feat(library/data/set/filter): work in material from Jacob Gross
This commit is contained in:
Jeremy Avigad
Leonardo de Moura
parent
41342f53df
commit
158acf878d
1 changed files with 178 additions and 3 deletions
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@ -34,12 +34,16 @@ P ∈ F
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theorem eventually_true (F : filter A) : eventually (λx, true) F :=
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!filter.univ_mem_sets
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theorem eventually_of_forall {P : A → Prop} (F : filter A) (H : ∀ x, P x) : eventually P F :=
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theorem eventually_of_forall (F : filter A) (H : ∀ x, P x) : eventually P F :=
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by rewrite [eq_univ_of_forall H]; apply eventually_true
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theorem eventually_mono (H₁ : eventually P F) (H₂ : ∀x, P x → Q x) : eventually Q F :=
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!filter.is_mono H₂ H₁
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theorem eventually_congr (H : ∀ x, P x ↔ Q x) (H' : eventually P F) : eventually Q F :=
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have P = Q, from ext H,
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using this, by rewrite -this; exact H'
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theorem eventually_and (H₁ : eventually P F) (H₂ : eventually Q F) :
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eventually (λ x, P x ∧ Q x) F :=
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!filter.inter_closed H₁ H₂
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@ -95,6 +99,54 @@ eventually_mono (eventually_and H₁ H₂) (λ x H, iff.mpr (and.left H) (and.ri
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theorem eventually_iff_iff (H : eventually (λ x, P x ↔ Q x) F) : eventually P F ↔ eventually Q F :=
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iff.intro (eventually_iff_mp H) (eventually_iff_mpr H)
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/- frequently -/
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definition frequently (P : A → Prop) (F : filter A) : Prop := ¬ eventually (λ x, ¬ P x) F
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theorem not_frequently_of_eventually : eventually (λ x, ¬ P x) F → ¬ frequently P F :=
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not_not_intro
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theorem frequently_mono (H₁ : frequently P F) (H₂ : ∀ x, P x → Q x) : frequently Q F :=
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not.mto (λ H, eventually_mono H ( λ x, not.mto (H₂ x))) H₁
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theorem frequently_mp (ev : eventually (λ x, P x → Q x) F) :
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frequently P F → frequently Q F :=
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not.mto (λ H, eventually_mp (eventually_mono ev (λ x HPQ, not.mto HPQ)) H)
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theorem not_frequently_false : ¬ frequently (λ x, false) F :=
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begin
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apply not_not_intro,
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apply eventually_congr,
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intro x, apply iff.symm not_false_iff,
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exact eventually_true F
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end
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section
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open classical
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theorem not_frequently_iff : ¬ frequently P F ↔ eventually (λ x, ¬ P x) F :=
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by unfold frequently; rewrite not_not_iff
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theorem exists_of_frequently : frequently P F → ∃ x, P x :=
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assume H, obtain x Hx, from !exists_not_of_not_eventually H,
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show _, from exists.intro x (not_not_elim Hx)
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theorem frequently_inl (H : frequently P F) : frequently (λx, P x ∨ Q x) F :=
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assume H' : eventually (λx, ¬ (P x ∨ Q x)) F,
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have (λx, ¬ (P x ∨ Q x)) = λ x, ¬ P x ∧ ¬ Q x,
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by apply funext; intro x; rewrite not_or_iff_not_and_not,
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show false, using this,
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by rewrite this at H'; exact H (eventually_of_eventually_and_left H')
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theorem frequently_inr (H : frequently Q F) : frequently (λx, P x ∨ Q x) F :=
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begin
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apply frequently_mp,
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apply eventually_of_forall,
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intro x, apply or.swap,
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exact frequently_inl H
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end
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end
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/- filters form a lattice under ⊇ -/
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protected theorem eq : sets F₁ = sets F₂ → F₁ = F₂ :=
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@ -257,9 +309,9 @@ protected definition complete_lattice [reducible] [trans_instance] : complete_la
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le_Sup := @complete_lattice.le_Sup A
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⦄
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protected theorem le_of_subset {F₁ F₂ : filter A} (H : F₁ ≤ F₂) : sets F₂ ⊆ sets F₁ := H
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protected theorem subset_of_le {F₁ F₂ : filter A} (H : F₁ ≤ F₂) : sets F₂ ⊆ sets F₁ := H
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protected theorem subset_of_le {F₁ F₂ : filter A} (H : sets F₂ ⊆ sets F₁) : F₁ ≤ F₂ := H
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protected theorem le_of_subset {F₁ F₂ : filter A} (H : sets F₂ ⊆ sets F₁) : F₁ ≤ F₂ := H
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theorem sets_Sup (S : set (filter A)) : sets (⨆ S) = ⋂ F ∈ S, sets F := rfl
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@ -379,5 +431,128 @@ by rewrite [↑eventually, sets_top_eq, mem_singleton_iff]; intro H x; rewrite H
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theorem eventually_top_iff (P : A → Prop) : eventually P top ↔ ∀ x, P x :=
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iff.intro forall_of_eventually_top eventually_top_of_forall
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/- filter generated by a collection of sets -/
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inductive sets_generated_by {A : Type} (B : set (set A)) : set A → Prop :=
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| generators_mem : ∀ ⦃s : set A⦄, s ∈ B → sets_generated_by B s
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| univ_mem : sets_generated_by B univ
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| inter_mem : ∀ {a b}, sets_generated_by B a → sets_generated_by B b → sets_generated_by B (a ∩ b)
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| mono : ∀ {a b}, a ⊆ b → sets_generated_by B a → sets_generated_by B b
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definition filter_generated_by [reducible] {A : Type} (B : set (set A)) : filter A :=
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⦃filter,
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sets := sets_generated_by B,
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univ_mem_sets := sets_generated_by.univ_mem B,
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inter_closed := @sets_generated_by.inter_mem A B,
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is_mono := @sets_generated_by.mono A B ⦄
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theorem generators_subset_generated_by (B : set (set A)) : B ⊆ filter_generated_by B :=
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λ s H, sets_generated_by.generators_mem H
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theorem sets_generated_by_initial {B : set (set A)} {F : filter A} (H : B ⊆ sets F) :
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sets_generated_by B ⊆ F :=
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begin
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intro s Hs,
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induction Hs with s sB s t os ot soA toA S SB SOA,
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{exact H sB},
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{exact filter.univ_mem_sets F},
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{exact filter.inter_closed F soA toA},
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exact filter.is_mono F SOA v_0
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end
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theorem le_filter_generated_by {B : set (set A)} {F : filter A}
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(H : B ⊆ sets F) : F ≤ filter_generated_by B :=
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sets_generated_by_initial H
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/- principal filter -/
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definition principal (s : set A) : filter A := filter_generated_by '{s}
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theorem mem_principal (s : set A) : s ∈ principal s :=
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!generators_subset_generated_by !mem_singleton
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theorem eventually_principal {s : set A} {P : A → Prop} (H : s ⊆ P) : eventually P (principal s) :=
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!filter.is_mono H !mem_principal
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theorem subset_of_eventually_principal {s : set A} {P : A → Prop}
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(H : eventually P (principal s)) : s ⊆ P :=
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begin
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induction H with s' Ps' a b Ha Hb ssuba ssubb a b asubb Ha ssuba,
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{rewrite [eq_of_mem_singleton Ps']; exact subset.refl s},
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{exact subset_univ s},
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{exact subset_inter ssuba ssubb},
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{exact subset.trans ssuba asubb}
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end
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theorem eventually_principal_iff (s P : set A) : eventually P (principal s) ↔ (s ⊆ P) :=
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iff.intro subset_of_eventually_principal eventually_principal
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theorem sets_principal (s : set A) : sets (principal s) = { t | s ⊆ t } :=
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ext (take P, eventually_principal_iff s P)
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theorem principal_empty : principal (∅ : set A) = ⊥ :=
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begin
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apply filter.eq,
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rewrite [sets_principal, sets_bot_eq],
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show { t | ∅ ⊆ t} = univ, from ext (take x, iff.intro
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(assume H, trivial)
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(assume H, empty_subset x))
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end
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theorem principal_univ_eq : principal (@univ A) = ⊤ :=
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begin
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apply filter.eq,
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rewrite [sets_principal, sets_top_eq],
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show { t | univ ⊆ t} = '{univ}, from ext (take x, iff.intro
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(assume H : univ ⊆ x, mem_singleton_of_eq (!eq_univ_of_univ_subset H))
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(assume H : x ∈ '{univ}, by rewrite [eq_of_mem_singleton H]; apply subset.refl))
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end
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theorem principal_le_principal {s t : set A} (H : s ⊆ t) : principal s ≤ principal t :=
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begin
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apply filter.le_of_subset,
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rewrite *sets_principal,
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show { u | t ⊆ u } ⊆ { u | s ⊆ u }, from
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take u, suppose t ⊆ u, subset.trans H this
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end
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theorem subset_of_principal_le_principal {s t : set A} (H : principal s ≤ principal t) : s ⊆ t :=
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begin
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note H' := filter.subset_of_le H,
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revert H', rewrite *sets_principal,
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intro H',
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exact H' (subset.refl t)
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end
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theorem principal_refines_principal_iff {s t : set A} : principal s ≤ principal t ↔ s ⊆ t :=
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iff.intro subset_of_principal_le_principal principal_le_principal
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theorem principal_sup_principal {s t : set A} : principal s ⊔ principal t = principal (s ∪ t) :=
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begin
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apply filter.eq,
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rewrite [sets_sup, *sets_principal],
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apply ext, intro u, apply iff.intro,
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exact (suppose s ⊆ u ∧ t ⊆ u,
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show s ∪ t ⊆ u, from union_subset (and.left this) (and.right this)),
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exact suppose s ∪ t ⊆ u,
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and.intro (subset.trans (subset_union_left _ _) this)
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(subset.trans (subset_union_right _ _) this)
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end
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theorem principal_inf_principal {s t : set A} : principal s ⊓ principal t = principal (s ∩ t) :=
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begin
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apply filter.eq,
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rewrite [sets_inf, *sets_principal],
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apply ext, intro u, apply iff.intro,
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{intro H,
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show s ∩ t ⊆ u, from
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obtain s' [(ss' : s ⊆ s') [t' [(tt' : t ⊆ t') (Hu : s' ∩ t' ⊆ u)]]], from H,
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take x, assume H', Hu (and.intro (ss' (and.left H')) (tt' (and.right H')))},
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{intro H',
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exact exists.intro s (and.intro (subset.refl s) (exists.intro t
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(and.intro (subset.refl t) H')))}
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end
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end filter
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end set
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