225 lines
7.3 KiB
Text
225 lines
7.3 KiB
Text
/-
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Copyright (c) 2015 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura, Jeremy Avigad
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Finite products on a monoid, and finite sums on an additive monoid. These are called
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algebra.list.prod
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algebra.finset.prod
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algebra.list.sum
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algebra.finset.sum
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So when we open algebra we have list.prod etc., and when we migrate to nat, we have
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nat.list.prod etc.
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We have to be careful with dependencies. This theory imports files from finset and list, which
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import basic files from nat. Then nat imports this file to instantiate finite products and sums.
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-/
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import .group data.list.basic data.list.perm data.finset.basic
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open algebra function binary quot subtype
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namespace algebra
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/- list.prod -/
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namespace list -- i.e. algebra.list
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open list -- i.e. ordinary lists
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variable {A : Type}
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section monoid
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variables {B : Type} [mB : monoid B]
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include mB
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definition mulf (f : A → B) : B → A → B :=
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λ b a, b * f a
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definition prod (l : list A) (f : A → B) : B :=
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list.foldl (mulf f) 1 l
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-- ∏ x ← l, f x
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notation `∏` binders `←` l, r:(scoped f, prod l f) := r
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private theorem foldl_const (f : A → B) :
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∀ (l : list A) (b : B), foldl (mulf f) b l = b * foldl (mulf f) 1 l
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| [] b := by rewrite [*foldl_nil, mul_one]
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| (a::l) b := by rewrite [*foldl_cons, foldl_const, {foldl _ (mulf f 1 a) _}foldl_const, ↑mulf,
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one_mul, mul.assoc]
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theorem prod_nil (f : A → B) : prod [] f = 1 := rfl
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theorem prod_cons (f : A → B) (a : A) (l : list A) : prod (a::l) f = f a * prod l f :=
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by rewrite [↑prod, foldl_cons, foldl_const, ↑mulf, one_mul]
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theorem prod_append :
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∀ (l₁ l₂ : list A) (f : A → B), prod (l₁++l₂) f = prod l₁ f * prod l₂ f
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| [] l₂ f := by rewrite [append_nil_left, prod_nil, one_mul]
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| (a::l) l₂ f := by rewrite [append_cons, *prod_cons, prod_append, mul.assoc]
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section decidable_eq
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variable [H : decidable_eq A]
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include H
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theorem prod_insert_of_mem (f : A → B) {a : A} {l : list A} : a ∈ l →
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prod (insert a l) f = prod l f :=
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assume ainl, by rewrite [insert_eq_of_mem ainl]
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theorem prod_insert_of_not_mem (f : A → B) {a : A} {l : list A} :
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a ∉ l → prod (insert a l) f = f a * prod l f :=
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assume nainl, by rewrite [insert_eq_of_not_mem nainl, prod_cons]
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theorem prod_union {l₁ l₂ : list A} (f : A → B) (d : disjoint l₁ l₂) :
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prod (union l₁ l₂) f = prod l₁ f * prod l₂ f :=
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by rewrite [union_eq_append d, prod_append]
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end decidable_eq
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end monoid
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section comm_monoid
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variables {B : Type} [cmB : comm_monoid B]
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include cmB
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theorem prod_mul (l : list A) (f g : A → B) : prod l (λx, f x * g x) = prod l f * prod l g :=
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list.induction_on l
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(by rewrite [*prod_nil, mul_one])
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(take a l,
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assume IH,
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by rewrite [*prod_cons, IH, *mul.assoc, mul.left_comm (prod l f)])
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end comm_monoid
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end list
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/- finset.prod -/
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namespace finset
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open finset
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variables {A B : Type}
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variable [cmB : comm_monoid B]
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include cmB
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theorem mulf_rcomm (f : A → B) : right_commutative (list.mulf f) :=
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right_commutative_compose_right (@has_mul.mul B cmB) f (@mul.right_comm B cmB)
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section
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open perm
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theorem listprod_of_perm (f : A → B) {l₁ l₂ : list A} :
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l₁ ~ l₂ → list.prod l₁ f = list.prod l₂ f :=
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λ p, foldl_eq_of_perm (mulf_rcomm f) p 1
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end
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definition prod (s : finset A) (f : A → B) : B :=
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quot.lift_on s
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(λ l, list.prod (elt_of l) f)
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(λ l₁ l₂ p, listprod_of_perm f p)
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-- ∏ x ∈ s, f x
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notation `∏` binders `∈` s, r:(scoped f, prod s f) := r
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theorem prod_empty (f : A → B) : prod ∅ f = 1 :=
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list.prod_nil f
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section decidable_eq
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variable [H : decidable_eq A]
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include H
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theorem prod_insert_of_mem (f : A → B) {a : A} {s : finset A} :
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a ∈ s → prod (insert a s) f = prod s f :=
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quot.induction_on s
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(λ l ainl, list.prod_insert_of_mem f ainl)
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theorem prod_insert_of_not_mem (f : A → B) {a : A} {s : finset A} :
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a ∉ s → prod (insert a s) f = f a * prod s f :=
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quot.induction_on s
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(λ l nainl, list.prod_insert_of_not_mem f nainl)
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theorem prod_union (f : A → B) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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prod (s₁ ∪ s₂) f = prod s₁ f * prod s₂ f :=
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have H1 : disjoint s₁ s₂ → prod (s₁ ∪ s₂) f = prod s₁ f * prod s₂ f, from
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quot.induction_on₂ s₁ s₂
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(λ l₁ l₂ d, list.prod_union f d),
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H1 (disjoint_of_inter_empty disj)
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end decidable_eq
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theorem prod_mul (s : finset A) (f g : A → B) : prod s (λx, f x * g x) = prod s f * prod s g :=
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quot.induction_on s (take u, !list.prod_mul)
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end finset
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/- list.sum -/
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namespace list
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open list
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variable {A : Type}
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section add_monoid
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variables {B : Type} [amB : add_monoid B]
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include amB
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local attribute add_monoid.to_monoid [instance]
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definition sum (l : list A) (f : A → B) : B := prod l f
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-- ∑ x ← l, f x
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notation `∑` binders `←` l, r:(scoped f, sum l f) := r
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theorem sum_nil (f : A → B) : sum [] f = 0 := prod_nil f
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theorem sum_cons (f : A → B) (a : A) (l : list A) : sum (a::l) f = f a + sum l f :=
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prod_cons f a l
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theorem sum_append (l₁ l₂ : list A) (f : A → B) : sum (l₁++l₂) f = sum l₁ f + sum l₂ f :=
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prod_append l₁ l₂ f
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section decidable_eq
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variable [H : decidable_eq A]
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include H
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theorem sum_insert_of_mem (f : A → B) {a : A} {l : list A} (H : a ∈ l) :
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sum (insert a l) f = sum l f := prod_insert_of_mem f H
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theorem sum_insert_of_not_mem (f : A → B) {a : A} {l : list A} (H : a ∉ l) :
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sum (insert a l) f = f a + sum l f := prod_insert_of_not_mem f H
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theorem sum_union {l₁ l₂ : list A} (f : A → B) (d : disjoint l₁ l₂) :
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sum (union l₁ l₂) f = sum l₁ f + sum l₂ f := prod_union f d
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end decidable_eq
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end add_monoid
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section add_comm_monoid
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variables {B : Type} [acmB : add_comm_monoid B]
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include acmB
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local attribute add_comm_monoid.to_comm_monoid [instance]
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theorem sum_add (l : list A) (f g : A → B) : sum l (λx, f x + g x) = sum l f + sum l g :=
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prod_mul l f g
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end add_comm_monoid
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end list
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/- finset.sum -/
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namespace finset
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open finset
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variable {A : Type}
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section add_comm_monoid
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variables {B : Type} [acmB : add_comm_monoid B]
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include acmB
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local attribute add_comm_monoid.to_comm_monoid [instance]
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definition sum (s : finset A) (f : A → B) : B := prod s f
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-- ∑ x ∈ s, f x
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notation `∑` binders `∈` s, r:(scoped f, sum s f) := r
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theorem sum_empty (f : A → B) : sum ∅ f = 0 := prod_empty f
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section decidable_eq
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variable [H : decidable_eq A]
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include H
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theorem sum_insert_of_mem (f : A → B) {a : A} {s : finset A} (H : a ∈ s) :
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sum (insert a s) f = sum s f := prod_insert_of_mem f H
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theorem sum_insert_of_not_mem (f : A → B) {a : A} {s : finset A} (H : a ∉ s) :
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sum (insert a s) f = f a + sum s f := prod_insert_of_not_mem f H
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theorem sum_union (f : A → B) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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sum (s₁ ∪ s₂) f = sum s₁ f + sum s₂ f := prod_union f disj
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end decidable_eq
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theorem sum_add (s : finset A) (f g : A → B) :
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sum s (λx, f x + g x) = sum s f + sum s g := prod_mul s f g
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end add_comm_monoid
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end finset
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end algebra
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