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 < X₃ ∧ X₁ < X₃
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₀
t₄: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₁
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₄, X₅, X₂, X₃, X₄, X₅)
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+1, X₁+1, X₂, X₃, X₄, X₅)
t₆: 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 X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location l5

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location l1

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location l4

Found invariant 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₄: l0(X₀, X₁, X₃, X₄, X₅) → l2(X₀, X₁, X₃, X₄, X₅)
t₁₅: l1(X₀, X₁, X₃, X₄, X₅) → l3(X₀, X₁, X₃, X₄, X₅) :|: X₀ < X₃ ∧ X₁ < X₃ ∧ X₅ ≤ X₁ ∧ X₄ ≤ X₀
t₁₆: l1(X₀, X₁, X₃, X₄, X₅) → l4(X₀, X₁, X₃, X₄, X₅) :|: X₃ ≤ X₀ ∧ X₅ ≤ X₁ ∧ X₄ ≤ X₀
t₁₇: l1(X₀, X₁, X₃, X₄, X₅) → l4(X₀, X₁, X₃, X₄, X₅) :|: X₃ ≤ X₁ ∧ X₅ ≤ X₁ ∧ X₄ ≤ X₀
t₁₈: l2(X₀, X₁, X₃, X₄, X₅) → l1(X₄, X₅, X₃, X₄, X₅)
t₁₉: l3(X₀, X₁, X₃, X₄, X₅) → l1(X₀+1, X₁+1, X₃, X₄, X₅) :|: 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃
t₂₀: l4(X₀, X₁, X₃, X₄, X₅) → l5(X₀, X₁, X₃, X₄, X₅) :|: X₅ ≤ X₁ ∧ X₄ ≤ X₀

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location l5

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location l1

Found invariant X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location l4

Found invariant 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₅ 2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}

TWN-Loops:

entry: t₁₈: l2(X₀, X₁, X₃, X₄, X₅) → l1(X₄, X₅, X₃, X₄, X₅)
results in twn-loop: twn:Inv: [X₅ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃] , (X₀,X₁,X₃,X₄,X₅) -> (X₀+1,X₁+1,X₃,X₄,X₅) :|: X₀ < X₃ ∧ X₁ < X₃
order: [X₀; X₁; X₃; X₄; X₅]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁ + [[n != 0]] * n^1
X₃: X₃
X₄: X₄
X₅: X₅

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ X₀ < X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₁ < X₃ ∧ X₀ < X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₁ < X₃
alphas_abs: X₁+X₃
M: 0
N: 1
Bound: 2⋅X₁+2⋅X₃+2 {O(n)}
Stabilization-Threshold for: X₀ < X₃
alphas_abs: X₀+X₃
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₃+2 {O(n)}

relevant size-bounds w.r.t. t₁₈:
X₀: X₄ {O(n)}
X₁: X₅ {O(n)}
X₃: X₃ {O(n)}
Runtime-bound of t₁₈: 1 {O(1)}
Results in: 2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}

2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₉ 2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}

relevant size-bounds w.r.t. t₁₈:
X₀: X₄ {O(n)}
X₁: X₅ {O(n)}
X₃: X₃ {O(n)}
Runtime-bound of t₁₈: 1 {O(1)}
Results in: 2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}

2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₄+4⋅X₅+8⋅X₃+17 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}
t₂₀: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₄+4⋅X₅+8⋅X₃+17 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 2⋅X₄+2⋅X₅+4⋅X₃+6 {O(n)}
t₂₀: 1 {O(1)}

Sizebounds

t₁₄, X₀: X₀ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₄, X₅: X₅ {O(n)}
t₁₅, X₀: 2⋅X₅+3⋅X₄+4⋅X₃+6 {O(n)}
t₁₅, X₁: 2⋅X₄+3⋅X₅+4⋅X₃+6 {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₅, X₄: X₄ {O(n)}
t₁₅, X₅: X₅ {O(n)}
t₁₆, X₀: 2⋅X₅+4⋅X₃+4⋅X₄+6 {O(n)}
t₁₆, X₁: 2⋅X₄+4⋅X₃+4⋅X₅+6 {O(n)}
t₁₆, X₃: 2⋅X₃ {O(n)}
t₁₆, X₄: 2⋅X₄ {O(n)}
t₁₆, X₅: 2⋅X₅ {O(n)}
t₁₇, X₀: 2⋅X₅+4⋅X₃+4⋅X₄+6 {O(n)}
t₁₇, X₁: 2⋅X₄+4⋅X₃+4⋅X₅+6 {O(n)}
t₁₇, X₃: 2⋅X₃ {O(n)}
t₁₇, X₄: 2⋅X₄ {O(n)}
t₁₇, X₅: 2⋅X₅ {O(n)}
t₁₈, X₀: X₄ {O(n)}
t₁₈, X₁: X₅ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₈, X₅: X₅ {O(n)}
t₁₉, X₀: 2⋅X₅+3⋅X₄+4⋅X₃+6 {O(n)}
t₁₉, X₁: 2⋅X₄+3⋅X₅+4⋅X₃+6 {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₁₉, X₅: X₅ {O(n)}
t₂₀, X₀: 4⋅X₅+8⋅X₃+8⋅X₄+12 {O(n)}
t₂₀, X₁: 4⋅X₄+8⋅X₃+8⋅X₅+12 {O(n)}
t₂₀, X₃: 4⋅X₃ {O(n)}
t₂₀, X₄: 4⋅X₄ {O(n)}
t₂₀, X₅: 4⋅X₅ {O(n)}