Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, 3⋅X₁+X₂, -6⋅X₁-2⋅X₂, X₃+2⋅(X₀)²) :|: 0 < X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂, X₃-1) :|: 0 < X₁+X₃

Preprocessing

Found invariant X₀ ≤ 0 for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, 3⋅X₁+X₂, -6⋅X₁-2⋅X₂, X₃+2⋅(X₀)²) :|: 0 < X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂, X₃-1) :|: 0 < X₁+X₃ ∧ X₀ ≤ 0

TWN. Size Bound: t₁: l1→l1 for X₃

cycle: [t₁: l1→l1]
loop: (0 < X₀,(X₀,X₃) -> (X₀-1,X₃+2⋅(X₀)²)
closed-form: X₃ + [[n != 0]] * 2⋅(X₀)² * n^1 + [[n != 0, n != 1]] * 2/3 * n^3 + [[n != 0, n != 1]] * -1-2⋅X₀ * n^2 + [[n != 0, n != 1]] * 1/3+2⋅X₀ * n^1
runtime bound: X₀+1 {O(n)}

TWN Size Bound - Lifting for t₁: l1→l1 and X₃: 5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+10⋅X₀+X₃+3 {O(n^3)}

Solv. Size Bound: t₁: l1→l1 for X₁

cycle: [t₁: l1→l1]
loop: (0 < X₀,(X₁,X₂) -> (3⋅X₁+X₂,-6⋅X₁-2⋅X₂)
overappr. closed-form: 2⋅X₂+6⋅X₁ {O(n)}
runtime bound: X₀+1 {O(n)}

Solv. Size Bound - Lifting for t₁: l1→l1 and X₁: 2⋅X₂+6⋅X₁ {O(n)}

Solv. Size Bound: t₁: l1→l1 for X₂

cycle: [t₁: l1→l1]
loop: (0 < X₀,(X₁,X₂) -> (3⋅X₁+X₂,-6⋅X₁-2⋅X₂)
overappr. closed-form: 12⋅X₁+4⋅X₂ {O(n)}
runtime bound: X₀+1 {O(n)}

Solv. Size Bound - Lifting for t₁: l1→l1 and X₂: 12⋅X₁+4⋅X₂ {O(n)}

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, 3⋅X₁+X₂, -6⋅X₁-2⋅X₂, X₃+2⋅Temp_Int₁₅₉) :|: 0 < X₀ ∧ 0 < Temp_Int₁₅₉ ∧ X₀ ≤ Temp_Int₁₅₉ of depth 1:

new bound:

X₀ {O(n)}

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

new bound:

5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+10⋅X₀+2⋅X₂+2⋅X₃+7⋅X₁+3 {O(n^3)}

Analysing control-flow refined program

Found invariant X₀ ≤ 0 for location l2

MPRF for transition t₅₂: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂, X₃-1) :|: 0 < X₁+X₃ ∧ X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ 0 of depth 1:

new bound:

5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+10⋅X₀+2⋅X₂+2⋅X₃+7⋅X₁+3 {O(n^3)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+11⋅X₀+2⋅X₂+2⋅X₃+7⋅X₁+5 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+10⋅X₀+2⋅X₂+2⋅X₃+7⋅X₁+3 {O(n^3)}

Costbounds

Overall costbound: 5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+11⋅X₀+2⋅X₂+2⋅X₃+7⋅X₁+5 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+10⋅X₀+2⋅X₂+2⋅X₃+7⋅X₁+3 {O(n^3)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 2⋅X₂+6⋅X₁ {O(n)}
t₁, X₂: 12⋅X₁+4⋅X₂ {O(n)}
t₁, X₃: 5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+10⋅X₀+X₃+3 {O(n^3)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₂+7⋅X₁ {O(n)}
t₂, X₂: 12⋅X₁+5⋅X₂ {O(n)}
t₂, X₃: 5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+10⋅X₀+2⋅X₃+3 {O(n^3)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 5⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+10⋅X₀+14⋅X₁+2⋅X₃+4⋅X₂+3 {O(n^3)}
t₃, X₂: 12⋅X₁+5⋅X₂ {O(n)}
t₃, X₃: 10⋅X₀⋅X₀⋅X₀+24⋅X₀⋅X₀+2⋅X₂+20⋅X₀+4⋅X₃+7⋅X₁+6 {O(n^3)}