Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 < X₂
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₁, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ 0 < X₀
t₅: l3(X₀, X₁, X₂, X₃) → l1(X₁, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₀ ≤ 0
t₆: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ 0 < X₀
t₇: l3(X₀, X₁, X₂, X₃) → l1(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ 0
t₈: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
Preprocessing
Cut unsatisfiable transition t₅: l3→l1
Cut unsatisfiable transition t₆: l3→l1
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l5
Found invariant X₂ ≤ X₁ for location l1
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l4
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 < X₂ ∧ X₂ ≤ X₁
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0 ∧ X₂ ≤ X₁
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₁, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₇: l3(X₀, X₁, X₂, X₃) → l1(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁
t₈: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0 ∧ X₂ ≤ X₁
MPRF for transition t₇: l3(X₀, X₁, X₂, X₃) → l1(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₁ {O(n)}
MPRF:
l3 [X₂ ]
l1 [X₂ ]
MPRF for transition t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃) :|: 0 < X₀ ∧ 0 < X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ of depth 1:
new bound:
X₁⋅X₁+X₁+X₃+1 {O(n^2)}
MPRF:
l3 [X₀+1 ]
l1 [X₀+1 ]
knowledge_propagation leads to new time bound X₁⋅X₁+2⋅X₁+X₃+2 {O(n^2)} for transition t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 < X₂ ∧ X₂ ≤ X₁
Analysing control-flow refined program
Cut unsatisfiable transition t₇₁: n_l1___2→l4
Cut unsatisfiable transition t₇₃: n_l1___6→l4
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l1___6
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l3___4
Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ for location n_l3___3
Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l1___2
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l5
Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l1___5
Found invariant X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ for location l1
Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location l4
Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___1
Found invariant X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₁ for location n_l3___7
MPRF for transition t₅₅: n_l1___2(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ 0 < X₂ ∧ 1 ≤ X₂ ∧ 0 < X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 0 < X₂ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁+1 {O(n)}
MPRF:
n_l3___1 [X₂-1 ]
n_l3___3 [X₂ ]
n_l1___2 [X₂ ]
n_l1___5 [X₂ ]
n_l3___4 [X₂-1 ]
n_l1___6 [X₂-1 ]
MPRF for transition t₅₆: n_l1___5(X₀, X₁, X₂, X₃) → n_l3___3(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ 0 < X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1+X₂ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 < X₂ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁+1 {O(n)}
MPRF:
n_l3___1 [X₂ ]
n_l3___3 [X₂ ]
n_l1___2 [X₂ ]
n_l1___5 [X₂+1 ]
n_l3___4 [X₂ ]
n_l1___6 [X₂ ]
MPRF for transition t₅₉: n_l3___1(X₀, X₁, X₂, X₃) → n_l1___6(X₀-1, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 < X₀ ∧ 0 < X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁+1 {O(n)}
MPRF:
n_l3___1 [X₂ ]
n_l3___3 [X₂ ]
n_l1___2 [X₂ ]
n_l1___5 [X₂ ]
n_l3___4 [X₂-1 ]
n_l1___6 [X₂-1 ]
MPRF for transition t₆₀: n_l3___3(X₀, X₁, X₂, X₃) → n_l1___2(X₀-1, X₁, X₂, X₃) :|: 1+X₂ ≤ X₀ ∧ 0 < X₂ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₁+1 {O(n)}
MPRF:
n_l3___1 [X₂-1 ]
n_l3___3 [X₂ ]
n_l1___2 [X₂-1 ]
n_l1___5 [X₂ ]
n_l3___4 [X₂-1 ]
n_l1___6 [X₂-1 ]
MPRF for transition t₆₁: n_l3___4(X₀, X₁, X₂, X₃) → n_l1___5(X₁, X₁, X₂-1, X₃) :|: X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₀ ≤ 0 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
2⋅X₁ {O(n)}
MPRF:
n_l3___1 [X₂ ]
n_l3___3 [X₂ ]
n_l1___2 [X₂ ]
n_l1___5 [X₂ ]
n_l3___4 [X₂ ]
n_l1___6 [X₂ ]
MPRF for transition t₅₇: n_l1___6(X₀, X₁, X₂, X₃) → n_l3___4(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ 0 < X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 0 < X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
4⋅X₁⋅X₁+X₁+X₃+1 {O(n^2)}
MPRF:
n_l1___5 [X₁ ]
n_l3___1 [X₀ ]
n_l3___3 [X₁ ]
n_l1___2 [X₀ ]
n_l3___4 [X₀ ]
n_l1___6 [X₀+1 ]
MPRF for transition t₆₂: n_l3___4(X₀, X₁, X₂, X₃) → n_l1___6(X₀-1, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 < X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
8⋅X₁⋅X₁+4⋅X₁+X₃+1 {O(n^2)}
MPRF:
n_l1___5 [X₀+X₁ ]
n_l3___1 [2⋅X₀ ]
n_l3___3 [X₀+X₁ ]
n_l1___2 [2⋅X₀ ]
n_l3___4 [X₀+X₁+1-X₂ ]
n_l1___6 [X₀+X₁+1-X₂ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:2⋅X₁⋅X₁+2⋅X₃+4⋅X₁+7 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₁⋅X₁+2⋅X₁+X₃+2 {O(n^2)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: X₁⋅X₁+X₁+X₃+1 {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}
Costbounds
Overall costbound: 2⋅X₁⋅X₁+2⋅X₃+4⋅X₁+7 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₁⋅X₁+2⋅X₁+X₃+2 {O(n^2)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: X₁⋅X₁+X₁+X₃+1 {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₂, X₀: X₁+X₃ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₁ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₁+X₃ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₁ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₁ {O(n)}
t₁, X₃: X₃ {O(n)}
t₄, X₀: X₁+X₃ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₁ {O(n)}
t₄, X₃: X₃ {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₁ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₁+X₃ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 2⋅X₁ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}