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

Preprocessing

Found invariant X₅ ≤ X₂ ∧ X₀ ≤ X₁ for location l5

Found invariant X₅ ≤ X₂ for location l1

Found invariant X₅ ≤ X₂ ∧ X₀ ≤ X₁ for location l4

Found invariant X₅ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location l3

Problem after Preprocessing

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

Found invariant X₅ ≤ X₂ ∧ X₀ ≤ X₁ for location l5

Found invariant X₅ ≤ X₂ for location l1

Found invariant X₅ ≤ X₂ ∧ X₀ ≤ X₁ for location l4

Found invariant X₅ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 4⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}

TWN-Loops:

entry: t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
results in twn-loop: twn:Inv: [X₅ ≤ X₂ ∧ X₅ ≤ X₂ ∧ 1+X₁ ≤ X₀] , (X₀,X₁,X₂,X₃,X₄,X₅) -> (X₁-X₀,X₂,X₂+1,X₃,X₄,X₅) :|: X₁ < X₀
order: [X₂; X₁; X₀; X₅]
closed-form:
X₂: X₂ + [[n != 0]] * 2 * n^1
X₁: [[n == 0]] * X₁ + [[n != 0]] * 1+X₂ + [[n != 0, n != 1]] * 2 * n^1 + [[n != 0, n != 1]] * -2
X₀: X₀ + [[n != 0]] * X₂ * n^1 + [[n != 0]] * -X₁ + [[n != 0, n != 1]] * n^2 + [[n != 0, n != 1]] * -2-X₂ * n^1 + [[n != 0, n != 1]] * 1+X₂ + [[n != 0, n != 1, n != 2]] * -1 * n^2 + [[n != 0, n != 1, n != 2]] * 3 * n^1 + [[n != 0, n != 1, n != 2]] * -4
X₅: X₅

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ X₁+2 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀+X₂ < X₁+2 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₀+X₂ < X₁+2 ∧ X₁+2 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

Stabilization-Threshold for: X₀+X₂ < X₁
alphas_abs: X₀+X₁+2+X₂
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2⋅X₂+6 {O(n)}
Stabilization-Threshold for: X₁ < X₀
alphas_abs: X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
X₁: X₄ {O(n)}
X₂: X₅ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 4⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}

4⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 4⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
X₁: X₄ {O(n)}
X₂: X₅ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 4⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}

4⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}

All Bounds

Timebounds

Overall timebound:16⋅X₃+16⋅X₄+8⋅X₅+46 {O(n)}
t₀: 1 {O(1)}
t₂: 4⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 4⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₃+16⋅X₄+8⋅X₅+46 {O(n)}
t₀: 1 {O(1)}
t₂: 4⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 4⋅X₅+8⋅X₃+8⋅X₄+21 {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)}
t₀, X₅: X₅ {O(n)}
t₂, X₀: 136⋅X₃⋅X₄+20⋅X₅⋅X₅+64⋅X₃⋅X₃+72⋅X₃⋅X₅+72⋅X₄⋅X₄+76⋅X₄⋅X₅+194⋅X₅+345⋅X₃+366⋅X₄+462 {O(n^2)}
t₂, X₁: 5⋅X₅+8⋅X₃+9⋅X₄+21 {O(n)}
t₂, X₂: 5⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: 136⋅X₃⋅X₄+20⋅X₅⋅X₅+64⋅X₃⋅X₃+72⋅X₃⋅X₅+72⋅X₄⋅X₄+76⋅X₄⋅X₅+194⋅X₅+346⋅X₃+366⋅X₄+462 {O(n^2)}
t₃, X₁: 5⋅X₅+8⋅X₃+9⋅X₄+21 {O(n)}
t₃, X₂: 6⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₅ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₄, X₀: 136⋅X₃⋅X₄+20⋅X₅⋅X₅+64⋅X₃⋅X₃+72⋅X₃⋅X₅+72⋅X₄⋅X₄+76⋅X₄⋅X₅+194⋅X₅+345⋅X₃+366⋅X₄+462 {O(n^2)}
t₄, X₁: 5⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}
t₄, X₂: 5⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: 136⋅X₃⋅X₄+20⋅X₅⋅X₅+64⋅X₃⋅X₃+72⋅X₃⋅X₅+72⋅X₄⋅X₄+76⋅X₄⋅X₅+194⋅X₅+346⋅X₃+366⋅X₄+462 {O(n^2)}
t₅, X₁: 5⋅X₅+8⋅X₃+9⋅X₄+21 {O(n)}
t₅, X₂: 6⋅X₅+8⋅X₃+8⋅X₄+21 {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}
t₅, X₅: 2⋅X₅ {O(n)}