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)
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₀+(X₁)² ≤ 0
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₀+(X₁)²
t₆: 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) :|: X₄ < 0
t₂: l3(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₄
t₅: l4(X₀, X₁, X₂, X₃, X₄) → l1(X₀+(X₁)²*X₄, X₁-2⋅(X₄)², X₂, X₃, X₄)

Preprocessing

Found invariant 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ for location l1

Found invariant 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ 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)
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₀+(X₁)² ≤ 0 ∧ 1+X₄ ≤ 0 ∧ X₁ ≤ X₃
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₀+(X₁)² ∧ 1+X₄ ≤ 0 ∧ X₁ ≤ X₃
t₆: 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) :|: X₄ < 0
t₂: l3(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₄
t₅: l4(X₀, X₁, X₂, X₃, X₄) → l1(X₀+(X₁)²*X₄, X₁-2⋅(X₄)², X₂, X₃, X₄) :|: 1+X₄ ≤ 0 ∧ X₁ ≤ X₃

Found invariant 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ for location l1

Found invariant 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ for location l4

Time-Bound by TWN-Loops:

TWN-Loops: t₃ 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}

TWN-Loops:

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

Termination: true
Formula:

0 < 4⋅(X₄)⁵
∨ 6⋅(X₄)⁵+6⋅X₁*(X₄)³ < 12⋅(X₄)⁴ ∧ 0 ≤ 4⋅(X₄)⁵ ∧ 4⋅(X₄)⁵ ≤ 0
∨ 12⋅X₁*(X₄)² < 3⋅(X₁)²*X₄+2⋅(X₄)⁵+6⋅X₁*(X₄)³ ∧ 0 ≤ 4⋅(X₄)⁵ ∧ 4⋅(X₄)⁵ ≤ 0 ∧ 6⋅(X₄)⁵+6⋅X₁*(X₄)³ ≤ 12⋅(X₄)⁴ ∧ 12⋅(X₄)⁴ ≤ 6⋅(X₄)⁵+6⋅X₁*(X₄)³
∨ 0 < 3⋅X₀+3⋅(X₁)² ∧ 0 ≤ 4⋅(X₄)⁵ ∧ 4⋅(X₄)⁵ ≤ 0 ∧ 6⋅(X₄)⁵+6⋅X₁*(X₄)³ ≤ 12⋅(X₄)⁴ ∧ 12⋅(X₄)⁴ ≤ 6⋅(X₄)⁵+6⋅X₁*(X₄)³ ∧ 12⋅X₁*(X₄)² ≤ 3⋅(X₁)²*X₄+2⋅(X₄)⁵+6⋅X₁*(X₄)³ ∧ 3⋅(X₁)²*X₄+2⋅(X₄)⁵+6⋅X₁*(X₄)³ ≤ 12⋅X₁*(X₄)²

Stabilization-Threshold for: 0 < X₀+(X₁)²
alphas_abs: 3⋅X₀+12⋅X₁*(X₄)²+6⋅X₁*(X₄)³+3⋅(X₁)²+12⋅(X₄)⁴+6⋅(X₄)⁵
M: 0
N: 3
Bound: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₁⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₁⋅X₄⋅X₄+6⋅X₁⋅X₁+6⋅X₀+4 {O(n^5)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}

12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}

12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}

Analysing control-flow refined program

Found invariant 1+X₄ ≤ 0 ∧ X₁ ≤ X₃ for location n_l4___2

Found invariant 1+X₄ ≤ 0 ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1

Found invariant 1+X₄ ≤ 0 for location n_l1___1

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:24⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄⋅X₄+48⋅X₄⋅X₄⋅X₄⋅X₄+48⋅X₃⋅X₄⋅X₄+12⋅X₃⋅X₃+12⋅X₂+17 {O(n^5)}
t₀: 1 {O(1)}
t₃: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}
t₄: 1 {O(1)}
t₆: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₅: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}

Costbounds

Overall costbound: 24⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄⋅X₄+48⋅X₄⋅X₄⋅X₄⋅X₄+48⋅X₃⋅X₄⋅X₄+12⋅X₃⋅X₃+12⋅X₂+17 {O(n^5)}
t₀: 1 {O(1)}
t₃: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}
t₄: 1 {O(1)}
t₆: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₅: 12⋅X₄⋅X₄⋅X₄⋅X₄⋅X₄+12⋅X₃⋅X₄⋅X₄⋅X₄+24⋅X₄⋅X₄⋅X₄⋅X₄+24⋅X₃⋅X₄⋅X₄+6⋅X₃⋅X₃+6⋅X₂+6 {O(n^5)}

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, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}
t₆, X₄: 3⋅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)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}