LMPO
MAYBE
We consider the following Problem:
Strict Trs:
{ and(false(), false()) -> false()
, and(true(), false()) -> false()
, and(false(), true()) -> false()
, and(true(), true()) -> true()
, eq(nil(), nil()) -> true()
, eq(cons(T, L), nil()) -> false()
, eq(nil(), cons(T, L)) -> false()
, eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
, eq(var(L), var(Lp)) -> eq(L, Lp)
, eq(var(L), apply(T, S)) -> false()
, eq(var(L), lambda(X, T)) -> false()
, eq(apply(T, S), var(L)) -> false()
, eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
, eq(apply(T, S), lambda(X, Tp)) -> false()
, eq(lambda(X, T), var(L)) -> false()
, eq(lambda(X, T), apply(Tp, Sp)) -> false()
, eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
, if(true(), var(K), var(L)) -> var(K)
, if(false(), var(K), var(L)) -> var(L)
, ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
, ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
, ren(X, Y, lambda(Z, T)) ->
lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
ren(X,
Y,
ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))}
StartTerms: basic terms
Strategy: innermost
Certificate: MAYBE
Proof:
The input cannot be shown compatible
Arrrr..
MPO
MAYBE
We consider the following Problem:
Strict Trs:
{ and(false(), false()) -> false()
, and(true(), false()) -> false()
, and(false(), true()) -> false()
, and(true(), true()) -> true()
, eq(nil(), nil()) -> true()
, eq(cons(T, L), nil()) -> false()
, eq(nil(), cons(T, L)) -> false()
, eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
, eq(var(L), var(Lp)) -> eq(L, Lp)
, eq(var(L), apply(T, S)) -> false()
, eq(var(L), lambda(X, T)) -> false()
, eq(apply(T, S), var(L)) -> false()
, eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
, eq(apply(T, S), lambda(X, Tp)) -> false()
, eq(lambda(X, T), var(L)) -> false()
, eq(lambda(X, T), apply(Tp, Sp)) -> false()
, eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
, if(true(), var(K), var(L)) -> var(K)
, if(false(), var(K), var(L)) -> var(L)
, ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
, ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
, ren(X, Y, lambda(Z, T)) ->
lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
ren(X,
Y,
ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))}
StartTerms: basic terms
Strategy: innermost
Certificate: MAYBE
Proof:
The input cannot be shown compatible
Arrrr..
POP*
MAYBE
We consider the following Problem:
Strict Trs:
{ and(false(), false()) -> false()
, and(true(), false()) -> false()
, and(false(), true()) -> false()
, and(true(), true()) -> true()
, eq(nil(), nil()) -> true()
, eq(cons(T, L), nil()) -> false()
, eq(nil(), cons(T, L)) -> false()
, eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
, eq(var(L), var(Lp)) -> eq(L, Lp)
, eq(var(L), apply(T, S)) -> false()
, eq(var(L), lambda(X, T)) -> false()
, eq(apply(T, S), var(L)) -> false()
, eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
, eq(apply(T, S), lambda(X, Tp)) -> false()
, eq(lambda(X, T), var(L)) -> false()
, eq(lambda(X, T), apply(Tp, Sp)) -> false()
, eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
, if(true(), var(K), var(L)) -> var(K)
, if(false(), var(K), var(L)) -> var(L)
, ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
, ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
, ren(X, Y, lambda(Z, T)) ->
lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
ren(X,
Y,
ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))}
StartTerms: basic terms
Strategy: innermost
Certificate: MAYBE
Proof:
The input cannot be shown compatible
Arrrr..
POP* (PS)
MAYBE
We consider the following Problem:
Strict Trs:
{ and(false(), false()) -> false()
, and(true(), false()) -> false()
, and(false(), true()) -> false()
, and(true(), true()) -> true()
, eq(nil(), nil()) -> true()
, eq(cons(T, L), nil()) -> false()
, eq(nil(), cons(T, L)) -> false()
, eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
, eq(var(L), var(Lp)) -> eq(L, Lp)
, eq(var(L), apply(T, S)) -> false()
, eq(var(L), lambda(X, T)) -> false()
, eq(apply(T, S), var(L)) -> false()
, eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
, eq(apply(T, S), lambda(X, Tp)) -> false()
, eq(lambda(X, T), var(L)) -> false()
, eq(lambda(X, T), apply(Tp, Sp)) -> false()
, eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
, if(true(), var(K), var(L)) -> var(K)
, if(false(), var(K), var(L)) -> var(L)
, ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
, ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
, ren(X, Y, lambda(Z, T)) ->
lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
ren(X,
Y,
ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))}
StartTerms: basic terms
Strategy: innermost
Certificate: MAYBE
Proof:
The input cannot be shown compatible
Arrrr..
Small POP*
MAYBE
We consider the following Problem:
Strict Trs:
{ and(false(), false()) -> false()
, and(true(), false()) -> false()
, and(false(), true()) -> false()
, and(true(), true()) -> true()
, eq(nil(), nil()) -> true()
, eq(cons(T, L), nil()) -> false()
, eq(nil(), cons(T, L)) -> false()
, eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
, eq(var(L), var(Lp)) -> eq(L, Lp)
, eq(var(L), apply(T, S)) -> false()
, eq(var(L), lambda(X, T)) -> false()
, eq(apply(T, S), var(L)) -> false()
, eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
, eq(apply(T, S), lambda(X, Tp)) -> false()
, eq(lambda(X, T), var(L)) -> false()
, eq(lambda(X, T), apply(Tp, Sp)) -> false()
, eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
, if(true(), var(K), var(L)) -> var(K)
, if(false(), var(K), var(L)) -> var(L)
, ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
, ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
, ren(X, Y, lambda(Z, T)) ->
lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
ren(X,
Y,
ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))}
StartTerms: basic terms
Strategy: innermost
Certificate: MAYBE
Proof:
The input cannot be shown compatible
Arrrr..
Small POP* (PS)
MAYBE
We consider the following Problem:
Strict Trs:
{ and(false(), false()) -> false()
, and(true(), false()) -> false()
, and(false(), true()) -> false()
, and(true(), true()) -> true()
, eq(nil(), nil()) -> true()
, eq(cons(T, L), nil()) -> false()
, eq(nil(), cons(T, L)) -> false()
, eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
, eq(var(L), var(Lp)) -> eq(L, Lp)
, eq(var(L), apply(T, S)) -> false()
, eq(var(L), lambda(X, T)) -> false()
, eq(apply(T, S), var(L)) -> false()
, eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
, eq(apply(T, S), lambda(X, Tp)) -> false()
, eq(lambda(X, T), var(L)) -> false()
, eq(lambda(X, T), apply(Tp, Sp)) -> false()
, eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
, if(true(), var(K), var(L)) -> var(K)
, if(false(), var(K), var(L)) -> var(L)
, ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
, ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
, ren(X, Y, lambda(Z, T)) ->
lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
ren(X,
Y,
ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))}
StartTerms: basic terms
Strategy: innermost
Certificate: MAYBE
Proof:
The input cannot be shown compatible
Arrrr..