%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT157^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n010.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 08:44:58 AM UTC 2026
% Result : Theorem 2.55s 1.54s
% Output : Refutation 2.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : DAT157^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.32 % Computer : n010.cluster.edu
% 0.13/0.32 % Model : x86_64 x86_64
% 0.13/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.32 % Memory : 8042.1875MB
% 0.13/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.32 % CPULimit : 300
% 0.13/0.32 % WCLimit : 300
% 0.13/0.32 % DateTime : Tue Feb 24 06:09:33 EST 2026
% 0.13/0.32 % CPUTime :
% 0.13/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.42 ---- Original DTF file ---
% 0.14/0.42 thf(spec,logic,$$dhol).
% 0.14/0.42 %------------------------------------------------------------------------------
% 0.14/0.42 % File : DAT157^1 : TPTP v9.2.1. Released v7.0.0.
% 0.14/0.42 % Domain : Data Structures
% 0.14/0.42 % Problem : Hamming stream 157
% 0.14/0.42 % Version : [Bla16] axioms : Especial.
% 0.14/0.42 % English :
% 0.14/0.42
% 0.14/0.42 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.14/0.42 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.14/0.42 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.14/0.42 % Source : [Bla16]
% 0.14/0.42 % Names : hamming_stream__157.p [Bla16]
% 0.14/0.42
% 0.14/0.42 % Status : Theorem
% 0.14/0.42 % Rating : 0.00 v7.1.0
% 0.14/0.42 % Syntax : Number of formulae : 310 ( 100 unt; 48 typ; 0 def)
% 0.14/0.42 % Number of atoms : 897 ( 173 equ; 0 cnn)
% 0.14/0.42 % Maximal formula atoms : 12 ( 3 avg)
% 0.14/0.42 % Number of connectives : 3016 ( 83 ~; 23 |; 37 &;2599 @)
% 0.14/0.42 % ( 0 <=>; 274 =>; 0 <=; 0 <~>)
% 0.14/0.42 % Maximal formula depth : 17 ( 7 avg)
% 0.14/0.42 % Number of types : 2 ( 1 usr)
% 0.14/0.42 % Number of type conns : 196 ( 196 >; 0 *; 0 +; 0 <<)
% 0.14/0.42 % Number of symbols : 48 ( 47 usr; 5 con; 0-6 aty)
% 0.14/0.42 % Number of variables : 767 ( 34 ^; 688 !; 4 ?; 767 :)
% 0.14/0.42 % ( 41 !>; 0 ?*; 0 @-; 0 @+)
% 0.14/0.42 % SPC : TH1_THM_EQU_NAR
% 0.14/0.42
% 0.14/0.42 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.14/0.42 % 2016-07-13 14:41:50.421
% 0.14/0.42 %------------------------------------------------------------------------------
% 0.14/0.42 %----Could-be-implicit typings (5)
% 0.14/0.42 thf(ty_t_Coinductive__List_Ollist,type,
% 0.14/0.42 coinductive_llist: $tType > $tType ).
% 0.14/0.42
% 0.14/0.42 thf(ty_t_List_Olist,type,
% 0.14/0.42 list: $tType > $tType ).
% 0.14/0.42
% 0.14/0.42 thf(ty_t_Set_Oset,type,
% 0.14/0.42 set: $tType > $tType ).
% 0.14/0.42
% 0.14/0.42 thf(ty_t_itself,type,
% 0.14/0.42 itself: $tType > $tType ).
% 0.14/0.42
% 0.14/0.42 thf(ty_tf_a,type,
% 0.14/0.42 a: $tType ).
% 0.14/0.42
% 0.14/0.42 %----Explicit typings (43)
% 0.14/0.42 thf(sy_cl_HOL_Otype,type,
% 0.14/0.42 type:
% 0.14/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_cl_Orderings_Oorder,type,
% 0.14/0.42 order:
% 0.14/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_cl_Orderings_Olinorder,type,
% 0.14/0.42 linorder:
% 0.14/0.42 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Coinductive__List_Oldistinct,type,
% 0.14/0.42 coindu351974385stinct:
% 0.14/0.42 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.14/0.42 coinductive_lfinite:
% 0.14/0.42 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 0.14/0.42 coinductive_LCons:
% 0.14/0.42 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.14/0.42 coinductive_lset:
% 0.14/0.42 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Coinductive__List_Oord_Olsorted,type,
% 0.14/0.42 coinductive_lsorted:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
% 0.14/0.42 condit2040224947_above:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
% 0.14/0.42 condit1201339847_below:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Finite__Set_Ofolding,type,
% 0.14/0.42 finite_folding:
% 0.14/0.42 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Groups_Osemigroup,type,
% 0.14/0.42 semigroup:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord_Olmerge,type,
% 0.14/0.42 hammin1328233080lmerge:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.14/0.42 inf_inf:
% 0.14/0.42 !>[A: $tType] : ( A > A > A ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Lattices_Osemilattice,type,
% 0.14/0.42 semilattice:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
% 0.14/0.42 lattic1693879045er_set:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_List_Olinorder_Oinsort__key,type,
% 0.14/0.42 insort_key:
% 0.14/0.42 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Orderings_Oord_OLeast,type,
% 0.14/0.42 least:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Orderings_Oord_Omin,type,
% 0.14/0.42 min:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.14/0.42 ord_less_eq:
% 0.14/0.42 !>[A: $tType] : ( A > A > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Orderings_Oorder_Oantimono,type,
% 0.14/0.42 antimono:
% 0.14/0.42 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Orderings_Oorder_Omono,type,
% 0.14/0.42 mono:
% 0.14/0.42 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Orderings_Oorder_Ostrict__mono,type,
% 0.14/0.42 strict_mono:
% 0.14/0.42 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Orderings_Oordering,type,
% 0.14/0.42 ordering:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Pure_Otype,type,
% 0.14/0.42 type2:
% 0.14/0.42 !>[A: $tType] : ( itself @ A ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Relation_Otransp,type,
% 0.14/0.42 transp:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Set_OCollect,type,
% 0.14/0.42 collect:
% 0.14/0.42 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Set__Interval_Oord_OatLeast,type,
% 0.14/0.42 set_atLeast:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
% 0.14/0.42 set_atLeastAtMost:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Set__Interval_Oord_OatLeastLessThan,type,
% 0.14/0.42 set_atLeastLessThan:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Set__Interval_Oord_OatMost,type,
% 0.14/0.42 set_atMost:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Set__Interval_Oord_OgreaterThan,type,
% 0.14/0.42 set_greaterThan:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
% 0.14/0.42 set_gr323396891AtMost:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
% 0.14/0.42 set_gr1161524159ssThan:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_Set__Interval_Oord_OlessThan,type,
% 0.14/0.42 set_lessThan:
% 0.14/0.42 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_c_member,type,
% 0.14/0.42 member:
% 0.14/0.42 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.14/0.42
% 0.14/0.42 thf(sy_v_less,type,
% 0.14/0.42 less: a > a > $o ).
% 0.14/0.42
% 0.14/0.42 thf(sy_v_less__eq,type,
% 0.14/0.42 less_eq: a > a > $o ).
% 0.14/0.42
% 0.14/0.42 thf(sy_v_thesis____,type,
% 0.14/0.42 thesis: $o ).
% 0.14/0.42
% 0.14/0.42 thf(sy_v_x,type,
% 0.14/0.42 x: a ).
% 0.14/0.42
% 0.14/0.42 thf(sy_v_x_H____,type,
% 0.14/0.42 x2: a ).
% 0.14/0.42
% 0.14/0.42 thf(sy_v_xsa____,type,
% 0.14/0.42 xsa: coinductive_llist @ a ).
% 0.14/0.42
% 0.14/0.42 thf(sy_v_ysa____,type,
% 0.14/0.42 ysa: coinductive_llist @ a ).
% 0.14/0.42
% 0.14/0.42 %----Relevant facts (256)
% 0.14/0.42 thf(fact_0_local_Oantisym__conv3,axiom,
% 0.14/0.42 ! [Y: a,X: a] :
% 0.14/0.42 ( ~ ( less @ Y @ X )
% 0.14/0.42 => ( ( ~ ( less @ X @ Y ) )
% 0.14/0.42 = ( X = Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.antisym_conv3
% 0.14/0.42 thf(fact_1_local_Odual__order_Oasym,axiom,
% 0.14/0.42 ! [B2: a,A2: a] :
% 0.14/0.42 ( ( less @ B2 @ A2 )
% 0.14/0.42 => ~ ( less @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.asym
% 0.14/0.42 thf(fact_2_local_Odual__order_Ostrict__implies__not__eq,axiom,
% 0.14/0.42 ! [B2: a,A2: a] :
% 0.14/0.42 ( ( less @ B2 @ A2 )
% 0.14/0.42 => ( A2 != B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.strict_implies_not_eq
% 0.14/0.42 thf(fact_3_local_Odual__order_Ostrict__trans,axiom,
% 0.14/0.42 ! [B2: a,A2: a,C: a] :
% 0.14/0.42 ( ( less @ B2 @ A2 )
% 0.14/0.42 => ( ( less @ C @ B2 )
% 0.14/0.42 => ( less @ C @ A2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.strict_trans
% 0.14/0.42 thf(fact_4_local_Oless__asym,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ~ ( less @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_asym
% 0.14/0.42 thf(fact_5_local_Oless__asym_H,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less @ A2 @ B2 )
% 0.14/0.42 => ~ ( less @ B2 @ A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_asym'
% 0.14/0.42 thf(fact_6_local_Oless__imp__neq,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ( X != Y ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_imp_neq
% 0.14/0.42 thf(fact_7_local_Oless__imp__not__eq,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ( X != Y ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_imp_not_eq
% 0.14/0.42 thf(fact_8_local_Oless__imp__not__eq2,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ( Y != X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_imp_not_eq2
% 0.14/0.42 thf(fact_9_local_Oless__imp__not__less,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ~ ( less @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_imp_not_less
% 0.14/0.42 thf(fact_10_local_Oless__imp__triv,axiom,
% 0.14/0.42 ! [X: a,Y: a,P: $o] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ( ( less @ Y @ X )
% 0.14/0.42 => P ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_imp_triv
% 0.14/0.42 thf(fact_11_local_Oless__irrefl,axiom,
% 0.14/0.42 ! [X: a] :
% 0.14/0.42 ~ ( less @ X @ X ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_irrefl
% 0.14/0.42 thf(fact_12_local_Oless__linear,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 | ( X = Y )
% 0.14/0.42 | ( less @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_linear
% 0.14/0.42 thf(fact_13_local_Oless__not__sym,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ~ ( less @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_not_sym
% 0.14/0.42 thf(fact_14_local_Oless__trans,axiom,
% 0.14/0.42 ! [X: a,Y: a,Z: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ( ( less @ Y @ Z )
% 0.14/0.42 => ( less @ X @ Z ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_trans
% 0.14/0.42 thf(fact_15_local_Olinorder__cases,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ~ ( less @ X @ Y )
% 0.14/0.42 => ( ( X != Y )
% 0.14/0.42 => ( less @ Y @ X ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.linorder_cases
% 0.14/0.42 thf(fact_16_local_OneqE,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( X != Y )
% 0.14/0.42 => ( ~ ( less @ X @ Y )
% 0.14/0.42 => ( less @ Y @ X ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.neqE
% 0.14/0.42 thf(fact_17_local_Oneq__iff,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( X != Y )
% 0.14/0.42 = ( ( less @ X @ Y )
% 0.14/0.42 | ( less @ Y @ X ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.neq_iff
% 0.14/0.42 thf(fact_18_local_Onot__less__iff__gr__or__eq,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( ~ ( less @ X @ Y ) )
% 0.14/0.42 = ( ( less @ Y @ X )
% 0.14/0.42 | ( X = Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.not_less_iff_gr_or_eq
% 0.14/0.42 thf(fact_19_local_Oord__eq__less__trans,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( A2 = B2 )
% 0.14/0.42 => ( ( less @ B2 @ C )
% 0.14/0.42 => ( less @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.ord_eq_less_trans
% 0.14/0.42 thf(fact_20_local_Oord__less__eq__trans,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less @ A2 @ B2 )
% 0.14/0.42 => ( ( B2 = C )
% 0.14/0.42 => ( less @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.ord_less_eq_trans
% 0.14/0.42 thf(fact_21_local_Oorder_Oasym,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less @ A2 @ B2 )
% 0.14/0.42 => ~ ( less @ B2 @ A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.asym
% 0.14/0.42 thf(fact_22_local_Oorder_Oirrefl,axiom,
% 0.14/0.42 ! [A2: a] :
% 0.14/0.42 ~ ( less @ A2 @ A2 ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.irrefl
% 0.14/0.42 thf(fact_23_local_Oorder_Ostrict__implies__not__eq,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less @ A2 @ B2 )
% 0.14/0.42 => ( A2 != B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.strict_implies_not_eq
% 0.14/0.42 thf(fact_24_local_Oorder_Ostrict__trans,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less @ A2 @ B2 )
% 0.14/0.42 => ( ( less @ B2 @ C )
% 0.14/0.42 => ( less @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.strict_trans
% 0.14/0.42 thf(fact_25_step_Ohyps_I2_J,axiom,
% 0.14/0.42 x != x2 ).
% 0.14/0.42
% 0.14/0.42 % step.hyps(2)
% 0.14/0.42 thf(fact_26_step_Oprems_I2_J,axiom,
% 0.14/0.42 ~ ( coinductive_lfinite @ a @ ysa ) ).
% 0.14/0.42
% 0.14/0.42 % step.prems(2)
% 0.14/0.42 thf(fact_27_step_Ohyps_I1_J,axiom,
% 0.14/0.42 member @ a @ x @ ( coinductive_lset @ a @ xsa ) ).
% 0.14/0.42
% 0.14/0.42 % step.hyps(1)
% 0.14/0.42 thf(fact_28_local_Oantisym,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ( less_eq @ Y @ X )
% 0.14/0.42 => ( X = Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.antisym
% 0.14/0.42 thf(fact_29_local_Oantisym__conv,axiom,
% 0.14/0.42 ! [Y: a,X: a] :
% 0.14/0.42 ( ( less_eq @ Y @ X )
% 0.14/0.42 => ( ( less_eq @ X @ Y )
% 0.14/0.42 = ( X = Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.antisym_conv
% 0.14/0.42 thf(fact_30_local_Odual__order_Oantisym,axiom,
% 0.14/0.42 ! [B2: a,A2: a] :
% 0.14/0.42 ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 => ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( A2 = B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.antisym
% 0.14/0.42 thf(fact_31_local_Odual__order_Otrans,axiom,
% 0.14/0.42 ! [B2: a,A2: a,C: a] :
% 0.14/0.42 ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 => ( ( less_eq @ C @ B2 )
% 0.14/0.42 => ( less_eq @ C @ A2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.trans
% 0.14/0.42 thf(fact_32_local_Oeq__iff,axiom,
% 0.14/0.42 ( ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 ) )
% 0.14/0.42 = ( ^ [X2: a,Y3: a] :
% 0.14/0.42 ( ( less_eq @ X2 @ Y3 )
% 0.14/0.42 & ( less_eq @ Y3 @ X2 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.eq_iff
% 0.14/0.42 thf(fact_33_local_Oeq__refl,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( X = Y )
% 0.14/0.42 => ( less_eq @ X @ Y ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.eq_refl
% 0.14/0.42 thf(fact_34_local_Ole__cases,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ~ ( less_eq @ X @ Y )
% 0.14/0.42 => ( less_eq @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.le_cases
% 0.14/0.42 thf(fact_35_local_Ole__cases3,axiom,
% 0.14/0.42 ! [X: a,Y: a,Z: a] :
% 0.14/0.42 ( ( ( less_eq @ X @ Y )
% 0.14/0.42 => ~ ( less_eq @ Y @ Z ) )
% 0.14/0.42 => ( ( ( less_eq @ Y @ X )
% 0.14/0.42 => ~ ( less_eq @ X @ Z ) )
% 0.14/0.42 => ( ( ( less_eq @ X @ Z )
% 0.14/0.42 => ~ ( less_eq @ Z @ Y ) )
% 0.14/0.42 => ( ( ( less_eq @ Z @ Y )
% 0.14/0.42 => ~ ( less_eq @ Y @ X ) )
% 0.14/0.42 => ( ( ( less_eq @ Y @ Z )
% 0.14/0.42 => ~ ( less_eq @ Z @ X ) )
% 0.14/0.42 => ~ ( ( less_eq @ Z @ X )
% 0.14/0.42 => ~ ( less_eq @ X @ Y ) ) ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.le_cases3
% 0.14/0.42 thf(fact_36_local_Olinear,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less_eq @ X @ Y )
% 0.14/0.42 | ( less_eq @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.linear
% 0.14/0.42 thf(fact_37_local_Olinorder__wlog,axiom,
% 0.14/0.42 ! [P: a > a > $o,A2: a,B2: a] :
% 0.14/0.42 ( ! [A3: a,B3: a] :
% 0.14/0.42 ( ( less_eq @ A3 @ B3 )
% 0.14/0.42 => ( P @ A3 @ B3 ) )
% 0.14/0.42 => ( ! [A3: a,B3: a] :
% 0.14/0.42 ( ( P @ B3 @ A3 )
% 0.14/0.42 => ( P @ A3 @ B3 ) )
% 0.14/0.42 => ( P @ A2 @ B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.linorder_wlog
% 0.14/0.42 thf(fact_38_local_Oord__eq__le__trans,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( A2 = B2 )
% 0.14/0.42 => ( ( less_eq @ B2 @ C )
% 0.14/0.42 => ( less_eq @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.ord_eq_le_trans
% 0.14/0.42 thf(fact_39_local_Oord__le__eq__trans,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( ( B2 = C )
% 0.14/0.42 => ( less_eq @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.ord_le_eq_trans
% 0.14/0.42 thf(fact_40_local_Oorder_Oantisym,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 => ( A2 = B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.antisym
% 0.14/0.42 thf(fact_41_local_Oorder_Otrans,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( ( less_eq @ B2 @ C )
% 0.14/0.42 => ( less_eq @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.trans
% 0.14/0.42 thf(fact_42_local_Oorder__trans,axiom,
% 0.14/0.42 ! [X: a,Y: a,Z: a] :
% 0.14/0.42 ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ( less_eq @ Y @ Z )
% 0.14/0.42 => ( less_eq @ X @ Z ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order_trans
% 0.14/0.42 thf(fact_43_step_Oprems_I3_J,axiom,
% 0.14/0.42 ? [X3: a] :
% 0.14/0.42 ( ( member @ a @ X3 @ ( coinductive_lset @ a @ ysa ) )
% 0.14/0.42 & ( less_eq @ x @ X3 ) ) ).
% 0.14/0.42
% 0.14/0.42 % step.prems(3)
% 0.14/0.42 thf(fact_44_local_Oantisym__conv1,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ~ ( less @ X @ Y )
% 0.14/0.42 => ( ( less_eq @ X @ Y )
% 0.14/0.42 = ( X = Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.antisym_conv1
% 0.14/0.42 thf(fact_45_mem__Collect__eq,axiom,
% 0.14/0.42 ! [A: $tType,A2: A,P: A > $o] :
% 0.14/0.42 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.14/0.42 = ( P @ A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % mem_Collect_eq
% 0.14/0.42 thf(fact_46_Collect__mem__eq,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A] :
% 0.14/0.42 ( ( collect @ A
% 0.14/0.42 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 0.14/0.42 = A4 ) ).
% 0.14/0.42
% 0.14/0.42 % Collect_mem_eq
% 0.14/0.42 thf(fact_47_Collect__cong,axiom,
% 0.14/0.42 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.14/0.42 ( ! [X3: A] :
% 0.14/0.42 ( ( P @ X3 )
% 0.14/0.42 = ( Q @ X3 ) )
% 0.14/0.42 => ( ( collect @ A @ P )
% 0.14/0.42 = ( collect @ A @ Q ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % Collect_cong
% 0.14/0.42 thf(fact_48_ext,axiom,
% 0.14/0.42 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.14/0.42 ( ! [X3: A] :
% 0.14/0.42 ( ( F @ X3 )
% 0.14/0.42 = ( G @ X3 ) )
% 0.14/0.42 => ( F = G ) ) ).
% 0.14/0.42
% 0.14/0.42 % ext
% 0.14/0.42 thf(fact_49_local_Oantisym__conv2,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ( ~ ( less @ X @ Y ) )
% 0.14/0.42 = ( X = Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.antisym_conv2
% 0.14/0.42 thf(fact_50_local_Odual__order_Onot__eq__order__implies__strict,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( A2 != B2 )
% 0.14/0.42 => ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 => ( less @ B2 @ A2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.not_eq_order_implies_strict
% 0.14/0.42 thf(fact_51_local_Odual__order_Oorder__iff__strict,axiom,
% 0.14/0.42 ! [B2: a,A2: a] :
% 0.14/0.42 ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 = ( ( less @ B2 @ A2 )
% 0.14/0.42 | ( A2 = B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.order_iff_strict
% 0.14/0.42 thf(fact_52_local_Odual__order_Ostrict__iff__order,axiom,
% 0.14/0.42 ! [B2: a,A2: a] :
% 0.14/0.42 ( ( less @ B2 @ A2 )
% 0.14/0.42 = ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 & ( A2 != B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.strict_iff_order
% 0.14/0.42 thf(fact_53_local_Odual__order_Ostrict__implies__order,axiom,
% 0.14/0.42 ! [B2: a,A2: a] :
% 0.14/0.42 ( ( less @ B2 @ A2 )
% 0.14/0.42 => ( less_eq @ B2 @ A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.strict_implies_order
% 0.14/0.42 thf(fact_54_local_Odual__order_Ostrict__trans1,axiom,
% 0.14/0.42 ! [B2: a,A2: a,C: a] :
% 0.14/0.42 ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 => ( ( less @ C @ B2 )
% 0.14/0.42 => ( less @ C @ A2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.strict_trans1
% 0.14/0.42 thf(fact_55_local_Odual__order_Ostrict__trans2,axiom,
% 0.14/0.42 ! [B2: a,A2: a,C: a] :
% 0.14/0.42 ( ( less @ B2 @ A2 )
% 0.14/0.42 => ( ( less_eq @ C @ B2 )
% 0.14/0.42 => ( less @ C @ A2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.strict_trans2
% 0.14/0.42 thf(fact_56_local_OleD,axiom,
% 0.14/0.42 ! [Y: a,X: a] :
% 0.14/0.42 ( ( less_eq @ Y @ X )
% 0.14/0.42 => ~ ( less @ X @ Y ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.leD
% 0.14/0.42 thf(fact_57_local_OleI,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ~ ( less @ X @ Y )
% 0.14/0.42 => ( less_eq @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.leI
% 0.14/0.42 thf(fact_58_local_Ole__imp__less__or__eq,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ( less @ X @ Y )
% 0.14/0.42 | ( X = Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.le_imp_less_or_eq
% 0.14/0.42 thf(fact_59_local_Ole__less,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less_eq @ X @ Y )
% 0.14/0.42 = ( ( less @ X @ Y )
% 0.14/0.42 | ( X = Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.le_less
% 0.14/0.42 thf(fact_60_local_Ole__less__linear,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less_eq @ X @ Y )
% 0.14/0.42 | ( less @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.le_less_linear
% 0.14/0.42 thf(fact_61_local_Ole__less__trans,axiom,
% 0.14/0.42 ! [X: a,Y: a,Z: a] :
% 0.14/0.42 ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ( less @ Y @ Z )
% 0.14/0.42 => ( less @ X @ Z ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.le_less_trans
% 0.14/0.42 thf(fact_62_local_Ole__neq__trans,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( ( A2 != B2 )
% 0.14/0.42 => ( less @ A2 @ B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.le_neq_trans
% 0.14/0.42 thf(fact_63_local_Oless__imp__le,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ( less_eq @ X @ Y ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_imp_le
% 0.14/0.42 thf(fact_64_local_Oless__le,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 = ( ( less_eq @ X @ Y )
% 0.14/0.42 & ( X != Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_le
% 0.14/0.42 thf(fact_65_local_Oless__le__not__le,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 = ( ( less_eq @ X @ Y )
% 0.14/0.42 & ~ ( less_eq @ Y @ X ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_le_not_le
% 0.14/0.42 thf(fact_66_local_Oless__le__trans,axiom,
% 0.14/0.42 ! [X: a,Y: a,Z: a] :
% 0.14/0.42 ( ( less @ X @ Y )
% 0.14/0.42 => ( ( less_eq @ Y @ Z )
% 0.14/0.42 => ( less @ X @ Z ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.less_le_trans
% 0.14/0.42 thf(fact_67_local_Onot__le,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( ~ ( less_eq @ X @ Y ) )
% 0.14/0.42 = ( less @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.not_le
% 0.14/0.42 thf(fact_68_local_Onot__le__imp__less,axiom,
% 0.14/0.42 ! [Y: a,X: a] :
% 0.14/0.42 ( ~ ( less_eq @ Y @ X )
% 0.14/0.42 => ( less @ X @ Y ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.not_le_imp_less
% 0.14/0.42 thf(fact_69_local_Onot__less,axiom,
% 0.14/0.42 ! [X: a,Y: a] :
% 0.14/0.42 ( ( ~ ( less @ X @ Y ) )
% 0.14/0.42 = ( less_eq @ Y @ X ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.not_less
% 0.14/0.42 thf(fact_70_local_Oorder_Onot__eq__order__implies__strict,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( A2 != B2 )
% 0.14/0.42 => ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( less @ A2 @ B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.not_eq_order_implies_strict
% 0.14/0.42 thf(fact_71_local_Oorder_Oorder__iff__strict,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 = ( ( less @ A2 @ B2 )
% 0.14/0.42 | ( A2 = B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.order_iff_strict
% 0.14/0.42 thf(fact_72_local_Oorder_Ostrict__iff__order,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less @ A2 @ B2 )
% 0.14/0.42 = ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 & ( A2 != B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.strict_iff_order
% 0.14/0.42 thf(fact_73_local_Oorder_Ostrict__implies__order,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less @ A2 @ B2 )
% 0.14/0.42 => ( less_eq @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.strict_implies_order
% 0.14/0.42 thf(fact_74_local_Oorder_Ostrict__trans1,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( ( less @ B2 @ C )
% 0.14/0.42 => ( less @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.strict_trans1
% 0.14/0.42 thf(fact_75_local_Oorder_Ostrict__trans2,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less @ A2 @ B2 )
% 0.14/0.42 => ( ( less_eq @ B2 @ C )
% 0.14/0.42 => ( less @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.strict_trans2
% 0.14/0.42 thf(fact_76_local_Oorder_Orefl,axiom,
% 0.14/0.42 ! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% 0.14/0.42
% 0.14/0.42 % local.order.refl
% 0.14/0.42 thf(fact_77_local_Oorder__refl,axiom,
% 0.14/0.42 ! [X: a] : ( less_eq @ X @ X ) ).
% 0.14/0.42
% 0.14/0.42 % local.order_refl
% 0.14/0.42 thf(fact_78_local_OLeastI2__order,axiom,
% 0.14/0.42 ! [P: a > $o,X: a,Q: a > $o] :
% 0.14/0.42 ( ( P @ X )
% 0.14/0.42 => ( ! [Y4: a] :
% 0.14/0.42 ( ( P @ Y4 )
% 0.14/0.42 => ( less_eq @ X @ Y4 ) )
% 0.14/0.42 => ( ! [X3: a] :
% 0.14/0.42 ( ( P @ X3 )
% 0.14/0.42 => ( ! [Y5: a] :
% 0.14/0.42 ( ( P @ Y5 )
% 0.14/0.42 => ( less_eq @ X3 @ Y5 ) )
% 0.14/0.42 => ( Q @ X3 ) ) )
% 0.14/0.42 => ( Q @ ( least @ a @ less_eq @ P ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.LeastI2_order
% 0.14/0.42 thf(fact_79_local_OLeast__equality,axiom,
% 0.14/0.42 ! [P: a > $o,X: a] :
% 0.14/0.42 ( ( P @ X )
% 0.14/0.42 => ( ! [Y4: a] :
% 0.14/0.42 ( ( P @ Y4 )
% 0.14/0.42 => ( less_eq @ X @ Y4 ) )
% 0.14/0.42 => ( ( least @ a @ less_eq @ P )
% 0.14/0.42 = X ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Least_equality
% 0.14/0.42 thf(fact_80_local_Olsorted_OLCons__LCons,axiom,
% 0.14/0.42 ! [X: a,Y: a,Xs: coinductive_llist @ a] :
% 0.14/0.42 ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ Y @ Xs ) )
% 0.14/0.42 => ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ ( coinductive_LCons @ a @ Y @ Xs ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.lsorted.LCons_LCons
% 0.14/0.42 thf(fact_81_local_Olsorted__LCons,axiom,
% 0.14/0.42 ! [X: a,Xs: coinductive_llist @ a] :
% 0.14/0.42 ( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ Xs ) )
% 0.14/0.42 = ( ( coinductive_lsorted @ a @ less_eq @ Xs )
% 0.14/0.42 & ! [X2: a] :
% 0.14/0.42 ( ( member @ a @ X2 @ ( coinductive_lset @ a @ Xs ) )
% 0.14/0.42 => ( less_eq @ X @ X2 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.lsorted_LCons
% 0.14/0.42 thf(fact_82_local_Oin__lset__lmergeD,axiom,
% 0.14/0.42 ! [X: a,Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.14/0.42 ( ( member @ a @ X @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) )
% 0.14/0.42 => ( ( member @ a @ X @ ( coinductive_lset @ a @ Xs ) )
% 0.14/0.42 | ( member @ a @ X @ ( coinductive_lset @ a @ Ys ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.in_lset_lmergeD
% 0.14/0.42 thf(fact_83_local_Obdd__above__def,axiom,
% 0.14/0.42 ! [A4: set @ a] :
% 0.14/0.42 ( ( condit2040224947_above @ a @ less_eq @ A4 )
% 0.14/0.42 = ( ? [M: a] :
% 0.14/0.42 ! [X2: a] :
% 0.14/0.42 ( ( member @ a @ X2 @ A4 )
% 0.14/0.42 => ( less_eq @ X2 @ M ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_def
% 0.14/0.42 thf(fact_84_local_Obdd__below__def,axiom,
% 0.14/0.42 ! [A4: set @ a] :
% 0.14/0.42 ( ( condit1201339847_below @ a @ less_eq @ A4 )
% 0.14/0.42 = ( ? [M2: a] :
% 0.14/0.42 ! [X2: a] :
% 0.14/0.42 ( ( member @ a @ X2 @ A4 )
% 0.14/0.42 => ( less_eq @ M2 @ X2 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_def
% 0.14/0.42 thf(fact_85_local_Oinsort__key__left__comm,axiom,
% 0.14/0.42 ! [B: $tType,F: B > a,X: B,Y: B,Xs: list @ B] :
% 0.14/0.42 ( ( ( F @ X )
% 0.14/0.42 != ( F @ Y ) )
% 0.14/0.42 => ( ( insort_key @ a @ B @ less_eq @ F @ Y @ ( insort_key @ a @ B @ less_eq @ F @ X @ Xs ) )
% 0.14/0.42 = ( insort_key @ a @ B @ less_eq @ F @ X @ ( insort_key @ a @ B @ less_eq @ F @ Y @ Xs ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.insort_key_left_comm
% 0.14/0.42 thf(fact_86_local_Otransp__le,axiom,
% 0.14/0.42 transp @ a @ less_eq ).
% 0.14/0.42
% 0.14/0.42 % local.transp_le
% 0.14/0.42 thf(fact_87_ord_Oin__lset__lmergeD,axiom,
% 0.14/0.42 ! [A: $tType,X: A,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.14/0.42 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
% 0.14/0.42 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.14/0.42 | ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % ord.in_lset_lmergeD
% 0.14/0.42 thf(fact_88_local_Olsorted__ldistinct__lset__unique,axiom,
% 0.14/0.42 ! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.14/0.42 ( ( coinductive_lsorted @ a @ less_eq @ Xs )
% 0.14/0.42 => ( ( coindu351974385stinct @ a @ Xs )
% 0.14/0.42 => ( ( coinductive_lsorted @ a @ less_eq @ Ys )
% 0.14/0.42 => ( ( coindu351974385stinct @ a @ Ys )
% 0.14/0.42 => ( ( ( coinductive_lset @ a @ Xs )
% 0.14/0.42 = ( coinductive_lset @ a @ Ys ) )
% 0.14/0.42 => ( Xs = Ys ) ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.lsorted_ldistinct_lset_unique
% 0.14/0.42 thf(fact_89_local_Olsorted__LCons__LCons,axiom,
% 0.14/0.42 ! [X: a,Y: a,Xs: coinductive_llist @ a] :
% 0.14/0.42 ( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ ( coinductive_LCons @ a @ Y @ Xs ) ) )
% 0.14/0.42 = ( ( less_eq @ X @ Y )
% 0.14/0.42 & ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ Y @ Xs ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.lsorted_LCons_LCons
% 0.14/0.42 thf(fact_90_local_Omin_Oabsorb1,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.14/0.42 = A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.absorb1
% 0.14/0.42 thf(fact_91_local_Omin_Oabsorb2,axiom,
% 0.14/0.42 ! [B2: a,A2: a] :
% 0.14/0.42 ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.14/0.42 = B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.absorb2
% 0.14/0.42 thf(fact_92_local_Omin_Oabsorb__iff1,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 = ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.14/0.42 = A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.absorb_iff1
% 0.14/0.42 thf(fact_93_local_Omin_Oabsorb__iff2,axiom,
% 0.14/0.42 ! [B2: a,A2: a] :
% 0.14/0.42 ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 = ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.14/0.42 = B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.absorb_iff2
% 0.14/0.42 thf(fact_94_local_Omin__le__iff__disj,axiom,
% 0.14/0.42 ! [X: a,Y: a,Z: a] :
% 0.14/0.42 ( ( less_eq @ ( min @ a @ less_eq @ X @ Y ) @ Z )
% 0.14/0.42 = ( ( less_eq @ X @ Z )
% 0.14/0.42 | ( less_eq @ Y @ Z ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min_le_iff_disj
% 0.14/0.42 thf(fact_95_local_Omin__def,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.14/0.42 = A2 ) )
% 0.14/0.42 & ( ~ ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.14/0.42 = B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min_def
% 0.14/0.42 thf(fact_96_local_Omin_Oorder__iff,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 = ( A2
% 0.14/0.42 = ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.order_iff
% 0.14/0.42 thf(fact_97_local_Omin_OorderI,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( A2
% 0.14/0.42 = ( min @ a @ less_eq @ A2 @ B2 ) )
% 0.14/0.42 => ( less_eq @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.orderI
% 0.14/0.42 thf(fact_98_local_Omin_OorderE,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( A2
% 0.14/0.42 = ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.orderE
% 0.14/0.42 thf(fact_99_local_Omin_Omono,axiom,
% 0.14/0.42 ! [A2: a,C: a,B2: a,D: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ C )
% 0.14/0.42 => ( ( less_eq @ B2 @ D )
% 0.14/0.42 => ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ ( min @ a @ less_eq @ C @ D ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.mono
% 0.14/0.42 thf(fact_100_local_Omin_Oleft__commute,axiom,
% 0.14/0.42 ! [B2: a,A2: a,C: a] :
% 0.14/0.42 ( ( min @ a @ less_eq @ B2 @ ( min @ a @ less_eq @ A2 @ C ) )
% 0.14/0.42 = ( min @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.left_commute
% 0.14/0.42 thf(fact_101_local_Omin_Ocommute,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( min @ a @ less_eq @ A2 @ B2 )
% 0.14/0.42 = ( min @ a @ less_eq @ B2 @ A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.commute
% 0.14/0.42 thf(fact_102_local_Omin_OcoboundedI2,axiom,
% 0.14/0.42 ! [B2: a,C: a,A2: a] :
% 0.14/0.42 ( ( less_eq @ B2 @ C )
% 0.14/0.42 => ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.coboundedI2
% 0.14/0.42 thf(fact_103_local_Omin_OcoboundedI1,axiom,
% 0.14/0.42 ! [A2: a,C: a,B2: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ C )
% 0.14/0.42 => ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.coboundedI1
% 0.14/0.42 thf(fact_104_local_Omin_Ocobounded2,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ B2 ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.cobounded2
% 0.14/0.42 thf(fact_105_local_Omin_Ocobounded1,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ A2 ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.cobounded1
% 0.14/0.42 thf(fact_106_local_Omin_OboundedI,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ( ( less_eq @ A2 @ C )
% 0.14/0.42 => ( less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.boundedI
% 0.14/0.42 thf(fact_107_local_Omin_OboundedE,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) )
% 0.14/0.42 => ~ ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 => ~ ( less_eq @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.boundedE
% 0.14/0.42 thf(fact_108_local_Omin_Oassoc,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( min @ a @ less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ C )
% 0.14/0.42 = ( min @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.assoc
% 0.14/0.42 thf(fact_109_local_Olmerge__simps,axiom,
% 0.14/0.42 ! [X: a,Y: a,Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.14/0.42 ( ( ( less @ X @ Y )
% 0.14/0.42 => ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ ( coinductive_LCons @ a @ Y @ Ys ) )
% 0.14/0.42 = ( coinductive_LCons @ a @ X @ ( hammin1328233080lmerge @ a @ less @ Xs @ ( coinductive_LCons @ a @ Y @ Ys ) ) ) ) )
% 0.14/0.42 & ( ~ ( less @ X @ Y )
% 0.14/0.42 => ( ( ( less @ Y @ X )
% 0.14/0.42 => ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ ( coinductive_LCons @ a @ Y @ Ys ) )
% 0.14/0.42 = ( coinductive_LCons @ a @ Y @ ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ Ys ) ) ) )
% 0.14/0.42 & ( ~ ( less @ Y @ X )
% 0.14/0.42 => ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ ( coinductive_LCons @ a @ Y @ Ys ) )
% 0.14/0.42 = ( coinductive_LCons @ a @ Y @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.lmerge_simps
% 0.14/0.42 thf(fact_110_local_Olinfinite__lmerge,axiom,
% 0.14/0.42 ! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.14/0.42 ( ~ ( coinductive_lfinite @ a @ Xs )
% 0.14/0.42 => ( ~ ( coinductive_lfinite @ a @ Ys )
% 0.14/0.42 => ~ ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.linfinite_lmerge
% 0.14/0.42 thf(fact_111_local_Olfinite__lmergeI,axiom,
% 0.14/0.42 ! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.14/0.42 ( ( coinductive_lfinite @ a @ Xs )
% 0.14/0.42 => ( ( coinductive_lfinite @ a @ Ys )
% 0.14/0.42 => ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.lfinite_lmergeI
% 0.14/0.42 thf(fact_112_local_Olfinite__lmergeD,axiom,
% 0.14/0.42 ! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 0.14/0.42 ( ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
% 0.14/0.42 => ( ( coinductive_lfinite @ a @ Xs )
% 0.14/0.42 | ( coinductive_lfinite @ a @ Ys ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.lfinite_lmergeD
% 0.14/0.42 thf(fact_113_local_Omin__less__iff__disj,axiom,
% 0.14/0.42 ! [X: a,Y: a,Z: a] :
% 0.14/0.42 ( ( less @ ( min @ a @ less_eq @ X @ Y ) @ Z )
% 0.14/0.42 = ( ( less @ X @ Z )
% 0.14/0.42 | ( less @ Y @ Z ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min_less_iff_disj
% 0.14/0.42 thf(fact_114_local_Omin_Ostrict__order__iff,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( less @ A2 @ B2 )
% 0.14/0.42 = ( ( A2
% 0.14/0.42 = ( min @ a @ less_eq @ A2 @ B2 ) )
% 0.14/0.42 & ( A2 != B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.strict_order_iff
% 0.14/0.42 thf(fact_115_local_Omin_Ostrict__coboundedI2,axiom,
% 0.14/0.42 ! [B2: a,C: a,A2: a] :
% 0.14/0.42 ( ( less @ B2 @ C )
% 0.14/0.42 => ( less @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.strict_coboundedI2
% 0.14/0.42 thf(fact_116_local_Omin_Ostrict__coboundedI1,axiom,
% 0.14/0.42 ! [A2: a,C: a,B2: a] :
% 0.14/0.42 ( ( less @ A2 @ C )
% 0.14/0.42 => ( less @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.strict_coboundedI1
% 0.14/0.42 thf(fact_117_local_Omin_Ostrict__boundedE,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less @ A2 @ ( min @ a @ less_eq @ B2 @ C ) )
% 0.14/0.42 => ~ ( ( less @ A2 @ B2 )
% 0.14/0.42 => ~ ( less @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.strict_boundedE
% 0.14/0.42 thf(fact_118_local_Omin_Oright__idem,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( min @ a @ less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ B2 )
% 0.14/0.42 = ( min @ a @ less_eq @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.right_idem
% 0.14/0.42 thf(fact_119_local_Omin_Oleft__idem,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( min @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ A2 @ B2 ) )
% 0.14/0.42 = ( min @ a @ less_eq @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.left_idem
% 0.14/0.42 thf(fact_120_local_Omin_Oidem,axiom,
% 0.14/0.42 ! [A2: a] :
% 0.14/0.42 ( ( min @ a @ less_eq @ A2 @ A2 )
% 0.14/0.42 = A2 ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.idem
% 0.14/0.42 thf(fact_121_local_Omin_Obounded__iff,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a] :
% 0.14/0.42 ( ( less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) )
% 0.14/0.42 = ( ( less_eq @ A2 @ B2 )
% 0.14/0.42 & ( less_eq @ A2 @ C ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.bounded_iff
% 0.14/0.42 thf(fact_122_local_Obdd__belowI,axiom,
% 0.14/0.42 ! [A4: set @ a,M3: a] :
% 0.14/0.42 ( ! [X3: a] :
% 0.14/0.42 ( ( member @ a @ X3 @ A4 )
% 0.14/0.42 => ( less_eq @ M3 @ X3 ) )
% 0.14/0.42 => ( condit1201339847_below @ a @ less_eq @ A4 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_belowI
% 0.14/0.42 thf(fact_123_local_Obdd__aboveI,axiom,
% 0.14/0.42 ! [A4: set @ a,M4: a] :
% 0.14/0.42 ( ! [X3: a] :
% 0.14/0.42 ( ( member @ a @ X3 @ A4 )
% 0.14/0.42 => ( less_eq @ X3 @ M4 ) )
% 0.14/0.42 => ( condit2040224947_above @ a @ less_eq @ A4 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_aboveI
% 0.14/0.42 thf(fact_124_local_Omin__less__iff__conj,axiom,
% 0.14/0.42 ! [Z: a,X: a,Y: a] :
% 0.14/0.42 ( ( less @ Z @ ( min @ a @ less_eq @ X @ Y ) )
% 0.14/0.42 = ( ( less @ Z @ X )
% 0.14/0.42 & ( less @ Z @ Y ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.min_less_iff_conj
% 0.14/0.42 thf(fact_125_local_Ostrict__mono__eq,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B,X: a,Y: a] :
% 0.14/0.42 ( ( strict_mono @ a @ B @ less @ F )
% 0.14/0.42 => ( ( ( F @ X )
% 0.14/0.42 = ( F @ Y ) )
% 0.14/0.42 = ( X = Y ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.strict_mono_eq
% 0.14/0.42 thf(fact_126_ord_Olinfinite__lmerge,axiom,
% 0.14/0.42 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.14/0.42 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.14/0.42 => ( ~ ( coinductive_lfinite @ A @ Ys )
% 0.14/0.42 => ~ ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % ord.linfinite_lmerge
% 0.14/0.42 thf(fact_127_ord_Olfinite__lmergeI,axiom,
% 0.14/0.42 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.14/0.42 ( ( coinductive_lfinite @ A @ Xs )
% 0.14/0.42 => ( ( coinductive_lfinite @ A @ Ys )
% 0.14/0.42 => ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % ord.lfinite_lmergeI
% 0.14/0.42 thf(fact_128_ord_Olfinite__lmergeD,axiom,
% 0.14/0.42 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.14/0.42 ( ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
% 0.14/0.42 => ( ( coinductive_lfinite @ A @ Xs )
% 0.14/0.42 | ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % ord.lfinite_lmergeD
% 0.14/0.42 thf(fact_129_ord_Olmerge__simps,axiom,
% 0.14/0.42 ! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.14/0.42 ( ( ( Less @ X @ Y )
% 0.14/0.42 => ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.14/0.42 = ( coinductive_LCons @ A @ X @ ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) )
% 0.14/0.42 & ( ~ ( Less @ X @ Y )
% 0.14/0.42 => ( ( ( Less @ Y @ X )
% 0.14/0.42 => ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.14/0.42 = ( coinductive_LCons @ A @ Y @ ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys ) ) ) )
% 0.14/0.42 & ( ~ ( Less @ Y @ X )
% 0.14/0.42 => ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.14/0.42 = ( coinductive_LCons @ A @ Y @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % ord.lmerge_simps
% 0.14/0.42 thf(fact_130_local_OMin_Osemilattice__order__set__axioms,axiom,
% 0.14/0.42 lattic1693879045er_set @ a @ ( min @ a @ less_eq ) @ less_eq @ less ).
% 0.14/0.42
% 0.14/0.42 % local.Min.semilattice_order_set_axioms
% 0.14/0.42 thf(fact_131_ldistinct__LCons,axiom,
% 0.14/0.42 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.14/0.42 ( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.14/0.42 = ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.14/0.42 & ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % ldistinct_LCons
% 0.14/0.42 thf(fact_132_local_OIoc__inj,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a,D: a] :
% 0.14/0.42 ( ( ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 )
% 0.14/0.42 = ( set_gr323396891AtMost @ a @ less_eq @ less @ C @ D ) )
% 0.14/0.42 = ( ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 & ( less_eq @ D @ C ) )
% 0.14/0.42 | ( ( A2 = C )
% 0.14/0.42 & ( B2 = D ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Ioc_inj
% 0.14/0.42 thf(fact_133_step_OIH,axiom,
% 0.14/0.42 ! [Ys: coinductive_llist @ a] :
% 0.14/0.42 ( ( coinductive_lsorted @ a @ less_eq @ xsa )
% 0.14/0.42 => ( ~ ( coinductive_lfinite @ a @ Ys )
% 0.14/0.42 => ( ? [X4: a] :
% 0.14/0.42 ( ( member @ a @ X4 @ ( coinductive_lset @ a @ Ys ) )
% 0.14/0.42 & ( less_eq @ x @ X4 ) )
% 0.14/0.42 => ( member @ a @ x @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ xsa @ Ys ) ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % step.IH
% 0.14/0.42 thf(fact_134_lfinite__code_I2_J,axiom,
% 0.14/0.42 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 0.14/0.42 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 0.14/0.42 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 0.14/0.42
% 0.14/0.42 % lfinite_code(2)
% 0.14/0.42 thf(fact_135_local_Oinsort__left__comm,axiom,
% 0.14/0.42 ! [X: a,Y: a,Xs: list @ a] :
% 0.14/0.42 ( ( insort_key @ a @ a @ less_eq
% 0.14/0.42 @ ^ [X2: a] : X2
% 0.14/0.42 @ X
% 0.14/0.42 @ ( insort_key @ a @ a @ less_eq
% 0.14/0.42 @ ^ [X2: a] : X2
% 0.14/0.42 @ Y
% 0.14/0.42 @ Xs ) )
% 0.14/0.42 = ( insort_key @ a @ a @ less_eq
% 0.14/0.42 @ ^ [X2: a] : X2
% 0.14/0.42 @ Y
% 0.14/0.42 @ ( insort_key @ a @ a @ less_eq
% 0.14/0.42 @ ^ [X2: a] : X2
% 0.14/0.42 @ X
% 0.14/0.42 @ Xs ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.insort_left_comm
% 0.14/0.42 thf(fact_136_step_Oprems_I1_J,axiom,
% 0.14/0.42 coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ x2 @ xsa ) ).
% 0.14/0.42
% 0.14/0.42 % step.prems(1)
% 0.14/0.42 thf(fact_137_llist_Oinject,axiom,
% 0.14/0.42 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 0.14/0.42 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 0.14/0.42 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 0.14/0.42 = ( ( X21 = Y21 )
% 0.14/0.42 & ( X22 = Y22 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % llist.inject
% 0.14/0.42 thf(fact_138_lfinite__LCons,axiom,
% 0.14/0.42 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.14/0.42 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.14/0.42 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.14/0.42
% 0.14/0.42 % lfinite_LCons
% 0.14/0.42 thf(fact_139_local_OgreaterThanAtMost__iff,axiom,
% 0.14/0.42 ! [I: a,L: a,U: a] :
% 0.14/0.42 ( ( member @ a @ I @ ( set_gr323396891AtMost @ a @ less_eq @ less @ L @ U ) )
% 0.14/0.42 = ( ( less @ L @ I )
% 0.14/0.42 & ( less_eq @ I @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.greaterThanAtMost_iff
% 0.14/0.42 thf(fact_140_local_Obdd__above__Ioc,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_Ioc
% 0.14/0.42 thf(fact_141_local_Obdd__below__Ioc,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_Ioc
% 0.14/0.42 thf(fact_142_local_Ostrict__mono__mono,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B] :
% 0.14/0.42 ( ( strict_mono @ a @ B @ less @ F )
% 0.14/0.42 => ( mono @ a @ B @ less_eq @ F ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.strict_mono_mono
% 0.14/0.42 thf(fact_143_lset__intros_I2_J,axiom,
% 0.14/0.42 ! [A: $tType,X: A,Xs: coinductive_llist @ A,X5: A] :
% 0.14/0.42 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.14/0.42 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % lset_intros(2)
% 0.14/0.42 thf(fact_144_lset__intros_I1_J,axiom,
% 0.14/0.42 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % lset_intros(1)
% 0.14/0.42 thf(fact_145_llist_Oset__intros_I2_J,axiom,
% 0.14/0.42 ! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
% 0.14/0.42 ( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
% 0.14/0.42 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % llist.set_intros(2)
% 0.14/0.42 thf(fact_146_llist_Oset__intros_I1_J,axiom,
% 0.14/0.42 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % llist.set_intros(1)
% 0.14/0.42 thf(fact_147_lset__cases,axiom,
% 0.14/0.42 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.14/0.42 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.14/0.42 => ( ! [Xs2: coinductive_llist @ A] :
% 0.14/0.42 ( Xs
% 0.14/0.42 != ( coinductive_LCons @ A @ X @ Xs2 ) )
% 0.14/0.42 => ~ ! [X6: A,Xs2: coinductive_llist @ A] :
% 0.14/0.42 ( ( Xs
% 0.14/0.42 = ( coinductive_LCons @ A @ X6 @ Xs2 ) )
% 0.14/0.42 => ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % lset_cases
% 0.14/0.42 thf(fact_148_lset__induct,axiom,
% 0.14/0.42 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.14/0.42 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.14/0.42 => ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
% 0.14/0.42 => ( ! [X6: A,Xs3: coinductive_llist @ A] :
% 0.14/0.42 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
% 0.14/0.42 => ( ( X != X6 )
% 0.14/0.42 => ( ( P @ Xs3 )
% 0.14/0.42 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs3 ) ) ) ) )
% 0.14/0.42 => ( P @ Xs ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % lset_induct
% 0.14/0.42 thf(fact_149_lset__induct_H,axiom,
% 0.14/0.42 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.14/0.42 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.14/0.42 => ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
% 0.14/0.42 => ( ! [X6: A,Xs3: coinductive_llist @ A] :
% 0.14/0.42 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
% 0.14/0.42 => ( ( P @ Xs3 )
% 0.14/0.42 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs3 ) ) ) )
% 0.14/0.42 => ( P @ Xs ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % lset_induct'
% 0.14/0.42 thf(fact_150_llist_Oset__cases,axiom,
% 0.14/0.42 ! [A: $tType,E: A,A2: coinductive_llist @ A] :
% 0.14/0.42 ( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
% 0.14/0.42 => ( ! [Z22: coinductive_llist @ A] :
% 0.14/0.42 ( A2
% 0.14/0.42 != ( coinductive_LCons @ A @ E @ Z22 ) )
% 0.14/0.42 => ~ ! [Z1: A,Z22: coinductive_llist @ A] :
% 0.14/0.42 ( ( A2
% 0.14/0.42 = ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 0.14/0.42 => ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % llist.set_cases
% 0.14/0.42 thf(fact_151_llist_Oset__induct,axiom,
% 0.14/0.42 ! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 0.14/0.42 ( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
% 0.14/0.42 => ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 0.14/0.42 => ( ! [Z1: A,Z22: coinductive_llist @ A,Xa: A] :
% 0.14/0.42 ( ( member @ A @ Xa @ ( coinductive_lset @ A @ Z22 ) )
% 0.14/0.42 => ( ( P @ Xa @ Z22 )
% 0.14/0.42 => ( P @ Xa @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
% 0.14/0.42 => ( P @ X @ A2 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % llist.set_induct
% 0.14/0.42 thf(fact_152_lfinite__LConsI,axiom,
% 0.14/0.42 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.14/0.42 ( ( coinductive_lfinite @ A @ Xs )
% 0.14/0.42 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % lfinite_LConsI
% 0.14/0.42 thf(fact_153_ord_Olsorted__LCons__LCons,axiom,
% 0.14/0.42 ! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.14/0.42 ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 0.14/0.42 = ( ( Less_eq @ X @ Y )
% 0.14/0.42 & ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % ord.lsorted_LCons_LCons
% 0.14/0.42 thf(fact_154_ord_OLCons__LCons,axiom,
% 0.14/0.42 ! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.14/0.42 ( ( Less_eq @ X @ Y )
% 0.14/0.42 => ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs ) )
% 0.14/0.42 => ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % ord.LCons_LCons
% 0.14/0.42 thf(fact_155_ldistinct_OLCons,axiom,
% 0.14/0.42 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.14/0.42 ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.14/0.42 => ( ( coindu351974385stinct @ A @ Xs )
% 0.14/0.42 => ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % ldistinct.LCons
% 0.14/0.42 thf(fact_156_local_Osorted__list__of__set_Ofolding__axioms,axiom,
% 0.14/0.42 ( finite_folding @ a @ ( list @ a )
% 0.14/0.42 @ ( insort_key @ a @ a @ less_eq
% 0.14/0.42 @ ^ [X2: a] : X2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.sorted_list_of_set.folding_axioms
% 0.14/0.42 thf(fact_157_local_OatLeast__def,axiom,
% 0.14/0.42 ! [L: a] :
% 0.14/0.42 ( ( set_atLeast @ a @ less_eq @ L )
% 0.14/0.42 = ( collect @ a @ ( less_eq @ L ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atLeast_def
% 0.14/0.42 thf(fact_158_local_Odual__order_Oordering__axioms,axiom,
% 0.14/0.42 ( ordering @ a
% 0.14/0.42 @ ^ [X2: a,Y3: a] : ( less_eq @ Y3 @ X2 )
% 0.14/0.42 @ ^ [X2: a,Y3: a] : ( less @ Y3 @ X2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.dual_order.ordering_axioms
% 0.14/0.42 thf(fact_159_local_OatMost__def,axiom,
% 0.14/0.42 ! [U: a] :
% 0.14/0.42 ( ( set_atMost @ a @ less_eq @ U )
% 0.14/0.42 = ( collect @ a
% 0.14/0.42 @ ^ [X2: a] : ( less_eq @ X2 @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atMost_def
% 0.14/0.42 thf(fact_160_local_Oorder_Oordering__axioms,axiom,
% 0.14/0.42 ordering @ a @ less_eq @ less ).
% 0.14/0.42
% 0.14/0.42 % local.order.ordering_axioms
% 0.14/0.42 thf(fact_161_local_OatMost__iff,axiom,
% 0.14/0.42 ! [I: a,K: a] :
% 0.14/0.42 ( ( member @ a @ I @ ( set_atMost @ a @ less_eq @ K ) )
% 0.14/0.42 = ( less_eq @ I @ K ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atMost_iff
% 0.14/0.42 thf(fact_162_local_OatLeast__iff,axiom,
% 0.14/0.42 ! [I: a,K: a] :
% 0.14/0.42 ( ( member @ a @ I @ ( set_atLeast @ a @ less_eq @ K ) )
% 0.14/0.42 = ( less_eq @ K @ I ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atLeast_iff
% 0.14/0.42 thf(fact_163_local_Obdd__above__Iic,axiom,
% 0.14/0.42 ! [B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atMost @ a @ less_eq @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_Iic
% 0.14/0.42 thf(fact_164_local_Obdd__below__Ici,axiom,
% 0.14/0.42 ! [A2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeast @ a @ less_eq @ A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_Ici
% 0.14/0.42 thf(fact_165_local_OgreaterThan__def,axiom,
% 0.14/0.42 ! [L: a] :
% 0.14/0.42 ( ( set_greaterThan @ a @ less @ L )
% 0.14/0.42 = ( collect @ a @ ( less @ L ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.greaterThan_def
% 0.14/0.42 thf(fact_166_local_Omin_Osemilattice__axioms,axiom,
% 0.14/0.42 semilattice @ a @ ( min @ a @ less_eq ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.semilattice_axioms
% 0.14/0.42 thf(fact_167_local_Omin_Osemigroup__axioms,axiom,
% 0.14/0.42 semigroup @ a @ ( min @ a @ less_eq ) ).
% 0.14/0.42
% 0.14/0.42 % local.min.semigroup_axioms
% 0.14/0.42 thf(fact_168_local_OgreaterThan__iff,axiom,
% 0.14/0.42 ! [I: a,K: a] :
% 0.14/0.42 ( ( member @ a @ I @ ( set_greaterThan @ a @ less @ K ) )
% 0.14/0.42 = ( less @ K @ I ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.greaterThan_iff
% 0.14/0.42 thf(fact_169_local_Obdd__below__Ioi,axiom,
% 0.14/0.42 ! [A2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_greaterThan @ a @ less @ A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_Ioi
% 0.14/0.42 thf(fact_170_local_OgreaterThanAtMost__def,axiom,
% 0.14/0.42 ! [L: a,U: a] :
% 0.14/0.42 ( ( set_gr323396891AtMost @ a @ less_eq @ less @ L @ U )
% 0.14/0.42 = ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ L ) @ ( set_atMost @ a @ less_eq @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.greaterThanAtMost_def
% 0.14/0.42 thf(fact_171_local_OIci__subset__Ioi__iff,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ A2 ) @ ( set_greaterThan @ a @ less @ B2 ) )
% 0.14/0.42 = ( less @ B2 @ A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Ici_subset_Ioi_iff
% 0.14/0.42 thf(fact_172_local_OIoi__le__Ico,axiom,
% 0.14/0.42 ! [A2: a] : ( ord_less_eq @ ( set @ a ) @ ( set_greaterThan @ a @ less @ A2 ) @ ( set_atLeast @ a @ less_eq @ A2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Ioi_le_Ico
% 0.14/0.42 thf(fact_173_local_Omono__def,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B] :
% 0.14/0.42 ( ( mono @ a @ B @ less_eq @ F )
% 0.14/0.42 = ( ! [X2: a,Y3: a] :
% 0.14/0.42 ( ( less_eq @ X2 @ Y3 )
% 0.14/0.42 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.mono_def
% 0.14/0.42 thf(fact_174_local_OmonoI,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B] :
% 0.14/0.42 ( ! [X3: a,Y4: a] :
% 0.14/0.42 ( ( less_eq @ X3 @ Y4 )
% 0.14/0.42 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.14/0.42 => ( mono @ a @ B @ less_eq @ F ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.monoI
% 0.14/0.42 thf(fact_175_local_OmonoE,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B,X: a,Y: a] :
% 0.14/0.42 ( ( mono @ a @ B @ less_eq @ F )
% 0.14/0.42 => ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.monoE
% 0.14/0.42 thf(fact_176_local_OmonoD,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B,X: a,Y: a] :
% 0.14/0.42 ( ( mono @ a @ B @ less_eq @ F )
% 0.14/0.42 => ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.monoD
% 0.14/0.42 thf(fact_177_local_Obdd__below__Int2,axiom,
% 0.14/0.42 ! [B4: set @ a,A4: set @ a] :
% 0.14/0.42 ( ( condit1201339847_below @ a @ less_eq @ B4 )
% 0.14/0.42 => ( condit1201339847_below @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_Int2
% 0.14/0.42 thf(fact_178_local_Obdd__below__Int1,axiom,
% 0.14/0.42 ! [A4: set @ a,B4: set @ a] :
% 0.14/0.42 ( ( condit1201339847_below @ a @ less_eq @ A4 )
% 0.14/0.42 => ( condit1201339847_below @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_Int1
% 0.14/0.42 thf(fact_179_local_Obdd__above__Int2,axiom,
% 0.14/0.42 ! [B4: set @ a,A4: set @ a] :
% 0.14/0.42 ( ( condit2040224947_above @ a @ less_eq @ B4 )
% 0.14/0.42 => ( condit2040224947_above @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_Int2
% 0.14/0.42 thf(fact_180_local_Obdd__above__Int1,axiom,
% 0.14/0.42 ! [A4: set @ a,B4: set @ a] :
% 0.14/0.42 ( ( condit2040224947_above @ a @ less_eq @ A4 )
% 0.14/0.42 => ( condit2040224947_above @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_Int1
% 0.14/0.42 thf(fact_181_local_Obdd__below__mono,axiom,
% 0.14/0.42 ! [B4: set @ a,A4: set @ a] :
% 0.14/0.42 ( ( condit1201339847_below @ a @ less_eq @ B4 )
% 0.14/0.42 => ( ( ord_less_eq @ ( set @ a ) @ A4 @ B4 )
% 0.14/0.42 => ( condit1201339847_below @ a @ less_eq @ A4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_mono
% 0.14/0.42 thf(fact_182_local_Obdd__above__mono,axiom,
% 0.14/0.42 ! [B4: set @ a,A4: set @ a] :
% 0.14/0.42 ( ( condit2040224947_above @ a @ less_eq @ B4 )
% 0.14/0.42 => ( ( ord_less_eq @ ( set @ a ) @ A4 @ B4 )
% 0.14/0.42 => ( condit2040224947_above @ a @ less_eq @ A4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_mono
% 0.14/0.42 thf(fact_183_local_Ostrict__mono__less__eq,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B,X: a,Y: a] :
% 0.14/0.42 ( ( strict_mono @ a @ B @ less @ F )
% 0.14/0.42 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
% 0.14/0.42 = ( less_eq @ X @ Y ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.strict_mono_less_eq
% 0.14/0.42 thf(fact_184_local_OIoc__subset__iff,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a,D: a] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ a ) @ ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 ) @ ( set_gr323396891AtMost @ a @ less_eq @ less @ C @ D ) )
% 0.14/0.42 = ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 | ( ( less_eq @ C @ A2 )
% 0.14/0.42 & ( less_eq @ B2 @ D ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Ioc_subset_iff
% 0.14/0.42 thf(fact_185_local_Oantimono__def,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B] :
% 0.14/0.42 ( ( antimono @ a @ B @ less_eq @ F )
% 0.14/0.42 = ( ! [X2: a,Y3: a] :
% 0.14/0.42 ( ( less_eq @ X2 @ Y3 )
% 0.14/0.42 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.antimono_def
% 0.14/0.42 thf(fact_186_local_OantimonoI,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B] :
% 0.14/0.42 ( ! [X3: a,Y4: a] :
% 0.14/0.42 ( ( less_eq @ X3 @ Y4 )
% 0.14/0.42 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 0.14/0.42 => ( antimono @ a @ B @ less_eq @ F ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.antimonoI
% 0.14/0.42 thf(fact_187_local_OantimonoE,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B,X: a,Y: a] :
% 0.14/0.42 ( ( antimono @ a @ B @ less_eq @ F )
% 0.14/0.42 => ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.antimonoE
% 0.14/0.42 thf(fact_188_local_OantimonoD,axiom,
% 0.14/0.42 ! [B: $tType] :
% 0.14/0.42 ( ( order @ B @ ( type2 @ B ) )
% 0.14/0.42 => ! [F: a > B,X: a,Y: a] :
% 0.14/0.42 ( ( antimono @ a @ B @ less_eq @ F )
% 0.14/0.42 => ( ( less_eq @ X @ Y )
% 0.14/0.42 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.antimonoD
% 0.14/0.42 thf(fact_189_local_OInt__atMost,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( inf_inf @ ( set @ a ) @ ( set_atMost @ a @ less_eq @ A2 ) @ ( set_atMost @ a @ less_eq @ B2 ) )
% 0.14/0.42 = ( set_atMost @ a @ less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Int_atMost
% 0.14/0.42 thf(fact_190_wlog__linorder__le,axiom,
% 0.14/0.42 ! [A: $tType] :
% 0.14/0.42 ( ( linorder @ A @ ( type2 @ A ) )
% 0.14/0.42 => ! [P: A > A > $o,B2: A,A2: A] :
% 0.14/0.42 ( ! [A3: A,B3: A] :
% 0.14/0.42 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.14/0.42 => ( P @ A3 @ B3 ) )
% 0.14/0.42 => ( ( ( P @ B2 @ A2 )
% 0.14/0.42 => ( P @ A2 @ B2 ) )
% 0.14/0.42 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % wlog_linorder_le
% 0.14/0.42 thf(fact_191_local_OatLeastLessThan__subset__iff,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a,D: a] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastLessThan @ a @ less_eq @ less @ A2 @ B2 ) @ ( set_atLeastLessThan @ a @ less_eq @ less @ C @ D ) )
% 0.14/0.42 => ( ( less_eq @ B2 @ A2 )
% 0.14/0.42 | ( ( less_eq @ C @ A2 )
% 0.14/0.42 & ( less_eq @ B2 @ D ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atLeastLessThan_subset_iff
% 0.14/0.42 thf(fact_192_local_OIic__subset__Iio__iff,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ a ) @ ( set_atMost @ a @ less_eq @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) )
% 0.14/0.42 = ( less @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Iic_subset_Iio_iff
% 0.14/0.42 thf(fact_193_local_OgreaterThan__Int__greaterThan,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( inf_inf @ ( set @ a ) @ ( set_lessThan @ a @ less @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) )
% 0.14/0.42 = ( set_lessThan @ a @ less @ ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.greaterThan_Int_greaterThan
% 0.14/0.42 thf(fact_194_local_OlessThan__def,axiom,
% 0.14/0.42 ! [U: a] :
% 0.14/0.42 ( ( set_lessThan @ a @ less @ U )
% 0.14/0.42 = ( collect @ a
% 0.14/0.42 @ ^ [X2: a] : ( less @ X2 @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.lessThan_def
% 0.14/0.42 thf(fact_195_local_OatLeastLessThan__def,axiom,
% 0.14/0.42 ! [L: a,U: a] :
% 0.14/0.42 ( ( set_atLeastLessThan @ a @ less_eq @ less @ L @ U )
% 0.14/0.42 = ( inf_inf @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ L ) @ ( set_lessThan @ a @ less @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atLeastLessThan_def
% 0.14/0.42 thf(fact_196_local_OlessThan__iff,axiom,
% 0.14/0.42 ! [I: a,K: a] :
% 0.14/0.42 ( ( member @ a @ I @ ( set_lessThan @ a @ less @ K ) )
% 0.14/0.42 = ( less @ I @ K ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.lessThan_iff
% 0.14/0.42 thf(fact_197_local_OatLeastLessThan__iff,axiom,
% 0.14/0.42 ! [I: a,L: a,U: a] :
% 0.14/0.42 ( ( member @ a @ I @ ( set_atLeastLessThan @ a @ less_eq @ less @ L @ U ) )
% 0.14/0.42 = ( ( less_eq @ L @ I )
% 0.14/0.42 & ( less @ I @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atLeastLessThan_iff
% 0.14/0.42 thf(fact_198_local_Obdd__above__Iio,axiom,
% 0.14/0.42 ! [B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_lessThan @ a @ less @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_Iio
% 0.14/0.42 thf(fact_199_local_Obdd__above__Ico,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atLeastLessThan @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_Ico
% 0.14/0.42 thf(fact_200_local_Obdd__below__Ico,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeastLessThan @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_Ico
% 0.14/0.42 thf(fact_201_local_OgreaterThanLessThan__eq,axiom,
% 0.14/0.42 ! [A2: a,B2: a] :
% 0.14/0.42 ( ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 )
% 0.14/0.42 = ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.greaterThanLessThan_eq
% 0.14/0.42 thf(fact_202_local_OgreaterThanLessThan__def,axiom,
% 0.14/0.42 ! [L: a,U: a] :
% 0.14/0.42 ( ( set_gr1161524159ssThan @ a @ less @ L @ U )
% 0.14/0.42 = ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ L ) @ ( set_lessThan @ a @ less @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.greaterThanLessThan_def
% 0.14/0.42 thf(fact_203_local_OgreaterThanLessThan__iff,axiom,
% 0.14/0.42 ! [I: a,L: a,U: a] :
% 0.14/0.42 ( ( member @ a @ I @ ( set_gr1161524159ssThan @ a @ less @ L @ U ) )
% 0.14/0.42 = ( ( less @ L @ I )
% 0.14/0.42 & ( less @ I @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.greaterThanLessThan_iff
% 0.14/0.42 thf(fact_204_local_Obdd__below__Ioo,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_Ioo
% 0.14/0.42 thf(fact_205_local_Obdd__above__Ioo,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_Ioo
% 0.14/0.42 thf(fact_206_local_OatLeastAtMost__def,axiom,
% 0.14/0.42 ! [L: a,U: a] :
% 0.14/0.42 ( ( set_atLeastAtMost @ a @ less_eq @ L @ U )
% 0.14/0.42 = ( inf_inf @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ L ) @ ( set_atMost @ a @ less_eq @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atLeastAtMost_def
% 0.14/0.42 thf(fact_207_Int__subset__iff,axiom,
% 0.14/0.42 ! [A: $tType,C2: set @ A,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 0.14/0.42 = ( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
% 0.14/0.42 & ( ord_less_eq @ ( set @ A ) @ C2 @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % Int_subset_iff
% 0.14/0.42 thf(fact_208_subset__antisym,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.14/0.42 => ( A4 = B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % subset_antisym
% 0.14/0.42 thf(fact_209_subsetI,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ! [X3: A] :
% 0.14/0.42 ( ( member @ A @ X3 @ A4 )
% 0.14/0.42 => ( member @ A @ X3 @ B4 ) )
% 0.14/0.42 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.14/0.42
% 0.14/0.42 % subsetI
% 0.14/0.42 thf(fact_210_Int__iff,axiom,
% 0.14/0.42 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 0.14/0.42 = ( ( member @ A @ C @ A4 )
% 0.14/0.42 & ( member @ A @ C @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % Int_iff
% 0.14/0.42 thf(fact_211_IntI,axiom,
% 0.14/0.42 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( member @ A @ C @ A4 )
% 0.14/0.42 => ( ( member @ A @ C @ B4 )
% 0.14/0.42 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % IntI
% 0.14/0.42 thf(fact_212_local_OatLeastAtMost__iff,axiom,
% 0.14/0.42 ! [I: a,L: a,U: a] :
% 0.14/0.42 ( ( member @ a @ I @ ( set_atLeastAtMost @ a @ less_eq @ L @ U ) )
% 0.14/0.42 = ( ( less_eq @ L @ I )
% 0.14/0.42 & ( less_eq @ I @ U ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atLeastAtMost_iff
% 0.14/0.42 thf(fact_213_local_OIcc__eq__Icc,axiom,
% 0.14/0.42 ! [L: a,H: a,L2: a,H2: a] :
% 0.14/0.42 ( ( ( set_atLeastAtMost @ a @ less_eq @ L @ H )
% 0.14/0.42 = ( set_atLeastAtMost @ a @ less_eq @ L2 @ H2 ) )
% 0.14/0.42 = ( ( ( L = L2 )
% 0.14/0.42 & ( H = H2 ) )
% 0.14/0.42 | ( ~ ( less_eq @ L @ H )
% 0.14/0.42 & ~ ( less_eq @ L2 @ H2 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Icc_eq_Icc
% 0.14/0.42 thf(fact_214_local_OatLeastatMost__subset__iff,axiom,
% 0.14/0.42 ! [A2: a,B2: a,C: a,D: a] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) @ ( set_atLeastAtMost @ a @ less_eq @ C @ D ) )
% 0.14/0.42 = ( ~ ( less_eq @ A2 @ B2 )
% 0.14/0.42 | ( ( less_eq @ C @ A2 )
% 0.14/0.42 & ( less_eq @ B2 @ D ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.atLeastatMost_subset_iff
% 0.14/0.42 thf(fact_215_local_Obdd__above__Icc,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_above_Icc
% 0.14/0.42 thf(fact_216_local_Obdd__below__Icc,axiom,
% 0.14/0.42 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.bdd_below_Icc
% 0.14/0.42 thf(fact_217_local_OIcc__subset__Iic__iff,axiom,
% 0.14/0.42 ! [L: a,H: a,H2: a] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atMost @ a @ less_eq @ H2 ) )
% 0.14/0.42 = ( ~ ( less_eq @ L @ H )
% 0.14/0.42 | ( less_eq @ H @ H2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Icc_subset_Iic_iff
% 0.14/0.42 thf(fact_218_local_OIcc__subset__Ici__iff,axiom,
% 0.14/0.42 ! [L: a,H: a,L2: a] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atLeast @ a @ less_eq @ L2 ) )
% 0.14/0.42 = ( ~ ( less_eq @ L @ H )
% 0.14/0.42 | ( less_eq @ L2 @ L ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Icc_subset_Ici_iff
% 0.14/0.42 thf(fact_219_local_OInt__atLeastAtMostL1,axiom,
% 0.14/0.42 ! [A2: a,B2: a,D: a] :
% 0.14/0.42 ( ( inf_inf @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) @ ( set_atMost @ a @ less_eq @ D ) )
% 0.14/0.42 = ( set_atLeastAtMost @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ D ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Int_atLeastAtMostL1
% 0.14/0.42 thf(fact_220_local_OInt__atLeastAtMostR1,axiom,
% 0.14/0.42 ! [B2: a,C: a,D: a] :
% 0.14/0.42 ( ( inf_inf @ ( set @ a ) @ ( set_atMost @ a @ less_eq @ B2 ) @ ( set_atLeastAtMost @ a @ less_eq @ C @ D ) )
% 0.14/0.42 = ( set_atLeastAtMost @ a @ less_eq @ C @ ( min @ a @ less_eq @ B2 @ D ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % local.Int_atLeastAtMostR1
% 0.14/0.42 thf(fact_221_Collect__mono__iff,axiom,
% 0.14/0.42 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.14/0.42 = ( ! [X2: A] :
% 0.14/0.42 ( ( P @ X2 )
% 0.14/0.42 => ( Q @ X2 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % Collect_mono_iff
% 0.14/0.42 thf(fact_222_contra__subsetD,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ( ~ ( member @ A @ C @ B4 )
% 0.14/0.42 => ~ ( member @ A @ C @ A4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % contra_subsetD
% 0.14/0.42 thf(fact_223_set__eq__subset,axiom,
% 0.14/0.42 ! [A: $tType] :
% 0.14/0.42 ( ( ^ [Y2: set @ A,Z2: set @ A] : ( Y2 = Z2 ) )
% 0.14/0.42 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.14/0.42 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % set_eq_subset
% 0.14/0.42 thf(fact_224_subset__trans,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
% 0.14/0.42 => ( ord_less_eq @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % subset_trans
% 0.14/0.42 thf(fact_225_Collect__mono,axiom,
% 0.14/0.42 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.14/0.42 ( ! [X3: A] :
% 0.14/0.42 ( ( P @ X3 )
% 0.14/0.42 => ( Q @ X3 ) )
% 0.14/0.42 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % Collect_mono
% 0.14/0.42 thf(fact_226_subset__refl,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 0.14/0.42
% 0.14/0.42 % subset_refl
% 0.14/0.42 thf(fact_227_rev__subsetD,axiom,
% 0.14/0.42 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( member @ A @ C @ A4 )
% 0.14/0.42 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ( member @ A @ C @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % rev_subsetD
% 0.14/0.42 thf(fact_228_subset__iff,axiom,
% 0.14/0.42 ! [A: $tType] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) )
% 0.14/0.42 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.14/0.42 ! [T: A] :
% 0.14/0.42 ( ( member @ A @ T @ A5 )
% 0.14/0.42 => ( member @ A @ T @ B5 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % subset_iff
% 0.14/0.42 thf(fact_229_set__rev__mp,axiom,
% 0.14/0.42 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( member @ A @ X @ A4 )
% 0.14/0.42 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ( member @ A @ X @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % set_rev_mp
% 0.14/0.42 thf(fact_230_equalityD2,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( A4 = B4 )
% 0.14/0.42 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 0.14/0.42
% 0.14/0.42 % equalityD2
% 0.14/0.42 thf(fact_231_equalityD1,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( A4 = B4 )
% 0.14/0.42 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.14/0.42
% 0.14/0.42 % equalityD1
% 0.14/0.42 thf(fact_232_subset__eq,axiom,
% 0.14/0.42 ! [A: $tType] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) )
% 0.14/0.42 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.14/0.42 ! [X2: A] :
% 0.14/0.42 ( ( member @ A @ X2 @ A5 )
% 0.14/0.42 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % subset_eq
% 0.14/0.42 thf(fact_233_equalityE,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( A4 = B4 )
% 0.14/0.42 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % equalityE
% 0.14/0.42 thf(fact_234_subsetCE,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ( ( member @ A @ C @ A4 )
% 0.14/0.42 => ( member @ A @ C @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % subsetCE
% 0.14/0.42 thf(fact_235_subsetD,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ( ( member @ A @ C @ A4 )
% 0.14/0.42 => ( member @ A @ C @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % subsetD
% 0.14/0.42 thf(fact_236_in__mono,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ( ( member @ A @ X @ A4 )
% 0.14/0.42 => ( member @ A @ X @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % in_mono
% 0.14/0.42 thf(fact_237_set__mp,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 0.14/0.42 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.42 => ( ( member @ A @ X @ A4 )
% 0.14/0.42 => ( member @ A @ X @ B4 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % set_mp
% 0.14/0.42 thf(fact_238_Int__left__commute,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
% 0.14/0.42 ( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) )
% 0.14/0.42 = ( inf_inf @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % Int_left_commute
% 0.14/0.42 thf(fact_239_Int__left__absorb,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.14/0.42 ( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 0.14/0.42 = ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ).
% 0.14/0.42
% 0.14/0.42 % Int_left_absorb
% 0.14/0.42 thf(fact_240_Int__commute,axiom,
% 0.14/0.42 ! [A: $tType] :
% 0.14/0.42 ( ( inf_inf @ ( set @ A ) )
% 0.14/0.42 = ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 0.14/0.42
% 0.14/0.42 % Int_commute
% 0.14/0.42 thf(fact_241_Int__absorb,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A] :
% 0.14/0.42 ( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
% 0.14/0.42 = A4 ) ).
% 0.14/0.42
% 0.14/0.42 % Int_absorb
% 0.14/0.42 thf(fact_242_Int__assoc,axiom,
% 0.14/0.42 ! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
% 0.14/0.42 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ C2 )
% 0.14/0.42 = ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% 0.14/0.43
% 0.14/0.43 % Int_assoc
% 0.14/0.43 thf(fact_243_IntD2,axiom,
% 0.14/0.43 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 0.14/0.43 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 0.14/0.43 => ( member @ A @ C @ B4 ) ) ).
% 0.14/0.43
% 0.14/0.43 % IntD2
% 0.14/0.43 thf(fact_244_IntD1,axiom,
% 0.14/0.43 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 0.14/0.43 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 0.14/0.43 => ( member @ A @ C @ A4 ) ) ).
% 0.14/0.43
% 0.14/0.43 % IntD1
% 0.14/0.43 thf(fact_245_IntE,axiom,
% 0.14/0.43 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 0.14/0.43 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 0.14/0.43 => ~ ( ( member @ A @ C @ A4 )
% 0.14/0.43 => ~ ( member @ A @ C @ B4 ) ) ) ).
% 0.14/0.43
% 0.14/0.43 % IntE
% 0.14/0.43 thf(fact_246_less__eq__set__def,axiom,
% 0.14/0.43 ! [A: $tType] :
% 0.14/0.43 ( ( ord_less_eq @ ( set @ A ) )
% 0.14/0.43 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.14/0.43 ( ord_less_eq @ ( A > $o )
% 0.14/0.43 @ ^ [X2: A] : ( member @ A @ X2 @ A5 )
% 0.14/0.43 @ ^ [X2: A] : ( member @ A @ X2 @ B5 ) ) ) ) ).
% 0.14/0.43
% 0.14/0.43 % less_eq_set_def
% 0.14/0.43 thf(fact_247_Int__def,axiom,
% 0.14/0.43 ! [A: $tType] :
% 0.14/0.43 ( ( inf_inf @ ( set @ A ) )
% 0.14/0.43 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.14/0.43 ( collect @ A
% 0.14/0.43 @ ^ [X2: A] :
% 0.14/0.43 ( ( member @ A @ X2 @ A5 )
% 0.14/0.43 & ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% 0.14/0.43
% 0.14/0.43 % Int_def
% 0.14/0.43 thf(fact_248_Int__Collect,axiom,
% 0.14/0.43 ! [A: $tType,X: A,A4: set @ A,P: A > $o] :
% 0.14/0.43 ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
% 0.14/0.43 = ( ( member @ A @ X @ A4 )
% 0.14/0.43 & ( P @ X ) ) ) ).
% 0.14/0.43
% 0.14/0.43 % Int_Collect
% 0.14/0.43 thf(fact_249_Collect__conj__eq,axiom,
% 0.14/0.43 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.14/0.43 ( ( collect @ A
% 0.14/0.43 @ ^ [X2: A] :
% 0.14/0.43 ( ( P @ X2 )
% 0.14/0.43 & ( Q @ X2 ) ) )
% 0.14/0.43 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.14/0.43
% 0.14/0.43 % Collect_conj_eq
% 0.14/0.43 thf(fact_250_Int__Collect__mono,axiom,
% 0.14/0.43 ! [A: $tType,A4: set @ A,B4: set @ A,P: A > $o,Q: A > $o] :
% 0.14/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.43 => ( ! [X3: A] :
% 0.14/0.43 ( ( member @ A @ X3 @ A4 )
% 0.14/0.43 => ( ( P @ X3 )
% 0.14/0.43 => ( Q @ X3 ) ) )
% 0.14/0.43 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B4 @ ( collect @ A @ Q ) ) ) ) ) ).
% 0.14/0.43
% 0.14/0.43 % Int_Collect_mono
% 0.14/0.43 thf(fact_251_Int__greatest,axiom,
% 0.14/0.43 ! [A: $tType,C2: set @ A,A4: set @ A,B4: set @ A] :
% 0.14/0.43 ( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
% 0.14/0.43 => ( ( ord_less_eq @ ( set @ A ) @ C2 @ B4 )
% 0.14/0.43 => ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% 0.14/0.43
% 0.14/0.43 % Int_greatest
% 0.14/0.43 thf(fact_252_Int__absorb2,axiom,
% 0.14/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 0.14/0.43 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.43 => ( ( inf_inf @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.43 = A4 ) ) ).
% 0.14/0.43
% 0.14/0.43 % Int_absorb2
% 0.14/0.43 thf(fact_253_Int__absorb1,axiom,
% 0.14/0.43 ! [A: $tType,B4: set @ A,A4: set @ A] :
% 0.14/0.43 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 0.14/0.43 => ( ( inf_inf @ ( set @ A ) @ A4 @ B4 )
% 0.14/0.43 = B4 ) ) ).
% 0.14/0.43
% 0.14/0.43 % Int_absorb1
% 0.14/0.43 thf(fact_254_Int__lower2,axiom,
% 0.14/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ B4 ) ).
% 0.14/0.43
% 0.14/0.43 % Int_lower2
% 0.14/0.43 thf(fact_255_Int__lower1,axiom,
% 0.14/0.43 ! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ A4 ) ).
% 0.14/0.43
% 0.14/0.43 % Int_lower1
% 0.14/0.43
% 0.14/0.43 %----Type constructors (4)
% 0.14/0.43 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.14/0.43 ! [A6: $tType,A7: $tType] :
% 0.14/0.43 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.14/0.43 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.14/0.43
% 0.14/0.43 thf(tcon_Set_Oset___Orderings_Oorder_1,axiom,
% 0.14/0.43 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.14/0.43
% 0.14/0.43 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.14/0.43 linorder @ $o @ ( type2 @ $o ) ).
% 0.14/0.43
% 0.14/0.43 thf(tcon_HOL_Obool___Orderings_Oorder_2,axiom,
% 0.14/0.43 order @ $o @ ( type2 @ $o ) ).
% 0.14/0.43
% 0.14/0.43 %----Conjectures (2)
% 0.14/0.43 thf(conj_0,hypothesis,
% 0.14/0.43 ! [Y5: a] :
% 0.14/0.43 ( ( member @ a @ Y5 @ ( coinductive_lset @ a @ ysa ) )
% 0.14/0.43 => ( ( less_eq @ x @ Y5 )
% 0.14/0.43 => thesis ) ) ).
% 0.14/0.43
% 0.14/0.43 thf(conj_1,conjecture,
% 0.14/0.43 thesis ).
% 0.14/0.43
% 0.14/0.43 %------------------------------------------------------------------------------
% 0.14/0.43 ------------------------
% 2.42/1.43 ---- Embedded in THF ---
% 2.42/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.42/1.43 %%% Generated on Tue Feb 24 06:09:34 EST 2026
% 2.42/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 2.42/1.43 %%% Logic specification used:
% 2.42/1.43 %%% thf(spec, logic, $$dhol).
% 2.42/1.43
% 2.42/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.0EB1vRga9H/DTF2DTF29649.p
% 2.42/1.43 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 2.42/1.43 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.42/1.43 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.42/1.43 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.42/1.43 thf(ty_tf_a, type, a: $tType).
% 2.42/1.43 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.42/1.43 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.42/1.43 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.42/1.43 thf(sy_c_Coinductive__List_Oldistinct, type, coindu351974385stinct: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.42/1.43 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.42/1.43 thf(sy_c_Coinductive__List_Ollist_OLCons, type, coinductive_LCons: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.42/1.43 thf(sy_c_Coinductive__List_Ollist_Olset, type, coinductive_lset: (!> [A:$tType]: (((coinductive_llist @ A) > (set @ A))))).
% 2.42/1.43 thf(sy_c_Coinductive__List_Oord_Olsorted, type, coinductive_lsorted: (!> [A:$tType]: (((A > (A > $o)) > ((coinductive_llist @ A) > $o))))).
% 2.42/1.43 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above, type, condit2040224947_above: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 2.42/1.43 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below, type, condit1201339847_below: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 2.42/1.43 thf(sy_c_Finite__Set_Ofolding, type, finite_folding: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 2.42/1.43 thf(sy_c_Groups_Osemigroup, type, semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.42/1.43 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord_Olmerge, type, hammin1328233080lmerge: (!> [A:$tType]: (((A > (A > $o)) > ((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A))))))).
% 2.42/1.43 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.42/1.43 thf(sy_c_Lattices_Osemilattice, type, semilattice: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.42/1.43 thf(sy_c_Lattices__Big_Osemilattice__order__set, type, lattic1693879045er_set: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > $o)) > $o)))))).
% 2.42/1.43 thf(sy_c_List_Olinorder_Oinsort__key, type, insort_key: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > A) > (B > ((list @ B) > (list @ B)))))))).
% 2.42/1.43 thf(sy_c_Orderings_Oord_OLeast, type, least: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > A))))).
% 2.42/1.43 thf(sy_c_Orderings_Oord_Omin, type, min: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 2.42/1.43 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.42/1.43 thf(sy_c_Orderings_Oorder_Oantimono, type, antimono: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > $o))))).
% 2.42/1.43 thf(sy_c_Orderings_Oorder_Omono, type, mono: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > $o))))).
% 2.42/1.43 thf(sy_c_Orderings_Oorder_Ostrict__mono, type, strict_mono: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > $o))))).
% 2.42/1.43 thf(sy_c_Orderings_Oordering, type, ordering: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 2.42/1.43 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.42/1.43 thf(sy_c_Relation_Otransp, type, transp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 2.42/1.43 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.42/1.43 thf(sy_c_Set__Interval_Oord_OatLeast, type, set_atLeast: (!> [A:$tType]: (((A > (A > $o)) > (A > (set @ A)))))).
% 2.42/1.43 thf(sy_c_Set__Interval_Oord_OatLeastAtMost, type, set_atLeastAtMost: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > (set @ A))))))).
% 2.42/1.43 thf(sy_c_Set__Interval_Oord_OatLeastLessThan, type, set_atLeastLessThan: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > (A > (A > (set @ A)))))))).
% 2.42/1.43 thf(sy_c_Set__Interval_Oord_OatMost, type, set_atMost: (!> [A:$tType]: (((A > (A > $o)) > (A > (set @ A)))))).
% 2.42/1.43 thf(sy_c_Set__Interval_Oord_OgreaterThan, type, set_greaterThan: (!> [A:$tType]: (((A > (A > $o)) > (A > (set @ A)))))).
% 2.42/1.43 thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost, type, set_gr323396891AtMost: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > (A > (A > (set @ A)))))))).
% 2.42/1.43 thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan, type, set_gr1161524159ssThan: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > (set @ A))))))).
% 2.42/1.43 thf(sy_c_Set__Interval_Oord_OlessThan, type, set_lessThan: (!> [A:$tType]: (((A > (A > $o)) > (A > (set @ A)))))).
% 2.42/1.43 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.42/1.43 thf(sy_v_less, type, less: (a > (a > $o))).
% 2.42/1.43 thf(sy_v_less__eq, type, less_eq: (a > (a > $o))).
% 2.42/1.43 thf(sy_v_thesis____, type, thesis: $o).
% 2.42/1.43 thf(sy_v_x, type, x: a).
% 2.42/1.43 thf(sy_v_x_H____, type, x2: a).
% 2.42/1.43 thf(sy_v_xsa____, type, xsa: (coinductive_llist @ a)).
% 2.42/1.43 thf(sy_v_ysa____, type, ysa: (coinductive_llist @ a)).
% 2.42/1.43 thf(fact_0_local_Oantisym__conv3, axiom, (! [Y:a,X:a]: (((~ (((less @ Y) @ X))) => ((~ (((less @ X) @ Y))) = (X = Y)))))).
% 2.42/1.43 thf(fact_1_local_Odual__order_Oasym, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) => (~ (((less @ A2) @ B2))))))).
% 2.42/1.43 thf(fact_2_local_Odual__order_Ostrict__implies__not__eq, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) => (A2 != B2))))).
% 2.42/1.43 thf(fact_3_local_Odual__order_Ostrict__trans, axiom, (! [B2:a,A2:a,C:a]: ((((less @ B2) @ A2) => (((less @ C) @ B2) => ((less @ C) @ A2)))))).
% 2.42/1.43 thf(fact_4_local_Oless__asym, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (~ (((less @ Y) @ X))))))).
% 2.42/1.43 thf(fact_5_local_Oless__asym_H, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => (~ (((less @ B2) @ A2))))))).
% 2.42/1.43 thf(fact_6_local_Oless__imp__neq, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (X != Y))))).
% 2.42/1.43 thf(fact_7_local_Oless__imp__not__eq, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (X != Y))))).
% 2.42/1.43 thf(fact_8_local_Oless__imp__not__eq2, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (Y != X))))).
% 2.42/1.43 thf(fact_9_local_Oless__imp__not__less, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (~ (((less @ Y) @ X))))))).
% 2.42/1.43 thf(fact_10_local_Oless__imp__triv, axiom, (! [X:a,Y:a,P:$o]: ((((less @ X) @ Y) => (((less @ Y) @ X) => P))))).
% 2.42/1.43 thf(fact_11_local_Oless__irrefl, axiom, (! [X:a]: ((~ (((less @ X) @ X)))))).
% 2.42/1.43 thf(fact_12_local_Oless__linear, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) | ((X = Y) | ((less @ Y) @ X)))))).
% 2.42/1.43 thf(fact_13_local_Oless__not__sym, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (~ (((less @ Y) @ X))))))).
% 2.42/1.43 thf(fact_14_local_Oless__trans, axiom, (! [X:a,Y:a,Z:a]: ((((less @ X) @ Y) => (((less @ Y) @ Z) => ((less @ X) @ Z)))))).
% 2.42/1.43 thf(fact_15_local_Olinorder__cases, axiom, (! [X:a,Y:a]: (((~ (((less @ X) @ Y))) => ((X != Y) => ((less @ Y) @ X)))))).
% 2.42/1.43 thf(fact_16_local_OneqE, axiom, (! [X:a,Y:a]: (((X != Y) => ((~ (((less @ X) @ Y))) => ((less @ Y) @ X)))))).
% 2.42/1.43 thf(fact_17_local_Oneq__iff, axiom, (! [X:a,Y:a]: (((X != Y) = (((less @ X) @ Y) | ((less @ Y) @ X)))))).
% 2.42/1.43 thf(fact_18_local_Onot__less__iff__gr__or__eq, axiom, (! [X:a,Y:a]: (((~ (((less @ X) @ Y))) = (((less @ Y) @ X) | (X = Y)))))).
% 2.42/1.43 thf(fact_19_local_Oord__eq__less__trans, axiom, (! [A2:a,B2:a,C:a]: (((A2 = B2) => (((less @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.42/1.43 thf(fact_20_local_Oord__less__eq__trans, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ B2) => ((B2 = C) => ((less @ A2) @ C)))))).
% 2.42/1.43 thf(fact_21_local_Oorder_Oasym, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => (~ (((less @ B2) @ A2))))))).
% 2.42/1.43 thf(fact_22_local_Oorder_Oirrefl, axiom, (! [A2:a]: ((~ (((less @ A2) @ A2)))))).
% 2.42/1.43 thf(fact_23_local_Oorder_Ostrict__implies__not__eq, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => (A2 != B2))))).
% 2.42/1.43 thf(fact_24_local_Oorder_Ostrict__trans, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ B2) => (((less @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.42/1.43 thf(fact_25_step_Ohyps_I2_J, axiom, (x != x2)).
% 2.42/1.43 thf(fact_26_step_Oprems_I2_J, axiom, (~ (((coinductive_lfinite @ a) @ ysa)))).
% 2.42/1.43 thf(fact_27_step_Ohyps_I1_J, axiom, (((member @ a) @ x) @ ((coinductive_lset @ a) @ xsa))).
% 2.42/1.43 thf(fact_28_local_Oantisym, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => (((less_eq @ Y) @ X) => (X = Y)))))).
% 2.42/1.43 thf(fact_29_local_Oantisym__conv, axiom, (! [Y:a,X:a]: ((((less_eq @ Y) @ X) => (((less_eq @ X) @ Y) = (X = Y)))))).
% 2.42/1.43 thf(fact_30_local_Odual__order_Oantisym, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (((less_eq @ A2) @ B2) => (A2 = B2)))))).
% 2.42/1.43 thf(fact_31_local_Odual__order_Otrans, axiom, (! [B2:a,A2:a,C:a]: ((((less_eq @ B2) @ A2) => (((less_eq @ C) @ B2) => ((less_eq @ C) @ A2)))))).
% 2.42/1.43 thf(fact_32_local_Oeq__iff, axiom, ((^ [Y2:a,Z2:a]: ((Y2 = Z2))) = (^ [X2:a,Y3:a]: ((((less_eq @ X2) @ Y3) & ((less_eq @ Y3) @ X2)))))).
% 2.42/1.43 thf(fact_33_local_Oeq__refl, axiom, (! [X:a,Y:a]: (((X = Y) => ((less_eq @ X) @ Y))))).
% 2.42/1.43 thf(fact_34_local_Ole__cases, axiom, (! [X:a,Y:a]: (((~ (((less_eq @ X) @ Y))) => ((less_eq @ Y) @ X))))).
% 2.42/1.43 thf(fact_35_local_Ole__cases3, axiom, (! [X:a,Y:a,Z:a]: (((((less_eq @ X) @ Y) => (~ (((less_eq @ Y) @ Z)))) => ((((less_eq @ Y) @ X) => (~ (((less_eq @ X) @ Z)))) => ((((less_eq @ X) @ Z) => (~ (((less_eq @ Z) @ Y)))) => ((((less_eq @ Z) @ Y) => (~ (((less_eq @ Y) @ X)))) => ((((less_eq @ Y) @ Z) => (~ (((less_eq @ Z) @ X)))) => (~ ((((less_eq @ Z) @ X) => (~ (((less_eq @ X) @ Y)))))))))))))).
% 2.42/1.43 thf(fact_36_local_Olinear, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) | ((less_eq @ Y) @ X))))).
% 2.42/1.43 thf(fact_37_local_Olinorder__wlog, axiom, (! [P:(a > (a > $o)),A2:a,B2:a]: (((! [A3:a,B3:a]: ((((less_eq @ A3) @ B3) => ((P @ A3) @ B3)))) => ((! [A3:a,B3:a]: ((((P @ B3) @ A3) => ((P @ A3) @ B3)))) => ((P @ A2) @ B2)))))).
% 2.42/1.43 thf(fact_38_local_Oord__eq__le__trans, axiom, (! [A2:a,B2:a,C:a]: (((A2 = B2) => (((less_eq @ B2) @ C) => ((less_eq @ A2) @ C)))))).
% 2.42/1.43 thf(fact_39_local_Oord__le__eq__trans, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => ((B2 = C) => ((less_eq @ A2) @ C)))))).
% 2.42/1.43 thf(fact_40_local_Oorder_Oantisym, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (((less_eq @ B2) @ A2) => (A2 = B2)))))).
% 2.42/1.43 thf(fact_41_local_Oorder_Otrans, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => (((less_eq @ B2) @ C) => ((less_eq @ A2) @ C)))))).
% 2.42/1.43 thf(fact_42_local_Oorder__trans, axiom, (! [X:a,Y:a,Z:a]: ((((less_eq @ X) @ Y) => (((less_eq @ Y) @ Z) => ((less_eq @ X) @ Z)))))).
% 2.42/1.43 thf(fact_43_step_Oprems_I3_J, axiom, (? [X3:a]: (((((member @ a) @ X3) @ ((coinductive_lset @ a) @ ysa)) & ((less_eq @ x) @ X3))))).
% 2.42/1.43 thf(fact_44_local_Oantisym__conv1, axiom, (! [X:a,Y:a]: (((~ (((less @ X) @ Y))) => (((less_eq @ X) @ Y) = (X = Y)))))).
% 2.42/1.43 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.42/1.43 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A4)))) = A4)))).
% 2.42/1.43 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.42/1.43 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.42/1.43 thf(fact_49_local_Oantisym__conv2, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => ((~ (((less @ X) @ Y))) = (X = Y)))))).
% 2.42/1.43 thf(fact_50_local_Odual__order_Onot__eq__order__implies__strict, axiom, (! [A2:a,B2:a]: (((A2 != B2) => (((less_eq @ B2) @ A2) => ((less @ B2) @ A2)))))).
% 2.42/1.43 thf(fact_51_local_Odual__order_Oorder__iff__strict, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) = (((less @ B2) @ A2) | (A2 = B2)))))).
% 2.42/1.43 thf(fact_52_local_Odual__order_Ostrict__iff__order, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) = (((less_eq @ B2) @ A2) & (A2 != B2)))))).
% 2.42/1.43 thf(fact_53_local_Odual__order_Ostrict__implies__order, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) => ((less_eq @ B2) @ A2))))).
% 2.42/1.43 thf(fact_54_local_Odual__order_Ostrict__trans1, axiom, (! [B2:a,A2:a,C:a]: ((((less_eq @ B2) @ A2) => (((less @ C) @ B2) => ((less @ C) @ A2)))))).
% 2.42/1.43 thf(fact_55_local_Odual__order_Ostrict__trans2, axiom, (! [B2:a,A2:a,C:a]: ((((less @ B2) @ A2) => (((less_eq @ C) @ B2) => ((less @ C) @ A2)))))).
% 2.42/1.43 thf(fact_56_local_OleD, axiom, (! [Y:a,X:a]: ((((less_eq @ Y) @ X) => (~ (((less @ X) @ Y))))))).
% 2.42/1.43 thf(fact_57_local_OleI, axiom, (! [X:a,Y:a]: (((~ (((less @ X) @ Y))) => ((less_eq @ Y) @ X))))).
% 2.42/1.43 thf(fact_58_local_Ole__imp__less__or__eq, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) => (((less @ X) @ Y) | (X = Y)))))).
% 2.42/1.43 thf(fact_59_local_Ole__less, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) = (((less @ X) @ Y) | (X = Y)))))).
% 2.42/1.43 thf(fact_60_local_Ole__less__linear, axiom, (! [X:a,Y:a]: ((((less_eq @ X) @ Y) | ((less @ Y) @ X))))).
% 2.42/1.43 thf(fact_61_local_Ole__less__trans, axiom, (! [X:a,Y:a,Z:a]: ((((less_eq @ X) @ Y) => (((less @ Y) @ Z) => ((less @ X) @ Z)))))).
% 2.42/1.43 thf(fact_62_local_Ole__neq__trans, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => ((A2 != B2) => ((less @ A2) @ B2)))))).
% 2.42/1.43 thf(fact_63_local_Oless__imp__le, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => ((less_eq @ X) @ Y))))).
% 2.42/1.43 thf(fact_64_local_Oless__le, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) = (((less_eq @ X) @ Y) & (X != Y)))))).
% 2.42/1.43 thf(fact_65_local_Oless__le__not__le, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) = (((less_eq @ X) @ Y) & (~ (((less_eq @ Y) @ X)))))))).
% 2.42/1.43 thf(fact_66_local_Oless__le__trans, axiom, (! [X:a,Y:a,Z:a]: ((((less @ X) @ Y) => (((less_eq @ Y) @ Z) => ((less @ X) @ Z)))))).
% 2.42/1.43 thf(fact_67_local_Onot__le, axiom, (! [X:a,Y:a]: (((~ (((less_eq @ X) @ Y))) = ((less @ Y) @ X))))).
% 2.42/1.43 thf(fact_68_local_Onot__le__imp__less, axiom, (! [Y:a,X:a]: (((~ (((less_eq @ Y) @ X))) => ((less @ X) @ Y))))).
% 2.42/1.43 thf(fact_69_local_Onot__less, axiom, (! [X:a,Y:a]: (((~ (((less @ X) @ Y))) = ((less_eq @ Y) @ X))))).
% 2.42/1.43 thf(fact_70_local_Oorder_Onot__eq__order__implies__strict, axiom, (! [A2:a,B2:a]: (((A2 != B2) => (((less_eq @ A2) @ B2) => ((less @ A2) @ B2)))))).
% 2.42/1.43 thf(fact_71_local_Oorder_Oorder__iff__strict, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (((less @ A2) @ B2) | (A2 = B2)))))).
% 2.42/1.43 thf(fact_72_local_Oorder_Ostrict__iff__order, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) = (((less_eq @ A2) @ B2) & (A2 != B2)))))).
% 2.42/1.43 thf(fact_73_local_Oorder_Ostrict__implies__order, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => ((less_eq @ A2) @ B2))))).
% 2.42/1.43 thf(fact_74_local_Oorder_Ostrict__trans1, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => (((less @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.42/1.43 thf(fact_75_local_Oorder_Ostrict__trans2, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ B2) => (((less_eq @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.42/1.43 thf(fact_76_local_Oorder_Orefl, axiom, (! [A2:a]: (((less_eq @ A2) @ A2)))).
% 2.42/1.43 thf(fact_77_local_Oorder__refl, axiom, (! [X:a]: (((less_eq @ X) @ X)))).
% 2.42/1.43 thf(fact_78_local_OLeastI2__order, axiom, (! [P:(a > $o),X:a,Q:(a > $o)]: (((P @ X) => ((! [Y4:a]: (((P @ Y4) => ((less_eq @ X) @ Y4)))) => ((! [X3:a]: (((P @ X3) => ((! [Y5:a]: (((P @ Y5) => ((less_eq @ X3) @ Y5)))) => (Q @ X3))))) => (Q @ (((least @ a) @ less_eq) @ P)))))))).
% 2.42/1.43 thf(fact_79_local_OLeast__equality, axiom, (! [P:(a > $o),X:a]: (((P @ X) => ((! [Y4:a]: (((P @ Y4) => ((less_eq @ X) @ Y4)))) => ((((least @ a) @ less_eq) @ P) = X)))))).
% 2.42/1.43 thf(fact_80_local_Olsorted_OLCons__LCons, axiom, (! [X:a,Y:a,Xs:(coinductive_llist @ a)]: ((((less_eq @ X) @ Y) => ((((coinductive_lsorted @ a) @ less_eq) @ (((coinductive_LCons @ a) @ Y) @ Xs)) => (((coinductive_lsorted @ a) @ less_eq) @ (((coinductive_LCons @ a) @ X) @ (((coinductive_LCons @ a) @ Y) @ Xs)))))))).
% 2.42/1.43 thf(fact_81_local_Olsorted__LCons, axiom, (! [X:a,Xs:(coinductive_llist @ a)]: (((((coinductive_lsorted @ a) @ less_eq) @ (((coinductive_LCons @ a) @ X) @ Xs)) = ((((coinductive_lsorted @ a) @ less_eq) @ Xs) & (! [X2:a]: (((((member @ a) @ X2) @ ((coinductive_lset @ a) @ Xs)) => ((less_eq @ X) @ X2))))))))).
% 2.42/1.43 thf(fact_82_local_Oin__lset__lmergeD, axiom, (! [X:a,Xs:(coinductive_llist @ a),Ys:(coinductive_llist @ a)]: (((((member @ a) @ X) @ ((coinductive_lset @ a) @ ((((hammin1328233080lmerge @ a) @ less) @ Xs) @ Ys))) => ((((member @ a) @ X) @ ((coinductive_lset @ a) @ Xs)) | (((member @ a) @ X) @ ((coinductive_lset @ a) @ Ys))))))).
% 2.42/1.43 thf(fact_83_local_Obdd__above__def, axiom, (! [A4:(set @ a)]: (((((condit2040224947_above @ a) @ less_eq) @ A4) = (? [M:a]: ((! [X2:a]: (((((member @ a) @ X2) @ A4) => ((less_eq @ X2) @ M)))))))))).
% 2.42/1.43 thf(fact_84_local_Obdd__below__def, axiom, (! [A4:(set @ a)]: (((((condit1201339847_below @ a) @ less_eq) @ A4) = (? [M2:a]: ((! [X2:a]: (((((member @ a) @ X2) @ A4) => ((less_eq @ M2) @ X2)))))))))).
% 2.42/1.43 thf(fact_85_local_Oinsort__key__left__comm, axiom, (! [B:$tType,F:(B > a),X:B,Y:B,Xs:(list @ B)]: ((((F @ X) != (F @ Y)) => (((((((insort_key @ a) @ B) @ less_eq) @ F) @ Y) @ ((((((insort_key @ a) @ B) @ less_eq) @ F) @ X) @ Xs)) = ((((((insort_key @ a) @ B) @ less_eq) @ F) @ X) @ ((((((insort_key @ a) @ B) @ less_eq) @ F) @ Y) @ Xs))))))).
% 2.42/1.43 thf(fact_86_local_Otransp__le, axiom, ((transp @ a) @ less_eq)).
% 2.42/1.43 thf(fact_87_ord_Oin__lset__lmergeD, axiom, (! [A:$tType,X:A,Less:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys))) => ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) | (((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys))))))).
% 2.42/1.43 thf(fact_88_local_Olsorted__ldistinct__lset__unique, axiom, (! [Xs:(coinductive_llist @ a),Ys:(coinductive_llist @ a)]: (((((coinductive_lsorted @ a) @ less_eq) @ Xs) => (((coindu351974385stinct @ a) @ Xs) => ((((coinductive_lsorted @ a) @ less_eq) @ Ys) => (((coindu351974385stinct @ a) @ Ys) => ((((coinductive_lset @ a) @ Xs) = ((coinductive_lset @ a) @ Ys)) => (Xs = Ys))))))))).
% 2.42/1.43 thf(fact_89_local_Olsorted__LCons__LCons, axiom, (! [X:a,Y:a,Xs:(coinductive_llist @ a)]: (((((coinductive_lsorted @ a) @ less_eq) @ (((coinductive_LCons @ a) @ X) @ (((coinductive_LCons @ a) @ Y) @ Xs))) = (((less_eq @ X) @ Y) & (((coinductive_lsorted @ a) @ less_eq) @ (((coinductive_LCons @ a) @ Y) @ Xs))))))).
% 2.42/1.43 thf(fact_90_local_Omin_Oabsorb1, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (((((min @ a) @ less_eq) @ A2) @ B2) = A2))))).
% 2.42/1.43 thf(fact_91_local_Omin_Oabsorb2, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) => (((((min @ a) @ less_eq) @ A2) @ B2) = B2))))).
% 2.42/1.43 thf(fact_92_local_Omin_Oabsorb__iff1, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (((((min @ a) @ less_eq) @ A2) @ B2) = A2))))).
% 2.42/1.43 thf(fact_93_local_Omin_Oabsorb__iff2, axiom, (! [B2:a,A2:a]: ((((less_eq @ B2) @ A2) = (((((min @ a) @ less_eq) @ A2) @ B2) = B2))))).
% 2.42/1.43 thf(fact_94_local_Omin__le__iff__disj, axiom, (! [X:a,Y:a,Z:a]: ((((less_eq @ ((((min @ a) @ less_eq) @ X) @ Y)) @ Z) = (((less_eq @ X) @ Z) | ((less_eq @ Y) @ Z)))))).
% 2.42/1.43 thf(fact_95_local_Omin__def, axiom, (! [A2:a,B2:a]: (((((less_eq @ A2) @ B2) => (((((min @ a) @ less_eq) @ A2) @ B2) = A2)) & ((~ (((less_eq @ A2) @ B2))) => (((((min @ a) @ less_eq) @ A2) @ B2) = B2)))))).
% 2.42/1.43 thf(fact_96_local_Omin_Oorder__iff, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) = (A2 = ((((min @ a) @ less_eq) @ A2) @ B2)))))).
% 2.42/1.43 thf(fact_97_local_Omin_OorderI, axiom, (! [A2:a,B2:a]: (((A2 = ((((min @ a) @ less_eq) @ A2) @ B2)) => ((less_eq @ A2) @ B2))))).
% 2.42/1.43 thf(fact_98_local_Omin_OorderE, axiom, (! [A2:a,B2:a]: ((((less_eq @ A2) @ B2) => (A2 = ((((min @ a) @ less_eq) @ A2) @ B2)))))).
% 2.42/1.43 thf(fact_99_local_Omin_Omono, axiom, (! [A2:a,C:a,B2:a,D:a]: ((((less_eq @ A2) @ C) => (((less_eq @ B2) @ D) => ((less_eq @ ((((min @ a) @ less_eq) @ A2) @ B2)) @ ((((min @ a) @ less_eq) @ C) @ D))))))).
% 2.42/1.43 thf(fact_100_local_Omin_Oleft__commute, axiom, (! [B2:a,A2:a,C:a]: ((((((min @ a) @ less_eq) @ B2) @ ((((min @ a) @ less_eq) @ A2) @ C)) = ((((min @ a) @ less_eq) @ A2) @ ((((min @ a) @ less_eq) @ B2) @ C)))))).
% 2.42/1.43 thf(fact_101_local_Omin_Ocommute, axiom, (! [A2:a,B2:a]: ((((((min @ a) @ less_eq) @ A2) @ B2) = ((((min @ a) @ less_eq) @ B2) @ A2))))).
% 2.42/1.43 thf(fact_102_local_Omin_OcoboundedI2, axiom, (! [B2:a,C:a,A2:a]: ((((less_eq @ B2) @ C) => ((less_eq @ ((((min @ a) @ less_eq) @ A2) @ B2)) @ C))))).
% 2.42/1.43 thf(fact_103_local_Omin_OcoboundedI1, axiom, (! [A2:a,C:a,B2:a]: ((((less_eq @ A2) @ C) => ((less_eq @ ((((min @ a) @ less_eq) @ A2) @ B2)) @ C))))).
% 2.42/1.43 thf(fact_104_local_Omin_Ocobounded2, axiom, (! [A2:a,B2:a]: (((less_eq @ ((((min @ a) @ less_eq) @ A2) @ B2)) @ B2)))).
% 2.42/1.43 thf(fact_105_local_Omin_Ocobounded1, axiom, (! [A2:a,B2:a]: (((less_eq @ ((((min @ a) @ less_eq) @ A2) @ B2)) @ A2)))).
% 2.42/1.43 thf(fact_106_local_Omin_OboundedI, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ B2) => (((less_eq @ A2) @ C) => ((less_eq @ A2) @ ((((min @ a) @ less_eq) @ B2) @ C))))))).
% 2.42/1.43 thf(fact_107_local_Omin_OboundedE, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ ((((min @ a) @ less_eq) @ B2) @ C)) => (~ ((((less_eq @ A2) @ B2) => (~ (((less_eq @ A2) @ C)))))))))).
% 2.42/1.43 thf(fact_108_local_Omin_Oassoc, axiom, (! [A2:a,B2:a,C:a]: ((((((min @ a) @ less_eq) @ ((((min @ a) @ less_eq) @ A2) @ B2)) @ C) = ((((min @ a) @ less_eq) @ A2) @ ((((min @ a) @ less_eq) @ B2) @ C)))))).
% 2.42/1.43 thf(fact_109_local_Olmerge__simps, axiom, (! [X:a,Y:a,Xs:(coinductive_llist @ a),Ys:(coinductive_llist @ a)]: (((((less @ X) @ Y) => (((((hammin1328233080lmerge @ a) @ less) @ (((coinductive_LCons @ a) @ X) @ Xs)) @ (((coinductive_LCons @ a) @ Y) @ Ys)) = (((coinductive_LCons @ a) @ X) @ ((((hammin1328233080lmerge @ a) @ less) @ Xs) @ (((coinductive_LCons @ a) @ Y) @ Ys))))) & ((~ (((less @ X) @ Y))) => ((((less @ Y) @ X) => (((((hammin1328233080lmerge @ a) @ less) @ (((coinductive_LCons @ a) @ X) @ Xs)) @ (((coinductive_LCons @ a) @ Y) @ Ys)) = (((coinductive_LCons @ a) @ Y) @ ((((hammin1328233080lmerge @ a) @ less) @ (((coinductive_LCons @ a) @ X) @ Xs)) @ Ys)))) & ((~ (((less @ Y) @ X))) => (((((hammin1328233080lmerge @ a) @ less) @ (((coinductive_LCons @ a) @ X) @ Xs)) @ (((coinductive_LCons @ a) @ Y) @ Ys)) = (((coinductive_LCons @ a) @ Y) @ ((((hammin1328233080lmerge @ a) @ less) @ Xs) @ Ys)))))))))).
% 2.42/1.43 thf(fact_110_local_Olinfinite__lmerge, axiom, (! [Xs:(coinductive_llist @ a),Ys:(coinductive_llist @ a)]: (((~ (((coinductive_lfinite @ a) @ Xs))) => ((~ (((coinductive_lfinite @ a) @ Ys))) => (~ (((coinductive_lfinite @ a) @ ((((hammin1328233080lmerge @ a) @ less) @ Xs) @ Ys))))))))).
% 2.42/1.43 thf(fact_111_local_Olfinite__lmergeI, axiom, (! [Xs:(coinductive_llist @ a),Ys:(coinductive_llist @ a)]: ((((coinductive_lfinite @ a) @ Xs) => (((coinductive_lfinite @ a) @ Ys) => ((coinductive_lfinite @ a) @ ((((hammin1328233080lmerge @ a) @ less) @ Xs) @ Ys))))))).
% 2.42/1.43 thf(fact_112_local_Olfinite__lmergeD, axiom, (! [Xs:(coinductive_llist @ a),Ys:(coinductive_llist @ a)]: ((((coinductive_lfinite @ a) @ ((((hammin1328233080lmerge @ a) @ less) @ Xs) @ Ys)) => (((coinductive_lfinite @ a) @ Xs) | ((coinductive_lfinite @ a) @ Ys)))))).
% 2.42/1.43 thf(fact_113_local_Omin__less__iff__disj, axiom, (! [X:a,Y:a,Z:a]: ((((less @ ((((min @ a) @ less_eq) @ X) @ Y)) @ Z) = (((less @ X) @ Z) | ((less @ Y) @ Z)))))).
% 2.42/1.43 thf(fact_114_local_Omin_Ostrict__order__iff, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) = ((A2 = ((((min @ a) @ less_eq) @ A2) @ B2)) & (A2 != B2)))))).
% 2.42/1.43 thf(fact_115_local_Omin_Ostrict__coboundedI2, axiom, (! [B2:a,C:a,A2:a]: ((((less @ B2) @ C) => ((less @ ((((min @ a) @ less_eq) @ A2) @ B2)) @ C))))).
% 2.42/1.43 thf(fact_116_local_Omin_Ostrict__coboundedI1, axiom, (! [A2:a,C:a,B2:a]: ((((less @ A2) @ C) => ((less @ ((((min @ a) @ less_eq) @ A2) @ B2)) @ C))))).
% 2.42/1.43 thf(fact_117_local_Omin_Ostrict__boundedE, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ ((((min @ a) @ less_eq) @ B2) @ C)) => (~ ((((less @ A2) @ B2) => (~ (((less @ A2) @ C)))))))))).
% 2.42/1.43 thf(fact_118_local_Omin_Oright__idem, axiom, (! [A2:a,B2:a]: ((((((min @ a) @ less_eq) @ ((((min @ a) @ less_eq) @ A2) @ B2)) @ B2) = ((((min @ a) @ less_eq) @ A2) @ B2))))).
% 2.42/1.43 thf(fact_119_local_Omin_Oleft__idem, axiom, (! [A2:a,B2:a]: ((((((min @ a) @ less_eq) @ A2) @ ((((min @ a) @ less_eq) @ A2) @ B2)) = ((((min @ a) @ less_eq) @ A2) @ B2))))).
% 2.42/1.43 thf(fact_120_local_Omin_Oidem, axiom, (! [A2:a]: ((((((min @ a) @ less_eq) @ A2) @ A2) = A2)))).
% 2.42/1.43 thf(fact_121_local_Omin_Obounded__iff, axiom, (! [A2:a,B2:a,C:a]: ((((less_eq @ A2) @ ((((min @ a) @ less_eq) @ B2) @ C)) = (((less_eq @ A2) @ B2) & ((less_eq @ A2) @ C)))))).
% 2.42/1.43 thf(fact_122_local_Obdd__belowI, axiom, (! [A4:(set @ a),M3:a]: (((! [X3:a]: (((((member @ a) @ X3) @ A4) => ((less_eq @ M3) @ X3)))) => (((condit1201339847_below @ a) @ less_eq) @ A4))))).
% 2.42/1.43 thf(fact_123_local_Obdd__aboveI, axiom, (! [A4:(set @ a),M4:a]: (((! [X3:a]: (((((member @ a) @ X3) @ A4) => ((less_eq @ X3) @ M4)))) => (((condit2040224947_above @ a) @ less_eq) @ A4))))).
% 2.42/1.43 thf(fact_124_local_Omin__less__iff__conj, axiom, (! [Z:a,X:a,Y:a]: ((((less @ Z) @ ((((min @ a) @ less_eq) @ X) @ Y)) = (((less @ Z) @ X) & ((less @ Z) @ Y)))))).
% 2.42/1.43 thf(fact_125_local_Ostrict__mono__eq, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B),X:a,Y:a]: ((((((strict_mono @ a) @ B) @ less) @ F) => (((F @ X) = (F @ Y)) = (X = Y))))))))).
% 2.42/1.43 thf(fact_126_ord_Olinfinite__lmerge, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Less:(A > (A > $o))]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((~ (((coinductive_lfinite @ A) @ Ys))) => (~ (((coinductive_lfinite @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys))))))))).
% 2.42/1.43 thf(fact_127_ord_Olfinite__lmergeI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Less:(A > (A > $o))]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_lfinite @ A) @ Ys) => ((coinductive_lfinite @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys))))))).
% 2.42/1.43 thf(fact_128_ord_Olfinite__lmergeD, axiom, (! [A:$tType,Less:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys)) => (((coinductive_lfinite @ A) @ Xs) | ((coinductive_lfinite @ A) @ Ys)))))).
% 2.42/1.43 thf(fact_129_ord_Olmerge__simps, axiom, (! [A:$tType,Less:(A > (A > $o)),X:A,Y:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((Less @ X) @ Y) => (((((hammin1328233080lmerge @ A) @ Less) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = (((coinductive_LCons @ A) @ X) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys))))) & ((~ (((Less @ X) @ Y))) => ((((Less @ Y) @ X) => (((((hammin1328233080lmerge @ A) @ Less) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = (((coinductive_LCons @ A) @ Y) @ ((((hammin1328233080lmerge @ A) @ Less) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ Ys)))) & ((~ (((Less @ Y) @ X))) => (((((hammin1328233080lmerge @ A) @ Less) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = (((coinductive_LCons @ A) @ Y) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys)))))))))).
% 2.42/1.43 thf(fact_130_local_OMin_Osemilattice__order__set__axioms, axiom, ((((lattic1693879045er_set @ a) @ ((min @ a) @ less_eq)) @ less_eq) @ less)).
% 2.42/1.43 thf(fact_131_ldistinct__LCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((coindu351974385stinct @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))) & ((coindu351974385stinct @ A) @ Xs)))))).
% 2.42/1.43 thf(fact_132_local_OIoc__inj, axiom, (! [A2:a,B2:a,C:a,D:a]: ((((((((set_gr323396891AtMost @ a) @ less_eq) @ less) @ A2) @ B2) = (((((set_gr323396891AtMost @ a) @ less_eq) @ less) @ C) @ D)) = ((((less_eq @ B2) @ A2) & ((less_eq @ D) @ C)) | ((A2 = C) & (B2 = D))))))).
% 2.42/1.43 thf(fact_133_step_OIH, axiom, (! [Ys:(coinductive_llist @ a)]: (((((coinductive_lsorted @ a) @ less_eq) @ xsa) => ((~ (((coinductive_lfinite @ a) @ Ys))) => ((? [X4:a]: (((((member @ a) @ X4) @ ((coinductive_lset @ a) @ Ys)) & ((less_eq @ x) @ X4)))) => (((member @ a) @ x) @ ((coinductive_lset @ a) @ ((((hammin1328233080lmerge @ a) @ less) @ xsa) @ Ys))))))))).
% 2.42/1.43 thf(fact_134_lfinite__code_I2_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B)]: ((((coinductive_lfinite @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) = ((coinductive_lfinite @ B) @ Xs))))).
% 2.42/1.43 thf(fact_135_local_Oinsort__left__comm, axiom, (! [X:a,Y:a,Xs:(list @ a)]: ((((((((insort_key @ a) @ a) @ less_eq) @ (^ [X2:a]: (X2))) @ X) @ ((((((insort_key @ a) @ a) @ less_eq) @ (^ [X2:a]: (X2))) @ Y) @ Xs)) = ((((((insort_key @ a) @ a) @ less_eq) @ (^ [X2:a]: (X2))) @ Y) @ ((((((insort_key @ a) @ a) @ less_eq) @ (^ [X2:a]: (X2))) @ X) @ Xs)))))).
% 2.42/1.43 thf(fact_136_step_Oprems_I1_J, axiom, (((coinductive_lsorted @ a) @ less_eq) @ (((coinductive_LCons @ a) @ x2) @ xsa))).
% 2.42/1.43 thf(fact_137_llist_Oinject, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A),Y21:A,Y22:(coinductive_llist @ A)]: ((((((coinductive_LCons @ A) @ X21) @ X22) = (((coinductive_LCons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 2.42/1.43 thf(fact_138_lfinite__LCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((coinductive_lfinite @ A) @ Xs))))).
% 2.42/1.43 thf(fact_139_local_OgreaterThanAtMost__iff, axiom, (! [I:a,L:a,U:a]: (((((member @ a) @ I) @ (((((set_gr323396891AtMost @ a) @ less_eq) @ less) @ L) @ U)) = (((less @ L) @ I) & ((less_eq @ I) @ U)))))).
% 2.42/1.43 thf(fact_140_local_Obdd__above__Ioc, axiom, (! [A2:a,B2:a]: ((((condit2040224947_above @ a) @ less_eq) @ (((((set_gr323396891AtMost @ a) @ less_eq) @ less) @ A2) @ B2))))).
% 2.42/1.43 thf(fact_141_local_Obdd__below__Ioc, axiom, (! [A2:a,B2:a]: ((((condit1201339847_below @ a) @ less_eq) @ (((((set_gr323396891AtMost @ a) @ less_eq) @ less) @ A2) @ B2))))).
% 2.42/1.43 thf(fact_142_local_Ostrict__mono__mono, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B)]: ((((((strict_mono @ a) @ B) @ less) @ F) => ((((mono @ a) @ B) @ less_eq) @ F)))))))).
% 2.42/1.43 thf(fact_143_lset__intros_I2_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),X5:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X5) @ Xs))))))).
% 2.42/1.44 thf(fact_144_lset__intros_I1_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)))))).
% 2.42/1.44 thf(fact_145_llist_Oset__intros_I2_J, axiom, (! [A:$tType,X:A,A22:(coinductive_llist @ A),A1:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ A22)) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22))))))).
% 2.42/1.44 thf(fact_146_llist_Oset__intros_I1_J, axiom, (! [A:$tType,A1:A,A22:(coinductive_llist @ A)]: ((((member @ A) @ A1) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22)))))).
% 2.42/1.44 thf(fact_147_lset__cases, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs2:(coinductive_llist @ A)]: ((Xs != (((coinductive_LCons @ A) @ X) @ Xs2)))) => (~ ((! [X6:A,Xs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ X6) @ Xs2)) => (~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs2)))))))))))))).
% 2.42/1.44 thf(fact_148_lset__induct, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs3:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X) @ Xs3)))) => ((! [X6:A,Xs3:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs3)) => ((X != X6) => ((P @ Xs3) => (P @ (((coinductive_LCons @ A) @ X6) @ Xs3))))))) => (P @ Xs))))))).
% 2.42/1.44 thf(fact_149_lset__induct_H, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs3:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X) @ Xs3)))) => ((! [X6:A,Xs3:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs3)) => ((P @ Xs3) => (P @ (((coinductive_LCons @ A) @ X6) @ Xs3)))))) => (P @ Xs))))))).
% 2.42/1.44 thf(fact_150_llist_Oset__cases, axiom, (! [A:$tType,E:A,A2:(coinductive_llist @ A)]: (((((member @ A) @ E) @ ((coinductive_lset @ A) @ A2)) => ((! [Z22:(coinductive_llist @ A)]: ((A2 != (((coinductive_LCons @ A) @ E) @ Z22)))) => (~ ((! [Z1:A,Z22:(coinductive_llist @ A)]: (((A2 = (((coinductive_LCons @ A) @ Z1) @ Z22)) => (~ ((((member @ A) @ E) @ ((coinductive_lset @ A) @ Z22)))))))))))))).
% 2.42/1.44 thf(fact_151_llist_Oset__induct, axiom, (! [A:$tType,X:A,A2:(coinductive_llist @ A),P:(A > ((coinductive_llist @ A) > $o))]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ A2)) => ((! [Z1:A,Z22:(coinductive_llist @ A)]: (((P @ Z1) @ (((coinductive_LCons @ A) @ Z1) @ Z22)))) => ((! [Z1:A,Z22:(coinductive_llist @ A),Xa:A]: (((((member @ A) @ Xa) @ ((coinductive_lset @ A) @ Z22)) => (((P @ Xa) @ Z22) => ((P @ Xa) @ (((coinductive_LCons @ A) @ Z1) @ Z22)))))) => ((P @ X) @ A2))))))).
% 2.42/1.44 thf(fact_152_lfinite__LConsI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)))))).
% 2.42/1.44 thf(fact_153_ord_Olsorted__LCons__LCons, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),X:A,Y:A,Xs:(coinductive_llist @ A)]: (((((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs))) = (((Less_eq @ X) @ Y) & (((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ Y) @ Xs))))))).
% 2.42/1.44 thf(fact_154_ord_OLCons__LCons, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),X:A,Y:A,Xs:(coinductive_llist @ A)]: ((((Less_eq @ X) @ Y) => ((((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ Y) @ Xs)) => (((coinductive_lsorted @ A) @ Less_eq) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs)))))))).
% 2.42/1.44 thf(fact_155_ldistinct_OLCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))) => (((coindu351974385stinct @ A) @ Xs) => ((coindu351974385stinct @ A) @ (((coinductive_LCons @ A) @ X) @ Xs))))))).
% 2.42/1.44 thf(fact_156_local_Osorted__list__of__set_Ofolding__axioms, axiom, (((finite_folding @ a) @ (list @ a)) @ ((((insort_key @ a) @ a) @ less_eq) @ (^ [X2:a]: (X2))))).
% 2.42/1.44 thf(fact_157_local_OatLeast__def, axiom, (! [L:a]: (((((set_atLeast @ a) @ less_eq) @ L) = ((collect @ a) @ (less_eq @ L)))))).
% 2.42/1.44 thf(fact_158_local_Odual__order_Oordering__axioms, axiom, (((ordering @ a) @ (^ [X2:a,Y3:a]: (((less_eq @ Y3) @ X2)))) @ (^ [X2:a,Y3:a]: (((less @ Y3) @ X2))))).
% 2.42/1.44 thf(fact_159_local_OatMost__def, axiom, (! [U:a]: (((((set_atMost @ a) @ less_eq) @ U) = ((collect @ a) @ (^ [X2:a]: (((less_eq @ X2) @ U)))))))).
% 2.42/1.44 thf(fact_160_local_Oorder_Oordering__axioms, axiom, (((ordering @ a) @ less_eq) @ less)).
% 2.42/1.44 thf(fact_161_local_OatMost__iff, axiom, (! [I:a,K:a]: (((((member @ a) @ I) @ (((set_atMost @ a) @ less_eq) @ K)) = ((less_eq @ I) @ K))))).
% 2.42/1.44 thf(fact_162_local_OatLeast__iff, axiom, (! [I:a,K:a]: (((((member @ a) @ I) @ (((set_atLeast @ a) @ less_eq) @ K)) = ((less_eq @ K) @ I))))).
% 2.42/1.44 thf(fact_163_local_Obdd__above__Iic, axiom, (! [B2:a]: ((((condit2040224947_above @ a) @ less_eq) @ (((set_atMost @ a) @ less_eq) @ B2))))).
% 2.42/1.44 thf(fact_164_local_Obdd__below__Ici, axiom, (! [A2:a]: ((((condit1201339847_below @ a) @ less_eq) @ (((set_atLeast @ a) @ less_eq) @ A2))))).
% 2.42/1.44 thf(fact_165_local_OgreaterThan__def, axiom, (! [L:a]: (((((set_greaterThan @ a) @ less) @ L) = ((collect @ a) @ (less @ L)))))).
% 2.42/1.44 thf(fact_166_local_Omin_Osemilattice__axioms, axiom, ((semilattice @ a) @ ((min @ a) @ less_eq))).
% 2.42/1.44 thf(fact_167_local_Omin_Osemigroup__axioms, axiom, ((semigroup @ a) @ ((min @ a) @ less_eq))).
% 2.42/1.44 thf(fact_168_local_OgreaterThan__iff, axiom, (! [I:a,K:a]: (((((member @ a) @ I) @ (((set_greaterThan @ a) @ less) @ K)) = ((less @ K) @ I))))).
% 2.42/1.44 thf(fact_169_local_Obdd__below__Ioi, axiom, (! [A2:a]: ((((condit1201339847_below @ a) @ less_eq) @ (((set_greaterThan @ a) @ less) @ A2))))).
% 2.42/1.44 thf(fact_170_local_OgreaterThanAtMost__def, axiom, (! [L:a,U:a]: (((((((set_gr323396891AtMost @ a) @ less_eq) @ less) @ L) @ U) = (((inf_inf @ (set @ a)) @ (((set_greaterThan @ a) @ less) @ L)) @ (((set_atMost @ a) @ less_eq) @ U)))))).
% 2.42/1.44 thf(fact_171_local_OIci__subset__Ioi__iff, axiom, (! [A2:a,B2:a]: (((((ord_less_eq @ (set @ a)) @ (((set_atLeast @ a) @ less_eq) @ A2)) @ (((set_greaterThan @ a) @ less) @ B2)) = ((less @ B2) @ A2))))).
% 2.42/1.44 thf(fact_172_local_OIoi__le__Ico, axiom, (! [A2:a]: ((((ord_less_eq @ (set @ a)) @ (((set_greaterThan @ a) @ less) @ A2)) @ (((set_atLeast @ a) @ less_eq) @ A2))))).
% 2.42/1.44 thf(fact_173_local_Omono__def, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B)]: ((((((mono @ a) @ B) @ less_eq) @ F) = (! [X2:a,Y3:a]: ((((less_eq @ X2) @ Y3) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y3)))))))))))).
% 2.42/1.44 thf(fact_174_local_OmonoI, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B)]: (((! [X3:a,Y4:a]: ((((less_eq @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y4))))) => ((((mono @ a) @ B) @ less_eq) @ F)))))))).
% 2.42/1.44 thf(fact_175_local_OmonoE, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B),X:a,Y:a]: ((((((mono @ a) @ B) @ less_eq) @ F) => (((less_eq @ X) @ Y) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)))))))))).
% 2.42/1.44 thf(fact_176_local_OmonoD, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B),X:a,Y:a]: ((((((mono @ a) @ B) @ less_eq) @ F) => (((less_eq @ X) @ Y) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)))))))))).
% 2.42/1.44 thf(fact_177_local_Obdd__below__Int2, axiom, (! [B4:(set @ a),A4:(set @ a)]: (((((condit1201339847_below @ a) @ less_eq) @ B4) => (((condit1201339847_below @ a) @ less_eq) @ (((inf_inf @ (set @ a)) @ A4) @ B4)))))).
% 2.42/1.44 thf(fact_178_local_Obdd__below__Int1, axiom, (! [A4:(set @ a),B4:(set @ a)]: (((((condit1201339847_below @ a) @ less_eq) @ A4) => (((condit1201339847_below @ a) @ less_eq) @ (((inf_inf @ (set @ a)) @ A4) @ B4)))))).
% 2.42/1.44 thf(fact_179_local_Obdd__above__Int2, axiom, (! [B4:(set @ a),A4:(set @ a)]: (((((condit2040224947_above @ a) @ less_eq) @ B4) => (((condit2040224947_above @ a) @ less_eq) @ (((inf_inf @ (set @ a)) @ A4) @ B4)))))).
% 2.42/1.44 thf(fact_180_local_Obdd__above__Int1, axiom, (! [A4:(set @ a),B4:(set @ a)]: (((((condit2040224947_above @ a) @ less_eq) @ A4) => (((condit2040224947_above @ a) @ less_eq) @ (((inf_inf @ (set @ a)) @ A4) @ B4)))))).
% 2.42/1.44 thf(fact_181_local_Obdd__below__mono, axiom, (! [B4:(set @ a),A4:(set @ a)]: (((((condit1201339847_below @ a) @ less_eq) @ B4) => ((((ord_less_eq @ (set @ a)) @ A4) @ B4) => (((condit1201339847_below @ a) @ less_eq) @ A4)))))).
% 2.42/1.44 thf(fact_182_local_Obdd__above__mono, axiom, (! [B4:(set @ a),A4:(set @ a)]: (((((condit2040224947_above @ a) @ less_eq) @ B4) => ((((ord_less_eq @ (set @ a)) @ A4) @ B4) => (((condit2040224947_above @ a) @ less_eq) @ A4)))))).
% 2.42/1.44 thf(fact_183_local_Ostrict__mono__less__eq, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B),X:a,Y:a]: ((((((strict_mono @ a) @ B) @ less) @ F) => ((((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)) = ((less_eq @ X) @ Y))))))))).
% 2.42/1.44 thf(fact_184_local_OIoc__subset__iff, axiom, (! [A2:a,B2:a,C:a,D:a]: (((((ord_less_eq @ (set @ a)) @ (((((set_gr323396891AtMost @ a) @ less_eq) @ less) @ A2) @ B2)) @ (((((set_gr323396891AtMost @ a) @ less_eq) @ less) @ C) @ D)) = (((less_eq @ B2) @ A2) | (((less_eq @ C) @ A2) & ((less_eq @ B2) @ D))))))).
% 2.42/1.44 thf(fact_185_local_Oantimono__def, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B)]: ((((((antimono @ a) @ B) @ less_eq) @ F) = (! [X2:a,Y3:a]: ((((less_eq @ X2) @ Y3) => (((ord_less_eq @ B) @ (F @ Y3)) @ (F @ X2)))))))))))).
% 2.42/1.44 thf(fact_186_local_OantimonoI, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B)]: (((! [X3:a,Y4:a]: ((((less_eq @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X3))))) => ((((antimono @ a) @ B) @ less_eq) @ F)))))))).
% 2.42/1.44 thf(fact_187_local_OantimonoE, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B),X:a,Y:a]: ((((((antimono @ a) @ B) @ less_eq) @ F) => (((less_eq @ X) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X)))))))))).
% 2.42/1.44 thf(fact_188_local_OantimonoD, axiom, (! [B:$tType]: ((((order @ B) @ (type2 @ B)) => (! [F:(a > B),X:a,Y:a]: ((((((antimono @ a) @ B) @ less_eq) @ F) => (((less_eq @ X) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X)))))))))).
% 2.42/1.44 thf(fact_189_local_OInt__atMost, axiom, (! [A2:a,B2:a]: (((((inf_inf @ (set @ a)) @ (((set_atMost @ a) @ less_eq) @ A2)) @ (((set_atMost @ a) @ less_eq) @ B2)) = (((set_atMost @ a) @ less_eq) @ ((((min @ a) @ less_eq) @ A2) @ B2)))))).
% 2.42/1.44 thf(fact_190_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B2:A,A2:A]: (((! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((P @ A3) @ B3)))) => ((((P @ B2) @ A2) => ((P @ A2) @ B2)) => ((P @ A2) @ B2))))))))).
% 2.42/1.44 thf(fact_191_local_OatLeastLessThan__subset__iff, axiom, (! [A2:a,B2:a,C:a,D:a]: (((((ord_less_eq @ (set @ a)) @ (((((set_atLeastLessThan @ a) @ less_eq) @ less) @ A2) @ B2)) @ (((((set_atLeastLessThan @ a) @ less_eq) @ less) @ C) @ D)) => (((less_eq @ B2) @ A2) | (((less_eq @ C) @ A2) & ((less_eq @ B2) @ D))))))).
% 2.42/1.44 thf(fact_192_local_OIic__subset__Iio__iff, axiom, (! [A2:a,B2:a]: (((((ord_less_eq @ (set @ a)) @ (((set_atMost @ a) @ less_eq) @ A2)) @ (((set_lessThan @ a) @ less) @ B2)) = ((less @ A2) @ B2))))).
% 2.42/1.44 thf(fact_193_local_OgreaterThan__Int__greaterThan, axiom, (! [A2:a,B2:a]: (((((inf_inf @ (set @ a)) @ (((set_lessThan @ a) @ less) @ A2)) @ (((set_lessThan @ a) @ less) @ B2)) = (((set_lessThan @ a) @ less) @ ((((min @ a) @ less_eq) @ A2) @ B2)))))).
% 2.42/1.44 thf(fact_194_local_OlessThan__def, axiom, (! [U:a]: (((((set_lessThan @ a) @ less) @ U) = ((collect @ a) @ (^ [X2:a]: (((less @ X2) @ U)))))))).
% 2.42/1.44 thf(fact_195_local_OatLeastLessThan__def, axiom, (! [L:a,U:a]: (((((((set_atLeastLessThan @ a) @ less_eq) @ less) @ L) @ U) = (((inf_inf @ (set @ a)) @ (((set_atLeast @ a) @ less_eq) @ L)) @ (((set_lessThan @ a) @ less) @ U)))))).
% 2.42/1.44 thf(fact_196_local_OlessThan__iff, axiom, (! [I:a,K:a]: (((((member @ a) @ I) @ (((set_lessThan @ a) @ less) @ K)) = ((less @ I) @ K))))).
% 2.42/1.44 thf(fact_197_local_OatLeastLessThan__iff, axiom, (! [I:a,L:a,U:a]: (((((member @ a) @ I) @ (((((set_atLeastLessThan @ a) @ less_eq) @ less) @ L) @ U)) = (((less_eq @ L) @ I) & ((less @ I) @ U)))))).
% 2.42/1.44 thf(fact_198_local_Obdd__above__Iio, axiom, (! [B2:a]: ((((condit2040224947_above @ a) @ less_eq) @ (((set_lessThan @ a) @ less) @ B2))))).
% 2.42/1.44 thf(fact_199_local_Obdd__above__Ico, axiom, (! [A2:a,B2:a]: ((((condit2040224947_above @ a) @ less_eq) @ (((((set_atLeastLessThan @ a) @ less_eq) @ less) @ A2) @ B2))))).
% 2.42/1.44 thf(fact_200_local_Obdd__below__Ico, axiom, (! [A2:a,B2:a]: ((((condit1201339847_below @ a) @ less_eq) @ (((((set_atLeastLessThan @ a) @ less_eq) @ less) @ A2) @ B2))))).
% 2.42/1.44 thf(fact_201_local_OgreaterThanLessThan__eq, axiom, (! [A2:a,B2:a]: ((((((set_gr1161524159ssThan @ a) @ less) @ A2) @ B2) = (((inf_inf @ (set @ a)) @ (((set_greaterThan @ a) @ less) @ A2)) @ (((set_lessThan @ a) @ less) @ B2)))))).
% 2.42/1.44 thf(fact_202_local_OgreaterThanLessThan__def, axiom, (! [L:a,U:a]: ((((((set_gr1161524159ssThan @ a) @ less) @ L) @ U) = (((inf_inf @ (set @ a)) @ (((set_greaterThan @ a) @ less) @ L)) @ (((set_lessThan @ a) @ less) @ U)))))).
% 2.42/1.44 thf(fact_203_local_OgreaterThanLessThan__iff, axiom, (! [I:a,L:a,U:a]: (((((member @ a) @ I) @ ((((set_gr1161524159ssThan @ a) @ less) @ L) @ U)) = (((less @ L) @ I) & ((less @ I) @ U)))))).
% 2.42/1.44 thf(fact_204_local_Obdd__below__Ioo, axiom, (! [A2:a,B2:a]: ((((condit1201339847_below @ a) @ less_eq) @ ((((set_gr1161524159ssThan @ a) @ less) @ A2) @ B2))))).
% 2.42/1.44 thf(fact_205_local_Obdd__above__Ioo, axiom, (! [A2:a,B2:a]: ((((condit2040224947_above @ a) @ less_eq) @ ((((set_gr1161524159ssThan @ a) @ less) @ A2) @ B2))))).
% 2.42/1.44 thf(fact_206_local_OatLeastAtMost__def, axiom, (! [L:a,U:a]: ((((((set_atLeastAtMost @ a) @ less_eq) @ L) @ U) = (((inf_inf @ (set @ a)) @ (((set_atLeast @ a) @ less_eq) @ L)) @ (((set_atMost @ a) @ less_eq) @ U)))))).
% 2.42/1.44 thf(fact_207_Int__subset__iff, axiom, (! [A:$tType,C2:(set @ A),A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C2) @ (((inf_inf @ (set @ A)) @ A4) @ B4)) = ((((ord_less_eq @ (set @ A)) @ C2) @ A4) & (((ord_less_eq @ (set @ A)) @ C2) @ B4)))))).
% 2.42/1.44 thf(fact_208_subset__antisym, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A4) => (A4 = B4)))))).
% 2.42/1.44 thf(fact_209_subsetI, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A4) => (((member @ A) @ X3) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.42/1.44 thf(fact_210_Int__iff, axiom, (! [A:$tType,C:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B4)) = ((((member @ A) @ C) @ A4) & (((member @ A) @ C) @ B4)))))).
% 2.42/1.44 thf(fact_211_IntI, axiom, (! [A:$tType,C:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ A4) => ((((member @ A) @ C) @ B4) => (((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B4))))))).
% 2.42/1.44 thf(fact_212_local_OatLeastAtMost__iff, axiom, (! [I:a,L:a,U:a]: (((((member @ a) @ I) @ ((((set_atLeastAtMost @ a) @ less_eq) @ L) @ U)) = (((less_eq @ L) @ I) & ((less_eq @ I) @ U)))))).
% 2.42/1.44 thf(fact_213_local_OIcc__eq__Icc, axiom, (! [L:a,H:a,L2:a,H2:a]: (((((((set_atLeastAtMost @ a) @ less_eq) @ L) @ H) = ((((set_atLeastAtMost @ a) @ less_eq) @ L2) @ H2)) = (((L = L2) & (H = H2)) | ((~ (((less_eq @ L) @ H))) & (~ (((less_eq @ L2) @ H2))))))))).
% 2.42/1.44 thf(fact_214_local_OatLeastatMost__subset__iff, axiom, (! [A2:a,B2:a,C:a,D:a]: (((((ord_less_eq @ (set @ a)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ C) @ D)) = ((~ (((less_eq @ A2) @ B2))) | (((less_eq @ C) @ A2) & ((less_eq @ B2) @ D))))))).
% 2.42/1.44 thf(fact_215_local_Obdd__above__Icc, axiom, (! [A2:a,B2:a]: ((((condit2040224947_above @ a) @ less_eq) @ ((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2))))).
% 2.42/1.44 thf(fact_216_local_Obdd__below__Icc, axiom, (! [A2:a,B2:a]: ((((condit1201339847_below @ a) @ less_eq) @ ((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2))))).
% 2.42/1.44 thf(fact_217_local_OIcc__subset__Iic__iff, axiom, (! [L:a,H:a,H2:a]: (((((ord_less_eq @ (set @ a)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ L) @ H)) @ (((set_atMost @ a) @ less_eq) @ H2)) = ((~ (((less_eq @ L) @ H))) | ((less_eq @ H) @ H2)))))).
% 2.42/1.44 thf(fact_218_local_OIcc__subset__Ici__iff, axiom, (! [L:a,H:a,L2:a]: (((((ord_less_eq @ (set @ a)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ L) @ H)) @ (((set_atLeast @ a) @ less_eq) @ L2)) = ((~ (((less_eq @ L) @ H))) | ((less_eq @ L2) @ L)))))).
% 2.42/1.44 thf(fact_219_local_OInt__atLeastAtMostL1, axiom, (! [A2:a,B2:a,D:a]: (((((inf_inf @ (set @ a)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ B2)) @ (((set_atMost @ a) @ less_eq) @ D)) = ((((set_atLeastAtMost @ a) @ less_eq) @ A2) @ ((((min @ a) @ less_eq) @ B2) @ D)))))).
% 2.42/1.44 thf(fact_220_local_OInt__atLeastAtMostR1, axiom, (! [B2:a,C:a,D:a]: (((((inf_inf @ (set @ a)) @ (((set_atMost @ a) @ less_eq) @ B2)) @ ((((set_atLeastAtMost @ a) @ less_eq) @ C) @ D)) = ((((set_atLeastAtMost @ a) @ less_eq) @ C) @ ((((min @ a) @ less_eq) @ B2) @ D)))))).
% 2.42/1.44 thf(fact_221_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X2:A]: (((P @ X2) => (Q @ X2)))))))).
% 2.42/1.44 thf(fact_222_contra__subsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((~ ((((member @ A) @ C) @ B4))) => (~ ((((member @ A) @ C) @ A4)))))))).
% 2.42/1.44 thf(fact_223_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z2:(set @ A)]: ((Y2 = Z2))) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))).
% 2.42/1.44 thf(fact_224_subset__trans, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C2) => (((ord_less_eq @ (set @ A)) @ A4) @ C2)))))).
% 2.42/1.44 thf(fact_225_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.42/1.44 thf(fact_226_subset__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A4) @ A4)))).
% 2.42/1.44 thf(fact_227_rev__subsetD, axiom, (! [A:$tType,C:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ A4) => ((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((member @ A) @ C) @ B4)))))).
% 2.42/1.44 thf(fact_228_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A5) => (((member @ A) @ T) @ B5)))))))))).
% 2.42/1.44 thf(fact_229_set__rev__mp, axiom, (! [A:$tType,X:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ X) @ A4) => ((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (((member @ A) @ X) @ B4)))))).
% 2.42/1.44 thf(fact_230_equalityD2, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A4))))).
% 2.42/1.44 thf(fact_231_equalityD1, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (((ord_less_eq @ (set @ A)) @ A4) @ B4))))).
% 2.42/1.44 thf(fact_232_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A5) => (((member @ A) @ X2) @ B5)))))))))).
% 2.42/1.44 thf(fact_233_equalityE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((A4 = B4) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))))).
% 2.42/1.44 thf(fact_234_subsetCE, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C) @ A4) => (((member @ A) @ C) @ B4)))))).
% 2.42/1.44 thf(fact_235_subsetD, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ C) @ A4) => (((member @ A) @ C) @ B4)))))).
% 2.42/1.44 thf(fact_236_in__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B4)))))).
% 2.42/1.44 thf(fact_237_set__mp, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B4)))))).
% 2.42/1.44 thf(fact_238_Int__left__commute, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ A4) @ (((inf_inf @ (set @ A)) @ B4) @ C2)) = (((inf_inf @ (set @ A)) @ B4) @ (((inf_inf @ (set @ A)) @ A4) @ C2)))))).
% 2.42/1.44 thf(fact_239_Int__left__absorb, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((inf_inf @ (set @ A)) @ A4) @ (((inf_inf @ (set @ A)) @ A4) @ B4)) = (((inf_inf @ (set @ A)) @ A4) @ B4))))).
% 2.42/1.44 thf(fact_240_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((((inf_inf @ (set @ A)) @ B5) @ A5))))))).
% 2.42/1.44 thf(fact_241_Int__absorb, axiom, (! [A:$tType,A4:(set @ A)]: (((((inf_inf @ (set @ A)) @ A4) @ A4) = A4)))).
% 2.42/1.44 thf(fact_242_Int__assoc, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),C2:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ B4)) @ C2) = (((inf_inf @ (set @ A)) @ A4) @ (((inf_inf @ (set @ A)) @ B4) @ C2)))))).
% 2.42/1.44 thf(fact_243_IntD2, axiom, (! [A:$tType,C:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B4)) => (((member @ A) @ C) @ B4))))).
% 2.42/1.44 thf(fact_244_IntD1, axiom, (! [A:$tType,C:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B4)) => (((member @ A) @ C) @ A4))))).
% 2.42/1.44 thf(fact_245_IntE, axiom, (! [A:$tType,C:A,A4:(set @ A),B4:(set @ A)]: (((((member @ A) @ C) @ (((inf_inf @ (set @ A)) @ A4) @ B4)) => (~ (((((member @ A) @ C) @ A4) => (~ ((((member @ A) @ C) @ B4)))))))))).
% 2.42/1.44 thf(fact_246_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X2:A]: ((((member @ A) @ X2) @ A5)))) @ (^ [X2:A]: ((((member @ A) @ X2) @ B5)))))))))).
% 2.42/1.44 thf(fact_247_Int__def, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((collect @ A) @ (^ [X2:A]: (((((member @ A) @ X2) @ A5) & (((member @ A) @ X2) @ B5))))))))))).
% 2.42/1.44 thf(fact_248_Int__Collect, axiom, (! [A:$tType,X:A,A4:(set @ A),P:(A > $o)]: (((((member @ A) @ X) @ (((inf_inf @ (set @ A)) @ A4) @ ((collect @ A) @ P))) = ((((member @ A) @ X) @ A4) & (P @ X)))))).
% 2.42/1.44 thf(fact_249_Collect__conj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X2:A]: (((P @ X2) & (Q @ X2))))) = (((inf_inf @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.42/1.44 thf(fact_250_Int__Collect__mono, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((! [X3:A]: (((((member @ A) @ X3) @ A4) => ((P @ X3) => (Q @ X3))))) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ ((collect @ A) @ P))) @ (((inf_inf @ (set @ A)) @ B4) @ ((collect @ A) @ Q)))))))).
% 2.42/1.44 thf(fact_251_Int__greatest, axiom, (! [A:$tType,C2:(set @ A),A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C2) @ A4) => ((((ord_less_eq @ (set @ A)) @ C2) @ B4) => (((ord_less_eq @ (set @ A)) @ C2) @ (((inf_inf @ (set @ A)) @ A4) @ B4))))))).
% 2.42/1.44 thf(fact_252_Int__absorb2, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) => ((((inf_inf @ (set @ A)) @ A4) @ B4) = A4))))).
% 2.42/1.44 thf(fact_253_Int__absorb1, axiom, (! [A:$tType,B4:(set @ A),A4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A4) => ((((inf_inf @ (set @ A)) @ A4) @ B4) = B4))))).
% 2.42/1.44 thf(fact_254_Int__lower2, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ B4)) @ B4)))).
% 2.42/1.44 thf(fact_255_Int__lower1, axiom, (! [A:$tType,A4:(set @ A),B4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A4) @ B4)) @ A4)))).
% 2.42/1.44 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.42/1.44 thf(tcon_Set_Oset___Orderings_Oorder_1, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.42/1.44 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.42/1.44 thf(tcon_HOL_Obool___Orderings_Oorder_2, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.42/1.44 thf(conj_0, hypothesis, (! [Y5:a]: (((((member @ a) @ Y5) @ ((coinductive_lset @ a) @ ysa)) => (((less_eq @ x) @ Y5) => thesis))))).
% 2.42/1.44 thf(conj_1, conjecture, thesis).
% 2.42/1.44 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.0EB1vRga9H/DTF2DTF29649.p
% 2.42/1.44 ------------------------
% 2.42/1.47 ---- Cleaned THF ---
% 2.42/1.47 thf(ty_t_Coinductive__List_Ollist,type,
% 2.42/1.47 coinductive_llist: $tType > $tType ).
% 2.42/1.47
% 2.42/1.47 thf(ty_t_List_Olist,type,
% 2.42/1.47 list: $tType > $tType ).
% 2.42/1.47
% 2.42/1.47 thf(ty_t_Set_Oset,type,
% 2.42/1.47 set: $tType > $tType ).
% 2.42/1.47
% 2.42/1.47 thf(ty_t_itself,type,
% 2.42/1.47 itself: $tType > $tType ).
% 2.42/1.47
% 2.42/1.47 thf(ty_tf_a,type,
% 2.42/1.47 a: $tType ).
% 2.42/1.47
% 2.42/1.47 thf(sy_cl_HOL_Otype,type,
% 2.42/1.47 type:
% 2.42/1.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_cl_Orderings_Oorder,type,
% 2.42/1.47 order:
% 2.42/1.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_cl_Orderings_Olinorder,type,
% 2.42/1.47 linorder:
% 2.42/1.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Coinductive__List_Oldistinct,type,
% 2.42/1.47 coindu351974385stinct:
% 2.42/1.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Coinductive__List_Olfinite,type,
% 2.42/1.47 coinductive_lfinite:
% 2.42/1.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 2.42/1.47 coinductive_LCons:
% 2.42/1.47 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 2.42/1.47 coinductive_lset:
% 2.42/1.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Coinductive__List_Oord_Olsorted,type,
% 2.42/1.47 coinductive_lsorted:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
% 2.42/1.47 condit2040224947_above:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
% 2.42/1.47 condit1201339847_below:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Finite__Set_Ofolding,type,
% 2.42/1.47 finite_folding:
% 2.42/1.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Groups_Osemigroup,type,
% 2.42/1.47 semigroup:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord_Olmerge,type,
% 2.42/1.47 hammin1328233080lmerge:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.42/1.47 inf_inf:
% 2.42/1.47 !>[A: $tType] : ( A > A > A ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Lattices_Osemilattice,type,
% 2.42/1.47 semilattice:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
% 2.42/1.47 lattic1693879045er_set:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_List_Olinorder_Oinsort__key,type,
% 2.42/1.47 insort_key:
% 2.42/1.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Orderings_Oord_OLeast,type,
% 2.42/1.47 least:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Orderings_Oord_Omin,type,
% 2.42/1.47 min:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.42/1.47 ord_less_eq:
% 2.42/1.47 !>[A: $tType] : ( A > A > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Orderings_Oorder_Oantimono,type,
% 2.42/1.47 antimono:
% 2.42/1.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Orderings_Oorder_Omono,type,
% 2.42/1.47 mono:
% 2.42/1.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Orderings_Oorder_Ostrict__mono,type,
% 2.42/1.47 strict_mono:
% 2.42/1.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Orderings_Oordering,type,
% 2.42/1.47 ordering:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Pure_Otype,type,
% 2.42/1.47 type2:
% 2.42/1.47 !>[A: $tType] : ( itself @ A ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Relation_Otransp,type,
% 2.42/1.47 transp:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Set_OCollect,type,
% 2.42/1.47 collect:
% 2.42/1.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Set__Interval_Oord_OatLeast,type,
% 2.42/1.47 set_atLeast:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
% 2.42/1.47 set_atLeastAtMost:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Set__Interval_Oord_OatLeastLessThan,type,
% 2.42/1.47 set_atLeastLessThan:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Set__Interval_Oord_OatMost,type,
% 2.42/1.47 set_atMost:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Set__Interval_Oord_OgreaterThan,type,
% 2.42/1.47 set_greaterThan:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
% 2.42/1.47 set_gr323396891AtMost:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
% 2.42/1.47 set_gr1161524159ssThan:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_Set__Interval_Oord_OlessThan,type,
% 2.42/1.47 set_lessThan:
% 2.42/1.47 !>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_c_member,type,
% 2.42/1.47 member:
% 2.42/1.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.42/1.47
% 2.42/1.47 thf(sy_v_less,type,
% 2.42/1.47 less: a > a > $o ).
% 2.42/1.47
% 2.42/1.47 thf(sy_v_less__eq,type,
% 2.42/1.47 less_eq: a > a > $o ).
% 2.42/1.47
% 2.42/1.47 thf(sy_v_thesis____,type,
% 2.42/1.47 thesis: $o ).
% 2.42/1.47
% 2.42/1.47 thf(sy_v_x,type,
% 2.42/1.47 x: a ).
% 2.42/1.47
% 2.42/1.47 thf(sy_v_x_H____,type,
% 2.42/1.47 x2: a ).
% 2.42/1.47
% 2.42/1.47 thf(sy_v_xsa____,type,
% 2.42/1.47 xsa: coinductive_llist @ a ).
% 2.42/1.47
% 2.42/1.47 thf(sy_v_ysa____,type,
% 2.42/1.47 ysa: coinductive_llist @ a ).
% 2.42/1.47
% 2.42/1.47 thf(fact_0_local_Oantisym__conv3,axiom,
% 2.42/1.47 ! [Y: a,X: a] :
% 2.42/1.47 ( ~ ( less @ Y @ X )
% 2.42/1.47 => ( ( ~ ( less @ X @ Y ) )
% 2.42/1.47 = ( X = Y ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_1_local_Odual__order_Oasym,axiom,
% 2.42/1.47 ! [B2: a,A2: a] :
% 2.42/1.47 ( ( less @ B2 @ A2 )
% 2.42/1.47 => ~ ( less @ A2 @ B2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_2_local_Odual__order_Ostrict__implies__not__eq,axiom,
% 2.42/1.47 ! [B2: a,A2: a] :
% 2.42/1.47 ( ( less @ B2 @ A2 )
% 2.42/1.47 => ( A2 != B2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_3_local_Odual__order_Ostrict__trans,axiom,
% 2.42/1.47 ! [B2: a,A2: a,C: a] :
% 2.42/1.47 ( ( less @ B2 @ A2 )
% 2.42/1.47 => ( ( less @ C @ B2 )
% 2.42/1.47 => ( less @ C @ A2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_4_local_Oless__asym,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ~ ( less @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_5_local_Oless__asym_H,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less @ A2 @ B2 )
% 2.42/1.47 => ~ ( less @ B2 @ A2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_6_local_Oless__imp__neq,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ( X != Y ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_7_local_Oless__imp__not__eq,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ( X != Y ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_8_local_Oless__imp__not__eq2,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ( Y != X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_9_local_Oless__imp__not__less,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ~ ( less @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_10_local_Oless__imp__triv,axiom,
% 2.42/1.47 ! [X: a,Y: a,P: $o] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ( ( less @ Y @ X )
% 2.42/1.47 => P ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_11_local_Oless__irrefl,axiom,
% 2.42/1.47 ! [X: a] :
% 2.42/1.47 ~ ( less @ X @ X ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_12_local_Oless__linear,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 | ( X = Y )
% 2.42/1.47 | ( less @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_13_local_Oless__not__sym,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ~ ( less @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_14_local_Oless__trans,axiom,
% 2.42/1.47 ! [X: a,Y: a,Z: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ( ( less @ Y @ Z )
% 2.42/1.47 => ( less @ X @ Z ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_15_local_Olinorder__cases,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ~ ( less @ X @ Y )
% 2.42/1.47 => ( ( X != Y )
% 2.42/1.47 => ( less @ Y @ X ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_16_local_OneqE,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( X != Y )
% 2.42/1.47 => ( ~ ( less @ X @ Y )
% 2.42/1.47 => ( less @ Y @ X ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_17_local_Oneq__iff,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( X != Y )
% 2.42/1.47 = ( ( less @ X @ Y )
% 2.42/1.47 | ( less @ Y @ X ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_18_local_Onot__less__iff__gr__or__eq,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( ~ ( less @ X @ Y ) )
% 2.42/1.47 = ( ( less @ Y @ X )
% 2.42/1.47 | ( X = Y ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_19_local_Oord__eq__less__trans,axiom,
% 2.42/1.47 ! [A2: a,B2: a,C: a] :
% 2.42/1.47 ( ( A2 = B2 )
% 2.42/1.47 => ( ( less @ B2 @ C )
% 2.42/1.47 => ( less @ A2 @ C ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_20_local_Oord__less__eq__trans,axiom,
% 2.42/1.47 ! [A2: a,B2: a,C: a] :
% 2.42/1.47 ( ( less @ A2 @ B2 )
% 2.42/1.47 => ( ( B2 = C )
% 2.42/1.47 => ( less @ A2 @ C ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_21_local_Oorder_Oasym,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less @ A2 @ B2 )
% 2.42/1.47 => ~ ( less @ B2 @ A2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_22_local_Oorder_Oirrefl,axiom,
% 2.42/1.47 ! [A2: a] :
% 2.42/1.47 ~ ( less @ A2 @ A2 ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_23_local_Oorder_Ostrict__implies__not__eq,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less @ A2 @ B2 )
% 2.42/1.47 => ( A2 != B2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_24_local_Oorder_Ostrict__trans,axiom,
% 2.42/1.47 ! [A2: a,B2: a,C: a] :
% 2.42/1.47 ( ( less @ A2 @ B2 )
% 2.42/1.47 => ( ( less @ B2 @ C )
% 2.42/1.47 => ( less @ A2 @ C ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_25_step_Ohyps_I2_J,axiom,
% 2.42/1.47 x != x2 ).
% 2.42/1.47
% 2.42/1.47 thf(fact_26_step_Oprems_I2_J,axiom,
% 2.42/1.47 ~ ( coinductive_lfinite @ a @ ysa ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_27_step_Ohyps_I1_J,axiom,
% 2.42/1.47 member @ a @ x @ ( coinductive_lset @ a @ xsa ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_28_local_Oantisym,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less_eq @ X @ Y )
% 2.42/1.47 => ( ( less_eq @ Y @ X )
% 2.42/1.47 => ( X = Y ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_29_local_Oantisym__conv,axiom,
% 2.42/1.47 ! [Y: a,X: a] :
% 2.42/1.47 ( ( less_eq @ Y @ X )
% 2.42/1.47 => ( ( less_eq @ X @ Y )
% 2.42/1.47 = ( X = Y ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_30_local_Odual__order_Oantisym,axiom,
% 2.42/1.47 ! [B2: a,A2: a] :
% 2.42/1.47 ( ( less_eq @ B2 @ A2 )
% 2.42/1.47 => ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( A2 = B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_31_local_Odual__order_Otrans,axiom,
% 2.42/1.47 ! [B2: a,A2: a,C: a] :
% 2.42/1.47 ( ( less_eq @ B2 @ A2 )
% 2.42/1.47 => ( ( less_eq @ C @ B2 )
% 2.42/1.47 => ( less_eq @ C @ A2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_32_local_Oeq__iff,axiom,
% 2.42/1.47 ( ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 ) )
% 2.42/1.47 = ( ^ [X2: a,Y3: a] :
% 2.42/1.47 ( ( less_eq @ X2 @ Y3 )
% 2.42/1.47 & ( less_eq @ Y3 @ X2 ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_33_local_Oeq__refl,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( X = Y )
% 2.42/1.47 => ( less_eq @ X @ Y ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_34_local_Ole__cases,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ~ ( less_eq @ X @ Y )
% 2.42/1.47 => ( less_eq @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_35_local_Ole__cases3,axiom,
% 2.42/1.47 ! [X: a,Y: a,Z: a] :
% 2.42/1.47 ( ( ( less_eq @ X @ Y )
% 2.42/1.47 => ~ ( less_eq @ Y @ Z ) )
% 2.42/1.47 => ( ( ( less_eq @ Y @ X )
% 2.42/1.47 => ~ ( less_eq @ X @ Z ) )
% 2.42/1.47 => ( ( ( less_eq @ X @ Z )
% 2.42/1.47 => ~ ( less_eq @ Z @ Y ) )
% 2.42/1.47 => ( ( ( less_eq @ Z @ Y )
% 2.42/1.47 => ~ ( less_eq @ Y @ X ) )
% 2.42/1.47 => ( ( ( less_eq @ Y @ Z )
% 2.42/1.47 => ~ ( less_eq @ Z @ X ) )
% 2.42/1.47 => ~ ( ( less_eq @ Z @ X )
% 2.42/1.47 => ~ ( less_eq @ X @ Y ) ) ) ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_36_local_Olinear,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less_eq @ X @ Y )
% 2.42/1.47 | ( less_eq @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_37_local_Olinorder__wlog,axiom,
% 2.42/1.47 ! [P: a > a > $o,A2: a,B2: a] :
% 2.42/1.47 ( ! [A3: a,B3: a] :
% 2.42/1.47 ( ( less_eq @ A3 @ B3 )
% 2.42/1.47 => ( P @ A3 @ B3 ) )
% 2.42/1.47 => ( ! [A3: a,B3: a] :
% 2.42/1.47 ( ( P @ B3 @ A3 )
% 2.42/1.47 => ( P @ A3 @ B3 ) )
% 2.42/1.47 => ( P @ A2 @ B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_38_local_Oord__eq__le__trans,axiom,
% 2.42/1.47 ! [A2: a,B2: a,C: a] :
% 2.42/1.47 ( ( A2 = B2 )
% 2.42/1.47 => ( ( less_eq @ B2 @ C )
% 2.42/1.47 => ( less_eq @ A2 @ C ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_39_local_Oord__le__eq__trans,axiom,
% 2.42/1.47 ! [A2: a,B2: a,C: a] :
% 2.42/1.47 ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( ( B2 = C )
% 2.42/1.47 => ( less_eq @ A2 @ C ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_40_local_Oorder_Oantisym,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( ( less_eq @ B2 @ A2 )
% 2.42/1.47 => ( A2 = B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_41_local_Oorder_Otrans,axiom,
% 2.42/1.47 ! [A2: a,B2: a,C: a] :
% 2.42/1.47 ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( ( less_eq @ B2 @ C )
% 2.42/1.47 => ( less_eq @ A2 @ C ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_42_local_Oorder__trans,axiom,
% 2.42/1.47 ! [X: a,Y: a,Z: a] :
% 2.42/1.47 ( ( less_eq @ X @ Y )
% 2.42/1.47 => ( ( less_eq @ Y @ Z )
% 2.42/1.47 => ( less_eq @ X @ Z ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_43_step_Oprems_I3_J,axiom,
% 2.42/1.47 ? [X3: a] :
% 2.42/1.47 ( ( member @ a @ X3 @ ( coinductive_lset @ a @ ysa ) )
% 2.42/1.47 & ( less_eq @ x @ X3 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_44_local_Oantisym__conv1,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ~ ( less @ X @ Y )
% 2.42/1.47 => ( ( less_eq @ X @ Y )
% 2.42/1.47 = ( X = Y ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_45_mem__Collect__eq,axiom,
% 2.42/1.47 ! [A: $tType,A2: A,P: A > $o] :
% 2.42/1.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.42/1.47 = ( P @ A2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_46_Collect__mem__eq,axiom,
% 2.42/1.47 ! [A: $tType,A4: set @ A] :
% 2.42/1.47 ( ( collect @ A
% 2.42/1.47 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 2.42/1.47 = A4 ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_47_Collect__cong,axiom,
% 2.42/1.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.42/1.47 ( ! [X3: A] :
% 2.42/1.47 ( ( P @ X3 )
% 2.42/1.47 = ( Q @ X3 ) )
% 2.42/1.47 => ( ( collect @ A @ P )
% 2.42/1.47 = ( collect @ A @ Q ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_48_ext,axiom,
% 2.42/1.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.42/1.47 ( ! [X3: A] :
% 2.42/1.47 ( ( F @ X3 )
% 2.42/1.47 = ( G @ X3 ) )
% 2.42/1.47 => ( F = G ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_49_local_Oantisym__conv2,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less_eq @ X @ Y )
% 2.42/1.47 => ( ( ~ ( less @ X @ Y ) )
% 2.42/1.47 = ( X = Y ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_50_local_Odual__order_Onot__eq__order__implies__strict,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( A2 != B2 )
% 2.42/1.47 => ( ( less_eq @ B2 @ A2 )
% 2.42/1.47 => ( less @ B2 @ A2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_51_local_Odual__order_Oorder__iff__strict,axiom,
% 2.42/1.47 ! [B2: a,A2: a] :
% 2.42/1.47 ( ( less_eq @ B2 @ A2 )
% 2.42/1.47 = ( ( less @ B2 @ A2 )
% 2.42/1.47 | ( A2 = B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_52_local_Odual__order_Ostrict__iff__order,axiom,
% 2.42/1.47 ! [B2: a,A2: a] :
% 2.42/1.47 ( ( less @ B2 @ A2 )
% 2.42/1.47 = ( ( less_eq @ B2 @ A2 )
% 2.42/1.47 & ( A2 != B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_53_local_Odual__order_Ostrict__implies__order,axiom,
% 2.42/1.47 ! [B2: a,A2: a] :
% 2.42/1.47 ( ( less @ B2 @ A2 )
% 2.42/1.47 => ( less_eq @ B2 @ A2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_54_local_Odual__order_Ostrict__trans1,axiom,
% 2.42/1.47 ! [B2: a,A2: a,C: a] :
% 2.42/1.47 ( ( less_eq @ B2 @ A2 )
% 2.42/1.47 => ( ( less @ C @ B2 )
% 2.42/1.47 => ( less @ C @ A2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_55_local_Odual__order_Ostrict__trans2,axiom,
% 2.42/1.47 ! [B2: a,A2: a,C: a] :
% 2.42/1.47 ( ( less @ B2 @ A2 )
% 2.42/1.47 => ( ( less_eq @ C @ B2 )
% 2.42/1.47 => ( less @ C @ A2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_56_local_OleD,axiom,
% 2.42/1.47 ! [Y: a,X: a] :
% 2.42/1.47 ( ( less_eq @ Y @ X )
% 2.42/1.47 => ~ ( less @ X @ Y ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_57_local_OleI,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ~ ( less @ X @ Y )
% 2.42/1.47 => ( less_eq @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_58_local_Ole__imp__less__or__eq,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less_eq @ X @ Y )
% 2.42/1.47 => ( ( less @ X @ Y )
% 2.42/1.47 | ( X = Y ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_59_local_Ole__less,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less_eq @ X @ Y )
% 2.42/1.47 = ( ( less @ X @ Y )
% 2.42/1.47 | ( X = Y ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_60_local_Ole__less__linear,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less_eq @ X @ Y )
% 2.42/1.47 | ( less @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_61_local_Ole__less__trans,axiom,
% 2.42/1.47 ! [X: a,Y: a,Z: a] :
% 2.42/1.47 ( ( less_eq @ X @ Y )
% 2.42/1.47 => ( ( less @ Y @ Z )
% 2.42/1.47 => ( less @ X @ Z ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_62_local_Ole__neq__trans,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( ( A2 != B2 )
% 2.42/1.47 => ( less @ A2 @ B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_63_local_Oless__imp__le,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ( less_eq @ X @ Y ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_64_local_Oless__le,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 = ( ( less_eq @ X @ Y )
% 2.42/1.47 & ( X != Y ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_65_local_Oless__le__not__le,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 = ( ( less_eq @ X @ Y )
% 2.42/1.47 & ~ ( less_eq @ Y @ X ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_66_local_Oless__le__trans,axiom,
% 2.42/1.47 ! [X: a,Y: a,Z: a] :
% 2.42/1.47 ( ( less @ X @ Y )
% 2.42/1.47 => ( ( less_eq @ Y @ Z )
% 2.42/1.47 => ( less @ X @ Z ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_67_local_Onot__le,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( ~ ( less_eq @ X @ Y ) )
% 2.42/1.47 = ( less @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_68_local_Onot__le__imp__less,axiom,
% 2.42/1.47 ! [Y: a,X: a] :
% 2.42/1.47 ( ~ ( less_eq @ Y @ X )
% 2.42/1.47 => ( less @ X @ Y ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_69_local_Onot__less,axiom,
% 2.42/1.47 ! [X: a,Y: a] :
% 2.42/1.47 ( ( ~ ( less @ X @ Y ) )
% 2.42/1.47 = ( less_eq @ Y @ X ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_70_local_Oorder_Onot__eq__order__implies__strict,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( A2 != B2 )
% 2.42/1.47 => ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( less @ A2 @ B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_71_local_Oorder_Oorder__iff__strict,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 = ( ( less @ A2 @ B2 )
% 2.42/1.47 | ( A2 = B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_72_local_Oorder_Ostrict__iff__order,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less @ A2 @ B2 )
% 2.42/1.47 = ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 & ( A2 != B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_73_local_Oorder_Ostrict__implies__order,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less @ A2 @ B2 )
% 2.42/1.47 => ( less_eq @ A2 @ B2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_74_local_Oorder_Ostrict__trans1,axiom,
% 2.42/1.47 ! [A2: a,B2: a,C: a] :
% 2.42/1.47 ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( ( less @ B2 @ C )
% 2.42/1.47 => ( less @ A2 @ C ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_75_local_Oorder_Ostrict__trans2,axiom,
% 2.42/1.47 ! [A2: a,B2: a,C: a] :
% 2.42/1.47 ( ( less @ A2 @ B2 )
% 2.42/1.47 => ( ( less_eq @ B2 @ C )
% 2.42/1.47 => ( less @ A2 @ C ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_76_local_Oorder_Orefl,axiom,
% 2.42/1.47 ! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_77_local_Oorder__refl,axiom,
% 2.42/1.47 ! [X: a] : ( less_eq @ X @ X ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_78_local_OLeastI2__order,axiom,
% 2.42/1.47 ! [P: a > $o,X: a,Q: a > $o] :
% 2.42/1.47 ( ( P @ X )
% 2.42/1.47 => ( ! [Y4: a] :
% 2.42/1.47 ( ( P @ Y4 )
% 2.42/1.47 => ( less_eq @ X @ Y4 ) )
% 2.42/1.47 => ( ! [X3: a] :
% 2.42/1.47 ( ( P @ X3 )
% 2.42/1.47 => ( ! [Y5: a] :
% 2.42/1.47 ( ( P @ Y5 )
% 2.42/1.47 => ( less_eq @ X3 @ Y5 ) )
% 2.42/1.47 => ( Q @ X3 ) ) )
% 2.42/1.47 => ( Q @ ( least @ a @ less_eq @ P ) ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_79_local_OLeast__equality,axiom,
% 2.42/1.47 ! [P: a > $o,X: a] :
% 2.42/1.47 ( ( P @ X )
% 2.42/1.47 => ( ! [Y4: a] :
% 2.42/1.47 ( ( P @ Y4 )
% 2.42/1.47 => ( less_eq @ X @ Y4 ) )
% 2.42/1.47 => ( ( least @ a @ less_eq @ P )
% 2.42/1.47 = X ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_80_local_Olsorted_OLCons__LCons,axiom,
% 2.42/1.47 ! [X: a,Y: a,Xs: coinductive_llist @ a] :
% 2.42/1.47 ( ( less_eq @ X @ Y )
% 2.42/1.47 => ( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ Y @ Xs ) )
% 2.42/1.47 => ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ ( coinductive_LCons @ a @ Y @ Xs ) ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_81_local_Olsorted__LCons,axiom,
% 2.42/1.47 ! [X: a,Xs: coinductive_llist @ a] :
% 2.42/1.47 ( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ Xs ) )
% 2.42/1.47 = ( ( coinductive_lsorted @ a @ less_eq @ Xs )
% 2.42/1.47 & ! [X2: a] :
% 2.42/1.47 ( ( member @ a @ X2 @ ( coinductive_lset @ a @ Xs ) )
% 2.42/1.47 => ( less_eq @ X @ X2 ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_82_local_Oin__lset__lmergeD,axiom,
% 2.42/1.47 ! [X: a,Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 2.42/1.47 ( ( member @ a @ X @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) )
% 2.42/1.47 => ( ( member @ a @ X @ ( coinductive_lset @ a @ Xs ) )
% 2.42/1.47 | ( member @ a @ X @ ( coinductive_lset @ a @ Ys ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_83_local_Obdd__above__def,axiom,
% 2.42/1.47 ! [A4: set @ a] :
% 2.42/1.47 ( ( condit2040224947_above @ a @ less_eq @ A4 )
% 2.42/1.47 = ( ? [M: a] :
% 2.42/1.47 ! [X2: a] :
% 2.42/1.47 ( ( member @ a @ X2 @ A4 )
% 2.42/1.47 => ( less_eq @ X2 @ M ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_84_local_Obdd__below__def,axiom,
% 2.42/1.47 ! [A4: set @ a] :
% 2.42/1.47 ( ( condit1201339847_below @ a @ less_eq @ A4 )
% 2.42/1.47 = ( ? [M2: a] :
% 2.42/1.47 ! [X2: a] :
% 2.42/1.47 ( ( member @ a @ X2 @ A4 )
% 2.42/1.47 => ( less_eq @ M2 @ X2 ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_85_local_Oinsort__key__left__comm,axiom,
% 2.42/1.47 ! [B: $tType,F: B > a,X: B,Y: B,Xs: list @ B] :
% 2.42/1.47 ( ( ( F @ X )
% 2.42/1.47 != ( F @ Y ) )
% 2.42/1.47 => ( ( insort_key @ a @ B @ less_eq @ F @ Y @ ( insort_key @ a @ B @ less_eq @ F @ X @ Xs ) )
% 2.42/1.47 = ( insort_key @ a @ B @ less_eq @ F @ X @ ( insort_key @ a @ B @ less_eq @ F @ Y @ Xs ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_86_local_Otransp__le,axiom,
% 2.42/1.47 transp @ a @ less_eq ).
% 2.42/1.47
% 2.42/1.47 thf(fact_87_ord_Oin__lset__lmergeD,axiom,
% 2.42/1.47 ! [A: $tType,X: A,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.42/1.47 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
% 2.42/1.47 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.42/1.47 | ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_88_local_Olsorted__ldistinct__lset__unique,axiom,
% 2.42/1.47 ! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 2.42/1.47 ( ( coinductive_lsorted @ a @ less_eq @ Xs )
% 2.42/1.47 => ( ( coindu351974385stinct @ a @ Xs )
% 2.42/1.47 => ( ( coinductive_lsorted @ a @ less_eq @ Ys )
% 2.42/1.47 => ( ( coindu351974385stinct @ a @ Ys )
% 2.42/1.47 => ( ( ( coinductive_lset @ a @ Xs )
% 2.42/1.47 = ( coinductive_lset @ a @ Ys ) )
% 2.42/1.47 => ( Xs = Ys ) ) ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_89_local_Olsorted__LCons__LCons,axiom,
% 2.42/1.47 ! [X: a,Y: a,Xs: coinductive_llist @ a] :
% 2.42/1.47 ( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ ( coinductive_LCons @ a @ Y @ Xs ) ) )
% 2.42/1.47 = ( ( less_eq @ X @ Y )
% 2.42/1.47 & ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ Y @ Xs ) ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_90_local_Omin_Oabsorb1,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 2.42/1.47 = A2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_91_local_Omin_Oabsorb2,axiom,
% 2.42/1.47 ! [B2: a,A2: a] :
% 2.42/1.47 ( ( less_eq @ B2 @ A2 )
% 2.42/1.47 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 2.42/1.47 = B2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_92_local_Omin_Oabsorb__iff1,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 = ( ( min @ a @ less_eq @ A2 @ B2 )
% 2.42/1.47 = A2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_93_local_Omin_Oabsorb__iff2,axiom,
% 2.42/1.47 ! [B2: a,A2: a] :
% 2.42/1.47 ( ( less_eq @ B2 @ A2 )
% 2.42/1.47 = ( ( min @ a @ less_eq @ A2 @ B2 )
% 2.42/1.47 = B2 ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_94_local_Omin__le__iff__disj,axiom,
% 2.42/1.47 ! [X: a,Y: a,Z: a] :
% 2.42/1.47 ( ( less_eq @ ( min @ a @ less_eq @ X @ Y ) @ Z )
% 2.42/1.47 = ( ( less_eq @ X @ Z )
% 2.42/1.47 | ( less_eq @ Y @ Z ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_95_local_Omin__def,axiom,
% 2.42/1.47 ! [A2: a,B2: a] :
% 2.42/1.47 ( ( ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 2.42/1.47 = A2 ) )
% 2.42/1.47 & ( ~ ( less_eq @ A2 @ B2 )
% 2.42/1.47 => ( ( min @ a @ less_eq @ A2 @ B2 )
% 2.42/1.47 = B2 ) ) ) ).
% 2.42/1.47
% 2.42/1.47 thf(fact_96_local_Omin_Oorder__iff,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( less_eq @ A2 @ B2 )
% 2.42/1.48 = ( A2
% 2.42/1.48 = ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_97_local_Omin_OorderI,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( A2
% 2.42/1.48 = ( min @ a @ less_eq @ A2 @ B2 ) )
% 2.42/1.48 => ( less_eq @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_98_local_Omin_OorderE,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( less_eq @ A2 @ B2 )
% 2.42/1.48 => ( A2
% 2.42/1.48 = ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_99_local_Omin_Omono,axiom,
% 2.42/1.48 ! [A2: a,C: a,B2: a,D: a] :
% 2.42/1.48 ( ( less_eq @ A2 @ C )
% 2.42/1.48 => ( ( less_eq @ B2 @ D )
% 2.42/1.48 => ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ ( min @ a @ less_eq @ C @ D ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_100_local_Omin_Oleft__commute,axiom,
% 2.42/1.48 ! [B2: a,A2: a,C: a] :
% 2.42/1.48 ( ( min @ a @ less_eq @ B2 @ ( min @ a @ less_eq @ A2 @ C ) )
% 2.42/1.48 = ( min @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_101_local_Omin_Ocommute,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( min @ a @ less_eq @ A2 @ B2 )
% 2.42/1.48 = ( min @ a @ less_eq @ B2 @ A2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_102_local_Omin_OcoboundedI2,axiom,
% 2.42/1.48 ! [B2: a,C: a,A2: a] :
% 2.42/1.48 ( ( less_eq @ B2 @ C )
% 2.42/1.48 => ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_103_local_Omin_OcoboundedI1,axiom,
% 2.42/1.48 ! [A2: a,C: a,B2: a] :
% 2.42/1.48 ( ( less_eq @ A2 @ C )
% 2.42/1.48 => ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_104_local_Omin_Ocobounded2,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ B2 ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_105_local_Omin_Ocobounded1,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ A2 ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_106_local_Omin_OboundedI,axiom,
% 2.42/1.48 ! [A2: a,B2: a,C: a] :
% 2.42/1.48 ( ( less_eq @ A2 @ B2 )
% 2.42/1.48 => ( ( less_eq @ A2 @ C )
% 2.42/1.48 => ( less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_107_local_Omin_OboundedE,axiom,
% 2.42/1.48 ! [A2: a,B2: a,C: a] :
% 2.42/1.48 ( ( less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) )
% 2.42/1.48 => ~ ( ( less_eq @ A2 @ B2 )
% 2.42/1.48 => ~ ( less_eq @ A2 @ C ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_108_local_Omin_Oassoc,axiom,
% 2.42/1.48 ! [A2: a,B2: a,C: a] :
% 2.42/1.48 ( ( min @ a @ less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ C )
% 2.42/1.48 = ( min @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_109_local_Olmerge__simps,axiom,
% 2.42/1.48 ! [X: a,Y: a,Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 2.42/1.48 ( ( ( less @ X @ Y )
% 2.42/1.48 => ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ ( coinductive_LCons @ a @ Y @ Ys ) )
% 2.42/1.48 = ( coinductive_LCons @ a @ X @ ( hammin1328233080lmerge @ a @ less @ Xs @ ( coinductive_LCons @ a @ Y @ Ys ) ) ) ) )
% 2.42/1.48 & ( ~ ( less @ X @ Y )
% 2.42/1.48 => ( ( ( less @ Y @ X )
% 2.42/1.48 => ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ ( coinductive_LCons @ a @ Y @ Ys ) )
% 2.42/1.48 = ( coinductive_LCons @ a @ Y @ ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ Ys ) ) ) )
% 2.42/1.48 & ( ~ ( less @ Y @ X )
% 2.42/1.48 => ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ ( coinductive_LCons @ a @ Y @ Ys ) )
% 2.42/1.48 = ( coinductive_LCons @ a @ Y @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_110_local_Olinfinite__lmerge,axiom,
% 2.42/1.48 ! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 2.42/1.48 ( ~ ( coinductive_lfinite @ a @ Xs )
% 2.42/1.48 => ( ~ ( coinductive_lfinite @ a @ Ys )
% 2.42/1.48 => ~ ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_111_local_Olfinite__lmergeI,axiom,
% 2.42/1.48 ! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 2.42/1.48 ( ( coinductive_lfinite @ a @ Xs )
% 2.42/1.48 => ( ( coinductive_lfinite @ a @ Ys )
% 2.42/1.48 => ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_112_local_Olfinite__lmergeD,axiom,
% 2.42/1.48 ! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
% 2.42/1.48 ( ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
% 2.42/1.48 => ( ( coinductive_lfinite @ a @ Xs )
% 2.42/1.48 | ( coinductive_lfinite @ a @ Ys ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_113_local_Omin__less__iff__disj,axiom,
% 2.42/1.48 ! [X: a,Y: a,Z: a] :
% 2.42/1.48 ( ( less @ ( min @ a @ less_eq @ X @ Y ) @ Z )
% 2.42/1.48 = ( ( less @ X @ Z )
% 2.42/1.48 | ( less @ Y @ Z ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_114_local_Omin_Ostrict__order__iff,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( less @ A2 @ B2 )
% 2.42/1.48 = ( ( A2
% 2.42/1.48 = ( min @ a @ less_eq @ A2 @ B2 ) )
% 2.42/1.48 & ( A2 != B2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_115_local_Omin_Ostrict__coboundedI2,axiom,
% 2.42/1.48 ! [B2: a,C: a,A2: a] :
% 2.42/1.48 ( ( less @ B2 @ C )
% 2.42/1.48 => ( less @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_116_local_Omin_Ostrict__coboundedI1,axiom,
% 2.42/1.48 ! [A2: a,C: a,B2: a] :
% 2.42/1.48 ( ( less @ A2 @ C )
% 2.42/1.48 => ( less @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_117_local_Omin_Ostrict__boundedE,axiom,
% 2.42/1.48 ! [A2: a,B2: a,C: a] :
% 2.42/1.48 ( ( less @ A2 @ ( min @ a @ less_eq @ B2 @ C ) )
% 2.42/1.48 => ~ ( ( less @ A2 @ B2 )
% 2.42/1.48 => ~ ( less @ A2 @ C ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_118_local_Omin_Oright__idem,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( min @ a @ less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ B2 )
% 2.42/1.48 = ( min @ a @ less_eq @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_119_local_Omin_Oleft__idem,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( min @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ A2 @ B2 ) )
% 2.42/1.48 = ( min @ a @ less_eq @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_120_local_Omin_Oidem,axiom,
% 2.42/1.48 ! [A2: a] :
% 2.42/1.48 ( ( min @ a @ less_eq @ A2 @ A2 )
% 2.42/1.48 = A2 ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_121_local_Omin_Obounded__iff,axiom,
% 2.42/1.48 ! [A2: a,B2: a,C: a] :
% 2.42/1.48 ( ( less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) )
% 2.42/1.48 = ( ( less_eq @ A2 @ B2 )
% 2.42/1.48 & ( less_eq @ A2 @ C ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_122_local_Obdd__belowI,axiom,
% 2.42/1.48 ! [A4: set @ a,M3: a] :
% 2.42/1.48 ( ! [X3: a] :
% 2.42/1.48 ( ( member @ a @ X3 @ A4 )
% 2.42/1.48 => ( less_eq @ M3 @ X3 ) )
% 2.42/1.48 => ( condit1201339847_below @ a @ less_eq @ A4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_123_local_Obdd__aboveI,axiom,
% 2.42/1.48 ! [A4: set @ a,M4: a] :
% 2.42/1.48 ( ! [X3: a] :
% 2.42/1.48 ( ( member @ a @ X3 @ A4 )
% 2.42/1.48 => ( less_eq @ X3 @ M4 ) )
% 2.42/1.48 => ( condit2040224947_above @ a @ less_eq @ A4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_124_local_Omin__less__iff__conj,axiom,
% 2.42/1.48 ! [Z: a,X: a,Y: a] :
% 2.42/1.48 ( ( less @ Z @ ( min @ a @ less_eq @ X @ Y ) )
% 2.42/1.48 = ( ( less @ Z @ X )
% 2.42/1.48 & ( less @ Z @ Y ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_125_local_Ostrict__mono__eq,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B,X: a,Y: a] :
% 2.42/1.48 ( ( strict_mono @ a @ B @ less @ F )
% 2.42/1.48 => ( ( ( F @ X )
% 2.42/1.48 = ( F @ Y ) )
% 2.42/1.48 = ( X = Y ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_126_ord_Olinfinite__lmerge,axiom,
% 2.42/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 2.42/1.48 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.42/1.48 => ( ~ ( coinductive_lfinite @ A @ Ys )
% 2.42/1.48 => ~ ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_127_ord_Olfinite__lmergeI,axiom,
% 2.42/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 2.42/1.48 ( ( coinductive_lfinite @ A @ Xs )
% 2.42/1.48 => ( ( coinductive_lfinite @ A @ Ys )
% 2.42/1.48 => ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_128_ord_Olfinite__lmergeD,axiom,
% 2.42/1.48 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.42/1.48 ( ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
% 2.42/1.48 => ( ( coinductive_lfinite @ A @ Xs )
% 2.42/1.48 | ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_129_ord_Olmerge__simps,axiom,
% 2.42/1.48 ! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.42/1.48 ( ( ( Less @ X @ Y )
% 2.42/1.48 => ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.42/1.48 = ( coinductive_LCons @ A @ X @ ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) )
% 2.42/1.48 & ( ~ ( Less @ X @ Y )
% 2.42/1.48 => ( ( ( Less @ Y @ X )
% 2.42/1.48 => ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.42/1.48 = ( coinductive_LCons @ A @ Y @ ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys ) ) ) )
% 2.42/1.48 & ( ~ ( Less @ Y @ X )
% 2.42/1.48 => ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.42/1.48 = ( coinductive_LCons @ A @ Y @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_130_local_OMin_Osemilattice__order__set__axioms,axiom,
% 2.42/1.48 lattic1693879045er_set @ a @ ( min @ a @ less_eq ) @ less_eq @ less ).
% 2.42/1.48
% 2.42/1.48 thf(fact_131_ldistinct__LCons,axiom,
% 2.42/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.42/1.48 ( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.42/1.48 = ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.42/1.48 & ( coindu351974385stinct @ A @ Xs ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_132_local_OIoc__inj,axiom,
% 2.42/1.48 ! [A2: a,B2: a,C: a,D: a] :
% 2.42/1.48 ( ( ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 )
% 2.42/1.48 = ( set_gr323396891AtMost @ a @ less_eq @ less @ C @ D ) )
% 2.42/1.48 = ( ( ( less_eq @ B2 @ A2 )
% 2.42/1.48 & ( less_eq @ D @ C ) )
% 2.42/1.48 | ( ( A2 = C )
% 2.42/1.48 & ( B2 = D ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_133_step_OIH,axiom,
% 2.42/1.48 ! [Ys: coinductive_llist @ a] :
% 2.42/1.48 ( ( coinductive_lsorted @ a @ less_eq @ xsa )
% 2.42/1.48 => ( ~ ( coinductive_lfinite @ a @ Ys )
% 2.42/1.48 => ( ? [X4: a] :
% 2.42/1.48 ( ( member @ a @ X4 @ ( coinductive_lset @ a @ Ys ) )
% 2.42/1.48 & ( less_eq @ x @ X4 ) )
% 2.42/1.48 => ( member @ a @ x @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ xsa @ Ys ) ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_134_lfinite__code_I2_J,axiom,
% 2.42/1.48 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 2.42/1.48 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 2.42/1.48 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_135_local_Oinsort__left__comm,axiom,
% 2.42/1.48 ! [X: a,Y: a,Xs: list @ a] :
% 2.42/1.48 ( ( insort_key @ a @ a @ less_eq
% 2.42/1.48 @ ^ [X2: a] : X2
% 2.42/1.48 @ X
% 2.42/1.48 @ ( insort_key @ a @ a @ less_eq
% 2.42/1.48 @ ^ [X2: a] : X2
% 2.42/1.48 @ Y
% 2.42/1.48 @ Xs ) )
% 2.42/1.48 = ( insort_key @ a @ a @ less_eq
% 2.42/1.48 @ ^ [X2: a] : X2
% 2.42/1.48 @ Y
% 2.42/1.48 @ ( insort_key @ a @ a @ less_eq
% 2.42/1.48 @ ^ [X2: a] : X2
% 2.42/1.48 @ X
% 2.42/1.48 @ Xs ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_136_step_Oprems_I1_J,axiom,
% 2.42/1.48 coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ x2 @ xsa ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_137_llist_Oinject,axiom,
% 2.42/1.48 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 2.42/1.48 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 2.42/1.48 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 2.42/1.48 = ( ( X21 = Y21 )
% 2.42/1.48 & ( X22 = Y22 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_138_lfinite__LCons,axiom,
% 2.42/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.42/1.48 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.42/1.48 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_139_local_OgreaterThanAtMost__iff,axiom,
% 2.42/1.48 ! [I: a,L: a,U: a] :
% 2.42/1.48 ( ( member @ a @ I @ ( set_gr323396891AtMost @ a @ less_eq @ less @ L @ U ) )
% 2.42/1.48 = ( ( less @ L @ I )
% 2.42/1.48 & ( less_eq @ I @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_140_local_Obdd__above__Ioc,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_141_local_Obdd__below__Ioc,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_142_local_Ostrict__mono__mono,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B] :
% 2.42/1.48 ( ( strict_mono @ a @ B @ less @ F )
% 2.42/1.48 => ( mono @ a @ B @ less_eq @ F ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_143_lset__intros_I2_J,axiom,
% 2.42/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,X5: A] :
% 2.42/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.42/1.48 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_144_lset__intros_I1_J,axiom,
% 2.42/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_145_llist_Oset__intros_I2_J,axiom,
% 2.42/1.48 ! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
% 2.42/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
% 2.42/1.48 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_146_llist_Oset__intros_I1_J,axiom,
% 2.42/1.48 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_147_lset__cases,axiom,
% 2.42/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.42/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.42/1.48 => ( ! [Xs2: coinductive_llist @ A] :
% 2.42/1.48 ( Xs
% 2.42/1.48 != ( coinductive_LCons @ A @ X @ Xs2 ) )
% 2.42/1.48 => ~ ! [X6: A,Xs2: coinductive_llist @ A] :
% 2.42/1.48 ( ( Xs
% 2.42/1.48 = ( coinductive_LCons @ A @ X6 @ Xs2 ) )
% 2.42/1.48 => ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_148_lset__induct,axiom,
% 2.42/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.42/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.42/1.48 => ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
% 2.42/1.48 => ( ! [X6: A,Xs3: coinductive_llist @ A] :
% 2.42/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
% 2.42/1.48 => ( ( X != X6 )
% 2.42/1.48 => ( ( P @ Xs3 )
% 2.42/1.48 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs3 ) ) ) ) )
% 2.42/1.48 => ( P @ Xs ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_149_lset__induct_H,axiom,
% 2.42/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.42/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.42/1.48 => ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
% 2.42/1.48 => ( ! [X6: A,Xs3: coinductive_llist @ A] :
% 2.42/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
% 2.42/1.48 => ( ( P @ Xs3 )
% 2.42/1.48 => ( P @ ( coinductive_LCons @ A @ X6 @ Xs3 ) ) ) )
% 2.42/1.48 => ( P @ Xs ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_150_llist_Oset__cases,axiom,
% 2.42/1.48 ! [A: $tType,E: A,A2: coinductive_llist @ A] :
% 2.42/1.48 ( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
% 2.42/1.48 => ( ! [Z22: coinductive_llist @ A] :
% 2.42/1.48 ( A2
% 2.42/1.48 != ( coinductive_LCons @ A @ E @ Z22 ) )
% 2.42/1.48 => ~ ! [Z1: A,Z22: coinductive_llist @ A] :
% 2.42/1.48 ( ( A2
% 2.42/1.48 = ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 2.42/1.48 => ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_151_llist_Oset__induct,axiom,
% 2.42/1.48 ! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 2.42/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
% 2.42/1.48 => ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
% 2.42/1.48 => ( ! [Z1: A,Z22: coinductive_llist @ A,Xa: A] :
% 2.42/1.48 ( ( member @ A @ Xa @ ( coinductive_lset @ A @ Z22 ) )
% 2.42/1.48 => ( ( P @ Xa @ Z22 )
% 2.42/1.48 => ( P @ Xa @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
% 2.42/1.48 => ( P @ X @ A2 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_152_lfinite__LConsI,axiom,
% 2.42/1.48 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 2.42/1.48 ( ( coinductive_lfinite @ A @ Xs )
% 2.42/1.48 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_153_ord_Olsorted__LCons__LCons,axiom,
% 2.42/1.48 ! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A] :
% 2.42/1.48 ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 2.42/1.48 = ( ( Less_eq @ X @ Y )
% 2.42/1.48 & ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_154_ord_OLCons__LCons,axiom,
% 2.42/1.48 ! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A] :
% 2.42/1.48 ( ( Less_eq @ X @ Y )
% 2.42/1.48 => ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs ) )
% 2.42/1.48 => ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_155_ldistinct_OLCons,axiom,
% 2.42/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.42/1.48 ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.42/1.48 => ( ( coindu351974385stinct @ A @ Xs )
% 2.42/1.48 => ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_156_local_Osorted__list__of__set_Ofolding__axioms,axiom,
% 2.42/1.48 ( finite_folding @ a @ ( list @ a )
% 2.42/1.48 @ ( insort_key @ a @ a @ less_eq
% 2.42/1.48 @ ^ [X2: a] : X2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_157_local_OatLeast__def,axiom,
% 2.42/1.48 ! [L: a] :
% 2.42/1.48 ( ( set_atLeast @ a @ less_eq @ L )
% 2.42/1.48 = ( collect @ a @ ( less_eq @ L ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_158_local_Odual__order_Oordering__axioms,axiom,
% 2.42/1.48 ( ordering @ a
% 2.42/1.48 @ ^ [X2: a,Y3: a] : ( less_eq @ Y3 @ X2 )
% 2.42/1.48 @ ^ [X2: a,Y3: a] : ( less @ Y3 @ X2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_159_local_OatMost__def,axiom,
% 2.42/1.48 ! [U: a] :
% 2.42/1.48 ( ( set_atMost @ a @ less_eq @ U )
% 2.42/1.48 = ( collect @ a
% 2.42/1.48 @ ^ [X2: a] : ( less_eq @ X2 @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_160_local_Oorder_Oordering__axioms,axiom,
% 2.42/1.48 ordering @ a @ less_eq @ less ).
% 2.42/1.48
% 2.42/1.48 thf(fact_161_local_OatMost__iff,axiom,
% 2.42/1.48 ! [I: a,K: a] :
% 2.42/1.48 ( ( member @ a @ I @ ( set_atMost @ a @ less_eq @ K ) )
% 2.42/1.48 = ( less_eq @ I @ K ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_162_local_OatLeast__iff,axiom,
% 2.42/1.48 ! [I: a,K: a] :
% 2.42/1.48 ( ( member @ a @ I @ ( set_atLeast @ a @ less_eq @ K ) )
% 2.42/1.48 = ( less_eq @ K @ I ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_163_local_Obdd__above__Iic,axiom,
% 2.42/1.48 ! [B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atMost @ a @ less_eq @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_164_local_Obdd__below__Ici,axiom,
% 2.42/1.48 ! [A2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeast @ a @ less_eq @ A2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_165_local_OgreaterThan__def,axiom,
% 2.42/1.48 ! [L: a] :
% 2.42/1.48 ( ( set_greaterThan @ a @ less @ L )
% 2.42/1.48 = ( collect @ a @ ( less @ L ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_166_local_Omin_Osemilattice__axioms,axiom,
% 2.42/1.48 semilattice @ a @ ( min @ a @ less_eq ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_167_local_Omin_Osemigroup__axioms,axiom,
% 2.42/1.48 semigroup @ a @ ( min @ a @ less_eq ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_168_local_OgreaterThan__iff,axiom,
% 2.42/1.48 ! [I: a,K: a] :
% 2.42/1.48 ( ( member @ a @ I @ ( set_greaterThan @ a @ less @ K ) )
% 2.42/1.48 = ( less @ K @ I ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_169_local_Obdd__below__Ioi,axiom,
% 2.42/1.48 ! [A2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_greaterThan @ a @ less @ A2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_170_local_OgreaterThanAtMost__def,axiom,
% 2.42/1.48 ! [L: a,U: a] :
% 2.42/1.48 ( ( set_gr323396891AtMost @ a @ less_eq @ less @ L @ U )
% 2.42/1.48 = ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ L ) @ ( set_atMost @ a @ less_eq @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_171_local_OIci__subset__Ioi__iff,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ A2 ) @ ( set_greaterThan @ a @ less @ B2 ) )
% 2.42/1.48 = ( less @ B2 @ A2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_172_local_OIoi__le__Ico,axiom,
% 2.42/1.48 ! [A2: a] : ( ord_less_eq @ ( set @ a ) @ ( set_greaterThan @ a @ less @ A2 ) @ ( set_atLeast @ a @ less_eq @ A2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_173_local_Omono__def,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B] :
% 2.42/1.48 ( ( mono @ a @ B @ less_eq @ F )
% 2.42/1.48 = ( ! [X2: a,Y3: a] :
% 2.42/1.48 ( ( less_eq @ X2 @ Y3 )
% 2.42/1.48 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_174_local_OmonoI,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B] :
% 2.42/1.48 ( ! [X3: a,Y4: a] :
% 2.42/1.48 ( ( less_eq @ X3 @ Y4 )
% 2.42/1.48 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.42/1.48 => ( mono @ a @ B @ less_eq @ F ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_175_local_OmonoE,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B,X: a,Y: a] :
% 2.42/1.48 ( ( mono @ a @ B @ less_eq @ F )
% 2.42/1.48 => ( ( less_eq @ X @ Y )
% 2.42/1.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_176_local_OmonoD,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B,X: a,Y: a] :
% 2.42/1.48 ( ( mono @ a @ B @ less_eq @ F )
% 2.42/1.48 => ( ( less_eq @ X @ Y )
% 2.42/1.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_177_local_Obdd__below__Int2,axiom,
% 2.42/1.48 ! [B4: set @ a,A4: set @ a] :
% 2.42/1.48 ( ( condit1201339847_below @ a @ less_eq @ B4 )
% 2.42/1.48 => ( condit1201339847_below @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_178_local_Obdd__below__Int1,axiom,
% 2.42/1.48 ! [A4: set @ a,B4: set @ a] :
% 2.42/1.48 ( ( condit1201339847_below @ a @ less_eq @ A4 )
% 2.42/1.48 => ( condit1201339847_below @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_179_local_Obdd__above__Int2,axiom,
% 2.42/1.48 ! [B4: set @ a,A4: set @ a] :
% 2.42/1.48 ( ( condit2040224947_above @ a @ less_eq @ B4 )
% 2.42/1.48 => ( condit2040224947_above @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_180_local_Obdd__above__Int1,axiom,
% 2.42/1.48 ! [A4: set @ a,B4: set @ a] :
% 2.42/1.48 ( ( condit2040224947_above @ a @ less_eq @ A4 )
% 2.42/1.48 => ( condit2040224947_above @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_181_local_Obdd__below__mono,axiom,
% 2.42/1.48 ! [B4: set @ a,A4: set @ a] :
% 2.42/1.48 ( ( condit1201339847_below @ a @ less_eq @ B4 )
% 2.42/1.48 => ( ( ord_less_eq @ ( set @ a ) @ A4 @ B4 )
% 2.42/1.48 => ( condit1201339847_below @ a @ less_eq @ A4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_182_local_Obdd__above__mono,axiom,
% 2.42/1.48 ! [B4: set @ a,A4: set @ a] :
% 2.42/1.48 ( ( condit2040224947_above @ a @ less_eq @ B4 )
% 2.42/1.48 => ( ( ord_less_eq @ ( set @ a ) @ A4 @ B4 )
% 2.42/1.48 => ( condit2040224947_above @ a @ less_eq @ A4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_183_local_Ostrict__mono__less__eq,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B,X: a,Y: a] :
% 2.42/1.48 ( ( strict_mono @ a @ B @ less @ F )
% 2.42/1.48 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
% 2.42/1.48 = ( less_eq @ X @ Y ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_184_local_OIoc__subset__iff,axiom,
% 2.42/1.48 ! [A2: a,B2: a,C: a,D: a] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ a ) @ ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 ) @ ( set_gr323396891AtMost @ a @ less_eq @ less @ C @ D ) )
% 2.42/1.48 = ( ( less_eq @ B2 @ A2 )
% 2.42/1.48 | ( ( less_eq @ C @ A2 )
% 2.42/1.48 & ( less_eq @ B2 @ D ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_185_local_Oantimono__def,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B] :
% 2.42/1.48 ( ( antimono @ a @ B @ less_eq @ F )
% 2.42/1.48 = ( ! [X2: a,Y3: a] :
% 2.42/1.48 ( ( less_eq @ X2 @ Y3 )
% 2.42/1.48 => ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_186_local_OantimonoI,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B] :
% 2.42/1.48 ( ! [X3: a,Y4: a] :
% 2.42/1.48 ( ( less_eq @ X3 @ Y4 )
% 2.42/1.48 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 2.42/1.48 => ( antimono @ a @ B @ less_eq @ F ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_187_local_OantimonoE,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B,X: a,Y: a] :
% 2.42/1.48 ( ( antimono @ a @ B @ less_eq @ F )
% 2.42/1.48 => ( ( less_eq @ X @ Y )
% 2.42/1.48 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_188_local_OantimonoD,axiom,
% 2.42/1.48 ! [B: $tType] :
% 2.42/1.48 ( ( order @ B @ ( type2 @ B ) )
% 2.42/1.48 => ! [F: a > B,X: a,Y: a] :
% 2.42/1.48 ( ( antimono @ a @ B @ less_eq @ F )
% 2.42/1.48 => ( ( less_eq @ X @ Y )
% 2.42/1.48 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_189_local_OInt__atMost,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( inf_inf @ ( set @ a ) @ ( set_atMost @ a @ less_eq @ A2 ) @ ( set_atMost @ a @ less_eq @ B2 ) )
% 2.42/1.48 = ( set_atMost @ a @ less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_190_wlog__linorder__le,axiom,
% 2.42/1.48 ! [A: $tType] :
% 2.42/1.48 ( ( linorder @ A @ ( type2 @ A ) )
% 2.42/1.48 => ! [P: A > A > $o,B2: A,A2: A] :
% 2.42/1.48 ( ! [A3: A,B3: A] :
% 2.42/1.48 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.42/1.48 => ( P @ A3 @ B3 ) )
% 2.42/1.48 => ( ( ( P @ B2 @ A2 )
% 2.42/1.48 => ( P @ A2 @ B2 ) )
% 2.42/1.48 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_191_local_OatLeastLessThan__subset__iff,axiom,
% 2.42/1.48 ! [A2: a,B2: a,C: a,D: a] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastLessThan @ a @ less_eq @ less @ A2 @ B2 ) @ ( set_atLeastLessThan @ a @ less_eq @ less @ C @ D ) )
% 2.42/1.48 => ( ( less_eq @ B2 @ A2 )
% 2.42/1.48 | ( ( less_eq @ C @ A2 )
% 2.42/1.48 & ( less_eq @ B2 @ D ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_192_local_OIic__subset__Iio__iff,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ a ) @ ( set_atMost @ a @ less_eq @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) )
% 2.42/1.48 = ( less @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_193_local_OgreaterThan__Int__greaterThan,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( inf_inf @ ( set @ a ) @ ( set_lessThan @ a @ less @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) )
% 2.42/1.48 = ( set_lessThan @ a @ less @ ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_194_local_OlessThan__def,axiom,
% 2.42/1.48 ! [U: a] :
% 2.42/1.48 ( ( set_lessThan @ a @ less @ U )
% 2.42/1.48 = ( collect @ a
% 2.42/1.48 @ ^ [X2: a] : ( less @ X2 @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_195_local_OatLeastLessThan__def,axiom,
% 2.42/1.48 ! [L: a,U: a] :
% 2.42/1.48 ( ( set_atLeastLessThan @ a @ less_eq @ less @ L @ U )
% 2.42/1.48 = ( inf_inf @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ L ) @ ( set_lessThan @ a @ less @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_196_local_OlessThan__iff,axiom,
% 2.42/1.48 ! [I: a,K: a] :
% 2.42/1.48 ( ( member @ a @ I @ ( set_lessThan @ a @ less @ K ) )
% 2.42/1.48 = ( less @ I @ K ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_197_local_OatLeastLessThan__iff,axiom,
% 2.42/1.48 ! [I: a,L: a,U: a] :
% 2.42/1.48 ( ( member @ a @ I @ ( set_atLeastLessThan @ a @ less_eq @ less @ L @ U ) )
% 2.42/1.48 = ( ( less_eq @ L @ I )
% 2.42/1.48 & ( less @ I @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_198_local_Obdd__above__Iio,axiom,
% 2.42/1.48 ! [B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_lessThan @ a @ less @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_199_local_Obdd__above__Ico,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atLeastLessThan @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_200_local_Obdd__below__Ico,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeastLessThan @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_201_local_OgreaterThanLessThan__eq,axiom,
% 2.42/1.48 ! [A2: a,B2: a] :
% 2.42/1.48 ( ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 )
% 2.42/1.48 = ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_202_local_OgreaterThanLessThan__def,axiom,
% 2.42/1.48 ! [L: a,U: a] :
% 2.42/1.48 ( ( set_gr1161524159ssThan @ a @ less @ L @ U )
% 2.42/1.48 = ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ L ) @ ( set_lessThan @ a @ less @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_203_local_OgreaterThanLessThan__iff,axiom,
% 2.42/1.48 ! [I: a,L: a,U: a] :
% 2.42/1.48 ( ( member @ a @ I @ ( set_gr1161524159ssThan @ a @ less @ L @ U ) )
% 2.42/1.48 = ( ( less @ L @ I )
% 2.42/1.48 & ( less @ I @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_204_local_Obdd__below__Ioo,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_205_local_Obdd__above__Ioo,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_206_local_OatLeastAtMost__def,axiom,
% 2.42/1.48 ! [L: a,U: a] :
% 2.42/1.48 ( ( set_atLeastAtMost @ a @ less_eq @ L @ U )
% 2.42/1.48 = ( inf_inf @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ L ) @ ( set_atMost @ a @ less_eq @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_207_Int__subset__iff,axiom,
% 2.42/1.48 ! [A: $tType,C2: set @ A,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 2.42/1.48 = ( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
% 2.42/1.48 & ( ord_less_eq @ ( set @ A ) @ C2 @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_208_subset__antisym,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 2.42/1.48 => ( A4 = B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_209_subsetI,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ! [X3: A] :
% 2.42/1.48 ( ( member @ A @ X3 @ A4 )
% 2.42/1.48 => ( member @ A @ X3 @ B4 ) )
% 2.42/1.48 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_210_Int__iff,axiom,
% 2.42/1.48 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 2.42/1.48 = ( ( member @ A @ C @ A4 )
% 2.42/1.48 & ( member @ A @ C @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_211_IntI,axiom,
% 2.42/1.48 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( member @ A @ C @ A4 )
% 2.42/1.48 => ( ( member @ A @ C @ B4 )
% 2.42/1.48 => ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_212_local_OatLeastAtMost__iff,axiom,
% 2.42/1.48 ! [I: a,L: a,U: a] :
% 2.42/1.48 ( ( member @ a @ I @ ( set_atLeastAtMost @ a @ less_eq @ L @ U ) )
% 2.42/1.48 = ( ( less_eq @ L @ I )
% 2.42/1.48 & ( less_eq @ I @ U ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_213_local_OIcc__eq__Icc,axiom,
% 2.42/1.48 ! [L: a,H: a,L2: a,H2: a] :
% 2.42/1.48 ( ( ( set_atLeastAtMost @ a @ less_eq @ L @ H )
% 2.42/1.48 = ( set_atLeastAtMost @ a @ less_eq @ L2 @ H2 ) )
% 2.42/1.48 = ( ( ( L = L2 )
% 2.42/1.48 & ( H = H2 ) )
% 2.42/1.48 | ( ~ ( less_eq @ L @ H )
% 2.42/1.48 & ~ ( less_eq @ L2 @ H2 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_214_local_OatLeastatMost__subset__iff,axiom,
% 2.42/1.48 ! [A2: a,B2: a,C: a,D: a] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) @ ( set_atLeastAtMost @ a @ less_eq @ C @ D ) )
% 2.42/1.48 = ( ~ ( less_eq @ A2 @ B2 )
% 2.42/1.48 | ( ( less_eq @ C @ A2 )
% 2.42/1.48 & ( less_eq @ B2 @ D ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_215_local_Obdd__above__Icc,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_216_local_Obdd__below__Icc,axiom,
% 2.42/1.48 ! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_217_local_OIcc__subset__Iic__iff,axiom,
% 2.42/1.48 ! [L: a,H: a,H2: a] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atMost @ a @ less_eq @ H2 ) )
% 2.42/1.48 = ( ~ ( less_eq @ L @ H )
% 2.42/1.48 | ( less_eq @ H @ H2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_218_local_OIcc__subset__Ici__iff,axiom,
% 2.42/1.48 ! [L: a,H: a,L2: a] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atLeast @ a @ less_eq @ L2 ) )
% 2.42/1.48 = ( ~ ( less_eq @ L @ H )
% 2.42/1.48 | ( less_eq @ L2 @ L ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_219_local_OInt__atLeastAtMostL1,axiom,
% 2.42/1.48 ! [A2: a,B2: a,D: a] :
% 2.42/1.48 ( ( inf_inf @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) @ ( set_atMost @ a @ less_eq @ D ) )
% 2.42/1.48 = ( set_atLeastAtMost @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ D ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_220_local_OInt__atLeastAtMostR1,axiom,
% 2.42/1.48 ! [B2: a,C: a,D: a] :
% 2.42/1.48 ( ( inf_inf @ ( set @ a ) @ ( set_atMost @ a @ less_eq @ B2 ) @ ( set_atLeastAtMost @ a @ less_eq @ C @ D ) )
% 2.42/1.48 = ( set_atLeastAtMost @ a @ less_eq @ C @ ( min @ a @ less_eq @ B2 @ D ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_221_Collect__mono__iff,axiom,
% 2.42/1.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.42/1.48 = ( ! [X2: A] :
% 2.42/1.48 ( ( P @ X2 )
% 2.42/1.48 => ( Q @ X2 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_222_contra__subsetD,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( ~ ( member @ A @ C @ B4 )
% 2.42/1.48 => ~ ( member @ A @ C @ A4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_223_set__eq__subset,axiom,
% 2.42/1.48 ! [A: $tType] :
% 2.42/1.48 ( ( ^ [Y2: set @ A,Z2: set @ A] : ( Y2 = Z2 ) )
% 2.42/1.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.42/1.48 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_224_subset__trans,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
% 2.42/1.48 => ( ord_less_eq @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_225_Collect__mono,axiom,
% 2.42/1.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.42/1.48 ( ! [X3: A] :
% 2.42/1.48 ( ( P @ X3 )
% 2.42/1.48 => ( Q @ X3 ) )
% 2.42/1.48 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_226_subset__refl,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_227_rev__subsetD,axiom,
% 2.42/1.48 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( member @ A @ C @ A4 )
% 2.42/1.48 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( member @ A @ C @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_228_subset__iff,axiom,
% 2.42/1.48 ! [A: $tType] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) )
% 2.42/1.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.42/1.48 ! [T: A] :
% 2.42/1.48 ( ( member @ A @ T @ A5 )
% 2.42/1.48 => ( member @ A @ T @ B5 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_229_set__rev__mp,axiom,
% 2.42/1.48 ! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( member @ A @ X @ A4 )
% 2.42/1.48 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( member @ A @ X @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_230_equalityD2,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( A4 = B4 )
% 2.42/1.48 => ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_231_equalityD1,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( A4 = B4 )
% 2.42/1.48 => ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_232_subset__eq,axiom,
% 2.42/1.48 ! [A: $tType] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) )
% 2.42/1.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.42/1.48 ! [X2: A] :
% 2.42/1.48 ( ( member @ A @ X2 @ A5 )
% 2.42/1.48 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_233_equalityE,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( A4 = B4 )
% 2.42/1.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_234_subsetCE,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( ( member @ A @ C @ A4 )
% 2.42/1.48 => ( member @ A @ C @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_235_subsetD,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( ( member @ A @ C @ A4 )
% 2.42/1.48 => ( member @ A @ C @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_236_in__mono,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( ( member @ A @ X @ A4 )
% 2.42/1.48 => ( member @ A @ X @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_237_set__mp,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( ( member @ A @ X @ A4 )
% 2.42/1.48 => ( member @ A @ X @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_238_Int__left__commute,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
% 2.42/1.48 ( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) )
% 2.42/1.48 = ( inf_inf @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_239_Int__left__absorb,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 2.42/1.48 = ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_240_Int__commute,axiom,
% 2.42/1.48 ! [A: $tType] :
% 2.42/1.48 ( ( inf_inf @ ( set @ A ) )
% 2.42/1.48 = ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_241_Int__absorb,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A] :
% 2.42/1.48 ( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
% 2.42/1.48 = A4 ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_242_Int__assoc,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
% 2.42/1.48 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ C2 )
% 2.42/1.48 = ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_243_IntD2,axiom,
% 2.42/1.48 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 2.42/1.48 => ( member @ A @ C @ B4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_244_IntD1,axiom,
% 2.42/1.48 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 2.42/1.48 => ( member @ A @ C @ A4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_245_IntE,axiom,
% 2.42/1.48 ! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
% 2.42/1.48 => ~ ( ( member @ A @ C @ A4 )
% 2.42/1.48 => ~ ( member @ A @ C @ B4 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_246_less__eq__set__def,axiom,
% 2.42/1.48 ! [A: $tType] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) )
% 2.42/1.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.42/1.48 ( ord_less_eq @ ( A > $o )
% 2.42/1.48 @ ^ [X2: A] : ( member @ A @ X2 @ A5 )
% 2.42/1.48 @ ^ [X2: A] : ( member @ A @ X2 @ B5 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_247_Int__def,axiom,
% 2.42/1.48 ! [A: $tType] :
% 2.42/1.48 ( ( inf_inf @ ( set @ A ) )
% 2.42/1.48 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.42/1.48 ( collect @ A
% 2.42/1.48 @ ^ [X2: A] :
% 2.42/1.48 ( ( member @ A @ X2 @ A5 )
% 2.42/1.48 & ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_248_Int__Collect,axiom,
% 2.42/1.48 ! [A: $tType,X: A,A4: set @ A,P: A > $o] :
% 2.42/1.48 ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
% 2.42/1.48 = ( ( member @ A @ X @ A4 )
% 2.42/1.48 & ( P @ X ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_249_Collect__conj__eq,axiom,
% 2.42/1.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.42/1.48 ( ( collect @ A
% 2.42/1.48 @ ^ [X2: A] :
% 2.42/1.48 ( ( P @ X2 )
% 2.42/1.48 & ( Q @ X2 ) ) )
% 2.42/1.48 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_250_Int__Collect__mono,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A,P: A > $o,Q: A > $o] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( ! [X3: A] :
% 2.42/1.48 ( ( member @ A @ X3 @ A4 )
% 2.42/1.48 => ( ( P @ X3 )
% 2.42/1.48 => ( Q @ X3 ) ) )
% 2.42/1.48 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B4 @ ( collect @ A @ Q ) ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_251_Int__greatest,axiom,
% 2.42/1.48 ! [A: $tType,C2: set @ A,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
% 2.42/1.48 => ( ( ord_less_eq @ ( set @ A ) @ C2 @ B4 )
% 2.42/1.48 => ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_252_Int__absorb2,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 => ( ( inf_inf @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 = A4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_253_Int__absorb1,axiom,
% 2.42/1.48 ! [A: $tType,B4: set @ A,A4: set @ A] :
% 2.42/1.48 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
% 2.42/1.48 => ( ( inf_inf @ ( set @ A ) @ A4 @ B4 )
% 2.42/1.48 = B4 ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_254_Int__lower2,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ B4 ) ).
% 2.42/1.48
% 2.42/1.48 thf(fact_255_Int__lower1,axiom,
% 2.42/1.48 ! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ A4 ) ).
% 2.42/1.48
% 2.42/1.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.42/1.48 ! [A6: $tType,A7: $tType] :
% 2.42/1.48 ( ( order @ A7 @ ( type2 @ A7 ) )
% 2.42/1.48 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(tcon_Set_Oset___Orderings_Oorder_1,axiom,
% 2.42/1.48 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.42/1.48 linorder @ $o @ ( type2 @ $o ) ).
% 2.42/1.48
% 2.42/1.48 thf(tcon_HOL_Obool___Orderings_Oorder_2,axiom,
% 2.42/1.48 order @ $o @ ( type2 @ $o ) ).
% 2.42/1.48
% 2.42/1.48 thf(conj_0,hypothesis,
% 2.42/1.48 ! [Y5: a] :
% 2.42/1.48 ( ( member @ a @ Y5 @ ( coinductive_lset @ a @ ysa ) )
% 2.42/1.48 => ( ( less_eq @ x @ Y5 )
% 2.42/1.48 => thesis ) ) ).
% 2.42/1.48
% 2.42/1.48 thf(conj_1,conjecture,
% 2.42/1.48 thesis ).
% 2.42/1.48 ------------------------
% 2.55/1.50 % (29752)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_29649 for (2999ds/183Mi)
% 2.55/1.51 % (29754)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_29649 for (2999ds/27Mi)
% 2.55/1.51 % (29756)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_29649 for (2999ds/2Mi)
% 2.55/1.51 % (29755)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_29649 for (2999ds/2Mi)
% 2.55/1.51 % (29758)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_29649 for (2999ds/18Mi)
% 2.55/1.51 % (29757)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_29649 for (2999ds/275Mi)
% 2.55/1.51 % (29753)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_29649 for (2999ds/4Mi)
% 2.55/1.51 % (29755)Instruction limit reached!
% 2.55/1.51 % (29755)------------------------------
% 2.55/1.51 % (29755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (29755)Termination reason: Unknown
% 2.55/1.51 % (29756)Instruction limit reached!
% 2.55/1.51 % (29756)------------------------------
% 2.55/1.51 % (29756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (29755)Termination phase: shuffling
% 2.55/1.51
% 2.55/1.51 % (29755)Memory used [KB]: 1279
% 2.55/1.51 % (29755)Time elapsed: 0.003 s
% 2.55/1.51 % (29755)Instructions burned: 3 (million)
% 2.55/1.51 % (29755)------------------------------
% 2.55/1.51 % (29755)------------------------------
% 2.55/1.51 % (29756)Termination reason: Unknown
% 2.55/1.51 % (29756)Termination phase: shuffling
% 2.55/1.51
% 2.55/1.51 % (29756)Memory used [KB]: 1279
% 2.55/1.51 % (29756)Time elapsed: 0.003 s
% 2.55/1.51 % (29756)Instructions burned: 4 (million)
% 2.55/1.51 % (29756)------------------------------
% 2.55/1.51 % (29756)------------------------------
% 2.55/1.51 % (29753)Instruction limit reached!
% 2.55/1.51 % (29753)------------------------------
% 2.55/1.51 % (29753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (29753)Termination reason: Unknown
% 2.55/1.51 % (29753)Termination phase: shuffling
% 2.55/1.51
% 2.55/1.51 % (29753)Memory used [KB]: 1407
% 2.55/1.51 % (29753)Time elapsed: 0.004 s
% 2.55/1.51 % (29753)Instructions burned: 6 (million)
% 2.55/1.51 % (29753)------------------------------
% 2.55/1.51 % (29753)------------------------------
% 2.55/1.52 % (29758)Instruction limit reached!
% 2.55/1.52 % (29758)------------------------------
% 2.55/1.52 % (29758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.52 % (29758)Termination reason: Unknown
% 2.55/1.52 % (29758)Termination phase: shuffling
% 2.55/1.52
% 2.55/1.52 % (29758)Memory used [KB]: 1535
% 2.55/1.52 % (29758)Time elapsed: 0.011 s
% 2.55/1.52 % (29758)Instructions burned: 19 (million)
% 2.55/1.52 % (29758)------------------------------
% 2.55/1.52 % (29758)------------------------------
% 2.55/1.52 % (29754)Instruction limit reached!
% 2.55/1.52 % (29754)------------------------------
% 2.55/1.52 % (29754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.52 % (29754)Termination reason: Unknown
% 2.55/1.52 % (29754)Termination phase: Property scanning
% 2.55/1.52
% 2.55/1.52 % (29754)Memory used [KB]: 1663
% 2.55/1.52 % (29754)Time elapsed: 0.015 s
% 2.55/1.52 % (29754)Instructions burned: 28 (million)
% 2.55/1.52 % (29754)------------------------------
% 2.55/1.52 % (29754)------------------------------
% 2.55/1.53 % (29760)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_29649 for (2999ds/37Mi)
% 2.55/1.53 % (29759)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_29649 for (2999ds/3Mi)
% 2.55/1.53 % (29761)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_29649 for (2999ds/15Mi)
% 2.55/1.53 % (29759)Instruction limit reached!
% 2.55/1.53 % (29759)------------------------------
% 2.55/1.53 % (29759)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (29759)Termination reason: Unknown
% 2.55/1.53 % (29759)Termination phase: shuffling
% 2.55/1.53
% 2.55/1.53 % (29759)Memory used [KB]: 1279
% 2.55/1.53 % (29759)Time elapsed: 0.003 s
% 2.55/1.53 % (29759)Instructions burned: 3 (million)
% 2.55/1.53 % (29759)------------------------------
% 2.55/1.53 % (29759)------------------------------
% 2.55/1.53 % (29757)First to succeed.
% 2.55/1.53 % (29762)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_29649 for (2999ds/3Mi)
% 2.55/1.53 % (29761)Instruction limit reached!
% 2.55/1.53 % (29761)------------------------------
% 2.55/1.53 % (29761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (29761)Termination reason: Unknown
% 2.55/1.53 % (29761)Termination phase: shuffling
% 2.55/1.53
% 2.55/1.53 % (29761)Memory used [KB]: 1535
% 2.55/1.53 % (29761)Time elapsed: 0.009 s
% 2.55/1.53 % (29761)Instructions burned: 15 (million)
% 2.55/1.53 % (29761)------------------------------
% 2.55/1.53 % (29761)------------------------------
% 2.55/1.53 % (29762)Instruction limit reached!
% 2.55/1.53 % (29762)------------------------------
% 2.55/1.53 % (29762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.54 % (29762)Termination reason: Unknown
% 2.55/1.54 % (29762)Termination phase: shuffling
% 2.55/1.54
% 2.55/1.54 % (29762)Memory used [KB]: 1279
% 2.55/1.54 % (29762)Time elapsed: 0.003 s
% 2.55/1.54 % (29762)Instructions burned: 4 (million)
% 2.55/1.54 % (29762)------------------------------
% 2.55/1.54 % (29762)------------------------------
% 2.55/1.54 % (29757)Refutation found. Thanks to Tanya!
% 2.55/1.54 % SZS status Theorem for DTF2THF_29649
% 2.55/1.54 % SZS output start Proof for DTF2THF_29649
% 2.55/1.54 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.55/1.54 thf(type_def_7, type, coinductive_llist: ($tType) > $tType).
% 2.55/1.54 thf(type_def_8, type, set: ($tType) > $tType).
% 2.55/1.54 thf(type_def_9, type, list: ($tType) > $tType).
% 2.55/1.54 thf(type_def_10, type, a: $tType).
% 2.55/1.54 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 2.55/1.54 thf(func_def_1, type, list: ($tType) > $tType).
% 2.55/1.54 thf(func_def_2, type, set: ($tType) > $tType).
% 2.55/1.54 thf(func_def_3, type, itself: ($tType) > $tType).
% 2.55/1.54 thf(func_def_4, type, a: $tType).
% 2.55/1.54 thf(func_def_5, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.55/1.54 thf(func_def_6, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.55/1.54 thf(func_def_7, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.55/1.54 thf(func_def_8, type, coindu351974385stinct: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.55/1.54 thf(func_def_9, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.55/1.54 thf(func_def_10, type, coinductive_LCons: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_11, type, coinductive_lset: !>[X0: $tType]:(coinductive_llist @ X0 > set @ X0)).
% 2.55/1.54 thf(func_def_12, type, coinductive_lsorted: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > $o)).
% 2.55/1.54 thf(func_def_13, type, condit2040224947_above: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 2.55/1.54 thf(func_def_14, type, condit1201339847_below: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 2.55/1.54 thf(func_def_15, type, finite_folding: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 2.55/1.54 thf(func_def_16, type, semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.55/1.54 thf(func_def_17, type, hammin1328233080lmerge: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_18, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.55/1.54 thf(func_def_19, type, semilattice: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.55/1.54 thf(func_def_20, type, lattic1693879045er_set: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 2.55/1.54 thf(func_def_21, type, insort_key: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > list @ X1 > list @ X1)).
% 2.55/1.54 thf(func_def_22, type, least: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > $o) > X0)).
% 2.55/1.54 thf(func_def_23, type, min: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 2.55/1.54 thf(func_def_24, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.55/1.54 thf(func_def_25, type, antimono: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > $o)).
% 2.55/1.54 thf(func_def_26, type, mono: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > $o)).
% 2.55/1.54 thf(func_def_27, type, strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > $o)).
% 2.55/1.54 thf(func_def_28, type, ordering: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 2.55/1.54 thf(func_def_29, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.55/1.54 thf(func_def_30, type, transp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 2.55/1.54 thf(func_def_31, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.55/1.54 thf(func_def_32, type, set_atLeast: !>[X0: $tType]:((X0 > X0 > $o) > X0 > set @ X0)).
% 2.55/1.54 thf(func_def_33, type, set_atLeastAtMost: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > set @ X0)).
% 2.55/1.54 thf(func_def_34, type, set_atLeastLessThan: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0 > X0 > set @ X0)).
% 2.55/1.54 thf(func_def_35, type, set_atMost: !>[X0: $tType]:((X0 > X0 > $o) > X0 > set @ X0)).
% 2.55/1.54 thf(func_def_36, type, set_greaterThan: !>[X0: $tType]:((X0 > X0 > $o) > X0 > set @ X0)).
% 2.55/1.54 thf(func_def_37, type, set_gr323396891AtMost: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0 > X0 > set @ X0)).
% 2.55/1.54 thf(func_def_38, type, set_gr1161524159ssThan: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > set @ X0)).
% 2.55/1.54 thf(func_def_39, type, set_lessThan: !>[X0: $tType]:((X0 > X0 > $o) > X0 > set @ X0)).
% 2.55/1.54 thf(func_def_40, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.55/1.54 thf(func_def_41, type, less: a > a > $o).
% 2.55/1.54 thf(func_def_42, type, less_eq: a > a > $o).
% 2.55/1.54 thf(func_def_44, type, x: a).
% 2.55/1.54 thf(func_def_45, type, x2: a).
% 2.55/1.54 thf(func_def_46, type, xsa: coinductive_llist @ a).
% 2.55/1.54 thf(func_def_47, type, ysa: coinductive_llist @ a).
% 2.55/1.54 thf(func_def_59, type, sP0: a > a > a > $o).
% 2.55/1.54 thf(func_def_60, type, sP1: a > a > a > $o).
% 2.55/1.54 thf(func_def_61, type, sP2: a > a > a > $o).
% 2.55/1.54 thf(func_def_62, type, sK3: a).
% 2.55/1.54 thf(func_def_63, type, sK4: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_64, type, sK5: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > X0)).
% 2.55/1.54 thf(func_def_65, type, sK6: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_66, type, sK7: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_67, type, sK8: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > X0)).
% 2.55/1.54 thf(func_def_68, type, sK9: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_69, type, sK10: (a > a > $o) > a).
% 2.55/1.54 thf(func_def_70, type, sK11: (a > a > $o) > a).
% 2.55/1.54 thf(func_def_71, type, sK12: (a > a > $o) > a).
% 2.55/1.54 thf(func_def_72, type, sK13: (a > a > $o) > a).
% 2.55/1.54 thf(func_def_73, type, sK14: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_74, type, sK15: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > X0)).
% 2.55/1.54 thf(func_def_75, type, sK16: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_76, type, sK17: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.55/1.54 thf(func_def_77, type, sK18: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > X0)).
% 2.55/1.54 thf(func_def_78, type, sK19: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_79, type, sK20: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > X0)).
% 2.55/1.54 thf(func_def_80, type, sK21: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_81, type, sK22: !>[X0: $tType]:((X0 > coinductive_llist @ X0 > $o) > X0)).
% 2.55/1.54 thf(func_def_82, type, sK23: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_83, type, sK24: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > X0)).
% 2.55/1.54 thf(func_def_84, type, sK25: !>[X0: $tType]:((coinductive_llist @ X0 > $o) > X0 > coinductive_llist @ X0)).
% 2.55/1.54 thf(func_def_86, type, ph27: !>[X0: $tType]:(X0)).
% 2.55/1.54 thf(f913,plain,(
% 2.55/1.54 $false),
% 2.55/1.54 inference(avatar_sat_refutation,[status(thm)],[f907,f908,f912])).
% 2.55/1.54 thf(f912,plain,(
% 2.55/1.54 ~spl26_2),
% 2.55/1.54 inference(avatar_contradiction_clause,[status(thm)],[f911])).
% 2.55/1.54 thf(f911,plain,(
% 2.55/1.54 $false | ~spl26_2),
% 2.55/1.54 inference(subsumption_resolution,[status(thm)],[f910,f821])).
% 2.55/1.54 thf(f821,plain,(
% 2.55/1.54 ((less_eq @ x @ sK3) = $true)),
% 2.55/1.54 inference(cnf_transformation,[status(thm)],[f788])).
% 2.55/1.54 thf(f788,plain,(
% 2.55/1.54 ((member @ a @ sK3 @ (coinductive_lset @ a @ ysa)) = $true) & ((less_eq @ x @ sK3) = $true)),
% 2.55/1.54 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3])],[f510,f787])).
% 2.55/1.54 thf(f787,plain,(
% 2.55/1.54 ? [X0 : a] : (((member @ a @ X0 @ (coinductive_lset @ a @ ysa)) = $true) & ((less_eq @ x @ X0) = $true)) => (((member @ a @ sK3 @ (coinductive_lset @ a @ ysa)) = $true) & ((less_eq @ x @ sK3) = $true))),
% 2.55/1.54 introduced(definition,[],[choice_axiom])).
% 2.55/1.54 thf(f510,plain,(
% 2.55/1.54 ? [X0 : a] : (((member @ a @ X0 @ (coinductive_lset @ a @ ysa)) = $true) & ((less_eq @ x @ X0) = $true))),
% 2.55/1.54 inference(fool_elimination,[status(thm)],[f509])).
% 2.55/1.54 thf(f509,plain,(
% 2.55/1.54 ? [X0 : a] : ((less_eq @ x @ X0) & (member @ a @ X0 @ (coinductive_lset @ a @ ysa)))),
% 2.55/1.54 inference(rectify,[status(thm)],[f44])).
% 2.55/1.54 thf(f44,axiom,(
% 2.55/1.54 ? [X15 : a] : ((less_eq @ x @ X15) & (member @ a @ X15 @ (coinductive_lset @ a @ ysa)))),
% 2.55/1.54 file('/export/starexec/sandbox2/tmp/tmp.0EB1vRga9H/DTF2THF_29649.p',fact_43_step_Oprems_I3_J)).
% 2.55/1.54 thf(f910,plain,(
% 2.55/1.54 ((less_eq @ x @ sK3) != $true) | ~spl26_2),
% 2.55/1.54 inference(trivial_inequality_removal,[status(thm)],[f909])).
% 2.55/1.54 thf(f909,plain,(
% 2.55/1.54 ($true != $true) | ((less_eq @ x @ sK3) != $true) | ~spl26_2),
% 2.55/1.54 inference(superposition,[status(thm)],[f906,f822])).
% 2.55/1.54 thf(f822,plain,(
% 2.55/1.54 ((member @ a @ sK3 @ (coinductive_lset @ a @ ysa)) = $true)),
% 2.55/1.54 inference(cnf_transformation,[status(thm)],[f788])).
% 2.55/1.54 thf(f906,plain,(
% 2.55/1.54 ( ! [X0 : a] : (((member @ a @ X0 @ (coinductive_lset @ a @ ysa)) != $true) | ((less_eq @ x @ X0) != $true)) ) | ~spl26_2),
% 2.55/1.54 inference(avatar_component_clause,[status(thm)],[f905])).
% 2.55/1.54 thf(f905,definition,(
% 2.55/1.54 spl26_2 <=> ! [X0 : a] : (((less_eq @ x @ X0) != $true) | ((member @ a @ X0 @ (coinductive_lset @ a @ ysa)) != $true))),
% 2.55/1.54 introduced(definition,[new_symbols(naming,[spl26_2])],[avatar_definition])).
% 2.55/1.54 thf(f908,plain,(
% 2.55/1.54 ~spl26_1),
% 2.55/1.54 inference(avatar_split_clause,[status(thm)],[f858,f901])).
% 2.55/1.54 thf(f901,definition,(
% 2.55/1.54 spl26_1 <=> (thesis = $true)),
% 2.55/1.54 introduced(definition,[new_symbols(naming,[spl26_1])],[avatar_definition])).
% 2.55/1.54 thf(f858,plain,(
% 2.55/1.54 (thesis != $true)),
% 2.55/1.54 inference(cnf_transformation,[status(thm)],[f721])).
% 2.55/1.54 thf(f721,plain,(
% 2.55/1.54 (thesis != $true)),
% 2.55/1.54 inference(flattening,[status(thm)],[f576])).
% 2.55/1.54 thf(f576,plain,(
% 2.55/1.54 ~(thesis = $true)),
% 2.55/1.54 inference(fool_elimination,[status(thm)],[f575])).
% 2.55/1.54 thf(f575,plain,(
% 2.55/1.54 ~thesis),
% 2.55/1.54 inference(rectify,[status(thm)],[f263])).
% 2.55/1.54 thf(f263,negated_conjecture,(
% 2.55/1.54 ~thesis),
% 2.55/1.54 inference(negated_conjecture,[status(cth)],[f262])).
% 2.55/1.54 thf(f262,conjecture,(
% 2.55/1.54 thesis),
% 2.55/1.54 file('/export/starexec/sandbox2/tmp/tmp.0EB1vRga9H/DTF2THF_29649.p',conj_1)).
% 2.55/1.54 thf(f907,plain,(
% 2.55/1.54 spl26_1 | spl26_2),
% 2.55/1.54 inference(avatar_split_clause,[status(thm)],[f861,f905,f901])).
% 2.55/1.54 thf(f861,plain,(
% 2.55/1.54 ( ! [X0 : a] : ((thesis = $true) | ((less_eq @ x @ X0) != $true) | ((member @ a @ X0 @ (coinductive_lset @ a @ ysa)) != $true)) )),
% 2.55/1.54 inference(cnf_transformation,[status(thm)],[f764])).
% 2.55/1.54 thf(f764,plain,(
% 2.55/1.54 ! [X0 : a] : (((member @ a @ X0 @ (coinductive_lset @ a @ ysa)) != $true) | (thesis = $true) | ((less_eq @ x @ X0) != $true))),
% 2.55/1.54 inference(flattening,[status(thm)],[f763])).
% 2.55/1.54 thf(f763,plain,(
% 2.55/1.54 ! [X0 : a] : (((thesis = $true) | ((less_eq @ x @ X0) != $true)) | ((member @ a @ X0 @ (coinductive_lset @ a @ ysa)) != $true))),
% 2.55/1.54 inference(ennf_transformation,[status(thm)],[f638])).
% 2.55/1.54 thf(f638,plain,(
% 2.55/1.54 ! [X0 : a] : (((member @ a @ X0 @ (coinductive_lset @ a @ ysa)) = $true) => (((less_eq @ x @ X0) = $true) => (thesis = $true)))),
% 2.55/1.54 inference(fool_elimination,[status(thm)],[f637])).
% 2.55/1.54 thf(f637,plain,(
% 2.55/1.54 ! [X0 : a] : ((member @ a @ X0 @ (coinductive_lset @ a @ ysa)) => ((less_eq @ x @ X0) => thesis))),
% 2.55/1.54 inference(rectify,[status(thm)],[f261])).
% 2.55/1.54 thf(f261,axiom,(
% 2.55/1.54 ! [X21 : a] : ((member @ a @ X21 @ (coinductive_lset @ a @ ysa)) => ((less_eq @ x @ X21) => thesis))),
% 2.55/1.54 file('/export/starexec/sandbox2/tmp/tmp.0EB1vRga9H/DTF2THF_29649.p',conj_0)).
% 2.55/1.54 % SZS output end Proof for DTF2THF_29649
% 2.55/1.54 % (29757)------------------------------
% 2.55/1.54 % (29757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.54 % (29757)Termination reason: Refutation
% 2.55/1.54
% 2.55/1.54 % (29757)Memory used [KB]: 6268
% 2.55/1.54 % (29757)Time elapsed: 0.027 s
% 2.55/1.54 % (29757)Instructions burned: 50 (million)
% 2.55/1.54 % (29757)------------------------------
% 2.55/1.54 % (29757)------------------------------
% 2.55/1.54 % (29751)Success in time 0.055 s
%------------------------------------------------------------------------------