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, X₂, X₃, X₄, X₅) :|: 0 < 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, X₂, X₂, 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(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄, X₅) :|: X₀ < (X₁)² ∧ 0 < X₀
Preprocessing
Found invariant 0 ≤ X₅ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 for location l2
Found invariant 0 ≤ 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, X₂, X₃, X₄, X₅) :|: 0 < X₅
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂, X₁, X₂-1, X₃, X₄, X₅) :|: 0 < X₂ ∧ 0 ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₂, X₄, X₅) :|: X₂ ≤ 0 ∧ 0 ≤ 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 ∧ 0 ≤ X₅ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄, X₅) :|: X₀ < (X₁)² ∧ 0 < X₀ ∧ 0 ≤ X₅ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0
MPRF for transition t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₂, X₄, X₅) :|: X₂ ≤ 0 ∧ 0 ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ of depth 1:
new bound:
X₅+1 {O(n)}
MPRF for transition t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅ ∧ 0 ≤ X₅ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 of depth 1:
new bound:
X₅+1 {O(n)}
TWN: t₁: l1→l1
cycle: [t₁: l1→l1]
loop: (0 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
loop: (0 < X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₄:
X₂: 2⋅X₅ {O(n)}
Runtime-bound of t₄: X₅+1 {O(n)}
Results in: 4⋅X₅⋅X₅+8⋅X₅+4 {O(n^2)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₂+4 {O(n)}
relevant size-bounds w.r.t. t₀:
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₂+4 {O(n)}
TWN: t₃: l2→l2
cycle: [t₃: l2→l2]
loop: (X₀ < (X₁)² ∧ 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)² ∧ 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)² ∧ 4⋅(X₁)² ≤ 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)² ∧ 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)² ∧ 4⋅(X₁)² ≤ 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)²
alphas_abs: 4⋅(X₁)²+(X₂)²
M: 11
N: 1
Bound: 2⋅X₂⋅X₂+8⋅X₁⋅X₁+12 {O(n^2)}
TWN - Lifting for t₃: l2→l2 of 4⋅X₂⋅X₂+8⋅X₁⋅X₁+16 {O(n^2)}
relevant size-bounds w.r.t. t₂:
X₁: 4⋅X₄ {O(n)}
X₂: 2⋅X₂+2⋅X₅ {O(n)}
Runtime-bound of t₂: X₅+1 {O(n)}
Results in: 128⋅X₄⋅X₄⋅X₅+16⋅X₂⋅X₂⋅X₅+16⋅X₅⋅X₅⋅X₅+32⋅X₂⋅X₅⋅X₅+128⋅X₄⋅X₄+16⋅X₂⋅X₂+16⋅X₅⋅X₅+32⋅X₂⋅X₅+16⋅X₅+16 {O(n^3)}
Analysing control-flow refined program
Found invariant 0 ≤ X₅ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 for location l2
Found invariant 0 ≤ X₅ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 for location n_l2___1
Found invariant 0 ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ for location l1
knowledge_propagation leads to new time bound X₅+1 {O(n)} for transition t₆₆: l2(X₀, X₁, X₂, X₃, X₄, X₅) → n_l2___1(NoDet0, 2⋅X₁, X₂, X₂, X₄, Arg5_P) :|: X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 ≤ Arg5_P ∧ 0 < X₀ ∧ X₅ ≤ Arg5_P ∧ Arg5_P ≤ X₅ ∧ X₂ ≤ 0 ∧ 0 ≤ X₅ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₅ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₅ ∧ X₃ ≤ 0 ∧ X₂ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₅ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0
MPRF for transition t₅₆: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂, X₁, X₂-1, X₃, X₁, X₅) :|: 0 < X₂ ∧ 0 < X₂ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 ≤ X₅ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ of depth 1:
new bound:
2⋅X₅⋅X₅+2⋅X₅+X₂ {O(n^2)}
MPRF for transition t₅₇: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+X₂, X₁, X₂-1, X₃, X₁, X₅) :|: 0 < X₂ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₅ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 ≤ X₅ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₄ ∧ X₄ ≤ X₁ ∧ 0 ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ of depth 1:
new bound:
2⋅X₅⋅X₅+2⋅X₅+X₂ {O(n^2)}
MPRF for transition t₆₉: n_l2___1(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₄, X₅, X₃, X₄, X₅-1) :|: 0 < X₅ ∧ 0 ≤ X₅ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 ∧ 0 ≤ X₅ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ X₃ ∧ X₂ ≤ 0 of depth 1:
new bound:
X₅+1 {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:128⋅X₄⋅X₄⋅X₅+16⋅X₂⋅X₂⋅X₅+16⋅X₅⋅X₅⋅X₅+32⋅X₂⋅X₅⋅X₅+128⋅X₄⋅X₄+16⋅X₂⋅X₂+20⋅X₅⋅X₅+32⋅X₂⋅X₅+2⋅X₂+26⋅X₅+27 {O(n^3)}
t₀: 1 {O(1)}
t₁: 4⋅X₅⋅X₅+2⋅X₂+8⋅X₅+8 {O(n^2)}
t₂: X₅+1 {O(n)}
t₃: 128⋅X₄⋅X₄⋅X₅+16⋅X₂⋅X₂⋅X₅+16⋅X₅⋅X₅⋅X₅+32⋅X₂⋅X₅⋅X₅+128⋅X₄⋅X₄+16⋅X₂⋅X₂+16⋅X₅⋅X₅+32⋅X₂⋅X₅+16⋅X₅+16 {O(n^3)}
t₄: X₅+1 {O(n)}
Costbounds
Overall costbound: 128⋅X₄⋅X₄⋅X₅+16⋅X₂⋅X₂⋅X₅+16⋅X₅⋅X₅⋅X₅+32⋅X₂⋅X₅⋅X₅+128⋅X₄⋅X₄+16⋅X₂⋅X₂+20⋅X₅⋅X₅+32⋅X₂⋅X₅+2⋅X₂+26⋅X₅+27 {O(n^3)}
t₀: 1 {O(1)}
t₁: 4⋅X₅⋅X₅+2⋅X₂+8⋅X₅+8 {O(n^2)}
t₂: X₅+1 {O(n)}
t₃: 128⋅X₄⋅X₄⋅X₅+16⋅X₂⋅X₂⋅X₅+16⋅X₅⋅X₅⋅X₅+32⋅X₂⋅X₅⋅X₅+128⋅X₄⋅X₄+16⋅X₂⋅X₂+16⋅X₅⋅X₅+32⋅X₂⋅X₅+16⋅X₅+16 {O(n^3)}
t₄: X₅+1 {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)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: 16⋅X₅⋅X₅⋅X₅+8⋅X₂⋅X₅⋅X₅+24⋅X₂⋅X₅+32⋅X₅⋅X₅+4⋅X₂⋅X₂+22⋅X₂+40⋅X₅+X₃ {O(n^3)}
t₁, X₁: 3⋅X₄ {O(n)}
t₁, X₂: 2⋅X₅+X₂ {O(n)}
t₁, X₃: 4⋅X₂+4⋅X₅+X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: 16⋅X₅⋅X₅⋅X₅+8⋅X₂⋅X₅⋅X₅+24⋅X₂⋅X₅+32⋅X₅⋅X₅+4⋅X₂⋅X₂+2⋅X₃+22⋅X₂+40⋅X₅ {O(n^3)}
t₂, X₁: 4⋅X₄ {O(n)}
t₂, X₂: 2⋅X₂+2⋅X₅ {O(n)}
t₂, X₃: 2⋅X₂+2⋅X₅ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₁: 2^(128⋅X₄⋅X₄⋅X₅+16⋅X₂⋅X₂⋅X₅+16⋅X₅⋅X₅⋅X₅+32⋅X₂⋅X₅⋅X₅+128⋅X₄⋅X₄+16⋅X₂⋅X₂+16⋅X₅⋅X₅+32⋅X₂⋅X₅+16⋅X₅+16)⋅4⋅X₄ {O(EXP)}
t₃, X₂: 2⋅X₂+2⋅X₅ {O(n)}
t₃, X₃: 2⋅X₂+2⋅X₅ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₀: 4⋅X₂+4⋅X₅ {O(n)}
t₄, X₁: 2⋅X₄ {O(n)}
t₄, X₂: 2⋅X₅ {O(n)}
t₄, X₃: 4⋅X₂+4⋅X₅ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}