Termination Proof using AProVE

Term Rewriting System R:
[x, y]
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Termination of R to be shown.

     ↳Overlay and local confluence Check

The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.

     ↳OC

       →TRS2

         ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

MIN(s(x), s(y)) -> MIN(x, y)
MAX(s(x), s(y)) -> MAX(x, y)
-'(s(x), s(y)) -> -'(x, y)
GCD(s(x), s(y)) -> GCD(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
GCD(s(x), s(y)) -> -'(s(max(x, y)), s(min(x, y)))
GCD(s(x), s(y)) -> MAX(x, y)
GCD(s(x), s(y)) -> MIN(x, y)

Furthermore, R contains four SCCs.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳Usable Rules (Innermost)

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

MIN(s(x), s(y)) -> MIN(x, y)

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

As we are in the innermost case, we can delete all 9 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

...

               →DP Problem 5

                 ↳Size-Change Principle

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

MIN(s(x), s(y)) -> MIN(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

MIN(s(x), s(y)) -> MIN(x, y)

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

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

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

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

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳Usable Rules (Innermost)

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

MAX(s(x), s(y)) -> MAX(x, y)

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

As we are in the innermost case, we can delete all 9 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

...

               →DP Problem 6

                 ↳Size-Change Principle

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

MAX(s(x), s(y)) -> MAX(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

MAX(s(x), s(y)) -> MAX(x, y)

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

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

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

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

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳Usable Rules (Innermost)

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

-'(s(x), s(y)) -> -'(x, y)

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

As we are in the innermost case, we can delete all 9 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

...

               →DP Problem 7

                 ↳Size-Change Principle

           →DP Problem 4

             ↳UsableRules

Dependency Pair:

-'(s(x), s(y)) -> -'(x, y)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

-'(s(x), s(y)) -> -'(x, y)

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

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

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

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

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳Usable Rules (Innermost)

Dependency Pair:

GCD(s(x), s(y)) -> GCD(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

As we are in the innermost case, we can delete all 1 non-usable-rules.

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 8

                 ↳Narrowing Transformation

Dependency Pair:

GCD(s(x), s(y)) -> GCD(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Rules:

max(s(x), s(y)) -> s(max(x, y))
max(x, 0) -> x
max(0, y) -> y
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
min(x, 0) -> 0
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x

Strategy:

innermost

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

GCD(s(x), s(y)) -> GCD(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

10 new Dependency Pairs are created:

GCD(s(x''), s(y'')) -> GCD(-(max(x'', y''), min(x'', y'')), s(min(x'', y'')))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(s(max(x'', y''))), s(min(s(x''), s(y'')))), s(min(s(x''), s(y''))))
GCD(s(x''), s(0)) -> GCD(-(s(x''), s(min(x'', 0))), s(min(x'', 0)))
GCD(s(0), s(y'')) -> GCD(-(s(y''), s(min(0, y''))), s(min(0, y'')))
GCD(s(0), s(y'')) -> GCD(-(s(max(0, y'')), s(0)), s(min(0, y'')))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(max(s(x''), s(y''))), s(s(min(x'', y'')))), s(min(s(x''), s(y''))))
GCD(s(x''), s(0)) -> GCD(-(s(max(x'', 0)), s(0)), s(min(x'', 0)))
GCD(s(0), s(y'')) -> GCD(-(s(max(0, y'')), s(min(0, y''))), s(0))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(max(s(x''), s(y''))), s(min(s(x''), s(y'')))), s(s(min(x'', y''))))
GCD(s(x''), s(0)) -> GCD(-(s(max(x'', 0)), s(min(x'', 0))), s(0))

The transformation is resulting in one new DP problem:

     ↳OC

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳UsableRules

           →DP Problem 2

             ↳UsableRules

           →DP Problem 3

             ↳UsableRules

           →DP Problem 4

             ↳UsableRules

...

               →DP Problem 9

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(max(s(x''), s(y''))), s(min(s(x''), s(y'')))), s(s(min(x'', y''))))
GCD(s(x''), s(0)) -> GCD(-(s(max(x'', 0)), s(min(x'', 0))), s(0))
GCD(s(0), s(y'')) -> GCD(-(s(max(0, y'')), s(min(0, y''))), s(0))
GCD(s(x''), s(0)) -> GCD(-(s(max(x'', 0)), s(0)), s(min(x'', 0)))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(max(s(x''), s(y''))), s(s(min(x'', y'')))), s(min(s(x''), s(y''))))
GCD(s(0), s(y'')) -> GCD(-(s(max(0, y'')), s(0)), s(min(0, y'')))
GCD(s(0), s(y'')) -> GCD(-(s(y''), s(min(0, y''))), s(min(0, y'')))
GCD(s(x''), s(0)) -> GCD(-(s(x''), s(min(x'', 0))), s(min(x'', 0)))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(s(max(x'', y''))), s(min(s(x''), s(y'')))), s(min(s(x''), s(y''))))
GCD(s(x''), s(y'')) -> GCD(-(max(x'', y''), min(x'', y'')), s(min(x'', y'')))

Rules:

max(s(x), s(y)) -> s(max(x, y))
max(x, 0) -> x
max(0, y) -> y
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
min(x, 0) -> 0
-(s(x), s(y)) -> -(x, y)
-(x, 0) -> x

Strategy:

innermost

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