The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 0th argument of plus: plus, times

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

times(x, s(y)) → plus(times(x, y), x)
plus(x, s(y)) → s(plus(x, y))
plus(x, 0) → x
times(x, 0) → 0

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, s(z1)) → plus(times(z0, z1), z0)
times(z0, 0) → 0
plus(z0, s(z1)) → s(plus(z0, z1))
plus(z0, 0) → z0

Tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))
TIMES(z0, 0) → c1
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
PLUS(z0, 0) → c3

S tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))
TIMES(z0, 0) → c1
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
PLUS(z0, 0) → c3

K tuples:none
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c, c1, c2, c3

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

Removed 2 trailing nodes:

PLUS(z0, 0) → c3
TIMES(z0, 0) → c1

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, s(z1)) → plus(times(z0, z1), z0)
times(z0, 0) → 0
plus(z0, s(z1)) → s(plus(z0, z1))
plus(z0, 0) → z0

Tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

S tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

K tuples:none
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c, c2

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

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

POL(0) = 0   
POL(PLUS(x₁, x₂)) = 0   
POL(TIMES(x₁, x₂)) = x₂   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(plus(x₁, x₂)) = 0   
POL(s(x₁)) = [1] + x₁   
POL(times(x₁, x₂)) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, s(z1)) → plus(times(z0, z1), z0)
times(z0, 0) → 0
plus(z0, s(z1)) → s(plus(z0, z1))
plus(z0, 0) → z0

Tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

S tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

K tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))

Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c, c2

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

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

POL(0) = 0   
POL(PLUS(x₁, x₂)) = [1] + x₂   
POL(TIMES(x₁, x₂)) = [2]x₂ + x₁·x₂   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(plus(x₁, x₂)) = [2]x₁ + [2]x₂ + [2]x₂² + x₁·x₂ + [2]x₁²   
POL(s(x₁)) = [2] + x₁   
POL(times(x₁, x₂)) = x₁ + [2]x₁·x₂ + [2]x₁²   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, s(z1)) → plus(times(z0, z1), z0)
times(z0, 0) → 0
plus(z0, s(z1)) → s(plus(z0, z1))
plus(z0, 0) → z0

Tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

S tuples:none
K tuples:

TIMES(z0, s(z1)) → c(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))

Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c, c2

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

The set S is empty

(12) BOUNDS(1, 1)