293 lines
11 KiB
Text
293 lines
11 KiB
Text
/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura, Jeremy Avigad
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Combinators for finite sets.
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-/
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import data.finset.basic logic.identities
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open list quot subtype decidable perm function
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namespace finset
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/- image (corresponds to map on list) -/
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section image
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variables {A B : Type}
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variable [h : decidable_eq B]
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include h
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definition image (f : A → B) (s : finset A) : finset B :=
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quot.lift_on s
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(λ l, to_finset (list.map f (elt_of l)))
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(λ l₁ l₂ p, quot.sound (perm_erase_dup_of_perm (perm_map _ p)))
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theorem image_empty (f : A → B) : image f ∅ = ∅ :=
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rfl
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theorem mem_image_of_mem (f : A → B) {s : finset A} {a : A} : a ∈ s → f a ∈ image f s :=
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quot.induction_on s (take l, assume H : a ∈ elt_of l, mem_to_finset (mem_map f H))
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theorem mem_image_of_mem_of_eq {f : A → B} {s : finset A} {a : A} {b : B}
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(H1 : a ∈ s) (H2 : f a = b) :
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b ∈ image f s :=
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eq.subst H2 (mem_image_of_mem f H1)
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theorem exists_of_mem_image {f : A → B} {s : finset A} {b : B} :
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b ∈ image f s → ∃a, a ∈ s ∧ f a = b :=
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quot.induction_on s
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(take l, assume H : b ∈ erase_dup (list.map f (elt_of l)),
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exists_of_mem_map (mem_of_mem_erase_dup H))
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theorem mem_image_iff (f : A → B) {s : finset A} {y : B} : y ∈ image f s ↔ ∃x, x ∈ s ∧ f x = y :=
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iff.intro exists_of_mem_image
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(assume H,
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obtain x (H₁ : x ∈ s) (H₂ : f x = y), from H,
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mem_image_of_mem_of_eq H₁ H₂)
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theorem mem_image_eq (f : A → B) {s : finset A} {y : B} : y ∈ image f s = ∃x, x ∈ s ∧ f x = y :=
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propext (mem_image_iff f)
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theorem mem_image_of_mem_image_of_subset {f : A → B} {s t : finset A} {y : B}
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(H1 : y ∈ image f s) (H2 : s ⊆ t) : y ∈ image f t :=
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obtain x (H3: x ∈ s) (H4 : f x = y), from exists_of_mem_image H1,
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have H5 : x ∈ t, from mem_of_subset_of_mem H2 H3,
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show y ∈ image f t, from mem_image_of_mem_of_eq H5 H4
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theorem image_insert [h' : decidable_eq A] (f : A → B) (s : finset A) (a : A) :
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image f (insert a s) = insert (f a) (image f s) :=
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ext (take y, iff.intro
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(assume H : y ∈ image f (insert a s),
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obtain x (H1l : x ∈ insert a s) (H1r :f x = y), from exists_of_mem_image H,
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have H2 : x = a ∨ x ∈ s, from eq_or_mem_of_mem_insert H1l,
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or.elim H2
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(assume H3 : x = a,
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have H4 : f a = y, from eq.subst H3 H1r,
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show y ∈ insert (f a) (image f s), from eq.subst H4 !mem_insert)
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(assume H3 : x ∈ s,
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have H5 : f x ∈ image f s, from mem_image_of_mem f H3,
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show y ∈ insert (f a) (image f s), from eq.subst H1r (mem_insert_of_mem _ H5)))
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(assume H : y ∈ insert (f a) (image f s),
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have H1 : y = f a ∨ y ∈ image f s, from eq_or_mem_of_mem_insert H,
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or.elim H1
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(assume H2 : y = f a,
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have H3 : f a ∈ image f (insert a s), from mem_image_of_mem f !mem_insert,
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show y ∈ image f (insert a s), from eq.subst (eq.symm H2) H3)
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(assume H2 : y ∈ image f s,
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show y ∈ image f (insert a s), from mem_image_of_mem_image_of_subset H2 !subset_insert)))
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end image
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/- filter and set-builder notation -/
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section filter
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variables {A : Type} [deceq : decidable_eq A]
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include deceq
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variables (p : A → Prop) [decp : decidable_pred p] (s : finset A) {x : A}
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include decp
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definition filter : finset A :=
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quot.lift_on s
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(λl, to_finset_of_nodup
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(list.filter p (subtype.elt_of l))
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(list.nodup_filter p (subtype.has_property l)))
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(λ l₁ l₂ u, quot.sound (perm.perm_filter u))
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notation `{` binders ∈ s `|` r:(scoped:1 p, filter p s) `}` := r
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theorem filter_empty : filter p ∅ = ∅ := rfl
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variables {p s}
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theorem of_mem_filter : x ∈ filter p s → p x :=
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quot.induction_on s (take l, list.of_mem_filter)
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theorem mem_of_mem_filter : x ∈ filter p s → x ∈ s :=
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quot.induction_on s (take l, list.mem_of_mem_filter)
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theorem mem_filter_of_mem {x : A} : x ∈ s → p x → x ∈ filter p s :=
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quot.induction_on s (take l, list.mem_filter_of_mem)
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variables (p s)
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theorem mem_filter_iff : x ∈ filter p s ↔ x ∈ s ∧ p x :=
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iff.intro
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(assume H, and.intro (mem_of_mem_filter H) (of_mem_filter H))
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(assume H, mem_filter_of_mem (and.left H) (and.right H))
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theorem mem_filter_eq : x ∈ filter p s = (x ∈ s ∧ p x) :=
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propext !mem_filter_iff
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end filter
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/- set difference -/
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section diff
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variables {A : Type} [deceq : decidable_eq A]
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include deceq
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definition diff (s t : finset A) : finset A := {x ∈ s | x ∉ t}
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infix `\`:70 := diff
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theorem mem_of_mem_diff {s t : finset A} {x : A} (H : x ∈ s \ t) : x ∈ s :=
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mem_of_mem_filter H
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theorem not_mem_of_mem_diff {s t : finset A} {x : A} (H : x ∈ s \ t) : x ∉ t :=
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of_mem_filter H
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theorem mem_diff {s t : finset A} {x : A} (H1 : x ∈ s) (H2 : x ∉ t) : x ∈ s \ t :=
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mem_filter_of_mem H1 H2
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theorem mem_diff_iff (s t : finset A) (x : A) : x ∈ s \ t ↔ x ∈ s ∧ x ∉ t :=
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iff.intro
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(assume H, and.intro (mem_of_mem_diff H) (not_mem_of_mem_diff H))
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(assume H, mem_diff (and.left H) (and.right H))
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theorem mem_diff_eq (s t : finset A) (x : A) : x ∈ s \ t = (x ∈ s ∧ x ∉ t) :=
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propext !mem_diff_iff
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theorem union_diff_cancel {s t : finset A} (H : s ⊆ t) : s ∪ (t \ s) = t :=
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ext (take x, iff.intro
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(assume H1 : x ∈ s ∪ (t \ s),
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or.elim (mem_or_mem_of_mem_union H1)
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(assume H2 : x ∈ s, mem_of_subset_of_mem H H2)
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(assume H2 : x ∈ t \ s, mem_of_mem_diff H2))
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(assume H1 : x ∈ t,
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decidable.by_cases
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(assume H2 : x ∈ s, mem_union_left _ H2)
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(assume H2 : x ∉ s, mem_union_right _ (mem_diff H1 H2))))
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theorem diff_union_cancel {s t : finset A} (H : s ⊆ t) : (t \ s) ∪ s = t :=
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eq.subst !union.comm (!union_diff_cancel H)
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end diff
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/- all -/
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section all
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variables {A : Type}
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definition all (s : finset A) (p : A → Prop) : Prop :=
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quot.lift_on s
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(λ l, all (elt_of l) p)
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(λ l₁ l₂ p, foldr_eq_of_perm (λ a₁ a₂ q, propext !and.left_comm) p true)
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theorem all_empty (p : A → Prop) : all ∅ p = true :=
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rfl
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theorem of_mem_of_all {p : A → Prop} {a : A} {s : finset A} : a ∈ s → all s p → p a :=
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quot.induction_on s (λ l i h, list.of_mem_of_all i h)
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theorem forall_of_all {p : A → Prop} {s : finset A} (H : all s p) : ∀{a}, a ∈ s → p a :=
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λ a H', of_mem_of_all H' H
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theorem all_of_forall {p : A → Prop} {s : finset A} : (∀a, a ∈ s → p a) → all s p :=
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quot.induction_on s (λ l H, list.all_of_forall H)
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theorem all_iff_forall (p : A → Prop) (s : finset A) : all s p ↔ (∀a, a ∈ s → p a) :=
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iff.intro forall_of_all all_of_forall
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definition decidable_all [instance] (p : A → Prop) [h : decidable_pred p] (s : finset A) :
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decidable (all s p) :=
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quot.rec_on_subsingleton s (λ l, list.decidable_all p (elt_of l))
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theorem all_implies {p q : A → Prop} {s : finset A} : all s p → (∀ x, p x → q x) → all s q :=
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quot.induction_on s (λ l h₁ h₂, list.all_implies h₁ h₂)
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variable [h : decidable_eq A]
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include h
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theorem all_union {p : A → Prop} {s₁ s₂ : finset A} : all s₁ p → all s₂ p → all (s₁ ∪ s₂) p :=
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quot.induction_on₂ s₁ s₂ (λ l₁ l₂ a₁ a₂, all_union a₁ a₂)
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theorem all_of_all_union_left {p : A → Prop} {s₁ s₂ : finset A} : all (s₁ ∪ s₂) p → all s₁ p :=
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quot.induction_on₂ s₁ s₂ (λ l₁ l₂ a, list.all_of_all_union_left a)
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theorem all_of_all_union_right {p : A → Prop} {s₁ s₂ : finset A} : all (s₁ ∪ s₂) p → all s₂ p :=
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quot.induction_on₂ s₁ s₂ (λ l₁ l₂ a, list.all_of_all_union_right a)
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theorem all_insert_of_all {p : A → Prop} {a : A} {s : finset A} : p a → all s p → all (insert a s) p :=
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quot.induction_on s (λ l h₁ h₂, list.all_insert_of_all h₁ h₂)
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theorem all_erase_of_all {p : A → Prop} (a : A) {s : finset A}: all s p → all (erase a s) p :=
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quot.induction_on s (λ l h, list.all_erase_of_all a h)
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theorem all_inter_of_all_left {p : A → Prop} {s₁ : finset A} (s₂ : finset A) : all s₁ p → all (s₁ ∩ s₂) p :=
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quot.induction_on₂ s₁ s₂ (λ l₁ l₂ h, list.all_inter_of_all_left _ h)
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theorem all_inter_of_all_right {p : A → Prop} {s₁ : finset A} (s₂ : finset A) : all s₂ p → all (s₁ ∩ s₂) p :=
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quot.induction_on₂ s₁ s₂ (λ l₁ l₂ h, list.all_inter_of_all_right _ h)
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theorem subset_iff_all (s t : finset A) : s ⊆ t ↔ all s (λ x, x ∈ t) :=
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iff.intro
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(assume H : s ⊆ t, all_of_forall (take x, assume H1, mem_of_subset_of_mem H H1))
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(assume H : all s (λ x, x ∈ t), subset_of_forall (take x, assume H1 : x ∈ s, of_mem_of_all H1 H))
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definition decidable_subset [instance] (s t : finset A) : decidable (s ⊆ t) :=
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decidable_of_decidable_of_iff _ (iff.symm !subset_iff_all)
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end all
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/- any -/
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section any
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variables {A : Type}
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definition any (s : finset A) (p : A → Prop) : Prop :=
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quot.lift_on s
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(λ l, any (elt_of l) p)
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(λ l₁ l₂ p, foldr_eq_of_perm (λ a₁ a₂ q, propext !or.left_comm) p false)
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theorem any_empty (p : A → Prop) : any ∅ p = false := rfl
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theorem exists_of_any {p : A → Prop} {s : finset A} : any s p → ∃a, a ∈ s ∧ p a :=
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quot.induction_on s (λ l H, list.exists_of_any H)
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theorem any_of_mem {p : A → Prop} {s : finset A} {a : A} : a ∈ s → p a → any s p :=
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quot.induction_on s (λ l H1 H2, list.any_of_mem H1 H2)
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theorem any_of_exists {p : A → Prop} {s : finset A} (H : ∃a, a ∈ s ∧ p a) : any s p :=
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obtain a H₁ H₂, from H,
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any_of_mem H₁ H₂
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theorem any_iff_exists (p : A → Prop) (s : finset A) : any s p ↔ (∃a, a ∈ s ∧ p a) :=
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iff.intro exists_of_any any_of_exists
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theorem any_of_insert [h : decidable_eq A] {p : A → Prop} (s : finset A) {a : A} (H : p a) :
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any (insert a s) p :=
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any_of_mem (mem_insert a s) H
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theorem any_of_insert_right [h : decidable_eq A] {p : A → Prop} {s : finset A} (a : A) (H : any s p) :
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any (insert a s) p :=
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obtain b (H₁ : b ∈ s) (H₂ : p b), from exists_of_any H,
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any_of_mem (mem_insert_of_mem a H₁) H₂
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definition decidable_any [instance] (p : A → Prop) [h : decidable_pred p] (s : finset A) :
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decidable (any s p) :=
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quot.rec_on_subsingleton s (λ l, list.decidable_any p (elt_of l))
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end any
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section product
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variables {A B : Type}
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definition product (s₁ : finset A) (s₂ : finset B) : finset (A × B) :=
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quot.lift_on₂ s₁ s₂
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(λ l₁ l₂,
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to_finset_of_nodup (product (elt_of l₁) (elt_of l₂))
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(nodup_product (has_property l₁) (has_property l₂)))
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(λ v₁ v₂ w₁ w₂ p₁ p₂, quot.sound (perm_product p₁ p₂))
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infix * := product
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theorem empty_product (s : finset B) : @empty A * s = ∅ :=
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quot.induction_on s (λ l, rfl)
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theorem mem_product {a : A} {b : B} {s₁ : finset A} {s₂ : finset B}
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: a ∈ s₁ → b ∈ s₂ → (a, b) ∈ s₁ * s₂ :=
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quot.induction_on₂ s₁ s₂ (λ l₁ l₂ i₁ i₂, list.mem_product i₁ i₂)
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theorem mem_of_mem_product_left {a : A} {b : B} {s₁ : finset A} {s₂ : finset B}
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: (a, b) ∈ s₁ * s₂ → a ∈ s₁ :=
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quot.induction_on₂ s₁ s₂ (λ l₁ l₂ i, list.mem_of_mem_product_left i)
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theorem mem_of_mem_product_right {a : A} {b : B} {s₁ : finset A} {s₂ : finset B}
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: (a, b) ∈ s₁ * s₂ → b ∈ s₂ :=
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quot.induction_on₂ s₁ s₂ (λ l₁ l₂ i, list.mem_of_mem_product_right i)
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theorem product_empty (s : finset A) : s * @empty B = ∅ :=
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ext (λ p,
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match p with
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| (a, b) := iff.intro
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(λ i, absurd (mem_of_mem_product_right i) !not_mem_empty)
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(λ i, absurd i !not_mem_empty)
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end)
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end product
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end finset
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