TPTP Problem File: ITP160^2.p
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%------------------------------------------------------------------------------
% File : ITP160^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_1493__3600276_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 : Refine_Basic/prob_1493__3600276_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 335 ( 167 unt; 58 typ; 0 def)
% Number of atoms : 542 ( 239 equ; 8 cnn)
% Maximal formula atoms : 13 ( 1 avg)
% Number of connectives : 8965 ( 48 ~; 5 |; 25 &;8654 @)
% ( 0 <=>; 233 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 10 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 2280 (2280 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 51 usr; 9 con; 0-7 aty)
% Number of variables : 1471 ( 70 ^;1311 !; 9 ?;1471 :)
% ( 81 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:20:45.194
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
refine1665802226e_nres: $tType > $tType ).
thf(ty_t_Autoref__Tagging_Oannot,type,
autoref_annot: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_d,type,
d: $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (48)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_c_Autoref__Tagging_OANNOT,type,
autoref_ANNOT:
!>[A: $tType] : ( A > autoref_annot > A ) ).
thf(sy_c_Autoref__Tagging_OAPP,type,
autoref_APP:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > A ) ).
thf(sy_c_Autoref__Tagging_OOP,type,
autoref_OP:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Autoref__Tagging_Orel__annot,type,
autoref_rel_annot:
!>[C: $tType,A: $tType] : ( ( set @ ( product_prod @ C @ A ) ) > autoref_annot ) ).
thf(sy_c_Autoref__Translate_OREMOVE__INTERNAL,type,
autore2013370551TERNAL:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Autoref__Translate_OREMOVE__INTERNAL__EQ,type,
autore813025987NAL_EQ:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List_Oall__interval__nat,type,
all_interval_nat: ( nat > $o ) > nat > nat > $o ).
thf(sy_c_Misc_Ouncurry,type,
uncurry:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Obool_Ocase__bool,type,
product_case_bool:
!>[A: $tType] : ( A > A > $o > A ) ).
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_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
refine1687780735RETURN:
!>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
refine463715084e_bind:
!>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Olift__assn,type,
refine1580981607t_assn:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( B > $o ) > A > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres__rel,type,
refine476890328es_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oop__nres__ASSERT__bnd,type,
refine171938416RT_bnd:
!>[A: $tType] : ( $o > ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Misc_Oless__than__bool,type,
refine1361002633n_bool: set @ ( product_prod @ $o @ $o ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relators_Ofun__rel,type,
fun_rel:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) ).
thf(sy_c_Relators_Oprod__rel,type,
prod_rel:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) ).
thf(sy_c_Relators_OrelAPP,type,
relAPP:
!>[C1: $tType,A1: $tType,A: $tType] : ( ( ( set @ ( product_prod @ C1 @ A1 ) ) > A ) > ( set @ ( product_prod @ C1 @ A1 ) ) > A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_R1,type,
r1: set @ ( product_prod @ a @ c ) ).
thf(sy_v_R2,type,
r2: set @ ( product_prod @ b @ d ) ).
% Relevant facts (252)
thf(fact_0_param__bind,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) @ ( product_Pair @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) @ ( refine463715084e_bind @ A @ B ) @ ( refine463715084e_bind @ C @ D ) ) @ ( relAPP @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) @ ( set @ ( product_prod @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) @ ( relAPP @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ C ) @ ( ( set @ ( product_prod @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) > ( set @ ( product_prod @ ( ( refine1665802226e_nres @ A ) > ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( refine1665802226e_nres @ C ) > ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) ) @ ( fun_rel @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ C ) @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) @ ( relAPP @ A @ C @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ C ) ) ) @ ( refine476890328es_rel @ A @ C ) @ Ra ) ) @ ( relAPP @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) @ ( set @ ( product_prod @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) @ ( relAPP @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) > ( set @ ( product_prod @ ( ( A > ( refine1665802226e_nres @ B ) ) > ( refine1665802226e_nres @ B ) ) @ ( ( C > ( refine1665802226e_nres @ D ) ) > ( refine1665802226e_nres @ D ) ) ) ) ) @ ( fun_rel @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) @ ( relAPP @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) @ ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) > ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ B ) ) @ ( C > ( refine1665802226e_nres @ D ) ) ) ) ) @ ( fun_rel @ A @ C @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) @ Ra ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) @ ( refine476890328es_rel @ B @ D ) @ Rb ) ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( refine1665802226e_nres @ B ) @ ( refine1665802226e_nres @ D ) ) ) @ ( refine476890328es_rel @ B @ D ) @ Rb ) ) ) ) ).
% param_bind
thf(fact_1_fun__relI,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A2: set @ ( product_prod @ A @ B ),F: A > C,F2: B > D,B2: set @ ( product_prod @ C @ D )] :
( ! [A3: A,A4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ A4 ) @ A2 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( F2 @ A4 ) ) @ B2 ) )
=> ( member @ ( product_prod @ ( A > C ) @ ( B > D ) ) @ ( product_Pair @ ( A > C ) @ ( B > D ) @ F @ F2 ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ A2 ) @ B2 ) ) ) ).
% fun_relI
thf(fact_2_autoref__Let,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D )] :
( member @ ( product_prod @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D ) )
@ ( product_Pair @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D )
@ ^ [S: A,F3: A > B] : ( F3 @ S )
@ ^ [S: C,F3: C > D] : ( F3 @ S ) )
@ ( relAPP @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) @ ( set @ ( product_prod @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) ) > ( set @ ( product_prod @ ( A > ( A > B ) > B ) @ ( C > ( C > D ) > D ) ) ) ) @ ( fun_rel @ A @ C @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) @ Ra ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) ) @ ( relAPP @ ( A > B ) @ ( C > D ) @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( ( A > B ) > B ) @ ( ( C > D ) > D ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( C > D ) @ B @ D ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) ) @ Rr ) ) ) ).
% autoref_Let
thf(fact_3_fun__relD1,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D ),X: A,R: B] :
( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) )
=> ( ( ( F @ X )
= R )
=> ! [X2: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X2 ) @ Ra )
=> ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ R @ ( F2 @ X2 ) ) @ Rr ) ) ) ) ).
% fun_relD1
thf(fact_4_fun__relD2,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D ),X3: C,R2: D] :
( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) )
=> ( ( ( F2 @ X3 )
= R2 )
=> ! [X4: A] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ X3 ) @ Ra )
=> ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X4 ) @ R2 ) @ Rr ) ) ) ) ).
% fun_relD2
thf(fact_5_tagged__fun__relD__lhs,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
=> ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% tagged_fun_relD_lhs
thf(fact_6_tagged__fun__relD__rhs,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
=> ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% tagged_fun_relD_rhs
thf(fact_7_tagged__fun__relD__both,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
=> ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% tagged_fun_relD_both
thf(fact_8_tagged__fun__relD__none,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F: A > B,F2: C > D,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D ),X: A,X3: C] :
( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ F @ F2 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ A2 )
=> ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( F @ X ) @ ( F2 @ X3 ) ) @ B2 ) ) ) ).
% tagged_fun_relD_none
thf(fact_9_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_10_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A5: A,B3: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A5 @ B3 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
= ( ( A5 = A6 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_11_autoref__APP,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,C2: A > B,A5: C > D,Ra: set @ ( product_prod @ A @ C ),Rr: set @ ( product_prod @ B @ D ),X: A,X3: C] :
( ( member @ ( product_prod @ ( A > B ) @ ( C > D ) ) @ ( product_Pair @ ( A > B ) @ ( C > D ) @ C2 @ A5 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( A > B ) @ ( C > D ) ) ) ) @ ( fun_rel @ A @ C @ B @ D ) @ Ra ) @ Rr ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ X3 ) @ Ra )
=> ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ ( autoref_APP @ A @ B @ C2 @ X ) @ ( autoref_APP @ C @ D @ A5 @ X3 ) ) @ Rr ) ) ) ).
% autoref_APP
thf(fact_12_lift__assnI,axiom,
! [B: $tType,A: $tType,S2: A,S3: B,R3: set @ ( product_prod @ A @ B ),Phi: B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S2 @ S3 ) @ R3 )
=> ( ( Phi @ S3 )
=> ( refine1580981607t_assn @ A @ B @ R3 @ Phi @ S2 ) ) ) ).
% lift_assnI
thf(fact_13_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A3: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_14_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A3: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A3 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_15_prod__induct7,axiom,
! [G: $tType,F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) )] :
( ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G ) @ E2 @ ( product_Pair @ F4 @ G @ F5 @ G2 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_16_prod__induct6,axiom,
! [F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_17_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( 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 ) ) )] :
( ! [A3: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_18_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A3: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_19_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A3: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_20_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) )] :
~ ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G ) @ E2 @ ( product_Pair @ F4 @ G @ F5 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_21_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A3: A,B5: B,C3: C,D2: D,E2: E,F5: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_22_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 ) ) )] :
~ ! [A3: A,B5: B,C3: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_23_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A3: A,B5: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_24_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A3: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_25_Pair__inject,axiom,
! [A: $tType,B: $tType,A5: A,B3: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A5 @ B3 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
=> ~ ( ( A5 = A6 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_26_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A3: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_27_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X5: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X5 @ Y3 ) ) ).
% surj_pair
thf(fact_28_relAPP__def,axiom,
! [A: $tType,A1: $tType,C1: $tType] :
( ( relAPP @ C1 @ A1 @ A )
= ( ^ [F3: ( set @ ( product_prod @ C1 @ A1 ) ) > A] : F3 ) ) ).
% relAPP_def
thf(fact_29_lift__assn__def,axiom,
! [B: $tType,A: $tType] :
( ( refine1580981607t_assn @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B ),Phi2: B > $o,S: A] :
? [S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S @ S4 ) @ R4 )
& ( Phi2 @ S4 ) ) ) ) ).
% lift_assn_def
thf(fact_30_autoref__RETURN,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) ) @ ( product_Pair @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) @ ( refine1687780735RETURN @ A ) @ ( refine1687780735RETURN @ B ) ) @ ( relAPP @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) > ( set @ ( product_prod @ ( A > ( refine1665802226e_nres @ A ) ) @ ( B > ( refine1665802226e_nres @ B ) ) ) ) ) @ ( fun_rel @ A @ B @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) @ ( refine476890328es_rel @ A @ B ) @ R3 ) ) ) ).
% autoref_RETURN
thf(fact_31_param__op__nres__ASSERT__bnd,axiom,
! [B: $tType,A: $tType,Phi3: $o,Phi: $o,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ B,R3: set @ ( product_prod @ A @ B )] :
( ( Phi3
=> Phi )
=> ( ( Phi3
=> ( Phi
=> ( member @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ M @ M2 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) @ ( refine476890328es_rel @ A @ B ) @ R3 ) ) ) )
=> ( member @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ ( refine171938416RT_bnd @ A @ Phi @ M ) @ ( refine171938416RT_bnd @ B @ Phi3 @ M2 ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) ) ) @ ( refine476890328es_rel @ A @ B ) @ R3 ) ) ) ) ).
% param_op_nres_ASSERT_bnd
thf(fact_32_autoref__If,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) @ ( product_Pair @ ( $o > A > A > A ) @ ( $o > B > B > B ) @ ( if @ A ) @ ( if @ B ) ) @ ( relAPP @ ( A > A > A ) @ ( B > B > B ) @ ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) > ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) ) @ ( fun_rel @ $o @ $o @ ( A > A > A ) @ ( B > B > B ) ) @ ( id2 @ $o ) ) @ ( relAPP @ ( A > A ) @ ( B > B ) @ ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) > ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > A ) @ ( B > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ) ) ).
% autoref_If
thf(fact_33_fun__relE2,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,F2: A > C,Rv: set @ ( product_prod @ B @ C ),T2: B,X: A] :
( ( member @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_Pair @ ( A > B ) @ ( A > C ) @ F @ F2 ) @ ( relAPP @ B @ C @ ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( fun_rel @ A @ A @ B @ C ) @ ( id2 @ A ) ) @ Rv ) )
=> ( ( T2
= ( F @ X ) )
=> ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ T2 @ ( F2 @ X ) ) @ Rv ) ) ) ).
% fun_relE2
thf(fact_34_fun__relE1,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,F2: A > C,Rv: set @ ( product_prod @ B @ C ),T3: C,X: A] :
( ( member @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_Pair @ ( A > B ) @ ( A > C ) @ F @ F2 ) @ ( relAPP @ B @ C @ ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( fun_rel @ A @ A @ B @ C ) @ ( id2 @ A ) ) @ Rv ) )
=> ( ( T3
= ( F2 @ X ) )
=> ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ ( F @ X ) @ T3 ) @ Rv ) ) ) ).
% fun_relE1
thf(fact_35_autoref__Let__cong,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,X3: A,X: B,Ra: set @ ( product_prod @ A @ B ),F2: A > C,F: B > D,Rr: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ X ) @ Ra )
=> ( ! [Y3: B] :
( ( autore2013370551TERNAL @ $o @ ( X = Y3 ) )
=> ! [Y4: A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y4 @ Y3 ) @ Ra )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F2 @ Y4 ) @ ( autoref_APP @ B @ D @ F @ Y3 ) ) @ Rr ) ) )
=> ( member @ ( product_prod @ C @ D )
@ ( product_Pair @ C @ D @ ( F2 @ X3 )
@ ( autoref_APP @ ( B > D ) @ D
@ ( autoref_APP @ B @ ( ( B > D ) > D )
@ ( autoref_ANNOT @ ( B > ( B > D ) > D )
@ ( autoref_OP @ ( B > ( B > D ) > D )
@ ^ [S: B,F3: B > D] : ( F3 @ S ) )
@ ( autoref_rel_annot @ ( A > ( A > C ) > C ) @ ( B > ( B > D ) > D ) @ ( relAPP @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ ( set @ ( product_prod @ ( A > ( A > C ) > C ) @ ( B > ( B > D ) > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) ) > ( set @ ( product_prod @ ( A > ( A > C ) > C ) @ ( B > ( B > D ) > D ) ) ) ) @ ( fun_rel @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) @ Ra ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) ) @ ( relAPP @ ( A > C ) @ ( B > D ) @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) ) ) ) @ ( fun_rel @ ( A > C ) @ ( B > D ) @ C @ D ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ Ra ) @ Rr ) ) @ Rr ) ) ) )
@ X )
@ F ) )
@ Rr ) ) ) ).
% autoref_Let_cong
thf(fact_36_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A5: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A5 @ B3 ) )
= ( C2 @ A5 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_37_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A5: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A5 @ B3 ) )
= ( F1 @ A5 @ B3 ) ) ).
% old.prod.rec
thf(fact_38_APP__def,axiom,
! [A: $tType,B: $tType] :
( ( autoref_APP @ B @ A )
= ( ^ [F3: B > A] : F3 ) ) ).
% APP_def
thf(fact_39_autoref__id,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > A ) @ ( B > B ) ) @ ( product_Pair @ ( A > A ) @ ( B > B ) @ ( id @ A ) @ ( id @ B ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ).
% autoref_id
thf(fact_40_autoref__REMOVE__INTERNAL__EQ,axiom,
! [B: $tType,A: $tType,C2: A,A5: B,R3: set @ ( product_prod @ A @ B ),C4: A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C2 @ A5 ) @ R3 )
=> ( ( autore813025987NAL_EQ @ A @ C2 @ C4 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C4 @ A5 ) @ R3 ) ) ) ).
% autoref_REMOVE_INTERNAL_EQ
thf(fact_41_nres__more__simps_I6_J,axiom,
! [A: $tType,X: A,Y: A] :
( ( ( refine1687780735RETURN @ A @ X )
= ( refine1687780735RETURN @ A @ Y ) )
= ( X = Y ) ) ).
% nres_more_simps(6)
thf(fact_42_mem__Collect__eq,axiom,
! [A: $tType,A5: A,P: A > $o] :
( ( member @ A @ A5 @ ( collect @ A @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X6: A] : ( member @ A @ X6 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G3: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G3 @ X5 ) )
=> ( F = G3 ) ) ).
% ext
thf(fact_46_nres__monad1,axiom,
! [A: $tType,B: $tType,X: B,F: B > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X ) @ F )
= ( F @ X ) ) ).
% nres_monad1
thf(fact_47_nres__monad2,axiom,
! [A: $tType,M3: refine1665802226e_nres @ A] :
( ( refine463715084e_bind @ A @ A @ M3 @ ( refine1687780735RETURN @ A ) )
= M3 ) ).
% nres_monad2
thf(fact_48_fun__rel__id__simp,axiom,
! [B: $tType,A: $tType] :
( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( fun_rel @ A @ A @ B @ B ) @ ( id2 @ A ) ) @ ( id2 @ B ) )
= ( id2 @ ( A > B ) ) ) ).
% fun_rel_id_simp
thf(fact_49_relprop__triggers_I3_J,axiom,
! [D: $tType,R3: set @ ( product_prod @ D @ D )] :
( ( R3
= ( id2 @ D ) )
=> ( ( id2 @ D )
= R3 ) ) ).
% relprop_triggers(3)
thf(fact_50_relprop__triggers_I2_J,axiom,
! [C: $tType,R3: set @ ( product_prod @ C @ C )] :
( ( R3
= ( id2 @ C ) )
=> ( R3
= ( id2 @ C ) ) ) ).
% relprop_triggers(2)
thf(fact_51_autoref__int_I6_J,axiom,
( member @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) )
@ ( product_Pair @ ( int > int > $o ) @ ( int > int > $o )
@ ^ [Y5: int,Z: int] : Y5 = Z
@ ^ [Y5: int,Z: int] : Y5 = Z )
@ ( relAPP @ ( int > $o ) @ ( int > $o ) @ ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) > ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) ) @ ( fun_rel @ int @ int @ ( int > $o ) @ ( int > $o ) ) @ ( id2 @ int ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) ) @ ( fun_rel @ int @ int @ $o @ $o ) @ ( id2 @ int ) ) @ ( id2 @ $o ) ) ) ) ).
% autoref_int(6)
thf(fact_52_autoref__nat_I7_J,axiom,
( member @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) )
@ ( product_Pair @ ( nat > nat > $o ) @ ( nat > nat > $o )
@ ^ [Y5: nat,Z: nat] : Y5 = Z
@ ^ [Y5: nat,Z: nat] : Y5 = Z )
@ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ) ).
% autoref_nat(7)
thf(fact_53_autoref__bool_I8_J,axiom,
member @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) @ ( product_Pair @ ( $o > $o > $o ) @ ( $o > $o > $o ) @ (=>) @ (=>) ) @ ( relAPP @ ( $o > $o ) @ ( $o > $o ) @ ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) > ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ ( $o > $o ) @ ( $o > $o ) ) @ ( id2 @ $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ) ).
% autoref_bool(8)
thf(fact_54_autoref__bool_I4_J,axiom,
member @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) @ ( product_Pair @ ( $o > $o ) @ ( $o > $o ) @ (~) @ (~) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ).
% autoref_bool(4)
thf(fact_55_autoref__bool_I3_J,axiom,
member @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) @ ( product_Pair @ ( $o > $o > $o ) @ ( $o > $o > $o ) @ (|) @ (|) ) @ ( relAPP @ ( $o > $o ) @ ( $o > $o ) @ ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) > ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ ( $o > $o ) @ ( $o > $o ) ) @ ( id2 @ $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ) ).
% autoref_bool(3)
thf(fact_56_autoref__bool_I2_J,axiom,
member @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) @ ( product_Pair @ ( $o > $o > $o ) @ ( $o > $o > $o ) @ (&) @ (&) ) @ ( relAPP @ ( $o > $o ) @ ( $o > $o ) @ ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) > ( set @ ( product_prod @ ( $o > $o > $o ) @ ( $o > $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ ( $o > $o ) @ ( $o > $o ) ) @ ( id2 @ $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( $o > $o ) @ ( $o > $o ) ) ) ) @ ( fun_rel @ $o @ $o @ $o @ $o ) @ ( id2 @ $o ) ) @ ( id2 @ $o ) ) ) ).
% autoref_bool(2)
thf(fact_57_autoref__bool_I1_J,axiom,
! [X: $o] : ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ X @ X ) @ ( id2 @ $o ) ) ).
% autoref_bool(1)
thf(fact_58_relprop__id__orient,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( R3
= ( id2 @ A ) )
=> ( ( id2 @ A )
= R3 ) ) ).
% relprop_id_orient
thf(fact_59_rel__ANNOT__eq,axiom,
! [A: $tType,B: $tType,T2: A,R3: set @ ( product_prod @ B @ A )] :
( T2
= ( autoref_ANNOT @ A @ T2 @ ( autoref_rel_annot @ B @ A @ R3 ) ) ) ).
% rel_ANNOT_eq
thf(fact_60_ANNOT__def,axiom,
! [A: $tType] :
( ( autoref_ANNOT @ A )
= ( ^ [X6: A,A7: autoref_annot] : X6 ) ) ).
% ANNOT_def
thf(fact_61_OP__def,axiom,
! [A: $tType] :
( ( autoref_OP @ A )
= ( ^ [X6: A] : X6 ) ) ).
% OP_def
thf(fact_62_rel__cong,axiom,
! [A: $tType,B: $tType,F: A > B,G3: A > B,X: A,Y: A] :
( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G3 ) @ ( id2 @ ( A > B ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id2 @ A ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( G3 @ Y ) ) @ ( id2 @ B ) ) ) ) ).
% rel_cong
thf(fact_63_rel__fun__cong,axiom,
! [A: $tType,B: $tType,F: A > B,G3: A > B,X: A] :
( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G3 ) @ ( id2 @ ( A > B ) ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( G3 @ X ) ) @ ( id2 @ B ) ) ) ).
% rel_fun_cong
thf(fact_64_REMOVE__INTERNAL__EQ__def,axiom,
! [A: $tType] :
( ( autore813025987NAL_EQ @ A )
= ( ^ [Y5: A,Z: A] : Y5 = Z ) ) ).
% REMOVE_INTERNAL_EQ_def
thf(fact_65_REMOVE__INTERNAL__def,axiom,
! [A: $tType] :
( ( autore2013370551TERNAL @ A )
= ( ^ [X6: A] : X6 ) ) ).
% REMOVE_INTERNAL_def
thf(fact_66_REMOVE__INTERNAL__EQI,axiom,
! [A: $tType,A5: A] : ( autore813025987NAL_EQ @ A @ A5 @ A5 ) ).
% REMOVE_INTERNAL_EQI
thf(fact_67_rel__arg__cong,axiom,
! [A: $tType,B: $tType,X: A,Y: A,F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id2 @ A ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ ( id2 @ B ) ) ) ).
% rel_arg_cong
thf(fact_68_fun__rel__id,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( R1
= ( id2 @ A ) )
=> ( ( R22
= ( id2 @ B ) )
=> ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( fun_rel @ A @ A @ B @ B ) @ R1 ) @ R22 )
= ( id2 @ ( A > B ) ) ) ) ) ).
% fun_rel_id
thf(fact_69_IdI,axiom,
! [A: $tType,A5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_70_pair__in__Id__conv,axiom,
! [A: $tType,A5: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ ( id2 @ A ) )
= ( A5 = B3 ) ) ).
% pair_in_Id_conv
thf(fact_71_autoref__bool_I6_J,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) @ ( product_Pair @ ( A > A > $o > A ) @ ( B > B > $o > B ) @ ( product_rec_bool @ A ) @ ( product_rec_bool @ B ) ) @ ( relAPP @ ( A > $o > A ) @ ( B > $o > B ) @ ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) > ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > $o > A ) @ ( B > $o > B ) ) @ R3 ) @ ( relAPP @ ( $o > A ) @ ( $o > B ) @ ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) > ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( $o > A ) @ ( $o > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) ) @ ( fun_rel @ $o @ $o @ A @ B ) @ ( id2 @ $o ) ) @ R3 ) ) ) ) ).
% autoref_bool(6)
thf(fact_72_autoref__If__cong,axiom,
! [B: $tType,A: $tType,C4: $o,C2: $o,T3: A,T2: B,R3: set @ ( product_prod @ A @ B ),E3: A,E4: B] :
( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ C4 @ C2 ) @ ( id2 @ $o ) )
=> ( ( ( autore2013370551TERNAL @ $o @ C2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ T3 @ T2 ) @ R3 ) )
=> ( ( ~ ( autore2013370551TERNAL @ $o @ C2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ E3 @ E4 ) @ R3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( if @ A @ C4 @ T3 @ E3 ) @ ( autoref_APP @ B @ B @ ( autoref_APP @ B @ ( B > B ) @ ( autoref_APP @ $o @ ( B > B > B ) @ ( autoref_ANNOT @ ( $o > B > B > B ) @ ( autoref_OP @ ( $o > B > B > B ) @ ( if @ B ) ) @ ( autoref_rel_annot @ ( $o > A > A > A ) @ ( $o > B > B > B ) @ ( relAPP @ ( A > A > A ) @ ( B > B > B ) @ ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) > ( set @ ( product_prod @ ( $o > A > A > A ) @ ( $o > B > B > B ) ) ) ) @ ( fun_rel @ $o @ $o @ ( A > A > A ) @ ( B > B > B ) ) @ ( id2 @ $o ) ) @ ( relAPP @ ( A > A ) @ ( B > B ) @ ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) > ( set @ ( product_prod @ ( A > A > A ) @ ( B > B > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > A ) @ ( B > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ) ) ) @ C2 ) @ T2 ) @ E4 ) ) @ R3 ) ) ) ) ).
% autoref_If_cong
thf(fact_73_param__fun__upd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Ra: set @ ( product_prod @ A @ B ),Rb: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ ( A > A > $o ) @ ( B > B > $o ) )
@ ( product_Pair @ ( A > A > $o ) @ ( B > B > $o )
@ ^ [Y5: A,Z: A] : Y5 = Z
@ ^ [Y5: B,Z: B] : Y5 = Z )
@ ( relAPP @ ( A > $o ) @ ( B > $o ) @ ( set @ ( product_prod @ ( A > A > $o ) @ ( B > B > $o ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) ) > ( set @ ( product_prod @ ( A > A > $o ) @ ( B > B > $o ) ) ) ) @ ( fun_rel @ A @ B @ ( A > $o ) @ ( B > $o ) ) @ Ra ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) ) ) @ ( fun_rel @ A @ B @ $o @ $o ) @ Ra ) @ ( id2 @ $o ) ) ) )
=> ( member @ ( product_prod @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) ) @ ( product_Pair @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) @ ( fun_upd @ A @ C ) @ ( fun_upd @ B @ D ) ) @ ( relAPP @ ( A > C > A > C ) @ ( B > D > B > D ) @ ( set @ ( product_prod @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) ) ) @ ( relAPP @ ( A > C ) @ ( B > D ) @ ( ( set @ ( product_prod @ ( A > C > A > C ) @ ( B > D > B > D ) ) ) > ( set @ ( product_prod @ ( ( A > C ) > A > C > A > C ) @ ( ( B > D ) > B > D > B > D ) ) ) ) @ ( fun_rel @ ( A > C ) @ ( B > D ) @ ( A > C > A > C ) @ ( B > D > B > D ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( C > A > C ) @ ( D > B > D ) @ ( set @ ( product_prod @ ( A > C > A > C ) @ ( B > D > B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( C > A > C ) @ ( D > B > D ) ) ) > ( set @ ( product_prod @ ( A > C > A > C ) @ ( B > D > B > D ) ) ) ) @ ( fun_rel @ A @ B @ ( C > A > C ) @ ( D > B > D ) ) @ Ra ) @ ( relAPP @ ( A > C ) @ ( B > D ) @ ( set @ ( product_prod @ ( C > A > C ) @ ( D > B > D ) ) ) @ ( relAPP @ C @ D @ ( ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) > ( set @ ( product_prod @ ( C > A > C ) @ ( D > B > D ) ) ) ) @ ( fun_rel @ C @ D @ ( A > C ) @ ( B > D ) ) @ Rb ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ Ra ) @ Rb ) ) ) ) ) ) ).
% param_fun_upd
thf(fact_74_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X6: A] : X6 ) ) ).
% id_apply
thf(fact_75_prod__refine_I3_J,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F4: $tType,E: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D ),Rr: set @ ( product_prod @ E @ F4 )] : ( member @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) @ ( product_Pair @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) @ ( product_rec_prod @ A @ B @ E ) @ ( product_rec_prod @ C @ D @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) @ ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) > ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) @ ( relAPP @ ( B > E ) @ ( D > F4 ) @ ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) ) @ ( fun_rel @ A @ C @ ( B > E ) @ ( D > F4 ) ) @ Ra ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) ) @ ( fun_rel @ B @ D @ E @ F4 ) @ Rb ) @ Rr ) ) ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ E @ F4 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Rr ) ) ) ).
% prod_refine(3)
thf(fact_76_autoref__bool_I5_J,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) @ ( product_Pair @ ( A > A > $o > A ) @ ( B > B > $o > B ) @ ( product_case_bool @ A ) @ ( product_case_bool @ B ) ) @ ( relAPP @ ( A > $o > A ) @ ( B > $o > B ) @ ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) > ( set @ ( product_prod @ ( A > A > $o > A ) @ ( B > B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( A > $o > A ) @ ( B > $o > B ) ) @ R3 ) @ ( relAPP @ ( $o > A ) @ ( $o > B ) @ ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) > ( set @ ( product_prod @ ( A > $o > A ) @ ( B > $o > B ) ) ) ) @ ( fun_rel @ A @ B @ ( $o > A ) @ ( $o > B ) ) @ R3 ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) @ ( relAPP @ $o @ $o @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( $o > A ) @ ( $o > B ) ) ) ) @ ( fun_rel @ $o @ $o @ A @ B ) @ ( id2 @ $o ) ) @ R3 ) ) ) ) ).
% autoref_bool(5)
thf(fact_77_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z2: B] :
( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z2 )
= ( fun_upd @ A @ B @ F @ X @ Z2 ) ) ).
% fun_upd_upd
thf(fact_78_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
= F ) ).
% fun_upd_triv
thf(fact_79_fun__upd__apply,axiom,
! [A: $tType,B: $tType] :
( ( fun_upd @ B @ A )
= ( ^ [F3: B > A,X6: B,Y6: A,Z3: B] : ( if @ A @ ( Z3 = X6 ) @ Y6 @ ( F3 @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_80_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_81_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_82_prod__rel__simp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A,B3: B,A6: C,B4: D,R1: set @ ( product_prod @ A @ C ),R22: set @ ( product_prod @ B @ D )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ C @ D @ A6 @ B4 ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ R1 ) @ R22 ) )
= ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A5 @ A6 ) @ R1 )
& ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ B3 @ B4 ) @ R22 ) ) ) ).
% prod_rel_simp
thf(fact_83_prod__rel__id__simp,axiom,
! [B: $tType,A: $tType] :
( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( prod_rel @ A @ A @ B @ B ) @ ( id2 @ A ) ) @ ( id2 @ B ) )
= ( id2 @ ( product_prod @ A @ B ) ) ) ).
% prod_rel_id_simp
thf(fact_84_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F3: A > B,A7: A,B6: B,X6: A] : ( if @ B @ ( X6 = A7 ) @ B6 @ ( F3 @ X6 ) ) ) ) ).
% fun_upd_def
thf(fact_85_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G3: A > B,Z2: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y )
= ( fun_upd @ A @ B @ G3 @ X @ Z2 ) )
=> ( Y = Z2 ) ) ).
% fun_upd_eqD
thf(fact_86_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
( ( ( F @ X )
= Y )
=> ( ( fun_upd @ B @ A @ F @ X @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_87_fun__upd__same,axiom,
! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
( ( fun_upd @ B @ A @ F @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_88_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z2: A,X: A,F: A > B,Y: B] :
( ( Z2 != X )
=> ( ( fun_upd @ A @ B @ F @ X @ Y @ Z2 )
= ( F @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_89_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A5: A,C2: A,M: A > B,B3: B,D3: B] :
( ( A5 != C2 )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A5 @ B3 ) @ C2 @ D3 )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C2 @ D3 ) @ A5 @ B3 ) ) ) ).
% fun_upd_twist
thf(fact_90_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y )
= F )
= ( ( F @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_91_rec__bool__is__case,axiom,
! [A: $tType] :
( ( product_rec_bool @ A )
= ( product_case_bool @ A ) ) ).
% rec_bool_is_case
thf(fact_92_bool_Osplit__sel__asm,axiom,
! [A: $tType,P: A > $o,F1: A,F22: A,Bool: $o] :
( ( P @ ( product_case_bool @ A @ F1 @ F22 @ Bool ) )
= ( ~ ( ( Bool
& ~ ( P @ F1 ) )
| ( ~ Bool
& ~ ( P @ F22 ) ) ) ) ) ).
% bool.split_sel_asm
thf(fact_93_bool_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F1: A,F22: A,Bool: $o] :
( ( H @ ( product_case_bool @ A @ F1 @ F22 @ Bool ) )
= ( product_case_bool @ B @ ( H @ F1 ) @ ( H @ F22 ) @ Bool ) ) ).
% bool.case_distrib
thf(fact_94_bool_Ocase__eq__if,axiom,
! [A: $tType] :
( ( product_case_bool @ A )
= ( ^ [F12: A,F23: A,Bool2: $o] : ( if @ A @ Bool2 @ F12 @ F23 ) ) ) ).
% bool.case_eq_if
thf(fact_95_bool_Osplit__sel,axiom,
! [A: $tType,P: A > $o,F1: A,F22: A,Bool: $o] :
( ( P @ ( product_case_bool @ A @ F1 @ F22 @ Bool ) )
= ( ( Bool
=> ( P @ F1 ) )
& ( ~ Bool
=> ( P @ F22 ) ) ) ) ).
% bool.split_sel
thf(fact_96_old_Obool_Osimps_I3_J,axiom,
! [A: $tType,F1: A,F22: A] :
( ( product_case_bool @ A @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(3)
thf(fact_97_old_Obool_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: A] :
( ( product_case_bool @ A @ F1 @ F22 @ $false )
= F22 ) ).
% old.bool.simps(4)
thf(fact_98_prod__relI,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A5: A,A6: B,R1: set @ ( product_prod @ A @ B ),B3: C,B4: D,R22: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A6 ) @ R1 )
=> ( ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ B3 @ B4 ) @ R22 )
=> ( member @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_Pair @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_Pair @ A @ C @ A5 @ B3 ) @ ( product_Pair @ B @ D @ A6 @ B4 ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) @ ( prod_rel @ A @ B @ C @ D ) @ R1 ) @ R22 ) ) ) ) ).
% prod_relI
thf(fact_99_prod__relE,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,P2: product_prod @ A @ B,P3: product_prod @ C @ D,R1: set @ ( product_prod @ A @ C ),R22: set @ ( product_prod @ B @ D )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ P2 @ P3 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ R1 ) @ R22 ) )
=> ~ ! [A3: A,B5: B] :
( ( P2
= ( product_Pair @ A @ B @ A3 @ B5 ) )
=> ! [A4: C,B7: D] :
( ( P3
= ( product_Pair @ C @ D @ A4 @ B7 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ A4 ) @ R1 )
=> ~ ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ B5 @ B7 ) @ R22 ) ) ) ) ) ).
% prod_relE
thf(fact_100_prod__refine_I1_J,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) ) @ ( product_Pair @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) @ ( product_Pair @ A @ B ) @ ( product_Pair @ C @ D ) ) @ ( relAPP @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) @ ( set @ ( product_prod @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) ) > ( set @ ( product_prod @ ( A > B > ( product_prod @ A @ B ) ) @ ( C > D > ( product_prod @ C @ D ) ) ) ) ) @ ( fun_rel @ A @ C @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) @ Ra ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) > ( set @ ( product_prod @ ( B > ( product_prod @ A @ B ) ) @ ( D > ( product_prod @ C @ D ) ) ) ) ) @ ( fun_rel @ B @ D @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ Rb ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) ) ) ).
% prod_refine(1)
thf(fact_101_prod__rel__id,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( R1
= ( id2 @ A ) )
=> ( ( R22
= ( id2 @ B ) )
=> ( ( relAPP @ B @ B @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( relAPP @ A @ A @ ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( prod_rel @ A @ A @ B @ B ) @ R1 ) @ R22 )
= ( id2 @ ( product_prod @ A @ B ) ) ) ) ) ).
% prod_rel_id
thf(fact_102_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X6: A] : X6 ) ) ).
% id_def
thf(fact_103_eq__id__iff,axiom,
! [A: $tType,F: A > A] :
( ( ! [X6: A] :
( ( F @ X6 )
= X6 ) )
= ( F
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_104_param__and__cong2,axiom,
! [A5: $o,A6: $o,B3: $o,B4: $o] :
( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ A5 @ A6 ) @ ( id2 @ $o ) )
=> ( ( A5
=> ( A6
=> ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ B3 @ B4 ) @ ( id2 @ $o ) ) ) )
=> ( member @ ( product_prod @ $o @ $o )
@ ( product_Pair @ $o @ $o
@ ( B3
& A5 )
@ ( B4
& A6 ) )
@ ( id2 @ $o ) ) ) ) ).
% param_and_cong2
thf(fact_105_param__and__cong1,axiom,
! [A5: $o,A6: $o,B3: $o,B4: $o] :
( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ A5 @ A6 ) @ ( id2 @ $o ) )
=> ( ( A5
=> ( A6
=> ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ B3 @ B4 ) @ ( id2 @ $o ) ) ) )
=> ( member @ ( product_prod @ $o @ $o )
@ ( product_Pair @ $o @ $o
@ ( A5
& B3 )
@ ( A6
& B4 ) )
@ ( id2 @ $o ) ) ) ) ).
% param_and_cong1
thf(fact_106_param__bool_I1_J,axiom,
member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ $true @ $true ) @ ( id2 @ $o ) ).
% param_bool(1)
thf(fact_107_param__bool_I2_J,axiom,
member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ $false @ $false ) @ ( id2 @ $o ) ).
% param_bool(2)
thf(fact_108_IdE,axiom,
! [A: $tType,P2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id2 @ A ) )
=> ~ ! [X5: A] :
( P2
!= ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ).
% IdE
thf(fact_109_param__if,axiom,
! [B: $tType,A: $tType,C2: $o,C4: $o,T2: A,T3: B,R3: set @ ( product_prod @ A @ B ),E4: A,E3: B] :
( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ C2 @ C4 ) @ ( id2 @ $o ) )
=> ( ( C2
=> ( C4
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ T2 @ T3 ) @ R3 ) ) )
=> ( ( ~ C2
=> ( ~ C4
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ E4 @ E3 ) @ R3 ) ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( if @ A @ C2 @ T2 @ E4 ) @ ( if @ B @ C4 @ T3 @ E3 ) ) @ R3 ) ) ) ) ).
% param_if
thf(fact_110_param__uncurry,axiom,
! [B: $tType,E: $tType,D: $tType,A: $tType,F4: $tType,C: $tType,A2: set @ ( product_prod @ A @ D ),B2: set @ ( product_prod @ B @ E ),C5: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) ) @ ( product_Pair @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) @ ( uncurry @ A @ B @ C ) @ ( uncurry @ D @ E @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) @ ( set @ ( product_prod @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) ) ) @ ( relAPP @ ( A > B > C ) @ ( D > E > F4 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) > ( set @ ( product_prod @ ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) @ ( ( D > E > F4 ) > ( product_prod @ D @ E ) > F4 ) ) ) ) @ ( fun_rel @ ( A > B > C ) @ ( D > E > F4 ) @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) @ ( relAPP @ ( B > C ) @ ( E > F4 ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) ) @ ( fun_rel @ A @ D @ ( B > C ) @ ( E > F4 ) ) @ A2 ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) @ ( relAPP @ B @ E @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) ) @ ( fun_rel @ B @ E @ C @ F4 ) @ B2 ) @ C5 ) ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F4 ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) ) @ ( prod_rel @ A @ D @ B @ E ) @ A2 ) @ B2 ) ) @ C5 ) ) ) ).
% param_uncurry
thf(fact_111_param__all__interval__nat,axiom,
member @ ( product_prod @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) ) @ ( product_Pair @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) @ all_interval_nat @ all_interval_nat ) @ ( relAPP @ ( nat > nat > $o ) @ ( nat > nat > $o ) @ ( set @ ( product_prod @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) > ( set @ ( product_prod @ ( ( nat > $o ) > nat > nat > $o ) @ ( ( nat > $o ) > nat > nat > $o ) ) ) ) @ ( fun_rel @ ( nat > $o ) @ ( nat > $o ) @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ) ).
% param_all_interval_nat
thf(fact_112_param__curry,axiom,
! [A: $tType,D: $tType,B: $tType,E: $tType,F4: $tType,C: $tType,Ra: set @ ( product_prod @ A @ D ),Rb: set @ ( product_prod @ B @ E ),Rc: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) ) @ ( product_Pair @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) @ ( product_curry @ A @ B @ C ) @ ( product_curry @ D @ E @ F4 ) ) @ ( relAPP @ ( A > B > C ) @ ( D > E > F4 ) @ ( set @ ( product_prod @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) @ ( ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) > ( set @ ( product_prod @ ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) @ ( ( ( product_prod @ D @ E ) > F4 ) > D > E > F4 ) ) ) ) @ ( fun_rel @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) @ ( A > B > C ) @ ( D > E > F4 ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F4 ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) ) ) ) @ ( prod_rel @ A @ D @ B @ E ) @ Ra ) @ Rb ) ) @ Rc ) ) @ ( relAPP @ ( B > C ) @ ( E > F4 ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > C ) @ ( D > E > F4 ) ) ) ) @ ( fun_rel @ A @ D @ ( B > C ) @ ( E > F4 ) ) @ Ra ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) @ ( relAPP @ B @ E @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( B > C ) @ ( E > F4 ) ) ) ) @ ( fun_rel @ B @ E @ C @ F4 ) @ Rb ) @ Rc ) ) ) ) ).
% param_curry
thf(fact_113_prod__refine_I2_J,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F4: $tType,E: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D ),Rr: set @ ( product_prod @ E @ F4 )] : ( member @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) @ ( product_Pair @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) @ ( product_case_prod @ A @ B @ E ) @ ( product_case_prod @ C @ D @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) @ ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) > ( set @ ( product_prod @ ( ( A > B > E ) > ( product_prod @ A @ B ) > E ) @ ( ( C > D > F4 ) > ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( A > B > E ) @ ( C > D > F4 ) @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) @ ( relAPP @ ( B > E ) @ ( D > F4 ) @ ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) > ( set @ ( product_prod @ ( A > B > E ) @ ( C > D > F4 ) ) ) ) @ ( fun_rel @ A @ C @ ( B > E ) @ ( D > F4 ) ) @ Ra ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( B > E ) @ ( D > F4 ) ) ) ) @ ( fun_rel @ B @ D @ E @ F4 ) @ Rb ) @ Rr ) ) ) @ ( relAPP @ E @ F4 @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ E @ F4 ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > E ) @ ( ( product_prod @ C @ D ) > F4 ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ E @ F4 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Rr ) ) ) ).
% prod_refine(2)
thf(fact_114_autoref__rec__nat,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) ) @ ( product_Pair @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) @ ( rec_nat @ A ) @ ( rec_nat @ B ) ) @ ( relAPP @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) @ ( set @ ( product_prod @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) ) > ( set @ ( product_prod @ ( A > ( nat > A > A ) > nat > A ) @ ( B > ( nat > B > B ) > nat > B ) ) ) ) @ ( fun_rel @ A @ B @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) @ R3 ) @ ( relAPP @ ( nat > A ) @ ( nat > B ) @ ( set @ ( product_prod @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) ) @ ( relAPP @ ( nat > A > A ) @ ( nat > B > B ) @ ( ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) > ( set @ ( product_prod @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) ) ) ) @ ( fun_rel @ ( nat > A > A ) @ ( nat > B > B ) @ ( nat > A ) @ ( nat > B ) ) @ ( relAPP @ ( A > A ) @ ( B > B ) @ ( set @ ( product_prod @ ( nat > A > A ) @ ( nat > B > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) > ( set @ ( product_prod @ ( nat > A > A ) @ ( nat > B > B ) ) ) ) @ ( fun_rel @ nat @ nat @ ( A > A ) @ ( B > B ) ) @ ( id2 @ nat ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( A > A ) @ ( B > B ) ) ) ) @ ( fun_rel @ A @ B @ A @ B ) @ R3 ) @ R3 ) ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) ) @ ( fun_rel @ nat @ nat @ A @ B ) @ ( id2 @ nat ) ) @ R3 ) ) ) ) ).
% autoref_rec_nat
thf(fact_115_autoref__int_I5_J,axiom,
member @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) @ ( product_Pair @ ( int > int > $o ) @ ( int > int > $o ) @ ( ord_less_eq @ int ) @ ( ord_less_eq @ int ) ) @ ( relAPP @ ( int > $o ) @ ( int > $o ) @ ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) > ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) ) @ ( fun_rel @ int @ int @ ( int > $o ) @ ( int > $o ) ) @ ( id2 @ int ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) ) @ ( fun_rel @ int @ int @ $o @ $o ) @ ( id2 @ int ) ) @ ( id2 @ $o ) ) ) ).
% autoref_int(5)
thf(fact_116_case__prod__Pair,axiom,
! [B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% case_prod_Pair
thf(fact_117_curryI,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A5: A,B3: B] :
( ( F @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( product_curry @ A @ B @ $o @ F @ A5 @ B3 ) ) ).
% curryI
thf(fact_118_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F3: ( product_prod @ B @ C ) > A,A7: B,B6: C] : ( F3 @ ( product_Pair @ B @ C @ A7 @ B6 ) ) ) ) ).
% curry_conv
thf(fact_119_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% case_prod_curry
thf(fact_120_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
= F ) ).
% curry_case_prod
thf(fact_121_lhs__step__If,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [B3: $o,T2: A,M: A,E4: A] :
( ( B3
=> ( ord_less_eq @ A @ T2 @ M ) )
=> ( ( ~ B3
=> ( ord_less_eq @ A @ E4 @ M ) )
=> ( ord_less_eq @ A @ ( if @ A @ B3 @ T2 @ E4 ) @ M ) ) ) ) ).
% lhs_step_If
thf(fact_122_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
=> ~ ! [X5: B,Y3: C] :
( ( P2
= ( product_Pair @ B @ C @ X5 @ Y3 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X5 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_123_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_124_param__case__prod_H_H,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F4: $tType,E: $tType,P2: product_prod @ A @ B,P3: product_prod @ C @ D,F: A > B > E,F2: C > D > F4,R3: set @ ( product_prod @ E @ F4 )] :
( ! [A3: A,B5: B,A4: C,B7: D] :
( ( P2
= ( product_Pair @ A @ B @ A3 @ B5 ) )
=> ( ( P3
= ( product_Pair @ C @ D @ A4 @ B7 ) )
=> ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( F @ A3 @ B5 ) @ ( F2 @ A4 @ B7 ) ) @ R3 ) ) )
=> ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( product_case_prod @ A @ B @ E @ F @ P2 ) @ ( product_case_prod @ C @ D @ F4 @ F2 @ P3 ) ) @ R3 ) ) ).
% param_case_prod''
thf(fact_125_rec__prod__is__case,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% rec_prod_is_case
thf(fact_126_curryD,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A5: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F @ A5 @ B3 )
=> ( F @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ).
% curryD
thf(fact_127_curryE,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A5: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F @ A5 @ B3 )
=> ( F @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ).
% curryE
thf(fact_128_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_129_param__case__prod_H,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F4: $tType,E: $tType,P2: product_prod @ A @ B,P3: product_prod @ C @ D,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D ),F: A > B > E,F2: C > D > F4,R3: set @ ( product_prod @ E @ F4 )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ P2 @ P3 ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) )
=> ( ! [A3: A,B5: B,A4: C,B7: D] :
( ( P2
= ( product_Pair @ A @ B @ A3 @ B5 ) )
=> ( ( P3
= ( product_Pair @ C @ D @ A4 @ B7 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ A4 ) @ Ra )
=> ( ( member @ ( product_prod @ B @ D ) @ ( product_Pair @ B @ D @ B5 @ B7 ) @ Rb )
=> ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( F @ A3 @ B5 ) @ ( F2 @ A4 @ B7 ) ) @ R3 ) ) ) ) )
=> ( member @ ( product_prod @ E @ F4 ) @ ( product_Pair @ E @ F4 @ ( product_case_prod @ A @ B @ E @ F @ P2 ) @ ( product_case_prod @ C @ D @ F4 @ F2 @ P3 ) ) @ R3 ) ) ) ).
% param_case_prod'
thf(fact_130_autoref__case__nat,axiom,
! [B: $tType,A: $tType,Ra: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) ) @ ( product_Pair @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) @ ( case_nat @ A ) @ ( case_nat @ B ) ) @ ( relAPP @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) @ ( set @ ( product_prod @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) ) > ( set @ ( product_prod @ ( A > ( nat > A ) > nat > A ) @ ( B > ( nat > B ) > nat > B ) ) ) ) @ ( fun_rel @ A @ B @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) @ Ra ) @ ( relAPP @ ( nat > A ) @ ( nat > B ) @ ( set @ ( product_prod @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) ) @ ( relAPP @ ( nat > A ) @ ( nat > B ) @ ( ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) > ( set @ ( product_prod @ ( ( nat > A ) > nat > A ) @ ( ( nat > B ) > nat > B ) ) ) ) @ ( fun_rel @ ( nat > A ) @ ( nat > B ) @ ( nat > A ) @ ( nat > B ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) ) @ ( fun_rel @ nat @ nat @ A @ B ) @ ( id2 @ nat ) ) @ Ra ) ) @ ( relAPP @ A @ B @ ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( nat > A ) @ ( nat > B ) ) ) ) @ ( fun_rel @ nat @ nat @ A @ B ) @ ( id2 @ nat ) ) @ Ra ) ) ) ) ).
% autoref_case_nat
thf(fact_131_uncurry__curry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% uncurry_curry_id
thf(fact_132_curry__uncurry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
= F ) ).
% curry_uncurry_id
thf(fact_133_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A5: B,B3: C] :
( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A5 @ B3 ) )
= ( F @ A5 @ B3 ) ) ).
% uncurry_apply
thf(fact_134_nres__order__simps_I20_J,axiom,
! [W: $tType,X: W,Y: W] :
( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X ) @ ( refine1687780735RETURN @ W @ Y ) )
= ( X = Y ) ) ).
% nres_order_simps(20)
thf(fact_135_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ).
% subrelI
thf(fact_136_fun__rel__mono,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: set @ ( product_prod @ A @ B ),R22: set @ ( product_prod @ A @ B ),R32: set @ ( product_prod @ C @ D ),R42: set @ ( product_prod @ C @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R1 @ R22 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ R32 @ R42 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ R22 ) @ R32 ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( A > C ) @ ( B > D ) ) ) ) @ ( fun_rel @ A @ B @ C @ D ) @ R1 ) @ R42 ) ) ) ) ).
% fun_rel_mono
thf(fact_137_prod__rel__mono,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R22: set @ ( product_prod @ A @ B ),R1: set @ ( product_prod @ A @ B ),R32: set @ ( product_prod @ C @ D ),R42: set @ ( product_prod @ C @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R22 @ R1 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ R32 @ R42 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) @ ( prod_rel @ A @ B @ C @ D ) @ R22 ) @ R32 ) @ ( relAPP @ C @ D @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) @ ( relAPP @ A @ B @ ( ( set @ ( product_prod @ C @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) @ ( prod_rel @ A @ B @ C @ D ) @ R1 ) @ R42 ) ) ) ) ).
% prod_rel_mono
thf(fact_138_bind__cong,axiom,
! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F2: A > ( refine1665802226e_nres @ B )] :
( ( M = M2 )
=> ( ! [X5: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X5 ) @ M2 )
=> ( ( F @ X5 )
= ( F2 @ X5 ) ) )
=> ( ( refine463715084e_bind @ A @ B @ M @ F )
= ( refine463715084e_bind @ A @ B @ M2 @ F2 ) ) ) ) ).
% bind_cong
thf(fact_139_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F2: A > ( refine1665802226e_nres @ B )] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M4 )
=> ( ! [X5: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X5 ) @ M3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X5 ) @ ( F2 @ X5 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M3 @ F ) @ ( refine463715084e_bind @ A @ B @ M4 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
thf(fact_140_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A5: A,B3: A,C2: A,D3: A] :
( ( A5 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ( C2 = D3 )
=> ( ord_less_eq @ A @ A5 @ D3 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_141_pairself_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A12: product_prod @ A @ A] :
( ! [F5: A > B,A3: A,B5: A] : ( P @ F5 @ ( product_Pair @ A @ A @ A3 @ B5 ) )
=> ( P @ A0 @ A12 ) ) ).
% pairself.induct
thf(fact_142_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F5: A > B,A3: A,B5: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F5 @ ( product_Pair @ A @ A @ A3 @ B5 ) ) ) ).
% pairself.cases
thf(fact_143_bex2I,axiom,
! [A: $tType,B: $tType,A5: A,B3: B,S5: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ S5 )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ S5 )
=> ( P @ A5 @ B3 ) )
=> ? [A3: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ S5 )
& ( P @ A3 @ B5 ) ) ) ) ).
% bex2I
thf(fact_144_autoref__nat_I6_J,axiom,
member @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) @ ( product_Pair @ ( nat > nat > $o ) @ ( nat > nat > $o ) @ ( ord_less_eq @ nat ) @ ( ord_less_eq @ nat ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ).
% autoref_nat(6)
thf(fact_145_autoref__int_I4_J,axiom,
member @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) @ ( product_Pair @ ( int > int > $o ) @ ( int > int > $o ) @ ( ord_less @ int ) @ ( ord_less @ int ) ) @ ( relAPP @ ( int > $o ) @ ( int > $o ) @ ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) > ( set @ ( product_prod @ ( int > int > $o ) @ ( int > int > $o ) ) ) ) @ ( fun_rel @ int @ int @ ( int > $o ) @ ( int > $o ) ) @ ( id2 @ int ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) @ ( relAPP @ int @ int @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( int > $o ) @ ( int > $o ) ) ) ) @ ( fun_rel @ int @ int @ $o @ $o ) @ ( id2 @ int ) ) @ ( id2 @ $o ) ) ) ).
% autoref_int(4)
thf(fact_146_autoref__nat_I5_J,axiom,
member @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) @ ( product_Pair @ ( nat > nat > $o ) @ ( nat > nat > $o ) @ ( ord_less @ nat ) @ ( ord_less @ nat ) ) @ ( relAPP @ ( nat > $o ) @ ( nat > $o ) @ ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) > ( set @ ( product_prod @ ( nat > nat > $o ) @ ( nat > nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ ( nat > $o ) @ ( nat > $o ) ) @ ( id2 @ nat ) ) @ ( relAPP @ $o @ $o @ ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) @ ( relAPP @ nat @ nat @ ( ( set @ ( product_prod @ $o @ $o ) ) > ( set @ ( product_prod @ ( nat > $o ) @ ( nat > $o ) ) ) ) @ ( fun_rel @ nat @ nat @ $o @ $o ) @ ( id2 @ nat ) ) @ ( id2 @ $o ) ) ) ).
% autoref_nat(5)
thf(fact_147_case__prod__mono,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ord @ C )
=> ! [P2: product_prod @ A @ B,F: A > B > C,F2: A > B > C] :
( ! [A3: A,B5: B] :
( ( P2
= ( product_Pair @ A @ B @ A3 @ B5 ) )
=> ( ord_less_eq @ C @ ( F @ A3 @ B5 ) @ ( F2 @ A3 @ B5 ) ) )
=> ( ord_less_eq @ C @ ( product_case_prod @ A @ B @ C @ F @ P2 ) @ ( product_case_prod @ A @ B @ C @ F2 @ P2 ) ) ) ) ).
% case_prod_mono
thf(fact_148_param__apsnd,axiom,
! [A: $tType,D: $tType,C: $tType,F4: $tType,E: $tType,B: $tType,Rb: set @ ( product_prod @ A @ D ),Rc: set @ ( product_prod @ B @ E ),Ra: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) @ ( product_Pair @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) @ ( product_apsnd @ A @ B @ C ) @ ( product_apsnd @ D @ E @ F4 ) ) @ ( relAPP @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) @ ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) @ ( relAPP @ ( A > B ) @ ( D > E ) @ ( ( set @ ( product_prod @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( D > E ) > ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( D > E ) @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) ) @ ( fun_rel @ A @ D @ B @ E ) @ Rb ) @ Rc ) ) @ ( relAPP @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) @ ( relAPP @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) @ ( ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ C @ A ) > ( product_prod @ C @ B ) ) @ ( ( product_prod @ F4 @ D ) > ( product_prod @ F4 @ E ) ) ) ) ) @ ( fun_rel @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) @ ( relAPP @ A @ D @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ A @ D ) ) > ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ F4 @ D ) ) ) ) @ ( prod_rel @ C @ F4 @ A @ D ) @ Ra ) @ Rb ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ F4 @ E ) ) ) ) @ ( prod_rel @ C @ F4 @ B @ E ) @ Ra ) @ Rc ) ) ) ) ).
% param_apsnd
thf(fact_149_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_150_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_151_relprop__triggers_I6_J,axiom,
! [I: $tType,R3: set @ I,R5: set @ I] :
( ( ord_less_eq @ ( set @ I ) @ R3 @ R5 )
=> ( ord_less_eq @ ( set @ I ) @ R3 @ R5 ) ) ).
% relprop_triggers(6)
thf(fact_152_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A2: A > B > $o,B2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B2 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_153_subset__Collect__conv,axiom,
! [A: $tType,S5: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S5 @ ( collect @ A @ P ) )
= ( ! [X6: A] :
( ( member @ A @ X6 @ S5 )
=> ( P @ X6 ) ) ) ) ).
% subset_Collect_conv
thf(fact_154_exists__leI,axiom,
! [N: nat,P: nat > $o] :
( ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ N )
=> ~ ( P @ N2 ) )
=> ( P @ N ) )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ N )
& ( P @ N3 ) ) ) ).
% exists_leI
thf(fact_155_Refine__Misc_Oif__mono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
( ( B3
=> ( ord_less_eq @ A @ M1 @ M12 ) )
=> ( ( ~ B3
=> ( ord_less_eq @ A @ M22 @ M23 ) )
=> ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_156_param__apfst,axiom,
! [D: $tType,A: $tType,B: $tType,E: $tType,F4: $tType,C: $tType,Ra: set @ ( product_prod @ A @ D ),Rb: set @ ( product_prod @ B @ E ),Rc: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) @ ( product_Pair @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) @ ( product_apfst @ A @ B @ C ) @ ( product_apfst @ D @ E @ F4 ) ) @ ( relAPP @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) @ ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) @ ( relAPP @ ( A > B ) @ ( D > E ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( D > E ) > ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( D > E ) @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) ) @ ( fun_rel @ A @ D @ B @ E ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) @ ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) @ ( relAPP @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) @ ( ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ C ) ) @ ( ( product_prod @ D @ F4 ) > ( product_prod @ E @ F4 ) ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ D @ F4 ) ) ) ) @ ( prod_rel @ A @ D @ C @ F4 ) @ Ra ) @ Rc ) ) @ ( relAPP @ C @ F4 @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) ) @ ( relAPP @ B @ E @ ( ( set @ ( product_prod @ C @ F4 ) ) > ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ E @ F4 ) ) ) ) @ ( prod_rel @ B @ E @ C @ F4 ) @ Rb ) @ Rc ) ) ) ) ).
% param_apfst
thf(fact_157_param__map__prod,axiom,
! [C: $tType,G: $tType,E: $tType,A: $tType,B: $tType,F4: $tType,H2: $tType,D: $tType,Ra: set @ ( product_prod @ A @ E ),Rb: set @ ( product_prod @ B @ F4 ),Rc: set @ ( product_prod @ C @ G ),Rd: set @ ( product_prod @ D @ H2 )] : ( member @ ( product_prod @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) @ ( product_Pair @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) @ ( product_map_prod @ A @ B @ C @ D ) @ ( product_map_prod @ E @ F4 @ G @ H2 ) ) @ ( relAPP @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) @ ( set @ ( product_prod @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) @ ( relAPP @ ( A > B ) @ ( E > F4 ) @ ( ( set @ ( product_prod @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( E > F4 ) > ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( E > F4 ) @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) @ ( relAPP @ B @ F4 @ ( set @ ( product_prod @ ( A > B ) @ ( E > F4 ) ) ) @ ( relAPP @ A @ E @ ( ( set @ ( product_prod @ B @ F4 ) ) > ( set @ ( product_prod @ ( A > B ) @ ( E > F4 ) ) ) ) @ ( fun_rel @ A @ E @ B @ F4 ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) @ ( set @ ( product_prod @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) @ ( relAPP @ ( C > D ) @ ( G > H2 ) @ ( ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) > ( set @ ( product_prod @ ( ( C > D ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( G > H2 ) > ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) ) @ ( fun_rel @ ( C > D ) @ ( G > H2 ) @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) @ ( relAPP @ D @ H2 @ ( set @ ( product_prod @ ( C > D ) @ ( G > H2 ) ) ) @ ( relAPP @ C @ G @ ( ( set @ ( product_prod @ D @ H2 ) ) > ( set @ ( product_prod @ ( C > D ) @ ( G > H2 ) ) ) ) @ ( fun_rel @ C @ G @ D @ H2 ) @ Rc ) @ Rd ) ) @ ( relAPP @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) @ ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) @ ( relAPP @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) @ ( ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ C ) > ( product_prod @ B @ D ) ) @ ( ( product_prod @ E @ G ) > ( product_prod @ F4 @ H2 ) ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) @ ( relAPP @ C @ G @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) ) ) @ ( relAPP @ A @ E @ ( ( set @ ( product_prod @ C @ G ) ) > ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ E @ G ) ) ) ) @ ( prod_rel @ A @ E @ C @ G ) @ Ra ) @ Rc ) ) @ ( relAPP @ D @ H2 @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) ) @ ( relAPP @ B @ F4 @ ( ( set @ ( product_prod @ D @ H2 ) ) > ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ F4 @ H2 ) ) ) ) @ ( prod_rel @ B @ F4 @ D @ H2 ) @ Rb ) @ Rd ) ) ) ) ) ).
% param_map_prod
thf(fact_158_param__prod__swap,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,A2: set @ ( product_prod @ A @ C ),B2: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) ) @ ( product_Pair @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) @ ( product_swap @ A @ B ) @ ( product_swap @ C @ D ) ) @ ( relAPP @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) @ ( ( product_prod @ C @ D ) > ( product_prod @ D @ C ) ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ A2 ) @ B2 ) ) @ ( relAPP @ A @ C @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) ) @ ( relAPP @ B @ D @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ D @ C ) ) ) ) @ ( prod_rel @ B @ D @ A @ C ) @ B2 ) @ A2 ) ) ) ).
% param_prod_swap
thf(fact_159_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_160_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > B,A5: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ ( product_Pair @ C @ D @ A5 @ B3 ) )
= ( product_Pair @ A @ B @ ( F @ A5 ) @ ( G3 @ B3 ) ) ) ).
% map_prod_simp
thf(fact_161_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
= ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_162_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_163_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_164_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F3: A > C] : ( product_map_prod @ A @ C @ B @ B @ F3 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_165_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G3: D > A,P2: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G3 @ P2 ) )
= ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F @ P2 ) ) ) ).
% apsnd_apfst_commute
thf(fact_166_apsnd__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apsnd @ B @ C @ A )
= ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% apsnd_def
thf(fact_167_prod_Omap__id0,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% prod.map_id0
thf(fact_168_autoref__comp,axiom,
! [A: $tType,D: $tType,C: $tType,F4: $tType,E: $tType,B: $tType,Ra: set @ ( product_prod @ A @ D ),Rb: set @ ( product_prod @ B @ E ),Rc: set @ ( product_prod @ C @ F4 )] : ( member @ ( product_prod @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) ) @ ( product_Pair @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) @ ( comp @ A @ B @ C ) @ ( comp @ D @ E @ F4 ) ) @ ( relAPP @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) @ ( set @ ( product_prod @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) ) ) @ ( relAPP @ ( A > B ) @ ( D > E ) @ ( ( set @ ( product_prod @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) ) > ( set @ ( product_prod @ ( ( A > B ) > ( C > A ) > C > B ) @ ( ( D > E ) > ( F4 > D ) > F4 > E ) ) ) ) @ ( fun_rel @ ( A > B ) @ ( D > E ) @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) @ ( relAPP @ A @ D @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( A > B ) @ ( D > E ) ) ) ) @ ( fun_rel @ A @ D @ B @ E ) @ Ra ) @ Rb ) ) @ ( relAPP @ ( C > B ) @ ( F4 > E ) @ ( set @ ( product_prod @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) ) @ ( relAPP @ ( C > A ) @ ( F4 > D ) @ ( ( set @ ( product_prod @ ( C > B ) @ ( F4 > E ) ) ) > ( set @ ( product_prod @ ( ( C > A ) > C > B ) @ ( ( F4 > D ) > F4 > E ) ) ) ) @ ( fun_rel @ ( C > A ) @ ( F4 > D ) @ ( C > B ) @ ( F4 > E ) ) @ ( relAPP @ A @ D @ ( set @ ( product_prod @ ( C > A ) @ ( F4 > D ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ A @ D ) ) > ( set @ ( product_prod @ ( C > A ) @ ( F4 > D ) ) ) ) @ ( fun_rel @ C @ F4 @ A @ D ) @ Rc ) @ Ra ) ) @ ( relAPP @ B @ E @ ( set @ ( product_prod @ ( C > B ) @ ( F4 > E ) ) ) @ ( relAPP @ C @ F4 @ ( ( set @ ( product_prod @ B @ E ) ) > ( set @ ( product_prod @ ( C > B ) @ ( F4 > E ) ) ) ) @ ( fun_rel @ C @ F4 @ B @ E ) @ Rc ) @ Rb ) ) ) ) ).
% autoref_comp
thf(fact_169_less__than__bool__iff,axiom,
! [X: $o,Y: $o] :
( ( member @ ( product_prod @ $o @ $o ) @ ( product_Pair @ $o @ $o @ X @ Y ) @ refine1361002633n_bool )
= ( ~ X
& Y ) ) ).
% less_than_bool_iff
thf(fact_170_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F3: B > A,G4: C > B,X6: C] : ( F3 @ ( G4 @ X6 ) ) ) ) ).
% comp_apply
thf(fact_171_comp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ A @ B @ A @ F @ ( id @ A ) )
= F ) ).
% comp_id
thf(fact_172_id__comp,axiom,
! [B: $tType,A: $tType,G3: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G3 )
= G3 ) ).
% id_comp
thf(fact_173_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_174_swap__comp__swap,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% swap_comp_swap
thf(fact_175_map__prod_Ocompositionality,axiom,
! [D: $tType,F4: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G3: D > F4,H: A > C,I2: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F4 @ F @ G3 @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F4 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F4 @ B @ G3 @ I2 ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_176_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F4: $tType,B: $tType,F1: E > C,F22: A > E,G1: F4 > D,G22: B > F4] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F4 @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F4 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F4 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F4 @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_177_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F4: $tType,D: $tType,B: $tType,F: C > E,G3: D > F4,H: A > C,I2: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F4 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F4 @ F @ G3 ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 ) )
= ( product_map_prod @ A @ E @ B @ F4 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F4 @ B @ G3 @ I2 ) ) ) ).
% map_prod.comp
thf(fact_178_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G3: D > C,X: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G3 @ X ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G3 ) @ X ) ) ).
% apfst_compose
thf(fact_179_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A5: C > B,B3: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A5 @ B3 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
=> ( ( A5 @ ( B3 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_180_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_181_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F: C > A] :
( ( X = Y )
=> ( ( comp @ A @ B @ C @ X @ F )
= ( comp @ A @ B @ C @ Y @ F ) ) ) ).
% comp_cong_left
thf(fact_182_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F: B > C] :
( ( X = Y )
=> ( ( comp @ B @ C @ A @ F @ X )
= ( comp @ B @ C @ A @ F @ Y ) ) ) ).
% comp_cong_right
thf(fact_183_fun__comp__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G3: A > C,Fg: A > B] :
( ( ( comp @ C @ B @ A @ F @ G3 )
= Fg )
= ( ! [X6: A] :
( ( F @ ( G3 @ X6 ) )
= ( Fg @ X6 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_184_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F3: B > C,G4: A > B,X6: A] : ( F3 @ ( G4 @ X6 ) ) ) ) ).
% comp_def
thf(fact_185_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G3: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G3 ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G3 @ H ) ) ) ).
% comp_assoc
thf(fact_186_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A5: C > B,B3: A > C,C2: D > B,D3: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A5 @ B3 )
= ( comp @ D @ B @ A @ C2 @ D3 ) )
=> ( ( A5 @ ( B3 @ V ) )
= ( C2 @ ( D3 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_187_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A5: C > B,B3: A > C,C2: D > B,D3: A > D] :
( ( ( comp @ C @ B @ A @ A5 @ B3 )
= ( comp @ D @ B @ A @ C2 @ D3 ) )
=> ! [V2: A] :
( ( A5 @ ( B3 @ V2 ) )
= ( C2 @ ( D3 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_188_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A5: C > B,B3: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A5 @ B3 )
= C2 )
=> ( ( A5 @ ( B3 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_189_fun__upd__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G3: A > C,X: A,Y: C] :
( ( comp @ C @ B @ A @ F @ ( fun_upd @ A @ C @ G3 @ X @ Y ) )
= ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F @ G3 ) @ X @ ( F @ Y ) ) ) ).
% fun_upd_comp
thf(fact_190_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G3: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G3 @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G3 ) @ X ) ) ).
% apsnd_compose
thf(fact_191_prod_Omap__id,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B @ ( id @ A ) @ ( id @ B ) @ T2 )
= T2 ) ).
% prod.map_id
thf(fact_192_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X5: A,Y3: B] :
( ( P @ X5 @ Y3 )
=> ( Q @ X5 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_193_prod__refine_I4_J,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) ) @ ( product_Pair @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) @ ( product_fst @ A @ B ) @ ( product_fst @ C @ D ) ) @ ( relAPP @ A @ C @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > A ) @ ( ( product_prod @ C @ D ) > C ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ A @ C ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Ra ) ) ).
% prod_refine(4)
thf(fact_194_prod__refine_I5_J,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,Ra: set @ ( product_prod @ A @ C ),Rb: set @ ( product_prod @ B @ D )] : ( member @ ( product_prod @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) ) @ ( product_Pair @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) @ ( product_snd @ A @ B ) @ ( product_snd @ C @ D ) ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) ) ) @ ( relAPP @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > B ) @ ( ( product_prod @ C @ D ) > D ) ) ) ) @ ( fun_rel @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ B @ D ) @ ( relAPP @ B @ D @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) @ ( relAPP @ A @ C @ ( ( set @ ( product_prod @ B @ D ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) ) ) ) @ ( prod_rel @ A @ C @ B @ D ) @ Ra ) @ Rb ) ) @ Rb ) ) ).
% prod_refine(5)
thf(fact_195_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G3 @ X ) )
= ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_196_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G3: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G3 @ X ) )
= ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_197_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
= ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_198_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G3: C > A] :
( ( ( product_apfst @ C @ A @ B @ F @ X )
= ( product_apfst @ C @ A @ B @ G3 @ X ) )
= ( ( F @ ( product_fst @ C @ B @ X ) )
= ( G3 @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_199_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_200_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_201_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
= ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_202_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G3: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F @ X )
= ( product_apsnd @ C @ B @ A @ G3 @ X ) )
= ( ( F @ ( product_snd @ A @ C @ X ) )
= ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_203_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_204_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_205_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_206_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_207_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_208_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G3: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G3 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_209_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G3: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G3 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G3 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_210_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_211_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_212_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G3 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_213_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G3: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_214_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_215_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_216_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_217_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_218_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_219_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A5: A,B3: B,P: B > $o] :
( ( X
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( ( P @ ( product_snd @ A @ B @ X ) )
=> ( P @ B3 ) ) ) ).
% sndE
thf(fact_220_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: A,B3: B,P: A > $o] :
( ( X
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( ( P @ ( product_fst @ A @ B @ X ) )
=> ( P @ A5 ) ) ) ).
% fstE
thf(fact_221_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A5: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A5 )
=> ( Y = A5 ) ) ).
% snd_eqD
thf(fact_222_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_223_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A5: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A5 )
=> ( X = A5 ) ) ).
% fst_eqD
thf(fact_224_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_225_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : Y5 = Z )
= ( ^ [S: product_prod @ A @ B,T4: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S )
= ( product_fst @ A @ B @ T4 ) )
& ( ( product_snd @ A @ B @ S )
= ( product_snd @ A @ B @ T4 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_226_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_227_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_228_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ! [A7: A,X7: B] : ( P @ A7 @ X7 ) )
= ( ! [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% All_prod_contract
thf(fact_229_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ? [A7: A,X7: B] : ( P @ A7 @ X7 ) )
= ( ? [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% Ex_prod_contract
thf(fact_230_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F3: B > C > A,P4: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P4 ) @ ( product_snd @ B @ C @ P4 ) ) ) ) ).
% case_prod_beta
thf(fact_231_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F3: A > B > C,Prod3: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% prod.case_eq_if
thf(fact_232_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A2: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A2 ) ) )
=> ( A2 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_233_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A5: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A5 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_234_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A5: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_235_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A5: A,F: B > A,B3: B,C2: B] :
( ( A5
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A5 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_236_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A5: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A5 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_237_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [X6: A,Y6: A] :
( ( ord_less_eq @ A @ X6 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X6 ) ) ) ) ) ).
% eq_iff
thf(fact_238_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_239_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_240_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_241_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_242_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% order.trans
thf(fact_243_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_244_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_245_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A7: A,B6: A] :
( ( ord_less_eq @ A @ A7 @ B6 )
& ( ord_less_eq @ A @ B6 @ A7 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_246_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( A5 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_247_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A5: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_248_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A5 )
=> ( A5 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_249_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_250_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% dual_order.refl
thf(fact_251_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A5: A,B3: A] :
( ! [A3: A,B5: A] :
( ( ord_less_eq @ A @ A3 @ B5 )
=> ( P @ A3 @ B5 ) )
=> ( ! [A3: A,B5: A] :
( ( P @ B5 @ A3 )
=> ( P @ A3 @ B5 ) )
=> ( P @ A5 @ B3 ) ) ) ) ).
% linorder_wlog
% Type constructors (21)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
order @ int ).
thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
ord @ int ).
thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_5,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_6,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_7,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_8,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_12,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
ord @ $o ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_15,axiom,
! [A8: $tType] : ( preorder @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_16,axiom,
! [A8: $tType] : ( order @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_17,axiom,
! [A8: $tType] : ( ord @ ( refine1665802226e_nres @ A8 ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $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 @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) @ ( product_Pair @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) @ ( refine463715084e_bind @ a @ b ) @ ( refine463715084e_bind @ c @ d ) ) @ ( relAPP @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) @ ( set @ ( product_prod @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) @ ( relAPP @ ( refine1665802226e_nres @ a ) @ ( refine1665802226e_nres @ c ) @ ( ( set @ ( product_prod @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) > ( set @ ( product_prod @ ( ( refine1665802226e_nres @ a ) > ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( refine1665802226e_nres @ c ) > ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) ) @ ( fun_rel @ ( refine1665802226e_nres @ a ) @ ( refine1665802226e_nres @ c ) @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) @ ( relAPP @ a @ c @ ( set @ ( product_prod @ ( refine1665802226e_nres @ a ) @ ( refine1665802226e_nres @ c ) ) ) @ ( refine476890328es_rel @ a @ c ) @ r1 ) ) @ ( relAPP @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) @ ( set @ ( product_prod @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) @ ( relAPP @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) > ( set @ ( product_prod @ ( ( a > ( refine1665802226e_nres @ b ) ) > ( refine1665802226e_nres @ b ) ) @ ( ( c > ( refine1665802226e_nres @ d ) ) > ( refine1665802226e_nres @ d ) ) ) ) ) @ ( fun_rel @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) @ ( relAPP @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) @ ( set @ ( product_prod @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) ) ) @ ( relAPP @ a @ c @ ( ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) > ( set @ ( product_prod @ ( a > ( refine1665802226e_nres @ b ) ) @ ( c > ( refine1665802226e_nres @ d ) ) ) ) ) @ ( fun_rel @ a @ c @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) @ r1 ) @ ( relAPP @ b @ d @ ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) @ ( refine476890328es_rel @ b @ d ) @ r2 ) ) ) @ ( relAPP @ b @ d @ ( set @ ( product_prod @ ( refine1665802226e_nres @ b ) @ ( refine1665802226e_nres @ d ) ) ) @ ( refine476890328es_rel @ b @ d ) @ r2 ) ) ) ).
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