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 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 X₅ ≤ X₁ ∧ X₄ ≤ X₀ 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:
t₁₂: l0(X₀, X₁, X₃, X₄, X₅) → l2(X₀, X₁, X₃, X₄, X₅)
t₁₃: l1(X₀, X₁, X₃, X₄, X₅) → l3(X₀, X₁, X₃, X₄, X₅) :|: X₀+X₁ < X₃ ∧ X₅ ≤ X₁ ∧ X₄ ≤ X₀
t₁₄: l1(X₀, X₁, X₃, X₄, X₅) → l4(X₀, X₁, X₃, X₄, X₅) :|: X₃ ≤ X₀+X₁ ∧ X₅ ≤ X₁ ∧ X₄ ≤ X₀
t₁₅: l2(X₀, X₁, X₃, X₄, X₅) → l1(X₄, X₅, X₃, X₄, X₅)
t₁₆: l3(X₀, X₁, X₃, X₄, X₅) → l1(X₀+1, X₁+1, X₃, X₄, X₅) :|: X₅ ≤ X₁ ∧ X₄ ≤ X₀
t₁₇: l4(X₀, X₁, X₃, X₄, X₅) → l5(X₀, X₁, X₃, X₄, X₅) :|: X₅ ≤ X₁ ∧ X₄ ≤ X₀

MPRF for transition t₁₃: l1(X₀, X₁, X₃, X₄, X₅) → l3(X₀, X₁, X₃, X₄, X₅) :|: X₀+X₁ < X₃ ∧ X₅ ≤ X₁ ∧ X₄ ≤ X₀ of depth 1:

new bound:

X₃+X₄+X₅ {O(n)}

MPRF:

l3 [X₃-X₁-X₄-1 ]
l1 [X₃-X₁-X₄ ]

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 X₅ ≤ X₁ ∧ X₄ ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₆ 2⋅X₃+2⋅X₄+2⋅X₅+4 {O(n)}

TWN-Loops:

entry: t₁₅: l2(X₀, X₁, X₃, X₄, X₅) → l1(X₄, X₅, X₃, X₄, X₅)
results in twn-loop: twn:Inv: [X₅ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₅ ≤ X₁ ∧ X₄ ≤ X₀] , (X₀,X₁,X₃,X₄,X₅) -> (X₀+1,X₁+1,X₃,X₄,X₅) :|: X₀+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:

2 < 0
∨ X₀+X₁ < X₃ ∧ 2 ≤ 0 ∧ 0 ≤ 2

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

relevant size-bounds w.r.t. t₁₅:
X₀: X₄ {O(n)}
X₁: X₅ {O(n)}
X₃: X₃ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 2⋅X₃+2⋅X₄+2⋅X₅+4 {O(n)}

2⋅X₃+2⋅X₄+2⋅X₅+4 {O(n)}

All Bounds

Timebounds

Overall timebound:3⋅X₃+3⋅X₄+3⋅X₅+8 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: X₃+X₄+X₅ {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₃+2⋅X₄+2⋅X₅+4 {O(n)}
t₁₇: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₃+3⋅X₄+3⋅X₅+8 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: X₃+X₄+X₅ {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₃+2⋅X₄+2⋅X₅+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)}
t₁₂, X₅: X₅ {O(n)}
t₁₃, X₀: 2⋅X₃+2⋅X₅+3⋅X₄+4 {O(n)}
t₁₃, X₁: 2⋅X₃+2⋅X₄+3⋅X₅+4 {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₃, X₄: X₄ {O(n)}
t₁₃, X₅: X₅ {O(n)}
t₁₄, X₀: 2⋅X₃+2⋅X₅+4⋅X₄+4 {O(n)}
t₁₄, X₁: 2⋅X₃+2⋅X₄+4⋅X₅+4 {O(n)}
t₁₄, X₃: 2⋅X₃ {O(n)}
t₁₄, X₄: 2⋅X₄ {O(n)}
t₁₄, X₅: 2⋅X₅ {O(n)}
t₁₅, X₀: X₄ {O(n)}
t₁₅, X₁: X₅ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₅, X₄: X₄ {O(n)}
t₁₅, X₅: X₅ {O(n)}
t₁₆, X₀: 2⋅X₃+2⋅X₅+3⋅X₄+4 {O(n)}
t₁₆, X₁: 2⋅X₃+2⋅X₄+3⋅X₅+4 {O(n)}
t₁₆, X₃: X₃ {O(n)}
t₁₆, X₄: X₄ {O(n)}
t₁₆, X₅: X₅ {O(n)}
t₁₇, X₀: 2⋅X₃+2⋅X₅+4⋅X₄+4 {O(n)}
t₁₇, X₁: 2⋅X₃+2⋅X₄+4⋅X₅+4 {O(n)}
t₁₇, X₃: 2⋅X₃ {O(n)}
t₁₇, X₄: 2⋅X₄ {O(n)}
t₁₇, X₅: 2⋅X₅ {O(n)}