| import OAI.Combinatorics.MatchingPSD.Model |
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| namespace OAI |
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| universe u_1 u_2 u_3 |
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| namespace MatchingSmoothing |
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| section CommutingInvolutions |
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| variable {ι : Type u_1} {V : Type u_2} [AddCommGroup V] [Module ℝ V] |
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| lemma eigen_add_commuting (A B : Module.End ℝ V) (hAB : Commute A B) |
| (v : V) (c : ℝ) (hv : A v = c • v) : |
| A (v + B v) = c • (v + B v) := by |
| have hc : A (B v) = B (A v) := congrArg (fun f : Module.End ℝ V => f v) hAB.eq |
| rw [map_add, hv, hc, hv, map_smul, smul_add] |
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| lemma eigen_sub_commuting (A B : Module.End ℝ V) (hAB : Commute A B) |
| (v : V) (c : ℝ) (hv : A v = c • v) : |
| A (v - B v) = c • (v - B v) := by |
| have hc : A (B v) = B (A v) := congrArg (fun f : Module.End ℝ V => f v) hAB.eq |
| rw [map_sub, hv, hc, hv, map_smul, smul_sub] |
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| lemma simultaneous_sign_eigenvector_on (A : ι → Module.End ℝ V) |
| (hcomm : ∀ i j, Commute (A i) (A j)) |
| (hinv : ∀ i v, A i (A i v) = v) |
| (W : Submodule ℝ V) (hstable : ∀ i v, v ∈ W → A i v ∈ W) |
| (hne : ∃ v ∈ W, v ≠ 0) (s : Finset ι) : |
| ∃ (χ : ι → ℝ) (v : V), v ∈ W ∧ v ≠ 0 ∧ |
| ∀ i ∈ s, (χ i = 1 ∨ χ i = -1) ∧ A i v = χ i • v := by |
| classical |
| induction s using Finset.induction_on with |
| | empty => |
| obtain ⟨v, hv, hv0⟩ := hne |
| exact ⟨fun _ => 1, v, hv, hv0, by simp⟩ |
| | @insert i s his ih => |
| obtain ⟨χ, v, hv, hv0, heig⟩ := ih |
| by_cases hp : v + A i v = 0 |
| · have hm : v - A i v ≠ 0 := by |
| intro hm |
| have htwo : (2 : ℝ) • v = 0 := by |
| rw [two_smul] |
| linear_combination (norm := abel) hp + hm |
| exact hv0 ((smul_eq_zero.mp htwo).resolve_left (by norm_num)) |
| refine ⟨Function.update χ i (-1), v - A i v, |
| W.sub_mem hv (hstable i v hv), hm, ?_⟩ |
| intro j hj |
| rcases Finset.mem_insert.mp hj with rfl | hj |
| · simp only [Function.update_self, neg_smul, one_smul] |
| refine ⟨by simp, ?_⟩ |
| rw [map_sub, hinv] |
| abel |
| · have hji : j ≠ i := by intro h; subst j; exact his hj |
| rw [Function.update_of_ne hji] |
| exact ⟨(heig j hj).1, eigen_sub_commuting (A j) (A i) (hcomm j i) |
| v (χ j) (heig j hj).2⟩ |
| · refine ⟨Function.update χ i 1, v + A i v, |
| W.add_mem hv (hstable i v hv), hp, ?_⟩ |
| intro j hj |
| rcases Finset.mem_insert.mp hj with rfl | hj |
| · simp only [Function.update_self, one_smul] |
| refine ⟨by simp, ?_⟩ |
| rw [map_add, hinv, add_comm] |
| · have hji : j ≠ i := by intro h; subst j; exact his hj |
| rw [Function.update_of_ne hji] |
| exact ⟨(heig j hj).1, eigen_add_commuting (A j) (A i) (hcomm j i) |
| v (χ j) (heig j hj).2⟩ |
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| lemma simultaneous_sign_eigenvector [Fintype ι] (A : ι → Module.End ℝ V) |
| (hcomm : ∀ i j, Commute (A i) (A j)) |
| (hinv : ∀ i v, A i (A i v) = v) |
| (W : Submodule ℝ V) (hstable : ∀ i v, v ∈ W → A i v ∈ W) |
| (hne : ∃ v ∈ W, v ≠ 0) : |
| ∃ (χ : ι → ℝ) (v : V), v ∈ W ∧ v ≠ 0 ∧ |
| ∀ i, (χ i = 1 ∨ χ i = -1) ∧ A i v = χ i • v := by |
| simpa using simultaneous_sign_eigenvector_on A hcomm hinv W hstable hne Finset.univ |
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| lemma simultaneous_sign_eigenvector_preserving [Fintype ι] |
| (A : ι → Module.End ℝ V) (hcomm : ∀ i j, Commute (A i) (A j)) |
| (hinv : ∀ i v, A i (A i v) = v) |
| (W : Submodule ℝ V) (hstable : ∀ i v, v ∈ W → A i v ∈ W) |
| (s : Finset ι) (χ : ι → ℝ) |
| (v : V) (hv : v ∈ W) (hv0 : v ≠ 0) (he : ∀ i ∈ s, A i v = χ i • v) : |
| ∃ (ψ : ι → ℝ) (w : V), w ∈ W ∧ w ≠ 0 ∧ |
| (∀ i, (ψ i = 1 ∨ ψ i = -1) ∧ A i w = ψ i • w) ∧ |
| ∀ i ∈ s, ψ i = χ i := by |
| let U : Submodule ℝ V := |
| { carrier := {w | w ∈ W ∧ ∀ i ∈ s, A i w = χ i • w} |
| zero_mem' := by simp |
| add_mem' := by |
| rintro a b ⟨ha, hea⟩ ⟨hb, heb⟩ |
| exact ⟨W.add_mem ha hb, fun i hi => by rw [map_add, hea i hi, heb i hi, smul_add]⟩ |
| smul_mem' := by |
| rintro c a ⟨ha, hea⟩ |
| exact ⟨W.smul_mem c ha, fun i hi => by rw [map_smul, hea i hi, smul_comm]⟩ } |
| have hU (j : ι) (w : V) (hw : w ∈ U) : A j w ∈ U := by |
| refine ⟨hstable j w hw.1, ?_⟩ |
| intro i hi |
| have hc := congrArg (fun f : Module.End ℝ V => f w) (hcomm i j).eq |
| change A i (A j w) = A j (A i w) at hc |
| rw [hc, hw.2 i hi, map_smul] |
| obtain ⟨ψ, w, hw, hw0, hew⟩ := simultaneous_sign_eigenvector A hcomm hinv U hU |
| ⟨v, ⟨hv, he⟩, hv0⟩ |
| refine ⟨ψ, w, hw.1, hw0, hew, ?_⟩ |
| intro i hi |
| exact smul_left_injective ℝ hw0 ((hew i).2.symm.trans (hw.2 i hi)) |
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| lemma independent_eigenprofiles {κ : Type u_3} (A : ι → Module.End ℝ V) |
| (hcomm : ∀ i j, Commute (A i) (A j)) (χ : κ → ι → ℝ) |
| (hinj : Function.Injective χ) (v : κ → V) (hv : ∀ k, v k ≠ 0) |
| (heig : ∀ k i, A i (v k) = χ k i • v k) : LinearIndependent ℝ v := by |
| have hi := Module.End.independent_iInf_maxGenEigenspace_of_forall_mapsTo A |
| (fun i j c => Module.End.mapsTo_maxGenEigenspace_of_comm (hcomm j i) c) |
| apply (hi.comp hinj).linearIndependent _ _ hv |
| intro k |
| change v k ∈ ⨅ i, (A i).maxGenEigenspace (χ k i) |
| rw [Submodule.mem_iInf] |
| intro i |
| exact Module.End.eigenspace_le_maxGenEigenspace |
| (Module.End.mem_eigenspace_iff.mpr (heig k i)) |
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| lemma binomial_le_finrank_of_eigenvector [Fintype ι] [DecidableEq ι] [FiniteDimensional ℝ V] |
| (A : ι → Module.End ℝ V) (hcomm : ∀ i j, Commute (A i) (A j)) |
| (P : Equiv.Perm ι → V ≃ₗ[ℝ] V) |
| (hconj : ∀ p i v, A (p i) (P p v) = P p (A i v)) |
| (s : Finset ι) (v : V) (hv : v ≠ 0) |
| (heig : ∀ i, A i v = (if i ∈ s then (-1 : ℝ) else 1) • v) : |
| (Fintype.card ι).choose s.card ≤ Module.finrank ℝ V := by |
| classical |
| let T := ↥((Finset.univ : Finset ι).powersetCard s.card) |
| have hp (t : T) : ∃ p : Equiv.Perm ι, s.map p.toEmbedding = t.1 := |
| Equiv.Perm.exists_map_finset_eq s t.1 |
| (Finset.mem_powersetCard.mp t.2).2.symm |
| choose p hp using hp |
| let χ (t : T) (i : ι) : ℝ := if i ∈ t.1 then -1 else 1 |
| have hχ : Function.Injective χ := by |
| intro t u h |
| apply Subtype.ext |
| ext i |
| have hh := congrFun h i |
| dsimp [χ] at hh |
| by_cases ht : i ∈ t.1 <;> by_cases hu : i ∈ u.1 <;> simp_all <;> norm_num at hh |
| have hprof (t : T) (i : ι) : A i (P (p t) v) = χ t i • P (p t) v := by |
| have hi : (p t).symm i ∈ s ↔ i ∈ t.1 := by |
| rw [← hp t] |
| simp |
| by_cases hii : i ∈ t.1 <;> |
| simpa [heig, χ, hi, hii] using hconj (p t) ((p t).symm i) v |
| have hli := independent_eigenprofiles A hcomm χ hχ |
| (fun t => P (p t) v) (fun t => by simpa using hv) hprof |
| have hc := hli.fintype_card_le_finrank |
| simpa only [T, Fintype.card_coe, Finset.card_powersetCard, Finset.card_univ] using hc |
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| lemma binomial_le_finrank_of_eigenvector_mem [Fintype ι] [DecidableEq ι] [FiniteDimensional ℝ V] |
| (A : ι → Module.End ℝ V) (hcomm : ∀ i j, Commute (A i) (A j)) |
| (P : Equiv.Perm ι → V ≃ₗ[ℝ] V) |
| (hconj : ∀ p i v, A (p i) (P p v) = P p (A i v)) |
| (W : Submodule ℝ V) (hW : ∀ p v, v ∈ W → P p v ∈ W) |
| (s : Finset ι) (v : V) (hvW : v ∈ W) (hv : v ≠ 0) |
| (heig : ∀ i, A i v = (if i ∈ s then (-1 : ℝ) else 1) • v) : |
| (Fintype.card ι).choose s.card ≤ Module.finrank ℝ W := by |
| classical |
| let T := ↥((Finset.univ : Finset ι).powersetCard s.card) |
| have hp (t : T) : ∃ p : Equiv.Perm ι, s.map p.toEmbedding = t.1 := |
| Equiv.Perm.exists_map_finset_eq s t.1 |
| (Finset.mem_powersetCard.mp t.2).2.symm |
| choose p hp using hp |
| let χ (t : T) (i : ι) : ℝ := if i ∈ t.1 then -1 else 1 |
| have hχ : Function.Injective χ := by |
| intro t u h |
| apply Subtype.ext |
| ext i |
| have hh := congrFun h i |
| dsimp [χ] at hh |
| by_cases ht : i ∈ t.1 <;> by_cases hu : i ∈ u.1 <;> simp_all <;> norm_num at hh |
| have hprof (t : T) (i : ι) : A i (P (p t) v) = χ t i • P (p t) v := by |
| have hi : (p t).symm i ∈ s ↔ i ∈ t.1 := by |
| rw [← hp t] |
| simp |
| by_cases hii : i ∈ t.1 <;> |
| simpa [heig, χ, hi, hii] using hconj (p t) ((p t).symm i) v |
| have hli := independent_eigenprofiles A hcomm χ hχ |
| (fun t => P (p t) v) (fun t => by simpa using hv) hprof |
| let vw (t : T) : W := ⟨P (p t) v, hW (p t) v hvW⟩ |
| have hliw : LinearIndependent ℝ vw := LinearIndependent.of_comp W.subtype hli |
| have hc := hliw.fintype_card_le_finrank |
| simpa only [T, Fintype.card_coe, Finset.card_powersetCard, Finset.card_univ] using hc |
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| end CommutingInvolutions |
| end MatchingSmoothing |
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| end OAI |
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