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import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
import Mathlib.LinearAlgebra.Charpoly.Basic

import MIL.Common



variable {K : Type*} [Field K] {V : Type*} [AddCommGroup V] [Module K V]


example (a : K) (u v : V) : a • (u + v) = a • u + a • v :=
  smul_add a u v

example (a b : K) (u : V) : (a + b) • u = a • u + b • u :=
  add_smul a b u

example (a b : K) (u : V) : a • b • u = b • a • u :=
  smul_comm a b u

def mkVectorSpace {K X : Type*} [Field K] [AddCommGroup X] (ρ : K →+* AddMonoid.End X) : Module K X where
  smul a x := ρ a x
  one_smul x := show ρ 1 x = x by simp
  mul_smul a b x := show ρ (a * b) _ = _ by rw [map_mul]; rfl
  smul_zero x := map_zero ..
  smul_add a x y := map_add ..
  add_smul a b x := show ρ (a + b) _ = _ by rw [map_add]; rfl
  zero_smul x := show ρ 0 _ = _ by simp

example {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M] :
    Module (Ideal R) (Submodule R M) :=
  inferInstance



variable {W : Type*} [AddCommGroup W] [Module K W]

variable (φ : V →ₗ[K] W)

example (a : K) (v : V) : φ (a • v) = a • φ v :=
  map_smul φ a v

example (v w : V) : φ (v + w) = φ v + φ w :=
  map_add φ v w


variable (ψ : V →ₗ[K] W)

#check (2 • φ + ψ : V →ₗ[K] W)


variable (θ : W →ₗ[K] V)

#check (φ.comp θ : W →ₗ[K] W)
#check (φ ∘ₗ θ : W →ₗ[K] W)


example : V →ₗ[K] V where
  toFun v := 3 • v
  map_add' _ _ := smul_add ..
  map_smul' _ _ := smul_comm ..



#check (φ.map_add' : ∀ x y : V, φ.toFun (x + y) = φ.toFun x + φ.toFun y)
#check (φ.map_add : ∀ x y : V, φ (x + y) = φ x + φ y)
#check (map_add φ : ∀ x y : V, φ (x + y) = φ x + φ y)



#check (LinearMap.lsmul K V 3 : V →ₗ[K] V)
#check (LinearMap.lsmul K V : K →ₗ[K] V →ₗ[K] V)


example (f : V ≃ₗ[K] W) : f ≪≫ₗ f.symm = LinearEquiv.refl K V :=
  f.self_trans_symm

noncomputable example (f : V →ₗ[K] W) (h : Function.Bijective f) : V ≃ₗ[K] W :=
  .ofBijective f h


section binary_product

variable {W : Type*} [AddCommGroup W] [Module K W]
variable {U : Type*} [AddCommGroup U] [Module K U]
variable {T : Type*} [AddCommGroup T] [Module K T]

-- First projection map
example : V × W →ₗ[K] V := LinearMap.fst K V W

-- Second projection map
example : V × W →ₗ[K] W := LinearMap.snd K V W

-- Universal property of the product
example (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : U →ₗ[K]  V × W := LinearMap.prod φ ψ

-- The product map does the expected thing, first component
example (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.fst K V W ∘ₗ LinearMap.prod φ ψ = φ := rfl

-- The product map does the expected thing, second component
example (φ : U →ₗ[K] V) (ψ : U →ₗ[K] W) : LinearMap.snd K V W ∘ₗ LinearMap.prod φ ψ = ψ := rfl

-- We can also combine maps in parallel
example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) : (V × W) →ₗ[K] (U × T) := φ.prodMap ψ

-- This is simply done by combining the projections with the universal property
example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] T) :
  φ.prodMap ψ = (φ ∘ₗ .fst K V W).prod (ψ ∘ₗ .snd K V W) := rfl

-- First inclusion map
example : V →ₗ[K] V × W := LinearMap.inl K V W

-- Second inclusion map
example : W →ₗ[K] V × W := LinearMap.inr K V W

-- Universal property of the sum (aka coproduct)
example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : V × W →ₗ[K] U := φ.coprod ψ

-- The coproduct map does the expected thing, first component
example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inl K V W = φ :=
  LinearMap.coprod_inl φ ψ

-- The coproduct map does the expected thing, second component
example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) : φ.coprod ψ ∘ₗ LinearMap.inr K V W = ψ :=
  LinearMap.coprod_inr φ ψ

-- The coproduct map is defined in the expected way
example (φ : V →ₗ[K] U) (ψ : W →ₗ[K] U) (v : V) (w : W) :
    φ.coprod ψ (v, w) = φ v + ψ w :=
  rfl

end binary_product


section families
open DirectSum

variable {ι : Type*} [DecidableEq ι]
         (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]

-- The universal property of the direct sum assembles maps from the summands to build
-- a map from the direct sum
example (φ : Π i, (V i →ₗ[K] W)) : (⨁ i, V i) →ₗ[K] W :=
  DirectSum.toModule K ι W φ

-- The universal property of the direct product assembles maps into the factors
-- to build a map into the direct product
example (φ : Π i, (W →ₗ[K] V i)) : W →ₗ[K] (Π i, V i) :=
  LinearMap.pi φ

-- The projection maps from the product
example (i : ι) : (Π j, V j) →ₗ[K] V i := LinearMap.proj i

-- The inclusion maps into the sum
example (i : ι) : V i →ₗ[K] (⨁ i, V i) := DirectSum.lof K ι V i

-- The inclusion maps into the product
example (i : ι) : V i →ₗ[K] (Π i, V i) := LinearMap.single K V i

-- In case `ι` is a finite type, there is an isomorphism between the sum and product.
example [Fintype ι] : (⨁ i, V i) ≃ₗ[K] (Π i, V i) :=
  linearEquivFunOnFintype K ι V

end families