TPTP Problem File: SWX024+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWX024+1 : TPTP v9.1.0. Released v9.1.0.
% Domain : Software Verification
% Problem : An element of the appended list belongs to at least one list
% Version : Especial.
% English :
% Refs : [MMP24] Mesnard et al. (2024), ATP for Prolog Verification
% Source : [Mes24] Mesnard (2024), Email to Geoff Sutcliffe
% Names : list-all57 [Mes24]
% Status : Theorem
% Rating : 0.85 v9.1.0
% Syntax : Number of formulae : 219 ( 17 unt; 0 def)
% Number of atoms : 696 ( 152 equ)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 553 ( 76 ~; 82 |; 196 &)
% ( 38 <=>; 161 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 37 ( 35 usr; 1 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 2 con; 0-2 aty)
% Number of variables : 596 ( 554 !; 42 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
fof(id1,axiom,
'0' != nil ).
fof(id2,axiom,
! [Xx3] : '0' != s(Xx3) ).
fof(id3,axiom,
! [Xx4,Xx5] : '0' != cons(Xx4,Xx5) ).
fof(id4,axiom,
! [Xx6] : nil != s(Xx6) ).
fof(id5,axiom,
! [Xx7,Xx8] : nil != cons(Xx7,Xx8) ).
fof(id6,axiom,
! [Xx9,Xx10] :
( s(Xx9) = s(Xx10)
=> Xx9 = Xx10 ) ).
fof(id7,axiom,
! [Xx11,Xx12,Xx13] : s(Xx11) != cons(Xx12,Xx13) ).
fof(id8,axiom,
! [Xx14,Xx15,Xx16,Xx17] :
( cons(Xx14,Xx15) = cons(Xx16,Xx17)
=> Xx15 = Xx17 ) ).
fof(id9,axiom,
! [Xx18,Xx19,Xx20,Xx21] :
( cons(Xx18,Xx19) = cons(Xx20,Xx21)
=> Xx18 = Xx20 ) ).
fof(id10,axiom,
gr('0') ).
fof(id11,axiom,
gr(nil) ).
fof(id12,axiom,
! [Xx22] :
( gr(Xx22)
<=> gr(s(Xx22)) ) ).
fof(id13,axiom,
! [Xx23,Xx24] :
( ( gr(Xx23)
& gr(Xx24) )
<=> gr(cons(Xx23,Xx24)) ) ).
fof(id14,axiom,
! [Xx25,Xx26,Xx27] :
~ ( delete_succeeds(Xx25,Xx26,Xx27)
& delete_fails(Xx25,Xx26,Xx27) ) ).
fof(id15,axiom,
! [Xx25,Xx26,Xx27] :
( delete_terminates(Xx25,Xx26,Xx27)
=> ( delete_succeeds(Xx25,Xx26,Xx27)
| delete_fails(Xx25,Xx26,Xx27) ) ) ).
fof(id16,axiom,
! [Xx28,Xx29] :
~ ( length_succeeds(Xx28,Xx29)
& length_fails(Xx28,Xx29) ) ).
fof(id17,axiom,
! [Xx28,Xx29] :
( length_terminates(Xx28,Xx29)
=> ( length_succeeds(Xx28,Xx29)
| length_fails(Xx28,Xx29) ) ) ).
fof(id18,axiom,
! [Xx30,Xx31,Xx32] :
~ ( append_succeeds(Xx30,Xx31,Xx32)
& append_fails(Xx30,Xx31,Xx32) ) ).
fof(id19,axiom,
! [Xx30,Xx31,Xx32] :
( append_terminates(Xx30,Xx31,Xx32)
=> ( append_succeeds(Xx30,Xx31,Xx32)
| append_fails(Xx30,Xx31,Xx32) ) ) ).
fof(id20,axiom,
! [Xx33,Xx34] :
~ ( member_succeeds(Xx33,Xx34)
& member_fails(Xx33,Xx34) ) ).
fof(id21,axiom,
! [Xx33,Xx34] :
( member_terminates(Xx33,Xx34)
=> ( member_succeeds(Xx33,Xx34)
| member_fails(Xx33,Xx34) ) ) ).
fof(id22,axiom,
! [Xx35] :
~ ( list_succeeds(Xx35)
& list_fails(Xx35) ) ).
fof(id23,axiom,
! [Xx35] :
( list_terminates(Xx35)
=> ( list_succeeds(Xx35)
| list_fails(Xx35) ) ) ).
fof(id24,axiom,
! [Xx36] :
~ ( nat_list_succeeds(Xx36)
& nat_list_fails(Xx36) ) ).
fof(id25,axiom,
! [Xx36] :
( nat_list_terminates(Xx36)
=> ( nat_list_succeeds(Xx36)
| nat_list_fails(Xx36) ) ) ).
fof(id26,axiom,
! [Xx37,Xx38,Xx39] :
~ ( times_succeeds(Xx37,Xx38,Xx39)
& times_fails(Xx37,Xx38,Xx39) ) ).
fof(id27,axiom,
! [Xx37,Xx38,Xx39] :
( times_terminates(Xx37,Xx38,Xx39)
=> ( times_succeeds(Xx37,Xx38,Xx39)
| times_fails(Xx37,Xx38,Xx39) ) ) ).
fof(id28,axiom,
! [Xx40,Xx41,Xx42] :
~ ( plus_succeeds(Xx40,Xx41,Xx42)
& plus_fails(Xx40,Xx41,Xx42) ) ).
fof(id29,axiom,
! [Xx40,Xx41,Xx42] :
( plus_terminates(Xx40,Xx41,Xx42)
=> ( plus_succeeds(Xx40,Xx41,Xx42)
| plus_fails(Xx40,Xx41,Xx42) ) ) ).
fof(id30,axiom,
! [Xx43,Xx44] :
~ ( '@=<_succeeds'(Xx43,Xx44)
& '@=<_fails'(Xx43,Xx44) ) ).
fof(id31,axiom,
! [Xx43,Xx44] :
( '@=<_terminates'(Xx43,Xx44)
=> ( '@=<_succeeds'(Xx43,Xx44)
| '@=<_fails'(Xx43,Xx44) ) ) ).
fof(id32,axiom,
! [Xx45,Xx46] :
~ ( '@<_succeeds'(Xx45,Xx46)
& '@<_fails'(Xx45,Xx46) ) ).
fof(id33,axiom,
! [Xx45,Xx46] :
( '@<_terminates'(Xx45,Xx46)
=> ( '@<_succeeds'(Xx45,Xx46)
| '@<_fails'(Xx45,Xx46) ) ) ).
fof(id34,axiom,
! [Xx47] :
~ ( nat_succeeds(Xx47)
& nat_fails(Xx47) ) ).
fof(id35,axiom,
! [Xx47] :
( nat_terminates(Xx47)
=> ( nat_succeeds(Xx47)
| nat_fails(Xx47) ) ) ).
fof(id36,axiom,
! [Xx1,Xx2,Xx3] :
( delete_succeeds(Xx1,Xx2,Xx3)
<=> ( ? [Xx4,Xx5,Xx6] :
( Xx2 = cons(Xx4,Xx5)
& Xx3 = cons(Xx4,Xx6)
& delete_succeeds(Xx1,Xx5,Xx6) )
| Xx2 = cons(Xx1,Xx3) ) ) ).
fof(id37,axiom,
! [Xx1,Xx2,Xx3] :
( delete_fails(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5,Xx6] :
( Xx2 != cons(Xx4,Xx5)
| Xx3 != cons(Xx4,Xx6)
| delete_fails(Xx1,Xx5,Xx6) )
& Xx2 != cons(Xx1,Xx3) ) ) ).
fof(id38,axiom,
! [Xx1,Xx2,Xx3] :
( delete_terminates(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5,Xx6] :
( $true
& ( Xx2 != cons(Xx4,Xx5)
| ( $true
& ( Xx3 != cons(Xx4,Xx6)
| delete_terminates(Xx1,Xx5,Xx6) ) ) ) )
& $true ) ) ).
fof(id39,axiom,
! [Xx1,Xx2] :
( length_succeeds(Xx1,Xx2)
<=> ( ? [Xx3,Xx4,Xx5] :
( Xx1 = cons(Xx3,Xx4)
& Xx2 = s(Xx5)
& length_succeeds(Xx4,Xx5) )
| ( Xx1 = nil
& Xx2 = '0' ) ) ) ).
fof(id40,axiom,
! [Xx1,Xx2] :
( length_fails(Xx1,Xx2)
<=> ( ! [Xx3,Xx4,Xx5] :
( Xx1 != cons(Xx3,Xx4)
| Xx2 != s(Xx5)
| length_fails(Xx4,Xx5) )
& ( Xx1 != nil
| Xx2 != '0' ) ) ) ).
fof(id41,axiom,
! [Xx1,Xx2] :
( length_terminates(Xx1,Xx2)
<=> ( ! [Xx3,Xx4,Xx5] :
( $true
& ( Xx1 != cons(Xx3,Xx4)
| ( $true
& ( Xx2 != s(Xx5)
| length_terminates(Xx4,Xx5) ) ) ) )
& $true
& ( Xx1 != nil
| $true ) ) ) ).
fof(id42,axiom,
! [Xx1,Xx2,Xx3] :
( append_succeeds(Xx1,Xx2,Xx3)
<=> ( ? [Xx4,Xx5,Xx6] :
( Xx1 = cons(Xx4,Xx5)
& Xx3 = cons(Xx4,Xx6)
& append_succeeds(Xx5,Xx2,Xx6) )
| ( Xx1 = nil
& Xx3 = Xx2 ) ) ) ).
fof(id43,axiom,
! [Xx1,Xx2,Xx3] :
( append_fails(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5,Xx6] :
( Xx1 != cons(Xx4,Xx5)
| Xx3 != cons(Xx4,Xx6)
| append_fails(Xx5,Xx2,Xx6) )
& ( Xx1 != nil
| Xx3 != Xx2 ) ) ) ).
fof(id44,axiom,
! [Xx1,Xx2,Xx3] :
( append_terminates(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5,Xx6] :
( $true
& ( Xx1 != cons(Xx4,Xx5)
| ( $true
& ( Xx3 != cons(Xx4,Xx6)
| append_terminates(Xx5,Xx2,Xx6) ) ) ) )
& $true
& ( Xx1 != nil
| $true ) ) ) ).
fof(id45,axiom,
! [Xx1,Xx2] :
( member_succeeds(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx2 = cons(Xx3,Xx4)
& member_succeeds(Xx1,Xx4) )
| ? [Xx5] : Xx2 = cons(Xx1,Xx5) ) ) ).
fof(id46,axiom,
! [Xx1,Xx2] :
( member_fails(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx2 != cons(Xx3,Xx4)
| member_fails(Xx1,Xx4) )
& ! [Xx5] : Xx2 != cons(Xx1,Xx5) ) ) ).
fof(id47,axiom,
! [Xx1,Xx2] :
( member_terminates(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( $true
& ( Xx2 != cons(Xx3,Xx4)
| member_terminates(Xx1,Xx4) ) )
& ! [Xx5] : $true ) ) ).
fof(id48,axiom,
! [Xx1] :
( list_succeeds(Xx1)
<=> ( ? [Xx2,Xx3] :
( Xx1 = cons(Xx2,Xx3)
& list_succeeds(Xx3) )
| Xx1 = nil ) ) ).
fof(id49,axiom,
! [Xx1] :
( list_fails(Xx1)
<=> ( ! [Xx2,Xx3] :
( Xx1 != cons(Xx2,Xx3)
| list_fails(Xx3) )
& Xx1 != nil ) ) ).
fof(id50,axiom,
! [Xx1] :
( list_terminates(Xx1)
<=> ( ! [Xx2,Xx3] :
( $true
& ( Xx1 != cons(Xx2,Xx3)
| list_terminates(Xx3) ) )
& $true ) ) ).
fof(id51,axiom,
! [Xx1] :
( nat_list_succeeds(Xx1)
<=> ( ? [Xx2,Xx3] :
( Xx1 = cons(Xx2,Xx3)
& nat_succeeds(Xx2)
& nat_list_succeeds(Xx3) )
| Xx1 = nil ) ) ).
fof(id52,axiom,
! [Xx1] :
( nat_list_fails(Xx1)
<=> ( ! [Xx2,Xx3] :
( Xx1 != cons(Xx2,Xx3)
| nat_fails(Xx2)
| nat_list_fails(Xx3) )
& Xx1 != nil ) ) ).
fof(id53,axiom,
! [Xx1] :
( nat_list_terminates(Xx1)
<=> ( ! [Xx2,Xx3] :
( $true
& ( Xx1 != cons(Xx2,Xx3)
| ( nat_terminates(Xx2)
& ( nat_fails(Xx2)
| nat_list_terminates(Xx3) ) ) ) )
& $true ) ) ).
fof(id54,axiom,
! [Xx1,Xx2,Xx3] :
( times_succeeds(Xx1,Xx2,Xx3)
<=> ( ? [Xx4,Xx5] :
( Xx1 = s(Xx4)
& times_succeeds(Xx4,Xx2,Xx5)
& plus_succeeds(Xx2,Xx5,Xx3) )
| ( Xx1 = '0'
& Xx3 = '0' ) ) ) ).
fof(id55,axiom,
! [Xx1,Xx2,Xx3] :
( times_fails(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5] :
( Xx1 != s(Xx4)
| times_fails(Xx4,Xx2,Xx5)
| plus_fails(Xx2,Xx5,Xx3) )
& ( Xx1 != '0'
| Xx3 != '0' ) ) ) ).
fof(id56,axiom,
! [Xx1,Xx2,Xx3] :
( times_terminates(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5] :
( $true
& ( Xx1 != s(Xx4)
| ( times_terminates(Xx4,Xx2,Xx5)
& ( times_fails(Xx4,Xx2,Xx5)
| plus_terminates(Xx2,Xx5,Xx3) ) ) ) )
& $true
& ( Xx1 != '0'
| $true ) ) ) ).
fof(id57,axiom,
! [Xx1,Xx2,Xx3] :
( plus_succeeds(Xx1,Xx2,Xx3)
<=> ( ? [Xx4,Xx5] :
( Xx1 = s(Xx4)
& Xx3 = s(Xx5)
& plus_succeeds(Xx4,Xx2,Xx5) )
| ( Xx1 = '0'
& Xx3 = Xx2 ) ) ) ).
fof(id58,axiom,
! [Xx1,Xx2,Xx3] :
( plus_fails(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5] :
( Xx1 != s(Xx4)
| Xx3 != s(Xx5)
| plus_fails(Xx4,Xx2,Xx5) )
& ( Xx1 != '0'
| Xx3 != Xx2 ) ) ) ).
fof(id59,axiom,
! [Xx1,Xx2,Xx3] :
( plus_terminates(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5] :
( $true
& ( Xx1 != s(Xx4)
| ( $true
& ( Xx3 != s(Xx5)
| plus_terminates(Xx4,Xx2,Xx5) ) ) ) )
& $true
& ( Xx1 != '0'
| $true ) ) ) ).
fof(id60,axiom,
! [Xx1,Xx2] :
( '@=<_succeeds'(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx1 = s(Xx3)
& Xx2 = s(Xx4)
& '@=<_succeeds'(Xx3,Xx4) )
| Xx1 = '0' ) ) ).
fof(id61,axiom,
! [Xx1,Xx2] :
( '@=<_fails'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx1 != s(Xx3)
| Xx2 != s(Xx4)
| '@=<_fails'(Xx3,Xx4) )
& Xx1 != '0' ) ) ).
fof(id62,axiom,
! [Xx1,Xx2] :
( '@=<_terminates'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( $true
& ( Xx1 != s(Xx3)
| ( $true
& ( Xx2 != s(Xx4)
| '@=<_terminates'(Xx3,Xx4) ) ) ) )
& $true ) ) ).
fof(id63,axiom,
! [Xx1,Xx2] :
( '@<_succeeds'(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx1 = s(Xx3)
& Xx2 = s(Xx4)
& '@<_succeeds'(Xx3,Xx4) )
| ? [Xx5] :
( Xx1 = '0'
& Xx2 = s(Xx5) ) ) ) ).
fof(id64,axiom,
! [Xx1,Xx2] :
( '@<_fails'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx1 != s(Xx3)
| Xx2 != s(Xx4)
| '@<_fails'(Xx3,Xx4) )
& ! [Xx5] :
( Xx1 != '0'
| Xx2 != s(Xx5) ) ) ) ).
fof(id65,axiom,
! [Xx1,Xx2] :
( '@<_terminates'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( $true
& ( Xx1 != s(Xx3)
| ( $true
& ( Xx2 != s(Xx4)
| '@<_terminates'(Xx3,Xx4) ) ) ) )
& ! [Xx5] :
( $true
& ( Xx1 != '0'
| $true ) ) ) ) ).
fof(id66,axiom,
! [Xx1] :
( nat_succeeds(Xx1)
<=> ( ? [Xx2] :
( Xx1 = s(Xx2)
& nat_succeeds(Xx2) )
| Xx1 = '0' ) ) ).
fof(id67,axiom,
! [Xx1] :
( nat_fails(Xx1)
<=> ( ! [Xx2] :
( Xx1 != s(Xx2)
| nat_fails(Xx2) )
& Xx1 != '0' ) ) ).
fof(id68,axiom,
! [Xx1] :
( nat_terminates(Xx1)
<=> ( ! [Xx2] :
( $true
& ( Xx1 != s(Xx2)
| nat_terminates(Xx2) ) )
& $true ) ) ).
fof('axiom-(nat:termination)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> nat_terminates(Xx) ) ).
fof('axiom-(nat:ground)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> gr(Xx) ) ).
fof('axiom-(plus:termination:1)',axiom,
! [Xx,Xy,Xz] :
( nat_succeeds(Xx)
=> plus_terminates(Xx,Xy,Xz) ) ).
fof('axiom-(plus:termination:2)',axiom,
! [Xx,Xy,Xz] :
( nat_succeeds(Xz)
=> plus_terminates(Xx,Xy,Xz) ) ).
fof('axiom-(plus:types:1)',axiom,
! [Xx,Xy,Xz] :
( plus_succeeds(Xx,Xy,Xz)
=> nat_succeeds(Xx) ) ).
fof('axiom-(plus:types:2)',axiom,
! [Xx,Xy,Xz] :
( ( plus_succeeds(Xx,Xy,Xz)
& nat_succeeds(Xy) )
=> nat_succeeds(Xz) ) ).
fof('axiom-(plus:types:3)',axiom,
! [Xx,Xy,Xz] :
( ( plus_succeeds(Xx,Xy,Xz)
& nat_succeeds(Xz) )
=> nat_succeeds(Xy) ) ).
fof('axiom-(plus:termination:3)',axiom,
! [Xx,Xy,Xz] :
( plus_succeeds(Xx,Xy,Xz)
=> plus_terminates(Xx,Xy,Xz) ) ).
fof('axiom-(plus:ground:1)',axiom,
! [Xx,Xy,Xz] :
( plus_succeeds(Xx,Xy,Xz)
=> gr(Xx) ) ).
fof('axiom-(plus:ground:2)',axiom,
! [Xx,Xy,Xz] :
( ( plus_succeeds(Xx,Xy,Xz)
& gr(Xy) )
=> gr(Xz) ) ).
fof('axiom-(plus:ground:3)',axiom,
! [Xx,Xy,Xz] :
( ( plus_succeeds(Xx,Xy,Xz)
& gr(Xz) )
=> gr(Xy) ) ).
fof('axiom-(plus:existence)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> ? [Xz] : plus_succeeds(Xx,Xy,Xz) ) ).
fof('axiom-(plus:uniqueness)',axiom,
! [Xx,Xy,Xz1,Xz2] :
( ( plus_succeeds(Xx,Xy,Xz1)
& plus_succeeds(Xx,Xy,Xz2) )
=> Xz1 = Xz2 ) ).
fof('axiom-(plus:zero:1)',axiom,
! [Xy] : '@+'('0',Xy) = Xy ).
fof('axiom-(plus:successor:1)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@+'(s(Xx),Xy) = s('@+'(Xx,Xy)) ) ).
fof('axiom-(plus:types)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> nat_succeeds('@+'(Xx,Xy)) ) ).
fof('axiom-(plus:associative)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@+'('@+'(Xx,Xy),Xz) = '@+'(Xx,'@+'(Xy,Xz)) ) ).
fof('axiom-(plus:zero:2)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@+'(Xx,'0') = Xx ) ).
fof('axiom-(plus:successor:2)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@+'(Xx,s(Xy)) = '@+'(s(Xx),Xy) ) ).
fof('axiom-(plus:commutative)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@+'(Xx,Xy) = '@+'(Xy,Xx) ) ).
fof('axiom-(plus:injective:second)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@+'(Xx,Xy) = '@+'(Xx,Xz) )
=> Xy = Xz ) ).
fof('axiom-(times:types:1)',axiom,
! [Xx,Xy,Xz] :
( times_succeeds(Xx,Xy,Xz)
=> nat_succeeds(Xx) ) ).
fof('axiom-(times:types:2)',axiom,
! [Xx,Xy,Xz] :
( ( times_succeeds(Xx,Xy,Xz)
& nat_succeeds(Xy) )
=> nat_succeeds(Xz) ) ).
fof('axiom-(times:ground:1)',axiom,
! [Xx,Xy,Xz] :
( times_succeeds(Xx,Xy,Xz)
=> gr(Xx) ) ).
fof('axiom-(times:ground:2)',axiom,
! [Xx,Xy,Xz] :
( ( times_succeeds(Xx,Xy,Xz)
& gr(Xy) )
=> gr(Xz) ) ).
fof('axiom-(times:termination)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> times_terminates(Xx,Xy,Xz) ) ).
fof('axiom-(times:existence)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> ? [Xz] : times_succeeds(Xx,Xy,Xz) ) ).
fof('axiom-(times:uniqueness)',axiom,
! [Xx,Xy,Xz1,Xz2] :
( ( times_succeeds(Xx,Xy,Xz1)
& times_succeeds(Xx,Xy,Xz2) )
=> Xz1 = Xz2 ) ).
fof('axiom-(times:zero:1)',axiom,
! [Xy] :
( nat_succeeds(Xy)
=> '@*'('0',Xy) = '0' ) ).
fof('axiom-(times:successor:1)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@*'(s(Xx),Xy) = '@+'(Xy,'@*'(Xx,Xy)) ) ).
fof('axiom-(times:types)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> nat_succeeds('@*'(Xx,Xy)) ) ).
fof('axiom-(plus:times:distributive:1)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@*'('@+'(Xx,Xy),Xz) = '@+'('@*'(Xx,Xz),'@*'(Xy,Xz)) ) ).
fof('axiom-(times:associative)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@*'('@*'(Xx,Xy),Xz) = '@*'(Xx,'@*'(Xy,Xz)) ) ).
fof('axiom-(times:zero:2)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@*'(Xx,'0') = '0' ) ).
fof('axiom-(times:successor:2)',axiom,
! [Xy,Xx] :
( ( nat_succeeds(Xy)
& nat_succeeds(Xx) )
=> '@+'('@*'(Xy,Xx),Xy) = '@*'(Xy,s(Xx)) ) ).
fof('axiom-(times:commutative)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@*'(Xx,Xy) = '@*'(Xy,Xx) ) ).
fof('axiom-(times:one:1)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@*'(s('0'),Xx) = Xx ) ).
fof('axiom-(times:one:2)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@*'(Xx,s('0')) = Xx ) ).
fof('axiom-(plus:times:distributive:2)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@*'(Xz,'@+'(Xx,Xy)) = '@+'('@*'(Xz,Xx),'@*'(Xz,Xy)) ) ).
fof('axiom-(less:termination:1)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@<_terminates'(Xx,Xy) ) ).
fof('axiom-(less:termination:2)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xy)
=> '@<_terminates'(Xx,Xy) ) ).
fof('axiom-(less:types)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> nat_succeeds(Xx) ) ).
fof('axiom-(less:successor)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> ? [Xz] : Xy = s(Xz) ) ).
fof('axiom-(less:transitive:successor)',axiom,
! [Xx,Xy,Xz] :
( ( '@<_succeeds'(Xx,Xy)
& '@<_succeeds'(Xy,s(Xz)) )
=> '@<_succeeds'(Xx,Xz) ) ).
fof('axiom-(less:weakening)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> '@<_succeeds'(Xx,s(Xy)) ) ).
fof('axiom-(less:transitive)',axiom,
! [Xx,Xy,Xz] :
( ( '@<_succeeds'(Xx,Xy)
& '@<_succeeds'(Xy,Xz) )
=> '@<_succeeds'(Xx,Xz) ) ).
fof('axiom-(less:failure)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@<_fails'(Xx,Xx) ) ).
fof('axiom-(less:strictness)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> ~ '@<_succeeds'(Xx,Xx) ) ).
fof('axiom-(less:one)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@<_succeeds'(Xx,s(Xx)) ) ).
fof('axiom-(less:axiom:successor)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xy)
& '@<_succeeds'(Xx,s(Xy)) )
=> ( '@<_succeeds'(Xx,Xy)
| Xx = Xy ) ) ).
fof('axiom-(less:totality)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> ( '@<_succeeds'(Xx,Xy)
| Xx = Xy
| '@<_succeeds'(Xy,Xx) ) ) ).
fof('axiom-(less:different:zero)',axiom,
! [Xx] :
( ( nat_succeeds(Xx)
& Xx != '0' )
=> '@<_succeeds'('0',Xx) ) ).
fof('axiom-(leq:termination:1)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@=<_terminates'(Xx,Xy) ) ).
fof('axiom-(leq:types)',axiom,
! [Xx,Xy] :
( '@=<_succeeds'(Xx,Xy)
=> nat_succeeds(Xx) ) ).
fof('axiom-(leq:plus:1)',axiom,
! [Xx,Xy] :
( '@=<_succeeds'(Xx,Xy)
=> ? [Xz] : plus_succeeds(Xx,Xz,Xy) ) ).
fof('axiom-(leq:plus:2)',axiom,
! [Xx,Xy] :
( '@=<_succeeds'(Xx,Xy)
=> ? [Xz] : '@+'(Xx,Xz) = Xy ) ).
fof('axiom-(less:plus:1)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> ? [Xz] : plus_succeeds(Xx,s(Xz),Xy) ) ).
fof('axiom-(less:plus:2)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> ? [Xz] : '@+'(Xx,s(Xz)) = Xy ) ).
fof('axiom-(less:leq)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> '@=<_succeeds'(Xx,Xy) ) ).
fof('axiom-(leq:reflexive)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@=<_succeeds'(Xx,Xx) ) ).
fof('axiom-(leq:totality)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> ( '@=<_succeeds'(Xx,Xy)
| '@=<_succeeds'(Xy,Xx) ) ) ).
fof('axiom-(less:leq:total)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> ( '@<_succeeds'(Xx,Xy)
| '@=<_succeeds'(Xy,Xx) ) ) ).
fof('axiom-(leq:failure)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& '@=<_fails'(Xx,Xy) )
=> '@=<_succeeds'(Xy,Xx) ) ).
fof('axiom-(leq:less)',axiom,
! [Xx,Xy] :
( ( '@=<_succeeds'(Xx,Xy)
& nat_succeeds(Xy) )
=> ( '@<_succeeds'(Xx,Xy)
| Xx = Xy ) ) ).
fof('axiom-(leq:less:transitive)',axiom,
! [Xx,Xy,Xz] :
( ( '@=<_succeeds'(Xx,Xy)
& '@<_succeeds'(Xy,Xz) )
=> '@<_succeeds'(Xx,Xz) ) ).
fof('axiom-(less:leq:transitive)',axiom,
! [Xx,Xy,Xz] :
( ( '@<_succeeds'(Xx,Xy)
& '@=<_succeeds'(Xy,Xz) )
=> '@<_succeeds'(Xx,Xz) ) ).
fof('axiom-(leq:transitive)',axiom,
! [Xx,Xy,Xz] :
( ( '@=<_succeeds'(Xx,Xy)
& '@=<_succeeds'(Xy,Xz) )
=> '@=<_succeeds'(Xx,Xz) ) ).
fof('axiom-(leq:antisymmetric)',axiom,
! [Xx,Xy] :
( ( '@=<_succeeds'(Xx,Xy)
& '@=<_succeeds'(Xy,Xx) )
=> Xx = Xy ) ).
fof('axiom-(leq:one:success)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@=<_succeeds'(Xx,s(Xx)) ) ).
fof('axiom-(leq:one:failure)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@=<_fails'(s(Xx),Xx) ) ).
fof('axiom-(less:plus:second:1)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@<_succeeds'(Xy,Xz) )
=> '@<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) ) ).
fof('axiom-(less:plus:second:2)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@<_succeeds'(Xx,'@+'(Xx,s(Xy))) ) ).
fof('axiom-(less:plus:first:1)',axiom,
! [Xx,Xy,Xz] :
( ( '@<_succeeds'(Xx,Xy)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@<_succeeds'('@+'(Xx,Xz),'@+'(Xy,Xz)) ) ).
fof('axiom-(less:plus:first:2)',axiom,
! [Xx,Xy] :
( ( '@<_succeeds'('0',Xy)
& nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@<_succeeds'(Xx,'@+'(Xy,Xx)) ) ).
fof('axiom-(leq:plus:second:1)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@=<_succeeds'(Xy,Xz) )
=> '@=<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) ) ).
fof('axiom-(leq:plus:first:1)',axiom,
! [Xx,Xy,Xz] :
( ( '@=<_succeeds'(Xx,Xy)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@=<_succeeds'('@+'(Xx,Xz),'@+'(Xy,Xz)) ) ).
fof('axiom-(leq:plus:first:2)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@=<_succeeds'(Xx,'@+'(Xx,Xy)) ) ).
fof('axiom-(leq:plus:second:2)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@=<_succeeds'(Xy,'@+'(Xx,Xy)) ) ).
fof('axiom-(less:plus:inverse:1)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) )
=> '@<_succeeds'(Xy,Xz) ) ).
fof('axiom-(less:plus:inverse:2)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz)
& '@<_succeeds'('@+'(Xx,Xz),'@+'(Xy,Xz)) )
=> '@<_succeeds'(Xx,Xy) ) ).
fof('axiom-(leq:plus:inverse)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@=<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) )
=> '@=<_succeeds'(Xy,Xz) ) ).
fof('axiom-(plus:leq:leq)',axiom,
! [Xx1,Xx2,Xy1,Xy2] :
( ( '@=<_succeeds'(Xx1,Xy1)
& '@=<_succeeds'(Xx2,Xy2)
& nat_succeeds(Xy1) )
=> '@=<_succeeds'('@+'(Xx1,Xx2),'@+'(Xy1,Xy2)) ) ).
fof('axiom-(plus:less:leq)',axiom,
! [Xx1,Xx2,Xy1,Xy2] :
( ( '@<_succeeds'(Xx1,Xy1)
& '@=<_succeeds'(Xx2,Xy2)
& nat_succeeds(Xy1) )
=> '@<_succeeds'('@+'(Xx1,Xx2),'@+'(Xy1,Xy2)) ) ).
fof('axiom-(plus:leq:less)',axiom,
! [Xx1,Xx2,Xy1,Xy2] :
( ( '@=<_succeeds'(Xx1,Xy1)
& '@<_succeeds'(Xx2,Xy2)
& nat_succeeds(Xy1) )
=> '@<_succeeds'('@+'(Xx1,Xx2),'@+'(Xy1,Xy2)) ) ).
fof('axiom-(plus:less:less)',axiom,
! [Xx1,Xx2,Xy1,Xy2] :
( ( '@<_succeeds'(Xx1,Xy1)
& '@<_succeeds'(Xx2,Xy2)
& nat_succeeds(Xy1) )
=> '@<_succeeds'('@+'(Xx1,Xx2),'@+'(Xy1,Xy2)) ) ).
fof('axiom-(times:leq:second)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@=<_succeeds'(Xy,Xz)
& nat_succeeds(Xz) )
=> '@=<_succeeds'('@*'(Xx,Xy),'@*'(Xx,Xz)) ) ).
fof('axiom-(times:leq:first)',axiom,
! [Xx,Xy,Xz] :
( ( '@=<_succeeds'(Xx,Xy)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@=<_succeeds'('@*'(Xx,Xz),'@*'(Xy,Xz)) ) ).
fof('axiom-(times:less:second)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& Xx != '0'
& '@<_succeeds'(Xy,Xz)
& nat_succeeds(Xz) )
=> '@<_succeeds'('@*'(Xx,Xy),'@*'(Xx,Xz)) ) ).
fof('axiom-(leq:times:inverse)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz)
& '@=<_succeeds'('@*'(s(Xx),Xy),'@*'(s(Xx),Xz)) )
=> '@=<_succeeds'(Xy,Xz) ) ).
fof('axiom-(plus:injective:first)',axiom,
! [Xx1,Xx2,Xy] :
( ( nat_succeeds(Xx1)
& nat_succeeds(Xx2)
& nat_succeeds(Xy)
& '@+'(Xx1,Xy) = '@+'(Xx2,Xy) )
=> Xx1 = Xx2 ) ).
fof('(**)/2',axiom,
! [Xl1,Xl2,Xl3] :
( list_succeeds(Xl1)
=> ( '**'(Xl1,Xl2) = Xl3
<=> append_succeeds(Xl1,Xl2,Xl3) ) ) ).
fof('lh/1',axiom,
! [Xl,Xn] :
( list_succeeds(Xl)
=> ( lh(Xl) = Xn
<=> length_succeeds(Xl,Xn) ) ) ).
fof('sub/2',axiom,
! [Xl1,Xl2] :
( sub(Xl1,Xl2)
<=> ! [Xx] :
( member_succeeds(Xx,Xl1)
=> member_succeeds(Xx,Xl2) ) ) ).
fof('lemma-(list:1)',axiom,
! [Xx] : list_succeeds(cons(Xx,nil)) ).
fof('lemma-(list:2)',axiom,
! [Xx,Xy] : list_succeeds(cons(Xx,cons(Xy,nil))) ).
fof('lemma-(list:3)',axiom,
! [Xx,Xy,Xz] : list_succeeds(cons(Xx,cons(Xy,cons(Xz,nil)))) ).
fof('lemma-(list:cons)',axiom,
! [Xx,Xl] :
( list_succeeds(cons(Xx,Xl))
=> list_succeeds(Xl) ) ).
fof('lemma-(list:termination)',axiom,
! [Xl] :
( list_succeeds(Xl)
=> list_terminates(Xl) ) ).
fof('lemma-(member:termination)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> member_terminates(Xx,Xl) ) ).
fof('corollary-(member:termination)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> ( member_succeeds(Xx,Xl)
| member_fails(Xx,Xl) ) ) ).
fof('lemma-(member:ground)',axiom,
! [Xx,Xl] :
( ( member_succeeds(Xx,Xl)
& gr(Xl) )
=> gr(Xx) ) ).
fof('lemma-(member:cons)',axiom,
! [Xx,Xy,Xz,Xl] :
( ( member_succeeds(Xx,cons(Xy,Xl))
& Xx != Xy )
=> member_succeeds(Xx,Xl) ) ).
fof('lemma-(append:types:1)',axiom,
! [Xl1,Xl2,Xl3] :
( append_succeeds(Xl1,Xl2,Xl3)
=> list_succeeds(Xl1) ) ).
fof('lemma-(append:types:2)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl2) )
=> list_succeeds(Xl3) ) ).
fof('lemma-(append:types:3)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> list_succeeds(Xl2) ) ).
fof('lemma-(append:types:4)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> ( list_succeeds(Xl1)
& list_succeeds(Xl2) ) ) ).
fof('lemma-(append:termination:1)',axiom,
! [Xl1,Xl2,Xl3] :
( list_succeeds(Xl1)
=> append_terminates(Xl1,Xl2,Xl3) ) ).
fof('lemma-(append:termination:2)',axiom,
! [Xl1,Xl2,Xl3] :
( list_succeeds(Xl3)
=> append_terminates(Xl1,Xl2,Xl3) ) ).
fof('lemma-(append:ground:1)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& gr(Xl1)
& gr(Xl2) )
=> gr(Xl3) ) ).
fof('lemma-(append:ground:2)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& gr(Xl3) )
=> ( gr(Xl1)
& gr(Xl2) ) ) ).
fof('lemma-(append:existence)',axiom,
! [Xl1,Xl2] :
( list_succeeds(Xl1)
=> ? [Xl3] : append_succeeds(Xl1,Xl2,Xl3) ) ).
fof('lemma-(append:uniqueness)',axiom,
! [Xl1,Xl2,Xl3,Xl4] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& append_succeeds(Xl1,Xl2,Xl4) )
=> Xl3 = Xl4 ) ).
fof('corollary-(app:nil)',axiom,
! [Xl] : '**'(nil,Xl) = Xl ).
fof('corollary-(app:cons)',axiom,
! [Xx,Xl1,Xl2] :
( list_succeeds(Xl1)
=> '**'(cons(Xx,Xl1),Xl2) = cons(Xx,'**'(Xl1,Xl2)) ) ).
fof('corollary-(app:types:1)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> list_succeeds('**'(Xl1,Xl2)) ) ).
fof('corollary-(app:types:2)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds('**'(Xl1,Xl2)) )
=> list_succeeds(Xl2) ) ).
fof('corollary-(app:ground:1)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& gr(Xl1)
& gr(Xl2) )
=> gr('**'(Xl1,Xl2)) ) ).
fof('corollary-(app:ground:2)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& gr('**'(Xl1,Xl2)) )
=> ( gr(Xl1)
& gr(Xl2) ) ) ).
fof('theorem-(app:associative)',axiom,
! [Xl1,Xl2,Xl3] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> '**'('**'(Xl1,Xl2),Xl3) = '**'(Xl1,'**'(Xl2,Xl3)) ) ).
fof('lemma-(app:nil)',axiom,
! [Xl] :
( list_succeeds(Xl)
=> '**'(Xl,nil) = Xl ) ).
fof('lemma-(length:types)',axiom,
! [Xl,Xn] :
( length_succeeds(Xl,Xn)
=> ( list_succeeds(Xl)
& nat_succeeds(Xn) ) ) ).
fof('lemma-(length:termination)',axiom,
! [Xl,Xn] :
( list_succeeds(Xl)
=> length_terminates(Xl,Xn) ) ).
fof('lemma-(length:ground)',axiom,
! [Xl,Xn] :
( length_succeeds(Xl,Xn)
=> gr(Xn) ) ).
fof('lemma-(length:existence)',axiom,
! [Xl] :
( list_succeeds(Xl)
=> ? [Xn] : length_succeeds(Xl,Xn) ) ).
fof('lemma-(length:uniqueness)',axiom,
! [Xl,Xm,Xn] :
( ( length_succeeds(Xl,Xm)
& length_succeeds(Xl,Xn) )
=> Xm = Xn ) ).
fof('corollary-(lh:nil)',axiom,
lh(nil) = '0' ).
fof('corollary-(lh:cons)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> lh(cons(Xx,Xl)) = s(lh(Xl)) ) ).
fof('corollary-(lh:types)',axiom,
! [Xl] :
( list_succeeds(Xl)
=> nat_succeeds(lh(Xl)) ) ).
fof('corollary-(lh:zero)',axiom,
! [Xl] :
( ( list_succeeds(Xl)
& lh(Xl) = '0' )
=> Xl = nil ) ).
fof('corollary-(lh:successor)',axiom,
! [Xn,Xl1] :
( ( list_succeeds(Xl1)
& lh(Xl1) = s(Xn) )
=> ? [Xx,Xl2] : Xl1 = cons(Xx,Xl2) ) ).
fof('theorem-(app:lh)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> lh('**'(Xl1,Xl2)) = '@+'(lh(Xl1),lh(Xl2)) ) ).
fof('corollary-(app:lh:leq:first)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> '@=<_succeeds'(lh(Xl1),lh('**'(Xl1,Xl2))) ) ).
fof('corollary-(app:lh:leq:second)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> '@=<_succeeds'(lh(Xl2),lh('**'(Xl1,Xl2))) ) ).
fof('corollary-(lh:cons:leq)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> '@=<_succeeds'(lh(Xl),lh(cons(Xx,Xl))) ) ).
fof('corollary-(append:lh)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> '@+'(lh(Xl1),lh(Xl2)) = lh(Xl3) ) ).
fof('corollary-(append:lh:leq:first)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> '@=<_succeeds'(lh(Xl1),lh(Xl3)) ) ).
fof('corollary-(append:lh:leq:second)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> '@=<_succeeds'(lh(Xl2),lh(Xl3)) ) ).
fof('lemma-(sub:cons)',axiom,
! [Xx,Xi] : sub(Xi,cons(Xx,Xi)) ).
fof('lemma-(sub:reflexive)',axiom,
! [Xl] : sub(Xl,Xl) ).
fof('lemma-(sub:transitive)',axiom,
! [Xi,Xj,Xk] :
( ( sub(Xi,Xj)
& sub(Xj,Xk) )
=> sub(Xi,Xk) ) ).
fof('lemma-(sub:nil)',axiom,
! [Xl] : sub(nil,Xl) ).
fof('lemma-(sub:member)',axiom,
! [Xx,Xi,Xj] :
( ( sub(Xi,Xj)
& member_succeeds(Xx,Xj) )
=> sub(cons(Xx,Xi),Xj) ) ).
fof('lemma-(sub:cons:both)',axiom,
! [Xx,Xi,Xj] :
( sub(Xi,Xj)
=> sub(cons(Xx,Xi),cons(Xx,Xj)) ) ).
fof('lemma-(member:append)',axiom,
! [Xx,Xl3] :
( member_succeeds(Xx,Xl3)
=> ? [Xl1,Xl2] : append_succeeds(Xl1,cons(Xx,Xl2),Xl3) ) ).
fof('lemma-(append:member:1)',axiom,
! [Xx,Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& member_succeeds(Xx,Xl1) )
=> member_succeeds(Xx,Xl3) ) ).
fof('lemma-(append:member:2)',axiom,
! [Xx,Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& member_succeeds(Xx,Xl2) )
=> member_succeeds(Xx,Xl3) ) ).
fof('corollary-(app:member:1)',axiom,
! [Xx,Xl1,Xl2] :
( ( member_succeeds(Xx,Xl1)
& list_succeeds(Xl1) )
=> member_succeeds(Xx,'**'(Xl1,Xl2)) ) ).
fof('corollary-(app:member:2)',axiom,
! [Xx,Xl1,Xl2] :
( ( member_succeeds(Xx,Xl2)
& list_succeeds(Xl1) )
=> member_succeeds(Xx,'**'(Xl1,Xl2)) ) ).
fof('corollary-(append:member)',axiom,
! [Xx,Xl1,Xl2,Xl3] :
( append_succeeds(Xl1,cons(Xx,Xl2),Xl3)
=> member_succeeds(Xx,Xl3) ) ).
fof(induction,axiom,
( ! [Xl1,Xl2,Xl3] :
( ( ? [Xx4,Xx5,Xx6] :
( Xl1 = cons(Xx4,Xx5)
& Xl3 = cons(Xx4,Xx6)
& append_succeeds(Xx5,Xl2,Xx6)
& ! [Xx] :
( member_succeeds(Xx,Xx6)
=> ( member_succeeds(Xx,Xx5)
| member_succeeds(Xx,Xl2) ) ) )
| ( Xl1 = nil
& Xl3 = Xl2 ) )
=> ! [Xx] :
( member_succeeds(Xx,Xl3)
=> ( member_succeeds(Xx,Xl1)
| member_succeeds(Xx,Xl2) ) ) )
=> ! [Xl1,Xl2,Xl3] :
( append_succeeds(Xl1,Xl2,Xl3)
=> ! [Xx] :
( member_succeeds(Xx,Xl3)
=> ( member_succeeds(Xx,Xl1)
| member_succeeds(Xx,Xl2) ) ) ) ) ).
fof('lemma-(append:member:3)',conjecture,
! [Xx,Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& member_succeeds(Xx,Xl3) )
=> ( member_succeeds(Xx,Xl1)
| member_succeeds(Xx,Xl2) ) ) ).
%------------------------------------------------------------------------------