TPTP Problem File: SWX031+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWX031+1 : TPTP v9.1.0. Released v9.1.0.
% Domain : Software Verification
% Problem : A link between split/3, permutation/2, and concatenation
% Version : Especial.
% English :
% Refs : [MMP24] Mesnard et al. (2024), ATP for Prolog Verification
% Source : [Mes24] Mesnard (2024), Email to Geoff Sutcliffe
% Names : mergesort-all3 [Mes24]
% Status : Theorem
% Rating : 0.91 v9.1.0
% Syntax : Number of formulae : 367 ( 30 unt; 0 def)
% Number of atoms : 1263 ( 293 equ)
% Maximal formula atoms : 31 ( 3 avg)
% Number of connectives : 1056 ( 160 ~; 182 |; 393 &)
% ( 71 <=>; 250 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 73 ( 71 usr; 1 prp; 0-3 aty)
% Number of functors : 9 ( 9 usr; 2 con; 0-2 aty)
% Number of variables : 1067 ( 980 !; 87 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
fof(id1,axiom,
! [Xx3] : '0' != s(Xx3) ).
fof(id2,axiom,
'0' != nil ).
fof(id3,axiom,
! [Xx4,Xx5] : '0' != cons(Xx4,Xx5) ).
fof(id4,axiom,
! [Xx6,Xx7] :
( s(Xx6) = s(Xx7)
=> Xx6 = Xx7 ) ).
fof(id5,axiom,
! [Xx8] : nil != s(Xx8) ).
fof(id6,axiom,
! [Xx9,Xx10,Xx11] : s(Xx9) != cons(Xx10,Xx11) ).
fof(id7,axiom,
! [Xx12,Xx13] : nil != 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,
! [Xx22] :
( gr(Xx22)
<=> gr(s(Xx22)) ) ).
fof(id12,axiom,
gr(nil) ).
fof(id13,axiom,
! [Xx23,Xx24] :
( ( gr(Xx23)
& gr(Xx24) )
<=> gr(cons(Xx23,Xx24)) ) ).
fof(id14,axiom,
! [Xx25] :
~ ( int_ordered_succeeds(Xx25)
& int_ordered_fails(Xx25) ) ).
fof(id15,axiom,
! [Xx25] :
( int_ordered_terminates(Xx25)
=> ( int_ordered_succeeds(Xx25)
| int_ordered_fails(Xx25) ) ) ).
fof(id16,axiom,
! [Xx26] :
~ ( int_list_succeeds(Xx26)
& int_list_fails(Xx26) ) ).
fof(id17,axiom,
! [Xx26] :
( int_list_terminates(Xx26)
=> ( int_list_succeeds(Xx26)
| int_list_fails(Xx26) ) ) ).
fof(id18,axiom,
! [Xx27,Xx28,Xx29] :
~ ( merge_succeeds(Xx27,Xx28,Xx29)
& merge_fails(Xx27,Xx28,Xx29) ) ).
fof(id19,axiom,
! [Xx27,Xx28,Xx29] :
( merge_terminates(Xx27,Xx28,Xx29)
=> ( merge_succeeds(Xx27,Xx28,Xx29)
| merge_fails(Xx27,Xx28,Xx29) ) ) ).
fof(id20,axiom,
! [Xx30,Xx31,Xx32] :
~ ( split_succeeds(Xx30,Xx31,Xx32)
& split_fails(Xx30,Xx31,Xx32) ) ).
fof(id21,axiom,
! [Xx30,Xx31,Xx32] :
( split_terminates(Xx30,Xx31,Xx32)
=> ( split_succeeds(Xx30,Xx31,Xx32)
| split_fails(Xx30,Xx31,Xx32) ) ) ).
fof(id22,axiom,
! [Xx33,Xx34] :
~ ( mergesort_succeeds(Xx33,Xx34)
& mergesort_fails(Xx33,Xx34) ) ).
fof(id23,axiom,
! [Xx33,Xx34] :
( mergesort_terminates(Xx33,Xx34)
=> ( mergesort_succeeds(Xx33,Xx34)
| mergesort_fails(Xx33,Xx34) ) ) ).
fof(id24,axiom,
! [Xx35,Xx36,Xx37] :
~ ( member2_succeeds(Xx35,Xx36,Xx37)
& member2_fails(Xx35,Xx36,Xx37) ) ).
fof(id25,axiom,
! [Xx35,Xx36,Xx37] :
( member2_terminates(Xx35,Xx36,Xx37)
=> ( member2_succeeds(Xx35,Xx36,Xx37)
| member2_fails(Xx35,Xx36,Xx37) ) ) ).
fof(id26,axiom,
! [Xx38,Xx39,Xx40] :
~ ( occ_succeeds(Xx38,Xx39,Xx40)
& occ_fails(Xx38,Xx39,Xx40) ) ).
fof(id27,axiom,
! [Xx38,Xx39,Xx40] :
( occ_terminates(Xx38,Xx39,Xx40)
=> ( occ_succeeds(Xx38,Xx39,Xx40)
| occ_fails(Xx38,Xx39,Xx40) ) ) ).
fof(id28,axiom,
! [Xx41,Xx42] :
~ ( not_same_occ_succeeds(Xx41,Xx42)
& not_same_occ_fails(Xx41,Xx42) ) ).
fof(id29,axiom,
! [Xx41,Xx42] :
( not_same_occ_terminates(Xx41,Xx42)
=> ( not_same_occ_succeeds(Xx41,Xx42)
| not_same_occ_fails(Xx41,Xx42) ) ) ).
fof(id30,axiom,
! [Xx43,Xx44] :
~ ( same_occ_succeeds(Xx43,Xx44)
& same_occ_fails(Xx43,Xx44) ) ).
fof(id31,axiom,
! [Xx43,Xx44] :
( same_occ_terminates(Xx43,Xx44)
=> ( same_occ_succeeds(Xx43,Xx44)
| same_occ_fails(Xx43,Xx44) ) ) ).
fof(id32,axiom,
! [Xx45,Xx46] :
~ ( permutation_succeeds(Xx45,Xx46)
& permutation_fails(Xx45,Xx46) ) ).
fof(id33,axiom,
! [Xx45,Xx46] :
( permutation_terminates(Xx45,Xx46)
=> ( permutation_succeeds(Xx45,Xx46)
| permutation_fails(Xx45,Xx46) ) ) ).
fof(id34,axiom,
! [Xx47,Xx48,Xx49] :
~ ( delete_succeeds(Xx47,Xx48,Xx49)
& delete_fails(Xx47,Xx48,Xx49) ) ).
fof(id35,axiom,
! [Xx47,Xx48,Xx49] :
( delete_terminates(Xx47,Xx48,Xx49)
=> ( delete_succeeds(Xx47,Xx48,Xx49)
| delete_fails(Xx47,Xx48,Xx49) ) ) ).
fof(id36,axiom,
! [Xx50,Xx51] :
~ ( length_succeeds(Xx50,Xx51)
& length_fails(Xx50,Xx51) ) ).
fof(id37,axiom,
! [Xx50,Xx51] :
( length_terminates(Xx50,Xx51)
=> ( length_succeeds(Xx50,Xx51)
| length_fails(Xx50,Xx51) ) ) ).
fof(id38,axiom,
! [Xx52,Xx53,Xx54] :
~ ( append_succeeds(Xx52,Xx53,Xx54)
& append_fails(Xx52,Xx53,Xx54) ) ).
fof(id39,axiom,
! [Xx52,Xx53,Xx54] :
( append_terminates(Xx52,Xx53,Xx54)
=> ( append_succeeds(Xx52,Xx53,Xx54)
| append_fails(Xx52,Xx53,Xx54) ) ) ).
fof(id40,axiom,
! [Xx55,Xx56] :
~ ( member_succeeds(Xx55,Xx56)
& member_fails(Xx55,Xx56) ) ).
fof(id41,axiom,
! [Xx55,Xx56] :
( member_terminates(Xx55,Xx56)
=> ( member_succeeds(Xx55,Xx56)
| member_fails(Xx55,Xx56) ) ) ).
fof(id42,axiom,
! [Xx57] :
~ ( list_succeeds(Xx57)
& list_fails(Xx57) ) ).
fof(id43,axiom,
! [Xx57] :
( list_terminates(Xx57)
=> ( list_succeeds(Xx57)
| list_fails(Xx57) ) ) ).
fof(id44,axiom,
! [Xx58] :
~ ( nat_list_succeeds(Xx58)
& nat_list_fails(Xx58) ) ).
fof(id45,axiom,
! [Xx58] :
( nat_list_terminates(Xx58)
=> ( nat_list_succeeds(Xx58)
| nat_list_fails(Xx58) ) ) ).
fof(id46,axiom,
! [Xx59,Xx60,Xx61] :
~ ( times_succeeds(Xx59,Xx60,Xx61)
& times_fails(Xx59,Xx60,Xx61) ) ).
fof(id47,axiom,
! [Xx59,Xx60,Xx61] :
( times_terminates(Xx59,Xx60,Xx61)
=> ( times_succeeds(Xx59,Xx60,Xx61)
| times_fails(Xx59,Xx60,Xx61) ) ) ).
fof(id48,axiom,
! [Xx62,Xx63,Xx64] :
~ ( plus_succeeds(Xx62,Xx63,Xx64)
& plus_fails(Xx62,Xx63,Xx64) ) ).
fof(id49,axiom,
! [Xx62,Xx63,Xx64] :
( plus_terminates(Xx62,Xx63,Xx64)
=> ( plus_succeeds(Xx62,Xx63,Xx64)
| plus_fails(Xx62,Xx63,Xx64) ) ) ).
fof(id50,axiom,
! [Xx65,Xx66] :
~ ( '@=<_succeeds'(Xx65,Xx66)
& '@=<_fails'(Xx65,Xx66) ) ).
fof(id51,axiom,
! [Xx65,Xx66] :
( '@=<_terminates'(Xx65,Xx66)
=> ( '@=<_succeeds'(Xx65,Xx66)
| '@=<_fails'(Xx65,Xx66) ) ) ).
fof(id52,axiom,
! [Xx67,Xx68] :
~ ( '@<_succeeds'(Xx67,Xx68)
& '@<_fails'(Xx67,Xx68) ) ).
fof(id53,axiom,
! [Xx67,Xx68] :
( '@<_terminates'(Xx67,Xx68)
=> ( '@<_succeeds'(Xx67,Xx68)
| '@<_fails'(Xx67,Xx68) ) ) ).
fof(id54,axiom,
! [Xx69] :
~ ( nat_succeeds(Xx69)
& nat_fails(Xx69) ) ).
fof(id55,axiom,
! [Xx69] :
( nat_terminates(Xx69)
=> ( nat_succeeds(Xx69)
| nat_fails(Xx69) ) ) ).
fof(id56,axiom,
! [Xx70,Xx71] :
~ ( '=<_succeeds'(Xx70,Xx71)
& '=<_fails'(Xx70,Xx71) ) ).
fof(id57,axiom,
! [Xx70,Xx71] :
( '=<_terminates'(Xx70,Xx71)
=> ( '=<_succeeds'(Xx70,Xx71)
| '=<_fails'(Xx70,Xx71) ) ) ).
fof(id58,axiom,
! [Xx72] :
~ ( integer_succeeds(Xx72)
& integer_fails(Xx72) ) ).
fof(id59,axiom,
! [Xx72] :
( integer_terminates(Xx72)
=> ( integer_succeeds(Xx72)
| integer_fails(Xx72) ) ) ).
fof(id60,axiom,
! [Xx73,Xx74] :
~ ( '=<_succeeds'(Xx73,Xx74)
& '=<_fails'(Xx73,Xx74) ) ).
fof(id61,axiom,
! [Xx73,Xx74] :
( '=<_terminates'(Xx73,Xx74)
=> ( '=<_succeeds'(Xx73,Xx74)
| '=<_fails'(Xx73,Xx74) ) ) ).
fof(id62,axiom,
! [Xx75,Xx76] :
~ ( '=<_succeeds'(Xx75,Xx76)
& '=<_fails'(Xx75,Xx76) ) ).
fof(id63,axiom,
! [Xx75,Xx76] :
( '=<_terminates'(Xx75,Xx76)
=> ( '=<_succeeds'(Xx75,Xx76)
| '=<_fails'(Xx75,Xx76) ) ) ).
fof(id64,axiom,
! [Xx1] :
( int_ordered_succeeds(Xx1)
<=> ( ? [Xx2,Xx3,Xx4] :
( Xx1 = cons(Xx2,cons(Xx3,Xx4))
& '=<_succeeds'(Xx2,Xx3)
& int_ordered_succeeds(cons(Xx3,Xx4)) )
| ? [Xx5] : Xx1 = cons(Xx5,nil)
| Xx1 = nil ) ) ).
fof(id65,axiom,
! [Xx1] :
( int_ordered_fails(Xx1)
<=> ( ! [Xx2,Xx3,Xx4] :
( Xx1 != cons(Xx2,cons(Xx3,Xx4))
| '=<_fails'(Xx2,Xx3)
| int_ordered_fails(cons(Xx3,Xx4)) )
& ! [Xx5] : Xx1 != cons(Xx5,nil)
& Xx1 != nil ) ) ).
fof(id66,axiom,
! [Xx1] :
( int_ordered_terminates(Xx1)
<=> ( ! [Xx2,Xx3,Xx4] :
( $true
& ( Xx1 != cons(Xx2,cons(Xx3,Xx4))
| ( '=<_terminates'(Xx2,Xx3)
& ( '=<_fails'(Xx2,Xx3)
| int_ordered_terminates(cons(Xx3,Xx4)) ) ) ) )
& ! [Xx5] : $true
& $true ) ) ).
fof(id67,axiom,
! [Xx1] :
( int_list_succeeds(Xx1)
<=> ( ? [Xx2,Xx3] :
( Xx1 = cons(Xx2,Xx3)
& integer_succeeds(Xx2)
& int_list_succeeds(Xx3) )
| Xx1 = nil ) ) ).
fof(id68,axiom,
! [Xx1] :
( int_list_fails(Xx1)
<=> ( ! [Xx2,Xx3] :
( Xx1 != cons(Xx2,Xx3)
| integer_fails(Xx2)
| int_list_fails(Xx3) )
& Xx1 != nil ) ) ).
fof(id69,axiom,
! [Xx1] :
( int_list_terminates(Xx1)
<=> ( ! [Xx2,Xx3] :
( $true
& ( Xx1 != cons(Xx2,Xx3)
| ( integer_terminates(Xx2)
& ( integer_fails(Xx2)
| int_list_terminates(Xx3) ) ) ) )
& $true ) ) ).
fof(id70,axiom,
! [Xx1,Xx2,Xx3] :
( merge_succeeds(Xx1,Xx2,Xx3)
<=> ( ? [Xx4,Xx5,Xx6,Xx7,Xx8,Xx9] :
( Xx1 = cons(Xx4,Xx5)
& Xx2 = cons(Xx6,Xx7)
& Xx3 = cons(Xx8,Xx9)
& '=<_fails'(Xx4,Xx6)
& Xx8 = Xx6
& merge_succeeds(cons(Xx4,Xx5),Xx7,Xx9) )
| ? [Xx10,Xx11,Xx12,Xx13,Xx14,Xx15] :
( Xx1 = cons(Xx10,Xx11)
& Xx2 = cons(Xx12,Xx13)
& Xx3 = cons(Xx14,Xx15)
& '=<_succeeds'(Xx10,Xx12)
& Xx14 = Xx10
& merge_succeeds(Xx11,cons(Xx12,Xx13),Xx15) )
| ( Xx2 = nil
& Xx3 = Xx1 )
| ( Xx1 = nil
& Xx3 = Xx2 ) ) ) ).
fof(id71,axiom,
! [Xx1,Xx2,Xx3] :
( merge_fails(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5,Xx6,Xx7,Xx8,Xx9] :
( Xx1 != cons(Xx4,Xx5)
| Xx2 != cons(Xx6,Xx7)
| Xx3 != cons(Xx8,Xx9)
| '=<_succeeds'(Xx4,Xx6)
| Xx8 != Xx6
| merge_fails(cons(Xx4,Xx5),Xx7,Xx9) )
& ! [Xx10,Xx11,Xx12,Xx13,Xx14,Xx15] :
( Xx1 != cons(Xx10,Xx11)
| Xx2 != cons(Xx12,Xx13)
| Xx3 != cons(Xx14,Xx15)
| '=<_fails'(Xx10,Xx12)
| Xx14 != Xx10
| merge_fails(Xx11,cons(Xx12,Xx13),Xx15) )
& ( Xx2 != nil
| Xx3 != Xx1 )
& ( Xx1 != nil
| Xx3 != Xx2 ) ) ) ).
fof(id72,axiom,
! [Xx1,Xx2,Xx3] :
( merge_terminates(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5,Xx6,Xx7,Xx8,Xx9] :
( $true
& ( Xx1 != cons(Xx4,Xx5)
| ( $true
& ( Xx2 != cons(Xx6,Xx7)
| ( $true
& ( Xx3 != cons(Xx8,Xx9)
| ( '=<_terminates'(Xx4,Xx6)
& gr(Xx4)
& gr(Xx6)
& ( '=<_succeeds'(Xx4,Xx6)
| ( $true
& ( Xx8 != Xx6
| merge_terminates(cons(Xx4,Xx5),Xx7,Xx9) ) ) ) ) ) ) ) ) ) )
& ! [Xx10,Xx11,Xx12,Xx13,Xx14,Xx15] :
( $true
& ( Xx1 != cons(Xx10,Xx11)
| ( $true
& ( Xx2 != cons(Xx12,Xx13)
| ( $true
& ( Xx3 != cons(Xx14,Xx15)
| ( '=<_terminates'(Xx10,Xx12)
& ( '=<_fails'(Xx10,Xx12)
| ( $true
& ( Xx14 != Xx10
| merge_terminates(Xx11,cons(Xx12,Xx13),Xx15) ) ) ) ) ) ) ) ) ) )
& $true
& ( Xx2 != nil
| $true )
& $true
& ( Xx1 != nil
| $true ) ) ) ).
fof(id73,axiom,
! [Xx1,Xx2,Xx3] :
( split_succeeds(Xx1,Xx2,Xx3)
<=> ( ? [Xx4,Xx5,Xx6] :
( Xx1 = cons(Xx4,Xx5)
& Xx2 = cons(Xx4,Xx6)
& split_succeeds(Xx5,Xx3,Xx6) )
| ( Xx1 = nil
& Xx2 = nil
& Xx3 = nil ) ) ) ).
fof(id74,axiom,
! [Xx1,Xx2,Xx3] :
( split_fails(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5,Xx6] :
( Xx1 != cons(Xx4,Xx5)
| Xx2 != cons(Xx4,Xx6)
| split_fails(Xx5,Xx3,Xx6) )
& ( Xx1 != nil
| Xx2 != nil
| Xx3 != nil ) ) ) ).
fof(id75,axiom,
! [Xx1,Xx2,Xx3] :
( split_terminates(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5,Xx6] :
( $true
& ( Xx1 != cons(Xx4,Xx5)
| ( $true
& ( Xx2 != cons(Xx4,Xx6)
| split_terminates(Xx5,Xx3,Xx6) ) ) ) )
& $true
& ( Xx1 != nil
| ( $true
& ( Xx2 != nil
| $true ) ) ) ) ) ).
fof(id76,axiom,
! [Xx1,Xx2] :
( mergesort_succeeds(Xx1,Xx2)
<=> ( ? [Xx3,Xx4,Xx5,Xx6,Xx7,Xx8,Xx9] :
( Xx1 = cons(Xx3,cons(Xx4,Xx5))
& split_succeeds(cons(Xx3,cons(Xx4,Xx5)),Xx6,Xx7)
& mergesort_succeeds(Xx6,Xx8)
& mergesort_succeeds(Xx7,Xx9)
& merge_succeeds(Xx8,Xx9,Xx2) )
| ? [Xx10] :
( Xx1 = cons(Xx10,nil)
& Xx2 = cons(Xx10,nil) )
| ( Xx1 = nil
& Xx2 = nil ) ) ) ).
fof(id77,axiom,
! [Xx1,Xx2] :
( mergesort_fails(Xx1,Xx2)
<=> ( ! [Xx3,Xx4,Xx5,Xx6,Xx7,Xx8,Xx9] :
( Xx1 != cons(Xx3,cons(Xx4,Xx5))
| split_fails(cons(Xx3,cons(Xx4,Xx5)),Xx6,Xx7)
| mergesort_fails(Xx6,Xx8)
| mergesort_fails(Xx7,Xx9)
| merge_fails(Xx8,Xx9,Xx2) )
& ! [Xx10] :
( Xx1 != cons(Xx10,nil)
| Xx2 != cons(Xx10,nil) )
& ( Xx1 != nil
| Xx2 != nil ) ) ) ).
fof(id78,axiom,
! [Xx1,Xx2] :
( mergesort_terminates(Xx1,Xx2)
<=> ( ! [Xx3,Xx4,Xx5,Xx6,Xx7,Xx8,Xx9] :
( $true
& ( Xx1 != cons(Xx3,cons(Xx4,Xx5))
| ( split_terminates(cons(Xx3,cons(Xx4,Xx5)),Xx6,Xx7)
& ( split_fails(cons(Xx3,cons(Xx4,Xx5)),Xx6,Xx7)
| ( mergesort_terminates(Xx6,Xx8)
& ( mergesort_fails(Xx6,Xx8)
| ( mergesort_terminates(Xx7,Xx9)
& ( mergesort_fails(Xx7,Xx9)
| merge_terminates(Xx8,Xx9,Xx2) ) ) ) ) ) ) ) )
& ! [Xx10] :
( $true
& ( Xx1 != cons(Xx10,nil)
| $true ) )
& $true
& ( Xx1 != nil
| $true ) ) ) ).
fof(id79,axiom,
! [Xx1,Xx2,Xx3] :
( member2_succeeds(Xx1,Xx2,Xx3)
<=> ( member_succeeds(Xx1,Xx3)
| member_succeeds(Xx1,Xx2) ) ) ).
fof(id80,axiom,
! [Xx1,Xx2,Xx3] :
( member2_fails(Xx1,Xx2,Xx3)
<=> ( member_fails(Xx1,Xx3)
& member_fails(Xx1,Xx2) ) ) ).
fof(id81,axiom,
! [Xx1,Xx2,Xx3] :
( member2_terminates(Xx1,Xx2,Xx3)
<=> ( member_terminates(Xx1,Xx3)
& member_terminates(Xx1,Xx2) ) ) ).
fof(id82,axiom,
! [Xx1,Xx2,Xx3] :
( occ_succeeds(Xx1,Xx2,Xx3)
<=> ( ? [Xx4,Xx5] :
( Xx2 = cons(Xx4,Xx5)
& Xx1 != Xx4
& occ_succeeds(Xx1,Xx5,Xx3) )
| ? [Xx6,Xx7] :
( Xx2 = cons(Xx1,Xx6)
& Xx3 = s(Xx7)
& occ_succeeds(Xx1,Xx6,Xx7) )
| ( Xx2 = nil
& Xx3 = '0' ) ) ) ).
fof(id83,axiom,
! [Xx1,Xx2,Xx3] :
( occ_fails(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5] :
( Xx2 != cons(Xx4,Xx5)
| Xx1 = Xx4
| occ_fails(Xx1,Xx5,Xx3) )
& ! [Xx6,Xx7] :
( Xx2 != cons(Xx1,Xx6)
| Xx3 != s(Xx7)
| occ_fails(Xx1,Xx6,Xx7) )
& ( Xx2 != nil
| Xx3 != '0' ) ) ) ).
fof(id84,axiom,
! [Xx1,Xx2,Xx3] :
( occ_terminates(Xx1,Xx2,Xx3)
<=> ( ! [Xx4,Xx5] :
( $true
& ( Xx2 != cons(Xx4,Xx5)
| ( $true
& gr(Xx1)
& gr(Xx4)
& ( Xx1 = Xx4
| occ_terminates(Xx1,Xx5,Xx3) ) ) ) )
& ! [Xx6,Xx7] :
( $true
& ( Xx2 != cons(Xx1,Xx6)
| ( $true
& ( Xx3 != s(Xx7)
| occ_terminates(Xx1,Xx6,Xx7) ) ) ) )
& $true
& ( Xx2 != nil
| $true ) ) ) ).
fof(id85,axiom,
! [Xx1,Xx2] :
( not_same_occ_succeeds(Xx1,Xx2)
<=> ? [Xx3,Xx4,Xx5] :
( member2_succeeds(Xx3,Xx1,Xx2)
& occ_succeeds(Xx3,Xx1,Xx4)
& occ_succeeds(Xx3,Xx2,Xx5)
& Xx4 != Xx5 ) ) ).
fof(id86,axiom,
! [Xx1,Xx2] :
( not_same_occ_fails(Xx1,Xx2)
<=> ! [Xx3,Xx4,Xx5] :
( member2_fails(Xx3,Xx1,Xx2)
| occ_fails(Xx3,Xx1,Xx4)
| occ_fails(Xx3,Xx2,Xx5)
| Xx4 = Xx5 ) ) ).
fof(id87,axiom,
! [Xx1,Xx2] :
( not_same_occ_terminates(Xx1,Xx2)
<=> ! [Xx3,Xx4,Xx5] :
( member2_terminates(Xx3,Xx1,Xx2)
& ( member2_fails(Xx3,Xx1,Xx2)
| ( occ_terminates(Xx3,Xx1,Xx4)
& ( occ_fails(Xx3,Xx1,Xx4)
| ( occ_terminates(Xx3,Xx2,Xx5)
& ( occ_fails(Xx3,Xx2,Xx5)
| ( $true
& gr(Xx4)
& gr(Xx5) ) ) ) ) ) ) ) ) ).
fof(id88,axiom,
! [Xx1,Xx2] :
( same_occ_succeeds(Xx1,Xx2)
<=> not_same_occ_fails(Xx1,Xx2) ) ).
fof(id89,axiom,
! [Xx1,Xx2] :
( same_occ_fails(Xx1,Xx2)
<=> not_same_occ_succeeds(Xx1,Xx2) ) ).
fof(id90,axiom,
! [Xx1,Xx2] :
( same_occ_terminates(Xx1,Xx2)
<=> ( not_same_occ_terminates(Xx1,Xx2)
& gr(Xx1)
& gr(Xx2) ) ) ).
fof(id91,axiom,
! [Xx1,Xx2] :
( permutation_succeeds(Xx1,Xx2)
<=> ( ? [Xx3,Xx4,Xx5] :
( Xx2 = cons(Xx3,Xx4)
& delete_succeeds(Xx3,Xx1,Xx5)
& permutation_succeeds(Xx5,Xx4) )
| ( Xx1 = nil
& Xx2 = nil ) ) ) ).
fof(id92,axiom,
! [Xx1,Xx2] :
( permutation_fails(Xx1,Xx2)
<=> ( ! [Xx3,Xx4,Xx5] :
( Xx2 != cons(Xx3,Xx4)
| delete_fails(Xx3,Xx1,Xx5)
| permutation_fails(Xx5,Xx4) )
& ( Xx1 != nil
| Xx2 != nil ) ) ) ).
fof(id93,axiom,
! [Xx1,Xx2] :
( permutation_terminates(Xx1,Xx2)
<=> ( ! [Xx3,Xx4,Xx5] :
( $true
& ( Xx2 != cons(Xx3,Xx4)
| ( delete_terminates(Xx3,Xx1,Xx5)
& ( delete_fails(Xx3,Xx1,Xx5)
| permutation_terminates(Xx5,Xx4) ) ) ) )
& $true
& ( Xx1 != nil
| $true ) ) ) ).
fof(id94,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(id95,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(id96,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(id97,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(id98,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(id99,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(id100,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(id101,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(id102,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(id103,axiom,
! [Xx1,Xx2] :
( member_succeeds(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx2 = cons(Xx3,Xx4)
& member_succeeds(Xx1,Xx4) )
| ? [Xx5] : Xx2 = cons(Xx1,Xx5) ) ) ).
fof(id104,axiom,
! [Xx1,Xx2] :
( member_fails(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx2 != cons(Xx3,Xx4)
| member_fails(Xx1,Xx4) )
& ! [Xx5] : Xx2 != cons(Xx1,Xx5) ) ) ).
fof(id105,axiom,
! [Xx1,Xx2] :
( member_terminates(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( $true
& ( Xx2 != cons(Xx3,Xx4)
| member_terminates(Xx1,Xx4) ) )
& ! [Xx5] : $true ) ) ).
fof(id106,axiom,
! [Xx1] :
( list_succeeds(Xx1)
<=> ( ? [Xx2,Xx3] :
( Xx1 = cons(Xx2,Xx3)
& list_succeeds(Xx3) )
| Xx1 = nil ) ) ).
fof(id107,axiom,
! [Xx1] :
( list_fails(Xx1)
<=> ( ! [Xx2,Xx3] :
( Xx1 != cons(Xx2,Xx3)
| list_fails(Xx3) )
& Xx1 != nil ) ) ).
fof(id108,axiom,
! [Xx1] :
( list_terminates(Xx1)
<=> ( ! [Xx2,Xx3] :
( $true
& ( Xx1 != cons(Xx2,Xx3)
| list_terminates(Xx3) ) )
& $true ) ) ).
fof(id109,axiom,
! [Xx1] :
( nat_list_succeeds(Xx1)
<=> ( ? [Xx2,Xx3] :
( Xx1 = cons(Xx2,Xx3)
& nat_succeeds(Xx2)
& nat_list_succeeds(Xx3) )
| Xx1 = nil ) ) ).
fof(id110,axiom,
! [Xx1] :
( nat_list_fails(Xx1)
<=> ( ! [Xx2,Xx3] :
( Xx1 != cons(Xx2,Xx3)
| nat_fails(Xx2)
| nat_list_fails(Xx3) )
& Xx1 != nil ) ) ).
fof(id111,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(id112,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(id113,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(id114,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(id115,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(id116,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(id117,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(id118,axiom,
! [Xx1,Xx2] :
( '@=<_succeeds'(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx1 = s(Xx3)
& Xx2 = s(Xx4)
& '@=<_succeeds'(Xx3,Xx4) )
| Xx1 = '0' ) ) ).
fof(id119,axiom,
! [Xx1,Xx2] :
( '@=<_fails'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx1 != s(Xx3)
| Xx2 != s(Xx4)
| '@=<_fails'(Xx3,Xx4) )
& Xx1 != '0' ) ) ).
fof(id120,axiom,
! [Xx1,Xx2] :
( '@=<_terminates'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( $true
& ( Xx1 != s(Xx3)
| ( $true
& ( Xx2 != s(Xx4)
| '@=<_terminates'(Xx3,Xx4) ) ) ) )
& $true ) ) ).
fof(id121,axiom,
! [Xx1,Xx2] :
( '@<_succeeds'(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx1 = s(Xx3)
& Xx2 = s(Xx4)
& '@<_succeeds'(Xx3,Xx4) )
| ? [Xx5] :
( Xx1 = '0'
& Xx2 = s(Xx5) ) ) ) ).
fof(id122,axiom,
! [Xx1,Xx2] :
( '@<_fails'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx1 != s(Xx3)
| Xx2 != s(Xx4)
| '@<_fails'(Xx3,Xx4) )
& ! [Xx5] :
( Xx1 != '0'
| Xx2 != s(Xx5) ) ) ) ).
fof(id123,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(id124,axiom,
! [Xx1] :
( nat_succeeds(Xx1)
<=> ( ? [Xx2] :
( Xx1 = s(Xx2)
& nat_succeeds(Xx2) )
| Xx1 = '0' ) ) ).
fof(id125,axiom,
! [Xx1] :
( nat_fails(Xx1)
<=> ( ! [Xx2] :
( Xx1 != s(Xx2)
| nat_fails(Xx2) )
& Xx1 != '0' ) ) ).
fof(id126,axiom,
! [Xx1] :
( nat_terminates(Xx1)
<=> ( ! [Xx2] :
( $true
& ( Xx1 != s(Xx2)
| nat_terminates(Xx2) ) )
& $true ) ) ).
fof(id127,axiom,
! [Xx77,Xx78] : ~ '=<_succeeds'(Xx77,Xx78) ).
fof(id128,axiom,
! [Xx79,Xx80] : '=<_fails'(Xx79,Xx80) ).
fof(id129,axiom,
! [Xx81,Xx82] : '=<_terminates'(Xx81,Xx82) ).
fof(id130,axiom,
! [Xx83] : ~ integer_succeeds(Xx83) ).
fof(id131,axiom,
! [Xx84] : integer_fails(Xx84) ).
fof(id132,axiom,
! [Xx85] : integer_terminates(Xx85) ).
fof(id133,axiom,
! [Xx86,Xx87] : ~ '=<_succeeds'(Xx86,Xx87) ).
fof(id134,axiom,
! [Xx88,Xx89] : '=<_fails'(Xx88,Xx89) ).
fof(id135,axiom,
! [Xx90,Xx91] : '=<_terminates'(Xx90,Xx91) ).
fof(id136,axiom,
! [Xx92,Xx93] : ~ '=<_succeeds'(Xx92,Xx93) ).
fof(id137,axiom,
! [Xx94,Xx95] : '=<_fails'(Xx94,Xx95) ).
fof(id138,axiom,
! [Xx96,Xx97] : '=<_terminates'(Xx96,Xx97) ).
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)',axiom,
! [Xy] : '@+'('0',Xy) = Xy ).
fof('axiom-(plus:successor)',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)_001',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@+'(Xx,'0') = Xx ) ).
fof('axiom-(plus:successor)_002',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)',axiom,
! [Xy] :
( nat_succeeds(Xy)
=> '@*'('0',Xy) = '0' ) ).
fof('axiom-(times:successor)',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)',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)_003',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@*'(Xx,'0') = '0' ) ).
fof('axiom-(times:successor)_004',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)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@*'(s('0'),Xx) = Xx ) ).
fof('axiom-(times:one)_005',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@*'(Xx,s('0')) = Xx ) ).
fof('axiom-(plus:times:distributive)_006',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)',axiom,
! [Xx,Xy] :
( '@=<_succeeds'(Xx,Xy)
=> ? [Xz] : plus_succeeds(Xx,Xz,Xy) ) ).
fof('axiom-(leq:plus)_007',axiom,
! [Xx,Xy] :
( '@=<_succeeds'(Xx,Xy)
=> ? [Xz] : '@+'(Xx,Xz) = Xy ) ).
fof('axiom-(less:plus)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> ? [Xz] : plus_succeeds(Xx,s(Xz),Xy) ) ).
fof('axiom-(less:plus)_008',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)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@<_succeeds'(Xy,Xz) )
=> '@<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) ) ).
fof('axiom-(less:plus:second)_009',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@<_succeeds'(Xx,'@+'(Xx,s(Xy))) ) ).
fof('axiom-(less:plus:first)',axiom,
! [Xx,Xy,Xz] :
( ( '@<_succeeds'(Xx,Xy)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@<_succeeds'('@+'(Xx,Xz),'@+'(Xy,Xz)) ) ).
fof('axiom-(less:plus:first)_010',axiom,
! [Xx,Xy] :
( ( '@<_succeeds'('0',Xy)
& nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@<_succeeds'(Xx,'@+'(Xy,Xx)) ) ).
fof('axiom-(leq:plus:second)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@=<_succeeds'(Xy,Xz) )
=> '@=<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) ) ).
fof('axiom-(leq:plus:first)',axiom,
! [Xx,Xy,Xz] :
( ( '@=<_succeeds'(Xx,Xy)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@=<_succeeds'('@+'(Xx,Xz),'@+'(Xy,Xz)) ) ).
fof('axiom-(leq:plus:first)_011',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@=<_succeeds'(Xx,'@+'(Xx,Xy)) ) ).
fof('axiom-(leq:plus:second)_012',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@=<_succeeds'(Xy,'@+'(Xx,Xy)) ) ).
fof('axiom-(less:plus:inverse)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) )
=> '@<_succeeds'(Xy,Xz) ) ).
fof('axiom-(less:plus:inverse)_013',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('axiom-(list:1)',axiom,
! [Xx] : list_succeeds(cons(Xx,nil)) ).
fof('axiom-(list:2)',axiom,
! [Xx,Xy] : list_succeeds(cons(Xx,cons(Xy,nil))) ).
fof('axiom-(list:3)',axiom,
! [Xx,Xy,Xz] : list_succeeds(cons(Xx,cons(Xy,cons(Xz,nil)))) ).
fof('axiom-(list:cons)',axiom,
! [Xx,Xl] :
( list_succeeds(cons(Xx,Xl))
=> list_succeeds(Xl) ) ).
fof('axiom-(list:termination)',axiom,
! [Xl] :
( list_succeeds(Xl)
=> list_terminates(Xl) ) ).
fof('axiom-(member:termination)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> member_terminates(Xx,Xl) ) ).
fof('axiom-(member:termination)_014',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> ( member_succeeds(Xx,Xl)
| member_fails(Xx,Xl) ) ) ).
fof('axiom-(member:ground)',axiom,
! [Xx,Xl] :
( ( member_succeeds(Xx,Xl)
& gr(Xl) )
=> gr(Xx) ) ).
fof('axiom-(member:cons)',axiom,
! [Xx,Xy,Xz,Xl] :
( ( member_succeeds(Xx,cons(Xy,Xl))
& Xx != Xy )
=> member_succeeds(Xx,Xl) ) ).
fof('axiom-(append:types:1)',axiom,
! [Xl1,Xl2,Xl3] :
( append_succeeds(Xl1,Xl2,Xl3)
=> list_succeeds(Xl1) ) ).
fof('axiom-(append:types:2)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl2) )
=> list_succeeds(Xl3) ) ).
fof('axiom-(append:types:3)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> list_succeeds(Xl2) ) ).
fof('axiom-(append:types:4)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> ( list_succeeds(Xl1)
& list_succeeds(Xl2) ) ) ).
fof('axiom-(append:termination:1)',axiom,
! [Xl1,Xl2,Xl3] :
( list_succeeds(Xl1)
=> append_terminates(Xl1,Xl2,Xl3) ) ).
fof('axiom-(append:termination:2)',axiom,
! [Xl1,Xl2,Xl3] :
( list_succeeds(Xl3)
=> append_terminates(Xl1,Xl2,Xl3) ) ).
fof('axiom-(append:ground:1)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& gr(Xl1)
& gr(Xl2) )
=> gr(Xl3) ) ).
fof('axiom-(append:ground:2)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& gr(Xl3) )
=> ( gr(Xl1)
& gr(Xl2) ) ) ).
fof('axiom-(append:existence)',axiom,
! [Xl1,Xl2] :
( list_succeeds(Xl1)
=> ? [Xl3] : append_succeeds(Xl1,Xl2,Xl3) ) ).
fof('axiom-(append:uniqueness)',axiom,
! [Xl1,Xl2,Xl3,Xl4] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& append_succeeds(Xl1,Xl2,Xl4) )
=> Xl3 = Xl4 ) ).
fof('axiom-(app:nil)',axiom,
! [Xl] : '**'(nil,Xl) = Xl ).
fof('axiom-(app:cons)',axiom,
! [Xx,Xl1,Xl2] :
( list_succeeds(Xl1)
=> '**'(cons(Xx,Xl1),Xl2) = cons(Xx,'**'(Xl1,Xl2)) ) ).
fof('axiom-(app:types:1)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> list_succeeds('**'(Xl1,Xl2)) ) ).
fof('axiom-(app:types:2)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds('**'(Xl1,Xl2)) )
=> list_succeeds(Xl2) ) ).
fof('axiom-(app:ground:1)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& gr(Xl1)
& gr(Xl2) )
=> gr('**'(Xl1,Xl2)) ) ).
fof('axiom-(app:ground:2)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& gr('**'(Xl1,Xl2)) )
=> ( gr(Xl1)
& gr(Xl2) ) ) ).
fof('axiom-(app:associative)',axiom,
! [Xl1,Xl2,Xl3] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> '**'('**'(Xl1,Xl2),Xl3) = '**'(Xl1,'**'(Xl2,Xl3)) ) ).
fof('axiom-(app:nil)_015',axiom,
! [Xl] :
( list_succeeds(Xl)
=> '**'(Xl,nil) = Xl ) ).
fof('axiom-(length:types)',axiom,
! [Xl,Xn] :
( length_succeeds(Xl,Xn)
=> ( list_succeeds(Xl)
& nat_succeeds(Xn) ) ) ).
fof('axiom-(length:termination)',axiom,
! [Xl,Xn] :
( list_succeeds(Xl)
=> length_terminates(Xl,Xn) ) ).
fof('axiom-(length:ground)',axiom,
! [Xl,Xn] :
( length_succeeds(Xl,Xn)
=> gr(Xn) ) ).
fof('axiom-(length:existence)',axiom,
! [Xl] :
( list_succeeds(Xl)
=> ? [Xn] : length_succeeds(Xl,Xn) ) ).
fof('axiom-(length:uniqueness)',axiom,
! [Xl,Xm,Xn] :
( ( length_succeeds(Xl,Xm)
& length_succeeds(Xl,Xn) )
=> Xm = Xn ) ).
fof('axiom-(lh:nil)',axiom,
lh(nil) = '0' ).
fof('axiom-(lh:cons)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> lh(cons(Xx,Xl)) = s(lh(Xl)) ) ).
fof('axiom-(lh:types)',axiom,
! [Xl] :
( list_succeeds(Xl)
=> nat_succeeds(lh(Xl)) ) ).
fof('axiom-(lh:zero)',axiom,
! [Xl] :
( ( list_succeeds(Xl)
& lh(Xl) = '0' )
=> Xl = nil ) ).
fof('axiom-(lh:successor)',axiom,
! [Xn,Xl1] :
( ( list_succeeds(Xl1)
& lh(Xl1) = s(Xn) )
=> ? [Xx,Xl2] : Xl1 = cons(Xx,Xl2) ) ).
fof('axiom-(app:lh)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> lh('**'(Xl1,Xl2)) = '@+'(lh(Xl1),lh(Xl2)) ) ).
fof('axiom-(app:lh:leq:first)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> '@=<_succeeds'(lh(Xl1),lh('**'(Xl1,Xl2))) ) ).
fof('axiom-(app:lh:leq:second)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> '@=<_succeeds'(lh(Xl2),lh('**'(Xl1,Xl2))) ) ).
fof('axiom-(lh:cons:leq)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> '@=<_succeeds'(lh(Xl),lh(cons(Xx,Xl))) ) ).
fof('axiom-(append:lh)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> '@+'(lh(Xl1),lh(Xl2)) = lh(Xl3) ) ).
fof('axiom-(append:lh:leq:first)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> '@=<_succeeds'(lh(Xl1),lh(Xl3)) ) ).
fof('axiom-(append:lh:leq:second)',axiom,
! [Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> '@=<_succeeds'(lh(Xl2),lh(Xl3)) ) ).
fof('axiom-(sub:cons)',axiom,
! [Xx,Xi] : sub(Xi,cons(Xx,Xi)) ).
fof('axiom-(sub:reflexive)',axiom,
! [Xl] : sub(Xl,Xl) ).
fof('axiom-(sub:transitive)',axiom,
! [Xi,Xj,Xk] :
( ( sub(Xi,Xj)
& sub(Xj,Xk) )
=> sub(Xi,Xk) ) ).
fof('axiom-(sub:nil)',axiom,
! [Xl] : sub(nil,Xl) ).
fof('axiom-(sub:member)',axiom,
! [Xx,Xi,Xj] :
( ( sub(Xi,Xj)
& member_succeeds(Xx,Xj) )
=> sub(cons(Xx,Xi),Xj) ) ).
fof('axiom-(sub:cons:both)',axiom,
! [Xx,Xi,Xj] :
( sub(Xi,Xj)
=> sub(cons(Xx,Xi),cons(Xx,Xj)) ) ).
fof('axiom-(member:append)',axiom,
! [Xx,Xl3] :
( member_succeeds(Xx,Xl3)
=> ? [Xl1,Xl2] : append_succeeds(Xl1,cons(Xx,Xl2),Xl3) ) ).
fof('axiom-(append:member:1)',axiom,
! [Xx,Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& member_succeeds(Xx,Xl1) )
=> member_succeeds(Xx,Xl3) ) ).
fof('axiom-(append:member:2)',axiom,
! [Xx,Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& member_succeeds(Xx,Xl2) )
=> member_succeeds(Xx,Xl3) ) ).
fof('axiom-(app:member:1)',axiom,
! [Xx,Xl1,Xl2] :
( ( member_succeeds(Xx,Xl1)
& list_succeeds(Xl1) )
=> member_succeeds(Xx,'**'(Xl1,Xl2)) ) ).
fof('axiom-(app:member:2)',axiom,
! [Xx,Xl1,Xl2] :
( ( member_succeeds(Xx,Xl2)
& list_succeeds(Xl1) )
=> member_succeeds(Xx,'**'(Xl1,Xl2)) ) ).
fof('axiom-(append:member)',axiom,
! [Xx,Xl1,Xl2,Xl3] :
( append_succeeds(Xl1,cons(Xx,Xl2),Xl3)
=> member_succeeds(Xx,Xl3) ) ).
fof('axiom-(append:member:3)',axiom,
! [Xx,Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& member_succeeds(Xx,Xl3) )
=> ( member_succeeds(Xx,Xl1)
| member_succeeds(Xx,Xl2) ) ) ).
fof('axiom-(app:member:3)',axiom,
! [Xx,Xl1,Xl2] :
( ( list_succeeds(Xl1)
& member_succeeds(Xx,'**'(Xl1,Xl2)) )
=> ( member_succeeds(Xx,Xl1)
| member_succeeds(Xx,Xl2) ) ) ).
fof('axiom-(sub:app:1)',axiom,
! [Xl1,Xl2] :
( list_succeeds(Xl1)
=> sub(Xl1,'**'(Xl1,Xl2)) ) ).
fof('axiom-(sub:app:2)',axiom,
! [Xl1,Xl2] :
( list_succeeds(Xl1)
=> sub(Xl2,'**'(Xl1,Xl2)) ) ).
fof('axiom-(append:cons:different)',axiom,
! [Xx,Xl1,Xl2,Xl3] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& list_succeeds(Xl3) )
=> Xl2 != cons(Xx,Xl3) ) ).
fof('axiom-(append:equal:nil)',axiom,
! [Xl1,Xl2] :
( ( append_succeeds(Xl1,Xl2,Xl2)
& list_succeeds(Xl2) )
=> Xl1 = nil ) ).
fof('axiom-(append:uniqueness:1)',axiom,
! [Xl1,Xl2,Xl3,Xl4] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& append_succeeds(Xl4,Xl2,Xl3)
& list_succeeds(Xl3) )
=> Xl1 = Xl4 ) ).
fof('axiom-(app:uniqueness:1)',axiom,
! [Xl1,Xl2,Xl3] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& list_succeeds(Xl3)
& '**'(Xl1,Xl3) = '**'(Xl2,Xl3) )
=> Xl1 = Xl2 ) ).
fof('axiom-(append:uniqueness:2)',axiom,
! [Xl1,Xl2,Xl3,Xl4] :
( ( append_succeeds(Xl1,Xl2,Xl3)
& append_succeeds(Xl1,Xl4,Xl3) )
=> Xl2 = Xl4 ) ).
fof('axiom-(nat_list:list)',axiom,
! [Xl] :
( nat_list_succeeds(Xl)
=> list_succeeds(Xl) ) ).
fof('axiom-(nat_list:termination)',axiom,
! [Xl] :
( nat_list_succeeds(Xl)
=> nat_list_terminates(Xl) ) ).
fof('axiom-(nat_list:ground)',axiom,
! [Xx] :
( nat_list_succeeds(Xx)
=> gr(Xx) ) ).
fof('axiom-(lh:cons:first)',axiom,
! [Xx,Xl1,Xl2,Xn] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& '@<_succeeds'('@+'(lh(cons(Xx,Xl1)),lh(Xl2)),s(Xn)) )
=> '@<_succeeds'('@+'(lh(Xl1),lh(Xl2)),Xn) ) ).
fof('axiom-(lh:cons:second)',axiom,
! [Xl1,Xy,Xl2,Xn] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& '@<_succeeds'('@+'(lh(Xl1),lh(cons(Xy,Xl2))),s(Xn)) )
=> '@<_succeeds'('@+'(lh(Xl1),lh(Xl2)),Xn) ) ).
fof('axiom-(delete:termination:1)',axiom,
! [Xx,Xl1,Xl2] :
( list_succeeds(Xl1)
=> delete_terminates(Xx,Xl1,Xl2) ) ).
fof('axiom-(delete:termination:2)',axiom,
! [Xx,Xl1,Xl2] :
( list_succeeds(Xl2)
=> delete_terminates(Xx,Xl1,Xl2) ) ).
fof('axiom-(delete:types:1)',axiom,
! [Xx,Xl1,Xl2] :
( ( delete_succeeds(Xx,Xl1,Xl2)
& list_succeeds(Xl1) )
=> list_succeeds(Xl2) ) ).
fof('axiom-(delete:types:2)',axiom,
! [Xx,Xl1,Xl2] :
( ( delete_succeeds(Xx,Xl1,Xl2)
& list_succeeds(Xl2) )
=> list_succeeds(Xl1) ) ).
fof('axiom-(delete:length)',axiom,
! [Xx,Xl1,Xl2] :
( ( delete_succeeds(Xx,Xl1,Xl2)
& list_succeeds(Xl1) )
=> lh(Xl1) = s(lh(Xl2)) ) ).
fof('axiom-(delete:app:1)',axiom,
! [Xx,Xl1,Xl2] :
( list_succeeds(Xl1)
=> delete_succeeds(Xx,'**'(Xl1,cons(Xx,Xl2)),'**'(Xl1,Xl2)) ) ).
fof('axiom-(delete:app:2)',axiom,
! [Xx,Xl1,Xl2] :
( delete_succeeds(Xx,Xl1,Xl2)
=> ? [Xl3,Xl4] :
( list_succeeds(Xl3)
& Xl1 = '**'(Xl3,cons(Xx,Xl4))
& Xl2 = '**'(Xl3,Xl4) ) ) ).
fof('axiom-(delete:nat_list)',axiom,
! [Xx,Xl1,Xl2] :
( ( delete_succeeds(Xx,Xl1,Xl2)
& nat_list_succeeds(Xl1) )
=> ( nat_succeeds(Xx)
& nat_list_succeeds(Xl2) ) ) ).
fof('axiom-(delete:ground)',axiom,
! [Xx,Xl1,Xl2] :
( ( delete_succeeds(Xx,Xl1,Xl2)
& gr(Xl1) )
=> ( gr(Xx)
& gr(Xl2) ) ) ).
fof('axiom-(delete:member:1)',axiom,
! [Xx,Xy,Xl1,Xl2] :
( ( delete_succeeds(Xx,Xl1,Xl2)
& member_succeeds(Xy,Xl1) )
=> ( member_succeeds(Xy,Xl2)
| Xy = Xx ) ) ).
fof('axiom-(delete:member:2)',axiom,
! [Xx,Xl1,Xl2] :
( delete_succeeds(Xx,Xl1,Xl2)
=> member_succeeds(Xx,Xl1) ) ).
fof('axiom-(delete:member:3)',axiom,
! [Xx,Xy,Xl1,Xl2] :
( ( delete_succeeds(Xx,Xl1,Xl2)
& member_succeeds(Xy,Xl2) )
=> member_succeeds(Xy,Xl1) ) ).
fof('axiom-(delete:member:existence)',axiom,
! [Xx,Xl1] :
( member_succeeds(Xx,Xl1)
=> ? [Xl2] : delete_succeeds(Xx,Xl1,Xl2) ) ).
fof('axiom-(delete:member:different)',axiom,
! [Xx,Xy,Xl1,Xl2] :
( ( delete_succeeds(Xx,Xl1,Xl2)
& member_succeeds(Xy,Xl1)
& Xx != Xy )
=> member_succeeds(Xy,Xl2) ) ).
fof('axiom-(permutation:types)',axiom,
! [Xl1,Xl2] :
( permutation_succeeds(Xl1,Xl2)
=> ( list_succeeds(Xl1)
& list_succeeds(Xl2) ) ) ).
fof('axiom-(permutation:termination)',axiom,
! [Xn,Xl1,Xl2] :
( ( nat_succeeds(Xn)
& list_succeeds(Xl1)
& lh(Xl1) = Xn )
=> permutation_terminates(Xl1,Xl2) ) ).
fof('axiom-(permutation:termination)_016',axiom,
! [Xl1,Xl2] :
( list_succeeds(Xl1)
=> permutation_terminates(Xl1,Xl2) ) ).
fof('axiom-(member2:termination)',axiom,
! [Xx,Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> member2_terminates(Xx,Xl1,Xl2) ) ).
fof('axiom-(occ:termination)',axiom,
! [Xx,Xl,Xn] :
( ( list_succeeds(Xl)
& gr(Xl)
& gr(Xx) )
=> occ_terminates(Xx,Xl,Xn) ) ).
fof('axiom-(member2:ground)',axiom,
! [Xx,Xl1,Xl2] :
( ( member2_succeeds(Xx,Xl1,Xl2)
& gr(Xl1)
& gr(Xl2) )
=> gr(Xx) ) ).
fof('axiom-(occ:ground)',axiom,
! [Xx,Xl,Xn] :
( occ_succeeds(Xx,Xl,Xn)
=> gr(Xn) ) ).
fof('axiom-(not_same_occ:termination)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& gr(Xl1)
& gr(Xl2) )
=> not_same_occ_terminates(Xl1,Xl2) ) ).
fof('axiom-(same_occ:termination)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& gr(Xl1)
& gr(Xl2) )
=> same_occ_terminates(Xl1,Xl2) ) ).
fof('axiom-(occ:types)',axiom,
! [Xx,Xl,Xn] :
( occ_succeeds(Xx,Xl,Xn)
=> ( list_succeeds(Xl)
& nat_succeeds(Xn) ) ) ).
fof('axiom-(occ:existence)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> ? [Xn] : occ_succeeds(Xx,Xl,Xn) ) ).
fof('axiom-(occ:uniqueness)',axiom,
! [Xx,Xl,Xm,Xn] :
( ( occ_succeeds(Xx,Xl,Xm)
& occ_succeeds(Xx,Xl,Xn) )
=> Xm = Xn ) ).
fof('axiom-(occ:nil)',axiom,
! [Xx] : occ(Xx,nil) = '0' ).
fof('axiom-(occ:cons:diff)',axiom,
! [Xx,Xy,Xl] :
( ( list_succeeds(Xl)
& Xx != Xy )
=> occ(Xx,cons(Xy,Xl)) = occ(Xx,Xl) ) ).
fof('axiom-(occ:cons:eq)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> occ(Xx,cons(Xx,Xl)) = s(occ(Xx,Xl)) ) ).
fof('axiom-(occ:types)_017',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> nat_succeeds(occ(Xx,Xl)) ) ).
fof('axiom-(occ:app)',axiom,
! [Xx,Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> occ(Xx,'**'(Xl1,Xl2)) = '@+'(occ(Xx,Xl1),occ(Xx,Xl2)) ) ).
fof('axiom-(delete:occ:diff)',axiom,
! [Xx,Xy,Xl1,Xl2] :
( ( list_succeeds(Xl1)
& delete_succeeds(Xx,Xl1,Xl2)
& Xx != Xy )
=> occ(Xy,Xl1) = occ(Xy,Xl2) ) ).
fof('axiom-(delete:occ:eq)',axiom,
! [Xx,Xl1,Xl2] :
( ( list_succeeds(Xl1)
& delete_succeeds(Xx,Xl1,Xl2) )
=> occ(Xx,Xl1) = s(occ(Xx,Xl2)) ) ).
fof('axiom-(permutation:occ)',axiom,
! [Xl1,Xl2] :
( permutation_succeeds(Xl1,Xl2)
=> ! [Xx] : occ(Xx,Xl1) = occ(Xx,Xl2) ) ).
fof('axiom-(permutation:soundness)',axiom,
! [Xl1,Xl2] :
( ( permutation_succeeds(Xl1,Xl2)
& gr(Xl1)
& gr(Xl2) )
=> same_occ_succeeds(Xl1,Xl2) ) ).
fof('axiom-(occ:zero)',axiom,
! [Xl] :
( ( list_succeeds(Xl)
& ! [Xx] : occ(Xx,Xl) = '0' )
=> Xl = nil ) ).
fof('axiom-(occ:successor)',axiom,
! [Xx,Xl1,Xn] :
( ( list_succeeds(Xl1)
& occ(Xx,Xl1) = s(Xn) )
=> ? [Xl2] : delete_succeeds(Xx,Xl1,Xl2) ) ).
fof('axiom-(permutation:completeness)',axiom,
! [Xl2] :
( list_succeeds(Xl2)
=> ! [Xl1] :
( ( list_succeeds(Xl1)
& ! [Xx] : occ(Xx,Xl1) = occ(Xx,Xl2) )
=> permutation_succeeds(Xl1,Xl2) ) ) ).
fof('axiom-(occ:permutation)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl2)
& list_succeeds(Xl1)
& ! [Xx] : occ(Xx,Xl1) = occ(Xx,Xl2) )
=> permutation_succeeds(Xl1,Xl2) ) ).
fof('axiom-(occ:member)',axiom,
! [Xx,Xl] :
( ( list_succeeds(Xl)
& member_fails(Xx,Xl) )
=> occ(Xx,Xl) = '0' ) ).
fof('axiom-(same_occ:success)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& same_occ_succeeds(Xl1,Xl2) )
=> ! [Xx] : occ(Xx,Xl1) = occ(Xx,Xl2) ) ).
fof('axiom-(permutation:completeness)_018',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& same_occ_succeeds(Xl1,Xl2) )
=> permutation_succeeds(Xl1,Xl2) ) ).
fof('axiom-(permutation:reflexive)',axiom,
! [Xl] :
( list_succeeds(Xl)
=> permutation_succeeds(Xl,Xl) ) ).
fof('axiom-(permutation:symmetric)',axiom,
! [Xl1,Xl2] :
( permutation_succeeds(Xl1,Xl2)
=> permutation_succeeds(Xl2,Xl1) ) ).
fof('axiom-(permutation:transitive)',axiom,
! [Xl1,Xl2,Xl3] :
( ( permutation_succeeds(Xl1,Xl2)
& permutation_succeeds(Xl2,Xl3) )
=> permutation_succeeds(Xl1,Xl3) ) ).
fof('axiom-(permutation:app)',axiom,
! [Xl1,Xl2,Xl3,Xl4] :
( ( permutation_succeeds(Xl1,Xl3)
& permutation_succeeds(Xl2,Xl4) )
=> permutation_succeeds('**'(Xl1,Xl2),'**'(Xl3,Xl4)) ) ).
fof('axiom-(permutation:app:commutative)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> permutation_succeeds('**'(Xl1,Xl2),'**'(Xl2,Xl1)) ) ).
fof('axiom-(permutation:nat_list)',axiom,
! [Xl1,Xl2] :
( ( permutation_succeeds(Xl1,Xl2)
& nat_list_succeeds(Xl1) )
=> nat_list_succeeds(Xl2) ) ).
fof('axiom-(occ:member:success)',axiom,
! [Xx,Xl,Xn] :
( ( list_succeeds(Xl)
& occ_succeeds(Xx,Xl,s(Xn)) )
=> member_succeeds(Xx,Xl) ) ).
fof('axiom-(occ:member:success)_019',axiom,
! [Xx,Xl,Xn] :
( ( list_succeeds(Xl)
& occ(Xx,Xl) = s(Xn) )
=> member_succeeds(Xx,Xl) ) ).
fof('axiom-(member:occ:success)',axiom,
! [Xx,Xl] :
( ( list_succeeds(Xl)
& member_succeeds(Xx,Xl) )
=> ? [Xn] : occ(Xx,Xl) = s(Xn) ) ).
fof('axiom-(permutation:member)',axiom,
! [Xx,Xl1,Xl2] :
( ( permutation_succeeds(Xl1,Xl2)
& member_succeeds(Xx,Xl1) )
=> member_succeeds(Xx,Xl2) ) ).
fof('axiom-(permutation:cons)',axiom,
! [Xx,Xl1,Xl2] :
( permutation_succeeds(cons(Xx,Xl1),cons(Xx,Xl2))
=> permutation_succeeds(Xl1,Xl2) ) ).
fof('axiom-(permutation:nil)',axiom,
! [Xl] :
( permutation_succeeds(nil,Xl)
=> Xl = nil ) ).
fof('axiom-(permutation:ground)',axiom,
! [Xl1,Xl2] :
( ( permutation_succeeds(Xl1,Xl2)
& gr(Xl1) )
=> gr(Xl2) ) ).
fof('axiom-(permutation:length)',axiom,
! [Xl1,Xl2] :
( permutation_succeeds(Xl1,Xl2)
=> lh(Xl1) = lh(Xl2) ) ).
fof('axiom-(leq:termination)',axiom,
! [Xx,Xy] :
( ( integer_succeeds(Xx)
& integer_succeeds(Xy) )
=> '=<_terminates'(Xx,Xy) ) ).
fof('axiom-(integer:gr)',axiom,
! [Xx] :
( integer_succeeds(Xx)
=> gr(Xx) ) ).
fof('axiom-(leq:total)',axiom,
! [Xx,Xy] :
( ( integer_succeeds(Xx)
& integer_succeeds(Xy)
& '=<_fails'(Xx,Xy) )
=> '=<_succeeds'(Xy,Xx) ) ).
fof('(@+)/2',axiom,
! [Xx,Xy,Xz] :
( nat_succeeds(Xx)
=> ( '@+'(Xx,Xy) = Xz
<=> plus_succeeds(Xx,Xy,Xz) ) ) ).
fof('(@*)/2',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> ( '@*'(Xx,Xy) = Xz
<=> times_succeeds(Xx,Xy,Xz) ) ) ).
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('occ/2',axiom,
! [Xx,Xl,Xm] :
( list_succeeds(Xl)
=> ( occ(Xx,Xl) = Xm
<=> occ_succeeds(Xx,Xl,Xm) ) ) ).
fof('lemma-(int_list:list)',axiom,
! [Xl] :
( int_list_succeeds(Xl)
=> list_succeeds(Xl) ) ).
fof('lemma-(split:types)',axiom,
! [Xl1,Xl2,Xl3] :
( split_succeeds(Xl1,Xl2,Xl3)
=> ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& list_succeeds(Xl3) ) ) ).
fof(induction,axiom,
( ! [Xl1,Xl2,Xl3] :
( ( ? [Xx4,Xx5,Xx6] :
( Xl1 = cons(Xx4,Xx5)
& Xl2 = cons(Xx4,Xx6)
& split_succeeds(Xx5,Xl3,Xx6)
& lh(Xx5) = '@+'(lh(Xl3),lh(Xx6)) )
| ( Xl1 = nil
& Xl2 = nil
& Xl3 = nil ) )
=> lh(Xl1) = '@+'(lh(Xl2),lh(Xl3)) )
=> ! [Xl1,Xl2,Xl3] :
( split_succeeds(Xl1,Xl2,Xl3)
=> lh(Xl1) = '@+'(lh(Xl2),lh(Xl3)) ) ) ).
fof('lemma-(split:length)',conjecture,
! [Xl1,Xl2,Xl3] :
( split_succeeds(Xl1,Xl2,Xl3)
=> lh(Xl1) = '@+'(lh(Xl2),lh(Xl3)) ) ).
%------------------------------------------------------------------------------