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) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀+X₁ ∧ X₀ ≤ X₄
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀+X₁ < 0
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₄ < X₀
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(2⋅X₀+X₁, X₁+1, X₂, X₃, X₄)
t₆: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant X₃ ≤ X₁ for location l5

Found invariant X₃ ≤ X₁ for location l1

Found invariant X₃ ≤ X₁ for location l4

Found invariant 0 ≤ X₁+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₀+X₁ for location l3

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) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 0 ≤ X₀+X₁ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀+X₁ < 0 ∧ X₃ ≤ X₁
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₄ < X₀ ∧ X₃ ≤ X₁
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(2⋅X₀+X₁, X₁+1, X₂, X₃, X₄) :|: 0 ≤ X₁+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₀+X₁
t₆: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₁

Found invariant X₃ ≤ X₁ for location l5

Found invariant X₃ ≤ X₁ for location l1

Found invariant X₃ ≤ X₁ for location l4

Found invariant 0 ≤ X₁+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₀+X₁ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₃+2⋅X₄+14 {O(n)}

TWN-Loops:

entry: t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄)
results in twn-loop: twn:Inv: [X₃ ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₀+X₁] , (X₀,X₁,X₂,X₃,X₄) -> (2⋅X₀+X₁,X₁+1,X₂,X₃,X₄) :|: 0 ≤ X₀+X₁ ∧ X₀ ≤ X₄
order: [X₁; X₀; X₃; X₄]
closed-form:
X₁: X₁ + [[n != 0]] * n^1
X₀: X₀ * 2^n + [[n != 0]] * X₁ * 2^n + [[n != 0]] * -X₁ + [[n != 0, n != 1]] * 2^n + [[n != 0, n != 1]] * -1 * n^1 + [[n != 0, n != 1]] * -1
X₃: X₃
X₄: X₄

Termination: true
Formula:

X₀+X₁+1 < 0 ∧ 0 < X₀+X₁+1
∨ X₀+X₁+1 < 0 ∧ 1 < 0 ∧ 0 ≤ X₀+X₁+1 ∧ X₀+X₁+1 ≤ 0
∨ X₀+X₁+1 < 0 ∧ 0 ≤ X₀+X₁+1 ∧ X₀+X₁+1 ≤ 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 1 ∧ X₀+X₁+1 ≤ 0 ∧ 0 ≤ X₀+X₁+1 ∧ 0 < X₀+X₁+1
∨ 0 < 1 ∧ 1 < 0 ∧ 0 ≤ X₀+X₁+1 ∧ X₀+X₁+1 ≤ 0
∨ 0 < 1 ∧ 0 ≤ X₀+X₁+1 ∧ X₀+X₁+1 ≤ 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < X₁+1+X₄ ∧ X₀+X₁+1 ≤ 0 ∧ 0 ≤ X₀+X₁+1 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 < X₀+X₁+1
∨ 0 < X₁+1+X₄ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < 0 ∧ 0 ≤ X₀+X₁+1 ∧ X₀+X₁+1 ≤ 0
∨ 0 < X₁+1+X₄ ∧ 0 ≤ X₀+X₁+1 ∧ X₀+X₁+1 ≤ 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀+X₁+1 ≤ 0 ∧ 0 ≤ X₀+X₁+1 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₁+1+X₄ ∧ X₁+1+X₄ ≤ 0 ∧ 0 < X₀+X₁+1
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₁+1+X₄ ∧ X₁+1+X₄ ≤ 0 ∧ 1 < 0 ∧ 0 ≤ X₀+X₁+1 ∧ X₀+X₁+1 ≤ 0
∨ 0 ≤ X₁+1+X₄ ∧ X₁+1+X₄ ≤ 0 ∧ 0 ≤ X₀+X₁+1 ∧ X₀+X₁+1 ≤ 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₀ ≤ X₄
alphas_abs: 1+X₁+X₄
M: 5
N: 2
Bound: 2⋅X₁+2⋅X₄+8 {O(n)}
Stabilization-Threshold for: 0 ≤ X₀+X₁
alphas_abs: 1
M: 0
N: 1
Bound: 4 {O(1)}

relevant size-bounds w.r.t. t₁:
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₃+2⋅X₄+14 {O(n)}

2⋅X₃+2⋅X₄+14 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 2⋅X₃+2⋅X₄+14 {O(n)}

relevant size-bounds w.r.t. t₁:
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₃+2⋅X₄+14 {O(n)}

2⋅X₃+2⋅X₄+14 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₃+4⋅X₄+33 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₃+2⋅X₄+14 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 2⋅X₃+2⋅X₄+14 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+4⋅X₄+33 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₃+2⋅X₄+14 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 2⋅X₃+2⋅X₄+14 {O(n)}
t₆: 1 {O(1)}

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)}
t₂, X₀: 10⋅2^(2⋅X₃+2⋅X₄+14)⋅X₃⋅X₄+210⋅2^(2⋅X₃+2⋅X₄+14)+2^(2⋅X₃+2⋅X₄+14)⋅4⋅X₄⋅X₄+2^(2⋅X₃+2⋅X₄+14)⋅58⋅X₄+2^(2⋅X₃+2⋅X₄+14)⋅6⋅X₃⋅X₃+2^(2⋅X₃+2⋅X₄+14)⋅73⋅X₃+2^(2⋅X₃+2⋅X₄+14)⋅X₂ {O(EXP)}
t₂, X₁: 2⋅X₄+3⋅X₃+14 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₄, X₀: 10⋅2^(2⋅X₃+2⋅X₄+14)⋅X₃⋅X₄+210⋅2^(2⋅X₃+2⋅X₄+14)+2^(2⋅X₃+2⋅X₄+14)⋅4⋅X₄⋅X₄+2^(2⋅X₃+2⋅X₄+14)⋅58⋅X₄+2^(2⋅X₃+2⋅X₄+14)⋅6⋅X₃⋅X₃+2^(2⋅X₃+2⋅X₄+14)⋅73⋅X₃+2^(2⋅X₃+2⋅X₄+14)⋅X₂+X₂ {O(EXP)}
t₄, X₁: 2⋅X₄+4⋅X₃+14 {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅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)}
t₅, X₀: 10⋅2^(2⋅X₃+2⋅X₄+14)⋅X₃⋅X₄+210⋅2^(2⋅X₃+2⋅X₄+14)+2^(2⋅X₃+2⋅X₄+14)⋅4⋅X₄⋅X₄+2^(2⋅X₃+2⋅X₄+14)⋅58⋅X₄+2^(2⋅X₃+2⋅X₄+14)⋅6⋅X₃⋅X₃+2^(2⋅X₃+2⋅X₄+14)⋅73⋅X₃+2^(2⋅X₃+2⋅X₄+14)⋅X₂ {O(EXP)}
t₅, X₁: 2⋅X₄+3⋅X₃+14 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: 10⋅2^(2⋅X₃+2⋅X₄+14)⋅X₃⋅X₄+210⋅2^(2⋅X₃+2⋅X₄+14)+2^(2⋅X₃+2⋅X₄+14)⋅4⋅X₄⋅X₄+2^(2⋅X₃+2⋅X₄+14)⋅58⋅X₄+2^(2⋅X₃+2⋅X₄+14)⋅6⋅X₃⋅X₃+2^(2⋅X₃+2⋅X₄+14)⋅73⋅X₃+2^(2⋅X₃+2⋅X₄+14)⋅X₂+2⋅X₂ {O(EXP)}
t₆, X₁: 2⋅X₄+5⋅X₃+14 {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}
t₆, X₄: 3⋅X₄ {O(n)}