(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(a, b) → +(b, a)
+(a, +(b, z0)) → +(b, +(a, z0))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
f(a, z0) → a
f(b, z0) → b
f(+(z0, z1), z2) → +(f(z0, z2), f(z1, z2))
Tuples:
+'(a, b) → c(+'(b, a))
+'(a, +(b, z0)) → c1(+'(b, +(a, z0)), +'(a, z0))
+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
F(+(z0, z1), z2) → c5(+'(f(z0, z2), f(z1, z2)), F(z0, z2), F(z1, z2))
S tuples:
+'(a, b) → c(+'(b, a))
+'(a, +(b, z0)) → c1(+'(b, +(a, z0)), +'(a, z0))
+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
F(+(z0, z1), z2) → c5(+'(f(z0, z2), f(z1, z2)), F(z0, z2), F(z1, z2))
K tuples:none
Defined Rule Symbols:
+, f
Defined Pair Symbols:
+', F
Compound Symbols:
c, c1, c2, c5
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
+'(a, +(b, z0)) → c1(+'(b, +(a, z0)), +'(a, z0))
+'(+(z0, z1), z2) → c2(+'(z0, +(z1, z2)), +'(z1, z2))
F(+(z0, z1), z2) → c5(+'(f(z0, z2), f(z1, z2)), F(z0, z2), F(z1, z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(a, b) → +(b, a)
+(a, +(b, z0)) → +(b, +(a, z0))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
f(a, z0) → a
f(b, z0) → b
f(+(z0, z1), z2) → +(f(z0, z2), f(z1, z2))
Tuples:
+'(a, b) → c(+'(b, a))
S tuples:
+'(a, b) → c(+'(b, a))
K tuples:none
Defined Rule Symbols:
+, f
Defined Pair Symbols:
+'
Compound Symbols:
c
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
+'(a, b) → c(+'(b, a))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(a, b) → +(b, a)
+(a, +(b, z0)) → +(b, +(a, z0))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
f(a, z0) → a
f(b, z0) → b
f(+(z0, z1), z2) → +(f(z0, z2), f(z1, z2))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
+, f
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))