Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ X₀
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂
t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1)
t₇: l3(X₀, X₁, X₂) → l4(X₀, X₁-1, X₂)
t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, 1) :|: 1 ≤ X₁
t₃: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 0
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₁: l6(X₀, X₁, X₂) → l4(X₁, X₀, X₂)

Preprocessing

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2

Found invariant X₁ ≤ 0 for location l7

Found invariant X₁ ≤ 0 for location l5

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l1

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₇: l3(X₀, X₁, X₂) → l4(X₀, X₁-1, X₂) :|: 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁
t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, 1) :|: 1 ≤ X₁
t₃: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 0
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: l6(X₀, X₁, X₂) → l4(X₁, X₀, X₂)

MPRF for transition t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l2 [X₁ ]
l3 [X₁-1 ]
l4 [X₁ ]
l1 [X₁ ]

MPRF for transition t₇: l3(X₀, X₁, X₂) → l4(X₀, X₁-1, X₂) :|: 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l2 [X₁ ]
l3 [X₁ ]
l4 [X₁ ]
l1 [X₁ ]

MPRF for transition t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, 1) :|: 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l2 [X₁-1 ]
l3 [X₁-1 ]
l4 [X₁ ]
l1 [X₁-1 ]

MPRF for transition t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀⋅X₁+2⋅X₀ {O(n^2)}

MPRF:

l2 [X₀-X₂ ]
l1 [X₀+1-X₂ ]
l3 [X₀-X₂ ]
l4 [X₀-X₂ ]

MPRF for transition t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀⋅X₁+2⋅X₀ {O(n^2)}

MPRF:

l2 [X₀+1-X₂ ]
l1 [X₀+1-X₂ ]
l3 [X₀-X₂ ]
l4 [X₀-X₂ ]

Analysing control-flow refined program

Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___2

Found invariant X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l2___1

Found invariant X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l2___3

Found invariant X₁ ≤ 0 for location l7

Found invariant X₁ ≤ 0 for location l5

Found invariant X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l1

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ for location l3

knowledge_propagation leads to new time bound X₀ {O(n)} for transition t₆₁: l1(X₀, X₁, X₂) → n_l2___3(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁

knowledge_propagation leads to new time bound X₀ {O(n)} for transition t₆₃: n_l2___3(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀

MPRF for transition t₆₀: n_l1___2(X₀, X₁, X₂) → n_l2___1(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀⋅X₁+4⋅X₀ {O(n^2)}

MPRF:

n_l2___3 [0 ]
l4 [0 ]
l1 [0 ]
l3 [0 ]
n_l2___1 [X₀+1-X₂ ]
n_l1___2 [X₀+2-X₂ ]

MPRF for transition t₆₇: n_l1___2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l4 [X₁ ]
l1 [X₁ ]
l3 [X₁-1 ]
n_l2___1 [X₁ ]
n_l2___3 [X₁ ]
n_l1___2 [X₁ ]

MPRF for transition t₆₂: n_l2___1(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀⋅X₁+3⋅X₀ {O(n^2)}

MPRF:

n_l2___3 [0 ]
l4 [0 ]
l1 [0 ]
l3 [0 ]
n_l2___1 [X₀+1-X₂ ]
n_l1___2 [X₀+1-X₂ ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅X₁+7⋅X₀+4 {O(n^2)}
t₀: 1 {O(1)}
t₄: X₀⋅X₁+2⋅X₀ {O(n^2)}
t₅: X₀ {O(n)}
t₆: X₀⋅X₁+2⋅X₀ {O(n^2)}
t₇: X₀ {O(n)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀⋅X₁+7⋅X₀+4 {O(n^2)}
t₀: 1 {O(1)}
t₄: X₀⋅X₁+2⋅X₀ {O(n^2)}
t₅: X₀ {O(n)}
t₆: X₀⋅X₁+2⋅X₀ {O(n^2)}
t₇: X₀ {O(n)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₀ {O(n)}
t₄, X₂: X₀⋅X₁+2⋅X₀+1 {O(n^2)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₀ {O(n)}
t₅, X₂: X₀⋅X₁+2⋅X₀+2 {O(n^2)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₀ {O(n)}
t₆, X₂: X₀⋅X₁+2⋅X₀+1 {O(n^2)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₀ {O(n)}
t₇, X₂: X₀⋅X₁+2⋅X₀+2 {O(n^2)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: 1 {O(1)}
t₃, X₀: 2⋅X₁ {O(n)}
t₃, X₁: 2⋅X₀ {O(n)}
t₃, X₂: X₀⋅X₁+2⋅X₀+X₂+2 {O(n^2)}
t₈, X₀: 2⋅X₁ {O(n)}
t₈, X₁: 2⋅X₀ {O(n)}
t₈, X₂: X₀⋅X₁+2⋅X₀+X₂+2 {O(n^2)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}