Runtime Complexity (innermost) proof of /tmp/tmpxC9ioY/gcd.xml

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 4 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtUsableRulesProof (⇔, 0 ms)

↳8 CdtProblem

↳9 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 131 ms)

↳10 CdtProblem

↳11 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 186 ms)

↳14 CdtProblem

↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 107 ms)

↳16 CdtProblem

↳17 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳18 BOUNDS(1, 1)

(0) 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:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(z0, s(z1)) → c(PRED(minus(z0, z1)), MINUS(z0, z1))
MINUS(z0, 0) → c1
PRED(s(z0)) → c2
LE(s(z0), s(z1)) → c3(LE(z0, z1))
LE(s(z0), 0) → c4
LE(0, z0) → c5
GCD(0, z0) → c6
GCD(s(z0), 0) → c7
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

MINUS(z0, s(z1)) → c(PRED(minus(z0, z1)), MINUS(z0, z1))
MINUS(z0, 0) → c1
PRED(s(z0)) → c2
LE(s(z0), s(z1)) → c3(LE(z0, z1))
LE(s(z0), 0) → c4
LE(0, z0) → c5
GCD(0, z0) → c6
GCD(s(z0), 0) → c7
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

K tuples:none
Defined Rule Symbols:

minus, pred, le, gcd, if

Defined Pair Symbols:

MINUS, PRED, LE, GCD, IF

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10

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

Removed 6 trailing nodes:

LE(s(z0), 0) → c4
GCD(s(z0), 0) → c7
PRED(s(z0)) → c2
GCD(0, z0) → c6
MINUS(z0, 0) → c1
LE(0, z0) → c5

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

MINUS(z0, s(z1)) → c(PRED(minus(z0, z1)), MINUS(z0, z1))
LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

MINUS(z0, s(z1)) → c(PRED(minus(z0, z1)), MINUS(z0, z1))
LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

K tuples:none
Defined Rule Symbols:

minus, pred, le, gcd, if

Defined Pair Symbols:

MINUS, LE, GCD, IF

Compound Symbols:

c, c3, c8, c9, c10

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

S tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

minus, pred, le, gcd, if

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

S tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

K tuples:none
Defined Rule Symbols:

le, minus, pred

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

(9) 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(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

We considered the (Usable) Rules:

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

And the Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

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

POL(0) = 0
POL(GCD(x₁, x₂)) = x₁ + x₂
POL(IF(x₁, x₂, x₃)) = x₂ + x₃
POL(LE(x₁, x₂)) = 0
POL(MINUS(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(false) = 0
POL(le(x₁, x₂)) = 0
POL(minus(x₁, x₂)) = x₁
POL(pred(x₁)) = x₁
POL(s(x₁)) = [1] + x₁
POL(true) = 0

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

S tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

K tuples:

IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

Defined Rule Symbols:

le, minus, pred

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

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

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

GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

S tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

K tuples:

IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))

Defined Rule Symbols:

le, minus, pred

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

(13) 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)) → c3(LE(z0, z1))

We considered the (Usable) Rules:

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

And the Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

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

POL(0) = [1]
POL(GCD(x₁, x₂)) = [2]x₂ + [2]x₁·x₂
POL(IF(x₁, x₂, x₃)) = [2]x₂·x₃
POL(LE(x₁, x₂)) = x₁
POL(MINUS(x₁, x₂)) = [2]
POL(c(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(false) = 0
POL(le(x₁, x₂)) = x₂²
POL(minus(x₁, x₂)) = x₁
POL(pred(x₁)) = x₁
POL(s(x₁)) = [2] + x₁
POL(true) = 0

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

S tuples:

MINUS(z0, s(z1)) → c(MINUS(z0, z1))

K tuples:

IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
LE(s(z0), s(z1)) → c3(LE(z0, z1))

Defined Rule Symbols:

le, minus, pred

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

(15) 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(z0, s(z1)) → c(MINUS(z0, z1))

We considered the (Usable) Rules:

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

And the Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

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

POL(0) = 0
POL(GCD(x₁, x₂)) = [2] + [2]x₂ + [2]x₁·x₂
POL(IF(x₁, x₂, x₃)) = [2]x₂·x₃
POL(LE(x₁, x₂)) = [2]
POL(MINUS(x₁, x₂)) = [2] + [2]x₂
POL(c(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(false) = 0
POL(le(x₁, x₂)) = 0
POL(minus(x₁, x₂)) = x₁
POL(pred(x₁)) = x₁
POL(s(x₁)) = [2] + x₁
POL(true) = 0

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

LE(s(z0), s(z1)) → c3(LE(z0, z1))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

S tuples:none
K tuples:

IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
LE(s(z0), s(z1)) → c3(LE(z0, z1))
MINUS(z0, s(z1)) → c(MINUS(z0, z1))

Defined Rule Symbols:

le, minus, pred

Defined Pair Symbols:

LE, GCD, IF, MINUS

Compound Symbols:

c3, c8, c9, c10, c

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

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CdtUsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) BOUNDS(1, 1)