Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ 100 ∧ X₁ ≤ X₂
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 100 < X₀
t₄: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₂ < X₁
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l1(X₃, X₄, X₅, X₃, X₄, X₅, X₆)
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l1(X₁, X₁+1, X₂-1, X₃, X₄, X₅, X₆)
t₆: l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

Preprocessing

Eliminate variables {X₆} that do not contribute to the problem

Found invariant X₂ ≤ X₅ ∧ X₄ ≤ X₁ for location l5

Found invariant X₂ ≤ X₅ ∧ X₄ ≤ X₁ for location l1

Found invariant X₂ ≤ X₅ ∧ X₄ ≤ X₁ for location l4

Found invariant X₄ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₁ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ 100 for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₄: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅)
t₁₅: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 100 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁
t₁₆: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: 100 < X₀ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁
t₁₇: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ < X₁ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁
t₁₈: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
t₁₉: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₁, X₁+1, X₂-1, X₃, X₄, X₅) :|: X₄ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₁ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ 100
t₂₀: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₅ ∧ X₄ ≤ X₁

Found invariant X₂ ≤ X₅ ∧ X₄ ≤ X₁ for location l5

Found invariant X₂ ≤ X₅ ∧ X₄ ≤ X₁ for location l1

Found invariant X₂ ≤ X₅ ∧ X₄ ≤ X₁ for location l4

Found invariant X₄ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₁ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ 100 for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₅ 2⋅X₅+4⋅X₄+208 {O(n)}

TWN-Loops:

entry: t₁₈: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
results in twn-loop: twn:Inv: [X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₁ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₀ ≤ 100] , (X₀,X₁,X₂,X₃,X₄,X₅) -> (X₁,X₁+1,X₂-1,X₃,X₄,X₅) :|: X₀ ≤ 100 ∧ X₁ ≤ X₂
order: [X₁; X₀; X₂; X₄; X₅]
closed-form:
X₁: X₁ + [[n != 0]] * n^1
X₀: [[n == 0]] * X₀ + [[n != 0]] * X₁ + [[n != 0, n != 1]] * n^1 + [[n != 0, n != 1]] * -1
X₂: X₂ + [[n != 0]] * -1 * n^1
X₄: X₄
X₅: X₅

Termination: true
Formula:

2 < 0 ∧ 1 < 0
∨ 2 < 0 ∧ 1 < 0 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 2 < 0 ∧ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁
∨ 2 < 0 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ 2 < 0 ∧ X₁ < 101 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 2 < 0 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁
∨ 2 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁ ∧ 1 < 0
∨ 2 < 0 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁ ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 2 < 0 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁
∨ X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 1 < 0
∨ X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 1 < 0 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁
∨ X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ < 101 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁
∨ X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁ ∧ 1 < 0
∨ X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁ ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 < 0
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 < 0 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₁ < 101 ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁ ∧ 1 < 0
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ 101 ∧ 101 ≤ X₁ ∧ X₁ < 101 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ 101 ∧ 101 ≤ X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 101 ∧ 101 ≤ X₁

Stabilization-Threshold for: X₁ ≤ X₂
alphas_abs: X₁+X₂
M: 0
N: 1
Bound: 2⋅X₁+2⋅X₂+2 {O(n)}
Stabilization-Threshold for: X₀ ≤ 100
alphas_abs: X₁+101
M: 0
N: 1
Bound: 2⋅X₁+204 {O(n)}

relevant size-bounds w.r.t. t₁₈:
X₁: X₄ {O(n)}
X₂: X₅ {O(n)}
Runtime-bound of t₁₈: 1 {O(1)}
Results in: 2⋅X₅+4⋅X₄+208 {O(n)}

2⋅X₅+4⋅X₄+208 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₉ 2⋅X₅+4⋅X₄+208 {O(n)}

relevant size-bounds w.r.t. t₁₈:
X₁: X₄ {O(n)}
X₂: X₅ {O(n)}
Runtime-bound of t₁₈: 1 {O(1)}
Results in: 2⋅X₅+4⋅X₄+208 {O(n)}

2⋅X₅+4⋅X₄+208 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₅+8⋅X₄+421 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₅+4⋅X₄+208 {O(n)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 2⋅X₅+4⋅X₄+208 {O(n)}
t₂₀: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₅+8⋅X₄+421 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₅+4⋅X₄+208 {O(n)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 2⋅X₅+4⋅X₄+208 {O(n)}
t₂₀: 1 {O(1)}

Sizebounds

t₁₄, X₀: X₀ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₄, X₅: X₅ {O(n)}
t₁₅, X₀: 2⋅X₅+5⋅X₄+X₃+208 {O(n)}
t₁₅, X₁: 2⋅X₅+5⋅X₄+208 {O(n)}
t₁₅, X₂: 3⋅X₅+4⋅X₄+208 {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₅, X₄: X₄ {O(n)}
t₁₅, X₅: X₅ {O(n)}
t₁₆, X₀: 2⋅X₅+5⋅X₄+X₃+208 {O(n)}
t₁₆, X₁: 2⋅X₅+6⋅X₄+208 {O(n)}
t₁₆, X₂: 4⋅X₄+4⋅X₅+208 {O(n)}
t₁₆, X₃: 2⋅X₃ {O(n)}
t₁₆, X₄: 2⋅X₄ {O(n)}
t₁₆, X₅: 2⋅X₅ {O(n)}
t₁₇, X₀: 2⋅X₅+5⋅X₄+X₃+208 {O(n)}
t₁₇, X₁: 2⋅X₅+6⋅X₄+208 {O(n)}
t₁₇, X₂: 4⋅X₄+4⋅X₅+208 {O(n)}
t₁₇, X₃: 2⋅X₃ {O(n)}
t₁₇, X₄: 2⋅X₄ {O(n)}
t₁₇, X₅: 2⋅X₅ {O(n)}
t₁₈, X₀: X₃ {O(n)}
t₁₈, X₁: X₄ {O(n)}
t₁₈, X₂: X₅ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₈, X₅: X₅ {O(n)}
t₁₉, X₀: 2⋅X₅+5⋅X₄+208 {O(n)}
t₁₉, X₁: 2⋅X₅+5⋅X₄+208 {O(n)}
t₁₉, X₂: 3⋅X₅+4⋅X₄+208 {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₁₉, X₅: X₅ {O(n)}
t₂₀, X₀: 10⋅X₄+2⋅X₃+4⋅X₅+416 {O(n)}
t₂₀, X₁: 12⋅X₄+4⋅X₅+416 {O(n)}
t₂₀, X₂: 8⋅X₄+8⋅X₅+416 {O(n)}
t₂₀, X₃: 4⋅X₃ {O(n)}
t₂₀, X₄: 4⋅X₄ {O(n)}
t₂₀, X₅: 4⋅X₅ {O(n)}