feat(library/data/{int,rat,real}/bigops): add bigops for int, rat, real
Because migrate does not handle parameters, we have to migrate by hand.
This commit is contained in:
Jeremy Avigad
parent
f97298394b
commit
4b39400439
10 changed files with 504 additions and 5 deletions
165
library/data/int/bigops.lean
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165
library/data/int/bigops.lean
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/-
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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: Jeremy Avigad
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Finite products and sums on the integers.
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-/
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import data.int.order algebra.group_bigops algebra.group_set_bigops
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open list
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namespace int
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open [classes] algebra
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local attribute int.decidable_linear_ordered_comm_ring [instance]
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variables {A : Type} [deceqA : decidable_eq A]
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/- Prodl -/
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definition Prodl (l : list A) (f : A → int) : int := algebra.Prodl l f
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notation `∏` binders `←` l, r:(scoped f, Prodl l f) := r
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theorem Prodl_nil (f : A → int) : Prodl [] f = 1 := algebra.Prodl_nil f
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theorem Prodl_cons (f : A → int) (a : A) (l : list A) : Prodl (a::l) f = f a * Prodl l f :=
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algebra.Prodl_cons f a l
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theorem Prodl_append (l₁ l₂ : list A) (f : A → int) : Prodl (l₁++l₂) f = Prodl l₁ f * Prodl l₂ f :=
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algebra.Prodl_append l₁ l₂ f
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theorem Prodl_mul (l : list A) (f g : A → int) :
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Prodl l (λx, f x * g x) = Prodl l f * Prodl l g := algebra.Prodl_mul l f g
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section deceqA
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include deceqA
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theorem Prodl_insert_of_mem (f : A → int) {a : A} {l : list A} (H : a ∈ l) :
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Prodl (insert a l) f = Prodl l f := algebra.Prodl_insert_of_mem f H
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theorem Prodl_insert_of_not_mem (f : A → int) {a : A} {l : list A} (H : a ∉ l) :
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Prodl (insert a l) f = f a * Prodl l f := algebra.Prodl_insert_of_not_mem f H
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theorem Prodl_union {l₁ l₂ : list A} (f : A → int) (d : disjoint l₁ l₂) :
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Prodl (union l₁ l₂) f = Prodl l₁ f * Prodl l₂ f := algebra.Prodl_union f d
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theorem Prodl_one (l : list A) : Prodl l (λ x, 1) = 1 := algebra.Prodl_one l
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end deceqA
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/- Prod over finset -/
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namespace finset
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open finset
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definition Prod (s : finset A) (f : A → int) : int := algebra.finset.Prod s f
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notation `∏` binders `∈` s, r:(scoped f, Prod s f) := r
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theorem Prod_empty (f : A → int) : Prod ∅ f = 1 := algebra.finset.Prod_empty f
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theorem Prod_mul (s : finset A) (f g : A → int) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
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algebra.finset.Prod_mul s f g
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section deceqA
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include deceqA
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theorem Prod_insert_of_mem (f : A → int) {a : A} {s : finset A} (H : a ∈ s) :
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Prod (insert a s) f = Prod s f := algebra.finset.Prod_insert_of_mem f H
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theorem Prod_insert_of_not_mem (f : A → int) {a : A} {s : finset A} (H : a ∉ s) :
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Prod (insert a s) f = f a * Prod s f := algebra.finset.Prod_insert_of_not_mem f H
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theorem Prod_union (f : A → int) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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Prod (s₁ ∪ s₂) f = Prod s₁ f * Prod s₂ f := algebra.finset.Prod_union f disj
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theorem Prod_ext {s : finset A} {f g : A → int} (H : ∀x, x ∈ s → f x = g x) :
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Prod s f = Prod s g := algebra.finset.Prod_ext H
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theorem Prod_one (s : finset A) : Prod s (λ x, 1) = 1 := algebra.finset.Prod_one s
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end deceqA
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end finset
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/- Prod over set -/
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namespace set
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open set
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noncomputable definition Prod (s : set A) (f : A → int) : int := algebra.set.Prod s f
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notation `∏` binders `∈` s, r:(scoped f, Prod s f) := r
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theorem Prod_empty (f : A → int) : Prod ∅ f = 1 := algebra.set.Prod_empty f
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theorem Prod_of_not_finite {s : set A} (nfins : ¬ finite s) (f : A → int) : Prod s f = 1 :=
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algebra.set.Prod_of_not_finite nfins f
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theorem Prod_mul (s : set A) (f g : A → int) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
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algebra.set.Prod_mul s f g
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theorem Prod_insert_of_mem (f : A → int) {a : A} {s : set A} (H : a ∈ s) :
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Prod (insert a s) f = Prod s f := algebra.set.Prod_insert_of_mem f H
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theorem Prod_insert_of_not_mem (f : A → int) {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 := algebra.set.Prod_insert_of_not_mem f H
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theorem Prod_union (f : A → int) {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 := algebra.set.Prod_union f disj
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theorem Prod_ext {s : set A} {f g : A → int} (H : ∀x, x ∈ s → f x = g x) :
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Prod s f = Prod s g := algebra.set.Prod_ext H
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theorem Prod_one (s : set A) : Prod s (λ x, 1) = 1 := algebra.set.Prod_one s
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end set
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/- Suml -/
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definition Suml (l : list A) (f : A → int) : int := algebra.Suml l f
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notation `∑` binders `←` l, r:(scoped f, Suml l f) := r
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theorem Suml_nil (f : A → int) : Suml [] f = 0 := algebra.Suml_nil f
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theorem Suml_cons (f : A → int) (a : A) (l : list A) : Suml (a::l) f = f a + Suml l f :=
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algebra.Suml_cons f a l
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theorem Suml_append (l₁ l₂ : list A) (f : A → int) : Suml (l₁++l₂) f = Suml l₁ f + Suml l₂ f :=
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algebra.Suml_append l₁ l₂ f
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theorem Suml_add (l : list A) (f g : A → int) : Suml l (λx, f x + g x) = Suml l f + Suml l g :=
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algebra.Suml_add l f g
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section deceqA
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include deceqA
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theorem Suml_insert_of_mem (f : A → int) {a : A} {l : list A} (H : a ∈ l) :
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Suml (insert a l) f = Suml l f := algebra.Suml_insert_of_mem f H
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theorem Suml_insert_of_not_mem (f : A → int) {a : A} {l : list A} (H : a ∉ l) :
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Suml (insert a l) f = f a + Suml l f := algebra.Suml_insert_of_not_mem f H
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theorem Suml_union {l₁ l₂ : list A} (f : A → int) (d : disjoint l₁ l₂) :
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Suml (union l₁ l₂) f = Suml l₁ f + Suml l₂ f := algebra.Suml_union f d
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theorem Suml_zero (l : list A) : Suml l (λ x, 0) = 0 := algebra.Suml_zero l
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end deceqA
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/- Sum over a finset -/
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namespace finset
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open finset
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definition Sum (s : finset A) (f : A → int) : int := algebra.finset.Sum s f
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notation `∑` binders `∈` s, r:(scoped f, Sum s f) := r
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theorem Sum_empty (f : A → int) : Sum ∅ f = 0 := algebra.finset.Sum_empty f
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theorem Sum_add (s : finset A) (f g : A → int) : Sum s (λx, f x + g x) = Sum s f + Sum s g :=
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algebra.finset.Sum_add s f g
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section deceqA
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include deceqA
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theorem Sum_insert_of_mem (f : A → int) {a : A} {s : finset A} (H : a ∈ s) :
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Sum (insert a s) f = Sum s f := algebra.finset.Sum_insert_of_mem f H
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theorem Sum_insert_of_not_mem (f : A → int) {a : A} {s : finset A} (H : a ∉ s) :
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Sum (insert a s) f = f a + Sum s f := algebra.finset.Sum_insert_of_not_mem f H
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theorem Sum_union (f : A → int) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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Sum (s₁ ∪ s₂) f = Sum s₁ f + Sum s₂ f := algebra.finset.Sum_union f disj
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theorem Sum_ext {s : finset A} {f g : A → int} (H : ∀x, x ∈ s → f x = g x) :
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Sum s f = Sum s g := algebra.finset.Sum_ext H
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theorem Sum_zero (s : finset A) : Sum s (λ x, 0) = 0 := algebra.finset.Sum_zero s
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end deceqA
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end finset
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/- Sum over a set -/
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namespace set
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open set
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noncomputable definition Sum (s : set A) (f : A → int) : int := algebra.set.Sum s f
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notation `∏` binders `∈` s, r:(scoped f, Sum s f) := r
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theorem Sum_empty (f : A → int) : Sum ∅ f = 0 := algebra.set.Sum_empty f
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theorem Sum_of_not_finite {s : set A} (nfins : ¬ finite s) (f : A → int) : Sum s f = 0 :=
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algebra.set.Sum_of_not_finite nfins f
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theorem Sum_add (s : set A) (f g : A → int) : Sum s (λx, f x + g x) = Sum s f + Sum s g :=
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algebra.set.Sum_add s f g
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theorem Sum_insert_of_mem (f : A → int) {a : A} {s : set A} (H : a ∈ s) :
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Sum (insert a s) f = Sum s f := algebra.set.Sum_insert_of_mem f H
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theorem Sum_insert_of_not_mem (f : A → int) {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 := algebra.set.Sum_insert_of_not_mem f H
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theorem Sum_union (f : A → int) {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 := algebra.set.Sum_union f disj
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theorem Sum_ext {s : set A} {f g : A → int} (H : ∀x, x ∈ s → f x = g x) :
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Sum s f = Sum s g := algebra.set.Sum_ext H
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theorem Sum_zero (s : set A) : Sum s (λ x, 0) = 0 := algebra.set.Sum_zero s
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end set
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end int
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@ -3,4 +3,4 @@ Copyright (c) 2014 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: Jeremy Avigad
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-/
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import .basic .order .div .power .gcd
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import .basic .order .div .power .gcd .bigops
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@ -8,3 +8,4 @@ The integers.
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* [div](div.lean) : div and mod
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* [power](power.lean)
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* [gcd](gcd.lean) : gcd, lcm, and coprime
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* [bigops](bigops.lean)
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165
library/data/rat/bigops.lean
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165
library/data/rat/bigops.lean
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@ -0,0 +1,165 @@
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/-
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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: Jeremy Avigad
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Finite products and sums on the rationals.
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-/
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import data.rat.order algebra.group_bigops algebra.group_set_bigops
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open list
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namespace rat
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open [classes] algebra
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local attribute rat.discrete_linear_ordered_field [instance]
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variables {A : Type} [deceqA : decidable_eq A]
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/- Prodl -/
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definition Prodl (l : list A) (f : A → rat) : rat := algebra.Prodl l f
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notation `∏` binders `←` l, r:(scoped f, Prodl l f) := r
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theorem Prodl_nil (f : A → rat) : Prodl [] f = 1 := algebra.Prodl_nil f
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theorem Prodl_cons (f : A → rat) (a : A) (l : list A) : Prodl (a::l) f = f a * Prodl l f :=
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algebra.Prodl_cons f a l
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theorem Prodl_append (l₁ l₂ : list A) (f : A → rat) : Prodl (l₁++l₂) f = Prodl l₁ f * Prodl l₂ f :=
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algebra.Prodl_append l₁ l₂ f
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theorem Prodl_mul (l : list A) (f g : A → rat) :
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Prodl l (λx, f x * g x) = Prodl l f * Prodl l g := algebra.Prodl_mul l f g
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section deceqA
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include deceqA
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theorem Prodl_insert_of_mem (f : A → rat) {a : A} {l : list A} (H : a ∈ l) :
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Prodl (insert a l) f = Prodl l f := algebra.Prodl_insert_of_mem f H
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theorem Prodl_insert_of_not_mem (f : A → rat) {a : A} {l : list A} (H : a ∉ l) :
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Prodl (insert a l) f = f a * Prodl l f := algebra.Prodl_insert_of_not_mem f H
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theorem Prodl_union {l₁ l₂ : list A} (f : A → rat) (d : disjoint l₁ l₂) :
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Prodl (union l₁ l₂) f = Prodl l₁ f * Prodl l₂ f := algebra.Prodl_union f d
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theorem Prodl_one (l : list A) : Prodl l (λ x, 1) = 1 := algebra.Prodl_one l
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end deceqA
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/- Prod over finset -/
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namespace finset
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open finset
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definition Prod (s : finset A) (f : A → rat) : rat := algebra.finset.Prod s f
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notation `∏` binders `∈` s, r:(scoped f, Prod s f) := r
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theorem Prod_empty (f : A → rat) : Prod ∅ f = 1 := algebra.finset.Prod_empty f
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theorem Prod_mul (s : finset A) (f g : A → rat) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
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algebra.finset.Prod_mul s f g
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section deceqA
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include deceqA
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theorem Prod_insert_of_mem (f : A → rat) {a : A} {s : finset A} (H : a ∈ s) :
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Prod (insert a s) f = Prod s f := algebra.finset.Prod_insert_of_mem f H
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theorem Prod_insert_of_not_mem (f : A → rat) {a : A} {s : finset A} (H : a ∉ s) :
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Prod (insert a s) f = f a * Prod s f := algebra.finset.Prod_insert_of_not_mem f H
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theorem Prod_union (f : A → rat) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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Prod (s₁ ∪ s₂) f = Prod s₁ f * Prod s₂ f := algebra.finset.Prod_union f disj
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theorem Prod_ext {s : finset A} {f g : A → rat} (H : ∀x, x ∈ s → f x = g x) :
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Prod s f = Prod s g := algebra.finset.Prod_ext H
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theorem Prod_one (s : finset A) : Prod s (λ x, 1) = 1 := algebra.finset.Prod_one s
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end deceqA
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end finset
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/- Prod over set -/
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namespace set
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open set
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noncomputable definition Prod (s : set A) (f : A → rat) : rat := algebra.set.Prod s f
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notation `∏` binders `∈` s, r:(scoped f, Prod s f) := r
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theorem Prod_empty (f : A → rat) : Prod ∅ f = 1 := algebra.set.Prod_empty f
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theorem Prod_of_not_finite {s : set A} (nfins : ¬ finite s) (f : A → rat) : Prod s f = 1 :=
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algebra.set.Prod_of_not_finite nfins f
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theorem Prod_mul (s : set A) (f g : A → rat) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
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algebra.set.Prod_mul s f g
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theorem Prod_insert_of_mem (f : A → rat) {a : A} {s : set A} (H : a ∈ s) :
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Prod (insert a s) f = Prod s f := algebra.set.Prod_insert_of_mem f H
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theorem Prod_insert_of_not_mem (f : A → rat) {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 := algebra.set.Prod_insert_of_not_mem f H
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theorem Prod_union (f : A → rat) {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 := algebra.set.Prod_union f disj
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theorem Prod_ext {s : set A} {f g : A → rat} (H : ∀x, x ∈ s → f x = g x) :
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Prod s f = Prod s g := algebra.set.Prod_ext H
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theorem Prod_one (s : set A) : Prod s (λ x, 1) = 1 := algebra.set.Prod_one s
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end set
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/- Suml -/
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definition Suml (l : list A) (f : A → rat) : rat := algebra.Suml l f
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notation `∑` binders `←` l, r:(scoped f, Suml l f) := r
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theorem Suml_nil (f : A → rat) : Suml [] f = 0 := algebra.Suml_nil f
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theorem Suml_cons (f : A → rat) (a : A) (l : list A) : Suml (a::l) f = f a + Suml l f :=
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algebra.Suml_cons f a l
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theorem Suml_append (l₁ l₂ : list A) (f : A → rat) : Suml (l₁++l₂) f = Suml l₁ f + Suml l₂ f :=
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algebra.Suml_append l₁ l₂ f
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theorem Suml_add (l : list A) (f g : A → rat) : Suml l (λx, f x + g x) = Suml l f + Suml l g :=
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algebra.Suml_add l f g
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section deceqA
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include deceqA
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theorem Suml_insert_of_mem (f : A → rat) {a : A} {l : list A} (H : a ∈ l) :
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Suml (insert a l) f = Suml l f := algebra.Suml_insert_of_mem f H
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theorem Suml_insert_of_not_mem (f : A → rat) {a : A} {l : list A} (H : a ∉ l) :
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Suml (insert a l) f = f a + Suml l f := algebra.Suml_insert_of_not_mem f H
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theorem Suml_union {l₁ l₂ : list A} (f : A → rat) (d : disjoint l₁ l₂) :
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Suml (union l₁ l₂) f = Suml l₁ f + Suml l₂ f := algebra.Suml_union f d
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theorem Suml_zero (l : list A) : Suml l (λ x, 0) = 0 := algebra.Suml_zero l
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end deceqA
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/- Sum over a finset -/
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namespace finset
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open finset
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definition Sum (s : finset A) (f : A → rat) : rat := algebra.finset.Sum s f
|
||||
notation `∑` binders `∈` s, r:(scoped f, Sum s f) := r
|
||||
|
||||
theorem Sum_empty (f : A → rat) : Sum ∅ f = 0 := algebra.finset.Sum_empty f
|
||||
theorem Sum_add (s : finset A) (f g : A → rat) : Sum s (λx, f x + g x) = Sum s f + Sum s g :=
|
||||
algebra.finset.Sum_add s f g
|
||||
section deceqA
|
||||
include deceqA
|
||||
theorem Sum_insert_of_mem (f : A → rat) {a : A} {s : finset A} (H : a ∈ s) :
|
||||
Sum (insert a s) f = Sum s f := algebra.finset.Sum_insert_of_mem f H
|
||||
theorem Sum_insert_of_not_mem (f : A → rat) {a : A} {s : finset A} (H : a ∉ s) :
|
||||
Sum (insert a s) f = f a + Sum s f := algebra.finset.Sum_insert_of_not_mem f H
|
||||
theorem Sum_union (f : A → rat) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
|
||||
Sum (s₁ ∪ s₂) f = Sum s₁ f + Sum s₂ f := algebra.finset.Sum_union f disj
|
||||
theorem Sum_ext {s : finset A} {f g : A → rat} (H : ∀x, x ∈ s → f x = g x) :
|
||||
Sum s f = Sum s g := algebra.finset.Sum_ext H
|
||||
theorem Sum_zero (s : finset A) : Sum s (λ x, 0) = 0 := algebra.finset.Sum_zero s
|
||||
end deceqA
|
||||
|
||||
end finset
|
||||
|
||||
/- Sum over a set -/
|
||||
|
||||
namespace set
|
||||
open set
|
||||
|
||||
noncomputable definition Sum (s : set A) (f : A → rat) : rat := algebra.set.Sum s f
|
||||
notation `∏` binders `∈` s, r:(scoped f, Sum s f) := r
|
||||
|
||||
theorem Sum_empty (f : A → rat) : Sum ∅ f = 0 := algebra.set.Sum_empty f
|
||||
theorem Sum_of_not_finite {s : set A} (nfins : ¬ finite s) (f : A → rat) : Sum s f = 0 :=
|
||||
algebra.set.Sum_of_not_finite nfins f
|
||||
theorem Sum_add (s : set A) (f g : A → rat) : Sum s (λx, f x + g x) = Sum s f + Sum s g :=
|
||||
algebra.set.Sum_add s f g
|
||||
theorem Sum_insert_of_mem (f : A → rat) {a : A} {s : set A} (H : a ∈ s) :
|
||||
Sum (insert a s) f = Sum s f := algebra.set.Sum_insert_of_mem f H
|
||||
theorem Sum_insert_of_not_mem (f : A → rat) {a : A} {s : set A} [fins : finite s] (H : a ∉ s) :
|
||||
Sum (insert a s) f = f a + Sum s f := algebra.set.Sum_insert_of_not_mem f H
|
||||
theorem Sum_union (f : A → rat) {s₁ s₂ : set A} [fins₁ : finite s₁] [fins₂ : finite s₂]
|
||||
(disj : s₁ ∩ s₂ = ∅) :
|
||||
Sum (s₁ ∪ s₂) f = Sum s₁ f + Sum s₂ f := algebra.set.Sum_union f disj
|
||||
theorem Sum_ext {s : set A} {f g : A → rat} (H : ∀x, x ∈ s → f x = g x) :
|
||||
Sum s f = Sum s g := algebra.set.Sum_ext H
|
||||
theorem Sum_zero (s : set A) : Sum s (λ x, 0) = 0 := algebra.set.Sum_zero s
|
||||
|
||||
end set
|
||||
|
||||
end rat
|
||||
|
|
@ -3,4 +3,4 @@ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Author: Jeremy Avigad
|
||||
-/
|
||||
import .basic .order
|
||||
import .basic .order .bigops
|
||||
|
|
|
|||
|
|
@ -5,3 +5,4 @@ The rational numbers.
|
|||
|
||||
* [basic](basic.lean) : the rationals as a field
|
||||
* [order](order.lean) : the order relations and the sign function
|
||||
* [bigops](bigops.lean)
|
||||
167
library/data/real/bigops.lean
Normal file
167
library/data/real/bigops.lean
Normal file
|
|
@ -0,0 +1,167 @@
|
|||
/-
|
||||
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Author: Jeremy Avigad
|
||||
|
||||
Finite products and sums on the reals.
|
||||
-/
|
||||
import data.real.division algebra.group_bigops algebra.group_set_bigops
|
||||
open list
|
||||
|
||||
namespace real
|
||||
open [classes] algebra
|
||||
local attribute real.ordered_ring [instance]
|
||||
local attribute real.comm_ring [instance]
|
||||
|
||||
variables {A : Type} [deceqA : decidable_eq A]
|
||||
|
||||
/- Prodl -/
|
||||
|
||||
definition Prodl (l : list A) (f : A → real) : real := algebra.Prodl l f
|
||||
notation `∏` binders `←` l, r:(scoped f, Prodl l f) := r
|
||||
|
||||
theorem Prodl_nil (f : A → real) : Prodl [] f = 1 := algebra.Prodl_nil f
|
||||
theorem Prodl_cons (f : A → real) (a : A) (l : list A) : Prodl (a::l) f = f a * Prodl l f :=
|
||||
algebra.Prodl_cons f a l
|
||||
theorem Prodl_append (l₁ l₂ : list A) (f : A → real) : Prodl (l₁++l₂) f = Prodl l₁ f * Prodl l₂ f :=
|
||||
algebra.Prodl_append l₁ l₂ f
|
||||
theorem Prodl_mul (l : list A) (f g : A → real) :
|
||||
Prodl l (λx, f x * g x) = Prodl l f * Prodl l g := algebra.Prodl_mul l f g
|
||||
section deceqA
|
||||
include deceqA
|
||||
theorem Prodl_insert_of_mem (f : A → real) {a : A} {l : list A} (H : a ∈ l) :
|
||||
Prodl (insert a l) f = Prodl l f := algebra.Prodl_insert_of_mem f H
|
||||
theorem Prodl_insert_of_not_mem (f : A → real) {a : A} {l : list A} (H : a ∉ l) :
|
||||
Prodl (insert a l) f = f a * Prodl l f := algebra.Prodl_insert_of_not_mem f H
|
||||
theorem Prodl_union {l₁ l₂ : list A} (f : A → real) (d : disjoint l₁ l₂) :
|
||||
Prodl (union l₁ l₂) f = Prodl l₁ f * Prodl l₂ f := algebra.Prodl_union f d
|
||||
theorem Prodl_one (l : list A) : Prodl l (λ x, 1) = 1 := algebra.Prodl_one l
|
||||
end deceqA
|
||||
|
||||
/- Prod over finset -/
|
||||
|
||||
namespace finset
|
||||
open finset
|
||||
|
||||
definition Prod (s : finset A) (f : A → real) : real := algebra.finset.Prod s f
|
||||
notation `∏` binders `∈` s, r:(scoped f, Prod s f) := r
|
||||
|
||||
theorem Prod_empty (f : A → real) : Prod ∅ f = 1 := algebra.finset.Prod_empty f
|
||||
theorem Prod_mul (s : finset A) (f g : A → real) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
|
||||
algebra.finset.Prod_mul s f g
|
||||
section deceqA
|
||||
include deceqA
|
||||
theorem Prod_insert_of_mem (f : A → real) {a : A} {s : finset A} (H : a ∈ s) :
|
||||
Prod (insert a s) f = Prod s f := algebra.finset.Prod_insert_of_mem f H
|
||||
theorem Prod_insert_of_not_mem (f : A → real) {a : A} {s : finset A} (H : a ∉ s) :
|
||||
Prod (insert a s) f = f a * Prod s f := algebra.finset.Prod_insert_of_not_mem f H
|
||||
theorem Prod_union (f : A → real) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
|
||||
Prod (s₁ ∪ s₂) f = Prod s₁ f * Prod s₂ f := algebra.finset.Prod_union f disj
|
||||
theorem Prod_ext {s : finset A} {f g : A → real} (H : ∀x, x ∈ s → f x = g x) :
|
||||
Prod s f = Prod s g := algebra.finset.Prod_ext H
|
||||
theorem Prod_one (s : finset A) : Prod s (λ x, 1) = 1 := algebra.finset.Prod_one s
|
||||
end deceqA
|
||||
|
||||
end finset
|
||||
|
||||
/- Prod over set -/
|
||||
|
||||
namespace set
|
||||
open set
|
||||
|
||||
noncomputable definition Prod (s : set A) (f : A → real) : real := algebra.set.Prod s f
|
||||
notation `∏` binders `∈` s, r:(scoped f, Prod s f) := r
|
||||
|
||||
theorem Prod_empty (f : A → real) : Prod ∅ f = 1 := algebra.set.Prod_empty f
|
||||
theorem Prod_of_not_finite {s : set A} (nfins : ¬ finite s) (f : A → real) : Prod s f = 1 :=
|
||||
algebra.set.Prod_of_not_finite nfins f
|
||||
theorem Prod_mul (s : set A) (f g : A → real) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
|
||||
algebra.set.Prod_mul s f g
|
||||
theorem Prod_insert_of_mem (f : A → real) {a : A} {s : set A} (H : a ∈ s) :
|
||||
Prod (insert a s) f = Prod s f := algebra.set.Prod_insert_of_mem f H
|
||||
theorem Prod_insert_of_not_mem (f : A → real) {a : A} {s : set A} [fins : finite s] (H : a ∉ s) :
|
||||
Prod (insert a s) f = f a * Prod s f := algebra.set.Prod_insert_of_not_mem f H
|
||||
theorem Prod_union (f : A → real) {s₁ s₂ : set A} [fins₁ : finite s₁] [fins₂ : finite s₂]
|
||||
(disj : s₁ ∩ s₂ = ∅) :
|
||||
Prod (s₁ ∪ s₂) f = Prod s₁ f * Prod s₂ f := algebra.set.Prod_union f disj
|
||||
theorem Prod_ext {s : set A} {f g : A → real} (H : ∀x, x ∈ s → f x = g x) :
|
||||
Prod s f = Prod s g := algebra.set.Prod_ext H
|
||||
theorem Prod_one (s : set A) : Prod s (λ x, 1) = 1 := algebra.set.Prod_one s
|
||||
|
||||
end set
|
||||
|
||||
/- Suml -/
|
||||
|
||||
definition Suml (l : list A) (f : A → real) : real := algebra.Suml l f
|
||||
notation `∑` binders `←` l, r:(scoped f, Suml l f) := r
|
||||
|
||||
theorem Suml_nil (f : A → real) : Suml [] f = 0 := algebra.Suml_nil f
|
||||
theorem Suml_cons (f : A → real) (a : A) (l : list A) : Suml (a::l) f = f a + Suml l f :=
|
||||
algebra.Suml_cons f a l
|
||||
theorem Suml_append (l₁ l₂ : list A) (f : A → real) : Suml (l₁++l₂) f = Suml l₁ f + Suml l₂ f :=
|
||||
algebra.Suml_append l₁ l₂ f
|
||||
theorem Suml_add (l : list A) (f g : A → real) : Suml l (λx, f x + g x) = Suml l f + Suml l g :=
|
||||
algebra.Suml_add l f g
|
||||
section deceqA
|
||||
include deceqA
|
||||
theorem Suml_insert_of_mem (f : A → real) {a : A} {l : list A} (H : a ∈ l) :
|
||||
Suml (insert a l) f = Suml l f := algebra.Suml_insert_of_mem f H
|
||||
theorem Suml_insert_of_not_mem (f : A → real) {a : A} {l : list A} (H : a ∉ l) :
|
||||
Suml (insert a l) f = f a + Suml l f := algebra.Suml_insert_of_not_mem f H
|
||||
theorem Suml_union {l₁ l₂ : list A} (f : A → real) (d : disjoint l₁ l₂) :
|
||||
Suml (union l₁ l₂) f = Suml l₁ f + Suml l₂ f := algebra.Suml_union f d
|
||||
theorem Suml_zero (l : list A) : Suml l (λ x, 0) = 0 := algebra.Suml_zero l
|
||||
end deceqA
|
||||
|
||||
/- Sum over a finset -/
|
||||
|
||||
namespace finset
|
||||
open finset
|
||||
definition Sum (s : finset A) (f : A → real) : real := algebra.finset.Sum s f
|
||||
notation `∑` binders `∈` s, r:(scoped f, Sum s f) := r
|
||||
|
||||
theorem Sum_empty (f : A → real) : Sum ∅ f = 0 := algebra.finset.Sum_empty f
|
||||
theorem Sum_add (s : finset A) (f g : A → real) : Sum s (λx, f x + g x) = Sum s f + Sum s g :=
|
||||
algebra.finset.Sum_add s f g
|
||||
section deceqA
|
||||
include deceqA
|
||||
theorem Sum_insert_of_mem (f : A → real) {a : A} {s : finset A} (H : a ∈ s) :
|
||||
Sum (insert a s) f = Sum s f := algebra.finset.Sum_insert_of_mem f H
|
||||
theorem Sum_insert_of_not_mem (f : A → real) {a : A} {s : finset A} (H : a ∉ s) :
|
||||
Sum (insert a s) f = f a + Sum s f := algebra.finset.Sum_insert_of_not_mem f H
|
||||
theorem Sum_union (f : A → real) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
|
||||
Sum (s₁ ∪ s₂) f = Sum s₁ f + Sum s₂ f := algebra.finset.Sum_union f disj
|
||||
theorem Sum_ext {s : finset A} {f g : A → real} (H : ∀x, x ∈ s → f x = g x) :
|
||||
Sum s f = Sum s g := algebra.finset.Sum_ext H
|
||||
theorem Sum_zero (s : finset A) : Sum s (λ x, 0) = 0 := algebra.finset.Sum_zero s
|
||||
end deceqA
|
||||
|
||||
end finset
|
||||
|
||||
/- Sum over a set -/
|
||||
|
||||
namespace set
|
||||
open set
|
||||
|
||||
noncomputable definition Sum (s : set A) (f : A → real) : real := algebra.set.Sum s f
|
||||
notation `∏` binders `∈` s, r:(scoped f, Sum s f) := r
|
||||
|
||||
theorem Sum_empty (f : A → real) : Sum ∅ f = 0 := algebra.set.Sum_empty f
|
||||
theorem Sum_of_not_finite {s : set A} (nfins : ¬ finite s) (f : A → real) : Sum s f = 0 :=
|
||||
algebra.set.Sum_of_not_finite nfins f
|
||||
theorem Sum_add (s : set A) (f g : A → real) : Sum s (λx, f x + g x) = Sum s f + Sum s g :=
|
||||
algebra.set.Sum_add s f g
|
||||
theorem Sum_insert_of_mem (f : A → real) {a : A} {s : set A} (H : a ∈ s) :
|
||||
Sum (insert a s) f = Sum s f := algebra.set.Sum_insert_of_mem f H
|
||||
theorem Sum_insert_of_not_mem (f : A → real) {a : A} {s : set A} [fins : finite s] (H : a ∉ s) :
|
||||
Sum (insert a s) f = f a + Sum s f := algebra.set.Sum_insert_of_not_mem f H
|
||||
theorem Sum_union (f : A → real) {s₁ s₂ : set A} [fins₁ : finite s₁] [fins₂ : finite s₂]
|
||||
(disj : s₁ ∩ s₂ = ∅) :
|
||||
Sum (s₁ ∪ s₂) f = Sum s₁ f + Sum s₂ f := algebra.set.Sum_union f disj
|
||||
theorem Sum_ext {s : set A} {f g : A → real} (H : ∀x, x ∈ s → f x = g x) :
|
||||
Sum s f = Sum s g := algebra.set.Sum_ext H
|
||||
theorem Sum_zero (s : set A) : Sum s (λ x, 0) = 0 := algebra.set.Sum_zero s
|
||||
|
||||
end set
|
||||
|
||||
end real
|
||||
|
|
@ -1011,7 +1011,6 @@ theorem under_lowest_bound : ∀ y : ℝ, ub y → sup_under ≤ y :=
|
|||
|
||||
theorem under_over_equiv : p_under_seq ≡ p_over_seq :=
|
||||
begin
|
||||
rewrite ↑equiv,
|
||||
intros,
|
||||
apply rat.le.trans,
|
||||
have H : p_under_seq n < p_over_seq n, from !under_seq_lt_over_seq,
|
||||
|
|
|
|||
|
|
@ -3,4 +3,4 @@ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Author: Robert Y. Lewis
|
||||
-/
|
||||
import .basic .order .division .complete
|
||||
import .basic .order .division .complete .bigops
|
||||
|
|
|
|||
|
|
@ -6,4 +6,5 @@ The real numbers: classically, as a quotient type; constructively, as a setoid.
|
|||
* [basic](basic.lean) : the reals as a commutative ring (constructive)
|
||||
* [order](order.lean) : the reals as an ordered ring (constructive)
|
||||
* [division](division.lean) : the reals as a discrete linear ordered field (classical)
|
||||
* [complete](complete.lean) : the reals are Cauchy complete (classical)
|
||||
* [complete](complete.lean) : the reals are Cauchy complete (classical)
|
||||
* [bigops](bigops.lean)
|
||||
Loading…
Reference in a new issue