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, 1, X₂) :|: X₀ ≤ X₂
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₂ < X₀
t₁: l2(X₀, X₁, X₂) → l1(1, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ < X₁
t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₁ ≤ X₂
t₈: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₇: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂)
t₆: l6(X₀, X₁, X₂) → l3(X₀, X₁+1, X₂)
Preprocessing
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l7
Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l5
Found invariant 1 ≤ X₀ for location l1
Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l4
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 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, 1, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₂ < X₀ ∧ 1 ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(1, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ < X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₈: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₀
t₇: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀
t₆: l6(X₀, X₁, X₂) → l3(X₀, X₁+1, X₂) :|: 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₂: l1(X₀, X₁, X₂) → l3(X₀, 1, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂+3 {O(n)}
MPRF:
l5 [X₁-X₀ ]
l1 [X₂+2-X₀ ]
l6 [X₂+1-X₀ ]
l3 [X₂+1-X₀ ]
MPRF for transition t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ < X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂+2 {O(n)}
MPRF:
l5 [X₂-X₀ ]
l1 [X₂+1-X₀ ]
l6 [X₂+1-X₀ ]
l3 [X₂+1-X₀ ]
MPRF for transition t₇: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂+2 {O(n)}
MPRF:
l5 [X₂+1-X₀ ]
l1 [X₂+1-X₀ ]
l6 [X₂+1-X₀ ]
l3 [X₂+1-X₀ ]
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l6
Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l7
Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l5
Found invariant 1 ≤ X₀ for location l1
Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l4
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
Time-Bound by TWN-Loops:
TWN-Loops: t₄ 2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
TWN-Loops:
entry: t₂: l1(X₀, X₁, X₂) → l3(X₀, 1, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀
results in twn-loop: twn:Inv: [1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁+1,X₂) :|: X₁ ≤ X₂
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
X₂: X₂
Termination: true
Formula:
1 < 0
∨ X₁ < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁
Stabilization-Threshold for: X₁ ≤ X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
relevant size-bounds w.r.t. t₂:
X₂: X₂ {O(n)}
Runtime-bound of t₂: X₂+3 {O(n)}
Results in: 2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
Time-Bound by TWN-Loops:
TWN-Loops: t₆ 2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
relevant size-bounds w.r.t. t₂:
X₂: X₂ {O(n)}
Runtime-bound of t₂: X₂+3 {O(n)}
Results in: 2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
Analysing control-flow refined program
Cut unsatisfiable transition t₅: l3→l5
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___3
Found invariant 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l6___1
Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l7
Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l5
Found invariant 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l3___2
Found invariant 1 ≤ X₀ for location l1
Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l4
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3
knowledge_propagation leads to new time bound X₂+3 {O(n)} for transition t₇₃: l3(X₀, X₁, X₂) → n_l6___3(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₂+3 {O(n)} for transition t₇₅: n_l6___3(X₀, X₁, X₂) → n_l3___2(X₀, X₁+1, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₇₂: n_l3___2(X₀, X₁, X₂) → n_l6___1(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 2 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂⋅X₂+6⋅X₂+9 {O(n^2)}
MPRF:
l3 [0 ]
n_l6___3 [0 ]
l1 [0 ]
l5 [X₂+1-X₁ ]
n_l6___1 [X₂-X₁ ]
n_l3___2 [X₂+1-X₁ ]
MPRF for transition t₇₉: n_l3___2(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ < X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂+2 {O(n)}
MPRF:
l3 [X₁+X₂-X₀ ]
l1 [X₂+1-X₀ ]
l5 [X₂-X₀ ]
n_l6___1 [X₂+1-X₀ ]
n_l6___3 [X₁+X₂-X₀ ]
n_l3___2 [X₂+1-X₀ ]
MPRF for transition t₇₄: n_l6___1(X₀, X₁, X₂) → n_l3___2(X₀, X₁+1, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
5⋅X₂⋅X₂+26⋅X₂+30 {O(n^2)}
MPRF:
l3 [2⋅X₂-2⋅X₀ ]
n_l6___3 [2⋅X₂-2⋅X₀ ]
l1 [2⋅X₂+1-2⋅X₀ ]
l5 [3⋅X₂-2⋅X₀-X₁ ]
n_l6___1 [3⋅X₂+1-2⋅X₀-X₁ ]
n_l3___2 [3⋅X₂+1-2⋅X₀-X₁ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:4⋅X₂⋅X₂+23⋅X₂+35 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₂+3 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
t₅: X₂+2 {O(n)}
t₈: 1 {O(1)}
t₇: X₂+2 {O(n)}
t₆: 2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
Costbounds
Overall costbound: 4⋅X₂⋅X₂+23⋅X₂+35 {O(n^2)}
t₀: 1 {O(1)}
t₂: X₂+3 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
t₅: X₂+2 {O(n)}
t₈: 1 {O(1)}
t₇: X₂+2 {O(n)}
t₆: 2⋅X₂⋅X₂+10⋅X₂+12 {O(n^2)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₂, X₀: X₂+3 {O(n)}
t₂, X₁: 1 {O(1)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₂+4 {O(n)}
t₃, X₁: 2⋅X₂⋅X₂+10⋅X₂+X₁+13 {O(n^2)}
t₃, X₂: 2⋅X₂ {O(n)}
t₁, X₀: 1 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₄, X₀: X₂+3 {O(n)}
t₄, X₁: 2⋅X₂⋅X₂+10⋅X₂+13 {O(n^2)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₂+3 {O(n)}
t₅, X₁: 2⋅X₂⋅X₂+10⋅X₂+13 {O(n^2)}
t₅, X₂: X₂ {O(n)}
t₈, X₀: X₂+4 {O(n)}
t₈, X₁: 2⋅X₂⋅X₂+10⋅X₂+X₁+13 {O(n^2)}
t₈, X₂: 2⋅X₂ {O(n)}
t₇, X₀: X₂+3 {O(n)}
t₇, X₁: 2⋅X₂⋅X₂+10⋅X₂+13 {O(n^2)}
t₇, X₂: X₂ {O(n)}
t₆, X₀: X₂+3 {O(n)}
t₆, X₁: 2⋅X₂⋅X₂+10⋅X₂+13 {O(n^2)}
t₆, X₂: X₂ {O(n)}