/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files


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Graph construction failed
Graph construction failed
Graph construction failed
MAYBE
proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

numvars(a,g,a)

w.r.t. the given Prolog program could not be shown:

(0) Prolog
(1) IntegerArithmeticTransformerProof [SOUND, 0 ms]
(2) Prolog
    (3) CutEliminatorProof [SOUND, 0 ms]
    (4) Prolog
    (5) UnifyTransformerProof [EQUIVALENT, 0 ms]
    (6) Prolog
    (7) UndefinedPredicateHandlerProof [SOUND, 0 ms]
    (8) Prolog
(9) CutEliminatorProof [SOUND, 0 ms]
(10) Prolog
    (11) UnifyTransformerProof [EQUIVALENT, 0 ms]
    (12) Prolog
    (13) UndefinedPredicateHandlerProof [SOUND, 0 ms]
    (14) Prolog


----------------------------------------

(0)
Obligation:
Clauses:

numvars(Term, N, N1) :- ','(var(Term), ','(!, ','(=(Term, /(var, N)), is(N1, +(N, 1))))).
numvars(Term, N, M) :- ','(=..(Term, .(_Functor, Args)), numberargs(Args, N, M)).
numberargs([], N, N) :- !.
numberargs(.(Arg, Args), N, M) :- ','(numvars(Arg, N, N1), numberargs(Args, N1, M)).
numvars2(var(N), N, N1) :- is(N1, +(N, 1)).
numvars2(Term, N, M) :- ','(nonvar(Term), ','(=..(Term, .(_Functor, Args)), numberargs2(Args, N, M))).
numberargs2([], N, N).
numberargs2(.(Arg, Args), N, M) :- ','(numvars2(Arg, N, N1), numberargs2(Args, N1, M)).


Query: numvars(a,g,a)
----------------------------------------

(1) IntegerArithmeticTransformerProof (SOUND)
Added definitions of predefined predicates [PROLOG].
----------------------------------------

(2)
Obligation:
Clauses:

numvars(Term, N, N1) :- ','(var(Term), ','(!, ','(=(Term, /(var, N)), ','(isPlus(N, succ(zero), U), =(N1, U))))).
numvars(Term, N, M) :- ','(=..(Term, .(_Functor, Args)), numberargs(Args, N, M)).
numberargs([], N, N) :- !.
numberargs(.(Arg, Args), N, M) :- ','(numvars(Arg, N, N1), numberargs(Args, N1, M)).
numvars2(var(N), N, N1) :- ','(isPlus(N, succ(zero), U), =(N1, U)).
numvars2(Term, N, M) :- ','(nonvar(Term), ','(=..(Term, .(_Functor, Args)), numberargs2(Args, N, M))).
numberargs2([], N, N).
numberargs2(.(Arg, Args), N, M) :- ','(numvars2(Arg, N, N1), numberargs2(Args, N1, M)).
isPlus(zero, X, X).
isPlus(succ(X), zero, succ(X)).
isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z).
isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), zero, pred(X)).
isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z).
isMinus(X, zero, X).
isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z).
isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z).
isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z).
isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z).
isTimes(X, zero, zero).
isTimes(zero, succ(Y), zero).
isTimes(zero, pred(Y), zero).
isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).
isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).
isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).
isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).
isDiv(zero, succ(Y), zero).
isDiv(zero, pred(Y), zero).
isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)).
isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).
isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).
isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)).
isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).
isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).
isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).


Query: numvars(a,g,a)
----------------------------------------

(3) CutEliminatorProof (SOUND)
Eliminated all cuts by simply ignoring them[PROLOG].
----------------------------------------

(4)
Obligation:
Clauses:

numvars(Term, N, N1) :- ','(var(Term), ','(=(Term, /(var, N)), ','(isPlus(N, succ(zero), U), =(N1, U)))).
numvars(Term, N, M) :- ','(=..(Term, .(_Functor, Args)), numberargs(Args, N, M)).
numberargs([], N, N).
numberargs(.(Arg, Args), N, M) :- ','(numvars(Arg, N, N1), numberargs(Args, N1, M)).
numvars2(var(N), N, N1) :- ','(isPlus(N, succ(zero), U), =(N1, U)).
numvars2(Term, N, M) :- ','(nonvar(Term), ','(=..(Term, .(_Functor, Args)), numberargs2(Args, N, M))).
numberargs2([], N, N).
numberargs2(.(Arg, Args), N, M) :- ','(numvars2(Arg, N, N1), numberargs2(Args, N1, M)).
isPlus(zero, X, X).
isPlus(succ(X), zero, succ(X)).
isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z).
isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), zero, pred(X)).
isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z).
isMinus(X, zero, X).
isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z).
isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z).
isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z).
isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z).
isTimes(X, zero, zero).
isTimes(zero, succ(Y), zero).
isTimes(zero, pred(Y), zero).
isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).
isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).
isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).
isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).
isDiv(zero, succ(Y), zero).
isDiv(zero, pred(Y), zero).
isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)).
isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).
isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).
isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)).
isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).
isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).
isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).


Query: numvars(a,g,a)
----------------------------------------

(5) UnifyTransformerProof (EQUIVALENT)
Added a fact for the built-in = predicate [PROLOG].
----------------------------------------

(6)
Obligation:
Clauses:

numvars(Term, N, N1) :- ','(var(Term), ','(=(Term, /(var, N)), ','(isPlus(N, succ(zero), U), =(N1, U)))).
numvars(Term, N, M) :- ','(=..(Term, .(_Functor, Args)), numberargs(Args, N, M)).
numberargs([], N, N).
numberargs(.(Arg, Args), N, M) :- ','(numvars(Arg, N, N1), numberargs(Args, N1, M)).
numvars2(var(N), N, N1) :- ','(isPlus(N, succ(zero), U), =(N1, U)).
numvars2(Term, N, M) :- ','(nonvar(Term), ','(=..(Term, .(_Functor, Args)), numberargs2(Args, N, M))).
numberargs2([], N, N).
numberargs2(.(Arg, Args), N, M) :- ','(numvars2(Arg, N, N1), numberargs2(Args, N1, M)).
isPlus(zero, X, X).
isPlus(succ(X), zero, succ(X)).
isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z).
isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), zero, pred(X)).
isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z).
isMinus(X, zero, X).
isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z).
isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z).
isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z).
isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z).
isTimes(X, zero, zero).
isTimes(zero, succ(Y), zero).
isTimes(zero, pred(Y), zero).
isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).
isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).
isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).
isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).
isDiv(zero, succ(Y), zero).
isDiv(zero, pred(Y), zero).
isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)).
isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).
isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).
isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)).
isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).
isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).
isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).
=(X, X).


Query: numvars(a,g,a)
----------------------------------------

(7) UndefinedPredicateHandlerProof (SOUND)
Added facts for all undefined predicates [PROLOG].
----------------------------------------

(8)
Obligation:
Clauses:

numvars(Term, N, N1) :- ','(var(Term), ','(=(Term, /(var, N)), ','(isPlus(N, succ(zero), U), =(N1, U)))).
numvars(Term, N, M) :- ','(=..(Term, .(_Functor, Args)), numberargs(Args, N, M)).
numberargs([], N, N).
numberargs(.(Arg, Args), N, M) :- ','(numvars(Arg, N, N1), numberargs(Args, N1, M)).
numvars2(var(N), N, N1) :- ','(isPlus(N, succ(zero), U), =(N1, U)).
numvars2(Term, N, M) :- ','(nonvar(Term), ','(=..(Term, .(_Functor, Args)), numberargs2(Args, N, M))).
numberargs2([], N, N).
numberargs2(.(Arg, Args), N, M) :- ','(numvars2(Arg, N, N1), numberargs2(Args, N1, M)).
isPlus(zero, X, X).
isPlus(succ(X), zero, succ(X)).
isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z).
isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), zero, pred(X)).
isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z).
isMinus(X, zero, X).
isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z).
isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z).
isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z).
isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z).
isTimes(X, zero, zero).
isTimes(zero, succ(Y), zero).
isTimes(zero, pred(Y), zero).
isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).
isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).
isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).
isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).
isDiv(zero, succ(Y), zero).
isDiv(zero, pred(Y), zero).
isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)).
isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).
isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).
isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)).
isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).
isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).
isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).
=(X, X).
var(X0).
=..(X0, X1).
nonvar(X0).


Query: numvars(a,g,a)
----------------------------------------

(9) CutEliminatorProof (SOUND)
Eliminated all cuts by simply ignoring them[PROLOG].
----------------------------------------

(10)
Obligation:
Clauses:

numvars(Term, N, N1) :- ','(var(Term), ','(=(Term, /(var, N)), is(N1, +(N, 1)))).
numvars(Term, N, M) :- ','(=..(Term, .(_Functor, Args)), numberargs(Args, N, M)).
numberargs([], N, N).
numberargs(.(Arg, Args), N, M) :- ','(numvars(Arg, N, N1), numberargs(Args, N1, M)).
numvars2(var(N), N, N1) :- is(N1, +(N, 1)).
numvars2(Term, N, M) :- ','(nonvar(Term), ','(=..(Term, .(_Functor, Args)), numberargs2(Args, N, M))).
numberargs2([], N, N).
numberargs2(.(Arg, Args), N, M) :- ','(numvars2(Arg, N, N1), numberargs2(Args, N1, M)).


Query: numvars(a,g,a)
----------------------------------------

(11) UnifyTransformerProof (EQUIVALENT)
Added a fact for the built-in = predicate [PROLOG].
----------------------------------------

(12)
Obligation:
Clauses:

numvars(Term, N, N1) :- ','(var(Term), ','(=(Term, /(var, N)), is(N1, +(N, 1)))).
numvars(Term, N, M) :- ','(=..(Term, .(_Functor, Args)), numberargs(Args, N, M)).
numberargs([], N, N).
numberargs(.(Arg, Args), N, M) :- ','(numvars(Arg, N, N1), numberargs(Args, N1, M)).
numvars2(var(N), N, N1) :- is(N1, +(N, 1)).
numvars2(Term, N, M) :- ','(nonvar(Term), ','(=..(Term, .(_Functor, Args)), numberargs2(Args, N, M))).
numberargs2([], N, N).
numberargs2(.(Arg, Args), N, M) :- ','(numvars2(Arg, N, N1), numberargs2(Args, N1, M)).
=(X, X).


Query: numvars(a,g,a)
----------------------------------------

(13) UndefinedPredicateHandlerProof (SOUND)
Added facts for all undefined predicates [PROLOG].
----------------------------------------

(14)
Obligation:
Clauses:

numvars(Term, N, N1) :- ','(var(Term), ','(=(Term, /(var, N)), is(N1, +(N, 1)))).
numvars(Term, N, M) :- ','(=..(Term, .(_Functor, Args)), numberargs(Args, N, M)).
numberargs([], N, N).
numberargs(.(Arg, Args), N, M) :- ','(numvars(Arg, N, N1), numberargs(Args, N1, M)).
numvars2(var(N), N, N1) :- is(N1, +(N, 1)).
numvars2(Term, N, M) :- ','(nonvar(Term), ','(=..(Term, .(_Functor, Args)), numberargs2(Args, N, M))).
numberargs2([], N, N).
numberargs2(.(Arg, Args), N, M) :- ','(numvars2(Arg, N, N1), numberargs2(Args, N1, M)).
=(X, X).
var(X0).
is(X0, X1).
=..(X0, X1).
nonvar(X0).


Query: numvars(a,g,a)