Termination Proof using AProVE

Term Rewriting System R:
[x, y]
even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(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:

EVEN(s(s(x))) -> EVEN(x)
HALF(s(s(x))) -> HALF(x)
PLUS(s(x), y) -> PLUS(x, y)
TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)
TIMES(s(x), y) -> EVEN(s(x))
IF_TIMES(true, s(x), y) -> PLUS(times(half(s(x)), y), times(half(s(x)), y))
IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)
IF_TIMES(true, s(x), y) -> HALF(s(x))
IF_TIMES(false, s(x), y) -> PLUS(y, times(x, y))
IF_TIMES(false, s(x), y) -> TIMES(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:

EVEN(s(s(x))) -> EVEN(x)

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Strategy:

innermost

As we are in the innermost case, we can delete all 11 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:

EVEN(s(s(x))) -> EVEN(x)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

EVEN(s(s(x))) -> EVEN(x)

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

{1}	,	{1}
1	>	1

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

{1}	,	{1}
1	>	1

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

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:

HALF(s(s(x))) -> HALF(x)

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Strategy:

innermost

As we are in the innermost case, we can delete all 11 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:

HALF(s(s(x))) -> HALF(x)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

HALF(s(s(x))) -> HALF(x)

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

{1}	,	{1}
1	>	1

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

{1}	,	{1}
1	>	1

D_P: empty set
Oriented Rules: none

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

trivial

with Argument Filtering System:

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:

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

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Strategy:

innermost

As we are in the innermost case, we can delete all 11 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:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

PLUS(s(x), y) -> PLUS(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 Pairs:

IF_TIMES(false, s(x), y) -> TIMES(x, y)
IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)
TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)

Rules:

even(0) -> true
even(s(0)) -> false
even(s(s(x))) -> even(x)
half(0) -> 0
half(s(s(x))) -> s(half(x))
plus(0, y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0, y) -> 0
times(s(x), y) -> if_times(even(s(x)), s(x), y)
if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
if_times(false, s(x), y) -> plus(y, times(x, y))

Strategy:

innermost

As we are in the innermost case, we can delete all 6 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

                 ↳Negative Polynomial Order

Dependency Pairs:

IF_TIMES(false, s(x), y) -> TIMES(x, y)
IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)
TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)

Rules:

half(0) -> 0
half(s(s(x))) -> s(half(x))
even(s(0)) -> false
even(s(s(x))) -> even(x)
even(0) -> true

Strategy:

innermost

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

IF_TIMES(false, s(x), y) -> TIMES(x, y)

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

half(0) -> 0
half(s(s(x))) -> s(half(x))
even(s(0)) -> false
even(s(s(x))) -> even(x)
even(0) -> true

Used ordering:
Polynomial Order with Interpretation:

POL( IF_TIMES(x₁, ..., x₃) ) = x₂

POL( s(x₁) ) = x₁ + 1

POL( TIMES(x₁, x₂) ) = x₁

POL( half(x₁) ) = x₁

POL( 0 ) = 0

POL( even(x₁) ) = 0

POL( false ) = 0

POL( true ) = 0

This results 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

                 ↳Narrowing Transformation

Dependency Pairs:

IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)
TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)

Rules:

half(0) -> 0
half(s(s(x))) -> s(half(x))
even(s(0)) -> false
even(s(s(x))) -> even(x)
even(0) -> true

Strategy:

innermost

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

IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)

one new Dependency Pair is created:

IF_TIMES(true, s(s(x'')), y) -> TIMES(s(half(x'')), y)

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 10

                 ↳Narrowing Transformation

Dependency Pairs:

IF_TIMES(true, s(s(x'')), y) -> TIMES(s(half(x'')), y)
TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)

Rules:

half(0) -> 0
half(s(s(x))) -> s(half(x))
even(s(0)) -> false
even(s(s(x))) -> even(x)
even(0) -> true

Strategy:

innermost

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

TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)

two new Dependency Pairs are created:

TIMES(s(0), y) -> IF_TIMES(false, s(0), y)
TIMES(s(s(x'')), y) -> IF_TIMES(even(x''), s(s(x'')), y)

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 11

                 ↳Negative Polynomial Order

Dependency Pairs:

TIMES(s(s(x'')), y) -> IF_TIMES(even(x''), s(s(x'')), y)
IF_TIMES(true, s(s(x'')), y) -> TIMES(s(half(x'')), y)

Rules:

half(0) -> 0
half(s(s(x))) -> s(half(x))
even(s(0)) -> false
even(s(s(x))) -> even(x)
even(0) -> true

Strategy:

innermost

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

TIMES(s(s(x'')), y) -> IF_TIMES(even(x''), s(s(x'')), y)

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

even(s(0)) -> false
even(s(s(x))) -> even(x)
even(0) -> true
half(0) -> 0
half(s(s(x))) -> s(half(x))

Used ordering:
Polynomial Order with Interpretation:

POL( TIMES(x₁, x₂) ) = x₁ + 1

POL( s(x₁) ) = x₁ + 1

POL( IF_TIMES(x₁, ..., x₃) ) = x₂

POL( half(x₁) ) = x₁

POL( even(x₁) ) = 0

POL( false ) = 0

POL( true ) = 0

POL( 0 ) = 0

This results 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 12

                 ↳Dependency Graph

Dependency Pair:

IF_TIMES(true, s(s(x'')), y) -> TIMES(s(half(x'')), y)

Rules:

half(0) -> 0
half(s(s(x))) -> s(half(x))
even(s(0)) -> false
even(s(s(x))) -> even(x)
even(0) -> true

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:07 minutes