Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 0 < X₁
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: l2(X₀, X₁, X₂) → l1(X₂, 1, X₂)
t₄: l3(X₀, X₁, X₂) → l1(X₀-1, 1, X₂) :|: 0 < X₀ ∧ 0 < X₀
t₅: l3(X₀, X₁, X₂) → l1(X₀, 1, X₂) :|: 0 < X₀ ∧ X₀ ≤ 0
t₆: l3(X₀, X₁, X₂) → l1(X₀-1, 0, X₂) :|: X₀ ≤ 0 ∧ 0 < X₀
t₇: l3(X₀, X₁, X₂) → l1(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ 0
t₈: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)

Preprocessing

Cut unsatisfiable transition t₅: l3→l1

Cut unsatisfiable transition t₆: l3→l1

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l5

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

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l4

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 0 < X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁
t₁: l2(X₀, X₁, X₂) → l1(X₂, 1, X₂)
t₄: l3(X₀, X₁, X₂) → l1(X₀-1, 1, X₂) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁
t₇: l3(X₀, X₁, X₂) → l1(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁
t₈: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l5

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

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l4

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₂+5 {O(n)}

TWN-Loops:

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

Termination: true
Formula:

1 < 0 ∧ 0 < 1
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1

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

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₂+5 {O(n)}

2⋅X₂+5 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 2⋅X₂+5 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₂+5 {O(n)}

2⋅X₂+5 {O(n)}

Found invariant 1 ≤ 0 for location l5

Found invariant 1 ≤ 0 for location l1

Found invariant 1 ≤ 0 for location l4

Found invariant 1 ≤ 0 for location l3

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l5

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

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l4

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₇ 6⋅X₂+18 {O(n)}

TWN-Loops:

entry: t₄: l3(X₀, X₁, X₂) → l1(X₀-1, 1, X₂) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁
results in twn-loop: twn:Inv: [X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁] , (X₀,X₁,X₂) -> (X₀,0,X₂) :|: 0 < X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ 0
entry: t₁: l2(X₀, X₁, X₂) → l1(X₂, 1, X₂)
results in twn-loop: twn:Inv: [X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁] , (X₀,X₁,X₂) -> (X₀,0,X₂) :|: 0 < X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ 0
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: [[n == 0]] * X₁
X₂: X₂

Termination: true
Formula:

relevant size-bounds w.r.t. t₄:
Runtime-bound of t₄: 2⋅X₂+5 {O(n)}
Results in: 6⋅X₂+15 {O(n)}

order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: [[n == 0]] * X₁
X₂: X₂

Termination: true
Formula:

relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 1 {O(1)}
Results in: 3 {O(1)}

6⋅X₂+18 {O(n)}

All Bounds

Timebounds

Overall timebound:10⋅X₂+32 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₂+5 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂+5 {O(n)}
t₇: 6⋅X₂+18 {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 10⋅X₂+32 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₂+5 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂+5 {O(n)}
t₇: 6⋅X₂+18 {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₂, X₀: X₂ {O(n)}
t₂, X₁: 1 {O(1)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: 0 {O(1)}
t₃, X₂: X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: 1 {O(1)}
t₁, X₂: X₂ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: 1 {O(1)}
t₄, X₂: X₂ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: 0 {O(1)}
t₇, X₂: X₂ {O(n)}
t₈, X₀: X₂ {O(n)}
t₈, X₁: 0 {O(1)}
t₈, X₂: X₂ {O(n)}