TPTP Problem File: COM103_5.p
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%------------------------------------------------------------------------------
% File : COM103_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 241
% 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_241 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 180 ( 52 unt; 50 typ; 0 def)
% Number of atoms : 281 ( 75 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 192 ( 41 ~; 8 |; 13 &)
% ( 16 <=>; 114 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 45 ( 24 >; 21 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-3 aty)
% Number of functors : 34 ( 34 usr; 10 con; 0-6 aty)
% Number of variables : 507 ( 456 !; 6 ?; 507 :)
% ( 45 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:24:02
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_PresArith_Oatom,type,
atom: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (44)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
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_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[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(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_ListVector_Oiprod,type,
iprod:
!>[A: $tType] : ( ( list(A) * list(A) ) > A ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) * list(A) ) > T ) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : ( list(A) > fun(A,bool) ) ).
tff(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_PresArith_OI_092_060_094isub_062Z,type,
i_Z: ( atom * list(int) ) > $o ).
tff(sy_c_PresArith_Oatom_OLe,type,
c_PresArith_Oatom_OLe: ( int * list(int) ) > atom ).
tff(sy_c_PresArith_Olbounds,type,
lbounds: list(atom) > list(product_prod(int,list(int))) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( ( A * B ) > product_prod(A,B) ) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(product_prod(A,B),C) * A * B ) > C ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * product_prod(A,B) ) > C ) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_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_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_li____,type,
li: int ).
tff(sy_v_lks____,type,
lks: list(int) ).
tff(sy_v_x____,type,
x: int ).
tff(sy_v_xs,type,
xs: list(int) ).
%----Relevant facts (99)
tff(fact_0__096a_A_058_Aset_Aas_096,axiom,
pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),a),set(atom,as))) ).
tff(fact_1_x,axiom,
! [X3: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X3),set(atom,as)))
=> i_Z(X3,cons(int,x,xs)) ) ).
tff(fact_2__096Le_Ali_A_I1_A_D_Alks_J_A_058_Aset_Aas_096,axiom,
pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),c_PresArith_Oatom_OLe(li,cons(int,one_one(int),lks))),set(atom,as))) ).
tff(fact_3__096EX_Ax_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_096,axiom,
? [X: int] :
! [Xa1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa1),set(atom,as)))
=> i_Z(Xa1,cons(int,X,xs)) ) ).
tff(fact_4__096I_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_096,axiom,
i_Z(a,cons(int,x,xs)) ).
tff(fact_5_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X1: A] : ord_less_eq(A,X1,X1) ) ).
tff(fact_6_iprod__left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys: list(A),Xs: list(A)] : iprod(A,minus_minus(list(A),Xs,Ys),Zs) = minus_minus(A,iprod(A,Xs,Zs),iprod(A,Ys,Zs)) ) ).
tff(fact_7_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D3: A,C1: A,B1: A,Aa: A] :
( ( minus_minus(A,Aa,B1) = minus_minus(A,C1,D3) )
=> ( ord_less_eq(A,Aa,B1)
<=> ord_less_eq(A,C1,D3) ) ) ) ).
tff(fact_8_I_092_060_094isub_062Z_Osimps_I1_J,axiom,
! [Xsa: list(int),Ksa: list(int),Ia: int] :
( i_Z(c_PresArith_Oatom_OLe(Ia,Ksa),Xsa)
<=> ord_less_eq(int,Ia,iprod(int,Ksa,Xsa)) ) ).
tff(fact_9_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( cl_Groups_Ominus(B)
=> ! [B3: fun(A,B),A2: fun(A,B),X3: A] : aa(A,B,minus_minus(fun(A,B),A2,B3),X3) = minus_minus(B,aa(A,B,A2,X3),aa(A,B,B3,X3)) ) ).
tff(fact_10_minus__apply,axiom,
! [A: $tType,B: $tType] :
( cl_Groups_Ominus(A)
=> ! [Xa: B,B3: fun(B,A),A2: fun(B,A)] : aa(B,A,minus_minus(fun(B,A),A2,B3),Xa) = minus_minus(A,aa(B,A,A2,Xa),aa(B,A,B3,Xa)) ) ).
tff(fact_11_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D3: A,C1: A,B1: A,Aa: A] :
( ( minus_minus(A,Aa,B1) = minus_minus(A,C1,D3) )
=> ( ( Aa = B1 )
<=> ( C1 = D3 ) ) ) ) ).
tff(fact_12_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
<=> ! [X2: A] : ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2)) ) ) ).
tff(fact_13_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
| ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_14_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y3: A,Xa: A] :
( ( Xa = Y3 )
<=> ( ord_less_eq(A,Xa,Y3)
& ord_less_eq(A,Y3,Xa) ) ) ) ).
tff(fact_15_atom_Osimps_I1_J,axiom,
! [List3: list(int),Int1: int,List: list(int),Int: int] :
( ( c_PresArith_Oatom_OLe(Int,List) = c_PresArith_Oatom_OLe(Int1,List3) )
<=> ( ( Int = Int1 )
& ( List = List3 ) ) ) ).
tff(fact_16_list__diff__Cons__Cons,axiom,
! [A: $tType] :
( ( cl_Groups_Ominus(A)
& zero(A) )
=> ! [Ys: list(A),Y1: A,Xs: list(A),X1: A] : minus_minus(list(A),cons(A,X1,Xs),cons(A,Y1,Ys)) = cons(A,minus_minus(A,X1,Y1),minus_minus(list(A),Xs,Ys)) ) ).
tff(fact_17__096_B_Bthesis_O_A_I_B_Bx_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [X: int] :
~ ! [Xa1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa1),set(atom,as)))
=> i_Z(Xa1,cons(int,X,xs)) ) ).
tff(fact_18_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [Xa: A] :
( ( one_one(A) = Xa )
<=> ( Xa = one_one(A) ) ) ) ).
tff(fact_19_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less_eq(A,X1,Y1)
=> ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_20_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,Xa),aa(A,B,G,Xa)) ) ) ).
tff(fact_21_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less_eq(A,X1,Z) ) ) ) ).
tff(fact_22_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less_eq(A,Y1,X1)
=> ( X1 = Y1 ) ) ) ) ).
tff(fact_23_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C4: A,B5: A,A4: A] :
( ord_less_eq(A,A4,B5)
=> ( ( B5 = C4 )
=> ord_less_eq(A,A4,C4) ) ) ) ).
tff(fact_24_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C4: A,B5: A,A4: A] :
( ( A4 = B5 )
=> ( ord_less_eq(A,B5,C4)
=> ord_less_eq(A,A4,C4) ) ) ) ).
tff(fact_25_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Xa: A,Y3: A] :
( ord_less_eq(A,Y3,Xa)
=> ( ord_less_eq(A,Xa,Y3)
<=> ( Xa = Y3 ) ) ) ) ).
tff(fact_26_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,Xa),aa(A,B,G,Xa)) ) ) ).
tff(fact_27_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ( X1 = Y1 )
=> ord_less_eq(A,X1,Y1) ) ) ).
tff(fact_28_list_Oinject,axiom,
! [A: $tType,List3: list(A),A6: A,List: list(A),Aa: A] :
( ( cons(A,Aa,List) = cons(A,A6,List3) )
<=> ( ( Aa = A6 )
& ( List = List3 ) ) ) ).
tff(fact_29_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ! [X: A] : ord_less_eq(B,aa(A,B,F,X),aa(A,B,G,X))
=> ord_less_eq(fun(A,B),F,G) ) ) ).
tff(fact_30_set__subset__Cons,axiom,
! [A: $tType,Xa: A,Xsa: list(A)] : ord_less_eq(fun(A,bool),set(A,Xsa),set(A,cons(A,Xa,Xsa))) ).
tff(fact_31_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),Xa: A,Y3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y3),set(A,cons(A,Xa,Xsa))))
=> ( ( Y3 = Xa )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y3),set(A,Xsa))) ) ) ).
tff(fact_32_int__le__induct,axiom,
! [P1: fun(int,bool),K: int,Ia: int] :
( ord_less_eq(int,Ia,K)
=> ( pp(aa(int,bool,P1,K))
=> ( ! [I: int] :
( ord_less_eq(int,I,K)
=> ( pp(aa(int,bool,P1,I))
=> pp(aa(int,bool,P1,minus_minus(int,I,one_one(int)))) ) )
=> pp(aa(int,bool,P1,Ia)) ) ) ) ).
tff(fact_33__096_Ili_M_Alks_J_A_058_Aset_A_Ilbounds_Aas_J_096,axiom,
pp(aa(fun(product_prod(int,list(int)),bool),bool,aa(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool),member(product_prod(int,list(int))),product_Pair(int,list(int),li,lks)),set(product_prod(int,list(int)),lbounds(as)))) ).
tff(fact_34__096lbounds_Aas_A_126_061_A_091_093_096,axiom,
lbounds(as) != nil(product_prod(int,list(int))) ).
tff(fact_35_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B1: B,F: fun(B,A),Aa: A] :
( ord_less_eq(A,Aa,aa(B,A,F,B1))
=> ( ord_less_eq(B,B1,C1)
=> ( ! [X: B,Y: B] :
( ord_less_eq(B,X,Y)
=> ord_less_eq(A,aa(B,A,F,X),aa(B,A,F,Y)) )
=> ord_less_eq(A,Aa,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_36_list_Osimps_I2_J,axiom,
! [A: $tType,List2: list(A),A5: A] : nil(A) != cons(A,A5,List2) ).
tff(fact_37_list_Osimps_I3_J,axiom,
! [A: $tType,List2: list(A),A5: A] : cons(A,A5,List2) != nil(A) ).
tff(fact_38_list__diff__Nil2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [Xs: list(A)] : minus_minus(list(A),Xs,nil(A)) = Xs ) ).
tff(fact_39_not__Cons__self2,axiom,
! [A: $tType,Xs: list(A),X1: A] : cons(A,X1,Xs) != Xs ).
tff(fact_40_not__Cons__self,axiom,
! [A: $tType,X1: A,Xs: list(A)] : Xs != cons(A,X1,Xs) ).
tff(fact_41_subsetD,axiom,
! [A: $tType,C1: A,B3: fun(A,bool),A2: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A2,B3)
=> ( 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),B3)) ) ) ).
tff(fact_42_equalityI,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A2,B3)
=> ( ord_less_eq(fun(A,bool),B3,A2)
=> ( A2 = B3 ) ) ) ).
tff(fact_43_Pair__eq,axiom,
! [A: $tType,B: $tType,B7: B,A6: A,B1: B,Aa: A] :
( ( product_Pair(A,B,Aa,B1) = product_Pair(A,B,A6,B7) )
<=> ( ( Aa = A6 )
& ( B1 = B7 ) ) ) ).
tff(fact_44_split__paired__All,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ! [A3: A,B4: B] : pp(aa(product_prod(A,B),bool,P1,product_Pair(A,B,A3,B4))) ) ).
tff(fact_45_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y2: A,Ys1: list(A)] : Xsa = cons(A,Y2,Ys1) ) ).
tff(fact_46_Pair__inject,axiom,
! [A: $tType,B: $tType,B6: B,A5: A,B5: B,A4: A] :
( ( product_Pair(A,B,A4,B5) = product_Pair(A,B,A5,B6) )
=> ~ ( ( A4 = A5 )
=> ( B5 != B6 ) ) ) ).
tff(fact_47_equalityE,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] :
( ( A2 = B3 )
=> ~ ( ord_less_eq(fun(A,bool),A2,B3)
=> ~ ord_less_eq(fun(A,bool),B3,A2) ) ) ).
tff(fact_48_double__diff,axiom,
! [A: $tType,C3: fun(A,bool),B3: fun(A,bool),A2: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A2,B3)
=> ( ord_less_eq(fun(A,bool),B3,C3)
=> ( minus_minus(fun(A,bool),B3,minus_minus(fun(A,bool),C3,A2)) = A2 ) ) ) ).
tff(fact_49_Diff__mono,axiom,
! [A: $tType,B3: fun(A,bool),D2: fun(A,bool),C3: fun(A,bool),A2: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A2,C3)
=> ( ord_less_eq(fun(A,bool),D2,B3)
=> ord_less_eq(fun(A,bool),minus_minus(fun(A,bool),A2,B3),minus_minus(fun(A,bool),C3,D2)) ) ) ).
tff(fact_50_subset__trans,axiom,
! [A: $tType,C3: fun(A,bool),B3: fun(A,bool),A2: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A2,B3)
=> ( ord_less_eq(fun(A,bool),B3,C3)
=> ord_less_eq(fun(A,bool),A2,C3) ) ) ).
tff(fact_51_set__mp,axiom,
! [A: $tType,Xa: A,B3: fun(A,bool),A2: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A2,B3)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),B3)) ) ) ).
tff(fact_52_set__rev__mp,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A2))
=> ( ord_less_eq(fun(A,bool),A2,B3)
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),B3)) ) ) ).
tff(fact_53_in__mono,axiom,
! [A: $tType,Xa: A,B3: fun(A,bool),A2: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A2,B3)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),B3)) ) ) ).
tff(fact_54_equalityD2,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] :
( ( A2 = B3 )
=> ord_less_eq(fun(A,bool),B3,A2) ) ).
tff(fact_55_equalityD1,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] :
( ( A2 = B3 )
=> ord_less_eq(fun(A,bool),A2,B3) ) ).
tff(fact_56_set__eq__subset,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] :
( ( A2 = B3 )
<=> ( ord_less_eq(fun(A,bool),A2,B3)
& ord_less_eq(fun(A,bool),B3,A2) ) ) ).
tff(fact_57_Diff__subset,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] : ord_less_eq(fun(A,bool),minus_minus(fun(A,bool),A2,B3),A2) ).
tff(fact_58_subset__refl,axiom,
! [A: $tType,A2: fun(A,bool)] : ord_less_eq(fun(A,bool),A2,A2) ).
tff(fact_59_subsetI,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] :
( ! [X: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),A2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),B3)) )
=> ord_less_eq(fun(A,bool),A2,B3) ) ).
tff(fact_60_split__paired__Ex,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ? [A3: A,B4: B] : pp(aa(product_prod(A,B),bool,P1,product_Pair(A,B,A3,B4))) ) ).
tff(fact_61_list_Oexhaust,axiom,
! [A: $tType,Y1: list(A)] :
( ( Y1 != nil(A) )
=> ~ ! [A1: A,List1: list(A)] : Y1 != cons(A,A1,List1) ) ).
tff(fact_62_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,Aa: B,F11: fun(B,fun(C,A))] : product_prod_rec(B,C,A,F11,product_Pair(B,C,Aa,B1)) = aa(C,A,aa(B,fun(C,A),F11,Aa),B1) ).
tff(fact_63_DiffE,axiom,
! [A: $tType,B3: 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),minus_minus(fun(A,bool),A2,B3)))
=> ~ ( 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),B3)) ) ) ).
tff(fact_64_DiffI,axiom,
! [A: $tType,B3: 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),B3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),minus_minus(fun(A,bool),A2,B3))) ) ) ).
tff(fact_65_Diff__idemp,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] : minus_minus(fun(A,bool),minus_minus(fun(A,bool),A2,B3),B3) = minus_minus(fun(A,bool),A2,B3) ).
tff(fact_66_Diff__iff,axiom,
! [A: $tType,B3: 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),minus_minus(fun(A,bool),A2,B3)))
<=> ( 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),B3)) ) ) ).
tff(fact_67_DiffD2,axiom,
! [A: $tType,B3: 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),minus_minus(fun(A,bool),A2,B3)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3)) ) ).
tff(fact_68_DiffD1,axiom,
! [A: $tType,B3: 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),minus_minus(fun(A,bool),A2,B3)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A2)) ) ).
tff(fact_69_set__diff__eq,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] : minus_minus(fun(A,bool),A2,B3) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,combc(A,fun(A,bool),bool,member(A),A2)),combb(bool,bool,A,fNot,combc(A,fun(A,bool),bool,member(A),B3)))) ).
tff(fact_70_insert__Nil,axiom,
! [A: $tType,X1: A] : insert(A,X1,nil(A)) = cons(A,X1,nil(A)) ).
tff(fact_71_splice_Osimps_I2_J,axiom,
! [A: $tType,Va: list(A),V: A] : splice(A,cons(A,V,Va),nil(A)) = cons(A,V,Va) ).
tff(fact_72_Collect__mono,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] :
( ! [X: A] :
( pp(aa(A,bool,P1,X))
=> pp(aa(A,bool,Q1,X)) )
=> ord_less_eq(fun(A,bool),collect(A,P1),collect(A,Q1)) ) ).
tff(fact_73_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys: list(A),Y1: A,Xs: list(A),X1: A] : splice(A,cons(A,X1,Xs),cons(A,Y1,Ys)) = cons(A,X1,cons(A,Y1,splice(A,Xs,Ys))) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X: A] : aa(A,B,F,X) = aa(A,B,G,X)
=> ( F = G ) ) ).
tff(fact_75_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A2))
<=> pp(aa(A,bool,A2,Xa)) ) ).
tff(fact_76_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_77_in__set__insert,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = Xsa ) ) ).
tff(fact_78_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : splice(A,nil(A),Ys) = Ys ).
tff(fact_79_splice__Nil2,axiom,
! [A: $tType,Xs: list(A)] : splice(A,Xs,nil(A)) = Xs ).
tff(fact_80_List_Oinsert__def,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = Xsa ) )
& ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = cons(A,Xa,Xsa) ) ) ) ).
tff(fact_81_not__in__set__insert,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = cons(A,Xa,Xsa) ) ) ).
tff(fact_82_list_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,F21: fun(B,fun(list(B),A)),F11: A] : list_case(A,B,F11,F21,nil(B)) = F11 ).
tff(fact_83_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),Aa: B,F21: fun(B,fun(list(B),A)),F11: A] : list_case(A,B,F11,F21,cons(B,Aa,List)) = aa(list(B),A,aa(B,fun(list(B),A),F21,Aa),List) ).
tff(fact_84_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y1: product_prod(A,B)] :
~ ! [A1: A,B2: B] : Y1 != product_Pair(A,B,A1,B2) ).
tff(fact_85_PairE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
~ ! [X: A,Y: B] : P2 != product_Pair(A,B,X,Y) ).
tff(fact_86_prod__induct6,axiom,
! [F1: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool)] :
( ! [A1: A,B2: B,C2: C,D1: D,E1: E,F2: F1] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,P1,product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),A1,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1))),B2,product_Pair(C,product_prod(D,product_prod(E,F1)),C2,product_Pair(D,product_prod(E,F1),D1,product_Pair(E,F1,E1,F2)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,P1,Xa)) ) ).
tff(fact_87_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F1: $tType,Y1: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))))] :
~ ! [A1: A,B2: B,C2: C,D1: D,E1: E,F2: F1] : Y1 != product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),A1,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1))),B2,product_Pair(C,product_prod(D,product_prod(E,F1)),C2,product_Pair(D,product_prod(E,F1),D1,product_Pair(E,F1,E1,F2))))) ).
tff(fact_88_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y1: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A1: A,B2: B,C2: C,D1: D,E1: E] : Y1 != product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E))),A1,product_Pair(B,product_prod(C,product_prod(D,E)),B2,product_Pair(C,product_prod(D,E),C2,product_Pair(D,E,D1,E1)))) ).
tff(fact_89_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool)] :
( ! [A1: A,B2: B,C2: C,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E))),A1,product_Pair(B,product_prod(C,product_prod(D,E)),B2,product_Pair(C,product_prod(D,E),C2,product_Pair(D,E,D1,E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,Xa)) ) ).
tff(fact_90_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A1: A,B2: B,C2: C,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,product_Pair(A,product_prod(B,product_prod(C,D)),A1,product_Pair(B,product_prod(C,D),B2,product_Pair(C,D,C2,D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,Xa)) ) ).
tff(fact_91_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y1: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A1: A,B2: B,C2: C,D1: D] : Y1 != product_Pair(A,product_prod(B,product_prod(C,D)),A1,product_Pair(B,product_prod(C,D),B2,product_Pair(C,D,C2,D1))) ).
tff(fact_92_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y1: product_prod(A,product_prod(B,C))] :
~ ! [A1: A,B2: B,C2: C] : Y1 != product_Pair(A,product_prod(B,C),A1,product_Pair(B,C,B2,C2)) ).
tff(fact_93_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,C)),P1: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A1: A,B2: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P1,product_Pair(A,product_prod(B,C),A1,product_Pair(B,C,B2,C2))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P1,Xa)) ) ).
tff(fact_94_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,Aa: B,C1: fun(B,fun(C,A))] : produc1605651328_split(B,C,A,C1,product_Pair(B,C,Aa,B1)) = aa(C,A,aa(B,fun(C,A),C1,Aa),B1) ).
tff(fact_95_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,B1: B,F: fun(B,A),Aa: A] :
( ( Aa = aa(B,A,F,B1) )
=> ( ord_less_eq(B,B1,C1)
=> ( ! [X: B,Y: B] :
( ord_less_eq(B,X,Y)
=> ord_less_eq(A,aa(B,A,F,X),aa(B,A,F,Y)) )
=> ord_less_eq(A,Aa,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_96_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,F: fun(A,B),B1: A,Aa: A] :
( ord_less_eq(A,Aa,B1)
=> ( ( aa(A,B,F,B1) = C1 )
=> ( ! [X: A,Y: A] :
( ord_less_eq(A,X,Y)
=> ord_less_eq(B,aa(A,B,F,X),aa(A,B,F,Y)) )
=> ord_less_eq(B,aa(A,B,F,Aa),C1) ) ) ) ) ).
tff(fact_97_order__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F: fun(A,B),B1: A,Aa: A] :
( ord_less_eq(A,Aa,B1)
=> ( ord_less_eq(B,aa(A,B,F,B1),C1)
=> ( ! [X: A,Y: A] :
( ord_less_eq(A,X,Y)
=> ord_less_eq(B,aa(A,B,F,X),aa(A,B,F,Y)) )
=> ord_less_eq(B,aa(A,B,F,Aa),C1) ) ) ) ) ).
tff(fact_98_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,B1: C,Aa: B,F: fun(product_prod(B,C),A)] : product_curry(B,C,A,F,Aa,B1) = aa(product_prod(B,C),A,F,product_Pair(B,C,Aa,B1)) ).
%----Arities (20)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(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___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
tff(arity_Int_Oint___Orderings_Oord,axiom,
ord(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_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(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)) ) ).
%----Helper facts (10)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,combb(B,C,A,P,Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_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) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
ord_less_eq(int,minus_minus(int,li,iprod(int,lks,xs)),x) ).
%------------------------------------------------------------------------------