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+1, X₁, X₂)
t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, 0) :|: 1+X₀ ≤ X₁
t₃: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ X₀
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₁: l6(X₀, X₁, X₂) → l4(0, X₁, X₂)

Preprocessing

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l2

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l7

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l5

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location l4

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

MPRF for transition t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l2 [X₁-X₀ ]
l3 [X₁-X₂ ]
l4 [X₁-X₀ ]
l1 [X₁-X₀ ]

MPRF for transition t₇: l3(X₀, X₁, X₂) → l4(X₀+1, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

l2 [X₁+1-X₀ ]
l3 [X₁+1-X₀ ]
l4 [X₁+1-X₀ ]
l1 [X₁+1-X₀ ]

MPRF for transition t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, 0) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l2 [X₁-X₀-1 ]
l3 [X₁-X₂ ]
l4 [X₁-X₀ ]
l1 [X₁-X₀-1 ]

MPRF for transition t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁⋅X₁+2⋅X₁ {O(n^2)}

MPRF:

l2 [X₀-X₂ ]
l1 [X₀+1-X₂ ]
l3 [X₀+1-X₂ ]
l4 [0 ]

MPRF for transition t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁⋅X₁+2⋅X₁ {O(n^2)}

MPRF:

l2 [X₀+1-X₂ ]
l1 [X₀+1-X₂ ]
l3 [0 ]
l4 [0 ]

Analysing control-flow refined program

Cut unsatisfiable transition t₅: l1→l3

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

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

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l2___3

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l7

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location l5

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Found invariant 0 ≤ X₀ for location l4

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l3

knowledge_propagation leads to new time bound X₁ {O(n)} for transition t₆₂: l1(X₀, X₁, X₂) → n_l2___3(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1+X₀ ≤ X₁ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀

knowledge_propagation leads to new time bound X₁ {O(n)} for transition t₆₄: n_l2___3(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1+X₀ ≤ X₁ ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀

MPRF for transition t₆₁: n_l1___2(X₀, X₁, X₂) → n_l2___1(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1+X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1+X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁⋅X₁+3⋅X₁ {O(n^2)}

MPRF:

n_l2___3 [X₁-X₀ ]
l4 [X₁-X₀ ]
l1 [X₁-X₀ ]
l3 [X₁-X₀ ]
n_l2___1 [X₁-X₂ ]
n_l1___2 [X₁+1-X₂ ]

MPRF for transition t₆₈: n_l1___2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l4 [X₁-X₀ ]
l1 [X₁-X₀ ]
l3 [X₁-X₀-1 ]
n_l2___1 [X₁-X₀ ]
n_l2___3 [X₁-X₀ ]
n_l1___2 [X₁-X₀ ]

MPRF for transition t₆₃: n_l2___1(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: 1+X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁⋅X₁+3⋅X₁ {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⋅X₁⋅X₁+7⋅X₁+5 {O(n^2)}
t₀: 1 {O(1)}
t₄: X₁⋅X₁+2⋅X₁ {O(n^2)}
t₅: X₁ {O(n)}
t₆: X₁⋅X₁+2⋅X₁ {O(n^2)}
t₇: X₁+1 {O(n)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁⋅X₁+7⋅X₁+5 {O(n^2)}
t₀: 1 {O(1)}
t₄: X₁⋅X₁+2⋅X₁ {O(n^2)}
t₅: X₁ {O(n)}
t₆: X₁⋅X₁+2⋅X₁ {O(n^2)}
t₇: X₁+1 {O(n)}
t₂: X₁ {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+1 {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₁⋅X₁+2⋅X₁ {O(n^2)}
t₅, X₀: X₁+1 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₁⋅X₁+2⋅X₁ {O(n^2)}
t₆, X₀: X₁+1 {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₁⋅X₁+2⋅X₁ {O(n^2)}
t₇, X₀: X₁+1 {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₁⋅X₁+2⋅X₁ {O(n^2)}
t₂, X₀: X₁+1 {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 0 {O(1)}
t₃, X₀: X₁+1 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: X₁⋅X₁+2⋅X₁+X₂ {O(n^2)}
t₈, X₀: X₁+1 {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: X₁⋅X₁+2⋅X₁+X₂ {O(n^2)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}