Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂, X₃, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₂
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂-1, X₃, X₄)
t₂: l2(X₀, X₁, X₂, X₃, X₄) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄) :|: X₀ < X₁ ∧ 0 < X₀
Preprocessing
Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ for location l2
Found invariant X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂, X₃, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₂ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂-1, X₃, X₄) :|: X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂
t₂: l2(X₀, X₁, X₂, X₃, X₄) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄) :|: X₀ < X₁ ∧ 0 < X₀ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂
TWN. Size Bound: t₂: l2→l2 for X₀
cycle: [t₂: l2→l2]
loop: (X₀ < X₁ ∧ 0 < X₀,(X₀,X₁,X₂,X₃,X₄) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂,X₃,X₄)
order: [X₂; X₀; X₁; X₃; X₄]
closed-form:
X₂: X₂
X₀: X₀ * 5^n + [[n != 0]] * 1/4⋅(X₂)² * 5^n + [[n != 0]] * -1/4⋅(X₂)²
X₁: X₁ * 2^n
X₃: X₃
X₄: X₄
Stabilization-Threshold for: 0 < X₀
alphas_abs: (X₂)²
M: 0
N: 1
Bound: 2⋅X₂⋅X₂+2 {O(n^2)}
Stabilization-Threshold for: X₀ < X₁
alphas_abs: 4⋅X₁+(X₂)²
M: 0
N: 1
Bound: 2⋅X₂⋅X₂+8⋅X₁+2 {O(n^2)}
loop: (X₀ < X₁ ∧ 0 < X₀,(X₀,X₂) -> (5⋅X₀+(X₂)²,X₂)
closed-form: X₀ * 5^n + [[n != 0]] * 1/4⋅(X₂)² * 5^n + [[n != 0]] * -1/4⋅(X₂)²
runtime bound: 4⋅X₂⋅X₂+8⋅X₁+6 {O(n^2)}
TWN Size Bound - Lifting for t₂: l2→l2 and X₀: 3⋅5^(4⋅X₂⋅X₂+24⋅X₄+6)⋅X₃+5^(4⋅X₂⋅X₂+24⋅X₄+6)⋅X₂⋅X₂+X₂⋅X₂ {O(EXP)}
MPRF for transition t₁: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₂ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:
new bound:
X₂ {O(n)}
MPRF for transition t₃: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂-1, X₃, X₄) :|: X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ of depth 1:
new bound:
X₂ {O(n)}
TWN: t₂: l2→l2
cycle: [t₂: l2→l2]
loop: (X₀ < X₁ ∧ 0 < X₀,(X₀,X₁,X₂) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂)
order: [X₂; X₀; X₁]
closed-form:
X₂: X₂
X₀: X₀ * 5^n + [[n != 0]] * 1/4⋅(X₂)² * 5^n + [[n != 0]] * -1/4⋅(X₂)²
X₁: X₁ * 2^n
Termination: true
Formula:
0 < 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² < 0
∨ 0 < 4⋅X₀+(X₂)² ∧ 0 < 4⋅X₁ ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)²
∨ 0 < 4⋅X₀+(X₂)² ∧ 0 < (X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 0 ≤ 4⋅X₁ ∧ 4⋅X₁ ≤ 0
∨ (X₂)² < 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 4⋅X₀+(X₂)² < 0
∨ (X₂)² < 0 ∧ 0 < 4⋅X₁ ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)²
∨ (X₂)² < 0 ∧ 0 < (X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 0 ≤ 4⋅X₁ ∧ 4⋅X₁ ≤ 0
Stabilization-Threshold for: 0 < X₀
alphas_abs: (X₂)²
M: 0
N: 1
Bound: 2⋅X₂⋅X₂+2 {O(n^2)}
Stabilization-Threshold for: X₀ < X₁
alphas_abs: 4⋅X₁+(X₂)²
M: 0
N: 1
Bound: 2⋅X₂⋅X₂+8⋅X₁+2 {O(n^2)}
TWN - Lifting for t₂: l2→l2 of 4⋅X₂⋅X₂+8⋅X₁+6 {O(n^2)}
relevant size-bounds w.r.t. t₁:
X₁: 3⋅X₄ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₁: X₂ {O(n)}
Results in: 4⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₄+6⋅X₂ {O(n^3)}
TWN Size Bound - Lifting for t₂: l2→l2 and X₀: 3⋅5^(4⋅X₂⋅X₂+24⋅X₄+6)⋅X₃+5^(4⋅X₂⋅X₂+24⋅X₄+6)⋅X₂⋅X₂+X₂⋅X₂ {O(EXP)}
Chain transitions t₃: l2→l1 and t₁: l1→l2 to t₃₆: l2→l2
Chain transitions t₀: l0→l1 and t₁: l1→l2 to t₃₇: l0→l2
Analysing control-flow refined program
Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ for location l2
Found invariant X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ for location l1
TWN Size Bound - Lifting for t₂: l2→l2 and X₀: inf {Infinity}
MPRF for transition t₃₆: l2(X₀, X₁, X₂, X₃, X₄) -{2}> l2(X₃, X₄, X₂-1, X₃, X₄) :|: 1 < X₂ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ of depth 1:
new bound:
X₂ {O(n)}
TWN Size Bound - Lifting for t₂: l2→l2 and X₀: inf {Infinity}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
Analysing control-flow refined program
Found invariant X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ for location l2
Found invariant 2 ≤ X₄ ∧ 3 ≤ X₃+X₄ ∧ 3 ≤ X₂+X₄ ∧ 5 ≤ X₁+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 4 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 3 ≤ X₁ for location n_l2___1
Found invariant X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ for location l1
knowledge_propagation leads to new time bound X₂ {O(n)} for transition t₆₆: l2(X₀, X₁, X₂, X₃, X₄) → n_l2___1(NoDet0, Arg1_P, X₂, X₃, Arg4_P) :|: 2⋅Arg4_P ≤ Arg1_P ∧ 2⋅X₀ < Arg1_P ∧ 0 < X₀ ∧ 2⋅X₁ ≤ Arg1_P ∧ Arg1_P ≤ 2⋅X₁ ∧ X₄ ≤ Arg4_P ∧ Arg4_P ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂
MPRF for transition t₆₉: n_l2___1(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂-1, X₃, X₄) :|: X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₄ ∧ 3 ≤ X₃+X₄ ∧ 3 ≤ X₂+X₄ ∧ 5 ≤ X₁+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 4 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 3 ≤ X₁ of depth 1:
new bound:
X₂ {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:4⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₄+8⋅X₂+1 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 4⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₄+6⋅X₂ {O(n^3)}
t₃: X₂ {O(n)}
Costbounds
Overall costbound: 4⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₄+8⋅X₂+1 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 4⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₄+6⋅X₂ {O(n^3)}
t₃: X₂ {O(n)}
Sizebounds
t₀, X₀: X₃ {O(n)}
t₀, X₁: X₄ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: 3⋅X₃ {O(n)}
t₁, X₁: 3⋅X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 3⋅5^(4⋅X₂⋅X₂+24⋅X₄+6)⋅X₃+5^(4⋅X₂⋅X₂+24⋅X₄+6)⋅X₂⋅X₂+X₂⋅X₂ {O(EXP)}
t₂, X₁: 2^(4⋅X₂⋅X₂⋅X₂+24⋅X₂⋅X₄+6⋅X₂)⋅3⋅X₄ {O(EXP)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 2⋅X₃ {O(n)}
t₃, X₁: 2⋅X₄ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}