Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z, X1, X2, X3]
active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVE(add(s(X), Y)) -> S(add(X, Y))
ACTIVE(add(s(X), Y)) -> ADD(X, Y)
ACTIVE(first(s(X), cons(Y, Z))) -> CONS(Y, first(X, Z))
ACTIVE(first(s(X), cons(Y, Z))) -> FIRST(X, Z)
ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(and(X1, X2)) -> AND(active(X1), X2)
ACTIVE(and(X1, X2)) -> ACTIVE(X1)
ACTIVE(if(X1, X2, X3)) -> IF(active(X1), X2, X3)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(active(X1), X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(first(X1, X2)) -> FIRST(active(X1), X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(first(X1, X2)) -> FIRST(X1, active(X2))
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
AND(mark(X1), X2) -> AND(X1, X2)
AND(ok(X1), ok(X2)) -> AND(X1, X2)
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
ADD(mark(X1), X2) -> ADD(X1, X2)
ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
FIRST(mark(X1), X2) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
PROPER(and(X1, X2)) -> AND(proper(X1), proper(X2))
PROPER(and(X1, X2)) -> PROPER(X1)
PROPER(and(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)
PROPER(add(X1, X2)) -> ADD(proper(X1), proper(X2))
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> FIRST(proper(X1), proper(X2))
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
S(ok(X)) -> S(X)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
FROM(ok(X)) -> FROM(X)
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.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(S(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

S(x₁) -> S(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 11

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pairs:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(ok(x₁)) = x₁

POL(ADD(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

ADD(x₁, x₂) -> ADD(x₁, x₂)
mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 12

             ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ok(x₁)) = 1 + x₁

POL(ADD(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

ADD(x₁, x₂) -> ADD(x₁, x₂)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 12

             ↳AFS

...

               →DP Problem 13

                 ↳Dependency Graph

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ok(x₁)) = 1 + x₁

POL(CONS(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

CONS(x₁, x₂) -> CONS(x₁, x₂)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 14

             ↳Dependency Graph

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pairs:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FIRST(x₁, x₂)) = x₁ + x₂

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

FIRST(x₁, x₂) -> FIRST(x₁, x₂)
ok(x₁) -> ok(x₁)
mark(x₁) -> mark(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 15

             ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pairs:

FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(mark(X1), X2) -> FIRST(X1, X2)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(mark(X1), X2) -> FIRST(X1, X2)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FIRST(x₁, x₂)) = x₁ + x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

FIRST(x₁, x₂) -> FIRST(x₁, x₂)
mark(x₁) -> mark(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 15

             ↳AFS

...

               →DP Problem 16

                 ↳Dependency Graph

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳Argument Filtering and Ordering

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

FROM(ok(X)) -> FROM(X)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

FROM(ok(X)) -> FROM(X)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FROM(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

FROM(x₁) -> FROM(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

           →DP Problem 17

             ↳Dependency Graph

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Argument Filtering and Ordering

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pairs:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

AND(mark(X1), X2) -> AND(X1, X2)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(ok(x₁)) = x₁

POL(AND(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

AND(x₁, x₂) -> AND(x₁, x₂)
mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

           →DP Problem 18

             ↳Argument Filtering and Ordering

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ok(x₁)) = 1 + x₁

POL(AND(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

AND(x₁, x₂) -> AND(x₁, x₂)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

           →DP Problem 18

             ↳AFS

...

               →DP Problem 19

                 ↳Dependency Graph

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳Argument Filtering and Ordering

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pairs:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

POL(IF(x₁, x₂, x₃)) = x₁ + x₂ + x₃

resulting in one new DP problem.
Used Argument Filtering System:

IF(x₁, x₂, x₃) -> IF(x₁, x₂, x₃)
ok(x₁) -> ok(x₁)
mark(x₁) -> mark(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

           →DP Problem 20

             ↳Argument Filtering and Ordering

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(IF(x₁, x₂, x₃)) = x₁ + x₂ + x₃

resulting in one new DP problem.
Used Argument Filtering System:

IF(x₁, x₂, x₃) -> IF(x₁, x₂, x₃)
mark(x₁) -> mark(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

           →DP Problem 20

             ↳AFS

...

               →DP Problem 21

                 ↳Dependency Graph

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳Argument Filtering and Ordering

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pairs:

ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(and(X1, X2)) -> ACTIVE(X1)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(and(X1, X2)) -> ACTIVE(X1)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(and(x₁, x₂)) = 1 + x₁ + x₂

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(first(x₁, x₂)) = x₁ + x₂

POL(add(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVE(x₁) -> ACTIVE(x₁)
and(x₁, x₂) -> and(x₁, x₂)
add(x₁, x₂) -> add(x₁, x₂)
first(x₁, x₂) -> first(x₁, x₂)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

           →DP Problem 22

             ↳Argument Filtering and Ordering

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pairs:

ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(add(X1, X2)) -> ACTIVE(X1)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(first(x₁, x₂)) = x₁ + x₂

POL(add(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVE(x₁) -> ACTIVE(x₁)
add(x₁, x₂) -> add(x₁, x₂)
first(x₁, x₂) -> first(x₁, x₂)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

           →DP Problem 22

             ↳AFS

...

               →DP Problem 23

                 ↳Argument Filtering and Ordering

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pairs:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(first(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVE(x₁) -> ACTIVE(x₁)
first(x₁, x₂) -> first(x₁, x₂)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

           →DP Problem 22

             ↳AFS

...

               →DP Problem 24

                 ↳Argument Filtering and Ordering

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

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

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVE(x₁) -> ACTIVE(x₁)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

           →DP Problem 22

             ↳AFS

...

               →DP Problem 25

                 ↳Dependency Graph

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Argument Filtering and Ordering

       →DP Problem 10

         ↳AFS

Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(and(x₁, x₂)) = x₁ + x₂

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(first(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(cons(x₁, x₂)) = 1 + x₁ + x₂

POL(s(x₁)) = x₁

POL(add(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
first(x₁, x₂) -> first(x₁, x₂)
from(x₁) -> from(x₁)
s(x₁) -> s(x₁)
and(x₁, x₂) -> and(x₁, x₂)
add(x₁, x₂) -> add(x₁, x₂)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

           →DP Problem 26

             ↳Argument Filtering and Ordering

       →DP Problem 10

         ↳AFS

Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(and(x₁, x₂)) = x₁ + x₂

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(first(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(s(x₁)) = x₁

POL(add(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
add(x₁, x₂) -> add(x₁, x₂)
first(x₁, x₂) -> first(x₁, x₂)
from(x₁) -> from(x₁)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)
and(x₁, x₂) -> and(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

           →DP Problem 26

             ↳AFS

...

               →DP Problem 27

                 ↳Argument Filtering and Ordering

       →DP Problem 10

         ↳AFS

Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(s(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(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(and(x₁, x₂)) = x₁ + x₂

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(first(x₁, x₂)) = 1 + x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(s(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
first(x₁, x₂) -> first(x₁, x₂)
from(x₁) -> from(x₁)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)
and(x₁, x₂) -> and(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

           →DP Problem 26

             ↳AFS

...

               →DP Problem 28

                 ↳Argument Filtering and Ordering

       →DP Problem 10

         ↳AFS

Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(s(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(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(from(X)) -> PROPER(X)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 1 + x₁

POL(and(x₁, x₂)) = x₁ + x₂

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(PROPER(x₁)) = x₁

POL(s(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
from(x₁) -> from(x₁)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)
and(x₁, x₂) -> and(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

           →DP Problem 26

             ↳AFS

...

               →DP Problem 29

                 ↳Argument Filtering and Ordering

       →DP Problem 10

         ↳AFS

Dependency Pairs:

PROPER(s(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(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(and(x₁, x₂)) = x₁ + x₂

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

POL(PROPER(x₁)) = x₁

POL(s(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)
and(x₁, x₂) -> and(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

           →DP Problem 26

             ↳AFS

...

               →DP Problem 30

                 ↳Argument Filtering and Ordering

       →DP Problem 10

         ↳AFS

Dependency Pairs:

PROPER(s(X)) -> PROPER(X)
PROPER(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(and(X1, X2)) -> PROPER(X2)
PROPER(and(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(and(x₁, x₂)) = 1 + x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(s(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
and(x₁, x₂) -> and(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

           →DP Problem 26

             ↳AFS

...

               →DP Problem 31

                 ↳Argument Filtering and Ordering

       →DP Problem 10

         ↳AFS

Dependency Pair:

PROPER(s(X)) -> PROPER(X)

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(s(X)) -> PROPER(X)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(PROPER(x₁)) = x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

           →DP Problem 26

             ↳AFS

...

               →DP Problem 32

                 ↳Dependency Graph

       →DP Problem 10

         ↳AFS

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Argument Filtering and Ordering

Dependency Pairs:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

The following usable rules for innermost w.r.t. to the AFS can be oriented:

proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
s(ok(X)) -> ok(s(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 1 + x₁

POL(proper(x₁)) = x₁

POL(and(x₁, x₂)) = 1 + x₁ + x₂

POL(false) = 1

POL(true) = 1

POL(mark(x₁)) = 1 + x₁

POL(TOP(x₁)) = x₁

POL(ok(x₁)) = x₁

POL(add(x₁, x₂)) = 1 + x₁ + x₂

POL(active(x₁)) = x₁

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(0) = 1

POL(first(x₁, x₂)) = x₁ + x₂

POL(nil) = 0

POL(s) = 1

resulting in one new DP problem.
Used Argument Filtering System:

TOP(x₁) -> TOP(x₁)
mark(x₁) -> mark(x₁)
proper(x₁) -> proper(x₁)
ok(x₁) -> ok(x₁)
active(x₁) -> active(x₁)
and(x₁, x₂) -> and(x₁, x₂)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)
add(x₁, x₂) -> add(x₁, x₂)
s(x₁) -> s
first(x₁, x₂) -> first(x₁, x₂)
cons(x₁, x₂) -> x₁
from(x₁) -> from(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

           →DP Problem 33

             ↳Argument Filtering and Ordering

Dependency Pair:

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

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

The following usable rules for innermost w.r.t. to the AFS can be oriented:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(and(x₁, x₂)) = x₁ + x₂

POL(false) = 0

POL(true) = 0

POL(mark(x₁)) = x₁

POL(TOP(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

POL(add(x₁, x₂)) = x₁ + x₂

POL(active(x₁)) = x₁

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(0) = 0

POL(first(x₁, x₂)) = x₁ + x₂

POL(nil) = 0

POL(s(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

TOP(x₁) -> TOP(x₁)
ok(x₁) -> ok(x₁)
active(x₁) -> active(x₁)
and(x₁, x₂) -> and(x₁, x₂)
mark(x₁) -> mark(x₁)
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)
add(x₁, x₂) -> add(x₁, x₂)
s(x₁) -> s(x₁)
first(x₁, x₂) -> first(x₁, x₂)
cons(x₁, x₂) -> x₁
from(x₁) -> from(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳AFS

           →DP Problem 33

             ↳AFS

...

               →DP Problem 34

                 ↳Dependency Graph

Dependency Pair:

Rules:

active(and(true, X)) -> mark(X)
active(and(false, Y)) -> mark(false)
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(and(X1, X2)) -> and(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
active(add(X1, X2)) -> add(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
and(mark(X1), X2) -> mark(and(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
add(mark(X1), X2) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:10 minutes

POL(mark(x₁))	= 1 + x₁
POL(ok(x₁))	= x₁
POL(ADD(x₁, x₂))	= x₁ + x₂

POL(ok(x₁))	= 1 + x₁
POL(CONS(x₁, x₂))	= x₁ + x₂

POL(FIRST(x₁, x₂))	= x₁ + x₂
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(ok(x₁))	= 1 + x₁
POL(AND(x₁, x₂))	= x₁ + x₂

POL(mark(x₁))	= 1 + x₁
POL(IF(x₁, x₂, x₃))	= x₁ + x₂ + x₃

POL(ACTIVE(x₁))	= x₁
POL(and(x₁, x₂))	= 1 + x₁ + x₂
POL(if(x₁, x₂, x₃))	= x₁ + x₂ + x₃
POL(first(x₁, x₂))	= x₁ + x₂
POL(add(x₁, x₂))	= x₁ + x₂

POL(from(x₁))	= x₁
POL(and(x₁, x₂))	= x₁ + x₂
POL(if(x₁, x₂, x₃))	= x₁ + x₂ + x₃
POL(first(x₁, x₂))	= x₁ + x₂
POL(PROPER(x₁))	= x₁
POL(cons(x₁, x₂))	= 1 + x₁ + x₂
POL(s(x₁))	= x₁
POL(add(x₁, x₂))	= x₁ + x₂