Initial Problem
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀, X₁+1) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁+1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁
t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 1 ≤ X₀
Preprocessing
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀, X₁+1) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁+1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 1 ≤ X₀
MPRF for transition t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
MPRF for transition t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
TWN: t₁: l1→l1
cycle: [t₁: l1→l1]
loop: (1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁+1 ≤ X₀,(X₀,X₁) -> (X₀,X₁+1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
Termination: true
Formula:
1 < 0 ∧ 0 < 1 ∧ 1 < X₀
∨ 1 < 0 ∧ 0 < 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ 1 < 0 ∧ 0 < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < X₀
∨ 1 < 0 ∧ 0 < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1 < X₀
∨ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₁+1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1 ∧ 1 < X₀
∨ X₁+1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₁+1 < X₀ ∧ 0 < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < X₀
∨ X₁+1 < X₀ ∧ 0 < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₁+1 < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1 < X₀
∨ X₁+1 < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ 0 < 1 ∧ 1 < X₀
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ 0 < 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ 0 < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 < X₀
∨ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ 0 < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1 < X₀
∨ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
Stabilization-Threshold for: X₁+1 ≤ X₀
alphas_abs: X₁+1+X₀
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+4 {O(n)}
Stabilization-Threshold for: 0 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₀+4⋅X₁+9 {O(n)}
relevant size-bounds w.r.t. t₀:
X₀: X₀ {O(n)}
X₁: 0 {O(1)}
Runtime-bound of t₀: X₀ {O(n)}
Results in: 2⋅X₀⋅X₀+9⋅X₀ {O(n^2)}
Chain transitions t₂: l1→l2 and t₀: l2→l1 to t₂₇: l1→l1
Chain transitions t₃: l0→l2 and t₀: l2→l1 to t₂₈: l0→l1
Analysing control-flow refined program
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1
knowledge_propagation leads to new time bound 2⋅X₀⋅X₀+9⋅X₀ {O(n^2)} for transition t₂₇: l1(X₀, X₁) -{2}> l1(X₀-1, 0) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
Analysing control-flow refined program
Cut unsatisfiable transition t₂: l1→l2
Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1
Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___1
knowledge_propagation leads to new time bound X₀ {O(n)} for transition t₅₃: l1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₅₂: n_l1___1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀⋅X₀+2⋅X₀ {O(n^2)}
MPRF for transition t₅₆: n_l1___1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:2⋅X₀⋅X₀+11⋅X₀+1 {O(n^2)}
t₀: X₀ {O(n)}
t₁: 2⋅X₀⋅X₀+9⋅X₀ {O(n^2)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
Costbounds
Overall costbound: 2⋅X₀⋅X₀+11⋅X₀+1 {O(n^2)}
t₀: X₀ {O(n)}
t₁: 2⋅X₀⋅X₀+9⋅X₀ {O(n^2)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: 0 {O(1)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 2⋅X₀⋅X₀+9⋅X₀ {O(n^2)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: 2⋅X₀⋅X₀+9⋅X₀ {O(n^2)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}