Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂)
t₁: l1(X₀, X₁, X₂) → l1(1+X₀, X₁, X₂) :|: 1+X₀ ≤ X₁
t₂: l1(X₀, X₁, X₂) → l1(1+X₀, X₀, X₂) :|: D ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₃: l1(X₀, X₁, X₂) → l2(X₀, X₁, D) :|: X₁ ≤ X₀ ∧ X₁+1 ≤ X₀
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, D) :|: X₁ ≤ X₀ ∧ X₀+1 ≤ X₁

Preprocessing

Cut unsatisfiable transition t₄: l1→l2

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

Found invariant 1+X₁ ≤ X₀ for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars: D
Locations: l0, l1, l2
Transitions:
t₁₅: l0(X₀, X₁) → l1(X₀, X₁)
t₁₆: l1(X₀, X₁) → l1(1+X₀, X₁) :|: 1+X₀ ≤ X₁
t₁₇: l1(X₀, X₁) → l1(1+X₀, X₀) :|: D ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₁₈: l1(X₀, X₁) → l2(X₀, X₁) :|: X₁ ≤ X₀ ∧ X₁+1 ≤ X₀

Found invariant 1+X₁ ≤ X₀ for location l2

Time-Bound by TWN-Loops:

TWN-Loops: t₁₆ 2⋅X₀+2⋅X₁+4 {O(n)}

TWN-Loops:

entry: t₁₅: l0(X₀, X₁) → l1(X₀, X₁)
results in twn-loop: twn: (X₀,X₁) -> (1+X₀,X₁) :|: 1+X₀ ≤ X₁
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁

Termination: true
Formula:

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

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

relevant size-bounds w.r.t. t₁₅:
X₀: X₀ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 2⋅X₀+2⋅X₁+4 {O(n)}

2⋅X₀+2⋅X₁+4 {O(n)}

knowledge_propagation leads to new time bound 2⋅X₀+2⋅X₁+5 {O(n)} for transition t₁₇: l1(X₀, X₁) → l1(1+X₀, X₀) :|: D ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀

All Bounds

Timebounds

Overall timebound:4⋅X₀+4⋅X₁+11 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₀+2⋅X₁+4 {O(n)}
t₁₇: 2⋅X₀+2⋅X₁+5 {O(n)}
t₁₈: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀+4⋅X₁+11 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₀+2⋅X₁+4 {O(n)}
t₁₇: 2⋅X₀+2⋅X₁+5 {O(n)}
t₁₈: 1 {O(1)}

Sizebounds

t₁₅, X₀: X₀ {O(n)}
t₁₅, X₁: X₁ {O(n)}
t₁₆, X₀: 2⋅X₁+3⋅X₀+4 {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₇, X₀: 2⋅X₁+4⋅X₀+6 {O(n)}
t₁₇, X₁: 2⋅X₁+4⋅X₀+4 {O(n)}
t₁₈, X₀: 2⋅X₁+5⋅X₀+6 {O(n)}
t₁₈, X₁: 3⋅X₁+4⋅X₀+4 {O(n)}