The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

0 CpxTRS
↳1 RcToIrcProof (BOTH BOUNDS(ID, ID), 61 ms)
↳2 CpxTRS
↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 3 ms)
↳8 CdtProblem
↳9 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 132 ms)
↳10 CdtProblem
↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 48 ms)
↳12 CdtProblem
↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)), 78 ms)
↳14 CdtProblem
↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 BOUNDS(1, 1)

(0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0
app(add(z0, z1), z2) → add(z0, app(z1, z2))
reverse(nil) → nil
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil))
shuffle(nil) → nil
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1)))
Tuples:

APP(nil, z0) → c
APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(nil) → c2
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) → c4
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
S tuples:

APP(nil, z0) → c
APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(nil) → c2
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) → c4
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
K tuples:none
Defined Rule Symbols:

app, reverse, shuffle

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c, c1, c2, c3, c4, c5

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

APP(nil, z0) → c
SHUFFLE(nil) → c4
REVERSE(nil) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0
app(add(z0, z1), z2) → add(z0, app(z1, z2))
reverse(nil) → nil
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil))
shuffle(nil) → nil
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1)))
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
S tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
K tuples:none
Defined Rule Symbols:

app, reverse, shuffle

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

shuffle(nil) → nil
shuffle(add(z0, z1)) → add(z0, shuffle(reverse(z1)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

reverse(nil) → nil
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil))
app(nil, z0) → z0
app(add(z0, z1), z2) → add(z0, app(z1, z2))
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
S tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
K tuples:none
Defined Rule Symbols:

reverse, app

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
We considered the (Usable) Rules:

reverse(nil) → nil
app(add(z0, z1), z2) → add(z0, app(z1, z2))
app(nil, z0) → z0
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil))
And the Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(APP(x1, x2)) = 0   
POL(REVERSE(x1)) = 0   
POL(SHUFFLE(x1)) = x1   
POL(add(x1, x2)) = [1] + x2   
POL(app(x1, x2)) = x1 + x2   
POL(c1(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(nil) = 0   
POL(reverse(x1)) = x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

reverse(nil) → nil
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil))
app(nil, z0) → z0
app(add(z0, z1), z2) → add(z0, app(z1, z2))
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
S tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
K tuples:

SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
Defined Rule Symbols:

reverse, app

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
We considered the (Usable) Rules:

reverse(nil) → nil
app(add(z0, z1), z2) → add(z0, app(z1, z2))
app(nil, z0) → z0
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil))
And the Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(APP(x1, x2)) = 0   
POL(REVERSE(x1)) = [1] + x1   
POL(SHUFFLE(x1)) = [2]x12   
POL(add(x1, x2)) = [1] + x1 + x2   
POL(app(x1, x2)) = x1 + x2   
POL(c1(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(nil) = 0   
POL(reverse(x1)) = x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

reverse(nil) → nil
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil))
app(nil, z0) → z0
app(add(z0, z1), z2) → add(z0, app(z1, z2))
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
S tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
K tuples:

SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
Defined Rule Symbols:

reverse, app

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(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) → c1(APP(z1, z2))
We considered the (Usable) Rules:

reverse(nil) → nil
app(add(z0, z1), z2) → add(z0, app(z1, z2))
app(nil, z0) → z0
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil))
And the Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(APP(x1, x2)) = x1 + x1·x2   
POL(REVERSE(x1)) = [1] + x12   
POL(SHUFFLE(x1)) = x1 + x12 + x13   
POL(add(x1, x2)) = [1] + x2   
POL(app(x1, x2)) = x1 + x2   
POL(c1(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(nil) = 0   
POL(reverse(x1)) = x1   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

reverse(nil) → nil
reverse(add(z0, z1)) → app(reverse(z1), add(z0, nil))
app(nil, z0) → z0
app(add(z0, z1), z2) → add(z0, app(z1, z2))
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
S tuples:none
K tuples:

SHUFFLE(add(z0, z1)) → c5(SHUFFLE(reverse(z1)), REVERSE(z1))
REVERSE(add(z0, z1)) → c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
APP(add(z0, z1), z2) → c1(APP(z1, z2))
Defined Rule Symbols:

reverse, app

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

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

The set S is empty

(16) BOUNDS(1, 1)