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:

a__g(X) → a__h(X)
a__c → d
a__h(d) → a__g(c)
mark(g(X)) → a__g(X)
mark(h(X)) → a__h(X)
mark(c) → a__c
mark(d) → d
a__g(X) → g(X)
a__h(X) → h(X)
a__c → c

Rewrite Strategy: INNERMOST

(1) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3.
The certificate found is represented by the following graph.

Start state: 1
Accept states: [2]
Transitions:
1→2[a__g_1|0, a__c|0, a__h_1|0, mark_1|0, a__h_1|1, g_1|1, d|1, c|1, h_1|1, a__g_1|1, a__c|1, a__h_1|2, g_1|2, d|2, c|2, h_1|2, h_1|3]
1→3[a__g_1|1, a__h_1|2, g_1|2, h_1|3]
2→2[d|0, c|0, g_1|0, h_1|0]
3→2[c|1]

(2) BOUNDS(1, n^1)

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__g(z0) → a__h(z0)
a__g(z0) → g(z0)
a__c → d
a__c → c
a__h(d) → a__g(c)
a__h(z0) → h(z0)
mark(g(z0)) → a__g(z0)
mark(h(z0)) → a__h(z0)
mark(c) → a__c
mark(d) → d

Tuples:

A__G(z0) → c1(A__H(z0))
A__G(z0) → c2
A__C → c3
A__C → c4
A__H(d) → c5(A__G(c))
A__H(z0) → c6
MARK(g(z0)) → c7(A__G(z0))
MARK(h(z0)) → c8(A__H(z0))
MARK(c) → c9(A__C)
MARK(d) → c10

S tuples:

A__G(z0) → c1(A__H(z0))
A__G(z0) → c2
A__C → c3
A__C → c4
A__H(d) → c5(A__G(c))
A__H(z0) → c6
MARK(g(z0)) → c7(A__G(z0))
MARK(h(z0)) → c8(A__H(z0))
MARK(c) → c9(A__C)
MARK(d) → c10

K tuples:none
Defined Rule Symbols:

a__g, a__c, a__h, mark

Defined Pair Symbols:

A__G, A__C, A__H, MARK

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

MARK(h(z0)) → c8(A__H(z0))
MARK(g(z0)) → c7(A__G(z0))

Removed 6 trailing nodes:

MARK(c) → c9(A__C)
A__H(z0) → c6
A__G(z0) → c2
A__C → c4
MARK(d) → c10
A__C → c3

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__g(z0) → a__h(z0)
a__g(z0) → g(z0)
a__c → d
a__c → c
a__h(d) → a__g(c)
a__h(z0) → h(z0)
mark(g(z0)) → a__g(z0)
mark(h(z0)) → a__h(z0)
mark(c) → a__c
mark(d) → d

Tuples:

A__G(z0) → c1(A__H(z0))
A__H(d) → c5(A__G(c))

S tuples:

A__G(z0) → c1(A__H(z0))
A__H(d) → c5(A__G(c))

K tuples:none
Defined Rule Symbols:

a__g, a__c, a__h, mark

Defined Pair Symbols:

A__G, A__H

Compound Symbols:

c1, c5

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

a__g(z0) → a__h(z0)
a__g(z0) → g(z0)
a__c → d
a__c → c
a__h(d) → a__g(c)
a__h(z0) → h(z0)
mark(g(z0)) → a__g(z0)
mark(h(z0)) → a__h(z0)
mark(c) → a__c
mark(d) → d

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

A__G(z0) → c1(A__H(z0))
A__H(d) → c5(A__G(c))

S tuples:

A__G(z0) → c1(A__H(z0))
A__H(d) → c5(A__G(c))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

A__G, A__H

Compound Symbols:

c1, c5

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)) transformation)

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

A__G(z0) → c1(A__H(z0))
A__H(d) → c5(A__G(c))

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

A__G(z0) → c1(A__H(z0))
A__H(d) → c5(A__G(c))

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

POL(A__G(x₁)) = [1] + x₁ + x₁² + x₁³   
POL(A__H(x₁)) = x₁ + x₁² + x₁³   
POL(c) = 0   
POL(c1(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(d) = [1]   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

A__G(z0) → c1(A__H(z0))
A__H(d) → c5(A__G(c))

S tuples:none
K tuples:

A__G(z0) → c1(A__H(z0))
A__H(d) → c5(A__G(c))

Defined Rule Symbols:none

Defined Pair Symbols:

A__G, A__H

Compound Symbols:

c1, c5

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

The set S is empty

(12) BOUNDS(1, 1)