Term Rewriting System R:
[X, Y, X1, X2, X3]
active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(fact(X)) -> IF(zero(X), s(0), prod(X, fact(p(X))))
ACTIVE(fact(X)) -> ZERO(X)
ACTIVE(fact(X)) -> S(0)
ACTIVE(fact(X)) -> PROD(X, fact(p(X)))
ACTIVE(fact(X)) -> FACT(p(X))
ACTIVE(fact(X)) -> P(X)
ACTIVE(add(s(X), Y)) -> S(add(X, Y))
ACTIVE(add(s(X), Y)) -> ADD(X, Y)
ACTIVE(prod(s(X), Y)) -> ADD(Y, prod(X, Y))
ACTIVE(prod(s(X), Y)) -> PROD(X, Y)
ACTIVE(fact(X)) -> FACT(active(X))
ACTIVE(fact(X)) -> ACTIVE(X)
ACTIVE(if(X1, X2, X3)) -> IF(active(X1), X2, X3)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(zero(X)) -> ZERO(active(X))
ACTIVE(zero(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(prod(X1, X2)) -> PROD(active(X1), X2)
ACTIVE(prod(X1, X2)) -> ACTIVE(X1)
ACTIVE(prod(X1, X2)) -> PROD(X1, active(X2))
ACTIVE(prod(X1, X2)) -> ACTIVE(X2)
ACTIVE(p(X)) -> P(active(X))
ACTIVE(p(X)) -> ACTIVE(X)
ACTIVE(add(X1, X2)) -> ADD(active(X1), X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(X1, active(X2))
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
FACT(mark(X)) -> FACT(X)
FACT(ok(X)) -> FACT(X)
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
ZERO(mark(X)) -> ZERO(X)
ZERO(ok(X)) -> ZERO(X)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
PROD(mark(X1), X2) -> PROD(X1, X2)
PROD(X1, mark(X2)) -> PROD(X1, X2)
PROD(ok(X1), ok(X2)) -> PROD(X1, X2)
P(mark(X)) -> P(X)
P(ok(X)) -> P(X)
ADD(mark(X1), X2) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
PROPER(fact(X)) -> FACT(proper(X))
PROPER(fact(X)) -> PROPER(X)
PROPER(if(X1, X2, X3)) -> IF(proper(X1), proper(X2), proper(X3))
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(zero(X)) -> ZERO(proper(X))
PROPER(zero(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(prod(X1, X2)) -> PROD(proper(X1), proper(X2))
PROPER(prod(X1, X2)) -> PROPER(X1)
PROPER(prod(X1, X2)) -> PROPER(X2)
PROPER(p(X)) -> P(proper(X))
PROPER(p(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> ADD(proper(X1), proper(X2))
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(add(X1, X2)) -> PROPER(X2)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains 10 SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
  2. IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1
2>2
3>3
{2, 1} , {2, 1}
1>1
2=2
3=3

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1
2=2
3=3
{2, 1} , {2, 1}
1>1
2>2
3>3

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Size-Change Principle
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

ZERO(ok(X)) -> ZERO(X)
ZERO(mark(X)) -> ZERO(X)


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ZERO(ok(X)) -> ZERO(X)
  2. ZERO(mark(X)) -> ZERO(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳Size-Change Principle
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

PROD(ok(X1), ok(X2)) -> PROD(X1, X2)
PROD(X1, mark(X2)) -> PROD(X1, X2)
PROD(mark(X1), X2) -> PROD(X1, X2)


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. PROD(ok(X1), ok(X2)) -> PROD(X1, X2)
  2. PROD(X1, mark(X2)) -> PROD(X1, X2)
  3. PROD(mark(X1), X2) -> PROD(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳Size-Change Principle
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

FACT(ok(X)) -> FACT(X)
FACT(mark(X)) -> FACT(X)


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. FACT(ok(X)) -> FACT(X)
  2. FACT(mark(X)) -> FACT(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳Size-Change Principle
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

P(ok(X)) -> P(X)
P(mark(X)) -> P(X)


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. P(ok(X)) -> P(X)
  2. P(mark(X)) -> P(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳Size-Change Principle
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

S(ok(X)) -> S(X)
S(mark(X)) -> S(X)


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. S(ok(X)) -> S(X)
  2. S(mark(X)) -> S(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳Size-Change Principle
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(mark(X1), X2) -> ADD(X1, X2)


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
  2. ADD(X1, mark(X2)) -> ADD(X1, X2)
  3. ADD(mark(X1), X2) -> ADD(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳Size-Change Principle
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(p(X)) -> ACTIVE(X)
ACTIVE(prod(X1, X2)) -> ACTIVE(X2)
ACTIVE(prod(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(zero(X)) -> ACTIVE(X)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(fact(X)) -> ACTIVE(X)


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ACTIVE(add(X1, X2)) -> ACTIVE(X2)
  2. ACTIVE(add(X1, X2)) -> ACTIVE(X1)
  3. ACTIVE(p(X)) -> ACTIVE(X)
  4. ACTIVE(prod(X1, X2)) -> ACTIVE(X2)
  5. ACTIVE(prod(X1, X2)) -> ACTIVE(X1)
  6. ACTIVE(s(X)) -> ACTIVE(X)
  7. ACTIVE(zero(X)) -> ACTIVE(X)
  8. ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
  9. ACTIVE(fact(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s):
{9, 8, 7, 6, 5, 4, 3, 2, 1} , {9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{9, 8, 7, 6, 5, 4, 3, 2, 1} , {9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
if(x1, x2, x3) -> if(x1, x2, x3)
prod(x1, x2) -> prod(x1, x2)
fact(x1) -> fact(x1)
s(x1) -> s(x1)
zero(x1) -> zero(x1)
p(x1) -> p(x1)
add(x1, x2) -> add(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Size-Change Principle
       →DP Problem 10
         ↳Nar


Dependency Pairs:

PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(p(X)) -> PROPER(X)
PROPER(prod(X1, X2)) -> PROPER(X2)
PROPER(prod(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(zero(X)) -> PROPER(X)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(fact(X)) -> PROPER(X)


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. PROPER(add(X1, X2)) -> PROPER(X2)
  2. PROPER(add(X1, X2)) -> PROPER(X1)
  3. PROPER(p(X)) -> PROPER(X)
  4. PROPER(prod(X1, X2)) -> PROPER(X2)
  5. PROPER(prod(X1, X2)) -> PROPER(X1)
  6. PROPER(s(X)) -> PROPER(X)
  7. PROPER(zero(X)) -> PROPER(X)
  8. PROPER(if(X1, X2, X3)) -> PROPER(X3)
  9. PROPER(if(X1, X2, X3)) -> PROPER(X2)
  10. PROPER(if(X1, X2, X3)) -> PROPER(X1)
  11. PROPER(fact(X)) -> PROPER(X)
and get the following Size-Change Graph(s):
{11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
if(x1, x2, x3) -> if(x1, x2, x3)
prod(x1, x2) -> prod(x1, x2)
fact(x1) -> fact(x1)
s(x1) -> s(x1)
zero(x1) -> zero(x1)
p(x1) -> p(x1)
add(x1, x2) -> add(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Narrowing Transformation


Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(mark(X)) -> TOP(proper(X))
10 new Dependency Pairs are created:

TOP(mark(fact(X''))) -> TOP(fact(proper(X'')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(zero(X''))) -> TOP(zero(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(prod(X1', X2'))) -> TOP(prod(proper(X1'), proper(X2')))
TOP(mark(p(X''))) -> TOP(p(proper(X'')))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))
TOP(mark(true)) -> TOP(ok(true))
TOP(mark(false)) -> TOP(ok(false))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar
           →DP Problem 11
             ↳Narrowing Transformation


Dependency Pairs:

TOP(mark(false)) -> TOP(ok(false))
TOP(mark(true)) -> TOP(ok(true))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))
TOP(mark(p(X''))) -> TOP(p(proper(X'')))
TOP(mark(prod(X1', X2'))) -> TOP(prod(proper(X1'), proper(X2')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(zero(X''))) -> TOP(zero(proper(X'')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(fact(X''))) -> TOP(fact(proper(X'')))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(ok(X)) -> TOP(active(X))
19 new Dependency Pairs are created:

TOP(ok(fact(X''))) -> TOP(mark(if(zero(X''), s(0), prod(X'', fact(p(X''))))))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(ok(prod(0, X''))) -> TOP(mark(0))
TOP(ok(prod(s(X''), Y'))) -> TOP(mark(add(Y', prod(X'', Y'))))
TOP(ok(if(true, X'', Y'))) -> TOP(mark(X''))
TOP(ok(if(false, X'', Y'))) -> TOP(mark(Y'))
TOP(ok(zero(0))) -> TOP(mark(true))
TOP(ok(zero(s(X'')))) -> TOP(mark(false))
TOP(ok(p(s(X'')))) -> TOP(mark(X''))
TOP(ok(fact(X''))) -> TOP(fact(active(X'')))
TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(zero(X''))) -> TOP(zero(active(X'')))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(prod(X1', X2'))) -> TOP(prod(active(X1'), X2'))
TOP(ok(prod(X1', X2'))) -> TOP(prod(X1', active(X2')))
TOP(ok(p(X''))) -> TOP(p(active(X'')))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar
           →DP Problem 11
             ↳Nar
             ...
               →DP Problem 12
                 ↳Negative Polynomial Order


Dependency Pairs:

TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(p(X''))) -> TOP(p(active(X'')))
TOP(ok(prod(X1', X2'))) -> TOP(prod(X1', active(X2')))
TOP(ok(prod(X1', X2'))) -> TOP(prod(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(zero(X''))) -> TOP(zero(active(X'')))
TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(fact(X''))) -> TOP(fact(active(X'')))
TOP(ok(p(s(X'')))) -> TOP(mark(X''))
TOP(ok(if(false, X'', Y'))) -> TOP(mark(Y'))
TOP(ok(if(true, X'', Y'))) -> TOP(mark(X''))
TOP(ok(prod(s(X''), Y'))) -> TOP(mark(add(Y', prod(X'', Y'))))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(fact(X''))) -> TOP(mark(if(zero(X''), s(0), prod(X'', fact(p(X''))))))
TOP(mark(p(X''))) -> TOP(p(proper(X'')))
TOP(mark(prod(X1', X2'))) -> TOP(prod(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(zero(X''))) -> TOP(zero(proper(X'')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(fact(X''))) -> TOP(fact(proper(X'')))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following Dependency Pair can be strictly oriented using the given order.

TOP(ok(p(s(X'')))) -> TOP(mark(X''))


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( p(x1) ) = x1 + 1

POL( s(x1) ) = x1

POL( mark(x1) ) = x1

POL( if(x1, ..., x3) ) = x2 + x3

POL( proper(x1) ) = x1

POL( active(x1) ) = x1

POL( fact(x1) ) = 0

POL( add(x1, x2) ) = x2

POL( zero(x1) ) = 0

POL( prod(x1, x2) ) = 0

POL( 0 ) = 0

POL( true ) = 0

POL( false ) = 0


This results in one new DP problem. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar
           →DP Problem 11
             ↳Nar
             ...
               →DP Problem 13
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(p(X''))) -> TOP(p(active(X'')))
TOP(ok(prod(X1', X2'))) -> TOP(prod(X1', active(X2')))
TOP(ok(prod(X1', X2'))) -> TOP(prod(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(zero(X''))) -> TOP(zero(active(X'')))
TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(fact(X''))) -> TOP(fact(active(X'')))
TOP(ok(if(false, X'', Y'))) -> TOP(mark(Y'))
TOP(ok(if(true, X'', Y'))) -> TOP(mark(X''))
TOP(ok(prod(s(X''), Y'))) -> TOP(mark(add(Y', prod(X'', Y'))))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(fact(X''))) -> TOP(mark(if(zero(X''), s(0), prod(X'', fact(p(X''))))))
TOP(mark(p(X''))) -> TOP(p(proper(X'')))
TOP(mark(prod(X1', X2'))) -> TOP(prod(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(zero(X''))) -> TOP(zero(proper(X'')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(fact(X''))) -> TOP(fact(proper(X'')))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))


Rules:


active(fact(X)) -> mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(prod(0, X)) -> mark(0)
active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(zero(0)) -> mark(true)
active(zero(s(X))) -> mark(false)
active(p(s(X))) -> mark(X)
active(fact(X)) -> fact(active(X))
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(zero(X)) -> zero(active(X))
active(s(X)) -> s(active(X))
active(prod(X1, X2)) -> prod(active(X1), X2)
active(prod(X1, X2)) -> prod(X1, active(X2))
active(p(X)) -> p(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fact(mark(X)) -> mark(fact(X))
fact(ok(X)) -> ok(fact(X))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
zero(mark(X)) -> mark(zero(X))
zero(ok(X)) -> ok(zero(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
prod(mark(X1), X2) -> mark(prod(X1, X2))
prod(X1, mark(X2)) -> mark(prod(X1, X2))
prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fact(X)) -> fact(proper(X))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) -> zero(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
proper(p(X)) -> p(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))




The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes