(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
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 Cpx (relative) TRS 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) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(g(h(a, b), c), d) → c2(F(.(b, g(h(a, b), c)), d), F(c, d'))
F(g(i(a, b, b'), c), d) → c1(F(.(b, c), d'), F(.(b', 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 (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)