a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__b → b
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__f'(X, g'(X), Y) → a__f'(Y, Y, Y)
a__g'(b') → c'
a__b' → c'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, X3)
mark'(g'(X)) → a__g'(mark'(X))
mark'(b') → a__b'
mark'(c') → c'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__g'(X) → g'(X)
a__b' → b'
Infered types.
Rules:
a__f'(X, g'(X), Y) → a__f'(Y, Y, Y)
a__g'(b') → c'
a__b' → c'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, X3)
mark'(g'(X)) → a__g'(mark'(X))
mark'(b') → a__b'
mark'(c') → c'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__g'(X) → g'(X)
a__b' → b'
Types:
a__f' :: g':b':c':f' → g':b':c':f' → g':b':c':f' → g':b':c':f'
g' :: g':b':c':f' → g':b':c':f'
a__g' :: g':b':c':f' → g':b':c':f'
b' :: g':b':c':f'
c' :: g':b':c':f'
a__b' :: g':b':c':f'
mark' :: g':b':c':f' → g':b':c':f'
f' :: g':b':c':f' → g':b':c':f' → g':b':c':f' → g':b':c':f'
_hole_g':b':c':f'1 :: g':b':c':f'
_gen_g':b':c':f'2 :: Nat → g':b':c':f'
Heuristically decided to analyse the following defined symbols:
a__f', mark'
They will be analysed ascendingly in the following order:
a__f' < mark'
Rules:
a__f'(X, g'(X), Y) → a__f'(Y, Y, Y)
a__g'(b') → c'
a__b' → c'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, X3)
mark'(g'(X)) → a__g'(mark'(X))
mark'(b') → a__b'
mark'(c') → c'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__g'(X) → g'(X)
a__b' → b'
Types:
a__f' :: g':b':c':f' → g':b':c':f' → g':b':c':f' → g':b':c':f'
g' :: g':b':c':f' → g':b':c':f'
a__g' :: g':b':c':f' → g':b':c':f'
b' :: g':b':c':f'
c' :: g':b':c':f'
a__b' :: g':b':c':f'
mark' :: g':b':c':f' → g':b':c':f'
f' :: g':b':c':f' → g':b':c':f' → g':b':c':f' → g':b':c':f'
_hole_g':b':c':f'1 :: g':b':c':f'
_gen_g':b':c':f'2 :: Nat → g':b':c':f'
Generator Equations:
_gen_g':b':c':f'2(0) ⇔ b'
_gen_g':b':c':f'2(+(x, 1)) ⇔ g'(_gen_g':b':c':f'2(x))
The following defined symbols remain to be analysed:
a__f', mark'
They will be analysed ascendingly in the following order:
a__f' < mark'
Could not prove a rewrite lemma for the defined symbol a__f'.
Rules:
a__f'(X, g'(X), Y) → a__f'(Y, Y, Y)
a__g'(b') → c'
a__b' → c'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, X3)
mark'(g'(X)) → a__g'(mark'(X))
mark'(b') → a__b'
mark'(c') → c'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__g'(X) → g'(X)
a__b' → b'
Types:
a__f' :: g':b':c':f' → g':b':c':f' → g':b':c':f' → g':b':c':f'
g' :: g':b':c':f' → g':b':c':f'
a__g' :: g':b':c':f' → g':b':c':f'
b' :: g':b':c':f'
c' :: g':b':c':f'
a__b' :: g':b':c':f'
mark' :: g':b':c':f' → g':b':c':f'
f' :: g':b':c':f' → g':b':c':f' → g':b':c':f' → g':b':c':f'
_hole_g':b':c':f'1 :: g':b':c':f'
_gen_g':b':c':f'2 :: Nat → g':b':c':f'
Generator Equations:
_gen_g':b':c':f'2(0) ⇔ b'
_gen_g':b':c':f'2(+(x, 1)) ⇔ g'(_gen_g':b':c':f'2(x))
The following defined symbols remain to be analysed:
mark'
Proved the following rewrite lemma:
mark'(_gen_g':b':c':f'2(+(1, _n116))) → _*3, rt ∈ Ω(n116)
Induction Base:
mark'(_gen_g':b':c':f'2(+(1, 0)))
Induction Step:
mark'(_gen_g':b':c':f'2(+(1, +(_$n117, 1)))) →RΩ(1)
a__g'(mark'(_gen_g':b':c':f'2(+(1, _$n117)))) →IH
a__g'(_*3)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
a__f'(X, g'(X), Y) → a__f'(Y, Y, Y)
a__g'(b') → c'
a__b' → c'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, X3)
mark'(g'(X)) → a__g'(mark'(X))
mark'(b') → a__b'
mark'(c') → c'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__g'(X) → g'(X)
a__b' → b'
Types:
a__f' :: g':b':c':f' → g':b':c':f' → g':b':c':f' → g':b':c':f'
g' :: g':b':c':f' → g':b':c':f'
a__g' :: g':b':c':f' → g':b':c':f'
b' :: g':b':c':f'
c' :: g':b':c':f'
a__b' :: g':b':c':f'
mark' :: g':b':c':f' → g':b':c':f'
f' :: g':b':c':f' → g':b':c':f' → g':b':c':f' → g':b':c':f'
_hole_g':b':c':f'1 :: g':b':c':f'
_gen_g':b':c':f'2 :: Nat → g':b':c':f'
Lemmas:
mark'(_gen_g':b':c':f'2(+(1, _n116))) → _*3, rt ∈ Ω(n116)
Generator Equations:
_gen_g':b':c':f'2(0) ⇔ b'
_gen_g':b':c':f'2(+(x, 1)) ⇔ g'(_gen_g':b':c':f'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_g':b':c':f'2(+(1, _n116))) → _*3, rt ∈ Ω(n116)