(0) Obligation:

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

i(x, x) → i(a, b)
g(x, x) → g(a, b)
h(s(f(x))) → h(f(x))
f(s(x)) → s(s(f(h(s(x)))))
f(g(s(x), y)) → f(g(x, s(y)))
h(g(x, s(y))) → h(g(s(x), y))
h(i(x, y)) → i(i(c, h(h(y))), x)
g(a, g(x, g(b, g(a, g(x, y))))) → g(a, g(a, g(a, g(x, g(b, g(b, y))))))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

I(z0, z0) → c1(I(a, b))
G(z0, z0) → c2(G(a, b))
G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1))
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
S tuples:

I(z0, z0) → c1(I(a, b))
G(z0, z0) → c2(G(a, b))
G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1))
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
K tuples:none
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

I, G, H, F

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8

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

Removed 2 trailing nodes:

I(z0, z0) → c1(I(a, b))
G(z0, z0) → c2(G(a, b))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1))
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
S tuples:

G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1))
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
K tuples:none
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

G, H, F

Compound Symbols:

c3, c4, c5, c6, c7, c8

(5) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1))
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1))
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
S tuples:

H(s(f(z0))) → c4(H(f(z0)), F(z0))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
K tuples:none
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

F, H

Compound Symbols:

c7, c8, c4

(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
S tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
K tuples:none
Defined Rule Symbols:

i, g, h, f

Defined Pair Symbols:

F, H

Compound Symbols:

c7, c8, c4

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

i(z0, z0) → i(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(s(f(z0))) → h(f(z0))
g(z0, z0) → g(a, b)
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
S tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
K tuples:none
Defined Rule Symbols:

h, g

Defined Pair Symbols:

F, H

Compound Symbols:

c7, c8, c4

(11) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

H(s(f(z0))) → c4(F(z0))
We considered the (Usable) Rules:

g(z0, z0) → g(a, b)
h(s(f(z0))) → h(f(z0))
And the Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1)) = [4]x1
POL(H(x1)) = [4]x1
POL(a) = [3]
POL(b) = 0
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(f(x1)) = x1
POL(g(x1, x2)) = 0
POL(h(x1)) = 0
POL(s(x1)) = [4] + x1

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(s(f(z0))) → h(f(z0))
g(z0, z0) → g(a, b)
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
S tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
K tuples:

H(s(f(z0))) → c4(F(z0))
Defined Rule Symbols:

h, g

Defined Pair Symbols:

F, H

Compound Symbols:

c7, c8, c4

(13) CdtRuleRemovalProof (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(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
We considered the (Usable) Rules:

g(z0, z0) → g(a, b)
h(s(f(z0))) → h(f(z0))
And the Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1)) = [2] + [4]x1
POL(H(x1)) = [4]x1
POL(a) = 0
POL(b) = 0
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(f(x1)) = x1
POL(g(x1, x2)) = [4]x1 + x2
POL(h(x1)) = 0
POL(s(x1)) = [4] + x1

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(s(f(z0))) → h(f(z0))
g(z0, z0) → g(a, b)
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
S tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
K tuples:

H(s(f(z0))) → c4(F(z0))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
Defined Rule Symbols:

h, g

Defined Pair Symbols:

F, H

Compound Symbols:

c7, c8, c4

(15) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
We considered the (Usable) Rules:

g(z0, z0) → g(a, b)
h(s(f(z0))) → h(f(z0))
And the Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1)) = [1] + [2]x1 + [3]x12
POL(H(x1)) = [3]x12
POL(a) = 0
POL(b) = 0
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(f(x1)) = x1
POL(g(x1, x2)) = 0
POL(h(x1)) = 0
POL(s(x1)) = [2] + x1

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(s(f(z0))) → h(f(z0))
g(z0, z0) → g(a, b)
Tuples:

F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
S tuples:none
K tuples:

H(s(f(z0))) → c4(F(z0))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
Defined Rule Symbols:

h, g

Defined Pair Symbols:

F, H

Compound Symbols:

c7, c8, c4

(17) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty