Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₂ < 0
t₁: l1(X₀, X₁, X₂) → l1(X₀+(X₁)²*X₂, X₁-2⋅(X₂)², X₂) :|: 0 < X₀+(X₁)²

Preprocessing

Found invariant 1+X₂ ≤ 0 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) :|: X₂ < 0
t₁: l1(X₀, X₁, X₂) → l1(X₀+(X₁)²*X₂, X₁-2⋅(X₂)², X₂) :|: 0 < X₀+(X₁)² ∧ 1+X₂ ≤ 0

TWN: t₁: l1→l1

cycle: [t₁: l1→l1]
loop: (0 < X₀+(X₁)²,(X₀,X₁,X₂) -> (X₀+(X₁)²*X₂,X₁-2⋅(X₂)²,X₂)
order: [X₂; X₁; X₀]
closed-form:
X₂: X₂
X₁: X₁ + [[n != 0]] * -2⋅(X₂)² * n^1
X₀: X₀ + [[n != 0]] * (X₁)²*X₂ * n^1 + [[n != 0, n != 1]] * 4/3⋅(X₂)⁵ * n^3 + [[n != 0, n != 1]] * (-2⋅(X₂)⁵-2⋅X₁*(X₂)³) * n^2 + [[n != 0, n != 1]] * (2/3⋅(X₂)⁵+2⋅X₁*(X₂)³) * n^1

Termination: true
Formula:

0 < 4⋅(X₂)⁵
∨ 6⋅(X₂)⁵+6⋅X₁*(X₂)³ < 12⋅(X₂)⁴ ∧ 0 ≤ 4⋅(X₂)⁵ ∧ 4⋅(X₂)⁵ ≤ 0
∨ 12⋅X₁*(X₂)² < 3⋅(X₁)²*X₂+2⋅(X₂)⁵+6⋅X₁*(X₂)³ ∧ 0 ≤ 4⋅(X₂)⁵ ∧ 4⋅(X₂)⁵ ≤ 0 ∧ 6⋅(X₂)⁵+6⋅X₁*(X₂)³ ≤ 12⋅(X₂)⁴ ∧ 12⋅(X₂)⁴ ≤ 6⋅(X₂)⁵+6⋅X₁*(X₂)³
∨ 0 < 3⋅X₀+3⋅(X₁)² ∧ 0 ≤ 4⋅(X₂)⁵ ∧ 4⋅(X₂)⁵ ≤ 0 ∧ 6⋅(X₂)⁵+6⋅X₁*(X₂)³ ≤ 12⋅(X₂)⁴ ∧ 12⋅(X₂)⁴ ≤ 6⋅(X₂)⁵+6⋅X₁*(X₂)³ ∧ 12⋅X₁*(X₂)² ≤ 3⋅(X₁)²*X₂+2⋅(X₂)⁵+6⋅X₁*(X₂)³ ∧ 3⋅(X₁)²*X₂+2⋅(X₂)⁵+6⋅X₁*(X₂)³ ≤ 12⋅X₁*(X₂)²

Stabilization-Threshold for: 0 < X₀+(X₁)²
alphas_abs: 3⋅X₀+12⋅X₁*(X₂)²+6⋅X₁*(X₂)³+3⋅(X₁)²+3⋅(X₁)²*X₂+12⋅(X₂)⁴+6⋅(X₂)⁵
M: 0
N: 3
Bound: 12⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₂⋅X₂⋅X₂+24⋅X₁⋅X₂⋅X₂+6⋅X₁⋅X₁⋅X₂+6⋅X₁⋅X₁+6⋅X₀+4 {O(n^5)}

TWN - Lifting for t₁: l1→l1 of 12⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₂⋅X₂⋅X₂+24⋅X₁⋅X₂⋅X₂+6⋅X₁⋅X₁⋅X₂+6⋅X₁⋅X₁+6⋅X₀+6 {O(n^5)}

relevant size-bounds w.r.t. t₀:
X₀: X₀ {O(n)}
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 12⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₂⋅X₂⋅X₂+24⋅X₁⋅X₂⋅X₂+6⋅X₁⋅X₁⋅X₂+6⋅X₁⋅X₁+6⋅X₀+6 {O(n^5)}

Analysing control-flow refined program

Eliminate variables {NoDet0,NoDet1,X₀,X₁} that do not contribute to the problem

Found invariant 1+X₀ ≤ 0 for location l1

Found invariant 1+X₀ ≤ 0 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:12⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₂⋅X₂⋅X₂+24⋅X₁⋅X₂⋅X₂+6⋅X₁⋅X₁⋅X₂+6⋅X₁⋅X₁+6⋅X₀+7 {O(n^5)}
t₀: 1 {O(1)}
t₁: 12⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₂⋅X₂⋅X₂+24⋅X₁⋅X₂⋅X₂+6⋅X₁⋅X₁⋅X₂+6⋅X₁⋅X₁+6⋅X₀+6 {O(n^5)}

Costbounds

Overall costbound: 12⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₂⋅X₂⋅X₂+24⋅X₁⋅X₂⋅X₂+6⋅X₁⋅X₁⋅X₂+6⋅X₁⋅X₁+6⋅X₀+7 {O(n^5)}
t₀: 1 {O(1)}
t₁: 12⋅X₂⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₂⋅X₂⋅X₂+24⋅X₁⋅X₂⋅X₂+6⋅X₁⋅X₁⋅X₂+6⋅X₁⋅X₁+6⋅X₀+6 {O(n^5)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₂: X₂ {O(n)}