YES
Left Termination proof of ../tpdb/LP/terminweb/old-terminweb/mergesort.pl
Left Termination of the query pattern
mergesort_in_2(g, a)
w.r.t. the given Prolog program could successfully be proven:
↳ Prolog
↳ PredefinedPredicateTransformerProof
Clauses:
mergesort([], []).
mergesort(.(X, []), .(X, [])).
mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))).
split([], [], []).
split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys).
merge([], Xs, Xs).
merge(Xs, [], Xs).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(=<(X, Y), merge(Xs, .(Y, Ys), Zs)).
merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(>(X, Y), merge(.(X, Xs), Ys, Zs)).
Queries:
mergesort(g,a).
Added definitions of predefined predicates.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
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)) :- ','(','(=(X1, X), ','(=(X2, Y), ;(=(X1, X2), isLess(X1, X2)))), merge(Xs, .(Y, Ys), Zs)).
merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(','(=(X1, X), ','(=(X2, Y), isGreater(X1, X2))), merge(.(X, Xs), Ys, Zs)).
isPlus(zero, X, X).
isPlus(succ(X), zero, succ(X)).
isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z).
isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), zero, pred(X)).
isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z).
isMinus(X, zero, X).
isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z).
isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z).
isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z).
isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z).
isTimes(X, zero, zero).
isTimes(zero, succ(Y), zero).
isTimes(zero, pred(Y), zero).
isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).
isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).
isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).
isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).
isDiv(zero, succ(Y), zero).
isDiv(zero, pred(Y), zero).
isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)).
isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).
isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).
isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)).
isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).
isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).
isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).
=(X, X).
isGreater(succ(X), zero).
isGreater(succ(X), pred(Y)).
isGreater(succ(X), succ(Y)) :- isGreater(X, Y).
isGreater(zero, pred(Y)).
isGreater(pred(X), pred(Y)) :- isGreater(X, Y).
isLess(pred(X), zero).
isLess(pred(X), succ(Y)).
isLess(pred(X), pred(Y)) :- isLess(X, Y).
isLess(zero, succ(Y)).
isLess(succ(X), succ(Y)) :- isLess(X, Y).
Queries:
mergesort(g,a).
Eliminated all or-constructs.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
Clauses:
mergesort([], []).
mergesort(.(X, []), .(X, [])).
mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys)))).
split([], [], []).
split(.(X, Xs), .(X, Ys), Zs) :- split(Xs, Zs, Ys).
merge([], Xs, Xs).
merge(Xs, [], Xs).
merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(=(X1, X), ','(=(X2, Y), ','(isGreater(X1, X2), merge(.(X, Xs), Ys, Zs)))).
isPlus(zero, X, X).
isPlus(succ(X), zero, succ(X)).
isPlus(succ(X), succ(Y), succ(succ(Z))) :- isPlus(X, Y, Z).
isPlus(succ(X), pred(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), zero, pred(X)).
isPlus(pred(X), succ(Y), Z) :- isPlus(X, Y, Z).
isPlus(pred(X), pred(Y), pred(pred(Z))) :- isPlus(X, Y, Z).
isMinus(X, zero, X).
isMinus(zero, succ(Y), pred(Z)) :- isMinus(zero, Y, Z).
isMinus(zero, pred(Y), succ(Z)) :- isMinus(zero, Y, Z).
isMinus(succ(X), succ(Y), Z) :- isMinus(X, Y, Z).
isMinus(succ(X), pred(Y), succ(succ(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), succ(Y), pred(pred(Z))) :- isMinus(X, Y, Z).
isMinus(pred(X), pred(Y), Z) :- isMinus(X, Y, Z).
isTimes(X, zero, zero).
isTimes(zero, succ(Y), zero).
isTimes(zero, pred(Y), zero).
isTimes(succ(X), succ(Y), Z) :- ','(isTimes(succ(X), Y, A), isPlus(A, succ(X), Z)).
isTimes(succ(X), pred(Y), Z) :- ','(isTimes(succ(X), Y, A), isMinus(A, succ(X), Z)).
isTimes(pred(X), succ(Y), Z) :- ','(isTimes(pred(X), Y, A), isPlus(A, pred(X), Z)).
isTimes(pred(X), pred(Y), Z) :- ','(isTimes(pred(X), Y, A), isMinus(A, pred(X), Z)).
isDiv(zero, succ(Y), zero).
isDiv(zero, pred(Y), zero).
isDiv(succ(X), succ(Y), zero) :- isMinus(succ(X), succ(Y), pred(Z)).
isDiv(succ(X), succ(Y), succ(Z)) :- ','(isMinus(succ(X), succ(Y), A), isDiv(A, succ(Y), Z)).
isDiv(succ(X), pred(Y), Z) :- ','(isMinus(zero, pred(Y), A), ','(isDiv(succ(X), A, B), isMinus(zero, B, Z))).
isDiv(pred(X), pred(Y), zero) :- isMinus(pred(X), pred(Y), succ(Z)).
isDiv(pred(X), pred(Y), succ(Z)) :- ','(isMinus(pred(X), pred(Y), A), isDiv(A, pred(Y), Z)).
isDiv(pred(X), succ(Y), Z) :- ','(isMinus(zero, pred(X), A), ','(isDiv(A, succ(Y), B), isMinus(zero, B, Z))).
isModulo(X, Y, Z) :- ','(isDiv(X, Y, A), ','(isTimes(A, Y, B), isMinus(X, B, Z))).
=(X, X).
isGreater(succ(X), zero).
isGreater(succ(X), pred(Y)).
isGreater(succ(X), succ(Y)) :- isGreater(X, Y).
isGreater(zero, pred(Y)).
isGreater(pred(X), pred(Y)) :- isGreater(X, Y).
isLess(pred(X), zero).
isLess(pred(X), succ(Y)).
isLess(pred(X), pred(Y)) :- isLess(X, Y).
isLess(zero, succ(Y)).
isLess(succ(X), succ(Y)) :- isLess(X, Y).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(=(X1, X), ','(=(X2, Y), ','(=(X1, X2), merge(Xs, .(Y, Ys), Zs)))).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(=(X1, X), ','(=(X2, Y), ','(isLess(X1, X2), merge(Xs, .(Y, Ys), Zs)))).
Queries:
mergesort(g,a).
We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort_in: (b,f)
split_in: (b,f,f)
merge_in: (b,b,f)
isGreater_in: (b,b)
isLess_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
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([], [], []) → 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), .(Y, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U7_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_gga(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U8_gga(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U47_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_gga(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_gga(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U50_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_gga(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_gga(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U52_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
.(x1, x2) = .(x1, 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(x1, 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(x1, x2, x3, x4, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U7_gga(x1, x2, x3, x4, x5, x6, x7) = U7_gga(x1, x2, x3, x4, x6, x7)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x3, x6)
U47_gga(x1, x2, x3, x4, x5, x6) = U47_gga(x1, x2, x3, x4, x6)
U48_gga(x1, x2, x3, x4, x5, x6, x7) = U48_gga(x1, x2, x3, x4, x6, x7)
U49_gga(x1, x2, x3, x4, x5, x6) = U49_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_gga(x1, x2, x3, x4, x5, x6) = U50_gga(x1, x6)
U51_gga(x1, x2, x3, x4, x5, x6) = U51_gga(x1, x2, x3, x4, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_gga(x1, x2, x3, x4, x5, x6) = U52_gga(x1, x6)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
Pi-finite rewrite system:
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([], [], []) → 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), .(Y, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U7_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_gga(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U8_gga(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U47_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_gga(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_gga(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U50_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_gga(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_gga(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U52_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
.(x1, x2) = .(x1, 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(x1, 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(x1, x2, x3, x4, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U7_gga(x1, x2, x3, x4, x5, x6, x7) = U7_gga(x1, x2, x3, x4, x6, x7)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x3, x6)
U47_gga(x1, x2, x3, x4, x5, x6) = U47_gga(x1, x2, x3, x4, x6)
U48_gga(x1, x2, x3, x4, x5, x6, x7) = U48_gga(x1, x2, x3, x4, x6, x7)
U49_gga(x1, x2, x3, x4, x5, x6) = U49_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_gga(x1, x2, x3, x4, x5, x6) = U50_gga(x1, x6)
U51_gga(x1, x2, x3, x4, x5, x6) = U51_gga(x1, x2, x3, x4, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_gga(x1, x2, x3, x4, x5, x6) = U52_gga(x1, 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_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), .(Y, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → =_IN_AG(X1, X)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_GGA(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → =_IN_AG(X2, Y)
U7_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_GGA(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
U7_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → ISGREATER_IN_GG(X1, X2)
ISGREATER_IN_GG(succ(X), succ(Y)) → U43_GG(X, Y, isGreater_in_gg(X, Y))
ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → U44_GG(X, Y, isGreater_in_gg(X, Y))
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)
U8_GGA(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U8_GGA(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_GGA(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AG(X1, X)
U47_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_GGA(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U47_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → =_IN_AG(X2, Y)
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → =_IN_GG(X1, X2)
U49_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U49_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_GGA(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → ISLESS_IN_GG(X1, X2)
ISLESS_IN_GG(pred(X), pred(Y)) → U45_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → U46_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
U51_GGA(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U51_GGA(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
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([], [], []) → 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), .(Y, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U7_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_gga(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U8_gga(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U47_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_gga(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_gga(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U50_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_gga(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_gga(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U52_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
.(x1, x2) = .(x1, 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(x1, 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(x1, x2, x3, x4, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U7_gga(x1, x2, x3, x4, x5, x6, x7) = U7_gga(x1, x2, x3, x4, x6, x7)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x3, x6)
U47_gga(x1, x2, x3, x4, x5, x6) = U47_gga(x1, x2, x3, x4, x6)
U48_gga(x1, x2, x3, x4, x5, x6, x7) = U48_gga(x1, x2, x3, x4, x6, x7)
U49_gga(x1, x2, x3, x4, x5, x6) = U49_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_gga(x1, x2, x3, x4, x5, x6) = U50_gga(x1, x6)
U51_gga(x1, x2, x3, x4, x5, x6) = U51_gga(x1, x2, x3, x4, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_gga(x1, x2, x3, x4, x5, x6) = U52_gga(x1, x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, x6)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x1, x5)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(x1, x2)
U48_GGA(x1, x2, x3, x4, x5, x6, x7) = U48_GGA(x1, x2, x3, x4, x6, x7)
=_IN_GG(x1, x2) = =_IN_GG(x1, x2)
ISLESS_IN_GG(x1, x2) = ISLESS_IN_GG(x1, x2)
U43_GG(x1, x2, x3) = U43_GG(x3)
U49_GGA(x1, x2, x3, x4, x5, x6) = U49_GGA(x1, x2, x3, x4, x6)
U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x3, x6)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
=_IN_AG(x1, x2) = =_IN_AG(x2)
U44_GG(x1, x2, x3) = U44_GG(x3)
U8_GGA(x1, x2, x3, x4, x5, x6) = U8_GGA(x1, x2, x3, x4, x6)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x5, x6)
U45_GG(x1, x2, x3) = U45_GG(x3)
U51_GGA(x1, x2, x3, x4, x5, x6) = U51_GGA(x1, x2, x3, x4, x6)
U46_GG(x1, x2, x3) = U46_GG(x3)
U47_GGA(x1, x2, x3, x4, x5, x6) = U47_GGA(x1, x2, x3, x4, x6)
U52_GGA(x1, x2, x3, x4, x5, x6) = U52_GGA(x1, x6)
U7_GGA(x1, x2, x3, x4, x5, x6, x7) = U7_GGA(x1, x2, x3, x4, x6, x7)
U50_GGA(x1, x2, x3, x4, x5, x6) = U50_GGA(x1, x6)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5)
SPLIT_IN_GAA(x1, x2, x3) = SPLIT_IN_GAA(x1)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
Pi DP problem:
The TRS P consists of the following rules:
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), .(Y, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → =_IN_AG(X1, X)
U6_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_GGA(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → =_IN_AG(X2, Y)
U7_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_GGA(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
U7_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → ISGREATER_IN_GG(X1, X2)
ISGREATER_IN_GG(succ(X), succ(Y)) → U43_GG(X, Y, isGreater_in_gg(X, Y))
ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → U44_GG(X, Y, isGreater_in_gg(X, Y))
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)
U8_GGA(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U8_GGA(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_GGA(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → =_IN_AG(X1, X)
U47_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_GGA(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U47_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → =_IN_AG(X2, Y)
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → =_IN_GG(X1, X2)
U49_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U49_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_GGA(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → ISLESS_IN_GG(X1, X2)
ISLESS_IN_GG(pred(X), pred(Y)) → U45_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → U46_GG(X, Y, isLess_in_gg(X, Y))
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
U51_GGA(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U51_GGA(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
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([], [], []) → 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), .(Y, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U7_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_gga(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U8_gga(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U47_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_gga(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_gga(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U50_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_gga(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_gga(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U52_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
.(x1, x2) = .(x1, 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(x1, 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(x1, x2, x3, x4, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U7_gga(x1, x2, x3, x4, x5, x6, x7) = U7_gga(x1, x2, x3, x4, x6, x7)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x3, x6)
U47_gga(x1, x2, x3, x4, x5, x6) = U47_gga(x1, x2, x3, x4, x6)
U48_gga(x1, x2, x3, x4, x5, x6, x7) = U48_gga(x1, x2, x3, x4, x6, x7)
U49_gga(x1, x2, x3, x4, x5, x6) = U49_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_gga(x1, x2, x3, x4, x5, x6) = U50_gga(x1, x6)
U51_gga(x1, x2, x3, x4, x5, x6) = U51_gga(x1, x2, x3, x4, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_gga(x1, x2, x3, x4, x5, x6) = U52_gga(x1, x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x5, x6)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x1, x5)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(x1, x2)
U48_GGA(x1, x2, x3, x4, x5, x6, x7) = U48_GGA(x1, x2, x3, x4, x6, x7)
=_IN_GG(x1, x2) = =_IN_GG(x1, x2)
ISLESS_IN_GG(x1, x2) = ISLESS_IN_GG(x1, x2)
U43_GG(x1, x2, x3) = U43_GG(x3)
U49_GGA(x1, x2, x3, x4, x5, x6) = U49_GGA(x1, x2, x3, x4, x6)
U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x3, x6)
MERGESORT_IN_GA(x1, x2) = MERGESORT_IN_GA(x1)
=_IN_AG(x1, x2) = =_IN_AG(x2)
U44_GG(x1, x2, x3) = U44_GG(x3)
U8_GGA(x1, x2, x3, x4, x5, x6) = U8_GGA(x1, x2, x3, x4, x6)
U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x5, x6)
U45_GG(x1, x2, x3) = U45_GG(x3)
U51_GGA(x1, x2, x3, x4, x5, x6) = U51_GGA(x1, x2, x3, x4, x6)
U46_GG(x1, x2, x3) = U46_GG(x3)
U47_GGA(x1, x2, x3, x4, x5, x6) = U47_GGA(x1, x2, x3, x4, x6)
U52_GGA(x1, x2, x3, x4, x5, x6) = U52_GGA(x1, x6)
U7_GGA(x1, x2, x3, x4, x5, x6, x7) = U7_GGA(x1, x2, x3, x4, x6, x7)
U50_GGA(x1, x2, x3, x4, x5, x6) = U50_GGA(x1, x6)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(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 19 less nodes.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
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([], [], []) → 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), .(Y, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U7_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_gga(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U8_gga(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U47_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_gga(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_gga(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U50_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_gga(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_gga(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U52_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
.(x1, x2) = .(x1, 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(x1, 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(x1, x2, x3, x4, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U7_gga(x1, x2, x3, x4, x5, x6, x7) = U7_gga(x1, x2, x3, x4, x6, x7)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x3, x6)
U47_gga(x1, x2, x3, x4, x5, x6) = U47_gga(x1, x2, x3, x4, x6)
U48_gga(x1, x2, x3, x4, x5, x6, x7) = U48_gga(x1, x2, x3, x4, x6, x7)
U49_gga(x1, x2, x3, x4, x5, x6) = U49_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_gga(x1, x2, x3, x4, x5, x6) = U50_gga(x1, x6)
U51_gga(x1, x2, x3, x4, x5, x6) = U51_gga(x1, x2, x3, x4, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_gga(x1, x2, x3, x4, x5, x6) = U52_gga(x1, x6)
ISLESS_IN_GG(x1, x2) = ISLESS_IN_GG(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
↳ OrTransformerProof
↳ Prolog
↳ 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:
ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
R is empty.
Pi is empty.
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
↳ OrTransformerProof
↳ Prolog
↳ 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:
ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
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:
- ISLESS_IN_GG(pred(X), pred(Y)) → ISLESS_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
- ISLESS_IN_GG(succ(X), succ(Y)) → ISLESS_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)
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([], [], []) → 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), .(Y, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U7_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_gga(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U8_gga(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U47_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_gga(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_gga(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U50_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_gga(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_gga(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U52_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
.(x1, x2) = .(x1, 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(x1, 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(x1, x2, x3, x4, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U7_gga(x1, x2, x3, x4, x5, x6, x7) = U7_gga(x1, x2, x3, x4, x6, x7)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x3, x6)
U47_gga(x1, x2, x3, x4, x5, x6) = U47_gga(x1, x2, x3, x4, x6)
U48_gga(x1, x2, x3, x4, x5, x6, x7) = U48_gga(x1, x2, x3, x4, x6, x7)
U49_gga(x1, x2, x3, x4, x5, x6) = U49_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_gga(x1, x2, x3, x4, x5, x6) = U50_gga(x1, x6)
U51_gga(x1, x2, x3, x4, x5, x6) = U51_gga(x1, x2, x3, x4, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_gga(x1, x2, x3, x4, x5, x6) = U52_gga(x1, x6)
ISGREATER_IN_GG(x1, x2) = ISGREATER_IN_GG(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
↳ OrTransformerProof
↳ Prolog
↳ 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:
ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)
R is empty.
Pi is empty.
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
↳ OrTransformerProof
↳ Prolog
↳ 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:
ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)
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:
- ISGREATER_IN_GG(succ(X), succ(Y)) → ISGREATER_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
- ISGREATER_IN_GG(pred(X), pred(Y)) → ISGREATER_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
U7_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_GGA(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
U51_GGA(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_GGA(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_GGA(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U47_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_GGA(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U8_GGA(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
U49_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_GGA(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
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([], [], []) → 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), .(Y, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U7_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_gga(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U8_gga(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U47_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_gga(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_gga(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U50_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_gga(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_gga(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U52_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
.(x1, x2) = .(x1, 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(x1, 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(x1, x2, x3, x4, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U7_gga(x1, x2, x3, x4, x5, x6, x7) = U7_gga(x1, x2, x3, x4, x6, x7)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x3, x6)
U47_gga(x1, x2, x3, x4, x5, x6) = U47_gga(x1, x2, x3, x4, x6)
U48_gga(x1, x2, x3, x4, x5, x6, x7) = U48_gga(x1, x2, x3, x4, x6, x7)
U49_gga(x1, x2, x3, x4, x5, x6) = U49_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_gga(x1, x2, x3, x4, x5, x6) = U50_gga(x1, x6)
U51_gga(x1, x2, x3, x4, x5, x6) = U51_gga(x1, x2, x3, x4, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_gga(x1, x2, x3, x4, x5, x6) = U52_gga(x1, x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U48_GGA(x1, x2, x3, x4, x5, x6, x7) = U48_GGA(x1, x2, x3, x4, x6, x7)
U49_GGA(x1, x2, x3, x4, x5, x6) = U49_GGA(x1, x2, x3, x4, x6)
U8_GGA(x1, x2, x3, x4, x5, x6) = U8_GGA(x1, x2, x3, x4, x6)
U51_GGA(x1, x2, x3, x4, x5, x6) = U51_GGA(x1, x2, x3, x4, x6)
U47_GGA(x1, x2, x3, x4, x5, x6) = U47_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6, x7) = U7_GGA(x1, x2, x3, x4, x6, x7)
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
↳ OrTransformerProof
↳ Prolog
↳ 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:
U7_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_GGA(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
U51_GGA(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_GGA(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_GGA(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U47_GGA(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_GGA(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U8_GGA(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)
U49_GGA(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_GGA(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U6_GGA(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U48_GGA(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_GGA(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
=_in_ag(X, X) → =_out_ag(X, X)
=_in_gg(X, X) → =_out_gg(X, X)
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U48_GGA(x1, x2, x3, x4, x5, x6, x7) = U48_GGA(x1, x2, x3, x4, x6, x7)
U49_GGA(x1, x2, x3, x4, x5, x6) = U49_GGA(x1, x2, x3, x4, x6)
U8_GGA(x1, x2, x3, x4, x5, x6) = U8_GGA(x1, x2, x3, x4, x6)
U51_GGA(x1, x2, x3, x4, x5, x6) = U51_GGA(x1, x2, x3, x4, x6)
U47_GGA(x1, x2, x3, x4, x5, x6) = U47_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6, x7) = U7_GGA(x1, x2, x3, x4, x6, x7)
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
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_in_ag(X))
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_in_ag(Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_in_ag(X))
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_GGA(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
=_in_ag(X) → =_out_ag(X)
=_in_gg(X, X) → =_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U44_gg(isGreater_out_gg) → isGreater_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U46_gg(isLess_out_gg) → isLess_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_ag(x0)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_in_ag(Y)) at position [5] we obtained the following new rules:
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_in_ag(X))
U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_in_ag(Y))
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_in_ag(X))
U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_GGA(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
=_in_ag(X) → =_out_ag(X)
=_in_gg(X, X) → =_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U44_gg(isGreater_out_gg) → isGreater_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U46_gg(isLess_out_gg) → isLess_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_ag(x0)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_in_ag(Y)) at position [5] we obtained the following new rules:
U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_in_ag(X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_in_ag(X))
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_GGA(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
=_in_ag(X) → =_out_ag(X)
=_in_gg(X, X) → =_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U44_gg(isGreater_out_gg) → isGreater_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U46_gg(isLess_out_gg) → isLess_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_ag(x0)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_in_ag(X)) at position [4] we obtained the following new rules:
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_in_ag(X))
U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_GGA(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
=_in_ag(X) → =_out_ag(X)
=_in_gg(X, X) → =_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U44_gg(isGreater_out_gg) → isGreater_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U46_gg(isLess_out_gg) → isLess_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_ag(x0)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_in_ag(X)) at position [4] we obtained the following new rules:
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_GGA(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
=_in_ag(X) → =_out_ag(X)
=_in_gg(X, X) → =_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U44_gg(isGreater_out_gg) → isGreater_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U46_gg(isLess_out_gg) → isLess_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_ag(x0)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(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
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_GGA(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
=_in_gg(X, X) → =_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_ag(x0)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(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.
=_in_ag(x0)
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_GGA(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
=_in_gg(X, X) → =_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U47_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y)) we obtained the following new rules:
U47_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U47_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_GGA(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
=_in_gg(X, X) → =_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_GGA(X, Xs, Y, Ys, isGreater_in_gg(X1, X2)) we obtained the following new rules:
U7_GGA(z0, z1, z2, z3, z4, =_out_ag(z2)) → U8_GGA(z0, z1, z2, z3, isGreater_in_gg(z4, z2))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U7_GGA(z0, z1, z2, z3, z4, =_out_ag(z2)) → U8_GGA(z0, z1, z2, z3, isGreater_in_gg(z4, z2))
U47_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
=_in_gg(X, X) → =_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U6_GGA(X, Xs, Y, Ys, =_out_ag(X1)) → U7_GGA(X, Xs, Y, Ys, X1, =_out_ag(Y)) we obtained the following new rules:
U6_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U7_GGA(z0, z1, z2, z3, z4, =_out_ag(z2)) → U8_GGA(z0, z1, z2, z3, isGreater_in_gg(z4, z2))
U47_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U6_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
=_in_gg(X, X) → =_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_GGA(X, Xs, Y, Ys, =_in_gg(X1, X2)) we obtained the following new rules:
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U49_GGA(z0, z1, z2, z3, =_in_gg(z0, z2))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U7_GGA(z0, z1, z2, z3, z4, =_out_ag(z2)) → U8_GGA(z0, z1, z2, z3, isGreater_in_gg(z4, z2))
U47_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U49_GGA(z0, z1, z2, z3, =_in_gg(z0, z2))
U6_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
=_in_gg(X, X) → =_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U48_GGA(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_GGA(X, Xs, Y, Ys, isLess_in_gg(X1, X2)) we obtained the following new rules:
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U51_GGA(z0, z1, z2, z3, isLess_in_gg(z0, z2))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U7_GGA(z0, z1, z2, z3, z4, =_out_ag(z2)) → U8_GGA(z0, z1, z2, z3, isGreater_in_gg(z4, z2))
U47_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U49_GGA(z0, z1, z2, z3, =_in_gg(z0, z2))
U6_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U51_GGA(z0, z1, z2, z3, isLess_in_gg(z0, z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
=_in_gg(X, X) → =_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U7_GGA(z0, z1, z2, z3, z4, =_out_ag(z2)) → U8_GGA(z0, z1, z2, z3, isGreater_in_gg(z4, z2)) we obtained the following new rules:
U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U8_GGA(z0, z1, z2, z3, isGreater_in_gg(z0, z2))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ QDPOrderProof
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U47_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U49_GGA(z0, z1, z2, z3, =_in_gg(z0, z2))
U6_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U8_GGA(z0, z1, z2, z3, isGreater_in_gg(z0, z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U51_GGA(z0, z1, z2, z3, isLess_in_gg(z0, z2))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
=_in_gg(X, X) → =_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(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.
U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U8_GGA(z0, z1, z2, z3, isGreater_in_gg(z0, z2))
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U51_GGA(z0, z1, z2, z3, isLess_in_gg(z0, z2))
U49_GGA(X, Xs, Y, Ys, =_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
The remaining pairs can at least be oriented weakly.
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U47_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U49_GGA(z0, z1, z2, z3, =_in_gg(z0, z2))
U6_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
Used ordering: Polynomial interpretation [25]:
POL(.(x1, x2)) = 1 + x2
POL(=_in_gg(x1, x2)) = 1
POL(=_out_ag(x1)) = 1
POL(=_out_gg) = 1
POL(MERGE_IN_GGA(x1, x2)) = x1 + x2
POL(U43_gg(x1)) = 0
POL(U44_gg(x1)) = 0
POL(U45_gg(x1)) = 0
POL(U46_gg(x1)) = 0
POL(U47_GGA(x1, x2, x3, x4, x5)) = 1 + x2 + x4 + x5
POL(U48_GGA(x1, x2, x3, x4, x5, x6)) = 1 + x2 + x4 + x6
POL(U49_GGA(x1, x2, x3, x4, x5)) = 1 + x2 + x4 + x5
POL(U51_GGA(x1, x2, x3, x4, x5)) = 1 + x2 + x4
POL(U6_GGA(x1, x2, x3, x4, x5)) = 1 + x2 + x4 + x5
POL(U7_GGA(x1, x2, x3, x4, x5, x6)) = 1 + x2 + x4 + x6
POL(U8_GGA(x1, x2, x3, x4, x5)) = 1 + x2 + x4
POL(isGreater_in_gg(x1, x2)) = 0
POL(isGreater_out_gg) = 0
POL(isLess_in_gg(x1, x2)) = 0
POL(isLess_out_gg) = 0
POL(pred(x1)) = 0
POL(succ(x1)) = 0
POL(zero) = 0
The following usable rules [17] were oriented:
=_in_gg(X, X) → =_out_gg
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
U8_GGA(X, Xs, Y, Ys, isGreater_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U47_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U6_GGA(X, Xs, Y, Ys, =_out_ag(X))
U48_GGA(z0, z1, z2, z3, z0, =_out_ag(z2)) → U49_GGA(z0, z1, z2, z3, =_in_gg(z0, z2))
U6_GGA(z0, z1, z2, z3, =_out_ag(z0)) → U7_GGA(z0, z1, z2, z3, z0, =_out_ag(z2))
U51_GGA(X, Xs, Y, Ys, isLess_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U47_GGA(X, Xs, Y, Ys, =_out_ag(X))
The TRS R consists of the following rules:
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
=_in_gg(X, X) → =_out_gg
The set Q consists of the following terms:
isGreater_in_gg(x0, x1)
isLess_in_gg(x0, x1)
=_in_gg(x0, x1)
U43_gg(x0)
U44_gg(x0)
U45_gg(x0)
U46_gg(x0)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 7 less nodes.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ 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_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), .(Y, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U7_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_gga(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U8_gga(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U47_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_gga(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_gga(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U50_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_gga(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_gga(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U52_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
.(x1, x2) = .(x1, 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(x1, 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(x1, x2, x3, x4, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U7_gga(x1, x2, x3, x4, x5, x6, x7) = U7_gga(x1, x2, x3, x4, x6, x7)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x3, x6)
U47_gga(x1, x2, x3, x4, x5, x6) = U47_gga(x1, x2, x3, x4, x6)
U48_gga(x1, x2, x3, x4, x5, x6, x7) = U48_gga(x1, x2, x3, x4, x6, x7)
U49_gga(x1, x2, x3, x4, x5, x6) = U49_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_gga(x1, x2, x3, x4, x5, x6) = U50_gga(x1, x6)
U51_gga(x1, x2, x3, x4, x5, x6) = U51_gga(x1, x2, x3, x4, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_gga(x1, x2, x3, x4, x5, x6) = U52_gga(x1, 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
↳ OrTransformerProof
↳ Prolog
↳ 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:
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) = .(x1, 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
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
SPLIT_IN_GAA(.(X, 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(.(X, Xs)) → SPLIT_IN_GAA(Xs)
The graph contains the following edges 1 > 1
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
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([], [], []) → 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), .(Y, Zs)) → U6_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
=_in_ag(X, X) → =_out_ag(X, X)
U6_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U7_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U7_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U8_gga(X, Xs, Y, Ys, Zs, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg(succ(X), zero)
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg(succ(X), pred(Y))
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(X, Y, isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg(zero, pred(Y))
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(X, Y, isGreater_in_gg(X, Y))
U44_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(pred(X), pred(Y))
U43_gg(X, Y, isGreater_out_gg(X, Y)) → isGreater_out_gg(succ(X), succ(Y))
U8_gga(X, Xs, Y, Ys, Zs, isGreater_out_gg(X1, X2)) → U9_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U47_gga(X, Xs, Y, Ys, Zs, =_in_ag(X1, X))
U47_gga(X, Xs, Y, Ys, Zs, =_out_ag(X1, X)) → U48_gga(X, Xs, Y, Ys, Zs, X1, =_in_ag(X2, Y))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U49_gga(X, Xs, Y, Ys, Zs, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg(X, X)
U49_gga(X, Xs, Y, Ys, Zs, =_out_gg(X1, X2)) → U50_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U50_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U48_gga(X, Xs, Y, Ys, Zs, X1, =_out_ag(X2, Y)) → U51_gga(X, Xs, Y, Ys, Zs, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg(pred(X), zero)
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg(pred(X), succ(Y))
isLess_in_gg(pred(X), pred(Y)) → U45_gg(X, Y, isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg(zero, succ(Y))
isLess_in_gg(succ(X), succ(Y)) → U46_gg(X, Y, isLess_in_gg(X, Y))
U46_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(succ(X), succ(Y))
U45_gg(X, Y, isLess_out_gg(X, Y)) → isLess_out_gg(pred(X), pred(Y))
U51_gga(X, Xs, Y, Ys, Zs, isLess_out_gg(X1, X2)) → U52_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U52_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))
U9_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U4_ga(X, Y, Xs, Ys, merge_out_gga(Y1s, Y2s, Ys)) → mergesort_out_ga(.(X, .(Y, Xs)), Ys)
The argument filtering Pi contains the following mapping:
mergesort_in_ga(x1, x2) = mergesort_in_ga(x1)
[] = []
mergesort_out_ga(x1, x2) = mergesort_out_ga(x2)
.(x1, x2) = .(x1, 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(x1, 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(x1, x2, x3, x4, x6)
=_in_ag(x1, x2) = =_in_ag(x2)
=_out_ag(x1, x2) = =_out_ag(x1)
U7_gga(x1, x2, x3, x4, x5, x6, x7) = U7_gga(x1, x2, x3, x4, x6, x7)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
isGreater_in_gg(x1, x2) = isGreater_in_gg(x1, x2)
succ(x1) = succ(x1)
zero = zero
isGreater_out_gg(x1, x2) = isGreater_out_gg
pred(x1) = pred(x1)
U43_gg(x1, x2, x3) = U43_gg(x3)
U44_gg(x1, x2, x3) = U44_gg(x3)
U9_gga(x1, x2, x3, x4, x5, x6) = U9_gga(x3, x6)
U47_gga(x1, x2, x3, x4, x5, x6) = U47_gga(x1, x2, x3, x4, x6)
U48_gga(x1, x2, x3, x4, x5, x6, x7) = U48_gga(x1, x2, x3, x4, x6, x7)
U49_gga(x1, x2, x3, x4, x5, x6) = U49_gga(x1, x2, x3, x4, x6)
=_in_gg(x1, x2) = =_in_gg(x1, x2)
=_out_gg(x1, x2) = =_out_gg
U50_gga(x1, x2, x3, x4, x5, x6) = U50_gga(x1, x6)
U51_gga(x1, x2, x3, x4, x5, x6) = U51_gga(x1, x2, x3, x4, x6)
isLess_in_gg(x1, x2) = isLess_in_gg(x1, x2)
isLess_out_gg(x1, x2) = isLess_out_gg
U45_gg(x1, x2, x3) = U45_gg(x3)
U46_gg(x1, x2, x3) = U46_gg(x3)
U52_gga(x1, x2, x3, x4, x5, x6) = U52_gga(x1, 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
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(split_in_gaa(.(X, .(Y, Xs))))
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))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(split_in_gaa(.(X, .(Y, Xs)))) at position [0] we obtained the following new rules:
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(U5_gaa(X, split_in_gaa(.(Y, Xs))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(U5_gaa(X, split_in_gaa(.(Y, Xs))))
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))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(U5_gaa(X, split_in_gaa(.(Y, Xs)))) at position [0,1] we obtained the following new rules:
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(U5_gaa(X, U5_gaa(Y, split_in_gaa(Xs))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
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(.(X, .(Y, Xs))) → U1_GA(U5_gaa(X, U5_gaa(Y, 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(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U1_GA(split_out_gaa(X1s, X2s)) → U2_GA(X2s, mergesort_in_ga(X1s)) at position [1] we obtained the following new rules:
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(split_in_gaa(.(x0, .(x1, x2)))))
U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(split_in_gaa(.(x0, .(x1, x2)))))
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, [])))
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(.(X, .(Y, Xs))) → U1_GA(U5_gaa(X, U5_gaa(Y, split_in_gaa(Xs))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(split_in_gaa(.(x0, .(x1, x2))))) at position [1,0] we obtained the following new rules:
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, split_in_gaa(.(x1, x2)))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, split_in_gaa(.(x1, x2)))))
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(U5_gaa(X, U5_gaa(Y, split_in_gaa(Xs))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, split_in_gaa(.(x1, x2))))) at position [1,0,1] we obtained the following new rules:
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(U5_gaa(X, U5_gaa(Y, split_in_gaa(Xs))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule MERGESORT_IN_GA(.(X, .(Y, Xs))) → U1_GA(U5_gaa(X, U5_gaa(Y, split_in_gaa(Xs)))) at position [0] we obtained the following new rules:
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(U5_gaa(y0, U5_gaa(y1, split_out_gaa([], []))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(U5_gaa(y0, U5_gaa(y1, split_out_gaa([], []))))
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(U5_gaa(y0, U5_gaa(y1, split_out_gaa([], [])))) at position [0,1] we obtained the following new rules:
MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(U5_gaa(y0, split_out_gaa(.(y1, []), [])))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, [])))
MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(U5_gaa(y0, split_out_gaa(.(y1, []), [])))
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(U5_gaa(y0, split_out_gaa(.(y1, []), []))) at position [0] we obtained the following new rules:
MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(split_out_gaa(.(y0, []), .(y1, [])))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s)
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(split_out_gaa(.(y0, []), .(y1, [])))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U1_GA(split_out_gaa(X1s, X2s)) → MERGESORT_IN_GA(X1s) we obtained the following new rules:
U1_GA(split_out_gaa(.(y_0, .(y_1, [])), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, [])))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa(.(y_0, .(y_1, [])), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, [])))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s)
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(split_out_gaa(.(y0, []), .(y1, [])))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U2_GA(X2s, mergesort_out_ga(Y1s)) → MERGESORT_IN_GA(X2s) we obtained the following new rules:
U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U2_GA(.(y_0, .(y_1, [])), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, [])))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U2_GA(.(y_0, .(y_1, [])), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, [])))
U1_GA(split_out_gaa(.(y_0, .(y_1, [])), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, [])))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(split_out_gaa(.(y0, []), .(y1, [])))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U1_GA(split_out_gaa(.(x0, []), y1)) → U2_GA(y1, mergesort_out_ga(.(x0, []))) we obtained the following new rules:
U1_GA(split_out_gaa(.(x0, []), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa(.(x0, []), .(y_0, .(y_1, [])))) → U2_GA(.(y_0, .(y_1, [])), mergesort_out_ga(.(x0, [])))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, []), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(y_0, .(y_1, [])), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, [])))
U2_GA(.(y_0, .(y_1, [])), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, [])))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
U1_GA(split_out_gaa(.(x0, []), .(y_0, .(y_1, [])))) → U2_GA(.(y_0, .(y_1, [])), mergesort_out_ga(.(x0, [])))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
MERGESORT_IN_GA(.(y0, .(y1, []))) → U1_GA(split_out_gaa(.(y0, []), .(y1, [])))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, []), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([]))
U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U1_GA(split_out_gaa([], y1)) → U2_GA(y1, mergesort_out_ga([])) we obtained the following new rules:
U1_GA(split_out_gaa([], .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga([]))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, []), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(.(x0, [])))
U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
U1_GA(split_out_gaa([], .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga([]))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(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
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, []), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(.(x0, [])))
U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U1_GA(split_out_gaa(.(x0, .(x1, x2)), y1)) → U2_GA(y1, U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2))))) we obtained the following new rules:
U1_GA(split_out_gaa(.(x0, .(x1, x2)), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(x0, []), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(.(x0, [])))
U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(x0, .(x1, x2)), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(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.
U1_GA(split_out_gaa(.(x0, []), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(.(x0, [])))
U1_GA(split_out_gaa(.(x0, .(x1, x2)), .(y_0, .(y_1, .(y_2, y_3))))) → U2_GA(.(y_0, .(y_1, .(y_2, y_3))), U1_ga(U5_gaa(x0, U5_gaa(x1, split_in_gaa(x2)))))
The remaining pairs can at least be oriented weakly.
U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
Used ordering: Polynomial interpretation [25]:
POL(.(x1, x2)) = 1 + x2
POL(=_in_ag(x1)) = 0
POL(=_in_gg(x1, x2)) = 0
POL(=_out_ag(x1)) = 0
POL(=_out_gg) = 0
POL(MERGESORT_IN_GA(x1)) = 1 + x1
POL(U1_GA(x1)) = x1
POL(U1_ga(x1)) = 0
POL(U2_GA(x1, x2)) = 1 + x1
POL(U2_ga(x1, x2)) = 0
POL(U3_ga(x1, x2)) = 0
POL(U43_gg(x1)) = 1 + x1
POL(U44_gg(x1)) = 1 + x1
POL(U45_gg(x1)) = x1
POL(U46_gg(x1)) = 1 + x1
POL(U47_gga(x1, x2, x3, x4, x5)) = 0
POL(U48_gga(x1, x2, x3, x4, x5, x6)) = 0
POL(U49_gga(x1, x2, x3, x4, x5)) = 0
POL(U4_ga(x1)) = 0
POL(U50_gga(x1, x2)) = 0
POL(U51_gga(x1, x2, x3, x4, x5)) = 0
POL(U52_gga(x1, x2)) = 0
POL(U5_gaa(x1, x2)) = 1 + x2
POL(U6_gga(x1, x2, x3, x4, x5)) = 0
POL(U7_gga(x1, x2, x3, x4, x5, x6)) = 0
POL(U8_gga(x1, x2, x3, x4, x5)) = 0
POL(U9_gga(x1, x2)) = 0
POL([]) = 0
POL(isGreater_in_gg(x1, x2)) = 1 + x1 + x2
POL(isGreater_out_gg) = 0
POL(isLess_in_gg(x1, x2)) = 1 + x1
POL(isLess_out_gg) = 1
POL(merge_in_gga(x1, x2)) = 0
POL(merge_out_gga(x1)) = 0
POL(mergesort_in_ga(x1)) = 0
POL(mergesort_out_ga(x1)) = 0
POL(pred(x1)) = 1 + x1
POL(split_in_gaa(x1)) = 1 + x1
POL(split_out_gaa(x1, x2)) = 1 + x1 + x2
POL(succ(x1)) = 1 + x1
POL(zero) = 1
The following usable rules [17] were oriented:
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U2_GA(.(y_0, .(y_1, .(y_2, y_3))), mergesort_out_ga(x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(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
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
mergesort_in_ga([]) → mergesort_out_ga([])
mergesort_in_ga(.(X, [])) → mergesort_out_ga(.(X, []))
mergesort_in_ga(.(X, .(Y, Xs))) → U1_ga(split_in_gaa(.(X, .(Y, Xs))))
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
U1_ga(split_out_gaa(X1s, X2s)) → U2_ga(X2s, mergesort_in_ga(X1s))
U2_ga(X2s, mergesort_out_ga(Y1s)) → U3_ga(Y1s, mergesort_in_ga(X2s))
U3_ga(Y1s, mergesort_out_ga(Y2s)) → U4_ga(merge_in_gga(Y1s, Y2s))
merge_in_gga([], Xs) → merge_out_gga(Xs)
merge_in_gga(Xs, []) → merge_out_gga(Xs)
merge_in_gga(.(X, Xs), .(Y, Ys)) → U6_gga(X, Xs, Y, Ys, =_in_ag(X))
=_in_ag(X) → =_out_ag(X)
U6_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U7_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U7_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U8_gga(X, Xs, Y, Ys, isGreater_in_gg(X1, X2))
isGreater_in_gg(succ(X), zero) → isGreater_out_gg
isGreater_in_gg(succ(X), pred(Y)) → isGreater_out_gg
isGreater_in_gg(succ(X), succ(Y)) → U43_gg(isGreater_in_gg(X, Y))
isGreater_in_gg(zero, pred(Y)) → isGreater_out_gg
isGreater_in_gg(pred(X), pred(Y)) → U44_gg(isGreater_in_gg(X, Y))
U44_gg(isGreater_out_gg) → isGreater_out_gg
U43_gg(isGreater_out_gg) → isGreater_out_gg
U8_gga(X, Xs, Y, Ys, isGreater_out_gg) → U9_gga(Y, merge_in_gga(.(X, Xs), Ys))
merge_in_gga(.(X, Xs), .(Y, Ys)) → U47_gga(X, Xs, Y, Ys, =_in_ag(X))
U47_gga(X, Xs, Y, Ys, =_out_ag(X1)) → U48_gga(X, Xs, Y, Ys, X1, =_in_ag(Y))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U49_gga(X, Xs, Y, Ys, =_in_gg(X1, X2))
=_in_gg(X, X) → =_out_gg
U49_gga(X, Xs, Y, Ys, =_out_gg) → U50_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U50_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U48_gga(X, Xs, Y, Ys, X1, =_out_ag(X2)) → U51_gga(X, Xs, Y, Ys, isLess_in_gg(X1, X2))
isLess_in_gg(pred(X), zero) → isLess_out_gg
isLess_in_gg(pred(X), succ(Y)) → isLess_out_gg
isLess_in_gg(pred(X), pred(Y)) → U45_gg(isLess_in_gg(X, Y))
isLess_in_gg(zero, succ(Y)) → isLess_out_gg
isLess_in_gg(succ(X), succ(Y)) → U46_gg(isLess_in_gg(X, Y))
U46_gg(isLess_out_gg) → isLess_out_gg
U45_gg(isLess_out_gg) → isLess_out_gg
U51_gga(X, Xs, Y, Ys, isLess_out_gg) → U52_gga(X, merge_in_gga(Xs, .(Y, Ys)))
U52_gga(X, merge_out_gga(Zs)) → merge_out_gga(.(X, Zs))
U9_gga(Y, merge_out_gga(Zs)) → merge_out_gga(.(Y, Zs))
U4_ga(merge_out_gga(Ys)) → mergesort_out_ga(Ys)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(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
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
The set Q consists of the following terms:
mergesort_in_ga(x0)
split_in_gaa(x0)
U5_gaa(x0, x1)
U1_ga(x0)
U2_ga(x0, x1)
U3_ga(x0, x1)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(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)
merge_in_gga(x0, x1)
=_in_ag(x0)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4, x5)
isGreater_in_gg(x0, x1)
U44_gg(x0)
U43_gg(x0)
U8_gga(x0, x1, x2, x3, x4)
U47_gga(x0, x1, x2, x3, x4)
U48_gga(x0, x1, x2, x3, x4, x5)
=_in_gg(x0, x1)
U49_gga(x0, x1, x2, x3, x4)
U50_gga(x0, x1)
isLess_in_gg(x0, x1)
U46_gg(x0)
U45_gg(x0)
U51_gga(x0, x1, x2, x3, x4)
U52_gga(x0, x1)
U9_gga(x0, x1)
U4_ga(x0)
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The TRS R consists of the following rules:
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
The set Q consists of the following terms:
split_in_gaa(x0)
U5_gaa(x0, x1)
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.
U1_GA(split_out_gaa(.(y_0, .(y_1, .(y_2, y_3))), x1)) → MERGESORT_IN_GA(.(y_0, .(y_1, .(y_2, y_3))))
MERGESORT_IN_GA(.(y0, .(y1, .(x0, x1)))) → U1_GA(U5_gaa(y0, U5_gaa(y1, U5_gaa(x0, split_in_gaa(x1)))))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( U5_gaa(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( split_out_gaa(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( .(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:
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
↳ Prolog
↳ PredefinedPredicateTransformerProof
↳ Prolog
↳ OrTransformerProof
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
split_in_gaa([]) → split_out_gaa([], [])
split_in_gaa(.(X, Xs)) → U5_gaa(X, split_in_gaa(Xs))
U5_gaa(X, split_out_gaa(Zs, Ys)) → split_out_gaa(.(X, Ys), Zs)
The set Q consists of the following terms:
split_in_gaa(x0)
U5_gaa(x0, x1)
We have to consider all (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.