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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
namespace OAI
/-! Multiplicity sequences and their arithmetic identities. -/
namespace Kaplansky
open scoped BigOperators
def testMultiplicity (m R j : ℕ) (i : Fin (j+1)) : ℕ :=
if i.val=0 then 64*m*4^(R-j) else 960*m*4^(R+2*i.val-j-2)
def testRank (m R j : ℕ) : ℕ := 4^j*(64*m*4^R)
theorem testRank_eq (m R j : ℕ) : testRank m R j = 64*m*4^(R+j) := by
simp only [testRank,pow_add]
ring
theorem testMultiplicity_pos {m : ℕ} (hm : 0<m) (R j : ℕ) (i : Fin (j+1)) :
0<testMultiplicity m R j i := by
unfold testMultiplicity
split <;> positivity
theorem testMultiplicity_quarter {m R j : ℕ} (hj : j<R) (i : Fin (j+1)) :
4*testMultiplicity m R (j+1) i.castSucc = testMultiplicity m R j i := by
unfold testMultiplicity
simp only [Fin.val_castSucc]
split_ifs with hi
· have he : R-j=(R-(j+1))+1 := by omega
rw [he,pow_succ]
ring
· have he : R+2*i.val-j-2=(R+2*i.val-(j+1)-2)+1 := by omega
rw [he,pow_succ]
ring
theorem testMultiplicity_fresh {m R j : ℕ} (hj : j<R) :
4*testMultiplicity m R (j+1) (Fin.last (j+1)) = 15*testRank m R j := by
have hi : (Fin.last (j+1)).val ≠ 0 := by simp
rw [testMultiplicity,ite_eq_right hi,testRank_eq]
have he : R+j=(R+2*(Fin.last (j+1)).val-(j+1)-2)+1 := by
simp only [Fin.val_last]
omega
rw [he,pow_succ]
ring
theorem sum_testMultiplicity {m R j : ℕ} (hj : j≤R) :
∑ i, testMultiplicity m R j i = testRank m R j := by
induction j with
| zero => simp [testMultiplicity,testRank]
| succ j ih =>
have hlt : j<R := by omega
have hprev := ih (by omega)
have hq : 4*(∑ i : Fin (j+1), testMultiplicity m R (j+1) i.castSucc) =
testRank m R j := by
rw [Finset.mul_sum]
simpa only [testMultiplicity_quarter hlt] using hprev
rw [Fin.sum_univ_castSucc]
have hf := testMultiplicity_fresh (m := m) hlt
have hr : testRank m R (j+1)=4*testRank m R j := by
simp only [testRank,pow_succ]
ring
rw [hr]
omega
theorem testMultiplicity_divisible {m R j : ℕ} (hj : j<R) (i : Fin (j+1)) :
4 ∣ testMultiplicity m R j i := ⟨_,(testMultiplicity_quarter hj i).symm⟩
theorem testMultiplicity_lower_bound {m R j : ℕ} (hj : j<R) (i : Fin (j+1)) :
256*m ≤ testMultiplicity m R j i := by
rw [← testMultiplicity_quarter hj i]
have h : 64*m ≤ testMultiplicity m R (j+1) i.castSucc := by
unfold testMultiplicity
split
· have hp : 1≤4^(R-(j+1)) := Nat.one_le_pow _ _ (by omega)
nlinarith
· have hp : 1≤4^(R+2*(i.castSucc).val-(j+1)-2) := Nat.one_le_pow _ _ (by omega)
nlinarith
omega
end Kaplansky
end OAI
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