(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:
not(x) → xor(x, true)
implies(x, y) → xor(and(x, y), xor(x, true))
or(x, y) → xor(and(x, y), xor(x, y))
=(x, y) → xor(x, xor(y, true))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
not(z0) → xor(z0, true)
implies(z0, z1) → xor(and(z0, z1), xor(z0, true))
or(z0, z1) → xor(and(z0, z1), xor(z0, z1))
=(z0, z1) → xor(z0, xor(z1, true))
Tuples:
NOT(z0) → c
IMPLIES(z0, z1) → c1
OR(z0, z1) → c2
='(z0, z1) → c3
S tuples:
NOT(z0) → c
IMPLIES(z0, z1) → c1
OR(z0, z1) → c2
='(z0, z1) → c3
K tuples:none
Defined Rule Symbols:
not, implies, or, =
Defined Pair Symbols:
NOT, IMPLIES, OR, ='
Compound Symbols:
c, c1, c2, c3
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
NOT(z0) → c
OR(z0, z1) → c2
IMPLIES(z0, z1) → c1
='(z0, z1) → c3
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
not(z0) → xor(z0, true)
implies(z0, z1) → xor(and(z0, z1), xor(z0, true))
or(z0, z1) → xor(and(z0, z1), xor(z0, z1))
=(z0, z1) → xor(z0, xor(z1, true))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
not, implies, or, =
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)