TPTP Problem File: SLH0926^1.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_00046_002352__11722506_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 378 ( 169 unt; 61 typ; 0 def)
% Number of atoms : 845 ( 495 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3451 ( 113 ~; 24 |; 40 &;2699 @)
% ( 0 <=>; 575 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 214 ( 214 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 51 usr; 12 con; 0-3 aty)
% Number of variables : 1042 ( 20 ^; 979 !; 43 ?;1042 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:09:57.941
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (51)
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Oappend_001tf__b,type,
append_b: list_b > list_b > list_b ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_OCons_001tf__b,type,
cons_b: b > list_b > list_b ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_ONil_001tf__b,type,
nil_b: list_b ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__ex1_001tf__b,type,
list_ex1_b: ( b > $o ) > list_b > $o ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__b_J,type,
size_size_list_b: list_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Stirling_OStirling,type,
stirling: nat > nat > nat ).
thf(sy_c_Stirling_Ostirling,type,
stirling2: nat > nat > nat ).
thf(sy_c_Stirling_Ostirling__row,type,
stirling_row: nat > list_nat ).
thf(sy_c_Stirling_Ostirling__row__aux_001t__Nat__Onat,type,
stirling_row_aux_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_v_P,type,
p: list_a > list_b > $o ).
thf(sy_v_x,type,
x: list_a ).
thf(sy_v_y,type,
y: list_b ).
% Relevant facts (311)
thf(fact_0_assms_I1_J,axiom,
( ( size_size_list_a @ x )
= ( size_size_list_b @ y ) ) ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
p @ nil_a @ nil_b ).
% assms(2)
thf(fact_2_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_a] :
( ( size_size_list_a @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_3_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_b] :
( ( size_size_list_b @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_4_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_5_neq__if__length__neq,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
!= ( size_size_list_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_6_neq__if__length__neq,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( ( size_size_list_b @ Xs2 )
!= ( size_size_list_b @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_7_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_8_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_9_size__neq__size__imp__neq,axiom,
! [X: list_b,Y: list_b] :
( ( ( size_size_list_b @ X )
!= ( size_size_list_b @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_10_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_11_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_12_assms_I3_J,axiom,
! [Xs2: list_a,Ys: list_b,X: a,Y: b] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( p @ Xs2 @ Ys )
=> ( p @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) @ ( append_b @ Ys @ ( cons_b @ Y @ nil_b ) ) ) ) ) ).
% assms(3)
thf(fact_13_length__0__conv,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_a ) ) ).
% length_0_conv
thf(fact_14_length__0__conv,axiom,
! [Xs2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_b ) ) ).
% length_0_conv
thf(fact_15_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_16_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_17_list__ex1__simps_I1_J,axiom,
! [P: b > $o] :
~ ( list_ex1_b @ P @ nil_b ) ).
% list_ex1_simps(1)
thf(fact_18_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_19_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_20_list_Osize_I3_J,axiom,
( ( size_size_list_b @ nil_b )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_21_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_22_list__induct2,axiom,
! [Xs2: list_int,Ys: list_int,P: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_int @ nil_int )
=> ( ! [X2: int,Xs: list_int,Y2: int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_23_list__induct2,axiom,
! [Xs2: list_int,Ys: list_a,P: list_int > list_a > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_int @ nil_a )
=> ( ! [X2: int,Xs: list_int,Y2: a,Ys2: list_a] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_24_list__induct2,axiom,
! [Xs2: list_int,Ys: list_b,P: list_int > list_b > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( P @ nil_int @ nil_b )
=> ( ! [X2: int,Xs: list_int,Y2: b,Ys2: list_b] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_25_list__induct2,axiom,
! [Xs2: list_int,Ys: list_nat,P: list_int > list_nat > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_int @ nil_nat )
=> ( ! [X2: int,Xs: list_int,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_26_list__induct2,axiom,
! [Xs2: list_a,Ys: list_int,P: list_a > list_int > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_a @ nil_int )
=> ( ! [X2: a,Xs: list_a,Y2: int,Ys2: list_int] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_27_list__induct2,axiom,
! [Xs2: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs: list_a,Y2: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_28_list__induct2,axiom,
! [Xs2: list_a,Ys: list_b,P: list_a > list_b > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( P @ nil_a @ nil_b )
=> ( ! [X2: a,Xs: list_a,Y2: b,Ys2: list_b] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_29_list__induct2,axiom,
! [Xs2: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs: list_a,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_30_list__induct2,axiom,
! [Xs2: list_b,Ys: list_int,P: list_b > list_int > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_b @ nil_int )
=> ( ! [X2: b,Xs: list_b,Y2: int,Ys2: list_int] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_31_list__induct2,axiom,
! [Xs2: list_b,Ys: list_a,P: list_b > list_a > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_b @ nil_a )
=> ( ! [X2: b,Xs: list_b,Y2: a,Ys2: list_a] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_32_list__induct3,axiom,
! [Xs2: list_int,Ys: list_int,Zs: list_int,P: list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_int )
=> ( ! [X2: int,Xs: list_int,Y2: int,Ys2: list_int,Z: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_33_list__induct3,axiom,
! [Xs2: list_int,Ys: list_int,Zs: list_a,P: list_int > list_int > list_a > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_a )
=> ( ! [X2: int,Xs: list_int,Y2: int,Ys2: list_int,Z: a,Zs2: list_a] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_34_list__induct3,axiom,
! [Xs2: list_int,Ys: list_int,Zs: list_b,P: list_int > list_int > list_b > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_b )
=> ( ! [X2: int,Xs: list_int,Y2: int,Ys2: list_int,Z: b,Zs2: list_b] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_b @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_35_list__induct3,axiom,
! [Xs2: list_int,Ys: list_int,Zs: list_nat,P: list_int > list_int > list_nat > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_nat )
=> ( ! [X2: int,Xs: list_int,Y2: int,Ys2: list_int,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_36_list__induct3,axiom,
! [Xs2: list_int,Ys: list_a,Zs: list_int,P: list_int > list_a > list_int > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_a @ nil_int )
=> ( ! [X2: int,Xs: list_int,Y2: a,Ys2: list_a,Z: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_37_list__induct3,axiom,
! [Xs2: list_int,Ys: list_a,Zs: list_a,P: list_int > list_a > list_a > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_int @ nil_a @ nil_a )
=> ( ! [X2: int,Xs: list_int,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_38_list__induct3,axiom,
! [Xs2: list_int,Ys: list_a,Zs: list_b,P: list_int > list_a > list_b > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P @ nil_int @ nil_a @ nil_b )
=> ( ! [X2: int,Xs: list_int,Y2: a,Ys2: list_a,Z: b,Zs2: list_b] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_b @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_39_list__induct3,axiom,
! [Xs2: list_int,Ys: list_a,Zs: list_nat,P: list_int > list_a > list_nat > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_int @ nil_a @ nil_nat )
=> ( ! [X2: int,Xs: list_int,Y2: a,Ys2: list_a,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_40_list__induct3,axiom,
! [Xs2: list_int,Ys: list_b,Zs: list_int,P: list_int > list_b > list_int > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_b @ nil_int )
=> ( ! [X2: int,Xs: list_int,Y2: b,Ys2: list_b,Z: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_41_list__induct3,axiom,
! [Xs2: list_int,Ys: list_b,Zs: list_a,P: list_int > list_b > list_a > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_int @ nil_b @ nil_a )
=> ( ! [X2: int,Xs: list_int,Y2: b,Ys2: list_b,Z: a,Zs2: list_a] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_42_list__induct4,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_b,Ws: list_int,P: list_b > list_b > list_b > list_int > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_b @ nil_int )
=> ( ! [X2: b,Xs: list_b,Y2: b,Ys2: list_b,Z: b,Zs2: list_b,W: int,Ws2: list_int] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( ( size_size_list_b @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_b @ Z @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_43_list__induct4,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_b,Ws: list_a,P: list_b > list_b > list_b > list_a > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_b @ nil_a )
=> ( ! [X2: b,Xs: list_b,Y2: b,Ys2: list_b,Z: b,Zs2: list_b,W: a,Ws2: list_a] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( ( size_size_list_b @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_b @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_44_list__induct4,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_b,Ws: list_b,P: list_b > list_b > list_b > list_b > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_b @ nil_b )
=> ( ! [X2: b,Xs: list_b,Y2: b,Ys2: list_b,Z: b,Zs2: list_b,W: b,Ws2: list_b] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( ( size_size_list_b @ Zs2 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_b @ Z @ Zs2 ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_45_list__induct4,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_b,Ws: list_nat,P: list_b > list_b > list_b > list_nat > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_b @ nil_nat )
=> ( ! [X2: b,Xs: list_b,Y2: b,Ys2: list_b,Z: b,Zs2: list_b,W: nat,Ws2: list_nat] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( ( size_size_list_b @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_b @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_46_list__induct4,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_nat,Ws: list_int,P: list_b > list_b > list_nat > list_int > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_nat @ nil_int )
=> ( ! [X2: b,Xs: list_b,Y2: b,Ys2: list_b,Z: nat,Zs2: list_nat,W: int,Ws2: list_int] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_47_list__induct4,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_nat,Ws: list_a,P: list_b > list_b > list_nat > list_a > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_nat @ nil_a )
=> ( ! [X2: b,Xs: list_b,Y2: b,Ys2: list_b,Z: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_48_list__induct4,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_nat,Ws: list_b,P: list_b > list_b > list_nat > list_b > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_nat @ nil_b )
=> ( ! [X2: b,Xs: list_b,Y2: b,Ys2: list_b,Z: nat,Zs2: list_nat,W: b,Ws2: list_b] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_b @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_49_list__induct4,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_nat,Ws: list_nat,P: list_b > list_b > list_nat > list_nat > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_b @ nil_b @ nil_nat @ nil_nat )
=> ( ! [X2: b,Xs: list_b,Y2: b,Ys2: list_b,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_b @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_50_list__induct4,axiom,
! [Xs2: list_b,Ys: list_nat,Zs: list_int,Ws: list_int,P: list_b > list_nat > list_int > list_int > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_int @ Ws ) )
=> ( ( P @ nil_b @ nil_nat @ nil_int @ nil_int )
=> ( ! [X2: b,Xs: list_b,Y2: nat,Ys2: list_nat,Z: int,Zs2: list_int,W: int,Ws2: list_int] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_int @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) @ ( cons_int @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_51_list__induct4,axiom,
! [Xs2: list_b,Ys: list_nat,Zs: list_int,Ws: list_a,P: list_b > list_nat > list_int > list_a > $o] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( ( size_size_list_int @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_b @ nil_nat @ nil_int @ nil_a )
=> ( ! [X2: b,Xs: list_b,Y2: nat,Ys2: list_nat,Z: int,Zs2: list_int,W: a,Ws2: list_a] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( ( size_size_list_int @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_b @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_52_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_53_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_54_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_55_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_56_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_57_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_58_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_59_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_60_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_61_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_62_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_63_diff__induct,axiom,
! [P: nat > nat > $o,M2: 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 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_64_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_65_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_66_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_67_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_68_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_69_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_70_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_71_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_72_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_73_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_74_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_75_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_76_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_77_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_78_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_79_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_80_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_81_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_82_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_83_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q ) ) ) ) ).
% less_natE
thf(fact_84_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_85_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_86_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_87_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_88_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_89_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_90_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_91_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_92_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_93_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_94_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_95_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_96_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_97_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_98_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_99_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_100_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_101_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_102_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_103_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_104_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_105_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_106_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_107_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_108_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_109_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_110_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_111_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_112_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_113_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_114_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_115_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_116_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_117_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_118_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_119_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_120_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_121_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_122_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_123_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_124_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_125_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_126_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_127_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_128_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_129_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_130_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_131_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_132_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_133_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_134_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_135_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_136_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_137_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_138_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_139_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_140_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_141_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
= ( P @ B2 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_142_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_143_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z2: int] :
? [N3: nat] :
( Z2
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_144_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_145_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_146_zadd__int__left,axiom,
! [M2: nat,N: nat,Z3: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_147_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_148_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_149_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_150_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_151_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_152_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_153_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_154_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_155_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_156_Stirling__1,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% Stirling_1
thf(fact_157_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_158_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_159_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_160_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_161_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_162_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_163_Stirling__same,axiom,
! [N: nat] :
( ( stirling @ N @ N )
= one_one_nat ) ).
% Stirling_same
thf(fact_164_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_165_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_166_Stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling @ N @ K )
= zero_zero_nat ) ) ).
% Stirling_less
thf(fact_167_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_168_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_169_zless__add1__eq,axiom,
! [W3: int,Z3: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W3 @ Z3 )
| ( W3 = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_170_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_171_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_172_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_173_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_174_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_175_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_176_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_177_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_178_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_179_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_180_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_181_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_182_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_183_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_184_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_185_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_186_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_187_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_188_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_189_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_190_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_191_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_192_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_193_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_194_Stirling_Osimps_I2_J,axiom,
! [K: nat] :
( ( stirling @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% Stirling.simps(2)
thf(fact_195_Stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% Stirling.simps(3)
thf(fact_196_Stirling_Osimps_I1_J,axiom,
( ( stirling @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% Stirling.simps(1)
thf(fact_197_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_198_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 ) )
| ? [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
& ~ ( P @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_199_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 ) )
& ! [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
=> ( P @ D ) ) ) ) ).
% nat_diff_split
thf(fact_200_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_201_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_202_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_203_Stirling_Oelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( stirling @ X @ Xa )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( ( Xa = zero_zero_nat )
=> ( Y != one_one_nat ) ) )
=> ( ( ( X = zero_zero_nat )
=> ( ? [K3: nat] :
( Xa
= ( suc @ K3 ) )
=> ( Y != zero_zero_nat ) ) )
=> ( ( ? [N2: nat] :
( X
= ( suc @ N2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y != zero_zero_nat ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ! [K3: nat] :
( ( Xa
= ( suc @ K3 ) )
=> ( Y
!= ( plus_plus_nat @ ( times_times_nat @ ( suc @ K3 ) @ ( stirling @ N2 @ ( suc @ K3 ) ) ) @ ( stirling @ N2 @ K3 ) ) ) ) ) ) ) ) ) ).
% Stirling.elims
thf(fact_204_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_205_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_206_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_207_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_208_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_209_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_210_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_211_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_212_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_213_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_214_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ M2 @ ( suc @ N ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_215_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_216_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_217_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_218_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_219_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_220_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_221_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_222_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_223_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_224_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_225_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_226_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_227_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_228_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_229_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_230_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_231_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_232_Stirling_Osimps_I4_J,axiom,
! [N: nat,K: nat] :
( ( stirling @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( suc @ K ) @ ( stirling @ N @ ( suc @ K ) ) ) @ ( stirling @ N @ K ) ) ) ).
% Stirling.simps(4)
thf(fact_233_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_234_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_235_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_236_int__distrib_I2_J,axiom,
! [W3: int,Z1: int,Z22: int] :
( ( times_times_int @ W3 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W3 @ Z1 ) @ ( times_times_int @ W3 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_237_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W3: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W3 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W3 ) @ ( times_times_int @ Z22 @ W3 ) ) ) ).
% int_distrib(1)
thf(fact_238_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_239_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_240_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_241_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_242_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_243_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_244_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_245_plusinfinity,axiom,
! [D2: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [X_1: int] : ( P2 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_246_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int,K3: int] :
( ( P1 @ X2 )
= ( P1 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P1 @ X2 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_247_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_248_stirling_Oelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( stirling2 @ X @ Xa )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( ( Xa = zero_zero_nat )
=> ( Y != one_one_nat ) ) )
=> ( ( ( X = zero_zero_nat )
=> ( ? [K3: nat] :
( Xa
= ( suc @ K3 ) )
=> ( Y != zero_zero_nat ) ) )
=> ( ( ? [N2: nat] :
( X
= ( suc @ N2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y != zero_zero_nat ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ! [K3: nat] :
( ( Xa
= ( suc @ K3 ) )
=> ( Y
!= ( plus_plus_nat @ ( times_times_nat @ N2 @ ( stirling2 @ N2 @ ( suc @ K3 ) ) ) @ ( stirling2 @ N2 @ K3 ) ) ) ) ) ) ) ) ) ).
% stirling.elims
thf(fact_249_stirling__same,axiom,
! [N: nat] :
( ( stirling2 @ N @ N )
= one_one_nat ) ).
% stirling_same
thf(fact_250_stirling__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( stirling2 @ N @ zero_zero_nat )
= zero_zero_nat ) ) ).
% stirling_0
thf(fact_251_stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling2 @ N @ K )
= zero_zero_nat ) ) ).
% stirling_less
thf(fact_252_stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling2 @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% stirling.simps(3)
thf(fact_253_stirling_Osimps_I2_J,axiom,
! [K: nat] :
( ( stirling2 @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% stirling.simps(2)
thf(fact_254_stirling_Osimps_I1_J,axiom,
( ( stirling2 @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% stirling.simps(1)
thf(fact_255_stirling_Osimps_I4_J,axiom,
! [N: nat,K: nat] :
( ( stirling2 @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( stirling2 @ N @ ( suc @ K ) ) ) @ ( stirling2 @ N @ K ) ) ) ).
% stirling.simps(4)
thf(fact_256_stirling__code,axiom,
( stirling2
= ( ^ [N3: nat,K2: nat] : ( if_nat @ ( K2 = zero_zero_nat ) @ ( if_nat @ ( N3 = zero_zero_nat ) @ one_one_nat @ zero_zero_nat ) @ ( if_nat @ ( ord_less_nat @ N3 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( K2 = N3 ) @ one_one_nat @ ( nth_nat @ ( stirling_row @ N3 ) @ K2 ) ) ) ) ) ) ).
% stirling_code
thf(fact_257_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_258_stirling__row__nonempty,axiom,
! [N: nat] :
( ( stirling_row @ N )
!= nil_nat ) ).
% stirling_row_nonempty
thf(fact_259_fact__less__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_260_stirling__Suc__n__1,axiom,
! [N: nat] :
( ( stirling2 @ ( suc @ N ) @ ( suc @ zero_zero_nat ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% stirling_Suc_n_1
thf(fact_261_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_262_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_263_incr__lemma,axiom,
! [D2: int,Z3: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ Z3 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z3 ) ) @ one_one_int ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_264_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( abs_abs_int @ Z3 ) @ one_one_int )
= ( Z3 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_265_decr__lemma,axiom,
! [D2: int,X: int,Z3: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z3 ) ) @ one_one_int ) @ D2 ) ) @ Z3 ) ) ).
% decr_lemma
thf(fact_266_cppi,axiom,
! [D3: int,P: int > $o,P2: int > $o,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ( ? [X3: int] : ( P @ X3 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ( P2 @ X4 ) )
| ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ A2 )
& ( P @ ( minus_minus_int @ Y3 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_267_cpmi,axiom,
! [D3: int,P: int > $o,P2: int > $o,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D3 ) ) ) )
=> ( ( ? [X3: int] : ( P @ X3 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ( P2 @ X4 ) )
| ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ? [Y3: int] :
( ( member_int @ Y3 @ B4 )
& ( P @ ( plus_plus_int @ Y3 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_268_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_269_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power_nat @ X @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_270_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_271_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_272_bset_I1_J,axiom,
! [D3: int,B4: set_int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q2 @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
& ( Q2 @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_273_bset_I2_J,axiom,
! [D3: int,B4: set_int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B4 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q2 @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
| ( Q2 @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_274_aset_I1_J,axiom,
! [D3: int,A2: set_int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q2 @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
& ( Q2 @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_275_aset_I2_J,axiom,
! [D3: int,A2: set_int,P: int > $o,Q2: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X2
!= ( minus_minus_int @ Xb @ Xa2 ) ) ) )
=> ( ( Q2 @ X2 )
=> ( Q2 @ ( plus_plus_int @ X2 @ D3 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q2 @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
| ( Q2 @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_276_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_277_bset_I3_J,axiom,
! [D3: int,T: int,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( minus_minus_int @ X5 @ D3 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_278_bset_I4_J,axiom,
! [D3: int,T: int,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( minus_minus_int @ X5 @ D3 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_279_bset_I5_J,axiom,
! [D3: int,B4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_280_bset_I7_J,axiom,
! [D3: int,T: int,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% bset(7)
thf(fact_281_aset_I3_J,axiom,
! [D3: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( plus_plus_int @ X5 @ D3 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_282_aset_I4_J,axiom,
! [D3: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( plus_plus_int @ X5 @ D3 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_283_aset_I5_J,axiom,
! [D3: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_284_aset_I7_J,axiom,
! [D3: int,A2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% aset(7)
thf(fact_285_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int,K3: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ( ? [X3: int] : ( P @ X3 ) )
= ( ? [X4: int] :
( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
& ( P @ X4 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_286_aset_I8_J,axiom,
! [D3: int,A2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% aset(8)
thf(fact_287_aset_I6_J,axiom,
! [D3: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_288_zle__add1__eq__le,axiom,
! [W3: int,Z3: int] :
( ( ord_less_int @ W3 @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W3 @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_289_zle__diff1__eq,axiom,
! [W3: int,Z3: int] :
( ( ord_less_eq_int @ W3 @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W3 @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_290_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_291_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z2: int] :
? [N3: nat] :
( Z2
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_292_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_293_add1__zle__eq,axiom,
! [W3: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z3 )
= ( ord_less_int @ W3 @ Z3 ) ) ).
% add1_zle_eq
thf(fact_294_zless__imp__add1__zle,axiom,
! [W3: int,Z3: int] :
( ( ord_less_int @ W3 @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W3 @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_295_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_296_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_297_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_298_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_299_bset_I6_J,axiom,
! [D3: int,B4: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_300_bset_I8_J,axiom,
! [D3: int,T: int,B4: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb2: int] :
( ( member_int @ Xb2 @ B4 )
=> ( X5
!= ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% bset(8)
thf(fact_301_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_302_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_303_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_304_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_305_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_306_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_307_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_308_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_309_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_310_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
% Helper facts (5)
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_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
p @ x @ y ).
%------------------------------------------------------------------------------