Termination Proof using AProVE

Term Rewriting System R:
[x, y]
p(0) -> s(s(0))
p(s(x)) -> x
p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))

Innermost Termination of R to be shown.

     ↳Removing Redundant Rules for Innermost Termination

Removing the following rules from R which left hand sides contain non normal subterms

p(p(s(x))) -> p(x)
le(p(s(x)), x) -> le(x, x)

     ↳RRRI

       →TRS2

         ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

LE(s(x), s(y)) -> LE(x, y)
MINUS(x, y) -> IF(le(x, y), x, y)
MINUS(x, y) -> LE(x, y)
IF(false, x, y) -> MINUS(p(x), y)
IF(false, x, y) -> P(x)

Furthermore, R contains two SCCs.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳Non-Overlappingness Check

           →DP Problem 2

             ↳NOC

Dependency Pair:

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

Rules:

p(0) -> s(s(0))
p(s(x)) -> x
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))

R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle:

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

...

               →DP Problem 3

                 ↳Usable Rules (Innermost)

           →DP Problem 2

             ↳NOC

Dependency Pair:

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

Rules:

p(0) -> s(s(0))
p(s(x)) -> x
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))

Strategy:

innermost

As we are in the innermost case, we can delete all 8 non-usable-rules.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

...

               →DP Problem 4

                 ↳Size-Change Principle

           →DP Problem 2

             ↳NOC

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

LE(s(x), s(y)) -> LE(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.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳Non-Overlappingness Check

Dependency Pairs:

IF(false, x, y) -> MINUS(p(x), y)
MINUS(x, y) -> IF(le(x, y), x, y)

Rules:

p(0) -> s(s(0))
p(s(x)) -> x
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))

R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle:

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 5

                 ↳Usable Rules (Innermost)

Dependency Pairs:

IF(false, x, y) -> MINUS(p(x), y)
MINUS(x, y) -> IF(le(x, y), x, y)

Rules:

p(0) -> s(s(0))
p(s(x)) -> x
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, y) -> if(le(x, y), x, y)
if(true, x, y) -> 0
if(false, x, y) -> s(minus(p(x), y))

Strategy:

innermost

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

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

IF(false, x, y) -> MINUS(p(x), y)
MINUS(x, y) -> IF(le(x, y), x, y)

Rules:

p(0) -> s(s(0))
p(s(x)) -> x
le(s(x), s(y)) -> le(x, y)
le(0, y) -> true
le(s(x), 0) -> false

Strategy:

innermost

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

MINUS(x, y) -> IF(le(x, y), x, y)

three new Dependency Pairs are created:

MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))
MINUS(0, y'') -> IF(true, 0, y'')
MINUS(s(x''), 0) -> IF(false, s(x''), 0)

The transformation is resulting in one new DP problem:

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pairs:

MINUS(s(x''), 0) -> IF(false, s(x''), 0)
MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))
IF(false, x, y) -> MINUS(p(x), y)

Rules:

p(0) -> s(s(0))
p(s(x)) -> x
le(s(x), s(y)) -> le(x, y)
le(0, y) -> true
le(s(x), 0) -> false

Strategy:

innermost

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

IF(false, x, y) -> MINUS(p(x), y)

two new Dependency Pairs are created:

IF(false, 0, y) -> MINUS(s(s(0)), y)
IF(false, s(x''), y) -> MINUS(x'', y)

The transformation is resulting in one new DP problem:

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 8

                 ↳Instantiation Transformation

Dependency Pairs:

MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))
IF(false, s(x''), y) -> MINUS(x'', y)
MINUS(s(x''), 0) -> IF(false, s(x''), 0)

Rules:

p(0) -> s(s(0))
p(s(x)) -> x
le(s(x), s(y)) -> le(x, y)
le(0, y) -> true
le(s(x), 0) -> false

Strategy:

innermost

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

IF(false, s(x''), y) -> MINUS(x'', y)

two new Dependency Pairs are created:

IF(false, s(x''''), s(y'''')) -> MINUS(x'''', s(y''''))
IF(false, s(x''''), 0) -> MINUS(x'''', 0)

The transformation is resulting in two new DP problems:

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 9

                 ↳Usable Rules (Innermost)

Dependency Pairs:

IF(false, s(x''''), s(y'''')) -> MINUS(x'''', s(y''''))
MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))

Rules:

p(0) -> s(s(0))
p(s(x)) -> x
le(s(x), s(y)) -> le(x, y)
le(0, y) -> true
le(s(x), 0) -> false

Strategy:

innermost

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

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 11

                 ↳Size-Change Principle

Dependency Pairs:

IF(false, s(x''''), s(y'''')) -> MINUS(x'''', s(y''''))
MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))

Rules:

le(s(x), s(y)) -> le(x, y)
le(0, y) -> true
le(s(x), 0) -> false

Strategy:

innermost

We number the DPs as follows:

IF(false, s(x''''), s(y'''')) -> MINUS(x'''', s(y''''))
MINUS(s(x''), s(y'')) -> IF(le(x'', y''), s(x''), s(y''))

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

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

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

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

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

{2}	,	{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.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 10

                 ↳Usable Rules (Innermost)

Dependency Pairs:

IF(false, s(x''''), 0) -> MINUS(x'''', 0)
MINUS(s(x''), 0) -> IF(false, s(x''), 0)

Rules:

p(0) -> s(s(0))
p(s(x)) -> x
le(s(x), s(y)) -> le(x, y)
le(0, y) -> true
le(s(x), 0) -> false

Strategy:

innermost

As we are in the innermost case, we can delete all 5 non-usable-rules.

     ↳RRRI

       →TRS2

         ↳DPs

           →DP Problem 1

             ↳NOC

           →DP Problem 2

             ↳NOC

...

               →DP Problem 12

                 ↳Size-Change Principle

Dependency Pairs:

IF(false, s(x''''), 0) -> MINUS(x'''', 0)
MINUS(s(x''), 0) -> IF(false, s(x''), 0)

Rule:

none

Strategy:

innermost

We number the DPs as follows:

IF(false, s(x''''), 0) -> MINUS(x'''', 0)
MINUS(s(x''), 0) -> IF(false, s(x''), 0)

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

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

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

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

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

{2}	,	{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.

Innermost Termination of R successfully shown.
Duration:
0:06 minutes