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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 | import OAI.Geometry.PinnedDistances.MixtureLaw
namespace OAI
universe u1 u2 u3 u4
section
/- The finite-place absolute values are norm-normalized. Dividing their logarithms
by [K:ℚ] gives the weighted contribution of locally normalized absolute values. -/
noncomputable section
open NumberField
open scoped BigOperators
namespace WeakPinned.Arithmetic
variable {K : Type u1} [Field K]
lemma totallyComplex_of_sq_neg_one (ii : K) (hii : ii ^ 2 = -1) :
NumberField.IsTotallyComplex K := by
constructor
intro v
rw [← InfinitePlace.not_isReal_iff_isComplex]
intro hv
let φ := InfinitePlace.embedding_of_isReal hv
have h := congrArg φ hii
simp only [map_pow, map_neg, map_one] at h
nlinarith [sq_nonneg (φ ii)]
variable [NumberField K]
lemma log_product_formula (z : K) (hz : z ≠ 0) :
(∑ v : InfinitePlace K, (v.mult : ℝ) * Real.log (v z)) +
∑ᶠ v : FinitePlace K, Real.log (v z) = 0 := by
have hi : ∀ v : InfinitePlace K, 0 < v z := fun _ => InfinitePlace.pos_iff.mpr hz
have hf : ∀ v : FinitePlace K, 0 < v z := fun _ => FinitePlace.pos_iff.mpr hz
have h := congrArg Real.log (NumberField.prod_abs_eq_one hz)
rw [Real.log_mul (Finset.prod_ne_zero_iff.mpr fun v _ => pow_ne_zero _ (hi v).ne')
(finprod_ne_zero fun v => (hf v).ne'), Real.log_prod (fun v _ => pow_ne_zero _ (hi v).ne'),
Real.log_finprod hf, Real.log_one] at h
simpa only [Real.log_pow] using h
def archWeight (v : InfinitePlace K) : ℝ :=
(v.mult : ℝ) / Module.finrank ℚ K
lemma sum_archWeight : ∑ v : InfinitePlace K, archWeight v = 1 := by
classical
unfold archWeight
rw [← Finset.sum_div, ← Nat.cast_sum, InfinitePlace.sum_mult_eq, div_self]
exact_mod_cast (Module.finrank_pos (R := ℚ) (M := K)).ne'
lemma archWeight_nonneg (v : InfinitePlace K) : 0 ≤ archWeight v := by
unfold archWeight
positivity
/-- The finitely many nonunit places for a given finite list of differences. -/
lemma finite_nonunit_places {ι : Type u2} [Finite ι] (z : ι → K)
(hz : ∀ i, z i ≠ 0) :
{v : FinitePlace K | ∃ i, v (z i) ≠ 1}.Finite := by
have he : {v : FinitePlace K | ∃ i, v (z i) ≠ 1} =
⋃ i, Function.mulSupport (fun v : FinitePlace K => v (z i)) := by
ext v
simp
rw [he]
exact Set.finite_iUnion (fun i => FinitePlace.hasFiniteMulSupport (hz i))
omit [NumberField K] in
/-- Multiplicativity gives the constant depth sum on each nonzero fiber at
every absolute value; neither the embeddings nor the field are fixed. -/
lemma raw_fiber_identity (v : AbsoluteValue K ℝ) (a b : K) (ha : a ≠ 0) (hb : b ≠ 0) :
-Real.log (v a) + -Real.log (v b) = -Real.log (v (a * b)) := by
rw [map_mul, Real.log_mul (v.pos ha).ne' (v.pos hb).ne']
ring
/-- The algebraic inversion identity used both by the unbounded tree argument
and the bounded Mobius argument. This is valid in any field. -/
lemma paired_coordinate {F : Type u3} [Field F] (a b c d t : F)
(ha : c ≠ a) (h : (c-a)*(d-b)=t) : d = b + t/(c-a) := by
have h' : d-b = t/(c-a) := (eq_div_iff (sub_ne_zero.mpr ha)).mpr
(by simpa [mul_comm] using h)
rw [← h']
ring
lemma paired_difference {F : Type u4} [Field F] (a b c d z w t : F)
(hc : c ≠ a) (hz : z ≠ a)
(hcd : (c-a)*(d-b)=t) (hzw : (z-a)*(w-b)=t) :
w-d = t * (c-z) / ((z-a)*(c-a)) := by
rw [paired_coordinate a b c d t hc hcd, paired_coordinate a b z w t hz hzw]
field_simp
ring
lemma finite_raw_ultra (v : FinitePlace K) {x y z : K}
(_hxy : x ≠ y) (_hyz : y ≠ z) (hxz : x ≠ z) :
min (-Real.log (v (x-y))) (-Real.log (v (y-z))) ≤ -Real.log (v (x-z)) := by
have h := v.add_le (x-y) (y-z)
rw [sub_add_sub_cancel] at h
have hp : 0 < v (x-z) := FinitePlace.pos_iff.mpr (sub_ne_zero.mpr hxz)
have hl := Real.log_le_log hp h
by_cases hc : v (x-y) ≤ v (y-z)
· rw [max_eq_right hc] at hl
exact (min_le_right _ _).trans (neg_le_neg hl)
· rw [max_eq_left (le_of_not_ge hc)] at hl
exact (min_le_left _ _).trans (neg_le_neg hl)
lemma normalized_log_product_formula (z : K) (hz : z ≠ 0) :
(∑ v : InfinitePlace K, archWeight v * (-Real.log (v z))) +
(Module.finrank ℚ K : ℝ)⁻¹ * ∑ᶠ v : FinitePlace K, -Real.log (v z) = 0 := by
have hh := log_product_formula z hz
simp only [archWeight, div_eq_mul_inv]
rw [finsum_neg_distrib]
have he : (∑ v : InfinitePlace K, (v.mult:ℝ) * (Module.finrank ℚ K : ℝ)⁻¹ * (-Real.log (v z))) =
-(Module.finrank ℚ K : ℝ)⁻¹ * ∑ v : InfinitePlace K, (v.mult:ℝ) * Real.log (v z) := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro v _
ring
rw [he]
linear_combination -(Module.finrank ℚ K : ℝ)⁻¹ * hh
end WeakPinned.Arithmetic
end
end
end OAI
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