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

(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:

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)
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y))
if_gcd(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:

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)
gcd(0, z0) → z0
gcd(s(z0), 0) → s(z0)
gcd(s(z0), s(z1)) → if_gcd(le(z1, z0), s(z0), s(z1))
if_gcd(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1))
if_gcd(false, s(z0), s(z1)) → gcd(minus(z1, z0), 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))
GCD(0, z0) → c5
GCD(s(z0), 0) → c6
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

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))
GCD(0, z0) → c5
GCD(s(z0), 0) → c6
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

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

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

Removed 5 trailing nodes:

MINUS(z0, 0) → c3
GCD(s(z0), 0) → c6
LE(s(z0), 0) → c1
GCD(0, z0) → c5
LE(0, z0) → c

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

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

K tuples:none
Defined Rule Symbols:

le, minus, gcd, if_gcd

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(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)

Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

K tuples:none
Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

(7) 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_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

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

POL(0) = 0   
POL(GCD(x₁, x₂)) = x₁ + x₂   
POL(IF_GCD(x₁, x₂, x₃)) = x₂ + x₃   
POL(LE(x₁, x₂)) = 0   
POL(MINUS(x₁, x₂)) = 0   
POL(c2(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
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(s(x₁)) = [1] + x₁   
POL(true) = 0   

(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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))

K tuples:

IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

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

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

GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

(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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

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

K tuples:

IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))

Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

(11) 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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

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

POL(0) = 0   
POL(GCD(x₁, x₂)) = x₂² + x₁²   
POL(IF_GCD(x₁, x₂, x₃)) = x₃² + x₂²   
POL(LE(x₁, x₂)) = 0   
POL(MINUS(x₁, x₂)) = [1] + x₁   
POL(c2(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
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(s(x₁)) = [2] + x₁   
POL(true) = 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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))

K tuples:

IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
MINUS(s(z0), s(z1)) → c4(MINUS(z0, z1))

Defined Rule Symbols:

le, minus

Defined Pair Symbols:

LE, MINUS, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

(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)) → 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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

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

POL(0) = [1]   
POL(GCD(x₁, x₂)) = [1] + [2]x₂ + [2]x₁·x₂   
POL(IF_GCD(x₁, x₂, x₃)) = [2]x₂·x₃   
POL(LE(x₁, x₂)) = x₁   
POL(MINUS(x₁, x₂)) = [1] + [2]x₂   
POL(c2(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
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(s(x₁)) = [2] + x₁   
POL(true) = 0   

(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))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))

S tuples:none
K tuples:

IF_GCD(true, s(z0), s(z1)) → c8(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1))
IF_GCD(false, s(z0), s(z1)) → c9(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
GCD(s(z0), s(z1)) → c7(IF_GCD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
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, GCD, IF_GCD

Compound Symbols:

c2, c4, c7, c8, c9

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

The set S is empty

(16) BOUNDS(1, 1)