Runtime Complexity (full) proof of /tmp/tmpZ1jo55/ma96.xml

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 991 ms)

↳2 BOUNDS(2^n, 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

    ↳9 RewriteLemmaProof (LOWER BOUND(ID), 4016 ms)

    ↳10 BEST

        ↳11 typed CpxTrs

            ↳12 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)

            ↳13 typed CpxTrs

            ↳14 LowerBoundsProof (⇔, 0 ms)

            ↳15 BOUNDS(n^1, INF)

        ↳16 typed CpxTrs

            ↳17 LowerBoundsProof (⇔, 0 ms)

            ↳18 BOUNDS(n^1, INF)

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

and(false, false) → false
and(true, false) → false
and(false, true) → false
and(true, true) → true
eq(nil, nil) → true
eq(cons(T, L), nil) → false
eq(nil, cons(T, L)) → false
eq(cons(T, L), cons(Tp, Lp)) → and(eq(T, Tp), eq(L, Lp))
eq(var(L), var(Lp)) → eq(L, Lp)
eq(var(L), apply(T, S)) → false
eq(var(L), lambda(X, T)) → false
eq(apply(T, S), var(L)) → false
eq(apply(T, S), apply(Tp, Sp)) → and(eq(T, Tp), eq(S, Sp))
eq(apply(T, S), lambda(X, Tp)) → false
eq(lambda(X, T), var(L)) → false
eq(lambda(X, T), apply(Tp, Sp)) → false
eq(lambda(X, T), lambda(Xp, Tp)) → and(eq(T, Tp), eq(X, Xp))
if(true, var(K), var(L)) → var(K)
if(false, var(K), var(L)) → var(L)
ren(var(L), var(K), var(Lp)) → if(eq(L, Lp), var(K), var(Lp))
ren(X, Y, apply(T, S)) → apply(ren(X, Y, T), ren(X, Y, S))
ren(X, Y, lambda(Z, T)) → lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
ren(X, Y, apply(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), S)) →⁺ apply(lambda(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), lambda(var(cons(X, cons(Y, cons(lambda(var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), ren(Z35986_0, var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), ren(Z37470_0, var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), T37471_0))), nil)))), ren(X, Y, ren(var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), var(cons(X, cons(Y, cons(lambda(var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), ren(Z35986_0, var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), ren(Z37470_0, var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), T37471_0))), nil)))), ren(Z35986_0, var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), ren(Z37470_0, var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), T37471_0)))))), ren(X, Y, S))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0,1,1,0,1,2].
The pumping substitution is [T37471_0 / apply(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), S)].
The result substitution is [X / Z37470_0, Y / var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil))))].

The rewrite sequence
ren(X, Y, apply(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), S)) →⁺ apply(lambda(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), lambda(var(cons(X, cons(Y, cons(lambda(var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), ren(Z35986_0, var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), ren(Z37470_0, var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), T37471_0))), nil)))), ren(X, Y, ren(var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), var(cons(X, cons(Y, cons(lambda(var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), ren(Z35986_0, var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), ren(Z37470_0, var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), T37471_0))), nil)))), ren(Z35986_0, var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), ren(Z37470_0, var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil)))), T37471_0)))))), ren(X, Y, S))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,1,2,1,0,1,1,0,1,2].
The pumping substitution is [T37471_0 / apply(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), S)].
The result substitution is [X / Z37470_0, Y / var(cons(Z35986_0, cons(var(cons(X, cons(Y, cons(lambda(Z35986_0, lambda(Z37470_0, T37471_0)), nil)))), cons(lambda(Z37470_0, T37471_0), nil))))].

(2) BOUNDS(2^n, 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:

and(false, false) → false
and(true, false) → false
and(false, true) → false
and(true, true) → true
eq(nil, nil) → true
eq(cons(T, L), nil) → false
eq(nil, cons(T, L)) → false
eq(cons(T, L), cons(Tp, Lp)) → and(eq(T, Tp), eq(L, Lp))
eq(var(L), var(Lp)) → eq(L, Lp)
eq(var(L), apply(T, S)) → false
eq(var(L), lambda(X, T)) → false
eq(apply(T, S), var(L)) → false
eq(apply(T, S), apply(Tp, Sp)) → and(eq(T, Tp), eq(S, Sp))
eq(apply(T, S), lambda(X, Tp)) → false
eq(lambda(X, T), var(L)) → false
eq(lambda(X, T), apply(Tp, Sp)) → false
eq(lambda(X, T), lambda(Xp, Tp)) → and(eq(T, Tp), eq(X, Xp))
if(true, var(K), var(L)) → var(K)
if(false, var(K), var(L)) → var(L)
ren(var(L), var(K), var(Lp)) → if(eq(L, Lp), var(K), var(Lp))
ren(X, Y, apply(T, S)) → apply(ren(X, Y, T), ren(X, Y, S))
ren(X, Y, lambda(Z, T)) → lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
and(false, false) → false
and(true, false) → false
and(false, true) → false
and(true, true) → true
eq(nil, nil) → true
eq(cons(T, L), nil) → false
eq(nil, cons(T, L)) → false
eq(cons(T, L), cons(Tp, Lp)) → and(eq(T, Tp), eq(L, Lp))
eq(var(L), var(Lp)) → eq(L, Lp)
eq(var(L), apply(T, S)) → false
eq(var(L), lambda(X, T)) → false
eq(apply(T, S), var(L)) → false
eq(apply(T, S), apply(Tp, Sp)) → and(eq(T, Tp), eq(S, Sp))
eq(apply(T, S), lambda(X, Tp)) → false
eq(lambda(X, T), var(L)) → false
eq(lambda(X, T), apply(Tp, Sp)) → false
eq(lambda(X, T), lambda(Xp, Tp)) → and(eq(T, Tp), eq(X, Xp))
if(true, var(K), var(L)) → var(K)
if(false, var(K), var(L)) → var(L)
ren(var(L), var(K), var(Lp)) → if(eq(L, Lp), var(K), var(Lp))
ren(X, Y, apply(T, S)) → apply(ren(X, Y, T), ren(X, Y, S))
ren(X, Y, lambda(Z, T)) → lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

Types:
and :: false:true → false:true → false:true
false :: false:true
true :: false:true
eq :: nil:cons:var:apply:lambda → nil:cons:var:apply:lambda → false:true
nil :: nil:cons:var:apply:lambda
cons :: nil:cons:var:apply:lambda → nil:cons:var:apply:lambda → nil:cons:var:apply:lambda
var :: nil:cons:var:apply:lambda → nil:cons:var:apply:lambda
apply :: nil:cons:var:apply:lambda → nil:cons:var:apply:lambda → nil:cons:var:apply:lambda
lambda :: nil:cons:var:apply:lambda → nil:cons:var:apply:lambda → nil:cons:var:apply:lambda
if :: false:true → nil:cons:var:apply:lambda → nil:cons:var:apply:lambda → nil:cons:var:apply:lambda
ren :: nil:cons:var:apply:lambda → nil:cons:var:apply:lambda → nil:cons:var:apply:lambda → nil:cons:var:apply:lambda
hole_false:true1_0 :: false:true
hole_nil:cons:var:apply:lambda2_0 :: nil:cons:var:apply:lambda
gen_nil:cons:var:apply:lambda3_0 :: Nat → nil:cons:var:apply:lambda

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
eq, ren

They will be analysed ascendingly in the following order:
eq < ren

(8) Obligation:

Generator Equations:
gen_nil:cons:var:apply:lambda3_0(0) ⇔ nil
gen_nil:cons:var:apply:lambda3_0(+(x, 1)) ⇔ cons(nil, gen_nil:cons:var:apply:lambda3_0(x))

The following defined symbols remain to be analysed:
eq, ren

They will be analysed ascendingly in the following order:
eq < ren

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
eq(gen_nil:cons:var:apply:lambda3_0(0), gen_nil:cons:var:apply:lambda3_0(0)) →_R^Ω(1)
true

Induction Step:
eq(gen_nil:cons:var:apply:lambda3_0(+(n5_0, 1)), gen_nil:cons:var:apply:lambda3_0(+(n5_0, 1))) →_R^Ω(1)
and(eq(nil, nil), eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0))) →_R^Ω(1)
and(true, eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0))) →_IH
and(true, true) →_R^Ω(1)
true

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

Lemmas:
eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:cons:var:apply:lambda3_0(0) ⇔ nil
gen_nil:cons:var:apply:lambda3_0(+(x, 1)) ⇔ cons(nil, gen_nil:cons:var:apply:lambda3_0(x))

The following defined symbols remain to be analysed:
ren

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol ren.

(13) Obligation:

Lemmas:
eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:cons:var:apply:lambda3_0(0) ⇔ nil
gen_nil:cons:var:apply:lambda3_0(+(x, 1)) ⇔ cons(nil, gen_nil:cons:var:apply:lambda3_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Lemmas:
eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:cons:var:apply:lambda3_0(0) ⇔ nil
gen_nil:cons:var:apply:lambda3_0(+(x, 1)) ⇔ cons(nil, gen_nil:cons:var:apply:lambda3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(2^n, INF)

(3) RenamingProof (EQUIVALENT transformation)

(4) Obligation:

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

(6) Obligation:

(7) OrderProof (LOWER BOUND(ID) transformation)

(8) Obligation:

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(10) Complex Obligation (BEST)

(11) Obligation:

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(13) Obligation:

(14) LowerBoundsProof (EQUIVALENT transformation)

(15) BOUNDS(n^1, INF)

(16) Obligation:

(17) LowerBoundsProof (EQUIVALENT transformation)

(18) BOUNDS(n^1, INF)