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2c7d975 | 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 | 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
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