TPTP Problem File: COM090_5.p
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%------------------------------------------------------------------------------
% File : COM090_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 214
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Nip08] Nipkow (2008), Linear Quantifier Elimination
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : qe_214 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 180 ( 48 unt; 47 typ; 0 def)
% Number of atoms : 272 ( 80 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 188 ( 49 ~; 18 |; 17 &)
% ( 14 <=>; 90 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 18 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 28 ( 18 >; 10 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 35 ( 35 usr; 7 con; 0-5 aty)
% Number of variables : 608 ( 546 !; 10 ?; 608 :)
% ( 52 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:37
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_PresArith_Oatom,type,
atom: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (41)
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ominus,type,
cl_Groups_Ominus:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(B,fun(A,C))) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( fun(A,bool) > $o ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_ListVector_Oiprod,type,
iprod:
!>[A: $tType] : fun(list(A),fun(list(A),A)) ).
tff(sy_c_ListVector_Ozipwith0,type,
zipwith0:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(list(A),fun(list(B),list(C))) ) ).
tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord2118332866rt_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) * list(A) ) > T ) ).
tff(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : ( list(A) > fun(A,bool) ) ).
tff(sy_c_PresArith_Olbounds,type,
lbounds: list(atom) > list(product_prod(int,list(int))) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( fun(product_prod(A,B),C) > fun(A,fun(B,C)) ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Oprod_Oprod__case,type,
product_prod_case:
!>[A: $tType,B: $tType,T: $tType] : fun(fun(A,fun(B,T)),fun(product_prod(A,B),T)) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,bool) ) > fun(B,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fEx,type,
fEx:
!>[A: $tType] : fun(fun(A,bool),bool) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_fimplies,type,
fimplies: bool > fun(bool,bool) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : fun(A,fun(fun(A,bool),bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_as,type,
as: list(atom) ).
tff(sy_v_xs,type,
xs: list(int) ).
%----Relevant facts (100)
tff(fact_0__096finite_A_123i_A_N_A_092_060langle_062ks_Mxs_092_060rangle_062_A_124ks_Ai_O_A_Ii_M_Aks_J_A_058_Aset_A_Ilbounds_Aas_J_125_096,axiom,
finite_finite(int,collect(int,aa(fun(int,fun(list(int),bool)),fun(int,bool),aa(fun(fun(list(int),bool),bool),fun(fun(int,fun(list(int),bool)),fun(int,bool)),combb(fun(list(int),bool),bool,int),fEx(list(int))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool)),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool))),combb(fun(list(int),fun(int,bool)),fun(list(int),bool),int),aa(fun(fun(int,bool),bool),fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),combb(fun(int,bool),bool,list(int)),fEx(int))),aa(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),aa(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),aa(fun(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))))),combb(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),int),combs(list(int),fun(int,bool),fun(int,bool))),aa(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),aa(fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool))))),combb(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool))),int),aa(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),list(int)),combs(int,bool,bool))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool)))),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool))))),combb(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool))),int),aa(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),list(int)),aa(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj))),aa(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),aa(fun(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))))),combb(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),int),combb(fun(int,int),fun(int,bool),list(int))),aa(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),aa(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),int),combb(int,bool,int)),fequal(int)))),aa(fun(list(int),int),fun(list(int),fun(int,int)),aa(fun(int,fun(int,int)),fun(fun(list(int),int),fun(list(int),fun(int,int))),combb(int,fun(int,int),list(int)),aa(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int))),aa(list(int),fun(list(int),int),aa(fun(list(int),fun(list(int),int)),fun(list(int),fun(list(int),int)),combc(list(int),list(int),int),iprod(int)),xs))))))),aa(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool)),aa(fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool))),combc(list(int),fun(product_prod(int,list(int)),bool),fun(int,bool)),aa(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),aa(fun(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)))),combb(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)),list(int)),combc(int,fun(product_prod(int,list(int)),bool),bool)),aa(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),aa(fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)))),combb(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),list(int)),aa(fun(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),combb(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool),int),member(product_prod(int,list(int))))),aa(fun(int,fun(list(int),product_prod(int,list(int)))),fun(list(int),fun(int,product_prod(int,list(int)))),combc(int,list(int),product_prod(int,list(int))),product_Pair(int,list(int)))))),set(product_prod(int,list(int)),lbounds(as)))))))) ).
tff(fact_1__096lbounds_Aas_A_126_061_A_091_093_096,axiom,
lbounds(as) != nil(product_prod(int,list(int))) ).
tff(fact_2_finite__set,axiom,
! [A: $tType,Xsa: list(A)] : finite_finite(A,set(A,Xsa)) ).
tff(fact_3_finite__Collect__disjI,axiom,
! [A: $tType,Q2: fun(A,bool),P1: fun(A,bool)] :
( finite_finite(A,collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fdisj),P1)),Q2)))
<=> ( finite_finite(A,collect(A,P1))
& finite_finite(A,collect(A,Q2)) ) ) ).
tff(fact_4_finite__Collect__conjI,axiom,
! [A: $tType,Q2: fun(A,bool),P1: fun(A,bool)] :
( ( finite_finite(A,collect(A,P1))
| finite_finite(A,collect(A,Q2)) )
=> finite_finite(A,collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),P1)),Q2))) ) ).
tff(fact_5__096_I_F_Ii_M_Aks_J_O_Ai_A_N_A_092_060langle_062ks_Mxs_092_060rangle_062_J_A_092_096_Aset_A_Ilbounds_Aas_J_A_061_123i_A_N_A_092_060langle_062ks_Mxs_092_060rangle_062_A_124ks_Ai_O_A_Ii_M_Aks_J_A_058_Aset_A_Ilbounds_Aas_J_125_096,axiom,
image(product_prod(int,list(int)),int,aa(fun(int,fun(list(int),int)),fun(product_prod(int,list(int)),int),product_prod_case(int,list(int),int),aa(fun(list(int),int),fun(int,fun(list(int),int)),aa(fun(int,fun(fun(list(int),int),fun(list(int),int))),fun(fun(list(int),int),fun(int,fun(list(int),int))),combc(int,fun(list(int),int),fun(list(int),int)),aa(fun(int,fun(int,int)),fun(int,fun(fun(list(int),int),fun(list(int),int))),aa(fun(fun(int,int),fun(fun(list(int),int),fun(list(int),int))),fun(fun(int,fun(int,int)),fun(int,fun(fun(list(int),int),fun(list(int),int)))),combb(fun(int,int),fun(fun(list(int),int),fun(list(int),int)),int),combb(int,int,list(int))),minus_minus(int))),aa(list(int),fun(list(int),int),aa(fun(list(int),fun(list(int),int)),fun(list(int),fun(list(int),int)),combc(list(int),list(int),int),iprod(int)),xs))),set(product_prod(int,list(int)),lbounds(as))) = collect(int,aa(fun(int,fun(list(int),bool)),fun(int,bool),aa(fun(fun(list(int),bool),bool),fun(fun(int,fun(list(int),bool)),fun(int,bool)),combb(fun(list(int),bool),bool,int),fEx(list(int))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool)),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool))),combb(fun(list(int),fun(int,bool)),fun(list(int),bool),int),aa(fun(fun(int,bool),bool),fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),combb(fun(int,bool),bool,list(int)),fEx(int))),aa(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),aa(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),aa(fun(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))))),combb(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),int),combs(list(int),fun(int,bool),fun(int,bool))),aa(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),aa(fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool))))),combb(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool))),int),aa(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),list(int)),combs(int,bool,bool))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool)))),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool))))),combb(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool))),int),aa(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),list(int)),aa(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj))),aa(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),aa(fun(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))))),combb(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),int),combb(fun(int,int),fun(int,bool),list(int))),aa(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),aa(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),int),combb(int,bool,int)),fequal(int)))),aa(fun(list(int),int),fun(list(int),fun(int,int)),aa(fun(int,fun(int,int)),fun(fun(list(int),int),fun(list(int),fun(int,int))),combb(int,fun(int,int),list(int)),aa(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int))),aa(list(int),fun(list(int),int),aa(fun(list(int),fun(list(int),int)),fun(list(int),fun(list(int),int)),combc(list(int),list(int),int),iprod(int)),xs))))))),aa(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool)),aa(fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool))),combc(list(int),fun(product_prod(int,list(int)),bool),fun(int,bool)),aa(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),aa(fun(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)))),combb(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)),list(int)),combc(int,fun(product_prod(int,list(int)),bool),bool)),aa(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),aa(fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)))),combb(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),list(int)),aa(fun(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),combb(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool),int),member(product_prod(int,list(int))))),aa(fun(int,fun(list(int),product_prod(int,list(int)))),fun(list(int),fun(int,product_prod(int,list(int)))),combc(int,list(int),product_prod(int,list(int))),product_Pair(int,list(int)))))),set(product_prod(int,list(int)),lbounds(as))))))) ).
tff(fact_6_finite__code,axiom,
! [A: $tType] :
( finite_finite1(A)
=> ! [A2: fun(A,bool)] : finite_finite(A,A2) ) ).
tff(fact_7_split__paired__All,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ! [A4: A,B4: B] : pp(aa(product_prod(A,B),bool,P1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))) ) ).
tff(fact_8_Pair__eq,axiom,
! [A: $tType,B: $tType,B7: B,A7: A,B1: B,A1: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
<=> ( ( A1 = A7 )
& ( B1 = B7 ) ) ) ).
tff(fact_9__096finite_A_I_I_F_Ii_M_Aks_J_O_Ai_A_N_A_092_060langle_062ks_Mxs_092_060rangle_062_J_A_092_096_Aset_A_Ilbounds_Aas_J_J_096,axiom,
finite_finite(int,image(product_prod(int,list(int)),int,aa(fun(int,fun(list(int),int)),fun(product_prod(int,list(int)),int),product_prod_case(int,list(int),int),aa(fun(list(int),int),fun(int,fun(list(int),int)),aa(fun(int,fun(fun(list(int),int),fun(list(int),int))),fun(fun(list(int),int),fun(int,fun(list(int),int))),combc(int,fun(list(int),int),fun(list(int),int)),aa(fun(int,fun(int,int)),fun(int,fun(fun(list(int),int),fun(list(int),int))),aa(fun(fun(int,int),fun(fun(list(int),int),fun(list(int),int))),fun(fun(int,fun(int,int)),fun(int,fun(fun(list(int),int),fun(list(int),int)))),combb(fun(int,int),fun(fun(list(int),int),fun(list(int),int)),int),combb(int,int,list(int))),minus_minus(int))),aa(list(int),fun(list(int),int),aa(fun(list(int),fun(list(int),int)),fun(list(int),fun(list(int),int)),combc(list(int),list(int),int),iprod(int)),xs))),set(product_prod(int,list(int)),lbounds(as)))) ).
tff(fact_10_finite__image__set,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P1: fun(A,bool)] :
( finite_finite(A,collect(A,P1))
=> finite_finite(B,collect(B,aa(fun(B,fun(A,bool)),fun(B,bool),aa(fun(fun(A,bool),bool),fun(fun(B,fun(A,bool)),fun(B,bool)),combb(fun(A,bool),bool,B),fEx(A)),aa(fun(A,bool),fun(B,fun(A,bool)),aa(fun(B,fun(fun(A,bool),fun(A,bool))),fun(fun(A,bool),fun(B,fun(A,bool))),combc(B,fun(A,bool),fun(A,bool)),aa(fun(B,fun(A,fun(bool,bool))),fun(B,fun(fun(A,bool),fun(A,bool))),aa(fun(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool))),fun(fun(B,fun(A,fun(bool,bool))),fun(B,fun(fun(A,bool),fun(A,bool)))),combb(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),B),combs(A,bool,bool)),aa(fun(B,fun(A,bool)),fun(B,fun(A,fun(bool,bool))),aa(fun(fun(A,bool),fun(A,fun(bool,bool))),fun(fun(B,fun(A,bool)),fun(B,fun(A,fun(bool,bool)))),combb(fun(A,bool),fun(A,fun(bool,bool)),B),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj)),aa(fun(A,B),fun(B,fun(A,bool)),aa(fun(B,fun(fun(A,B),fun(A,bool))),fun(fun(A,B),fun(B,fun(A,bool))),combc(B,fun(A,B),fun(A,bool)),aa(fun(B,fun(B,bool)),fun(B,fun(fun(A,B),fun(A,bool))),aa(fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),fun(fun(B,fun(B,bool)),fun(B,fun(fun(A,B),fun(A,bool)))),combb(fun(B,bool),fun(fun(A,B),fun(A,bool)),B),combb(B,bool,A)),fequal(B))),F)))),P1)))) ) ).
tff(fact_11_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,Q2: fun(B,fun(A,bool)),P1: fun(A,bool)] :
( finite_finite(A,collect(A,P1))
=> ( finite_finite(B,collect(B,aa(fun(B,fun(A,bool)),fun(B,bool),aa(fun(fun(A,bool),bool),fun(fun(B,fun(A,bool)),fun(B,bool)),combb(fun(A,bool),bool,B),fEx(A)),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(fun(A,bool),fun(A,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),combb(fun(A,bool),fun(A,bool),B),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),P1))),Q2))))
<=> ! [Y3: A] :
( pp(aa(A,bool,P1,Y3))
=> finite_finite(B,collect(B,aa(A,fun(B,bool),aa(fun(B,fun(A,bool)),fun(A,fun(B,bool)),combc(B,A,bool),Q2),Y3))) ) ) ) ).
tff(fact_12_iprod__left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys: list(A),Xs1: list(A)] : aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs1),Ys)),Zs) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Xs1),Zs)),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Ys),Zs)) ) ).
tff(fact_13_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S: fun(product_prod(A,B),bool),R1: fun(product_prod(A,B),bool)] :
( ! [X1: A,Xa2: B] :
( pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Xa2)),R1))
<=> pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Xa2)),S)) )
<=> ( R1 = S ) ) ).
tff(fact_14_finite__Diff,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] :
( finite_finite(A,A2)
=> finite_finite(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)) ) ).
tff(fact_15_finite__imageI,axiom,
! [B: $tType,A: $tType,H: fun(A,B),F3: fun(A,bool)] :
( finite_finite(A,F3)
=> finite_finite(B,image(A,B,H,F3)) ) ).
tff(fact_16_split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A1: B,F: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A1),B1)) = aa(C,A,aa(B,fun(C,A),F,A1),B1) ).
tff(fact_17_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),A2: fun(product_prod(A,B),bool),B1: B,A1: A] :
( pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1)),A2))
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),aa(B,C,aa(A,fun(B,C),F,A1),B1)),image(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F),A2))) ) ).
tff(fact_18_list__diff__Nil2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [Xs1: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs1),nil(A)) = Xs1 ) ).
tff(fact_19_split__twice,axiom,
! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,P3: product_prod(D,E),G: fun(D,fun(E,product_prod(B,C))),F: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F),aa(product_prod(D,E),product_prod(B,C),aa(fun(D,fun(E,product_prod(B,C))),fun(product_prod(D,E),product_prod(B,C)),product_prod_case(D,E,product_prod(B,C)),G),P3)) = aa(product_prod(D,E),A,aa(fun(D,fun(E,A)),fun(product_prod(D,E),A),product_prod_case(D,E,A),aa(fun(D,fun(E,product_prod(B,C))),fun(D,fun(E,A)),aa(fun(fun(E,product_prod(B,C)),fun(E,A)),fun(fun(D,fun(E,product_prod(B,C))),fun(D,fun(E,A))),combb(fun(E,product_prod(B,C)),fun(E,A),D),aa(fun(product_prod(B,C),A),fun(fun(E,product_prod(B,C)),fun(E,A)),combb(product_prod(B,C),A,E),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F))),G)),P3) ).
tff(fact_20_split__weak__cong,axiom,
! [C: $tType,B: $tType,A: $tType,C2: fun(A,fun(B,C)),Q3: product_prod(A,B),P3: product_prod(A,B)] :
( ( P3 = Q3 )
=> ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),C2),P3) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),C2),Q3) ) ) ).
tff(fact_21_finite__Diff2,axiom,
! [A: $tType,A2: fun(A,bool),B3: fun(A,bool)] :
( finite_finite(A,B3)
=> ( finite_finite(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3))
<=> finite_finite(A,A2) ) ) ).
tff(fact_22_prod_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A1: B,F11: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F11),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A1),B1)) = aa(C,A,aa(B,fun(C,A),F11,A1),B1) ).
tff(fact_23_split__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),aa(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C)),aa(fun(fun(B,product_prod(A,B)),fun(B,C)),fun(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C))),combb(fun(B,product_prod(A,B)),fun(B,C),A),aa(fun(product_prod(A,B),C),fun(fun(B,product_prod(A,B)),fun(B,C)),combb(product_prod(A,B),C,B),F)),product_Pair(A,B))) = F ).
tff(fact_24_Pair__inject,axiom,
! [A: $tType,B: $tType,B6: B,A6: A,B5: B,A5: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B5) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6) )
=> ~ ( ( A5 = A6 )
=> ( B5 != B6 ) ) ) ).
tff(fact_25_finite,axiom,
! [A: $tType] :
( finite_finite1(A)
=> ! [A2: fun(A,bool)] : finite_finite(A,A2) ) ).
tff(fact_26_split__paired__Ex,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ? [A4: A,B4: B] : pp(aa(product_prod(A,B),bool,P1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))) ) ).
tff(fact_27_cond__split__eta,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(product_prod(A,B),C),F: fun(A,fun(B,C))] :
( ! [X3: A,Y1: B] : aa(B,C,aa(A,fun(B,C),F,X3),Y1) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y1))
=> ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F) = G ) ) ).
tff(fact_28_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A2: fun(A,bool)] :
( ~ finite_finite(A,A2)
=> ( finite_finite(B,image(A,B,F,A2))
=> ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A2))
& ~ finite_finite(A,collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),A2))),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),combc(A,B,bool),aa(fun(A,B),fun(A,fun(B,bool)),aa(fun(B,fun(B,bool)),fun(fun(A,B),fun(A,fun(B,bool))),combb(B,fun(B,bool),A),fequal(B)),F)),aa(A,B,F,X3))))) ) ) ) ).
tff(fact_29_image__eqI,axiom,
! [A: $tType,B: $tType,A2: fun(B,bool),Xa: B,F: fun(B,A),B1: A] :
( ( B1 = aa(B,A,F,Xa) )
=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Xa),A2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),image(B,A,F,A2))) ) ) ).
tff(fact_30_splitE2,axiom,
! [B: $tType,A: $tType,C: $tType,Z: product_prod(B,C),P1: fun(B,fun(C,A)),Q2: fun(A,bool)] :
( pp(aa(A,bool,Q2,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),P1),Z)))
=> ~ ! [X3: B,Y1: C] :
( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y1) )
=> ~ pp(aa(A,bool,Q2,aa(C,A,aa(B,fun(C,A),P1,X3),Y1))) ) ) ).
tff(fact_31_finite__list,axiom,
! [A: $tType,A2: fun(A,bool)] :
( finite_finite(A,A2)
=> ? [Xs: list(A)] : set(A,Xs) = A2 ) ).
tff(fact_32_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A1: B,F11: fun(B,fun(C,A))] : product_prod_rec(B,C,A,F11,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A1),B1)) = aa(C,A,aa(B,fun(C,A),F11,A1),B1) ).
tff(fact_33_internal__split__def,axiom,
! [C: $tType,B: $tType,A: $tType] : produc1605651328_split(A,B,C) = product_prod_case(A,B,C) ).
tff(fact_34_image__image,axiom,
! [B: $tType,A: $tType,C: $tType,A2: fun(C,bool),G: fun(C,B),F: fun(B,A)] : image(B,A,F,image(C,B,G,A2)) = image(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),combb(B,A,C),F),G),A2) ).
tff(fact_35_splitI,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F,A1),B1))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1))) ) ).
tff(fact_36_prod__caseI,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F11: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F11,A1),B1))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F11),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1))) ) ).
tff(fact_37_mem__splitI,axiom,
! [A: $tType,B: $tType,C: $tType,B1: C,A1: B,C2: fun(B,fun(C,fun(A,bool))),Z: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C2,A1),B1)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(product_prod(B,C),fun(A,bool),aa(fun(B,fun(C,fun(A,bool))),fun(product_prod(B,C),fun(A,bool)),product_prod_case(B,C,fun(A,bool)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A1),B1)))) ) ).
tff(fact_38_Diff__iff,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A2))
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B3)) ) ) ).
tff(fact_39_Diff__idemp,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] : aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)),B3) = aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3) ).
tff(fact_40_DiffI,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A2))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3))) ) ) ).
tff(fact_41_DiffE,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)))
=> ~ ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B3)) ) ) ).
tff(fact_42_set__diff__eq,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] : aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),A2))),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),combb(bool,bool,A),fNot),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),B3)))) ).
tff(fact_43_DiffD1,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A2)) ) ).
tff(fact_44_DiffD2,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B3)) ) ).
tff(fact_45_splitD,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1)))
=> pp(aa(B,bool,aa(A,fun(B,bool),F,A1),B1)) ) ).
tff(fact_46_splitD_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: C,B1: B,A1: A,R1: fun(A,fun(B,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),R1),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1)),C2))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R1,A1),B1),C2)) ) ).
tff(fact_47_rev__image__eqI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B1: B,A2: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A2))
=> ( ( B1 = aa(A,B,F,Xa) )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B1),image(A,B,F,A2))) ) ) ).
tff(fact_48_imageI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A2: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A2))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,Xa)),image(A,B,F,A2))) ) ).
tff(fact_49_image__iff,axiom,
! [A: $tType,B: $tType,A2: fun(B,bool),F: fun(B,A),Z: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),image(B,A,F,A2)))
<=> ? [X1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),A2))
& ( Z = aa(B,A,F,X1) ) ) ) ).
tff(fact_50_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A1: B,C2: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc1605651328_split(B,C,A),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A1),B1)) = aa(C,A,aa(B,fun(C,A),C2,A1),B1) ).
tff(fact_51_image__ident,axiom,
! [A: $tType,Y4: fun(A,bool)] : image(A,A,combi(A),Y4) = Y4 ).
tff(fact_52_imageE,axiom,
! [A: $tType,B: $tType,A2: fun(B,bool),F: fun(B,A),B1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),image(B,A,F,A2)))
=> ~ ! [X3: B] :
( ( B1 = aa(B,A,F,X3) )
=> ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A2)) ) ) ).
tff(fact_53_splitI2,axiom,
! [B: $tType,A: $tType,C2: fun(A,fun(B,bool)),P3: product_prod(A,B)] :
( ! [A3: A,B2: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> pp(aa(B,bool,aa(A,fun(B,bool),C2,A3),B2)) )
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),C2),P3)) ) ).
tff(fact_54_splitE,axiom,
! [A: $tType,B: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),C2),P3))
=> ~ ! [X3: A,Y1: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y1) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),C2,X3),Y1)) ) ) ).
tff(fact_55_mem__splitI2,axiom,
! [C: $tType,B: $tType,A: $tType,C2: fun(A,fun(B,fun(C,bool))),Z: C,P3: product_prod(A,B)] :
( ! [A3: A,B2: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),Z),aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,A3),B2))) )
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),Z),aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C2),P3))) ) ).
tff(fact_56_splitI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,Xa: C,C2: fun(A,fun(B,fun(C,bool))),P3: product_prod(A,B)] :
( ! [A3: A,B2: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = P3 )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,A3),B2),Xa)) )
=> pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C2),P3),Xa)) ) ).
tff(fact_57_split__part,axiom,
! [B: $tType,A: $tType,Q2: fun(A,fun(B,bool)),P1: bool,X2: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(fun(B,bool),fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),combb(fun(B,bool),fun(B,bool),A),aa(fun(bool,bool),fun(fun(B,bool),fun(B,bool)),combb(bool,bool,B),aa(bool,fun(bool,bool),fconj,P1))),Q2)),X2))
<=> ( pp(P1)
& pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),Q2),X2)) ) ) ).
tff(fact_58_splitE_H,axiom,
! [B: $tType,A: $tType,C: $tType,Z: C,P3: product_prod(A,B),C2: fun(A,fun(B,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C2),P3),Z))
=> ~ ! [X3: A,Y1: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y1) )
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,X3),Y1),Z)) ) ) ).
tff(fact_59_mem__splitE,axiom,
! [B: $tType,A: $tType,C: $tType,P3: product_prod(B,C),C2: fun(B,fun(C,fun(A,bool))),Z: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(product_prod(B,C),fun(A,bool),aa(fun(B,fun(C,fun(A,bool))),fun(product_prod(B,C),fun(A,bool)),product_prod_case(B,C,fun(A,bool)),C2),P3)))
=> ~ ! [X3: B,Y1: C] :
( ( P3 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y1) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C2,X3),Y1))) ) ) ).
tff(fact_60_minus__apply,axiom,
! [A: $tType,B: $tType] :
( cl_Groups_Ominus(A)
=> ! [Xa: B,B3: fun(B,A),A2: fun(B,A)] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),minus_minus(fun(B,A)),A2),B3),Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A2,Xa)),aa(B,A,B3,Xa)) ) ).
tff(fact_61_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( cl_Groups_Ominus(B)
=> ! [B3: fun(A,B),A2: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A2),B3),X2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,X2)),aa(A,B,B3,X2)) ) ).
tff(fact_62_image__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),N: fun(A,bool),M: fun(A,bool)] :
( ( M = N )
=> ( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),N))
=> ( aa(A,B,F,X3) = aa(A,B,G,X3) ) )
=> ( image(A,B,F,M) = image(A,B,G,N) ) ) ) ).
tff(fact_63_list_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,F21: fun(B,fun(list(B),A)),F11: A] : list_case(A,B,F11,F21,nil(B)) = F11 ).
tff(fact_64_prod__induct6,axiom,
! [F1: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool)] :
( ! [A3: A,B2: B,C1: C,D1: D,E1: E,F2: F1] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,P1,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),A3),aa(product_prod(C,product_prod(D,product_prod(E,F1))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F1))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1)))),B2),aa(product_prod(D,product_prod(E,F1)),product_prod(C,product_prod(D,product_prod(E,F1))),aa(C,fun(product_prod(D,product_prod(E,F1)),product_prod(C,product_prod(D,product_prod(E,F1)))),product_Pair(C,product_prod(D,product_prod(E,F1))),C1),aa(product_prod(E,F1),product_prod(D,product_prod(E,F1)),aa(D,fun(product_prod(E,F1),product_prod(D,product_prod(E,F1))),product_Pair(D,product_prod(E,F1)),D1),aa(F1,product_prod(E,F1),aa(E,fun(F1,product_prod(E,F1)),product_Pair(E,F1),E1),F2)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,P1,Xa)) ) ).
tff(fact_65_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F1: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))))] :
~ ! [A3: A,B2: B,C1: C,D1: D,E1: E,F2: F1] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),A3),aa(product_prod(C,product_prod(D,product_prod(E,F1))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F1))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1)))),B2),aa(product_prod(D,product_prod(E,F1)),product_prod(C,product_prod(D,product_prod(E,F1))),aa(C,fun(product_prod(D,product_prod(E,F1)),product_prod(C,product_prod(D,product_prod(E,F1)))),product_Pair(C,product_prod(D,product_prod(E,F1))),C1),aa(product_prod(E,F1),product_prod(D,product_prod(E,F1)),aa(D,fun(product_prod(E,F1),product_prod(D,product_prod(E,F1))),product_Pair(D,product_prod(E,F1)),D1),aa(F1,product_prod(E,F1),aa(E,fun(F1,product_prod(E,F1)),product_Pair(E,F1),E1),F2))))) ).
tff(fact_66_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,B2: B,C1: C,D1: D,E1: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A3),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B2),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C1),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1)))) ).
tff(fact_67_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool)] :
( ! [A3: A,B2: B,C1: C,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A3),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B2),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C1),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,Xa)) ) ).
tff(fact_68_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A3: A,B2: B,C1: C,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A3),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C1),D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,Xa)) ) ).
tff(fact_69_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A3: A,B2: B,C1: C,D1: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A3),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C1),D1))) ).
tff(fact_70_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,C)),P1: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A3: A,B2: B,C1: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P1,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C1))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P1,Xa)) ) ).
tff(fact_71_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A3: A,B2: B,C1: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C1)) ).
tff(fact_72_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A3: A,B2: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) ).
tff(fact_73_PairE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
~ ! [X3: A,Y1: B] : P2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y1) ).
tff(fact_74_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,B1: C,A1: B,F: fun(product_prod(B,C),A)] : aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F),A1),B1) = aa(product_prod(B,C),A,F,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A1),B1)) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X3: A] : aa(A,B,F,X3) = aa(A,B,G,X3)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A2))
<=> pp(aa(A,bool,A2,Xa)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_78_list__diff__def,axiom,
! [A: $tType] :
( ( cl_Groups_Ominus(A)
& zero(A) )
=> ( minus_minus(list(A)) = zipwith0(A,A,A,minus_minus(A)) ) ) ).
tff(fact_79_curryI,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1)))
=> pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A1),B1)) ) ).
tff(fact_80_zipwith0_Osimps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( zero(B)
& zero(C) )
=> ! [F: fun(B,fun(C,A))] : aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F),nil(B)),nil(C)) = nil(A) ) ).
tff(fact_81_curryE,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A1),B1))
=> pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1))) ) ).
tff(fact_82_curryD,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A1),B1))
=> pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1))) ) ).
tff(fact_83_curry__split,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : product_curry(A,B,C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F)) = F ).
tff(fact_84_split__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),product_curry(A,B,C,F)) = F ).
tff(fact_85_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X2: fun(product_prod(A,B),C),Xa1: A,Xb: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X2),Xa1),Xb) = aa(product_prod(A,B),C,X2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa1),Xb)) ).
tff(fact_86_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_87_The__split__eq,axiom,
! [A: $tType,B: $tType,Y2: B,Xa: A] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aa(fun(A,fun(fun(B,bool),fun(B,bool))),fun(fun(B,bool),fun(A,fun(B,bool))),combc(A,fun(B,bool),fun(B,bool)),aa(fun(A,fun(bool,bool)),fun(A,fun(fun(B,bool),fun(B,bool))),aa(fun(fun(bool,bool),fun(fun(B,bool),fun(B,bool))),fun(fun(A,fun(bool,bool)),fun(A,fun(fun(B,bool),fun(B,bool)))),combb(fun(bool,bool),fun(fun(B,bool),fun(B,bool)),A),combb(bool,bool,B)),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(A,fun(A,bool),fequal(A),Xa)))),aa(B,fun(B,bool),fequal(B),Y2)))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y2) ).
tff(fact_88_list__ex1__iff,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( list_ex1(A,P1,Xsa)
<=> ? [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Xsa)))
& pp(aa(A,bool,P1,X1))
& ! [Y3: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y3),set(A,Xsa)))
& pp(aa(A,bool,P1,Y3)) )
=> ( Y3 = X1 ) ) ) ) ).
tff(fact_89_split__paired__The,axiom,
! [B: $tType,A: $tType,P1: fun(product_prod(A,B),bool)] : the(product_prod(A,B),P1) = the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),aa(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,bool)),aa(fun(fun(B,product_prod(A,B)),fun(B,bool)),fun(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,bool))),combb(fun(B,product_prod(A,B)),fun(B,bool),A),aa(fun(product_prod(A,B),bool),fun(fun(B,product_prod(A,B)),fun(B,bool)),combb(product_prod(A,B),bool,B),P1)),product_Pair(A,B)))) ).
tff(fact_90_If__def,axiom,
! [A: $tType,Y2: A,Xa: A,P1: bool] :
( ( pp(P1)
=> ( Xa = the(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),combb(bool,bool,A),fimplies(P1)),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),Xa)))),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),combb(bool,bool,A),fimplies(aa(bool,bool,fNot,P1))),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),Y2)))) ) )
& ( ~ pp(P1)
=> ( Y2 = the(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),combb(bool,bool,A),fimplies(P1)),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),Xa)))),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),combb(bool,bool,A),fimplies(aa(bool,bool,fNot,P1))),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),Y2)))) ) ) ) ).
tff(fact_91_the__eq__trivial,axiom,
! [A: $tType,A1: A] : the(A,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A1)) = A1 ).
tff(fact_92_the__sym__eq__trivial,axiom,
! [A: $tType,Xa: A] : the(A,aa(A,fun(A,bool),fequal(A),Xa)) = Xa ).
tff(fact_93_the__equality,axiom,
! [A: $tType,A1: A,P1: fun(A,bool)] :
( pp(aa(A,bool,P1,A1))
=> ( ! [X3: A] :
( pp(aa(A,bool,P1,X3))
=> ( X3 = A1 ) )
=> ( the(A,P1) = A1 ) ) ) ).
tff(fact_94_theI,axiom,
! [A: $tType,A1: A,P1: fun(A,bool)] :
( pp(aa(A,bool,P1,A1))
=> ( ! [X3: A] :
( pp(aa(A,bool,P1,X3))
=> ( X3 = A1 ) )
=> pp(aa(A,bool,P1,the(A,P1))) ) ) ).
tff(fact_95_theI_H,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ? [X2: A] :
( pp(aa(A,bool,P1,X2))
& ! [Y1: A] :
( pp(aa(A,bool,P1,Y1))
=> ( Y1 = X2 ) ) )
=> pp(aa(A,bool,P1,the(A,P1))) ) ).
tff(fact_96_the1__equality,axiom,
! [A: $tType,A1: A,P1: fun(A,bool)] :
( ? [X2: A] :
( pp(aa(A,bool,P1,X2))
& ! [Y1: A] :
( pp(aa(A,bool,P1,Y1))
=> ( Y1 = X2 ) ) )
=> ( pp(aa(A,bool,P1,A1))
=> ( the(A,P1) = A1 ) ) ) ).
tff(fact_97_Ex__def,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ? [X11: A] : pp(aa(A,bool,P1,X11))
<=> ! [Q1: bool] :
( ! [X1: A] :
( pp(aa(A,bool,P1,X1))
=> pp(Q1) )
=> pp(Q1) ) ) ).
tff(fact_98_insort__insert__key__triv,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Xsa: list(B),Xa: B,F: fun(B,A)] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),aa(B,A,F,Xa)),image(B,A,F,set(B,Xsa))))
=> ( linord2118332866rt_key(B,A,F,Xa,Xsa) = Xsa ) ) ) ).
tff(fact_99_insort__insert__triv,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xsa: list(A),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( linord2118332866rt_key(A,A,combi(A),Xa,Xsa) = Xsa ) ) ) ).
%----Arities (12)
tff(arity_fun___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite1(T_2)
& finite_finite1(T_1) )
=> finite_finite1(fun(T_1,T_2)) ) ).
tff(arity_fun___Groups_Ominus,axiom,
! [T_1: $tType,T_2: $tType] :
( cl_Groups_Ominus(T_2)
=> cl_Groups_Ominus(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(arity_Int_Oint___Groups_Ominus,axiom,
cl_Groups_Ominus(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite1(bool) ).
tff(arity_HOL_Obool___Groups_Ominus,axiom,
cl_Groups_Ominus(bool) ).
tff(arity_List_Olist___Groups_Ominus,axiom,
! [T_1: $tType] :
( ( zero(T_1)
& cl_Groups_Ominus(T_1) )
=> cl_Groups_Ominus(list(T_1)) ) ).
tff(arity_prod___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite1(T_2)
& finite_finite1(T_1) )
=> finite_finite1(product_prod(T_1,T_2)) ) ).
%----Helper facts (20)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fEx_1_1_U,axiom,
! [A: $tType,X: A,P: fun(A,bool)] :
( ~ pp(aa(A,bool,P,X))
| pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),combb(B,C,A),P),Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,aa(B,fun(A,C),aa(fun(A,fun(B,C)),fun(B,fun(A,C)),combc(A,B,C),P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,B),fun(A,C)),combs(A,B,C),P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q))
| pp(P)
| pp(Q) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ( X != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).
tff(help_fimplies_1_1_U,axiom,
! [Q: bool,P: bool] :
( pp(P)
| pp(aa(bool,bool,fimplies(P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,fimplies(P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,fimplies(P),Q))
| ~ pp(P)
| pp(Q) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
finite_finite(int,collect(int,aa(fun(int,fun(list(int),bool)),fun(int,bool),aa(fun(fun(list(int),bool),bool),fun(fun(int,fun(list(int),bool)),fun(int,bool)),combb(fun(list(int),bool),bool,int),fEx(list(int))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool)),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool))),combb(fun(list(int),fun(int,bool)),fun(list(int),bool),int),aa(fun(fun(int,bool),bool),fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),combb(fun(int,bool),bool,list(int)),fEx(int))),aa(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),aa(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),aa(fun(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))))),combb(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),int),combs(list(int),fun(int,bool),fun(int,bool))),aa(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),aa(fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool))))),combb(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool))),int),aa(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),list(int)),combs(int,bool,bool))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool)))),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool))))),combb(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool))),int),aa(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),list(int)),aa(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj))),aa(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),aa(fun(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))))),combb(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),int),combb(fun(int,int),fun(int,bool),list(int))),aa(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),aa(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),int),combb(int,bool,int)),fequal(int)))),aa(fun(list(int),int),fun(list(int),fun(int,int)),aa(fun(int,fun(int,int)),fun(fun(list(int),int),fun(list(int),fun(int,int))),combb(int,fun(int,int),list(int)),aa(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int))),aa(list(int),fun(list(int),int),aa(fun(list(int),fun(list(int),int)),fun(list(int),fun(list(int),int)),combc(list(int),list(int),int),iprod(int)),xs))))))),aa(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool)),aa(fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool))),combc(list(int),fun(product_prod(int,list(int)),bool),fun(int,bool)),aa(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),aa(fun(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)))),combb(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)),list(int)),combc(int,fun(product_prod(int,list(int)),bool),bool)),aa(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),aa(fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)))),combb(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),list(int)),aa(fun(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),combb(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool),int),member(product_prod(int,list(int))))),aa(fun(int,fun(list(int),product_prod(int,list(int)))),fun(list(int),fun(int,product_prod(int,list(int)))),combc(int,list(int),product_prod(int,list(int))),product_Pair(int,list(int)))))),set(product_prod(int,list(int)),lbounds(as)))))))) ).
%------------------------------------------------------------------------------