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

Innermost Termination of R to be shown. 



   R
     ↳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) -> IFTIMES(even(s(x)), s(x), y)
TIMES(s(x), y) -> EVEN(s(x))
IFTIMES(true, s(x), y) -> PLUS(times(half(s(x)), y), times(half(s(x)), y))
IFTIMES(true, s(x), y) -> TIMES(half(s(x)), y)
IFTIMES(true, s(x), y) -> HALF(s(x))
IFTIMES(false, s(x), y) -> PLUS(y, times(x, y))
IFTIMES(false, s(x), y) -> TIMES(x, y)

Furthermore, R contains four SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
EVEN(x1) -> EVEN(x1)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 5
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS


Dependency Pair:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
HALF(x1) -> HALF(x1)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 6
             ↳Dependency Graph
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS


Dependency Pair:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Argument Filtering and Ordering
       →DP Problem 4
         ↳AFS


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
PLUS(x1, x2) -> PLUS(x1, x2)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
           →DP Problem 7
             ↳Dependency Graph
       →DP Problem 4
         ↳AFS


Dependency Pair:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳Argument Filtering and Ordering


Dependency Pairs:

IFTIMES(false, s(x), y) -> TIMES(x, y)
IFTIMES(true, s(x), y) -> TIMES(half(s(x)), y)
TIMES(s(x), y) -> IFTIMES(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) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rules for innermost can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
0 > true
{even, false}

resulting in one new DP problem.
Used Argument Filtering System:
IFTIMES(x1, x2, x3) -> x2
s(x1) -> s(x1)
TIMES(x1, x2) -> x1
half(x1) -> x1
even(x1) -> even(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
       →DP Problem 4
         ↳AFS
           →DP Problem 8
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

IFTIMES(true, s(x), y) -> TIMES(half(s(x)), y)
TIMES(s(x), y) -> IFTIMES(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) -> iftimes(even(s(x)), s(x), y)
iftimes(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
iftimes(false, s(x), y) -> plus(y, times(x, y))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes