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) → l4(X₀, X₁, X₂, X₃)
t₄: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 2+X₁ < X₀
t₅: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 2+X₀ < X₁
t₆: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁+2 ∧ X₁ ≤ X₀+2
t₇: l2(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, X₂, X₃) :|: X₀ < X₁ ∧ X₀ < X₁
t₈: l2(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃) :|: X₀ < X₁ ∧ X₁ ≤ X₀
t₉: l2(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁+1, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ < X₁
t₁₀: l2(X₀, X₁, X₂, X₃) → l1(X₀, X₁+1, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₁ ≤ X₀
t₁₁: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
t₃: l4(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃) :|: 0 ≤ X₂ ∧ 0 ≤ X₃
t₁: l4(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₂ < 0
t₂: l4(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₃ < 0

Preprocessing

Cut unsatisfiable transition t₈: l2→l1

Cut unsatisfiable transition t₉: l2→l1

Found invariant X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2

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

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) → l4(X₀, X₁, X₂, X₃)
t₄: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 2+X₁ < X₀ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₅: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 2+X₀ < X₁ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₆: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁+2 ∧ X₁ ≤ X₀+2 ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₇: l2(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, X₂, X₃) :|: X₀ < X₁ ∧ X₀ < X₁ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₁₀: l2(X₀, X₁, X₂, X₃) → l1(X₀, X₁+1, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₁₁: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
t₃: l4(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃) :|: 0 ≤ X₂ ∧ 0 ≤ X₃
t₁: l4(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₂ < 0
t₂: l4(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₃ < 0

Found invariant X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2

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

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 4⋅X₂+6 {O(n)}

TWN-Loops:

entry: t₃: l4(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃) :|: 0 ≤ X₂ ∧ 0 ≤ X₃
results in twn-loop: twn:Inv: [X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀,X₁+1,X₂,X₃) :|: 2+X₁ < X₀ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀
order: [X₀; X₁; X₂; X₃]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
X₂: X₂
X₃: X₃

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ 2+X₁ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₁ < X₀ ∧ 2+X₁ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 < 0
∨ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 2+X₁ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

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

relevant size-bounds w.r.t. t₃:
X₀: X₂ {O(n)}
Runtime-bound of t₃: 1 {O(1)}
Results in: 4⋅X₂+6 {O(n)}

4⋅X₂+6 {O(n)}

Found invariant X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2

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

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 4⋅X₃+6 {O(n)}

TWN-Loops:

entry: t₃: l4(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃) :|: 0 ≤ X₂ ∧ 0 ≤ X₃
results in twn-loop: twn:Inv: [X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀+1,X₁,X₂,X₃) :|: 2+X₀ < X₁ ∧ X₀ < X₁ ∧ X₀ < X₁
order: [X₀; X₁; X₂; X₃]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁
X₂: X₂
X₃: X₃

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ 2+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₀ < X₁ ∧ 2+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₀ < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
Stabilization-Threshold for: 2+X₀ < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}

relevant size-bounds w.r.t. t₃:
X₁: X₃ {O(n)}
Runtime-bound of t₃: 1 {O(1)}
Results in: 4⋅X₃+6 {O(n)}

4⋅X₃+6 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₇ 4⋅X₃+6 {O(n)}

relevant size-bounds w.r.t. t₃:
X₁: X₃ {O(n)}
Runtime-bound of t₃: 1 {O(1)}
Results in: 4⋅X₃+6 {O(n)}

4⋅X₃+6 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₀ 4⋅X₂+6 {O(n)}

relevant size-bounds w.r.t. t₃:
X₀: X₂ {O(n)}
Runtime-bound of t₃: 1 {O(1)}
Results in: 4⋅X₂+6 {O(n)}

4⋅X₂+6 {O(n)}

All Bounds

Timebounds

Overall timebound:8⋅X₂+8⋅X₃+30 {O(n)}
t₀: 1 {O(1)}
t₄: 4⋅X₂+6 {O(n)}
t₅: 4⋅X₃+6 {O(n)}
t₆: 1 {O(1)}
t₇: 4⋅X₃+6 {O(n)}
t₁₀: 4⋅X₂+6 {O(n)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₂+8⋅X₃+30 {O(n)}
t₀: 1 {O(1)}
t₄: 4⋅X₂+6 {O(n)}
t₅: 4⋅X₃+6 {O(n)}
t₆: 1 {O(1)}
t₇: 4⋅X₃+6 {O(n)}
t₁₀: 4⋅X₂+6 {O(n)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: 4⋅X₂+X₃+6 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 4⋅X₃+X₂+6 {O(n)}
t₅, X₁: X₃ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: 3⋅X₂+4⋅X₃+6 {O(n)}
t₆, X₁: 3⋅X₃+4⋅X₂+6 {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}
t₇, X₀: 4⋅X₃+X₂+6 {O(n)}
t₇, X₁: X₃ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₁₀, X₀: X₂ {O(n)}
t₁₀, X₁: 4⋅X₂+X₃+6 {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: X₃ {O(n)}
t₁₁, X₀: 2⋅X₀+3⋅X₂+4⋅X₃+6 {O(n)}
t₁₁, X₁: 2⋅X₁+3⋅X₃+4⋅X₂+6 {O(n)}
t₁₁, X₂: 5⋅X₂ {O(n)}
t₁₁, X₃: 5⋅X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}