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 ≤ X₁
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₁+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, X₀) :|: 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 ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 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 ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 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 ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+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 ≤ X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₁+1 ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₇: l3(X₀, X₁, X₂) → l4(X₀+1, X₁, X₂) :|: X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+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, X₀) :|: 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) :|: X₁+1 ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 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 ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+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, X₀) :|: 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 ]
MPRF for transition t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁⋅X₁+9⋅X₁+10 {O(n^2)}
MPRF:
l2 [X₁-X₂ ]
l1 [X₁+1-X₂ ]
l3 [X₁-X₂ ]
l4 [-1 ]
MPRF for transition t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁⋅X₁+9⋅X₁+10 {O(n^2)}
MPRF:
l2 [X₁+1-X₂ ]
l1 [X₁+1-X₂ ]
l3 [X₁-X₂ ]
l4 [-1 ]
Analysing control-flow refined program
Cut unsatisfiable transition t₅: l1→l3
Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l1___2
Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l2___1
Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 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 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 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 ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+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 t₆₃: l1(X₀, X₁, X₂) → n_l2___3(X₀, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₁+2 {O(n)} for transition t₆₅: n_l2___3(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
MPRF for transition t₆₂: n_l1___2(X₀, X₁, X₂) → n_l2___1(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 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:
2⋅X₁⋅X₁+11⋅X₁+15 {O(n^2)}
MPRF:
n_l2___3 [-X₂ ]
l4 [-1 ]
l1 [-X₂ ]
l3 [X₁-X₂ ]
n_l2___1 [X₁+1-X₂ ]
n_l1___2 [X₁+2-X₂ ]
MPRF for transition t₆₉: n_l1___2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₁+1 ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ 1+X₁ ∧ 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:
l4 [X₁+1-X₀ ]
l1 [X₁+1-X₂ ]
l3 [X₁-X₀ ]
n_l2___1 [X₁+1-X₀ ]
n_l2___3 [X₁+1-X₀ ]
n_l1___2 [X₁+1-X₀ ]
MPRF for transition t₆₄: n_l2___1(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂+1) :|: X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁⋅X₁+10⋅X₁+13 {O(n^2)}
MPRF:
n_l2___3 [-X₂ ]
l4 [-X₀ ]
l1 [-X₀ ]
l3 [2⋅X₁-2⋅X₂ ]
n_l2___1 [X₁+1-X₂ ]
n_l1___2 [X₁+1-X₂ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:4⋅X₁⋅X₁+21⋅X₁+30 {O(n^2)}
t₀: 1 {O(1)}
t₄: 2⋅X₁⋅X₁+9⋅X₁+10 {O(n^2)}
t₅: X₁+2 {O(n)}
t₆: 2⋅X₁⋅X₁+9⋅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: 4⋅X₁⋅X₁+21⋅X₁+30 {O(n^2)}
t₀: 1 {O(1)}
t₄: 2⋅X₁⋅X₁+9⋅X₁+10 {O(n^2)}
t₅: X₁+2 {O(n)}
t₆: 2⋅X₁⋅X₁+9⋅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: 2⋅X₁⋅X₁+10⋅X₁+14 {O(n^2)}
t₅, X₀: X₁+3 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 2⋅X₁⋅X₁+10⋅X₁+14 {O(n^2)}
t₆, X₀: X₁+3 {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: 2⋅X₁⋅X₁+10⋅X₁+14 {O(n^2)}
t₇, X₀: X₁+3 {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 2⋅X₁⋅X₁+10⋅X₁+14 {O(n^2)}
t₂, X₀: X₁+3 {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₁+4 {O(n)}
t₃, X₀: X₁+4 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₁⋅X₁+10⋅X₁+X₂+14 {O(n^2)}
t₈, X₀: X₁+4 {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 2⋅X₁⋅X₁+10⋅X₁+X₂+14 {O(n^2)}
t₁, X₀: 1 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}