a__f(a, b, X) → a__f(X, X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(X1, X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__f'(a', b', X) → a__f'(X, X, mark'(X))
a__c' → a'
a__c' → b'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, mark'(X3))
mark'(c') → a__c'
mark'(a') → a'
mark'(b') → b'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__c' → c'
Infered types.
Rules:
a__f'(a', b', X) → a__f'(X, X, mark'(X))
a__c' → a'
a__c' → b'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, mark'(X3))
mark'(c') → a__c'
mark'(a') → a'
mark'(b') → b'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__c' → c'
Types:
a__f' :: a':b':f':c' → a':b':f':c' → a':b':f':c' → a':b':f':c'
a' :: a':b':f':c'
b' :: a':b':f':c'
mark' :: a':b':f':c' → a':b':f':c'
a__c' :: a':b':f':c'
f' :: a':b':f':c' → a':b':f':c' → a':b':f':c' → a':b':f':c'
c' :: a':b':f':c'
_hole_a':b':f':c'1 :: a':b':f':c'
_gen_a':b':f':c'2 :: Nat → a':b':f':c'
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'(a', b', X) → a__f'(X, X, mark'(X))
a__c' → a'
a__c' → b'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, mark'(X3))
mark'(c') → a__c'
mark'(a') → a'
mark'(b') → b'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__c' → c'
Types:
a__f' :: a':b':f':c' → a':b':f':c' → a':b':f':c' → a':b':f':c'
a' :: a':b':f':c'
b' :: a':b':f':c'
mark' :: a':b':f':c' → a':b':f':c'
a__c' :: a':b':f':c'
f' :: a':b':f':c' → a':b':f':c' → a':b':f':c' → a':b':f':c'
c' :: a':b':f':c'
_hole_a':b':f':c'1 :: a':b':f':c'
_gen_a':b':f':c'2 :: Nat → a':b':f':c'
Generator Equations:
_gen_a':b':f':c'2(0) ⇔ a'
_gen_a':b':f':c'2(+(x, 1)) ⇔ f'(a', a', _gen_a':b':f':c'2(x))
The following defined symbols remain to be analysed:
mark', a__f'
They will be analysed ascendingly in the following order:
a__f' = mark'
Proved the following rewrite lemma:
mark'(_gen_a':b':f':c'2(_n4)) → _gen_a':b':f':c'2(_n4), rt ∈ Ω(1 + n4)
Induction Base:
mark'(_gen_a':b':f':c'2(0)) →RΩ(1)
a'
Induction Step:
mark'(_gen_a':b':f':c'2(+(_$n5, 1))) →RΩ(1)
a__f'(a', a', mark'(_gen_a':b':f':c'2(_$n5))) →IH
a__f'(a', a', _gen_a':b':f':c'2(_$n5)) →RΩ(1)
f'(a', a', _gen_a':b':f':c'2(_$n5))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
a__f'(a', b', X) → a__f'(X, X, mark'(X))
a__c' → a'
a__c' → b'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, mark'(X3))
mark'(c') → a__c'
mark'(a') → a'
mark'(b') → b'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__c' → c'
Types:
a__f' :: a':b':f':c' → a':b':f':c' → a':b':f':c' → a':b':f':c'
a' :: a':b':f':c'
b' :: a':b':f':c'
mark' :: a':b':f':c' → a':b':f':c'
a__c' :: a':b':f':c'
f' :: a':b':f':c' → a':b':f':c' → a':b':f':c' → a':b':f':c'
c' :: a':b':f':c'
_hole_a':b':f':c'1 :: a':b':f':c'
_gen_a':b':f':c'2 :: Nat → a':b':f':c'
Lemmas:
mark'(_gen_a':b':f':c'2(_n4)) → _gen_a':b':f':c'2(_n4), rt ∈ Ω(1 + n4)
Generator Equations:
_gen_a':b':f':c'2(0) ⇔ a'
_gen_a':b':f':c'2(+(x, 1)) ⇔ f'(a', a', _gen_a':b':f':c'2(x))
The following defined symbols remain to be analysed:
a__f'
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'(a', b', X) → a__f'(X, X, mark'(X))
a__c' → a'
a__c' → b'
mark'(f'(X1, X2, X3)) → a__f'(X1, X2, mark'(X3))
mark'(c') → a__c'
mark'(a') → a'
mark'(b') → b'
a__f'(X1, X2, X3) → f'(X1, X2, X3)
a__c' → c'
Types:
a__f' :: a':b':f':c' → a':b':f':c' → a':b':f':c' → a':b':f':c'
a' :: a':b':f':c'
b' :: a':b':f':c'
mark' :: a':b':f':c' → a':b':f':c'
a__c' :: a':b':f':c'
f' :: a':b':f':c' → a':b':f':c' → a':b':f':c' → a':b':f':c'
c' :: a':b':f':c'
_hole_a':b':f':c'1 :: a':b':f':c'
_gen_a':b':f':c'2 :: Nat → a':b':f':c'
Lemmas:
mark'(_gen_a':b':f':c'2(_n4)) → _gen_a':b':f':c'2(_n4), rt ∈ Ω(1 + n4)
Generator Equations:
_gen_a':b':f':c'2(0) ⇔ a'
_gen_a':b':f':c'2(+(x, 1)) ⇔ f'(a', a', _gen_a':b':f':c'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_a':b':f':c'2(_n4)) → _gen_a':b':f':c'2(_n4), rt ∈ Ω(1 + n4)