FenchelNielsenZomorrodian.Discrete.Arithmetic.FinsetLcm
This module studies finset lcm for fenchel nielsen zomorrodian. If a finite set of periods contains an equal pair in the required position, then twice the lcm is bounded by the product.
theorem two_mul_lcm_le_prod_of_equal_pair
{ι : Type*} {periods : ι → ℕ} {s : Finset ι}
(hge : ∀ i ∈ s, 2 ≤ periods i)
{i j : ι} (hi : i ∈ s) (hj : j ∈ s)
(hji : j ≠ i) (hjEq : periods j = periods i) :
2 * s.lcm periods ≤ s.prod periodsIf a finite set of periods contains an equal pair in the required position, then twice the lcm is bounded by the product.
Show proof
by
classical
have hjmem : j ∈ s.erase i := by
exact Finset.mem_erase.mpr ⟨hji, hj⟩
have hlcmDivProdErase :
s.lcm periods ∣ (s.erase i).prod periods := by
apply Finset.lcm_dvd
intro k hk
by_cases hki : k = i
· subst hki
rw [← hjEq]
exact Finset.dvd_prod_of_mem periods hjmem
· have hkmem : k ∈ s.erase i := by
exact Finset.mem_erase.mpr ⟨hki, hk⟩
exact Finset.dvd_prod_of_mem periods hkmem
have hProdErasePos : 0 < (s.erase i).prod periods := by
exact Finset.prod_pos
(fun k hk => lt_of_lt_of_le (by decide : 0 < 2)
(hge k (Finset.mem_of_mem_erase hk)))
have hlcmLeProdErase : s.lcm periods ≤ (s.erase i).prod periods :=
Nat.le_of_dvd hProdErasePos hlcmDivProdErase
calc
2 * s.lcm periods
≤ periods i * (s.erase i).prod periods :=
Nat.mul_le_mul (hge i hi) hlcmLeProdErase
_ = s.prod periods := by
rw [mul_comm]
exact Finset.prod_erase_mul (s := s) (f := periods) (a := i) hiProof. Use the Fenchel--Nielsen--Zomorrodian presentation with elliptic generators of prescribed periods, handle and boundary generators, and the defining product relation. The proof checks the images of the named generators and verifies that every presentation relator is preserved in the target quotient, abelianization, or reduced presentation. For period-class statements, the relevant lcm or gcd divisibility is converted into a scalar multiple of the abelianized elliptic generator, and the defining period relation makes that multiple vanish. Schreier-rewriting steps are performed with the chosen transversal; the letter-by-letter formula sends each relator to the corresponding canonical word in the subgroup presentation. Because the generator images satisfy all defining relations and agree with the displayed quotient data, the presentation universal property yields the asserted identity. The reductions preserve the ordering of the canonical generators and the period data, so the image of every relator can be read directly in the target presentation. In the abelianized arguments, commutators vanish and only the period coefficients remain. The divisibility or lcm calculation then gives the required scalar multiple, while the presentation relation supplies its vanishing. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked. For the Fenchel--Nielsen--Zomorrodian relations, the period calculation checks that each elliptic generator has image of the prescribed order, and the product relation is verified by multiplying the images in the stated order. Divisibility, lcm, or gcd hypotheses are used precisely to make the period relators vanish in the target quotient.
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