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 ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -X₁-1, X₂, X₃) :|: 0 ≤ X₁
t₅: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -X₁, X₂, X₃) :|: X₁ < 0
t₆: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
Preprocessing
Found invariant 1+X₀ ≤ 0 for location l5
Found invariant 1+X₀ ≤ 0 for location l4
Found invariant 0 ≤ X₀ 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 ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -X₁-1, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀
t₅: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -X₁, X₂, X₃) :|: X₁ < 0 ∧ 0 ≤ X₀
t₆: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0
MPRF for transition t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -X₁-1, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
2⋅X₂+X₃+1 {O(n)}
MPRF:
l3 [2⋅X₀+X₁+1 ]
l1 [2⋅X₀+X₁+1 ]
MPRF for transition t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ of depth 1:
new bound:
10⋅X₂⋅X₃⋅X₃+16⋅X₂⋅X₂⋅X₃+2⋅X₃⋅X₃⋅X₃+8⋅X₂⋅X₂⋅X₂+22⋅X₂⋅X₂+29⋅X₂⋅X₃+9⋅X₃⋅X₃+12⋅X₃+20⋅X₂+7 {O(n^3)}
MPRF:
l3 [X₀+1 ]
l1 [X₀+2 ]
MPRF for transition t₅: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, -X₁, X₂, X₃) :|: X₁ < 0 ∧ 0 ≤ X₀ of depth 1:
new bound:
2⋅X₃⋅X₃+4⋅X₂⋅X₂+6⋅X₂⋅X₃+5⋅X₃+6⋅X₂+3 {O(n^2)}
MPRF:
l3 [1-X₁ ]
l1 [1-X₁ ]
knowledge_propagation leads to new time bound 2⋅X₃⋅X₃+4⋅X₂⋅X₂+6⋅X₂⋅X₃+6⋅X₃+8⋅X₂+5 {O(n^2)} for transition t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀
Analysing control-flow refined program
Cut unsatisfiable transition t₆₉: n_l1___6→l4
Found invariant 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location n_l1___6
Found invariant 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location n_l3___4
Found invariant 1 ≤ X₁+X₃ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀+X₁ for location n_l1___3
Found invariant 1+X₀ ≤ 0 for location l5
Found invariant 1+X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₁ ∧ X₁+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀+X₁ for location n_l1___5
Found invariant 1 ≤ X₁+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l3___2
Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1
Found invariant 1+X₀ ≤ 0 for location l4
Found invariant 1+X₃ ≤ 0 ∧ 2+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₁ ∧ X₁+X₃ ≤ 0 ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l3___1
Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location n_l3___7
MPRF for transition t₅₂: n_l1___3(X₀, X₁, X₂, X₃) → n_l3___2(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀+X₁ of depth 1:
new bound:
2⋅X₂+5⋅X₃+4 {O(n)}
MPRF:
n_l3___2 [X₀ ]
n_l1___6 [X₀+X₁+1 ]
n_l3___4 [X₀+X₁+1 ]
n_l1___3 [X₀+1 ]
MPRF for transition t₅₄: n_l1___6(X₀, X₁, X₂, X₃) → n_l3___4(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₁ < 0 ∧ 0 ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 1+X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
2⋅X₂+5⋅X₃+6 {O(n)}
MPRF:
n_l3___2 [X₀+1 ]
n_l1___6 [X₀+X₁+2 ]
n_l3___4 [X₀+X₁+1 ]
n_l1___3 [X₀+1 ]
MPRF for transition t₅₇: n_l3___2(X₀, X₁, X₂, X₃) → n_l1___6(X₀+X₁, -X₁-1, X₂, X₃) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
2⋅X₂+5⋅X₃+4 {O(n)}
MPRF:
n_l3___2 [X₀+1 ]
n_l1___6 [X₀+X₁+1 ]
n_l3___4 [X₀+X₁+1 ]
n_l1___3 [X₀+1 ]
MPRF for transition t₅₈: n_l3___4(X₀, X₁, X₂, X₃) → n_l1___3(X₀+X₁, -X₁, X₂, X₃) :|: 1+X₁ ≤ 0 ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ ∧ X₁ < 0 ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
2⋅X₂+5⋅X₃+6 {O(n)}
MPRF:
n_l3___2 [X₀+1 ]
n_l1___6 [X₀+X₁+2 ]
n_l3___4 [X₀+X₁+2 ]
n_l1___3 [X₀+1 ]
CFR: Improvement to new bound with the following program:
new bound:
20⋅X₃+8⋅X₂+20 {O(n)}
cfr-program:
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l4, l5, n_l1___3, n_l1___5, n_l1___6, n_l3___1, n_l3___2, n_l3___4, n_l3___7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0 ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₅₅: l1(X₀, X₁, X₂, X₃) → n_l3___7(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₃ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₆: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ 1+X₀ ≤ 0
t₆₇: n_l1___3(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0 ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀+X₁
t₅₂: n_l1___3(X₀, X₁, X₂, X₃) → n_l3___2(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀+X₁
t₆₈: n_l1___5(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0 ∧ 1+X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₁ ∧ X₁+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀+X₁
t₅₃: n_l1___5(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₁ ∧ X₁+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀+X₁
t₅₄: n_l1___6(X₀, X₁, X₂, X₃) → n_l3___4(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₁ < 0 ∧ 0 ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 1+X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀
t₅₆: n_l3___1(X₀, X₁, X₂, X₃) → n_l1___6(X₀+X₁, -X₁-1, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ 1+X₃ ≤ 0 ∧ 2+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₁ ∧ X₁+X₃ ≤ 0 ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₅₇: n_l3___2(X₀, X₁, X₂, X₃) → n_l1___6(X₀+X₁, -X₁-1, X₂, X₃) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₅₈: n_l3___4(X₀, X₁, X₂, X₃) → n_l1___3(X₀+X₁, -X₁, X₂, X₃) :|: 1+X₁ ≤ 0 ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ ∧ X₁ < 0 ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀
t₅₉: n_l3___7(X₀, X₁, X₂, X₃) → n_l1___5(X₀+X₁, -X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₁ < 0 ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀
t₆₀: n_l3___7(X₀, X₁, X₂, X₃) → n_l1___6(X₀+X₁, -X₁-1, X₂, X₃) :|: 0 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀
All Bounds
Timebounds
Overall timebound:20⋅X₃+8⋅X₂+31 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₅₅: 1 {O(1)}
t₁: 1 {O(1)}
t₆: 1 {O(1)}
t₅₂: 2⋅X₂+5⋅X₃+4 {O(n)}
t₆₇: 1 {O(1)}
t₅₃: 1 {O(1)}
t₆₈: 1 {O(1)}
t₅₄: 2⋅X₂+5⋅X₃+6 {O(n)}
t₅₆: 1 {O(1)}
t₅₇: 2⋅X₂+5⋅X₃+4 {O(n)}
t₅₈: 2⋅X₂+5⋅X₃+6 {O(n)}
t₅₉: 1 {O(1)}
t₆₀: 1 {O(1)}
Costbounds
Overall costbound: 20⋅X₃+8⋅X₂+31 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₅₅: 1 {O(1)}
t₁: 1 {O(1)}
t₆: 1 {O(1)}
t₅₂: 2⋅X₂+5⋅X₃+4 {O(n)}
t₆₇: 1 {O(1)}
t₅₃: 1 {O(1)}
t₆₈: 1 {O(1)}
t₅₄: 2⋅X₂+5⋅X₃+6 {O(n)}
t₅₆: 1 {O(1)}
t₅₇: 2⋅X₂+5⋅X₃+4 {O(n)}
t₅₈: 2⋅X₂+5⋅X₃+6 {O(n)}
t₅₉: 1 {O(1)}
t₆₀: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₅₅, X₀: X₂ {O(n)}
t₅₅, X₁: X₃ {O(n)}
t₅₅, X₂: X₂ {O(n)}
t₅₅, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₆, X₀: 2⋅X₃⋅X₃+4⋅X₂⋅X₂+6⋅X₂⋅X₃+10⋅X₂+7⋅X₃+3 {O(n^2)}
t₆, X₁: 2⋅X₂+9⋅X₃+6 {O(n)}
t₆, X₂: 4⋅X₂ {O(n)}
t₆, X₃: 4⋅X₃ {O(n)}
t₅₂, X₀: 24⋅X₂⋅X₃+35⋅X₃⋅X₃+4⋅X₂⋅X₂+24⋅X₂+68⋅X₃+31 {O(n^2)}
t₅₂, X₁: 2⋅X₂+7⋅X₃+6 {O(n)}
t₅₂, X₂: 2⋅X₂ {O(n)}
t₅₂, X₃: 2⋅X₃ {O(n)}
t₆₇, X₀: 24⋅X₂⋅X₃+35⋅X₃⋅X₃+4⋅X₂⋅X₂+24⋅X₂+68⋅X₃+31 {O(n^2)}
t₆₇, X₁: 2⋅X₂+7⋅X₃+6 {O(n)}
t₆₇, X₂: 2⋅X₂ {O(n)}
t₆₇, X₃: 2⋅X₃ {O(n)}
t₅₃, X₀: X₂+X₃ {O(n)}
t₅₃, X₁: X₃ {O(n)}
t₅₃, X₂: X₂ {O(n)}
t₅₃, X₃: X₃ {O(n)}
t₆₈, X₀: X₂+X₃ {O(n)}
t₆₈, X₁: X₃ {O(n)}
t₆₈, X₂: X₂ {O(n)}
t₆₈, X₃: X₃ {O(n)}
t₅₄, X₀: 24⋅X₂⋅X₃+35⋅X₃⋅X₃+4⋅X₂⋅X₂+24⋅X₂+68⋅X₃+31 {O(n^2)}
t₅₄, X₁: 2⋅X₂+7⋅X₃+6 {O(n)}
t₅₄, X₂: 2⋅X₂ {O(n)}
t₅₄, X₃: 2⋅X₃ {O(n)}
t₅₆, X₀: 2⋅X₃+X₂ {O(n)}
t₅₆, X₁: X₃+1 {O(n)}
t₅₆, X₂: X₂ {O(n)}
t₅₆, X₃: X₃ {O(n)}
t₅₇, X₀: 24⋅X₂⋅X₃+35⋅X₃⋅X₃+4⋅X₂⋅X₂+24⋅X₂+68⋅X₃+31 {O(n^2)}
t₅₇, X₁: 2⋅X₂+7⋅X₃+6 {O(n)}
t₅₇, X₂: 2⋅X₂ {O(n)}
t₅₇, X₃: 2⋅X₃ {O(n)}
t₅₈, X₀: 24⋅X₂⋅X₃+35⋅X₃⋅X₃+4⋅X₂⋅X₂+24⋅X₂+68⋅X₃+31 {O(n^2)}
t₅₈, X₁: 2⋅X₂+7⋅X₃+6 {O(n)}
t₅₈, X₂: 2⋅X₂ {O(n)}
t₅₈, X₃: 2⋅X₃ {O(n)}
t₅₉, X₀: X₂+X₃ {O(n)}
t₅₉, X₁: X₃ {O(n)}
t₅₉, X₂: X₂ {O(n)}
t₅₉, X₃: X₃ {O(n)}
t₆₀, X₀: X₂+X₃ {O(n)}
t₆₀, X₁: X₃+1 {O(n)}
t₆₀, X₂: X₂ {O(n)}
t₆₀, X₃: X₃ {O(n)}