TPTP Problem File: ITP134^2.p
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%------------------------------------------------------------------------------
% File : ITP134^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Number_Partition problem prob_94__5324562_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_94__5324562_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 334 ( 74 unt; 44 typ; 0 def)
% Number of atoms : 929 ( 217 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3092 ( 55 ~; 15 |; 57 &;2532 @)
% ( 0 <=>; 433 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 115 ( 115 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 43 usr; 2 con; 0-3 aty)
% Number of variables : 943 ( 87 ^; 802 !; 14 ?; 943 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:27:30.035
%------------------------------------------------------------------------------
% Could-be-implicit typings (2)
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Number__Partition__Mirabelle__ihnzjotehb_Opartitions,type,
number2016821345itions: ( nat > nat ) > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (253)
thf(fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [F: nat > nat] :
( ! [I: nat] : ( ord_less_eq @ nat @ ( F @ I ) @ n )
& ! [I: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ I )
=> ( ( F @ I )
= ( zero_zero @ nat ) ) ) ) ) ) ).
% \<open>finite {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>n + 1. f i = 0)}\<close>
thf(fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
( ord_less_eq @ ( set @ ( nat > nat ) )
@ ( collect @ ( nat > nat )
@ ^ [P: nat > nat] : ( number2016821345itions @ P @ n ) )
@ ( collect @ ( nat > nat )
@ ^ [F: nat > nat] :
( ! [I: nat] : ( ord_less_eq @ nat @ ( F @ I ) @ n )
& ! [I: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) @ I )
=> ( ( F @ I )
= ( zero_zero @ nat ) ) ) ) ) ) ).
% \<open>{p. p partitions n} \<subseteq> {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>n + 1. f i = 0)}\<close>
thf(fact_2_finite__Collect__conjI,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P2 @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3_finite__Collect__disjI,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P2 @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_4_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A2: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_5_partitions__zero,axiom,
! [P3: nat > nat] :
( ( number2016821345itions @ P3 @ ( zero_zero @ nat ) )
= ( P3
= ( ^ [I: nat] : ( zero_zero @ nat ) ) ) ) ).
% partitions_zero
thf(fact_6_partitions__bounds,axiom,
! [P3: nat > nat,N: nat,I2: nat] :
( ( number2016821345itions @ P3 @ N )
=> ( ord_less_eq @ nat @ ( P3 @ I2 ) @ N ) ) ).
% partitions_bounds
thf(fact_7_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B,D: B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F: A > B] :
! [X: A] :
( ( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F @ X ) @ B2 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( F @ X )
= D ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_8_not__finite__existsD,axiom,
! [A: $tType,P2: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ? [X_1: A] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_9_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ B2 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A4: A] :
( ( member @ A @ A4 @ A3 )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_10_finite__Collect__subsets,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_11_finite__set__choice,axiom,
! [B: $tType,A: $tType,A3: set @ A,P2: A > B > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ? [X_12: B] : ( P2 @ X2 @ X_12 ) )
=> ? [F2: A > B] :
! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( P2 @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% finite_set_choice
thf(fact_12_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% finite
thf(fact_13_bound__domain__and__range__impl__finitely__many__functions,axiom,
! [N: nat,M: nat] :
( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [F: nat > nat] :
( ! [I: nat] : ( ord_less_eq @ nat @ ( F @ I ) @ N )
& ! [I: nat] :
( ( ord_less_eq @ nat @ M @ I )
=> ( ( F @ I )
= ( zero_zero @ nat ) ) ) ) ) ) ).
% bound_domain_and_range_impl_finitely_many_functions
thf(fact_14_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_15_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A5: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A5 @ A3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X2 @ A5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_16_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A5: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A5 @ A3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ A5 @ X2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A3 )
=> ( ( ord_less_eq @ A @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_17_rev__finite__subset,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_18_infinite__super,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T ) ) ) ).
% infinite_super
thf(fact_19_finite__subset,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_20_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B4: A,A5: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B4 @ A5 ) @ B4 )
= ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_21_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ B4 ) @ B4 )
= ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_22_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ ( plus_plus @ A @ A5 @ B4 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% le_add_same_cancel1
thf(fact_23_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ ( plus_plus @ A @ B4 @ A5 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% le_add_same_cancel2
thf(fact_24_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A5: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ A5 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_25_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A5: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ A5 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A5 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_26_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_27_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_28_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_29_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_30_bot__nat__0_Oextremum,axiom,
! [A5: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A5 ) ).
% bot_nat_0.extremum
thf(fact_31_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B4: A,A5: A,C: A] :
( ( ( plus_plus @ A @ B4 @ A5 )
= ( plus_plus @ A @ C @ A5 ) )
= ( B4 = C ) ) ) ).
% add_right_cancel
thf(fact_32_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ( plus_plus @ A @ A5 @ B4 )
= ( plus_plus @ A @ A5 @ C ) )
= ( B4 = C ) ) ) ).
% add_left_cancel
thf(fact_33_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_34_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A5: A,C: A,B4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) )
= ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% add_le_cancel_right
thf(fact_35_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A5: A,B4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) )
= ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% add_le_cancel_left
thf(fact_36_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X4: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X4 @ Y ) )
= ( ( X4
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_37_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X4: A,Y: A] :
( ( ( plus_plus @ A @ X4 @ Y )
= ( zero_zero @ A ) )
= ( ( X4
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_38_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A5: A,B4: A] :
( ( A5
= ( plus_plus @ A @ A5 @ B4 ) )
= ( B4
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_39_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A5: A,B4: A] :
( ( A5
= ( plus_plus @ A @ B4 @ A5 ) )
= ( B4
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_40_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A5: A,B4: A] :
( ( ( plus_plus @ A @ A5 @ B4 )
= A5 )
= ( B4
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_41_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [B4: A,A5: A] :
( ( ( plus_plus @ A @ B4 @ A5 )
= A5 )
= ( B4
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_42_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A5: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A5 @ A5 ) )
= ( A5
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A5: A,P2: A > $o] :
( ( member @ A @ A5 @ ( collect @ A @ P2 ) )
= ( P2 @ A5 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A5: A] :
( ( ( plus_plus @ A @ A5 @ A5 )
= ( zero_zero @ A ) )
= ( A5
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_47_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A5: A] :
( ( plus_plus @ A @ A5 @ ( zero_zero @ A ) )
= A5 ) ) ).
% add.right_neutral
thf(fact_48_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A5: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A5 )
= A5 ) ) ).
% add.left_neutral
thf(fact_49_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X4: A] :
( ( ( zero_zero @ A )
= X4 )
= ( X4
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_50_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B4: A,A5: A,C: A] :
( ( ( plus_plus @ A @ B4 @ A5 )
= ( plus_plus @ A @ C @ A5 ) )
=> ( B4 = C ) ) ) ).
% add_right_imp_eq
thf(fact_51_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ( plus_plus @ A @ A5 @ B4 )
= ( plus_plus @ A @ A5 @ C ) )
=> ( B4 = C ) ) ) ).
% add_left_imp_eq
thf(fact_52_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B4: A,A5: A,C: A] :
( ( plus_plus @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) )
= ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% add.left_commute
thf(fact_53_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B5: A] : ( plus_plus @ A @ B5 @ A4 ) ) ) ) ).
% add.commute
thf(fact_54_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B4: A,A5: A,C: A] :
( ( ( plus_plus @ A @ B4 @ A5 )
= ( plus_plus @ A @ C @ A5 ) )
= ( B4 = C ) ) ) ).
% add.right_cancel
thf(fact_55_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ( plus_plus @ A @ A5 @ B4 )
= ( plus_plus @ A @ A5 @ C ) )
= ( B4 = C ) ) ) ).
% add.left_cancel
thf(fact_56_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A5: A,B4: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C )
= ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% add.assoc
thf(fact_57_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: A,K: A,B4: A,A5: A] :
( ( B2
= ( plus_plus @ A @ K @ B4 ) )
=> ( ( plus_plus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_58_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A5: A,B4: A] :
( ( A3
= ( plus_plus @ A @ K @ A5 ) )
=> ( ( plus_plus @ A @ A3 @ B4 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_59_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I2 @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_60_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A5: A,B4: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C )
= ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_61_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X4: A] :
( ( ( one_one @ A )
= X4 )
= ( X4
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_62_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B4: nat] :
( ( P2 @ K )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B4 ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_63_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_64_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_65_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_66_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_67_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_68_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) ) ).
% zero_le
thf(fact_69_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A5: A,C: A,B4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) )
=> ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% add_le_imp_le_right
thf(fact_70_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A5: A,B4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) )
=> ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% add_le_imp_le_left
thf(fact_71_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B5: A] :
? [C2: A] :
( B5
= ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% le_iff_add
thf(fact_72_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% add_right_mono
thf(fact_73_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ~ ! [C3: A] :
( B4
!= ( plus_plus @ A @ A5 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_74_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) ) ) ) ).
% add_left_mono
thf(fact_75_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A5: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ D ) ) ) ) ) ).
% add_mono
thf(fact_76_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_77_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_78_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_79_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A5: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A5 )
= A5 ) ) ).
% add.group_left_neutral
thf(fact_80_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: A] :
( ( plus_plus @ A @ A5 @ ( zero_zero @ A ) )
= A5 ) ) ).
% add.comm_neutral
thf(fact_81_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A5 )
= A5 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_82_bot__nat__0_Oextremum__uniqueI,axiom,
! [A5: nat] :
( ( ord_less_eq @ nat @ A5 @ ( zero_zero @ nat ) )
=> ( A5
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_83_bot__nat__0_Oextremum__unique,axiom,
! [A5: nat] :
( ( ord_less_eq @ nat @ A5 @ ( zero_zero @ nat ) )
= ( A5
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_84_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_85_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_86_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_87_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_88_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_89_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_90_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_91_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_92_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_93_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_94_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_95_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_96_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_97_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_98_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_99_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X4 @ Y )
= ( zero_zero @ A ) )
= ( ( X4
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_100_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X4: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X4 @ Y )
= ( zero_zero @ A ) )
= ( ( X4
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_101_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B4 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_102_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_103_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C: A,B4: A,A5: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ( ord_less_eq @ A @ B4 @ A5 )
=> ( ord_less_eq @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) ) ) ) ) ).
% add_increasing2
thf(fact_104_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C: A,A5: A,B4: A] :
( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ B4 ) ) ) ) ).
% add_decreasing2
thf(fact_105_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A5 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less_eq @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) ) ) ) ) ).
% add_increasing
thf(fact_106_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A5: A,C: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C @ B4 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C ) @ B4 ) ) ) ) ).
% add_decreasing
thf(fact_107_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I3: set @ B,X4: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I3 )
& ( ( X4 @ I )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I3 )
& ( ( Y @ I )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I3 )
& ( ( plus_plus @ A @ ( X4 @ I ) @ ( Y @ I ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_108_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_109_zero__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_le_one
thf(fact_110_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F3 @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ ( F3 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_111_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_112_subsetI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% subsetI
thf(fact_113_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_refl
thf(fact_114_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N4: set @ nat] :
? [M2: nat] :
! [X: nat] :
( ( member @ nat @ X @ N4 )
=> ( ord_less_eq @ nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_115_Euclid__induct,axiom,
! [P2: nat > nat > $o,A5: nat,B4: nat] :
( ! [A6: nat,B6: nat] :
( ( P2 @ A6 @ B6 )
= ( P2 @ B6 @ A6 ) )
=> ( ! [A6: nat] : ( P2 @ A6 @ ( zero_zero @ nat ) )
=> ( ! [A6: nat,B6: nat] :
( ( P2 @ A6 @ B6 )
=> ( P2 @ A6 @ ( plus_plus @ nat @ A6 @ B6 ) ) )
=> ( P2 @ A5 @ B4 ) ) ) ) ).
% Euclid_induct
thf(fact_116_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funD
thf(fact_117_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funE
thf(fact_118_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% le_funI
thf(fact_119_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_120_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A5: A,F3: B > A,B4: B,C: B] :
( ( ord_less_eq @ A @ A5 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_121_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A5: A,B4: A,F3: A > C4,C: C4] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( ord_less_eq @ C4 @ ( F3 @ B4 ) @ C )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ C4 @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ C4 @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_122_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A5: A,F3: B > A,B4: B,C: B] :
( ( A5
= ( F3 @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_123_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A5: A,B4: A,F3: A > B,C: B] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( ( F3 @ B4 )
= C )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_124_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_125_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X4 )
=> ( X4 = Y ) ) ) ) ).
% antisym
thf(fact_126_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
| ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% linear
thf(fact_127_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y: A] :
( ( X4 = Y )
=> ( ord_less_eq @ A @ X4 @ Y ) ) ) ).
% eq_refl
thf(fact_128_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y: A] :
( ~ ( ord_less_eq @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% le_cases
thf(fact_129_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less_eq @ A @ A5 @ C ) ) ) ) ).
% order.trans
thf(fact_130_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X4 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X4 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_131_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X4: A] :
( ( ord_less_eq @ A @ Y @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ Y )
= ( X4 = Y ) ) ) ) ).
% antisym_conv
thf(fact_132_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A4: A,B5: A] :
( ( ord_less_eq @ A @ A4 @ B5 )
& ( ord_less_eq @ A @ B5 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_133_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A5: A,B4: A,C: A] :
( ( A5 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less_eq @ A @ A5 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_134_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq @ A @ A5 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_135_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_136_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_137_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% dual_order.refl
thf(fact_138_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A5: A,B4: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P2 @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P2 @ B6 @ A6 )
=> ( P2 @ A6 @ B6 ) )
=> ( P2 @ A5 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_139_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A5: A,C: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
=> ( ( ord_less_eq @ A @ C @ B4 )
=> ( ord_less_eq @ A @ C @ A5 ) ) ) ) ).
% dual_order.trans
thf(fact_140_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A4: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A4 )
& ( ord_less_eq @ A @ A4 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_141_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
=> ( ( ord_less_eq @ A @ A5 @ B4 )
=> ( A5 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_142_in__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_143_subsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B2 ) ) ) ).
% subsetD
thf(fact_144_equalityE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_145_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A2: set @ A,B3: set @ A] :
! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_146_equalityD1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_147_equalityD2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_148_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A2: set @ A,B3: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A2 )
=> ( member @ A @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_149_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_150_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_151_subset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% subset_trans
thf(fact_152_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z: set @ A] : Y4 = Z )
= ( ^ [A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_153_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_154_bounded__Max__nat,axiom,
! [P2: nat > $o,X4: nat,M3: nat] :
( ( P2 @ X4 )
=> ( ! [X2: nat] :
( ( P2 @ X2 )
=> ( ord_less_eq @ nat @ X2 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_155_Collect__subset,axiom,
! [A: $tType,A3: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P2 @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_156_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A2: set @ A,B3: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A2 )
@ ^ [X: A] : ( member @ A @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_157_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_158_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_159_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A5: A] :
( ( plus_plus @ A @ A5 @ ( zero_zero @ A ) )
= A5 ) ) ).
% verit_sum_simplify
thf(fact_160_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [B4: A,A5: A] :
( ( B4
= ( plus_plus @ A @ B4 @ A5 ) )
= ( A5
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_161_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_162_predicate1I,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% predicate1I
thf(fact_163_predicate1D,axiom,
! [A: $tType,P2: A > $o,Q: A > $o,X4: A] :
( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ).
% predicate1D
thf(fact_164_rev__predicate1D,axiom,
! [A: $tType,P2: A > $o,X4: A,Q: A > $o] :
( ( P2 @ X4 )
=> ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( Q @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_165_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: A,B4: A] :
( ( A5 = B4 )
| ~ ( ord_less_eq @ A @ A5 @ B4 )
| ~ ( ord_less_eq @ A @ B4 @ A5 ) ) ) ).
% verit_la_disequality
thf(fact_166_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A5: A,B4: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B4 ) @ C )
= ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B4 @ C ) ) ) ) ).
% is_num_normalize(1)
thf(fact_167_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R )
@ ^ [X: A] : ( member @ A @ X @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_168_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_169_conj__subset__def,axiom,
! [A: $tType,A3: set @ A,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3
@ ( collect @ A
@ ^ [X: A] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P2 ) )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_170_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X: A] : ( plus_plus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_171_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A3: set @ A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P2 @ X ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ B2 )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_172_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A3: set @ A,Q: A > $o,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( ( Q @ X2 )
=> ( P2 @ X2 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ B2 )
& ( Q @ X ) ) )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A3 )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_173_Collect__restrict,axiom,
! [A: $tType,X5: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_174_prop__restrict,axiom,
! [A: $tType,X4: A,Z3: set @ A,X5: set @ A,P2: A > $o] :
( ( member @ A @ X4 @ Z3 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z3
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X4 ) ) ) ).
% prop_restrict
thf(fact_175_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C4: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C4,G: C4] :
( ( F3
= ( ^ [X: A > B] : G ) )
=> ( ( F3
@ ^ [X: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_176_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A2: set @ A] :
( collect @ ( set @ A )
@ ^ [X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ A2 )
& ( finite_finite2 @ A @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_177_partitions__imp__finite__elements,axiom,
! [P3: nat > nat,N: nat] :
( ( number2016821345itions @ P3 @ N )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( P3 @ I ) ) ) ) ) ).
% partitions_imp_finite_elements
thf(fact_178_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_179_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A5: A,C: A,B4: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ C ) )
= ( ord_less @ A @ A5 @ B4 ) ) ) ).
% add_less_cancel_right
thf(fact_180_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A5: A,B4: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A5 ) @ ( plus_plus @ A @ C @ B4 ) )
= ( ord_less @ A @ A5 @ B4 ) ) ) ).
% add_less_cancel_left
thf(fact_181_bot__nat__0_Onot__eq__extremum,axiom,
! [A5: nat] :
( ( A5
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A5 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_182_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_183_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_184_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_185_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_186_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B4: A,A5: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B4 @ A5 ) @ B4 )
= ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_187_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A5: A,B4: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A5 @ B4 ) @ B4 )
= ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_188_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ ( plus_plus @ A @ A5 @ B4 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% less_add_same_cancel1
thf(fact_189_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ ( plus_plus @ A @ B4 @ A5 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B4 ) ) ) ).
% less_add_same_cancel2
thf(fact_190_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A5: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A5 @ A5 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_191_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ A5 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A5 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_192_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_193_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_194_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A5: A,B4: A,C: A,D: A] :
( ( ord_less @ A @ A5 @ B4 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_195_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A5: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A5 @ C ) @ ( plus_plus @ A @ B4 @ D ) ) ) ) ) ).
% add_le_less_mono
thf(fact_196_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_197_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_198_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ B4 @ ( plus_plus @ A @ A5 @ C ) ) ) ) ) ).
% pos_add_strict
thf(fact_199_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
=> ~ ! [C3: A] :
( ( B4
= ( plus_plus @ A @ A5 @ C3 ) )
=> ( C3
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_200_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A5: A,B4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A5 @ B4 ) ) ) ) ) ).
% add_pos_pos
thf(fact_201_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B4 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_202_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X4: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X4 @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X4 @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_203_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B4: A] :
( ( A5 != B4 )
=> ( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ord_less @ A @ A5 @ B4 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_204_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
=> ( ord_less_eq @ A @ B4 @ A5 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_205_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A4: A] :
( ( ord_less_eq @ A @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_206_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A4: A] :
( ( ord_less @ A @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_207_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
=> ( ord_less_eq @ A @ A5 @ B4 ) ) ) ).
% order.strict_implies_order
thf(fact_208_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X4: A,Y: A,Z2: A] :
( ( ord_less @ A @ X4 @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X4 @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_209_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X4: A,Y: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X4 )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_210_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A5: A,C: A] :
( ( ord_less @ A @ B4 @ A5 )
=> ( ( ord_less_eq @ A @ C @ B4 )
=> ( ord_less @ A @ C @ A5 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_211_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A5: A,C: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
=> ( ( ord_less @ A @ C @ B4 )
=> ( ord_less @ A @ C @ A5 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_212_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B5: A] :
( ( ord_less_eq @ A @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_213_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B5: A] :
( ( ord_less @ A @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_214_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ord_less @ A @ A5 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less @ A @ A5 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_215_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ A5 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_216_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X4: A] :
( ~ ( ord_less_eq @ A @ Y @ X4 )
=> ( ord_less @ A @ X4 @ Y ) ) ) ).
% not_le_imp_less
thf(fact_217_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ~ ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% less_le_not_le
thf(fact_218_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( ( ord_less @ A @ X4 @ Y )
| ( X4 = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_219_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
| ( ord_less @ A @ Y @ X4 ) ) ) ).
% le_less_linear
thf(fact_220_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_221_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_222_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y: A,Z2: A] :
( ( ord_less @ A @ X4 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_223_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_224_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y: A] :
( ( ord_less @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ X4 @ Y ) ) ) ).
% less_imp_le
thf(fact_225_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( ( ~ ( ord_less @ A @ X4 @ Y ) )
= ( X4 = Y ) ) ) ) ).
% antisym_conv2
thf(fact_226_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y: A] :
( ~ ( ord_less @ A @ X4 @ Y )
=> ( ( ord_less_eq @ A @ X4 @ Y )
= ( X4 = Y ) ) ) ) ).
% antisym_conv1
thf(fact_227_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( A5 != B4 )
=> ( ord_less @ A @ A5 @ B4 ) ) ) ) ).
% le_neq_trans
thf(fact_228_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y: A] :
( ( ~ ( ord_less @ A @ X4 @ Y ) )
= ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% not_less
thf(fact_229_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X4 @ Y ) )
= ( ord_less @ A @ Y @ X4 ) ) ) ).
% not_le
thf(fact_230_order__less__le__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A5: A,B4: A,F3: A > C4,C: C4] :
( ( ord_less @ A @ A5 @ B4 )
=> ( ( ord_less_eq @ C4 @ ( F3 @ B4 ) @ C )
=> ( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ C4 @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C4 @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_231_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A5: A,F3: B > A,B4: B,C: B] :
( ( ord_less @ A @ A5 @ ( F3 @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_232_order__le__less__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A5: A,B4: A,F3: A > C4,C: C4] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ( ord_less @ C4 @ ( F3 @ B4 ) @ C )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ C4 @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C4 @ ( F3 @ A5 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_233_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A5: A,F3: B > A,B4: B,C: B] :
( ( ord_less_eq @ A @ A5 @ ( F3 @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C )
=> ( ! [X2: B,Y2: B] :
( ( ord_less @ B @ X2 @ Y2 )
=> ( ord_less @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A5 @ ( F3 @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_234_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ( X != Y5 ) ) ) ) ) ).
% less_le
thf(fact_235_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y5: A] :
( ( ord_less @ A @ X @ Y5 )
| ( X = Y5 ) ) ) ) ) ).
% le_less
thf(fact_236_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y: A] :
( ~ ( ord_less @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% leI
thf(fact_237_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X4: A] :
( ( ord_less_eq @ A @ Y @ X4 )
=> ~ ( ord_less @ A @ X4 @ Y ) ) ) ).
% leD
thf(fact_238_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_239_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_240_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_241_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_242_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_243_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_244_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I2: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less @ nat @ I4 @ J2 )
=> ( ord_less @ nat @ ( F3 @ I4 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( F3 @ I2 ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_245_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A] :
~ ( ord_less @ A @ A5 @ A5 ) ) ).
% verit_comp_simplify1(1)
thf(fact_246_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_247_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B7: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) )
= ( ord_less @ B @ A7 @ B7 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_248_bounded__nat__set__is__finite,axiom,
! [N5: set @ nat,N: nat] :
( ! [X2: nat] :
( ( member @ nat @ X2 @ N5 )
=> ( ord_less @ nat @ X2 @ N ) )
=> ( finite_finite2 @ nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_249_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N4: set @ nat] :
? [M2: nat] :
! [X: nat] :
( ( member @ nat @ X @ N4 )
=> ( ord_less @ nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_250_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( P2 @ K2 )
& ( ord_less @ nat @ K2 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_251_infinite__descent__measure,axiom,
! [A: $tType,P2: A > $o,V: A > nat,X4: A] :
( ! [X2: A] :
( ~ ( P2 @ X2 )
=> ? [Y3: A] :
( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
& ~ ( P2 @ Y3 ) ) )
=> ( P2 @ X4 ) ) ).
% infinite_descent_measure
thf(fact_252_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P2: A > $o,A5: A] :
( ! [X2: A] :
( ! [Y3: A] :
( ( ord_less @ B @ ( F3 @ Y3 ) @ ( F3 @ X2 ) )
=> ( P2 @ Y3 ) )
=> ( P2 @ X2 ) )
=> ( P2 @ A5 ) ) ) ).
% measure_induct_rule
% Type constructors (36)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_10,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
ord @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
( finite_finite2 @ ( nat > nat )
@ ( collect @ ( nat > nat )
@ ^ [P: nat > nat] : ( number2016821345itions @ P @ n ) ) ) ).
%------------------------------------------------------------------------------