### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

implies(not(z0), z1) → or(z0, z1)
implies(not(z0), or(z1, z2)) → implies(z1, or(z0, z2))
implies(z0, or(z1, z2)) → or(z1, implies(z0, z2))
Tuples:

IMPLIES(not(z0), z1) → c
IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
S tuples:

IMPLIES(not(z0), z1) → c
IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
K tuples:none
Defined Rule Symbols:

implies

Defined Pair Symbols:

IMPLIES

Compound Symbols:

c, c1, c2

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

Removed 1 trailing nodes:

IMPLIES(not(z0), z1) → c

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

implies(not(z0), z1) → or(z0, z1)
implies(not(z0), or(z1, z2)) → implies(z1, or(z0, z2))
implies(z0, or(z1, z2)) → or(z1, implies(z0, z2))
Tuples:

IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
S tuples:

IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
K tuples:none
Defined Rule Symbols:

implies

Defined Pair Symbols:

IMPLIES

Compound Symbols:

c1, c2

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

implies(not(z0), z1) → or(z0, z1)
implies(not(z0), or(z1, z2)) → implies(z1, or(z0, z2))
implies(z0, or(z1, z2)) → or(z1, implies(z0, z2))

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
S tuples:

IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

IMPLIES

Compound Symbols:

c1, c2

### (7) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:

IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(IMPLIES(x1, x2)) = [4]x1 + [4]x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(not(x1)) = [4] + x1
POL(or(x1, x2)) = [4] + x1 + x2

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
S tuples:none
K tuples:

IMPLIES(not(z0), or(z1, z2)) → c1(IMPLIES(z1, or(z0, z2)))
IMPLIES(z0, or(z1, z2)) → c2(IMPLIES(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

IMPLIES

Compound Symbols:

c1, c2

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

The set S is empty