Termination Proof using AProVE

Term Rewriting System R:
[X, X1, X2, X3]
active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
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(a, b, X)) -> F(X, X, X)
ACTIVE(f(X1, X2, X3)) -> F(active(X1), X2, X3)
ACTIVE(f(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(f(X1, X2, X3)) -> F(X1, X2, active(X3))
ACTIVE(f(X1, X2, X3)) -> ACTIVE(X3)
F(mark(X1), X2, X3) -> F(X1, X2, X3)
F(X1, X2, mark(X3)) -> F(X1, X2, X3)
F(ok(X1), ok(X2), ok(X3)) -> F(X1, X2, X3)
PROPER(f(X1, X2, X3)) -> F(proper(X1), proper(X2), proper(X3))
PROPER(f(X1, X2, X3)) -> PROPER(X1)
PROPER(f(X1, X2, X3)) -> PROPER(X2)
PROPER(f(X1, X2, X3)) -> PROPER(X3)
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 four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Usable Rules (Innermost)

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

Dependency Pairs:

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

Rules:

active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
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 14 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

           →DP Problem 5

             ↳Size-Change Principle

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

Dependency Pairs:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

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

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

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

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

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

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

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

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

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

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳Usable Rules (Innermost)

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

Dependency Pairs:

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

Rules:

active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
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 14 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

           →DP Problem 6

             ↳Size-Change Principle

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

Dependency Pairs:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

ACTIVE(f(X1, X2, X3)) -> ACTIVE(X3)
ACTIVE(f(X1, X2, X3)) -> 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:

f(x₁, x₂, x₃) -> f(x₁, x₂, x₃)

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

Dependency Pairs:

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

Rules:

active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
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 14 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

           →DP Problem 7

             ↳Size-Change Principle

       →DP Problem 4

         ↳UsableRules

Dependency Pairs:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

PROPER(f(X1, X2, X3)) -> PROPER(X3)
PROPER(f(X1, X2, X3)) -> PROPER(X2)
PROPER(f(X1, X2, X3)) -> 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:

f(x₁, x₂, x₃) -> f(x₁, x₂, x₃)

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)

Dependency Pairs:

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

Rules:

active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
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 8

             ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(c) -> mark(a)
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(c) -> ok(c)
proper(a) -> ok(a)
proper(b) -> ok(b)

Strategy:

innermost

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

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

five new Dependency Pairs are created:

TOP(ok(f(a, b, X''))) -> TOP(mark(f(X'', X'', X'')))
TOP(ok(c)) -> TOP(mark(b))
TOP(ok(f(X1', X2', X3'))) -> TOP(f(X1', X2', active(X3')))
TOP(ok(f(X1', X2', X3'))) -> TOP(f(active(X1'), X2', X3'))
TOP(ok(c)) -> TOP(mark(a))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

           →DP Problem 8

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

TOP(ok(c)) -> TOP(mark(a))
TOP(ok(f(X1', X2', X3'))) -> TOP(f(active(X1'), X2', X3'))
TOP(ok(f(X1', X2', X3'))) -> TOP(f(X1', X2', active(X3')))
TOP(ok(c)) -> TOP(mark(b))
TOP(ok(f(a, b, X''))) -> TOP(mark(f(X'', X'', X'')))
TOP(mark(X)) -> TOP(proper(X))

Rules:

f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(c) -> mark(a)
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(c) -> ok(c)
proper(a) -> ok(a)
proper(b) -> ok(b)

Strategy:

innermost

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

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

four new Dependency Pairs are created:

TOP(mark(f(X1', X2', X3'))) -> TOP(f(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(c)) -> TOP(ok(c))
TOP(mark(a)) -> TOP(ok(a))
TOP(mark(b)) -> TOP(ok(b))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳UsableRules

       →DP Problem 4

         ↳UsableRules

           →DP Problem 8

             ↳Nar

...

               →DP Problem 10

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

TOP(ok(f(X1', X2', X3'))) -> TOP(f(X1', X2', active(X3')))
TOP(mark(f(X1', X2', X3'))) -> TOP(f(proper(X1'), proper(X2'), proper(X3')))
TOP(ok(f(a, b, X''))) -> TOP(mark(f(X'', X'', X'')))
TOP(ok(f(X1', X2', X3'))) -> TOP(f(active(X1'), X2', X3'))

Rules:

f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(c) -> mark(a)
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(c) -> ok(c)
proper(a) -> ok(a)
proper(b) -> ok(b)

Strategy:

innermost

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