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) :|: X₀+X₁ < X₂
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 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+1, X₂, X₃, X₄)
t₅: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)

Preprocessing

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

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

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}

TWN-Loops:

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

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)}

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

2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}

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

2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₂+4⋅X₃+4⋅X₄+12 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₂+4⋅X₃+4⋅X₄+12 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂+2⋅X₃+2⋅X₄+4 {O(n)}
t₅: 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⋅X₂+2⋅X₄+3⋅X₃+4 {O(n)}
t₂, X₁: 2⋅X₂+2⋅X₃+3⋅X₄+4 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 2⋅X₂+2⋅X₄+4⋅X₃+4 {O(n)}
t₃, X₁: 2⋅X₂+2⋅X₃+4⋅X₄+4 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₄, X₀: 2⋅X₂+2⋅X₄+3⋅X₃+4 {O(n)}
t₄, X₁: 2⋅X₂+2⋅X₃+3⋅X₄+4 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: 2⋅X₂+2⋅X₄+4⋅X₃+4 {O(n)}
t₅, X₁: 2⋅X₂+2⋅X₃+4⋅X₄+4 {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}