Left Termination of the query pattern
mergesort_in_2(g, a)
w.r.t. the given Prolog program could successfully be proven:
↳ Prolog
↳ PredefinedPredicateTransformerProof
Clauses:
mergesort([], []).
mergesort(.(X, []), .(X, [])).
mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))).
split([], [], []).
split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys).
merge([], Xs, Xs).
merge(Xs, [], Xs).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(=(X, Y), merge(.(X, Xs), Ys, Zs)).
Queries:
mergesort(g,a).
Added definitions of predefined predicates.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
Clauses:
mergesort([], []).
mergesort(.(X, []), .(X, [])).
mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))).
split([], [], []).
split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys).
merge([], Xs, Xs).
merge(Xs, [], Xs).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(=(X, Y), merge(.(X, Xs), Ys, Zs)).
=(X, X).
Queries:
mergesort(g,a).
We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
mergesort_in(.(X, []), .(X, [])) → mergesort_out(.(X, []), .(X, []))
mergesort_in([], []) → mergesort_out([], [])
U2(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, mergesort_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, =_in(X, Y))
=_in(X, X) → =_out(X, X)
U6(X, Xs, Y, Ys, Zs, =_out(X, Y)) → U7(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → mergesort_out(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in(x1, x2) = mergesort_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4, x5) = U1(x5)
split_in(x1, x2, x3) = split_in(x1)
U5(x1, x2, x3, x4, x5) = U5(x1, x5)
[] = []
split_out(x1, x2, x3) = split_out(x2, x3)
U2(x1, x2, x3, x4, x5, x6) = U2(x5, x6)
mergesort_out(x1, x2) = mergesort_out(x2)
U3(x1, x2, x3, x4, x5, x6) = U3(x5, x6)
U4(x1, x2, x3, x4, x5) = U4(x5)
merge_in(x1, x2, x3) = merge_in(x1, x2)
U6(x1, x2, x3, x4, x5, x6) = U6(x1, x2, x4, x6)
=_in(x1, x2) = =_in(x1, x2)
=_out(x1, x2) = =_out
U7(x1, x2, x3, x4, x5, x6) = U7(x1, x6)
merge_out(x1, x2, x3) = merge_out(x3)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
Pi-finite rewrite system:
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
mergesort_in(.(X, []), .(X, [])) → mergesort_out(.(X, []), .(X, []))
mergesort_in([], []) → mergesort_out([], [])
U2(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, mergesort_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, =_in(X, Y))
=_in(X, X) → =_out(X, X)
U6(X, Xs, Y, Ys, Zs, =_out(X, Y)) → U7(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → mergesort_out(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in(x1, x2) = mergesort_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4, x5) = U1(x5)
split_in(x1, x2, x3) = split_in(x1)
U5(x1, x2, x3, x4, x5) = U5(x1, x5)
[] = []
split_out(x1, x2, x3) = split_out(x2, x3)
U2(x1, x2, x3, x4, x5, x6) = U2(x5, x6)
mergesort_out(x1, x2) = mergesort_out(x2)
U3(x1, x2, x3, x4, x5, x6) = U3(x5, x6)
U4(x1, x2, x3, x4, x5) = U4(x5)
merge_in(x1, x2, x3) = merge_in(x1, x2)
U6(x1, x2, x3, x4, x5, x6) = U6(x1, x2, x4, x6)
=_in(x1, x2) = =_in(x1, x2)
=_out(x1, x2) = =_out
U7(x1, x2, x3, x4, x5, x6) = U7(x1, x6)
merge_out(x1, x2, x3) = merge_out(x3)
Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
MERGESORT_IN(.(X, .(Y, Xs)), Ys) → U11(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN(.(X, .(Y, Xs)), Ys) → SPLIT_IN(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → U51(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U21(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN(X1s, Y1s)
U21(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → U31(X, Y, Xs, Ys, Y1s, mergesort_in(X2s, Y2s))
U21(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → MERGESORT_IN(X2s, Y2s)
U31(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → U41(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
U31(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → MERGE_IN(Y1s, Y2s, Ys)
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, =_in(X, Y))
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN(X, Y)
U61(X, Xs, Y, Ys, Zs, =_out(X, Y)) → U71(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
U61(X, Xs, Y, Ys, Zs, =_out(X, Y)) → MERGE_IN(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
mergesort_in(.(X, []), .(X, [])) → mergesort_out(.(X, []), .(X, []))
mergesort_in([], []) → mergesort_out([], [])
U2(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, mergesort_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, =_in(X, Y))
=_in(X, X) → =_out(X, X)
U6(X, Xs, Y, Ys, Zs, =_out(X, Y)) → U7(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → mergesort_out(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in(x1, x2) = mergesort_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4, x5) = U1(x5)
split_in(x1, x2, x3) = split_in(x1)
U5(x1, x2, x3, x4, x5) = U5(x1, x5)
[] = []
split_out(x1, x2, x3) = split_out(x2, x3)
U2(x1, x2, x3, x4, x5, x6) = U2(x5, x6)
mergesort_out(x1, x2) = mergesort_out(x2)
U3(x1, x2, x3, x4, x5, x6) = U3(x5, x6)
U4(x1, x2, x3, x4, x5) = U4(x5)
merge_in(x1, x2, x3) = merge_in(x1, x2)
U6(x1, x2, x3, x4, x5, x6) = U6(x1, x2, x4, x6)
=_in(x1, x2) = =_in(x1, x2)
=_out(x1, x2) = =_out
U7(x1, x2, x3, x4, x5, x6) = U7(x1, x6)
merge_out(x1, x2, x3) = merge_out(x3)
SPLIT_IN(x1, x2, x3) = SPLIT_IN(x1)
=_IN(x1, x2) = =_IN(x1, x2)
MERGE_IN(x1, x2, x3) = MERGE_IN(x1, x2)
U51(x1, x2, x3, x4, x5) = U51(x1, x5)
U41(x1, x2, x3, x4, x5) = U41(x5)
U21(x1, x2, x3, x4, x5, x6) = U21(x5, x6)
U11(x1, x2, x3, x4, x5) = U11(x5)
U31(x1, x2, x3, x4, x5, x6) = U31(x5, x6)
U71(x1, x2, x3, x4, x5, x6) = U71(x1, x6)
U61(x1, x2, x3, x4, x5, x6) = U61(x1, x2, x4, x6)
MERGESORT_IN(x1, x2) = MERGESORT_IN(x1)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
Pi DP problem:
The TRS P consists of the following rules:
MERGESORT_IN(.(X, .(Y, Xs)), Ys) → U11(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN(.(X, .(Y, Xs)), Ys) → SPLIT_IN(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → U51(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U21(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN(X1s, Y1s)
U21(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → U31(X, Y, Xs, Ys, Y1s, mergesort_in(X2s, Y2s))
U21(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → MERGESORT_IN(X2s, Y2s)
U31(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → U41(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
U31(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → MERGE_IN(Y1s, Y2s, Ys)
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, =_in(X, Y))
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN(X, Y)
U61(X, Xs, Y, Ys, Zs, =_out(X, Y)) → U71(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
U61(X, Xs, Y, Ys, Zs, =_out(X, Y)) → MERGE_IN(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
mergesort_in(.(X, []), .(X, [])) → mergesort_out(.(X, []), .(X, []))
mergesort_in([], []) → mergesort_out([], [])
U2(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, mergesort_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, =_in(X, Y))
=_in(X, X) → =_out(X, X)
U6(X, Xs, Y, Ys, Zs, =_out(X, Y)) → U7(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → mergesort_out(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in(x1, x2) = mergesort_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4, x5) = U1(x5)
split_in(x1, x2, x3) = split_in(x1)
U5(x1, x2, x3, x4, x5) = U5(x1, x5)
[] = []
split_out(x1, x2, x3) = split_out(x2, x3)
U2(x1, x2, x3, x4, x5, x6) = U2(x5, x6)
mergesort_out(x1, x2) = mergesort_out(x2)
U3(x1, x2, x3, x4, x5, x6) = U3(x5, x6)
U4(x1, x2, x3, x4, x5) = U4(x5)
merge_in(x1, x2, x3) = merge_in(x1, x2)
U6(x1, x2, x3, x4, x5, x6) = U6(x1, x2, x4, x6)
=_in(x1, x2) = =_in(x1, x2)
=_out(x1, x2) = =_out
U7(x1, x2, x3, x4, x5, x6) = U7(x1, x6)
merge_out(x1, x2, x3) = merge_out(x3)
SPLIT_IN(x1, x2, x3) = SPLIT_IN(x1)
=_IN(x1, x2) = =_IN(x1, x2)
MERGE_IN(x1, x2, x3) = MERGE_IN(x1, x2)
U51(x1, x2, x3, x4, x5) = U51(x1, x5)
U41(x1, x2, x3, x4, x5) = U41(x5)
U21(x1, x2, x3, x4, x5, x6) = U21(x5, x6)
U11(x1, x2, x3, x4, x5) = U11(x5)
U31(x1, x2, x3, x4, x5, x6) = U31(x5, x6)
U71(x1, x2, x3, x4, x5, x6) = U71(x1, x6)
U61(x1, x2, x3, x4, x5, x6) = U61(x1, x2, x4, x6)
MERGESORT_IN(x1, x2) = MERGESORT_IN(x1)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 7 less nodes.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, =_in(X, Y))
U61(X, Xs, Y, Ys, Zs, =_out(X, Y)) → MERGE_IN(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
mergesort_in(.(X, []), .(X, [])) → mergesort_out(.(X, []), .(X, []))
mergesort_in([], []) → mergesort_out([], [])
U2(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, mergesort_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, =_in(X, Y))
=_in(X, X) → =_out(X, X)
U6(X, Xs, Y, Ys, Zs, =_out(X, Y)) → U7(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → mergesort_out(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in(x1, x2) = mergesort_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4, x5) = U1(x5)
split_in(x1, x2, x3) = split_in(x1)
U5(x1, x2, x3, x4, x5) = U5(x1, x5)
[] = []
split_out(x1, x2, x3) = split_out(x2, x3)
U2(x1, x2, x3, x4, x5, x6) = U2(x5, x6)
mergesort_out(x1, x2) = mergesort_out(x2)
U3(x1, x2, x3, x4, x5, x6) = U3(x5, x6)
U4(x1, x2, x3, x4, x5) = U4(x5)
merge_in(x1, x2, x3) = merge_in(x1, x2)
U6(x1, x2, x3, x4, x5, x6) = U6(x1, x2, x4, x6)
=_in(x1, x2) = =_in(x1, x2)
=_out(x1, x2) = =_out
U7(x1, x2, x3, x4, x5, x6) = U7(x1, x6)
merge_out(x1, x2, x3) = merge_out(x3)
MERGE_IN(x1, x2, x3) = MERGE_IN(x1, x2)
U61(x1, x2, x3, x4, x5, x6) = U61(x1, x2, x4, x6)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U61(X, Xs, Y, Ys, Zs, =_in(X, Y))
U61(X, Xs, Y, Ys, Zs, =_out(X, Y)) → MERGE_IN(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
=_in(X, X) → =_out(X, X)
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
=_in(x1, x2) = =_in(x1, x2)
=_out(x1, x2) = =_out
MERGE_IN(x1, x2, x3) = MERGE_IN(x1, x2)
U61(x1, x2, x3, x4, x5, x6) = U61(x1, x2, x4, x6)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
MERGE_IN(.(X, Xs), .(Y, Ys)) → U61(X, Xs, Ys, =_in(X, Y))
U61(X, Xs, Ys, =_out) → MERGE_IN(.(X, Xs), Ys)
The TRS R consists of the following rules:
=_in(X, X) → =_out
The set Q consists of the following terms:
=_in(x0, x1)
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U61(X, Xs, Ys, =_out) → MERGE_IN(.(X, Xs), Ys)
The graph contains the following edges 3 >= 2
- MERGE_IN(.(X, Xs), .(Y, Ys)) → U61(X, Xs, Ys, =_in(X, Y))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
mergesort_in(.(X, []), .(X, [])) → mergesort_out(.(X, []), .(X, []))
mergesort_in([], []) → mergesort_out([], [])
U2(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, mergesort_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, =_in(X, Y))
=_in(X, X) → =_out(X, X)
U6(X, Xs, Y, Ys, Zs, =_out(X, Y)) → U7(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → mergesort_out(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in(x1, x2) = mergesort_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4, x5) = U1(x5)
split_in(x1, x2, x3) = split_in(x1)
U5(x1, x2, x3, x4, x5) = U5(x1, x5)
[] = []
split_out(x1, x2, x3) = split_out(x2, x3)
U2(x1, x2, x3, x4, x5, x6) = U2(x5, x6)
mergesort_out(x1, x2) = mergesort_out(x2)
U3(x1, x2, x3, x4, x5, x6) = U3(x5, x6)
U4(x1, x2, x3, x4, x5) = U4(x5)
merge_in(x1, x2, x3) = merge_in(x1, x2)
U6(x1, x2, x3, x4, x5, x6) = U6(x1, x2, x4, x6)
=_in(x1, x2) = =_in(x1, x2)
=_out(x1, x2) = =_out
U7(x1, x2, x3, x4, x5, x6) = U7(x1, x6)
merge_out(x1, x2, x3) = merge_out(x3)
SPLIT_IN(x1, x2, x3) = SPLIT_IN(x1)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN(Xs, Zs, Ys)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SPLIT_IN(x1, x2, x3) = SPLIT_IN(x1)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
SPLIT_IN(.(X, Xs)) → SPLIT_IN(Xs)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- SPLIT_IN(.(X, Xs)) → SPLIT_IN(Xs)
The graph contains the following edges 1 > 1
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
Pi DP problem:
The TRS P consists of the following rules:
MERGESORT_IN(.(X, .(Y, Xs)), Ys) → U11(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN(X1s, Y1s)
U21(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → MERGESORT_IN(X2s, Y2s)
U11(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U21(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs)), Ys) → U1(X, Y, Xs, Ys, split_in(.(X, .(Y, Xs)), X1s, X2s))
split_in(.(X, Xs), .(X, Ys), Zs) → U5(X, Xs, Ys, Zs, split_in(Xs, Zs, Ys))
split_in([], [], []) → split_out([], [], [])
U5(X, Xs, Ys, Zs, split_out(Xs, Zs, Ys)) → split_out(.(X, Xs), .(X, Ys), Zs)
U1(X, Y, Xs, Ys, split_out(.(X, .(Y, Xs)), X1s, X2s)) → U2(X, Y, Xs, Ys, X2s, mergesort_in(X1s, Y1s))
mergesort_in(.(X, []), .(X, [])) → mergesort_out(.(X, []), .(X, []))
mergesort_in([], []) → mergesort_out([], [])
U2(X, Y, Xs, Ys, X2s, mergesort_out(X1s, Y1s)) → U3(X, Y, Xs, Ys, Y1s, mergesort_in(X2s, Y2s))
U3(X, Y, Xs, Ys, Y1s, mergesort_out(X2s, Y2s)) → U4(X, Y, Xs, Ys, merge_in(Y1s, Y2s, Ys))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6(X, Xs, Y, Ys, Zs, =_in(X, Y))
=_in(X, X) → =_out(X, X)
U6(X, Xs, Y, Ys, Zs, =_out(X, Y)) → U7(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(Xs, [], Xs) → merge_out(Xs, [], Xs)
merge_in([], Xs, Xs) → merge_out([], Xs, Xs)
U7(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Y, Xs, Ys, merge_out(Y1s, Y2s, Ys)) → mergesort_out(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in(x1, x2) = mergesort_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4, x5) = U1(x5)
split_in(x1, x2, x3) = split_in(x1)
U5(x1, x2, x3, x4, x5) = U5(x1, x5)
[] = []
split_out(x1, x2, x3) = split_out(x2, x3)
U2(x1, x2, x3, x4, x5, x6) = U2(x5, x6)
mergesort_out(x1, x2) = mergesort_out(x2)
U3(x1, x2, x3, x4, x5, x6) = U3(x5, x6)
U4(x1, x2, x3, x4, x5) = U4(x5)
merge_in(x1, x2, x3) = merge_in(x1, x2)
U6(x1, x2, x3, x4, x5, x6) = U6(x1, x2, x4, x6)
=_in(x1, x2) = =_in(x1, x2)
=_out(x1, x2) = =_out
U7(x1, x2, x3, x4, x5, x6) = U7(x1, x6)
merge_out(x1, x2, x3) = merge_out(x3)
U21(x1, x2, x3, x4, x5, x6) = U21(x5, x6)
U11(x1, x2, x3, x4, x5) = U11(x5)
MERGESORT_IN(x1, x2) = MERGESORT_IN(x1)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
U21(X2s, mergesort_out(Y1s)) → MERGESORT_IN(X2s)
U11(split_out(X1s, X2s)) → MERGESORT_IN(X1s)
MERGESORT_IN(.(X, .(Y, Xs))) → U11(split_in(.(X, .(Y, Xs))))
U11(split_out(X1s, X2s)) → U21(X2s, mergesort_in(X1s))
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs))) → U1(split_in(.(X, .(Y, Xs))))
split_in(.(X, Xs)) → U5(X, split_in(Xs))
split_in([]) → split_out([], [])
U5(X, split_out(Zs, Ys)) → split_out(.(X, Ys), Zs)
U1(split_out(X1s, X2s)) → U2(X2s, mergesort_in(X1s))
mergesort_in(.(X, [])) → mergesort_out(.(X, []))
mergesort_in([]) → mergesort_out([])
U2(X2s, mergesort_out(Y1s)) → U3(Y1s, mergesort_in(X2s))
U3(Y1s, mergesort_out(Y2s)) → U4(merge_in(Y1s, Y2s))
merge_in(.(X, Xs), .(Y, Ys)) → U6(X, Xs, Ys, =_in(X, Y))
=_in(X, X) → =_out
U6(X, Xs, Ys, =_out) → U7(X, merge_in(.(X, Xs), Ys))
merge_in(Xs, []) → merge_out(Xs)
merge_in([], Xs) → merge_out(Xs)
U7(X, merge_out(Zs)) → merge_out(.(X, Zs))
U4(merge_out(Ys)) → mergesort_out(Ys)
The set Q consists of the following terms:
mergesort_in(x0)
split_in(x0)
U5(x0, x1)
U1(x0)
U2(x0, x1)
U3(x0, x1)
merge_in(x0, x1)
=_in(x0, x1)
U6(x0, x1, x2, x3)
U7(x0, x1)
U4(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN(.(X, .(Y, Xs))) → U11(split_in(.(X, .(Y, Xs)))) at position [0] we obtained the following new rules:
MERGESORT_IN(.(X, .(Y, Xs))) → U11(U5(X, split_in(.(Y, Xs))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
U11(split_out(X1s, X2s)) → MERGESORT_IN(X1s)
U21(X2s, mergesort_out(Y1s)) → MERGESORT_IN(X2s)
U11(split_out(X1s, X2s)) → U21(X2s, mergesort_in(X1s))
MERGESORT_IN(.(X, .(Y, Xs))) → U11(U5(X, split_in(.(Y, Xs))))
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs))) → U1(split_in(.(X, .(Y, Xs))))
split_in(.(X, Xs)) → U5(X, split_in(Xs))
split_in([]) → split_out([], [])
U5(X, split_out(Zs, Ys)) → split_out(.(X, Ys), Zs)
U1(split_out(X1s, X2s)) → U2(X2s, mergesort_in(X1s))
mergesort_in(.(X, [])) → mergesort_out(.(X, []))
mergesort_in([]) → mergesort_out([])
U2(X2s, mergesort_out(Y1s)) → U3(Y1s, mergesort_in(X2s))
U3(Y1s, mergesort_out(Y2s)) → U4(merge_in(Y1s, Y2s))
merge_in(.(X, Xs), .(Y, Ys)) → U6(X, Xs, Ys, =_in(X, Y))
=_in(X, X) → =_out
U6(X, Xs, Ys, =_out) → U7(X, merge_in(.(X, Xs), Ys))
merge_in(Xs, []) → merge_out(Xs)
merge_in([], Xs) → merge_out(Xs)
U7(X, merge_out(Zs)) → merge_out(.(X, Zs))
U4(merge_out(Ys)) → mergesort_out(Ys)
The set Q consists of the following terms:
mergesort_in(x0)
split_in(x0)
U5(x0, x1)
U1(x0)
U2(x0, x1)
U3(x0, x1)
merge_in(x0, x1)
=_in(x0, x1)
U6(x0, x1, x2, x3)
U7(x0, x1)
U4(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN(.(X, .(Y, Xs))) → U11(U5(X, split_in(.(Y, Xs)))) at position [0,1] we obtained the following new rules:
MERGESORT_IN(.(X, .(Y, Xs))) → U11(U5(X, U5(Y, split_in(Xs))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
U21(X2s, mergesort_out(Y1s)) → MERGESORT_IN(X2s)
U11(split_out(X1s, X2s)) → MERGESORT_IN(X1s)
U11(split_out(X1s, X2s)) → U21(X2s, mergesort_in(X1s))
MERGESORT_IN(.(X, .(Y, Xs))) → U11(U5(X, U5(Y, split_in(Xs))))
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs))) → U1(split_in(.(X, .(Y, Xs))))
split_in(.(X, Xs)) → U5(X, split_in(Xs))
split_in([]) → split_out([], [])
U5(X, split_out(Zs, Ys)) → split_out(.(X, Ys), Zs)
U1(split_out(X1s, X2s)) → U2(X2s, mergesort_in(X1s))
mergesort_in(.(X, [])) → mergesort_out(.(X, []))
mergesort_in([]) → mergesort_out([])
U2(X2s, mergesort_out(Y1s)) → U3(Y1s, mergesort_in(X2s))
U3(Y1s, mergesort_out(Y2s)) → U4(merge_in(Y1s, Y2s))
merge_in(.(X, Xs), .(Y, Ys)) → U6(X, Xs, Ys, =_in(X, Y))
=_in(X, X) → =_out
U6(X, Xs, Ys, =_out) → U7(X, merge_in(.(X, Xs), Ys))
merge_in(Xs, []) → merge_out(Xs)
merge_in([], Xs) → merge_out(Xs)
U7(X, merge_out(Zs)) → merge_out(.(X, Zs))
U4(merge_out(Ys)) → mergesort_out(Ys)
The set Q consists of the following terms:
mergesort_in(x0)
split_in(x0)
U5(x0, x1)
U1(x0)
U2(x0, x1)
U3(x0, x1)
merge_in(x0, x1)
=_in(x0, x1)
U6(x0, x1, x2, x3)
U7(x0, x1)
U4(x0)
We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MERGESORT_IN(.(X, .(Y, Xs))) → U11(U5(X, U5(Y, split_in(Xs))))
The remaining pairs can at least be oriented weakly.
U21(X2s, mergesort_out(Y1s)) → MERGESORT_IN(X2s)
U11(split_out(X1s, X2s)) → MERGESORT_IN(X1s)
U11(split_out(X1s, X2s)) → U21(X2s, mergesort_in(X1s))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( U3(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( .(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( split_out(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U2(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( merge_out(x1) ) = | | + | | · | x1 |
| M( U5(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( =_in(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U7(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( merge_in(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( mergesort_in(x1) ) = | | + | | · | x1 |
| M( mergesort_out(x1) ) = | | + | | · | x1 |
| M( U6(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
Tuple symbols:
| M( MERGESORT_IN(x1) ) = | 0 | + | | · | x1 |
| M( U21(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
U5(X, split_out(Zs, Ys)) → split_out(.(X, Ys), Zs)
split_in([]) → split_out([], [])
split_in(.(X, Xs)) → U5(X, split_in(Xs))
=_in(X, X) → =_out
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U11(split_out(X1s, X2s)) → MERGESORT_IN(X1s)
U21(X2s, mergesort_out(Y1s)) → MERGESORT_IN(X2s)
U11(split_out(X1s, X2s)) → U21(X2s, mergesort_in(X1s))
The TRS R consists of the following rules:
mergesort_in(.(X, .(Y, Xs))) → U1(split_in(.(X, .(Y, Xs))))
split_in(.(X, Xs)) → U5(X, split_in(Xs))
split_in([]) → split_out([], [])
U5(X, split_out(Zs, Ys)) → split_out(.(X, Ys), Zs)
U1(split_out(X1s, X2s)) → U2(X2s, mergesort_in(X1s))
mergesort_in(.(X, [])) → mergesort_out(.(X, []))
mergesort_in([]) → mergesort_out([])
U2(X2s, mergesort_out(Y1s)) → U3(Y1s, mergesort_in(X2s))
U3(Y1s, mergesort_out(Y2s)) → U4(merge_in(Y1s, Y2s))
merge_in(.(X, Xs), .(Y, Ys)) → U6(X, Xs, Ys, =_in(X, Y))
=_in(X, X) → =_out
U6(X, Xs, Ys, =_out) → U7(X, merge_in(.(X, Xs), Ys))
merge_in(Xs, []) → merge_out(Xs)
merge_in([], Xs) → merge_out(Xs)
U7(X, merge_out(Zs)) → merge_out(.(X, Zs))
U4(merge_out(Ys)) → mergesort_out(Ys)
The set Q consists of the following terms:
mergesort_in(x0)
split_in(x0)
U5(x0, x1)
U1(x0)
U2(x0, x1)
U3(x0, x1)
merge_in(x0, x1)
=_in(x0, x1)
U6(x0, x1, x2, x3)
U7(x0, x1)
U4(x0)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.