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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 | import Mathlib.Algebra.Field.ZMod
import Mathlib.LinearAlgebra.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.FieldTheory.Finiteness
import Mathlib.Tactic.Ring
namespace OAI
/-! # Binary Matrices -/
noncomputable section
namespace HadwigerCounterexample
open scoped BigOperators
open Matrix
abbrev F := ZMod 2
abbrev Vec (I : Type) := I → F
abbrev Mat (I J : Type) := Matrix I J F
abbrev Space (V : Type) [AddCommGroup V] [Module F V] := Submodule F V
theorem bit_cases (x : F) : x = 0 ∨ x = 1 :=
(by decide : ∀ x : ZMod 2, x = 0 ∨ x = 1) x
theorem bit_sq (x : F) : x*x = x := by rcases bit_cases x with e|e <;> rw [e] <;> simp
theorem char_two {V : Type*} [AddCommGroup V] [Module F V] (x : V) : x+x=0 :=
ZModModule.add_self x
theorem odd_smul_bit {n : ℕ} (hn : Odd n) (x : F) : n • x = x := by
obtain ⟨m,rfl⟩ := hn
rw [add_smul, one_smul, mul_smul, two_smul,char_two,zero_add]
section Linear
variable {I J K L R S : Type} [Fintype I] [Fintype J]
def evmat (M : Mat I J) (x : Vec I) (y : Vec J) := dotProduct x (M *ᵥ y)
def mpair (M : Mat I J) : Mat I J →ₗ[F] F where
toFun A := ∑ i, ∑ j, M i j * A i j
map_add' A B := by simp only [Matrix.add_apply,mul_add,Finset.sum_add_distrib]
map_smul' b A := by
simp only [Matrix.smul_apply, smul_eq_mul,RingHom.id_apply,Finset.mul_sum,mul_left_comm b]
theorem mpair_comm (M A : Mat I J) : mpair M A = mpair A M := by
dsimp [mpair]
simp only [mul_comm]
theorem mpair_outer (M : Mat I J) (x : Vec I) (y : Vec J) :
mpair M (vecMulVec x y) = evmat M x y := by
dsimp [mpair,evmat,mulVec,dotProduct,vecMulVec]
simp only [Finset.mul_sum,mul_left_comm]
/-- Apply frame maps to the two factors of every tensor. -/
def liftM (P : Mat K I) (Q : Mat L J) : Mat I J →ₗ[F] Mat K L where
toFun A := P*A*Qᵀ
map_add' A B := by rw [Matrix.mul_add,Matrix.add_mul]
map_smul' b A := by simp only [Matrix.mul_smul, Matrix.smul_mul,RingHom.id_apply]
theorem liftM_def (P : Mat K I) (Q : Mat L J) (A : Mat I J) :
liftM P Q A = P*A*Qᵀ := rfl
theorem liftM_outer (P : Mat K I) (Q : Mat L J) (x : Vec I) (y : Vec J) :
liftM P Q (vecMulVec x y) = vecMulVec (P *ᵥ x) (Q *ᵥ y) := by
rw [liftM_def,mul_vecMulVec,vecMulVec_mul,vecMul_transpose]
variable [Fintype K] [Fintype L]
theorem liftM_comp (P : Mat K I) (Q : Mat L J) (P' : Mat R K) (Q' : Mat S L)
(A : Mat I J) :
liftM P' Q' (liftM P Q A) = liftM (P'*P) (Q'*Q) A := by
simp only [liftM_def, Matrix.transpose_mul,Matrix.mul_assoc]
theorem mpair_liftM (P : Mat K I) (Q : Mat L J) (M : Mat K L) (A : Mat I J) :
mpair M (liftM P Q A) = mpair (liftM Pᵀ Qᵀ M) A := by
simp only [liftM_def, mpair, transpose_transpose, LinearMap.coe_mk,AddHom.coe_mk,
Matrix.mul_apply, Finset.sum_mul, Finset.mul_sum, transpose_apply]
conv_lhs =>
arg 2
intro k
rw [Finset.sum_comm]
rw [Finset.sum_comm]
conv_rhs => rw [Finset.sum_comm]
refine Finset.sum_congr rfl fun j _ => ?_
conv_lhs =>
arg 2
intro k
rw [Finset.sum_comm]
rw [Finset.sum_comm]
refine Finset.sum_congr rfl fun i _ => ?_
rw [Finset.sum_comm]
refine Finset.sum_congr rfl fun l _ => Finset.sum_congr rfl fun k _ => ?_
ring
theorem mat_left_inv [DecidableEq I] {P : Mat K I} (hp : Function.Injective P.mulVec) :
∃ Q : Mat I K, Q*P = 1 := by
classical
have hp' : Function.Injective (P.mulVecLin) := hp
obtain ⟨f,hf⟩ := P.mulVecLin.exists_leftInverse_of_injective (LinearMap.ker_eq_bot.mpr hp')
refine ⟨LinearMap.toMatrix' f, ?_⟩
apply Matrix.toLin'.injective
rw [Matrix.toLin'_mul]
simp only [Matrix.toLin'_one]
change ((LinearMap.toMatrix').symm _) ∘ₗ _ = _
rw [LinearEquiv.symm_apply_apply]
exact hf
theorem liftM_inj {P : Mat K I} {Q : Mat L J} (hp : Function.Injective P.mulVec)
(hq : Function.Injective Q.mulVec) : Function.Injective (liftM P Q) := by
classical
obtain ⟨P', hP⟩ := mat_left_inv hp
obtain ⟨Q', hQ⟩ := mat_left_inv hq
intro A B hab
have hh := congrArg (liftM P' Q') hab
rw [liftM_comp,liftM_comp,hP,hQ] at hh
simpa only [liftM_def, Matrix.one_mul, Matrix.transpose_one,
Matrix.mul_one] using hh
end Linear
end HadwigerCounterexample
end
end OAI
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