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:

p(0) → 0
p(s(x)) → x
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, y) → if(le(x, y), x, y)
if(true, x, y) → 0
if(false, x, y) → s(minus(p(x), y))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(z0, z1) → c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1), P(z0))
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(z0, z1) → c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1), P(z0))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c7

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(z0, z1) → c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1))
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(z0, z1) → c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c7

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

Use narrowing to replace MINUS(z0, z1) → c5(IF(le(z0, z1), z0, z1), LE(z0, z1)) by

MINUS(0, z0) → c5(IF(true, 0, z0), LE(0, z0))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0), LE(s(z0), 0))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1))
MINUS(0, z0) → c5(IF(true, 0, z0), LE(0, z0))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0), LE(s(z0), 0))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1))
MINUS(0, z0) → c5(IF(true, 0, z0), LE(0, z0))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0), LE(s(z0), 0))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, IF, MINUS

Compound Symbols:

c4, c7, c5

(7) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 5 dangling nodes:

MINUS(0, z0) → c5(IF(true, 0, z0), LE(0, z0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0), LE(s(z0), 0))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0), LE(s(z0), 0))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, IF, MINUS

Compound Symbols:

c4, c7, c5

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, IF, MINUS

Compound Symbols:

c4, c7, c5, c5

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

Use narrowing to replace IF(false, z0, z1) → c7(MINUS(p(z0), z1)) by

IF(false, 0, x1) → c7(MINUS(0, x1))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, 0, x1) → c7(MINUS(0, x1))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, 0, x1) → c7(MINUS(0, x1))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c5, c7

(13) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 5 dangling nodes:

IF(false, 0, x1) → c7(MINUS(0, x1))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c5, c7

(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(false, s(z0), x1) → c7(MINUS(z0, x1))
We considered the (Usable) Rules:

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

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(IF(x1, x2, x3)) = x2   
POL(LE(x1, x2)) = 0   
POL(MINUS(x1, x2)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c5(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(false) = 0   
POL(le(x1, x2)) = [3] + [4]x1 + [2]x2   
POL(s(x1)) = [2] + x1   
POL(true) = [2]   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
K tuples:

IF(false, s(z0), x1) → c7(MINUS(z0, x1))
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c5, c7

(17) CdtKnowledgeProof (EQUIVALENT transformation)

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

MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
K tuples:

IF(false, s(z0), x1) → c7(MINUS(z0, x1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c5, c7

(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.

LE(s(z0), s(z1)) → c4(LE(z0, z1))
We considered the (Usable) Rules:

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

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(IF(x1, x2, x3)) = [2]x2·x3   
POL(LE(x1, x2)) = x2   
POL(MINUS(x1, x2)) = [2]x2 + [2]x1·x2   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c5(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(false) = 0   
POL(le(x1, x2)) = 0   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0
p(s(z0)) → z0
le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, z1) → if(le(z0, z1), z0, z1)
if(true, z0, z1) → 0
if(false, z0, z1) → s(minus(p(z0), z1))
Tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1))
S tuples:none
K tuples:

IF(false, s(z0), x1) → c7(MINUS(z0, x1))
MINUS(s(z0), s(z1)) → c5(IF(le(z0, z1), s(z0), s(z1)), LE(s(z0), s(z1)))
MINUS(s(z0), 0) → c5(IF(false, s(z0), 0))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c5, c7

(21) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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