Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁) → l1(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀-1, X₁+X₀) :|: 1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ 0
t₃: l2(X₀, X₁) → l2(X₀, X₁-1) :|: 1 ≤ X₁

Preprocessing

Found invariant X₀ ≤ 0 for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁) → l1(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀-1, X₁+X₀) :|: 1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ 0
t₃: l2(X₀, X₁) → l2(X₀, X₁-1) :|: 1 ≤ X₁ ∧ X₀ ≤ 0

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

new bound:

X₀ {O(n)}

MPRF:

l1 [X₀ ]

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

new bound:

2⋅X₀⋅X₀+2⋅X₀+2⋅X₁ {O(n^2)}

MPRF:

l2 [X₁ ]

Analysing control-flow refined program

Found invariant X₀ ≤ 0 for location l2

Found invariant 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l2___1

MPRF for transition t₃₃: n_l2___1(X₀, X₁) → n_l2___1(X₀, X₁-1) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 of depth 1:

new bound:

2⋅X₀⋅X₀+2⋅X₀+2⋅X₁+1 {O(n^2)}

MPRF:

n_l2___1 [X₁+1 ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅X₀+2⋅X₁+3⋅X₀+2 {O(n^2)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₀⋅X₀+2⋅X₀+2⋅X₁ {O(n^2)}

Costbounds

Overall costbound: 2⋅X₀⋅X₀+2⋅X₁+3⋅X₀+2 {O(n^2)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₀⋅X₀+2⋅X₀+2⋅X₁ {O(n^2)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 2⋅X₀⋅X₀+2⋅X₀+X₁ {O(n^2)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₀⋅X₀+2⋅X₀+2⋅X₁ {O(n^2)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 2⋅X₀⋅X₀+2⋅X₀+2⋅X₁ {O(n^2)}