### (0) Obligation:

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

eq(0, 0) → true
eq(0, s(X)) → false
eq(s(X), 0) → false
eq(s(X), s(Y)) → eq(X, Y)
rm(N, nil) → nil
ifrm(true, N, add(M, X)) → rm(N, X)
purge(nil) → nil

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
eq(s(X), s(Y)) →+ eq(X, Y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X / s(X), Y / s(Y)].
The result substitution is [ ].

### (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:

eq(0', 0') → true
eq(0', s(X)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
rm(N, nil) → nil
ifrm(true, N, add(M, X)) → rm(N, X)
purge(nil) → nil

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(X)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
rm(N, nil) → nil
ifrm(true, N, add(M, X)) → rm(N, X)
purge(nil) → nil

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
eq, rm, purge

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

### (8) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(X)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
rm(N, nil) → nil
ifrm(true, N, add(M, X)) → rm(N, X)
purge(nil) → nil

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
eq, rm, purge

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

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

Proved the following rewrite lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Induction Base:
eq(gen_0':s4_0(0), gen_0':s4_0(0)) →RΩ(1)
true

Induction Step:
eq(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →RΩ(1)
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →IH
true

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

### (11) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(X)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
rm(N, nil) → nil
ifrm(true, N, add(M, X)) → rm(N, X)
purge(nil) → nil

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
rm, purge

They will be analysed ascendingly in the following order:
rm < purge

### (12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:

Induction Base:
nil

Induction Step:

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

### (14) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(X)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
rm(N, nil) → nil
ifrm(true, N, add(M, X)) → rm(N, X)
purge(nil) → nil

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
purge

### (15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (16) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(X)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
rm(N, nil) → nil
ifrm(true, N, add(M, X)) → rm(N, X)
purge(nil) → nil

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_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_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

### (19) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(X)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
rm(N, nil) → nil
ifrm(true, N, add(M, X)) → rm(N, X)
purge(nil) → nil

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

### (22) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(X)) → false
eq(s(X), 0') → false
eq(s(X), s(Y)) → eq(X, Y)
rm(N, nil) → nil
ifrm(true, N, add(M, X)) → rm(N, X)
purge(nil) → nil

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))