| import OAI.Combinatorics.HadwigerCounterexample.SingletonPhaseTests |
|
|
| namespace OAI |
|
|
| /-! # Additional Probe Bounds -/ |
|
|
| noncomputable section |
| namespace HadwigerCounterexample |
| open Matrix Module Submodule Function Finset |
| open scoped BigOperators |
| variable {I J K : Type} [Fintype I] [Fintype J] [Fintype K] |
|
|
| private def moreMatrix {r : ℕ} (A : Mat I (Fin r)) (v : Vec I) : Mat I (Fin (r+1)) := |
| Matrix.of (fun i => Fin.cons (v i) (A i)) |
|
|
| omit [Fintype I] in |
| private theorem moreMatrix_act {r : ℕ} (A : Mat I (Fin r)) (v : Vec I) |
| (x : Vec (Fin (r+1))) : |
| moreMatrix A v *ᵥ x=x 0 • v + A *ᵥ (fun i => x i.succ) := by |
| funext i; simp only [Matrix.mulVec,dotProduct,moreMatrix,Matrix.of_apply, |
| Fin.sum_univ_succ,Pi.add_apply,Pi.smul_apply,smul_eq_mul,Fin.cons_zero,Fin.cons_succ] |
| congr 1; exact mul_comm _ _ |
|
|
| omit [Fintype I] in |
| private theorem moreMatrix_inj {r : ℕ} (A : Mat I (Fin r)) (v : Vec I) |
| (ha : Injective A.mulVec) (h : v∉colSp A) : |
| Injective (moreMatrix A v).mulVec := by |
| rw [← Matrix.coe_mulVecLin,← LinearMap.ker_eq_bot, Matrix.ker_mulVecLin_eq_bot_iff] |
| intro x hx |
| rw [moreMatrix_act] at hx |
| let b (i : Fin r) := x i.succ |
| rcases bit_cases (x 0) with he|he |
| · have hh : b=0 := by |
| apply ha |
| rw [he,zero_smul,zero_add] at hx |
| exact hx.trans (Matrix.mulVec_zero _).symm |
| funext i |
| refine Fin.cases he (fun k => ?_) i |
| exact congr_fun hh k |
| · rw [he,one_smul] at hx |
| have heq : v=A *ᵥ b := add_right_cancel (hx.trans (char_two _).symm) |
| exact (h ⟨b,heq.symm⟩).elim |
|
|
| omit [Fintype J] in |
| private theorem moreMatrix_mul {r} (B : Mat J I) (A : Mat I (Fin r)) (v) : |
| B*moreMatrix A v=moreMatrix (B*A) (B *ᵥ v) := by |
| ext i j |
| refine Fin.cases ?_ (fun j => ?_) j |
| · rfl |
| simp only [moreMatrix,Matrix.mul_apply,Matrix.of_apply,Fin.cons_succ] |
|
|
| variable {p : Shape} {n N r : ℕ} {T : Cov p N} |
| variable {q : Place p → ℕ} |
|
|
| private def growFmt (q : Place p→ℕ) (z : Place p) := fun i => |
| q i+(if i=z then 1 else 0) |
|
|
| private theorem grow_sum (q : Place p→ℕ) (z : Place p) : |
| ctot p (growFmt (p:=p) q z)=ctot p q+1 := by simp [growFmt,ctot,Finset.sum_add_distrib] |
|
|
| private def growing (A : DTuple (p:=p) q N) (z : Place p) (v : Vec (Fin N)) : |
| DTuple (p:=p) (growFmt (p:=p) q z) N := fun i => by |
| change Mat (Fin N) (Fin (q i+ if i=z then 1 else 0)) |
| by_cases h : i=z |
| · rw [ite_eq_left h]; exact moreMatrix (A i) v |
| rw [ite_eq_right h]; exact A i |
|
|
| private theorem castMat_inj {r n m : ℕ} (e : r=n) |
| (A : Mat (Fin m) (Fin r)) (h : Injective A.mulVec) : |
| Injective (cast (show Mat (Fin m) (Fin r)=Mat (Fin m) (Fin n) from congrArg (fun t => Mat (Fin m) (Fin t)) e) A).mulVec := by |
| subst n |
| exact h |
| private theorem castMat_prod {r n m : ℕ} (e : r=n) |
| (A : Mat (Fin m) (Fin r)) (B : Mat (Fin m) (Fin m)) : |
| B*(cast (show Mat (Fin m) (Fin r)=Mat (Fin m) (Fin n) from congrArg (fun t => Mat (Fin m) (Fin t)) e) A) = |
| cast (show Mat (Fin m) (Fin r)=Mat (Fin m) (Fin n) from congrArg (fun t => Mat (Fin m) (Fin t)) e) (B*A) := by |
| subst n; rfl |
|
|
| private theorem growth_inj (A : DTuple (p:=p) q N) (z : Place p) (v : Vec (Fin N)) |
| (hv : v∉colSp (A z)) (h : ∀ i, Injective (A i).mulVec) : |
| ∀ i,Injective (growing A z v i).mulVec := by |
| intro i |
| by_cases he : i=z |
| · subst i |
| simp [growing,growFmt] |
| apply castMat_inj _ (moreMatrix (A z) v) (moreMatrix_inj (A z) v (h z) hv) |
| simp |
| simp [growing,growFmt,he] |
| apply castMat_inj _ (A i) (h i) |
| simp [he] |
|
|
| private theorem growth_value (A : DTuple (p:=p) q N) (z : Place p) (v : Vec (Fin N)) |
| (x : UnitType p n N) (S : Party) : |
| thermal p x S (growing A z v)=growing (thermal p x S A) z (heat p x S z *ᵥ v) := by |
| funext i |
| by_cases he : i=z |
| · subst i |
| simp [thermal,growing,growFmt] |
| rw [←moreMatrix_mul] |
| apply castMat_prod |
| simp |
| simp [thermal,growing,growFmt,he] |
| apply castMat_prod |
| simp [he] |
|
|
| theorem column_observe (A : DTuple (p:=p) q N) (z : Place p) (j : Fin (q z)) |
| (x : UnitType p n N) (S : Party) : |
| heat p x S z *ᵥ (A z).col j=(thermal p x S A z).col j := rfl |
|
|
| namespace Bazaar |
| variable (s : Bazaar p n r T) |
|
|
| abbrev Catalog (p : Shape) {N} (T : Cov p N) (r : ℕ) (q : Place p→ℕ) := |
| (M : Atlas p r T) → Batch M N → Finset (DTuple (p:=p) q N) |
|
|
| def atList (s : Bazaar p n r T) (S : Party) (j : Bool) |
| (A : DTuple (p:=p) q N) (F : Catalog p T r q) (x : UnitType p n N) := |
| let M := s.ix x |
| thermal p x S A∈ F M (s.bag M j x) |
|
|
| def probed (p : Shape) {N} {q : Place p→ℕ} (A : DTuple (p:=p) q N) : Cov p N := |
| fun z => colSp (A z) |
|
|
| theorem extend_probe (w : Wgt (UnitType p n N)) (S : Party) (j : Bool) {k : ℕ} |
| (A : DTuple (p:=p) q N) (hq : ctot p q=k) |
| (h : ∀ i, Injective (A i).mulVec) |
| (F : Catalog p T r q) (hF : ∀ M Z,(↑(F M Z).card:ℝ) ≤ bp (0.51*k*N)) |
| (G : Atlas p r T) (hg : ¬ offsetCov G ≤ probed p A) |
| (hb : ∀ M Z,(↑(s.bids w S j G M Z).card:ℝ) ≤ bp (0.51*N)) : |
| ∃ (q : Place p→ℕ) (A' : DTuple (p:=p) q N) |
| (F' : Catalog p T r q), ctot p q=k+1 ∧ |
| (∀ i, Injective (A' i).mulVec) ∧ |
| (∀ M Z,(↑(F' M Z).card:ℝ) ≤ bp (0.51*(k+1)*N)) ∧ |
| ∀ x,s.atList S j A F x → s.frequent w S j G (s.ix x) x → |
| s.atList S j A' F' x := by |
| classical |
| obtain ⟨z,i,hg⟩ : |
| ∃ (z : Place p) (i : Fin (breadth G z)),(src G z).col i∉colSp (A z) := by |
| by_contra! hh |
| apply hg |
| intro z |
| change colSp (src G z) ≤ _ |
| rw [colSp,range_mulVecLin] |
| exact Submodule.span_le.mpr (Set.range_subset_iff.mpr (hh z)) |
| let q' := growFmt (p:=p) q z |
| let v := (src G z).col i |
| let f (M Z) : Finset (DTuple (p:=p) q' N) := |
| ((F M Z) ×ˢ (s.bids w S j G M Z)).image |
| (fun a => growing a.1 z ((a.2 z).col i)) |
| refine ⟨q',growing A z v, f,(grow_sum q z).trans (by rw [hq]), |
| growth_inj A z v hg h,fun M Z => ?_,fun x hx hi => ?_⟩ |
| · have he := bp_pos (0.51*(k:ℝ)*N) |
| have h1 : |
| (↑(f M Z).card:ℝ) ≤ (↑(F M Z).card:ℝ)*(↑(s.bids w S j G M Z).card:ℝ) := by |
| exact_mod_cast (show (f M Z).card ≤ _*_ from |
| Finset.card_image_le.trans_eq (Finset.card_product ..)) |
| apply (h1.trans (mul_le_mul (hF M Z) (hb M Z) (Nat.cast_nonneg _) he.le)).trans_eq |
| rw [←bp_add]; congr 1; ring |
| · rw [atList,growth_value] |
| |
| refine Finset.mem_image.mpr ⟨(thermal p x S A,thermal p x S (src G)),?_,?_⟩ |
| · exact Finset.mem_product.mpr ⟨hx,(s.bids_mem ..).mp hi⟩ |
| · change growing _ _ _ = _ |
| rw [show heat p x S z *ᵥ v=_ from column_observe (src G) z i x S] |
| |
| end Bazaar |
| end HadwigerCounterexample |
| |
| end |
| |
| end OAI |
| |
