TPTP Problem File: ITP133^1.p
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%------------------------------------------------------------------------------
% File : ITP133^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Number_Partition problem prob_28__5323650_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Number_Partition/prob_28__5323650_1 [Des21]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 283 ( 117 unt; 21 typ; 0 def)
% Number of atoms : 777 ( 256 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 1961 ( 82 ~; 23 |; 46 &;1469 @)
% ( 0 <=>; 341 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 107 ( 107 >; 0 *; 0 +; 0 <<)
% Number of symbols : 19 ( 18 usr; 2 con; 0-2 aty)
% Number of variables : 719 ( 97 ^; 616 !; 6 ?; 719 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:41:56.283
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
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__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (18)
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
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__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups1842438620at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
number1551313001itions: ( nat > nat ) > nat > $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__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $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__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or1086813439et_nat: set_nat > set_set_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $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_p,type,
p: nat > nat ).
% Relevant facts (261)
thf(fact_0_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups1842438620at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_1_partitionsE,axiom,
! [P: nat > nat,N: nat] :
( ( number1551313001itions @ P @ N )
=> ~ ( ! [I: nat] :
( ( ( P @ I )
!= zero_zero_nat )
=> ( ( ord_less_eq_nat @ one_one_nat @ I )
& ( ord_less_eq_nat @ I @ N ) ) )
=> ( ( groups1842438620at_nat
@ ^ [I2: nat] : ( times_times_nat @ ( P @ I2 ) @ I2 )
@ ( set_ord_atMost_nat @ N ) )
!= N ) ) ) ).
% partitionsE
thf(fact_2_partitionsI,axiom,
! [P: nat > nat,N: nat] :
( ! [I3: nat] :
( ( ( P @ I3 )
!= zero_zero_nat )
=> ( ( ord_less_eq_nat @ one_one_nat @ I3 )
& ( ord_less_eq_nat @ I3 @ N ) ) )
=> ( ( ( groups1842438620at_nat
@ ^ [I2: nat] : ( times_times_nat @ ( P @ I2 ) @ I2 )
@ ( set_ord_atMost_nat @ N ) )
= N )
=> ( number1551313001itions @ P @ N ) ) ) ).
% partitionsI
thf(fact_3_partitions__def,axiom,
( number1551313001itions
= ( ^ [P2: nat > nat,N2: nat] :
( ! [I2: nat] :
( ( ( P2 @ I2 )
!= zero_zero_nat )
=> ( ( ord_less_eq_nat @ one_one_nat @ I2 )
& ( ord_less_eq_nat @ I2 @ N2 ) ) )
& ( ( groups1842438620at_nat
@ ^ [I2: nat] : ( times_times_nat @ ( P2 @ I2 ) @ I2 )
@ ( set_ord_atMost_nat @ N2 ) )
= N2 ) ) ) ) ).
% partitions_def
thf(fact_4_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_5_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_6_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le1613022364et_nat @ ( set_or1086813439et_nat @ X ) @ ( set_or1086813439et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_7_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_8_atMost__iff,axiom,
! [I4: set_nat,K: set_nat] :
( ( member_set_nat @ I4 @ ( set_or1086813439et_nat @ K ) )
= ( ord_less_eq_set_nat @ I4 @ K ) ) ).
% atMost_iff
thf(fact_9_atMost__iff,axiom,
! [I4: nat,K: nat] :
( ( member_nat @ I4 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I4 @ K ) ) ).
% atMost_iff
thf(fact_10_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_11_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_12_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_13_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_14_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_15_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_16_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_17_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_18_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_19_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_20_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_21_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_22_bounded__Max__nat,axiom,
! [P3: nat > $o,X: nat,M2: nat] :
( ( P3 @ X )
=> ( ! [X2: nat] :
( ( P3 @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P3 @ M3 )
=> ~ ! [X3: nat] :
( ( P3 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_23_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K: nat,B: nat] :
( ( P3 @ K )
=> ( ! [Y2: nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P3 @ X2 )
& ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_24_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_25_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_26_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_27_le__trans,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I4 @ K ) ) ) ).
% le_trans
thf(fact_28_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_29_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I4: nat > nat,J: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I4 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat @ ( J @ A3 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J @ ( I4 @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_nat @ ( I4 @ B2 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups1842438620at_nat @ G @ S )
= ( groups1842438620at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_30_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups1842438620at_nat @ Phi @ A )
= ( groups1842438620at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_31_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups1842438620at_nat @ Phi @ A )
= ( groups1842438620at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_32_sum_Ocong,axiom,
! [A: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1842438620at_nat @ G @ A )
= ( groups1842438620at_nat @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_33_sum_Oswap,axiom,
! [G: nat > nat > nat,B3: set_nat,A: set_nat] :
( ( groups1842438620at_nat
@ ^ [I2: nat] : ( groups1842438620at_nat @ ( G @ I2 ) @ B3 )
@ A )
= ( groups1842438620at_nat
@ ^ [J2: nat] :
( groups1842438620at_nat
@ ^ [I2: nat] : ( G @ I2 @ J2 )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_34_mult__right__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_35_mult__left__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_36_no__zero__divisors,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_37_divisors__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_38_mult__not__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_39_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_40_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups1842438620at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_41_sum_Oneutral,axiom,
! [A: set_nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups1842438620at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_42_mem__Collect__eq,axiom,
! [A2: nat,P3: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_44_Collect__cong,axiom,
! [P3: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P3 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P3 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_45_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_46_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_47_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_48_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_49_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_50_mult__le__mono2,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I4 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_51_mult__le__mono1,axiom,
! [I4: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_52_mult__le__mono,axiom,
! [I4: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I4 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_53_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_54_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_55_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_56_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_57_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_58_lambda__one,axiom,
( ( ^ [X4: nat] : X4 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_59_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ K2 ) @ ( groups1842438620at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_60_sum__distrib__right,axiom,
! [F: nat > nat,A: set_nat,R: nat] :
( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ R )
= ( groups1842438620at_nat
@ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
@ A ) ) ).
% sum_distrib_right
thf(fact_61_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A: set_nat] :
( ( times_times_nat @ R @ ( groups1842438620at_nat @ F @ A ) )
= ( groups1842438620at_nat
@ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
@ A ) ) ).
% sum_distrib_left
thf(fact_62_sum__product,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B3: set_nat] :
( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ ( groups1842438620at_nat @ G @ B3 ) )
= ( groups1842438620at_nat
@ ^ [I2: nat] :
( groups1842438620at_nat
@ ^ [J2: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J2 ) )
@ B3 )
@ A ) ) ).
% sum_product
thf(fact_63_atMost__def,axiom,
( set_or1086813439et_nat
= ( ^ [U: set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_64_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_65_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_66_mult__nonneg__nonpos2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_67_mult__nonpos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_68_mult__nonneg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_69_mult__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_70_split__mult__neg__le,axiom,
! [A2: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_71_mult__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_72_mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_73_mult__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_74_mult__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_75_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_76_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_le_one
thf(fact_77_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_78_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1842438620at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_79_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_80_mult__left__le,axiom,
! [C: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_81_mult__le__one,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_82_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_83_mult_Oleft__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult.left_neutral
thf(fact_84_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_85_subsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B3 ) )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% subsetI
thf(fact_86_subset__antisym,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_87_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_88_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_89_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_90_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_91_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_92_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_93_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_94_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_95_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_96_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_97_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_98_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_99_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_100_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_103_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_104_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_105_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_106_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_108_eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : Y4 = Z )
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_109_eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_110_antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_111_antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_112_linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linear
thf(fact_113_eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% eq_refl
thf(fact_114_eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% eq_refl
thf(fact_115_le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% le_cases
thf(fact_116_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_117_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_118_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_119_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 ) ) ) ).
% antisym_conv
thf(fact_120_antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_121_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : Y4 = Z )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_122_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_123_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_124_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_125_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_126_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_127_order__class_Oorder_Oantisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_class.order.antisym
thf(fact_128_order__class_Oorder_Oantisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_class.order.antisym
thf(fact_129_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_130_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_131_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_132_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_133_linorder__wlog,axiom,
! [P3: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P3 @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P3 @ B2 @ A3 )
=> ( P3 @ A3 @ B2 ) )
=> ( P3 @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_134_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_135_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_136_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : Y4 = Z )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_137_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_138_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_139_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_140_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_141_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_142_mult_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_143_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% mult.commute
thf(fact_144_mult_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_145_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_146_Collect__mono__iff,axiom,
! [P3: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P3 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_147_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : Y4 = Z )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_148_subset__trans,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_149_Collect__mono,axiom,
! [P3: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_150_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_151_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_152_equalityD2,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ( ord_less_eq_set_nat @ B3 @ A ) ) ).
% equalityD2
thf(fact_153_equalityD1,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% equalityD1
thf(fact_154_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( member_nat @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_155_equalityE,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B3 )
=> ~ ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% equalityE
thf(fact_156_subsetD,axiom,
! [A: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_157_in__mono,axiom,
! [A: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_158_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_159_Collect__subset,axiom,
! [A: set_nat,P3: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P3 @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_160_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_161_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_162_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_163_subset__Collect__iff,axiom,
! [B3: set_nat,A: set_nat,P3: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ( ord_less_eq_set_nat @ B3
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P3 @ X4 ) ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ B3 )
=> ( P3 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_164_subset__CollectI,axiom,
! [B3: set_nat,A: set_nat,Q: nat > $o,P3: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( Q @ X2 )
=> ( P3 @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( Q @ X4 ) ) )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P3 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_165_Collect__restrict,axiom,
! [X5: set_nat,P3: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P3 @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_166_prop__restrict,axiom,
! [X: nat,Z3: set_nat,X5: set_nat,P3: nat > $o] :
( ( member_nat @ X @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P3 @ X4 ) ) ) )
=> ( P3 @ X ) ) ) ).
% prop_restrict
thf(fact_167_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
@ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_168_conj__subset__def,axiom,
! [A: set_nat,P3: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X4: nat] :
( ( P3 @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P3 ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_169_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_170_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_171_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_172_psubsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_set_nat @ A @ B3 ) ) ) ).
% psubsetI
thf(fact_173_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_174_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_175_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_176_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_177_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_178_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_179_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_180_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_181_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_182_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_183_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_184_linorder__less__wlog,axiom,
! [P3: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P3 @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P3 @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P3 @ B2 @ A3 )
=> ( P3 @ A3 @ B2 ) )
=> ( P3 @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_185_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
? [N2: nat] :
( ( P5 @ N2 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ~ ( P5 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_186_less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_187_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_188_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_189_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_190_less__imp__triv,axiom,
! [X: nat,Y: nat,P3: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P3 ) ) ).
% less_imp_triv
thf(fact_191_less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_192_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_193_less__induct,axiom,
! [P3: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P3 @ Y3 ) )
=> ( P3 @ X2 ) )
=> ( P3 @ A2 ) ) ).
% less_induct
thf(fact_194_less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_not_sym
thf(fact_195_less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_196_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_197_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_198_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_199_less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% less_irrefl
thf(fact_200_less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% less_linear
thf(fact_201_less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% less_trans
thf(fact_202_less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% less_asym'
thf(fact_203_less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_asym
thf(fact_204_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_205_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_206_neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% neq_iff
thf(fact_207_neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% neqE
thf(fact_208_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_209_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_210_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_211_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_212_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_213_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_214_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_215_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_216_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_217_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
& ( M4 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_218_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_219_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
| ( M4 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_220_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_221_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_222_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I4: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I4 @ J )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_223_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_224_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_225_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_226_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_227_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_228_infinite__descent0,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P3 @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P3 @ M5 ) ) ) )
=> ( P3 @ N ) ) ) ).
% infinite_descent0
thf(fact_229_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_230_psubsetE,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% psubsetE
thf(fact_231_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_232_psubset__imp__subset,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% psubset_imp_subset
thf(fact_233_psubset__subset__trans,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_234_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_235_subset__psubset__trans,axiom,
! [A: set_nat,B3: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_236_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_237_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_238_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_239_order_Onot__eq__order__implies__strict,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_240_order_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_241_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_242_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_243_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_244_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_245_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_246_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_247_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_248_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_249_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_250_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_251_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_252_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_253_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_254_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_255_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_256_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_257_order_Ostrict__trans2,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_258_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_259_order_Ostrict__trans1,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_260_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
% Conjectures (1)
thf(conj_0,conjecture,
( ( ! [I2: nat] :
( ( ( p @ I2 )
!= zero_zero_nat )
=> ( ( ord_less_eq_nat @ one_one_nat @ I2 )
& ( ord_less_eq_nat @ I2 @ zero_zero_nat ) ) )
& ( ( groups1842438620at_nat
@ ^ [I2: nat] : ( times_times_nat @ ( p @ I2 ) @ I2 )
@ ( set_ord_atMost_nat @ zero_zero_nat ) )
= zero_zero_nat ) )
= ( p
= ( ^ [I2: nat] : zero_zero_nat ) ) ) ).
%------------------------------------------------------------------------------