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) :|: 1 ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃-1) :|: 1 ≤ X₃

Preprocessing

Eliminate variables [X₁; X₂] that do not contribute to the problem

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₉: l0(X₀, X₁) → l1(X₀, X₁)
t₁₀: l1(X₀, X₁) → l1(X₀-1, (X₀)²+X₁) :|: 1 ≤ X₀
t₁₁: l1(X₀, X₁) → l2(X₀, X₁)
t₁₂: l2(X₀, X₁) → l2(X₀, X₁-1) :|: 1 ≤ X₁

MPRF for transition t₁₀: l1(X₀, X₁) → l1(X₀-1, Temp_Int₃₆+X₁) :|: 1 ≤ X₀ ∧ 1 ≤ Temp_Int₃₆ ∧ X₀ ≤ Temp_Int₃₆ of depth 1:

new bound:

X₀ {O(n)}

• l1: [X₀]

MPRF for transition t₁₂: l2(X₀, X₁) → l2(X₀, X₁-1) :|: 1 ≤ X₁ of depth 1:

new bound:

3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₁+6 {O(n^3)}

• l2: [X₁]

Found invariant 0 ≤ X₁ for location l2_v1

All Bounds

Timebounds

Overall timebound:3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+15⋅X₀+2⋅X₁+8 {O(n^3)}
t₉: 1 {O(1)}
t₁₀: X₀ {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₁+6 {O(n^3)}

Costbounds

Overall costbound: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+15⋅X₀+2⋅X₁+8 {O(n^3)}
t₉: 1 {O(1)}
t₁₀: X₀ {O(n)}
t₁₁: 1 {O(1)}
t₁₂: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₁+6 {O(n^3)}

Sizebounds

t₉, X₀: X₀ {O(n)}
t₉, X₁: X₁ {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+X₁+6 {O(n^3)}
t₁₁, X₀: 2⋅X₀ {O(n)}
t₁₁, X₁: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₁+6 {O(n^3)}
t₁₂, X₀: 2⋅X₀ {O(n)}
t₁₂, X₁: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₁+6 {O(n^3)}