TPTP Problem File: SWX054+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SWX054+1 : TPTP v9.1.0. Released v9.1.0.
% Domain : Software Verification
% Problem : If occ is invariant, permutation/2 succeeds
% Version : Especial.
% English :
% Refs : [MMP24] Mesnard et al. (2024), ATP for Prolog Verification
% Source : [Mes24] Mesnard (2024), Email to Geoff Sutcliffe
% Names : permutation-all24 [Mes24]
% Status : Theorem
% Rating : 1.00 v9.1.0
% Syntax : Number of formulae : 298 ( 18 unt; 0 def)
% Number of atoms : 977 ( 209 equ)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 780 ( 101 ~; 112 |; 291 &)
% ( 56 <=>; 220 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 52 ( 50 usr; 1 prp; 0-3 aty)
% Number of functors : 9 ( 9 usr; 2 con; 0-2 aty)
% Number of variables : 840 ( 784 !; 56 ?)
% 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] :
~ ( member2_succeeds(Xx25,Xx26,Xx27)
& member2_fails(Xx25,Xx26,Xx27) ) ).
fof(id15,axiom,
! [Xx25,Xx26,Xx27] :
( member2_terminates(Xx25,Xx26,Xx27)
=> ( member2_succeeds(Xx25,Xx26,Xx27)
| member2_fails(Xx25,Xx26,Xx27) ) ) ).
fof(id16,axiom,
! [Xx28,Xx29,Xx30] :
~ ( occ_succeeds(Xx28,Xx29,Xx30)
& occ_fails(Xx28,Xx29,Xx30) ) ).
fof(id17,axiom,
! [Xx28,Xx29,Xx30] :
( occ_terminates(Xx28,Xx29,Xx30)
=> ( occ_succeeds(Xx28,Xx29,Xx30)
| occ_fails(Xx28,Xx29,Xx30) ) ) ).
fof(id18,axiom,
! [Xx31,Xx32] :
~ ( not_same_occ_succeeds(Xx31,Xx32)
& not_same_occ_fails(Xx31,Xx32) ) ).
fof(id19,axiom,
! [Xx31,Xx32] :
( not_same_occ_terminates(Xx31,Xx32)
=> ( not_same_occ_succeeds(Xx31,Xx32)
| not_same_occ_fails(Xx31,Xx32) ) ) ).
fof(id20,axiom,
! [Xx33,Xx34] :
~ ( same_occ_succeeds(Xx33,Xx34)
& same_occ_fails(Xx33,Xx34) ) ).
fof(id21,axiom,
! [Xx33,Xx34] :
( same_occ_terminates(Xx33,Xx34)
=> ( same_occ_succeeds(Xx33,Xx34)
| same_occ_fails(Xx33,Xx34) ) ) ).
fof(id22,axiom,
! [Xx35,Xx36] :
~ ( permutation_succeeds(Xx35,Xx36)
& permutation_fails(Xx35,Xx36) ) ).
fof(id23,axiom,
! [Xx35,Xx36] :
( permutation_terminates(Xx35,Xx36)
=> ( permutation_succeeds(Xx35,Xx36)
| permutation_fails(Xx35,Xx36) ) ) ).
fof(id24,axiom,
! [Xx37,Xx38,Xx39] :
~ ( delete_succeeds(Xx37,Xx38,Xx39)
& delete_fails(Xx37,Xx38,Xx39) ) ).
fof(id25,axiom,
! [Xx37,Xx38,Xx39] :
( delete_terminates(Xx37,Xx38,Xx39)
=> ( delete_succeeds(Xx37,Xx38,Xx39)
| delete_fails(Xx37,Xx38,Xx39) ) ) ).
fof(id26,axiom,
! [Xx40,Xx41] :
~ ( length_succeeds(Xx40,Xx41)
& length_fails(Xx40,Xx41) ) ).
fof(id27,axiom,
! [Xx40,Xx41] :
( length_terminates(Xx40,Xx41)
=> ( length_succeeds(Xx40,Xx41)
| length_fails(Xx40,Xx41) ) ) ).
fof(id28,axiom,
! [Xx42,Xx43,Xx44] :
~ ( append_succeeds(Xx42,Xx43,Xx44)
& append_fails(Xx42,Xx43,Xx44) ) ).
fof(id29,axiom,
! [Xx42,Xx43,Xx44] :
( append_terminates(Xx42,Xx43,Xx44)
=> ( append_succeeds(Xx42,Xx43,Xx44)
| append_fails(Xx42,Xx43,Xx44) ) ) ).
fof(id30,axiom,
! [Xx45,Xx46] :
~ ( member_succeeds(Xx45,Xx46)
& member_fails(Xx45,Xx46) ) ).
fof(id31,axiom,
! [Xx45,Xx46] :
( member_terminates(Xx45,Xx46)
=> ( member_succeeds(Xx45,Xx46)
| member_fails(Xx45,Xx46) ) ) ).
fof(id32,axiom,
! [Xx47] :
~ ( list_succeeds(Xx47)
& list_fails(Xx47) ) ).
fof(id33,axiom,
! [Xx47] :
( list_terminates(Xx47)
=> ( list_succeeds(Xx47)
| list_fails(Xx47) ) ) ).
fof(id34,axiom,
! [Xx48] :
~ ( nat_list_succeeds(Xx48)
& nat_list_fails(Xx48) ) ).
fof(id35,axiom,
! [Xx48] :
( nat_list_terminates(Xx48)
=> ( nat_list_succeeds(Xx48)
| nat_list_fails(Xx48) ) ) ).
fof(id36,axiom,
! [Xx49,Xx50,Xx51] :
~ ( times_succeeds(Xx49,Xx50,Xx51)
& times_fails(Xx49,Xx50,Xx51) ) ).
fof(id37,axiom,
! [Xx49,Xx50,Xx51] :
( times_terminates(Xx49,Xx50,Xx51)
=> ( times_succeeds(Xx49,Xx50,Xx51)
| times_fails(Xx49,Xx50,Xx51) ) ) ).
fof(id38,axiom,
! [Xx52,Xx53,Xx54] :
~ ( plus_succeeds(Xx52,Xx53,Xx54)
& plus_fails(Xx52,Xx53,Xx54) ) ).
fof(id39,axiom,
! [Xx52,Xx53,Xx54] :
( plus_terminates(Xx52,Xx53,Xx54)
=> ( plus_succeeds(Xx52,Xx53,Xx54)
| plus_fails(Xx52,Xx53,Xx54) ) ) ).
fof(id40,axiom,
! [Xx55,Xx56] :
~ ( '@=<_succeeds'(Xx55,Xx56)
& '@=<_fails'(Xx55,Xx56) ) ).
fof(id41,axiom,
! [Xx55,Xx56] :
( '@=<_terminates'(Xx55,Xx56)
=> ( '@=<_succeeds'(Xx55,Xx56)
| '@=<_fails'(Xx55,Xx56) ) ) ).
fof(id42,axiom,
! [Xx57,Xx58] :
~ ( '@<_succeeds'(Xx57,Xx58)
& '@<_fails'(Xx57,Xx58) ) ).
fof(id43,axiom,
! [Xx57,Xx58] :
( '@<_terminates'(Xx57,Xx58)
=> ( '@<_succeeds'(Xx57,Xx58)
| '@<_fails'(Xx57,Xx58) ) ) ).
fof(id44,axiom,
! [Xx59] :
~ ( nat_succeeds(Xx59)
& nat_fails(Xx59) ) ).
fof(id45,axiom,
! [Xx59] :
( nat_terminates(Xx59)
=> ( nat_succeeds(Xx59)
| nat_fails(Xx59) ) ) ).
fof(id46,axiom,
! [Xx1,Xx2,Xx3] :
( member2_succeeds(Xx1,Xx2,Xx3)
<=> ( member_succeeds(Xx1,Xx3)
| member_succeeds(Xx1,Xx2) ) ) ).
fof(id47,axiom,
! [Xx1,Xx2,Xx3] :
( member2_fails(Xx1,Xx2,Xx3)
<=> ( member_fails(Xx1,Xx3)
& member_fails(Xx1,Xx2) ) ) ).
fof(id48,axiom,
! [Xx1,Xx2,Xx3] :
( member2_terminates(Xx1,Xx2,Xx3)
<=> ( member_terminates(Xx1,Xx3)
& member_terminates(Xx1,Xx2) ) ) ).
fof(id49,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(id50,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(id51,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(id52,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(id53,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(id54,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(id55,axiom,
! [Xx1,Xx2] :
( same_occ_succeeds(Xx1,Xx2)
<=> not_same_occ_fails(Xx1,Xx2) ) ).
fof(id56,axiom,
! [Xx1,Xx2] :
( same_occ_fails(Xx1,Xx2)
<=> not_same_occ_succeeds(Xx1,Xx2) ) ).
fof(id57,axiom,
! [Xx1,Xx2] :
( same_occ_terminates(Xx1,Xx2)
<=> ( not_same_occ_terminates(Xx1,Xx2)
& gr(Xx1)
& gr(Xx2) ) ) ).
fof(id58,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(id59,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(id60,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(id61,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(id62,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(id63,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(id64,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(id65,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(id66,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(id67,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(id68,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(id69,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(id70,axiom,
! [Xx1,Xx2] :
( member_succeeds(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx2 = cons(Xx3,Xx4)
& member_succeeds(Xx1,Xx4) )
| ? [Xx5] : Xx2 = cons(Xx1,Xx5) ) ) ).
fof(id71,axiom,
! [Xx1,Xx2] :
( member_fails(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx2 != cons(Xx3,Xx4)
| member_fails(Xx1,Xx4) )
& ! [Xx5] : Xx2 != cons(Xx1,Xx5) ) ) ).
fof(id72,axiom,
! [Xx1,Xx2] :
( member_terminates(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( $true
& ( Xx2 != cons(Xx3,Xx4)
| member_terminates(Xx1,Xx4) ) )
& ! [Xx5] : $true ) ) ).
fof(id73,axiom,
! [Xx1] :
( list_succeeds(Xx1)
<=> ( ? [Xx2,Xx3] :
( Xx1 = cons(Xx2,Xx3)
& list_succeeds(Xx3) )
| Xx1 = nil ) ) ).
fof(id74,axiom,
! [Xx1] :
( list_fails(Xx1)
<=> ( ! [Xx2,Xx3] :
( Xx1 != cons(Xx2,Xx3)
| list_fails(Xx3) )
& Xx1 != nil ) ) ).
fof(id75,axiom,
! [Xx1] :
( list_terminates(Xx1)
<=> ( ! [Xx2,Xx3] :
( $true
& ( Xx1 != cons(Xx2,Xx3)
| list_terminates(Xx3) ) )
& $true ) ) ).
fof(id76,axiom,
! [Xx1] :
( nat_list_succeeds(Xx1)
<=> ( ? [Xx2,Xx3] :
( Xx1 = cons(Xx2,Xx3)
& nat_succeeds(Xx2)
& nat_list_succeeds(Xx3) )
| Xx1 = nil ) ) ).
fof(id77,axiom,
! [Xx1] :
( nat_list_fails(Xx1)
<=> ( ! [Xx2,Xx3] :
( Xx1 != cons(Xx2,Xx3)
| nat_fails(Xx2)
| nat_list_fails(Xx3) )
& Xx1 != nil ) ) ).
fof(id78,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(id79,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(id80,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(id81,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(id82,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(id83,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(id84,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(id85,axiom,
! [Xx1,Xx2] :
( '@=<_succeeds'(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx1 = s(Xx3)
& Xx2 = s(Xx4)
& '@=<_succeeds'(Xx3,Xx4) )
| Xx1 = '0' ) ) ).
fof(id86,axiom,
! [Xx1,Xx2] :
( '@=<_fails'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx1 != s(Xx3)
| Xx2 != s(Xx4)
| '@=<_fails'(Xx3,Xx4) )
& Xx1 != '0' ) ) ).
fof(id87,axiom,
! [Xx1,Xx2] :
( '@=<_terminates'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( $true
& ( Xx1 != s(Xx3)
| ( $true
& ( Xx2 != s(Xx4)
| '@=<_terminates'(Xx3,Xx4) ) ) ) )
& $true ) ) ).
fof(id88,axiom,
! [Xx1,Xx2] :
( '@<_succeeds'(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx1 = s(Xx3)
& Xx2 = s(Xx4)
& '@<_succeeds'(Xx3,Xx4) )
| ? [Xx5] :
( Xx1 = '0'
& Xx2 = s(Xx5) ) ) ) ).
fof(id89,axiom,
! [Xx1,Xx2] :
( '@<_fails'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx1 != s(Xx3)
| Xx2 != s(Xx4)
| '@<_fails'(Xx3,Xx4) )
& ! [Xx5] :
( Xx1 != '0'
| Xx2 != s(Xx5) ) ) ) ).
fof(id90,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(id91,axiom,
! [Xx1] :
( nat_succeeds(Xx1)
<=> ( ? [Xx2] :
( Xx1 = s(Xx2)
& nat_succeeds(Xx2) )
| Xx1 = '0' ) ) ).
fof(id92,axiom,
! [Xx1] :
( nat_fails(Xx1)
<=> ( ! [Xx2] :
( Xx1 != s(Xx2)
| nat_fails(Xx2) )
& Xx1 != '0' ) ) ).
fof(id93,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)',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('(@+)/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-(permutation:types)',axiom,
! [Xl1,Xl2] :
( permutation_succeeds(Xl1,Xl2)
=> ( list_succeeds(Xl1)
& list_succeeds(Xl2) ) ) ).
fof('lemma-(permutation:termination)',axiom,
! [Xn,Xl1,Xl2] :
( ( nat_succeeds(Xn)
& list_succeeds(Xl1)
& lh(Xl1) = Xn )
=> permutation_terminates(Xl1,Xl2) ) ).
fof('theorem-(permutation:termination)',axiom,
! [Xl1,Xl2] :
( list_succeeds(Xl1)
=> permutation_terminates(Xl1,Xl2) ) ).
fof('lemma-(member2:termination)',axiom,
! [Xx,Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> member2_terminates(Xx,Xl1,Xl2) ) ).
fof('lemma-(occ:termination)',axiom,
! [Xx,Xl,Xn] :
( ( list_succeeds(Xl)
& gr(Xl)
& gr(Xx) )
=> occ_terminates(Xx,Xl,Xn) ) ).
fof('lemma-(member2:ground)',axiom,
! [Xx,Xl1,Xl2] :
( ( member2_succeeds(Xx,Xl1,Xl2)
& gr(Xl1)
& gr(Xl2) )
=> gr(Xx) ) ).
fof('lemma-(occ:ground)',axiom,
! [Xx,Xl,Xn] :
( occ_succeeds(Xx,Xl,Xn)
=> gr(Xn) ) ).
fof('lemma-(not_same_occ:termination)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& gr(Xl1)
& gr(Xl2) )
=> not_same_occ_terminates(Xl1,Xl2) ) ).
fof('theorem-(same_occ:termination)',axiom,
! [Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2)
& gr(Xl1)
& gr(Xl2) )
=> same_occ_terminates(Xl1,Xl2) ) ).
fof('lemma-(occ:types)',axiom,
! [Xx,Xl,Xn] :
( occ_succeeds(Xx,Xl,Xn)
=> ( list_succeeds(Xl)
& nat_succeeds(Xn) ) ) ).
fof('lemma-(occ:existence)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> ? [Xn] : occ_succeeds(Xx,Xl,Xn) ) ).
fof('lemma-(occ:uniqueness)',axiom,
! [Xx,Xl,Xm,Xn] :
( ( occ_succeeds(Xx,Xl,Xm)
& occ_succeeds(Xx,Xl,Xn) )
=> Xm = Xn ) ).
fof('lemma-(occ:nil)',axiom,
! [Xx] : occ(Xx,nil) = '0' ).
fof('lemma-(occ:cons:diff)',axiom,
! [Xx,Xy,Xl] :
( ( list_succeeds(Xl)
& Xx != Xy )
=> occ(Xx,cons(Xy,Xl)) = occ(Xx,Xl) ) ).
fof('lemma-(occ:cons:eq)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> occ(Xx,cons(Xx,Xl)) = s(occ(Xx,Xl)) ) ).
fof('corollary-(occ:types)',axiom,
! [Xx,Xl] :
( list_succeeds(Xl)
=> nat_succeeds(occ(Xx,Xl)) ) ).
fof('lemma-(occ:app)',axiom,
! [Xx,Xl1,Xl2] :
( ( list_succeeds(Xl1)
& list_succeeds(Xl2) )
=> occ(Xx,'**'(Xl1,Xl2)) = '@+'(occ(Xx,Xl1),occ(Xx,Xl2)) ) ).
fof('lemma-(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('lemma-(delete:occ:eq)',axiom,
! [Xx,Xl1,Xl2] :
( ( list_succeeds(Xl1)
& delete_succeeds(Xx,Xl1,Xl2) )
=> occ(Xx,Xl1) = s(occ(Xx,Xl2)) ) ).
fof('theorem-(permutation:occ)',axiom,
! [Xl1,Xl2] :
( permutation_succeeds(Xl1,Xl2)
=> ! [Xx] : occ(Xx,Xl1) = occ(Xx,Xl2) ) ).
fof('theorem-(permutation:soundness)',axiom,
! [Xl1,Xl2] :
( ( permutation_succeeds(Xl1,Xl2)
& gr(Xl1)
& gr(Xl2) )
=> same_occ_succeeds(Xl1,Xl2) ) ).
fof('lemma-(occ:zero)',axiom,
! [Xl] :
( ( list_succeeds(Xl)
& ! [Xx] : occ(Xx,Xl) = '0' )
=> Xl = nil ) ).
fof('lemma-(occ:successor)',axiom,
! [Xx,Xl1,Xn] :
( ( list_succeeds(Xl1)
& occ(Xx,Xl1) = s(Xn) )
=> ? [Xl2] : delete_succeeds(Xx,Xl1,Xl2) ) ).
fof(induction,axiom,
( ! [Xl2] :
( ( ? [Xx2,Xx3] :
( Xl2 = cons(Xx2,Xx3)
& list_succeeds(Xx3)
& ! [Xl1] :
( ( list_succeeds(Xl1)
& ! [Xx] : occ(Xx,Xl1) = occ(Xx,Xx3) )
=> permutation_succeeds(Xl1,Xx3) ) )
| Xl2 = nil )
=> ! [Xl1] :
( ( list_succeeds(Xl1)
& ! [Xx] : occ(Xx,Xl1) = occ(Xx,Xl2) )
=> permutation_succeeds(Xl1,Xl2) ) )
=> ! [Xl2] :
( list_succeeds(Xl2)
=> ! [Xl1] :
( ( list_succeeds(Xl1)
& ! [Xx] : occ(Xx,Xl1) = occ(Xx,Xl2) )
=> permutation_succeeds(Xl1,Xl2) ) ) ) ).
fof('lemma-(permutation:completeness)',conjecture,
! [Xl2] :
( list_succeeds(Xl2)
=> ! [Xl1] :
( ( list_succeeds(Xl1)
& ! [Xx] : occ(Xx,Xl1) = occ(Xx,Xl2) )
=> permutation_succeeds(Xl1,Xl2) ) ) ).
%------------------------------------------------------------------------------