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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 | import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import OAI.Algebra.Finitistic.Selection.TestModules
namespace OAI
namespace LittleFinitistic
open CategoryTheory
/-- The image of the central idempotent, expressed by its fixed vectors. -/
noncomputable def selectedSubmodule (M : ModuleCat SelectionRing) : Submodule SelectionRing M where
carrier := {x | selectionIdempotent 0 • x = x}
zero_mem' := by simp
add_mem' := by
intro x y hx hy
change selectionIdempotent 0 • x = x at hx
change selectionIdempotent 0 • y = y at hy
change selectionIdempotent 0 • (x+y) = x+y
rw [smul_add,hx,hy]
smul_mem' := by
intro r x hx
change selectionIdempotent 0 • (r • x) = r • x
rw [← mul_smul,(selectionIdempotent_central 0 r).eq,mul_smul,hx]
theorem selectedSubmodule_eq_image (M : ModuleCat SelectionRing) (x : M) :
x ∈ selectedSubmodule M ↔ ∃ y : M, selectionIdempotent 0 • y = x := by
constructor
· intro hx; exact ⟨x,hx⟩
· rintro ⟨y,rfl⟩
change selectionIdempotent 0 • (selectionIdempotent 0 • y) = _
rw [← mul_smul,selectionIdempotent_square]
/-- Exactly H(Y) = restriction along alpha of eY from the manuscript. -/
noncomputable def selectionStep (M : ModuleCat SelectionRing) : ModuleCat SelectionRing :=
(ModuleCat.restrictScalars (selectionAutomorphism 1).toRingHom).obj
(ModuleCat.of SelectionRing (selectedSubmodule M))
noncomputable def selectionIterate (M : ModuleCat SelectionRing) : ℕ → ModuleCat SelectionRing
| 0 => M
| n+1 => selectionStep (selectionIterate M n)
def IdentityAt (M : ModuleCat SelectionRing) (q : ℕ) : Prop :=
∀ x : M, selectionIdempotent q • x = x
def ZeroAt (M : ModuleCat SelectionRing) (q : ℕ) : Prop :=
∀ x : M, selectionIdempotent q • x = 0
theorem identityAt_step (M : ModuleCat SelectionRing) (q : ℕ)
(h : IdentityAt M (q+1)) : IdentityAt (selectionStep M) q := by
intro x
apply Subtype.ext
change selectionAutomorphism 1 (selectionIdempotent q) • x.val = x.val
rw [selectionAutomorphism_idempotent]
exact_mod_cast h x.val
theorem zeroAt_step (M : ModuleCat SelectionRing) (q : ℕ)
(h : ZeroAt M (q+1)) : ZeroAt (selectionStep M) q := by
intro x
apply Subtype.ext
change selectionAutomorphism 1 (selectionIdempotent q) • x.val = 0
rw [selectionAutomorphism_idempotent]
exact_mod_cast h x.val
theorem selectionStep_nontrivial (M : ModuleCat SelectionRing) [Nontrivial M]
(h : IdentityAt M 0) : Nontrivial (selectionStep M) := by
obtain ⟨x,y,hxy⟩ := exists_pair_ne M
exact ⟨⟨⟨x,h x⟩,⟨y,h y⟩,fun he => hxy (congrArg Subtype.val he)⟩⟩
theorem selectionStep_subsingleton (M : ModuleCat SelectionRing)
(h : ZeroAt M 0) : Subsingleton (selectionStep M) := by
refine ⟨fun x y => Subtype.ext ?_⟩
have hx : x.val = 0 := (x.property).symm.trans (h x.val)
have hy : y.val = 0 := (y.property).symm.trans (h y.val)
exact hx.trans hy.symm
theorem identityAt_iterate (M : ModuleCat SelectionRing) (n q : ℕ)
(h : IdentityAt M (q+n)) : IdentityAt (selectionIterate M n) q := by
induction n generalizing q with
| zero => simpa [selectionIterate] using h
| succ n ih =>
apply identityAt_step
apply ih
simpa [Nat.add_assoc,Nat.add_comm,Nat.add_left_comm] using h
theorem zeroAt_iterate (M : ModuleCat SelectionRing) (n q : ℕ)
(h : ZeroAt M (q+n)) : ZeroAt (selectionIterate M n) q := by
induction n generalizing q with
| zero => simpa [selectionIterate] using h
| succ n ih =>
apply zeroAt_step
apply ih
simpa [Nat.add_assoc,Nat.add_comm,Nat.add_left_comm] using h
noncomputable def testObject (m : ℕ) [NeZero m] : ModuleCat SelectionRing :=
ModuleCat.of SelectionRing (TestModule m)
theorem testObject_identityAt (m : ℕ) [NeZero m] (q : ℕ) (hq : q < m-1) :
IdentityAt (testObject m) q := by
intro x
let i : Fin m := ⟨q,by omega⟩
have hi : i ≠ lastIndex m := by
intro h
have hv := congrArg Fin.val h
change q = m-1 at hv
omega
exact (testModule_idempotent_action m i x).trans (ite_eq_right hi)
theorem testObject_zeroAt (m : ℕ) [NeZero m] : ZeroAt (testObject m) (m-1) := by
intro x
exact (testModule_idempotent_action m (lastIndex m) x).trans (ite_eq_left rfl)
theorem selectionIterate_test_nontrivial (m : ℕ) [NeZero m] (n : ℕ) (hn : n < m) :
Nontrivial (selectionIterate (testObject m) n) := by
induction n with
| zero => exact testModule_nontrivial m
| succ n ih =>
have := ih (by omega)
apply selectionStep_nontrivial
apply identityAt_iterate
simpa only [Nat.zero_add] using testObject_identityAt m n (by omega)
theorem selectionIterate_test_subsingleton (m : ℕ) [NeZero m] :
Subsingleton (selectionIterate (testObject m) m) := by
have hm : m = (m-1)+1 := by have := NeZero.ne m; omega
have hzero : ZeroAt (selectionIterate (testObject m) (m-1)) 0 :=
zeroAt_iterate (testObject m) (m-1) 0 (by simpa using testObject_zeroAt m)
have h := selectionStep_subsingleton _ hzero
change Subsingleton (selectionIterate (testObject m) ((m-1)+1)) at h
rwa [← hm] at h
/-- Exact extinction under the manuscript's actual selection operation H:
every earlier iterate is nonzero and the m-th iterate is zero. -/
theorem selection_extinction (m : ℕ) [NeZero m] :
(∀ n < m, Nontrivial (selectionIterate (testObject m) n)) ∧
Subsingleton (selectionIterate (testObject m) m) :=
⟨selectionIterate_test_nontrivial m,selectionIterate_test_subsingleton m⟩
noncomputable def complexObject (M : ModuleCat SelectionRing) : ModuleCat ℂ :=
(ModuleCat.restrictScalars (algebraMap ℂ SelectionRing)).obj M
noncomputable def selectionStep_inclusion (M : ModuleCat SelectionRing) :
complexObject (selectionStep M) →ₗ[ℂ] complexObject M where
toFun x := x.val
map_add' _ _ := rfl
map_smul' c x := by
change selectionAutomorphism 1 (algebraMap ℂ SelectionRing c) • x.val =
algebraMap ℂ SelectionRing c • x.val
rw [AlgEquiv.commutes]
theorem selectionStep_finiteDimensional (M : ModuleCat SelectionRing)
[FiniteDimensional ℂ (complexObject M)] :
FiniteDimensional ℂ (complexObject (selectionStep M)) :=
FiniteDimensional.of_injective (selectionStep_inclusion M) Subtype.val_injective
theorem testObject_finiteDimensional (m : ℕ) [NeZero m] :
FiniteDimensional ℂ (complexObject (testObject m)) := by
let f : complexObject (testObject m) →ₗ[ℂ] TestModule m :=
{ toFun := fun x => x
map_add' := fun _ _ => rfl
map_smul' := fun c x => by
have h : ∀ y : TestModule m, algebraMap ℂ SelectionRing c • y = c • y :=
fun y => algebraMap_smul SelectionRing c y
exact h x }
exact FiniteDimensional.of_injective f (fun _ _ h => h)
theorem selectionIterate_test_finiteDimensional (m : ℕ) [NeZero m] (n : ℕ) :
FiniteDimensional ℂ (complexObject (selectionIterate (testObject m) n)) := by
induction n with
| zero => exact testObject_finiteDimensional m
| succ n ih =>
have := ih
exact selectionStep_finiteDimensional _
end LittleFinitistic
end OAI
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