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) → l2(X₀, X₁, X₂, X₃, X₄, X₅)
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₄: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ < X₀
t₅: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ < X₁
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₅, X₄, X₂, X₃, X₄, X₅) :|: X₄ < X₅
t₂: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₅, X₅, X₂, X₃, X₄, X₅) :|: X₅ ≤ X₄
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+1, X₄, X₂, X₃, X₄, X₅) :|: X₄ < X₀+1
t₇: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+1, X₀+1, X₂, X₃, X₄, X₅) :|: X₀+1 ≤ X₄
t₈: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Cut unsatisfiable transition t₅: l1→l4

Eliminate variables {X₂,X₃} that do not contribute to the problem

Found invariant X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ 1+X₁ ≤ X₀ for location l5

Found invariant X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₀ for location l1

Found invariant X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ 1+X₁ ≤ X₀ for location l4

Found invariant X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₂₂: l0(X₀, X₁, X₄, X₅) → l2(X₀, X₁, X₄, X₅)
t₂₃: l1(X₀, X₁, X₄, X₅) → l3(X₀, X₁, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₀
t₂₄: l1(X₀, X₁, X₄, X₅) → l4(X₀, X₁, X₄, X₅) :|: X₁ < X₀ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₀
t₂₅: l2(X₀, X₁, X₄, X₅) → l1(X₅, X₄, X₄, X₅) :|: X₄ < X₅
t₂₆: l2(X₀, X₁, X₄, X₅) → l1(X₅, X₅, X₄, X₅) :|: X₅ ≤ X₄
t₂₇: l3(X₀, X₁, X₄, X₅) → l1(X₀+1, X₄, X₄, X₅) :|: X₄ < X₀+1 ∧ X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₂₈: l3(X₀, X₁, X₄, X₅) → l1(X₀+1, X₀+1, X₄, X₅) :|: X₀+1 ≤ X₄ ∧ X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₂₉: l4(X₀, X₁, X₄, X₅) → l5(X₀, X₁, X₄, X₅) :|: X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ 1+X₁ ≤ X₀

Found invariant X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ 1+X₁ ≤ X₀ for location l5

Found invariant X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₀ for location l1

Found invariant X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ 1+X₁ ≤ X₀ for location l4

Found invariant X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂₃ 2⋅X₄+2⋅X₅+6 {O(n)}

TWN-Loops:

entry: t₂₆: l2(X₀, X₁, X₄, X₅) → l1(X₅, X₅, X₄, X₅) :|: X₅ ≤ X₄
results in twn-loop: twn:Inv: [X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₀ ∧ X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁] , (X₀,X₁,X₄,X₅) -> (X₀+1,X₀+1,X₄,X₅) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀+1 ≤ X₄
order: [X₀; X₁; X₄; X₅]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: [[n == 0]] * X₁ + [[n != 0]] * X₀+1 + [[n != 0, n != 1]] * n^1 + [[n != 0, n != 1]] * -1
X₄: X₄
X₅: X₅

Termination: true
Formula:

1 < 0 ∧ 0 < 1
∨ 1 < 0 ∧ X₅ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅
∨ 1 < 0 ∧ X₀ < X₄ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ 1 < 0 ∧ X₀ < X₄ ∧ X₅ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ 1 < 0 ∧ X₀ < X₄ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₀ ∧ 0 < 1
∨ 1 < 0 ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₀ ∧ X₅ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ 1 < 0 ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅
∨ X₀+1 < X₄ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ 0 < 1
∨ X₀+1 < X₄ ∧ 1 < 0 ∧ X₅ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀+1 < X₄ ∧ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅
∨ X₀+1 < X₄ ∧ X₀ < X₄ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ X₀+1 < X₄ ∧ X₀ < X₄ ∧ X₅ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀+1 < X₄ ∧ X₀ < X₄ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅
∨ X₀+1 < X₄ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₀ ∧ 0 < 1
∨ X₀+1 < X₄ ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₀ ∧ X₅ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀+1 < X₄ ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀+1 ≤ X₄ ∧ X₄ ≤ X₀+1 ∧ 1 < 0 ∧ 0 < 1
∨ X₀+1 ≤ X₄ ∧ X₄ ≤ X₀+1 ∧ 1 < 0 ∧ X₅ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀+1 ≤ X₄ ∧ X₄ ≤ X₀+1 ∧ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅
∨ X₀+1 ≤ X₄ ∧ X₄ ≤ X₀+1 ∧ X₀ < X₄ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1
∨ X₀+1 ≤ X₄ ∧ X₄ ≤ X₀+1 ∧ X₀ < X₄ ∧ X₅ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀+1 ≤ X₄ ∧ X₄ ≤ X₀+1 ∧ X₀ < X₄ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅
∨ X₀+1 ≤ X₄ ∧ X₄ ≤ X₀+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₀ ∧ 0 < 1
∨ X₀+1 ≤ X₄ ∧ X₄ ≤ X₀+1 ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₀ ∧ X₅ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0
∨ X₀+1 ≤ X₄ ∧ X₄ ≤ X₀+1 ∧ X₀ ≤ X₄ ∧ X₄ ≤ X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅

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

relevant size-bounds w.r.t. t₂₆:
X₀: X₅ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₂₆: 1 {O(1)}
Results in: 2⋅X₄+2⋅X₅+6 {O(n)}

2⋅X₄+2⋅X₅+6 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₂₈ 2⋅X₄+2⋅X₅+6 {O(n)}

relevant size-bounds w.r.t. t₂₆:
X₀: X₅ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₂₆: 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+6 {O(n)} for transition t₂₇: l3(X₀, X₁, X₄, X₅) → l1(X₀+1, X₄, X₄, X₅) :|: X₄ < X₀+1 ∧ X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁

All Bounds

Timebounds

Overall timebound:6⋅X₄+6⋅X₅+23 {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 2⋅X₄+2⋅X₅+6 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}
t₂₆: 1 {O(1)}
t₂₇: 2⋅X₄+2⋅X₅+6 {O(n)}
t₂₈: 2⋅X₄+2⋅X₅+6 {O(n)}
t₂₉: 1 {O(1)}

Costbounds

Overall costbound: 6⋅X₄+6⋅X₅+23 {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 2⋅X₄+2⋅X₅+6 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}
t₂₆: 1 {O(1)}
t₂₇: 2⋅X₄+2⋅X₅+6 {O(n)}
t₂₈: 2⋅X₄+2⋅X₅+6 {O(n)}
t₂₉: 1 {O(1)}

Sizebounds

t₂₂, X₀: X₀ {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₂, X₅: X₅ {O(n)}
t₂₃, X₀: 2⋅X₄+3⋅X₅+6 {O(n)}
t₂₃, X₁: 2⋅X₄+4⋅X₅+7 {O(n)}
t₂₃, X₄: X₄ {O(n)}
t₂₃, X₅: X₅ {O(n)}
t₂₄, X₀: 2⋅X₄+4⋅X₅+7 {O(n)}
t₂₄, X₁: 3⋅X₄+4⋅X₅+7 {O(n)}
t₂₄, X₄: 2⋅X₄ {O(n)}
t₂₄, X₅: 2⋅X₅ {O(n)}
t₂₅, X₀: X₅ {O(n)}
t₂₅, X₁: X₄ {O(n)}
t₂₅, X₄: X₄ {O(n)}
t₂₅, X₅: X₅ {O(n)}
t₂₆, X₀: X₅ {O(n)}
t₂₆, X₁: X₅ {O(n)}
t₂₆, X₄: X₄ {O(n)}
t₂₆, X₅: X₅ {O(n)}
t₂₇, X₀: 2⋅X₄+3⋅X₅+7 {O(n)}
t₂₇, X₁: 2⋅X₄+4⋅X₅+7 {O(n)}
t₂₇, X₄: X₄ {O(n)}
t₂₇, X₅: X₅ {O(n)}
t₂₈, X₀: 2⋅X₄+3⋅X₅+6 {O(n)}
t₂₈, X₁: 2⋅X₄+3⋅X₅+7 {O(n)}
t₂₈, X₄: X₄ {O(n)}
t₂₈, X₅: X₅ {O(n)}
t₂₉, X₀: 2⋅X₄+4⋅X₅+7 {O(n)}
t₂₉, X₁: 3⋅X₄+4⋅X₅+7 {O(n)}
t₂₉, X₄: 2⋅X₄ {O(n)}
t₂₉, X₅: 2⋅X₅ {O(n)}