TPTP Problem File: COM073_5.p
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%------------------------------------------------------------------------------
% File : COM073_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 166
% 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_166 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 187 ( 60 unt; 54 typ; 0 def)
% Number of atoms : 253 ( 111 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 181 ( 61 ~; 22 |; 11 &)
% ( 30 <=>; 57 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 4 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 39 ( 24 >; 15 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 1 prp; 0-3 aty)
% Number of functors : 38 ( 38 usr; 9 con; 0-5 aty)
% Number of variables : 329 ( 285 !; 4 ?; 329 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:20:46
%------------------------------------------------------------------------------
%----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_Nat_Onat,type,
nat: $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 (47)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[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_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[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_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_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > 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:
!>[T1: $tType,A: $tType] : ( ( T1 * fun(A,fun(list(A),T1)) * list(A) ) > T1 ) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : ( list(A) > fun(A,bool) ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
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_Oatom_OLe,type,
c_PresArith_Oatom_OLe: ( int * list(int) ) > atom ).
tff(sy_c_PresArith_Oatom_Oatom__case,type,
atom_case:
!>[T1: $tType] : ( ( fun(int,fun(list(int),T1)) * fun(int,fun(int,fun(list(int),T1))) * fun(int,fun(int,fun(list(int),T1))) * atom ) > T1 ) ).
tff(sy_c_PresArith_Oatom_Oatom__size,type,
atom_size: atom > nat ).
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_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_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( fun(A,bool) > A ) ).
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_a____,type,
a: atom ).
tff(sy_v_as,type,
as: list(atom) ).
tff(sy_v_thesis____,type,
thesis: $o ).
%----Relevant facts (96)
tff(fact_0_atom_Osimps_I1_J,axiom,
! [List3: list(int),Int2: int,List2: list(int),Int1: int] :
( ( c_PresArith_Oatom_OLe(Int1,List2) = c_PresArith_Oatom_OLe(Int2,List3) )
<=> ( ( Int1 = Int2 )
& ( List2 = List3 ) ) ) ).
tff(fact_1_list_Oinject,axiom,
! [A: $tType,List3: list(A),A3: A,List2: list(A),Aa: A] :
( ( cons(A,Aa,List2) = cons(A,A3,List3) )
<=> ( ( Aa = A3 )
& ( List2 = List3 ) ) ) ).
tff(fact_2_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y2: int,X1: int] :
( ( number_number_of(A,X1) = number_number_of(A,Y2) )
<=> ( X1 = Y2 ) ) ) ).
tff(fact_3_number__of__is__id,axiom,
! [K1: int] : number_number_of(int,K1) = K1 ).
tff(fact_4_not__Cons__self,axiom,
! [A: $tType,X: A,Xs: list(A)] : Xs != cons(A,X,Xs) ).
tff(fact_5_not__Cons__self2,axiom,
! [A: $tType,Xs: list(A),X: A] : cons(A,X,Xs) != Xs ).
tff(fact_6_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X1: A,W1: int] :
( ( number_number_of(A,W1) = X1 )
<=> ( X1 = number_number_of(A,W1) ) ) ) ).
tff(fact_7_atom_Osimps_I10_J,axiom,
! [A: $tType,List2: list(int),Int1: 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))] : atom_case(A,F1,F2,F3,c_PresArith_Oatom_OLe(Int1,List2)) = aa(list(int),A,aa(int,fun(list(int),A),F1,Int1),List2) ).
tff(fact_8_rel__simps_I7_J,axiom,
~ ord_less(int,min,min) ).
tff(fact_9_of__int__m1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( ring_1_of_int(A,number_number_of(int,min)) = number_number_of(A,min) ) ) ).
tff(fact_10_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int,W1: int] :
( ( ring_1_of_int(A,W1) = ring_1_of_int(A,Z) )
<=> ( W1 = Z ) ) ) ).
tff(fact_11_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y2: int,X1: int] :
( ord_less(A,number_number_of(A,X1),number_number_of(A,Y2))
<=> ord_less(int,X1,Y2) ) ) ).
tff(fact_12_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int,W1: int] :
( ord_less(A,ring_1_of_int(A,W1),ring_1_of_int(A,Z))
<=> ord_less(int,W1,Z) ) ) ).
tff(fact_13_of__int__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [V: int] : ring_1_of_int(A,number_number_of(int,V)) = number_number_of(A,V) ) ).
tff(fact_14_number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [K1: int] : number_number_of(A,K1) = ring_1_of_int(A,K1) ) ).
tff(fact_15_int__number__of__def,axiom,
! [W: int] : number_number_of(int,W) = ring_1_of_int(int,W) ).
tff(fact_16_less__number__of__int__code,axiom,
! [L1: int,K: int] :
( ord_less(int,number_number_of(int,K),number_number_of(int,L1))
<=> ord_less(int,K,L1) ) ).
tff(fact_17_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List2: list(B),Aa: B,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,cons(B,Aa,List2)) = aa(list(B),A,aa(B,fun(list(B),A),F2,Aa),List2) ).
tff(fact_18_of__int__0__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less(A,zero_zero(A),ring_1_of_int(A,Z))
<=> ord_less(int,zero_zero(int),Z) ) ) ).
tff(fact_19_of__int__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less(A,ring_1_of_int(A,Z),zero_zero(A))
<=> ord_less(int,Z,zero_zero(int)) ) ) ).
tff(fact_20_bin__less__0__simps_I2_J,axiom,
ord_less(int,min,zero_zero(int)) ).
tff(fact_21_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),X1: A,Y2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),set(A,cons(A,X1,Xsa))))
=> ( ( Y2 = X1 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),set(A,Xsa))) ) ) ).
tff(fact_22_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_23_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( zero_zero(A) = ring_1_of_int(A,Z) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_24_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( ring_1_of_int(A,Z) = zero_zero(A) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_25_of__int__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,one_one(int)) = one_one(A) ) ) ).
tff(fact_26_norm,axiom,
! [X2: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X2),set(atom,as)))
=> ( divisor(X2) != zero_zero(int) ) ) ).
tff(fact_27_List_Oset_Osimps_I2_J,axiom,
! [A: $tType,Xsa: list(A),X1: A] : set(A,cons(A,X1,Xsa)) = insert(A,X1,set(A,Xsa)) ).
tff(fact_28_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less(A,zero_zero(A),one_one(A)) ) ).
tff(fact_29_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less(A,one_one(A),zero_zero(A)) ) ).
tff(fact_30_atom_Osize_I1_J,axiom,
! [List1: list(int),Int: int] : atom_size(c_PresArith_Oatom_OLe(Int,List1)) = zero_zero(nat) ).
tff(fact_31_divisor_Osimps_I1_J,axiom,
! [Ks2: list(int),I1: int] : divisor(c_PresArith_Oatom_OLe(I1,Ks2)) = one_one(int) ).
tff(fact_32_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_33_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_34_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X1: A] :
( ( one_one(A) = X1 )
<=> ( X1 = one_one(A) ) ) ) ).
tff(fact_35_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_36_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_37__096lbounds_Aas_A_061_A_091_093_096,axiom,
lbounds(as) = nil(product_prod(int,list(int))) ).
tff(fact_38_atom_Osize_I4_J,axiom,
! [List1: list(int),Int: int] : size_size(atom,c_PresArith_Oatom_OLe(Int,List1)) = zero_zero(nat) ).
tff(fact_39_insert__absorb2,axiom,
! [A: $tType,A1: fun(A,bool),X1: A] : insert(A,X1,insert(A,X1,A1)) = insert(A,X1,A1) ).
tff(fact_40_insert__iff,axiom,
! [A: $tType,A1: fun(A,bool),B1: A,Aa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),insert(A,B1,A1)))
<=> ( ( Aa = B1 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),A1)) ) ) ).
tff(fact_41_insertE,axiom,
! [A: $tType,A1: fun(A,bool),B1: A,Aa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),insert(A,B1,A1)))
=> ( ( Aa != B1 )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),A1)) ) ) ).
tff(fact_42_insertCI,axiom,
! [A: $tType,B1: A,B2: fun(A,bool),Aa: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),B2))
=> ( Aa = B1 ) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),insert(A,B1,B2))) ) ).
tff(fact_43_list_Osimps_I2_J,axiom,
! [A: $tType,List: list(A),A2: A] : nil(A) != cons(A,A2,List) ).
tff(fact_44_list_Osimps_I3_J,axiom,
! [A: $tType,List: list(A),A2: A] : cons(A,A2,List) != nil(A) ).
tff(fact_45_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_46_insert__absorb,axiom,
! [A: $tType,A1: fun(A,bool),Aa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),A1))
=> ( insert(A,Aa,A1) = A1 ) ) ).
tff(fact_47_insertI2,axiom,
! [A: $tType,B1: A,B2: fun(A,bool),Aa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),B2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),insert(A,B1,B2))) ) ).
tff(fact_48_insert__eq__iff,axiom,
! [A: $tType,B2: fun(A,bool),B1: A,A1: fun(A,bool),Aa: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),B2))
=> ( ( insert(A,Aa,A1) = insert(A,B1,B2) )
<=> ( ( ( Aa = B1 )
=> ( A1 = B2 ) )
& ( ( Aa != B1 )
=> ? [C2: fun(A,bool)] :
( ( A1 = insert(A,B1,C2) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),C2))
& ( B2 = insert(A,Aa,C2) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),C2)) ) ) ) ) ) ) ).
tff(fact_49_insert__ident,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool),X1: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),B2))
=> ( ( insert(A,X1,A1) = insert(A,X1,B2) )
<=> ( A1 = B2 ) ) ) ) ).
tff(fact_50_insert__code,axiom,
! [A: $tType,X1: A,A1: fun(A,bool),Y2: A] :
( pp(aa(A,bool,insert(A,Y2,A1),X1))
<=> ( ( Y2 = X1 )
| pp(aa(A,bool,A1,X1)) ) ) ).
tff(fact_51_insert__commute,axiom,
! [A: $tType,A1: fun(A,bool),Y2: A,X1: A] : insert(A,X1,insert(A,Y2,A1)) = insert(A,Y2,insert(A,X1,A1)) ).
tff(fact_52_insert__Collect,axiom,
! [A: $tType,P1: fun(A,bool),Aa: A] : insert(A,Aa,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),Aa))),P1)) ).
tff(fact_53_insert__compr,axiom,
! [A: $tType,B2: fun(A,bool),Aa: A] : insert(A,Aa,B2) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,A,bool,fequal(A),Aa)),combc(A,fun(A,bool),bool,member(A),B2))) ).
tff(fact_54_insertI1,axiom,
! [A: $tType,B2: fun(A,bool),Aa: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),insert(A,Aa,B2))) ).
tff(fact_55_List_Oset_Osimps_I1_J,axiom,
! [A: $tType] : set(A,nil(A)) = bot_bot(fun(A,bool)) ).
tff(fact_56_set__empty2,axiom,
! [A: $tType,Xsa: list(A)] :
( ( bot_bot(fun(A,bool)) = set(A,Xsa) )
<=> ( Xsa = nil(A) ) ) ).
tff(fact_57_set__empty,axiom,
! [A: $tType,Xsa: list(A)] :
( ( set(A,Xsa) = bot_bot(fun(A,bool)) )
<=> ( Xsa = nil(A) ) ) ).
tff(fact_58_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_59_all__not__in__conv,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [X4: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A1))
<=> ( A1 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_60_empty__Collect__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P1) )
<=> ! [X4: A] : ~ pp(aa(A,bool,P1,X4)) ) ).
tff(fact_61_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_62_Collect__empty__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( collect(A,P1) = bot_bot(fun(A,bool)) )
<=> ! [X4: A] : ~ pp(aa(A,bool,P1,X4)) ) ).
tff(fact_63_emptyE,axiom,
! [A: $tType,Aa: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),bot_bot(fun(A,bool)))) ).
tff(fact_64_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_65_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_66_equals0D,axiom,
! [A: $tType,Aa: A,A1: fun(A,bool)] :
( ( A1 = bot_bot(fun(A,bool)) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Aa),A1)) ) ).
tff(fact_67_ex__in__conv,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ? [X4: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A1))
<=> ( A1 != bot_bot(fun(A,bool)) ) ) ).
tff(fact_68_empty__def,axiom,
! [A: $tType] : bot_bot(fun(A,bool)) = collect(A,aa(bool,fun(A,bool),combk(bool,A),fFalse)) ).
tff(fact_69_not__psubset__empty,axiom,
! [A: $tType,A1: fun(A,bool)] : ~ ord_less(fun(A,bool),A1,bot_bot(fun(A,bool))) ).
tff(fact_70_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_71_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_72_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A1))
<=> pp(aa(A,bool,A1,X1)) ) ).
tff(fact_73_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_74_nat__neq__iff,axiom,
! [Na: nat,M1: nat] :
( ( M1 != Na )
<=> ( ord_less(nat,M1,Na)
| ord_less(nat,Na,M1) ) ) ).
tff(fact_75_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_76_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_77_less__not__refl2,axiom,
! [M: nat,N: nat] :
( ord_less(nat,N,M)
=> ( M != N ) ) ).
tff(fact_78_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_79_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),Na: nat,M1: nat] :
( ( ord_less(nat,M1,Na)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) )
=> ( ( ( M1 = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) )
=> ( ( ord_less(nat,Na,M1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) ) ) ) ).
tff(fact_80_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_81_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_82_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_83_empty__not__insert,axiom,
! [A: $tType,A1: fun(A,bool),Aa: A] : bot_bot(fun(A,bool)) != insert(A,Aa,A1) ).
tff(fact_84_insert__not__empty,axiom,
! [A: $tType,A1: fun(A,bool),Aa: A] : insert(A,Aa,A1) != bot_bot(fun(A,bool)) ).
tff(fact_85_singleton__iff,axiom,
! [A: $tType,Aa: A,B1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),insert(A,Aa,bot_bot(fun(A,bool)))))
<=> ( B1 = Aa ) ) ).
tff(fact_86_doubleton__eq__iff,axiom,
! [A: $tType,D: A,C1: A,B1: A,Aa: A] :
( ( insert(A,Aa,insert(A,B1,bot_bot(fun(A,bool)))) = insert(A,C1,insert(A,D,bot_bot(fun(A,bool)))) )
<=> ( ( ( Aa = C1 )
& ( B1 = D ) )
| ( ( Aa = D )
& ( B1 = C1 ) ) ) ) ).
tff(fact_87_singletonE,axiom,
! [A: $tType,Aa: A,B1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),insert(A,Aa,bot_bot(fun(A,bool)))))
=> ( B1 = Aa ) ) ).
tff(fact_88_singleton__inject,axiom,
! [A: $tType,B1: A,Aa: A] :
( ( insert(A,Aa,bot_bot(fun(A,bool))) = insert(A,B1,bot_bot(fun(A,bool))) )
=> ( Aa = B1 ) ) ).
tff(fact_89_hd,axiom,
! [X2: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X2),set(atom,as)))
=> pp(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(X2)),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(X2)),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(X2)),one_one(int))))),X2)) ) ).
tff(fact_90_the__elem__eq,axiom,
! [A: $tType,X1: A] : the_elem(A,insert(A,X1,bot_bot(fun(A,bool)))) = X1 ).
tff(fact_91_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y1: A,Ys: list(A)] : Xsa = cons(A,Y1,Ys) ) ).
tff(fact_92_singleton__conv2,axiom,
! [A: $tType,Aa: A] : collect(A,aa(A,fun(A,bool),fequal(A),Aa)) = insert(A,Aa,bot_bot(fun(A,bool))) ).
tff(fact_93_singleton__conv,axiom,
! [A: $tType,Aa: A] : collect(A,combc(A,A,bool,fequal(A),Aa)) = insert(A,Aa,bot_bot(fun(A,bool))) ).
tff(fact_94_hd__coeff_Osimps_I1_J,axiom,
! [Ks1: list(int),I: int] : hd_coeff(c_PresArith_Oatom_OLe(I,Ks1)) = list_case(int,int,zero_zero(int),combk(int,list(int)),Ks1) ).
tff(fact_95_Collect__conv__if2,axiom,
! [A: $tType,Aa: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,Aa))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,aa(A,fun(A,bool),fequal(A),Aa)),P1)) = insert(A,Aa,bot_bot(fun(A,bool))) ) )
& ( ~ pp(aa(A,bool,P1,Aa))
=> ( collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,aa(A,fun(A,bool),fequal(A),Aa)),P1)) = bot_bot(fun(A,bool)) ) ) ) ).
%----Arities (14)
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(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___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
%----Helper facts (21)
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_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_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 (2)
tff(conj_0,hypothesis,
! [L: int,Ks: list(int)] :
( ( a = c_PresArith_Oatom_OLe(L,cons(int,number_number_of(int,min),Ks)) )
=> thesis ) ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------