(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:
terms(N) → cons(recip(sqr(N)))
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
add(0, X) → X
add(s, Y) → s
first(0, X) → nil
first(s, cons(Y)) → cons(Y)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
terms(z0) → cons(recip(sqr(z0)))
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
add(0, z0) → z0
add(s, z0) → s
first(0, z0) → nil
first(s, cons(z0)) → cons(z0)
Tuples:
TERMS(z0) → c(SQR(z0))
SQR(0) → c1
SQR(s) → c2
DBL(0) → c3
DBL(s) → c4
ADD(0, z0) → c5
ADD(s, z0) → c6
FIRST(0, z0) → c7
FIRST(s, cons(z0)) → c8
S tuples:
TERMS(z0) → c(SQR(z0))
SQR(0) → c1
SQR(s) → c2
DBL(0) → c3
DBL(s) → c4
ADD(0, z0) → c5
ADD(s, z0) → c6
FIRST(0, z0) → c7
FIRST(s, cons(z0)) → c8
K tuples:none
Defined Rule Symbols:
terms, sqr, dbl, add, first
Defined Pair Symbols:
TERMS, SQR, DBL, ADD, FIRST
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 9 trailing nodes:
SQR(0) → c1
FIRST(s, cons(z0)) → c8
FIRST(0, z0) → c7
ADD(s, z0) → c6
ADD(0, z0) → c5
DBL(0) → c3
TERMS(z0) → c(SQR(z0))
SQR(s) → c2
DBL(s) → c4
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
terms(z0) → cons(recip(sqr(z0)))
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
add(0, z0) → z0
add(s, z0) → s
first(0, z0) → nil
first(s, cons(z0)) → cons(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
terms, sqr, dbl, add, first
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)