The Runtime Complexity (innermost) of the given CpxTRS could be proven to be 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: 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:

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: INNERMOST

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

Infered types.

(4) Obligation:

Innermost 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

(5) 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

(6) Obligation:

Innermost 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

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

(7) 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).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost 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

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

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost 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

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.

(12) 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)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost 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

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.

(15) 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)

(16) BOUNDS(n^1, INF)