Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: U, V
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, U, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃+1, X₄) :|: 1 ≤ X₃ ∧ X₃ ≤ 3 ∧ V ≤ 0 ∧ 0 ≤ V
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V
t₂: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄-1)
t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄) :|: X₀ < (X₁)² ∧ 0 < X₀

Preprocessing

Found invariant 1 ≤ X₄ for location l2

Found invariant 1 ≤ X₄ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: U, V
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, U, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃+1, X₄) :|: 1 ≤ X₃ ∧ X₃ ≤ 3 ∧ V ≤ 0 ∧ 0 ≤ V
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V
t₂: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄-1) :|: 1 ≤ X₄
t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₄
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄) :|: X₀ < (X₁)² ∧ 0 < X₀ ∧ 1 ≤ X₄

MPRF for transition t₃: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V of depth 1:

new bound:

X₄ {O(n)}

MPRF:

l2 [X₄-1 ]
l1 [X₄ ]

MPRF for transition t₂: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄-1) :|: 1 ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

l2 [X₄ ]
l1 [X₄ ]

Analysing control-flow refined program

Found invariant 1 ≤ X₄ for location l2

Found invariant X₃ ≤ 4 ∧ 2 ≤ X₃ for location n_l1___2

Found invariant 1 ≤ X₄ for location l3

Found invariant 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ for location n_l1___1

knowledge_propagation leads to new time bound X₄+1 {O(n)} for transition t₆₄: l1(X₀, X₁, X₂, X₃, X₄) → n_l1___1(X₀, X₁, X₂, Arg3_P, X₄) :|: 0 ≤ X₄ ∧ Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃

knowledge_propagation leads to new time bound X₄+1 {O(n)} for transition t₆₅: l1(X₀, X₁, X₂, X₃, X₄) → n_l1___2(X₀, X₁, X₂, Arg3_P, X₄) :|: Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃

MPRF for transition t₆₂: n_l1___1(X₀, X₁, X₂, X₃, X₄) → n_l1___1(X₀, X₁, X₂, Arg3_P, X₄) :|: 1 ≤ X₃ ∧ 0 ≤ X₄ ∧ 2 ≤ X₃ ∧ X₃ ≤ 4 ∧ Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃ ∧ 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ of depth 1:

new bound:

6⋅X₄+7 {O(n)}

MPRF:

l1 [6⋅X₄+7 ]
n_l1___1 [6⋅X₄+13-3⋅X₃ ]
n_l1___2 [6⋅X₄+13-3⋅X₃ ]
l2 [6⋅X₄+1 ]

MPRF for transition t₆₈: n_l1___1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V ∧ 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

l1 [X₄+1 ]
n_l1___1 [X₄+1 ]
n_l1___2 [X₄ ]
l2 [X₄ ]

MPRF for transition t₆₃: n_l1___2(X₀, X₁, X₂, X₃, X₄) → n_l1___2(X₀, X₁, X₂, Arg3_P, X₄) :|: 1 ≤ X₃ ∧ 2 ≤ X₃ ∧ X₃ ≤ 4 ∧ Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ of depth 1:

new bound:

16⋅X₄+16 {O(n)}

MPRF:

l1 [16 ]
n_l1___1 [16 ]
n_l1___2 [34-9⋅X₃ ]
l2 [16-18⋅X₄ ]

MPRF for transition t₆₉: n_l1___2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

l1 [X₄ ]
n_l1___1 [X₄ ]
n_l1___2 [X₄ ]
l2 [X₄-1 ]

CFR: Improvement to new bound with the following program:

new bound:

28⋅X₄+26 {O(n)}

cfr-program:

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: Arg3_P, U, V
Locations: l0, l1, l2, l3, n_l1___1, n_l1___2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, U, X₄)
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V
t₆₄: l1(X₀, X₁, X₂, X₃, X₄) → n_l1___1(X₀, X₁, X₂, Arg3_P, X₄) :|: 0 ≤ X₄ ∧ Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃
t₆₅: l1(X₀, X₁, X₂, X₃, X₄) → n_l1___2(X₀, X₁, X₂, Arg3_P, X₄) :|: Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃
t₂: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄-1) :|: 1 ≤ X₄ ∧ 1 ≤ X₄
t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₄ ∧ 1 ≤ X₄
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄) :|: X₀ < (X₁)² ∧ 0 < X₀ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄
t₆₈: n_l1___1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V ∧ 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃
t₆₂: n_l1___1(X₀, X₁, X₂, X₃, X₄) → n_l1___1(X₀, X₁, X₂, Arg3_P, X₄) :|: 1 ≤ X₃ ∧ 0 ≤ X₄ ∧ 2 ≤ X₃ ∧ X₃ ≤ 4 ∧ Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃ ∧ 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃
t₆₉: n_l1___2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V ∧ X₃ ≤ 4 ∧ 2 ≤ X₃
t₆₃: n_l1___2(X₀, X₁, X₂, X₃, X₄) → n_l1___2(X₀, X₁, X₂, Arg3_P, X₄) :|: 1 ≤ X₃ ∧ 2 ≤ X₃ ∧ X₃ ≤ 4 ∧ Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃

Found invariant 1 ≤ X₄ for location l2

Found invariant X₃ ≤ 4 ∧ 2 ≤ X₃ for location n_l1___2

Found invariant 1 ≤ X₄ for location l3

Found invariant 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ for location n_l1___1

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 324⋅X₂⋅X₂+648⋅X₁⋅X₁+16 {O(n^2)}

TWN-Loops:

entry: t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₄ ∧ 1 ≤ X₄
results in twn-loop: twn:Inv: [1 ≤ X₄] , (X₀,X₁,X₂,X₃,X₄) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂,X₃,X₄) :|: X₀ < (X₁)² ∧ 0 < X₀
order: [X₂; X₀; X₁; 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
X₄: X₄

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)}

relevant size-bounds w.r.t. t₄:
X₁: 9⋅X₁ {O(n)}
X₂: 9⋅X₂ {O(n)}
Runtime-bound of t₄: 1 {O(1)}
Results in: 324⋅X₂⋅X₂+648⋅X₁⋅X₁+16 {O(n^2)}

324⋅X₂⋅X₂+648⋅X₁⋅X₁+16 {O(n^2)}

Analysing control-flow refined program

Eliminate variables {X₁,X₂} that do not contribute to the problem

Found invariant 1 ≤ X₄ for location l2

Found invariant X₃ ≤ 4 ∧ 2 ≤ X₃ for location n_l1___2

Found invariant 1 ≤ X₄ for location l3

Found invariant 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ for location n_l1___1

MPRF for transition t₁₆₈: l1(X₀, X₃, X₄) → l2(X₀, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V of depth 1:

new bound:

X₄ {O(n)}

MPRF:

l1 [X₄ ]
n_l1___1 [X₄ ]
n_l1___2 [X₄ ]
l2 [X₄-1 ]

MPRF for transition t₁₆₉: l1(X₀, X₃, X₄) → n_l1___1(X₀, Arg3_P, X₄) :|: 0 ≤ X₄ ∧ Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

l1 [X₄+1 ]
n_l1___1 [X₄ ]
n_l1___2 [X₄ ]
l2 [X₄ ]

MPRF for transition t₁₇₁: l2(X₀, X₃, X₄) → l1(X₀, X₃, X₄-1) :|: 1 ≤ X₄ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

l1 [X₄ ]
n_l1___1 [X₄ ]
n_l1___2 [X₄ ]
l2 [X₄ ]

MPRF for transition t₁₇₄: n_l1___1(X₀, X₃, X₄) → n_l1___1(X₀, Arg3_P, X₄) :|: 1 ≤ X₃ ∧ 0 ≤ X₄ ∧ 2 ≤ X₃ ∧ X₃ ≤ 4 ∧ Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃ ∧ 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ ∧ 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ of depth 1:

new bound:

2⋅X₄+3 {O(n)}

MPRF:

l1 [2⋅X₄+3 ]
n_l1___1 [2⋅X₄+5-X₃ ]
n_l1___2 [2⋅X₄+1 ]
l2 [2⋅X₄+1 ]

MPRF for transition t₁₇₅: n_l1___1(X₀, X₃, X₄) → l2(X₀, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V ∧ 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ ∧ 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

l1 [X₄+1 ]
n_l1___1 [X₄+1 ]
n_l1___2 [X₄ ]
l2 [X₄ ]

MPRF for transition t₁₇₇: n_l1___2(X₀, X₃, X₄) → l2(X₀, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ of depth 1:

new bound:

10⋅X₄ {O(n)}

MPRF:

l1 [10⋅X₄ ]
n_l1___1 [10⋅X₄+6-3⋅X₃ ]
n_l1___2 [10⋅X₄-5 ]
l2 [10⋅X₄-6 ]

knowledge_propagation leads to new time bound X₄+1 {O(n)} for transition t₁₇₀: l1(X₀, X₃, X₄) → n_l1___2(X₀, Arg3_P, X₄) :|: Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃

MPRF for transition t₁₇₆: n_l1___2(X₀, X₃, X₄) → n_l1___2(X₀, Arg3_P, X₄) :|: 1 ≤ X₃ ∧ 2 ≤ X₃ ∧ X₃ ≤ 4 ∧ Arg3_P ≤ 4 ∧ 2 ≤ Arg3_P ∧ X₃+1 ≤ Arg3_P ∧ Arg3_P ≤ 1+X₃ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ of depth 1:

new bound:

20⋅X₄+2 {O(n)}

MPRF:

l1 [2 ]
n_l1___1 [2 ]
l2 [2 ]
n_l1___2 [4-X₃ ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:324⋅X₂⋅X₂+648⋅X₁⋅X₁+28⋅X₄+44 {O(n^2)}
t₀: 1 {O(1)}
t₃: X₄ {O(n)}
t₆₄: X₄+1 {O(n)}
t₆₅: X₄+1 {O(n)}
t₂: X₄ {O(n)}
t₄: 1 {O(1)}
t₅: 324⋅X₂⋅X₂+648⋅X₁⋅X₁+16 {O(n^2)}
t₆₂: 6⋅X₄+7 {O(n)}
t₆₈: X₄+1 {O(n)}
t₆₃: 16⋅X₄+16 {O(n)}
t₆₉: X₄ {O(n)}

Costbounds

Overall costbound: 324⋅X₂⋅X₂+648⋅X₁⋅X₁+28⋅X₄+44 {O(n^2)}
t₀: 1 {O(1)}
t₃: X₄ {O(n)}
t₆₄: X₄+1 {O(n)}
t₆₅: X₄+1 {O(n)}
t₂: X₄ {O(n)}
t₄: 1 {O(1)}
t₅: 324⋅X₂⋅X₂+648⋅X₁⋅X₁+16 {O(n^2)}
t₆₂: 6⋅X₄+7 {O(n)}
t₆₈: X₄+1 {O(n)}
t₆₃: 16⋅X₄+16 {O(n)}
t₆₉: X₄ {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₄: X₄ {O(n)}
t₃, X₀: 3⋅X₀ {O(n)}
t₃, X₁: 3⋅X₁ {O(n)}
t₃, X₂: 3⋅X₂ {O(n)}
t₃, X₄: 3⋅X₄ {O(n)}
t₆₄, X₀: 3⋅X₀ {O(n)}
t₆₄, X₁: 3⋅X₁ {O(n)}
t₆₄, X₂: 3⋅X₂ {O(n)}
t₆₄, X₃: 4 {O(1)}
t₆₄, X₄: 3⋅X₄ {O(n)}
t₆₅, X₀: 3⋅X₀ {O(n)}
t₆₅, X₁: 3⋅X₁ {O(n)}
t₆₅, X₂: 3⋅X₂ {O(n)}
t₆₅, X₃: 4 {O(1)}
t₆₅, X₄: 3⋅X₄ {O(n)}
t₂, X₀: 3⋅X₀ {O(n)}
t₂, X₁: 3⋅X₁ {O(n)}
t₂, X₂: 3⋅X₂ {O(n)}
t₂, X₄: 3⋅X₄ {O(n)}
t₄, X₀: 9⋅X₀ {O(n)}
t₄, X₁: 9⋅X₁ {O(n)}
t₄, X₂: 9⋅X₂ {O(n)}
t₄, X₄: 9⋅X₄ {O(n)}
t₅, X₁: 2^(324⋅X₂⋅X₂+648⋅X₁⋅X₁+16)⋅9⋅X₁ {O(EXP)}
t₅, X₂: 9⋅X₂ {O(n)}
t₅, X₄: 9⋅X₄ {O(n)}
t₆₂, X₀: 3⋅X₀ {O(n)}
t₆₂, X₁: 3⋅X₁ {O(n)}
t₆₂, X₂: 3⋅X₂ {O(n)}
t₆₂, X₃: 4 {O(1)}
t₆₂, X₄: 3⋅X₄ {O(n)}
t₆₈, X₀: 3⋅X₀ {O(n)}
t₆₈, X₁: 3⋅X₁ {O(n)}
t₆₈, X₂: 3⋅X₂ {O(n)}
t₆₈, X₃: 4 {O(1)}
t₆₈, X₄: 3⋅X₄ {O(n)}
t₆₃, X₀: 3⋅X₀ {O(n)}
t₆₃, X₁: 3⋅X₁ {O(n)}
t₆₃, X₂: 3⋅X₂ {O(n)}
t₆₃, X₃: 4 {O(1)}
t₆₃, X₄: 3⋅X₄ {O(n)}
t₆₉, X₀: 3⋅X₀ {O(n)}
t₆₉, X₁: 3⋅X₁ {O(n)}
t₆₉, X₂: 3⋅X₂ {O(n)}
t₆₉, X₃: 4 {O(1)}
t₆₉, X₄: 3⋅X₄ {O(n)}