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fcab081 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | import OAI.Combinatorics.HadwigerCounterexample.BilinearCompression
namespace OAI
/-! # Channel Factorization -/
noncomputable section
namespace HadwigerCounterexample
open Matrix LinearMap Submodule Module
open scoped BigOperators
variable {p : Shape} {n : ℕ}
variable {I J : Type} [Fintype I] [Fintype J]
/-- Pairing rank after imposing orthogonality constraints. -/
theorem biorth_exists (S T : Space (Vec I)) (r : ℕ)
(h : r+dim S+dim T≤Fintype.card I) :
∃ A B : Mat I (Fin r), colSp A≤ annih S ∧ colSp B≤ annih T ∧ Aᵀ*B=1 := by
classical
let V := annih S
let W := annih T
let M := (pack V)ᵀ * pack W
let t := M.rank
have ht : r≤t := by
have h1 := rank_ge_mul (pack V)ᵀ (pack W)
rw [Matrix.rank_transpose,mrk_inj.mp (pack_inj V),mrk_inj.mp (pack_inj W),
Fintype.card_fin,Fintype.card_fin] at h1
have h2 : dim S+dim V=Fintype.card I := annih_dim S
have h3 : dim T+dim W=Fintype.card I := annih_dim T
have he (a b c d t r e : ℕ) (h : r+a+b ≤ e) (hx : a+c=e) (hy : b+d=e) (ht : c+d≤t+e) :
r≤t := by omega
exact he _ _ _ _ _ _ _ h h2 h3 h1
obtain ⟨A,B,hA,hB,hM⟩ := rank_product M
obtain ⟨L,hL⟩ := mat_left_inv hA
obtain ⟨R,hR⟩ := mat_left_inv (P:=Bᵀ) (transpose_surj_iff.mp (by rw [Matrix.transpose_transpose]; exact hB))
let E := embedMat (Fin.castLE ht)
refine ⟨pack V * (Lᵀ * E),pack W*(Rᵀ*E),?_,?_,?_⟩
· rintro x ⟨x,rfl⟩
rw [Matrix.mulVecLin_apply,← Matrix.mulVec_mulVec]
exact (pack_mem_iff ..).mpr ⟨_,rfl⟩
· rintro x ⟨x,rfl⟩
rw [Matrix.mulVecLin_apply,← Matrix.mulVec_mulVec]
exact (pack_mem_iff ..).mpr ⟨_,rfl⟩
have he : L*M*Rᵀ=1 := by
have hg : B*Rᵀ=1 := by
simpa only [Matrix.transpose_mul, Matrix.transpose_transpose, Matrix.transpose_one] using
congrArg Matrix.transpose hR
refine (congrArg (fun X => L*X*Rᵀ) hM).symm.trans ?_
rw [← Matrix.mul_assoc L,hL,Matrix.one_mul,hg]
have heq : Eᵀ * E=1 := embedMat_self (Fin.castLE_injective ht)
rw [Matrix.transpose_mul,Matrix.transpose_mul,Matrix.transpose_transpose]
calc _ = Eᵀ*(L*M*Rᵀ)*E := by simp only [M,Matrix.mul_assoc]
_ = _ := by rw [he,Matrix.mul_one,heq]
/-- split desired respective bounded-rank bilinear form -/
theorem pure_chan_split (M : Mat J J) (S T : Space (Vec I)) (X Y : Space (Vec J))
(h : M∈box (annih X) (annih Y)) (ht : M.rank+dim S+dim T ≤ Fintype.card I) :
∃ A ∈ box (annih S) (annih X),∃ B∈box (annih T) (annih Y), Aᵀ * B=M := by
classical
obtain ⟨P,Q,hP,hQ,hM⟩ := rank_product M
obtain ⟨A,B,hA,hB,he⟩ := biorth_exists S T M.rank ht
obtain ⟨hx,hy⟩ := box_kill.mp h
refine ⟨A * Pᵀ,?_,B*Q,?_,?_⟩
· rw [box_kill]
constructor
· intro x hx; rw [Matrix.transpose_mul,← Matrix.mulVec_mulVec,(colSp_kill ..).mp hA x hx,
Matrix.mulVec_zero]
intro x hv
have hv' : Pᵀ *ᵥ x=0 := by
have hqi := (transpose_surj_iff (P:=Qᵀ)).mp (by rw [Matrix.transpose_transpose]; exact hQ)
apply hqi
rw [Matrix.mulVec_mulVec,← Matrix.transpose_mul,hM,hx x hv,Matrix.mulVec_zero]
rw [←Matrix.mulVec_mulVec,hv',mulVec_zero]
· rw [box_kill]
constructor
· intro x hv; rw [Matrix.transpose_mul,←Matrix.mulVec_mulVec,(colSp_kill ..).mp hB x hv,
Matrix.mulVec_zero]
intro y hv
have hv' : Q *ᵥ y=0 := by
apply hP
rw [Matrix.mulVec_mulVec,hM,hy y hv,mulVec_zero]
rw [←Matrix.mulVec_mulVec,hv',mulVec_zero]
· rw [Matrix.transpose_mul,Matrix.transpose_transpose,← Matrix.mul_assoc _ B, Matrix.mul_assoc P,
he,Matrix.mul_one,hM]
/-- The final channel change for fixed B endpoint. -/
theorem chan_factor {f b : Plan p n} {P : PureForms p n} {C : ChanOut p n} {j : Bool}
(H T K : ℕ) (hc : outSmall H C) (hK : ∀ z,dim (hPin b z j)≤K)
(hh : T+2*(K+2*H)≤p.h)
(hP : P∈pureSpace p f) (ht : ∀ e,(P e).rank≤T) :
∃ D∈deltaSpace p f b j,obsPair p (chanInner p (C+D) (C+D))=
obsPair p (chanInner p C C)+obsPair p P := by
classical
let S (z : Place p) := hPin b (flipZ p z) j ⊔ priOut p C (flipZ p z)
have h (z : Place p) : dim (S z) ≤ K+2*H :=
(Submodule.finrank_add_le_finrank_add_finrank ..).trans
(add_le_add (hK _) (dim_priOut hc _))
choose A hA B hB hQ using fun e : Comp p.g =>
pure_chan_split (P e) (S (e,true)) (S (e,false)) (f (e,true)) (f (e,false))
((pureSpace_mem _).mp hP e) ((add_le_add
(add_le_add (ht e) (h (e,true))) (h (e,false))).trans
(by
rw [Fintype.card_fin]
rwa [two_mul, ← add_assoc] at hh))
let D : ChanOut p n := fun z => if z.2 then A z.1 else B z.1
have h1 (z : Place p) : D z∈box (annih (S z)) (annih (f z)) := by
cases z with | mk e c => cases c; exact hB e; exact hA e
refine ⟨D,(deltaSpace_mem _).mpr ?_,?_⟩
· intro z
refine box_le ?_ le_rfl (h1 z)
intro x hx
rw [annih_mem]
exact fun y hy => (annih_mem ..).mp hx y (Submodule.mem_sup_left hy)
have hg : chanInner p D D=P := funext (fun e => hQ e)
have he (z : Place p) (i l : Bool) (x y : Ray p n) :
dotProduct ((C (flipZ p z)*IP p n i) *ᵥ x) ((D z*IP p n l)*ᵥ y)=0 := by
have hq : ((C (flipZ p z)*IP p n i) *ᵥ x) ∈ S z :=
Submodule.mem_sup_right (le_priOut C _ i ⟨x,rfl⟩)
have hl : ((D z * IP p n l)*ᵥ y) ∈ annih (S z) := by
rw [←Matrix.mulVec_mulVec]
exact ((box_mem ..).mp (h1 z)).1 ⟨_,rfl⟩
rw [dotProduct_comm]
exact (annih_mem ..).mp hl _ hq
have hd : obsPair p (chanDeriv p C D)=0 := by
rw [←obsI_each]
have hz : eachF p (chanDeriv p C D)=0 := by
unfold chanDeriv; rw [map_add]
funext i e
rw [Pi.add_apply,Pi.add_apply,each_chan_val,each_chan_val]
refine evmat_ext fun x y => ?_
rw [evmat_addM,ev_duom,ev_duom]
have hh1 := he (e,false) i i x y
have hh2 := he (e,true) i i y x
simp only [flipZ,Bool.not_false,Bool.not_true] at hh1 hh2
rw [hh1,
dotProduct_comm ((D (e,true) * IP p n i) *ᵥ _),hh2]
simp [evmat]
rw [hz,map_zero]
rw [obsPair_chan_change,hd,add_zero,hg]
end HadwigerCounterexample
end
end OAI
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