Runtime Complexity (innermost) proof of /scratch/tmpCWATby/#4.30.xml

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))

↳2 CdtProblem

↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

↳4 CdtProblem

↳5 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳6 CdtProblem

↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

↳8 CdtProblem

↳9 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳10 CdtProblem

↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

↳12 CdtProblem

↳13 CdtKnowledgeProof (⇔)

↳14 CdtProblem

↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳16 CdtProblem

↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳18 CdtProblem

↳19 SIsEmptyProof (⇔)

↳20 BOUNDS(O(1), O(1))

(0) Obligation:

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

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, s(y)) → if_quot(le(s(y), x), x, s(y))
if_quot(true, x, y) → s(quot(minus(x, y), y))
if_quot(false, x, y) → 0

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(z0, s(z1)) → c5(IF_QUOT(le(s(z1), z0), z0, s(z1)), LE(s(z1), z0))
IF_QUOT(true, z0, z1) → c6(QUOT(minus(z0, z1), z1), MINUS(z0, z1))

S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(z0, s(z1)) → c5(IF_QUOT(le(s(z1), z0), z0, s(z1)), LE(s(z1), z0))
IF_QUOT(true, z0, z1) → c6(QUOT(minus(z0, z1), z1), MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

minus, le, quot, if_quot

Defined Pair Symbols:

MINUS, LE, QUOT, IF_QUOT

Compound Symbols:

c1, c4, c5, c6

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

Use narrowing to replace QUOT(z0, s(z1)) → c5(IF_QUOT(le(s(z1), z0), z0, s(z1)), LE(s(z1), z0)) by

QUOT(0, s(z0)) → c5(IF_QUOT(false, 0, s(z0)), LE(s(z0), 0))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF_QUOT(true, z0, z1) → c6(QUOT(minus(z0, z1), z1), MINUS(z0, z1))
QUOT(0, s(z0)) → c5(IF_QUOT(false, 0, s(z0)), LE(s(z0), 0))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))

S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF_QUOT(true, z0, z1) → c6(QUOT(minus(z0, z1), z1), MINUS(z0, z1))
QUOT(0, s(z0)) → c5(IF_QUOT(false, 0, s(z0)), LE(s(z0), 0))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

minus, le, quot, if_quot

Defined Pair Symbols:

MINUS, LE, IF_QUOT, QUOT

Compound Symbols:

c1, c4, c6, c5

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 5 dangling nodes:

QUOT(0, s(z0)) → c5(IF_QUOT(false, 0, s(z0)), LE(s(z0), 0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF_QUOT(true, z0, z1) → c6(QUOT(minus(z0, z1), z1), MINUS(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))

S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
IF_QUOT(true, z0, z1) → c6(QUOT(minus(z0, z1), z1), MINUS(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

minus, le, quot, if_quot

Defined Pair Symbols:

MINUS, LE, IF_QUOT, QUOT

Compound Symbols:

c1, c4, c6, c5

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

Use narrowing to replace IF_QUOT(true, z0, z1) → c6(QUOT(minus(z0, z1), z1), MINUS(z0, z1)) by

IF_QUOT(true, z0, 0) → c6(QUOT(z0, 0), MINUS(z0, 0))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, z0, 0) → c6(QUOT(z0, 0), MINUS(z0, 0))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, z0, 0) → c6(QUOT(z0, 0), MINUS(z0, 0))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

minus, le, quot, if_quot

Defined Pair Symbols:

MINUS, LE, QUOT, IF_QUOT

Compound Symbols:

c1, c4, c5, c6

(9) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 5 dangling nodes:

IF_QUOT(true, z0, 0) → c6(QUOT(z0, 0), MINUS(z0, 0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

K tuples:none
Defined Rule Symbols:

minus, le, quot, if_quot

Defined Pair Symbols:

MINUS, LE, QUOT, IF_QUOT

Compound Symbols:

c1, c4, c5, c6

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

We considered the (Usable) Rules:

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

And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(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_QUOT(x₁, x₂, x₃)) = [1] + [2]x₂
POL(LE(x₁, x₂)) = [2]
POL(MINUS(x₁, x₂)) = [1]
POL(QUOT(x₁, x₂)) = [4] + [2]x₁
POL(c1(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(false) = [3]
POL(le(x₁, x₂)) = x₁ + [3]x₂
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = [4] + x₁
POL(true) = 0

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))

K tuples:

IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

Defined Rule Symbols:

minus, le, quot, if_quot

Defined Pair Symbols:

MINUS, LE, QUOT, IF_QUOT

Compound Symbols:

c1, c4, c5, c6

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))

K tuples:

IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))

Defined Rule Symbols:

minus, le, quot, if_quot

Defined Pair Symbols:

MINUS, LE, QUOT, IF_QUOT

Compound Symbols:

c1, c4, c5, c6

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(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)) → c1(MINUS(z0, z1))

We considered the (Usable) Rules:

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

And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(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_QUOT(x₁, x₂, x₃)) = [2]x₂ + x₂²
POL(LE(x₁, x₂)) = 0
POL(MINUS(x₁, x₂)) = [1] + [2]x₁
POL(QUOT(x₁, x₂)) = [2]x₁ + x₁²
POL(c1(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(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

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:

LE(s(z0), s(z1)) → c4(LE(z0, z1))

K tuples:

IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))

Defined Rule Symbols:

minus, le, quot, if_quot

Defined Pair Symbols:

MINUS, LE, QUOT, IF_QUOT

Compound Symbols:

c1, c4, c5, c6

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(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)) → c4(LE(z0, z1))

We considered the (Usable) Rules:

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

And the Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(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_QUOT(x₁, x₂, x₃)) = [2]x₂²
POL(LE(x₁, x₂)) = x₂
POL(MINUS(x₁, x₂)) = 0
POL(QUOT(x₁, x₂)) = x₁ + [2]x₁²
POL(c1(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(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

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))

S tuples:none
K tuples:

IF_QUOT(true, s(z0), s(z1)) → c6(QUOT(minus(z0, z1), s(z1)), MINUS(s(z0), s(z1)))
QUOT(s(z1), s(z0)) → c5(IF_QUOT(le(z0, z1), s(z1), s(z0)), LE(s(z0), s(z1)))
MINUS(s(z0), s(z1)) → c1(MINUS(z0, z1))
LE(s(z0), s(z1)) → c4(LE(z0, z1))

Defined Rule Symbols:

minus, le, quot, if_quot

Defined Pair Symbols:

MINUS, LE, QUOT, IF_QUOT

Compound Symbols:

c1, c4, c5, c6

(19) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(6) Obligation:

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

(8) Obligation:

(9) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(10) Obligation:

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(12) Obligation:

(13) CdtKnowledgeProof (EQUIVALENT transformation)

(14) Obligation:

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(16) Obligation:

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(18) Obligation:

(19) SIsEmptyProof (EQUIVALENT transformation)

(20) BOUNDS(O(1), O(1))