Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₁ < X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₁
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂, X₃, X₄)
t₄: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀+1, X₁+2, X₂, X₃, X₄)
t₅: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)

Preprocessing

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

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

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

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

Found invariant X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₂: l0(X₀, X₁, X₃, X₄) → l2(X₀, X₁, X₃, X₄)
t₁₃: l1(X₀, X₁, X₃, X₄) → l3(X₀, X₁, X₃, X₄) :|: X₁ < X₀ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀
t₁₄: l1(X₀, X₁, X₃, X₄) → l4(X₀, X₁, X₃, X₄) :|: X₀ ≤ X₁ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀
t₁₅: l2(X₀, X₁, X₃, X₄) → l1(X₃, X₄, X₃, X₄)
t₁₆: l3(X₀, X₁, X₃, X₄) → l1(X₀+1, X₁+2, X₃, X₄) :|: X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₀
t₁₇: l4(X₀, X₁, X₃, X₄) → l5(X₀, X₁, X₃, X₄) :|: X₄ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁

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

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

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

Found invariant X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₃ 2⋅X₃+2⋅X₄+4 {O(n)}

TWN-Loops:

entry: t₁₅: l2(X₀, X₁, X₃, X₄) → l1(X₃, X₄, X₃, X₄)
results in twn-loop: twn:Inv: [X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₀] , (X₀,X₁,X₃,X₄) -> (X₀+1,X₁+2,X₃,X₄) :|: X₁ < X₀
order: [X₀; X₁; X₃; X₄]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁ + [[n != 0]] * 2 * n^1
X₃: X₃
X₄: X₄

Termination: true
Formula:

1 < 0
∨ X₁ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

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

relevant size-bounds w.r.t. t₁₅:
X₀: X₃ {O(n)}
X₁: X₄ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 2⋅X₃+2⋅X₄+4 {O(n)}

2⋅X₃+2⋅X₄+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₆ 2⋅X₃+2⋅X₄+4 {O(n)}

relevant size-bounds w.r.t. t₁₅:
X₀: X₃ {O(n)}
X₁: X₄ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 2⋅X₃+2⋅X₄+4 {O(n)}

2⋅X₃+2⋅X₄+4 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₃+4⋅X₄+12 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 2⋅X₃+2⋅X₄+4 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₃+2⋅X₄+4 {O(n)}
t₁₇: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+4⋅X₄+12 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 2⋅X₃+2⋅X₄+4 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₃+2⋅X₄+4 {O(n)}
t₁₇: 1 {O(1)}

Sizebounds

t₁₂, X₀: X₀ {O(n)}
t₁₂, X₁: X₁ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₂, X₄: X₄ {O(n)}
t₁₃, X₀: 2⋅X₄+3⋅X₃+4 {O(n)}
t₁₃, X₁: 4⋅X₃+5⋅X₄+8 {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₃, X₄: X₄ {O(n)}
t₁₄, X₀: 2⋅X₄+4⋅X₃+4 {O(n)}
t₁₄, X₁: 4⋅X₃+6⋅X₄+8 {O(n)}
t₁₄, X₃: 2⋅X₃ {O(n)}
t₁₄, X₄: 2⋅X₄ {O(n)}
t₁₅, X₀: X₃ {O(n)}
t₁₅, X₁: X₄ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₅, X₄: X₄ {O(n)}
t₁₆, X₀: 2⋅X₄+3⋅X₃+4 {O(n)}
t₁₆, X₁: 4⋅X₃+5⋅X₄+8 {O(n)}
t₁₆, X₃: X₃ {O(n)}
t₁₆, X₄: X₄ {O(n)}
t₁₇, X₀: 2⋅X₄+4⋅X₃+4 {O(n)}
t₁₇, X₁: 4⋅X₃+6⋅X₄+8 {O(n)}
t₁₇, X₃: 2⋅X₃ {O(n)}
t₁₇, X₄: 2⋅X₄ {O(n)}