The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

rev(a) → a
rev(b) → b
rev(++(z0, z1)) → ++(rev(z1), rev(z0))
rev(++(z0, z0)) → rev(z0)

Tuples:

REV(a) → c
REV(b) → c1
REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

S tuples:

REV(a) → c
REV(b) → c1
REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

K tuples:none
Defined Rule Symbols:

rev

Defined Pair Symbols:

REV

Compound Symbols:

c, c1, c2, c3

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

Removed 2 trailing nodes:

REV(b) → c1
REV(a) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

rev(a) → a
rev(b) → b
rev(++(z0, z1)) → ++(rev(z1), rev(z0))
rev(++(z0, z0)) → rev(z0)

Tuples:

REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

S tuples:

REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

K tuples:none
Defined Rule Symbols:

rev

Defined Pair Symbols:

REV

Compound Symbols:

c2, c3

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

rev(a) → a
rev(b) → b
rev(++(z0, z1)) → ++(rev(z1), rev(z0))
rev(++(z0, z0)) → rev(z0)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

S tuples:

REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

REV

Compound Symbols:

c2, c3

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

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

REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

We considered the (Usable) Rules:none
And the Tuples:

REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(++(x₁, x₂)) = [1] + x₁ + x₂   
POL(REV(x₁)) = x₁ + x₁²   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁)) = x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

S tuples:none
K tuples:

REV(++(z0, z1)) → c2(REV(z1), REV(z0))
REV(++(z0, z0)) → c3(REV(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

REV

Compound Symbols:

c2, c3

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

The set S is empty

(12) BOUNDS(1, 1)