Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃)
t₄: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₃
t₅: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₃+1 ≤ X₂
t₆: l2(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+1, X₃)
t₇: l3(X₀, X₁, X₂, X₃) → l4(X₀+1, X₁, X₂, X₃)
t₂: l4(X₀, X₁, X₂, X₃) → l1(X₀, X₁, 1, X₃) :|: X₀ ≤ X₁
t₃: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁+1 ≤ X₀
t₈: l5(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₁: l6(X₀, X₁, X₂, X₃) → l4(1, X₁, X₂, X₃)
Preprocessing
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ 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 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1
Found invariant 1 ≤ X₀ for location l4
Found invariant 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃)
t₄: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₅: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₃+1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₆: l2(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+1, X₃) :|: 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₇: l3(X₀, X₁, X₂, X₃) → l4(X₀+1, X₁, X₂, X₃) :|: 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₂: l4(X₀, X₁, X₂, X₃) → l1(X₀, X₁, 1, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀
t₃: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁+1 ≤ X₀ ∧ 1 ≤ X₀
t₈: l5(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀
t₁: l6(X₀, X₁, X₂, X₃) → l4(1, X₁, X₂, X₃)
MPRF for transition t₅: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₃+1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ 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, X₃) → l4(X₀+1, X₁, X₂, X₃) :|: 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ 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, X₃) → l1(X₀, X₁, 1, X₃) :|: 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 ]
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ 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 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1
Found invariant 1 ≤ X₀ for location l4
Found invariant 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l3
Time-Bound by TWN-Loops:
TWN-Loops: t₄ 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {O(n^2)}
TWN-Loops:
entry: t₂: l4(X₀, X₁, X₂, X₃) → l1(X₀, X₁, 1, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀
results in twn-loop: twn:Inv: [1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀,X₁,X₂+1,X₃) :|: X₂ ≤ X₃
order: [X₀; X₁; X₂; X₃]
closed-form:
X₀: X₀
X₁: X₁
X₂: X₂ + [[n != 0]] * n^1
X₃: X₃
Termination: true
Formula:
1 < 0
∨ X₂ < X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂
Stabilization-Threshold for: X₂ ≤ X₃
alphas_abs: X₃
M: 0
N: 1
Bound: 2⋅X₃+2 {O(n)}
relevant size-bounds w.r.t. t₂:
X₃: X₃ {O(n)}
Runtime-bound of t₂: X₁+2 {O(n)}
Results in: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {O(n^2)}
2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {O(n^2)}
Time-Bound by TWN-Loops:
TWN-Loops: t₆ 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {O(n^2)}
relevant size-bounds w.r.t. t₂:
X₃: X₃ {O(n)}
Runtime-bound of t₂: X₁+2 {O(n)}
Results in: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {O(n^2)}
2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {O(n^2)}
Analysing control-flow refined program
Found invariant 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l1___2
Found invariant 2 ≤ X₃ ∧ 4 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l2___1
Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ 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 ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1
Found invariant 1 ≤ X₀ for location l4
Found invariant 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ 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 t₇₃: l1(X₀, X₁, X₂, X₃) → n_l2___3(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₁+2 {O(n)} for transition t₇₅: n_l2___3(X₀, X₁, X₂, X₃) → n_l1___2(X₀, X₁, X₂+1, X₃) :|: 1 ≤ X₃ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
MPRF for transition t₇₂: n_l1___2(X₀, X₁, X₂, X₃) → n_l2___1(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₂ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₁⋅X₃+2⋅X₃+3⋅X₁+6 {O(n^2)}
MPRF:
n_l2___3 [0 ]
l4 [0 ]
l1 [0 ]
l3 [0 ]
n_l2___1 [X₃-X₂ ]
n_l1___2 [X₃+1-X₂ ]
MPRF for transition t₇₉: n_l1___2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₃+1 ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ 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+X₂-X₀ ]
l3 [X₁-X₀ ]
n_l2___1 [X₁+1-X₀ ]
n_l2___3 [X₁+1-X₀ ]
n_l1___2 [X₁+1-X₀ ]
MPRF for transition t₇₄: n_l2___1(X₀, X₁, X₂, X₃) → n_l1___2(X₀, X₁, X₂+1, X₃) :|: X₂ ≤ X₃ ∧ 2 ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₃ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₃ ∧ 4 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₁⋅X₃+2⋅X₃+3⋅X₁+6 {O(n^2)}
MPRF:
n_l2___3 [0 ]
l4 [0 ]
l1 [0 ]
l3 [0 ]
n_l2___1 [X₃+1-X₂ ]
n_l1___2 [X₃+1-X₂ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:4⋅X₁⋅X₃+11⋅X₁+8⋅X₃+26 {O(n^2)}
t₀: 1 {O(1)}
t₄: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {O(n^2)}
t₅: X₁+2 {O(n)}
t₆: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {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⋅X₃+11⋅X₁+8⋅X₃+26 {O(n^2)}
t₀: 1 {O(1)}
t₄: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {O(n^2)}
t₅: X₁+2 {O(n)}
t₆: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+8 {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)}
t₀, X₃: X₃ {O(n)}
t₄, X₀: X₁+3 {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+9 {O(n^2)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₁+3 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+10 {O(n^2)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₁+3 {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+9 {O(n^2)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₁+3 {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+10 {O(n^2)}
t₇, X₃: X₃ {O(n)}
t₂, X₀: X₁+3 {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 1 {O(1)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₁+4 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+X₂+10 {O(n^2)}
t₃, X₃: 2⋅X₃ {O(n)}
t₈, X₀: X₁+4 {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 2⋅X₁⋅X₃+4⋅X₁+4⋅X₃+X₂+10 {O(n^2)}
t₈, X₃: 2⋅X₃ {O(n)}
t₁, X₀: 1 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}