Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₂-1) :|: 2 < X₂
t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ 2
t₁: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₁)
t₄: l3(X₀, X₁, X₂) → l4(X₀, X₁, X₂)

Preprocessing

Eliminate variables {X₀} that do not contribute to the problem

Found invariant X₂ ≤ X₁ for location l1

Found invariant X₂ ≤ 2 ∧ X₂ ≤ X₁ for location l4

Found invariant X₂ ≤ 2 ∧ X₂ ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4
Transitions:
t₁₀: l0(X₁, X₂) → l2(X₁, X₂)
t₁₁: l1(X₁, X₂) → l1(X₁, X₂-1) :|: 2 < X₂ ∧ X₂ ≤ X₁
t₁₂: l1(X₁, X₂) → l3(X₁, X₂) :|: X₂ ≤ 2 ∧ X₂ ≤ X₁
t₁₃: l2(X₁, X₂) → l1(X₁, X₁)
t₁₄: l3(X₁, X₂) → l4(X₁, X₂) :|: X₂ ≤ 2 ∧ X₂ ≤ X₁

Found invariant X₂ ≤ X₁ for location l1

Found invariant X₂ ≤ 2 ∧ X₂ ≤ X₁ for location l4

Found invariant X₂ ≤ 2 ∧ X₂ ≤ X₁ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₁ 2⋅X₁+4 {O(n)}

TWN-Loops:

entry: t₁₃: l2(X₁, X₂) → l1(X₁, X₁)
results in twn-loop: twn:Inv: [X₂ ≤ X₁] , (X₁,X₂) -> (X₁,X₂-1) :|: 2 < X₂
order: [X₁; X₂]
closed-form:
X₁: X₁
X₂: X₂ + [[n != 0]] * -1 * n^1

Termination: true
Formula:

1 < 0
∨ 2 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1

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

relevant size-bounds w.r.t. t₁₃:
X₂: X₁ {O(n)}
Runtime-bound of t₁₃: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

2⋅X₁+4 {O(n)}

All Bounds

Timebounds

Overall timebound:2⋅X₁+8 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: 2⋅X₁+4 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁+8 {O(n)}
t₁₀: 1 {O(1)}
t₁₁: 2⋅X₁+4 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}

Sizebounds

t₁₀, X₁: X₁ {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₁, X₁: X₁ {O(n)}
t₁₁, X₂: X₁ {O(n)}
t₁₂, X₁: 2⋅X₁ {O(n)}
t₁₂, X₂: 2⋅X₁ {O(n)}
t₁₃, X₁: X₁ {O(n)}
t₁₃, X₂: X₁ {O(n)}
t₁₄, X₁: 2⋅X₁ {O(n)}
t₁₄, X₂: 2⋅X₁ {O(n)}