Left Termination of the query pattern
mergesort_in_2(a, g)
w.r.t. the given Prolog program could not be shown:
↳ 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(a,g).
Added definitions of predefined predicates.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ 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(a,g).
We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort_in: (f,b) (b,f)
split_in: (f,f,f) (b,f,f)
merge_in: (b,b,f) (b,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x6)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PrologToPiTRSProof
Pi-finite rewrite system:
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x6)
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_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_GAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → U5_GAA(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_GA(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_GA(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGA(Y1s, Y2s, Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGG(Y1s, Y2s, Ys)
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGG(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_GGG(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGG(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x6)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x2, x4, x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, x6)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x5)
U6_GGG(x1, x2, x3, x4, x5, x6) = U6_GGG(x2, x4, x5, x6)
U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x5, x6)
=_IN_AA(x1, x2) = =_IN_AA
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
U7_GGG(x1, x2, x3, x4, x5, x6) = U7_GGG(x6)
MERGE_IN_GGG(x1, x2, x3) = MERGE_IN_GGG(x1, x2, x3)
MERGESORT_IN_AG(x1, x2) = MERGESORT_IN_AG(x2)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x3, x4, x5, x6)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x3, x5)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x3, x4, x5, x6)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_GAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → U5_GAA(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_GA(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_GA(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGA(Y1s, Y2s, Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGG(Y1s, Y2s, Ys)
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGG(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_GGG(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGG(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x6)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x2, x4, x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, x6)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x5)
U6_GGG(x1, x2, x3, x4, x5, x6) = U6_GGG(x2, x4, x5, x6)
U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x5, x6)
=_IN_AA(x1, x2) = =_IN_AA
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
U7_GGG(x1, x2, x3, x4, x5, x6) = U7_GGG(x6)
MERGE_IN_GGG(x1, x2, x3) = MERGE_IN_GGG(x1, x2, x3)
MERGESORT_IN_AG(x1, x2) = MERGESORT_IN_AG(x2)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x3, x4, x5, x6)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x3, x5)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x3, x4, x5, x6)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 5 SCCs with 18 less nodes.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGG(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGG(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x6)
U6_GGG(x1, x2, x3, x4, x5, x6) = U6_GGG(x2, x4, x5, x6)
MERGE_IN_GGG(x1, x2, x3) = MERGE_IN_GGG(x1, x2, x3)
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
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGG(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGG(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
=_in_aa(X, X) → =_out_aa(X, X)
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U6_GGG(x1, x2, x3, x4, x5, x6) = U6_GGG(x2, x4, x5, x6)
MERGE_IN_GGG(x1, x2, x3) = MERGE_IN_GGG(x1, x2, x3)
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
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U6_GGG(Xs, Ys, Zs, =_out_aa) → MERGE_IN_GGG(.(Xs), Ys, Zs)
MERGE_IN_GGG(.(Xs), .(Ys), .(Zs)) → U6_GGG(Xs, Ys, Zs, =_in_aa)
The TRS R consists of the following rules:
=_in_aa → =_out_aa
The set Q consists of the following terms:
=_in_aa
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:
- U6_GGG(Xs, Ys, Zs, =_out_aa) → MERGE_IN_GGG(.(Xs), Ys, Zs)
The graph contains the following edges 2 >= 2, 3 >= 3
- MERGE_IN_GGG(.(Xs), .(Ys), .(Zs)) → U6_GGG(Xs, Ys, Zs, =_in_aa)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x2, x4, x6)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
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
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
The TRS R consists of the following rules:
=_in_aa(X, X) → =_out_aa(X, X)
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x2, x4, x6)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
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
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U6_GGA(Xs, Ys, =_out_aa) → MERGE_IN_GGA(.(Xs), Ys)
MERGE_IN_GGA(.(Xs), .(Ys)) → U6_GGA(Xs, Ys, =_in_aa)
The TRS R consists of the following rules:
=_in_aa → =_out_aa
The set Q consists of the following terms:
=_in_aa
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:
- MERGE_IN_GGA(.(Xs), .(Ys)) → U6_GGA(Xs, Ys, =_in_aa)
The graph contains the following edges 1 > 1, 2 > 2
- U6_GGA(Xs, Ys, =_out_aa) → MERGE_IN_GGA(.(Xs), Ys)
The graph contains the following edges 2 >= 2
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x6)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(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
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(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
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
SPLIT_IN_GAA(.(Xs)) → SPLIT_IN_GAA(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_GAA(.(Xs)) → SPLIT_IN_GAA(Xs)
The graph contains the following edges 1 > 1
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, x6)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)
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
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
=_in_aa(X, X) → =_out_aa(X, X)
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, x6)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)
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
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ RuleRemovalProof
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
U1_GA(split_out_gaa(X1s, X2s)) → U2_GA(X2s, mergesort_in_ga(X1s))
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(split_in_gaa(.(.(Xs))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.([])) → mergesort_out_ga(.([]))
mergesort_in_ga(.(.(Xs))) → U1_ga(split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [])
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(merge_out_gga(Zs)) → merge_out_gga(.(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0)
We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
mergesort_in_ga(.([])) → mergesort_out_ga(.([]))
Used ordering: POLO with Polynomial interpretation [25]:
POL(.(x1)) = 2 + x1
POL(=_in_aa) = 2
POL(=_out_aa) = 2
POL(MERGESORT_IN_GA(x1)) = 2·x1
POL(U1_GA(x1)) = 2·x1
POL(U1_ga(x1)) = 2·x1
POL(U2_GA(x1, x2)) = 2·x1 + x2
POL(U2_ga(x1, x2)) = 2·x1 + x2
POL(U3_ga(x1, x2)) = x1 + x2
POL(U4_ga(x1)) = x1
POL(U5_gaa(x1)) = 2 + x1
POL(U6_gga(x1, x2, x3)) = x1 + x2 + 2·x3
POL(U7_gga(x1)) = 2 + x1
POL([]) = 0
POL(merge_in_gga(x1, x2)) = x1 + x2
POL(merge_out_gga(x1)) = x1
POL(mergesort_in_ga(x1)) = 2·x1
POL(mergesort_out_ga(x1)) = x1
POL(split_in_gaa(x1)) = x1
POL(split_out_gaa(x1, x2)) = x1 + x2
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Rewriting
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
U1_GA(split_out_gaa(X1s, X2s)) → U2_GA(X2s, mergesort_in_ga(X1s))
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(split_in_gaa(.(.(Xs))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(.(Xs))) → U1_ga(split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [])
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(merge_out_gga(Zs)) → merge_out_gga(.(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GA(.(.(Xs))) → U1_GA(split_in_gaa(.(.(Xs)))) at position [0] we obtained the following new rules:
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(split_in_gaa(.(Xs))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(split_in_gaa(.(Xs))))
U1_GA(split_out_gaa(X1s, X2s)) → U2_GA(X2s, mergesort_in_ga(X1s))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(.(Xs))) → U1_ga(split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [])
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(merge_out_gga(Zs)) → merge_out_gga(.(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(split_in_gaa(.(Xs)))) at position [0,0] we obtained the following new rules:
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(U5_gaa(split_in_gaa(Xs))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(U5_gaa(split_in_gaa(Xs))))
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
U1_GA(split_out_gaa(X1s, X2s)) → U2_GA(X2s, mergesort_in_ga(X1s))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(.(Xs))) → U1_ga(split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [])
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(merge_out_gga(Zs)) → merge_out_gga(.(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(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.
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
The remaining pairs can at least be oriented weakly.
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(U5_gaa(split_in_gaa(Xs))))
U1_GA(split_out_gaa(X1s, X2s)) → U2_GA(X2s, mergesort_in_ga(X1s))
Used ordering: Polynomial interpretation [25]:
POL(.(x1)) = 0
POL(=_in_aa) = 0
POL(=_out_aa) = 0
POL(MERGESORT_IN_GA(x1)) = 0
POL(U1_GA(x1)) = x1
POL(U1_ga(x1)) = x1
POL(U2_GA(x1, x2)) = x2
POL(U2_ga(x1, x2)) = x2
POL(U3_ga(x1, x2)) = 1
POL(U4_ga(x1)) = 1
POL(U5_gaa(x1)) = 0
POL(U6_gga(x1, x2, x3)) = 0
POL(U7_gga(x1)) = 0
POL([]) = 1
POL(merge_in_gga(x1, x2)) = 0
POL(merge_out_gga(x1)) = 0
POL(mergesort_in_ga(x1)) = x1
POL(mergesort_out_ga(x1)) = 1
POL(split_in_gaa(x1)) = 0
POL(split_out_gaa(x1, x2)) = x1
The following usable rules [17] were oriented:
mergesort_in_ga(.(.(Xs))) → U1_ga(split_in_gaa(.(.(Xs))))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
mergesort_in_ga([]) → mergesort_out_ga([])
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(U5_gaa(split_in_gaa(Xs))))
U1_GA(split_out_gaa(X1s, X2s)) → U2_GA(X2s, mergesort_in_ga(X1s))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(.(Xs))) → U1_ga(split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [])
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(merge_out_gga(Zs)) → merge_out_gga(.(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(U5_gaa(split_in_gaa(Xs))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(.(Xs))) → U1_ga(split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [])
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(merge_out_gga(Zs)) → merge_out_gga(.(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0)
We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(U5_gaa(split_in_gaa(Xs))))
The TRS R consists of the following rules:
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
split_in_gaa([]) → split_out_gaa([], [])
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0)
U5_gaa(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0)
We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
mergesort_in_ga(x0)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
U4_ga(x0)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0)
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(U5_gaa(U5_gaa(split_in_gaa(Xs))))
The TRS R consists of the following rules:
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
split_in_gaa([]) → split_out_gaa([], [])
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
The set Q consists of the following terms:
split_in_gaa(x0)
U5_gaa(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_GA(.(.(Xs))) → U1_GA(U5_gaa(U5_gaa(split_in_gaa(Xs))))
The remaining pairs can at least be oriented weakly.
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( split_out_gaa(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( split_in_gaa(x1) ) = | | + | | · | x1 |
Tuple symbols:
| M( MERGESORT_IN_GA(x1) ) = | 0 | + | | · | x1 |
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:
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
split_in_gaa([]) → split_out_gaa([], [])
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ PiDP
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
The TRS R consists of the following rules:
split_in_gaa(.(Xs)) → U5_gaa(split_in_gaa(Xs))
split_in_gaa([]) → split_out_gaa([], [])
U5_gaa(split_out_gaa(Zs, Ys)) → split_out_gaa(.(Ys), Zs)
The set Q consists of the following terms:
split_in_gaa(x0)
U5_gaa(x0)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x6)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
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
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PrologToPiTRSProof
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
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
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
↳ PrologToPiTRSProof
Q DP problem:
The TRS P consists of the following rules:
SPLIT_IN_AAA → SPLIT_IN_AAA
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
SPLIT_IN_AAA → SPLIT_IN_AAA
The TRS R consists of the following rules:none
s = SPLIT_IN_AAA evaluates to t =SPLIT_IN_AAA
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from SPLIT_IN_AAA to SPLIT_IN_AAA.
We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort_in: (f,b) (b,f)
split_in: (f,f,f) (b,f,f)
merge_in: (b,b,f) (b,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1, x2)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x4, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x2, x4, x5, x6)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
Pi-finite rewrite system:
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1, x2)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x4, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x2, x4, x5, x6)
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_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_GAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → U5_GAA(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_GA(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_GA(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGA(Y1s, Y2s, Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGG(Y1s, Y2s, Ys)
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGG(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_GGG(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGG(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1, x2)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x4, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x2, x4, x5, x6)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x2, x4, x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x2, x4, x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x5, x6)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x2, x5)
U6_GGG(x1, x2, x3, x4, x5, x6) = U6_GGG(x2, x4, x5, x6)
U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x3, x5, x6)
=_IN_AA(x1, x2) = =_IN_AA
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x3, x5)
U7_GGG(x1, x2, x3, x4, x5, x6) = U7_GGG(x2, x4, x5, x6)
MERGE_IN_GGG(x1, x2, x3) = MERGE_IN_GGG(x1, x2, x3)
MERGESORT_IN_AG(x1, x2) = MERGESORT_IN_AG(x2)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x3, x4, x5, x6)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x3, x4, x5)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x3, x4, x5, x6)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x3, x5)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
Pi DP problem:
The TRS P consists of the following rules:
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → U1_AG(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_AG(.(X, .(Y, Xs)), Ys) → SPLIT_IN_AAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → U5_AAA(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_AG(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_AG(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → SPLIT_IN_GAA(.(X, .(Y, Xs)), X1s, X2s)
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → U5_GAA(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_GA(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_GA(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U3_GA(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGA(Y1s, Y2s, Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_AG(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U2_AG(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_AG(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
U3_AG(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → MERGE_IN_GGG(Y1s, Y2s, Ys)
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGG(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AA(X, Y)
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_GGG(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGG(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1, x2)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x4, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x2, x4, x5, x6)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x2, x4, x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x2, x4, x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x5, x6)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x2, x5)
U6_GGG(x1, x2, x3, x4, x5, x6) = U6_GGG(x2, x4, x5, x6)
U5_AAA(x1, x2, x3, x4, x5) = U5_AAA(x5)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x3, x5, x6)
=_IN_AA(x1, x2) = =_IN_AA
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x3, x5)
U7_GGG(x1, x2, x3, x4, x5, x6) = U7_GGG(x2, x4, x5, x6)
MERGE_IN_GGG(x1, x2, x3) = MERGE_IN_GGG(x1, x2, x3)
MERGESORT_IN_AG(x1, x2) = MERGESORT_IN_AG(x2)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
U3_AG(x1, x2, x3, x4, x5, x6) = U3_AG(x3, x4, x5, x6)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x3, x4, x5)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U2_AG(x1, x2, x3, x4, x5, x6) = U2_AG(x3, x4, x5, x6)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x3, x5)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 5 SCCs with 18 less nodes.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGG(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGG(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1, x2)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x4, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x2, x4, x5, x6)
U6_GGG(x1, x2, x3, x4, x5, x6) = U6_GGG(x2, x4, x5, x6)
MERGE_IN_GGG(x1, x2, x3) = MERGE_IN_GGG(x1, x2, x3)
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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGG(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGG(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_GGG(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGG(.(X, Xs), Ys, Zs)
The TRS R consists of the following rules:
=_in_aa(X, X) → =_out_aa(X, X)
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U6_GGG(x1, x2, x3, x4, x5, x6) = U6_GGG(x2, x4, x5, x6)
MERGE_IN_GGG(x1, x2, x3) = MERGE_IN_GGG(x1, x2, x3)
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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U6_GGG(Xs, Ys, Zs, =_out_aa) → MERGE_IN_GGG(.(Xs), Ys, Zs)
MERGE_IN_GGG(.(Xs), .(Ys), .(Zs)) → U6_GGG(Xs, Ys, Zs, =_in_aa)
The TRS R consists of the following rules:
=_in_aa → =_out_aa
The set Q consists of the following terms:
=_in_aa
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:
- U6_GGG(Xs, Ys, Zs, =_out_aa) → MERGE_IN_GGG(.(Xs), Ys, Zs)
The graph contains the following edges 2 >= 2, 3 >= 3
- MERGE_IN_GGG(.(Xs), .(Ys), .(Zs)) → U6_GGG(Xs, Ys, Zs, =_in_aa)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1, x2)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x4, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x2, x4, x5, x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x2, x4, x6)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
U6_GGA(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
The TRS R consists of the following rules:
=_in_aa(X, X) → =_out_aa(X, X)
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x2, x4, x6)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U6_GGA(Xs, Ys, =_out_aa) → MERGE_IN_GGA(.(Xs), Ys)
MERGE_IN_GGA(.(Xs), .(Ys)) → U6_GGA(Xs, Ys, =_in_aa)
The TRS R consists of the following rules:
=_in_aa → =_out_aa
The set Q consists of the following terms:
=_in_aa
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:
- MERGE_IN_GGA(.(Xs), .(Ys)) → U6_GGA(Xs, Ys, =_in_aa)
The graph contains the following edges 1 > 1, 2 > 2
- U6_GGA(Xs, Ys, =_out_aa) → MERGE_IN_GGA(.(Xs), Ys)
The graph contains the following edges 2 >= 2
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1, x2)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x4, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x2, x4, x5, x6)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN_GAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_GAA(Xs, Zs, Ys)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
SPLIT_IN_GAA(.(Xs)) → SPLIT_IN_GAA(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_GAA(.(Xs)) → SPLIT_IN_GAA(Xs)
The graph contains the following edges 1 > 1
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1, x2)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x4, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x2, x4, x5, x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x5, x6)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x3, x5)
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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_GA(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U1_GA(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s, Y1s)
MERGESORT_IN_GA(.(X, .(Y, Xs)), Ys) → U1_GA(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
U2_GA(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s, Y2s)
The TRS R consists of the following rules:
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
=_in_aa(X, X) → =_out_aa(X, X)
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x2)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x5, x6)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x3, x5)
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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → U2_GA(Xs, X2s, mergesort_in_ga(X1s))
U2_GA(Xs, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(Xs, split_in_gaa(.(.(Xs))))
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s)
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([], [])
mergesort_in_ga(.([])) → mergesort_out_ga(.([]), .([]))
mergesort_in_ga(.(.(Xs))) → U1_ga(Xs, split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(Xs, split_in_gaa(Xs))
U1_ga(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → U2_ga(Xs, X2s, mergesort_in_ga(X1s))
U5_gaa(Xs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(Xs), .(Ys), Zs)
U2_ga(Xs, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(Xs, Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [], [])
U3_ga(Xs, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(Xs, merge_in_gga(Y1s, Y2s))
U4_ga(Xs, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(.(Xs)), Ys)
merge_in_gga([], Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(Xs, Ys, merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(Xs, Ys, merge_out_gga(.(Xs), Ys, Zs)) → merge_out_gga(.(Xs), .(Ys), .(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
U4_ga(x0, x1)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GA(.(.(Xs))) → U1_GA(Xs, split_in_gaa(.(.(Xs)))) at position [1] we obtained the following new rules:
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(Xs, U5_gaa(.(Xs), split_in_gaa(.(Xs))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → U2_GA(Xs, X2s, mergesort_in_ga(X1s))
U2_GA(Xs, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s)
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s)
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(Xs, U5_gaa(.(Xs), split_in_gaa(.(Xs))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([], [])
mergesort_in_ga(.([])) → mergesort_out_ga(.([]), .([]))
mergesort_in_ga(.(.(Xs))) → U1_ga(Xs, split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(Xs, split_in_gaa(Xs))
U1_ga(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → U2_ga(Xs, X2s, mergesort_in_ga(X1s))
U5_gaa(Xs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(Xs), .(Ys), Zs)
U2_ga(Xs, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(Xs, Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [], [])
U3_ga(Xs, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(Xs, merge_in_gga(Y1s, Y2s))
U4_ga(Xs, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(.(Xs)), Ys)
merge_in_gga([], Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(Xs, Ys, merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(Xs, Ys, merge_out_gga(.(Xs), Ys, Zs)) → merge_out_gga(.(Xs), .(Ys), .(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
U4_ga(x0, x1)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GA(.(.(Xs))) → U1_GA(Xs, U5_gaa(.(Xs), split_in_gaa(.(Xs)))) at position [1,1] we obtained the following new rules:
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(Xs, U5_gaa(.(Xs), U5_gaa(Xs, split_in_gaa(Xs))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
MERGESORT_IN_GA(.(.(Xs))) → U1_GA(Xs, U5_gaa(.(Xs), U5_gaa(Xs, split_in_gaa(Xs))))
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → U2_GA(Xs, X2s, mergesort_in_ga(X1s))
U2_GA(Xs, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s)
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s)
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([], [])
mergesort_in_ga(.([])) → mergesort_out_ga(.([]), .([]))
mergesort_in_ga(.(.(Xs))) → U1_ga(Xs, split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(Xs, split_in_gaa(Xs))
U1_ga(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → U2_ga(Xs, X2s, mergesort_in_ga(X1s))
U5_gaa(Xs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(Xs), .(Ys), Zs)
U2_ga(Xs, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(Xs, Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [], [])
U3_ga(Xs, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(Xs, merge_in_gga(Y1s, Y2s))
U4_ga(Xs, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(.(Xs)), Ys)
merge_in_gga([], Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(Xs, Ys, merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(Xs, Ys, merge_out_gga(.(Xs), Ys, Zs)) → merge_out_gga(.(Xs), .(Ys), .(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
U4_ga(x0, x1)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0, x1, x2)
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_GA(.(.(Xs))) → U1_GA(Xs, U5_gaa(.(Xs), U5_gaa(Xs, split_in_gaa(Xs))))
The remaining pairs can at least be oriented weakly.
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → U2_GA(Xs, X2s, mergesort_in_ga(X1s))
U2_GA(Xs, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s)
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( mergesort_out_ga(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( mergesort_in_ga(x1) ) = | | + | | · | x1 |
| M( U7_gga(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( merge_in_gga(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U5_gaa(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U6_gga(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( split_out_gaa(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( U1_ga(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U4_ga(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( merge_out_gga(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( U3_ga(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( U2_ga(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( split_in_gaa(x1) ) = | | + | | · | x1 |
Tuple symbols:
| M( U1_GA(x1, x2) ) = | 1 | + | | · | x1 | + | | · | x2 |
| M( U2_GA(x1, ..., x3) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( MERGESORT_IN_GA(x1) ) = | 1 | + | | · | x1 |
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:
split_in_gaa([]) → split_out_gaa([], [], [])
split_in_gaa(.(Xs)) → U5_gaa(Xs, split_in_gaa(Xs))
U5_gaa(Xs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(Xs), .(Ys), Zs)
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → U2_GA(Xs, X2s, mergesort_in_ga(X1s))
U2_GA(Xs, X2s, mergesort_out_ga(X1s, Y1s)) → MERGESORT_IN_GA(X2s)
U1_GA(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → MERGESORT_IN_GA(X1s)
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([], [])
mergesort_in_ga(.([])) → mergesort_out_ga(.([]), .([]))
mergesort_in_ga(.(.(Xs))) → U1_ga(Xs, split_in_gaa(.(.(Xs))))
split_in_gaa(.(Xs)) → U5_gaa(Xs, split_in_gaa(Xs))
U1_ga(Xs, split_out_gaa(.(.(Xs)), X1s, X2s)) → U2_ga(Xs, X2s, mergesort_in_ga(X1s))
U5_gaa(Xs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(Xs), .(Ys), Zs)
U2_ga(Xs, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(Xs, Y1s, mergesort_in_ga(X2s))
split_in_gaa([]) → split_out_gaa([], [], [])
U3_ga(Xs, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(Xs, merge_in_gga(Y1s, Y2s))
U4_ga(Xs, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(.(Xs)), Ys)
merge_in_gga([], Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(Xs), .(Ys)) → U6_gga(Xs, Ys, =_in_aa)
U6_gga(Xs, Ys, =_out_aa) → U7_gga(Xs, Ys, merge_in_gga(.(Xs), Ys))
=_in_aa → =_out_aa
U7_gga(Xs, Ys, merge_out_gga(.(Xs), Ys, Zs)) → merge_out_gga(.(Xs), .(Ys), .(Zs))
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U1_ga(x0, x1)
U5_gaa(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
U4_ga(x0, x1)
merge_in_gga(x0, x1)
U6_gga(x0, x1, x2)
=_in_aa
U7_gga(x0, x1, x2)
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.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
The TRS R consists of the following rules:
mergesort_in_ag([], []) → mergesort_out_ag([], [])
mergesort_in_ag(.(X, []), .(X, [])) → mergesort_out_ag(.(X, []), .(X, []))
mergesort_in_ag(.(X, .(Y, Xs)), Ys) → U1_ag(X, Y, Xs, Ys, split_in_aaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_aaa([], [], []) → split_out_aaa([], [], [])
split_in_aaa(.(X, Xs), .(X, Ys), Zs) → U5_aaa(X, Xs, Ys, Zs, split_in_aaa(Xs, Zs, Ys))
U5_aaa(X, Xs, Ys, Zs, split_out_aaa(Xs, Zs, Ys)) → split_out_aaa(.(X, Xs), .(X, Ys), Zs)
U1_ag(X, Y, Xs, Ys, split_out_aaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ag(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
mergesort_in_ga([], []) → mergesort_out_ga([], [])
mergesort_in_ga(.(X, []), .(X, [])) → mergesort_out_ga(.(X, []), .(X, []))
mergesort_in_ga(.(X, .(Y, Xs)), Ys) → U1_ga(X, Y, Xs, Ys, split_in_gaa(.(X, .(Y, Xs)), X1s, X2s))
split_in_gaa([], [], []) → split_out_gaa([], [], [])
split_in_gaa(.(X, Xs), .(X, Ys), Zs) → U5_gaa(X, Xs, Ys, Zs, split_in_gaa(Xs, Zs, Ys))
U5_gaa(X, Xs, Ys, Zs, split_out_gaa(Xs, Zs, Ys)) → split_out_gaa(.(X, Xs), .(X, Ys), Zs)
U1_ga(X, Y, Xs, Ys, split_out_gaa(.(X, .(Y, Xs)), X1s, X2s)) → U2_ga(X, Y, Xs, Ys, X2s, mergesort_in_ga(X1s, Y1s))
U2_ga(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ga(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ga(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ga(X, Y, Xs, Ys, merge_in_gga(Y1s, Y2s, Ys))
merge_in_gga([], Xs, Xs) → merge_out_gga([], Xs, Xs)
merge_in_gga(Xs, [], Xs) → merge_out_gga(Xs, [], Xs)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
=_in_aa(X, X) → =_out_aa(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U7_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
U2_ag(X, Y, Xs, Ys, X2s, mergesort_out_ga(X1s, Y1s)) → U3_ag(X, Y, Xs, Ys, Y1s, mergesort_in_ga(X2s, Y2s))
U3_ag(X, Y, Xs, Ys, Y1s, mergesort_out_ga(X2s, Y2s)) → U4_ag(X, Y, Xs, Ys, merge_in_ggg(Y1s, Y2s, Ys))
merge_in_ggg([], Xs, Xs) → merge_out_ggg([], Xs, Xs)
merge_in_ggg(Xs, [], Xs) → merge_out_ggg(Xs, [], Xs)
merge_in_ggg(.(X, Xs), .(Y, Ys), .(X, Zs)) → U6_ggg(X, Xs, Y, Ys, Zs, =_in_aa(X, Y))
U6_ggg(X, Xs, Y, Ys, Zs, =_out_aa(X, Y)) → U7_ggg(X, Xs, Y, Ys, Zs, merge_in_ggg(.(X, Xs), Ys, Zs))
U7_ggg(X, Xs, Y, Ys, Zs, merge_out_ggg(.(X, Xs), Ys, Zs)) → merge_out_ggg(.(X, Xs), .(Y, Ys), .(X, Zs))
U4_ag(X, Y, Xs, Ys, merge_out_ggg(Y1s, Y2s, Ys)) → mergesort_out_ag(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ag(x1, x2) = mergesort_in_ag(x2)
[] = []
mergesort_out_ag(x1, x2) = mergesort_out_ag(x1, x2)
.(x1, x2) = .(x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
split_in_aaa(x1, x2, x3) = split_in_aaa
split_out_aaa(x1, x2, x3) = split_out_aaa(x1, x2, x3)
U5_aaa(x1, x2, x3, x4, x5) = U5_aaa(x5)
U2_ag(x1, x2, x3, x4, x5, x6) = U2_ag(x3, x4, x5, x6)
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
mergesort_out_ga(x1, x2) = mergesort_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x3, x5)
split_in_gaa(x1, x2, x3) = split_in_gaa(x1)
split_out_gaa(x1, x2, x3) = split_out_gaa(x1, x2, x3)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x5, x6)
U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x3, x5, x6)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x3, x5)
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x2, x4, x6)
=_in_aa(x1, x2) = =_in_aa
=_out_aa(x1, x2) = =_out_aa
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x2, x4, x6)
U3_ag(x1, x2, x3, x4, x5, x6) = U3_ag(x3, x4, x5, x6)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x3, x4, x5)
merge_in_ggg(x1, x2, x3) = merge_in_ggg(x1, x2, x3)
merge_out_ggg(x1, x2, x3) = merge_out_ggg(x1, x2, x3)
U6_ggg(x1, x2, x3, x4, x5, x6) = U6_ggg(x2, x4, x5, x6)
U7_ggg(x1, x2, x3, x4, x5, x6) = U7_ggg(x2, x4, x5, x6)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
Pi DP problem:
The TRS P consists of the following rules:
SPLIT_IN_AAA(.(X, Xs), .(X, Ys), Zs) → SPLIT_IN_AAA(Xs, Zs, Ys)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
SPLIT_IN_AAA(x1, x2, x3) = SPLIT_IN_AAA
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
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
Q DP problem:
The TRS P consists of the following rules:
SPLIT_IN_AAA → SPLIT_IN_AAA
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
SPLIT_IN_AAA → SPLIT_IN_AAA
The TRS R consists of the following rules:none
s = SPLIT_IN_AAA evaluates to t =SPLIT_IN_AAA
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from SPLIT_IN_AAA to SPLIT_IN_AAA.