(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:
*(i(x), x) → 1
*(1, y) → y
*(x, 0) → 0
*(*(x, y), z) → *(x, *(y, z))
Rewrite Strategy: FULL
(1) 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:
*(*(x, y), z) → *(x, *(y, z))
(2) 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:
*(i(x), x) → 1
*(x, 0) → 0
*(1, y) → y
Rewrite Strategy: FULL
(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS does not nest defined symbols, we have rc = irc.
(4) 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:
*(i(x), x) → 1
*(x, 0) → 0
*(1, y) → y
Rewrite Strategy: INNERMOST
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
*(i(z0), z0) → 1
*(z0, 0) → 0
*(1, z0) → z0
Tuples:
*'(i(z0), z0) → c
*'(z0, 0) → c1
*'(1, z0) → c2
S tuples:
*'(i(z0), z0) → c
*'(z0, 0) → c1
*'(1, z0) → c2
K tuples:none
Defined Rule Symbols:
*
Defined Pair Symbols:
*'
Compound Symbols:
c, c1, c2
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
*'(1, z0) → c2
*'(z0, 0) → c1
*'(i(z0), z0) → c
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
*(i(z0), z0) → 1
*(z0, 0) → 0
*(1, z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
*
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)