Options:

Batch mode:

/usr/home/aoto/ttt/sttrs803.ttt
/usr/home/aoto/ttt/sttrs102.ttt
/usr/home/aoto/ttt/sttrs103.ttt
/usr/home/aoto/ttt/sttrs104.ttt
/usr/home/aoto/ttt/sttrs105.ttt
/usr/home/aoto/ttt/sttrs106.ttt
/usr/home/aoto/ttt/sttrs107.ttt
/usr/home/aoto/ttt/sttrs108.ttt
/usr/home/aoto/ttt/sttrs109.ttt
/usr/home/aoto/ttt/sttrs110.ttt
/usr/home/aoto/ttt/sttrs111.ttt
/usr/home/aoto/ttt/sttrs112.ttt
/usr/home/aoto/ttt/sttrs113.ttt
/usr/home/aoto/ttt/sttrs114.ttt
/usr/home/aoto/ttt/sttrs115.ttt
/usr/home/aoto/ttt/sttrs116.ttt
/usr/home/aoto/ttt/sttrs117.ttt
/usr/home/aoto/ttt/sttrs118.ttt
/usr/home/aoto/ttt/sttrs119.ttt
/usr/home/aoto/ttt/sttrs120.ttt
/usr/home/aoto/ttt/sttrs121.ttt
/usr/home/aoto/ttt/sttrs122.ttt
/usr/home/aoto/ttt/sttrs123.ttt
/usr/home/aoto/ttt/sttrs124.ttt
/usr/home/aoto/ttt/sttrs125.ttt
/usr/home/aoto/ttt/sttrs126.ttt
/usr/home/aoto/ttt/sttrs127.ttt
/usr/home/aoto/ttt/sttrs128.ttt
/usr/home/aoto/ttt/sttrs129.ttt
/usr/home/aoto/ttt/sttrs130.ttt
/usr/home/aoto/ttt/sttrs131.ttt
/usr/home/aoto/ttt/sttrs132.ttt
/usr/home/aoto/ttt/sttrs133.ttt
/usr/home/aoto/ttt/sttrs134.ttt
/usr/home/aoto/ttt/sttrs135.ttt
/usr/home/aoto/ttt/sttrs136.ttt
/usr/home/aoto/ttt/sttrs137.ttt
/usr/home/aoto/ttt/sttrs138.ttt
/usr/home/aoto/ttt/sttrs139.ttt
/usr/home/aoto/ttt/sttrs140.ttt
/usr/home/aoto/ttt/sttrs201.ttt
/usr/home/aoto/ttt/sttrs202.ttt
/usr/home/aoto/ttt/sttrs203.ttt
/usr/home/aoto/ttt/sttrs204.ttt
/usr/home/aoto/ttt/sttrs205.ttt
/usr/home/aoto/ttt/sttrs206.ttt
/usr/home/aoto/ttt/sttrs207.ttt
/usr/home/aoto/ttt/sttrs208.ttt
/usr/home/aoto/ttt/sttrs209.ttt
/usr/home/aoto/ttt/sttrs210.ttt
/usr/home/aoto/ttt/sttrs211.ttt
/usr/home/aoto/ttt/sttrs212.ttt
/usr/home/aoto/ttt/sttrs213.ttt
/usr/home/aoto/ttt/sttrs214.ttt
/usr/home/aoto/ttt/sttrs215.ttt
/usr/home/aoto/ttt/sttrs216.ttt
/usr/home/aoto/ttt/sttrs217.ttt
/usr/home/aoto/ttt/sttrs218.ttt
/usr/home/aoto/ttt/sttrs219.ttt
/usr/home/aoto/ttt/sttrs220.ttt
/usr/home/aoto/ttt/sttrs221.ttt
/usr/home/aoto/ttt/sttrs301.ttt
/usr/home/aoto/ttt/sttrs302.ttt
/usr/home/aoto/ttt/sttrs303.ttt
/usr/home/aoto/ttt/sttrs304.ttt
/usr/home/aoto/ttt/sttrs305.ttt
/usr/home/aoto/ttt/sttrs306.ttt
/usr/home/aoto/ttt/sttrs307.ttt
/usr/home/aoto/ttt/sttrs308.ttt
/usr/home/aoto/ttt/sttrs309.ttt
/usr/home/aoto/ttt/sttrs310.ttt
/usr/home/aoto/ttt/sttrs311.ttt
/usr/home/aoto/ttt/sttrs312.ttt
/usr/home/aoto/ttt/sttrs313.ttt
/usr/home/aoto/ttt/sttrs314.ttt
/usr/home/aoto/ttt/sttrs315.ttt
/usr/home/aoto/ttt/sttrs316.ttt
/usr/home/aoto/ttt/sttrs317.ttt
/usr/home/aoto/ttt/sttrs318.ttt
/usr/home/aoto/ttt/sttrs319.ttt
/usr/home/aoto/ttt/sttrs320.ttt
/usr/home/aoto/ttt/sttrs321.ttt
/usr/home/aoto/ttt/sttrs322.ttt
/usr/home/aoto/ttt/sttrs323.ttt
/usr/home/aoto/ttt/sttrs324.ttt
/usr/home/aoto/ttt/sttrs325.ttt
/usr/home/aoto/ttt/sttrs326.ttt
/usr/home/aoto/ttt/sttrs327.ttt
/usr/home/aoto/ttt/sttrs328.ttt
/usr/home/aoto/ttt/sttrs401.ttt
/usr/home/aoto/ttt/sttrs402.ttt
/usr/home/aoto/ttt/sttrs403.ttt
/usr/home/aoto/ttt/sttrs404.ttt
/usr/home/aoto/ttt/sttrs405.ttt
/usr/home/aoto/ttt/sttrs406.ttt
/usr/home/aoto/ttt/sttrs407.ttt
/usr/home/aoto/ttt/sttrs408.ttt
/usr/home/aoto/ttt/sttrs409.ttt
/usr/home/aoto/ttt/sttrs410.ttt
/usr/home/aoto/ttt/sttrs411.ttt
/usr/home/aoto/ttt/sttrs412.ttt
/usr/home/aoto/ttt/sttrs413.ttt
/usr/home/aoto/ttt/sttrs414.ttt
/usr/home/aoto/ttt/sttrs415.ttt
/usr/home/aoto/ttt/sttrs501.ttt
/usr/home/aoto/ttt/sttrs502.ttt
/usr/home/aoto/ttt/sttrs504.ttt
/usr/home/aoto/ttt/sttrs505.ttt
/usr/home/aoto/ttt/sttrs506.ttt
/usr/home/aoto/ttt/sttrs507.ttt
/usr/home/aoto/ttt/sttrs601.ttt
/usr/home/aoto/ttt/sttrs602.ttt
/usr/home/aoto/ttt/sttrs603.ttt
/usr/home/aoto/ttt/sttrs604.ttt
/usr/home/aoto/ttt/sttrs605.ttt
/usr/home/aoto/ttt/sttrs702.ttt
/usr/home/aoto/ttt/sttrs703.ttt
/usr/home/aoto/ttt/sttrs704.ttt
/usr/home/aoto/ttt/sttrs705.ttt
/usr/home/aoto/ttt/sttrs706.ttt
/usr/home/aoto/ttt/sttrs801.ttt
/usr/home/aoto/ttt/sttrs101.ttt

File: /usr/home/aoto/ttt/sttrs803.ttt

Term rewriting system R:

[y, x, f, ys, xs, g]
a3(gt, 0, y) -> false
a3(gt, x, 0) -> true
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a3(le, 0, y) -> true
a3(le, x, 0) -> false
a3(le, a2(s, x), a2(s, y)) -> a3(le, x, y)
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
a3(high, x, xs) -> a2(a2(filter, a2(a2(curry, gt), x)), xs)
a3(low, x, xs) -> a2(a2(filter, a2(a2(curry, le), x)), xs)
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
a2(qsort, nil) -> nil
a2(qsort, a3(cons, x, xs)) -> a3(append, a2(qsort, a3(high, x, xs)), a3(cons, x, a2(qsort, a3(low, x, xs))))
Dependency Pairs for R
A3(low, x, xs) -> A2(a2(filter, a2(a2(curry, le), x)), xs)
A3(low, x, xs) -> A2(filter, a2(a2(curry, le), x))
A3(low, x, xs) -> A2(a2(curry, le), x)
A3(low, x, xs) -> A2(curry, le)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A3(cons, y, a2(a2(filter, f), ys))
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(filter, f)
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
A3(high, x, xs) -> A2(a2(filter, a2(a2(curry, gt), x)), xs)
A3(high, x, xs) -> A2(filter, a2(a2(curry, gt), x))
A3(high, x, xs) -> A2(a2(curry, gt), x)
A3(high, x, xs) -> A2(curry, gt)
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(filter, f)
A3(le, a2(s, x), a2(s, y)) -> A3(le, x, y)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(filtersub, a2(f, y))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(cons, y, ys)
A2(qsort, a3(cons, x, xs)) -> A3(append, a2(qsort, a3(high, x, xs)), a3(cons, x, a2(qsort, a3(low, x, xs))))
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(high, x, xs))
A2(qsort, a3(cons, x, xs)) -> A3(high, x, xs)
A2(qsort, a3(cons, x, xs)) -> A3(cons, x, a2(qsort, a3(low, x, xs)))
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(low, x, xs))
A2(qsort, a3(cons, x, xs)) -> A3(low, x, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 4 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  1  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

2
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(gt)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

3
A3(le, a2(s, x), a2(s, y)) -> A3(le, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(le)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

4
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(high, x, xs) -> A2(a2(curry, gt), x)
A2(qsort, a3(cons, x, xs)) -> A3(low, x, xs)
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(low, x, xs))
A3(high, x, xs) -> A2(a2(filter, a2(a2(curry, gt), x)), xs)
A2(qsort, a3(cons, x, xs)) -> A3(high, x, xs)
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(high, x, xs))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A3(low, x, xs) -> A2(a2(curry, le), x)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A3(low, x, xs) -> A2(a2(filter, a2(a2(curry, le), x)), xs)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 3.55 sec.

File: /usr/home/aoto/ttt/sttrs102.ttt

Term rewriting system R:

[f, g, x]
a2(a3(comp, f, g), x) -> a2(f, a2(g, x))
Dependency Pairs for R
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
A2(a3(comp, f, g), x) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(comp, f, g), x) -> A2(g, x)
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
Oriented Rule(s):

a2(a3(comp, f, g), x) -> a2(f, a2(g, x))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A2(x1, x2))=  x1 + x2  
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(comp)=  0  
  POL(a2(x1, x2))=  x1 + x2  

Termination of R successfully proved!


Duration: 0.04 sec.

File: /usr/home/aoto/ttt/sttrs103.ttt

Term rewriting system R:

[f, g, x]
a2(a2(a2(comp, f), g), x) -> a2(f, a2(g, x))
Dependency Pairs for R
A2(a2(a2(comp, f), g), x) -> A2(f, a2(g, x))
A2(a2(a2(comp, f), g), x) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(comp, f), g), x) -> A2(g, x)
A2(a2(a2(comp, f), g), x) -> A2(f, a2(g, x))
Oriented Rule(s):

a2(a2(a2(comp, f), g), x) -> a2(f, a2(g, x))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(comp)=  1  
  POL(a2(x1, x2))=  x1 + x2  
  POL(A2(x1, x2))=  1 + x1 + x2  

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs104.ttt

Term rewriting system R:

[f, x, y]
a2(a2(a2(curry, f), x), y) -> a3(f, x, y)
There are no Dependency Pairs for R.

The Dependency Pair Graph for R contains no SCCs!

Termination of R successfully proved!


Duration: 0.01 sec.

File: /usr/home/aoto/ttt/sttrs105.ttt

Term rewriting system R:

[f, y, ys]
a3(filter, f, nil) -> nil
a3(filter, f, a3(cons, y, ys)) -> a4(filtersub, a2(f, y), f, a3(cons, y, ys))
a4(filtersub, true, f, a3(cons, y, ys)) -> a3(cons, y, a3(filter, f, ys))
a4(filtersub, false, f, a3(cons, y, ys)) -> a3(filter, f, ys)
Dependency Pairs for R
A3(filter, f, a3(cons, y, ys)) -> A4(filtersub, a2(f, y), f, a3(cons, y, ys))
A3(filter, f, a3(cons, y, ys)) -> A3(cons, y, ys)
A4(filtersub, false, f, a3(cons, y, ys)) -> A3(filter, f, ys)
A4(filtersub, true, f, a3(cons, y, ys)) -> A3(cons, y, a3(filter, f, ys))
A4(filtersub, true, f, a3(cons, y, ys)) -> A3(filter, f, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains no SCCs!

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs106.ttt

Term rewriting system R:

[f, y, ys]
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
Dependency Pairs for R
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(filter, f)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A3(cons, y, a2(a2(filter, f), ys))
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(filter, f)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(filtersub, a2(f, y))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(cons, y, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
Oriented Rule(s):

a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a2(a2(filter, f), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(nil)=  0  
  POL(false)=  0  
  POL(true)=  0  
  POL(filter)=  1  
  POL(filtersub)=  0  
  POL(a2(x1, x2))=  x1 + x2  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  x3  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 1.51 sec.

File: /usr/home/aoto/ttt/sttrs107.ttt

Term rewriting system R:

[f, y, ys]
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a2(a2(cons, y), ys)) -> a2(a2(a2(filtersub, a2(f, y)), f), a2(a2(cons, y), ys))
a2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> a2(a2(cons, y), a2(a2(filter, f), ys))
a2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> a2(a2(filter, f), ys)
Dependency Pairs for R
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(filter, f), ys))
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(filter, f)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(f, y)), f), a2(a2(cons, y), ys))
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(f, y)), f)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(filtersub, a2(f, y))
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> A2(filter, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(f, y)), f)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(f, y)), f), a2(a2(cons, y), ys))
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(filter, f), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.3 sec.

File: /usr/home/aoto/ttt/sttrs108.ttt

Term rewriting system R:

[f, x, y, ys]
a4(filter, f, x, nil) -> nil
a4(filter, f, x, a3(cons, y, ys)) -> a5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
a5(filtersub, true, f, x, a3(cons, y, ys)) -> a3(cons, y, a4(filter, f, x, ys))
a5(filtersub, false, f, x, a3(cons, y, ys)) -> a4(filter, f, x, ys)
Dependency Pairs for R
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A5(filtersub, false, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A4(filter, f, x, a3(cons, y, ys)) -> A5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains no SCCs!

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs109.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a3(filter, f, x), nil) -> nil
a2(a3(filter, f, x), a3(cons, y, ys)) -> a4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
a4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> a3(cons, y, a2(a3(filter, f, x), ys))
a4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> a2(a3(filter, f, x), ys)
Dependency Pairs for R
A2(a3(filter, f, x), a3(cons, y, ys)) -> A4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
A2(a3(filter, f, x), a3(cons, y, ys)) -> A2(filtersub, a3(f, x, y))
A4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A2(a3(filter, f, x), a3(cons, y, ys)) -> A4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a4(x1, x2, x3, x4))=  0  
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(nil)=  0  
  POL(false)=  0  
  POL(true)=  0  
  POL(filter)=  0  
  POL(filtersub)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  1  
  POL(A4(x1, x2, x3, x4))=  x4  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.18 sec.

File: /usr/home/aoto/ttt/sttrs110.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a2(a2(filter, f), x), nil) -> nil
a2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(a2(filtersub, a2(a2(f, x), y)), f), x), a2(a2(cons, y), ys))
a2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> a2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
a2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(filter, f), x), ys)
Dependency Pairs for R
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(filter, f)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(a2(filtersub, a2(a2(f, x), y)), f), x), a2(a2(cons, y), ys))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(a2(f, x), y)), f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(a2(f, x), y)), f)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(filtersub, a2(a2(f, x), y))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(filter, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(a2(f, x), y)), f)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(a2(f, x), y)), f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(a2(filtersub, a2(a2(f, x), y)), f), x), a2(a2(cons, y), ys))
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.53 sec.

File: /usr/home/aoto/ttt/sttrs111.ttt

Term rewriting system R:

[f, x, y, ys]
a4(fold, f, x, nil) -> x
a4(fold, f, x, a3(cons, y, ys)) -> a3(f, y, a4(fold, f, x, ys))
Dependency Pairs for R
A4(fold, f, x, a3(cons, y, ys)) -> A4(fold, f, x, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(fold, f, x, a3(cons, y, ys)) -> A4(fold, f, x, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(fold)=  a6  
  POL(A4(x1, x2, x3, x4))=  a1 + x1 + x2 + x3 + x4  
  POL(cons)=  1  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  

Termination of R successfully proved!


Duration: 0.02 sec.

File: /usr/home/aoto/ttt/sttrs112.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a3(f, y, a2(a3(fold, f, x), ys))
Dependency Pairs for R
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(cons)=  0  
  POL(fold)=  a8  
  POL(A2(x1, x2))=  a1 + x1 + x2  

Termination of R successfully proved!


Duration: 0.02 sec.

File: /usr/home/aoto/ttt/sttrs113.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a2(a2(f, y), a2(a3(fold, f, x), ys))
Dependency Pairs for R
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a2(f, y), a2(a3(fold, f, x), ys))
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(f, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(f, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a2(f, y), a2(a3(fold, f, x), ys))
Oriented Rule(s):

a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a2(a2(f, y), a2(a3(fold, f, x), ys))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(nil)=  0  
  POL(a2(x1, x2))=  x1  
  POL(cons)=  0  
  POL(fold)=  1  
  POL(A2(x1, x2))=  1 + x1  

Need to check 1 sub cycle of this SCC.

1.1
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(cons)=  0  
  POL(fold)=  a8  
  POL(A2(x1, x2))=  a1 + x1 + x2  

Termination of R successfully proved!


Duration: 0.14 sec.

File: /usr/home/aoto/ttt/sttrs114.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> a2(a2(f, y), a2(a2(a2(fold, f), x), ys))
Dependency Pairs for R
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(fold, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.09 sec.

File: /usr/home/aoto/ttt/sttrs115.ttt

Term rewriting system R:

[f, x, y, l, r]
a4(foldbt, f, x, a2(leaf, y)) -> a3(f, x, y)
a4(foldbt, f, x, a4(branch, y, l, r)) -> a4(foldbt, f, a4(foldbt, f, a3(f, x, y), l), r)
Dependency Pairs for R
A4(foldbt, f, x, a4(branch, y, l, r)) -> A4(foldbt, f, a4(foldbt, f, a3(f, x, y), l), r)
A4(foldbt, f, x, a4(branch, y, l, r)) -> A4(foldbt, f, a3(f, x, y), l)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(foldbt, f, x, a4(branch, y, l, r)) -> A4(foldbt, f, a3(f, x, y), l)
A4(foldbt, f, x, a4(branch, y, l, r)) -> A4(foldbt, f, a4(foldbt, f, a3(f, x, y), l), r)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(leaf)=  0  
  POL(foldbt)=  0  
  POL(A4(x1, x2, x3, x4))=  x4  
  POL(branch)=  1  
  POL(a4(x1, x2, x3, x4))=  x1 + x3 + x4  
  POL(a3(x1, x2, x3))=  0  
  POL(a2(x1, x2))=  0  

Termination of R successfully proved!


Duration: 0.07 sec.

File: /usr/home/aoto/ttt/sttrs116.ttt

Term rewriting system R:

[f, x, y, ys]
a4(foldl, f, x, nil) -> x
a4(foldl, f, x, a3(cons, y, ys)) -> a4(foldl, f, a3(f, x, y), ys)
Dependency Pairs for R
A4(foldl, f, x, a3(cons, y, ys)) -> A4(foldl, f, a3(f, x, y), ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(foldl, f, x, a3(cons, y, ys)) -> A4(foldl, f, a3(f, x, y), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(foldl)=  a6  
  POL(A4(x1, x2, x3, x4))=  a1 + a2·x1 + a3·x2 + x4  
  POL(cons)=  1  

Termination of R successfully proved!


Duration: 0.04 sec.

File: /usr/home/aoto/ttt/sttrs117.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a3(foldl, f, x), nil) -> x
a2(a3(foldl, f, x), a3(cons, y, ys)) -> a2(a3(foldl, f, a3(f, x, y)), ys)
Dependency Pairs for R
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a3(f, x, y)), ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a3(f, x, y)), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(foldl)=  a8  
  POL(cons)=  1  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs118.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a3(foldl, f, x), nil) -> x
a2(a3(foldl, f, x), a3(cons, y, ys)) -> a2(a3(foldl, f, a2(a2(f, x), y)), ys)
Dependency Pairs for R
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a2(a2(f, x), y)), ys)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a2(f, x), y)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(f, x)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a2(f, x), y)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a2(a2(f, x), y)), ys)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.05 sec.

File: /usr/home/aoto/ttt/sttrs119.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a2(a2(foldl, f), x), nil) -> x
a2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(foldl, f), a2(a2(f, x), y)), ys)
Dependency Pairs for R
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(foldl, f), a2(a2(f, x), y)), ys)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(foldl, f), a2(a2(f, x), y))
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(foldl, f)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(foldl, f), a2(a2(f, x), y))
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(foldl, f), a2(a2(f, x), y)), ys)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.09 sec.

File: /usr/home/aoto/ttt/sttrs120.ttt

Term rewriting system R:

[f, x]
a3(iterate, f, x) -> a3(cons, x, a3(iterate, f, a2(f, x)))
Dependency Pairs for R
A3(iterate, f, x) -> A3(cons, x, a3(iterate, f, a2(f, x)))
A3(iterate, f, x) -> A3(iterate, f, a2(f, x))

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(iterate, f, x) -> A3(iterate, f, a2(f, x))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs121.ttt

Term rewriting system R:

[f, x, xs]
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
Dependency Pairs for R
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  a5  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(cons)=  1  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  

Termination of R successfully proved!


Duration: 0.02 sec.

File: /usr/home/aoto/ttt/sttrs122.ttt

Term rewriting system R:

[f, x, xs]
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a3(cons, x, xs)) -> a3(cons, a2(f, x), a2(a2(map, f), xs))
Dependency Pairs for R
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(map, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
Oriented Rule(s):

a2(a2(map, f), a3(cons, x, xs)) -> a3(cons, a2(f, x), a2(a2(map, f), xs))
a2(a2(map, f), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  1  
  POL(a3(x1, x2, x3))=  0  
  POL(nil)=  0  
  POL(a2(x1, x2))=  x1 + x2  
  POL(cons)=  0  
  POL(A2(x1, x2))=  1 + x1  

Need to check 1 sub cycle of this SCC.

1.1
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  0  
  POL(a3(x1, x2, x3))=  1 + x3  
  POL(nil)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.1 sec.

File: /usr/home/aoto/ttt/sttrs123.ttt

Term rewriting system R:

[f, x, xs]
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a2(a2(cons, x), xs)) -> a2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
Dependency Pairs for R
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(cons, a2(f, x))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(map, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.1 sec.

File: /usr/home/aoto/ttt/sttrs124.ttt

Term rewriting system R:

[f, x, l, r]
a3(mapbt, f, a2(leaf, x)) -> a2(leaf, a2(f, x))
a3(mapbt, f, a4(branch, x, l, r)) -> a4(branch, a2(f, x), a3(mapbt, f, l), a3(mapbt, f, r))
Dependency Pairs for R
A3(mapbt, f, a4(branch, x, l, r)) -> A3(mapbt, f, l)
A3(mapbt, f, a4(branch, x, l, r)) -> A3(mapbt, f, r)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(mapbt, f, a4(branch, x, l, r)) -> A3(mapbt, f, r)
A3(mapbt, f, a4(branch, x, l, r)) -> A3(mapbt, f, l)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(mapbt)=  a5  
  POL(branch)=  1  
  POL(a4(x1, x2, x3, x4))=  x1 + x2 + x3 + x4  

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs125.ttt

Term rewriting system R:

[f, x, l, r]
a2(a2(mapbt, f), a2(leaf, x)) -> a2(leaf, a2(f, x))
a2(a2(mapbt, f), a4(branch, x, l, r)) -> a4(branch, a2(f, x), a2(a2(mapbt, f), l), a2(a2(mapbt, f), r))
Dependency Pairs for R
A2(a2(mapbt, f), a2(leaf, x)) -> A2(leaf, a2(f, x))
A2(a2(mapbt, f), a2(leaf, x)) -> A2(f, x)
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(f, x)
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(a2(mapbt, f), l)
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(mapbt, f)
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(a2(mapbt, f), r)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(a2(mapbt, f), r)
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(a2(mapbt, f), l)
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(f, x)
A2(a2(mapbt, f), a2(leaf, x)) -> A2(f, x)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(leaf)=  0  
  POL(branch)=  0  
  POL(mapbt)=  0  
  POL(a4(x1, x2, x3, x4))=  x2 + x3 + x4  
  POL(a2(x1, x2))=  1 + x2  
  POL(A2(x1, x2))=  x2  

Need to check 1 sub cycle of this SCC.

1.1
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(a2(mapbt, f), l)
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(f, x)
A2(a2(mapbt, f), a4(branch, x, l, r)) -> A2(a2(mapbt, f), r)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(leaf)=  0  
  POL(branch)=  0  
  POL(mapbt)=  0  
  POL(a4(x1, x2, x3, x4))=  1 + x2 + x3 + x4  
  POL(a2(x1, x2))=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.15 sec.

File: /usr/home/aoto/ttt/sttrs126.ttt

Term rewriting system R:

[f, x, xs]
a3(mapt, f, a2(leaf, x)) -> a2(leaf, a2(f, x))
a3(mapt, f, a2(node, xs)) -> a2(node, a3(maptlist, f, xs))
a3(maptlist, f, nil) -> nil
a3(maptlist, f, a3(cons, x, xs)) -> a3(cons, a3(mapt, f, x), a3(maptlist, f, xs))
Dependency Pairs for R
A3(mapt, f, a2(node, xs)) -> A3(maptlist, f, xs)
A3(maptlist, f, a3(cons, x, xs)) -> A3(cons, a3(mapt, f, x), a3(maptlist, f, xs))
A3(maptlist, f, a3(cons, x, xs)) -> A3(mapt, f, x)
A3(maptlist, f, a3(cons, x, xs)) -> A3(maptlist, f, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(maptlist, f, a3(cons, x, xs)) -> A3(maptlist, f, xs)
A3(maptlist, f, a3(cons, x, xs)) -> A3(mapt, f, x)
A3(mapt, f, a2(node, xs)) -> A3(maptlist, f, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(maptlist)=  0  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(mapt)=  0  
  POL(node)=  0  
  POL(A3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(a2(x1, x2))=  x1 + x2  
  POL(cons)=  1  

Termination of R successfully proved!


Duration: 0.05 sec.

File: /usr/home/aoto/ttt/sttrs127.ttt

Term rewriting system R:

[y, x]
a2(a2(max, 0), y) -> y
a2(a2(max, x), 0) -> x
a2(a2(max, a2(s, x)), a2(s, y)) -> a2(s, a2(a2(max, x), y))
Dependency Pairs for R
A2(a2(max, a2(s, x)), a2(s, y)) -> A2(s, a2(a2(max, x), y))
A2(a2(max, a2(s, x)), a2(s, y)) -> A2(a2(max, x), y)
A2(a2(max, a2(s, x)), a2(s, y)) -> A2(max, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(max, a2(s, x)), a2(s, y)) -> A2(a2(max, x), y)
Oriented Rule(s):

a2(a2(max, a2(s, x)), a2(s, y)) -> a2(s, a2(a2(max, x), y))
a2(a2(max, 0), y) -> y
a2(a2(max, x), 0) -> x

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(max)=  1  
  POL(0)=  0  
  POL(a2(x1, x2))=  x1 + x2  
  POL(A2(x1, x2))=  1 + x1 + x2  

where we removed the following rules (MRR):

a2(a2(max, 0), y) -> y
a2(a2(max, x), 0) -> x
Need to check 1 sub cycle of this SCC.

1.1
A2(a2(max, a2(s, x)), a2(s, y)) -> A2(a2(max, x), y)
Oriented Rule(s):

a2(a2(max, a2(s, x)), a2(s, y)) -> a2(s, a2(a2(max, x), y))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(max)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A2(x1, x2))=  x1 + x2  

Termination of R successfully proved!


Duration: 0.05 sec.

File: /usr/home/aoto/ttt/sttrs128.ttt

Term rewriting system R:

[y, x]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
Dependency Pairs for R
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
Oriented Rule(s):

a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(0)=  1  
  POL(a2(x1, x2))=  x1 + x2  
  POL(plus)=  0  
  POL(A2(x1, x2))=  1 + x1 + x2  

where we removed the following rules (MRR):

a2(a2(plus, 0), y) -> y
Need to check 1 sub cycle of this SCC.

1.1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
Oriented Rule(s):

a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(plus)=  0  
  POL(A2(x1, x2))=  1 + x1 + x2  

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs129.ttt

Term rewriting system R:

[f, x, y]
a4(rec, f, x, 0) -> x
a4(rec, f, x, a2(s, y)) -> a3(f, a2(s, y), a4(rec, f, x, y))
Dependency Pairs for R
A4(rec, f, x, a2(s, y)) -> A4(rec, f, x, y)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(rec, f, x, a2(s, y)) -> A4(rec, f, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(rec)=  a6  
  POL(a2(x1, x2))=  x1 + x2  
  POL(A4(x1, x2, x3, x4))=  a1 + x1 + x2 + x3 + x4  
  POL(s)=  1  

Termination of R successfully proved!


Duration: 0.01 sec.

File: /usr/home/aoto/ttt/sttrs130.ttt

Term rewriting system R:

[f, x, y]
a2(a3(rec, f, x), 0) -> x
a2(a3(rec, f, x), a2(s, y)) -> a3(f, a2(s, y), a2(a3(rec, f, x), y))
Dependency Pairs for R
A2(a3(rec, f, x), a2(s, y)) -> A2(s, y)
A2(a3(rec, f, x), a2(s, y)) -> A2(a3(rec, f, x), y)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(rec, f, x), a2(s, y)) -> A2(a3(rec, f, x), y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  1  
  POL(a3(x1, x2, x3))=  a4 + x1 + x2 + x3  
  POL(rec)=  a8  
  POL(a2(x1, x2))=  x1 + x2  
  POL(A2(x1, x2))=  a1 + x1 + x2  

Termination of R successfully proved!


Duration: 0.02 sec.

File: /usr/home/aoto/ttt/sttrs131.ttt

Term rewriting system R:

[f, x, y]
a2(a3(rec, f, x), 0) -> x
a2(a3(rec, f, x), a2(s, y)) -> a2(a2(f, a2(s, y)), a2(a3(rec, f, x), y))
Dependency Pairs for R
A2(a3(rec, f, x), a2(s, y)) -> A2(a2(f, a2(s, y)), a2(a3(rec, f, x), y))
A2(a3(rec, f, x), a2(s, y)) -> A2(f, a2(s, y))
A2(a3(rec, f, x), a2(s, y)) -> A2(s, y)
A2(a3(rec, f, x), a2(s, y)) -> A2(a3(rec, f, x), y)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(rec, f, x), a2(s, y)) -> A2(a3(rec, f, x), y)
A2(a3(rec, f, x), a2(s, y)) -> A2(f, a2(s, y))
A2(a3(rec, f, x), a2(s, y)) -> A2(a2(f, a2(s, y)), a2(a3(rec, f, x), y))
Oriented Rule(s):

a2(a3(rec, f, x), a2(s, y)) -> a2(a2(f, a2(s, y)), a2(a3(rec, f, x), y))
a2(a3(rec, f, x), 0) -> x

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(rec)=  0  
  POL(0)=  0  
  POL(a2(x1, x2))=  x1  
  POL(A2(x1, x2))=  1 + x1  

Need to check 1 sub cycle of this SCC.

1.1
A2(a3(rec, f, x), a2(s, y)) -> A2(a3(rec, f, x), y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  1  
  POL(a3(x1, x2, x3))=  a4 + x1 + x2 + x3  
  POL(rec)=  a8  
  POL(a2(x1, x2))=  x1 + x2  
  POL(A2(x1, x2))=  a1 + x1 + x2  

Termination of R successfully proved!


Duration: 0.12 sec.

File: /usr/home/aoto/ttt/sttrs132.ttt

Term rewriting system R:

[f, x, y]
a2(a2(a2(rec, f), x), 0) -> x
a2(a2(a2(rec, f), x), a2(s, y)) -> a2(a2(f, a2(s, y)), a2(a2(a2(rec, f), x), y))
Dependency Pairs for R
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(f, a2(s, y)), a2(a2(a2(rec, f), x), y))
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(f, a2(s, y))
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(s, y)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(a2(rec, f), x), y)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(rec, f), x)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(rec, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(rec, f), x)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(a2(rec, f), x), y)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(f, a2(s, y))
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(f, a2(s, y)), a2(a2(a2(rec, f), x), y))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.09 sec.

File: /usr/home/aoto/ttt/sttrs133.ttt

Term rewriting system R:

[f, x, xs, y, ys]
a4(scan, f, x, xs) -> a3(cons, x, a4(scansub, f, x, xs))
a4(scansub, f, x, nil) -> nil
a4(scansub, f, x, a3(cons, y, ys)) -> a4(scan, f, a3(f, x, y), ys)
Dependency Pairs for R
A4(scansub, f, x, a3(cons, y, ys)) -> A4(scan, f, a3(f, x, y), ys)
A4(scan, f, x, xs) -> A4(scansub, f, x, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(scan, f, x, xs) -> A4(scansub, f, x, xs)
A4(scansub, f, x, a3(cons, y, ys)) -> A4(scan, f, a3(f, x, y), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x3  
  POL(A4(x1, x2, x3, x4))=  x4  
  POL(scan)=  0  
  POL(scansub)=  0  
  POL(cons)=  0  

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs134.ttt

Term rewriting system R:

[f, x, xs, y, ys]
a2(a3(scan, f, x), xs) -> a3(cons, x, a2(a3(scansub, f, x), xs))
a2(a3(scansub, f, x), nil) -> nil
a2(a3(scansub, f, x), a3(cons, y, ys)) -> a2(a3(scan, f, a3(f, x, y)), ys)
Dependency Pairs for R
A2(a3(scan, f, x), xs) -> A2(a3(scansub, f, x), xs)
A2(a3(scansub, f, x), a3(cons, y, ys)) -> A2(a3(scan, f, a3(f, x, y)), ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(scansub, f, x), a3(cons, y, ys)) -> A2(a3(scan, f, a3(f, x, y)), ys)
A2(a3(scan, f, x), xs) -> A2(a3(scansub, f, x), xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x3  
  POL(scan)=  0  
  POL(scansub)=  0  
  POL(cons)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs135.ttt

Term rewriting system R:

[f, x, xs, y, ys]
a2(a3(scan, f, x), xs) -> a3(cons, x, a2(a3(scansub, f, x), xs))
a2(a3(scansub, f, x), nil) -> nil
a2(a3(scansub, f, x), a3(cons, y, ys)) -> a2(a3(scan, f, a2(a2(f, x), y)), ys)
Dependency Pairs for R
A2(a3(scan, f, x), xs) -> A2(a3(scansub, f, x), xs)
A2(a3(scansub, f, x), a3(cons, y, ys)) -> A2(a3(scan, f, a2(a2(f, x), y)), ys)
A2(a3(scansub, f, x), a3(cons, y, ys)) -> A2(a2(f, x), y)
A2(a3(scansub, f, x), a3(cons, y, ys)) -> A2(f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(scansub, f, x), a3(cons, y, ys)) -> A2(f, x)
A2(a3(scansub, f, x), a3(cons, y, ys)) -> A2(a2(f, x), y)
A2(a3(scansub, f, x), a3(cons, y, ys)) -> A2(a3(scan, f, a2(a2(f, x), y)), ys)
A2(a3(scan, f, x), xs) -> A2(a3(scansub, f, x), xs)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.08 sec.

File: /usr/home/aoto/ttt/sttrs136.ttt

Term rewriting system R:

[f, x, xs, y, ys]
a2(a2(a2(scan, f), x), xs) -> a2(a2(cons, x), a2(a2(a2(scansub, f), x), xs))
a2(a2(a2(scansub, f), x), nil) -> nil
a2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(scan, f), a2(a2(f, x), y)), ys)
Dependency Pairs for R
A2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(scan, f), a2(a2(f, x), y)), ys)
A2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> A2(a2(scan, f), a2(a2(f, x), y))
A2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> A2(scan, f)
A2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(scan, f), x), xs) -> A2(a2(cons, x), a2(a2(a2(scansub, f), x), xs))
A2(a2(a2(scan, f), x), xs) -> A2(cons, x)
A2(a2(a2(scan, f), x), xs) -> A2(a2(a2(scansub, f), x), xs)
A2(a2(a2(scan, f), x), xs) -> A2(a2(scansub, f), x)
A2(a2(a2(scan, f), x), xs) -> A2(scansub, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(scan, f), x), xs) -> A2(a2(scansub, f), x)
A2(a2(a2(scan, f), x), xs) -> A2(a2(a2(scansub, f), x), xs)
A2(a2(a2(scan, f), x), xs) -> A2(a2(cons, x), a2(a2(a2(scansub, f), x), xs))
A2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> A2(a2(scan, f), a2(a2(f, x), y))
A2(a2(a2(scansub, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(scan, f), a2(a2(f, x), y)), ys)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.19 sec.

File: /usr/home/aoto/ttt/sttrs137.ttt

Term rewriting system R:

[f, x, y]
a3(a2(uncurry, f), x, y) -> a2(a2(f, x), y)
There are no Dependency Pairs for R.

The Dependency Pair Graph for R contains no SCCs!

Termination of R successfully proved!


Duration: 0.0 sec.

File: /usr/home/aoto/ttt/sttrs138.ttt

Term rewriting system R:

[p, f, x]
a4(until, p, f, x) -> a5(untilsub, a2(p, x), p, f, x)
a5(untilsub, true, p, f, x) -> x
a5(untilsub, false, p, f, x) -> a4(until, p, f, a2(f, x))
Dependency Pairs for R
A5(untilsub, false, p, f, x) -> A4(until, p, f, a2(f, x))
A4(until, p, f, x) -> A5(untilsub, a2(p, x), p, f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains no SCCs!

Termination of R successfully proved!


Duration: 0.01 sec.

File: /usr/home/aoto/ttt/sttrs139.ttt

Term rewriting system R:

[f, x, xs, y, ys]
a4(zipwith, f, nil, nil) -> nil
a4(zipwith, f, a3(cons, x, xs), nil) -> nil
a4(zipwith, f, nil, a3(cons, y, ys)) -> nil
a4(zipwith, f, a3(cons, x, xs), a3(cons, y, ys)) -> a3(cons, a3(f, x, y), a4(zipwith, f, xs, ys))
Dependency Pairs for R
A4(zipwith, f, a3(cons, x, xs), a3(cons, y, ys)) -> A4(zipwith, f, xs, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(zipwith, f, a3(cons, x, xs), a3(cons, y, ys)) -> A4(zipwith, f, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(zipwith)=  a6  
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(A4(x1, x2, x3, x4))=  x1 + x2 + x3 + x4  
  POL(cons)=  0  

Termination of R successfully proved!


Duration: 0.02 sec.

File: /usr/home/aoto/ttt/sttrs140.ttt

Term rewriting system R:

[f, x, xs, y, ys]
a3(a2(zipwith, f), nil, nil) -> nil
a3(a2(zipwith, f), a3(cons, x, xs), nil) -> nil
a3(a2(zipwith, f), nil, a3(cons, y, ys)) -> nil
a3(a2(zipwith, f), a3(cons, x, xs), a3(cons, y, ys)) -> a3(cons, a3(f, x, y), a3(a2(zipwith, f), xs, ys))
Dependency Pairs for R
A3(a2(zipwith, f), a3(cons, x, xs), a3(cons, y, ys)) -> A3(cons, a3(f, x, y), a3(a2(zipwith, f), xs, ys))
A3(a2(zipwith, f), a3(cons, x, xs), a3(cons, y, ys)) -> A3(f, x, y)
A3(a2(zipwith, f), a3(cons, x, xs), a3(cons, y, ys)) -> A3(a2(zipwith, f), xs, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(a2(zipwith, f), a3(cons, x, xs), a3(cons, y, ys)) -> A3(a2(zipwith, f), xs, ys)
A3(a2(zipwith, f), a3(cons, x, xs), a3(cons, y, ys)) -> A3(f, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  0  
  POL(zipwith)=  0  

Need to check 1 sub cycle of this SCC.

1.1
A3(a2(zipwith, f), a3(cons, x, xs), a3(cons, y, ys)) -> A3(a2(zipwith, f), xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a2(x1, x2))=  a5 + x1 + x2  
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  0  
  POL(zipwith)=  a8  

Termination of R successfully proved!


Duration: 0.05 sec.

File: /usr/home/aoto/ttt/sttrs201.ttt

Term rewriting system R:

[y, x]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(double, x) -> a2(a2(plus, x), x)
Dependency Pairs for R
A2(double, x) -> A2(a2(plus, x), x)
A2(double, x) -> A2(plus, x)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(double, x) -> A2(a2(plus, x), x)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.04 sec.

File: /usr/home/aoto/ttt/sttrs202.ttt

Term rewriting system R:

[x, ys, f, xs]
a4(consif, true, x, ys) -> a3(cons, x, ys)
a4(consif, false, x, ys) -> ys
a3(filter, f, nil) -> nil
a3(filter, f, a3(cons, x, xs)) -> a4(consif, a2(f, x), x, a3(filter, f, xs))
Dependency Pairs for R
A3(filter, f, a3(cons, x, xs)) -> A4(consif, a2(f, x), x, a3(filter, f, xs))
A3(filter, f, a3(cons, x, xs)) -> A3(filter, f, xs)
A4(consif, true, x, ys) -> A3(cons, x, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(filter, f, a3(cons, x, xs)) -> A3(filter, f, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(cons)=  1  
  POL(filter)=  a5  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  

Termination of R successfully proved!


Duration: 0.02 sec.

File: /usr/home/aoto/ttt/sttrs203.ttt

Term rewriting system R:

[x, y, f, xs]
a4(if, true, x, y) -> x
a4(if, false, x, y) -> y
a3(filter, f, nil) -> nil
a3(filter, f, a3(cons, x, xs)) -> a4(if, a2(f, x), a3(cons, x, a3(filter, f, xs)), a3(filter, f, xs))
Dependency Pairs for R
A3(filter, f, a3(cons, x, xs)) -> A4(if, a2(f, x), a3(cons, x, a3(filter, f, xs)), a3(filter, f, xs))
A3(filter, f, a3(cons, x, xs)) -> A3(cons, x, a3(filter, f, xs))
A3(filter, f, a3(cons, x, xs)) -> A3(filter, f, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(filter, f, a3(cons, x, xs)) -> A3(filter, f, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(filter)=  a5  
  POL(cons)=  1  

Termination of R successfully proved!


Duration: 0.02 sec.

File: /usr/home/aoto/ttt/sttrs204.ttt

Term rewriting system R:

[x, ys, f, xs]
a2(a2(a2(consif, true), x), ys) -> a2(a2(cons, x), ys)
a2(a2(a2(consif, false), x), ys) -> ys
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a2(a2(cons, x), xs)) -> a2(a2(a2(consif, a2(f, x)), x), a2(a2(filter, f), xs))
Dependency Pairs for R
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(a2(consif, a2(f, x)), x), a2(a2(filter, f), xs))
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(consif, a2(f, x)), x)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(consif, a2(f, x))
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(filter, f), xs)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(filter, f)
A2(a2(a2(consif, true), x), ys) -> A2(a2(cons, x), ys)
A2(a2(a2(consif, true), x), ys) -> A2(cons, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(consif, true), x), ys) -> A2(a2(cons, x), ys)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(filter, f), xs)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(consif, a2(f, x)), x)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(a2(consif, a2(f, x)), x), a2(a2(filter, f), xs))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.14 sec.

File: /usr/home/aoto/ttt/sttrs205.ttt

Term rewriting system R:

[x, y, f, xs]
a2(a2(a2(if, true), x), y) -> x
a2(a2(a2(if, false), x), y) -> y
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a2(a2(cons, x), xs)) -> a2(a2(a2(if, a2(f, x)), a2(a2(cons, x), a2(a2(filter, f), xs))), a2(a2(filter, f), xs))
Dependency Pairs for R
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(a2(if, a2(f, x)), a2(a2(cons, x), a2(a2(filter, f), xs))), a2(a2(filter, f), xs))
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(if, a2(f, x)), a2(a2(cons, x), a2(a2(filter, f), xs)))
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(if, a2(f, x))
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(cons, x), a2(a2(filter, f), xs))
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(cons, x)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(filter, f), xs)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(filter, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(filter, f), xs)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(cons, x), a2(a2(filter, f), xs))
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(if, a2(f, x)), a2(a2(cons, x), a2(a2(filter, f), xs)))
A2(a2(filter, f), a2(a2(cons, x), xs)) -> A2(a2(a2(if, a2(f, x)), a2(a2(cons, x), a2(a2(filter, f), xs))), a2(a2(filter, f), xs))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.17 sec.

File: /usr/home/aoto/ttt/sttrs206.ttt

Term rewriting system R:

[x, xs, f, y, ys]
a4(consif, true, x, xs) -> a3(cons, x, xs)
a4(consif, false, x, xs) -> xs
a4(filter, f, x, nil) -> nil
a4(filter, f, x, a3(cons, y, ys)) -> a4(consif, a2(f, x, y), y, a4(filter, f, x, ys))
Dependency Pairs for R
A4(filter, f, x, a3(cons, y, ys)) -> A4(consif, a2(f, x, y), y, a4(filter, f, x, ys))
A4(filter, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(filter, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(cons)=  1  
  POL(filter)=  a6  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A4(x1, x2, x3, x4))=  a1 + x1 + x2 + x3 + x4  

Termination of R successfully proved!


Duration: 0.02 sec.

File: /usr/home/aoto/ttt/sttrs207.ttt

Term rewriting system R:

[x, y, f, ys]
a4(if, true, x, y) -> x
a4(if, false, x, y) -> y
a4(filter, f, x, nil) -> nil
a4(filter, f, x, a3(cons, y, ys)) -> a4(if, a2(f, x, y), a3(cons, y, a4(filter, f, x, ys)), a4(filter, f, x, ys))
Dependency Pairs for R
A4(filter, f, x, a3(cons, y, ys)) -> A4(if, a2(f, x, y), a3(cons, y, a4(filter, f, x, ys)), a4(filter, f, x, ys))
A4(filter, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(filter, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(filter)=  a6  
  POL(A4(x1, x2, x3, x4))=  a1 + x1 + x2 + x3 + x4  
  POL(cons)=  1  

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs208.ttt

Term rewriting system R:

[y, ys, f, x]
a2(a2(a2(consif, true), y), ys) -> a2(a2(cons, y), ys)
a2(a2(a2(consif, false), y), ys) -> ys
a2(a2(a2(filter, f), x), nil) -> nil
a2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(consif, a2(a2(f, x), y)), y), a2(a2(a2(filter, f), x), ys))
Dependency Pairs for R
A2(a2(a2(consif, true), y), ys) -> A2(a2(cons, y), ys)
A2(a2(a2(consif, true), y), ys) -> A2(cons, y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(consif, a2(a2(f, x), y)), y), a2(a2(a2(filter, f), x), ys))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(consif, a2(a2(f, x), y)), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(consif, a2(a2(f, x), y))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(filter, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(consif, a2(a2(f, x), y)), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(consif, a2(a2(f, x), y)), y), a2(a2(a2(filter, f), x), ys))
A2(a2(a2(consif, true), y), ys) -> A2(a2(cons, y), ys)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.22 sec.

File: /usr/home/aoto/ttt/sttrs209.ttt

Term rewriting system R:

[x, y, f, ys]
a2(a2(a2(if, true), x), y) -> x
a2(a2(a2(if, false), x), y) -> y
a2(a2(a2(filter, f), x), nil) -> nil
a2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(if, a2(a2(f, x), y)), a2(a2(cons, y), a2(a2(a2(filter, f), x), ys))), a2(a2(a2(filter, f), x), ys))
Dependency Pairs for R
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(if, a2(a2(f, x), y)), a2(a2(cons, y), a2(a2(a2(filter, f), x), ys))), a2(a2(a2(filter, f), x), ys))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(if, a2(a2(f, x), y)), a2(a2(cons, y), a2(a2(a2(filter, f), x), ys)))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(if, a2(a2(f, x), y))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(filter, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(if, a2(a2(f, x), y)), a2(a2(cons, y), a2(a2(a2(filter, f), x), ys)))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(if, a2(a2(f, x), y)), a2(a2(cons, y), a2(a2(a2(filter, f), x), ys))), a2(a2(a2(filter, f), x), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.27 sec.

File: /usr/home/aoto/ttt/sttrs210.ttt

Term rewriting system R:

[ys, x, xs, f]
a3(append, nil, ys) -> ys
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a3(flatwith, f, a2(leaf, x)) -> a3(cons, a2(f, x), nil)
a3(flatwith, f, a2(node, xs)) -> a3(flatwithsub, f, xs)
a3(flatwithsub, f, nil) -> nil
a3(flatwithsub, f, a3(cons, x, xs)) -> a3(append, a3(flatwith, f, x), a3(flatwithsub, f, xs))
Dependency Pairs for R
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A3(flatwith, f, a2(node, xs)) -> A3(flatwithsub, f, xs)
A3(flatwith, f, a2(leaf, x)) -> A3(cons, a2(f, x), nil)
A3(flatwithsub, f, a3(cons, x, xs)) -> A3(append, a3(flatwith, f, x), a3(flatwithsub, f, xs))
A3(flatwithsub, f, a3(cons, x, xs)) -> A3(flatwith, f, x)
A3(flatwithsub, f, a3(cons, x, xs)) -> A3(flatwithsub, f, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(cons)=  1  

2
A3(flatwithsub, f, a3(cons, x, xs)) -> A3(flatwithsub, f, xs)
A3(flatwithsub, f, a3(cons, x, xs)) -> A3(flatwith, f, x)
A3(flatwith, f, a2(node, xs)) -> A3(flatwithsub, f, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(flatwith)=  0  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(node)=  0  
  POL(A3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(a2(x1, x2))=  x1 + x2  
  POL(cons)=  1  
  POL(flatwithsub)=  0  

Termination of R successfully proved!


Duration: 0.06 sec.

File: /usr/home/aoto/ttt/sttrs211.ttt

Term rewriting system R:

[ys, x, xs, f]
a3(append, nil, ys) -> ys
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a2(a2(flatwith, f), a2(leaf, x)) -> a3(cons, a2(f, x), nil)
a2(a2(flatwith, f), a2(node, xs)) -> a2(a2(flatwithsub, f), xs)
a2(a2(flatwithsub, f), nil) -> nil
a2(a2(flatwithsub, f), a3(cons, x, xs)) -> a3(append, a2(a2(flatwith, f), x), a2(a2(flatwithsub, f), xs))
Dependency Pairs for R
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A2(a2(flatwith, f), a2(leaf, x)) -> A3(cons, a2(f, x), nil)
A2(a2(flatwith, f), a2(leaf, x)) -> A2(f, x)
A2(a2(flatwith, f), a2(node, xs)) -> A2(a2(flatwithsub, f), xs)
A2(a2(flatwith, f), a2(node, xs)) -> A2(flatwithsub, f)
A2(a2(flatwithsub, f), a3(cons, x, xs)) -> A3(append, a2(a2(flatwith, f), x), a2(a2(flatwithsub, f), xs))
A2(a2(flatwithsub, f), a3(cons, x, xs)) -> A2(a2(flatwith, f), x)
A2(a2(flatwithsub, f), a3(cons, x, xs)) -> A2(flatwith, f)
A2(a2(flatwithsub, f), a3(cons, x, xs)) -> A2(a2(flatwithsub, f), xs)
A2(a2(flatwithsub, f), a3(cons, x, xs)) -> A2(flatwithsub, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(cons)=  1  

2
A2(a2(flatwithsub, f), a3(cons, x, xs)) -> A2(a2(flatwithsub, f), xs)
A2(a2(flatwithsub, f), a3(cons, x, xs)) -> A2(a2(flatwith, f), x)
A2(a2(flatwith, f), a2(node, xs)) -> A2(a2(flatwithsub, f), xs)
A2(a2(flatwith, f), a2(leaf, x)) -> A2(f, x)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A2(x1, x2))=  x2  
  POL(append)=  0  
  POL(a3(x1, x2, x3))=  x2 + x3  
  POL(node)=  0  
  POL(flatwith)=  0  
  POL(leaf)=  0  
  POL(flatwithsub)=  0  
  POL(a2(x1, x2))=  1 + x2  
  POL(cons)=  0  
  POL(nil)=  0  

Need to check 1 sub cycle of this SCC.

2.1
A2(a2(flatwithsub, f), a3(cons, x, xs)) -> A2(a2(flatwith, f), x)
A2(a2(flatwithsub, f), a3(cons, x, xs)) -> A2(a2(flatwithsub, f), xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A2(x1, x2))=  x2  
  POL(append)=  0  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(node)=  0  
  POL(flatwith)=  0  
  POL(leaf)=  0  
  POL(flatwithsub)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  1  
  POL(nil)=  0  

Termination of R successfully proved!


Duration: 0.24 sec.

File: /usr/home/aoto/ttt/sttrs212.ttt

Term rewriting system R:

[ys, x, xs, f]
a2(a2(append, nil), ys) -> ys
a2(a2(append, a2(a2(cons, x), xs)), ys) -> a2(a2(cons, x), a2(a2(append, xs), ys))
a2(a2(flatwith, f), a2(leaf, x)) -> a2(a2(cons, a2(f, x)), nil)
a2(a2(flatwith, f), a2(node, xs)) -> a2(a2(flatwithsub, f), xs)
a2(a2(flatwithsub, f), nil) -> nil
a2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> a2(a2(append, a2(a2(flatwith, f), x)), a2(a2(flatwithsub, f), xs))
Dependency Pairs for R
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(cons, x)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(append, xs)
A2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> A2(a2(append, a2(a2(flatwith, f), x)), a2(a2(flatwithsub, f), xs))
A2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> A2(append, a2(a2(flatwith, f), x))
A2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> A2(a2(flatwith, f), x)
A2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> A2(flatwith, f)
A2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> A2(a2(flatwithsub, f), xs)
A2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> A2(flatwithsub, f)
A2(a2(flatwith, f), a2(node, xs)) -> A2(a2(flatwithsub, f), xs)
A2(a2(flatwith, f), a2(node, xs)) -> A2(flatwithsub, f)
A2(a2(flatwith, f), a2(leaf, x)) -> A2(a2(cons, a2(f, x)), nil)
A2(a2(flatwith, f), a2(leaf, x)) -> A2(cons, a2(f, x))
A2(a2(flatwith, f), a2(leaf, x)) -> A2(f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(flatwith, f), a2(leaf, x)) -> A2(f, x)
A2(a2(flatwith, f), a2(leaf, x)) -> A2(a2(cons, a2(f, x)), nil)
A2(a2(flatwith, f), a2(node, xs)) -> A2(a2(flatwithsub, f), xs)
A2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> A2(a2(flatwithsub, f), xs)
A2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> A2(a2(flatwith, f), x)
A2(a2(flatwithsub, f), a2(a2(cons, x), xs)) -> A2(a2(append, a2(a2(flatwith, f), x)), a2(a2(flatwithsub, f), xs))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.28 sec.

File: /usr/home/aoto/ttt/sttrs213.ttt

Term rewriting system R:

[x, y]
a2(id, x) -> x
a2(plus, 0) -> id
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
Dependency Pairs for R
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
Oriented Rule(s):

a2(plus, 0) -> id
a2(id, x) -> x
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(id)=  0  
  POL(0)=  1  
  POL(a2(x1, x2))=  x1 + x2  
  POL(plus)=  0  
  POL(A2(x1, x2))=  1 + x1 + x2  

where we removed the following rules (MRR):

a2(plus, 0) -> id
Need to check 1 sub cycle of this SCC.

1.1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
Oriented Rule(s):

a2(id, x) -> x
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(id)=  1  
  POL(a2(x1, x2))=  x1 + x2  
  POL(plus)=  0  
  POL(A2(x1, x2))=  1 + x1 + x2  

where we removed the following rules (MRR):

a2(id, x) -> x
Need to check 1 sub cycle of this SCC.

1.1.1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
Oriented Rule(s):

a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(plus)=  0  
  POL(A2(x1, x2))=  1 + x1 + x2  

Termination of R successfully proved!


Duration: 0.05 sec.

File: /usr/home/aoto/ttt/sttrs214.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a3(f, y, a2(a3(fold, f, x), ys))
reverse -> a3(fold, cons, nil)
Dependency Pairs for R
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(cons)=  0  
  POL(fold)=  a8  
  POL(A2(x1, x2))=  a1 + x1 + x2  

Termination of R successfully proved!


Duration: 0.01 sec.

File: /usr/home/aoto/ttt/sttrs215.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a3(cons, y, ys)) -> a3(f, y, a2(a2(a2(fold, f), x), ys))
reverse -> a2(a2(fold, cons), nil)
Dependency Pairs for R
REVERSE -> A2(a2(fold, cons), nil)
REVERSE -> A2(fold, cons)
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(fold, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(a2(a2(fold, f), x), ys)
Oriented Rule(s):

a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a3(cons, y, ys)) -> a3(f, y, a2(a2(a2(fold, f), x), ys))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(fold)=  0  
  POL(nil)=  1  
  POL(a2(x1, x2))=  x1 + x2  
  POL(cons)=  0  
  POL(a3(x1, x2, x3))=  1 + x3  
  POL(A2(x1, x2))=  x1 + x2  

Termination of R successfully proved!


Duration: 0.07 sec.

File: /usr/home/aoto/ttt/sttrs216.ttt

Term rewriting system R:

[f, x, y, ys]
a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> a2(a2(f, y), a2(a2(a2(fold, f), x), ys))
reverse -> a2(a2(fold, cons), nil)
Dependency Pairs for R
REVERSE -> A2(a2(fold, cons), nil)
REVERSE -> A2(fold, cons)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(fold, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.1 sec.

File: /usr/home/aoto/ttt/sttrs217.ttt

Term rewriting system R:

[y, x, f, xs]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(sumwith, f), nil) -> nil
a2(a2(sumwith, f), a2(a2(cons, x), xs)) -> a2(a2(plus, a2(f, x)), a2(a2(sumwith, f), xs))
Dependency Pairs for R
A2(a2(sumwith, f), a2(a2(cons, x), xs)) -> A2(a2(plus, a2(f, x)), a2(a2(sumwith, f), xs))
A2(a2(sumwith, f), a2(a2(cons, x), xs)) -> A2(plus, a2(f, x))
A2(a2(sumwith, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(sumwith, f), a2(a2(cons, x), xs)) -> A2(a2(sumwith, f), xs)
A2(a2(sumwith, f), a2(a2(cons, x), xs)) -> A2(sumwith, f)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(sumwith, f), a2(a2(cons, x), xs)) -> A2(a2(sumwith, f), xs)
A2(a2(sumwith, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(sumwith, f), a2(a2(cons, x), xs)) -> A2(a2(plus, a2(f, x)), a2(a2(sumwith, f), xs))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.15 sec.

File: /usr/home/aoto/ttt/sttrs218.ttt

Term rewriting system R:

[y, x]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(times, 0), y) -> 0
a2(a2(times, a2(s, x)), y) -> a2(a2(plus, a2(a2(times, x), y)), y)
Dependency Pairs for R
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(times, a2(s, x)), y) -> A2(plus, a2(a2(times, x), y))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(times, x)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.07 sec.

File: /usr/home/aoto/ttt/sttrs219.ttt

Term rewriting system R:

[f, g, x]
a2(a3(comp, f, g), x) -> a2(f, a2(g, x))
a2(twice, f) -> a3(comp, f, f)
Dependency Pairs for R
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
A2(a3(comp, f, g), x) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(comp, f, g), x) -> A2(g, x)
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(a2(x1, x2))=  0  
  POL(comp)=  0  
  POL(twice)=  0  
  POL(A2(x1, x2))=  x1  

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs220.ttt

Term rewriting system R:

[f, g, x]
a2(a2(a2(comp, f), g), x) -> a2(f, a2(g, x))
a2(twice, f) -> a2(a2(comp, f), f)
Dependency Pairs for R
A2(twice, f) -> A2(a2(comp, f), f)
A2(twice, f) -> A2(comp, f)
A2(a2(a2(comp, f), g), x) -> A2(f, a2(g, x))
A2(a2(a2(comp, f), g), x) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(comp, f), g), x) -> A2(g, x)
A2(a2(a2(comp, f), g), x) -> A2(f, a2(g, x))
A2(twice, f) -> A2(a2(comp, f), f)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.04 sec.

File: /usr/home/aoto/ttt/sttrs221.ttt

Term rewriting system R:

[x, y, p, f]
a4(if, true, x, y) -> x
a4(if, false, x, y) -> y
a4(until, p, f, x) -> a4(if, a2(p, x), x, a4(until, p, f, a2(f, x)))
Dependency Pairs for R
A4(until, p, f, x) -> A4(if, a2(p, x), x, a4(until, p, f, a2(f, x)))
A4(until, p, f, x) -> A4(until, p, f, a2(f, x))

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A4(until, p, f, x) -> A4(until, p, f, a2(f, x))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.01 sec.

File: /usr/home/aoto/ttt/sttrs301.ttt

Term rewriting system R:

[y, x, f]
a3(plus, 0, y) -> y
a3(plus, a2(s, x), y) -> a2(s, a3(plus, x, y))
a2(a2(a2(curry, f), x), y) -> a3(f, x, y)
add -> a2(curry, plus)
Dependency Pairs for R
A2(a2(a2(curry, f), x), y) -> A3(f, x, y)
A3(plus, a2(s, x), y) -> A2(s, a3(plus, x, y))
A3(plus, a2(s, x), y) -> A3(plus, x, y)
ADD -> A2(curry, plus)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(plus, a2(s, x), y) -> A3(plus, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(plus)=  a5  

Termination of R successfully proved!


Duration: 0.02 sec.

File: /usr/home/aoto/ttt/sttrs302.ttt

Term rewriting system R:

[f, x, y, ys]
a3(and, true, true) -> true
a3(and, true, false) -> false
a3(and, false, true) -> false
a3(and, false, false) -> false
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a3(f, y, a2(a3(fold, f, x), ys))
andl -> a3(fold, and, true)
Dependency Pairs for R
A2(a3(fold, f, x), a3(cons, y, ys)) -> A3(f, y, a2(a3(fold, f, x), ys))
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A3(fold, f, x)
ANDL -> A3(fold, and, true)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
Oriented Rule(s):

a3(and, true, false) -> false
a3(and, true, true) -> true
a3(and, false, true) -> false
a3(and, false, false) -> false

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(fold)=  0  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(false)=  0  
  POL(true)=  0  
  POL(and)=  0  
  POL(cons)=  1  
  POL(A2(x1, x2))=  x1 + x2  

Termination of R successfully proved!


Duration: 0.04 sec.

File: /usr/home/aoto/ttt/sttrs303.ttt

Term rewriting system R:

[x, y, f, ys]
a2(a2(and, true), true) -> true
a2(a2(and, x), false) -> false
a2(a2(and, false), y) -> false
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a2(a2(f, y), a2(a3(fold, f, x), ys))
andl -> a3(fold, and, true)
Dependency Pairs for R
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a2(f, y), a2(a3(fold, f, x), ys))
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(f, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(f, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a2(f, y), a2(a3(fold, f, x), ys))
Oriented Rule(s):

a2(a2(and, x), false) -> false
a2(a2(and, false), y) -> false
a2(a3(fold, f, x), nil) -> x
a2(a2(and, true), true) -> true
a2(a3(fold, f, x), a3(cons, y, ys)) -> a2(a2(f, y), a2(a3(fold, f, x), ys))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(false)=  0  
  POL(nil)=  0  
  POL(true)=  0  
  POL(and)=  0  
  POL(a2(x1, x2))=  x1  
  POL(cons)=  0  
  POL(fold)=  1  
  POL(A2(x1, x2))=  1 + x1  

Need to check 1 sub cycle of this SCC.

1.1
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(cons)=  0  
  POL(fold)=  a8  
  POL(A2(x1, x2))=  a1 + x1 + x2  

Termination of R successfully proved!


Duration: 0.14 sec.

File: /usr/home/aoto/ttt/sttrs304.ttt

Term rewriting system R:

[x, y, f, ys]
a2(a2(and, true), true) -> true
a2(a2(and, x), false) -> false
a2(a2(and, false), y) -> false
a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> a2(a2(f, y), a2(a2(a2(fold, f), x), ys))
andl -> a2(a2(fold, and), true)
Dependency Pairs for R
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(fold, f)
ANDL -> A2(a2(fold, and), true)
ANDL -> A2(fold, and)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.12 sec.

File: /usr/home/aoto/ttt/sttrs305.ttt

Term rewriting system R:

[xs, x, ys, f, y]
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a3(f, y, a2(a3(fold, f, x), ys))
concat -> a3(fold, append, nil)
Dependency Pairs for R
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
CONCAT -> A3(fold, append, nil)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A3(f, y, a2(a3(fold, f, x), ys))
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A3(fold, f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  1  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

2
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
Oriented Rule(s):

a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(fold)=  0  
  POL(append)=  0  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(nil)=  1  
  POL(cons)=  0  
  POL(A2(x1, x2))=  x1 + x2  

where we removed the following rules (MRR):

a3(append, nil, xs) -> xs
Need to check 1 sub cycle of this SCC.

2.1
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
Oriented Rule(s):

a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(fold)=  0  
  POL(append)=  0  
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(cons)=  0  
  POL(A2(x1, x2))=  x1 + x2  

Termination of R successfully proved!


Duration: 0.05 sec.

File: /usr/home/aoto/ttt/sttrs306.ttt

Term rewriting system R:

[xs, x, ys, f, y]
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a3(cons, y, ys)) -> a3(f, y, a2(a2(a2(fold, f), x), ys))
concat -> a2(a2(fold, append), nil)
Dependency Pairs for R
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A3(f, y, a2(a2(a2(fold, f), x), ys))
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(fold, f)
CONCAT -> A2(a2(fold, append), nil)
CONCAT -> A2(fold, append)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  1  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

2
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a3(cons, y, ys)) -> A2(a2(a2(fold, f), x), ys)
Oriented Rule(s):

a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a3(cons, y, ys)) -> a3(f, y, a2(a2(a2(fold, f), x), ys))
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(fold)=  0  
  POL(append)=  0  
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(nil)=  1  
  POL(a2(x1, x2))=  x1 + x2  
  POL(cons)=  0  
  POL(A2(x1, x2))=  x1 + x2  

Termination of R successfully proved!


Duration: 0.09 sec.

File: /usr/home/aoto/ttt/sttrs307.ttt

Term rewriting system R:

[xs, x, ys, f, y]
a2(a2(append, nil), xs) -> xs
a2(a2(append, a2(a2(cons, x), xs)), ys) -> a2(a2(cons, x), a2(a2(append, xs), ys))
a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> a2(a2(f, y), a2(a2(a2(fold, f), x), ys))
concat -> a2(a2(fold, append), nil)
Dependency Pairs for R
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(cons, x)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(append, xs)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(fold, f)
CONCAT -> A2(a2(fold, append), nil)
CONCAT -> A2(fold, append)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.18 sec.

File: /usr/home/aoto/ttt/sttrs308.ttt

Term rewriting system R:

[y, x, xs, f, ys]
a3(gt, 0, y) -> false
a3(gt, x, 0) -> true
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a3(high, x, xs) -> a4(filter, gt, x, xs)
a4(filter, f, x, nil) -> nil
a4(filter, f, x, a3(cons, y, ys)) -> a5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
a5(filtersub, true, f, x, a3(cons, y, ys)) -> a3(cons, y, a4(filter, f, x, ys))
a5(filtersub, false, f, x, a3(cons, y, ys)) -> a4(filter, f, x, ys)
Dependency Pairs for R
A3(high, x, xs) -> A4(filter, gt, x, xs)
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
A4(filter, f, x, a3(cons, y, ys)) -> A5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
A4(filter, f, x, a3(cons, y, ys)) -> A3(f, x, y)
A4(filter, f, x, a3(cons, y, ys)) -> A3(cons, y, ys)
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A3(cons, y, a4(filter, f, x, ys))
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A5(filtersub, false, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(gt)=  a5  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  

2
A5(filtersub, false, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A4(filter, f, x, a3(cons, y, ys)) -> A3(f, x, y)
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A4(filter, f, x, a3(cons, y, ys)) -> A5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
A3(high, x, xs) -> A4(filter, gt, x, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a4(x1, x2, x3, x4))=  0  
  POL(filtersub)=  0  
  POL(A4(x1, x2, x3, x4))=  x2  
  POL(gt)=  0  
  POL(0)=  0  
  POL(A5(x1, x2, x3, x4, x5))=  x3  
  POL(true)=  0  
  POL(nil)=  0  
  POL(a3(x1, x2, x3))=  0  
  POL(false)=  0  
  POL(A3(x1, x2, x3))=  x1  
  POL(cons)=  0  
  POL(a5(x1, x2, x3, x4, x5))=  0  
  POL(s)=  0  
  POL(filter)=  0  
  POL(a2(x1, x2))=  0  
  POL(high)=  1  

Need to check 1 sub cycle of this SCC.

2.1
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A4(filter, f, x, a3(cons, y, ys)) -> A5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
A5(filtersub, false, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
Oriented Rule(s):

a4(filter, f, x, a3(cons, y, ys)) -> a5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
a4(filter, f, x, nil) -> nil
a3(high, x, xs) -> a4(filter, gt, x, xs)
a3(gt, x, 0) -> true
a3(gt, 0, y) -> false
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a5(filtersub, true, f, x, a3(cons, y, ys)) -> a3(cons, y, a4(filter, f, x, ys))
a5(filtersub, false, f, x, a3(cons, y, ys)) -> a4(filter, f, x, ys)

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a4(x1, x2, x3, x4))=  x4  
  POL(filtersub)=  0  
  POL(A4(x1, x2, x3, x4))=  x4  
  POL(gt)=  0  
  POL(0)=  0  
  POL(A5(x1, x2, x3, x4, x5))=  x5  
  POL(true)=  0  
  POL(nil)=  0  
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(false)=  0  
  POL(cons)=  1  
  POL(s)=  0  
  POL(a5(x1, x2, x3, x4, x5))=  x5  
  POL(filter)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(high)=  0  

Termination of R successfully proved!


Duration: 1.77 sec.

File: /usr/home/aoto/ttt/sttrs309.ttt

Term rewriting system R:

[y, x, xs, f, ys]
a3(gt, 0, y) -> false
a3(gt, x, 0) -> true
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a3(high, x, xs) -> a2(a3(filter, gt, x), xs)
a2(a3(filter, f, x), nil) -> nil
a2(a3(filter, f, x), a3(cons, y, ys)) -> a4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
a4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> a3(cons, y, a2(a3(filter, f, x), ys))
a4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> a2(a3(filter, f, x), ys)
Dependency Pairs for R
A2(a3(filter, f, x), a3(cons, y, ys)) -> A4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
A2(a3(filter, f, x), a3(cons, y, ys)) -> A2(filtersub, a3(f, x, y))
A2(a3(filter, f, x), a3(cons, y, ys)) -> A3(f, x, y)
A2(a3(filter, f, x), a3(cons, y, ys)) -> A3(cons, y, ys)
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
A3(high, x, xs) -> A2(a3(filter, gt, x), xs)
A3(high, x, xs) -> A3(filter, gt, x)
A4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> A3(filter, f, x)
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A3(cons, y, a2(a3(filter, f, x), ys))
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A3(filter, f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(gt)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

2
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A3(high, x, xs) -> A2(a3(filter, gt, x), xs)
A2(a3(filter, f, x), a3(cons, y, ys)) -> A3(f, x, y)
A4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A2(a3(filter, f, x), a3(cons, y, ys)) -> A4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
Oriented Rule(s):

a4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> a2(a3(filter, f, x), ys)
a4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> a3(cons, y, a2(a3(filter, f, x), ys))
a3(gt, x, 0) -> true
a3(gt, 0, y) -> false
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a3(high, x, xs) -> a2(a3(filter, gt, x), xs)
a2(a3(filter, f, x), a3(cons, y, ys)) -> a4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
a2(a3(filter, f, x), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x2 + x3  
  POL(false)=  0  
  POL(gt)=  0  
  POL(filter)=  0  
  POL(filtersub)=  0  
  POL(a2(x1, x2))=  x2  
  POL(A2(x1, x2))=  x1  
  POL(A4(x1, x2, x3, x4))=  x2 + x3  
  POL(s)=  0  
  POL(a4(x1, x2, x3, x4))=  x4  
  POL(nil)=  0  
  POL(true)=  0  
  POL(0)=  0  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  x1 + x2  
  POL(high)=  1  

Need to check 1 sub cycle of this SCC.

2.1
A4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A2(a3(filter, f, x), a3(cons, y, ys)) -> A4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
Oriented Rule(s):

a4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> a2(a3(filter, f, x), ys)
a4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> a3(cons, y, a2(a3(filter, f, x), ys))
a3(gt, x, 0) -> true
a3(gt, 0, y) -> false
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a3(high, x, xs) -> a2(a3(filter, gt, x), xs)
a2(a3(filter, f, x), a3(cons, y, ys)) -> a4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
a2(a3(filter, f, x), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(false)=  0  
  POL(gt)=  0  
  POL(filter)=  0  
  POL(filtersub)=  0  
  POL(a2(x1, x2))=  x2  
  POL(A2(x1, x2))=  x2  
  POL(A4(x1, x2, x3, x4))=  x4  
  POL(s)=  0  
  POL(a4(x1, x2, x3, x4))=  x4  
  POL(nil)=  0  
  POL(true)=  0  
  POL(0)=  0  
  POL(cons)=  1  
  POL(high)=  0  

Termination of R successfully proved!


Duration: 2.34 sec.

File: /usr/home/aoto/ttt/sttrs310.ttt

Term rewriting system R:

[y, x, f, ys, xs]
a2(a2(gt, 0), y) -> false
a2(a2(gt, x), 0) -> true
a2(a2(gt, a2(s, x)), a2(s, y)) -> a2(a2(gt, x), y)
a2(a2(a2(filter, f), x), nil) -> nil
a2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(a2(filtersub, a2(a2(f, x), y)), f), x), a2(a2(cons, y), ys))
a2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> a2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
a2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(filter, f), x), ys)
a2(a2(high, x), xs) -> a2(a2(a2(filter, gt), x), xs)
Dependency Pairs for R
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(filter, f)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(a2(filtersub, a2(a2(f, x), y)), f), x), a2(a2(cons, y), ys))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(a2(f, x), y)), f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(a2(f, x), y)), f)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(filtersub, a2(a2(f, x), y))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(high, x), xs) -> A2(a2(a2(filter, gt), x), xs)
A2(a2(high, x), xs) -> A2(a2(filter, gt), x)
A2(a2(high, x), xs) -> A2(filter, gt)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(a2(gt, x), y)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(gt, x)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(filter, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(a2(gt, x), y)
A2(a2(high, x), xs) -> A2(a2(filter, gt), x)
A2(a2(high, x), xs) -> A2(a2(a2(filter, gt), x), xs)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(a2(f, x), y)), f)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(a2(f, x), y)), f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(a2(filtersub, a2(a2(f, x), y)), f), x), a2(a2(cons, y), ys))
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.59 sec.

File: /usr/home/aoto/ttt/sttrs311.ttt

Term rewriting system R:

[y, x, xs, f]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(inc, xs) -> a3(map, a2(plus, a2(s, 0)), xs)
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
Dependency Pairs for R
A2(inc, xs) -> A3(map, a2(plus, a2(s, 0)), xs)
A2(inc, xs) -> A2(plus, a2(s, 0))
A2(inc, xs) -> A2(s, 0)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A2(inc, xs) -> A3(map, a2(plus, a2(s, 0)), xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(nil)=  0  
  POL(map)=  0  
  POL(0)=  0  
  POL(inc)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  0  
  POL(plus)=  0  
  POL(A3(x1, x2, x3))=  x3  
  POL(A2(x1, x2))=  x2  

Need to check 1 sub cycle of this SCC.

1.1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
Oriented Rule(s):

a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
a3(map, f, nil) -> nil
a2(inc, xs) -> a3(map, a2(plus, a2(s, 0)), xs)
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  1  
  POL(a3(x1, x2, x3))=  0  
  POL(nil)=  0  
  POL(map)=  0  
  POL(0)=  0  
  POL(inc)=  0  
  POL(a2(x1, x2))=  x1 + x2  
  POL(cons)=  0  
  POL(plus)=  0  
  POL(A2(x1, x2))=  1 + x1  

Termination of R successfully proved!


Duration: 0.2 sec.

File: /usr/home/aoto/ttt/sttrs312.ttt

Term rewriting system R:

[y, x, f, xs]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a2(a2(cons, x), xs)) -> a2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
inc -> a2(map, a2(plus, a2(s, 0)))
Dependency Pairs for R
INC -> A2(map, a2(plus, a2(s, 0)))
INC -> A2(plus, a2(s, 0))
INC -> A2(s, 0)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(cons, a2(f, x))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(map, f)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.15 sec.

File: /usr/home/aoto/ttt/sttrs313.ttt

Term rewriting system R:

[y, x, f, ys]
a2(a2(neq, 0), 0) -> false
a2(a2(neq, 0), a2(s, y)) -> true
a2(a2(neq, a2(s, x)), 0) -> true
a2(a2(neq, a2(s, x)), a2(s, y)) -> a2(a2(neq, x), y)
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
nonzero -> a2(filter, a2(neq, 0))
Dependency Pairs for R
A2(a2(neq, a2(s, x)), a2(s, y)) -> A2(a2(neq, x), y)
A2(a2(neq, a2(s, x)), a2(s, y)) -> A2(neq, x)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(filtersub, a2(f, y))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(cons, y, ys)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(filter, f)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A3(cons, y, a2(a2(filter, f), ys))
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(filter, f)
NONZERO -> A2(filter, a2(neq, 0))
NONZERO -> A2(neq, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A2(a2(neq, a2(s, x)), a2(s, y)) -> A2(a2(neq, x), y)
Oriented Rule(s):

a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a2(a2(neq, a2(s, x)), 0) -> true
a2(a2(neq, a2(s, x)), a2(s, y)) -> a2(a2(neq, x), y)
a2(a2(neq, 0), a2(s, y)) -> true
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a2(a2(neq, 0), 0) -> false
a2(a2(filter, f), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x2 + x3  
  POL(false)=  0  
  POL(filter)=  0  
  POL(filtersub)=  0  
  POL(a2(x1, x2))=  x1 + x2  
  POL(A2(x1, x2))=  x2  
  POL(neq)=  0  
  POL(s)=  1  
  POL(nil)=  0  
  POL(true)=  0  
  POL(0)=  0  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  x3  

Need to check 1 sub cycle of this SCC.

1.1
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
Oriented Rule(s):

a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a2(a2(neq, a2(s, x)), 0) -> true
a2(a2(neq, a2(s, x)), a2(s, y)) -> a2(a2(neq, x), y)
a2(a2(neq, 0), a2(s, y)) -> true
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a2(a2(neq, 0), 0) -> false
a2(a2(filter, f), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(false)=  0  
  POL(filter)=  1  
  POL(filtersub)=  0  
  POL(a2(x1, x2))=  x1 + x2  
  POL(A2(x1, x2))=  x2  
  POL(neq)=  0  
  POL(s)=  0  
  POL(nil)=  0  
  POL(true)=  0  
  POL(0)=  0  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  x3  

Termination of R successfully proved!


Duration: 5.41 sec.

File: /usr/home/aoto/ttt/sttrs314.ttt

Term rewriting system R:

[y, x, f, ys]
a2(a2(neq, 0), 0) -> false
a2(a2(neq, 0), a2(s, y)) -> true
a2(a2(neq, a2(s, x)), 0) -> true
a2(a2(neq, a2(s, x)), a2(s, y)) -> a2(a2(neq, x), y)
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a2(a2(cons, y), ys)) -> a2(a2(a2(filtersub, a2(f, y)), f), a2(a2(cons, y), ys))
a2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> a2(a2(cons, y), a2(a2(filter, f), ys))
a2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> a2(a2(filter, f), ys)
nonzero -> a2(filter, a2(neq, 0))
Dependency Pairs for R
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(filter, f), ys))
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(filter, f)
A2(a2(neq, a2(s, x)), a2(s, y)) -> A2(a2(neq, x), y)
A2(a2(neq, a2(s, x)), a2(s, y)) -> A2(neq, x)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(f, y)), f), a2(a2(cons, y), ys))
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(f, y)), f)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(filtersub, a2(f, y))
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> A2(filter, f)
NONZERO -> A2(filter, a2(neq, 0))
NONZERO -> A2(neq, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(f, y)), f)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(f, y)), f), a2(a2(cons, y), ys))
A2(a2(neq, a2(s, x)), a2(s, y)) -> A2(a2(neq, x), y)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(filter, f), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.32 sec.

File: /usr/home/aoto/ttt/sttrs315.ttt

Term rewriting system R:

[f, x, y, ys, xs]
a2(a3(foldl, f, x), nil) -> x
a2(a3(foldl, f, x), a3(cons, y, ys)) -> a2(a3(foldl, f, a3(f, x, y)), ys)
a3(snoc, xs, x) -> a3(cons, x, xs)
reverse -> a3(foldl, snoc, nil)
Dependency Pairs for R
A3(snoc, xs, x) -> A3(cons, x, xs)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a3(f, x, y)), ys)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A3(foldl, f, a3(f, x, y))
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A3(f, x, y)
REVERSE -> A3(foldl, snoc, nil)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a3(f, x, y)), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x3  
  POL(foldl)=  0  
  POL(snoc)=  0  
  POL(cons)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.03 sec.

File: /usr/home/aoto/ttt/sttrs316.ttt

Term rewriting system R:

[f, x, y, ys, xs]
a2(a3(foldl, f, x), nil) -> x
a2(a3(foldl, f, x), a3(cons, y, ys)) -> a2(a3(foldl, f, a2(a2(f, x), y)), ys)
a2(a2(snoc, xs), x) -> a3(cons, x, xs)
reverse -> a3(foldl, snoc, nil)
Dependency Pairs for R
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a2(a2(f, x), y)), ys)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a2(f, x), y)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(f, x)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a2(f, x), y)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a2(a2(f, x), y)), ys)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.06 sec.

File: /usr/home/aoto/ttt/sttrs317.ttt

Term rewriting system R:

[f, x, y, ys, xs]
a2(a2(a2(foldl, f), x), nil) -> x
a2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(foldl, f), a2(a2(f, x), y)), ys)
a2(a2(snoc, xs), x) -> a2(a2(cons, x), xs)
reverse -> a2(a2(foldl, snoc), nil)
Dependency Pairs for R
A2(a2(snoc, xs), x) -> A2(a2(cons, x), xs)
A2(a2(snoc, xs), x) -> A2(cons, x)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(foldl, f), a2(a2(f, x), y)), ys)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(foldl, f), a2(a2(f, x), y))
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(foldl, f)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
REVERSE -> A2(a2(foldl, snoc), nil)
REVERSE -> A2(foldl, snoc)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(foldl, f), a2(a2(f, x), y))
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(foldl, f), a2(a2(f, x), y)), ys)
A2(a2(snoc, xs), x) -> A2(a2(cons, x), xs)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.11 sec.

File: /usr/home/aoto/ttt/sttrs318.ttt

Term rewriting system R:

[xs, x, ys, f]
a2(a2(append, nil), xs) -> xs
a2(a2(append, a2(a2(cons, x), xs)), ys) -> a2(a2(cons, x), a2(a2(append, xs), ys))
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a2(a2(cons, x), xs)) -> a2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
a2(sublists, nil) -> a2(a2(cons, nil), nil)
a2(sublists, a2(a2(cons, x), xs)) -> a2(a2(append, a2(a2(map, a2(cons, x)), a2(sublists, xs))), a2(sublists, xs))
Dependency Pairs for R
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(cons, x)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(append, xs)
A2(sublists, a2(a2(cons, x), xs)) -> A2(a2(append, a2(a2(map, a2(cons, x)), a2(sublists, xs))), a2(sublists, xs))
A2(sublists, a2(a2(cons, x), xs)) -> A2(append, a2(a2(map, a2(cons, x)), a2(sublists, xs)))
A2(sublists, a2(a2(cons, x), xs)) -> A2(a2(map, a2(cons, x)), a2(sublists, xs))
A2(sublists, a2(a2(cons, x), xs)) -> A2(map, a2(cons, x))
A2(sublists, a2(a2(cons, x), xs)) -> A2(cons, x)
A2(sublists, a2(a2(cons, x), xs)) -> A2(sublists, xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(cons, a2(f, x))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(map, f)
A2(sublists, nil) -> A2(a2(cons, nil), nil)
A2(sublists, nil) -> A2(cons, nil)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
A2(sublists, nil) -> A2(a2(cons, nil), nil)
A2(sublists, a2(a2(cons, x), xs)) -> A2(sublists, xs)
A2(sublists, a2(a2(cons, x), xs)) -> A2(a2(map, a2(cons, x)), a2(sublists, xs))
A2(sublists, a2(a2(cons, x), xs)) -> A2(a2(append, a2(a2(map, a2(cons, x)), a2(sublists, xs))), a2(sublists, xs))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.3 sec.

File: /usr/home/aoto/ttt/sttrs319.ttt

Term rewriting system R:

[y, x, f, ys]
a3(plus, 0, y) -> y
a3(plus, a2(s, x), y) -> a2(s, a3(plus, x, y))
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a3(f, y, a2(a3(fold, f, x), ys))
sum -> a3(fold, plus, 0)
Dependency Pairs for R
A3(plus, a2(s, x), y) -> A2(s, a3(plus, x, y))
A3(plus, a2(s, x), y) -> A3(plus, x, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A3(f, y, a2(a3(fold, f, x), ys))
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A3(fold, f, x)
SUM -> A3(fold, plus, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(plus, a2(s, x), y) -> A3(plus, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(plus)=  a5  

2
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(fold)=  0  
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(nil)=  0  
  POL(0)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  1  
  POL(plus)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.06 sec.

File: /usr/home/aoto/ttt/sttrs320.ttt

Term rewriting system R:

[y, x, f, ys]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a2(a2(f, y), a2(a3(fold, f, x), ys))
sum -> a3(fold, plus, 0)
Dependency Pairs for R
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a2(f, y), a2(a3(fold, f, x), ys))
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(f, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(f, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a2(f, y), a2(a3(fold, f, x), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.15 sec.

File: /usr/home/aoto/ttt/sttrs321.ttt

Term rewriting system R:

[y, x, f, ys]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> a2(a2(f, y), a2(a2(a2(fold, f), x), ys))
sum -> a2(a2(fold, plus), 0)
Dependency Pairs for R
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(fold, f)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
SUM -> A2(a2(fold, plus), 0)
SUM -> A2(fold, plus)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.16 sec.

File: /usr/home/aoto/ttt/sttrs322.ttt

Term rewriting system R:

[y, x, f, ys]
a3(plus, 0, y) -> y
a3(plus, a2(s, x), y) -> a2(s, a3(plus, x, y))
a2(a3(foldl, f, x), nil) -> x
a2(a3(foldl, f, x), a3(cons, y, ys)) -> a2(a3(foldl, f, a3(f, x, y)), ys)
sum -> a3(foldl, plus, 0)
Dependency Pairs for R
A3(plus, a2(s, x), y) -> A2(s, a3(plus, x, y))
A3(plus, a2(s, x), y) -> A3(plus, x, y)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a3(f, x, y)), ys)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A3(foldl, f, a3(f, x, y))
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A3(f, x, y)
SUM -> A3(foldl, plus, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(plus, a2(s, x), y) -> A3(plus, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(plus)=  a5  

2
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a3(f, x, y)), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(nil)=  0  
  POL(foldl)=  0  
  POL(0)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  1  
  POL(plus)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.06 sec.

File: /usr/home/aoto/ttt/sttrs323.ttt

Term rewriting system R:

[y, x, f, ys]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a3(foldl, f, x), nil) -> x
a2(a3(foldl, f, x), a3(cons, y, ys)) -> a2(a3(foldl, f, a2(a2(f, x), y)), ys)
sum -> a3(foldl, plus, 0)
Dependency Pairs for R
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a2(a2(f, x), y)), ys)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a2(f, x), y)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(f, x)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a2(f, x), y)
A2(a3(foldl, f, x), a3(cons, y, ys)) -> A2(a3(foldl, f, a2(a2(f, x), y)), ys)
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.08 sec.

File: /usr/home/aoto/ttt/sttrs324.ttt

Term rewriting system R:

[y, x, f, ys]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(a2(foldl, f), x), nil) -> x
a2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(foldl, f), a2(a2(f, x), y)), ys)
sum -> a2(a2(foldl, plus), 0)
Dependency Pairs for R
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(foldl, f), a2(a2(f, x), y)), ys)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(foldl, f), a2(a2(f, x), y))
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(foldl, f)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
SUM -> A2(a2(foldl, plus), 0)
SUM -> A2(foldl, plus)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(foldl, f), a2(a2(f, x), y))
A2(a2(a2(foldl, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(foldl, f), a2(a2(f, x), y)), ys)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.12 sec.

File: /usr/home/aoto/ttt/sttrs325.ttt

Term rewriting system R:

[x, y]
a2(id, x) -> x
a2(plus, 0) -> id
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(times, 0), y) -> 0
a2(a2(times, a2(s, x)), y) -> a2(a2(plus, a2(a2(times, x), y)), y)
Dependency Pairs for R
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(times, a2(s, x)), y) -> A2(plus, a2(a2(times, x), y))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(times, x)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.07 sec.

File: /usr/home/aoto/ttt/sttrs326.ttt

Term rewriting system R:

[f, x, xs, g]
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
a2(a3(comp, f, g), x) -> a2(f, a2(g, x))
a2(twice, f) -> a3(comp, f, f)
Dependency Pairs for R
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(twice, f) -> A3(comp, f, f)
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
A2(a3(comp, f, g), x) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A2(a3(comp, f, g), x) -> A2(g, x)
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A2(x1, x2))=  x1  
  POL(map)=  0  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(comp)=  1  
  POL(a2(x1, x2))=  0  
  POL(cons)=  0  
  POL(nil)=  0  
  POL(twice)=  0  

2
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(cons)=  1  

Termination of R successfully proved!


Duration: 0.06 sec.

File: /usr/home/aoto/ttt/sttrs327.ttt

Term rewriting system R:

[f, x, xs, g]
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a3(cons, x, xs)) -> a3(cons, a2(f, x), a2(a2(map, f), xs))
a2(a3(comp, f, g), x) -> a2(f, a2(g, x))
a2(twice, f) -> a3(comp, f, f)
Dependency Pairs for R
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(map, f)
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
A2(a3(comp, f, g), x) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(comp, f, g), x) -> A2(g, x)
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.13 sec.

File: /usr/home/aoto/ttt/sttrs328.ttt

Term rewriting system R:

[f, x, xs, g]
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a2(a2(cons, x), xs)) -> a2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
a2(a2(a2(comp, f), g), x) -> a2(f, a2(g, x))
a2(twice, f) -> a2(a2(comp, f), f)
Dependency Pairs for R
A2(twice, f) -> A2(a2(comp, f), f)
A2(twice, f) -> A2(comp, f)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(cons, a2(f, x))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(map, f)
A2(a2(a2(comp, f), g), x) -> A2(f, a2(g, x))
A2(a2(a2(comp, f), g), x) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(comp, f), g), x) -> A2(g, x)
A2(a2(a2(comp, f), g), x) -> A2(f, a2(g, x))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
A2(twice, f) -> A2(a2(comp, f), f)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.13 sec.

File: /usr/home/aoto/ttt/sttrs401.ttt

Term rewriting system R:

[xs, x, ys, f, g, y]
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
a3(cartprod, nil, ys) -> nil
a3(cartprod, a3(cons, x, xs), ys) -> a3(append, a3(map, a2(a2(curry, pair), x), ys), a3(cartprod, xs, ys))
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
Dependency Pairs for R
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A3(cartprod, a3(cons, x, xs), ys) -> A3(append, a3(map, a2(a2(curry, pair), x), ys), a3(cartprod, xs, ys))
A3(cartprod, a3(cons, x, xs), ys) -> A3(map, a2(a2(curry, pair), x), ys)
A3(cartprod, a3(cons, x, xs), ys) -> A2(a2(curry, pair), x)
A3(cartprod, a3(cons, x, xs), ys) -> A2(curry, pair)
A3(cartprod, a3(cons, x, xs), ys) -> A3(cartprod, xs, ys)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  1  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

2
A3(cartprod, a3(cons, x, xs), ys) -> A3(cartprod, xs, ys)
A3(cartprod, a3(cons, x, xs), ys) -> A2(a2(curry, pair), x)
A3(cartprod, a3(cons, x, xs), ys) -> A3(map, a2(a2(curry, pair), x), ys)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.28 sec.

File: /usr/home/aoto/ttt/sttrs402.ttt

Term rewriting system R:

[y, x, f, xs]
a3(plus, 0, y) -> y
a3(plus, a2(s, x), y) -> a2(s, a3(plus, x, y))
a3(times, 0, y) -> 0
a3(times, a2(s, x), y) -> a3(plus, a3(times, x, y), y)
a4(rec, f, x, 0) -> x
a4(rec, f, x, a2(s, y)) -> a3(f, a2(s, y), a4(rec, f, x, y))
a2(fact, xs) -> a4(rec, times, a2(s, 0), xs)
Dependency Pairs for R
A3(times, a2(s, x), y) -> A3(plus, a3(times, x, y), y)
A3(times, a2(s, x), y) -> A3(times, x, y)
A3(plus, a2(s, x), y) -> A2(s, a3(plus, x, y))
A3(plus, a2(s, x), y) -> A3(plus, x, y)
A2(fact, xs) -> A4(rec, times, a2(s, 0), xs)
A2(fact, xs) -> A2(s, 0)
A4(rec, f, x, a2(s, y)) -> A3(f, a2(s, y), a4(rec, f, x, y))
A4(rec, f, x, a2(s, y)) -> A2(s, y)
A4(rec, f, x, a2(s, y)) -> A4(rec, f, x, y)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 3 SCCs:

1
A3(plus, a2(s, x), y) -> A3(plus, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(s)=  0  
  POL(plus)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  

2
A3(times, a2(s, x), y) -> A3(times, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(times)=  a5  
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  

3
A4(rec, f, x, a2(s, y)) -> A4(rec, f, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A4(x1, x2, x3, x4))=  a1 + x1 + x2 + x3 + x4  
  POL(rec)=  a6  
  POL(s)=  1  
  POL(a2(x1, x2))=  x1 + x2  

Termination of R successfully proved!


Duration: 0.05 sec.

File: /usr/home/aoto/ttt/sttrs403.ttt

Term rewriting system R:

[y, x, f]
a3(plus, 0, y) -> y
a3(plus, a2(s, x), y) -> a2(s, a3(plus, x, y))
a3(times, 0, y) -> 0
a3(times, a2(s, x), y) -> a3(plus, a3(times, x, y), y)
a2(a3(rec, f, x), 0) -> x
a2(a3(rec, f, x), a2(s, y)) -> a3(f, a2(s, y), a2(a3(rec, f, x), y))
fact -> a3(rec, times, a2(s, 0))
Dependency Pairs for R
A3(times, a2(s, x), y) -> A3(plus, a3(times, x, y), y)
A3(times, a2(s, x), y) -> A3(times, x, y)
A3(plus, a2(s, x), y) -> A2(s, a3(plus, x, y))
A3(plus, a2(s, x), y) -> A3(plus, x, y)
A2(a3(rec, f, x), a2(s, y)) -> A3(f, a2(s, y), a2(a3(rec, f, x), y))
A2(a3(rec, f, x), a2(s, y)) -> A2(s, y)
A2(a3(rec, f, x), a2(s, y)) -> A2(a3(rec, f, x), y)
A2(a3(rec, f, x), a2(s, y)) -> A3(rec, f, x)
FACT -> A3(rec, times, a2(s, 0))
FACT -> A2(s, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 3 SCCs:

1
A3(plus, a2(s, x), y) -> A3(plus, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(plus)=  a5  

2
A3(times, a2(s, x), y) -> A3(times, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(times)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

3
A2(a3(rec, f, x), a2(s, y)) -> A2(a3(rec, f, x), y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a3(x1, x2, x3))=  0  
  POL(rec)=  0  
  POL(0)=  0  
  POL(times)=  0  
  POL(a2(x1, x2))=  1 + x2  
  POL(plus)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.09 sec.

File: /usr/home/aoto/ttt/sttrs404.ttt

Term rewriting system R:

[y, x, f]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(times, 0), y) -> 0
a2(a2(times, a2(s, x)), y) -> a2(a2(plus, a2(a2(times, x), y)), y)
a2(a3(rec, f, x), 0) -> x
a2(a3(rec, f, x), a2(s, y)) -> a2(a2(f, a2(s, y)), a2(a3(rec, f, x), y))
fact -> a3(rec, times, a2(s, 0))
Dependency Pairs for R
A2(a3(rec, f, x), a2(s, y)) -> A2(a2(f, a2(s, y)), a2(a3(rec, f, x), y))
A2(a3(rec, f, x), a2(s, y)) -> A2(f, a2(s, y))
A2(a3(rec, f, x), a2(s, y)) -> A2(s, y)
A2(a3(rec, f, x), a2(s, y)) -> A2(a3(rec, f, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(times, a2(s, x)), y) -> A2(plus, a2(a2(times, x), y))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(times, x)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
FACT -> A2(s, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a3(rec, f, x), a2(s, y)) -> A2(a3(rec, f, x), y)
A2(a3(rec, f, x), a2(s, y)) -> A2(f, a2(s, y))
A2(a3(rec, f, x), a2(s, y)) -> A2(a2(f, a2(s, y)), a2(a3(rec, f, x), y))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.26 sec.

File: /usr/home/aoto/ttt/sttrs405.ttt

Term rewriting system R:

[y, x, f]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(times, 0), y) -> 0
a2(a2(times, a2(s, x)), y) -> a2(a2(plus, a2(a2(times, x), y)), y)
a2(a2(a2(rec, f), x), 0) -> x
a2(a2(a2(rec, f), x), a2(s, y)) -> a2(a2(f, a2(s, y)), a2(a2(a2(rec, f), x), y))
fact -> a2(a2(rec, times), a2(s, 0))
Dependency Pairs for R
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(f, a2(s, y)), a2(a2(a2(rec, f), x), y))
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(f, a2(s, y))
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(s, y)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(a2(rec, f), x), y)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(rec, f), x)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(rec, f)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(times, a2(s, x)), y) -> A2(plus, a2(a2(times, x), y))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(times, x)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
FACT -> A2(a2(rec, times), a2(s, 0))
FACT -> A2(rec, times)
FACT -> A2(s, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(rec, f), x)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(a2(rec, f), x), y)
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(f, a2(s, y))
A2(a2(a2(rec, f), x), a2(s, y)) -> A2(a2(f, a2(s, y)), a2(a2(a2(rec, f), x), y))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.23 sec.

File: /usr/home/aoto/ttt/sttrs406.ttt

Term rewriting system R:

[p, x, xs]
a3(and, true, true) -> true
a3(and, true, false) -> false
a3(and, false, true) -> false
a3(and, false, false) -> false
a3(or, true, true) -> true
a3(or, true, false) -> true
a3(or, false, true) -> true
a3(or, false, false) -> false
a3(forall, p, nil) -> true
a3(forall, p, a3(cons, x, xs)) -> a3(and, a2(p, x), a3(forall, p, xs))
a3(forsome, p, nil) -> false
a3(forsome, p, a3(cons, x, xs)) -> a3(or, a2(p, x), a3(forsome, p, xs))
Dependency Pairs for R
A3(forsome, p, a3(cons, x, xs)) -> A3(or, a2(p, x), a3(forsome, p, xs))
A3(forsome, p, a3(cons, x, xs)) -> A3(forsome, p, xs)
A3(forall, p, a3(cons, x, xs)) -> A3(and, a2(p, x), a3(forall, p, xs))
A3(forall, p, a3(cons, x, xs)) -> A3(forall, p, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(forsome, p, a3(cons, x, xs)) -> A3(forsome, p, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(cons)=  1  
  POL(forsome)=  a5  

2
A3(forall, p, a3(cons, x, xs)) -> A3(forall, p, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(forall)=  a5  
  POL(cons)=  1  

Termination of R successfully proved!


Duration: 0.07 sec.

File: /usr/home/aoto/ttt/sttrs407.ttt

Term rewriting system R:

[x, y, p, xs]
a2(a2(and, true), true) -> true
a2(a2(and, x), false) -> false
a2(a2(and, false), y) -> false
a2(a2(or, true), y) -> true
a2(a2(or, x), true) -> true
a2(a2(or, false), false) -> false
a2(a2(forall, p), nil) -> true
a2(a2(forall, p), a3(cons, x, xs)) -> a2(a2(and, a2(p, x)), a2(a2(forall, p), xs))
a2(a2(forsome, p), nil) -> false
a2(a2(forsome, p), a3(cons, x, xs)) -> a2(a2(or, a2(p, x)), a2(a2(forsome, p), xs))
Dependency Pairs for R
A2(a2(forsome, p), a3(cons, x, xs)) -> A2(a2(or, a2(p, x)), a2(a2(forsome, p), xs))
A2(a2(forsome, p), a3(cons, x, xs)) -> A2(or, a2(p, x))
A2(a2(forsome, p), a3(cons, x, xs)) -> A2(p, x)
A2(a2(forsome, p), a3(cons, x, xs)) -> A2(a2(forsome, p), xs)
A2(a2(forsome, p), a3(cons, x, xs)) -> A2(forsome, p)
A2(a2(forall, p), a3(cons, x, xs)) -> A2(a2(and, a2(p, x)), a2(a2(forall, p), xs))
A2(a2(forall, p), a3(cons, x, xs)) -> A2(and, a2(p, x))
A2(a2(forall, p), a3(cons, x, xs)) -> A2(p, x)
A2(a2(forall, p), a3(cons, x, xs)) -> A2(a2(forall, p), xs)
A2(a2(forall, p), a3(cons, x, xs)) -> A2(forall, p)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(forall, p), a3(cons, x, xs)) -> A2(a2(forall, p), xs)
A2(a2(forall, p), a3(cons, x, xs)) -> A2(p, x)
A2(a2(forsome, p), a3(cons, x, xs)) -> A2(a2(forsome, p), xs)
A2(a2(forsome, p), a3(cons, x, xs)) -> A2(p, x)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(false)=  0  
  POL(nil)=  0  
  POL(true)=  0  
  POL(or)=  0  
  POL(and)=  0  
  POL(forall)=  0  
  POL(a2(x1, x2))=  0  
  POL(forsome)=  0  
  POL(cons)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.14 sec.

File: /usr/home/aoto/ttt/sttrs408.ttt

Term rewriting system R:

[y, x, f, xs, g]
a3(plus, 0, y) -> y
a3(plus, a2(s, x), y) -> a2(s, a3(plus, x, y))
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a3(cons, x, xs)) -> a3(cons, a2(f, x), a2(a2(map, f), xs))
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
inc -> a2(map, a2(a2(curry, plus), a2(s, 0)))
Dependency Pairs for R
A3(plus, a2(s, x), y) -> A2(s, a3(plus, x, y))
A3(plus, a2(s, x), y) -> A3(plus, x, y)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A2(a2(map, f), a3(cons, x, xs)) -> A3(cons, a2(f, x), a2(a2(map, f), xs))
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(map, f)
INC -> A2(map, a2(a2(curry, plus), a2(s, 0)))
INC -> A2(a2(curry, plus), a2(s, 0))
INC -> A2(curry, plus)
INC -> A2(s, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(plus, a2(s, x), y) -> A3(plus, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(plus)=  a5  

2
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(map)=  0  
  POL(curry)=  0  
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(nil)=  0  
  POL(0)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  0  
  POL(plus)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.09 sec.

File: /usr/home/aoto/ttt/sttrs409.ttt

Term rewriting system R:

[y, x, f, ys, g]
a3(neq, 0, 0) -> false
a3(neq, 0, a2(s, y)) -> true
a3(neq, a2(s, x), 0) -> true
a3(neq, a2(s, x), a2(s, y)) -> a3(neq, x, y)
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
nonzero -> a2(filter, a2(a2(curry, neq), 0))
Dependency Pairs for R
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(filter, f)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A3(cons, y, a2(a2(filter, f), ys))
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(filter, f)
A3(neq, a2(s, x), a2(s, y)) -> A3(neq, x, y)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(filtersub, a2(f, y))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(cons, y, ys)
NONZERO -> A2(filter, a2(a2(curry, neq), 0))
NONZERO -> A2(a2(curry, neq), 0)
NONZERO -> A2(curry, neq)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(neq, a2(s, x), a2(s, y)) -> A3(neq, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(neq)=  a5  
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

2
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
Oriented Rule(s):

a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
a3(neq, a2(s, x), 0) -> true
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a3(neq, 0, a2(s, y)) -> true
a3(neq, 0, 0) -> false
a3(neq, a2(s, x), a2(s, y)) -> a3(neq, x, y)
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a2(a2(filter, f), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x3  
  POL(curry)=  0  
  POL(false)=  0  
  POL(filter)=  0  
  POL(filtersub)=  0  
  POL(a2(x1, x2))=  1 + x2  
  POL(A2(x1, x2))=  x1  
  POL(neq)=  0  
  POL(s)=  0  
  POL(nil)=  0  
  POL(true)=  0  
  POL(0)=  0  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  1 + x2  

Need to check 1 sub cycle of this SCC.

2.1
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
Oriented Rule(s):

a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
a3(neq, a2(s, x), 0) -> true
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a3(neq, 0, a2(s, y)) -> true
a3(neq, 0, 0) -> false
a3(neq, a2(s, x), a2(s, y)) -> a3(neq, x, y)
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a2(a2(filter, f), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(curry)=  1  
  POL(a3(x1, x2, x3))=  0  
  POL(false)=  0  
  POL(filter)=  0  
  POL(filtersub)=  0  
  POL(a2(x1, x2))=  x1  
  POL(A2(x1, x2))=  x1  
  POL(neq)=  0  
  POL(s)=  0  
  POL(nil)=  0  
  POL(true)=  0  
  POL(0)=  0  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  0  

Need to check 1 sub cycle of this SCC.

2.1.1
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
Oriented Rule(s):

a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
a3(neq, a2(s, x), 0) -> true
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a3(neq, 0, a2(s, y)) -> true
a3(neq, 0, 0) -> false
a3(neq, a2(s, x), a2(s, y)) -> a3(neq, x, y)
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a2(a2(filter, f), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(curry)=  1  
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(false)=  0  
  POL(filter)=  1  
  POL(filtersub)=  0  
  POL(a2(x1, x2))=  x1 + x2  
  POL(A2(x1, x2))=  x2  
  POL(neq)=  0  
  POL(s)=  0  
  POL(nil)=  0  
  POL(true)=  0  
  POL(0)=  0  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  x3  

Termination of R successfully proved!


Duration: 4.12 sec.

File: /usr/home/aoto/ttt/sttrs410.ttt

Term rewriting system R:

[xs, x, ys, f, g, y]
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
a2(sublists, nil) -> a3(cons, nil, nil)
a2(sublists, a3(cons, x, xs)) -> a3(append, a3(map, a2(a2(curry, cons), x), a2(sublists, xs)), a2(sublists, xs))
Dependency Pairs for R
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A2(sublists, nil) -> A3(cons, nil, nil)
A2(sublists, a3(cons, x, xs)) -> A3(append, a3(map, a2(a2(curry, cons), x), a2(sublists, xs)), a2(sublists, xs))
A2(sublists, a3(cons, x, xs)) -> A3(map, a2(a2(curry, cons), x), a2(sublists, xs))
A2(sublists, a3(cons, x, xs)) -> A2(a2(curry, cons), x)
A2(sublists, a3(cons, x, xs)) -> A2(curry, cons)
A2(sublists, a3(cons, x, xs)) -> A2(sublists, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  1  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

2
A2(sublists, a3(cons, x, xs)) -> A2(sublists, xs)
A2(sublists, a3(cons, x, xs)) -> A2(a2(curry, cons), x)
A2(sublists, a3(cons, x, xs)) -> A3(map, a2(a2(curry, cons), x), a2(sublists, xs))
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 3.33 sec.

File: /usr/home/aoto/ttt/sttrs411.ttt

Term rewriting system R:

[y, x, f, g]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(times, 0), y) -> 0
a2(a2(times, a2(s, x)), y) -> a2(a2(plus, a2(a2(times, x), y)), y)
a2(a3(comp, f, g), x) -> a2(f, a2(g, x))
a2(twice, f) -> a3(comp, f, f)
Dependency Pairs for R
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(times, a2(s, x)), y) -> A2(plus, a2(a2(times, x), y))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(times, x)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
A2(a3(comp, f, g), x) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a3(comp, f, g), x) -> A2(g, x)
A2(a3(comp, f, g), x) -> A2(f, a2(g, x))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.14 sec.

File: /usr/home/aoto/ttt/sttrs412.ttt

Term rewriting system R:

[x, xs, y, ys, f, g]
a3(zip, nil, nil) -> nil
a3(zip, a3(cons, x, xs), nil) -> nil
a3(zip, nil, a3(cons, y, ys)) -> nil
a3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> a3(cons, a3(pair, x, y), a3(zip, xs, ys))
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
a2(a2(applypair, g), a3(pair, x, y)) -> a3(g, x, y)
a4(zipwith, g, xs, ys) -> a3(map, a2(applypair, g), a3(zip, xs, ys))
Dependency Pairs for R
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(cons, a3(pair, x, y), a3(zip, xs, ys))
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(pair, x, y)
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(zip, xs, ys)
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(a2(applypair, g), a3(pair, x, y)) -> A3(g, x, y)
A4(zipwith, g, xs, ys) -> A3(map, a2(applypair, g), a3(zip, xs, ys))
A4(zipwith, g, xs, ys) -> A2(applypair, g)
A4(zipwith, g, xs, ys) -> A3(zip, xs, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(zip, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(zip)=  a5  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

2
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(a2(applypair, g), a3(pair, x, y)) -> A3(g, x, y)
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  0  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(applypair)=  0  
  POL(pair)=  0  
  POL(cons)=  1  
  POL(a2(x1, x2))=  x1 + x2  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A2(x1, x2))=  x1 + x2  

Termination of R successfully proved!


Duration: 0.08 sec.

File: /usr/home/aoto/ttt/sttrs413.ttt

Term rewriting system R:

[x, xs, y, ys, g, f]
a3(zip, nil, nil) -> nil
a3(zip, a3(cons, x, xs), nil) -> nil
a3(zip, nil, a3(cons, y, ys)) -> nil
a3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> a3(cons, a3(pair, x, y), a3(zip, xs, ys))
a3(a2(zipwith, g), xs, ys) -> a2(a2(map, a2(applypair, g)), a3(zip, xs, ys))
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a3(cons, x, xs)) -> a3(cons, a2(f, x), a2(a2(map, f), xs))
a2(a2(applypair, g), a3(pair, x, y)) -> a3(g, x, y)
Dependency Pairs for R
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(cons, a3(pair, x, y), a3(zip, xs, ys))
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(pair, x, y)
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(zip, xs, ys)
A3(a2(zipwith, g), xs, ys) -> A2(a2(map, a2(applypair, g)), a3(zip, xs, ys))
A3(a2(zipwith, g), xs, ys) -> A2(map, a2(applypair, g))
A3(a2(zipwith, g), xs, ys) -> A2(applypair, g)
A3(a2(zipwith, g), xs, ys) -> A3(zip, xs, ys)
A2(a2(map, f), a3(cons, x, xs)) -> A3(cons, a2(f, x), a2(a2(map, f), xs))
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(map, f)
A2(a2(applypair, g), a3(pair, x, y)) -> A3(g, x, y)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(zip, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(zip)=  a5  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

2
A2(a2(applypair, g), a3(pair, x, y)) -> A3(g, x, y)
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
A3(a2(zipwith, g), xs, ys) -> A2(a2(map, a2(applypair, g)), a3(zip, xs, ys))
Oriented Rule(s):

a3(zip, a3(cons, x, xs), nil) -> nil
a3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> a3(cons, a3(pair, x, y), a3(zip, xs, ys))
a3(zip, nil, a3(cons, y, ys)) -> nil
a3(zip, nil, nil) -> nil
a3(a2(zipwith, g), xs, ys) -> a2(a2(map, a2(applypair, g)), a3(zip, xs, ys))
a2(a2(map, f), a3(cons, x, xs)) -> a3(cons, a2(f, x), a2(a2(map, f), xs))
a2(a2(applypair, g), a3(pair, x, y)) -> a3(g, x, y)
a2(a2(map, f), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  0  
  POL(zipwith)=  1  
  POL(a3(x1, x2, x3))=  x1  
  POL(applypair)=  0  
  POL(nil)=  0  
  POL(zip)=  0  
  POL(pair)=  0  
  POL(a2(x1, x2))=  x1 + x2  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  x1  
  POL(A2(x1, x2))=  x1  

Need to check 1 sub cycle of this SCC.

2.1
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
Oriented Rule(s):

a3(zip, a3(cons, x, xs), nil) -> nil
a3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> a3(cons, a3(pair, x, y), a3(zip, xs, ys))
a3(zip, nil, a3(cons, y, ys)) -> nil
a3(zip, nil, nil) -> nil
a3(a2(zipwith, g), xs, ys) -> a2(a2(map, a2(applypair, g)), a3(zip, xs, ys))
a2(a2(map, f), a3(cons, x, xs)) -> a3(cons, a2(f, x), a2(a2(map, f), xs))
a2(a2(applypair, g), a3(pair, x, y)) -> a3(g, x, y)
a2(a2(map, f), nil) -> nil

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  1  
  POL(zipwith)=  1  
  POL(a3(x1, x2, x3))=  x1  
  POL(applypair)=  0  
  POL(nil)=  0  
  POL(zip)=  0  
  POL(pair)=  0  
  POL(a2(x1, x2))=  x1 + x2  
  POL(cons)=  0  
  POL(A2(x1, x2))=  1 + x1  

Need to check 1 sub cycle of this SCC.

2.1.1
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  0  
  POL(zipwith)=  0  
  POL(a3(x1, x2, x3))=  1 + x3  
  POL(applypair)=  0  
  POL(nil)=  0  
  POL(zip)=  0  
  POL(pair)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.47 sec.

File: /usr/home/aoto/ttt/sttrs414.ttt

Term rewriting system R:

[x, xs, y, ys, f, g]
a3(zip, nil, nil) -> nil
a3(zip, a3(cons, x, xs), nil) -> nil
a3(zip, nil, a3(cons, y, ys)) -> nil
a3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> a3(cons, a3(pair, x, y), a3(zip, xs, ys))
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
a2(a2(uncurry, g), a3(pair, x, y)) -> a2(a2(g, x), y)
a4(zipwith, g, xs, ys) -> a3(map, a2(uncurry, g), a3(zip, xs, ys))
Dependency Pairs for R
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(cons, a3(pair, x, y), a3(zip, xs, ys))
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(pair, x, y)
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(zip, xs, ys)
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A4(zipwith, g, xs, ys) -> A3(map, a2(uncurry, g), a3(zip, xs, ys))
A4(zipwith, g, xs, ys) -> A2(uncurry, g)
A4(zipwith, g, xs, ys) -> A3(zip, xs, ys)
A2(a2(uncurry, g), a3(pair, x, y)) -> A2(a2(g, x), y)
A2(a2(uncurry, g), a3(pair, x, y)) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 3 SCCs:

1
A2(a2(uncurry, g), a3(pair, x, y)) -> A2(g, x)
A2(a2(uncurry, g), a3(pair, x, y)) -> A2(a2(g, x), y)
Oriented Rule(s):

a2(a2(uncurry, g), a3(pair, x, y)) -> a2(a2(g, x), y)

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A2(x1, x2))=  1 + x1 + x2  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(uncurry)=  0  
  POL(pair)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  

Need to check 1 sub cycle of this SCC.

1.1
A2(a2(uncurry, g), a3(pair, x, y)) -> A2(a2(g, x), y)
Oriented Rule(s):

a2(a2(uncurry, g), a3(pair, x, y)) -> a2(a2(g, x), y)

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(A2(x1, x2))=  x1 + x2  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(uncurry)=  1  
  POL(pair)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  

2
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(zip, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(zip)=  a5  
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  0  

3
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(cons)=  1  

Termination of R successfully proved!


Duration: 0.09 sec.

File: /usr/home/aoto/ttt/sttrs415.ttt

Term rewriting system R:

[x, xs, y, ys, g, f]
a3(zip, nil, nil) -> nil
a3(zip, a3(cons, x, xs), nil) -> nil
a3(zip, nil, a3(cons, y, ys)) -> nil
a3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> a3(cons, a3(pair, x, y), a3(zip, xs, ys))
a3(a2(zipwith, g), xs, ys) -> a2(a2(map, a2(uncurry, g)), a3(zip, xs, ys))
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a3(cons, x, xs)) -> a3(cons, a2(f, x), a2(a2(map, f), xs))
a2(a2(uncurry, g), a3(pair, x, y)) -> a2(a2(g, x), y)
Dependency Pairs for R
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(cons, a3(pair, x, y), a3(zip, xs, ys))
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(pair, x, y)
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(zip, xs, ys)
A3(a2(zipwith, g), xs, ys) -> A2(a2(map, a2(uncurry, g)), a3(zip, xs, ys))
A3(a2(zipwith, g), xs, ys) -> A2(map, a2(uncurry, g))
A3(a2(zipwith, g), xs, ys) -> A2(uncurry, g)
A3(a2(zipwith, g), xs, ys) -> A3(zip, xs, ys)
A2(a2(map, f), a3(cons, x, xs)) -> A3(cons, a2(f, x), a2(a2(map, f), xs))
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(map, f)
A2(a2(uncurry, g), a3(pair, x, y)) -> A2(a2(g, x), y)
A2(a2(uncurry, g), a3(pair, x, y)) -> A2(g, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(zip, a3(cons, x, xs), a3(cons, y, ys)) -> A3(zip, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(zip)=  a5  
  POL(cons)=  0  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

2
A2(a2(uncurry, g), a3(pair, x, y)) -> A2(g, x)
A2(a2(uncurry, g), a3(pair, x, y)) -> A2(a2(g, x), y)
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(map)=  0  
  POL(zipwith)=  0  
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(nil)=  0  
  POL(zip)=  0  
  POL(pair)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  0  
  POL(uncurry)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.18 sec.

File: /usr/home/aoto/ttt/sttrs501.ttt

Term rewriting system R:

[y, x, xs, f]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(times, 0), y) -> 0
a2(a2(times, a2(s, x)), y) -> a2(a2(plus, a2(a2(times, x), y)), y)
a2(inc, xs) -> a3(map, a2(plus, a2(s, 0)), xs)
a2(double, xs) -> a3(map, a2(times, a2(s, a2(s, 0))), xs)
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
Dependency Pairs for R
A2(inc, xs) -> A3(map, a2(plus, a2(s, 0)), xs)
A2(inc, xs) -> A2(plus, a2(s, 0))
A2(inc, xs) -> A2(s, 0)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(times, a2(s, x)), y) -> A2(plus, a2(a2(times, x), y))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(times, x)
A2(double, xs) -> A3(map, a2(times, a2(s, a2(s, 0))), xs)
A2(double, xs) -> A2(times, a2(s, a2(s, 0)))
A2(double, xs) -> A2(s, a2(s, 0))
A2(double, xs) -> A2(s, 0)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(double, xs) -> A3(map, a2(times, a2(s, a2(s, 0))), xs)
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A2(inc, xs) -> A3(map, a2(plus, a2(s, 0)), xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(map)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  0  
  POL(plus)=  0  
  POL(double)=  0  
  POL(A2(x1, x2))=  x2  
  POL(s)=  0  
  POL(nil)=  0  
  POL(0)=  0  
  POL(inc)=  0  
  POL(times)=  0  
  POL(A3(x1, x2, x3))=  x3  

Need to check 1 sub cycle of this SCC.

1.1
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.42 sec.

File: /usr/home/aoto/ttt/sttrs502.ttt

Term rewriting system R:

[y, x, f, xs]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(times, 0), y) -> 0
a2(a2(times, a2(s, x)), y) -> a2(a2(plus, a2(a2(times, x), y)), y)
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a2(a2(cons, x), xs)) -> a2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
inc -> a2(map, a2(plus, a2(s, 0)))
double -> a2(map, a2(times, a2(s, a2(s, 0))))
Dependency Pairs for R
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(cons, a2(f, x))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(map, f)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(times, a2(s, x)), y) -> A2(plus, a2(a2(times, x), y))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(times, x)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
INC -> A2(map, a2(plus, a2(s, 0)))
INC -> A2(plus, a2(s, 0))
INC -> A2(s, 0)
DOUBLE -> A2(map, a2(times, a2(s, a2(s, 0))))
DOUBLE -> A2(times, a2(s, a2(s, 0)))
DOUBLE -> A2(s, a2(s, 0))
DOUBLE -> A2(s, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.2 sec.

File: /usr/home/aoto/ttt/sttrs504.ttt

Term rewriting system R:

[xs, x, ys, f, y]
a2(a2(append, nil), xs) -> xs
a2(a2(append, a2(a2(cons, x), xs)), ys) -> a2(a2(cons, x), a2(a2(append, xs), ys))
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a2(a2(cons, x), xs)) -> a2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
a2(a2(ins, x), nil) -> a2(a2(cons, a2(a2(cons, x), nil)), nil)
a2(a2(ins, x), a2(a2(cons, y), ys)) -> a2(a2(cons, a2(a2(cons, x), a2(a2(cons, y), ys))), a2(a2(map, a2(cons, y)), a2(a2(ins, x), ys)))
a2(a2(insall, x), nil) -> nil
a2(a2(insall, x), a2(a2(cons, y), ys)) -> a2(a2(append, a2(a2(ins, x), y)), a2(a2(insall, x), ys))
a2(perm, nil) -> a2(a2(cons, nil), nil)
a2(perm, a2(a2(cons, x), xs)) -> a2(a2(insall, x), a2(perm, xs))
Dependency Pairs for R
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(cons, x)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(append, xs)
A2(perm, a2(a2(cons, x), xs)) -> A2(a2(insall, x), a2(perm, xs))
A2(perm, a2(a2(cons, x), xs)) -> A2(insall, x)
A2(perm, a2(a2(cons, x), xs)) -> A2(perm, xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(cons, a2(f, x))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(map, f)
A2(perm, nil) -> A2(a2(cons, nil), nil)
A2(perm, nil) -> A2(cons, nil)
A2(a2(ins, x), nil) -> A2(a2(cons, a2(a2(cons, x), nil)), nil)
A2(a2(ins, x), nil) -> A2(cons, a2(a2(cons, x), nil))
A2(a2(ins, x), nil) -> A2(a2(cons, x), nil)
A2(a2(ins, x), nil) -> A2(cons, x)
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(cons, a2(a2(cons, x), a2(a2(cons, y), ys))), a2(a2(map, a2(cons, y)), a2(a2(ins, x), ys)))
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(cons, a2(a2(cons, x), a2(a2(cons, y), ys)))
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(cons, x), a2(a2(cons, y), ys))
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(cons, x)
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(map, a2(cons, y)), a2(a2(ins, x), ys))
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(map, a2(cons, y))
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(ins, x), ys)
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(ins, x)
A2(a2(insall, x), a2(a2(cons, y), ys)) -> A2(a2(append, a2(a2(ins, x), y)), a2(a2(insall, x), ys))
A2(a2(insall, x), a2(a2(cons, y), ys)) -> A2(append, a2(a2(ins, x), y))
A2(a2(insall, x), a2(a2(cons, y), ys)) -> A2(a2(ins, x), y)
A2(a2(insall, x), a2(a2(cons, y), ys)) -> A2(ins, x)
A2(a2(insall, x), a2(a2(cons, y), ys)) -> A2(a2(insall, x), ys)
A2(a2(insall, x), a2(a2(cons, y), ys)) -> A2(insall, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(insall, x), a2(a2(cons, y), ys)) -> A2(a2(insall, x), ys)
A2(a2(insall, x), a2(a2(cons, y), ys)) -> A2(a2(ins, x), y)
A2(a2(insall, x), a2(a2(cons, y), ys)) -> A2(a2(append, a2(a2(ins, x), y)), a2(a2(insall, x), ys))
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(ins, x), ys)
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(map, a2(cons, y)), a2(a2(ins, x), ys))
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(cons, x), a2(a2(cons, y), ys))
A2(a2(ins, x), a2(a2(cons, y), ys)) -> A2(a2(cons, a2(a2(cons, x), a2(a2(cons, y), ys))), a2(a2(map, a2(cons, y)), a2(a2(ins, x), ys)))
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(map, f), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(f, x)), a2(a2(map, f), xs))
A2(a2(ins, x), nil) -> A2(a2(cons, x), nil)
A2(a2(ins, x), nil) -> A2(a2(cons, a2(a2(cons, x), nil)), nil)
A2(perm, nil) -> A2(a2(cons, nil), nil)
A2(perm, a2(a2(cons, x), xs)) -> A2(perm, xs)
A2(perm, a2(a2(cons, x), xs)) -> A2(a2(insall, x), a2(perm, xs))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.87 sec.

File: /usr/home/aoto/ttt/sttrs505.ttt

Term rewriting system R:

[y, x, f, ys]
a3(plus, 0, y) -> y
a3(plus, a2(s, x), y) -> a2(s, a3(plus, x, y))
a3(times, 0, y) -> 0
a3(times, a2(s, x), y) -> a3(plus, a3(times, x, y), y)
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a3(f, y, a2(a3(fold, f, x), ys))
sum -> a3(fold, plus, 0)
product -> a3(fold, times, a2(s, 0))
Dependency Pairs for R
A3(times, a2(s, x), y) -> A3(plus, a3(times, x, y), y)
A3(times, a2(s, x), y) -> A3(times, x, y)
A3(plus, a2(s, x), y) -> A2(s, a3(plus, x, y))
A3(plus, a2(s, x), y) -> A3(plus, x, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A3(f, y, a2(a3(fold, f, x), ys))
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A3(fold, f, x)
SUM -> A3(fold, plus, 0)
PRODUCT -> A3(fold, times, a2(s, 0))
PRODUCT -> A2(s, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 3 SCCs:

1
A3(plus, a2(s, x), y) -> A3(plus, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(plus)=  a5  

2
A3(times, a2(s, x), y) -> A3(times, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(times)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

3
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(fold)=  0  
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(nil)=  0  
  POL(0)=  0  
  POL(times)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  1  
  POL(plus)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.09 sec.

File: /usr/home/aoto/ttt/sttrs506.ttt

Term rewriting system R:

[y, x, f, ys]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(times, 0), y) -> 0
a2(a2(times, a2(s, x)), y) -> a2(a2(plus, a2(a2(times, x), y)), y)
a2(a3(fold, f, x), nil) -> x
a2(a3(fold, f, x), a3(cons, y, ys)) -> a2(a2(f, y), a2(a3(fold, f, x), ys))
sum -> a3(fold, plus, 0)
product -> a3(fold, times, a2(s, 0))
Dependency Pairs for R
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(times, a2(s, x)), y) -> A2(plus, a2(a2(times, x), y))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(times, x)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a2(f, y), a2(a3(fold, f, x), ys))
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(f, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
PRODUCT -> A2(s, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a3(fold, f, x), ys)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(f, y)
A2(a3(fold, f, x), a3(cons, y, ys)) -> A2(a2(f, y), a2(a3(fold, f, x), ys))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.21 sec.

File: /usr/home/aoto/ttt/sttrs507.ttt

Term rewriting system R:

[y, x, f, ys]
a2(a2(plus, 0), y) -> y
a2(a2(plus, a2(s, x)), y) -> a2(s, a2(a2(plus, x), y))
a2(a2(times, 0), y) -> 0
a2(a2(times, a2(s, x)), y) -> a2(a2(plus, a2(a2(times, x), y)), y)
a2(a2(a2(fold, f), x), nil) -> x
a2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> a2(a2(f, y), a2(a2(a2(fold, f), x), ys))
sum -> a2(a2(fold, plus), 0)
product -> a2(a2(fold, times), a2(s, 0))
Dependency Pairs for R
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(fold, f)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(times, a2(s, x)), y) -> A2(plus, a2(a2(times, x), y))
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(times, x)
A2(a2(plus, a2(s, x)), y) -> A2(s, a2(a2(plus, x), y))
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(plus, a2(s, x)), y) -> A2(plus, x)
SUM -> A2(a2(fold, plus), 0)
SUM -> A2(fold, plus)
PRODUCT -> A2(a2(fold, times), a2(s, 0))
PRODUCT -> A2(fold, times)
PRODUCT -> A2(s, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(plus, a2(s, x)), y) -> A2(a2(plus, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(times, x), y)
A2(a2(times, a2(s, x)), y) -> A2(a2(plus, a2(a2(times, x), y)), y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(fold, f), x)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(fold, f), x), ys)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(a2(fold, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, y), a2(a2(a2(fold, f), x), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.23 sec.

File: /usr/home/aoto/ttt/sttrs601.ttt

Term rewriting system R:

[y, x, f, g, xs]
a3(max, 0, y) -> y
a3(max, x, 0) -> x
a3(max, a2(s, x), a2(s, y)) -> a2(s, a3(max, x, y))
a3(min, 0, y) -> 0
a3(min, x, 0) -> 0
a3(min, a2(s, x), a2(s, y)) -> a2(s, a3(min, x, y))
a2(a4(ins, f, g, y), nil) -> a3(cons, y, nil)
a2(a4(ins, f, g, y), a3(cons, x, xs)) -> a3(cons, a3(f, x, y), a2(a4(ins, f, g, a3(g, x, y)), xs))
a2(a3(sort, f, g), nil) -> nil
a2(a3(sort, f, g), a3(cons, x, xs)) -> a2(a4(ins, f, g, x), a2(a3(sort, f, g), xs))
ascendingsort -> a3(sort, min, max)
descendingsort -> a3(sort, max, min)
Dependency Pairs for R
A3(max, a2(s, x), a2(s, y)) -> A2(s, a3(max, x, y))
A3(max, a2(s, x), a2(s, y)) -> A3(max, x, y)
A3(min, a2(s, x), a2(s, y)) -> A2(s, a3(min, x, y))
A3(min, a2(s, x), a2(s, y)) -> A3(min, x, y)
A2(a4(ins, f, g, y), nil) -> A3(cons, y, nil)
A2(a3(sort, f, g), a3(cons, x, xs)) -> A2(a4(ins, f, g, x), a2(a3(sort, f, g), xs))
A2(a3(sort, f, g), a3(cons, x, xs)) -> A2(a3(sort, f, g), xs)
A2(a3(sort, f, g), a3(cons, x, xs)) -> A3(sort, f, g)
A2(a4(ins, f, g, y), a3(cons, x, xs)) -> A3(cons, a3(f, x, y), a2(a4(ins, f, g, a3(g, x, y)), xs))
A2(a4(ins, f, g, y), a3(cons, x, xs)) -> A3(f, x, y)
A2(a4(ins, f, g, y), a3(cons, x, xs)) -> A2(a4(ins, f, g, a3(g, x, y)), xs)
A2(a4(ins, f, g, y), a3(cons, x, xs)) -> A3(g, x, y)
DESCENDINGSORT -> A3(sort, max, min)
ASCENDINGSORT -> A3(sort, min, max)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 4 SCCs:

1
A3(max, a2(s, x), a2(s, y)) -> A3(max, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(max)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

2
A3(min, a2(s, x), a2(s, y)) -> A3(min, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(min)=  a5  
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

3
A2(a4(ins, f, g, y), a3(cons, x, xs)) -> A2(a4(ins, f, g, a3(g, x, y)), xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(min)=  0  
  POL(sort)=  0  
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(max)=  0  
  POL(nil)=  0  
  POL(a4(x1, x2, x3, x4))=  0  
  POL(0)=  0  
  POL(ins)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  1  
  POL(A2(x1, x2))=  x2  

4
A2(a3(sort, f, g), a3(cons, x, xs)) -> A2(a3(sort, f, g), xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(min)=  0  
  POL(sort)=  0  
  POL(a3(x1, x2, x3))=  x1 + x3  
  POL(max)=  0  
  POL(nil)=  0  
  POL(a4(x1, x2, x3, x4))=  0  
  POL(0)=  0  
  POL(ins)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  1  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.25 sec.

File: /usr/home/aoto/ttt/sttrs602.ttt

Term rewriting system R:

[y, x, f, g, xs]
a2(a2(max, 0), y) -> y
a2(a2(max, x), 0) -> x
a2(a2(max, a2(s, x)), a2(s, y)) -> a2(s, a2(a2(max, x), y))
a2(a2(min, 0), y) -> 0
a2(a2(min, x), 0) -> 0
a2(a2(min, a2(s, x)), a2(s, y)) -> a2(s, a2(a2(min, x), y))
a2(a2(a2(a2(ins, f), g), y), nil) -> a2(a2(cons, y), nil)
a2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> a2(a2(cons, a2(a2(f, x), y)), a2(a2(a2(a2(ins, f), g), a2(a2(g, x), y)), xs))
a2(a2(a2(sort, f), g), nil) -> nil
a2(a2(a2(sort, f), g), a2(a2(cons, x), xs)) -> a2(a2(a2(a2(ins, f), g), x), a2(a2(a2(sort, f), g), xs))
ascendingsort -> a2(a2(sort, min), max)
descendingsort -> a2(a2(sort, max), min)
Dependency Pairs for R
A2(a2(max, a2(s, x)), a2(s, y)) -> A2(s, a2(a2(max, x), y))
A2(a2(max, a2(s, x)), a2(s, y)) -> A2(a2(max, x), y)
A2(a2(max, a2(s, x)), a2(s, y)) -> A2(max, x)
A2(a2(a2(a2(ins, f), g), y), nil) -> A2(a2(cons, y), nil)
A2(a2(a2(a2(ins, f), g), y), nil) -> A2(cons, y)
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(a2(cons, a2(a2(f, x), y)), a2(a2(a2(a2(ins, f), g), a2(a2(g, x), y)), xs))
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(cons, a2(a2(f, x), y))
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(a2(f, x), y)
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(f, x)
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(a2(a2(a2(ins, f), g), a2(a2(g, x), y)), xs)
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(a2(a2(ins, f), g), a2(a2(g, x), y))
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(a2(ins, f), g)
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(ins, f)
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(a2(g, x), y)
A2(a2(a2(a2(ins, f), g), y), a2(a2(cons, x), xs)) -> A2(g, x)
A2(a2(a2(sort, f), g), a2(a2(cons, x), xs)) -> A2(a2(a2(a2(ins, f), g), x), a2(a2(a2(sort, f), g), xs))
A2(a2(a2(sort, f), g), a2(a2(cons, x), xs)) -> A2(a2(a2(ins, f), g), x)
A2(a2(a2(sort, f), g), a2(a2(cons, x), xs)) -> A2(a2(ins, f), g)
A2(a2(a2(sort, f), g), a2(a2(cons, x), xs)) -> A2(ins, f)
A2(a2(a2(sort, f), g), a2(a2(cons, x), xs)) -> A2(a2(a2(sort, f), g), xs)
A2(a2(a2(sort, f), g), a2(a2(cons, x), xs)) -> A2(a2(sort, f), g)
A2(a2(a2(sort, f), g), a2(a2(cons, x), xs)) -> A2(sort, f)
A2(a2(min, a2(s, x)), a2(s, y)) -> A2(s, a2(a2(min, x), y))
A2(a2(min, a2(s, x)), a2(s, y)) -> A2(a2(min, x), y)
A2(a2(min, a2(s, x)), a2(s, y)) -> A2(min, x)
DESCENDINGSORT -> A2(a2(sort, max), min)
DESCENDINGSORT -> A2(sort, max)
ASCENDINGSORT -> A2(a2(sort, min), max)
ASCENDINGSORT -> A2(sort, min)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A2(a2(a2(a2(ins, f), g), y), nil) -> A2(a2(cons, y), nil)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.52 sec.

File: /usr/home/aoto/ttt/sttrs603.ttt

Term rewriting system R:

[y, x, g, f, xs]
a3(plus, 0, y) -> y
a3(plus, a2(s, x), y) -> a2(s, a3(plus, x, y))
a3(times, 0, y) -> 0
a3(times, a2(s, x), y) -> a3(plus, a3(times, x, y), y)
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
a2(a2(map, f), nil) -> nil
a2(a2(map, f), a3(cons, x, xs)) -> a3(cons, a2(f, x), a2(a2(map, f), xs))
inc -> a2(map, a2(a2(curry, plus), a2(s, 0)))
double -> a2(map, a2(a2(curry, times), a2(s, a2(s, 0))))
Dependency Pairs for R
A3(times, a2(s, x), y) -> A3(plus, a3(times, x, y), y)
A3(times, a2(s, x), y) -> A3(times, x, y)
A3(plus, a2(s, x), y) -> A2(s, a3(plus, x, y))
A3(plus, a2(s, x), y) -> A3(plus, x, y)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A2(a2(map, f), a3(cons, x, xs)) -> A3(cons, a2(f, x), a2(a2(map, f), xs))
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(map, f)
INC -> A2(map, a2(a2(curry, plus), a2(s, 0)))
INC -> A2(a2(curry, plus), a2(s, 0))
INC -> A2(curry, plus)
INC -> A2(s, 0)
DOUBLE -> A2(map, a2(a2(curry, times), a2(s, a2(s, 0))))
DOUBLE -> A2(a2(curry, times), a2(s, a2(s, 0)))
DOUBLE -> A2(curry, times)
DOUBLE -> A2(s, a2(s, 0))
DOUBLE -> A2(s, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 3 SCCs:

1
A3(plus, a2(s, x), y) -> A3(plus, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(plus)=  a5  

2
A3(times, a2(s, x), y) -> A3(times, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(times)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

3
A2(a2(map, f), a3(cons, x, xs)) -> A2(a2(map, f), xs)
A2(a2(map, f), a3(cons, x, xs)) -> A2(f, x)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(map)=  0  
  POL(curry)=  0  
  POL(a3(x1, x2, x3))=  1 + x2 + x3  
  POL(nil)=  0  
  POL(0)=  0  
  POL(a2(x1, x2))=  0  
  POL(cons)=  0  
  POL(times)=  0  
  POL(plus)=  0  
  POL(A2(x1, x2))=  x2  

Termination of R successfully proved!


Duration: 0.13 sec.

File: /usr/home/aoto/ttt/sttrs604.ttt

Term rewriting system R:

[xs, x, ys, f, y, g]
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
a3(ins, x, nil) -> a3(cons, a3(cons, x, nil), nil)
a3(ins, x, a3(cons, y, ys)) -> a3(cons, a3(cons, x, a3(cons, y, ys)), a3(map, a2(a2(curry, cons), y), a3(ins, x, ys)))
a3(insall, x, nil) -> nil
a3(insall, x, a3(cons, y, ys)) -> a3(append, a3(ins, x, y), a3(insall, x, ys))
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
a2(perm, nil) -> a3(cons, nil, nil)
a2(perm, a3(cons, x, xs)) -> a3(insall, x, a2(perm, xs))
Dependency Pairs for R
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A3(ins, x, nil) -> A3(cons, a3(cons, x, nil), nil)
A3(ins, x, nil) -> A3(cons, x, nil)
A3(insall, x, a3(cons, y, ys)) -> A3(append, a3(ins, x, y), a3(insall, x, ys))
A3(insall, x, a3(cons, y, ys)) -> A3(ins, x, y)
A3(insall, x, a3(cons, y, ys)) -> A3(insall, x, ys)
A3(ins, x, a3(cons, y, ys)) -> A3(cons, a3(cons, x, a3(cons, y, ys)), a3(map, a2(a2(curry, cons), y), a3(ins, x, ys)))
A3(ins, x, a3(cons, y, ys)) -> A3(cons, x, a3(cons, y, ys))
A3(ins, x, a3(cons, y, ys)) -> A3(cons, y, ys)
A3(ins, x, a3(cons, y, ys)) -> A3(map, a2(a2(curry, cons), y), a3(ins, x, ys))
A3(ins, x, a3(cons, y, ys)) -> A2(a2(curry, cons), y)
A3(ins, x, a3(cons, y, ys)) -> A2(curry, cons)
A3(ins, x, a3(cons, y, ys)) -> A3(ins, x, ys)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A2(perm, a3(cons, x, xs)) -> A3(insall, x, a2(perm, xs))
A2(perm, a3(cons, x, xs)) -> A2(perm, xs)
A2(perm, nil) -> A3(cons, nil, nil)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  1  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

2
A3(ins, x, a3(cons, y, ys)) -> A3(ins, x, ys)
A2(perm, a3(cons, x, xs)) -> A2(perm, xs)
A3(insall, x, a3(cons, y, ys)) -> A3(insall, x, ys)
A2(perm, a3(cons, x, xs)) -> A3(insall, x, a2(perm, xs))
A3(ins, x, a3(cons, y, ys)) -> A2(a2(curry, cons), y)
A3(ins, x, a3(cons, y, ys)) -> A3(map, a2(a2(curry, cons), y), a3(ins, x, ys))
A3(insall, x, a3(cons, y, ys)) -> A3(ins, x, y)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.71 sec.

File: /usr/home/aoto/ttt/sttrs605.ttt

Term rewriting system R:

[y, x, f, xs, g, ys]
a3(plus, 0, y) -> y
a3(plus, a2(s, x), y) -> a2(s, a3(plus, x, y))
a3(times, 0, y) -> 0
a3(times, a2(s, x), y) -> a3(plus, a3(times, x, y), y)
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
a2(square, x) -> a3(times, x, x)
a2(a3(fold, g, x), nil) -> x
a2(a3(fold, g, x), a3(cons, y, ys)) -> a3(g, y, a2(a3(fold, g, x), ys))
a2(sqsum, xs) -> a2(sum, a3(map, square, xs))
sum -> a3(fold, plus, 0)
Dependency Pairs for R
A3(times, a2(s, x), y) -> A3(plus, a3(times, x, y), y)
A3(times, a2(s, x), y) -> A3(times, x, y)
A3(plus, a2(s, x), y) -> A2(s, a3(plus, x, y))
A3(plus, a2(s, x), y) -> A3(plus, x, y)
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(sqsum, xs) -> A2(sum, a3(map, square, xs))
A2(sqsum, xs) -> SUM
A2(sqsum, xs) -> A3(map, square, xs)
A2(square, x) -> A3(times, x, x)
A2(a3(fold, g, x), a3(cons, y, ys)) -> A3(g, y, a2(a3(fold, g, x), ys))
A2(a3(fold, g, x), a3(cons, y, ys)) -> A2(a3(fold, g, x), ys)
A2(a3(fold, g, x), a3(cons, y, ys)) -> A3(fold, g, x)
SUM -> A3(fold, plus, 0)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 3 SCCs:

1
A3(plus, a2(s, x), y) -> A3(plus, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(plus)=  a5  

2
A3(times, a2(s, x), y) -> A3(times, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(times)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

3
A2(a3(fold, g, x), a3(cons, y, ys)) -> A2(a3(fold, g, x), ys)
A2(a3(fold, g, x), a3(cons, y, ys)) -> A3(g, y, a2(a3(fold, g, x), ys))
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(sqsum, xs) -> A3(map, square, xs)
A2(sqsum, xs) -> A2(sum, a3(map, square, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.31 sec.

File: /usr/home/aoto/ttt/sttrs702.ttt

Term rewriting system R:

[y, x, f, ys, xs]
a2(a2(gt, 0), y) -> false
a2(a2(gt, x), 0) -> true
a2(a2(gt, a2(s, x)), a2(s, y)) -> a2(a2(gt, x), y)
a2(a2(le, 0), y) -> true
a2(a2(le, x), 0) -> false
a2(a2(le, a2(s, x)), a2(s, y)) -> a2(a2(le, x), y)
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a3(cons, y, ys)) -> a3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
a2(qsort, nil) -> nil
a2(qsort, a3(cons, x, xs)) -> a3(append, a2(qsort, a3(low, x, xs)), a3(cons, x, a2(qsort, a3(high, x, xs))))
a3(a2(filtersub, true), f, a3(cons, y, ys)) -> a3(cons, y, a2(a2(filter, f), ys))
a3(a2(filtersub, false), f, a3(cons, y, ys)) -> a2(a2(filter, f), ys)
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a3(high, x, xs) -> a2(a2(filter, a2(gt, x)), xs)
a3(low, x, xs) -> a2(a2(filter, a2(le, x)), xs)
Dependency Pairs for R
A2(qsort, a3(cons, x, xs)) -> A3(append, a2(qsort, a3(low, x, xs)), a3(cons, x, a2(qsort, a3(high, x, xs))))
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(low, x, xs))
A2(qsort, a3(cons, x, xs)) -> A3(low, x, xs)
A2(qsort, a3(cons, x, xs)) -> A3(cons, x, a2(qsort, a3(high, x, xs)))
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(high, x, xs))
A2(qsort, a3(cons, x, xs)) -> A3(high, x, xs)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(a2(gt, x), y)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(gt, x)
A2(a2(le, a2(s, x)), a2(s, y)) -> A2(a2(le, x), y)
A2(a2(le, a2(s, x)), a2(s, y)) -> A2(le, x)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(filtersub, a2(f, y))
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(cons, y, ys)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A3(cons, y, a2(a2(filter, f), ys))
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(filter, f)
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(filter, f)
A3(low, x, xs) -> A2(a2(filter, a2(le, x)), xs)
A3(low, x, xs) -> A2(filter, a2(le, x))
A3(low, x, xs) -> A2(le, x)
A3(high, x, xs) -> A2(a2(filter, a2(gt, x)), xs)
A3(high, x, xs) -> A2(filter, a2(gt, x))
A3(high, x, xs) -> A2(gt, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 2 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  1  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

2
A3(a2(filtersub, false), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a3(cons, y, ys)) -> A2(f, y)
A3(a2(filtersub, true), f, a3(cons, y, ys)) -> A2(a2(filter, f), ys)
A2(a2(filter, f), a3(cons, y, ys)) -> A3(a2(filtersub, a2(f, y)), f, a3(cons, y, ys))
A2(a2(le, a2(s, x)), a2(s, y)) -> A2(a2(le, x), y)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(a2(gt, x), y)
A3(high, x, xs) -> A2(a2(filter, a2(gt, x)), xs)
A2(qsort, a3(cons, x, xs)) -> A3(high, x, xs)
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(high, x, xs))
A3(low, x, xs) -> A2(a2(filter, a2(le, x)), xs)
A2(qsort, a3(cons, x, xs)) -> A3(low, x, xs)
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(low, x, xs))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 5.71 sec.

File: /usr/home/aoto/ttt/sttrs703.ttt

Term rewriting system R:

[y, x, xs, ys, f]
a3(gt, 0, y) -> false
a3(gt, x, 0) -> true
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a3(le, 0, y) -> true
a3(le, x, 0) -> false
a3(le, a2(s, x), a2(s, y)) -> a3(le, x, y)
a3(high, x, xs) -> a4(filter, gt, x, xs)
a3(low, x, xs) -> a4(filter, le, x, xs)
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a4(filter, f, x, nil) -> nil
a4(filter, f, x, a3(cons, y, ys)) -> a5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
a5(filtersub, true, f, x, a3(cons, y, ys)) -> a3(cons, y, a4(filter, f, x, ys))
a5(filtersub, false, f, x, a3(cons, y, ys)) -> a4(filter, f, x, ys)
a2(qsort, nil) -> nil
a2(qsort, a3(cons, x, xs)) -> a3(append, a2(qsort, a3(low, x, xs)), a3(cons, x, a2(qsort, a3(high, x, xs))))
Dependency Pairs for R
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
A3(low, x, xs) -> A4(filter, le, x, xs)
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A3(high, x, xs) -> A4(filter, gt, x, xs)
A3(le, a2(s, x), a2(s, y)) -> A3(le, x, y)
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A3(cons, y, a4(filter, f, x, ys))
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A5(filtersub, false, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A4(filter, f, x, a3(cons, y, ys)) -> A5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
A4(filter, f, x, a3(cons, y, ys)) -> A3(f, x, y)
A4(filter, f, x, a3(cons, y, ys)) -> A3(cons, y, ys)
A2(qsort, a3(cons, x, xs)) -> A3(append, a2(qsort, a3(low, x, xs)), a3(cons, x, a2(qsort, a3(high, x, xs))))
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(low, x, xs))
A2(qsort, a3(cons, x, xs)) -> A3(low, x, xs)
A2(qsort, a3(cons, x, xs)) -> A3(cons, x, a2(qsort, a3(high, x, xs)))
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(high, x, xs))
A2(qsort, a3(cons, x, xs)) -> A3(high, x, xs)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 5 SCCs:

1
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(gt)=  a5  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  

2
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  
  POL(cons)=  1  

3
A3(le, a2(s, x), a2(s, y)) -> A3(le, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(le)=  a5  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  

4
A5(filtersub, false, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A3(high, x, xs) -> A4(filter, gt, x, xs)
A4(filter, f, x, a3(cons, y, ys)) -> A3(f, x, y)
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A4(filter, f, x, a3(cons, y, ys)) -> A5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
A3(low, x, xs) -> A4(filter, le, x, xs)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a4(x1, x2, x3, x4))=  0  
  POL(append)=  0  
  POL(filtersub)=  0  
  POL(A4(x1, x2, x3, x4))=  x2  
  POL(gt)=  0  
  POL(0)=  0  
  POL(A5(x1, x2, x3, x4, x5))=  x3  
  POL(true)=  0  
  POL(nil)=  0  
  POL(a3(x1, x2, x3))=  0  
  POL(le)=  0  
  POL(false)=  0  
  POL(A3(x1, x2, x3))=  x1  
  POL(cons)=  0  
  POL(low)=  1  
  POL(s)=  0  
  POL(a5(x1, x2, x3, x4, x5))=  0  
  POL(filter)=  0  
  POL(a2(x1, x2))=  0  
  POL(high)=  0  

Need to check 1 sub cycle of this SCC.

4.1
A3(high, x, xs) -> A4(filter, gt, x, xs)
A4(filter, f, x, a3(cons, y, ys)) -> A3(f, x, y)
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A4(filter, f, x, a3(cons, y, ys)) -> A5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
A5(filtersub, false, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a4(x1, x2, x3, x4))=  0  
  POL(append)=  0  
  POL(filtersub)=  0  
  POL(A4(x1, x2, x3, x4))=  x2  
  POL(gt)=  0  
  POL(0)=  0  
  POL(A5(x1, x2, x3, x4, x5))=  x3  
  POL(true)=  0  
  POL(nil)=  0  
  POL(a3(x1, x2, x3))=  0  
  POL(le)=  0  
  POL(false)=  0  
  POL(A3(x1, x2, x3))=  x1  
  POL(cons)=  0  
  POL(low)=  0  
  POL(s)=  0  
  POL(a5(x1, x2, x3, x4, x5))=  0  
  POL(filter)=  0  
  POL(a2(x1, x2))=  0  
  POL(high)=  1  

Need to check 1 sub cycle of this SCC.

4.1.1
A5(filtersub, false, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
A4(filter, f, x, a3(cons, y, ys)) -> A5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
A5(filtersub, true, f, x, a3(cons, y, ys)) -> A4(filter, f, x, ys)
Oriented Rule(s):

a4(filter, f, x, a3(cons, y, ys)) -> a5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
a4(filter, f, x, nil) -> nil
a3(le, x, 0) -> false
a3(le, 0, y) -> true
a3(gt, 0, y) -> false
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a3(low, x, xs) -> a4(filter, le, x, xs)
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a3(append, nil, xs) -> xs
a3(high, x, xs) -> a4(filter, gt, x, xs)
a3(gt, x, 0) -> true
a3(le, a2(s, x), a2(s, y)) -> a3(le, x, y)
a5(filtersub, true, f, x, a3(cons, y, ys)) -> a3(cons, y, a4(filter, f, x, ys))
a5(filtersub, false, f, x, a3(cons, y, ys)) -> a4(filter, f, x, ys)

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a4(x1, x2, x3, x4))=  x4  
  POL(append)=  0  
  POL(filtersub)=  0  
  POL(A4(x1, x2, x3, x4))=  x4  
  POL(gt)=  0  
  POL(0)=  0  
  POL(A5(x1, x2, x3, x4, x5))=  x5  
  POL(true)=  0  
  POL(nil)=  0  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(le)=  0  
  POL(false)=  0  
  POL(cons)=  1  
  POL(low)=  0  
  POL(s)=  1  
  POL(a5(x1, x2, x3, x4, x5))=  x5  
  POL(filter)=  0  
  POL(a2(x1, x2))=  1 + x2  
  POL(high)=  0  

5
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(high, x, xs))
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(low, x, xs))
Oriented Rule(s):

a4(filter, f, x, a3(cons, y, ys)) -> a5(filtersub, a3(f, x, y), f, x, a3(cons, y, ys))
a4(filter, f, x, nil) -> nil
a3(le, x, 0) -> false
a3(le, 0, y) -> true
a3(gt, 0, y) -> false
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a3(low, x, xs) -> a4(filter, le, x, xs)
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a3(append, nil, xs) -> xs
a3(high, x, xs) -> a4(filter, gt, x, xs)
a3(gt, x, 0) -> true
a3(le, a2(s, x), a2(s, y)) -> a3(le, x, y)
a5(filtersub, true, f, x, a3(cons, y, ys)) -> a3(cons, y, a4(filter, f, x, ys))
a5(filtersub, false, f, x, a3(cons, y, ys)) -> a4(filter, f, x, ys)

Ordering: Polynomial ordering
Polynomial interpretation:
  POL(a4(x1, x2, x3, x4))=  x4  
  POL(A2(x1, x2))=  1 + x1 + x2  
  POL(append)=  0  
  POL(filtersub)=  0  
  POL(gt)=  0  
  POL(0)=  0  
  POL(true)=  0  
  POL(qsort)=  0  
  POL(nil)=  0  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(le)=  0  
  POL(false)=  0  
  POL(cons)=  1  
  POL(low)=  0  
  POL(s)=  0  
  POL(a5(x1, x2, x3, x4, x5))=  x5  
  POL(filter)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(high)=  0  

Termination of R successfully proved!


Duration: 3.86 sec.

File: /usr/home/aoto/ttt/sttrs704.ttt

Term rewriting system R:

[y, x, xs, ys, f]
a3(gt, 0, y) -> false
a3(gt, x, 0) -> true
a3(gt, a2(s, x), a2(s, y)) -> a3(gt, x, y)
a3(le, 0, y) -> true
a3(le, x, 0) -> false
a3(le, a2(s, x), a2(s, y)) -> a3(le, x, y)
a3(high, x, xs) -> a2(a3(filter, gt, x), xs)
a3(low, x, xs) -> a2(a3(filter, le, x), xs)
a3(append, nil, xs) -> xs
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a2(a3(filter, f, x), nil) -> nil
a2(a3(filter, f, x), a3(cons, y, ys)) -> a4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
a2(qsort, nil) -> nil
a2(qsort, a3(cons, x, xs)) -> a3(append, a2(qsort, a3(low, x, xs)), a3(cons, x, a2(qsort, a3(high, x, xs))))
a4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> a3(cons, y, a2(a3(filter, f, x), ys))
a4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> a2(a3(filter, f, x), ys)
Dependency Pairs for R
A2(a3(filter, f, x), a3(cons, y, ys)) -> A4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
A2(a3(filter, f, x), a3(cons, y, ys)) -> A2(filtersub, a3(f, x, y))
A2(a3(filter, f, x), a3(cons, y, ys)) -> A3(f, x, y)
A2(a3(filter, f, x), a3(cons, y, ys)) -> A3(cons, y, ys)
A2(qsort, a3(cons, x, xs)) -> A3(append, a2(qsort, a3(low, x, xs)), a3(cons, x, a2(qsort, a3(high, x, xs))))
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(low, x, xs))
A2(qsort, a3(cons, x, xs)) -> A3(low, x, xs)
A2(qsort, a3(cons, x, xs)) -> A3(cons, x, a2(qsort, a3(high, x, xs)))
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(high, x, xs))
A2(qsort, a3(cons, x, xs)) -> A3(high, x, xs)
A3(low, x, xs) -> A2(a3(filter, le, x), xs)
A3(low, x, xs) -> A3(filter, le, x)
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A3(le, a2(s, x), a2(s, y)) -> A3(le, x, y)
A3(high, x, xs) -> A2(a3(filter, gt, x), xs)
A3(high, x, xs) -> A3(filter, gt, x)
A4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> A3(filter, f, x)
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A3(cons, y, a2(a3(filter, f, x), ys))
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A3(filter, f, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 4 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  1  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

2
A3(gt, a2(s, x), a2(s, y)) -> A3(gt, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(gt)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

3
A3(le, a2(s, x), a2(s, y)) -> A3(le, x, y)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(le)=  a5  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(A3(x1, x2, x3))=  x1 + x2 + x3  

4
A4(a2(filtersub, true), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A3(high, x, xs) -> A2(a3(filter, gt, x), xs)
A2(qsort, a3(cons, x, xs)) -> A3(high, x, xs)
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(high, x, xs))
A2(qsort, a3(cons, x, xs)) -> A3(low, x, xs)
A2(qsort, a3(cons, x, xs)) -> A2(qsort, a3(low, x, xs))
A3(low, x, xs) -> A2(a3(filter, le, x), xs)
A2(a3(filter, f, x), a3(cons, y, ys)) -> A3(f, x, y)
A4(a2(filtersub, false), f, x, a3(cons, y, ys)) -> A2(a3(filter, f, x), ys)
A2(a3(filter, f, x), a3(cons, y, ys)) -> A4(a2(filtersub, a3(f, x, y)), f, x, a3(cons, y, ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 12.4 sec.

File: /usr/home/aoto/ttt/sttrs705.ttt

Term rewriting system R:

[y, x, f, ys, xs]
a2(a2(gt, 0), y) -> false
a2(a2(gt, x), 0) -> true
a2(a2(gt, a2(s, x)), a2(s, y)) -> a2(a2(gt, x), y)
a2(a2(le, 0), y) -> true
a2(a2(le, x), 0) -> false
a2(a2(le, a2(s, x)), a2(s, y)) -> a2(a2(le, x), y)
a2(a2(filter, f), nil) -> nil
a2(a2(filter, f), a2(a2(cons, y), ys)) -> a2(a2(a2(filtersub, a2(f, y)), f), a2(a2(cons, y), ys))
a2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> a2(a2(cons, y), a2(a2(filter, f), ys))
a2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> a2(a2(filter, f), ys)
a2(a2(append, nil), xs) -> xs
a2(a2(append, a2(a2(cons, x), xs)), ys) -> a2(a2(cons, x), a2(a2(append, xs), ys))
a2(a2(high, x), xs) -> a2(a2(filter, a2(gt, x)), xs)
a2(a2(low, x), xs) -> a2(a2(filter, a2(le, x)), xs)
a2(qsort, nil) -> nil
a2(qsort, a2(a2(cons, x), xs)) -> a2(a2(append, a2(qsort, a2(a2(low, x), xs))), a2(a2(cons, x), a2(qsort, a2(a2(high, x), xs))))
Dependency Pairs for R
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(cons, x)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(append, xs)
A2(a2(low, x), xs) -> A2(a2(filter, a2(le, x)), xs)
A2(a2(low, x), xs) -> A2(filter, a2(le, x))
A2(a2(low, x), xs) -> A2(le, x)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(a2(gt, x), y)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(gt, x)
A2(a2(high, x), xs) -> A2(a2(filter, a2(gt, x)), xs)
A2(a2(high, x), xs) -> A2(filter, a2(gt, x))
A2(a2(high, x), xs) -> A2(gt, x)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(filter, f), ys))
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(filter, f)
A2(a2(le, a2(s, x)), a2(s, y)) -> A2(a2(le, x), y)
A2(a2(le, a2(s, x)), a2(s, y)) -> A2(le, x)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(f, y)), f), a2(a2(cons, y), ys))
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(f, y)), f)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(filtersub, a2(f, y))
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(append, a2(qsort, a2(a2(low, x), xs))), a2(a2(cons, x), a2(qsort, a2(a2(high, x), xs))))
A2(qsort, a2(a2(cons, x), xs)) -> A2(append, a2(qsort, a2(a2(low, x), xs)))
A2(qsort, a2(a2(cons, x), xs)) -> A2(qsort, a2(a2(low, x), xs))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(low, x), xs)
A2(qsort, a2(a2(cons, x), xs)) -> A2(low, x)
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(cons, x), a2(qsort, a2(a2(high, x), xs)))
A2(qsort, a2(a2(cons, x), xs)) -> A2(cons, x)
A2(qsort, a2(a2(cons, x), xs)) -> A2(qsort, a2(a2(high, x), xs))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(high, x), xs)
A2(qsort, a2(a2(cons, x), xs)) -> A2(high, x)
A2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> A2(filter, f)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(a2(a2(filtersub, false), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(high, x), xs)
A2(qsort, a2(a2(cons, x), xs)) -> A2(qsort, a2(a2(high, x), xs))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(cons, x), a2(qsort, a2(a2(high, x), xs)))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(low, x), xs)
A2(qsort, a2(a2(cons, x), xs)) -> A2(qsort, a2(a2(low, x), xs))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(append, a2(qsort, a2(a2(low, x), xs))), a2(a2(cons, x), a2(qsort, a2(a2(high, x), xs))))
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(f, y)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(f, y)), f)
A2(a2(filter, f), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(f, y)), f), a2(a2(cons, y), ys))
A2(a2(le, a2(s, x)), a2(s, y)) -> A2(a2(le, x), y)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(filter, f), ys)
A2(a2(a2(filtersub, true), f), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(filter, f), ys))
A2(a2(high, x), xs) -> A2(a2(filter, a2(gt, x)), xs)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(a2(gt, x), y)
A2(a2(low, x), xs) -> A2(a2(filter, a2(le, x)), xs)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 0.92 sec.

File: /usr/home/aoto/ttt/sttrs706.ttt

Term rewriting system R:

[y, x, f, ys, xs]
a2(a2(gt, 0), y) -> false
a2(a2(gt, x), 0) -> true
a2(a2(gt, a2(s, x)), a2(s, y)) -> a2(a2(gt, x), y)
a2(a2(le, 0), y) -> true
a2(a2(le, x), 0) -> false
a2(a2(le, a2(s, x)), a2(s, y)) -> a2(a2(le, x), y)
a2(a2(a2(filter, f), x), nil) -> nil
a2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(a2(filtersub, a2(a2(f, x), y)), f), x), a2(a2(cons, y), ys))
a2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> a2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
a2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> a2(a2(a2(filter, f), x), ys)
a2(a2(high, x), xs) -> a2(a2(a2(filter, gt), x), xs)
a2(a2(low, x), xs) -> a2(a2(a2(filter, le), x), xs)
a2(a2(append, nil), xs) -> xs
a2(a2(append, a2(a2(cons, x), xs)), ys) -> a2(a2(cons, x), a2(a2(append, xs), ys))
a2(qsort, nil) -> nil
a2(qsort, a2(a2(cons, x), xs)) -> a2(a2(append, a2(qsort, a2(a2(low, x), xs))), a2(a2(cons, x), a2(qsort, a2(a2(high, x), xs))))
Dependency Pairs for R
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(cons, x)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(append, xs)
A2(a2(high, x), xs) -> A2(a2(a2(filter, gt), x), xs)
A2(a2(high, x), xs) -> A2(a2(filter, gt), x)
A2(a2(high, x), xs) -> A2(filter, gt)
A2(a2(low, x), xs) -> A2(a2(a2(filter, le), x), xs)
A2(a2(low, x), xs) -> A2(a2(filter, le), x)
A2(a2(low, x), xs) -> A2(filter, le)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(a2(gt, x), y)
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(gt, x)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(filter, f)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(filter, f)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(a2(filtersub, a2(a2(f, x), y)), f), x), a2(a2(cons, y), ys))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(a2(f, x), y)), f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(a2(f, x), y)), f)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(filtersub, a2(a2(f, x), y))
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(cons, y)
A2(a2(le, a2(s, x)), a2(s, y)) -> A2(a2(le, x), y)
A2(a2(le, a2(s, x)), a2(s, y)) -> A2(le, x)
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(append, a2(qsort, a2(a2(low, x), xs))), a2(a2(cons, x), a2(qsort, a2(a2(high, x), xs))))
A2(qsort, a2(a2(cons, x), xs)) -> A2(append, a2(qsort, a2(a2(low, x), xs)))
A2(qsort, a2(a2(cons, x), xs)) -> A2(qsort, a2(a2(low, x), xs))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(low, x), xs)
A2(qsort, a2(a2(cons, x), xs)) -> A2(low, x)
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(cons, x), a2(qsort, a2(a2(high, x), xs)))
A2(qsort, a2(a2(cons, x), xs)) -> A2(cons, x)
A2(qsort, a2(a2(cons, x), xs)) -> A2(qsort, a2(a2(high, x), xs))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(high, x), xs)
A2(qsort, a2(a2(cons, x), xs)) -> A2(high, x)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 1 SCC:

1
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(high, x), xs)
A2(qsort, a2(a2(cons, x), xs)) -> A2(qsort, a2(a2(high, x), xs))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(cons, x), a2(qsort, a2(a2(high, x), xs)))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(low, x), xs)
A2(qsort, a2(a2(cons, x), xs)) -> A2(qsort, a2(a2(low, x), xs))
A2(qsort, a2(a2(cons, x), xs)) -> A2(a2(append, a2(qsort, a2(a2(low, x), xs))), a2(a2(cons, x), a2(qsort, a2(a2(high, x), xs))))
A2(a2(le, a2(s, x)), a2(s, y)) -> A2(a2(le, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), ys)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(f, x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(f, x), y)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(filtersub, a2(a2(f, x), y)), f)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filtersub, a2(a2(f, x), y)), f), x)
A2(a2(a2(filter, f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(a2(filtersub, a2(a2(f, x), y)), f), x), a2(a2(cons, y), ys))
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, false), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(filter, f), x)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(a2(filter, f), x), ys)
A2(a2(a2(a2(filtersub, true), f), x), a2(a2(cons, y), ys)) -> A2(a2(cons, y), a2(a2(a2(filter, f), x), ys))
A2(a2(gt, a2(s, x)), a2(s, y)) -> A2(a2(gt, x), y)
A2(a2(low, x), xs) -> A2(a2(filter, le), x)
A2(a2(low, x), xs) -> A2(a2(a2(filter, le), x), xs)
A2(a2(high, x), xs) -> A2(a2(filter, gt), x)
A2(a2(high, x), xs) -> A2(a2(a2(filter, gt), x), xs)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(append, xs), ys)
A2(a2(append, a2(a2(cons, x), xs)), ys) -> A2(a2(cons, x), a2(a2(append, xs), ys))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 1.3 sec.

File: /usr/home/aoto/ttt/sttrs801.ttt

Term rewriting system R:

[x, g, y, ys, xs, f]
a2(declist, 0) -> a3(cons, 0, nil)
a2(declist, a2(s, x)) -> a3(cons, a2(s, x), a2(declist, x))
a2(a2(a2(curry, g), x), y) -> a3(g, x, y)
a2(genpairs, x) -> a3(map, a2(a2(curry, pair), x), a2(declist, x))
a2(genpairlist, x) -> a3(flatmap, genpairs, a2(declist, x))
a3(append, nil, ys) -> ys
a3(append, a3(cons, x, xs), ys) -> a3(cons, x, a3(append, xs, ys))
a3(map, f, nil) -> nil
a3(map, f, a3(cons, x, xs)) -> a3(cons, a2(f, x), a3(map, f, xs))
a3(flatmap, f, xs) -> a4(fold, append, nil, a3(map, f, xs))
a4(fold, g, x, nil) -> x
a4(fold, g, x, a3(cons, y, ys)) -> a3(g, y, a4(fold, g, x, ys))
Dependency Pairs for R
A3(append, a3(cons, x, xs), ys) -> A3(cons, x, a3(append, xs, ys))
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
A3(flatmap, f, xs) -> A4(fold, append, nil, a3(map, f, xs))
A3(flatmap, f, xs) -> A3(map, f, xs)
A3(map, f, a3(cons, x, xs)) -> A3(cons, a2(f, x), a3(map, f, xs))
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A2(genpairs, x) -> A3(map, a2(a2(curry, pair), x), a2(declist, x))
A2(genpairs, x) -> A2(a2(curry, pair), x)
A2(genpairs, x) -> A2(curry, pair)
A2(genpairs, x) -> A2(declist, x)
A2(genpairlist, x) -> A3(flatmap, genpairs, a2(declist, x))
A2(genpairlist, x) -> A2(declist, x)
A2(declist, a2(s, x)) -> A3(cons, a2(s, x), a2(declist, x))
A2(declist, a2(s, x)) -> A2(s, x)
A2(declist, a2(s, x)) -> A2(declist, x)
A2(declist, 0) -> A3(cons, 0, nil)
A4(fold, g, x, a3(cons, y, ys)) -> A3(g, y, a4(fold, g, x, ys))
A4(fold, g, x, a3(cons, y, ys)) -> A4(fold, g, x, ys)

Termination of R to be shown using SCCs of the Estimated Dependency Pair Graph.

The Dependency Pair Graph for R contains 3 SCCs:

1
A3(append, a3(cons, x, xs), ys) -> A3(append, xs, ys)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(append)=  a5  
  POL(a3(x1, x2, x3))=  x1 + x2 + x3  
  POL(cons)=  1  
  POL(A3(x1, x2, x3))=  a1 + x1 + x2 + x3  

2
A2(declist, a2(s, x)) -> A2(declist, x)
Oriented Rule(s): none
Ordering: Polynomial ordering
Polynomial interpretation:
  POL(s)=  0  
  POL(a2(x1, x2))=  1 + x1 + x2  
  POL(declist)=  a4  
  POL(A2(x1, x2))=  a1 + x1 + x2  

3
A4(fold, g, x, a3(cons, y, ys)) -> A4(fold, g, x, ys)
A2(genpairlist, x) -> A3(flatmap, genpairs, a2(declist, x))
A2(genpairs, x) -> A2(a2(curry, pair), x)
A2(genpairs, x) -> A3(map, a2(a2(curry, pair), x), a2(declist, x))
A3(map, f, a3(cons, x, xs)) -> A3(map, f, xs)
A2(a2(a2(curry, g), x), y) -> A3(g, x, y)
A3(map, f, a3(cons, x, xs)) -> A2(f, x)
A3(flatmap, f, xs) -> A3(map, f, xs)
A4(fold, g, x, a3(cons, y, ys)) -> A3(g, y, a4(fold, g, x, ys))
A3(flatmap, f, xs) -> A4(fold, append, nil, a3(map, f, xs))
Could not order constraints for this SCC.
Termination of R could not be shown!

Duration: 1.7 sec.

File: /usr/home/aoto/ttt/sttrs101.ttt

Term rewriting system R:

[f, x]
a3(apply, f, x) -> a2(f, x)
There are no Dependency Pairs for R.

The Dependency Pair Graph for R contains no SCCs!

Termination of R successfully proved!


Duration: 0.01 sec.

The following files succeeded:

/usr/home/aoto/ttt/sttrs102.ttt 0.04 sec.
/usr/home/aoto/ttt/sttrs103.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs104.ttt 0.01 sec.
/usr/home/aoto/ttt/sttrs105.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs106.ttt 1.51 sec.
/usr/home/aoto/ttt/sttrs108.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs109.ttt 0.18 sec.
/usr/home/aoto/ttt/sttrs111.ttt 0.02 sec.
/usr/home/aoto/ttt/sttrs112.ttt 0.02 sec.
/usr/home/aoto/ttt/sttrs113.ttt 0.14 sec.
/usr/home/aoto/ttt/sttrs115.ttt 0.07 sec.
/usr/home/aoto/ttt/sttrs116.ttt 0.04 sec.
/usr/home/aoto/ttt/sttrs117.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs121.ttt 0.02 sec.
/usr/home/aoto/ttt/sttrs122.ttt 0.1 sec.
/usr/home/aoto/ttt/sttrs124.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs125.ttt 0.15 sec.
/usr/home/aoto/ttt/sttrs126.ttt 0.05 sec.
/usr/home/aoto/ttt/sttrs127.ttt 0.05 sec.
/usr/home/aoto/ttt/sttrs128.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs129.ttt 0.01 sec.
/usr/home/aoto/ttt/sttrs130.ttt 0.02 sec.
/usr/home/aoto/ttt/sttrs131.ttt 0.12 sec.
/usr/home/aoto/ttt/sttrs133.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs134.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs137.ttt 0.0 sec.
/usr/home/aoto/ttt/sttrs138.ttt 0.01 sec.
/usr/home/aoto/ttt/sttrs139.ttt 0.02 sec.
/usr/home/aoto/ttt/sttrs140.ttt 0.05 sec.
/usr/home/aoto/ttt/sttrs202.ttt 0.02 sec.
/usr/home/aoto/ttt/sttrs203.ttt 0.02 sec.
/usr/home/aoto/ttt/sttrs206.ttt 0.02 sec.
/usr/home/aoto/ttt/sttrs207.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs210.ttt 0.06 sec.
/usr/home/aoto/ttt/sttrs211.ttt 0.24 sec.
/usr/home/aoto/ttt/sttrs213.ttt 0.05 sec.
/usr/home/aoto/ttt/sttrs214.ttt 0.01 sec.
/usr/home/aoto/ttt/sttrs215.ttt 0.07 sec.
/usr/home/aoto/ttt/sttrs219.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs301.ttt 0.02 sec.
/usr/home/aoto/ttt/sttrs302.ttt 0.04 sec.
/usr/home/aoto/ttt/sttrs303.ttt 0.14 sec.
/usr/home/aoto/ttt/sttrs305.ttt 0.05 sec.
/usr/home/aoto/ttt/sttrs306.ttt 0.09 sec.
/usr/home/aoto/ttt/sttrs308.ttt 1.77 sec.
/usr/home/aoto/ttt/sttrs309.ttt 2.34 sec.
/usr/home/aoto/ttt/sttrs311.ttt 0.2 sec.
/usr/home/aoto/ttt/sttrs313.ttt 5.41 sec.
/usr/home/aoto/ttt/sttrs315.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs319.ttt 0.06 sec.
/usr/home/aoto/ttt/sttrs322.ttt 0.06 sec.
/usr/home/aoto/ttt/sttrs326.ttt 0.06 sec.
/usr/home/aoto/ttt/sttrs402.ttt 0.05 sec.
/usr/home/aoto/ttt/sttrs403.ttt 0.09 sec.
/usr/home/aoto/ttt/sttrs406.ttt 0.07 sec.
/usr/home/aoto/ttt/sttrs407.ttt 0.14 sec.
/usr/home/aoto/ttt/sttrs408.ttt 0.09 sec.
/usr/home/aoto/ttt/sttrs409.ttt 4.12 sec.
/usr/home/aoto/ttt/sttrs412.ttt 0.08 sec.
/usr/home/aoto/ttt/sttrs413.ttt 0.47 sec.
/usr/home/aoto/ttt/sttrs414.ttt 0.09 sec.
/usr/home/aoto/ttt/sttrs415.ttt 0.18 sec.
/usr/home/aoto/ttt/sttrs505.ttt 0.09 sec.
/usr/home/aoto/ttt/sttrs601.ttt 0.25 sec.
/usr/home/aoto/ttt/sttrs603.ttt 0.13 sec.
/usr/home/aoto/ttt/sttrs703.ttt 3.86 sec.
/usr/home/aoto/ttt/sttrs101.ttt 0.01 sec.

The following files failed:

/usr/home/aoto/ttt/sttrs803.ttt 3.55 sec.
/usr/home/aoto/ttt/sttrs107.ttt 0.3 sec.
/usr/home/aoto/ttt/sttrs110.ttt 0.53 sec.
/usr/home/aoto/ttt/sttrs114.ttt 0.09 sec.
/usr/home/aoto/ttt/sttrs118.ttt 0.05 sec.
/usr/home/aoto/ttt/sttrs119.ttt 0.09 sec.
/usr/home/aoto/ttt/sttrs120.ttt 0.03 sec.
/usr/home/aoto/ttt/sttrs123.ttt 0.1 sec.
/usr/home/aoto/ttt/sttrs132.ttt 0.09 sec.
/usr/home/aoto/ttt/sttrs135.ttt 0.08 sec.
/usr/home/aoto/ttt/sttrs136.ttt 0.19 sec.
/usr/home/aoto/ttt/sttrs201.ttt 0.04 sec.
/usr/home/aoto/ttt/sttrs204.ttt 0.14 sec.
/usr/home/aoto/ttt/sttrs205.ttt 0.17 sec.
/usr/home/aoto/ttt/sttrs208.ttt 0.22 sec.
/usr/home/aoto/ttt/sttrs209.ttt 0.27 sec.
/usr/home/aoto/ttt/sttrs212.ttt 0.28 sec.
/usr/home/aoto/ttt/sttrs216.ttt 0.1 sec.
/usr/home/aoto/ttt/sttrs217.ttt 0.15 sec.
/usr/home/aoto/ttt/sttrs218.ttt 0.07 sec.
/usr/home/aoto/ttt/sttrs220.ttt 0.04 sec.
/usr/home/aoto/ttt/sttrs221.ttt 0.01 sec.
/usr/home/aoto/ttt/sttrs304.ttt 0.12 sec.
/usr/home/aoto/ttt/sttrs307.ttt 0.18 sec.
/usr/home/aoto/ttt/sttrs310.ttt 0.59 sec.
/usr/home/aoto/ttt/sttrs312.ttt 0.15 sec.
/usr/home/aoto/ttt/sttrs314.ttt 0.32 sec.
/usr/home/aoto/ttt/sttrs316.ttt 0.06 sec.
/usr/home/aoto/ttt/sttrs317.ttt 0.11 sec.
/usr/home/aoto/ttt/sttrs318.ttt 0.3 sec.
/usr/home/aoto/ttt/sttrs320.ttt 0.15 sec.
/usr/home/aoto/ttt/sttrs321.ttt 0.16 sec.
/usr/home/aoto/ttt/sttrs323.ttt 0.08 sec.
/usr/home/aoto/ttt/sttrs324.ttt 0.12 sec.
/usr/home/aoto/ttt/sttrs325.ttt 0.07 sec.
/usr/home/aoto/ttt/sttrs327.ttt 0.13 sec.
/usr/home/aoto/ttt/sttrs328.ttt 0.13 sec.
/usr/home/aoto/ttt/sttrs401.ttt 0.28 sec.
/usr/home/aoto/ttt/sttrs404.ttt 0.26 sec.
/usr/home/aoto/ttt/sttrs405.ttt 0.23 sec.
/usr/home/aoto/ttt/sttrs410.ttt 3.33 sec.
/usr/home/aoto/ttt/sttrs411.ttt 0.14 sec.
/usr/home/aoto/ttt/sttrs501.ttt 0.42 sec.
/usr/home/aoto/ttt/sttrs502.ttt 0.2 sec.
/usr/home/aoto/ttt/sttrs504.ttt 0.87 sec.
/usr/home/aoto/ttt/sttrs506.ttt 0.21 sec.
/usr/home/aoto/ttt/sttrs507.ttt 0.23 sec.
/usr/home/aoto/ttt/sttrs602.ttt 0.52 sec.
/usr/home/aoto/ttt/sttrs604.ttt 0.71 sec.
/usr/home/aoto/ttt/sttrs605.ttt 0.31 sec.
/usr/home/aoto/ttt/sttrs702.ttt 5.71 sec.
/usr/home/aoto/ttt/sttrs704.ttt 12.4 sec.
/usr/home/aoto/ttt/sttrs705.ttt 0.92 sec.
/usr/home/aoto/ttt/sttrs706.ttt 1.3 sec.
/usr/home/aoto/ttt/sttrs801.ttt 1.7 sec.

The following files timed out:

/usr/home/aoto/ttt/sttrs503.ttt 30.0 sec.
/usr/home/aoto/ttt/sttrs701.ttt 30.0 sec.
/usr/home/aoto/ttt/sttrs802.ttt 30.0 sec.

Duration of 67 successful proofs: 23.39 sec.
Duration of 55 failed proofs: 39.0 sec.
Duration of 3 time outs: 90.0 sec.

Total duration: 152.39 sec.