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

(0) Obligation:

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

fact(X) → if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X))))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
prod(0, X) → 0
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
zero(0) → true
zero(s(X)) → false
p(s(X)) → X
s(X) → n__s(X)
0 → n__0
prod(X1, X2) → n__prod(X1, X2)
fact(X) → n__fact(X)
p(X) → n__p(X)
activate(n__s(X)) → s(activate(X))
activate(n__0) → 0
activate(n__prod(X1, X2)) → prod(activate(X1), activate(X2))
activate(n__fact(X)) → fact(activate(X))
activate(n__p(X)) → p(activate(X))
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

fact(z0) → if(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0))))
fact(z0) → n__fact(z0)
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
prod(0, z0) → 0
prod(s(z0), z1) → add(z1, prod(z0, z1))
prod(z0, z1) → n__prod(z0, z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
zero(0) → true
zero(s(z0)) → false
p(s(z0)) → z0
p(z0) → n__p(z0)
s(z0) → n__s(z0)
0 → n__0
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(n__prod(z0, z1)) → prod(activate(z0), activate(z1))
activate(n__fact(z0)) → fact(activate(z0))
activate(n__p(z0)) → p(activate(z0))
activate(z0) → z0

Tuples:

FACT(z0) → c(IF(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0)))), ZERO(z0))
ADD(s(z0), z1) → c3(S(add(z0, z1)), ADD(z0, z1))
PROD(0, z0) → c4(0')
PROD(s(z0), z1) → c5(ADD(z1, prod(z0, z1)), PROD(z0, z1))
IF(true, z0, z1) → c7(ACTIVATE(z0))
IF(false, z0, z1) → c8(ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c15(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c16(0')
ACTIVATE(n__prod(z0, z1)) → c17(PROD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(FACT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(P(activate(z0)), ACTIVATE(z0))

S tuples:

FACT(z0) → c(IF(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0)))), ZERO(z0))
ADD(s(z0), z1) → c3(S(add(z0, z1)), ADD(z0, z1))
PROD(0, z0) → c4(0')
PROD(s(z0), z1) → c5(ADD(z1, prod(z0, z1)), PROD(z0, z1))
IF(true, z0, z1) → c7(ACTIVATE(z0))
IF(false, z0, z1) → c8(ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c15(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c16(0')
ACTIVATE(n__prod(z0, z1)) → c17(PROD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(FACT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(P(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

fact, add, prod, if, zero, p, s, 0, activate

Defined Pair Symbols:

FACT, ADD, PROD, IF, ACTIVATE

Compound Symbols:

c, c3, c4, c5, c7, c8, c15, c16, c17, c18, c19

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

ADD(s(z0), z1) → c3(S(add(z0, z1)), ADD(z0, z1))
PROD(0, z0) → c4(0')
PROD(s(z0), z1) → c5(ADD(z1, prod(z0, z1)), PROD(z0, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

fact(z0) → if(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0))))
fact(z0) → n__fact(z0)
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
prod(0, z0) → 0
prod(s(z0), z1) → add(z1, prod(z0, z1))
prod(z0, z1) → n__prod(z0, z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
zero(0) → true
zero(s(z0)) → false
p(s(z0)) → z0
p(z0) → n__p(z0)
s(z0) → n__s(z0)
0 → n__0
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(n__prod(z0, z1)) → prod(activate(z0), activate(z1))
activate(n__fact(z0)) → fact(activate(z0))
activate(n__p(z0)) → p(activate(z0))
activate(z0) → z0

Tuples:

FACT(z0) → c(IF(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0)))), ZERO(z0))
IF(true, z0, z1) → c7(ACTIVATE(z0))
IF(false, z0, z1) → c8(ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c15(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c16(0')
ACTIVATE(n__prod(z0, z1)) → c17(PROD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(FACT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(P(activate(z0)), ACTIVATE(z0))

S tuples:

FACT(z0) → c(IF(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0)))), ZERO(z0))
IF(true, z0, z1) → c7(ACTIVATE(z0))
IF(false, z0, z1) → c8(ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c15(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c16(0')
ACTIVATE(n__prod(z0, z1)) → c17(PROD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(FACT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(P(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

fact, add, prod, if, zero, p, s, 0, activate

Defined Pair Symbols:

FACT, IF, ACTIVATE

Compound Symbols:

c, c7, c8, c15, c16, c17, c18, c19

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

IF(true, z0, z1) → c7(ACTIVATE(z0))
IF(false, z0, z1) → c8(ACTIVATE(z1))

Removed 2 trailing nodes:

ACTIVATE(n__0) → c16(0')
FACT(z0) → c(IF(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0)))), ZERO(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

fact(z0) → if(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0))))
fact(z0) → n__fact(z0)
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
prod(0, z0) → 0
prod(s(z0), z1) → add(z1, prod(z0, z1))
prod(z0, z1) → n__prod(z0, z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
zero(0) → true
zero(s(z0)) → false
p(s(z0)) → z0
p(z0) → n__p(z0)
s(z0) → n__s(z0)
0 → n__0
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(n__prod(z0, z1)) → prod(activate(z0), activate(z1))
activate(n__fact(z0)) → fact(activate(z0))
activate(n__p(z0)) → p(activate(z0))
activate(z0) → z0

Tuples:

ACTIVATE(n__s(z0)) → c15(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(PROD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(FACT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(P(activate(z0)), ACTIVATE(z0))

S tuples:

ACTIVATE(n__s(z0)) → c15(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(PROD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(FACT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(P(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

fact, add, prod, if, zero, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c17, c18, c19

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

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

fact(z0) → if(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0))))
fact(z0) → n__fact(z0)
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
prod(0, z0) → 0
prod(s(z0), z1) → add(z1, prod(z0, z1))
prod(z0, z1) → n__prod(z0, z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
zero(0) → true
zero(s(z0)) → false
p(s(z0)) → z0
p(z0) → n__p(z0)
s(z0) → n__s(z0)
0 → n__0
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(n__prod(z0, z1)) → prod(activate(z0), activate(z1))
activate(n__fact(z0)) → fact(activate(z0))
activate(n__p(z0)) → p(activate(z0))
activate(z0) → z0

Tuples:

ACTIVATE(n__s(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(ACTIVATE(z0))

S tuples:

ACTIVATE(n__s(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

fact, add, prod, if, zero, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c17, c18, c19

(9) 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.

ACTIVATE(n__fact(z0)) → c18(ACTIVATE(z0))

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

ACTIVATE(n__s(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(ACTIVATE(z0))

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

POL(ACTIVATE(x₁)) = [2]x₁   
POL(c15(x₁)) = x₁   
POL(c17(x₁, x₂)) = x₁ + x₂   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(n__fact(x₁)) = [2] + x₁   
POL(n__p(x₁)) = x₁   
POL(n__prod(x₁, x₂)) = x₁ + x₂   
POL(n__s(x₁)) = x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

fact(z0) → if(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0))))
fact(z0) → n__fact(z0)
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
prod(0, z0) → 0
prod(s(z0), z1) → add(z1, prod(z0, z1))
prod(z0, z1) → n__prod(z0, z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
zero(0) → true
zero(s(z0)) → false
p(s(z0)) → z0
p(z0) → n__p(z0)
s(z0) → n__s(z0)
0 → n__0
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(n__prod(z0, z1)) → prod(activate(z0), activate(z1))
activate(n__fact(z0)) → fact(activate(z0))
activate(n__p(z0)) → p(activate(z0))
activate(z0) → z0

Tuples:

ACTIVATE(n__s(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(ACTIVATE(z0))

S tuples:

ACTIVATE(n__s(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__p(z0)) → c19(ACTIVATE(z0))

K tuples:

ACTIVATE(n__fact(z0)) → c18(ACTIVATE(z0))

Defined Rule Symbols:

fact, add, prod, if, zero, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c17, c18, c19

(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.

ACTIVATE(n__s(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__p(z0)) → c19(ACTIVATE(z0))

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

ACTIVATE(n__s(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(ACTIVATE(z0))

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

POL(ACTIVATE(x₁)) = [2] + [4]x₁   
POL(c15(x₁)) = x₁   
POL(c17(x₁, x₂)) = x₁ + x₂   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(n__fact(x₁)) = x₁   
POL(n__p(x₁)) = [4] + x₁   
POL(n__prod(x₁, x₂)) = [5] + x₁ + x₂   
POL(n__s(x₁)) = [5] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

fact(z0) → if(zero(z0), n__s(n__0), n__prod(z0, n__fact(n__p(z0))))
fact(z0) → n__fact(z0)
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
prod(0, z0) → 0
prod(s(z0), z1) → add(z1, prod(z0, z1))
prod(z0, z1) → n__prod(z0, z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
zero(0) → true
zero(s(z0)) → false
p(s(z0)) → z0
p(z0) → n__p(z0)
s(z0) → n__s(z0)
0 → n__0
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(n__prod(z0, z1)) → prod(activate(z0), activate(z1))
activate(n__fact(z0)) → fact(activate(z0))
activate(n__p(z0)) → p(activate(z0))
activate(z0) → z0

Tuples:

ACTIVATE(n__s(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__fact(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__p(z0)) → c19(ACTIVATE(z0))

S tuples:none
K tuples:

ACTIVATE(n__fact(z0)) → c18(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__prod(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__p(z0)) → c19(ACTIVATE(z0))

Defined Rule Symbols:

fact, add, prod, if, zero, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c17, c18, c19

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))