| import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings |
| import OAI.Algebra.Finitistic.Selection.TestModules |
|
|
| namespace OAI |
| namespace LittleFinitistic |
| open CategoryTheory |
|
|
| / |
| 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] |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
|
|
