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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳4 CdtProblem
↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳6 CdtProblem
↳7 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳8 CdtProblem
↳9 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳10 CdtProblem
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳12 CdtProblem
↳13 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳14 CdtProblem
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳16 CdtProblem
↳17 CdtKnowledgeProof (⇔)
↳18 CdtProblem
↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳20 CdtProblem
↳21 SIsEmptyProof (⇔)
↳22 BOUNDS(O(1), O(1))

(0) Obligation:

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

m(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(m(p(x), y))
if(false, x, y) → 0
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
p(0) → 0
p(s(x)) → x

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

M(z0, z1) → c(IF(gt(z0, z1), z0, z1), GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1), P(z0))
GT(s(z0), s(z1)) → c5(GT(z0, z1))
S tuples:

M(z0, z1) → c(IF(gt(z0, z1), z0, z1), GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1), P(z0))
GT(s(z0), s(z1)) → c5(GT(z0, z1))
K tuples:none
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

M, IF, GT

Compound Symbols:

c, c1, c5

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

M(z0, z1) → c(IF(gt(z0, z1), z0, z1), GT(z0, z1))
GT(s(z0), s(z1)) → c5(GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1))
S tuples:

M(z0, z1) → c(IF(gt(z0, z1), z0, z1), GT(z0, z1))
GT(s(z0), s(z1)) → c5(GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1))
K tuples:none
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

M, GT, IF

Compound Symbols:

c, c5, c1

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

Use narrowing to replace M(z0, z1) → c(IF(gt(z0, z1), z0, z1), GT(z0, z1)) by

M(0, z0) → c(IF(false, 0, z0), GT(0, z0))
M(s(z0), 0) → c(IF(true, s(z0), 0), GT(s(z0), 0))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1))
M(0, z0) → c(IF(false, 0, z0), GT(0, z0))
M(s(z0), 0) → c(IF(true, s(z0), 0), GT(s(z0), 0))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
S tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1))
M(0, z0) → c(IF(false, 0, z0), GT(0, z0))
M(s(z0), 0) → c(IF(true, s(z0), 0), GT(s(z0), 0))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

GT, IF, M

Compound Symbols:

c5, c1, c

(7) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 5 dangling nodes:

M(0, z0) → c(IF(false, 0, z0), GT(0, z0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1))
M(s(z0), 0) → c(IF(true, s(z0), 0), GT(s(z0), 0))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
S tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1))
M(s(z0), 0) → c(IF(true, s(z0), 0), GT(s(z0), 0))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

GT, IF, M

Compound Symbols:

c5, c1, c

(9) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
S tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
IF(true, z0, z1) → c1(M(p(z0), z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
K tuples:none
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

GT, IF, M

Compound Symbols:

c5, c1, c, c

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

Use narrowing to replace IF(true, z0, z1) → c1(M(p(z0), z1)) by

IF(true, 0, x1) → c1(M(0, x1))
IF(true, s(z0), x1) → c1(M(z0, x1))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, 0, x1) → c1(M(0, x1))
IF(true, s(z0), x1) → c1(M(z0, x1))
S tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, 0, x1) → c1(M(0, x1))
IF(true, s(z0), x1) → c1(M(z0, x1))
K tuples:none
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

GT, M, IF

Compound Symbols:

c5, c, c, c1

(13) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 5 dangling nodes:

IF(true, 0, x1) → c1(M(0, x1))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, s(z0), x1) → c1(M(z0, x1))
S tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, s(z0), x1) → c1(M(z0, x1))
K tuples:none
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

GT, M, IF

Compound Symbols:

c5, c, c, c1

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

IF(true, s(z0), x1) → c1(M(z0, x1))
We considered the (Usable) Rules:

gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
And the Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, s(z0), x1) → c1(M(z0, x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(GT(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = x2   
POL(M(x1, x2)) = x1   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1(x1)) = x1   
POL(c5(x1)) = x1   
POL(false) = [2]   
POL(gt(x1, x2)) = [3] + [4]x1 + [2]x2   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, s(z0), x1) → c1(M(z0, x1))
S tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
K tuples:

IF(true, s(z0), x1) → c1(M(z0, x1))
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

GT, M, IF

Compound Symbols:

c5, c, c, c1

(17) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, s(z0), x1) → c1(M(z0, x1))
IF(true, s(z0), x1) → c1(M(z0, x1))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, s(z0), x1) → c1(M(z0, x1))
S tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
K tuples:

IF(true, s(z0), x1) → c1(M(z0, x1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

GT, M, IF

Compound Symbols:

c5, c, c, c1

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

GT(s(z0), s(z1)) → c5(GT(z0, z1))
We considered the (Usable) Rules:

gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
And the Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, s(z0), x1) → c1(M(z0, x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(GT(x1, x2)) = x2   
POL(IF(x1, x2, x3)) = [2]x2·x3   
POL(M(x1, x2)) = [2]x2 + [2]x1·x2   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1(x1)) = x1   
POL(c5(x1)) = x1   
POL(false) = 0   
POL(gt(x1, x2)) = 0   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

m(z0, z1) → if(gt(z0, z1), z0, z1)
if(true, z0, z1) → s(m(p(z0), z1))
if(false, z0, z1) → 0
gt(0, z0) → false
gt(s(z0), 0) → true
gt(s(z0), s(z1)) → gt(z0, z1)
p(0) → 0
p(s(z0)) → z0
Tuples:

GT(s(z0), s(z1)) → c5(GT(z0, z1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
IF(true, s(z0), x1) → c1(M(z0, x1))
S tuples:none
K tuples:

IF(true, s(z0), x1) → c1(M(z0, x1))
M(s(z0), s(z1)) → c(IF(gt(z0, z1), s(z0), s(z1)), GT(s(z0), s(z1)))
M(s(z0), 0) → c(IF(true, s(z0), 0))
GT(s(z0), s(z1)) → c5(GT(z0, z1))
Defined Rule Symbols:

m, if, gt, p

Defined Pair Symbols:

GT, M, IF

Compound Symbols:

c5, c, c, c1

(21) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(22) BOUNDS(O(1), O(1))