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

(0) Obligation:

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


The TRS R consists of the following rules:

fst(0, Z) → nil
fst(s, cons(Y)) → cons(Y)
from(X) → cons(X)
add(0, X) → X
add(s, Y) → s
len(nil) → 0
len(cons(X)) → s

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, 1).


The TRS R consists of the following rules:

fst(0, Z) → nil
fst(s, cons(Y)) → cons(Y)
from(X) → cons(X)
add(0, X) → X
add(s, Y) → s
len(nil) → 0
len(cons(X)) → s

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

fst(0, z0) → nil
fst(s, cons(z0)) → cons(z0)
from(z0) → cons(z0)
add(0, z0) → z0
add(s, z0) → s
len(nil) → 0
len(cons(z0)) → s

Tuples:

FST(0, z0) → c
FST(s, cons(z0)) → c1
FROM(z0) → c2
ADD(0, z0) → c3
ADD(s, z0) → c4
LEN(nil) → c5
LEN(cons(z0)) → c6

S tuples:

FST(0, z0) → c
FST(s, cons(z0)) → c1
FROM(z0) → c2
ADD(0, z0) → c3
ADD(s, z0) → c4
LEN(nil) → c5
LEN(cons(z0)) → c6

K tuples:none
Defined Rule Symbols:

fst, from, add, len

Defined Pair Symbols:

FST, FROM, ADD, LEN

Compound Symbols:

c, c1, c2, c3, c4, c5, c6

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

Removed 7 trailing nodes:

FROM(z0) → c2
LEN(cons(z0)) → c6
LEN(nil) → c5
ADD(s, z0) → c4
ADD(0, z0) → c3
FST(0, z0) → c
FST(s, cons(z0)) → c1

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

fst(0, z0) → nil
fst(s, cons(z0)) → cons(z0)
from(z0) → cons(z0)
add(0, z0) → z0
add(s, z0) → s
len(nil) → 0
len(cons(z0)) → s

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

fst, from, add, len

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(8) BOUNDS(1, 1)