Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₂
t₁: l1(X₀, X₁, X₂) → l1(4⋅X₀, 9⋅X₁-8⋅(X₂)³, X₂) :|: (X₀)²+(X₂)⁵ < X₁ ∧ X₀ < 0
t₂: l1(X₀, X₁, X₂) → l1(4⋅X₀, 9⋅X₁-8⋅(X₂)³, X₂) :|: (X₀)²+(X₂)⁵ < X₁ ∧ 0 < X₀

Preprocessing

Found invariant 1 ≤ X₂ for location l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₂
t₁: l1(X₀, X₁, X₂) → l1(4⋅X₀, 9⋅X₁-8⋅(X₂)³, X₂) :|: (X₀)²+(X₂)⁵ < X₁ ∧ X₀ < 0 ∧ 1 ≤ X₂
t₂: l1(X₀, X₁, X₂) → l1(4⋅X₀, 9⋅X₁-8⋅(X₂)³, X₂) :|: (X₀)²+(X₂)⁵ < X₁ ∧ 0 < X₀ ∧ 1 ≤ X₂

Found invariant 1 ≤ X₂ for location l1

Time-Bound by TWN-Loops:

TWN-Loops: t₁ 2⋅X₁+6 {O(n)}

TWN-Loops:

entry: t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₂
results in twn-loop: twn:Inv: [1 ≤ X₂ ∧ 1 ≤ X₂] , (X₀,X₁,X₂) -> (4⋅X₀,9⋅X₁-8⋅(X₂)³,X₂) :|: (X₀)²+(X₂)⁵ < X₁ ∧ X₀ < 0 ∨ (X₀)²+(X₂)⁵ < X₁ ∧ 0 < X₀
order: [X₀; X₂; X₁]
closed-form:
X₀: X₀ * 4^n
X₂: X₂
X₁: X₁ * 9^n + [[n != 0]] * -(X₂)³ * 9^n + [[n != 0]] * (X₂)³

Termination: true
Formula:

X₀ < 0 ∧ (X₀)² < 0
∨ X₀ < 0 ∧ (X₂)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ X₀ < 0 ∧ (X₂)⁵ < (X₂)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₂)³ ≤ X₁ ∧ X₁ ≤ (X₂)³
∨ 0 < X₀ ∧ (X₀)² < 0
∨ 0 < X₀ ∧ (X₂)³ < X₁ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)²
∨ 0 < X₀ ∧ (X₂)⁵ < (X₂)³ ∧ (X₀)² ≤ 0 ∧ 0 ≤ (X₀)² ∧ (X₂)³ ≤ X₁ ∧ X₁ ≤ (X₂)³

Stabilization-Threshold for: (X₀)²+(X₂)⁵ < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}

relevant size-bounds w.r.t. t₀:
X₁: X₁ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₁+6 {O(n)}

2⋅X₁+6 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₁+6 {O(n)}

relevant size-bounds w.r.t. t₀:
X₁: X₁ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₁+6 {O(n)}

2⋅X₁+6 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₁+13 {O(n)}
t₀: 1 {O(1)}
t₁: 2⋅X₁+6 {O(n)}
t₂: 2⋅X₁+6 {O(n)}

Costbounds

Overall costbound: 4⋅X₁+13 {O(n)}
t₀: 1 {O(1)}
t₁: 2⋅X₁+6 {O(n)}
t₂: 2⋅X₁+6 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 4^(2⋅X₁+6)⋅X₀ {O(EXP)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 4^(2⋅X₁+6)⋅X₀ {O(EXP)}
t₂, X₂: X₂ {O(n)}