Initial Problem

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

Preprocessing

Found invariant X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1

Problem after Preprocessing

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

MPRF for transition t₂: l1(X₀, X₁) → l2(X₀-2, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀+1 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

l2 [X₀+1 ]
l1 [X₀ ]

MPRF for transition t₀: l2(X₀, X₁) → l1(X₀+1, 1) :|: 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

l2 [X₀+1 ]
l1 [X₀-1 ]

Found invariant X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1

MPRF for transition t₁: l1(X₀, X₁) → l1(X₀, X₁+1) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

4⋅X₀⋅X₀+7⋅X₀+3 {O(n^2)}

MPRF:

l1 [2⋅X₀-X₁ ]
l2 [X₀+1 ]

Analysing control-flow refined program

Cut unsatisfiable transition t₂: l1→l2

Found invariant X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1

Found invariant X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___1

knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₄₀: l1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀

MPRF for transition t₃₉: n_l1___1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: 1 ≤ X₁ ∧ 2 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}

MPRF:

l1 [X₀-X₁ ]
n_l1___1 [X₀+1-X₁ ]
l2 [-X₁ ]

MPRF for transition t₄₃: n_l1___1(X₀, X₁) → l2(X₀-2, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀+1 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀+2 {O(n)}

MPRF:

l1 [X₀+1 ]
n_l1___1 [X₀+1 ]
l2 [X₀+2 ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:4⋅X₀⋅X₀+9⋅X₀+6 {O(n^2)}
t₃: 1 {O(1)}
t₁: 4⋅X₀⋅X₀+7⋅X₀+3 {O(n^2)}
t₂: X₀+1 {O(n)}
t₀: X₀+1 {O(n)}

Costbounds

Overall costbound: 4⋅X₀⋅X₀+9⋅X₀+6 {O(n^2)}
t₃: 1 {O(1)}
t₁: 4⋅X₀⋅X₀+7⋅X₀+3 {O(n^2)}
t₂: X₀+1 {O(n)}
t₀: X₀+1 {O(n)}

Sizebounds

t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₁, X₀: 2⋅X₀+1 {O(n)}
t₁, X₁: 4⋅X₀⋅X₀+7⋅X₀+4 {O(n^2)}
t₂, X₀: 2⋅X₀+1 {O(n)}
t₂, X₁: 4⋅X₀⋅X₀+7⋅X₀+4 {O(n^2)}
t₀, X₀: 2⋅X₀+1 {O(n)}
t₀, X₁: 1 {O(1)}