TPTP Problem File: ITP139^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP139^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Paraconsistency problem prob_881__3274616_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 : Paraconsistency/prob_881__3274616_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 333 ( 61 unt; 45 typ; 0 def)
% Number of atoms : 982 ( 226 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3572 ( 61 ~; 12 |; 58 &;2921 @)
% ( 0 <=>; 520 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 182 ( 182 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 43 usr; 0 con; 1-7 aty)
% Number of variables : 1161 ( 56 ^;1040 !; 15 ?;1161 :)
% ( 50 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:24:02.098
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
paraco415392788lle_tv: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (41)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[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_Oorder__top,type,
order_top:
!>[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_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_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[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_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : $o ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
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_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ochange__tv,type,
paraco1920534163nge_tv: ( nat > nat ) > paraco415392788lle_tv > paraco415392788lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_OIndet,type,
paraco676387099_Indet: nat > paraco415392788lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
paraco152590079rec_tv:
!>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: nat > nat ).
% Relevant facts (256)
thf(fact_0__092_060open_062_092_060And_062tv2_Atv1_O_Achange__tv_Af_Atv1_A_061_Achange__tv_Af_Atv2_A_092_060Longrightarrow_062_Atv1_A_061_Atv2_092_060close_062,axiom,
! [Tv1: paraco415392788lle_tv,Tv2: paraco415392788lle_tv] :
( ( ( paraco1920534163nge_tv @ f @ Tv1 )
= ( paraco1920534163nge_tv @ f @ Tv2 ) )
=> ( Tv1 = Tv2 ) ) ).
% \<open>\<And>tv2 tv1. change_tv f tv1 = change_tv f tv2 \<Longrightarrow> tv1 = tv2\<close>
thf(fact_1_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_2_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_3_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_4_injD,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_5_injI,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ! [X3: A,Y2: A] :
( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% injI
thf(fact_6_inj__eq,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_7_inj__def,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( ! [X2: A,Y3: A] :
( ( ( F @ X2 )
= ( F @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inj_def
thf(fact_8_inj__onD,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member @ A @ X @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_9_inj__onI,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y2 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( X3 = Y2 ) ) ) )
=> ( inj_on @ A @ B @ F @ A2 ) ) ).
% inj_onI
thf(fact_10_inj__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F2: A > B,A3: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y3 ) )
=> ( X2 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_11_inj__on__cong,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B] :
( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on @ A @ B @ F @ A2 )
= ( inj_on @ A @ B @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_12_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_13_assms,axiom,
inj_on @ nat @ nat @ f @ ( top_top @ ( set @ nat ) ) ).
% assms
thf(fact_14_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_15_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_16_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_17_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A2: set @ A,G: B > A,F: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on @ A @ B @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_18_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( X != Y )
=> ( ( member @ A @ X @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_19_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_20_the__inv__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_21_strict__mono__on__imp__inj__on,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F: A > B,A2: set @ A] :
( ( strict_mono_on @ A @ B @ F @ A2 )
=> ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% strict_mono_on_imp_inj_on
thf(fact_22_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_23_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A5: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A5 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_24_change__tv_Osimps_I2_J,axiom,
! [F: nat > nat,N: nat] :
( ( paraco1920534163nge_tv @ F @ ( paraco676387099_Indet @ N ) )
= ( paraco676387099_Indet @ ( F @ N ) ) ) ).
% change_tv.simps(2)
thf(fact_25_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_26_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A5: A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F @ A5 ) @ ( image @ A @ B @ F @ A2 ) )
= ( member @ A @ A5 @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_27_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image @ A @ B @ F @ A2 )
= ( image @ A @ B @ F @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_28_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F: A > B,B3: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ B3 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
= ( ? [X2: A] :
( ( B3
= ( F @ X2 ) )
& ! [Y3: A] :
( ( B3
= ( F @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_29_strict__mono__imp__inj__on,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,A2: set @ A] :
( ( order_strict_mono @ A @ B @ F )
=> ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% strict_mono_imp_inj_on
thf(fact_30_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A2: set @ B] :
( ( B3
= ( F @ X ) )
=> ( ( member @ B @ X @ A2 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_31_tv_Oinject_I2_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( paraco676387099_Indet @ X22 )
= ( paraco676387099_Indet @ Y22 ) )
= ( X22 = Y22 ) ) ).
% tv.inject(2)
thf(fact_32_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_33_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A5 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_34_the__inv__into__onto,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_35_imageI,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,F: A > B] :
( ( member @ A @ X @ A2 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% imageI
thf(fact_36_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A2: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A2 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_37_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_38_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
( ( M = N2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_39_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_40_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A2: set @ A,B3: B,F: A > B] :
( ( member @ A @ X @ A2 )
=> ( ( B3
= ( F @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_41_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F: A > B,A2: set @ A,R: A,S: A] :
( ( strict_mono_on @ A @ B @ F @ A2 )
=> ( ( member @ A @ R @ A2 )
=> ( ( member @ A @ S @ A2 )
=> ( ( ord_less @ A @ R @ S )
=> ( ord_less @ B @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_42_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A2: set @ A,F: A > B] :
( ! [R2: A,S2: A] :
( ( member @ A @ R2 @ A2 )
=> ( ( member @ A @ S2 @ A2 )
=> ( ( ord_less @ A @ R2 @ S2 )
=> ( ord_less @ B @ ( F @ R2 ) @ ( F @ S2 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F @ A2 ) ) ) ).
% strict_mono_onI
thf(fact_43_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F2: A > B,A3: set @ A] :
! [R3: A,S3: A] :
( ( ( member @ A @ R3 @ A3 )
& ( member @ A @ S3 @ A3 )
& ( ord_less @ A @ R3 @ S3 ) )
=> ( ord_less @ B @ ( F2 @ R3 ) @ ( F2 @ S3 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_44_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A5: A,F: B > A,B3: B,C2: B] :
( ( A5
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A5: A,P: A > $o] :
( ( member @ A @ A5 @ ( collect @ A @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A5: A,B3: A,F: A > B,C2: B] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ B @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_50_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A5: A,F: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A5 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_51_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A5: A,B3: A,F: A > C,C2: C] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_52_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_53_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_54_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_55_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_56_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A5 ) ) ) ).
% order.asym
thf(fact_57_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z2: A] :
( ( ord_less @ A @ X @ Z2 )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% dense
thf(fact_58_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_59_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_60_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( order_strict_mono @ A @ B @ F ) ) ) ).
% strict_monoI
thf(fact_61_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_62_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: A,B3: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A5 ) ) ) ).
% less_asym'
thf(fact_63_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_trans
thf(fact_64_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_65_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_66_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( A5 = B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_67_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_68_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A5: A] :
( ( ord_less @ A @ B3 @ A5 )
=> ~ ( ord_less @ A @ A5 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_69_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_70_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F2: A > B] :
! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_71_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_72_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A5: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A5 ) ) ) ).
% less_induct
thf(fact_73_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_74_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_75_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_76_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_77_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_78_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A] :
~ ( ord_less @ A @ A5 @ A5 ) ) ).
% dual_order.irrefl
thf(fact_79_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_80_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_81_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_82_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X5: A] : ( P2 @ X5 ) )
= ( ^ [P3: A > $o] :
? [N3: A] :
( ( P3 @ N3 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N3 )
=> ~ ( P3 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_83_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A5: A,B3: A] :
( ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A] : ( P @ A4 @ A4 )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A5 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_84_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A5: A,C2: A] :
( ( ord_less @ A @ B3 @ A5 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A5 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_85_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_86_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( A5 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_87_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A5: A] :
( ( ord_less @ A @ B3 @ A5 )
=> ( A5 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_88_strict__mono__on__eqD,axiom,
! [D: $tType,C: $tType] :
( ( ( linorder @ C )
& ( preorder @ D ) )
=> ! [F: C > D,A2: set @ C,X: C,Y: C] :
( ( strict_mono_on @ C @ D @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member @ C @ X @ A2 )
=> ( ( member @ C @ Y @ A2 )
=> ( Y = X ) ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_89_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F: A > B,B2: set @ A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_90_inj__on__the__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_91_f__the__inv__into__f,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,Y: B] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ B @ Y @ ( image @ A @ B @ F @ A2 ) )
=> ( ( F @ ( the_inv_into @ A @ B @ A2 @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_92_strict__mono__inv,axiom,
! [A: $tType,B: $tType] :
( ( ( linorder @ B )
& ( linorder @ A ) )
=> ! [F: A > B,G: B > A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ! [X3: A] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% strict_mono_inv
thf(fact_93_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X3: B] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_94_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X3: B] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_95_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X3: A] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_96_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_97_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y3: A] :
? [X2: B] :
( Y3
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_98_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X: B] :
( ( B3
= ( F @ X ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_99_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A2: set @ A,G: A > B,F: A > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on @ A @ A @ F @ A2 )
=> ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A2 ) )
= ( inj_on @ A @ B @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_100_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A5: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A5 ) ) ).
% top.extremum_strict
thf(fact_101_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A5: A] :
( ( A5
!= ( top_top @ A ) )
= ( ord_less @ A @ A5 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_102_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A5: A,A2: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A5 ) @ A2 ) ) ).
% inj_on_add
thf(fact_103_the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ( the_inv_into @ A @ B @ A2 @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_104_the__inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: A,Y: B] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ( F @ X )
= Y )
=> ( ( member @ A @ X @ A2 )
=> ( ( the_inv_into @ A @ B @ A2 @ F @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_105_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A5: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less @ A @ A5 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_106_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A5: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A5 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_107_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A5: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A5 )
= ( plus_plus @ A @ C2 @ A5 ) )
= ( B3 = C2 ) ) ) ).
% add_right_cancel
thf(fact_108_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A5 @ B3 )
= ( plus_plus @ A @ A5 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add_left_cancel
thf(fact_109_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_110_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A5: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less @ A @ A5 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_111_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A5: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less @ A @ A5 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_112_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_113_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_114_psubsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_115_psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_trans
thf(fact_116_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P4: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P4 ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P4 ) ) ) ).
% apsnd_apfst_commute
thf(fact_117_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B3 ) @ C2 )
= ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_118_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_119_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A,K: A,A5: A,B3: A] :
( ( A2
= ( plus_plus @ A @ K @ A5 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B3 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_120_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: A,K: A,B3: A,A5: A] :
( ( B2
= ( plus_plus @ A @ K @ B3 ) )
=> ( ( plus_plus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A5 @ B3 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_121_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A5 @ B3 ) @ C2 )
= ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_122_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A5 @ B3 )
= ( plus_plus @ A @ A5 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add.left_cancel
thf(fact_123_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B3: A,A5: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A5 )
= ( plus_plus @ A @ C2 @ A5 ) )
= ( B3 = C2 ) ) ) ).
% add.right_cancel
thf(fact_124_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_125_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B3: A,A5: A,C2: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A5 @ C2 ) )
= ( plus_plus @ A @ A5 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_126_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A5 @ B3 )
= ( plus_plus @ A @ A5 @ C2 ) )
=> ( B3 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_127_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A5: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A5 )
= ( plus_plus @ A @ C2 @ A5 ) )
=> ( B3 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_128_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_129_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_130_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_131_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A )
=> ! [A5: A,B3: A,C2: A,D2: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_132_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_133_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_134_tv_Osimps_I8_J,axiom,
! [A: $tType,F1: $o > A,F22: nat > A,X22: nat] :
( ( paraco152590079rec_tv @ A @ F1 @ F22 @ ( paraco676387099_Indet @ X22 ) )
= ( F22 @ X22 ) ) ).
% tv.simps(8)
thf(fact_135_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X: B,B2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ B @ X @ ( image @ A @ B @ F @ A2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A2 @ F @ X ) @ B2 ) ) ) ) ).
% the_inv_into_into
thf(fact_136_prod_Oinj__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F22: B > D] :
( ( inj_on @ A @ C @ F1 @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ B @ D @ F22 @ ( top_top @ ( set @ B ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% prod.inj_map
thf(fact_137_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_138_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_139_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_140_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A5: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_141_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A5: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_142_psubsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_143_psubsetE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_144_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
& ( A3 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_145_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_146_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_147_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_148_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_149_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B6 )
| ( A3 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_150_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_151_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_152_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_153_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A5: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_154_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_left_mono
thf(fact_155_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ~ ! [C4: A] :
( B3
!= ( plus_plus @ A @ A5 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_156_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_157_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
? [C5: A] :
( B5
= ( plus_plus @ A @ A6 @ C5 ) ) ) ) ) ).
% le_iff_add
thf(fact_158_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A5: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A5 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_159_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A5: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_160_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A] :
( ( A5 != B3 )
=> ( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ord_less @ A @ A5 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_161_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A5: A] :
( ( ord_less @ A @ B3 @ A5 )
=> ( ord_less_eq @ A @ B3 @ A5 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_162_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( A6 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_163_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less @ A @ B5 @ A6 )
| ( A6 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_164_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ord_less_eq @ A @ A5 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_165_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_166_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_167_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A5: A,C2: A] :
( ( ord_less @ A @ B3 @ A5 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A5 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_168_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A5: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A5 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A5 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_169_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( A6 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_170_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
| ( A6 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_171_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_172_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A5 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_173_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_174_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_175_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_176_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_177_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_178_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_179_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_le_trans
thf(fact_180_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% le_less_trans
thf(fact_181_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_182_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_183_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_184_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( A5 != B3 )
=> ( ord_less @ A @ A5 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_185_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_186_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_187_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A5: A,B3: A,F: A > C,C2: C] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_188_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A5: A,F: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A5 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_189_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A5: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_190_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A5: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A5 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_191_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_192_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_193_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_194_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_195_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
& ( B2
= ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_196_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A2 )
=> ( member @ A @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_197_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [C6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C6 @ A2 )
=> ( B2
!= ( image @ B @ A @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_198_image__subsetI,axiom,
! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ B @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_199_image__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% image_mono
thf(fact_200_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A5: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A5 )
=> ( A5
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_201_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A5: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A5 )
= ( A5
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_202_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A5: A] : ( ord_less_eq @ A @ A5 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_203_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_204_subset__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,B2: set @ A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_205_inj__on__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( inj_on @ A @ B @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_206_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_207_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_208_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_209_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_210_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_211_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_212_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B6: set @ A] :
! [T: A] :
( ( member @ A @ T @ A3 )
=> ( member @ A @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_213_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_214_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_215_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_216_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_217_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_218_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_219_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z3: set @ A] : Y5 = Z3 )
= ( ^ [A3: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_220_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_221_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_222_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A5: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A5 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_223_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A5: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_224_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A5: A,F: B > A,B3: B,C2: B] :
( ( A5
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_225_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A5: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_226_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_227_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_228_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_229_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_230_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_231_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% order.trans
thf(fact_232_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_233_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_234_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_235_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( A5 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_236_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_237_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A5 )
=> ( A5 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_238_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_239_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% dual_order.refl
thf(fact_240_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A5: A,B3: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A5 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_241_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A5: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A5 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A5 ) ) ) ) ).
% dual_order.trans
thf(fact_242_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_243_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A5: A] :
( ( ord_less_eq @ A @ B3 @ A5 )
=> ( ( ord_less_eq @ A @ A5 @ B3 )
=> ( A5 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_244_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_245_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F: A > B,A2: set @ A,X: A,Y: A] :
( ( strict_mono_on @ A @ B @ F @ A2 )
=> ( ( member @ A @ X @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_246_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A5: A,B3: A,C2: A,D2: A] :
( ( ord_less @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_247_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A5: A,B3: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A5 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_248_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_249_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_250_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,F: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y2 @ A2 )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( member @ A @ Y2 @ A2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F @ A2 ) ) ) ) ).
% linorder_inj_onI
thf(fact_251_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,B2: set @ A,A5: A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ B2 )
=> ( ( member @ A @ A5 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ B @ ( F @ A5 ) @ ( image @ A @ B @ F @ A2 ) )
= ( member @ A @ A5 @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_252_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,C3: set @ A,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ( ( image @ A @ B @ F @ A2 )
= ( image @ A @ B @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_253_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_254_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_255_subset__image__inj,axiom,
! [A: $tType,B: $tType,S4: set @ A,F: B > A,T2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S4 @ ( image @ B @ A @ F @ T2 ) )
= ( ? [U: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U @ T2 )
& ( inj_on @ B @ A @ F @ U )
& ( S4
= ( image @ B @ A @ F @ U ) ) ) ) ) ).
% subset_image_inj
% Type constructors (31)
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A7: $tType,A8: $tType] :
( ( order_top @ A8 )
=> ( order_top @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 )
=> ( top @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ 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___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_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_Set_Oset___Orderings_Oorder__top_4,axiom,
! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_13,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
ord @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
inj_on @ paraco415392788lle_tv @ paraco415392788lle_tv @ ( paraco1920534163nge_tv @ f ) @ ( top_top @ ( set @ paraco415392788lle_tv ) ) ).
%------------------------------------------------------------------------------