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)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ < 0
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 < X₀
t₄: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ 0 ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(1, X₄, X₂, X₃, X₄, X₅)
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(0, X₁+1, X₂, X₃, X₄, X₅) :|: X₅ ≤ X₁
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁+1, X₂, X₃, X₄, X₅) :|: X₁ < X₅
t₇: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

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

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l5

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location l1

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l4

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location l3

Cut unsatisfiable transition t₁₆: l1→l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₅: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₁₇: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀
t₁₈: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀
t₁₉: l2(X₀, X₁, X₂, X₃) → l1(1, X₂, X₂, X₃)
t₂₀: l3(X₀, X₁, X₂, X₃) → l1(0, X₁+1, X₂, X₃) :|: X₃ ≤ X₁ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₂₁: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁+1, X₂, X₃) :|: X₁ < X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₂₂: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

MPRF for transition t₂₀: l3(X₀, X₁, X₂, X₃) → l1(0, X₁+1, X₂, X₃) :|: X₃ ≤ X₁ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:

new bound:

1 {O(1)}

MPRF for transition t₂₁: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁+1, X₂, X₃) :|: X₁ < X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂+X₃ {O(n)}

TWN: t₁₇: l1→l3

cycle: [t₁₇: l1→l3; t₂₁: l3→l1]
loop: (0 < X₀ ∧ X₁ < X₃,(X₀,X₁,X₃) -> (X₀,X₁+1,X₃)
order: [X₀; X₁; X₃]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
X₃: X₃

Termination: true
Formula:

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

Stabilization-Threshold for: X₁ < X₃
alphas_abs: X₁+X₃
M: 0
N: 1
Bound: 2⋅X₁+2⋅X₃+2 {O(n)}

TWN - Lifting for t₁₇: l1→l3 of 2⋅X₁+2⋅X₃+5 {O(n)}

relevant size-bounds w.r.t. t₁₉:
X₁: X₂ {O(n)}
X₃: X₃ {O(n)}
Runtime-bound of t₁₉: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+5 {O(n)}

All Bounds

Timebounds

Overall timebound:3⋅X₂+3⋅X₃+10 {O(n)}
t₁₅: 1 {O(1)}
t₁₇: 2⋅X₂+2⋅X₃+5 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₂+X₃ {O(n)}
t₂₂: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₂+3⋅X₃+10 {O(n)}
t₁₅: 1 {O(1)}
t₁₇: 2⋅X₂+2⋅X₃+5 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₂+X₃ {O(n)}
t₂₂: 1 {O(1)}

Sizebounds

t₁₅, X₀: X₀ {O(n)}
t₁₅, X₁: X₁ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₇, X₀: 1 {O(1)}
t₁₇, X₁: 2⋅X₂+X₃ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₈, X₀: 0 {O(1)}
t₁₈, X₁: 2⋅X₂+X₃+1 {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₉, X₀: 1 {O(1)}
t₁₉, X₁: X₂ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₂₀, X₀: 0 {O(1)}
t₂₀, X₁: 2⋅X₂+X₃+1 {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₁, X₀: 1 {O(1)}
t₂₁, X₁: 2⋅X₂+X₃ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: 0 {O(1)}
t₂₂, X₁: 2⋅X₂+X₃+1 {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}