The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

rw(Val(n), c) → Op(Val(n), rewrite(c))
rewrite(Op(x, y)) → rw(x, y)
rw(Op(x, y), c) → rw[Let](Op(x, y), c, rewrite(x))
rewrite(Val(n)) → Val(n)
second(Op(x, y)) → y
isOp(Val(n)) → False
isOp(Op(x, y)) → True
first(Val(n)) → Val(n)
first(Op(x, y)) → x
assrewrite(exp) → rewrite(exp)

The (relative) TRS S consists of the following rules:

rw[Let](Op(x, y), c, a1) → rw[Let][Let](Op(x, y), c, a1, rewrite(y))
rw[Let][Let](ab, c, a1, b1) → rw[Let][Let][Let](c, a1, b1, rewrite(c))
rw[Let][Let][Let](c, a1, b1, c1) → rw(a1, Op(b1, c1))

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

rw(Val(n), c) → Op(Val(n), rewrite(c))
rewrite(Op(x, y)) → rw(x, y)
rw(Op(x, y), c) → rw[Let](Op(x, y), c, rewrite(x))
rewrite(Val(n)) → Val(n)
second(Op(x, y)) → y
isOp(Val(n)) → False
isOp(Op(x, y)) → True
first(Val(n)) → Val(n)
first(Op(x, y)) → x
assrewrite(exp) → rewrite(exp)

The (relative) TRS S consists of the following rules:

rw[Let](Op(x, y), c, a1) → rw[Let][Let](Op(x, y), c, a1, rewrite(y))
rw[Let][Let](ab, c, a1, b1) → rw[Let][Let][Let](c, a1, b1, rewrite(c))
rw[Let][Let][Let](c, a1, b1, c1) → rw(a1, Op(b1, c1))

Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Val/0
rw[Let][Let]/0
rw[Let][Let][Let]/0

(4) Obligation:

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

rw(Val, c) → Op(Val, rewrite(c))
rewrite(Op(x, y)) → rw(x, y)
rw(Op(x, y), c) → rw[Let](Op(x, y), c, rewrite(x))
rewrite(Val) → Val
second(Op(x, y)) → y
isOp(Val) → False
isOp(Op(x, y)) → True
first(Val) → Val
first(Op(x, y)) → x
assrewrite(exp) → rewrite(exp)

The (relative) TRS S consists of the following rules:

rw[Let](Op(x, y), c, a1) → rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) → rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) → rw(a1, Op(b1, c1))

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
rw(Val, c) → Op(Val, rewrite(c))
rewrite(Op(x, y)) → rw(x, y)
rw(Op(x, y), c) → rw[Let](Op(x, y), c, rewrite(x))
rewrite(Val) → Val
second(Op(x, y)) → y
isOp(Val) → False
isOp(Op(x, y)) → True
first(Val) → Val
first(Op(x, y)) → x
assrewrite(exp) → rewrite(exp)
rw[Let](Op(x, y), c, a1) → rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) → rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) → rw(a1, Op(b1, c1))

Types:
rw :: Val:Op → Val:Op → Val:Op
Val :: Val:Op
Op :: Val:Op → Val:Op → Val:Op
rewrite :: Val:Op → Val:Op
rw[Let] :: Val:Op → Val:Op → Val:Op → Val:Op
second :: Val:Op → Val:Op
isOp :: Val:Op → False:True
False :: False:True
True :: False:True
first :: Val:Op → Val:Op
assrewrite :: Val:Op → Val:Op
rw[Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
rw[Let][Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
hole_Val:Op1_0 :: Val:Op
hole_False:True2_0 :: False:True
gen_Val:Op3_0 :: Nat → Val:Op

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

Heuristically decided to analyse the following defined symbols:
rw, rewrite

They will be analysed ascendingly in the following order:
rw = rewrite

(8) Obligation:

Innermost TRS:
Rules:
rw(Val, c) → Op(Val, rewrite(c))
rewrite(Op(x, y)) → rw(x, y)
rw(Op(x, y), c) → rw[Let](Op(x, y), c, rewrite(x))
rewrite(Val) → Val
second(Op(x, y)) → y
isOp(Val) → False
isOp(Op(x, y)) → True
first(Val) → Val
first(Op(x, y)) → x
assrewrite(exp) → rewrite(exp)
rw[Let](Op(x, y), c, a1) → rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) → rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) → rw(a1, Op(b1, c1))

Types:
rw :: Val:Op → Val:Op → Val:Op
Val :: Val:Op
Op :: Val:Op → Val:Op → Val:Op
rewrite :: Val:Op → Val:Op
rw[Let] :: Val:Op → Val:Op → Val:Op → Val:Op
second :: Val:Op → Val:Op
isOp :: Val:Op → False:True
False :: False:True
True :: False:True
first :: Val:Op → Val:Op
assrewrite :: Val:Op → Val:Op
rw[Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
rw[Let][Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
hole_Val:Op1_0 :: Val:Op
hole_False:True2_0 :: False:True
gen_Val:Op3_0 :: Nat → Val:Op

Generator Equations:
gen_Val:Op3_0(0) ⇔ Val
gen_Val:Op3_0(+(x, 1)) ⇔ Op(Val, gen_Val:Op3_0(x))

The following defined symbols remain to be analysed:
rewrite, rw

They will be analysed ascendingly in the following order:
rw = rewrite

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

Proved the following rewrite lemma:
rewrite(gen_Val:Op3_0(n5_0)) → gen_Val:Op3_0(n5_0), rt ∈ Ω(1 + n5₀)

Induction Base:
rewrite(gen_Val:Op3_0(0)) →_R^Ω(1)
Val

Induction Step:
rewrite(gen_Val:Op3_0(+(n5_0, 1))) →_R^Ω(1)
rw(Val, gen_Val:Op3_0(n5_0)) →_R^Ω(1)
Op(Val, rewrite(gen_Val:Op3_0(n5_0))) →_IH
Op(Val, gen_Val:Op3_0(c6_0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
rw(Val, c) → Op(Val, rewrite(c))
rewrite(Op(x, y)) → rw(x, y)
rw(Op(x, y), c) → rw[Let](Op(x, y), c, rewrite(x))
rewrite(Val) → Val
second(Op(x, y)) → y
isOp(Val) → False
isOp(Op(x, y)) → True
first(Val) → Val
first(Op(x, y)) → x
assrewrite(exp) → rewrite(exp)
rw[Let](Op(x, y), c, a1) → rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) → rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) → rw(a1, Op(b1, c1))

Types:
rw :: Val:Op → Val:Op → Val:Op
Val :: Val:Op
Op :: Val:Op → Val:Op → Val:Op
rewrite :: Val:Op → Val:Op
rw[Let] :: Val:Op → Val:Op → Val:Op → Val:Op
second :: Val:Op → Val:Op
isOp :: Val:Op → False:True
False :: False:True
True :: False:True
first :: Val:Op → Val:Op
assrewrite :: Val:Op → Val:Op
rw[Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
rw[Let][Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
hole_Val:Op1_0 :: Val:Op
hole_False:True2_0 :: False:True
gen_Val:Op3_0 :: Nat → Val:Op

Lemmas:
rewrite(gen_Val:Op3_0(n5_0)) → gen_Val:Op3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Val:Op3_0(0) ⇔ Val
gen_Val:Op3_0(+(x, 1)) ⇔ Op(Val, gen_Val:Op3_0(x))

The following defined symbols remain to be analysed:
rw

They will be analysed ascendingly in the following order:
rw = rewrite

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

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

(13) Obligation:

Innermost TRS:
Rules:
rw(Val, c) → Op(Val, rewrite(c))
rewrite(Op(x, y)) → rw(x, y)
rw(Op(x, y), c) → rw[Let](Op(x, y), c, rewrite(x))
rewrite(Val) → Val
second(Op(x, y)) → y
isOp(Val) → False
isOp(Op(x, y)) → True
first(Val) → Val
first(Op(x, y)) → x
assrewrite(exp) → rewrite(exp)
rw[Let](Op(x, y), c, a1) → rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) → rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) → rw(a1, Op(b1, c1))

Types:
rw :: Val:Op → Val:Op → Val:Op
Val :: Val:Op
Op :: Val:Op → Val:Op → Val:Op
rewrite :: Val:Op → Val:Op
rw[Let] :: Val:Op → Val:Op → Val:Op → Val:Op
second :: Val:Op → Val:Op
isOp :: Val:Op → False:True
False :: False:True
True :: False:True
first :: Val:Op → Val:Op
assrewrite :: Val:Op → Val:Op
rw[Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
rw[Let][Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
hole_Val:Op1_0 :: Val:Op
hole_False:True2_0 :: False:True
gen_Val:Op3_0 :: Nat → Val:Op

Lemmas:
rewrite(gen_Val:Op3_0(n5_0)) → gen_Val:Op3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Val:Op3_0(0) ⇔ Val
gen_Val:Op3_0(+(x, 1)) ⇔ Op(Val, gen_Val:Op3_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rewrite(gen_Val:Op3_0(n5_0)) → gen_Val:Op3_0(n5_0), rt ∈ Ω(1 + n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
rw(Val, c) → Op(Val, rewrite(c))
rewrite(Op(x, y)) → rw(x, y)
rw(Op(x, y), c) → rw[Let](Op(x, y), c, rewrite(x))
rewrite(Val) → Val
second(Op(x, y)) → y
isOp(Val) → False
isOp(Op(x, y)) → True
first(Val) → Val
first(Op(x, y)) → x
assrewrite(exp) → rewrite(exp)
rw[Let](Op(x, y), c, a1) → rw[Let][Let](c, a1, rewrite(y))
rw[Let][Let](c, a1, b1) → rw[Let][Let][Let](a1, b1, rewrite(c))
rw[Let][Let][Let](a1, b1, c1) → rw(a1, Op(b1, c1))

Types:
rw :: Val:Op → Val:Op → Val:Op
Val :: Val:Op
Op :: Val:Op → Val:Op → Val:Op
rewrite :: Val:Op → Val:Op
rw[Let] :: Val:Op → Val:Op → Val:Op → Val:Op
second :: Val:Op → Val:Op
isOp :: Val:Op → False:True
False :: False:True
True :: False:True
first :: Val:Op → Val:Op
assrewrite :: Val:Op → Val:Op
rw[Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
rw[Let][Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
hole_Val:Op1_0 :: Val:Op
hole_False:True2_0 :: False:True
gen_Val:Op3_0 :: Nat → Val:Op

Lemmas:
rewrite(gen_Val:Op3_0(n5_0)) → gen_Val:Op3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Val:Op3_0(0) ⇔ Val
gen_Val:Op3_0(+(x, 1)) ⇔ Op(Val, gen_Val:Op3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rewrite(gen_Val:Op3_0(n5_0)) → gen_Val:Op3_0(n5_0), rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)