Runtime Complexity (innermost) proof of /tmp/tmpvXTtyc/Ex49_GM04

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

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

↳4 CdtProblem

↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 205 ms)

↳6 CdtProblem

↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 83 ms)

↳8 CdtProblem

↳9 SIsEmptyProof (⇔, 0 ms)

↳10 BOUNDS(O(1), O(1))

(0) Obligation:

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

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1))
geq(z0, n__0) → true
geq(n__0, n__s(z0)) → false
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1))
div(0, n__s(z0)) → 0
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__s(z0)) → s(z0)
activate(z0) → z0

Tuples:

MINUS(n__0, z0) → c(0')
MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
DIV(0, n__s(z0)) → c5(0')
DIV(s(z0), n__s(z1)) → c6(IF(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0), GEQ(z0, activate(z1)), ACTIVATE(z1), DIV(minus(z0, activate(z1)), n__s(activate(z1))), MINUS(z0, activate(z1)), ACTIVATE(z1), ACTIVATE(z1))
IF(true, z0, z1) → c7(ACTIVATE(z0))
IF(false, z0, z1) → c8(ACTIVATE(z1))
ACTIVATE(n__0) → c11(0')
ACTIVATE(n__s(z0)) → c12(S(z0))

S tuples:

MINUS(n__0, z0) → c(0')
MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
DIV(0, n__s(z0)) → c5(0')
DIV(s(z0), n__s(z1)) → c6(IF(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0), GEQ(z0, activate(z1)), ACTIVATE(z1), DIV(minus(z0, activate(z1)), n__s(activate(z1))), MINUS(z0, activate(z1)), ACTIVATE(z1), ACTIVATE(z1))
IF(true, z0, z1) → c7(ACTIVATE(z0))
IF(false, z0, z1) → c8(ACTIVATE(z1))
ACTIVATE(n__0) → c11(0')
ACTIVATE(n__s(z0)) → c12(S(z0))

K tuples:none
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ, DIV, IF, ACTIVATE

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c11, c12

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

Removed 7 trailing nodes:

DIV(s(z0), n__s(z1)) → c6(IF(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0), GEQ(z0, activate(z1)), ACTIVATE(z1), DIV(minus(z0, activate(z1)), n__s(activate(z1))), MINUS(z0, activate(z1)), ACTIVATE(z1), ACTIVATE(z1))
IF(true, z0, z1) → c7(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c12(S(z0))
ACTIVATE(n__0) → c11(0')
MINUS(n__0, z0) → c(0')
DIV(0, n__s(z0)) → c5(0')
IF(false, z0, z1) → c8(ACTIVATE(z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1))
geq(z0, n__0) → true
geq(n__0, n__s(z0)) → false
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1))
div(0, n__s(z0)) → 0
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__s(z0)) → s(z0)
activate(z0) → z0

Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

K tuples:none
Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ

Compound Symbols:

c1, c4

(5) 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.

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__s(z0)) → s(z0)
activate(z0) → z0
s(z0) → n__s(z0)
0 → n__0

And the Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

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

POL(0) = [3]
POL(ACTIVATE(x₁)) = 0
POL(GEQ(x₁, x₂)) = 0
POL(MINUS(x₁, x₂)) = [4]x₁
POL(activate(x₁)) = x₁
POL(c1(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(n__0) = [3]
POL(n__s(x₁)) = [2] + x₁
POL(s(x₁)) = [2] + x₁

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1))
geq(z0, n__0) → true
geq(n__0, n__s(z0)) → false
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1))
div(0, n__s(z0)) → 0
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__s(z0)) → s(z0)
activate(z0) → z0

Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

K tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ

Compound Symbols:

c1, c4

(7) 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.

GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__s(z0)) → s(z0)
activate(z0) → z0
s(z0) → n__s(z0)
0 → n__0

And the Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

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

POL(0) = [3]
POL(ACTIVATE(x₁)) = 0
POL(GEQ(x₁, x₂)) = [4]x₂
POL(MINUS(x₁, x₂)) = 0
POL(activate(x₁)) = [2] + x₁
POL(c1(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(n__0) = [3]
POL(n__s(x₁)) = [3] + x₁
POL(s(x₁)) = [5] + x₁

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(n__0, z0) → 0
minus(n__s(z0), n__s(z1)) → minus(activate(z0), activate(z1))
geq(z0, n__0) → true
geq(n__0, n__s(z0)) → false
geq(n__s(z0), n__s(z1)) → geq(activate(z0), activate(z1))
div(0, n__s(z0)) → 0
div(s(z0), n__s(z1)) → if(geq(z0, activate(z1)), n__s(div(minus(z0, activate(z1)), n__s(activate(z1)))), n__0)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__s(z0)) → s(z0)
activate(z0) → z0

Tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:none
K tuples:

MINUS(n__s(z0), n__s(z1)) → c1(MINUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
GEQ(n__s(z0), n__s(z1)) → c4(GEQ(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

Defined Rule Symbols:

minus, geq, div, if, 0, s, activate

Defined Pair Symbols:

MINUS, GEQ

Compound Symbols:

c1, c4

(9) SIsEmptyProof (EQUIVALENT 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) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(6) Obligation:

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

(8) Obligation:

(9) SIsEmptyProof (EQUIVALENT transformation)

(10) BOUNDS(O(1), O(1))