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

(0) Obligation:

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


The TRS R consists of the following rules:

d(x) → e(u(x))
d(u(x)) → c(x)
c(u(x)) → b(x)
v(e(x)) → x
b(u(x)) → a(e(x))

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 1.
The certificate found is represented by the following graph.

Start state: 1
Accept states: [2]
Transitions:
1→2[d_1|0, c_1|0, v_1|0, b_1|0, c_1|1, b_1|1, e_1|1, u_1|1, a_1|1]
1→3[e_1|1]
2→2[e_1|0, u_1|0, a_1|0, e_1|1]
3→2[u_1|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:

d(z0) → e(u(z0))
d(u(z0)) → c(z0)
c(u(z0)) → b(z0)
v(e(z0)) → z0
b(u(z0)) → a(e(z0))

Tuples:

D(z0) → c1
D(u(z0)) → c2(C(z0))
C(u(z0)) → c3(B(z0))
V(e(z0)) → c4
B(u(z0)) → c5

S tuples:

D(z0) → c1
D(u(z0)) → c2(C(z0))
C(u(z0)) → c3(B(z0))
V(e(z0)) → c4
B(u(z0)) → c5

K tuples:none
Defined Rule Symbols:

d, c, v, b

Defined Pair Symbols:

D, C, V, B

Compound Symbols:

c1, c2, c3, c4, c5

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

Removed 5 trailing nodes:

D(z0) → c1
V(e(z0)) → c4
C(u(z0)) → c3(B(z0))
D(u(z0)) → c2(C(z0))
B(u(z0)) → c5

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

d(z0) → e(u(z0))
d(u(z0)) → c(z0)
c(u(z0)) → b(z0)
v(e(z0)) → z0
b(u(z0)) → a(e(z0))

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

d, c, v, b

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(8) BOUNDS(1, 1)