TPTP Problem File: ITP119^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP119^1 : 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 : 0.30 v8.2.0, 0.23 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 425 ( 211 unt; 67 typ; 0 def)
% Number of atoms : 818 ( 483 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3280 ( 180 ~; 24 |; 99 &;2599 @)
% ( 0 <=>; 378 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 485 ( 485 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 63 usr; 7 con; 0-6 aty)
% Number of variables : 1176 ( 18 ^;1063 !; 95 ?;1176 :)
% SPC : TH0_CAX_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:43:54.694
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (63)
thf(sy_c_Finite__Set_Ocomp__fun__idem_001tf__a_001tf__a,type,
finite40241356em_a_a: ( a > a > a ) > $o ).
thf(sy_c_Groups_Oabel__semigroup_001t__List__Olist_Itf__a_J,type,
abel_s1075322906list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_Groups_Oabel__semigroup_001tf__a,type,
abel_semigroup_a: ( a > a > a ) > $o ).
thf(sy_c_Groups_Oabel__semigroup__axioms_001t__List__Olist_Itf__a_J,type,
abel_s374399671list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_Groups_Oabel__semigroup__axioms_001tf__a,type,
abel_s2117090225ioms_a: ( a > a > a ) > $o ).
thf(sy_c_Groups_Osemigroup_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
semigr656274520list_a: ( list_list_a > list_list_a > list_list_a ) > $o ).
thf(sy_c_Groups_Osemigroup_001t__List__Olist_Itf__a_J,type,
semigroup_list_a: ( list_a > list_a > list_a ) > $o ).
thf(sy_c_Groups_Osemigroup_001tf__a,type,
semigroup_a: ( a > a > a ) > $o ).
thf(sy_c_Lattices_Osemilattice_001tf__a,type,
semilattice_a: ( a > a > a ) > $o ).
thf(sy_c_Lattices_Osemilattice__axioms_001tf__a,type,
semilattice_axioms_a: ( a > a > a ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001tf__a,type,
lattic1885654924_set_a: ( a > a > a ) > $o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
append_list_list_a: list_list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
bind_list_a_list_a: list_list_a > ( list_a > list_list_a ) > list_list_a ).
thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001tf__a,type,
bind_list_a_a: list_list_a > ( list_a > list_a ) > list_a ).
thf(sy_c_List_Obind_001tf__a_001t__List__Olist_Itf__a_J,type,
bind_a_list_a: list_a > ( a > list_list_a ) > list_list_a ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001t__List__Olist_Itf__a_J,type,
butlast_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Oconcat_001t__List__Olist_Itf__a_J,type,
concat_list_a: list_list_list_a > list_list_a ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001t__List__Olist_Itf__a_J,type,
last_list_a: list_list_a > list_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
cons_list_list_a: list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_Itf__a_J,type,
list_ex1_list_a: ( list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
maps_list_a_list_a: ( list_a > list_list_a ) > list_list_a > list_list_a ).
thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001tf__a,type,
maps_list_a_a: ( list_a > list_a ) > list_list_a > list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__List__Olist_Itf__a_J,type,
maps_a_list_a: ( a > list_list_a ) > list_a > list_list_a ).
thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
maps_a_a: ( a > list_a ) > list_a > list_a ).
thf(sy_c_List_Oord_Olexordp_001t__List__Olist_Itf__a_J,type,
lexordp_list_a: ( list_a > list_a > $o ) > list_list_a > list_list_a > $o ).
thf(sy_c_List_Oord_Olexordp_001tf__a,type,
lexordp_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001t__List__Olist_Itf__a_J,type,
lexordp_eq_list_a: ( list_a > list_a > $o ) > list_list_a > list_list_a > $o ).
thf(sy_c_List_Oord_Olexordp__eq_001tf__a,type,
lexordp_eq_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_Itf__a_J,type,
product_lists_list_a: list_list_list_a > list_list_list_a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Orotate1_001t__List__Olist_Itf__a_J,type,
rotate1_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Osubseqs_001t__List__Olist_Itf__a_J,type,
subseqs_list_a: list_list_a > list_list_list_a ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_OM5__lattice_001tf__a,type,
modula1376131916tice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_ON5__lattice_001tf__a,type,
modula397570059tice_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oa__aux_001tf__a,type,
modula17988509_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Ob__aux_001tf__a,type,
modula1373251614_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oc__aux_001tf__a,type,
modula581031071_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Od__aux_001tf__a,type,
modula1936294176_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Oe__aux_001tf__a,type,
modula1144073633_aux_a: ( a > a > a ) > ( a > a > a ) > a > a > a > a ).
thf(sy_c_Modular__Distrib__Lattice__Mirabelle__ybbibajlty_Olattice_Ono__distrib_001tf__a,type,
modula1962211574trib_a: ( a > a > a ) > ( a > a > $o ) > ( a > a > a ) > a > a > a > $o ).
thf(sy_c_Relation_Otransp_001tf__a,type,
transp_a: ( 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 (352)
thf(fact_0_local_Odual__order_Oasym,axiom,
! [B: a,A: a] :
( ( less @ B @ A )
=> ~ ( less @ A @ B ) ) ).
% local.dual_order.asym
thf(fact_1_local_Odual__order_Ostrict__implies__not__eq,axiom,
! [B: a,A: a] :
( ( less @ B @ A )
=> ( A != B ) ) ).
% local.dual_order.strict_implies_not_eq
thf(fact_2_local_Odual__order_Ostrict__trans,axiom,
! [B: a,A: a,C: a] :
( ( less @ B @ A )
=> ( ( less @ C @ B )
=> ( less @ C @ A ) ) ) ).
% 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,
! [A: a,B: a] :
( ( less @ A @ B )
=> ~ ( less @ B @ A ) ) ).
% 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,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( less @ B @ C )
=> ( less @ A @ C ) ) ) ).
% local.ord_eq_less_trans
thf(fact_14_local_Oord__less__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( less @ A @ B )
=> ( ( B = C )
=> ( less @ A @ C ) ) ) ).
% local.ord_less_eq_trans
thf(fact_15_local_Oorder_Oasym,axiom,
! [A: a,B: a] :
( ( less @ A @ B )
=> ~ ( less @ B @ A ) ) ).
% local.order.asym
thf(fact_16_local_Oorder_Oirrefl,axiom,
! [A: a] :
~ ( less @ A @ A ) ).
% local.order.irrefl
thf(fact_17_local_Oorder_Ostrict__implies__not__eq,axiom,
! [A: a,B: a] :
( ( less @ A @ B )
=> ( A != B ) ) ).
% local.order.strict_implies_not_eq
thf(fact_18_local_Oorder_Ostrict__trans,axiom,
! [A: a,B: a,C: a] :
( ( less @ A @ B )
=> ( ( less @ B @ C )
=> ( less @ A @ C ) ) ) ).
% local.order.strict_trans
thf(fact_19_local_Oinf_Oassoc,axiom,
! [A: a,B: a,C: a] :
( ( inf @ ( inf @ A @ B ) @ C )
= ( inf @ A @ ( inf @ B @ C ) ) ) ).
% local.inf.assoc
thf(fact_20_local_Oinf_Ocommute,axiom,
! [A: a,B: a] :
( ( inf @ A @ B )
= ( inf @ B @ A ) ) ).
% local.inf.commute
thf(fact_21_local_Oinf_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( inf @ B @ ( inf @ A @ C ) )
= ( inf @ A @ ( inf @ B @ 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,
! [A: a,B: a,C: a] :
( ( sup @ ( sup @ A @ B ) @ C )
= ( sup @ A @ ( sup @ B @ C ) ) ) ).
% local.sup.assoc
thf(fact_26_local_Osup_Ocommute,axiom,
! [A: a,B: a] :
( ( sup @ A @ B )
= ( sup @ B @ A ) ) ).
% local.sup.commute
thf(fact_27_local_Osup_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( sup @ B @ ( sup @ A @ C ) )
= ( sup @ A @ ( sup @ B @ 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,
! [A: a,B: a,C: a] :
( ( less @ A @ ( inf @ B @ C ) )
=> ~ ( ( less @ A @ B )
=> ~ ( less @ A @ C ) ) ) ).
% local.inf.strict_boundedE
thf(fact_32_local_Oinf_Ostrict__coboundedI1,axiom,
! [A: a,C: a,B: a] :
( ( less @ A @ C )
=> ( less @ ( inf @ A @ B ) @ C ) ) ).
% local.inf.strict_coboundedI1
thf(fact_33_local_Oinf_Ostrict__coboundedI2,axiom,
! [B: a,C: a,A: a] :
( ( less @ B @ C )
=> ( less @ ( inf @ A @ B ) @ C ) ) ).
% local.inf.strict_coboundedI2
thf(fact_34_local_Oinf_Ostrict__order__iff,axiom,
! [A: a,B: a] :
( ( less @ A @ B )
= ( ( A
= ( inf @ A @ B ) )
& ( A != B ) ) ) ).
% local.inf.strict_order_iff
thf(fact_35_local_Oless__infI1,axiom,
! [A: a,X: a,B: a] :
( ( less @ A @ X )
=> ( less @ ( inf @ A @ B ) @ X ) ) ).
% local.less_infI1
thf(fact_36_local_Oless__infI2,axiom,
! [B: a,X: a,A: a] :
( ( less @ B @ X )
=> ( less @ ( inf @ A @ B ) @ X ) ) ).
% local.less_infI2
thf(fact_37_local_Oless__supI1,axiom,
! [X: a,A: a,B: a] :
( ( less @ X @ A )
=> ( less @ X @ ( sup @ A @ B ) ) ) ).
% local.less_supI1
thf(fact_38_local_Oless__supI2,axiom,
! [X: a,B: a,A: a] :
( ( less @ X @ B )
=> ( less @ X @ ( sup @ A @ B ) ) ) ).
% local.less_supI2
thf(fact_39_local_Osup_Ostrict__boundedE,axiom,
! [B: a,C: a,A: a] :
( ( less @ ( sup @ B @ C ) @ A )
=> ~ ( ( less @ B @ A )
=> ~ ( less @ C @ A ) ) ) ).
% local.sup.strict_boundedE
thf(fact_40_local_Osup_Ostrict__coboundedI1,axiom,
! [C: a,A: a,B: a] :
( ( less @ C @ A )
=> ( less @ C @ ( sup @ A @ B ) ) ) ).
% local.sup.strict_coboundedI1
thf(fact_41_local_Osup_Ostrict__coboundedI2,axiom,
! [C: a,B: a,A: a] :
( ( less @ C @ B )
=> ( less @ C @ ( sup @ A @ B ) ) ) ).
% local.sup.strict_coboundedI2
thf(fact_42_local_Osup_Ostrict__order__iff,axiom,
! [B: a,A: a] :
( ( less @ B @ A )
= ( ( A
= ( sup @ A @ B ) )
& ( A != B ) ) ) ).
% 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_local_Oinf_Oidem,axiom,
! [A: a] :
( ( inf @ A @ A )
= A ) ).
% local.inf.idem
thf(fact_46_local_Oinf_Oleft__idem,axiom,
! [A: a,B: a] :
( ( inf @ A @ ( inf @ A @ B ) )
= ( inf @ A @ B ) ) ).
% local.inf.left_idem
thf(fact_47_local_Oinf_Oright__idem,axiom,
! [A: a,B: a] :
( ( inf @ ( inf @ A @ B ) @ B )
= ( inf @ A @ B ) ) ).
% local.inf.right_idem
thf(fact_48_local_Oinf__idem,axiom,
! [X: a] :
( ( inf @ X @ X )
= X ) ).
% local.inf_idem
thf(fact_49_local_Oinf__left__idem,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( inf @ X @ Y ) )
= ( inf @ X @ Y ) ) ).
% local.inf_left_idem
thf(fact_50_local_Oinf__right__idem,axiom,
! [X: a,Y: a] :
( ( inf @ ( inf @ X @ Y ) @ Y )
= ( inf @ X @ Y ) ) ).
% local.inf_right_idem
thf(fact_51_local_Osup_Oidem,axiom,
! [A: a] :
( ( sup @ A @ A )
= A ) ).
% local.sup.idem
thf(fact_52_local_Osup_Oleft__idem,axiom,
! [A: a,B: a] :
( ( sup @ A @ ( sup @ A @ B ) )
= ( sup @ A @ B ) ) ).
% local.sup.left_idem
thf(fact_53_local_Osup_Oright__idem,axiom,
! [A: a,B: a] :
( ( sup @ ( sup @ A @ B ) @ B )
= ( sup @ A @ B ) ) ).
% local.sup.right_idem
thf(fact_54_local_Osup__idem,axiom,
! [X: a] :
( ( sup @ X @ X )
= X ) ).
% local.sup_idem
thf(fact_55_local_Osup__left__idem,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( sup @ X @ Y ) )
= ( sup @ X @ Y ) ) ).
% local.sup_left_idem
thf(fact_56_M5__lattice__def,axiom,
! [A: a,B: a,C: a] :
( ( modula1376131916tice_a @ inf @ less @ sup @ A @ B @ C )
= ( ( ( inf @ A @ B )
= ( inf @ B @ C ) )
& ( ( inf @ C @ A )
= ( inf @ B @ C ) )
& ( ( sup @ A @ B )
= ( sup @ B @ C ) )
& ( ( sup @ C @ A )
= ( sup @ B @ C ) )
& ( less @ ( inf @ A @ B ) @ ( sup @ A @ B ) ) ) ) ).
% M5_lattice_def
thf(fact_57_N5__lattice__def,axiom,
! [A: a,B: a,C: a] :
( ( modula397570059tice_a @ inf @ less @ sup @ A @ B @ C )
= ( ( ( inf @ A @ C )
= ( inf @ B @ C ) )
& ( less @ A @ B )
& ( ( sup @ A @ C )
= ( sup @ B @ C ) ) ) ) ).
% N5_lattice_def
thf(fact_58_no__distrib__def,axiom,
! [A: a,B: a,C: a] :
( ( modula1962211574trib_a @ inf @ less @ sup @ A @ B @ C )
= ( less @ ( sup @ ( inf @ A @ B ) @ ( inf @ C @ A ) ) @ ( inf @ A @ ( sup @ B @ C ) ) ) ) ).
% no_distrib_def
thf(fact_59_local_Oinf__sup__absorb,axiom,
! [X: a,Y: a] :
( ( inf @ X @ ( sup @ X @ Y ) )
= X ) ).
% local.inf_sup_absorb
thf(fact_60_local_Osup__inf__absorb,axiom,
! [X: a,Y: a] :
( ( sup @ X @ ( inf @ X @ Y ) )
= X ) ).
% local.sup_inf_absorb
thf(fact_61_local_Oa__join__d,axiom,
! [A: a,B: a,C: a] :
( ( sup @ A @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) )
= ( sup @ A @ ( inf @ B @ C ) ) ) ).
% local.a_join_d
thf(fact_62_local_Ob__join__d,axiom,
! [B: a,A: a,C: a] :
( ( sup @ B @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) )
= ( sup @ B @ ( inf @ C @ A ) ) ) ).
% local.b_join_d
thf(fact_63_local_Od__aux__def,axiom,
! [A: a,B: a,C: a] :
( ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C )
= ( sup @ ( sup @ ( inf @ A @ B ) @ ( inf @ B @ C ) ) @ ( inf @ C @ A ) ) ) ).
% local.d_aux_def
thf(fact_64_local_Od__b__c__a,axiom,
! [B: a,C: a,A: a] :
( ( modula1936294176_aux_a @ inf @ sup @ B @ C @ A )
= ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% local.d_b_c_a
thf(fact_65_local_Od__c__a__b,axiom,
! [C: a,A: a,B: a] :
( ( modula1936294176_aux_a @ inf @ sup @ C @ A @ B )
= ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% local.d_c_a_b
thf(fact_66_local_Oa__meet__e,axiom,
! [A: a,B: a,C: a] :
( ( inf @ A @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) )
= ( inf @ A @ ( sup @ B @ C ) ) ) ).
% local.a_meet_e
thf(fact_67_local_Ob__meet__e,axiom,
! [B: a,A: a,C: a] :
( ( inf @ B @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) )
= ( inf @ B @ ( sup @ C @ A ) ) ) ).
% local.b_meet_e
thf(fact_68_local_Oc__meet__e,axiom,
! [C: a,A: a,B: a] :
( ( inf @ C @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) )
= ( inf @ C @ ( sup @ A @ B ) ) ) ).
% local.c_meet_e
thf(fact_69_local_Oe__aux__def,axiom,
! [A: a,B: a,C: a] :
( ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C )
= ( inf @ ( inf @ ( sup @ A @ B ) @ ( sup @ B @ C ) ) @ ( sup @ C @ A ) ) ) ).
% local.e_aux_def
thf(fact_70_local_Oe__b__c__a,axiom,
! [B: a,C: a,A: a] :
( ( modula1144073633_aux_a @ inf @ sup @ B @ C @ A )
= ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% local.e_b_c_a
thf(fact_71_local_Oe__c__a__b,axiom,
! [C: a,A: a,B: a] :
( ( modula1144073633_aux_a @ inf @ sup @ C @ A @ B )
= ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) ).
% local.e_c_a_b
thf(fact_72_local_Ocomp__fun__idem__sup,axiom,
finite40241356em_a_a @ sup ).
% local.comp_fun_idem_sup
thf(fact_73_local_Ocomp__fun__idem__inf,axiom,
finite40241356em_a_a @ inf ).
% local.comp_fun_idem_inf
thf(fact_74_local_Oc__aux__def,axiom,
! [A: a,B: a,C: a] :
( ( modula581031071_aux_a @ inf @ sup @ A @ B @ C )
= ( sup @ ( inf @ C @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ) ).
% local.c_aux_def
thf(fact_75_local_Ob__aux__def,axiom,
! [A: a,B: a,C: a] :
( ( modula1373251614_aux_a @ inf @ sup @ A @ B @ C )
= ( sup @ ( inf @ B @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ) ).
% local.b_aux_def
thf(fact_76_local_Oa__aux__def,axiom,
! [A: a,B: a,C: a] :
( ( modula17988509_aux_a @ inf @ sup @ A @ B @ C )
= ( sup @ ( inf @ A @ ( modula1144073633_aux_a @ inf @ sup @ A @ B @ C ) ) @ ( modula1936294176_aux_a @ inf @ sup @ A @ B @ C ) ) ) ).
% local.a_aux_def
thf(fact_77_lattice_Ono__distrib_Ocong,axiom,
modula1962211574trib_a = modula1962211574trib_a ).
% lattice.no_distrib.cong
thf(fact_78_lattice_ON5__lattice_Ocong,axiom,
modula397570059tice_a = modula397570059tice_a ).
% lattice.N5_lattice.cong
thf(fact_79_lattice_OM5__lattice_Ocong,axiom,
modula1376131916tice_a = modula1376131916tice_a ).
% lattice.M5_lattice.cong
thf(fact_80_lattice_Oe__aux_Ocong,axiom,
modula1144073633_aux_a = modula1144073633_aux_a ).
% lattice.e_aux.cong
thf(fact_81_lattice_Od__aux_Ocong,axiom,
modula1936294176_aux_a = modula1936294176_aux_a ).
% lattice.d_aux.cong
thf(fact_82_local_Oc__a,axiom,
! [A: a,B: a,C: a] :
( ( modula581031071_aux_a @ inf @ sup @ A @ B @ C )
= ( modula17988509_aux_a @ inf @ sup @ C @ A @ B ) ) ).
% local.c_a
thf(fact_83_local_Ob__a,axiom,
! [A: a,B: a,C: a] :
( ( modula1373251614_aux_a @ inf @ sup @ A @ B @ C )
= ( modula17988509_aux_a @ inf @ sup @ B @ C @ A ) ) ).
% local.b_a
thf(fact_84_local_Osup_Osemigroup__axioms,axiom,
semigroup_a @ sup ).
% local.sup.semigroup_axioms
thf(fact_85_local_Oinf_Osemigroup__axioms,axiom,
semigroup_a @ inf ).
% local.inf.semigroup_axioms
thf(fact_86_local_Osup_Osemilattice__axioms,axiom,
semilattice_a @ sup ).
% local.sup.semilattice_axioms
thf(fact_87_local_Oinf_Osemilattice__axioms,axiom,
semilattice_a @ inf ).
% local.inf.semilattice_axioms
thf(fact_88_lattice_Oa__aux_Ocong,axiom,
modula17988509_aux_a = modula17988509_aux_a ).
% lattice.a_aux.cong
thf(fact_89_lattice_Ob__aux_Ocong,axiom,
modula1373251614_aux_a = modula1373251614_aux_a ).
% lattice.b_aux.cong
thf(fact_90_lattice_Oc__aux_Ocong,axiom,
modula581031071_aux_a = modula581031071_aux_a ).
% lattice.c_aux.cong
thf(fact_91_local_Osup_Oabel__semigroup__axioms,axiom,
abel_semigroup_a @ sup ).
% local.sup.abel_semigroup_axioms
thf(fact_92_local_Oinf_Oabel__semigroup__axioms,axiom,
abel_semigroup_a @ inf ).
% local.inf.abel_semigroup_axioms
thf(fact_93_local_OSup__fin_Osemilattice__set__axioms,axiom,
lattic1885654924_set_a @ sup ).
% local.Sup_fin.semilattice_set_axioms
thf(fact_94_local_OInf__fin_Osemilattice__set__axioms,axiom,
lattic1885654924_set_a @ inf ).
% local.Inf_fin.semilattice_set_axioms
thf(fact_95_local_Olexordp__eq__refl,axiom,
! [Xs: list_a] : ( lexordp_eq_a @ less @ Xs @ Xs ) ).
% local.lexordp_eq_refl
thf(fact_96_local_Olexordp__eq_ONil,axiom,
! [Ys: list_a] : ( lexordp_eq_a @ less @ nil_a @ Ys ) ).
% local.lexordp_eq.Nil
thf(fact_97_abel__semigroup_Oaxioms_I1_J,axiom,
! [F: list_a > list_a > list_a] :
( ( abel_s1075322906list_a @ F )
=> ( semigroup_list_a @ F ) ) ).
% abel_semigroup.axioms(1)
thf(fact_98_abel__semigroup_Oaxioms_I1_J,axiom,
! [F: a > a > a] :
( ( abel_semigroup_a @ F )
=> ( semigroup_a @ F ) ) ).
% abel_semigroup.axioms(1)
thf(fact_99_semilattice__set_Oaxioms,axiom,
! [F: a > a > a] :
( ( lattic1885654924_set_a @ F )
=> ( semilattice_a @ F ) ) ).
% semilattice_set.axioms
thf(fact_100_semilattice__set_Ointro,axiom,
! [F: a > a > a] :
( ( semilattice_a @ F )
=> ( lattic1885654924_set_a @ F ) ) ).
% semilattice_set.intro
thf(fact_101_semilattice__set__def,axiom,
lattic1885654924_set_a = semilattice_a ).
% semilattice_set_def
thf(fact_102_local_Olexordp__eq__simps_I1_J,axiom,
! [Ys: list_a] : ( lexordp_eq_a @ less @ nil_a @ Ys ) ).
% local.lexordp_eq_simps(1)
thf(fact_103_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_104_abel__semigroup_Oleft__commute,axiom,
! [F: a > a > a,B: a,A: a,C: a] :
( ( abel_semigroup_a @ F )
=> ( ( F @ B @ ( F @ A @ C ) )
= ( F @ A @ ( F @ B @ C ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_105_abel__semigroup_Ocommute,axiom,
! [F: a > a > a,A: a,B: a] :
( ( abel_semigroup_a @ F )
=> ( ( F @ A @ B )
= ( F @ B @ A ) ) ) ).
% abel_semigroup.commute
thf(fact_106_semigroup_Ointro,axiom,
! [F: list_a > list_a > list_a] :
( ! [A2: list_a,B2: list_a,C2: list_a] :
( ( F @ ( F @ A2 @ B2 ) @ C2 )
= ( F @ A2 @ ( F @ B2 @ C2 ) ) )
=> ( semigroup_list_a @ F ) ) ).
% semigroup.intro
thf(fact_107_semigroup_Ointro,axiom,
! [F: a > a > a] :
( ! [A2: a,B2: a,C2: a] :
( ( F @ ( F @ A2 @ B2 ) @ C2 )
= ( F @ A2 @ ( F @ B2 @ C2 ) ) )
=> ( semigroup_a @ F ) ) ).
% semigroup.intro
thf(fact_108_semigroup_Oassoc,axiom,
! [F: list_a > list_a > list_a,A: list_a,B: list_a,C: list_a] :
( ( semigroup_list_a @ F )
=> ( ( F @ ( F @ A @ B ) @ C )
= ( F @ A @ ( F @ B @ C ) ) ) ) ).
% semigroup.assoc
thf(fact_109_semigroup_Oassoc,axiom,
! [F: a > a > a,A: a,B: a,C: a] :
( ( semigroup_a @ F )
=> ( ( F @ ( F @ A @ B ) @ C )
= ( F @ A @ ( F @ B @ C ) ) ) ) ).
% semigroup.assoc
thf(fact_110_semigroup__def,axiom,
( semigroup_list_a
= ( ^ [F2: list_a > list_a > list_a] :
! [A3: list_a,B3: list_a,C3: list_a] :
( ( F2 @ ( F2 @ A3 @ B3 ) @ C3 )
= ( F2 @ A3 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% semigroup_def
thf(fact_111_semigroup__def,axiom,
( semigroup_a
= ( ^ [F2: a > a > a] :
! [A3: a,B3: a,C3: a] :
( ( F2 @ ( F2 @ A3 @ B3 ) @ C3 )
= ( F2 @ A3 @ ( F2 @ B3 @ C3 ) ) ) ) ) ).
% semigroup_def
thf(fact_112_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: list_a > list_a > $o,Xs: list_list_a] :
( ( lexordp_eq_list_a @ Less @ Xs @ nil_list_a )
= ( Xs = nil_list_a ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_113_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: a > a > $o,Xs: list_a] :
( ( lexordp_eq_a @ Less @ Xs @ nil_a )
= ( Xs = nil_a ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_114_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: list_a > list_a > $o,Ys: list_list_a] : ( lexordp_eq_list_a @ Less @ nil_list_a @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_115_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_116_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_117_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_118_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_119_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_120_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_121_list_Oinject,axiom,
! [X21: list_a,X222: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X222 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_122_list_Oinject,axiom,
! [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,
! [Less: list_a > list_a > $o,X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( lexordp_eq_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_list_a @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_124_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: a > a > $o,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 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_125_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: list_a > list_a > $o,X: list_a,Xs: list_list_a] :
~ ( lexordp_eq_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ nil_list_a ) ).
% ord.lexordp_eq_simps(3)
thf(fact_126_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: a > a > $o,X: a,Xs: list_a] :
~ ( lexordp_eq_a @ Less @ ( cons_a @ X @ Xs ) @ nil_a ) ).
% ord.lexordp_eq_simps(3)
thf(fact_127_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_128_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_129_map__tailrec__rev_Oinduct,axiom,
! [P: ( list_a > a ) > list_list_a > list_a > $o,A0: list_a > a,A1: list_list_a,A22: list_a] :
( ! [F3: list_a > a,X_1: list_a] : ( P @ F3 @ nil_list_a @ X_1 )
=> ( ! [F3: list_a > a,A2: list_a,As: list_list_a,Bs: list_a] :
( ( P @ F3 @ As @ ( cons_a @ ( F3 @ A2 ) @ Bs ) )
=> ( P @ F3 @ ( cons_list_a @ A2 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_130_map__tailrec__rev_Oinduct,axiom,
! [P: ( a > list_a ) > list_a > list_list_a > $o,A0: a > list_a,A1: list_a,A22: list_list_a] :
( ! [F3: a > list_a,X_1: list_list_a] : ( P @ F3 @ nil_a @ X_1 )
=> ( ! [F3: a > list_a,A2: a,As: list_a,Bs: list_list_a] :
( ( P @ F3 @ As @ ( cons_list_a @ ( F3 @ A2 ) @ Bs ) )
=> ( P @ F3 @ ( cons_a @ A2 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_131_map__tailrec__rev_Oinduct,axiom,
! [P: ( list_a > list_a ) > list_list_a > list_list_a > $o,A0: list_a > list_a,A1: list_list_a,A22: list_list_a] :
( ! [F3: list_a > list_a,X_1: list_list_a] : ( P @ F3 @ nil_list_a @ X_1 )
=> ( ! [F3: list_a > list_a,A2: list_a,As: list_list_a,Bs: list_list_a] :
( ( P @ F3 @ As @ ( cons_list_a @ ( F3 @ A2 ) @ Bs ) )
=> ( P @ F3 @ ( cons_list_a @ A2 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_132_map__tailrec__rev_Oinduct,axiom,
! [P: ( a > a ) > list_a > list_a > $o,A0: a > a,A1: list_a,A22: list_a] :
( ! [F3: a > a,X_1: list_a] : ( P @ F3 @ nil_a @ X_1 )
=> ( ! [F3: a > a,A2: a,As: list_a,Bs: list_a] :
( ( P @ F3 @ As @ ( cons_a @ ( F3 @ A2 ) @ Bs ) )
=> ( P @ F3 @ ( cons_a @ A2 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_133_list__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Xs3: list_list_a] :
( ( Xs3 != nil_list_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_134_list__nonempty__induct,axiom,
! [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_135_successively_Oinduct,axiom,
! [P: ( list_a > list_a > $o ) > list_list_a > $o,A0: list_a > list_a > $o,A1: list_list_a] :
( ! [P2: list_a > list_a > $o] : ( P @ P2 @ nil_list_a )
=> ( ! [P2: list_a > list_a > $o,X2: list_a] : ( P @ P2 @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [P2: list_a > list_a > $o,X2: list_a,Y2: list_a,Xs3: list_list_a] :
( ( P @ P2 @ ( cons_list_a @ Y2 @ Xs3 ) )
=> ( P @ P2 @ ( cons_list_a @ X2 @ ( cons_list_a @ Y2 @ Xs3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_136_successively_Oinduct,axiom,
! [P: ( a > a > $o ) > list_a > $o,A0: a > a > $o,A1: list_a] :
( ! [P2: a > a > $o] : ( P @ P2 @ nil_a )
=> ( ! [P2: a > a > $o,X2: a] : ( P @ P2 @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [P2: a > a > $o,X2: a,Y2: a,Xs3: list_a] :
( ( P @ P2 @ ( cons_a @ Y2 @ Xs3 ) )
=> ( P @ P2 @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_137_remdups__adj_Oinduct,axiom,
! [P: list_list_a > $o,A0: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Y2: list_a,Xs3: list_list_a] :
( ( ( X2 = Y2 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) ) )
=> ( ( ( X2 != Y2 )
=> ( P @ ( cons_list_a @ Y2 @ Xs3 ) ) )
=> ( P @ ( cons_list_a @ X2 @ ( cons_list_a @ Y2 @ Xs3 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_138_remdups__adj_Oinduct,axiom,
! [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_139_sorted__wrt_Oinduct,axiom,
! [P: ( list_a > list_a > $o ) > list_list_a > $o,A0: list_a > list_a > $o,A1: list_list_a] :
( ! [P2: list_a > list_a > $o] : ( P @ P2 @ nil_list_a )
=> ( ! [P2: list_a > list_a > $o,X2: list_a,Ys3: list_list_a] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons_list_a @ X2 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_140_sorted__wrt_Oinduct,axiom,
! [P: ( a > a > $o ) > list_a > $o,A0: a > a > $o,A1: list_a] :
( ! [P2: a > a > $o] : ( P @ P2 @ nil_a )
=> ( ! [P2: a > a > $o,X2: a,Ys3: list_a] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons_a @ X2 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_141_remdups__adj_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [X2: list_a] :
( X
!= ( cons_list_a @ X2 @ nil_list_a ) )
=> ~ ! [X2: list_a,Y2: list_a,Xs3: list_list_a] :
( X
!= ( cons_list_a @ X2 @ ( cons_list_a @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_142_remdups__adj_Ocases,axiom,
! [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_143_transpose_Ocases,axiom,
! [X: list_list_list_a] :
( ( X != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X2: list_a,Xs3: list_list_a,Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ ( cons_list_a @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_144_transpose_Ocases,axiom,
! [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_145_shuffles_Oinduct,axiom,
! [P: list_list_a > list_list_a > $o,A0: list_list_a,A1: list_list_a] :
( ! [X_1: list_list_a] : ( P @ nil_list_a @ X_1 )
=> ( ! [Xs3: list_list_a] : ( P @ Xs3 @ nil_list_a )
=> ( ! [X2: list_a,Xs3: list_list_a,Y2: list_a,Ys3: list_list_a] :
( ( P @ Xs3 @ ( cons_list_a @ Y2 @ Ys3 ) )
=> ( ( P @ ( cons_list_a @ X2 @ Xs3 ) @ Ys3 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) @ ( cons_list_a @ Y2 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_146_shuffles_Oinduct,axiom,
! [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_147_induct__list012,axiom,
! [P: list_list_a > $o,Xs: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Y2: list_a,Zs: list_list_a] :
( ( P @ Zs )
=> ( ( P @ ( cons_list_a @ Y2 @ Zs ) )
=> ( P @ ( cons_list_a @ X2 @ ( cons_list_a @ Y2 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_148_induct__list012,axiom,
! [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_149_splice_Oinduct,axiom,
! [P: list_list_a > list_list_a > $o,A0: list_list_a,A1: list_list_a] :
( ! [X_1: list_list_a] : ( P @ nil_list_a @ X_1 )
=> ( ! [X2: list_a,Xs3: list_list_a,Ys3: list_list_a] :
( ( P @ Ys3 @ Xs3 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_150_splice_Oinduct,axiom,
! [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_151_list__induct2_H,axiom,
! [P: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P @ nil_a @ nil_list_a )
=> ( ! [X2: a,Xs3: list_a] : ( P @ ( cons_a @ X2 @ Xs3 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys3: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y2 @ Ys3 ) )
=> ( ! [X2: a,Xs3: list_a,Y2: list_a,Ys3: list_list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_list_a @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_152_list__induct2_H,axiom,
! [P: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P @ nil_list_a @ nil_a )
=> ( ! [X2: list_a,Xs3: list_list_a] : ( P @ ( cons_list_a @ X2 @ Xs3 ) @ nil_a )
=> ( ! [Y2: a,Ys3: list_a] : ( P @ nil_list_a @ ( cons_a @ Y2 @ Ys3 ) )
=> ( ! [X2: list_a,Xs3: list_list_a,Y2: a,Ys3: list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_153_list__induct2_H,axiom,
! [P: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X2: list_a,Xs3: list_list_a] : ( P @ ( cons_list_a @ X2 @ Xs3 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys3: list_list_a] : ( P @ nil_list_a @ ( cons_list_a @ Y2 @ Ys3 ) )
=> ( ! [X2: list_a,Xs3: list_list_a,Y2: list_a,Ys3: list_list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) @ ( cons_list_a @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_154_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs3: list_a] : ( P @ ( cons_a @ X2 @ Xs3 ) @ nil_a )
=> ( ! [Y2: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys3 ) )
=> ( ! [X2: a,Xs3: list_a,Y2: a,Ys3: list_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_155_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y3: list_a,Ys2: list_list_a] :
( Xs
= ( cons_list_a @ Y3 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_156_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_157_list_Oinducts,axiom,
! [P: list_list_a > $o,List: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X12: list_a,X23: list_list_a] :
( ( P @ X23 )
=> ( P @ ( cons_list_a @ X12 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_158_list_Oinducts,axiom,
! [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_159_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X212: list_a,X223: list_list_a] :
( Y
!= ( cons_list_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_160_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_161_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X222: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X222 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_162_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_163_list_Odistinct_I1_J,axiom,
! [X21: list_a,X222: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_164_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_165_not__Cons__self2,axiom,
! [X: list_a,Xs: list_list_a] :
( ( cons_list_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_166_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_167_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: list_a > list_a > $o,X: list_a,Y: list_a,Xs: list_list_a,Ys: list_list_a] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_list_a @ Less @ Xs @ Ys )
=> ( lexordp_eq_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_168_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: a > a > $o,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 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_169_ord_Olexordp__eq_OCons,axiom,
! [Less: list_a > list_a > $o,X: list_a,Y: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( Less @ X @ Y )
=> ( lexordp_eq_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_170_ord_Olexordp__eq_OCons,axiom,
! [Less: a > a > $o,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 ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_171_ord_Olexordp__eq_Oinducts,axiom,
! [Less: list_a > list_a > $o,X1: list_list_a,X22: list_list_a,P: list_list_a > list_list_a > $o] :
( ( lexordp_eq_list_a @ Less @ X1 @ X22 )
=> ( ! [X_1: list_list_a] : ( P @ nil_list_a @ X_1 )
=> ( ! [X2: list_a,Y2: list_a,Xs3: list_list_a,Ys3: list_list_a] :
( ( Less @ X2 @ Y2 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) @ ( cons_list_a @ Y2 @ Ys3 ) ) )
=> ( ! [X2: list_a,Y2: list_a,Xs3: list_list_a,Ys3: list_list_a] :
( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ( ( lexordp_eq_list_a @ Less @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) @ ( cons_list_a @ Y2 @ Ys3 ) ) ) ) ) )
=> ( P @ X1 @ X22 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_172_ord_Olexordp__eq_Oinducts,axiom,
! [Less: a > a > $o,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 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_173_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_list_a
= ( ^ [Less2: list_a > list_a > $o,A12: list_list_a,A23: list_list_a] :
( ? [Ys2: list_list_a] :
( ( A12 = nil_list_a )
& ( A23 = Ys2 ) )
| ? [X3: list_a,Y3: list_a,Xs2: list_list_a,Ys2: list_list_a] :
( ( A12
= ( cons_list_a @ X3 @ Xs2 ) )
& ( A23
= ( cons_list_a @ Y3 @ Ys2 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: list_a,Y3: list_a,Xs2: list_list_a,Ys2: list_list_a] :
( ( A12
= ( cons_list_a @ X3 @ Xs2 ) )
& ( A23
= ( cons_list_a @ Y3 @ Ys2 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_eq_list_a @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_174_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_a
= ( ^ [Less2: 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 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: a,Y3: a,Xs2: list_a,Ys2: list_a] :
( ( A12
= ( cons_a @ X3 @ Xs2 ) )
& ( A23
= ( cons_a @ Y3 @ Ys2 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_eq_a @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_175_ord_Olexordp__eq_Ocases,axiom,
! [Less: list_a > list_a > $o,A1: list_list_a,A22: list_list_a] :
( ( lexordp_eq_list_a @ Less @ A1 @ A22 )
=> ( ( A1 != nil_list_a )
=> ( ! [X2: list_a] :
( ? [Xs3: list_list_a] :
( A1
= ( cons_list_a @ X2 @ Xs3 ) )
=> ! [Y2: list_a] :
( ? [Ys3: list_list_a] :
( A22
= ( cons_list_a @ Y2 @ Ys3 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: list_a,Y2: list_a,Xs3: list_list_a] :
( ( A1
= ( cons_list_a @ X2 @ Xs3 ) )
=> ! [Ys3: list_list_a] :
( ( A22
= ( cons_list_a @ Y2 @ Ys3 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexordp_eq_list_a @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_176_ord_Olexordp__eq_Ocases,axiom,
! [Less: a > a > $o,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 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_177_ord_Olexordp__eq_Ocong,axiom,
lexordp_eq_a = lexordp_eq_a ).
% ord.lexordp_eq.cong
thf(fact_178_ord_Olexordp__eq__refl,axiom,
! [Less: a > a > $o,Xs: list_a] : ( lexordp_eq_a @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_179_ord_Olexordp__eq_ONil,axiom,
! [Less: list_a > list_a > $o,Ys: list_list_a] : ( lexordp_eq_list_a @ Less @ nil_list_a @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_180_ord_Olexordp__eq_ONil,axiom,
! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_181_local_Olexordp_ONil,axiom,
! [Y: a,Ys: list_a] : ( lexordp_a @ less @ nil_a @ ( cons_a @ Y @ Ys ) ) ).
% local.lexordp.Nil
thf(fact_182_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_183_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_184_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_185_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_186_local_Olexordp__irreflexive,axiom,
! [Xs: list_a] :
( ! [X2: a] :
~ ( less @ X2 @ X2 )
=> ~ ( lexordp_a @ less @ Xs @ Xs ) ) ).
% local.lexordp_irreflexive
thf(fact_187_local_Olexordp__irreflexive_H,axiom,
! [Xs: list_a] :
~ ( lexordp_a @ less @ Xs @ Xs ) ).
% local.lexordp_irreflexive'
thf(fact_188_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_189_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_190_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_191_ord_Olexordp__simps_I3_J,axiom,
! [Less: list_a > list_a > $o,X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( lexordp_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_list_a @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_192_ord_Olexordp__simps_I3_J,axiom,
! [Less: a > a > $o,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 ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_193_ord_Olexordp__simps_I2_J,axiom,
! [Less: list_a > list_a > $o,Xs: list_list_a] :
~ ( lexordp_list_a @ Less @ Xs @ nil_list_a ) ).
% ord.lexordp_simps(2)
thf(fact_194_ord_Olexordp__simps_I2_J,axiom,
! [Less: a > a > $o,Xs: list_a] :
~ ( lexordp_a @ Less @ Xs @ nil_a ) ).
% ord.lexordp_simps(2)
thf(fact_195_ord_Olexordp__simps_I1_J,axiom,
! [Less: list_a > list_a > $o,Ys: list_list_a] :
( ( lexordp_list_a @ Less @ nil_list_a @ Ys )
= ( Ys != nil_list_a ) ) ).
% ord.lexordp_simps(1)
thf(fact_196_ord_Olexordp__simps_I1_J,axiom,
! [Less: a > a > $o,Ys: list_a] :
( ( lexordp_a @ Less @ nil_a @ Ys )
= ( Ys != nil_a ) ) ).
% ord.lexordp_simps(1)
thf(fact_197_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_198_local_Olexordp__simps_I1_J,axiom,
! [Ys: list_a] :
( ( lexordp_a @ less @ nil_a @ Ys )
= ( Ys != nil_a ) ) ).
% local.lexordp_simps(1)
thf(fact_199_local_Olexordp__simps_I2_J,axiom,
! [Xs: list_a] :
~ ( lexordp_a @ less @ Xs @ nil_a ) ).
% local.lexordp_simps(2)
thf(fact_200_ord_Olexordp_Ocong,axiom,
lexordp_a = lexordp_a ).
% ord.lexordp.cong
thf(fact_201_ord_Olexordp__irreflexive,axiom,
! [Less: a > a > $o,Xs: list_a] :
( ! [X2: a] :
~ ( Less @ X2 @ X2 )
=> ~ ( lexordp_a @ Less @ Xs @ Xs ) ) ).
% ord.lexordp_irreflexive
thf(fact_202_ord_Olexordp_OCons__eq,axiom,
! [Less: list_a > list_a > $o,X: list_a,Y: list_a,Xs: list_list_a,Ys: list_list_a] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_list_a @ Less @ Xs @ Ys )
=> ( lexordp_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_203_ord_Olexordp_OCons__eq,axiom,
! [Less: a > a > $o,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 ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_204_ord_Olexordp_OCons,axiom,
! [Less: list_a > list_a > $o,X: list_a,Y: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( Less @ X @ Y )
=> ( lexordp_list_a @ Less @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_205_ord_Olexordp_OCons,axiom,
! [Less: a > a > $o,X: a,Y: a,Xs: list_a,Ys: list_a] :
( ( Less @ X @ Y )
=> ( lexordp_a @ Less @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_206_ord_Olexordp__into__lexordp__eq,axiom,
! [Less: a > a > $o,Xs: list_a,Ys: list_a] :
( ( lexordp_a @ Less @ Xs @ Ys )
=> ( lexordp_eq_a @ Less @ Xs @ Ys ) ) ).
% ord.lexordp_into_lexordp_eq
thf(fact_207_ord_Olexordp_ONil,axiom,
! [Less: list_a > list_a > $o,Y: list_a,Ys: list_list_a] : ( lexordp_list_a @ Less @ nil_list_a @ ( cons_list_a @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_208_ord_Olexordp_ONil,axiom,
! [Less: a > a > $o,Y: a,Ys: list_a] : ( lexordp_a @ Less @ nil_a @ ( cons_a @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_209_ord_Olexordp_Ocases,axiom,
! [Less: list_a > list_a > $o,A1: list_list_a,A22: list_list_a] :
( ( lexordp_list_a @ Less @ A1 @ A22 )
=> ( ( ( A1 = nil_list_a )
=> ! [Y2: list_a,Ys3: list_list_a] :
( A22
!= ( cons_list_a @ Y2 @ Ys3 ) ) )
=> ( ! [X2: list_a] :
( ? [Xs3: list_list_a] :
( A1
= ( cons_list_a @ X2 @ Xs3 ) )
=> ! [Y2: list_a] :
( ? [Ys3: list_list_a] :
( A22
= ( cons_list_a @ Y2 @ Ys3 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: list_a,Y2: list_a,Xs3: list_list_a] :
( ( A1
= ( cons_list_a @ X2 @ Xs3 ) )
=> ! [Ys3: list_list_a] :
( ( A22
= ( cons_list_a @ Y2 @ Ys3 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexordp_list_a @ Less @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_210_ord_Olexordp_Ocases,axiom,
! [Less: a > a > $o,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 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_211_ord_Olexordp_Osimps,axiom,
( lexordp_list_a
= ( ^ [Less2: list_a > list_a > $o,A12: list_list_a,A23: list_list_a] :
( ? [Y3: list_a,Ys2: list_list_a] :
( ( A12 = nil_list_a )
& ( A23
= ( cons_list_a @ Y3 @ Ys2 ) ) )
| ? [X3: list_a,Y3: list_a,Xs2: list_list_a,Ys2: list_list_a] :
( ( A12
= ( cons_list_a @ X3 @ Xs2 ) )
& ( A23
= ( cons_list_a @ Y3 @ Ys2 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: list_a,Y3: list_a,Xs2: list_list_a,Ys2: list_list_a] :
( ( A12
= ( cons_list_a @ X3 @ Xs2 ) )
& ( A23
= ( cons_list_a @ Y3 @ Ys2 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_list_a @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_212_ord_Olexordp_Osimps,axiom,
( lexordp_a
= ( ^ [Less2: 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 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: a,Y3: a,Xs2: list_a,Ys2: list_a] :
( ( A12
= ( cons_a @ X3 @ Xs2 ) )
& ( A23
= ( cons_a @ Y3 @ Ys2 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_a @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_213_ord_Olexordp_Oinducts,axiom,
! [Less: list_a > list_a > $o,X1: list_list_a,X22: list_list_a,P: list_list_a > list_list_a > $o] :
( ( lexordp_list_a @ Less @ X1 @ X22 )
=> ( ! [Y2: list_a,Ys3: list_list_a] : ( P @ nil_list_a @ ( cons_list_a @ Y2 @ Ys3 ) )
=> ( ! [X2: list_a,Y2: list_a,Xs3: list_list_a,Ys3: list_list_a] :
( ( Less @ X2 @ Y2 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) @ ( cons_list_a @ Y2 @ Ys3 ) ) )
=> ( ! [X2: list_a,Y2: list_a,Xs3: list_list_a,Ys3: list_list_a] :
( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ( ( lexordp_list_a @ Less @ Xs3 @ Ys3 )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_a @ X2 @ Xs3 ) @ ( cons_list_a @ Y2 @ Ys3 ) ) ) ) ) )
=> ( P @ X1 @ X22 ) ) ) ) ) ).
% ord.lexordp.inducts
thf(fact_214_ord_Olexordp_Oinducts,axiom,
! [Less: a > a > $o,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 ) ) ) ) ) ).
% ord.lexordp.inducts
thf(fact_215_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_216_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_217_semilattice_Oaxioms_I1_J,axiom,
! [F: a > a > a] :
( ( semilattice_a @ F )
=> ( abel_semigroup_a @ F ) ) ).
% semilattice.axioms(1)
thf(fact_218_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 ) )
=> ( ! [A2: a] :
~ ( less @ A2 @ A2 )
=> ( lexordp_a @ less @ Us @ Vs ) ) ) ).
% local.lexordp_append_leftD
thf(fact_219_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_220_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_221_same__append__eq,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs2: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_222_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs2: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_223_append__same__eq,axiom,
! [Ys: list_list_a,Xs: list_list_a,Zs2: list_list_a] :
( ( ( append_list_a @ Ys @ Xs )
= ( append_list_a @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_224_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs2: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_225_append__assoc,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs2: list_list_a] :
( ( append_list_a @ ( append_list_a @ Xs @ Ys ) @ Zs2 )
= ( append_list_a @ Xs @ ( append_list_a @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_226_append__assoc,axiom,
! [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_227_append_Oassoc,axiom,
! [A: list_list_a,B: list_list_a,C: list_list_a] :
( ( append_list_a @ ( append_list_a @ A @ B ) @ C )
= ( append_list_a @ A @ ( append_list_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_228_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_229_append_Oright__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ A @ nil_list_a )
= A ) ).
% append.right_neutral
thf(fact_230_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_231_append__is__Nil__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= nil_list_a )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_232_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_233_Nil__is__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs @ Ys ) )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_234_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_235_self__append__conv2,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( Ys
= ( append_list_a @ Xs @ Ys ) )
= ( Xs = nil_list_a ) ) ).
% self_append_conv2
thf(fact_236_self__append__conv2,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys
= ( append_a @ Xs @ Ys ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_237_append__self__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_a ) ) ).
% append_self_conv2
thf(fact_238_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_239_self__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs
= ( append_list_a @ Xs @ Ys ) )
= ( Ys = nil_list_a ) ) ).
% self_append_conv
thf(fact_240_self__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs
= ( append_a @ Xs @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_241_append__self__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_242_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_243_append__Nil2,axiom,
! [Xs: list_list_a] :
( ( append_list_a @ Xs @ nil_list_a )
= Xs ) ).
% append_Nil2
thf(fact_244_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_245_append1__eq__conv,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_246_append1__eq__conv,axiom,
! [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_247_append_Oleft__neutral,axiom,
! [A: list_list_a] :
( ( append_list_a @ nil_list_a @ A )
= A ) ).
% append.left_neutral
thf(fact_248_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_249_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_250_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_251_eq__Nil__appendI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_252_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_253_Cons__eq__appendI,axiom,
! [X: list_a,Xs1: list_list_a,Ys: list_list_a,Xs: list_list_a,Zs2: list_list_a] :
( ( ( cons_list_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_a @ Xs1 @ Zs2 ) )
=> ( ( cons_list_a @ X @ Xs )
= ( append_list_a @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_254_Cons__eq__appendI,axiom,
! [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_255_append__Cons,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( append_list_a @ ( cons_list_a @ X @ Xs ) @ Ys )
= ( cons_list_a @ X @ ( append_list_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_256_append__Cons,axiom,
! [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_257_ord_Olexordp__append__leftD,axiom,
! [Less: list_a > list_a > $o,Xs: list_list_a,Us: list_list_a,Vs: list_list_a] :
( ( lexordp_list_a @ Less @ ( append_list_a @ Xs @ Us ) @ ( append_list_a @ Xs @ Vs ) )
=> ( ! [A2: list_a] :
~ ( Less @ A2 @ A2 )
=> ( lexordp_list_a @ Less @ Us @ Vs ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_258_ord_Olexordp__append__leftD,axiom,
! [Less: a > a > $o,Xs: list_a,Us: list_a,Vs: list_a] :
( ( lexordp_a @ Less @ ( append_a @ Xs @ Us ) @ ( append_a @ Xs @ Vs ) )
=> ( ! [A2: a] :
~ ( Less @ A2 @ A2 )
=> ( lexordp_a @ Less @ Us @ Vs ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_259_ord_Olexordp__append__leftI,axiom,
! [Less: list_a > list_a > $o,Us: list_list_a,Vs: list_list_a,Xs: list_list_a] :
( ( lexordp_list_a @ Less @ Us @ Vs )
=> ( lexordp_list_a @ Less @ ( append_list_a @ Xs @ Us ) @ ( append_list_a @ Xs @ Vs ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_260_ord_Olexordp__append__leftI,axiom,
! [Less: a > a > $o,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 ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_261_ord_Olexordp__eq__pref,axiom,
! [Less: list_a > list_a > $o,U: list_list_a,V: list_list_a] : ( lexordp_eq_list_a @ Less @ U @ ( append_list_a @ U @ V ) ) ).
% ord.lexordp_eq_pref
thf(fact_262_ord_Olexordp__eq__pref,axiom,
! [Less: a > a > $o,U: list_a,V: list_a] : ( lexordp_eq_a @ Less @ U @ ( append_a @ U @ V ) ) ).
% ord.lexordp_eq_pref
thf(fact_263_append__eq__append__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs2: list_list_a,Ts: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs2 @ Ts ) )
= ( ? [Us2: list_list_a] :
( ( ( Xs
= ( append_list_a @ Zs2 @ Us2 ) )
& ( ( append_list_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_list_a @ Xs @ Us2 )
= Zs2 )
& ( Ys
= ( append_list_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_264_append__eq__append__conv2,axiom,
! [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_265_append__eq__appendI,axiom,
! [Xs: list_list_a,Xs1: list_list_a,Zs2: list_list_a,Ys: list_list_a,Us: list_list_a] :
( ( ( append_list_a @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append_list_a @ Xs1 @ Us ) )
=> ( ( append_list_a @ Xs @ Ys )
= ( append_list_a @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_266_append__eq__appendI,axiom,
! [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_267_append_Osemigroup__axioms,axiom,
semigr656274520list_a @ append_list_a ).
% append.semigroup_axioms
thf(fact_268_append_Osemigroup__axioms,axiom,
semigroup_list_a @ append_a ).
% append.semigroup_axioms
thf(fact_269_rev__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X2: list_a] : ( P @ ( cons_list_a @ X2 @ nil_list_a ) )
=> ( ! [X2: list_a,Xs3: list_list_a] :
( ( Xs3 != nil_list_a )
=> ( ( P @ Xs3 )
=> ( P @ ( append_list_a @ Xs3 @ ( cons_list_a @ X2 @ nil_list_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_270_rev__nonempty__induct,axiom,
! [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_271_append__eq__Cons__conv,axiom,
! [Ys: list_list_a,Zs2: list_list_a,X: list_a,Xs: list_list_a] :
( ( ( append_list_a @ Ys @ Zs2 )
= ( cons_list_a @ X @ Xs ) )
= ( ( ( Ys = nil_list_a )
& ( Zs2
= ( cons_list_a @ X @ Xs ) ) )
| ? [Ys4: list_list_a] :
( ( Ys
= ( cons_list_a @ X @ Ys4 ) )
& ( ( append_list_a @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_272_append__eq__Cons__conv,axiom,
! [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_273_Cons__eq__append__conv,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,Zs2: list_list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( append_list_a @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_list_a )
& ( ( cons_list_a @ X @ Xs )
= Zs2 ) )
| ? [Ys4: list_list_a] :
( ( ( cons_list_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_list_a @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_274_Cons__eq__append__conv,axiom,
! [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_275_rev__exhaust,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ~ ! [Ys3: list_list_a,Y2: list_a] :
( Xs
!= ( append_list_a @ Ys3 @ ( cons_list_a @ Y2 @ nil_list_a ) ) ) ) ).
% rev_exhaust
thf(fact_276_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_277_rev__induct,axiom,
! [P: list_list_a > $o,Xs: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X2: list_a,Xs3: list_list_a] :
( ( P @ Xs3 )
=> ( P @ ( append_list_a @ Xs3 @ ( cons_list_a @ X2 @ nil_list_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_278_rev__induct,axiom,
! [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_279_ord_Olexordp__append__left__rightI,axiom,
! [Less: list_a > list_a > $o,X: list_a,Y: list_a,Us: list_list_a,Xs: list_list_a,Ys: list_list_a] :
( ( Less @ X @ Y )
=> ( lexordp_list_a @ Less @ ( append_list_a @ Us @ ( cons_list_a @ X @ Xs ) ) @ ( append_list_a @ Us @ ( cons_list_a @ Y @ Ys ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_280_ord_Olexordp__append__left__rightI,axiom,
! [Less: a > a > $o,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 ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_281_ord_Olexordp__append__rightI,axiom,
! [Ys: list_list_a,Less: list_a > list_a > $o,Xs: list_list_a] :
( ( Ys != nil_list_a )
=> ( lexordp_list_a @ Less @ Xs @ ( append_list_a @ Xs @ Ys ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_282_ord_Olexordp__append__rightI,axiom,
! [Ys: list_a,Less: a > a > $o,Xs: list_a] :
( ( Ys != nil_a )
=> ( lexordp_a @ Less @ Xs @ ( append_a @ Xs @ Ys ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_283_semilattice_Oidem,axiom,
! [F: a > a > a,A: a] :
( ( semilattice_a @ F )
=> ( ( F @ A @ A )
= A ) ) ).
% semilattice.idem
thf(fact_284_semilattice_Oleft__idem,axiom,
! [F: a > a > a,A: a,B: a] :
( ( semilattice_a @ F )
=> ( ( F @ A @ ( F @ A @ B ) )
= ( F @ A @ B ) ) ) ).
% semilattice.left_idem
thf(fact_285_semilattice_Oright__idem,axiom,
! [F: a > a > a,A: a,B: a] :
( ( semilattice_a @ F )
=> ( ( F @ ( F @ A @ B ) @ B )
= ( F @ A @ B ) ) ) ).
% semilattice.right_idem
thf(fact_286_product__lists_Osimps_I1_J,axiom,
( ( product_lists_list_a @ nil_list_list_a )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).
% product_lists.simps(1)
thf(fact_287_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_288_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_list_a] :
( ( bind_a_list_a @ ( cons_a @ X @ Xs ) @ F )
= ( append_list_a @ ( F @ X ) @ ( bind_a_list_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_289_bind__simps_I2_J,axiom,
! [X: list_a,Xs: list_list_a,F: list_a > list_a] :
( ( bind_list_a_a @ ( cons_list_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_list_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_290_bind__simps_I2_J,axiom,
! [X: list_a,Xs: list_list_a,F: list_a > list_list_a] :
( ( bind_list_a_list_a @ ( cons_list_a @ X @ Xs ) @ F )
= ( append_list_a @ ( F @ X ) @ ( bind_list_a_list_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_291_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_292_bind__simps_I1_J,axiom,
! [F: a > list_list_a] :
( ( bind_a_list_a @ nil_a @ F )
= nil_list_a ) ).
% bind_simps(1)
thf(fact_293_bind__simps_I1_J,axiom,
! [F: list_a > list_a] :
( ( bind_list_a_a @ nil_list_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_294_bind__simps_I1_J,axiom,
! [F: list_a > list_list_a] :
( ( bind_list_a_list_a @ nil_list_a @ F )
= nil_list_a ) ).
% bind_simps(1)
thf(fact_295_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_296_abel__semigroup__def,axiom,
( abel_s1075322906list_a
= ( ^ [F2: list_a > list_a > list_a] :
( ( semigroup_list_a @ F2 )
& ( abel_s374399671list_a @ F2 ) ) ) ) ).
% abel_semigroup_def
thf(fact_297_abel__semigroup__def,axiom,
( abel_semigroup_a
= ( ^ [F2: a > a > a] :
( ( semigroup_a @ F2 )
& ( abel_s2117090225ioms_a @ F2 ) ) ) ) ).
% abel_semigroup_def
thf(fact_298_abel__semigroup_Ointro,axiom,
! [F: list_a > list_a > list_a] :
( ( semigroup_list_a @ F )
=> ( ( abel_s374399671list_a @ F )
=> ( abel_s1075322906list_a @ F ) ) ) ).
% abel_semigroup.intro
thf(fact_299_abel__semigroup_Ointro,axiom,
! [F: a > a > a] :
( ( semigroup_a @ F )
=> ( ( abel_s2117090225ioms_a @ F )
=> ( abel_semigroup_a @ F ) ) ) ).
% abel_semigroup.intro
thf(fact_300_abel__semigroup__axioms_Ointro,axiom,
! [F: a > a > a] :
( ! [A2: a,B2: a] :
( ( F @ A2 @ B2 )
= ( F @ B2 @ A2 ) )
=> ( abel_s2117090225ioms_a @ F ) ) ).
% abel_semigroup_axioms.intro
thf(fact_301_abel__semigroup__axioms__def,axiom,
( abel_s2117090225ioms_a
= ( ^ [F2: a > a > a] :
! [A3: a,B3: a] :
( ( F2 @ A3 @ B3 )
= ( F2 @ B3 @ A3 ) ) ) ) ).
% abel_semigroup_axioms_def
thf(fact_302_abel__semigroup_Oaxioms_I2_J,axiom,
! [F: a > a > a] :
( ( abel_semigroup_a @ F )
=> ( abel_s2117090225ioms_a @ F ) ) ).
% abel_semigroup.axioms(2)
thf(fact_303_semilattice_Ointro,axiom,
! [F: a > a > a] :
( ( abel_semigroup_a @ F )
=> ( ( semilattice_axioms_a @ F )
=> ( semilattice_a @ F ) ) ) ).
% semilattice.intro
thf(fact_304_semilattice__def,axiom,
( semilattice_a
= ( ^ [F2: a > a > a] :
( ( abel_semigroup_a @ F2 )
& ( semilattice_axioms_a @ F2 ) ) ) ) ).
% semilattice_def
thf(fact_305_semilattice_Oaxioms_I2_J,axiom,
! [F: a > a > a] :
( ( semilattice_a @ F )
=> ( semilattice_axioms_a @ F ) ) ).
% semilattice.axioms(2)
thf(fact_306_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_a @ nil_list_a )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).
% subseqs.simps(1)
thf(fact_307_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_308_maps__simps_I1_J,axiom,
! [F: a > list_list_a,X: a,Xs: list_a] :
( ( maps_a_list_a @ F @ ( cons_a @ X @ Xs ) )
= ( append_list_a @ ( F @ X ) @ ( maps_a_list_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_309_maps__simps_I1_J,axiom,
! [F: list_a > list_a,X: list_a,Xs: list_list_a] :
( ( maps_list_a_a @ F @ ( cons_list_a @ X @ Xs ) )
= ( append_a @ ( F @ X ) @ ( maps_list_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_310_maps__simps_I1_J,axiom,
! [F: list_a > list_list_a,X: list_a,Xs: list_list_a] :
( ( maps_list_a_list_a @ F @ ( cons_list_a @ X @ Xs ) )
= ( append_list_a @ ( F @ X ) @ ( maps_list_a_list_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_311_maps__simps_I1_J,axiom,
! [F: a > list_a,X: a,Xs: list_a] :
( ( maps_a_a @ F @ ( cons_a @ X @ Xs ) )
= ( append_a @ ( F @ X ) @ ( maps_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_312_maps__simps_I2_J,axiom,
! [F: a > list_list_a] :
( ( maps_a_list_a @ F @ nil_a )
= nil_list_a ) ).
% maps_simps(2)
thf(fact_313_maps__simps_I2_J,axiom,
! [F: list_a > list_a] :
( ( maps_list_a_a @ F @ nil_list_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_314_maps__simps_I2_J,axiom,
! [F: list_a > list_list_a] :
( ( maps_list_a_list_a @ F @ nil_list_a )
= nil_list_a ) ).
% maps_simps(2)
thf(fact_315_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_316_local_Otransp__less,axiom,
transp_a @ less ).
% local.transp_less
thf(fact_317_insert__Nil,axiom,
! [X: list_a] :
( ( insert_list_a @ X @ nil_list_a )
= ( cons_list_a @ X @ nil_list_a ) ) ).
% insert_Nil
thf(fact_318_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_319_concat__eq__append__conv,axiom,
! [Xss2: list_list_list_a,Ys: list_list_a,Zs2: list_list_a] :
( ( ( concat_list_a @ Xss2 )
= ( append_list_a @ Ys @ Zs2 ) )
= ( ( ( Xss2 = nil_list_list_a )
=> ( ( Ys = nil_list_a )
& ( Zs2 = nil_list_a ) ) )
& ( ( Xss2 != nil_list_list_a )
=> ? [Xss1: list_list_list_a,Xs2: list_list_a,Xs4: list_list_a,Xss22: list_list_list_a] :
( ( Xss2
= ( append_list_list_a @ Xss1 @ ( cons_list_list_a @ ( append_list_a @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_a @ ( concat_list_a @ Xss1 ) @ Xs2 ) )
& ( Zs2
= ( append_list_a @ Xs4 @ ( concat_list_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_320_concat__eq__append__conv,axiom,
! [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_321_rotate1_Osimps_I2_J,axiom,
! [X: list_a,Xs: list_list_a] :
( ( rotate1_list_a @ ( cons_list_a @ X @ Xs ) )
= ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) ) ).
% rotate1.simps(2)
thf(fact_322_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_323_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( ( rotate1_list_a @ Xs )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_324_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_325_concat__append,axiom,
! [Xs: list_list_list_a,Ys: list_list_list_a] :
( ( concat_list_a @ ( append_list_list_a @ Xs @ Ys ) )
= ( append_list_a @ ( concat_list_a @ Xs ) @ ( concat_list_a @ Ys ) ) ) ).
% concat_append
thf(fact_326_concat__append,axiom,
! [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_327_concat_Osimps_I1_J,axiom,
( ( concat_list_a @ nil_list_list_a )
= nil_list_a ) ).
% concat.simps(1)
thf(fact_328_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_329_concat_Osimps_I2_J,axiom,
! [X: list_list_a,Xs: list_list_list_a] :
( ( concat_list_a @ ( cons_list_list_a @ X @ Xs ) )
= ( append_list_a @ X @ ( concat_list_a @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_330_concat_Osimps_I2_J,axiom,
! [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_331_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_a @ nil_list_a )
= nil_list_a ) ).
% rotate1.simps(1)
thf(fact_332_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_333_concat__eq__appendD,axiom,
! [Xss2: list_list_list_a,Ys: list_list_a,Zs2: list_list_a] :
( ( ( concat_list_a @ Xss2 )
= ( append_list_a @ Ys @ Zs2 ) )
=> ( ( Xss2 != nil_list_list_a )
=> ? [Xss12: list_list_list_a,Xs3: list_list_a,Xs5: list_list_a,Xss23: list_list_list_a] :
( ( Xss2
= ( append_list_list_a @ Xss12 @ ( cons_list_list_a @ ( append_list_a @ Xs3 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append_list_a @ ( concat_list_a @ Xss12 ) @ Xs3 ) )
& ( Zs2
= ( append_list_a @ Xs5 @ ( concat_list_a @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_334_concat__eq__appendD,axiom,
! [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_335_butlast__snoc,axiom,
! [Xs: list_list_a,X: list_a] :
( ( butlast_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_336_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_337_list__ex1__simps_I1_J,axiom,
! [P: list_a > $o] :
~ ( list_ex1_list_a @ P @ nil_list_a ) ).
% list_ex1_simps(1)
thf(fact_338_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_339_butlast__append,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( ( Ys = nil_list_a )
=> ( ( butlast_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( butlast_list_a @ Xs ) ) )
& ( ( Ys != nil_list_a )
=> ( ( butlast_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_list_a @ Xs @ ( butlast_list_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_340_butlast__append,axiom,
! [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_341_butlast_Osimps_I1_J,axiom,
( ( butlast_list_a @ nil_list_a )
= nil_list_a ) ).
% butlast.simps(1)
thf(fact_342_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_343_butlast_Osimps_I2_J,axiom,
! [Xs: list_list_a,X: list_a] :
( ( ( Xs = nil_list_a )
=> ( ( butlast_list_a @ ( cons_list_a @ X @ Xs ) )
= nil_list_a ) )
& ( ( Xs != nil_list_a )
=> ( ( butlast_list_a @ ( cons_list_a @ X @ Xs ) )
= ( cons_list_a @ X @ ( butlast_list_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_344_butlast_Osimps_I2_J,axiom,
! [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_345_append__butlast__last__id,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( append_list_a @ ( butlast_list_a @ Xs ) @ ( cons_list_a @ ( last_list_a @ Xs ) @ nil_list_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_346_append__butlast__last__id,axiom,
! [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_347_snoc__eq__iff__butlast,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) )
= Ys )
= ( ( Ys != nil_list_a )
& ( ( butlast_list_a @ Ys )
= Xs )
& ( ( last_list_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_348_snoc__eq__iff__butlast,axiom,
! [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_349_last__appendR,axiom,
! [Ys: list_list_a,Xs: list_list_a] :
( ( Ys != nil_list_a )
=> ( ( last_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( last_list_a @ Ys ) ) ) ).
% last_appendR
thf(fact_350_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_351_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
% 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 ).
%------------------------------------------------------------------------------