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

Termination of R to be shown.



   R
Overlay and local confluence Check



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


   R
OC
       →TRS2
Dependency Pair Analysis



R contains the following Dependency Pairs:

MINUS(s(x), s(y)) -> MINUS(x, y)
LE(s(x), s(y)) -> LE(x, y)
QUOT(x, s(y)) -> IFQUOT(le(s(y), x), x, s(y))
QUOT(x, s(y)) -> LE(s(y), x)
IFQUOT(true, x, y) -> QUOT(minus(x, y), y)
IFQUOT(true, x, y) -> MINUS(x, y)

Furthermore, R contains three SCCs.


   R
OC
       →TRS2
DPs
           →DP Problem 1
Usable Rules (Innermost)
           →DP Problem 2
UsableRules
           →DP Problem 3
UsableRules


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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


   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
             ...
               →DP Problem 4
Size-Change Principle
           →DP Problem 2
UsableRules
           →DP Problem 3
UsableRules


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. MINUS(s(x), s(y)) -> MINUS(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

DP: 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.


   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
Usable Rules (Innermost)
           →DP Problem 3
UsableRules


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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


   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
             ...
               →DP Problem 5
Size-Change Principle
           →DP Problem 3
UsableRules


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. 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

DP: 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.


   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
           →DP Problem 3
Usable Rules (Innermost)


Dependency Pairs:

IFQUOT(true, x, y) -> QUOT(minus(x, y), y)
QUOT(x, s(y)) -> IFQUOT(le(s(y), x), x, s(y))


Rules:


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


Strategy:

innermost




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


   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
           →DP Problem 3
UsableRules
             ...
               →DP Problem 6
Narrowing Transformation


Dependency Pairs:

IFQUOT(true, x, y) -> QUOT(minus(x, y), y)
QUOT(x, s(y)) -> IFQUOT(le(s(y), x), x, s(y))


Rules:


minus(s(x), s(y)) -> minus(x, y)
minus(x, 0) -> 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

QUOT(x, s(y)) -> IFQUOT(le(s(y), x), x, s(y))
two new Dependency Pairs are created:

QUOT(s(y''), s(y0)) -> IFQUOT(le(y0, y''), s(y''), s(y0))
QUOT(0, s(y')) -> IFQUOT(false, 0, s(y'))

The transformation is resulting in one new DP problem:



   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
           →DP Problem 3
UsableRules
             ...
               →DP Problem 7
Narrowing Transformation


Dependency Pairs:

QUOT(s(y''), s(y0)) -> IFQUOT(le(y0, y''), s(y''), s(y0))
IFQUOT(true, x, y) -> QUOT(minus(x, y), y)


Rules:


minus(s(x), s(y)) -> minus(x, y)
minus(x, 0) -> 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

IFQUOT(true, x, y) -> QUOT(minus(x, y), y)
two new Dependency Pairs are created:

IFQUOT(true, s(x''), s(y'')) -> QUOT(minus(x'', y''), s(y''))
IFQUOT(true, x'', 0) -> QUOT(x'', 0)

The transformation is resulting in one new DP problem:



   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
           →DP Problem 3
UsableRules
             ...
               →DP Problem 8
Negative Polynomial Order


Dependency Pairs:

IFQUOT(true, s(x''), s(y'')) -> QUOT(minus(x'', y''), s(y''))
QUOT(s(y''), s(y0)) -> IFQUOT(le(y0, y''), s(y''), s(y0))


Rules:


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


Strategy:

innermost




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

IFQUOT(true, s(x''), s(y'')) -> QUOT(minus(x'', y''), s(y''))


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

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


Used ordering:
Polynomial Order with Interpretation:

POL( IFQUOT(x1, ..., x3) ) = x2

POL( s(x1) ) = x1 + 1

POL( QUOT(x1, x2) ) = x1

POL( minus(x1, x2) ) = x1

POL( le(x1, x2) ) = 0

POL( true ) = 0

POL( false ) = 0


This results in one new DP problem.


   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
           →DP Problem 3
UsableRules
             ...
               →DP Problem 9
Dependency Graph


Dependency Pair:

QUOT(s(y''), s(y0)) -> IFQUOT(le(y0, y''), s(y''), s(y0))


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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