Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l10, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂)
t₁: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂)
t₂: l3(X₀, X₁, X₂) → l1(X₀, X₁, X₂)
t₄: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)
t₅: l5(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₆: l6(X₀, X₁, X₂) → l7(X₀, X₁, X₀)
t₇: l7(X₀, X₁, X₂) → l8(X₀, X₁, X₂) :|: X₂ ≤ X₁
t₈: l7(X₀, X₁, X₂) → l9(X₀, X₁, X₂) :|: X₁ < X₂
t₉: l8(X₀, X₁, X₂) → l7(X₀, X₁, X₂+1)
t₁₀: l9(X₀, X₁, X₂) → l10(X₀, X₁, X₂)

Preprocessing

Found invariant X₀ ≤ X₂ for location l7

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₁ for location l8

Found invariant 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ for location l10

Found invariant 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ for location l9

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l10, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂)
t₁: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂)
t₂: l3(X₀, X₁, X₂) → l1(X₀, X₁, X₂)
t₄: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)
t₅: l5(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₆: l6(X₀, X₁, X₂) → l7(X₀, X₁, X₀)
t₇: l7(X₀, X₁, X₂) → l8(X₀, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₀ ≤ X₂
t₈: l7(X₀, X₁, X₂) → l9(X₀, X₁, X₂) :|: X₁ < X₂ ∧ X₀ ≤ X₂
t₉: l8(X₀, X₁, X₂) → l7(X₀, X₁, X₂+1) :|: X₂ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₁
t₁₀: l9(X₀, X₁, X₂) → l10(X₀, X₁, X₂) :|: 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂

Found invariant X₀ ≤ X₂ for location l7

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₁ for location l8

Found invariant 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ for location l10

Found invariant 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ for location l9

Time-Bound by TWN-Loops:

TWN-Loops: t₇ 2⋅X₀+2⋅X₁+4 {O(n)}

TWN-Loops:

entry: t₆: l6(X₀, X₁, X₂) → l7(X₀, X₁, X₀)
results in twn-loop: twn:Inv: [X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₁] , (X₀,X₁,X₂) -> (X₀,X₁,X₂+1) :|: X₂ ≤ X₁
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁
X₂: X₂ + [[n != 0]] * n^1

Termination: true
Formula:

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

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

relevant size-bounds w.r.t. t₆:
X₁: X₁ {O(n)}
X₂: X₀ {O(n)}
Runtime-bound of t₆: 1 {O(1)}
Results in: 2⋅X₀+2⋅X₁+4 {O(n)}

2⋅X₀+2⋅X₁+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₉ 2⋅X₀+2⋅X₁+4 {O(n)}

relevant size-bounds w.r.t. t₆:
X₁: X₁ {O(n)}
X₂: X₀ {O(n)}
Runtime-bound of t₆: 1 {O(1)}
Results in: 2⋅X₀+2⋅X₁+4 {O(n)}

2⋅X₀+2⋅X₁+4 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₀+4⋅X₁+17 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 2⋅X₀+2⋅X₁+4 {O(n)}
t₈: 1 {O(1)}
t₉: 2⋅X₀+2⋅X₁+4 {O(n)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀+4⋅X₁+17 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 2⋅X₀+2⋅X₁+4 {O(n)}
t₈: 1 {O(1)}
t₉: 2⋅X₀+2⋅X₁+4 {O(n)}
t₁₀: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₀ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 2⋅X₁+3⋅X₀+4 {O(n)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 2⋅X₁+4⋅X₀+4 {O(n)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: 2⋅X₁+3⋅X₀+4 {O(n)}
t₁₀, X₀: 2⋅X₀ {O(n)}
t₁₀, X₁: 2⋅X₁ {O(n)}
t₁₀, X₂: 2⋅X₁+4⋅X₀+4 {O(n)}