TPTP Problem File: COM058_5.p
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%------------------------------------------------------------------------------
% File : COM058_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 123
% 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_123 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 186 ( 66 unt; 57 typ; 0 def)
% Number of atoms : 245 ( 114 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 164 ( 48 ~; 23 |; 24 &)
% ( 30 <=>; 39 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 53 ( 29 >; 24 *; 0 +; 0 <<)
% Number of predicates : 11 ( 10 usr; 0 prp; 1-5 aty)
% Number of functors : 44 ( 44 usr; 10 con; 0-5 aty)
% Number of variables : 419 ( 369 !; 7 ?; 419 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:18:45
%------------------------------------------------------------------------------
%----Should-be-implicit typings (7)
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_Logic_Ofm,type,
fm: $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 (50)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__ring,type,
number_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[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_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : fun(A,fun(B,A)) ).
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_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Int_OMin,type,
min: int ).
tff(sy_c_Int_Onumber__class_Onumber__of,type,
number_number_of:
!>[A: $tType] : ( int > 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_Olinorder__class_Oinsort__insert__key,type,
linord2118332866rt_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) * list(A) ) > T ) ).
tff(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Omember,type,
member1:
!>[A: $tType] : fun(list(A),fun(A,bool)) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : fun(list(A),fun(A,bool)) ).
tff(sy_c_Logic_Ointerpret,type,
interpret:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(list(B),bool)) * fm(A) * list(B) ) > $o ) ).
tff(sy_c_Logic_Olist__conj,type,
list_conj:
!>[A: $tType] : ( list(fm(A)) > fm(A) ) ).
tff(sy_c_Logic_Olist__disj,type,
list_disj:
!>[A: $tType] : ( list(fm(A)) > fm(A) ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_PresArith_OI_092_060_094isub_062Z,type,
i_Z: fun(atom,fun(list(int),bool)) ).
tff(sy_c_PresArith_Oatom_ODvd,type,
dvd: ( int * int * list(int) ) > atom ).
tff(sy_c_PresArith_Oatom_OLe,type,
c_PresArith_Oatom_OLe: ( int * list(int) ) > atom ).
tff(sy_c_PresArith_Oatom_ONDvd,type,
nDvd: ( int * int * list(int) ) > atom ).
tff(sy_c_PresArith_Oatom_Oatom__case,type,
atom_case:
!>[T: $tType] : ( ( fun(int,fun(list(int),T)) * fun(int,fun(int,fun(list(int),T))) * fun(int,fun(int,fun(list(int),T))) ) > fun(atom,T) ) ).
tff(sy_c_PresArith_Odivisor,type,
divisor: atom > int ).
tff(sy_c_PresArith_Ohd__coeff,type,
hd_coeff: atom > int ).
tff(sy_c_PresArith_Olbounds,type,
lbounds: list(atom) > list(product_prod(int,list(int))) ).
tff(sy_c_QEpres__Mirabelle__iocckttzyp_Oqe__pres_092_060_094isub_0621,type,
qEpres896714165pres_1: list(atom) > fm(atom) ).
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_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_fimplies,type,
fimplies: 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_xs,type,
xs: list(int) ).
%----Relevant facts (98)
tff(fact_0_norm,axiom,
! [X4: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X4),aa(list(atom),fun(atom,bool),set(atom),as)))
=> ( divisor(X4) != zero_zero(int) ) ) ).
tff(fact_1__096lbounds_Aas_A_061_A_091_093_096,axiom,
lbounds(as) = nil(product_prod(int,list(int))) ).
tff(fact_2__096Z_OI_A_Iqe__pres_092_060_094isub_0621_Aas_J_Axs_096,axiom,
interpret(atom,int,i_Z,qEpres896714165pres_1(as),xs) ).
tff(fact_3_insort__insert__triv,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xsa: list(A),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),Xsa)))
=> ( linord2118332866rt_key(A,A,combi(A),X2,Xsa) = Xsa ) ) ) ).
tff(fact_4_atom_Osimps_I10_J,axiom,
! [A: $tType,List2: list(int),Int3: int,F3: fun(int,fun(int,fun(list(int),A))),F2: fun(int,fun(int,fun(list(int),A))),F1: fun(int,fun(list(int),A))] : aa(atom,A,atom_case(A,F1,F2,F3),c_PresArith_Oatom_OLe(Int3,List2)) = aa(list(int),A,aa(int,fun(list(int),A),F1,Int3),List2) ).
tff(fact_5_atom_Osimps_I12_J,axiom,
! [A: $tType,List2: list(int),Int22: int,Int12: int,F3: fun(int,fun(int,fun(list(int),A))),F2: fun(int,fun(int,fun(list(int),A))),F1: fun(int,fun(list(int),A))] : aa(atom,A,atom_case(A,F1,F2,F3),nDvd(Int12,Int22,List2)) = aa(list(int),A,aa(int,fun(list(int),A),aa(int,fun(int,fun(list(int),A)),F3,Int12),Int22),List2) ).
tff(fact_6_atom_Osimps_I11_J,axiom,
! [A: $tType,List2: list(int),Int22: int,Int12: int,F3: fun(int,fun(int,fun(list(int),A))),F2: fun(int,fun(int,fun(list(int),A))),F1: fun(int,fun(list(int),A))] : aa(atom,A,atom_case(A,F1,F2,F3),dvd(Int12,Int22,List2)) = aa(list(int),A,aa(int,fun(list(int),A),aa(int,fun(int,fun(list(int),A)),F2,Int12),Int22),List2) ).
tff(fact_7_hd,axiom,
! [X4: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X4),aa(list(atom),fun(atom,bool),set(atom),as)))
=> pp(aa(atom,bool,atom_case(bool,aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),aa(fun(int,bool),bool,aa(int,fun(fun(int,bool),bool),member(int),hd_coeff(X4)),insert(int,one_one(int),insert(int,number_number_of(int,min),bot_bot(fun(int,bool))))))),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),aa(int,bool,aa(int,fun(int,bool),fequal(int),hd_coeff(X4)),one_one(int))))),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),aa(int,bool,aa(int,fun(int,bool),fequal(int),hd_coeff(X4)),one_one(int)))))),X4)) ) ).
tff(fact_8_member__set,axiom,
! [A: $tType] : member1(A) = set(A) ).
tff(fact_9_list__ex1__iff,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( list_ex1(A,P1,Xsa)
<=> ? [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),aa(list(A),fun(A,bool),set(A),Xsa)))
& pp(aa(A,bool,P1,X1))
& ! [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 = X1 ) ) ) ) ).
tff(fact_10_List_Omember__def,axiom,
! [A: $tType,X2: A,Xsa: list(A)] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),Xsa),X2))
<=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),aa(list(A),fun(A,bool),set(A),Xsa))) ) ).
tff(fact_11_ListMem__iff,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( listMem(A,X2,Xsa)
<=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),aa(list(A),fun(A,bool),set(A),Xsa))) ) ).
tff(fact_12_atom_Osimps_I2_J,axiom,
! [List3: list(int),Int23: int,Int13: int,List2: list(int),Int22: int,Int12: int] :
( ( dvd(Int12,Int22,List2) = dvd(Int13,Int23,List3) )
<=> ( ( Int12 = Int13 )
& ( Int22 = Int23 )
& ( List2 = List3 ) ) ) ).
tff(fact_13_atom_Osimps_I1_J,axiom,
! [List3: list(int),Int4: int,List2: list(int),Int3: int] :
( ( c_PresArith_Oatom_OLe(Int3,List2) = c_PresArith_Oatom_OLe(Int4,List3) )
<=> ( ( Int3 = Int4 )
& ( List2 = List3 ) ) ) ).
tff(fact_14_atom_Osimps_I3_J,axiom,
! [List3: list(int),Int23: int,Int13: int,List2: list(int),Int22: int,Int12: int] :
( ( nDvd(Int12,Int22,List2) = nDvd(Int13,Int23,List3) )
<=> ( ( Int12 = Int13 )
& ( Int22 = Int23 )
& ( List2 = List3 ) ) ) ).
tff(fact_15_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_16_set__empty,axiom,
! [A: $tType,Xsa: list(A)] :
( ( aa(list(A),fun(A,bool),set(A),Xsa) = bot_bot(fun(A,bool)) )
<=> ( Xsa = nil(A) ) ) ).
tff(fact_17_set__empty2,axiom,
! [A: $tType,Xsa: list(A)] :
( ( bot_bot(fun(A,bool)) = aa(list(A),fun(A,bool),set(A),Xsa) )
<=> ( Xsa = nil(A) ) ) ).
tff(fact_18_List_Oset_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),fun(A,bool),set(A),nil(A)) = bot_bot(fun(A,bool)) ).
tff(fact_19_divisor_Osimps_I1_J,axiom,
! [Ks1: list(int),I1: int] : divisor(c_PresArith_Oatom_OLe(I1,Ks1)) = one_one(int) ).
tff(fact_20_atom_Osimps_I4_J,axiom,
! [List: list(int),Int2: int,Int1: int,List1: list(int),Int: int] : c_PresArith_Oatom_OLe(Int,List1) != dvd(Int1,Int2,List) ).
tff(fact_21_atom_Osimps_I6_J,axiom,
! [List: list(int),Int2: int,Int1: int,List1: list(int),Int: int] : c_PresArith_Oatom_OLe(Int,List1) != nDvd(Int1,Int2,List) ).
tff(fact_22_atom_Osimps_I5_J,axiom,
! [List1: list(int),Int: int,List: list(int),Int2: int,Int1: int] : dvd(Int1,Int2,List) != c_PresArith_Oatom_OLe(Int,List1) ).
tff(fact_23_atom_Osimps_I7_J,axiom,
! [List1: list(int),Int: int,List: list(int),Int2: int,Int1: int] : nDvd(Int1,Int2,List) != c_PresArith_Oatom_OLe(Int,List1) ).
tff(fact_24_atom_Osimps_I8_J,axiom,
! [List: list(int),Int2: int,Int1: int,List1: list(int),Int21: int,Int11: int] : dvd(Int11,Int21,List1) != nDvd(Int1,Int2,List) ).
tff(fact_25_atom_Osimps_I9_J,axiom,
! [List1: list(int),Int21: int,Int11: int,List: list(int),Int2: int,Int1: int] : nDvd(Int1,Int2,List) != dvd(Int11,Int21,List1) ).
tff(fact_26_divisor_Osimps_I2_J,axiom,
! [Ks1: list(int),I1: int,D1: int] : divisor(dvd(D1,I1,Ks1)) = D1 ).
tff(fact_27_divisor_Osimps_I3_J,axiom,
! [Ks1: list(int),I1: int,D1: int] : divisor(nDvd(D1,I1,Ks1)) = D1 ).
tff(fact_28_member__rec_I2_J,axiom,
! [A: $tType,Y: A] : ~ pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),nil(A)),Y)) ).
tff(fact_29_set__insort__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xsa: list(A),X2: A] : aa(list(A),fun(A,bool),set(A),linord2118332866rt_key(A,A,combi(A),X2,Xsa)) = insert(A,X2,aa(list(A),fun(A,bool),set(A),Xsa)) ) ).
tff(fact_30_in__set__member,axiom,
! [A: $tType,Xsa: list(A),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),Xsa)))
<=> pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),Xsa),X2)) ) ).
tff(fact_31_singleton__conv2,axiom,
! [A: $tType,A1: A] : collect(A,aa(A,fun(A,bool),fequal(A),A1)) = insert(A,A1,bot_bot(fun(A,bool))) ).
tff(fact_32_singleton__conv,axiom,
! [A: $tType,A1: A] : collect(A,combc(A,A,bool,fequal(A),A1)) = insert(A,A1,bot_bot(fun(A,bool))) ).
tff(fact_33_list_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,nil(B)) = F1 ).
tff(fact_34_hd__coeff_Osimps_I1_J,axiom,
! [Ks: list(int),I: int] : hd_coeff(c_PresArith_Oatom_OLe(I,Ks)) = list_case(int,int,zero_zero(int),combk(int,list(int)),Ks) ).
tff(fact_35_hd__coeff_Osimps_I3_J,axiom,
! [Ks: list(int),I: int,D: int] : hd_coeff(nDvd(D,I,Ks)) = list_case(int,int,zero_zero(int),combk(int,list(int)),Ks) ).
tff(fact_36_hd__coeff_Osimps_I2_J,axiom,
! [Ks: list(int),I: int,D: int] : hd_coeff(dvd(D,I,Ks)) = list_case(int,int,zero_zero(int),combk(int,list(int)),Ks) ).
tff(fact_37_Z_OI__list__conj,axiom,
! [Xsa: list(int),Fs: list(fm(atom))] :
( interpret(atom,int,i_Z,list_conj(atom,Fs),Xsa)
<=> ! [X1: fm(atom)] :
( pp(aa(fun(fm(atom),bool),bool,aa(fm(atom),fun(fun(fm(atom),bool),bool),member(fm(atom)),X1),aa(list(fm(atom)),fun(fm(atom),bool),set(fm(atom)),Fs)))
=> interpret(atom,int,i_Z,X1,Xsa) ) ) ).
tff(fact_38_insert__absorb2,axiom,
! [A: $tType,A2: fun(A,bool),X2: A] : insert(A,X2,insert(A,X2,A2)) = insert(A,X2,A2) ).
tff(fact_39_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_40_Collect__empty__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( collect(A,P1) = bot_bot(fun(A,bool)) )
<=> ! [X1: A] : ~ pp(aa(A,bool,P1,X1)) ) ).
tff(fact_41_empty__iff,axiom,
! [A: $tType,C1: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),bot_bot(fun(A,bool)))) ).
tff(fact_42_empty__Collect__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P1) )
<=> ! [X1: A] : ~ pp(aa(A,bool,P1,X1)) ) ).
tff(fact_43_all__not__in__conv,axiom,
! [A: $tType,A2: fun(A,bool)] :
( ! [X1: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A2))
<=> ( A2 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_44_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y1: int,X2: int] :
( ( number_number_of(A,X2) = number_number_of(A,Y1) )
<=> ( X2 = Y1 ) ) ) ).
tff(fact_45_insertCI,axiom,
! [A: $tType,B2: A,B1: fun(A,bool),A1: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),B1))
=> ( A1 = B2 ) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),insert(A,B2,B1))) ) ).
tff(fact_46_insertE,axiom,
! [A: $tType,A2: fun(A,bool),B2: A,A1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),insert(A,B2,A2)))
=> ( ( A1 != B2 )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),A2)) ) ) ).
tff(fact_47_insert__iff,axiom,
! [A: $tType,A2: fun(A,bool),B2: A,A1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),insert(A,B2,A2)))
<=> ( ( A1 = B2 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),A2)) ) ) ).
tff(fact_48_equals0D,axiom,
! [A: $tType,A1: A,A2: fun(A,bool)] :
( ( A2 = bot_bot(fun(A,bool)) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),A2)) ) ).
tff(fact_49_ex__in__conv,axiom,
! [A: $tType,A2: fun(A,bool)] :
( ? [X1: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A2))
<=> ( A2 != bot_bot(fun(A,bool)) ) ) ).
tff(fact_50_empty__def,axiom,
! [A: $tType] : bot_bot(fun(A,bool)) = collect(A,aa(bool,fun(A,bool),combk(bool,A),fFalse)) ).
tff(fact_51_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X2: A,W: int] :
( ( number_number_of(A,W) = X2 )
<=> ( X2 = number_number_of(A,W) ) ) ) ).
tff(fact_52_insert__absorb,axiom,
! [A: $tType,A2: fun(A,bool),A1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),A2))
=> ( insert(A,A1,A2) = A2 ) ) ).
tff(fact_53_insertI2,axiom,
! [A: $tType,B2: A,B1: fun(A,bool),A1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),B1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),insert(A,B2,B1))) ) ).
tff(fact_54_insert__eq__iff,axiom,
! [A: $tType,B1: fun(A,bool),B2: A,A2: fun(A,bool),A1: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),A2))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B2),B1))
=> ( ( insert(A,A1,A2) = insert(A,B2,B1) )
<=> ( ( ( A1 = B2 )
=> ( A2 = B1 ) )
& ( ( A1 != B2 )
=> ? [C3: fun(A,bool)] :
( ( A2 = insert(A,B2,C3) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B2),C3))
& ( B1 = insert(A,A1,C3) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),C3)) ) ) ) ) ) ) ).
tff(fact_55_insert__ident,axiom,
! [A: $tType,B1: fun(A,bool),A2: fun(A,bool),X2: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A2))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),B1))
=> ( ( insert(A,X2,A2) = insert(A,X2,B1) )
<=> ( A2 = B1 ) ) ) ) ).
tff(fact_56_insert__code,axiom,
! [A: $tType,X2: A,A2: fun(A,bool),Y1: A] :
( pp(aa(A,bool,insert(A,Y1,A2),X2))
<=> ( ( Y1 = X2 )
| pp(aa(A,bool,A2,X2)) ) ) ).
tff(fact_57_insert__commute,axiom,
! [A: $tType,A2: fun(A,bool),Y1: A,X2: A] : insert(A,X2,insert(A,Y1,A2)) = insert(A,Y1,insert(A,X2,A2)) ).
tff(fact_58_insert__Collect,axiom,
! [A: $tType,P1: fun(A,bool),A1: A] : insert(A,A1,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),A1))),P1)) ).
tff(fact_59_insert__compr,axiom,
! [A: $tType,B1: fun(A,bool),A1: A] : insert(A,A1,B1) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,A,bool,fequal(A),A1)),combc(A,fun(A,bool),bool,member(A),B1))) ).
tff(fact_60_insertI1,axiom,
! [A: $tType,B1: fun(A,bool),A1: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A1),insert(A,A1,B1))) ).
tff(fact_61_number__of__is__id,axiom,
! [K1: int] : number_number_of(int,K1) = K1 ).
tff(fact_62_insert__compr__raw,axiom,
! [A: $tType,X4: A,Xa: fun(A,bool)] : insert(A,X4,Xa) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,A,bool,fequal(A),X4)),combc(A,fun(A,bool),bool,member(A),Xa))) ).
tff(fact_63_singleton__inject,axiom,
! [A: $tType,B2: A,A1: A] :
( ( insert(A,A1,bot_bot(fun(A,bool))) = insert(A,B2,bot_bot(fun(A,bool))) )
=> ( A1 = B2 ) ) ).
tff(fact_64_singletonE,axiom,
! [A: $tType,A1: A,B2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B2),insert(A,A1,bot_bot(fun(A,bool)))))
=> ( B2 = A1 ) ) ).
tff(fact_65_doubleton__eq__iff,axiom,
! [A: $tType,D: A,C1: A,B2: A,A1: A] :
( ( insert(A,A1,insert(A,B2,bot_bot(fun(A,bool)))) = insert(A,C1,insert(A,D,bot_bot(fun(A,bool)))) )
<=> ( ( ( A1 = C1 )
& ( B2 = D ) )
| ( ( A1 = D )
& ( B2 = C1 ) ) ) ) ).
tff(fact_66_singleton__iff,axiom,
! [A: $tType,A1: A,B2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B2),insert(A,A1,bot_bot(fun(A,bool)))))
<=> ( B2 = A1 ) ) ).
tff(fact_67_insert__not__empty,axiom,
! [A: $tType,A2: fun(A,bool),A1: A] : insert(A,A1,A2) != bot_bot(fun(A,bool)) ).
tff(fact_68_empty__not__insert,axiom,
! [A: $tType,A2: fun(A,bool),A1: A] : bot_bot(fun(A,bool)) != insert(A,A1,A2) ).
tff(fact_69_Collect__conv__if,axiom,
! [A: $tType,A1: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,A1))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,combc(A,A,bool,fequal(A),A1)),P1)) = insert(A,A1,bot_bot(fun(A,bool))) ) )
& ( ~ pp(aa(A,bool,P1,A1))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,combc(A,A,bool,fequal(A),A1)),P1)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_70_Collect__conv__if2,axiom,
! [A: $tType,A1: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,A1))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,aa(A,fun(A,bool),fequal(A),A1)),P1)) = insert(A,A1,bot_bot(fun(A,bool))) ) )
& ( ~ pp(aa(A,bool,P1,A1))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,aa(A,fun(A,bool),fequal(A),A1)),P1)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_71_rel__simps_I7_J,axiom,
~ ord_less(int,min,min) ).
tff(fact_72_Un__insert__left,axiom,
! [A: $tType,C2: fun(A,bool),B1: fun(A,bool),A1: A] : sup_sup(fun(A,bool),insert(A,A1,B1),C2) = insert(A,A1,sup_sup(fun(A,bool),B1,C2)) ).
tff(fact_73_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X3: A] : aa(A,B,F,X3) = aa(A,B,G,X3)
=> ( F = G ) ) ).
tff(fact_74_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A2))
<=> pp(aa(A,bool,A2,X2)) ) ).
tff(fact_75_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_76_Un__insert__right,axiom,
! [A: $tType,B1: fun(A,bool),A1: A,A2: fun(A,bool)] : sup_sup(fun(A,bool),A2,insert(A,A1,B1)) = insert(A,A1,sup_sup(fun(A,bool),A2,B1)) ).
tff(fact_77_Un__empty,axiom,
! [A: $tType,B1: fun(A,bool),A2: fun(A,bool)] :
( ( sup_sup(fun(A,bool),A2,B1) = bot_bot(fun(A,bool)) )
<=> ( ( A2 = bot_bot(fun(A,bool)) )
& ( B1 = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_78_Z_OI__list__disj,axiom,
! [Xsa: list(int),Fs: list(fm(atom))] :
( interpret(atom,int,i_Z,list_disj(atom,Fs),Xsa)
<=> ? [X1: fm(atom)] :
( pp(aa(fun(fm(atom),bool),bool,aa(fm(atom),fun(fun(fm(atom),bool),bool),member(fm(atom)),X1),aa(list(fm(atom)),fun(fm(atom),bool),set(fm(atom)),Fs)))
& interpret(atom,int,i_Z,X1,Xsa) ) ) ).
tff(fact_79_Un__iff,axiom,
! [A: $tType,B1: fun(A,bool),A2: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A2,B1)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A2))
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1)) ) ) ).
tff(fact_80_UnE,axiom,
! [A: $tType,B1: fun(A,bool),A2: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A2,B1)))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1)) ) ) ).
tff(fact_81_UnCI,axiom,
! [A: $tType,A2: fun(A,bool),B1: fun(A,bool),C1: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A2)) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A2,B1))) ) ).
tff(fact_82_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y1: int,X2: int] :
( ord_less(A,number_number_of(A,X2),number_number_of(A,Y1))
<=> ord_less(int,X2,Y1) ) ) ).
tff(fact_83_Un__absorb,axiom,
! [A: $tType,A2: fun(A,bool)] : sup_sup(fun(A,bool),A2,A2) = A2 ).
tff(fact_84_Un__def,axiom,
! [A: $tType,B1: fun(A,bool),A2: fun(A,bool)] : sup_sup(fun(A,bool),A2,B1) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,fun(A,bool),bool,member(A),A2)),combc(A,fun(A,bool),bool,member(A),B1))) ).
tff(fact_85_Un__commute,axiom,
! [A: $tType,B1: fun(A,bool),A2: fun(A,bool)] : sup_sup(fun(A,bool),A2,B1) = sup_sup(fun(A,bool),B1,A2) ).
tff(fact_86_Un__left__absorb,axiom,
! [A: $tType,B1: fun(A,bool),A2: fun(A,bool)] : sup_sup(fun(A,bool),A2,sup_sup(fun(A,bool),A2,B1)) = sup_sup(fun(A,bool),A2,B1) ).
tff(fact_87_Un__left__commute,axiom,
! [A: $tType,C2: fun(A,bool),B1: fun(A,bool),A2: fun(A,bool)] : sup_sup(fun(A,bool),A2,sup_sup(fun(A,bool),B1,C2)) = sup_sup(fun(A,bool),B1,sup_sup(fun(A,bool),A2,C2)) ).
tff(fact_88_Un__assoc,axiom,
! [A: $tType,C2: fun(A,bool),B1: fun(A,bool),A2: fun(A,bool)] : sup_sup(fun(A,bool),sup_sup(fun(A,bool),A2,B1),C2) = sup_sup(fun(A,bool),A2,sup_sup(fun(A,bool),B1,C2)) ).
tff(fact_89_bex__Un,axiom,
! [A: $tType,P1: fun(A,bool),B1: fun(A,bool),A2: fun(A,bool)] :
( ? [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),sup_sup(fun(A,bool),A2,B1)))
& pp(aa(A,bool,P1,X1)) )
<=> ( ? [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A2))
& pp(aa(A,bool,P1,X1)) )
| ? [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),B1))
& pp(aa(A,bool,P1,X1)) ) ) ) ).
tff(fact_90_ball__Un,axiom,
! [A: $tType,P1: fun(A,bool),B1: fun(A,bool),A2: fun(A,bool)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),sup_sup(fun(A,bool),A2,B1)))
=> pp(aa(A,bool,P1,X1)) )
<=> ( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A2))
=> pp(aa(A,bool,P1,X1)) )
& ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),B1))
=> pp(aa(A,bool,P1,X1)) ) ) ) ).
tff(fact_91_UnI1,axiom,
! [A: $tType,B1: fun(A,bool),A2: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A2,B1))) ) ).
tff(fact_92_UnI2,axiom,
! [A: $tType,A2: fun(A,bool),B1: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A2,B1))) ) ).
tff(fact_93_Collect__disj__eq,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] : collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,P1),Q1)) = sup_sup(fun(A,bool),collect(A,P1),collect(A,Q1)) ).
tff(fact_94_Un__empty__left,axiom,
! [A: $tType,B1: fun(A,bool)] : sup_sup(fun(A,bool),bot_bot(fun(A,bool)),B1) = B1 ).
tff(fact_95_Un__empty__right,axiom,
! [A: $tType,A2: fun(A,bool)] : sup_sup(fun(A,bool),A2,bot_bot(fun(A,bool))) = A2 ).
tff(fact_96_less__number__of__int__code,axiom,
! [L: int,K: int] :
( ord_less(int,number_number_of(int,K),number_number_of(int,L))
<=> ord_less(int,K,L) ) ).
tff(fact_97_insert__def,axiom,
! [A: $tType,B1: fun(A,bool),A1: A] : insert(A,A1,B1) = sup_sup(fun(A,bool),collect(A,combc(A,A,bool,fequal(A),A1)),B1) ).
%----Arities (6)
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
%----Helper facts (24)
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_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,aa(A,fun(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,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
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)) ) ).
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,
aa(list(atom),fun(atom,bool),set(atom),as) = collect(atom,combs(atom,bool,bool,combb(bool,fun(bool,bool),atom,fconj,combc(atom,fun(atom,bool),bool,member(atom),aa(list(atom),fun(atom,bool),set(atom),as))),atom_case(bool,aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fFalse)),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fTrue))),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fTrue)))))) ).
%------------------------------------------------------------------------------