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2fca478 | 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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import OAI.Geometry.PeriodicTiling.ForwardDifference
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Basic.Real.Basic
import Mathlib.RingTheory.Binomial
import Mathlib.Topology.Instances.AddCircle.Defs
namespace OAI
universe uA
noncomputable section
namespace PeriodicTilingThree
open scoped BigOperators
section AddCommGroup
variable {A : Type uA} [AddCommGroup A]
theorem intSequence_eq_of_forwardDiff_eq {f g : ℤ → A}
(hd : forwardDiff f = forwardDiff g) (hzero : f 0 = g 0) : f = g := by
have hsucc (n : ℤ) (hn : f n = g n) : f (n + 1) = g (n + 1) := by
have h := congrFun hd n
change f (n + 1) - f n = g (n + 1) - g n at h
rw [hn] at h
exact sub_left_inj.mp h
have hpred (n : ℤ) (hn : f n = g n) : f (n - 1) = g (n - 1) := by
have h := congrFun hd (n - 1)
change f (n - 1 + 1) - f (n - 1) = g (n - 1 + 1) - g (n - 1) at h
rw [sub_add_cancel, hn] at h
exact sub_right_inj.mp h
funext n
refine Int.induction_on n hzero ?_ ?_
· intro m hm
exact hsucc (m : ℤ) hm
· intro m hm
exact hpred (-(m : ℤ)) hm
def newtonSequence {k : ℕ} (c : Fin k → A) (n : ℤ) : A :=
∑ i : Fin k, Ring.choose n i.val • c i
theorem newtonSequence_cons {k : ℕ} (a : A) (c : Fin k → A) (n : ℤ) :
newtonSequence (Fin.cons a c) n =
a + ∑ i : Fin k, Ring.choose n (i.val + 1) • c i := by
rw [newtonSequence, Fin.sum_univ_succ]
simp only [Fin.val_zero, Ring.choose_zero_right, one_zsmul,
Fin.cons_zero, Fin.val_succ, Fin.cons_succ]
@[simp] theorem newtonSequence_cons_zero {k : ℕ} (a : A) (c : Fin k → A) :
newtonSequence (Fin.cons a c) 0 = a := by
simp [newtonSequence_cons]
theorem forwardDiff_newtonSequence_cons {k : ℕ} (a : A) (c : Fin k → A) :
forwardDiff (newtonSequence (Fin.cons a c)) = newtonSequence c := by
funext n
simp only [forwardDiff_apply, newtonSequence_cons]
unfold newtonSequence
calc
(a + ∑ i : Fin k, Ring.choose (n + 1) (i.val + 1) • c i) -
(a + ∑ i : Fin k, Ring.choose n (i.val + 1) • c i) =
∑ i : Fin k,
(Ring.choose (n + 1) (i.val + 1) • c i -
Ring.choose n (i.val + 1) • c i) := by
rw [Finset.sum_sub_distrib]
abel
_ = ∑ i : Fin k, Ring.choose n i.val • c i := by
apply Finset.sum_congr rfl
intro i hi
rw [Ring.choose_succ_succ, add_zsmul, add_sub_cancel_right]
theorem exists_newtonSequence_of_iter_forwardDiff_eq_zero
(k : ℕ) (f : ℤ → A) (hk : (forwardDiff^[k]) f = 0) :
∃ c : Fin k → A, f = newtonSequence c := by
induction k generalizing f with
| zero =>
have hf : f = 0 := by simpa using hk
refine ⟨Fin.elim0, ?_⟩
subst f
funext n
simp [newtonSequence]
| succ k ih =>
have hdiff : (forwardDiff^[k]) (forwardDiff f) = 0 := by
simpa only [Function.iterate_succ_apply] using hk
obtain ⟨c, hc⟩ := ih (forwardDiff f) hdiff
refine ⟨Fin.cons (f 0) c, intSequence_eq_of_forwardDiff_eq ?_ ?_⟩
· rw [forwardDiff_newtonSequence_cons]
exact hc
· simp
end AddCommGroup
def newtonBasisPolynomial (k : ℕ) : Polynomial ℝ :=
Polynomial.C ((k.factorial : ℝ)⁻¹) * descPochhammer ℝ k
theorem descPochhammer_eval_int_eq_factorial_mul_choose (k : ℕ) (n : ℤ) :
(descPochhammer ℝ k).eval (n : ℝ) =
(k.factorial : ℝ) * ((Ring.choose n k : ℤ) : ℝ) := by
have h : (descPochhammer ℤ k).eval n = (k.factorial : ℤ) * Ring.choose n k := by
simpa only [Polynomial.eval_eq_smeval, nsmul_eq_mul] using
(Ring.descPochhammer_eq_factorial_smul_choose n k)
rw [← descPochhammer_eval_cast ℝ k n, h, Int.cast_mul, Int.cast_natCast]
@[simp] theorem newtonBasisPolynomial_eval_int (k : ℕ) (n : ℤ) :
(newtonBasisPolynomial k).eval (n : ℝ) = ((Ring.choose n k : ℤ) : ℝ) := by
have hk : (k.factorial : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero k)
rw [newtonBasisPolynomial, Polynomial.eval_mul, Polynomial.eval_C,
descPochhammer_eval_int_eq_factorial_mul_choose, ← mul_assoc, inv_mul_cancel₀ hk, one_mul]
def newtonPolynomial {k : ℕ} (c : Fin k → ℝ) : Polynomial ℝ :=
∑ i : Fin k, Polynomial.C (c i) * newtonBasisPolynomial i.val
theorem newtonPolynomial_eval_int {k : ℕ} (c : Fin k → ℝ) (n : ℤ) :
(newtonPolynomial c).eval (n : ℝ) =
∑ i : Fin k, Ring.choose n i.val • c i := by
rw [newtonPolynomial, Polynomial.eval_finsetSum]
apply Finset.sum_congr rfl
intro i hi
simp only [Polynomial.eval_mul, Polynomial.eval_C,
newtonBasisPolynomial_eval_int, zsmul_eq_mul]
exact mul_comm (c i) ((Ring.choose n i.val : ℤ) : ℝ)
theorem exists_polynomial_lift_of_iter_forwardDiff_eq_zero
(f : ℤ → AddCircle (1 : ℝ)) (k : ℕ) (hk : (forwardDiff^[k]) f = 0) :
∃ P : Polynomial ℝ, ∀ n : ℤ, ((P.eval (n : ℝ) : ℝ) : AddCircle (1 : ℝ)) = f n := by
obtain ⟨c, hc⟩ := exists_newtonSequence_of_iter_forwardDiff_eq_zero k f hk
have hrepr : ∀ i : Fin k, ∃ x : ℝ, (x : AddCircle (1 : ℝ)) = c i := by
intro i
exact QuotientAddGroup.mk_surjective (c i)
choose a ha using hrepr
refine ⟨newtonPolynomial a, ?_⟩
intro n
change QuotientAddGroup.mk' (AddSubgroup.zmultiples (1 : ℝ))
((newtonPolynomial a).eval (n : ℝ)) = f n
rw [newtonPolynomial_eval_int, map_sum, hc]
simp only [newtonSequence, map_zsmul, QuotientAddGroup.mk'_apply, ha]
end PeriodicTilingThree
end
end OAI
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