Termination Proof using AProVE

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.

     ↳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.

     ↳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:

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

D_P: 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.

     ↳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:

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

D_P: 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.

     ↳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:

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

D_P: 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.

     ↳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:

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

D_P: 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.

     ↳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:

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

D_P: 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.

     ↳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:

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

D_P: 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.

     ↳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:

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)

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

D_P: 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.

     ↳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:

     ↳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:

     ↳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.

     ↳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.

     ↳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.

     ↳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:

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.

     ↳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

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