TPTP Problem File: SLH0177^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : FSM_Tests/0056_Prefix_Tree/prob_01700_054572__20580138_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1380 ( 459 unt; 109 typ; 0 def)
% Number of atoms : 3775 (1021 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11744 ( 389 ~; 75 |; 213 &;9120 @)
% ( 0 <=>;1947 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 419 ( 419 >; 0 *; 0 +; 0 <<)
% Number of symbols : 100 ( 97 usr; 19 con; 0-3 aty)
% Number of variables : 3501 ( 100 ^;3250 !; 151 ?;3501 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:29:35.695
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Option__Ooption_It__Prefix____Tree__Oprefix____tree_Itf__a_J_J,type,
option7782433257363429738tree_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Prefix____Tree__Oprefix____tree_Itf__a_J,type,
prefix_prefix_tree_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (97)
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
gbinomial_int: int > nat > int ).
thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: nat > nat > nat ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
semiri1406184849735516958ct_int: nat > int ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
finite_card_int: set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin_001t__Int__Oint,type,
semiri4256215615220890538in_int: set_int > int ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin_001t__Nat__Onat,type,
semiri4258706085729940814in_nat: set_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_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
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_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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
ring_1_Ints_int: set_int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Int__Oint,type,
lattic8443796201974363763nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Nat__Onat,type,
lattic8446286672483414039nt_nat: ( int > nat ) > set_int > int ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Int__Oint,type,
lattic7444442490073309207at_int: ( nat > int ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
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_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Prefix____Tree__Oprefix____tree_Itf__a_J,type,
size_s5139796252398215440tree_a: prefix_prefix_tree_a > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_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_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Prefix__Tree_Oafter_001tf__a,type,
prefix_after_a: prefix_prefix_tree_a > list_a > prefix_prefix_tree_a ).
thf(sy_c_Prefix__Tree_Ocombine_001tf__a,type,
prefix_combine_a: prefix_prefix_tree_a > prefix_prefix_tree_a > prefix_prefix_tree_a ).
thf(sy_c_Prefix__Tree_Ocombine__after_001tf__a,type,
prefix3285631374902996868fter_a: prefix_prefix_tree_a > list_a > prefix_prefix_tree_a > prefix_prefix_tree_a ).
thf(sy_c_Prefix__Tree_Oempty_001tf__a,type,
prefix_empty_a: prefix_prefix_tree_a ).
thf(sy_c_Prefix__Tree_Ofinite__tree_001tf__a,type,
prefix_finite_tree_a: prefix_prefix_tree_a > $o ).
thf(sy_c_Prefix__Tree_Ofrom__list_001tf__a,type,
prefix_from_list_a: list_list_a > prefix_prefix_tree_a ).
thf(sy_c_Prefix__Tree_Oheight_001tf__a,type,
prefix_height_a: prefix_prefix_tree_a > nat ).
thf(sy_c_Prefix__Tree_Oinsert_001tf__a,type,
prefix_insert_a: prefix_prefix_tree_a > list_a > prefix_prefix_tree_a ).
thf(sy_c_Prefix__Tree_Oisin_001tf__a,type,
prefix_isin_a: prefix_prefix_tree_a > list_a > $o ).
thf(sy_c_Prefix__Tree_Oprefix__tree_OPT_001tf__a,type,
prefix_prefix_PT_a: ( a > option7782433257363429738tree_a ) > prefix_prefix_tree_a ).
thf(sy_c_Prefix__Tree_Oprefix__tree_Osize__prefix__tree_001tf__a,type,
prefix5161986780453196784tree_a: ( a > nat ) > prefix_prefix_tree_a > nat ).
thf(sy_c_Prefix__Tree_Oset_001tf__a,type,
prefix_set_a: prefix_prefix_tree_a > set_list_a ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Int__Oint_J,type,
set_or4447831506222458678et_int: set_int > set_int > set_set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or8625682525731655386et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_k____,type,
k: nat ).
thf(sy_v_m1____,type,
m1: a > option7782433257363429738tree_a ).
thf(sy_v_m2____,type,
m2: a > option7782433257363429738tree_a ).
thf(sy_v_t1,type,
t1: prefix_prefix_tree_a ).
thf(sy_v_t1a____,type,
t1a: prefix_prefix_tree_a ).
thf(sy_v_t2,type,
t2: prefix_prefix_tree_a ).
thf(sy_v_t2a____,type,
t2a: prefix_prefix_tree_a ).
thf(sy_v_thesis____,type,
thesis: $o ).
% Relevant facts (1266)
thf(fact_0__092_060open_062m1_A_092_060noteq_062_Am2_092_060close_062,axiom,
m1 != m2 ).
% \<open>m1 \<noteq> m2\<close>
thf(fact_1__092_060open_062t2_A_061_APT_Am2_092_060close_062,axiom,
( t2a
= ( prefix_prefix_PT_a @ m2 ) ) ).
% \<open>t2 = PT m2\<close>
thf(fact_2_assms_I2_J,axiom,
prefix_finite_tree_a @ t1 ).
% assms(2)
thf(fact_3__092_060open_062t1_A_061_APT_Am1_092_060close_062,axiom,
( t1a
= ( prefix_prefix_PT_a @ m1 ) ) ).
% \<open>t1 = PT m1\<close>
thf(fact_4__092_060open_062t1_A_092_060noteq_062_At2_092_060close_062,axiom,
t1a != t2a ).
% \<open>t1 \<noteq> t2\<close>
thf(fact_5_assms_I1_J,axiom,
( ( prefix_set_a @ t1 )
= ( prefix_set_a @ t2 ) ) ).
% assms(1)
thf(fact_6_less_Oprems_I2_J,axiom,
prefix_finite_tree_a @ t1a ).
% less.prems(2)
thf(fact_7_less_Oprems_I1_J,axiom,
( ( prefix_set_a @ t1a )
= ( prefix_set_a @ t2a ) ) ).
% less.prems(1)
thf(fact_8_prefix__tree_Oinject,axiom,
! [X: a > option7782433257363429738tree_a,Ya: a > option7782433257363429738tree_a] :
( ( ( prefix_prefix_PT_a @ X )
= ( prefix_prefix_PT_a @ Ya ) )
= ( X = Ya ) ) ).
% prefix_tree.inject
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m1_Am2_O_A_092_060lbrakk_062t1_A_061_APT_Am1_059_At2_A_061_APT_Am2_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ? [M1: a > option7782433257363429738tree_a] :
( t1a
= ( prefix_prefix_PT_a @ M1 ) )
=> ! [M2: a > option7782433257363429738tree_a] :
( t2a
!= ( prefix_prefix_PT_a @ M2 ) ) ) ).
% \<open>\<And>thesis. (\<And>m1 m2. \<lbrakk>t1 = PT m1; t2 = PT m2\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_10_finite__tree_Ocases,axiom,
! [X: prefix_prefix_tree_a] :
~ ! [M: a > option7782433257363429738tree_a] :
( X
!= ( prefix_prefix_PT_a @ M ) ) ).
% finite_tree.cases
thf(fact_11_prefix__tree_Oexhaust,axiom,
! [Y: prefix_prefix_tree_a] :
~ ! [X2: a > option7782433257363429738tree_a] :
( Y
!= ( prefix_prefix_PT_a @ X2 ) ) ).
% prefix_tree.exhaust
thf(fact_12_less_Ohyps,axiom,
! [T1: prefix_prefix_tree_a,T2: prefix_prefix_tree_a] :
( ( ord_less_nat @ ( prefix_height_a @ T1 ) @ ( prefix_height_a @ t1a ) )
=> ( ( ( prefix_set_a @ T1 )
= ( prefix_set_a @ T2 ) )
=> ( ( prefix_finite_tree_a @ T1 )
=> ( T1 = T2 ) ) ) ) ).
% less.hyps
thf(fact_13_Suc,axiom,
( ( prefix_height_a @ t1a )
= ( suc @ k ) ) ).
% Suc
thf(fact_14_from__list__finite__tree,axiom,
! [Xs: list_list_a] : ( prefix_finite_tree_a @ ( prefix_from_list_a @ Xs ) ) ).
% from_list_finite_tree
thf(fact_15_finite__tree__iff,axiom,
( prefix_finite_tree_a
= ( ^ [T: prefix_prefix_tree_a] : ( finite_finite_list_a @ ( prefix_set_a @ T ) ) ) ) ).
% finite_tree_iff
thf(fact_16_combine__after__finite__tree,axiom,
! [T1: prefix_prefix_tree_a,T2: prefix_prefix_tree_a,Alpha: list_a] :
( ( prefix_finite_tree_a @ T1 )
=> ( ( prefix_finite_tree_a @ T2 )
=> ( prefix_finite_tree_a @ ( prefix3285631374902996868fter_a @ T1 @ Alpha @ T2 ) ) ) ) ).
% combine_after_finite_tree
thf(fact_17_combine__finite__tree,axiom,
! [T1: prefix_prefix_tree_a,T2: prefix_prefix_tree_a] :
( ( prefix_finite_tree_a @ T1 )
=> ( ( prefix_finite_tree_a @ T2 )
=> ( prefix_finite_tree_a @ ( prefix_combine_a @ T1 @ T2 ) ) ) ) ).
% combine_finite_tree
thf(fact_18_insert__finite__tree,axiom,
! [T3: prefix_prefix_tree_a,Xs: list_a] :
( ( prefix_finite_tree_a @ T3 )
=> ( prefix_finite_tree_a @ ( prefix_insert_a @ T3 @ Xs ) ) ) ).
% insert_finite_tree
thf(fact_19_insert__isin,axiom,
! [Xs: list_a,T3: prefix_prefix_tree_a] : ( member_list_a @ Xs @ ( prefix_set_a @ ( prefix_insert_a @ T3 @ Xs ) ) ) ).
% insert_isin
thf(fact_20_empty__finite__tree,axiom,
prefix_finite_tree_a @ prefix_empty_a ).
% empty_finite_tree
thf(fact_21_set_Oelims,axiom,
! [X: prefix_prefix_tree_a,Y: set_list_a] :
( ( ( prefix_set_a @ X )
= Y )
=> ( Y
= ( collect_list_a @ ( prefix_isin_a @ X ) ) ) ) ).
% set.elims
thf(fact_22_set_Osimps,axiom,
( prefix_set_a
= ( ^ [T: prefix_prefix_tree_a] : ( collect_list_a @ ( prefix_isin_a @ T ) ) ) ) ).
% set.simps
thf(fact_23_insert__isin__other,axiom,
! [T3: prefix_prefix_tree_a,Xs: list_a,Xs2: list_a] :
( ( prefix_isin_a @ T3 @ Xs )
=> ( prefix_isin_a @ ( prefix_insert_a @ T3 @ Xs2 ) @ Xs ) ) ).
% insert_isin_other
thf(fact_24_Suc__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_less_eq
thf(fact_25_Suc__mono,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_26_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_27_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_int,N: nat,M3: nat] :
( ! [N2: nat] : ( ord_less_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_set_int @ ( F @ N ) @ ( F @ M3 ) )
= ( ord_less_nat @ N @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_28_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_nat,N: nat,M3: nat] :
( ! [N2: nat] : ( ord_less_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_set_nat @ ( F @ N ) @ ( F @ M3 ) )
= ( ord_less_nat @ N @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_29_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M3 ) )
= ( ord_less_nat @ N @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_30_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M3 ) )
= ( ord_less_nat @ N @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_31_lift__Suc__mono__less,axiom,
! [F: nat > set_int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_set_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_32_lift__Suc__mono__less,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_set_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_33_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_34_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_35_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_36_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_37_not__less__less__Suc__eq,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less_nat @ N @ M3 )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_38_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_39_finite__psubset__induct,axiom,
! [A: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A )
=> ( ! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ! [B: set_list_a] :
( ( ord_less_set_list_a @ B @ A2 )
=> ( P @ B ) )
=> ( P @ A2 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_40_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( P @ B ) )
=> ( P @ A2 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_41_finite__psubset__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ! [B: set_int] :
( ( ord_less_set_int @ B @ A2 )
=> ( P @ B ) )
=> ( P @ A2 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_42_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_43_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_44_nat__neq__iff,axiom,
! [M3: nat,N: nat] :
( ( M3 != N )
= ( ( ord_less_nat @ M3 @ N )
| ( ord_less_nat @ N @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_45_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A3: int,P: int > $o] :
( ( member_int @ A3 @ ( collect_int @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A3: list_a,P: list_a > $o] :
( ( member_list_a @ A3 @ ( collect_list_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_51_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X2: list_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_52_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_53_less__not__refl2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( M3 != N ) ) ).
% less_not_refl2
thf(fact_54_less__not__refl3,axiom,
! [S: nat,T3: nat] :
( ( ord_less_nat @ S @ T3 )
=> ( S != T3 ) ) ).
% less_not_refl3
thf(fact_55_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_56_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_57_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_58_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_59_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_60_Suc__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_lessD
thf(fact_61_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_62_Suc__lessI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ( suc @ M3 )
!= N )
=> ( ord_less_nat @ ( suc @ M3 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_63_less__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M3 @ N )
=> ( M3 = N ) ) ) ).
% less_SucE
thf(fact_64_less__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_65_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_66_less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) ) ) ).
% less_Suc_eq
thf(fact_67_not__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_nat @ M3 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_68_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_69_Suc__less__eq2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M3 )
= ( ? [M5: nat] :
( ( M3
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_70_less__antisym,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less_nat @ N @ M3 )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
=> ( M3 = N ) ) ) ).
% less_antisym
thf(fact_71_Suc__less__SucD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_less_SucD
thf(fact_72_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_73_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,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_74_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_75_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M6: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_76_minf_I7_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ~ ( ord_less_nat @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_77_minf_I7_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ~ ( ord_less_int @ T3 @ X4 ) ) ).
% minf(7)
thf(fact_78_minf_I5_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ord_less_nat @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_79_minf_I5_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ord_less_int @ X4 @ T3 ) ) ).
% minf(5)
thf(fact_80_minf_I4_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_81_minf_I4_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( X4 != T3 ) ) ).
% minf(4)
thf(fact_82_minf_I3_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_83_minf_I3_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( X4 != T3 ) ) ).
% minf(3)
thf(fact_84_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_85_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_86_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_87_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_88_pinf_I7_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ord_less_nat @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_89_pinf_I7_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ord_less_int @ T3 @ X4 ) ) ).
% pinf(7)
thf(fact_90_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_91_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_92_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_93_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_94_pinf_I3_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_95_pinf_I3_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(3)
thf(fact_96_pinf_I4_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_97_pinf_I4_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( X4 != T3 ) ) ).
% pinf(4)
thf(fact_98_pinf_I5_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ~ ( ord_less_nat @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_99_pinf_I5_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ~ ( ord_less_int @ X4 @ T3 ) ) ).
% pinf(5)
thf(fact_100_psubset__trans,axiom,
! [A: set_int,B2: set_int,C: set_int] :
( ( ord_less_set_int @ A @ B2 )
=> ( ( ord_less_set_int @ B2 @ C )
=> ( ord_less_set_int @ A @ C ) ) ) ).
% psubset_trans
thf(fact_101_psubset__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% psubset_trans
thf(fact_102_psubsetD,axiom,
! [A: set_list_a,B2: set_list_a,C2: list_a] :
( ( ord_less_set_list_a @ A @ B2 )
=> ( ( member_list_a @ C2 @ A )
=> ( member_list_a @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_103_psubsetD,axiom,
! [A: set_int,B2: set_int,C2: int] :
( ( ord_less_set_int @ A @ B2 )
=> ( ( member_int @ C2 @ A )
=> ( member_int @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_104_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_105_order__less__imp__not__less,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ~ ( ord_less_set_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_106_order__less__imp__not__less,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_107_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_108_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_109_order__less__imp__not__eq2,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_110_order__less__imp__not__eq2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_111_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_112_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_113_order__less__imp__not__eq,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_114_order__less__imp__not__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_115_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_116_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_117_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_118_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_119_order__less__imp__triv,axiom,
! [X: set_int,Y: set_int,P: $o] :
( ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_set_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_120_order__less__imp__triv,axiom,
! [X: set_nat,Y: set_nat,P: $o] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_121_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_122_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_123_order__less__not__sym,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ~ ( ord_less_set_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_124_order__less__not__sym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_125_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_126_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_127_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_128_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_129_order__less__subst2,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_130_order__less__subst2,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_131_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > set_int,C2: set_int] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_set_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_132_order__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_133_order__less__subst2,axiom,
! [A3: int,B3: int,F: int > set_int,C2: set_int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_set_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_134_order__less__subst2,axiom,
! [A3: int,B3: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_set_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_135_order__less__subst2,axiom,
! [A3: set_int,B3: set_int,F: set_int > nat,C2: nat] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_set_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_136_order__less__subst2,axiom,
! [A3: set_int,B3: set_int,F: set_int > int,C2: int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_set_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_137_order__less__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_138_order__less__subst1,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_139_order__less__subst1,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_140_order__less__subst1,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_141_order__less__subst1,axiom,
! [A3: nat,F: set_int > nat,B3: set_int,C2: set_int] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_set_int @ B3 @ C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_set_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_142_order__less__subst1,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C2: set_nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_set_nat @ B3 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_143_order__less__subst1,axiom,
! [A3: int,F: set_int > int,B3: set_int,C2: set_int] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_set_int @ B3 @ C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_set_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_144_order__less__subst1,axiom,
! [A3: int,F: set_nat > int,B3: set_nat,C2: set_nat] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_set_nat @ B3 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_145_order__less__subst1,axiom,
! [A3: set_int,F: nat > set_int,B3: nat,C2: nat] :
( ( ord_less_set_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_146_order__less__subst1,axiom,
! [A3: set_int,F: int > set_int,B3: int,C2: int] :
( ( ord_less_set_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_147_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_148_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_149_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_150_less__imp__neq,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_151_less__imp__neq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_152_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_153_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_154_order_Oasym,axiom,
! [A3: set_int,B3: set_int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ~ ( ord_less_set_int @ B3 @ A3 ) ) ).
% order.asym
thf(fact_155_order_Oasym,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ~ ( ord_less_set_nat @ B3 @ A3 ) ) ).
% order.asym
thf(fact_156_order_Oasym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order.asym
thf(fact_157_order_Oasym,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ~ ( ord_less_int @ B3 @ A3 ) ) ).
% order.asym
thf(fact_158_ord__eq__less__trans,axiom,
! [A3: set_int,B3: set_int,C2: set_int] :
( ( A3 = B3 )
=> ( ( ord_less_set_int @ B3 @ C2 )
=> ( ord_less_set_int @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_159_ord__eq__less__trans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( A3 = B3 )
=> ( ( ord_less_set_nat @ B3 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_160_ord__eq__less__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( A3 = B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_161_ord__eq__less__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( A3 = B3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_162_ord__less__eq__trans,axiom,
! [A3: set_int,B3: set_int,C2: set_int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_set_int @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_163_ord__less__eq__trans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_164_ord__less__eq__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_165_ord__less__eq__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_166_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_167_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_168_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_169_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_170_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_171_dual__order_Oasym,axiom,
! [B3: set_int,A3: set_int] :
( ( ord_less_set_int @ B3 @ A3 )
=> ~ ( ord_less_set_int @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_172_dual__order_Oasym,axiom,
! [B3: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
=> ~ ( ord_less_set_nat @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_173_dual__order_Oasym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_174_dual__order_Oasym,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ~ ( ord_less_int @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_175_dual__order_Oirrefl,axiom,
! [A3: set_int] :
~ ( ord_less_set_int @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_176_dual__order_Oirrefl,axiom,
! [A3: set_nat] :
~ ( ord_less_set_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_177_dual__order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_178_dual__order_Oirrefl,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% dual_order.irrefl
thf(fact_179_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N6: nat] :
( ( P4 @ N6 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N6 )
=> ~ ( P4 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_180_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_181_linorder__less__wlog,axiom,
! [P: int > int > $o,A3: int,B3: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_182_order_Ostrict__trans,axiom,
! [A3: set_int,B3: set_int,C2: set_int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( ( ord_less_set_int @ B3 @ C2 )
=> ( ord_less_set_int @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_183_order_Ostrict__trans,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C2 )
=> ( ord_less_set_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_184_order_Ostrict__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_185_order_Ostrict__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_186_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_187_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_188_dual__order_Ostrict__trans,axiom,
! [B3: set_int,A3: set_int,C2: set_int] :
( ( ord_less_set_int @ B3 @ A3 )
=> ( ( ord_less_set_int @ C2 @ B3 )
=> ( ord_less_set_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_189_dual__order_Ostrict__trans,axiom,
! [B3: set_nat,A3: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
=> ( ( ord_less_set_nat @ C2 @ B3 )
=> ( ord_less_set_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_190_dual__order_Ostrict__trans,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_191_dual__order_Ostrict__trans,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_int @ C2 @ B3 )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_192_order_Ostrict__implies__not__eq,axiom,
! [A3: set_int,B3: set_int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_193_order_Ostrict__implies__not__eq,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_194_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_195_order_Ostrict__implies__not__eq,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_196_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: set_int,A3: set_int] :
( ( ord_less_set_int @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_197_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_198_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_199_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_200_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_201_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_202_order__less__asym,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ~ ( ord_less_set_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_203_order__less__asym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_204_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_205_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_206_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_207_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_208_order__less__asym_H,axiom,
! [A3: set_int,B3: set_int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ~ ( ord_less_set_int @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_209_order__less__asym_H,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ~ ( ord_less_set_nat @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_210_order__less__asym_H,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_211_order__less__asym_H,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ~ ( ord_less_int @ B3 @ A3 ) ) ).
% order_less_asym'
thf(fact_212_order__less__trans,axiom,
! [X: set_int,Y: set_int,Z3: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_set_int @ Y @ Z3 )
=> ( ord_less_set_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_213_order__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z3 )
=> ( ord_less_set_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_214_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_215_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_216_ord__eq__less__subst,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_217_ord__eq__less__subst,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_218_ord__eq__less__subst,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_219_ord__eq__less__subst,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_220_ord__eq__less__subst,axiom,
! [A3: set_int,F: nat > set_int,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_221_ord__eq__less__subst,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_222_ord__eq__less__subst,axiom,
! [A3: set_int,F: int > set_int,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_223_ord__eq__less__subst,axiom,
! [A3: set_nat,F: int > set_nat,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_224_ord__eq__less__subst,axiom,
! [A3: nat,F: set_int > nat,B3: set_int,C2: set_int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_set_int @ B3 @ C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_set_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_225_ord__eq__less__subst,axiom,
! [A3: int,F: set_int > int,B3: set_int,C2: set_int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_set_int @ B3 @ C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_set_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_226_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_227_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_228_ord__less__eq__subst,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_229_ord__less__eq__subst,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_230_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > set_int,C2: set_int] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_231_ord__less__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_232_ord__less__eq__subst,axiom,
! [A3: int,B3: int,F: int > set_int,C2: set_int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_233_ord__less__eq__subst,axiom,
! [A3: int,B3: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_234_ord__less__eq__subst,axiom,
! [A3: set_int,B3: set_int,F: set_int > nat,C2: nat] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_set_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_235_ord__less__eq__subst,axiom,
! [A3: set_int,B3: set_int,F: set_int > int,C2: int] :
( ( ord_less_set_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_set_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_236_order__less__irrefl,axiom,
! [X: set_int] :
~ ( ord_less_set_int @ X @ X ) ).
% order_less_irrefl
thf(fact_237_order__less__irrefl,axiom,
! [X: set_nat] :
~ ( ord_less_set_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_238_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_239_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_240_verit__comp__simplify1_I1_J,axiom,
! [A3: set_int] :
~ ( ord_less_set_int @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_241_verit__comp__simplify1_I1_J,axiom,
! [A3: set_nat] :
~ ( ord_less_set_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_242_verit__comp__simplify1_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_243_verit__comp__simplify1_I1_J,axiom,
! [A3: int] :
~ ( ord_less_int @ A3 @ A3 ) ).
% verit_comp_simplify1(1)
thf(fact_244_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_245_combine__after_Osimps_I1_J,axiom,
! [T1: prefix_prefix_tree_a,T2: prefix_prefix_tree_a] :
( ( prefix3285631374902996868fter_a @ T1 @ nil_a @ T2 )
= ( prefix_combine_a @ T1 @ T2 ) ) ).
% combine_after.simps(1)
thf(fact_246_psubset__card__mono,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_less_set_list_a @ A @ B2 )
=> ( ord_less_nat @ ( finite_card_list_a @ A ) @ ( finite_card_list_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_247_psubset__card__mono,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_248_psubset__card__mono,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_set_int @ A @ B2 )
=> ( ord_less_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_249_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_250_greaterThanLessThan__iff,axiom,
! [I: set_int,L: set_int,U: set_int] :
( ( member_set_int @ I @ ( set_or4447831506222458678et_int @ L @ U ) )
= ( ( ord_less_set_int @ L @ I )
& ( ord_less_set_int @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_251_greaterThanLessThan__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or8625682525731655386et_nat @ L @ U ) )
= ( ( ord_less_set_nat @ L @ I )
& ( ord_less_set_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_252_greaterThanLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or5832277885323065728an_int @ L @ U ) )
= ( ( ord_less_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_253_greaterThanLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_254_set__Nil,axiom,
! [T3: prefix_prefix_tree_a] : ( member_list_a @ nil_a @ ( prefix_set_a @ T3 ) ) ).
% set_Nil
thf(fact_255_isin_Osimps_I1_J,axiom,
! [T3: prefix_prefix_tree_a] : ( prefix_isin_a @ T3 @ nil_a ) ).
% isin.simps(1)
thf(fact_256_insert_Osimps_I1_J,axiom,
! [T3: prefix_prefix_tree_a] :
( ( prefix_insert_a @ T3 @ nil_a )
= T3 ) ).
% insert.simps(1)
thf(fact_257_card__psubset,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_less_nat @ ( finite_card_list_a @ A ) @ ( finite_card_list_a @ B2 ) )
=> ( ord_less_set_list_a @ A @ B2 ) ) ) ) ).
% card_psubset
thf(fact_258_card__psubset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ) ).
% card_psubset
thf(fact_259_card__psubset,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( ord_less_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B2 ) )
=> ( ord_less_set_int @ A @ B2 ) ) ) ) ).
% card_psubset
thf(fact_260_card__insert__disjoint,axiom,
! [A: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A )
=> ( ~ ( member_list_a @ X @ A )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A ) )
= ( suc @ ( finite_card_list_a @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_261_card__insert__disjoint,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_262_card__insert__disjoint,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( finite_card_int @ ( insert_int @ X @ A ) )
= ( suc @ ( finite_card_int @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_263_card__ge__0__finite,axiom,
! [A: set_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_a @ A ) )
=> ( finite_finite_list_a @ A ) ) ).
% card_ge_0_finite
thf(fact_264_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_265_card__ge__0__finite,axiom,
! [A: set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
=> ( finite_finite_int @ A ) ) ).
% card_ge_0_finite
thf(fact_266_after__set__Cons,axiom,
! [Gamma: list_a,T4: prefix_prefix_tree_a,Alpha: list_a] :
( ( member_list_a @ Gamma @ ( prefix_set_a @ ( prefix_after_a @ T4 @ Alpha ) ) )
=> ( ( Gamma != nil_a )
=> ( member_list_a @ Alpha @ ( prefix_set_a @ T4 ) ) ) ) ).
% after_set_Cons
thf(fact_267_card__less__sym__Diff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_list_a @ A ) @ ( finite_card_list_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ B2 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_268_card__less__sym__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_269_card__less__sym__Diff,axiom,
! [A: set_int,B2: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B2 )
=> ( ( ord_less_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B2 ) )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_270_card__Suc__eq__finite,axiom,
! [A: set_list_a,K: nat] :
( ( ( finite_card_list_a @ A )
= ( suc @ K ) )
= ( ? [B5: list_a,B6: set_list_a] :
( ( A
= ( insert_list_a @ B5 @ B6 ) )
& ~ ( member_list_a @ B5 @ B6 )
& ( ( finite_card_list_a @ B6 )
= K )
& ( finite_finite_list_a @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_271_card__Suc__eq__finite,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B5: nat,B6: set_nat] :
( ( A
= ( insert_nat @ B5 @ B6 ) )
& ~ ( member_nat @ B5 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( finite_finite_nat @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_272_card__Suc__eq__finite,axiom,
! [A: set_int,K: nat] :
( ( ( finite_card_int @ A )
= ( suc @ K ) )
= ( ? [B5: int,B6: set_int] :
( ( A
= ( insert_int @ B5 @ B6 ) )
& ~ ( member_int @ B5 @ B6 )
& ( ( finite_card_int @ B6 )
= K )
& ( finite_finite_int @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_273_card__insert__if,axiom,
! [A: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A )
=> ( ( ( member_list_a @ X @ A )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A ) )
= ( finite_card_list_a @ A ) ) )
& ( ~ ( member_list_a @ X @ A )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A ) )
= ( suc @ ( finite_card_list_a @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_274_card__insert__if,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( finite_card_nat @ A ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_275_card__insert__if,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( ( member_int @ X @ A )
=> ( ( finite_card_int @ ( insert_int @ X @ A ) )
= ( finite_card_int @ A ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( finite_card_int @ ( insert_int @ X @ A ) )
= ( suc @ ( finite_card_int @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_276_card_Oinfinite,axiom,
! [A: set_list_a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_card_list_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_277_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_278_card_Oinfinite,axiom,
! [A: set_int] :
( ~ ( finite_finite_int @ A )
=> ( ( finite_card_int @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_279_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_280_dual__order_Orefl,axiom,
! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_281_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_282_dual__order_Orefl,axiom,
! [A3: set_list_a] : ( ord_le8861187494160871172list_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_283_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_284_dual__order_Orefl,axiom,
! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_285_order__refl,axiom,
! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% order_refl
thf(fact_286_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_287_order__refl,axiom,
! [X: set_list_a] : ( ord_le8861187494160871172list_a @ X @ X ) ).
% order_refl
thf(fact_288_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_289_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_290_subset__antisym,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_291_subset__antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_292_subset__antisym,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_293_subsetI,axiom,
! [A: set_int,B2: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ X2 @ B2 ) )
=> ( ord_less_eq_set_int @ A @ B2 ) ) ).
% subsetI
thf(fact_294_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_295_subsetI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% subsetI
thf(fact_296_insert__absorb2,axiom,
! [X: int,A: set_int] :
( ( insert_int @ X @ ( insert_int @ X @ A ) )
= ( insert_int @ X @ A ) ) ).
% insert_absorb2
thf(fact_297_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
= ( insert_nat @ X @ A ) ) ).
% insert_absorb2
thf(fact_298_insert__absorb2,axiom,
! [X: list_a,A: set_list_a] :
( ( insert_list_a @ X @ ( insert_list_a @ X @ A ) )
= ( insert_list_a @ X @ A ) ) ).
% insert_absorb2
thf(fact_299_insert__iff,axiom,
! [A3: list_a,B3: list_a,A: set_list_a] :
( ( member_list_a @ A3 @ ( insert_list_a @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_list_a @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_300_insert__iff,axiom,
! [A3: nat,B3: nat,A: set_nat] :
( ( member_nat @ A3 @ ( insert_nat @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_nat @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_301_insert__iff,axiom,
! [A3: int,B3: int,A: set_int] :
( ( member_int @ A3 @ ( insert_int @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_int @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_302_insertCI,axiom,
! [A3: list_a,B2: set_list_a,B3: list_a] :
( ( ~ ( member_list_a @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member_list_a @ A3 @ ( insert_list_a @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_303_insertCI,axiom,
! [A3: nat,B2: set_nat,B3: nat] :
( ( ~ ( member_nat @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member_nat @ A3 @ ( insert_nat @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_304_insertCI,axiom,
! [A3: int,B2: set_int,B3: int] :
( ( ~ ( member_int @ A3 @ B2 )
=> ( A3 = B3 ) )
=> ( member_int @ A3 @ ( insert_int @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_305_Diff__idemp,axiom,
! [A: set_int,B2: set_int] :
( ( minus_minus_set_int @ ( minus_minus_set_int @ A @ B2 ) @ B2 )
= ( minus_minus_set_int @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_306_Diff__idemp,axiom,
! [A: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_307_Diff__iff,axiom,
! [C2: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( ( member_list_a @ C2 @ A )
& ~ ( member_list_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_308_Diff__iff,axiom,
! [C2: int,A: set_int,B2: set_int] :
( ( member_int @ C2 @ ( minus_minus_set_int @ A @ B2 ) )
= ( ( member_int @ C2 @ A )
& ~ ( member_int @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_309_Diff__iff,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C2 @ A )
& ~ ( member_nat @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_310_DiffI,axiom,
! [C2: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C2 @ A )
=> ( ~ ( member_list_a @ C2 @ B2 )
=> ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_311_DiffI,axiom,
! [C2: int,A: set_int,B2: set_int] :
( ( member_int @ C2 @ A )
=> ( ~ ( member_int @ C2 @ B2 )
=> ( member_int @ C2 @ ( minus_minus_set_int @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_312_DiffI,axiom,
! [C2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ~ ( member_nat @ C2 @ B2 )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_313_finite__insert,axiom,
! [A3: list_a,A: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A3 @ A ) )
= ( finite_finite_list_a @ A ) ) ).
% finite_insert
thf(fact_314_finite__insert,axiom,
! [A3: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A3 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_315_finite__insert,axiom,
! [A3: int,A: set_int] :
( ( finite_finite_int @ ( insert_int @ A3 @ A ) )
= ( finite_finite_int @ A ) ) ).
% finite_insert
thf(fact_316_insert__subset,axiom,
! [X: int,A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( insert_int @ X @ A ) @ B2 )
= ( ( member_int @ X @ B2 )
& ( ord_less_eq_set_int @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_317_insert__subset,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_318_insert__subset,axiom,
! [X: list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X @ A ) @ B2 )
= ( ( member_list_a @ X @ B2 )
& ( ord_le8861187494160871172list_a @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_319_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_320_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_321_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_322_finite__Diff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_323_finite__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_324_finite__Diff,axiom,
! [A: set_int,B2: set_int] :
( ( finite_finite_int @ A )
=> ( finite_finite_int @ ( minus_minus_set_int @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_325_finite__Diff2,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( finite_finite_list_a @ A ) ) ) ).
% finite_Diff2
thf(fact_326_finite__Diff2,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_327_finite__Diff2,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( finite_finite_int @ ( minus_minus_set_int @ A @ B2 ) )
= ( finite_finite_int @ A ) ) ) ).
% finite_Diff2
thf(fact_328_insert__Diff1,axiom,
! [X: list_a,B2: set_list_a,A: set_list_a] :
( ( member_list_a @ X @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A ) @ B2 )
= ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_329_insert__Diff1,axiom,
! [X: int,B2: set_int,A: set_int] :
( ( member_int @ X @ B2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B2 )
= ( minus_minus_set_int @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_330_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_331_Diff__insert0,axiom,
! [X: list_a,A: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X @ A )
=> ( ( minus_646659088055828811list_a @ A @ ( insert_list_a @ X @ B2 ) )
= ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_332_Diff__insert0,axiom,
! [X: int,A: set_int,B2: set_int] :
( ~ ( member_int @ X @ A )
=> ( ( minus_minus_set_int @ A @ ( insert_int @ X @ B2 ) )
= ( minus_minus_set_int @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_333_Diff__insert0,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B2 ) )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_334_psubsetI,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_int @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_335_psubsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_336_psubsetI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_list_a @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_337_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_338_finite__Diff__insert,axiom,
! [A: set_list_a,A3: list_a,B2: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A3 @ B2 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_339_finite__Diff__insert,axiom,
! [A: set_nat,A3: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_340_finite__Diff__insert,axiom,
! [A: set_int,A3: int,B2: set_int] :
( ( finite_finite_int @ ( minus_minus_set_int @ A @ ( insert_int @ A3 @ B2 ) ) )
= ( finite_finite_int @ ( minus_minus_set_int @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_341_lift__Suc__mono__le,axiom,
! [F: nat > set_int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_342_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_343_lift__Suc__mono__le,axiom,
! [F: nat > set_list_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le8861187494160871172list_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le8861187494160871172list_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_344_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_345_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_346_lift__Suc__antimono__le,axiom,
! [F: nat > set_int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_347_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_348_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le8861187494160871172list_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le8861187494160871172list_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_349_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_350_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_351_order__antisym__conv,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_352_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_353_order__antisym__conv,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( ord_le8861187494160871172list_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_354_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_355_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_356_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_357_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_358_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_359_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_360_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_361_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_362_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > set_int,C2: set_int] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_363_ord__le__eq__subst,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_364_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F: int > set_int,C2: set_int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_365_ord__le__eq__subst,axiom,
! [A3: int,B3: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_366_ord__le__eq__subst,axiom,
! [A3: set_int,B3: set_int,F: set_int > nat,C2: nat] :
( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_367_ord__le__eq__subst,axiom,
! [A3: set_int,B3: set_int,F: set_int > int,C2: int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: set_int,Y3: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_368_ord__eq__le__subst,axiom,
! [A3: set_list_a,F: set_nat > set_list_a,B3: set_nat,C2: set_nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_369_ord__eq__le__subst,axiom,
! [A3: nat,F: set_list_a > nat,B3: set_list_a,C2: set_list_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_370_ord__eq__le__subst,axiom,
! [A3: int,F: set_list_a > int,B3: set_list_a,C2: set_list_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_371_ord__eq__le__subst,axiom,
! [A3: set_int,F: set_list_a > set_int,B3: set_list_a,C2: set_list_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ord_less_eq_set_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_372_ord__eq__le__subst,axiom,
! [A3: set_nat,F: set_list_a > set_nat,B3: set_list_a,C2: set_list_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_373_ord__eq__le__subst,axiom,
! [A3: set_list_a,F: set_list_a > set_list_a,B3: set_list_a,C2: set_list_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ord_le8861187494160871172list_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le8861187494160871172list_a @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_374_ord__eq__le__subst,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_375_ord__eq__le__subst,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_376_ord__eq__le__subst,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_377_ord__eq__le__subst,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_378_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_379_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_380_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_381_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_382_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_383_order__subst2,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_384_order__subst2,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_385_order__subst2,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_386_order__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_387_order__subst1,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_388_order__subst1,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_389_order__subst1,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_390_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_391_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_392_antisym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_393_antisym,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% antisym
thf(fact_394_dual__order_Otrans,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_395_dual__order_Otrans,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B3 )
=> ( ord_less_eq_int @ C2 @ A3 ) ) ) ).
% dual_order.trans
thf(fact_396_dual__order_Oantisym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_397_dual__order_Oantisym,axiom,
! [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_398_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_399_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_400_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_401_linorder__wlog,axiom,
! [P: int > int > $o,A3: int,B3: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_402_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_403_order__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_404_order_Otrans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_405_order_Otrans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% order.trans
thf(fact_406_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_407_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_408_ord__le__eq__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_409_ord__le__eq__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_410_ord__eq__le__trans,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_411_ord__eq__le__trans,axiom,
! [A3: int,B3: int,C2: int] :
( ( A3 = B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ A3 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_412_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_413_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
= ( ^ [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
& ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_414_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_415_le__cases3,axiom,
! [X: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_416_nle__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_417_nle__le,axiom,
! [A3: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A3 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_418_subset__after__subset,axiom,
! [T22: prefix_prefix_tree_a,T12: prefix_prefix_tree_a,Xs: list_a] :
( ( ord_le8861187494160871172list_a @ ( prefix_set_a @ T22 ) @ ( prefix_set_a @ T12 ) )
=> ( ord_le8861187494160871172list_a @ ( prefix_set_a @ ( prefix_after_a @ T22 @ Xs ) ) @ ( prefix_set_a @ ( prefix_after_a @ T12 @ Xs ) ) ) ) ).
% subset_after_subset
thf(fact_419_finite_OinsertI,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A3 @ A ) ) ) ).
% finite.insertI
thf(fact_420_finite_OinsertI,axiom,
! [A: set_int,A3: int] :
( ( finite_finite_int @ A )
=> ( finite_finite_int @ ( insert_int @ A3 @ A ) ) ) ).
% finite.insertI
thf(fact_421_Diff__infinite__finite,axiom,
! [T4: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T4 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_422_Diff__infinite__finite,axiom,
! [T4: set_int,S2: set_int] :
( ( finite_finite_int @ T4 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_423_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_424_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_425_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_426_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_427_nless__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_428_nless__le,axiom,
! [A3: int,B3: int] :
( ( ~ ( ord_less_int @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_429_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_430_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_431_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_432_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_433_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_434_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_435_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_436_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_437_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_438_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_439_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_440_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_441_order_Ostrict__trans1,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_442_order_Ostrict__trans1,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_443_order_Ostrict__trans2,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_nat @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_444_order_Ostrict__trans2,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_int @ A3 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_445_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_446_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_447_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_448_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_int @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_449_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_450_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_451_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_452_dual__order_Ostrict__trans1,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A3 )
=> ( ( ord_less_int @ C2 @ B3 )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_453_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A3: nat,C2: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_454_dual__order_Ostrict__trans2,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_eq_int @ C2 @ B3 )
=> ( ord_less_int @ C2 @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_455_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_456_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_457_order_Ostrict__implies__order,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_458_order_Ostrict__implies__order,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_459_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_460_dual__order_Ostrict__implies__order,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ord_less_eq_int @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_461_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_nat @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_462_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y6: int] :
( ( ord_less_int @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_463_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_464_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_465_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_466_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_467_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_468_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_469_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_470_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_471_order__le__neq__trans,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_472_order__le__neq__trans,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_473_order__neq__le__trans,axiom,
! [A3: nat,B3: nat] :
( ( A3 != B3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_474_order__neq__le__trans,axiom,
! [A3: int,B3: int] :
( ( A3 != B3 )
=> ( ( ord_less_eq_int @ A3 @ B3 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_475_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_476_order__le__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_477_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_478_order__less__le__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_479_order__le__less__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_480_order__le__less__subst1,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_481_order__le__less__subst1,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_482_order__le__less__subst1,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( ord_less_eq_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_483_order__le__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_484_order__le__less__subst2,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_485_order__le__less__subst2,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_486_order__le__less__subst2,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_487_order__less__le__subst1,axiom,
! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_488_order__less__le__subst1,axiom,
! [A3: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_489_order__less__le__subst1,axiom,
! [A3: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_490_order__less__le__subst1,axiom,
! [A3: int,F: int > int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A3 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_491_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_492_order__less__le__subst2,axiom,
! [A3: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_493_order__less__le__subst2,axiom,
! [A3: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_494_order__less__le__subst2,axiom,
! [A3: int,B3: int,F: int > int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A3 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_495_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_496_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_497_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_498_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_499_minf_I8_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_500_minf_I8_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ~ ( ord_less_eq_int @ T3 @ X4 ) ) ).
% minf(8)
thf(fact_501_minf_I6_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ord_less_eq_nat @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_502_minf_I6_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z )
=> ( ord_less_eq_int @ X4 @ T3 ) ) ).
% minf(6)
thf(fact_503_pinf_I8_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ord_less_eq_nat @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_504_pinf_I8_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ( ord_less_eq_int @ T3 @ X4 ) ) ).
% pinf(8)
thf(fact_505_pinf_I6_J,axiom,
! [T3: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_506_pinf_I6_J,axiom,
! [T3: int] :
? [Z: int] :
! [X4: int] :
( ( ord_less_int @ Z @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T3 ) ) ).
% pinf(6)
thf(fact_507_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_508_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
= ( ord_less_int @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_509_complete__interval,axiom,
! [A3: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A3 @ C3 )
& ( ord_less_eq_nat @ C3 @ B3 )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A3 @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A3 @ X2 )
& ( ord_less_nat @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_510_complete__interval,axiom,
! [A3: int,B3: int,P: int > $o] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C3: int] :
( ( ord_less_eq_int @ A3 @ C3 )
& ( ord_less_eq_int @ C3 @ B3 )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A3 @ X4 )
& ( ord_less_int @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A3 @ X2 )
& ( ord_less_int @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_511_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_512_finite__has__maximal2,axiom,
! [A: set_int,A3: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A3 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ A3 @ X2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_513_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_514_finite__has__minimal2,axiom,
! [A: set_int,A3: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A3 @ A )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( ord_less_eq_int @ X2 @ A3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_515_finite__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_516_finite__subset,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( finite_finite_int @ B2 )
=> ( finite_finite_int @ A ) ) ) ).
% finite_subset
thf(fact_517_infinite__super,axiom,
! [S2: set_nat,T4: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T4 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ).
% infinite_super
thf(fact_518_infinite__super,axiom,
! [S2: set_int,T4: set_int] :
( ( ord_less_eq_set_int @ S2 @ T4 )
=> ( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ T4 ) ) ) ).
% infinite_super
thf(fact_519_rev__finite__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_520_rev__finite__subset,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( finite_finite_int @ A ) ) ) ).
% rev_finite_subset
thf(fact_521_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_522_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_523_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_524_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_525_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_526_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_527_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M3 @ N ) ) ) ) ).
% diff_induct
thf(fact_528_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_529_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_530_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_531_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_532_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_533_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_534_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_535_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_536_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_537_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_538_gr__implies__not0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_539_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_540_combine__after__after__subset,axiom,
! [T22: prefix_prefix_tree_a,T12: prefix_prefix_tree_a,Xs: list_a] : ( ord_le8861187494160871172list_a @ ( prefix_set_a @ T22 ) @ ( prefix_set_a @ ( prefix_after_a @ ( prefix3285631374902996868fter_a @ T12 @ Xs @ T22 ) @ Xs ) ) ) ).
% combine_after_after_subset
thf(fact_541_card__subset__eq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_542_card__subset__eq,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( ( finite_card_int @ A )
= ( finite_card_int @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_543_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N )
& ( ord_less_eq_set_nat @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_544_infinite__arbitrarily__large,axiom,
! [A: set_int,N: nat] :
( ~ ( finite_finite_int @ A )
=> ? [B8: set_int] :
( ( finite_finite_int @ B8 )
& ( ( finite_card_int @ B8 )
= N )
& ( ord_less_eq_set_int @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_545_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_546_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_547_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_548_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_549_less__Suc__eq__0__disj,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ( M3 = zero_zero_nat )
| ? [J3: nat] :
( ( M3
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_550_diff__gt__0__iff__gt,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B3 ) )
= ( ord_less_int @ B3 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_551_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_552_prefix__tree_Osize__gen,axiom,
! [Xa2: a > nat,X: a > option7782433257363429738tree_a] :
( ( prefix5161986780453196784tree_a @ Xa2 @ ( prefix_prefix_PT_a @ X ) )
= zero_zero_nat ) ).
% prefix_tree.size_gen
thf(fact_553_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_554_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_555_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_556_Suc__diff__diff,axiom,
! [M3: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_557_diff__Suc__Suc,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_Suc_Suc
thf(fact_558_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_559_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_560_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_561_zero__less__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M3 ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% zero_less_diff
thf(fact_562_Suc__le__mono,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ).
% Suc_le_mono
thf(fact_563_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_564_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_565_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_566_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_567_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_568_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_569_finite__set__min__param__ex,axiom,
! [XS: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ XS )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ XS )
=> ? [K3: nat] :
! [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
=> ( P @ X2 @ K4 ) ) )
=> ? [K2: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ XS )
=> ( P @ X4 @ K2 ) ) ) ) ).
% finite_set_min_param_ex
thf(fact_570_finite__set__min__param__ex,axiom,
! [XS: set_int,P: int > nat > $o] :
( ( finite_finite_int @ XS )
=> ( ! [X2: int] :
( ( member_int @ X2 @ XS )
=> ? [K3: nat] :
! [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
=> ( P @ X2 @ K4 ) ) )
=> ? [K2: nat] :
! [X4: int] :
( ( member_int @ X4 @ XS )
=> ( P @ X4 @ K2 ) ) ) ) ).
% finite_set_min_param_ex
thf(fact_571_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M7 ) )
=> ~ ! [M: nat] :
( ( P @ M )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_572_Suc__diff__le,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_573_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_574_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
| ( ord_less_eq_nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_575_diff__le__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_576_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A3 ) @ ( minus_minus_nat @ C2 @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_577_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_578_diff__le__mono,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_579_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_580_le__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_581_eq__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_582_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_583_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_584_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_585_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_586_less__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M3 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_587_diff__less__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C2 ) @ ( minus_minus_nat @ B3 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_588_minimal__fixpoint__helper_I1_J,axiom,
! [F: nat > nat,P: nat > $o,K: nat,X: nat] :
( ( F
= ( ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ X3 @ ( F @ ( suc @ X3 ) ) ) ) )
=> ( ! [X2: nat] :
( ( ord_less_eq_nat @ K @ X2 )
=> ( P @ X2 ) )
=> ( P @ ( F @ X ) ) ) ) ).
% minimal_fixpoint_helper(1)
thf(fact_589_recursion__renaming__helper,axiom,
! [F1: nat > nat,P: nat > $o,F2: nat > nat,K: nat] :
( ( F1
= ( ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ X3 @ ( F1 @ ( suc @ X3 ) ) ) ) )
=> ( ( F2
= ( ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ X3 @ ( F2 @ ( suc @ X3 ) ) ) ) )
=> ( ! [X2: nat] :
( ( ord_less_eq_nat @ K @ X2 )
=> ( P @ X2 ) )
=> ( F1 = F2 ) ) ) ) ).
% recursion_renaming_helper
thf(fact_590_monotone__function__with__limit__witness__helper,axiom,
! [F: nat > nat,K: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( F @ I2 )
= ( F @ J2 ) )
=> ! [M: nat] :
( ( ord_less_eq_nat @ J2 @ M )
=> ( ( F @ I2 )
= ( F @ M ) ) ) ) )
=> ( ! [I2: nat] : ( ord_less_eq_nat @ ( F @ I2 ) @ K )
=> ~ ! [X2: nat] :
( ( ( F @ ( suc @ X2 ) )
= ( F @ X2 ) )
=> ~ ( ord_less_eq_nat @ X2 @ ( minus_minus_nat @ K @ ( F @ zero_zero_nat ) ) ) ) ) ) ) ).
% monotone_function_with_limit_witness_helper
thf(fact_591_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_592_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_593_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M3 )
= zero_zero_nat )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_594_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_595_diff__less__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_596_transitive__stepwise__le,axiom,
! [M3: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z: nat] :
( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X2 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M3 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_597_nat__induct__at__least,axiom,
! [M3: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( P @ M3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_598_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_599_not__less__eq__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_600_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_601_le__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M3 @ N )
| ( M3
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_602_Suc__le__D,axiom,
! [N: nat,M8: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M8 )
=> ? [M: nat] :
( M8
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_603_le__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_604_le__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N )
=> ( M3
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_605_Suc__leD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% Suc_leD
thf(fact_606_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_607_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_608_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_609_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_610_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_611_le__neq__implies__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( M3 != N )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_612_less__or__eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_613_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N6: nat] :
( ( ord_less_nat @ M6 @ N6 )
| ( M6 = N6 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_614_less__imp__le__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_imp_le_nat
thf(fact_615_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N6: nat] :
( ( ord_less_eq_nat @ M6 @ N6 )
& ( M6 != N6 ) ) ) ) ).
% nat_less_le
thf(fact_616_minimal__fixpoint__helper_I2_J,axiom,
! [F: nat > nat,P: nat > $o,K: nat,X: nat,X6: nat] :
( ( F
= ( ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ X3 @ ( F @ ( suc @ X3 ) ) ) ) )
=> ( ! [X2: nat] :
( ( ord_less_eq_nat @ K @ X2 )
=> ( P @ X2 ) )
=> ( ( ord_less_eq_nat @ X @ X6 )
=> ( ( ord_less_nat @ X6 @ ( F @ X ) )
=> ~ ( P @ X6 ) ) ) ) ) ).
% minimal_fixpoint_helper(2)
thf(fact_617_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M6: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_618_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_619_diff__card__le__card__Diff,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ A @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_620_diff__less__Suc,axiom,
! [M3: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ ( suc @ M3 ) ) ).
% diff_less_Suc
thf(fact_621_Suc__diff__Suc,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( ( suc @ ( minus_minus_nat @ M3 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_622_diff__less,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ) ) ).
% diff_less
thf(fact_623_finite__subset__mapping__limit,axiom,
! [F: nat > set_nat] :
( ( finite_finite_nat @ ( F @ zero_zero_nat ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_set_nat @ ( F @ J2 ) @ ( F @ I2 ) ) )
=> ~ ! [K2: nat] :
~ ! [K5: nat] :
( ( ord_less_eq_nat @ K2 @ K5 )
=> ( ( F @ K5 )
= ( F @ K2 ) ) ) ) ) ).
% finite_subset_mapping_limit
thf(fact_624_finite__subset__mapping__limit,axiom,
! [F: nat > set_int] :
( ( finite_finite_int @ ( F @ zero_zero_nat ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_set_int @ ( F @ J2 ) @ ( F @ I2 ) ) )
=> ~ ! [K2: nat] :
~ ! [K5: nat] :
( ( ord_less_eq_nat @ K2 @ K5 )
=> ( ( F @ K5 )
= ( F @ K2 ) ) ) ) ) ).
% finite_subset_mapping_limit
thf(fact_625_le__imp__less__Suc,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_626_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N6: nat] : ( ord_less_eq_nat @ ( suc @ N6 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_627_less__Suc__eq__le,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_Suc_eq_le
thf(fact_628_le__less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_629_Suc__le__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_lessD
thf(fact_630_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_631_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_632_Suc__le__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_eq
thf(fact_633_Suc__leI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_leI
thf(fact_634_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_635_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( R2 @ A1 @ B4 )
=> ( ( R2 @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_636_card__le__if__inj__on__rel,axiom,
! [B2: set_int,A: set_nat,R2: nat > int > $o] :
( ( finite_finite_int @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ? [B9: int] :
( ( member_int @ B9 @ B2 )
& ( R2 @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B4: int] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_int @ B4 @ B2 )
=> ( ( R2 @ A1 @ B4 )
=> ( ( R2 @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_int @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_637_card__insert__le,axiom,
! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) ) ).
% card_insert_le
thf(fact_638_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_639_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_640_card__mono,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_641_card__mono,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B2 ) ) ) ) ).
% card_mono
thf(fact_642_card__seteq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_643_card__seteq,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ B2 ) @ ( finite_card_int @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_644_exists__subset__between,axiom,
! [A: set_nat,N: nat,C: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C ) )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( finite_finite_nat @ C )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C )
& ( ( finite_card_nat @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_645_exists__subset__between,axiom,
! [A: set_int,N: nat,C: set_int] :
( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_int @ C ) )
=> ( ( ord_less_eq_set_int @ A @ C )
=> ( ( finite_finite_int @ C )
=> ? [B8: set_int] :
( ( ord_less_eq_set_int @ A @ B8 )
& ( ord_less_eq_set_int @ B8 @ C )
& ( ( finite_card_int @ B8 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_646_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T5: set_nat] :
( ( ord_less_eq_set_nat @ T5 @ S2 )
=> ( ( ( finite_card_nat @ T5 )
= N )
=> ~ ( finite_finite_nat @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_647_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_int] :
( ( ord_less_eq_nat @ N @ ( finite_card_int @ S2 ) )
=> ~ ! [T5: set_int] :
( ( ord_less_eq_set_int @ T5 @ S2 )
=> ( ( ( finite_card_int @ T5 )
= N )
=> ~ ( finite_finite_int @ T5 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_648_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat,C: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F3 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C ) ) )
=> ( ( finite_finite_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_649_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_int,C: nat] :
( ! [G: set_int] :
( ( ord_less_eq_set_int @ G @ F3 )
=> ( ( finite_finite_int @ G )
=> ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C ) ) )
=> ( ( finite_finite_int @ F3 )
& ( ord_less_eq_nat @ ( finite_card_int @ F3 ) @ C ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_650_card__le__sym__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_651_card__le__sym__Diff,axiom,
! [A: set_int,B2: set_int] :
( ( finite_finite_int @ A )
=> ( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_652_card__Diff__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_653_card__Diff__subset,axiom,
! [B2: set_int,A: set_int] :
( ( finite_finite_int @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A @ B2 ) )
= ( minus_minus_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_654_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y6: nat] :
( ( member_nat @ Y6 @ A )
=> ( X3 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_655_card__le__Suc0__iff__eq,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ! [Y6: int] :
( ( member_int @ Y6 @ A )
=> ( X3 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_656_card__le__Suc__iff,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A ) )
= ( ? [A5: nat,B6: set_nat] :
( ( A
= ( insert_nat @ A5 @ B6 ) )
& ~ ( member_nat @ A5 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
& ( finite_finite_nat @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_657_card__le__Suc__iff,axiom,
! [N: nat,A: set_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_int @ A ) )
= ( ? [A5: int,B6: set_int] :
( ( A
= ( insert_int @ A5 @ B6 ) )
& ~ ( member_int @ A5 @ B6 )
& ( ord_less_eq_nat @ N @ ( finite_card_int @ B6 ) )
& ( finite_finite_int @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_658_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_659_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_660_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_661_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_662_diff__strict__mono,axiom,
! [A3: int,B3: int,D2: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ D2 @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C2 ) @ ( minus_minus_int @ B3 @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_663_diff__eq__diff__less,axiom,
! [A3: int,B3: int,C2: int,D2: int] :
( ( ( minus_minus_int @ A3 @ B3 )
= ( minus_minus_int @ C2 @ D2 ) )
=> ( ( ord_less_int @ A3 @ B3 )
= ( ord_less_int @ C2 @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_664_diff__strict__left__mono,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A3 ) @ ( minus_minus_int @ C2 @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_665_diff__strict__right__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( minus_minus_int @ A3 @ C2 ) @ ( minus_minus_int @ B3 @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_666_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M3: 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 @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_667_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_668_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_669_prefix__tree_Osize_I2_J,axiom,
! [X: a > option7782433257363429738tree_a] :
( ( size_s5139796252398215440tree_a @ ( prefix_prefix_PT_a @ X ) )
= zero_zero_nat ) ).
% prefix_tree.size(2)
thf(fact_670_card__Diff1__less,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_671_card__Diff1__less,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_672_card__Diff2__less,axiom,
! [A: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_673_card__Diff2__less,axiom,
! [A: set_int,X: int,Y: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( insert_int @ Y @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_674_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_675_greaterThanLessThan__empty,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ L @ K )
=> ( ( set_or5832277885323065728an_int @ K @ L )
= bot_bot_set_int ) ) ).
% greaterThanLessThan_empty
thf(fact_676_greaterThanLessThan__empty,axiom,
! [L: nat,K: nat] :
( ( ord_less_eq_nat @ L @ K )
=> ( ( set_or5834768355832116004an_nat @ K @ L )
= bot_bot_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_677_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_678_card__0__eq,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( ( finite_card_int @ A )
= zero_zero_nat )
= ( A = bot_bot_set_int ) ) ) ).
% card_0_eq
thf(fact_679_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_680_bot_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
=> ( A3 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_681_bot_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
= ( A3 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_682_bot_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).
% bot.extremum
thf(fact_683_bot_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_684_bot_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A3 ) ) ).
% bot.not_eq_extremum
thf(fact_685_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_686_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_687_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_688_infinite__imp__nonempty,axiom,
! [S2: set_int] :
( ~ ( finite_finite_int @ S2 )
=> ( S2 != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_689_infinite__growing,axiom,
! [X7: set_nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X7 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X7 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_690_infinite__growing,axiom,
! [X7: set_int] :
( ( X7 != bot_bot_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X7 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X7 )
& ( ord_less_int @ X2 @ Xa ) ) )
=> ~ ( finite_finite_int @ X7 ) ) ) ).
% infinite_growing
thf(fact_691_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_692_ex__min__if__finite,axiom,
! [S2: set_int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ S2 )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S2 )
& ( ord_less_int @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_693_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X2 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_694_finite__ranking__induct,axiom,
! [S2: set_int,P: set_int > $o,F: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,S3: set_int] :
( ( finite_finite_int @ S3 )
=> ( ! [Y4: int] :
( ( member_int @ Y4 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_int @ X2 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_695_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ S3 )
=> ( ord_less_eq_int @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X2 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_696_finite__ranking__induct,axiom,
! [S2: set_int,P: set_int > $o,F: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,S3: set_int] :
( ( finite_finite_int @ S3 )
=> ( ! [Y4: int] :
( ( member_int @ Y4 @ S3 )
=> ( ord_less_eq_int @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_int @ X2 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_697_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_nat @ X4 @ B4 ) )
=> ( ( P @ A2 )
=> ( P @ ( insert_nat @ B4 @ A2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_698_finite__linorder__max__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [B4: int,A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ord_less_int @ X4 @ B4 ) )
=> ( ( P @ A2 )
=> ( P @ ( insert_int @ B4 @ A2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_699_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_nat @ B4 @ X4 ) )
=> ( ( P @ A2 )
=> ( P @ ( insert_nat @ B4 @ A2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_700_finite__linorder__min__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [B4: int,A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ord_less_int @ B4 @ X4 ) )
=> ( ( P @ A2 )
=> ( P @ ( insert_int @ B4 @ A2 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_701_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_702_finite__has__maximal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_703_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_704_finite__has__minimal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_705_finite_Ocases,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ~ ! [A2: set_nat] :
( ? [A4: nat] :
( A3
= ( insert_nat @ A4 @ A2 ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ) ).
% finite.cases
thf(fact_706_finite_Ocases,axiom,
! [A3: set_int] :
( ( finite_finite_int @ A3 )
=> ( ( A3 != bot_bot_set_int )
=> ~ ! [A2: set_int] :
( ? [A4: int] :
( A3
= ( insert_int @ A4 @ A2 ) )
=> ~ ( finite_finite_int @ A2 ) ) ) ) ).
% finite.cases
thf(fact_707_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A7: set_nat,B5: nat] :
( ( A5
= ( insert_nat @ B5 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_708_finite_Osimps,axiom,
( finite_finite_int
= ( ^ [A5: set_int] :
( ( A5 = bot_bot_set_int )
| ? [A7: set_int,B5: int] :
( ( A5
= ( insert_int @ B5 @ A7 ) )
& ( finite_finite_int @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_709_finite__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_710_finite__induct,axiom,
! [F3: set_int,P: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ~ ( member_int @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_711_finite__ne__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_712_finite__ne__induct,axiom,
! [F3: set_int,P: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( F3 != bot_bot_set_int )
=> ( ! [X2: int] : ( P @ ( insert_int @ X2 @ bot_bot_set_int ) )
=> ( ! [X2: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( F4 != bot_bot_set_int )
=> ( ~ ( member_int @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_713_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( P @ A2 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_714_infinite__finite__induct,axiom,
! [P: set_int > $o,A: set_int] :
( ! [A2: set_int] :
( ~ ( finite_finite_int @ A2 )
=> ( P @ A2 ) )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ~ ( member_int @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ X2 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_715_finite__subset__induct_H,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A4 @ A )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ~ ( member_nat @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_716_finite__subset__induct_H,axiom,
! [F3: set_int,A: set_int,P: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( ord_less_eq_set_int @ F3 @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A4: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( member_int @ A4 @ A )
=> ( ( ord_less_eq_set_int @ F4 @ A )
=> ( ~ ( member_int @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_717_finite__subset__induct,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_718_finite__subset__induct,axiom,
! [F3: set_int,A: set_int,P: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( ord_less_eq_set_int @ F3 @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A4: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( member_int @ A4 @ A )
=> ( ~ ( member_int @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_719_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A4: nat,A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A4 @ A2 )
=> ( ( P @ A2 )
=> ( P @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_720_finite__empty__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ( P @ A )
=> ( ! [A4: int,A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A4 @ A2 )
=> ( ( P @ A2 )
=> ( P @ ( minus_minus_set_int @ A2 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
=> ( P @ bot_bot_set_int ) ) ) ) ).
% finite_empty_induct
thf(fact_721_infinite__coinduct,axiom,
! [X7: set_nat > $o,A: set_nat] :
( ( X7 @ A )
=> ( ! [A2: set_nat] :
( ( X7 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( X7 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_722_infinite__coinduct,axiom,
! [X7: set_int > $o,A: set_int] :
( ( X7 @ A )
=> ( ! [A2: set_int] :
( ( X7 @ A2 )
=> ? [X4: int] :
( ( member_int @ X4 @ A2 )
& ( ( X7 @ ( minus_minus_set_int @ A2 @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
| ~ ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) )
=> ~ ( finite_finite_int @ A ) ) ) ).
% infinite_coinduct
thf(fact_723_infinite__remove,axiom,
! [S2: set_nat,A3: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_724_infinite__remove,axiom,
! [S2: set_int,A3: int] :
( ~ ( finite_finite_int @ S2 )
=> ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ).
% infinite_remove
thf(fact_725_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_726_card__eq__0__iff,axiom,
! [A: set_int] :
( ( ( finite_card_int @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_int )
| ~ ( finite_finite_int @ A ) ) ) ).
% card_eq_0_iff
thf(fact_727_Prefix__Tree_Oset__empty,axiom,
( ( prefix_set_a @ prefix_empty_a )
= ( insert_list_a @ nil_a @ bot_bot_set_list_a ) ) ).
% Prefix_Tree.set_empty
thf(fact_728_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A2 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_729_finite__remove__induct,axiom,
! [B2: set_int,P: set_int > $o] :
( ( finite_finite_int @ B2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( P @ ( minus_minus_set_int @ A2 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
=> ( P @ A2 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_730_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A2 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_731_remove__induct,axiom,
! [P: set_int > $o,B2: set_int] :
( ( P @ bot_bot_set_int )
=> ( ( ~ ( finite_finite_int @ B2 )
=> ( P @ B2 ) )
=> ( ! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( P @ ( minus_minus_set_int @ A2 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
=> ( P @ A2 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_732_card__gt__0__iff,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
= ( ( A != bot_bot_set_nat )
& ( finite_finite_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_733_card__gt__0__iff,axiom,
! [A: set_int] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
= ( ( A != bot_bot_set_int )
& ( finite_finite_int @ A ) ) ) ).
% card_gt_0_iff
thf(fact_734_card__Suc__eq,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B5: nat,B6: set_nat] :
( ( A
= ( insert_nat @ B5 @ B6 ) )
& ~ ( member_nat @ B5 @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_735_card__eq__SucD,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
=> ? [B4: nat,B8: set_nat] :
( ( A
= ( insert_nat @ B4 @ B8 ) )
& ~ ( member_nat @ B4 @ B8 )
& ( ( finite_card_nat @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_736_card__1__singleton__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: nat] :
( A
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_737_finite__induct__select,axiom,
! [S2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T5: set_nat] :
( ( ord_less_set_nat @ T5 @ S2 )
=> ( ( P @ T5 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ S2 @ T5 ) )
& ( P @ ( insert_nat @ X4 @ T5 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_738_finite__induct__select,axiom,
! [S2: set_int,P: set_int > $o] :
( ( finite_finite_int @ S2 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [T5: set_int] :
( ( ord_less_set_int @ T5 @ S2 )
=> ( ( P @ T5 )
=> ? [X4: int] :
( ( member_int @ X4 @ ( minus_minus_set_int @ S2 @ T5 ) )
& ( P @ ( insert_int @ X4 @ T5 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_739_card__Diff1__le,axiom,
! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ).
% card_Diff1_le
thf(fact_740_card__Suc__Diff1,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_741_card__Suc__Diff1,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
= ( finite_card_int @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_742_card_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_743_card_Oinsert__remove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( finite_card_int @ ( insert_int @ X @ A ) )
= ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_744_card_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ A )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_745_card_Oremove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( finite_card_int @ A )
= ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% card.remove
thf(fact_746_card__Diff1__less__iff,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
= ( ( finite_finite_nat @ A )
& ( member_nat @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_747_card__Diff1__less__iff,axiom,
! [A: set_int,X: int] :
( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) )
= ( ( finite_finite_int @ A )
& ( member_int @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_748_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat @ X4 @ S2 )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_749_arg__min__if__finite_I2_J,axiom,
! [S2: set_int,F: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ~ ? [X4: int] :
( ( member_int @ X4 @ S2 )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_750_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat @ X4 @ S2 )
& ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic7444442490073309207at_int @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_751_arg__min__if__finite_I2_J,axiom,
! [S2: set_int,F: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ~ ? [X4: int] :
( ( member_int @ X4 @ S2 )
& ( ord_less_int @ ( F @ X4 ) @ ( F @ ( lattic8443796201974363763nt_int @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_752_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_753_arg__min__least,axiom,
! [S2: set_int,Y: int,F: int > nat] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( member_int @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_754_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_int @ ( F @ ( lattic7444442490073309207at_int @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_755_arg__min__least,axiom,
! [S2: set_int,Y: int,F: int > int] :
( ( finite_finite_int @ S2 )
=> ( ( S2 != bot_bot_set_int )
=> ( ( member_int @ Y @ S2 )
=> ( ord_less_eq_int @ ( F @ ( lattic8443796201974363763nt_int @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_756_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_757_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_758_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_759_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_760_card__Diff__insert,axiom,
! [A3: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ A3 @ A )
=> ( ~ ( member_nat @ A3 @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_761_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A7: set_nat] :
( ( finite_card_nat @ A7 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_762_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_763_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_764_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_765_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_766_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_767_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_768_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_769_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_770_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_771_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_772_diff__Suc__eq__diff__pred,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ M3 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_773_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_774_card__1__singletonE,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= one_one_nat )
=> ~ ! [X2: nat] :
( A
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_775_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( minus_minus_nat @ M3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_776_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_777_card__Diff__singleton__if,axiom,
! [X: nat,A: set_nat] :
( ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_778_card__Diff__singleton,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_779_le__div__geq,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( ( divide_divide_nat @ M3 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_780_div__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( divide_divide_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_781_div__by__Suc__0,axiom,
! [M3: nat] :
( ( divide_divide_nat @ M3 @ ( suc @ zero_zero_nat ) )
= M3 ) ).
% div_by_Suc_0
thf(fact_782_div__if,axiom,
( divide_divide_nat
= ( ^ [M6: nat,N6: nat] :
( if_nat
@ ( ( ord_less_nat @ M6 @ N6 )
| ( N6 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N6 ) @ N6 ) ) ) ) ) ).
% div_if
thf(fact_783_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M3: nat,N: nat] :
( ( ( divide_divide_nat @ M3 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M3 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_784_Suc__div__le__mono,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M3 @ N ) @ ( divide_divide_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_785_div__greater__zero__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M3 @ N ) )
= ( ( ord_less_eq_nat @ N @ M3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_786_div__le__mono2,axiom,
! [M3: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M3 ) ) ) ) ).
% div_le_mono2
thf(fact_787_div__eq__dividend__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ( divide_divide_nat @ M3 @ N )
= M3 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_788_div__less__dividend,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( divide_divide_nat @ M3 @ N ) @ M3 ) ) ) ).
% div_less_dividend
thf(fact_789_split__div_H,axiom,
! [P: nat > $o,M3: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M3 @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M3 )
& ( ord_less_nat @ M3 @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
& ( P @ Q3 ) ) ) ) ).
% split_div'
thf(fact_790_of__nat__eq__iff,axiom,
! [M3: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M3 = N ) ) ).
% of_nat_eq_iff
thf(fact_791_mult__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M3 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M3 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_792_mult__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ( times_times_nat @ K @ M3 )
= ( times_times_nat @ K @ N ) )
= ( ( M3 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_793_mult__0__right,axiom,
! [M3: nat] :
( ( times_times_nat @ M3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_794_mult__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_795_nat__mult__eq__1__iff,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= one_one_nat )
= ( ( M3 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_796_nat__1__eq__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M3 @ N ) )
= ( ( M3 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_797_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_798_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_799_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_800_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_801_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1316708129612266289at_nat @ M3 )
= zero_zero_nat )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_802_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= zero_zero_int )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_803_of__nat__le__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% of_nat_le_iff
thf(fact_804_of__nat__le__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% of_nat_le_iff
thf(fact_805_of__nat__less__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_iff
thf(fact_806_of__nat__less__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_iff
thf(fact_807_one__eq__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M3 @ N ) )
= ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_808_mult__eq__1__iff,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_809_of__nat__mult,axiom,
! [M3: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M3 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_810_of__nat__mult,axiom,
! [M3: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M3 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_811_mult__less__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M3 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_812_nat__0__less__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_813_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_814_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_815_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_816_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_817_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_818_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_819_of__nat__le__0__iff,axiom,
! [M3: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ zero_zero_nat )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_820_of__nat__le__0__iff,axiom,
! [M3: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ zero_zero_int )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_821_one__le__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M3 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M3 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_822_mult__le__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_823_div__mult__self1__is__m,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M3 ) @ N )
= M3 ) ) ).
% div_mult_self1_is_m
thf(fact_824_div__mult__self__is__m,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M3 @ N ) @ N )
= M3 ) ) ).
% div_mult_self_is_m
thf(fact_825_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_826_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_827_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_828_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_829_diff__mult__distrib2,axiom,
! [K: nat,M3: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M3 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_830_diff__mult__distrib,axiom,
! [M3: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M3 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_831_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_832_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X2: int,K2: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X4: int,K3: int] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) )
& ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_833_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X2: int,K2: int] :
( ( P @ X2 )
= ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( Q @ X2 )
= ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X4: int,K3: int] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) )
| ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_834_Suc__mult__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M3 )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M3 = N ) ) ).
% Suc_mult_cancel1
thf(fact_835_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_836_le__cube,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).
% le_cube
thf(fact_837_le__square,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ).
% le_square
thf(fact_838_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_839_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_840_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_841_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_842_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_843_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_844_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_845_of__nat__less__0__iff,axiom,
! [M3: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_846_of__nat__less__0__iff,axiom,
! [M3: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_847_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_848_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_849_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_850_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_851_of__nat__less__imp__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_852_of__nat__less__imp__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_853_less__imp__of__nat__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_854_less__imp__of__nat__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_855_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_856_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_857_mult__less__cancel__right__disj,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A3 @ B3 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_858_mult__strict__right__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_859_mult__strict__right__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_860_mult__strict__right__mono__neg,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_861_mult__less__cancel__left__disj,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A3 @ B3 ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_862_mult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_863_mult__strict__left__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_864_mult__strict__left__mono__neg,axiom,
! [B3: int,A3: int,C2: int] :
( ( ord_less_int @ B3 @ A3 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_865_mult__less__cancel__left__pos,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_866_mult__less__cancel__left__neg,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_int @ B3 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_867_zero__less__mult__pos2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_868_zero__less__mult__pos2,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B3 @ A3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_869_zero__less__mult__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_870_zero__less__mult__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_871_zero__less__mult__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ zero_zero_int @ B3 ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ B3 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_872_mult__pos__neg2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_873_mult__pos__neg2,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B3 @ A3 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_874_mult__pos__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% mult_pos_pos
thf(fact_875_mult__pos__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).
% mult_pos_pos
thf(fact_876_mult__pos__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_877_mult__pos__neg,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_878_mult__neg__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_879_mult__neg__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_880_mult__less__0__iff,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_int @ B3 @ zero_zero_int ) )
| ( ( ord_less_int @ A3 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% mult_less_0_iff
thf(fact_881_not__square__less__zero,axiom,
! [A3: int] :
~ ( ord_less_int @ ( times_times_int @ A3 @ A3 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_882_mult__neg__neg,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).
% mult_neg_neg
thf(fact_883_less__1__mult,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M3 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M3 @ N ) ) ) ) ).
% less_1_mult
thf(fact_884_less__1__mult,axiom,
! [M3: int,N: int] :
( ( ord_less_int @ one_one_int @ M3 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M3 @ N ) ) ) ) ).
% less_1_mult
thf(fact_885_Suc__mult__less__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M3 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_886_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_887_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_888_Suc__mult__le__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M3 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_889_mult__eq__self__implies__10,axiom,
! [M3: nat,N: nat] :
( ( M3
= ( times_times_nat @ M3 @ N ) )
=> ( ( N = one_one_nat )
| ( M3 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_890_less__mult__imp__div__less,axiom,
! [M3: nat,I: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M3 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_891_of__nat__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M3 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_892_of__nat__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M3 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_893_mult__less__le__imp__less,axiom,
! [A3: nat,B3: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_894_mult__less__le__imp__less,axiom,
! [A3: int,B3: int,C2: int,D2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_895_mult__le__less__imp__less,axiom,
! [A3: nat,B3: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_896_mult__le__less__imp__less,axiom,
! [A3: int,B3: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_897_mult__right__le__imp__le,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_898_mult__right__le__imp__le,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_899_mult__left__le__imp__le,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_900_mult__left__le__imp__le,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_901_mult__le__cancel__left__pos,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_902_mult__le__cancel__left__neg,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ord_less_eq_int @ B3 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_903_mult__less__cancel__right,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ B3 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_904_mult__strict__mono_H,axiom,
! [A3: nat,B3: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_905_mult__strict__mono_H,axiom,
! [A3: int,B3: int,C2: int,D2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_906_mult__right__less__imp__less,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ C2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_907_mult__right__less__imp__less,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_908_mult__less__cancel__left,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ B3 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_909_mult__strict__mono,axiom,
! [A3: nat,B3: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C2 ) @ ( times_times_nat @ B3 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_910_mult__strict__mono,axiom,
! [A3: int,B3: int,C2: int,D2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_911_mult__left__less__imp__less,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ C2 @ A3 ) @ ( times_times_nat @ C2 @ B3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_912_mult__left__less__imp__less,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_913_mult__le__cancel__right,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ ( times_times_int @ B3 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ B3 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_914_mult__le__cancel__left,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ ( times_times_int @ C2 @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ B3 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_915_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_916_n__less__n__mult__m,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M3 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_917_n__less__m__mult__n,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M3 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_918_one__less__mult,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M3 @ N ) ) ) ) ).
% one_less_mult
thf(fact_919_div__less__iff__less__mult,axiom,
! [Q4: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M3 @ Q4 ) @ N )
= ( ord_less_nat @ M3 @ ( times_times_nat @ N @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_920_mult__less__cancel__right2,axiom,
! [A3: int,C2: int] :
( ( ord_less_int @ ( times_times_int @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_921_mult__less__cancel__right1,axiom,
! [C2: int,B3: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ B3 @ C2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B3 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_922_mult__less__cancel__left2,axiom,
! [C2: int,A3: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ A3 @ one_one_int ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_923_mult__less__cancel__left1,axiom,
! [C2: int,B3: int] :
( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B3 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ one_one_int @ B3 ) )
& ( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_924_mult__le__cancel__right2,axiom,
! [A3: int,C2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_925_mult__le__cancel__right1,axiom,
! [C2: int,B3: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ B3 @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B3 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_926_mult__le__cancel__left2,axiom,
! [C2: int,A3: int] :
( ( ord_less_eq_int @ ( times_times_int @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ A3 @ one_one_int ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_927_mult__le__cancel__left1,axiom,
! [C2: int,B3: int] :
( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B3 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_eq_int @ one_one_int @ B3 ) )
& ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_928_div__nat__eqI,axiom,
! [N: nat,Q4: nat,M3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M3 )
=> ( ( ord_less_nat @ M3 @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
=> ( ( divide_divide_nat @ M3 @ N )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_929_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q4 )
=> ( ( ord_less_eq_nat @ M3 @ ( divide_divide_nat @ N @ Q4 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M3 @ Q4 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_930_nat__mult__le__cancel__disj,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_931_nat__mult__less__cancel__disj,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M3 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_932_nat__mult__div__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M3 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_933_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 )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_934_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int,K2: int] :
( ( P1 @ X2 )
= ( P1 @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P1 @ X2 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_935_plusinfinity,axiom,
! [D2: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int,K2: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D2 ) ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [X_12: int] : ( P2 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_936_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_937_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_938_nat__mult__eq__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M3 )
= ( times_times_nat @ K @ N ) )
= ( M3 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_939_nat__mult__less__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_940_nat__mult__le__cancel1,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_941_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_942_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_943_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_944_mult__le__cancel__iff1,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_945_mult__le__cancel__iff2,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z3 @ X ) @ ( times_times_int @ Z3 @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_946_mult__less__iff1,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_947_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1406184849735516958ct_int @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_948_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1408675320244567234ct_nat @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_949_int__power__div__base,axiom,
! [M3: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M3 ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_950_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_951_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_952_fact__Suc__0,axiom,
( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% fact_Suc_0
thf(fact_953_fact__Suc__0,axiom,
( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% fact_Suc_0
thf(fact_954_fact__Suc,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_Suc
thf(fact_955_fact__Suc,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_Suc
thf(fact_956_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% fact_not_neg
thf(fact_957_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% fact_not_neg
thf(fact_958_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_gt_zero
thf(fact_959_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_gt_zero
thf(fact_960_fact__less__mono__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_961_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_Suc_0_nat
thf(fact_962_fact__less__mono,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ord_less_nat @ M3 @ N )
=> ( ord_less_int @ ( semiri1406184849735516958ct_int @ M3 ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% fact_less_mono
thf(fact_963_fact__less__mono,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_964_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_965_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_966_power__decreasing__iff,axiom,
! [B3: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ B3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ M3 ) @ ( power_power_nat @ B3 @ N ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ) ) ).
% power_decreasing_iff
thf(fact_967_power__decreasing__iff,axiom,
! [B3: int,M3: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ B3 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B3 @ M3 ) @ ( power_power_int @ B3 @ N ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ) ) ).
% power_decreasing_iff
thf(fact_968_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_969_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M3: nat] :
( ( ( power_power_nat @ X @ M3 )
= ( suc @ zero_zero_nat ) )
= ( ( M3 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_970_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_971_power__inject__exp,axiom,
! [A3: nat,M3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ( ( power_power_nat @ A3 @ M3 )
= ( power_power_nat @ A3 @ N ) )
= ( M3 = N ) ) ) ).
% power_inject_exp
thf(fact_972_power__inject__exp,axiom,
! [A3: int,M3: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ( ( power_power_int @ A3 @ M3 )
= ( power_power_int @ A3 @ N ) )
= ( M3 = N ) ) ) ).
% power_inject_exp
thf(fact_973_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_974_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_975_power__Suc0__right,axiom,
! [A3: int] :
( ( power_power_int @ A3 @ ( suc @ zero_zero_nat ) )
= A3 ) ).
% power_Suc0_right
thf(fact_976_power__Suc0__right,axiom,
! [A3: nat] :
( ( power_power_nat @ A3 @ ( suc @ zero_zero_nat ) )
= A3 ) ).
% power_Suc0_right
thf(fact_977_power__strict__increasing__iff,axiom,
! [B3: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B3 )
=> ( ( ord_less_nat @ ( power_power_nat @ B3 @ X ) @ ( power_power_nat @ B3 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_978_power__strict__increasing__iff,axiom,
! [B3: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B3 )
=> ( ( ord_less_int @ ( power_power_int @ B3 @ X ) @ ( power_power_int @ B3 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_979_power__eq__0__iff,axiom,
! [A3: int,N: nat] :
( ( ( power_power_int @ A3 @ N )
= zero_zero_int )
= ( ( A3 = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_980_power__eq__0__iff,axiom,
! [A3: nat,N: nat] :
( ( ( power_power_nat @ A3 @ N )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_981_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B3 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_982_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B3: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B3 @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_983_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B3: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B3 @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_984_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B3: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B3 @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_985_power__strict__decreasing__iff,axiom,
! [B3: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ B3 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B3 @ M3 ) @ ( power_power_nat @ B3 @ N ) )
= ( ord_less_nat @ N @ M3 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_986_power__strict__decreasing__iff,axiom,
! [B3: int,M3: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ( ord_less_int @ B3 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B3 @ M3 ) @ ( power_power_int @ B3 @ N ) )
= ( ord_less_nat @ N @ M3 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_987_power__increasing__iff,axiom,
! [B3: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ X ) @ ( power_power_nat @ B3 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_988_power__increasing__iff,axiom,
! [B3: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B3 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B3 @ X ) @ ( power_power_int @ B3 @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_989_power__mono__iff,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B3 @ N ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ) ).
% power_mono_iff
thf(fact_990_power__mono__iff,axiom,
! [A3: int,B3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) )
= ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ) ).
% power_mono_iff
thf(fact_991_nat__power__less__imp__less,axiom,
! [I: nat,M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M3 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_992_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_993_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_994_zero__less__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A3 @ N ) ) ) ).
% zero_less_power
thf(fact_995_zero__less__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ N ) ) ) ).
% zero_less_power
thf(fact_996_power__Suc,axiom,
! [A3: nat,N: nat] :
( ( power_power_nat @ A3 @ ( suc @ N ) )
= ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ).
% power_Suc
thf(fact_997_power__Suc,axiom,
! [A3: int,N: nat] :
( ( power_power_int @ A3 @ ( suc @ N ) )
= ( times_times_int @ A3 @ ( power_power_int @ A3 @ N ) ) ) ).
% power_Suc
thf(fact_998_power__Suc2,axiom,
! [A3: nat,N: nat] :
( ( power_power_nat @ A3 @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A3 @ N ) @ A3 ) ) ).
% power_Suc2
thf(fact_999_power__Suc2,axiom,
! [A3: int,N: nat] :
( ( power_power_int @ A3 @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A3 @ N ) @ A3 ) ) ).
% power_Suc2
thf(fact_1000_power__less__imp__less__base,axiom,
! [A3: nat,N: nat,B3: nat] :
( ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B3 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% power_less_imp_less_base
thf(fact_1001_power__less__imp__less__base,axiom,
! [A3: int,N: nat,B3: int] :
( ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ A3 @ B3 ) ) ) ).
% power_less_imp_less_base
thf(fact_1002_power__le__imp__le__base,axiom,
! [A3: nat,N: nat,B3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ ( power_power_nat @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% power_le_imp_le_base
thf(fact_1003_power__le__imp__le__base,axiom,
! [A3: int,N: nat,B3: int] :
( ( ord_less_eq_int @ ( power_power_int @ A3 @ ( suc @ N ) ) @ ( power_power_int @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_eq_int @ A3 @ B3 ) ) ) ).
% power_le_imp_le_base
thf(fact_1004_power__inject__base,axiom,
! [A3: nat,N: nat,B3: nat] :
( ( ( power_power_nat @ A3 @ ( suc @ N ) )
= ( power_power_nat @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% power_inject_base
thf(fact_1005_power__inject__base,axiom,
! [A3: int,N: nat,B3: int] :
( ( ( power_power_int @ A3 @ ( suc @ N ) )
= ( power_power_int @ B3 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% power_inject_base
thf(fact_1006_power__gt1__lemma,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1007_power__gt1__lemma,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1008_power__less__power__Suc,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1009_power__less__power__Suc,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1010_power__gt1,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1011_power__gt1,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A3 @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1012_power__strict__increasing,axiom,
! [N: nat,N4: nat,A3: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ A3 @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1013_power__strict__increasing,axiom,
! [N: nat,N4: nat,A3: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ one_one_int @ A3 )
=> ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ A3 @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1014_power__less__imp__less__exp,axiom,
! [A3: nat,M3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ( ord_less_nat @ ( power_power_nat @ A3 @ M3 ) @ ( power_power_nat @ A3 @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1015_power__less__imp__less__exp,axiom,
! [A3: int,M3: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ( ord_less_int @ ( power_power_int @ A3 @ M3 ) @ ( power_power_int @ A3 @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1016_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_1017_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1018_power__Suc__less,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1019_power__Suc__less,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N ) ) @ ( power_power_int @ A3 @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1020_power__Suc__le__self,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_1021_power__Suc__le__self,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ A3 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ).
% power_Suc_le_self
thf(fact_1022_power__Suc__less__one,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_1023_power__Suc__less__one,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A3 @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_1024_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A3: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ A3 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N4 ) @ ( power_power_nat @ A3 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1025_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A3: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ A3 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A3 @ N4 ) @ ( power_power_int @ A3 @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1026_power__le__imp__le__exp,axiom,
! [A3: nat,M3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ M3 ) @ ( power_power_nat @ A3 @ N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1027_power__le__imp__le__exp,axiom,
! [A3: int,M3: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A3 @ M3 ) @ ( power_power_int @ A3 @ N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1028_power__eq__imp__eq__base,axiom,
! [A3: nat,N: nat,B3: nat] :
( ( ( power_power_nat @ A3 @ N )
= ( power_power_nat @ B3 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A3 = B3 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1029_power__eq__imp__eq__base,axiom,
! [A3: int,N: nat,B3: int] :
( ( ( power_power_int @ A3 @ N )
= ( power_power_int @ B3 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A3 = B3 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1030_power__eq__iff__eq__base,axiom,
! [N: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ( power_power_nat @ A3 @ N )
= ( power_power_nat @ B3 @ N ) )
= ( A3 = B3 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1031_power__eq__iff__eq__base,axiom,
! [N: nat,A3: int,B3: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ( ( power_power_int @ A3 @ N )
= ( power_power_int @ B3 @ N ) )
= ( A3 = B3 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1032_self__le__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A3 @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% self_le_power
thf(fact_1033_self__le__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A3 @ ( power_power_int @ A3 @ N ) ) ) ) ).
% self_le_power
thf(fact_1034_one__less__power,axiom,
! [A3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A3 @ N ) ) ) ) ).
% one_less_power
thf(fact_1035_one__less__power,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ one_one_int @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A3 @ N ) ) ) ) ).
% one_less_power
thf(fact_1036_power__strict__mono,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A3 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1037_power__strict__mono,axiom,
! [A3: int,B3: int,N: nat] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A3 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1038_power__minus__mult,axiom,
! [N: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A3 )
= ( power_power_nat @ A3 @ N ) ) ) ).
% power_minus_mult
thf(fact_1039_power__minus__mult,axiom,
! [N: nat,A3: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A3 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A3 )
= ( power_power_int @ A3 @ N ) ) ) ).
% power_minus_mult
thf(fact_1040_add__Suc__right,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% add_Suc_right
thf(fact_1041_add__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1042_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1043_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1044_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1045_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_1046_add__less__cancel__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_1047_add__less__cancel__right,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ C2 ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_cancel_right
thf(fact_1048_add__less__cancel__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_1049_add__less__cancel__left,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_cancel_left
thf(fact_1050_of__nat__add,axiom,
! [M3: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_1051_of__nat__add,axiom,
! [M3: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M3 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_1052_add__gr__0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1053_mult__Suc__right,axiom,
! [M3: nat,N: nat] :
( ( times_times_nat @ M3 @ ( suc @ N ) )
= ( plus_plus_nat @ M3 @ ( times_times_nat @ M3 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1054_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_1055_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_1056_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
thf(fact_1057_add__less__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1058_add__less__same__cancel1,axiom,
! [B3: int,A3: int] :
( ( ord_less_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1059_add__less__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1060_add__less__same__cancel2,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1061_less__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_1062_less__add__same__cancel1,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
= ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% less_add_same_cancel1
thf(fact_1063_less__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_1064_less__add__same__cancel2,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ ( plus_plus_int @ B3 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% less_add_same_cancel2
thf(fact_1065_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1066_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
= ( ord_less_int @ zero_zero_int @ A3 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1067_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1068_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1069_of__nat__Suc,axiom,
! [M3: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M3 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M3 ) ) ) ).
% of_nat_Suc
thf(fact_1070_of__nat__Suc,axiom,
! [M3: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M3 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M3 ) ) ) ).
% of_nat_Suc
thf(fact_1071_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1072_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1073_less__add__eq__less,axiom,
! [K: nat,L: nat,M3: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1074_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_1075_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_1076_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_1077_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1078_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1079_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_1080_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_1081_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1082_add__eq__self__zero,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= M3 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1083_add__mono__thms__linordered__field_I5_J,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_mono_thms_linordered_field(5)
thf(fact_1084_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1085_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1086_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1087_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1088_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1089_add__strict__mono,axiom,
! [A3: nat,B3: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1090_add__strict__mono,axiom,
! [A3: int,B3: int,C2: int,D2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1091_add__strict__left__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_1092_add__strict__left__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B3 ) ) ) ).
% add_strict_left_mono
thf(fact_1093_add__strict__right__mono,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1094_add__strict__right__mono,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_1095_add__less__imp__less__left,axiom,
! [C2: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A3 ) @ ( plus_plus_nat @ C2 @ B3 ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_1096_add__less__imp__less__left,axiom,
! [C2: int,A3: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A3 ) @ ( plus_plus_int @ C2 @ B3 ) )
=> ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_imp_less_left
thf(fact_1097_add__less__imp__less__right,axiom,
! [A3: nat,C2: nat,B3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ C2 ) )
=> ( ord_less_nat @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_1098_add__less__imp__less__right,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ C2 ) )
=> ( ord_less_int @ A3 @ B3 ) ) ).
% add_less_imp_less_right
thf(fact_1099_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1100_diff__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_cancel2
thf(fact_1101_diff__add__inverse,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M3 ) @ N )
= M3 ) ).
% diff_add_inverse
thf(fact_1102_diff__add__inverse2,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N ) @ N )
= M3 ) ).
% diff_add_inverse2
thf(fact_1103_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1104_pos__add__strict,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1105_pos__add__strict,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1106_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ! [C3: nat] :
( ( B3
= ( plus_plus_nat @ A3 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1107_add__pos__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_1108_add__pos__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% add_pos_pos
thf(fact_1109_add__neg__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1110_add__neg__neg,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1111_add__mono1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( plus_plus_nat @ B3 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1112_add__mono1,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( plus_plus_int @ B3 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1113_less__add__one,axiom,
! [A3: nat] : ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ).
% less_add_one
thf(fact_1114_less__add__one,axiom,
! [A3: int] : ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ one_one_int ) ) ).
% less_add_one
thf(fact_1115_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: nat,B3: nat] :
( ~ ( ord_less_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1116_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: int,B3: int] :
( ~ ( ord_less_int @ A3 @ B3 )
=> ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A3 @ B3 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1117_less__diff__eq,axiom,
! [A3: int,C2: int,B3: int] :
( ( ord_less_int @ A3 @ ( minus_minus_int @ C2 @ B3 ) )
= ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ C2 ) ) ).
% less_diff_eq
thf(fact_1118_diff__less__eq,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ C2 )
= ( ord_less_int @ A3 @ ( plus_plus_int @ C2 @ B3 ) ) ) ).
% diff_less_eq
thf(fact_1119_add__is__1,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1120_one__is__add,axiom,
! [M3: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1121_less__natE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ~ ! [Q5: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M3 @ Q5 ) ) ) ) ).
% less_natE
thf(fact_1122_less__add__Suc1,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M3 ) ) ) ).
% less_add_Suc1
thf(fact_1123_less__add__Suc2,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M3 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1124_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N6: nat] :
? [K6: nat] :
( N6
= ( suc @ ( plus_plus_nat @ M6 @ K6 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1125_less__imp__Suc__add,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1126_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1127_diff__add__0,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1128_add__diff__inverse__nat,axiom,
! [M3: nat,N: nat] :
( ~ ( ord_less_nat @ M3 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M3 @ N ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1129_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_1130_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1131_add__Suc,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% add_Suc
thf(fact_1132_add__Suc__shift,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N )
= ( plus_plus_nat @ M3 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1133_Suc__eq__plus1,axiom,
( suc
= ( ^ [N6: nat] : ( plus_plus_nat @ N6 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1134_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1135_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1136_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1137_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1138_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1139_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1140_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1141_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N6: nat] :
? [K6: nat] :
( N6
= ( plus_plus_nat @ M6 @ K6 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1142_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_1143_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1144_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1145_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1146_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1147_add__leD2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1148_add__leD1,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% add_leD1
thf(fact_1149_le__add2,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M3 @ N ) ) ).
% le_add2
thf(fact_1150_le__add1,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M3 ) ) ).
% le_add1
thf(fact_1151_add__leE,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M3 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1152_mult__Suc,axiom,
! [M3: nat,N: nat] :
( ( times_times_nat @ ( suc @ M3 ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M3 @ N ) ) ) ).
% mult_Suc
thf(fact_1153_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( F @ M ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1154_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1155_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1156_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1157_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1158_add__le__less__mono,axiom,
! [A3: nat,B3: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1159_add__le__less__mono,axiom,
! [A3: int,B3: int,C2: int,D2: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1160_add__less__le__mono,axiom,
! [A3: nat,B3: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C2 ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1161_add__less__le__mono,axiom,
! [A3: int,B3: int,C2: int,D2: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( ( ord_less_eq_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ C2 ) @ ( plus_plus_int @ B3 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1162_add__mult__distrib2,axiom,
! [K: nat,M3: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M3 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1163_add__mult__distrib,axiom,
! [M3: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M3 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1164_add__neg__nonpos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1165_add__neg__nonpos,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1166_add__nonneg__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1167_add__nonneg__pos,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1168_add__nonpos__neg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1169_add__nonpos__neg,axiom,
! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ zero_zero_int )
=> ( ( ord_less_int @ B3 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1170_add__pos__nonneg,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1171_add__pos__nonneg,axiom,
! [A3: int,B3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1172_add__strict__increasing,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1173_add__strict__increasing,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_1174_add__strict__increasing2,axiom,
! [A3: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1175_add__strict__increasing2,axiom,
! [A3: int,B3: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1176_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_1177_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1178_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1179_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1180_less__add__iff2,axiom,
! [A3: int,E: int,C2: int,B3: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E ) @ D2 ) )
= ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A3 ) @ E ) @ D2 ) ) ) ).
% less_add_iff2
thf(fact_1181_less__add__iff1,axiom,
! [A3: int,E: int,C2: int,B3: int,D2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E ) @ D2 ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ E ) @ C2 ) @ D2 ) ) ).
% less_add_iff1
thf(fact_1182_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
= ( ( ( ord_less_nat @ A3 @ B3 )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1183_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B3: nat] :
( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B3 )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A3
= ( plus_plus_nat @ B3 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1184_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1185_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N6: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N6 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1186_Ints__odd__less__0,axiom,
! [A3: int] :
( ( member_int @ A3 @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A3 ) @ A3 ) @ zero_zero_int )
= ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).
% Ints_odd_less_0
thf(fact_1187_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N6 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N6 ) ) ) ) ) ).
% add_eq_if
thf(fact_1188_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M3 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1189_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M3 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M3 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1190_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 )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1191_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M6: nat,N6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N6 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N6 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1192_dividend__less__times__div,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M3 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M3 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1193_dividend__less__div__times,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M3 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1194_split__div,axiom,
! [P: nat > $o,M3: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M3 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M3
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_div
thf(fact_1195_convex__bound__lt,axiom,
! [X: int,A3: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A3 )
=> ( ( ord_less_int @ Y @ A3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A3 ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1196_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1197_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_1198_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_1199_zero__less__abs__iff,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A3 ) )
= ( A3 != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_1200_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_1201_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_1202_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_1203_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_1204_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_1205_zero__less__power__abs__iff,axiom,
! [A3: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A3 ) @ N ) )
= ( ( A3 != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_1206_Suc__times__binomial__add,axiom,
! [A3: nat,B3: nat] :
( ( times_times_nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A3 @ B3 ) ) @ ( suc @ A3 ) ) )
= ( times_times_nat @ ( suc @ B3 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A3 @ B3 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_1207_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_1208_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_1209_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_1210_abs__mult__less,axiom,
! [A3: int,C2: int,B3: int,D2: int] :
( ( ord_less_int @ ( abs_abs_int @ A3 ) @ C2 )
=> ( ( ord_less_int @ ( abs_abs_int @ B3 ) @ D2 )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) @ ( times_times_int @ C2 @ D2 ) ) ) ) ).
% abs_mult_less
thf(fact_1211_abs__of__pos,axiom,
! [A3: int] :
( ( ord_less_int @ zero_zero_int @ A3 )
=> ( ( abs_abs_int @ A3 )
= A3 ) ) ).
% abs_of_pos
thf(fact_1212_abs__not__less__zero,axiom,
! [A3: int] :
~ ( ord_less_int @ ( abs_abs_int @ A3 ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1213_abs__diff__less__iff,axiom,
! [X: int,A3: int,R2: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A3 ) ) @ R2 )
= ( ( ord_less_int @ ( minus_minus_int @ A3 @ R2 ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A3 @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1214_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_1215_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_1216_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1217_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_1218_Ints__nonzero__abs__less1,axiom,
! [X: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
=> ( X = zero_zero_int ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_1219_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorption
thf(fact_1220_Ints__eq__abs__less1,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( member_int @ Y @ ring_1_Ints_int )
=> ( ( X = Y )
= ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_1221_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_1222_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_1223_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_1224_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_1225_nat__intermed__int__val,axiom,
! [M3: nat,N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M3 @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_int @ ( F @ M3 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M3 @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1226_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1227_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1228_verit__le__mono__div,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B2 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B2 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1229_mod__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( modulo_modulo_nat @ M3 @ N )
= M3 ) ) ).
% mod_less
thf(fact_1230_mod__by__Suc__0,axiom,
! [M3: nat] :
( ( modulo_modulo_nat @ M3 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1231_Suc__mod__mult__self1,axiom,
! [M3: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M3 @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1232_Suc__mod__mult__self2,axiom,
! [M3: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M3 @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1233_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M3: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M3 ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1234_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M3: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M3 ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1235_mod__le__divisor,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M3 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1236_div__Suc,axiom,
! [M3: nat,N: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M3 ) @ N )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( divide_divide_nat @ M3 @ N ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M3 ) @ N )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M3 ) @ N )
= ( divide_divide_nat @ M3 @ N ) ) ) ) ).
% div_Suc
thf(fact_1237_div__less__mono,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B2 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1238_mod__Suc,axiom,
! [M3: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M3 @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M3 ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M3 @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( modulo_modulo_nat @ M3 @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1239_mod__induct,axiom,
! [P: nat > $o,N: nat,P5: nat,M3: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P5 )
=> ( ( ord_less_nat @ M3 @ P5 )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ P5 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P5 ) ) ) )
=> ( P @ M3 ) ) ) ) ) ).
% mod_induct
thf(fact_1240_mod__less__divisor,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M3 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_1241_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M6: nat,N6: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N6 ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N6 ) @ N6 ) ) ) ) ).
% mod_if
thf(fact_1242_mod__Suc__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M3 @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M3 ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_1243_mod__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M3 @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M3 ) @ N ) ) ).
% mod_Suc_eq
thf(fact_1244_mod__Suc__le__divisor,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M3 @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_1245_split__mod,axiom,
! [Q: nat > $o,M3: nat,N: nat] :
( ( Q @ ( modulo_modulo_nat @ M3 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( Q @ M3 ) )
& ( ( N != zero_zero_nat )
=> ! [I3: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M3
= ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) ) )
=> ( Q @ J3 ) ) ) ) ) ).
% split_mod
thf(fact_1246_Suc__times__mod__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M3 )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M3 @ N ) ) @ M3 )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1247_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M3: nat,N: nat] :
( ! [M: nat] : ( P @ M @ zero_zero_nat )
=> ( ! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo_nat @ M @ N2 ) )
=> ( P @ M @ N2 ) ) )
=> ( P @ M3 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1248_Gcd__fin__0__iff,axiom,
! [A: set_nat] :
( ( ( semiri4258706085729940814in_nat @ A )
= zero_zero_nat )
= ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
& ( finite_finite_nat @ A ) ) ) ).
% Gcd_fin_0_iff
thf(fact_1249_Gcd__fin__0__iff,axiom,
! [A: set_int] :
( ( ( semiri4256215615220890538in_int @ A )
= zero_zero_int )
= ( ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
& ( finite_finite_int @ A ) ) ) ).
% Gcd_fin_0_iff
thf(fact_1250_of__int__less__iff,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ).
% of_int_less_iff
thf(fact_1251_Gcd__fin_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( semiri4258706085729940814in_nat @ A )
= one_one_nat ) ) ).
% Gcd_fin.infinite
thf(fact_1252_Gcd__fin_Oinfinite,axiom,
! [A: set_int] :
( ~ ( finite_finite_int @ A )
=> ( ( semiri4256215615220890538in_int @ A )
= one_one_int ) ) ).
% Gcd_fin.infinite
thf(fact_1253_of__int__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1254_of__int__0__less__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_less_iff
thf(fact_1255_of__int__less__1__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ one_one_int )
= ( ord_less_int @ Z3 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_1256_of__int__1__less__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% of_int_1_less_iff
thf(fact_1257_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B3: int,W: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B3 @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_1258_of__int__less__of__int__power__cancel__iff,axiom,
! [B3: int,W: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B3 @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_1259_of__int__pos,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_int_pos
thf(fact_1260_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_1261_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_1262_cpmi,axiom,
! [D3: int,P: int > $o,P2: int > $o,B2: set_int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B2 )
=> ( X2
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus_int @ X2 @ D3 ) ) ) )
=> ( ! [X2: int,K2: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ( P2 @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
& ? [Y6: int] :
( ( member_int @ Y6 @ B2 )
& ( P @ ( plus_plus_int @ Y6 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_1263_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_1264_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_1265_atLeastAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_eq_int @ I @ U ) ) ) ).
% atLeastAtMost_iff
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
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 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [X4: a] :
( ( ( m1 @ X4 )
!= ( m2 @ X4 ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------