The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(0, y) → 0
mod(s(x), 0) → 0
mod(s(x), s(y)) → if_mod(le(y, x), s(x), s(y))
if_mod(true, x, y) → mod(minus(x, y), y)
if_mod(false, s(x), s(y)) → s(x)

Rewrite Strategy: FULL

(1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

The duplicating contexts are:
mod(s(x), s([]))
mod(s([]), s(y))
if_mod(true, x, [])

The defined contexts are:
if_mod([], s(x1), s(x2))
mod([], x1)
le(x0, [])
if_mod(x0, s([]), s(x2))
minus([], x1)

[] just represents basic- or constructor-terms in the following defined contexts:
if_mod([], s(x1), s(x2))
mod([], x1)

As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(0, y) → 0
mod(s(x), 0) → 0
mod(s(x), s(y)) → if_mod(le(y, x), s(x), s(y))
if_mod(true, x, y) → mod(minus(x, y), y)
if_mod(false, s(x), s(y)) → s(x)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)

Tuples:

LE(0, z0) → c
LE(s(z0), 0) → c1
LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(z0, 0) → c3
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(0, z0) → c5
MOD(s(z0), 0) → c6
MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) → c9

S tuples:

LE(0, z0) → c
LE(s(z0), 0) → c1
LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(z0, 0) → c3
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(0, z0) → c5
MOD(s(z0), 0) → c6
MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
IF_MOD(false, s(z0), s(z1)) → c9

K tuples:none
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

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

Removed 6 trailing nodes:

MOD(0, z0) → c5
MINUS(z0, 0) → c3
LE(s(z0), 0) → c1
MOD(s(z0), 0) → c6
LE(0, z0) → c
IF_MOD(false, s(z0), s(z1)) → c9

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)
mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

le, minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c8

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

mod(0, z0) → 0
mod(s(z0), 0) → 0
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1))
if_mod(true, z0, z1) → mod(minus(z0, z1), z1)
if_mod(false, s(z0), s(z1)) → s(z0)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c8

(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MOD(s(z0), s(z1)) → c7(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) by

MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(0), s(s(z0))) → c7(IF_MOD(false, s(0), s(s(z0))), LE(s(z0), 0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(0), s(s(z0))) → c7(IF_MOD(false, s(0), s(s(z0))), LE(s(z0), 0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(0), s(s(z0))) → c7(IF_MOD(false, s(0), s(s(z0))), LE(s(z0), 0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, IF_MOD, MOD

Compound Symbols:

c2, c4, c8, c7

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

MOD(s(0), s(s(z0))) → c7(IF_MOD(false, s(0), s(s(z0))), LE(s(z0), 0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)), LE(0, z0))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, IF_MOD, MOD

Compound Symbols:

c2, c4, c8, c7

(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))

K tuples:none
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, IF_MOD, MOD

Compound Symbols:

c2, c4, c8, c7, c7

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace IF_MOD(true, z0, z1) → c8(MOD(minus(z0, z1), z1), MINUS(z0, z1)) by

IF_MOD(true, z0, 0) → c8(MOD(z0, 0), MINUS(z0, 0))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, z0, 0) → c8(MOD(z0, 0), MINUS(z0, 0))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, z0, 0) → c8(MOD(z0, 0), MINUS(z0, 0))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(17) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

IF_MOD(true, z0, 0) → c8(MOD(z0, 0), MINUS(z0, 0))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0

And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(IF_MOD(x₁, x₂, x₃)) = x₂   
POL(LE(x₁, x₂)) = [1]   
POL(MINUS(x₁, x₂)) = 0   
POL(MOD(x₁, x₂)) = [1] + x₁   
POL(c2(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(false) = 0   
POL(le(x₁, x₂)) = 0   
POL(minus(x₁, x₂)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

K tuples:

MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))

Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0

And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(IF_MOD(x₁, x₂, x₃)) = x₂   
POL(LE(x₁, x₂)) = 0   
POL(MINUS(x₁, x₂)) = 0   
POL(MOD(x₁, x₂)) = x₁   
POL(c2(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(false) = 0   
POL(le(x₁, x₂)) = 0   
POL(minus(x₁, x₂)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))

K tuples:

MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(23) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))

K tuples:

MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))

Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0

And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(IF_MOD(x₁, x₂, x₃)) = [2]x₂²   
POL(LE(x₁, x₂)) = 0   
POL(MINUS(x₁, x₂)) = [2] + x₁   
POL(MOD(x₁, x₂)) = [2]x₁²   
POL(c2(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(false) = [1]   
POL(le(x₁, x₂)) = x₂ + [2]x₂² + x₁²   
POL(minus(x₁, x₂)) = x₁   
POL(s(x₁)) = [2] + x₁   
POL(true) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))

K tuples:

MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))

Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(27) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

LE(s(z0), s(z1)) → c2(LE(z0, z1))

We considered the (Usable) Rules:

minus(s(z0), s(z1)) → minus(z0, z1)
minus(z0, 0) → z0

And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]   
POL(IF_MOD(x₁, x₂, x₃)) = x₂²   
POL(LE(x₁, x₂)) = x₂   
POL(MINUS(x₁, x₂)) = [1]   
POL(MOD(x₁, x₂)) = x₁ + x₁²   
POL(c2(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(false) = 0   
POL(le(x₁, x₂)) = 0   
POL(minus(x₁, x₂)) = x₁   
POL(s(x₁)) = [1] + x₁   
POL(true) = 0   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true
le(s(z0), 0) → false
le(s(z0), s(z1)) → le(z0, z1)
minus(z0, 0) → z0
minus(s(z0), s(z1)) → minus(z0, z1)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:none
K tuples:

MOD(s(z0), s(0)) → c7(IF_MOD(true, s(z0), s(0)))
IF_MOD(true, s(z0), s(z1)) → c8(MOD(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
MOD(s(s(z1)), s(s(z0))) → c7(IF_MOD(le(z0, z1), s(s(z1)), s(s(z0))), LE(s(z0), s(z1)))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
LE(s(z0), s(z1)) → c2(LE(z0, z1))

Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c7, c7, c8

(29) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(30) BOUNDS(1, 1)