YES(O(1), O(n^3)) 23.62/6.76 YES(O(1), O(n^3)) 23.62/6.78 23.62/6.78 23.62/6.78 23.62/6.78 23.62/6.78 23.62/6.78 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 23.62/6.78 23.62/6.78 23.62/6.78
23.62/6.78
23.62/6.78
23.62/6.78
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS
23.62/6.78 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
23.62/6.78 
↳2 CdtProblem
23.62/6.78 
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
23.62/6.78 
↳4 CdtProblem
23.62/6.78 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
23.62/6.78 
↳6 CdtProblem
23.62/6.78 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
23.62/6.78 
↳8 CdtProblem
23.62/6.78 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
23.62/6.78 
↳10 CdtProblem
23.62/6.78 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
23.62/6.78 
↳12 CdtProblem
23.62/6.78 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
23.62/6.78 
↳14 CdtProblem
23.62/6.78 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))))
23.62/6.78 
↳16 CdtProblem
23.62/6.78 
↳17 SIsEmptyProof (BOTH BOUNDS(ID, ID))
23.62/6.78 
↳18 BOUNDS(O(1), O(1))
23.62/6.78
23.62/6.78 23.62/6.78
23.62/6.78
23.62/6.78

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

minus(x, 0) → x 23.62/6.78
minus(s(x), s(y)) → minus(x, y) 23.62/6.78
quot(0, s(y)) → 0 23.62/6.78
quot(s(x), s(y)) → s(quot(minus(x, y), s(y))) 23.62/6.78
app(nil, y) → y 23.62/6.78
app(add(n, x), y) → add(n, app(x, y)) 23.62/6.78
reverse(nil) → nil 23.62/6.78
reverse(add(n, x)) → app(reverse(x), add(n, nil)) 23.62/6.78
shuffle(nil) → nil 23.62/6.78
shuffle(add(n, x)) → add(n, shuffle(reverse(x))) 23.62/6.78
concat(leaf, y) → y 23.62/6.78
concat(cons(u, v), y) → cons(u, concat(v, y)) 23.62/6.78
less_leaves(x, leaf) → false 23.62/6.78
less_leaves(leaf, cons(w, z)) → true 23.62/6.78
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Rewrite Strategy: INNERMOST
23.62/6.78
23.62/6.78

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
23.62/6.78
23.62/6.78

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, 0) → z0 23.62/6.78
minus(s(z0), s(z1)) → minus(z0, z1) 23.62/6.78
quot(0, s(z0)) → 0 23.62/6.79
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 23.62/6.79
app(nil, z0) → z0 23.62/6.79
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.62/6.79
reverse(nil) → nil 23.62/6.79
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.62/6.79
shuffle(nil) → nil 23.62/6.79
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1))) 23.62/6.79
concat(leaf, z0) → z0 23.62/6.79
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.62/6.79
less_leaves(z0, leaf) → false 23.62/6.79
less_leaves(leaf, cons(z0, z1)) → true 23.62/6.79
less_leaves(cons(z0, z1), cons(z2, z3)) → less_leaves(concat(z0, z1), concat(z2, z3))
Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.62/6.79
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.62/6.79
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.62/6.79
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.62/6.79
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.62/6.79
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.62/6.79
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.62/6.79
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.62/6.79
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.62/6.79
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.62/6.79
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.62/6.79
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.62/6.79
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
K tuples:none
Defined Rule Symbols:

minus, quot, app, reverse, shuffle, concat, less_leaves

Defined Pair Symbols:

MINUS, QUOT, APP, REVERSE, SHUFFLE, CONCAT, LESS_LEAVES

Compound Symbols:

c1, c3, c5, c7, c9, c11, c14

23.62/6.79
23.62/6.79

(3) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
We considered the (Usable) Rules:

concat(leaf, z0) → z0 23.62/6.79
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.62/6.79
reverse(nil) → nil 23.62/6.79
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.62/6.79
app(nil, z0) → z0 23.62/6.79
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.62/6.79
minus(z0, 0) → z0 23.62/6.79
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.62/6.79
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.62/6.79
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.62/6.79
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.62/6.79
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.62/6.79
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.62/6.79
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.62/6.79

POL(0) = 0    23.62/6.79
POL(APP(x1, x2)) = 0    23.62/6.79
POL(CONCAT(x1, x2)) = 0    23.62/6.79
POL(LESS_LEAVES(x1, x2)) = 0    23.62/6.79
POL(MINUS(x1, x2)) = 0    23.62/6.79
POL(QUOT(x1, x2)) = x1    23.62/6.79
POL(REVERSE(x1)) = 0    23.62/6.79
POL(SHUFFLE(x1)) = 0    23.62/6.79
POL(add(x1, x2)) = 0    23.62/6.79
POL(app(x1, x2)) = [2] + x2    23.62/6.79
POL(c1(x1)) = x1    23.62/6.79
POL(c11(x1)) = x1    23.62/6.79
POL(c14(x1, x2, x3)) = x1 + x2 + x3    23.62/6.79
POL(c3(x1, x2)) = x1 + x2    23.62/6.79
POL(c5(x1)) = x1    23.62/6.79
POL(c7(x1, x2)) = x1 + x2    23.62/6.79
POL(c9(x1, x2)) = x1 + x2    23.62/6.79
POL(concat(x1, x2)) = [2] + [2]x1 + [2]x2    23.62/6.79
POL(cons(x1, x2)) = [4]    23.62/6.79
POL(leaf) = 0    23.62/6.79
POL(minus(x1, x2)) = x1    23.62/6.79
POL(nil) = 0    23.62/6.79
POL(reverse(x1)) = [4]    23.62/6.79
POL(s(x1)) = [1] + x1   
23.62/6.79
23.62/6.79

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, 0) → z0 23.62/6.79
minus(s(z0), s(z1)) → minus(z0, z1) 23.62/6.79
quot(0, s(z0)) → 0 23.62/6.79
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 23.62/6.79
app(nil, z0) → z0 23.62/6.79
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.62/6.79
reverse(nil) → nil 23.62/6.79
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.62/6.79
shuffle(nil) → nil 23.62/6.79
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1))) 23.62/6.79
concat(leaf, z0) → z0 23.62/6.79
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.62/6.79
less_leaves(z0, leaf) → false 23.62/6.79
less_leaves(leaf, cons(z0, z1)) → true 23.62/6.79
less_leaves(cons(z0, z1), cons(z2, z3)) → less_leaves(concat(z0, z1), concat(z2, z3))
Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.62/6.79
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.62/6.79
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.62/6.79
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
K tuples:

QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
Defined Rule Symbols:

minus, quot, app, reverse, shuffle, concat, less_leaves

Defined Pair Symbols:

MINUS, QUOT, APP, REVERSE, SHUFFLE, CONCAT, LESS_LEAVES

Compound Symbols:

c1, c3, c5, c7, c9, c11, c14

23.96/6.82
23.96/6.82

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1))
We considered the (Usable) Rules:

concat(leaf, z0) → z0 23.96/6.82
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.82
reverse(nil) → nil 23.96/6.82
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.82
app(nil, z0) → z0 23.96/6.82
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.82
minus(z0, 0) → z0 23.96/6.82
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.96/6.82

POL(0) = 0    23.96/6.82
POL(APP(x1, x2)) = 0    23.96/6.82
POL(CONCAT(x1, x2)) = 0    23.96/6.82
POL(LESS_LEAVES(x1, x2)) = 0    23.96/6.82
POL(MINUS(x1, x2)) = 0    23.96/6.82
POL(QUOT(x1, x2)) = 0    23.96/6.82
POL(REVERSE(x1)) = [3]    23.96/6.82
POL(SHUFFLE(x1)) = [4]x1    23.96/6.82
POL(add(x1, x2)) = [1] + x2    23.96/6.82
POL(app(x1, x2)) = x1 + x2    23.96/6.82
POL(c1(x1)) = x1    23.96/6.82
POL(c11(x1)) = x1    23.96/6.82
POL(c14(x1, x2, x3)) = x1 + x2 + x3    23.96/6.82
POL(c3(x1, x2)) = x1 + x2    23.96/6.82
POL(c5(x1)) = x1    23.96/6.82
POL(c7(x1, x2)) = x1 + x2    23.96/6.82
POL(c9(x1, x2)) = x1 + x2    23.96/6.82
POL(concat(x1, x2)) = 0    23.96/6.82
POL(cons(x1, x2)) = 0    23.96/6.82
POL(leaf) = 0    23.96/6.82
POL(minus(x1, x2)) = [1] + [3]x2    23.96/6.82
POL(nil) = 0    23.96/6.82
POL(reverse(x1)) = x1    23.96/6.82
POL(s(x1)) = x1   
23.96/6.82
23.96/6.82

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, 0) → z0 23.96/6.82
minus(s(z0), s(z1)) → minus(z0, z1) 23.96/6.82
quot(0, s(z0)) → 0 23.96/6.82
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 23.96/6.82
app(nil, z0) → z0 23.96/6.82
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.82
reverse(nil) → nil 23.96/6.82
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.82
shuffle(nil) → nil 23.96/6.82
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1))) 23.96/6.82
concat(leaf, z0) → z0 23.96/6.82
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.82
less_leaves(z0, leaf) → false 23.96/6.82
less_leaves(leaf, cons(z0, z1)) → true 23.96/6.82
less_leaves(cons(z0, z1), cons(z2, z3)) → less_leaves(concat(z0, z1), concat(z2, z3))
Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
K tuples:

QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1))
Defined Rule Symbols:

minus, quot, app, reverse, shuffle, concat, less_leaves

Defined Pair Symbols:

MINUS, QUOT, APP, REVERSE, SHUFFLE, CONCAT, LESS_LEAVES

Compound Symbols:

c1, c3, c5, c7, c9, c11, c14

23.96/6.82
23.96/6.82

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
We considered the (Usable) Rules:

concat(leaf, z0) → z0 23.96/6.82
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.82
reverse(nil) → nil 23.96/6.82
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.82
app(nil, z0) → z0 23.96/6.82
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.82
minus(z0, 0) → z0 23.96/6.82
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.96/6.82

POL(0) = 0    23.96/6.82
POL(APP(x1, x2)) = 0    23.96/6.82
POL(CONCAT(x1, x2)) = 0    23.96/6.82
POL(LESS_LEAVES(x1, x2)) = [4]x2    23.96/6.82
POL(MINUS(x1, x2)) = 0    23.96/6.82
POL(QUOT(x1, x2)) = 0    23.96/6.82
POL(REVERSE(x1)) = 0    23.96/6.82
POL(SHUFFLE(x1)) = 0    23.96/6.82
POL(add(x1, x2)) = 0    23.96/6.82
POL(app(x1, x2)) = [3]    23.96/6.82
POL(c1(x1)) = x1    23.96/6.82
POL(c11(x1)) = x1    23.96/6.82
POL(c14(x1, x2, x3)) = x1 + x2 + x3    23.96/6.82
POL(c3(x1, x2)) = x1 + x2    23.96/6.82
POL(c5(x1)) = x1    23.96/6.82
POL(c7(x1, x2)) = x1 + x2    23.96/6.82
POL(c9(x1, x2)) = x1 + x2    23.96/6.82
POL(concat(x1, x2)) = x1 + x2    23.96/6.82
POL(cons(x1, x2)) = [2] + x1 + x2    23.96/6.82
POL(leaf) = 0    23.96/6.82
POL(minus(x1, x2)) = [1] + [3]x2    23.96/6.82
POL(nil) = 0    23.96/6.82
POL(reverse(x1)) = 0    23.96/6.82
POL(s(x1)) = 0   
23.96/6.82
23.96/6.82

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, 0) → z0 23.96/6.82
minus(s(z0), s(z1)) → minus(z0, z1) 23.96/6.82
quot(0, s(z0)) → 0 23.96/6.82
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 23.96/6.82
app(nil, z0) → z0 23.96/6.82
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.82
reverse(nil) → nil 23.96/6.82
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.82
shuffle(nil) → nil 23.96/6.82
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1))) 23.96/6.82
concat(leaf, z0) → z0 23.96/6.82
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.82
less_leaves(z0, leaf) → false 23.96/6.82
less_leaves(leaf, cons(z0, z1)) → true 23.96/6.82
less_leaves(cons(z0, z1), cons(z2, z3)) → less_leaves(concat(z0, z1), concat(z2, z3))
Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2))
K tuples:

QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
Defined Rule Symbols:

minus, quot, app, reverse, shuffle, concat, less_leaves

Defined Pair Symbols:

MINUS, QUOT, APP, REVERSE, SHUFFLE, CONCAT, LESS_LEAVES

Compound Symbols:

c1, c3, c5, c7, c9, c11, c14

23.96/6.82
23.96/6.82

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
We considered the (Usable) Rules:

concat(leaf, z0) → z0 23.96/6.82
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.82
reverse(nil) → nil 23.96/6.82
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.82
app(nil, z0) → z0 23.96/6.82
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.82
minus(z0, 0) → z0 23.96/6.82
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.96/6.82

POL(0) = [3]    23.96/6.82
POL(APP(x1, x2)) = 0    23.96/6.82
POL(CONCAT(x1, x2)) = 0    23.96/6.82
POL(LESS_LEAVES(x1, x2)) = 0    23.96/6.82
POL(MINUS(x1, x2)) = [2]x1    23.96/6.82
POL(QUOT(x1, x2)) = x12    23.96/6.82
POL(REVERSE(x1)) = 0    23.96/6.82
POL(SHUFFLE(x1)) = 0    23.96/6.82
POL(add(x1, x2)) = 0    23.96/6.82
POL(app(x1, x2)) = [3] + [3]x22 + [3]x1·x2 + [3]x12    23.96/6.82
POL(c1(x1)) = x1    23.96/6.82
POL(c11(x1)) = x1    23.96/6.82
POL(c14(x1, x2, x3)) = x1 + x2 + x3    23.96/6.82
POL(c3(x1, x2)) = x1 + x2    23.96/6.82
POL(c5(x1)) = x1    23.96/6.82
POL(c7(x1, x2)) = x1 + x2    23.96/6.82
POL(c9(x1, x2)) = x1 + x2    23.96/6.82
POL(concat(x1, x2)) = 0    23.96/6.82
POL(cons(x1, x2)) = 0    23.96/6.82
POL(leaf) = [3]    23.96/6.82
POL(minus(x1, x2)) = [1] + x1    23.96/6.82
POL(nil) = 0    23.96/6.82
POL(reverse(x1)) = 0    23.96/6.82
POL(s(x1)) = [2] + x1   
23.96/6.82
23.96/6.82

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, 0) → z0 23.96/6.82
minus(s(z0), s(z1)) → minus(z0, z1) 23.96/6.82
quot(0, s(z0)) → 0 23.96/6.82
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 23.96/6.82
app(nil, z0) → z0 23.96/6.82
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.82
reverse(nil) → nil 23.96/6.82
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.82
shuffle(nil) → nil 23.96/6.82
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1))) 23.96/6.82
concat(leaf, z0) → z0 23.96/6.82
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.82
less_leaves(z0, leaf) → false 23.96/6.82
less_leaves(leaf, cons(z0, z1)) → true 23.96/6.82
less_leaves(cons(z0, z1), cons(z2, z3)) → less_leaves(concat(z0, z1), concat(z2, z3))
Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
S tuples:

APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2))
K tuples:

QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3)) 23.96/6.82
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
Defined Rule Symbols:

minus, quot, app, reverse, shuffle, concat, less_leaves

Defined Pair Symbols:

MINUS, QUOT, APP, REVERSE, SHUFFLE, CONCAT, LESS_LEAVES

Compound Symbols:

c1, c3, c5, c7, c9, c11, c14

23.96/6.82
23.96/6.82

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2))
We considered the (Usable) Rules:

concat(leaf, z0) → z0 23.96/6.82
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.82
reverse(nil) → nil 23.96/6.82
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.82
app(nil, z0) → z0 23.96/6.82
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.82
minus(z0, 0) → z0 23.96/6.82
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.96/6.82

POL(0) = [3]    23.96/6.82
POL(APP(x1, x2)) = 0    23.96/6.82
POL(CONCAT(x1, x2)) = [1] + [2]x1    23.96/6.82
POL(LESS_LEAVES(x1, x2)) = [2]x1·x2 + x12    23.96/6.82
POL(MINUS(x1, x2)) = 0    23.96/6.82
POL(QUOT(x1, x2)) = x1 + x12    23.96/6.82
POL(REVERSE(x1)) = 0    23.96/6.82
POL(SHUFFLE(x1)) = 0    23.96/6.82
POL(add(x1, x2)) = 0    23.96/6.82
POL(app(x1, x2)) = [3] + [3]x22 + [3]x1·x2 + [3]x12    23.96/6.82
POL(c1(x1)) = x1    23.96/6.82
POL(c11(x1)) = x1    23.96/6.82
POL(c14(x1, x2, x3)) = x1 + x2 + x3    23.96/6.82
POL(c3(x1, x2)) = x1 + x2    23.96/6.82
POL(c5(x1)) = x1    23.96/6.82
POL(c7(x1, x2)) = x1 + x2    23.96/6.82
POL(c9(x1, x2)) = x1 + x2    23.96/6.82
POL(concat(x1, x2)) = [1] + x1 + x2    23.96/6.82
POL(cons(x1, x2)) = [2] + x1 + x2    23.96/6.82
POL(leaf) = [3]    23.96/6.82
POL(minus(x1, x2)) = x1    23.96/6.82
POL(nil) = 0    23.96/6.82
POL(reverse(x1)) = 0    23.96/6.82
POL(s(x1)) = [1] + x1   
23.96/6.82
23.96/6.82

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, 0) → z0 23.96/6.82
minus(s(z0), s(z1)) → minus(z0, z1) 23.96/6.82
quot(0, s(z0)) → 0 23.96/6.82
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 23.96/6.82
app(nil, z0) → z0 23.96/6.82
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.82
reverse(nil) → nil 23.96/6.82
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.82
shuffle(nil) → nil 23.96/6.82
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1))) 23.96/6.82
concat(leaf, z0) → z0 23.96/6.82
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.82
less_leaves(z0, leaf) → false 23.96/6.82
less_leaves(leaf, cons(z0, z1)) → true 23.96/6.82
less_leaves(cons(z0, z1), cons(z2, z3)) → less_leaves(concat(z0, z1), concat(z2, z3))
Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
S tuples:

APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
K tuples:

QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3)) 23.96/6.82
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2))
Defined Rule Symbols:

minus, quot, app, reverse, shuffle, concat, less_leaves

Defined Pair Symbols:

MINUS, QUOT, APP, REVERSE, SHUFFLE, CONCAT, LESS_LEAVES

Compound Symbols:

c1, c3, c5, c7, c9, c11, c14

23.96/6.82
23.96/6.82

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
We considered the (Usable) Rules:

concat(leaf, z0) → z0 23.96/6.82
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.82
reverse(nil) → nil 23.96/6.82
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.82
app(nil, z0) → z0 23.96/6.82
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.82
minus(z0, 0) → z0 23.96/6.82
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.82
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.82
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.82
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.82
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.82
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.82
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.96/6.83

POL(0) = [3]    23.96/6.83
POL(APP(x1, x2)) = 0    23.96/6.83
POL(CONCAT(x1, x2)) = 0    23.96/6.83
POL(LESS_LEAVES(x1, x2)) = x1·x2    23.96/6.83
POL(MINUS(x1, x2)) = 0    23.96/6.83
POL(QUOT(x1, x2)) = 0    23.96/6.83
POL(REVERSE(x1)) = x1    23.96/6.83
POL(SHUFFLE(x1)) = [2]x12    23.96/6.83
POL(add(x1, x2)) = [1] + x1 + x2    23.96/6.83
POL(app(x1, x2)) = x1 + x2    23.96/6.83
POL(c1(x1)) = x1    23.96/6.83
POL(c11(x1)) = x1    23.96/6.83
POL(c14(x1, x2, x3)) = x1 + x2 + x3    23.96/6.83
POL(c3(x1, x2)) = x1 + x2    23.96/6.83
POL(c5(x1)) = x1    23.96/6.83
POL(c7(x1, x2)) = x1 + x2    23.96/6.83
POL(c9(x1, x2)) = x1 + x2    23.96/6.83
POL(concat(x1, x2)) = x1 + x2    23.96/6.83
POL(cons(x1, x2)) = [2] + x1 + x2    23.96/6.83
POL(leaf) = [3]    23.96/6.83
POL(minus(x1, x2)) = 0    23.96/6.83
POL(nil) = 0    23.96/6.83
POL(reverse(x1)) = x1    23.96/6.83
POL(s(x1)) = 0   
23.96/6.83
23.96/6.83

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, 0) → z0 23.96/6.83
minus(s(z0), s(z1)) → minus(z0, z1) 23.96/6.83
quot(0, s(z0)) → 0 23.96/6.83
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 23.96/6.83
app(nil, z0) → z0 23.96/6.83
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.83
reverse(nil) → nil 23.96/6.83
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.83
shuffle(nil) → nil 23.96/6.83
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1))) 23.96/6.83
concat(leaf, z0) → z0 23.96/6.83
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.83
less_leaves(z0, leaf) → false 23.96/6.83
less_leaves(leaf, cons(z0, z1)) → true 23.96/6.83
less_leaves(cons(z0, z1), cons(z2, z3)) → less_leaves(concat(z0, z1), concat(z2, z3))
Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.83
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.83
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.83
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.83
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.83
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.83
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
S tuples:

APP(add(z0, z1), z2) → c5(APP(z1, z2))
K tuples:

QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.83
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.83
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3)) 23.96/6.83
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.83
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.83
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
Defined Rule Symbols:

minus, quot, app, reverse, shuffle, concat, less_leaves

Defined Pair Symbols:

MINUS, QUOT, APP, REVERSE, SHUFFLE, CONCAT, LESS_LEAVES

Compound Symbols:

c1, c3, c5, c7, c9, c11, c14

23.96/6.83
23.96/6.83

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

APP(add(z0, z1), z2) → c5(APP(z1, z2))
We considered the (Usable) Rules:

concat(leaf, z0) → z0 23.96/6.83
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.83
reverse(nil) → nil 23.96/6.83
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.83
app(nil, z0) → z0 23.96/6.83
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.83
minus(z0, 0) → z0 23.96/6.83
minus(s(z0), s(z1)) → minus(z0, z1)
And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.83
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.83
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.83
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.83
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.83
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.83
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation : 23.96/6.83

POL(0) = [1]    23.96/6.83
POL(APP(x1, x2)) = x1    23.96/6.83
POL(CONCAT(x1, x2)) = 0    23.96/6.83
POL(LESS_LEAVES(x1, x2)) = 0    23.96/6.83
POL(MINUS(x1, x2)) = 0    23.96/6.83
POL(QUOT(x1, x2)) = 0    23.96/6.83
POL(REVERSE(x1)) = x12    23.96/6.83
POL(SHUFFLE(x1)) = x12 + x13    23.96/6.83
POL(add(x1, x2)) = [1] + x2    23.96/6.83
POL(app(x1, x2)) = x1 + x2    23.96/6.83
POL(c1(x1)) = x1    23.96/6.83
POL(c11(x1)) = x1    23.96/6.83
POL(c14(x1, x2, x3)) = x1 + x2 + x3    23.96/6.83
POL(c3(x1, x2)) = x1 + x2    23.96/6.83
POL(c5(x1)) = x1    23.96/6.83
POL(c7(x1, x2)) = x1 + x2    23.96/6.83
POL(c9(x1, x2)) = x1 + x2    23.96/6.83
POL(concat(x1, x2)) = x13 + x12·x2 + x1·x22 + x23    23.96/6.83
POL(cons(x1, x2)) = 0    23.96/6.83
POL(leaf) = [1]    23.96/6.83
POL(minus(x1, x2)) = x23    23.96/6.83
POL(nil) = 0    23.96/6.83
POL(reverse(x1)) = x1    23.96/6.83
POL(s(x1)) = 0   
23.96/6.83
23.96/6.83

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(z0, 0) → z0 23.96/6.83
minus(s(z0), s(z1)) → minus(z0, z1) 23.96/6.83
quot(0, s(z0)) → 0 23.96/6.83
quot(s(z0), s(z1)) → s(quot(minus(z0, z1), s(z1))) 23.96/6.83
app(nil, z0) → z0 23.96/6.83
app(add(z0, z1), z2) → add(z0, app(z1, z2)) 23.96/6.83
reverse(nil) → nil 23.96/6.83
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil)) 23.96/6.83
shuffle(nil) → nil 23.96/6.83
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1))) 23.96/6.83
concat(leaf, z0) → z0 23.96/6.83
concat(cons(z0, z1), z2) → cons(z0, concat(z1, z2)) 23.96/6.83
less_leaves(z0, leaf) → false 23.96/6.83
less_leaves(leaf, cons(z0, z1)) → true 23.96/6.83
less_leaves(cons(z0, z1), cons(z2, z3)) → less_leaves(concat(z0, z1), concat(z2, z3))
Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.83
QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.83
APP(add(z0, z1), z2) → c5(APP(z1, z2)) 23.96/6.83
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.83
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.83
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.83
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3))
S tuples:none
K tuples:

QUOT(s(z0), s(z1)) → c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) 23.96/6.83
SHUFFLE(add(z0, z1)) → c9(SHUFFLE(reverse(z1)), REVERSE(z1)) 23.96/6.83
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) → c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1), CONCAT(z2, z3)) 23.96/6.83
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1)) 23.96/6.83
CONCAT(cons(z0, z1), z2) → c11(CONCAT(z1, z2)) 23.96/6.83
REVERSE(add(z0, z1)) → c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) 23.96/6.83
APP(add(z0, z1), z2) → c5(APP(z1, z2))
Defined Rule Symbols:

minus, quot, app, reverse, shuffle, concat, less_leaves

Defined Pair Symbols:

MINUS, QUOT, APP, REVERSE, SHUFFLE, CONCAT, LESS_LEAVES

Compound Symbols:

c1, c3, c5, c7, c9, c11, c14

23.96/6.83
23.96/6.83

(17) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
23.96/6.83
23.96/6.83

(18) BOUNDS(O(1), O(1))

23.96/6.83
23.96/6.83
23.96/6.88 EOF