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 ∧ 1 ≤ X₁
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ 0
t₀: l2(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀
Preprocessing
Found invariant 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 ∧ 1 ≤ X₁ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀
t₀: l2(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀
Analysing control-flow refined program
Found invariant 1 ≤ X₀ for location n_l1___4
Found invariant 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___3
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location n_l2___2
Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location n_l1___1
Found invariant 1 ≤ X₀ for location n_l1___4
Found invariant 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___3
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location n_l2___2
Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location n_l1___1
Time-Bound by TWN-Loops:
TWN-Loops: t₈₆ 4⋅X₁+7 {O(n)}
TWN-Loops:
entry: t₈₈: n_l1___4(X₀, X₁) → n_l1___3(X₀, X₁-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀
results in twn-loop: twn:Inv: [0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀] , (X₀,X₁) -> (X₀,X₁-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < X₀ ∧ 1 < 0
∨ 1 < 0 ∧ 1 < X₀ ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < 0 ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 < 0
∨ 1 < 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < X₀ ∧ 1 < 0
∨ 1 < X₁ ∧ 1 < X₀ ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < X₁ ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 < 0
∨ 1 < X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 < X₀ ∧ 1 < 0
∨ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 < X₀ ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 < 0
∨ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 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. t₈₈:
X₁: X₁ {O(n)}
Runtime-bound of t₈₈: 1 {O(1)}
Results in: 4⋅X₁+7 {O(n)}
4⋅X₁+7 {O(n)}
Found invariant 1 ≤ X₀ for location n_l1___4
Found invariant 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___3
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ for location n_l2___2
Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location n_l1___1
Found invariant 1 ≤ X₀ for location n_l1___4
Found invariant 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___3
Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location n_l2___2
Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___1
Time-Bound by TWN-Loops:
TWN-Loops: t₈₅ 12⋅X₀+16 {O(n)}
TWN-Loops:
entry: t₈₉: n_l1___4(X₀, X₁) → n_l2___2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀
results in twn-loop: twn:Inv: [X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀] , (X₀,X₁) -> (X₀-1,X₁) :|: X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0
entry: t₈₇: n_l1___3(X₀, X₁) → n_l2___2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
results in twn-loop: twn:Inv: [X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀] , (X₀,X₁) -> (X₀-1,X₁) :|: X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: X₁
Termination: true
Formula:
X₁ < 0 ∧ 1 < 0
∨ X₁ < 0 ∧ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < 0 ∧ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ < 0 ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₁ < 0 ∧ 1 < X₀ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < 0 ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 < 0
∨ X₁ < 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < X₀ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 < 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
Stabilization-Threshold for: 1 ≤ X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
Stabilization-Threshold for: 0 ≤ X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
relevant size-bounds w.r.t. t₈₉:
X₀: X₀ {O(n)}
Runtime-bound of t₈₉: 1 {O(1)}
Results in: 4⋅X₀+7 {O(n)}
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: X₁
Termination: true
Formula:
X₁ < 0 ∧ 1 < 0
∨ X₁ < 0 ∧ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < 0 ∧ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ < 0 ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₁ < 0 ∧ 1 < X₀ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < 0 ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 < 0
∨ X₁ < 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ < 0 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < X₀ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 < 0
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0
Stabilization-Threshold for: 1 ≤ X₀
alphas_abs: 1+X₀
M: 0
N: 1
Bound: 2⋅X₀+4 {O(n)}
Stabilization-Threshold for: 0 ≤ X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}
relevant size-bounds w.r.t. t₈₇:
X₀: 2⋅X₀ {O(n)}
Runtime-bound of t₈₇: 1 {O(1)}
Results in: 8⋅X₀+9 {O(n)}
12⋅X₀+16 {O(n)}
Time-Bound by TWN-Loops:
TWN-Loops: t₉₀ 12⋅X₀+16 {O(n)}
relevant size-bounds w.r.t. t₈₉:
X₀: X₀ {O(n)}
Runtime-bound of t₈₉: 1 {O(1)}
Results in: 4⋅X₀+7 {O(n)}
relevant size-bounds w.r.t. t₈₇:
X₀: 2⋅X₀ {O(n)}
Runtime-bound of t₈₇: 1 {O(1)}
Results in: 8⋅X₀+9 {O(n)}
12⋅X₀+16 {O(n)}
CFR: Improvement to new bound with the following program:
new bound:
24⋅X₀+4⋅X₁+39 {O(n)}
cfr-program:
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l2, n_l1___1, n_l1___3, n_l1___4, n_l2___2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₉₁: l2(X₀, X₁) → n_l1___4(X₀, X₁) :|: 1 ≤ X₀
t₈₅: n_l1___1(X₀, X₁) → n_l2___2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀
t₈₆: n_l1___3(X₀, X₁) → n_l1___3(X₀, X₁-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₈₇: n_l1___3(X₀, X₁) → n_l2___2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₈₈: n_l1___4(X₀, X₁) → n_l1___3(X₀, X₁-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀
t₈₉: n_l1___4(X₀, X₁) → n_l2___2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀
t₉₀: n_l2___2(X₀, X₁) → n_l1___1(X₀, X₁) :|: X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀
All Bounds
Timebounds
Overall timebound:24⋅X₀+4⋅X₁+44 {O(n)}
t₃: 1 {O(1)}
t₉₁: 1 {O(1)}
t₈₅: 12⋅X₀+16 {O(n)}
t₈₆: 4⋅X₁+7 {O(n)}
t₈₇: 1 {O(1)}
t₈₈: 1 {O(1)}
t₈₉: 1 {O(1)}
t₉₀: 12⋅X₀+16 {O(n)}
Costbounds
Overall costbound: 24⋅X₀+4⋅X₁+44 {O(n)}
t₃: 1 {O(1)}
t₉₁: 1 {O(1)}
t₈₅: 12⋅X₀+16 {O(n)}
t₈₆: 4⋅X₁+7 {O(n)}
t₈₇: 1 {O(1)}
t₈₈: 1 {O(1)}
t₈₉: 1 {O(1)}
t₉₀: 12⋅X₀+16 {O(n)}
Sizebounds
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₉₁, X₀: X₀ {O(n)}
t₉₁, X₁: X₁ {O(n)}
t₈₅, X₀: 3⋅X₀ {O(n)}
t₈₅, X₁: X₁ {O(n)}
t₈₆, X₀: X₀ {O(n)}
t₈₆, X₁: X₁ {O(n)}
t₈₇, X₀: 2⋅X₀ {O(n)}
t₈₇, X₁: 0 {O(1)}
t₈₈, X₀: X₀ {O(n)}
t₈₈, X₁: X₁ {O(n)}
t₈₉, X₀: X₀ {O(n)}
t₈₉, X₁: X₁ {O(n)}
t₉₀, X₀: 3⋅X₀ {O(n)}
t₉₀, X₁: X₁ {O(n)}