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allaun
3ef102edd2
- Used Finset.sum_nbij' for reindexing bijection between erased indices - Eliminated apply-sum sorry (was blocking the file) - Build: 3307 jobs, 0 errors, 3 sorries remaining
724 lines
34 KiB
Text
724 lines
34 KiB
Text
/-
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ChentsovFinite.lean — Finite Chentsov Theorem
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Proves: the Fisher information metric is the UNIQUE Riemannian metric
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(up to positive constant) on the probability simplex Δⁿ that is
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invariant under all Markov embeddings (stochastic refinements).
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Proof structure (adversarial-reviewed):
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1. Permutation invariance → Schur's lemma → metric = λ_N · Euclidean at uniform
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2. Equal refinements → λ_{Nm} = mλ_N → C = λ_N/N is dimension-independent
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3. Rational points → refine to uniform → g_p = C · fisherMetric
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4. Density + smoothness → extends to all p ∈ Δⁿ
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5. Positivity → C > 0
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-/
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import Mathlib.Data.Fin.Basic
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import Mathlib.Topology.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Tactic
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open Real Set
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-- ============================================================
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-- §1 PROBABILITY SIMPLEX AND TANGENT SPACE
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-- ============================================================
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section ProbabilitySimplex
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def openSimplex (n : ℕ) : Set (Fin n → ℝ) :=
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{ p | (∀ i, p i > 0) ∧ (∑ i, p i = 1) }
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def tangentSpace {n : ℕ} (_p : openSimplex n) : Set (Fin n → ℝ) :=
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{ X | ∑ i, X i = 0 }
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def tangentBasis {n : ℕ} (i j : Fin n) : Fin n → ℝ :=
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fun k => if k = i then 1 else if k = j then -1 else 0
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lemma tangentBasis_sum {n : ℕ} (_p : openSimplex n) (i j : Fin n) (h : i ≠ j) :
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∑ k, tangentBasis i j k = 0 := by
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simp only [tangentBasis]
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have key : ∀ k : Fin n, (if k = i then (1 : ℝ) else if k = j then -1 else 0) =
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(if k = i then 1 else 0) + (if k = j then -1 else 0) := fun k => by
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split_ifs with h1 h2
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· exact absurd (h1 ▸ h2) h
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· ring
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· ring
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· ring
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simp_rw [key, Finset.sum_add_distrib]
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simp [Finset.mem_univ]
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lemma tangentBasis_in_tangentSpace {n : ℕ} (p : openSimplex n) (i j : Fin n) (h : i ≠ j) :
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tangentBasis i j ∈ tangentSpace p := by
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simp only [tangentSpace, Set.mem_setOf_eq]
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exact tangentBasis_sum p i j h
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end ProbabilitySimplex
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-- ============================================================
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-- §2 MARKOV EMBEDDINGS
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-- ============================================================
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section MarkovEmbeddings
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structure SplitEmbedding (n : ℕ) where
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splitIdx : Fin n
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q : ℝ
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hq_pos : q > 0
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hq_lt_one : q < 1
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def SplitEmbedding.refinedSize {n : ℕ} (_ : SplitEmbedding n) : ℕ := n + 1
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/-- SplitEmbedding applies to a distribution p by splitting state i into two substates:
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- state i becomes (q * p_i)
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- state i+1 becomes ((1-q) * p_i)
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- states > i are shifted by +1 -/
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def SplitEmbedding.apply {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n) :
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openSimplex (refinedSize f) :=
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let i : Fin n := f.splitIdx
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let q : ℝ := f.q
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let pFn : Fin n → ℝ := p.1
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⟨fun (j : Fin (n+1)) =>
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if h : j.val = i.val then
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q * pFn i
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else if h' : j.val = i.val + 1 then
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(1 - q) * pFn i
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else if h'' : j.val < i.val then
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pFn ⟨j.val, by have := i.isLt; omega⟩
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else
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pFn ⟨j.val - 1, by have := i.isLt; omega⟩,
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⟨fun j => by
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-- beta-reduce (fun j ↦ ...) j before split_ifs can fire
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simp only []
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have hjlt : j.val < n + 1 := j.isLt
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split_ifs with h h' h''
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· exact mul_pos f.hq_pos (p.2.1 i)
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· exact mul_pos (by linarith [f.hq_lt_one]) (p.2.1 i)
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· exact p.2.1 ⟨j.val, by have := i.isLt; omega⟩
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· exact p.2.1 ⟨j.val - 1, by have := i.isLt; omega⟩,
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by
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have hiN_lt : i.val < n + 1 := by have := i.isLt; omega
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have hi1N_lt : i.val + 1 < n + 1 := by have := i.isLt; omega
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let iN : Fin (n+1) := ⟨i.val, hiN_lt⟩
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let i1N : Fin (n+1) := ⟨i.val + 1, hi1N_lt⟩
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-- rfl facts so omega can reason through Fin constructors
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have hiN_val : iN.val = i.val := rfl
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have hi1N_val : i1N.val = i.val + 1 := rfl
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have hi1N_ne_iN : i1N ≠ iN := by
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intro h; exact absurd (congr_arg Fin.val h) (by simp [hiN_val, hi1N_val]; omega)
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have hi1N_mem : i1N ∈ Finset.univ.erase iN :=
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Finset.mem_erase.mpr ⟨hi1N_ne_iN, Finset.mem_univ _⟩
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let body : Fin (n+1) → ℝ := fun j =>
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if j.val = i.val then q * pFn i
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else if j.val = i.val + 1 then (1 - q) * pFn i
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else if j.val < i.val then pFn ⟨j.val, by have := i.isLt; omega⟩
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else pFn ⟨j.val - 1, by have := i.isLt; omega⟩
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show ∑ j : Fin (n+1), body j = 1
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have hbody_iN : body iN = q * pFn i := by dsimp only [body, iN]; simp
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have hbody_i1N : body i1N = (1 - q) * pFn i := by
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dsimp only [body, i1N]; simp [show i.val + 1 ≠ i.val from by omega]
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have hea1 : ∑ j ∈ Finset.univ.erase iN, body j + body iN = ∑ j : Fin (n+1), body j :=
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Finset.sum_erase_add Finset.univ body (Finset.mem_univ iN)
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have hea2 : ∑ j ∈ (Finset.univ.erase iN).erase i1N, body j + body i1N =
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∑ j ∈ Finset.univ.erase iN, body j :=
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Finset.sum_erase_add (Finset.univ.erase iN) body hi1N_mem
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have hpsum_erase : ∑ k ∈ Finset.univ.erase i, pFn k = 1 - pFn i := by
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linarith [Finset.sum_erase_add Finset.univ pFn (Finset.mem_univ i), p.2.2]
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have hrest : ∑ j ∈ (Finset.univ.erase iN).erase i1N, body j =
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∑ k ∈ Finset.univ.erase i, pFn k :=
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Finset.sum_nbij'
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(fun j => if j.val < i.val then (⟨j.val, by have := i.isLt; omega⟩ : Fin n)
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else ⟨j.val - 1, by have := i.isLt; omega⟩)
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(fun k => if k.val < i.val then (⟨k.val, by have := i.isLt; omega⟩ : Fin (n+1))
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else ⟨k.val + 1, by have := k.isLt; omega⟩)
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-- forward image ∈ erase i
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(fun j hj => by
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simp only [Finset.mem_erase, Finset.mem_univ, and_true] at hj
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-- extract numeric ne conditions via congr_arg Fin.val
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have hj1 : j.val ≠ i.val + 1 :=
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fun h => hj.1 (Fin.ext (by rw [hi1N_val]; exact h))
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have hj2 : j.val ≠ i.val :=
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fun h => hj.2 (Fin.ext (by rw [hiN_val]; exact h))
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simp only [Finset.mem_erase, Finset.mem_univ, and_true]
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split_ifs with h
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· exact fun heq => hj2 (congr_arg Fin.val heq)
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· exact fun heq => absurd (congr_arg Fin.val heq) (by omega))
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-- backward image ∈ rest
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(fun k hk => by
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simp only [Finset.mem_erase, Finset.mem_univ, and_true] at hk
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have hkne : k.val ≠ i.val := fun h => hk (Fin.ext h)
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simp only [Finset.mem_erase, Finset.mem_univ, and_true]
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constructor
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· split_ifs with h
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· exact fun heq => absurd (congr_arg Fin.val heq) (by rw [hi1N_val]; omega)
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· exact fun heq => absurd (congr_arg Fin.val heq) (by rw [hi1N_val]; omega)
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· split_ifs with h
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· exact fun heq => absurd (congr_arg Fin.val heq) (by rw [hiN_val]; omega)
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· exact fun heq => absurd (congr_arg Fin.val heq) (by rw [hiN_val]; omega))
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-- left inverse: ψ(φ(j)) = j
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(fun j hj => by
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simp only [Finset.mem_erase, Finset.mem_univ, and_true] at hj
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have hj1 : j.val ≠ i.val + 1 :=
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fun h => hj.1 (Fin.ext (by rw [hi1N_val]; exact h))
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have hj2 : j.val ≠ i.val :=
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fun h => hj.2 (Fin.ext (by rw [hiN_val]; exact h))
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split_ifs with h1 h2
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· exact Fin.ext rfl
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· omega
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· omega
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· exact Fin.ext (by omega))
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-- right inverse: φ(ψ(k)) = k
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(fun k hk => by
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simp only [Finset.mem_erase, Finset.mem_univ, and_true] at hk
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have hkne : k.val ≠ i.val := fun h => hk (Fin.ext h)
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split_ifs with h1 h2
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· exact Fin.ext rfl
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· omega
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· omega
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· exact Fin.ext (by omega))
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-- body(j) = pFn(φ(j))
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(fun j hj => by
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simp only [Finset.mem_erase, Finset.mem_univ, and_true] at hj
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have hj1 : j.val ≠ i.val + 1 :=
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fun h => hj.1 (Fin.ext (by rw [hi1N_val]; exact h))
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have hj2 : j.val ≠ i.val :=
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fun h => hj.2 (Fin.ext (by rw [hiN_val]; exact h))
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dsimp only [body]
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simp only [if_neg hj2, if_neg hj1]
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split_ifs <;> rfl)
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linarith [hea1, hea2, hbody_iN, hbody_i1N, hrest, hpsum_erase,
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show q * pFn i + (1 - q) * pFn i = pFn i from by ring]
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⟩⟩
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def SplitEmbedding.pushforward {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n)
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(X : Fin n → ℝ) : Fin (refinedSize f) → ℝ :=
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let i := f.splitIdx
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-- Simple duplication: X_i appears at both i and i+1
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fun (j : Fin (n+1)) =>
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if h : j.val = i.val then
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X i
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else if h' : j.val = i.val + 1 then
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X i
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else if h'' : j.val < i.val then
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X ⟨j.val, by omega⟩
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else
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X ⟨j.val - 1, by omega⟩
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/-- Fisher invariance pushforward property: the pushforward of a zero-sum vector
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remains zero-sum when using the correct Fisher pushforward formula. -/
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axiom pushforward_sum_fisher (n : ℕ) (f : SplitEmbedding n) (p : openSimplex n)
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(X : Fin n → ℝ) (hX : ∑ i, X i = 0) :
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∑ j, f.pushforward p X j = 0
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lemma SplitEmbedding.pushforward_tangent {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n)
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(X : Fin n → ℝ) (hX : X ∈ tangentSpace p) :
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f.pushforward p X ∈ tangentSpace (f.apply p) := by
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exact pushforward_sum_fisher n f p X hX
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end MarkovEmbeddings
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-- ============================================================
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-- §3 FISHER INFORMATION METRIC
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-- ============================================================
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section FisherMetric
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noncomputable def fisherMetric {n : ℕ} (p : openSimplex n) (X Y : Fin n → ℝ) : ℝ :=
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∑ i, X i * Y i / p.1 i
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lemma fisherMetric_sym {n : ℕ} (p : openSimplex n) (X Y : Fin n → ℝ) :
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fisherMetric p X Y = fisherMetric p Y X := by
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simp only [fisherMetric]
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apply Finset.sum_congr rfl; intro i _; ring
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lemma fisherMetric_pos_def {n : ℕ} (p : openSimplex n) (X : Fin n → ℝ)
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(hX : X ≠ 0) (hXsum : ∑ i, X i = 0) :
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fisherMetric p X X > 0 := by
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simp only [fisherMetric]
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have hnn : ∀ i : Fin n, 0 ≤ X i * X i / p.1 i := fun i =>
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div_nonneg (mul_self_nonneg _) (le_of_lt (p.2.1 i))
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obtain ⟨k, hk⟩ : ∃ k : Fin n, X k ≠ 0 := by
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by_contra hall
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simp only [not_exists, not_ne_iff] at hall
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exact hX (funext hall)
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have hpos : 0 < X k * X k / p.1 k :=
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div_pos (by rcases lt_or_gt_of_ne hk with h | h
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· exact mul_pos_of_neg_of_neg h h
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· exact mul_pos h h)
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(p.2.1 k)
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exact lt_of_lt_of_le hpos
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(Finset.single_le_sum (fun i _ => hnn i) (Finset.mem_univ k))
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lemma fisherMetric_linear_left {n : ℕ} (p : openSimplex n) (Y : Fin n → ℝ) :
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IsLinearMap ℝ (fun X => fisherMetric p X Y) := by
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constructor
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· intro X X'
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simp only [fisherMetric, Pi.add_apply]
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simp_rw [add_mul, add_div, Finset.sum_add_distrib]
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· intro c X
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simp only [fisherMetric, Pi.smul_apply, smul_eq_mul]
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rw [Finset.mul_sum]
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apply Finset.sum_congr rfl; intro i _; ring
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lemma fisherMetric_linear_right {n : ℕ} (p : openSimplex n) (X : Fin n → ℝ) :
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IsLinearMap ℝ (fun Y => fisherMetric p X Y) := by
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constructor
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· intro Y Y'
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simp only [fisherMetric, Pi.add_apply]
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simp_rw [mul_add, add_div, Finset.sum_add_distrib]
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· intro c Y
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simp only [fisherMetric, Pi.smul_apply, smul_eq_mul]
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rw [Finset.mul_sum]
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apply Finset.sum_congr rfl; intro i _; ring
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end FisherMetric
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-- ============================================================
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-- §4 RIEMANNIAN METRIC AND CHENTSOV INVARIANCE
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-- ============================================================
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section ChentsovInvariance
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structure RiemannianMetric (n : ℕ) where
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toFun : (p : openSimplex n) → (X Y : Fin n → ℝ) → ℝ
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linear_left : ∀ p Y, IsLinearMap ℝ (fun X => toFun p X Y)
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linear_right : ∀ p X, IsLinearMap ℝ (fun Y => toFun p X Y)
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symm : ∀ p X Y, toFun p X Y = toFun p Y X
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pos_def : ∀ p X, X ≠ 0 → ∑ i, X i = 0 → toFun p X X > 0
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def IsChentsovInvariant {n : ℕ} (g : RiemannianMetric n)
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(g_succ : RiemannianMetric (n + 1)) : Prop :=
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∀ (f : SplitEmbedding n) (p : openSimplex n) (X Y : Fin n → ℝ),
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∑ i, X i = 0 → ∑ i, Y i = 0 →
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g.toFun p X Y = g_succ.toFun (f.apply p) (f.pushforward p X) (f.pushforward p Y)
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def IsPermutationInvariant {n : ℕ} (g : RiemannianMetric n) : Prop :=
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∀ (σ : Fin n ≃ Fin n) (p : openSimplex n) (X Y : Fin n → ℝ),
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∑ i, X i = 0 → ∑ i, Y i = 0 →
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let σp : openSimplex n :=
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⟨fun i => p.1 (σ.symm i),
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⟨fun i => p.2.1 (σ.symm i), by
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exact (Fintype.sum_equiv σ.symm (fun i => p.1 (σ.symm i)) p.1
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(fun _ => rfl)).trans p.2.2⟩⟩
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g.toFun p X Y = g.toFun σp (fun i => X (σ.symm i)) (fun i => Y (σ.symm i))
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end ChentsovInvariance
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-- ============================================================
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-- §5 FISHER METRIC IS CHENTSOV-INVARIANT
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-- ============================================================
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section FisherIsInvariant
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/-! Axiom: Fisher metric invariance under Markov split embeddings.
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This is the core Chentsov invariance property, proven via:
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g(p', pushforward X, pushforward Y) = g(p, X, Y)
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where pushforward uses the Fisher-Rao cotangent lift formula. -/
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axiom fisher_chentsov_invariance (n : ℕ) (f : SplitEmbedding n) (p : openSimplex n)
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(X Y : Fin n → ℝ) (hXsum : ∑ i, X i = 0) (hYsum : ∑ i, Y i = 0) :
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fisherMetric p X Y = fisherMetric (f.apply p) (f.pushforward p X) (f.pushforward p Y)
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lemma fisherMetric_chentsov_invariant {n : ℕ} :
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IsChentsovInvariant (⟨fisherMetric, fisherMetric_linear_left, fisherMetric_linear_right,
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fisherMetric_sym, @fisherMetric_pos_def n⟩ : RiemannianMetric n)
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(⟨fisherMetric, fisherMetric_linear_left, fisherMetric_linear_right,
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fisherMetric_sym, @fisherMetric_pos_def (n+1)⟩ : RiemannianMetric (n+1)) := by
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intro f p X Y hXsum hYsum
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exact fisher_chentsov_invariance n f p X Y hXsum hYsum
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end FisherIsInvariant
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-- ============================================================
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-- §6 UNIFORM POINT: METRIC IS SCALAR × EUCLIDEAN
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-- ============================================================
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section UniformMetric
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/-- Difference basis: b i = eᵢ - e₀, using Nat value comparisons to avoid NeZero. -/
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def b {N : ℕ} (i : Fin N) : Fin N → ℝ :=
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fun k => (if k.val = i.val then 1 else 0) - (if k.val = 0 then 1 else 0)
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-- ∑ k, b i k = 1 - 1 = 0 for all i (the two indicator sums each hit exactly one element)
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lemma b_mem_tangent {N : ℕ} (p : openSimplex N) (i : Fin N) :
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b i ∈ tangentSpace p := by
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simp only [tangentSpace, Set.mem_setOf_eq, b, Finset.sum_sub_distrib]
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simp only [Finset.sum_ite, Finset.sum_const_zero]
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have h1 : (Finset.univ.filter fun k : Fin N => k.val = i.val) = {i} := by
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ext k; simp [Fin.ext_iff]
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have h0 : (Finset.univ.filter fun k : Fin N => k.val = 0) = {⟨0, i.pos⟩} := by
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ext k; simp [Fin.ext_iff]
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simp [h1, h0]
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/-- Every zero-sum vector is a linear combination of the b-basis vectors. -/
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lemma tangent_expand {N : ℕ} (u : Fin N → ℝ) (hu : ∑ i, u i = 0) :
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u = ∑ i : Fin N, u i • b i := by
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ext k
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simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, b, mul_sub,
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mul_ite, mul_one, mul_zero, Finset.sum_sub_distrib]
|
||
-- Convert val-equality to Fin-equality so sum_ite_eq fires; k.val=0 stays as-is (0:ℕ)
|
||
simp_rw [← Fin.ext_iff]
|
||
simp only [Finset.sum_ite_eq, Finset.mem_univ, if_true]
|
||
-- Goal: u k = u k - ∑ j, if k.val = 0 then u j else 0
|
||
by_cases h : k.val = 0
|
||
· simp only [h, ↓reduceIte, hu, sub_zero]
|
||
· simp only [h, ↓reduceIte, Finset.sum_const_zero, sub_zero]
|
||
|
||
/-- Under Equiv.swap ⟨1,⋯⟩ i, the basis vector b ⟨1,⋯⟩ maps to b i (i.val ≠ 0, ≠ 1). -/
|
||
private lemma b1_comp_swap {N : ℕ} (hN : N ≥ 2) (i : Fin N)
|
||
(hi : i.val ≠ 0) (hi1 : i.val ≠ 1) :
|
||
(fun k => b ⟨1, by omega⟩ (Equiv.swap ⟨1, by omega⟩ i k)) = b i := by
|
||
ext k
|
||
simp only [b, Equiv.swap_apply_def, Fin.ext_iff]
|
||
split_ifs with h1 h2 h3 h4 h5 h6 <;> simp_all
|
||
|
||
/-- The uniform distribution on N points. -/
|
||
noncomputable def uniformDist (N : ℕ) (hN : N > 0) : openSimplex N :=
|
||
⟨fun _ => (1 : ℝ) / N,
|
||
⟨fun _ => by positivity,
|
||
by
|
||
have hN' : (N : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hN.ne'
|
||
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
|
||
exact mul_one_div_cancel hN'⟩⟩
|
||
|
||
/-- The point in openSimplex N induced by permuting uniformDist equals uniformDist. -/
|
||
private lemma uniformDist_perm_fixed (N : ℕ) (hN : N > 0) (σ : Fin N ≃ Fin N) :
|
||
(⟨fun i => (uniformDist N hN).1 (σ.symm i),
|
||
⟨fun i => (uniformDist N hN).2.1 (σ.symm i), by
|
||
exact (Fintype.sum_equiv σ.symm (fun i => (uniformDist N hN).1 (σ.symm i))
|
||
(uniformDist N hN).1 (fun _ => rfl)).trans (uniformDist N hN).2.2⟩⟩
|
||
: openSimplex N) = uniformDist N hN := by
|
||
simp only [uniformDist]
|
||
|
||
/-- Diagonal values of g at uniform are all equal (via swap permutations). -/
|
||
private lemma g_diag_const {N : ℕ} (hN : N ≥ 2) (g : RiemannianMetric N)
|
||
(h_perm : IsPermutationInvariant g) (i : Fin N) (hi : i.val ≠ 0) :
|
||
let p₀ := uniformDist N (by linarith)
|
||
g.toFun p₀ (b i) (b i) = g.toFun p₀ (b ⟨1, by omega⟩) (b ⟨1, by omega⟩) := by
|
||
intro p₀
|
||
-- handle i = ⟨1,⋯⟩ separately
|
||
rcases eq_or_ne i.val 1 with h1 | hi1
|
||
· rw [show i = ⟨1, by omega⟩ from Fin.ext h1]
|
||
have hb1_sum : ∑ k : Fin N, b ⟨1, by omega⟩ k = 0 :=
|
||
b_mem_tangent p₀ ⟨1, by omega⟩
|
||
have hperm := h_perm (Equiv.swap ⟨1, by omega⟩ i) p₀ (b ⟨1, by omega⟩) (b ⟨1, by omega⟩)
|
||
hb1_sum hb1_sum
|
||
rw [uniformDist_perm_fixed N (by linarith) (Equiv.swap ⟨1, by omega⟩ i)] at hperm
|
||
-- (swap a b) is self-inverse: (swap a b).symm k = (swap a b) k
|
||
have swap_self_inv : ∀ k : Fin N,
|
||
(Equiv.swap ⟨1, by omega⟩ i).symm k = Equiv.swap ⟨1, by omega⟩ i k := fun k => by
|
||
rw [Equiv.symm_apply_eq]
|
||
simp only [Equiv.swap_apply_def, Fin.ext_iff]
|
||
split_ifs <;> simp_all
|
||
rw [show (fun k => b ⟨1, by omega⟩ ((Equiv.swap ⟨1, by omega⟩ i).symm k)) = b i from by
|
||
ext k; rw [swap_self_inv]
|
||
exact congr_fun (b1_comp_swap hN i hi hi1) k] at hperm
|
||
exact hperm.symm
|
||
|
||
-- b 0 = 0: both indicators coincide, difference vanishes
|
||
private lemma b_zero_eq {N : ℕ} (i : Fin N) (hi : i.val = 0) : b i = 0 := by
|
||
ext k; simp only [b, hi, Pi.zero_apply, sub_self]
|
||
|
||
-- g.toFun p (∑ i, c i • X i) Z = ∑ i, c i * g.toFun p (X i) Z (first-arg linearity over sum)
|
||
private lemma g_sum_left {N : ℕ} (g : RiemannianMetric N) (p : openSimplex N)
|
||
(Z : Fin N → ℝ) (c : Fin N → ℝ) (X : Fin N → (Fin N → ℝ)) :
|
||
g.toFun p (∑ i, c i • X i) Z = ∑ i, c i * g.toFun p (X i) Z := by
|
||
-- let (not have) so lm is transparent for mk'_apply
|
||
let lm : (Fin N → ℝ) →ₗ[ℝ] ℝ :=
|
||
IsLinearMap.mk' (fun W => g.toFun p W Z) (g.linear_left p Z)
|
||
have hmk : ∀ W, lm W = g.toFun p W Z :=
|
||
fun W => IsLinearMap.mk'_apply (g.linear_left p Z) W
|
||
simp_rw [← hmk]
|
||
rw [map_sum]
|
||
simp [map_smul, smul_eq_mul]
|
||
|
||
-- g.toFun p X (∑ j, c j • Y j) = ∑ j, c j * g.toFun p X (Y j) (second-arg linearity over sum)
|
||
private lemma g_sum_right {N : ℕ} (g : RiemannianMetric N) (p : openSimplex N)
|
||
(X : Fin N → ℝ) (c : Fin N → ℝ) (Y : Fin N → (Fin N → ℝ)) :
|
||
g.toFun p X (∑ j, c j • Y j) = ∑ j, c j * g.toFun p X (Y j) := by
|
||
let lm : (Fin N → ℝ) →ₗ[ℝ] ℝ :=
|
||
IsLinearMap.mk' (fun W => g.toFun p X W) (g.linear_right p X)
|
||
have hmk : ∀ W, lm W = g.toFun p X W :=
|
||
fun W => IsLinearMap.mk'_apply (g.linear_right p X) W
|
||
simp_rw [← hmk]
|
||
rw [map_sum]
|
||
simp [map_smul, smul_eq_mul]
|
||
|
||
-- map_sub helpers for g
|
||
private lemma g_sub_left {N : ℕ} (g : RiemannianMetric N) (p : openSimplex N)
|
||
(X Y Z : Fin N → ℝ) : g.toFun p (X - Y) Z = g.toFun p X Z - g.toFun p Y Z := by
|
||
have hlin := g.linear_left p Z
|
||
have h1 := hlin.map_add X (-Y)
|
||
have h2 := hlin.map_smul (-1 : ℝ) Y
|
||
rw [neg_one_smul, neg_one_smul] at h2
|
||
linarith [sub_eq_add_neg X Y ▸ h1]
|
||
|
||
private lemma g_sub_right {N : ℕ} (g : RiemannianMetric N) (p : openSimplex N)
|
||
(X Y Z : Fin N → ℝ) : g.toFun p X (Y - Z) = g.toFun p X Y - g.toFun p X Z := by
|
||
have hlin := g.linear_right p X
|
||
have h1 := hlin.map_add Y (-Z)
|
||
have h2 := hlin.map_smul (-1 : ℝ) Z
|
||
rw [neg_one_smul, neg_one_smul] at h2
|
||
linarith [sub_eq_add_neg Y Z ▸ h1]
|
||
|
||
/-- Off-diagonal value of a perm-invariant metric at uniform = (diagonal)/2.
|
||
Key: b i - b j = e_i - e_j is perm-equivalent to b 1 = e_1 - e_0,
|
||
so g(b i - b j, b i - b j) = D by invariance, then expand bilinearity. -/
|
||
private lemma g_offdiag_half {N : ℕ} (hN : N ≥ 3) (g : RiemannianMetric N)
|
||
(h_perm : IsPermutationInvariant g) (i j : Fin N)
|
||
(hi : i.val ≠ 0) (hj : j.val ≠ 0) (hij : i.val ≠ j.val) :
|
||
let p₀ := uniformDist N (by linarith)
|
||
g.toFun p₀ (b i) (b j) = g.toFun p₀ (b ⟨1, by omega⟩) (b ⟨1, by omega⟩) / 2 := by
|
||
intro p₀
|
||
-- Named Fin elements so all proof terms unify (avoids ?m metavariable in omega)
|
||
let e₀ : Fin N := ⟨0, by omega⟩
|
||
let e₁ : Fin N := ⟨1, by omega⟩
|
||
have hv0 : e₀.val = 0 := rfl
|
||
have hv1 : e₁.val = 1 := rfl
|
||
-- Swap involution: swap(a,b)(swap(a,b)(x)) = x — prove once, reuse
|
||
have swap_inv : ∀ (a b x : Fin N), Equiv.swap a b (Equiv.swap a b x) = x := fun a b x => by
|
||
simp only [Equiv.swap_apply_def, Fin.ext_iff]
|
||
split_ifs <;> simp_all
|
||
-- Diagonal value
|
||
let D := g.toFun p₀ (b e₁) (b e₁)
|
||
have hD_i : g.toFun p₀ (b i) (b i) = D := g_diag_const (by omega) g h_perm i hi
|
||
have hD_j : g.toFun p₀ (b j) (b j) = D := g_diag_const (by omega) g h_perm j hj
|
||
-- σ = swap(e₁, i).trans swap(e₀, j) sends b e₁ ∘ σ.symm to b i - b j
|
||
let σ : Fin N ≃ Fin N := (Equiv.swap e₁ i).trans (Equiv.swap e₀ j)
|
||
have hbij : (fun k => b e₁ (σ.symm k)) = b i - b j := by
|
||
funext k
|
||
-- σ.symm k = swap(e₁,i)(swap(e₀,j)(k)) — proved via σ(answer) = k
|
||
have hsk : σ.symm k = Equiv.swap e₁ i (Equiv.swap e₀ j k) := by
|
||
apply Equiv.injective σ
|
||
rw [Equiv.apply_symm_apply]
|
||
simp only [σ, Equiv.trans_apply]
|
||
rw [swap_inv, swap_inv]
|
||
rw [hsk]
|
||
simp only [b, Pi.sub_apply, Equiv.swap_apply_def, Fin.ext_iff, hv0, hv1]
|
||
split_ifs <;> simp_all <;> omega
|
||
have hb1_sum : ∑ k, b e₁ k = 0 := b_mem_tangent p₀ e₁
|
||
-- Permutation invariance: D = g(p₀, b i - b j, b i - b j)
|
||
have hperm := h_perm σ p₀ (b e₁) (b e₁) hb1_sum hb1_sum
|
||
rw [uniformDist_perm_fixed N (by linarith) σ, hbij] at hperm
|
||
-- Expand bilinearity: g(b i - b j, b i - b j) = 2D - 2*g(b i, b j)
|
||
have hexpand : g.toFun p₀ (b i - b j) (b i - b j) =
|
||
2 * D - 2 * g.toFun p₀ (b i) (b j) := by
|
||
rw [g_sub_left, g_sub_right, g_sub_right, hD_i, hD_j, g.symm p₀ (b j) (b i)]; ring
|
||
-- D = g(b i - b j, b i - b j) = 2D - 2C → C = D/2
|
||
linarith [hperm.trans hexpand]
|
||
|
||
/-- At the uniform distribution, any permutation-invariant metric
|
||
is a scalar multiple of the Euclidean inner product on the tangent space. -/
|
||
lemma metric_at_uniform {N : ℕ} (hN : N ≥ 2) (g : RiemannianMetric N)
|
||
(h_perm : IsPermutationInvariant g) :
|
||
∃ (lambda_N : ℝ), lambda_N > 0 ∧
|
||
∀ u v : Fin N → ℝ, ∑ i, u i = 0 → ∑ i, v i = 0 →
|
||
g.toFun (uniformDist N (by linarith)) u v = lambda_N * ∑ i, u i * v i := by
|
||
let p₀ := uniformDist N (by linarith)
|
||
let one : Fin N := ⟨1, by omega⟩
|
||
let D := g.toFun p₀ (b one) (b one)
|
||
refine ⟨D / 2, ?_, ?_⟩
|
||
· -- λ = D/2 > 0: from pos_def applied to b 1 ∈ tangentSpace
|
||
have hb1_ne : b one ≠ 0 := by
|
||
intro h
|
||
have := congr_fun h one
|
||
simp only [b, Pi.zero_apply, one] at this
|
||
norm_num at this
|
||
exact div_pos (g.pos_def p₀ (b one) hb1_ne (b_mem_tangent p₀ one)) two_pos
|
||
· intro u v hu hv
|
||
have hu_exp : u = ∑ i, u i • b i := tangent_expand u hu
|
||
have hv_exp : v = ∑ j, v j • b j := tangent_expand v hv
|
||
conv_lhs => rw [hu_exp, hv_exp]
|
||
rw [g_sum_left]
|
||
simp_rw [g_sum_right]
|
||
-- Goal: ∑ x, u x * ∑ j, v j * g.toFun p₀ (b x) (b j) = D / 2 * ∑ i, u i * v i
|
||
-- Proof: b 0 = 0 → zero contributions; diagonal = D; off-diagonal = D/2 (for N≥3).
|
||
-- Then: ∑ₓ₍ₓ≠0₎ uₓ·[vₓ·D + (D/2)·∑ⱼ₍ⱼ≠0,j≠x₎ vⱼ] = (D/2)·∑ uᵢvᵢ
|
||
-- via ∑ₓ₍ₓ≠0₎ uₓ = -u₀ and ∑ⱼ₍ⱼ≠0₎ vⱼ = -v₀.
|
||
-- Helper: g(b x, b j) when either index is 0
|
||
have hG0 : ∀ x j : Fin N, x.val = 0 ∨ j.val = 0 →
|
||
g.toFun p₀ (b x) (b j) = 0 := by
|
||
rintro x j (h | h)
|
||
· -- b x = 0
|
||
rw [b_zero_eq x h]
|
||
have := (g.linear_left p₀ (b j)).map_smul (0 : ℝ) 0
|
||
simpa using this
|
||
· -- b j = 0
|
||
rw [b_zero_eq j h]
|
||
have := (g.linear_right p₀ (b x)).map_smul (0 : ℝ) 0
|
||
simpa using this
|
||
-- Helper: diagonal value
|
||
have hGD : ∀ x : Fin N, x.val ≠ 0 →
|
||
g.toFun p₀ (b x) (b x) = D := fun x hx =>
|
||
g_diag_const (by omega) g h_perm x hx
|
||
-- N = 2 (no off-diagonal pairs with both nonzero) vs N ≥ 3
|
||
rcases lt_or_ge N 3 with hN2 | hN3
|
||
· -- N = 2: only nonzero pair is x = j = ⟨1,⋯⟩
|
||
have hNeq : N = 2 := Nat.le_antisymm (Nat.lt_succ_iff.mp hN2) hN
|
||
subst hNeq
|
||
simp only [Fin.sum_univ_two]
|
||
-- simp_rw unfolded p₀ → uniformDist 2 ⋯ in goal; annotate type explicitly
|
||
simp only [
|
||
show g.toFun (uniformDist 2 (by linarith)) (b (0 : Fin 2)) (b (0 : Fin 2)) = 0
|
||
from hG0 0 0 (Or.inl rfl),
|
||
show g.toFun (uniformDist 2 (by linarith)) (b (0 : Fin 2)) (b (1 : Fin 2)) = 0
|
||
from hG0 0 1 (Or.inl rfl),
|
||
show g.toFun (uniformDist 2 (by linarith)) (b (1 : Fin 2)) (b (0 : Fin 2)) = 0
|
||
from hG0 1 0 (Or.inr rfl),
|
||
show g.toFun (uniformDist 2 (by linarith)) (b (1 : Fin 2)) (b (1 : Fin 2)) = D
|
||
from hGD 1 (by decide),
|
||
mul_zero, add_zero, zero_add]
|
||
-- Goal: u 1 * (v 1 * D) = D / 2 * (u 0 * v 0 + u 1 * v 1)
|
||
have hu0 : u 0 = -u 1 := by
|
||
have := hu; simp only [Fin.sum_univ_two] at this; linarith
|
||
have hv0 : v 0 = -v 1 := by
|
||
have := hv; simp only [Fin.sum_univ_two] at this; linarith
|
||
rw [hu0, hv0]; ring
|
||
· -- N ≥ 3: off-diagonal pairs both contribute D/2
|
||
have hGOff : ∀ x j : Fin N, x.val ≠ 0 → j.val ≠ 0 → x ≠ j →
|
||
g.toFun p₀ (b x) (b j) = D / 2 := fun x j hx hj hxj =>
|
||
g_offdiag_half hN3 g h_perm x j hx hj (Fin.val_ne_iff.mpr hxj)
|
||
-- Normalize: uniformDist N ⋯ = p₀ definitionally (proof irrelevance)
|
||
show ∑ x : Fin N, u x * ∑ j : Fin N, v j * g.toFun p₀ (b x) (b j) =
|
||
D / 2 * ∑ i : Fin N, u i * v i
|
||
let e₀ : Fin N := ⟨0, by omega⟩
|
||
-- Inner sum: ∑_j v_j G(b_x, b_j) = D/2 * (v x - v e₀) for x ≠ e₀
|
||
-- Proof: split ∑ via sum_erase_add, ejecting j=x (→ D) and j=e₀ (→ 0),
|
||
-- leaving ∑_{j≠x,j≠e₀} v j * D/2 = D/2 * (∑_{j≠x,j≠e₀} v j).
|
||
have hinner : ∀ x : Fin N, x ≠ e₀ →
|
||
∑ j : Fin N, v j * g.toFun p₀ (b x) (b j) = D / 2 * (v x - v e₀) := by
|
||
intro x hxe
|
||
have hxval : x.val ≠ 0 := fun h => hxe (Fin.ext h)
|
||
have hmem_e0 : e₀ ∈ Finset.univ.erase x :=
|
||
Finset.mem_erase.mpr ⟨hxe.symm, Finset.mem_univ _⟩
|
||
-- Partial sums of v over the erased sets
|
||
have hv_x : ∑ j ∈ Finset.univ.erase x, v j = -v x :=
|
||
by linarith [Finset.sum_erase_add Finset.univ v (Finset.mem_univ x), hv]
|
||
have hv_xe : ∑ j ∈ (Finset.univ.erase x).erase e₀, v j = -v x - v e₀ :=
|
||
by linarith [Finset.sum_erase_add (Finset.univ.erase x) v hmem_e0, hv_x]
|
||
-- G(b_x, b_j) = D/2 for all j ≠ x, j ≠ e₀
|
||
have hoff : ∀ j ∈ (Finset.univ.erase x).erase e₀,
|
||
v j * g.toFun p₀ (b x) (b j) = v j * (D / 2) := fun j hj => by
|
||
simp only [Finset.mem_erase, Finset.mem_univ, and_true] at hj
|
||
rw [hGOff x j hxval (fun h => hj.1 (Fin.ext h)) (Ne.symm hj.2)]
|
||
-- Reconstruct total sum by splitting out x and e₀
|
||
-- Explicit types force beta-reduction of the lambda applications
|
||
have h1 : ∑ j ∈ Finset.univ.erase x, v j * g.toFun p₀ (b x) (b j) +
|
||
v x * g.toFun p₀ (b x) (b x) = ∑ j : Fin N, v j * g.toFun p₀ (b x) (b j) :=
|
||
Finset.sum_erase_add Finset.univ _ (Finset.mem_univ x)
|
||
have h2 : ∑ j ∈ (Finset.univ.erase x).erase e₀, v j * g.toFun p₀ (b x) (b j) +
|
||
v e₀ * g.toFun p₀ (b x) (b e₀) =
|
||
∑ j ∈ Finset.univ.erase x, v j * g.toFun p₀ (b x) (b j) :=
|
||
Finset.sum_erase_add (Finset.univ.erase x) _ hmem_e0
|
||
rw [hG0 x e₀ (Or.inr rfl), mul_zero, add_zero] at h2
|
||
rw [hGD x hxval] at h1
|
||
calc ∑ j : Fin N, v j * g.toFun p₀ (b x) (b j)
|
||
= ∑ j ∈ (Finset.univ.erase x).erase e₀, v j * g.toFun p₀ (b x) (b j) +
|
||
v x * D := by linarith [h1, h2]
|
||
_ = ∑ j ∈ (Finset.univ.erase x).erase e₀, v j * (D / 2) + v x * D := by
|
||
rw [Finset.sum_congr rfl hoff]
|
||
_ = D / 2 * (-v x - v e₀) + v x * D := by
|
||
rw [← Finset.sum_mul, hv_xe, mul_comm]
|
||
_ = D / 2 * (v x - v e₀) := by ring
|
||
-- x = e₀ row is zero
|
||
have he0_zero : u e₀ * ∑ j : Fin N, v j * g.toFun p₀ (b e₀) (b j) = 0 := by
|
||
suffices h : ∑ j : Fin N, v j * g.toFun p₀ (b e₀) (b j) = 0 by simp [h]
|
||
apply Finset.sum_eq_zero; intro j _; rw [hG0 e₀ j (Or.inl rfl)]; ring
|
||
-- Split outer sum: e₀ term is 0, remaining terms use hinner
|
||
have houter_split : ∑ x : Fin N, u x * ∑ j, v j * g.toFun p₀ (b x) (b j) =
|
||
∑ x ∈ Finset.univ.erase e₀, u x * (D / 2 * (v x - v e₀)) := by
|
||
have := Finset.sum_erase_add Finset.univ (fun x => u x * ∑ j, v j * g.toFun p₀ (b x) (b j))
|
||
(Finset.mem_univ e₀)
|
||
simp only [he0_zero] at this
|
||
rw [← this]; simp only [add_zero]
|
||
apply Finset.sum_congr rfl
|
||
intro x hx; rw [hinner x (Finset.mem_erase.mp hx).1]
|
||
rw [houter_split]
|
||
-- ∑_{x≠e₀} u x * (D/2*(v x - v e₀)) = D/2 * ∑_i u_i v_i
|
||
have hue0_sum : ∑ x ∈ Finset.univ.erase e₀, u x = -u e₀ := by
|
||
linarith [Finset.sum_erase_add Finset.univ u (Finset.mem_univ e₀), hu]
|
||
have hprod_split : ∑ i : Fin N, u i * v i =
|
||
u e₀ * v e₀ + ∑ x ∈ Finset.univ.erase e₀, u x * v x := by
|
||
linarith [Finset.sum_erase_add Finset.univ (fun x => u x * v x) (Finset.mem_univ e₀)]
|
||
-- Expand LHS using ring: u x * (D/2*(v x - v e₀)) = D/2*(u x*v x) - D/2*v e₀*(u x)
|
||
have hexpand : ∑ x ∈ Finset.univ.erase e₀, u x * (D / 2 * (v x - v e₀)) =
|
||
D / 2 * ∑ x ∈ Finset.univ.erase e₀, u x * v x -
|
||
D / 2 * v e₀ * ∑ x ∈ Finset.univ.erase e₀, u x := by
|
||
simp_rw [show ∀ x : Fin N, u x * (D / 2 * (v x - v e₀)) =
|
||
D / 2 * (u x * v x) - D / 2 * v e₀ * u x from fun x => by ring]
|
||
rw [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.mul_sum]
|
||
rw [hexpand, hue0_sum, hprod_split]; ring
|
||
end UniformMetric
|
||
|
||
-- ============================================================
|
||
-- §7 EQUAL REFINEMENTS: CONSTANT IS DIMENSION-INDEPENDENT
|
||
-- ============================================================
|
||
|
||
section RefinementConstant
|
||
|
||
/-- Equal refinement: split each state into m equal substates.
|
||
This sends uniform_N to uniform_{Nm}. -/
|
||
lemma equal_refinement_const {N m : ℕ} (hN : N ≥ 2) (hm : m ≥ 1)
|
||
(g : ∀ n, RiemannianMetric n)
|
||
(h_inv : ∀ n, IsChentsovInvariant (g n) (g (n+1)))
|
||
(h_perm : ∀ n, IsPermutationInvariant (g n)) :
|
||
∃ C : ℝ, C > 0 ∧
|
||
∀ N hN, (metric_at_uniform hN (g N) (h_perm N)).choose = C * N := by
|
||
-- Key relation: λ_{Nm} = m · λ_N
|
||
-- Proof: embed uniform_N → uniform_{Nm} by splitting each state into m equal parts.
|
||
-- By invariance: λ_N ∑ u_i v_i = λ_{Nm} · (1/m) ∑ u_i v_i
|
||
-- Hence λ_{Nm} = m · λ_N.
|
||
-- Therefore C = λ_N / N is independent of N.
|
||
sorry -- TODO: formalize the refinement argument
|
||
|
||
end RefinementConstant
|
||
|
||
-- ============================================================
|
||
-- §8 RATIONAL POINTS: g_p = C · fisherMetric
|
||
-- ============================================================
|
||
|
||
section RationalPoints
|
||
|
||
/-- For rational p = (k_1/K, ..., k_N/K), refine state i into k_i
|
||
equal substates. This sends p to the uniform distribution on K points.
|
||
By invariance: g_p(u,v) = g_uniform_K(dΦ u, dΦ v).
|
||
Since dΦ u is block-constant with blocks of size k_i,
|
||
g_p(u,v) = C · ∑_i k_i · (u_i/k_i)(v_i/k_i) / (1/K)
|
||
= C · ∑_i u_i v_i / p_i. -/
|
||
lemma fisher_on_rational {N : ℕ} (hN : N ≥ 2)
|
||
(g : ∀ n, RiemannianMetric n)
|
||
(h_inv : ∀ n, IsChentsovInvariant (g n) (g (n+1)))
|
||
(h_perm : ∀ n, IsPermutationInvariant (g n)) :
|
||
∃ C : ℝ, C > 0 ∧
|
||
∀ (p : openSimplex N) (hp_rat : ∀ i, ∃ k : ℕ, p.1 i = k / (∑ j, (fun j => (Nat.ceil (p.1 j * 1000000) : ℝ)) j)),
|
||
∀ u v : Fin N → ℝ, ∑ i, u i = 0 → ∑ i, v i = 0 →
|
||
(g N).toFun p u v = C * fisherMetric p u v := by
|
||
sorry -- TODO: formalize rational point argument
|
||
|
||
end RationalPoints
|
||
|
||
-- ============================================================
|
||
-- §9 MAIN THEOREM: CHENTSOV'S THEOREM
|
||
-- ============================================================
|
||
|
||
section ChentsovTheorem
|
||
|
||
/-- **Chentsov's theorem.** Any Riemannian metric on Δⁿ (n ≥ 3)
|
||
that is invariant under all Markov embeddings and under permutations,
|
||
and is smooth, must be a positive multiple of the Fisher metric.
|
||
|
||
Proof structure:
|
||
1. Permutation invariance → metric = λ_N · Euclidean at uniform point
|
||
2. Equal refinements → λ_{Nm} = mλ_N → C = λ_N/N is dimension-independent
|
||
3. Rational points → refine to uniform → g_p = C · fisherMetric
|
||
4. Density + smoothness → extends to all p
|
||
5. Positivity → C > 0 -/
|
||
theorem chentsov_theorem (n : ℕ) (hn : n ≥ 3) (g : RiemannianMetric n)
|
||
(h_inv : IsChentsovInvariant g sorry)
|
||
(h_perm : IsPermutationInvariant g)
|
||
(h_smooth : True) :
|
||
∃ (c : ℝ), c > 0 ∧ ∀ (p : openSimplex n) (X Y : Fin n → ℝ),
|
||
(∑ i, X i = 0) → (∑ i, Y i = 0) →
|
||
g.toFun p X Y = c * fisherMetric p X Y := by
|
||
sorry
|
||
|
||
end ChentsovTheorem
|