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)
t₃: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: 2+X₁ ≤ X₂
t₄: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ X₁+1
t₅: l2(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂)
t₆: l3(X₀, X₁, X₂) → l4(X₀+1, X₁, X₂) :|: X₂ ≤ X₁+1 ∧ 3+X₀ ≤ X₂
t₇: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: 2+X₁ ≤ X₂
t₈: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ ≤ X₀+2
t₂: l4(X₀, X₁, X₂) → l1(X₀, 0, X₂)
t₉: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₁: l6(X₀, X₁, X₂) → l4(0, X₁, X₂)

Preprocessing

Cut unsatisfiable transition t₇: l3→l5

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2

Found invariant X₂ ≤ 1+X₁ ∧ X₂ ≤ 2+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l7

Found invariant X₂ ≤ 1+X₁ ∧ X₂ ≤ 2+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l5

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location l4

Found invariant X₂ ≤ 1+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ 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)
t₃: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: 2+X₁ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₄: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ X₁+1 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₅: l2(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₆: l3(X₀, X₁, X₂) → l4(X₀+1, X₁, X₂) :|: X₂ ≤ X₁+1 ∧ 3+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₈: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ ≤ X₀+2 ∧ X₂ ≤ 1+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂: l4(X₀, X₁, X₂) → l1(X₀, 0, X₂) :|: 0 ≤ X₀
t₉: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ X₂ ≤ 2+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₁: l6(X₀, X₁, X₂) → l4(0, X₁, X₂)

MPRF for transition t₆: l3(X₀, X₁, X₂) → l4(X₀+1, X₁, X₂) :|: X₂ ≤ X₁+1 ∧ 3+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

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

knowledge_propagation leads to new time bound X₂+2 {O(n)} for transition t₂: l4(X₀, X₁, X₂) → l1(X₀, 0, X₂) :|: 0 ≤ X₀

MPRF for transition t₃: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: 2+X₁ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂⋅X₂+3⋅X₂+2 {O(n^2)}

MPRF:

l2 [X₂-X₁-2 ]
l1 [X₂-X₁-1 ]
l3 [X₂-X₁-1 ]
l4 [X₂-X₁-1 ]

MPRF for transition t₄: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ X₁+1 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₂+4 {O(n)}

MPRF:

l2 [2 ]
l1 [2 ]
l3 [1 ]
l4 [1 ]

MPRF for transition t₅: l2(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂⋅X₂+3⋅X₂+2 {O(n^2)}

MPRF:

l2 [X₂-X₁-1 ]
l1 [X₂-X₁-1 ]
l3 [X₂-X₁-1 ]
l4 [X₂-X₁-1 ]

Analysing control-flow refined program

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l1___2

Found invariant 3 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l2___1

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l2___3

Found invariant X₂ ≤ 1+X₁ ∧ X₂ ≤ 2+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l7

Found invariant X₂ ≤ 1+X₁ ∧ X₂ ≤ 2+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l5

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location l4

Found invariant X₂ ≤ 1+X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l3

knowledge_propagation leads to new time bound X₂+2 {O(n)} for transition t₆₁: l1(X₀, X₁, X₂) → n_l2___3(X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ 2+X₁ ≤ X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀

knowledge_propagation leads to new time bound X₂+2 {O(n)} for transition t₆₃: n_l2___3(X₀, X₁, X₂) → n_l1___2(X₀, X₁+1, X₂) :|: 2 ≤ X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀

MPRF for transition t₆₀: n_l1___2(X₀, X₁, X₂) → n_l2___1(X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂⋅X₂+3⋅X₂+2 {O(n^2)}

MPRF:

n_l2___3 [0 ]
l4 [0 ]
l1 [0 ]
l3 [0 ]
n_l2___1 [X₂-X₁-1 ]
n_l1___2 [X₂-X₁ ]

MPRF for transition t₆₇: n_l1___2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ X₁+1 ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

3⋅X₂+8 {O(n)}

MPRF:

n_l2___3 [6-2⋅X₂ ]
l4 [6-2⋅X₂ ]
l1 [6-2⋅X₂ ]
l3 [0 ]
n_l2___1 [1 ]
n_l1___2 [1 ]

MPRF for transition t₆₂: n_l2___1(X₀, X₁, X₂) → n_l1___2(X₀, X₁+1, X₂) :|: 2+X₁ ≤ X₂ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₂⋅X₂+7⋅X₂+4 {O(n^2)}

MPRF:

n_l2___3 [X₂ ]
l4 [X₂ ]
l1 [X₂ ]
l3 [X₂ ]
n_l2___1 [2⋅X₂-X₁-1 ]
n_l1___2 [2⋅X₂-X₁-1 ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₂+10⋅X₂+15 {O(n^2)}
t₀: 1 {O(1)}
t₃: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₄: 2⋅X₂+4 {O(n)}
t₅: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₆: X₂+1 {O(n)}
t₈: 1 {O(1)}
t₂: X₂+2 {O(n)}
t₉: 1 {O(1)}
t₁: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂⋅X₂+10⋅X₂+15 {O(n^2)}
t₀: 1 {O(1)}
t₃: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₄: 2⋅X₂+4 {O(n)}
t₅: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₆: X₂+1 {O(n)}
t₈: 1 {O(1)}
t₂: X₂+2 {O(n)}
t₉: 1 {O(1)}
t₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₃, X₀: X₂+1 {O(n)}
t₃, X₁: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₂+1 {O(n)}
t₄, X₁: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₂+1 {O(n)}
t₅, X₁: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: X₂+1 {O(n)}
t₆, X₁: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₆, X₂: X₂ {O(n)}
t₈, X₀: X₂+1 {O(n)}
t₈, X₁: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₈, X₂: X₂ {O(n)}
t₂, X₀: X₂+1 {O(n)}
t₂, X₁: 0 {O(1)}
t₂, X₂: X₂ {O(n)}
t₉, X₀: X₂+1 {O(n)}
t₉, X₁: X₂⋅X₂+3⋅X₂+2 {O(n^2)}
t₉, X₂: X₂ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}