(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
activate(n__f(X)) →+ f(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__f(X)].
The result substitution is [ ].
(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X
Types:
f :: c:n__true:n__f:false → c:n__true:n__f:false
if :: c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false
c :: c:n__true:n__f:false
n__f :: c:n__true:n__f:false → c:n__true:n__f:false
n__true :: c:n__true:n__f:false
true :: c:n__true:n__f:false
false :: c:n__true:n__f:false
activate :: c:n__true:n__f:false → c:n__true:n__f:false
hole_c:n__true:n__f:false1_0 :: c:n__true:n__f:false
gen_c:n__true:n__f:false2_0 :: Nat → c:n__true:n__f:false
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
f,
activateThey will be analysed ascendingly in the following order:
f = activate
(8) Obligation:
TRS:
Rules:
f(
X) →
if(
X,
c,
n__f(
n__true))
if(
true,
X,
Y) →
Xif(
false,
X,
Y) →
activate(
Y)
f(
X) →
n__f(
X)
true →
n__trueactivate(
n__f(
X)) →
f(
activate(
X))
activate(
n__true) →
trueactivate(
X) →
XTypes:
f :: c:n__true:n__f:false → c:n__true:n__f:false
if :: c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false
c :: c:n__true:n__f:false
n__f :: c:n__true:n__f:false → c:n__true:n__f:false
n__true :: c:n__true:n__f:false
true :: c:n__true:n__f:false
false :: c:n__true:n__f:false
activate :: c:n__true:n__f:false → c:n__true:n__f:false
hole_c:n__true:n__f:false1_0 :: c:n__true:n__f:false
gen_c:n__true:n__f:false2_0 :: Nat → c:n__true:n__f:false
Generator Equations:
gen_c:n__true:n__f:false2_0(0) ⇔ n__true
gen_c:n__true:n__f:false2_0(+(x, 1)) ⇔ n__f(gen_c:n__true:n__f:false2_0(x))
The following defined symbols remain to be analysed:
activate, f
They will be analysed ascendingly in the following order:
f = activate
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol activate.
(10) Obligation:
TRS:
Rules:
f(
X) →
if(
X,
c,
n__f(
n__true))
if(
true,
X,
Y) →
Xif(
false,
X,
Y) →
activate(
Y)
f(
X) →
n__f(
X)
true →
n__trueactivate(
n__f(
X)) →
f(
activate(
X))
activate(
n__true) →
trueactivate(
X) →
XTypes:
f :: c:n__true:n__f:false → c:n__true:n__f:false
if :: c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false
c :: c:n__true:n__f:false
n__f :: c:n__true:n__f:false → c:n__true:n__f:false
n__true :: c:n__true:n__f:false
true :: c:n__true:n__f:false
false :: c:n__true:n__f:false
activate :: c:n__true:n__f:false → c:n__true:n__f:false
hole_c:n__true:n__f:false1_0 :: c:n__true:n__f:false
gen_c:n__true:n__f:false2_0 :: Nat → c:n__true:n__f:false
Generator Equations:
gen_c:n__true:n__f:false2_0(0) ⇔ n__true
gen_c:n__true:n__f:false2_0(+(x, 1)) ⇔ n__f(gen_c:n__true:n__f:false2_0(x))
The following defined symbols remain to be analysed:
f
They will be analysed ascendingly in the following order:
f = activate
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol f.
(12) Obligation:
TRS:
Rules:
f(
X) →
if(
X,
c,
n__f(
n__true))
if(
true,
X,
Y) →
Xif(
false,
X,
Y) →
activate(
Y)
f(
X) →
n__f(
X)
true →
n__trueactivate(
n__f(
X)) →
f(
activate(
X))
activate(
n__true) →
trueactivate(
X) →
XTypes:
f :: c:n__true:n__f:false → c:n__true:n__f:false
if :: c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false
c :: c:n__true:n__f:false
n__f :: c:n__true:n__f:false → c:n__true:n__f:false
n__true :: c:n__true:n__f:false
true :: c:n__true:n__f:false
false :: c:n__true:n__f:false
activate :: c:n__true:n__f:false → c:n__true:n__f:false
hole_c:n__true:n__f:false1_0 :: c:n__true:n__f:false
gen_c:n__true:n__f:false2_0 :: Nat → c:n__true:n__f:false
Generator Equations:
gen_c:n__true:n__f:false2_0(0) ⇔ n__true
gen_c:n__true:n__f:false2_0(+(x, 1)) ⇔ n__f(gen_c:n__true:n__f:false2_0(x))
No more defined symbols left to analyse.