(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) 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)))
MINUS(x0, x1) → c5

(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))
IF(false, z0, z1) → c7(MINUS(p(z0), z1), P(z0))
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)))
MINUS(x0, x1) → c5
S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF(false, z0, z1) → c7(MINUS(p(z0), z1), P(z0))
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)))
MINUS(x0, x1) → c5
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, IF, MINUS

Compound Symbols:

c4, c7, c5, c5

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

Removed 2 trailing nodes:

MINUS(0, z0) → c5(IF(true, 0, z0), LE(0, z0))
MINUS(x0, x1) → c5

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

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

IF(false, 0, x1) → c7(MINUS(0, x1), P(0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1), P(s(z0)))
IF(false, x0, x1) → c7

(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))
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)))
IF(false, 0, x1) → c7(MINUS(0, x1), P(0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1), P(s(z0)))
IF(false, x0, x1) → c7
S tuples:

LE(s(z0), s(z1)) → c4(LE(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)))
IF(false, 0, x1) → c7(MINUS(0, x1), P(0))
IF(false, s(z0), x1) → c7(MINUS(z0, x1), P(s(z0)))
IF(false, x0, x1) → c7
K tuples:none
Defined Rule Symbols:

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c7, c7

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

Removed 2 trailing nodes:

IF(false, 0, x1) → c7(MINUS(0, x1), P(0))
IF(false, x0, x1) → c7

(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))
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)))
IF(false, s(z0), x1) → c7(MINUS(z0, x1), P(s(z0)))
S tuples:

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

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c7

(11) 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), P(s(z0)))
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), 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)))
IF(false, s(z0), x1) → c7(MINUS(z0, x1), P(s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(IF(x1, x2, x3)) = [2]x2   
POL(LE(x1, x2)) = 0   
POL(MINUS(x1, x2)) = [2]x1   
POL(P(x1)) = 0   
POL(c4(x1)) = x1   
POL(c5(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(false) = 0   
POL(le(x1, x2)) = [5]x1   
POL(s(x1)) = [4] + x1   
POL(true) = [2]   

(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), 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)))
IF(false, s(z0), x1) → c7(MINUS(z0, x1), P(s(z0)))
S tuples:

LE(s(z0), s(z1)) → c4(LE(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:

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

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c7

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

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

(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), 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)))
IF(false, s(z0), x1) → c7(MINUS(z0, x1), P(s(z0)))
S tuples:

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

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

p, le, minus, if

Defined Pair Symbols:

LE, MINUS, IF

Compound Symbols:

c4, c5, c7

(15) 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), 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)))
IF(false, s(z0), x1) → c7(MINUS(z0, x1), P(s(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(IF(x1, x2, x3)) = [2]x22   
POL(LE(x1, x2)) = [2]x1   
POL(MINUS(x1, x2)) = [2] + [2]x1 + [2]x12   
POL(P(x1)) = 0   
POL(c4(x1)) = x1   
POL(c5(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(false) = 0   
POL(le(x1, x2)) = 0   
POL(s(x1)) = [1] + x1   
POL(true) = 0   

(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), 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)))
IF(false, s(z0), x1) → c7(MINUS(z0, x1), P(s(z0)))
S tuples:none
K tuples:

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

(17) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(18) BOUNDS(O(1), O(1))