TPTP Problem File: ITP157^2.p
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%------------------------------------------------------------------------------
% File : ITP157^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Preferences problem prob_86__6247862_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_86__6247862_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 346 ( 152 unt; 56 typ; 0 def)
% Number of atoms : 645 ( 287 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4486 ( 48 ~; 7 |; 67 &;4129 @)
% ( 0 <=>; 235 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 301 ( 301 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 54 usr; 6 con; 0-7 aty)
% Number of variables : 1409 ( 156 ^;1157 !; 16 ?;1409 :)
% ( 80 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:27:40.388
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (52)
thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
ordere890947078_space:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[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_Groups_Ogroup__add,type,
group_add:
!>[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_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > 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_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
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_Oas__good__as,type,
prefer951318096ood_as:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ 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_Ono__better__than,type,
prefer1532642881r_than:
!>[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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( 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_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( T > T > $o > T > $o ) ).
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_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
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_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
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_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_B,type,
b: set @ a ).
thf(sy_v_Pr,type,
pr: set @ ( product_prod @ a @ a ) ).
thf(sy_v_y,type,
y: a ).
thf(sy_v_z,type,
z: a ).
% Relevant facts (256)
thf(fact_0_assms,axiom,
member @ a @ z @ ( prefer310429814s_good @ a @ y @ b @ pr ) ).
% assms
thf(fact_1_at__lst__asgd__ge,axiom,
! [A: $tType,X: A,Y: A,B2: set @ A,Pr: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( prefer310429814s_good @ A @ Y @ B2 @ Pr ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Pr ) ) ).
% at_lst_asgd_ge
thf(fact_2_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_3_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B3: B,A3: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B3 )
= ( product_Pair @ A @ B @ A3 @ B4 ) )
= ( ( A2 = A3 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_4_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_5_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A4: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B5 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_6_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B3: B,A3: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B3 )
= ( product_Pair @ A @ B @ A3 @ B4 ) )
=> ~ ( ( A2 = A3 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_7_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B5: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_8_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B5: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_9_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,B5: 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 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_10_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,B5: 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 ) ) ) @ B5 @ ( 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_11_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,B5: 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 ) ) ) ) @ B5 @ ( 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_12_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,B5: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_13_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B5 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_14_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_15_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,B5: 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 ) ) ) ) @ B5 @ ( 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_16_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,B5: 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 ) ) ) @ B5 @ ( 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_17_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,B5: 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 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct5
thf(fact_18_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,B5: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct4
thf(fact_19_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B3 ) )
= ( F1 @ A2 @ B3 ) ) ).
% old.prod.rec
thf(fact_20_at__least__as__good__def,axiom,
! [A: $tType] :
( ( prefer310429814s_good @ A )
= ( ^ [X4: A,B6: set @ A,P3: set @ ( product_prod @ A @ A )] :
( collect @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ B6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% at_least_as_good_def
thf(fact_21_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B3 ) )
= ( C3 @ A2 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_22_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_23_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F3: ( product_prod @ B @ C ) > A,A5: B,B7: C] : ( F3 @ ( product_Pair @ B @ C @ A5 @ B7 ) ) ) ) ).
% curry_conv
thf(fact_24_curryI,axiom,
! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B3: B] :
( ( F4 @ ( product_Pair @ A @ B @ A2 @ B3 ) )
=> ( product_curry @ A @ B @ $o @ F4 @ A2 @ B3 ) ) ).
% curryI
thf(fact_25_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_26_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_27_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_28_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_29_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_30_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_31_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C3: C] :
( ( product_curry @ A @ B @ C
@ ^ [X4: product_prod @ A @ B] : C3 )
= ( ^ [X4: A,Y4: B] : C3 ) ) ).
% curry_K
thf(fact_32_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C4: ( product_prod @ A @ B ) > C,X4: A,Y4: B] : ( C4 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ).
% curry_def
thf(fact_33_curryD,axiom,
! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B3 )
=> ( F4 @ ( product_Pair @ A @ B @ A2 @ B3 ) ) ) ).
% curryD
thf(fact_34_curryE,axiom,
! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A2: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F4 @ A2 @ B3 )
=> ( F4 @ ( product_Pair @ A @ B @ A2 @ B3 ) ) ) ).
% curryE
thf(fact_35_no__better__than__def,axiom,
! [A: $tType] :
( ( prefer1532642881r_than @ A )
= ( ^ [X4: A,B6: set @ A,P3: set @ ( product_prod @ A @ A )] :
( collect @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ B6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% no_better_than_def
thf(fact_36_as__good__as__def,axiom,
! [A: $tType] :
( ( prefer951318096ood_as @ A )
= ( ^ [X4: A,B6: set @ A,P3: set @ ( product_prod @ A @ A )] :
( collect @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ B6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ P3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% as_good_as_def
thf(fact_37_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) ) )
= ( R2 = S3 ) ) ).
% pred_equals_eq2
thf(fact_38_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_39_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_40_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_41_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_42_strict__contour__is__diff,axiom,
! [A: $tType,B2: set @ A,Y: A,Pr: set @ ( product_prod @ A @ A )] :
( ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ Y ) @ Pr )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ A5 ) @ Pr ) ) )
= ( minus_minus @ ( set @ A ) @ ( prefer310429814s_good @ A @ Y @ B2 @ Pr ) @ ( prefer951318096ood_as @ A @ Y @ B2 @ Pr ) ) ) ).
% strict_contour_is_diff
thf(fact_43_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_prod_def
thf(fact_44_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F23: T,X4: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X4 ) ) ) ) ).
% old.rec_bool_def
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,F4: A > B,G3: A > B] :
( ! [X3: A] :
( ( F4 @ X3 )
= ( G3 @ X3 ) )
=> ( F4 = G3 ) ) ).
% ext
thf(fact_49_diff__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( minus @ B )
& ( minus @ A ) )
=> ! [A2: A,B3: B,C3: A,D3: B] :
( ( minus_minus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
= ( product_Pair @ A @ B @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ B @ B3 @ D3 ) ) ) ) ).
% diff_Pair
thf(fact_50_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_51_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X4: A] : X4 = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_52_the__equality,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( P2 @ A2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A2 ) )
=> ( ( the @ A @ P2 )
= A2 ) ) ) ).
% the_equality
thf(fact_53_image__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : X4
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_54_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ S )
@ ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_55_DiffI,axiom,
! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A6 )
=> ( ~ ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) ) ) ) ).
% DiffI
thf(fact_56_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F4: B > A,X: B,A6: set @ B] :
( ( B3
= ( F4 @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% image_eqI
thf(fact_57_Diff__idemp,axiom,
! [A: $tType,A6: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A6 @ B2 ) ) ).
% Diff_idemp
thf(fact_58_Diff__iff,axiom,
! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
= ( ( member @ A @ C3 @ A6 )
& ~ ( member @ A @ C3 @ B2 ) ) ) ).
% Diff_iff
thf(fact_59_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_60_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,B3: B,F4: A > B] :
( ( member @ A @ X @ A6 )
=> ( ( B3
= ( F4 @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_61_ball__imageD,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F4 @ A6 ) )
=> ( P2 @ X3 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( P2 @ ( F4 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_62_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F4: A > B,G3: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F4 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image @ A @ B @ F4 @ M )
= ( image @ A @ B @ G3 @ N ) ) ) ) ).
% image_cong
thf(fact_63_bex__imageD,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F4 @ A6 ) )
& ( P2 @ X5 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A6 )
& ( P2 @ ( F4 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_64_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F4: B > A,A6: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F4 @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( Z
= ( F4 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_65_imageI,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,F4: A > B] :
( ( member @ A @ X @ A6 )
=> ( member @ B @ ( F4 @ X ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ).
% imageI
thf(fact_66_DiffD2,axiom,
! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
=> ~ ( member @ A @ C3 @ B2 ) ) ).
% DiffD2
thf(fact_67_DiffD1,axiom,
! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
=> ( member @ A @ C3 @ A6 ) ) ).
% DiffD1
thf(fact_68_DiffE,axiom,
! [A: $tType,C3: A,A6: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) )
=> ~ ( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% DiffE
thf(fact_69_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F4: B > A,A6: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F4 @ A6 ) )
=> ~ ! [X3: B] :
( ( B3
= ( F4 @ X3 ) )
=> ~ ( member @ B @ X3 @ A6 ) ) ) ).
% imageE
thf(fact_70_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F4: B > A,G3: C > B,A6: set @ C] :
( ( image @ B @ A @ F4 @ ( image @ C @ B @ G3 @ A6 ) )
= ( image @ C @ A
@ ^ [X4: C] : ( F4 @ ( G3 @ X4 ) )
@ A6 ) ) ).
% image_image
thf(fact_71_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F4 @ A6 ) )
& ( P2 @ X4 ) ) )
= ( image @ B @ A @ F4
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( P2 @ ( F4 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_72_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
& ~ ( member @ A @ X4 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_73_theI,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( P2 @ A2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A2 ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ) ).
% theI
thf(fact_74_theI_H,axiom,
! [A: $tType,P2: A > $o] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( P2 @ ( the @ A @ P2 ) ) ) ).
% theI'
thf(fact_75_theI2,axiom,
! [A: $tType,P2: A > $o,A2: A,Q: A > $o] :
( ( P2 @ A2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A2 ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P2 ) ) ) ) ) ).
% theI2
thf(fact_76_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X4: A,Y4: A] :
( the @ A
@ ^ [Z3: A] :
( ( P3
=> ( Z3 = X4 ) )
& ( ~ P3
=> ( Z3 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_77_the1I2,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P2 ) ) ) ) ).
% the1I2
thf(fact_78_the1__equality,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( ( P2 @ A2 )
=> ( ( the @ A @ P2 )
= A2 ) ) ) ).
% the1_equality
thf(fact_79_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B6: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B6 @ X4 ) ) ) ) ) ).
% minus_apply
thf(fact_80_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A6: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A6 ) )
= ( Inf @ A6 ) ) ).
% Inf.INF_identity_eq
thf(fact_81_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A6: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A6 ) )
= ( Sup @ A6 ) ) ).
% Sup.SUP_identity_eq
thf(fact_82_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_unit_def
thf(fact_83_theI__unique,axiom,
! [A: $tType,P2: A > $o,X: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( ( P2 @ X )
= ( X
= ( the @ A @ P2 ) ) ) ) ).
% theI_unique
thf(fact_84_Nitpick_OThe__psimp,axiom,
! [A: $tType,P2: A > $o,X: A] :
( ( P2
= ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X ) )
=> ( ( the @ A @ P2 )
= X ) ) ).
% Nitpick.The_psimp
thf(fact_85_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_86_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_87_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_88_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_89_UNIV__eq__I,axiom,
! [A: $tType,A6: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A6 )
=> ( ( top_top @ ( set @ A ) )
= A6 ) ) ).
% UNIV_eq_I
thf(fact_90_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_91_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F4: B > A,X: B] :
( ( B3
= ( F4 @ X ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_92_surj__def,axiom,
! [B: $tType,A: $tType,F4: B > A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y4: A] :
? [X4: B] :
( Y4
= ( F4 @ X4 ) ) ) ) ).
% surj_def
thf(fact_93_rangeI,axiom,
! [A: $tType,B: $tType,F4: B > A,X: B] : ( member @ A @ ( F4 @ X ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_94_surjI,axiom,
! [B: $tType,A: $tType,G3: B > A,F4: A > B] :
( ! [X3: A] :
( ( G3 @ ( F4 @ X3 ) )
= X3 )
=> ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_95_surjE,axiom,
! [A: $tType,B: $tType,F4: B > A,Y: A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X3: B] :
( Y
!= ( F4 @ X3 ) ) ) ).
% surjE
thf(fact_96_surjD,axiom,
! [A: $tType,B: $tType,F4: B > A,Y: A] :
( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X3: B] :
( Y
= ( F4 @ X3 ) ) ) ).
% surjD
thf(fact_97_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F4: C > A,G3: B > C] :
( ( image @ B @ A
@ ^ [X4: B] : ( F4 @ ( G3 @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F4 @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_98_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F4: B > A] :
( ( member @ A @ B3 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B3
!= ( F4 @ X3 ) ) ) ).
% rangeE
thf(fact_99_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B6: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B6 @ X4 ) ) ) ) ) ).
% fun_diff_def
thf(fact_100_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B2: set @ B,C5: B > A,D4: B > A,Inf: ( set @ A ) > A] :
( ( A6 = B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ( C5 @ X3 )
= ( D4 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C5 @ A6 ) )
= ( Inf @ ( image @ B @ A @ D4 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_101_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B2: set @ B,C5: B > A,D4: B > A,Sup: ( set @ A ) > A] :
( ( A6 = B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ( C5 @ X3 )
= ( D4 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C5 @ A6 ) )
= ( Sup @ ( image @ B @ A @ D4 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_102_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_103_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_104_same__fstI,axiom,
! [B: $tType,A: $tType,P2: A > $o,X: A,Y7: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P2 @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R2 @ 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 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_105_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_106_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F4: A > B,G3: C > D] :
( ( ( image @ A @ B @ F4 @ ( 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 @ F4 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_107_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_108_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [Y4: B] : Y4 )
= ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% map_prod_ident
thf(fact_109_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F4: C > A,G3: D > B,A2: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ ( product_Pair @ C @ D @ A2 @ B3 ) )
= ( product_Pair @ A @ B @ ( F4 @ A2 ) @ ( G3 @ B3 ) ) ) ).
% map_prod_simp
thf(fact_110_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ X ) )
= ( F4 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_111_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B3: B,R2: set @ ( product_prod @ A @ B ),F4: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ R2 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F4 @ A2 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F4 @ G3 ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_112_fst__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( minus @ A )
& ( minus @ B ) )
=> ! [X: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( product_fst @ A @ B @ ( minus_minus @ ( product_prod @ A @ B ) @ X @ Y ) )
= ( minus_minus @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% fst_diff
thf(fact_113_fst__top,axiom,
! [B: $tType,A: $tType] :
( ( ( top @ A )
& ( top @ B ) )
=> ( ( product_fst @ A @ B @ ( top_top @ ( product_prod @ A @ B ) ) )
= ( top_top @ A ) ) ) ).
% fst_top
thf(fact_114_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_115_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_116_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_117_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_118_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F4: C > A,G3: D > B,R2: 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 @ F4 @ G3 ) @ R2 ) )
=> ~ ! [X3: C,Y3: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F4 @ X3 ) @ ( G3 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_119_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_120_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P ) )
= ( ? [B7: B] :
( P
= ( product_Pair @ A @ B @ A2 @ B7 ) ) ) ) ).
% eq_fst_iff
thf(fact_121_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_122_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [X4: B] : X4
@ T2 )
= T2 ) ).
% prod.map_ident
thf(fact_123_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_124_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_125_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_126_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_127_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > B,G3: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F4 @ G3 @ X ) )
= ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_128_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_129_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_130_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_131_snd__diff,axiom,
! [A: $tType,B: $tType] :
( ( ( minus @ B )
& ( minus @ A ) )
=> ! [X: product_prod @ B @ A,Y: product_prod @ B @ A] :
( ( product_snd @ B @ A @ ( minus_minus @ ( product_prod @ B @ A ) @ X @ Y ) )
= ( minus_minus @ A @ ( product_snd @ B @ A @ X ) @ ( product_snd @ B @ A @ Y ) ) ) ) ).
% snd_diff
thf(fact_132_snd__top,axiom,
! [B: $tType,A: $tType] :
( ( ( top @ A )
& ( top @ B ) )
=> ( ( product_snd @ B @ A @ ( top_top @ ( product_prod @ B @ A ) ) )
= ( top_top @ A ) ) ) ).
% snd_top
thf(fact_133_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_134_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_135_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y5 = Z2 )
= ( ^ [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_136_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B3: A,P: product_prod @ B @ A] :
( ( B3
= ( product_snd @ B @ A @ P ) )
= ( ? [A5: B] :
( P
= ( product_Pair @ B @ A @ A5 @ B3 ) ) ) ) ).
% eq_snd_iff
thf(fact_137_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_138_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_139_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_140_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_141_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_142_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P4: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P4 ) @ ( product_fst @ A @ B @ P4 ) ) ) ) ).
% prod.swap_def
thf(fact_143_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_144_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_145_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_146_top__conj_I2_J,axiom,
! [A: $tType,P2: $o,X: A] :
( ( P2
& ( top_top @ ( A > $o ) @ X ) )
= P2 ) ).
% top_conj(2)
thf(fact_147_top__conj_I1_J,axiom,
! [A: $tType,X: A,P2: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P2 )
= P2 ) ).
% top_conj(1)
thf(fact_148_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F3: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( G4 @ ( product_fst @ B @ C @ ( F3 @ X4 ) ) @ ( product_snd @ B @ C @ ( F3 @ X4 ) ) ) ) ) ).
% scomp_unfold
thf(fact_149_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P3: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P3 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_150_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P3: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P3 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_151_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_152_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: $tType,E: $tType,F4: A > ( product_prod @ E @ F ),G3: E > F > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F @ ( product_prod @ C @ D ) @ F4 @ G3 ) @ H )
= ( product_scomp @ A @ E @ F @ B @ F4
@ ^ [X4: E] : ( product_scomp @ F @ C @ D @ B @ ( G3 @ X4 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_153_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F4: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F4 )
= ( F4 @ X ) ) ).
% Pair_scomp
thf(fact_154_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B3: A,F4: B > A,X: B,C3: C,G3: B > C,A6: set @ B] :
( ( B3
= ( F4 @ X ) )
=> ( ( C3
= ( G3 @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F4 @ G3 ) ) ) ) ) ).
% image2_eqI
thf(fact_155_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F4: B > A,A6: set @ B,A8: set @ A,G3: D > C,B2: set @ D,B8: set @ C] :
( ( ( image @ B @ A @ F4 @ A6 )
= A8 )
=> ( ( ( image @ D @ C @ G3 @ B2 )
= B8 )
=> ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F4 @ G3 )
@ ( product_Sigma @ B @ D @ A6
@ ^ [Uu: B] : B2 ) )
= ( product_Sigma @ A @ C @ A8
@ ^ [Uu: A] : B8 ) ) ) ) ).
% map_prod_surj_on
thf(fact_156_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B3: B,A3: A,B4: B,R: 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 @ B3 ) @ ( product_Pair @ A @ B @ A3 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R )
| ( ( A2 = A3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_157_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
= ( ( member @ A @ A2 @ A6 )
& ( member @ B @ B3 @ ( B2 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_158_SigmaI,axiom,
! [B: $tType,A: $tType,A2: A,A6: set @ A,B3: B,B2: A > ( set @ B )] :
( ( member @ A @ A2 @ A6 )
=> ( ( member @ B @ B3 @ ( B2 @ A2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_159_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_160_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B2: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A6 @ B2 )
@ ^ [Uu: A] : C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu: A] : C5 ) ) ) ).
% Times_Diff_distrib1
thf(fact_161_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I: set @ A,A6: A > ( set @ B ),B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I
@ ^ [I2: A] : ( minus_minus @ ( set @ B ) @ ( A6 @ I2 ) @ ( B2 @ I2 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ A6 ) @ ( product_Sigma @ A @ B @ I @ B2 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_162_Sigma__cong,axiom,
! [B: $tType,A: $tType,A6: set @ A,B2: set @ A,C5: A > ( set @ B ),D4: A > ( set @ B )] :
( ( A6 = B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( C5 @ X3 )
= ( D4 @ X3 ) ) )
=> ( ( product_Sigma @ A @ B @ A6 @ C5 )
= ( product_Sigma @ A @ B @ B2 @ D4 ) ) ) ) ).
% Sigma_cong
thf(fact_163_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A6: set @ B,B2: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : C5 )
= ( product_Sigma @ B @ A @ B2
@ ^ [Uu: B] : C5 ) )
= ( A6 = B2 ) ) ) ).
% Times_eq_cancel2
thf(fact_164_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I: set @ A,J: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I @ J ) @ C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I @ C5 ) @ ( product_Sigma @ A @ B @ J @ C5 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_165_SigmaE2,axiom,
! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
=> ~ ( ( member @ A @ A2 @ A6 )
=> ~ ( member @ B @ B3 @ ( B2 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_166_SigmaD2,axiom,
! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
=> ( member @ B @ B3 @ ( B2 @ A2 ) ) ) ).
% SigmaD2
thf(fact_167_SigmaD1,axiom,
! [B: $tType,A: $tType,A2: A,B3: B,A6: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
=> ( member @ A @ A2 @ A6 ) ) ).
% SigmaD1
thf(fact_168_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod @ A @ B,A6: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A6 @ B2 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ! [Y3: B] :
( ( member @ B @ Y3 @ ( B2 @ X3 ) )
=> ( C3
!= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_169_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A6: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B2 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A6 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B2 ) ) ) ).
% mem_Times_iff
thf(fact_170_product__swap,axiom,
! [B: $tType,A: $tType,A6: set @ B,B2: set @ A] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B2 ) )
= ( product_Sigma @ A @ B @ B2
@ ^ [Uu: A] : A6 ) ) ).
% product_swap
thf(fact_171_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A7: set @ A,B6: set @ B] :
( product_Sigma @ A @ B @ A7
@ ^ [Uu: A] : B6 ) ) ) ).
% Product_Type.product_def
thf(fact_172_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A6 @ B2 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B2 ) ) ) ).
% member_product
thf(fact_173_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6
@ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 )
@ ^ [Uu: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) ) ) ).
% subset_fst_snd
thf(fact_174_subset__antisym,axiom,
! [A: $tType,A6: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A6 )
=> ( A6 = B2 ) ) ) ).
% subset_antisym
thf(fact_175_subsetI,axiom,
! [A: $tType,A6: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B2 ) ) ).
% subsetI
thf(fact_176_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B3: B,C3: A,D3: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
= ( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ B @ B3 @ D3 ) ) ) ) ).
% Pair_le
thf(fact_177_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A6: set @ B,B2: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : C5 )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu: B] : C5 ) )
= ( ord_less_eq @ ( set @ B ) @ A6 @ B2 ) ) ) ).
% Times_subset_cancel2
thf(fact_178_Sigma__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,C5: set @ A,B2: A > ( set @ B ),D4: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ X3 ) @ ( D4 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A6 @ B2 ) @ ( product_Sigma @ A @ B @ C5 @ D4 ) ) ) ) ).
% Sigma_mono
thf(fact_179_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_180_image__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B2: set @ A,F4: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B2 ) ) ) ).
% image_mono
thf(fact_181_image__subsetI,axiom,
! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B2: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ B @ ( F4 @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B2 ) ) ).
% image_subsetI
thf(fact_182_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F4: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F4 @ A6 ) )
=> ~ ! [C6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C6 @ A6 )
=> ( B2
!= ( image @ B @ A @ F4 @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_183_image__subset__iff,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_184_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F4: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F4 @ A6 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
& ( B2
= ( image @ B @ A @ F4 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_185_subrelI,axiom,
! [B: $tType,A: $tType,R: 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 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_186_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_187_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2 )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_188_subset__trans,axiom,
! [A: $tType,A6: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ C5 ) ) ) ).
% subset_trans
thf(fact_189_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_190_subset__refl,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% subset_refl
thf(fact_191_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_192_equalityD2,axiom,
! [A: $tType,A6: set @ A,B2: set @ A] :
( ( A6 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A6 ) ) ).
% equalityD2
thf(fact_193_equalityD1,axiom,
! [A: $tType,A6: set @ A,B2: set @ A] :
( ( A6 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B2 ) ) ).
% equalityD1
thf(fact_194_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ A @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_195_equalityE,axiom,
! [A: $tType,A6: set @ A,B2: set @ A] :
( ( A6 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A6 ) ) ) ).
% equalityE
thf(fact_196_subsetD,axiom,
! [A: $tType,A6: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
=> ( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_197_in__mono,axiom,
! [A: $tType,A6: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
=> ( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_198_Diff__mono,axiom,
! [A: $tType,A6: set @ A,C5: set @ A,D4: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_199_Diff__subset,axiom,
! [A: $tType,A6: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B2 ) @ A6 ) ).
% Diff_subset
thf(fact_200_double__diff,axiom,
! [A: $tType,A6: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C5 @ A6 ) )
= A6 ) ) ) ).
% double_diff
thf(fact_201_subset__UNIV,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_202_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_203_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_204_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_205_prop__restrict,axiom,
! [A: $tType,X: A,Z4: set @ A,X7: set @ A,P2: A > $o] :
( ( member @ A @ X @ Z4 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z4
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_206_Collect__restrict,axiom,
! [A: $tType,X7: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( P2 @ X4 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_207_Collect__subset,axiom,
! [A: $tType,A6: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P2 @ X4 ) ) )
@ A6 ) ).
% Collect_subset
thf(fact_208_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P2: A > $o,F4: A > B,B2: set @ B] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( member @ B @ ( F4 @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ ( collect @ A @ P2 ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_209_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A6: set @ A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_210_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A6: set @ A,Q: A > $o,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( Q @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_211_image__diff__subset,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ ( image @ B @ A @ F4 @ B2 ) ) @ ( image @ B @ A @ F4 @ ( minus_minus @ ( set @ B ) @ A6 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_212_subset__snd__imageI,axiom,
! [B: $tType,A: $tType,A6: set @ A,B2: set @ B,S3: set @ ( product_prod @ A @ B ),X: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B2 )
@ S3 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ B2 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S3 ) ) ) ) ).
% subset_snd_imageI
thf(fact_213_subset__fst__imageI,axiom,
! [B: $tType,A: $tType,A6: set @ A,B2: set @ B,S3: set @ ( product_prod @ A @ B ),Y: B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B2 )
@ S3 )
=> ( ( member @ B @ Y @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) ) ) ) ).
% subset_fst_imageI
thf(fact_214_range__subsetD,axiom,
! [B: $tType,A: $tType,F4: B > A,B2: set @ A,I3: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) @ B2 )
=> ( member @ A @ ( F4 @ I3 ) @ B2 ) ) ).
% range_subsetD
thf(fact_215_pred__subset__eq,axiom,
! [A: $tType,R2: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R2 )
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R2 @ S3 ) ) ).
% pred_subset_eq
thf(fact_216_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_217_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_218_Gr__incl,axiom,
! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B2: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A6 @ F4 )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B2 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B2 ) ) ).
% Gr_incl
thf(fact_219_all__subset__image,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A6 )
=> ( P2 @ ( image @ B @ A @ F4 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_220_eq__subset,axiom,
! [A: $tType,P2: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ ^ [A5: A,B7: A] :
( ( P2 @ A5 @ B7 )
| ( A5 = B7 ) ) ) ).
% eq_subset
thf(fact_221_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,A6: set @ A,F4: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F4 ) )
=> ( member @ A @ X @ A6 ) ) ).
% GrD1
thf(fact_222_GrD2,axiom,
! [A: $tType,B: $tType,X: A,Fx: B,A6: set @ A,F4: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F4 ) )
=> ( ( F4 @ X )
= Fx ) ) ).
% GrD2
thf(fact_223_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B3: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B3 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
= ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_224_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B3 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_225_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C3: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B3 ) @ C3 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_226_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B3: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B3 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( A2 = B3 )
= ( C3 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_227_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B3: A,D3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ D3 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_228_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_229_snd__image__times,axiom,
! [B: $tType,A: $tType,A6: set @ B,B2: set @ A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B2 ) )
= B2 ) ) ) ).
% snd_image_times
thf(fact_230_fst__image__times,axiom,
! [B: $tType,A: $tType,B2: set @ B,A6: set @ A] :
( ( ( B2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B2 ) )
= A6 ) ) ) ).
% fst_image_times
thf(fact_231_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X4: A] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_232_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_233_all__not__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_234_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_235_image__is__empty,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
( ( ( image @ B @ A @ F4 @ A6 )
= ( bot_bot @ ( set @ A ) ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_236_empty__is__image,axiom,
! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F4 @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_237_image__empty,axiom,
! [B: $tType,A: $tType,F4: B > A] :
( ( image @ B @ A @ F4 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_238_subset__empty,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_239_empty__subsetI,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 ) ).
% empty_subsetI
thf(fact_240_Diff__empty,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= A6 ) ).
% Diff_empty
thf(fact_241_empty__Diff,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_242_Diff__cancel,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_243_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_244_Collect__const,axiom,
! [A: $tType,P2: $o] :
( ( P2
=> ( ( collect @ A
@ ^ [S4: A] : P2 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P2
=> ( ( collect @ A
@ ^ [S4: A] : P2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_245_Times__empty,axiom,
! [A: $tType,B: $tType,A6: set @ A,B2: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_246_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A6: set @ A] :
( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_247_Diff__UNIV,axiom,
! [A: $tType,A6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_248_Diff__eq__empty__iff,axiom,
! [A: $tType,A6: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A6 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_249_times__eq__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,B2: set @ B,C5: set @ A,D4: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B2 )
= ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : D4 ) )
= ( ( ( A6 = C5 )
& ( B2 = D4 ) )
| ( ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( D4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_250_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I: set @ A,X7: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I @ X7 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ I )
=> ( ( X7 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_251_bot__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( bot @ B )
& ( bot @ A ) )
=> ( ( bot_bot @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( bot_bot @ A ) @ ( bot_bot @ B ) ) ) ) ).
% bot_prod_def
thf(fact_252_ex__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A6 ) )
= ( A6
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_253_equals0I,axiom,
! [A: $tType,A6: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A6 )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_254_equals0D,axiom,
! [A: $tType,A6: set @ A,A2: A] :
( ( A6
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A6 ) ) ).
% equals0D
thf(fact_255_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
% Type constructors (30)
thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( ordere890947078_space @ A10 ) )
=> ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A9: $tType,A10: $tType] :
( ( order_top @ A10 )
=> ( order_top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A9: $tType,A10: $tType] :
( ( top @ A10 )
=> ( top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A9: $tType,A10: $tType] :
( ( bot @ A10 )
=> ( bot @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A9: $tType,A10: $tType] :
( ( minus @ A10 )
=> ( minus @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_4,axiom,
! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_5,axiom,
! [A9: $tType] : ( minus @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_7,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_9,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_10,axiom,
minus @ $o ).
thf(tcon_Product__Type_Oprod___Groups_Ocancel__ab__semigroup__add,axiom,
! [A9: $tType,A10: $tType] :
( ( ( cancel146912293up_add @ A9 )
& ( cancel146912293up_add @ A10 ) )
=> ( cancel146912293up_add @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( ordere890947078_space @ A10 ) )
=> ( ordered_ab_group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_11,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order_top @ A9 )
& ( order_top @ A10 ) )
=> ( order_top @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oab__group__add,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ab_group_add @ A9 )
& ( ab_group_add @ A10 ) )
=> ( ab_group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ogroup__add,axiom,
! [A9: $tType,A10: $tType] :
( ( ( group_add @ A9 )
& ( group_add @ A10 ) )
=> ( group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Otop_12,axiom,
! [A9: $tType,A10: $tType] :
( ( ( top @ A9 )
& ( top @ A10 ) )
=> ( top @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_13,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ord @ A9 )
& ( ord @ A10 ) )
=> ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Obot_14,axiom,
! [A9: $tType,A10: $tType] :
( ( ( bot @ A9 )
& ( bot @ A10 ) )
=> ( bot @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ominus_15,axiom,
! [A9: $tType,A10: $tType] :
( ( ( minus @ A9 )
& ( minus @ A10 ) )
=> ( minus @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_16,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_17,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_18,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_19,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_20,axiom,
minus @ product_unit ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ z @ y ) @ pr ).
%------------------------------------------------------------------------------