Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂, X₁, X₂-1, X₃, X₄, X₅) :|: 0 < X₂
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₅, X₃, X₄, X₅) :|: X₂ ≤ 0
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅
t₃: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, 3⋅X₁-4⋅X₂, 4⋅X₁-3⋅X₂, 5⋅X₃, 5⋅X₄-(X₀)², X₅) :|: 1 < (X₁)² ∧ 0 < X₀*X₂+2⋅X₀
Preprocessing
Found invariant X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂, X₁, X₂-1, X₃, X₄, X₅) :|: 0 < X₂ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₅, X₃, X₄, X₅) :|: X₂ ≤ 0 ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅
t₃: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, 3⋅X₁-4⋅X₂, 4⋅X₁-3⋅X₂, 5⋅X₃, 5⋅X₄-(X₀)², X₅) :|: 1 < (X₁)² ∧ 0 < X₀*X₂+2⋅X₀
MPRF for transition t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅ of depth 1:
new bound:
X₅ {O(n)}
MPRF:
l2 [X₅ ]
l1 [X₅ ]
Found invariant 1 ≤ 0 for location l2
Found invariant 1 ≤ 0 for location l1
Found invariant X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ for location l1
Time-Bound by TWN-Loops:
TWN-Loops: t₁ 8⋅X₅⋅X₅+6⋅X₅+4 {O(n^2)}
TWN-Loops:
entry: t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅
results in twn-loop: twn:Inv: [X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀] , (X₀,X₁,X₂,X₃,X₄,X₅) -> (X₀+X₂,X₁,X₂-1,X₃,X₄,X₅) :|: 0 < X₂
entry: t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅)
results in twn-loop: twn:Inv: [X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀] , (X₀,X₁,X₂,X₃,X₄,X₅) -> (X₀+X₂,X₁,X₂-1,X₃,X₄,X₅) :|: 0 < X₂
order: [X₂; X₀; X₁; X₃; X₄; X₅]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
X₀: X₀ + [[n != 0]] * X₂ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1
X₁: X₁
X₃: X₃
X₄: X₄
X₅: X₅
Termination: true
Formula:
1 < 0
∨ 1 < 0 ∧ 0 < 2⋅X₂+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < 0 ∧ 2⋅X₃ < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0 ∧ 2⋅X₃ ≤ 2⋅X₀ ∧ 2⋅X₀ ≤ 2⋅X₃
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ 0 < X₂ ∧ 0 < 2⋅X₂+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < X₂ ∧ 2⋅X₃ < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0 ∧ 2⋅X₃ ≤ 2⋅X₀ ∧ 2⋅X₀ ≤ 2⋅X₃
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
relevant size-bounds w.r.t. t₄:
X₂: 4⋅X₅ {O(n)}
Runtime-bound of t₄: X₅ {O(n)}
Results in: 8⋅X₅⋅X₅+4⋅X₅ {O(n^2)}
order: [X₂; X₀; X₁; X₃; X₄; X₅]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
X₀: X₀ + [[n != 0]] * X₂ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1
X₁: X₁
X₃: X₃
X₄: X₄
X₅: X₅
Termination: true
Formula:
1 < 0
∨ 1 < 0 ∧ 0 < 2⋅X₂+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < 0 ∧ 2⋅X₃ < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0 ∧ 2⋅X₃ ≤ 2⋅X₀ ∧ 2⋅X₀ ≤ 2⋅X₃
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ 0 < X₂ ∧ 0 < 2⋅X₂+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < X₂ ∧ 2⋅X₃ < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0 ∧ 2⋅X₃ ≤ 2⋅X₀ ∧ 2⋅X₀ ≤ 2⋅X₃
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
relevant size-bounds w.r.t. t₀:
X₂: X₅ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₅+4 {O(n)}
8⋅X₅⋅X₅+6⋅X₅+4 {O(n^2)}
knowledge_propagation leads to new time bound 8⋅X₅⋅X₅+6⋅X₅+5 {O(n^2)} for transition t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₅, X₃, X₄, X₅) :|: X₂ ≤ 0 ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
Analysing control-flow refined program
Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ for location n_l2___6
Found invariant 1+X₅ ≤ X₂ ∧ 0 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ for location n_l1___4
Found invariant X₅ ≤ X₂ ∧ 1 ≤ X₅ ∧ 2 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ for location n_l2___5
Found invariant 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ for location n_l1___7
Found invariant 0 ≤ X₅ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ for location n_l1___2
Found invariant 1 ≤ X₅ for location n_l2___3
Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ for location l1
MPRF for transition t₉₈: n_l1___7(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___7(X₀+X₂, X₁, X₂-1, X₃, X₁, X₅) :|: X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 < X₅ ∧ X₂ ≤ 1+X₅ ∧ 0 < X₂ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:
new bound:
X₅+1 {O(n)}
MPRF:
n_l1___7 [X₂+1 ]
MPRF for transition t₉₆: n_l1___2(X₀, X₁, X₂, X₃, X₄, X₅) → n_l2___6(X₀, X₁, X₅, X₃, X₁, X₅) :|: X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₂ ≤ 0 ∧ X₂ ≤ 1+X₅ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 ≤ X₅ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF:
n_l1___2 [X₅+1 ]
n_l1___4 [X₅+1 ]
n_l2___6 [X₅ ]
n_l2___1 [X₅ ]
MPRF for transition t₉₇: n_l1___4(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___2(X₀+X₂, X₁, X₂-1, X₃, X₁, X₅) :|: 0 < X₂ ∧ X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₂ ≤ 1+X₅ ∧ 0 < X₂ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 1+X₅ ≤ X₂ ∧ 0 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF:
n_l1___2 [X₅ ]
n_l1___4 [X₅+1 ]
n_l2___6 [X₅ ]
n_l2___1 [X₅ ]
MPRF for transition t₁₀₂: n_l2___1(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___4(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF:
n_l1___2 [X₅ ]
n_l1___4 [X₂-1 ]
n_l2___6 [X₂ ]
n_l2___1 [X₅ ]
MPRF for transition t₁₀₈: n_l2___6(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___4(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: X₂ ≤ X₅ ∧ X₅ ≤ X₂ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 < X₅ ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ of depth 1:
new bound:
7⋅X₅+2 {O(n)}
MPRF:
n_l1___2 [X₅ ]
n_l1___4 [X₂-1 ]
n_l2___6 [X₂ ]
n_l2___1 [X₅ ]
Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ for location n_l2___6
Found invariant 1+X₅ ≤ X₂ ∧ 0 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ X₂ ≤ 1+X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ for location n_l1___4
Found invariant X₅ ≤ X₂ ∧ 1 ≤ X₅ ∧ 2 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ for location n_l2___5
Found invariant 1 ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ for location n_l1___7
Found invariant 0 ≤ X₅ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ for location n_l1___2
Found invariant 1 ≤ X₅ for location n_l2___3
Found invariant X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ for location l1
knowledge_propagation leads to new time bound 7⋅X₅+3 {O(n)} for transition t₁₀₉: n_l2___6(X₀, X₁, X₂, X₃, X₄, X₅) → n_l2___1(X₀, Arg1_P, Arg2_P, 5⋅X₃, NoDet0, X₅) :|: X₂ ≤ X₅ ∧ X₅ ≤ X₂ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 4⋅X₁ ≤ 3⋅X₂+Arg2_P ∧ 3⋅X₂+Arg2_P ≤ 4⋅X₁ ∧ 7⋅X₁+3⋅Arg1_P ≤ 4⋅Arg2_P ∧ 4⋅Arg2_P ≤ 7⋅X₁+3⋅Arg1_P ∧ X₅ ≤ X₂ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
MPRF for transition t₉₅: n_l1___2(X₀, X₁, X₂, X₃, X₄, X₅) → n_l1___2(X₀+X₂, X₁, X₂-1, X₃, X₁, X₅) :|: X₃ ≤ X₀ ∧ X₂ ≤ 1+X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ X₂ ≤ 1+X₅ ∧ 0 < X₂ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 ≤ X₅ ∧ 0 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:
new bound:
147⋅X₅⋅X₅+55⋅X₅ {O(n^2)}
MPRF:
n_l2___6 [1 ]
n_l1___2 [X₂+1 ]
n_l1___4 [X₂ ]
n_l2___1 [X₅ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 8⋅X₅⋅X₅+6⋅X₅+4 {O(n^2)}
t₂: 8⋅X₅⋅X₅+6⋅X₅+5 {O(n^2)}
t₃: inf {Infinity}
t₄: X₅ {O(n)}
Costbounds
Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 8⋅X₅⋅X₅+6⋅X₅+4 {O(n^2)}
t₂: 8⋅X₅⋅X₅+6⋅X₅+5 {O(n^2)}
t₃: inf {Infinity}
t₄: X₅ {O(n)}
Sizebounds
t₀, X₀: X₃ {O(n)}
t₀, X₁: X₄ {O(n)}
t₀, X₂: X₅ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₁, X₂: 5⋅X₅ {O(n)}
t₁, X₅: 2⋅X₅ {O(n)}
t₂, X₂: 3⋅X₅ {O(n)}
t₂, X₅: 2⋅X₅ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₄, X₂: 4⋅X₅ {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}