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

Preprocessing

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

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l7

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l5

Found invariant X₂ ≤ 1+X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1

Found invariant 1 ≤ X₀ for location l4

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

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

new bound:

X₁+2 {O(n)}

MPRF:

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

MPRF for transition t₇: l3(X₀, X₁, X₂) → l4(X₀+1, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁+2 {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, 1) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁+2 {O(n)}

MPRF:

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

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

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l7

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l5

Found invariant X₂ ≤ 1+X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1

Found invariant 1 ≤ X₀ for location l4

Found invariant X₂ ≤ 1+X₁ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l3

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

new bound:

X₁⋅X₁+7⋅X₁+10 {O(n^2)}

MPRF:

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

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

new bound:

X₁⋅X₁+7⋅X₁+10 {O(n^2)}

MPRF:

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

Analysing control-flow refined program

Cut unsatisfiable transition t₅: l1→l3

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

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 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 ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l2___3

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l7

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l5

Found invariant X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1

Found invariant 1 ≤ X₀ for location l4

Found invariant X₂ ≤ 1+X₁ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l3

knowledge_propagation leads to new time bound X₁+2 {O(n)} for transition t₇₆: l1(X₀, X₁, X₂) → n_l2___3(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀

knowledge_propagation leads to new time bound X₁+2 {O(n)} for transition t₇₈: n_l2___3(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀

MPRF for transition t₇₅: n_l1___2(X₀, X₁, X₂) → n_l2___1(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ 1+X₁ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁⋅X₁+8⋅X₁+13 {O(n^2)}

MPRF:

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

MPRF for transition t₈₂: n_l1___2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀+1 ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ 1+X₁ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁+2 {O(n)}

MPRF:

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

MPRF for transition t₇₇: n_l2___1(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₁⋅X₁+8⋅X₁+12 {O(n^2)}

MPRF:

n_l2___3 [0 ]
l4 [0 ]
l1 [0 ]
l3 [X₀+1-X₂ ]
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+17⋅X₁+30 {O(n^2)}
t₀: 1 {O(1)}
t₄: X₁⋅X₁+7⋅X₁+10 {O(n^2)}
t₅: X₁+2 {O(n)}
t₆: X₁⋅X₁+7⋅X₁+10 {O(n^2)}
t₇: X₁+2 {O(n)}
t₂: X₁+2 {O(n)}
t₃: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁⋅X₁+17⋅X₁+30 {O(n^2)}
t₀: 1 {O(1)}
t₄: X₁⋅X₁+7⋅X₁+10 {O(n^2)}
t₅: X₁+2 {O(n)}
t₆: X₁⋅X₁+7⋅X₁+10 {O(n^2)}
t₇: X₁+2 {O(n)}
t₂: X₁+2 {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+3 {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₁⋅X₁+7⋅X₁+11 {O(n^2)}
t₅, X₀: X₁+3 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₁⋅X₁+7⋅X₁+11 {O(n^2)}
t₆, X₀: X₁+3 {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₁⋅X₁+7⋅X₁+11 {O(n^2)}
t₇, X₀: X₁+3 {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₁⋅X₁+7⋅X₁+11 {O(n^2)}
t₂, X₀: X₁+3 {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 1 {O(1)}
t₃, X₀: X₁+4 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: X₁⋅X₁+7⋅X₁+X₂+11 {O(n^2)}
t₈, X₀: X₁+4 {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: X₁⋅X₁+7⋅X₁+X₂+11 {O(n^2)}
t₁, X₀: 1 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}