Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂+X₃, X₃+X₄, X₄-1) :|: 1 ≤ X₄
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₄ < 1
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂-1, X₃, X₄) :|: 1 ≤ X₂
t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₂ < 1
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(X₀+X₁, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₀

Preprocessing

Found invariant X₄ ≤ 0 for location l2

Found invariant X₄ ≤ 0 ∧ X₂+X₄ ≤ 0 ∧ X₂ ≤ 0 for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂+X₃, X₃+X₄, X₄-1) :|: 1 ≤ X₄
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₄ < 1
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂-1, X₃, X₄) :|: 1 ≤ X₂ ∧ X₄ ≤ 0
t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₂ < 1 ∧ X₄ ≤ 0
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(X₀+X₁, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ X₄ ≤ 0 ∧ X₂+X₄ ≤ 0 ∧ X₂ ≤ 0

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂+X₃, X₃+X₄, X₄-1) :|: 1 ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF for transition t₃: l2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂-1, X₃, X₄) :|: 1 ≤ X₂ ∧ X₄ ≤ 0 of depth 1:

new bound:

2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}

Analysing control-flow refined program

Found invariant X₄ ≤ 0 for location l2

Found invariant X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ 0 ≤ X₂ for location n_l2___1

Found invariant X₄ ≤ 0 ∧ X₂+X₄ ≤ 0 ∧ X₂ ≤ 0 for location l3

MPRF for transition t₄₃: n_l2___1(X₀, X₁, X₂, X₃, X₄) → n_l2___1(X₀, X₁, X₂-1, X₃, X₄) :|: 0 ≤ X₂ ∧ 1 ≤ X₂ ∧ X₄ ≤ 0 ∧ X₄ ≤ 0 ∧ X₄ ≤ 0 ∧ X₄ ≤ 0 ∧ X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄+1 {O(n^3)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

TWN: t₅: l3→l3

cycle: [t₅: l3→l3]
loop: (1 ≤ X₀,(X₀,X₁) -> (X₀+X₁,X₁-1)
order: [X₁; X₀]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
X₀: X₀ + [[n != 0]] * X₁ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1

Termination: true
Formula:

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

Stabilization-Threshold for: 1 ≤ X₀
alphas_abs: 2+2⋅X₀+2⋅X₁
M: 0
N: 2
Bound: 4⋅X₀+4⋅X₁+7 {O(n)}

TWN - Lifting for t₅: l3→l3 of 4⋅X₀+4⋅X₁+9 {O(n)}

relevant size-bounds w.r.t. t₄:
X₀: 4⋅X₀ {O(n)}
X₁: 4⋅X₁ {O(n)}
Runtime-bound of t₄: 1 {O(1)}
Results in: 16⋅X₀+16⋅X₁+9 {O(n)}

All Bounds

Timebounds

Overall timebound:2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+16⋅X₀+16⋅X₁+2⋅X₂+2⋅X₃+3⋅X₄+12 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₄ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}
t₄: 1 {O(1)}
t₅: 16⋅X₀+16⋅X₁+9 {O(n)}

Costbounds

Overall costbound: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+16⋅X₀+16⋅X₁+2⋅X₂+2⋅X₃+3⋅X₄+12 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₄ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}
t₄: 1 {O(1)}
t₅: 16⋅X₀+16⋅X₁+9 {O(n)}

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)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₃+2⋅X₄+X₂ {O(n^3)}
t₁, X₃: 2⋅X₄⋅X₄+2⋅X₄+X₃ {O(n^2)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}
t₂, X₃: 2⋅X₄⋅X₄+2⋅X₃+2⋅X₄ {O(n^2)}
t₂, X₄: 2⋅X₄ {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}
t₃, X₃: 2⋅X₄⋅X₄+2⋅X₃+2⋅X₄ {O(n^2)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 4⋅X₀ {O(n)}
t₄, X₁: 4⋅X₁ {O(n)}
t₄, X₂: 4⋅X₄⋅X₄⋅X₄+4⋅X₃⋅X₄+8⋅X₄⋅X₄+4⋅X₂+4⋅X₃+4⋅X₄ {O(n^3)}
t₄, X₃: 4⋅X₄⋅X₄+4⋅X₃+4⋅X₄ {O(n^2)}
t₄, X₄: 4⋅X₄ {O(n)}
t₅, X₀: 256⋅X₀⋅X₀+384⋅X₁⋅X₁+640⋅X₀⋅X₁+308⋅X₀+384⋅X₁+90 {O(n^2)}
t₅, X₁: 16⋅X₀+20⋅X₁+9 {O(n)}
t₅, X₂: 4⋅X₄⋅X₄⋅X₄+4⋅X₃⋅X₄+8⋅X₄⋅X₄+4⋅X₂+4⋅X₃+4⋅X₄ {O(n^3)}
t₅, X₃: 4⋅X₄⋅X₄+4⋅X₃+4⋅X₄ {O(n^2)}
t₅, X₄: 4⋅X₄ {O(n)}