TPTP Problem File: SLH0048^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00283_010897__11938278_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1538 ( 621 unt; 266 typ; 0 def)
% Number of atoms : 3359 (1813 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11699 ( 499 ~; 97 |; 293 &;9485 @)
% ( 0 <=>;1325 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 50 ( 49 usr)
% Number of type conns : 666 ( 666 >; 0 *; 0 +; 0 <<)
% Number of symbols : 220 ( 217 usr; 21 con; 0-3 aty)
% Number of variables : 3781 ( 99 ^;3442 !; 240 ?;3781 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:01.623
%------------------------------------------------------------------------------
% Could-be-implicit typings (49)
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (217)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
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thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
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thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Nat__Onat,type,
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thf(sy_c_Equivalence__Relation__Enumeration_Orgf,type,
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thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit,type,
equiva5889994315859557365_limit: list_nat > nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit__rel,type,
equiva5575797544161152836it_rel: list_nat > list_nat > $o ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set_Pr8693737435421807431at_nat ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set_Pr8693737435421807431at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member3919319858682911658at_nat: produc7148259590854449153at_nat > set_Pr846279711151727201at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_x,type,
x: list_nat ).
thf(sy_v_xa____,type,
xa: nat ).
thf(sy_v_xs____,type,
xs: list_nat ).
thf(sy_v_y,type,
y: list_nat ).
thf(sy_v_ya____,type,
ya: nat ).
thf(sy_v_ys____,type,
ys: list_nat ).
% Relevant facts (1262)
thf(fact_0_Cons_Ohyps_I1_J,axiom,
( ( size_size_list_nat @ xs )
= ( size_size_list_nat @ ys ) ) ).
% Cons.hyps(1)
thf(fact_1_b,axiom,
xs = ys ).
% b
thf(fact_2_assms_I1_J,axiom,
( ( size_size_list_nat @ x )
= ( size_size_list_nat @ y ) ) ).
% assms(1)
thf(fact_3_assms_I4_J,axiom,
( ( equiva2048684438135499664of_nat @ x )
= ( equiva2048684438135499664of_nat @ y ) ) ).
% assms(4)
thf(fact_4_Cons_Oprems_I3_J,axiom,
( ( equiva2048684438135499664of_nat @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) )
= ( equiva2048684438135499664of_nat @ ( append_nat @ ys @ ( cons_nat @ ya @ nil_nat ) ) ) ) ).
% Cons.prems(3)
thf(fact_5_a,axiom,
( ( equiva2048684438135499664of_nat @ xs )
= ( equiva2048684438135499664of_nat @ ys ) ) ).
% a
thf(fact_6_assms_I3_J,axiom,
equiva3371634703666331078on_rgf @ y ).
% assms(3)
thf(fact_7_assms_I2_J,axiom,
equiva3371634703666331078on_rgf @ x ).
% assms(2)
thf(fact_8_Cons_Ohyps_I2_J,axiom,
( ( equiva3371634703666331078on_rgf @ xs )
=> ( ( equiva3371634703666331078on_rgf @ ys )
=> ( ( ( equiva2048684438135499664of_nat @ xs )
= ( equiva2048684438135499664of_nat @ ys ) )
=> ( xs = ys ) ) ) ) ).
% Cons.hyps(2)
thf(fact_9_Cons_Oprems_I1_J,axiom,
equiva3371634703666331078on_rgf @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ).
% Cons.prems(1)
thf(fact_10_Cons_Oprems_I2_J,axiom,
equiva3371634703666331078on_rgf @ ( append_nat @ ys @ ( cons_nat @ ya @ nil_nat ) ) ).
% Cons.prems(2)
thf(fact_11_i__l,axiom,
ord_less_nat @ i @ ( size_size_list_nat @ xs ) ).
% i_l
thf(fact_12_rgf__limit_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) ) ) ).
% rgf_limit.cases
thf(fact_13_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_14_kernel__of__eq__len,axiom,
! [X: list_list_nat,Y: list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_s3023201423986296836st_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_15_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_16_kernel__of__eq__len,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
=> ( ( size_s3023201423986296836st_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_17_kernel__of__eq__len,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_nat] :
( ( ( equiva5289778660929030465at_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_s5460976970255530739at_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_18_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_P6011104703257516679at_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva5289778660929030465at_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_s5460976970255530739at_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_19_kernel__of__eq__len,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_list_nat] :
( ( ( equiva5289778660929030465at_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
=> ( ( size_s5460976970255530739at_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_20_kernel__of__eq__len,axiom,
! [X: list_list_nat,Y: list_P6011104703257516679at_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva5289778660929030465at_nat @ Y ) )
=> ( ( size_s3023201423986296836st_nat @ X )
= ( size_s5460976970255530739at_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_21_kernel__of__eq__len,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ( ( equiva5289778660929030465at_nat @ X )
= ( equiva5289778660929030465at_nat @ Y ) )
=> ( ( size_s5460976970255530739at_nat @ X )
= ( size_s5460976970255530739at_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_22_kernel__of__eq__len,axiom,
! [X: list_P8469869581646625389at_nat,Y: list_nat] :
( ( ( equiva5029308436158967975at_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_s3679842834875189465at_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_23_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_24_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_list_nat,P: list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_25_list__induct__2__rev,axiom,
! [X: list_list_nat,Y: list_nat,P: list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_26_list__induct__2__rev,axiom,
! [X: list_nat_nat,Y: list_nat,P: list_nat_nat > list_nat > $o] :
( ( ( size_s8208510060688613859at_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_nat_nat @ nil_nat )
=> ( ! [X2: nat > nat,Xs: list_nat_nat,Y2: nat,Ys: list_nat] :
( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_nat_nat @ Xs @ ( cons_nat_nat @ X2 @ nil_nat_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_27_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_nat_nat,P: list_nat > list_nat_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_s8208510060688613859at_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_nat_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat > nat,Ys: list_nat_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s8208510060688613859at_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat_nat @ Ys @ ( cons_nat_nat @ Y2 @ nil_nat_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_28_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_P6011104703257516679at_nat,P: list_nat > list_P6011104703257516679at_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_s5460976970255530739at_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y2 @ nil_Pr5478986624290739719at_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_29_list__induct__2__rev,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_nat,P: list_P6011104703257516679at_nat > list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y2: nat,Ys: list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_30_list__induct__2__rev,axiom,
! [X: list_list_nat,Y: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
=> ( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_31_list__induct__2__rev,axiom,
! [X: list_nat_nat,Y: list_list_nat,P: list_nat_nat > list_list_nat > $o] :
( ( ( size_s8208510060688613859at_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
=> ( ( P @ nil_nat_nat @ nil_list_nat )
=> ( ! [X2: nat > nat,Xs: list_nat_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_nat_nat @ Xs @ ( cons_nat_nat @ X2 @ nil_nat_nat ) ) @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_32_list__induct__2__rev,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_list_nat,P: list_P6011104703257516679at_nat > list_list_nat > $o] :
( ( ( size_s5460976970255530739at_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
=> ( ( P @ nil_Pr5478986624290739719at_nat @ nil_list_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) ) @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_33_d_I2_J,axiom,
equiva3371634703666331078on_rgf @ ys ).
% d(2)
thf(fact_34_d_I1_J,axiom,
equiva3371634703666331078on_rgf @ xs ).
% d(1)
thf(fact_35_calculation,axiom,
( ( ( nth_nat @ xs @ i )
= xa )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ i @ ( size_size_list_nat @ xs ) ) @ ( equiva2048684438135499664of_nat @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ) ) ) ).
% calculation
thf(fact_36_append1__eq__conv,axiom,
! [Xs2: list_list_nat,X: list_nat,Ys2: list_list_nat,Y: list_nat] :
( ( ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) )
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_37_append1__eq__conv,axiom,
! [Xs2: list_P9162950289778280392at_nat,X: product_prod_nat_nat > nat,Ys2: list_P9162950289778280392at_nat,Y: product_prod_nat_nat > nat] :
( ( ( append299529829995958237at_nat @ Xs2 @ ( cons_P4861729644591583992at_nat @ X @ nil_Pr2865493887535707976at_nat ) )
= ( append299529829995958237at_nat @ Ys2 @ ( cons_P4861729644591583992at_nat @ Y @ nil_Pr2865493887535707976at_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_38_append1__eq__conv,axiom,
! [Xs2: list_nat_nat,X: nat > nat,Ys2: list_nat_nat,Y: nat > nat] :
( ( ( append_nat_nat @ Xs2 @ ( cons_nat_nat @ X @ nil_nat_nat ) )
= ( append_nat_nat @ Ys2 @ ( cons_nat_nat @ Y @ nil_nat_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_39_append1__eq__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) )
= ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_40_append1__eq__conv,axiom,
! [Xs2: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat,Ys2: list_P8469869581646625389at_nat,Y: produc859450856879609959at_nat] :
( ( ( append8751754712269456642at_nat @ Xs2 @ ( cons_P8732206157123786781at_nat @ X @ nil_Pr2582115297535392877at_nat ) )
= ( append8751754712269456642at_nat @ Ys2 @ ( cons_P8732206157123786781at_nat @ Y @ nil_Pr2582115297535392877at_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_41_append1__eq__conv,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_42_append__eq__append__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Us: list_P6011104703257516679at_nat,Vs: list_P6011104703257516679at_nat] :
( ( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
| ( ( size_s5460976970255530739at_nat @ Us )
= ( size_s5460976970255530739at_nat @ Vs ) ) )
=> ( ( ( append985823374593552924at_nat @ Xs2 @ Us )
= ( append985823374593552924at_nat @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_43_append__eq__append__conv,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Us: list_list_nat,Vs: list_list_nat] :
( ( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
| ( ( size_s3023201423986296836st_nat @ Us )
= ( size_s3023201423986296836st_nat @ Vs ) ) )
=> ( ( ( append_list_nat @ Xs2 @ Us )
= ( append_list_nat @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_44_append__eq__append__conv,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat,Us: list_P8469869581646625389at_nat,Vs: list_P8469869581646625389at_nat] :
( ( ( ( size_s3679842834875189465at_nat @ Xs2 )
= ( size_s3679842834875189465at_nat @ Ys2 ) )
| ( ( size_s3679842834875189465at_nat @ Us )
= ( size_s3679842834875189465at_nat @ Vs ) ) )
=> ( ( ( append8751754712269456642at_nat @ Xs2 @ Us )
= ( append8751754712269456642at_nat @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_45_append__eq__append__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs2 @ Us )
= ( append_nat @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_46_append_Oright__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ A @ nil_list_nat )
= A ) ).
% append.right_neutral
thf(fact_47_append_Oright__neutral,axiom,
! [A: list_P9162950289778280392at_nat] :
( ( append299529829995958237at_nat @ A @ nil_Pr2865493887535707976at_nat )
= A ) ).
% append.right_neutral
thf(fact_48_append_Oright__neutral,axiom,
! [A: list_nat_nat] :
( ( append_nat_nat @ A @ nil_nat_nat )
= A ) ).
% append.right_neutral
thf(fact_49_append_Oright__neutral,axiom,
! [A: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ A @ nil_Pr5478986624290739719at_nat )
= A ) ).
% append.right_neutral
thf(fact_50_append_Oright__neutral,axiom,
! [A: list_P8469869581646625389at_nat] :
( ( append8751754712269456642at_nat @ A @ nil_Pr2582115297535392877at_nat )
= A ) ).
% append.right_neutral
thf(fact_51_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_52_append__Nil2,axiom,
! [Xs2: list_list_nat] :
( ( append_list_nat @ Xs2 @ nil_list_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_53_append__Nil2,axiom,
! [Xs2: list_P9162950289778280392at_nat] :
( ( append299529829995958237at_nat @ Xs2 @ nil_Pr2865493887535707976at_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_54_append__Nil2,axiom,
! [Xs2: list_nat_nat] :
( ( append_nat_nat @ Xs2 @ nil_nat_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_55_append__Nil2,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ Xs2 @ nil_Pr5478986624290739719at_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_56_append__Nil2,axiom,
! [Xs2: list_P8469869581646625389at_nat] :
( ( append8751754712269456642at_nat @ Xs2 @ nil_Pr2582115297535392877at_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_57_append__Nil2,axiom,
! [Xs2: list_nat] :
( ( append_nat @ Xs2 @ nil_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_58_append__self__conv,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_list_nat ) ) ).
% append_self_conv
thf(fact_59_append__self__conv,axiom,
! [Xs2: list_P9162950289778280392at_nat,Ys2: list_P9162950289778280392at_nat] :
( ( ( append299529829995958237at_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_Pr2865493887535707976at_nat ) ) ).
% append_self_conv
thf(fact_60_append__self__conv,axiom,
! [Xs2: list_nat_nat,Ys2: list_nat_nat] :
( ( ( append_nat_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_nat_nat ) ) ).
% append_self_conv
thf(fact_61_append__self__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_Pr5478986624290739719at_nat ) ) ).
% append_self_conv
thf(fact_62_append__self__conv,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( ( append8751754712269456642at_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_Pr2582115297535392877at_nat ) ) ).
% append_self_conv
thf(fact_63_append__self__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_nat ) ) ).
% append_self_conv
thf(fact_64_self__append__conv,axiom,
! [Y: list_list_nat,Ys2: list_list_nat] :
( ( Y
= ( append_list_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_list_nat ) ) ).
% self_append_conv
thf(fact_65_self__append__conv,axiom,
! [Y: list_P9162950289778280392at_nat,Ys2: list_P9162950289778280392at_nat] :
( ( Y
= ( append299529829995958237at_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_Pr2865493887535707976at_nat ) ) ).
% self_append_conv
thf(fact_66_self__append__conv,axiom,
! [Y: list_nat_nat,Ys2: list_nat_nat] :
( ( Y
= ( append_nat_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_nat_nat ) ) ).
% self_append_conv
thf(fact_67_self__append__conv,axiom,
! [Y: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( Y
= ( append985823374593552924at_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_Pr5478986624290739719at_nat ) ) ).
% self_append_conv
thf(fact_68_self__append__conv,axiom,
! [Y: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( Y
= ( append8751754712269456642at_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_Pr2582115297535392877at_nat ) ) ).
% self_append_conv
thf(fact_69_self__append__conv,axiom,
! [Y: list_nat,Ys2: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_nat ) ) ).
% self_append_conv
thf(fact_70_append__self__conv2,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_71_append__self__conv2,axiom,
! [Xs2: list_P9162950289778280392at_nat,Ys2: list_P9162950289778280392at_nat] :
( ( ( append299529829995958237at_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_Pr2865493887535707976at_nat ) ) ).
% append_self_conv2
thf(fact_72_append__self__conv2,axiom,
! [Xs2: list_nat_nat,Ys2: list_nat_nat] :
( ( ( append_nat_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_nat_nat ) ) ).
% append_self_conv2
thf(fact_73_append__self__conv2,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_Pr5478986624290739719at_nat ) ) ).
% append_self_conv2
thf(fact_74_append__self__conv2,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( ( append8751754712269456642at_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_Pr2582115297535392877at_nat ) ) ).
% append_self_conv2
thf(fact_75_append__self__conv2,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_nat ) ) ).
% append_self_conv2
thf(fact_76_self__append__conv2,axiom,
! [Y: list_list_nat,Xs2: list_list_nat] :
( ( Y
= ( append_list_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_77_self__append__conv2,axiom,
! [Y: list_P9162950289778280392at_nat,Xs2: list_P9162950289778280392at_nat] :
( ( Y
= ( append299529829995958237at_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_Pr2865493887535707976at_nat ) ) ).
% self_append_conv2
thf(fact_78_self__append__conv2,axiom,
! [Y: list_nat_nat,Xs2: list_nat_nat] :
( ( Y
= ( append_nat_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_nat_nat ) ) ).
% self_append_conv2
thf(fact_79_self__append__conv2,axiom,
! [Y: list_P6011104703257516679at_nat,Xs2: list_P6011104703257516679at_nat] :
( ( Y
= ( append985823374593552924at_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_Pr5478986624290739719at_nat ) ) ).
% self_append_conv2
thf(fact_80_self__append__conv2,axiom,
! [Y: list_P8469869581646625389at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( Y
= ( append8751754712269456642at_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_Pr2582115297535392877at_nat ) ) ).
% self_append_conv2
thf(fact_81_self__append__conv2,axiom,
! [Y: list_nat,Xs2: list_nat] :
( ( Y
= ( append_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_nat ) ) ).
% self_append_conv2
thf(fact_82_Nil__is__append__conv,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_list_nat )
& ( Ys2 = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_83_Nil__is__append__conv,axiom,
! [Xs2: list_P9162950289778280392at_nat,Ys2: list_P9162950289778280392at_nat] :
( ( nil_Pr2865493887535707976at_nat
= ( append299529829995958237at_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_Pr2865493887535707976at_nat )
& ( Ys2 = nil_Pr2865493887535707976at_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_84_Nil__is__append__conv,axiom,
! [Xs2: list_nat_nat,Ys2: list_nat_nat] :
( ( nil_nat_nat
= ( append_nat_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_nat_nat )
& ( Ys2 = nil_nat_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_85_Nil__is__append__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( append985823374593552924at_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_Pr5478986624290739719at_nat )
& ( Ys2 = nil_Pr5478986624290739719at_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_86_Nil__is__append__conv,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( nil_Pr2582115297535392877at_nat
= ( append8751754712269456642at_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_Pr2582115297535392877at_nat )
& ( Ys2 = nil_Pr2582115297535392877at_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_87_Nil__is__append__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( nil_nat
= ( append_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_88_list_Oinject,axiom,
! [X21: list_nat,X22: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X22 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_89_list_Oinject,axiom,
! [X21: product_prod_nat_nat > nat,X22: list_P9162950289778280392at_nat,Y21: product_prod_nat_nat > nat,Y22: list_P9162950289778280392at_nat] :
( ( ( cons_P4861729644591583992at_nat @ X21 @ X22 )
= ( cons_P4861729644591583992at_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_90_list_Oinject,axiom,
! [X21: nat > nat,X22: list_nat_nat,Y21: nat > nat,Y22: list_nat_nat] :
( ( ( cons_nat_nat @ X21 @ X22 )
= ( cons_nat_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_91_list_Oinject,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,Y21: product_prod_nat_nat,Y22: list_P6011104703257516679at_nat] :
( ( ( cons_P6512896166579812791at_nat @ X21 @ X22 )
= ( cons_P6512896166579812791at_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_92_list_Oinject,axiom,
! [X21: produc859450856879609959at_nat,X22: list_P8469869581646625389at_nat,Y21: produc859450856879609959at_nat,Y22: list_P8469869581646625389at_nat] :
( ( ( cons_P8732206157123786781at_nat @ X21 @ X22 )
= ( cons_P8732206157123786781at_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_93_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_94_same__append__eq,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs2 @ Ys2 )
= ( append985823374593552924at_nat @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_95_same__append__eq,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= ( append_list_nat @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_96_same__append__eq,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat,Zs: list_P8469869581646625389at_nat] :
( ( ( append8751754712269456642at_nat @ Xs2 @ Ys2 )
= ( append8751754712269456642at_nat @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_97_same__append__eq,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= ( append_nat @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_98_append__same__eq,axiom,
! [Ys2: list_P6011104703257516679at_nat,Xs2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Ys2 @ Xs2 )
= ( append985823374593552924at_nat @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_99_append__same__eq,axiom,
! [Ys2: list_list_nat,Xs2: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Ys2 @ Xs2 )
= ( append_list_nat @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_100_append__same__eq,axiom,
! [Ys2: list_P8469869581646625389at_nat,Xs2: list_P8469869581646625389at_nat,Zs: list_P8469869581646625389at_nat] :
( ( ( append8751754712269456642at_nat @ Ys2 @ Xs2 )
= ( append8751754712269456642at_nat @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_101_append__same__eq,axiom,
! [Ys2: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys2 @ Xs2 )
= ( append_nat @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_102_append__assoc,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ ( append985823374593552924at_nat @ Xs2 @ Ys2 ) @ Zs )
= ( append985823374593552924at_nat @ Xs2 @ ( append985823374593552924at_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_103_append__assoc,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ Zs )
= ( append_list_nat @ Xs2 @ ( append_list_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_104_append__assoc,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat,Zs: list_P8469869581646625389at_nat] :
( ( append8751754712269456642at_nat @ ( append8751754712269456642at_nat @ Xs2 @ Ys2 ) @ Zs )
= ( append8751754712269456642at_nat @ Xs2 @ ( append8751754712269456642at_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_105_append__assoc,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs2 @ Ys2 ) @ Zs )
= ( append_nat @ Xs2 @ ( append_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_106_append_Oassoc,axiom,
! [A: list_P6011104703257516679at_nat,B: list_P6011104703257516679at_nat,C: list_P6011104703257516679at_nat] :
( ( append985823374593552924at_nat @ ( append985823374593552924at_nat @ A @ B ) @ C )
= ( append985823374593552924at_nat @ A @ ( append985823374593552924at_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_107_append_Oassoc,axiom,
! [A: list_list_nat,B: list_list_nat,C: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ A @ B ) @ C )
= ( append_list_nat @ A @ ( append_list_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_108_append_Oassoc,axiom,
! [A: list_P8469869581646625389at_nat,B: list_P8469869581646625389at_nat,C: list_P8469869581646625389at_nat] :
( ( append8751754712269456642at_nat @ ( append8751754712269456642at_nat @ A @ B ) @ C )
= ( append8751754712269456642at_nat @ A @ ( append8751754712269456642at_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_109_append_Oassoc,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A @ B ) @ C )
= ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_110_append__is__Nil__conv,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= nil_list_nat )
= ( ( Xs2 = nil_list_nat )
& ( Ys2 = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_111_append__is__Nil__conv,axiom,
! [Xs2: list_P9162950289778280392at_nat,Ys2: list_P9162950289778280392at_nat] :
( ( ( append299529829995958237at_nat @ Xs2 @ Ys2 )
= nil_Pr2865493887535707976at_nat )
= ( ( Xs2 = nil_Pr2865493887535707976at_nat )
& ( Ys2 = nil_Pr2865493887535707976at_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_112_append__is__Nil__conv,axiom,
! [Xs2: list_nat_nat,Ys2: list_nat_nat] :
( ( ( append_nat_nat @ Xs2 @ Ys2 )
= nil_nat_nat )
= ( ( Xs2 = nil_nat_nat )
& ( Ys2 = nil_nat_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_113_append__is__Nil__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs2 @ Ys2 )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs2 = nil_Pr5478986624290739719at_nat )
& ( Ys2 = nil_Pr5478986624290739719at_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_114_append__is__Nil__conv,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( ( append8751754712269456642at_nat @ Xs2 @ Ys2 )
= nil_Pr2582115297535392877at_nat )
= ( ( Xs2 = nil_Pr2582115297535392877at_nat )
& ( Ys2 = nil_Pr2582115297535392877at_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_115_append__is__Nil__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= nil_nat )
= ( ( Xs2 = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_116_nth__append__length,axiom,
! [Xs2: list_P9162950289778280392at_nat,X: product_prod_nat_nat > nat,Ys2: list_P9162950289778280392at_nat] :
( ( nth_Pr5595005939758809033at_nat @ ( append299529829995958237at_nat @ Xs2 @ ( cons_P4861729644591583992at_nat @ X @ Ys2 ) ) @ ( size_s4562453492649449012at_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_117_nth__append__length,axiom,
! [Xs2: list_nat_nat,X: nat > nat,Ys2: list_nat_nat] :
( ( nth_nat_nat @ ( append_nat_nat @ Xs2 @ ( cons_nat_nat @ X @ Ys2 ) ) @ ( size_s8208510060688613859at_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_118_nth__append__length,axiom,
! [Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( nth_Pr7617993195940197384at_nat @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X @ Ys2 ) ) @ ( size_s5460976970255530739at_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_119_nth__append__length,axiom,
! [Xs2: list_list_nat,X: list_nat,Ys2: list_list_nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ Ys2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_120_nth__append__length,axiom,
! [Xs2: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( nth_Pr6744343527793145070at_nat @ ( append8751754712269456642at_nat @ Xs2 @ ( cons_P8732206157123786781at_nat @ X @ Ys2 ) ) @ ( size_s3679842834875189465at_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_121_nth__append__length,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_122_nth__equalityI,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I )
= ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_123_nth__equalityI,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ Xs2 @ I )
= ( nth_list_nat @ Ys2 @ I ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_124_nth__equalityI,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( ( size_s3679842834875189465at_nat @ Xs2 )
= ( size_s3679842834875189465at_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs2 ) )
=> ( ( nth_Pr6744343527793145070at_nat @ Xs2 @ I )
= ( nth_Pr6744343527793145070at_nat @ Ys2 @ I ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_125_nth__equalityI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Ys2 @ I ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_126_Skolem__list__nth,axiom,
! [K: nat,P: nat > product_prod_nat_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X3: product_prod_nat_nat] : ( P @ I2 @ X3 ) ) )
= ( ? [Xs3: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_Pr7617993195940197384at_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_127_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X3: list_nat] : ( P @ I2 @ X3 ) ) )
= ( ? [Xs3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_list_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_128_Skolem__list__nth,axiom,
! [K: nat,P: nat > produc859450856879609959at_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X3: produc859450856879609959at_nat] : ( P @ I2 @ X3 ) ) )
= ( ? [Xs3: list_P8469869581646625389at_nat] :
( ( ( size_s3679842834875189465at_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_Pr6744343527793145070at_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_129_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X3: nat] : ( P @ I2 @ X3 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_130_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_P6011104703257516679at_nat,Z: list_P6011104703257516679at_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs3 )
= ( size_s5460976970255530739at_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs3 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs3 @ I2 )
= ( nth_Pr7617993195940197384at_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_131_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_list_nat,Z: list_list_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_list_nat,Ys3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( ( nth_list_nat @ Xs3 @ I2 )
= ( nth_list_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_132_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_P8469869581646625389at_nat,Z: list_P8469869581646625389at_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_P8469869581646625389at_nat,Ys3: list_P8469869581646625389at_nat] :
( ( ( size_s3679842834875189465at_nat @ Xs3 )
= ( size_s3679842834875189465at_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Xs3 ) )
=> ( ( nth_Pr6744343527793145070at_nat @ Xs3 @ I2 )
= ( nth_Pr6744343527793145070at_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_133_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_134_length__induct,axiom,
! [P: list_P6011104703257516679at_nat > $o,Xs2: list_P6011104703257516679at_nat] :
( ! [Xs: list_P6011104703257516679at_nat] :
( ! [Ys4: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ys4 ) @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_135_length__induct,axiom,
! [P: list_list_nat > $o,Xs2: list_list_nat] :
( ! [Xs: list_list_nat] :
( ! [Ys4: list_list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys4 ) @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_136_length__induct,axiom,
! [P: list_P8469869581646625389at_nat > $o,Xs2: list_P8469869581646625389at_nat] :
( ! [Xs: list_P8469869581646625389at_nat] :
( ! [Ys4: list_P8469869581646625389at_nat] :
( ( ord_less_nat @ ( size_s3679842834875189465at_nat @ Ys4 ) @ ( size_s3679842834875189465at_nat @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_137_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs: list_nat] :
( ! [Ys4: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys4 ) @ ( size_size_list_nat @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_138_kernel__of__eq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
= ( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_139_kernel__of__eq,axiom,
! [X: list_list_nat,Y: list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
= ( ( ( size_s3023201423986296836st_nat @ X )
= ( size_size_list_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_list_nat @ X @ I2 )
= ( nth_list_nat @ X @ J ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_140_kernel__of__eq,axiom,
! [X: list_nat,Y: list_list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
= ( ( ( size_size_list_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J ) )
= ( ( nth_list_nat @ Y @ I2 )
= ( nth_list_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_141_kernel__of__eq,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
= ( ( ( size_s3023201423986296836st_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_list_nat @ X @ I2 )
= ( nth_list_nat @ X @ J ) )
= ( ( nth_list_nat @ Y @ I2 )
= ( nth_list_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_142_kernel__of__eq,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_nat] :
( ( ( equiva5289778660929030465at_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
= ( ( ( size_s5460976970255530739at_nat @ X )
= ( size_size_list_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s5460976970255530739at_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_Pr7617993195940197384at_nat @ X @ I2 )
= ( nth_Pr7617993195940197384at_nat @ X @ J ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_143_kernel__of__eq,axiom,
! [X: list_nat,Y: list_P6011104703257516679at_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva5289778660929030465at_nat @ Y ) )
= ( ( ( size_size_list_nat @ X )
= ( size_s5460976970255530739at_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J ) )
= ( ( nth_Pr7617993195940197384at_nat @ Y @ I2 )
= ( nth_Pr7617993195940197384at_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_144_kernel__of__eq,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_list_nat] :
( ( ( equiva5289778660929030465at_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
= ( ( ( size_s5460976970255530739at_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s5460976970255530739at_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_Pr7617993195940197384at_nat @ X @ I2 )
= ( nth_Pr7617993195940197384at_nat @ X @ J ) )
= ( ( nth_list_nat @ Y @ I2 )
= ( nth_list_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_145_kernel__of__eq,axiom,
! [X: list_list_nat,Y: list_P6011104703257516679at_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva5289778660929030465at_nat @ Y ) )
= ( ( ( size_s3023201423986296836st_nat @ X )
= ( size_s5460976970255530739at_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_list_nat @ X @ I2 )
= ( nth_list_nat @ X @ J ) )
= ( ( nth_Pr7617993195940197384at_nat @ Y @ I2 )
= ( nth_Pr7617993195940197384at_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_146_kernel__of__eq,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ( ( equiva5289778660929030465at_nat @ X )
= ( equiva5289778660929030465at_nat @ Y ) )
= ( ( ( size_s5460976970255530739at_nat @ X )
= ( size_s5460976970255530739at_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s5460976970255530739at_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_Pr7617993195940197384at_nat @ X @ I2 )
= ( nth_Pr7617993195940197384at_nat @ X @ J ) )
= ( ( nth_Pr7617993195940197384at_nat @ Y @ I2 )
= ( nth_Pr7617993195940197384at_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_147_kernel__of__eq,axiom,
! [X: list_P8469869581646625389at_nat,Y: list_nat] :
( ( ( equiva5029308436158967975at_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
= ( ( ( size_s3679842834875189465at_nat @ X )
= ( size_size_list_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3679842834875189465at_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_Pr6744343527793145070at_nat @ X @ I2 )
= ( nth_Pr6744343527793145070at_nat @ X @ J ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_148_not__Cons__self2,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( cons_list_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_149_not__Cons__self2,axiom,
! [X: product_prod_nat_nat > nat,Xs2: list_P9162950289778280392at_nat] :
( ( cons_P4861729644591583992at_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_150_not__Cons__self2,axiom,
! [X: nat > nat,Xs2: list_nat_nat] :
( ( cons_nat_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_151_not__Cons__self2,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( cons_P6512896166579812791at_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_152_not__Cons__self2,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( cons_P8732206157123786781at_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_153_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_154_neq__if__length__neq,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
!= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_155_neq__if__length__neq,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
!= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_156_neq__if__length__neq,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( ( size_s3679842834875189465at_nat @ Xs2 )
!= ( size_s3679842834875189465at_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_157_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_158_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_P6011104703257516679at_nat] :
( ( size_s5460976970255530739at_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_159_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_160_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_P8469869581646625389at_nat] :
( ( size_s3679842834875189465at_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_161_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_162_append__eq__append__conv2,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,Ts: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs2 @ Ys2 )
= ( append985823374593552924at_nat @ Zs @ Ts ) )
= ( ? [Us2: list_P6011104703257516679at_nat] :
( ( ( Xs2
= ( append985823374593552924at_nat @ Zs @ Us2 ) )
& ( ( append985823374593552924at_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append985823374593552924at_nat @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append985823374593552924at_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_163_append__eq__append__conv2,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat,Ts: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= ( append_list_nat @ Zs @ Ts ) )
= ( ? [Us2: list_list_nat] :
( ( ( Xs2
= ( append_list_nat @ Zs @ Us2 ) )
& ( ( append_list_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_list_nat @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append_list_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_164_append__eq__append__conv2,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat,Zs: list_P8469869581646625389at_nat,Ts: list_P8469869581646625389at_nat] :
( ( ( append8751754712269456642at_nat @ Xs2 @ Ys2 )
= ( append8751754712269456642at_nat @ Zs @ Ts ) )
= ( ? [Us2: list_P8469869581646625389at_nat] :
( ( ( Xs2
= ( append8751754712269456642at_nat @ Zs @ Us2 ) )
& ( ( append8751754712269456642at_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append8751754712269456642at_nat @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append8751754712269456642at_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_165_append__eq__append__conv2,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs2
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_nat @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_166_mem__Collect__eq,axiom,
! [A: produc2487518378626728076at_nat,P: produc2487518378626728076at_nat > $o] :
( ( member5754182594613576739at_nat @ A @ ( collec3941131003887891809at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_167_mem__Collect__eq,axiom,
! [A: produc7248412053542808358at_nat,P: produc7248412053542808358at_nat > $o] :
( ( member2223272150424702269at_nat @ A @ ( collec5903703980526211963at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_168_mem__Collect__eq,axiom,
! [A: list_P8469869581646625389at_nat,P: list_P8469869581646625389at_nat > $o] :
( ( member3799944675974059798at_nat @ A @ ( collec6247604700435595864at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_169_mem__Collect__eq,axiom,
! [A: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > $o] :
( ( member3067507820990806192at_nat @ A @ ( collec3343600615725829874at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_170_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_171_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_172_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_173_mem__Collect__eq,axiom,
! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_174_Collect__mem__eq,axiom,
! [A2: set_Pr938417207581201730at_nat] :
( ( collec3941131003887891809at_nat
@ ^ [X4: produc2487518378626728076at_nat] : ( member5754182594613576739at_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_175_Collect__mem__eq,axiom,
! [A2: set_Pr7717912310451564380at_nat] :
( ( collec5903703980526211963at_nat
@ ^ [X4: produc7248412053542808358at_nat] : ( member2223272150424702269at_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_176_Collect__mem__eq,axiom,
! [A2: set_li3197816953174176717at_nat] :
( ( collec6247604700435595864at_nat
@ ^ [X4: list_P8469869581646625389at_nat] : ( member3799944675974059798at_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_177_Collect__mem__eq,axiom,
! [A2: set_li5450038453877631591at_nat] :
( ( collec3343600615725829874at_nat
@ ^ [X4: list_P6011104703257516679at_nat] : ( member3067507820990806192at_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_178_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_179_Collect__mem__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_180_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_181_Collect__mem__eq,axiom,
! [A2: set_Pr8693737435421807431at_nat] :
( ( collec7088162979684241874at_nat
@ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_182_Collect__cong,axiom,
! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
( ! [X2: produc859450856879609959at_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec7088162979684241874at_nat @ P )
= ( collec7088162979684241874at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_183_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_184_Collect__cong,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X2: product_prod_nat_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec3392354462482085612at_nat @ P )
= ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_185_append__eq__appendI,axiom,
! [Xs2: list_P6011104703257516679at_nat,Xs1: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Us: list_P6011104703257516679at_nat] :
( ( ( append985823374593552924at_nat @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append985823374593552924at_nat @ Xs1 @ Us ) )
=> ( ( append985823374593552924at_nat @ Xs2 @ Ys2 )
= ( append985823374593552924at_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_186_append__eq__appendI,axiom,
! [Xs2: list_list_nat,Xs1: list_list_nat,Zs: list_list_nat,Ys2: list_list_nat,Us: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_list_nat @ Xs1 @ Us ) )
=> ( ( append_list_nat @ Xs2 @ Ys2 )
= ( append_list_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_187_append__eq__appendI,axiom,
! [Xs2: list_P8469869581646625389at_nat,Xs1: list_P8469869581646625389at_nat,Zs: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat,Us: list_P8469869581646625389at_nat] :
( ( ( append8751754712269456642at_nat @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append8751754712269456642at_nat @ Xs1 @ Us ) )
=> ( ( append8751754712269456642at_nat @ Xs2 @ Ys2 )
= ( append8751754712269456642at_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_188_append__eq__appendI,axiom,
! [Xs2: list_nat,Xs1: list_nat,Zs: list_nat,Ys2: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs2 @ Ys2 )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_189_list__nonempty__induct,axiom,
! [Xs2: list_list_nat,P: list_list_nat > $o] :
( ( Xs2 != nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_190_list__nonempty__induct,axiom,
! [Xs2: list_P9162950289778280392at_nat,P: list_P9162950289778280392at_nat > $o] :
( ( Xs2 != nil_Pr2865493887535707976at_nat )
=> ( ! [X2: product_prod_nat_nat > nat] : ( P @ ( cons_P4861729644591583992at_nat @ X2 @ nil_Pr2865493887535707976at_nat ) )
=> ( ! [X2: product_prod_nat_nat > nat,Xs: list_P9162950289778280392at_nat] :
( ( Xs != nil_Pr2865493887535707976at_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_P4861729644591583992at_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_191_list__nonempty__induct,axiom,
! [Xs2: list_nat_nat,P: list_nat_nat > $o] :
( ( Xs2 != nil_nat_nat )
=> ( ! [X2: nat > nat] : ( P @ ( cons_nat_nat @ X2 @ nil_nat_nat ) )
=> ( ! [X2: nat > nat,Xs: list_nat_nat] :
( ( Xs != nil_nat_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_nat_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_192_list__nonempty__induct,axiom,
! [Xs2: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > $o] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_193_list__nonempty__induct,axiom,
! [Xs2: list_P8469869581646625389at_nat,P: list_P8469869581646625389at_nat > $o] :
( ( Xs2 != nil_Pr2582115297535392877at_nat )
=> ( ! [X2: produc859450856879609959at_nat] : ( P @ ( cons_P8732206157123786781at_nat @ X2 @ nil_Pr2582115297535392877at_nat ) )
=> ( ! [X2: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
( ( Xs != nil_Pr2582115297535392877at_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_P8732206157123786781at_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_194_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_195_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_196_list__induct2_H,axiom,
! [P: list_nat > list_list_nat > $o,Xs2: list_nat,Ys2: list_list_nat] :
( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y2 @ Ys ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_197_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs2: list_list_nat,Ys2: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y2 @ Ys ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_198_list__induct2_H,axiom,
! [P: list_nat > list_nat_nat > $o,Xs2: list_nat,Ys2: list_nat_nat] :
( ( P @ nil_nat @ nil_nat_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat_nat )
=> ( ! [Y2: nat > nat,Ys: list_nat_nat] : ( P @ nil_nat @ ( cons_nat_nat @ Y2 @ Ys ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat > nat,Ys: list_nat_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_199_list__induct2_H,axiom,
! [P: list_nat > list_P6011104703257516679at_nat > $o,Xs2: list_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_Pr5478986624290739719at_nat )
=> ( ! [Y2: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] : ( P @ nil_nat @ ( cons_P6512896166579812791at_nat @ Y2 @ Ys ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_200_list__induct2_H,axiom,
! [P: list_list_nat > list_list_nat > $o,Xs2: list_list_nat,Ys2: list_list_nat] :
( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y2 @ Ys ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_201_list__induct2_H,axiom,
! [P: list_nat_nat > list_nat > $o,Xs2: list_nat_nat,Ys2: list_nat] :
( ( P @ nil_nat_nat @ nil_nat )
=> ( ! [X2: nat > nat,Xs: list_nat_nat] : ( P @ ( cons_nat_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_nat_nat @ ( cons_nat @ Y2 @ Ys ) )
=> ( ! [X2: nat > nat,Xs: list_nat_nat,Y2: nat,Ys: list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_202_list__induct2_H,axiom,
! [P: list_P6011104703257516679at_nat > list_nat > $o,Xs2: list_P6011104703257516679at_nat,Ys2: list_nat] :
( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_nat @ Y2 @ Ys ) )
=> ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y2: nat,Ys: list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_203_list__induct2_H,axiom,
! [P: list_list_nat > list_nat_nat > $o,Xs2: list_list_nat,Ys2: list_nat_nat] :
( ( P @ nil_list_nat @ nil_nat_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_nat_nat )
=> ( ! [Y2: nat > nat,Ys: list_nat_nat] : ( P @ nil_list_nat @ ( cons_nat_nat @ Y2 @ Ys ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat > nat,Ys: list_nat_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_204_list__induct2_H,axiom,
! [P: list_list_nat > list_P6011104703257516679at_nat > $o,Xs2: list_list_nat,Ys2: list_P6011104703257516679at_nat] :
( ( P @ nil_list_nat @ nil_Pr5478986624290739719at_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_Pr5478986624290739719at_nat )
=> ( ! [Y2: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] : ( P @ nil_list_nat @ ( cons_P6512896166579812791at_nat @ Y2 @ Ys ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_205_neq__Nil__conv,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
= ( ? [Y4: list_nat,Ys3: list_list_nat] :
( Xs2
= ( cons_list_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_206_neq__Nil__conv,axiom,
! [Xs2: list_P9162950289778280392at_nat] :
( ( Xs2 != nil_Pr2865493887535707976at_nat )
= ( ? [Y4: product_prod_nat_nat > nat,Ys3: list_P9162950289778280392at_nat] :
( Xs2
= ( cons_P4861729644591583992at_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_207_neq__Nil__conv,axiom,
! [Xs2: list_nat_nat] :
( ( Xs2 != nil_nat_nat )
= ( ? [Y4: nat > nat,Ys3: list_nat_nat] :
( Xs2
= ( cons_nat_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_208_neq__Nil__conv,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
= ( ? [Y4: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( Xs2
= ( cons_P6512896166579812791at_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_209_neq__Nil__conv,axiom,
! [Xs2: list_P8469869581646625389at_nat] :
( ( Xs2 != nil_Pr2582115297535392877at_nat )
= ( ? [Y4: produc859450856879609959at_nat,Ys3: list_P8469869581646625389at_nat] :
( Xs2
= ( cons_P8732206157123786781at_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_210_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y4: nat,Ys3: list_nat] :
( Xs2
= ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_211_remdups__adj_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [X2: list_nat] :
( X
!= ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ~ ! [X2: list_nat,Y2: list_nat,Xs: list_list_nat] :
( X
!= ( cons_list_nat @ X2 @ ( cons_list_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_212_remdups__adj_Ocases,axiom,
! [X: list_P9162950289778280392at_nat] :
( ( X != nil_Pr2865493887535707976at_nat )
=> ( ! [X2: product_prod_nat_nat > nat] :
( X
!= ( cons_P4861729644591583992at_nat @ X2 @ nil_Pr2865493887535707976at_nat ) )
=> ~ ! [X2: product_prod_nat_nat > nat,Y2: product_prod_nat_nat > nat,Xs: list_P9162950289778280392at_nat] :
( X
!= ( cons_P4861729644591583992at_nat @ X2 @ ( cons_P4861729644591583992at_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_213_remdups__adj_Ocases,axiom,
! [X: list_nat_nat] :
( ( X != nil_nat_nat )
=> ( ! [X2: nat > nat] :
( X
!= ( cons_nat_nat @ X2 @ nil_nat_nat ) )
=> ~ ! [X2: nat > nat,Y2: nat > nat,Xs: list_nat_nat] :
( X
!= ( cons_nat_nat @ X2 @ ( cons_nat_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_214_remdups__adj_Ocases,axiom,
! [X: list_P6011104703257516679at_nat] :
( ( X != nil_Pr5478986624290739719at_nat )
=> ( ! [X2: product_prod_nat_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) )
=> ~ ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( X
!= ( cons_P6512896166579812791at_nat @ X2 @ ( cons_P6512896166579812791at_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_215_remdups__adj_Ocases,axiom,
! [X: list_P8469869581646625389at_nat] :
( ( X != nil_Pr2582115297535392877at_nat )
=> ( ! [X2: produc859450856879609959at_nat] :
( X
!= ( cons_P8732206157123786781at_nat @ X2 @ nil_Pr2582115297535392877at_nat ) )
=> ~ ! [X2: produc859450856879609959at_nat,Y2: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
( X
!= ( cons_P8732206157123786781at_nat @ X2 @ ( cons_P8732206157123786781at_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_216_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_217_transpose_Ocases,axiom,
! [X: list_l3264859301627795341at_nat] :
( ( X != nil_li8973309667444810893at_nat )
=> ( ! [Xss: list_l3264859301627795341at_nat] :
( X
!= ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ Xss ) )
=> ~ ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Xss: list_l3264859301627795341at_nat] :
( X
!= ( cons_l7612840610449961021at_nat @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_218_transpose_Ocases,axiom,
! [X: list_l3298181151656792051at_nat] :
( ( X != nil_li3705895014200927091at_nat )
=> ( ! [Xss: list_l3298181151656792051at_nat] :
( X
!= ( cons_l3196404044853765923at_nat @ nil_Pr2582115297535392877at_nat @ Xss ) )
=> ~ ! [X2: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,Xss: list_l3298181151656792051at_nat] :
( X
!= ( cons_l3196404044853765923at_nat @ ( cons_P8732206157123786781at_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_219_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_220_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_221_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_222_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_223_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_224_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys2: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_225_append__Cons,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs2 ) @ Ys2 )
= ( cons_nat @ X @ ( append_nat @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_226_eq__Nil__appendI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_nat @ nil_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_227_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_228_append__Nil,axiom,
! [Ys2: list_nat] :
( ( append_nat @ nil_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_229_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_230_list__induct3,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_231_list__induct2,axiom,
! [Xs2: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_232_rev__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_233_append__eq__Cons__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,X: nat,Xs2: list_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs2 ) ) )
| ? [Ys5: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Ys5 ) )
& ( ( append_nat @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_234_Cons__eq__append__conv,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_nat )
& ( ( cons_nat @ X @ Xs2 )
= Zs ) )
| ? [Ys5: list_nat] :
( ( ( cons_nat @ X @ Ys5 )
= Ys2 )
& ( Xs2
= ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_235_rev__exhaust,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ~ ! [Ys: list_nat,Y2: nat] :
( Xs2
!= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_236_rev__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] :
( ( P @ Xs )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_237_same__length__different,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 != Ys2 )
=> ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ? [Pre: list_nat,X2: nat,Xs4: list_nat,Y2: nat,Ys6: list_nat] :
( ( X2 != Y2 )
& ( Xs2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X2 @ nil_nat ) @ Xs4 ) ) )
& ( Ys2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_238_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A3: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_239_old_Oprod_Oinject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A @ B )
= ( produc6161850002892822231at_nat @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_240_prod_Oinject,axiom,
! [X1: nat,X23: nat,Y1: nat,Y23: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X23 )
= ( product_Pair_nat_nat @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_241_prod_Oinject,axiom,
! [X1: product_prod_nat_nat,X23: product_prod_nat_nat,Y1: product_prod_nat_nat,Y23: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ X1 @ X23 )
= ( produc6161850002892822231at_nat @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_242_bind__simps_I2_J,axiom,
! [X: nat,Xs2: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs2 ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_243_in__measures_I2_J,axiom,
! [X: nat,Y: nat,F: nat > nat,Fs: list_nat_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_244_in__measures_I2_J,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F: product_prod_nat_nat > nat,Fs: list_P9162950289778280392at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_245_Cons__lenlex__iff,axiom,
! [M: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ M @ Ms ) @ ( cons_P6512896166579812791at_nat @ N @ Ns ) ) @ ( lenlex325483962726685836at_nat @ R ) )
= ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ms ) @ ( size_s5460976970255530739at_nat @ Ns ) )
| ( ( ( size_s5460976970255530739at_nat @ Ms )
= ( size_s5460976970255530739at_nat @ Ns ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ M @ N ) @ R ) )
| ( ( M = N )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ms @ Ns ) @ ( lenlex325483962726685836at_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_246_Cons__lenlex__iff,axiom,
! [M: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ R ) )
| ( ( M = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_247_Cons__in__lex,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) @ ( lex_Pr8571645452597969515at_nat @ R ) )
= ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
& ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) ) )
| ( ( X = Y )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( lex_Pr8571645452597969515at_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_248_Cons__in__lex,axiom,
! [X: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( lex_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( lex_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_249_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs2: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs2 ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_250_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_251_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P2: nat > $o,Xs3: list_nat] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs3 ) )
& ( P2 @ ( nth_nat @ Xs3 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_252_listrel__iff__nth,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( listre818007680106770737at_nat @ R ) )
= ( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N2 ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_253_listrel__iff__nth,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ N2 ) @ ( nth_nat @ Ys2 @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_254_list__ex__simps_I1_J,axiom,
! [P: nat > $o,X: nat,Xs2: list_nat] :
( ( list_ex_nat @ P @ ( cons_nat @ X @ Xs2 ) )
= ( ( P @ X )
| ( list_ex_nat @ P @ Xs2 ) ) ) ).
% list_ex_simps(1)
thf(fact_255_list__ex__simps_I2_J,axiom,
! [P: nat > $o] :
~ ( list_ex_nat @ P @ nil_nat ) ).
% list_ex_simps(2)
thf(fact_256_list__ex__append,axiom,
! [P: nat > $o,Xs2: list_nat,Ys2: list_nat] :
( ( list_ex_nat @ P @ ( append_nat @ Xs2 @ Ys2 ) )
= ( ( list_ex_nat @ P @ Xs2 )
| ( list_ex_nat @ P @ Ys2 ) ) ) ).
% list_ex_append
thf(fact_257_in__measures_I1_J,axiom,
! [X: nat,Y: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ nil_nat_nat ) ) ).
% in_measures(1)
thf(fact_258_in__measures_I1_J,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ nil_Pr2865493887535707976at_nat ) ) ).
% in_measures(1)
thf(fact_259_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_260_Nil__lenlex__iff1,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
= ( Ns != nil_nat ) ) ).
% Nil_lenlex_iff1
thf(fact_261_listrel__Nil2,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( listrel_nat_nat @ R ) )
=> ( Xs2 = nil_nat ) ) ).
% listrel_Nil2
thf(fact_262_listrel__Nil1,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs2 ) @ ( listrel_nat_nat @ R ) )
=> ( Xs2 = nil_nat ) ) ).
% listrel_Nil1
thf(fact_263_listrel_ONil,axiom,
! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).
% listrel.Nil
thf(fact_264_listrel__eq__len,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% listrel_eq_len
thf(fact_265_Nil2__notin__lex,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( lex_nat @ R ) ) ).
% Nil2_notin_lex
thf(fact_266_Nil__notin__lex,axiom,
! [Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) @ ( lex_nat @ R ) ) ).
% Nil_notin_lex
thf(fact_267_lex__append__leftI,axiom,
! [Ys2: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lex_nat @ R ) ) ) ).
% lex_append_leftI
thf(fact_268_Nil__lenlex__iff2,axiom,
! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_269_lex__append__rightI,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( lex_nat @ R ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Us ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Us ) @ ( append_nat @ Ys2 @ Vs ) ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_270_lenlex__append1,axiom,
! [Us: list_nat,Xs2: list_nat,R2: set_Pr1261947904930325089at_nat,Vs: list_nat,Ys2: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Xs2 ) @ ( lenlex_nat @ R2 ) )
=> ( ( ( size_size_list_nat @ Vs )
= ( size_size_list_nat @ Ys2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us @ Vs ) @ ( append_nat @ Xs2 @ Ys2 ) ) @ ( lenlex_nat @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_271_longest__common__prefix_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( ! [Uv: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Uv ) )
=> ~ ! [Uu: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Uu @ nil_nat ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_272_lenlex__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
( ! [X2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Xs2 ) @ ( lenlex_nat @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_273_lenlex__irreflexive,axiom,
! [R: set_Pr8693737435421807431at_nat,Xs2: list_P6011104703257516679at_nat] :
( ! [X2: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ R )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Xs2 ) @ ( lenlex325483962726685836at_nat @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_274_sorted__wrt_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ~ ! [P3: nat > nat > $o,X2: nat,Ys: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X2 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_275_successively_Ocases,axiom,
! [X: produc254973753779126261st_nat] :
( ! [P3: nat > nat > $o] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ nil_nat ) )
=> ( ! [P3: nat > nat > $o,X2: nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X2 @ nil_nat ) ) )
=> ~ ! [P3: nat > nat > $o,X2: nat,Y2: nat,Xs: list_nat] :
( X
!= ( produc4727192421694094319st_nat @ P3 @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_276_splice_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
=> ~ ! [X2: nat,Xs: list_nat,Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) ) ) ).
% splice.cases
thf(fact_277_shuffles_Ocases,axiom,
! [X: produc1828647624359046049st_nat] :
( ! [Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
=> ( ! [Xs: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ Xs @ nil_nat ) )
=> ~ ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( X
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_278_listrel_OCons,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_279_listrel_OCons,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( listre818007680106770737at_nat @ R ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) @ ( listre818007680106770737at_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_280_listrel__Cons1,axiom,
! [Y: nat,Ys2: list_nat,Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys2 ) @ Xs2 ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [Y2: nat,Ys: list_nat] :
( ( Xs2
= ( cons_nat @ Y2 @ Ys ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y2 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_281_listrel__Cons1,axiom,
! [Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Xs2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) @ Xs2 ) @ ( listre818007680106770737at_nat @ R ) )
=> ~ ! [Y2: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
( ( Xs2
= ( cons_P6512896166579812791at_nat @ Y2 @ Ys ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Y2 ) @ R )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys2 @ Ys ) @ ( listre818007680106770737at_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_282_listrel__Cons2,axiom,
! [Xs2: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel_nat_nat @ R ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( Xs2
= ( cons_nat @ X2 @ Xs ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_283_listrel__Cons2,axiom,
! [Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) @ ( listre818007680106770737at_nat @ R ) )
=> ~ ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( Xs2
= ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y ) @ R )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys2 ) @ ( listre818007680106770737at_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_284_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_285_lex__append__leftD,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ! [X2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lex_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_286_lex__append__leftD,axiom,
! [R: set_Pr8693737435421807431at_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ! [X2: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ R )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs2 @ Ys2 ) @ ( append985823374593552924at_nat @ Xs2 @ Zs ) ) @ ( lex_Pr8571645452597969515at_nat @ R ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys2 @ Zs ) @ ( lex_Pr8571645452597969515at_nat @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_287_lex__append__left__iff,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ! [X2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lex_nat @ R ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lex_nat @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_288_lex__append__left__iff,axiom,
! [R: set_Pr8693737435421807431at_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ! [X2: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ R )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs2 @ Ys2 ) @ ( append985823374593552924at_nat @ Xs2 @ Zs ) ) @ ( lex_Pr8571645452597969515at_nat @ R ) )
= ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys2 @ Zs ) @ ( lex_Pr8571645452597969515at_nat @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_289_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A4: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_290_old_Oprod_Oexhaust,axiom,
! [Y: produc859450856879609959at_nat] :
~ ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
( Y
!= ( produc6161850002892822231at_nat @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_291_surj__pair,axiom,
! [P4: product_prod_nat_nat] :
? [X2: nat,Y2: nat] :
( P4
= ( product_Pair_nat_nat @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_292_surj__pair,axiom,
! [P4: produc859450856879609959at_nat] :
? [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( P4
= ( produc6161850002892822231at_nat @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_293_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
( ! [A4: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_294_prod__cases,axiom,
! [P: produc859450856879609959at_nat > $o,P4: produc859450856879609959at_nat] :
( ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A4 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_295_Pair__inject,axiom,
! [A: nat,B: nat,A3: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_296_Pair__inject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A @ B )
= ( produc6161850002892822231at_nat @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_297_prod__cases3,axiom,
! [Y: produc859450856879609959at_nat] :
~ ! [A4: product_prod_nat_nat,B3: nat,C2: nat] :
( Y
!= ( produc6161850002892822231at_nat @ A4 @ ( product_Pair_nat_nat @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_298_prod__induct3,axiom,
! [P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
( ! [A4: product_prod_nat_nat,B3: nat,C2: nat] : ( P @ ( produc6161850002892822231at_nat @ A4 @ ( product_Pair_nat_nat @ B3 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_299_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs ) )
=> ! [Ys: list_nat] :
( ( A22
= ( cons_nat @ Y2 @ Ys ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_300_listrel_Ocases,axiom,
! [A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A1 @ A22 ) @ ( listre818007680106770737at_nat @ R ) )
=> ( ( ( A1 = nil_Pr5478986624290739719at_nat )
=> ( A22 != nil_Pr5478986624290739719at_nat ) )
=> ~ ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( A1
= ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
=> ! [Ys: list_P6011104703257516679at_nat] :
( ( A22
= ( cons_P6512896166579812791at_nat @ Y2 @ Ys ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ R )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre818007680106770737at_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_301_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_nat ) )
| ? [X4: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X4 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_302_listrel_Osimps,axiom,
! [A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A1 @ A22 ) @ ( listre818007680106770737at_nat @ R ) )
= ( ( ( A1 = nil_Pr5478986624290739719at_nat )
& ( A22 = nil_Pr5478986624290739719at_nat ) )
| ? [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
( ( A1
= ( cons_P6512896166579812791at_nat @ X4 @ Xs3 ) )
& ( A22
= ( cons_P6512896166579812791at_nat @ Y4 @ Ys3 ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs3 @ Ys3 ) @ ( listre818007680106770737at_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_303_measures__less,axiom,
! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_304_measures__less,axiom,
! [F: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_305_prefixes__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs2 ) @ ( cons_list_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_306_prefixes__eq__snoc,axiom,
! [Ys2: list_nat,Xs2: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys2 )
= ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys2 = nil_nat )
& ( Xs2 = nil_list_nat ) )
| ? [Z3: nat,Zs3: list_nat] :
( ( Ys2
= ( append_nat @ Zs3 @ ( cons_nat @ Z3 @ nil_nat ) ) )
& ( Xs2
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_307_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_308_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_309_append__one__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( ( ord_less_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) )
=> ( prefix_nat @ ( append_nat @ Xs2 @ ( cons_nat @ ( nth_nat @ Ys2 @ ( size_size_list_nat @ Xs2 ) ) @ nil_nat ) ) @ Ys2 ) ) ) ).
% append_one_prefix
thf(fact_310_lex__take__index,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( lex_Pr8571645452597969515at_nat @ R ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( ( take_P2173866234530122223at_nat @ I @ Xs2 )
= ( take_P2173866234530122223at_nat @ I @ Ys2 ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_311_lex__take__index,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( lex_nat @ R ) )
=> ~ ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( ( take_nat @ I @ Xs2 )
= ( take_nat @ I @ Ys2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Ys2 @ I ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_312_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_313_snoc__listrel1__snoc__iff,axiom,
! [Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) @ ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) ) @ ( listre4828114922151135584at_nat @ R ) )
= ( ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( listre4828114922151135584at_nat @ R ) )
& ( X = Y ) )
| ( ( Xs2 = Ys2 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_314_suffixes__eq__snoc,axiom,
! [Ys2: list_nat,Xs2: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys2 )
= ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys2 = nil_nat )
& ( Xs2 = nil_list_nat ) )
| ? [Z3: nat,Zs3: list_nat] :
( ( Ys2
= ( cons_nat @ Z3 @ Zs3 ) )
& ( Xs2
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_315_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_316_prefix__order_Odual__order_Orefl,axiom,
! [A: list_nat] : ( prefix_nat @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_317_prefix__order_Oorder__refl,axiom,
! [X: list_nat] : ( prefix_nat @ X @ X ) ).
% prefix_order.order_refl
thf(fact_318_Cons__prefix__Cons,axiom,
! [X: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( prefix_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) )
= ( ( X = Y )
& ( prefix_nat @ Xs2 @ Ys2 ) ) ) ).
% Cons_prefix_Cons
thf(fact_319_prefix__Nil,axiom,
! [Xs2: list_nat] :
( ( prefix_nat @ Xs2 @ nil_nat )
= ( Xs2 = nil_nat ) ) ).
% prefix_Nil
thf(fact_320_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_321_prefix__code_I1_J,axiom,
! [Xs2: list_nat] : ( prefix_nat @ nil_nat @ Xs2 ) ).
% prefix_code(1)
thf(fact_322_same__prefix__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) )
= ( prefix_nat @ Ys2 @ Zs ) ) ).
% same_prefix_prefix
thf(fact_323_same__prefix__nil,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs2 @ Ys2 ) @ Xs2 )
= ( Ys2 = nil_nat ) ) ).
% same_prefix_nil
thf(fact_324_nth__take,axiom,
! [I3: nat,N: nat,Xs2: list_nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs2 ) @ I3 )
= ( nth_nat @ Xs2 @ I3 ) ) ) ).
% nth_take
thf(fact_325_prefix__snoc,axiom,
! [Xs2: list_nat,Ys2: list_nat,Y: nat] :
( ( prefix_nat @ Xs2 @ ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
| ( prefix_nat @ Xs2 @ Ys2 ) ) ) ).
% prefix_snoc
thf(fact_326_Cons__listrel1__Cons,axiom,
! [X: nat,Xs2: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_327_Cons__listrel1__Cons,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) @ ( listre4828114922151135584at_nat @ R ) )
= ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( listre4828114922151135584at_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_328_prefix__same__cases,axiom,
! [Xs_1: list_nat,Ys2: list_nat,Xs_2: list_nat] :
( ( prefix_nat @ Xs_1 @ Ys2 )
=> ( ( prefix_nat @ Xs_2 @ Ys2 )
=> ( ( prefix_nat @ Xs_1 @ Xs_2 )
| ( prefix_nat @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_329_take__is__prefix,axiom,
! [N: nat,Xs2: list_nat] : ( prefix_nat @ ( take_nat @ N @ Xs2 ) @ Xs2 ) ).
% take_is_prefix
thf(fact_330_prefix__order_Odual__order_Oantisym,axiom,
! [B: list_nat,A: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ A @ B )
=> ( A = B ) ) ) ).
% prefix_order.dual_order.antisym
thf(fact_331_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [A5: list_nat,B4: list_nat] :
( ( prefix_nat @ B4 @ A5 )
& ( prefix_nat @ A5 @ B4 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_332_prefix__order_Odual__order_Otrans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ C @ B )
=> ( prefix_nat @ C @ A ) ) ) ).
% prefix_order.dual_order.trans
thf(fact_333_prefix__order_Oord__le__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( B = C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_le_eq_trans
thf(fact_334_prefix__order_Oord__eq__le__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_eq_le_trans
thf(fact_335_prefix__order_Oorder__antisym,axiom,
! [X: list_nat,Y: list_nat] :
( ( prefix_nat @ X @ Y )
=> ( ( prefix_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% prefix_order.order_antisym
thf(fact_336_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [X4: list_nat,Y4: list_nat] :
( ( prefix_nat @ X4 @ Y4 )
& ( prefix_nat @ Y4 @ X4 ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_337_prefix__order_Oantisym__conv,axiom,
! [Y: list_nat,X: list_nat] :
( ( prefix_nat @ Y @ X )
=> ( ( prefix_nat @ X @ Y )
= ( X = Y ) ) ) ).
% prefix_order.antisym_conv
thf(fact_338_prefix__order_Oorder__trans,axiom,
! [X: list_nat,Y: list_nat,Z4: list_nat] :
( ( prefix_nat @ X @ Y )
=> ( ( prefix_nat @ Y @ Z4 )
=> ( prefix_nat @ X @ Z4 ) ) ) ).
% prefix_order.order_trans
thf(fact_339_prefix__order_Oeq__refl,axiom,
! [X: list_nat,Y: list_nat] :
( ( X = Y )
=> ( prefix_nat @ X @ Y ) ) ).
% prefix_order.eq_refl
thf(fact_340_prefix__order_Oantisym,axiom,
! [A: list_nat,B: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ A )
=> ( A = B ) ) ) ).
% prefix_order.antisym
thf(fact_341_prefix__order_Oeq__iff,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [A5: list_nat,B4: list_nat] :
( ( prefix_nat @ A5 @ B4 )
& ( prefix_nat @ B4 @ A5 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_342_prefix__order_Otrans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.trans
thf(fact_343_take__equalityI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ! [I: nat] :
( ( take_nat @ I @ Xs2 )
= ( take_nat @ I @ Ys2 ) )
=> ( Xs2 = Ys2 ) ) ).
% take_equalityI
thf(fact_344_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_345_Nil__prefix,axiom,
! [Xs2: list_nat] : ( prefix_nat @ nil_nat @ Xs2 ) ).
% Nil_prefix
thf(fact_346_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_347_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_348_append__prefixD,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs2 @ Ys2 ) @ Zs )
=> ( prefix_nat @ Xs2 @ Zs ) ) ).
% append_prefixD
thf(fact_349_prefix__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( prefix_nat @ Xs2 @ ( append_nat @ Ys2 @ Zs ) ) ) ).
% prefix_prefix
thf(fact_350_prefix__append,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs2 @ ( append_nat @ Ys2 @ Zs ) )
= ( ( prefix_nat @ Xs2 @ Ys2 )
| ? [Us2: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ Us2 ) )
& ( prefix_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_351_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
? [Zs3: list_nat] :
( Ys3
= ( append_nat @ Xs3 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_352_prefixI,axiom,
! [Ys2: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( Ys2
= ( append_nat @ Xs2 @ Zs ) )
=> ( prefix_nat @ Xs2 @ Ys2 ) ) ).
% prefixI
thf(fact_353_prefixE,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ~ ! [Zs2: list_nat] :
( Ys2
!= ( append_nat @ Xs2 @ Zs2 ) ) ) ).
% prefixE
thf(fact_354_not__prefix__induct,axiom,
! [Ps: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( X2 != Y2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( X2 = Y2 )
=> ( ~ ( prefix_nat @ Xs @ Ys )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_355_not__prefix__cases,axiom,
! [Ps: list_nat,Ls: list_nat] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_nat )
=> ( Ls != nil_nat ) )
=> ( ! [A4: nat,As: list_nat] :
( ( Ps
= ( cons_nat @ A4 @ As ) )
=> ! [X2: nat,Xs: list_nat] :
( ( Ls
= ( cons_nat @ X2 @ Xs ) )
=> ( ( X2 = A4 )
=> ( prefix_nat @ As @ Xs ) ) ) )
=> ~ ! [A4: nat] :
( ? [As: list_nat] :
( Ps
= ( cons_nat @ A4 @ As ) )
=> ! [X2: nat] :
( ? [Xs: list_nat] :
( Ls
= ( cons_nat @ X2 @ Xs ) )
=> ( X2 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_356_prefix__Cons,axiom,
! [Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) )
= ( ( Xs2 = nil_nat )
| ? [Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Y @ Zs3 ) )
& ( prefix_nat @ Zs3 @ Ys2 ) ) ) ) ).
% prefix_Cons
thf(fact_357_prefix__code_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X @ Xs2 ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_358_listrel1I2,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,X: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ X @ Ys2 ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I2
thf(fact_359_not__Nil__listrel1,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs2 ) @ ( listrel1_nat @ R ) ) ).
% not_Nil_listrel1
thf(fact_360_not__listrel1__Nil,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( listrel1_nat @ R ) ) ).
% not_listrel1_Nil
thf(fact_361_listrel1__eq__len,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_362_append__listrel1I,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R ) ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Us ) @ ( append_nat @ Ys2 @ Vs ) ) @ ( listrel1_nat @ R ) ) ) ).
% append_listrel1I
thf(fact_363_Cons__listrel1E2,axiom,
! [Xs2: list_nat,Y: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel1_nat @ R ) )
=> ( ! [X2: nat] :
( ( Xs2
= ( cons_nat @ X2 @ Ys2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R ) )
=> ~ ! [Zs2: list_nat] :
( ( Xs2
= ( cons_nat @ Y @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_364_Cons__listrel1E2,axiom,
! [Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) ) @ ( listre4828114922151135584at_nat @ R ) )
=> ( ! [X2: product_prod_nat_nat] :
( ( Xs2
= ( cons_P6512896166579812791at_nat @ X2 @ Ys2 ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y ) @ R ) )
=> ~ ! [Zs2: list_P6011104703257516679at_nat] :
( ( Xs2
= ( cons_P6512896166579812791at_nat @ Y @ Zs2 ) )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Zs2 @ Ys2 ) @ ( listre4828114922151135584at_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_365_Cons__listrel1E1,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ( ! [Y2: nat] :
( ( Ys2
= ( cons_nat @ Y2 @ Xs2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R ) )
=> ~ ! [Zs2: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Zs2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_366_Cons__listrel1E1,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ Ys2 ) @ ( listre4828114922151135584at_nat @ R ) )
=> ( ! [Y2: product_prod_nat_nat] :
( ( Ys2
= ( cons_P6512896166579812791at_nat @ Y2 @ Xs2 ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y2 ) @ R ) )
=> ~ ! [Zs2: list_P6011104703257516679at_nat] :
( ( Ys2
= ( cons_P6512896166579812791at_nat @ X @ Zs2 ) )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Zs2 ) @ ( listre4828114922151135584at_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_367_listrel1I1,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Xs2 ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I1
thf(fact_368_listrel1I1,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Xs2 ) ) @ ( listre4828114922151135584at_nat @ R ) ) ) ).
% listrel1I1
thf(fact_369_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_370_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs2 ) )
= ( append_list_nat @ ( suffixes_nat @ Xs2 ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs2 ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_371_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_372_listrel1I,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Us: list_nat,Vs: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( ( Xs2
= ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
=> ( ( Ys2
= ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_373_listrel1I,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,Xs2: list_P6011104703257516679at_nat,Us: list_P6011104703257516679at_nat,Vs: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
=> ( ( Xs2
= ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ X @ Vs ) ) )
=> ( ( Ys2
= ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ Y @ Vs ) ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( listre4828114922151135584at_nat @ R ) ) ) ) ) ).
% listrel1I
thf(fact_374_listrel1E,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
=> ~ ! [X2: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
=> ! [Us3: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us3 @ ( cons_nat @ X2 @ Vs2 ) ) )
=> ( Ys2
!= ( append_nat @ Us3 @ ( cons_nat @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_375_listrel1E,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( listre4828114922151135584at_nat @ R ) )
=> ~ ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ R )
=> ! [Us3: list_P6011104703257516679at_nat,Vs2: list_P6011104703257516679at_nat] :
( ( Xs2
= ( append985823374593552924at_nat @ Us3 @ ( cons_P6512896166579812791at_nat @ X2 @ Vs2 ) ) )
=> ( Ys2
!= ( append985823374593552924at_nat @ Us3 @ ( cons_P6512896166579812791at_nat @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_376_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs2: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( take_nat @ ( suc @ I3 ) @ Xs2 )
= ( append_nat @ ( take_nat @ I3 @ Xs2 ) @ ( cons_nat @ ( nth_nat @ Xs2 @ I3 ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_377_listrel1__iff__update,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( listre4828114922151135584at_nat @ R ) )
= ( ? [Y4: product_prod_nat_nat,N2: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N2 ) @ Y4 ) @ R )
& ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
& ( Ys2
= ( list_u6180841689913720943at_nat @ Xs2 @ N2 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_378_listrel1__iff__update,axiom,
! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
= ( ? [Y4: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ N2 ) @ Y4 ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
& ( Ys2
= ( list_update_nat @ Xs2 @ N2 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_379_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys2 @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys2 = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs3: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_380_lexord__append__left__rightI,axiom,
! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,U: list_nat,X: list_nat,Y: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y ) ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_381_lexord__append__left__rightI,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,U: list_P6011104703257516679at_nat,X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ U @ ( cons_P6512896166579812791at_nat @ A @ X ) ) @ ( append985823374593552924at_nat @ U @ ( cons_P6512896166579812791at_nat @ B @ Y ) ) ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_382_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys2 @ Zs ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs: list_nat,Xs4: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs @ Xs4 ) @ Xss23 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs ) )
& ( Zs
= ( append_nat @ Xs4 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_383_prefix__bot_Omin__bot2,axiom,
! [X: list_nat] :
( ( min_list_nat @ prefix_nat @ X @ nil_nat )
= nil_nat ) ).
% prefix_bot.min_bot2
thf(fact_384_prefix__bot_Omax__bot2,axiom,
! [X: list_nat] :
( ( max_list_nat @ prefix_nat @ X @ nil_nat )
= X ) ).
% prefix_bot.max_bot2
thf(fact_385_prefix__bot_Omin__bot,axiom,
! [X: list_nat] :
( ( min_list_nat @ prefix_nat @ nil_nat @ X )
= nil_nat ) ).
% prefix_bot.min_bot
thf(fact_386_prefix__bot_Omax__bot,axiom,
! [X: list_nat] :
( ( max_list_nat @ prefix_nat @ nil_nat @ X )
= X ) ).
% prefix_bot.max_bot
thf(fact_387_list__update__nonempty,axiom,
! [Xs2: list_nat,K: nat,X: nat] :
( ( ( list_update_nat @ Xs2 @ K @ X )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% list_update_nonempty
thf(fact_388_length__list__update,axiom,
! [Xs2: list_nat,I3: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_389_nth__list__update__neq,axiom,
! [I3: nat,J2: nat,Xs2: list_nat,X: nat] :
( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ J2 )
= ( nth_nat @ Xs2 @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_390_list__update__id,axiom,
! [Xs2: list_nat,I3: nat] :
( ( list_update_nat @ Xs2 @ I3 @ ( nth_nat @ Xs2 @ I3 ) )
= Xs2 ) ).
% list_update_id
thf(fact_391_nth__Cons__Suc,axiom,
! [X: nat,Xs2: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_nat @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_392_take__Suc__Cons,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( take_nat @ N @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_393_length__suffixes,axiom,
! [Xs2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs2 ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_suffixes
thf(fact_394_concat__append,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( concat_nat @ Xs2 ) @ ( concat_nat @ Ys2 ) ) ) ).
% concat_append
thf(fact_395_list__update__length,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs2 ) @ Y )
= ( append_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_396_nth__list__update__eq,axiom,
! [I3: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ I3 )
= X ) ) ).
% nth_list_update_eq
thf(fact_397_lexord__cons__cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R ) )
= ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
| ( ( A = B )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_398_lexord__cons__cons,axiom,
! [A: product_prod_nat_nat,X: list_P6011104703257516679at_nat,B: product_prod_nat_nat,Y: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ A @ X ) @ ( cons_P6512896166579812791at_nat @ B @ Y ) ) @ ( lexord2841853652668343668at_nat @ R ) )
= ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
| ( ( A = B )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X @ Y ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_399_lexord__Nil__left,axiom,
! [Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y ) @ ( lexord_nat @ R ) )
= ( ? [A5: nat,X4: list_nat] :
( Y
= ( cons_nat @ A5 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_400_list__update__code_I3_J,axiom,
! [X: nat,Xs2: list_nat,I3: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ I3 ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs2 @ I3 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_401_list__update__code_I1_J,axiom,
! [I3: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I3 @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_402_list__update_Osimps_I1_J,axiom,
! [I3: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I3 @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_403_take__update__swap,axiom,
! [M: nat,Xs2: list_nat,N: nat,X: nat] :
( ( take_nat @ M @ ( list_update_nat @ Xs2 @ N @ X ) )
= ( list_update_nat @ ( take_nat @ M @ Xs2 ) @ N @ X ) ) ).
% take_update_swap
thf(fact_404_prefix__order_Omax__def,axiom,
! [A: list_nat,B: list_nat] :
( ( ( prefix_nat @ A @ B )
=> ( ( max_list_nat @ prefix_nat @ A @ B )
= B ) )
& ( ~ ( prefix_nat @ A @ B )
=> ( ( max_list_nat @ prefix_nat @ A @ B )
= A ) ) ) ).
% prefix_order.max_def
thf(fact_405_prefix__order_Omin__def,axiom,
! [A: list_nat,B: list_nat] :
( ( ( prefix_nat @ A @ B )
=> ( ( min_list_nat @ prefix_nat @ A @ B )
= A ) )
& ( ~ ( prefix_nat @ A @ B )
=> ( ( min_list_nat @ prefix_nat @ A @ B )
= B ) ) ) ).
% prefix_order.min_def
thf(fact_406_Suc__length__conv,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs2 ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_407_length__Suc__conv,axiom,
! [Xs2: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_408_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_nat @ X @ ( concat_nat @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_409_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_410_lexord__linear,axiom,
! [R: set_Pr1261947904930325089at_nat,X: list_nat,Y: list_nat] :
( ! [A4: nat,B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ R )
| ( A4 = B3 )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A4 ) @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) )
| ( X = Y )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_linear
thf(fact_411_lexord__linear,axiom,
! [R: set_Pr8693737435421807431at_nat,X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ B3 ) @ R )
| ( A4 = B3 )
| ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ A4 ) @ R ) )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X @ Y ) @ ( lexord2841853652668343668at_nat @ R ) )
| ( X = Y )
| ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Y @ X ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ) ).
% lexord_linear
thf(fact_412_lexord__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
( ! [X2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Xs2 ) @ ( lexord_nat @ R ) ) ) ).
% lexord_irreflexive
thf(fact_413_lexord__irreflexive,axiom,
! [R: set_Pr8693737435421807431at_nat,Xs2: list_P6011104703257516679at_nat] :
( ! [X2: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ R )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Xs2 ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ).
% lexord_irreflexive
thf(fact_414_lexord__Nil__right,axiom,
! [X: list_nat,R: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ nil_nat ) @ ( lexord_nat @ R ) ) ).
% lexord_Nil_right
thf(fact_415_lexord__append__leftI,axiom,
! [U: list_nat,V: list_nat,R: set_Pr1261947904930325089at_nat,X: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V ) @ ( lexord_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_leftI
thf(fact_416_list__update__append1,axiom,
! [I3: nat,Xs2: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( list_update_nat @ ( append_nat @ Xs2 @ Ys2 ) @ I3 @ X )
= ( append_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_417_list__update__same__conv,axiom,
! [I3: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( list_update_nat @ Xs2 @ I3 @ X )
= Xs2 )
= ( ( nth_nat @ Xs2 @ I3 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_418_nth__list__update,axiom,
! [I3: nat,Xs2: list_nat,J2: nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ J2 )
= X ) )
& ( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ J2 )
= ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_419_lexord__append__leftD,axiom,
! [X: list_nat,U: list_nat,V: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V ) ) @ ( lexord_nat @ R ) )
=> ( ! [A4: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_420_lexord__append__leftD,axiom,
! [X: list_P6011104703257516679at_nat,U: list_P6011104703257516679at_nat,V: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ X @ U ) @ ( append985823374593552924at_nat @ X @ V ) ) @ ( lexord2841853652668343668at_nat @ R ) )
=> ( ! [A4: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ A4 ) @ R )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ U @ V ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_421_lexord__append__rightI,axiom,
! [Y: list_nat,X: list_nat,R: set_Pr1261947904930325089at_nat] :
( ? [B5: nat,Z5: list_nat] :
( Y
= ( cons_nat @ B5 @ Z5 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ ( append_nat @ X @ Y ) ) @ ( lexord_nat @ R ) ) ) ).
% lexord_append_rightI
thf(fact_422_lexord__sufE,axiom,
! [Xs2: list_nat,Zs: list_nat,Ys2: list_nat,Qs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Zs ) @ ( append_nat @ Ys2 @ Qs ) ) @ ( lexord_nat @ R ) )
=> ( ( Xs2 != Ys2 )
=> ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Qs ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( lexord_nat @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_423_lexord__lex,axiom,
! [X: list_nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lex_nat @ R ) )
= ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) )
& ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ) ).
% lexord_lex
thf(fact_424_length__Suc__conv__rev,axiom,
! [Xs2: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_425_length__append__singleton,axiom,
! [Xs2: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_426_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_427_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_428_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_429_length__Cons,axiom,
! [X: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs2 ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_Cons
thf(fact_430_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_431_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_432_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_433_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_434_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_435_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_436_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_437_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_438_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_439_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_440_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_441_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_442_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_443_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_444_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_445_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_446_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_447_strict__inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I: nat] :
( ( J2
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_448_less__Suc__induct,axiom,
! [I3: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I @ K2 ) ) ) ) )
=> ( P @ I3 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_449_less__trans__Suc,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_450_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_451_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_452_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_453_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_454_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_455_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_456_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_457_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_458_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_459_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_460_Suc__lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_461_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_462_Nat_OlessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( K
!= ( suc @ I3 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_463_lexord__take__index__conv,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X @ Y ) @ ( lexord2841853652668343668at_nat @ R ) )
= ( ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ X ) @ ( size_s5460976970255530739at_nat @ Y ) )
& ( ( take_P2173866234530122223at_nat @ ( size_s5460976970255530739at_nat @ X ) @ Y )
= X ) )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( ord_min_nat @ ( size_s5460976970255530739at_nat @ X ) @ ( size_s5460976970255530739at_nat @ Y ) ) )
& ( ( take_P2173866234530122223at_nat @ I2 @ X )
= ( take_P2173866234530122223at_nat @ I2 @ Y ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ X @ I2 ) @ ( nth_Pr7617993195940197384at_nat @ Y @ I2 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_464_lexord__take__index__conv,axiom,
! [X: list_nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) )
= ( ( ( ord_less_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) )
& ( ( take_nat @ ( size_size_list_nat @ X ) @ Y )
= X ) )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( ord_min_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) ) )
& ( ( take_nat @ I2 @ X )
= ( take_nat @ I2 @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X @ I2 ) @ ( nth_nat @ Y @ I2 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_465_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs2: list_nat,A: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( list_update_nat @ Xs2 @ I3 @ A )
= ( append_nat @ ( take_nat @ I3 @ Xs2 ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I3 ) @ Xs2 ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_466_SuccI,axiom,
! [Kl: list_P6011104703257516679at_nat,K: product_prod_nat_nat,Kl2: set_li5450038453877631591at_nat] :
( ( member3067507820990806192at_nat @ ( append985823374593552924at_nat @ Kl @ ( cons_P6512896166579812791at_nat @ K @ nil_Pr5478986624290739719at_nat ) ) @ Kl2 )
=> ( member8440522571783428010at_nat @ K @ ( bNF_Gr5363859321595349404at_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_467_SuccI,axiom,
! [Kl: list_P8469869581646625389at_nat,K: produc859450856879609959at_nat,Kl2: set_li3197816953174176717at_nat] :
( ( member3799944675974059798at_nat @ ( append8751754712269456642at_nat @ Kl @ ( cons_P8732206157123786781at_nat @ K @ nil_Pr2582115297535392877at_nat ) ) @ Kl2 )
=> ( member8206827879206165904at_nat @ K @ ( bNF_Gr8603829661560577154at_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_468_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_469_SuccD,axiom,
! [K: product_prod_nat_nat,Kl2: set_li5450038453877631591at_nat,Kl: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ K @ ( bNF_Gr5363859321595349404at_nat @ Kl2 @ Kl ) )
=> ( member3067507820990806192at_nat @ ( append985823374593552924at_nat @ Kl @ ( cons_P6512896166579812791at_nat @ K @ nil_Pr5478986624290739719at_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_470_SuccD,axiom,
! [K: produc859450856879609959at_nat,Kl2: set_li3197816953174176717at_nat,Kl: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ K @ ( bNF_Gr8603829661560577154at_nat @ Kl2 @ Kl ) )
=> ( member3799944675974059798at_nat @ ( append8751754712269456642at_nat @ Kl @ ( cons_P8732206157123786781at_nat @ K @ nil_Pr2582115297535392877at_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_471_SuccD,axiom,
! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
=> ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_472_id__take__nth__drop,axiom,
! [I3: nat,Xs2: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( Xs2
= ( append_nat @ ( take_nat @ I3 @ Xs2 ) @ ( cons_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs2 ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_473_enumerate__simps_I2_J,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs2 ) ) ) ).
% enumerate_simps(2)
thf(fact_474_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_475_take__take,axiom,
! [N: nat,M: nat,Xs2: list_nat] :
( ( take_nat @ N @ ( take_nat @ M @ Xs2 ) )
= ( take_nat @ ( ord_min_nat @ N @ M ) @ Xs2 ) ) ).
% take_take
thf(fact_476_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_nat @ N @ nil_nat )
= nil_Pr5478986624290739719at_nat ) ).
% enumerate_simps(1)
thf(fact_477_length__enumerate,axiom,
! [N: nat,Xs2: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_enumerate
thf(fact_478_drop__Suc__Cons,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs2 ) )
= ( drop_nat @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_479_append__take__drop__id,axiom,
! [N: nat,Xs2: list_nat] :
( ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( drop_nat @ N @ Xs2 ) )
= Xs2 ) ).
% append_take_drop_id
thf(fact_480_drop__update__cancel,axiom,
! [N: nat,M: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs2 @ N @ X ) )
= ( drop_nat @ M @ Xs2 ) ) ) ).
% drop_update_cancel
thf(fact_481_length__take,axiom,
! [N: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( take_nat @ N @ Xs2 ) )
= ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% length_take
thf(fact_482_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_483_nth__via__drop,axiom,
! [N: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( drop_nat @ N @ Xs2 )
= ( cons_nat @ Y @ Ys2 ) )
=> ( ( nth_nat @ Xs2 @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_484_append__eq__conv__conj,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= Zs )
= ( ( Xs2
= ( take_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) )
& ( Ys2
= ( drop_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_485_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs2: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( cons_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs2 ) )
= ( drop_nat @ I3 @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_486_min__less__iff__conj,axiom,
! [Z4: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z4 @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z4 @ X )
& ( ord_less_nat @ Z4 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_487_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_488_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_489_take__hd__drop,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs2 ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_hd_drop
thf(fact_490_empty__Shift,axiom,
! [Kl2: set_li5450038453877631591at_nat,K: product_prod_nat_nat] :
( ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ Kl2 )
=> ( ( member8440522571783428010at_nat @ K @ ( bNF_Gr5363859321595349404at_nat @ Kl2 @ nil_Pr5478986624290739719at_nat ) )
=> ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ ( bNF_Gr3130287167067265568at_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_491_empty__Shift,axiom,
! [Kl2: set_li3197816953174176717at_nat,K: produc859450856879609959at_nat] :
( ( member3799944675974059798at_nat @ nil_Pr2582115297535392877at_nat @ Kl2 )
=> ( ( member8206827879206165904at_nat @ K @ ( bNF_Gr8603829661560577154at_nat @ Kl2 @ nil_Pr2582115297535392877at_nat ) )
=> ( member3799944675974059798at_nat @ nil_Pr2582115297535392877at_nat @ ( bNF_Gr9155129491085760262at_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_492_empty__Shift,axiom,
! [Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl2 )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_493_Succ__Shift,axiom,
! [Kl2: set_list_nat,K: nat,Kl: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ ( cons_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_494_hd__prefixes,axiom,
! [Xs2: list_nat] :
( ( hd_list_nat @ ( prefixes_nat @ Xs2 ) )
= nil_nat ) ).
% hd_prefixes
thf(fact_495_hd__suffixes,axiom,
! [Xs2: list_nat] :
( ( hd_list_nat @ ( suffixes_nat @ Xs2 ) )
= nil_nat ) ).
% hd_suffixes
thf(fact_496_hd__append2,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( hd_nat @ Xs2 ) ) ) ).
% hd_append2
thf(fact_497_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_498_hd__concat,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( ( hd_list_nat @ Xs2 )
!= nil_nat )
=> ( ( hd_nat @ ( concat_nat @ Xs2 ) )
= ( hd_nat @ ( hd_list_nat @ Xs2 ) ) ) ) ) ).
% hd_concat
thf(fact_499_longest__common__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
? [Ps2: list_nat,Xs4: list_nat,Ys6: list_nat] :
( ( Xs2
= ( append_nat @ Ps2 @ Xs4 ) )
& ( Ys2
= ( append_nat @ Ps2 @ Ys6 ) )
& ( ( Xs4 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( hd_nat @ Xs4 )
!= ( hd_nat @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_500_hd__append,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( Xs2 = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( hd_nat @ Ys2 ) ) )
& ( ( Xs2 != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( hd_nat @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_501_ShiftD,axiom,
! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ Kl @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) )
=> ( member_list_nat @ ( cons_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_502_hd__drop__conv__nth,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( hd_nat @ ( drop_nat @ N @ Xs2 ) )
= ( nth_nat @ Xs2 @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_503_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z4 )
= ( ( ord_less_nat @ X @ Z4 )
| ( ord_less_nat @ Y @ Z4 ) ) ) ).
% min_less_iff_disj
thf(fact_504_min_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_505_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( A5
= ( ord_min_nat @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% min.strict_order_iff
thf(fact_506_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_507_min_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_508_nth__enumerate__eq,axiom,
! [M: nat,Xs2: list_nat,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs2 ) @ M )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs2 @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_509_take__Suc,axiom,
! [Xs2: list_nat,N: nat] :
( ( Xs2 != nil_nat )
=> ( ( take_nat @ ( suc @ N ) @ Xs2 )
= ( cons_nat @ ( hd_nat @ Xs2 ) @ ( take_nat @ N @ ( tl_nat @ Xs2 ) ) ) ) ) ).
% take_Suc
thf(fact_510_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs2 ) ) ).
% gen_length_code(2)
thf(fact_511_set__swap,axiom,
! [I3: nat,Xs2: list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I3 @ ( nth_nat @ Xs2 @ J2 ) ) @ J2 @ ( nth_nat @ Xs2 @ I3 ) ) )
= ( set_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_512_distinct__swap,axiom,
! [I3: nat,Xs2: list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I3 @ ( nth_nat @ Xs2 @ J2 ) ) @ J2 @ ( nth_nat @ Xs2 @ I3 ) ) )
= ( distinct_nat @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_513_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_514_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_515_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
=> ( X4 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_516_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
=> ( X4 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_517_drop__drop,axiom,
! [N: nat,M: nat,Xs2: list_nat] :
( ( drop_nat @ N @ ( drop_nat @ M @ Xs2 ) )
= ( drop_nat @ ( plus_plus_nat @ N @ M ) @ Xs2 ) ) ).
% drop_drop
thf(fact_518_in__set__prefixes,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( prefixes_nat @ Ys2 ) ) )
= ( prefix_nat @ Xs2 @ Ys2 ) ) ).
% in_set_prefixes
thf(fact_519_in__set__insert,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ( insert8944034826898310173at_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_520_in__set__insert,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ( insert4532235091570160003at_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_521_in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_522_distinct__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ( distinct_nat @ ( insert_nat @ X @ Xs2 ) )
= ( distinct_nat @ Xs2 ) ) ).
% distinct_insert
thf(fact_523_length__append,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_append
thf(fact_524_tl__append2,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( tl_nat @ Xs2 ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_525_not__in__set__insert,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ( insert8944034826898310173at_nat @ X @ Xs2 )
= ( cons_P6512896166579812791at_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_526_not__in__set__insert,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ~ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ( insert4532235091570160003at_nat @ X @ Xs2 )
= ( cons_P8732206157123786781at_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_527_not__in__set__insert,axiom,
! [X: nat,Xs2: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X @ Xs2 )
= ( cons_nat @ X @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_528_nth__append__length__plus,axiom,
! [Xs2: list_nat,Ys2: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) )
= ( nth_nat @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_529_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_530_hd__Cons__tl,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs2 ) @ ( tl_nat @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_531_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs3: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_532_list_Oset__sel_I2_J,axiom,
! [A: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( A != nil_Pr5478986624290739719at_nat )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( tl_Pro4228036916689694320at_nat @ A ) ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_533_list_Oset__sel_I2_J,axiom,
! [A: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
( ( A != nil_Pr2582115297535392877at_nat )
=> ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( tl_Pro5060323577808663382at_nat @ A ) ) )
=> ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_534_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X: nat] :
( ( A != nil_nat )
=> ( ( member_nat @ X @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat @ X @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_535_distinct__product__lists,axiom,
! [Xss2: list_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss2 ) )
=> ( distinct_nat @ X2 ) )
=> ( distinct_list_nat @ ( product_lists_nat @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_536_subseqs__distinctD,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
=> ( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ Ys2 ) ) ) ).
% subseqs_distinctD
thf(fact_537_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ Xs2 ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_538_distinct__tl,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ ( tl_nat @ Xs2 ) ) ) ).
% distinct_tl
thf(fact_539_not__distinct__conv__prefix,axiom,
! [As2: list_P6011104703257516679at_nat] :
( ( ~ ( distin6923225563576452346at_nat @ As2 ) )
= ( ? [Xs3: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ Y4 @ ( set_Pr5648618587558075414at_nat @ Xs3 ) )
& ( distin6923225563576452346at_nat @ Xs3 )
& ( As2
= ( append985823374593552924at_nat @ Xs3 @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_540_not__distinct__conv__prefix,axiom,
! [As2: list_P8469869581646625389at_nat] :
( ( ~ ( distin6906083803243959008at_nat @ As2 ) )
= ( ? [Xs3: list_P8469869581646625389at_nat,Y4: produc859450856879609959at_nat,Ys3: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ Y4 @ ( set_Pr5518436109238095868at_nat @ Xs3 ) )
& ( distin6906083803243959008at_nat @ Xs3 )
& ( As2
= ( append8751754712269456642at_nat @ Xs3 @ ( cons_P8732206157123786781at_nat @ Y4 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_541_not__distinct__conv__prefix,axiom,
! [As2: list_nat] :
( ( ~ ( distinct_nat @ As2 ) )
= ( ? [Xs3: list_nat,Y4: nat,Ys3: list_nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Xs3 ) )
& ( distinct_nat @ Xs3 )
& ( As2
= ( append_nat @ Xs3 @ ( cons_nat @ Y4 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_542_distinct_Osimps_I2_J,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( distin6923225563576452346at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) )
= ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
& ( distin6923225563576452346at_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_543_distinct_Osimps_I2_J,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( distin6906083803243959008at_nat @ ( cons_P8732206157123786781at_nat @ X @ Xs2 ) )
= ( ~ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
& ( distin6906083803243959008at_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_544_distinct_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( distinct_nat @ ( cons_nat @ X @ Xs2 ) )
= ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_545_distinct__length__2__or__more,axiom,
! [A: nat,B: nat,Xs2: list_nat] :
( ( distinct_nat @ ( cons_nat @ A @ ( cons_nat @ B @ Xs2 ) ) )
= ( ( A != B )
& ( distinct_nat @ ( cons_nat @ A @ Xs2 ) )
& ( distinct_nat @ ( cons_nat @ B @ Xs2 ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_546_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_547_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ X22 ) )
=> ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_548_list_Oset__intros_I2_J,axiom,
! [Y: produc859450856879609959at_nat,X22: list_P8469869581646625389at_nat,X21: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ X22 ) )
=> ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_549_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_550_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] : ( member8440522571783428010at_nat @ X21 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_551_list_Oset__intros_I1_J,axiom,
! [X21: produc859450856879609959at_nat,X22: list_P8469869581646625389at_nat] : ( member8206827879206165904at_nat @ X21 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_552_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_553_list_Oset__cases,axiom,
! [E: product_prod_nat_nat,A: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ A ) )
=> ( ! [Z22: list_P6011104703257516679at_nat] :
( A
!= ( cons_P6512896166579812791at_nat @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_nat_nat,Z22: list_P6011104703257516679at_nat] :
( ( A
= ( cons_P6512896166579812791at_nat @ Z1 @ Z22 ) )
=> ~ ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_554_list_Oset__cases,axiom,
! [E: produc859450856879609959at_nat,A: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ E @ ( set_Pr5518436109238095868at_nat @ A ) )
=> ( ! [Z22: list_P8469869581646625389at_nat] :
( A
!= ( cons_P8732206157123786781at_nat @ E @ Z22 ) )
=> ~ ! [Z1: produc859450856879609959at_nat,Z22: list_P8469869581646625389at_nat] :
( ( A
= ( cons_P8732206157123786781at_nat @ Z1 @ Z22 ) )
=> ~ ( member8206827879206165904at_nat @ E @ ( set_Pr5518436109238095868at_nat @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_555_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_556_set__ConsD,axiom,
! [Y: product_prod_nat_nat,X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_557_set__ConsD,axiom,
! [Y: produc859450856879609959at_nat,X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_558_set__ConsD,axiom,
! [Y: nat,X: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_559_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_560_distinct__take,axiom,
! [Xs2: list_nat,I3: nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ ( take_nat @ I3 @ Xs2 ) ) ) ).
% distinct_take
thf(fact_561_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_562_distinct__drop,axiom,
! [Xs2: list_nat,I3: nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ ( drop_nat @ I3 @ Xs2 ) ) ) ).
% distinct_drop
thf(fact_563_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_564_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_565_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_566_add__lessD1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_567_add__less__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_568_not__add__less1,axiom,
! [I3: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ I3 ) ).
% not_add_less1
thf(fact_569_not__add__less2,axiom,
! [J2: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_570_add__less__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_571_trans__less__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_572_trans__less__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_573_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_574_in__set__takeD,axiom,
! [X: product_prod_nat_nat,N: nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( take_P2173866234530122223at_nat @ N @ Xs2 ) ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_575_in__set__takeD,axiom,
! [X: produc859450856879609959at_nat,N: nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( take_P5254422574997664853at_nat @ N @ Xs2 ) ) )
=> ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_576_in__set__takeD,axiom,
! [X: nat,N: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_577_in__set__dropD,axiom,
! [X: product_prod_nat_nat,N: nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( drop_P8868858903918902087at_nat @ N @ Xs2 ) ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_578_in__set__dropD,axiom,
! [X: produc859450856879609959at_nat,N: nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( drop_P4867432431663989165at_nat @ N @ Xs2 ) ) )
=> ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_579_in__set__dropD,axiom,
! [X: nat,N: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_580_tl__drop,axiom,
! [N: nat,Xs2: list_nat] :
( ( tl_nat @ ( drop_nat @ N @ Xs2 ) )
= ( drop_nat @ N @ ( tl_nat @ Xs2 ) ) ) ).
% tl_drop
thf(fact_581_distinct__Ex1,axiom,
! [Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( distin6923225563576452346at_nat @ Xs2 )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ X2 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ Y5 )
= X ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_582_distinct__Ex1,axiom,
! [Xs2: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
( ( distin6906083803243959008at_nat @ Xs2 )
=> ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s3679842834875189465at_nat @ Xs2 ) )
& ( ( nth_Pr6744343527793145070at_nat @ Xs2 @ X2 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_s3679842834875189465at_nat @ Xs2 ) )
& ( ( nth_Pr6744343527793145070at_nat @ Xs2 @ Y5 )
= X ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_583_distinct__Ex1,axiom,
! [Xs2: list_nat,X: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ X2 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ Y5 )
= X ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_584_list__ex__cong,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,F: product_prod_nat_nat > $o,G: product_prod_nat_nat > $o] :
( ( Xs2 = Ys2 )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Ys2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_e7689525607045846085at_nat @ F @ Xs2 )
= ( list_e7689525607045846085at_nat @ G @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_585_list__ex__cong,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat,F: produc859450856879609959at_nat > $o,G: produc859450856879609959at_nat > $o] :
( ( Xs2 = Ys2 )
=> ( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Ys2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_e7227117345903044011at_nat @ F @ Xs2 )
= ( list_e7227117345903044011at_nat @ G @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_586_list__ex__cong,axiom,
! [Xs2: list_nat,Ys2: list_nat,F: nat > $o,G: nat > $o] :
( ( Xs2 = Ys2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_ex_nat @ F @ Xs2 )
= ( list_ex_nat @ G @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_587_distinct__singleton,axiom,
! [X: nat] : ( distinct_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% distinct_singleton
thf(fact_588_split__list,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ? [Ys: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
( Xs2
= ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_589_split__list,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ? [Ys: list_P8469869581646625389at_nat,Zs2: list_P8469869581646625389at_nat] :
( Xs2
= ( append8751754712269456642at_nat @ Ys @ ( cons_P8732206157123786781at_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_590_split__list,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys: list_nat,Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_591_split__list__last,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ? [Ys: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
( ( Xs2
= ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X @ Zs2 ) ) )
& ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_592_split__list__last,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ? [Ys: list_P8469869581646625389at_nat,Zs2: list_P8469869581646625389at_nat] :
( ( Xs2
= ( append8751754712269456642at_nat @ Ys @ ( cons_P8732206157123786781at_nat @ X @ Zs2 ) ) )
& ~ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_593_split__list__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys: list_nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_594_split__list__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_595_split__list__first,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ? [Ys: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
( ( Xs2
= ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X @ Zs2 ) ) )
& ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys ) ) ) ) ).
% split_list_first
thf(fact_596_split__list__first,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ? [Ys: list_P8469869581646625389at_nat,Zs2: list_P8469869581646625389at_nat] :
( ( Xs2
= ( append8751754712269456642at_nat @ Ys @ ( cons_P8732206157123786781at_nat @ X @ Zs2 ) ) )
& ~ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Ys ) ) ) ) ).
% split_list_first
thf(fact_597_split__list__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys: list_nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_598_split__list__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_599_append__Cons__eq__iff,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Xs6: list_P6011104703257516679at_nat,Ys7: list_P6011104703257516679at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys2 ) )
=> ( ( ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X @ Ys2 ) )
= ( append985823374593552924at_nat @ Xs6 @ ( cons_P6512896166579812791at_nat @ X @ Ys7 ) ) )
= ( ( Xs2 = Xs6 )
& ( Ys2 = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_600_append__Cons__eq__iff,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat,Xs6: list_P8469869581646625389at_nat,Ys7: list_P8469869581646625389at_nat] :
( ~ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ~ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Ys2 ) )
=> ( ( ( append8751754712269456642at_nat @ Xs2 @ ( cons_P8732206157123786781at_nat @ X @ Ys2 ) )
= ( append8751754712269456642at_nat @ Xs6 @ ( cons_P8732206157123786781at_nat @ X @ Ys7 ) ) )
= ( ( Xs2 = Xs6 )
& ( Ys2 = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_601_append__Cons__eq__iff,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat,Xs6: list_nat,Ys7: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
=> ( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys2 ) )
= ( append_nat @ Xs6 @ ( cons_nat @ X @ Ys7 ) ) )
= ( ( Xs2 = Xs6 )
& ( Ys2 = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_602_in__set__conv__decomp,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
= ( ? [Ys3: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
( Xs2
= ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_603_in__set__conv__decomp,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
= ( ? [Ys3: list_P8469869581646625389at_nat,Zs3: list_P8469869581646625389at_nat] :
( Xs2
= ( append8751754712269456642at_nat @ Ys3 @ ( cons_P8732206157123786781at_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_604_in__set__conv__decomp,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_605_split__list__last__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys: list_nat,X2: nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_606_split__list__first__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_607_split__list__last__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys: list_nat,X2: nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_608_split__list__first__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_609_in__set__conv__decomp__last,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
= ( ? [Ys3: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
( ( Xs2
= ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) )
& ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_610_in__set__conv__decomp__last,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
= ( ? [Ys3: list_P8469869581646625389at_nat,Zs3: list_P8469869581646625389at_nat] :
( ( Xs2
= ( append8751754712269456642at_nat @ Ys3 @ ( cons_P8732206157123786781at_nat @ X @ Zs3 ) ) )
& ~ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_611_in__set__conv__decomp__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_612_in__set__conv__decomp__first,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
= ( ? [Ys3: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
( ( Xs2
= ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) )
& ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_613_in__set__conv__decomp__first,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
= ( ? [Ys3: list_P8469869581646625389at_nat,Zs3: list_P8469869581646625389at_nat] :
( ( Xs2
= ( append8751754712269456642at_nat @ Ys3 @ ( cons_P8732206157123786781at_nat @ X @ Zs3 ) ) )
& ~ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_614_in__set__conv__decomp__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_615_split__list__last__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_nat,X4: nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_616_split__list__first__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_nat,X4: nat] :
( ? [Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_617_Nil__tl,axiom,
! [Xs2: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs2 ) )
= ( ( Xs2 = nil_nat )
| ? [X4: nat] :
( Xs2
= ( cons_nat @ X4 @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_618_tl__Nil,axiom,
! [Xs2: list_nat] :
( ( ( tl_nat @ Xs2 )
= nil_nat )
= ( ( Xs2 = nil_nat )
| ? [X4: nat] :
( Xs2
= ( cons_nat @ X4 @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_619_tl__append__if,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( Xs2 = nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( tl_nat @ Ys2 ) ) )
& ( ( Xs2 != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( tl_nat @ Xs2 ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_620_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_621_less__add__Suc1,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M ) ) ) ).
% less_add_Suc1
thf(fact_622_less__add__Suc2,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_623_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_624_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_625_hd__in__set,axiom,
! [Xs2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( member8440522571783428010at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs2 ) @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_626_hd__in__set,axiom,
! [Xs2: list_P8469869581646625389at_nat] :
( ( Xs2 != nil_Pr2582115297535392877at_nat )
=> ( member8206827879206165904at_nat @ ( hd_Pro8462011474880202578at_nat @ Xs2 ) @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_627_hd__in__set,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs2 ) @ ( set_nat2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_628_list_Oset__sel_I1_J,axiom,
! [A: list_P6011104703257516679at_nat] :
( ( A != nil_Pr5478986624290739719at_nat )
=> ( member8440522571783428010at_nat @ ( hd_Pro3460610213475200108at_nat @ A ) @ ( set_Pr5648618587558075414at_nat @ A ) ) ) ).
% list.set_sel(1)
thf(fact_629_list_Oset__sel_I1_J,axiom,
! [A: list_P8469869581646625389at_nat] :
( ( A != nil_Pr2582115297535392877at_nat )
=> ( member8206827879206165904at_nat @ ( hd_Pro8462011474880202578at_nat @ A ) @ ( set_Pr5518436109238095868at_nat @ A ) ) ) ).
% list.set_sel(1)
thf(fact_630_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_631_take__tl,axiom,
! [N: nat,Xs2: list_nat] :
( ( take_nat @ N @ ( tl_nat @ Xs2 ) )
= ( tl_nat @ ( take_nat @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_tl
thf(fact_632_drop__Suc,axiom,
! [N: nat,Xs2: list_nat] :
( ( drop_nat @ ( suc @ N ) @ Xs2 )
= ( drop_nat @ N @ ( tl_nat @ Xs2 ) ) ) ).
% drop_Suc
thf(fact_633_list_Oexpand,axiom,
! [List: list_nat,List2: list_nat] :
( ( ( List = nil_nat )
= ( List2 = nil_nat ) )
=> ( ( ( List != nil_nat )
=> ( ( List2 != nil_nat )
=> ( ( ( hd_nat @ List )
= ( hd_nat @ List2 ) )
& ( ( tl_nat @ List )
= ( tl_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_634_take__drop,axiom,
! [N: nat,M: nat,Xs2: list_nat] :
( ( take_nat @ N @ ( drop_nat @ M @ Xs2 ) )
= ( drop_nat @ M @ ( take_nat @ ( plus_plus_nat @ N @ M ) @ Xs2 ) ) ) ).
% take_drop
thf(fact_635_Cons__in__subseqsD,axiom,
! [Y: nat,Ys2: list_nat,Xs2: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y @ Ys2 ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
=> ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_636_List_Oinsert__def,axiom,
( insert8944034826898310173at_nat
= ( ^ [X4: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat] : ( if_lis9186351972506106189at_nat @ ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs3 ) ) @ Xs3 @ ( cons_P6512896166579812791at_nat @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_637_List_Oinsert__def,axiom,
( insert4532235091570160003at_nat
= ( ^ [X4: produc859450856879609959at_nat,Xs3: list_P8469869581646625389at_nat] : ( if_lis7763640049307703347at_nat @ ( member8206827879206165904at_nat @ X4 @ ( set_Pr5518436109238095868at_nat @ Xs3 ) ) @ Xs3 @ ( cons_P8732206157123786781at_nat @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_638_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X4: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X4 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_639_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ ( distinct_nat @ Ws )
=> ? [Xs: list_nat,Ys: list_nat,Zs2: list_nat,Y2: nat] :
( Ws
= ( append_nat @ Xs @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ ( append_nat @ Ys @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_640_nth__eq__iff__index__eq,axiom,
! [Xs2: list_nat,I3: nat,J2: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Xs2 @ J2 ) )
= ( I3 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_641_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs3: list_nat] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs3 ) )
=> ( ( I2 != J )
=> ( ( nth_nat @ Xs3 @ I2 )
!= ( nth_nat @ Xs3 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_642_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_643_all__set__conv__all__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X4 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_644_all__nth__imp__all__set,axiom,
! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I ) ) )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_645_all__nth__imp__all__set,axiom,
! [Xs2: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3679842834875189465at_nat @ Xs2 ) )
=> ( P @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ I ) ) )
=> ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_646_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P: nat > $o,X: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_647_in__set__conv__nth,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_648_in__set__conv__nth,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Xs2 ) )
& ( ( nth_Pr6744343527793145070at_nat @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_649_in__set__conv__nth,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_650_list__ball__nth,axiom,
! [N: nat,Xs2: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_651_nth__mem,axiom,
! [N: nat,Xs2: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% nth_mem
thf(fact_652_nth__mem,axiom,
! [N: nat,Xs2: list_P8469869581646625389at_nat] :
( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs2 ) )
=> ( member8206827879206165904at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ N ) @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) ) ).
% nth_mem
thf(fact_653_nth__mem,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_654_set__update__memI,axiom,
! [N: nat,Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_655_set__update__memI,axiom,
! [N: nat,Xs2: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs2 ) )
=> ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( list_u5003261594476800725at_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_656_set__update__memI,axiom,
! [N: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_657_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_658_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_659_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_660_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_661_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_662_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_663_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_664_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_665_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_666_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_667_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_668_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_669_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_670_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_671_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_672_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_673_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_674_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_675_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_676_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_677_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_678_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_679_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_680_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X6: nat] : ( P5 @ X6 ) )
= ( ^ [P2: nat > $o] :
? [N2: nat] :
( ( P2 @ N2 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ~ ( P2 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_681_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_682_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_683_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_684_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_685_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_686_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_687_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_688_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_689_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_690_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_691_lexord__partial__trans,axiom,
! [Xs2: list_P8469869581646625389at_nat,R: set_Pr553994874890374343at_nat,Ys2: list_P8469869581646625389at_nat,Zs: list_P8469869581646625389at_nat] :
( ! [X2: produc859450856879609959at_nat,Y2: produc859450856879609959at_nat,Z2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X2 @ Y2 ) @ R )
=> ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y2 @ Z2 ) @ R )
=> ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X2 @ Z2 ) @ R ) ) ) )
=> ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs2 @ Ys2 ) @ ( lexord5831005462426227802at_nat @ R ) )
=> ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Ys2 @ Zs ) @ ( lexord5831005462426227802at_nat @ R ) )
=> ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs2 @ Zs ) @ ( lexord5831005462426227802at_nat @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_692_lexord__partial__trans,axiom,
! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat,Ys2: list_nat,Zs: list_nat] :
( ! [X2: nat,Y2: nat,Z2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z2 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R ) ) ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( lexord_nat @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lexord_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Zs ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_693_lexord__partial__trans,axiom,
! [Xs2: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat,Ys2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ R )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y2 @ Z2 ) @ R )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Z2 ) @ R ) ) ) )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( lexord2841853652668343668at_nat @ R ) )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys2 @ Zs ) @ ( lexord2841853652668343668at_nat @ R ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Zs ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_694_take__add,axiom,
! [I3: nat,J2: nat,Xs2: list_nat] :
( ( take_nat @ ( plus_plus_nat @ I3 @ J2 ) @ Xs2 )
= ( append_nat @ ( take_nat @ I3 @ Xs2 ) @ ( take_nat @ J2 @ ( drop_nat @ I3 @ Xs2 ) ) ) ) ).
% take_add
thf(fact_695_enumerate__append__eq,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat] :
( ( enumerate_nat @ N @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs2 ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_696_in__set__product__lists__length,axiom,
! [Xs2: list_nat,Xss2: list_list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_697_nth__tl,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( tl_nat @ Xs2 ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_698_lexord__same__pref__iff,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lexord_nat @ R ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R ) )
| ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_699_lexord__same__pref__iff,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,R: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs2 @ Ys2 ) @ ( append985823374593552924at_nat @ Xs2 @ Zs ) ) @ ( lexord2841853652668343668at_nat @ R ) )
= ( ? [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R ) )
| ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys2 @ Zs ) @ ( lexord2841853652668343668at_nat @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_700_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_701_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_702_distinct__union,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs2 @ Ys2 ) )
= ( distinct_nat @ Ys2 ) ) ).
% distinct_union
thf(fact_703_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_704_rotate1__hd__tl,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( rotate1_nat @ Xs2 )
= ( append_nat @ ( tl_nat @ Xs2 ) @ ( cons_nat @ ( hd_nat @ Xs2 ) @ nil_nat ) ) ) ) ).
% rotate1_hd_tl
thf(fact_705_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_706_rotate1__is__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( ( rotate1_nat @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_707_set__rotate1,axiom,
! [Xs2: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_708_length__rotate1,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_709_distinct1__rotate,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ ( rotate1_nat @ Xs2 ) )
= ( distinct_nat @ Xs2 ) ) ).
% distinct1_rotate
thf(fact_710_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_711_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs2 ) )
= ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_712_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_713_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( I3 = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_714_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_715_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_716_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_717_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_718_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_719_nth__zip,axiom,
! [I3: nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( nth_Pr6744343527793145070at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) @ I3 )
= ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I3 ) @ ( nth_Pr7617993195940197384at_nat @ Ys2 @ I3 ) ) ) ) ) ).
% nth_zip
thf(fact_720_nth__zip,axiom,
! [I3: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) @ I3 )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ).
% nth_zip
thf(fact_721_length__prefixes,axiom,
! [Xs2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( prefixes_nat @ Xs2 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_722_Nil__eq__zip__iff,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( zip_nat_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_nat )
| ( Ys2 = nil_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_723_zip__eq__Nil__iff,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( zip_nat_nat @ Xs2 @ Ys2 )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs2 = nil_nat )
| ( Ys2 = nil_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_724_zip__Cons__Cons,axiom,
! [X: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( zip_nat_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_725_zip__Cons__Cons,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
( ( zip_Pr4664179122662387191at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys2 ) )
= ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_726_zip__append,axiom,
! [Xs2: list_nat,Us: list_nat,Ys2: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Us @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs2 @ Us ) @ ( zip_nat_nat @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_727_length__zip,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( size_s5460976970255530739at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) )
= ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_zip
thf(fact_728_take__zip,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat] :
( ( take_P2173866234530122223at_nat @ N @ ( zip_nat_nat @ Xs2 @ Ys2 ) )
= ( zip_nat_nat @ ( take_nat @ N @ Xs2 ) @ ( take_nat @ N @ Ys2 ) ) ) ).
% take_zip
thf(fact_729_drop__zip,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat] :
( ( drop_P8868858903918902087at_nat @ N @ ( zip_nat_nat @ Xs2 @ Ys2 ) )
= ( zip_nat_nat @ ( drop_nat @ N @ Xs2 ) @ ( drop_nat @ N @ Ys2 ) ) ) ).
% drop_zip
thf(fact_730_set__zip__rightD,axiom,
! [X: nat,Y: nat,Xs2: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) ) )
=> ( member_nat @ Y @ ( set_nat2 @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_731_set__zip__rightD,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) ) )
=> ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_732_set__zip__leftD,axiom,
! [X: nat,Y: nat,Xs2: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% set_zip_leftD
thf(fact_733_set__zip__leftD,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% set_zip_leftD
thf(fact_734_in__set__zipE,axiom,
! [X: product_prod_nat_nat,Y: produc859450856879609959at_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ X @ Y ) @ ( set_Pr7842263693884001378at_nat @ ( zip_Pr7973478175215168605at_nat @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ~ ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_735_in__set__zipE,axiom,
! [X: produc859450856879609959at_nat,Y: product_prod_nat_nat,Xs2: list_P8469869581646625389at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ X @ Y ) @ ( set_Pr5728078514408733718at_nat @ ( zip_Pr4337087602547434257at_nat @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ~ ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_736_in__set__zipE,axiom,
! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ ( set_Pr1322963821424748924at_nat @ ( zip_Pr935030979083031159at_nat @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ~ ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_737_in__set__zipE,axiom,
! [X: product_prod_nat_nat,Y: nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_nat] :
( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ X @ Y ) @ ( set_Pr7836445846575771563at_nat @ ( zip_Pr6869450617852699226at_nat @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ~ ( member_nat @ Y @ ( set_nat2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_738_in__set__zipE,axiom,
! [X: produc859450856879609959at_nat,Y: nat,Xs2: list_P8469869581646625389at_nat,Ys2: list_nat] :
( ( member5422927253076579709at_nat @ ( produc1143935811122074848at_nat @ X @ Y ) @ ( set_Pr1996455851702135825at_nat @ ( zip_Pr4184619967206331840at_nat @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ~ ( member_nat @ Y @ ( set_nat2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_739_in__set__zipE,axiom,
! [X: nat,Y: product_prod_nat_nat,Xs2: list_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X @ Y ) @ ( set_Pr6710958862608470481at_nat @ ( zip_na1006125974040638520at_nat @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_740_in__set__zipE,axiom,
! [X: nat,Y: produc859450856879609959at_nat,Xs2: list_nat,Ys2: list_P8469869581646625389at_nat] :
( ( member5754182594613576739at_nat @ ( produc7482930922861554750at_nat @ X @ Y ) @ ( set_Pr2327711193239132855at_nat @ ( zip_na1300243042091035934at_nat @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_741_in__set__zipE,axiom,
! [X: nat,Y: nat,Xs2: list_nat,Ys2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( member_nat @ Y @ ( set_nat2 @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_742_in__set__zipE,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ~ ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_743_zip__same,axiom,
! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ ( set_Pr1322963821424748924at_nat @ ( zip_Pr935030979083031159at_nat @ Xs2 @ Xs2 ) ) )
= ( ( member8206827879206165904at_nat @ A @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_744_zip__same,axiom,
! [A: nat,B: nat,Xs2: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Xs2 ) ) )
= ( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_745_zip__same,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Xs2 ) ) )
= ( ( member8440522571783428010at_nat @ A @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
& ( A = B ) ) ) ).
% zip_same
thf(fact_746_zip__update,axiom,
! [Xs2: list_nat,I3: nat,X: nat,Ys2: list_nat,Y: nat] :
( ( zip_nat_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ ( list_update_nat @ Ys2 @ I3 @ Y ) )
= ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) @ I3 @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).
% zip_update
thf(fact_747_zip__update,axiom,
! [Xs2: list_P6011104703257516679at_nat,I3: nat,X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( zip_Pr4664179122662387191at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I3 @ X ) @ ( list_u6180841689913720943at_nat @ Ys2 @ I3 @ Y ) )
= ( list_u5003261594476800725at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) @ I3 @ ( produc6161850002892822231at_nat @ X @ Y ) ) ) ).
% zip_update
thf(fact_748_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_749_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_750_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_751_in__set__impl__in__set__zip1,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ~ ! [Y2: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y2 ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_752_in__set__impl__in__set__zip1,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_nat,X: product_prod_nat_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ~ ! [Y2: nat] :
~ ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ X @ Y2 ) @ ( set_Pr7836445846575771563at_nat @ ( zip_Pr6869450617852699226at_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_753_in__set__impl__in__set__zip1,axiom,
! [Xs2: list_P8469869581646625389at_nat,Ys2: list_nat,X: produc859450856879609959at_nat] :
( ( ( size_s3679842834875189465at_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ~ ! [Y2: nat] :
~ ( member5422927253076579709at_nat @ ( produc1143935811122074848at_nat @ X @ Y2 ) @ ( set_Pr1996455851702135825at_nat @ ( zip_Pr4184619967206331840at_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_754_in__set__impl__in__set__zip1,axiom,
! [Xs2: list_nat,Ys2: list_nat,X: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ! [Y2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_755_in__set__impl__in__set__zip2,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys2 ) )
=> ~ ! [X2: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_756_in__set__impl__in__set__zip2,axiom,
! [Xs2: list_nat,Ys2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys2 ) )
=> ~ ! [X2: nat] :
~ ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X2 @ Y ) @ ( set_Pr6710958862608470481at_nat @ ( zip_na1006125974040638520at_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_757_in__set__impl__in__set__zip2,axiom,
! [Xs2: list_nat,Ys2: list_P8469869581646625389at_nat,Y: produc859450856879609959at_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3679842834875189465at_nat @ Ys2 ) )
=> ( ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ Ys2 ) )
=> ~ ! [X2: nat] :
~ ( member5754182594613576739at_nat @ ( produc7482930922861554750at_nat @ X2 @ Y ) @ ( set_Pr2327711193239132855at_nat @ ( zip_na1300243042091035934at_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_758_in__set__impl__in__set__zip2,axiom,
! [Xs2: list_nat,Ys2: list_nat,Y: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( member_nat @ Y @ ( set_nat2 @ Ys2 ) )
=> ~ ! [X2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_759_hd__zip,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( Ys2 != nil_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) )
= ( product_Pair_nat_nat @ ( hd_nat @ Xs2 ) @ ( hd_nat @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_760_hd__zip,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( ( Ys2 != nil_Pr5478986624290739719at_nat )
=> ( ( hd_Pro8462011474880202578at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) )
= ( produc6161850002892822231at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs2 ) @ ( hd_Pro3460610213475200108at_nat @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_761_zip__obtain__same__length,axiom,
! [Xs2: list_nat,Ys2: list_nat,P: list_P6011104703257516679at_nat > $o] :
( ! [Zs2: list_nat,Ws2: list_nat,N3: nat] :
( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( N3
= ( ord_min_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) )
=> ( ( Zs2
= ( take_nat @ N3 @ Xs2 ) )
=> ( ( Ws2
= ( take_nat @ N3 @ Ys2 ) )
=> ( P @ ( zip_nat_nat @ Zs2 @ Ws2 ) ) ) ) ) )
=> ( P @ ( zip_nat_nat @ Xs2 @ Ys2 ) ) ) ).
% zip_obtain_same_length
thf(fact_762_zip__eq__ConsE,axiom,
! [Xs2: list_nat,Ys2: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
( ( ( zip_nat_nat @ Xs2 @ Ys2 )
= ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
=> ~ ! [X2: nat,Xs4: list_nat] :
( ( Xs2
= ( cons_nat @ X2 @ Xs4 ) )
=> ! [Y2: nat,Ys6: list_nat] :
( ( Ys2
= ( cons_nat @ Y2 @ Ys6 ) )
=> ( ( Xy
= ( product_Pair_nat_nat @ X2 @ Y2 ) )
=> ( Xys
!= ( zip_nat_nat @ Xs4 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_763_zip__eq__ConsE,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Xy: produc859450856879609959at_nat,Xys: list_P8469869581646625389at_nat] :
( ( ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 )
= ( cons_P8732206157123786781at_nat @ Xy @ Xys ) )
=> ~ ! [X2: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat] :
( ( Xs2
= ( cons_P6512896166579812791at_nat @ X2 @ Xs4 ) )
=> ! [Y2: product_prod_nat_nat,Ys6: list_P6011104703257516679at_nat] :
( ( Ys2
= ( cons_P6512896166579812791at_nat @ Y2 @ Ys6 ) )
=> ( ( Xy
= ( produc6161850002892822231at_nat @ X2 @ Y2 ) )
=> ( Xys
!= ( zip_Pr4664179122662387191at_nat @ Xs4 @ Ys6 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_764_zip__append1,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( zip_nat_nat @ ( append_nat @ Xs2 @ Ys2 ) @ Zs )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs2 @ ( take_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) @ ( zip_nat_nat @ Ys2 @ ( drop_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) ) ) ).
% zip_append1
thf(fact_765_zip__append2,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( zip_nat_nat @ Xs2 @ ( append_nat @ Ys2 @ Zs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ ( take_nat @ ( size_size_list_nat @ Ys2 ) @ Xs2 ) @ Ys2 ) @ ( zip_nat_nat @ ( drop_nat @ ( size_size_list_nat @ Ys2 ) @ Xs2 ) @ Zs ) ) ) ).
% zip_append2
thf(fact_766_rgf__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( equiva3371634703666331078on_rgf @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= ( ( equiva3371634703666331078on_rgf @ Xs2 )
& ( ord_less_nat @ X @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ Xs2 ) @ one_one_nat ) ) ) ) ).
% rgf_snoc
thf(fact_767_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_768_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_769_rgf__limit__ge,axiom,
! [Y: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ Y @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ).
% rgf_limit_ge
thf(fact_770_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F2: nat > nat > nat,A4: nat,B3: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_771_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_772_rgf__limit__snoc,axiom,
! [X: list_nat,Y: nat] :
( ( equiva5889994315859557365_limit @ ( append_nat @ X @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ord_max_nat @ ( plus_plus_nat @ Y @ one_one_nat ) @ ( equiva5889994315859557365_limit @ X ) ) ) ).
% rgf_limit_snoc
thf(fact_773_rgf__def,axiom,
( equiva3371634703666331078on_rgf
= ( ^ [X4: list_nat] :
! [Ys3: list_nat,Y4: nat] :
( ( prefix_nat @ ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) @ X4 )
=> ( ord_less_eq_nat @ Y4 @ ( equiva5889994315859557365_limit @ Ys3 ) ) ) ) ) ).
% rgf_def
thf(fact_774_nth__drop,axiom,
! [N: nat,Xs2: list_nat,I3: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs2 ) @ I3 )
= ( nth_nat @ Xs2 @ ( plus_plus_nat @ N @ I3 ) ) ) ) ).
% nth_drop
thf(fact_775_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_776_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_777_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_778_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_779_max__less__iff__conj,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z4 )
= ( ( ord_less_nat @ X @ Z4 )
& ( ord_less_nat @ Y @ Z4 ) ) ) ).
% max_less_iff_conj
thf(fact_780_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_781_take__all__iff,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( take_nat @ N @ Xs2 )
= Xs2 )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% take_all_iff
thf(fact_782_take__all,axiom,
! [Xs2: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
=> ( ( take_nat @ N @ Xs2 )
= Xs2 ) ) ).
% take_all
thf(fact_783_list__update__beyond,axiom,
! [Xs2: list_nat,I3: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I3 )
=> ( ( list_update_nat @ Xs2 @ I3 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_784_take__update__cancel,axiom,
! [N: nat,M: nat,Xs2: list_nat,Y: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs2 @ M @ Y ) )
= ( take_nat @ N @ Xs2 ) ) ) ).
% take_update_cancel
thf(fact_785_drop__all,axiom,
! [Xs2: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
=> ( ( drop_nat @ N @ Xs2 )
= nil_nat ) ) ).
% drop_all
thf(fact_786_drop__eq__Nil,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( drop_nat @ N @ Xs2 )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_787_drop__eq__Nil2,axiom,
! [N: nat,Xs2: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs2 ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_788_rotate1__length01,axiom,
! [Xs2: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_789_less__max__iff__disj,axiom,
! [Z4: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z4 @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z4 @ X )
| ( ord_less_nat @ Z4 @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_790_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_791_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( A5
= ( ord_max_nat @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ).
% max.strict_order_iff
thf(fact_792_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_793_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_794_nat__add__max__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_795_nat__add__max__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_796_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_797_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_798_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_799_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_800_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_801_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_802_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_803_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_804_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_805_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_806_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_807_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_808_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_809_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_810_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_811_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_812_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_813_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_814_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_815_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_816_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_817_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_818_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_819_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_820_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_821_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_822_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_823_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_824_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_825_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_826_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_827_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_828_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_829_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_830_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z2: nat] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z2 )
=> ( R2 @ X2 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_831_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_832_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_833_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_834_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_835_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_836_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_837_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_838_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_839_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_840_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_841_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_842_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_843_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_844_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_845_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_846_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_847_trans__le__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_848_trans__le__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_849_add__le__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_850_add__le__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_851_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_852_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_853_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_854_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_855_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_856_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_857_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_858_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_859_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_860_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_861_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_862_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_863_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_864_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_865_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_866_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_867_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_868_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_869_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_870_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_871_dec__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ I3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_872_inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_873_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_874_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_875_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_876_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_877_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_878_impossible__Cons,axiom,
! [Xs2: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs2
!= ( cons_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_879_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_880_prefix__order_Olift__Suc__antimono__le,axiom,
! [F: nat > list_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( prefix_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( prefix_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% prefix_order.lift_Suc_antimono_le
thf(fact_881_prefix__order_Olift__Suc__mono__le,axiom,
! [F: nat > list_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( prefix_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( prefix_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% prefix_order.lift_Suc_mono_le
thf(fact_882_prefix__length__prefix,axiom,
! [Ps: list_nat,Xs2: list_nat,Qs: list_nat] :
( ( prefix_nat @ Ps @ Xs2 )
=> ( ( prefix_nat @ Qs @ Xs2 )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
=> ( prefix_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_883_prefix__length__le,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% prefix_length_le
thf(fact_884_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
= ( ? [X4: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_885_lenlex__length,axiom,
! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).
% lenlex_length
thf(fact_886_rgf__limit_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( equiva5889994315859557365_limit @ ( cons_nat @ X @ Xs2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ).
% rgf_limit.simps(2)
thf(fact_887_nth__take__lemma,axiom,
! [K: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Ys2 @ I ) ) )
=> ( ( take_nat @ K @ Xs2 )
= ( take_nat @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_888_append__eq__append__conv__if,axiom,
! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
( ( ( append_nat @ Xs_1 @ Xs_2 )
= ( append_nat @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( Xs_1
= ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_889_lexord__sufI,axiom,
! [U: list_nat,W2: list_nat,R: set_Pr1261947904930325089at_nat,V: list_nat,Z4: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ W2 ) @ ( lexord_nat @ R ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ W2 ) @ ( size_size_list_nat @ U ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ V ) @ ( append_nat @ W2 @ Z4 ) ) @ ( lexord_nat @ R ) ) ) ) ).
% lexord_sufI
thf(fact_890_measures__lesseq,axiom,
! [F: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_891_measures__lesseq,axiom,
! [F: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_892_rgf__limit_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( equiva5889994315859557365_limit @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs ) )
=> ( Y
!= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) ) ) ) ) ).
% rgf_limit.elims
thf(fact_893_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_894_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_895_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_896_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_897_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_898_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_899_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_900_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_901_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_902_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X4: list_nat] : X4 ) ) ).
% drop0
thf(fact_903_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_904_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_905_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_906_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_907_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_908_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_909_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_910_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_911_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_912_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_913_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_914_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_915_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_916_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_917_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_918_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_919_nth__Cons__0,axiom,
! [X: nat,Xs2: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_920_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_921_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs3: list_nat] : nil_nat ) ) ).
% take0
thf(fact_922_take__eq__Nil,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( take_nat @ N @ Xs2 )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_923_take__eq__Nil2,axiom,
! [N: nat,Xs2: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs2 ) )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_924_length__greater__0__conv,axiom,
! [Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
= ( Xs2 != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_925_hd__take,axiom,
! [J2: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ J2 )
=> ( ( hd_nat @ ( take_nat @ J2 @ Xs2 ) )
= ( hd_nat @ Xs2 ) ) ) ).
% hd_take
thf(fact_926_subset__code_I1_J,axiom,
! [Xs2: list_P6011104703257516679at_nat,B6: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ B6 )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ).
% subset_code(1)
thf(fact_927_subset__code_I1_J,axiom,
! [Xs2: list_P8469869581646625389at_nat,B6: set_Pr8693737435421807431at_nat] :
( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs2 ) @ B6 )
= ( ! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( member8206827879206165904at_nat @ X4 @ B6 ) ) ) ) ).
% subset_code(1)
thf(fact_928_subset__code_I1_J,axiom,
! [Xs2: list_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B6 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X4 @ B6 ) ) ) ) ).
% subset_code(1)
thf(fact_929_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_930_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_931_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_932_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_933_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_934_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_935_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_936_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_937_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_938_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_939_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_940_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_941_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_942_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_943_enum__rgfs_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% enum_rgfs.cases
thf(fact_944_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_945_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_946_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_947_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_948_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_949_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_950_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_951_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_952_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_953_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_954_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_955_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_956_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_957_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_958_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_959_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_960_drop__0,axiom,
! [Xs2: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs2 )
= Xs2 ) ).
% drop_0
thf(fact_961_set__subset__Cons,axiom,
! [Xs2: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_962_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_963_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_964_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_965_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_966_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_967_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_968_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_969_set__mono__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys2 ) ) ) ).
% set_mono_prefix
thf(fact_970_set__take__subset,axiom,
! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% set_take_subset
thf(fact_971_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_972_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_973_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_974_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_975_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_976_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_977_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_978_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_979_set__drop__subset,axiom,
! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% set_drop_subset
thf(fact_980_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_981_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I3 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_982_set__update__subsetI,axiom,
! [Xs2: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I3: nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
=> ( ( member8440522571783428010at_nat @ X @ A2 )
=> ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I3 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_983_set__update__subsetI,axiom,
! [Xs2: list_P8469869581646625389at_nat,A2: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,I3: nat] :
( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs2 ) @ A2 )
=> ( ( member8206827879206165904at_nat @ X @ A2 )
=> ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ ( list_u5003261594476800725at_nat @ Xs2 @ I3 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_984_set__update__subsetI,axiom,
! [Xs2: list_nat,A2: set_nat,X: nat,I3: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I3 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_985_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_986_take__0,axiom,
! [Xs2: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs2 )
= nil_nat ) ).
% take_0
thf(fact_987_list__update__code_I2_J,axiom,
! [X: nat,Xs2: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_988_rgf__limit_Osimps_I1_J,axiom,
( ( equiva5889994315859557365_limit @ nil_nat )
= zero_zero_nat ) ).
% rgf_limit.simps(1)
thf(fact_989_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_990_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_991_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_992_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_993_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_994_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_995_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_996_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_997_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs2 ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) ) ) ).
% set_take_subset_set_take
thf(fact_998_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M @ Xs2 ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_999_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1000_length__pos__if__in__set,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1001_length__pos__if__in__set,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3679842834875189465at_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1002_length__pos__if__in__set,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1003_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1004_hd__conv__nth,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ Xs2 )
= ( nth_nat @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1005_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1006_nth__equal__first__eq,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,N: nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1007_nth__equal__first__eq,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat,N: nat] :
( ~ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3679842834875189465at_nat @ Xs2 ) )
=> ( ( ( nth_Pr6744343527793145070at_nat @ ( cons_P8732206157123786781at_nat @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1008_nth__equal__first__eq,axiom,
! [X: nat,Xs2: list_nat,N: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1009_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs3: list_nat] : ( if_nat @ ( Xs3 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_1010_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_1011_nat__gcd_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [X2: nat,Y2: nat] :
( X
!= ( product_Pair_nat_nat @ X2 @ Y2 ) ) ).
% nat_gcd.cases
thf(fact_1012_length__n__lists__elem,axiom,
! [Ys2: list_nat,N: nat,Xs2: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) ) )
=> ( ( size_size_list_nat @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_1013_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs2 )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_1014_stirling_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ( X
!= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) )
=> ( ! [K2: nat] :
( X
!= ( product_Pair_nat_nat @ zero_zero_nat @ ( suc @ K2 ) ) )
=> ( ! [N3: nat] :
( X
!= ( product_Pair_nat_nat @ ( suc @ N3 ) @ zero_zero_nat ) )
=> ~ ! [N3: nat,K2: nat] :
( X
!= ( product_Pair_nat_nat @ ( suc @ N3 ) @ ( suc @ K2 ) ) ) ) ) ) ).
% stirling.cases
thf(fact_1015_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_nat
= ( ^ [F3: nat > nat,Xs3: list_nat] : ( if_nat @ ( Xs3 = nil_nat ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F3 @ ( hd_nat @ Xs3 ) ) @ ( size_list_nat @ F3 @ ( tl_nat @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_1016_size__list__append,axiom,
! [F: nat > nat,Xs2: list_nat,Ys2: list_nat] :
( ( size_list_nat @ F @ ( append_nat @ Xs2 @ Ys2 ) )
= ( plus_plus_nat @ ( size_list_nat @ F @ Xs2 ) @ ( size_list_nat @ F @ Ys2 ) ) ) ).
% size_list_append
thf(fact_1017_list_Osize__gen_I1_J,axiom,
! [X: nat > nat] :
( ( size_list_nat @ X @ nil_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_1018_size__list__estimation,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y: nat,F: product_prod_nat_nat > nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ( ord_less_nat @ Y @ ( F @ X ) )
=> ( ord_less_nat @ Y @ ( size_l3005607024265538313at_nat @ F @ Xs2 ) ) ) ) ).
% size_list_estimation
thf(fact_1019_size__list__estimation,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat,Y: nat,F: produc859450856879609959at_nat > nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ( ord_less_nat @ Y @ ( F @ X ) )
=> ( ord_less_nat @ Y @ ( size_l916709973823129199at_nat @ F @ Xs2 ) ) ) ) ).
% size_list_estimation
thf(fact_1020_size__list__estimation,axiom,
! [X: nat,Xs2: list_nat,Y: nat,F: nat > nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( ord_less_nat @ Y @ ( F @ X ) )
=> ( ord_less_nat @ Y @ ( size_list_nat @ F @ Xs2 ) ) ) ) ).
% size_list_estimation
thf(fact_1021_size__list__pointwise,axiom,
! [Xs2: list_P6011104703257516679at_nat,F: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat] :
( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( size_l3005607024265538313at_nat @ F @ Xs2 ) @ ( size_l3005607024265538313at_nat @ G @ Xs2 ) ) ) ).
% size_list_pointwise
thf(fact_1022_size__list__pointwise,axiom,
! [Xs2: list_P8469869581646625389at_nat,F: produc859450856879609959at_nat > nat,G: produc859450856879609959at_nat > nat] :
( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( size_l916709973823129199at_nat @ F @ Xs2 ) @ ( size_l916709973823129199at_nat @ G @ Xs2 ) ) ) ).
% size_list_pointwise
thf(fact_1023_size__list__pointwise,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( size_list_nat @ F @ Xs2 ) @ ( size_list_nat @ G @ Xs2 ) ) ) ).
% size_list_pointwise
thf(fact_1024_size__list__estimation_H,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y: nat,F: product_prod_nat_nat > nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
=> ( ord_less_eq_nat @ Y @ ( size_l3005607024265538313at_nat @ F @ Xs2 ) ) ) ) ).
% size_list_estimation'
thf(fact_1025_size__list__estimation_H,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat,Y: nat,F: produc859450856879609959at_nat > nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
=> ( ord_less_eq_nat @ Y @ ( size_l916709973823129199at_nat @ F @ Xs2 ) ) ) ) ).
% size_list_estimation'
thf(fact_1026_size__list__estimation_H,axiom,
! [X: nat,Xs2: list_nat,Y: nat,F: nat > nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_nat @ F @ Xs2 ) ) ) ) ).
% size_list_estimation'
thf(fact_1027_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_1028_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_1029_psubsetD,axiom,
! [A2: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le7866589430770878221at_nat @ A2 @ B6 )
=> ( ( member8440522571783428010at_nat @ C @ A2 )
=> ( member8440522571783428010at_nat @ C @ B6 ) ) ) ).
% psubsetD
thf(fact_1030_psubsetD,axiom,
! [A2: set_nat,B6: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B6 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B6 ) ) ) ).
% psubsetD
thf(fact_1031_psubsetD,axiom,
! [A2: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
( ( ord_le6428140832669894131at_nat @ A2 @ B6 )
=> ( ( member8206827879206165904at_nat @ C @ A2 )
=> ( member8206827879206165904at_nat @ C @ B6 ) ) ) ).
% psubsetD
thf(fact_1032_stirling__row__nonempty,axiom,
! [N: nat] :
( ( stirling_row @ N )
!= nil_nat ) ).
% stirling_row_nonempty
thf(fact_1033_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1034_stirling__row__code_I2_J,axiom,
! [N: nat] :
( ( stirling_row @ ( suc @ N ) )
= ( stirling_row_aux_nat @ N @ zero_zero_nat @ ( stirling_row @ N ) ) ) ).
% stirling_row_code(2)
thf(fact_1035_stirling__row__aux_Osimps_I1_J,axiom,
! [N: nat,Y: nat] :
( ( stirling_row_aux_nat @ N @ Y @ nil_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_aux.simps(1)
thf(fact_1036_stirling__code,axiom,
( stirling2
= ( ^ [N2: nat,K3: nat] : ( if_nat @ ( K3 = zero_zero_nat ) @ ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat @ zero_zero_nat ) @ ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( K3 = N2 ) @ one_one_nat @ ( nth_nat @ ( stirling_row @ N2 ) @ K3 ) ) ) ) ) ) ).
% stirling_code
thf(fact_1037_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_1038_subrelI,axiom,
! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ R )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ S ) )
=> ( ord_le3000389064537975527at_nat @ R @ S ) ) ).
% subrelI
thf(fact_1039_stirling__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( stirling2 @ N @ zero_zero_nat )
= zero_zero_nat ) ) ).
% stirling_0
thf(fact_1040_stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling2 @ N @ K )
= zero_zero_nat ) ) ).
% stirling_less
thf(fact_1041_stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling2 @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% stirling.simps(3)
thf(fact_1042_stirling_Osimps_I2_J,axiom,
! [K: nat] :
( ( stirling2 @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% stirling.simps(2)
thf(fact_1043_nth__stirling__row,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( nth_nat @ ( stirling_row @ N ) @ K )
= ( stirling2 @ N @ K ) ) ) ).
% nth_stirling_row
thf(fact_1044_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1045_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_1046_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_1047_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1048_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1049_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_1050_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_1051_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_1052_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_1053_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P6 @ X5 )
& ( Q4 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1054_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P6 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P6 @ X5 )
| ( Q4 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1055_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_1056_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_1057_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_1058_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_1059_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1060_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1061_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A3: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A3 ) )
= ( ord_less_nat @ A3 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1062_rgf__limit_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( equiva5889994315859557365_limit @ X )
= Y )
=> ( ( accp_list_nat @ equiva5575797544161152836it_rel @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ nil_nat ) ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs ) )
=> ( ( Y
= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ ( cons_nat @ X2 @ Xs ) ) ) ) ) ) ) ).
% rgf_limit.pelims
thf(fact_1063_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1064_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1065_pair__lessI2,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_1066_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1067_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1068_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1069_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1070_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1071_diff__diff__left,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1072_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1073_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1074_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1075_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1076_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1077_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1078_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1079_length__drop,axiom,
! [N: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_1080_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1081_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1082_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1083_take__append,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_1084_drop__append,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( drop_nat @ N @ Xs2 ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_1085_length__tl,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( tl_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1086_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z4 ) ) @ fun_pair_less )
= ( ord_less_nat @ Y @ Z4 ) ) ).
% pair_less_iff1
thf(fact_1087_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1088_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I3: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1089_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1090_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1091_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1092_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1093_min__diff,axiom,
! [M: nat,I3: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I3 ) @ ( minus_minus_nat @ N @ I3 ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I3 ) ) ).
% min_diff
thf(fact_1094_diff__commute,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_1095_drop__take,axiom,
! [N: nat,M: nat,Xs2: list_nat] :
( ( drop_nat @ N @ ( take_nat @ M @ Xs2 ) )
= ( take_nat @ ( minus_minus_nat @ M @ N ) @ ( drop_nat @ N @ Xs2 ) ) ) ).
% drop_take
thf(fact_1096_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_1097_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1098_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I3 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1099_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1100_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1101_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1102_le__diff__conv,axiom,
! [J2: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1103_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1104_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1105_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1106_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1107_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1108_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1109_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1110_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1111_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1112_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1113_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1114_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1115_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1116_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1117_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1118_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1119_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1120_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1121_less__diff__conv,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1122_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1123_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1124_diff__Suc__less,axiom,
! [N: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1125_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1126_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1127_less__diff__conv2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1128_drop__update__swap,axiom,
! [M: nat,N: nat,Xs2: list_nat,X: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs2 @ N @ X ) )
= ( list_update_nat @ ( drop_nat @ M @ Xs2 ) @ ( minus_minus_nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_1129_tl__take,axiom,
! [N: nat,Xs2: list_nat] :
( ( tl_nat @ ( take_nat @ N @ Xs2 ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_nat @ Xs2 ) ) ) ).
% tl_take
thf(fact_1130_pair__lessI1,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_1131_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1132_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1133_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N2: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1134_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1135_nth__append,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys2 ) @ N )
= ( nth_nat @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys2 ) @ N )
= ( nth_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_1136_drop__Cons_H,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ Xs2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_1137_list__update__append,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( list_update_nat @ ( append_nat @ Xs2 @ Ys2 ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs2 @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( list_update_nat @ ( append_nat @ Xs2 @ Ys2 ) @ N @ X )
= ( append_nat @ Xs2 @ ( list_update_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1138_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs2: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1139_take__Cons_H,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ) ).
% take_Cons'
thf(fact_1140_pair__leqI2,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ S @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_1141_pair__leqI1,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_1142_psubset__imp__ex__mem,axiom,
! [A2: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A2 @ B6 )
=> ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B6 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1143_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B6: set_nat] :
( ( ord_less_set_nat @ A2 @ B6 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B6 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1144_psubset__imp__ex__mem,axiom,
! [A2: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
( ( ord_le6428140832669894131at_nat @ A2 @ B6 )
=> ? [B3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ B6 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1145_butlast__take,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( butlast_nat @ ( take_nat @ N @ Xs2 ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_take
thf(fact_1146_last__list__update,axiom,
! [Xs2: list_nat,K: nat,X: nat] :
( ( Xs2 != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
= ( last_nat @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_1147_last__appendR,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ).
% last_appendR
thf(fact_1148_last__appendL,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_appendL
thf(fact_1149_last__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_1150_butlast__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_1151_length__butlast,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1152_last__drop,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( last_nat @ ( drop_nat @ N @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_drop
thf(fact_1153_append__butlast__last__id,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs2 ) @ ( cons_nat @ ( last_nat @ Xs2 ) @ nil_nat ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_1154_butlast__append,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( butlast_nat @ Xs2 ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ Xs2 @ ( butlast_nat @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_1155_in__set__butlast__appendI,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( butlas5569151904373679443at_nat @ Xs2 ) ) )
| ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( butlas5569151904373679443at_nat @ Ys2 ) ) ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( butlas5569151904373679443at_nat @ ( append985823374593552924at_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1156_in__set__butlast__appendI,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat,Ys2: list_P8469869581646625389at_nat] :
( ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( butlas8493624449043409337at_nat @ Xs2 ) ) )
| ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( butlas8493624449043409337at_nat @ Ys2 ) ) ) )
=> ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( butlas8493624449043409337at_nat @ ( append8751754712269456642at_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1157_in__set__butlast__appendI,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
| ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys2 ) ) ) )
=> ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1158_longest__common__suffix,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
? [Ss: list_nat,Xs4: list_nat,Ys6: list_nat] :
( ( Xs2
= ( append_nat @ Xs4 @ Ss ) )
& ( Ys2
= ( append_nat @ Ys6 @ Ss ) )
& ( ( Xs4 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( last_nat @ Xs4 )
!= ( last_nat @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_1159_last__append,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Xs2 ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ) ).
% last_append
thf(fact_1160_snoc__eq__iff__butlast,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
= Ys2 )
= ( ( Ys2 != nil_nat )
& ( ( butlast_nat @ Ys2 )
= Xs2 )
& ( ( last_nat @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1161_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_1162_distinct__butlast,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ ( butlast_nat @ Xs2 ) ) ) ).
% distinct_butlast
thf(fact_1163_drop__butlast,axiom,
! [N: nat,Xs2: list_nat] :
( ( drop_nat @ N @ ( butlast_nat @ Xs2 ) )
= ( butlast_nat @ ( drop_nat @ N @ Xs2 ) ) ) ).
% drop_butlast
thf(fact_1164_butlast__tl,axiom,
! [Xs2: list_nat] :
( ( butlast_nat @ ( tl_nat @ Xs2 ) )
= ( tl_nat @ ( butlast_nat @ Xs2 ) ) ) ).
% butlast_tl
thf(fact_1165_prefixeq__butlast,axiom,
! [Xs2: list_nat] : ( prefix_nat @ ( butlast_nat @ Xs2 ) @ Xs2 ) ).
% prefixeq_butlast
thf(fact_1166_in__set__butlastD,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( butlas5569151904373679443at_nat @ Xs2 ) ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_1167_in__set__butlastD,axiom,
! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( butlas8493624449043409337at_nat @ Xs2 ) ) )
=> ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_1168_in__set__butlastD,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_1169_last__in__set,axiom,
! [As2: list_P6011104703257516679at_nat] :
( ( As2 != nil_Pr5478986624290739719at_nat )
=> ( member8440522571783428010at_nat @ ( last_P6484183829340986144at_nat @ As2 ) @ ( set_Pr5648618587558075414at_nat @ As2 ) ) ) ).
% last_in_set
thf(fact_1170_last__in__set,axiom,
! [As2: list_P8469869581646625389at_nat] :
( ( As2 != nil_Pr2582115297535392877at_nat )
=> ( member8206827879206165904at_nat @ ( last_P5835599730086418438at_nat @ As2 ) @ ( set_Pr5518436109238095868at_nat @ As2 ) ) ) ).
% last_in_set
thf(fact_1171_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_1172_last__ConsR,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_1173_last__ConsL,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_1174_last_Osimps,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_1175_butlast_Osimps_I2_J,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs2 ) )
= nil_nat ) )
& ( ( Xs2 != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1176_last__tl,axiom,
! [Xs2: list_nat] :
( ( ( Xs2 = nil_nat )
| ( ( tl_nat @ Xs2 )
!= nil_nat ) )
=> ( ( last_nat @ ( tl_nat @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_tl
thf(fact_1177_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_1178_last__zip,axiom,
! [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( Xs2 != nil_Pr5478986624290739719at_nat )
=> ( ( Ys2 != nil_Pr5478986624290739719at_nat )
=> ( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
=> ( ( last_P5835599730086418438at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys2 ) )
= ( produc6161850002892822231at_nat @ ( last_P6484183829340986144at_nat @ Xs2 ) @ ( last_P6484183829340986144at_nat @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_1179_last__zip,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( Ys2 != nil_nat )
=> ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( last_P6484183829340986144at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) )
= ( product_Pair_nat_nat @ ( last_nat @ Xs2 ) @ ( last_nat @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_1180_nth__butlast,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs2 ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ N ) ) ) ).
% nth_butlast
thf(fact_1181_take__butlast,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs2 ) )
= ( take_nat @ N @ Xs2 ) ) ) ).
% take_butlast
thf(fact_1182_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs3: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_1183_butlast__list__update,axiom,
! [K: nat,Xs2: list_nat,X: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
= ( butlast_nat @ Xs2 ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
= ( list_update_nat @ ( butlast_nat @ Xs2 ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_1184_last__conv__nth,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ Xs2 )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1185_upt__Suc,axiom,
! [I3: nat,J2: nat] :
( ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I3 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ ( suc @ J2 ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_1186_upt__Suc__append,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I3 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_1187_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl_nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_1188_hd__upt,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( hd_nat @ ( upt @ I3 @ J2 ) )
= I3 ) ) ).
% hd_upt
thf(fact_1189_drop__upt,axiom,
! [M: nat,I3: nat,J2: nat] :
( ( drop_nat @ M @ ( upt @ I3 @ J2 ) )
= ( upt @ ( plus_plus_nat @ I3 @ M ) @ J2 ) ) ).
% drop_upt
thf(fact_1190_take__upt,axiom,
! [I3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I3 @ N ) )
= ( upt @ I3 @ ( plus_plus_nat @ I3 @ M ) ) ) ) ).
% take_upt
thf(fact_1191_upt__conv__Nil,axiom,
! [J2: nat,I3: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( upt @ I3 @ J2 )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1192_length__upt,axiom,
! [I3: nat,J2: nat] :
( ( size_size_list_nat @ ( upt @ I3 @ J2 ) )
= ( minus_minus_nat @ J2 @ I3 ) ) ).
% length_upt
thf(fact_1193_last__upt,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( last_nat @ ( upt @ I3 @ J2 ) )
= ( minus_minus_nat @ J2 @ one_one_nat ) ) ) ).
% last_upt
thf(fact_1194_upt__eq__Nil__conv,axiom,
! [I3: nat,J2: nat] :
( ( ( upt @ I3 @ J2 )
= nil_nat )
= ( ( J2 = zero_zero_nat )
| ( ord_less_eq_nat @ J2 @ I3 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1195_nth__upt,axiom,
! [I3: nat,K: nat,J2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 )
=> ( ( nth_nat @ ( upt @ I3 @ J2 ) @ K )
= ( plus_plus_nat @ I3 @ K ) ) ) ).
% nth_upt
thf(fact_1196_upt__0,axiom,
! [I3: nat] :
( ( upt @ I3 @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1197_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q3: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q3 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_1198_distinct__upt,axiom,
! [I3: nat,J2: nat] : ( distinct_nat @ ( upt @ I3 @ J2 ) ) ).
% distinct_upt
thf(fact_1199_upt__conv__Cons,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ J2 )
= ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J2 ) ) ) ) ).
% upt_conv_Cons
thf(fact_1200_enumerate__eq__zip,axiom,
( enumerate_nat
= ( ^ [N2: nat,Xs3: list_nat] : ( zip_nat_nat @ ( upt @ N2 @ ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% enumerate_eq_zip
thf(fact_1201_upt__add__eq__append,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ ( plus_plus_nat @ J2 @ K ) )
= ( append_nat @ ( upt @ I3 @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_1202_upt__eq__Cons__conv,axiom,
! [I3: nat,J2: nat,X: nat,Xs2: list_nat] :
( ( ( upt @ I3 @ J2 )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ord_less_nat @ I3 @ J2 )
& ( I3 = X )
& ( ( upt @ ( plus_plus_nat @ I3 @ one_one_nat ) @ J2 )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1203_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_1204_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_1205_map__upt__eqI,axiom,
! [Xs2: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I )
= ( F @ ( plus_plus_nat @ M @ I ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_1206_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_1207_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1208_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1209_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1210_map__eq__conv,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1211_length__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_1212_map__append,axiom,
! [F: nat > nat,Xs2: list_nat,Ys2: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_1213_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_1214_nth__Cons__numeral,axiom,
! [X: nat,Xs2: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1215_take__Cons__numeral,axiom,
! [V: num,X: nat,Xs2: list_nat] :
( ( take_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs2 ) ) ) ).
% take_Cons_numeral
thf(fact_1216_drop__Cons__numeral,axiom,
! [V: num,X: nat,Xs2: list_nat] :
( ( drop_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs2 ) )
= ( drop_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs2 ) ) ).
% drop_Cons_numeral
thf(fact_1217_last__map,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( F @ ( last_nat @ Xs2 ) ) ) ) ).
% last_map
thf(fact_1218_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_1219_map__equality__iff,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1220_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1221_map__update,axiom,
! [F: nat > nat,Xs2: list_nat,K: nat,Y: nat] :
( ( map_nat_nat @ F @ ( list_update_nat @ Xs2 @ K @ Y ) )
= ( list_update_nat @ ( map_nat_nat @ F @ Xs2 ) @ K @ ( F @ Y ) ) ) ).
% map_update
thf(fact_1222_drop__map,axiom,
! [N: nat,F: nat > nat,Xs2: list_nat] :
( ( drop_nat @ N @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( drop_nat @ N @ Xs2 ) ) ) ).
% drop_map
thf(fact_1223_take__map,axiom,
! [N: nat,F: nat > nat,Xs2: list_nat] :
( ( take_nat @ N @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( take_nat @ N @ Xs2 ) ) ) ).
% take_map
thf(fact_1224_append__eq__map__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,F: nat > nat,Xs2: list_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( map_nat_nat @ F @ Xs2 ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys2
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1225_map__eq__append__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( append_nat @ Ys2 @ Zs ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys2
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1226_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1227_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1228_Cons__eq__map__D,axiom,
! [X: nat,Xs2: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F @ Ys2 ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Ys2
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1229_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys2 ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Xs2
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_1230_Cons__eq__map__conv,axiom,
! [X: nat,Xs2: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F @ Ys2 ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Ys2
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1231_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys2 ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1232_rotate1__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_1233_map__tl,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( map_nat_nat @ F @ ( tl_nat @ Xs2 ) )
= ( tl_nat @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% map_tl
thf(fact_1234_map__concat,axiom,
! [F: nat > nat,Xs2: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs2 ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs2 ) ) ) ).
% map_concat
thf(fact_1235_map__mono__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat,F: nat > nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( prefix_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ F @ Ys2 ) ) ) ).
% map_mono_prefix
thf(fact_1236_prefix__map__rightE,axiom,
! [Xs2: list_nat,F: nat > nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ ( map_nat_nat @ F @ Ys2 ) )
=> ? [Xs4: list_nat] :
( ( prefix_nat @ Xs4 @ Ys2 )
& ( Xs2
= ( map_nat_nat @ F @ Xs4 ) ) ) ) ).
% prefix_map_rightE
thf(fact_1237_ex__map__conv,axiom,
! [Ys2: list_nat,F: nat > nat] :
( ( ? [Xs3: list_nat] :
( Ys2
= ( map_nat_nat @ F @ Xs3 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys2 ) )
=> ? [Y4: nat] :
( X4
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1238_map__cong,axiom,
! [Xs2: list_nat,Ys2: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs2 = Ys2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys2 ) ) ) ) ).
% map_cong
thf(fact_1239_map__idI,axiom,
! [Xs2: list_P6011104703257516679at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_Pr8058819605623181956at_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_1240_map__idI,axiom,
! [Xs2: list_P8469869581646625389at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat] :
( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_Pr7106542193144342532at_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_1241_map__idI,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_1242_map__ext,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_1243_list_Omap__ident__strong,axiom,
! [T: list_P6011104703257516679at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
( ! [Z2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Z2 @ ( set_Pr5648618587558075414at_nat @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_Pr8058819605623181956at_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1244_list_Omap__ident__strong,axiom,
! [T: list_P8469869581646625389at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat] :
( ! [Z2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Z2 @ ( set_Pr5518436109238095868at_nat @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_Pr7106542193144342532at_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1245_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1246_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z2: nat,Za: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1247_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_1248_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1249_list_Omap__sel_I2_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( tl_nat @ ( map_nat_nat @ F @ A ) )
= ( map_nat_nat @ F @ ( tl_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_1250_list_Omap__sel_I1_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ A ) )
= ( F @ ( hd_nat @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1251_hd__map,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( F @ ( hd_nat @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_1252_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1253_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1254_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1255_map__butlast,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs2 ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% map_butlast
thf(fact_1256_nth__map__upt,axiom,
! [I3: nat,N: nat,M: nat,F: nat > nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M @ N ) ) @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) ) ).
% nth_map_upt
thf(fact_1257_Stirling__1,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% Stirling_1
thf(fact_1258_distinct__adj__append__iff,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( ( distinct_adj_nat @ Xs2 )
& ( distinct_adj_nat @ Ys2 )
& ( ( Xs2 = nil_nat )
| ( Ys2 = nil_nat )
| ( ( last_nat @ Xs2 )
!= ( hd_nat @ Ys2 ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_1259_distinct__adj__Cons__Cons,axiom,
! [X: nat,Y: nat,Xs2: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs2 ) ) )
= ( ( X != Y )
& ( distinct_adj_nat @ ( cons_nat @ Y @ Xs2 ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_1260_Stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling @ N @ K )
= zero_zero_nat ) ) ).
% Stirling_less
thf(fact_1261_distinct__set__subseqs,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_set_nat @ ( map_list_nat_set_nat @ set_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ).
% distinct_set_subseqs
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ( if_lis9186351972506106189at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
( ( if_lis9186351972506106189at_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
( ( if_lis7763640049307703347at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
! [X: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
( ( if_lis7763640049307703347at_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ i @ ( size_size_list_nat @ xs ) ) @ ( equiva2048684438135499664of_nat @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ i @ ( size_size_list_nat @ xs ) ) @ ( equiva2048684438135499664of_nat @ ( append_nat @ ys @ ( cons_nat @ ya @ nil_nat ) ) ) ) ) ).
%------------------------------------------------------------------------------