TPTP Problem File: COM051_5.p
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%------------------------------------------------------------------------------
% File : COM051_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 107
% 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_107 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 175 ( 58 unt; 46 typ; 0 def)
% Number of atoms : 248 ( 96 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 158 ( 39 ~; 19 |; 16 &)
% ( 20 <=>; 64 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 39 ( 22 >; 17 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 33 ( 33 usr; 7 con; 0-5 aty)
% Number of variables : 403 ( 356 !; 7 ?; 403 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:17:32
%------------------------------------------------------------------------------
%----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_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_PresArith_Oatom,type,
atom: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (40)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_List_OListMem,type,
listMem:
!>[A: $tType] : ( ( A * list(A) ) > $o ) ).
tff(sy_c_List_Oinsert,type,
insert1:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * list(B) ) > list(B) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > 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__ex,type,
list_ex:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Omember,type,
member1:
!>[A: $tType] : fun(list(A),fun(A,bool)) ).
tff(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > fun(list(A),list(A)) ) ).
tff(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : fun(list(A),fun(A,bool)) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType,B: $tType] : fun(nat,fun(fun(A,B),fun(A,B))) ).
tff(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_PresArith_Odivisor,type,
divisor: atom > int ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * fun(A,bool) ) > fun(A,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
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_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: fun(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_x____,type,
x: atom ).
%----Relevant facts (98)
tff(fact_0_norm,axiom,
! [X1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X1),aa(list(atom),fun(atom,bool),set(atom),as)))
=> ( divisor(X1) != zero_zero(int) ) ) ).
tff(fact_1_member__set,axiom,
! [A: $tType] : member1(A) = set(A) ).
tff(fact_2_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),aa(list(A),fun(A,bool),set(A),Xsa)))
& pp(aa(A,bool,P1,X3))
& ! [Y2: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),aa(list(A),fun(A,bool),set(A),Xsa)))
& pp(aa(A,bool,P1,Y2)) )
=> ( Y2 = X3 ) ) ) ) ).
tff(fact_3_List_Omember__def,axiom,
! [A: $tType,Xa: A,Xsa: list(A)] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),Xsa),Xa))
<=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),aa(list(A),fun(A,bool),set(A),Xsa))) ) ).
tff(fact_4_ListMem__iff,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( listMem(A,Xa,Xsa)
<=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),aa(list(A),fun(A,bool),set(A),Xsa))) ) ).
tff(fact_5_in__set__member,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),aa(list(A),fun(A,bool),set(A),Xsa)))
<=> pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),Xsa),Xa)) ) ).
tff(fact_6_set__rotate1,axiom,
! [A: $tType,Xsa: list(A)] : aa(list(A),fun(A,bool),set(A),aa(list(A),list(A),rotate1(A),Xsa)) = aa(list(A),fun(A,bool),set(A),Xsa) ).
tff(fact_7_set__sort,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Xsa: list(A),F: fun(A,B)] : aa(list(A),fun(A,bool),set(A),linorder_sort_key(A,B,F,Xsa)) = aa(list(A),fun(A,bool),set(A),Xsa) ) ).
tff(fact_8_in__set__insert,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),aa(list(A),fun(A,bool),set(A),Xsa)))
=> ( insert1(A,Xa,Xsa) = Xsa ) ) ).
tff(fact_9_set__rotate,axiom,
! [A: $tType,Xsa: list(A),N: nat] : aa(list(A),fun(A,bool),set(A),aa(list(A),list(A),rotate(A,N),Xsa)) = aa(list(A),fun(A,bool),set(A),Xsa) ).
tff(fact_10_list__ex__iff,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( list_ex(A,P1,Xsa)
<=> ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),aa(list(A),fun(A,bool),set(A),Xsa)))
& pp(aa(A,bool,P1,X3)) ) ) ).
tff(fact_11_set__remdups,axiom,
! [A: $tType,Xsa: list(A)] : aa(list(A),fun(A,bool),set(A),remdups(A,Xsa)) = aa(list(A),fun(A,bool),set(A),Xsa) ).
tff(fact_12_remdups__remdups,axiom,
! [A: $tType,Xs: list(A)] : remdups(A,remdups(A,Xs)) = remdups(A,Xs) ).
tff(fact_13_rotate1__rotate__swap,axiom,
! [A: $tType,Xs: list(A),N1: nat] : aa(list(A),list(A),rotate1(A),aa(list(A),list(A),rotate(A,N1),Xs)) = aa(list(A),list(A),rotate(A,N1),aa(list(A),list(A),rotate1(A),Xs)) ).
tff(fact_14_insert__remdups,axiom,
! [A: $tType,Xs: list(A),X: A] : insert1(A,X,remdups(A,Xs)) = remdups(A,insert1(A,X,Xs)) ).
tff(fact_15_list__any__cong,axiom,
! [A: $tType,G: fun(A,bool),F: fun(A,bool),Ys: list(A),Xsa: list(A)] :
( ( Xsa = Ys )
=> ( ! [X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),aa(list(A),fun(A,bool),set(A),Ys)))
=> ( pp(aa(A,bool,F,X2))
<=> pp(aa(A,bool,G,X2)) ) )
=> ( list_ex(A,F,Xsa)
<=> list_ex(A,G,Ys) ) ) ) ).
tff(fact_16_rotate__def,axiom,
! [A: $tType,N: nat] : rotate(A,N) = aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(nat,fun(fun(list(A),list(A)),fun(list(A),list(A))),compow(list(A),list(A)),N),rotate1(A)) ).
tff(fact_17_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa: A] :
( ( zero_zero(A) = Xa )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_18_rotate__Suc,axiom,
! [A: $tType,Xs: list(A),N1: nat] : aa(list(A),list(A),rotate(A,suc(N1)),Xs) = aa(list(A),list(A),rotate1(A),aa(list(A),list(A),rotate(A,N1),Xs)) ).
tff(fact_19_List_Oset__insert,axiom,
! [A: $tType,Xsa: list(A),Xa: A] : aa(list(A),fun(A,bool),set(A),insert1(A,Xa,Xsa)) = insert(A,Xa,aa(list(A),fun(A,bool),set(A),Xsa)) ).
tff(fact_20_remdups_Osimps_I2_J,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),aa(list(A),fun(A,bool),set(A),Xsa)))
=> ( remdups(A,cons(A,Xa,Xsa)) = remdups(A,Xsa) ) )
& ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),aa(list(A),fun(A,bool),set(A),Xsa)))
=> ( remdups(A,cons(A,Xa,Xsa)) = cons(A,Xa,remdups(A,Xsa)) ) ) ) ).
tff(fact_21_List_Oinsert__def,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),aa(list(A),fun(A,bool),set(A),Xsa)))
=> ( insert1(A,Xa,Xsa) = Xsa ) )
& ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),aa(list(A),fun(A,bool),set(A),Xsa)))
=> ( insert1(A,Xa,Xsa) = cons(A,Xa,Xsa) ) ) ) ).
tff(fact_22_not__in__set__insert,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),aa(list(A),fun(A,bool),set(A),Xsa)))
=> ( insert1(A,Xa,Xsa) = cons(A,Xa,Xsa) ) ) ).
tff(fact_23_list_Oinject,axiom,
! [A: $tType,List1: list(A),A4: A,List: list(A),A3: A] :
( ( cons(A,A3,List) = cons(A,A4,List1) )
<=> ( ( A3 = A4 )
& ( List = List1 ) ) ) ).
tff(fact_24_list__ex__simps_I1_J,axiom,
! [A: $tType,Xsa: list(A),Xa: A,P1: fun(A,bool)] :
( list_ex(A,P1,cons(A,Xa,Xsa))
<=> ( pp(aa(A,bool,P1,Xa))
| list_ex(A,P1,Xsa) ) ) ).
tff(fact_25_List_Oset_Osimps_I2_J,axiom,
! [A: $tType,Xsa: list(A),Xa: A] : aa(list(A),fun(A,bool),set(A),cons(A,Xa,Xsa)) = insert(A,Xa,aa(list(A),fun(A,bool),set(A),Xsa)) ).
tff(fact_26_not__Cons__self,axiom,
! [A: $tType,X: A,Xs: list(A)] : Xs != cons(A,X,Xs) ).
tff(fact_27_not__Cons__self2,axiom,
! [A: $tType,Xs: list(A),X: A] : cons(A,X,Xs) != Xs ).
tff(fact_28_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),Xa: A,Y1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y1),aa(list(A),fun(A,bool),set(A),cons(A,Xa,Xsa))))
=> ( ( Y1 = Xa )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y1),aa(list(A),fun(A,bool),set(A),Xsa))) ) ) ).
tff(fact_29_insert,axiom,
! [A: $tType,Y: A,Xs: list(A),X: A] :
( listMem(A,X,Xs)
=> listMem(A,X,cons(A,Y,Xs)) ) ).
tff(fact_30_member__rec_I1_J,axiom,
! [A: $tType,Y1: A,Xsa: list(A),Xa: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),cons(A,Xa,Xsa)),Y1))
<=> ( ( Xa = Y1 )
| pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),Xsa),Y1)) ) ) ).
tff(fact_31_elem,axiom,
! [A: $tType,Xs: list(A),X: A] : listMem(A,X,cons(A,X,Xs)) ).
tff(fact_32_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),A3: B,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,cons(B,A3,List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A3),List) ).
tff(fact_33_insertCI,axiom,
! [A: $tType,B4: A,B1: fun(A,bool),A3: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),B1))
=> ( A3 = B4 ) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,B4,B1))) ) ).
tff(fact_34_insertE,axiom,
! [A: $tType,A1: fun(A,bool),B4: A,A3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,B4,A1)))
=> ( ( A3 != B4 )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A1)) ) ) ).
tff(fact_35_insert__iff,axiom,
! [A: $tType,A1: fun(A,bool),B4: A,A3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,B4,A1)))
<=> ( ( A3 = B4 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A1)) ) ) ).
tff(fact_36_insert__absorb2,axiom,
! [A: $tType,A1: fun(A,bool),Xa: A] : insert(A,Xa,insert(A,Xa,A1)) = insert(A,Xa,A1) ).
tff(fact_37_nat_Oinject,axiom,
! [Nat3: nat,Nat2: nat] :
( ( suc(Nat2) = suc(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_38_n__not__Suc__n,axiom,
! [N1: nat] : N1 != suc(N1) ).
tff(fact_39_Suc__n__not__n,axiom,
! [N1: nat] : suc(N1) != N1 ).
tff(fact_40_Suc__inject,axiom,
! [Y: nat,X: nat] :
( ( suc(X) = suc(Y) )
=> ( X = Y ) ) ).
tff(fact_41_Suc__neq__Zero,axiom,
! [M: nat] : suc(M) != zero_zero(nat) ).
tff(fact_42_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero(nat) != suc(M) ).
tff(fact_43_nat_Osimps_I3_J,axiom,
! [Nat1: nat] : suc(Nat1) != zero_zero(nat) ).
tff(fact_44_Suc__not__Zero,axiom,
! [M: nat] : suc(M) != zero_zero(nat) ).
tff(fact_45_nat_Osimps_I2_J,axiom,
! [Nat: nat] : zero_zero(nat) != suc(Nat) ).
tff(fact_46_Zero__not__Suc,axiom,
! [M: nat] : zero_zero(nat) != suc(M) ).
tff(fact_47_insertI1,axiom,
! [A: $tType,B1: fun(A,bool),A3: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,A3,B1))) ).
tff(fact_48_insert__compr,axiom,
! [A: $tType,B1: fun(A,bool),A3: A] : insert(A,A3,B1) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,A,bool,fequal(A),A3)),combc(A,fun(A,bool),bool,member(A),B1))) ).
tff(fact_49_insert__Collect,axiom,
! [A: $tType,P1: fun(A,bool),A3: A] : insert(A,A3,collect(A,P1)) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fimplies,combb(bool,bool,A,fNot,combc(A,A,bool,fequal(A),A3))),P1)) ).
tff(fact_50_insert__commute,axiom,
! [A: $tType,A1: fun(A,bool),Y1: A,Xa: A] : insert(A,Xa,insert(A,Y1,A1)) = insert(A,Y1,insert(A,Xa,A1)) ).
tff(fact_51_insert__code,axiom,
! [A: $tType,Xa: A,A1: fun(A,bool),Y1: A] :
( pp(aa(A,bool,insert(A,Y1,A1),Xa))
<=> ( ( Y1 = Xa )
| pp(aa(A,bool,A1,Xa)) ) ) ).
tff(fact_52_insert__ident,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),Xa: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),B1))
=> ( ( insert(A,Xa,A1) = insert(A,Xa,B1) )
<=> ( A1 = B1 ) ) ) ) ).
tff(fact_53_insert__eq__iff,axiom,
! [A: $tType,B1: fun(A,bool),B4: A,A1: fun(A,bool),A3: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B4),B1))
=> ( ( insert(A,A3,A1) = insert(A,B4,B1) )
<=> ( ( ( A3 = B4 )
=> ( A1 = B1 ) )
& ( ( A3 != B4 )
=> ? [C4: fun(A,bool)] :
( ( A1 = insert(A,B4,C4) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B4),C4))
& ( B1 = insert(A,A3,C4) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),C4)) ) ) ) ) ) ) ).
tff(fact_54_insertI2,axiom,
! [A: $tType,B4: A,B1: fun(A,bool),A3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),B1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,B4,B1))) ) ).
tff(fact_55_insert__absorb,axiom,
! [A: $tType,A1: fun(A,bool),A3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A1))
=> ( insert(A,A3,A1) = A1 ) ) ).
tff(fact_56_funpow__swap1,axiom,
! [A: $tType,Xa: A,N: nat,F: fun(A,A)] : aa(A,A,F,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(A,A),N),F),Xa)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(A,A),N),F),aa(A,A,F,Xa)) ).
tff(fact_57_Un__insert__right,axiom,
! [A: $tType,B1: fun(A,bool),A3: A,A1: fun(A,bool)] : sup_sup(fun(A,bool),A1,insert(A,A3,B1)) = insert(A,A3,sup_sup(fun(A,bool),A1,B1)) ).
tff(fact_58_Un__insert__left,axiom,
! [A: $tType,C2: fun(A,bool),B1: fun(A,bool),A3: A] : sup_sup(fun(A,bool),insert(A,A3,B1),C2) = insert(A,A3,sup_sup(fun(A,bool),B1,C2)) ).
tff(fact_59_funpow__code__def,axiom,
! [A: $tType] : funpow(A) = compow(A,A) ).
tff(fact_60_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [I: A,N: nat,Inc: fun(A,A)] : semiri532925092at_aux(A,Inc,suc(N),I) = semiri532925092at_aux(A,Inc,N,aa(A,A,Inc,I)) ) ).
tff(fact_61_mk__disjoint__insert,axiom,
! [A: $tType,A1: fun(A,bool),A3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A1))
=> ? [B3: fun(A,bool)] :
( ( A1 = insert(A,A3,B3) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),B3)) ) ) ).
tff(fact_62_UnCI,axiom,
! [A: $tType,A1: fun(A,bool),B1: fun(A,bool),C3: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),B1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),A1)) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),sup_sup(fun(A,bool),A1,B1))) ) ).
tff(fact_63_UnE,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),sup_sup(fun(A,bool),A1,B1)))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),B1)) ) ) ).
tff(fact_64_Un__iff,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),sup_sup(fun(A,bool),A1,B1)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),A1))
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),B1)) ) ) ).
tff(fact_65_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [I: A,Inc: fun(A,A)] : semiri532925092at_aux(A,Inc,zero_zero(nat),I) = I ) ).
tff(fact_66_UnI2,axiom,
! [A: $tType,A1: fun(A,bool),B1: fun(A,bool),C3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),B1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),sup_sup(fun(A,bool),A1,B1))) ) ).
tff(fact_67_UnI1,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),sup_sup(fun(A,bool),A1,B1))) ) ).
tff(fact_68_ball__Un,axiom,
! [A: $tType,P1: fun(A,bool),B1: fun(A,bool),A1: fun(A,bool)] :
( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),sup_sup(fun(A,bool),A1,B1)))
=> pp(aa(A,bool,P1,X3)) )
<=> ( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A1))
=> pp(aa(A,bool,P1,X3)) )
& ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),B1))
=> pp(aa(A,bool,P1,X3)) ) ) ) ).
tff(fact_69_bex__Un,axiom,
! [A: $tType,P1: fun(A,bool),B1: fun(A,bool),A1: fun(A,bool)] :
( ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),sup_sup(fun(A,bool),A1,B1)))
& pp(aa(A,bool,P1,X3)) )
<=> ( ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A1))
& pp(aa(A,bool,P1,X3)) )
| ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),B1))
& pp(aa(A,bool,P1,X3)) ) ) ) ).
tff(fact_70_Un__assoc,axiom,
! [A: $tType,C2: fun(A,bool),B1: fun(A,bool),A1: fun(A,bool)] : sup_sup(fun(A,bool),sup_sup(fun(A,bool),A1,B1),C2) = sup_sup(fun(A,bool),A1,sup_sup(fun(A,bool),B1,C2)) ).
tff(fact_71_Un__left__commute,axiom,
! [A: $tType,C2: fun(A,bool),B1: fun(A,bool),A1: fun(A,bool)] : sup_sup(fun(A,bool),A1,sup_sup(fun(A,bool),B1,C2)) = sup_sup(fun(A,bool),B1,sup_sup(fun(A,bool),A1,C2)) ).
tff(fact_72_Un__left__absorb,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] : sup_sup(fun(A,bool),A1,sup_sup(fun(A,bool),A1,B1)) = sup_sup(fun(A,bool),A1,B1) ).
tff(fact_73_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
tff(fact_74_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A1))
<=> pp(aa(A,bool,A1,Xa)) ) ).
tff(fact_75_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_76_Un__commute,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] : sup_sup(fun(A,bool),A1,B1) = sup_sup(fun(A,bool),B1,A1) ).
tff(fact_77_Un__def,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] : sup_sup(fun(A,bool),A1,B1) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,fun(A,bool),bool,member(A),A1)),combc(A,fun(A,bool),bool,member(A),B1))) ).
tff(fact_78_Un__absorb,axiom,
! [A: $tType,A1: fun(A,bool)] : sup_sup(fun(A,bool),A1,A1) = A1 ).
tff(fact_79_sup1CI,axiom,
! [A: $tType,A1: fun(A,bool),Xa: A,B1: fun(A,bool)] :
( ( ~ pp(aa(A,bool,B1,Xa))
=> pp(aa(A,bool,A1,Xa)) )
=> pp(aa(A,bool,sup_sup(fun(A,bool),A1,B1),Xa)) ) ).
tff(fact_80_sup1E,axiom,
! [A: $tType,Xa: A,B1: fun(A,bool),A1: fun(A,bool)] :
( pp(aa(A,bool,sup_sup(fun(A,bool),A1,B1),Xa))
=> ( ~ pp(aa(A,bool,A1,Xa))
=> pp(aa(A,bool,B1,Xa)) ) ) ).
tff(fact_81_sup__left__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A] : sup_sup(A,X,sup_sup(A,X,Y)) = sup_sup(A,X,Y) ) ).
tff(fact_82_sup_Oleft__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A2: A] : sup_sup(A,A2,sup_sup(A,A2,B2)) = sup_sup(A,A2,B2) ) ).
tff(fact_83_sup_Oidem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A2: A] : sup_sup(A,A2,A2) = A2 ) ).
tff(fact_84_sup__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A] : sup_sup(A,X,X) = X ) ).
tff(fact_85_sup_Ocommute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A2: A] : sup_sup(A,A2,B2) = sup_sup(A,B2,A2) ) ).
tff(fact_86_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X: A] : sup_sup(A,X,Y) = sup_sup(A,Y,X) ) ).
tff(fact_87_sup__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A] : sup_sup(A,X,Y) = sup_sup(A,Y,X) ) ).
tff(fact_88_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X: A] : sup_sup(A,X,sup_sup(A,X,Y)) = sup_sup(A,X,Y) ) ).
tff(fact_89_sup_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C1: A,A2: A,B2: A] : sup_sup(A,B2,sup_sup(A,A2,C1)) = sup_sup(A,A2,sup_sup(A,B2,C1)) ) ).
tff(fact_90_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Z: A,Y: A,X: A] : sup_sup(A,X,sup_sup(A,Y,Z)) = sup_sup(A,Y,sup_sup(A,X,Z)) ) ).
tff(fact_91_sup__left__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Z: A,Y: A,X: A] : sup_sup(A,X,sup_sup(A,Y,Z)) = sup_sup(A,Y,sup_sup(A,X,Z)) ) ).
tff(fact_92_sup_Oassoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C1: A,B2: A,A2: A] : sup_sup(A,sup_sup(A,A2,B2),C1) = sup_sup(A,A2,sup_sup(A,B2,C1)) ) ).
tff(fact_93_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Z: A,Y: A,X: A] : sup_sup(A,sup_sup(A,X,Y),Z) = sup_sup(A,X,sup_sup(A,Y,Z)) ) ).
tff(fact_94_sup__assoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Z: A,Y: A,X: A] : sup_sup(A,sup_sup(A,X,Y),Z) = sup_sup(A,X,sup_sup(A,Y,Z)) ) ).
tff(fact_95_sup1I1,axiom,
! [A: $tType,B1: fun(A,bool),Xa: A,A1: fun(A,bool)] :
( pp(aa(A,bool,A1,Xa))
=> pp(aa(A,bool,sup_sup(fun(A,bool),A1,B1),Xa)) ) ).
tff(fact_96_sup1I2,axiom,
! [A: $tType,A1: fun(A,bool),Xa: A,B1: fun(A,bool)] :
( pp(aa(A,bool,B1,Xa))
=> pp(aa(A,bool,sup_sup(fun(A,bool),A1,B1),Xa)) ) ).
tff(fact_97_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( lattice(B)
=> ! [G: fun(A,B),F: fun(A,B),X1: A] : aa(A,B,sup_sup(fun(A,B),F,G),X1) = sup_sup(B,aa(A,B,F,X1),aa(A,B,G,X1)) ) ).
%----Arities (15)
tff(arity_fun___Lattices_Osemilattice__sup,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> semilattice_sup(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Olattice,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> lattice(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Lattices_Osemilattice__sup,axiom,
semilattice_sup(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Lattices_Olattice,axiom,
lattice(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Nat_Onat___Lattices_Osemilattice__sup,axiom,
semilattice_sup(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Lattices_Olattice,axiom,
lattice(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Lattices_Osemilattice__sup,axiom,
semilattice_sup(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Lattices_Olattice,axiom,
lattice(bool) ).
%----Helper facts (15)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
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,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,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
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,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
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,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
| ~ pp(P)
| pp(Q) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),x),aa(list(atom),fun(atom,bool),set(atom),as))) ).
%------------------------------------------------------------------------------