Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅)
t₆: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ < 0 ∧ X₁ < 0
t₄: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀
t₅: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₁
t₈: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅)
t₁: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₄, X₅, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₃ ∧ X₃ ≤ X₂
t₂: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ < X₃
t₃: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ < X₂
t₇: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂-X₃-1, X₁+X₃-X₂-1, X₂, X₃, X₄, X₅)

Preprocessing

Found invariant X₃ ≤ X₂ ∧ X₂ ≤ X₃ for location l1

Found invariant X₃ ≤ X₂ ∧ X₂ ≤ X₃ for location l4

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅)
t₆: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ < 0 ∧ X₁ < 0 ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
t₄: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
t₅: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃
t₈: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅)
t₁: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₄, X₅, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₃ ∧ X₃ ≤ X₂
t₂: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ < X₃
t₃: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ < X₂
t₇: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂-X₃-1, X₁+X₃-X₂-1, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₂ ∧ X₂ ≤ X₃

Found invariant X₃ ≤ X₂ ∧ X₂ ≤ X₃ for location l1

Found invariant X₃ ≤ X₂ ∧ X₂ ≤ X₃ for location l4

Time-Bound by TWN-Loops:

TWN-Loops: t₇ 2⋅X₄+2⋅X₅+6 {O(n)}

TWN-Loops:

entry: t₁: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₄, X₅, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₃ ∧ X₃ ≤ X₂
results in twn-loop: twn:Inv: [X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃] , (X₀,X₁,X₂,X₃,X₄,X₅) -> (X₀+X₂-X₃-1,X₁+X₃-X₂-1,X₂,X₃,X₄,X₅) :|: 0 ≤ X₀ ∨ 0 ≤ X₁
order: [X₂; X₃; X₀; X₁]
closed-form:
X₂: X₂
X₃: X₃
X₀: X₀ + [[n != 0]] * (X₂-X₃-1) * n^1
X₁: X₁ + [[n != 0]] * (X₃-X₂-1) * n^1

Termination: true
Formula:

X₃+1 < X₂
∨ 0 < X₀ ∧ X₃+1 ≤ X₂ ∧ X₂ ≤ X₃+1
∨ X₃+1 ≤ X₂ ∧ X₂ ≤ X₃+1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₂+1 < X₃
∨ 0 < X₁ ∧ X₂+1 ≤ X₃ ∧ X₃ ≤ X₂+1
∨ X₂+1 ≤ X₃ ∧ X₃ ≤ X₂+1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0

Stabilization-Threshold for: 0 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
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: X₄ {O(n)}
X₁: X₅ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₄+2⋅X₅+6 {O(n)}

2⋅X₄+2⋅X₅+6 {O(n)}

knowledge_propagation leads to new time bound 2⋅X₄+2⋅X₅+7 {O(n)} for transition t₄: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃

knowledge_propagation leads to new time bound 2⋅X₄+2⋅X₅+7 {O(n)} for transition t₅: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃

All Bounds

Timebounds

Overall timebound:6⋅X₄+6⋅X₅+26 {O(n)}
t₀: 1 {O(1)}
t₄: 2⋅X₄+2⋅X₅+7 {O(n)}
t₅: 2⋅X₄+2⋅X₅+7 {O(n)}
t₆: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₇: 2⋅X₄+2⋅X₅+6 {O(n)}

Costbounds

Overall costbound: 6⋅X₄+6⋅X₅+26 {O(n)}
t₀: 1 {O(1)}
t₄: 2⋅X₄+2⋅X₅+7 {O(n)}
t₅: 2⋅X₄+2⋅X₅+7 {O(n)}
t₆: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₇: 2⋅X₄+2⋅X₅+6 {O(n)}

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)}
t₀, X₅: X₅ {O(n)}
t₄, X₀: 2⋅X₅+4⋅X₄+6 {O(n)}
t₄, X₁: 2⋅X₄+4⋅X₅+6 {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}
t₅, X₀: 2⋅X₅+4⋅X₄+6 {O(n)}
t₅, X₁: 2⋅X₄+4⋅X₅+6 {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}
t₅, X₅: 2⋅X₅ {O(n)}
t₆, X₀: 2⋅X₅+5⋅X₄+6 {O(n)}
t₆, X₁: 2⋅X₄+5⋅X₅+6 {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}
t₆, X₄: 3⋅X₄ {O(n)}
t₆, X₅: 3⋅X₅ {O(n)}
t₈, X₀: 2⋅X₀+2⋅X₅+5⋅X₄+6 {O(n)}
t₈, X₁: 2⋅X₁+2⋅X₄+5⋅X₅+6 {O(n)}
t₈, X₂: 5⋅X₂ {O(n)}
t₈, X₃: 5⋅X₃ {O(n)}
t₈, X₄: 5⋅X₄ {O(n)}
t₈, X₅: 5⋅X₅ {O(n)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₅ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {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)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₇, X₀: 2⋅X₅+4⋅X₄+6 {O(n)}
t₇, X₁: 2⋅X₄+4⋅X₅+6 {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₃ {O(n)}
t₇, X₄: 2⋅X₄ {O(n)}
t₇, X₅: 2⋅X₅ {O(n)}