TPTP Problem File: COM093_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : COM093_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 220
% 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_220 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 202 ( 53 unt; 60 typ; 0 def)
% Number of atoms : 272 ( 91 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 178 ( 48 ~; 15 |; 14 &)
% ( 24 <=>; 77 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 21 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 38 ( 23 >; 15 *; 0 +; 0 <<)
% Number of predicates : 20 ( 19 usr; 1 prp; 0-5 aty)
% Number of functors : 38 ( 38 usr; 8 con; 0-6 aty)
% Number of variables : 507 ( 438 !; 7 ?; 507 :)
% ( 62 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:23:04
%------------------------------------------------------------------------------
%----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 (54)
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oplus,type,
cl_Groups_Oplus:
!>[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_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord2061991079up_add:
!>[A: $tType] : $o ).
tff(sy_c_Big__Operators_Olinorder__class_OMax,type,
big_linorder_Max:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_Big__Operators_Olinorder__class_OMin,type,
big_linorder_Min:
!>[A: $tType] : ( fun(A,bool) > A ) ).
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_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,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_Enum_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite1:
!>[A: $tType] : ( fun(A,bool) > $o ) ).
tff(sy_c_FunDef_Oin__rel,type,
in_rel:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(A,B),bool) * A * B ) > $o ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : fun(A,fun(A,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_Osorted__list__of__set,type,
linord467138063of_set:
!>[A: $tType] : ( fun(A,bool) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
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_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_PresArith_OI_092_060_094isub_062Z,type,
i_Z: ( atom * list(int) ) > $o ).
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) * A * B ) > C ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * product_prod(A,B) ) > C ) ).
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_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_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_thesis____,type,
thesis: $o ).
tff(sy_v_x____,type,
x: int ).
tff(sy_v_xs,type,
xs: list(int) ).
%----Relevant facts (100)
tff(fact_0__096Max_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_058_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,
pp(aa(fun(int,bool),bool,aa(int,fun(fun(int,bool),bool),member(int),big_linorder_Max(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))))))))),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__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_finite1(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_3__096_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_A_126_061_A_123_125_096,axiom,
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))))))) != bot_bot(fun(int,bool)) ).
tff(fact_4_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))
<=> ! [A6: A,B5: 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),A6),B5))) ) ).
tff(fact_5_Pair__eq,axiom,
! [A: $tType,B: $tType,B7: B,A2: 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),A2),B7) )
<=> ( ( A1 = A2 )
& ( B1 = B7 ) ) ) ).
tff(fact_6_iprod__left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys: list(A),Xs: 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)),Xs),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),Xs),Zs)),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Ys),Zs)) ) ).
tff(fact_7_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S: fun(product_prod(A,B),bool),R1: fun(product_prod(A,B),bool)] :
( ! [X3: 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),X3),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),X3),Xa2)),S)) )
<=> ( R1 = S ) ) ).
tff(fact_8_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( cl_Groups_Ominus(B)
=> ! [B4: fun(A,B),A3: 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)),A3),B4),X2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,X2)),aa(A,B,B4,X2)) ) ).
tff(fact_9_minus__apply,axiom,
! [A: $tType,B: $tType] :
( cl_Groups_Ominus(A)
=> ! [Xa1: B,B4: fun(B,A),A3: 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)),A3),B4),Xa1) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A3,Xa1)),aa(B,A,B4,Xa1)) ) ).
tff(fact_10_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D2: A,C2: A,B1: A,A1: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A1),B1) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
=> ( ( A1 = B1 )
<=> ( C2 = D2 ) ) ) ) ).
tff(fact_11_Pair__inject,axiom,
! [A: $tType,B: $tType,B6: B,A7: A,B2: B,A4: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B6) )
=> ~ ( ( A4 = A7 )
=> ( B2 != B6 ) ) ) ).
tff(fact_12_in__rel__def,axiom,
! [B: $tType,A: $tType,Y3: B,Xa1: A,R1: fun(product_prod(A,B),bool)] :
( in_rel(A,B,R1,Xa1,Y3)
<=> 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),Xa1),Y3)),R1)) ) ).
tff(fact_13_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))
<=> ? [A6: A,B5: 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),A6),B5))) ) ).
tff(fact_14_bot__empty__eq,axiom,
! [A: $tType,X2: A] :
( pp(aa(A,bool,bot_bot(fun(A,bool)),X2))
<=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),bot_bot(fun(A,bool)))) ) ).
tff(fact_15_list__diff__Nil2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs),nil(A)) = Xs ) ).
tff(fact_16_List_Oset_Osimps_I1_J,axiom,
! [A: $tType] : set(A,nil(A)) = bot_bot(fun(A,bool)) ).
tff(fact_17_set__empty2,axiom,
! [A: $tType,Xsa: list(A)] :
( ( bot_bot(fun(A,bool)) = set(A,Xsa) )
<=> ( Xsa = nil(A) ) ) ).
tff(fact_18_set__empty,axiom,
! [A: $tType,Xsa: list(A)] :
( ( set(A,Xsa) = bot_bot(fun(A,bool)) )
<=> ( Xsa = nil(A) ) ) ).
tff(fact_19_finite__set,axiom,
! [A: $tType,Xsa: list(A)] : finite_finite1(A,set(A,Xsa)) ).
tff(fact_20_finite_OemptyI,axiom,
! [A: $tType] : finite_finite1(A,bot_bot(fun(A,bool))) ).
tff(fact_21_finite__Collect__disjI,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] :
( finite_finite1(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)),Q1)))
<=> ( finite_finite1(A,collect(A,P1))
& finite_finite1(A,collect(A,Q1)) ) ) ).
tff(fact_22_finite__Collect__conjI,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] :
( ( finite_finite1(A,collect(A,P1))
| finite_finite1(A,collect(A,Q1)) )
=> finite_finite1(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)),Q1))) ) ).
tff(fact_23_Max__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: fun(A,bool)] :
( finite_finite1(A,A3)
=> ( ( A3 != bot_bot(fun(A,bool)) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),big_linorder_Max(A,A3)),A3)) ) ) ) ).
tff(fact_24_finite__code,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [A3: fun(A,bool)] : finite_finite1(A,A3) ) ).
tff(fact_25_finite__Diff,axiom,
! [A: $tType,B4: fun(A,bool),A3: fun(A,bool)] :
( finite_finite1(A,A3)
=> finite_finite1(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A3),B4)) ) ).
tff(fact_26_finite__Diff2,axiom,
! [A: $tType,A3: fun(A,bool),B4: fun(A,bool)] :
( finite_finite1(A,B4)
=> ( finite_finite1(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A3),B4))
<=> finite_finite1(A,A3) ) ) ).
tff(fact_27_finite,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [A3: fun(A,bool)] : finite_finite1(A,A3) ) ).
tff(fact_28_finite__image__set,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P1: fun(A,bool)] :
( finite_finite1(A,collect(A,P1))
=> finite_finite1(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_29_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,Q1: fun(B,fun(A,bool)),P1: fun(A,bool)] :
( finite_finite1(A,collect(A,P1))
=> ( finite_finite1(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))),Q1))))
<=> ! [Y1: A] :
( pp(aa(A,bool,P1,Y1))
=> finite_finite1(B,collect(B,aa(A,fun(B,bool),aa(fun(B,fun(A,bool)),fun(A,fun(B,bool)),combc(B,A,bool),Q1),Y1))) ) ) ) ).
tff(fact_30_emptyE,axiom,
! [A: $tType,A1: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),bot_bot(fun(A,bool)))) ).
tff(fact_31_Collect__empty__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( collect(A,P1) = bot_bot(fun(A,bool)) )
<=> ! [X3: A] : ~ pp(aa(A,bool,P1,X3)) ) ).
tff(fact_32_Diff__cancel,axiom,
! [A: $tType,A3: 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)),A3),A3) = bot_bot(fun(A,bool)) ).
tff(fact_33_empty__iff,axiom,
! [A: $tType,C2: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),bot_bot(fun(A,bool)))) ).
tff(fact_34_empty__Collect__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P1) )
<=> ! [X3: A] : ~ pp(aa(A,bool,P1,X3)) ) ).
tff(fact_35_empty__Diff,axiom,
! [A: $tType,A3: 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)),bot_bot(fun(A,bool))),A3) = bot_bot(fun(A,bool)) ).
tff(fact_36_all__not__in__conv,axiom,
! [A: $tType,A3: fun(A,bool)] :
( ! [X3: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A3))
<=> ( A3 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_37_DiffE,axiom,
! [A: $tType,B4: fun(A,bool),A3: 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)),A3),B4)))
=> ~ ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B4)) ) ) ).
tff(fact_38_DiffI,axiom,
! [A: $tType,B4: fun(A,bool),A3: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A3))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B4))
=> 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)),A3),B4))) ) ) ).
tff(fact_39_Diff__idemp,axiom,
! [A: $tType,B4: fun(A,bool),A3: 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)),A3),B4)),B4) = aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A3),B4) ).
tff(fact_40_Diff__iff,axiom,
! [A: $tType,B4: fun(A,bool),A3: 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)),A3),B4)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A3))
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B4)) ) ) ).
tff(fact_41_DiffD2,axiom,
! [A: $tType,B4: fun(A,bool),A3: 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)),A3),B4)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B4)) ) ).
tff(fact_42_DiffD1,axiom,
! [A: $tType,B4: fun(A,bool),A3: 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)),A3),B4)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A3)) ) ).
tff(fact_43_set__diff__eq,axiom,
! [A: $tType,B4: fun(A,bool),A3: 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)),A3),B4) = 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)),A3))),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)),B4)))) ).
tff(fact_44_empty__def,axiom,
! [A: $tType] : bot_bot(fun(A,bool)) = collect(A,combk(bool,A,fFalse)) ).
tff(fact_45_ex__in__conv,axiom,
! [A: $tType,A3: fun(A,bool)] :
( ? [X3: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A3))
<=> ( A3 != bot_bot(fun(A,bool)) ) ) ).
tff(fact_46_Diff__empty,axiom,
! [A: $tType,A3: 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)),A3),bot_bot(fun(A,bool))) = A3 ).
tff(fact_47_equals0D,axiom,
! [A: $tType,A1: A,A3: fun(A,bool)] :
( ( A3 = bot_bot(fun(A,bool)) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),A3)) ) ).
tff(fact_48_finite__list,axiom,
! [A: $tType,A3: fun(A,bool)] :
( finite_finite1(A,A3)
=> ? [Xs1: list(A)] : set(A,Xs1) = A3 ) ).
tff(fact_49_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_50_bot__fun__def,axiom,
! [A: $tType,B: $tType] :
( bot(B)
=> ! [X2: A] : aa(A,B,bot_bot(fun(A,B)),X2) = bot_bot(B) ) ).
tff(fact_51_bot__apply,axiom,
! [B: $tType,A: $tType] :
( bot(A)
=> ! [Xa1: B] : aa(B,A,bot_bot(fun(B,A)),Xa1) = bot_bot(A) ) ).
tff(fact_52_sorted__list__of__set__empty,axiom,
! [A: $tType] :
( linorder(A)
=> ( linord467138063of_set(A,bot_bot(fun(A,bool))) = nil(A) ) ) ).
tff(fact_53_bot__empty__eq2,axiom,
! [B: $tType,A: $tType,X2: A,Xa: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X2),Xa))
<=> 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),X2),Xa)),bot_bot(fun(product_prod(A,B),bool)))) ) ).
tff(fact_54_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_55_equals0I,axiom,
! [A: $tType,A3: fun(A,bool)] :
( ! [Y2: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),A3))
=> ( A3 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_56_prod__induct6,axiom,
! [F1: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa1: 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)] :
( ! [A5: A,B3: B,C3: 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))))),A5),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)))),B3),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))),C3),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,Xa1)) ) ).
tff(fact_57_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)))))] :
~ ! [A5: A,B3: B,C3: 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))))),A5),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)))),B3),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))),C3),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_58_bot2E,axiom,
! [A: $tType,B: $tType,Y3: B,Xa1: A] : ~ pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),Xa1),Y3)) ).
tff(fact_59_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))))] :
~ ! [A5: A,B3: B,C3: 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)))),A5),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))),B3),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)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1)))) ).
tff(fact_60_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa1: 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)] :
( ! [A5: A,B3: B,C3: 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)))),A5),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))),B3),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)),C3),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,Xa1)) ) ).
tff(fact_61_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A5: A,B3: B,C3: 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))),A5),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)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D1))) ).
tff(fact_62_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Xa1: product_prod(A,product_prod(B,product_prod(C,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A5: A,B3: B,C3: 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))),A5),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)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,Xa1)) ) ).
tff(fact_63_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,Xa1: product_prod(A,product_prod(B,C)),P1: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A5: A,B3: B,C3: 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)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P1,Xa1)) ) ).
tff(fact_64_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A5: A,B3: B,C3: 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)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)) ).
tff(fact_65_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A5: A,B3: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) ).
tff(fact_66_PairE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
~ ! [X1: A,Y2: B] : P2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Y2) ).
tff(fact_67_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A1: B,C2: fun(B,fun(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_68_add__Max__commute,axiom,
! [A: $tType] :
( linord2061991079up_add(A)
=> ! [K: A,N: fun(A,bool)] :
( finite_finite1(A,N)
=> ( ( N != bot_bot(fun(A,bool)) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),K),big_linorder_Max(A,N)) = big_linorder_Max(A,collect(A,aa(fun(A,fun(A,bool)),fun(A,bool),aa(fun(fun(A,bool),bool),fun(fun(A,fun(A,bool)),fun(A,bool)),combb(fun(A,bool),bool,A),fEx(A)),aa(fun(A,bool),fun(A,fun(A,bool)),aa(fun(A,fun(fun(A,bool),fun(A,bool))),fun(fun(A,bool),fun(A,fun(A,bool))),combc(A,fun(A,bool),fun(A,bool)),aa(fun(A,fun(A,fun(bool,bool))),fun(A,fun(fun(A,bool),fun(A,bool))),aa(fun(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool))),fun(fun(A,fun(A,fun(bool,bool))),fun(A,fun(fun(A,bool),fun(A,bool)))),combb(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),A),combs(A,bool,bool)),aa(fun(A,fun(A,bool)),fun(A,fun(A,fun(bool,bool))),aa(fun(fun(A,bool),fun(A,fun(bool,bool))),fun(fun(A,fun(A,bool)),fun(A,fun(A,fun(bool,bool)))),combb(fun(A,bool),fun(A,fun(bool,bool)),A),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,A),fun(A,fun(A,bool)),aa(fun(A,fun(fun(A,A),fun(A,bool))),fun(fun(A,A),fun(A,fun(A,bool))),combc(A,fun(A,A),fun(A,bool)),aa(fun(A,fun(A,bool)),fun(A,fun(fun(A,A),fun(A,bool))),aa(fun(fun(A,bool),fun(fun(A,A),fun(A,bool))),fun(fun(A,fun(A,bool)),fun(A,fun(fun(A,A),fun(A,bool)))),combb(fun(A,bool),fun(fun(A,A),fun(A,bool)),A),combb(A,bool,A)),fequal(A))),aa(A,fun(A,A),plus_plus(A),K))))),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)),N))))) ) ) ) ) ).
tff(fact_69_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,B1: C,A1: B,F: fun(product_prod(B,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_70_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_71_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(product_curry(A,B,bool,F,A1,B1)) ) ).
tff(fact_72_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C2: A,A1: A,B1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),A1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A1) )
<=> ( B1 = C2 ) ) ) ).
tff(fact_73_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C2: A,B1: A,A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C2) )
<=> ( B1 = C2 ) ) ) ).
tff(fact_74_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_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : aa(A,B,F,X1) = aa(A,B,G,X1)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A3: fun(A,bool),Xa1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa1),A3))
<=> pp(aa(A,bool,A3,Xa1)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_78_diff__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A4),B2)),B2) = A4 ) ).
tff(fact_79_add__diff__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A4: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B2)),B2) = A4 ) ).
tff(fact_80_list__add__Nil2,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs),nil(A)) = Xs ) ).
tff(fact_81_list__add__Nil,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),nil(A)),Xs) = Xs ) ).
tff(fact_82_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A4: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),A4) )
=> ( B2 = C1 ) ) ) ).
tff(fact_83_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C1: A,B2: A,A4: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C1) )
=> ( B2 = C1 ) ) ) ).
tff(fact_84_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B2: A,A4: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C1) )
=> ( B2 = C1 ) ) ) ).
tff(fact_85_iprod__left__add__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys: list(A),Xs: 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)),plus_plus(list(A)),Xs),Ys)),Zs) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Xs),Zs)),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Ys),Zs)) ) ).
tff(fact_86_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C1: A,B2: A,A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B2)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C1)) ) ).
tff(fact_87_list__add__def,axiom,
! [A: $tType] :
( ( cl_Groups_Oplus(A)
& zero(A) )
=> ( plus_plus(list(A)) = zipwith0(A,A,A,plus_plus(A)) ) ) ).
tff(fact_88_curryE,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(product_prod(A,B),bool)] :
( pp(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_89_curryD,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(product_prod(A,B),bool)] :
( pp(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_90_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X2: fun(product_prod(A,B),C),Xa: A,Xb: B] : product_curry(A,B,C,X2,Xa,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),Xa),Xb)) ).
tff(fact_91_add__Min__commute,axiom,
! [A: $tType] :
( linord2061991079up_add(A)
=> ! [K: A,N: fun(A,bool)] :
( finite_finite1(A,N)
=> ( ( N != bot_bot(fun(A,bool)) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),K),big_linorder_Min(A,N)) = big_linorder_Min(A,collect(A,aa(fun(A,fun(A,bool)),fun(A,bool),aa(fun(fun(A,bool),bool),fun(fun(A,fun(A,bool)),fun(A,bool)),combb(fun(A,bool),bool,A),fEx(A)),aa(fun(A,bool),fun(A,fun(A,bool)),aa(fun(A,fun(fun(A,bool),fun(A,bool))),fun(fun(A,bool),fun(A,fun(A,bool))),combc(A,fun(A,bool),fun(A,bool)),aa(fun(A,fun(A,fun(bool,bool))),fun(A,fun(fun(A,bool),fun(A,bool))),aa(fun(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool))),fun(fun(A,fun(A,fun(bool,bool))),fun(A,fun(fun(A,bool),fun(A,bool)))),combb(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),A),combs(A,bool,bool)),aa(fun(A,fun(A,bool)),fun(A,fun(A,fun(bool,bool))),aa(fun(fun(A,bool),fun(A,fun(bool,bool))),fun(fun(A,fun(A,bool)),fun(A,fun(A,fun(bool,bool)))),combb(fun(A,bool),fun(A,fun(bool,bool)),A),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,A),fun(A,fun(A,bool)),aa(fun(A,fun(fun(A,A),fun(A,bool))),fun(fun(A,A),fun(A,fun(A,bool))),combc(A,fun(A,A),fun(A,bool)),aa(fun(A,fun(A,bool)),fun(A,fun(fun(A,A),fun(A,bool))),aa(fun(fun(A,bool),fun(fun(A,A),fun(A,bool))),fun(fun(A,fun(A,bool)),fun(A,fun(fun(A,A),fun(A,bool)))),combb(fun(A,bool),fun(fun(A,A),fun(A,bool)),A),combb(A,bool,A)),fequal(A))),aa(A,fun(A,A),plus_plus(A),K))))),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)),N))))) ) ) ) ) ).
tff(fact_92_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_93_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list(B)] : product(A,B,nil(A),Uu) = nil(product_prod(A,B)) ).
tff(fact_94_list__add__assoc,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Zs: list(A),Ys: list(A),Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs),Ys)),Zs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Ys),Zs)) ) ).
tff(fact_95_Min__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: fun(A,bool)] :
( finite_finite1(A,A3)
=> ( ( A3 != bot_bot(fun(A,bool)) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),big_linorder_Min(A,A3)),A3)) ) ) ) ).
tff(fact_96_list__ex1__iff,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( list_ex1(A,P1,Xsa)
<=> ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),set(A,Xsa)))
& pp(aa(A,bool,P1,X3))
& ! [Y1: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y1),set(A,Xsa)))
& pp(aa(A,bool,P1,Y1)) )
=> ( Y1 = X3 ) ) ) ) ).
tff(fact_97_x,axiom,
! [X2: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X2),set(atom,as)))
=> i_Z(X2,cons(int,x,xs)) ) ).
tff(fact_98__096EX_Ax_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_096,axiom,
? [X1: int] :
! [Xa: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa),set(atom,as)))
=> i_Z(Xa,cons(int,X1,xs)) ) ).
tff(fact_99_list_Oinject,axiom,
! [A: $tType,List1: list(A),A2: A,List: list(A),A1: A] :
( ( cons(A,A1,List) = cons(A,A2,List1) )
<=> ( ( A1 = A2 )
& ( List = List1 ) ) ) ).
%----Arities (21)
tff(arity_fun___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite(T_2)
& finite_finite(T_1) )
=> finite_finite(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Obot,axiom,
! [T_1: $tType,T_2: $tType] :
( bot(T_2)
=> bot(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___Groups_Olinordered__ab__semigroup__add,axiom,
linord2061991079up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(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___Groups_Oplus,axiom,
cl_Groups_Oplus(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_finite(bool) ).
tff(arity_HOL_Obool___Orderings_Obot,axiom,
bot(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_List_Olist___Groups_Oplus,axiom,
! [T_1: $tType] :
( ( zero(T_1)
& cl_Groups_Oplus(T_1) )
=> cl_Groups_Oplus(list(T_1)) ) ).
tff(arity_prod___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite(T_2)
& finite_finite(T_1) )
=> finite_finite(product_prod(T_1,T_2)) ) ).
%----Helper facts (19)
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_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = 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_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
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)) ) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
! [Li: int,Lks: list(int)] :
( pp(aa(fun(product_prod(int,list(int)),bool),bool,aa(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool),member(product_prod(int,list(int))),aa(list(int),product_prod(int,list(int)),aa(int,fun(list(int),product_prod(int,list(int))),product_Pair(int,list(int)),Li),Lks)),set(product_prod(int,list(int)),lbounds(as))))
=> ( ( big_linorder_Max(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)))))))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),Li),aa(list(int),int,aa(list(int),fun(list(int),int),iprod(int),Lks),xs)) )
=> thesis ) ) ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------