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

Preprocessing

Found invariant X₂ ≤ X₁ ∧ X₁ ≤ X₂ for location l1

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

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃)
t₅: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₀ < 0 ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₄: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₇: l2(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
t₁: l3(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₂, X₃) :|: X₁ ≤ X₂ ∧ X₂ ≤ X₁
t₂: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₁ < X₂
t₃: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₂ < X₁
t₆: l4(X₀, X₁, X₂, X₃) → l1(X₀+X₁-X₂-1, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀

Found invariant X₂ ≤ X₁ ∧ X₁ ≤ X₂ for location l1

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

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 2⋅X₃+4 {O(n)}

TWN-Loops:

entry: t₁: l3(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₂, X₃) :|: X₁ ≤ X₂ ∧ X₂ ≤ X₁
results in twn-loop: twn:Inv: [X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀+X₁-X₂-1,X₁,X₂,X₃) :|: 0 ≤ X₀
order: [X₁; X₂; X₀]
closed-form:
X₁: X₁
X₂: X₂
X₀: X₀ + [[n != 0]] * (X₁-X₂-1) * n^1

Termination: true
Formula:

X₂+1 < X₁
∨ 0 < X₀ ∧ X₂+1 ≤ X₁ ∧ X₁ ≤ X₂+1
∨ X₂+1 ≤ X₁ ∧ X₁ ≤ X₂+1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0

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

relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₃+4 {O(n)}

2⋅X₃+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₆ 2⋅X₃+4 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₃+4 {O(n)}

2⋅X₃+4 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₃+14 {O(n)}
t₀: 1 {O(1)}
t₄: 2⋅X₃+4 {O(n)}
t₅: 1 {O(1)}
t₇: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₆: 2⋅X₃+4 {O(n)}

Costbounds

Overall costbound: 4⋅X₃+14 {O(n)}
t₀: 1 {O(1)}
t₄: 2⋅X₃+4 {O(n)}
t₅: 1 {O(1)}
t₇: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₆: 2⋅X₃+4 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₄, X₀: X₃+1 {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 2⋅X₃+1 {O(n)}
t₅, X₁: 2⋅X₁ {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₇, X₀: 2⋅X₀+2⋅X₃+1 {O(n)}
t₇, X₁: 4⋅X₁ {O(n)}
t₇, X₂: 4⋅X₂ {O(n)}
t₇, X₃: 4⋅X₃ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₆, X₀: X₃+1 {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}