TPTP Problem File: DAT243^1.p
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%------------------------------------------------------------------------------
% File : DAT243^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Red-black trees 2056
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : rbt_impl__2056.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 334 ( 135 unt; 72 typ; 0 def)
% Number of atoms : 705 ( 413 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 9090 ( 181 ~; 26 |; 78 &;8415 @)
% ( 0 <=>; 390 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 10 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 835 ( 835 >; 0 *; 0 +; 0 <<)
% Number of symbols : 69 ( 68 usr; 0 con; 1-7 aty)
% Number of variables : 1605 ( 9 ^;1444 !; 49 ?;1605 :)
% ( 103 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:49:36.456
%------------------------------------------------------------------------------
%----Could-be-implicit typings (9)
thf(ty_t_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor,type,
rBT_Im1923302023_color: $tType ).
thf(ty_t_RBT__Impl__Mirabelle__msmaddcmtr_Orbt,type,
rBT_Im246033960le_rbt: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (63)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ 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_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
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_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ 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_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev__rel,type,
map_tailrec_rev_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_Onull,type,
null:
!>[A: $tType] : ( ( 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_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
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_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Obalance,type,
rBT_Im1648453169alance:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oentries,type,
rBT_Im954575269ntries:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ofoldi,type,
rBT_Im501988381_foldi:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > $o ) > ( A > B > C > C ) > ( rBT_Im246033960le_rbt @ A @ B ) > C > C ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ogen__entries,type,
rBT_Im1841113099ntries:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ ( product_prod @ A @ B ) @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) > ( rBT_Im246033960le_rbt @ A @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ogen__keys,type,
rBT_Im1235880025n_keys:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) > ( rBT_Im246033960le_rbt @ A @ B ) > ( list @ A ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Okeys,type,
rBT_Im380146495e_keys:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( list @ A ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Omap,type,
rBT_Im1728190513le_map:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ C ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Ois__rbt,type,
rBT_Im862805236is_rbt:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__map__entry,type,
rBT_Im2018130356_entry:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( B > B ) > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__sorted,type,
rBT_Im759614907sorted:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Osinter__with,type,
rBT_Im319793781r_with:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Osinter__with__rel,type,
rBT_Im735314116th_rel:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) > ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Osunion__with,type,
rBT_Im1445157352n_with:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Osunion__with__rel,type,
rBT_Im753582353th_rel:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) > ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Osinter__with,type,
rBT_Im393311009r_with:
!>[A: $tType,B: $tType] : ( ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Osinter__with__rel,type,
rBT_Im310218520th_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) > ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Osunion__with,type,
rBT_Im1518674580n_with:
!>[A: $tType,B: $tType] : ( ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Osunion__with__rel,type,
rBT_Im328486757th_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) > ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OBranch,type,
rBT_Im480247531Branch:
!>[A: $tType,B: $tType] : ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OEmpty,type,
rBT_Im418718756_Empty:
!>[A: $tType,B: $tType] : ( rBT_Im246033960le_rbt @ A @ B ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Ocase__rbt,type,
rBT_Im858806507se_rbt:
!>[C: $tType,A: $tType,B: $tType] : ( C > ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > C ) > ( rBT_Im246033960le_rbt @ A @ B ) > C ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Orec__rbt,type,
rBT_Im1947144893ec_rbt:
!>[E: $tType,A: $tType,B: $tType] : ( E > ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > E > E > E ) > ( rBT_Im246033960le_rbt @ A @ B ) > E ) ).
thf(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
%----Relevant facts (255)
thf(fact_0_gen__keys__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1235880025n_keys @ A @ B @ ( nil @ ( product_prod @ A @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( nil @ A ) ) ).
% gen_keys_simps(1)
thf(fact_1_keys__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im380146495e_keys @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( nil @ A ) ) ).
% keys_simps(1)
thf(fact_2_map__keys,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C > B,T2: rBT_Im246033960le_rbt @ A @ C] :
( ( rBT_Im380146495e_keys @ A @ B @ ( rBT_Im1728190513le_map @ A @ C @ B @ F @ T2 ) )
= ( rBT_Im380146495e_keys @ A @ C @ T2 ) ) ).
% map_keys
thf(fact_3_non__empty__rbt__keys,axiom,
! [B: $tType,A: $tType,T2: rBT_Im246033960le_rbt @ A @ B] :
( ( T2
!= ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ( rBT_Im380146495e_keys @ A @ B @ T2 )
!= ( nil @ A ) ) ) ).
% non_empty_rbt_keys
thf(fact_4_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list @ B] :
( ( product @ A @ B @ ( nil @ A ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% product.simps(1)
thf(fact_5_ord_Osinter__with_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,Uv: list @ ( product_prod @ A @ B )] :
( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ Uv @ ( nil @ ( product_prod @ A @ B ) ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% ord.sinter_with.simps(3)
thf(fact_6_ord_Osinter__with_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,Uu: list @ ( product_prod @ A @ B )] :
( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( nil @ ( product_prod @ A @ B ) ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% ord.sinter_with.simps(2)
thf(fact_7_ord_Osunion__with_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,As: list @ ( product_prod @ A @ B )] :
( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ As @ ( nil @ ( product_prod @ A @ B ) ) )
= As ) ).
% ord.sunion_with.simps(3)
thf(fact_8_ord_Osunion__with_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,Bs: list @ ( product_prod @ A @ B )] :
( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( nil @ ( product_prod @ A @ B ) ) @ Bs )
= Bs ) ).
% ord.sunion_with.simps(2)
thf(fact_9_sunion__with_Osimps_I3_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,As: list @ ( product_prod @ A @ B )] :
( ( rBT_Im1518674580n_with @ A @ B @ F @ As @ ( nil @ ( product_prod @ A @ B ) ) )
= As ) ) ).
% sunion_with.simps(3)
thf(fact_10_sunion__with_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,Bs: list @ ( product_prod @ A @ B )] :
( ( rBT_Im1518674580n_with @ A @ B @ F @ ( nil @ ( product_prod @ A @ B ) ) @ Bs )
= Bs ) ) ).
% sunion_with.simps(2)
thf(fact_11_map_Osimps_I1_J,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B > C] :
( ( rBT_Im1728190513le_map @ A @ B @ C @ F @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ C ) ) ).
% map.simps(1)
thf(fact_12_gen__entries__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1841113099ntries @ A @ B @ ( nil @ ( product_prod @ ( product_prod @ A @ B ) @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% gen_entries_simps(1)
thf(fact_13_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_14_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_15_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_16_foldi_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [C2: C > $o,F: A > B > C > C,S: C] :
( ( rBT_Im501988381_foldi @ C @ A @ B @ C2 @ F @ ( rBT_Im418718756_Empty @ A @ B ) @ S )
= S ) ) ).
% foldi.simps(1)
thf(fact_17_gen__length__code_I1_J,axiom,
! [A: $tType,N: nat] :
( ( gen_length @ A @ N @ ( nil @ A ) )
= N ) ).
% gen_length_code(1)
thf(fact_18_ord_Omap__is__rbt,axiom,
! [B: $tType,C: $tType,A: $tType,Less: A > A > $o,F: A > C > B,T2: rBT_Im246033960le_rbt @ A @ C] :
( ( rBT_Im862805236is_rbt @ A @ B @ Less @ ( rBT_Im1728190513le_map @ A @ C @ B @ F @ T2 ) )
= ( rBT_Im862805236is_rbt @ A @ C @ Less @ T2 ) ) ).
% ord.map_is_rbt
thf(fact_19_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_20_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_21_eq__Nil__null,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
= ( nil @ A ) )
= ( null @ A @ Xs ) ) ).
% eq_Nil_null
thf(fact_22_null__rec_I2_J,axiom,
! [B: $tType] : ( null @ B @ ( nil @ B ) ) ).
% null_rec(2)
thf(fact_23_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_24_ord_OEmpty__is__rbt,axiom,
! [B: $tType,A: $tType,Less: A > A > $o] : ( rBT_Im862805236is_rbt @ A @ B @ Less @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.Empty_is_rbt
thf(fact_25_entries__code,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im954575269ntries @ A @ B )
= ( rBT_Im1841113099ntries @ A @ B @ ( nil @ ( product_prod @ ( product_prod @ A @ B ) @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) ) ) ).
% entries_code
thf(fact_26_ord_Orbt__map__entry__is__rbt,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K: A,F: B > B,T2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im862805236is_rbt @ A @ B @ Less @ ( rBT_Im2018130356_entry @ A @ B @ Less @ K @ F @ T2 ) )
= ( rBT_Im862805236is_rbt @ A @ B @ Less @ T2 ) ) ).
% ord.rbt_map_entry_is_rbt
thf(fact_27_ord_Ois__rbt__rbt__sorted,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,T2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im862805236is_rbt @ A @ B @ Less @ T2 )
=> ( rBT_Im759614907sorted @ A @ B @ Less @ T2 ) ) ).
% ord.is_rbt_rbt_sorted
thf(fact_28_rbt_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F1: C,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > C] :
( ( rBT_Im858806507se_rbt @ C @ A @ B @ F1 @ F2 @ ( rBT_Im418718756_Empty @ A @ B ) )
= F1 ) ).
% rbt.simps(4)
thf(fact_29_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_30_zip__Nil,axiom,
! [B: $tType,A: $tType,Ys: list @ B] :
( ( zip @ A @ B @ ( nil @ A ) @ Ys )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip_Nil
thf(fact_31_splice_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V: A,Va: list @ A] :
( ( X
= ( cons @ A @ V @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V @ Va ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( Y
!= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ ( splice @ A @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% splice.elims
thf(fact_32_splice_Osimps_I2_J,axiom,
! [A: $tType,V2: A,Va2: list @ A] :
( ( splice @ A @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) )
= ( cons @ A @ V2 @ Va2 ) ) ).
% splice.simps(2)
thf(fact_33_list__ex__simps_I2_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex @ A @ P @ ( nil @ A ) ) ).
% list_ex_simps(2)
thf(fact_34_rbt_Osimps_I6_J,axiom,
! [A: $tType,B: $tType,E: $tType,F1: E,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > E > E > E] :
( ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ ( rBT_Im418718756_Empty @ A @ B ) )
= F1 ) ).
% rbt.simps(6)
thf(fact_35_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_36_list__ex__simps_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( list_ex @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( P @ X )
| ( list_ex @ A @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_37_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,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_38_ord_Orbt__map__entry__rbt__sorted,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K: A,F: B > B,T2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im759614907sorted @ A @ B @ Less @ ( rBT_Im2018130356_entry @ A @ B @ Less @ K @ F @ T2 ) )
= ( rBT_Im759614907sorted @ A @ B @ Less @ T2 ) ) ).
% ord.rbt_map_entry_rbt_sorted
thf(fact_39_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_40_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A2: list @ B] :
( ! [F3: A > B,X1: list @ B] : ( P @ F3 @ ( nil @ A ) @ X1 )
=> ( ! [F3: A > B,A3: A,As2: list @ A,Bs2: list @ B] :
( ( P @ F3 @ As2 @ ( cons @ B @ ( F3 @ A3 ) @ Bs2 ) )
=> ( P @ F3 @ ( cons @ A @ A3 @ As2 ) @ Bs2 ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_41_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,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_42_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,Xs2: list @ A] :
( ( ( X2 = Y2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y2 )
=> ( P @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_43_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member2 @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_49_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,Xs2: list @ A] : ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys2 ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_50_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_51_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_52_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_53_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_54_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_55_splice_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ X @ ( cons @ A @ Y @ ( splice @ A @ Xs @ Ys ) ) ) ) ).
% splice.simps(3)
thf(fact_56_null__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
~ ( null @ A @ ( cons @ A @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_57_member__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_58_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_59_ord_Orbt__sorted_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o] : ( rBT_Im759614907sorted @ A @ B @ Less @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_sorted.simps(1)
thf(fact_60_ord_Orbt__map__entry_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K: A,F: B > B] :
( ( rBT_Im2018130356_entry @ A @ B @ Less @ K @ F @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_map_entry.simps(1)
thf(fact_61_zip_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Xs: list @ A] :
( ( zip @ A @ B @ Xs @ ( nil @ B ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip.simps(1)
thf(fact_62_entries_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im954575269ntries @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% entries.simps(1)
thf(fact_63_ord_Omap__rbt__sorted,axiom,
! [B: $tType,C: $tType,A: $tType,Less: A > A > $o,F: A > C > B,T2: rBT_Im246033960le_rbt @ A @ C] :
( ( rBT_Im759614907sorted @ A @ B @ Less @ ( rBT_Im1728190513le_map @ A @ C @ B @ F @ T2 ) )
= ( rBT_Im759614907sorted @ A @ C @ Less @ T2 ) ) ).
% ord.map_rbt_sorted
thf(fact_64_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_65_gen__keys__simps_I2_J,axiom,
! [D: $tType,C: $tType,K: C,T2: rBT_Im246033960le_rbt @ C @ D,Kts: list @ ( product_prod @ C @ ( rBT_Im246033960le_rbt @ C @ D ) )] :
( ( rBT_Im1235880025n_keys @ C @ D @ ( cons @ ( product_prod @ C @ ( rBT_Im246033960le_rbt @ C @ D ) ) @ ( product_Pair @ C @ ( rBT_Im246033960le_rbt @ C @ D ) @ K @ T2 ) @ Kts ) @ ( rBT_Im418718756_Empty @ C @ D ) )
= ( cons @ C @ K @ ( rBT_Im1235880025n_keys @ C @ D @ Kts @ T2 ) ) ) ).
% gen_keys_simps(2)
thf(fact_66_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,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_67_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A3: A,As2: list @ A] :
( ( Xa
= ( cons @ A @ A3 @ As2 ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X @ As2 @ ( cons @ B @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_68_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A1: list @ A,A2: list @ B] :
( ( listrelp @ A @ B @ R @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A2
!= ( nil @ B ) ) )
=> ~ ! [X2: A,Y2: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ B] :
( ( A2
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( R @ X2 @ Y2 )
=> ~ ( listrelp @ A @ B @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_69_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R2: A > B > $o,A12: list @ A,A22: list @ B] :
( ( ( A12
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X3: A,Y3: B,Xs3: list @ A,Ys3: list @ B] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y3 @ Ys3 ) )
& ( R2 @ X3 @ Y3 )
& ( listrelp @ A @ B @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_70_listrelp_Oinducts,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X12: list @ A,X24: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( listrelp @ A @ B @ R @ X12 @ X24 )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Y2: B,Xs2: list @ A,Ys2: list @ B] :
( ( R @ X2 @ Y2 )
=> ( ( listrelp @ A @ B @ R @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ).
% listrelp.inducts
thf(fact_71_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_72_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_73_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_74_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_75_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_76_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_77_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_78_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_79_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_80_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_81_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_82_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_83_lexordp__eq__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ) ).
% lexordp_eq_simps(2)
thf(fact_84_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq_simps(1)
thf(fact_85_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,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_86_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_87_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_88_list__ex__append,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( list_ex @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( list_ex @ A @ P @ Xs )
| ( list_ex @ A @ P @ Ys ) ) ) ).
% list_ex_append
thf(fact_89_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_90_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_91_gen__entries__simps_I2_J,axiom,
! [D: $tType,C: $tType,Kv: product_prod @ C @ D,T2: rBT_Im246033960le_rbt @ C @ D,Kvts: list @ ( product_prod @ ( product_prod @ C @ D ) @ ( rBT_Im246033960le_rbt @ C @ D ) )] :
( ( rBT_Im1841113099ntries @ C @ D @ ( cons @ ( product_prod @ ( product_prod @ C @ D ) @ ( rBT_Im246033960le_rbt @ C @ D ) ) @ ( product_Pair @ ( product_prod @ C @ D ) @ ( rBT_Im246033960le_rbt @ C @ D ) @ Kv @ T2 ) @ Kvts ) @ ( rBT_Im418718756_Empty @ C @ D ) )
= ( cons @ ( product_prod @ C @ D ) @ Kv @ ( rBT_Im1841113099ntries @ C @ D @ Kvts @ T2 ) ) ) ).
% gen_entries_simps(2)
thf(fact_92_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_93_ord_Osunion__with_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) )] :
( ! [F3: A > B > B > B,K2: A,V: B,As2: list @ ( product_prod @ A @ B ),K3: A,V3: B,Bs2: list @ ( product_prod @ A @ B )] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ! [F3: A > B > B > B,Bs2: list @ ( product_prod @ A @ B )] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Bs2 ) ) )
=> ~ ! [F3: A > B > B > B,As2: list @ ( product_prod @ A @ B )] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ As2 @ ( nil @ ( product_prod @ A @ B ) ) ) ) ) ) ) ).
% ord.sunion_with.cases
thf(fact_94_sunion__with_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) )] :
( ! [F3: A > B > B > B,K2: A,V: B,As2: list @ ( product_prod @ A @ B ),K3: A,V3: B,Bs2: list @ ( product_prod @ A @ B )] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ! [F3: A > B > B > B,Bs2: list @ ( product_prod @ A @ B )] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Bs2 ) ) )
=> ~ ! [F3: A > B > B > B,As2: list @ ( product_prod @ A @ B )] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ As2 @ ( nil @ ( product_prod @ A @ B ) ) ) ) ) ) ) ) ).
% sunion_with.cases
thf(fact_95_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_96_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_97_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_98_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_99_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Zs )
=> ( ord_lexordp_eq @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_eq_trans
thf(fact_100_lexordp__eq__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
| ( ord_lexordp_eq @ A @ Ys @ Xs ) ) ) ).
% lexordp_eq_linear
thf(fact_101_lexordp__eq__antisym,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% lexordp_eq_antisym
thf(fact_102_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_103_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_104_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_105_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_106_ord_Osunion__with_Oinduct,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,P: ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > $o,A0: A > B > B > B,A1: list @ ( product_prod @ A @ B ),A2: list @ ( product_prod @ A @ B )] :
( ! [F3: A > B > B > B,K2: A,V: B,As2: list @ ( product_prod @ A @ B ),K3: A,V3: B,Bs2: list @ ( product_prod @ A @ B )] :
( ( ( Less @ K3 @ K2 )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ Bs2 ) )
=> ( ( ~ ( Less @ K3 @ K2 )
=> ( ( Less @ K2 @ K3 )
=> ( P @ F3 @ As2 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ~ ( Less @ K3 @ K2 )
=> ( ~ ( Less @ K2 @ K3 )
=> ( P @ F3 @ As2 @ Bs2 ) ) )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) ) )
=> ( ! [F3: A > B > B > B,X1: list @ ( product_prod @ A @ B )] : ( P @ F3 @ ( nil @ ( product_prod @ A @ B ) ) @ X1 )
=> ( ! [F3: A > B > B > B,As2: list @ ( product_prod @ A @ B )] : ( P @ F3 @ As2 @ ( nil @ ( product_prod @ A @ B ) ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ) ).
% ord.sunion_with.induct
thf(fact_107_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq.Nil
thf(fact_108_ord_Osunion__with_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K4: A,K: A,F: A > B > B > B,V2: B,As: list @ ( product_prod @ A @ B ),V4: B,Bs: list @ ( product_prod @ A @ B )] :
( ( ( Less @ K4 @ K )
=> ( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ Bs ) ) ) )
& ( ~ ( Less @ K4 @ K )
=> ( ( ( Less @ K @ K4 )
=> ( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ As @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) ) )
& ( ~ ( Less @ K @ K4 )
=> ( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ ( F @ K @ V2 @ V4 ) ) @ ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ As @ Bs ) ) ) ) ) ) ) ).
% ord.sunion_with.simps(1)
thf(fact_109_ord_Osinter__with_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K4: A,K: A,F: A > B > B > B,V2: B,As: list @ ( product_prod @ A @ B ),V4: B,Bs: list @ ( product_prod @ A @ B )] :
( ( ( Less @ K4 @ K )
=> ( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ Bs ) ) )
& ( ~ ( Less @ K4 @ K )
=> ( ( ( Less @ K @ K4 )
=> ( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( rBT_Im319793781r_with @ A @ B @ Less @ F @ As @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) )
& ( ~ ( Less @ K @ K4 )
=> ( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ ( F @ K @ V2 @ V4 ) ) @ ( rBT_Im319793781r_with @ A @ B @ Less @ F @ As @ Bs ) ) ) ) ) ) ) ).
% ord.sinter_with.simps(1)
thf(fact_110_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,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_111_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,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_112_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_113_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( R @ X @ Y )
=> ( ( listrelp @ A @ B @ R @ Xs @ Ys )
=> ( listrelp @ A @ B @ R @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_114_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( listrelp @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_115_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_116_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys2: list @ A,Y2: A] :
( Xs
!= ( append @ A @ Ys2 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_117_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys4: list @ A] :
( ( ( cons @ A @ X @ Ys4 )
= Ys )
& ( Xs
= ( append @ A @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_118_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
& ( ( append @ A @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_119_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,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_120_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A4 @ As ) @ Bs )
= ( map_tailrec_rev @ A @ B @ F @ As @ ( cons @ B @ ( F @ A4 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_121_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F: A > B,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( nil @ A ) @ Bs )
= Bs ) ).
% map_tailrec_rev.simps(1)
thf(fact_122_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_123_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X12: list @ A,X24: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X12 @ X24 )
=> ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [X2: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less @ X2 @ Y2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X2: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_124_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A22: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22 = Ys3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_125_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A2: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys2: list @ A] : A2 != Ys2 )
=> ( ! [X2: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A2
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A2
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_126_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A4: A,B2: B,A6: A,B3: B] :
( ( ( product_Pair @ A @ B @ A4 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B3 ) )
= ( ( A4 = A6 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_127_prod_Oinject,axiom,
! [A: $tType,B: $tType,X12: A,X24: B,Y1: A,Y23: B] :
( ( ( product_Pair @ A @ B @ X12 @ X24 )
= ( product_Pair @ A @ B @ Y1 @ Y23 ) )
= ( ( X12 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.inject
thf(fact_128_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_129_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_130_gen__keys__simps_I3_J,axiom,
! [D: $tType,C: $tType,Kts: list @ ( product_prod @ C @ ( rBT_Im246033960le_rbt @ C @ D ) ),C2: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ C @ D,K: C,V2: D,R: rBT_Im246033960le_rbt @ C @ D] :
( ( rBT_Im1235880025n_keys @ C @ D @ Kts @ ( rBT_Im480247531Branch @ C @ D @ C2 @ L @ K @ V2 @ R ) )
= ( rBT_Im1235880025n_keys @ C @ D @ ( cons @ ( product_prod @ C @ ( rBT_Im246033960le_rbt @ C @ D ) ) @ ( product_Pair @ C @ ( rBT_Im246033960le_rbt @ C @ D ) @ K @ R ) @ Kts ) @ L ) ) ).
% gen_keys_simps(3)
thf(fact_131_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_132_rbt_Oinject,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X232: A,X242: B,X25: rBT_Im246033960le_rbt @ A @ B,Y21: rBT_Im1923302023_color,Y22: rBT_Im246033960le_rbt @ A @ B,Y232: A,Y24: B,Y25: rBT_Im246033960le_rbt @ A @ B] :
( ( ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X232 @ X242 @ X25 )
= ( rBT_Im480247531Branch @ A @ B @ Y21 @ Y22 @ Y232 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X232 = Y232 )
& ( X242 = Y24 )
& ( X25 = Y25 ) ) ) ).
% rbt.inject
thf(fact_133_keys__simps_I2_J,axiom,
! [D: $tType,C: $tType,C2: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ C @ D,K: C,V2: D,R: rBT_Im246033960le_rbt @ C @ D] :
( ( rBT_Im380146495e_keys @ C @ D @ ( rBT_Im480247531Branch @ C @ D @ C2 @ L @ K @ V2 @ R ) )
= ( append @ C @ ( rBT_Im380146495e_keys @ C @ D @ L ) @ ( cons @ C @ K @ ( rBT_Im380146495e_keys @ C @ D @ R ) ) ) ) ).
% keys_simps(2)
thf(fact_134_gen__entries__simps_I3_J,axiom,
! [D: $tType,C: $tType,Kvts: list @ ( product_prod @ ( product_prod @ C @ D ) @ ( rBT_Im246033960le_rbt @ C @ D ) ),C2: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ C @ D,K: C,V2: D,R: rBT_Im246033960le_rbt @ C @ D] :
( ( rBT_Im1841113099ntries @ C @ D @ Kvts @ ( rBT_Im480247531Branch @ C @ D @ C2 @ L @ K @ V2 @ R ) )
= ( rBT_Im1841113099ntries @ C @ D @ ( cons @ ( product_prod @ ( product_prod @ C @ D ) @ ( rBT_Im246033960le_rbt @ C @ D ) ) @ ( product_Pair @ ( product_prod @ C @ D ) @ ( rBT_Im246033960le_rbt @ C @ D ) @ ( product_Pair @ C @ D @ K @ V2 ) @ R ) @ Kvts ) @ L ) ) ).
% gen_entries_simps(3)
thf(fact_135_foldi_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: product_prod @ ( C > $o ) @ ( product_prod @ ( A > B > C > C ) @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ C ) )] :
( ! [C3: C > $o,F3: A > B > C > C,S2: C] :
( X
!= ( product_Pair @ ( C > $o ) @ ( product_prod @ ( A > B > C > C ) @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ C ) ) @ C3 @ ( product_Pair @ ( A > B > C > C ) @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ C ) @ F3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ A @ B ) @ C @ ( rBT_Im418718756_Empty @ A @ B ) @ S2 ) ) ) )
=> ~ ! [C3: C > $o,F3: A > B > C > C,Col: rBT_Im1923302023_color,L2: rBT_Im246033960le_rbt @ A @ B,K2: A,V: B,R3: rBT_Im246033960le_rbt @ A @ B,S2: C] :
( X
!= ( product_Pair @ ( C > $o ) @ ( product_prod @ ( A > B > C > C ) @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ C ) ) @ C3 @ ( product_Pair @ ( A > B > C > C ) @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ C ) @ F3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ A @ B ) @ C @ ( rBT_Im480247531Branch @ A @ B @ Col @ L2 @ K2 @ V @ R3 ) @ S2 ) ) ) ) ) ) ).
% foldi.cases
thf(fact_136_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F3: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F3: A > B,A3: A,As2: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A3 @ As2 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_137_rbt_Odistinct_I1_J,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X232: A,X242: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im418718756_Empty @ A @ B )
!= ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X232 @ X242 @ X25 ) ) ).
% rbt.distinct(1)
thf(fact_138_rbt_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( rBT_Im246033960le_rbt @ A @ B ) > $o,Rbt: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [X1: rBT_Im1923302023_color,X23: rBT_Im246033960le_rbt @ A @ B,X32: A,X4: B,X5: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ X23 )
=> ( ( P @ X5 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ X1 @ X23 @ X32 @ X4 @ X5 ) ) ) )
=> ( P @ Rbt ) ) ) ).
% rbt.induct
thf(fact_139_rbt_Oexhaust,axiom,
! [B: $tType,A: $tType,Y: rBT_Im246033960le_rbt @ A @ B] :
( ( Y
!= ( rBT_Im418718756_Empty @ A @ B ) )
=> ~ ! [X212: rBT_Im1923302023_color,X222: rBT_Im246033960le_rbt @ A @ B,X233: A,X243: B,X252: rBT_Im246033960le_rbt @ A @ B] :
( Y
!= ( rBT_Im480247531Branch @ A @ B @ X212 @ X222 @ X233 @ X243 @ X252 ) ) ) ).
% rbt.exhaust
thf(fact_140_rbt_Osimps_I5_J,axiom,
! [C: $tType,B: $tType,A: $tType,F1: C,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > C,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X232: A,X242: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im858806507se_rbt @ C @ A @ B @ F1 @ F2 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X232 @ X242 @ X25 ) )
= ( F2 @ X21 @ X22 @ X232 @ X242 @ X25 ) ) ).
% rbt.simps(5)
thf(fact_141_map_Osimps_I2_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,C2: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ B,K: A,V2: B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1728190513le_map @ A @ B @ C @ F @ ( rBT_Im480247531Branch @ A @ B @ C2 @ Lt @ K @ V2 @ Rt ) )
= ( rBT_Im480247531Branch @ A @ C @ C2 @ ( rBT_Im1728190513le_map @ A @ B @ C @ F @ Lt ) @ K @ ( F @ K @ V2 ) @ ( rBT_Im1728190513le_map @ A @ B @ C @ F @ Rt ) ) ) ).
% map.simps(2)
thf(fact_142_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_143_ord_Orbt__map__entry_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K: A,X: A,F: B > B,C2: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ B,V2: B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( ( Less @ K @ X )
=> ( ( rBT_Im2018130356_entry @ A @ B @ Less @ K @ F @ ( rBT_Im480247531Branch @ A @ B @ C2 @ Lt @ X @ V2 @ Rt ) )
= ( rBT_Im480247531Branch @ A @ B @ C2 @ ( rBT_Im2018130356_entry @ A @ B @ Less @ K @ F @ Lt ) @ X @ V2 @ Rt ) ) )
& ( ~ ( Less @ K @ X )
=> ( ( ( Less @ X @ K )
=> ( ( rBT_Im2018130356_entry @ A @ B @ Less @ K @ F @ ( rBT_Im480247531Branch @ A @ B @ C2 @ Lt @ X @ V2 @ Rt ) )
= ( rBT_Im480247531Branch @ A @ B @ C2 @ Lt @ X @ V2 @ ( rBT_Im2018130356_entry @ A @ B @ Less @ K @ F @ Rt ) ) ) )
& ( ~ ( Less @ X @ K )
=> ( ( rBT_Im2018130356_entry @ A @ B @ Less @ K @ F @ ( rBT_Im480247531Branch @ A @ B @ C2 @ Lt @ X @ V2 @ Rt ) )
= ( rBT_Im480247531Branch @ A @ B @ C2 @ Lt @ X @ ( F @ V2 ) @ Rt ) ) ) ) ) ) ).
% ord.rbt_map_entry.simps(2)
thf(fact_144_rbt_Osimps_I7_J,axiom,
! [E: $tType,B: $tType,A: $tType,F1: E,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > E > E > E,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X232: A,X242: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X232 @ X242 @ X25 ) )
= ( F2 @ X21 @ X22 @ X232 @ X242 @ X25 @ ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ X22 ) @ ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ X25 ) ) ) ).
% rbt.simps(7)
thf(fact_145_splice_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( ! [V: A,Va: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V @ Va ) @ ( nil @ A ) ) )
=> ~ ! [X2: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ).
% splice.cases
thf(fact_146_rbt__del_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: product_prod @ A @ ( rBT_Im246033960le_rbt @ A @ B )] :
( ! [X2: A] :
( X
!= ( product_Pair @ A @ ( rBT_Im246033960le_rbt @ A @ B ) @ X2 @ ( rBT_Im418718756_Empty @ A @ B ) ) )
=> ~ ! [X2: A,C3: rBT_Im1923302023_color,A3: rBT_Im246033960le_rbt @ A @ B,Y2: A,S2: B,B4: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( rBT_Im246033960le_rbt @ A @ B ) @ X2 @ ( rBT_Im480247531Branch @ A @ B @ C3 @ A3 @ Y2 @ S2 @ B4 ) ) ) ) ) ).
% rbt_del.cases
thf(fact_147_ord_Orbt__del_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ ( rBT_Im246033960le_rbt @ A @ B )] :
( ! [X2: A] :
( X
!= ( product_Pair @ A @ ( rBT_Im246033960le_rbt @ A @ B ) @ X2 @ ( rBT_Im418718756_Empty @ A @ B ) ) )
=> ~ ! [X2: A,C3: rBT_Im1923302023_color,A3: rBT_Im246033960le_rbt @ A @ B,Y2: A,S2: B,B4: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( rBT_Im246033960le_rbt @ A @ B ) @ X2 @ ( rBT_Im480247531Branch @ A @ B @ C3 @ A3 @ Y2 @ S2 @ B4 ) ) ) ) ).
% ord.rbt_del.cases
thf(fact_148_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_149_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_150_entries_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Uu: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V2: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im954575269ntries @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ Uu @ L @ K @ V2 @ R ) )
= ( append @ ( product_prod @ A @ B ) @ ( rBT_Im954575269ntries @ A @ B @ L ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( rBT_Im954575269ntries @ A @ B @ R ) ) ) ) ).
% entries.simps(2)
thf(fact_151_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X2: A,Y2: B] :
( P2
= ( product_Pair @ A @ B @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_152_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A3: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_153_Pair__inject,axiom,
! [A: $tType,B: $tType,A4: A,B2: B,A6: A,B3: B] :
( ( ( product_Pair @ A @ B @ A4 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B3 ) )
=> ~ ( ( A4 = A6 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_154_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A3: A,B4: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) ) ).
% prod_cases3
thf(fact_155_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A3: A,B4: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_156_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A3: A,B4: B,C3: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_157_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A3: A,B4: B,C3: C,D2: D,E2: E,F3: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_158_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
~ ! [A3: A,B4: B,C3: C,D2: D,E2: E,F3: F4,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_159_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A3: A,B4: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_160_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A3: A,B4: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_161_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A3: A,B4: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_162_prod__induct6,axiom,
! [F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A3: A,B4: B,C3: C,D2: D,E2: E,F3: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_163_prod__induct7,axiom,
! [G2: $tType,F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
( ! [A3: A,B4: B,C3: C,D2: D,E2: E,F3: F4,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_164_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A3: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A3 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_165_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A3: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_166_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A4: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A4 @ B2 ) )
= ( F1 @ A4 @ B2 ) ) ).
% old.prod.rec
thf(fact_167_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_168_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_169_lexord__append__left__rightI,axiom,
! [A: $tType,A4: A,B2: A,R: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B2 ) @ R )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A4 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_170_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
thf(fact_171_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_172_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_173_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_174_lexord__cons__cons,axiom,
! [A: $tType,A4: A,X: list @ A,B2: A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A4 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R ) )
= ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B2 ) @ R )
| ( ( A4 = B2 )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_175_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R ) )
= ( ? [A7: A,X3: list @ A] :
( Y
= ( cons @ A @ A7 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_176_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I2
thf(fact_177_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% not_listrel1_Nil
thf(fact_178_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R ) ) ).
% not_Nil_listrel1
thf(fact_179_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% append_listrel1I
thf(fact_180_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_181_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_182_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_183_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_184_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs4: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Xs4 @ Ss ) )
& ( Ys
= ( append @ A @ Ys5 @ Ss ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( last @ A @ Xs4 )
!= ( last @ A @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_185_lexord__linear,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A3: A,B4: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B4 ) @ R )
| ( A3 = B4 )
| ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A3 ) @ R ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
| ( X = Y )
| ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_linear
thf(fact_186_lexord__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X2: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irreflexive
thf(fact_187_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% lexord_Nil_right
thf(fact_188_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V2: list @ A,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_leftI
thf(fact_189_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R ) )
=> ( ! [X2: A] :
( ( Xs
= ( cons @ A @ X2 @ Ys ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs2 ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_190_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( ! [Y2: A] :
( ( Ys
= ( cons @ A @ Y2 @ Xs ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs2 ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_191_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I1
thf(fact_192_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R ) )
=> ( ! [A3: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_193_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R: set @ ( product_prod @ A @ A )] :
( ? [B5: A,Z: list @ A] :
( Y
= ( cons @ A @ B5 @ Z ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_rightI
thf(fact_194_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% listrel1I
thf(fact_195_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ~ ! [X2: A,Y2: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R )
=> ! [Us3: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X2 @ Vs2 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_196_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_197_entries__balance,axiom,
! [B: $tType,A: $tType,L: rBT_Im246033960le_rbt @ A @ B,K: A,V2: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im954575269ntries @ A @ B @ ( rBT_Im1648453169alance @ A @ B @ L @ K @ V2 @ R ) )
= ( append @ ( product_prod @ A @ B ) @ ( rBT_Im954575269ntries @ A @ B @ L ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( rBT_Im954575269ntries @ A @ B @ R ) ) ) ) ).
% entries_balance
thf(fact_198_lexord__asymmetric,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A4: list @ A,B2: list @ A] :
( ( asym @ A @ R4 )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A4 @ B2 ) @ ( lexord @ A @ R4 ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B2 @ A4 ) @ ( lexord @ A @ R4 ) ) ) ) ).
% lexord_asymmetric
thf(fact_199_listrel_Oinducts,axiom,
! [A: $tType,B: $tType,X12: list @ A,X24: list @ B,R: set @ ( product_prod @ A @ B ),P: ( list @ A ) > ( list @ B ) > $o] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X12 @ X24 ) @ ( listrel @ A @ B @ R ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Y2: B,Xs2: list @ A,Ys2: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ).
% listrel.inducts
thf(fact_200_keys__balance,axiom,
! [B: $tType,A: $tType,L: rBT_Im246033960le_rbt @ A @ B,K: A,V2: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im380146495e_keys @ A @ B @ ( rBT_Im1648453169alance @ A @ B @ L @ K @ V2 @ R ) )
= ( append @ A @ ( rBT_Im380146495e_keys @ A @ B @ L ) @ ( cons @ A @ K @ ( rBT_Im380146495e_keys @ A @ B @ R ) ) ) ) ).
% keys_balance
thf(fact_201_lexord__asym,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R4 )
=> ( asym @ ( list @ A ) @ ( lexord @ A @ R4 ) ) ) ).
% lexord_asym
thf(fact_202_listrel_ONil,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] : ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) ) ).
% listrel.Nil
thf(fact_203_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_204_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_205_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrel.Cons
thf(fact_206_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [Y2: B,Ys2: list @ B] :
( ( Xs
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y2 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_207_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_208_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A2
!= ( nil @ B ) ) )
=> ~ ! [X2: A,Y2: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ B] :
( ( A2
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_209_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A2 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A2
= ( nil @ B ) ) )
| ? [X3: A,Y3: B,Xs3: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X3 @ Xs3 ) )
& ( A2
= ( cons @ B @ Y3 @ Ys3 ) )
& ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_210_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A4: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A4 @ B2 ) )
= ( C2 @ A4 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_211_map__tailrec__rev_Opelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y = Xb )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xb ) ) ) ) )
=> ~ ! [A3: A,As2: list @ A] :
( ( Xa
= ( cons @ A @ A3 @ As2 ) )
=> ( ( Y
= ( map_tailrec_rev @ A @ B @ X @ As2 @ ( cons @ B @ ( X @ A3 ) @ Xb ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A3 @ As2 ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_212_ord_Osunion__with_Opsimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,K: A,V2: B,As: list @ ( product_prod @ A @ B ),K4: A,V4: B,Bs: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im753582353th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) )
=> ( ( ( Less @ K4 @ K )
=> ( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ Bs ) ) ) )
& ( ~ ( Less @ K4 @ K )
=> ( ( ( Less @ K @ K4 )
=> ( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ As @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) ) )
& ( ~ ( Less @ K @ K4 )
=> ( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ ( F @ K @ V2 @ V4 ) ) @ ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ As @ Bs ) ) ) ) ) ) ) ) ).
% ord.sunion_with.psimps(1)
thf(fact_213_ord_Osinter__with_Opsimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,K: A,V2: B,As: list @ ( product_prod @ A @ B ),K4: A,V4: B,Bs: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im735314116th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) )
=> ( ( ( Less @ K4 @ K )
=> ( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ Bs ) ) )
& ( ~ ( Less @ K4 @ K )
=> ( ( ( Less @ K @ K4 )
=> ( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( rBT_Im319793781r_with @ A @ B @ Less @ F @ As @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) )
& ( ~ ( Less @ K @ K4 )
=> ( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ ( F @ K @ V2 @ V4 ) ) @ ( rBT_Im319793781r_with @ A @ B @ Less @ F @ As @ Bs ) ) ) ) ) ) ) ) ).
% ord.sinter_with.psimps(1)
thf(fact_214_ord_Osinter__with_Opsimps_I3_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,Uv: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im735314116th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ Uv @ ( nil @ ( product_prod @ A @ B ) ) ) ) )
=> ( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ Uv @ ( nil @ ( product_prod @ A @ B ) ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% ord.sinter_with.psimps(3)
thf(fact_215_ord_Osinter__with_Opsimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,Uu: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im735314116th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Uu ) ) )
=> ( ( rBT_Im319793781r_with @ A @ B @ Less @ F @ ( nil @ ( product_prod @ A @ B ) ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% ord.sinter_with.psimps(2)
thf(fact_216_ord_Osunion__with_Opsimps_I3_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,As: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im753582353th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ As @ ( nil @ ( product_prod @ A @ B ) ) ) ) )
=> ( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ As @ ( nil @ ( product_prod @ A @ B ) ) )
= As ) ) ).
% ord.sunion_with.psimps(3)
thf(fact_217_ord_Osunion__with_Opsimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,Bs: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im753582353th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Bs ) ) )
=> ( ( rBT_Im1445157352n_with @ A @ B @ Less @ F @ ( nil @ ( product_prod @ A @ B ) ) @ Bs )
= Bs ) ) ).
% ord.sunion_with.psimps(2)
thf(fact_218_ord_Osunion__with_Opinduct,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,A0: A > B > B > B,A1: list @ ( product_prod @ A @ B ),A2: list @ ( product_prod @ A @ B ),P: ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > $o] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im753582353th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ A0 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ A1 @ A2 ) ) )
=> ( ! [F3: A > B > B > B,K2: A,V: B,As2: list @ ( product_prod @ A @ B ),K3: A,V3: B,Bs2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im753582353th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ( Less @ K3 @ K2 )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ Bs2 ) )
=> ( ( ~ ( Less @ K3 @ K2 )
=> ( ( Less @ K2 @ K3 )
=> ( P @ F3 @ As2 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ~ ( Less @ K3 @ K2 )
=> ( ~ ( Less @ K2 @ K3 )
=> ( P @ F3 @ As2 @ Bs2 ) ) )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) ) ) )
=> ( ! [F3: A > B > B > B,Bs2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im753582353th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Bs2 ) ) )
=> ( P @ F3 @ ( nil @ ( product_prod @ A @ B ) ) @ Bs2 ) )
=> ( ! [F3: A > B > B > B,As2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im753582353th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ As2 @ ( nil @ ( product_prod @ A @ B ) ) ) ) )
=> ( P @ F3 @ As2 @ ( nil @ ( product_prod @ A @ B ) ) ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ) ) ).
% ord.sunion_with.pinduct
thf(fact_219_ord_Osinter__with_Opinduct,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,A0: A > B > B > B,A1: list @ ( product_prod @ A @ B ),A2: list @ ( product_prod @ A @ B ),P: ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > $o] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im735314116th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ A0 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ A1 @ A2 ) ) )
=> ( ! [F3: A > B > B > B,K2: A,V: B,As2: list @ ( product_prod @ A @ B ),K3: A,V3: B,Bs2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im735314116th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ( Less @ K3 @ K2 )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ Bs2 ) )
=> ( ( ~ ( Less @ K3 @ K2 )
=> ( ( Less @ K2 @ K3 )
=> ( P @ F3 @ As2 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ~ ( Less @ K3 @ K2 )
=> ( ~ ( Less @ K2 @ K3 )
=> ( P @ F3 @ As2 @ Bs2 ) ) )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) ) ) )
=> ( ! [F3: A > B > B > B,Uu2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im735314116th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Uu2 ) ) )
=> ( P @ F3 @ ( nil @ ( product_prod @ A @ B ) ) @ Uu2 ) )
=> ( ! [F3: A > B > B > B,Uv2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im735314116th_rel @ A @ B @ Less ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ Uv2 @ ( nil @ ( product_prod @ A @ B ) ) ) ) )
=> ( P @ F3 @ Uv2 @ ( nil @ ( product_prod @ A @ B ) ) ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ) ) ).
% ord.sinter_with.pinduct
thf(fact_220_sunion__with_Opsimps_I3_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,As: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im328486757th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ As @ ( nil @ ( product_prod @ A @ B ) ) ) ) )
=> ( ( rBT_Im1518674580n_with @ A @ B @ F @ As @ ( nil @ ( product_prod @ A @ B ) ) )
= As ) ) ) ).
% sunion_with.psimps(3)
thf(fact_221_sunion__with_Opsimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,Bs: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im328486757th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Bs ) ) )
=> ( ( rBT_Im1518674580n_with @ A @ B @ F @ ( nil @ ( product_prod @ A @ B ) ) @ Bs )
= Bs ) ) ) ).
% sunion_with.psimps(2)
thf(fact_222_sunion__with_Opsimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,K: A,V2: B,As: list @ ( product_prod @ A @ B ),K4: A,V4: B,Bs: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im328486757th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) )
=> ( ( ( ord_less @ A @ K4 @ K )
=> ( ( rBT_Im1518674580n_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ ( rBT_Im1518674580n_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ Bs ) ) ) )
& ( ~ ( ord_less @ A @ K4 @ K )
=> ( ( ( ord_less @ A @ K @ K4 )
=> ( ( rBT_Im1518674580n_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( rBT_Im1518674580n_with @ A @ B @ F @ As @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) ) )
& ( ~ ( ord_less @ A @ K @ K4 )
=> ( ( rBT_Im1518674580n_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ ( F @ K @ V2 @ V4 ) ) @ ( rBT_Im1518674580n_with @ A @ B @ F @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% sunion_with.psimps(1)
thf(fact_223_sunion__with_Opinduct,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A0: A > B > B > B,A1: list @ ( product_prod @ A @ B ),A2: list @ ( product_prod @ A @ B ),P: ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > $o] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im328486757th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ A0 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ A1 @ A2 ) ) )
=> ( ! [F3: A > B > B > B,K2: A,V: B,As2: list @ ( product_prod @ A @ B ),K3: A,V3: B,Bs2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im328486757th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ( ord_less @ A @ K3 @ K2 )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ Bs2 ) )
=> ( ( ~ ( ord_less @ A @ K3 @ K2 )
=> ( ( ord_less @ A @ K2 @ K3 )
=> ( P @ F3 @ As2 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ~ ( ord_less @ A @ K3 @ K2 )
=> ( ~ ( ord_less @ A @ K2 @ K3 )
=> ( P @ F3 @ As2 @ Bs2 ) ) )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) ) ) )
=> ( ! [F3: A > B > B > B,Bs2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im328486757th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Bs2 ) ) )
=> ( P @ F3 @ ( nil @ ( product_prod @ A @ B ) ) @ Bs2 ) )
=> ( ! [F3: A > B > B > B,As2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im328486757th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ As2 @ ( nil @ ( product_prod @ A @ B ) ) ) ) )
=> ( P @ F3 @ As2 @ ( nil @ ( product_prod @ A @ B ) ) ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ) ) ) ).
% sunion_with.pinduct
thf(fact_224_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp_eq @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_225_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_226_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_227_sunion__with_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [P: ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > $o,A0: A > B > B > B,A1: list @ ( product_prod @ A @ B ),A2: list @ ( product_prod @ A @ B )] :
( ! [F3: A > B > B > B,K2: A,V: B,As2: list @ ( product_prod @ A @ B ),K3: A,V3: B,Bs2: list @ ( product_prod @ A @ B )] :
( ( ( ord_less @ A @ K3 @ K2 )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ Bs2 ) )
=> ( ( ~ ( ord_less @ A @ K3 @ K2 )
=> ( ( ord_less @ A @ K2 @ K3 )
=> ( P @ F3 @ As2 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ~ ( ord_less @ A @ K3 @ K2 )
=> ( ~ ( ord_less @ A @ K2 @ K3 )
=> ( P @ F3 @ As2 @ Bs2 ) ) )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) ) )
=> ( ! [F3: A > B > B > B,X1: list @ ( product_prod @ A @ B )] : ( P @ F3 @ ( nil @ ( product_prod @ A @ B ) ) @ X1 )
=> ( ! [F3: A > B > B > B,As2: list @ ( product_prod @ A @ B )] : ( P @ F3 @ As2 @ ( nil @ ( product_prod @ A @ B ) ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ) ) ).
% sunion_with.induct
thf(fact_228_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A1: list @ A,A2: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys2: list @ A] : A2 != Ys2 )
=> ( ! [X2: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A2
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( ord_less @ A @ X2 @ Y2 ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A2
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X2 )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_229_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A22: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22 = Ys3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X3 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X3 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_230_lexordp__eq_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X12: list @ A,X24: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp_eq @ A @ X12 @ X24 )
=> ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [X2: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X2: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X2 )
=> ( ( ord_lexordp_eq @ A @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ) ) ).
% lexordp_eq.inducts
thf(fact_231_sunion__with_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K4: A,K: A,F: A > B > B > B,V2: B,As: list @ ( product_prod @ A @ B ),V4: B,Bs: list @ ( product_prod @ A @ B )] :
( ( ( ord_less @ A @ K4 @ K )
=> ( ( rBT_Im1518674580n_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ ( rBT_Im1518674580n_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ Bs ) ) ) )
& ( ~ ( ord_less @ A @ K4 @ K )
=> ( ( ( ord_less @ A @ K @ K4 )
=> ( ( rBT_Im1518674580n_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( rBT_Im1518674580n_with @ A @ B @ F @ As @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) ) )
& ( ~ ( ord_less @ A @ K @ K4 )
=> ( ( rBT_Im1518674580n_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ ( F @ K @ V2 @ V4 ) ) @ ( rBT_Im1518674580n_with @ A @ B @ F @ As @ Bs ) ) ) ) ) ) ) ) ).
% sunion_with.simps(1)
thf(fact_232_sinter__with_Opinduct,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A0: A > B > B > B,A1: list @ ( product_prod @ A @ B ),A2: list @ ( product_prod @ A @ B ),P: ( A > B > B > B ) > ( list @ ( product_prod @ A @ B ) ) > ( list @ ( product_prod @ A @ B ) ) > $o] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im310218520th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ A0 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ A1 @ A2 ) ) )
=> ( ! [F3: A > B > B > B,K2: A,V: B,As2: list @ ( product_prod @ A @ B ),K3: A,V3: B,Bs2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im310218520th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ( ord_less @ A @ K3 @ K2 )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ Bs2 ) )
=> ( ( ~ ( ord_less @ A @ K3 @ K2 )
=> ( ( ord_less @ A @ K2 @ K3 )
=> ( P @ F3 @ As2 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) )
=> ( ( ~ ( ord_less @ A @ K3 @ K2 )
=> ( ~ ( ord_less @ A @ K2 @ K3 )
=> ( P @ F3 @ As2 @ Bs2 ) ) )
=> ( P @ F3 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ As2 ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V3 ) @ Bs2 ) ) ) ) ) )
=> ( ! [F3: A > B > B > B,Uu2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im310218520th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Uu2 ) ) )
=> ( P @ F3 @ ( nil @ ( product_prod @ A @ B ) ) @ Uu2 ) )
=> ( ! [F3: A > B > B > B,Uv2: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im310218520th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F3 @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ Uv2 @ ( nil @ ( product_prod @ A @ B ) ) ) ) )
=> ( P @ F3 @ Uv2 @ ( nil @ ( product_prod @ A @ B ) ) ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ) ) ) ).
% sinter_with.pinduct
thf(fact_233_sinter__with_Opsimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,K: A,V2: B,As: list @ ( product_prod @ A @ B ),K4: A,V4: B,Bs: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im310218520th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) )
=> ( ( ( ord_less @ A @ K4 @ K )
=> ( ( rBT_Im393311009r_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( rBT_Im393311009r_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ Bs ) ) )
& ( ~ ( ord_less @ A @ K4 @ K )
=> ( ( ( ord_less @ A @ K @ K4 )
=> ( ( rBT_Im393311009r_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( rBT_Im393311009r_with @ A @ B @ F @ As @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) )
& ( ~ ( ord_less @ A @ K @ K4 )
=> ( ( rBT_Im393311009r_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ ( F @ K @ V2 @ V4 ) ) @ ( rBT_Im393311009r_with @ A @ B @ F @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% sinter_with.psimps(1)
thf(fact_234_sinter__with_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,Uu: list @ ( product_prod @ A @ B )] :
( ( rBT_Im393311009r_with @ A @ B @ F @ ( nil @ ( product_prod @ A @ B ) ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% sinter_with.simps(2)
thf(fact_235_sinter__with_Osimps_I3_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,Uv: list @ ( product_prod @ A @ B )] :
( ( rBT_Im393311009r_with @ A @ B @ F @ Uv @ ( nil @ ( product_prod @ A @ B ) ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% sinter_with.simps(3)
thf(fact_236_sinter__with_Opsimps_I3_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,Uv: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im310218520th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ Uv @ ( nil @ ( product_prod @ A @ B ) ) ) ) )
=> ( ( rBT_Im393311009r_with @ A @ B @ F @ Uv @ ( nil @ ( product_prod @ A @ B ) ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ) ) ).
% sinter_with.psimps(3)
thf(fact_237_sinter__with_Opsimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,Uu: list @ ( product_prod @ A @ B )] :
( ( accp @ ( product_prod @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) ) @ ( rBT_Im310218520th_rel @ A @ B ) @ ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) ) @ F @ ( product_Pair @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ ( product_prod @ A @ B ) ) @ ( nil @ ( product_prod @ A @ B ) ) @ Uu ) ) )
=> ( ( rBT_Im393311009r_with @ A @ B @ F @ ( nil @ ( product_prod @ A @ B ) ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ) ) ).
% sinter_with.psimps(2)
thf(fact_238_sinter__with_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K4: A,K: A,F: A > B > B > B,V2: B,As: list @ ( product_prod @ A @ B ),V4: B,Bs: list @ ( product_prod @ A @ B )] :
( ( ( ord_less @ A @ K4 @ K )
=> ( ( rBT_Im393311009r_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( rBT_Im393311009r_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ Bs ) ) )
& ( ~ ( ord_less @ A @ K4 @ K )
=> ( ( ( ord_less @ A @ K @ K4 )
=> ( ( rBT_Im393311009r_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( rBT_Im393311009r_with @ A @ B @ F @ As @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) ) ) )
& ( ~ ( ord_less @ A @ K @ K4 )
=> ( ( rBT_Im393311009r_with @ A @ B @ F @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ As ) @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K4 @ V4 ) @ Bs ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ ( F @ K @ V2 @ V4 ) ) @ ( rBT_Im393311009r_with @ A @ B @ F @ As @ Bs ) ) ) ) ) ) ) ) ).
% sinter_with.simps(1)
thf(fact_239_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_240_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Us: list @ B,Ys: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Us ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ ( append @ B @ Us @ Vs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Us ) @ ( zip @ A @ B @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_241_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_242_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_243_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A4: list @ A] :
( ( ( map @ A @ B @ F @ A4 )
= ( nil @ B ) )
= ( A4
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_244_length__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_245_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_246_map__append,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F @ Xs ) @ ( map @ B @ A @ F @ Ys ) ) ) ).
% map_append
thf(fact_247_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_248_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_249_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_250_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_251_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( map @ A @ B @ F @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_252_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: list @ A] :
( ( map @ A @ B @ F @ ( cons @ A @ X21 @ X22 ) )
= ( cons @ B @ ( F @ X21 ) @ ( map @ A @ B @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_253_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F @ Ys ) )
=> ? [Z2: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map @ B @ A @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_254_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z2: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map @ B @ A @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
%----Type constructors (6)
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_1,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_3,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( rBT_Im380146495e_keys @ a @ b )
= ( rBT_Im1235880025n_keys @ a @ b @ ( nil @ ( product_prod @ a @ ( rBT_Im246033960le_rbt @ a @ b ) ) ) ) ) ).
%------------------------------------------------------------------------------