Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂)
t₀: l1(X₀, X₁, X₂) → l1(1+X₀, 1+X₁, X₂) :|: X₀ ≤ 1
t₁: l1(X₀, X₁, X₂) → l1(1+X₀, 1+X₁, X₂) :|: X₁ ≤ 2 ∧ 2 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, D) :|: 3 ≤ X₁ ∧ 2 ≤ X₀

Preprocessing

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

Found invariant 3 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₁₁: l0(X₀, X₁) → l1(X₀, X₁)
t₁₂: l1(X₀, X₁) → l1(1+X₀, 1+X₁) :|: X₀ ≤ 1
t₁₃: l1(X₀, X₁) → l1(1+X₀, 1+X₁) :|: X₁ ≤ 2 ∧ 2 ≤ X₀
t₁₄: l1(X₀, X₁) → l2(X₀, X₁) :|: 3 ≤ X₁ ∧ 2 ≤ X₀

Found invariant 3 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l2

Time-Bound by TWN-Loops:

TWN-Loops: t₁₂ 2⋅X₁+4⋅X₀+18 {O(n)}

TWN-Loops:

entry: t₁₁: l0(X₀, X₁) → l1(X₀, X₁)
results in twn-loop: twn: (X₀,X₁) -> (1+X₀,1+X₁) :|: X₀ ≤ 1 ∨ X₁ ≤ 2 ∧ 2 ≤ X₀
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0
∨ X₀ < 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
∨ 0 < 1 ∧ 1 < 0
∨ 0 < 1 ∧ X₁ < 2 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 2 ∧ 2 ≤ X₁
∨ 2 < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < 0
∨ 2 < X₀ ∧ X₁ < 2 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 2 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 2 ∧ 2 ≤ X₁
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ X₀ ∧ X₀ ≤ 2 ∧ 1 < 0
∨ 2 ≤ X₀ ∧ X₀ ≤ 2 ∧ X₁ < 2 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 2 ≤ X₀ ∧ X₀ ≤ 2 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ 2 ∧ 2 ≤ X₁

Stabilization-Threshold for: 2 ≤ X₀
alphas_abs: 2+X₀
M: 0
N: 1
Bound: 2⋅X₀+6 {O(n)}
Stabilization-Threshold for: X₁ ≤ 2
alphas_abs: X₁+2
M: 0
N: 1
Bound: 2⋅X₁+6 {O(n)}
Stabilization-Threshold for: X₀ ≤ 1
alphas_abs: X₀+1
M: 0
N: 1
Bound: 2⋅X₀+4 {O(n)}

relevant size-bounds w.r.t. t₁₁:
X₀: X₀ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₁₁: 1 {O(1)}
Results in: 2⋅X₁+4⋅X₀+18 {O(n)}

2⋅X₁+4⋅X₀+18 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₃ 2⋅X₁+4⋅X₀+18 {O(n)}

relevant size-bounds w.r.t. t₁₁:
X₀: X₀ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₁₁: 1 {O(1)}
Results in: 2⋅X₁+4⋅X₀+18 {O(n)}

2⋅X₁+4⋅X₀+18 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₁+8⋅X₀+38 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 2⋅X₁+4⋅X₀+18 {O(n)}
t₁₃: 2⋅X₁+4⋅X₀+18 {O(n)}
t₁₄: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁+8⋅X₀+38 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 2⋅X₁+4⋅X₀+18 {O(n)}
t₁₃: 2⋅X₁+4⋅X₀+18 {O(n)}
t₁₄: 1 {O(1)}

Sizebounds

t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₁ {O(n)}
t₁₂, X₀: 2⋅X₁+5⋅X₀+18 {O(n)}
t₁₂, X₁: 3⋅X₁+4⋅X₀+18 {O(n)}
t₁₃, X₀: 10⋅X₀+4⋅X₁+36 {O(n)}
t₁₃, X₁: 6⋅X₁+8⋅X₀+36 {O(n)}
t₁₄, X₀: 16⋅X₀+6⋅X₁+54 {O(n)}
t₁₄, X₁: 10⋅X₁+12⋅X₀+54 {O(n)}