Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀+X₁ < X₃
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₀+X₁
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₄, X₅, X₂, X₃, X₄, X₅)
t₄: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+1, X₁+1, X₂, X₃, X₄, X₅)
t₅: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

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

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location l5

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location l1

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location l4

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₁: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₁₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₀+X₁ < X₂ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀
t₁₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀+X₁ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀
t₁₄: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂, X₃, X₄)
t₁₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀+1, X₁+1, X₂, X₃, X₄) :|: X₄ ≤ X₁ ∧ X₃ ≤ X₀
t₁₆: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ X₁ ∧ X₃ ≤ X₀

MPRF for transition t₁₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₀+X₁ < X₂ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ of depth 1:

new bound:

X₂+X₃+X₄ {O(n)}

TWN: t₁₅: l3→l1

cycle: [t₁₅: l3→l1; t₁₂: l1→l3]
loop: (X₀+X₁ < X₂,(X₀,X₁,X₂) -> (X₀+1,X₁+1,X₂)
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁ + [[n != 0]] * n^1
X₂: X₂

Termination: true
Formula:

2 < 0
∨ X₀+X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2

Stabilization-Threshold for: X₀+X₁ < X₂
alphas_abs: X₀+X₁+X₂
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2⋅X₂+2 {O(n)}

TWN - Lifting for t₁₅: l3→l1 of 2⋅X₀+2⋅X₁+2⋅X₂+4 {O(n)}

relevant size-bounds w.r.t. t₁₄:
X₀: X₃ {O(n)}
X₁: X₄ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₁₄: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}

All Bounds

Timebounds

Overall timebound:3⋅X₂+3⋅X₃+3⋅X₄+8 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₂+X₃+X₄ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₂+3⋅X₃+3⋅X₄+8 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₂+X₃+X₄ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₁₆: 1 {O(1)}

Sizebounds

t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₁ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₁, X₄: X₄ {O(n)}
t₁₂, X₀: 2⋅X₂+2⋅X₄+3⋅X₃+4 {O(n)}
t₁₂, X₁: 2⋅X₂+2⋅X₃+3⋅X₄+4 {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₂, X₄: X₄ {O(n)}
t₁₃, X₀: 2⋅X₂+2⋅X₄+4⋅X₃+4 {O(n)}
t₁₃, X₁: 2⋅X₂+2⋅X₃+4⋅X₄+4 {O(n)}
t₁₃, X₂: 2⋅X₂ {O(n)}
t₁₃, X₃: 2⋅X₃ {O(n)}
t₁₃, X₄: 2⋅X₄ {O(n)}
t₁₄, X₀: X₃ {O(n)}
t₁₄, X₁: X₄ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₅, X₀: 2⋅X₂+2⋅X₄+3⋅X₃+4 {O(n)}
t₁₅, X₁: 2⋅X₂+2⋅X₃+3⋅X₄+4 {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₅, X₄: X₄ {O(n)}
t₁₆, X₀: 2⋅X₂+2⋅X₄+4⋅X₃+4 {O(n)}
t₁₆, X₁: 2⋅X₂+2⋅X₃+4⋅X₄+4 {O(n)}
t₁₆, X₂: 2⋅X₂ {O(n)}
t₁₆, X₃: 2⋅X₃ {O(n)}
t₁₆, X₄: 2⋅X₄ {O(n)}