TPTP Problem File: COM061_5.p
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%------------------------------------------------------------------------------
% File : COM061_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 134
% 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_134 [Bla13]
% Status : Theorem
% Rating : 1.00 v8.1.0, 0.67 v7.5.0, 1.00 v6.4.0
% Syntax : Number of formulae : 187 ( 48 unt; 56 typ; 0 def)
% Number of atoms : 264 ( 87 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 176 ( 43 ~; 16 |; 15 &)
% ( 21 <=>; 81 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 19 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 33 ( 23 >; 10 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-5 aty)
% Number of functors : 43 ( 43 usr; 9 con; 0-4 aty)
% Number of variables : 404 ( 350 !; 8 ?; 404 :)
% ( 46 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:19:01
%------------------------------------------------------------------------------
%----Should-be-implicit typings (8)
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_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 (48)
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ominus,type,
cl_Groups_Ominus:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(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] : ( 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_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( fun(A,bool) > $o ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : 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_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( list(fun(A,bool)) > fun(list(A),bool) ) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : ( list(A) > fun(A,bool) ) ).
tff(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( fun(A,bool) * fun(list(A),bool) ) > fun(list(A),bool) ) ).
tff(sy_c_List_Otl,type,
tl:
!>[A: $tType] : fun(list(A),list(A)) ).
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_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_PresArith_OI_092_060_094isub_062Z,type,
i_Z: fun(atom,fun(list(int),bool)) ).
tff(sy_c_PresArith_Oatom_OLe,type,
c_PresArith_Oatom_OLe: fun(int,fun(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))) ) > fun(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_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_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,bool) ) > fun(B,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fEx,type,
fEx:
!>[A: $tType] : fun(fun(A,bool),bool) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_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 (99)
tff(fact_0_norm,axiom,
! [X1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X1),set(atom,as)))
=> ( divisor(X1) != zero_zero(int) ) ) ).
tff(fact_1_finite__imageI,axiom,
! [B: $tType,A: $tType,H: fun(A,B),F4: fun(A,bool)] :
( finite_finite(A,F4)
=> finite_finite(B,image(A,B,H,F4)) ) ).
tff(fact_2_finite__set,axiom,
! [A: $tType,Xsa: list(A)] : finite_finite(A,set(A,Xsa)) ).
tff(fact_3_list__diff__Cons__Cons,axiom,
! [A: $tType] :
( ( cl_Groups_Ominus(A)
& zero(A) )
=> ! [Ys2: list(A),Y: A,Xs1: list(A),X: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs1),Ys2)) ) ).
tff(fact_4_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A4),A4) = zero_zero(A) ) ).
tff(fact_5__096lbounds_Aas_A_061_A_091_093_096,axiom,
lbounds(as) = nil(product_prod(int,list(int))) ).
tff(fact_6_finite__Collect__disjI,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] :
( finite_finite(A,collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fdisj),P1)),Q1)))
<=> ( finite_finite(A,collect(A,P1))
& finite_finite(A,collect(A,Q1)) ) ) ).
tff(fact_7_finite__Collect__conjI,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] :
( ( finite_finite(A,collect(A,P1))
| finite_finite(A,collect(A,Q1)) )
=> finite_finite(A,collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),P1)),Q1))) ) ).
tff(fact_8_less__bin__lemma,axiom,
! [L1: int,K1: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K1),L1))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K1),L1)),zero_zero(int))) ) ).
tff(fact_9_atom_Osimps_I1_J,axiom,
! [List4: list(int),Int2: int,List3: list(int),Int1: int] :
( ( aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int1),List3) = aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int2),List4) )
<=> ( ( Int1 = Int2 )
& ( List3 = List4 ) ) ) ).
tff(fact_10_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A))) ) ) ).
tff(fact_11_atom_Osimps_I10_J,axiom,
! [A: $tType,List3: list(int),Int1: int,F3: fun(int,fun(int,fun(list(int),A))),F2: fun(int,fun(int,fun(list(int),A))),F11: fun(int,fun(list(int),A))] : aa(atom,A,atom_case(A,F11,F2,F3),aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int1),List3)) = aa(list(int),A,aa(int,fun(list(int),A),F11,Int1),List3) ).
tff(fact_12_image__eqI,axiom,
! [A: $tType,B: $tType,A1: fun(B,bool),X2: B,F1: fun(B,A),B2: A] :
( ( B2 = aa(B,A,F1,X2) )
=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B2),image(B,A,F1,A1))) ) ) ).
tff(fact_13_finite__Diff,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( finite_finite(A,A1)
=> finite_finite(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B1)) ) ).
tff(fact_14_finite__code,axiom,
! [A: $tType] :
( finite_finite1(A)
=> ! [A1: fun(A,bool)] : finite_finite(A,A1) ) ).
tff(fact_15_list_Oinject,axiom,
! [A: $tType,List4: list(A),A6: A,List3: list(A),A3: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),List3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),List4) )
<=> ( ( A3 = A6 )
& ( List3 = List4 ) ) ) ).
tff(fact_16_tl_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).
tff(fact_17_iprod__Nil2,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xs1: list(A)] : aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Xs1),nil(A)) = zero_zero(A) ) ).
tff(fact_18_iprod__Nil,axiom,
! [A: $tType] :
( ring(A)
=> ! [Ys2: list(A)] : aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),nil(A)),Ys2) = zero_zero(A) ) ).
tff(fact_19_list__diff__Nil2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [Xs1: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs1),nil(A)) = Xs1 ) ).
tff(fact_20_nat__seg__image__imp__finite,axiom,
! [A: $tType,Na: nat,F1: fun(nat,A),A1: fun(A,bool)] :
( ( A1 = image(nat,A,F1,collect(nat,aa(nat,fun(nat,bool),combc(nat,nat,bool,ord_less(nat)),Na))) )
=> finite_finite(A,A1) ) ).
tff(fact_21_finite__Diff2,axiom,
! [A: $tType,A1: fun(A,bool),B1: fun(A,bool)] :
( finite_finite(A,B1)
=> ( finite_finite(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B1))
<=> finite_finite(A,A1) ) ) ).
tff(fact_22_list_Osimps_I3_J,axiom,
! [A: $tType,List2: list(A),A5: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List2) != nil(A) ).
tff(fact_23_list_Osimps_I2_J,axiom,
! [A: $tType,List2: list(A),A5: A] : nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List2) ).
tff(fact_24_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_25_finite,axiom,
! [A: $tType] :
( finite_finite1(A)
=> ! [A1: fun(A,bool)] : finite_finite(A,A1) ) ).
tff(fact_26_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D: A,C1: A,B2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C1),D) )
=> ( ( A3 = B2 )
<=> ( C1 = D ) ) ) ) ).
tff(fact_27_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) != Xs1 ).
tff(fact_28_not__Cons__self,axiom,
! [A: $tType,X: A,Xs1: list(A)] : Xs1 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) ).
tff(fact_29_rev__image__eqI,axiom,
! [B: $tType,A: $tType,F1: fun(A,B),B2: B,A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
=> ( ( B2 = aa(A,B,F1,X2) )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),image(A,B,F1,A1))) ) ) ).
tff(fact_30_imageI,axiom,
! [B: $tType,A: $tType,F1: fun(A,B),A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F1,X2)),image(A,B,F1,A1))) ) ).
tff(fact_31_image__iff,axiom,
! [A: $tType,B: $tType,A1: fun(B,bool),F1: fun(B,A),Z: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),image(B,A,F1,A1)))
<=> ? [X4: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),A1))
& ( Z = aa(B,A,F1,X4) ) ) ) ).
tff(fact_32_image__ident,axiom,
! [A: $tType,Y3: fun(A,bool)] : image(A,A,combi(A),Y3) = Y3 ).
tff(fact_33_image__image,axiom,
! [B: $tType,A: $tType,C: $tType,A1: fun(C,bool),G: fun(C,B),F1: fun(B,A)] : image(B,A,F1,image(C,B,G,A1)) = image(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),combb(B,A,C),F1),G),A1) ).
tff(fact_34_right__minus__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = zero_zero(A) )
<=> ( A3 = B2 ) ) ) ).
tff(fact_35_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A3: A] :
( ( A3 = B2 )
<=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = zero_zero(A) ) ) ) ).
tff(fact_36_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A4),zero_zero(A)) = A4 ) ).
tff(fact_37_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D: A,C1: A,B2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C1),D) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C1),D)) ) ) ) ).
tff(fact_38_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),X2: A,Y1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y1),set(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa))))
=> ( ( Y1 = X2 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y1),set(A,Xsa))) ) ) ).
tff(fact_39_tl_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list(A),X: A] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)) = Xs1 ).
tff(fact_40_iprod__left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys2: list(A),Xs1: list(A)] : aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs1),Ys2)),Zs) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Xs1),Zs)),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Ys2),Zs)) ) ).
tff(fact_41_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,Q1: fun(B,fun(A,bool)),P1: fun(A,bool)] :
( finite_finite(A,collect(A,P1))
=> ( finite_finite(B,collect(B,aa(fun(B,fun(A,bool)),fun(B,bool),aa(fun(fun(A,bool),bool),fun(fun(B,fun(A,bool)),fun(B,bool)),combb(fun(A,bool),bool,B),fEx(A)),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(fun(A,bool),fun(A,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),combb(fun(A,bool),fun(A,bool),B),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),P1))),Q1))))
<=> ! [Y2: A] :
( pp(aa(A,bool,P1,Y2))
=> finite_finite(B,collect(B,aa(A,fun(B,bool),combc(B,A,bool,Q1),Y2))) ) ) ) ).
tff(fact_42_finite__image__set,axiom,
! [B: $tType,A: $tType,F1: fun(A,B),P1: fun(A,bool)] :
( finite_finite(A,collect(A,P1))
=> finite_finite(B,collect(B,aa(fun(B,fun(A,bool)),fun(B,bool),aa(fun(fun(A,bool),bool),fun(fun(B,fun(A,bool)),fun(B,bool)),combb(fun(A,bool),bool,B),fEx(A)),aa(fun(A,bool),fun(B,fun(A,bool)),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)),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))),F1)))),P1)))) ) ).
tff(fact_43_imageE,axiom,
! [A: $tType,B: $tType,A1: fun(B,bool),F1: fun(B,A),B2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B2),image(B,A,F1,A1)))
=> ~ ! [X3: B] :
( ( B2 = aa(B,A,F1,X3) )
=> ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A1)) ) ) ).
tff(fact_44_iprod0__if__coeffs0,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xsa: list(A),Cs: list(A)] :
( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),set(A,Cs)))
=> ( X3 = zero_zero(A) ) )
=> ( aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Cs),Xsa) = zero_zero(A) ) ) ) ).
tff(fact_45_set__Cons__def,axiom,
! [A: $tType,XS: fun(list(A),bool),A1: fun(A,bool)] : set_Cons(A,A1,XS) = collect(list(A),aa(fun(list(A),fun(A,bool)),fun(list(A),bool),aa(fun(fun(A,bool),bool),fun(fun(list(A),fun(A,bool)),fun(list(A),bool)),combb(fun(A,bool),bool,list(A)),fEx(A)),aa(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,bool)),aa(fun(fun(A,fun(list(A),bool)),fun(A,bool)),fun(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,bool))),combb(fun(A,fun(list(A),bool)),fun(A,bool),list(A)),aa(fun(fun(list(A),bool),bool),fun(fun(A,fun(list(A),bool)),fun(A,bool)),combb(fun(list(A),bool),bool,A),fEx(list(A)))),aa(fun(A,fun(list(A),bool)),fun(list(A),fun(A,fun(list(A),bool))),combc(list(A),fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool)),aa(fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),fun(list(A),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool)))),aa(fun(fun(A,fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool)))),fun(fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),fun(list(A),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool))))),combb(fun(A,fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool))),list(A)),combs(A,fun(list(A),bool),fun(list(A),bool))),aa(fun(list(A),fun(A,fun(list(A),fun(bool,bool)))),fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),aa(fun(fun(A,fun(list(A),fun(bool,bool))),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),fun(fun(list(A),fun(A,fun(list(A),fun(bool,bool)))),fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool))))),combb(fun(A,fun(list(A),fun(bool,bool))),fun(A,fun(fun(list(A),bool),fun(list(A),bool))),list(A)),aa(fun(fun(list(A),fun(bool,bool)),fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(list(A),fun(bool,bool))),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),combb(fun(list(A),fun(bool,bool)),fun(fun(list(A),bool),fun(list(A),bool)),A),combs(list(A),bool,bool))),aa(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,fun(list(A),fun(bool,bool)))),aa(fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),fun(bool,bool)))),fun(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,fun(list(A),fun(bool,bool))))),combb(fun(A,fun(list(A),bool)),fun(A,fun(list(A),fun(bool,bool))),list(A)),aa(fun(fun(list(A),bool),fun(list(A),fun(bool,bool))),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),fun(bool,bool)))),combb(fun(list(A),bool),fun(list(A),fun(bool,bool)),A),aa(fun(bool,fun(bool,bool)),fun(fun(list(A),bool),fun(list(A),fun(bool,bool))),combb(bool,fun(bool,bool),list(A)),fconj))),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(A,fun(list(A),bool))),combc(list(A),fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool)),aa(fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool))),fun(list(A),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool)))),aa(fun(fun(fun(list(A),list(A)),fun(list(A),bool)),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool)))),fun(fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool))),fun(list(A),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool))))),combb(fun(fun(list(A),list(A)),fun(list(A),bool)),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool))),list(A)),combb(fun(list(A),list(A)),fun(list(A),bool),A)),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool))),aa(fun(fun(list(A),bool),fun(fun(list(A),list(A)),fun(list(A),bool))),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool)))),combb(fun(list(A),bool),fun(fun(list(A),list(A)),fun(list(A),bool)),list(A)),combb(list(A),bool,list(A))),fequal(list(A))))),cons(A)))))),aa(fun(list(A),bool),fun(A,fun(list(A),bool)),combc(A,fun(list(A),bool),fun(list(A),bool),aa(fun(A,fun(bool,bool)),fun(A,fun(fun(list(A),bool),fun(list(A),bool))),aa(fun(fun(bool,bool),fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(bool,bool)),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),combb(fun(bool,bool),fun(fun(list(A),bool),fun(list(A),bool)),A),combb(bool,bool,list(A))),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),combc(A,fun(A,bool),bool,member(A)),A1)))),aa(fun(list(A),bool),fun(list(A),bool),combc(list(A),fun(list(A),bool),bool,member(list(A))),XS)))))) ).
tff(fact_46_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,F1: fun(A,B),A1: fun(A,bool)] :
( ~ finite_finite(A,A1)
=> ( finite_finite(B,image(A,B,F1,A1))
=> ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A1))
& ~ finite_finite(A,collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(fun(A,bool),fun(A,bool),combc(A,fun(A,bool),bool,member(A)),A1))),aa(B,fun(A,bool),combc(A,B,bool,aa(fun(A,B),fun(A,fun(B,bool)),aa(fun(B,fun(B,bool)),fun(fun(A,B),fun(A,fun(B,bool))),combb(B,fun(B,bool),A),fequal(B)),F1)),aa(A,B,F1,X3))))) ) ) ) ).
tff(fact_47_atom_Osize_I1_J,axiom,
! [List: list(int),Int: int] : atom_size(aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int),List)) = zero_zero(nat) ).
tff(fact_48_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A2: A,List1: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),List1) ) ).
tff(fact_49_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y2: A,Ys1: list(A)] : Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys1) ) ).
tff(fact_50_finite__list,axiom,
! [A: $tType,A1: fun(A,bool)] :
( finite_finite(A,A1)
=> ? [Xs: list(A)] : set(A,Xs) = A1 ) ).
tff(fact_51_zipwith0_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Xsa: list(B),X2: B,F1: fun(B,fun(C,A))] : aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xsa)),nil(C)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(C,A,aa(B,fun(C,A),F1,X2),zero_zero(C))),aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F1),Xsa),nil(C))) ) ).
tff(fact_52_Diff__iff,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B1)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A1))
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1)) ) ) ).
tff(fact_53_Diff__idemp,axiom,
! [A: $tType,B1: fun(A,bool),A1: 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)),A1),B1)),B1) = aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B1) ).
tff(fact_54_DiffI,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A1))
=> ( ~ 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),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B1))) ) ) ).
tff(fact_55_DiffE,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B1)))
=> ~ ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1)) ) ) ).
tff(fact_56_finite__Collect__less__nat,axiom,
! [K1: nat] : finite_finite(nat,collect(nat,aa(nat,fun(nat,bool),combc(nat,nat,bool,ord_less(nat)),K1))) ).
tff(fact_57_zipwith0_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Ys: list(C),Y1: C,Xsa: list(B),X2: B,F1: fun(B,fun(C,A))] : aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xsa)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y1),Ys)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(C,A,aa(B,fun(C,A),F1,X2),Y1)),aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F1),Xsa),Ys)) ) ).
tff(fact_58_zipwith0_Osimps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( zero(B)
& zero(C) )
=> ! [F1: fun(B,fun(C,A))] : aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F1),nil(B)),nil(C)) = nil(A) ) ).
tff(fact_59_zipwith0_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Ys: list(C),Y1: C,F1: fun(B,fun(C,A))] : aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F1),nil(B)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y1),Ys)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(C,A,aa(B,fun(C,A),F1,zero_zero(B)),Y1)),aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F1),nil(B)),Ys)) ) ).
tff(fact_60_set__diff__eq,axiom,
! [A: $tType,B1: fun(A,bool),A1: 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)),A1),B1) = 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),combc(A,fun(A,bool),bool,member(A)),A1))),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),combc(A,fun(A,bool),bool,member(A)),B1)))) ).
tff(fact_61_DiffD1,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B1)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A1)) ) ).
tff(fact_62_DiffD2,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B1)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1)) ) ).
tff(fact_63_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_64_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).
tff(fact_65_zero__less__diff,axiom,
! [M2: nat,Na: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M2)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),Na)) ) ).
tff(fact_66_less__nat__zero__code,axiom,
! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),zero_zero(nat))) ).
tff(fact_67_less__zeroE,axiom,
! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),zero_zero(nat))) ).
tff(fact_68_diff__self__eq__0,axiom,
! [M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M1),M1) = zero_zero(nat) ).
tff(fact_69_diff__0__eq__0,axiom,
! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N2) = zero_zero(nat) ).
tff(fact_70_diff__commute,axiom,
! [K: nat,J: nat,I1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I1),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I1),K)),J) ).
tff(fact_71_diffs0__imp__equal,axiom,
! [N2: nat,M1: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M1),N2) = zero_zero(nat) )
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),M1) = zero_zero(nat) )
=> ( M1 = N2 ) ) ) ).
tff(fact_72_minus__nat_Odiff__0,axiom,
! [M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M1),zero_zero(nat)) = M1 ).
tff(fact_73_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),Na: nat,M2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),Na))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) )
=> ( ( ( M2 = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Na),M2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) ) ) ) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F1: fun(A,B)] :
( ! [X3: A] : aa(A,B,F1,X3) = aa(A,B,G,X3)
=> ( F1 = G ) ) ).
tff(fact_75_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
<=> pp(aa(A,bool,A1,X2)) ) ).
tff(fact_76_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_77_diff__less__mono2,axiom,
! [L: nat,N2: nat,M1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),L))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M1))) ) ) ).
tff(fact_78_less__imp__diff__less,axiom,
! [N2: nat,K: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),N2)),K)) ) ).
tff(fact_79_less__not__refl3,axiom,
! [T: nat,S: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T))
=> ( S != T ) ) ).
tff(fact_80_less__not__refl2,axiom,
! [M1: nat,N2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M1))
=> ( M1 != N2 ) ) ).
tff(fact_81_less__irrefl__nat,axiom,
! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N2)) ).
tff(fact_82_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).
tff(fact_83_nat__neq__iff,axiom,
! [Na: nat,M2: nat] :
( ( M2 != Na )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),Na))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Na),M2)) ) ) ).
tff(fact_84_less__not__refl,axiom,
! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N2)) ).
tff(fact_85_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero(nat) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ).
tff(fact_86_gr__implies__not0,axiom,
! [N2: nat,M1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N2))
=> ( N2 != zero_zero(nat) ) ) ).
tff(fact_87_diff__less,axiom,
! [M1: nat,N2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M1))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M1),N2)),M1)) ) ) ).
tff(fact_88_not__less0,axiom,
! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),zero_zero(nat))) ).
tff(fact_89_finite__conv__nat__seg__image,axiom,
! [A: $tType,A1: fun(A,bool)] :
( finite_finite(A,A1)
<=> ? [N1: nat,F: fun(nat,A)] : A1 = image(nat,A,F,collect(nat,aa(nat,fun(nat,bool),combc(nat,nat,bool,ord_less(nat)),N1))) ) ).
tff(fact_90_listset_Osimps_I2_J,axiom,
! [A: $tType,As: list(fun(A,bool)),A1: fun(A,bool)] : listset(A,aa(list(fun(A,bool)),list(fun(A,bool)),aa(fun(A,bool),fun(list(fun(A,bool)),list(fun(A,bool))),cons(fun(A,bool)),A1),As)) = set_Cons(A,A1,listset(A,As)) ).
tff(fact_91_atom_Osize_I4_J,axiom,
! [List: list(int),Int: int] : size_size(atom,aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int),List)) = zero_zero(nat) ).
tff(fact_92_finite__M__bounded__by__nat,axiom,
! [I: nat,P1: fun(nat,bool)] : finite_finite(nat,collect(nat,aa(fun(nat,bool),fun(nat,bool),aa(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),aa(fun(nat,bool),fun(nat,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),P1)),aa(nat,fun(nat,bool),combc(nat,nat,bool,ord_less(nat)),I)))) ).
tff(fact_93_finite__nat__set__iff__bounded,axiom,
! [N: fun(nat,bool)] :
( finite_finite(nat,N)
<=> ? [M: nat] :
! [X4: nat] :
( pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),X4),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),M)) ) ) ).
tff(fact_94_bounded__nat__set__is__finite,axiom,
! [Na: nat,N: fun(nat,bool)] :
( ! [X3: nat] :
( pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),X3),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),Na)) )
=> finite_finite(nat,N) ) ).
tff(fact_95_minus__apply,axiom,
! [A: $tType,B: $tType] :
( cl_Groups_Ominus(A)
=> ! [X2: B,B1: fun(B,A),A1: 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)),A1),B1),X2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A1,X2)),aa(B,A,B1,X2)) ) ).
tff(fact_96_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( cl_Groups_Ominus(B)
=> ! [B1: fun(A,B),A1: fun(A,B),X1: 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)),A1),B1),X1) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A1,X1)),aa(A,B,B1,X1)) ) ).
tff(fact_97_insert__Nil,axiom,
! [A: $tType,X: A] : insert(A,X,nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).
tff(fact_98__096Z_OI_A_Iqe__pres_092_060_094isub_0621_Aas_J_Axs_096,axiom,
interpret(atom,int,i_Z,qEpres896714165pres_1(as),xs) ).
%----Arities (13)
tff(arity_fun___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite1(T_2)
& finite_finite1(T_1) )
=> finite_finite1(fun(T_1,T_2)) ) ).
tff(arity_fun___Groups_Ominus,axiom,
! [T_1: $tType,T_2: $tType] :
( cl_Groups_Ominus(T_2)
=> cl_Groups_Ominus(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_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___Rings_Oring,axiom,
ring(int) ).
tff(arity_Nat_Onat___Groups_Ominus,axiom,
cl_Groups_Ominus(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite1(bool) ).
tff(arity_HOL_Obool___Groups_Ominus,axiom,
cl_Groups_Ominus(bool) ).
tff(arity_List_Olist___Groups_Ominus,axiom,
! [T_1: $tType] :
( ( zero(T_1)
& cl_Groups_Ominus(T_1) )
=> cl_Groups_Ominus(list(T_1)) ) ).
tff(arity_prod___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite1(T_2)
& finite_finite1(T_1) )
=> finite_finite1(product_prod(T_1,T_2)) ) ).
%----Helper facts (18)
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),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,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_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 (1)
tff(conj_0,conjecture,
finite_finite(int,image(atom,int,atom_case(int,combc(list(int),int,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)),minus_minus(int)),aa(list(int),fun(list(int),int),combc(list(int),list(int),int,aa(fun(list(int),list(int)),fun(list(int),fun(list(int),int)),aa(fun(list(int),fun(list(int),int)),fun(fun(list(int),list(int)),fun(list(int),fun(list(int),int))),combb(list(int),fun(list(int),int),list(int)),iprod(int)),tl(int))),xs))),combk(fun(int,fun(list(int),int)),int,combk(fun(list(int),int),int,combk(int,list(int),undefined(int)))),combk(fun(int,fun(list(int),int)),int,combk(fun(list(int),int),int,combk(int,list(int),undefined(int))))),collect(atom,aa(fun(atom,bool),fun(atom,bool),aa(fun(atom,fun(bool,bool)),fun(fun(atom,bool),fun(atom,bool)),combs(atom,bool,bool),aa(fun(atom,bool),fun(atom,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(atom,bool),fun(atom,fun(bool,bool))),combb(bool,fun(bool,bool),atom),fconj),aa(fun(atom,bool),fun(atom,bool),combc(atom,fun(atom,bool),bool,member(atom)),set(atom,as)))),aa(fun(atom,fun(int,bool)),fun(atom,bool),aa(fun(fun(int,bool),bool),fun(fun(atom,fun(int,bool)),fun(atom,bool)),combb(fun(int,bool),bool,atom),fEx(int)),aa(fun(atom,fun(int,fun(int,bool))),fun(atom,fun(int,bool)),aa(fun(fun(int,fun(int,bool)),fun(int,bool)),fun(fun(atom,fun(int,fun(int,bool))),fun(atom,fun(int,bool))),combb(fun(int,fun(int,bool)),fun(int,bool),atom),aa(fun(fun(int,bool),bool),fun(fun(int,fun(int,bool)),fun(int,bool)),combb(fun(int,bool),bool,int),fEx(int))),aa(fun(atom,fun(int,fun(int,fun(list(int),bool)))),fun(atom,fun(int,fun(int,bool))),aa(fun(fun(int,fun(int,fun(list(int),bool))),fun(int,fun(int,bool))),fun(fun(atom,fun(int,fun(int,fun(list(int),bool)))),fun(atom,fun(int,fun(int,bool)))),combb(fun(int,fun(int,fun(list(int),bool))),fun(int,fun(int,bool)),atom),aa(fun(fun(int,fun(list(int),bool)),fun(int,bool)),fun(fun(int,fun(int,fun(list(int),bool))),fun(int,fun(int,bool))),combb(fun(int,fun(list(int),bool)),fun(int,bool),int),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(atom,fun(int,fun(int,fun(list(int),bool)))),fun(atom,fun(int,fun(int,fun(list(int),bool)))),aa(fun(fun(int,fun(int,fun(list(int),bool))),fun(int,fun(int,fun(list(int),bool)))),fun(fun(atom,fun(int,fun(int,fun(list(int),bool)))),fun(atom,fun(int,fun(int,fun(list(int),bool))))),combb(fun(int,fun(int,fun(list(int),bool))),fun(int,fun(int,fun(list(int),bool))),atom),aa(fun(fun(int,fun(list(int),bool)),fun(int,fun(list(int),bool))),fun(fun(int,fun(int,fun(list(int),bool))),fun(int,fun(int,fun(list(int),bool)))),combb(fun(int,fun(list(int),bool)),fun(int,fun(list(int),bool)),int),aa(fun(int,fun(fun(list(int),bool),fun(list(int),bool))),fun(fun(int,fun(list(int),bool)),fun(int,fun(list(int),bool))),combs(int,fun(list(int),bool),fun(list(int),bool)),aa(fun(int,fun(bool,bool)),fun(int,fun(fun(list(int),bool),fun(list(int),bool))),aa(fun(fun(bool,bool),fun(fun(list(int),bool),fun(list(int),bool))),fun(fun(int,fun(bool,bool)),fun(int,fun(fun(list(int),bool),fun(list(int),bool)))),combb(fun(bool,bool),fun(fun(list(int),bool),fun(list(int),bool)),int),combb(bool,bool,list(int))),aa(fun(int,bool),fun(int,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),aa(int,fun(int,bool),combc(int,int,bool,ord_less(int)),zero_zero(int))))))),aa(fun(int,fun(int,fun(list(int),atom))),fun(atom,fun(int,fun(int,fun(list(int),bool)))),combc(atom,fun(int,fun(int,fun(list(int),atom))),fun(int,fun(int,fun(list(int),bool))),aa(fun(atom,fun(fun(int,fun(list(int),atom)),fun(int,fun(list(int),bool)))),fun(atom,fun(fun(int,fun(int,fun(list(int),atom))),fun(int,fun(int,fun(list(int),bool))))),aa(fun(fun(fun(int,fun(list(int),atom)),fun(int,fun(list(int),bool))),fun(fun(int,fun(int,fun(list(int),atom))),fun(int,fun(int,fun(list(int),bool))))),fun(fun(atom,fun(fun(int,fun(list(int),atom)),fun(int,fun(list(int),bool)))),fun(atom,fun(fun(int,fun(int,fun(list(int),atom))),fun(int,fun(int,fun(list(int),bool)))))),combb(fun(fun(int,fun(list(int),atom)),fun(int,fun(list(int),bool))),fun(fun(int,fun(int,fun(list(int),atom))),fun(int,fun(int,fun(list(int),bool)))),atom),combb(fun(int,fun(list(int),atom)),fun(int,fun(list(int),bool)),int)),aa(fun(atom,fun(fun(list(int),atom),fun(list(int),bool))),fun(atom,fun(fun(int,fun(list(int),atom)),fun(int,fun(list(int),bool)))),aa(fun(fun(fun(list(int),atom),fun(list(int),bool)),fun(fun(int,fun(list(int),atom)),fun(int,fun(list(int),bool)))),fun(fun(atom,fun(fun(list(int),atom),fun(list(int),bool))),fun(atom,fun(fun(int,fun(list(int),atom)),fun(int,fun(list(int),bool))))),combb(fun(fun(list(int),atom),fun(list(int),bool)),fun(fun(int,fun(list(int),atom)),fun(int,fun(list(int),bool))),atom),combb(fun(list(int),atom),fun(list(int),bool),int)),aa(fun(atom,fun(atom,bool)),fun(atom,fun(fun(list(int),atom),fun(list(int),bool))),aa(fun(fun(atom,bool),fun(fun(list(int),atom),fun(list(int),bool))),fun(fun(atom,fun(atom,bool)),fun(atom,fun(fun(list(int),atom),fun(list(int),bool)))),combb(fun(atom,bool),fun(fun(list(int),atom),fun(list(int),bool)),atom),combb(atom,bool,list(int))),fequal(atom))))),aa(fun(int,fun(list(int),list(int))),fun(int,fun(int,fun(list(int),atom))),combc(int,fun(int,fun(list(int),list(int))),fun(int,fun(list(int),atom)),aa(fun(int,fun(fun(list(int),list(int)),fun(list(int),atom))),fun(int,fun(fun(int,fun(list(int),list(int))),fun(int,fun(list(int),atom)))),aa(fun(fun(fun(list(int),list(int)),fun(list(int),atom)),fun(fun(int,fun(list(int),list(int))),fun(int,fun(list(int),atom)))),fun(fun(int,fun(fun(list(int),list(int)),fun(list(int),atom))),fun(int,fun(fun(int,fun(list(int),list(int))),fun(int,fun(list(int),atom))))),combb(fun(fun(list(int),list(int)),fun(list(int),atom)),fun(fun(int,fun(list(int),list(int))),fun(int,fun(list(int),atom))),int),combb(fun(list(int),list(int)),fun(list(int),atom),int)),aa(fun(int,fun(list(int),atom)),fun(int,fun(fun(list(int),list(int)),fun(list(int),atom))),aa(fun(fun(list(int),atom),fun(fun(list(int),list(int)),fun(list(int),atom))),fun(fun(int,fun(list(int),atom)),fun(int,fun(fun(list(int),list(int)),fun(list(int),atom)))),combb(fun(list(int),atom),fun(fun(list(int),list(int)),fun(list(int),atom)),int),combb(list(int),atom,list(int))),c_PresArith_Oatom_OLe))),cons(int))))))))))) ).
%------------------------------------------------------------------------------