Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀-(X₁)², X₁+(X₂)², X₂) :|: 0 < X₀ ∧ X₂ < 0
t₂: l1(X₀, X₁, X₂) → l1(X₀-(X₁)², X₁+(X₂)², X₂) :|: 0 < X₀ ∧ 0 < X₂
t₃: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ ≤ 0
t₄: l2(X₀, X₁, X₂) → l2(X₀, X₁-1, X₂) :|: 0 < X₁

Preprocessing

Found invariant X₀ ≤ 0 for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀-(X₁)², X₁+(X₂)², X₂) :|: 0 < X₀ ∧ X₂ < 0
t₂: l1(X₀, X₁, X₂) → l1(X₀-(X₁)², X₁+(X₂)², X₂) :|: 0 < X₀ ∧ 0 < X₂
t₃: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ ≤ 0
t₄: l2(X₀, X₁, X₂) → l2(X₀, X₁-1, X₂) :|: 0 < X₁ ∧ X₀ ≤ 0

TWN: t₁: l1→l1

cycle: [t₁: l1→l1; t₂: l1→l1]
loop: (0 < X₀ ∧ X₂ < 0 ∨ 0 < X₀ ∧ 0 < X₂,(X₀,X₁,X₂) -> (X₀-(X₁)²,X₁+(X₂)²,X₂)
order: [X₂; X₁; X₀]
closed-form:
X₂: X₂
X₁: X₁ + [[n != 0]] * (X₂)² * n^1
X₀: X₀ + [[n != 0]] * -(X₁)² * n^1 + [[n != 0, n != 1]] * -1/3⋅(X₂)⁴ * n^3 + [[n != 0, n != 1]] * (1/2⋅(X₂)⁴-X₁*(X₂)²) * n^2 + [[n != 0, n != 1]] * (X₁*(X₂)²-1/6⋅(X₂)⁴) * n^1

Termination: true
Formula:

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

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

TWN - Lifting for t₁: l1→l1 of 6⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂+12⋅X₁⋅X₁+12⋅X₀+8 {O(n^4)}

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: 6⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂+12⋅X₁⋅X₁+12⋅X₀+8 {O(n^4)}

TWN: t₂: l1→l1

TWN - Lifting for t₂: l1→l1 of 6⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂+12⋅X₁⋅X₁+12⋅X₀+8 {O(n^4)}

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: 6⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂+12⋅X₁⋅X₁+12⋅X₀+8 {O(n^4)}

Analysing control-flow refined program

Found invariant X₀ ≤ 0 for location l2

Found invariant 1+X₂ ≤ 0 for location n_l1___2

Found invariant 1 ≤ X₂ for location n_l1___1

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

Analysing control-flow refined program

Found invariant X₀ ≤ 0 for location l2

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 6⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂+12⋅X₁⋅X₁+12⋅X₀+8 {O(n^4)}
t₂: 6⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂+12⋅X₁⋅X₁+12⋅X₀+8 {O(n^4)}
t₃: 1 {O(1)}
t₄: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 6⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂+12⋅X₁⋅X₁+12⋅X₀+8 {O(n^4)}
t₂: 6⋅X₂⋅X₂⋅X₂⋅X₂+12⋅X₁⋅X₂⋅X₂+12⋅X₁⋅X₁+12⋅X₀+8 {O(n^4)}
t₃: 1 {O(1)}
t₄: inf {Infinity}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₂: 3⋅X₂ {O(n)}
t₄, X₂: 3⋅X₂ {O(n)}