Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l10, l11, 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)
t₁₄: l10(X₀, X₁, X₂) → l11(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, X₂)
t₇: l7(X₀, X₁, X₂) → l8(X₂, 1, X₂)
t₉: l8(X₀, X₁, X₂) → l10(X₀, X₁, X₂) :|: X₁ ≤ 0
t₈: l8(X₀, X₁, X₂) → l9(X₀, X₁, X₂) :|: 0 < X₁
t₁₀: l9(X₀, X₁, X₂) → l8(X₀-1, 1, X₂) :|: 0 < X₀ ∧ 0 < X₀
t₁₁: l9(X₀, X₁, X₂) → l8(X₀, 1, X₂) :|: 0 < X₀ ∧ X₀ ≤ 0
t₁₂: l9(X₀, X₁, X₂) → l8(X₀-1, 0, X₂) :|: X₀ ≤ 0 ∧ 0 < X₀
t₁₃: l9(X₀, X₁, X₂) → l8(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ 0

Preprocessing

Cut unsatisfiable transition t₁₁: l9→l8

Cut unsatisfiable transition t₁₂: l9→l8

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l11

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁ for location l8

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l10

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

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l10, l11, 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)
t₁₄: l10(X₀, X₁, X₂) → l11(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁
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, X₂)
t₇: l7(X₀, X₁, X₂) → l8(X₂, 1, X₂)
t₉: l8(X₀, X₁, X₂) → l10(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁
t₈: l8(X₀, X₁, X₂) → l9(X₀, X₁, X₂) :|: 0 < X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁
t₁₀: l9(X₀, X₁, X₂) → l8(X₀-1, 1, X₂) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁
t₁₃: l9(X₀, X₁, X₂) → l8(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁

MPRF for transition t₁₀: l9(X₀, X₁, X₂) → l8(X₀-1, 1, X₂) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l9 [X₀ ]
l8 [X₀ ]

MPRF for transition t₁₃: l9(X₀, X₁, X₂) → l8(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ of depth 1:

new bound:

1 {O(1)}

MPRF:

l9 [1 ]
l8 [X₁ ]

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l11

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁ for location l8

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location l10

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

Time-Bound by TWN-Loops:

TWN-Loops: t₈ 2⋅X₂+5 {O(n)}

TWN-Loops:

entry: t₇: l7(X₀, X₁, X₂) → l8(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. t₇:
X₀: X₂ {O(n)}
Runtime-bound of t₇: 1 {O(1)}
Results in: 2⋅X₂+5 {O(n)}

2⋅X₂+5 {O(n)}

All Bounds

Timebounds

Overall timebound:3⋅X₂+16 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+5 {O(n)}
t₉: 1 {O(1)}
t₁₀: X₂ {O(n)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₂+16 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+5 {O(n)}
t₉: 1 {O(1)}
t₁₀: X₂ {O(n)}
t₁₃: 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)}
t₁₄, X₀: X₂ {O(n)}
t₁₄, X₁: 0 {O(1)}
t₁₄, 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: X₂ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: 1 {O(1)}
t₇, X₂: X₂ {O(n)}
t₈, X₀: X₂ {O(n)}
t₈, X₁: 1 {O(1)}
t₈, X₂: X₂ {O(n)}
t₉, X₀: X₂ {O(n)}
t₉, X₁: 0 {O(1)}
t₉, X₂: X₂ {O(n)}
t₁₀, X₀: X₂ {O(n)}
t₁₀, X₁: 1 {O(1)}
t₁₀, X₂: X₂ {O(n)}
t₁₃, X₀: X₂ {O(n)}
t₁₃, X₁: 0 {O(1)}
t₁₃, X₂: X₂ {O(n)}