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) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₁) :|: X₁ ≤ X₀
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ < X₁
t₁: l2(X₀, X₁, X₂) → l1(X₀, 1, X₂)
t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ < X₂
t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₂ ≤ X₀
t₈: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₇: l5(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂)
t₆: l6(X₀, X₁, X₂) → l3(X₀, X₁, X₂+1)
Preprocessing
Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location l7
Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5
Found invariant 1 ≤ X₁ for location l1
Found invariant 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location l4
Found invariant X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ 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) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₁) :|: X₁ ≤ X₀ ∧ 1 ≤ X₁
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ < X₁ ∧ 1 ≤ X₁
t₁: l2(X₀, X₁, X₂) → l1(X₀, 1, X₂)
t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ < X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₈: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ 1+X₀ ≤ X₁
t₇: l5(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₆: l6(X₀, X₁, X₂) → l3(X₀, X₁, X₂+1) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₁) :|: X₁ ≤ X₀ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₀+2 {O(n)}
MPRF:
l5 [X₂-X₁-1 ]
l1 [X₀+1-X₁ ]
l6 [X₀-X₁ ]
l3 [X₀-X₁ ]
MPRF for transition t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ < X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+2 {O(n)}
MPRF:
l5 [X₀-X₁ ]
l1 [X₀+1-X₁ ]
l6 [X₀+1-X₁ ]
l3 [X₀+1-X₁ ]
MPRF for transition t₇: l5(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+2 {O(n)}
MPRF:
l5 [X₀+1-X₁ ]
l1 [X₀+1-X₁ ]
l6 [X₀+1-X₁ ]
l3 [X₀+1-X₁ ]
MPRF for transition t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀⋅X₀+3⋅X₀ {O(n^2)}
MPRF:
l1 [X₀ ]
l5 [X₀-X₂ ]
l6 [X₀-X₂ ]
l3 [X₀+1-X₂ ]
MPRF for transition t₆: l6(X₀, X₁, X₂) → l3(X₀, X₁, X₂+1) :|: X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀⋅X₀+3⋅X₀ {O(n^2)}
MPRF:
l1 [X₀ ]
l5 [X₀-X₂ ]
l6 [X₀+1-X₂ ]
l3 [X₀+1-X₂ ]
Analysing control-flow refined program
Cut unsatisfiable transition t₅: l3→l5
Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___3
Found invariant X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l6___1
Found invariant 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location l7
Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l5
Found invariant X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l3___2
Found invariant 1 ≤ X₁ for location l1
Found invariant 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location l4
Found invariant X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
knowledge_propagation leads to new time bound X₀+2 {O(n)} for transition t₆₁: l3(X₀, X₁, X₂) → n_l6___3(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₀+2 {O(n)} for transition t₆₃: n_l6___3(X₀, X₁, X₂) → n_l3___2(X₀, X₁, X₂+1) :|: X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₆₀: n_l3___2(X₀, X₁, X₂) → n_l6___1(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+11⋅X₀+14 {O(n^2)}
MPRF:
l3 [0 ]
n_l6___3 [0 ]
l1 [0 ]
l5 [X₀+1-X₂ ]
n_l6___1 [X₀+1-X₂ ]
n_l3___2 [X₀+2-X₂ ]
MPRF for transition t₆₇: n_l3___2(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ < X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₀+1 {O(n)}
MPRF:
l3 [2⋅X₀-X₂ ]
l1 [2⋅X₀-X₁ ]
l5 [2⋅X₀-X₁-1 ]
n_l6___1 [2⋅X₀-X₁ ]
n_l6___3 [2⋅X₀-X₁ ]
n_l3___2 [2⋅X₀-X₁ ]
MPRF for transition t₆₂: n_l6___1(X₀, X₁, X₂) → n_l3___2(X₀, X₁, X₂+1) :|: X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+11⋅X₀+12 {O(n^2)}
MPRF:
l3 [-X₀ ]
n_l6___3 [-X₀ ]
l1 [-X₀ ]
l5 [2-X₂ ]
n_l6___1 [X₀+1-X₂ ]
n_l3___2 [X₀+1-X₂ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:2⋅X₀⋅X₀+9⋅X₀+10 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₀+2 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: X₀⋅X₀+3⋅X₀ {O(n^2)}
t₅: X₀+2 {O(n)}
t₈: 1 {O(1)}
t₇: X₀+2 {O(n)}
t₆: X₀⋅X₀+3⋅X₀ {O(n^2)}
Costbounds
Overall costbound: 2⋅X₀⋅X₀+9⋅X₀+10 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₀+2 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: X₀⋅X₀+3⋅X₀ {O(n^2)}
t₅: X₀+2 {O(n)}
t₈: 1 {O(1)}
t₇: X₀+2 {O(n)}
t₆: X₀⋅X₀+3⋅X₀ {O(n^2)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₀+3 {O(n)}
t₂, X₂: X₀+4 {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: X₀+4 {O(n)}
t₃, X₂: X₀⋅X₀+4⋅X₀+X₂+4 {O(n^2)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 1 {O(1)}
t₁, X₂: X₂ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₀+3 {O(n)}
t₄, X₂: X₀⋅X₀+4⋅X₀+4 {O(n^2)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₀+3 {O(n)}
t₅, X₂: X₀⋅X₀+4⋅X₀+4 {O(n^2)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: X₀+4 {O(n)}
t₈, X₂: X₀⋅X₀+4⋅X₀+X₂+4 {O(n^2)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀+3 {O(n)}
t₇, X₂: X₀⋅X₀+4⋅X₀+4 {O(n^2)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₀+3 {O(n)}
t₆, X₂: X₀⋅X₀+4⋅X₀+4 {O(n^2)}