106 lines
3.9 KiB
Text
106 lines
3.9 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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Author: Jeremy Avigad
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Set-based version of group_bigops.
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-/
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import .group_bigops data.set.finite
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open set classical
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namespace algebra
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namespace set
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variables {A B : Type}
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/- Prod: product indexed by a set -/
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section Prod
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variable [cmB : comm_monoid B]
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include cmB
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noncomputable definition Prod (s : set A) (f : A → B) : B := algebra.finset.Prod (to_finset s) f
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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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by rewrite [↑Prod, to_finset_empty]
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theorem Prod_of_not_finite {s : set A} (nfins : ¬ finite s) (f : A → B) : Prod s f = 1 :=
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by rewrite [↑Prod, to_finset_of_not_finite nfins]
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theorem Prod_mul (s : set A) (f g : A → B) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
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by rewrite [↑Prod, finset.Prod_mul]
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theorem Prod_insert_of_mem (f : A → B) {a : A} {s : set A} (H : a ∈ s) :
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Prod (insert a s) f = Prod s f :=
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by_cases
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(suppose finite s,
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assert (#finset a ∈ set.to_finset s), by rewrite mem_to_finset_eq; apply H,
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by rewrite [↑Prod, to_finset_insert, finset.Prod_insert_of_mem f this])
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(assume nfs : ¬ finite s,
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assert ¬ finite (insert a s), from assume H, nfs (finite_of_finite_insert H),
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by rewrite [Prod_of_not_finite nfs, Prod_of_not_finite this])
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theorem Prod_insert_of_not_mem (f : A → B) {a : A} {s : set A} [fins : finite s] (H : a ∉ s) :
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Prod (insert a s) f = f a * Prod s f :=
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assert (#finset a ∉ set.to_finset s), by rewrite mem_to_finset_eq; apply H,
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by rewrite [↑Prod, to_finset_insert, finset.Prod_insert_of_not_mem f this]
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theorem Prod_union (f : A → B) {s₁ s₂ : set A} [fins₁ : finite s₁] [fins₂ : finite s₂]
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(disj : s₁ ∩ s₂ = ∅) :
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Prod (s₁ ∪ s₂) f = Prod s₁ f * Prod s₂ f :=
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begin
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rewrite [↑Prod, to_finset_union],
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apply finset.Prod_union,
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apply finset.eq_of_to_set_eq_to_set,
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rewrite [finset.to_set_inter, *to_set_to_finset, finset.to_set_empty, disj]
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end
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theorem Prod_ext {s : set A} {f g : A → B} (H : ∀{x}, x ∈ s → f x = g x) : Prod s f = Prod s g :=
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by_cases
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(suppose finite s,
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by esimp [Prod]; apply finset.Prod_ext; intro x; rewrite [mem_to_finset_eq]; apply H)
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(assume nfs : ¬ finite s,
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by rewrite [*Prod_of_not_finite nfs])
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theorem Prod_one (s : set A) : Prod s (λ x, 1) = (1:B) :=
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by rewrite [↑Prod, finset.Prod_one]
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end Prod
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/- Sum -/
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section Sum
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variable [acmB : add_comm_monoid B]
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include acmB
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local attribute add_comm_monoid.to_comm_monoid [trans_instance]
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noncomputable definition Sum (s : set 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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theorem Sum_of_not_finite {s : set A} (nfins : ¬ finite s) (f : A → B) : Sum s f = 0 :=
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Prod_of_not_finite nfins f
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theorem Sum_add (s : set 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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theorem Sum_insert_of_mem (f : A → B) {a : A} {s : set 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 : set A} [fins : finite s] (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 : set A} [fins₁ : finite s₁] [fins₂ : finite s₂]
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(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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theorem Sum_ext {s : set A} {f g : A → B} (H : ∀x, x ∈ s → f x = g x) :
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Sum s f = Sum s g := Prod_ext H
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theorem Sum_zero (s : set A) : Sum s (λ x, 0) = (0:B) := Prod_one s
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end Sum
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end set
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end algebra
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