TPTP Problem File: ITP152^2.p
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%------------------------------------------------------------------------------
% File : ITP152^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_107__6248646_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 : Preferences/prob_107__6248646_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 353 ( 137 unt; 62 typ; 0 def)
% Number of atoms : 704 ( 293 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 5539 ( 68 ~; 13 |; 78 &;5111 @)
% ( 0 <=>; 269 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 241 ( 241 >; 0 *; 0 +; 0 <<)
% Number of symbols : 64 ( 61 usr; 6 con; 0-7 aty)
% Number of variables : 1315 ( 49 ^;1164 !; 23 ?;1315 :)
% ( 79 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:27:54.280
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (58)
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[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_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V1076094709vector:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Multiset_Oadd__mset,type,
add_mset:
!>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset,type,
comm_monoid_sum_mset:
!>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ofold__mset,type,
fold_mset:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
thf(sy_c_Multiset_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omult,type,
mult:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omult1,type,
mult1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omultp,type,
multp:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Ostrict__linear__order__on,type,
order_215145569der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Preferences__Mirabelle__stygcjuplb_Oat__least__as__good,type,
prefer310429814s_good:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Preferences__Mirabelle__stygcjuplb_Opreference,type,
prefer199794634erence:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference,type,
prefer1997167224erence:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Preferences__Mirabelle__stygcjuplb_Orational__preference__axioms,type,
prefer1801827867axioms:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > 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_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
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_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_carrier,type,
carrier: set @ a ).
thf(sy_v_relation,type,
relation: set @ ( product_prod @ a @ a ) ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (256)
thf(fact_0_indiff__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
=> ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
& ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% indiff_trans
thf(fact_1_strict__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
=> ( ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ Z ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ Y ) @ relation ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Z ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Z @ X ) @ relation ) ) ) ) ).
% strict_trans
thf(fact_2_rational__preference__axioms,axiom,
prefer1997167224erence @ a @ carrier @ relation ).
% rational_preference_axioms
thf(fact_3_compl,axiom,
! [X2: a] :
( ( member @ a @ X2 @ carrier )
=> ! [Xa: a] :
( ( member @ a @ Xa @ carrier )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X2 @ Xa ) @ relation )
| ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Xa @ X2 ) @ relation ) ) ) ) ).
% compl
thf(fact_4_not__outside,axiom,
! [X: a,Y: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
=> ( member @ a @ X @ carrier ) ) ).
% not_outside
thf(fact_5_strict__not__refl__weak,axiom,
! [X: a,Y: a] :
( ( ( member @ a @ X @ carrier )
& ( member @ a @ Y @ carrier ) )
=> ( ( ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) )
= ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X @ Y ) @ relation )
& ~ ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y @ X ) @ relation ) ) ) ) ).
% strict_not_refl_weak
thf(fact_6_preference__axioms,axiom,
prefer199794634erence @ a @ carrier @ relation ).
% preference_axioms
thf(fact_7_trans__refl,axiom,
order_preorder_on @ a @ carrier @ relation ).
% trans_refl
thf(fact_8_total,axiom,
total_on @ a @ carrier @ relation ).
% total
thf(fact_9_reflexivity,axiom,
refl_on @ a @ carrier @ relation ).
% reflexivity
thf(fact_10_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_12_transitivity,axiom,
trans @ a @ relation ).
% transitivity
thf(fact_13_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_14_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_15_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_16_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_17_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_18_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_19_rational__preference_Oaxioms_I1_J,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( prefer199794634erence @ A @ Carrier @ Relation ) ) ).
% rational_preference.axioms(1)
thf(fact_20_preference_Otrans__refl,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( order_preorder_on @ A @ Carrier @ Relation ) ) ).
% preference.trans_refl
thf(fact_21_preference_Oreflexivity,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( refl_on @ A @ Carrier @ Relation ) ) ).
% preference.reflexivity
thf(fact_22_preference_Otransitivity,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( trans @ A @ Relation ) ) ).
% preference.transitivity
thf(fact_23_rational__preference_Ototal,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( total_on @ A @ Carrier @ Relation ) ) ).
% rational_preference.total
thf(fact_24_preference_Ointro,axiom,
! [A: $tType,Relation: set @ ( product_prod @ A @ A ),Carrier: set @ A] :
( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Relation )
=> ( member @ A @ X3 @ Carrier ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Relation )
=> ( member @ A @ Y3 @ Carrier ) )
=> ( ( order_preorder_on @ A @ Carrier @ Relation )
=> ( prefer199794634erence @ A @ Carrier @ Relation ) ) ) ) ).
% preference.intro
thf(fact_25_preference__def,axiom,
! [A: $tType] :
( ( prefer199794634erence @ A )
= ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ Relation2 )
=> ( member @ A @ X4 @ Carrier2 ) )
& ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ Relation2 )
=> ( member @ A @ Y4 @ Carrier2 ) )
& ( order_preorder_on @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% preference_def
thf(fact_26_preference_Oindiff__trans,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% preference.indiff_trans
thf(fact_27_preference_Onot__outside,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
=> ( member @ A @ X @ Carrier ) ) ) ).
% preference.not_outside
thf(fact_28_rational__preference_Ostrict__not__refl__weak,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ A @ X @ Carrier )
& ( member @ A @ Y @ Carrier ) )
=> ( ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_not_refl_weak
thf(fact_29_rational__preference_Ostrict__trans,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Relation ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Relation ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ Relation )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ X ) @ Relation ) ) ) ) ) ).
% rational_preference.strict_trans
thf(fact_30_rational__preference_Ocompl,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ! [X2: A] :
( ( member @ A @ X2 @ Carrier )
=> ! [Xa: A] :
( ( member @ A @ Xa @ Carrier )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa ) @ Relation )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% rational_preference.compl
thf(fact_31_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_32_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_33_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct7
thf(fact_34_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct6
thf(fact_35_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct5
thf(fact_36_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct4
thf(fact_37_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_38_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_39_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_40_preorder__on__def,axiom,
! [A: $tType] :
( ( order_preorder_on @ A )
= ( ^ [A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R )
& ( trans @ A @ R ) ) ) ) ).
% preorder_on_def
thf(fact_41_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_42_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A5: set @ A,R: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( ( X4 != Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_43_total__onI,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ( X3 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A6 @ R2 ) ) ).
% total_onI
thf(fact_44_refl__on__domain,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ A @ A2 @ A6 )
& ( member @ A @ B2 @ A6 ) ) ) ) ).
% refl_on_domain
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G3 @ X3 ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_49_refl__onD2,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ Y @ A6 ) ) ) ).
% refl_onD2
thf(fact_50_refl__onD1,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ X @ A6 ) ) ) ).
% refl_onD1
thf(fact_51_refl__onD,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
( ( refl_on @ A @ A6 @ R2 )
=> ( ( member @ A @ A2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_52_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [X4: A,Y4: A,Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% trans_def
thf(fact_53_transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R2 ) ) )
=> ( trans @ A @ R2 ) ) ).
% transI
thf(fact_54_transD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% transD
thf(fact_55_transE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 ) ) ) ) ).
% transE
thf(fact_56_rational__preference_Ointro,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier @ Relation )
=> ( ( prefer1801827867axioms @ A @ Carrier @ Relation )
=> ( prefer1997167224erence @ A @ Carrier @ Relation ) ) ) ).
% rational_preference.intro
thf(fact_57_rational__preference__def,axiom,
! [A: $tType] :
( ( prefer1997167224erence @ A )
= ( ^ [Carrier2: set @ A,Relation2: set @ ( product_prod @ A @ A )] :
( ( prefer199794634erence @ A @ Carrier2 @ Relation2 )
& ( prefer1801827867axioms @ A @ Carrier2 @ Relation2 ) ) ) ) ).
% rational_preference_def
thf(fact_58_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_59_rational__preference_Oaxioms_I2_J,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( prefer1997167224erence @ A @ Carrier @ Relation )
=> ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% rational_preference.axioms(2)
thf(fact_60_rational__preference__axioms_Ointro,axiom,
! [A: $tType,Carrier: set @ A,Relation: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ Carrier @ Relation )
=> ( prefer1801827867axioms @ A @ Carrier @ Relation ) ) ).
% rational_preference_axioms.intro
thf(fact_61_rational__preference__axioms__def,axiom,
! [A: $tType] :
( ( prefer1801827867axioms @ A )
= ( total_on @ A ) ) ).
% rational_preference_axioms_def
thf(fact_62_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_63_at__lst__asgd__ge,axiom,
! [A: $tType,X: A,Y: A,B5: set @ A,Pr: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ B5 @ Pr ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_64_at__least__as__goodD,axiom,
! [A: $tType,Z: A,Y: A,B5: set @ A,Pr: set @ ( product_prod @ A @ A )] :
( ( member @ A @ Z @ ( prefer310429814s_good @ A @ Y @ B5 @ Pr ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ Pr ) ) ).
% at_least_as_goodD
thf(fact_65_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F4: ( product_prod @ B @ C ) > A,A7: B,B6: C] : ( F4 @ ( product_Pair @ B @ C @ A7 @ B6 ) ) ) ) ).
% curry_conv
thf(fact_66_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R: A > A > $o,F4: B > A,X4: B,Y4: B] : ( R @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_67_curryI,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 ) ) ).
% curryI
thf(fact_68_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R2 @ F3 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_69_strict__linear__order__on__def,axiom,
! [A: $tType] :
( ( order_215145569der_on @ A )
= ( ^ [A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R )
& ( irrefl @ A @ R )
& ( total_on @ A @ A5 @ R ) ) ) ) ).
% strict_linear_order_on_def
thf(fact_70_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_71_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc2004651681e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_72_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
= P ) ).
% swap_swap
thf(fact_73_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F3 ) )
= F3 ) ).
% case_prod_curry
thf(fact_74_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F3 ) )
= F3 ) ).
% curry_case_prod
thf(fact_75_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P ) )
=> ~ ! [X3: B,Y3: C] :
( ( P
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z @ ( C3 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_76_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F3 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_77_irreflI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [A4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R3 )
=> ( irrefl @ A @ R3 ) ) ).
% irreflI
thf(fact_78_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [A7: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A7 ) @ R ) ) ) ).
% irrefl_def
thf(fact_79_curryD,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
=> ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_80_curryE,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A2 @ B2 )
=> ( F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_81_trans__inv__image,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),F3: B > A] :
( ( trans @ A @ R2 )
=> ( trans @ B @ ( inv_image @ A @ B @ R2 @ F3 ) ) ) ).
% trans_inv_image
thf(fact_82_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q2 )
=> ( ( F3 @ X3 @ Y3 )
= ( G3 @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F3 @ P )
= ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_83_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A6: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% pair_in_swap_image
thf(fact_84_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F4: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_apply
thf(fact_85_multp__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P2: A > A > $o,N: multiset @ A,M: multiset @ A] :
( ( irrefl @ A @ R3 )
=> ( ( trans @ A @ R3 )
=> ( ! [X3: A,Y3: A] :
( ( P2 @ X3 @ Y3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) )
=> ( ( multp @ A @ P2 @ N @ M )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M ) @ ( mult @ A @ R3 ) ) ) ) ) ) ).
% multp_iff
thf(fact_86_asym__inv__image,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),F3: B > A] :
( ( asym @ A @ R3 )
=> ( asym @ B @ ( inv_image @ A @ B @ R3 @ F3 ) ) ) ).
% asym_inv_image
thf(fact_87_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_88_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $false )
= F22 ) ).
% old.bool.simps(6)
thf(fact_89_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_90_asym_Oinducts,axiom,
! [A: $tType,X: set @ ( product_prod @ A @ A ),P2: ( set @ ( product_prod @ A @ A ) ) > $o] :
( ( asym @ A @ X )
=> ( ! [R4: set @ ( product_prod @ A @ A )] :
( ! [A8: A,B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B7 ) @ R4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A8 ) @ R4 ) )
=> ( P2 @ R4 ) )
=> ( P2 @ X ) ) ) ).
% asym.inducts
thf(fact_91_asym_Ointros,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [A4: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R3 ) )
=> ( asym @ A @ R3 ) ) ).
% asym.intros
thf(fact_92_asym_Osimps,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [A7: set @ ( product_prod @ A @ A )] :
? [R5: set @ ( product_prod @ A @ A )] :
( ( A7 = R5 )
& ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R5 ) ) ) ) ) ).
% asym.simps
thf(fact_93_asym_Ocases,axiom,
! [A: $tType,A2: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ A2 )
=> ! [A8: A,B7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A8 @ B7 ) @ A2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A8 ) @ A2 ) ) ) ).
% asym.cases
thf(fact_94_asym__iff,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R5 ) ) ) ) ).
% asym_iff
thf(fact_95_asymD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( asym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R3 ) ) ) ).
% asymD
thf(fact_96_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F3: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F3 )
= ( F3 @ X ) ) ).
% Pair_scomp
thf(fact_97_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_def
thf(fact_98_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A6: set @ B] :
( ( B2
= ( F3 @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A6 ) ) ) ) ).
% image_eqI
thf(fact_99_mult__cancel,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),X5: multiset @ A,Z4: multiset @ A,Y5: multiset @ A] :
( ( trans @ A @ S )
=> ( ( irrefl @ A @ S )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X5 @ Z4 ) @ ( plus_plus @ ( multiset @ A ) @ Y5 @ Z4 ) ) @ ( mult @ A @ S ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X5 @ Y5 ) @ ( mult @ A @ S ) ) ) ) ) ).
% mult_cancel
thf(fact_100_mult__cancel__add__mset,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),Uu: A,X5: multiset @ A,Y5: multiset @ A] :
( ( trans @ A @ S )
=> ( ( irrefl @ A @ S )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu @ X5 ) @ ( add_mset @ A @ Uu @ Y5 ) ) @ ( mult @ A @ S ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X5 @ Y5 ) @ ( mult @ A @ S ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_101_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_102_imageI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,F3: A > B] :
( ( member @ A @ X @ A6 )
=> ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A6 ) ) ) ).
% imageI
thf(fact_103_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F3: B > A,A6: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( Z
= ( F3 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_104_add__mset__add__mset__same__iff,axiom,
! [A: $tType,A2: A,A6: multiset @ A,B5: multiset @ A] :
( ( ( add_mset @ A @ A2 @ A6 )
= ( add_mset @ A @ A2 @ B5 ) )
= ( A6 = B5 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_105_multi__self__add__other__not__self,axiom,
! [A: $tType,M: multiset @ A,X: A] :
( M
!= ( add_mset @ A @ X @ M ) ) ).
% multi_self_add_other_not_self
thf(fact_106_union__mset__add__mset__left,axiom,
! [A: $tType,A2: A,A6: multiset @ A,B5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ A6 ) @ B5 )
= ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A6 @ B5 ) ) ) ).
% union_mset_add_mset_left
thf(fact_107_union__mset__add__mset__right,axiom,
! [A: $tType,A6: multiset @ A,A2: A,B5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ A6 @ ( add_mset @ A @ A2 @ B5 ) )
= ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A6 @ B5 ) ) ) ).
% union_mset_add_mset_right
thf(fact_108_rangeI,axiom,
! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_109_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X: B] :
( ( B2
= ( F3 @ X ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_110_union__assoc,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A,K: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M @ N ) @ K )
= ( plus_plus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ K ) ) ) ).
% union_assoc
thf(fact_111_union__lcomm,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A,K: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ K ) )
= ( plus_plus @ ( multiset @ A ) @ N @ ( plus_plus @ ( multiset @ A ) @ M @ K ) ) ) ).
% union_lcomm
thf(fact_112_union__commute,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( ^ [M2: multiset @ A,N2: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_113_add__eq__conv__ex,axiom,
! [A: $tType,A2: A,M: multiset @ A,B2: A,N: multiset @ A] :
( ( ( add_mset @ A @ A2 @ M )
= ( add_mset @ A @ B2 @ N ) )
= ( ( ( M = N )
& ( A2 = B2 ) )
| ? [K2: multiset @ A] :
( ( M
= ( add_mset @ A @ B2 @ K2 ) )
& ( N
= ( add_mset @ A @ A2 @ K2 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_114_add__mset__commute,axiom,
! [A: $tType,X: A,Y: A,M: multiset @ A] :
( ( add_mset @ A @ X @ ( add_mset @ A @ Y @ M ) )
= ( add_mset @ A @ Y @ ( add_mset @ A @ X @ M ) ) ) ).
% add_mset_commute
thf(fact_115_union__left__cancel,axiom,
! [A: $tType,K: multiset @ A,M: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ K @ M )
= ( plus_plus @ ( multiset @ A ) @ K @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_116_union__right__cancel,axiom,
! [A: $tType,M: multiset @ A,K: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M @ K )
= ( plus_plus @ ( multiset @ A ) @ N @ K ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_117_multi__union__self__other__eq,axiom,
! [A: $tType,A6: multiset @ A,X5: multiset @ A,Y5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A6 @ X5 )
= ( plus_plus @ ( multiset @ A ) @ A6 @ Y5 ) )
=> ( X5 = Y5 ) ) ).
% multi_union_self_other_eq
thf(fact_118_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,B2: B,F3: A > B] :
( ( member @ A @ X @ A6 )
=> ( ( B2
= ( F3 @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_119_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,P2: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A6 ) )
=> ( P2 @ X3 ) )
=> ! [X2: B] :
( ( member @ B @ X2 @ A6 )
=> ( P2 @ ( F3 @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_120_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F3: A > B,G3: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image @ A @ B @ F3 @ M )
= ( image @ A @ B @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_121_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,P2: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F3 @ A6 ) )
& ( P2 @ X2 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A6 )
& ( P2 @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_122_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_123_add__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( plus @ B )
& ( plus @ A ) )
=> ! [A2: A,B2: B,C3: A,D3: B] :
( ( plus_plus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
= ( product_Pair @ A @ B @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ B @ B2 @ D3 ) ) ) ) ).
% add_Pair
thf(fact_124_top__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( top @ B )
& ( top @ A ) )
=> ( ( top_top @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% top_prod_def
thf(fact_125_surjD,axiom,
! [A: $tType,B: $tType,F3: B > A,Y: A] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X3: B] :
( Y
= ( F3 @ X3 ) ) ) ).
% surjD
thf(fact_126_surjE,axiom,
! [A: $tType,B: $tType,F3: B > A,Y: A] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X3: B] :
( Y
!= ( F3 @ X3 ) ) ) ).
% surjE
thf(fact_127_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_128_surj__def,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y4: A] :
? [X4: B] :
( Y4
= ( F3 @ X4 ) ) ) ) ).
% surj_def
thf(fact_129_surjI,axiom,
! [B: $tType,A: $tType,G3: B > A,F3: A > B] :
( ! [X3: A] :
( ( G3 @ ( F3 @ X3 ) )
= X3 )
=> ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_130_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G3: C > D] :
( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_131_total__lex__prod,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ S )
=> ( total_on @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( lex_prod @ A @ B @ R2 @ S ) ) ) ) ).
% total_lex_prod
thf(fact_132_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_133_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_134_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,A2: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( F3 @ A2 ) @ ( G3 @ B2 ) ) ) ).
% map_prod_simp
thf(fact_135_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ X ) )
= ( F3 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_136_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > B,G3: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F3 @ G3 @ X ) )
= ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_137_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A3 @ B3 ) ) @ ( lex_prod @ A @ B @ R2 @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R2 )
| ( ( A2 = A3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_138_trans__lex__prod,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( trans @ A @ R1 )
=> ( ( trans @ B @ R22 )
=> ( trans @ ( product_prod @ A @ B ) @ ( lex_prod @ A @ B @ R1 @ R22 ) ) ) ) ).
% trans_lex_prod
thf(fact_139_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_140_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_141_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_142_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R3: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A2 ) @ ( G3 @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_143_asym__lex__prod,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( asym @ A @ R3 )
=> ( ( asym @ B @ S3 )
=> ( asym @ ( product_prod @ A @ B ) @ ( lex_prod @ A @ B @ R3 @ S3 ) ) ) ) ).
% asym_lex_prod
thf(fact_144_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F4: B > C > A,P3: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_145_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,Prod2: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% prod.case_eq_if
thf(fact_146_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
=> ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_147_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_148_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_149_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P2 @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_150_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P2 @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_151_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_152_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod3 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod3 ) ) )
=> ( Prod = Prod3 ) ) ).
% prod.expand
thf(fact_153_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y6: product_prod @ A @ B,Z5: product_prod @ A @ B] : Y6 = Z5 )
= ( ^ [S4: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S4 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S4 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_154_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_155_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_156_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_157_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_158_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_159_plus__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( plus @ A )
& ( plus @ B ) )
=> ( ( plus_plus @ ( product_prod @ A @ B ) )
= ( ^ [X4: product_prod @ A @ B,Y4: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( plus_plus @ A @ ( product_fst @ A @ B @ X4 ) @ ( product_fst @ A @ B @ Y4 ) ) @ ( plus_plus @ B @ ( product_snd @ A @ B @ X4 ) @ ( product_snd @ A @ B @ Y4 ) ) ) ) ) ) ).
% plus_prod_def
thf(fact_160_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G3: D > B,R3: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 ) @ R3 ) )
=> ~ ! [X3: C,Y3: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F3 @ X3 ) @ ( G3 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_161_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y: A,X: B] :
( ( P2 @ Y @ X )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_162_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_163_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
( ( P2 @ X @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_164_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X )
= Z ) ) ).
% sndI
thf(fact_165_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_166_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P ) )
= ( ? [A7: B] :
( P
= ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_167_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P ) )
= ( ? [B6: B] :
( P
= ( product_Pair @ A @ B @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_168_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X5: set @ A,A6: set @ ( product_prod @ A @ B ),Y5: set @ B,P2: A > B > $o,Q: A > B > $o] :
( ( X5
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
=> ( ( Y5
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ! [Xa2: B] :
( ( member @ B @ Xa2 @ Y5 )
=> ( ( P2 @ X3 @ Xa2 )
=> ( Q @ X3 @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_169_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: B,C3: A,D3: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
= ( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ B @ B2 @ D3 ) ) ) ) ).
% Pair_le
thf(fact_170_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_171_Pair__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [X: A,X6: A,Y: B,Y7: B] :
( ( ord_less_eq @ A @ X @ X6 )
=> ( ( ord_less_eq @ B @ Y @ Y7 )
=> ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X6 @ Y7 ) ) ) ) ) ).
% Pair_mono
thf(fact_172_mono__mult,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R6: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R2 ) @ ( mult @ A @ R6 ) ) ) ).
% mono_mult
thf(fact_173_image__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B5: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A6 ) @ ( image @ A @ B @ F3 @ B5 ) ) ) ).
% image_mono
thf(fact_174_image__subsetI,axiom,
! [A: $tType,B: $tType,A6: set @ A,F3: A > B,B5: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ B @ ( F3 @ X3 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A6 ) @ B5 ) ) ).
% image_subsetI
thf(fact_175_subset__imageE,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A6 ) )
=> ~ ! [C4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C4 @ A6 )
=> ( B5
!= ( image @ B @ A @ F3 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_176_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A6 ) @ B5 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( member @ A @ ( F3 @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_177_subset__image__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A6 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
& ( B5
= ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_178_subset__translation__eq,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) )
= ( ord_less_eq @ ( set @ A ) @ S @ T2 ) ) ) ).
% subset_translation_eq
thf(fact_179_range__subsetD,axiom,
! [B: $tType,A: $tType,F3: B > A,B5: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) @ B5 )
=> ( member @ A @ ( F3 @ I ) @ B5 ) ) ).
% range_subsetD
thf(fact_180_all__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A6: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ ( image @ B @ A @ F3 @ A6 ) )
=> ( P2 @ B8 ) ) )
= ( ! [B8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B8 @ A6 )
=> ( P2 @ ( image @ B @ A @ F3 @ B8 ) ) ) ) ) ).
% all_subset_image
thf(fact_181_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,B5: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_182_mono__mult1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R6: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R2 ) @ ( mult1 @ A @ R6 ) ) ) ).
% mono_mult1
thf(fact_183_same__fstI,axiom,
! [B: $tType,A: $tType,P2: A > $o,X: A,Y7: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P2 @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R3 @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P2 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_184_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
! [I2: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R )
=> ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_185_mult1__union,axiom,
! [A: $tType,B5: multiset @ A,D4: multiset @ A,R2: set @ ( product_prod @ A @ A ),C5: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B5 @ D4 ) @ ( mult1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) @ ( mult1 @ A @ R2 ) ) ) ).
% mult1_union
thf(fact_186_mult1E,axiom,
! [A: $tType,N: multiset @ A,M: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M ) @ ( mult1 @ A @ R2 ) )
=> ~ ! [A4: A,M0: multiset @ A] :
( ( M
= ( add_mset @ A @ A4 @ M0 ) )
=> ! [K3: multiset @ A] :
( ( N
= ( plus_plus @ ( multiset @ A ) @ M0 @ K3 ) )
=> ~ ! [B7: A] :
( ( member @ A @ B7 @ ( set_mset @ A @ K3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A4 ) @ R2 ) ) ) ) ) ).
% mult1E
thf(fact_187_mult1I,axiom,
! [A: $tType,M: multiset @ A,A2: A,M02: multiset @ A,N: multiset @ A,K: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( M
= ( add_mset @ A @ A2 @ M02 ) )
=> ( ( N
= ( plus_plus @ ( multiset @ A ) @ M02 @ K ) )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( set_mset @ A @ K ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M ) @ ( mult1 @ A @ R2 ) ) ) ) ) ).
% mult1I
thf(fact_188_less__add,axiom,
! [A: $tType,N: multiset @ A,A2: A,M02: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ ( add_mset @ A @ A2 @ M02 ) ) @ ( mult1 @ A @ R2 ) )
=> ( ? [M3: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M3 @ M02 ) @ ( mult1 @ A @ R2 ) )
& ( N
= ( add_mset @ A @ A2 @ M3 ) ) )
| ? [K3: multiset @ A] :
( ! [B7: A] :
( ( member @ A @ B7 @ ( set_mset @ A @ K3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B7 @ A2 ) @ R2 ) )
& ( N
= ( plus_plus @ ( multiset @ A ) @ M02 @ K3 ) ) ) ) ) ).
% less_add
thf(fact_189_mset__add,axiom,
! [A: $tType,A2: A,A6: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ A6 ) )
=> ~ ! [B9: multiset @ A] :
( A6
!= ( add_mset @ A @ A2 @ B9 ) ) ) ).
% mset_add
thf(fact_190_multi__member__split,axiom,
! [A: $tType,X: A,M: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M ) )
=> ? [A9: multiset @ A] :
( M
= ( add_mset @ A @ X @ A9 ) ) ) ).
% multi_member_split
thf(fact_191_insert__noteq__member,axiom,
! [A: $tType,B2: A,B5: multiset @ A,C3: A,C5: multiset @ A] :
( ( ( add_mset @ A @ B2 @ B5 )
= ( add_mset @ A @ C3 @ C5 ) )
=> ( ( B2 != C3 )
=> ( member @ A @ C3 @ ( set_mset @ A @ B5 ) ) ) ) ).
% insert_noteq_member
thf(fact_192_union__single__eq__member,axiom,
! [A: $tType,X: A,M: multiset @ A,N: multiset @ A] :
( ( ( add_mset @ A @ X @ M )
= N )
=> ( member @ A @ X @ ( set_mset @ A @ N ) ) ) ).
% union_single_eq_member
thf(fact_193_union__iff,axiom,
! [A: $tType,A2: A,A6: multiset @ A,B5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A6 @ B5 ) ) )
= ( ( member @ A @ A2 @ ( set_mset @ A @ A6 ) )
| ( member @ A @ A2 @ ( set_mset @ A @ B5 ) ) ) ) ).
% union_iff
thf(fact_194_mult__implies__one__step,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),M: multiset @ A,N: multiset @ A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M @ N ) @ ( mult @ A @ R2 ) )
=> ? [I3: multiset @ A,J2: multiset @ A] :
( ( N
= ( plus_plus @ ( multiset @ A ) @ I3 @ J2 ) )
& ? [K3: multiset @ A] :
( ( M
= ( plus_plus @ ( multiset @ A ) @ I3 @ K3 ) )
& ( J2
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ K3 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R2 ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_195_one__step__implies__mult,axiom,
! [A: $tType,J3: multiset @ A,K: multiset @ A,R2: set @ ( product_prod @ A @ A ),I4: multiset @ A] :
( ( J3
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ K ) )
=> ? [Xa: A] :
( ( member @ A @ Xa @ ( set_mset @ A @ J3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa ) @ R2 ) ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ K ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J3 ) ) @ ( mult @ A @ R2 ) ) ) ) ).
% one_step_implies_mult
thf(fact_196_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ X @ Y ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_197_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_198_union__eq__empty,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M @ N )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% union_eq_empty
thf(fact_199_empty__eq__union,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ M @ N ) )
= ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% empty_eq_union
thf(fact_200_add__mset__eq__singleton__iff,axiom,
! [A: $tType,X: A,M: multiset @ A,Y: A] :
( ( ( add_mset @ A @ X @ M )
= ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_201_single__eq__add__mset,axiom,
! [A: $tType,A2: A,B2: A,M: multiset @ A] :
( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= ( add_mset @ A @ B2 @ M ) )
= ( ( B2 = A2 )
& ( M
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% single_eq_add_mset
thf(fact_202_add__mset__eq__single,axiom,
! [A: $tType,B2: A,M: multiset @ A,A2: A] :
( ( ( add_mset @ A @ B2 @ M )
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( B2 = A2 )
& ( M
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% add_mset_eq_single
thf(fact_203_single__eq__single,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( A2 = B2 ) ) ).
% single_eq_single
thf(fact_204_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
= S3 ) ) ).
% image_add_0
thf(fact_205_multiset__nonemptyE,axiom,
! [A: $tType,A6: multiset @ A] :
( ( A6
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ ( set_mset @ A @ A6 ) ) ) ).
% multiset_nonemptyE
thf(fact_206_zero__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( zero @ B )
& ( zero @ A ) )
=> ( ( zero_zero @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% zero_prod_def
thf(fact_207_empty__neutral_I1_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X )
= X ) ).
% empty_neutral(1)
thf(fact_208_empty__neutral_I2_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X @ ( zero_zero @ ( multiset @ A ) ) )
= X ) ).
% empty_neutral(2)
thf(fact_209_multi__nonempty__split,axiom,
! [A: $tType,M: multiset @ A] :
( ( M
!= ( zero_zero @ ( multiset @ A ) ) )
=> ? [A9: multiset @ A,A4: A] :
( M
= ( add_mset @ A @ A4 @ A9 ) ) ) ).
% multi_nonempty_split
thf(fact_210_empty__not__add__mset,axiom,
! [A: $tType,A2: A,A6: multiset @ A] :
( ( zero_zero @ ( multiset @ A ) )
!= ( add_mset @ A @ A2 @ A6 ) ) ).
% empty_not_add_mset
thf(fact_211_multiset__induct2,axiom,
! [A: $tType,B: $tType,P2: ( multiset @ A ) > ( multiset @ B ) > $o,M: multiset @ A,N: multiset @ B] :
( ( P2 @ ( zero_zero @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ B ) ) )
=> ( ! [A4: A,M3: multiset @ A,N3: multiset @ B] :
( ( P2 @ M3 @ N3 )
=> ( P2 @ ( add_mset @ A @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: B,M3: multiset @ A,N3: multiset @ B] :
( ( P2 @ M3 @ N3 )
=> ( P2 @ M3 @ ( add_mset @ B @ A4 @ N3 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_212_multiset__induct,axiom,
! [A: $tType,P2: ( multiset @ A ) > $o,M: multiset @ A] :
( ( P2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M3: multiset @ A] :
( ( P2 @ M3 )
=> ( P2 @ ( add_mset @ A @ X3 @ M3 ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct
thf(fact_213_multiset__cases,axiom,
! [A: $tType,M: multiset @ A] :
( ( M
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [X3: A,N3: multiset @ A] :
( M
!= ( add_mset @ A @ X3 @ N3 ) ) ) ).
% multiset_cases
thf(fact_214_single__is__union,axiom,
! [A: $tType,A2: A,M: multiset @ A,N: multiset @ A] :
( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= ( plus_plus @ ( multiset @ A ) @ M @ N ) )
= ( ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= M )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= N ) ) ) ) ).
% single_is_union
thf(fact_215_union__is__single,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A,A2: A] :
( ( ( plus_plus @ ( multiset @ A ) @ M @ N )
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( ( M
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( N
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% union_is_single
thf(fact_216_add__mset__add__single,axiom,
! [A: $tType] :
( ( add_mset @ A )
= ( ^ [A7: A,A5: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ A7 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% add_mset_add_single
thf(fact_217_multi__member__last,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( set_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% multi_member_last
thf(fact_218_not__less__empty,axiom,
! [A: $tType,M: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M @ ( zero_zero @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R2 ) ) ).
% not_less_empty
thf(fact_219_multiset__induct__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: ( multiset @ A ) > $o,M: multiset @ A] :
( ( P2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M3: multiset @ A] :
( ( P2 @ M3 )
=> ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set_mset @ A @ M3 ) )
=> ( ord_less_eq @ A @ Xa @ X3 ) )
=> ( P2 @ ( add_mset @ A @ X3 @ M3 ) ) ) )
=> ( P2 @ M ) ) ) ) ).
% multiset_induct_max
thf(fact_220_multiset__induct__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: ( multiset @ A ) > $o,M: multiset @ A] :
( ( P2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M3: multiset @ A] :
( ( P2 @ M3 )
=> ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set_mset @ A @ M3 ) )
=> ( ord_less_eq @ A @ X3 @ Xa ) )
=> ( P2 @ ( add_mset @ A @ X3 @ M3 ) ) ) )
=> ( P2 @ M ) ) ) ) ).
% multiset_induct_min
thf(fact_221_multi__member__skip,axiom,
! [A: $tType,X: A,XS: multiset @ A,Y: A] :
( ( member @ A @ X @ ( set_mset @ A @ XS ) )
=> ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_222_multi__member__this,axiom,
! [A: $tType,X: A,XS: multiset @ A] : ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ).
% multi_member_this
thf(fact_223_Multiset_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: multiset @ A] :
( A5
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Multiset.is_empty_def
thf(fact_224_subset__mset_Osum__mset__0__iff,axiom,
! [A: $tType,M: multiset @ ( multiset @ A )] :
( ( ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ M )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ ( set_mset @ ( multiset @ A ) @ M ) )
=> ( X4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_225_comm__monoid__add_Osum__mset_Ocong,axiom,
! [A: $tType] :
( ( comm_monoid_sum_mset @ A )
= ( comm_monoid_sum_mset @ A ) ) ).
% comm_monoid_add.sum_mset.cong
thf(fact_226_in__mset__fold__plus__iff,axiom,
! [A: $tType,X: A,M: multiset @ A,NN: multiset @ ( multiset @ A )] :
( ( member @ A @ X @ ( set_mset @ A @ ( fold_mset @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ M @ NN ) ) )
= ( ( member @ A @ X @ ( set_mset @ A @ M ) )
| ? [N2: multiset @ A] :
( ( member @ ( multiset @ A ) @ N2 @ ( set_mset @ ( multiset @ A ) @ NN ) )
& ( member @ A @ X @ ( set_mset @ A @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_227_insert__DiffM2,axiom,
! [A: $tType,X: A,M: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M ) )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= M ) ) ).
% insert_DiffM2
thf(fact_228_diff__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( minus @ B )
& ( minus @ A ) )
=> ! [A2: A,B2: B,C3: A,D3: B] :
( ( minus_minus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
= ( product_Pair @ A @ B @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ B @ B2 @ D3 ) ) ) ) ).
% diff_Pair
thf(fact_229_diff__diff__add__mset,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A,P2: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M @ N ) @ P2 )
= ( minus_minus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ P2 ) ) ) ).
% diff_diff_add_mset
thf(fact_230_fold__mset__empty,axiom,
! [B: $tType,A: $tType,F3: B > A > A,S: A] :
( ( fold_mset @ B @ A @ F3 @ S @ ( zero_zero @ ( multiset @ B ) ) )
= S ) ).
% fold_mset_empty
thf(fact_231_add__mset__remove__trivial,axiom,
! [A: $tType,X: A,M: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ X @ M ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= M ) ).
% add_mset_remove_trivial
thf(fact_232_diff__add__mset__swap,axiom,
! [A: $tType,B2: A,A6: multiset @ A,M: multiset @ A] :
( ~ ( member @ A @ B2 @ ( set_mset @ A @ A6 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ M ) @ A6 )
= ( add_mset @ A @ B2 @ ( minus_minus @ ( multiset @ A ) @ M @ A6 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_233_insert__DiffM,axiom,
! [A: $tType,X: A,M: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M ) )
=> ( ( add_mset @ A @ X @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_234_diff__union__swap2,axiom,
! [A: $tType,Y: A,M: multiset @ A,X: A] :
( ( member @ A @ Y @ ( set_mset @ A @ M ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ X @ M ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( add_mset @ A @ X @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% diff_union_swap2
thf(fact_235_diff__empty,axiom,
! [A: $tType,M: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ M @ ( zero_zero @ ( multiset @ A ) ) )
= M )
& ( ( minus_minus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ M )
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% diff_empty
thf(fact_236_Multiset_Odiff__cancel,axiom,
! [A: $tType,A6: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ A6 @ A6 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Multiset.diff_cancel
thf(fact_237_union__single__eq__diff,axiom,
! [A: $tType,X: A,M: multiset @ A,N: multiset @ A] :
( ( ( add_mset @ A @ X @ M )
= N )
=> ( M
= ( minus_minus @ ( multiset @ A ) @ N @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% union_single_eq_diff
thf(fact_238_add__eq__conv__diff,axiom,
! [A: $tType,A2: A,M: multiset @ A,B2: A,N: multiset @ A] :
( ( ( add_mset @ A @ A2 @ M )
= ( add_mset @ A @ B2 @ N ) )
= ( ( ( M = N )
& ( A2 = B2 ) )
| ( ( M
= ( add_mset @ A @ B2 @ ( minus_minus @ ( multiset @ A ) @ N @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
& ( N
= ( add_mset @ A @ A2 @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_239_diff__union__swap,axiom,
! [A: $tType,A2: A,B2: A,M: multiset @ A] :
( ( A2 != B2 )
=> ( ( add_mset @ A @ B2 @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ M ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% diff_union_swap
thf(fact_240_in__diffD,axiom,
! [A: $tType,A2: A,M: multiset @ A,N: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M @ N ) ) )
=> ( member @ A @ A2 @ ( set_mset @ A @ M ) ) ) ).
% in_diffD
thf(fact_241_minus__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( minus @ A )
& ( minus @ B ) )
=> ( ( minus_minus @ ( product_prod @ A @ B ) )
= ( ^ [X4: product_prod @ A @ B,Y4: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( minus_minus @ A @ ( product_fst @ A @ B @ X4 ) @ ( product_fst @ A @ B @ Y4 ) ) @ ( minus_minus @ B @ ( product_snd @ A @ B @ X4 ) @ ( product_snd @ A @ B @ Y4 ) ) ) ) ) ) ).
% minus_prod_def
thf(fact_242_diff__union__cancelR,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M @ N ) @ N )
= M ) ).
% diff_union_cancelR
thf(fact_243_diff__union__cancelL,axiom,
! [A: $tType,N: multiset @ A,M: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ N @ M ) @ N )
= M ) ).
% diff_union_cancelL
thf(fact_244_Multiset_Odiff__add,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A,Q: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ Q ) )
= ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_245_add__mset__diff__bothsides,axiom,
! [A: $tType,A2: A,M: multiset @ A,A6: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ M ) @ ( add_mset @ A @ A2 @ A6 ) )
= ( minus_minus @ ( multiset @ A ) @ M @ A6 ) ) ).
% add_mset_diff_bothsides
thf(fact_246_Multiset_Odiff__right__commute,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A,Q: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M @ N ) @ Q )
= ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M @ Q ) @ N ) ) ).
% Multiset.diff_right_commute
thf(fact_247_union__fold__mset__add__mset,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( fold_mset @ A @ ( multiset @ A ) @ ( add_mset @ A ) ) ) ).
% union_fold_mset_add_mset
thf(fact_248_diff__single__trivial,axiom,
! [A: $tType,X: A,M: multiset @ A] :
( ~ ( member @ A @ X @ ( set_mset @ A @ M ) )
=> ( ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= M ) ) ).
% diff_single_trivial
thf(fact_249_diff__single__eq__union,axiom,
! [A: $tType,X: A,M: multiset @ A,N: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= N )
= ( M
= ( add_mset @ A @ X @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_250_multi__drop__mem__not__eq,axiom,
! [A: $tType,C3: A,B5: multiset @ A] :
( ( member @ A @ C3 @ ( set_mset @ A @ B5 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ C3 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= B5 ) ) ).
% multi_drop_mem_not_eq
thf(fact_251_add__mset__remove__trivial__If,axiom,
! [A: $tType,A2: A,N: multiset @ A] :
( ( ( member @ A @ A2 @ ( set_mset @ A @ N ) )
=> ( ( add_mset @ A @ A2 @ ( minus_minus @ ( multiset @ A ) @ N @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= N ) )
& ( ~ ( member @ A @ A2 @ ( set_mset @ A @ N ) )
=> ( ( add_mset @ A @ A2 @ ( minus_minus @ ( multiset @ A ) @ N @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( add_mset @ A @ A2 @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_252_add__mset__remove__trivial__eq,axiom,
! [A: $tType,N: multiset @ A,A2: A] :
( ( N
= ( add_mset @ A @ A2 @ ( minus_minus @ ( multiset @ A ) @ N @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
= ( member @ A @ A2 @ ( set_mset @ A @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_253_multiset__add__sub__el__shuffle,axiom,
! [A: $tType,C3: A,B5: multiset @ A,B2: A] :
( ( member @ A @ C3 @ ( set_mset @ A @ B5 ) )
=> ( ( B2 != C3 )
=> ( ( add_mset @ A @ B2 @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ C3 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ B5 ) @ ( add_mset @ A @ C3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_254_more__than__one__mset__mset__diff,axiom,
! [A: $tType,A2: A,M: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( set_mset @ A @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_255_diff__union__single__conv,axiom,
! [A: $tType,A2: A,J3: multiset @ A,I4: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ J3 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J3 ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ I4 @ ( minus_minus @ ( multiset @ A ) @ J3 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% diff_union_single_conv
% Type constructors (31)
thf(tcon_Product__Type_Oprod___Orderings_Opreorder,axiom,
! [A10: $tType,A11: $tType] :
( ( ( preorder @ A10 )
& ( preorder @ A11 ) )
=> ( preorder @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_1,axiom,
! [A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( multiset @ A10 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_2,axiom,
preorder @ $o ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A10: $tType] : ( preorder @ ( set @ A10 ) ) ).
thf(tcon_fun___Orderings_Opreorder_4,axiom,
! [A10: $tType,A11: $tType] :
( ( preorder @ A11 )
=> ( preorder @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A10: $tType,A11: $tType] :
( ( top @ A11 )
=> ( top @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A10: $tType,A11: $tType] :
( ( ord @ A11 )
=> ( ord @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A10: $tType,A11: $tType] :
( ( minus @ A11 )
=> ( minus @ ( A10 > A11 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__add,axiom,
! [A10: $tType] :
( ( comm_monoid_add @ A10 )
=> ( comm_monoid_add @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_5,axiom,
! [A10: $tType] : ( top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A10: $tType] : ( ord @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_7,axiom,
! [A10: $tType] : ( minus @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A10: $tType] :
( ( zero @ A10 )
=> ( zero @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oplus,axiom,
! [A10: $tType] :
( ( plus @ A10 )
=> ( plus @ ( set @ A10 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_8,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_10,axiom,
minus @ $o ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_11,axiom,
! [A10: $tType] : ( comm_monoid_add @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_12,axiom,
! [A10: $tType] :
( ( preorder @ A10 )
=> ( ord @ ( multiset @ A10 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_13,axiom,
! [A10: $tType] : ( minus @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_14,axiom,
! [A10: $tType] : ( zero @ ( multiset @ A10 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_15,axiom,
! [A10: $tType] : ( plus @ ( multiset @ A10 ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__vector,axiom,
! [A10: $tType,A11: $tType] :
( ( ( real_V1076094709vector @ A10 )
& ( real_V1076094709vector @ A11 ) )
=> ( real_V1076094709vector @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ocomm__monoid__add_16,axiom,
! [A10: $tType,A11: $tType] :
( ( ( comm_monoid_add @ A10 )
& ( comm_monoid_add @ A11 ) )
=> ( comm_monoid_add @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oab__group__add,axiom,
! [A10: $tType,A11: $tType] :
( ( ( ab_group_add @ A10 )
& ( ab_group_add @ A11 ) )
=> ( ab_group_add @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Otop_17,axiom,
! [A10: $tType,A11: $tType] :
( ( ( top @ A10 )
& ( top @ A11 ) )
=> ( top @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_18,axiom,
! [A10: $tType,A11: $tType] :
( ( ( ord @ A10 )
& ( ord @ A11 ) )
=> ( ord @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ominus_19,axiom,
! [A10: $tType,A11: $tType] :
( ( ( minus @ A10 )
& ( minus @ A11 ) )
=> ( minus @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ozero_20,axiom,
! [A10: $tType,A11: $tType] :
( ( ( zero @ A10 )
& ( zero @ A11 ) )
=> ( zero @ ( product_prod @ A10 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oplus_21,axiom,
! [A10: $tType,A11: $tType] :
( ( ( plus @ A10 )
& ( plus @ A11 ) )
=> ( plus @ ( product_prod @ A10 @ A11 ) ) ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
member @ a @ x @ carrier ).
thf(conj_1,hypothesis,
member @ a @ y @ carrier ).
thf(conj_2,hypothesis,
x != y ).
thf(conj_3,conjecture,
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ x @ y ) @ relation )
| ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ y @ x ) @ relation ) ) ).
%------------------------------------------------------------------------------