Initial Problem
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁) → l1(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀-1, X₁+X₀) :|: 1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ 0
t₃: l2(X₀, X₁) → l2(X₀, X₁-1) :|: 1 ≤ X₁
Preprocessing
Found invariant X₀ ≤ 0 for location l2
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁) → l1(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀-1, X₁+X₀) :|: 1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ 0
t₃: l2(X₀, X₁) → l2(X₀, X₁-1) :|: 1 ≤ X₁ ∧ X₀ ≤ 0
Found invariant X₀ ≤ 0 for location l2
Time-Bound by TWN-Loops:
TWN-Loops: t₁ 2⋅X₀+4 {O(n)}
TWN-Loops:
entry: t₀: l0(X₀, X₁) → l1(X₀, X₁)
results in twn-loop: twn: (X₀,X₁) -> (X₀-1,X₁+X₀) :|: 1 ≤ X₀
order: [X₀]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
Stabilization-Threshold for: 1 ≤ X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
relevant size-bounds w.r.t. t₀:
X₀: X₀ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₀+4 {O(n)}
2⋅X₀+4 {O(n)}
Found invariant X₀ ≤ 0 for location l2
Time-Bound by TWN-Loops:
TWN-Loops: t₃ 8⋅X₀⋅X₀+20⋅X₀+4⋅X₁+4 {O(n^2)}
TWN-Loops:
entry: t₂: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ 0
results in twn-loop: twn:Inv: [X₀ ≤ 0] , (X₀,X₁) -> (X₀,X₁-1) :|: 1 ≤ X₁
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1
Stabilization-Threshold for: 1 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
relevant size-bounds w.r.t. t₂:
X₁: 4⋅X₀⋅X₀+10⋅X₀+2⋅X₁ {O(n^2)}
Runtime-bound of t₂: 1 {O(1)}
Results in: 8⋅X₀⋅X₀+20⋅X₀+4⋅X₁+4 {O(n^2)}
8⋅X₀⋅X₀+20⋅X₀+4⋅X₁+4 {O(n^2)}
Analysing control-flow refined program
Found invariant X₀ ≤ 0 for location l2
Found invariant 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l2___1
Found invariant X₀ ≤ 0 for location l2
Found invariant 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 for location n_l2___1
Time-Bound by TWN-Loops:
TWN-Loops: t₄₁ 16⋅X₀⋅X₀+40⋅X₀+8⋅X₁+7 {O(n^2)}
TWN-Loops:
entry: t₄₂: l2(X₀, X₁) → n_l2___1(X₀, X₁-1) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ 0
results in twn-loop: twn:Inv: [0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0] , (X₀,X₁) -> (X₀,X₁-1) :|: X₀ ≤ 0 ∧ X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ X₀ ≤ 0
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
X₀ < 0 ∧ 1 < 0
∨ X₀ < 0 ∧ 1 < 0 ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀ < 0 ∧ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ X₀ < 0 ∧ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₀ < 0 ∧ 1 < X₁ ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀ < 0 ∧ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ X₀ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 < 0
∨ X₀ < 0 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀ < 0 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 < 0
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 < 0 ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 < X₁ ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 < 0
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
Stabilization-Threshold for: 1 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
Stabilization-Threshold for: 0 ≤ X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
relevant size-bounds w.r.t. t₄₂:
X₁: 4⋅X₀⋅X₀+10⋅X₀+2⋅X₁ {O(n^2)}
Runtime-bound of t₄₂: 1 {O(1)}
Results in: 16⋅X₀⋅X₀+40⋅X₀+8⋅X₁+7 {O(n^2)}
16⋅X₀⋅X₀+40⋅X₀+8⋅X₁+7 {O(n^2)}
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:8⋅X₀⋅X₀+22⋅X₀+4⋅X₁+10 {O(n^2)}
t₀: 1 {O(1)}
t₁: 2⋅X₀+4 {O(n)}
t₂: 1 {O(1)}
t₃: 8⋅X₀⋅X₀+20⋅X₀+4⋅X₁+4 {O(n^2)}
Costbounds
Overall costbound: 8⋅X₀⋅X₀+22⋅X₀+4⋅X₁+10 {O(n^2)}
t₀: 1 {O(1)}
t₁: 2⋅X₀+4 {O(n)}
t₂: 1 {O(1)}
t₃: 8⋅X₀⋅X₀+20⋅X₀+4⋅X₁+4 {O(n^2)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 4⋅X₀⋅X₀+10⋅X₀+X₁ {O(n^2)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 4⋅X₀⋅X₀+10⋅X₀+2⋅X₁ {O(n^2)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 4⋅X₀⋅X₀+10⋅X₀+2⋅X₁ {O(n^2)}