Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄)
t₅: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ (X₀)²+(X₄)⁵
t₆: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ 0 ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: (X₀)²+(X₄)⁵ < X₁ ∧ X₀ < 0
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: (X₀)²+(X₄)⁵ < X₁ ∧ 0 < X₀
t₈: l2(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)
t₁: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄) :|: 0 < X₄
t₂: l3(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ 0
t₇: l4(X₀, X₁, X₂, X₃, X₄) → l1(-2⋅X₀, 3⋅X₁-2⋅(X₄)³, X₂, X₃, X₄)

Preprocessing

Found invariant 1 ≤ X₄ for location l1

Found invariant 1 ≤ X₄ for location l4

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄)
t₅: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ (X₀)²+(X₄)⁵ ∧ 1 ≤ X₄
t₆: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 ≤ X₄
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: (X₀)²+(X₄)⁵ < X₁ ∧ X₀ < 0 ∧ 1 ≤ X₄
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: (X₀)²+(X₄)⁵ < X₁ ∧ 0 < X₀ ∧ 1 ≤ X₄
t₈: l2(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)
t₁: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄) :|: 0 < X₄
t₂: l3(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ 0
t₇: l4(X₀, X₁, X₂, X₃, X₄) → l1(-2⋅X₀, 3⋅X₁-2⋅(X₄)³, X₂, X₃, X₄) :|: 1 ≤ X₄

Found invariant 1 ≤ X₄ for location l1

Found invariant 1 ≤ X₄ for location l4

Time-Bound by TWN-Loops:

TWN-Loops: t₇ 16⋅X₃+21 {O(n)}

TWN-Loops:

entry: t₁: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄) :|: 0 < X₄
results in twn-loop: twn:Inv: [1 ≤ X₄ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄] , (X₀,X₁,X₂,X₃,X₄) -> (-2⋅X₀,3⋅X₁-2⋅(X₄)³,X₂,X₃,X₄) :|: (X₀)²+(X₄)⁵ < X₁ ∧ X₀ < 0 ∨ (X₀)²+(X₄)⁵ < X₁ ∧ 0 < X₀
order: [X₀; X₄; X₁]
closed-form:
X₀: X₀ * 4^n
X₄: X₄
X₁: X₁ * 9^n + [[n != 0]] * -(X₄)³ * 9^n + [[n != 0]] * (X₄)³

Termination: true
Formula:

0 < 2⋅X₀ ∧ 4⋅(X₀)² < 0 ∧ X₀ < 0 ∧ (X₀)² < 0
∨ 0 < 2⋅X₀ ∧ 4⋅(X₀)² < 0 ∧ X₀ < 0 ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 0 < 2⋅X₀ ∧ 4⋅(X₀)² < 0 ∧ X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 0 < 2⋅X₀ ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ X₀ < 0 ∧ (X₀)² < 0
∨ 0 < 2⋅X₀ ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ X₀ < 0 ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 0 < 2⋅X₀ ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 0 < 2⋅X₀ ∧ (X₄)⁵ < (X₄)³ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ X₀ < 0 ∧ (X₀)² < 0
∨ 0 < 2⋅X₀ ∧ (X₄)⁵ < (X₄)³ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ X₀ < 0 ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 0 < 2⋅X₀ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 2⋅X₀ < 0 ∧ 4⋅(X₀)² < 0 ∧ X₀ < 0 ∧ (X₀)² < 0
∨ 2⋅X₀ < 0 ∧ 4⋅(X₀)² < 0 ∧ X₀ < 0 ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 2⋅X₀ < 0 ∧ 4⋅(X₀)² < 0 ∧ X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 2⋅X₀ < 0 ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ X₀ < 0 ∧ (X₀)² < 0
∨ 2⋅X₀ < 0 ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ X₀ < 0 ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 2⋅X₀ < 0 ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 2⋅X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ X₀ < 0 ∧ (X₀)² < 0
∨ 2⋅X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ X₀ < 0 ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 2⋅X₀ < 0 ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 0 < 2⋅X₀ ∧ 4⋅(X₀)² < 0 ∧ 0 < X₀ ∧ (X₀)² < 0
∨ 0 < 2⋅X₀ ∧ 4⋅(X₀)² < 0 ∧ 0 < X₀ ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 0 < 2⋅X₀ ∧ 4⋅(X₀)² < 0 ∧ 0 < X₀ ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 0 < 2⋅X₀ ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 0 < X₀ ∧ (X₀)² < 0
∨ 0 < 2⋅X₀ ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 0 < X₀ ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 0 < 2⋅X₀ ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 0 < X₀ ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 0 < 2⋅X₀ ∧ (X₄)⁵ < (X₄)³ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ 0 < X₀ ∧ (X₀)² < 0
∨ 0 < 2⋅X₀ ∧ (X₄)⁵ < (X₄)³ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ 0 < X₀ ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 0 < 2⋅X₀ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ 0 < X₀ ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 2⋅X₀ < 0 ∧ 4⋅(X₀)² < 0 ∧ 0 < X₀ ∧ (X₀)² < 0
∨ 2⋅X₀ < 0 ∧ 4⋅(X₀)² < 0 ∧ 0 < X₀ ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 2⋅X₀ < 0 ∧ 4⋅(X₀)² < 0 ∧ 0 < X₀ ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 2⋅X₀ < 0 ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 0 < X₀ ∧ (X₀)² < 0
∨ 2⋅X₀ < 0 ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 0 < X₀ ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 2⋅X₀ < 0 ∧ 3⋅(X₄)³ < 3⋅X₁ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 0 < X₀ ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³
∨ 2⋅X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ 0 < X₀ ∧ (X₀)² < 0
∨ 2⋅X₀ < 0 ∧ (X₄)⁵ < (X₄)³ ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ 0 < X₀ ∧ (X₄)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 2⋅X₀ < 0 ∧ 4⋅(X₀)² ≤ 0 ∧ 0 ≤ 4⋅(X₀)² ∧ 3⋅(X₄)³ ≤ 3⋅X₁ ∧ 3⋅X₁ ≤ 3⋅(X₄)³ ∧ 0 < X₀ ∧ (X₄)⁵ < (X₄)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₄)³ ≤ X₁ ∧ X₁ ≤ (X₄)³

Stabilization-Threshold for: 4⋅(X₀)²+(X₄)⁵+2⋅(X₄)³ < 3⋅X₁
alphas_abs: 3⋅X₁
M: 0
N: 1
Bound: 6⋅X₁+2 {O(n)}
Stabilization-Threshold for: (X₀)²+(X₄)⁵ < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}

relevant size-bounds w.r.t. t₁:
X₁: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 16⋅X₃+21 {O(n)}

16⋅X₃+21 {O(n)}

knowledge_propagation leads to new time bound 16⋅X₃+22 {O(n)} for transition t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: (X₀)²+(X₄)⁵ < X₁ ∧ X₀ < 0 ∧ 1 ≤ X₄

knowledge_propagation leads to new time bound 16⋅X₃+22 {O(n)} for transition t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: (X₀)²+(X₄)⁵ < X₁ ∧ 0 < X₀ ∧ 1 ≤ X₄

All Bounds

Timebounds

Overall timebound:48⋅X₃+71 {O(n)}
t₀: 1 {O(1)}
t₃: 16⋅X₃+22 {O(n)}
t₄: 16⋅X₃+22 {O(n)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₇: 16⋅X₃+21 {O(n)}

Costbounds

Overall costbound: 48⋅X₃+71 {O(n)}
t₀: 1 {O(1)}
t₃: 16⋅X₃+22 {O(n)}
t₄: 16⋅X₃+22 {O(n)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₇: 16⋅X₃+21 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₃, X₀: 2⋅2^(16⋅X₃+21)⋅X₂ {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 2⋅2^(16⋅X₃+21)⋅X₂ {O(EXP)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₅, X₀: 2⋅2^(16⋅X₃+21)⋅X₂+X₂ {O(EXP)}
t₅, X₂: 3⋅X₂ {O(n)}
t₅, X₃: 3⋅X₃ {O(n)}
t₅, X₄: 3⋅X₄ {O(n)}
t₆, X₀: 0 {O(1)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}
t₆, X₄: 3⋅X₄ {O(n)}
t₈, X₀: 2⋅2^(16⋅X₃+21)⋅X₂+X₀+X₂ {O(EXP)}
t₈, X₂: 7⋅X₂ {O(n)}
t₈, X₃: 7⋅X₃ {O(n)}
t₈, X₄: 7⋅X₄ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₇, X₀: 2⋅2^(16⋅X₃+21)⋅X₂ {O(EXP)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₃ {O(n)}
t₇, X₄: 2⋅X₄ {O(n)}