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

(0) Obligation:

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


The TRS R consists of the following rules:

flatten(nil) → nil
flatten(unit(x)) → flatten(x)
flatten(++(x, y)) → ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) → ++(flatten(x), flatten(y))
flatten(flatten(x)) → flatten(x)
rev(nil) → nil
rev(unit(x)) → unit(x)
rev(++(x, y)) → ++(rev(y), rev(x))
rev(rev(x)) → x
++(x, nil) → x
++(nil, y) → y
++(++(x, y), z) → ++(x, ++(y, z))

Rewrite Strategy: INNERMOST

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

The following rules are not reachable from basic terms in the dependency graph and can be removed:
rev(++(x, y)) → ++(rev(y), rev(x))
rev(rev(x)) → x

(2) Obligation:

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


The TRS R consists of the following rules:

flatten(nil) → nil
rev(unit(x)) → unit(x)
flatten(++(unit(x), y)) → ++(flatten(x), flatten(y))
flatten(++(x, y)) → ++(flatten(x), flatten(y))
++(++(x, y), z) → ++(x, ++(y, z))
rev(nil) → nil
flatten(flatten(x)) → flatten(x)
flatten(unit(x)) → flatten(x)
++(x, nil) → x
++(nil, y) → y

Rewrite Strategy: INNERMOST

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
flatten(++(unit(x), y)) → ++(flatten(x), flatten(y))
flatten(++(x, y)) → ++(flatten(x), flatten(y))
++(++(x, y), z) → ++(x, ++(y, z))
flatten(flatten(x)) → flatten(x)

(4) Obligation:

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


The TRS R consists of the following rules:

flatten(nil) → nil
rev(unit(x)) → unit(x)
rev(nil) → nil
flatten(unit(x)) → flatten(x)
++(x, nil) → x
++(nil, y) → y

Rewrite Strategy: INNERMOST

(5) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3]
transitions:
nil0() → 0
unit0(0) → 0
flatten0(0) → 1
rev0(0) → 2
++0(0, 0) → 3
nil1() → 1
unit1(0) → 2
nil1() → 2
flatten1(0) → 1
0 → 3

(6) BOUNDS(1, n^1)

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

Converted Cpx (relative) TRS to CDT

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

flatten(nil) → nil
flatten(unit(z0)) → flatten(z0)
rev(unit(z0)) → unit(z0)
rev(nil) → nil
++(z0, nil) → z0
++(nil, z0) → z0

Tuples:

FLATTEN(nil) → c
FLATTEN(unit(z0)) → c1(FLATTEN(z0))
REV(unit(z0)) → c2
REV(nil) → c3
++'(z0, nil) → c4
++'(nil, z0) → c5

S tuples:

FLATTEN(nil) → c
FLATTEN(unit(z0)) → c1(FLATTEN(z0))
REV(unit(z0)) → c2
REV(nil) → c3
++'(z0, nil) → c4
++'(nil, z0) → c5

K tuples:none
Defined Rule Symbols:

flatten, rev, ++

Defined Pair Symbols:

FLATTEN, REV, ++'

Compound Symbols:

c, c1, c2, c3, c4, c5

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

Removed 5 trailing nodes:

++'(nil, z0) → c5
++'(z0, nil) → c4
FLATTEN(nil) → c
REV(unit(z0)) → c2
REV(nil) → c3

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

flatten(nil) → nil
flatten(unit(z0)) → flatten(z0)
rev(unit(z0)) → unit(z0)
rev(nil) → nil
++(z0, nil) → z0
++(nil, z0) → z0

Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0))

S tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0))

K tuples:none
Defined Rule Symbols:

flatten, rev, ++

Defined Pair Symbols:

FLATTEN

Compound Symbols:

c1

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

flatten(nil) → nil
flatten(unit(z0)) → flatten(z0)
rev(unit(z0)) → unit(z0)
rev(nil) → nil
++(z0, nil) → z0
++(nil, z0) → z0

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0))

S tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

FLATTEN

Compound Symbols:

c1

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

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

FLATTEN(unit(z0)) → c1(FLATTEN(z0))

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

FLATTEN(unit(z0)) → c1(FLATTEN(z0))

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

POL(FLATTEN(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(unit(x₁)) = [1] + x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0))

S tuples:none
K tuples:

FLATTEN(unit(z0)) → c1(FLATTEN(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

FLATTEN

Compound Symbols:

c1

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

The set S is empty

(16) BOUNDS(1, 1)