KILLEDRuntime Complexity (innermost) proof of /tmp/tmpU1HIO7/anchored.xml
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 DecreasingLoopProof (⇔, 191 ms)↳2 BOUNDS(n^1, INF)↳3 RenamingProof (⇔, 0 ms)↳4 CpxRelTRS↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)↳6 typed CpxTrs↳7 OrderProof (LOWER BOUND(ID), 0 ms)↳8 typed CpxTrs(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
anchored(Cons(x, xs), y) → anchored(xs, Cons(Cons(Nil, Nil), y))
anchored(Nil, y) → y
goal(x, y) → anchored(x, y)
Rewrite Strategy: INNERMOST(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
anchored(Cons(x, xs), y) →+ anchored(xs, Cons(Cons(Nil, Nil), y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [xs / Cons(x, xs)].
The result substitution is [y / Cons(Cons(Nil, Nil), y)].(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:
anchored(Cons(x, xs), y) → anchored(xs, Cons(Cons(Nil, Nil), y))
anchored(Nil, y) → y
goal(x, y) → anchored(x, y)
S is empty.
Rewrite Strategy: INNERMOST(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(6) Obligation:
Innermost TRS:
Rules:
anchored(Cons(x, xs), y) → anchored(xs, Cons(Cons(Nil, Nil), y))
anchored(Nil, y) → y
goal(x, y) → anchored(x, y)
Types:
anchored :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
anchored(8) Obligation:
Innermost TRS:
Rules:
anchored(Cons(x, xs), y) → anchored(xs, Cons(Cons(Nil, Nil), y))
anchored(Nil, y) → y
goal(x, y) → anchored(x, y)
Types:
anchored :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:NilGenerator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil2_0(x))The following defined symbols remain to be analysed:
anchored