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:

rewrite(Op(Val(n), y)) → Op(rewrite(y), Val(n))
rewrite(Op(Op(x, y), y')) → rewrite[Let](Op(Op(x, y), y'), Op(x, y), 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:

rewrite[Let](exp, Op(x, y), a1) → rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y))
rewrite[Let][Let](Op(x, y), opab, a1, b1) → rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y))
rewrite[Let][Let][Let](exp, a1, b1, c1) → rewrite(Op(a1, Op(b1, rewrite(c1))))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
rewrite(Op(Val(n), y)) →⁺ Op(rewrite(y), Val(n))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [y / Op(Val(n), y)].
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:

rewrite(Op(Val(n), y)) → Op(rewrite(y), Val(n))
rewrite(Op(Op(x, y), y')) → rewrite[Let](Op(Op(x, y), y'), Op(x, y), 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:

rewrite[Let](exp, Op(x, y), a1) → rewrite[Let][Let](exp, Op(x, y), a1, rewrite(y))
rewrite[Let][Let](Op(x, y), opab, a1, b1) → rewrite[Let][Let][Let](Op(x, y), a1, b1, rewrite(y))
rewrite[Let][Let][Let](exp, a1, b1, c1) → rewrite(Op(a1, Op(b1, rewrite(c1))))

Rewrite Strategy: INNERMOST

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Val/0
rewrite[Let][Let]/1
rewrite[Let][Let][Let]/0

(6) Obligation:

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

rewrite(Op(Val, y)) → Op(rewrite(y), Val)
rewrite(Op(Op(x, y), y')) → rewrite[Let](Op(Op(x, y), y'), Op(x, y), 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:

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

Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

Innermost TRS:
Rules:
rewrite(Op(Val, y)) → Op(rewrite(y), Val)
rewrite(Op(Op(x, y), y')) → rewrite[Let](Op(Op(x, y), y'), Op(x, y), 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)
rewrite[Let](exp, Op(x, y), a1) → rewrite[Let][Let](exp, a1, rewrite(y))
rewrite[Let][Let](Op(x, y), a1, b1) → rewrite[Let][Let][Let](a1, b1, rewrite(y))
rewrite[Let][Let][Let](a1, b1, c1) → rewrite(Op(a1, Op(b1, rewrite(c1))))

Types:
rewrite :: Val:Op → Val:Op
Op :: Val:Op → Val:Op → Val:Op
Val :: Val:Op
rewrite[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
rewrite[Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
rewrite[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

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
rewrite

(10) Obligation:

Innermost TRS:
Rules:
rewrite(Op(Val, y)) → Op(rewrite(y), Val)
rewrite(Op(Op(x, y), y')) → rewrite[Let](Op(Op(x, y), y'), Op(x, y), 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)
rewrite[Let](exp, Op(x, y), a1) → rewrite[Let][Let](exp, a1, rewrite(y))
rewrite[Let][Let](Op(x, y), a1, b1) → rewrite[Let][Let][Let](a1, b1, rewrite(y))
rewrite[Let][Let][Let](a1, b1, c1) → rewrite(Op(a1, Op(b1, rewrite(c1))))

Types:
rewrite :: Val:Op → Val:Op
Op :: Val:Op → Val:Op → Val:Op
Val :: Val:Op
rewrite[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
rewrite[Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
rewrite[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

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
rewrite(gen_Val:Op3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
rewrite(gen_Val:Op3_0(+(1, 0)))

Induction Step:
rewrite(gen_Val:Op3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
Op(rewrite(gen_Val:Op3_0(+(1, n5_0))), Val) →_IH
Op(*4_0, Val)

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
rewrite(Op(Val, y)) → Op(rewrite(y), Val)
rewrite(Op(Op(x, y), y')) → rewrite[Let](Op(Op(x, y), y'), Op(x, y), 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)
rewrite[Let](exp, Op(x, y), a1) → rewrite[Let][Let](exp, a1, rewrite(y))
rewrite[Let][Let](Op(x, y), a1, b1) → rewrite[Let][Let][Let](a1, b1, rewrite(y))
rewrite[Let][Let][Let](a1, b1, c1) → rewrite(Op(a1, Op(b1, rewrite(c1))))

Types:
rewrite :: Val:Op → Val:Op
Op :: Val:Op → Val:Op → Val:Op
Val :: Val:Op
rewrite[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
rewrite[Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
rewrite[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(+(1, n5_0))) → *4_0, rt ∈ Ω(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(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
rewrite(Op(Val, y)) → Op(rewrite(y), Val)
rewrite(Op(Op(x, y), y')) → rewrite[Let](Op(Op(x, y), y'), Op(x, y), 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)
rewrite[Let](exp, Op(x, y), a1) → rewrite[Let][Let](exp, a1, rewrite(y))
rewrite[Let][Let](Op(x, y), a1, b1) → rewrite[Let][Let][Let](a1, b1, rewrite(y))
rewrite[Let][Let][Let](a1, b1, c1) → rewrite(Op(a1, Op(b1, rewrite(c1))))

Types:
rewrite :: Val:Op → Val:Op
Op :: Val:Op → Val:Op → Val:Op
Val :: Val:Op
rewrite[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
rewrite[Let][Let] :: Val:Op → Val:Op → Val:Op → Val:Op
rewrite[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(+(1, n5_0))) → *4_0, rt ∈ Ω(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(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(18) BOUNDS(n^1, INF)