TPTP Problem File: SWX044+1.p
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%------------------------------------------------------------------------------
% File : SWX044+1 : TPTP v9.1.0. Released v9.1.0.
% Domain : Software Verification
% Problem : Right simplification of an inequality wrt the addition
% Version : Especial.
% English :
% Refs : [MMP24] Mesnard et al. (2024), ATP for Prolog Verification
% Source : [Mes24] Mesnard (2024), Email to Geoff Sutcliffe
% Names : nat80 [Mes24]
% Status : Theorem
% Rating : 0.97 v9.1.0
% Syntax : Number of formulae : 112 ( 3 unt; 0 def)
% Number of atoms : 356 ( 78 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 279 ( 35 ~; 41 |; 92 &)
% ( 18 <=>; 93 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-3 aty)
% Number of functors : 4 ( 4 usr; 1 con; 0-2 aty)
% Number of variables : 289 ( 271 !; 18 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
fof(id1,axiom,
! [Xx3] : '0' != s(Xx3) ).
fof(id2,axiom,
! [Xx4,Xx5] :
( s(Xx4) = s(Xx5)
=> Xx4 = Xx5 ) ).
fof(id3,axiom,
gr('0') ).
fof(id4,axiom,
! [Xx6] :
( gr(Xx6)
<=> gr(s(Xx6)) ) ).
fof(id5,axiom,
! [Xx7,Xx8,Xx9] :
~ ( times_succeeds(Xx7,Xx8,Xx9)
& times_fails(Xx7,Xx8,Xx9) ) ).
fof(id6,axiom,
! [Xx7,Xx8,Xx9] :
( times_terminates(Xx7,Xx8,Xx9)
=> ( times_succeeds(Xx7,Xx8,Xx9)
| times_fails(Xx7,Xx8,Xx9) ) ) ).
fof(id7,axiom,
! [Xx10,Xx11,Xx12] :
~ ( plus_succeeds(Xx10,Xx11,Xx12)
& plus_fails(Xx10,Xx11,Xx12) ) ).
fof(id8,axiom,
! [Xx10,Xx11,Xx12] :
( plus_terminates(Xx10,Xx11,Xx12)
=> ( plus_succeeds(Xx10,Xx11,Xx12)
| plus_fails(Xx10,Xx11,Xx12) ) ) ).
fof(id9,axiom,
! [Xx13,Xx14] :
~ ( '@=<_succeeds'(Xx13,Xx14)
& '@=<_fails'(Xx13,Xx14) ) ).
fof(id10,axiom,
! [Xx13,Xx14] :
( '@=<_terminates'(Xx13,Xx14)
=> ( '@=<_succeeds'(Xx13,Xx14)
| '@=<_fails'(Xx13,Xx14) ) ) ).
fof(id11,axiom,
! [Xx15,Xx16] :
~ ( '@<_succeeds'(Xx15,Xx16)
& '@<_fails'(Xx15,Xx16) ) ).
fof(id12,axiom,
! [Xx15,Xx16] :
( '@<_terminates'(Xx15,Xx16)
=> ( '@<_succeeds'(Xx15,Xx16)
| '@<_fails'(Xx15,Xx16) ) ) ).
fof(id13,axiom,
! [Xx17] :
~ ( nat_succeeds(Xx17)
& nat_fails(Xx17) ) ).
fof(id14,axiom,
! [Xx17] :
( nat_terminates(Xx17)
=> ( nat_succeeds(Xx17)
| nat_fails(Xx17) ) ) ).
fof(id15,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(id16,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(id17,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(id18,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(id19,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(id20,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(id21,axiom,
! [Xx1,Xx2] :
( '@=<_succeeds'(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx1 = s(Xx3)
& Xx2 = s(Xx4)
& '@=<_succeeds'(Xx3,Xx4) )
| Xx1 = '0' ) ) ).
fof(id22,axiom,
! [Xx1,Xx2] :
( '@=<_fails'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx1 != s(Xx3)
| Xx2 != s(Xx4)
| '@=<_fails'(Xx3,Xx4) )
& Xx1 != '0' ) ) ).
fof(id23,axiom,
! [Xx1,Xx2] :
( '@=<_terminates'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( $true
& ( Xx1 != s(Xx3)
| ( $true
& ( Xx2 != s(Xx4)
| '@=<_terminates'(Xx3,Xx4) ) ) ) )
& $true ) ) ).
fof(id24,axiom,
! [Xx1,Xx2] :
( '@<_succeeds'(Xx1,Xx2)
<=> ( ? [Xx3,Xx4] :
( Xx1 = s(Xx3)
& Xx2 = s(Xx4)
& '@<_succeeds'(Xx3,Xx4) )
| ? [Xx5] :
( Xx1 = '0'
& Xx2 = s(Xx5) ) ) ) ).
fof(id25,axiom,
! [Xx1,Xx2] :
( '@<_fails'(Xx1,Xx2)
<=> ( ! [Xx3,Xx4] :
( Xx1 != s(Xx3)
| Xx2 != s(Xx4)
| '@<_fails'(Xx3,Xx4) )
& ! [Xx5] :
( Xx1 != '0'
| Xx2 != s(Xx5) ) ) ) ).
fof(id26,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(id27,axiom,
! [Xx1] :
( nat_succeeds(Xx1)
<=> ( ? [Xx2] :
( Xx1 = s(Xx2)
& nat_succeeds(Xx2) )
| Xx1 = '0' ) ) ).
fof(id28,axiom,
! [Xx1] :
( nat_fails(Xx1)
<=> ( ! [Xx2] :
( Xx1 != s(Xx2)
| nat_fails(Xx2) )
& Xx1 != '0' ) ) ).
fof(id29,axiom,
! [Xx1] :
( nat_terminates(Xx1)
<=> ( ! [Xx2] :
( $true
& ( Xx1 != s(Xx2)
| nat_terminates(Xx2) ) )
& $true ) ) ).
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('lemma-(nat:termination)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> nat_terminates(Xx) ) ).
fof('lemma-(nat:ground)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> gr(Xx) ) ).
fof('lemma-(plus:termination:1)',axiom,
! [Xx,Xy,Xz] :
( nat_succeeds(Xx)
=> plus_terminates(Xx,Xy,Xz) ) ).
fof('lemma-(plus:termination:2)',axiom,
! [Xx,Xy,Xz] :
( nat_succeeds(Xz)
=> plus_terminates(Xx,Xy,Xz) ) ).
fof('lemma-(plus:types:1)',axiom,
! [Xx,Xy,Xz] :
( plus_succeeds(Xx,Xy,Xz)
=> nat_succeeds(Xx) ) ).
fof('lemma-(plus:types:2)',axiom,
! [Xx,Xy,Xz] :
( ( plus_succeeds(Xx,Xy,Xz)
& nat_succeeds(Xy) )
=> nat_succeeds(Xz) ) ).
fof('lemma-(plus:types:3)',axiom,
! [Xx,Xy,Xz] :
( ( plus_succeeds(Xx,Xy,Xz)
& nat_succeeds(Xz) )
=> nat_succeeds(Xy) ) ).
fof('lemma-(plus:termination:3)',axiom,
! [Xx,Xy,Xz] :
( plus_succeeds(Xx,Xy,Xz)
=> plus_terminates(Xx,Xy,Xz) ) ).
fof('lemma-(plus:ground:1)',axiom,
! [Xx,Xy,Xz] :
( plus_succeeds(Xx,Xy,Xz)
=> gr(Xx) ) ).
fof('lemma-(plus:ground:2)',axiom,
! [Xx,Xy,Xz] :
( ( plus_succeeds(Xx,Xy,Xz)
& gr(Xy) )
=> gr(Xz) ) ).
fof('lemma-(plus:ground:3)',axiom,
! [Xx,Xy,Xz] :
( ( plus_succeeds(Xx,Xy,Xz)
& gr(Xz) )
=> gr(Xy) ) ).
fof('lemma-(plus:existence)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> ? [Xz] : plus_succeeds(Xx,Xy,Xz) ) ).
fof('lemma-(plus:uniqueness)',axiom,
! [Xx,Xy,Xz1,Xz2] :
( ( plus_succeeds(Xx,Xy,Xz1)
& plus_succeeds(Xx,Xy,Xz2) )
=> Xz1 = Xz2 ) ).
fof('corollary-(plus:zero)',axiom,
! [Xy] : '@+'('0',Xy) = Xy ).
fof('corollary-(plus:successor)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@+'(s(Xx),Xy) = s('@+'(Xx,Xy)) ) ).
fof('corollary-(plus:types)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> nat_succeeds('@+'(Xx,Xy)) ) ).
fof('theorem-(plus:associative)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@+'('@+'(Xx,Xy),Xz) = '@+'(Xx,'@+'(Xy,Xz)) ) ).
fof('lemma-(plus:zero)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@+'(Xx,'0') = Xx ) ).
fof('lemma-(plus:successor)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@+'(Xx,s(Xy)) = '@+'(s(Xx),Xy) ) ).
fof('theorem-(plus:commutative)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@+'(Xx,Xy) = '@+'(Xy,Xx) ) ).
fof('lemma-(plus:injective:second)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@+'(Xx,Xy) = '@+'(Xx,Xz) )
=> Xy = Xz ) ).
fof('lemma-(times:types:1)',axiom,
! [Xx,Xy,Xz] :
( times_succeeds(Xx,Xy,Xz)
=> nat_succeeds(Xx) ) ).
fof('lemma-(times:types:2)',axiom,
! [Xx,Xy,Xz] :
( ( times_succeeds(Xx,Xy,Xz)
& nat_succeeds(Xy) )
=> nat_succeeds(Xz) ) ).
fof('lemma-(times:ground:1)',axiom,
! [Xx,Xy,Xz] :
( times_succeeds(Xx,Xy,Xz)
=> gr(Xx) ) ).
fof('lemma-(times:ground:2)',axiom,
! [Xx,Xy,Xz] :
( ( times_succeeds(Xx,Xy,Xz)
& gr(Xy) )
=> gr(Xz) ) ).
fof('lemma-(times:termination)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> times_terminates(Xx,Xy,Xz) ) ).
fof('lemma-(times:existence)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> ? [Xz] : times_succeeds(Xx,Xy,Xz) ) ).
fof('lemma-(times:uniqueness)',axiom,
! [Xx,Xy,Xz1,Xz2] :
( ( times_succeeds(Xx,Xy,Xz1)
& times_succeeds(Xx,Xy,Xz2) )
=> Xz1 = Xz2 ) ).
fof('corollary-(times:zero)',axiom,
! [Xy] :
( nat_succeeds(Xy)
=> '@*'('0',Xy) = '0' ) ).
fof('corollary-(times:successor)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@*'(s(Xx),Xy) = '@+'(Xy,'@*'(Xx,Xy)) ) ).
fof('corollary-(times:types)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> nat_succeeds('@*'(Xx,Xy)) ) ).
fof('theorem-(plus:times:distributive)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@*'('@+'(Xx,Xy),Xz) = '@+'('@*'(Xx,Xz),'@*'(Xy,Xz)) ) ).
fof('theorem-(times:associative)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@*'('@*'(Xx,Xy),Xz) = '@*'(Xx,'@*'(Xy,Xz)) ) ).
fof('lemma-(times:zero)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@*'(Xx,'0') = '0' ) ).
fof('lemma-(times:successor)',axiom,
! [Xy,Xx] :
( ( nat_succeeds(Xy)
& nat_succeeds(Xx) )
=> '@+'('@*'(Xy,Xx),Xy) = '@*'(Xy,s(Xx)) ) ).
fof('theorem-(times:commutative)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@*'(Xx,Xy) = '@*'(Xy,Xx) ) ).
fof('lemma-(times:one)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@*'(s('0'),Xx) = Xx ) ).
fof('corollary-(times:one)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@*'(Xx,s('0')) = Xx ) ).
fof('corollary-(plus:times:distributive)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@*'(Xz,'@+'(Xx,Xy)) = '@+'('@*'(Xz,Xx),'@*'(Xz,Xy)) ) ).
fof('lemma-(less:termination:1)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@<_terminates'(Xx,Xy) ) ).
fof('lemma-(less:termination:2)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xy)
=> '@<_terminates'(Xx,Xy) ) ).
fof('lemma-(less:types)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> nat_succeeds(Xx) ) ).
fof('lemma-(less:successor)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> ? [Xz] : Xy = s(Xz) ) ).
fof('lemma-(less:transitive:successor)',axiom,
! [Xx,Xy,Xz] :
( ( '@<_succeeds'(Xx,Xy)
& '@<_succeeds'(Xy,s(Xz)) )
=> '@<_succeeds'(Xx,Xz) ) ).
fof('lemma-(less:weakening)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> '@<_succeeds'(Xx,s(Xy)) ) ).
fof('theorem-(less:transitive)',axiom,
! [Xx,Xy,Xz] :
( ( '@<_succeeds'(Xx,Xy)
& '@<_succeeds'(Xy,Xz) )
=> '@<_succeeds'(Xx,Xz) ) ).
fof('lemma-(less:failure)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@<_fails'(Xx,Xx) ) ).
fof('theorem-(less:strictness)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> ~ '@<_succeeds'(Xx,Xx) ) ).
fof('lemma-(less:one)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@<_succeeds'(Xx,s(Xx)) ) ).
fof('lemma-(less:axiom:successor)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xy)
& '@<_succeeds'(Xx,s(Xy)) )
=> ( '@<_succeeds'(Xx,Xy)
| Xx = Xy ) ) ).
fof('theorem-(less:totality)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> ( '@<_succeeds'(Xx,Xy)
| Xx = Xy
| '@<_succeeds'(Xy,Xx) ) ) ).
fof('lemma-(less:different:zero)',axiom,
! [Xx] :
( ( nat_succeeds(Xx)
& Xx != '0' )
=> '@<_succeeds'('0',Xx) ) ).
fof('lemma-(leq:termination:1)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@=<_terminates'(Xx,Xy) ) ).
fof('lemma-(leq:termination:2)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xy)
=> '@=<_terminates'(Xx,Xy) ) ).
fof('lemma-(leq:types)',axiom,
! [Xx,Xy] :
( '@=<_succeeds'(Xx,Xy)
=> nat_succeeds(Xx) ) ).
fof('lemma-(leq:plus)',axiom,
! [Xx,Xy] :
( '@=<_succeeds'(Xx,Xy)
=> ? [Xz] : plus_succeeds(Xx,Xz,Xy) ) ).
fof('corollary-(leq:plus)',axiom,
! [Xx,Xy] :
( '@=<_succeeds'(Xx,Xy)
=> ? [Xz] : '@+'(Xx,Xz) = Xy ) ).
fof('lemma-(less:plus)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> ? [Xz] : plus_succeeds(Xx,s(Xz),Xy) ) ).
fof('corollary-(less:plus)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> ? [Xz] : '@+'(Xx,s(Xz)) = Xy ) ).
fof('lemma-(less:leq)',axiom,
! [Xx,Xy] :
( '@<_succeeds'(Xx,Xy)
=> '@=<_succeeds'(Xx,Xy) ) ).
fof('theorem-(leq:reflexive)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@=<_succeeds'(Xx,Xx) ) ).
fof('theorem-(leq:totality)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> ( '@=<_succeeds'(Xx,Xy)
| '@=<_succeeds'(Xy,Xx) ) ) ).
fof('corollary-(less:leq:total)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> ( '@<_succeeds'(Xx,Xy)
| '@=<_succeeds'(Xy,Xx) ) ) ).
fof('corollary-(leq:failure)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy)
& '@=<_fails'(Xx,Xy) )
=> '@=<_succeeds'(Xy,Xx) ) ).
fof('lemma-(leq:less)',axiom,
! [Xx,Xy] :
( ( '@=<_succeeds'(Xx,Xy)
& nat_succeeds(Xy) )
=> ( '@<_succeeds'(Xx,Xy)
| Xx = Xy ) ) ).
fof('theorem-(leq:less:transitive)',axiom,
! [Xx,Xy,Xz] :
( ( '@=<_succeeds'(Xx,Xy)
& '@<_succeeds'(Xy,Xz) )
=> '@<_succeeds'(Xx,Xz) ) ).
fof('theorem-(less:leq:transitive)',axiom,
! [Xx,Xy,Xz] :
( ( '@<_succeeds'(Xx,Xy)
& '@=<_succeeds'(Xy,Xz) )
=> '@<_succeeds'(Xx,Xz) ) ).
fof('theorem-(leq:transitive)',axiom,
! [Xx,Xy,Xz] :
( ( '@=<_succeeds'(Xx,Xy)
& '@=<_succeeds'(Xy,Xz) )
=> '@=<_succeeds'(Xx,Xz) ) ).
fof('theorem-(leq:antisymmetric)',axiom,
! [Xx,Xy] :
( ( '@=<_succeeds'(Xx,Xy)
& '@=<_succeeds'(Xy,Xx) )
=> Xx = Xy ) ).
fof('lemma-(leq:one:success)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@=<_succeeds'(Xx,s(Xx)) ) ).
fof('lemma-(leq:one:failure)',axiom,
! [Xx] :
( nat_succeeds(Xx)
=> '@=<_fails'(s(Xx),Xx) ) ).
fof('lemma-(less:plus:second)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@<_succeeds'(Xy,Xz) )
=> '@<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) ) ).
fof('corollary-(less:plus:second)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@<_succeeds'(Xx,'@+'(Xx,s(Xy))) ) ).
fof('lemma-(less:plus:first)',axiom,
! [Xx,Xy,Xz] :
( ( '@<_succeeds'(Xx,Xy)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@<_succeeds'('@+'(Xx,Xz),'@+'(Xy,Xz)) ) ).
fof('corollary-(less:plus:first)',axiom,
! [Xx,Xy] :
( ( '@<_succeeds'('0',Xy)
& nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@<_succeeds'(Xx,'@+'(Xy,Xx)) ) ).
fof('lemma-(leq:plus:second)',axiom,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@=<_succeeds'(Xy,Xz) )
=> '@=<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) ) ).
fof('lemma-(leq:plus:first)',axiom,
! [Xx,Xy,Xz] :
( ( '@=<_succeeds'(Xx,Xy)
& nat_succeeds(Xy)
& nat_succeeds(Xz) )
=> '@=<_succeeds'('@+'(Xx,Xz),'@+'(Xy,Xz)) ) ).
fof('corollary-(leq:plus:first)',axiom,
! [Xx,Xy] :
( nat_succeeds(Xx)
=> '@=<_succeeds'(Xx,'@+'(Xx,Xy)) ) ).
fof('corollary-(leq:plus:second)',axiom,
! [Xx,Xy] :
( ( nat_succeeds(Xx)
& nat_succeeds(Xy) )
=> '@=<_succeeds'(Xy,'@+'(Xx,Xy)) ) ).
fof(induction,axiom,
( ! [Xx] :
( ( ? [Xx2] :
( Xx = s(Xx2)
& nat_succeeds(Xx2)
& ! [Xy,Xz] :
( '@<_succeeds'('@+'(Xx2,Xy),'@+'(Xx2,Xz))
=> '@<_succeeds'(Xy,Xz) ) )
| Xx = '0' )
=> ! [Xy,Xz] :
( '@<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz))
=> '@<_succeeds'(Xy,Xz) ) )
=> ! [Xx] :
( nat_succeeds(Xx)
=> ! [Xy,Xz] :
( '@<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz))
=> '@<_succeeds'(Xy,Xz) ) ) ) ).
fof('lemma-(less:plus:inverse)',conjecture,
! [Xx,Xy,Xz] :
( ( nat_succeeds(Xx)
& '@<_succeeds'('@+'(Xx,Xy),'@+'(Xx,Xz)) )
=> '@<_succeeds'(Xy,Xz) ) ).
%------------------------------------------------------------------------------