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

(0) Obligation:

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

del(.(x, .(y, z))) → f(=(x, y), x, y, z)
f(true, x, y, z) → del(.(y, z))
f(false, x, y, z) → .(x, del(.(y, z)))
=(nil, nil) → true
=(.(x, y), nil) → false
=(nil, .(y, z)) → false
=(.(x, y), .(u, v)) → and(=(x, u), =(y, v))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

del(.(z0, .(z1, z2))) → f(=(z0, z1), z0, z1, z2)
f(true, z0, z1, z2) → del(.(z1, z2))
f(false, z0, z1, z2) → .(z0, del(.(z1, z2)))
=(nil, nil) → true
=(.(z0, z1), nil) → false
=(nil, .(z0, z1)) → false
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))

Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))
='(.(z0, z1), .(u, v)) → c6(='(z0, u), ='(z1, v))

S tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))
='(.(z0, z1), .(u, v)) → c6(='(z0, u), ='(z1, v))

K tuples:none
Defined Rule Symbols:

del, f, =

Defined Pair Symbols:

DEL, F, ='

Compound Symbols:

c, c1, c2, c6

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

Removed 1 trailing nodes:

='(.(z0, z1), .(u, v)) → c6(='(z0, u), ='(z1, v))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

del(.(z0, .(z1, z2))) → f(=(z0, z1), z0, z1, z2)
f(true, z0, z1, z2) → del(.(z1, z2))
f(false, z0, z1, z2) → .(z0, del(.(z1, z2)))
=(nil, nil) → true
=(.(z0, z1), nil) → false
=(nil, .(z0, z1)) → false
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))

Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))

S tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))

K tuples:none
Defined Rule Symbols:

del, f, =

Defined Pair Symbols:

DEL, F

Compound Symbols:

c, c1, c2

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))

We considered the (Usable) Rules:

=(nil, nil) → true
=(.(z0, z1), nil) → false
=(nil, .(z0, z1)) → false
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))

And the Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [4] + x₂   
POL(=(x₁, x₂)) = [3]   
POL(='(x₁, x₂)) = 0   
POL(DEL(x₁)) = [2]x₁   
POL(F(x₁, x₂, x₃, x₄)) = [4] + [4]x₁ + [2]x₄   
POL(and(x₁, x₂)) = 0   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(false) = [2]   
POL(nil) = [3]   
POL(true) = [1]   
POL(u) = 0   
POL(v) = [2]   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

del(.(z0, .(z1, z2))) → f(=(z0, z1), z0, z1, z2)
f(true, z0, z1, z2) → del(.(z1, z2))
f(false, z0, z1, z2) → .(z0, del(.(z1, z2)))
=(nil, nil) → true
=(.(z0, z1), nil) → false
=(nil, .(z0, z1)) → false
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))

Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))

S tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))

K tuples:

F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))

Defined Rule Symbols:

del, f, =

Defined Pair Symbols:

DEL, F

Compound Symbols:

c, c1, c2

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))

We considered the (Usable) Rules:

=(nil, nil) → true
=(.(z0, z1), nil) → false
=(nil, .(z0, z1)) → false
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))

And the Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [1] + x₁ + x₂   
POL(=(x₁, x₂)) = [3] + [3]x₁ + x₂   
POL(='(x₁, x₂)) = x₁   
POL(DEL(x₁)) = x₁   
POL(F(x₁, x₂, x₃, x₄)) = [2] + x₃ + x₄   
POL(and(x₁, x₂)) = [2]   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(false) = [4]   
POL(nil) = [1]   
POL(true) = [3]   
POL(u) = 0   
POL(v) = [5]   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

del(.(z0, .(z1, z2))) → f(=(z0, z1), z0, z1, z2)
f(true, z0, z1, z2) → del(.(z1, z2))
f(false, z0, z1, z2) → .(z0, del(.(z1, z2)))
=(nil, nil) → true
=(.(z0, z1), nil) → false
=(nil, .(z0, z1)) → false
=(.(z0, z1), .(u, v)) → and(=(z0, u), =(z1, v))

Tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))

S tuples:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))

K tuples:

F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))

Defined Rule Symbols:

del, f, =

Defined Pair Symbols:

DEL, F

Compound Symbols:

c, c1, c2

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

DEL(.(z0, .(z1, z2))) → c(F(=(z0, z1), z0, z1, z2), ='(z0, z1))
F(true, z0, z1, z2) → c1(DEL(.(z1, z2)))
F(false, z0, z1, z2) → c2(DEL(.(z1, z2)))

Now S is empty

(10) BOUNDS(O(1), O(1))