Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₄, X₂, X₃, X₄)
t₄: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁-1, X₂, X₃, X₄) :|: 0 < X₁ ∧ 0 < X₁
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, X₁-1, X₂, X₃, X₄) :|: 0 < X₁ ∧ X₁ ≤ 0
t₆: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₂, X₂, X₃, X₄) :|: X₁ ≤ 0 ∧ 0 < X₁
t₇: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, X₂, X₂, X₃, X₄) :|: X₁ ≤ 0 ∧ X₁ ≤ 0
t₈: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)
Preprocessing
Cut unsatisfiable transition t₅: l3→l1
Cut unsatisfiable transition t₆: l3→l1
Eliminate variables {X₃} that do not contribute to the problem
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l5
Found invariant X₀ ≤ X₂ for location l1
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₇: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₁₈: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 < X₀ ∧ X₀ ≤ X₂
t₁₉: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂
t₂₀: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₂₁: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁-1, X₂, X₃) :|: 0 < X₁ ∧ 0 < X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀
t₂₂: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₂, X₂, X₃) :|: X₁ ≤ 0 ∧ X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀
t₂₃: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₀ ≤ 0
MPRF for transition t₂₂: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₂, X₂, X₃) :|: X₁ ≤ 0 ∧ X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂ {O(n)}
TWN: t₁₈: l1→l3
cycle: [t₁₈: l1→l3; t₂₁: l3→l1]
loop: (0 < X₀ ∧ 0 < X₁ ∧ 0 < X₁,(X₀,X₁) -> (X₀,X₁-1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0 ∧ 0 < X₀
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₀
Stabilization-Threshold for: 0 < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
loop: (0 < X₀ ∧ 0 < X₁ ∧ 0 < X₁,(X₀,X₁) -> (X₀,X₁-1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0 ∧ 0 < X₀
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₀
Stabilization-Threshold for: 0 < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
TWN - Lifting for t₁₈: l1→l3 of 2⋅X₁+5 {O(n)}
relevant size-bounds w.r.t. t₂₂:
X₁: X₂ {O(n)}
Runtime-bound of t₂₂: X₂ {O(n)}
Results in: 2⋅X₂⋅X₂+5⋅X₂ {O(n^2)}
TWN - Lifting for t₁₈: l1→l3 of 2⋅X₁+5 {O(n)}
relevant size-bounds w.r.t. t₂₀:
X₁: X₃ {O(n)}
Runtime-bound of t₂₀: 1 {O(1)}
Results in: 2⋅X₃+5 {O(n)}
TWN: t₂₁: l3→l1
TWN - Lifting for t₂₁: l3→l1 of 2⋅X₁+5 {O(n)}
relevant size-bounds w.r.t. t₂₂:
X₁: X₂ {O(n)}
Runtime-bound of t₂₂: X₂ {O(n)}
Results in: 2⋅X₂⋅X₂+5⋅X₂ {O(n^2)}
TWN - Lifting for t₂₁: l3→l1 of 2⋅X₁+5 {O(n)}
relevant size-bounds w.r.t. t₂₀:
X₁: X₃ {O(n)}
Runtime-bound of t₂₀: 1 {O(1)}
Results in: 2⋅X₃+5 {O(n)}
Chain transitions t₂₂: l3→l1 and t₁₉: l1→l4 to t₅₃: l3→l4
Chain transitions t₂₁: l3→l1 and t₁₉: l1→l4 to t₅₄: l3→l4
Chain transitions t₂₁: l3→l1 and t₁₈: l1→l3 to t₅₅: l3→l3
Chain transitions t₂₂: l3→l1 and t₁₈: l1→l3 to t₅₆: l3→l3
Chain transitions t₂₀: l2→l1 and t₁₈: l1→l3 to t₅₇: l2→l3
Chain transitions t₂₀: l2→l1 and t₁₉: l1→l4 to t₅₈: l2→l4
Analysing control-flow refined program
Cut unsatisfiable transition t₅₄: l3→l4
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l5
Found invariant X₀ ≤ X₂ for location l1
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location l3
MPRF for transition t₅₆: l3(X₀, X₁, X₂, X₃) -{2}> l3(X₀-1, X₂, X₂, X₃) :|: X₁ ≤ 0 ∧ X₁ ≤ 0 ∧ 1 < X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂ {O(n)}
TWN: t₅₅: l3→l3
cycle: [t₅₅: l3→l3]
loop: (0 < X₁ ∧ 0 < X₁ ∧ 0 < X₀,(X₀,X₁) -> (X₀,X₁-1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
0 < X₀ ∧ 1 < 0
∨ 0 < X₀ ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
loop: (0 < X₁ ∧ 0 < X₁ ∧ 0 < X₀,(X₀,X₁) -> (X₀,X₁-1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
0 < X₀ ∧ 1 < 0
∨ 0 < X₀ ∧ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 0 < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
TWN - Lifting for t₅₅: l3→l3 of 2⋅X₁+5 {O(n)}
relevant size-bounds w.r.t. t₅₆:
X₁: 2⋅X₂ {O(n)}
Runtime-bound of t₅₆: X₂ {O(n)}
Results in: 4⋅X₂⋅X₂+5⋅X₂ {O(n^2)}
TWN - Lifting for t₅₅: l3→l3 of 2⋅X₁+5 {O(n)}
relevant size-bounds w.r.t. t₅₇:
X₁: X₃ {O(n)}
Runtime-bound of t₅₇: 1 {O(1)}
Results in: 2⋅X₃+5 {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
Analysing control-flow refined program
Cut unsatisfiable transition t₁₄₂: n_l1___2→l4
Cut unsatisfiable transition t₁₄₄: n_l1___6→l4
Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___6
Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l3___4
Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___3
Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l1___2
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l5
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___5
Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1
Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l4
Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___1
Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location n_l3___7
MPRF for transition t₁₂₆: n_l1___2(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃) :|: 0 < X₀ ∧ 0 < X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₂+1 {O(n)}
MPRF for transition t₁₂₇: n_l1___5(X₀, X₁, X₂, X₃) → n_l3___3(X₀, X₁, X₂, X₃) :|: 0 < X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
2⋅X₂+1 {O(n)}
MPRF for transition t₁₃₀: n_l3___1(X₀, X₁, X₂, X₃) → n_l1___6(X₀, X₁-1, X₂, X₃) :|: 0 < X₁ ∧ 0 < X₁ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₂+1 {O(n)}
MPRF for transition t₁₃₁: n_l3___3(X₀, X₁, X₂, X₃) → n_l1___2(X₀, X₁-1, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ 0 < X₀ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 < X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₂+1 {O(n)}
MPRF for transition t₁₃₂: n_l3___4(X₀, X₁, X₂, X₃) → n_l1___5(X₀-1, X₂, X₂, X₃) :|: X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₂ {O(n)}
TWN: t₁₂₈: n_l1___6→n_l3___4
cycle: [t₁₃₃: n_l3___4→n_l1___6; t₁₂₈: n_l1___6→n_l3___4]
loop: (0 < X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 < X₀ ∧ 0 < X₁,(X₀,X₁) -> (X₀,X₁-1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0 ∧ 0 < X₀ ∧ 1 < X₀
∨ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₀ ∧ 1 < X₀
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
Stabilization-Threshold for: 0 < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
loop: (0 < X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 < X₀ ∧ 0 < X₁,(X₀,X₁) -> (X₀,X₁-1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0 ∧ 0 < X₀ ∧ 1 < X₀
∨ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₀ ∧ 1 < X₀
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
Stabilization-Threshold for: 0 < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
TWN - Lifting for t₁₂₈: n_l1___6→n_l3___4 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₁₃₅:
X₁: X₃ {O(n)}
Runtime-bound of t₁₃₅: 1 {O(1)}
Results in: 2⋅X₃+6 {O(n)}
TWN - Lifting for t₁₂₈: n_l1___6→n_l3___4 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₁₃₀:
X₁: 3⋅X₂ {O(n)}
Runtime-bound of t₁₃₀: 2⋅X₂+1 {O(n)}
Results in: 12⋅X₂⋅X₂+18⋅X₂+6 {O(n^2)}
TWN: t₁₃₃: n_l3___4→n_l1___6
TWN - Lifting for t₁₃₃: n_l3___4→n_l1___6 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₁₃₅:
X₁: X₃ {O(n)}
Runtime-bound of t₁₃₅: 1 {O(1)}
Results in: 2⋅X₃+6 {O(n)}
TWN - Lifting for t₁₃₃: n_l3___4→n_l1___6 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₁₃₀:
X₁: 3⋅X₂ {O(n)}
Runtime-bound of t₁₃₀: 2⋅X₂+1 {O(n)}
Results in: 12⋅X₂⋅X₂+18⋅X₂+6 {O(n^2)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:4⋅X₂⋅X₂+11⋅X₂+4⋅X₃+14 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₂⋅X₂+2⋅X₃+5⋅X₂+5 {O(n^2)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 2⋅X₂⋅X₂+2⋅X₃+5⋅X₂+5 {O(n^2)}
t₂₂: X₂ {O(n)}
t₂₃: 1 {O(1)}
Costbounds
Overall costbound: 4⋅X₂⋅X₂+11⋅X₂+4⋅X₃+14 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₂⋅X₂+2⋅X₃+5⋅X₂+5 {O(n^2)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 2⋅X₂⋅X₂+2⋅X₃+5⋅X₂+5 {O(n^2)}
t₂₂: X₂ {O(n)}
t₂₃: 1 {O(1)}
Sizebounds
t₁₇, X₀: X₀ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₈, X₀: X₂ {O(n)}
t₁₈, X₁: X₂+X₃ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₉, X₀: 2⋅X₂ {O(n)}
t₁₉, X₁: X₂+X₃ {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₂₀, X₀: X₂ {O(n)}
t₂₀, X₁: X₃ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₁, X₀: X₂ {O(n)}
t₂₁, X₁: X₂+X₃ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: X₂ {O(n)}
t₂₂, X₁: X₂ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₃, X₀: 2⋅X₂ {O(n)}
t₂₃, X₁: X₂+X₃ {O(n)}
t₂₃, X₂: 2⋅X₂ {O(n)}
t₂₃, X₃: 2⋅X₃ {O(n)}