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+2⋅X₂, -5⋅X₁-3⋅X₂, (X₀)²+X₃) :|: 0 < X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂-1, X₃-1) :|: 0 < X₁+X₂+X₃

Preprocessing

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+2⋅X₂, -5⋅X₁-3⋅X₂, (X₀)²+X₃) :|: 0 < X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂-1, X₃-1) :|: 0 < X₁+X₂+X₃

TWN. Size Bound: t₁: l1→l1 for X₃

cycle: [t₁: l1→l1]
loop: (0 < X₀,(X₀,X₃) -> (X₀-1,(X₀)²+X₃)
closed-form: X₃ + [[n != 0]] * (X₀)² * n^1 + [[n != 0, n != 1]] * 1/3 * n^3 + [[n != 0, n != 1]] * -1/2-X₀ * n^2 + [[n != 0, n != 1]] * 1/6+X₀ * n^1
runtime bound: X₀+1 {O(n)}

TWN Size Bound - Lifting for t₁: l1→l1 and X₃: 3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+8⋅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+2⋅X₂,-5⋅X₁-3⋅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+2⋅X₂,-5⋅X₁-3⋅X₂)
overappr. closed-form: 6⋅X₁+6⋅X₂ {O(n)}
runtime bound: X₀+1 {O(n)}

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

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, Temp_Int₁₃₉+X₃) :|: 0 < X₀ ∧ 0 < Temp_Int₁₃₉ ∧ X₀ ≤ Temp_Int₁₃₉ of depth 1:

new bound:

X₀ {O(n)}

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

new bound:

3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+13⋅X₁+2⋅X₃+8⋅X₀+9⋅X₂+3 {O(n^3)}

Analysing control-flow refined program

MPRF for transition t₄₇: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂-1, X₃-1) :|: 0 < X₁+X₂+X₃ of depth 1:

new bound:

3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+13⋅X₁+2⋅X₃+8⋅X₀+9⋅X₂+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:3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+13⋅X₁+2⋅X₃+9⋅X₀+9⋅X₂+5 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+13⋅X₁+2⋅X₃+8⋅X₀+9⋅X₂+3 {O(n^3)}

Costbounds

Overall costbound: 3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+13⋅X₁+2⋅X₃+9⋅X₀+9⋅X₂+5 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+13⋅X₁+2⋅X₃+8⋅X₀+9⋅X₂+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: 6⋅X₁+6⋅X₂ {O(n)}
t₁, X₃: 3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+8⋅X₀+X₃+3 {O(n^3)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₂+7⋅X₁ {O(n)}
t₂, X₂: 6⋅X₁+7⋅X₂ {O(n)}
t₂, X₃: 3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+2⋅X₃+8⋅X₀+3 {O(n^3)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+11⋅X₂+2⋅X₃+20⋅X₁+8⋅X₀+3 {O(n^3)}
t₃, X₂: 3⋅X₀⋅X₀⋅X₀+8⋅X₀⋅X₀+16⋅X₂+19⋅X₁+2⋅X₃+8⋅X₀+3 {O(n^3)}
t₃, X₃: 6⋅X₀⋅X₀⋅X₀+16⋅X₀⋅X₀+13⋅X₁+16⋅X₀+4⋅X₃+9⋅X₂+6 {O(n^3)}