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refactor(linear_algebra/basic): move the lattice structure to its own…
… file (#6767) The entire lattice structure is thoroughly uninteresting. By moving it to its own shorter file, it should be easier to unify with the lattice of `submonoid` I'd hope in future we can generate this automatically for any `subobject A` with an injection into `set A`.
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| /- | ||
| Copyright (c) 2017 Johannes Hölzl. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov | ||
| -/ | ||
| import algebra.module.submodule | ||
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| /-! | ||
| # The lattice structure on `submodule`s | ||
| This file defines the lattice structure on submodules, `submodule.complete_lattice`, with `⊥` | ||
| defined as `{0}` and `⊓` defined as intersection of the underlying carrier. | ||
| If `p` and `q` are submodules of a module, `p ≤ q` means that `p ⊆ q`. | ||
| Many results about operations on this lattice structure are defined in `linear_algebra/basic.lean`, | ||
| most notably those which use `span`. | ||
| ## Implementation notes | ||
| This structure should match the `add_submonoid.complete_lattice` structure, and we should try | ||
| to unify the APIs where possible. | ||
| -/ | ||
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| variables {R : Type*} {M : Type*} | ||
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| section add_comm_monoid | ||
| variables [semiring R] [add_comm_monoid M] [semimodule R M] | ||
| variables {p q : submodule R M} | ||
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| namespace submodule | ||
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| instance : partial_order (submodule R M) := | ||
| { le := λ p q, ∀ ⦃x⦄, x ∈ p → x ∈ q, | ||
| ..partial_order.lift (coe : submodule R M → set M) coe_injective } | ||
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| lemma le_def : p ≤ q ↔ (p : set M) ⊆ q := iff.rfl | ||
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| @[simp, norm_cast] lemma coe_subset_coe : (p : set M) ⊆ q ↔ p ≤ q := iff.rfl | ||
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| lemma le_def' : p ≤ q ↔ ∀ x ∈ p, x ∈ q := iff.rfl | ||
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| lemma lt_def : p < q ↔ (p : set M) ⊂ q := iff.rfl | ||
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| lemma not_le_iff_exists : ¬ (p ≤ q) ↔ ∃ x ∈ p, x ∉ q := set.not_subset | ||
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| lemma exists_of_lt {p q : submodule R M} : p < q → ∃ x ∈ q, x ∉ p := set.exists_of_ssubset | ||
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| lemma lt_iff_le_and_exists : p < q ↔ p ≤ q ∧ ∃ x ∈ q, x ∉ p := | ||
| by rw [lt_iff_le_not_le, not_le_iff_exists] | ||
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| /-- The set `{0}` is the bottom element of the lattice of submodules. -/ | ||
| instance : has_bot (submodule R M) := | ||
| ⟨{ carrier := {0}, smul_mem' := by simp { contextual := tt }, .. (⊥ : add_submonoid M)}⟩ | ||
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| instance inhabited' : inhabited (submodule R M) := ⟨⊥⟩ | ||
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| @[simp] lemma bot_coe : ((⊥ : submodule R M) : set M) = {0} := rfl | ||
| @[simp] lemma bot_to_add_submonoid : (⊥ : submodule R M).to_add_submonoid = ⊥ := rfl | ||
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| section | ||
| variables (R) | ||
| @[simp] lemma mem_bot {x : M} : x ∈ (⊥ : submodule R M) ↔ x = 0 := set.mem_singleton_iff | ||
| end | ||
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| instance unique_bot : unique (⊥ : submodule R M) := | ||
| ⟨infer_instance, λ x, subtype.ext $ (mem_bot R).1 x.mem⟩ | ||
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| lemma nonzero_mem_of_bot_lt {I : submodule R M} (bot_lt : ⊥ < I) : ∃ a : I, a ≠ 0 := | ||
| begin | ||
| have h := (submodule.lt_iff_le_and_exists.1 bot_lt).2, | ||
| tidy, | ||
| end | ||
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| instance : order_bot (submodule R M) := | ||
| { bot := ⊥, | ||
| bot_le := λ p x, by simp {contextual := tt}, | ||
| ..submodule.partial_order } | ||
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| protected lemma eq_bot_iff (p : submodule R M) : p = ⊥ ↔ ∀ x ∈ p, x = (0 : M) := | ||
| ⟨ λ h, h.symm ▸ λ x hx, (mem_bot R).mp hx, | ||
| λ h, eq_bot_iff.mpr (λ x hx, (mem_bot R).mpr (h x hx)) ⟩ | ||
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| protected lemma ne_bot_iff (p : submodule R M) : p ≠ ⊥ ↔ ∃ x ∈ p, x ≠ (0 : M) := | ||
| by { haveI := classical.prop_decidable, simp_rw [ne.def, p.eq_bot_iff, not_forall] } | ||
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| /-- The universal set is the top element of the lattice of submodules. -/ | ||
| instance : has_top (submodule R M) := | ||
| ⟨{ carrier := set.univ, smul_mem' := λ _ _ _, trivial, .. (⊤ : add_submonoid M)}⟩ | ||
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| @[simp] lemma top_coe : ((⊤ : submodule R M) : set M) = set.univ := rfl | ||
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| @[simp] lemma top_to_add_submonoid : (⊤ : submodule R M).to_add_submonoid = ⊤ := rfl | ||
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| @[simp] lemma mem_top {x : M} : x ∈ (⊤ : submodule R M) := trivial | ||
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| instance : order_top (submodule R M) := | ||
| { top := ⊤, | ||
| le_top := λ p x _, trivial, | ||
| ..submodule.partial_order } | ||
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| lemma eq_top_iff' {p : submodule R M} : p = ⊤ ↔ ∀ x, x ∈ p := | ||
| eq_top_iff.trans ⟨λ h x, h trivial, λ h x _, h x⟩ | ||
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| instance : has_Inf (submodule R M) := | ||
| ⟨λ S, { | ||
| carrier := ⋂ s ∈ S, (s : set M), | ||
| zero_mem' := by simp, | ||
| add_mem' := by simp [add_mem] {contextual := tt}, | ||
| smul_mem' := by simp [smul_mem] {contextual := tt} }⟩ | ||
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| private lemma Inf_le' {S : set (submodule R M)} {p} : p ∈ S → Inf S ≤ p := | ||
| set.bInter_subset_of_mem | ||
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| private lemma le_Inf' {S : set (submodule R M)} {p} : (∀q ∈ S, p ≤ q) → p ≤ Inf S := | ||
| set.subset_bInter | ||
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| instance : has_inf (submodule R M) := | ||
| ⟨λ p q, { | ||
| carrier := p ∩ q, | ||
| zero_mem' := by simp, | ||
| add_mem' := by simp [add_mem] {contextual := tt}, | ||
| smul_mem' := by simp [smul_mem] {contextual := tt} }⟩ | ||
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| instance : complete_lattice (submodule R M) := | ||
| { sup := λ a b, Inf {x | a ≤ x ∧ b ≤ x}, | ||
| le_sup_left := λ a b, le_Inf' $ λ x ⟨ha, hb⟩, ha, | ||
| le_sup_right := λ a b, le_Inf' $ λ x ⟨ha, hb⟩, hb, | ||
| sup_le := λ a b c h₁ h₂, Inf_le' ⟨h₁, h₂⟩, | ||
| inf := (⊓), | ||
| le_inf := λ a b c, set.subset_inter, | ||
| inf_le_left := λ a b, set.inter_subset_left _ _, | ||
| inf_le_right := λ a b, set.inter_subset_right _ _, | ||
| Sup := λtt, Inf {t | ∀t'∈tt, t' ≤ t}, | ||
| le_Sup := λ s p hs, le_Inf' $ λ q hq, hq _ hs, | ||
| Sup_le := λ s p hs, Inf_le' hs, | ||
| Inf := Inf, | ||
| le_Inf := λ s a, le_Inf', | ||
| Inf_le := λ s a, Inf_le', | ||
| ..submodule.order_top, | ||
| ..submodule.order_bot } | ||
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| @[simp] theorem inf_coe : (p ⊓ q : set M) = p ∩ q := rfl | ||
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| @[simp] theorem mem_inf {p q : submodule R M} {x : M} : | ||
| x ∈ p ⊓ q ↔ x ∈ p ∧ x ∈ q := iff.rfl | ||
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| @[simp] theorem Inf_coe (P : set (submodule R M)) : (↑(Inf P) : set M) = ⋂ p ∈ P, ↑p := rfl | ||
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| @[simp] theorem infi_coe {ι} (p : ι → submodule R M) : | ||
| (↑⨅ i, p i : set M) = ⋂ i, ↑(p i) := | ||
| by rw [infi, Inf_coe]; ext a; simp; exact | ||
| ⟨λ h i, h _ i rfl, λ h i x e, e ▸ h _⟩ | ||
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| @[simp] lemma mem_Inf {S : set (submodule R M)} {x : M} : x ∈ Inf S ↔ ∀ p ∈ S, x ∈ p := | ||
| set.mem_bInter_iff | ||
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| @[simp] theorem mem_infi {ι} (p : ι → submodule R M) {x} : | ||
| x ∈ (⨅ i, p i) ↔ ∀ i, x ∈ p i := | ||
| by rw [← mem_coe, infi_coe, set.mem_Inter]; refl | ||
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| lemma mem_sup_left {S T : submodule R M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T := | ||
| show S ≤ S ⊔ T, from le_sup_left | ||
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| lemma mem_sup_right {S T : submodule R M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T := | ||
| show T ≤ S ⊔ T, from le_sup_right | ||
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| lemma mem_supr_of_mem {ι : Sort*} {b : M} {p : ι → submodule R M} (i : ι) (h : b ∈ p i) : | ||
| b ∈ (⨆i, p i) := | ||
| have p i ≤ (⨆i, p i) := le_supr p i, | ||
| @this b h | ||
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| /-! Note that `submodule.mem_supr` is provided in `linear_algebra/basic.lean`. -/ | ||
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| lemma mem_Sup_of_mem {S : set (submodule R M)} {s : submodule R M} | ||
| (hs : s ∈ S) : ∀ {x : M}, x ∈ s → x ∈ Sup S := | ||
| show s ≤ Sup S, from le_Sup hs | ||
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| end submodule | ||
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| end add_comm_monoid |
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