Term Rewriting System R:
[Y, X, X1, X2, X3]
active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
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(minus(s(X), s(Y))) -> MINUS(X, Y)
ACTIVE(geq(s(X), s(Y))) -> GEQ(X, Y)
ACTIVE(div(s(X), s(Y))) -> IF(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
ACTIVE(div(s(X), s(Y))) -> GEQ(X, Y)
ACTIVE(div(s(X), s(Y))) -> S(div(minus(X, Y), s(Y)))
ACTIVE(div(s(X), s(Y))) -> DIV(minus(X, Y), s(Y))
ACTIVE(div(s(X), s(Y))) -> MINUS(X, Y)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(div(X1, X2)) -> DIV(active(X1), X2)
ACTIVE(div(X1, X2)) -> ACTIVE(X1)
ACTIVE(if(X1, X2, X3)) -> IF(active(X1), X2, X3)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
DIV(mark(X1), X2) -> DIV(X1, X2)
DIV(ok(X1), ok(X2)) -> DIV(X1, X2)
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
PROPER(minus(X1, X2)) -> MINUS(proper(X1), proper(X2))
PROPER(minus(X1, X2)) -> PROPER(X1)
PROPER(minus(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(geq(X1, X2)) -> GEQ(proper(X1), proper(X2))
PROPER(geq(X1, X2)) -> PROPER(X1)
PROPER(geq(X1, X2)) -> PROPER(X2)
PROPER(div(X1, X2)) -> DIV(proper(X1), proper(X2))
PROPER(div(X1, X2)) -> PROPER(X1)
PROPER(div(X1, X2)) -> PROPER(X2)
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)
MINUS(ok(X1), ok(X2)) -> MINUS(X1, X2)
GEQ(ok(X1), ok(X2)) -> GEQ(X1, 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 eight 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
Nar


Dependency Pair:

MINUS(ok(X1), ok(X2)) -> MINUS(X1, X2)


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

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


Dependency Pair:

GEQ(ok(X1), ok(X2)) -> GEQ(X1, X2)


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

which lead(s) to this/these maximal multigraph(s):
{1} , {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:
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
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(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
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
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
Nar


Dependency Pairs:

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


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
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
Size-Change Principle
       →DP Problem 6
SCP
       →DP Problem 7
SCP
       →DP Problem 8
Nar


Dependency Pairs:

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


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1
2=2
{2, 1} , {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
Size-Change Principle
       →DP Problem 7
SCP
       →DP Problem 8
Nar


Dependency Pairs:

ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(div(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
  2. ACTIVE(div(X1, X2)) -> ACTIVE(X1)
  3. ACTIVE(s(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {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)
s(x1) -> s(x1)
div(x1, x2) -> div(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
Size-Change Principle
       →DP Problem 8
Nar


Dependency Pairs:

PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(div(X1, X2)) -> PROPER(X2)
PROPER(div(X1, X2)) -> PROPER(X1)
PROPER(geq(X1, X2)) -> PROPER(X2)
PROPER(geq(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(minus(X1, X2)) -> PROPER(X2)
PROPER(minus(X1, X2)) -> PROPER(X1)


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

which lead(s) to this/these maximal multigraph(s):
{10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {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)
geq(x1, x2) -> geq(x1, x2)
minus(x1, x2) -> minus(x1, x2)
s(x1) -> s(x1)
div(x1, x2) -> div(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
Narrowing Transformation


Dependency Pairs:

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


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
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))
eight new Dependency Pairs are created:

TOP(mark(minus(X1', X2'))) -> TOP(minus(proper(X1'), proper(X2')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(geq(X1', X2'))) -> TOP(geq(proper(X1'), proper(X2')))
TOP(mark(true)) -> TOP(ok(true))
TOP(mark(false)) -> TOP(ok(false))
TOP(mark(div(X1', X2'))) -> TOP(div(proper(X1'), proper(X2')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))

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
Nar
           →DP Problem 9
Narrowing Transformation


Dependency Pairs:

TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(div(X1', X2'))) -> TOP(div(proper(X1'), proper(X2')))
TOP(mark(false)) -> TOP(ok(false))
TOP(mark(true)) -> TOP(ok(true))
TOP(mark(geq(X1', X2'))) -> TOP(geq(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(minus(X1', X2'))) -> TOP(minus(proper(X1'), proper(X2')))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
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))
12 new Dependency Pairs are created:

TOP(ok(minus(0, Y'))) -> TOP(mark(0))
TOP(ok(minus(s(X''), s(Y')))) -> TOP(mark(minus(X'', Y')))
TOP(ok(geq(X'', 0))) -> TOP(mark(true))
TOP(ok(geq(0, s(Y')))) -> TOP(mark(false))
TOP(ok(geq(s(X''), s(Y')))) -> TOP(mark(geq(X'', Y')))
TOP(ok(div(0, s(Y')))) -> TOP(mark(0))
TOP(ok(div(s(X''), s(Y')))) -> TOP(mark(if(geq(X'', Y'), s(div(minus(X'', Y'), s(Y'))), 0)))
TOP(ok(if(true, X'', Y'))) -> TOP(mark(X''))
TOP(ok(if(false, X'', Y'))) -> TOP(mark(Y'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(div(X1', X2'))) -> TOP(div(active(X1'), X2'))
TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))

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
Nar
           →DP Problem 9
Nar
             ...
               →DP Problem 10
Negative Polynomial Order


Dependency Pairs:

TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(div(X1', X2'))) -> TOP(div(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(if(false, X'', Y'))) -> TOP(mark(Y'))
TOP(ok(if(true, X'', Y'))) -> TOP(mark(X''))
TOP(ok(div(s(X''), s(Y')))) -> TOP(mark(if(geq(X'', Y'), s(div(minus(X'', Y'), s(Y'))), 0)))
TOP(ok(geq(s(X''), s(Y')))) -> TOP(mark(geq(X'', Y')))
TOP(ok(minus(s(X''), s(Y')))) -> TOP(mark(minus(X'', Y')))
TOP(mark(div(X1', X2'))) -> TOP(div(proper(X1'), proper(X2')))
TOP(mark(geq(X1', X2'))) -> TOP(geq(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(minus(X1', X2'))) -> TOP(minus(proper(X1'), proper(X2')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

TOP(ok(if(false, X'', Y'))) -> TOP(mark(Y'))
TOP(ok(if(true, X'', Y'))) -> TOP(mark(X''))


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

if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( if(x1, ..., x3) ) = x2 + x3 + 1

POL( mark(x1) ) = x1

POL( proper(x1) ) = x1

POL( s(x1) ) = 0

POL( div(x1, x2) ) = 1

POL( 0 ) = 0

POL( geq(x1, x2) ) = 0

POL( minus(x1, x2) ) = 0

POL( active(x1) ) = x1

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
Nar
           →DP Problem 9
Nar
             ...
               →DP Problem 11
Negative Polynomial Order


Dependency Pairs:

TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(div(X1', X2'))) -> TOP(div(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(div(s(X''), s(Y')))) -> TOP(mark(if(geq(X'', Y'), s(div(minus(X'', Y'), s(Y'))), 0)))
TOP(ok(geq(s(X''), s(Y')))) -> TOP(mark(geq(X'', Y')))
TOP(ok(minus(s(X''), s(Y')))) -> TOP(mark(minus(X'', Y')))
TOP(mark(div(X1', X2'))) -> TOP(div(proper(X1'), proper(X2')))
TOP(mark(geq(X1', X2'))) -> TOP(geq(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(minus(X1', X2'))) -> TOP(minus(proper(X1'), proper(X2')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
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(div(s(X''), s(Y')))) -> TOP(mark(if(geq(X'', Y'), s(div(minus(X'', Y'), s(Y'))), 0)))


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

if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( div(x1, x2) ) = 1

POL( mark(x1) ) = x1

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

POL( s(x1) ) = 0

POL( 0 ) = 0

POL( proper(x1) ) = x1

POL( minus(x1, x2) ) = 0

POL( geq(x1, x2) ) = 0

POL( active(x1) ) = x1

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
Nar
           →DP Problem 9
Nar
             ...
               →DP Problem 12
Negative Polynomial Order


Dependency Pairs:

TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(div(X1', X2'))) -> TOP(div(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(geq(s(X''), s(Y')))) -> TOP(mark(geq(X'', Y')))
TOP(ok(minus(s(X''), s(Y')))) -> TOP(mark(minus(X'', Y')))
TOP(mark(div(X1', X2'))) -> TOP(div(proper(X1'), proper(X2')))
TOP(mark(geq(X1', X2'))) -> TOP(geq(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(minus(X1', X2'))) -> TOP(minus(proper(X1'), proper(X2')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
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(geq(s(X''), s(Y')))) -> TOP(mark(geq(X'', Y')))


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

if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( geq(x1, x2) ) = x2

POL( s(x1) ) = x1 + 1

POL( mark(x1) ) = x1

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

POL( proper(x1) ) = x1

POL( minus(x1, x2) ) = 0

POL( active(x1) ) = x1

POL( div(x1, x2) ) = x1

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
Nar
           →DP Problem 9
Nar
             ...
               →DP Problem 13
Negative Polynomial Order


Dependency Pairs:

TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(div(X1', X2'))) -> TOP(div(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(minus(s(X''), s(Y')))) -> TOP(mark(minus(X'', Y')))
TOP(mark(div(X1', X2'))) -> TOP(div(proper(X1'), proper(X2')))
TOP(mark(geq(X1', X2'))) -> TOP(geq(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(minus(X1', X2'))) -> TOP(minus(proper(X1'), proper(X2')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
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(minus(s(X''), s(Y')))) -> TOP(mark(minus(X'', Y')))


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

if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( minus(x1, x2) ) = x1

POL( s(x1) ) = x1 + 1

POL( mark(x1) ) = x1

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

POL( proper(x1) ) = x1

POL( active(x1) ) = x1

POL( div(x1, x2) ) = x1

POL( geq(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
Nar
           →DP Problem 9
Nar
             ...
               →DP Problem 14
Negative Polynomial Order


Dependency Pairs:

TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(div(X1', X2'))) -> TOP(div(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(mark(div(X1', X2'))) -> TOP(div(proper(X1'), proper(X2')))
TOP(mark(geq(X1', X2'))) -> TOP(geq(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(minus(X1', X2'))) -> TOP(minus(proper(X1'), proper(X2')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

TOP(mark(geq(X1', X2'))) -> TOP(geq(proper(X1'), proper(X2')))
TOP(mark(minus(X1', X2'))) -> TOP(minus(proper(X1'), proper(X2')))


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

if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( mark(x1) ) = 1

POL( geq(x1, x2) ) = 0

POL( if(x1, ..., x3) ) = 1

POL( minus(x1, x2) ) = 0

POL( s(x1) ) = 1

POL( ok(x1) ) = x1

POL( div(x1, x2) ) = 1

POL( active(x1) ) = 1

POL( proper(x1) ) = 1

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
Nar
           →DP Problem 9
Nar
             ...
               →DP Problem 15
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(if(X1', X2', X3'))) -> TOP(if(active(X1'), X2', X3'))
TOP(ok(div(X1', X2'))) -> TOP(div(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(mark(div(X1', X2'))) -> TOP(div(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(if(X1', X2', X3'))) -> TOP(if(proper(X1'), proper(X2'), proper(X3')))


Rules:


active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
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