%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP119^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n004.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:42 AM UTC 2026
% Result : Theorem 2.65s 1.59s
% Output : Refutation 2.65s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP119^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n004.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:26:41 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.27/0.46 ---- Original DTF file ---
% 0.27/0.46 thf(spec,logic,$$dhol).
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % File : ITP119^2 : TPTP v9.2.1. Released v7.5.0.
% 0.27/0.46 % Domain : Interactive Theorem Proving
% 0.27/0.46 % Problem : Sledgehammer Modular_Distrib_Lattice problem prob_144__3260118_1
% 0.27/0.46 % Version : Especial.
% 0.27/0.46 % English :
% 0.27/0.46
% 0.27/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.27/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.27/0.46 % Source : [Des21]
% 0.27/0.46 % Names : Modular_Distrib_Lattice/prob_144__3260118_1 [Des21]
% 0.27/0.46
% 0.27/0.46 % Status : ContradictoryAxioms
% 0.27/0.46 % Rating : 0.33 v9.1.0, 0.00 v8.1.0, 0.25 v7.5.0
% 0.27/0.46 % Syntax : Number of formulae : 308 ( 154 unt; 44 typ; 0 def)
% 0.27/0.46 % Number of atoms : 620 ( 313 equ; 0 cnn)
% 0.27/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.27/0.46 % Number of connectives : 3325 ( 126 ~; 15 |; 70 &;2860 @)
% 0.27/0.46 % ( 0 <=>; 254 =>; 0 <=; 0 <~>)
% 0.27/0.46 % Maximal formula depth : 23 ( 7 avg)
% 0.27/0.46 % Number of types : 2 ( 1 usr)
% 0.27/0.46 % Number of type conns : 331 ( 331 >; 0 *; 0 +; 0 <<)
% 0.27/0.46 % Number of symbols : 45 ( 43 usr; 4 con; 0-7 aty)
% 0.27/0.46 % Number of variables : 981 ( 18 ^; 859 !; 63 ?; 981 :)
% 0.27/0.46 % ( 41 !>; 0 ?*; 0 @-; 0 @+)
% 0.27/0.46 % SPC : TH1_THM_EQU_NAR
% 0.27/0.46
% 0.27/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:29:06.337
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % Could-be-implicit typings (2)
% 0.27/0.46 thf(ty_t_List_Olist,type,
% 0.27/0.46 list: $tType > $tType ).
% 0.27/0.46
% 0.27/0.46 thf(ty_tf_a,type,
% 0.27/0.46 a: $tType ).
% 0.27/0.46
% 0.27/0.46 % Explicit typings (42)
% 0.27/0.46 thf(sy_cl_HOL_Otype,type,
% 0.27/0.46 type:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.27/0.46 ord:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.27/0.46 linorder:
% 0.27/0.46 !>[A: $tType] : $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
% 0.27/0.46 finite_comp_fun_idem:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Groups_Oabel__semigroup,type,
% 0.27/0.46 abel_semigroup:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Groups_Oabel__semigroup__axioms,type,
% 0.27/0.46 abel_s1917375468axioms:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Groups_Osemigroup,type,
% 0.27/0.46 semigroup:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Lattices_Osemilattice,type,
% 0.27/0.46 semilattice:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Lattices_Osemilattice__axioms,type,
% 0.27/0.46 semilattice_axioms:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Lattices__Big_Oord_Ois__arg__max,type,
% 0.27/0.46 lattices_is_arg_max:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Lattices__Big_Oord_Ois__arg__min,type,
% 0.27/0.46 lattices_is_arg_min:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 0.27/0.46 lattic35693393ce_set:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Oappend,type,
% 0.27/0.46 append:
% 0.27/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Obind,type,
% 0.27/0.46 bind:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Obutlast,type,
% 0.27/0.46 butlast:
% 0.27/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Oconcat,type,
% 0.27/0.46 concat:
% 0.27/0.46 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Oinsert,type,
% 0.27/0.46 insert:
% 0.27/0.46 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Olast,type,
% 0.27/0.46 last:
% 0.27/0.46 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Olist_OCons,type,
% 0.27/0.46 cons:
% 0.27/0.46 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Olist_ONil,type,
% 0.27/0.46 nil:
% 0.27/0.46 !>[A: $tType] : ( list @ A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Olist__ex1,type,
% 0.27/0.46 list_ex1:
% 0.27/0.46 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Omaps,type,
% 0.27/0.46 maps:
% 0.27/0.46 !>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Oord_Olexordp,type,
% 0.27/0.46 lexordp:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Oord_Olexordp__eq,type,
% 0.27/0.46 lexordp_eq:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Oproduct__lists,type,
% 0.27/0.46 product_lists:
% 0.27/0.46 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Orotate1,type,
% 0.27/0.46 rotate1:
% 0.27/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_List_Osubseqs,type,
% 0.27/0.46 subseqs:
% 0.27/0.46 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_OM5__lattice,type,
% 0.27/0.46 modula1618706180attice:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_ON5__lattice,type,
% 0.27/0.46 modula1192512645attice:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oa__aux,type,
% 0.27/0.46 modula105138637_a_aux:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ob__aux,type,
% 0.27/0.46 modula477145868_b_aux:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oc__aux,type,
% 0.27/0.46 modula849153099_c_aux:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux,type,
% 0.27/0.46 modula1221160330_d_aux:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux,type,
% 0.27/0.46 modula1593167561_e_aux:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ono__distrib,type,
% 0.27/0.46 modula141148890istrib:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_c_Relation_Otransp,type,
% 0.27/0.46 transp:
% 0.27/0.46 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_a,type,
% 0.27/0.46 a2: a ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_b,type,
% 0.27/0.46 b: a ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_c,type,
% 0.27/0.46 c: a ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_inf,type,
% 0.27/0.46 inf: a > a > a ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_less,type,
% 0.27/0.46 less: a > a > $o ).
% 0.27/0.46
% 0.27/0.46 thf(sy_v_sup,type,
% 0.27/0.46 sup: a > a > a ).
% 0.27/0.46
% 0.27/0.46 % Relevant facts (255)
% 0.27/0.46 thf(fact_0_local_Odual__order_Oasym,axiom,
% 0.27/0.46 ! [B2: a,A2: a] :
% 0.27/0.46 ( ( less @ B2 @ A2 )
% 0.27/0.46 => ~ ( less @ A2 @ B2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.dual_order.asym
% 0.27/0.46 thf(fact_1_local_Odual__order_Ostrict__implies__not__eq,axiom,
% 0.27/0.46 ! [B2: a,A2: a] :
% 0.27/0.46 ( ( less @ B2 @ A2 )
% 0.27/0.46 => ( A2 != B2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.dual_order.strict_implies_not_eq
% 0.27/0.46 thf(fact_2_local_Odual__order_Ostrict__trans,axiom,
% 0.27/0.46 ! [B2: a,A2: a,C: a] :
% 0.27/0.46 ( ( less @ B2 @ A2 )
% 0.27/0.46 => ( ( less @ C @ B2 )
% 0.27/0.46 => ( less @ C @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.dual_order.strict_trans
% 0.27/0.46 thf(fact_3_local_Oless__asym,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( less @ X @ Y )
% 0.27/0.46 => ~ ( less @ Y @ X ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_asym
% 0.27/0.46 thf(fact_4_local_Oless__asym_H,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( less @ A2 @ B2 )
% 0.27/0.46 => ~ ( less @ B2 @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_asym'
% 0.27/0.46 thf(fact_5_local_Oless__imp__neq,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( less @ X @ Y )
% 0.27/0.46 => ( X != Y ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_imp_neq
% 0.27/0.46 thf(fact_6_local_Oless__imp__not__eq,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( less @ X @ Y )
% 0.27/0.46 => ( X != Y ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_imp_not_eq
% 0.27/0.46 thf(fact_7_local_Oless__imp__not__eq2,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( less @ X @ Y )
% 0.27/0.46 => ( Y != X ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_imp_not_eq2
% 0.27/0.46 thf(fact_8_local_Oless__imp__not__less,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( less @ X @ Y )
% 0.27/0.46 => ~ ( less @ Y @ X ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_imp_not_less
% 0.27/0.46 thf(fact_9_local_Oless__imp__triv,axiom,
% 0.27/0.46 ! [X: a,Y: a,P: $o] :
% 0.27/0.46 ( ( less @ X @ Y )
% 0.27/0.46 => ( ( less @ Y @ X )
% 0.27/0.46 => P ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_imp_triv
% 0.27/0.46 thf(fact_10_local_Oless__irrefl,axiom,
% 0.27/0.46 ! [X: a] :
% 0.27/0.46 ~ ( less @ X @ X ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_irrefl
% 0.27/0.46 thf(fact_11_local_Oless__not__sym,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( less @ X @ Y )
% 0.27/0.46 => ~ ( less @ Y @ X ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_not_sym
% 0.27/0.46 thf(fact_12_local_Oless__trans,axiom,
% 0.27/0.46 ! [X: a,Y: a,Z: a] :
% 0.27/0.46 ( ( less @ X @ Y )
% 0.27/0.46 => ( ( less @ Y @ Z )
% 0.27/0.46 => ( less @ X @ Z ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_trans
% 0.27/0.46 thf(fact_13_local_Oord__eq__less__trans,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( A2 = B2 )
% 0.27/0.46 => ( ( less @ B2 @ C )
% 0.27/0.46 => ( less @ A2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.ord_eq_less_trans
% 0.27/0.46 thf(fact_14_local_Oord__less__eq__trans,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( less @ A2 @ B2 )
% 0.27/0.46 => ( ( B2 = C )
% 0.27/0.46 => ( less @ A2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.ord_less_eq_trans
% 0.27/0.46 thf(fact_15_local_Oorder_Oasym,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( less @ A2 @ B2 )
% 0.27/0.46 => ~ ( less @ B2 @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.order.asym
% 0.27/0.46 thf(fact_16_local_Oorder_Oirrefl,axiom,
% 0.27/0.46 ! [A2: a] :
% 0.27/0.46 ~ ( less @ A2 @ A2 ) ).
% 0.27/0.46
% 0.27/0.46 % local.order.irrefl
% 0.27/0.46 thf(fact_17_local_Oorder_Ostrict__implies__not__eq,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( less @ A2 @ B2 )
% 0.27/0.46 => ( A2 != B2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.order.strict_implies_not_eq
% 0.27/0.46 thf(fact_18_local_Oorder_Ostrict__trans,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( less @ A2 @ B2 )
% 0.27/0.46 => ( ( less @ B2 @ C )
% 0.27/0.46 => ( less @ A2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.order.strict_trans
% 0.27/0.46 thf(fact_19_local_Oinf_Oassoc,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( inf @ ( inf @ A2 @ B2 ) @ C )
% 0.27/0.46 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.assoc
% 0.27/0.46 thf(fact_20_local_Oinf_Ocommute,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( inf @ A2 @ B2 )
% 0.27/0.46 = ( inf @ B2 @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.commute
% 0.27/0.46 thf(fact_21_local_Oinf_Oleft__commute,axiom,
% 0.27/0.46 ! [B2: a,A2: a,C: a] :
% 0.27/0.46 ( ( inf @ B2 @ ( inf @ A2 @ C ) )
% 0.27/0.46 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.left_commute
% 0.27/0.46 thf(fact_22_local_Oinf__assoc,axiom,
% 0.27/0.46 ! [X: a,Y: a,Z: a] :
% 0.27/0.46 ( ( inf @ ( inf @ X @ Y ) @ Z )
% 0.27/0.46 = ( inf @ X @ ( inf @ Y @ Z ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf_assoc
% 0.27/0.46 thf(fact_23_local_Oinf__commute,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( inf @ X @ Y )
% 0.27/0.46 = ( inf @ Y @ X ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf_commute
% 0.27/0.46 thf(fact_24_local_Oinf__left__commute,axiom,
% 0.27/0.46 ! [X: a,Y: a,Z: a] :
% 0.27/0.46 ( ( inf @ X @ ( inf @ Y @ Z ) )
% 0.27/0.46 = ( inf @ Y @ ( inf @ X @ Z ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf_left_commute
% 0.27/0.46 thf(fact_25_local_Osup_Oassoc,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( sup @ ( sup @ A2 @ B2 ) @ C )
% 0.27/0.46 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.assoc
% 0.27/0.46 thf(fact_26_local_Osup_Ocommute,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( sup @ A2 @ B2 )
% 0.27/0.46 = ( sup @ B2 @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.commute
% 0.27/0.46 thf(fact_27_local_Osup_Oleft__commute,axiom,
% 0.27/0.46 ! [B2: a,A2: a,C: a] :
% 0.27/0.46 ( ( sup @ B2 @ ( sup @ A2 @ C ) )
% 0.27/0.46 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.left_commute
% 0.27/0.46 thf(fact_28_local_Osup__assoc,axiom,
% 0.27/0.46 ! [X: a,Y: a,Z: a] :
% 0.27/0.46 ( ( sup @ ( sup @ X @ Y ) @ Z )
% 0.27/0.46 = ( sup @ X @ ( sup @ Y @ Z ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup_assoc
% 0.27/0.46 thf(fact_29_local_Osup__commute,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( sup @ X @ Y )
% 0.27/0.46 = ( sup @ Y @ X ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup_commute
% 0.27/0.46 thf(fact_30_local_Osup__left__commute,axiom,
% 0.27/0.46 ! [X: a,Y: a,Z: a] :
% 0.27/0.46 ( ( sup @ X @ ( sup @ Y @ Z ) )
% 0.27/0.46 = ( sup @ Y @ ( sup @ X @ Z ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup_left_commute
% 0.27/0.46 thf(fact_31_local_Oinf_Ostrict__boundedE,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( less @ A2 @ ( inf @ B2 @ C ) )
% 0.27/0.46 => ~ ( ( less @ A2 @ B2 )
% 0.27/0.46 => ~ ( less @ A2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.strict_boundedE
% 0.27/0.46 thf(fact_32_local_Oinf_Ostrict__coboundedI1,axiom,
% 0.27/0.46 ! [A2: a,C: a,B2: a] :
% 0.27/0.46 ( ( less @ A2 @ C )
% 0.27/0.46 => ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.strict_coboundedI1
% 0.27/0.46 thf(fact_33_local_Oinf_Ostrict__coboundedI2,axiom,
% 0.27/0.46 ! [B2: a,C: a,A2: a] :
% 0.27/0.46 ( ( less @ B2 @ C )
% 0.27/0.46 => ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.strict_coboundedI2
% 0.27/0.46 thf(fact_34_local_Oinf_Ostrict__order__iff,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( less @ A2 @ B2 )
% 0.27/0.46 = ( ( A2
% 0.27/0.46 = ( inf @ A2 @ B2 ) )
% 0.27/0.46 & ( A2 != B2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.strict_order_iff
% 0.27/0.46 thf(fact_35_local_Oless__infI1,axiom,
% 0.27/0.46 ! [A2: a,X: a,B2: a] :
% 0.27/0.46 ( ( less @ A2 @ X )
% 0.27/0.46 => ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_infI1
% 0.27/0.46 thf(fact_36_local_Oless__infI2,axiom,
% 0.27/0.46 ! [B2: a,X: a,A2: a] :
% 0.27/0.46 ( ( less @ B2 @ X )
% 0.27/0.46 => ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_infI2
% 0.27/0.46 thf(fact_37_local_Oless__supI1,axiom,
% 0.27/0.46 ! [X: a,A2: a,B2: a] :
% 0.27/0.46 ( ( less @ X @ A2 )
% 0.27/0.46 => ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_supI1
% 0.27/0.46 thf(fact_38_local_Oless__supI2,axiom,
% 0.27/0.46 ! [X: a,B2: a,A2: a] :
% 0.27/0.46 ( ( less @ X @ B2 )
% 0.27/0.46 => ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.less_supI2
% 0.27/0.46 thf(fact_39_local_Osup_Ostrict__boundedE,axiom,
% 0.27/0.46 ! [B2: a,C: a,A2: a] :
% 0.27/0.46 ( ( less @ ( sup @ B2 @ C ) @ A2 )
% 0.27/0.46 => ~ ( ( less @ B2 @ A2 )
% 0.27/0.46 => ~ ( less @ C @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.strict_boundedE
% 0.27/0.46 thf(fact_40_local_Osup_Ostrict__coboundedI1,axiom,
% 0.27/0.46 ! [C: a,A2: a,B2: a] :
% 0.27/0.46 ( ( less @ C @ A2 )
% 0.27/0.46 => ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.strict_coboundedI1
% 0.27/0.46 thf(fact_41_local_Osup_Ostrict__coboundedI2,axiom,
% 0.27/0.46 ! [C: a,B2: a,A2: a] :
% 0.27/0.46 ( ( less @ C @ B2 )
% 0.27/0.46 => ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.strict_coboundedI2
% 0.27/0.46 thf(fact_42_local_Osup_Ostrict__order__iff,axiom,
% 0.27/0.46 ! [B2: a,A2: a] :
% 0.27/0.46 ( ( less @ B2 @ A2 )
% 0.27/0.46 = ( ( A2
% 0.27/0.46 = ( sup @ A2 @ B2 ) )
% 0.27/0.46 & ( A2 != B2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.strict_order_iff
% 0.27/0.46 thf(fact_43_local_Odistrib__imp1,axiom,
% 0.27/0.46 ! [X: a,Y: a,Z: a] :
% 0.27/0.46 ( ! [X2: a,Y2: a,Z2: a] :
% 0.27/0.46 ( ( inf @ X2 @ ( sup @ Y2 @ Z2 ) )
% 0.27/0.46 = ( sup @ ( inf @ X2 @ Y2 ) @ ( inf @ X2 @ Z2 ) ) )
% 0.27/0.46 => ( ( sup @ X @ ( inf @ Y @ Z ) )
% 0.27/0.46 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.distrib_imp1
% 0.27/0.46 thf(fact_44_local_Odistrib__imp2,axiom,
% 0.27/0.46 ! [X: a,Y: a,Z: a] :
% 0.27/0.46 ( ! [X2: a,Y2: a,Z2: a] :
% 0.27/0.46 ( ( sup @ X2 @ ( inf @ Y2 @ Z2 ) )
% 0.27/0.46 = ( inf @ ( sup @ X2 @ Y2 ) @ ( sup @ X2 @ Z2 ) ) )
% 0.27/0.46 => ( ( inf @ X @ ( sup @ Y @ Z ) )
% 0.27/0.46 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.distrib_imp2
% 0.27/0.46 thf(fact_45_ext,axiom,
% 0.27/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.27/0.46 ( ! [X2: A] :
% 0.27/0.46 ( ( F @ X2 )
% 0.27/0.46 = ( G @ X2 ) )
% 0.27/0.46 => ( F = G ) ) ).
% 0.27/0.46
% 0.27/0.46 % ext
% 0.27/0.46 thf(fact_46_local_Oinf_Oidem,axiom,
% 0.27/0.46 ! [A2: a] :
% 0.27/0.46 ( ( inf @ A2 @ A2 )
% 0.27/0.46 = A2 ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.idem
% 0.27/0.46 thf(fact_47_local_Oinf_Oleft__idem,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
% 0.27/0.46 = ( inf @ A2 @ B2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.left_idem
% 0.27/0.46 thf(fact_48_local_Oinf_Oright__idem,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
% 0.27/0.46 = ( inf @ A2 @ B2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf.right_idem
% 0.27/0.46 thf(fact_49_local_Oinf__idem,axiom,
% 0.27/0.46 ! [X: a] :
% 0.27/0.46 ( ( inf @ X @ X )
% 0.27/0.46 = X ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf_idem
% 0.27/0.46 thf(fact_50_local_Oinf__left__idem,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( inf @ X @ ( inf @ X @ Y ) )
% 0.27/0.46 = ( inf @ X @ Y ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf_left_idem
% 0.27/0.46 thf(fact_51_local_Oinf__right__idem,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 0.27/0.46 = ( inf @ X @ Y ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf_right_idem
% 0.27/0.46 thf(fact_52_local_Osup_Oidem,axiom,
% 0.27/0.46 ! [A2: a] :
% 0.27/0.46 ( ( sup @ A2 @ A2 )
% 0.27/0.46 = A2 ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.idem
% 0.27/0.46 thf(fact_53_local_Osup_Oleft__idem,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
% 0.27/0.46 = ( sup @ A2 @ B2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.left_idem
% 0.27/0.46 thf(fact_54_local_Osup_Oright__idem,axiom,
% 0.27/0.46 ! [A2: a,B2: a] :
% 0.27/0.46 ( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
% 0.27/0.46 = ( sup @ A2 @ B2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup.right_idem
% 0.27/0.46 thf(fact_55_local_Osup__idem,axiom,
% 0.27/0.46 ! [X: a] :
% 0.27/0.46 ( ( sup @ X @ X )
% 0.27/0.46 = X ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup_idem
% 0.27/0.46 thf(fact_56_local_Osup__left__idem,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( sup @ X @ ( sup @ X @ Y ) )
% 0.27/0.46 = ( sup @ X @ Y ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup_left_idem
% 0.27/0.46 thf(fact_57_local_Ois__arg__max__def,axiom,
% 0.27/0.46 ! [B: $tType,F: B > a,P: B > $o,X: B] :
% 0.27/0.46 ( ( lattices_is_arg_max @ a @ B @ less @ F @ P @ X )
% 0.27/0.46 = ( ( P @ X )
% 0.27/0.46 & ~ ? [Y3: B] :
% 0.27/0.46 ( ( P @ Y3 )
% 0.27/0.46 & ( less @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.is_arg_max_def
% 0.27/0.46 thf(fact_58_local_Ois__arg__min__def,axiom,
% 0.27/0.46 ! [B: $tType,F: B > a,P: B > $o,X: B] :
% 0.27/0.46 ( ( lattices_is_arg_min @ a @ B @ less @ F @ P @ X )
% 0.27/0.46 = ( ( P @ X )
% 0.27/0.46 & ~ ? [Y3: B] :
% 0.27/0.46 ( ( P @ Y3 )
% 0.27/0.46 & ( less @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.is_arg_min_def
% 0.27/0.46 thf(fact_59_M5__lattice__def,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula1618706180attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( ( ( inf @ A2 @ B2 )
% 0.27/0.46 = ( inf @ B2 @ C ) )
% 0.27/0.46 & ( ( inf @ C @ A2 )
% 0.27/0.46 = ( inf @ B2 @ C ) )
% 0.27/0.46 & ( ( sup @ A2 @ B2 )
% 0.27/0.46 = ( sup @ B2 @ C ) )
% 0.27/0.46 & ( ( sup @ C @ A2 )
% 0.27/0.46 = ( sup @ B2 @ C ) )
% 0.27/0.46 & ( less @ ( inf @ A2 @ B2 ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % M5_lattice_def
% 0.27/0.46 thf(fact_60_N5__lattice__def,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula1192512645attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( ( ( inf @ A2 @ C )
% 0.27/0.46 = ( inf @ B2 @ C ) )
% 0.27/0.46 & ( less @ A2 @ B2 )
% 0.27/0.46 & ( ( sup @ A2 @ C )
% 0.27/0.46 = ( sup @ B2 @ C ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % N5_lattice_def
% 0.27/0.46 thf(fact_61_no__distrib__def,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula141148890istrib @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( less @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ C @ A2 ) ) @ ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % no_distrib_def
% 0.27/0.46 thf(fact_62_local_Oinf__sup__absorb,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( inf @ X @ ( sup @ X @ Y ) )
% 0.27/0.46 = X ) ).
% 0.27/0.46
% 0.27/0.46 % local.inf_sup_absorb
% 0.27/0.46 thf(fact_63_local_Osup__inf__absorb,axiom,
% 0.27/0.46 ! [X: a,Y: a] :
% 0.27/0.46 ( ( sup @ X @ ( inf @ X @ Y ) )
% 0.27/0.46 = X ) ).
% 0.27/0.46
% 0.27/0.46 % local.sup_inf_absorb
% 0.27/0.46 thf(fact_64_local_Oa__join__d,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( sup @ A2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.27/0.46 = ( sup @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.a_join_d
% 0.27/0.46 thf(fact_65_local_Ob__join__d,axiom,
% 0.27/0.46 ! [B2: a,A2: a,C: a] :
% 0.27/0.46 ( ( sup @ B2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.27/0.46 = ( sup @ B2 @ ( inf @ C @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.b_join_d
% 0.27/0.46 thf(fact_66_local_Od__aux__def,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( sup @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ B2 @ C ) ) @ ( inf @ C @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.d_aux_def
% 0.27/0.46 thf(fact_67_local_Od__b__c__a,axiom,
% 0.27/0.46 ! [B2: a,C: a,A2: a] :
% 0.27/0.46 ( ( modula1221160330_d_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 0.27/0.46 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.d_b_c_a
% 0.27/0.46 thf(fact_68_local_Od__c__a__b,axiom,
% 0.27/0.46 ! [C: a,A2: a,B2: a] :
% 0.27/0.46 ( ( modula1221160330_d_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 0.27/0.46 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.d_c_a_b
% 0.27/0.46 thf(fact_69_local_Oa__meet__e,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.27/0.46 = ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.a_meet_e
% 0.27/0.46 thf(fact_70_local_Ob__meet__e,axiom,
% 0.27/0.46 ! [B2: a,A2: a,C: a] :
% 0.27/0.46 ( ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.27/0.46 = ( inf @ B2 @ ( sup @ C @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.b_meet_e
% 0.27/0.46 thf(fact_71_local_Oc__meet__e,axiom,
% 0.27/0.46 ! [C: a,A2: a,B2: a] :
% 0.27/0.46 ( ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 0.27/0.46 = ( inf @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.c_meet_e
% 0.27/0.46 thf(fact_72_local_Oe__aux__def,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( inf @ ( inf @ ( sup @ A2 @ B2 ) @ ( sup @ B2 @ C ) ) @ ( sup @ C @ A2 ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.e_aux_def
% 0.27/0.46 thf(fact_73_local_Oe__b__c__a,axiom,
% 0.27/0.46 ! [B2: a,C: a,A2: a] :
% 0.27/0.46 ( ( modula1593167561_e_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 0.27/0.46 = ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.e_b_c_a
% 0.27/0.46 thf(fact_74_local_Oe__c__a__b,axiom,
% 0.27/0.46 ! [C: a,A2: a,B2: a] :
% 0.27/0.46 ( ( modula1593167561_e_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 0.27/0.46 = ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.e_c_a_b
% 0.27/0.46 thf(fact_75_local_Ocomp__fun__idem__sup,axiom,
% 0.27/0.46 finite_comp_fun_idem @ a @ a @ sup ).
% 0.27/0.46
% 0.27/0.46 % local.comp_fun_idem_sup
% 0.27/0.46 thf(fact_76_local_Ocomp__fun__idem__inf,axiom,
% 0.27/0.46 finite_comp_fun_idem @ a @ a @ inf ).
% 0.27/0.46
% 0.27/0.46 % local.comp_fun_idem_inf
% 0.27/0.46 thf(fact_77_local_Oc__aux__def,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( sup @ ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.c_aux_def
% 0.27/0.46 thf(fact_78_local_Ob__aux__def,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( sup @ ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.b_aux_def
% 0.27/0.46 thf(fact_79_local_Oa__aux__def,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula105138637_a_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( sup @ ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.a_aux_def
% 0.27/0.46 thf(fact_80_lattice_Ono__distrib_Ocong,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( modula141148890istrib @ A )
% 0.27/0.46 = ( modula141148890istrib @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lattice.no_distrib.cong
% 0.27/0.46 thf(fact_81_lattice_ON5__lattice_Ocong,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( modula1192512645attice @ A )
% 0.27/0.46 = ( modula1192512645attice @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lattice.N5_lattice.cong
% 0.27/0.46 thf(fact_82_lattice_OM5__lattice_Ocong,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( modula1618706180attice @ A )
% 0.27/0.46 = ( modula1618706180attice @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lattice.M5_lattice.cong
% 0.27/0.46 thf(fact_83_lattice_Oe__aux_Ocong,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( modula1593167561_e_aux @ A )
% 0.27/0.46 = ( modula1593167561_e_aux @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lattice.e_aux.cong
% 0.27/0.46 thf(fact_84_lattice_Od__aux_Ocong,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( modula1221160330_d_aux @ A )
% 0.27/0.46 = ( modula1221160330_d_aux @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lattice.d_aux.cong
% 0.27/0.46 thf(fact_85_local_Oc__a,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( modula105138637_a_aux @ a @ inf @ sup @ C @ A2 @ B2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.c_a
% 0.27/0.46 thf(fact_86_local_Ob__a,axiom,
% 0.27/0.46 ! [A2: a,B2: a,C: a] :
% 0.27/0.46 ( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 0.27/0.46 = ( modula105138637_a_aux @ a @ inf @ sup @ B2 @ C @ A2 ) ) ).
% 0.27/0.46
% 0.27/0.46 % local.b_a
% 0.27/0.46 thf(fact_87_local_Osup_Osemigroup__axioms,axiom,
% 0.27/0.46 semigroup @ a @ sup ).
% 0.27/0.46
% 0.27/0.46 % local.sup.semigroup_axioms
% 0.27/0.46 thf(fact_88_local_Oinf_Osemigroup__axioms,axiom,
% 0.27/0.46 semigroup @ a @ inf ).
% 0.27/0.46
% 0.27/0.46 % local.inf.semigroup_axioms
% 0.27/0.46 thf(fact_89_local_Osup_Osemilattice__axioms,axiom,
% 0.27/0.46 semilattice @ a @ sup ).
% 0.27/0.46
% 0.27/0.46 % local.sup.semilattice_axioms
% 0.27/0.46 thf(fact_90_local_Oinf_Osemilattice__axioms,axiom,
% 0.27/0.46 semilattice @ a @ inf ).
% 0.27/0.46
% 0.27/0.46 % local.inf.semilattice_axioms
% 0.27/0.46 thf(fact_91_lattice_Oa__aux_Ocong,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( modula105138637_a_aux @ A )
% 0.27/0.46 = ( modula105138637_a_aux @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lattice.a_aux.cong
% 0.27/0.46 thf(fact_92_lattice_Ob__aux_Ocong,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( modula477145868_b_aux @ A )
% 0.27/0.46 = ( modula477145868_b_aux @ A ) ) ).
% 0.27/0.46
% 0.27/0.46 % lattice.b_aux.cong
% 0.27/0.46 thf(fact_93_lattice_Oc__aux_Ocong,axiom,
% 0.27/0.46 ! [A: $tType] :
% 0.27/0.46 ( ( modula849153099_c_aux @ A )
% 0.27/0.47 = ( modula849153099_c_aux @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % lattice.c_aux.cong
% 0.27/0.47 thf(fact_94_local_Osup_Oabel__semigroup__axioms,axiom,
% 0.27/0.47 abel_semigroup @ a @ sup ).
% 0.27/0.47
% 0.27/0.47 % local.sup.abel_semigroup_axioms
% 0.27/0.47 thf(fact_95_local_Oinf_Oabel__semigroup__axioms,axiom,
% 0.27/0.47 abel_semigroup @ a @ inf ).
% 0.27/0.47
% 0.27/0.47 % local.inf.abel_semigroup_axioms
% 0.27/0.47 thf(fact_96_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 0.27/0.47 lattic35693393ce_set @ a @ sup ).
% 0.27/0.47
% 0.27/0.47 % local.Sup_fin.semilattice_set_axioms
% 0.27/0.47 thf(fact_97_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 0.27/0.47 lattic35693393ce_set @ a @ inf ).
% 0.27/0.47
% 0.27/0.47 % local.Inf_fin.semilattice_set_axioms
% 0.27/0.47 thf(fact_98_local_Olexordp__eq__refl,axiom,
% 0.27/0.47 ! [Xs: list @ a] : ( lexordp_eq @ a @ less @ Xs @ Xs ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq_refl
% 0.27/0.47 thf(fact_99_local_Olexordp__eq_ONil,axiom,
% 0.27/0.47 ! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq.Nil
% 0.27/0.47 thf(fact_100_abel__semigroup_Oaxioms_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ( abel_semigroup @ A @ F )
% 0.27/0.47 => ( semigroup @ A @ F ) ) ).
% 0.27/0.47
% 0.27/0.47 % abel_semigroup.axioms(1)
% 0.27/0.47 thf(fact_101_semilattice__set_Oaxioms,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ( lattic35693393ce_set @ A @ F )
% 0.27/0.47 => ( semilattice @ A @ F ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice_set.axioms
% 0.27/0.47 thf(fact_102_semilattice__set_Ointro,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ( semilattice @ A @ F )
% 0.27/0.47 => ( lattic35693393ce_set @ A @ F ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice_set.intro
% 0.27/0.47 thf(fact_103_semilattice__set__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lattic35693393ce_set @ A )
% 0.27/0.47 = ( semilattice @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice_set_def
% 0.27/0.47 thf(fact_104_local_Olexordp__eq__simps_I1_J,axiom,
% 0.27/0.47 ! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq_simps(1)
% 0.27/0.47 thf(fact_105_local_Olexordp__eq__simps_I2_J,axiom,
% 0.27/0.47 ! [Xs: list @ a] :
% 0.27/0.47 ( ( lexordp_eq @ a @ less @ Xs @ ( nil @ a ) )
% 0.27/0.47 = ( Xs
% 0.27/0.47 = ( nil @ a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq_simps(2)
% 0.27/0.47 thf(fact_106_abel__semigroup_Oleft__commute,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A,B2: A,A2: A,C: A] :
% 0.27/0.47 ( ( abel_semigroup @ A @ F )
% 0.27/0.47 => ( ( F @ B2 @ ( F @ A2 @ C ) )
% 0.27/0.47 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % abel_semigroup.left_commute
% 0.27/0.47 thf(fact_107_abel__semigroup_Ocommute,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.27/0.47 ( ( abel_semigroup @ A @ F )
% 0.27/0.47 => ( ( F @ A2 @ B2 )
% 0.27/0.47 = ( F @ B2 @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % abel_semigroup.commute
% 0.27/0.47 thf(fact_108_semigroup_Ointro,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ! [A3: A,B3: A,C2: A] :
% 0.27/0.47 ( ( F @ ( F @ A3 @ B3 ) @ C2 )
% 0.27/0.47 = ( F @ A3 @ ( F @ B3 @ C2 ) ) )
% 0.27/0.47 => ( semigroup @ A @ F ) ) ).
% 0.27/0.47
% 0.27/0.47 % semigroup.intro
% 0.27/0.47 thf(fact_109_semigroup_Oassoc,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
% 0.27/0.47 ( ( semigroup @ A @ F )
% 0.27/0.47 => ( ( F @ ( F @ A2 @ B2 ) @ C )
% 0.27/0.47 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % semigroup.assoc
% 0.27/0.47 thf(fact_110_semigroup__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semigroup @ A )
% 0.27/0.47 = ( ^ [F2: A > A > A] :
% 0.27/0.47 ! [A4: A,B4: A,C3: A] :
% 0.27/0.47 ( ( F2 @ ( F2 @ A4 @ B4 ) @ C3 )
% 0.27/0.47 = ( F2 @ A4 @ ( F2 @ B4 @ C3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % semigroup_def
% 0.27/0.47 thf(fact_111_ord_Ois__arg__min_Ocong,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( lattices_is_arg_min @ A @ B )
% 0.27/0.47 = ( lattices_is_arg_min @ A @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.is_arg_min.cong
% 0.27/0.47 thf(fact_112_ord_Ois__arg__max_Ocong,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( lattices_is_arg_max @ A @ B )
% 0.27/0.47 = ( lattices_is_arg_max @ A @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.is_arg_max.cong
% 0.27/0.47 thf(fact_113_ord_Ois__arg__min__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( lattices_is_arg_min @ A @ B )
% 0.27/0.47 = ( ^ [Less: A > A > $o,F2: B > A,P2: B > $o,X3: B] :
% 0.27/0.47 ( ( P2 @ X3 )
% 0.27/0.47 & ~ ? [Y3: B] :
% 0.27/0.47 ( ( P2 @ Y3 )
% 0.27/0.47 & ( Less @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.is_arg_min_def
% 0.27/0.47 thf(fact_114_ord_Ois__arg__max__def,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( lattices_is_arg_max @ A @ B )
% 0.27/0.47 = ( ^ [Less: A > A > $o,F2: B > A,P2: B > $o,X3: B] :
% 0.27/0.47 ( ( P2 @ X3 )
% 0.27/0.47 & ~ ? [Y3: B] :
% 0.27/0.47 ( ( P2 @ Y3 )
% 0.27/0.47 & ( Less @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.is_arg_max_def
% 0.27/0.47 thf(fact_115_ord_Olexordp__eq__simps_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
% 0.27/0.47 ( ( lexordp_eq @ A @ Less2 @ Xs @ ( nil @ A ) )
% 0.27/0.47 = ( Xs
% 0.27/0.47 = ( nil @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq_simps(2)
% 0.27/0.47 thf(fact_116_ord_Olexordp__eq__simps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less2 @ ( nil @ A ) @ Ys ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq_simps(1)
% 0.27/0.47 thf(fact_117_local_Olexordp__eq_Osimps,axiom,
% 0.27/0.47 ! [A1: list @ a,A22: list @ a] :
% 0.27/0.47 ( ( lexordp_eq @ a @ less @ A1 @ A22 )
% 0.27/0.47 = ( ? [Ys2: list @ a] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( nil @ a ) )
% 0.27/0.47 & ( A22 = Ys2 ) )
% 0.27/0.47 | ? [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( cons @ a @ X3 @ Xs2 ) )
% 0.27/0.47 & ( A22
% 0.27/0.47 = ( cons @ a @ Y3 @ Ys2 ) )
% 0.27/0.47 & ( less @ X3 @ Y3 ) )
% 0.27/0.47 | ? [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( cons @ a @ X3 @ Xs2 ) )
% 0.27/0.47 & ( A22
% 0.27/0.47 = ( cons @ a @ Y3 @ Ys2 ) )
% 0.27/0.47 & ~ ( less @ X3 @ Y3 )
% 0.27/0.47 & ~ ( less @ Y3 @ X3 )
% 0.27/0.47 & ( lexordp_eq @ a @ less @ Xs2 @ Ys2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq.simps
% 0.27/0.47 thf(fact_118_local_Olexordp__eq_Oinducts,axiom,
% 0.27/0.47 ! [X1: list @ a,X22: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
% 0.27/0.47 ( ( lexordp_eq @ a @ less @ X1 @ X22 )
% 0.27/0.47 => ( ! [X_1: list @ a] : ( P @ ( nil @ a ) @ X_1 )
% 0.27/0.47 => ( ! [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
% 0.27/0.47 ( ( less @ X2 @ Y2 )
% 0.27/0.47 => ( P @ ( cons @ a @ X2 @ Xs3 ) @ ( cons @ a @ Y2 @ Ys3 ) ) )
% 0.27/0.47 => ( ! [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
% 0.27/0.47 ( ~ ( less @ X2 @ Y2 )
% 0.27/0.47 => ( ~ ( less @ Y2 @ X2 )
% 0.27/0.47 => ( ( lexordp_eq @ a @ less @ Xs3 @ Ys3 )
% 0.27/0.47 => ( ( P @ Xs3 @ Ys3 )
% 0.27/0.47 => ( P @ ( cons @ a @ X2 @ Xs3 ) @ ( cons @ a @ Y2 @ Ys3 ) ) ) ) ) )
% 0.27/0.47 => ( P @ X1 @ X22 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq.inducts
% 0.27/0.47 thf(fact_119_local_Olexordp__eq_Ocases,axiom,
% 0.27/0.47 ! [A1: list @ a,A22: list @ a] :
% 0.27/0.47 ( ( lexordp_eq @ a @ less @ A1 @ A22 )
% 0.27/0.47 => ( ( A1
% 0.27/0.47 != ( nil @ a ) )
% 0.27/0.47 => ( ! [X2: a] :
% 0.27/0.47 ( ? [Xs3: list @ a] :
% 0.27/0.47 ( A1
% 0.27/0.47 = ( cons @ a @ X2 @ Xs3 ) )
% 0.27/0.47 => ! [Y2: a] :
% 0.27/0.47 ( ? [Ys3: list @ a] :
% 0.27/0.47 ( A22
% 0.27/0.47 = ( cons @ a @ Y2 @ Ys3 ) )
% 0.27/0.47 => ~ ( less @ X2 @ Y2 ) ) )
% 0.27/0.47 => ~ ! [X2: a,Y2: a,Xs3: list @ a] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( cons @ a @ X2 @ Xs3 ) )
% 0.27/0.47 => ! [Ys3: list @ a] :
% 0.27/0.47 ( ( A22
% 0.27/0.47 = ( cons @ a @ Y2 @ Ys3 ) )
% 0.27/0.47 => ( ~ ( less @ X2 @ Y2 )
% 0.27/0.47 => ( ~ ( less @ Y2 @ X2 )
% 0.27/0.47 => ~ ( lexordp_eq @ a @ less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq.cases
% 0.27/0.47 thf(fact_120_local_Olexordp__eq_OCons__eq,axiom,
% 0.27/0.47 ! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
% 0.27/0.47 ( ~ ( less @ X @ Y )
% 0.27/0.47 => ( ~ ( less @ Y @ X )
% 0.27/0.47 => ( ( lexordp_eq @ a @ less @ Xs @ Ys )
% 0.27/0.47 => ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq.Cons_eq
% 0.27/0.47 thf(fact_121_local_Olexordp__eq_OCons,axiom,
% 0.27/0.47 ! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
% 0.27/0.47 ( ( less @ X @ Y )
% 0.27/0.47 => ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq.Cons
% 0.27/0.47 thf(fact_122_list_Oinject,axiom,
% 0.27/0.47 ! [A: $tType,X21: A,X222: list @ A,Y21: A,Y22: list @ A] :
% 0.27/0.47 ( ( ( cons @ A @ X21 @ X222 )
% 0.27/0.47 = ( cons @ A @ Y21 @ Y22 ) )
% 0.27/0.47 = ( ( X21 = Y21 )
% 0.27/0.47 & ( X222 = Y22 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % list.inject
% 0.27/0.47 thf(fact_123_ord_Olexordp__eq__simps_I4_J,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
% 0.27/0.47 ( ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
% 0.27/0.47 = ( ( Less2 @ X @ Y )
% 0.27/0.47 | ( ~ ( Less2 @ Y @ X )
% 0.27/0.47 & ( lexordp_eq @ A @ Less2 @ Xs @ Ys ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq_simps(4)
% 0.27/0.47 thf(fact_124_ord_Olexordp__eq__simps_I3_J,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X: A,Xs: list @ A] :
% 0.27/0.47 ~ ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq_simps(3)
% 0.27/0.47 thf(fact_125_local_Olexordp__eq__simps_I4_J,axiom,
% 0.27/0.47 ! [X: a,Xs: list @ a,Y: a,Ys: list @ a] :
% 0.27/0.47 ( ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) )
% 0.27/0.47 = ( ( less @ X @ Y )
% 0.27/0.47 | ( ~ ( less @ Y @ X )
% 0.27/0.47 & ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq_simps(4)
% 0.27/0.47 thf(fact_126_local_Olexordp__eq__simps_I3_J,axiom,
% 0.27/0.47 ! [X: a,Xs: list @ a] :
% 0.27/0.47 ~ ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( nil @ a ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq_simps(3)
% 0.27/0.47 thf(fact_127_strict__sorted_Oinduct,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 0.27/0.47 ( ( P @ ( nil @ A ) )
% 0.27/0.47 => ( ! [X2: A,Ys3: list @ A] :
% 0.27/0.47 ( ( P @ Ys3 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Ys3 ) ) )
% 0.27/0.47 => ( P @ A0 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % strict_sorted.induct
% 0.27/0.47 thf(fact_128_strict__sorted_Ocases,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( linorder @ A )
% 0.27/0.47 => ! [X: list @ A] :
% 0.27/0.47 ( ( X
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ~ ! [X2: A,Ys3: list @ A] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( cons @ A @ X2 @ Ys3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % strict_sorted.cases
% 0.27/0.47 thf(fact_129_map__tailrec__rev_Oinduct,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
% 0.27/0.47 ( ! [F3: A > B,X_1: list @ B] : ( P @ F3 @ ( nil @ A ) @ X_1 )
% 0.27/0.47 => ( ! [F3: A > B,A3: A,As: list @ A,Bs: list @ B] :
% 0.27/0.47 ( ( P @ F3 @ As @ ( cons @ B @ ( F3 @ A3 ) @ Bs ) )
% 0.27/0.47 => ( P @ F3 @ ( cons @ A @ A3 @ As ) @ Bs ) )
% 0.27/0.47 => ( P @ A0 @ A1 @ A22 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % map_tailrec_rev.induct
% 0.27/0.47 thf(fact_130_list__nonempty__induct,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.27/0.47 ( ( Xs
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.27/0.47 => ( ! [X2: A,Xs3: list @ A] :
% 0.27/0.47 ( ( Xs3
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ( P @ Xs3 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) )
% 0.27/0.47 => ( P @ Xs ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % list_nonempty_induct
% 0.27/0.47 thf(fact_131_successively_Oinduct,axiom,
% 0.27/0.47 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.27/0.47 ( ! [P3: A > A > $o] : ( P @ P3 @ ( nil @ A ) )
% 0.27/0.47 => ( ! [P3: A > A > $o,X2: A] : ( P @ P3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.27/0.47 => ( ! [P3: A > A > $o,X2: A,Y2: A,Xs3: list @ A] :
% 0.27/0.47 ( ( P @ P3 @ ( cons @ A @ Y2 @ Xs3 ) )
% 0.27/0.47 => ( P @ P3 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) )
% 0.27/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % successively.induct
% 0.27/0.47 thf(fact_132_arg__min__list_Oinduct,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType] :
% 0.27/0.47 ( ( linorder @ B )
% 0.27/0.47 => ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
% 0.27/0.47 ( ! [F3: A > B,X2: A] : ( P @ F3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.27/0.47 => ( ! [F3: A > B,X2: A,Y2: A,Zs: list @ A] :
% 0.27/0.47 ( ( P @ F3 @ ( cons @ A @ Y2 @ Zs ) )
% 0.27/0.47 => ( P @ F3 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs ) ) ) )
% 0.27/0.47 => ( ! [A3: A > B] : ( P @ A3 @ ( nil @ A ) )
% 0.27/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % arg_min_list.induct
% 0.27/0.47 thf(fact_133_remdups__adj_Oinduct,axiom,
% 0.27/0.47 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 0.27/0.47 ( ( P @ ( nil @ A ) )
% 0.27/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.27/0.47 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.27/0.47 ( ( ( X2 = Y2 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 0.27/0.47 => ( ( ( X2 != Y2 )
% 0.27/0.47 => ( P @ ( cons @ A @ Y2 @ Xs3 ) ) )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
% 0.27/0.47 => ( P @ A0 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % remdups_adj.induct
% 0.27/0.47 thf(fact_134_sorted__wrt_Oinduct,axiom,
% 0.27/0.47 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 0.27/0.47 ( ! [P3: A > A > $o] : ( P @ P3 @ ( nil @ A ) )
% 0.27/0.47 => ( ! [P3: A > A > $o,X2: A,Ys3: list @ A] :
% 0.27/0.47 ( ( P @ P3 @ Ys3 )
% 0.27/0.47 => ( P @ P3 @ ( cons @ A @ X2 @ Ys3 ) ) )
% 0.27/0.47 => ( P @ A0 @ A1 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % sorted_wrt.induct
% 0.27/0.47 thf(fact_135_remdups__adj_Ocases,axiom,
% 0.27/0.47 ! [A: $tType,X: list @ A] :
% 0.27/0.47 ( ( X
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ! [X2: A] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.27/0.47 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % remdups_adj.cases
% 0.27/0.47 thf(fact_136_transpose_Ocases,axiom,
% 0.27/0.47 ! [A: $tType,X: list @ ( list @ A )] :
% 0.27/0.47 ( ( X
% 0.27/0.47 != ( nil @ ( list @ A ) ) )
% 0.27/0.47 => ( ! [Xss: list @ ( list @ A )] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 0.27/0.47 => ~ ! [X2: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % transpose.cases
% 0.27/0.47 thf(fact_137_shuffles_Oinduct,axiom,
% 0.27/0.47 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.27/0.47 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.27/0.47 => ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
% 0.27/0.47 => ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
% 0.27/0.47 ( ( P @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) )
% 0.27/0.47 => ( ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) )
% 0.27/0.47 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % shuffles.induct
% 0.27/0.47 thf(fact_138_min__list_Oinduct,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord @ A )
% 0.27/0.47 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 0.27/0.47 ( ! [X2: A,Xs3: list @ A] :
% 0.27/0.47 ( ! [X212: A,X223: list @ A] :
% 0.27/0.47 ( ( Xs3
% 0.27/0.47 = ( cons @ A @ X212 @ X223 ) )
% 0.27/0.47 => ( P @ Xs3 ) )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 0.27/0.47 => ( ( P @ ( nil @ A ) )
% 0.27/0.47 => ( P @ A0 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % min_list.induct
% 0.27/0.47 thf(fact_139_min__list_Ocases,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( ord @ A )
% 0.27/0.47 => ! [X: list @ A] :
% 0.27/0.47 ( ! [X2: A,Xs3: list @ A] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( cons @ A @ X2 @ Xs3 ) )
% 0.27/0.47 => ( X
% 0.27/0.47 = ( nil @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % min_list.cases
% 0.27/0.47 thf(fact_140_induct__list012,axiom,
% 0.27/0.47 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.27/0.47 ( ( P @ ( nil @ A ) )
% 0.27/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.27/0.47 => ( ! [X2: A,Y2: A,Zs: list @ A] :
% 0.27/0.47 ( ( P @ Zs )
% 0.27/0.47 => ( ( P @ ( cons @ A @ Y2 @ Zs ) )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs ) ) ) ) )
% 0.27/0.47 => ( P @ Xs ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % induct_list012
% 0.27/0.47 thf(fact_141_splice_Oinduct,axiom,
% 0.27/0.47 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 0.27/0.47 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.27/0.47 => ( ! [X2: A,Xs3: list @ A,Ys3: list @ A] :
% 0.27/0.47 ( ( P @ Ys3 @ Xs3 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) )
% 0.27/0.47 => ( P @ A0 @ A1 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % splice.induct
% 0.27/0.47 thf(fact_142_list__induct2_H,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
% 0.27/0.47 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.27/0.47 => ( ! [X2: A,Xs3: list @ A] : ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( nil @ B ) )
% 0.27/0.47 => ( ! [Y2: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys3 ) )
% 0.27/0.47 => ( ! [X2: A,Xs3: list @ A,Y2: B,Ys3: list @ B] :
% 0.27/0.47 ( ( P @ Xs3 @ Ys3 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y2 @ Ys3 ) ) )
% 0.27/0.47 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % list_induct2'
% 0.27/0.47 thf(fact_143_neq__Nil__conv,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A] :
% 0.27/0.47 ( ( Xs
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 = ( ? [Y3: A,Ys2: list @ A] :
% 0.27/0.47 ( Xs
% 0.27/0.47 = ( cons @ A @ Y3 @ Ys2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % neq_Nil_conv
% 0.27/0.47 thf(fact_144_list_Oinducts,axiom,
% 0.27/0.47 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 0.27/0.47 ( ( P @ ( nil @ A ) )
% 0.27/0.47 => ( ! [X12: A,X23: list @ A] :
% 0.27/0.47 ( ( P @ X23 )
% 0.27/0.47 => ( P @ ( cons @ A @ X12 @ X23 ) ) )
% 0.27/0.47 => ( P @ List ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % list.inducts
% 0.27/0.47 thf(fact_145_list_Oexhaust,axiom,
% 0.27/0.47 ! [A: $tType,Y: list @ A] :
% 0.27/0.47 ( ( Y
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ~ ! [X213: A,X224: list @ A] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( cons @ A @ X213 @ X224 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % list.exhaust
% 0.27/0.47 thf(fact_146_list_OdiscI,axiom,
% 0.27/0.47 ! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
% 0.27/0.47 ( ( List
% 0.27/0.47 = ( cons @ A @ X21 @ X222 ) )
% 0.27/0.47 => ( List
% 0.27/0.47 != ( nil @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % list.discI
% 0.27/0.47 thf(fact_147_list_Odistinct_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,X21: A,X222: list @ A] :
% 0.27/0.47 ( ( nil @ A )
% 0.27/0.47 != ( cons @ A @ X21 @ X222 ) ) ).
% 0.27/0.47
% 0.27/0.47 % list.distinct(1)
% 0.27/0.47 thf(fact_148_not__Cons__self2,axiom,
% 0.27/0.47 ! [A: $tType,X: A,Xs: list @ A] :
% 0.27/0.47 ( ( cons @ A @ X @ Xs )
% 0.27/0.47 != Xs ) ).
% 0.27/0.47
% 0.27/0.47 % not_Cons_self2
% 0.27/0.47 thf(fact_149_ord_Olexordp__eq_OCons__eq,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ~ ( Less2 @ X @ Y )
% 0.27/0.47 => ( ~ ( Less2 @ Y @ X )
% 0.27/0.47 => ( ( lexordp_eq @ A @ Less2 @ Xs @ Ys )
% 0.27/0.47 => ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq.Cons_eq
% 0.27/0.47 thf(fact_150_ord_Olexordp__eq_OCons,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( Less2 @ X @ Y )
% 0.27/0.47 => ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq.Cons
% 0.27/0.47 thf(fact_151_ord_Olexordp__eq_Oinducts,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X1: list @ A,X22: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.27/0.47 ( ( lexordp_eq @ A @ Less2 @ X1 @ X22 )
% 0.27/0.47 => ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 0.27/0.47 => ( ! [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
% 0.27/0.47 ( ( Less2 @ X2 @ Y2 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 0.27/0.47 => ( ! [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
% 0.27/0.47 ( ~ ( Less2 @ X2 @ Y2 )
% 0.27/0.47 => ( ~ ( Less2 @ Y2 @ X2 )
% 0.27/0.47 => ( ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 )
% 0.27/0.47 => ( ( P @ Xs3 @ Ys3 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) )
% 0.27/0.47 => ( P @ X1 @ X22 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq.inducts
% 0.27/0.47 thf(fact_152_ord_Olexordp__eq_Osimps,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lexordp_eq @ A )
% 0.27/0.47 = ( ^ [Less: A > A > $o,A12: list @ A,A23: list @ A] :
% 0.27/0.47 ( ? [Ys2: list @ A] :
% 0.27/0.47 ( ( A12
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 & ( A23 = Ys2 ) )
% 0.27/0.47 | ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
% 0.27/0.47 ( ( A12
% 0.27/0.47 = ( cons @ A @ X3 @ Xs2 ) )
% 0.27/0.47 & ( A23
% 0.27/0.47 = ( cons @ A @ Y3 @ Ys2 ) )
% 0.27/0.47 & ( Less @ X3 @ Y3 ) )
% 0.27/0.47 | ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
% 0.27/0.47 ( ( A12
% 0.27/0.47 = ( cons @ A @ X3 @ Xs2 ) )
% 0.27/0.47 & ( A23
% 0.27/0.47 = ( cons @ A @ Y3 @ Ys2 ) )
% 0.27/0.47 & ~ ( Less @ X3 @ Y3 )
% 0.27/0.47 & ~ ( Less @ Y3 @ X3 )
% 0.27/0.47 & ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq.simps
% 0.27/0.47 thf(fact_153_ord_Olexordp__eq_Ocases,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,A1: list @ A,A22: list @ A] :
% 0.27/0.47 ( ( lexordp_eq @ A @ Less2 @ A1 @ A22 )
% 0.27/0.47 => ( ( A1
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ! [X2: A] :
% 0.27/0.47 ( ? [Xs3: list @ A] :
% 0.27/0.47 ( A1
% 0.27/0.47 = ( cons @ A @ X2 @ Xs3 ) )
% 0.27/0.47 => ! [Y2: A] :
% 0.27/0.47 ( ? [Ys3: list @ A] :
% 0.27/0.47 ( A22
% 0.27/0.47 = ( cons @ A @ Y2 @ Ys3 ) )
% 0.27/0.47 => ~ ( Less2 @ X2 @ Y2 ) ) )
% 0.27/0.47 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( cons @ A @ X2 @ Xs3 ) )
% 0.27/0.47 => ! [Ys3: list @ A] :
% 0.27/0.47 ( ( A22
% 0.27/0.47 = ( cons @ A @ Y2 @ Ys3 ) )
% 0.27/0.47 => ( ~ ( Less2 @ X2 @ Y2 )
% 0.27/0.47 => ( ~ ( Less2 @ Y2 @ X2 )
% 0.27/0.47 => ~ ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq.cases
% 0.27/0.47 thf(fact_154_ord_Olexordp__eq_Ocong,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lexordp_eq @ A )
% 0.27/0.47 = ( lexordp_eq @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq.cong
% 0.27/0.47 thf(fact_155_ord_Olexordp__eq__refl,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less2 @ Xs @ Xs ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq_refl
% 0.27/0.47 thf(fact_156_ord_Olexordp__eq_ONil,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less2 @ ( nil @ A ) @ Ys ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq.Nil
% 0.27/0.47 thf(fact_157_local_Olexordp_ONil,axiom,
% 0.27/0.47 ! [Y: a,Ys: list @ a] : ( lexordp @ a @ less @ ( nil @ a ) @ ( cons @ a @ Y @ Ys ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp.Nil
% 0.27/0.47 thf(fact_158_local_Olexordp_Ocases,axiom,
% 0.27/0.47 ! [A1: list @ a,A22: list @ a] :
% 0.27/0.47 ( ( lexordp @ a @ less @ A1 @ A22 )
% 0.27/0.47 => ( ( ( A1
% 0.27/0.47 = ( nil @ a ) )
% 0.27/0.47 => ! [Y2: a,Ys3: list @ a] :
% 0.27/0.47 ( A22
% 0.27/0.47 != ( cons @ a @ Y2 @ Ys3 ) ) )
% 0.27/0.47 => ( ! [X2: a] :
% 0.27/0.47 ( ? [Xs3: list @ a] :
% 0.27/0.47 ( A1
% 0.27/0.47 = ( cons @ a @ X2 @ Xs3 ) )
% 0.27/0.47 => ! [Y2: a] :
% 0.27/0.47 ( ? [Ys3: list @ a] :
% 0.27/0.47 ( A22
% 0.27/0.47 = ( cons @ a @ Y2 @ Ys3 ) )
% 0.27/0.47 => ~ ( less @ X2 @ Y2 ) ) )
% 0.27/0.47 => ~ ! [X2: a,Y2: a,Xs3: list @ a] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( cons @ a @ X2 @ Xs3 ) )
% 0.27/0.47 => ! [Ys3: list @ a] :
% 0.27/0.47 ( ( A22
% 0.27/0.47 = ( cons @ a @ Y2 @ Ys3 ) )
% 0.27/0.47 => ( ~ ( less @ X2 @ Y2 )
% 0.27/0.47 => ( ~ ( less @ Y2 @ X2 )
% 0.27/0.47 => ~ ( lexordp @ a @ less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp.cases
% 0.27/0.47 thf(fact_159_local_Olexordp_Oinducts,axiom,
% 0.27/0.47 ! [X1: list @ a,X22: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
% 0.27/0.47 ( ( lexordp @ a @ less @ X1 @ X22 )
% 0.27/0.47 => ( ! [Y2: a,Ys3: list @ a] : ( P @ ( nil @ a ) @ ( cons @ a @ Y2 @ Ys3 ) )
% 0.27/0.47 => ( ! [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
% 0.27/0.47 ( ( less @ X2 @ Y2 )
% 0.27/0.47 => ( P @ ( cons @ a @ X2 @ Xs3 ) @ ( cons @ a @ Y2 @ Ys3 ) ) )
% 0.27/0.47 => ( ! [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
% 0.27/0.47 ( ~ ( less @ X2 @ Y2 )
% 0.27/0.47 => ( ~ ( less @ Y2 @ X2 )
% 0.27/0.47 => ( ( lexordp @ a @ less @ Xs3 @ Ys3 )
% 0.27/0.47 => ( ( P @ Xs3 @ Ys3 )
% 0.27/0.47 => ( P @ ( cons @ a @ X2 @ Xs3 ) @ ( cons @ a @ Y2 @ Ys3 ) ) ) ) ) )
% 0.27/0.47 => ( P @ X1 @ X22 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp.inducts
% 0.27/0.47 thf(fact_160_local_Olexordp_Osimps,axiom,
% 0.27/0.47 ! [A1: list @ a,A22: list @ a] :
% 0.27/0.47 ( ( lexordp @ a @ less @ A1 @ A22 )
% 0.27/0.47 = ( ? [Y3: a,Ys2: list @ a] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( nil @ a ) )
% 0.27/0.47 & ( A22
% 0.27/0.47 = ( cons @ a @ Y3 @ Ys2 ) ) )
% 0.27/0.47 | ? [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( cons @ a @ X3 @ Xs2 ) )
% 0.27/0.47 & ( A22
% 0.27/0.47 = ( cons @ a @ Y3 @ Ys2 ) )
% 0.27/0.47 & ( less @ X3 @ Y3 ) )
% 0.27/0.47 | ? [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( cons @ a @ X3 @ Xs2 ) )
% 0.27/0.47 & ( A22
% 0.27/0.47 = ( cons @ a @ Y3 @ Ys2 ) )
% 0.27/0.47 & ~ ( less @ X3 @ Y3 )
% 0.27/0.47 & ~ ( less @ Y3 @ X3 )
% 0.27/0.47 & ( lexordp @ a @ less @ Xs2 @ Ys2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp.simps
% 0.27/0.47 thf(fact_161_local_Olexordp__antisym,axiom,
% 0.27/0.47 ! [Xs: list @ a,Ys: list @ a] :
% 0.27/0.47 ( ( lexordp @ a @ less @ Xs @ Ys )
% 0.27/0.47 => ~ ( lexordp @ a @ less @ Ys @ Xs ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_antisym
% 0.27/0.47 thf(fact_162_local_Olexordp__irreflexive,axiom,
% 0.27/0.47 ! [Xs: list @ a] :
% 0.27/0.47 ( ! [X2: a] :
% 0.27/0.47 ~ ( less @ X2 @ X2 )
% 0.27/0.47 => ~ ( lexordp @ a @ less @ Xs @ Xs ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_irreflexive
% 0.27/0.47 thf(fact_163_local_Olexordp__irreflexive_H,axiom,
% 0.27/0.47 ! [Xs: list @ a] :
% 0.27/0.47 ~ ( lexordp @ a @ less @ Xs @ Xs ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_irreflexive'
% 0.27/0.47 thf(fact_164_local_Olexordp_OCons__eq,axiom,
% 0.27/0.47 ! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
% 0.27/0.47 ( ~ ( less @ X @ Y )
% 0.27/0.47 => ( ~ ( less @ Y @ X )
% 0.27/0.47 => ( ( lexordp @ a @ less @ Xs @ Ys )
% 0.27/0.47 => ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp.Cons_eq
% 0.27/0.47 thf(fact_165_local_Olexordp_OCons,axiom,
% 0.27/0.47 ! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
% 0.27/0.47 ( ( less @ X @ Y )
% 0.27/0.47 => ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp.Cons
% 0.27/0.47 thf(fact_166_local_Olexordp__into__lexordp__eq,axiom,
% 0.27/0.47 ! [Xs: list @ a,Ys: list @ a] :
% 0.27/0.47 ( ( lexordp @ a @ less @ Xs @ Ys )
% 0.27/0.47 => ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_into_lexordp_eq
% 0.27/0.47 thf(fact_167_ord_Olexordp__simps_I3_J,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
% 0.27/0.47 ( ( lexordp @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
% 0.27/0.47 = ( ( Less2 @ X @ Y )
% 0.27/0.47 | ( ~ ( Less2 @ Y @ X )
% 0.27/0.47 & ( lexordp @ A @ Less2 @ Xs @ Ys ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_simps(3)
% 0.27/0.47 thf(fact_168_ord_Olexordp__simps_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
% 0.27/0.47 ~ ( lexordp @ A @ Less2 @ Xs @ ( nil @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_simps(2)
% 0.27/0.47 thf(fact_169_ord_Olexordp__simps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Ys: list @ A] :
% 0.27/0.47 ( ( lexordp @ A @ Less2 @ ( nil @ A ) @ Ys )
% 0.27/0.47 = ( Ys
% 0.27/0.47 != ( nil @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_simps(1)
% 0.27/0.47 thf(fact_170_local_Olexordp__simps_I3_J,axiom,
% 0.27/0.47 ! [X: a,Xs: list @ a,Y: a,Ys: list @ a] :
% 0.27/0.47 ( ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) )
% 0.27/0.47 = ( ( less @ X @ Y )
% 0.27/0.47 | ( ~ ( less @ Y @ X )
% 0.27/0.47 & ( lexordp @ a @ less @ Xs @ Ys ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_simps(3)
% 0.27/0.47 thf(fact_171_local_Olexordp__simps_I1_J,axiom,
% 0.27/0.47 ! [Ys: list @ a] :
% 0.27/0.47 ( ( lexordp @ a @ less @ ( nil @ a ) @ Ys )
% 0.27/0.47 = ( Ys
% 0.27/0.47 != ( nil @ a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_simps(1)
% 0.27/0.47 thf(fact_172_local_Olexordp__simps_I2_J,axiom,
% 0.27/0.47 ! [Xs: list @ a] :
% 0.27/0.47 ~ ( lexordp @ a @ less @ Xs @ ( nil @ a ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_simps(2)
% 0.27/0.47 thf(fact_173_ord_Olexordp_Ocong,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lexordp @ A )
% 0.27/0.47 = ( lexordp @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp.cong
% 0.27/0.47 thf(fact_174_ord_Olexordp__irreflexive,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
% 0.27/0.47 ( ! [X2: A] :
% 0.27/0.47 ~ ( Less2 @ X2 @ X2 )
% 0.27/0.47 => ~ ( lexordp @ A @ Less2 @ Xs @ Xs ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_irreflexive
% 0.27/0.47 thf(fact_175_ord_Olexordp_OCons__eq,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ~ ( Less2 @ X @ Y )
% 0.27/0.47 => ( ~ ( Less2 @ Y @ X )
% 0.27/0.47 => ( ( lexordp @ A @ Less2 @ Xs @ Ys )
% 0.27/0.47 => ( lexordp @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp.Cons_eq
% 0.27/0.47 thf(fact_176_ord_Olexordp_OCons,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( Less2 @ X @ Y )
% 0.27/0.47 => ( lexordp @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp.Cons
% 0.27/0.47 thf(fact_177_ord_Olexordp__into__lexordp__eq,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( lexordp @ A @ Less2 @ Xs @ Ys )
% 0.27/0.47 => ( lexordp_eq @ A @ Less2 @ Xs @ Ys ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_into_lexordp_eq
% 0.27/0.47 thf(fact_178_ord_Olexordp_ONil,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Y: A,Ys: list @ A] : ( lexordp @ A @ Less2 @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp.Nil
% 0.27/0.47 thf(fact_179_ord_Olexordp_Ocases,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,A1: list @ A,A22: list @ A] :
% 0.27/0.47 ( ( lexordp @ A @ Less2 @ A1 @ A22 )
% 0.27/0.47 => ( ( ( A1
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 => ! [Y2: A,Ys3: list @ A] :
% 0.27/0.47 ( A22
% 0.27/0.47 != ( cons @ A @ Y2 @ Ys3 ) ) )
% 0.27/0.47 => ( ! [X2: A] :
% 0.27/0.47 ( ? [Xs3: list @ A] :
% 0.27/0.47 ( A1
% 0.27/0.47 = ( cons @ A @ X2 @ Xs3 ) )
% 0.27/0.47 => ! [Y2: A] :
% 0.27/0.47 ( ? [Ys3: list @ A] :
% 0.27/0.47 ( A22
% 0.27/0.47 = ( cons @ A @ Y2 @ Ys3 ) )
% 0.27/0.47 => ~ ( Less2 @ X2 @ Y2 ) ) )
% 0.27/0.47 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 0.27/0.47 ( ( A1
% 0.27/0.47 = ( cons @ A @ X2 @ Xs3 ) )
% 0.27/0.47 => ! [Ys3: list @ A] :
% 0.27/0.47 ( ( A22
% 0.27/0.47 = ( cons @ A @ Y2 @ Ys3 ) )
% 0.27/0.47 => ( ~ ( Less2 @ X2 @ Y2 )
% 0.27/0.47 => ( ~ ( Less2 @ Y2 @ X2 )
% 0.27/0.47 => ~ ( lexordp @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp.cases
% 0.27/0.47 thf(fact_180_ord_Olexordp_Osimps,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( lexordp @ A )
% 0.27/0.47 = ( ^ [Less: A > A > $o,A12: list @ A,A23: list @ A] :
% 0.27/0.47 ( ? [Y3: A,Ys2: list @ A] :
% 0.27/0.47 ( ( A12
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 & ( A23
% 0.27/0.47 = ( cons @ A @ Y3 @ Ys2 ) ) )
% 0.27/0.47 | ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
% 0.27/0.47 ( ( A12
% 0.27/0.47 = ( cons @ A @ X3 @ Xs2 ) )
% 0.27/0.47 & ( A23
% 0.27/0.47 = ( cons @ A @ Y3 @ Ys2 ) )
% 0.27/0.47 & ( Less @ X3 @ Y3 ) )
% 0.27/0.47 | ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
% 0.27/0.47 ( ( A12
% 0.27/0.47 = ( cons @ A @ X3 @ Xs2 ) )
% 0.27/0.47 & ( A23
% 0.27/0.47 = ( cons @ A @ Y3 @ Ys2 ) )
% 0.27/0.47 & ~ ( Less @ X3 @ Y3 )
% 0.27/0.47 & ~ ( Less @ Y3 @ X3 )
% 0.27/0.47 & ( lexordp @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp.simps
% 0.27/0.47 thf(fact_181_ord_Olexordp_Oinducts,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X1: list @ A,X22: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.27/0.47 ( ( lexordp @ A @ Less2 @ X1 @ X22 )
% 0.27/0.47 => ( ! [Y2: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys3 ) )
% 0.27/0.47 => ( ! [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
% 0.27/0.47 ( ( Less2 @ X2 @ Y2 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 0.27/0.47 => ( ! [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
% 0.27/0.47 ( ~ ( Less2 @ X2 @ Y2 )
% 0.27/0.47 => ( ~ ( Less2 @ Y2 @ X2 )
% 0.27/0.47 => ( ( lexordp @ A @ Less2 @ Xs3 @ Ys3 )
% 0.27/0.47 => ( ( P @ Xs3 @ Ys3 )
% 0.27/0.47 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) )
% 0.27/0.47 => ( P @ X1 @ X22 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp.inducts
% 0.27/0.47 thf(fact_182_local_Olexordp__append__rightI,axiom,
% 0.27/0.47 ! [Ys: list @ a,Xs: list @ a] :
% 0.27/0.47 ( ( Ys
% 0.27/0.47 != ( nil @ a ) )
% 0.27/0.47 => ( lexordp @ a @ less @ Xs @ ( append @ a @ Xs @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_append_rightI
% 0.27/0.47 thf(fact_183_local_Olexordp__append__left__rightI,axiom,
% 0.27/0.47 ! [X: a,Y: a,Us: list @ a,Xs: list @ a,Ys: list @ a] :
% 0.27/0.47 ( ( less @ X @ Y )
% 0.27/0.47 => ( lexordp @ a @ less @ ( append @ a @ Us @ ( cons @ a @ X @ Xs ) ) @ ( append @ a @ Us @ ( cons @ a @ Y @ Ys ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_append_left_rightI
% 0.27/0.47 thf(fact_184_semilattice_Oaxioms_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ( semilattice @ A @ F )
% 0.27/0.47 => ( abel_semigroup @ A @ F ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice.axioms(1)
% 0.27/0.47 thf(fact_185_local_Olexordp__append__leftD,axiom,
% 0.27/0.47 ! [Xs: list @ a,Us: list @ a,Vs: list @ a] :
% 0.27/0.47 ( ( lexordp @ a @ less @ ( append @ a @ Xs @ Us ) @ ( append @ a @ Xs @ Vs ) )
% 0.27/0.47 => ( ! [A3: a] :
% 0.27/0.47 ~ ( less @ A3 @ A3 )
% 0.27/0.47 => ( lexordp @ a @ less @ Us @ Vs ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_append_leftD
% 0.27/0.47 thf(fact_186_local_Olexordp__append__leftI,axiom,
% 0.27/0.47 ! [Us: list @ a,Vs: list @ a,Xs: list @ a] :
% 0.27/0.47 ( ( lexordp @ a @ less @ Us @ Vs )
% 0.27/0.47 => ( lexordp @ a @ less @ ( append @ a @ Xs @ Us ) @ ( append @ a @ Xs @ Vs ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_append_leftI
% 0.27/0.47 thf(fact_187_local_Olexordp__eq__pref,axiom,
% 0.27/0.47 ! [U: list @ a,V: list @ a] : ( lexordp_eq @ a @ less @ U @ ( append @ a @ U @ V ) ) ).
% 0.27/0.47
% 0.27/0.47 % local.lexordp_eq_pref
% 0.27/0.47 thf(fact_188_same__append__eq,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
% 0.27/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.27/0.47 = ( append @ A @ Xs @ Zs2 ) )
% 0.27/0.47 = ( Ys = Zs2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % same_append_eq
% 0.27/0.47 thf(fact_189_append__same__eq,axiom,
% 0.27/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
% 0.27/0.47 ( ( ( append @ A @ Ys @ Xs )
% 0.27/0.47 = ( append @ A @ Zs2 @ Xs ) )
% 0.27/0.47 = ( Ys = Zs2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_same_eq
% 0.27/0.47 thf(fact_190_append__assoc,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
% 0.27/0.47 ( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs2 )
% 0.27/0.47 = ( append @ A @ Xs @ ( append @ A @ Ys @ Zs2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_assoc
% 0.27/0.47 thf(fact_191_append_Oassoc,axiom,
% 0.27/0.47 ! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
% 0.27/0.47 ( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C )
% 0.27/0.47 = ( append @ A @ A2 @ ( append @ A @ B2 @ C ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append.assoc
% 0.27/0.47 thf(fact_192_append_Oright__neutral,axiom,
% 0.27/0.47 ! [A: $tType,A2: list @ A] :
% 0.27/0.47 ( ( append @ A @ A2 @ ( nil @ A ) )
% 0.27/0.47 = A2 ) ).
% 0.27/0.47
% 0.27/0.47 % append.right_neutral
% 0.27/0.47 thf(fact_193_append__is__Nil__conv,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 = ( ( Xs
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 & ( Ys
% 0.27/0.47 = ( nil @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_is_Nil_conv
% 0.27/0.47 thf(fact_194_Nil__is__append__conv,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( ( nil @ A )
% 0.27/0.47 = ( append @ A @ Xs @ Ys ) )
% 0.27/0.47 = ( ( Xs
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 & ( Ys
% 0.27/0.47 = ( nil @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Nil_is_append_conv
% 0.27/0.47 thf(fact_195_self__append__conv2,axiom,
% 0.27/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 0.27/0.47 ( ( Ys
% 0.27/0.47 = ( append @ A @ Xs @ Ys ) )
% 0.27/0.47 = ( Xs
% 0.27/0.47 = ( nil @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % self_append_conv2
% 0.27/0.47 thf(fact_196_append__self__conv2,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.27/0.47 = Ys )
% 0.27/0.47 = ( Xs
% 0.27/0.47 = ( nil @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_self_conv2
% 0.27/0.47 thf(fact_197_self__append__conv,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( Xs
% 0.27/0.47 = ( append @ A @ Xs @ Ys ) )
% 0.27/0.47 = ( Ys
% 0.27/0.47 = ( nil @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % self_append_conv
% 0.27/0.47 thf(fact_198_append__self__conv,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.27/0.47 = Xs )
% 0.27/0.47 = ( Ys
% 0.27/0.47 = ( nil @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_self_conv
% 0.27/0.47 thf(fact_199_append__Nil2,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A] :
% 0.27/0.47 ( ( append @ A @ Xs @ ( nil @ A ) )
% 0.27/0.47 = Xs ) ).
% 0.27/0.47
% 0.27/0.47 % append_Nil2
% 0.27/0.47 thf(fact_200_append1__eq__conv,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
% 0.27/0.47 ( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.27/0.47 = ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
% 0.27/0.47 = ( ( Xs = Ys )
% 0.27/0.47 & ( X = Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append1_eq_conv
% 0.27/0.47 thf(fact_201_append_Oleft__neutral,axiom,
% 0.27/0.47 ! [A: $tType,A2: list @ A] :
% 0.27/0.47 ( ( append @ A @ ( nil @ A ) @ A2 )
% 0.27/0.47 = A2 ) ).
% 0.27/0.47
% 0.27/0.47 % append.left_neutral
% 0.27/0.47 thf(fact_202_append__Nil,axiom,
% 0.27/0.47 ! [A: $tType,Ys: list @ A] :
% 0.27/0.47 ( ( append @ A @ ( nil @ A ) @ Ys )
% 0.27/0.47 = Ys ) ).
% 0.27/0.47
% 0.27/0.47 % append_Nil
% 0.27/0.47 thf(fact_203_eq__Nil__appendI,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( Xs = Ys )
% 0.27/0.47 => ( Xs
% 0.27/0.47 = ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_Nil_appendI
% 0.27/0.47 thf(fact_204_Cons__eq__appendI,axiom,
% 0.27/0.47 ! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
% 0.27/0.47 ( ( ( cons @ A @ X @ Xs1 )
% 0.27/0.47 = Ys )
% 0.27/0.47 => ( ( Xs
% 0.27/0.47 = ( append @ A @ Xs1 @ Zs2 ) )
% 0.27/0.47 => ( ( cons @ A @ X @ Xs )
% 0.27/0.47 = ( append @ A @ Ys @ Zs2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Cons_eq_appendI
% 0.27/0.47 thf(fact_205_append__Cons,axiom,
% 0.27/0.47 ! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
% 0.27/0.47 = ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_Cons
% 0.27/0.47 thf(fact_206_ord_Olexordp__append__leftD,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Xs: list @ A,Us: list @ A,Vs: list @ A] :
% 0.27/0.47 ( ( lexordp @ A @ Less2 @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
% 0.27/0.47 => ( ! [A3: A] :
% 0.27/0.47 ~ ( Less2 @ A3 @ A3 )
% 0.27/0.47 => ( lexordp @ A @ Less2 @ Us @ Vs ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_append_leftD
% 0.27/0.47 thf(fact_207_ord_Olexordp__append__leftI,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,Us: list @ A,Vs: list @ A,Xs: list @ A] :
% 0.27/0.47 ( ( lexordp @ A @ Less2 @ Us @ Vs )
% 0.27/0.47 => ( lexordp @ A @ Less2 @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_append_leftI
% 0.27/0.47 thf(fact_208_ord_Olexordp__eq__pref,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,U: list @ A,V: list @ A] : ( lexordp_eq @ A @ Less2 @ U @ ( append @ A @ U @ V ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_eq_pref
% 0.27/0.47 thf(fact_209_append__eq__append__conv2,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,Ts: list @ A] :
% 0.27/0.47 ( ( ( append @ A @ Xs @ Ys )
% 0.27/0.47 = ( append @ A @ Zs2 @ Ts ) )
% 0.27/0.47 = ( ? [Us2: list @ A] :
% 0.27/0.47 ( ( ( Xs
% 0.27/0.47 = ( append @ A @ Zs2 @ Us2 ) )
% 0.27/0.47 & ( ( append @ A @ Us2 @ Ys )
% 0.27/0.47 = Ts ) )
% 0.27/0.47 | ( ( ( append @ A @ Xs @ Us2 )
% 0.27/0.47 = Zs2 )
% 0.27/0.47 & ( Ys
% 0.27/0.47 = ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_eq_append_conv2
% 0.27/0.47 thf(fact_210_append__eq__appendI,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys: list @ A,Us: list @ A] :
% 0.27/0.47 ( ( ( append @ A @ Xs @ Xs1 )
% 0.27/0.47 = Zs2 )
% 0.27/0.47 => ( ( Ys
% 0.27/0.47 = ( append @ A @ Xs1 @ Us ) )
% 0.27/0.47 => ( ( append @ A @ Xs @ Ys )
% 0.27/0.47 = ( append @ A @ Zs2 @ Us ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_eq_appendI
% 0.27/0.47 thf(fact_211_append_Osemigroup__axioms,axiom,
% 0.27/0.47 ! [A: $tType] : ( semigroup @ ( list @ A ) @ ( append @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % append.semigroup_axioms
% 0.27/0.47 thf(fact_212_rev__nonempty__induct,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 0.27/0.47 ( ( Xs
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.27/0.47 => ( ! [X2: A,Xs3: list @ A] :
% 0.27/0.47 ( ( Xs3
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ( P @ Xs3 )
% 0.27/0.47 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 0.27/0.47 => ( P @ Xs ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rev_nonempty_induct
% 0.27/0.47 thf(fact_213_append__eq__Cons__conv,axiom,
% 0.27/0.47 ! [A: $tType,Ys: list @ A,Zs2: list @ A,X: A,Xs: list @ A] :
% 0.27/0.47 ( ( ( append @ A @ Ys @ Zs2 )
% 0.27/0.47 = ( cons @ A @ X @ Xs ) )
% 0.27/0.47 = ( ( ( Ys
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 & ( Zs2
% 0.27/0.47 = ( cons @ A @ X @ Xs ) ) )
% 0.27/0.47 | ? [Ys4: list @ A] :
% 0.27/0.47 ( ( Ys
% 0.27/0.47 = ( cons @ A @ X @ Ys4 ) )
% 0.27/0.47 & ( ( append @ A @ Ys4 @ Zs2 )
% 0.27/0.47 = Xs ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_eq_Cons_conv
% 0.27/0.47 thf(fact_214_Cons__eq__append__conv,axiom,
% 0.27/0.47 ! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
% 0.27/0.47 ( ( ( cons @ A @ X @ Xs )
% 0.27/0.47 = ( append @ A @ Ys @ Zs2 ) )
% 0.27/0.47 = ( ( ( Ys
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 & ( ( cons @ A @ X @ Xs )
% 0.27/0.47 = Zs2 ) )
% 0.27/0.47 | ? [Ys4: list @ A] :
% 0.27/0.47 ( ( ( cons @ A @ X @ Ys4 )
% 0.27/0.47 = Ys )
% 0.27/0.47 & ( Xs
% 0.27/0.47 = ( append @ A @ Ys4 @ Zs2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Cons_eq_append_conv
% 0.27/0.47 thf(fact_215_rev__exhaust,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A] :
% 0.27/0.47 ( ( Xs
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ~ ! [Ys3: list @ A,Y2: A] :
% 0.27/0.47 ( Xs
% 0.27/0.47 != ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rev_exhaust
% 0.27/0.47 thf(fact_216_rev__induct,axiom,
% 0.27/0.47 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.27/0.47 ( ( P @ ( nil @ A ) )
% 0.27/0.47 => ( ! [X2: A,Xs3: list @ A] :
% 0.27/0.47 ( ( P @ Xs3 )
% 0.27/0.47 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 0.27/0.47 => ( P @ Xs ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rev_induct
% 0.27/0.47 thf(fact_217_ord_Olexordp__append__left__rightI,axiom,
% 0.27/0.47 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
% 0.27/0.47 ( ( Less2 @ X @ Y )
% 0.27/0.47 => ( lexordp @ A @ Less2 @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_append_left_rightI
% 0.27/0.47 thf(fact_218_ord_Olexordp__append__rightI,axiom,
% 0.27/0.47 ! [A: $tType,Ys: list @ A,Less2: A > A > $o,Xs: list @ A] :
% 0.27/0.47 ( ( Ys
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( lexordp @ A @ Less2 @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord.lexordp_append_rightI
% 0.27/0.47 thf(fact_219_semilattice_Oidem,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A,A2: A] :
% 0.27/0.47 ( ( semilattice @ A @ F )
% 0.27/0.47 => ( ( F @ A2 @ A2 )
% 0.27/0.47 = A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice.idem
% 0.27/0.47 thf(fact_220_semilattice_Oleft__idem,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.27/0.47 ( ( semilattice @ A @ F )
% 0.27/0.47 => ( ( F @ A2 @ ( F @ A2 @ B2 ) )
% 0.27/0.47 = ( F @ A2 @ B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice.left_idem
% 0.27/0.47 thf(fact_221_semilattice_Oright__idem,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 0.27/0.47 ( ( semilattice @ A @ F )
% 0.27/0.47 => ( ( F @ ( F @ A2 @ B2 ) @ B2 )
% 0.27/0.47 = ( F @ A2 @ B2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice.right_idem
% 0.27/0.47 thf(fact_222_product__lists_Osimps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
% 0.27/0.47 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % product_lists.simps(1)
% 0.27/0.47 thf(fact_223_bind__simps_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,X: B,Xs: list @ B,F: B > ( list @ A )] :
% 0.27/0.47 ( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F )
% 0.27/0.47 = ( append @ A @ ( F @ X ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % bind_simps(2)
% 0.27/0.47 thf(fact_224_bind__simps_I1_J,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 0.27/0.47 ( ( bind @ B @ A @ ( nil @ B ) @ F )
% 0.27/0.47 = ( nil @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % bind_simps(1)
% 0.27/0.47 thf(fact_225_abel__semigroup__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( abel_semigroup @ A )
% 0.27/0.47 = ( ^ [F2: A > A > A] :
% 0.27/0.47 ( ( semigroup @ A @ F2 )
% 0.27/0.47 & ( abel_s1917375468axioms @ A @ F2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % abel_semigroup_def
% 0.27/0.47 thf(fact_226_abel__semigroup_Ointro,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ( semigroup @ A @ F )
% 0.27/0.47 => ( ( abel_s1917375468axioms @ A @ F )
% 0.27/0.47 => ( abel_semigroup @ A @ F ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % abel_semigroup.intro
% 0.27/0.47 thf(fact_227_abel__semigroup__axioms_Ointro,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ! [A3: A,B3: A] :
% 0.27/0.47 ( ( F @ A3 @ B3 )
% 0.27/0.47 = ( F @ B3 @ A3 ) )
% 0.27/0.47 => ( abel_s1917375468axioms @ A @ F ) ) ).
% 0.27/0.47
% 0.27/0.47 % abel_semigroup_axioms.intro
% 0.27/0.47 thf(fact_228_abel__semigroup__axioms__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( abel_s1917375468axioms @ A )
% 0.27/0.47 = ( ^ [F2: A > A > A] :
% 0.27/0.47 ! [A4: A,B4: A] :
% 0.27/0.47 ( ( F2 @ A4 @ B4 )
% 0.27/0.47 = ( F2 @ B4 @ A4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % abel_semigroup_axioms_def
% 0.27/0.47 thf(fact_229_abel__semigroup_Oaxioms_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ( abel_semigroup @ A @ F )
% 0.27/0.47 => ( abel_s1917375468axioms @ A @ F ) ) ).
% 0.27/0.47
% 0.27/0.47 % abel_semigroup.axioms(2)
% 0.27/0.47 thf(fact_230_semilattice_Ointro,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ( abel_semigroup @ A @ F )
% 0.27/0.47 => ( ( semilattice_axioms @ A @ F )
% 0.27/0.47 => ( semilattice @ A @ F ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice.intro
% 0.27/0.47 thf(fact_231_semilattice__def,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( semilattice @ A )
% 0.27/0.47 = ( ^ [F2: A > A > A] :
% 0.27/0.47 ( ( abel_semigroup @ A @ F2 )
% 0.27/0.47 & ( semilattice_axioms @ A @ F2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice_def
% 0.27/0.47 thf(fact_232_semilattice_Oaxioms_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,F: A > A > A] :
% 0.27/0.47 ( ( semilattice @ A @ F )
% 0.27/0.47 => ( semilattice_axioms @ A @ F ) ) ).
% 0.27/0.47
% 0.27/0.47 % semilattice.axioms(2)
% 0.27/0.47 thf(fact_233_subseqs_Osimps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( subseqs @ A @ ( nil @ A ) )
% 0.27/0.47 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subseqs.simps(1)
% 0.27/0.47 thf(fact_234_maps__simps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,B: $tType,F: B > ( list @ A ),X: B,Xs: list @ B] :
% 0.27/0.47 ( ( maps @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
% 0.27/0.47 = ( append @ A @ ( F @ X ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % maps_simps(1)
% 0.27/0.47 thf(fact_235_maps__simps_I2_J,axiom,
% 0.27/0.47 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 0.27/0.47 ( ( maps @ B @ A @ F @ ( nil @ B ) )
% 0.27/0.47 = ( nil @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % maps_simps(2)
% 0.27/0.47 thf(fact_236_local_Otransp__less,axiom,
% 0.27/0.47 transp @ a @ less ).
% 0.27/0.47
% 0.27/0.47 % local.transp_less
% 0.27/0.47 thf(fact_237_insert__Nil,axiom,
% 0.27/0.47 ! [A: $tType,X: A] :
% 0.27/0.47 ( ( insert @ A @ X @ ( nil @ A ) )
% 0.27/0.47 = ( cons @ A @ X @ ( nil @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_Nil
% 0.27/0.47 thf(fact_238_concat__eq__append__conv,axiom,
% 0.27/0.47 ! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
% 0.27/0.47 ( ( ( concat @ A @ Xss2 )
% 0.27/0.47 = ( append @ A @ Ys @ Zs2 ) )
% 0.27/0.47 = ( ( ( Xss2
% 0.27/0.47 = ( nil @ ( list @ A ) ) )
% 0.27/0.47 => ( ( Ys
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 & ( Zs2
% 0.27/0.47 = ( nil @ A ) ) ) )
% 0.27/0.47 & ( ( Xss2
% 0.27/0.47 != ( nil @ ( list @ A ) ) )
% 0.27/0.47 => ? [Xss1: list @ ( list @ A ),Xs2: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] :
% 0.27/0.47 ( ( Xss2
% 0.27/0.47 = ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs4 ) @ Xss22 ) ) )
% 0.27/0.47 & ( Ys
% 0.27/0.47 = ( append @ A @ ( concat @ A @ Xss1 ) @ Xs2 ) )
% 0.27/0.47 & ( Zs2
% 0.27/0.47 = ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % concat_eq_append_conv
% 0.27/0.47 thf(fact_239_rotate1_Osimps_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,X: A,Xs: list @ A] :
% 0.27/0.47 ( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
% 0.27/0.47 = ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rotate1.simps(2)
% 0.27/0.47 thf(fact_240_rotate1__is__Nil__conv,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A] :
% 0.27/0.47 ( ( ( rotate1 @ A @ Xs )
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 = ( Xs
% 0.27/0.47 = ( nil @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % rotate1_is_Nil_conv
% 0.27/0.47 thf(fact_241_concat__append,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
% 0.27/0.47 ( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys ) )
% 0.27/0.47 = ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % concat_append
% 0.27/0.47 thf(fact_242_concat_Osimps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( concat @ A @ ( nil @ ( list @ A ) ) )
% 0.27/0.47 = ( nil @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % concat.simps(1)
% 0.27/0.47 thf(fact_243_concat_Osimps_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,X: list @ A,Xs: list @ ( list @ A )] :
% 0.27/0.47 ( ( concat @ A @ ( cons @ ( list @ A ) @ X @ Xs ) )
% 0.27/0.47 = ( append @ A @ X @ ( concat @ A @ Xs ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % concat.simps(2)
% 0.27/0.47 thf(fact_244_rotate1_Osimps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( rotate1 @ A @ ( nil @ A ) )
% 0.27/0.47 = ( nil @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % rotate1.simps(1)
% 0.27/0.47 thf(fact_245_concat__eq__appendD,axiom,
% 0.27/0.47 ! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
% 0.27/0.47 ( ( ( concat @ A @ Xss2 )
% 0.27/0.47 = ( append @ A @ Ys @ Zs2 ) )
% 0.27/0.47 => ( ( Xss2
% 0.27/0.47 != ( nil @ ( list @ A ) ) )
% 0.27/0.47 => ? [Xss12: list @ ( list @ A ),Xs3: list @ A,Xs5: list @ A,Xss23: list @ ( list @ A )] :
% 0.27/0.47 ( ( Xss2
% 0.27/0.47 = ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs5 ) @ Xss23 ) ) )
% 0.27/0.47 & ( Ys
% 0.27/0.47 = ( append @ A @ ( concat @ A @ Xss12 ) @ Xs3 ) )
% 0.27/0.47 & ( Zs2
% 0.27/0.47 = ( append @ A @ Xs5 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % concat_eq_appendD
% 0.27/0.47 thf(fact_246_butlast__snoc,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,X: A] :
% 0.27/0.47 ( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 0.27/0.47 = Xs ) ).
% 0.27/0.47
% 0.27/0.47 % butlast_snoc
% 0.27/0.47 thf(fact_247_list__ex1__simps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType,P: A > $o] :
% 0.27/0.47 ~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % list_ex1_simps(1)
% 0.27/0.47 thf(fact_248_butlast__append,axiom,
% 0.27/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 0.27/0.47 ( ( ( Ys
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 => ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
% 0.27/0.47 = ( butlast @ A @ Xs ) ) )
% 0.27/0.47 & ( ( Ys
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
% 0.27/0.47 = ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % butlast_append
% 0.27/0.47 thf(fact_249_butlast_Osimps_I1_J,axiom,
% 0.27/0.47 ! [A: $tType] :
% 0.27/0.47 ( ( butlast @ A @ ( nil @ A ) )
% 0.27/0.47 = ( nil @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % butlast.simps(1)
% 0.27/0.47 thf(fact_250_butlast_Osimps_I2_J,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,X: A] :
% 0.27/0.47 ( ( ( Xs
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 => ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
% 0.27/0.47 = ( nil @ A ) ) )
% 0.27/0.47 & ( ( Xs
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
% 0.27/0.47 = ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % butlast.simps(2)
% 0.27/0.47 thf(fact_251_append__butlast__last__id,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A] :
% 0.27/0.47 ( ( Xs
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
% 0.27/0.47 = Xs ) ) ).
% 0.27/0.47
% 0.27/0.47 % append_butlast_last_id
% 0.27/0.47 thf(fact_252_snoc__eq__iff__butlast,axiom,
% 0.27/0.47 ! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
% 0.27/0.47 ( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.27/0.47 = Ys )
% 0.27/0.47 = ( ( Ys
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 & ( ( butlast @ A @ Ys )
% 0.27/0.47 = Xs )
% 0.27/0.47 & ( ( last @ A @ Ys )
% 0.27/0.47 = X ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % snoc_eq_iff_butlast
% 0.27/0.47 thf(fact_253_last__appendR,axiom,
% 0.27/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 0.27/0.47 ( ( Ys
% 0.27/0.47 != ( nil @ A ) )
% 0.27/0.47 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 0.27/0.47 = ( last @ A @ Ys ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % last_appendR
% 0.27/0.47 thf(fact_254_last__appendL,axiom,
% 0.27/0.47 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 0.27/0.47 ( ( Ys
% 0.27/0.47 = ( nil @ A ) )
% 0.27/0.47 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 0.27/0.47 = ( last @ A @ Xs ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % last_appendL
% 0.27/0.47
% 0.27/0.47 % Type constructors (3)
% 0.27/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.27/0.47 ! [A5: $tType,A6: $tType] :
% 0.27/0.47 ( ( ord @ A6 )
% 0.27/0.47 => ( ord @ ( A5 > A6 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.27/0.47 linorder @ $o ).
% 0.27/0.47
% 0.27/0.47 thf(tcon_HOL_Obool___Orderings_Oord_1,axiom,
% 0.27/0.47 ord @ $o ).
% 0.27/0.47
% 0.27/0.47 % Conjectures (6)
% 0.27/0.47 thf(conj_0,hypothesis,
% 0.27/0.47 ( b
% 0.27/0.47 = ( inf @ b @ c ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(conj_1,hypothesis,
% 0.27/0.47 ( ( inf @ c @ ( sup @ b @ c ) )
% 0.27/0.47 = b ) ).
% 0.27/0.47
% 0.27/0.47 thf(conj_2,hypothesis,
% 0.27/0.47 ( a2
% 0.27/0.47 = ( sup @ b @ c ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(conj_3,hypothesis,
% 0.27/0.47 less @ b @ ( sup @ b @ c ) ).
% 0.27/0.47
% 0.27/0.47 thf(conj_4,hypothesis,
% 0.27/0.47 ( ( inf @ c @ ( sup @ b @ c ) )
% 0.27/0.47 = c ) ).
% 0.27/0.47
% 0.27/0.47 thf(conj_5,conjecture,
% 0.27/0.47 $false ).
% 0.27/0.47
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 ------------------------
% 2.40/1.47 ---- Embedded in THF ---
% 2.40/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.40/1.48 %%% Generated on Tue Feb 24 08:26:42 EST 2026
% 2.40/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.40/1.48 %%% Logic specification used:
% 2.40/1.48 %%% thf(spec, logic, $$dhol).
% 2.40/1.48
% 2.40/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.41GWJZtw0l/DTF2DTF4588.p
% 2.40/1.48 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.40/1.48 thf(ty_tf_a, type, a: $tType).
% 2.40/1.48 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.40/1.48 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.40/1.48 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.40/1.48 thf(sy_c_Finite__Set_Ocomp__fun__idem, type, finite_comp_fun_idem: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 2.40/1.48 thf(sy_c_Groups_Oabel__semigroup, type, abel_semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.40/1.48 thf(sy_c_Groups_Oabel__semigroup__axioms, type, abel_s1917375468axioms: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.40/1.48 thf(sy_c_Groups_Osemigroup, type, semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.40/1.48 thf(sy_c_Lattices_Osemilattice, type, semilattice: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.40/1.48 thf(sy_c_Lattices_Osemilattice__axioms, type, semilattice_axioms: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.40/1.48 thf(sy_c_Lattices__Big_Oord_Ois__arg__max, type, lattices_is_arg_max: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > A) > ((B > $o) > (B > $o))))))).
% 2.40/1.48 thf(sy_c_Lattices__Big_Oord_Ois__arg__min, type, lattices_is_arg_min: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > A) > ((B > $o) > (B > $o))))))).
% 2.40/1.48 thf(sy_c_Lattices__Big_Osemilattice__set, type, lattic35693393ce_set: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 2.40/1.48 thf(sy_c_List_Oappend, type, append: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 2.40/1.48 thf(sy_c_List_Obind, type, bind: (!> [A:$tType,B:$tType]: (((list @ A) > ((A > (list @ B)) > (list @ B)))))).
% 2.40/1.48 thf(sy_c_List_Obutlast, type, butlast: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.40/1.48 thf(sy_c_List_Oconcat, type, concat: (!> [A:$tType]: (((list @ (list @ A)) > (list @ A))))).
% 2.40/1.48 thf(sy_c_List_Oinsert, type, insert: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 2.40/1.48 thf(sy_c_List_Olast, type, last: (!> [A:$tType]: (((list @ A) > A)))).
% 2.40/1.48 thf(sy_c_List_Olist_OCons, type, cons: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 2.40/1.48 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 2.40/1.48 thf(sy_c_List_Olist__ex1, type, list_ex1: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 2.40/1.48 thf(sy_c_List_Omaps, type, maps: (!> [A:$tType,B:$tType]: (((A > (list @ B)) > ((list @ A) > (list @ B)))))).
% 2.40/1.48 thf(sy_c_List_Oord_Olexordp, type, lexordp: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > $o)))))).
% 2.40/1.48 thf(sy_c_List_Oord_Olexordp__eq, type, lexordp_eq: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > $o)))))).
% 2.40/1.48 thf(sy_c_List_Oproduct__lists, type, product_lists: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))).
% 2.40/1.48 thf(sy_c_List_Orotate1, type, rotate1: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 2.40/1.48 thf(sy_c_List_Osubseqs, type, subseqs: (!> [A:$tType]: (((list @ A) > (list @ (list @ A)))))).
% 2.40/1.48 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_OM5__lattice, type, modula1618706180attice: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > A)) > (A > (A > (A > $o))))))))).
% 2.40/1.48 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_ON5__lattice, type, modula1192512645attice: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > A)) > (A > (A > (A > $o))))))))).
% 2.40/1.48 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oa__aux, type, modula105138637_a_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.40/1.48 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ob__aux, type, modula477145868_b_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.40/1.48 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oc__aux, type, modula849153099_c_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.40/1.48 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux, type, modula1221160330_d_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.40/1.48 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux, type, modula1593167561_e_aux: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > (A > (A > (A > A)))))))).
% 2.40/1.48 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ono__distrib, type, modula141148890istrib: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > $o)) > ((A > (A > A)) > (A > (A > (A > $o))))))))).
% 2.40/1.48 thf(sy_c_Relation_Otransp, type, transp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 2.40/1.48 thf(sy_v_a, type, a2: a).
% 2.40/1.48 thf(sy_v_b, type, b: a).
% 2.40/1.48 thf(sy_v_c, type, c: a).
% 2.40/1.48 thf(sy_v_inf, type, inf: (a > (a > a))).
% 2.40/1.48 thf(sy_v_less, type, less: (a > (a > $o))).
% 2.40/1.48 thf(sy_v_sup, type, sup: (a > (a > a))).
% 2.40/1.48 thf(fact_0_local_Odual__order_Oasym, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) => (~ (((less @ A2) @ B2))))))).
% 2.40/1.48 thf(fact_1_local_Odual__order_Ostrict__implies__not__eq, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) => (A2 != B2))))).
% 2.40/1.48 thf(fact_2_local_Odual__order_Ostrict__trans, axiom, (! [B2:a,A2:a,C:a]: ((((less @ B2) @ A2) => (((less @ C) @ B2) => ((less @ C) @ A2)))))).
% 2.40/1.48 thf(fact_3_local_Oless__asym, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (~ (((less @ Y) @ X))))))).
% 2.40/1.48 thf(fact_4_local_Oless__asym_H, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => (~ (((less @ B2) @ A2))))))).
% 2.40/1.48 thf(fact_5_local_Oless__imp__neq, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (X != Y))))).
% 2.40/1.48 thf(fact_6_local_Oless__imp__not__eq, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (X != Y))))).
% 2.40/1.48 thf(fact_7_local_Oless__imp__not__eq2, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (Y != X))))).
% 2.40/1.48 thf(fact_8_local_Oless__imp__not__less, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (~ (((less @ Y) @ X))))))).
% 2.40/1.48 thf(fact_9_local_Oless__imp__triv, axiom, (! [X:a,Y:a,P:$o]: ((((less @ X) @ Y) => (((less @ Y) @ X) => P))))).
% 2.40/1.48 thf(fact_10_local_Oless__irrefl, axiom, (! [X:a]: ((~ (((less @ X) @ X)))))).
% 2.40/1.48 thf(fact_11_local_Oless__not__sym, axiom, (! [X:a,Y:a]: ((((less @ X) @ Y) => (~ (((less @ Y) @ X))))))).
% 2.40/1.48 thf(fact_12_local_Oless__trans, axiom, (! [X:a,Y:a,Z:a]: ((((less @ X) @ Y) => (((less @ Y) @ Z) => ((less @ X) @ Z)))))).
% 2.40/1.48 thf(fact_13_local_Oord__eq__less__trans, axiom, (! [A2:a,B2:a,C:a]: (((A2 = B2) => (((less @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.40/1.48 thf(fact_14_local_Oord__less__eq__trans, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ B2) => ((B2 = C) => ((less @ A2) @ C)))))).
% 2.40/1.48 thf(fact_15_local_Oorder_Oasym, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => (~ (((less @ B2) @ A2))))))).
% 2.40/1.48 thf(fact_16_local_Oorder_Oirrefl, axiom, (! [A2:a]: ((~ (((less @ A2) @ A2)))))).
% 2.40/1.48 thf(fact_17_local_Oorder_Ostrict__implies__not__eq, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) => (A2 != B2))))).
% 2.40/1.48 thf(fact_18_local_Oorder_Ostrict__trans, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ B2) => (((less @ B2) @ C) => ((less @ A2) @ C)))))).
% 2.40/1.48 thf(fact_19_local_Oinf_Oassoc, axiom, (! [A2:a,B2:a,C:a]: ((((inf @ ((inf @ A2) @ B2)) @ C) = ((inf @ A2) @ ((inf @ B2) @ C)))))).
% 2.40/1.48 thf(fact_20_local_Oinf_Ocommute, axiom, (! [A2:a,B2:a]: ((((inf @ A2) @ B2) = ((inf @ B2) @ A2))))).
% 2.40/1.48 thf(fact_21_local_Oinf_Oleft__commute, axiom, (! [B2:a,A2:a,C:a]: ((((inf @ B2) @ ((inf @ A2) @ C)) = ((inf @ A2) @ ((inf @ B2) @ C)))))).
% 2.40/1.48 thf(fact_22_local_Oinf__assoc, axiom, (! [X:a,Y:a,Z:a]: ((((inf @ ((inf @ X) @ Y)) @ Z) = ((inf @ X) @ ((inf @ Y) @ Z)))))).
% 2.40/1.48 thf(fact_23_local_Oinf__commute, axiom, (! [X:a,Y:a]: ((((inf @ X) @ Y) = ((inf @ Y) @ X))))).
% 2.40/1.48 thf(fact_24_local_Oinf__left__commute, axiom, (! [X:a,Y:a,Z:a]: ((((inf @ X) @ ((inf @ Y) @ Z)) = ((inf @ Y) @ ((inf @ X) @ Z)))))).
% 2.40/1.48 thf(fact_25_local_Osup_Oassoc, axiom, (! [A2:a,B2:a,C:a]: ((((sup @ ((sup @ A2) @ B2)) @ C) = ((sup @ A2) @ ((sup @ B2) @ C)))))).
% 2.40/1.48 thf(fact_26_local_Osup_Ocommute, axiom, (! [A2:a,B2:a]: ((((sup @ A2) @ B2) = ((sup @ B2) @ A2))))).
% 2.40/1.48 thf(fact_27_local_Osup_Oleft__commute, axiom, (! [B2:a,A2:a,C:a]: ((((sup @ B2) @ ((sup @ A2) @ C)) = ((sup @ A2) @ ((sup @ B2) @ C)))))).
% 2.40/1.48 thf(fact_28_local_Osup__assoc, axiom, (! [X:a,Y:a,Z:a]: ((((sup @ ((sup @ X) @ Y)) @ Z) = ((sup @ X) @ ((sup @ Y) @ Z)))))).
% 2.40/1.48 thf(fact_29_local_Osup__commute, axiom, (! [X:a,Y:a]: ((((sup @ X) @ Y) = ((sup @ Y) @ X))))).
% 2.40/1.48 thf(fact_30_local_Osup__left__commute, axiom, (! [X:a,Y:a,Z:a]: ((((sup @ X) @ ((sup @ Y) @ Z)) = ((sup @ Y) @ ((sup @ X) @ Z)))))).
% 2.40/1.48 thf(fact_31_local_Oinf_Ostrict__boundedE, axiom, (! [A2:a,B2:a,C:a]: ((((less @ A2) @ ((inf @ B2) @ C)) => (~ ((((less @ A2) @ B2) => (~ (((less @ A2) @ C)))))))))).
% 2.40/1.48 thf(fact_32_local_Oinf_Ostrict__coboundedI1, axiom, (! [A2:a,C:a,B2:a]: ((((less @ A2) @ C) => ((less @ ((inf @ A2) @ B2)) @ C))))).
% 2.40/1.48 thf(fact_33_local_Oinf_Ostrict__coboundedI2, axiom, (! [B2:a,C:a,A2:a]: ((((less @ B2) @ C) => ((less @ ((inf @ A2) @ B2)) @ C))))).
% 2.40/1.48 thf(fact_34_local_Oinf_Ostrict__order__iff, axiom, (! [A2:a,B2:a]: ((((less @ A2) @ B2) = ((A2 = ((inf @ A2) @ B2)) & (A2 != B2)))))).
% 2.40/1.48 thf(fact_35_local_Oless__infI1, axiom, (! [A2:a,X:a,B2:a]: ((((less @ A2) @ X) => ((less @ ((inf @ A2) @ B2)) @ X))))).
% 2.40/1.48 thf(fact_36_local_Oless__infI2, axiom, (! [B2:a,X:a,A2:a]: ((((less @ B2) @ X) => ((less @ ((inf @ A2) @ B2)) @ X))))).
% 2.40/1.48 thf(fact_37_local_Oless__supI1, axiom, (! [X:a,A2:a,B2:a]: ((((less @ X) @ A2) => ((less @ X) @ ((sup @ A2) @ B2)))))).
% 2.40/1.48 thf(fact_38_local_Oless__supI2, axiom, (! [X:a,B2:a,A2:a]: ((((less @ X) @ B2) => ((less @ X) @ ((sup @ A2) @ B2)))))).
% 2.40/1.48 thf(fact_39_local_Osup_Ostrict__boundedE, axiom, (! [B2:a,C:a,A2:a]: ((((less @ ((sup @ B2) @ C)) @ A2) => (~ ((((less @ B2) @ A2) => (~ (((less @ C) @ A2)))))))))).
% 2.40/1.48 thf(fact_40_local_Osup_Ostrict__coboundedI1, axiom, (! [C:a,A2:a,B2:a]: ((((less @ C) @ A2) => ((less @ C) @ ((sup @ A2) @ B2)))))).
% 2.40/1.48 thf(fact_41_local_Osup_Ostrict__coboundedI2, axiom, (! [C:a,B2:a,A2:a]: ((((less @ C) @ B2) => ((less @ C) @ ((sup @ A2) @ B2)))))).
% 2.40/1.48 thf(fact_42_local_Osup_Ostrict__order__iff, axiom, (! [B2:a,A2:a]: ((((less @ B2) @ A2) = ((A2 = ((sup @ A2) @ B2)) & (A2 != B2)))))).
% 2.40/1.48 thf(fact_43_local_Odistrib__imp1, axiom, (! [X:a,Y:a,Z:a]: (((! [X2:a,Y2:a,Z2:a]: ((((inf @ X2) @ ((sup @ Y2) @ Z2)) = ((sup @ ((inf @ X2) @ Y2)) @ ((inf @ X2) @ Z2))))) => (((sup @ X) @ ((inf @ Y) @ Z)) = ((inf @ ((sup @ X) @ Y)) @ ((sup @ X) @ Z))))))).
% 2.40/1.48 thf(fact_44_local_Odistrib__imp2, axiom, (! [X:a,Y:a,Z:a]: (((! [X2:a,Y2:a,Z2:a]: ((((sup @ X2) @ ((inf @ Y2) @ Z2)) = ((inf @ ((sup @ X2) @ Y2)) @ ((sup @ X2) @ Z2))))) => (((inf @ X) @ ((sup @ Y) @ Z)) = ((sup @ ((inf @ X) @ Y)) @ ((inf @ X) @ Z))))))).
% 2.40/1.48 thf(fact_45_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.40/1.48 thf(fact_46_local_Oinf_Oidem, axiom, (! [A2:a]: ((((inf @ A2) @ A2) = A2)))).
% 2.40/1.48 thf(fact_47_local_Oinf_Oleft__idem, axiom, (! [A2:a,B2:a]: ((((inf @ A2) @ ((inf @ A2) @ B2)) = ((inf @ A2) @ B2))))).
% 2.40/1.48 thf(fact_48_local_Oinf_Oright__idem, axiom, (! [A2:a,B2:a]: ((((inf @ ((inf @ A2) @ B2)) @ B2) = ((inf @ A2) @ B2))))).
% 2.40/1.48 thf(fact_49_local_Oinf__idem, axiom, (! [X:a]: ((((inf @ X) @ X) = X)))).
% 2.40/1.48 thf(fact_50_local_Oinf__left__idem, axiom, (! [X:a,Y:a]: ((((inf @ X) @ ((inf @ X) @ Y)) = ((inf @ X) @ Y))))).
% 2.40/1.48 thf(fact_51_local_Oinf__right__idem, axiom, (! [X:a,Y:a]: ((((inf @ ((inf @ X) @ Y)) @ Y) = ((inf @ X) @ Y))))).
% 2.40/1.48 thf(fact_52_local_Osup_Oidem, axiom, (! [A2:a]: ((((sup @ A2) @ A2) = A2)))).
% 2.40/1.48 thf(fact_53_local_Osup_Oleft__idem, axiom, (! [A2:a,B2:a]: ((((sup @ A2) @ ((sup @ A2) @ B2)) = ((sup @ A2) @ B2))))).
% 2.40/1.48 thf(fact_54_local_Osup_Oright__idem, axiom, (! [A2:a,B2:a]: ((((sup @ ((sup @ A2) @ B2)) @ B2) = ((sup @ A2) @ B2))))).
% 2.40/1.48 thf(fact_55_local_Osup__idem, axiom, (! [X:a]: ((((sup @ X) @ X) = X)))).
% 2.40/1.48 thf(fact_56_local_Osup__left__idem, axiom, (! [X:a,Y:a]: ((((sup @ X) @ ((sup @ X) @ Y)) = ((sup @ X) @ Y))))).
% 2.40/1.48 thf(fact_57_local_Ois__arg__max__def, axiom, (! [B:$tType,F:(B > a),P:(B > $o),X:B]: ((((((((lattices_is_arg_max @ a) @ B) @ less) @ F) @ P) @ X) = ((P @ X) & (~ ((? [Y3:B]: (((P @ Y3) & ((less @ (F @ X)) @ (F @ Y3)))))))))))).
% 2.40/1.48 thf(fact_58_local_Ois__arg__min__def, axiom, (! [B:$tType,F:(B > a),P:(B > $o),X:B]: ((((((((lattices_is_arg_min @ a) @ B) @ less) @ F) @ P) @ X) = ((P @ X) & (~ ((? [Y3:B]: (((P @ Y3) & ((less @ (F @ Y3)) @ (F @ X)))))))))))).
% 2.40/1.48 thf(fact_59_M5__lattice__def, axiom, (! [A2:a,B2:a,C:a]: (((((((((modula1618706180attice @ a) @ inf) @ less) @ sup) @ A2) @ B2) @ C) = ((((inf @ A2) @ B2) = ((inf @ B2) @ C)) & ((((inf @ C) @ A2) = ((inf @ B2) @ C)) & ((((sup @ A2) @ B2) = ((sup @ B2) @ C)) & ((((sup @ C) @ A2) = ((sup @ B2) @ C)) & ((less @ ((inf @ A2) @ B2)) @ ((sup @ A2) @ B2)))))))))).
% 2.40/1.48 thf(fact_60_N5__lattice__def, axiom, (! [A2:a,B2:a,C:a]: (((((((((modula1192512645attice @ a) @ inf) @ less) @ sup) @ A2) @ B2) @ C) = ((((inf @ A2) @ C) = ((inf @ B2) @ C)) & (((less @ A2) @ B2) & (((sup @ A2) @ C) = ((sup @ B2) @ C)))))))).
% 2.40/1.48 thf(fact_61_no__distrib__def, axiom, (! [A2:a,B2:a,C:a]: (((((((((modula141148890istrib @ a) @ inf) @ less) @ sup) @ A2) @ B2) @ C) = ((less @ ((sup @ ((inf @ A2) @ B2)) @ ((inf @ C) @ A2))) @ ((inf @ A2) @ ((sup @ B2) @ C))))))).
% 2.40/1.48 thf(fact_62_local_Oinf__sup__absorb, axiom, (! [X:a,Y:a]: ((((inf @ X) @ ((sup @ X) @ Y)) = X)))).
% 2.40/1.48 thf(fact_63_local_Osup__inf__absorb, axiom, (! [X:a,Y:a]: ((((sup @ X) @ ((inf @ X) @ Y)) = X)))).
% 2.40/1.48 thf(fact_64_local_Oa__join__d, axiom, (! [A2:a,B2:a,C:a]: ((((sup @ A2) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((sup @ A2) @ ((inf @ B2) @ C)))))).
% 2.40/1.48 thf(fact_65_local_Ob__join__d, axiom, (! [B2:a,A2:a,C:a]: ((((sup @ B2) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((sup @ B2) @ ((inf @ C) @ A2)))))).
% 2.40/1.48 thf(fact_66_local_Od__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((sup @ ((sup @ ((inf @ A2) @ B2)) @ ((inf @ B2) @ C))) @ ((inf @ C) @ A2)))))).
% 2.40/1.48 thf(fact_67_local_Od__b__c__a, axiom, (! [B2:a,C:a,A2:a]: ((((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ B2) @ C) @ A2) = ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 2.40/1.48 thf(fact_68_local_Od__c__a__b, axiom, (! [C:a,A2:a,B2:a]: ((((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ C) @ A2) @ B2) = ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 2.40/1.48 thf(fact_69_local_Oa__meet__e, axiom, (! [A2:a,B2:a,C:a]: ((((inf @ A2) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((inf @ A2) @ ((sup @ B2) @ C)))))).
% 2.40/1.48 thf(fact_70_local_Ob__meet__e, axiom, (! [B2:a,A2:a,C:a]: ((((inf @ B2) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((inf @ B2) @ ((sup @ C) @ A2)))))).
% 2.40/1.48 thf(fact_71_local_Oc__meet__e, axiom, (! [C:a,A2:a,B2:a]: ((((inf @ C) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)) = ((inf @ C) @ ((sup @ A2) @ B2)))))).
% 2.40/1.48 thf(fact_72_local_Oe__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((inf @ ((inf @ ((sup @ A2) @ B2)) @ ((sup @ B2) @ C))) @ ((sup @ C) @ A2)))))).
% 2.40/1.48 thf(fact_73_local_Oe__b__c__a, axiom, (! [B2:a,C:a,A2:a]: ((((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ B2) @ C) @ A2) = ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 2.40/1.48 thf(fact_74_local_Oe__c__a__b, axiom, (! [C:a,A2:a,B2:a]: ((((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ C) @ A2) @ B2) = ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))))).
% 2.40/1.48 thf(fact_75_local_Ocomp__fun__idem__sup, axiom, (((finite_comp_fun_idem @ a) @ a) @ sup)).
% 2.40/1.48 thf(fact_76_local_Ocomp__fun__idem__inf, axiom, (((finite_comp_fun_idem @ a) @ a) @ inf)).
% 2.40/1.48 thf(fact_77_local_Oc__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula849153099_c_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((sup @ ((inf @ C) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)))))).
% 2.40/1.48 thf(fact_78_local_Ob__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula477145868_b_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((sup @ ((inf @ B2) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)))))).
% 2.40/1.48 thf(fact_79_local_Oa__aux__def, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula105138637_a_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((sup @ ((inf @ A2) @ ((((((modula1593167561_e_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C))) @ ((((((modula1221160330_d_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C)))))).
% 2.40/1.48 thf(fact_80_lattice_Ono__distrib_Ocong, axiom, (! [A:$tType]: (((modula141148890istrib @ A) = (modula141148890istrib @ A))))).
% 2.40/1.48 thf(fact_81_lattice_ON5__lattice_Ocong, axiom, (! [A:$tType]: (((modula1192512645attice @ A) = (modula1192512645attice @ A))))).
% 2.40/1.48 thf(fact_82_lattice_OM5__lattice_Ocong, axiom, (! [A:$tType]: (((modula1618706180attice @ A) = (modula1618706180attice @ A))))).
% 2.40/1.48 thf(fact_83_lattice_Oe__aux_Ocong, axiom, (! [A:$tType]: (((modula1593167561_e_aux @ A) = (modula1593167561_e_aux @ A))))).
% 2.40/1.48 thf(fact_84_lattice_Od__aux_Ocong, axiom, (! [A:$tType]: (((modula1221160330_d_aux @ A) = (modula1221160330_d_aux @ A))))).
% 2.40/1.48 thf(fact_85_local_Oc__a, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula849153099_c_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((((((modula105138637_a_aux @ a) @ inf) @ sup) @ C) @ A2) @ B2))))).
% 2.40/1.48 thf(fact_86_local_Ob__a, axiom, (! [A2:a,B2:a,C:a]: ((((((((modula477145868_b_aux @ a) @ inf) @ sup) @ A2) @ B2) @ C) = ((((((modula105138637_a_aux @ a) @ inf) @ sup) @ B2) @ C) @ A2))))).
% 2.40/1.48 thf(fact_87_local_Osup_Osemigroup__axioms, axiom, ((semigroup @ a) @ sup)).
% 2.40/1.48 thf(fact_88_local_Oinf_Osemigroup__axioms, axiom, ((semigroup @ a) @ inf)).
% 2.40/1.48 thf(fact_89_local_Osup_Osemilattice__axioms, axiom, ((semilattice @ a) @ sup)).
% 2.40/1.48 thf(fact_90_local_Oinf_Osemilattice__axioms, axiom, ((semilattice @ a) @ inf)).
% 2.40/1.48 thf(fact_91_lattice_Oa__aux_Ocong, axiom, (! [A:$tType]: (((modula105138637_a_aux @ A) = (modula105138637_a_aux @ A))))).
% 2.40/1.48 thf(fact_92_lattice_Ob__aux_Ocong, axiom, (! [A:$tType]: (((modula477145868_b_aux @ A) = (modula477145868_b_aux @ A))))).
% 2.40/1.48 thf(fact_93_lattice_Oc__aux_Ocong, axiom, (! [A:$tType]: (((modula849153099_c_aux @ A) = (modula849153099_c_aux @ A))))).
% 2.40/1.48 thf(fact_94_local_Osup_Oabel__semigroup__axioms, axiom, ((abel_semigroup @ a) @ sup)).
% 2.40/1.48 thf(fact_95_local_Oinf_Oabel__semigroup__axioms, axiom, ((abel_semigroup @ a) @ inf)).
% 2.40/1.48 thf(fact_96_local_OSup__fin_Osemilattice__set__axioms, axiom, ((lattic35693393ce_set @ a) @ sup)).
% 2.40/1.48 thf(fact_97_local_OInf__fin_Osemilattice__set__axioms, axiom, ((lattic35693393ce_set @ a) @ inf)).
% 2.40/1.48 thf(fact_98_local_Olexordp__eq__refl, axiom, (! [Xs:(list @ a)]: (((((lexordp_eq @ a) @ less) @ Xs) @ Xs)))).
% 2.40/1.48 thf(fact_99_local_Olexordp__eq_ONil, axiom, (! [Ys:(list @ a)]: (((((lexordp_eq @ a) @ less) @ (nil @ a)) @ Ys)))).
% 2.40/1.48 thf(fact_100_abel__semigroup_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => ((semigroup @ A) @ F))))).
% 2.40/1.48 thf(fact_101_semilattice__set_Oaxioms, axiom, (! [A:$tType,F:(A > (A > A))]: ((((lattic35693393ce_set @ A) @ F) => ((semilattice @ A) @ F))))).
% 2.40/1.48 thf(fact_102_semilattice__set_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((lattic35693393ce_set @ A) @ F))))).
% 2.40/1.48 thf(fact_103_semilattice__set__def, axiom, (! [A:$tType]: (((lattic35693393ce_set @ A) = (semilattice @ A))))).
% 2.40/1.48 thf(fact_104_local_Olexordp__eq__simps_I1_J, axiom, (! [Ys:(list @ a)]: (((((lexordp_eq @ a) @ less) @ (nil @ a)) @ Ys)))).
% 2.40/1.48 thf(fact_105_local_Olexordp__eq__simps_I2_J, axiom, (! [Xs:(list @ a)]: ((((((lexordp_eq @ a) @ less) @ Xs) @ (nil @ a)) = (Xs = (nil @ a)))))).
% 2.40/1.48 thf(fact_106_abel__semigroup_Oleft__commute, axiom, (! [A:$tType,F:(A > (A > A)),B2:A,A2:A,C:A]: ((((abel_semigroup @ A) @ F) => (((F @ B2) @ ((F @ A2) @ C)) = ((F @ A2) @ ((F @ B2) @ C))))))).
% 2.40/1.48 thf(fact_107_abel__semigroup_Ocommute, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((abel_semigroup @ A) @ F) => (((F @ A2) @ B2) = ((F @ B2) @ A2)))))).
% 2.40/1.48 thf(fact_108_semigroup_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: (((! [A3:A,B3:A,C2:A]: ((((F @ ((F @ A3) @ B3)) @ C2) = ((F @ A3) @ ((F @ B3) @ C2))))) => ((semigroup @ A) @ F))))).
% 2.40/1.48 thf(fact_109_semigroup_Oassoc, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A,C:A]: ((((semigroup @ A) @ F) => (((F @ ((F @ A2) @ B2)) @ C) = ((F @ A2) @ ((F @ B2) @ C))))))).
% 2.40/1.48 thf(fact_110_semigroup__def, axiom, (! [A:$tType]: (((semigroup @ A) = (^ [F2:(A > (A > A))]: ((! [A4:A,B4:A,C3:A]: ((((F2 @ ((F2 @ A4) @ B4)) @ C3) = ((F2 @ A4) @ ((F2 @ B4) @ C3))))))))))).
% 2.40/1.48 thf(fact_111_ord_Ois__arg__min_Ocong, axiom, (! [B:$tType,A:$tType]: ((((lattices_is_arg_min @ A) @ B) = ((lattices_is_arg_min @ A) @ B))))).
% 2.40/1.48 thf(fact_112_ord_Ois__arg__max_Ocong, axiom, (! [B:$tType,A:$tType]: ((((lattices_is_arg_max @ A) @ B) = ((lattices_is_arg_max @ A) @ B))))).
% 2.40/1.48 thf(fact_113_ord_Ois__arg__min__def, axiom, (! [B:$tType,A:$tType]: ((((lattices_is_arg_min @ A) @ B) = (^ [Less:(A > (A > $o)),F2:(B > A),P2:(B > $o),X3:B]: (((P2 @ X3) & (~ ((? [Y3:B]: (((P2 @ Y3) & ((Less @ (F2 @ Y3)) @ (F2 @ X3)))))))))))))).
% 2.40/1.48 thf(fact_114_ord_Ois__arg__max__def, axiom, (! [B:$tType,A:$tType]: ((((lattices_is_arg_max @ A) @ B) = (^ [Less:(A > (A > $o)),F2:(B > A),P2:(B > $o),X3:B]: (((P2 @ X3) & (~ ((? [Y3:B]: (((P2 @ Y3) & ((Less @ (F2 @ X3)) @ (F2 @ Y3)))))))))))))).
% 2.40/1.48 thf(fact_115_ord_Olexordp__eq__simps_I2_J, axiom, (! [A:$tType,Less2:(A > (A > $o)),Xs:(list @ A)]: ((((((lexordp_eq @ A) @ Less2) @ Xs) @ (nil @ A)) = (Xs = (nil @ A)))))).
% 2.40/1.48 thf(fact_116_ord_Olexordp__eq__simps_I1_J, axiom, (! [A:$tType,Less2:(A > (A > $o)),Ys:(list @ A)]: (((((lexordp_eq @ A) @ Less2) @ (nil @ A)) @ Ys)))).
% 2.40/1.48 thf(fact_117_local_Olexordp__eq_Osimps, axiom, (! [A1:(list @ a),A22:(list @ a)]: ((((((lexordp_eq @ a) @ less) @ A1) @ A22) = ((? [Ys2:(list @ a)]: (((A1 = (nil @ a)) & (A22 = Ys2)))) | ((? [X3:a,Y3:a,Xs2:(list @ a),Ys2:(list @ a)]: (((A1 = (((cons @ a) @ X3) @ Xs2)) & ((A22 = (((cons @ a) @ Y3) @ Ys2)) & ((less @ X3) @ Y3))))) | (? [X3:a,Y3:a,Xs2:(list @ a),Ys2:(list @ a)]: (((A1 = (((cons @ a) @ X3) @ Xs2)) & ((A22 = (((cons @ a) @ Y3) @ Ys2)) & ((~ (((less @ X3) @ Y3))) & ((~ (((less @ Y3) @ X3))) & ((((lexordp_eq @ a) @ less) @ Xs2) @ Ys2))))))))))))).
% 2.40/1.48 thf(fact_118_local_Olexordp__eq_Oinducts, axiom, (! [X1:(list @ a),X22:(list @ a),P:((list @ a) > ((list @ a) > $o))]: ((((((lexordp_eq @ a) @ less) @ X1) @ X22) => ((! [X_1:(list @ a)]: (((P @ (nil @ a)) @ X_1))) => ((! [X2:a,Y2:a,Xs3:(list @ a),Ys3:(list @ a)]: ((((less @ X2) @ Y2) => ((P @ (((cons @ a) @ X2) @ Xs3)) @ (((cons @ a) @ Y2) @ Ys3))))) => ((! [X2:a,Y2:a,Xs3:(list @ a),Ys3:(list @ a)]: (((~ (((less @ X2) @ Y2))) => ((~ (((less @ Y2) @ X2))) => (((((lexordp_eq @ a) @ less) @ Xs3) @ Ys3) => (((P @ Xs3) @ Ys3) => ((P @ (((cons @ a) @ X2) @ Xs3)) @ (((cons @ a) @ Y2) @ Ys3)))))))) => ((P @ X1) @ X22)))))))).
% 2.40/1.48 thf(fact_119_local_Olexordp__eq_Ocases, axiom, (! [A1:(list @ a),A22:(list @ a)]: ((((((lexordp_eq @ a) @ less) @ A1) @ A22) => ((A1 != (nil @ a)) => ((! [X2:a]: (((? [Xs3:(list @ a)]: ((A1 = (((cons @ a) @ X2) @ Xs3)))) => (! [Y2:a]: (((? [Ys3:(list @ a)]: ((A22 = (((cons @ a) @ Y2) @ Ys3)))) => (~ (((less @ X2) @ Y2))))))))) => (~ ((! [X2:a,Y2:a,Xs3:(list @ a)]: (((A1 = (((cons @ a) @ X2) @ Xs3)) => (! [Ys3:(list @ a)]: (((A22 = (((cons @ a) @ Y2) @ Ys3)) => ((~ (((less @ X2) @ Y2))) => ((~ (((less @ Y2) @ X2))) => (~ (((((lexordp_eq @ a) @ less) @ Xs3) @ Ys3))))))))))))))))))).
% 2.40/1.48 thf(fact_120_local_Olexordp__eq_OCons__eq, axiom, (! [X:a,Y:a,Xs:(list @ a),Ys:(list @ a)]: (((~ (((less @ X) @ Y))) => ((~ (((less @ Y) @ X))) => (((((lexordp_eq @ a) @ less) @ Xs) @ Ys) => ((((lexordp_eq @ a) @ less) @ (((cons @ a) @ X) @ Xs)) @ (((cons @ a) @ Y) @ Ys)))))))).
% 2.40/1.48 thf(fact_121_local_Olexordp__eq_OCons, axiom, (! [X:a,Y:a,Xs:(list @ a),Ys:(list @ a)]: ((((less @ X) @ Y) => ((((lexordp_eq @ a) @ less) @ (((cons @ a) @ X) @ Xs)) @ (((cons @ a) @ Y) @ Ys)))))).
% 2.40/1.48 thf(fact_122_list_Oinject, axiom, (! [A:$tType,X21:A,X222:(list @ A),Y21:A,Y22:(list @ A)]: ((((((cons @ A) @ X21) @ X222) = (((cons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X222 = Y22)))))).
% 2.40/1.48 thf(fact_123_ord_Olexordp__eq__simps_I4_J, axiom, (! [A:$tType,Less2:(A > (A > $o)),X:A,Xs:(list @ A),Y:A,Ys:(list @ A)]: ((((((lexordp_eq @ A) @ Less2) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys)) = (((Less2 @ X) @ Y) | ((~ (((Less2 @ Y) @ X))) & ((((lexordp_eq @ A) @ Less2) @ Xs) @ Ys))))))).
% 2.40/1.48 thf(fact_124_ord_Olexordp__eq__simps_I3_J, axiom, (! [A:$tType,Less2:(A > (A > $o)),X:A,Xs:(list @ A)]: ((~ (((((lexordp_eq @ A) @ Less2) @ (((cons @ A) @ X) @ Xs)) @ (nil @ A))))))).
% 2.40/1.48 thf(fact_125_local_Olexordp__eq__simps_I4_J, axiom, (! [X:a,Xs:(list @ a),Y:a,Ys:(list @ a)]: ((((((lexordp_eq @ a) @ less) @ (((cons @ a) @ X) @ Xs)) @ (((cons @ a) @ Y) @ Ys)) = (((less @ X) @ Y) | ((~ (((less @ Y) @ X))) & ((((lexordp_eq @ a) @ less) @ Xs) @ Ys))))))).
% 2.40/1.48 thf(fact_126_local_Olexordp__eq__simps_I3_J, axiom, (! [X:a,Xs:(list @ a)]: ((~ (((((lexordp_eq @ a) @ less) @ (((cons @ a) @ X) @ Xs)) @ (nil @ a))))))).
% 2.40/1.48 thf(fact_127_strict__sorted_Oinduct, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:((list @ A) > $o),A0:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A,Ys3:(list @ A)]: (((P @ Ys3) => (P @ (((cons @ A) @ X2) @ Ys3))))) => (P @ A0))))))))).
% 2.40/1.48 thf(fact_128_strict__sorted_Ocases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(list @ A)]: (((X != (nil @ A)) => (~ ((! [X2:A,Ys3:(list @ A)]: ((X != (((cons @ A) @ X2) @ Ys3))))))))))))).
% 2.40/1.48 thf(fact_129_map__tailrec__rev_Oinduct, axiom, (! [A:$tType,B:$tType,P:((A > B) > ((list @ A) > ((list @ B) > $o))),A0:(A > B),A1:(list @ A),A22:(list @ B)]: (((! [F3:(A > B),X_1:(list @ B)]: ((((P @ F3) @ (nil @ A)) @ X_1))) => ((! [F3:(A > B),A3:A,As:(list @ A),Bs:(list @ B)]: (((((P @ F3) @ As) @ (((cons @ B) @ (F3 @ A3)) @ Bs)) => (((P @ F3) @ (((cons @ A) @ A3) @ As)) @ Bs)))) => (((P @ A0) @ A1) @ A22)))))).
% 2.40/1.48 thf(fact_130_list__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Xs3:(list @ A)]: (((Xs3 != (nil @ A)) => ((P @ Xs3) => (P @ (((cons @ A) @ X2) @ Xs3)))))) => (P @ Xs))))))).
% 2.40/1.48 thf(fact_131_successively_Oinduct, axiom, (! [A:$tType,P:((A > (A > $o)) > ((list @ A) > $o)),A0:(A > (A > $o)),A1:(list @ A)]: (((! [P3:(A > (A > $o))]: (((P @ P3) @ (nil @ A)))) => ((! [P3:(A > (A > $o)),X2:A]: (((P @ P3) @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [P3:(A > (A > $o)),X2:A,Y2:A,Xs3:(list @ A)]: ((((P @ P3) @ (((cons @ A) @ Y2) @ Xs3)) => ((P @ P3) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3)))))) => ((P @ A0) @ A1))))))).
% 2.40/1.48 thf(fact_132_arg__min__list_Oinduct, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [P:((A > B) > ((list @ A) > $o)),A0:(A > B),A1:(list @ A)]: (((! [F3:(A > B),X2:A]: (((P @ F3) @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [F3:(A > B),X2:A,Y2:A,Zs:(list @ A)]: ((((P @ F3) @ (((cons @ A) @ Y2) @ Zs)) => ((P @ F3) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Zs)))))) => ((! [A3:(A > B)]: (((P @ A3) @ (nil @ A)))) => ((P @ A0) @ A1)))))))))).
% 2.40/1.48 thf(fact_133_remdups__adj_Oinduct, axiom, (! [A:$tType,P:((list @ A) > $o),A0:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Y2:A,Xs3:(list @ A)]: ((((X2 = Y2) => (P @ (((cons @ A) @ X2) @ Xs3))) => (((X2 != Y2) => (P @ (((cons @ A) @ Y2) @ Xs3))) => (P @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3))))))) => (P @ A0))))))).
% 2.40/1.48 thf(fact_134_sorted__wrt_Oinduct, axiom, (! [A:$tType,P:((A > (A > $o)) > ((list @ A) > $o)),A0:(A > (A > $o)),A1:(list @ A)]: (((! [P3:(A > (A > $o))]: (((P @ P3) @ (nil @ A)))) => ((! [P3:(A > (A > $o)),X2:A,Ys3:(list @ A)]: ((((P @ P3) @ Ys3) => ((P @ P3) @ (((cons @ A) @ X2) @ Ys3))))) => ((P @ A0) @ A1)))))).
% 2.40/1.48 thf(fact_135_remdups__adj_Ocases, axiom, (! [A:$tType,X:(list @ A)]: (((X != (nil @ A)) => ((! [X2:A]: ((X != (((cons @ A) @ X2) @ (nil @ A))))) => (~ ((! [X2:A,Y2:A,Xs3:(list @ A)]: ((X != (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Xs3)))))))))))).
% 2.40/1.48 thf(fact_136_transpose_Ocases, axiom, (! [A:$tType,X:(list @ (list @ A))]: (((X != (nil @ (list @ A))) => ((! [Xss:(list @ (list @ A))]: ((X != (((cons @ (list @ A)) @ (nil @ A)) @ Xss)))) => (~ ((! [X2:A,Xs3:(list @ A),Xss:(list @ (list @ A))]: ((X != (((cons @ (list @ A)) @ (((cons @ A) @ X2) @ Xs3)) @ Xss))))))))))).
% 2.40/1.48 thf(fact_137_shuffles_Oinduct, axiom, (! [A:$tType,P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X_1:(list @ A)]: (((P @ (nil @ A)) @ X_1))) => ((! [Xs3:(list @ A)]: (((P @ Xs3) @ (nil @ A)))) => ((! [X2:A,Xs3:(list @ A),Y2:A,Ys3:(list @ A)]: ((((P @ Xs3) @ (((cons @ A) @ Y2) @ Ys3)) => (((P @ (((cons @ A) @ X2) @ Xs3)) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys3)))))) => ((P @ A0) @ A1))))))).
% 2.40/1.48 thf(fact_138_min__list_Oinduct, axiom, (! [A:$tType]: (((ord @ A) => (! [P:((list @ A) > $o),A0:(list @ A)]: (((! [X2:A,Xs3:(list @ A)]: (((! [X212:A,X223:(list @ A)]: (((Xs3 = (((cons @ A) @ X212) @ X223)) => (P @ Xs3)))) => (P @ (((cons @ A) @ X2) @ Xs3))))) => ((P @ (nil @ A)) => (P @ A0))))))))).
% 2.40/1.48 thf(fact_139_min__list_Ocases, axiom, (! [A:$tType]: (((ord @ A) => (! [X:(list @ A)]: (((! [X2:A,Xs3:(list @ A)]: ((X != (((cons @ A) @ X2) @ Xs3)))) => (X = (nil @ A))))))))).
% 2.40/1.48 thf(fact_140_induct__list012, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Y2:A,Zs:(list @ A)]: (((P @ Zs) => ((P @ (((cons @ A) @ Y2) @ Zs)) => (P @ (((cons @ A) @ X2) @ (((cons @ A) @ Y2) @ Zs))))))) => (P @ Xs))))))).
% 2.40/1.48 thf(fact_141_splice_Oinduct, axiom, (! [A:$tType,P:((list @ A) > ((list @ A) > $o)),A0:(list @ A),A1:(list @ A)]: (((! [X_1:(list @ A)]: (((P @ (nil @ A)) @ X_1))) => ((! [X2:A,Xs3:(list @ A),Ys3:(list @ A)]: ((((P @ Ys3) @ Xs3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ Ys3)))) => ((P @ A0) @ A1)))))).
% 2.40/1.48 thf(fact_142_list__induct2_H, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),Xs:(list @ A),Ys:(list @ B)]: ((((P @ (nil @ A)) @ (nil @ B)) => ((! [X2:A,Xs3:(list @ A)]: (((P @ (((cons @ A) @ X2) @ Xs3)) @ (nil @ B)))) => ((! [Y2:B,Ys3:(list @ B)]: (((P @ (nil @ A)) @ (((cons @ B) @ Y2) @ Ys3)))) => ((! [X2:A,Xs3:(list @ A),Y2:B,Ys3:(list @ B)]: ((((P @ Xs3) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ B) @ Y2) @ Ys3))))) => ((P @ Xs) @ Ys)))))))).
% 2.40/1.48 thf(fact_143_neq__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) = (? [Y3:A,Ys2:(list @ A)]: ((Xs = (((cons @ A) @ Y3) @ Ys2)))))))).
% 2.40/1.48 thf(fact_144_list_Oinducts, axiom, (! [A:$tType,P:((list @ A) > $o),List:(list @ A)]: (((P @ (nil @ A)) => ((! [X12:A,X23:(list @ A)]: (((P @ X23) => (P @ (((cons @ A) @ X12) @ X23))))) => (P @ List)))))).
% 2.40/1.48 thf(fact_145_list_Oexhaust, axiom, (! [A:$tType,Y:(list @ A)]: (((Y != (nil @ A)) => (~ ((! [X213:A,X224:(list @ A)]: ((Y != (((cons @ A) @ X213) @ X224)))))))))).
% 2.40/1.48 thf(fact_146_list_OdiscI, axiom, (! [A:$tType,List:(list @ A),X21:A,X222:(list @ A)]: (((List = (((cons @ A) @ X21) @ X222)) => (List != (nil @ A)))))).
% 2.40/1.48 thf(fact_147_list_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X222:(list @ A)]: (((nil @ A) != (((cons @ A) @ X21) @ X222))))).
% 2.40/1.48 thf(fact_148_not__Cons__self2, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((cons @ A) @ X) @ Xs) != Xs)))).
% 2.40/1.48 thf(fact_149_ord_Olexordp__eq_OCons__eq, axiom, (! [A:$tType,Less2:(A > (A > $o)),X:A,Y:A,Xs:(list @ A),Ys:(list @ A)]: (((~ (((Less2 @ X) @ Y))) => ((~ (((Less2 @ Y) @ X))) => (((((lexordp_eq @ A) @ Less2) @ Xs) @ Ys) => ((((lexordp_eq @ A) @ Less2) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys)))))))).
% 2.40/1.48 thf(fact_150_ord_Olexordp__eq_OCons, axiom, (! [A:$tType,Less2:(A > (A > $o)),X:A,Y:A,Xs:(list @ A),Ys:(list @ A)]: ((((Less2 @ X) @ Y) => ((((lexordp_eq @ A) @ Less2) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys)))))).
% 2.40/1.48 thf(fact_151_ord_Olexordp__eq_Oinducts, axiom, (! [A:$tType,Less2:(A > (A > $o)),X1:(list @ A),X22:(list @ A),P:((list @ A) > ((list @ A) > $o))]: ((((((lexordp_eq @ A) @ Less2) @ X1) @ X22) => ((! [X_1:(list @ A)]: (((P @ (nil @ A)) @ X_1))) => ((! [X2:A,Y2:A,Xs3:(list @ A),Ys3:(list @ A)]: ((((Less2 @ X2) @ Y2) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys3))))) => ((! [X2:A,Y2:A,Xs3:(list @ A),Ys3:(list @ A)]: (((~ (((Less2 @ X2) @ Y2))) => ((~ (((Less2 @ Y2) @ X2))) => (((((lexordp_eq @ A) @ Less2) @ Xs3) @ Ys3) => (((P @ Xs3) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys3)))))))) => ((P @ X1) @ X22)))))))).
% 2.40/1.48 thf(fact_152_ord_Olexordp__eq_Osimps, axiom, (! [A:$tType]: (((lexordp_eq @ A) = (^ [Less:(A > (A > $o)),A12:(list @ A),A23:(list @ A)]: (((? [Ys2:(list @ A)]: (((A12 = (nil @ A)) & (A23 = Ys2)))) | ((? [X3:A,Y3:A,Xs2:(list @ A),Ys2:(list @ A)]: (((A12 = (((cons @ A) @ X3) @ Xs2)) & ((A23 = (((cons @ A) @ Y3) @ Ys2)) & ((Less @ X3) @ Y3))))) | (? [X3:A,Y3:A,Xs2:(list @ A),Ys2:(list @ A)]: (((A12 = (((cons @ A) @ X3) @ Xs2)) & ((A23 = (((cons @ A) @ Y3) @ Ys2)) & ((~ (((Less @ X3) @ Y3))) & ((~ (((Less @ Y3) @ X3))) & ((((lexordp_eq @ A) @ Less) @ Xs2) @ Ys2))))))))))))))).
% 2.40/1.48 thf(fact_153_ord_Olexordp__eq_Ocases, axiom, (! [A:$tType,Less2:(A > (A > $o)),A1:(list @ A),A22:(list @ A)]: ((((((lexordp_eq @ A) @ Less2) @ A1) @ A22) => ((A1 != (nil @ A)) => ((! [X2:A]: (((? [Xs3:(list @ A)]: ((A1 = (((cons @ A) @ X2) @ Xs3)))) => (! [Y2:A]: (((? [Ys3:(list @ A)]: ((A22 = (((cons @ A) @ Y2) @ Ys3)))) => (~ (((Less2 @ X2) @ Y2))))))))) => (~ ((! [X2:A,Y2:A,Xs3:(list @ A)]: (((A1 = (((cons @ A) @ X2) @ Xs3)) => (! [Ys3:(list @ A)]: (((A22 = (((cons @ A) @ Y2) @ Ys3)) => ((~ (((Less2 @ X2) @ Y2))) => ((~ (((Less2 @ Y2) @ X2))) => (~ (((((lexordp_eq @ A) @ Less2) @ Xs3) @ Ys3))))))))))))))))))).
% 2.40/1.48 thf(fact_154_ord_Olexordp__eq_Ocong, axiom, (! [A:$tType]: (((lexordp_eq @ A) = (lexordp_eq @ A))))).
% 2.40/1.48 thf(fact_155_ord_Olexordp__eq__refl, axiom, (! [A:$tType,Less2:(A > (A > $o)),Xs:(list @ A)]: (((((lexordp_eq @ A) @ Less2) @ Xs) @ Xs)))).
% 2.40/1.48 thf(fact_156_ord_Olexordp__eq_ONil, axiom, (! [A:$tType,Less2:(A > (A > $o)),Ys:(list @ A)]: (((((lexordp_eq @ A) @ Less2) @ (nil @ A)) @ Ys)))).
% 2.40/1.48 thf(fact_157_local_Olexordp_ONil, axiom, (! [Y:a,Ys:(list @ a)]: (((((lexordp @ a) @ less) @ (nil @ a)) @ (((cons @ a) @ Y) @ Ys))))).
% 2.40/1.48 thf(fact_158_local_Olexordp_Ocases, axiom, (! [A1:(list @ a),A22:(list @ a)]: ((((((lexordp @ a) @ less) @ A1) @ A22) => (((A1 = (nil @ a)) => (! [Y2:a,Ys3:(list @ a)]: ((A22 != (((cons @ a) @ Y2) @ Ys3))))) => ((! [X2:a]: (((? [Xs3:(list @ a)]: ((A1 = (((cons @ a) @ X2) @ Xs3)))) => (! [Y2:a]: (((? [Ys3:(list @ a)]: ((A22 = (((cons @ a) @ Y2) @ Ys3)))) => (~ (((less @ X2) @ Y2))))))))) => (~ ((! [X2:a,Y2:a,Xs3:(list @ a)]: (((A1 = (((cons @ a) @ X2) @ Xs3)) => (! [Ys3:(list @ a)]: (((A22 = (((cons @ a) @ Y2) @ Ys3)) => ((~ (((less @ X2) @ Y2))) => ((~ (((less @ Y2) @ X2))) => (~ (((((lexordp @ a) @ less) @ Xs3) @ Ys3))))))))))))))))))).
% 2.40/1.48 thf(fact_159_local_Olexordp_Oinducts, axiom, (! [X1:(list @ a),X22:(list @ a),P:((list @ a) > ((list @ a) > $o))]: ((((((lexordp @ a) @ less) @ X1) @ X22) => ((! [Y2:a,Ys3:(list @ a)]: (((P @ (nil @ a)) @ (((cons @ a) @ Y2) @ Ys3)))) => ((! [X2:a,Y2:a,Xs3:(list @ a),Ys3:(list @ a)]: ((((less @ X2) @ Y2) => ((P @ (((cons @ a) @ X2) @ Xs3)) @ (((cons @ a) @ Y2) @ Ys3))))) => ((! [X2:a,Y2:a,Xs3:(list @ a),Ys3:(list @ a)]: (((~ (((less @ X2) @ Y2))) => ((~ (((less @ Y2) @ X2))) => (((((lexordp @ a) @ less) @ Xs3) @ Ys3) => (((P @ Xs3) @ Ys3) => ((P @ (((cons @ a) @ X2) @ Xs3)) @ (((cons @ a) @ Y2) @ Ys3)))))))) => ((P @ X1) @ X22)))))))).
% 2.40/1.48 thf(fact_160_local_Olexordp_Osimps, axiom, (! [A1:(list @ a),A22:(list @ a)]: ((((((lexordp @ a) @ less) @ A1) @ A22) = ((? [Y3:a,Ys2:(list @ a)]: (((A1 = (nil @ a)) & (A22 = (((cons @ a) @ Y3) @ Ys2))))) | ((? [X3:a,Y3:a,Xs2:(list @ a),Ys2:(list @ a)]: (((A1 = (((cons @ a) @ X3) @ Xs2)) & ((A22 = (((cons @ a) @ Y3) @ Ys2)) & ((less @ X3) @ Y3))))) | (? [X3:a,Y3:a,Xs2:(list @ a),Ys2:(list @ a)]: (((A1 = (((cons @ a) @ X3) @ Xs2)) & ((A22 = (((cons @ a) @ Y3) @ Ys2)) & ((~ (((less @ X3) @ Y3))) & ((~ (((less @ Y3) @ X3))) & ((((lexordp @ a) @ less) @ Xs2) @ Ys2))))))))))))).
% 2.40/1.48 thf(fact_161_local_Olexordp__antisym, axiom, (! [Xs:(list @ a),Ys:(list @ a)]: ((((((lexordp @ a) @ less) @ Xs) @ Ys) => (~ (((((lexordp @ a) @ less) @ Ys) @ Xs))))))).
% 2.40/1.48 thf(fact_162_local_Olexordp__irreflexive, axiom, (! [Xs:(list @ a)]: (((! [X2:a]: ((~ (((less @ X2) @ X2))))) => (~ (((((lexordp @ a) @ less) @ Xs) @ Xs))))))).
% 2.40/1.48 thf(fact_163_local_Olexordp__irreflexive_H, axiom, (! [Xs:(list @ a)]: ((~ (((((lexordp @ a) @ less) @ Xs) @ Xs)))))).
% 2.40/1.48 thf(fact_164_local_Olexordp_OCons__eq, axiom, (! [X:a,Y:a,Xs:(list @ a),Ys:(list @ a)]: (((~ (((less @ X) @ Y))) => ((~ (((less @ Y) @ X))) => (((((lexordp @ a) @ less) @ Xs) @ Ys) => ((((lexordp @ a) @ less) @ (((cons @ a) @ X) @ Xs)) @ (((cons @ a) @ Y) @ Ys)))))))).
% 2.40/1.48 thf(fact_165_local_Olexordp_OCons, axiom, (! [X:a,Y:a,Xs:(list @ a),Ys:(list @ a)]: ((((less @ X) @ Y) => ((((lexordp @ a) @ less) @ (((cons @ a) @ X) @ Xs)) @ (((cons @ a) @ Y) @ Ys)))))).
% 2.40/1.48 thf(fact_166_local_Olexordp__into__lexordp__eq, axiom, (! [Xs:(list @ a),Ys:(list @ a)]: ((((((lexordp @ a) @ less) @ Xs) @ Ys) => ((((lexordp_eq @ a) @ less) @ Xs) @ Ys))))).
% 2.40/1.48 thf(fact_167_ord_Olexordp__simps_I3_J, axiom, (! [A:$tType,Less2:(A > (A > $o)),X:A,Xs:(list @ A),Y:A,Ys:(list @ A)]: ((((((lexordp @ A) @ Less2) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys)) = (((Less2 @ X) @ Y) | ((~ (((Less2 @ Y) @ X))) & ((((lexordp @ A) @ Less2) @ Xs) @ Ys))))))).
% 2.40/1.48 thf(fact_168_ord_Olexordp__simps_I2_J, axiom, (! [A:$tType,Less2:(A > (A > $o)),Xs:(list @ A)]: ((~ (((((lexordp @ A) @ Less2) @ Xs) @ (nil @ A))))))).
% 2.40/1.48 thf(fact_169_ord_Olexordp__simps_I1_J, axiom, (! [A:$tType,Less2:(A > (A > $o)),Ys:(list @ A)]: ((((((lexordp @ A) @ Less2) @ (nil @ A)) @ Ys) = (Ys != (nil @ A)))))).
% 2.40/1.48 thf(fact_170_local_Olexordp__simps_I3_J, axiom, (! [X:a,Xs:(list @ a),Y:a,Ys:(list @ a)]: ((((((lexordp @ a) @ less) @ (((cons @ a) @ X) @ Xs)) @ (((cons @ a) @ Y) @ Ys)) = (((less @ X) @ Y) | ((~ (((less @ Y) @ X))) & ((((lexordp @ a) @ less) @ Xs) @ Ys))))))).
% 2.40/1.48 thf(fact_171_local_Olexordp__simps_I1_J, axiom, (! [Ys:(list @ a)]: ((((((lexordp @ a) @ less) @ (nil @ a)) @ Ys) = (Ys != (nil @ a)))))).
% 2.40/1.48 thf(fact_172_local_Olexordp__simps_I2_J, axiom, (! [Xs:(list @ a)]: ((~ (((((lexordp @ a) @ less) @ Xs) @ (nil @ a))))))).
% 2.40/1.48 thf(fact_173_ord_Olexordp_Ocong, axiom, (! [A:$tType]: (((lexordp @ A) = (lexordp @ A))))).
% 2.40/1.48 thf(fact_174_ord_Olexordp__irreflexive, axiom, (! [A:$tType,Less2:(A > (A > $o)),Xs:(list @ A)]: (((! [X2:A]: ((~ (((Less2 @ X2) @ X2))))) => (~ (((((lexordp @ A) @ Less2) @ Xs) @ Xs))))))).
% 2.40/1.48 thf(fact_175_ord_Olexordp_OCons__eq, axiom, (! [A:$tType,Less2:(A > (A > $o)),X:A,Y:A,Xs:(list @ A),Ys:(list @ A)]: (((~ (((Less2 @ X) @ Y))) => ((~ (((Less2 @ Y) @ X))) => (((((lexordp @ A) @ Less2) @ Xs) @ Ys) => ((((lexordp @ A) @ Less2) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys)))))))).
% 2.40/1.48 thf(fact_176_ord_Olexordp_OCons, axiom, (! [A:$tType,Less2:(A > (A > $o)),X:A,Y:A,Xs:(list @ A),Ys:(list @ A)]: ((((Less2 @ X) @ Y) => ((((lexordp @ A) @ Less2) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys)))))).
% 2.40/1.48 thf(fact_177_ord_Olexordp__into__lexordp__eq, axiom, (! [A:$tType,Less2:(A > (A > $o)),Xs:(list @ A),Ys:(list @ A)]: ((((((lexordp @ A) @ Less2) @ Xs) @ Ys) => ((((lexordp_eq @ A) @ Less2) @ Xs) @ Ys))))).
% 2.40/1.48 thf(fact_178_ord_Olexordp_ONil, axiom, (! [A:$tType,Less2:(A > (A > $o)),Y:A,Ys:(list @ A)]: (((((lexordp @ A) @ Less2) @ (nil @ A)) @ (((cons @ A) @ Y) @ Ys))))).
% 2.40/1.48 thf(fact_179_ord_Olexordp_Ocases, axiom, (! [A:$tType,Less2:(A > (A > $o)),A1:(list @ A),A22:(list @ A)]: ((((((lexordp @ A) @ Less2) @ A1) @ A22) => (((A1 = (nil @ A)) => (! [Y2:A,Ys3:(list @ A)]: ((A22 != (((cons @ A) @ Y2) @ Ys3))))) => ((! [X2:A]: (((? [Xs3:(list @ A)]: ((A1 = (((cons @ A) @ X2) @ Xs3)))) => (! [Y2:A]: (((? [Ys3:(list @ A)]: ((A22 = (((cons @ A) @ Y2) @ Ys3)))) => (~ (((Less2 @ X2) @ Y2))))))))) => (~ ((! [X2:A,Y2:A,Xs3:(list @ A)]: (((A1 = (((cons @ A) @ X2) @ Xs3)) => (! [Ys3:(list @ A)]: (((A22 = (((cons @ A) @ Y2) @ Ys3)) => ((~ (((Less2 @ X2) @ Y2))) => ((~ (((Less2 @ Y2) @ X2))) => (~ (((((lexordp @ A) @ Less2) @ Xs3) @ Ys3))))))))))))))))))).
% 2.40/1.48 thf(fact_180_ord_Olexordp_Osimps, axiom, (! [A:$tType]: (((lexordp @ A) = (^ [Less:(A > (A > $o)),A12:(list @ A),A23:(list @ A)]: (((? [Y3:A,Ys2:(list @ A)]: (((A12 = (nil @ A)) & (A23 = (((cons @ A) @ Y3) @ Ys2))))) | ((? [X3:A,Y3:A,Xs2:(list @ A),Ys2:(list @ A)]: (((A12 = (((cons @ A) @ X3) @ Xs2)) & ((A23 = (((cons @ A) @ Y3) @ Ys2)) & ((Less @ X3) @ Y3))))) | (? [X3:A,Y3:A,Xs2:(list @ A),Ys2:(list @ A)]: (((A12 = (((cons @ A) @ X3) @ Xs2)) & ((A23 = (((cons @ A) @ Y3) @ Ys2)) & ((~ (((Less @ X3) @ Y3))) & ((~ (((Less @ Y3) @ X3))) & ((((lexordp @ A) @ Less) @ Xs2) @ Ys2))))))))))))))).
% 2.40/1.48 thf(fact_181_ord_Olexordp_Oinducts, axiom, (! [A:$tType,Less2:(A > (A > $o)),X1:(list @ A),X22:(list @ A),P:((list @ A) > ((list @ A) > $o))]: ((((((lexordp @ A) @ Less2) @ X1) @ X22) => ((! [Y2:A,Ys3:(list @ A)]: (((P @ (nil @ A)) @ (((cons @ A) @ Y2) @ Ys3)))) => ((! [X2:A,Y2:A,Xs3:(list @ A),Ys3:(list @ A)]: ((((Less2 @ X2) @ Y2) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys3))))) => ((! [X2:A,Y2:A,Xs3:(list @ A),Ys3:(list @ A)]: (((~ (((Less2 @ X2) @ Y2))) => ((~ (((Less2 @ Y2) @ X2))) => (((((lexordp @ A) @ Less2) @ Xs3) @ Ys3) => (((P @ Xs3) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs3)) @ (((cons @ A) @ Y2) @ Ys3)))))))) => ((P @ X1) @ X22)))))))).
% 2.40/1.48 thf(fact_182_local_Olexordp__append__rightI, axiom, (! [Ys:(list @ a),Xs:(list @ a)]: (((Ys != (nil @ a)) => ((((lexordp @ a) @ less) @ Xs) @ (((append @ a) @ Xs) @ Ys)))))).
% 2.40/1.48 thf(fact_183_local_Olexordp__append__left__rightI, axiom, (! [X:a,Y:a,Us:(list @ a),Xs:(list @ a),Ys:(list @ a)]: ((((less @ X) @ Y) => ((((lexordp @ a) @ less) @ (((append @ a) @ Us) @ (((cons @ a) @ X) @ Xs))) @ (((append @ a) @ Us) @ (((cons @ a) @ Y) @ Ys))))))).
% 2.40/1.48 thf(fact_184_semilattice_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((abel_semigroup @ A) @ F))))).
% 2.40/1.48 thf(fact_185_local_Olexordp__append__leftD, axiom, (! [Xs:(list @ a),Us:(list @ a),Vs:(list @ a)]: ((((((lexordp @ a) @ less) @ (((append @ a) @ Xs) @ Us)) @ (((append @ a) @ Xs) @ Vs)) => ((! [A3:a]: ((~ (((less @ A3) @ A3))))) => ((((lexordp @ a) @ less) @ Us) @ Vs)))))).
% 2.40/1.48 thf(fact_186_local_Olexordp__append__leftI, axiom, (! [Us:(list @ a),Vs:(list @ a),Xs:(list @ a)]: ((((((lexordp @ a) @ less) @ Us) @ Vs) => ((((lexordp @ a) @ less) @ (((append @ a) @ Xs) @ Us)) @ (((append @ a) @ Xs) @ Vs)))))).
% 2.40/1.48 thf(fact_187_local_Olexordp__eq__pref, axiom, (! [U:(list @ a),V:(list @ a)]: (((((lexordp_eq @ a) @ less) @ U) @ (((append @ a) @ U) @ V))))).
% 2.40/1.48 thf(fact_188_same__append__eq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Xs) @ Zs2)) = (Ys = Zs2))))).
% 2.40/1.48 thf(fact_189_append__same__eq, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A),Zs2:(list @ A)]: ((((((append @ A) @ Ys) @ Xs) = (((append @ A) @ Zs2) @ Xs)) = (Ys = Zs2))))).
% 2.40/1.48 thf(fact_190_append__assoc, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: (((((append @ A) @ (((append @ A) @ Xs) @ Ys)) @ Zs2) = (((append @ A) @ Xs) @ (((append @ A) @ Ys) @ Zs2)))))).
% 2.40/1.48 thf(fact_191_append_Oassoc, axiom, (! [A:$tType,A2:(list @ A),B2:(list @ A),C:(list @ A)]: (((((append @ A) @ (((append @ A) @ A2) @ B2)) @ C) = (((append @ A) @ A2) @ (((append @ A) @ B2) @ C)))))).
% 2.40/1.48 thf(fact_192_append_Oright__neutral, axiom, (! [A:$tType,A2:(list @ A)]: (((((append @ A) @ A2) @ (nil @ A)) = A2)))).
% 2.40/1.48 thf(fact_193_append__is__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (nil @ A)) = ((Xs = (nil @ A)) & (Ys = (nil @ A))))))).
% 2.40/1.48 thf(fact_194_Nil__is__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((nil @ A) = (((append @ A) @ Xs) @ Ys)) = ((Xs = (nil @ A)) & (Ys = (nil @ A))))))).
% 2.40/1.48 thf(fact_195_self__append__conv2, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A)]: (((Ys = (((append @ A) @ Xs) @ Ys)) = (Xs = (nil @ A)))))).
% 2.40/1.48 thf(fact_196_append__self__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = Ys) = (Xs = (nil @ A)))))).
% 2.40/1.48 thf(fact_197_self__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs = (((append @ A) @ Xs) @ Ys)) = (Ys = (nil @ A)))))).
% 2.40/1.48 thf(fact_198_append__self__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = Xs) = (Ys = (nil @ A)))))).
% 2.40/1.48 thf(fact_199_append__Nil2, axiom, (! [A:$tType,Xs:(list @ A)]: (((((append @ A) @ Xs) @ (nil @ A)) = Xs)))).
% 2.40/1.48 thf(fact_200_append1__eq__conv, axiom, (! [A:$tType,Xs:(list @ A),X:A,Ys:(list @ A),Y:A]: ((((((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A))) = (((append @ A) @ Ys) @ (((cons @ A) @ Y) @ (nil @ A)))) = ((Xs = Ys) & (X = Y)))))).
% 2.40/1.48 thf(fact_201_append_Oleft__neutral, axiom, (! [A:$tType,A2:(list @ A)]: (((((append @ A) @ (nil @ A)) @ A2) = A2)))).
% 2.40/1.48 thf(fact_202_append__Nil, axiom, (! [A:$tType,Ys:(list @ A)]: (((((append @ A) @ (nil @ A)) @ Ys) = Ys)))).
% 2.40/1.48 thf(fact_203_eq__Nil__appendI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs = Ys) => (Xs = (((append @ A) @ (nil @ A)) @ Ys)))))).
% 2.40/1.48 thf(fact_204_Cons__eq__appendI, axiom, (! [A:$tType,X:A,Xs1:(list @ A),Ys:(list @ A),Xs:(list @ A),Zs2:(list @ A)]: ((((((cons @ A) @ X) @ Xs1) = Ys) => ((Xs = (((append @ A) @ Xs1) @ Zs2)) => ((((cons @ A) @ X) @ Xs) = (((append @ A) @ Ys) @ Zs2))))))).
% 2.40/1.48 thf(fact_205_append__Cons, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys:(list @ A)]: (((((append @ A) @ (((cons @ A) @ X) @ Xs)) @ Ys) = (((cons @ A) @ X) @ (((append @ A) @ Xs) @ Ys)))))).
% 2.40/1.48 thf(fact_206_ord_Olexordp__append__leftD, axiom, (! [A:$tType,Less2:(A > (A > $o)),Xs:(list @ A),Us:(list @ A),Vs:(list @ A)]: ((((((lexordp @ A) @ Less2) @ (((append @ A) @ Xs) @ Us)) @ (((append @ A) @ Xs) @ Vs)) => ((! [A3:A]: ((~ (((Less2 @ A3) @ A3))))) => ((((lexordp @ A) @ Less2) @ Us) @ Vs)))))).
% 2.40/1.48 thf(fact_207_ord_Olexordp__append__leftI, axiom, (! [A:$tType,Less2:(A > (A > $o)),Us:(list @ A),Vs:(list @ A),Xs:(list @ A)]: ((((((lexordp @ A) @ Less2) @ Us) @ Vs) => ((((lexordp @ A) @ Less2) @ (((append @ A) @ Xs) @ Us)) @ (((append @ A) @ Xs) @ Vs)))))).
% 2.40/1.48 thf(fact_208_ord_Olexordp__eq__pref, axiom, (! [A:$tType,Less2:(A > (A > $o)),U:(list @ A),V:(list @ A)]: (((((lexordp_eq @ A) @ Less2) @ U) @ (((append @ A) @ U) @ V))))).
% 2.40/1.48 thf(fact_209_append__eq__append__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A),Ts:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Zs2) @ Ts)) = (? [Us2:(list @ A)]: ((((Xs = (((append @ A) @ Zs2) @ Us2)) & ((((append @ A) @ Us2) @ Ys) = Ts)) | (((((append @ A) @ Xs) @ Us2) = Zs2) & (Ys = (((append @ A) @ Us2) @ Ts)))))))))).
% 2.40/1.48 thf(fact_210_append__eq__appendI, axiom, (! [A:$tType,Xs:(list @ A),Xs1:(list @ A),Zs2:(list @ A),Ys:(list @ A),Us:(list @ A)]: ((((((append @ A) @ Xs) @ Xs1) = Zs2) => ((Ys = (((append @ A) @ Xs1) @ Us)) => ((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Zs2) @ Us))))))).
% 2.40/1.48 thf(fact_211_append_Osemigroup__axioms, axiom, (! [A:$tType]: (((semigroup @ (list @ A)) @ (append @ A))))).
% 2.40/1.48 thf(fact_212_rev__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X2:A,Xs3:(list @ A)]: (((Xs3 != (nil @ A)) => ((P @ Xs3) => (P @ (((append @ A) @ Xs3) @ (((cons @ A) @ X2) @ (nil @ A)))))))) => (P @ Xs))))))).
% 2.40/1.48 thf(fact_213_append__eq__Cons__conv, axiom, (! [A:$tType,Ys:(list @ A),Zs2:(list @ A),X:A,Xs:(list @ A)]: ((((((append @ A) @ Ys) @ Zs2) = (((cons @ A) @ X) @ Xs)) = (((Ys = (nil @ A)) & (Zs2 = (((cons @ A) @ X) @ Xs))) | (? [Ys4:(list @ A)]: (((Ys = (((cons @ A) @ X) @ Ys4)) & ((((append @ A) @ Ys4) @ Zs2) = Xs))))))))).
% 2.40/1.48 thf(fact_214_Cons__eq__append__conv, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((((cons @ A) @ X) @ Xs) = (((append @ A) @ Ys) @ Zs2)) = (((Ys = (nil @ A)) & ((((cons @ A) @ X) @ Xs) = Zs2)) | (? [Ys4:(list @ A)]: ((((((cons @ A) @ X) @ Ys4) = Ys) & (Xs = (((append @ A) @ Ys4) @ Zs2)))))))))).
% 2.40/1.48 thf(fact_215_rev__exhaust, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (~ ((! [Ys3:(list @ A),Y2:A]: ((Xs != (((append @ A) @ Ys3) @ (((cons @ A) @ Y2) @ (nil @ A)))))))))))).
% 2.40/1.48 thf(fact_216_rev__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A,Xs3:(list @ A)]: (((P @ Xs3) => (P @ (((append @ A) @ Xs3) @ (((cons @ A) @ X2) @ (nil @ A))))))) => (P @ Xs)))))).
% 2.40/1.48 thf(fact_217_ord_Olexordp__append__left__rightI, axiom, (! [A:$tType,Less2:(A > (A > $o)),X:A,Y:A,Us:(list @ A),Xs:(list @ A),Ys:(list @ A)]: ((((Less2 @ X) @ Y) => ((((lexordp @ A) @ Less2) @ (((append @ A) @ Us) @ (((cons @ A) @ X) @ Xs))) @ (((append @ A) @ Us) @ (((cons @ A) @ Y) @ Ys))))))).
% 2.40/1.48 thf(fact_218_ord_Olexordp__append__rightI, axiom, (! [A:$tType,Ys:(list @ A),Less2:(A > (A > $o)),Xs:(list @ A)]: (((Ys != (nil @ A)) => ((((lexordp @ A) @ Less2) @ Xs) @ (((append @ A) @ Xs) @ Ys)))))).
% 2.40/1.48 thf(fact_219_semilattice_Oidem, axiom, (! [A:$tType,F:(A > (A > A)),A2:A]: ((((semilattice @ A) @ F) => (((F @ A2) @ A2) = A2))))).
% 2.40/1.48 thf(fact_220_semilattice_Oleft__idem, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((semilattice @ A) @ F) => (((F @ A2) @ ((F @ A2) @ B2)) = ((F @ A2) @ B2)))))).
% 2.40/1.48 thf(fact_221_semilattice_Oright__idem, axiom, (! [A:$tType,F:(A > (A > A)),A2:A,B2:A]: ((((semilattice @ A) @ F) => (((F @ ((F @ A2) @ B2)) @ B2) = ((F @ A2) @ B2)))))).
% 2.40/1.48 thf(fact_222_product__lists_Osimps_I1_J, axiom, (! [A:$tType]: ((((product_lists @ A) @ (nil @ (list @ A))) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 2.40/1.48 thf(fact_223_bind__simps_I2_J, axiom, (! [A:$tType,B:$tType,X:B,Xs:(list @ B),F:(B > (list @ A))]: ((((((bind @ B) @ A) @ (((cons @ B) @ X) @ Xs)) @ F) = (((append @ A) @ (F @ X)) @ ((((bind @ B) @ A) @ Xs) @ F)))))).
% 2.40/1.48 thf(fact_224_bind__simps_I1_J, axiom, (! [B:$tType,A:$tType,F:(B > (list @ A))]: ((((((bind @ B) @ A) @ (nil @ B)) @ F) = (nil @ A))))).
% 2.40/1.48 thf(fact_225_abel__semigroup__def, axiom, (! [A:$tType]: (((abel_semigroup @ A) = (^ [F2:(A > (A > A))]: ((((semigroup @ A) @ F2) & ((abel_s1917375468axioms @ A) @ F2)))))))).
% 2.40/1.48 thf(fact_226_abel__semigroup_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semigroup @ A) @ F) => (((abel_s1917375468axioms @ A) @ F) => ((abel_semigroup @ A) @ F)))))).
% 2.40/1.48 thf(fact_227_abel__semigroup__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: (((! [A3:A,B3:A]: ((((F @ A3) @ B3) = ((F @ B3) @ A3)))) => ((abel_s1917375468axioms @ A) @ F))))).
% 2.40/1.48 thf(fact_228_abel__semigroup__axioms__def, axiom, (! [A:$tType]: (((abel_s1917375468axioms @ A) = (^ [F2:(A > (A > A))]: ((! [A4:A,B4:A]: ((((F2 @ A4) @ B4) = ((F2 @ B4) @ A4)))))))))).
% 2.40/1.48 thf(fact_229_abel__semigroup_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => ((abel_s1917375468axioms @ A) @ F))))).
% 2.40/1.48 thf(fact_230_semilattice_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => (((semilattice_axioms @ A) @ F) => ((semilattice @ A) @ F)))))).
% 2.40/1.48 thf(fact_231_semilattice__def, axiom, (! [A:$tType]: (((semilattice @ A) = (^ [F2:(A > (A > A))]: ((((abel_semigroup @ A) @ F2) & ((semilattice_axioms @ A) @ F2)))))))).
% 2.40/1.48 thf(fact_232_semilattice_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semilattice @ A) @ F) => ((semilattice_axioms @ A) @ F))))).
% 2.40/1.48 thf(fact_233_subseqs_Osimps_I1_J, axiom, (! [A:$tType]: ((((subseqs @ A) @ (nil @ A)) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))).
% 2.40/1.48 thf(fact_234_maps__simps_I1_J, axiom, (! [A:$tType,B:$tType,F:(B > (list @ A)),X:B,Xs:(list @ B)]: ((((((maps @ B) @ A) @ F) @ (((cons @ B) @ X) @ Xs)) = (((append @ A) @ (F @ X)) @ ((((maps @ B) @ A) @ F) @ Xs)))))).
% 2.40/1.48 thf(fact_235_maps__simps_I2_J, axiom, (! [B:$tType,A:$tType,F:(B > (list @ A))]: ((((((maps @ B) @ A) @ F) @ (nil @ B)) = (nil @ A))))).
% 2.40/1.48 thf(fact_236_local_Otransp__less, axiom, ((transp @ a) @ less)).
% 2.40/1.48 thf(fact_237_insert__Nil, axiom, (! [A:$tType,X:A]: (((((insert @ A) @ X) @ (nil @ A)) = (((cons @ A) @ X) @ (nil @ A)))))).
% 2.40/1.48 thf(fact_238_concat__eq__append__conv, axiom, (! [A:$tType,Xss2:(list @ (list @ A)),Ys:(list @ A),Zs2:(list @ A)]: (((((concat @ A) @ Xss2) = (((append @ A) @ Ys) @ Zs2)) = (((Xss2 = (nil @ (list @ A))) => ((Ys = (nil @ A)) & (Zs2 = (nil @ A)))) & ((Xss2 != (nil @ (list @ A))) => (? [Xss1:(list @ (list @ A)),Xs2:(list @ A),Xs4:(list @ A),Xss22:(list @ (list @ A))]: (((Xss2 = (((append @ (list @ A)) @ Xss1) @ (((cons @ (list @ A)) @ (((append @ A) @ Xs2) @ Xs4)) @ Xss22))) & ((Ys = (((append @ A) @ ((concat @ A) @ Xss1)) @ Xs2)) & (Zs2 = (((append @ A) @ Xs4) @ ((concat @ A) @ Xss22))))))))))))).
% 2.40/1.48 thf(fact_239_rotate1_Osimps_I2_J, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: ((((rotate1 @ A) @ (((cons @ A) @ X) @ Xs)) = (((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A))))))).
% 2.40/1.48 thf(fact_240_rotate1__is__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((rotate1 @ A) @ Xs) = (nil @ A)) = (Xs = (nil @ A)))))).
% 2.40/1.48 thf(fact_241_concat__append, axiom, (! [A:$tType,Xs:(list @ (list @ A)),Ys:(list @ (list @ A))]: ((((concat @ A) @ (((append @ (list @ A)) @ Xs) @ Ys)) = (((append @ A) @ ((concat @ A) @ Xs)) @ ((concat @ A) @ Ys)))))).
% 2.40/1.48 thf(fact_242_concat_Osimps_I1_J, axiom, (! [A:$tType]: ((((concat @ A) @ (nil @ (list @ A))) = (nil @ A))))).
% 2.40/1.48 thf(fact_243_concat_Osimps_I2_J, axiom, (! [A:$tType,X:(list @ A),Xs:(list @ (list @ A))]: ((((concat @ A) @ (((cons @ (list @ A)) @ X) @ Xs)) = (((append @ A) @ X) @ ((concat @ A) @ Xs)))))).
% 2.40/1.48 thf(fact_244_rotate1_Osimps_I1_J, axiom, (! [A:$tType]: ((((rotate1 @ A) @ (nil @ A)) = (nil @ A))))).
% 2.40/1.48 thf(fact_245_concat__eq__appendD, axiom, (! [A:$tType,Xss2:(list @ (list @ A)),Ys:(list @ A),Zs2:(list @ A)]: (((((concat @ A) @ Xss2) = (((append @ A) @ Ys) @ Zs2)) => ((Xss2 != (nil @ (list @ A))) => (? [Xss12:(list @ (list @ A)),Xs3:(list @ A),Xs5:(list @ A),Xss23:(list @ (list @ A))]: (((Xss2 = (((append @ (list @ A)) @ Xss12) @ (((cons @ (list @ A)) @ (((append @ A) @ Xs3) @ Xs5)) @ Xss23))) & ((Ys = (((append @ A) @ ((concat @ A) @ Xss12)) @ Xs3)) & (Zs2 = (((append @ A) @ Xs5) @ ((concat @ A) @ Xss23)))))))))))).
% 2.40/1.48 thf(fact_246_butlast__snoc, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((butlast @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A)))) = Xs)))).
% 2.40/1.48 thf(fact_247_list__ex1__simps_I1_J, axiom, (! [A:$tType,P:(A > $o)]: ((~ ((((list_ex1 @ A) @ P) @ (nil @ A))))))).
% 2.40/1.48 thf(fact_248_butlast__append, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A)]: ((((Ys = (nil @ A)) => (((butlast @ A) @ (((append @ A) @ Xs) @ Ys)) = ((butlast @ A) @ Xs))) & ((Ys != (nil @ A)) => (((butlast @ A) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ Xs) @ ((butlast @ A) @ Ys)))))))).
% 2.40/1.48 thf(fact_249_butlast_Osimps_I1_J, axiom, (! [A:$tType]: ((((butlast @ A) @ (nil @ A)) = (nil @ A))))).
% 2.40/1.48 thf(fact_250_butlast_Osimps_I2_J, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((Xs = (nil @ A)) => (((butlast @ A) @ (((cons @ A) @ X) @ Xs)) = (nil @ A))) & ((Xs != (nil @ A)) => (((butlast @ A) @ (((cons @ A) @ X) @ Xs)) = (((cons @ A) @ X) @ ((butlast @ A) @ Xs)))))))).
% 2.40/1.48 thf(fact_251_append__butlast__last__id, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => ((((append @ A) @ ((butlast @ A) @ Xs)) @ (((cons @ A) @ ((last @ A) @ Xs)) @ (nil @ A))) = Xs))))).
% 2.40/1.48 thf(fact_252_snoc__eq__iff__butlast, axiom, (! [A:$tType,Xs:(list @ A),X:A,Ys:(list @ A)]: ((((((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A))) = Ys) = ((Ys != (nil @ A)) & ((((butlast @ A) @ Ys) = Xs) & (((last @ A) @ Ys) = X))))))).
% 2.40/1.48 thf(fact_253_last__appendR, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A)]: (((Ys != (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys)) = ((last @ A) @ Ys)))))).
% 2.40/1.48 thf(fact_254_last__appendL, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A)]: (((Ys = (nil @ A)) => (((last @ A) @ (((append @ A) @ Xs) @ Ys)) = ((last @ A) @ Xs)))))).
% 2.40/1.48 thf(tcon_fun___Orderings_Oord, axiom, (! [A5:$tType,A6:$tType]: (((ord @ A6) => (ord @ (A5 > A6)))))).
% 2.40/1.48 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.40/1.48 thf(tcon_HOL_Obool___Orderings_Oord_1, axiom, (ord @ $o)).
% 2.40/1.48 thf(conj_0, hypothesis, (b = ((inf @ b) @ c))).
% 2.40/1.48 thf(conj_1, hypothesis, (((inf @ c) @ ((sup @ b) @ c)) = b)).
% 2.40/1.48 thf(conj_2, hypothesis, (a2 = ((sup @ b) @ c))).
% 2.40/1.48 thf(conj_3, hypothesis, ((less @ b) @ ((sup @ b) @ c))).
% 2.40/1.48 thf(conj_4, hypothesis, (((inf @ c) @ ((sup @ b) @ c)) = c)).
% 2.40/1.48 thf(conj_5, conjecture, $false).
% 2.40/1.48 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.41GWJZtw0l/DTF2DTF4588.p
% 2.40/1.48 ------------------------
% 2.65/1.51 ---- Cleaned THF ---
% 2.65/1.52 thf(ty_t_List_Olist,type,
% 2.65/1.52 list: $tType > $tType ).
% 2.65/1.52
% 2.65/1.52 thf(ty_tf_a,type,
% 2.65/1.52 a: $tType ).
% 2.65/1.52
% 2.65/1.52 thf(sy_cl_HOL_Otype,type,
% 2.65/1.52 type:
% 2.65/1.52 !>[A: $tType] : $o ).
% 2.65/1.52
% 2.65/1.52 thf(sy_cl_Orderings_Oord,type,
% 2.65/1.52 ord:
% 2.65/1.52 !>[A: $tType] : $o ).
% 2.65/1.52
% 2.65/1.52 thf(sy_cl_Orderings_Olinorder,type,
% 2.65/1.52 linorder:
% 2.65/1.52 !>[A: $tType] : $o ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
% 2.65/1.52 finite_comp_fun_idem:
% 2.65/1.52 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Groups_Oabel__semigroup,type,
% 2.65/1.52 abel_semigroup:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Groups_Oabel__semigroup__axioms,type,
% 2.65/1.52 abel_s1917375468axioms:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Groups_Osemigroup,type,
% 2.65/1.52 semigroup:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Lattices_Osemilattice,type,
% 2.65/1.52 semilattice:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Lattices_Osemilattice__axioms,type,
% 2.65/1.52 semilattice_axioms:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Lattices__Big_Oord_Ois__arg__max,type,
% 2.65/1.52 lattices_is_arg_max:
% 2.65/1.52 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Lattices__Big_Oord_Ois__arg__min,type,
% 2.65/1.52 lattices_is_arg_min:
% 2.65/1.52 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( B > $o ) > B > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 2.65/1.52 lattic35693393ce_set:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Oappend,type,
% 2.65/1.52 append:
% 2.65/1.52 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Obind,type,
% 2.65/1.52 bind:
% 2.65/1.52 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Obutlast,type,
% 2.65/1.52 butlast:
% 2.65/1.52 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Oconcat,type,
% 2.65/1.52 concat:
% 2.65/1.52 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Oinsert,type,
% 2.65/1.52 insert:
% 2.65/1.52 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Olast,type,
% 2.65/1.52 last:
% 2.65/1.52 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Olist_OCons,type,
% 2.65/1.52 cons:
% 2.65/1.52 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Olist_ONil,type,
% 2.65/1.52 nil:
% 2.65/1.52 !>[A: $tType] : ( list @ A ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Olist__ex1,type,
% 2.65/1.52 list_ex1:
% 2.65/1.52 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Omaps,type,
% 2.65/1.52 maps:
% 2.65/1.52 !>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Oord_Olexordp,type,
% 2.65/1.52 lexordp:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Oord_Olexordp__eq,type,
% 2.65/1.52 lexordp_eq:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Oproduct__lists,type,
% 2.65/1.52 product_lists:
% 2.65/1.52 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Orotate1,type,
% 2.65/1.52 rotate1:
% 2.65/1.52 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_List_Osubseqs,type,
% 2.65/1.52 subseqs:
% 2.65/1.52 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_OM5__lattice,type,
% 2.65/1.52 modula1618706180attice:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_ON5__lattice,type,
% 2.65/1.52 modula1192512645attice:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oa__aux,type,
% 2.65/1.52 modula105138637_a_aux:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ob__aux,type,
% 2.65/1.52 modula477145868_b_aux:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oc__aux,type,
% 2.65/1.52 modula849153099_c_aux:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Od__aux,type,
% 2.65/1.52 modula1221160330_d_aux:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Oe__aux,type,
% 2.65/1.52 modula1593167561_e_aux:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > A > A > A > A ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Modular__Distrib__Lattice__Mirabelle__sluygzlgpl_Olattice_Ono__distrib,type,
% 2.65/1.52 modula141148890istrib:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > A ) > A > A > A > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_c_Relation_Otransp,type,
% 2.65/1.52 transp:
% 2.65/1.52 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 2.65/1.52
% 2.65/1.52 thf(sy_v_a,type,
% 2.65/1.52 a2: a ).
% 2.65/1.52
% 2.65/1.52 thf(sy_v_b,type,
% 2.65/1.52 b: a ).
% 2.65/1.52
% 2.65/1.52 thf(sy_v_c,type,
% 2.65/1.52 c: a ).
% 2.65/1.52
% 2.65/1.52 thf(sy_v_inf,type,
% 2.65/1.52 inf: a > a > a ).
% 2.65/1.52
% 2.65/1.52 thf(sy_v_less,type,
% 2.65/1.52 less: a > a > $o ).
% 2.65/1.52
% 2.65/1.52 thf(sy_v_sup,type,
% 2.65/1.52 sup: a > a > a ).
% 2.65/1.52
% 2.65/1.52 thf(fact_0_local_Odual__order_Oasym,axiom,
% 2.65/1.52 ! [B2: a,A2: a] :
% 2.65/1.52 ( ( less @ B2 @ A2 )
% 2.65/1.52 => ~ ( less @ A2 @ B2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_1_local_Odual__order_Ostrict__implies__not__eq,axiom,
% 2.65/1.52 ! [B2: a,A2: a] :
% 2.65/1.52 ( ( less @ B2 @ A2 )
% 2.65/1.52 => ( A2 != B2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_2_local_Odual__order_Ostrict__trans,axiom,
% 2.65/1.52 ! [B2: a,A2: a,C: a] :
% 2.65/1.52 ( ( less @ B2 @ A2 )
% 2.65/1.52 => ( ( less @ C @ B2 )
% 2.65/1.52 => ( less @ C @ A2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_3_local_Oless__asym,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ~ ( less @ Y @ X ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_4_local_Oless__asym_H,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( less @ A2 @ B2 )
% 2.65/1.52 => ~ ( less @ B2 @ A2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_5_local_Oless__imp__neq,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ( X != Y ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_6_local_Oless__imp__not__eq,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ( X != Y ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_7_local_Oless__imp__not__eq2,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ( Y != X ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_8_local_Oless__imp__not__less,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ~ ( less @ Y @ X ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_9_local_Oless__imp__triv,axiom,
% 2.65/1.52 ! [X: a,Y: a,P: $o] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ( ( less @ Y @ X )
% 2.65/1.52 => P ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_10_local_Oless__irrefl,axiom,
% 2.65/1.52 ! [X: a] :
% 2.65/1.52 ~ ( less @ X @ X ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_11_local_Oless__not__sym,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ~ ( less @ Y @ X ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_12_local_Oless__trans,axiom,
% 2.65/1.52 ! [X: a,Y: a,Z: a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ( ( less @ Y @ Z )
% 2.65/1.52 => ( less @ X @ Z ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_13_local_Oord__eq__less__trans,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( A2 = B2 )
% 2.65/1.52 => ( ( less @ B2 @ C )
% 2.65/1.52 => ( less @ A2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_14_local_Oord__less__eq__trans,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( less @ A2 @ B2 )
% 2.65/1.52 => ( ( B2 = C )
% 2.65/1.52 => ( less @ A2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_15_local_Oorder_Oasym,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( less @ A2 @ B2 )
% 2.65/1.52 => ~ ( less @ B2 @ A2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_16_local_Oorder_Oirrefl,axiom,
% 2.65/1.52 ! [A2: a] :
% 2.65/1.52 ~ ( less @ A2 @ A2 ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_17_local_Oorder_Ostrict__implies__not__eq,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( less @ A2 @ B2 )
% 2.65/1.52 => ( A2 != B2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_18_local_Oorder_Ostrict__trans,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( less @ A2 @ B2 )
% 2.65/1.52 => ( ( less @ B2 @ C )
% 2.65/1.52 => ( less @ A2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_19_local_Oinf_Oassoc,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( inf @ ( inf @ A2 @ B2 ) @ C )
% 2.65/1.52 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_20_local_Oinf_Ocommute,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( inf @ A2 @ B2 )
% 2.65/1.52 = ( inf @ B2 @ A2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_21_local_Oinf_Oleft__commute,axiom,
% 2.65/1.52 ! [B2: a,A2: a,C: a] :
% 2.65/1.52 ( ( inf @ B2 @ ( inf @ A2 @ C ) )
% 2.65/1.52 = ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_22_local_Oinf__assoc,axiom,
% 2.65/1.52 ! [X: a,Y: a,Z: a] :
% 2.65/1.52 ( ( inf @ ( inf @ X @ Y ) @ Z )
% 2.65/1.52 = ( inf @ X @ ( inf @ Y @ Z ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_23_local_Oinf__commute,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( inf @ X @ Y )
% 2.65/1.52 = ( inf @ Y @ X ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_24_local_Oinf__left__commute,axiom,
% 2.65/1.52 ! [X: a,Y: a,Z: a] :
% 2.65/1.52 ( ( inf @ X @ ( inf @ Y @ Z ) )
% 2.65/1.52 = ( inf @ Y @ ( inf @ X @ Z ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_25_local_Osup_Oassoc,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( sup @ ( sup @ A2 @ B2 ) @ C )
% 2.65/1.52 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_26_local_Osup_Ocommute,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( sup @ A2 @ B2 )
% 2.65/1.52 = ( sup @ B2 @ A2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_27_local_Osup_Oleft__commute,axiom,
% 2.65/1.52 ! [B2: a,A2: a,C: a] :
% 2.65/1.52 ( ( sup @ B2 @ ( sup @ A2 @ C ) )
% 2.65/1.52 = ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_28_local_Osup__assoc,axiom,
% 2.65/1.52 ! [X: a,Y: a,Z: a] :
% 2.65/1.52 ( ( sup @ ( sup @ X @ Y ) @ Z )
% 2.65/1.52 = ( sup @ X @ ( sup @ Y @ Z ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_29_local_Osup__commute,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( sup @ X @ Y )
% 2.65/1.52 = ( sup @ Y @ X ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_30_local_Osup__left__commute,axiom,
% 2.65/1.52 ! [X: a,Y: a,Z: a] :
% 2.65/1.52 ( ( sup @ X @ ( sup @ Y @ Z ) )
% 2.65/1.52 = ( sup @ Y @ ( sup @ X @ Z ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_31_local_Oinf_Ostrict__boundedE,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( less @ A2 @ ( inf @ B2 @ C ) )
% 2.65/1.52 => ~ ( ( less @ A2 @ B2 )
% 2.65/1.52 => ~ ( less @ A2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_32_local_Oinf_Ostrict__coboundedI1,axiom,
% 2.65/1.52 ! [A2: a,C: a,B2: a] :
% 2.65/1.52 ( ( less @ A2 @ C )
% 2.65/1.52 => ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_33_local_Oinf_Ostrict__coboundedI2,axiom,
% 2.65/1.52 ! [B2: a,C: a,A2: a] :
% 2.65/1.52 ( ( less @ B2 @ C )
% 2.65/1.52 => ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_34_local_Oinf_Ostrict__order__iff,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( less @ A2 @ B2 )
% 2.65/1.52 = ( ( A2
% 2.65/1.52 = ( inf @ A2 @ B2 ) )
% 2.65/1.52 & ( A2 != B2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_35_local_Oless__infI1,axiom,
% 2.65/1.52 ! [A2: a,X: a,B2: a] :
% 2.65/1.52 ( ( less @ A2 @ X )
% 2.65/1.52 => ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_36_local_Oless__infI2,axiom,
% 2.65/1.52 ! [B2: a,X: a,A2: a] :
% 2.65/1.52 ( ( less @ B2 @ X )
% 2.65/1.52 => ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_37_local_Oless__supI1,axiom,
% 2.65/1.52 ! [X: a,A2: a,B2: a] :
% 2.65/1.52 ( ( less @ X @ A2 )
% 2.65/1.52 => ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_38_local_Oless__supI2,axiom,
% 2.65/1.52 ! [X: a,B2: a,A2: a] :
% 2.65/1.52 ( ( less @ X @ B2 )
% 2.65/1.52 => ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_39_local_Osup_Ostrict__boundedE,axiom,
% 2.65/1.52 ! [B2: a,C: a,A2: a] :
% 2.65/1.52 ( ( less @ ( sup @ B2 @ C ) @ A2 )
% 2.65/1.52 => ~ ( ( less @ B2 @ A2 )
% 2.65/1.52 => ~ ( less @ C @ A2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_40_local_Osup_Ostrict__coboundedI1,axiom,
% 2.65/1.52 ! [C: a,A2: a,B2: a] :
% 2.65/1.52 ( ( less @ C @ A2 )
% 2.65/1.52 => ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_41_local_Osup_Ostrict__coboundedI2,axiom,
% 2.65/1.52 ! [C: a,B2: a,A2: a] :
% 2.65/1.52 ( ( less @ C @ B2 )
% 2.65/1.52 => ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_42_local_Osup_Ostrict__order__iff,axiom,
% 2.65/1.52 ! [B2: a,A2: a] :
% 2.65/1.52 ( ( less @ B2 @ A2 )
% 2.65/1.52 = ( ( A2
% 2.65/1.52 = ( sup @ A2 @ B2 ) )
% 2.65/1.52 & ( A2 != B2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_43_local_Odistrib__imp1,axiom,
% 2.65/1.52 ! [X: a,Y: a,Z: a] :
% 2.65/1.52 ( ! [X2: a,Y2: a,Z2: a] :
% 2.65/1.52 ( ( inf @ X2 @ ( sup @ Y2 @ Z2 ) )
% 2.65/1.52 = ( sup @ ( inf @ X2 @ Y2 ) @ ( inf @ X2 @ Z2 ) ) )
% 2.65/1.52 => ( ( sup @ X @ ( inf @ Y @ Z ) )
% 2.65/1.52 = ( inf @ ( sup @ X @ Y ) @ ( sup @ X @ Z ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_44_local_Odistrib__imp2,axiom,
% 2.65/1.52 ! [X: a,Y: a,Z: a] :
% 2.65/1.52 ( ! [X2: a,Y2: a,Z2: a] :
% 2.65/1.52 ( ( sup @ X2 @ ( inf @ Y2 @ Z2 ) )
% 2.65/1.52 = ( inf @ ( sup @ X2 @ Y2 ) @ ( sup @ X2 @ Z2 ) ) )
% 2.65/1.52 => ( ( inf @ X @ ( sup @ Y @ Z ) )
% 2.65/1.52 = ( sup @ ( inf @ X @ Y ) @ ( inf @ X @ Z ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_45_ext,axiom,
% 2.65/1.52 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.65/1.52 ( ! [X2: A] :
% 2.65/1.52 ( ( F @ X2 )
% 2.65/1.52 = ( G @ X2 ) )
% 2.65/1.52 => ( F = G ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_46_local_Oinf_Oidem,axiom,
% 2.65/1.52 ! [A2: a] :
% 2.65/1.52 ( ( inf @ A2 @ A2 )
% 2.65/1.52 = A2 ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_47_local_Oinf_Oleft__idem,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
% 2.65/1.52 = ( inf @ A2 @ B2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_48_local_Oinf_Oright__idem,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
% 2.65/1.52 = ( inf @ A2 @ B2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_49_local_Oinf__idem,axiom,
% 2.65/1.52 ! [X: a] :
% 2.65/1.52 ( ( inf @ X @ X )
% 2.65/1.52 = X ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_50_local_Oinf__left__idem,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( inf @ X @ ( inf @ X @ Y ) )
% 2.65/1.52 = ( inf @ X @ Y ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_51_local_Oinf__right__idem,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( inf @ ( inf @ X @ Y ) @ Y )
% 2.65/1.52 = ( inf @ X @ Y ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_52_local_Osup_Oidem,axiom,
% 2.65/1.52 ! [A2: a] :
% 2.65/1.52 ( ( sup @ A2 @ A2 )
% 2.65/1.52 = A2 ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_53_local_Osup_Oleft__idem,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
% 2.65/1.52 = ( sup @ A2 @ B2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_54_local_Osup_Oright__idem,axiom,
% 2.65/1.52 ! [A2: a,B2: a] :
% 2.65/1.52 ( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
% 2.65/1.52 = ( sup @ A2 @ B2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_55_local_Osup__idem,axiom,
% 2.65/1.52 ! [X: a] :
% 2.65/1.52 ( ( sup @ X @ X )
% 2.65/1.52 = X ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_56_local_Osup__left__idem,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( sup @ X @ ( sup @ X @ Y ) )
% 2.65/1.52 = ( sup @ X @ Y ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_57_local_Ois__arg__max__def,axiom,
% 2.65/1.52 ! [B: $tType,F: B > a,P: B > $o,X: B] :
% 2.65/1.52 ( ( lattices_is_arg_max @ a @ B @ less @ F @ P @ X )
% 2.65/1.52 = ( ( P @ X )
% 2.65/1.52 & ~ ? [Y3: B] :
% 2.65/1.52 ( ( P @ Y3 )
% 2.65/1.52 & ( less @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_58_local_Ois__arg__min__def,axiom,
% 2.65/1.52 ! [B: $tType,F: B > a,P: B > $o,X: B] :
% 2.65/1.52 ( ( lattices_is_arg_min @ a @ B @ less @ F @ P @ X )
% 2.65/1.52 = ( ( P @ X )
% 2.65/1.52 & ~ ? [Y3: B] :
% 2.65/1.52 ( ( P @ Y3 )
% 2.65/1.52 & ( less @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_59_M5__lattice__def,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula1618706180attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( ( ( inf @ A2 @ B2 )
% 2.65/1.52 = ( inf @ B2 @ C ) )
% 2.65/1.52 & ( ( inf @ C @ A2 )
% 2.65/1.52 = ( inf @ B2 @ C ) )
% 2.65/1.52 & ( ( sup @ A2 @ B2 )
% 2.65/1.52 = ( sup @ B2 @ C ) )
% 2.65/1.52 & ( ( sup @ C @ A2 )
% 2.65/1.52 = ( sup @ B2 @ C ) )
% 2.65/1.52 & ( less @ ( inf @ A2 @ B2 ) @ ( sup @ A2 @ B2 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_60_N5__lattice__def,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula1192512645attice @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( ( ( inf @ A2 @ C )
% 2.65/1.52 = ( inf @ B2 @ C ) )
% 2.65/1.52 & ( less @ A2 @ B2 )
% 2.65/1.52 & ( ( sup @ A2 @ C )
% 2.65/1.52 = ( sup @ B2 @ C ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_61_no__distrib__def,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula141148890istrib @ a @ inf @ less @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( less @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ C @ A2 ) ) @ ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_62_local_Oinf__sup__absorb,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( inf @ X @ ( sup @ X @ Y ) )
% 2.65/1.52 = X ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_63_local_Osup__inf__absorb,axiom,
% 2.65/1.52 ! [X: a,Y: a] :
% 2.65/1.52 ( ( sup @ X @ ( inf @ X @ Y ) )
% 2.65/1.52 = X ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_64_local_Oa__join__d,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( sup @ A2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.65/1.52 = ( sup @ A2 @ ( inf @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_65_local_Ob__join__d,axiom,
% 2.65/1.52 ! [B2: a,A2: a,C: a] :
% 2.65/1.52 ( ( sup @ B2 @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.65/1.52 = ( sup @ B2 @ ( inf @ C @ A2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_66_local_Od__aux__def,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( sup @ ( sup @ ( inf @ A2 @ B2 ) @ ( inf @ B2 @ C ) ) @ ( inf @ C @ A2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_67_local_Od__b__c__a,axiom,
% 2.65/1.52 ! [B2: a,C: a,A2: a] :
% 2.65/1.52 ( ( modula1221160330_d_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 2.65/1.52 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_68_local_Od__c__a__b,axiom,
% 2.65/1.52 ! [C: a,A2: a,B2: a] :
% 2.65/1.52 ( ( modula1221160330_d_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 2.65/1.52 = ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_69_local_Oa__meet__e,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.65/1.52 = ( inf @ A2 @ ( sup @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_70_local_Ob__meet__e,axiom,
% 2.65/1.52 ! [B2: a,A2: a,C: a] :
% 2.65/1.52 ( ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.65/1.52 = ( inf @ B2 @ ( sup @ C @ A2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_71_local_Oc__meet__e,axiom,
% 2.65/1.52 ! [C: a,A2: a,B2: a] :
% 2.65/1.52 ( ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) )
% 2.65/1.52 = ( inf @ C @ ( sup @ A2 @ B2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_72_local_Oe__aux__def,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( inf @ ( inf @ ( sup @ A2 @ B2 ) @ ( sup @ B2 @ C ) ) @ ( sup @ C @ A2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_73_local_Oe__b__c__a,axiom,
% 2.65/1.52 ! [B2: a,C: a,A2: a] :
% 2.65/1.52 ( ( modula1593167561_e_aux @ a @ inf @ sup @ B2 @ C @ A2 )
% 2.65/1.52 = ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_74_local_Oe__c__a__b,axiom,
% 2.65/1.52 ! [C: a,A2: a,B2: a] :
% 2.65/1.52 ( ( modula1593167561_e_aux @ a @ inf @ sup @ C @ A2 @ B2 )
% 2.65/1.52 = ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_75_local_Ocomp__fun__idem__sup,axiom,
% 2.65/1.52 finite_comp_fun_idem @ a @ a @ sup ).
% 2.65/1.52
% 2.65/1.52 thf(fact_76_local_Ocomp__fun__idem__inf,axiom,
% 2.65/1.52 finite_comp_fun_idem @ a @ a @ inf ).
% 2.65/1.52
% 2.65/1.52 thf(fact_77_local_Oc__aux__def,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( sup @ ( inf @ C @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_78_local_Ob__aux__def,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( sup @ ( inf @ B2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_79_local_Oa__aux__def,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula105138637_a_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( sup @ ( inf @ A2 @ ( modula1593167561_e_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) @ ( modula1221160330_d_aux @ a @ inf @ sup @ A2 @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_80_lattice_Ono__distrib_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( modula141148890istrib @ A )
% 2.65/1.52 = ( modula141148890istrib @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_81_lattice_ON5__lattice_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( modula1192512645attice @ A )
% 2.65/1.52 = ( modula1192512645attice @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_82_lattice_OM5__lattice_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( modula1618706180attice @ A )
% 2.65/1.52 = ( modula1618706180attice @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_83_lattice_Oe__aux_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( modula1593167561_e_aux @ A )
% 2.65/1.52 = ( modula1593167561_e_aux @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_84_lattice_Od__aux_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( modula1221160330_d_aux @ A )
% 2.65/1.52 = ( modula1221160330_d_aux @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_85_local_Oc__a,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula849153099_c_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( modula105138637_a_aux @ a @ inf @ sup @ C @ A2 @ B2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_86_local_Ob__a,axiom,
% 2.65/1.52 ! [A2: a,B2: a,C: a] :
% 2.65/1.52 ( ( modula477145868_b_aux @ a @ inf @ sup @ A2 @ B2 @ C )
% 2.65/1.52 = ( modula105138637_a_aux @ a @ inf @ sup @ B2 @ C @ A2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_87_local_Osup_Osemigroup__axioms,axiom,
% 2.65/1.52 semigroup @ a @ sup ).
% 2.65/1.52
% 2.65/1.52 thf(fact_88_local_Oinf_Osemigroup__axioms,axiom,
% 2.65/1.52 semigroup @ a @ inf ).
% 2.65/1.52
% 2.65/1.52 thf(fact_89_local_Osup_Osemilattice__axioms,axiom,
% 2.65/1.52 semilattice @ a @ sup ).
% 2.65/1.52
% 2.65/1.52 thf(fact_90_local_Oinf_Osemilattice__axioms,axiom,
% 2.65/1.52 semilattice @ a @ inf ).
% 2.65/1.52
% 2.65/1.52 thf(fact_91_lattice_Oa__aux_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( modula105138637_a_aux @ A )
% 2.65/1.52 = ( modula105138637_a_aux @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_92_lattice_Ob__aux_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( modula477145868_b_aux @ A )
% 2.65/1.52 = ( modula477145868_b_aux @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_93_lattice_Oc__aux_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( modula849153099_c_aux @ A )
% 2.65/1.52 = ( modula849153099_c_aux @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_94_local_Osup_Oabel__semigroup__axioms,axiom,
% 2.65/1.52 abel_semigroup @ a @ sup ).
% 2.65/1.52
% 2.65/1.52 thf(fact_95_local_Oinf_Oabel__semigroup__axioms,axiom,
% 2.65/1.52 abel_semigroup @ a @ inf ).
% 2.65/1.52
% 2.65/1.52 thf(fact_96_local_OSup__fin_Osemilattice__set__axioms,axiom,
% 2.65/1.52 lattic35693393ce_set @ a @ sup ).
% 2.65/1.52
% 2.65/1.52 thf(fact_97_local_OInf__fin_Osemilattice__set__axioms,axiom,
% 2.65/1.52 lattic35693393ce_set @ a @ inf ).
% 2.65/1.52
% 2.65/1.52 thf(fact_98_local_Olexordp__eq__refl,axiom,
% 2.65/1.52 ! [Xs: list @ a] : ( lexordp_eq @ a @ less @ Xs @ Xs ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_99_local_Olexordp__eq_ONil,axiom,
% 2.65/1.52 ! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_100_abel__semigroup_Oaxioms_I1_J,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ( abel_semigroup @ A @ F )
% 2.65/1.52 => ( semigroup @ A @ F ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_101_semilattice__set_Oaxioms,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ( lattic35693393ce_set @ A @ F )
% 2.65/1.52 => ( semilattice @ A @ F ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_102_semilattice__set_Ointro,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ( semilattice @ A @ F )
% 2.65/1.52 => ( lattic35693393ce_set @ A @ F ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_103_semilattice__set__def,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( lattic35693393ce_set @ A )
% 2.65/1.52 = ( semilattice @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_104_local_Olexordp__eq__simps_I1_J,axiom,
% 2.65/1.52 ! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_105_local_Olexordp__eq__simps_I2_J,axiom,
% 2.65/1.52 ! [Xs: list @ a] :
% 2.65/1.52 ( ( lexordp_eq @ a @ less @ Xs @ ( nil @ a ) )
% 2.65/1.52 = ( Xs
% 2.65/1.52 = ( nil @ a ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_106_abel__semigroup_Oleft__commute,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A,B2: A,A2: A,C: A] :
% 2.65/1.52 ( ( abel_semigroup @ A @ F )
% 2.65/1.52 => ( ( F @ B2 @ ( F @ A2 @ C ) )
% 2.65/1.52 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_107_abel__semigroup_Ocommute,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 2.65/1.52 ( ( abel_semigroup @ A @ F )
% 2.65/1.52 => ( ( F @ A2 @ B2 )
% 2.65/1.52 = ( F @ B2 @ A2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_108_semigroup_Ointro,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ! [A3: A,B3: A,C2: A] :
% 2.65/1.52 ( ( F @ ( F @ A3 @ B3 ) @ C2 )
% 2.65/1.52 = ( F @ A3 @ ( F @ B3 @ C2 ) ) )
% 2.65/1.52 => ( semigroup @ A @ F ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_109_semigroup_Oassoc,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
% 2.65/1.52 ( ( semigroup @ A @ F )
% 2.65/1.52 => ( ( F @ ( F @ A2 @ B2 ) @ C )
% 2.65/1.52 = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_110_semigroup__def,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( semigroup @ A )
% 2.65/1.52 = ( ^ [F2: A > A > A] :
% 2.65/1.52 ! [A4: A,B4: A,C3: A] :
% 2.65/1.52 ( ( F2 @ ( F2 @ A4 @ B4 ) @ C3 )
% 2.65/1.52 = ( F2 @ A4 @ ( F2 @ B4 @ C3 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_111_ord_Ois__arg__min_Ocong,axiom,
% 2.65/1.52 ! [B: $tType,A: $tType] :
% 2.65/1.52 ( ( lattices_is_arg_min @ A @ B )
% 2.65/1.52 = ( lattices_is_arg_min @ A @ B ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_112_ord_Ois__arg__max_Ocong,axiom,
% 2.65/1.52 ! [B: $tType,A: $tType] :
% 2.65/1.52 ( ( lattices_is_arg_max @ A @ B )
% 2.65/1.52 = ( lattices_is_arg_max @ A @ B ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_113_ord_Ois__arg__min__def,axiom,
% 2.65/1.52 ! [B: $tType,A: $tType] :
% 2.65/1.52 ( ( lattices_is_arg_min @ A @ B )
% 2.65/1.52 = ( ^ [Less: A > A > $o,F2: B > A,P2: B > $o,X3: B] :
% 2.65/1.52 ( ( P2 @ X3 )
% 2.65/1.52 & ~ ? [Y3: B] :
% 2.65/1.52 ( ( P2 @ Y3 )
% 2.65/1.52 & ( Less @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_114_ord_Ois__arg__max__def,axiom,
% 2.65/1.52 ! [B: $tType,A: $tType] :
% 2.65/1.52 ( ( lattices_is_arg_max @ A @ B )
% 2.65/1.52 = ( ^ [Less: A > A > $o,F2: B > A,P2: B > $o,X3: B] :
% 2.65/1.52 ( ( P2 @ X3 )
% 2.65/1.52 & ~ ? [Y3: B] :
% 2.65/1.52 ( ( P2 @ Y3 )
% 2.65/1.52 & ( Less @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_115_ord_Olexordp__eq__simps_I2_J,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
% 2.65/1.52 ( ( lexordp_eq @ A @ Less2 @ Xs @ ( nil @ A ) )
% 2.65/1.52 = ( Xs
% 2.65/1.52 = ( nil @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_116_ord_Olexordp__eq__simps_I1_J,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less2 @ ( nil @ A ) @ Ys ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_117_local_Olexordp__eq_Osimps,axiom,
% 2.65/1.52 ! [A1: list @ a,A22: list @ a] :
% 2.65/1.52 ( ( lexordp_eq @ a @ less @ A1 @ A22 )
% 2.65/1.52 = ( ? [Ys2: list @ a] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( nil @ a ) )
% 2.65/1.52 & ( A22 = Ys2 ) )
% 2.65/1.52 | ? [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( cons @ a @ X3 @ Xs2 ) )
% 2.65/1.52 & ( A22
% 2.65/1.52 = ( cons @ a @ Y3 @ Ys2 ) )
% 2.65/1.52 & ( less @ X3 @ Y3 ) )
% 2.65/1.52 | ? [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( cons @ a @ X3 @ Xs2 ) )
% 2.65/1.52 & ( A22
% 2.65/1.52 = ( cons @ a @ Y3 @ Ys2 ) )
% 2.65/1.52 & ~ ( less @ X3 @ Y3 )
% 2.65/1.52 & ~ ( less @ Y3 @ X3 )
% 2.65/1.52 & ( lexordp_eq @ a @ less @ Xs2 @ Ys2 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_118_local_Olexordp__eq_Oinducts,axiom,
% 2.65/1.52 ! [X1: list @ a,X22: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
% 2.65/1.52 ( ( lexordp_eq @ a @ less @ X1 @ X22 )
% 2.65/1.52 => ( ! [X_1: list @ a] : ( P @ ( nil @ a ) @ X_1 )
% 2.65/1.52 => ( ! [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
% 2.65/1.52 ( ( less @ X2 @ Y2 )
% 2.65/1.52 => ( P @ ( cons @ a @ X2 @ Xs3 ) @ ( cons @ a @ Y2 @ Ys3 ) ) )
% 2.65/1.52 => ( ! [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
% 2.65/1.52 ( ~ ( less @ X2 @ Y2 )
% 2.65/1.52 => ( ~ ( less @ Y2 @ X2 )
% 2.65/1.52 => ( ( lexordp_eq @ a @ less @ Xs3 @ Ys3 )
% 2.65/1.52 => ( ( P @ Xs3 @ Ys3 )
% 2.65/1.52 => ( P @ ( cons @ a @ X2 @ Xs3 ) @ ( cons @ a @ Y2 @ Ys3 ) ) ) ) ) )
% 2.65/1.52 => ( P @ X1 @ X22 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_119_local_Olexordp__eq_Ocases,axiom,
% 2.65/1.52 ! [A1: list @ a,A22: list @ a] :
% 2.65/1.52 ( ( lexordp_eq @ a @ less @ A1 @ A22 )
% 2.65/1.52 => ( ( A1
% 2.65/1.52 != ( nil @ a ) )
% 2.65/1.52 => ( ! [X2: a] :
% 2.65/1.52 ( ? [Xs3: list @ a] :
% 2.65/1.52 ( A1
% 2.65/1.52 = ( cons @ a @ X2 @ Xs3 ) )
% 2.65/1.52 => ! [Y2: a] :
% 2.65/1.52 ( ? [Ys3: list @ a] :
% 2.65/1.52 ( A22
% 2.65/1.52 = ( cons @ a @ Y2 @ Ys3 ) )
% 2.65/1.52 => ~ ( less @ X2 @ Y2 ) ) )
% 2.65/1.52 => ~ ! [X2: a,Y2: a,Xs3: list @ a] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( cons @ a @ X2 @ Xs3 ) )
% 2.65/1.52 => ! [Ys3: list @ a] :
% 2.65/1.52 ( ( A22
% 2.65/1.52 = ( cons @ a @ Y2 @ Ys3 ) )
% 2.65/1.52 => ( ~ ( less @ X2 @ Y2 )
% 2.65/1.52 => ( ~ ( less @ Y2 @ X2 )
% 2.65/1.52 => ~ ( lexordp_eq @ a @ less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_120_local_Olexordp__eq_OCons__eq,axiom,
% 2.65/1.52 ! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
% 2.65/1.52 ( ~ ( less @ X @ Y )
% 2.65/1.52 => ( ~ ( less @ Y @ X )
% 2.65/1.52 => ( ( lexordp_eq @ a @ less @ Xs @ Ys )
% 2.65/1.52 => ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_121_local_Olexordp__eq_OCons,axiom,
% 2.65/1.52 ! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_122_list_Oinject,axiom,
% 2.65/1.52 ! [A: $tType,X21: A,X222: list @ A,Y21: A,Y22: list @ A] :
% 2.65/1.52 ( ( ( cons @ A @ X21 @ X222 )
% 2.65/1.52 = ( cons @ A @ Y21 @ Y22 ) )
% 2.65/1.52 = ( ( X21 = Y21 )
% 2.65/1.52 & ( X222 = Y22 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_123_ord_Olexordp__eq__simps_I4_J,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
% 2.65/1.52 ( ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
% 2.65/1.52 = ( ( Less2 @ X @ Y )
% 2.65/1.52 | ( ~ ( Less2 @ Y @ X )
% 2.65/1.52 & ( lexordp_eq @ A @ Less2 @ Xs @ Ys ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_124_ord_Olexordp__eq__simps_I3_J,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X: A,Xs: list @ A] :
% 2.65/1.52 ~ ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_125_local_Olexordp__eq__simps_I4_J,axiom,
% 2.65/1.52 ! [X: a,Xs: list @ a,Y: a,Ys: list @ a] :
% 2.65/1.52 ( ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) )
% 2.65/1.52 = ( ( less @ X @ Y )
% 2.65/1.52 | ( ~ ( less @ Y @ X )
% 2.65/1.52 & ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_126_local_Olexordp__eq__simps_I3_J,axiom,
% 2.65/1.52 ! [X: a,Xs: list @ a] :
% 2.65/1.52 ~ ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( nil @ a ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_127_strict__sorted_Oinduct,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( linorder @ A )
% 2.65/1.52 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 2.65/1.52 ( ( P @ ( nil @ A ) )
% 2.65/1.52 => ( ! [X2: A,Ys3: list @ A] :
% 2.65/1.52 ( ( P @ Ys3 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Ys3 ) ) )
% 2.65/1.52 => ( P @ A0 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_128_strict__sorted_Ocases,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( linorder @ A )
% 2.65/1.52 => ! [X: list @ A] :
% 2.65/1.52 ( ( X
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ~ ! [X2: A,Ys3: list @ A] :
% 2.65/1.52 ( X
% 2.65/1.52 != ( cons @ A @ X2 @ Ys3 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_129_map__tailrec__rev_Oinduct,axiom,
% 2.65/1.52 ! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
% 2.65/1.52 ( ! [F3: A > B,X_1: list @ B] : ( P @ F3 @ ( nil @ A ) @ X_1 )
% 2.65/1.52 => ( ! [F3: A > B,A3: A,As: list @ A,Bs: list @ B] :
% 2.65/1.52 ( ( P @ F3 @ As @ ( cons @ B @ ( F3 @ A3 ) @ Bs ) )
% 2.65/1.52 => ( P @ F3 @ ( cons @ A @ A3 @ As ) @ Bs ) )
% 2.65/1.52 => ( P @ A0 @ A1 @ A22 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_130_list__nonempty__induct,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 2.65/1.52 ( ( Xs
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.65/1.52 => ( ! [X2: A,Xs3: list @ A] :
% 2.65/1.52 ( ( Xs3
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ( P @ Xs3 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) )
% 2.65/1.52 => ( P @ Xs ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_131_successively_Oinduct,axiom,
% 2.65/1.52 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 2.65/1.52 ( ! [P3: A > A > $o] : ( P @ P3 @ ( nil @ A ) )
% 2.65/1.52 => ( ! [P3: A > A > $o,X2: A] : ( P @ P3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.65/1.52 => ( ! [P3: A > A > $o,X2: A,Y2: A,Xs3: list @ A] :
% 2.65/1.52 ( ( P @ P3 @ ( cons @ A @ Y2 @ Xs3 ) )
% 2.65/1.52 => ( P @ P3 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) )
% 2.65/1.52 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_132_arg__min__list_Oinduct,axiom,
% 2.65/1.52 ! [B: $tType,A: $tType] :
% 2.65/1.52 ( ( linorder @ B )
% 2.65/1.52 => ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
% 2.65/1.52 ( ! [F3: A > B,X2: A] : ( P @ F3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.65/1.52 => ( ! [F3: A > B,X2: A,Y2: A,Zs: list @ A] :
% 2.65/1.52 ( ( P @ F3 @ ( cons @ A @ Y2 @ Zs ) )
% 2.65/1.52 => ( P @ F3 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs ) ) ) )
% 2.65/1.52 => ( ! [A3: A > B] : ( P @ A3 @ ( nil @ A ) )
% 2.65/1.52 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_133_remdups__adj_Oinduct,axiom,
% 2.65/1.52 ! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
% 2.65/1.52 ( ( P @ ( nil @ A ) )
% 2.65/1.52 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.65/1.52 => ( ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.65/1.52 ( ( ( X2 = Y2 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 2.65/1.52 => ( ( ( X2 != Y2 )
% 2.65/1.52 => ( P @ ( cons @ A @ Y2 @ Xs3 ) ) )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
% 2.65/1.52 => ( P @ A0 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_134_sorted__wrt_Oinduct,axiom,
% 2.65/1.52 ! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
% 2.65/1.52 ( ! [P3: A > A > $o] : ( P @ P3 @ ( nil @ A ) )
% 2.65/1.52 => ( ! [P3: A > A > $o,X2: A,Ys3: list @ A] :
% 2.65/1.52 ( ( P @ P3 @ Ys3 )
% 2.65/1.52 => ( P @ P3 @ ( cons @ A @ X2 @ Ys3 ) ) )
% 2.65/1.52 => ( P @ A0 @ A1 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_135_remdups__adj_Ocases,axiom,
% 2.65/1.52 ! [A: $tType,X: list @ A] :
% 2.65/1.52 ( ( X
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ! [X2: A] :
% 2.65/1.52 ( X
% 2.65/1.52 != ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.65/1.52 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.65/1.52 ( X
% 2.65/1.52 != ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_136_transpose_Ocases,axiom,
% 2.65/1.52 ! [A: $tType,X: list @ ( list @ A )] :
% 2.65/1.52 ( ( X
% 2.65/1.52 != ( nil @ ( list @ A ) ) )
% 2.65/1.52 => ( ! [Xss: list @ ( list @ A )] :
% 2.65/1.52 ( X
% 2.65/1.52 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 2.65/1.52 => ~ ! [X2: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
% 2.65/1.52 ( X
% 2.65/1.52 != ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_137_shuffles_Oinduct,axiom,
% 2.65/1.52 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.65/1.52 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 2.65/1.52 => ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
% 2.65/1.52 => ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
% 2.65/1.52 ( ( P @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) )
% 2.65/1.52 => ( ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) )
% 2.65/1.52 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_138_min__list_Oinduct,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( ord @ A )
% 2.65/1.52 => ! [P: ( list @ A ) > $o,A0: list @ A] :
% 2.65/1.52 ( ! [X2: A,Xs3: list @ A] :
% 2.65/1.52 ( ! [X212: A,X223: list @ A] :
% 2.65/1.52 ( ( Xs3
% 2.65/1.52 = ( cons @ A @ X212 @ X223 ) )
% 2.65/1.52 => ( P @ Xs3 ) )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
% 2.65/1.52 => ( ( P @ ( nil @ A ) )
% 2.65/1.52 => ( P @ A0 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_139_min__list_Ocases,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( ord @ A )
% 2.65/1.52 => ! [X: list @ A] :
% 2.65/1.52 ( ! [X2: A,Xs3: list @ A] :
% 2.65/1.52 ( X
% 2.65/1.52 != ( cons @ A @ X2 @ Xs3 ) )
% 2.65/1.52 => ( X
% 2.65/1.52 = ( nil @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_140_induct__list012,axiom,
% 2.65/1.52 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.65/1.52 ( ( P @ ( nil @ A ) )
% 2.65/1.52 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.65/1.52 => ( ! [X2: A,Y2: A,Zs: list @ A] :
% 2.65/1.52 ( ( P @ Zs )
% 2.65/1.52 => ( ( P @ ( cons @ A @ Y2 @ Zs ) )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs ) ) ) ) )
% 2.65/1.52 => ( P @ Xs ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_141_splice_Oinduct,axiom,
% 2.65/1.52 ! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
% 2.65/1.52 ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 2.65/1.52 => ( ! [X2: A,Xs3: list @ A,Ys3: list @ A] :
% 2.65/1.52 ( ( P @ Ys3 @ Xs3 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) )
% 2.65/1.52 => ( P @ A0 @ A1 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_142_list__induct2_H,axiom,
% 2.65/1.52 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
% 2.65/1.52 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 2.65/1.52 => ( ! [X2: A,Xs3: list @ A] : ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( nil @ B ) )
% 2.65/1.52 => ( ! [Y2: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys3 ) )
% 2.65/1.52 => ( ! [X2: A,Xs3: list @ A,Y2: B,Ys3: list @ B] :
% 2.65/1.52 ( ( P @ Xs3 @ Ys3 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B @ Y2 @ Ys3 ) ) )
% 2.65/1.52 => ( P @ Xs @ Ys ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_143_neq__Nil__conv,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A] :
% 2.65/1.52 ( ( Xs
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 = ( ? [Y3: A,Ys2: list @ A] :
% 2.65/1.52 ( Xs
% 2.65/1.52 = ( cons @ A @ Y3 @ Ys2 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_144_list_Oinducts,axiom,
% 2.65/1.52 ! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
% 2.65/1.52 ( ( P @ ( nil @ A ) )
% 2.65/1.52 => ( ! [X12: A,X23: list @ A] :
% 2.65/1.52 ( ( P @ X23 )
% 2.65/1.52 => ( P @ ( cons @ A @ X12 @ X23 ) ) )
% 2.65/1.52 => ( P @ List ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_145_list_Oexhaust,axiom,
% 2.65/1.52 ! [A: $tType,Y: list @ A] :
% 2.65/1.52 ( ( Y
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ~ ! [X213: A,X224: list @ A] :
% 2.65/1.52 ( Y
% 2.65/1.52 != ( cons @ A @ X213 @ X224 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_146_list_OdiscI,axiom,
% 2.65/1.52 ! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
% 2.65/1.52 ( ( List
% 2.65/1.52 = ( cons @ A @ X21 @ X222 ) )
% 2.65/1.52 => ( List
% 2.65/1.52 != ( nil @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_147_list_Odistinct_I1_J,axiom,
% 2.65/1.52 ! [A: $tType,X21: A,X222: list @ A] :
% 2.65/1.52 ( ( nil @ A )
% 2.65/1.52 != ( cons @ A @ X21 @ X222 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_148_not__Cons__self2,axiom,
% 2.65/1.52 ! [A: $tType,X: A,Xs: list @ A] :
% 2.65/1.52 ( ( cons @ A @ X @ Xs )
% 2.65/1.52 != Xs ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_149_ord_Olexordp__eq_OCons__eq,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ~ ( Less2 @ X @ Y )
% 2.65/1.52 => ( ~ ( Less2 @ Y @ X )
% 2.65/1.52 => ( ( lexordp_eq @ A @ Less2 @ Xs @ Ys )
% 2.65/1.52 => ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_150_ord_Olexordp__eq_OCons,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( Less2 @ X @ Y )
% 2.65/1.52 => ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_151_ord_Olexordp__eq_Oinducts,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X1: list @ A,X22: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 2.65/1.52 ( ( lexordp_eq @ A @ Less2 @ X1 @ X22 )
% 2.65/1.52 => ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
% 2.65/1.52 => ( ! [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
% 2.65/1.52 ( ( Less2 @ X2 @ Y2 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 2.65/1.52 => ( ! [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
% 2.65/1.52 ( ~ ( Less2 @ X2 @ Y2 )
% 2.65/1.52 => ( ~ ( Less2 @ Y2 @ X2 )
% 2.65/1.52 => ( ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 )
% 2.65/1.52 => ( ( P @ Xs3 @ Ys3 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) )
% 2.65/1.52 => ( P @ X1 @ X22 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_152_ord_Olexordp__eq_Osimps,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( lexordp_eq @ A )
% 2.65/1.52 = ( ^ [Less: A > A > $o,A12: list @ A,A23: list @ A] :
% 2.65/1.52 ( ? [Ys2: list @ A] :
% 2.65/1.52 ( ( A12
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 & ( A23 = Ys2 ) )
% 2.65/1.52 | ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
% 2.65/1.52 ( ( A12
% 2.65/1.52 = ( cons @ A @ X3 @ Xs2 ) )
% 2.65/1.52 & ( A23
% 2.65/1.52 = ( cons @ A @ Y3 @ Ys2 ) )
% 2.65/1.52 & ( Less @ X3 @ Y3 ) )
% 2.65/1.52 | ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
% 2.65/1.52 ( ( A12
% 2.65/1.52 = ( cons @ A @ X3 @ Xs2 ) )
% 2.65/1.52 & ( A23
% 2.65/1.52 = ( cons @ A @ Y3 @ Ys2 ) )
% 2.65/1.52 & ~ ( Less @ X3 @ Y3 )
% 2.65/1.52 & ~ ( Less @ Y3 @ X3 )
% 2.65/1.52 & ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_153_ord_Olexordp__eq_Ocases,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,A1: list @ A,A22: list @ A] :
% 2.65/1.52 ( ( lexordp_eq @ A @ Less2 @ A1 @ A22 )
% 2.65/1.52 => ( ( A1
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ! [X2: A] :
% 2.65/1.52 ( ? [Xs3: list @ A] :
% 2.65/1.52 ( A1
% 2.65/1.52 = ( cons @ A @ X2 @ Xs3 ) )
% 2.65/1.52 => ! [Y2: A] :
% 2.65/1.52 ( ? [Ys3: list @ A] :
% 2.65/1.52 ( A22
% 2.65/1.52 = ( cons @ A @ Y2 @ Ys3 ) )
% 2.65/1.52 => ~ ( Less2 @ X2 @ Y2 ) ) )
% 2.65/1.52 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( cons @ A @ X2 @ Xs3 ) )
% 2.65/1.52 => ! [Ys3: list @ A] :
% 2.65/1.52 ( ( A22
% 2.65/1.52 = ( cons @ A @ Y2 @ Ys3 ) )
% 2.65/1.52 => ( ~ ( Less2 @ X2 @ Y2 )
% 2.65/1.52 => ( ~ ( Less2 @ Y2 @ X2 )
% 2.65/1.52 => ~ ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_154_ord_Olexordp__eq_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( lexordp_eq @ A )
% 2.65/1.52 = ( lexordp_eq @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_155_ord_Olexordp__eq__refl,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less2 @ Xs @ Xs ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_156_ord_Olexordp__eq_ONil,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less2 @ ( nil @ A ) @ Ys ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_157_local_Olexordp_ONil,axiom,
% 2.65/1.52 ! [Y: a,Ys: list @ a] : ( lexordp @ a @ less @ ( nil @ a ) @ ( cons @ a @ Y @ Ys ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_158_local_Olexordp_Ocases,axiom,
% 2.65/1.52 ! [A1: list @ a,A22: list @ a] :
% 2.65/1.52 ( ( lexordp @ a @ less @ A1 @ A22 )
% 2.65/1.52 => ( ( ( A1
% 2.65/1.52 = ( nil @ a ) )
% 2.65/1.52 => ! [Y2: a,Ys3: list @ a] :
% 2.65/1.52 ( A22
% 2.65/1.52 != ( cons @ a @ Y2 @ Ys3 ) ) )
% 2.65/1.52 => ( ! [X2: a] :
% 2.65/1.52 ( ? [Xs3: list @ a] :
% 2.65/1.52 ( A1
% 2.65/1.52 = ( cons @ a @ X2 @ Xs3 ) )
% 2.65/1.52 => ! [Y2: a] :
% 2.65/1.52 ( ? [Ys3: list @ a] :
% 2.65/1.52 ( A22
% 2.65/1.52 = ( cons @ a @ Y2 @ Ys3 ) )
% 2.65/1.52 => ~ ( less @ X2 @ Y2 ) ) )
% 2.65/1.52 => ~ ! [X2: a,Y2: a,Xs3: list @ a] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( cons @ a @ X2 @ Xs3 ) )
% 2.65/1.52 => ! [Ys3: list @ a] :
% 2.65/1.52 ( ( A22
% 2.65/1.52 = ( cons @ a @ Y2 @ Ys3 ) )
% 2.65/1.52 => ( ~ ( less @ X2 @ Y2 )
% 2.65/1.52 => ( ~ ( less @ Y2 @ X2 )
% 2.65/1.52 => ~ ( lexordp @ a @ less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_159_local_Olexordp_Oinducts,axiom,
% 2.65/1.52 ! [X1: list @ a,X22: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
% 2.65/1.52 ( ( lexordp @ a @ less @ X1 @ X22 )
% 2.65/1.52 => ( ! [Y2: a,Ys3: list @ a] : ( P @ ( nil @ a ) @ ( cons @ a @ Y2 @ Ys3 ) )
% 2.65/1.52 => ( ! [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
% 2.65/1.52 ( ( less @ X2 @ Y2 )
% 2.65/1.52 => ( P @ ( cons @ a @ X2 @ Xs3 ) @ ( cons @ a @ Y2 @ Ys3 ) ) )
% 2.65/1.52 => ( ! [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
% 2.65/1.52 ( ~ ( less @ X2 @ Y2 )
% 2.65/1.52 => ( ~ ( less @ Y2 @ X2 )
% 2.65/1.52 => ( ( lexordp @ a @ less @ Xs3 @ Ys3 )
% 2.65/1.52 => ( ( P @ Xs3 @ Ys3 )
% 2.65/1.52 => ( P @ ( cons @ a @ X2 @ Xs3 ) @ ( cons @ a @ Y2 @ Ys3 ) ) ) ) ) )
% 2.65/1.52 => ( P @ X1 @ X22 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_160_local_Olexordp_Osimps,axiom,
% 2.65/1.52 ! [A1: list @ a,A22: list @ a] :
% 2.65/1.52 ( ( lexordp @ a @ less @ A1 @ A22 )
% 2.65/1.52 = ( ? [Y3: a,Ys2: list @ a] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( nil @ a ) )
% 2.65/1.52 & ( A22
% 2.65/1.52 = ( cons @ a @ Y3 @ Ys2 ) ) )
% 2.65/1.52 | ? [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( cons @ a @ X3 @ Xs2 ) )
% 2.65/1.52 & ( A22
% 2.65/1.52 = ( cons @ a @ Y3 @ Ys2 ) )
% 2.65/1.52 & ( less @ X3 @ Y3 ) )
% 2.65/1.52 | ? [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( cons @ a @ X3 @ Xs2 ) )
% 2.65/1.52 & ( A22
% 2.65/1.52 = ( cons @ a @ Y3 @ Ys2 ) )
% 2.65/1.52 & ~ ( less @ X3 @ Y3 )
% 2.65/1.52 & ~ ( less @ Y3 @ X3 )
% 2.65/1.52 & ( lexordp @ a @ less @ Xs2 @ Ys2 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_161_local_Olexordp__antisym,axiom,
% 2.65/1.52 ! [Xs: list @ a,Ys: list @ a] :
% 2.65/1.52 ( ( lexordp @ a @ less @ Xs @ Ys )
% 2.65/1.52 => ~ ( lexordp @ a @ less @ Ys @ Xs ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_162_local_Olexordp__irreflexive,axiom,
% 2.65/1.52 ! [Xs: list @ a] :
% 2.65/1.52 ( ! [X2: a] :
% 2.65/1.52 ~ ( less @ X2 @ X2 )
% 2.65/1.52 => ~ ( lexordp @ a @ less @ Xs @ Xs ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_163_local_Olexordp__irreflexive_H,axiom,
% 2.65/1.52 ! [Xs: list @ a] :
% 2.65/1.52 ~ ( lexordp @ a @ less @ Xs @ Xs ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_164_local_Olexordp_OCons__eq,axiom,
% 2.65/1.52 ! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
% 2.65/1.52 ( ~ ( less @ X @ Y )
% 2.65/1.52 => ( ~ ( less @ Y @ X )
% 2.65/1.52 => ( ( lexordp @ a @ less @ Xs @ Ys )
% 2.65/1.52 => ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_165_local_Olexordp_OCons,axiom,
% 2.65/1.52 ! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_166_local_Olexordp__into__lexordp__eq,axiom,
% 2.65/1.52 ! [Xs: list @ a,Ys: list @ a] :
% 2.65/1.52 ( ( lexordp @ a @ less @ Xs @ Ys )
% 2.65/1.52 => ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_167_ord_Olexordp__simps_I3_J,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
% 2.65/1.52 ( ( lexordp @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
% 2.65/1.52 = ( ( Less2 @ X @ Y )
% 2.65/1.52 | ( ~ ( Less2 @ Y @ X )
% 2.65/1.52 & ( lexordp @ A @ Less2 @ Xs @ Ys ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_168_ord_Olexordp__simps_I2_J,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
% 2.65/1.52 ~ ( lexordp @ A @ Less2 @ Xs @ ( nil @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_169_ord_Olexordp__simps_I1_J,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Ys: list @ A] :
% 2.65/1.52 ( ( lexordp @ A @ Less2 @ ( nil @ A ) @ Ys )
% 2.65/1.52 = ( Ys
% 2.65/1.52 != ( nil @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_170_local_Olexordp__simps_I3_J,axiom,
% 2.65/1.52 ! [X: a,Xs: list @ a,Y: a,Ys: list @ a] :
% 2.65/1.52 ( ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) )
% 2.65/1.52 = ( ( less @ X @ Y )
% 2.65/1.52 | ( ~ ( less @ Y @ X )
% 2.65/1.52 & ( lexordp @ a @ less @ Xs @ Ys ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_171_local_Olexordp__simps_I1_J,axiom,
% 2.65/1.52 ! [Ys: list @ a] :
% 2.65/1.52 ( ( lexordp @ a @ less @ ( nil @ a ) @ Ys )
% 2.65/1.52 = ( Ys
% 2.65/1.52 != ( nil @ a ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_172_local_Olexordp__simps_I2_J,axiom,
% 2.65/1.52 ! [Xs: list @ a] :
% 2.65/1.52 ~ ( lexordp @ a @ less @ Xs @ ( nil @ a ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_173_ord_Olexordp_Ocong,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( lexordp @ A )
% 2.65/1.52 = ( lexordp @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_174_ord_Olexordp__irreflexive,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
% 2.65/1.52 ( ! [X2: A] :
% 2.65/1.52 ~ ( Less2 @ X2 @ X2 )
% 2.65/1.52 => ~ ( lexordp @ A @ Less2 @ Xs @ Xs ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_175_ord_Olexordp_OCons__eq,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ~ ( Less2 @ X @ Y )
% 2.65/1.52 => ( ~ ( Less2 @ Y @ X )
% 2.65/1.52 => ( ( lexordp @ A @ Less2 @ Xs @ Ys )
% 2.65/1.52 => ( lexordp @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_176_ord_Olexordp_OCons,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( Less2 @ X @ Y )
% 2.65/1.52 => ( lexordp @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_177_ord_Olexordp__into__lexordp__eq,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( lexordp @ A @ Less2 @ Xs @ Ys )
% 2.65/1.52 => ( lexordp_eq @ A @ Less2 @ Xs @ Ys ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_178_ord_Olexordp_ONil,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Y: A,Ys: list @ A] : ( lexordp @ A @ Less2 @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_179_ord_Olexordp_Ocases,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,A1: list @ A,A22: list @ A] :
% 2.65/1.52 ( ( lexordp @ A @ Less2 @ A1 @ A22 )
% 2.65/1.52 => ( ( ( A1
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 => ! [Y2: A,Ys3: list @ A] :
% 2.65/1.52 ( A22
% 2.65/1.52 != ( cons @ A @ Y2 @ Ys3 ) ) )
% 2.65/1.52 => ( ! [X2: A] :
% 2.65/1.52 ( ? [Xs3: list @ A] :
% 2.65/1.52 ( A1
% 2.65/1.52 = ( cons @ A @ X2 @ Xs3 ) )
% 2.65/1.52 => ! [Y2: A] :
% 2.65/1.52 ( ? [Ys3: list @ A] :
% 2.65/1.52 ( A22
% 2.65/1.52 = ( cons @ A @ Y2 @ Ys3 ) )
% 2.65/1.52 => ~ ( Less2 @ X2 @ Y2 ) ) )
% 2.65/1.52 => ~ ! [X2: A,Y2: A,Xs3: list @ A] :
% 2.65/1.52 ( ( A1
% 2.65/1.52 = ( cons @ A @ X2 @ Xs3 ) )
% 2.65/1.52 => ! [Ys3: list @ A] :
% 2.65/1.52 ( ( A22
% 2.65/1.52 = ( cons @ A @ Y2 @ Ys3 ) )
% 2.65/1.52 => ( ~ ( Less2 @ X2 @ Y2 )
% 2.65/1.52 => ( ~ ( Less2 @ Y2 @ X2 )
% 2.65/1.52 => ~ ( lexordp @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_180_ord_Olexordp_Osimps,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( lexordp @ A )
% 2.65/1.52 = ( ^ [Less: A > A > $o,A12: list @ A,A23: list @ A] :
% 2.65/1.52 ( ? [Y3: A,Ys2: list @ A] :
% 2.65/1.52 ( ( A12
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 & ( A23
% 2.65/1.52 = ( cons @ A @ Y3 @ Ys2 ) ) )
% 2.65/1.52 | ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
% 2.65/1.52 ( ( A12
% 2.65/1.52 = ( cons @ A @ X3 @ Xs2 ) )
% 2.65/1.52 & ( A23
% 2.65/1.52 = ( cons @ A @ Y3 @ Ys2 ) )
% 2.65/1.52 & ( Less @ X3 @ Y3 ) )
% 2.65/1.52 | ? [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
% 2.65/1.52 ( ( A12
% 2.65/1.52 = ( cons @ A @ X3 @ Xs2 ) )
% 2.65/1.52 & ( A23
% 2.65/1.52 = ( cons @ A @ Y3 @ Ys2 ) )
% 2.65/1.52 & ~ ( Less @ X3 @ Y3 )
% 2.65/1.52 & ~ ( Less @ Y3 @ X3 )
% 2.65/1.52 & ( lexordp @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_181_ord_Olexordp_Oinducts,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X1: list @ A,X22: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 2.65/1.52 ( ( lexordp @ A @ Less2 @ X1 @ X22 )
% 2.65/1.52 => ( ! [Y2: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys3 ) )
% 2.65/1.52 => ( ! [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
% 2.65/1.52 ( ( Less2 @ X2 @ Y2 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
% 2.65/1.52 => ( ! [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
% 2.65/1.52 ( ~ ( Less2 @ X2 @ Y2 )
% 2.65/1.52 => ( ~ ( Less2 @ Y2 @ X2 )
% 2.65/1.52 => ( ( lexordp @ A @ Less2 @ Xs3 @ Ys3 )
% 2.65/1.52 => ( ( P @ Xs3 @ Ys3 )
% 2.65/1.52 => ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) )
% 2.65/1.52 => ( P @ X1 @ X22 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_182_local_Olexordp__append__rightI,axiom,
% 2.65/1.52 ! [Ys: list @ a,Xs: list @ a] :
% 2.65/1.52 ( ( Ys
% 2.65/1.52 != ( nil @ a ) )
% 2.65/1.52 => ( lexordp @ a @ less @ Xs @ ( append @ a @ Xs @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_183_local_Olexordp__append__left__rightI,axiom,
% 2.65/1.52 ! [X: a,Y: a,Us: list @ a,Xs: list @ a,Ys: list @ a] :
% 2.65/1.52 ( ( less @ X @ Y )
% 2.65/1.52 => ( lexordp @ a @ less @ ( append @ a @ Us @ ( cons @ a @ X @ Xs ) ) @ ( append @ a @ Us @ ( cons @ a @ Y @ Ys ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_184_semilattice_Oaxioms_I1_J,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ( semilattice @ A @ F )
% 2.65/1.52 => ( abel_semigroup @ A @ F ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_185_local_Olexordp__append__leftD,axiom,
% 2.65/1.52 ! [Xs: list @ a,Us: list @ a,Vs: list @ a] :
% 2.65/1.52 ( ( lexordp @ a @ less @ ( append @ a @ Xs @ Us ) @ ( append @ a @ Xs @ Vs ) )
% 2.65/1.52 => ( ! [A3: a] :
% 2.65/1.52 ~ ( less @ A3 @ A3 )
% 2.65/1.52 => ( lexordp @ a @ less @ Us @ Vs ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_186_local_Olexordp__append__leftI,axiom,
% 2.65/1.52 ! [Us: list @ a,Vs: list @ a,Xs: list @ a] :
% 2.65/1.52 ( ( lexordp @ a @ less @ Us @ Vs )
% 2.65/1.52 => ( lexordp @ a @ less @ ( append @ a @ Xs @ Us ) @ ( append @ a @ Xs @ Vs ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_187_local_Olexordp__eq__pref,axiom,
% 2.65/1.52 ! [U: list @ a,V: list @ a] : ( lexordp_eq @ a @ less @ U @ ( append @ a @ U @ V ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_188_same__append__eq,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
% 2.65/1.52 ( ( ( append @ A @ Xs @ Ys )
% 2.65/1.52 = ( append @ A @ Xs @ Zs2 ) )
% 2.65/1.52 = ( Ys = Zs2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_189_append__same__eq,axiom,
% 2.65/1.52 ! [A: $tType,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
% 2.65/1.52 ( ( ( append @ A @ Ys @ Xs )
% 2.65/1.52 = ( append @ A @ Zs2 @ Xs ) )
% 2.65/1.52 = ( Ys = Zs2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_190_append__assoc,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
% 2.65/1.52 ( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs2 )
% 2.65/1.52 = ( append @ A @ Xs @ ( append @ A @ Ys @ Zs2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_191_append_Oassoc,axiom,
% 2.65/1.52 ! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
% 2.65/1.52 ( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C )
% 2.65/1.52 = ( append @ A @ A2 @ ( append @ A @ B2 @ C ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_192_append_Oright__neutral,axiom,
% 2.65/1.52 ! [A: $tType,A2: list @ A] :
% 2.65/1.52 ( ( append @ A @ A2 @ ( nil @ A ) )
% 2.65/1.52 = A2 ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_193_append__is__Nil__conv,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( ( append @ A @ Xs @ Ys )
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 = ( ( Xs
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 & ( Ys
% 2.65/1.52 = ( nil @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_194_Nil__is__append__conv,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( ( nil @ A )
% 2.65/1.52 = ( append @ A @ Xs @ Ys ) )
% 2.65/1.52 = ( ( Xs
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 & ( Ys
% 2.65/1.52 = ( nil @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_195_self__append__conv2,axiom,
% 2.65/1.52 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 2.65/1.52 ( ( Ys
% 2.65/1.52 = ( append @ A @ Xs @ Ys ) )
% 2.65/1.52 = ( Xs
% 2.65/1.52 = ( nil @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_196_append__self__conv2,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( ( append @ A @ Xs @ Ys )
% 2.65/1.52 = Ys )
% 2.65/1.52 = ( Xs
% 2.65/1.52 = ( nil @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_197_self__append__conv,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( Xs
% 2.65/1.52 = ( append @ A @ Xs @ Ys ) )
% 2.65/1.52 = ( Ys
% 2.65/1.52 = ( nil @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_198_append__self__conv,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( ( append @ A @ Xs @ Ys )
% 2.65/1.52 = Xs )
% 2.65/1.52 = ( Ys
% 2.65/1.52 = ( nil @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_199_append__Nil2,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A] :
% 2.65/1.52 ( ( append @ A @ Xs @ ( nil @ A ) )
% 2.65/1.52 = Xs ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_200_append1__eq__conv,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
% 2.65/1.52 ( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
% 2.65/1.52 = ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
% 2.65/1.52 = ( ( Xs = Ys )
% 2.65/1.52 & ( X = Y ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_201_append_Oleft__neutral,axiom,
% 2.65/1.52 ! [A: $tType,A2: list @ A] :
% 2.65/1.52 ( ( append @ A @ ( nil @ A ) @ A2 )
% 2.65/1.52 = A2 ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_202_append__Nil,axiom,
% 2.65/1.52 ! [A: $tType,Ys: list @ A] :
% 2.65/1.52 ( ( append @ A @ ( nil @ A ) @ Ys )
% 2.65/1.52 = Ys ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_203_eq__Nil__appendI,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( Xs = Ys )
% 2.65/1.52 => ( Xs
% 2.65/1.52 = ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_204_Cons__eq__appendI,axiom,
% 2.65/1.52 ! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
% 2.65/1.52 ( ( ( cons @ A @ X @ Xs1 )
% 2.65/1.52 = Ys )
% 2.65/1.52 => ( ( Xs
% 2.65/1.52 = ( append @ A @ Xs1 @ Zs2 ) )
% 2.65/1.52 => ( ( cons @ A @ X @ Xs )
% 2.65/1.52 = ( append @ A @ Ys @ Zs2 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_205_append__Cons,axiom,
% 2.65/1.52 ! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
% 2.65/1.52 = ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_206_ord_Olexordp__append__leftD,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Xs: list @ A,Us: list @ A,Vs: list @ A] :
% 2.65/1.52 ( ( lexordp @ A @ Less2 @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
% 2.65/1.52 => ( ! [A3: A] :
% 2.65/1.52 ~ ( Less2 @ A3 @ A3 )
% 2.65/1.52 => ( lexordp @ A @ Less2 @ Us @ Vs ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_207_ord_Olexordp__append__leftI,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,Us: list @ A,Vs: list @ A,Xs: list @ A] :
% 2.65/1.52 ( ( lexordp @ A @ Less2 @ Us @ Vs )
% 2.65/1.52 => ( lexordp @ A @ Less2 @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_208_ord_Olexordp__eq__pref,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,U: list @ A,V: list @ A] : ( lexordp_eq @ A @ Less2 @ U @ ( append @ A @ U @ V ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_209_append__eq__append__conv2,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,Ts: list @ A] :
% 2.65/1.52 ( ( ( append @ A @ Xs @ Ys )
% 2.65/1.52 = ( append @ A @ Zs2 @ Ts ) )
% 2.65/1.52 = ( ? [Us2: list @ A] :
% 2.65/1.52 ( ( ( Xs
% 2.65/1.52 = ( append @ A @ Zs2 @ Us2 ) )
% 2.65/1.52 & ( ( append @ A @ Us2 @ Ys )
% 2.65/1.52 = Ts ) )
% 2.65/1.52 | ( ( ( append @ A @ Xs @ Us2 )
% 2.65/1.52 = Zs2 )
% 2.65/1.52 & ( Ys
% 2.65/1.52 = ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_210_append__eq__appendI,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys: list @ A,Us: list @ A] :
% 2.65/1.52 ( ( ( append @ A @ Xs @ Xs1 )
% 2.65/1.52 = Zs2 )
% 2.65/1.52 => ( ( Ys
% 2.65/1.52 = ( append @ A @ Xs1 @ Us ) )
% 2.65/1.52 => ( ( append @ A @ Xs @ Ys )
% 2.65/1.52 = ( append @ A @ Zs2 @ Us ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_211_append_Osemigroup__axioms,axiom,
% 2.65/1.52 ! [A: $tType] : ( semigroup @ ( list @ A ) @ ( append @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_212_rev__nonempty__induct,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
% 2.65/1.52 ( ( Xs
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 2.65/1.52 => ( ! [X2: A,Xs3: list @ A] :
% 2.65/1.52 ( ( Xs3
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ( P @ Xs3 )
% 2.65/1.52 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 2.65/1.52 => ( P @ Xs ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_213_append__eq__Cons__conv,axiom,
% 2.65/1.52 ! [A: $tType,Ys: list @ A,Zs2: list @ A,X: A,Xs: list @ A] :
% 2.65/1.52 ( ( ( append @ A @ Ys @ Zs2 )
% 2.65/1.52 = ( cons @ A @ X @ Xs ) )
% 2.65/1.52 = ( ( ( Ys
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 & ( Zs2
% 2.65/1.52 = ( cons @ A @ X @ Xs ) ) )
% 2.65/1.52 | ? [Ys4: list @ A] :
% 2.65/1.52 ( ( Ys
% 2.65/1.52 = ( cons @ A @ X @ Ys4 ) )
% 2.65/1.52 & ( ( append @ A @ Ys4 @ Zs2 )
% 2.65/1.52 = Xs ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_214_Cons__eq__append__conv,axiom,
% 2.65/1.52 ! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
% 2.65/1.52 ( ( ( cons @ A @ X @ Xs )
% 2.65/1.52 = ( append @ A @ Ys @ Zs2 ) )
% 2.65/1.52 = ( ( ( Ys
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 & ( ( cons @ A @ X @ Xs )
% 2.65/1.52 = Zs2 ) )
% 2.65/1.52 | ? [Ys4: list @ A] :
% 2.65/1.52 ( ( ( cons @ A @ X @ Ys4 )
% 2.65/1.52 = Ys )
% 2.65/1.52 & ( Xs
% 2.65/1.52 = ( append @ A @ Ys4 @ Zs2 ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_215_rev__exhaust,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A] :
% 2.65/1.52 ( ( Xs
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ~ ! [Ys3: list @ A,Y2: A] :
% 2.65/1.52 ( Xs
% 2.65/1.52 != ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_216_rev__induct,axiom,
% 2.65/1.52 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 2.65/1.52 ( ( P @ ( nil @ A ) )
% 2.65/1.52 => ( ! [X2: A,Xs3: list @ A] :
% 2.65/1.52 ( ( P @ Xs3 )
% 2.65/1.52 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 2.65/1.52 => ( P @ Xs ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_217_ord_Olexordp__append__left__rightI,axiom,
% 2.65/1.52 ! [A: $tType,Less2: A > A > $o,X: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
% 2.65/1.52 ( ( Less2 @ X @ Y )
% 2.65/1.52 => ( lexordp @ A @ Less2 @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_218_ord_Olexordp__append__rightI,axiom,
% 2.65/1.52 ! [A: $tType,Ys: list @ A,Less2: A > A > $o,Xs: list @ A] :
% 2.65/1.52 ( ( Ys
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( lexordp @ A @ Less2 @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_219_semilattice_Oidem,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A,A2: A] :
% 2.65/1.52 ( ( semilattice @ A @ F )
% 2.65/1.52 => ( ( F @ A2 @ A2 )
% 2.65/1.52 = A2 ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_220_semilattice_Oleft__idem,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 2.65/1.52 ( ( semilattice @ A @ F )
% 2.65/1.52 => ( ( F @ A2 @ ( F @ A2 @ B2 ) )
% 2.65/1.52 = ( F @ A2 @ B2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_221_semilattice_Oright__idem,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A,A2: A,B2: A] :
% 2.65/1.52 ( ( semilattice @ A @ F )
% 2.65/1.52 => ( ( F @ ( F @ A2 @ B2 ) @ B2 )
% 2.65/1.52 = ( F @ A2 @ B2 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_222_product__lists_Osimps_I1_J,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
% 2.65/1.52 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_223_bind__simps_I2_J,axiom,
% 2.65/1.52 ! [A: $tType,B: $tType,X: B,Xs: list @ B,F: B > ( list @ A )] :
% 2.65/1.52 ( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F )
% 2.65/1.52 = ( append @ A @ ( F @ X ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_224_bind__simps_I1_J,axiom,
% 2.65/1.52 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 2.65/1.52 ( ( bind @ B @ A @ ( nil @ B ) @ F )
% 2.65/1.52 = ( nil @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_225_abel__semigroup__def,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( abel_semigroup @ A )
% 2.65/1.52 = ( ^ [F2: A > A > A] :
% 2.65/1.52 ( ( semigroup @ A @ F2 )
% 2.65/1.52 & ( abel_s1917375468axioms @ A @ F2 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_226_abel__semigroup_Ointro,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ( semigroup @ A @ F )
% 2.65/1.52 => ( ( abel_s1917375468axioms @ A @ F )
% 2.65/1.52 => ( abel_semigroup @ A @ F ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_227_abel__semigroup__axioms_Ointro,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ! [A3: A,B3: A] :
% 2.65/1.52 ( ( F @ A3 @ B3 )
% 2.65/1.52 = ( F @ B3 @ A3 ) )
% 2.65/1.52 => ( abel_s1917375468axioms @ A @ F ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_228_abel__semigroup__axioms__def,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( abel_s1917375468axioms @ A )
% 2.65/1.52 = ( ^ [F2: A > A > A] :
% 2.65/1.52 ! [A4: A,B4: A] :
% 2.65/1.52 ( ( F2 @ A4 @ B4 )
% 2.65/1.52 = ( F2 @ B4 @ A4 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_229_abel__semigroup_Oaxioms_I2_J,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ( abel_semigroup @ A @ F )
% 2.65/1.52 => ( abel_s1917375468axioms @ A @ F ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_230_semilattice_Ointro,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ( abel_semigroup @ A @ F )
% 2.65/1.52 => ( ( semilattice_axioms @ A @ F )
% 2.65/1.52 => ( semilattice @ A @ F ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_231_semilattice__def,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( semilattice @ A )
% 2.65/1.52 = ( ^ [F2: A > A > A] :
% 2.65/1.52 ( ( abel_semigroup @ A @ F2 )
% 2.65/1.52 & ( semilattice_axioms @ A @ F2 ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_232_semilattice_Oaxioms_I2_J,axiom,
% 2.65/1.52 ! [A: $tType,F: A > A > A] :
% 2.65/1.52 ( ( semilattice @ A @ F )
% 2.65/1.52 => ( semilattice_axioms @ A @ F ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_233_subseqs_Osimps_I1_J,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( subseqs @ A @ ( nil @ A ) )
% 2.65/1.52 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_234_maps__simps_I1_J,axiom,
% 2.65/1.52 ! [A: $tType,B: $tType,F: B > ( list @ A ),X: B,Xs: list @ B] :
% 2.65/1.52 ( ( maps @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
% 2.65/1.52 = ( append @ A @ ( F @ X ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_235_maps__simps_I2_J,axiom,
% 2.65/1.52 ! [B: $tType,A: $tType,F: B > ( list @ A )] :
% 2.65/1.52 ( ( maps @ B @ A @ F @ ( nil @ B ) )
% 2.65/1.52 = ( nil @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_236_local_Otransp__less,axiom,
% 2.65/1.52 transp @ a @ less ).
% 2.65/1.52
% 2.65/1.52 thf(fact_237_insert__Nil,axiom,
% 2.65/1.52 ! [A: $tType,X: A] :
% 2.65/1.52 ( ( insert @ A @ X @ ( nil @ A ) )
% 2.65/1.52 = ( cons @ A @ X @ ( nil @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_238_concat__eq__append__conv,axiom,
% 2.65/1.52 ! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
% 2.65/1.52 ( ( ( concat @ A @ Xss2 )
% 2.65/1.52 = ( append @ A @ Ys @ Zs2 ) )
% 2.65/1.52 = ( ( ( Xss2
% 2.65/1.52 = ( nil @ ( list @ A ) ) )
% 2.65/1.52 => ( ( Ys
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 & ( Zs2
% 2.65/1.52 = ( nil @ A ) ) ) )
% 2.65/1.52 & ( ( Xss2
% 2.65/1.52 != ( nil @ ( list @ A ) ) )
% 2.65/1.52 => ? [Xss1: list @ ( list @ A ),Xs2: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] :
% 2.65/1.52 ( ( Xss2
% 2.65/1.52 = ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs4 ) @ Xss22 ) ) )
% 2.65/1.52 & ( Ys
% 2.65/1.52 = ( append @ A @ ( concat @ A @ Xss1 ) @ Xs2 ) )
% 2.65/1.52 & ( Zs2
% 2.65/1.52 = ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_239_rotate1_Osimps_I2_J,axiom,
% 2.65/1.52 ! [A: $tType,X: A,Xs: list @ A] :
% 2.65/1.52 ( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
% 2.65/1.52 = ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_240_rotate1__is__Nil__conv,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A] :
% 2.65/1.52 ( ( ( rotate1 @ A @ Xs )
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 = ( Xs
% 2.65/1.52 = ( nil @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_241_concat__append,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
% 2.65/1.52 ( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys ) )
% 2.65/1.52 = ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_242_concat_Osimps_I1_J,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( concat @ A @ ( nil @ ( list @ A ) ) )
% 2.65/1.52 = ( nil @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_243_concat_Osimps_I2_J,axiom,
% 2.65/1.52 ! [A: $tType,X: list @ A,Xs: list @ ( list @ A )] :
% 2.65/1.52 ( ( concat @ A @ ( cons @ ( list @ A ) @ X @ Xs ) )
% 2.65/1.52 = ( append @ A @ X @ ( concat @ A @ Xs ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_244_rotate1_Osimps_I1_J,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( rotate1 @ A @ ( nil @ A ) )
% 2.65/1.52 = ( nil @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_245_concat__eq__appendD,axiom,
% 2.65/1.52 ! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
% 2.65/1.52 ( ( ( concat @ A @ Xss2 )
% 2.65/1.52 = ( append @ A @ Ys @ Zs2 ) )
% 2.65/1.52 => ( ( Xss2
% 2.65/1.52 != ( nil @ ( list @ A ) ) )
% 2.65/1.52 => ? [Xss12: list @ ( list @ A ),Xs3: list @ A,Xs5: list @ A,Xss23: list @ ( list @ A )] :
% 2.65/1.52 ( ( Xss2
% 2.65/1.52 = ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs5 ) @ Xss23 ) ) )
% 2.65/1.52 & ( Ys
% 2.65/1.52 = ( append @ A @ ( concat @ A @ Xss12 ) @ Xs3 ) )
% 2.65/1.52 & ( Zs2
% 2.65/1.52 = ( append @ A @ Xs5 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_246_butlast__snoc,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,X: A] :
% 2.65/1.52 ( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 2.65/1.52 = Xs ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_247_list__ex1__simps_I1_J,axiom,
% 2.65/1.52 ! [A: $tType,P: A > $o] :
% 2.65/1.52 ~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_248_butlast__append,axiom,
% 2.65/1.52 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 2.65/1.52 ( ( ( Ys
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 => ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
% 2.65/1.52 = ( butlast @ A @ Xs ) ) )
% 2.65/1.52 & ( ( Ys
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
% 2.65/1.52 = ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_249_butlast_Osimps_I1_J,axiom,
% 2.65/1.52 ! [A: $tType] :
% 2.65/1.52 ( ( butlast @ A @ ( nil @ A ) )
% 2.65/1.52 = ( nil @ A ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_250_butlast_Osimps_I2_J,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,X: A] :
% 2.65/1.52 ( ( ( Xs
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 => ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
% 2.65/1.52 = ( nil @ A ) ) )
% 2.65/1.52 & ( ( Xs
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
% 2.65/1.52 = ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_251_append__butlast__last__id,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A] :
% 2.65/1.52 ( ( Xs
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
% 2.65/1.52 = Xs ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_252_snoc__eq__iff__butlast,axiom,
% 2.65/1.52 ! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
% 2.65/1.52 ( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
% 2.65/1.52 = Ys )
% 2.65/1.52 = ( ( Ys
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 & ( ( butlast @ A @ Ys )
% 2.65/1.52 = Xs )
% 2.65/1.52 & ( ( last @ A @ Ys )
% 2.65/1.52 = X ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_253_last__appendR,axiom,
% 2.65/1.52 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 2.65/1.52 ( ( Ys
% 2.65/1.52 != ( nil @ A ) )
% 2.65/1.52 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 2.65/1.52 = ( last @ A @ Ys ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(fact_254_last__appendL,axiom,
% 2.65/1.52 ! [A: $tType,Ys: list @ A,Xs: list @ A] :
% 2.65/1.52 ( ( Ys
% 2.65/1.52 = ( nil @ A ) )
% 2.65/1.52 => ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
% 2.65/1.52 = ( last @ A @ Xs ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(tcon_fun___Orderings_Oord,axiom,
% 2.65/1.52 ! [A5: $tType,A6: $tType] :
% 2.65/1.52 ( ( ord @ A6 )
% 2.65/1.52 => ( ord @ ( A5 > A6 ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.65/1.52 linorder @ $o ).
% 2.65/1.52
% 2.65/1.52 thf(tcon_HOL_Obool___Orderings_Oord_1,axiom,
% 2.65/1.52 ord @ $o ).
% 2.65/1.52
% 2.65/1.52 thf(conj_0,hypothesis,
% 2.65/1.52 ( b
% 2.65/1.52 = ( inf @ b @ c ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(conj_1,hypothesis,
% 2.65/1.52 ( ( inf @ c @ ( sup @ b @ c ) )
% 2.65/1.52 = b ) ).
% 2.65/1.52
% 2.65/1.52 thf(conj_2,hypothesis,
% 2.65/1.52 ( a2
% 2.65/1.52 = ( sup @ b @ c ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(conj_3,hypothesis,
% 2.65/1.52 less @ b @ ( sup @ b @ c ) ).
% 2.65/1.52
% 2.65/1.52 thf(conj_4,hypothesis,
% 2.65/1.52 ( ( inf @ c @ ( sup @ b @ c ) )
% 2.65/1.52 = c ) ).
% 2.65/1.52
% 2.65/1.52 thf(conj_5,conjecture,
% 2.65/1.52 $false ).
% 2.65/1.52 ------------------------
% 2.65/1.54 % (4867)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/18Mi)
% 2.65/1.54 % (4861)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/183Mi)
% 2.65/1.54 % (4863)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_4588 for (2999ds/27Mi)
% 2.65/1.54 % (4867)Instruction limit reached!
% 2.65/1.54 % (4867)------------------------------
% 2.65/1.54 % (4867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.54 % (4867)Termination reason: Unknown
% 2.65/1.54 % (4867)Termination phase: Property scanning
% 2.65/1.54
% 2.65/1.54 % (4867)Memory used [KB]: 1535
% 2.65/1.54 % (4867)Time elapsed: 0.008 s
% 2.65/1.54 % (4867)Instructions burned: 21 (million)
% 2.65/1.54 % (4867)------------------------------
% 2.65/1.54 % (4867)------------------------------
% 2.65/1.54 % (4862)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_4588 for (2999ds/4Mi)
% 2.65/1.54 % (4864)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/2Mi)
% 2.65/1.54 % (4866)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_4588 for (2999ds/275Mi)
% 2.65/1.54 % (4865)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_4588 for (2999ds/2Mi)
% 2.65/1.55 % (4864)Instruction limit reached!
% 2.65/1.55 % (4864)------------------------------
% 2.65/1.55 % (4864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.55 % (4864)Termination reason: Unknown
% 2.65/1.55 % (4864)Termination phase: shuffling
% 2.65/1.55
% 2.65/1.55 % (4864)Memory used [KB]: 1279
% 2.65/1.55 % (4864)Time elapsed: 0.004 s
% 2.65/1.55 % (4864)Instructions burned: 3 (million)
% 2.65/1.55 % (4864)------------------------------
% 2.65/1.55 % (4864)------------------------------
% 2.65/1.55 % (4865)Instruction limit reached!
% 2.65/1.55 % (4865)------------------------------
% 2.65/1.55 % (4865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.55 % (4865)Termination reason: Unknown
% 2.65/1.55 % (4865)Termination phase: shuffling
% 2.65/1.55
% 2.65/1.55 % (4865)Memory used [KB]: 1279
% 2.65/1.55 % (4865)Time elapsed: 0.004 s
% 2.65/1.55 % (4865)Instructions burned: 3 (million)
% 2.65/1.55 % (4865)------------------------------
% 2.65/1.55 % (4865)------------------------------
% 2.65/1.55 % (4862)Instruction limit reached!
% 2.65/1.55 % (4862)------------------------------
% 2.65/1.55 % (4862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.55 % (4862)Termination reason: Unknown
% 2.65/1.55 % (4862)Termination phase: shuffling
% 2.65/1.55
% 2.65/1.55 % (4862)Memory used [KB]: 1279
% 2.65/1.55 % (4862)Time elapsed: 0.005 s
% 2.65/1.55 % (4863)Instruction limit reached!
% 2.65/1.55 % (4863)------------------------------
% 2.65/1.55 % (4863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.55 % (4862)Instructions burned: 5 (million)
% 2.65/1.55 % (4863)Termination reason: Unknown
% 2.65/1.55 % (4862)------------------------------
% 2.65/1.55 % (4862)------------------------------
% 2.65/1.55 % (4863)Termination phase: Property scanning
% 2.65/1.55
% 2.65/1.55 % (4863)Memory used [KB]: 1663
% 2.65/1.55 % (4863)Time elapsed: 0.011 s
% 2.65/1.55 % (4863)Instructions burned: 30 (million)
% 2.65/1.55 % (4863)------------------------------
% 2.65/1.55 % (4863)------------------------------
% 2.65/1.55 % (4868)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/3Mi)
% 2.65/1.55 % (4868)Instruction limit reached!
% 2.65/1.55 % (4868)------------------------------
% 2.65/1.55 % (4868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.55 % (4868)Termination reason: Unknown
% 2.65/1.55 % (4868)Termination phase: shuffling
% 2.65/1.55
% 2.65/1.55 % (4868)Memory used [KB]: 1279
% 2.65/1.55 % (4868)Time elapsed: 0.002 s
% 2.65/1.55 % (4868)Instructions burned: 3 (million)
% 2.65/1.55 % (4868)------------------------------
% 2.65/1.55 % (4868)------------------------------
% 2.65/1.56 % (4872)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/1041Mi)
% 2.65/1.56 % (4873)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/7Mi)
% 2.65/1.56 % (4869)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_4588 for (2999ds/37Mi)
% 2.65/1.56 % (4870)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_4588 for (2999ds/15Mi)
% 2.65/1.56 % (4871)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_4588 for (2999ds/3Mi)
% 2.65/1.57 % (4873)Instruction limit reached!
% 2.65/1.57 % (4873)------------------------------
% 2.65/1.57 % (4873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.57 % (4873)Termination reason: Unknown
% 2.65/1.57 % (4873)Termination phase: shuffling
% 2.65/1.57
% 2.65/1.57 % (4873)Memory used [KB]: 1407
% 2.65/1.57 % (4873)Time elapsed: 0.004 s
% 2.65/1.57 % (4873)Instructions burned: 9 (million)
% 2.65/1.57 % (4873)------------------------------
% 2.65/1.57 % (4873)------------------------------
% 2.65/1.57 % (4871)Instruction limit reached!
% 2.65/1.57 % (4871)------------------------------
% 2.65/1.57 % (4871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.57 % (4871)Termination reason: Unknown
% 2.65/1.57 % (4871)Termination phase: shuffling
% 2.65/1.57
% 2.65/1.57 % (4871)Memory used [KB]: 1279
% 2.65/1.57 % (4871)Time elapsed: 0.004 s
% 2.65/1.57 % (4871)Instructions burned: 3 (million)
% 2.65/1.57 % (4871)------------------------------
% 2.65/1.57 % (4871)------------------------------
% 2.65/1.57 % (4870)Instruction limit reached!
% 2.65/1.57 % (4870)------------------------------
% 2.65/1.57 % (4870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.57 % (4870)Termination reason: Unknown
% 2.65/1.57 % (4870)Termination phase: shuffling
% 2.65/1.57
% 2.65/1.57 % (4870)Memory used [KB]: 1535
% 2.65/1.57 % (4870)Time elapsed: 0.012 s
% 2.65/1.57 % (4870)Instructions burned: 16 (million)
% 2.65/1.57 % (4870)------------------------------
% 2.65/1.57 % (4870)------------------------------
% 2.65/1.58 % (4874)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/16Mi)
% 2.65/1.58 % (4874)Instruction limit reached!
% 2.65/1.58 % (4874)------------------------------
% 2.65/1.58 % (4874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.58 % (4874)Termination reason: Unknown
% 2.65/1.58 % (4874)Termination phase: shuffling
% 2.65/1.58
% 2.65/1.58 % (4874)Memory used [KB]: 1535
% 2.65/1.58 % (4874)Time elapsed: 0.006 s
% 2.65/1.58 % (4874)Instructions burned: 16 (million)
% 2.65/1.58 % (4874)------------------------------
% 2.65/1.58 % (4874)------------------------------
% 2.65/1.58 % (4869)Instruction limit reached!
% 2.65/1.58 % (4869)------------------------------
% 2.65/1.58 % (4869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.58 % (4869)Termination reason: Unknown
% 2.65/1.58 % (4869)Termination phase: shuffling
% 2.65/1.58
% 2.65/1.58 % (4869)Memory used [KB]: 2046
% 2.65/1.58 % (4869)Time elapsed: 0.021 s
% 2.65/1.58 % (4869)Instructions burned: 37 (million)
% 2.65/1.58 % (4869)------------------------------
% 2.65/1.58 % (4869)------------------------------
% 2.65/1.58 % (4875)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/3Mi)
% 2.65/1.59 % (4875)Instruction limit reached!
% 2.65/1.59 % (4875)------------------------------
% 2.65/1.59 % (4875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.59 % (4875)Termination reason: Unknown
% 2.65/1.59 % (4875)Termination phase: shuffling
% 2.65/1.59
% 2.65/1.59 % (4875)Memory used [KB]: 1279
% 2.65/1.59 % (4875)Time elapsed: 0.004 s
% 2.65/1.59 % (4875)Instructions burned: 3 (million)
% 2.65/1.59 % (4875)------------------------------
% 2.65/1.59 % (4875)------------------------------
% 2.65/1.59 % (4877)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/7Mi)
% 2.65/1.59 % (4866)First to succeed.
% 2.65/1.59 % (4876)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4588 for (2999ds/3Mi)
% 2.65/1.59 % (4877)Instruction limit reached!
% 2.65/1.59 % (4877)------------------------------
% 2.65/1.59 % (4877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.59 % (4877)Termination reason: Unknown
% 2.65/1.59 % (4877)Termination phase: shuffling
% 2.65/1.59
% 2.65/1.59 % (4877)Memory used [KB]: 1407
% 2.65/1.59 % (4877)Time elapsed: 0.004 s
% 2.65/1.59 % (4877)Instructions burned: 10 (million)
% 2.65/1.59 % (4877)------------------------------
% 2.65/1.59 % (4877)------------------------------
% 2.65/1.59 % (4861)Instruction limit reached!
% 2.65/1.59 % (4861)------------------------------
% 2.65/1.59 % (4861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.59 % (4861)Termination reason: Unknown
% 2.65/1.59 % (4861)Termination phase: Saturation
% 2.65/1.59
% 2.65/1.59 % (4861)Memory used [KB]: 7164
% 2.65/1.59 % (4861)Time elapsed: 0.057 s
% 2.65/1.59 % (4861)Instructions burned: 183 (million)
% 2.65/1.59 % (4861)------------------------------
% 2.65/1.59 % (4861)------------------------------
% 2.65/1.59 % (4876)Instruction limit reached!
% 2.65/1.59 % (4876)------------------------------
% 2.65/1.59 % (4876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.59 % (4866)Refutation found. Thanks to Tanya!
% 2.65/1.59 % SZS status Theorem for DTF2THF_4588
% 2.65/1.59 % SZS output start Proof for DTF2THF_4588
% 2.65/1.59 thf(type_def_6, type, list: ($tType) > $tType).
% 2.65/1.59 thf(type_def_7, type, a: $tType).
% 2.65/1.59 thf(func_def_0, type, list: ($tType) > $tType).
% 2.65/1.59 thf(func_def_1, type, a: $tType).
% 2.65/1.59 thf(func_def_2, type, type: !>[X0: $tType]:($o)).
% 2.65/1.59 thf(func_def_3, type, ord: !>[X0: $tType]:($o)).
% 2.65/1.59 thf(func_def_4, type, linorder: !>[X0: $tType]:($o)).
% 2.65/1.59 thf(func_def_5, type, finite_comp_fun_idem: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 2.65/1.59 thf(func_def_6, type, abel_semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.65/1.59 thf(func_def_7, type, abel_s1917375468axioms: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.65/1.59 thf(func_def_8, type, semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.65/1.59 thf(func_def_9, type, semilattice: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.65/1.59 thf(func_def_10, type, semilattice_axioms: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.65/1.59 thf(func_def_11, type, lattices_is_arg_max: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > (X1 > $o) > X1 > $o)).
% 2.65/1.59 thf(func_def_12, type, lattices_is_arg_min: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > (X1 > $o) > X1 > $o)).
% 2.65/1.59 thf(func_def_13, type, lattic35693393ce_set: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 2.65/1.59 thf(func_def_14, type, append: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 2.65/1.59 thf(func_def_15, type, bind: !>[X0: $tType, X1: $tType]:(list @ X0 > (X0 > list @ X1) > list @ X1)).
% 2.65/1.59 thf(func_def_16, type, butlast: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.65/1.59 thf(func_def_17, type, concat: !>[X0: $tType]:(list @ list @ X0 > list @ X0)).
% 2.65/1.59 thf(func_def_18, type, insert: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 2.65/1.59 thf(func_def_19, type, last: !>[X0: $tType]:(list @ X0 > X0)).
% 2.65/1.59 thf(func_def_20, type, cons: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 2.65/1.59 thf(func_def_21, type, nil: !>[X0: $tType]:(list @ X0)).
% 2.65/1.59 thf(func_def_22, type, list_ex1: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 2.65/1.59 thf(func_def_23, type, maps: !>[X0: $tType, X1: $tType]:((X0 > list @ X1) > list @ X0 > list @ X1)).
% 2.65/1.59 thf(func_def_24, type, lexordp: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > list @ X0 > $o)).
% 2.65/1.59 thf(func_def_25, type, lexordp_eq: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > list @ X0 > $o)).
% 2.65/1.59 thf(func_def_26, type, product_lists: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 2.65/1.59 thf(func_def_27, type, rotate1: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 2.65/1.59 thf(func_def_28, type, subseqs: !>[X0: $tType]:(list @ X0 > list @ list @ X0)).
% 2.65/1.59 thf(func_def_29, type, modula1618706180attice: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > X0) > X0 > X0 > X0 > $o)).
% 2.65/1.59 thf(func_def_30, type, modula1192512645attice: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > X0) > X0 > X0 > X0 > $o)).
% 2.65/1.59 thf(func_def_31, type, modula105138637_a_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.65/1.59 thf(func_def_32, type, modula477145868_b_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.65/1.59 thf(func_def_33, type, modula849153099_c_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.65/1.59 thf(func_def_34, type, modula1221160330_d_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.65/1.59 thf(func_def_35, type, modula1593167561_e_aux: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > X0 > X0 > X0 > X0)).
% 2.65/1.59 thf(func_def_36, type, modula141148890istrib: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > $o) > (X0 > X0 > X0) > X0 > X0 > X0 > $o)).
% 2.65/1.59 thf(func_def_37, type, transp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 2.65/1.59 thf(func_def_38, type, a2: a).
% 2.65/1.59 thf(func_def_39, type, b: a).
% 2.65/1.59 thf(func_def_40, type, c: a).
% 2.65/1.59 thf(func_def_41, type, inf: a > a > a).
% 2.65/1.59 thf(func_def_42, type, less: a > a > $o).
% 2.65/1.59 thf(func_def_43, type, sup: a > a > a).
% 2.65/1.59 thf(func_def_61, type, sK0: a).
% 2.65/1.59 thf(func_def_62, type, sK1: a).
% 2.65/1.59 thf(func_def_63, type, sK2: a).
% 2.65/1.59 thf(func_def_64, type, sK3: a).
% 2.65/1.59 thf(func_def_65, type, sK4: a).
% 2.65/1.59 thf(func_def_66, type, sK5: a).
% 2.65/1.59 thf(func_def_67, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.65/1.59 thf(f953,plain,(
% 2.65/1.59 $false),
% 2.65/1.59 inference(avatar_sat_refutation,[status(thm)],[f938,f952])).
% 2.65/1.59 thf(f952,plain,(
% 2.65/1.59 ~spl7_5),
% 2.65/1.59 inference(avatar_contradiction_clause,[status(thm)],[f951])).
% 2.65/1.59 thf(f951,plain,(
% 2.65/1.59 $false | ~spl7_5),
% 2.65/1.59 inference(subsumption_resolution,[status(thm)],[f950,f709])).
% 2.65/1.59 thf(f709,plain,(
% 2.65/1.59 ( ! [X0 : a] : (((less @ X0 @ X0) != $true)) )),
% 2.65/1.59 inference(cnf_transformation,[status(thm)],[f626])).
% 2.65/1.59 thf(f626,plain,(
% 2.65/1.59 ! [X0 : a] : ((less @ X0 @ X0) != $true)),
% 2.65/1.59 inference(flattening,[status(thm)],[f544])).
% 2.65/1.59 thf(f544,plain,(
% 2.65/1.59 ! [X0 : a] : ~((less @ X0 @ X0) = $true)),
% 2.65/1.59 inference(fool_elimination,[status(thm)],[f543])).
% 2.65/1.59 thf(f543,plain,(
% 2.65/1.59 ! [X0 : a] : ~(less @ X0 @ X0)),
% 2.65/1.59 inference(rectify,[status(thm)],[f17])).
% 2.65/1.59 thf(f17,axiom,(
% 2.65/1.59 ! [X3 : a] : ~(less @ X3 @ X3)),
% 2.65/1.59 file('/export/starexec/sandbox2/tmp/tmp.41GWJZtw0l/DTF2THF_4588.p',fact_16_local_Oorder_Oirrefl)).
% 2.65/1.59 thf(f950,plain,(
% 2.65/1.59 ((less @ b @ b) = $true) | ~spl7_5),
% 2.65/1.59 inference(forward_demodulation,[status(thm)],[f945,f687])).
% 2.65/1.59 thf(f687,plain,(
% 2.65/1.59 ( ! [X0 : a] : (((sup @ X0 @ X0) = X0)) )),
% 2.65/1.59 inference(cnf_transformation,[status(thm)],[f618])).
% 2.65/1.59 thf(f618,plain,(
% 2.65/1.59 ! [X0 : a] : ((sup @ X0 @ X0) = X0)),
% 2.65/1.59 inference(rectify,[status(thm)],[f56])).
% 2.65/1.59 thf(f56,axiom,(
% 2.65/1.59 ! [X5 : a] : ((sup @ X5 @ X5) = X5)),
% 2.65/1.59 file('/export/starexec/sandbox2/tmp/tmp.41GWJZtw0l/DTF2THF_4588.p',fact_55_local_Osup__idem)).
% 2.65/1.59 thf(f945,plain,(
% 2.65/1.59 ((less @ b @ (sup @ b @ b)) = $true) | ~spl7_5),
% 2.65/1.59 inference(superposition,[status(thm)],[f705,f820])).
% 2.65/1.59 thf(f820,plain,(
% 2.65/1.59 (b = c) | ~spl7_5),
% 2.65/1.59 inference(avatar_component_clause,[status(thm)],[f818])).
% 2.65/1.59 thf(f818,definition,(
% 2.65/1.59 spl7_5 <=> (b = c)),
% 2.65/1.59 introduced(definition,[new_symbols(naming,[spl7_5])],[avatar_definition])).
% 2.65/1.59 thf(f705,plain,(
% 2.65/1.59 ((less @ b @ (sup @ b @ c)) = $true)),
% 2.65/1.59 inference(cnf_transformation,[status(thm)],[f520])).
% 2.65/1.59 thf(f520,plain,(
% 2.65/1.59 ((less @ b @ (sup @ b @ c)) = $true)),
% 2.65/1.59 inference(fool_elimination,[status(thm)],[f519])).
% 2.65/1.59 thf(f519,plain,(
% 2.65/1.59 (less @ b @ (sup @ b @ c))),
% 2.65/1.59 inference(rectify,[status(thm)],[f262])).
% 2.65/1.59 thf(f262,axiom,(
% 2.65/1.59 (less @ b @ (sup @ b @ c))),
% 2.65/1.59 file('/export/starexec/sandbox2/tmp/tmp.41GWJZtw0l/DTF2THF_4588.p',conj_3)).
% 2.65/1.59 thf(f938,plain,(
% 2.65/1.59 spl7_5),
% 2.65/1.59 inference(avatar_split_clause,[status(thm)],[f915,f818])).
% 2.65/1.59 thf(f915,plain,(
% 2.65/1.59 (b = c)),
% 2.65/1.59 inference(superposition,[status(thm)],[f718,f686])).
% 2.65/1.59 thf(f686,plain,(
% 2.65/1.59 (b = (inf @ c @ (sup @ b @ c)))),
% 2.65/1.59 inference(cnf_transformation,[status(thm)],[f260])).
% 2.65/1.59 thf(f260,axiom,(
% 2.65/1.59 (b = (inf @ c @ (sup @ b @ c)))),
% 2.65/1.59 file('/export/starexec/sandbox2/tmp/tmp.41GWJZtw0l/DTF2THF_4588.p',conj_1)).
% 2.65/1.59 thf(f718,plain,(
% 2.65/1.59 (c = (inf @ c @ (sup @ b @ c)))),
% 2.65/1.59 inference(cnf_transformation,[status(thm)],[f263])).
% 2.65/1.59 thf(f263,axiom,(
% 2.65/1.59 (c = (inf @ c @ (sup @ b @ c)))),
% 2.65/1.59 file('/export/starexec/sandbox2/tmp/tmp.41GWJZtw0l/DTF2THF_4588.p',conj_4)).
% 2.65/1.59 % SZS output end Proof for DTF2THF_4588
% 2.65/1.59 % (4866)------------------------------
% 2.65/1.59 % (4866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.59 % (4866)Termination reason: Refutation
% 2.65/1.59
% 2.65/1.59 % (4866)Memory used [KB]: 6396
% 2.65/1.59 % (4866)Time elapsed: 0.049 s
% 2.65/1.59 % (4866)Instructions burned: 79 (million)
% 2.65/1.59 % (4866)------------------------------
% 2.65/1.59 % (4866)------------------------------
% 2.65/1.59 % (4860)Success in time 0.068 s
%------------------------------------------------------------------------------