Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2]
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(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(f(g(X), Y)) -> F(X, f(g(X), Y))
ACTIVE(f(X1, X2)) -> F(active(X1), X2)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)
ACTIVE(g(X)) -> G(active(X))
ACTIVE(g(X)) -> ACTIVE(X)
F(mark(X1), X2) -> F(X1, X2)
F(ok(X1), ok(X2)) -> F(X1, X2)
G(mark(X)) -> G(X)
G(ok(X)) -> G(X)
PROPER(f(X1, X2)) -> F(proper(X1), proper(X2))
PROPER(f(X1, X2)) -> PROPER(X1)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(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 five SCCs.

     ↳DPs

       →DP Problem 1

         ↳Usable Rules (Innermost)

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

Dependency Pairs:

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

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

As we are in the innermost case, we can delete all 11 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

           →DP Problem 6

             ↳Size-Change Principle

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

Dependency Pairs:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

and get the following Size-Change Graph(s):

{1, 2}	,	{1, 2}
1	>	1
2	>	2

{1, 2}	,	{1, 2}
1	>	1
2	=	2

which lead(s) to this/these maximal multigraph(s):

{1, 2}	,	{1, 2}
1	>	1
2	=	2

{1, 2}	,	{1, 2}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳Usable Rules (Innermost)

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

Dependency Pairs:

G(ok(X)) -> G(X)
G(mark(X)) -> G(X)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

As we are in the innermost case, we can delete all 11 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

           →DP Problem 7

             ↳Size-Change Principle

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

Dependency Pairs:

G(ok(X)) -> G(X)
G(mark(X)) -> G(X)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

G(ok(X)) -> G(X)
G(mark(X)) -> G(X)

and get the following Size-Change Graph(s):

{1, 2}	,	{1, 2}
1	>	1

which lead(s) to this/these maximal multigraph(s):

{1, 2}	,	{1, 2}
1	>	1

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Usable Rules (Innermost)

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

Dependency Pairs:

ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

As we are in the innermost case, we can delete all 11 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

           →DP Problem 8

             ↳Size-Change Principle

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

Dependency Pairs:

ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(f(X1, X2)) -> ACTIVE(X1)

and get the following Size-Change Graph(s):

{1, 2}	,	{1, 2}
1	>	1

which lead(s) to this/these maximal multigraph(s):

{1, 2}	,	{1, 2}
1	>	1

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

g(x₁) -> g(x₁)
f(x₁, x₂) -> f(x₁, x₂)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳Usable Rules (Innermost)

       →DP Problem 5

         ↳UsableRules

Dependency Pairs:

PROPER(g(X)) -> PROPER(X)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

As we are in the innermost case, we can delete all 11 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

           →DP Problem 9

             ↳Size-Change Principle

       →DP Problem 5

         ↳UsableRules

Dependency Pairs:

PROPER(g(X)) -> PROPER(X)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

PROPER(g(X)) -> PROPER(X)
PROPER(f(X1, X2)) -> PROPER(X2)
PROPER(f(X1, X2)) -> PROPER(X1)

and get the following Size-Change Graph(s):

{1, 2, 3}	,	{1, 2, 3}
1	>	1

which lead(s) to this/these maximal multigraph(s):

{1, 2, 3}	,	{1, 2, 3}
1	>	1

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

g(x₁) -> g(x₁)
f(x₁, x₂) -> f(x₁, x₂)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳Usable Rules (Innermost)

Dependency Pairs:

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

Rules:

active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

As we are in the innermost case, we can delete all 2 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

           →DP Problem 10

             ↳Negative Polynomial Order

Dependency Pairs:

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

Rules:

f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
active(g(X)) -> g(active(X))
active(f(X1, X2)) -> f(active(X1), X2)
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))

Strategy:

innermost

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

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

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

f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
active(g(X)) -> g(active(X))
active(f(X1, X2)) -> f(active(X1), X2)
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))

Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x₁) ) = x₁

POL( ok(x₁) ) = 1

POL( active(x₁) ) = 0

POL( mark(x₁) ) = 0

POL( proper(x₁) ) = 0

POL( f(x₁, x₂) ) = x₁

POL( g(x₁) ) = x₁

This results in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

           →DP Problem 10

             ↳Neg POLO

...

               →DP Problem 11

                 ↳Usable Rules (Innermost)

Dependency Pair:

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

Rules:

f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
active(g(X)) -> g(active(X))
active(f(X1, X2)) -> f(active(X1), X2)
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))

Strategy:

innermost

As we are in the innermost case, we can delete all 3 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

       →DP Problem 5

         ↳UsableRules

           →DP Problem 10

             ↳Neg POLO

...

               →DP Problem 12

                 ↳Modular Removal of Rules

Dependency Pair:

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

Rules:

f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))

Strategy:

innermost

We have the following set of usable rules:

f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))

To remove rules and DPs from this DP problem we used the following monotonic and C_E-compatible order: Polynomial ordering.
Polynomial interpretation:

POL(proper(x₁)) = x₁

POL(g(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

POL(TOP(x₁)) = x₁

POL(ok(x₁)) = x₁

POL(f(x₁, x₂)) = x₁ + x₂

We have the following set D of usable symbols: {proper, g, mark, TOP, ok, f}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

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

No Rules can be deleted.

After the removal, there are no SCCs in the dependency graph which results in no DP problems which have to be solved.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes

POL(proper(x₁))	= x₁
POL(g(x₁))	= x₁
POL(mark(x₁))	= 1 + x₁
POL(TOP(x₁))	= x₁
POL(ok(x₁))	= x₁
POL(f(x₁, x₂))	= x₁ + x₂