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ae00908 | 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 | import Mathlib.LinearAlgebra.Prod
import Mathlib.Tactic.FinCases
import OAI.Algebra.Finitistic.Encoding.QuadraticEncoding
namespace OAI
namespace LittleFinitistic.CornerSpaces
open ThreeVertexAlgebra
universe u
variable {k V W U : Type u} [Field k]
[AddCommGroup V] [Module k V] [AddCommGroup W] [Module k W]
[AddCommGroup U] [Module k U]
variable (μ : W →ₗ[k] V →ₗ[k] U)
/-- The actual two-sided corner e_i B e_j as a vector subspace. -/
def corner (i j : Fin 3) : Submodule k (ThreeVertexAlgebra μ) where
carrier := {x | vertex μ i * x * vertex μ j = x}
zero_mem' := by
change vertex μ i * (0 : ThreeVertexAlgebra μ) * vertex μ j = 0
rw [mul_zero,zero_mul]
add_mem' := by
intro x y hx hy
change vertex μ i * x * vertex μ j = x at hx
change vertex μ i * y * vertex μ j = y at hy
change vertex μ i * (x+y) * vertex μ j = x+y
rw [mul_add,add_mul,hx,hy]
smul_mem' := by
intro r x hx
change vertex μ i * x * vertex μ j = x at hx
change vertex μ i * (r • x) * vertex μ j = r • x
rw [mul_smul_comm,smul_mul_assoc,hx]
noncomputable def corner10Equiv : corner μ 1 0 ≃ₗ[k] V where
toFun x := x.val.2.2.2.1
invFun v := ⟨(0,0,0,v,0,0),by
change mul μ (mul μ (vertex μ 1) (0,0,0,v,0,0)) (vertex μ 0) = _
simp [vertex,mul]⟩
left_inv x := by
apply Subtype.ext
have hx := x.property
change mul μ (mul μ (vertex μ 1) x.val) (vertex μ 0) = x.val at hx
simpa [vertex,mul] using hx
right_inv _ := rfl
map_add' _ _ := rfl
map_smul' _ _ := rfl
noncomputable def corner21Equiv : corner μ 2 1 ≃ₗ[k] W where
toFun x := x.val.2.2.2.2.1
invFun w := ⟨(0,0,0,0,w,0),by
change mul μ (mul μ (vertex μ 2) (0,0,0,0,w,0)) (vertex μ 1) = _
simp [vertex,mul]⟩
left_inv x := by
apply Subtype.ext
have hx := x.property
change mul μ (mul μ (vertex μ 2) x.val) (vertex μ 1) = x.val at hx
simpa [vertex,mul] using hx
right_inv _ := rfl
map_add' _ _ := rfl
map_smul' _ _ := rfl
noncomputable def corner20Equiv : corner μ 2 0 ≃ₗ[k] U where
toFun x := x.val.2.2.2.2.2
invFun u := ⟨(0,0,0,0,0,u),by
change mul μ (mul μ (vertex μ 2) (0,0,0,0,0,u)) (vertex μ 0) = _
simp [vertex,mul]⟩
left_inv x := by
apply Subtype.ext
have hx := x.property
change mul μ (mul μ (vertex μ 2) x.val) (vertex μ 0) = x.val at hx
simpa [vertex,mul] using hx
right_inv _ := rfl
map_add' _ _ := rfl
map_smul' _ _ := rfl
/-- A diagonal corner has only its scalar trivial path. -/
noncomputable def diagonalEquiv (i : Fin 3) : corner μ i i ≃ₗ[k] k where
toFun x := if i = 0 then x.val.1 else if i = 1 then x.val.2.1 else x.val.2.2.1
invFun c := ⟨c • vertex μ i,by
change vertex μ i * (c • vertex μ i) * vertex μ i = c • vertex μ i
rw [mul_smul_comm,smul_mul_assoc]
have hi := (directedSystem μ).idem i
change vertex μ i * vertex μ i = vertex μ i at hi
rw [hi,hi]⟩
left_inv x := by
apply Subtype.ext
have hx := x.property
change mul μ (mul μ (vertex μ i) x.val) (vertex μ i) = x.val at hx
fin_cases i <;> simpa [vertex,mul,smul_eq μ] using hx
right_inv c := by fin_cases i <;> simp [vertex,smul_eq μ]
map_add' x y := by fin_cases i <;> rfl
map_smul' c x := by fin_cases i <;> rfl
/-- No path points backwards in the actual algebra. -/
theorem corner_zero_of_lt (i j : Fin 3) (hij : i < j) (x : corner μ i j) : x = 0 := by
apply Subtype.ext
have hx := x.property
change (directedSystem μ).e i * x.val * (directedSystem μ).e j = x.val at hx
rw [(directedSystem μ).directed j i hij x.val] at hx
exact hx.symm
end LittleFinitistic.CornerSpaces
end OAI
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