### (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:

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: 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:

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

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

Converted Cpx (relative) TRS to CDT

### (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:

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

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

Removed 4 trailing nodes:

NOT(z0) → c
IMPLIES(z0, z1) → c1
OR(z0, z1) → c2
='(z0, z1) → c3

### (6) 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

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

The set S is empty