 % (6935)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_6822 for (2999ds/4Mi)
% 3.50/1.90 % (6938)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_6822 for (2999ds/2Mi)
% 3.50/1.91 % (6937)Instruction limit reached!
% 3.50/1.91 % (6937)------------------------------
% 3.50/1.91 % (6937)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.91 % (6937)Termination reason: Unknown
% 3.50/1.91 % (6937)Termination phase: shuffling
% 3.50/1.91
% 3.50/1.91 % (6937)Memory used [KB]: 1407
% 3.50/1.91 % (6937)Time elapsed: 0.003 s
% 3.50/1.91 % (6937)Instructions burned: 2 (million)
% 3.50/1.91 % (6937)------------------------------
% 3.50/1.91 % (6937)------------------------------
% 3.50/1.91 % (6938)Instruction limit reached!
% 3.50/1.91 % (6938)------------------------------
% 3.50/1.91 % (6938)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.91 % (6938)Termination reason: Unknown
% 3.50/1.91 % (6938)Termination phase: shuffling
% 3.50/1.91
% 3.50/1.91 % (6938)Memory used [KB]: 1407
% 3.50/1.91 % (6938)Time elapsed: 0.003 s
% 3.50/1.91 % (6938)Instructions burned: 2 (million)
% 3.50/1.91 % (6938)------------------------------
% 3.50/1.91 % (6938)------------------------------
% 3.50/1.91 % (6935)Instruction limit reached!
% 3.50/1.91 % (6935)------------------------------
% 3.50/1.91 % (6935)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.91 % (6935)Termination reason: Unknown
% 3.50/1.91 % (6935)Termination phase: shuffling
% 3.50/1.91
% 3.50/1.91 % (6935)Memory used [KB]: 1407
% 3.50/1.91 % (6935)Time elapsed: 0.004 s
% 3.50/1.91 % (6935)Instructions burned: 4 (million)
% 3.50/1.91 % (6935)------------------------------
% 3.50/1.91 % (6935)------------------------------
% 3.50/1.91 % (6940)Instruction limit reached!
% 3.50/1.91 % (6940)------------------------------
% 3.50/1.91 % (6940)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.91 % (6940)Termination reason: Unknown
% 3.50/1.91 % (6940)Termination phase: shuffling
% 3.50/1.91
% 3.50/1.91 % (6940)Memory used [KB]: 1791
% 3.50/1.91 % (6940)Time elapsed: 0.013 s
% 3.50/1.91 % (6940)Instructions burned: 19 (million)
% 3.50/1.91 % (6940)------------------------------
% 3.50/1.91 % (6940)------------------------------
% 3.50/1.92 % (6936)Instruction limit reached!
% 3.50/1.92 % (6936)------------------------------
% 3.50/1.92 % (6936)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.92 % (6936)Termination reason: Unknown
% 3.50/1.92 % (6936)Termination phase: Property scanning
% 3.50/1.92
% 3.50/1.92 % (6936)Memory used [KB]: 1791
% 3.50/1.92 % (6936)Time elapsed: 0.016 s
% 3.50/1.92 % (6936)Instructions burned: 27 (million)
% 3.50/1.92 % (6936)------------------------------
% 3.50/1.92 % (6936)------------------------------
% 3.50/1.92 % (6942)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_6822 for (2999ds/37Mi)
% 3.50/1.92 % (6941)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_6822 for (2999ds/3Mi)
% 3.50/1.92 % (6943)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_6822 for (2999ds/15Mi)
% 3.50/1.92 % (6941)Instruction limit reached!
% 3.50/1.92 % (6941)------------------------------
% 3.50/1.92 % (6941)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.92 % (6941)Termination reason: Unknown
% 3.50/1.92 % (6941)Termination phase: shuffling
% 3.50/1.92
% 3.50/1.92 % (6941)Memory used [KB]: 1407
% 3.50/1.92 % (6941)Time elapsed: 0.004 s
% 3.50/1.92 % (6941)Instructions burned: 4 (million)
% 3.50/1.92 % (6941)------------------------------
% 3.50/1.92 % (6941)------------------------------
% 3.50/1.93 % (6943)Instruction limit reached!
% 3.50/1.93 % (6943)------------------------------
% 3.50/1.93 % (6943)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.93 % (6943)Termination reason: Unknown
% 3.50/1.93 % (6943)Termination phase: shuffling
% 3.50/1.93
% 3.50/1.93 % (6943)Memory used [KB]: 1663
% 3.50/1.93 % (6943)Time elapsed: 0.010 s
% 3.50/1.93 % (6943)Instructions burned: 15 (million)
% 3.50/1.93 % (6943)------------------------------
% 3.50/1.93 % (6943)------------------------------
% 3.50/1.93 % (6944)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_6822 for (2999ds/3Mi)
% 3.50/1.93 % (6944)Instruction limit reached!
% 3.50/1.93 % (6944)------------------------------
% 3.50/1.93 % (6944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.93 % (6944)Termination reason: Unknown
% 3.50/1.93 % (6944)Termination phase: shuffling
% 3.50/1.93
% 3.50/1.93 % (6944)Memory used [KB]: 1407
% 3.50/1.93 % (6944)Time elapsed: 0.004 s
% 3.50/1.93 % (6944)Instructions burned: 4 (million)
% 3.50/1.93 % (6944)------------------------------
% 3.50/1.93 % (6944)------------------------------
% 3.50/1.93 % (6945)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_6822 for (2999ds/1041Mi)
% 3.50/1.93 % (6939)First to succeed.
% 3.50/1.94 % (6939)Refutation found. Thanks to Tanya!
% 3.50/1.94 % SZS status Theorem for DTF2THF_6822
% 3.50/1.94 % SZS output start Proof for DTF2THF_6822
% 3.50/1.94 thf(type_def_5, type, countable_Set_cset: ($tType) > $tType).
% 3.50/1.94 thf(type_def_7, type, monomo1453983439ondetT: ($tType * $tType) > $tType).
% 3.50/1.94 thf(type_def_8, type, c: $tType).
% 3.50/1.94 thf(type_def_9, type, a: $tType).
% 3.50/1.94 thf(type_def_10, type, m: $tType).
% 3.50/1.94 thf(func_def_0, type, monomo1453983439ondetT: ($tType * $tType) > $tType).
% 3.50/1.94 thf(func_def_1, type, countable_Set_cset: ($tType) > $tType).
% 3.50/1.94 thf(func_def_2, type, m: $tType).
% 3.50/1.94 thf(func_def_3, type, c: $tType).
% 3.50/1.94 thf(func_def_4, type, a: $tType).
% 3.50/1.94 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_6, type, minus: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_7, type, bot: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_10, type, group_add: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_11, type, no_bot: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_12, type, no_top: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_13, type, linorder: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_14, type, preorder: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_15, type, order_bot: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_16, type, wellorder: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_17, type, dense_order: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_18, type, dense_linorder: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_19, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_20, type, cancel146912293up_add: !>[X0: $tType]:($o)).
% 3.50/1.94 thf(func_def_21, type, countable_Set_cin: !>[X0: $tType]:(X0 > countable_Set_cset @ X0 > $o)).
% 3.50/1.94 thf(func_def_22, type, counta2111716221insert: !>[X0: $tType]:(X0 > countable_Set_cset @ X0 > countable_Set_cset @ X0)).
% 3.50/1.94 thf(func_def_23, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 3.50/1.94 thf(func_def_24, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.50/1.94 thf(func_def_25, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.50/1.94 thf(func_def_26, type, monomo2011693592ondetM: !>[X0: $tType, X1: $tType, X2: $tType]:((countable_Set_cset @ X0 > X1) > (X1 > (countable_Set_cset @ X0 > X1) > X1) > (countable_Set_cset @ X0 > (X0 > X1) > X1) > (countable_Set_cset @ X2 > (X2 > X1) > X1) > $o)).
% 3.50/1.94 thf(func_def_27, type, monomo759963067axioms: !>[X0: $tType, X1: $tType, X2: $tType]:((countable_Set_cset @ X0 > X1) > (X1 > (countable_Set_cset @ X0 > X1) > X1) > (countable_Set_cset @ X0 > (X0 > X1) > X1) > (countable_Set_cset @ X2 > (X2 > X1) > X1) > $o)).
% 3.50/1.94 thf(func_def_28, type, monomo955616121nondet: !>[X0: $tType, X1: $tType, X2: $tType]:((countable_Set_cset @ X0 > (X0 > X1) > X1) > countable_Set_cset @ X0 > (X0 > monomo1453983439ondetT @ X2 @ X1) > monomo1453983439ondetT @ X2 @ X1)).
% 3.50/1.94 thf(func_def_29, type, monomo439771545d_altc: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X1 > (X0 > X1) > X1) > (countable_Set_cset @ X2 > (X2 > X1) > X1) > $o)).
% 3.50/1.94 thf(func_def_30, type, monomo1425736922_altc3: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X1 > (X0 > X1) > X1) > (countable_Set_cset @ X2 > (X2 > X1) > X1) > $o)).
% 3.50/1.94 thf(func_def_31, type, monomo555039548axioms: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > (X1 > X0) > X0) > (countable_Set_cset @ X2 > (X2 > X0) > X0) > $o)).
% 3.50/1.94 thf(func_def_32, type, monomo109450930_state: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X1 > (X0 > X1) > X1) > ((X2 > X1) > X1) > (X2 > X1 > X1) > $o)).
% 3.50/1.94 thf(func_def_33, type, monomo1036387116e_altc: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X1 > (X0 > X1) > X1) > ((X2 > X1) > X1) > (X2 > X1 > X1) > (countable_Set_cset @ X3 > (X3 > X1) > X1) > $o)).
% 3.50/1.94 thf(func_def_34, type, monomo1324021455axioms: !>[X0: $tType, X1: $tType, X2: $tType]:(((X0 > X1) > X1) > (X0 > X1 > X1) > (countable_Set_cset @ X2 > (X2 > X1) > X1) > $o)).
% 3.50/1.94 thf(func_def_35, type, monomo410456285nondet: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X1) > X0 > monomo1453983439ondetT @ X2 @ X1 > monomo1453983439ondetT @ X2 @ X1)).
% 3.50/1.94 thf(func_def_36, type, monomo1975189248ondetT: !>[X0: $tType, X1: $tType]:(X0 > monomo1453983439ondetT @ X1 @ X0)).
% 3.50/1.94 thf(func_def_37, type, monomo1564381680ondetT: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > monomo1453983439ondetT @ X2 @ X0 > X1)).
% 3.50/1.94 thf(func_def_38, type, monomo1291740684ondetT: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > monomo1453983439ondetT @ X2 @ X0 > X1)).
% 3.50/1.94 thf(func_def_39, type, monomo603716163nondet: !>[X0: $tType, X1: $tType]:(monomo1453983439ondetT @ X0 @ X1 > X1)).
% 3.50/1.94 thf(func_def_40, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.50/1.94 thf(func_def_41, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.50/1.94 thf(func_def_42, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.50/1.94 thf(func_def_43, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.50/1.94 thf(func_def_44, type, f: c > monomo1453983439ondetT @ a @ m).
% 3.50/1.94 thf(func_def_45, type, mergec: countable_Set_cset @ c > (c > m) > m).
% 3.50/1.94 thf(func_def_46, type, x: c).
% 3.50/1.94 thf(func_def_63, type, sK0: !>[X0: $tType]:(countable_Set_cset @ X0 > X0)).
% 3.50/1.94 thf(func_def_64, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X0) > (X1 > X0) > X2 > X1)).
% 3.50/1.94 thf(func_def_65, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X0) > (X1 > X0) > X2)).
% 3.50/1.94 thf(func_def_66, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType]:(X2 > (X2 > X0) > (X1 > X0) > X1)).
% 3.50/1.94 thf(func_def_67, type, sK4: !>[X0: $tType, X1: $tType]:((monomo1453983439ondetT @ X1 @ X0 > $o) > X0)).
% 3.50/1.94 thf(func_def_68, type, sK5: !>[X0: $tType]:(countable_Set_cset @ X0 > X0 > countable_Set_cset @ X0)).
% 3.50/1.94 thf(func_def_69, type, sK6: !>[X0: $tType]:(countable_Set_cset @ X0 > countable_Set_cset @ X0 > X0)).
% 3.50/1.94 thf(func_def_70, type, sK7: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.50/1.94 thf(func_def_71, type, sK8: !>[X0: $tType]:(countable_Set_cset @ X0 > countable_Set_cset @ X0 > X0)).
% 3.50/1.94 thf(func_def_72, type, sK9: !>[X0: $tType]:(countable_Set_cset @ X0 > X0)).
% 3.50/1.94 thf(func_def_73, type, sK10: !>[X0: $tType]:(countable_Set_cset @ X0 > X0 > countable_Set_cset @ X0)).
% 3.50/1.94 thf(func_def_74, type, sK11: !>[X0: $tType, X1: $tType]:(monomo1453983439ondetT @ X0 @ X1 > X1)).
% 3.50/1.94 thf(func_def_75, type, sK12: !>[X0: $tType]:(countable_Set_cset @ X0 > X0)).
% 3.50/1.94 thf(func_def_76, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X0 > X2) > X2 > X1)).
% 3.50/1.94 thf(func_def_77, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X0 > X2) > X0)).
% 3.50/1.94 thf(func_def_78, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X0 > X2) > X0)).
% 3.50/1.94 thf(func_def_80, type, ph17: !>[X0: $tType]:(X0)).
% 3.50/1.94 thf(f1095,plain,(
% 3.50/1.94 $false),
% 3.50/1.94 inference(subsumption_resolution,[status(thm)],[f1094,f918])).
% 3.50/1.94 thf(f918,plain,(
% 3.50/1.94 ( ! [X1 : $tType,X0 : $tType,X2 : monomo1453983439ondetT @ X1 @ X0] : (((monomo1975189248ondetT @ X0 @ X1 @ (monomo603716163nondet @ X1 @ X0 @ X2)) = X2)) )),
% 3.50/1.94 inference(cnf_transformation,[status(thm)],[f795])).
% 3.50/1.94 thf(f795,plain,(
% 3.50/1.94 ! [X0 : $tType,X1 : $tType,X2 : monomo1453983439ondetT @ X1 @ X0] : ((monomo1975189248ondetT @ X0 @ X1 @ (monomo603716163nondet @ X1 @ X0 @ X2)) = X2)),
% 3.50/1.94 inference(rectify,[status(thm)],[f32])).
% 3.50/1.94 thf(f32,axiom,(
% 3.50/1.94 ! [X3 : $tType,X0 : $tType,X21 : monomo1453983439ondetT @ X0 @ X3] : ((monomo1975189248ondetT @ X3 @ X0 @ (monomo603716163nondet @ X0 @ X3 @ X21)) = X21)),
% 3.50/1.94 file('/export/starexec/sandbox/tmp/tmp.fjdGAlfEYi/DTF2THF_6822.p',fact_31_nondetT_Oexhaust__sel)).
% 3.50/1.94 thf(f1094,plain,(
% 3.50/1.94 ((f @ x) != (monomo1975189248ondetT @ m @ a @ (monomo603716163nondet @ a @ m @ (f @ x))))),
% 3.50/1.94 inference(beta_eta_normalization,[status(thm)],[f1093])).
% 3.50/1.94 thf(f1093,plain,(
% 3.50/1.94 ((f @ x) != (monomo1975189248ondetT @ m @ a @ ((^[Y0 : c]: (monomo603716163nondet @ a @ m @ (f @ Y0))) @ x)))),
% 3.50/1.94 inference(forward_demodulation,[status(thm)],[f1092,f983])).
% 3.50/1.94 thf(f983,plain,(
% 3.50/1.94 ( ! [X0 : c,X1 : c > m] : (((X1 @ X0) = (mergec @ (counta2111716221insert @ c @ X0 @ bot_bot @ countable_Set_cset @ c) @ X1))) )),
% 3.50/1.94 inference(cnf_transformation,[status(thm)],[f760])).
% 3.50/1.94 thf(f760,plain,(
% 3.50/1.94 ! [X0 : c,X1 : c > m] : ((X1 @ X0) = (mergec @ (counta2111716221insert @ c @ X0 @ bot_bot @ countable_Set_cset @ c) @ X1))),
% 3.50/1.94 inference(rectify,[status(thm)],[f1])).
% 3.50/1.94 thf(f1,axiom,(
% 3.50/1.94 ! [X7 : c,X8 : c > m] : ((mergec @ (counta2111716221insert @ c @ X7 @ bot_bot @ countable_Set_cset @ c) @ X8) = (X8 @ X7))),
% 3.50/1.94 file('/export/starexec/sandbox/tmp/tmp.fjdGAlfEYi/DTF2THF_6822.p',fact_0_mergec__single)).
% 3.50/1.94 thf(f1092,plain,(
% 3.50/1.94 ((f @ x) != (monomo1975189248ondetT @ m @ a @ (mergec @ (counta2111716221insert @ c @ x @ bot_bot @ countable_Set_cset @ c) @ (^[Y0 : c]: (monomo603716163nondet @ a @ m @ (f @ Y0))))))),
% 3.50/1.94 inference(beta_eta_normalization,[status(thm)],[f1020])).
% 3.50/1.94 thf(f1020,plain,(
% 3.50/1.94 ((f @ x) != ((^[Y0 : countable_Set_cset @ c > (c > m) > m]: ((^[Y1 : countable_Set_cset @ c]: ((^[Y2 : c > monomo1453983439ondetT @ a @ m]: (monomo1975189248ondetT @ m @ a @ (Y0 @ Y1 @ ((^[Y3 : monomo1453983439ondetT @ a @ m > m]: ((^[Y4 : c > monomo1453983439ondetT @ a @ m]: ((^[Y5 : c]: (Y3 @ (Y4 @ Y5))))))) @ monomo603716163nondet @ a @ m @ Y2)))))))) @ mergec @ (counta2111716221insert @ c @ x @ bot_bot @ countable_Set_cset @ c) @ f))),
% 3.50/1.94 inference(definition_unfolding,[status(thm)],[f920,f1015])).
% 3.50/1.94 thf(f1015,plain,(
% 3.50/1.94 ( ! [X1 : $tType,X0 : $tType,X2 : $tType] : ((monomo955616121nondet @ X2 @ X1 @ X0 = (^[Y0 : countable_Set_cset @ X2 > (X2 > X1) > X1]: ((^[Y1 : countable_Set_cset @ X2]: ((^[Y2 : X2 > monomo1453983439ondetT @ X0 @ X1]: (monomo1975189248ondetT @ X1 @ X0 @ (Y0 @ Y1 @ ((^[Y3 : monomo1453983439ondetT @ X0 @ X1 > X1]: ((^[Y4 : X2 > monomo1453983439ondetT @ X0 @ X1]: ((^[Y5 : X2]: (Y3 @ (Y4 @ Y5))))))) @ monomo603716163nondet @ X0 @ X1 @ Y2)))))))))) )),
% 3.50/1.94 inference(definition_unfolding,[status(thm)],[f947,f908])).
% 3.50/1.94 thf(f908,plain,(
% 3.50/1.94 ( ! [X1 : $tType,X0 : $tType,X2 : $tType] : ((comp @ X2 @ X1 @ X0 = (^[Y0 : X2 > X1]: ((^[Y1 : X0 > X2]: ((^[Y2 : X0]: (Y0 @ (Y1 @ Y2))))))))) )),
% 3.50/1.94 inference(cnf_transformation,[status(thm)],[f647])).
% 3.50/1.94 thf(f647,plain,(
% 3.50/1.94 ! [X0 : $tType,X1 : $tType,X2 : $tType] : (comp @ X2 @ X1 @ X0 = (^[Y0 : X2 > X1]: ((^[Y1 : X0 > X2]: ((^[Y2 : X0]: (Y0 @ (Y1 @ Y2))))))))),
% 3.50/1.94 inference(fool_elimination,[status(thm)],[f646])).
% 3.50/1.94 thf(f646,plain,(
% 3.50/1.94 ! [X0 : $tType,X1 : $tType,X2 : $tType] : ((^[X3 : X2 > X1, X4 : X0 > X2, X5 : X0] : (X3 @ (X4 @ X5))) = comp @ X2 @ X1 @ X0)),
% 3.50/1.94 inference(rectify,[status(thm)],[f21])).
% 3.50/1.94 thf(f21,axiom,(
% 3.50/1.94 ! [X0 : $tType,X2 : $tType,X1 : $tType] : ((^[X23 : X1 > X2, X24 : X0 > X1, X10 : X0] : (X23 @ (X24 @ X10))) = comp @ X1 @ X2 @ X0)),
% 3.50/1.94 file('/export/starexec/sandbox/tmp/tmp.fjdGAlfEYi/DTF2THF_6822.p',fact_20_comp__def)).
% 3.50/1.94 thf(f947,plain,(
% 3.50/1.94 ( ! [X1 : $tType,X0 : $tType,X2 : $tType] : ((monomo955616121nondet @ X2 @ X1 @ X0 = (^[Y0 : countable_Set_cset @ X2 > (X2 > X1) > X1]: ((^[Y1 : countable_Set_cset @ X2]: ((^[Y2 : X2 > monomo1453983439ondetT @ X0 @ X1]: (monomo1975189248ondetT @ X1 @ X0 @ (Y0 @ Y1 @ (comp @ monomo1453983439ondetT @ X0 @ X1 @ X1 @ X2 @ monomo603716163nondet @ X0 @ X1 @ Y2)))))))))) )),
% 3.50/1.94 inference(cnf_transformation,[status(thm)],[f665])).
% 3.50/1.94 thf(f665,plain,(
% 3.50/1.94 ! [X0 : $tType,X1 : $tType,X2 : $tType] : (monomo955616121nondet @ X2 @ X1 @ X0 = (^[Y0 : countable_Set_cset @ X2 > (X2 > X1) > X1]: ((^[Y1 : countable_Set_cset @ X2]: ((^[Y2 : X2 > monomo1453983439ondetT @ X0 @ X1]: (monomo1975189248ondetT @ X1 @ X0 @ (Y0 @ Y1 @ (comp @ monomo1453983439ondetT @ X0 @ X1 @ X1 @ X2 @ monomo603716163nondet @ X0 @ X1 @ Y2)))))))))),
% 3.50/1.94 inference(fool_elimination,[status(thm)],[f664])).
% 3.50/1.94 thf(f664,plain,(
% 3.50/1.94 ! [X0 : $tType,X1 : $tType,X2 : $tType] : (monomo955616121nondet @ X2 @ X1 @ X0 = (^[X3 : countable_Set_cset @ X2 > (X2 > X1) > X1, X4 : countable_Set_cset @ X2, X5 : X2 > monomo1453983439ondetT @ X0 @ X1] : (monomo1975189248ondetT @ X1 @ X0 @ (X3 @ X4 @ (comp @ monomo1453983439ondetT @ X0 @ X1 @ X1 @ X2 @ monomo603716163nondet @ X0 @ X1 @ X5)))))),
% 3.50/1.94 inference(rectify,[status(thm)],[f19])).
% 3.50/1.94 thf(f19,axiom,(
% 3.50/1.94 ! [X0 : $tType,X3 : $tType,X2 : $tType] : (monomo955616121nondet @ X2 @ X3 @ X0 = (^[X25 : countable_Set_cset @ X2 > (X2 > X3) > X3, X26 : countable_Set_cset @ X2, X23 : X2 > monomo1453983439ondetT @ X0 @ X3] : (monomo1975189248ondetT @ X3 @ X0 @ (X25 @ X26 @ (comp @ monomo1453983439ondetT @ X0 @ X3 @ X3 @ X2 @ monomo603716163nondet @ X0 @ X3 @ X23)))))),
% 3.50/1.94 file('/export/starexec/sandbox/tmp/tmp.fjdGAlfEYi/DTF2THF_6822.p',fact_18_cset__nondetM__base_Oaltc__nondet__def)).
% 3.50/1.94 thf(f920,plain,(
% 3.50/1.94 ((monomo955616121nondet @ c @ m @ a @ mergec @ (counta2111716221insert @ c @ x @ bot_bot @ countable_Set_cset @ c) @ f) != (f @ x))),
% 3.50/1.94 inference(cnf_transformation,[status(thm)],[f756])).
% 3.50/1.94 thf(f756,plain,(
% 3.50/1.94 ((monomo955616121nondet @ c @ m @ a @ mergec @ (counta2111716221insert @ c @ x @ bot_bot @ countable_Set_cset @ c) @ f) != (f @ x))),
% 3.50/1.94 inference(flattening,[status(thm)],[f280])).
% 3.50/1.94 thf(f280,negated_conjecture,(
% 3.50/1.94 ~((monomo955616121nondet @ c @ m @ a @ mergec @ (counta2111716221insert @ c @ x @ bot_bot @ countable_Set_cset @ c) @ f) = (f @ x))),
% 3.50/1.94 inference(negated_conjecture,[status(cth)],[f279])).
% 3.50/1.94 thf(f279,conjecture,(
% 3.50/1.94 ((monomo955616121nondet @ c @ m @ a @ mergec @ (counta2111716221insert @ c @ x @ bot_bot @ countable_Set_cset @ c) @ f) = (f @ x))),
% 3.50/1.94 file('/export/starexec/sandbox/tmp/tmp.fjdGAlfEYi/DTF2THF_6822.p',conj_0)).
% 3.50/1.94 % SZS output end Proof for DTF2THF_6822
% 3.50/1.94 % (6939)------------------------------
% 3.50/1.94 % (6939)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.50/1.94 % (6939)Termination reason: Refutation
% 3.50/1.94
% 3.50/1.94 % (6939)Memory used [KB]: 6652
% 3.50/1.94 % (6939)Time elapsed: 0.034 s
% 3.50/1.94 % (6939)Instructions burned: 58 (million)
% 3.50/1.94 % (6939)------------------------------
% 3.50/1.94 % (6939)------------------------------
% 3.50/1.94 % (6933)Success in time 0.062 s
%------------------------------------------------------------------------------