TPTP Problem File: ITP119^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP119^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Modular_Distrib_Lattice problem prob_144__3260118_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Modular_Distrib_Lattice/prob_144__3260118_1 [Des21]
% Status : ContradictoryAxioms
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 308 ( 154 unt; 44 typ; 0 def)
% Number of atoms : 620 ( 313 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3325 ( 126 ~; 15 |; 70 &;2860 @)
% ( 0 <=>; 254 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 331 ( 331 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 43 usr; 4 con; 0-7 aty)
% Number of variables : 981 ( 18 ^; 859 !; 63 ?; 981 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:29:06.337
%------------------------------------------------------------------------------
% Could-be-implicit typings (2)
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
finite_comp_fun_idem:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Groups_Oabel__semigroup,type,
abel_semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Oabel__semigroup__axioms,type,
abel_s1917375468axioms:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__axioms,type,
semilattice_axioms:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
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 ) ).
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 ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic35693393ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
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 ) ).
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 ) ).
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 ) ).
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 ) ).
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 ) ).
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 ) ).
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 ) ).
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 ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b: a ).
thf(sy_v_c,type,
c: a ).
thf(sy_v_inf,type,
inf: a > a > a ).
thf(sy_v_less,type,
less: a > a > $o ).
thf(sy_v_sup,type,
sup: a > a > a ).
% Relevant facts (255)
thf(fact_0_local_Odual__order_Oasym,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ~ ( less @ A2 @ B2 ) ) ).
% local.dual_order.asym
thf(fact_1_local_Odual__order_Ostrict__implies__not__eq,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% local.dual_order.strict_implies_not_eq
thf(fact_2_local_Odual__order_Ostrict__trans,axiom,
! [B2: a,A2: a,C: a] :
( ( less @ B2 @ A2 )
=> ( ( less @ C @ B2 )
=> ( less @ C @ A2 ) ) ) ).
% local.dual_order.strict_trans
thf(fact_3_local_Oless__asym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_asym
thf(fact_4_local_Oless__asym_H,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.less_asym'
thf(fact_5_local_Oless__imp__neq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_neq
thf(fact_6_local_Oless__imp__not__eq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_not_eq
thf(fact_7_local_Oless__imp__not__eq2,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( Y != X ) ) ).
% local.less_imp_not_eq2
thf(fact_8_local_Oless__imp__not__less,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_imp_not_less
thf(fact_9_local_Oless__imp__triv,axiom,
! [X: a,Y: a,P: $o] :
( ( less @ X @ Y )
=> ( ( less @ Y @ X )
=> P ) ) ).
% local.less_imp_triv
thf(fact_10_local_Oless__irrefl,axiom,
! [X: a] :
~ ( less @ X @ X ) ).
% local.less_irrefl
thf(fact_11_local_Oless__not__sym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_not_sym
thf(fact_12_local_Oless__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( less @ X @ Y )
=> ( ( less @ Y @ Z )
=> ( less @ X @ Z ) ) ) ).
% local.less_trans
thf(fact_13_local_Oord__eq__less__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_eq_less_trans
thf(fact_14_local_Oord__less__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( B2 = C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_less_eq_trans
thf(fact_15_local_Oorder_Oasym,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.order.asym
thf(fact_16_local_Oorder_Oirrefl,axiom,
! [A2: a] :
~ ( less @ A2 @ A2 ) ).
% local.order.irrefl
thf(fact_17_local_Oorder_Ostrict__implies__not__eq,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% local.order.strict_implies_not_eq
thf(fact_18_local_Oorder_Ostrict__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.order.strict_trans
thf(fact_19_local_Oinf_Oassoc,axiom,
! [A2: a,B2: a,C: a] :
( ( inf @ ( inf @ A2 @ B2 ) @ C )
= ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% local.inf.assoc
thf(fact_20_local_Oinf_Ocommute,axiom,
! [A2: a,B2: a] :
( ( inf @ A2 @ B2 )
= ( inf @ B2 @ A2 ) ) ).
% local.inf.commute
thf(fact_21_local_Oinf_Oleft__commute,axiom,
! [B2: a,A2: a,C: a] :
( ( inf @ B2 @ ( inf @ A2 @ C ) )
= ( inf @ A2 @ ( inf @ B2 @ C ) ) ) ).
% local.inf.left_commute
thf(fact_22_local_Oinf__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( inf @ ( inf @ X @ Y ) @ Z )
= ( inf @ X @ ( inf @ Y @ Z ) ) ) ).
% local.inf_assoc
thf(fact_23_local_Oinf__commute,axiom,
! [X: a,Y: a] :
( ( inf @ X @ Y )
= ( inf @ Y @ X ) ) ).
% local.inf_commute
thf(fact_24_local_Oinf__left__commute,axiom,
! [X: a,Y: a,Z: a] :
( ( inf @ X @ ( inf @ Y @ Z ) )
= ( inf @ Y @ ( inf @ X @ Z ) ) ) ).
% local.inf_left_commute
thf(fact_25_local_Osup_Oassoc,axiom,
! [A2: a,B2: a,C: a] :
( ( sup @ ( sup @ A2 @ B2 ) @ C )
= ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% local.sup.assoc
thf(fact_26_local_Osup_Ocommute,axiom,
! [A2: a,B2: a] :
( ( sup @ A2 @ B2 )
= ( sup @ B2 @ A2 ) ) ).
% local.sup.commute
thf(fact_27_local_Osup_Oleft__commute,axiom,
! [B2: a,A2: a,C: a] :
( ( sup @ B2 @ ( sup @ A2 @ C ) )
= ( sup @ A2 @ ( sup @ B2 @ C ) ) ) ).
% local.sup.left_commute
thf(fact_28_local_Osup__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( sup @ ( sup @ X @ Y ) @ Z )
= ( sup @ X @ ( sup @ Y @ Z ) ) ) ).
% local.sup_assoc
thf(fact_29_local_Osup__commute,axiom,
! [X: a,Y: a] :
( ( sup @ X @ Y )
= ( sup @ Y @ X ) ) ).
% local.sup_commute
thf(fact_30_local_Osup__left__commute,axiom,
! [X: a,Y: a,Z: a] :
( ( sup @ X @ ( sup @ Y @ Z ) )
= ( sup @ Y @ ( sup @ X @ Z ) ) ) ).
% local.sup_left_commute
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 ) ) ) ).
% local.inf.strict_boundedE
thf(fact_32_local_Oinf_Ostrict__coboundedI1,axiom,
! [A2: a,C: a,B2: a] :
( ( less @ A2 @ C )
=> ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% local.inf.strict_coboundedI1
thf(fact_33_local_Oinf_Ostrict__coboundedI2,axiom,
! [B2: a,C: a,A2: a] :
( ( less @ B2 @ C )
=> ( less @ ( inf @ A2 @ B2 ) @ C ) ) ).
% local.inf.strict_coboundedI2
thf(fact_34_local_Oinf_Ostrict__order__iff,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
= ( ( A2
= ( inf @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ).
% local.inf.strict_order_iff
thf(fact_35_local_Oless__infI1,axiom,
! [A2: a,X: a,B2: a] :
( ( less @ A2 @ X )
=> ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% local.less_infI1
thf(fact_36_local_Oless__infI2,axiom,
! [B2: a,X: a,A2: a] :
( ( less @ B2 @ X )
=> ( less @ ( inf @ A2 @ B2 ) @ X ) ) ).
% local.less_infI2
thf(fact_37_local_Oless__supI1,axiom,
! [X: a,A2: a,B2: a] :
( ( less @ X @ A2 )
=> ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% local.less_supI1
thf(fact_38_local_Oless__supI2,axiom,
! [X: a,B2: a,A2: a] :
( ( less @ X @ B2 )
=> ( less @ X @ ( sup @ A2 @ B2 ) ) ) ).
% local.less_supI2
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 ) ) ) ).
% local.sup.strict_boundedE
thf(fact_40_local_Osup_Ostrict__coboundedI1,axiom,
! [C: a,A2: a,B2: a] :
( ( less @ C @ A2 )
=> ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% local.sup.strict_coboundedI1
thf(fact_41_local_Osup_Ostrict__coboundedI2,axiom,
! [C: a,B2: a,A2: a] :
( ( less @ C @ B2 )
=> ( less @ C @ ( sup @ A2 @ B2 ) ) ) ).
% local.sup.strict_coboundedI2
thf(fact_42_local_Osup_Ostrict__order__iff,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
= ( ( A2
= ( sup @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ).
% local.sup.strict_order_iff
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 ) ) ) ) ).
% local.distrib_imp1
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 ) ) ) ) ).
% local.distrib_imp2
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_local_Oinf_Oidem,axiom,
! [A2: a] :
( ( inf @ A2 @ A2 )
= A2 ) ).
% local.inf.idem
thf(fact_47_local_Oinf_Oleft__idem,axiom,
! [A2: a,B2: a] :
( ( inf @ A2 @ ( inf @ A2 @ B2 ) )
= ( inf @ A2 @ B2 ) ) ).
% local.inf.left_idem
thf(fact_48_local_Oinf_Oright__idem,axiom,
! [A2: a,B2: a] :
( ( inf @ ( inf @ A2 @ B2 ) @ B2 )
= ( inf @ A2 @ B2 ) ) ).
% local.inf.right_idem
thf(fact_49_local_Oinf__idem,axiom,
! [X: a] :
( ( inf @ X @ X )
= X ) ).
% local.inf_idem
thf(fact_50_local_Oinf__left__idem,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( inf @ X @ Y ) )
= ( inf @ X @ Y ) ) ).
% local.inf_left_idem
thf(fact_51_local_Oinf__right__idem,axiom,
! [X: a,Y: a] :
( ( inf @ ( inf @ X @ Y ) @ Y )
= ( inf @ X @ Y ) ) ).
% local.inf_right_idem
thf(fact_52_local_Osup_Oidem,axiom,
! [A2: a] :
( ( sup @ A2 @ A2 )
= A2 ) ).
% local.sup.idem
thf(fact_53_local_Osup_Oleft__idem,axiom,
! [A2: a,B2: a] :
( ( sup @ A2 @ ( sup @ A2 @ B2 ) )
= ( sup @ A2 @ B2 ) ) ).
% local.sup.left_idem
thf(fact_54_local_Osup_Oright__idem,axiom,
! [A2: a,B2: a] :
( ( sup @ ( sup @ A2 @ B2 ) @ B2 )
= ( sup @ A2 @ B2 ) ) ).
% local.sup.right_idem
thf(fact_55_local_Osup__idem,axiom,
! [X: a] :
( ( sup @ X @ X )
= X ) ).
% local.sup_idem
thf(fact_56_local_Osup__left__idem,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( sup @ X @ Y ) )
= ( sup @ X @ Y ) ) ).
% local.sup_left_idem
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 ) ) ) ) ) ).
% local.is_arg_max_def
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 ) ) ) ) ) ).
% local.is_arg_min_def
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 ) ) ) ) ).
% M5_lattice_def
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 ) ) ) ) ).
% N5_lattice_def
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 ) ) ) ) ).
% no_distrib_def
thf(fact_62_local_Oinf__sup__absorb,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( sup @ X @ Y ) )
= X ) ).
% local.inf_sup_absorb
thf(fact_63_local_Osup__inf__absorb,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( inf @ X @ Y ) )
= X ) ).
% local.sup_inf_absorb
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 ) ) ) ).
% local.a_join_d
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 ) ) ) ).
% local.b_join_d
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 ) ) ) ).
% local.d_aux_def
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 ) ) ).
% local.d_b_c_a
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 ) ) ).
% local.d_c_a_b
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 ) ) ) ).
% local.a_meet_e
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 ) ) ) ).
% local.b_meet_e
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 ) ) ) ).
% local.c_meet_e
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 ) ) ) ).
% local.e_aux_def
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 ) ) ).
% local.e_b_c_a
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 ) ) ).
% local.e_c_a_b
thf(fact_75_local_Ocomp__fun__idem__sup,axiom,
finite_comp_fun_idem @ a @ a @ sup ).
% local.comp_fun_idem_sup
thf(fact_76_local_Ocomp__fun__idem__inf,axiom,
finite_comp_fun_idem @ a @ a @ inf ).
% local.comp_fun_idem_inf
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 ) ) ) ).
% local.c_aux_def
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 ) ) ) ).
% local.b_aux_def
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 ) ) ) ).
% local.a_aux_def
thf(fact_80_lattice_Ono__distrib_Ocong,axiom,
! [A: $tType] :
( ( modula141148890istrib @ A )
= ( modula141148890istrib @ A ) ) ).
% lattice.no_distrib.cong
thf(fact_81_lattice_ON5__lattice_Ocong,axiom,
! [A: $tType] :
( ( modula1192512645attice @ A )
= ( modula1192512645attice @ A ) ) ).
% lattice.N5_lattice.cong
thf(fact_82_lattice_OM5__lattice_Ocong,axiom,
! [A: $tType] :
( ( modula1618706180attice @ A )
= ( modula1618706180attice @ A ) ) ).
% lattice.M5_lattice.cong
thf(fact_83_lattice_Oe__aux_Ocong,axiom,
! [A: $tType] :
( ( modula1593167561_e_aux @ A )
= ( modula1593167561_e_aux @ A ) ) ).
% lattice.e_aux.cong
thf(fact_84_lattice_Od__aux_Ocong,axiom,
! [A: $tType] :
( ( modula1221160330_d_aux @ A )
= ( modula1221160330_d_aux @ A ) ) ).
% lattice.d_aux.cong
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 ) ) ).
% local.c_a
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 ) ) ).
% local.b_a
thf(fact_87_local_Osup_Osemigroup__axioms,axiom,
semigroup @ a @ sup ).
% local.sup.semigroup_axioms
thf(fact_88_local_Oinf_Osemigroup__axioms,axiom,
semigroup @ a @ inf ).
% local.inf.semigroup_axioms
thf(fact_89_local_Osup_Osemilattice__axioms,axiom,
semilattice @ a @ sup ).
% local.sup.semilattice_axioms
thf(fact_90_local_Oinf_Osemilattice__axioms,axiom,
semilattice @ a @ inf ).
% local.inf.semilattice_axioms
thf(fact_91_lattice_Oa__aux_Ocong,axiom,
! [A: $tType] :
( ( modula105138637_a_aux @ A )
= ( modula105138637_a_aux @ A ) ) ).
% lattice.a_aux.cong
thf(fact_92_lattice_Ob__aux_Ocong,axiom,
! [A: $tType] :
( ( modula477145868_b_aux @ A )
= ( modula477145868_b_aux @ A ) ) ).
% lattice.b_aux.cong
thf(fact_93_lattice_Oc__aux_Ocong,axiom,
! [A: $tType] :
( ( modula849153099_c_aux @ A )
= ( modula849153099_c_aux @ A ) ) ).
% lattice.c_aux.cong
thf(fact_94_local_Osup_Oabel__semigroup__axioms,axiom,
abel_semigroup @ a @ sup ).
% local.sup.abel_semigroup_axioms
thf(fact_95_local_Oinf_Oabel__semigroup__axioms,axiom,
abel_semigroup @ a @ inf ).
% local.inf.abel_semigroup_axioms
thf(fact_96_local_OSup__fin_Osemilattice__set__axioms,axiom,
lattic35693393ce_set @ a @ sup ).
% local.Sup_fin.semilattice_set_axioms
thf(fact_97_local_OInf__fin_Osemilattice__set__axioms,axiom,
lattic35693393ce_set @ a @ inf ).
% local.Inf_fin.semilattice_set_axioms
thf(fact_98_local_Olexordp__eq__refl,axiom,
! [Xs: list @ a] : ( lexordp_eq @ a @ less @ Xs @ Xs ) ).
% local.lexordp_eq_refl
thf(fact_99_local_Olexordp__eq_ONil,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq.Nil
thf(fact_100_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( semigroup @ A @ F ) ) ).
% abel_semigroup.axioms(1)
thf(fact_101_semilattice__set_Oaxioms,axiom,
! [A: $tType,F: A > A > A] :
( ( lattic35693393ce_set @ A @ F )
=> ( semilattice @ A @ F ) ) ).
% semilattice_set.axioms
thf(fact_102_semilattice__set_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( lattic35693393ce_set @ A @ F ) ) ).
% semilattice_set.intro
thf(fact_103_semilattice__set__def,axiom,
! [A: $tType] :
( ( lattic35693393ce_set @ A )
= ( semilattice @ A ) ) ).
% semilattice_set_def
thf(fact_104_local_Olexordp__eq__simps_I1_J,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq_simps(1)
thf(fact_105_local_Olexordp__eq__simps_I2_J,axiom,
! [Xs: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ ( nil @ a ) )
= ( Xs
= ( nil @ a ) ) ) ).
% local.lexordp_eq_simps(2)
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 ) ) ) ) ).
% abel_semigroup.left_commute
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 ) ) ) ).
% abel_semigroup.commute
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 ) ) ).
% semigroup.intro
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 ) ) ) ) ).
% semigroup.assoc
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 ) ) ) ) ) ).
% semigroup_def
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 ) ) ).
% ord.is_arg_min.cong
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 ) ) ).
% ord.is_arg_max.cong
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 ) ) ) ) ) ) ).
% ord.is_arg_min_def
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 ) ) ) ) ) ) ).
% ord.is_arg_max_def
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 ) ) ) ).
% ord.lexordp_eq_simps(2)
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 ) ).
% ord.lexordp_eq_simps(1)
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 ) ) ) ) ).
% local.lexordp_eq.simps
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 ) ) ) ) ) ).
% local.lexordp_eq.inducts
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 ) ) ) ) ) ) ) ) ).
% local.lexordp_eq.cases
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 ) ) ) ) ) ).
% local.lexordp_eq.Cons_eq
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 ) ) ) ).
% local.lexordp_eq.Cons
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 ) ) ) ).
% list.inject
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 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
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 ) ) ).
% ord.lexordp_eq_simps(3)
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 ) ) ) ) ).
% local.lexordp_eq_simps(4)
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 ) ) ).
% local.lexordp_eq_simps(3)
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 ) ) ) ) ).
% strict_sorted.induct
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 ) ) ) ) ).
% strict_sorted.cases
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 ) ) ) ).
% map_tailrec_rev.induct
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 ) ) ) ) ).
% list_nonempty_induct
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 ) ) ) ) ).
% successively.induct
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 ) ) ) ) ) ).
% arg_min_list.induct
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 ) ) ) ) ).
% remdups_adj.induct
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 ) ) ) ).
% sorted_wrt.induct
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 ) ) ) ) ) ).
% remdups_adj.cases
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 ) ) ) ) ).
% transpose.cases
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 ) ) ) ) ).
% shuffles.induct
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 ) ) ) ) ).
% min_list.induct
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 ) ) ) ) ).
% min_list.cases
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 ) ) ) ) ).
% induct_list012
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 ) ) ) ).
% splice.induct
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 ) ) ) ) ) ).
% list_induct2'
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 ) ) ) ) ).
% neq_Nil_conv
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 ) ) ) ).
% list.inducts
thf(fact_145_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X213: A,X224: list @ A] :
( Y
!= ( cons @ A @ X213 @ X224 ) ) ) ).
% list.exhaust
thf(fact_146_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
( ( List
= ( cons @ A @ X21 @ X222 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_147_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_148_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
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 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
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 ) ) ) ).
% ord.lexordp_eq.Cons
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 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
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 ) ) ) ) ) ).
% ord.lexordp_eq.simps
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 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_154_ord_Olexordp__eq_Ocong,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( lexordp_eq @ A ) ) ).
% ord.lexordp_eq.cong
thf(fact_155_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less2: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less2 @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_156_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less2: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less2 @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_157_local_Olexordp_ONil,axiom,
! [Y: a,Ys: list @ a] : ( lexordp @ a @ less @ ( nil @ a ) @ ( cons @ a @ Y @ Ys ) ) ).
% local.lexordp.Nil
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 ) ) ) ) ) ) ) ) ).
% local.lexordp.cases
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 ) ) ) ) ) ).
% local.lexordp.inducts
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 ) ) ) ) ).
% local.lexordp.simps
thf(fact_161_local_Olexordp__antisym,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ~ ( lexordp @ a @ less @ Ys @ Xs ) ) ).
% local.lexordp_antisym
thf(fact_162_local_Olexordp__irreflexive,axiom,
! [Xs: list @ a] :
( ! [X2: a] :
~ ( less @ X2 @ X2 )
=> ~ ( lexordp @ a @ less @ Xs @ Xs ) ) ).
% local.lexordp_irreflexive
thf(fact_163_local_Olexordp__irreflexive_H,axiom,
! [Xs: list @ a] :
~ ( lexordp @ a @ less @ Xs @ Xs ) ).
% local.lexordp_irreflexive'
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 ) ) ) ) ) ).
% local.lexordp.Cons_eq
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 ) ) ) ).
% local.lexordp.Cons
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 ) ) ).
% local.lexordp_into_lexordp_eq
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 ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_168_ord_Olexordp__simps_I2_J,axiom,
! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
~ ( lexordp @ A @ Less2 @ Xs @ ( nil @ A ) ) ).
% ord.lexordp_simps(2)
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 ) ) ) ).
% ord.lexordp_simps(1)
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 ) ) ) ) ).
% local.lexordp_simps(3)
thf(fact_171_local_Olexordp__simps_I1_J,axiom,
! [Ys: list @ a] :
( ( lexordp @ a @ less @ ( nil @ a ) @ Ys )
= ( Ys
!= ( nil @ a ) ) ) ).
% local.lexordp_simps(1)
thf(fact_172_local_Olexordp__simps_I2_J,axiom,
! [Xs: list @ a] :
~ ( lexordp @ a @ less @ Xs @ ( nil @ a ) ) ).
% local.lexordp_simps(2)
thf(fact_173_ord_Olexordp_Ocong,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( lexordp @ A ) ) ).
% ord.lexordp.cong
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 ) ) ).
% ord.lexordp_irreflexive
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 ) ) ) ) ) ).
% ord.lexordp.Cons_eq
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 ) ) ) ).
% ord.lexordp.Cons
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 ) ) ).
% ord.lexordp_into_lexordp_eq
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 ) ) ).
% ord.lexordp.Nil
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 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
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 ) ) ) ) ) ).
% ord.lexordp.simps
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 ) ) ) ) ) ).
% ord.lexordp.inducts
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 ) ) ) ).
% local.lexordp_append_rightI
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 ) ) ) ) ).
% local.lexordp_append_left_rightI
thf(fact_184_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ).
% semilattice.axioms(1)
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 ) ) ) ).
% local.lexordp_append_leftD
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 ) ) ) ).
% local.lexordp_append_leftI
thf(fact_187_local_Olexordp__eq__pref,axiom,
! [U: list @ a,V: list @ a] : ( lexordp_eq @ a @ less @ U @ ( append @ a @ U @ V ) ) ).
% local.lexordp_eq_pref
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 ) ) ).
% same_append_eq
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 ) ) ).
% append_same_eq
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 ) ) ) ).
% append_assoc
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 ) ) ) ).
% append.assoc
thf(fact_192_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
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 ) ) ) ) ).
% append_is_Nil_conv
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 ) ) ) ) ).
% Nil_is_append_conv
thf(fact_195_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_196_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_197_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_198_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_199_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
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 ) ) ) ).
% append1_eq_conv
thf(fact_201_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_202_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_203_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
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 ) ) ) ) ).
% Cons_eq_appendI
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 ) ) ) ).
% append_Cons
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 ) ) ) ).
% ord.lexordp_append_leftD
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 ) ) ) ).
% ord.lexordp_append_leftI
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 ) ) ).
% ord.lexordp_eq_pref
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 ) ) ) ) ) ) ).
% append_eq_append_conv2
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 ) ) ) ) ).
% append_eq_appendI
thf(fact_211_append_Osemigroup__axioms,axiom,
! [A: $tType] : ( semigroup @ ( list @ A ) @ ( append @ A ) ) ).
% append.semigroup_axioms
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 ) ) ) ) ).
% rev_nonempty_induct
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 ) ) ) ) ).
% append_eq_Cons_conv
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 ) ) ) ) ) ).
% Cons_eq_append_conv
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 ) ) ) ) ) ).
% rev_exhaust
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 ) ) ) ).
% rev_induct
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 ) ) ) ) ).
% ord.lexordp_append_left_rightI
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 ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_219_semilattice_Oidem,axiom,
! [A: $tType,F: A > A > A,A2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ A2 @ A2 )
= A2 ) ) ).
% semilattice.idem
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 ) ) ) ).
% semilattice.left_idem
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 ) ) ) ).
% semilattice.right_idem
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 ) ) ) ) ).
% product_lists.simps(1)
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 ) ) ) ).
% bind_simps(2)
thf(fact_224_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_225_abel__semigroup__def,axiom,
! [A: $tType] :
( ( abel_semigroup @ A )
= ( ^ [F2: A > A > A] :
( ( semigroup @ A @ F2 )
& ( abel_s1917375468axioms @ A @ F2 ) ) ) ) ).
% abel_semigroup_def
thf(fact_226_abel__semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( semigroup @ A @ F )
=> ( ( abel_s1917375468axioms @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ) ).
% abel_semigroup.intro
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 ) ) ).
% abel_semigroup_axioms.intro
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 ) ) ) ) ).
% abel_semigroup_axioms_def
thf(fact_229_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( abel_s1917375468axioms @ A @ F ) ) ).
% abel_semigroup.axioms(2)
thf(fact_230_semilattice_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( ( semilattice_axioms @ A @ F )
=> ( semilattice @ A @ F ) ) ) ).
% semilattice.intro
thf(fact_231_semilattice__def,axiom,
! [A: $tType] :
( ( semilattice @ A )
= ( ^ [F2: A > A > A] :
( ( abel_semigroup @ A @ F2 )
& ( semilattice_axioms @ A @ F2 ) ) ) ) ).
% semilattice_def
thf(fact_232_semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( semilattice_axioms @ A @ F ) ) ).
% semilattice.axioms(2)
thf(fact_233_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
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 ) ) ) ).
% maps_simps(1)
thf(fact_235_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( maps @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_236_local_Otransp__less,axiom,
transp @ a @ less ).
% local.transp_less
thf(fact_237_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
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 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
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 ) ) ) ) ).
% rotate1.simps(2)
thf(fact_240_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
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 ) ) ) ).
% concat_append
thf(fact_242_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
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 ) ) ) ).
% concat.simps(2)
thf(fact_244_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
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 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_246_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_247_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
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 ) ) ) ) ) ).
% butlast_append
thf(fact_249_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
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 ) ) ) ) ) ).
% butlast.simps(2)
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 ) ) ).
% append_butlast_last_id
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 ) ) ) ).
% snoc_eq_iff_butlast
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 ) ) ) ).
% last_appendR
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 ) ) ) ).
% last_appendL
% Type constructors (3)
thf(tcon_fun___Orderings_Oord,axiom,
! [A5: $tType,A6: $tType] :
( ( ord @ A6 )
=> ( ord @ ( A5 > A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_1,axiom,
ord @ $o ).
% Conjectures (6)
thf(conj_0,hypothesis,
( b
= ( inf @ b @ c ) ) ).
thf(conj_1,hypothesis,
( ( inf @ c @ ( sup @ b @ c ) )
= b ) ).
thf(conj_2,hypothesis,
( a2
= ( sup @ b @ c ) ) ).
thf(conj_3,hypothesis,
less @ b @ ( sup @ b @ c ) ).
thf(conj_4,hypothesis,
( ( inf @ c @ ( sup @ b @ c ) )
= c ) ).
thf(conj_5,conjecture,
$false ).
%------------------------------------------------------------------------------