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

TWN: t₁: l1→l1

cycle: [t₁: l1→l1; t₂: l1→l1]
loop: ((X₀)²+(X₂)⁵ < X₁ ∧ X₀ < 0 ∨ (X₀)²+(X₂)⁵ < X₁ ∧ 0 < X₀,(X₀,X₁,X₂) -> (4⋅X₀,9⋅X₁-8⋅(X₂)³,X₂)
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+(X₂)³+(X₂)⁵
M: 0
N: 1
Bound: 2⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+2⋅X₂⋅X₂⋅X₂+2⋅X₁+2 {O(n^5)}

TWN - Lifting for t₁: l1→l1 of 2⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+2⋅X₂⋅X₂⋅X₂+2⋅X₁+6 {O(n^5)}

relevant size-bounds w.r.t. t₀:
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+2⋅X₂⋅X₂⋅X₂+2⋅X₁+6 {O(n^5)}

TWN: t₂: l1→l1

TWN - Lifting for t₂: l1→l1 of 2⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+2⋅X₂⋅X₂⋅X₂+2⋅X₁+6 {O(n^5)}

relevant size-bounds w.r.t. t₀:
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+2⋅X₂⋅X₂⋅X₂+2⋅X₁+6 {O(n^5)}

Analysing control-flow refined program

Eliminate variables {NoDet0,X₁} that do not contribute to the problem

Found invariant 1 ≤ X₁ ∧ 5+X₀ ≤ X₁ ∧ 4+X₀ ≤ 0 for location n_l1___2

Found invariant 1 ≤ X₁ for location l1

Found invariant 1 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 4 ≤ X₀ for location n_l1___1

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:4⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+4⋅X₂⋅X₂⋅X₂+4⋅X₁+13 {O(n^5)}
t₀: 1 {O(1)}
t₁: 2⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+2⋅X₂⋅X₂⋅X₂+2⋅X₁+6 {O(n^5)}
t₂: 2⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+2⋅X₂⋅X₂⋅X₂+2⋅X₁+6 {O(n^5)}

Costbounds

Overall costbound: 4⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+4⋅X₂⋅X₂⋅X₂+4⋅X₁+13 {O(n^5)}
t₀: 1 {O(1)}
t₁: 2⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+2⋅X₂⋅X₂⋅X₂+2⋅X₁+6 {O(n^5)}
t₂: 2⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+2⋅X₂⋅X₂⋅X₂+2⋅X₁+6 {O(n^5)}

Sizebounds

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