| import Mathlib.Algebra.Group.TransferInstance |
| import Mathlib.Algebra.Module.TransferInstance |
| import Mathlib.Algebra.Algebra.Defs |
| import Mathlib.Algebra.Algebra.Opposite |
| import Mathlib.LinearAlgebra.Dimension.Constructions |
| import Mathlib.RingTheory.Finiteness.Prod |
|
|
| namespace OAI |
|
|
| namespace ArExplicit |
|
|
| @[ext] structure Triangular (R F : Type*) where |
| top : R |
| bottom : R |
| off : F |
|
|
| namespace Triangular |
|
|
| variable {R F : Type*} |
|
|
| def coordinates : Triangular R F ≃ (R × R) × F where |
| toFun x := ((x.top, x.bottom), x.off) |
| invFun x := ⟨x.1.1, x.1.2, x.2⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
| |
| instance [AddCommGroup R] [AddCommGroup F] : AddCommGroup (Triangular R F) := |
| coordinates.addCommGroup |
| |
| def coordinateAddEquiv [AddCommGroup R] [AddCommGroup F] : |
| Triangular R F ≃+ (R × R) × F where |
| toEquiv := coordinates |
| map_add' _ _ := rfl |
|
|
| variable [Ring R] [AddCommGroup F] |
| variable [Module R F] [Module Rᵐᵒᵖ F] |
| |
| instance : One (Triangular R F) := ⟨⟨1, 1, 0⟩⟩ |
| |
| instance : Mul (Triangular R F) := |
| ⟨fun x y => ⟨x.top * y.top, x.bottom * y.bottom, |
| x.bottom • y.off + MulOpposite.op y.top • x.off⟩⟩ |
| |
| section BasicCoordinates |
| |
| omit [Module R F] [Module Rᵐᵒᵖ F] |
| |
| @[simp] theorem top_zero : (0 : Triangular R F).top = 0 := rfl |
| @[simp] theorem bottom_zero : (0 : Triangular R F).bottom = 0 := rfl |
| @[simp] theorem off_zero : (0 : Triangular R F).off = 0 := rfl |
| @[simp] theorem top_one : (1 : Triangular R F).top = 1 := rfl |
| @[simp] theorem bottom_one : (1 : Triangular R F).bottom = 1 := rfl |
| @[simp] theorem off_one : (1 : Triangular R F).off = 0 := rfl |
| @[simp] theorem top_add (x y : Triangular R F) : (x + y).top = x.top + y.top := rfl |
| @[simp] theorem bottom_add (x y : Triangular R F) : |
| (x + y).bottom = x.bottom + y.bottom := rfl |
| @[simp] theorem off_add (x y : Triangular R F) : (x + y).off = x.off + y.off := rfl |
| end BasicCoordinates |
| |
| @[simp] theorem top_mul (x y : Triangular R F) : (x * y).top = x.top * y.top := rfl |
| @[simp] theorem bottom_mul (x y : Triangular R F) : |
| (x * y).bottom = x.bottom * y.bottom := rfl |
| @[simp] theorem off_mul (x y : Triangular R F) : |
| (x * y).off = x.bottom • y.off + MulOpposite.op y.top • x.off := rfl |
| |
| variable [SMulCommClass R Rᵐᵒᵖ F] |
| |
| instance : Ring (Triangular R F) where |
| __ := (inferInstance : AddCommGroup (Triangular R F)) |
| mul_assoc x y z := by |
| ext |
| · exact mul_assoc .. |
| · exact mul_assoc .. |
| · simp only [off_mul, bottom_mul, top_mul, mul_smul, smul_add, MulOpposite.op_mul] |
| rw [← smul_comm x.bottom (MulOpposite.op z.top) y.off] |
| exact (add_assoc ..).symm |
| one_mul x := by ext <;> simp |
| mul_one x := by ext <;> simp |
| left_distrib x y z := by ext <;> simp [mul_add, smul_add, add_smul, add_add_add_comm] |
| right_distrib x y z := by ext <;> simp [add_mul, smul_add, add_smul, add_add_add_comm] |
| zero_mul x := by ext <;> simp |
| mul_zero x := by ext <;> simp |
|
|
| def diagonal : Triangular R F →+* R × R where |
| toFun x := (x.top, x.bottom) |
| map_one' := rfl |
| map_mul' _ _ := rfl |
| map_zero' := rfl |
| map_add' _ _ := rfl |
|
|
| theorem diagonal_surjective : Function.Surjective (diagonal (R := R) (F := F)) := by |
| intro x |
| exact ⟨⟨x.1, x.2, 0⟩, rfl⟩ |
|
|
| def offDiagonal (f : F) : Triangular R F := ⟨0, 0, f⟩ |
|
|
| @[simp] theorem offDiagonal_mul_offDiagonal (f g : F) : |
| offDiagonal (R := R) f * offDiagonal g = 0 := by |
| ext <;> simp [offDiagonal] |
|
|
| section Scalars |
|
|
| variable (k : Type*) [CommRing k] [Algebra k R] [Module k F] |
| |
| instance : Module k (Triangular R F) := coordinateAddEquiv.module k |
| |
| @[simp] theorem top_smul (c : k) (x : Triangular R F) : (c • x).top = c • x.top := rfl |
| @[simp] theorem bottom_smul (c : k) (x : Triangular R F) : |
| (c • x).bottom = c • x.bottom := rfl |
| @[simp] theorem off_smul (c : k) (x : Triangular R F) : (c • x).off = c • x.off := rfl |
| |
| variable [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] |
| variable [SMulCommClass k R F] [SMulCommClass k Rᵐᵒᵖ F] |
| |
| instance : Algebra k (Triangular R F) := Algebra.ofModule |
| (by |
| intro c x y |
| ext |
| · exact smul_mul_assoc c x.top y.top |
| · exact smul_mul_assoc c x.bottom y.bottom |
| · simp only [off_mul, off_smul, bottom_smul, smul_assoc, smul_add] |
| rw [← smul_comm c (MulOpposite.op y.top) x.off]) |
| (by |
| intro c x y |
| ext |
| · exact mul_smul_comm c x.top y.top |
| · exact mul_smul_comm c x.bottom y.bottom |
| · simp only [off_mul, off_smul, top_smul, MulOpposite.op_smul, |
| smul_assoc, smul_add] |
| rw [← smul_comm c x.bottom y.off]) |
| |
| def coordinateLinearEquiv : Triangular R F ≃ₗ[k] (R × R) × F := |
| (coordinateAddEquiv (R := R) (F := F)).linearEquiv k |
| |
| instance [Module.Finite k R] [Module.Finite k F] : Module.Finite k (Triangular R F) := |
| Module.Finite.equiv (coordinateLinearEquiv (R := R) (F := F) k).symm |
| |
| end Scalars |
| |
| end Triangular |
| end ArExplicit |
| |
| end OAI |
| |
