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

(0) Obligation:

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

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

Tuples:

F(g(i(a, b, b'), c), d) → c1(F(.(b, c), d'), F(.(b', c), d'))
F(g(h(a, b), c), d) → c2(F(.(b, g(h(a, b), c)), d), F(c, d'))

S tuples:

F(g(i(a, b, b'), c), d) → c1(F(.(b, c), d'), F(.(b', c), d'))
F(g(h(a, b), c), d) → c2(F(.(b, g(h(a, b), c)), d), F(c, d'))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 2 dangling nodes:

F(g(i(a, b, b'), c), d) → c1(F(.(b, c), d'), F(.(b', c), d'))
F(g(h(a, b), c), d) → c2(F(.(b, g(h(a, b), c)), d), F(c, d'))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))