TPTP Problem File: ITP056^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP056^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer FLPTheorem problem prob_1320__3306644_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 : FLPTheorem/prob_1320__3306644_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 342 ( 70 unt; 49 typ; 0 def)
% Number of atoms : 841 ( 210 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3320 ( 66 ~; 13 |; 35 &;2740 @)
% ( 0 <=>; 466 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 94 ( 94 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 44 usr; 2 con; 0-4 aty)
% Number of variables : 1025 ( 36 ^; 911 !; 39 ?;1025 :)
% ( 39 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:17:54.899
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_t_AsynchronousSystem_Oconfiguration_Oconfiguration__ext,type,
configuration_ext: $tType > $tType > $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_v,type,
v: $tType ).
thf(ty_tf_s,type,
s: $tType ).
thf(ty_tf_p,type,
p: $tType ).
% Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[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_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[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_Groups_Oab__group__add,type,
ab_group_add:
!>[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_Orderings_Odense__order,type,
dense_order:
!>[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_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_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_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__cancel__comm__monoid__diff,type,
ordere623563068d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
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_ListUtilities_OprefixList,type,
prefixList:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
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_v_fe____,type,
fe: nat > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ).
thf(sy_v_n0____,type,
n0: nat ).
thf(sy_v_n____,type,
n: nat ).
% Relevant facts (254)
thf(fact_0_FeNNotEmpty,axiom,
( ( fe @ n )
!= ( nil @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) ).
% FeNNotEmpty
thf(fact_1_LastIsLastIndex,axiom,
! [A2: $tType,L: list @ A2] :
( ( L
!= ( nil @ A2 ) )
=> ( ( last @ A2 @ L )
= ( nth @ A2 @ L @ ( minus_minus @ nat @ ( size_size @ ( list @ A2 ) @ L ) @ ( one_one @ nat ) ) ) ) ) ).
% LastIsLastIndex
thf(fact_2_Fou2,axiom,
ord_less @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) ).
% Fou2
thf(fact_3_AssmNoDecided_I1_J,axiom,
ord_less @ nat @ n0 @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) ).
% AssmNoDecided(1)
thf(fact_4_last__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_5_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A3: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% minus_apply
thf(fact_6_Fou,axiom,
ord_less_eq @ nat @ n0 @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) ).
% Fou
thf(fact_7_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_8_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_9_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_10_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_11_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X2: A,Y: A] :
( ( ( size_size @ A @ X2 )
!= ( size_size @ A @ Y ) )
=> ( X2 != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_12_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A3: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% fun_diff_def
thf(fact_13_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_14_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_15_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_16_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_17_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_18_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_19_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_20_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_21_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_22_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_23_diff__less__mono,axiom,
! [A4: nat,B3: nat,C: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( ( ord_less_eq @ nat @ C @ A4 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A4 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_24_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_25_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_26_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y2: A] :
( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% measure_induct
thf(fact_27_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_28_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_29_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_30_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_31_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_32_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_33_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less @ nat @ X2 @ Y )
=> ( ord_less @ nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_34_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_35_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_36_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y2: A] :
( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% measure_induct_rule
thf(fact_37_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_38_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X2: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y2: A] :
( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
& ~ ( P @ Y2 ) ) )
=> ( P @ X2 ) ) ).
% infinite_descent_measure
thf(fact_39_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_40_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A,C: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A4 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_41_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C: A,D: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less @ A @ A4 @ B3 )
= ( ord_less @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_42_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,D: A,C: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ D @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_43_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_44_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C: A,D: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
= ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_45_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_46_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A4 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_47_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,D: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ D @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_48_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_49_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_50_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_51_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_52_le__diff__iff_H,axiom,
! [A4: nat,C: nat,B3: nat] :
( ( ord_less_eq @ nat @ A4 @ C )
=> ( ( ord_less_eq @ nat @ B3 @ C )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A4 ) @ ( minus_minus @ nat @ C @ B3 ) )
= ( ord_less_eq @ nat @ B3 @ A4 ) ) ) ) ).
% le_diff_iff'
thf(fact_53_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_54_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_55_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_56_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_57_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_58_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z: list @ A] : Y4 = Z )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I3 )
= ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_59_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ? [X4: A] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( P @ I3 @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_60_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_61_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X2: A] :
( ( ( one_one @ A )
= X2 )
= ( X2
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_62_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A4: A,C: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_63_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C: A,D: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( A4 = B3 )
= ( C = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_64_Length,axiom,
! [N4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ N4 ) ) ) ).
% Length
thf(fact_65_AllArePrefixesExec,axiom,
! [M3: nat,N4: nat] :
( ( ord_less @ nat @ M3 @ N4 )
=> ( prefixList @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ M3 ) @ ( fe @ N4 ) ) ) ).
% AllArePrefixesExec
thf(fact_66_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_67_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_68_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F @ Y3 ) @ B3 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_69_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_70_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_71_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% minf(8)
thf(fact_72_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% minf(6)
thf(fact_73_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% pinf(8)
thf(fact_74_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% pinf(6)
thf(fact_75_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C ) )
= ( B3 = C ) ) ) ).
% add_left_cancel
thf(fact_76_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A4: A,C: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C @ A4 ) )
= ( B3 = C ) ) ) ).
% add_right_cancel
thf(fact_77_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A4: A,C: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_78_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_79_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A4: A,C: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_80_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_81_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ).
% add_diff_cancel
thf(fact_82_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ).
% diff_add_cancel
thf(fact_83_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [C: A,A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
= ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% add_diff_cancel_left
thf(fact_84_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ A4 )
= B3 ) ) ).
% add_diff_cancel_left'
thf(fact_85_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A4: A,C: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
= ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% add_diff_cancel_right
thf(fact_86_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ).
% add_diff_cancel_right'
thf(fact_87_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_88_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_89_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_90_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_91_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_92_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_93_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_94_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_95_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: A,K: A,A4: A,B3: A] :
( ( A5
= ( plus_plus @ A @ K @ A4 ) )
=> ( ( plus_plus @ A @ A5 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_96_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: A,K: A,B3: A,A4: A] :
( ( B4
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( plus_plus @ A @ A4 @ B4 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_97_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% add.assoc
thf(fact_98_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C ) )
= ( B3 = C ) ) ) ).
% add.left_cancel
thf(fact_99_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B3: A,A4: A,C: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C @ A4 ) )
= ( B3 = C ) ) ) ).
% add.right_cancel
thf(fact_100_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% add.commute
thf(fact_101_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B3: A,A4: A,C: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A4 @ C ) )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% add.left_commute
thf(fact_102_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C ) )
=> ( B3 = C ) ) ) ).
% add_left_imp_eq
thf(fact_103_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A4: A,C: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C @ A4 ) )
=> ( B3 = C ) ) ) ).
% add_right_imp_eq
thf(fact_104_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A4: A,B3: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% is_num_normalize(1)
thf(fact_105_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A4: A,C: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_106_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_107_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
? [C2: A] :
( B5
= ( plus_plus @ A @ A6 @ C2 ) ) ) ) ) ).
% le_iff_add
thf(fact_108_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% add_right_mono
thf(fact_109_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ~ ! [C3: A] :
( B3
!= ( plus_plus @ A @ A4 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_110_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% add_left_mono
thf(fact_111_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A4: A,B3: A,C: A,D: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% add_mono
thf(fact_112_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_113_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_114_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_115_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A4: A,C: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_116_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_117_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_118_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A4 ) @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_119_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A )
=> ! [A4: A,B3: A,C: A,D: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_120_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_121_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_122_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_123_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K: A,A4: A,B3: A] :
( ( A5
= ( plus_plus @ A @ K @ A4 ) )
=> ( ( minus_minus @ A @ A5 @ B3 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_124_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= C )
= ( A4
= ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% diff_eq_eq
thf(fact_125_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,C: A,B3: A] :
( ( A4
= ( minus_minus @ A @ C @ B3 ) )
= ( ( plus_plus @ A @ A4 @ B3 )
= C ) ) ) ).
% eq_diff_eq
thf(fact_126_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ C ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C ) ) ) ).
% add_diff_eq
thf(fact_127_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( minus_minus @ A @ A4 @ ( minus_minus @ A @ B3 @ C ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C ) @ B3 ) ) ) ).
% diff_diff_eq2
thf(fact_128_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C ) @ B3 ) ) ) ).
% diff_add_eq
thf(fact_129_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C ) @ B3 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_130_diff__diff__add,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
= ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C ) ) ) ) ).
% diff_diff_add
thf(fact_131_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [C: A,B3: A,A4: A] :
( ( ( plus_plus @ A @ C @ B3 )
= A4 )
=> ( C
= ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% add_implies_diff
thf(fact_132_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_133_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_134_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_135_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_136_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_137_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_138_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_139_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_140_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_141_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_142_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_143_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_144_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_145_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_146_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_147_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_148_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_149_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_150_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_151_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_152_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_153_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_154_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_155_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y2: A] :
( ( P @ Y2 )
& ~ ( ord_less_eq @ nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less @ nat @ ( F @ Y3 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_156_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A4: A,B3: A,C: A,D: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_157_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A4: A,B3: A,C: A,D: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C ) @ ( plus_plus @ A @ B3 @ D ) ) ) ) ) ).
% add_le_less_mono
thf(fact_158_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_159_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_160_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( minus_minus @ A @ B3 @ A4 )
= C )
= ( B3
= ( plus_plus @ A @ C @ A4 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_161_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ A4 ) )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_162_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C @ A4 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_163_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C ) @ A4 )
= ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_164_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C )
= ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C ) @ A4 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_165_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C @ B3 ) @ A4 )
= ( plus_plus @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_166_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C @ B3 ) @ A4 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_167_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C @ ( minus_minus @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A4 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_168_le__add__diff,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ C @ ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C ) @ A4 ) ) ) ) ).
% le_add_diff
thf(fact_169_diff__add,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ A4 )
= B3 ) ) ) ).
% diff_add
thf(fact_170_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,C: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( minus_minus @ A @ C @ B3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C ) ) ) ).
% le_diff_eq
thf(fact_171_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
= ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% diff_le_eq
thf(fact_172_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,C: A,B3: A] :
( ( ord_less @ A @ A4 @ ( minus_minus @ A @ C @ B3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C ) ) ) ).
% less_diff_eq
thf(fact_173_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C )
= ( ord_less @ A @ A4 @ ( plus_plus @ A @ C @ B3 ) ) ) ) ).
% diff_less_eq
thf(fact_174_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
=> ( ord_less @ nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_175_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_176_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_177_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( minus_minus @ nat @ J @ I )
= K )
= ( J
= ( plus_plus @ nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_178_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_179_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_180_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_181_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_182_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_183_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_184_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_185_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_186_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A7: A,B6: A] :
( ( ord_less_eq @ A @ A7 @ B6 )
=> ( P @ A7 @ B6 ) )
=> ( ! [A7: A,B6: A] :
( ( P @ B6 @ A7 )
=> ( P @ A7 @ B6 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_187_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_188_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% order_trans
thf(fact_189_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_190_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_191_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_192_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_193_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_194_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_195_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% order.trans
thf(fact_196_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_197_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_198_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_199_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_200_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_201_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_202_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_203_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A4: A,B3: A,F: A > C4,C: C4] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C4 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C4 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C4 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_204_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_205_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_206_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_207_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_208_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_209_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( A4 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_210_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( A4 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_211_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( ( ord_less @ A @ Y @ X2 )
| ( X2 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_212_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A,C: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_213_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A7: A,B6: A] :
( ( ord_less @ A @ A7 @ B6 )
=> ( P @ A7 @ B6 ) )
=> ( ! [A7: A] : ( P @ A7 @ A7 )
=> ( ! [A7: A,B6: A] :
( ( P @ B6 @ A7 )
=> ( P @ A7 @ B6 ) )
=> ( P @ A4 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_214_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X4: A] : ( P2 @ X4 ) )
= ( ^ [P3: A > $o] :
? [N2: A] :
( ( P3 @ N2 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_215_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_imp_not_less
thf(fact_216_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_217_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_218_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_cases
thf(fact_219_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,P: $o] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ X2 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_220_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( Y != X2 ) ) ) ).
% less_imp_not_eq2
thf(fact_221_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ~ ( ord_less @ A @ Y @ X2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( X2 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_222_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X3 )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_223_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_not_sym
thf(fact_224_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_not_eq
thf(fact_225_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_226_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_227_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C: A] :
( ( A4 = B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_228_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 ) ) ).
% less_irrefl
thf(fact_229_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_linear
thf(fact_230_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z3: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X2 @ Z3 ) ) ) ) ).
% less_trans
thf(fact_231_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% less_asym'
thf(fact_232_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_asym
thf(fact_233_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_neq
thf(fact_234_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ? [Z2: A] :
( ( ord_less @ A @ X2 @ Z2 )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% dense
thf(fact_235_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% order.asym
thf(fact_236_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ( ord_less @ A @ X2 @ Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neq_iff
thf(fact_237_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neqE
thf(fact_238_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X2: A] :
? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% gt_ex
thf(fact_239_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X2: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ).
% lt_ex
thf(fact_240_order__less__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A4: A,B3: A,F: A > C4,C: C4] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ C4 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C4 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C4 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_241_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_242_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F: A > B,C: B] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_243_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_244_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_245_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_246_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(3)
thf(fact_247_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(4)
thf(fact_248_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% pinf(5)
thf(fact_249_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less @ A @ T @ X5 ) ) ) ).
% pinf(7)
thf(fact_250_pinf_I11_J,axiom,
! [C4: $tType,D2: $tType] :
( ( ord @ C4 )
=> ! [F3: D2] :
? [Z2: C4] :
! [X5: C4] :
( ( ord_less @ C4 @ Z2 @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_251_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_252_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_253_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( X5 != T ) ) ) ).
% minf(3)
% Type constructors (38)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A2: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A2 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A2: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A2 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A2: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A2 > A8 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A2: $tType,A8: $tType] :
( ( minus @ A8 )
=> ( minus @ ( A2 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
ordere623563068d_diff @ 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_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_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_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___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ 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_Ominus_4,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_9,axiom,
minus @ $o ).
thf(tcon_List_Olist___Nat_Osize_10,axiom,
! [A2: $tType] : ( size @ ( list @ A2 ) ) ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_11,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_12,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_13,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_14,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_15,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_16,axiom,
minus @ product_unit ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( last @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) )
= ( nth @ ( configuration_ext @ p @ v @ s @ product_unit ) @ ( fe @ n ) @ ( minus_minus @ nat @ ( size_size @ ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) @ ( fe @ n ) ) @ ( one_one @ nat ) ) ) ) ).
%------------------------------------------------------------------------------