Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -2⋅X₁-1, X₂, X₃)
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant 1+X₀ ≤ 0 for location l5

Found invariant 1+X₀ ≤ 0 for location l4

Found invariant 0 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -2⋅X₁-1, X₂, X₃) :|: 0 ≤ X₀
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0

Found invariant 1+X₀ ≤ 0 for location l5

Found invariant 1+X₀ ≤ 0 for location l4

Found invariant 0 ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 24⋅X₃+72⋅X₂+65 {O(n)}

TWN-Loops:

entry: t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
results in twn-loop: twn:Inv: [0 ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀+X₁,-2⋅X₁-1,X₂,X₃) :|: 0 ≤ X₀
order: [X₁; X₀]
closed-form:
X₁: X₁ * 4^n + [[n != 0]] * 1/3 * 4^n + [[n != 0]] * -1/3
X₀: X₀ + [[n != 0]] * -1/3⋅X₁ * 4^n + [[n != 0]] * -1 * n^1 + [[n != 0]] * 1/3⋅X₁ + [[n != 0, n != 1]] * -1/9 * 4^n + [[n != 0, n != 1]] * 1/3 * n^1 + [[n != 0, n != 1]] * 1/9

Termination: true
Formula:

0 < 6⋅X₁+2 ∧ 3⋅X₁+1 < 0
∨ 0 < 6⋅X₁+2 ∧ 6 < 0 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1
∨ 0 < 6⋅X₁+2 ∧ 0 < 9⋅X₀+3⋅X₁+1 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1 ∧ 6 ≤ 0 ∧ 0 ≤ 6
∨ 0 < 6⋅X₁+2 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1 ∧ 6 ≤ 0 ∧ 0 ≤ 6 ∧ 0 ≤ 9⋅X₀+3⋅X₁+1 ∧ 9⋅X₀+3⋅X₁+1 ≤ 0
∨ 6 < 0 ∧ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 3⋅X₁+1 < 0
∨ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 6 < 0 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1
∨ 6 < 0 ∧ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 0 < 9⋅X₀+3⋅X₁+1 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1 ∧ 6 ≤ 0 ∧ 0 ≤ 6
∨ 6 < 0 ∧ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1 ∧ 6 ≤ 0 ∧ 0 ≤ 6 ∧ 0 ≤ 9⋅X₀+3⋅X₁+1 ∧ 9⋅X₀+3⋅X₁+1 ≤ 0
∨ 2 < 9⋅X₀+3⋅X₁ ∧ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 6 ≤ 0 ∧ 0 ≤ 6 ∧ 3⋅X₁+1 < 0
∨ 2 < 9⋅X₀+3⋅X₁ ∧ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 6 ≤ 0 ∧ 0 ≤ 6 ∧ 6 < 0 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1
∨ 2 < 9⋅X₀+3⋅X₁ ∧ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 0 < 9⋅X₀+3⋅X₁+1 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1 ∧ 6 ≤ 0 ∧ 0 ≤ 6
∨ 2 < 9⋅X₀+3⋅X₁ ∧ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1 ∧ 6 ≤ 0 ∧ 0 ≤ 6 ∧ 0 ≤ 9⋅X₀+3⋅X₁+1 ∧ 9⋅X₀+3⋅X₁+1 ≤ 0
∨ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 6 ≤ 0 ∧ 0 ≤ 6 ∧ 2 ≤ 9⋅X₀+3⋅X₁ ∧ 9⋅X₀+3⋅X₁ ≤ 2 ∧ 3⋅X₁+1 < 0
∨ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 6 ≤ 0 ∧ 0 ≤ 6 ∧ 2 ≤ 9⋅X₀+3⋅X₁ ∧ 9⋅X₀+3⋅X₁ ≤ 2 ∧ 6 < 0 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1
∨ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 2 ≤ 9⋅X₀+3⋅X₁ ∧ 9⋅X₀+3⋅X₁ ≤ 2 ∧ 0 < 9⋅X₀+3⋅X₁+1 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1 ∧ 6 ≤ 0 ∧ 0 ≤ 6
∨ 0 ≤ 6⋅X₁+2 ∧ 6⋅X₁+2 ≤ 0 ∧ 2 ≤ 9⋅X₀+3⋅X₁ ∧ 9⋅X₀+3⋅X₁ ≤ 2 ∧ 3⋅X₁+1 ≤ 0 ∧ 0 ≤ 3⋅X₁+1 ∧ 6 ≤ 0 ∧ 0 ≤ 6 ∧ 0 ≤ 9⋅X₀+3⋅X₁+1 ∧ 9⋅X₀+3⋅X₁+1 ≤ 0

Stabilization-Threshold for: 0 ≤ X₀+X₁
alphas_abs: 6+9⋅X₀+3⋅X₁
M: 0
N: 2
Bound: 18⋅X₀+6⋅X₁+15 {O(n)}
Stabilization-Threshold for: 0 ≤ X₀
alphas_abs: 6+9⋅X₀+3⋅X₁
M: 0
N: 2
Bound: 18⋅X₀+6⋅X₁+15 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 24⋅X₃+72⋅X₂+65 {O(n)}

24⋅X₃+72⋅X₂+65 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 24⋅X₃+72⋅X₂+65 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 24⋅X₃+72⋅X₂+65 {O(n)}

24⋅X₃+72⋅X₂+65 {O(n)}

All Bounds

Timebounds

Overall timebound:144⋅X₂+48⋅X₃+134 {O(n)}
t₀: 1 {O(1)}
t₂: 24⋅X₃+72⋅X₂+65 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 24⋅X₃+72⋅X₂+65 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 144⋅X₂+48⋅X₃+134 {O(n)}
t₀: 1 {O(1)}
t₂: 24⋅X₃+72⋅X₂+65 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 24⋅X₃+72⋅X₂+65 {O(n)}
t₅: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₂, X₀: 25⋅2^(24⋅X₃+72⋅X₂+65)⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅3185⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅3528⋅X₂⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅4290+2^(24⋅X₃+72⋅X₂+65)⋅5184⋅X₂⋅X₂+2^(24⋅X₃+72⋅X₂+65)⋅600⋅X₃⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅72⋅X₂+2^(24⋅X₃+72⋅X₂+65)⋅9360⋅X₂+X₂ {O(EXP)}
t₂, X₁: 25⋅2^(24⋅X₃+72⋅X₂+65)⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅65+2^(24⋅X₃+72⋅X₂+65)⋅72⋅X₂ {O(EXP)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 25⋅2^(24⋅X₃+72⋅X₂+65)⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅3185⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅3528⋅X₂⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅4290+2^(24⋅X₃+72⋅X₂+65)⋅5184⋅X₂⋅X₂+2^(24⋅X₃+72⋅X₂+65)⋅600⋅X₃⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅72⋅X₂+2^(24⋅X₃+72⋅X₂+65)⋅9360⋅X₂+2⋅X₂ {O(EXP)}
t₃, X₁: 25⋅2^(24⋅X₃+72⋅X₂+65)⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅65+2^(24⋅X₃+72⋅X₂+65)⋅72⋅X₂+X₃ {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₄, X₀: 25⋅2^(24⋅X₃+72⋅X₂+65)⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅3185⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅3528⋅X₂⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅4290+2^(24⋅X₃+72⋅X₂+65)⋅5184⋅X₂⋅X₂+2^(24⋅X₃+72⋅X₂+65)⋅600⋅X₃⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅72⋅X₂+2^(24⋅X₃+72⋅X₂+65)⋅9360⋅X₂+X₂ {O(EXP)}
t₄, X₁: 25⋅2^(24⋅X₃+72⋅X₂+65)⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅65+2^(24⋅X₃+72⋅X₂+65)⋅72⋅X₂ {O(EXP)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 25⋅2^(24⋅X₃+72⋅X₂+65)⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅3185⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅3528⋅X₂⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅4290+2^(24⋅X₃+72⋅X₂+65)⋅5184⋅X₂⋅X₂+2^(24⋅X₃+72⋅X₂+65)⋅600⋅X₃⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅72⋅X₂+2^(24⋅X₃+72⋅X₂+65)⋅9360⋅X₂+2⋅X₂ {O(EXP)}
t₅, X₁: 25⋅2^(24⋅X₃+72⋅X₂+65)⋅X₃+2^(24⋅X₃+72⋅X₂+65)⋅65+2^(24⋅X₃+72⋅X₂+65)⋅72⋅X₂+X₃ {O(EXP)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}