Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₀+1 ≤ X₂
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₂ < X₀+1
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, X₂, X₃)
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)

Preprocessing

Eliminate variables {X₁} that do not contribute to the problem

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location l5

Found invariant X₃ ≤ X₀ for location l1

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location l4

Found invariant 1+X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 1+X₀ ≤ X₂ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₂: l0(X₀, X₂, X₃) → l2(X₀, X₂, X₃)
t₁₃: l1(X₀, X₂, X₃) → l3(X₀, X₂, X₃) :|: X₀+1 ≤ X₂ ∧ X₃ ≤ X₀
t₁₄: l1(X₀, X₂, X₃) → l4(X₀, X₂, X₃) :|: X₂ < X₀+1 ∧ X₃ ≤ X₀
t₁₅: l2(X₀, X₂, X₃) → l1(X₃, X₂, X₃)
t₁₆: l3(X₀, X₂, X₃) → l1(X₀+1, X₂, X₃) :|: 1+X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 1+X₀ ≤ X₂
t₁₇: l4(X₀, X₂, X₃) → l5(X₀, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₂ ≤ X₀

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location l5

Found invariant X₃ ≤ X₀ for location l1

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location l4

Found invariant 1+X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 1+X₀ ≤ X₂ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₃ 2⋅X₂+2⋅X₃+4 {O(n)}

TWN-Loops:

entry: t₁₅: l2(X₀, X₂, X₃) → l1(X₃, X₂, X₃)
results in twn-loop: twn:Inv: [X₃ ≤ X₀ ∧ 1+X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 1+X₀ ≤ X₂] , (X₀,X₂,X₃) -> (X₀+1,X₂,X₃) :|: X₀+1 ≤ X₂
order: [X₀; X₂; X₃]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₂: X₂
X₃: X₃

Termination: true
Formula:

1 < 0
∨ X₀+1 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀+1 ≤ X₂ ∧ X₂ ≤ X₀+1

Stabilization-Threshold for: X₀+1 ≤ X₂
alphas_abs: X₀+X₂
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₂+2 {O(n)}

relevant size-bounds w.r.t. t₁₅:
X₀: X₃ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+4 {O(n)}

2⋅X₂+2⋅X₃+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₆ 2⋅X₂+2⋅X₃+4 {O(n)}

relevant size-bounds w.r.t. t₁₅:
X₀: X₃ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+4 {O(n)}

2⋅X₂+2⋅X₃+4 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₂+4⋅X₃+12 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 2⋅X₂+2⋅X₃+4 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₂+2⋅X₃+4 {O(n)}
t₁₇: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₂+4⋅X₃+12 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 2⋅X₂+2⋅X₃+4 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₂+2⋅X₃+4 {O(n)}
t₁₇: 1 {O(1)}

Sizebounds

t₁₂, X₀: X₀ {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₀: 2⋅X₂+3⋅X₃+4 {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₄, X₀: 2⋅X₂+4⋅X₃+4 {O(n)}
t₁₄, X₂: 2⋅X₂ {O(n)}
t₁₄, X₃: 2⋅X₃ {O(n)}
t₁₅, X₀: X₃ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₆, X₀: 2⋅X₂+3⋅X₃+4 {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₆, X₃: X₃ {O(n)}
t₁₇, X₀: 2⋅X₂+4⋅X₃+4 {O(n)}
t₁₇, X₂: 2⋅X₂ {O(n)}
t₁₇, X₃: 2⋅X₃ {O(n)}