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) :|: X₄ < X₀ ∧ X₄ < X₁
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₄
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₄
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, X₁-1, X₂, X₃, X₄)
t₆: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location l5

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location l1

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location l4

Found invariant 1+X₄ ≤ X₃ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₂ 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:
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) :|: X₄ < X₀ ∧ X₄ < X₁ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₂
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₄ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₂
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₄ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₂
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, X₁-1, X₂, X₃, X₄) :|: 1+X₄ ≤ X₃ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₂
t₆: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₃ ∧ X₀ ≤ X₂

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location l5

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location l1

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location l4

Found invariant 1+X₄ ≤ X₃ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}

TWN-Loops:

entry: t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄)
results in twn-loop: twn:Inv: [X₁ ≤ X₃ ∧ X₀ ≤ X₂ ∧ 1+X₄ ≤ X₃ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₀ ≤ X₂] , (X₀,X₁,X₂,X₃,X₄) -> (X₀-1,X₁-1,X₂,X₃,X₄) :|: X₄ < X₀ ∧ X₄ < X₁
order: [X₀; X₁; X₂; X₃; X₄]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: X₁ + [[n != 0]] * -1 * n^1
X₂: X₂
X₃: X₃
X₄: X₄

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ X₄ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₄ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₄ < X₁ ∧ X₄ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₄ < X₁
alphas_abs: X₄+X₁
M: 0
N: 1
Bound: 2⋅X₁+2⋅X₄+2 {O(n)}
Stabilization-Threshold for: X₄ < X₀
alphas_abs: X₄+X₀
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₄+2 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}

2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}

2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₂+4⋅X₃+8⋅X₄+17 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₂+4⋅X₃+8⋅X₄+17 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 2⋅X₂+2⋅X₃+4⋅X₄+6 {O(n)}
t₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₂, X₀: 2⋅X₃+3⋅X₂+4⋅X₄+6 {O(n)}
t₂, X₁: 2⋅X₂+3⋅X₃+4⋅X₄+6 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 2⋅X₃+4⋅X₂+4⋅X₄+6 {O(n)}
t₃, X₁: 2⋅X₂+4⋅X₃+4⋅X₄+6 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 2⋅X₃+4⋅X₂+4⋅X₄+6 {O(n)}
t₄, X₁: 2⋅X₂+4⋅X₃+4⋅X₄+6 {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₅, X₀: 2⋅X₃+3⋅X₂+4⋅X₄+6 {O(n)}
t₅, X₁: 2⋅X₂+3⋅X₃+4⋅X₄+6 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: 4⋅X₃+8⋅X₂+8⋅X₄+12 {O(n)}
t₆, X₁: 4⋅X₂+8⋅X₃+8⋅X₄+12 {O(n)}
t₆, X₂: 4⋅X₂ {O(n)}
t₆, X₃: 4⋅X₃ {O(n)}
t₆, X₄: 4⋅X₄ {O(n)}