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085ff1a30c78b9c3851d2cdf123f27847e32058c | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /src/Init/Data/List/Basic.lean | e4786a9bac2d3b8522bc8437069c9f1649995241 | [
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"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 29,106 | lean | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
prelude
import Init.SimpLemmas
import Init.Data.Nat.Basic
set_option linter.missingDocs true -- keep it documented
open Decidable List
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
namespace List
instance : GetElem (List α) Nat α fun as i => i < as.length where
getElem as i h := as.get ⟨i, h⟩
@[simp] theorem cons_getElem_zero (a : α) (as : List α) (h : 0 < (a :: as).length) : getElem (a :: as) 0 h = a := by
rfl
@[simp] theorem cons_getElem_succ (a : α) (as : List α) (i : Nat) (h : i + 1 < (a :: as).length) : getElem (a :: as) (i+1) h = getElem as i (Nat.lt_of_succ_lt_succ h) := by
rfl
theorem length_add_eq_lengthTRAux (as : List α) (n : Nat) : as.length + n = as.lengthTRAux n := by
induction as generalizing n with
| nil => simp [length, lengthTRAux]
| cons a as ih =>
simp [length, lengthTRAux, ← ih, Nat.succ_add]
rfl
@[csimp] theorem length_eq_lengthTR : @List.length = @List.lengthTR := by
apply funext; intro α; apply funext; intro as
simp [lengthTR, ← length_add_eq_lengthTRAux]
@[simp] theorem length_nil : length ([] : List α) = 0 :=
rfl
/-- Auxiliary for `List.reverse`. `List.reverseAux l r = l.reverse ++ r`, but it is defined directly. -/
def reverseAux : List α → List α → List α
| [], r => r
| a::l, r => reverseAux l (a::r)
/--
`O(|as|)`. Reverse of a list:
* `[1, 2, 3, 4].reverse = [4, 3, 2, 1]`
Note that because of the "functional but in place" optimization implemented by Lean's compiler,
this function works without any allocations provided that the input list is unshared:
it simply walks the linked list and reverses all the node pointers.
-/
def reverse (as : List α) : List α :=
reverseAux as []
theorem reverseAux_reverseAux_nil (as bs : List α) : reverseAux (reverseAux as bs) [] = reverseAux bs as := by
induction as generalizing bs with
| nil => rfl
| cons a as ih => simp [reverseAux, ih]
theorem reverseAux_reverseAux (as bs cs : List α) : reverseAux (reverseAux as bs) cs = reverseAux bs (reverseAux (reverseAux as []) cs) := by
induction as generalizing bs cs with
| nil => rfl
| cons a as ih => simp [reverseAux, ih (a::bs), ih [a]]
@[simp] theorem reverse_reverse (as : List α) : as.reverse.reverse = as := by
simp [reverse]; rw [reverseAux_reverseAux_nil]; rfl
/--
`O(|xs|)`: append two lists. `[1, 2, 3] ++ [4, 5] = [1, 2, 3, 4, 5]`.
It takes time proportional to the first list.
-/
protected def append : (xs ys : List α) → List α
| [], bs => bs
| a::as, bs => a :: List.append as bs
/-- Tail-recursive version of `List.append`. -/
def appendTR (as bs : List α) : List α :=
reverseAux as.reverse bs
@[csimp] theorem append_eq_appendTR : @List.append = @appendTR := by
apply funext; intro α; apply funext; intro as; apply funext; intro bs
simp [appendTR, reverse]
induction as with
| nil => rfl
| cons a as ih =>
simp [reverseAux, List.append, ih, reverseAux_reverseAux]
instance : Append (List α) := ⟨List.append⟩
@[simp] theorem nil_append (as : List α) : [] ++ as = as := rfl
@[simp] theorem append_nil (as : List α) : as ++ [] = as := by
induction as with
| nil => rfl
| cons a as ih =>
simp_all [HAppend.hAppend, Append.append, List.append]
@[simp] theorem cons_append (a : α) (as bs : List α) : (a::as) ++ bs = a::(as ++ bs) := rfl
@[simp] theorem append_eq (as bs : List α) : List.append as bs = as ++ bs := rfl
theorem append_assoc (as bs cs : List α) : (as ++ bs) ++ cs = as ++ (bs ++ cs) := by
induction as with
| nil => rfl
| cons a as ih => simp [ih]
theorem append_cons (as : List α) (b : α) (bs : List α) : as ++ b :: bs = as ++ [b] ++ bs := by
induction as with
| nil => simp
| cons a as ih => simp [ih]
instance : EmptyCollection (List α) := ⟨List.nil⟩
/--
`O(|l|)`. `erase l a` removes the first occurrence of `a` from `l`.
* `erase [1, 5, 3, 2, 5] 5 = [1, 3, 2, 5]`
* `erase [1, 5, 3, 2, 5] 6 = [1, 5, 3, 2, 5]`
-/
protected def erase {α} [BEq α] : List α → α → List α
| [], _ => []
| a::as, b => match a == b with
| true => as
| false => a :: List.erase as b
/--
`O(i)`. `eraseIdx l i` removes the `i`'th element of the list `l`.
* `erase [a, b, c, d, e] 0 = [b, c, d, e]`
* `erase [a, b, c, d, e] 1 = [a, c, d, e]`
* `erase [a, b, c, d, e] 5 = [a, b, c, d, e]`
-/
def eraseIdx : List α → Nat → List α
| [], _ => []
| _::as, 0 => as
| a::as, n+1 => a :: eraseIdx as n
/--
`O(1)`. `isEmpty l` is true if the list is empty.
* `isEmpty [] = true`
* `isEmpty [a] = false`
* `isEmpty [a, b] = false`
-/
def isEmpty : List α → Bool
| [] => true
| _ :: _ => false
/--
`O(|l|)`. `map f l` applies `f` to each element of the list.
* `map f [a, b, c] = [f a, f b, f c]`
-/
@[specialize] def map (f : α → β) : List α → List β
| [] => []
| a::as => f a :: map f as
/-- Tail-recursive version of `List.map`. -/
@[inline] def mapTR (f : α → β) (as : List α) : List β :=
loop as []
where
@[specialize] loop : List α → List β → List β
| [], bs => bs.reverse
| a::as, bs => loop as (f a :: bs)
theorem reverseAux_eq_append (as bs : List α) : reverseAux as bs = reverseAux as [] ++ bs := by
induction as generalizing bs with
| nil => simp [reverseAux]
| cons a as ih =>
simp [reverseAux]
rw [ih (a :: bs), ih [a], append_assoc]
rfl
@[simp] theorem reverse_nil : reverse ([] : List α) = [] :=
rfl
@[simp] theorem reverse_cons (a : α) (as : List α) : reverse (a :: as) = reverse as ++ [a] := by
simp [reverse, reverseAux]
rw [← reverseAux_eq_append]
@[simp] theorem reverse_append (as bs : List α) : (as ++ bs).reverse = bs.reverse ++ as.reverse := by
induction as generalizing bs with
| nil => simp
| cons a as ih => simp [ih]; rw [append_assoc]
theorem mapTR_loop_eq (f : α → β) (as : List α) (bs : List β) :
mapTR.loop f as bs = bs.reverse ++ map f as := by
induction as generalizing bs with
| nil => simp [mapTR.loop, map]
| cons a as ih =>
simp [mapTR.loop, map]
rw [ih (f a :: bs), reverse_cons, append_assoc]
rfl
@[csimp] theorem map_eq_mapTR : @map = @mapTR :=
funext fun α => funext fun β => funext fun f => funext fun as => by
simp [mapTR, mapTR_loop_eq]
/--
`O(|join L|)`. `join L` concatenates all the lists in `L` into one list.
* `join [[a], [], [b, c], [d, e, f]] = [a, b, c, d, e, f]`
-/
def join : List (List α) → List α
| [] => []
| a :: as => a ++ join as
/--
`O(|l|)`. `filterMap f l` takes a function `f : α → Option β` and applies it to each element of `l`;
the resulting non-`none` values are collected to form the output list.
```
filterMap
(fun x => if x > 2 then some (2 * x) else none)
[1, 2, 5, 2, 7, 7]
= [10, 14, 14]
```
-/
@[specialize] def filterMap (f : α → Option β) : List α → List β
| [] => []
| a::as =>
match f a with
| none => filterMap f as
| some b => b :: filterMap f as
/--
`O(|l|)`. `filter f l` returns the list of elements in `l` for which `f` returns true.
```
filter (· > 2) [1, 2, 5, 2, 7, 7] = [5, 7, 7]
```
-/
def filter (p : α → Bool) : List α → List α
| [] => []
| a::as => match p a with
| true => a :: filter p as
| false => filter p as
/-- Tail-recursive version of `List.filter`. -/
@[inline] def filterTR (p : α → Bool) (as : List α) : List α :=
loop as []
where
@[specialize] loop : List α → List α → List α
| [], rs => rs.reverse
| a::as, rs => match p a with
| true => loop as (a::rs)
| false => loop as rs
theorem filterTR_loop_eq (p : α → Bool) (as bs : List α) :
filterTR.loop p as bs = bs.reverse ++ filter p as := by
induction as generalizing bs with
| nil => simp [filterTR.loop, filter]
| cons a as ih =>
simp [filterTR.loop, filter]
split
next => rw [ih, reverse_cons, append_assoc]; simp
next => rw [ih]
@[csimp] theorem filter_eq_filterTR : @filter = @filterTR := by
apply funext; intro α; apply funext; intro p; apply funext; intro as
simp [filterTR, filterTR_loop_eq]
/--
`O(|l|)`. `partition p l` calls `p` on each element of `l`, partitioning the list into two lists
`(l_true, l_false)` where `l_true` has the elements where `p` was true
and `l_false` has the elements where `p` is false.
`partition p l = (filter p l, filter (not ∘ p) l)`, but it is slightly more efficient
since it only has to do one pass over the list.
```
partition (· > 2) [1, 2, 5, 2, 7, 7] = ([5, 7, 7], [1, 2, 2])
```
-/
@[inline] def partition (p : α → Bool) (as : List α) : List α × List α :=
loop as ([], [])
where
@[specialize] loop : List α → List α × List α → List α × List α
| [], (bs, cs) => (bs.reverse, cs.reverse)
| a::as, (bs, cs) =>
match p a with
| true => loop as (a::bs, cs)
| false => loop as (bs, a::cs)
/--
`O(|l|)`. `dropWhile p l` removes elements from the list until it finds the first element
for which `p` returns false; this element and everything after it is returned.
```
dropWhile (· < 4) [1, 3, 2, 4, 2, 7, 4] = [4, 2, 7, 4]
```
-/
def dropWhile (p : α → Bool) : List α → List α
| [] => []
| a::l => match p a with
| true => dropWhile p l
| false => a::l
/--
`O(|l|)`. `find? p l` returns the first element for which `p` returns true,
or `none` if no such element is found.
* `find? (· < 5) [7, 6, 5, 8, 1, 2, 6] = some 1`
* `find? (· < 1) [7, 6, 5, 8, 1, 2, 6] = none`
-/
def find? (p : α → Bool) : List α → Option α
| [] => none
| a::as => match p a with
| true => some a
| false => find? p as
/--
`O(|l|)`. `findSome? f l` applies `f` to each element of `l`, and returns the first non-`none` result.
* `findSome? (fun x => if x < 5 then some (10 * x) else none) [7, 6, 5, 8, 1, 2, 6] = some 10`
-/
def findSome? (f : α → Option β) : List α → Option β
| [] => none
| a::as => match f a with
| some b => some b
| none => findSome? f as
/--
`O(|l|)`. `replace l a b` replaces the first element in the list equal to `a` with `b`.
* `replace [1, 4, 2, 3, 3, 7] 3 6 = [1, 4, 2, 6, 3, 7]`
* `replace [1, 4, 2, 3, 3, 7] 5 6 = [1, 4, 2, 3, 3, 7]`
-/
def replace [BEq α] : List α → α → α → List α
| [], _, _ => []
| a::as, b, c => match a == b with
| true => c::as
| false => a :: replace as b c
/--
`O(|l|)`. `elem a l` or `l.contains a` is true if there is an element in `l` equal to `a`.
* `elem 3 [1, 4, 2, 3, 3, 7] = true`
* `elem 5 [1, 4, 2, 3, 3, 7] = false`
-/
def elem [BEq α] (a : α) : List α → Bool
| [] => false
| b::bs => match a == b with
| true => true
| false => elem a bs
/-- `notElem a l` is `!(elem a l)`. -/
def notElem [BEq α] (a : α) (as : List α) : Bool :=
!(as.elem a)
@[inherit_doc elem] abbrev contains [BEq α] (as : List α) (a : α) : Bool :=
elem a as
/--
`a ∈ l` is a predicate which asserts that `a` is in the list `l`.
Unlike `elem`, this uses `=` instead of `==` and is suited for mathematical reasoning.
* `a ∈ [x, y, z] ↔ a = x ∨ a = y ∨ a = z`
-/
inductive Mem (a : α) : List α → Prop
/-- The head of a list is a member: `a ∈ a :: as`. -/
| head (as : List α) : Mem a (a::as)
/-- A member of the tail of a list is a member of the list: `a ∈ l → a ∈ b :: l`. -/
| tail (b : α) {as : List α} : Mem a as → Mem a (b::as)
instance : Membership α (List α) where
mem := Mem
theorem mem_of_elem_eq_true [DecidableEq α] {a : α} {as : List α} : elem a as = true → a ∈ as := by
match as with
| [] => simp [elem]
| a'::as =>
simp [elem]
split
next h => intros; simp [BEq.beq] at h; subst h; apply Mem.head
next _ => intro h; exact Mem.tail _ (mem_of_elem_eq_true h)
theorem elem_eq_true_of_mem [DecidableEq α] {a : α} {as : List α} (h : a ∈ as) : elem a as = true := by
induction h with
| head _ => simp [elem]
| tail _ _ ih => simp [elem]; split; rfl; assumption
instance [DecidableEq α] (a : α) (as : List α) : Decidable (a ∈ as) :=
decidable_of_decidable_of_iff (Iff.intro mem_of_elem_eq_true elem_eq_true_of_mem)
theorem mem_append_of_mem_left {a : α} {as : List α} (bs : List α) : a ∈ as → a ∈ as ++ bs := by
intro h
induction h with
| head => apply Mem.head
| tail => apply Mem.tail; assumption
theorem mem_append_of_mem_right {b : α} {bs : List α} (as : List α) : b ∈ bs → b ∈ as ++ bs := by
intro h
induction as with
| nil => simp [h]
| cons => apply Mem.tail; assumption
/-- `O(|l|^2)`. Erase duplicated elements in the list.
Keeps the first occurrence of duplicated elements.
* `eraseDups [1, 3, 2, 2, 3, 5] = [1, 3, 2, 5]`
-/
def eraseDups {α} [BEq α] (as : List α) : List α :=
loop as []
where
loop : List α → List α → List α
| [], bs => bs.reverse
| a::as, bs => match bs.elem a with
| true => loop as bs
| false => loop as (a::bs)
/--
`O(|l|)`. Erase repeated adjacent elements. Keeps the first occurrence of each run.
* `eraseReps [1, 3, 2, 2, 2, 3, 5] = [1, 3, 2, 3, 5]`
-/
def eraseReps {α} [BEq α] : List α → List α
| [] => []
| a::as => loop a as []
where
loop {α} [BEq α] : α → List α → List α → List α
| a, [], rs => (a::rs).reverse
| a, a'::as, rs => match a == a' with
| true => loop a as rs
| false => loop a' as (a::rs)
/--
`O(|l|)`. `span p l` splits the list `l` into two parts, where the first part
contains the longest initial segment for which `p` returns true
and the second part is everything else.
* `span (· > 5) [6, 8, 9, 5, 2, 9] = ([6, 8, 9], [5, 2, 9])`
* `span (· > 10) [6, 8, 9, 5, 2, 9] = ([6, 8, 9, 5, 2, 9], [])`
-/
@[inline] def span (p : α → Bool) (as : List α) : List α × List α :=
loop as []
where
@[specialize] loop : List α → List α → List α × List α
| [], rs => (rs.reverse, [])
| a::as, rs => match p a with
| true => loop as (a::rs)
| false => (rs.reverse, a::as)
/--
`O(|l|)`. `groupBy R l` splits `l` into chains of elements
such that adjacent elements are related by `R`.
* `groupBy (·==·) [1, 1, 2, 2, 2, 3, 2] = [[1, 1], [2, 2, 2], [3], [2]]`
* `groupBy (·<·) [1, 2, 5, 4, 5, 1, 4] = [[1, 2, 5], [4, 5], [1, 4]]`
-/
@[specialize] def groupBy (R : α → α → Bool) : List α → List (List α)
| [] => []
| a::as => loop as a [] []
where
@[specialize] loop : List α → α → List α → List (List α) → List (List α)
| a::as, ag, g, gs => match R ag a with
| true => loop as a (ag::g) gs
| false => loop as a [] ((ag::g).reverse::gs)
| [], ag, g, gs => ((ag::g).reverse::gs).reverse
/--
`O(|l|)`. `lookup a l` treats `l : List (α × β)` like an association list,
and returns the first `β` value corresponding to an `α` value in the list equal to `a`.
* `lookup 3 [(1, 2), (3, 4), (3, 5)] = some 4`
* `lookup 2 [(1, 2), (3, 4), (3, 5)] = none`
-/
def lookup [BEq α] : α → List (α × β) → Option β
| _, [] => none
| a, (k,b)::es => match a == k with
| true => some b
| false => lookup a es
/-- `O(|xs|)`. Computes the "set difference" of lists,
by filtering out all elements of `xs` which are also in `ys`.
* `removeAll [1, 1, 5, 1, 2, 4, 5] [1, 2, 2] = [5, 4, 5]`
-/
def removeAll [BEq α] (xs ys : List α) : List α :=
xs.filter (fun x => ys.notElem x)
/--
`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
-/
def drop : Nat → List α → List α
| 0, a => a
| _+1, [] => []
| n+1, _::as => drop n as
@[simp] theorem drop_nil : ([] : List α).drop i = [] := by
cases i <;> rfl
theorem get_drop_eq_drop (as : List α) (i : Nat) (h : i < as.length) : as[i] :: as.drop (i+1) = as.drop i :=
match as, i with
| _::_, 0 => rfl
| _::_, i+1 => get_drop_eq_drop _ i _
/--
`O(min n |xs|)`. Returns the first `n` elements of `xs`, or the whole list if `n` is too large.
* `take 0 [a, b, c, d, e] = []`
* `take 3 [a, b, c, d, e] = [a, b, c]`
* `take 6 [a, b, c, d, e] = [a, b, c, d, e]`
-/
def take : Nat → List α → List α
| 0, _ => []
| _+1, [] => []
| n+1, a::as => a :: take n as
/--
`O(|xs|)`. Returns the longest initial segment of `xs` for which `p` returns true.
* `takeWhile (· > 5) [7, 6, 4, 8] = [7, 6]`
* `takeWhile (· > 5) [7, 6, 6, 8] = [7, 6, 6, 8]`
-/
def takeWhile (p : α → Bool) : (xs : List α) → List α
| [] => []
| hd :: tl => match p hd with
| true => hd :: takeWhile p tl
| false => []
/--
`O(|l|)`. Applies function `f` to all of the elements of the list, from right to left.
* `foldr f init [a, b, c] = f a <| f b <| f c <| init`
-/
@[specialize] def foldr (f : α → β → β) (init : β) : List α → β
| [] => init
| a :: l => f a (foldr f init l)
/--
`O(|l|)`. Returns true if `p` is true for any element of `l`.
* `any p [a, b, c] = p a || p b || p c`
-/
@[inline] def any (l : List α) (p : α → Bool) : Bool :=
foldr (fun a r => p a || r) false l
/--
`O(|l|)`. Returns true if `p` is true for every element of `l`.
* `all p [a, b, c] = p a && p b && p c`
-/
@[inline] def all (l : List α) (p : α → Bool) : Bool :=
foldr (fun a r => p a && r) true l
/--
`O(|l|)`. Returns true if `true` is an element of the list of booleans `l`.
* `or [a, b, c] = a || b || c`
-/
def or (bs : List Bool) : Bool := bs.any id
/--
`O(|l|)`. Returns true if every element of `l` is the value `true`.
* `and [a, b, c] = a && b && c`
-/
def and (bs : List Bool) : Bool := bs.all id
/--
`O(min |xs| |ys|)`. Applies `f` to the two lists in parallel, stopping at the shorter list.
* `zipWith f [x₁, x₂, x₃] [y₁, y₂, y₃, y₄] = [f x₁ y₁, f x₂ y₂, f x₃ y₃]`
-/
@[specialize] def zipWith (f : α → β → γ) : (xs : List α) → (ys : List β) → List γ
| x::xs, y::ys => f x y :: zipWith f xs ys
| _, _ => []
/--
`O(min |xs| |ys|)`. Combines the two lists into a list of pairs, with one element from each list.
The longer list is truncated to match the shorter list.
* `zip [x₁, x₂, x₃] [y₁, y₂, y₃, y₄] = [(x₁, y₁), (x₂, y₂), (x₃, y₃)]`
-/
def zip : List α → List β → List (Prod α β) :=
zipWith Prod.mk
/--
`O(|l|)`. Separates a list of pairs into two lists containing the first components and second components.
* `unzip [(x₁, y₁), (x₂, y₂), (x₃, y₃)] = ([x₁, x₂, x₃], [y₁, y₂, y₃])`
-/
def unzip : List (α × β) → List α × List β
| [] => ([], [])
| (a, b) :: t => match unzip t with | (al, bl) => (a::al, b::bl)
/--
`O(n)`. `range n` is the numbers from `0` to `n` exclusive, in increasing order.
* `range 5 = [0, 1, 2, 3, 4]`
-/
def range (n : Nat) : List Nat :=
loop n []
where
loop : Nat → List Nat → List Nat
| 0, ns => ns
| n+1, ns => loop n (n::ns)
/--
`O(n)`. `iota n` is the numbers from `1` to `n` inclusive, in decreasing order.
* `iota 5 = [5, 4, 3, 2, 1]`
-/
def iota : Nat → List Nat
| 0 => []
| m@(n+1) => m :: iota n
/-- Tail-recursive version of `iota`. -/
def iotaTR (n : Nat) : List Nat :=
let rec go : Nat → List Nat → List Nat
| 0, r => r.reverse
| m@(n+1), r => go n (m::r)
go n []
@[csimp]
theorem iota_eq_iotaTR : @iota = @iotaTR :=
have aux (n : Nat) (r : List Nat) : iotaTR.go n r = r.reverse ++ iota n := by
induction n generalizing r with
| zero => simp [iota, iotaTR.go]
| succ n ih => simp [iota, iotaTR.go, ih, append_assoc]
funext fun n => by simp [iotaTR, aux]
/--
`O(|l|)`. `enumFrom n l` is like `enum` but it allows you to specify the initial index.
* `enumFrom 5 [a, b, c] = [(5, a), (6, b), (7, c)]`
-/
def enumFrom : Nat → List α → List (Nat × α)
| _, [] => nil
| n, x :: xs => (n, x) :: enumFrom (n + 1) xs
/--
`O(|l|)`. `enum l` pairs up each element with its index in the list.
* `enum [a, b, c] = [(0, a), (1, b), (2, c)]`
-/
def enum : List α → List (Nat × α) := enumFrom 0
/--
`O(|l|)`. `intersperse sep l` alternates `sep` and the elements of `l`:
* `intersperse sep [] = []`
* `intersperse sep [a] = [a]`
* `intersperse sep [a, b] = [a, sep, b]`
* `intersperse sep [a, b, c] = [a, sep, b, sep, c]`
-/
def intersperse (sep : α) : List α → List α
| [] => []
| [x] => [x]
| x::xs => x :: sep :: intersperse sep xs
/--
`O(|xs|)`. `intercalate sep xs` alternates `sep` and the elements of `xs`:
* `intercalate sep [] = []`
* `intercalate sep [a] = a`
* `intercalate sep [a, b] = a ++ sep ++ b`
* `intercalate sep [a, b, c] = a ++ sep ++ b ++ sep ++ c`
-/
def intercalate (sep : List α) (xs : List (List α)) : List α :=
join (intersperse sep xs)
/--
`bind xs f` is the bind operation of the list monad. It applies `f` to each element of `xs`
to get a list of lists, and then concatenates them all together.
* `[2, 3, 2].bind range = [0, 1, 0, 1, 2, 0, 1]`
-/
@[inline] protected def bind {α : Type u} {β : Type v} (a : List α) (b : α → List β) : List β := join (map b a)
/-- `pure x = [x]` is the `pure` operation of the list monad. -/
@[inline] protected def pure {α : Type u} (a : α) : List α := [a]
/--
The lexicographic order on lists.
`[] < a::as`, and `a::as < b::bs` if `a < b` or if `a` and `b` are equivalent and `as < bs`.
-/
inductive lt [LT α] : List α → List α → Prop where
/-- `[]` is the smallest element in the order. -/
| nil (b : α) (bs : List α) : lt [] (b::bs)
/-- If `a < b` then `a::as < b::bs`. -/
| head {a : α} (as : List α) {b : α} (bs : List α) : a < b → lt (a::as) (b::bs)
/-- If `a` and `b` are equivalent and `as < bs`, then `a::as < b::bs`. -/
| tail {a : α} {as : List α} {b : α} {bs : List α} : ¬ a < b → ¬ b < a → lt as bs → lt (a::as) (b::bs)
instance [LT α] : LT (List α) := ⟨List.lt⟩
instance hasDecidableLt [LT α] [h : DecidableRel (α:=α) (·<·)] : (l₁ l₂ : List α) → Decidable (l₁ < l₂)
| [], [] => isFalse (fun h => nomatch h)
| [], _::_ => isTrue (List.lt.nil _ _)
| _::_, [] => isFalse (fun h => nomatch h)
| a::as, b::bs =>
match h a b with
| isTrue h₁ => isTrue (List.lt.head _ _ h₁)
| isFalse h₁ =>
match h b a with
| isTrue h₂ => isFalse (fun h => match h with
| List.lt.head _ _ h₁' => absurd h₁' h₁
| List.lt.tail _ h₂' _ => absurd h₂ h₂')
| isFalse h₂ =>
match hasDecidableLt as bs with
| isTrue h₃ => isTrue (List.lt.tail h₁ h₂ h₃)
| isFalse h₃ => isFalse (fun h => match h with
| List.lt.head _ _ h₁' => absurd h₁' h₁
| List.lt.tail _ _ h₃' => absurd h₃' h₃)
/-- The lexicographic order on lists. -/
@[reducible] protected def le [LT α] (a b : List α) : Prop := ¬ b < a
instance [LT α] : LE (List α) := ⟨List.le⟩
instance [LT α] [DecidableRel ((· < ·) : α → α → Prop)] : (l₁ l₂ : List α) → Decidable (l₁ ≤ l₂) :=
fun _ _ => inferInstanceAs (Decidable (Not _))
/-- `isPrefixOf l₁ l₂` returns `true` Iff `l₁` is a prefix of `l₂`.
That is, there exists a `t` such that `l₂ == l₁ ++ t`. -/
def isPrefixOf [BEq α] : List α → List α → Bool
| [], _ => true
| _, [] => false
| a::as, b::bs => a == b && isPrefixOf as bs
/-- `isPrefixOf? l₁ l₂` returns `some t` when `l₂ == l₁ ++ t`. -/
def isPrefixOf? [BEq α] : List α → List α → Option (List α)
| [], l₂ => some l₂
| _, [] => none
| (x₁ :: l₁), (x₂ :: l₂) =>
if x₁ == x₂ then isPrefixOf? l₁ l₂ else none
/-- `isSuffixOf l₁ l₂` returns `true` Iff `l₁` is a suffix of `l₂`.
That is, there exists a `t` such that `l₂ == t ++ l₁`. -/
def isSuffixOf [BEq α] (l₁ l₂ : List α) : Bool :=
isPrefixOf l₁.reverse l₂.reverse
/-- `isSuffixOf? l₁ l₂` returns `some t` when `l₂ == t ++ l₁`.-/
def isSuffixOf? [BEq α] (l₁ l₂ : List α) : Option (List α) :=
Option.map List.reverse <| isPrefixOf? l₁.reverse l₂.reverse
/--
`O(min |as| |bs|)`. Returns true if `as` and `bs` have the same length,
and they are pairwise related by `eqv`.
-/
@[specialize] def isEqv : (as bs : List α) → (eqv : α → α → Bool) → Bool
| [], [], _ => true
| a::as, b::bs, eqv => eqv a b && isEqv as bs eqv
| _, _, _ => false
/--
The equality relation on lists asserts that they have the same length
and they are pairwise `BEq`.
-/
protected def beq [BEq α] : List α → List α → Bool
| [], [] => true
| a::as, b::bs => a == b && List.beq as bs
| _, _ => false
instance [BEq α] : BEq (List α) := ⟨List.beq⟩
/--
`replicate n a` is `n` copies of `a`:
* `replicate 5 a = [a, a, a, a, a]`
-/
@[simp] def replicate : (n : Nat) → (a : α) → List α
| 0, _ => []
| n+1, a => a :: replicate n a
/-- Tail-recursive version of `List.replicate`. -/
def replicateTR {α : Type u} (n : Nat) (a : α) : List α :=
let rec loop : Nat → List α → List α
| 0, as => as
| n+1, as => loop n (a::as)
loop n []
theorem replicateTR_loop_replicate_eq (a : α) (m n : Nat) :
replicateTR.loop a n (replicate m a) = replicate (n + m) a := by
induction n generalizing m with simp [replicateTR.loop]
| succ n ih => simp [Nat.succ_add]; exact ih (m+1)
@[csimp] theorem replicate_eq_replicateTR : @List.replicate = @List.replicateTR := by
apply funext; intro α; apply funext; intro n; apply funext; intro a
exact (replicateTR_loop_replicate_eq _ 0 n).symm
/--
Removes the last element of the list.
* `dropLast [] = []`
* `dropLast [a] = []`
* `dropLast [a, b, c] = [a, b]`
-/
def dropLast {α} : List α → List α
| [] => []
| [_] => []
| a::as => a :: dropLast as
@[simp] theorem length_replicate (n : Nat) (a : α) : (replicate n a).length = n := by
induction n <;> simp_all
@[simp] theorem length_concat (as : List α) (a : α) : (concat as a).length = as.length + 1 := by
induction as with
| nil => rfl
| cons _ xs ih => simp [concat, ih]
@[simp] theorem length_set (as : List α) (i : Nat) (a : α) : (as.set i a).length = as.length := by
induction as generalizing i with
| nil => rfl
| cons x xs ih =>
cases i with
| zero => rfl
| succ i => simp [set, ih]
@[simp] theorem length_dropLast_cons (a : α) (as : List α) : (a :: as).dropLast.length = as.length := by
match as with
| [] => rfl
| b::bs =>
have ih := length_dropLast_cons b bs
simp[dropLast, ih]
@[simp] theorem length_append (as bs : List α) : (as ++ bs).length = as.length + bs.length := by
induction as with
| nil => simp
| cons _ as ih => simp [ih, Nat.succ_add]
@[simp] theorem length_map (as : List α) (f : α → β) : (as.map f).length = as.length := by
induction as with
| nil => simp [List.map]
| cons _ as ih => simp [List.map, ih]
@[simp] theorem length_reverse (as : List α) : (as.reverse).length = as.length := by
induction as with
| nil => rfl
| cons a as ih => simp [ih]
/--
Returns the largest element of the list, if it is not empty.
* `[].maximum? = none`
* `[4].maximum? = some 4`
* `[1, 4, 2, 10, 6].maximum? = some 10`
-/
def maximum? [Max α] : List α → Option α
| [] => none
| a::as => some <| as.foldl max a
/--
Returns the smallest element of the list, if it is not empty.
* `[].minimum? = none`
* `[4].minimum? = some 4`
* `[1, 4, 2, 10, 6].minimum? = some 1`
-/
def minimum? [Min α] : List α → Option α
| [] => none
| a::as => some <| as.foldl min a
instance [BEq α] [LawfulBEq α] : LawfulBEq (List α) where
eq_of_beq {as bs} := by
induction as generalizing bs with
| nil => intro h; cases bs <;> first | rfl | contradiction
| cons a as ih =>
cases bs with
| nil => intro h; contradiction
| cons b bs =>
simp [show (a::as == b::bs) = (a == b && as == bs) from rfl]
intro ⟨h₁, h₂⟩
exact ⟨h₁, ih h₂⟩
rfl {as} := by
induction as with
| nil => rfl
| cons a as ih => simp [BEq.beq, List.beq, LawfulBEq.rfl]; exact ih
theorem of_concat_eq_concat {as bs : List α} {a b : α} (h : as.concat a = bs.concat b) : as = bs ∧ a = b := by
match as, bs with
| [], [] => simp [concat] at h; simp [h]
| [_], [] => simp [concat] at h
| _::_::_, [] => simp [concat] at h
| [], [_] => simp [concat] at h
| [], _::_::_ => simp [concat] at h
| _::_, _::_ => simp [concat] at h; simp [h]; apply of_concat_eq_concat h.2
theorem concat_eq_append (as : List α) (a : α) : as.concat a = as ++ [a] := by
induction as <;> simp [concat, *]
theorem drop_eq_nil_of_le {as : List α} {i : Nat} (h : as.length ≤ i) : as.drop i = [] := by
match as, i with
| [], i => simp
| _::_, 0 => simp at h
| _::as, i+1 => simp at h; exact @drop_eq_nil_of_le as i (Nat.le_of_succ_le_succ h)
end List
|
9939ef1cbdddad31b1825b2960705545c59b816e | 9b9a16fa2cb737daee6b2785474678b6fa91d6d4 | /src/category_theory/functor.lean | a50fe7f41d916e916f08db585047c4cf492321c0 | [
"Apache-2.0"
] | permissive | johoelzl/mathlib | 253f46daa30b644d011e8e119025b01ad69735c4 | 592e3c7a2dfbd5826919b4605559d35d4d75938f | refs/heads/master | 1,625,657,216,488 | 1,551,374,946,000 | 1,551,374,946,000 | 98,915,829 | 0 | 0 | Apache-2.0 | 1,522,917,267,000 | 1,501,524,499,000 | Lean | UTF-8 | Lean | false | false | 3,116 | lean | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison
Defines a functor between categories.
(As it is a 'bundled' object rather than the `is_functorial` typeclass parametrised
by the underlying function on objects, the name is capitalised.)
Introduces notations
`C ⥤ D` for the type of all functors from `C` to `D`.
(I would like a better arrow here, unfortunately ⇒ (`\functor`) is taken by core.)
-/
import category_theory.category
import tactic.tidy
namespace category_theory
universes v v₁ v₂ v₃ u u₁ u₂ u₃ -- declare the `v`'s first; see `category_theory.category` for an explanation
/--
`functor C D` represents a functor between categories `C` and `D`.
To apply a functor `F` to an object use `F.obj X`, and to a morphism use `F.map f`.
The axiom `map_id_lemma` expresses preservation of identities, and
`map_comp_lemma` expresses functoriality.
-/
structure functor (C : Type u₁) [category.{v₁} C] (D : Type u₂) [category.{v₂} D] :
Type (max u₁ v₁ u₂ v₂) :=
(obj : C → D)
(map : Π {X Y : C}, (X ⟶ Y) → ((obj X) ⟶ (obj Y)))
(map_id' : ∀ (X : C), map (𝟙 X) = 𝟙 (obj X) . obviously)
(map_comp' : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = (map f) ≫ (map g) . obviously)
-- A functor is basically a function, so give ⥤ a similar precedence to → (25).
-- For example, `C × D ⥤ E` should parse as `(C × D) ⥤ E` not `C × (D ⥤ E)`.
infixr ` ⥤ `:26 := functor -- type as \func --
restate_axiom functor.map_id'
attribute [simp] functor.map_id
restate_axiom functor.map_comp'
attribute [simp] functor.map_comp
namespace functor
section
variables (C : Type u₁) [𝒞 : category.{v₁} C]
include 𝒞
/-- `functor.id C` is the identity functor on a category `C`. -/
protected def id : C ⥤ C :=
{ obj := λ X, X,
map := λ _ _ f, f }
variable {C}
@[simp] lemma id_obj (X : C) : (functor.id C).obj X = X := rfl
@[simp] lemma id_map {X Y : C} (f : X ⟶ Y) : (functor.id C).map f = f := rfl
end
section
variables {C : Type u₁} [𝒞 : category.{v₁} C]
{D : Type u₂} [𝒟 : category.{v₂} D]
{E : Type u₃} [ℰ : category.{v₃} E]
include 𝒞 𝒟 ℰ
/--
`F ⋙ G` is the composition of a functor `F` and a functor `G` (`F` first, then `G`).
-/
def comp (F : C ⥤ D) (G : D ⥤ E) : C ⥤ E :=
{ obj := λ X, G.obj (F.obj X),
map := λ _ _ f, G.map (F.map f) }
infixr ` ⋙ `:80 := comp
@[simp] lemma comp_obj (F : C ⥤ D) (G : D ⥤ E) (X : C) : (F ⋙ G).obj X = G.obj (F.obj X) := rfl
@[simp] lemma comp_map (F : C ⥤ D) (G : D ⥤ E) (X Y : C) (f : X ⟶ Y) :
(F ⋙ G).map f = G.map (F.map f) := rfl
end
section
variables (C : Type u₁) [𝒞 : category.{v₁} C]
include 𝒞
@[simp] def ulift_down : (ulift.{u₂} C) ⥤ C :=
{ obj := λ X, X.down,
map := λ X Y f, f }
@[simp] def ulift_up : C ⥤ (ulift.{u₂} C) :=
{ obj := λ X, ⟨ X ⟩,
map := λ X Y f, f }
end
end functor
end category_theory
|
d9f7642433c45fe525db5c1143d97e45eb4980e1 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/topology/algebra/open_subgroup.lean | 61c5049e76d7a1f8d2404c32a41d20faadd63806 | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 10,138 | lean | /-
Copyright (c) 2019 Johan Commelin All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import topology.algebra.ring
import topology.algebra.filter_basis
import topology.sets.opens
/-!
# Open subgroups of a topological groups
This files builds the lattice `open_subgroup G` of open subgroups in a topological group `G`,
and its additive version `open_add_subgroup`. This lattice has a top element, the subgroup of all
elements, but no bottom element in general. The trivial subgroup which is the natural candidate
bottom has no reason to be open (this happens only in discrete groups).
Note that this notion is especially relevant in a non-archimedean context, for instance for
`p`-adic groups.
## Main declarations
* `open_subgroup.is_closed`: An open subgroup is automatically closed.
* `subgroup.is_open_mono`: A subgroup containing an open subgroup is open.
There are also versions for additive groups, submodules and ideals.
* `open_subgroup.comap`: Open subgroups can be pulled back by a continuous group morphism.
## TODO
* Prove that the identity component of a locally path connected group is an open subgroup.
Up to now this file is really geared towards non-archimedean algebra, not Lie groups.
-/
open topological_space
open_locale topological_space
/-- The type of open subgroups of a topological additive group. -/
@[ancestor add_subgroup]
structure open_add_subgroup (G : Type*) [add_group G] [topological_space G]
extends add_subgroup G :=
(is_open' : is_open carrier)
/-- The type of open subgroups of a topological group. -/
@[ancestor subgroup, to_additive]
structure open_subgroup (G : Type*) [group G] [topological_space G] extends subgroup G :=
(is_open' : is_open carrier)
/-- Reinterpret an `open_subgroup` as a `subgroup`. -/
add_decl_doc open_subgroup.to_subgroup
/-- Reinterpret an `open_add_subgroup` as an `add_subgroup`. -/
add_decl_doc open_add_subgroup.to_add_subgroup
namespace open_subgroup
open function topological_space
variables {G : Type*} [group G] [topological_space G]
variables {U V : open_subgroup G} {g : G}
@[to_additive]
instance has_coe_set : has_coe_t (open_subgroup G) (set G) := ⟨λ U, U.1⟩
@[to_additive]
instance : has_mem G (open_subgroup G) := ⟨λ g U, g ∈ (U : set G)⟩
@[to_additive]
instance has_coe_subgroup : has_coe_t (open_subgroup G) (subgroup G) := ⟨to_subgroup⟩
@[to_additive]
instance has_coe_opens : has_coe_t (open_subgroup G) (opens G) := ⟨λ U, ⟨U, U.is_open'⟩⟩
@[simp, norm_cast, to_additive] lemma mem_coe : g ∈ (U : set G) ↔ g ∈ U := iff.rfl
@[simp, norm_cast, to_additive] lemma mem_coe_opens : g ∈ (U : opens G) ↔ g ∈ U := iff.rfl
@[simp, norm_cast, to_additive]
lemma mem_coe_subgroup : g ∈ (U : subgroup G) ↔ g ∈ U := iff.rfl
@[to_additive] lemma coe_injective : injective (coe : open_subgroup G → set G) :=
by { rintros ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨h⟩, congr, }
@[ext, to_additive]
lemma ext (h : ∀ x, x ∈ U ↔ x ∈ V) : (U = V) := coe_injective $ set.ext h
@[to_additive]
lemma ext_iff : (U = V) ↔ (∀ x, x ∈ U ↔ x ∈ V) := ⟨λ h x, h ▸ iff.rfl, ext⟩
variable (U)
@[to_additive]
protected lemma is_open : is_open (U : set G) := U.is_open'
@[to_additive]
protected lemma one_mem : (1 : G) ∈ U := U.one_mem'
@[to_additive]
protected lemma inv_mem {g : G} (h : g ∈ U) : g⁻¹ ∈ U := U.inv_mem' h
@[to_additive]
protected lemma mul_mem {g₁ g₂ : G} (h₁ : g₁ ∈ U) (h₂ : g₂ ∈ U) : g₁ * g₂ ∈ U := U.mul_mem' h₁ h₂
@[to_additive]
lemma mem_nhds_one : (U : set G) ∈ 𝓝 (1 : G) :=
is_open.mem_nhds U.is_open U.one_mem
variable {U}
@[to_additive]
instance : has_top (open_subgroup G) := ⟨{ is_open' := is_open_univ, .. (⊤ : subgroup G) }⟩
@[to_additive]
instance : inhabited (open_subgroup G) := ⟨⊤⟩
@[to_additive]
lemma is_closed [has_continuous_mul G] (U : open_subgroup G) : is_closed (U : set G) :=
begin
apply is_open_compl_iff.1,
refine is_open_iff_forall_mem_open.2 (λ x hx, ⟨(λ y, y * x⁻¹) ⁻¹' U, _, _, _⟩),
{ intros u hux,
simp only [set.mem_preimage, set.mem_compl_iff, mem_coe] at hux hx ⊢,
refine mt (λ hu, _) hx,
convert U.mul_mem (U.inv_mem hux) hu,
simp },
{ exact U.is_open.preimage (continuous_mul_right _) },
{ simp [U.one_mem] }
end
section
variables {H : Type*} [group H] [topological_space H]
/-- The product of two open subgroups as an open subgroup of the product group. -/
@[to_additive "The product of two open subgroups as an open subgroup of the product group."]
def prod (U : open_subgroup G) (V : open_subgroup H) : open_subgroup (G × H) :=
{ carrier := U ×ˢ V,
is_open' := U.is_open.prod V.is_open,
.. (U : subgroup G).prod (V : subgroup H) }
end
@[to_additive]
instance : partial_order (open_subgroup G) :=
{ le := λ U V, ∀ ⦃x⦄, x ∈ U → x ∈ V,
.. partial_order.lift (coe : open_subgroup G → set G) coe_injective }
@[to_additive]
instance : semilattice_inf (open_subgroup G) :=
{ inf := λ U V, { is_open' := is_open.inter U.is_open V.is_open, .. (U : subgroup G) ⊓ V },
inf_le_left := λ U V, set.inter_subset_left _ _,
inf_le_right := λ U V, set.inter_subset_right _ _,
le_inf := λ U V W hV hW, set.subset_inter hV hW,
..open_subgroup.partial_order }
@[to_additive]
instance : order_top (open_subgroup G) :=
{ top := ⊤,
le_top := λ U, set.subset_univ _ }
@[simp, norm_cast, to_additive] lemma coe_inf : (↑(U ⊓ V) : set G) = (U : set G) ∩ V := rfl
@[simp, norm_cast, to_additive] lemma coe_subset : (U : set G) ⊆ V ↔ U ≤ V := iff.rfl
@[simp, norm_cast, to_additive] lemma coe_subgroup_le :
(U : subgroup G) ≤ (V : subgroup G) ↔ U ≤ V := iff.rfl
variables {N : Type*} [group N] [topological_space N]
/-- The preimage of an `open_subgroup` along a continuous `monoid` homomorphism
is an `open_subgroup`. -/
@[to_additive "The preimage of an `open_add_subgroup` along a continuous `add_monoid` homomorphism
is an `open_add_subgroup`."]
def comap (f : G →* N)
(hf : continuous f) (H : open_subgroup N) : open_subgroup G :=
{ is_open' := H.is_open.preimage hf,
.. (H : subgroup N).comap f }
@[simp, to_additive]
lemma coe_comap (H : open_subgroup N) (f : G →* N) (hf : continuous f) :
(H.comap f hf : set G) = f ⁻¹' H := rfl
@[simp, to_additive]
lemma mem_comap {H : open_subgroup N} {f : G →* N} {hf : continuous f} {x : G} :
x ∈ H.comap f hf ↔ f x ∈ H := iff.rfl
@[to_additive]
lemma comap_comap {P : Type*} [group P] [topological_space P]
(K : open_subgroup P) (f₂ : N →* P) (hf₂ : continuous f₂) (f₁ : G →* N) (hf₁ : continuous f₁) :
(K.comap f₂ hf₂).comap f₁ hf₁ = K.comap (f₂.comp f₁) (hf₂.comp hf₁) :=
rfl
end open_subgroup
namespace subgroup
variables {G : Type*} [group G] [topological_space G] [has_continuous_mul G] (H : subgroup G)
@[to_additive]
lemma is_open_of_mem_nhds {g : G} (hg : (H : set G) ∈ 𝓝 g) :
is_open (H : set G) :=
begin
simp only [is_open_iff_mem_nhds, set_like.mem_coe] at hg ⊢,
intros x hx,
have : filter.tendsto (λ y, y * (x⁻¹ * g)) (𝓝 x) (𝓝 $ x * (x⁻¹ * g)) :=
(continuous_id.mul continuous_const).tendsto _,
rw [mul_inv_cancel_left] at this,
have := filter.mem_map'.1 (this hg),
replace hg : g ∈ H := set_like.mem_coe.1 (mem_of_mem_nhds hg),
simp only [set_like.mem_coe, H.mul_mem_cancel_right (H.mul_mem (H.inv_mem hx) hg)] at this,
exact this
end
@[to_additive]
lemma is_open_of_open_subgroup {U : open_subgroup G} (h : U.1 ≤ H) :
is_open (H : set G) :=
H.is_open_of_mem_nhds (filter.mem_of_superset U.mem_nhds_one h)
/-- If a subgroup of a topological group has `1` in its interior, then it is open. -/
@[to_additive "If a subgroup of an additive topological group has `0` in its interior, then it is
open."]
lemma is_open_of_one_mem_interior {G : Type*} [group G] [topological_space G]
[topological_group G] {H : subgroup G} (h_1_int : (1 : G) ∈ interior (H : set G)) :
is_open (H : set G) :=
begin
have h : 𝓝 1 ≤ filter.principal (H : set G) :=
nhds_le_of_le h_1_int (is_open_interior) (filter.principal_mono.2 interior_subset),
rw is_open_iff_nhds,
intros g hg,
rw (show 𝓝 g = filter.map ⇑(homeomorph.mul_left g) (𝓝 1), by simp),
convert filter.map_mono h,
simp only [homeomorph.coe_mul_left, filter.map_principal, set.image_mul_left,
filter.principal_eq_iff_eq],
ext,
simp [H.mul_mem_cancel_left (H.inv_mem hg)],
end
@[to_additive]
lemma is_open_mono {H₁ H₂ : subgroup G} (h : H₁ ≤ H₂) (h₁ : is_open (H₁ : set G)) :
is_open (H₂ : set G) :=
@is_open_of_open_subgroup _ _ _ _ H₂ { is_open' := h₁, .. H₁ } h
end subgroup
namespace open_subgroup
variables {G : Type*} [group G] [topological_space G] [has_continuous_mul G]
@[to_additive]
instance : semilattice_sup (open_subgroup G) :=
{ sup := λ U V,
{ is_open' := show is_open (((U : subgroup G) ⊔ V : subgroup G) : set G),
from subgroup.is_open_mono le_sup_left U.is_open,
.. ((U : subgroup G) ⊔ V) },
le_sup_left := λ U V, coe_subgroup_le.1 le_sup_left,
le_sup_right := λ U V, coe_subgroup_le.1 le_sup_right,
sup_le := λ U V W hU hV, coe_subgroup_le.1 (sup_le hU hV),
..open_subgroup.semilattice_inf }
@[to_additive]
instance : lattice (open_subgroup G) :=
{ ..open_subgroup.semilattice_sup, ..open_subgroup.semilattice_inf }
end open_subgroup
namespace submodule
open open_add_subgroup
variables {R : Type*} {M : Type*} [comm_ring R]
variables [add_comm_group M] [topological_space M] [topological_add_group M] [module R M]
lemma is_open_mono {U P : submodule R M} (h : U ≤ P) (hU : is_open (U : set M)) :
is_open (P : set M) :=
@add_subgroup.is_open_mono M _ _ _ U.to_add_subgroup P.to_add_subgroup h hU
end submodule
namespace ideal
variables {R : Type*} [comm_ring R]
variables [topological_space R] [topological_ring R]
lemma is_open_of_open_subideal {U I : ideal R} (h : U ≤ I) (hU : is_open (U : set R)) :
is_open (I : set R) :=
submodule.is_open_mono h hU
end ideal
|
4068654f68fb3f5e551bc93128df898e06e771b7 | e030b0259b777fedcdf73dd966f3f1556d392178 | /library/init/data/subtype/basic.lean | eaa6e18733fc5ab0d3f8a11df3c71aae096a0775 | [
"Apache-2.0"
] | permissive | fgdorais/lean | 17b46a095b70b21fa0790ce74876658dc5faca06 | c3b7c54d7cca7aaa25328f0a5660b6b75fe26055 | refs/heads/master | 1,611,523,590,686 | 1,484,412,902,000 | 1,484,412,902,000 | 38,489,734 | 0 | 0 | null | 1,435,923,380,000 | 1,435,923,379,000 | null | UTF-8 | Lean | false | false | 841 | lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Jeremy Avigad
-/
prelude
import init.logic
open decidable
universe variables u
structure subtype {α : Type u} (p : α → Prop) :=
tag :: (elt_of : α) (has_property : p elt_of)
namespace subtype
def exists_of_subtype {α : Type u} {p : α → Prop} : { x // p x } → ∃ x, p x
| ⟨a, h⟩ := ⟨a, h⟩
variables {α : Type u} {p : α → Prop}
lemma tag_irrelevant {a : α} (h1 h2 : p a) : tag a h1 = tag a h2 :=
rfl
protected lemma eq : ∀ {a1 a2 : {x // p x}}, elt_of a1 = elt_of a2 → a1 = a2
| ⟨x, h1⟩ ⟨.x, h2⟩ rfl := rfl
end subtype
open subtype
instance {α : Type u} {p : α → Prop} {a : α} (h : p a) : inhabited {x // p x} :=
⟨⟨a, h⟩⟩
|
4d3155bc253ffde3c11bb0cafe63d87960762e3b | a7dd8b83f933e72c40845fd168dde330f050b1c9 | /src/algebra/ordered_group.lean | 5b10a7de959ca054c78bbeb44d034a6a482b6c5b | [
"Apache-2.0"
] | permissive | NeilStrickland/mathlib | 10420e92ee5cb7aba1163c9a01dea2f04652ed67 | 3efbd6f6dff0fb9b0946849b43b39948560a1ffe | refs/heads/master | 1,589,043,046,346 | 1,558,938,706,000 | 1,558,938,706,000 | 181,285,984 | 0 | 0 | Apache-2.0 | 1,568,941,848,000 | 1,555,233,833,000 | Lean | UTF-8 | Lean | false | false | 24,671 | lean | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
Ordered monoids and groups.
-/
import algebra.group order.bounded_lattice tactic.basic
universe u
variable {α : Type u}
section old_structure_cmd
set_option old_structure_cmd true
/-- An ordered (additive) commutative monoid is a commutative monoid
with a partial order such that addition is an order embedding, i.e.
`a + b ≤ a + c ↔ b ≤ c`. These monoids are automatically cancellative. -/
class ordered_comm_monoid (α : Type*) extends add_comm_monoid α, partial_order α :=
(add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b)
(lt_of_add_lt_add_left : ∀ a b c : α, a + b < a + c → b < c)
/-- A canonically ordered monoid is an ordered commutative monoid
in which the ordering coincides with the divisibility relation,
which is to say, `a ≤ b` iff there exists `c` with `b = a + c`.
This is satisfied by the natural numbers, for example, but not
the integers or other ordered groups. -/
class canonically_ordered_monoid (α : Type*) extends ordered_comm_monoid α, lattice.order_bot α :=
(le_iff_exists_add : ∀a b:α, a ≤ b ↔ ∃c, b = a + c)
end old_structure_cmd
section ordered_comm_monoid
variables [ordered_comm_monoid α] {a b c d : α}
lemma add_le_add_left' (h : a ≤ b) : c + a ≤ c + b :=
ordered_comm_monoid.add_le_add_left a b h c
lemma add_le_add_right' (h : a ≤ b) : a + c ≤ b + c :=
add_comm c a ▸ add_comm c b ▸ add_le_add_left' h
lemma lt_of_add_lt_add_left' : a + b < a + c → b < c :=
ordered_comm_monoid.lt_of_add_lt_add_left a b c
lemma add_le_add' (h₁ : a ≤ b) (h₂ : c ≤ d) : a + c ≤ b + d :=
le_trans (add_le_add_right' h₁) (add_le_add_left' h₂)
lemma le_add_of_nonneg_right' (h : b ≥ 0) : a ≤ a + b :=
have a + b ≥ a + 0, from add_le_add_left' h,
by rwa add_zero at this
lemma le_add_of_nonneg_left' (h : b ≥ 0) : a ≤ b + a :=
have 0 + a ≤ b + a, from add_le_add_right' h,
by rwa zero_add at this
lemma lt_of_add_lt_add_right' (h : a + b < c + b) : a < c :=
lt_of_add_lt_add_left'
(show b + a < b + c, begin rw [add_comm b a, add_comm b c], assumption end)
-- here we start using properties of zero.
lemma le_add_of_nonneg_of_le' (ha : 0 ≤ a) (hbc : b ≤ c) : b ≤ a + c :=
zero_add b ▸ add_le_add' ha hbc
lemma le_add_of_le_of_nonneg' (hbc : b ≤ c) (ha : 0 ≤ a) : b ≤ c + a :=
add_zero b ▸ add_le_add' hbc ha
lemma add_nonneg' (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a + b :=
le_add_of_nonneg_of_le' ha hb
lemma add_pos_of_pos_of_nonneg' (ha : 0 < a) (hb : 0 ≤ b) : 0 < a + b :=
lt_of_lt_of_le ha $ le_add_of_nonneg_right' hb
lemma add_pos' (ha : 0 < a) (hb : 0 < b) : 0 < a + b :=
add_pos_of_pos_of_nonneg' ha $ le_of_lt hb
lemma add_pos_of_nonneg_of_pos' (ha : 0 ≤ a) (hb : 0 < b) : 0 < a + b :=
lt_of_lt_of_le hb $ le_add_of_nonneg_left' ha
lemma add_nonpos' (ha : a ≤ 0) (hb : b ≤ 0) : a + b ≤ 0 :=
zero_add (0:α) ▸ (add_le_add' ha hb)
lemma add_le_of_nonpos_of_le' (ha : a ≤ 0) (hbc : b ≤ c) : a + b ≤ c :=
zero_add c ▸ add_le_add' ha hbc
lemma add_le_of_le_of_nonpos' (hbc : b ≤ c) (ha : a ≤ 0) : b + a ≤ c :=
add_zero c ▸ add_le_add' hbc ha
lemma add_neg_of_neg_of_nonpos' (ha : a < 0) (hb : b ≤ 0) : a + b < 0 :=
lt_of_le_of_lt (add_le_of_le_of_nonpos' (le_refl _) hb) ha
lemma add_neg_of_nonpos_of_neg' (ha : a ≤ 0) (hb : b < 0) : a + b < 0 :=
lt_of_le_of_lt (add_le_of_nonpos_of_le' ha (le_refl _)) hb
lemma add_neg' (ha : a < 0) (hb : b < 0) : a + b < 0 :=
add_neg_of_nonpos_of_neg' (le_of_lt ha) hb
lemma lt_add_of_nonneg_of_lt' (ha : 0 ≤ a) (hbc : b < c) : b < a + c :=
lt_of_lt_of_le hbc $ le_add_of_nonneg_left' ha
lemma lt_add_of_lt_of_nonneg' (hbc : b < c) (ha : 0 ≤ a) : b < c + a :=
lt_of_lt_of_le hbc $ le_add_of_nonneg_right' ha
lemma lt_add_of_pos_of_lt' (ha : 0 < a) (hbc : b < c) : b < a + c :=
lt_add_of_nonneg_of_lt' (le_of_lt ha) hbc
lemma lt_add_of_lt_of_pos' (hbc : b < c) (ha : 0 < a) : b < c + a :=
lt_add_of_lt_of_nonneg' hbc (le_of_lt ha)
lemma add_lt_of_nonpos_of_lt' (ha : a ≤ 0) (hbc : b < c) : a + b < c :=
lt_of_le_of_lt (add_le_of_nonpos_of_le' ha (le_refl _)) hbc
lemma add_lt_of_lt_of_nonpos' (hbc : b < c) (ha : a ≤ 0) : b + a < c :=
lt_of_le_of_lt (add_le_of_le_of_nonpos' (le_refl _) ha) hbc
lemma add_lt_of_neg_of_lt' (ha : a < 0) (hbc : b < c) : a + b < c :=
add_lt_of_nonpos_of_lt' (le_of_lt ha) hbc
lemma add_lt_of_lt_of_neg' (hbc : b < c) (ha : a < 0) : b + a < c :=
add_lt_of_lt_of_nonpos' hbc (le_of_lt ha)
lemma add_eq_zero_iff' (ha : 0 ≤ a) (hb : 0 ≤ b) : a + b = 0 ↔ a = 0 ∧ b = 0 :=
iff.intro
(assume hab : a + b = 0,
have a ≤ 0, from hab ▸ le_add_of_le_of_nonneg' (le_refl _) hb,
have a = 0, from le_antisymm this ha,
have b ≤ 0, from hab ▸ le_add_of_nonneg_of_le' ha (le_refl _),
have b = 0, from le_antisymm this hb,
and.intro ‹a = 0› ‹b = 0›)
(assume ⟨ha', hb'⟩, by rw [ha', hb', add_zero])
lemma bit0_pos {a : α} (h : 0 < a) : 0 < bit0 a :=
add_pos' h h
end ordered_comm_monoid
namespace units
instance [monoid α] [i : preorder α] : preorder (units α) :=
preorder.lift (coe : units α → α) i
@[simp] theorem coe_le_coe [monoid α] [preorder α] {a b : units α} :
(a : α) ≤ b ↔ a ≤ b := iff.rfl
@[simp] theorem coe_lt_coe [monoid α] [preorder α] {a b : units α} :
(a : α) < b ↔ a < b := iff.rfl
instance [monoid α] [i : partial_order α] : partial_order (units α) :=
partial_order.lift (coe : units α → α) (by ext) i
instance [monoid α] [i : linear_order α] : linear_order (units α) :=
linear_order.lift (coe : units α → α) (by ext) i
instance [monoid α] [i : decidable_linear_order α] : decidable_linear_order (units α) :=
decidable_linear_order.lift (coe : units α → α) (by ext) i
theorem max_coe [monoid α] [decidable_linear_order α] {a b : units α} :
(↑(max a b) : α) = max a b :=
by by_cases a ≤ b; simp [max, h]
theorem min_coe [monoid α] [decidable_linear_order α] {a b : units α} :
(↑(min a b) : α) = min a b :=
by by_cases a ≤ b; simp [min, h]
end units
namespace with_zero
open lattice
instance [preorder α] : preorder (with_zero α) := with_bot.preorder
instance [partial_order α] : partial_order (with_zero α) := with_bot.partial_order
instance [partial_order α] : order_bot (with_zero α) := with_bot.order_bot
instance [lattice α] : lattice (with_zero α) := with_bot.lattice
instance [linear_order α] : linear_order (with_zero α) := with_bot.linear_order
instance [decidable_linear_order α] :
decidable_linear_order (with_zero α) := with_bot.decidable_linear_order
def ordered_comm_monoid [ordered_comm_monoid α]
(zero_le : ∀ a : α, 0 ≤ a) : ordered_comm_monoid (with_zero α) :=
begin
suffices, refine {
add_le_add_left := this,
..with_zero.partial_order,
..with_zero.add_comm_monoid, .. },
{ intros a b c h,
have h' := lt_iff_le_not_le.1 h,
rw lt_iff_le_not_le at ⊢,
refine ⟨λ b h₂, _, λ h₂, h'.2 $ this _ _ h₂ _⟩,
cases h₂, cases c with c,
{ cases h'.2 (this _ _ bot_le a) },
{ refine ⟨_, rfl, _⟩,
cases a with a,
{ exact with_bot.some_le_some.1 h'.1 },
{ exact le_of_lt (lt_of_add_lt_add_left' $
with_bot.some_lt_some.1 h), } } },
{ intros a b h c ca h₂,
cases b with b,
{ rw le_antisymm h bot_le at h₂,
exact ⟨_, h₂, le_refl _⟩ },
cases a with a,
{ change c + 0 = some ca at h₂,
simp at h₂, simp [h₂],
exact ⟨_, rfl, by simpa using add_le_add_left' (zero_le b)⟩ },
{ simp at h,
cases c with c; change some _ = _ at h₂;
simp [-add_comm] at h₂; subst ca; refine ⟨_, rfl, _⟩,
{ exact h },
{ exact add_le_add_left' h } } }
end
end with_zero
namespace with_top
open lattice
instance [add_semigroup α] : add_semigroup (with_top α) :=
{ add := λ o₁ o₂, o₁.bind (λ a, o₂.map (λ b, a + b)),
..@additive.add_semigroup _ $ @with_zero.semigroup (multiplicative α) _ }
lemma coe_add [add_semigroup α] {a b : α} : ((a + b : α) : with_top α) = a + b := rfl
instance [add_comm_semigroup α] : add_comm_semigroup (with_top α) :=
{ ..@additive.add_comm_semigroup _ $
@with_zero.comm_semigroup (multiplicative α) _ }
instance [add_monoid α] : add_monoid (with_top α) :=
{ zero := some 0,
add := (+),
..@additive.add_monoid _ $ @with_zero.monoid (multiplicative α) _ }
instance [add_comm_monoid α] : add_comm_monoid (with_top α) :=
{ zero := 0,
add := (+),
..@additive.add_comm_monoid _ $
@with_zero.comm_monoid (multiplicative α) _ }
instance [ordered_comm_monoid α] : ordered_comm_monoid (with_top α) :=
begin
suffices, refine {
add_le_add_left := this,
..with_top.partial_order,
..with_top.add_comm_monoid, ..},
{ intros a b c h,
refine ⟨λ c h₂, _, λ h₂, h.2 $ this _ _ h₂ _⟩,
cases h₂, cases a with a,
{ exact (not_le_of_lt h).elim le_top },
cases b with b,
{ exact (not_le_of_lt h).elim le_top },
{ exact ⟨_, rfl, le_of_lt (lt_of_add_lt_add_left' $
with_top.some_lt_some.1 h)⟩ } },
{ intros a b h c cb h₂,
cases c with c, {cases h₂},
cases b with b; cases h₂,
cases a with a, {cases le_antisymm h le_top},
simp at h,
exact ⟨_, rfl, add_le_add_left' h⟩, }
end
@[simp] lemma zero_lt_top [ordered_comm_monoid α] : (0 : with_top α) < ⊤ :=
coe_lt_top 0
@[simp] lemma zero_lt_coe [ordered_comm_monoid α] (a : α) : (0 : with_top α) < a ↔ 0 < a :=
coe_lt_coe
@[simp] lemma add_top [ordered_comm_monoid α] : ∀{a : with_top α}, a + ⊤ = ⊤
| none := rfl
| (some a) := rfl
@[simp] lemma top_add [ordered_comm_monoid α] {a : with_top α} : ⊤ + a = ⊤ := rfl
lemma add_eq_top [ordered_comm_monoid α] (a b : with_top α) : a + b = ⊤ ↔ a = ⊤ ∨ b = ⊤ :=
by cases a; cases b; simp [none_eq_top, some_eq_coe, coe_add.symm]
lemma add_lt_top [ordered_comm_monoid α] (a b : with_top α) : a + b < ⊤ ↔ a < ⊤ ∧ b < ⊤ :=
begin
apply not_iff_not.1,
simp [lt_top_iff_ne_top, add_eq_top],
finish,
apply classical.dec _,
apply classical.dec _,
end
instance [canonically_ordered_monoid α] : canonically_ordered_monoid (with_top α) :=
{ le_iff_exists_add := assume a b,
match a, b with
| a, none := show a ≤ ⊤ ↔ ∃c, ⊤ = a + c, by simp; refine ⟨⊤, _⟩; cases a; refl
| (some a), (some b) := show (a:with_top α) ≤ ↑b ↔ ∃c:with_top α, ↑b = ↑a + c,
begin
simp [canonically_ordered_monoid.le_iff_exists_add, -add_comm],
split,
{ rintro ⟨c, rfl⟩, refine ⟨c, _⟩, simp [coe_add] },
{ exact assume h, match b, h with _, ⟨some c, rfl⟩ := ⟨_, rfl⟩ end }
end
| none, some b := show (⊤ : with_top α) ≤ b ↔ ∃c:with_top α, ↑b = ⊤ + c, by simp
end,
.. with_top.order_bot,
.. with_top.ordered_comm_monoid }
end with_top
namespace with_bot
open lattice
instance [add_semigroup α] : add_semigroup (with_bot α) := with_top.add_semigroup
instance [add_comm_semigroup α] : add_comm_semigroup (with_bot α) := with_top.add_comm_semigroup
instance [add_monoid α] : add_monoid (with_bot α) := with_top.add_monoid
instance [add_comm_monoid α] : add_comm_monoid (with_bot α) := with_top.add_comm_monoid
instance [ordered_comm_monoid α] : ordered_comm_monoid (with_bot α) :=
begin
suffices, refine {
add_le_add_left := this,
..with_bot.partial_order,
..with_bot.add_comm_monoid, ..},
{ intros a b c h,
have h' := h,
rw lt_iff_le_not_le at h' ⊢,
refine ⟨λ b h₂, _, λ h₂, h'.2 $ this _ _ h₂ _⟩,
cases h₂, cases a with a,
{ exact (not_le_of_lt h).elim bot_le },
cases c with c,
{ exact (not_le_of_lt h).elim bot_le },
{ exact ⟨_, rfl, le_of_lt (lt_of_add_lt_add_left' $
with_bot.some_lt_some.1 h)⟩ } },
{ intros a b h c ca h₂,
cases c with c, {cases h₂},
cases a with a; cases h₂,
cases b with b, {cases le_antisymm h bot_le},
simp at h,
exact ⟨_, rfl, add_le_add_left' h⟩, }
end
@[simp] lemma coe_zero [add_monoid α] : ((0 : α) : with_bot α) = 0 := rfl
@[simp] lemma coe_add [add_semigroup α] (a b : α) : ((a + b : α) : with_bot α) = a + b := rfl
@[simp] lemma bot_add [ordered_comm_monoid α] (a : with_bot α) : ⊥ + a = ⊥ := rfl
@[simp] lemma add_bot [ordered_comm_monoid α] (a : with_bot α) : a + ⊥ = ⊥ := by cases a; refl
instance has_one [has_one α] : has_one (with_bot α) := ⟨(1 : α)⟩
@[simp] lemma coe_one [has_one α] : ((1 : α) : with_bot α) = 1 := rfl
end with_bot
section canonically_ordered_monoid
variables [canonically_ordered_monoid α] {a b c d : α}
lemma le_iff_exists_add : a ≤ b ↔ ∃c, b = a + c :=
canonically_ordered_monoid.le_iff_exists_add a b
@[simp] lemma zero_le (a : α) : 0 ≤ a := le_iff_exists_add.mpr ⟨a, by simp⟩
lemma bot_eq_zero : (⊥ : α) = 0 :=
le_antisymm lattice.bot_le (zero_le ⊥)
@[simp] lemma add_eq_zero_iff : a + b = 0 ↔ a = 0 ∧ b = 0 :=
add_eq_zero_iff' (zero_le _) (zero_le _)
@[simp] lemma le_zero_iff_eq : a ≤ 0 ↔ a = 0 :=
iff.intro
(assume h, le_antisymm h (zero_le a))
(assume h, h ▸ le_refl a)
protected lemma zero_lt_iff_ne_zero : 0 < a ↔ a ≠ 0 :=
iff.intro ne_of_gt $ assume hne, lt_of_le_of_ne (zero_le _) hne.symm
lemma le_add_left (h : a ≤ c) : a ≤ b + c :=
calc a = 0 + a : by simp
... ≤ b + c : add_le_add' (zero_le _) h
lemma le_add_right (h : a ≤ b) : a ≤ b + c :=
calc a = a + 0 : by simp
... ≤ b + c : add_le_add' h (zero_le _)
instance with_zero.canonically_ordered_monoid :
canonically_ordered_monoid (with_zero α) :=
{ le_iff_exists_add := λ a b, begin
cases a with a,
{ exact iff_of_true lattice.bot_le ⟨b, (zero_add b).symm⟩ },
cases b with b,
{ exact iff_of_false
(mt (le_antisymm lattice.bot_le) (by simp))
(λ ⟨c, h⟩, by cases c; cases h) },
{ simp [le_iff_exists_add, -add_comm],
split; intro h; rcases h with ⟨c, h⟩,
{ exact ⟨some c, congr_arg some h⟩ },
{ cases c; cases h,
{ exact ⟨_, (add_zero _).symm⟩ },
{ exact ⟨_, rfl⟩ } } }
end,
bot := 0,
bot_le := assume a a' h, option.no_confusion h,
.. with_zero.ordered_comm_monoid zero_le }
end canonically_ordered_monoid
instance ordered_cancel_comm_monoid.to_ordered_comm_monoid
[H : ordered_cancel_comm_monoid α] : ordered_comm_monoid α :=
{ lt_of_add_lt_add_left := @lt_of_add_lt_add_left _ _, ..H }
section ordered_cancel_comm_monoid
variables [ordered_cancel_comm_monoid α] {a b c : α}
@[simp] lemma add_le_add_iff_left (a : α) {b c : α} : a + b ≤ a + c ↔ b ≤ c :=
⟨le_of_add_le_add_left, λ h, add_le_add_left h _⟩
@[simp] lemma add_le_add_iff_right (c : α) : a + c ≤ b + c ↔ a ≤ b :=
add_comm c a ▸ add_comm c b ▸ add_le_add_iff_left c
@[simp] lemma add_lt_add_iff_left (a : α) {b c : α} : a + b < a + c ↔ b < c :=
⟨lt_of_add_lt_add_left, λ h, add_lt_add_left h _⟩
@[simp] lemma add_lt_add_iff_right (c : α) : a + c < b + c ↔ a < b :=
add_comm c a ▸ add_comm c b ▸ add_lt_add_iff_left c
@[simp] lemma le_add_iff_nonneg_right (a : α) {b : α} : a ≤ a + b ↔ 0 ≤ b :=
have a + 0 ≤ a + b ↔ 0 ≤ b, from add_le_add_iff_left a,
by rwa add_zero at this
@[simp] lemma le_add_iff_nonneg_left (a : α) {b : α} : a ≤ b + a ↔ 0 ≤ b :=
by rw [add_comm, le_add_iff_nonneg_right]
@[simp] lemma lt_add_iff_pos_right (a : α) {b : α} : a < a + b ↔ 0 < b :=
have a + 0 < a + b ↔ 0 < b, from add_lt_add_iff_left a,
by rwa add_zero at this
@[simp] lemma lt_add_iff_pos_left (a : α) {b : α} : a < b + a ↔ 0 < b :=
by rw [add_comm, lt_add_iff_pos_right]
lemma add_eq_zero_iff_eq_zero_of_nonneg
(ha : 0 ≤ a) (hb : 0 ≤ b) : a + b = 0 ↔ a = 0 ∧ b = 0 :=
⟨λ hab : a + b = 0,
by split; apply le_antisymm; try {assumption};
rw ← hab; simp [ha, hb],
λ ⟨ha', hb'⟩, by rw [ha', hb', add_zero]⟩
lemma with_top.add_lt_add_iff_left :
∀{a b c : with_top α}, a < ⊤ → (a + c < a + b ↔ c < b)
| none := assume b c h, (lt_irrefl ⊤ h).elim
| (some a) :=
begin
assume b c h,
cases b; cases c;
simp [with_top.none_eq_top, with_top.some_eq_coe, with_top.coe_lt_top, with_top.coe_lt_coe],
{ rw [← with_top.coe_add], exact with_top.coe_lt_top _ },
{ rw [← with_top.coe_add, ← with_top.coe_add, with_top.coe_lt_coe],
exact add_lt_add_iff_left _ }
end
lemma with_top.add_lt_add_iff_right
{a b c : with_top α} : a < ⊤ → (c + a < b + a ↔ c < b) :=
by simpa [add_comm] using @with_top.add_lt_add_iff_left _ _ a b c
end ordered_cancel_comm_monoid
section ordered_comm_group
variables [ordered_comm_group α] {a b c : α}
lemma neg_neg_iff_pos {α : Type} [_inst_1 : ordered_comm_group α] {a : α} : -a < 0 ↔ 0 < a :=
⟨ pos_of_neg_neg, neg_neg_of_pos ⟩
@[simp] lemma neg_le_neg_iff : -a ≤ -b ↔ b ≤ a :=
have a + b + -a ≤ a + b + -b ↔ -a ≤ -b, from add_le_add_iff_left _,
by simp at this; simp [this]
lemma neg_le : -a ≤ b ↔ -b ≤ a :=
have -a ≤ -(-b) ↔ -b ≤ a, from neg_le_neg_iff,
by rwa neg_neg at this
lemma le_neg : a ≤ -b ↔ b ≤ -a :=
have -(-a) ≤ -b ↔ b ≤ -a, from neg_le_neg_iff,
by rwa neg_neg at this
@[simp] lemma neg_nonpos : -a ≤ 0 ↔ 0 ≤ a :=
have -a ≤ -0 ↔ 0 ≤ a, from neg_le_neg_iff,
by rwa neg_zero at this
@[simp] lemma neg_nonneg : 0 ≤ -a ↔ a ≤ 0 :=
have -0 ≤ -a ↔ a ≤ 0, from neg_le_neg_iff,
by rwa neg_zero at this
@[simp] lemma neg_lt_neg_iff : -a < -b ↔ b < a :=
have a + b + -a < a + b + -b ↔ -a < -b, from add_lt_add_iff_left _,
by simp at this; simp [this]
lemma neg_lt_zero : -a < 0 ↔ 0 < a :=
have -a < -0 ↔ 0 < a, from neg_lt_neg_iff,
by rwa neg_zero at this
lemma neg_pos : 0 < -a ↔ a < 0 :=
have -0 < -a ↔ a < 0, from neg_lt_neg_iff,
by rwa neg_zero at this
lemma neg_lt : -a < b ↔ -b < a :=
have -a < -(-b) ↔ -b < a, from neg_lt_neg_iff,
by rwa neg_neg at this
lemma lt_neg : a < -b ↔ b < -a :=
have -(-a) < -b ↔ b < -a, from neg_lt_neg_iff,
by rwa neg_neg at this
lemma sub_le_sub_iff_left (a : α) {b c : α} : a - b ≤ a - c ↔ c ≤ b :=
(add_le_add_iff_left _).trans neg_le_neg_iff
lemma sub_le_sub_iff_right (c : α) : a - c ≤ b - c ↔ a ≤ b :=
add_le_add_iff_right _
lemma sub_lt_sub_iff_left (a : α) {b c : α} : a - b < a - c ↔ c < b :=
(add_lt_add_iff_left _).trans neg_lt_neg_iff
lemma sub_lt_sub_iff_right (c : α) : a - c < b - c ↔ a < b :=
add_lt_add_iff_right _
@[simp] lemma sub_nonneg : 0 ≤ a - b ↔ b ≤ a :=
have a - a ≤ a - b ↔ b ≤ a, from sub_le_sub_iff_left a,
by rwa sub_self at this
@[simp] lemma sub_nonpos : a - b ≤ 0 ↔ a ≤ b :=
have a - b ≤ b - b ↔ a ≤ b, from sub_le_sub_iff_right b,
by rwa sub_self at this
@[simp] lemma sub_pos : 0 < a - b ↔ b < a :=
have a - a < a - b ↔ b < a, from sub_lt_sub_iff_left a,
by rwa sub_self at this
@[simp] lemma sub_lt_zero : a - b < 0 ↔ a < b :=
have a - b < b - b ↔ a < b, from sub_lt_sub_iff_right b,
by rwa sub_self at this
lemma le_neg_add_iff_add_le : b ≤ -a + c ↔ a + b ≤ c :=
have -a + (a + b) ≤ -a + c ↔ a + b ≤ c, from add_le_add_iff_left _,
by rwa neg_add_cancel_left at this
lemma le_sub_iff_add_le' : b ≤ c - a ↔ a + b ≤ c :=
by rw [sub_eq_add_neg, add_comm, le_neg_add_iff_add_le]
lemma le_sub_iff_add_le : a ≤ c - b ↔ a + b ≤ c :=
by rw [le_sub_iff_add_le', add_comm]
@[simp] lemma neg_add_le_iff_le_add : -b + a ≤ c ↔ a ≤ b + c :=
have -b + a ≤ -b + (b + c) ↔ a ≤ b + c, from add_le_add_iff_left _,
by rwa neg_add_cancel_left at this
lemma sub_le_iff_le_add' : a - b ≤ c ↔ a ≤ b + c :=
by rw [sub_eq_add_neg, add_comm, neg_add_le_iff_le_add]
lemma sub_le_iff_le_add : a - c ≤ b ↔ a ≤ b + c :=
by rw [sub_le_iff_le_add', add_comm]
@[simp] lemma add_neg_le_iff_le_add : a + -c ≤ b ↔ a ≤ b + c :=
sub_le_iff_le_add
@[simp] lemma add_neg_le_iff_le_add' : a + -b ≤ c ↔ a ≤ b + c :=
sub_le_iff_le_add'
lemma neg_add_le_iff_le_add' : -c + a ≤ b ↔ a ≤ b + c :=
by rw [neg_add_le_iff_le_add, add_comm]
@[simp] lemma neg_le_sub_iff_le_add : -b ≤ a - c ↔ c ≤ a + b :=
le_sub_iff_add_le.trans neg_add_le_iff_le_add'
lemma neg_le_sub_iff_le_add' : -a ≤ b - c ↔ c ≤ a + b :=
by rw [neg_le_sub_iff_le_add, add_comm]
lemma sub_le : a - b ≤ c ↔ a - c ≤ b :=
sub_le_iff_le_add'.trans sub_le_iff_le_add.symm
theorem le_sub : a ≤ b - c ↔ c ≤ b - a :=
le_sub_iff_add_le'.trans le_sub_iff_add_le.symm
@[simp] lemma lt_neg_add_iff_add_lt : b < -a + c ↔ a + b < c :=
have -a + (a + b) < -a + c ↔ a + b < c, from add_lt_add_iff_left _,
by rwa neg_add_cancel_left at this
lemma lt_sub_iff_add_lt' : b < c - a ↔ a + b < c :=
by rw [sub_eq_add_neg, add_comm, lt_neg_add_iff_add_lt]
lemma lt_sub_iff_add_lt : a < c - b ↔ a + b < c :=
by rw [lt_sub_iff_add_lt', add_comm]
@[simp] lemma neg_add_lt_iff_lt_add : -b + a < c ↔ a < b + c :=
have -b + a < -b + (b + c) ↔ a < b + c, from add_lt_add_iff_left _,
by rwa neg_add_cancel_left at this
lemma sub_lt_iff_lt_add' : a - b < c ↔ a < b + c :=
by rw [sub_eq_add_neg, add_comm, neg_add_lt_iff_lt_add]
lemma sub_lt_iff_lt_add : a - c < b ↔ a < b + c :=
by rw [sub_lt_iff_lt_add', add_comm]
lemma neg_add_lt_iff_lt_add_right : -c + a < b ↔ a < b + c :=
by rw [neg_add_lt_iff_lt_add, add_comm]
@[simp] lemma neg_lt_sub_iff_lt_add : -b < a - c ↔ c < a + b :=
lt_sub_iff_add_lt.trans neg_add_lt_iff_lt_add_right
lemma neg_lt_sub_iff_lt_add' : -a < b - c ↔ c < a + b :=
by rw [neg_lt_sub_iff_lt_add, add_comm]
lemma sub_lt : a - b < c ↔ a - c < b :=
sub_lt_iff_lt_add'.trans sub_lt_iff_lt_add.symm
theorem lt_sub : a < b - c ↔ c < b - a :=
lt_sub_iff_add_lt'.trans lt_sub_iff_add_lt.symm
lemma sub_le_self_iff (a : α) {b : α} : a - b ≤ a ↔ 0 ≤ b :=
sub_le_iff_le_add'.trans (le_add_iff_nonneg_left _)
lemma sub_lt_self_iff (a : α) {b : α} : a - b < a ↔ 0 < b :=
sub_lt_iff_lt_add'.trans (lt_add_iff_pos_left _)
end ordered_comm_group
set_option old_structure_cmd true
/-- This is not so much a new structure as a construction mechanism
for ordered groups, by specifying only the "positive cone" of the group. -/
class nonneg_comm_group (α : Type*) extends add_comm_group α :=
(nonneg : α → Prop)
(pos : α → Prop := λ a, nonneg a ∧ ¬ nonneg (neg a))
(pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬ nonneg (-a) . order_laws_tac)
(zero_nonneg : nonneg 0)
(add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b))
(nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0)
namespace nonneg_comm_group
variable [s : nonneg_comm_group α]
include s
@[reducible] instance to_ordered_comm_group : ordered_comm_group α :=
{ le := λ a b, nonneg (b - a),
lt := λ a b, pos (b - a),
lt_iff_le_not_le := λ a b, by simp; rw [pos_iff]; simp,
le_refl := λ a, by simp [zero_nonneg],
le_trans := λ a b c nab nbc, by simp [-sub_eq_add_neg];
rw ← sub_add_sub_cancel; exact add_nonneg nbc nab,
le_antisymm := λ a b nab nba, eq_of_sub_eq_zero $
nonneg_antisymm nba (by rw neg_sub; exact nab),
add_le_add_left := λ a b nab c, by simpa [(≤), preorder.le] using nab,
add_lt_add_left := λ a b nab c, by simpa [(<), preorder.lt] using nab, ..s }
theorem nonneg_def {a : α} : nonneg a ↔ 0 ≤ a :=
show _ ↔ nonneg _, by simp
theorem pos_def {a : α} : pos a ↔ 0 < a :=
show _ ↔ pos _, by simp
theorem not_zero_pos : ¬ pos (0 : α) :=
mt pos_def.1 (lt_irrefl _)
theorem zero_lt_iff_nonneg_nonneg {a : α} :
0 < a ↔ nonneg a ∧ ¬ nonneg (-a) :=
pos_def.symm.trans (pos_iff α _)
theorem nonneg_total_iff :
(∀ a : α, nonneg a ∨ nonneg (-a)) ↔
(∀ a b : α, a ≤ b ∨ b ≤ a) :=
⟨λ h a b, by have := h (b - a); rwa [neg_sub] at this,
λ h a, by rw [nonneg_def, nonneg_def, neg_nonneg]; apply h⟩
def to_decidable_linear_ordered_comm_group
[decidable_pred (@nonneg α _)]
(nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a))
: decidable_linear_ordered_comm_group α :=
{ le := (≤),
lt := (<),
lt_iff_le_not_le := @lt_iff_le_not_le _ _,
le_refl := @le_refl _ _,
le_trans := @le_trans _ _,
le_antisymm := @le_antisymm _ _,
le_total := nonneg_total_iff.1 nonneg_total,
decidable_le := by apply_instance,
decidable_eq := by apply_instance,
decidable_lt := by apply_instance,
..@nonneg_comm_group.to_ordered_comm_group _ s }
end nonneg_comm_group
|
1742e76fa2265698cd396934b2aef9871432a7e6 | d1a52c3f208fa42c41df8278c3d280f075eb020c | /tests/lean/run/536.lean | c62a665682b413767af10198235e25eaf7218468 | [
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"NCSA",
"LGPL-3.0-only",
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"SunPro",
"CMU-Mach"... | permissive | cipher1024/lean4 | 6e1f98bb58e7a92b28f5364eb38a14c8d0aae393 | 69114d3b50806264ef35b57394391c3e738a9822 | refs/heads/master | 1,642,227,983,603 | 1,642,011,696,000 | 1,642,011,696,000 | 228,607,691 | 0 | 0 | Apache-2.0 | 1,576,584,269,000 | 1,576,584,268,000 | null | UTF-8 | Lean | false | false | 100 | lean | variable (C : Type) [Inhabited C]
example : C := arbitrary
variable {C}
example : C := arbitrary
|
23025d582ff0d909179c5adc648705100938c3ff | aa3f8992ef7806974bc1ffd468baa0c79f4d6643 | /library/standard/logic/connectives/basic.lean | 5ac020c7ac1bc9bb8eb81a80934683606ded58f7 | [
"Apache-2.0"
] | permissive | codyroux/lean | 7f8dff750722c5382bdd0a9a9275dc4bb2c58dd3 | 0cca265db19f7296531e339192e9b9bae4a31f8b | refs/heads/master | 1,610,909,964,159 | 1,407,084,399,000 | 1,416,857,075,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 6,477 | lean | ----------------------------------------------------------------------------------------------------
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Leonardo de Moura, Jeremy Avigad
----------------------------------------------------------------------------------------------------
import .prop
-- implication
-- -----------
abbreviation imp (a b : Prop) : Prop := a → b
-- true and false
-- --------------
inductive false : Prop
theorem false_elim (c : Prop) (H : false) : c :=
false_rec c H
inductive true : Prop :=
| trivial : true
abbreviation not (a : Prop) := a → false
prefix `¬`:40 := not
notation `assume` binders `,` r:(scoped f, f) := r
notation `take` binders `,` r:(scoped f, f) := r
-- not
-- ---
theorem not_intro {a : Prop} (H : a → false) : ¬a := H
theorem not_elim {a : Prop} (H1 : ¬a) (H2 : a) : false := H1 H2
theorem absurd {a : Prop} (H1 : a) (H2 : ¬a) : false := H2 H1
theorem not_not_intro {a : Prop} (Ha : a) : ¬¬a :=
assume Hna : ¬a, absurd Ha Hna
theorem mt {a b : Prop} (H1 : a → b) (H2 : ¬b) : ¬a :=
assume Ha : a, absurd (H1 Ha) H2
theorem contrapos {a b : Prop} (H : a → b) : ¬b → ¬a :=
assume Hnb : ¬b, mt H Hnb
theorem absurd_elim {a : Prop} (b : Prop) (H1 : a) (H2 : ¬a) : b :=
false_elim b (absurd H1 H2)
theorem absurd_not_true (H : ¬true) : false :=
absurd trivial H
theorem not_false_trivial : ¬false :=
assume H : false, H
theorem not_implies_left {a b : Prop} (H : ¬(a → b)) : ¬¬a :=
assume Hna : ¬a, absurd (assume Ha : a, absurd_elim b Ha Hna) H
theorem not_implies_right {a b : Prop} (H : ¬(a → b)) : ¬b :=
assume Hb : b, absurd (assume Ha : a, Hb) H
-- and
-- ---
inductive and (a b : Prop) : Prop :=
| and_intro : a → b → and a b
infixr `/\`:35 := and
infixr `∧`:35 := and
theorem and_elim {a b c : Prop} (H1 : a ∧ b) (H2 : a → b → c) : c :=
and_rec H2 H1
theorem and_elim_left {a b : Prop} (H : a ∧ b) : a :=
and_rec (λa b, a) H
theorem and_elim_right {a b : Prop} (H : a ∧ b) : b :=
and_rec (λa b, b) H
theorem and_swap {a b : Prop} (H : a ∧ b) : b ∧ a :=
and_intro (and_elim_right H) (and_elim_left H)
theorem and_not_left {a : Prop} (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
assume H : a ∧ b, absurd (and_elim_left H) Hna
theorem and_not_right (a : Prop) {b : Prop} (Hnb : ¬b) : ¬(a ∧ b) :=
assume H : a ∧ b, absurd (and_elim_right H) Hnb
theorem and_imp_and {a b c d : Prop} (H1 : a ∧ b) (H2 : a → c) (H3 : b → d) : c ∧ d :=
and_elim H1 (assume Ha : a, assume Hb : b, and_intro (H2 Ha) (H3 Hb))
theorem imp_and_left {a b c : Prop} (H1 : a ∧ c) (H : a → b) : b ∧ c :=
and_elim H1 (assume Ha : a, assume Hc : c, and_intro (H Ha) Hc)
theorem imp_and_right {a b c : Prop} (H1 : c ∧ a) (H : a → b) : c ∧ b :=
and_elim H1 (assume Hc : c, assume Ha : a, and_intro Hc (H Ha))
-- or
-- --
inductive or (a b : Prop) : Prop :=
| or_intro_left : a → or a b
| or_intro_right : b → or a b
infixr `\/`:30 := or
infixr `∨`:30 := or
theorem or_inl {a b : Prop} (Ha : a) : a ∨ b := or_intro_left b Ha
theorem or_inr {a b : Prop} (Hb : b) : a ∨ b := or_intro_right a Hb
theorem or_elim {a b c : Prop} (H1 : a ∨ b) (H2 : a → c) (H3 : b → c) : c :=
or_rec H2 H3 H1
theorem resolve_right {a b : Prop} (H1 : a ∨ b) (H2 : ¬a) : b :=
or_elim H1 (assume Ha, absurd_elim b Ha H2) (assume Hb, Hb)
theorem resolve_left {a b : Prop} (H1 : a ∨ b) (H2 : ¬b) : a :=
or_elim H1 (assume Ha, Ha) (assume Hb, absurd_elim a Hb H2)
theorem or_swap {a b : Prop} (H : a ∨ b) : b ∨ a :=
or_elim H (assume Ha, or_inr Ha) (assume Hb, or_inl Hb)
theorem or_not_intro {a b : Prop} (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=
assume H : a ∨ b, or_elim H
(assume Ha, absurd_elim _ Ha Hna)
(assume Hb, absurd_elim _ Hb Hnb)
theorem or_imp_or {a b c d : Prop} (H1 : a ∨ b) (H2 : a → c) (H3 : b → d) : c ∨ d :=
or_elim H1
(assume Ha : a, or_inl (H2 Ha))
(assume Hb : b, or_inr (H3 Hb))
theorem imp_or_left {a b c : Prop} (H1 : a ∨ c) (H : a → b) : b ∨ c :=
or_elim H1
(assume H2 : a, or_inl (H H2))
(assume H2 : c, or_inr H2)
theorem imp_or_right {a b c : Prop} (H1 : c ∨ a) (H : a → b) : c ∨ b :=
or_elim H1
(assume H2 : c, or_inl H2)
(assume H2 : a, or_inr (H H2))
-- iff
-- ---
definition iff (a b : Prop) := (a → b) ∧ (b → a)
infix `<->`:25 := iff
infix `↔`:25 := iff
theorem iff_intro {a b : Prop} (H1 : a → b) (H2 : b → a) : a ↔ b := and_intro H1 H2
theorem iff_elim {a b c : Prop} (H1 : (a → b) → (b → a) → c) (H2 : a ↔ b) : c := and_rec H1 H2
theorem iff_elim_left {a b : Prop} (H : a ↔ b) : a → b :=
iff_elim (assume H1 H2, H1) H
theorem iff_elim_right {a b : Prop} (H : a ↔ b) : b → a :=
iff_elim (assume H1 H2, H2) H
theorem iff_flip_sign {a b : Prop} (H1 : a ↔ b) : ¬a ↔ ¬b :=
iff_intro
(assume Hna, mt (iff_elim_right H1) Hna)
(assume Hnb, mt (iff_elim_left H1) Hnb)
theorem iff_refl (a : Prop) : a ↔ a :=
iff_intro (assume H, H) (assume H, H)
theorem iff_trans {a b c : Prop} (H1 : a ↔ b) (H2 : b ↔ c) : a ↔ c :=
iff_intro
(assume Ha, iff_elim_left H2 (iff_elim_left H1 Ha))
(assume Hc, iff_elim_right H1 (iff_elim_right H2 Hc))
theorem iff_symm {a b : Prop} (H : a ↔ b) : b ↔ a :=
iff_intro
(assume Hb, iff_elim_right H Hb)
(assume Ha, iff_elim_left H Ha)
calc_trans iff_trans
-- comm and assoc for and / or
-- ---------------------------
theorem and_comm (a b : Prop) : a ∧ b ↔ b ∧ a :=
iff_intro (λH, and_swap H) (λH, and_swap H)
theorem and_assoc (a b c : Prop) : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
iff_intro
(assume H, and_intro
(and_elim_left (and_elim_left H))
(and_intro (and_elim_right (and_elim_left H)) (and_elim_right H)))
(assume H, and_intro
(and_intro (and_elim_left H) (and_elim_left (and_elim_right H)))
(and_elim_right (and_elim_right H)))
theorem or_comm (a b : Prop) : a ∨ b ↔ b ∨ a :=
iff_intro (λH, or_swap H) (λH, or_swap H)
theorem or_assoc (a b c : Prop) : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
iff_intro
(assume H, or_elim H
(assume H1, or_elim H1
(assume Ha, or_inl Ha)
(assume Hb, or_inr (or_inl Hb)))
(assume Hc, or_inr (or_inr Hc)))
(assume H, or_elim H
(assume Ha, (or_inl (or_inl Ha)))
(assume H1, or_elim H1
(assume Hb, or_inl (or_inr Hb))
(assume Hc, or_inr Hc)))
|
d652cf4abcf938dfb347593f6093ef0be62d74e2 | 957a80ea22c5abb4f4670b250d55534d9db99108 | /tests/lean/run/sufficies.lean | 0dcff7247b79b339aa3614f02ac7baba54614cf9 | [
"Apache-2.0"
] | permissive | GaloisInc/lean | aa1e64d604051e602fcf4610061314b9a37ab8cd | f1ec117a24459b59c6ff9e56a1d09d9e9e60a6c0 | refs/heads/master | 1,592,202,909,807 | 1,504,624,387,000 | 1,504,624,387,000 | 75,319,626 | 2 | 1 | Apache-2.0 | 1,539,290,164,000 | 1,480,616,104,000 | C++ | UTF-8 | Lean | false | false | 264 | lean | open nat
protected theorem one_le_bit0 (n : ℕ) : n ≠ 0 → 1 ≤ bit0 n :=
nat.cases_on n
(λ h, absurd rfl h)
(λ n h,
suffices 1 ≤ succ (succ (bit0 n)), from
eq.symm (nat.bit0_succ_eq n) ▸ this,
succ_le_succ (zero_le (succ (bit0 n))))
|
bfb26e7b644af4b2462caa61f1eb65a24c8dc00e | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/logic/nontrivial.lean | 14ad9c3c2116d39465015f2b52904100e3a5df6b | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 7,084 | lean | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import data.prod.basic
import data.subtype
import logic.function.basic
import logic.unique
/-!
# Nontrivial types
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
A type is *nontrivial* if it contains at least two elements. This is useful in particular for rings
(where it is equivalent to the fact that zero is different from one) and for vector spaces
(where it is equivalent to the fact that the dimension is positive).
We introduce a typeclass `nontrivial` formalizing this property.
-/
variables {α : Type*} {β : Type*}
open_locale classical
/-- Predicate typeclass for expressing that a type is not reduced to a single element. In rings,
this is equivalent to `0 ≠ 1`. In vector spaces, this is equivalent to positive dimension. -/
class nontrivial (α : Type*) : Prop :=
(exists_pair_ne : ∃ (x y : α), x ≠ y)
lemma nontrivial_iff : nontrivial α ↔ ∃ (x y : α), x ≠ y :=
⟨λ h, h.exists_pair_ne, λ h, ⟨h⟩⟩
lemma exists_pair_ne (α : Type*) [nontrivial α] : ∃ (x y : α), x ≠ y :=
nontrivial.exists_pair_ne
-- See Note [decidable namespace]
protected lemma decidable.exists_ne [nontrivial α] [decidable_eq α] (x : α) : ∃ y, y ≠ x :=
begin
rcases exists_pair_ne α with ⟨y, y', h⟩,
by_cases hx : x = y,
{ rw ← hx at h,
exact ⟨y', h.symm⟩ },
{ exact ⟨y, ne.symm hx⟩ }
end
lemma exists_ne [nontrivial α] (x : α) : ∃ y, y ≠ x :=
by classical; exact decidable.exists_ne x
-- `x` and `y` are explicit here, as they are often needed to guide typechecking of `h`.
lemma nontrivial_of_ne (x y : α) (h : x ≠ y) : nontrivial α :=
⟨⟨x, y, h⟩⟩
-- `x` and `y` are explicit here, as they are often needed to guide typechecking of `h`.
lemma nontrivial_of_lt [preorder α] (x y : α) (h : x < y) : nontrivial α :=
⟨⟨x, y, ne_of_lt h⟩⟩
lemma exists_pair_lt (α : Type*) [nontrivial α] [linear_order α] : ∃ (x y : α), x < y :=
begin
rcases exists_pair_ne α with ⟨x, y, hxy⟩,
cases lt_or_gt_of_ne hxy;
exact ⟨_, _, h⟩
end
lemma nontrivial_iff_lt [linear_order α] : nontrivial α ↔ ∃ (x y : α), x < y :=
⟨λ h, @exists_pair_lt α h _, λ ⟨x, y, h⟩, nontrivial_of_lt x y h⟩
lemma nontrivial_iff_exists_ne (x : α) : nontrivial α ↔ ∃ y, y ≠ x :=
⟨λ h, @exists_ne α h x, λ ⟨y, hy⟩, nontrivial_of_ne _ _ hy⟩
lemma subtype.nontrivial_iff_exists_ne (p : α → Prop) (x : subtype p) :
nontrivial (subtype p) ↔ ∃ (y : α) (hy : p y), y ≠ x :=
by simp only [nontrivial_iff_exists_ne x, subtype.exists, ne.def, subtype.ext_iff, subtype.coe_mk]
instance : nontrivial Prop := ⟨⟨true, false, true_ne_false⟩⟩
/--
See Note [lower instance priority]
Note that since this and `nonempty_of_inhabited` are the most "obvious" way to find a nonempty
instance if no direct instance can be found, we give this a higher priority than the usual `100`.
-/
@[priority 500]
instance nontrivial.to_nonempty [nontrivial α] : nonempty α :=
let ⟨x, _⟩ := exists_pair_ne α in ⟨x⟩
attribute [instance, priority 500] nonempty_of_inhabited
/-- An inhabited type is either nontrivial, or has a unique element. -/
noncomputable def nontrivial_psum_unique (α : Type*) [inhabited α] :
psum (nontrivial α) (unique α) :=
if h : nontrivial α then psum.inl h else psum.inr
{ default := default,
uniq := λ (x : α),
begin
change x = default,
contrapose! h,
use [x, default]
end }
lemma subsingleton_iff : subsingleton α ↔ ∀ (x y : α), x = y :=
⟨by { introsI h, exact subsingleton.elim }, λ h, ⟨h⟩⟩
lemma not_nontrivial_iff_subsingleton : ¬(nontrivial α) ↔ subsingleton α :=
by { rw [nontrivial_iff, subsingleton_iff], push_neg, refl }
lemma not_nontrivial (α) [subsingleton α] : ¬nontrivial α :=
λ ⟨⟨x, y, h⟩⟩, h $ subsingleton.elim x y
lemma not_subsingleton (α) [h : nontrivial α] : ¬subsingleton α :=
let ⟨⟨x, y, hxy⟩⟩ := h in λ ⟨h'⟩, hxy $ h' x y
/-- A type is either a subsingleton or nontrivial. -/
lemma subsingleton_or_nontrivial (α : Type*) : subsingleton α ∨ nontrivial α :=
by { rw [← not_nontrivial_iff_subsingleton, or_comm], exact classical.em _ }
lemma false_of_nontrivial_of_subsingleton (α : Type*) [nontrivial α] [subsingleton α] : false :=
let ⟨x, y, h⟩ := exists_pair_ne α in h $ subsingleton.elim x y
instance option.nontrivial [nonempty α] : nontrivial (option α) :=
by { inhabit α, use [none, some default] }
/-- Pushforward a `nontrivial` instance along an injective function. -/
protected lemma function.injective.nontrivial [nontrivial α]
{f : α → β} (hf : function.injective f) : nontrivial β :=
let ⟨x, y, h⟩ := exists_pair_ne α in ⟨⟨f x, f y, hf.ne h⟩⟩
/-- Pullback a `nontrivial` instance along a surjective function. -/
protected lemma function.surjective.nontrivial [nontrivial β]
{f : α → β} (hf : function.surjective f) : nontrivial α :=
begin
rcases exists_pair_ne β with ⟨x, y, h⟩,
rcases hf x with ⟨x', hx'⟩,
rcases hf y with ⟨y', hy'⟩,
have : x' ≠ y', by { contrapose! h, rw [← hx', ← hy', h] },
exact ⟨⟨x', y', this⟩⟩
end
/-- An injective function from a nontrivial type has an argument at
which it does not take a given value. -/
protected lemma function.injective.exists_ne [nontrivial α] {f : α → β}
(hf : function.injective f) (y : β) : ∃ x, f x ≠ y :=
begin
rcases exists_pair_ne α with ⟨x₁, x₂, hx⟩,
by_cases h : f x₂ = y,
{ exact ⟨x₁, (hf.ne_iff' h).2 hx⟩ },
{ exact ⟨x₂, h⟩ }
end
instance nontrivial_prod_right [nonempty α] [nontrivial β] : nontrivial (α × β) :=
prod.snd_surjective.nontrivial
instance nontrivial_prod_left [nontrivial α] [nonempty β] : nontrivial (α × β) :=
prod.fst_surjective.nontrivial
namespace pi
variables {I : Type*} {f : I → Type*}
/-- A pi type is nontrivial if it's nonempty everywhere and nontrivial somewhere. -/
lemma nontrivial_at (i' : I) [inst : Π i, nonempty (f i)] [nontrivial (f i')] :
nontrivial (Π i : I, f i) :=
by classical; exact
(function.update_injective (λ i, classical.choice (inst i)) i').nontrivial
/--
As a convenience, provide an instance automatically if `(f default)` is nontrivial.
If a different index has the non-trivial type, then use `haveI := nontrivial_at that_index`.
-/
instance nontrivial [inhabited I] [inst : Π i, nonempty (f i)] [nontrivial (f default)] :
nontrivial (Π i : I, f i) := nontrivial_at default
end pi
instance function.nontrivial [h : nonempty α] [nontrivial β] : nontrivial (α → β) :=
h.elim $ λ a, pi.nontrivial_at a
@[nontriviality]
protected lemma subsingleton.le [preorder α] [subsingleton α] (x y : α) : x ≤ y :=
le_of_eq (subsingleton.elim x y)
namespace bool
instance : nontrivial bool := ⟨⟨tt,ff, tt_eq_ff_eq_false⟩⟩
end bool
|
9a9b481df92165700eca71ea3725cd23c297d07f | 08a8ee10652ba4f8592710ceb654b37e951d9082 | /src/hott/init/logic.lean | bb2109d1aca3acd56e1e3172ede73c1286e910d4 | [
"Apache-2.0"
] | permissive | felixwellen/hott3 | e9f299c84d30a782a741c40d38741ec024d391fb | 8ac87a2699ab94c23ea7984b4a5fbd5a7052575c | refs/heads/master | 1,619,972,899,098 | 1,509,047,351,000 | 1,518,040,986,000 | 120,676,559 | 0 | 0 | null | 1,518,040,503,000 | 1,518,040,503,000 | null | UTF-8 | Lean | false | false | 22,040 | lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Floris van Doorn
-/
import .path .meta.rewrite
universes u v w
hott_theory
/- prod -/
@[reducible] def pair := @prod.mk
@[hott] protected def prod.elim {a b c} (H₁ : a × b) (H₂ : a → b → c) : c :=
prod.rec H₂ H₁
@[hott] def prod.flip {α : Type u} {β : Type v} : α × β → β × α
| ⟨a,b⟩ := ⟨b,a⟩
@[hott] def prod.swap {α : Type u} {β : Type v} : α × β → β × α
| ⟨a,b⟩ := ⟨b,a⟩
@[hott, elab_as_eliminator] def prod.destruct {α : Type u} {β : Type v} {C} :=
@prod.cases_on α β C
/- sum -/
infixr ` ⊎ `:30 := sum
@[hott] protected def sum.elim {a b c} (H₁ : a ⊎ b) (H₂ : a → c) (H₃ : b → c) : c :=
sum.rec H₂ H₃ H₁
@[hott] def sum.swap {a b} : a ⊎ b → b ⊎ a
| (sum.inl ha) := sum.inr ha
| (sum.inr hb) := sum.inl hb
@[hott] def sum.functor {A A' B B'} (f : A → A') (g : B → B') : A ⊎ B → A' ⊎ B'
| (sum.inl a) := sum.inl (f a)
| (sum.inr b) := sum.inr (g b)
@[hott] def sum.functor_right (A) {B B'} (g : B → B') : A ⊎ B → A ⊎ B' :=
sum.functor id g
@[hott] def sum.functor_left {A A'} (B) (f : A → A') : A ⊎ B → A' ⊎ B :=
sum.functor f id
namespace hott
open unit
/- not -/
@[hott] def not (a : Type _) := a → empty
hott_theory_cmd "local prefix ¬ := hott.not"
@[hott] def absurd {a b : Type _} (H₁ : a) (H₂ : ¬a) : b :=
empty.rec (λ e, b) (H₂ H₁)
@[hott] def mt {a b : Type _} (H₁ : a → b) (H₂ : ¬b) : ¬a :=
assume Ha : a, absurd (H₁ Ha) H₂
@[hott] def not_empty : ¬empty :=
assume H : empty, H
@[hott] def non_contradictory (a : Type _) : Type _ := ¬¬a
@[hott] def non_contradictory_intro {a : Type _} (Ha : a) : ¬¬a :=
assume Hna : ¬a, absurd Ha Hna
@[hott] def not.intro {a : Type _} (H : a → empty) : ¬a := H
/- empty -/
@[hott] def empty.elim {c : Type _} (H : empty) : c :=
empty.rec _ H
@[hott, reducible] def ne {A : Type _} (a b : A) := ¬(a = b)
hott_theory_cmd "local notation a ≠ b := hott.ne a b"
namespace ne
variable {A : Type _}
variables {a b : A}
@[hott] def intro (H : a = b → empty) : a ≠ b := H
@[hott] def elim (H : a ≠ b) : a = b → empty := H
@[hott] def irrefl (H : a ≠ a) : empty := H rfl
@[hott] def symm (H : a ≠ b) : b ≠ a :=
assume (H₁ : b = a), H (H₁⁻¹)
end ne
@[hott] def empty_of_ne {A : Type _} {a : A} : a ≠ a → empty := ne.irrefl
section
variables {p : Type}
@[hott] def ne_empty_of_self : p → p ≠ empty :=
assume (Hp : p) (Heq : p = empty), Heq ▸ Hp
@[hott] def ne_unit_of_not : ¬p → p ≠ unit :=
assume (Hnp : ¬p) (Heq : p = unit), (Heq ▸ Hnp) star
@[hott] def unit_ne_empty : ¬unit = empty :=
ne_empty_of_self star
end
@[hott] def hott.non_contradictory_em (a : Type _) : ¬¬(a ⊎ ¬a) :=
assume not_em : ¬(a ⊎ ¬a),
have neg_a : ¬a, from
assume pos_a : a, absurd (sum.inl pos_a) not_em,
absurd (sum.inr neg_a) not_em
variables {a : Type _} {b : Type _} {c : Type _} {d : Type _}
/- iff -/
@[hott] def iff (a b : Type _) := (a → b) × (b → a)
hott_theory_cmd "local notation a <-> b := hott.iff a b"
hott_theory_cmd "local notation a ↔ b := hott.iff a b"
@[hott, intro!] def iff.intro : (a → b) → (b → a) → (a ↔ b) := prod.mk
@[hott] def iff.elim : ((a → b) → (b → a) → c) → (a ↔ b) → c :=
λ H₁ H₂, prod.rec H₁ H₂
@[hott] def iff.elim_left : (a ↔ b) → a → b := prod.fst
@[hott] def iff.mp := @iff.elim_left
@[hott] def iff.elim_right : (a ↔ b) → b → a := prod.snd
@[hott] def iff.mpr := @iff.elim_right
@[hott, refl] def iff.refl (a : Type _) : a ↔ a :=
iff.intro (assume H, H) (assume H, H)
@[hott] def iff.rfl {a : Type _} : a ↔ a :=
iff.refl a
@[hott] def iff.of_eq {a b : Type _} (H : a = b) : a ↔ b :=
eq.rec_on H iff.rfl
@[hott, trans] def iff.trans (H₁ : a ↔ b) (H₂ : b ↔ c) : a ↔ c :=
iff.intro
(assume Ha, iff.mp H₂ (iff.mp H₁ Ha))
(assume Hc, iff.mpr H₁ (iff.mpr H₂ Hc))
@[hott, trans] def iff.trans_eq {a b c : Type _} (H₁ : a ↔ b) (H₂ : b = c) : a ↔ c :=
iff.trans H₁ (iff.of_eq H₂)
@[hott, trans] def iff.eq_trans {a b c : Type _} (H₁ : a = b) (H₂ : b ↔ c) : a ↔ c :=
iff.trans (iff.of_eq H₁) H₂
@[hott, symm] def iff.symm (H : a ↔ b) : b ↔ a :=
iff.intro (iff.elim_right H) (iff.elim_left H)
@[hott] def iff.comm : (a ↔ b) ↔ (b ↔ a) :=
iff.intro iff.symm iff.symm
@[hott] def not_iff_not_of_iff (H₁ : a ↔ b) : ¬a ↔ ¬b :=
iff.intro
(assume (Hna : ¬ a) (Hb : b), Hna (iff.elim_right H₁ Hb))
(assume (Hnb : ¬ b) (Ha : a), Hnb (iff.elim_left H₁ Ha))
@[hott] def of_iff_unit (H : a ↔ unit) : a :=
iff.mp (iff.symm H) ()
@[hott] def not_of_iff_empty : (a ↔ empty) → ¬a := iff.mp
@[hott] def iff_unit_intro (H : a) : a ↔ unit :=
iff.intro
(λ Hl, ())
(λ Hr, H)
@[hott] def iff_empty_intro (H : ¬a) : a ↔ empty :=
iff.intro H (empty.rec _)
@[hott] def not_non_contradictory_iff_absurd (a : Type _) : ¬¬¬a ↔ ¬a :=
iff.intro
(λ (Hl : ¬¬¬a) (Ha : a), Hl (non_contradictory_intro Ha))
absurd
@[hott, congr] def imp_congr (H1 : a ↔ c) (H2 : b ↔ d) : (a → b) ↔ (c → d) :=
iff.intro
(λHab Hc, iff.mp H2 (Hab (iff.mpr H1 Hc)))
(λHcd Ha, iff.mpr H2 (Hcd (iff.mp H1 Ha)))
@[hott] def not_not_intro (Ha : a) : ¬¬a :=
assume Hna : ¬a, Hna Ha
@[hott] def not_of_not_not_not (H : ¬¬¬a) : ¬a :=
λ Ha, absurd (not_not_intro Ha) H
@[hott, hsimp] def not_unit : (¬ unit) ↔ empty :=
iff_empty_intro (not_not_intro ())
@[hott, hsimp] def not_empty_iff : (¬ empty) ↔ unit :=
iff_unit_intro not_empty
@[hott] def not_congr (H : a ↔ b) : ¬a ↔ ¬b :=
iff.intro (λ H₁ H₂, H₁ (iff.mpr H H₂)) (λ H₁ H₂, H₁ (iff.mp H H₂))
@[hott, hsimp] def ne_self_iff_empty {A : Type _} (a : A) : (not (a = a)) ↔ empty :=
iff.intro empty_of_ne empty.elim
@[hott, hsimp] def eq_self_iff_unit {A : Type _} (a : A) : (a = a) ↔ unit :=
iff_unit_intro rfl
@[hott, hsimp] def iff_not_self (a : Type _) : (a ↔ ¬a) ↔ empty :=
iff_empty_intro (λ H,
have H' : ¬a, from (λ Ha, (iff.mp H Ha) Ha),
H' (iff.mpr H H'))
@[hott, hsimp] def not_iff_self (a : Type _) : (¬a ↔ a) ↔ empty :=
iff_empty_intro (λ H,
have H' : ¬a, from (λ Ha, (iff.mpr H Ha) Ha),
H' (iff.mp H H'))
@[hott, hsimp] def unit_iff_empty : (unit ↔ empty) ↔ empty :=
iff_empty_intro (λ H, iff.mp H ())
@[hott, hsimp] def empty_iff_unit : (empty ↔ unit) ↔ empty :=
iff_empty_intro (λ H, iff.mpr H ())
@[hott] def empty_of_unit_iff_empty : (unit ↔ empty) → empty :=
assume H, iff.mp H ()
/- prod simp rules -/
@[hott] def prod.imp {a b c d} (H₂ : a → c) (H₃ : b → d) : a × b → c × d
| ⟨ha, hb⟩ := ⟨H₂ ha, H₃ hb⟩
@[hott, congr] def prod_congr (H1 : a ↔ c) (H2 : b ↔ d) : (a × b) ↔ (c × d) :=
iff.intro (prod.imp (iff.mp H1) (iff.mp H2)) (prod.imp (iff.mpr H1) (iff.mpr H2))
@[hott, hsimp] def prod.comm : a × b ↔ b × a :=
iff.intro prod.swap prod.swap
@[hott, hsimp] def prod.assoc : (a × b) × c ↔ a × (b × c) :=
iff.intro
(λ ⟨⟨Ha, Hb⟩, Hc⟩, ⟨Ha, ⟨Hb, Hc⟩⟩)
(λ ⟨Ha, ⟨Hb, Hc⟩⟩, ⟨⟨Ha, Hb⟩, Hc⟩)
@[hott, hsimp] def prod.fst_comm : a × (b × c) ↔ b × (a × c) :=
iff.intro
(λ ⟨Ha, ⟨Hb, Hc⟩⟩, ⟨Hb, ⟨Ha, Hc⟩⟩)
(λ ⟨Hb, ⟨Ha, Hc⟩⟩, ⟨Ha, ⟨Hb, Hc⟩⟩)
@[hott] def prod_iff_left {a b : Type _} (Hb : b) : (a × b) ↔ a :=
iff.intro prod.fst (λHa, prod.mk Ha Hb)
@[hott] def prod_iff_right {a b : Type _} (Ha : a) : (a × b) ↔ b :=
iff.intro prod.snd (prod.mk Ha)
@[hott, hsimp] def prod_unit (a : Type _) : a × unit ↔ a :=
prod_iff_left ()
@[hott, hsimp] def unit_prod (a : Type _) : unit × a ↔ a :=
prod_iff_right ()
@[hott, hsimp] def prod_empty (a : Type _) : a × empty ↔ empty :=
iff_empty_intro prod.snd
@[hott, hsimp] def empty_prod (a : Type _) : empty × a ↔ empty :=
iff_empty_intro prod.fst
@[hott, hsimp] def not_prod_self (a : Type _) : (¬a × a) ↔ empty :=
iff_empty_intro (λ H, prod.elim H (λ H₁ H₂, absurd H₂ H₁))
@[hott, hsimp] def prod_not_self (a : Type _) : (a × ¬a) ↔ empty :=
iff_empty_intro (λ H, prod.elim H (λ H₁ H₂, absurd H₁ H₂))
@[hott, hsimp] def prod_self (a : Type _) : a × a ↔ a :=
iff.intro prod.fst (assume H, prod.mk H H)
/- sum simp rules -/
@[hott] def sum.imp (H₂ : a → c) (H₃ : b → d) : a ⊎ b → c ⊎ d
| (sum.inl H) := sum.inl (H₂ H)
| (sum.inr H) := sum.inr (H₃ H)
@[hott] def sum.imp_left (H : a → b) : a ⊎ c → b ⊎ c :=
sum.imp H id
@[hott] def sum.imp_right (H : a → b) : c ⊎ a → c ⊎ b :=
sum.imp id H
@[hott, congr] def sum_congr (H1 : a ↔ c) (H2 : b ↔ d) : (a ⊎ b) ↔ (c ⊎ d) :=
iff.intro (sum.imp (iff.mp H1) (iff.mp H2)) (sum.imp (iff.mpr H1) (iff.mpr H2))
@[hott, hsimp] def sum.comm : a ⊎ b ↔ b ⊎ a := iff.intro sum.swap sum.swap
@[hott, hsimp] def sum.assoc : (a ⊎ b) ⊎ c ↔ a ⊎ (b ⊎ c) :=
iff.intro
(λ Habc, match Habc with
| sum.inl (sum.inl Ha) := sum.inl Ha
| sum.inl (sum.inr Hb) := sum.inr (sum.inl Hb)
| sum.inr Hc := sum.inr (sum.inr Hc)
end)
(λ Habc, match Habc with
| sum.inl Ha := sum.inl (sum.inl Ha)
| sum.inr (sum.inl Hb) := sum.inl (sum.inr Hb)
| sum.inr (sum.inr Hc) := sum.inr Hc
end)
@[hott, hsimp] def sum.left_comm : a ⊎ (b ⊎ c) ↔ b ⊎ (a ⊎ c) :=
begin
transitivity, {symmetry, exact sum.assoc},
transitivity, {apply sum_congr, apply sum.comm, refl},
apply sum.assoc
end
@[hott, hsimp] def sum_unit (a : Type _) : a ⊎ unit ↔ unit :=
iff_unit_intro (sum.inr ())
@[hott, hsimp] def unit_sum (a : Type _) : unit ⊎ a ↔ unit :=
iff_unit_intro (sum.inl ())
@[hott, hsimp] def sum_empty (a : Type _) : a ⊎ empty ↔ a :=
iff.intro (λ Hae, match Hae with sum.inl Ha := Ha end) sum.inl
@[hott, hsimp] def empty_sum (a : Type _) : empty ⊎ a ↔ a :=
iff.trans sum.comm (sum_empty _)
@[hott, hsimp] def sum_self (a : Type _) : a ⊎ a ↔ a :=
iff.intro (λ Haa, Haa.elim id id) sum.inl
/- sum resolution rulse -/
@[hott] def sum.resolve_left {a b : Type _} (H : a ⊎ b) (na : ¬ a) : b :=
sum.elim H (λ Ha, absurd Ha na) id
@[hott] def sum.neg_resolve_left {a b : Type _} (H : ¬ a ⊎ b) (Ha : a) : b :=
sum.elim H (λ na, absurd Ha na) id
@[hott] def sum.resolve_right {a b : Type _} (H : a ⊎ b) (nb : ¬ b) : a :=
sum.elim H id (λ Hb, absurd Hb nb)
@[hott] def sum.neg_resolve_right {a b : Type _} (H : a ⊎ ¬ b) (Hb : b) : a :=
sum.elim H id (λ nb, absurd Hb nb)
/- iff simp rules -/
@[hott, hsimp] def iff_unit (a : Type _) : (a ↔ unit) ↔ a :=
iff.intro (assume H, iff.mpr H star) iff_unit_intro
@[hott, hsimp] def unit_iff (a : Type _) : (unit ↔ a) ↔ a :=
iff.trans iff.comm (iff_unit _)
@[hott, hsimp] def iff_empty (a : Type _) : (a ↔ empty) ↔ ¬ a :=
iff.intro prod.fst iff_empty_intro
@[hott, hsimp] def empty_iff (a : Type _) : (empty ↔ a) ↔ ¬ a :=
iff.trans iff.comm (iff_empty _)
@[hott, hsimp] def iff_self (a : Type _) : (a ↔ a) ↔ unit :=
iff_unit_intro iff.rfl
@[hott, congr] def iff_congr (H1 : a ↔ c) (H2 : b ↔ d) : (a ↔ b) ↔ (c ↔ d) :=
prod_congr (imp_congr H1 H2) (imp_congr H2 H1)
/- decidable -/
set_option pp.all true
set_option pp.notation false
set_option pp.full_names true
class inductive decidable (p : Type u) : Type u
| inl : p → decidable
| inr : ¬p → decidable
@[hott, instance] def decidable_unit : decidable unit :=
decidable.inl ()
@[hott, instance] def decidable_empty : decidable empty :=
decidable.inr not_empty
-- We use "dependent" if-then-else to be able to communicate the if-then-else condition
-- to the branches
@[hott] def dite (c : Type _) [H : decidable c] {A : Type _} (Hp : c → A) (Hn : ¬ c → A): A :=
decidable.rec_on H Hp Hn
/- if-then-else -/
@[hott] def ite (c : Type _) [H : decidable c] {A : Type _} (t e : A) : A :=
decidable.rec_on H (λ Hc, t) (λ Hnc, e)
hott_theory_cmd "local notation `if' ` c ` then ` t:45 ` else ` e:45 := hott.ite c t e"
hott_theory_cmd "local notation `if ` binder ` :: ` c ` then ` t:scoped ` else ` e:scoped := hott.dite c t e"
namespace decidable
variables {p : Type _} {q : Type _}
@[hott] def by_cases {q : Type _} [C : decidable p] : (p → q) → (¬p → q) → q := dite _
@[hott] theorem em (p : Type _) [H : decidable p] : p ⊎ ¬p := by_cases sum.inl sum.inr
@[hott] theorem by_contradiction [Hp : decidable p] (H : ¬p → empty) : p :=
if H1 :: p then H1 else empty.rec _ (H H1)
end decidable
section
variables {p : Type _} {q : Type _}
open hott.decidable
@[hott] def decidable_of_decidable_of_iff (Hp : decidable p) (H : p ↔ q) : decidable q :=
if Hp :: p then inl (iff.mp H Hp)
else inr (iff.mp (not_iff_not_of_iff H) Hp)
@[hott] def decidable_of_decidable_of_eq {p q : Type _} (Hp : decidable p) (H : p = q)
: decidable q :=
decidable_of_decidable_of_iff Hp (iff.of_eq H)
@[hott] protected def sum.by_cases [Hp : decidable p] [Hq : decidable q] {A : Type _}
(h : p ⊎ q) (h₁ : p → A) (h₂ : q → A) : A :=
if hp :: p then h₁ hp else
if hq :: q then h₂ hq else
empty.rec _ (sum.elim h hp hq)
end
section
variables {p : Type _} {q : Type _}
open hott.decidable
@[hott, instance] def decidable_prod [Hp : decidable p] [Hq : decidable q] : decidable (p × q) :=
if hp :: p then
if hq :: q then inl (prod.mk hp hq)
else inr (assume H : p × q, hq H.snd)
else inr (assume H : p × q, hp H.fst)
@[hott, instance] def decidable_sum [Hp : decidable p] [Hq : decidable q] : decidable (p ⊎ q) :=
if hp :: p then inl (sum.inl hp) else
if hq :: q then inl (sum.inr hq) else
inr (λ hpq, sum.rec hp hq hpq)
@[hott, instance] def decidable_not [Hp : decidable p] : decidable (¬p) :=
if hp :: p then inr (absurd hp) else inl hp
@[hott, instance] def decidable_implies [Hp : decidable p] [Hq : decidable q] : decidable (p → q) :=
if hp :: p then
if hq :: q then inl (assume H, hq)
else inr (assume H : p → q, absurd (H hp) hq)
else inl (assume Hp, absurd Hp hp)
@[hott, instance] def decidable_iff [Hp : decidable p] [Hq : decidable q] : decidable (p ↔ q) :=
decidable_prod
end
@[hott, reducible] def decidable_pred {A : Type _} (R : A → Type _) := Π (a : A), decidable (R a)
@[hott, reducible] def decidable_rel {A : Type _} (R : A → A → Type _) := Π (a b : A), decidable (R a b)
@[hott, reducible] def decidable_eq (A : Type _) := decidable_rel (@eq A)
@[hott, reducible, instance] def decidable_ne {A : Type _} [H : decidable_eq A] (a b : A) : decidable (a ≠ b) :=
decidable_implies
namespace bool
protected def no_confusion {P : Sort u} {v1 v2 : bool} (H12 : v1 = v2) : bool.no_confusion_type P v1 v2 :=
eq.rec (λ (H11 : v1 = v1), bool.cases_on v1 (λ (a : P), a) (λ (a : P), a)) H12 H12
@[hott] def ff_ne_tt : ff = tt → empty :=
bool.no_confusion
end bool
open hott.bool
@[hott] def is_dec_eq {A : Type _} (p : A → A → bool) : Type _ := Π ⦃x y : A⦄, p x y = tt → x = y
@[hott] def is_dec_refl {A : Type _} (p : A → A → bool) : Type _ := Πx, p x x = tt
open hott.decidable
@[hott, instance] protected def bool.has_decidable_eq : Πa b : bool, decidable (a = b)
| ff ff := inl rfl
| ff tt := inr ff_ne_tt
| tt ff := inr (ne.symm ff_ne_tt)
| tt tt := inl rfl
@[hott] def decidable_eq_of_bool_pred {A : Type _} {p : A → A → bool} (H₁ : is_dec_eq p) (H₂ : is_dec_refl p) : decidable_eq A :=
λ x y : A, if Hp :: p x y = tt then inl (H₁ Hp)
else inr begin intro Hxy, apply Hp, rwr Hxy, apply H₂ end
/- inhabited -/
@[hott] protected def inhabited.value {A : Type _} : inhabited A → A
| ⟨v⟩ := v
@[hott] protected def inhabited.destruct {A : Type _} {B : Type _} (H1 : inhabited A) (H2 : A → B) : B :=
inhabited.rec H2 H1
/- subsingleton -/
class subsingleton (A : Type _) :=
(prop : (Π a b : A, a = b))
@[hott] protected def subsingleton.elim {A : Type _} [H : subsingleton A] : Π(a b : A), a = b :=
subsingleton.rec (λp, p) H
@[hott] protected theorem rec_subsingleton {p : Type _} [H : decidable p]
{H1 : p → Type _} {H2 : ¬p → Type _}
[H3 : Π(h : p), subsingleton (H1 h)] [H4 : Π(h : ¬p), subsingleton (H2 h)]
: subsingleton (decidable.rec_on H H1 H2) :=
decidable.rec_on H (λh, H3 h) (λh, H4 h) --this can be proven using dependent version of "by_cases"
@[hott] def if_pos {c : Type _} [H : decidable c] (Hc : c) {A : Type _} {t e : A} : (ite c t e) = t :=
decidable.rec
(λ Hc : c, eq.refl (@ite c (decidable.inl Hc) A t e))
(λ Hnc : ¬c, absurd Hc Hnc)
H
@[hott] def if_neg {c : Type _} [H : decidable c] (Hnc : ¬c) {A : Type _} {t e : A} : (ite c t e) = e :=
decidable.rec
(λ Hc : c, absurd Hc Hnc)
(λ Hnc : ¬c, eq.refl (@ite c (decidable.inr Hnc) A t e))
H
@[hott, hsimp] theorem if_t_t (c : Type _) [H : decidable c] {A : Type _} (t : A) : (ite c t t) = t :=
decidable.rec
(λ Hc : c, eq.refl (@ite c (decidable.inl Hc) A t t))
(λ Hnc : ¬c, eq.refl (@ite c (decidable.inr Hnc) A t t))
H
@[hott] theorem implies_of_if_pos {c t e : Type _} [H : decidable c] (h : ite c t e) : c → t :=
assume Hc, transport id (if_pos Hc) h
@[hott] theorem implies_of_if_neg {c t e : Type _} [H : decidable c] (h : ite c t e) : ¬c → e :=
assume Hc, transport id (if_neg Hc) h
@[hott] theorem if_ctx_congr {A : Type _} {b c : Type _} [dec_b : decidable b] [dec_c : decidable c]
{x y u v : A}
(h_c : b ↔ c) (h_t : c → x = u) (h_e : ¬c → y = v) :
ite b x y = ite c u v :=
if hp :: b then
calc
ite b x y = x : if_pos hp
... = u : h_t (iff.mp h_c hp)
... = ite c u v : (if_pos (iff.mp h_c hp)).inverse
else
calc
ite b x y = y : if_neg hp
... = v : h_e (iff.mp (not_iff_not_of_iff h_c) hp)
... = ite c u v : (if_neg (iff.mp (not_iff_not_of_iff h_c) hp)).inverse
@[hott, congr] theorem if_congr {A : Type _} {b c : Type _} [dec_b : decidable b] [dec_c : decidable c]
{x y u v : A}
(h_c : b ↔ c) (h_t : x = u) (h_e : y = v) :
ite b x y = ite c u v :=
@if_ctx_congr A b c dec_b dec_c x y u v h_c (λ h, h_t) (λ h, h_e)
@[hott, congr] theorem if_ctx_simp_congr {A : Type _} {b c : Type _} [dec_b : decidable b] {x y u v : A}
(h_c : b ↔ c) (h_t : c → x = u) (h_e : ¬c → y = v) :
ite b x y = (@ite c (decidable_of_decidable_of_iff dec_b h_c) A u v) :=
@if_ctx_congr A b c dec_b (decidable_of_decidable_of_iff dec_b h_c) x y u v h_c h_t h_e
@[hott, congr] theorem if_simp_congr {A : Type _} {b c : Type _} [dec_b : decidable b] {x y u v : A}
(h_c : b ↔ c) (h_t : x = u) (h_e : y = v) :
ite b x y = (@ite c (decidable_of_decidable_of_iff dec_b h_c) A u v) :=
@if_ctx_simp_congr A b c dec_b x y u v h_c (λ h, h_t) (λ h, h_e)
@[hott, hsimp] def if_unit {A : Type _} (t e : A) : (if' unit then t else e) = t :=
if_pos star
@[hott, hsimp] def if_empty {A : Type _} (t e : A) : (if' empty then t else e) = e :=
if_neg not_empty
@[hott] theorem if_ctx_congr_prop {b c x y u v : Type _} [dec_b : decidable b] [dec_c : decidable c]
(h_c : b ↔ c) (h_t : c → (x ↔ u)) (h_e : ¬c → (y ↔ v)) :
ite b x y ↔ ite c u v :=
if hp :: b then
calc
ite b x y ↔ x : iff.of_eq (if_pos hp)
... ↔ u : h_t (iff.mp h_c hp)
... ↔ ite c u v : iff.of_eq (if_pos (iff.mp h_c hp)).inverse
else
calc
ite b x y ↔ y : iff.of_eq (if_neg hp)
... ↔ v : h_e (iff.mp (not_iff_not_of_iff h_c) hp)
... ↔ ite c u v : iff.of_eq (if_neg (iff.mp (not_iff_not_of_iff h_c) hp)).inverse
@[hott, congr] theorem if_congr_prop {b c x y u v : Type _} [dec_b : decidable b] [dec_c : decidable c]
(h_c : b ↔ c) (h_t : x ↔ u) (h_e : y ↔ v) :
ite b x y ↔ ite c u v :=
if_ctx_congr_prop h_c (λ h, h_t) (λ h, h_e)
@[hott] theorem if_ctx_simp_congr_prop {b c x y u v : Type _} [dec_b : decidable b]
(h_c : b ↔ c) (h_t : c → (x ↔ u)) (h_e : ¬c → (y ↔ v)) :
ite b x y ↔ (@ite c (decidable_of_decidable_of_iff dec_b h_c) _ u v) :=
@if_ctx_congr_prop b c x y u v dec_b (decidable_of_decidable_of_iff dec_b h_c) h_c h_t h_e
@[hott, congr] theorem if_simp_congr_prop {b c x y u v : Type _} [dec_b : decidable b]
(h_c : b ↔ c) (h_t : x ↔ u) (h_e : y ↔ v) :
ite b x y ↔ (@ite c (decidable_of_decidable_of_iff dec_b h_c) _ u v) :=
@if_ctx_simp_congr_prop b c x y u v dec_b h_c (λ h, h_t) (λ h, h_e)
-- Remark: dite and ite are "definitionally equal" when we ignore the proofs.
@[hott] theorem dite_ite_eq (c : Type _) [H : decidable c] {A : Type _} (t : A) (e : A) :
dite c (λh, t) (λh, e) = ite c t e :=
by refl
@[hott] def is_unit (c : Type _) [H : decidable c] : Type :=
if' c then unit else empty
@[hott] def is_empty (c : Type _) [H : decidable c] : Type :=
if' c then empty else unit
@[hott] def of_is_unit {c : Type _} [H₁ : decidable c] (H₂ : is_unit c) : c :=
if Hc :: c then Hc else begin induction H₁, assumption, cases H₂ end
notation `dec_star` := of_is_unit star
@[hott] theorem not_of_not_is_unit {c : Type _} [H₁ : decidable c] (H₂ : ¬ is_unit c) : ¬ c :=
if Hc :: c then absurd star (if_pos Hc ▸ H₂) else Hc
@[hott] theorem not_of_is_empty {c : Type _} [H₁ : decidable c] (H₂ : is_empty c) : ¬ c :=
if Hc :: c then begin induction H₁, cases H₂, assumption end else Hc
@[hott] theorem of_not_is_empty {c : Type _} [H₁ : decidable c] (H₂ : ¬ is_empty c) : c :=
if Hc :: c then Hc else absurd star (if_neg Hc ▸ H₂)
end hott
|
da71798dc770987afadccd642f5f70ad592bc322 | cffb3e591122726ebf5d0b11644a27c70d1afb93 | /Gröbner.lean | 16dae1281cfdbe532772ddec442c592af859131f | [] | no_license | lean-forward/ring_equational_reasoner | 6db875c5969697d31d1913bc4a5a3447888a4b80 | 6da1443393449f8efd2b9a5b75dd1ca2474c7078 | refs/heads/master | 1,598,984,144,058 | 1,573,740,296,000 | 1,573,740,296,000 | 219,024,384 | 0 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 29,626 | lean | import group_theory.coset ring_theory.ideals algebra.gcd_domain algebra.euclidean_domain data.int.modeq group_theory.quotient_group data.equiv.algebra group_theory.subgroup tactic.ring tactic.fin_cases tactic.tidy algebra.ring algebra.field linear_algebra.multivariate_polynomial
open tactic prod native environment interactive lean.parser ideal classical lattice declaration rbtree
infixl ` ⬝ ` := has_mul.mul
notation x`²`:99 := x⬝x
notation ` ` binders ` ↦ ` f:(scoped f, f) := f
notation `⟮` binders ` ↦ ` f:(scoped f, f) `⟯`:= f
def flip_over_2{A B C D}(f: A→B→C→D) := z x y ↦ f x y z
infixl ` @₁`:99 := flip
infixl ` @₂`:99 := flip_over_2
infixr ` ∘₂ `:80 := (∘)∘(∘)
instance decidable_bool{b:bool}: decidable b := by{cases b, apply is_false, simp, apply is_true, simp}
instance{X}: monoid(list X) := {
one := [],
mul := (++),
one_mul := by safe,
mul_one := list.append_nil,
mul_assoc := list.append_assoc,
}
instance endomonoid{t}: monoid(t → t) := {
one := id,
mul := (∘),
mul_assoc := function.comp.assoc,
one_mul := function.comp.left_id,
mul_one := function.comp.right_id,
}
--option.cases_on has typing problems.
def option.maybe{A B}(no: B)(yes: A→B): option A → B
| none := no
| (some a) := yes a
def ifoldl_help{S T}(f: ℕ→S→T→T): ℕ → list S → T → T
| i [] r := r
| i (x::s) r := ifoldl_help (i+1) s (f i x r)
def list.ifoldl{S T}(f: ℕ→S→T→T) := ifoldl_help f 0
def list.imap{S T}(f: ℕ→S→T)(s: list S) := (s.ifoldl(list.cons ∘₂ f) []).reverse
universe U
def list.mfilter_map{A B : Type U}{M}[monad M](f: A → M(option B)): list A → M(list B)
| [] := pure[]
| (a::s) := do
fa ← f a,
s' ← s.mfilter_map,
pure(fa.maybe s' ⟮b ↦ b::s'⟯)
@[simp] def map₂_default{X Z}(d)(f: X → X → Z): list X → list X → list Z
| [] [] := []
--| s l := f ((s.nth 0).get_or_else d) ((l.nth 0).get_or_else d) :: map₂_default s.tail l.tail
| [] (y::ys) := f d y :: map₂_default [] ys
| (x::xs) [] := f x d :: map₂_default xs []
| (x::xs) (y::ys) := f x y :: map₂_default xs ys
def trim_tail{X}[decidable_eq X](x)(s: list X) := (s.reverse.drop_while(=x)).reverse
--unique_pairs[xⱼ | j] = [(xᵢ,xⱼ) | i<j]
def list.unique_pairs{T}: list T → list(T×T)
| [] := []
| (x::xs) := xs.map⟮y↦ (x,y)⟯ ++ xs.unique_pairs
@[priority 0]instance to_string_of_repr{X}[has_repr X]: has_to_string X := ⟨repr⟩
@[priority 0]meta instance format_of_repr{X}[has_repr X]: has_to_tactic_format X := ⟨has_to_tactic_format.to_tactic_format ∘ repr⟩
/--
`nat.mk_numeral n` embeds `n` as a numeral expression inside a type with 0, 1, and +.
`type`: an expression representing the target type
`has_zero`, `has_one`, `has_add`: expressions of the type `has_zero %%type`, etc.
-/
meta def nat.mk_numeral (type has_zero has_one has_add : expr) : ℕ → expr :=
let z : expr := `(@has_zero.zero.{0} %%type %%has_zero),
o : expr := `(@has_one.one.{0} %%type %%has_one) in
nat.binary_rec z
(λ b n e, if n = 0 then o else
if b then `(@bit1.{0} %%type %%has_one %%has_add %%e)
else `(@bit0.{0} %%type %%has_add %%e))
meta def int.mk_numeral (type has_zero has_one has_add has_neg : expr) : ℤ → expr
| (int.of_nat n) := n.mk_numeral type has_zero has_one has_add
| -[1+n] := let ne := (n+1).mk_numeral type has_zero has_one has_add in
`(@has_neg.neg.{0} %%type %%has_neg %%ne)
meta def rat.mk_numeral (type has_zero has_one has_add has_neg has_div : expr) : ℚ → expr
| ⟨num, denom, _, _⟩ :=
let nume := num.mk_numeral type has_zero has_one has_add has_neg in
if denom = 1 then nume else
let dene := denom.mk_numeral type has_zero has_one has_add in
`(@has_div.div.{0} %%type %%has_div %%nume %%dene)
meta def rat.reflect : ℚ → expr :=
rat.mk_numeral `(ℚ) `((infer_instance : has_zero ℚ))
`((infer_instance : has_one ℚ))`((infer_instance : has_add ℚ))
`((infer_instance : has_neg ℚ)) `(infer_instance : has_div ℚ)
section
local attribute [semireducible] reflected
meta instance ℚ_has_reflect: has_reflect ℚ := rat.reflect
end
-- infixr ` ∷ `:67 := (++)∘repr
-- --set_option pp.all true
-- def replace_1_: list char → string
-- --| ('+'::' '::'-'::s) := "- "++ replace_minus s
-- | [] := ""
-- | (a::s) := match if a≠' ' then none else match s with
-- | [] := none
-- | (b::s) := if b≠'1' then none else match s with
-- | [] := none
-- | (c::s) := if c≠' ' then none else
-- have hint: s.sizeof < a.val+(1+s.sizeof), from by{simp[has_lt.lt,nat.lt], rw(_:nat.succ _=0+_), rw(_:1+_=nat.succ _), apply nat.add_le_add_right, tidy, rw nat.add_comm},
-- some(' '∷ replace_1_ s)
-- end end with
-- | none := a ∷ replace_1_ s
-- | some s := s
--–Monomials are now represented by lists, which are thought to be zero extended. Trailing zeros are normalized away. (This type should have been parameterised by type for the indeterminates—this mistake made the selection of monomial orders cubersome and reduced a lot the reuseability of the custom pattern matching simplifier tactic skeleton.)
def monomial := list ℕ
namespace monomial
def deg(m: monomial) := list.sum m
def variable_name: ℕ → string
| 0 := "x" | 1 := "y" | 2 := "z"
| n := (char.of_nat(121-n)).to_string
def rise_digit(n: char) := ("⁰¹²³⁴⁵⁶⁷⁸⁹".to_list.nth n.to_string.to_nat).get_or_else '?'
def rise_number(n: string) := (n.to_list.map rise_digit).as_string
instance monomial.has_repr: has_repr monomial := ⟨ m ↦ m.ifoldl⟮j e ↦ ite(e=0) id (++ variable_name j ++ ite(e=1) "" (rise_number(repr e)))⟯ "" ⟩
instance: has_one monomial := ⟨[]⟩
instance: has_mul monomial := ⟨map₂_default 0 (+)⟩ --no need to trim
instance: has_div monomial := ⟨trim_tail 0 ∘₂ map₂_default 0 ⟮x y ↦ x-y⟯⟩
def gcd(n m : monomial): monomial := trim_tail 0 (list.map₂ min n m) --no need to extend
def lcm(n m : monomial): monomial := map₂_default 0 max n m --no need to trim
def dvd': monomial → monomial → bool
| [] _ := tt
| (n::ns) m := n ≤ (m.nth 0).get_or_else 0 ∧ dvd' ns m.tail
def dvd := n m ↦ (dvd' n m : Prop)
instance: has_dvd monomial := ⟨dvd⟩
instance: decidable_rel dvd := by unfold dvd; apply_instance
--–Orders should be admissible (unit least and multiplication monotonous).
class order :=
(lt: monomial → monomial → Prop)
(decidable: decidable_rel lt)
def lex: order := {
lt := list.lex(<),
decidable := infer_instance,
}
def deg_lex: order := {
lt := n m ↦ deg n < deg m ∨ (deg n = deg m ∧ list.lex(<) n m),
decidable := _ _↦ by apply or.decidable,
}
end monomial
@[reducible] private def mo := monomial.order
--–Polynomials (this ended up essentially reimplementing very basics of rbmap equivalently, because I didn't find rbmap early enough)
--Reverse order to have the leading term first.
def poly.less[mo]{K}(x y : K×monomial) := monomial.order.lt y.snd x.snd
def poly[mo](K)[ring K] := rbtree (K×monomial) poly.less
namespace poly
--K ∈ Type 0 because otherwise combination with tactics gets problematic.
variables{K: Type}[ring K][decidable_eq K][o:mo](P R : poly K)
instance[mo]: has_lt monomial := ⟨monomial.order.lt⟩
instance[mo]: has_le monomial := ⟨ n m ↦ ¬n>m ⟩
instance decidable_lt[mo]: @decidable_rel monomial (<) := monomial.order.decidable
instance decidable_le[mo]: @decidable_rel monomial (≤) := by apply_instance
instance decidable_less: decidable_rel(@less o K) := _ _↦ by unfold less; apply_instance
def coef(m) := ((P.find(0,m)).get_or_else(0,m)).fst
--Value 0 should not be inserted, but rbtree lacks removal rutines. Since full simplification will rebuild a polynomial from scratch, extra zeros should not add up too badly.
def update(m)(f: K→K): poly K := let k := f(coef P m) in P.insert(k,m)
def monom[mo](m): poly K := rbtree_of[m]less
instance[mo]: has_coe monomial (poly K) := ⟨ m↦ monom(1,m) ⟩
--Let f' = (f; 0↦id). Then combine@₂f maps Σ pⱼ⬝mⱼ and Σ rⱼ⬝mⱼ to Σ f' pⱼ rⱼ ⬝ mⱼ (...assuming unsoundly that there's no explicit 0 coefficients...exact behavior depends on what the representation happens to be).
def combine(f: K→K→K): poly K := P.fold⟮p R' ↦ update R' p.snd (f p.fst)⟯ R
def map_poly(f: K→K): poly K := combine P P _↦f
instance[mo]: has_zero(poly K) := ⟨rbtree_of[]less⟩
instance[mo]: has_one(poly K) := ⟨monom(1,1)⟩
instance[mo]: has_add(poly K) := ⟨combine @₂(+)⟩
instance[mo]: has_neg(poly K) := ⟨map_poly @₁ k↦-k⟩
instance[mo]: has_sub(poly K) := ⟨ P R ↦ P + -R ⟩
instance[mo]: has_mul(poly K) := ⟨ P↦ fold⟮m↦ P.fold⟮n↦ (+monom(m⬝n))⟯⟯ @₁0 ⟩
instance[mo]: has_scalar K (poly K) := ⟨ k↦ map_poly @₁(⬝k) ⟩
instance[mo]: has_pow(poly K) ℕ := ⟨ P n ↦ (list.repeat P n).foldl(⬝)1 ⟩
instance[mo][has_repr K]: has_repr(poly K) := ⟨ P↦ match P.to_list.filter⟮p:K×_ ↦ p.fst ≠ 0⟯ with
| [] := "0"
| (m::ms) := ms.foldl⟮s p ↦ s ++" + "++ repr p.fst ++" "++ repr p.snd⟯ (repr m.fst ++" "++ repr m.snd)
end⟩
def lead_term := (P.fold⟮p:K×_ ↦ option.maybe (if p.fst = 0 then none else some p) some⟯ none).maybe (0,1) id
def lead_coef := P.lead_term.fst
def lead_mono := P.lead_term.snd
def is0 := lead_coef P = 0
instance decidable_is0: decidable P.is0 := by unfold is0; apply_instance
def is_const := lead_mono P = 1
instance decidable_is_const: decidable P.is_const := by unfold is_const; apply_instance
--This is ridiculous!
instance rbnode_eq{X}[eqX: decidable_eq X]: decidable_eq(rbnode X)
| rbnode.leaf rbnode.leaf := is_true rfl
| (rbnode.red_node l1 v1 r1) (rbnode.red_node l2 v2 r2) :=
match eqX v1 v2 with
| is_false v := is_false(by{by_contra a, injection a, contradiction})
| is_true v :=
match rbnode_eq l1 l2 with
| is_false l := is_false(by{by_contra a, injection a, contradiction})
| is_true l :=
match rbnode_eq r1 r2 with
| is_false r := is_false(by{by_contra a, injection a, contradiction})
| is_true r := is_true(by rw[l,v,r])
end
end
end
| (rbnode.black_node l1 v1 r1) (rbnode.black_node l2 v2 r2) :=
match eqX v1 v2 with
| is_false v := is_false(by{by_contra a, injection a, contradiction})
| is_true v :=
match rbnode_eq l1 l2 with
| is_false l := is_false(by{by_contra a, injection a, contradiction})
| is_true l :=
match rbnode_eq r1 r2 with
| is_false r := is_false(by{by_contra a, injection a, contradiction})
| is_true r := is_true(by rw[l,v,r])
end
end
end
| rbnode.leaf (rbnode.red_node l1 v1 r1) := is_false(by by_contra; injection a)
| rbnode.leaf (rbnode.black_node l1 v1 r1) := is_false(by by_contra; injection a)
| (rbnode.red_node l1 v1 r1) rbnode.leaf := is_false(by by_contra; injection a)
| (rbnode.red_node l1 v1 r1) (rbnode.black_node l2 v2 r2) := is_false(by by_contra; injection a)
| (rbnode.black_node l1 v1 r1) rbnode.leaf := is_false(by by_contra; injection a)
| (rbnode.black_node l1 v1 r1) (rbnode.red_node l2 v2 r2) := is_false(by by_contra; injection a)
instance[mo]: decidable_eq(poly K) := by apply_instance
instance[mo]: inhabited(poly K) := ⟨0⟩
variable [has_repr K]
def see{X Y}[has_repr X][has_repr Y](m:Y)(x:X) := _root_.trace (repr m ++ repr x) x
private def proof[mo](K)[ring K] := list(poly K)
def poly_mem[mo](K)[ring K] := poly K × proof K
def polys[mo](K)[ring K] := list(poly_mem K)
instance hrp[mo]: has_repr(proof K) := by unfold proof; apply_instance--TODO remove after debug
def poly_mem.is0[mo](P: poly_mem K) := is0 P.fst
instance[mo](P: poly_mem K): decidable P.is0 := by unfold poly_mem.is0; apply_instance
instance evvk[mo]: inhabited(poly_mem K) := ⟨(0,[])⟩
--Construct trivial proof by cloning a proof from non-empty list.
def proof_triv[mo](B: polys K. assumption): proof K := B.head.snd.map⟮_↦0⟯
def is_triv[mo]: proof K → bool
| [] := tt
| (p::ps) := is0 p ∧ is_triv ps
def proof_add[mo](p₁ p₂ : proof K)(f: poly K → poly K) := p₁.map₂(+) (p₂.map f)
--Compute the S-polynomial of monic polynomials with membership proof.
def monicS[mo]: poly_mem K → poly_mem K → poly_mem K | (P,pP) (R,pR) :=
let p:= lead_mono P, r:= lead_mono R, m:= p.lcm r, mp:= monom((1:K), m/p), mr:= monom((1:K), m/r)
in (P⬝mp - R⬝mr, proof_add (pP.map(⬝mp)) pR(⬝(-mr)))
--Accumulates proof to show that if P→R then R - P ∈ ⟨B⟩.
meta def simplify_leading_loop[mo](B: polys K): poly_mem K → poly_mem K |(P, proof) :=
let p := P.lead_mono in match B.filter((∣p)∘lead_mono∘fst) with
| [] := (P, proof)
| (b,prf)::_ := let c := -monom(lead_coef P, p / lead_mono b) in simplify_leading_loop(P + b⬝c, proof_add proof prf(⬝c))
end
--B must be a non-empty list of monic (and ≠0) polynomials.
meta def simplify_leading[mo](B: polys K)(P: poly_mem K): poly_mem K :=
match B.filter(is_const ∘ fst) with
| (_,prf)::_ := (0, proof_add P.snd prf(⬝(-P.fst)))
| _ := simplify_leading_loop B P
end
meta def simplify_loop[mo](B): poly K → poly_mem K → poly_mem K | R P :=
if P.is0 then (R, P.snd) else let (P,prf) := simplify_leading B P, p := monom P.lead_term in simplify_loop (R+p) (P-p, prf)
--Return fully simplified R←P and proof that R - P ∈ ⟨B⟩. Input P comes without membership proof, because simplification should be applicable to arbitrary polynomials.
meta def simplify[mo](B: polys K)(P) := simplify_loop B 0 (P, proof_triv)
variable[field K]
--scale_monic 0 := 0
def scale_monic[mo]: poly_mem K → poly_mem K | (P,prf) := let c := P.lead_coef ⁻¹ in (c•P, prf.map((•)c))
meta def simplify_basis_loop[mo](simp: polys K → poly_mem K → poly_mem K): ℕ → polys K → polys K
| 0 B := B
| l [] := sorry -- l ≤ B.length
| l (P::B) := let
P' := simp B P,
B' := ite P'.is0 B (B++[scale_monic P'])
in simplify_basis_loop(if is_triv(P.snd.map₂⟮x y ↦ x-y⟯ P'.snd) then l-1 else B'.length) B'
--For each element of B, if S simplifies the leading term, then simplify additionally with other elements of the basis.
meta def simplify_basis_by[mo](S: poly_mem K)(B: polys K) := simplify_basis_loop⟮B P ↦ let P' := simplify_leading [S] P in if is_triv P'.snd then P else simplify_leading B P'⟯ B.length B
--Interreduce B.
meta def simplify_basis[mo](B: polys K) := simplify_basis_loop(simplify_loop @₁0) B.length B
--main loop
private meta def go[mo]: polys K → list(poly_mem K × poly_mem K) → polys K
| G [] := G
| G ((p₁,p₂)::ps) := let S := scale_monic(simplify_leading G (monicS p₁ p₂))
in if S.is0 then go G ps else let G := simplify_basis_by S G in go (S::G) (ps ++ G.map⟮P↦ (P,S)⟯)
meta def «Gröbner basis of»[mo](B: list(poly K)) := let B := B.filter(not∘is0), B1 := B.imap⟮i b ↦ scale_monic(b, (B.map⟮_↦(0: poly K)⟯).update_nth i 1)⟯ in simplify_basis(go B1 B1.unique_pairs)
notation `Gröbner_basis_of` := «Gröbner basis of»
--Lean's letter recognition is broken! It is not just an implementation mistake, but it is even specified in an adhoc way – see https://leanprover.github.io/reference/lexical_structure.html#identifiers – which is incompatible with Unicode. Not only a huge number of letters is ignored but also some non-letters included (though correctly called just letterlike):
def ℡⅋℀ := "Telephone sign ǝʇ “account” are letters only in Lean!"
--Inclusion of non-letters means that Lean can't be said to support a subset of Unicode. FYI: Unicode is about semantics of code points (numbers). UTF-8 is a character encoding (mapping between bytes and numbers) that Lean does use. Observations here hold at the time of writing and hopefully not in the future.
--Tästä voisi johtaa tyyliin ringa-taktiikan. Sievennetään kaikkia ... hmm, miten sievennyskelpoiset lausekkeet määrätään? Kertoimien tulee olla kunnasta, mutta muuttujia saa käsitellä vain rengasoperaatioilla. Teoreettisesti nättiä olisi yleistää hieman ja ratkaista renkaiden ehdolliset sanaongelmat, mutta sehän edellyttäisi paljon lisää koodausta!
--Sitten pitää ratkaista, miten termien triviaali sievennys kuten x+y-x=y hoidetaan. Koska tulos tiedetään aina, voidaan turvallisesti turvautua ring-taktiikkaan.
--Luetaan tavoitetta kunnes vastaan tulee +,⬝,- (tärkeää on, että löydetään maksimaalinen termi sievennettäväksi—tämän pitäisi riittää siihen, koska tietenkään päälioperaatio ei tällöin voi olla jokin sellainen, jota ei osata käsitellä). Huom. - voi esiintyä sekä unaarisena että binäärisenä. Seuraavaksi tarkistetaan, että alitermi on kuntatyyppiä...mutta tämä rajoittaa käytettävyyttä melko merkittävästi. Teoriassa voisi vaatia, että tyyppi on vaihdannainen rengas ja K-moduli jonkin kunnan K suhteen, ja K:lta vaaditaan lisäksi päätettävä yhtyvyys. Jotta tästä teoriasta tulee käytäntöä, lienee vaadittava, että käyttäjä syöttää kunnan (ℚ voi olla oletusarvo).
meta def 𝔼(pre) := to_expr pre tt ff
--meta def childs(e: expr) := (e.mfoldl⟮c s ↦ [list.cons s c]⟯ []).head
meta def childs: expr → list expr
| (expr.app f p) := [f,p]
| (expr.pi _ _ S T) := [S,T]
| (expr.elet _ t v b) := [t,v,b] --Does this work with infer_type?
| (expr.macro _ cs) := cs
| _ := []
notation `~`x := pure x
notation `ᵘᵖ ` m := monad_lift m
@[reducible] meta def ST := state_t (list expr) tactic
--Run reaction if state is not [].
meta def if_not_found(reaction: ST unit): ST unit := do s ← state_t.get, when(s=[]) reaction
meta def fbs_loop{T}(t)(test: (expr → ST T) → expr → ST T)(atoms: rb_set expr): bool → expr → ST unit | layer's_top e :=
when(¬ atoms.contains e) (test⟮x ↦ t<$
if x≠e then fbs_loop ff x
else (childs x).mmap'(if_not_found ∘ fbs_loop tt)
⟯ e >> if_not_found(when layer's_top (test⟮e' ↦ when(e≠e') (state_t.put[e]) $>t⟯ e $>())))
--Finds some minimal subterm accepted by test while treating terms in the set atoms as such. Parameter t is just an inhabitance proof.
meta def find_bottom_simplifiable{T}(t:T)(test)(atoms): expr → tactic(option expr)
| e := prod.fst <$> (do
fbs_loop t test atoms tt e,
g ← state_t.get,
~ g.nth 0
).run[]
meta def prepare_loop{T}(var: ℕ→T)(test: (expr → ST T) → expr → ST T): expr → ST T | e :=
test⟮x ↦ if x≠e then prepare_loop x else do
vs ← state_t.get,
let i := vs.index_of x,
when(i = vs.length) (state_t.put(vs++[x])) $> var i
⟯e
--Transform (top layer of) e to its T-representation according to test, with alien subterms replaced by variables generated from var with syntactic equality preserved. Second component of the return value is list of the replaced alien terms.
meta def prepare{T}(var: ℕ→T)(test)(e) := (prepare_loop var test e).run[]
--Like mapping the above, but naming of the alien terms is consistent.
meta def prepares{T}(var: ℕ → T)(test)(es: list expr) := (es.mmap(prepare_loop var test)).run[]
meta def simplify_by_loop{T}(var: ℕ→T)(test: (expr → ST T) → expr → ST T)(simp: expr → T → tactic expr): rb_set expr → tactic unit | simplified := do
e ← target,
x ← find_bottom_simplifiable (var 0) test simplified e,
x.maybe(~())⟮x ↦ do
(x',g) ← prepare var test x,
proof ← simp x x',
rewrite_target proof,
`(%%_ = %%s) ← infer_type proof,
simplify_by_loop(simplified.insert s)⟯
--Warning: in practise this interface turned out to behave ugly! This is a simplifier skeleton whose advantage over simp is that pattern matching and rule selection can be done programmatically. (For example simp could often do nothing with a Gröbner basis represented as simplifying equations, because non-syntactic pattern matching is needed to use them.) This functions like simp only [...]. The actual (single step) simplifier is given as a parameter. Its operation is extended by searching the proof target for simplifiable terms and calling the simplifier for all of these bottom up.
--Parameters
--T: an auxiliarity term representation that the simplifier may choose as it likes.
--var: a stream of distinct variables.
--test recursor ∈ expr → a_monad T : transforms a top operation of an input term into T-representation calling recursor for the childs, or for the whole term if it is alien. See test_poly for an example.
--simp: from original expression E and its T-representation produce a proof that E = simplified E. Failed: Variable var i should be represented by a metavariable whose name ends with mk_numeral i, (or var 0, ..., var n may be represented by quantifying ∀x₀...∀xₙ ???).
meta def simplify_by{T}(var: ℕ→T)(test)(simp) := simplify_by_loop var test simp mk_rb_set
--Notes: Examining term structure and mapping it to T was combined into test. Original term is given to simp, because T-representation may be lossy. However if orig. rep. is needed (Gröbner bases avoid it by using ring), examination of it usually repeats. It even turned out that due to consistent alien term naming the second parameter of simp is practically useless. Currently test can work in ST, though safer would be to require polymorphicity over monad transformer on top of tactic. The tedious part (in addition to the core simplification in T) is producing the proof term in simp. Could this be simplified in a suitable monad?
meta def test_instance(i) := (𝔼 i >>= mk_instance) $> tt <|> ~ff
meta def test_poly[mo][reflected K](r: expr → ST(poly K))(e: expr): ST(poly K) := match e with
| `(%%x ⬝ %%y) := (⬝) <$> r x <*> r y
| `(%%x + %%y) := (+) <$> r x <*> r y
| `(%%x - %%y) := ⟮x y ↦ x-y⟯ <$> r x <*> r y
| `(- %%x) := ⟮x ↦ -x⟯ <$> r x
| `(%%x ^ %%n) := do
N ←ᵘᵖ infer_type n,
if N ≠ `(ℕ) ∨ n.has_var ∨ n.has_local then r e
else do n ←ᵘᵖ eval_expr ℕ n, (^n) <$> r x
| e := do
E ←ᵘᵖ infer_type e,
if E ≠ reflect K ∨ e.has_var ∨ e.has_local then r e
else do k ←ᵘᵖ eval_expr K e, ~monom(k,[])
end
meta def test_poly_typed[mo][reflected K](M: option expr)(r: expr → ST(poly K))(e): ST(poly K) := do
E ←ᵘᵖ infer_type e,
ok ← match M with some M := ~(E=M : bool)
| _ :=ᵘᵖ band <$> test_instance``(ring %%E) <*> test_instance``(module %%(reflect K) %%E) end,
(ite ok test_poly id) r e
--X' i is the iᵗʰ variable "Xᵢ".
def X'[mo](i:ℕ): poly K := monom(1, ((list.cons 0)^i) [1])
meta def represent_mono[mo](r: has_reflect K)(M:expr)(vs: list expr)(m: monomial): tactic expr :=
m.ifoldl ⟮x p e ↦ if p=0 then e else do e←e, 𝔼``(%%e ⬝ %%(vs.nth x).iget ^ %%(reflect p))⟯ (𝔼``(1:%%M))
meta def represent_poly[mo][r: has_reflect K](M:expr)(vs: list expr)(P: poly K): tactic expr :=
P.fold ⟮m e ↦ let c:= m.fst in if c=0 then e else do e←e, mono ← represent_mono r M vs m.snd, 𝔼``(%%e + %%(reflect c)⬝%%mono)⟯ (𝔼``(0:%%M))
--TODO Use module product • to multiply mono by c. Problem is that then ring doesn't work!
meta def local_equations_of_type(M) := do
ls ← local_context,
ls.mfilter⟮a ↦ do b ← infer_type a, match b with `(%%x = %%y) := do Y ← infer_type y, ~Y=M | _:=~ff end⟯
--Return a proof of goal found by the given tactic solve.
meta def prove_by(solve: tactic unit)(goal: tactic expr) := do
n ← get_unused_name,
goal >>= assert n,
solve,
proof ← get_local n,
--clear proof, --TODO How to clean the local context while keeping proof usable?
~proof
namespace proof_building_blocks
lemma mul_sub_is_0{M}[ring M][module K M]{x y O : M}(c: M)(o0: O=0)(h: x=y): O - c⬝(x-y) = 0 := by simp[*]
lemma combines{M}[add_comm_group M][module K M]{P R O : M}(pr: P-R = O)(o0: O = 0): P = R := by rw(by simp : P = P-R + R);simp[*]
lemma triv_simp{M}[ring M](IisO: (1:M)=0):∀x:M, x=0 := by intro; rw[← one_mul x, IisO, zero_mul]
end proof_building_blocks
open proof_building_blocks
meta def zero_all(M: expr)(_1_0: expr): expr → tactic unit | e := do
E ← infer_type e,
if E ≠ M then (childs e).mmap' zero_all else
𝔼``(triv_simp %%_1_0 %%e) >>= simp_lemmas.add simp_lemmas.mk >>= simp_target
--Compute a Gröbner basis from non-empty set of polynomial equations E and return a reducer suitable for simplify_by that uses the computed basis.
meta def verifying_reducer[mo][reflected K][r: has_reflect K](M)(E: list expr): tactic(option(expr → poly K → tactic expr)) := do
let test: (expr → ST(poly K)) → _ := test_poly_typed(option.some M),
be ← E.mmap⟮p ↦ do e ← infer_type p, match e with `(%%x = %%y) := 𝔼``(%%x - %%y) | _:=sorry end⟯,
((B: list(poly K)), vs) ← prepares X' test be,
let G := Gröbner_basis_of B,
let reducer: expr → poly K → tactic expr := ⟮pe _ ↦ do
(P, vs) ← (prepare_loop X' test pe).run vs,
let (R, coef) := simplify G P,
--R = P + coef•(fⱼ - gⱼ)ⱼ
--P - R =ʳⁱⁿᵍ= -coef•(fⱼ - gⱼ)ⱼ = “coef•0” = 0 ⟹ P=R
re ← represent_poly M vs R,
ce ← coef.mmap(represent_poly M vs),
K0is0 ← 𝔼``(rfl : (0:%%(reflect K)) = 0),
step2 ← (ce.zip E).mfoldl ⟮prf cb ↦ 𝔼``(@mul_sub_is_0
%%(reflect K) infer_instance infer_instance infer_instance infer_instance
%%M infer_instance infer_instance
_ _ _ %%cb.fst %%prf %%cb.snd)⟯ K0is0,
`(%%ce_be = %%_) ← infer_type step2,
ring_step ← prove_by`[{ring}] (𝔼``(%%pe - %%re = %%ce_be)),
𝔼``(@combines
%%(reflect K) infer_instance infer_instance infer_instance infer_instance
%%M infer_instance infer_instance
_ _ _ %%ring_step %%step2)⟯,
--Note: E is non-empty and hence G is non-empty.
if ¬G.head.fst.is_const then ~ some reducer else do
I ← 𝔼``(1:%%M),
IisO ← reducer I 1,
target >>= zero_all M IisO,
~none
meta def exactℚ := `[exact ℚ]
meta def ringa(K:Type. exactℚ)[reflected K][has_reflect K][field K][decidable_eq K][has_repr K/-debug-/][mo]: tactic unit := do
t ← target,
--"find_top_simplifiable" would be more expected behavior...if it existed
e ← find_bottom_simplifiable (0: poly K) (test_poly_typed none) mk_rb_set t,
if e=none then fail"nothing to simplify in target" else do
M ← infer_type e.iget,
B ← local_equations_of_type M,
if B=[] then `[ring] else do --Do not fail to preserve composability.
reducer ← verifying_reducer M B,
match reducer with none := ~() | some reducer :=
simplify_by (X': ℕ → poly K) (test_poly_typed(some M)) reducer
end,
`[try{ring}]
#check Gröbner_basis_of
--Test cases
instance use_this_order := monomial.deg_lex
variables{v x y z : ℚ}{f: ℚ→ℚ}
--In algebras over ℚ
open polynomial
variables{F: polynomial ℚ}
--example(_: X² - X - 1 = (0: polynomial ℚ))(_: F = 1-X): F² = F+1 := by ringa
example: (F+1)² = F² + 2⬝F + 1 := by ring
example: (F+1)² = F² + (2:ℚ)•F + 1 := by ring
--These delegate to ring tactic
example: (x+y)⬝(x-y) = x² - y² := by ringa
example: f(2⬝x) = f(x+x) := by ringa
--These don't
example(_:v=z): (x+y)⬝(x-y) = x² - y² := by ringa
example(_:v=z): f(2⬝x) = f(x+x) := by ringa
--Core functionality tests
example(_: x+y = z)(_: x² + y² = z²): x⬝y = 0 := by ringa
example(_: x⬝y² = x+y)(_: x²⬝y = x² + y²): y^5 = (2⬝x-1)⬝(2⬝y-1)/2 - 1/2 := by ringa
example(_: x⬝y² = x+y)(_: x²⬝y = x+1): y = x² := by ringa
example(_: z⬝x=y)(_: y=x²)(_: v²=2): x⬝(2⬝z-x-x) = 0 := by ringa
example(_: x²⬝y = x²)(_: x⬝y² = y²): (x+y)⬝(x-y) + x² = x^3 := by ringa
--Iteration tests
example(_: x=y): x⬝f(2⬝x² - y²) - y⬝f(x⬝y) = 0 := by ringa
example(_: x=y): x⬝f(x-y) - y⬝f(x-x) = 0 := by ringa
example(_: x=y+1): (x-1)⬝f(2⬝x-1) - y⬝f(x² - y²) = 0 := by ringa
example(_: x²+y² = z²)(_: x^3 + y^3 = z^3)(_: x⬝y = 1): f(x + y + f(2/3)) = f(f(z²) - 2⬝z) := by ringa
--Inconsistent axioms get special inconsistent treatment.
example(_: x² = -1)(_: y^3 = -1)(_: x⬝y = 2): x-x = 1 := by ringa
example(_: x^4 ⬝ y = 1)(_: x ⬝ y^3 = 2)(_: x² + y² = 0): x-x = 1 := by ringa
example(_: x²+3⬝x+1 = 0)(_: y²+3⬝y+1 = 0)(_: x^5 + y^5 = 0): x-y = 1 := by ringa
end poly
open poly
instance{K:Type}[field K][decidable_eq K][has_repr K][mo]: has_repr(poly_mem K) :=
--by unfold poly_mem; apply_instance
⟨ x ↦ repr x.fst ⟩
instance{K:Type}[field K][decidable_eq K][has_repr K][mo]: has_repr(polys K) := by unfold polys; apply_instance
instance use_this := monomial.deg_lex
def lm(m: list ℕ): poly ℚ := monom(1,m)
def a := lm[2] +3⬝lm[1]+lm[]
def b := lm[0,2] +3⬝lm[0,1]+lm[]
def c := lm[5] + lm[0,5]
#eval simplify(Gröbner_basis_of[a,b]) c
--#eval Gröbner_basis_of[a,b,c] --Time out just because the algorithm is so slow!
def α := lm[0,0,2] + lm[0,2] - lm[2]
def β := lm[0,0,3] + lm[0,3] - lm[3]
def γ := lm[0,1,1] - 1
#eval (Gröbner_basis_of[α,β,γ])
#eval simplify(Gröbner_basis_of[α,β,γ]) (lm[2])
def P := lm[2,1] + ((1:ℚ)/2)•lm[2]
def R := lm[1,2] + lm[0,2]
def S := -lm[2,2] + lm[1,1] + lm[2]
#eval (Gröbner_basis_of[P,R])
#eval simplify (Gröbner_basis_of[P,R]) (S²)
def B := [lm [3] -1, lm[2]+lm[1,1], lm[2]+lm[1,0,1]+lm[0,0,2]]
#eval B
#eval Gröbner_basis_of B
#eval P²⬝R
#eval Gröbner_basis_of$ [lm [3] -1, lm[2]+lm[1,1], lm[2]+lm[1,0,1]+lm[0,0,2], lm [0,3] -1, lm[0,2]+lm[0,1,1]+lm[0,0,2], lm [0,0,3] -1].map(⬝1)
#eval Gröbner_basis_of[lm [3] -1, lm[2]+lm[1,1]+lm[0,2], lm[2]+lm[1,0,1]+lm[0,0,2], lm [0,3] -1, lm[0,2]+lm[0,1,1]+lm[0,0,2], lm [0,0,3] -1]
#eval Gröbner_basis_of(B++[P²⬝R])
--#eval Gröbner_basis_of(P²⬝R::B)
#eval (lm[] + 0).lead_coef
#eval (2⬝lm[4,2] + lm[3,4])².fold((++)∘repr) ""
#eval (2⬝lm[4,2] + lm[3,4])².lead_mono
#eval (P + 3⬝P² + P²)²
|
cb1e6d723f9f74facf2c861881cd1105ef3ca379 | 31f556cdeb9239ffc2fad8f905e33987ff4feab9 | /stage0/src/Lean/Util/Recognizers.lean | 98125141f31da4b8db0bd17b29c50c93cfc49b2f | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | tobiasgrosser/lean4 | ce0fd9cca0feba1100656679bf41f0bffdbabb71 | ebdbdc10436a4d9d6b66acf78aae7a23f5bd073f | refs/heads/master | 1,673,103,412,948 | 1,664,930,501,000 | 1,664,930,501,000 | 186,870,185 | 0 | 0 | Apache-2.0 | 1,665,129,237,000 | 1,557,939,901,000 | Lean | UTF-8 | Lean | false | false | 4,688 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Environment
namespace Lean
namespace Expr
@[inline] def const? (e : Expr) : Option (Name × List Level) :=
match e with
| Expr.const n us => some (n, us)
| _ => none
@[inline] def app1? (e : Expr) (fName : Name) : Option Expr :=
if e.isAppOfArity fName 1 then
some e.appArg!
else
none
@[inline] def app2? (e : Expr) (fName : Name) : Option (Expr × Expr) :=
if e.isAppOfArity fName 2 then
some (e.appFn!.appArg!, e.appArg!)
else
none
@[inline] def app3? (e : Expr) (fName : Name) : Option (Expr × Expr × Expr) :=
if e.isAppOfArity fName 3 then
some (e.appFn!.appFn!.appArg!, e.appFn!.appArg!, e.appArg!)
else
none
@[inline] def app4? (e : Expr) (fName : Name) : Option (Expr × Expr × Expr × Expr) :=
if e.isAppOfArity fName 4 then
some (e.appFn!.appFn!.appFn!.appArg!, e.appFn!.appFn!.appArg!, e.appFn!.appArg!, e.appArg!)
else
none
@[inline] def eq? (p : Expr) : Option (Expr × Expr × Expr) :=
p.app3? ``Eq
@[inline] def ne? (p : Expr) : Option (Expr × Expr × Expr) :=
p.app3? ``Ne
@[inline] def iff? (p : Expr) : Option (Expr × Expr) :=
p.app2? ``Iff
@[inline] def not? (p : Expr) : Option Expr :=
p.app1? ``Not
@[inline] def notNot? (p : Expr) : Option Expr :=
match p.not? with
| some p => p.not?
| none => none
@[inline] def and? (p : Expr) : Option (Expr × Expr) :=
p.app2? ``And
@[inline] def heq? (p : Expr) : Option (Expr × Expr × Expr × Expr) :=
p.app4? ``HEq
def natAdd? (e : Expr) : Option (Expr × Expr) :=
e.app2? ``Nat.add
@[inline] def arrow? : Expr → Option (Expr × Expr)
| Expr.forallE _ α β _ => if β.hasLooseBVars then none else some (α, β)
| _ => none
def isEq (e : Expr) :=
e.isAppOfArity ``Eq 3
def isHEq (e : Expr) :=
e.isAppOfArity ``HEq 4
def isIte (e : Expr) :=
e.isAppOfArity ``ite 5
def isDIte (e : Expr) :=
e.isAppOfArity ``dite 5
partial def listLit? (e : Expr) : Option (Expr × List Expr) :=
let rec loop (e : Expr) (acc : List Expr) :=
if e.isAppOfArity' ``List.nil 1 then
some (e.appArg!', acc.reverse)
else if e.isAppOfArity' ``List.cons 3 then
loop e.appArg!' (e.appFn!'.appArg!' :: acc)
else
none
loop e []
def arrayLit? (e : Expr) : Option (Expr × List Expr) :=
if e.isAppOfArity' ``List.toArray 2 then
listLit? e.appArg!'
else
none
/-- Recognize `α × β` -/
def prod? (e : Expr) : Option (Expr × Expr) :=
e.app2? ``Prod
private def getConstructorVal? (env : Environment) (ctorName : Name) : Option ConstructorVal :=
match env.find? ctorName with
| some (ConstantInfo.ctorInfo v) => v
| _ => none
def isConstructorApp? (env : Environment) (e : Expr) : Option ConstructorVal :=
match e with
| Expr.lit (Literal.natVal n) => if n == 0 then getConstructorVal? env `Nat.zero else getConstructorVal? env `Nat.succ
| _ =>
match e.getAppFn with
| Expr.const n _ => match getConstructorVal? env n with
| some v => if v.numParams + v.numFields == e.getAppNumArgs then some v else none
| none => none
| _ => none
def isConstructorApp (env : Environment) (e : Expr) : Bool :=
e.isConstructorApp? env |>.isSome
/--
If `e` is a constructor application, return a pair containing the corresponding `ConstructorVal` and the constructor
application arguments.
This function treats numerals as constructors. For example, if `e` is the numeral `2`, the result pair
is `ConstructorVal` for `Nat.succ`, and the array `#[1]`. The parameter `useRaw` controls how the resulting
numeral is represented. If `useRaw := false`, then `mkNatLit` is used, otherwise `mkRawNatLit`.
Recall that `mkNatLit` uses the `OfNat.ofNat` application which is the canonical way of representing numerals
in the elaborator and tactic framework. We `useRaw := false` in the compiler (aka code generator).
-/
def constructorApp? (env : Environment) (e : Expr) (useRaw := false) : Option (ConstructorVal × Array Expr) := do
match e with
| Expr.lit (Literal.natVal n) =>
if n == 0 then do
let v ← getConstructorVal? env `Nat.zero
pure (v, #[])
else do
let v ← getConstructorVal? env `Nat.succ
pure (v, #[if useRaw then mkRawNatLit (n-1) else mkNatLit (n-1)])
| _ =>
match e.getAppFn with
| Expr.const n _ => do
let v ← getConstructorVal? env n
if v.numParams + v.numFields == e.getAppNumArgs then
pure (v, e.getAppArgs)
else
none
| _ => none
end Lean.Expr
|
d3fc82c3dcebd074dc96ea27c67b44135ba9d82e | 82e44445c70db0f03e30d7be725775f122d72f3e | /archive/100-theorems-list/70_perfect_numbers.lean | 131612c411dcc524e5f1e3621084aed818023b1a | [
"Apache-2.0"
] | permissive | stjordanis/mathlib | 51e286d19140e3788ef2c470bc7b953e4991f0c9 | 2568d41bca08f5d6bf39d915434c8447e21f42ee | refs/heads/master | 1,631,748,053,501 | 1,627,938,886,000 | 1,627,938,886,000 | 228,728,358 | 0 | 0 | Apache-2.0 | 1,576,630,588,000 | 1,576,630,587,000 | null | UTF-8 | Lean | false | false | 4,955 | lean | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import number_theory.arithmetic_function
import number_theory.lucas_lehmer
import algebra.geom_sum
import ring_theory.multiplicity
/-!
# Perfect Numbers
This file proves Theorem 70 from the [100 Theorems List](https://www.cs.ru.nl/~freek/100/).
The theorem characterizes even perfect numbers.
Euclid proved that if `2 ^ (k + 1) - 1` is prime (these primes are known as Mersenne primes),
then `2 ^ k * 2 ^ (k + 1) - 1` is perfect.
Euler proved the converse, that if `n` is even and perfect, then there exists `k` such that
`n = 2 ^ k * 2 ^ (k + 1) - 1` and `2 ^ (k + 1) - 1` is prime.
## References
https://en.wikipedia.org/wiki/Euclid%E2%80%93Euler_theorem
-/
lemma odd_mersenne_succ (k : ℕ) : ¬ 2 ∣ mersenne (k + 1) :=
by simp [← even_iff_two_dvd, ← nat.even_succ, nat.succ_eq_add_one] with parity_simps
namespace nat
open arithmetic_function finset
open_locale arithmetic_function
lemma sigma_two_pow_eq_mersenne_succ (k : ℕ) : σ 1 (2 ^ k) = mersenne (k + 1) :=
by simpa [mersenne, prime_two, ← geom_sum_mul_add 1 (k+1)]
/-- Euclid's theorem that Mersenne primes induce perfect numbers -/
theorem perfect_two_pow_mul_mersenne_of_prime (k : ℕ) (pr : (mersenne (k + 1)).prime) :
perfect ((2 ^ k) * mersenne (k + 1)) :=
begin
rw [perfect_iff_sum_divisors_eq_two_mul, ← mul_assoc, ← pow_succ, ← sigma_one_apply, mul_comm,
is_multiplicative_sigma.map_mul_of_coprime
(nat.prime_two.coprime_pow_of_not_dvd (odd_mersenne_succ _)),
sigma_two_pow_eq_mersenne_succ],
{ simp [pr, nat.prime_two] },
{ apply mul_pos (pow_pos _ k) (mersenne_pos (nat.succ_pos k)),
norm_num }
end
lemma ne_zero_of_prime_mersenne (k : ℕ) (pr : (mersenne (k + 1)).prime) :
k ≠ 0 :=
begin
rintro rfl,
simpa [mersenne, not_prime_one] using pr,
end
theorem even_two_pow_mul_mersenne_of_prime (k : ℕ) (pr : (mersenne (k + 1)).prime) :
even ((2 ^ k) * mersenne (k + 1)) :=
by simp [ne_zero_of_prime_mersenne k pr] with parity_simps
lemma eq_two_pow_mul_odd {n : ℕ} (hpos : 0 < n) :
∃ (k m : ℕ), n = 2 ^ k * m ∧ ¬ even m :=
begin
have h := (multiplicity.finite_nat_iff.2 ⟨nat.prime_two.ne_one, hpos⟩),
cases multiplicity.pow_multiplicity_dvd h with m hm,
use [(multiplicity 2 n).get h, m],
refine ⟨hm, _⟩,
rw even_iff_two_dvd,
have hg := multiplicity.is_greatest' h (nat.lt_succ_self _),
contrapose! hg,
rcases hg with ⟨k, rfl⟩,
apply dvd.intro k,
rw [pow_succ', mul_assoc, ← hm],
end
/-- **Perfect Number Theorem**: Euler's theorem that even perfect numbers can be factored as a
power of two times a Mersenne prime. -/
theorem eq_two_pow_mul_prime_mersenne_of_even_perfect {n : ℕ} (ev : even n) (perf : perfect n) :
∃ (k : ℕ), prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1) :=
begin
have hpos := perf.2,
rcases eq_two_pow_mul_odd hpos with ⟨k, m, rfl, hm⟩,
use k,
rw [perfect_iff_sum_divisors_eq_two_mul hpos, ← sigma_one_apply,
is_multiplicative_sigma.map_mul_of_coprime (nat.prime_two.coprime_pow_of_not_dvd hm).symm,
sigma_two_pow_eq_mersenne_succ, ← mul_assoc, ← pow_succ] at perf,
rcases nat.coprime.dvd_of_dvd_mul_left
(nat.prime_two.coprime_pow_of_not_dvd (odd_mersenne_succ _)) (dvd.intro _ perf) with ⟨j, rfl⟩,
rw [← mul_assoc, mul_comm _ (mersenne _), mul_assoc] at perf,
have h := mul_left_cancel' (ne_of_gt (mersenne_pos (nat.succ_pos _))) perf,
rw [sigma_one_apply, sum_divisors_eq_sum_proper_divisors_add_self, ← succ_mersenne, add_mul,
one_mul, add_comm] at h,
have hj := add_left_cancel h,
cases sum_proper_divisors_dvd (by { rw hj, apply dvd.intro_left (mersenne (k + 1)) rfl }),
{ have j1 : j = 1 := eq.trans hj.symm h_1,
rw [j1, mul_one, sum_proper_divisors_eq_one_iff_prime] at h_1,
simp [h_1, j1] },
{ have jcon := eq.trans hj.symm h_1,
rw [← one_mul j, ← mul_assoc, mul_one] at jcon,
have jcon2 := mul_right_cancel' _ jcon,
{ exfalso,
cases k,
{ apply hm,
rw [← jcon2, pow_zero, one_mul, one_mul] at ev,
rw [← jcon2, one_mul],
exact ev },
apply ne_of_lt _ jcon2,
rw [mersenne, ← nat.pred_eq_sub_one, lt_pred_iff, ← pow_one (nat.succ 1)],
apply pow_lt_pow (nat.lt_succ_self 1) (nat.succ_lt_succ (nat.succ_pos k)) },
contrapose! hm,
simp [hm] }
end
/-- The Euclid-Euler theorem characterizing even perfect numbers -/
theorem even_and_perfect_iff {n : ℕ} :
(even n ∧ perfect n) ↔ ∃ (k : ℕ), prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1) :=
begin
split,
{ rintro ⟨ev, perf⟩,
exact eq_two_pow_mul_prime_mersenne_of_even_perfect ev perf },
{ rintro ⟨k, pr, rfl⟩,
exact ⟨even_two_pow_mul_mersenne_of_prime k pr, perfect_two_pow_mul_mersenne_of_prime k pr⟩ }
end
end nat
|
941d25fb466bdcf12feb4d62eab3941e1f836624 | c777c32c8e484e195053731103c5e52af26a25d1 | /src/analysis/convex/slope.lean | e6ddba269e7dcfdda1ac91003c6de5bf703439d3 | [
"Apache-2.0"
] | permissive | kbuzzard/mathlib | 2ff9e85dfe2a46f4b291927f983afec17e946eb8 | 58537299e922f9c77df76cb613910914a479c1f7 | refs/heads/master | 1,685,313,702,744 | 1,683,974,212,000 | 1,683,974,212,000 | 128,185,277 | 1 | 0 | null | 1,522,920,600,000 | 1,522,920,600,000 | null | UTF-8 | Lean | false | false | 11,468 | lean | /-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import analysis.convex.function
import tactic.field_simp
import tactic.linarith
/-!
# Slopes of convex functions
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file relates convexity/concavity of functions in a linearly ordered field and the monotonicity
of their slopes.
The main use is to show convexity/concavity from monotonicity of the derivative.
-/
variables {𝕜 : Type*} [linear_ordered_field 𝕜] {s : set 𝕜} {f : 𝕜 → 𝕜}
/-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
lemma convex_on.slope_mono_adjacent (hf : convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f y - f x) / (y - x) ≤ (f z - f y) / (z - y) :=
begin
have hxz := hxy.trans hyz,
rw ←sub_pos at hxy hxz hyz,
suffices : f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y),
{ ring_nf at this ⊢, linarith },
set a := (z - y) / (z - x),
set b := (y - x) / (z - x),
have hy : a • x + b • z = y, by { field_simp, rw div_eq_iff; [ring, linarith] },
have key, from
hf.2 hx hz
(show 0 ≤ a, by apply div_nonneg; linarith)
(show 0 ≤ b, by apply div_nonneg; linarith)
(show a + b = 1, by { field_simp, rw div_eq_iff; [ring, linarith] }),
rw hy at key,
replace key := mul_le_mul_of_nonneg_left key hxz.le,
field_simp [hxy.ne', hyz.ne', hxz.ne', mul_comm (z - x) _] at key ⊢,
rw div_le_div_right,
{ linarith },
{ nlinarith }
end
/-- If `f : 𝕜 → 𝕜` is concave, then for any three points `x < y < z` the slope of the secant line of
`f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/
lemma concave_on.slope_anti_adjacent (hf : concave_on 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
(hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f z - f y) / (z - y) ≤ (f y - f x) / (y - x) :=
begin
rw [←neg_le_neg_iff, ←neg_sub_neg (f x), ←neg_sub_neg (f y)],
simp_rw [←pi.neg_apply, ←neg_div, neg_sub],
exact convex_on.slope_mono_adjacent hf.neg hx hz hxy hyz,
end
/-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
lemma strict_convex_on.slope_strict_mono_adjacent (hf : strict_convex_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f y - f x) / (y - x) < (f z - f y) / (z - y) :=
begin
have hxz := hxy.trans hyz,
have hxz' := hxz.ne,
rw ←sub_pos at hxy hxz hyz,
suffices : f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y),
{ ring_nf at this ⊢, linarith },
set a := (z - y) / (z - x),
set b := (y - x) / (z - x),
have hy : a • x + b • z = y, by { field_simp, rw div_eq_iff; [ring, linarith] },
have key, from
hf.2 hx hz hxz' (div_pos hyz hxz) (div_pos hxy hxz)
(show a + b = 1, by { field_simp, rw div_eq_iff; [ring, linarith] }),
rw hy at key,
replace key := mul_lt_mul_of_pos_left key hxz,
field_simp [hxy.ne', hyz.ne', hxz.ne', mul_comm (z - x) _] at key ⊢,
rw div_lt_div_right,
{ linarith },
{ nlinarith }
end
/-- If `f : 𝕜 → 𝕜` is strictly concave, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly greater than the slope of the secant line of `f` on
`[x, z]`. -/
lemma strict_concave_on.slope_anti_adjacent (hf : strict_concave_on 𝕜 s f)
{x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
(f z - f y) / (z - y) < (f y - f x) / (y - x) :=
begin
rw [←neg_lt_neg_iff, ←neg_sub_neg (f x), ←neg_sub_neg (f y)],
simp_rw [←pi.neg_apply, ←neg_div, neg_sub],
exact strict_convex_on.slope_strict_mono_adjacent hf.neg hx hz hxy hyz,
end
/-- If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is
less than the slope of the secant line of `f` on `[x, z]`, then `f` is convex. -/
lemma convex_on_of_slope_mono_adjacent (hs : convex 𝕜 s)
(hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
(f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) :
convex_on 𝕜 s f :=
linear_order.convex_on_of_lt hs $ λ x hx z hz hxz a b ha hb hab,
begin
let y := a * x + b * z,
have hxy : x < y,
{ rw [← one_mul x, ← hab, add_mul],
exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _ },
have hyz : y < z,
{ rw [← one_mul z, ← hab, add_mul],
exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _ },
have : (f y - f x) * (z - y) ≤ (f z - f y) * (y - x),
from (div_le_div_iff (sub_pos.2 hxy) (sub_pos.2 hyz)).1 (hf hx hz hxy hyz),
have hxz : 0 < z - x, from sub_pos.2 (hxy.trans hyz),
have ha : (z - y) / (z - x) = a,
{ rw [eq_comm, ← sub_eq_iff_eq_add'] at hab,
simp_rw [div_eq_iff hxz.ne', y, ←hab], ring },
have hb : (y - x) / (z - x) = b,
{ rw [eq_comm, ← sub_eq_iff_eq_add] at hab,
simp_rw [div_eq_iff hxz.ne', y, ←hab], ring },
rwa [sub_mul, sub_mul, sub_le_iff_le_add', ← add_sub_assoc, le_sub_iff_add_le, ← mul_add,
sub_add_sub_cancel, ← le_div_iff hxz, add_div, mul_div_assoc, mul_div_assoc, mul_comm (f x),
mul_comm (f z), ha, hb] at this,
end
/-- If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is
greater than the slope of the secant line of `f` on `[x, z]`, then `f` is concave. -/
lemma concave_on_of_slope_anti_adjacent (hs : convex 𝕜 s)
(hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
(f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) : concave_on 𝕜 s f :=
begin
rw ←neg_convex_on_iff,
refine convex_on_of_slope_mono_adjacent hs (λ x y z hx hz hxy hyz, _),
rw ←neg_le_neg_iff,
simp_rw [←neg_div, neg_sub, pi.neg_apply, neg_sub_neg],
exact hf hx hz hxy hyz,
end
/-- If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is
strictly less than the slope of the secant line of `f` on `[x, z]`, then `f` is strictly convex. -/
lemma strict_convex_on_of_slope_strict_mono_adjacent (hs : convex 𝕜 s)
(hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
(f y - f x) / (y - x) < (f z - f y) / (z - y)) :
strict_convex_on 𝕜 s f :=
linear_order.strict_convex_on_of_lt hs $ λ x hx z hz hxz a b ha hb hab,
begin
let y := a * x + b * z,
have hxy : x < y,
{ rw [← one_mul x, ← hab, add_mul],
exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _ },
have hyz : y < z,
{ rw [← one_mul z, ← hab, add_mul],
exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _ },
have : (f y - f x) * (z - y) < (f z - f y) * (y - x),
from (div_lt_div_iff (sub_pos.2 hxy) (sub_pos.2 hyz)).1 (hf hx hz hxy hyz),
have hxz : 0 < z - x, from sub_pos.2 (hxy.trans hyz),
have ha : (z - y) / (z - x) = a,
{ rw [eq_comm, ← sub_eq_iff_eq_add'] at hab,
simp_rw [div_eq_iff hxz.ne', y, ←hab], ring },
have hb : (y - x) / (z - x) = b,
{ rw [eq_comm, ← sub_eq_iff_eq_add] at hab,
simp_rw [div_eq_iff hxz.ne', y, ←hab], ring },
rwa [sub_mul, sub_mul, sub_lt_iff_lt_add', ← add_sub_assoc, lt_sub_iff_add_lt, ← mul_add,
sub_add_sub_cancel, ← lt_div_iff hxz, add_div, mul_div_assoc, mul_div_assoc, mul_comm (f x),
mul_comm (f z), ha, hb] at this,
end
/-- If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is
strictly greater than the slope of the secant line of `f` on `[x, z]`, then `f` is strictly concave.
-/
lemma strict_concave_on_of_slope_strict_anti_adjacent (hs : convex 𝕜 s)
(hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
(f z - f y) / (z - y) < (f y - f x) / (y - x)) : strict_concave_on 𝕜 s f :=
begin
rw ←neg_strict_convex_on_iff,
refine strict_convex_on_of_slope_strict_mono_adjacent hs (λ x y z hx hz hxy hyz, _),
rw ←neg_lt_neg_iff,
simp_rw [←neg_div, neg_sub, pi.neg_apply, neg_sub_neg],
exact hf hx hz hxy hyz,
end
/-- A function `f : 𝕜 → 𝕜` is convex iff for any three points `x < y < z` the slope of the secant
line of `f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/
lemma convex_on_iff_slope_mono_adjacent :
convex_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z →
(f y - f x) / (y - x) ≤ (f z - f y) / (z - y) :=
⟨λ h, ⟨h.1, λ x y z, h.slope_mono_adjacent⟩, λ h, convex_on_of_slope_mono_adjacent h.1 h.2⟩
/-- A function `f : 𝕜 → 𝕜` is concave iff for any three points `x < y < z` the slope of the secant
line of `f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/
lemma concave_on_iff_slope_anti_adjacent :
concave_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z →
(f z - f y) / (z - y) ≤ (f y - f x) / (y - x) :=
⟨λ h, ⟨h.1, λ x y z, h.slope_anti_adjacent⟩, λ h, concave_on_of_slope_anti_adjacent h.1 h.2⟩
/-- A function `f : 𝕜 → 𝕜` is strictly convex iff for any three points `x < y < z` the slope of
the secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
`[x, z]`. -/
lemma strict_convex_on_iff_slope_strict_mono_adjacent :
strict_convex_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z →
(f y - f x) / (y - x) < (f z - f y) / (z - y) :=
⟨λ h, ⟨h.1, λ x y z, h.slope_strict_mono_adjacent⟩,
λ h, strict_convex_on_of_slope_strict_mono_adjacent h.1 h.2⟩
/-- A function `f : 𝕜 → 𝕜` is strictly concave iff for any three points `x < y < z` the slope of
the secant line of `f` on `[x, y]` is strictly greater than the slope of the secant line of `f` on
`[x, z]`. -/
lemma strict_concave_on_iff_slope_strict_anti_adjacent :
strict_concave_on 𝕜 s f ↔ convex 𝕜 s ∧
∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z →
(f z - f y) / (z - y) < (f y - f x) / (y - x) :=
⟨λ h, ⟨h.1, λ x y z, h.slope_anti_adjacent⟩,
λ h, strict_concave_on_of_slope_strict_anti_adjacent h.1 h.2⟩
/-- If `f` is convex on a set `s` in a linearly ordered field, and `f x < f y` for two points
`x < y` in `s`, then `f` is strictly monotone on `s ∩ [y, ∞)`. -/
lemma convex_on.strict_mono_of_lt (hf : convex_on 𝕜 s f)
{x y : 𝕜} (hx : x ∈ s) (hxy : x < y) (hxy' : f x < f y) :
strict_mono_on f (s ∩ set.Ici y) :=
begin
intros u hu v hv huv,
have step1 : ∀ {z : 𝕜}, z ∈ s ∩ set.Ioi y → f y < f z,
{ refine λ z hz, hf.lt_right_of_left_lt hx hz.1 _ hxy',
rw open_segment_eq_Ioo (hxy.trans hz.2),
exact ⟨hxy, hz.2⟩ },
rcases eq_or_lt_of_le hu.2 with rfl | hu2,
{ exact step1 ⟨hv.1, huv⟩ },
{ refine hf.lt_right_of_left_lt _ hv.1 _ (step1 ⟨hu.1, hu2⟩),
{ apply hf.1.segment_subset hx hu.1,
rw segment_eq_Icc (hxy.le.trans hu.2),
exact ⟨hxy.le, hu.2⟩ },
{ rw open_segment_eq_Ioo (hu2.trans huv),
exact ⟨hu2, huv⟩ } },
end
|
e0f8e5d065c15dd7f37dbbaf96df127b3cd18a28 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/category_theory/limits/shapes/constructions/over/connected.lean | f4fdbb9edf374a5496dd1d2b1fecae207fd1ecd6 | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 2,780 | lean | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Reid Barton, Bhavik Mehta
-/
import category_theory.over
import category_theory.limits.connected
import category_theory.limits.creates
/-!
# `forget : over B ⥤ C` creates connected limits.
-/
universes v u -- declare the `v`'s first; see `category_theory.category` for an explanation
noncomputable theory
open category_theory category_theory.limits
variables {J : Type v} [small_category J]
variables {C : Type u} [category.{v} C]
variable {X : C}
namespace category_theory.over
namespace creates_connected
/--
(Impl) Given a diagram in the over category, produce a natural transformation from the
diagram legs to the specific object.
-/
def nat_trans_in_over {B : C} (F : J ⥤ over B) :
F ⋙ forget B ⟶ (category_theory.functor.const J).obj B :=
{ app := λ j, (F.obj j).hom }
local attribute [tidy] tactic.case_bash
/--
(Impl) Given a cone in the base category, raise it to a cone in the over category. Note this is
where the connected assumption is used.
-/
@[simps]
def raise_cone [is_connected J] {B : C} {F : J ⥤ over B} (c : cone (F ⋙ forget B)) :
cone F :=
{ X := over.mk (c.π.app (classical.arbitrary J) ≫ (F.obj (classical.arbitrary J)).hom),
π :=
{ app := λ j,
over.hom_mk (c.π.app j) (nat_trans_from_is_connected (c.π ≫ nat_trans_in_over F) j _) } }
lemma raised_cone_lowers_to_original [is_connected J] {B : C} {F : J ⥤ over B}
(c : cone (F ⋙ forget B)) (t : is_limit c) :
(forget B).map_cone (raise_cone c) = c :=
by tidy
/-- (Impl) Show that the raised cone is a limit. -/
def raised_cone_is_limit [is_connected J] {B : C} {F : J ⥤ over B} {c : cone (F ⋙ forget B)} (t : is_limit c) :
is_limit (raise_cone c) :=
{ lift := λ s, over.hom_mk (t.lift ((forget B).map_cone s)) (by { dsimp, simp }),
uniq' := λ s m K, by { ext1, apply t.hom_ext, intro j, simp [← K j] } }
end creates_connected
/-- The forgetful functor from the over category creates any connected limit. -/
instance forget_creates_connected_limits
[is_connected J] {B : C} : creates_limits_of_shape J (forget B) :=
{ creates_limit := λ K,
creates_limit_of_reflects_iso (λ c t,
{ lifted_cone := creates_connected.raise_cone c,
valid_lift := eq_to_iso (creates_connected.raised_cone_lowers_to_original c t),
makes_limit := creates_connected.raised_cone_is_limit t } ) }
/-- The over category has any connected limit which the original category has. -/
instance has_connected_limits
{B : C} [is_connected J] [has_limits_of_shape J C] : has_limits_of_shape J (over B) :=
{ has_limit := λ F, has_limit_of_created F (forget B) }
end category_theory.over
|
757c73f40341581a981d3ec6f5fad53288862d32 | 843cffbd2a25fd01677509247a92e9e8b33bec48 | /xnat.lean | d493ec47e1f04ca1c4f7d3795095d58e055e633c | [] | no_license | AlexandruBosinta/MyLeanPlayground | 9a78eba4a5c7a2b82edf84e1794a966f53f06c4f | 5dc50a590d784bfc27e7fb37b6361a6dcc1b2790 | refs/heads/master | 1,586,230,010,351 | 1,542,487,422,000 | 1,542,487,422,000 | 158,016,626 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 394 | lean | namespace xena
inductive xnat
| zero : xnat
| succ : xnat → xnat
open xnat
definition lt : xnat → xnat → Prop
| zero zero := false
| (succ m) zero := false
| zero (succ p) := true
| (succ m) (succ p) := lt m p
notation a < b := lt a b
#print int
theorem inequality_A2 (a b c : xnat) : a < b → b < c → a < c := λ hab hbc, sorry
end xena
#print prod |
10e780de647771b8a5a69fc85cd00f49d45fab5d | 5883d9218e6f144e20eee6ca1dab8529fa1a97c0 | /src/exp/map.lean | c28b9f1dad7c811fb8df1f32c716d8ac40596fe8 | [] | no_license | spl/alpha-conversion-is-easy | 0d035bc570e52a6345d4890e4d0c9e3f9b8126c1 | ed937fe85d8495daffd9412a5524c77b9fcda094 | refs/heads/master | 1,607,649,280,020 | 1,517,380,240,000 | 1,517,380,240,000 | 52,174,747 | 4 | 0 | null | 1,456,052,226,000 | 1,456,001,163,000 | Lean | UTF-8 | Lean | false | false | 2,872 | lean | /-
This files contains a collection of core definitions and properties for `exp`.
-/
import .type
namespace acie -----------------------------------------------------------------
namespace exp ------------------------------------------------------------------
variables {V : Type} [decidable_eq V] -- Type of variable names
variables {a b : V} -- Variable names
variables {vs : Type → Type} [vset vs V] -- Type of variable name sets
variables {X Y Z : vs V} -- Variable name sets
variables {p : X ⊆ Y} {q : Y ⊆ Z} -- Subset properties
-- Given proof `p : X ⊆ Y`, `map p e` maps the free variables from `X` to `Y` in
-- `e : exp X`.
def map : ∀ {X Y : vs V}, X ⊆ Y → exp X → exp Y
| X Y p (var x) := var (vname.map_of_subset p x)
| X Y p (app f e) := app (map p f) (map p e)
| X Y p (lam' a e) := lam (map (vset.prop_insert_of_subset a p) e)
namespace map ------------------------------------------------------------------
-- Identity of `map`.
theorem id : ∀ {X : vs V} (e : exp X), e = map (vset.prop_subset_refl X) e
| X (var x) := by cases x; reflexivity
| X (app f e) := by rw map; conv {to_lhs, rw [id f, id e]}
| X (lam e) := by rw map; conv {to_lhs, rw [id e]}; reflexivity
-- Composition of `map`.
theorem comp : ∀ {X Y Z : vs V} {p : X ⊆ Y} {q : Y ⊆ Z} (e : exp X),
map q (map p e) = map (vset.prop_subset_trans p q) e
| X Y Z p q (var x) := by repeat {rw [map]}
| X Y Z p q (app f e) := by repeat {rw [map]}; rw [comp f, comp e]
| X Y Z p q (lam e) := by repeat {rw [map]}; rw [comp e]; reflexivity
end /- namespace -/ map --------------------------------------------------------
-- A weakening property that allows increasing the free variable set without
-- changing the structure of an expression.
def insert_var (a : V) : exp X → exp (insert a X) :=
map (vset.prop_subset_insert_self a X)
namespace insert_var -----------------------------------------------------------
/- Equational lemmas -/
protected
theorem var {x : ν∈ X} : insert_var a (var x) = var (vname.insert a x) :=
rfl
protected
theorem app {f e : exp X}
: insert_var a (app f e) = app (insert_var a f) (insert_var a e) :=
rfl
protected
theorem lam {e : exp (insert a X)}
: insert_var b (lam e) = lam (map (vset.prop_insert_of_subset a (vset.prop_subset_insert_self b X)) e) :=
rfl
theorem lam_comp (e : exp (insert a X))
: insert_var b (lam e) = lam (map (vset.prop_subset_of_insert_comm b a X) (insert_var b e)) :=
by unfold insert_var; rw [map.comp e]; reflexivity
theorem lam_self (e : exp (insert a X))
: insert_var a (lam e) = lam (insert_var a e) :=
rfl
end /- namespace -/ insert_var -------------------------------------------------
end /- namespace -/ exp --------------------------------------------------------
end /- namespace -/ acie -------------------------------------------------------
|
17bd6a41f26bc9b86674b165dd23e017bf0434ad | 83c8119e3298c0bfc53fc195c41a6afb63d01513 | /library/init/meta/type_context.lean | cad0bcfeec4fcee5a064f1b3917ce84447d831c1 | [
"Apache-2.0"
] | permissive | anfelor/lean | 584b91c4e87a6d95f7630c2a93fb082a87319ed0 | 31cfc2b6bf7d674f3d0f73848b842c9c9869c9f1 | refs/heads/master | 1,610,067,141,310 | 1,585,992,232,000 | 1,585,992,232,000 | 251,683,543 | 0 | 0 | Apache-2.0 | 1,585,676,570,000 | 1,585,676,569,000 | null | UTF-8 | Lean | false | false | 6,606 | lean | prelude
import init.category init.meta.local_context init.meta.tactic init.meta.fun_info
namespace tactic.unsafe
/-- A monad that exposes the functionality of the C++ class `type_context_old`.
The idea is that the methods to `type_context` are more powerful but _unsafe_ in the
sense that you can create terms that do not typecheck or that are infinitely descending.
Under the hood, `type_context` is implemented as a reader monad, with a mutable `type_context` object.
-/
meta constant type_context : Type → Type
namespace type_context
variables {α β : Type}
protected meta constant bind : type_context α → (α → type_context β) → type_context β
protected meta constant pure : α → type_context α
protected meta constant fail : format → type_context α
protected meta def failure : type_context α := type_context.fail ""
meta instance : monad type_context := {bind := @type_context.bind, pure := @type_context.pure}
meta instance : monad_fail type_context := {fail := λ α, type_context.fail ∘ to_fmt}
meta constant get_env : type_context environment
meta constant whnf : expr → type_context expr
meta constant is_def_eq (e₁ e₂ : expr) (approx := ff) : type_context bool
meta constant unify (e₁ e₂ : expr) (approx := ff) : type_context bool
/-- Infer the type of the given expr. Inferring the type does not mean that it typechecks.
Will fail if type can't be inferred. -/
meta constant infer : expr → type_context expr
/-- A stuck expression `e` is an expression that _would_ reduce,
except that a metavariable is present that prevents the reduction.
Returns the metavariable which is causing the stuckage.
For example, `@has_add.add nat ?m a b` can't project because `?m` is not given. -/
meta constant is_stuck : expr → type_context (option expr)
/-- Add a local to the tc local context. -/
meta constant push_local (pp_name : name) (type : expr) (bi := binder_info.default) : type_context expr
meta constant pop_local : type_context unit
/-- Get the local context of the type_context. -/
meta constant get_local_context : type_context local_context
/-- Create and declare a new metavariable. If the local context is not given then it will use the current local context. -/
meta constant mk_mvar (pp_name : name) (type : expr) (context : option local_context := none) : type_context expr
/-- Iterate over all mvars in the mvar context. -/
meta constant fold_mvars {α : Type} (f : α → expr → type_context α) : α → type_context α
meta def list_mvars : type_context (list expr) := fold_mvars (λ l x, pure $ x :: l) []
/-- Set the mvar to the following assignments.
Works for temporary metas too.
[WARNING] `assign` does not perform certain checks:
- No typecheck is done before assignment.
- If the metavariable is already assigned this will clobber the assignment.
- It will not stop you from assigning an metavariable to itself or creating cycles of metavariable assignments.
These will manifest as 'deep recursion' exceptions when `instantiate_mvars` is used.
- It is not checked whether the assignment uses local constants outside the declaration's context.
You can avoid the unsafety by using `unify` instead.
-/
meta constant assign (mvar : expr) (assignment : expr) : type_context unit
/-- Assigns a given level metavariable. -/
meta constant level.assign (mvar : level) (assignment : level) : type_context unit
/-- Returns true if the given expression is a declared local constant or a declared metavariable. -/
meta constant is_declared (e : expr) : type_context bool
meta constant is_assigned (mvar : expr) : type_context bool
/-- Given a metavariable, returns the local context that the metavariable was declared with. -/
meta constant get_context (mvar : expr) : type_context local_context
/-- Get the expression that is assigned to the given mvar expression. Fails if given a -/
meta constant get_assignment (mvar : expr) : type_context expr
meta constant instantiate_mvars : expr → type_context expr
meta constant level.instantiate_mvars : level → type_context level
meta constant is_tmp_mvar (mvar : expr) : type_context bool
meta constant is_regular_mvar (mvar : expr) : type_context bool
/-- Run the given `type_context` monad in a temporary mvar scope.
Doing this twice will push the old tmp_mvar assignments to a stack.
So it is safe to do this whether or not you are already in tmp mode. -/
meta constant tmp_mode (n_uvars n_mvars : nat) : type_context α → type_context α
/-- Returns true when in temp mode. -/
meta constant in_tmp_mode : type_context bool
meta constant tmp_is_assigned : nat → type_context bool
meta constant tmp_get_assignment : nat → type_context expr
meta constant level.tmp_is_assigned : nat → type_context bool
meta constant level.tmp_get_assignment : nat → type_context level
/-- Replace each metavariable in the given expression with a temporary metavariable. -/
meta constant to_tmp_mvars : expr → type_context (expr × list level × list expr)
meta constant mk_tmp_mvar (index : nat) (type : expr): expr
meta constant level.mk_tmp_mvar (index : nat) : level
/-- Run the provided type_context within a backtracking scope.
This means that any changes to the metavariable context will not be committed if the inner monad fails.
[warning]: the local context modified by `push_local` and `pop_local` is not affected by `try`.
Any unpopped locals will be present after the `try` even if the inner `type_context` failed.
-/
meta constant try {α : Type} : type_context α → type_context (option α)
meta def orelse {α : Type} : type_context α → type_context α → type_context α
| x y := try x >>= λ x, option.rec y pure x
meta instance type_context_alternative : alternative type_context := {
failure := λ α, type_context.fail "failed",
orelse := λ α x y, type_context.orelse x y
}
/-- Runs the given type_context monad using the type context of the current tactic state.
You can use this to perform unsafe operations such as direct metavariable assignment and the use of temporary metavariables.
-/
meta constant run (inner : type_context α) (tr := tactic.transparency.semireducible) : tactic α
meta def trace {α} [has_to_format α] : α → type_context unit | a :=
pure $ _root_.trace_fmt (to_fmt a) (λ u, ())
meta def print_mvars : type_context unit := do
mvs ← list_mvars,
mvs ← pure $ mvs.map (λ x, match x with (expr.mvar _ pp _) := to_fmt pp | _ := "" end),
trace mvs
/-- Same as tactic.get_fun_info -/
meta constant get_fun_info (f : expr) (nargs : option nat := none) : type_context fun_info
end type_context
end tactic.unsafe
|
a243a45090479fd4232f2865ef8394d27ab514c2 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/data/int/associated.lean | 1aae01eef87ac0833510aae910cdbbae498e1084 | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 912 | lean | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import algebra.associated
import data.int.units
/-!
# Associated elements and the integers
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file contains some results on equality up to units in the integers.
## Main results
* `int.nat_abs_eq_iff_associated`: the absolute value is equal iff integers are associated
-/
lemma int.nat_abs_eq_iff_associated {a b : ℤ} :
a.nat_abs = b.nat_abs ↔ associated a b :=
begin
refine int.nat_abs_eq_nat_abs_iff.trans _,
split,
{ rintro (rfl | rfl),
{ refl },
{ exact ⟨-1, by simp⟩ } },
{ rintro ⟨u, rfl⟩,
obtain (rfl | rfl) := int.units_eq_one_or u,
{ exact or.inl (by simp) },
{ exact or.inr (by simp) } }
end
|
4c758fa0801651715cd9e4e974deddd16ab0d8b0 | 7c2dd01406c42053207061adb11703dc7ce0b5e5 | /src/exercises/09_limits_final.lean | dd198eca8200e821ccea06aaa6990bc35772a091 | [
"Apache-2.0"
] | permissive | leanprover-community/tutorials | 50ec79564cbf2ad1afd1ac43d8ee3c592c2883a8 | 79a6872a755c4ae0c2aca57e1adfdac38b1d8bb1 | refs/heads/master | 1,687,466,144,386 | 1,672,061,276,000 | 1,672,061,276,000 | 189,169,918 | 186 | 81 | Apache-2.0 | 1,686,350,300,000 | 1,559,113,678,000 | Lean | UTF-8 | Lean | false | false | 7,276 | lean | import tuto_lib
set_option pp.beta true
set_option pp.coercions false
/-
This is the final file in the series. Here we use everything covered
in previous files to prove a couple of famous theorems from
elementary real analysis. Of course they all have more general versions
in mathlib.
As usual, keep in mind the following:
abs_le {x y : ℝ} : |x| ≤ y ↔ -y ≤ x ∧ x ≤ y
ge_max_iff (p q r) : r ≥ max p q ↔ r ≥ p ∧ r ≥ q
le_max_left p q : p ≤ max p q
le_max_right p q : q ≤ max p q
as well as a lemma from the previous file:
le_of_le_add_all : (∀ ε > 0, y ≤ x + ε) → y ≤ x
Let's start with a variation on a known exercise.
-/
-- 0071
lemma le_lim {x y : ℝ} {u : ℕ → ℝ} (hu : seq_limit u x)
(ineg : ∃ N, ∀ n ≥ N, y ≤ u n) : y ≤ x :=
begin
sorry
end
/-
Let's now return to the result proved in the `00_` file of this series,
and prove again the sequential characterization of upper bounds (with a slighly
different proof).
For this, and other exercises below, we'll need many things that we proved in previous files,
and a couple of extras.
From the 5th file:
limit_const (x : ℝ) : seq_limit (λ n, x) x
squeeze (lim_u : seq_limit u l) (lim_w : seq_limit w l)
(hu : ∀ n, u n ≤ v n) (hw : ∀ n, v n ≤ w n) : seq_limit v l
From the 8th:
def upper_bound (A : set ℝ) (x : ℝ) := ∀ a ∈ A, a ≤ x
def is_sup (A : set ℝ) (x : ℝ) := upper_bound A x ∧ ∀ y, upper_bound A y → x ≤ y
lt_sup (hx : is_sup A x) : ∀ y, y < x → ∃ a ∈ A, y < a :=
You can also use:
nat.one_div_pos_of_nat {n : ℕ} : 0 < 1 / (n + 1 : ℝ)
inv_succ_le_all : ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, 1/(n + 1 : ℝ) ≤ ε
and their easy consequences:
limit_of_sub_le_inv_succ (h : ∀ n, |u n - x| ≤ 1/(n+1)) : seq_limit u x
limit_const_add_inv_succ (x : ℝ) : seq_limit (λ n, x + 1/(n+1)) x
limit_const_sub_inv_succ (x : ℝ) : seq_limit (λ n, x - 1/(n+1)) x
as well as:
lim_le (hu : seq_limit u x) (ineg : ∀ n, u n ≤ y) : x ≤ y
The structure of the proof is offered. It features a new tactic:
`choose` which invokes the axiom of choice (observing the tactic state before and
after using it should be enough to understand everything).
-/
-- 0072
lemma is_sup_iff (A : set ℝ) (x : ℝ) :
(is_sup A x) ↔ (upper_bound A x ∧ ∃ u : ℕ → ℝ, seq_limit u x ∧ ∀ n, u n ∈ A ) :=
begin
split,
{ intro h,
split,
{
sorry
},
{ have : ∀ n : ℕ, ∃ a ∈ A, x - 1/(n+1) < a,
{ intros n,
have : 1/(n+1 : ℝ) > 0,
exact nat.one_div_pos_of_nat,
sorry
},
choose u hu using this,
sorry
} },
{ rintro ⟨maj, u, limu, u_in⟩,
sorry
},
end
/-- Continuity of a function at a point -/
def continuous_at_pt (f : ℝ → ℝ) (x₀ : ℝ) : Prop :=
∀ ε > 0, ∃ δ > 0, ∀ x, |x - x₀| ≤ δ → |f x - f x₀| ≤ ε
variables {f : ℝ → ℝ} {x₀ : ℝ} {u : ℕ → ℝ}
-- 0073
lemma seq_continuous_of_continuous (hf : continuous_at_pt f x₀)
(hu : seq_limit u x₀) : seq_limit (f ∘ u) (f x₀) :=
begin
sorry
end
-- 0074
example :
(∀ u : ℕ → ℝ, seq_limit u x₀ → seq_limit (f ∘ u) (f x₀)) →
continuous_at_pt f x₀ :=
begin
sorry
end
/-
Recall from the 6th file:
def extraction (φ : ℕ → ℕ) := ∀ n m, n < m → φ n < φ m
def cluster_point (u : ℕ → ℝ) (a : ℝ) :=
∃ φ, extraction φ ∧ seq_limit (u ∘ φ) a
id_le_extraction : extraction φ → ∀ n, n ≤ φ n
and from the 8th file:
def tendsto_infinity (u : ℕ → ℝ) := ∀ A, ∃ N, ∀ n ≥ N, u n ≥ A
not_seq_limit_of_tendstoinfinity : tendsto_infinity u → ∀ l, ¬ seq_limit u l
-/
variables {φ : ℕ → ℕ}
-- 0075
lemma subseq_tendstoinfinity
(h : tendsto_infinity u) (hφ : extraction φ) :
tendsto_infinity (u ∘ φ) :=
begin
sorry
end
-- 0076
lemma squeeze_infinity {u v : ℕ → ℝ} (hu : tendsto_infinity u)
(huv : ∀ n, u n ≤ v n) : tendsto_infinity v :=
begin
sorry
end
/-
We will use segments: Icc a b := { x | a ≤ x ∧ x ≤ b }
The notation stands for Interval-closed-closed. Variations exist with
o or i instead of c, where o stands for open and i for infinity.
We will use the following version of Bolzano-Weierstrass
bolzano_weierstrass (h : ∀ n, u n ∈ [a, b]) :
∃ c ∈ [a, b], cluster_point u c
as well as the obvious
seq_limit_id : tendsto_infinity (λ n, n)
-/
open set
-- 0077
lemma bdd_above_segment {f : ℝ → ℝ} {a b : ℝ} (hf : ∀ x ∈ Icc a b, continuous_at_pt f x) :
∃ M, ∀ x ∈ Icc a b, f x ≤ M :=
begin
sorry
end
/-
In the next exercise, we can use:
abs_neg x : |-x| = |x|
-/
-- 0078
lemma continuous_opposite {f : ℝ → ℝ} {x₀ : ℝ} (h : continuous_at_pt f x₀) :
continuous_at_pt (λ x, -f x) x₀ :=
begin
sorry
end
/-
Now let's combine the two exercises above
-/
-- 0079
lemma bdd_below_segment {f : ℝ → ℝ} {a b : ℝ} (hf : ∀ x ∈ Icc a b, continuous_at_pt f x) :
∃ m, ∀ x ∈ Icc a b, m ≤ f x :=
begin
sorry
end
/-
Remember from the 5th file:
unique_limit : seq_limit u l → seq_limit u l' → l = l'
and from the 6th one:
subseq_tendsto_of_tendsto (h : seq_limit u l) (hφ : extraction φ) :
seq_limit (u ∘ φ) l
We now admit the following version of the least upper bound theorem
(that cannot be proved without discussing the construction of real numbers
or admitting another strong theorem).
sup_segment {a b : ℝ} {A : set ℝ} (hnonvide : ∃ x, x ∈ A) (h : A ⊆ Icc a b) :
∃ x ∈ Icc a b, is_sup A x
In the next exercise, it can be useful to prove inclusions of sets of real number.
By definition, A ⊆ B means : ∀ x, x ∈ A → x ∈ B.
Hence one can start a proof of A ⊆ B by `intros x x_in`,
which brings `x : ℝ` and `x_in : x ∈ A` in the local context,
and then prove `x ∈ B`.
Note also the use of
{x | P x}
which denotes the set of x satisfying predicate P.
Hence `x' ∈ { x | P x} ↔ P x'`, by definition.
-/
-- 0080
example {a b : ℝ} (hab : a ≤ b) (hf : ∀ x ∈ Icc a b, continuous_at_pt f x) :
∃ x₀ ∈ Icc a b, ∀ x ∈ Icc a b, f x ≤ f x₀ :=
begin
sorry
end
lemma stupid {a b x : ℝ} (h : x ∈ Icc a b) (h' : x ≠ b) : x < b :=
lt_of_le_of_ne h.right h'
/-
And now the final boss...
-/
def I := (Icc 0 1 : set ℝ) -- the type ascription makes sure 0 and 1 are real numbers here
-- 0081
example (f : ℝ → ℝ) (hf : ∀ x, continuous_at_pt f x) (h₀ : f 0 < 0) (h₁ : f 1 > 0) :
∃ x₀ ∈ I, f x₀ = 0 :=
begin
let A := { x | x ∈ I ∧ f x < 0},
have ex_x₀ : ∃ x₀ ∈ I, is_sup A x₀,
{
sorry
},
rcases ex_x₀ with ⟨x₀, x₀_in, x₀_sup⟩,
use [x₀, x₀_in],
have : f x₀ ≤ 0,
{
sorry
},
have x₀_1: x₀ < 1,
{
sorry
},
have : f x₀ ≥ 0,
{ have in_I : ∃ N : ℕ, ∀ n ≥ N, x₀ + 1/(n+1) ∈ I,
{ have : ∃ N : ℕ, ∀ n≥ N, 1/(n+1 : ℝ) ≤ 1-x₀,
{
sorry
},
sorry
},
have not_in : ∀ n : ℕ, x₀ + 1/(n+1) ∉ A,
-- By definition, x ∉ A means ¬ (x ∈ A).
{
sorry
},
dsimp [A] at not_in,
sorry
},
linarith,
end
|
a068185f4051280cdb07c4f1d48fb70b81c1c11a | 2272e503179a58556187901b8698b789fad7e0c4 | /src/free_group.lean | 982c0955f63b0eb5651a5645e18818c84bfbab5c | [] | no_license | kckennylau/category-theory | f15e582be862379453a5341d83b8cd5ebc686729 | b24962838c7370b5257e38b7648040aec95922bb | refs/heads/master | 1,583,605,043,449 | 1,525,154,882,000 | 1,525,154,882,000 | 127,633,452 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 15,120 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
Three-step construction of free group of a type.
Based on https://math.stackexchange.com/a/852661/328173
-/
import data.list.basic algebra.group
universes u v
class inv_type (IT : Type u) extends has_inv IT :=
(inv_inv : ∀ x : IT, x⁻¹⁻¹ = x)
class involution_monoid (M : Type u) extends monoid M, inv_type M :=
(mul_inv : ∀ x y : M, (x*y)⁻¹ = y⁻¹ * x⁻¹)
(one_inv : (1:M)⁻¹ = 1)
instance group.to_involution_monoid (G : Type u) [group G] : involution_monoid G :=
{ mul_inv := mul_inv_rev,
one_inv := one_inv,
inv_inv := inv_inv }
class involution_monoid.is_hom {M : Type u} {N : Type v}
[involution_monoid M] [involution_monoid N] (f : M → N) : Prop :=
(mul : ∀ x y, f (x * y) = f x * f y)
(one : f 1 = 1)
(inv : ∀ x, f x⁻¹ = (f x)⁻¹)
namespace involution_monoid.to_group
variables (M : Type u) [involution_monoid M]
inductive rel : M → M → Prop
| refl : ∀ x, rel x x
| symm : ∀ x y, rel x y → rel y x
| trans : ∀ x y z, rel x y → rel y z → rel x z
| mul : ∀ x y z w, rel x z → rel y w → rel (x * y) (z * w)
| inv : ∀ x y, rel x y → rel x⁻¹ y⁻¹
| mul_left_inv : ∀ x, rel (x⁻¹ * x) 1
instance quotient_rel : setoid M :=
⟨rel M, rel.refl, rel.symm, rel.trans⟩
end involution_monoid.to_group
@[reducible] def involution_monoid.to_group (M : Type u) [involution_monoid M] : Type u :=
quotient (involution_monoid.to_group.quotient_rel M)
namespace involution_monoid.to_group
instance (M : Type u) [involution_monoid M] : group (involution_monoid.to_group M) :=
{ mul := λ x y, quotient.lift_on₂ x y (λ m n, quotient.mk (m * n))
(λ x y z w hxz hyw, quotient.sound $ rel.mul x y z w hxz hyw),
mul_assoc := λ x y z, quotient.induction_on₃ x y z $
λ a b c, quotient.sound $ by rw mul_assoc,
one := quotient.mk 1,
mul_one := λ x, quotient.induction_on x $
λ a, quotient.sound $ by rw mul_one,
one_mul := λ x, quotient.induction_on x $
λ a, quotient.sound $ by rw one_mul,
inv := λ x, quotient.lift_on x (λ m, quotient.mk m⁻¹)
(λ x y hxy, quotient.sound $ rel.inv x y hxy),
mul_left_inv := λ x, quotient.induction_on x $
λ a, quotient.sound $ rel.mul_left_inv a }
def of_involution_monoid {M : Type u} [involution_monoid M] : M → involution_monoid.to_group M :=
quotient.mk
@[simp] lemma of_involution_monoid.mul {M : Type u} [involution_monoid M] {x y : M} :
of_involution_monoid (x * y) = of_involution_monoid x * of_involution_monoid y := rfl
@[simp] lemma of_involution_monoid.one {M : Type u} [involution_monoid M] :
of_involution_monoid (1:M) = 1 := rfl
@[simp] lemma of_involution_monoid.inv {M : Type u} [involution_monoid M] {x : M} :
of_involution_monoid x⁻¹ = (of_involution_monoid x)⁻¹ := rfl
section left_adjoint
parameters (M : Type u) [involution_monoid M]
parameters (G : Type v) [group G]
parameters (f : M → G) -- M → Forgetful(G)
parameters [Hf : involution_monoid.is_hom f]
include Hf
def left_adjoint : involution_monoid.to_group M → G := -- Free(M) → G
λ x, quotient.lift_on x f $ λ a b hab, begin
induction hab with h c d h ih c d e h1 h2 ih1 ih2
c d p q h1 h2 ih1 ih2 c d h ih c,
case involution_monoid.to_group.rel.refl
{ refl },
case involution_monoid.to_group.rel.symm
{ exact ih.symm },
case involution_monoid.to_group.rel.trans
{ exact ih1.trans ih2 },
case involution_monoid.to_group.rel.mul
{ rw [Hf.mul, Hf.mul, ih1, ih2] },
case involution_monoid.to_group.rel.inv
{ rw [Hf.inv, Hf.inv, ih] },
case involution_monoid.to_group.rel.mul_left_inv
{ rw [Hf.mul, Hf.inv, mul_left_inv, Hf.one] }
end
theorem left_adjoint.is_group_hom : is_group_hom left_adjoint :=
λ x y, quotient.induction_on₂ x y Hf.mul
theorem left_adjoint.commutes (x) : left_adjoint (of_involution_monoid x) = f x := rfl
parameters (g : involution_monoid.to_group M → G)
parameters (Hg : ∀ x, g (of_involution_monoid x) = f x)
theorem left_adjoint.unique : ∀ x, g x = left_adjoint x :=
λ x, quotient.induction_on x $ λ m, Hg m
end left_adjoint
end involution_monoid.to_group
@[reducible] def inv_type.to_involution_monoid (IT : Type u) [inv_type IT] :=
list IT
namespace inv_type.to_involution_monoid
instance (IT : Type u) [inv_type IT] : involution_monoid (inv_type.to_involution_monoid IT) :=
{ mul := (++),
one := [],
inv := λ x, x.reverse.map has_inv.inv,
mul_assoc := list.append_assoc,
one_mul := list.nil_append,
mul_one := list.append_nil,
mul_inv := λ x y, show (x ++ y).reverse.map has_inv.inv
= y.reverse.map has_inv.inv ++ x.reverse.map has_inv.inv,
by rw [list.reverse_append, list.map_append],
one_inv := rfl,
inv_inv := λ x, have h1 : (has_inv.inv ∘ has_inv.inv) = @id IT,
from funext $ inv_type.inv_inv,
by dsimp; rw [list.map_reverse, list.map_reverse];
rw [list.map_reverse, list.reverse_reverse];
rw [list.map_map, h1, list.map_id] }
def of_inv_type {IT : Type u} [inv_type IT] : IT → inv_type.to_involution_monoid IT :=
λ x, [x]
@[simp] lemma of_inv_type.inv {IT : Type u} [inv_type IT] {x : IT} :
of_inv_type x⁻¹ = (of_inv_type x)⁻¹ := rfl
section left_adjoint
parameters (IT : Type u) [inv_type IT]
parameters (M : Type v) [involution_monoid M]
parameters (f : IT → M) -- IT → Forgetful(M)
parameters (Hinv : ∀ x, f x⁻¹ = (f x)⁻¹)
def left_adjoint : inv_type.to_involution_monoid IT → M := -- Free(IT) → M
λ L, list.prod (L.map f)
theorem left_adjoint.append (x y) : left_adjoint (x ++ y) = left_adjoint x * left_adjoint y :=
by dsimp [left_adjoint]; rw [list.map_append, list.prod_append]
theorem left_adjoint.mul (x y) : left_adjoint (x * y) = left_adjoint x * left_adjoint y :=
left_adjoint.append x y
theorem left_adjoint.one : left_adjoint 1 = 1 := rfl
include Hinv
theorem left_adjoint.inv : ∀ x, left_adjoint x⁻¹ = (left_adjoint x)⁻¹
| [] := eq.symm $ involution_monoid.one_inv M
| (h::IT) := have _ := left_adjoint.inv IT,
by dsimp [left_adjoint, has_inv.inv] at this ⊢;
rw [list.map_reverse, list.map_reverse] at this ⊢;
rw [list.map_cons, list.map_cons, list.prod_cons, involution_monoid.mul_inv, ← this];
rw [list.reverse_cons', list.prod_append, list.prod_cons, list.prod_nil];
rw [mul_one, Hinv]
omit Hinv
instance left_adjoint.hom : involution_monoid.is_hom left_adjoint :=
⟨left_adjoint.mul, left_adjoint.one, left_adjoint.inv⟩
theorem left_adjoint.commutes (x) : left_adjoint (of_inv_type x) = f x :=
one_mul _
parameters (g : inv_type.to_involution_monoid IT → M)
parameters (Hg1 : ∀ x y, g (x * y) = g x * g y)
parameters (Hg2 : g 1 = 1)
parameters (Hg3 : ∀ x, g (of_inv_type x) = f x)
include Hg1 Hg3
theorem left_adjoint.unique : ∀ x, g x = left_adjoint x
| [] := Hg2
| (h::t) := eq.trans (Hg1 (of_inv_type h) t) $
by rw [left_adjoint.unique t]; dsimp [left_adjoint]; rw [Hg3, list.prod_cons]
end left_adjoint
end inv_type.to_involution_monoid
@[reducible] def to_inv_type (T : Type u) :=
sum T T
namespace to_inv_type
def inv (T : Type u) : to_inv_type T → to_inv_type T
| (sum.inl x) := sum.inr x
| (sum.inr x) := sum.inl x
theorem inv.inv (T : Type u) : ∀ x : to_inv_type T, inv T (inv T x) = x
| (sum.inl x) := rfl
| (sum.inr x) := rfl
instance (T : Type u) : inv_type (to_inv_type T) :=
{ inv := inv T,
inv_inv := inv.inv T }
def of_inv_type {T : Type u} : T → to_inv_type T :=
sum.inl
section left_adjoint
parameters (T : Type u)
parameters (IT : Type v) [inv_type IT]
parameters (f : T → IT) -- T → Forgetful(IT)
def left_adjoint : to_inv_type T → IT -- Free(T) → IT
| (sum.inl x) := f x
| (sum.inr x) := (f x)⁻¹
theorem left_adjoint.inv : ∀ x, left_adjoint x⁻¹ = (left_adjoint x)⁻¹
| (sum.inl x) := rfl
| (sum.inr x) := eq.symm $ inv_type.inv_inv _
theorem left_adjoint.commutes (x) : left_adjoint (of_inv_type x) = f x := rfl
parameters (g : to_inv_type T → IT)
parameters (Hg1 : ∀ x, g x⁻¹ = (g x)⁻¹)
parameters (Hg2 : ∀ x, g (of_inv_type x) = f x)
theorem left_adjoint.unique : ∀ x, g x = left_adjoint x
| (sum.inl x) := Hg2 x
| (sum.inr x) := (Hg1 (sum.inl x)).trans $ congr_arg _ $ Hg2 x
end left_adjoint
end to_inv_type
@[reducible] def free_group.word (S : Type u) : Type u :=
inv_type.to_involution_monoid $ to_inv_type S
@[reducible] def free_group (S : Type u) : Type u :=
involution_monoid.to_group $ free_group.word S
namespace free_group
variables (S : Type u)
instance : group (free_group S) :=
involution_monoid.to_group.group _
def of_type : S → free_group S :=
λ x, ⟦[sum.inl x]⟧
variables (G : Type v) [group G]
variables (f : S → G) -- S → Forgetful(G)
def to_group : free_group S → G := -- Free(S) → G
involution_monoid.to_group.left_adjoint _ G
(λ y, inv_type.to_involution_monoid.left_adjoint (to_inv_type S) _
(to_inv_type.left_adjoint S _ f) y)
def to_group.is_group_hom : is_group_hom (to_group S G f) :=
involution_monoid.to_group.left_adjoint.is_group_hom _ _ _ _ _ _
def to_group.commutes (x) : to_group S G f (of_type S x) = f x :=
mul_one _
variables (g : free_group S → G)
variables (Hg1 : is_group_hom g)
variables (Hg2 : ∀ x, g (of_type S x) = f x)
include Hg1 Hg2
def to_group.unique : ∀ x, g x = to_group S G f x :=
λ x, quotient.induction_on x $ λ m, list.rec_on m (is_group_hom.one g) $
λ h t ih, sum.rec_on h
(λ a, show g (of_type S a * ⟦t⟧) = _ , by rw [Hg1, Hg2, ih]; refl)
(λ a, show g ((of_type S a)⁻¹ * ⟦t⟧) = _ , by rw [Hg1, is_group_hom.inv g, Hg2, ih]; refl)
omit Hg1 Hg2
variable [decidable_eq S]
def reduce : word S → word S
| ((sum.inl x)::(sum.inr y)::t) := if x = y then reduce t else ((sum.inl x)::(sum.inr y)::reduce t)
| ((sum.inr x)::(sum.inl y)::t) := if x = y then reduce t else ((sum.inr x)::(sum.inl y)::reduce t)
| (h1::h2::t) := h1::h2::reduce t
| w := w
theorem reduce.sound : ∀ w : word S, ⟦w⟧ = ⟦reduce S w⟧
| [] := rfl
| ((sum.inl x)::[]) := rfl
| ((sum.inr x)::[]) := rfl
| ((sum.inl x)::(sum.inl y)::t) :=
show ⟦[sum.inl x, sum.inl y]⟧ * ⟦t⟧ = ⟦[sum.inl x, sum.inl y]⟧ * ⟦reduce S t⟧,
from congr_arg _ $ reduce.sound t
| ((sum.inr x)::(sum.inr y)::t) :=
show ⟦[sum.inr x, sum.inr y]⟧ * ⟦t⟧ = ⟦[sum.inr x, sum.inr y]⟧ * ⟦reduce S t⟧,
from congr_arg _ $ reduce.sound t
| ((sum.inl x)::(sum.inr y)::t) := begin
dsimp [reduce],
by_cases x = y; simp [h],
{ change ⟦[sum.inl y]⟧ * ⟦[sum.inr y]⟧ * ⟦t⟧ = ⟦reduce S t⟧,
have h1 : ⟦[sum.inl y]⟧ * ⟦[sum.inr y]⟧ = 1,
{ exact mul_inv_self _ },
rw [h1, one_mul, reduce.sound t] },
{ change ⟦[sum.inl x, sum.inr y]⟧ * ⟦t⟧ = ⟦[sum.inl x, sum.inr y]⟧ * ⟦reduce S t⟧,
rw reduce.sound t }
end
| ((sum.inr x)::(sum.inl y)::t) := begin
dsimp [reduce],
by_cases x = y; simp [h],
{ change ⟦[sum.inr y]⟧ * ⟦[sum.inl y]⟧ * ⟦t⟧ = ⟦reduce S t⟧,
have h1 : ⟦[sum.inr y]⟧ * ⟦[sum.inl y]⟧ = 1,
{ exact mul_inv_self _ },
rw [h1, one_mul, reduce.sound t] },
{ change ⟦[sum.inr x, sum.inl y]⟧ * ⟦t⟧ = ⟦[sum.inr x, sum.inl y]⟧ * ⟦reduce S t⟧,
rw reduce.sound t }
end
theorem reduce.exact.mul.nil : ∀ p q : word S, [] = reduce S p →
reduce S q = reduce S (p * q)
| [] q h := rfl
| ((sum.inl x)::(sum.inr y)::t) q h :=
show reduce S q = reduce S (sum.inl x :: sum.inr y :: t * q),
begin
dsimp [reduce] at h ⊢,
by_cases H : x = y,
{ rw [if_pos H] at h ⊢,
exact reduce.exact.mul.nil _ _ h },
{ rw [if_neg H] at h ⊢,
exact list.no_confusion h },
end
| ((sum.inr x)::(sum.inl y)::t) q h :=
show reduce S q = reduce S (sum.inr x :: sum.inl y :: t * q),
begin
dsimp [reduce] at h ⊢,
by_cases H : x = y,
{ rw [if_pos H] at h ⊢,
exact reduce.exact.mul.nil _ _ h },
{ rw [if_neg H] at h ⊢,
exact list.no_confusion h },
end
-- TODO
-- theorem reduce.exact : ∀ v w : word S, v ≈ w → reduce S v = reduce S w :=
theorem reduce.min : ∀ w : word S, (reduce S w).length ≤ w.length
| [] := dec_trivial
| ((sum.inl x)::[]) := dec_trivial
| ((sum.inr x)::[]) := dec_trivial
| ((sum.inl x)::(sum.inl y)::t) :=
add_le_add_right (add_le_add_right (reduce.min t) 1) 1
| ((sum.inr x)::(sum.inr y)::t) :=
add_le_add_right (add_le_add_right (reduce.min t) 1) 1
| ((sum.inl x)::(sum.inr y)::t) :=
match (by apply_instance : decidable (x = y)) with
| (decidable.is_true h) := nat.le_succ_of_le $
nat.le_succ_of_le $ by dsimp [reduce]; rw [if_pos h];
exact reduce.min t
| (decidable.is_false h) := by dsimp [reduce]; rw [if_neg h];
exact add_le_add_right (add_le_add_right (reduce.min t) 1) 1
end
| ((sum.inr x)::(sum.inl y)::t) :=
match (by apply_instance : decidable (x = y)) with
| (decidable.is_true h) := nat.le_succ_of_le $
nat.le_succ_of_le $ by dsimp [reduce];
rw [if_pos h]; exact reduce.min t
| (decidable.is_false h) := by dsimp [reduce]; rw [if_neg h];
exact add_le_add_right (add_le_add_right (reduce.min t) 1) 1
end
theorem reduce.idem : ∀ w : word S, reduce S (reduce S w) = reduce S w
| [] := rfl
| ((sum.inl x)::[]) := rfl
| ((sum.inr x)::[]) := rfl
| ((sum.inl x)::(sum.inl y)::t) :=
show sum.inl x :: sum.inl y :: reduce.{u} S (reduce S t)
= sum.inl x :: sum.inl y :: reduce S t,
from congr_arg _ $ congr_arg _ $ reduce.idem t
| ((sum.inr x)::(sum.inr y)::t) :=
show sum.inr x :: sum.inr y :: reduce.{u} S (reduce S t)
= sum.inr x :: sum.inr y :: reduce S t,
from congr_arg _ $ congr_arg _ $ reduce.idem t
| ((sum.inl x)::(sum.inr y)::t) :=
match (by apply_instance : decidable (x = y)) with
| (decidable.is_true h) := by dsimp [reduce];
rw [if_pos h]; exact reduce.idem t
| (decidable.is_false h) := by dsimp [reduce]; rw [if_neg h];
dsimp [reduce]; rw [if_neg h];
from congr_arg _ (congr_arg _ (reduce.idem t))
end
| ((sum.inr x)::(sum.inl y)::t) :=
match (by apply_instance : decidable (x = y)) with
| (decidable.is_true h) := by dsimp [reduce];
rw [if_pos h]; exact reduce.idem t
| (decidable.is_false h) := by dsimp [reduce]; rw [if_neg h];
dsimp [reduce]; rw [if_neg h];
from congr_arg _ (congr_arg _ (reduce.idem t))
end
-- TODO
-- def to_word : free_group S → word S :=
end free_group
|
527a042159c7cabc6bfb5fb0032c0e837ff23b0b | e21db629d2e37a833531fdcb0b37ce4d71825408 | /src/parlang/lemmas_active_map.lean | bdf057d673eec40924577bddb1a7206f74e03ee1 | [] | no_license | fischerman/GPU-transformation-verifier | 614a28cb4606a05a0eb27e8d4eab999f4f5ea60c | 75a5016f05382738ff93ce5859c4cfa47ccb63c1 | refs/heads/master | 1,586,985,789,300 | 1,579,290,514,000 | 1,579,290,514,000 | 165,031,073 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 7,801 | lean | import parlang.defs
namespace parlang
variables {n : ℕ} {σ : Type} {ι : Type} {τ : ι → Type} [decidable_eq ι]
variables {s t u : state n σ τ} {ac : vector bool n} {f f' : σ → bool}
lemma all_threads_active_nth : ∀ {n} {ac : vector bool n}, all_threads_active ac → ∀ i, ac.nth i
| 0 ⟨[], refl⟩ h i := by apply (vector.nat_le_zero i.is_lt).elim
| (n + 1) ⟨ a :: as, hp⟩ h ⟨i, hi⟩ := begin
rw vector.nth,
cases i,
case nat.zero {
simp [all_threads_active, vector.to_list, list.all] at h,
exact h.left,
},
case nat.succ {
simp,
simp [all_threads_active, vector.to_list, list.all] at h,
have hi' : i < n := begin
rw [← nat.add_one, nat.add_comm i 1, nat.add_comm n 1] at hi,
apply lt_of_add_lt_add_left hi,
end,
have hp' : list.length as = n := by exact nat.add_right_cancel hp,
specialize @all_threads_active_nth n ⟨as, hp'⟩,
simp [all_threads_active, list.all] at all_threads_active_nth,
specialize all_threads_active_nth h.right ⟨i, hi'⟩,
exact all_threads_active_nth,
}
end
lemma all_threads_active_nth_zero (ac : vector bool (nat.succ n)) : all_threads_active ac → ac.nth 0
| h := all_threads_active_nth h 0
lemma no_threads_active_nth : ∀ {n} {ac : vector bool n}, no_thread_active ac → ∀ i, ¬ac.nth i
| n ⟨[], h'⟩ h tid := (by subst h'; apply fin_zero_elim tid)
| (n + 1) ⟨t :: ts, h'⟩ h ⟨tid, h''⟩ := begin
simp [no_thread_active, vector.to_list, list.any, not_or_distrib] at h,
cases tid,
{
simp [vector.nth, list.nth_le],
exact h.left,
}, {
simp [vector.nth, list.nth_le],
specialize @no_threads_active_nth n ⟨ts, nat.succ_inj h'⟩,
simp [no_thread_active, list.any] at no_threads_active_nth,
specialize no_threads_active_nth h.right ⟨tid, _⟩,
exact no_threads_active_nth,
exact nat.succ_lt_succ_iff.mp h'',
}
end
lemma no_threads_active_nth_zero (ac : vector bool (nat.succ n)) : no_thread_active ac → ¬ac.nth 0
| h := no_threads_active_nth h 0
lemma all_threads_active_repeat (n : ℕ) : all_threads_active (vector.repeat tt n) := begin
induction n,
{ rw vector.repeat, simp [all_threads_active], },
{
rw vector.repeat,
simp [all_threads_active],
apply n_ih,
}
end
lemma no_threads_active_not_all_threads {ac : vector bool n} (hl : 0 < n) : no_thread_active ac → ¬↥(all_threads_active ac) := begin
cases n,
case nat.zero {
have : ¬ 0 < 0 := by simp,
contradiction,
},
case nat.succ {
intros a b,
have : ↥(ac.nth ⟨0, hl⟩) := begin
apply all_threads_active_nth_zero,
assumption,
end,
have : ¬↥(ac.nth ⟨0, hl⟩) := begin
apply no_threads_active_nth_zero,
assumption,
end,
contradiction,
},
end
lemma no_threads_active_no_active_thread {ac : vector bool n} : no_thread_active ac → ¬any_thread_active ac := begin
induction n,
case nat.zero {
cases ac,
cases ac_val,
case list.nil {
rw no_thread_active,
rw any_thread_active,
simp,
},
case list.cons {
contradiction,
}
},
case nat.succ {
unfold no_thread_active any_thread_active,
simp,
}
end
lemma deactivate_threads_alive {f : σ → bool} {ac : vector bool n} {s : state n σ τ} {i} : (deactivate_threads f ac s).nth i → ac.nth i := begin
intro hd,
simp[deactivate_threads] at hd,
exact hd.left,
end
lemma deactivate_threads_deactivate_inactive_thread {f : σ → bool} {ac : vector bool n} {s : state n σ τ} {i} : ¬ac.nth i → ¬(deactivate_threads f ac s).nth i
| a b := a (deactivate_threads_alive b) -- is there a more elegant way for contraposition?
lemma active_map_deactivate_threads {ac : vector bool n} {i} {f : σ → bool} {s : state n σ τ} :
ac.nth i → f (s.threads.nth i).tlocal → (deactivate_threads (bnot ∘ f) ac s).nth i :=
begin
intros hac hf,
simp [deactivate_threads],
exact ⟨hac, hf⟩,
end
lemma active_map_deactivate_threads' {ac : vector bool n} {i} {f : σ → bool} {s : state n σ τ} :
ac.nth i → ¬f (s.threads.nth i).tlocal → (deactivate_threads f ac s).nth i := begin
intros hac hf,
simp [deactivate_threads],
simp at hf,
rw bool.bnot_ff,
simp,
exact ⟨hac, hf⟩,
end
lemma deactivate_threads_condition {f : σ → bool} {ac : vector bool n} {s : state n σ τ} {i} :
vector.nth (deactivate_threads (bnot ∘ f) ac s) i → f ((vector.nth (s.threads) i).tlocal) := by simp [deactivate_threads]
lemma deactivate_threads_condition' {f : σ → bool} {ac : vector bool n} {s : state n σ τ} {i} :
vector.nth (deactivate_threads (f) ac s) i → ¬f ((vector.nth (s.threads) i).tlocal) := begin
simp [deactivate_threads],
intros _ h,
rw ← bnot_eq_true_eq_eq_ff,
exact h,
end
lemma deactivate_threads_complement {f : σ → bool} {ac : vector bool n} {s : state n σ τ} {i} :
vector.nth (deactivate_threads (bnot ∘ f) ac s) i → ¬↥(vector.nth (deactivate_threads (f) ac s) i) := begin
intro h,
simp [deactivate_threads] at ⊢ h,
intro _,
cases h,
apply h_right,
end
lemma ac_sub_deac {f : σ → bool} : ac ≥ (deactivate_threads (bnot ∘ f) ac s) := begin
intros t h₁ h₂,
apply h₁,
unfold deactivate_threads at h₂,
rw vector.nth_map₂ at h₂,
rw band_coe_iff at h₂,
cases h₂,
assumption,
end
lemma ac_deac_comm : deactivate_threads f (deactivate_threads f' ac s) t = deactivate_threads f' (deactivate_threads f ac t) s := begin
apply vector.eq_element_wise,
unfold deactivate_threads,
simp [vector.nth_map₂],
end
lemma ac_trans {ac' ac'' : vector bool n} : ac ≥ ac' → ac' ≥ ac'' → ac ≥ ac'' := begin
intros h₁ h₂ t hna ha,
specialize h₁ t hna,
specialize h₂ t h₁,
contradiction,
end
instance : is_trans (vector bool n) ac_ge := ⟨begin intros a b c h₁ h₂, apply ac_trans, assumption, assumption, end⟩
lemma ac_deac_ge (h : deactivate_threads f ac s ≥ deactivate_threads f' ac t) : deactivate_threads f' (deactivate_threads f ac s) t = deactivate_threads f' ac t := begin
apply vector.eq_element_wise,
intro i,
specialize h i,
unfold deactivate_threads,
simp [vector.nth_map₂],
by_cases eq : vector.nth ac i = tt,
{
rw eq,
simp,
by_cases eq₂ : bnot (f' ((vector.nth (t.threads) i).tlocal)) = tt,
{
rw eq₂,
simp,
unfold deactivate_threads at h,
simp [vector.nth_map₂] at h,
simp [*] at h,
rw ← eq_ff_eq_not_eq_tt,
intro,
specialize h a,
rw h at eq₂,
cases eq₂,
}, {
simp at eq₂,
simp [*],
}
}, {
simp at eq,
simp [*],
}
end
lemma deac_ff : (deactivate_threads (λ_, ff) ac s) = ac := begin
apply vector.eq_element_wise,
intro,
unfold deactivate_threads,
simp [vector.nth_map₂],
end
lemma ac_deac_ge' (h : deactivate_threads f ac s ≥ ac) : deactivate_threads f ac s = ac := begin
rw ← @deac_ff _ _ _ _ _ _ ac,
rw ac_deac_comm,
apply ac_deac_ge,
rw ← @deac_ff _ _ _ _ _ _ ac at h { occs := occurrences.pos [2] },
exact h,
exact s,
end
lemma no_thread_active_ge {ac' : vector bool n} (h : no_thread_active ac) : ac' ≥ ac := begin
intros i h',
apply no_threads_active_nth h,
end
lemma all_threads_active_eq {ac' : vector bool n} (h : all_threads_active ac) (h' : all_threads_active ac') : ac = ac' := begin
apply vector.eq_element_wise,
intro i,
have := all_threads_active_nth h i,
unfold coe_sort has_coe_to_sort.coe at this,
rw this,
have := all_threads_active_nth h' i,
unfold coe_sort has_coe_to_sort.coe at this,
rw this,
end
end parlang |
5048cefa20dee36fba3c9e8991c5ecb8a6bf5051 | 5e3548e65f2c037cb94cd5524c90c623fbd6d46a | /src_icannos_totilas/reduction/cpge_reduction_012_d.lean | 55bc3e203a3207059174cce081dc8fcfd83c7066 | [] | no_license | ahayat16/lean_exos | d4f08c30adb601a06511a71b5ffb4d22d12ef77f | 682f2552d5b04a8c8eb9e4ab15f875a91b03845c | refs/heads/main | 1,693,101,073,585 | 1,636,479,336,000 | 1,636,479,336,000 | 415,000,441 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,023 | lean | import data.real.basic
import data.polynomial.derivative
import linear_algebra.basic
import linear_algebra.finite_dimensional
-- On se place dans le R -espace vectoriel E = R[X] .
-- (a) Soit H un sous-espace vectoriel de dimension finie et f un endomorphisme de H. Montrer qu'il existe p ∈ N tel que
-- ∀k ≥ p, Ker f k+1 = Ker f k.
--
-- Soit F un sous-espace vectoriel de E stable par l'opérateur D de dérivation.
-- (b) On suppose que F est de dimension fi nie non nulle. Montrer que l'endomorphisme induit par D sur R n [X] est nilpotent pour tout n ∈ N .
-- Montrer qu'il existe m ∈ N tel que F = R m [X] .
-- (c) Montrer que si F est de dimension in nie alors F = R[X] .
-- (d) Soit g ∈ L(E) tel que g^2 = kId + D avec k ∈ R . Quel est le signe de k ?
theorem d:
forall g: (polynomial real) →ₗ[real] (polynomial real),
(exists k: real, (g^2 = k • linear_map.id + polynomial.derivative))
-> (forall k: real, (g^2 = k • linear_map.id + polynomial.derivative) -> k > 0)
:= sorry
|
90aa648c14ecc86520e0624bda719c722a44bc07 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/algebra/order/group/units.lean | db8aec98bda150e80cc79ff62327059b7a5ceb54 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 846 | lean | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import algebra.order.group.defs
import algebra.order.monoid.defs
import algebra.order.monoid.units
/-!
# Adjoining a top element to a `linear_ordered_add_comm_group_with_top`.
-/
variable {α : Type*}
/-- The units of an ordered commutative monoid form an ordered commutative group. -/
@[to_additive "The units of an ordered commutative additive monoid form an ordered commutative
additive group."]
instance units.ordered_comm_group [ordered_comm_monoid α] : ordered_comm_group αˣ :=
{ mul_le_mul_left := λ a b h c, (mul_le_mul_left' (h : (a : α) ≤ b) _ : (c : α) * a ≤ c * b),
.. units.partial_order,
.. units.comm_group }
|
4264ec3becb843b660e8b4ffc0206c340894d02d | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/category_theory/limits/cofinal.lean | 78752f19a8a10c41c89707bb7433052ea76bdacd | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 16,428 | lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import category_theory.punit
import category_theory.structured_arrow
import category_theory.is_connected
import category_theory.limits.yoneda
import category_theory.limits.types
/-!
# Cofinal functors
A functor `F : C ⥤ D` is cofinal if for every `d : D`,
the comma category of morphisms `d ⟶ F.obj c` is connected.
We prove the following three statements are equivalent:
1. `F : C ⥤ D` is cofinal.
2. Every functor `G : D ⥤ E` has a colimit if and only if `F ⋙ G` does,
and these colimits are isomorphic via `colimit.pre G F`.
3. `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit`.
Starting at 1. we show (in `cocones_equiv`) that
the categories of cocones over `G : D ⥤ E` and over `F ⋙ G` are equivalent.
(In fact, via an equivalence which does not change the cocone point.)
This readily implies 2., as `comp_has_colimit`, `has_colimit_of_comp`, and `colimit_iso`.
From 2. we can specialize to `G = coyoneda.obj (op d)` to obtain 3., as `colimit_comp_coyoneda_iso`.
From 3., we prove 1. directly in `cofinal_of_colimit_comp_coyoneda_iso_punit`.
We also show these conditions imply:
4. Every functor `H : Dᵒᵖ ⥤ E` has a limit if and only if `F.op ⋙ H` does,
and these limits are isomorphic via `limit.pre H F.op`.
## Naming
There is some discrepancy in the literature about naming; some say 'final' instead of 'cofinal'.
The explanation for this is that the 'co' prefix here is *not* the usual category-theoretic one
indicating duality, but rather indicating the sense of "along with".
While the trend seems to be towards using 'final', for now we go with the bulk of the literature
and use 'cofinal'.
## References
* https://stacks.math.columbia.edu/tag/09WN
* https://ncatlab.org/nlab/show/final+functor
* Borceux, Handbook of Categorical Algebra I, Section 2.11.
(Note he reverses the roles of definition and main result relative to here!)
-/
noncomputable theory
universes v u
namespace category_theory
open opposite
open category_theory.limits
variables {C : Type v} [small_category C]
variables {D : Type v} [small_category D]
/--
A functor `F : C ⥤ D` is cofinal if for every `d : D`, the comma category of morphisms `d ⟶ F.obj c`
is connected.
See https://stacks.math.columbia.edu/tag/04E6
-/
class cofinal (F : C ⥤ D) : Prop :=
(out (d : D) : is_connected (structured_arrow d F))
attribute [instance] cofinal.out
namespace cofinal
variables (F : C ⥤ D) [cofinal F]
instance (d : D) : nonempty (structured_arrow d F) := is_connected.is_nonempty
variables {E : Type u} [category.{v} E] (G : D ⥤ E)
/--
When `F : C ⥤ D` is cofinal, we denote by `lift F d` an arbitrary choice of object in `C` such that
there exists a morphism `d ⟶ F.obj (lift F d)`.
-/
def lift (d : D) : C :=
(classical.arbitrary (structured_arrow d F)).right
/--
When `F : C ⥤ D` is cofinal, we denote by `hom_to_lift` an arbitrary choice of morphism
`d ⟶ F.obj (lift F d)`.
-/
def hom_to_lift (d : D) : d ⟶ F.obj (lift F d) :=
(classical.arbitrary (structured_arrow d F)).hom
/--
We provide an induction principle for reasoning about `lift` and `hom_to_lift`.
We want to perform some construction (usually just a proof) about
the particular choices `lift F d` and `hom_to_lift F d`,
it suffices to perform that construction for some other pair of choices
(denoted `X₀ : C` and `k₀ : d ⟶ F.obj X₀` below),
and to show that how to transport such a construction
*both* directions along a morphism between such choices.
-/
lemma induction {d : D} (Z : Π (X : C) (k : d ⟶ F.obj X), Prop)
(h₁ : Π X₁ X₂ (k₁ : d ⟶ F.obj X₁) (k₂ : d ⟶ F.obj X₂) (f : X₁ ⟶ X₂),
(k₁ ≫ F.map f = k₂) → Z X₁ k₁ → Z X₂ k₂)
(h₂ : Π X₁ X₂ (k₁ : d ⟶ F.obj X₁) (k₂ : d ⟶ F.obj X₂) (f : X₁ ⟶ X₂),
(k₁ ≫ F.map f = k₂) → Z X₂ k₂ → Z X₁ k₁)
{X₀ : C} {k₀ : d ⟶ F.obj X₀} (z : Z X₀ k₀) : Z (lift F d) (hom_to_lift F d) :=
begin
apply nonempty.some,
apply @is_preconnected_induction _ _ _
(λ (Y : structured_arrow d F), Z Y.right Y.hom) _ _ { right := X₀, hom := k₀, } z,
{ intros j₁ j₂ f a, fapply h₁ _ _ _ _ f.right _ a, convert f.w.symm, dsimp, simp, },
{ intros j₁ j₂ f a, fapply h₂ _ _ _ _ f.right _ a, convert f.w.symm, dsimp, simp, },
end
variables {F G}
/--
Given a cocone over `F ⋙ G`, we can construct a `cocone G` with the same cocone point.
-/
@[simps]
def extend_cocone : cocone (F ⋙ G) ⥤ cocone G :=
{ obj := λ c,
{ X := c.X,
ι :=
{ app := λ X, G.map (hom_to_lift F X) ≫ c.ι.app (lift F X),
naturality' := λ X Y f,
begin
dsimp, simp,
-- This would be true if we'd chosen `lift F X` to be `lift F Y`
-- and `hom_to_lift F X` to be `f ≫ hom_to_lift F Y`.
apply induction F
(λ Z k, G.map f ≫ G.map (hom_to_lift F Y) ≫ c.ι.app (lift F Y) = G.map k ≫ c.ι.app Z),
{ intros Z₁ Z₂ k₁ k₂ g a z,
rw [←a, functor.map_comp, category.assoc, ←functor.comp_map, c.w, z], },
{ intros Z₁ Z₂ k₁ k₂ g a z,
rw [←a, functor.map_comp, category.assoc, ←functor.comp_map, c.w] at z,
rw z, },
{ rw [←functor.map_comp_assoc], },
end } },
map := λ X Y f,
{ hom := f.hom, } }
@[simp]
lemma colimit_cocone_comp_aux (s : cocone (F ⋙ G)) (j : C) :
G.map (hom_to_lift F (F.obj j)) ≫ s.ι.app (lift F (F.obj j)) =
s.ι.app j :=
begin
-- This point is that this would be true if we took `lift (F.obj j)` to just be `j`
-- and `hom_to_lift (F.obj j)` to be `𝟙 (F.obj j)`.
apply induction F (λ X k, G.map k ≫ s.ι.app X = (s.ι.app j : _)),
{ intros j₁ j₂ k₁ k₂ f w h, rw ←w, rw ← s.w f at h, simpa using h, },
{ intros j₁ j₂ k₁ k₂ f w h, rw ←w at h, rw ← s.w f, simpa using h, },
{ exact s.w (𝟙 _), },
end
variables {H : Dᵒᵖ ⥤ E}
/-- An auxiliary construction for `extend_cone`, moving `op` around. -/
@[simps]
def extend_cone_cone_to_cocone {F : C ⥤ D} {H : Dᵒᵖ ⥤ E} (c : cone (F.op ⋙ H)) :
cocone (F ⋙ H.right_op) :=
{ X := op c.X,
ι :=
{ app := λ j, (c.π.app (op j)).op,
naturality' := λ j j' f,
begin apply quiver.hom.unop_inj, dsimp, simp only [category.id_comp], exact c.w f.op, end }}
/-- An auxiliary construction for `extend_cone`, moving `op` around. -/
@[simps]
def extend_cone_cocone_to_cone (c : cocone H.right_op) : cone H :=
{ X := unop c.X,
π :=
{ app := λ j, (c.ι.app (unop j)).unop,
naturality' := λ j j' f,
begin
apply quiver.hom.op_inj,
dsimp,
simp only [category.comp_id],
exact (c.w f.unop).symm,
end }}
/--
Given a cone over `F.op ⋙ H`, we can construct a `cone H` with the same cone point.
-/
@[simps]
def extend_cone : cone (F.op ⋙ H) ⥤ cone H :=
{ obj := λ c, extend_cone_cocone_to_cone (extend_cocone.obj (extend_cone_cone_to_cocone c)),
map := λ X Y f, { hom := f.hom, } }
@[simp]
lemma limit_cone_comp_aux (s : cone (F.op ⋙ H)) (j : Cᵒᵖ) :
s.π.app (op (lift F (F.obj (unop j)))) ≫ H.map (hom_to_lift F (F.obj (unop j))).op =
s.π.app j :=
begin
apply quiver.hom.op_inj,
exact colimit_cocone_comp_aux (extend_cone_cone_to_cocone s) (unop j)
end
variables (F G H)
/--
If `F` is cofinal,
the category of cocones on `F ⋙ G` is equivalent to the category of cocones on `G`,
for any `G : D ⥤ E`.
-/
@[simps]
def cocones_equiv : cocone (F ⋙ G) ≌ cocone G :=
{ functor := extend_cocone,
inverse := cocones.whiskering F,
unit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy),
counit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy), }.
/--
If `F` is cofinal,
the category of cones on `F.op ⋙ H` is equivalent to the category of cones on `H`,
for any `H : Dᵒᵖ ⥤ E`.
-/
@[simps]
def cones_equiv : cone (F.op ⋙ H) ≌ cone H :=
{ functor := extend_cone,
inverse := cones.whiskering F.op,
unit_iso := nat_iso.of_components (λ c, cones.ext (iso.refl _) (by tidy)) (by tidy),
counit_iso := nat_iso.of_components (λ c, cones.ext (iso.refl _) (by tidy)) (by tidy), }.
-- We could have done this purely formally in terms of `cocones_equiv`,
-- without having defined `extend_cone` at all,
-- but it comes at the cost of moving a *lot* of opposites around:
-- (((cones.functoriality_equivalence _ (op_op_equivalence E)).symm.trans
-- ((((cocone_equivalence_op_cone_op _).symm.trans
-- (cocones_equiv F (unop_unop _ ⋙ H.op))).trans
-- (cocone_equivalence_op_cone_op _)).unop)).trans
-- (cones.functoriality_equivalence _ (op_op_equivalence E))).trans
-- (cones.postcompose_equivalence (nat_iso.of_components (λ X, iso.refl _) (by tidy) :
-- H ≅ (unop_unop D ⋙ H.op).op ⋙ (op_op_equivalence E).functor)).symm
variables {G H}
/--
When `F : C ⥤ D` is cofinal, and `t : cocone G` for some `G : D ⥤ E`,
`t.whisker F` is a colimit cocone exactly when `t` is.
-/
def is_colimit_whisker_equiv (t : cocone G) : is_colimit (t.whisker F) ≃ is_colimit t :=
is_colimit.of_cocone_equiv (cocones_equiv F G).symm
/--
When `F : C ⥤ D` is cofinal, and `t : cone H` for some `H : Dᵒᵖ ⥤ E`,
`t.whisker F.op` is a limit cone exactly when `t` is.
-/
def is_limit_whisker_equiv (t : cone H) : is_limit (t.whisker F.op) ≃ is_limit t :=
is_limit.of_cone_equiv (cones_equiv F H).symm
/--
When `F` is cofinal, and `t : cocone (F ⋙ G)`,
`extend_cocone.obj t` is a colimit coconne exactly when `t` is.
-/
def is_colimit_extend_cocone_equiv (t : cocone (F ⋙ G)) :
is_colimit (extend_cocone.obj t) ≃ is_colimit t :=
is_colimit.of_cocone_equiv (cocones_equiv F G)
/--
When `F` is cofinal, and `t : cone (F.op ⋙ H)`,
`extend_cone.obj t` is a limit conne exactly when `t` is.
-/
def is_limit_extend_cone_equiv (t : cone (F.op ⋙ H)) :
is_limit (extend_cone.obj t) ≃ is_limit t :=
is_limit.of_cone_equiv (cones_equiv F H)
/-- Given a colimit cocone over `G : D ⥤ E` we can construct a colimit cocone over `F ⋙ G`. -/
@[simps]
def colimit_cocone_comp (t : colimit_cocone G) :
colimit_cocone (F ⋙ G) :=
{ cocone := _,
is_colimit := (is_colimit_whisker_equiv F _).symm (t.is_colimit) }
/-- Given a limit cone over `H : Dᵒᵖ ⥤ E` we can construct a limit cone over `F.op ⋙ H`. -/
@[simps]
def limit_cone_comp (t : limit_cone H) :
limit_cone (F.op ⋙ H) :=
{ cone := _,
is_limit := (is_limit_whisker_equiv F _).symm (t.is_limit) }
@[priority 100]
instance comp_has_colimit [has_colimit G] :
has_colimit (F ⋙ G) :=
has_colimit.mk (colimit_cocone_comp F (get_colimit_cocone G))
@[priority 100]
instance comp_has_limit [has_limit H] :
has_limit (F.op ⋙ H) :=
has_limit.mk (limit_cone_comp F (get_limit_cone H))
lemma colimit_pre_is_iso_aux {t : cocone G} (P : is_colimit t) :
((is_colimit_whisker_equiv F _).symm P).desc (t.whisker F) = 𝟙 t.X :=
begin
dsimp [is_colimit_whisker_equiv],
apply P.hom_ext,
intro j,
dsimp, simp, dsimp, simp, -- See library note [dsimp, simp].
end
instance colimit_pre_is_iso [has_colimit G] :
is_iso (colimit.pre G F) :=
begin
rw colimit.pre_eq (colimit_cocone_comp F (get_colimit_cocone G)) (get_colimit_cocone G),
erw colimit_pre_is_iso_aux,
dsimp,
apply_instance,
end
lemma limit_pre_is_iso_aux {t : cone H} (P : is_limit t) :
((is_limit_whisker_equiv F _).symm P).lift (t.whisker F.op) = 𝟙 t.X :=
begin
dsimp [is_limit_whisker_equiv],
apply P.hom_ext,
intro j,
simp, refl,
end
instance limit_pre_is_iso [has_limit H] :
is_iso (limit.pre H F.op) :=
begin
rw limit.pre_eq (limit_cone_comp F (get_limit_cone H)) (get_limit_cone H),
erw limit_pre_is_iso_aux,
dsimp,
apply_instance,
end
section
variables (G H)
/--
When `F : C ⥤ D` is cofinal, and `G : D ⥤ E` has a colimit, then `F ⋙ G` has a colimit also and
`colimit (F ⋙ G) ≅ colimit G`
https://stacks.math.columbia.edu/tag/04E7
-/
def colimit_iso [has_colimit G] : colimit (F ⋙ G) ≅ colimit G := as_iso (colimit.pre G F)
/--
When `F : C ⥤ D` is cofinal, and `H : Dᵒᵖ ⥤ E` has a limit, then `F.op ⋙ H` has a limit also and
`limit (F.op ⋙ H) ≅ limit H`
https://stacks.math.columbia.edu/tag/04E7
-/
def limit_iso [has_limit H] : limit (F.op ⋙ H) ≅ limit H := (as_iso (limit.pre H F.op)).symm
end
/-- Given a colimit cocone over `F ⋙ G` we can construct a colimit cocone over `G`. -/
@[simps]
def colimit_cocone_of_comp (t : colimit_cocone (F ⋙ G)) :
colimit_cocone G :=
{ cocone := extend_cocone.obj t.cocone,
is_colimit := (is_colimit_extend_cocone_equiv F _).symm (t.is_colimit), }
/-- Given a limit cone over `F.op ⋙ H` we can construct a limit cone over `H`. -/
@[simps]
def limit_cone_of_comp (t : limit_cone (F.op ⋙ H)) :
limit_cone H :=
{ cone := extend_cone.obj t.cone,
is_limit := (is_limit_extend_cone_equiv F _).symm (t.is_limit), }
/--
When `F` is cofinal, and `F ⋙ G` has a colimit, then `G` has a colimit also.
We can't make this an instance, because `F` is not determined by the goal.
(Even if this weren't a problem, it would cause a loop with `comp_has_colimit`.)
-/
lemma has_colimit_of_comp [has_colimit (F ⋙ G)] :
has_colimit G :=
has_colimit.mk (colimit_cocone_of_comp F (get_colimit_cocone (F ⋙ G)))
/--
When `F` is cofinal, and `F.op ⋙ H` has a limit, then `H` has a limit also.
We can't make this an instance, because `F` is not determined by the goal.
(Even if this weren't a problem, it would cause a loop with `comp_has_limit`.)
-/
lemma has_limit_of_comp [has_limit (F.op ⋙ H)] :
has_limit H :=
has_limit.mk (limit_cone_of_comp F (get_limit_cone (F.op ⋙ H)))
section
local attribute [instance] has_colimit_of_comp has_limit_of_comp
/--
When `F` is cofinal, and `F ⋙ G` has a colimit, then `G` has a colimit also and
`colimit (F ⋙ G) ≅ colimit G`
https://stacks.math.columbia.edu/tag/04E7
-/
def colimit_iso' [has_colimit (F ⋙ G)] : colimit (F ⋙ G) ≅ colimit G := as_iso (colimit.pre G F)
/--
When `F` is cofinal, and `F.op ⋙ H` has a limit, then `H` has a limit also and
`limit (F.op ⋙ H) ≅ limit H`
https://stacks.math.columbia.edu/tag/04E7
-/
def limit_iso' [has_limit (F.op ⋙ H)] : limit (F.op ⋙ H) ≅ limit H :=
(as_iso (limit.pre H F.op)).symm
end
/--
If the universal morphism `colimit (F ⋙ coyoneda.obj (op d)) ⟶ colimit (coyoneda.obj (op d))`
is an isomorphism (as it always is when `F` is cofinal),
then `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit`
(simply because `colimit (coyoneda.obj (op d)) ≅ punit`).
-/
def colimit_comp_coyoneda_iso (d : D) [is_iso (colimit.pre (coyoneda.obj (op d)) F)] :
colimit (F ⋙ coyoneda.obj (op d)) ≅ punit :=
as_iso (colimit.pre (coyoneda.obj (op d)) F) ≪≫ coyoneda.colimit_coyoneda_iso (op d)
lemma zigzag_of_eqv_gen_quot_rel {F : C ⥤ D} {d : D} {f₁ f₂ : Σ X, d ⟶ F.obj X}
(t : eqv_gen (types.quot.rel (F ⋙ coyoneda.obj (op d))) f₁ f₂) :
zigzag (structured_arrow.mk f₁.2) (structured_arrow.mk f₂.2) :=
begin
induction t,
case eqv_gen.rel : x y r
{ obtain ⟨f, w⟩ := r,
fconstructor,
swap 2, fconstructor,
left, fsplit,
exact { right := f, } },
case eqv_gen.refl
{ fconstructor, },
case eqv_gen.symm : x y h ih
{ apply zigzag_symmetric,
exact ih, },
case eqv_gen.trans : x y z h₁ h₂ ih₁ ih₂
{ apply relation.refl_trans_gen.trans,
exact ih₁, exact ih₂, }
end
/--
If `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit` for all `d : D`, then `F` is cofinal.
-/
lemma cofinal_of_colimit_comp_coyoneda_iso_punit
(I : Π d, colimit (F ⋙ coyoneda.obj (op d)) ≅ punit) : cofinal F :=
⟨λ d, begin
haveI : nonempty (structured_arrow d F) := by
{ have := (I d).inv punit.star,
obtain ⟨j, y, rfl⟩ := limits.types.jointly_surjective' this,
exact ⟨structured_arrow.mk y⟩, },
apply zigzag_is_connected,
rintros ⟨⟨⟩,X₁,f₁⟩ ⟨⟨⟩,X₂,f₂⟩,
dsimp at *,
let y₁ := colimit.ι (F ⋙ coyoneda.obj (op d)) X₁ f₁,
let y₂ := colimit.ι (F ⋙ coyoneda.obj (op d)) X₂ f₂,
have e : y₁ = y₂,
{ apply (I d).to_equiv.injective, ext, },
have t := types.colimit_eq e,
clear e y₁ y₂,
exact zigzag_of_eqv_gen_quot_rel t,
end⟩
end cofinal
end category_theory
|
ffec91543bae8a5e6cbe84eaa016ac322c56700c | 32025d5c2d6e33ad3b6dd8a3c91e1e838066a7f7 | /stage0/src/Init/System/ST.lean | 0930afb4270ada7bac1ed407fe6bb29c08f975e0 | [
"Apache-2.0"
] | permissive | walterhu1015/lean4 | b2c71b688975177402758924eaa513475ed6ce72 | 2214d81e84646a905d0b20b032c89caf89c737ad | refs/heads/master | 1,671,342,096,906 | 1,599,695,985,000 | 1,599,695,985,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 4,350 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.Control.EState
import Init.Control.Reader
def EST (ε : Type) (σ : Type) : Type → Type := EStateM ε σ
abbrev ST (σ : Type) := EST Empty σ
instance (ε σ : Type) : Monad (EST ε σ) := inferInstanceAs (Monad (EStateM _ _))
instance (ε σ : Type) : MonadExceptOf ε (EST ε σ) := inferInstanceAs (MonadExceptOf ε (EStateM _ _))
instance {ε σ : Type} {α : Type} [Inhabited ε] : Inhabited (EST ε σ α) := inferInstanceAs (Inhabited (EStateM _ _ _))
instance (σ : Type) : Monad (ST σ) := inferInstanceAs (Monad (EST _ _))
-- Auxiliary class for inferring the "state" of `EST` and `ST` monads
class STWorld (σ : outParam Type) (m : Type → Type)
instance STWorld.trans {σ m n} [STWorld σ m] [MonadLift m n] : STWorld σ n := ⟨⟩
instance STWorld.base {ε σ} : STWorld σ (EST ε σ) := ⟨⟩
def runEST {ε α : Type} (x : forall (σ : Type), EST ε σ α) : Except ε α :=
match x Unit () with
| EStateM.Result.ok a _ => Except.ok a
| EStateM.Result.error ex _ => Except.error ex
def runST {α : Type} (x : forall (σ : Type), ST σ α) : α :=
match x Unit () with
| EStateM.Result.ok a _ => a
| EStateM.Result.error ex _ => Empty.rec _ ex
instance st2est {ε σ} : MonadLift (ST σ) (EST ε σ) :=
⟨fun α x s => match x s with
| EStateM.Result.ok a s => EStateM.Result.ok a s
| EStateM.Result.error ex _ => Empty.rec _ ex⟩
namespace ST
/- References -/
constant RefPointed : PointedType.{0} := arbitrary _
structure Ref (σ : Type) (α : Type) : Type :=
(ref : RefPointed.type) (h : Nonempty α)
instance Ref.inhabited {σ α} [Inhabited α] : Inhabited (Ref σ α) :=
⟨{ ref := RefPointed.val, h := Nonempty.intro $ arbitrary _}⟩
namespace Prim
/- Auxiliary definition for showing that `ST σ α` is inhabited when we have a `Ref σ α` -/
private noncomputable def inhabitedFromRef {σ α} (r : Ref σ α) : ST σ α :=
pure $ (Classical.inhabitedOfNonempty r.h).default
@[extern "lean_st_mk_ref"]
constant mkRef {σ α} (a : α) : ST σ (Ref σ α) := pure { ref := RefPointed.val, h := Nonempty.intro a }
@[extern "lean_st_ref_get"]
constant Ref.get {σ α} (r : @& Ref σ α) : ST σ α := inhabitedFromRef r
@[extern "lean_st_ref_set"]
constant Ref.set {σ α} (r : @& Ref σ α) (a : α) : ST σ Unit := arbitrary _
@[extern "lean_st_ref_swap"]
constant Ref.swap {σ α} (r : @& Ref σ α) (a : α) : ST σ α := inhabitedFromRef r
@[extern "lean_st_ref_take"]
unsafe constant Ref.take {σ α} (r : @& Ref σ α) : ST σ α := inhabitedFromRef r
@[extern "lean_st_ref_ptr_eq"]
constant Ref.ptrEq {σ α} (r1 r2 : @& Ref σ α) : ST σ Bool := arbitrary _
@[inline] unsafe def Ref.modifyUnsafe {σ α : Type} (r : Ref σ α) (f : α → α) : ST σ Unit := do
v ← Ref.take r;
Ref.set r (f v)
@[inline] unsafe def Ref.modifyGetUnsafe {σ α β : Type} (r : Ref σ α) (f : α → β × α) : ST σ β := do
v ← Ref.take r;
let (b, a) := f v;
Ref.set r a;
pure b
@[implementedBy Ref.modifyUnsafe]
def Ref.modify {σ α : Type} (r : Ref σ α) (f : α → α) : ST σ Unit := do
v ← Ref.get r;
Ref.set r (f v)
@[implementedBy Ref.modifyGetUnsafe]
def Ref.modifyGet {σ α β : Type} (r : Ref σ α) (f : α → β × α) : ST σ β := do
v ← Ref.get r;
let (b, a) := f v;
Ref.set r a;
pure b
end Prim
section
variables {σ : Type} {m : Type → Type} [Monad m] [MonadLiftT (ST σ) m]
@[inline] def mkRef {α : Type} (a : α) : m (Ref σ α) := liftM $ Prim.mkRef a
@[inline] def Ref.get {α : Type} (r : Ref σ α) : m α := liftM $ Prim.Ref.get r
@[inline] def Ref.set {α : Type} (r : Ref σ α) (a : α) : m Unit := liftM $ Prim.Ref.set r a
@[inline] def Ref.swap {α : Type} (r : Ref σ α) (a : α) : m α := liftM $ Prim.Ref.swap r a
@[inline] unsafe def Ref.take {α : Type} (r : Ref σ α) : m α := liftM $ Prim.Ref.take r
@[inline] def Ref.ptrEq {α : Type} (r1 r2 : Ref σ α) : m Bool := liftM $ Prim.Ref.ptrEq r1 r2
@[inline] def Ref.modify {α : Type} (r : Ref σ α) (f : α → α) : m Unit := liftM $ Prim.Ref.modify r f
@[inline] def Ref.modifyGet {α : Type} {β : Type} (r : Ref σ α) (f : α → β × α) : m β := liftM $ Prim.Ref.modifyGet r f
end
end ST
|
a667dd3f0bef718700b51c0524a904fec8e10714 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/group_theory/presented_group.lean | 27dd9b037d0a053238beecb69ae3428a3da8b0f8 | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 3,095 | lean | /-
Copyright (c) 2019 Michael Howes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Howes
-/
import group_theory.free_group
import group_theory.quotient_group
/-!
# Defining a group given by generators and relations
Given a subset `rels` of relations of the free group on a type `α`, this file constructs the group
given by generators `x : α` and relations `r ∈ rels`.
## Main definitions
* `presented_group rels`: the quotient group of the free group on a type `α` by a subset `rels` of
relations of the free group on `α`.
* `of`: The canonical map from `α` to a presented group with generators `α`.
* `to_group f`: the canonical group homomorphism `presented_group rels → G`, given a function
`f : α → G` from a type `α` to a group `G` which satisfies the relations `rels`.
## Tags
generators, relations, group presentations
-/
variables {α : Type}
/-- Given a set of relations, rels, over a type `α`, presented_group constructs the group with
generators `x : α` and relations `rels` as a quotient of free_group `α`.-/
def presented_group (rels : set (free_group α)) : Type :=
quotient_group.quotient $ subgroup.normal_closure rels
namespace presented_group
instance (rels : set (free_group α)) : group (presented_group (rels)) :=
quotient_group.quotient.group _
/-- `of` is the canonical map from `α` to a presented group with generators `x : α`. The term `x` is
mapped to the equivalence class of the image of `x` in `free_group α`. -/
def of {rels : set (free_group α)} (x : α) : presented_group rels :=
quotient_group.mk (free_group.of x)
section to_group
/-
Presented groups satisfy a universal property. If `G` is a group and `f : α → G` is a map such that
the images of `f` satisfy all the given relations, then `f` extends uniquely to a group homomorphism
from `presented_group rels` to `G`.
-/
variables {G : Type} [group G] {f : α → G} {rels : set (free_group α)}
local notation `F` := free_group.lift f
variable (h : ∀ r ∈ rels, F r = 1)
lemma closure_rels_subset_ker : subgroup.normal_closure rels ≤ monoid_hom.ker F :=
subgroup.normal_closure_le_normal (λ x w, (monoid_hom.mem_ker _).2 (h x w))
lemma to_group_eq_one_of_mem_closure : ∀ x ∈ subgroup.normal_closure rels, F x = 1 :=
λ x w, (monoid_hom.mem_ker _).1 $ closure_rels_subset_ker h w
/-- The extension of a map `f : α → G` that satisfies the given relations to a group homomorphism
from `presented_group rels → G`. -/
def to_group : presented_group rels →* G :=
quotient_group.lift (subgroup.normal_closure rels) F (to_group_eq_one_of_mem_closure h)
@[simp] lemma to_group.of {x : α} : to_group h (of x) = f x := free_group.lift.of
theorem to_group.unique (g : presented_group rels →* G)
(hg : ∀ x : α, g (of x) = f x) : ∀ {x}, g x = to_group h x :=
λ x, quotient_group.induction_on x
(λ _, free_group.lift.unique (g.comp (quotient_group.mk' _)) hg)
end to_group
instance (rels : set (free_group α)) : inhabited (presented_group rels) := ⟨1⟩
end presented_group
|
e4e2b677c3cff83111ac0f366e4f44fb24c4614d | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/analysis/inner_product_space/adjoint.lean | 044b0de10850d3ecb44b6ffc70f7649d0b389e5d | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 18,841 | lean | /-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import analysis.inner_product_space.dual
import analysis.inner_product_space.pi_L2
/-!
# Adjoint of operators on Hilbert spaces
Given an operator `A : E →L[𝕜] F`, where `E` and `F` are Hilbert spaces, its adjoint
`adjoint A : F →L[𝕜] E` is the unique operator such that `⟪x, A y⟫ = ⟪adjoint A x, y⟫` for all
`x` and `y`.
We then use this to put a C⋆-algebra structure on `E →L[𝕜] E` with the adjoint as the star
operation.
This construction is used to define an adjoint for linear maps (i.e. not continuous) between
finite dimensional spaces.
## Main definitions
* `continuous_linear_map.adjoint : (E →L[𝕜] F) ≃ₗᵢ⋆[𝕜] (F →L[𝕜] E)`: the adjoint of a continuous
linear map, bundled as a conjugate-linear isometric equivalence.
* `linear_map.adjoint : (E →ₗ[𝕜] F) ≃ₗ⋆[𝕜] (F →ₗ[𝕜] E)`: the adjoint of a linear map between
finite-dimensional spaces, this time only as a conjugate-linear equivalence, since there is no
norm defined on these maps.
## Implementation notes
* The continuous conjugate-linear version `adjoint_aux` is only an intermediate
definition and is not meant to be used outside this file.
## Tags
adjoint
-/
noncomputable theory
open is_R_or_C
open_locale complex_conjugate
variables {𝕜 E F G : Type*} [is_R_or_C 𝕜]
variables [inner_product_space 𝕜 E] [inner_product_space 𝕜 F] [inner_product_space 𝕜 G]
local notation `⟪`x`, `y`⟫` := @inner 𝕜 _ _ x y
/-! ### Adjoint operator -/
open inner_product_space
namespace continuous_linear_map
variables [complete_space E] [complete_space G]
/-- The adjoint, as a continuous conjugate-linear map. This is only meant as an auxiliary
definition for the main definition `adjoint`, where this is bundled as a conjugate-linear isometric
equivalence. -/
def adjoint_aux : (E →L[𝕜] F) →L⋆[𝕜] (F →L[𝕜] E) :=
(continuous_linear_map.compSL _ _ _ _ _ ((to_dual 𝕜 E).symm : normed_space.dual 𝕜 E →L⋆[𝕜] E)).comp
(to_sesq_form : (E →L[𝕜] F) →L[𝕜] F →L⋆[𝕜] normed_space.dual 𝕜 E)
@[simp] lemma adjoint_aux_apply (A : E →L[𝕜] F) (x : F) :
adjoint_aux A x = ((to_dual 𝕜 E).symm : (normed_space.dual 𝕜 E) → E) ((to_sesq_form A) x) := rfl
lemma adjoint_aux_inner_left (A : E →L[𝕜] F) (x : E) (y : F) : ⟪adjoint_aux A y, x⟫ = ⟪y, A x⟫ :=
by { simp only [adjoint_aux_apply, to_dual_symm_apply, to_sesq_form_apply_coe, coe_comp',
innerSL_apply_coe]}
lemma adjoint_aux_inner_right (A : E →L[𝕜] F) (x : E) (y : F) : ⟪x, adjoint_aux A y⟫ = ⟪A x, y⟫ :=
by rw [←inner_conj_sym, adjoint_aux_inner_left, inner_conj_sym]
variables [complete_space F]
lemma adjoint_aux_adjoint_aux (A : E →L[𝕜] F) : adjoint_aux (adjoint_aux A) = A :=
begin
ext v,
refine ext_inner_left 𝕜 (λ w, _),
rw [adjoint_aux_inner_right, adjoint_aux_inner_left],
end
@[simp] lemma adjoint_aux_norm (A : E →L[𝕜] F) : ‖adjoint_aux A‖ = ‖A‖ :=
begin
refine le_antisymm _ _,
{ refine continuous_linear_map.op_norm_le_bound _ (norm_nonneg _) (λ x, _),
rw [adjoint_aux_apply, linear_isometry_equiv.norm_map],
exact to_sesq_form_apply_norm_le },
{ nth_rewrite_lhs 0 [←adjoint_aux_adjoint_aux A],
refine continuous_linear_map.op_norm_le_bound _ (norm_nonneg _) (λ x, _),
rw [adjoint_aux_apply, linear_isometry_equiv.norm_map],
exact to_sesq_form_apply_norm_le }
end
/-- The adjoint of a bounded operator from Hilbert space E to Hilbert space F. -/
def adjoint : (E →L[𝕜] F) ≃ₗᵢ⋆[𝕜] (F →L[𝕜] E) :=
linear_isometry_equiv.of_surjective
{ norm_map' := adjoint_aux_norm,
..adjoint_aux }
(λ A, ⟨adjoint_aux A, adjoint_aux_adjoint_aux A⟩)
localized "postfix (name := adjoint) `†`:1000 := continuous_linear_map.adjoint" in inner_product
/-- The fundamental property of the adjoint. -/
lemma adjoint_inner_left (A : E →L[𝕜] F) (x : E) (y : F) : ⟪A† y, x⟫ = ⟪y, A x⟫ :=
adjoint_aux_inner_left A x y
/-- The fundamental property of the adjoint. -/
lemma adjoint_inner_right (A : E →L[𝕜] F) (x : E) (y : F) : ⟪x, A† y⟫ = ⟪A x, y⟫ :=
adjoint_aux_inner_right A x y
/-- The adjoint is involutive -/
@[simp] lemma adjoint_adjoint (A : E →L[𝕜] F) : A†† = A :=
adjoint_aux_adjoint_aux A
/-- The adjoint of the composition of two operators is the composition of the two adjoints
in reverse order. -/
@[simp] lemma adjoint_comp (A : F →L[𝕜] G) (B : E →L[𝕜] F) : (A ∘L B)† = B† ∘L A† :=
begin
ext v,
refine ext_inner_left 𝕜 (λ w, _),
simp only [adjoint_inner_right, continuous_linear_map.coe_comp', function.comp_app],
end
lemma apply_norm_sq_eq_inner_adjoint_left (A : E →L[𝕜] E) (x : E) : ‖A x‖^2 = re ⟪(A† * A) x, x⟫ :=
have h : ⟪(A† * A) x, x⟫ = ⟪A x, A x⟫ := by { rw [←adjoint_inner_left], refl },
by rw [h, ←inner_self_eq_norm_sq _]
lemma apply_norm_eq_sqrt_inner_adjoint_left (A : E →L[𝕜] E) (x : E) :
‖A x‖ = real.sqrt (re ⟪(A† * A) x, x⟫) :=
by rw [←apply_norm_sq_eq_inner_adjoint_left, real.sqrt_sq (norm_nonneg _)]
lemma apply_norm_sq_eq_inner_adjoint_right (A : E →L[𝕜] E) (x : E) : ‖A x‖^2 = re ⟪x, (A† * A) x⟫ :=
have h : ⟪x, (A† * A) x⟫ = ⟪A x, A x⟫ := by { rw [←adjoint_inner_right], refl },
by rw [h, ←inner_self_eq_norm_sq _]
lemma apply_norm_eq_sqrt_inner_adjoint_right (A : E →L[𝕜] E) (x : E) :
‖A x‖ = real.sqrt (re ⟪x, (A† * A) x⟫) :=
by rw [←apply_norm_sq_eq_inner_adjoint_right, real.sqrt_sq (norm_nonneg _)]
/-- The adjoint is unique: a map `A` is the adjoint of `B` iff it satisfies `⟪A x, y⟫ = ⟪x, B y⟫`
for all `x` and `y`. -/
lemma eq_adjoint_iff (A : E →L[𝕜] F) (B : F →L[𝕜] E) :
A = B† ↔ (∀ x y, ⟪A x, y⟫ = ⟪x, B y⟫) :=
begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, _⟩,
ext x,
exact ext_inner_right 𝕜 (λ y, by simp only [adjoint_inner_left, h x y])
end
@[simp] lemma adjoint_id : (continuous_linear_map.id 𝕜 E).adjoint = continuous_linear_map.id 𝕜 E :=
begin
refine eq.symm _,
rw eq_adjoint_iff,
simp,
end
lemma _root_.submodule.adjoint_subtypeL (U : submodule 𝕜 E)
[complete_space U] :
(U.subtypeL)† = orthogonal_projection U :=
begin
symmetry,
rw eq_adjoint_iff,
intros x u,
rw [U.coe_inner, inner_orthogonal_projection_left_eq_right,
orthogonal_projection_mem_subspace_eq_self],
refl
end
lemma _root_.submodule.adjoint_orthogonal_projection (U : submodule 𝕜 E)
[complete_space U] :
(orthogonal_projection U : E →L[𝕜] U)† = U.subtypeL :=
by rw [← U.adjoint_subtypeL, adjoint_adjoint]
/-- `E →L[𝕜] E` is a star algebra with the adjoint as the star operation. -/
instance : has_star (E →L[𝕜] E) := ⟨adjoint⟩
instance : has_involutive_star (E →L[𝕜] E) := ⟨adjoint_adjoint⟩
instance : star_semigroup (E →L[𝕜] E) := ⟨adjoint_comp⟩
instance : star_ring (E →L[𝕜] E) := ⟨linear_isometry_equiv.map_add adjoint⟩
instance : star_module 𝕜 (E →L[𝕜] E) := ⟨linear_isometry_equiv.map_smulₛₗ adjoint⟩
lemma star_eq_adjoint (A : E →L[𝕜] E) : star A = A† := rfl
/-- A continuous linear operator is self-adjoint iff it is equal to its adjoint. -/
lemma is_self_adjoint_iff' {A : E →L[𝕜] E} : is_self_adjoint A ↔ A.adjoint = A := iff.rfl
instance : cstar_ring (E →L[𝕜] E) :=
⟨begin
intros A,
rw [star_eq_adjoint],
refine le_antisymm _ _,
{ calc ‖A† * A‖ ≤ ‖A†‖ * ‖A‖ : op_norm_comp_le _ _
... = ‖A‖ * ‖A‖ : by rw [linear_isometry_equiv.norm_map] },
{ rw [←sq, ←real.sqrt_le_sqrt_iff (norm_nonneg _), real.sqrt_sq (norm_nonneg _)],
refine op_norm_le_bound _ (real.sqrt_nonneg _) (λ x, _),
have := calc
re ⟪(A† * A) x, x⟫ ≤ ‖(A† * A) x‖ * ‖x‖ : re_inner_le_norm _ _
... ≤ ‖A† * A‖ * ‖x‖ * ‖x‖ : mul_le_mul_of_nonneg_right
(le_op_norm _ _) (norm_nonneg _),
calc ‖A x‖ = real.sqrt (re ⟪(A† * A) x, x⟫) : by rw [apply_norm_eq_sqrt_inner_adjoint_left]
... ≤ real.sqrt (‖A† * A‖ * ‖x‖ * ‖x‖) : real.sqrt_le_sqrt this
... = real.sqrt (‖A† * A‖) * ‖x‖
: by rw [mul_assoc, real.sqrt_mul (norm_nonneg _), real.sqrt_mul_self (norm_nonneg _)] }
end⟩
section real
variables {E' : Type*} {F' : Type*} [inner_product_space ℝ E'] [inner_product_space ℝ F']
variables [complete_space E'] [complete_space F']
-- Todo: Generalize this to `is_R_or_C`.
lemma is_adjoint_pair_inner (A : E' →L[ℝ] F') :
linear_map.is_adjoint_pair (sesq_form_of_inner : E' →ₗ[ℝ] E' →ₗ[ℝ] ℝ)
(sesq_form_of_inner : F' →ₗ[ℝ] F' →ₗ[ℝ] ℝ) A (A†) :=
λ x y, by simp only [sesq_form_of_inner_apply_apply, adjoint_inner_left, to_linear_map_eq_coe,
coe_coe]
end real
end continuous_linear_map
/-! ### Self-adjoint operators -/
namespace is_self_adjoint
open continuous_linear_map
variables [complete_space E] [complete_space F]
lemma adjoint_eq {A : E →L[𝕜] E} (hA : is_self_adjoint A) : A.adjoint = A := hA
/-- Every self-adjoint operator on an inner product space is symmetric. -/
lemma is_symmetric {A : E →L[𝕜] E} (hA : is_self_adjoint A) :
(A : E →ₗ[𝕜] E).is_symmetric :=
λ x y, by rw_mod_cast [←A.adjoint_inner_right, hA.adjoint_eq]
/-- Conjugating preserves self-adjointness -/
lemma conj_adjoint {T : E →L[𝕜] E} (hT : is_self_adjoint T) (S : E →L[𝕜] F) :
is_self_adjoint (S ∘L T ∘L S.adjoint) :=
begin
rw is_self_adjoint_iff' at ⊢ hT,
simp only [hT, adjoint_comp, adjoint_adjoint],
exact continuous_linear_map.comp_assoc _ _ _,
end
/-- Conjugating preserves self-adjointness -/
lemma adjoint_conj {T : E →L[𝕜] E} (hT : is_self_adjoint T) (S : F →L[𝕜] E) :
is_self_adjoint (S.adjoint ∘L T ∘L S) :=
begin
rw is_self_adjoint_iff' at ⊢ hT,
simp only [hT, adjoint_comp, adjoint_adjoint],
exact continuous_linear_map.comp_assoc _ _ _,
end
lemma _root_.continuous_linear_map.is_self_adjoint_iff_is_symmetric {A : E →L[𝕜] E} :
is_self_adjoint A ↔ (A : E →ₗ[𝕜] E).is_symmetric :=
⟨λ hA, hA.is_symmetric, λ hA, ext $ λ x, ext_inner_right 𝕜 $
λ y, (A.adjoint_inner_left y x).symm ▸ (hA x y).symm⟩
lemma _root_.linear_map.is_symmetric.is_self_adjoint {A : E →L[𝕜] E}
(hA : (A : E →ₗ[𝕜] E).is_symmetric) : is_self_adjoint A :=
by rwa ←continuous_linear_map.is_self_adjoint_iff_is_symmetric at hA
/-- The orthogonal projection is self-adjoint. -/
lemma _root_.orthogonal_projection_is_self_adjoint (U : submodule 𝕜 E)
[complete_space U] :
is_self_adjoint (U.subtypeL ∘L orthogonal_projection U) :=
(orthogonal_projection_is_symmetric U).is_self_adjoint
lemma conj_orthogonal_projection {T : E →L[𝕜] E}
(hT : is_self_adjoint T) (U : submodule 𝕜 E) [complete_space U] :
is_self_adjoint (U.subtypeL ∘L orthogonal_projection U ∘L T ∘L U.subtypeL ∘L
orthogonal_projection U) :=
begin
rw ←continuous_linear_map.comp_assoc,
nth_rewrite 0 ←(orthogonal_projection_is_self_adjoint U).adjoint_eq,
refine hT.adjoint_conj _,
end
end is_self_adjoint
namespace linear_map
variables [complete_space E]
variables {T : E →ₗ[𝕜] E}
/-- The **Hellinger--Toeplitz theorem**: Construct a self-adjoint operator from an everywhere
defined symmetric operator.-/
def is_symmetric.to_self_adjoint (hT : is_symmetric T) : self_adjoint (E →L[𝕜] E) :=
⟨⟨T, hT.continuous⟩, continuous_linear_map.is_self_adjoint_iff_is_symmetric.mpr hT⟩
lemma is_symmetric.coe_to_self_adjoint (hT : is_symmetric T) :
(hT.to_self_adjoint : E →ₗ[𝕜] E) = T := rfl
lemma is_symmetric.to_self_adjoint_apply (hT : is_symmetric T) {x : E} :
hT.to_self_adjoint x = T x := rfl
end linear_map
namespace linear_map
variables [finite_dimensional 𝕜 E] [finite_dimensional 𝕜 F] [finite_dimensional 𝕜 G]
local attribute [instance, priority 20] finite_dimensional.complete
/-- The adjoint of an operator from the finite-dimensional inner product space E to the finite-
dimensional inner product space F. -/
def adjoint : (E →ₗ[𝕜] F) ≃ₗ⋆[𝕜] (F →ₗ[𝕜] E) :=
((linear_map.to_continuous_linear_map : (E →ₗ[𝕜] F) ≃ₗ[𝕜] (E →L[𝕜] F)).trans
continuous_linear_map.adjoint.to_linear_equiv).trans
linear_map.to_continuous_linear_map.symm
lemma adjoint_to_continuous_linear_map (A : E →ₗ[𝕜] F) :
A.adjoint.to_continuous_linear_map = A.to_continuous_linear_map.adjoint := rfl
lemma adjoint_eq_to_clm_adjoint (A : E →ₗ[𝕜] F) :
A.adjoint = A.to_continuous_linear_map.adjoint := rfl
/-- The fundamental property of the adjoint. -/
lemma adjoint_inner_left (A : E →ₗ[𝕜] F) (x : E) (y : F) : ⟪adjoint A y, x⟫ = ⟪y, A x⟫ :=
begin
rw [←coe_to_continuous_linear_map A, adjoint_eq_to_clm_adjoint],
exact continuous_linear_map.adjoint_inner_left _ x y,
end
/-- The fundamental property of the adjoint. -/
lemma adjoint_inner_right (A : E →ₗ[𝕜] F) (x : E) (y : F) : ⟪x, adjoint A y⟫ = ⟪A x, y⟫ :=
begin
rw [←coe_to_continuous_linear_map A, adjoint_eq_to_clm_adjoint],
exact continuous_linear_map.adjoint_inner_right _ x y,
end
/-- The adjoint is involutive -/
@[simp] lemma adjoint_adjoint (A : E →ₗ[𝕜] F) : A.adjoint.adjoint = A :=
begin
ext v,
refine ext_inner_left 𝕜 (λ w, _),
rw [adjoint_inner_right, adjoint_inner_left],
end
/-- The adjoint of the composition of two operators is the composition of the two adjoints
in reverse order. -/
@[simp] lemma adjoint_comp (A : F →ₗ[𝕜] G) (B : E →ₗ[𝕜] F) :
(A ∘ₗ B).adjoint = B.adjoint ∘ₗ A.adjoint :=
begin
ext v,
refine ext_inner_left 𝕜 (λ w, _),
simp only [adjoint_inner_right, linear_map.coe_comp, function.comp_app],
end
/-- The adjoint is unique: a map `A` is the adjoint of `B` iff it satisfies `⟪A x, y⟫ = ⟪x, B y⟫`
for all `x` and `y`. -/
lemma eq_adjoint_iff (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
A = B.adjoint ↔ (∀ x y, ⟪A x, y⟫ = ⟪x, B y⟫) :=
begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, _⟩,
ext x,
exact ext_inner_right 𝕜 (λ y, by simp only [adjoint_inner_left, h x y])
end
/-- The adjoint is unique: a map `A` is the adjoint of `B` iff it satisfies `⟪A x, y⟫ = ⟪x, B y⟫`
for all basis vectors `x` and `y`. -/
lemma eq_adjoint_iff_basis {ι₁ : Type*} {ι₂ : Type*} (b₁ : basis ι₁ 𝕜 E) (b₂ : basis ι₂ 𝕜 F)
(A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
A = B.adjoint ↔ (∀ (i₁ : ι₁) (i₂ : ι₂), ⟪A (b₁ i₁), b₂ i₂⟫ = ⟪b₁ i₁, B (b₂ i₂)⟫) :=
begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, _⟩,
refine basis.ext b₁ (λ i₁, _),
exact ext_inner_right_basis b₂ (λ i₂, by simp only [adjoint_inner_left, h i₁ i₂]),
end
lemma eq_adjoint_iff_basis_left {ι : Type*} (b : basis ι 𝕜 E) (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
A = B.adjoint ↔ (∀ i y, ⟪A (b i), y⟫ = ⟪b i, B y⟫) :=
begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, basis.ext b (λ i, _)⟩,
exact ext_inner_right 𝕜 (λ y, by simp only [h i, adjoint_inner_left]),
end
lemma eq_adjoint_iff_basis_right {ι : Type*} (b : basis ι 𝕜 F) (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
A = B.adjoint ↔ (∀ i x, ⟪A x, b i⟫ = ⟪x, B (b i)⟫) :=
begin
refine ⟨λ h x y, by rw [h, adjoint_inner_left], λ h, _⟩,
ext x,
refine ext_inner_right_basis b (λ i, by simp only [h i, adjoint_inner_left]),
end
/-- `E →ₗ[𝕜] E` is a star algebra with the adjoint as the star operation. -/
instance : has_star (E →ₗ[𝕜] E) := ⟨adjoint⟩
instance : has_involutive_star (E →ₗ[𝕜] E) := ⟨adjoint_adjoint⟩
instance : star_semigroup (E →ₗ[𝕜] E) := ⟨adjoint_comp⟩
instance : star_ring (E →ₗ[𝕜] E) := ⟨linear_equiv.map_add adjoint⟩
instance : star_module 𝕜 (E →ₗ[𝕜] E) := ⟨linear_equiv.map_smulₛₗ adjoint⟩
lemma star_eq_adjoint (A : E →ₗ[𝕜] E) : star A = A.adjoint := rfl
/-- A continuous linear operator is self-adjoint iff it is equal to its adjoint. -/
lemma is_self_adjoint_iff' {A : E →ₗ[𝕜] E} : is_self_adjoint A ↔ A.adjoint = A := iff.rfl
lemma is_symmetric_iff_is_self_adjoint (A : E →ₗ[𝕜] E) :
is_symmetric A ↔ is_self_adjoint A :=
by { rw [is_self_adjoint_iff', is_symmetric, ← linear_map.eq_adjoint_iff], exact eq_comm }
section real
variables {E' : Type*} {F' : Type*} [inner_product_space ℝ E'] [inner_product_space ℝ F']
variables [finite_dimensional ℝ E'] [finite_dimensional ℝ F']
-- Todo: Generalize this to `is_R_or_C`.
lemma is_adjoint_pair_inner (A : E' →ₗ[ℝ] F') :
is_adjoint_pair (sesq_form_of_inner : E' →ₗ[ℝ] E' →ₗ[ℝ] ℝ)
(sesq_form_of_inner : F' →ₗ[ℝ] F' →ₗ[ℝ] ℝ) A A.adjoint :=
λ x y, by simp only [sesq_form_of_inner_apply_apply, adjoint_inner_left]
end real
/-- The Gram operator T†T is symmetric. -/
lemma is_symmetric_adjoint_mul_self (T : E →ₗ[𝕜] E) : is_symmetric (T.adjoint * T) :=
λ x y, by simp only [mul_apply, adjoint_inner_left, adjoint_inner_right]
/-- The Gram operator T†T is a positive operator. -/
lemma re_inner_adjoint_mul_self_nonneg (T : E →ₗ[𝕜] E) (x : E) :
0 ≤ re ⟪ x, (T.adjoint * T) x ⟫ := by {simp only [mul_apply, adjoint_inner_right,
inner_self_eq_norm_sq_to_K], norm_cast, exact sq_nonneg _}
@[simp] lemma im_inner_adjoint_mul_self_eq_zero (T : E →ₗ[𝕜] E) (x : E) :
im ⟪ x, linear_map.adjoint T (T x) ⟫ = 0 := by {simp only [mul_apply,
adjoint_inner_right, inner_self_eq_norm_sq_to_K], norm_cast}
end linear_map
namespace matrix
variables {m n : Type*} [fintype m] [decidable_eq m] [fintype n] [decidable_eq n]
open_locale complex_conjugate
/-- The adjoint of the linear map associated to a matrix is the linear map associated to the
conjugate transpose of that matrix. -/
lemma conj_transpose_eq_adjoint (A : matrix m n 𝕜) :
to_lin' A.conj_transpose =
@linear_map.adjoint _ (euclidean_space 𝕜 n) (euclidean_space 𝕜 m) _ _ _ _ _ (to_lin' A) :=
begin
rw @linear_map.eq_adjoint_iff _ (euclidean_space 𝕜 m) (euclidean_space 𝕜 n),
intros x y,
convert dot_product_assoc (conj ∘ (id x : m → 𝕜)) y A using 1,
simp [dot_product, mul_vec, ring_hom.map_sum, ← star_ring_end_apply, mul_comm],
end
end matrix
|
1210ca57f0690cec6c158eed4cee568ce3472270 | 367134ba5a65885e863bdc4507601606690974c1 | /src/topology/G_delta.lean | 0b306a82ba5b86acf54b55e0c3ef73987a960bc0 | [
"Apache-2.0"
] | permissive | kodyvajjha/mathlib | 9bead00e90f68269a313f45f5561766cfd8d5cad | b98af5dd79e13a38d84438b850a2e8858ec21284 | refs/heads/master | 1,624,350,366,310 | 1,615,563,062,000 | 1,615,563,062,000 | 162,666,963 | 0 | 0 | Apache-2.0 | 1,545,367,651,000 | 1,545,367,651,000 | null | UTF-8 | Lean | false | false | 5,033 | lean | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import topology.metric_space.emetric_space
/-!
# `Gδ` sets
In this file we define `Gδ` sets and prove their basic properties.
## Main definitions
* `is_Gδ`: a set `s` is a `Gδ` set if it can be represented as an intersection
of countably many open sets;
* `residual`: the filter of residual sets. A set `s` is called *residual* if it includes a dense
`Gδ` set. In a Baire space (e.g., in a complete (e)metric space), residual sets form a filter.
For technical reasons, we define `residual` in any topological space but the definition agrees
with the description above only in Baire spaces.
## Main results
We prove that finite or countable intersections of Gδ sets is a Gδ set. We also prove that the
continuity set of a function from a topological space to an (e)metric space is a Gδ set.
## Tags
Gδ set, residual set
-/
noncomputable theory
open_locale classical topological_space filter
open filter encodable set
variables {α : Type*} {β : Type*} {γ : Type*} {ι : Type*}
section is_Gδ
variable [topological_space α]
/-- A Gδ set is a countable intersection of open sets. -/
def is_Gδ (s : set α) : Prop :=
∃T : set (set α), (∀t ∈ T, is_open t) ∧ countable T ∧ s = (⋂₀ T)
/-- An open set is a Gδ set. -/
lemma is_open.is_Gδ {s : set α} (h : is_open s) : is_Gδ s :=
⟨{s}, by simp [h], countable_singleton _, (set.sInter_singleton _).symm⟩
lemma is_Gδ_univ : is_Gδ (univ : set α) := is_open_univ.is_Gδ
lemma is_Gδ_bInter_of_open {I : set ι} (hI : countable I) {f : ι → set α}
(hf : ∀i ∈ I, is_open (f i)) : is_Gδ (⋂i∈I, f i) :=
⟨f '' I, by rwa ball_image_iff, hI.image _, by rw sInter_image⟩
lemma is_Gδ_Inter_of_open [encodable ι] {f : ι → set α}
(hf : ∀i, is_open (f i)) : is_Gδ (⋂i, f i) :=
⟨range f, by rwa forall_range_iff, countable_range _, by rw sInter_range⟩
/-- A countable intersection of Gδ sets is a Gδ set. -/
lemma is_Gδ_sInter {S : set (set α)} (h : ∀s∈S, is_Gδ s) (hS : countable S) : is_Gδ (⋂₀ S) :=
begin
choose T hT using h,
refine ⟨_, _, _, (sInter_bUnion (λ s hs, (hT s hs).2.2)).symm⟩,
{ simp only [mem_Union],
rintros t ⟨s, hs, tTs⟩,
exact (hT s hs).1 t tTs },
{ exact hS.bUnion (λs hs, (hT s hs).2.1) },
end
lemma is_Gδ_Inter [encodable ι] {s : ι → set α} (hs : ∀ i, is_Gδ (s i)) : is_Gδ (⋂ i, s i) :=
is_Gδ_sInter (forall_range_iff.2 hs) $ countable_range s
lemma is_Gδ_bInter {s : set ι} (hs : countable s) {t : Π i ∈ s, set α}
(ht : ∀ i ∈ s, is_Gδ (t i ‹_›)) : is_Gδ (⋂ i ∈ s, t i ‹_›) :=
begin
rw [bInter_eq_Inter],
haveI := hs.to_encodable,
exact is_Gδ_Inter (λ x, ht x x.2)
end
lemma is_Gδ.inter {s t : set α} (hs : is_Gδ s) (ht : is_Gδ t) : is_Gδ (s ∩ t) :=
by { rw inter_eq_Inter, exact is_Gδ_Inter (bool.forall_bool.2 ⟨ht, hs⟩) }
/-- The union of two Gδ sets is a Gδ set. -/
lemma is_Gδ.union {s t : set α} (hs : is_Gδ s) (ht : is_Gδ t) : is_Gδ (s ∪ t) :=
begin
rcases hs with ⟨S, Sopen, Scount, rfl⟩,
rcases ht with ⟨T, Topen, Tcount, rfl⟩,
rw [sInter_union_sInter],
apply is_Gδ_bInter_of_open (countable_prod Scount Tcount),
rintros ⟨a, b⟩ hab,
exact is_open_union (Sopen a hab.1) (Topen b hab.2)
end
end is_Gδ
section continuous_at
open topological_space
open_locale uniformity
variables [topological_space α]
lemma is_Gδ_set_of_continuous_at_of_countably_generated_uniformity
[uniform_space β] (hU : is_countably_generated (𝓤 β)) (f : α → β) :
is_Gδ {x | continuous_at f x} :=
begin
rcases hU.exists_antimono_subbasis uniformity_has_basis_open_symmetric with ⟨U, hUo, hU⟩,
simp only [uniform.continuous_at_iff_prod, nhds_prod_eq],
simp only [(nhds_basis_opens _).prod_self.tendsto_iff hU.to_has_basis, forall_prop_of_true,
set_of_forall, id],
refine is_Gδ_Inter (λ k, is_open.is_Gδ $ is_open_iff_mem_nhds.2 $ λ x, _),
rintros ⟨s, ⟨hsx, hso⟩, hsU⟩,
filter_upwards [mem_nhds_sets hso hsx],
intros y hy,
exact ⟨s, ⟨hy, hso⟩, hsU⟩
end
/-- The set of points where a function is continuous is a Gδ set. -/
lemma is_Gδ_set_of_continuous_at [emetric_space β] (f : α → β) :
is_Gδ {x | continuous_at f x} :=
is_Gδ_set_of_continuous_at_of_countably_generated_uniformity
emetric.uniformity_has_countable_basis _
end continuous_at
/-- A set `s` is called *residual* if it includes a dense `Gδ` set. If `α` is a Baire space
(e.g., a complete metric space), then residual sets form a filter, see `mem_residual`.
For technical reasons we define the filter `residual` in any topological space but in a non-Baire
space it is not useful because it may contain some non-residual sets. -/
def residual (α : Type*) [topological_space α] : filter α :=
⨅ t (ht : is_Gδ t) (ht' : dense t), 𝓟 t
|
a68b83a162e18167296a615781d997c46c365772 | 5749d8999a76f3a8fddceca1f6941981e33aaa96 | /src/topology/metric_space/gromov_hausdorff.lean | 63a19de126a4c8ebbcb96476928bdb1ca17ea45a | [
"Apache-2.0"
] | permissive | jdsalchow/mathlib | 13ab43ef0d0515a17e550b16d09bd14b76125276 | 497e692b946d93906900bb33a51fd243e7649406 | refs/heads/master | 1,585,819,143,348 | 1,580,072,892,000 | 1,580,072,892,000 | 154,287,128 | 0 | 0 | Apache-2.0 | 1,540,281,610,000 | 1,540,281,609,000 | null | UTF-8 | Lean | false | false | 56,693 | lean | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Sébastien Gouëzel
-/
import topology.metric_space.closeds set_theory.cardinal topology.metric_space.gromov_hausdorff_realized
topology.metric_space.completion
/-!
# Gromov-Hausdorff distance
This file defines the Gromov-Hausdorff distance on the space of nonempty compact metric spaces
up to isometry.
We introduce the space of all nonempty compact metric spaces, up to isometry,
called `GH_space`, and endow it with a metric space structure. The distance,
known as the Gromov-Hausdorff distance, is defined as follows: given two
nonempty compact spaces `X` and `Y`, their distance is the minimum Hausdorff distance
between all possible isometric embeddings of `X` and `Y` in all metric spaces.
To define properly the Gromov-Hausdorff space, we consider the non-empty
compact subsets of `ℓ^∞(ℝ)` up to isometry, which is a well-defined type,
and define the distance as the infimum of the Hausdorff distance over all
embeddings in `ℓ^∞(ℝ)`. We prove that this coincides with the previous description,
as all separable metric spaces embed isometrically into `ℓ^∞(ℝ)`, through an
embedding called the Kuratowski embedding.
To prove that we have a distance, we should show that if spaces can be coupled
to be arbitrarily close, then they are isometric. More generally, the Gromov-Hausdorff
distance is realized, i.e., there is a coupling for which the Hausdorff distance
is exactly the Gromov-Hausdorff distance. This follows from a compactness
argument, essentially following from Arzela-Ascoli.
## Main results
We prove the most important properties of the Gromov-Hausdorff space: it is a polish space,
i.e., it is complete and second countable. We also prove the Gromov compactness criterion.
-/
noncomputable theory
open_locale classical topological_space
universes u v w
open classical lattice set function topological_space filter metric quotient
open bounded_continuous_function nat Kuratowski_embedding
open sum (inl inr)
set_option class.instance_max_depth 50
local attribute [instance] metric_space_sum
namespace Gromov_Hausdorff
section GH_space
/- In this section, we define the Gromov-Hausdorff space, denoted `GH_space` as the quotient
of nonempty compact subsets of `ℓ^∞(ℝ)` by identifying isometric sets.
Using the Kuratwoski embedding, we get a canonical map `to_GH_space` mapping any nonempty
compact type to `GH_space`. -/
/-- Equivalence relation identifying two nonempty compact sets which are isometric -/
private definition isometry_rel : nonempty_compacts ℓ_infty_ℝ → nonempty_compacts ℓ_infty_ℝ → Prop :=
λx y, nonempty (x.val ≃ᵢ y.val)
/-- This is indeed an equivalence relation -/
private lemma is_equivalence_isometry_rel : equivalence isometry_rel :=
⟨λx, ⟨isometric.refl _⟩, λx y ⟨e⟩, ⟨e.symm⟩, λ x y z ⟨e⟩ ⟨f⟩, ⟨e.trans f⟩⟩
/-- setoid instance identifying two isometric nonempty compact subspaces of ℓ^∞(ℝ) -/
instance isometry_rel.setoid : setoid (nonempty_compacts ℓ_infty_ℝ) :=
setoid.mk isometry_rel is_equivalence_isometry_rel
/-- The Gromov-Hausdorff space -/
definition GH_space : Type := quotient (isometry_rel.setoid)
/-- Map any nonempty compact type to `GH_space` -/
definition to_GH_space (α : Type u) [metric_space α] [compact_space α] [nonempty α] : GH_space :=
⟦nonempty_compacts.Kuratowski_embedding α⟧
/-- A metric space representative of any abstract point in `GH_space` -/
definition GH_space.rep (p : GH_space) : Type := (quot.out p).val
lemma eq_to_GH_space_iff {α : Type u} [metric_space α] [compact_space α] [nonempty α] {p : nonempty_compacts ℓ_infty_ℝ} :
⟦p⟧ = to_GH_space α ↔ ∃Ψ : α → ℓ_infty_ℝ, isometry Ψ ∧ range Ψ = p.val :=
begin
simp only [to_GH_space, quotient.eq],
split,
{ assume h,
rcases setoid.symm h with ⟨e⟩,
have f := (Kuratowski_embedding.isometry α).isometric_on_range.trans e,
use λx, f x,
split,
{ apply isometry_subtype_val.comp f.isometry },
{ rw [range_comp, f.range_coe, set.image_univ, set.range_coe_subtype] } },
{ rintros ⟨Ψ, ⟨isomΨ, rangeΨ⟩⟩,
have f := ((Kuratowski_embedding.isometry α).isometric_on_range.symm.trans
isomΨ.isometric_on_range).symm,
have E : (range Ψ ≃ᵢ (nonempty_compacts.Kuratowski_embedding α).val) = (p.val ≃ᵢ range (Kuratowski_embedding α)),
by { dunfold nonempty_compacts.Kuratowski_embedding, rw [rangeΨ]; refl },
have g := cast E f,
exact ⟨g⟩ }
end
lemma eq_to_GH_space {p : nonempty_compacts ℓ_infty_ℝ} : ⟦p⟧ = to_GH_space p.val :=
begin
refine eq_to_GH_space_iff.2 ⟨((λx, x) : p.val → ℓ_infty_ℝ), _, subtype.val_range⟩,
apply isometry_subtype_val
end
section
local attribute [reducible] GH_space.rep
instance rep_GH_space_metric_space {p : GH_space} : metric_space (p.rep) :=
by apply_instance
instance rep_GH_space_compact_space {p : GH_space} : compact_space (p.rep) :=
by apply_instance
instance rep_GH_space_nonempty {p : GH_space} : nonempty (p.rep) :=
by apply_instance
end
lemma GH_space.to_GH_space_rep (p : GH_space) : to_GH_space (p.rep) = p :=
begin
change to_GH_space (quot.out p).val = p,
rw ← eq_to_GH_space,
exact quot.out_eq p
end
/-- Two nonempty compact spaces have the same image in `GH_space` if and only if they are isometric -/
lemma to_GH_space_eq_to_GH_space_iff_isometric {α : Type u} [metric_space α] [compact_space α] [nonempty α]
{β : Type u} [metric_space β] [compact_space β] [nonempty β] :
to_GH_space α = to_GH_space β ↔ nonempty (α ≃ᵢ β) :=
⟨begin
simp only [to_GH_space, quotient.eq],
assume h,
rcases h with e,
have I : ((nonempty_compacts.Kuratowski_embedding α).val ≃ᵢ (nonempty_compacts.Kuratowski_embedding β).val)
= ((range (Kuratowski_embedding α)) ≃ᵢ (range (Kuratowski_embedding β))),
by { dunfold nonempty_compacts.Kuratowski_embedding, refl },
have e' := cast I e,
have f := (Kuratowski_embedding.isometry α).isometric_on_range,
have g := (Kuratowski_embedding.isometry β).isometric_on_range.symm,
have h := (f.trans e').trans g,
exact ⟨h⟩
end,
begin
rintros ⟨e⟩,
simp only [to_GH_space, quotient.eq],
have f := (Kuratowski_embedding.isometry α).isometric_on_range.symm,
have g := (Kuratowski_embedding.isometry β).isometric_on_range,
have h := (f.trans e).trans g,
have I : ((range (Kuratowski_embedding α)) ≃ᵢ (range (Kuratowski_embedding β))) =
((nonempty_compacts.Kuratowski_embedding α).val ≃ᵢ (nonempty_compacts.Kuratowski_embedding β).val),
by { dunfold nonempty_compacts.Kuratowski_embedding, refl },
have h' := cast I h,
exact ⟨h'⟩
end⟩
/-- Distance on `GH_space`: the distance between two nonempty compact spaces is the infimum
Hausdorff distance between isometric copies of the two spaces in a metric space. For the definition,
we only consider embeddings in `ℓ^∞(ℝ)`, but we will prove below that it works for all spaces. -/
instance : has_dist (GH_space) :=
{ dist := λx y, Inf ((λp : nonempty_compacts ℓ_infty_ℝ × nonempty_compacts ℓ_infty_ℝ, Hausdorff_dist p.1.val p.2.val) ''
(set.prod {a | ⟦a⟧ = x} {b | ⟦b⟧ = y})) }
/-- The Gromov-Hausdorff distance between two nonempty compact metric spaces, equal by definition to
the distance of the equivalence classes of these spaces in the Gromov-Hausdorff space. -/
def GH_dist (α : Type u) (β : Type v) [metric_space α] [nonempty α] [compact_space α]
[metric_space β] [nonempty β] [compact_space β] : ℝ := dist (to_GH_space α) (to_GH_space β)
lemma dist_GH_dist (p q : GH_space) : dist p q = GH_dist (p.rep) (q.rep) :=
by rw [GH_dist, p.to_GH_space_rep, q.to_GH_space_rep]
/-- The Gromov-Hausdorff distance between two spaces is bounded by the Hausdorff distance
of isometric copies of the spaces, in any metric space. -/
theorem GH_dist_le_Hausdorff_dist {α : Type u} [metric_space α] [compact_space α] [nonempty α]
{β : Type v} [metric_space β] [compact_space β] [nonempty β]
{γ : Type w} [metric_space γ] {Φ : α → γ} {Ψ : β → γ} (ha : isometry Φ) (hb : isometry Ψ) :
GH_dist α β ≤ Hausdorff_dist (range Φ) (range Ψ) :=
begin
/- For the proof, we want to embed `γ` in `ℓ^∞(ℝ)`, to say that the Hausdorff distance is realized
in `ℓ^∞(ℝ)` and therefore bounded below by the Gromov-Hausdorff-distance. However, `γ` is not
separable in general. We restrict to the union of the images of `α` and `β` in `γ`, which is
separable and therefore embeddable in `ℓ^∞(ℝ)`. -/
rcases exists_mem_of_nonempty α with ⟨xα, _⟩,
letI : inhabited α := ⟨xα⟩,
letI : inhabited β := classical.inhabited_of_nonempty (by assumption),
let s : set γ := (range Φ) ∪ (range Ψ),
let Φ' : α → subtype s := λy, ⟨Φ y, mem_union_left _ (mem_range_self _)⟩,
let Ψ' : β → subtype s := λy, ⟨Ψ y, mem_union_right _ (mem_range_self _)⟩,
have IΦ' : isometry Φ' := λx y, ha x y,
have IΨ' : isometry Ψ' := λx y, hb x y,
have : compact s, from (compact_range ha.continuous).union (compact_range hb.continuous),
letI : metric_space (subtype s) := by apply_instance,
haveI : compact_space (subtype s) := ⟨compact_iff_compact_univ.1 ‹compact s›⟩,
haveI : nonempty (subtype s) := ⟨Φ' xα⟩,
have ΦΦ' : Φ = subtype.val ∘ Φ', by { funext, refl },
have ΨΨ' : Ψ = subtype.val ∘ Ψ', by { funext, refl },
have : Hausdorff_dist (range Φ) (range Ψ) = Hausdorff_dist (range Φ') (range Ψ'),
{ rw [ΦΦ', ΨΨ', range_comp, range_comp],
exact Hausdorff_dist_image (isometry_subtype_val) },
rw this,
-- Embed `s` in `ℓ^∞(ℝ)` through its Kuratowski embedding
let F := Kuratowski_embedding (subtype s),
have : Hausdorff_dist (F '' (range Φ')) (F '' (range Ψ')) = Hausdorff_dist (range Φ') (range Ψ') :=
Hausdorff_dist_image (Kuratowski_embedding.isometry _),
rw ← this,
-- Let `A` and `B` be the images of `α` and `β` under this embedding. They are in `ℓ^∞(ℝ)`, and
-- their Hausdorff distance is the same as in the original space.
let A : nonempty_compacts ℓ_infty_ℝ := ⟨F '' (range Φ'), ⟨by simp,
(compact_range IΦ'.continuous).image (Kuratowski_embedding.isometry _).continuous⟩⟩,
let B : nonempty_compacts ℓ_infty_ℝ := ⟨F '' (range Ψ'), ⟨by simp,
(compact_range IΨ'.continuous).image (Kuratowski_embedding.isometry _).continuous⟩⟩,
have Aα : ⟦A⟧ = to_GH_space α,
{ rw eq_to_GH_space_iff,
exact ⟨λx, F (Φ' x), ⟨(Kuratowski_embedding.isometry _).comp IΦ', by rw range_comp⟩⟩ },
have Bβ : ⟦B⟧ = to_GH_space β,
{ rw eq_to_GH_space_iff,
exact ⟨λx, F (Ψ' x), ⟨(Kuratowski_embedding.isometry _).comp IΨ', by rw range_comp⟩⟩ },
refine cInf_le ⟨0,
begin simp [lower_bounds], assume t _ _ _ _ ht, rw ← ht, exact Hausdorff_dist_nonneg end⟩ _,
apply (mem_image _ _ _).2,
existsi (⟨A, B⟩ : nonempty_compacts ℓ_infty_ℝ × nonempty_compacts ℓ_infty_ℝ),
simp [Aα, Bβ]
end
local attribute [instance, priority 10] inhabited_of_nonempty'
/-- The optimal coupling constructed above realizes exactly the Gromov-Hausdorff distance,
essentially by design. -/
lemma Hausdorff_dist_optimal {α : Type u} [metric_space α] [compact_space α] [nonempty α]
{β : Type v} [metric_space β] [compact_space β] [nonempty β] :
Hausdorff_dist (range (optimal_GH_injl α β)) (range (optimal_GH_injr α β)) = GH_dist α β :=
begin
/- we only need to check the inequality `≤`, as the other one follows from the previous lemma.
As the Gromov-Hausdorff distance is an infimum, we need to check that the Hausdorff distance
in the optimal coupling is smaller than the Hausdorff distance of any coupling.
First, we check this for couplings which already have small Hausdorff distance: in this
case, the induced "distance" on `α ⊕ β` belongs to the candidates family introduced in the
definition of the optimal coupling, and the conclusion follows from the optimality
of the optimal coupling within this family.
-/
have A : ∀p q : nonempty_compacts (ℓ_infty_ℝ), ⟦p⟧ = to_GH_space α → ⟦q⟧ = to_GH_space β →
Hausdorff_dist (p.val) (q.val) < diam (univ : set α) + 1 + diam (univ : set β) →
Hausdorff_dist (range (optimal_GH_injl α β)) (range (optimal_GH_injr α β)) ≤ Hausdorff_dist (p.val) (q.val),
{ assume p q hp hq bound,
rcases eq_to_GH_space_iff.1 hp with ⟨Φ, ⟨Φisom, Φrange⟩⟩,
rcases eq_to_GH_space_iff.1 hq with ⟨Ψ, ⟨Ψisom, Ψrange⟩⟩,
have I : diam (range Φ ∪ range Ψ) ≤ 2 * diam (univ : set α) + 1 + 2 * diam (univ : set β),
{ rcases exists_mem_of_nonempty α with ⟨xα, _⟩,
have : ∃y ∈ range Ψ, dist (Φ xα) y < diam (univ : set α) + 1 + diam (univ : set β),
{ rw Ψrange,
have : Φ xα ∈ p.val := begin rw ← Φrange, exact mem_range_self _ end,
exact exists_dist_lt_of_Hausdorff_dist_lt this bound
(Hausdorff_edist_ne_top_of_ne_empty_of_bounded p.2.1 q.2.1 p.2.2.bounded q.2.2.bounded) },
rcases this with ⟨y, hy, dy⟩,
rcases mem_range.1 hy with ⟨z, hzy⟩,
rw ← hzy at dy,
have DΦ : diam (range Φ) = diam (univ : set α) :=
begin rw [← image_univ], apply metric.isometry.diam_image Φisom end,
have DΨ : diam (range Ψ) = diam (univ : set β) :=
begin rw [← image_univ], apply metric.isometry.diam_image Ψisom end,
calc
diam (range Φ ∪ range Ψ) ≤ diam (range Φ) + dist (Φ xα) (Ψ z) + diam (range Ψ) :
diam_union (mem_range_self _) (mem_range_self _)
... ≤ diam (univ : set α) + (diam (univ : set α) + 1 + diam (univ : set β)) + diam (univ : set β) :
by { rw [DΦ, DΨ], apply add_le_add (add_le_add (le_refl _) (le_of_lt dy)) (le_refl _) }
... = 2 * diam (univ : set α) + 1 + 2 * diam (univ : set β) : by ring },
let f : α ⊕ β → ℓ_infty_ℝ := λx, match x with | inl y := Φ y | inr z := Ψ z end,
let F : (α ⊕ β) × (α ⊕ β) → ℝ := λp, dist (f p.1) (f p.2),
-- check that the induced "distance" is a candidate
have Fgood : F ∈ candidates α β,
{ simp only [candidates, forall_const, and_true, add_comm, eq_self_iff_true, dist_eq_zero,
and_self, set.mem_set_of_eq],
repeat {split},
{ exact λx y, calc
F (inl x, inl y) = dist (Φ x) (Φ y) : rfl
... = dist x y : Φisom.dist_eq },
{ exact λx y, calc
F (inr x, inr y) = dist (Ψ x) (Ψ y) : rfl
... = dist x y : Ψisom.dist_eq },
{ exact λx y, dist_comm _ _ },
{ exact λx y z, dist_triangle _ _ _ },
{ exact λx y, calc
F (x, y) ≤ diam (range Φ ∪ range Ψ) :
begin
have A : ∀z : α ⊕ β, f z ∈ range Φ ∪ range Ψ,
{ assume z,
cases z,
{ apply mem_union_left, apply mem_range_self },
{ apply mem_union_right, apply mem_range_self } },
refine dist_le_diam_of_mem _ (A _) (A _),
rw [Φrange, Ψrange],
exact (p.2.2.union q.2.2).bounded,
end
... ≤ 2 * diam (univ : set α) + 1 + 2 * diam (univ : set β) : I } },
let Fb := candidates_b_of_candidates F Fgood,
have : Hausdorff_dist (range (optimal_GH_injl α β)) (range (optimal_GH_injr α β)) ≤ HD Fb :=
Hausdorff_dist_optimal_le_HD _ _ (candidates_b_of_candidates_mem F Fgood),
refine le_trans this (le_of_forall_le_of_dense (λr hr, _)),
have I1 : ∀x : α, infi (λy:β, Fb (inl x, inr y)) ≤ r,
{ assume x,
have : f (inl x) ∈ p.val, by { rw [← Φrange], apply mem_range_self },
rcases exists_dist_lt_of_Hausdorff_dist_lt this hr
(Hausdorff_edist_ne_top_of_ne_empty_of_bounded p.2.1 q.2.1 p.2.2.bounded q.2.2.bounded)
with ⟨z, zq, hz⟩,
have : z ∈ range Ψ, by rwa [← Ψrange] at zq,
rcases mem_range.1 this with ⟨y, hy⟩,
calc infi (λy:β, Fb (inl x, inr y)) ≤ Fb (inl x, inr y) :
cinfi_le (by simpa using HD_below_aux1 0)
... = dist (Φ x) (Ψ y) : rfl
... = dist (f (inl x)) z : by rw hy
... ≤ r : le_of_lt hz },
have I2 : ∀y : β, infi (λx:α, Fb (inl x, inr y)) ≤ r,
{ assume y,
have : f (inr y) ∈ q.val, by { rw [← Ψrange], apply mem_range_self },
rcases exists_dist_lt_of_Hausdorff_dist_lt' this hr
(Hausdorff_edist_ne_top_of_ne_empty_of_bounded p.2.1 q.2.1 p.2.2.bounded q.2.2.bounded)
with ⟨z, zq, hz⟩,
have : z ∈ range Φ, by rwa [← Φrange] at zq,
rcases mem_range.1 this with ⟨x, hx⟩,
calc infi (λx:α, Fb (inl x, inr y)) ≤ Fb (inl x, inr y) :
cinfi_le (by simpa using HD_below_aux2 0)
... = dist (Φ x) (Ψ y) : rfl
... = dist z (f (inr y)) : by rw hx
... ≤ r : le_of_lt hz },
simp [HD, csupr_le I1, csupr_le I2] },
/- Get the same inequality for any coupling. If the coupling is quite good, the desired
inequality has been proved above. If it is bad, then the inequality is obvious. -/
have B : ∀p q : nonempty_compacts (ℓ_infty_ℝ), ⟦p⟧ = to_GH_space α → ⟦q⟧ = to_GH_space β →
Hausdorff_dist (range (optimal_GH_injl α β)) (range (optimal_GH_injr α β)) ≤ Hausdorff_dist (p.val) (q.val),
{ assume p q hp hq,
by_cases h : Hausdorff_dist (p.val) (q.val) < diam (univ : set α) + 1 + diam (univ : set β),
{ exact A p q hp hq h },
{ calc Hausdorff_dist (range (optimal_GH_injl α β)) (range (optimal_GH_injr α β)) ≤ HD (candidates_b_dist α β) :
Hausdorff_dist_optimal_le_HD _ _ (candidates_b_dist_mem_candidates_b)
... ≤ diam (univ : set α) + 1 + diam (univ : set β) : HD_candidates_b_dist_le
... ≤ Hausdorff_dist (p.val) (q.val) : not_lt.1 h } },
refine le_antisymm _ _,
{ apply le_cInf,
{ rw ne_empty_iff_exists_mem,
simp only [set.mem_image, nonempty_of_inhabited, set.mem_set_of_eq, prod.exists],
existsi [Hausdorff_dist (nonempty_compacts.Kuratowski_embedding α).val (nonempty_compacts.Kuratowski_embedding β).val,
nonempty_compacts.Kuratowski_embedding α, nonempty_compacts.Kuratowski_embedding β],
simp [to_GH_space, -quotient.eq] },
{ rintro b ⟨⟨p, q⟩, ⟨hp, hq⟩, rfl⟩,
exact B p q hp hq } },
{ exact GH_dist_le_Hausdorff_dist (isometry_optimal_GH_injl α β) (isometry_optimal_GH_injr α β) }
end
/-- The Gromov-Hausdorff distance can also be realized by a coupling in `ℓ^∞(ℝ)`, by embedding
the optimal coupling through its Kuratowski embedding. -/
theorem GH_dist_eq_Hausdorff_dist (α : Type u) [metric_space α] [compact_space α] [nonempty α]
(β : Type v) [metric_space β] [compact_space β] [nonempty β] :
∃Φ : α → ℓ_infty_ℝ, ∃Ψ : β → ℓ_infty_ℝ, isometry Φ ∧ isometry Ψ ∧
GH_dist α β = Hausdorff_dist (range Φ) (range Ψ) :=
begin
let F := Kuratowski_embedding (optimal_GH_coupling α β),
let Φ := F ∘ optimal_GH_injl α β,
let Ψ := F ∘ optimal_GH_injr α β,
refine ⟨Φ, Ψ, _, _, _⟩,
{ exact (Kuratowski_embedding.isometry _).comp (isometry_optimal_GH_injl α β) },
{ exact (Kuratowski_embedding.isometry _).comp (isometry_optimal_GH_injr α β) },
{ rw [← image_univ, ← image_univ, image_comp F, image_univ, image_comp F (optimal_GH_injr α β),
image_univ, ← Hausdorff_dist_optimal],
exact (Hausdorff_dist_image (Kuratowski_embedding.isometry _)).symm },
end
-- without the next two lines, `{ exact closed_of_compact (range Φ) hΦ }` in the next
-- proof is very slow, as the `t2_space` instance is very hard to find
local attribute [instance, priority 10] order_topology.t2_space
local attribute [instance, priority 10] order_closed_topology.to_t2_space
/-- The Gromov-Hausdorff distance defines a genuine distance on the Gromov-Hausdorff space. -/
instance GH_space_metric_space : metric_space GH_space :=
{ dist_self := λx, begin
rcases exists_rep x with ⟨y, hy⟩,
refine le_antisymm _ _,
{ apply cInf_le,
{ exact ⟨0, by { rintro b ⟨⟨u, v⟩, ⟨hu, hv⟩, rfl⟩, exact Hausdorff_dist_nonneg } ⟩},
{ simp, existsi [y, y], simpa } },
{ apply le_cInf,
{ simp only [set.image_eq_empty, ne.def],
apply ne_empty_iff_exists_mem.2,
existsi (⟨y, y⟩ : nonempty_compacts ℓ_infty_ℝ × nonempty_compacts ℓ_infty_ℝ),
simpa },
{ rintro b ⟨⟨u, v⟩, ⟨hu, hv⟩, rfl⟩, exact Hausdorff_dist_nonneg } },
end,
dist_comm := λx y, begin
have A : (λ (p : nonempty_compacts ℓ_infty_ℝ × nonempty_compacts ℓ_infty_ℝ),
Hausdorff_dist ((p.fst).val) ((p.snd).val)) '' (set.prod {a | ⟦a⟧ = x} {b | ⟦b⟧ = y})
= ((λ (p : nonempty_compacts ℓ_infty_ℝ × nonempty_compacts ℓ_infty_ℝ),
Hausdorff_dist ((p.fst).val) ((p.snd).val)) ∘ prod.swap) '' (set.prod {a | ⟦a⟧ = x} {b | ⟦b⟧ = y}) :=
by { congr, funext, simp, rw Hausdorff_dist_comm },
simp only [dist, A, image_comp, prod.swap, image_swap_prod],
end,
eq_of_dist_eq_zero := λx y hxy, begin
/- To show that two spaces at zero distance are isometric, we argue that the distance
is realized by some coupling. In this coupling, the two spaces are at zero Hausdorff distance,
i.e., they coincide. Therefore, the original spaces are isometric. -/
rcases GH_dist_eq_Hausdorff_dist x.rep y.rep with ⟨Φ, Ψ, Φisom, Ψisom, DΦΨ⟩,
rw [← dist_GH_dist, hxy] at DΦΨ,
have : range Φ = range Ψ,
{ have hΦ : compact (range Φ) := compact_range Φisom.continuous,
have hΨ : compact (range Ψ) := compact_range Ψisom.continuous,
apply (Hausdorff_dist_zero_iff_eq_of_closed _ _ _).1 (DΦΨ.symm),
{ exact closed_of_compact (range Φ) hΦ },
{ exact closed_of_compact (range Ψ) hΨ },
{ exact Hausdorff_edist_ne_top_of_ne_empty_of_bounded (by simp [-nonempty_subtype])
(by simp [-nonempty_subtype]) hΦ.bounded hΨ.bounded } },
have T : ((range Ψ) ≃ᵢ y.rep) = ((range Φ) ≃ᵢ y.rep), by rw this,
have eΨ := cast T Ψisom.isometric_on_range.symm,
have e := Φisom.isometric_on_range.trans eΨ,
rw [← x.to_GH_space_rep, ← y.to_GH_space_rep, to_GH_space_eq_to_GH_space_iff_isometric],
exact ⟨e⟩
end,
dist_triangle := λx y z, begin
/- To show the triangular inequality between `X`, `Y` and `Z`, realize an optimal coupling
between `X` and `Y` in a space `γ1`, and an optimal coupling between `Y`and `Z` in a space `γ2`.
Then, glue these metric spaces along `Y`. We get a new space `γ` in which `X` and `Y` are
optimally coupled, as well as `Y` and `Z`. Apply the triangle inequality for the Hausdorff
distance in `γ` to conclude. -/
let X := x.rep,
let Y := y.rep,
let Z := z.rep,
let γ1 := optimal_GH_coupling X Y,
let γ2 := optimal_GH_coupling Y Z,
let Φ : Y → γ1 := optimal_GH_injr X Y,
have hΦ : isometry Φ := isometry_optimal_GH_injr X Y,
let Ψ : Y → γ2 := optimal_GH_injl Y Z,
have hΨ : isometry Ψ := isometry_optimal_GH_injl Y Z,
let γ := glue_space hΦ hΨ,
letI : metric_space γ := metric.metric_space_glue_space hΦ hΨ,
have Comm : (to_glue_l hΦ hΨ) ∘ (optimal_GH_injr X Y) = (to_glue_r hΦ hΨ) ∘ (optimal_GH_injl Y Z) :=
to_glue_commute hΦ hΨ,
calc dist x z = dist (to_GH_space X) (to_GH_space Z) :
by rw [x.to_GH_space_rep, z.to_GH_space_rep]
... ≤ Hausdorff_dist (range ((to_glue_l hΦ hΨ) ∘ (optimal_GH_injl X Y)))
(range ((to_glue_r hΦ hΨ) ∘ (optimal_GH_injr Y Z))) :
GH_dist_le_Hausdorff_dist
((to_glue_l_isometry hΦ hΨ).comp (isometry_optimal_GH_injl X Y))
((to_glue_r_isometry hΦ hΨ).comp (isometry_optimal_GH_injr Y Z))
... ≤ Hausdorff_dist (range ((to_glue_l hΦ hΨ) ∘ (optimal_GH_injl X Y)))
(range ((to_glue_l hΦ hΨ) ∘ (optimal_GH_injr X Y)))
+ Hausdorff_dist (range ((to_glue_l hΦ hΨ) ∘ (optimal_GH_injr X Y)))
(range ((to_glue_r hΦ hΨ) ∘ (optimal_GH_injr Y Z))) :
begin
refine Hausdorff_dist_triangle (Hausdorff_edist_ne_top_of_ne_empty_of_bounded
(by simp [-nonempty_subtype]) (by simp [-nonempty_subtype]) _ _),
{ exact (compact_range (isometry.continuous ((to_glue_l_isometry hΦ hΨ).comp
(isometry_optimal_GH_injl X Y)))).bounded },
{ exact (compact_range (isometry.continuous ((to_glue_l_isometry hΦ hΨ).comp
(isometry_optimal_GH_injr X Y)))).bounded }
end
... = Hausdorff_dist ((to_glue_l hΦ hΨ) '' (range (optimal_GH_injl X Y)))
((to_glue_l hΦ hΨ) '' (range (optimal_GH_injr X Y)))
+ Hausdorff_dist ((to_glue_r hΦ hΨ) '' (range (optimal_GH_injl Y Z)))
((to_glue_r hΦ hΨ) '' (range (optimal_GH_injr Y Z))) :
by simp only [eq.symm range_comp, Comm, eq_self_iff_true, add_right_inj]
... = Hausdorff_dist (range (optimal_GH_injl X Y))
(range (optimal_GH_injr X Y))
+ Hausdorff_dist (range (optimal_GH_injl Y Z))
(range (optimal_GH_injr Y Z)) :
by rw [Hausdorff_dist_image (to_glue_l_isometry hΦ hΨ),
Hausdorff_dist_image (to_glue_r_isometry hΦ hΨ)]
... = dist (to_GH_space X) (to_GH_space Y) + dist (to_GH_space Y) (to_GH_space Z) :
by rw [Hausdorff_dist_optimal, Hausdorff_dist_optimal, GH_dist, GH_dist]
... = dist x y + dist y z:
by rw [x.to_GH_space_rep, y.to_GH_space_rep, z.to_GH_space_rep]
end }
end GH_space --section
end Gromov_Hausdorff
/-- In particular, nonempty compacts of a metric space map to `GH_space`. We register this
in the topological_space namespace to take advantage of the notation `p.to_GH_space`. -/
definition topological_space.nonempty_compacts.to_GH_space {α : Type u} [metric_space α]
(p : nonempty_compacts α) : Gromov_Hausdorff.GH_space := Gromov_Hausdorff.to_GH_space p.val
open topological_space
namespace Gromov_Hausdorff
section nonempty_compacts
variables {α : Type u} [metric_space α]
theorem GH_dist_le_nonempty_compacts_dist (p q : nonempty_compacts α) :
dist p.to_GH_space q.to_GH_space ≤ dist p q :=
begin
have ha : isometry (subtype.val : p.val → α) := isometry_subtype_val,
have hb : isometry (subtype.val : q.val → α) := isometry_subtype_val,
have A : dist p q = Hausdorff_dist p.val q.val := rfl,
have I : p.val = range (subtype.val : p.val → α), by simp,
have J : q.val = range (subtype.val : q.val → α), by simp,
rw [I, J] at A,
rw A,
exact GH_dist_le_Hausdorff_dist ha hb
end
lemma to_GH_space_lipschitz :
lipschitz_with 1 (nonempty_compacts.to_GH_space : nonempty_compacts α → GH_space) :=
lipschitz_with.one_mk GH_dist_le_nonempty_compacts_dist
lemma to_GH_space_continuous :
continuous (nonempty_compacts.to_GH_space : nonempty_compacts α → GH_space) :=
to_GH_space_lipschitz.to_continuous
end nonempty_compacts
section
/- In this section, we show that if two metric spaces are isometric up to `ε₂`, then their
Gromov-Hausdorff distance is bounded by `ε₂ / 2`. More generally, if there are subsets which are
`ε₁`-dense and `ε₃`-dense in two spaces, and isometric up to `ε₂`, then the Gromov-Hausdorff distance
between the spaces is bounded by `ε₁ + ε₂/2 + ε₃`. For this, we construct a suitable coupling between
the two spaces, by gluing them (approximately) along the two matching subsets. -/
variables {α : Type u} [metric_space α] [compact_space α] [nonempty α]
{β : Type v} [metric_space β] [compact_space β] [nonempty β]
-- we want to ignore these instances in the following theorem
local attribute [instance, priority 10] sum.topological_space sum.uniform_space
/-- If there are subsets which are `ε₁`-dense and `ε₃`-dense in two spaces, and
isometric up to `ε₂`, then the Gromov-Hausdorff distance between the spaces is bounded by
`ε₁ + ε₂/2 + ε₃`. -/
theorem GH_dist_le_of_approx_subsets {s : set α} (Φ : s → β) {ε₁ ε₂ ε₃ : ℝ}
(hs : ∀x : α, ∃y ∈ s, dist x y ≤ ε₁) (hs' : ∀x : β, ∃y : s, dist x (Φ y) ≤ ε₃)
(H : ∀x y : s, abs (dist x y - dist (Φ x) (Φ y)) ≤ ε₂) :
GH_dist α β ≤ ε₁ + ε₂ / 2 + ε₃ :=
begin
refine real.le_of_forall_epsilon_le (λδ δ0, _),
rcases exists_mem_of_nonempty α with ⟨xα, _⟩,
rcases hs xα with ⟨xs, hxs, Dxs⟩,
have sne : s ≠ ∅ := ne_empty_of_mem hxs,
letI : nonempty (subtype s) := ⟨⟨xs, hxs⟩⟩,
have : 0 ≤ ε₂ := le_trans (abs_nonneg _) (H ⟨xs, hxs⟩ ⟨xs, hxs⟩),
have : ∀ p q : s, abs (dist p q - dist (Φ p) (Φ q)) ≤ 2 * (ε₂/2 + δ) := λp q, calc
abs (dist p q - dist (Φ p) (Φ q)) ≤ ε₂ : H p q
... ≤ 2 * (ε₂/2 + δ) : by linarith,
-- glue `α` and `β` along the almost matching subsets
letI : metric_space (α ⊕ β) := glue_metric_approx (@subtype.val α s) (λx, Φ x) (ε₂/2 + δ) (by linarith) this,
let Fl := @sum.inl α β,
let Fr := @sum.inr α β,
have Il : isometry Fl := isometry_emetric_iff_metric.2 (λx y, rfl),
have Ir : isometry Fr := isometry_emetric_iff_metric.2 (λx y, rfl),
/- The proof goes as follows : the `GH_dist` is bounded by the Hausdorff distance of the images in the
coupling, which is bounded (using the triangular inequality) by the sum of the Hausdorff distances
of `α` and `s` (in the coupling or, equivalently in the original space), of `s` and `Φ s`, and of
`Φ s` and `β` (in the coupling or, equivalently, in the original space). The first term is bounded
by `ε₁`, by `ε₁`-density. The third one is bounded by `ε₃`. And the middle one is bounded by `ε₂/2`
as in the coupling the points `x` and `Φ x` are at distance `ε₂/2` by construction of the coupling
(in fact `ε₂/2 + δ` where `δ` is an arbitrarily small positive constant where positivity is used
to ensure that the coupling is really a metric space and not a premetric space on `α ⊕ β`). -/
have : GH_dist α β ≤ Hausdorff_dist (range Fl) (range Fr) :=
GH_dist_le_Hausdorff_dist Il Ir,
have : Hausdorff_dist (range Fl) (range Fr) ≤ Hausdorff_dist (range Fl) (Fl '' s)
+ Hausdorff_dist (Fl '' s) (range Fr),
{ have B : bounded (range Fl) := (compact_range Il.continuous).bounded,
exact Hausdorff_dist_triangle (Hausdorff_edist_ne_top_of_ne_empty_of_bounded (by simpa) (by simpa)
B (bounded.subset (image_subset_range _ _) B)) },
have : Hausdorff_dist (Fl '' s) (range Fr) ≤ Hausdorff_dist (Fl '' s) (Fr '' (range Φ))
+ Hausdorff_dist (Fr '' (range Φ)) (range Fr),
{ have B : bounded (range Fr) := (compact_range Ir.continuous).bounded,
exact Hausdorff_dist_triangle' (Hausdorff_edist_ne_top_of_ne_empty_of_bounded
(by simpa [-nonempty_subtype]) (by simpa) (bounded.subset (image_subset_range _ _) B) B) },
have : Hausdorff_dist (range Fl) (Fl '' s) ≤ ε₁,
{ rw [← image_univ, Hausdorff_dist_image Il],
have : 0 ≤ ε₁ := le_trans dist_nonneg Dxs,
refine Hausdorff_dist_le_of_mem_dist this (λx hx, hs x)
(λx hx, ⟨x, mem_univ _, by simpa⟩) },
have : Hausdorff_dist (Fl '' s) (Fr '' (range Φ)) ≤ ε₂/2 + δ,
{ refine Hausdorff_dist_le_of_mem_dist (by linarith) _ _,
{ assume x' hx',
rcases (set.mem_image _ _ _).1 hx' with ⟨x, ⟨x_in_s, xx'⟩⟩,
rw ← xx',
use [Fr (Φ ⟨x, x_in_s⟩), mem_image_of_mem Fr (mem_range_self _)],
exact le_of_eq (glue_dist_glued_points (@subtype.val α s) Φ (ε₂/2 + δ) ⟨x, x_in_s⟩) },
{ assume x' hx',
rcases (set.mem_image _ _ _).1 hx' with ⟨y, ⟨y_in_s', yx'⟩⟩,
rcases mem_range.1 y_in_s' with ⟨x, xy⟩,
use [Fl x, mem_image_of_mem _ x.2],
rw [← yx', ← xy, dist_comm],
exact le_of_eq (glue_dist_glued_points (@subtype.val α s) Φ (ε₂/2 + δ) x) } },
have : Hausdorff_dist (Fr '' (range Φ)) (range Fr) ≤ ε₃,
{ rw [← @image_univ _ _ Fr, Hausdorff_dist_image Ir],
rcases exists_mem_of_nonempty β with ⟨xβ, _⟩,
rcases hs' xβ with ⟨xs', Dxs'⟩,
have : 0 ≤ ε₃ := le_trans dist_nonneg Dxs',
refine Hausdorff_dist_le_of_mem_dist this (λx hx, ⟨x, mem_univ _, by simpa⟩) (λx _, _),
rcases hs' x with ⟨y, Dy⟩,
exact ⟨Φ y, mem_range_self _, Dy⟩ },
linarith
end
end --section
/-- The Gromov-Hausdorff space is second countable. -/
instance second_countable : second_countable_topology GH_space :=
begin
refine second_countable_of_countable_discretization (λδ δpos, _),
let ε := (2/5) * δ,
have εpos : 0 < ε := mul_pos (by norm_num) δpos,
have : ∀p:GH_space, ∃s : set (p.rep), finite s ∧ (univ ⊆ (⋃x∈s, ball x ε)) :=
λp, by simpa using finite_cover_balls_of_compact (@compact_univ p.rep _ _) εpos,
-- for each `p`, `s p` is a finite `ε`-dense subset of `p` (or rather the metric space
-- `p.rep` representing `p`)
choose s hs using this,
have : ∀p:GH_space, ∀t:set (p.rep), finite t → ∃n:ℕ, ∃e:equiv t (fin n), true,
{ assume p t ht,
letI : fintype t := finite.fintype ht,
rcases fintype.exists_equiv_fin t with ⟨n, hn⟩,
rcases hn with e,
exact ⟨n, e, trivial⟩ },
choose N e hne using this,
-- cardinality of the nice finite subset `s p` of `p.rep`, called `N p`
let N := λp:GH_space, N p (s p) (hs p).1,
-- equiv from `s p`, a nice finite subset of `p.rep`, to `fin (N p)`, called `E p`
let E := λp:GH_space, e p (s p) (hs p).1,
-- A function `F` associating to `p : GH_space` the data of all distances between points
-- in the `ε`-dense set `s p`.
let F : GH_space → Σn:ℕ, (fin n → fin n → ℤ) :=
λp, ⟨N p, λa b, floor (ε⁻¹ * dist ((E p).inv_fun a) ((E p).inv_fun b))⟩,
refine ⟨_, by apply_instance, F, λp q hpq, _⟩,
/- As the target space of F is countable, it suffices to show that two points
`p` and `q` with `F p = F q` are at distance `≤ δ`.
For this, we construct a map `Φ` from `s p ⊆ p.rep` (representing `p`)
to `q.rep` (representing `q`) which is almost an isometry on `s p`, and
with image `s q`. For this, we compose the identification of `s p` with `fin (N p)`
and the inverse of the identification of `s q` with `fin (N q)`. Together with
the fact that `N p = N q`, this constructs `Ψ` between `s p` and `s q`, and then
composing with the canonical inclusion we get `Φ`. -/
have Npq : N p = N q := (sigma.mk.inj_iff.1 hpq).1,
let Ψ : s p → s q := λx, (E q).inv_fun (fin.cast Npq ((E p).to_fun x)),
let Φ : s p → q.rep := λx, Ψ x,
-- Use the almost isometry `Φ` to show that `p.rep` and `q.rep`
-- are within controlled Gromov-Hausdorff distance.
have main : GH_dist p.rep q.rep ≤ ε + ε/2 + ε,
{ refine GH_dist_le_of_approx_subsets Φ _ _ _,
show ∀x : p.rep, ∃ (y : p.rep) (H : y ∈ s p), dist x y ≤ ε,
{ -- by construction, `s p` is `ε`-dense
assume x,
have : x ∈ ⋃y∈(s p), ball y ε := (hs p).2 (mem_univ _),
rcases mem_bUnion_iff.1 this with ⟨y, ys, hy⟩,
exact ⟨y, ys, le_of_lt hy⟩ },
show ∀x : q.rep, ∃ (z : s p), dist x (Φ z) ≤ ε,
{ -- by construction, `s q` is `ε`-dense, and it is the range of `Φ`
assume x,
have : x ∈ ⋃y∈(s q), ball y ε := (hs q).2 (mem_univ _),
rcases mem_bUnion_iff.1 this with ⟨y, ys, hy⟩,
let i := ((E q).to_fun ⟨y, ys⟩).1,
let hi := ((E q).to_fun ⟨y, ys⟩).2,
have ihi_eq : (⟨i, hi⟩ : fin (N q)) = (E q).to_fun ⟨y, ys⟩, by rw fin.ext_iff,
have hiq : i < N q := hi,
have hip : i < N p, { rwa Npq.symm at hiq },
let z := (E p).inv_fun ⟨i, hip⟩,
use z,
have C1 : (E p).to_fun z = ⟨i, hip⟩ := (E p).right_inv ⟨i, hip⟩,
have C2 : fin.cast Npq ⟨i, hip⟩ = ⟨i, hi⟩ := rfl,
have C3 : (E q).inv_fun ⟨i, hi⟩ = ⟨y, ys⟩, by { rw ihi_eq, exact (E q).left_inv ⟨y, ys⟩ },
have : Φ z = y :=
by { simp only [Φ, Ψ], rw [C1, C2, C3], refl },
rw this,
exact le_of_lt hy },
show ∀x y : s p, abs (dist x y - dist (Φ x) (Φ y)) ≤ ε,
{ /- the distance between `x` and `y` is encoded in `F p`, and the distance between
`Φ x` and `Φ y` (two points of `s q`) is encoded in `F q`, all this up to `ε`.
As `F p = F q`, the distances are almost equal. -/
assume x y,
have : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y) := rfl,
rw this,
-- introduce `i`, that codes both `x` and `Φ x` in `fin (N p) = fin (N q)`
let i := ((E p).to_fun x).1,
have hip : i < N p := ((E p).to_fun x).2,
have hiq : i < N q, by rwa Npq at hip,
have i' : i = ((E q).to_fun (Ψ x)).1, by { simp [Ψ, (E q).right_inv _] },
-- introduce `j`, that codes both `y` and `Φ y` in `fin (N p) = fin (N q)`
let j := ((E p).to_fun y).1,
have hjp : j < N p := ((E p).to_fun y).2,
have hjq : j < N q, by rwa Npq at hjp,
have j' : j = ((E q).to_fun (Ψ y)).1, by { simp [Ψ, (E q).right_inv _] },
-- Express `dist x y` in terms of `F p`
have : (F p).2 ((E p).to_fun x) ((E p).to_fun y) = floor (ε⁻¹ * dist x y),
by simp only [F, (E p).left_inv _],
have Ap : (F p).2 ⟨i, hip⟩ ⟨j, hjp⟩ = floor (ε⁻¹ * dist x y),
by { rw ← this, congr; apply (fin.ext_iff _ _).2; refl },
-- Express `dist (Φ x) (Φ y)` in terms of `F q`
have : (F q).2 ((E q).to_fun (Ψ x)) ((E q).to_fun (Ψ y)) = floor (ε⁻¹ * dist (Ψ x) (Ψ y)),
by simp only [F, (E q).left_inv _],
have Aq : (F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩ = floor (ε⁻¹ * dist (Ψ x) (Ψ y)),
by { rw ← this, congr; apply (fin.ext_iff _ _).2; [exact i', exact j'] },
-- use the equality between `F p` and `F q` to deduce that the distances have equal
-- integer parts
have : (F p).2 ⟨i, hip⟩ ⟨j, hjp⟩ = (F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩,
{ -- we want to `subst hpq` where `hpq : F p = F q`, except that `subst` only works
-- with a constant, so replace `F q` (and everything that depends on it) by a constant `f`
-- then `subst`
revert hiq hjq,
change N q with (F q).1,
generalize_hyp : F q = f at hpq ⊢,
subst hpq,
intros,
refl },
rw [Ap, Aq] at this,
-- deduce that the distances coincide up to `ε`, by a straightforward computation
-- that should be automated
have I := calc
abs (ε⁻¹) * abs (dist x y - dist (Ψ x) (Ψ y)) =
abs (ε⁻¹ * (dist x y - dist (Ψ x) (Ψ y))) : (abs_mul _ _).symm
... = abs ((ε⁻¹ * dist x y) - (ε⁻¹ * dist (Ψ x) (Ψ y))) : by { congr, ring }
... ≤ 1 : le_of_lt (abs_sub_lt_one_of_floor_eq_floor this),
calc
abs (dist x y - dist (Ψ x) (Ψ y)) = (ε * ε⁻¹) * abs (dist x y - dist (Ψ x) (Ψ y)) :
by rw [mul_inv_cancel (ne_of_gt εpos), one_mul]
... = ε * (abs (ε⁻¹) * abs (dist x y - dist (Ψ x) (Ψ y))) :
by rw [abs_of_nonneg (le_of_lt (inv_pos εpos)), mul_assoc]
... ≤ ε * 1 : mul_le_mul_of_nonneg_left I (le_of_lt εpos)
... = ε : mul_one _ } },
calc dist p q = GH_dist (p.rep) (q.rep) : dist_GH_dist p q
... ≤ ε + ε/2 + ε : main
... = δ : by { simp [ε], ring }
end
/-- Compactness criterion: a closed set of compact metric spaces is compact if the spaces have
a uniformly bounded diameter, and for all `ε` the number of balls of radius `ε` required
to cover the spaces is uniformly bounded. This is an equivalence, but we only prove the
interesting direction that these conditions imply compactness. -/
lemma totally_bounded {t : set GH_space} {C : ℝ} {u : ℕ → ℝ} {K : ℕ → ℕ}
(ulim : tendsto u at_top (𝓝 0))
(hdiam : ∀p ∈ t, diam (univ : set (GH_space.rep p)) ≤ C)
(hcov : ∀p ∈ t, ∀n:ℕ, ∃s : set (GH_space.rep p), cardinal.mk s ≤ K n ∧ univ ⊆ ⋃x∈s, ball x (u n)) :
totally_bounded t :=
begin
/- Let `δ>0`, and `ε = δ/5`. For each `p`, we construct a finite subset `s p` of `p`, which
is `ε`-dense and has cardinality at most `K n`. Encoding the mutual distances of points in `s p`,
up to `ε`, we will get a map `F` associating to `p` finitely many data, and making it possible to
reconstruct `p` up to `ε`. This is enough to prove total boundedness. -/
refine metric.totally_bounded_of_finite_discretization (λδ δpos, _),
let ε := (1/5) * δ,
have εpos : 0 < ε := mul_pos (by norm_num) δpos,
-- choose `n` for which `u n < ε`
rcases metric.tendsto_at_top.1 ulim ε εpos with ⟨n, hn⟩,
have u_le_ε : u n ≤ ε,
{ have := hn n (le_refl _),
simp only [real.dist_eq, add_zero, sub_eq_add_neg, neg_zero] at this,
exact le_of_lt (lt_of_le_of_lt (le_abs_self _) this) },
-- construct a finite subset `s p` of `p` which is `ε`-dense and has cardinal `≤ K n`
have : ∀p:GH_space, ∃s : set (p.rep), ∃N ≤ K n, ∃E : equiv s (fin N),
p ∈ t → univ ⊆ ⋃x∈s, ball x (u n),
{ assume p,
by_cases hp : p ∉ t,
{ have : nonempty (equiv (∅ : set (p.rep)) (fin 0)),
{ rw ← fintype.card_eq, simp },
use [∅, 0, bot_le, choice (this)] },
{ rcases hcov _ (set.not_not_mem.1 hp) n with ⟨s, ⟨scard, scover⟩⟩,
rcases cardinal.lt_omega.1 (lt_of_le_of_lt scard (cardinal.nat_lt_omega _)) with ⟨N, hN⟩,
rw [hN, cardinal.nat_cast_le] at scard,
have : cardinal.mk s = cardinal.mk (fin N), by rw [hN, cardinal.mk_fin],
cases quotient.exact this with E,
use [s, N, scard, E],
simp [hp, scover] } },
choose s N hN E hs using this,
-- Define a function `F` taking values in a finite type and associating to `p` enough data
-- to reconstruct it up to `ε`, namely the (discretized) distances between elements of `s p`.
let M := (floor (ε⁻¹ * max C 0)).to_nat,
let F : GH_space → (Σk:fin ((K n).succ), (fin k → fin k → fin (M.succ))) :=
λp, ⟨⟨N p, lt_of_le_of_lt (hN p) (nat.lt_succ_self _)⟩,
λa b, ⟨min M (floor (ε⁻¹ * dist ((E p).inv_fun a) ((E p).inv_fun b))).to_nat,
lt_of_le_of_lt ( min_le_left _ _) (nat.lt_succ_self _) ⟩ ⟩,
refine ⟨_, by apply_instance, (λp, F p), _⟩,
-- It remains to show that if `F p = F q`, then `p` and `q` are `ε`-close
rintros ⟨p, pt⟩ ⟨q, qt⟩ hpq,
have Npq : N p = N q := (fin.ext_iff _ _).1 (sigma.mk.inj_iff.1 hpq).1,
let Ψ : s p → s q := λx, (E q).inv_fun (fin.cast Npq ((E p).to_fun x)),
let Φ : s p → q.rep := λx, Ψ x,
have main : GH_dist (p.rep) (q.rep) ≤ ε + ε/2 + ε,
{ -- to prove the main inequality, argue that `s p` is `ε`-dense in `p`, and `s q` is `ε`-dense
-- in `q`, and `s p` and `s q` are almost isometric. Then closeness follows
-- from `GH_dist_le_of_approx_subsets`
refine GH_dist_le_of_approx_subsets Φ _ _ _,
show ∀x : p.rep, ∃ (y : p.rep) (H : y ∈ s p), dist x y ≤ ε,
{ -- by construction, `s p` is `ε`-dense
assume x,
have : x ∈ ⋃y∈(s p), ball y (u n) := (hs p pt) (mem_univ _),
rcases mem_bUnion_iff.1 this with ⟨y, ys, hy⟩,
exact ⟨y, ys, le_trans (le_of_lt hy) u_le_ε⟩ },
show ∀x : q.rep, ∃ (z : s p), dist x (Φ z) ≤ ε,
{ -- by construction, `s q` is `ε`-dense, and it is the range of `Φ`
assume x,
have : x ∈ ⋃y∈(s q), ball y (u n) := (hs q qt) (mem_univ _),
rcases mem_bUnion_iff.1 this with ⟨y, ys, hy⟩,
let i := ((E q).to_fun ⟨y, ys⟩).1,
let hi := ((E q).to_fun ⟨y, ys⟩).2,
have ihi_eq : (⟨i, hi⟩ : fin (N q)) = (E q).to_fun ⟨y, ys⟩, by rw fin.ext_iff,
have hiq : i < N q := hi,
have hip : i < N p, { rwa Npq.symm at hiq },
let z := (E p).inv_fun ⟨i, hip⟩,
use z,
have C1 : (E p).to_fun z = ⟨i, hip⟩ := (E p).right_inv ⟨i, hip⟩,
have C2 : fin.cast Npq ⟨i, hip⟩ = ⟨i, hi⟩ := rfl,
have C3 : (E q).inv_fun ⟨i, hi⟩ = ⟨y, ys⟩, by { rw ihi_eq, exact (E q).left_inv ⟨y, ys⟩ },
have : Φ z = y :=
by { simp only [Φ, Ψ], rw [C1, C2, C3], refl },
rw this,
exact le_trans (le_of_lt hy) u_le_ε },
show ∀x y : s p, abs (dist x y - dist (Φ x) (Φ y)) ≤ ε,
{ /- the distance between `x` and `y` is encoded in `F p`, and the distance between
`Φ x` and `Φ y` (two points of `s q`) is encoded in `F q`, all this up to `ε`.
As `F p = F q`, the distances are almost equal. -/
assume x y,
have : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y) := rfl,
rw this,
-- introduce `i`, that codes both `x` and `Φ x` in `fin (N p) = fin (N q)`
let i := ((E p).to_fun x).1,
have hip : i < N p := ((E p).to_fun x).2,
have hiq : i < N q, by rwa Npq at hip,
have i' : i = ((E q).to_fun (Ψ x)).1, by { simp [Ψ, (E q).right_inv _] },
-- introduce `j`, that codes both `y` and `Φ y` in `fin (N p) = fin (N q)`
let j := ((E p).to_fun y).1,
have hjp : j < N p := ((E p).to_fun y).2,
have hjq : j < N q, by rwa Npq at hjp,
have j' : j = ((E q).to_fun (Ψ y)).1, by { simp [Ψ, (E q).right_inv _] },
-- Express `dist x y` in terms of `F p`
have Ap : ((F p).2 ⟨i, hip⟩ ⟨j, hjp⟩).1 = (floor (ε⁻¹ * dist x y)).to_nat := calc
((F p).2 ⟨i, hip⟩ ⟨j, hjp⟩).1 = ((F p).2 ((E p).to_fun x) ((E p).to_fun y)).1 :
by { congr; apply (fin.ext_iff _ _).2; refl }
... = min M (floor (ε⁻¹ * dist x y)).to_nat :
by simp only [F, (E p).left_inv _]
... = (floor (ε⁻¹ * dist x y)).to_nat :
begin
refine min_eq_right (int.to_nat_le_to_nat (floor_mono _)),
refine mul_le_mul_of_nonneg_left (le_trans _ (le_max_left _ _)) (le_of_lt (inv_pos εpos)),
change dist (x : p.rep) y ≤ C,
refine le_trans (dist_le_diam_of_mem compact_univ.bounded (mem_univ _) (mem_univ _)) _,
exact hdiam p pt
end,
-- Express `dist (Φ x) (Φ y)` in terms of `F q`
have Aq : ((F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩).1 = (floor (ε⁻¹ * dist (Ψ x) (Ψ y))).to_nat := calc
((F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩).1 = ((F q).2 ((E q).to_fun (Ψ x)) ((E q).to_fun (Ψ y))).1 :
by { congr; apply (fin.ext_iff _ _).2; [exact i', exact j'] }
... = min M (floor (ε⁻¹ * dist (Ψ x) (Ψ y))).to_nat :
by simp only [F, (E q).left_inv _]
... = (floor (ε⁻¹ * dist (Ψ x) (Ψ y))).to_nat :
begin
refine min_eq_right (int.to_nat_le_to_nat (floor_mono _)),
refine mul_le_mul_of_nonneg_left (le_trans _ (le_max_left _ _)) (le_of_lt (inv_pos εpos)),
change dist (Ψ x : q.rep) (Ψ y) ≤ C,
refine le_trans (dist_le_diam_of_mem compact_univ.bounded (mem_univ _) (mem_univ _)) _,
exact hdiam q qt
end,
-- use the equality between `F p` and `F q` to deduce that the distances have equal
-- integer parts
have : ((F p).2 ⟨i, hip⟩ ⟨j, hjp⟩).1 = ((F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩).1,
{ -- we want to `subst hpq` where `hpq : F p = F q`, except that `subst` only works
-- with a constant, so replace `F q` (and everything that depends on it) by a constant `f`
-- then `subst`
revert hiq hjq,
change N q with (F q).1,
generalize_hyp : F q = f at hpq ⊢,
subst hpq,
intros,
refl },
have : floor (ε⁻¹ * dist x y) = floor (ε⁻¹ * dist (Ψ x) (Ψ y)),
{ rw [Ap, Aq] at this,
have D : 0 ≤ floor (ε⁻¹ * dist x y) :=
floor_nonneg.2 (mul_nonneg (le_of_lt (inv_pos εpos)) dist_nonneg),
have D' : floor (ε⁻¹ * dist (Ψ x) (Ψ y)) ≥ 0 :=
floor_nonneg.2 (mul_nonneg (le_of_lt (inv_pos εpos)) dist_nonneg),
rw [← int.to_nat_of_nonneg D, ← int.to_nat_of_nonneg D', this] },
-- deduce that the distances coincide up to `ε`, by a straightforward computation
-- that should be automated
have I := calc
abs (ε⁻¹) * abs (dist x y - dist (Ψ x) (Ψ y)) =
abs (ε⁻¹ * (dist x y - dist (Ψ x) (Ψ y))) : (abs_mul _ _).symm
... = abs ((ε⁻¹ * dist x y) - (ε⁻¹ * dist (Ψ x) (Ψ y))) : by { congr, ring }
... ≤ 1 : le_of_lt (abs_sub_lt_one_of_floor_eq_floor this),
calc
abs (dist x y - dist (Ψ x) (Ψ y)) = (ε * ε⁻¹) * abs (dist x y - dist (Ψ x) (Ψ y)) :
by rw [mul_inv_cancel (ne_of_gt εpos), one_mul]
... = ε * (abs (ε⁻¹) * abs (dist x y - dist (Ψ x) (Ψ y))) :
by rw [abs_of_nonneg (le_of_lt (inv_pos εpos)), mul_assoc]
... ≤ ε * 1 : mul_le_mul_of_nonneg_left I (le_of_lt εpos)
... = ε : mul_one _ } },
calc dist p q = GH_dist (p.rep) (q.rep) : dist_GH_dist p q
... ≤ ε + ε/2 + ε : main
... = δ/2 : by { simp [ε], ring }
... < δ : half_lt_self δpos
end
section complete
/- We will show that a sequence `u n` of compact metric spaces satisfying
`dist (u n) (u (n+1)) < 1/2^n` converges, which implies completeness of the Gromov-Hausdorff space.
We need to exhibit the limiting compact metric space. For this, start from
a sequence `X n` of representatives of `u n`, and glue in an optimal way `X n` to `X (n+1)`
for all `n`, in a common metric space. Formally, this is done as follows.
Start from `Y 0 = X 0`. Then, glue `X 0` to `X 1` in an optimal way, yielding a space
`Y 1` (with an embedding of `X 1`). Then, consider an optimal gluing of `X 1` and `X 2`, and
glue it to `Y 1` along their common subspace `X 1`. This gives a new space `Y 2`, with an
embedding of `X 2`. Go on, to obtain a sequence of spaces `Y n`. Let `Z0` be the inductive
limit of the `Y n`, and finally let `Z` be the completion of `Z0`.
The images `X2 n` of `X n` in `Z` are at Hausdorff distance `< 1/2^n` by construction, hence they
form a Cauchy sequence for the Hausdorff distance. By completeness (of `Z`, and therefore of its
set of nonempty compact subsets), they converge to a limit `L`. This is the nonempty
compact metric space we are looking for. -/
variables (X : ℕ → Type) [∀n, metric_space (X n)] [∀n, compact_space (X n)] [∀n, nonempty (X n)]
/-- Auxiliary structure used to glue metric spaces below, recording an isometric embedding
of a type `A` in another metric space. -/
structure aux_gluing_struct (A : Type) [metric_space A] : Type 1 :=
(space : Type)
(metric : metric_space space)
(embed : A → space)
(isom : isometry embed)
/-- Auxiliary sequence of metric spaces, containing copies of `X 0`, ..., `X n`, where each
`X i` is glued to `X (i+1)` in an optimal way. The space at step `n+1` is obtained from the space
at step `n` by adding `X (n+1)`, glued in an optimal way to the `X n` already sitting there. -/
def aux_gluing (n : ℕ) : aux_gluing_struct (X n) := nat.rec_on n
{ space := X 0,
metric := by apply_instance,
embed := id,
isom := λx y, rfl }
(λn a, by letI : metric_space a.space := a.metric; exact
{ space := glue_space a.isom (isometry_optimal_GH_injl (X n) (X n.succ)),
metric := metric.metric_space_glue_space a.isom (isometry_optimal_GH_injl (X n) (X n.succ)),
embed := (to_glue_r a.isom (isometry_optimal_GH_injl (X n) (X n.succ)))
∘ (optimal_GH_injr (X n) (X n.succ)),
isom := (to_glue_r_isometry _ _).comp (isometry_optimal_GH_injr (X n) (X n.succ)) })
/-- The Gromov-Hausdorff space is complete. -/
instance : complete_space (GH_space) :=
begin
have : ∀ (n : ℕ), 0 < ((1:ℝ) / 2) ^ n, by { apply _root_.pow_pos, norm_num },
-- start from a sequence of nonempty compact metric spaces within distance `1/2^n` of each other
refine metric.complete_of_convergent_controlled_sequences (λn, (1/2)^n) this (λu hu, _),
-- `X n` is a representative of `u n`
let X := λn, (u n).rep,
-- glue them together successively in an optimal way, getting a sequence of metric spaces `Y n`
let Y := aux_gluing X,
letI : ∀n, metric_space (Y n).space := λn, (Y n).metric,
have E : ∀n:ℕ, glue_space (Y n).isom (isometry_optimal_GH_injl (X n) (X n.succ)) = (Y n.succ).space :=
λn, by { simp [Y, aux_gluing], refl },
let c := λn, cast (E n),
have ic : ∀n, isometry (c n) := λn x y, rfl,
-- there is a canonical embedding of `Y n` in `Y (n+1)`, by construction
let f : Πn, (Y n).space → (Y n.succ).space :=
λn, (c n) ∘ (to_glue_l (aux_gluing X n).isom (isometry_optimal_GH_injl (X n) (X n.succ))),
have I : ∀n, isometry (f n),
{ assume n,
apply isometry.comp,
{ assume x y, refl },
{ apply to_glue_l_isometry } },
-- consider the inductive limit `Z0` of the `Y n`, and then its completion `Z`
let Z0 := metric.inductive_limit I,
let Z := uniform_space.completion Z0,
let Φ := to_inductive_limit I,
let coeZ := (coe : Z0 → Z),
-- let `X2 n` be the image of `X n` in the space `Z`
let X2 := λn, range (coeZ ∘ (Φ n) ∘ (Y n).embed),
have isom : ∀n, isometry (coeZ ∘ (Φ n) ∘ (Y n).embed),
{ assume n,
apply isometry.comp completion.coe_isometry _,
apply isometry.comp _ (Y n).isom,
apply to_inductive_limit_isometry },
-- The Hausdorff distance of `X2 n` and `X2 (n+1)` is by construction the distance between
-- `u n` and `u (n+1)`, therefore bounded by `1/2^n`
have D2 : ∀n, Hausdorff_dist (X2 n) (X2 n.succ) < (1/2)^n,
{ assume n,
have X2n : X2 n = range ((coeZ ∘ (Φ n.succ) ∘ (c n)
∘ (to_glue_r (Y n).isom (isometry_optimal_GH_injl (X n) (X n.succ))))
∘ (optimal_GH_injl (X n) (X n.succ))),
{ change X2 n = range (coeZ ∘ (Φ n.succ) ∘ (c n)
∘ (to_glue_r (Y n).isom (isometry_optimal_GH_injl (X n) (X n.succ)))
∘ (optimal_GH_injl (X n) (X n.succ))),
simp only [X2, Φ],
rw [← to_inductive_limit_commute I],
simp only [f],
rw ← to_glue_commute },
rw range_comp at X2n,
have X2nsucc : X2 n.succ = range ((coeZ ∘ (Φ n.succ) ∘ (c n)
∘ (to_glue_r (Y n).isom (isometry_optimal_GH_injl (X n) (X n.succ))))
∘ (optimal_GH_injr (X n) (X n.succ))), by refl,
rw range_comp at X2nsucc,
rw [X2n, X2nsucc, Hausdorff_dist_image, Hausdorff_dist_optimal, ← dist_GH_dist],
{ exact hu n n n.succ (le_refl n) (le_succ n) },
{ apply isometry.comp completion.coe_isometry _,
apply isometry.comp _ ((ic n).comp (to_glue_r_isometry _ _)),
apply to_inductive_limit_isometry } },
-- consider `X2 n` as a member `X3 n` of the type of nonempty compact subsets of `Z`, which
-- is a metric space
let X3 : ℕ → nonempty_compacts Z := λn, ⟨X2 n,
⟨by { simp only [X2, set.range_eq_empty, not_not, ne.def], apply_instance },
compact_range (isom n).continuous ⟩⟩,
-- `X3 n` is a Cauchy sequence by construction, as the successive distances are
-- bounded by `(1/2)^n`
have : cauchy_seq X3,
{ refine cauchy_seq_of_le_geometric (1/2) 1 (by norm_num) (λn, _),
rw one_mul,
exact le_of_lt (D2 n) },
-- therefore, it converges to a limit `L`
rcases cauchy_seq_tendsto_of_complete this with ⟨L, hL⟩,
-- the images of `X3 n` in the Gromov-Hausdorff space converge to the image of `L`
have M : tendsto (λn, (X3 n).to_GH_space) at_top (𝓝 L.to_GH_space) :=
tendsto.comp (to_GH_space_continuous.tendsto _) hL,
-- By construction, the image of `X3 n` in the Gromov-Hausdorff space is `u n`.
have : ∀n, (X3 n).to_GH_space = u n,
{ assume n,
rw [nonempty_compacts.to_GH_space, ← (u n).to_GH_space_rep,
to_GH_space_eq_to_GH_space_iff_isometric],
constructor,
convert (isom n).isometric_on_range.symm,
},
-- Finally, we have proved the convergence of `u n`
exact ⟨L.to_GH_space, by simpa [this] using M⟩
end
end complete--section
end Gromov_Hausdorff --namespace
|
06b39edd52d669fe576a1b8cce2b85f125429ef7 | 82e44445c70db0f03e30d7be725775f122d72f3e | /src/algebra/ring/basic.lean | 62a21c5b347d85135f2e9a162ba720419af4d2c9 | [
"Apache-2.0"
] | permissive | stjordanis/mathlib | 51e286d19140e3788ef2c470bc7b953e4991f0c9 | 2568d41bca08f5d6bf39d915434c8447e21f42ee | refs/heads/master | 1,631,748,053,501 | 1,627,938,886,000 | 1,627,938,886,000 | 228,728,358 | 0 | 0 | Apache-2.0 | 1,576,630,588,000 | 1,576,630,587,000 | null | UTF-8 | Lean | false | false | 46,448 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland
-/
import algebra.divisibility
import data.set.basic
/-!
# Properties and homomorphisms of semirings and rings
This file proves simple properties of semirings, rings and domains and their unit groups. It also
defines bundled homomorphisms of semirings and rings. As with monoid and groups, we use the same
structure `ring_hom a β`, a.k.a. `α →+* β`, for both homomorphism types.
The unbundled homomorphisms are defined in `deprecated/ring`. They are deprecated and the plan is to
slowly remove them from mathlib.
## Main definitions
ring_hom, nonzero, domain, integral_domain
## Notations
→+* for bundled ring homs (also use for semiring homs)
## Implementation notes
There's a coercion from bundled homs to fun, and the canonical
notation is to use the bundled hom as a function via this coercion.
There is no `semiring_hom` -- the idea is that `ring_hom` is used.
The constructor for a `ring_hom` between semirings needs a proof of `map_zero`, `map_one` and
`map_add` as well as `map_mul`; a separate constructor `ring_hom.mk'` will construct ring homs
between rings from monoid homs given only a proof that addition is preserved.
## Tags
`ring_hom`, `semiring_hom`, `semiring`, `comm_semiring`, `ring`, `comm_ring`, `domain`,
`integral_domain`, `nonzero`, `units`
-/
universes u v w x
variables {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
set_option old_structure_cmd true
open function
/-!
### `distrib` class
-/
/-- A typeclass stating that multiplication is left and right distributive
over addition. -/
@[protect_proj, ancestor has_mul has_add]
class distrib (R : Type*) extends has_mul R, has_add R :=
(left_distrib : ∀ a b c : R, a * (b + c) = (a * b) + (a * c))
(right_distrib : ∀ a b c : R, (a + b) * c = (a * c) + (b * c))
lemma left_distrib [distrib R] (a b c : R) : a * (b + c) = a * b + a * c :=
distrib.left_distrib a b c
alias left_distrib ← mul_add
lemma right_distrib [distrib R] (a b c : R) : (a + b) * c = a * c + b * c :=
distrib.right_distrib a b c
alias right_distrib ← add_mul
/-- Pullback a `distrib` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.distrib {S} [has_mul R] [has_add R] [distrib S]
(f : R → S) (hf : injective f) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) :
distrib R :=
{ mul := (*),
add := (+),
left_distrib := λ x y z, hf $ by simp only [*, left_distrib],
right_distrib := λ x y z, hf $ by simp only [*, right_distrib] }
/-- Pushforward a `distrib` instance along a surjective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.surjective.distrib {S} [distrib R] [has_add S] [has_mul S]
(f : R → S) (hf : surjective f) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) :
distrib S :=
{ mul := (*),
add := (+),
left_distrib := hf.forall₃.2 $ λ x y z, by simp only [← add, ← mul, left_distrib],
right_distrib := hf.forall₃.2 $ λ x y z, by simp only [← add, ← mul, right_distrib] }
/-!
### Semirings
-/
/-- A not-necessarily-unital, not-necessarily-associative semiring. -/
@[protect_proj, ancestor add_comm_monoid distrib mul_zero_class]
class non_unital_non_assoc_semiring (α : Type u) extends
add_comm_monoid α, distrib α, mul_zero_class α
/-- An associative but not-necessarily unital semiring. -/
@[protect_proj, ancestor non_unital_non_assoc_semiring semigroup_with_zero]
class non_unital_semiring (α : Type u) extends
non_unital_non_assoc_semiring α, semigroup_with_zero α
/-- A unital but not-necessarily-associative semiring. -/
@[protect_proj, ancestor non_unital_non_assoc_semiring mul_zero_one_class]
class non_assoc_semiring (α : Type u) extends
non_unital_non_assoc_semiring α, mul_zero_one_class α
/-- A semiring is a type with the following structures: additive commutative monoid
(`add_comm_monoid`), multiplicative monoid (`monoid`), distributive laws (`distrib`), and
multiplication by zero law (`mul_zero_class`). The actual definition extends `monoid_with_zero`
instead of `monoid` and `mul_zero_class`. -/
@[protect_proj, ancestor non_unital_semiring non_assoc_semiring monoid_with_zero]
class semiring (α : Type u) extends non_unital_semiring α, non_assoc_semiring α, monoid_with_zero α
section injective_surjective_maps
variables [has_zero β] [has_add β] [has_mul β]
/-- Pullback a `non_unital_non_assoc_semiring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.non_unital_non_assoc_semiring
{α : Type u} [non_unital_non_assoc_semiring α]
(f : β → α) (hf : injective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
non_unital_non_assoc_semiring β :=
{ .. hf.mul_zero_class f zero mul, .. hf.add_comm_monoid f zero add, .. hf.distrib f add mul }
/-- Pullback a `non_unital_semiring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.non_unital_semiring
{α : Type u} [non_unital_semiring α]
(f : β → α) (hf : injective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
non_unital_semiring β :=
{ .. hf.non_unital_non_assoc_semiring f zero add mul, .. hf.semigroup_with_zero f zero mul }
/-- Pullback a `non_assoc_semiring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.non_assoc_semiring
{α : Type u} [non_assoc_semiring α] [has_one β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
non_assoc_semiring β :=
{ .. hf.non_unital_non_assoc_semiring f zero add mul, .. hf.mul_one_class f one mul }
/-- Pullback a `semiring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.semiring
{α : Type u} [semiring α] [has_one β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
semiring β :=
{ .. hf.monoid_with_zero f zero one mul, .. hf.add_comm_monoid f zero add,
.. hf.distrib f add mul }
/-- Pushforward a `non_unital_non_assoc_semiring` instance along a surjective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.surjective.non_unital_non_assoc_semiring
{α : Type u} [non_unital_non_assoc_semiring α]
(f : α → β) (hf : surjective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
non_unital_non_assoc_semiring β :=
{ .. hf.mul_zero_class f zero mul, .. hf.add_comm_monoid f zero add, .. hf.distrib f add mul }
/-- Pushforward a `non_unital_semiring` instance along a surjective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.surjective.non_unital_semiring
{α : Type u} [non_unital_semiring α]
(f : α → β) (hf : surjective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
non_unital_semiring β :=
{ .. hf.non_unital_non_assoc_semiring f zero add mul, .. hf.semigroup_with_zero f zero mul }
/-- Pushforward a `non_assoc_semiring` instance along a surjective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.surjective.non_assoc_semiring
{α : Type u} [non_assoc_semiring α] [has_one β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
non_assoc_semiring β :=
{ .. hf.non_unital_non_assoc_semiring f zero add mul, .. hf.mul_one_class f one mul }
/-- Pushforward a `semiring` instance along a surjective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.surjective.semiring
{α : Type u} [semiring α] [has_one β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
semiring β :=
{ .. hf.monoid_with_zero f zero one mul, .. hf.add_comm_monoid f zero add,
.. hf.distrib f add mul }
end injective_surjective_maps
section semiring
variables [semiring α]
lemma one_add_one_eq_two : 1 + 1 = (2 : α) :=
by unfold bit0
theorem two_mul (n : α) : 2 * n = n + n :=
eq.trans (right_distrib 1 1 n) (by simp)
lemma distrib_three_right (a b c d : α) : (a + b + c) * d = a * d + b * d + c * d :=
by simp [right_distrib]
theorem mul_two (n : α) : n * 2 = n + n :=
(left_distrib n 1 1).trans (by simp)
theorem bit0_eq_two_mul (n : α) : bit0 n = 2 * n :=
(two_mul _).symm
@[to_additive] lemma mul_ite {α} [has_mul α] (P : Prop) [decidable P] (a b c : α) :
a * (if P then b else c) = if P then a * b else a * c :=
by split_ifs; refl
@[to_additive] lemma ite_mul {α} [has_mul α] (P : Prop) [decidable P] (a b c : α) :
(if P then a else b) * c = if P then a * c else b * c :=
by split_ifs; refl
-- We make `mul_ite` and `ite_mul` simp lemmas,
-- but not `add_ite` or `ite_add`.
-- The problem we're trying to avoid is dealing with
-- summations of the form `∑ x in s, (f x + ite P 1 0)`,
-- in which `add_ite` followed by `sum_ite` would needlessly slice up
-- the `f x` terms according to whether `P` holds at `x`.
-- There doesn't appear to be a corresponding difficulty so far with
-- `mul_ite` and `ite_mul`.
attribute [simp] mul_ite ite_mul
@[simp] lemma mul_boole {α} [non_assoc_semiring α] (P : Prop) [decidable P] (a : α) :
a * (if P then 1 else 0) = if P then a else 0 :=
by simp
@[simp] lemma boole_mul {α} [non_assoc_semiring α] (P : Prop) [decidable P] (a : α) :
(if P then 1 else 0) * a = if P then a else 0 :=
by simp
lemma ite_mul_zero_left {α : Type*} [mul_zero_class α] (P : Prop) [decidable P] (a b : α) :
ite P (a * b) 0 = ite P a 0 * b :=
by { by_cases h : P; simp [h], }
lemma ite_mul_zero_right {α : Type*} [mul_zero_class α] (P : Prop) [decidable P] (a b : α) :
ite P (a * b) 0 = a * ite P b 0 :=
by { by_cases h : P; simp [h], }
/-- An element `a` of a semiring is even if there exists `k` such `a = 2*k`. -/
def even (a : α) : Prop := ∃ k, a = 2*k
lemma even_iff_two_dvd {a : α} : even a ↔ 2 ∣ a := iff.rfl
@[simp] lemma range_two_mul (α : Type*) [semiring α] :
set.range (λ x : α, 2 * x) = {a | even a} :=
by { ext x, simp [even, eq_comm] }
@[simp] lemma even_bit0 (a : α) : even (bit0 a) :=
⟨a, by rw [bit0, two_mul]⟩
/-- An element `a` of a semiring is odd if there exists `k` such `a = 2*k + 1`. -/
def odd (a : α) : Prop := ∃ k, a = 2*k + 1
@[simp] lemma odd_bit1 (a : α) : odd (bit1 a) :=
⟨a, by rw [bit1, bit0, two_mul]⟩
@[simp] lemma range_two_mul_add_one (α : Type*) [semiring α] :
set.range (λ x : α, 2 * x + 1) = {a | odd a} :=
by { ext x, simp [odd, eq_comm] }
theorem dvd_add {a b c : α} (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b + c :=
dvd.elim h₁ (λ d hd, dvd.elim h₂ (λ e he, dvd.intro (d + e) (by simp [left_distrib, hd, he])))
end semiring
namespace add_monoid_hom
/-- Left multiplication by an element of a (semi)ring is an `add_monoid_hom` -/
def mul_left {R : Type*} [non_unital_non_assoc_semiring R] (r : R) : R →+ R :=
{ to_fun := (*) r,
map_zero' := mul_zero r,
map_add' := mul_add r }
@[simp] lemma coe_mul_left {R : Type*} [non_unital_non_assoc_semiring R] (r : R) :
⇑(mul_left r) = (*) r := rfl
/-- Right multiplication by an element of a (semi)ring is an `add_monoid_hom` -/
def mul_right {R : Type*} [non_unital_non_assoc_semiring R] (r : R) : R →+ R :=
{ to_fun := λ a, a * r,
map_zero' := zero_mul r,
map_add' := λ _ _, add_mul _ _ r }
@[simp] lemma coe_mul_right {R : Type*} [non_unital_non_assoc_semiring R] (r : R) :
⇑(mul_right r) = (* r) := rfl
lemma mul_right_apply {R : Type*} [non_unital_non_assoc_semiring R] (a r : R) :
mul_right r a = a * r := rfl
end add_monoid_hom
/-- Bundled semiring homomorphisms; use this for bundled ring homomorphisms too.
This extends from both `monoid_hom` and `monoid_with_zero_hom` in order to put the fields in a
sensible order, even though `monoid_with_zero_hom` already extends `monoid_hom`. -/
structure ring_hom (α : Type*) (β : Type*) [non_assoc_semiring α] [non_assoc_semiring β]
extends monoid_hom α β, add_monoid_hom α β, monoid_with_zero_hom α β
infixr ` →+* `:25 := ring_hom
/-- Reinterpret a ring homomorphism `f : R →+* S` as a `monoid_with_zero_hom R S`.
The `simp`-normal form is `(f : monoid_with_zero_hom R S)`. -/
add_decl_doc ring_hom.to_monoid_with_zero_hom
/-- Reinterpret a ring homomorphism `f : R →+* S` as a monoid homomorphism `R →* S`.
The `simp`-normal form is `(f : R →* S)`. -/
add_decl_doc ring_hom.to_monoid_hom
/-- Reinterpret a ring homomorphism `f : R →+* S` as an additive monoid homomorphism `R →+ S`.
The `simp`-normal form is `(f : R →+ S)`. -/
add_decl_doc ring_hom.to_add_monoid_hom
namespace ring_hom
section coe
/-!
Throughout this section, some `semiring` arguments are specified with `{}` instead of `[]`.
See note [implicit instance arguments].
-/
variables {rα : non_assoc_semiring α} {rβ : non_assoc_semiring β}
include rα rβ
instance : has_coe_to_fun (α →+* β) := ⟨_, ring_hom.to_fun⟩
initialize_simps_projections ring_hom (to_fun → apply)
@[simp] lemma to_fun_eq_coe (f : α →+* β) : f.to_fun = f := rfl
@[simp] lemma coe_mk (f : α → β) (h₁ h₂ h₃ h₄) : ⇑(⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) = f := rfl
instance has_coe_monoid_hom : has_coe (α →+* β) (α →* β) := ⟨ring_hom.to_monoid_hom⟩
@[simp, norm_cast] lemma coe_monoid_hom (f : α →+* β) : ⇑(f : α →* β) = f := rfl
@[simp] lemma to_monoid_hom_eq_coe (f : α →+* β) : f.to_monoid_hom = f := rfl
@[simp] lemma to_monoid_with_zero_hom_eq_coe (f : α →+* β) :
(f.to_monoid_with_zero_hom : α → β) = f := rfl
@[simp] lemma coe_monoid_hom_mk (f : α → β) (h₁ h₂ h₃ h₄) :
((⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) : α →* β) = ⟨f, h₁, h₂⟩ :=
rfl
instance has_coe_add_monoid_hom : has_coe (α →+* β) (α →+ β) := ⟨ring_hom.to_add_monoid_hom⟩
@[simp, norm_cast] lemma coe_add_monoid_hom (f : α →+* β) : ⇑(f : α →+ β) = f := rfl
@[simp] lemma to_add_monoid_hom_eq_coe (f : α →+* β) : f.to_add_monoid_hom = f := rfl
@[simp] lemma coe_add_monoid_hom_mk (f : α → β) (h₁ h₂ h₃ h₄) :
((⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨f, h₃, h₄⟩ :=
rfl
end coe
variables [rα : non_assoc_semiring α] [rβ : non_assoc_semiring β]
section
include rα rβ
variables (f : α →+* β) {x y : α} {rα rβ}
theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x :=
congr_arg (λ h : α →+* β, h x) h
theorem congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y :=
congr_arg (λ x : α, f x) h
theorem coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g :=
by cases f; cases g; cases h; refl
@[ext] theorem ext ⦃f g : α →+* β⦄ (h : ∀ x, f x = g x) : f = g :=
coe_inj (funext h)
theorem ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x :=
⟨λ h x, h ▸ rfl, λ h, ext h⟩
@[simp] lemma mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : ring_hom.mk f h₁ h₂ h₃ h₄ = f :=
ext $ λ _, rfl
theorem coe_add_monoid_hom_injective : function.injective (coe : (α →+* β) → (α →+ β)) :=
λ f g h, ext (λ x, add_monoid_hom.congr_fun h x)
theorem coe_monoid_hom_injective : function.injective (coe : (α →+* β) → (α →* β)) :=
λ f g h, ext (λ x, monoid_hom.congr_fun h x)
/-- Ring homomorphisms map zero to zero. -/
@[simp] lemma map_zero (f : α →+* β) : f 0 = 0 := f.map_zero'
/-- Ring homomorphisms map one to one. -/
@[simp] lemma map_one (f : α →+* β) : f 1 = 1 := f.map_one'
/-- Ring homomorphisms preserve addition. -/
@[simp] lemma map_add (f : α →+* β) (a b : α) : f (a + b) = f a + f b := f.map_add' a b
/-- Ring homomorphisms preserve multiplication. -/
@[simp] lemma map_mul (f : α →+* β) (a b : α) : f (a * b) = f a * f b := f.map_mul' a b
/-- Ring homomorphisms preserve `bit0`. -/
@[simp] lemma map_bit0 (f : α →+* β) (a : α) : f (bit0 a) = bit0 (f a) := map_add _ _ _
/-- Ring homomorphisms preserve `bit1`. -/
@[simp] lemma map_bit1 (f : α →+* β) (a : α) : f (bit1 a) = bit1 (f a) :=
by simp [bit1]
/-- `f : R →+* S` has a trivial codomain iff `f 1 = 0`. -/
lemma codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 :=
by rw [map_one, eq_comm]
/-- `f : R →+* S` has a trivial codomain iff it has a trivial range. -/
lemma codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ (∀ x, f x = 0) :=
f.codomain_trivial_iff_map_one_eq_zero.trans
⟨λ h x, by rw [←mul_one x, map_mul, h, mul_zero], λ h, h 1⟩
/-- `f : R →+* S` has a trivial codomain iff its range is `{0}`. -/
lemma codomain_trivial_iff_range_eq_singleton_zero : (0 : β) = 1 ↔ set.range f = {0} :=
f.codomain_trivial_iff_range_trivial.trans
⟨ λ h, set.ext (λ y, ⟨λ ⟨x, hx⟩, by simp [←hx, h x], λ hy, ⟨0, by simpa using hy.symm⟩⟩),
λ h x, set.mem_singleton_iff.mp (h ▸ set.mem_range_self x)⟩
/-- `f : R →+* S` doesn't map `1` to `0` if `S` is nontrivial -/
lemma map_one_ne_zero [nontrivial β] : f 1 ≠ 0 :=
mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one
/-- If there is a homomorphism `f : R →+* S` and `S` is nontrivial, then `R` is nontrivial. -/
lemma domain_nontrivial [nontrivial β] : nontrivial α :=
⟨⟨1, 0, mt (λ h, show f 1 = 0, by rw [h, map_zero]) f.map_one_ne_zero⟩⟩
end
lemma is_unit_map [semiring α] [semiring β] (f : α →+* β) {a : α} (h : is_unit a) : is_unit (f a) :=
h.map f.to_monoid_hom
/-- The identity ring homomorphism from a semiring to itself. -/
def id (α : Type*) [non_assoc_semiring α] : α →+* α :=
by refine {to_fun := id, ..}; intros; refl
include rα
instance : inhabited (α →+* α) := ⟨id α⟩
@[simp] lemma id_apply (x : α) : ring_hom.id α x = x := rfl
@[simp] lemma coe_add_monoid_hom_id : (id α : α →+ α) = add_monoid_hom.id α := rfl
@[simp] lemma coe_monoid_hom_id : (id α : α →* α) = monoid_hom.id α := rfl
variable {rγ : non_assoc_semiring γ}
include rβ rγ
/-- Composition of ring homomorphisms is a ring homomorphism. -/
def comp (hnp : β →+* γ) (hmn : α →+* β) : α →+* γ :=
{ to_fun := hnp ∘ hmn,
map_zero' := by simp,
map_one' := by simp,
map_add' := λ x y, by simp,
map_mul' := λ x y, by simp}
/-- Composition of semiring homomorphisms is associative. -/
lemma comp_assoc {δ} {rδ: non_assoc_semiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) :
(h.comp g).comp f = h.comp (g.comp f) := rfl
@[simp] lemma coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn := rfl
lemma comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x =
(hnp (hmn x)) := rfl
omit rγ
@[simp] lemma comp_id (f : α →+* β) : f.comp (id α) = f := ext $ λ x, rfl
@[simp] lemma id_comp (f : α →+* β) : (id β).comp f = f := ext $ λ x, rfl
omit rβ
instance : monoid (α →+* α) :=
{ one := id α,
mul := comp,
mul_one := comp_id,
one_mul := id_comp,
mul_assoc := λ f g h, comp_assoc _ _ _ }
lemma one_def : (1 : α →+* α) = id α := rfl
@[simp] lemma coe_one : ⇑(1 : α →+* α) = _root_.id := rfl
lemma mul_def (f g : α →+* α) : f * g = f.comp g := rfl
@[simp] lemma coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g := rfl
include rβ rγ
lemma cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨λ h, ring_hom.ext $ (forall_iff_forall_surj hf).1 (ext_iff.1 h), λ h, h ▸ rfl⟩
lemma cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨λ h, ring_hom.ext $ λ x, hg $ by rw [← comp_apply, h, comp_apply], λ h, h ▸ rfl⟩
omit rα rβ rγ
end ring_hom
section semiring
variables [semiring α] {a : α}
@[simp] theorem two_dvd_bit0 : 2 ∣ bit0 a := ⟨a, bit0_eq_two_mul _⟩
end semiring
/-- A commutative semiring is a `semiring` with commutative multiplication. In other words, it is a
type with the following structures: additive commutative monoid (`add_comm_monoid`), multiplicative
commutative monoid (`comm_monoid`), distributive laws (`distrib`), and multiplication by zero law
(`mul_zero_class`). -/
@[protect_proj, ancestor semiring comm_monoid]
class comm_semiring (α : Type u) extends semiring α, comm_monoid α
@[priority 100] -- see Note [lower instance priority]
instance comm_semiring.to_comm_monoid_with_zero [comm_semiring α] : comm_monoid_with_zero α :=
{ .. comm_semiring.to_comm_monoid α, .. comm_semiring.to_semiring α }
section comm_semiring
variables [comm_semiring α] [comm_semiring β] {a b c : α}
/-- Pullback a `semiring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.comm_semiring [has_zero γ] [has_one γ] [has_add γ] [has_mul γ]
(f : γ → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
comm_semiring γ :=
{ .. hf.semiring f zero one add mul, .. hf.comm_semigroup f mul }
/-- Pullback a `semiring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.surjective.comm_semiring [has_zero γ] [has_one γ] [has_add γ] [has_mul γ]
(f : α → γ) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
comm_semiring γ :=
{ .. hf.semiring f zero one add mul, .. hf.comm_semigroup f mul }
lemma add_mul_self_eq (a b : α) : (a + b) * (a + b) = a*a + 2*a*b + b*b :=
by simp only [two_mul, add_mul, mul_add, add_assoc, mul_comm b]
lemma ring_hom.map_dvd (f : α →+* β) {a b : α} : a ∣ b → f a ∣ f b :=
λ ⟨z, hz⟩, ⟨f z, by rw [hz, f.map_mul]⟩
end comm_semiring
/-!
### Rings
-/
/-- A ring is a type with the following structures: additive commutative group (`add_comm_group`),
multiplicative monoid (`monoid`), and distributive laws (`distrib`). Equivalently, a ring is a
`semiring` with a negation operation making it an additive group. -/
@[protect_proj, ancestor add_comm_group monoid distrib]
class ring (α : Type u) extends add_comm_group α, monoid α, distrib α
section ring
variables [ring α] {a b c d e : α}
/- The instance from `ring` to `semiring` happens often in linear algebra, for which all the basic
definitions are given in terms of semirings, but many applications use rings or fields. We increase
a little bit its priority above 100 to try it quickly, but remaining below the default 1000 so that
more specific instances are tried first. -/
@[priority 200]
instance ring.to_semiring : semiring α :=
{ zero_mul := λ a, add_left_cancel $ show 0 * a + 0 * a = 0 * a + 0,
by rw [← add_mul, zero_add, add_zero],
mul_zero := λ a, add_left_cancel $ show a * 0 + a * 0 = a * 0 + 0,
by rw [← mul_add, add_zero, add_zero],
..‹ring α› }
/-- Pullback a `ring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
ring β :=
{ .. hf.add_comm_group f zero add neg sub, .. hf.monoid f one mul, .. hf.distrib f add mul }
/-- Pullback a `ring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.surjective.ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
ring β :=
{ .. hf.add_comm_group f zero add neg sub, .. hf.monoid f one mul, .. hf.distrib f add mul }
lemma neg_mul_eq_neg_mul (a b : α) : -(a * b) = -a * b :=
neg_eq_of_add_eq_zero
begin rw [← right_distrib, add_right_neg, zero_mul] end
lemma neg_mul_eq_mul_neg (a b : α) : -(a * b) = a * -b :=
neg_eq_of_add_eq_zero
begin rw [← left_distrib, add_right_neg, mul_zero] end
@[simp] lemma neg_mul_eq_neg_mul_symm (a b : α) : - a * b = - (a * b) :=
eq.symm (neg_mul_eq_neg_mul a b)
@[simp] lemma mul_neg_eq_neg_mul_symm (a b : α) : a * - b = - (a * b) :=
eq.symm (neg_mul_eq_mul_neg a b)
lemma neg_mul_neg (a b : α) : -a * -b = a * b :=
by simp
lemma neg_mul_comm (a b : α) : -a * b = a * -b :=
by simp
theorem neg_eq_neg_one_mul (a : α) : -a = -1 * a :=
by simp
lemma mul_sub_left_distrib (a b c : α) : a * (b - c) = a * b - a * c :=
by simpa only [sub_eq_add_neg, neg_mul_eq_mul_neg] using mul_add a b (-c)
alias mul_sub_left_distrib ← mul_sub
lemma mul_sub_right_distrib (a b c : α) : (a - b) * c = a * c - b * c :=
by simpa only [sub_eq_add_neg, neg_mul_eq_neg_mul] using add_mul a (-b) c
alias mul_sub_right_distrib ← sub_mul
/-- An element of a ring multiplied by the additive inverse of one is the element's additive
inverse. -/
lemma mul_neg_one (a : α) : a * -1 = -a := by simp
/-- The additive inverse of one multiplied by an element of a ring is the element's additive
inverse. -/
lemma neg_one_mul (a : α) : -1 * a = -a := by simp
/-- An iff statement following from right distributivity in rings and the definition
of subtraction. -/
theorem mul_add_eq_mul_add_iff_sub_mul_add_eq : a * e + c = b * e + d ↔ (a - b) * e + c = d :=
calc
a * e + c = b * e + d ↔ a * e + c = d + b * e : by simp [add_comm]
... ↔ a * e + c - b * e = d : iff.intro (λ h, begin rw h, simp end) (λ h,
begin rw ← h, simp end)
... ↔ (a - b) * e + c = d : begin simp [sub_mul, sub_add_eq_add_sub] end
/-- A simplification of one side of an equation exploiting right distributivity in rings
and the definition of subtraction. -/
theorem sub_mul_add_eq_of_mul_add_eq_mul_add : a * e + c = b * e + d → (a - b) * e + c = d :=
assume h,
calc
(a - b) * e + c = (a * e + c) - b * e : begin simp [sub_mul, sub_add_eq_add_sub] end
... = d : begin rw h, simp [@add_sub_cancel α] end
end ring
namespace units
variables [ring α] {a b : α}
/-- Each element of the group of units of a ring has an additive inverse. -/
instance : has_neg (units α) := ⟨λu, ⟨-↑u, -↑u⁻¹, by simp, by simp⟩ ⟩
/-- Representing an element of a ring's unit group as an element of the ring commutes with
mapping this element to its additive inverse. -/
@[simp, norm_cast] protected theorem coe_neg (u : units α) : (↑-u : α) = -u := rfl
@[simp, norm_cast] protected theorem coe_neg_one : ((-1 : units α) : α) = -1 := rfl
/-- Mapping an element of a ring's unit group to its inverse commutes with mapping this element
to its additive inverse. -/
@[simp] protected theorem neg_inv (u : units α) : (-u)⁻¹ = -u⁻¹ := rfl
/-- An element of a ring's unit group equals the additive inverse of its additive inverse. -/
@[simp] protected theorem neg_neg (u : units α) : - -u = u :=
units.ext $ neg_neg _
/-- Multiplication of elements of a ring's unit group commutes with mapping the first
argument to its additive inverse. -/
@[simp] protected theorem neg_mul (u₁ u₂ : units α) : -u₁ * u₂ = -(u₁ * u₂) :=
units.ext $ neg_mul_eq_neg_mul_symm _ _
/-- Multiplication of elements of a ring's unit group commutes with mapping the second argument
to its additive inverse. -/
@[simp] protected theorem mul_neg (u₁ u₂ : units α) : u₁ * -u₂ = -(u₁ * u₂) :=
units.ext $ (neg_mul_eq_mul_neg _ _).symm
/-- Multiplication of the additive inverses of two elements of a ring's unit group equals
multiplication of the two original elements. -/
@[simp] protected theorem neg_mul_neg (u₁ u₂ : units α) : -u₁ * -u₂ = u₁ * u₂ := by simp
/-- The additive inverse of an element of a ring's unit group equals the additive inverse of
one times the original element. -/
protected theorem neg_eq_neg_one_mul (u : units α) : -u = -1 * u := by simp
end units
lemma is_unit.neg [ring α] {a : α} : is_unit a → is_unit (-a)
| ⟨x, hx⟩ := hx ▸ (-x).is_unit
namespace ring_hom
/-- Ring homomorphisms preserve additive inverse. -/
@[simp] theorem map_neg {α β} [ring α] [ring β] (f : α →+* β) (x : α) : f (-x) = -(f x) :=
(f : α →+ β).map_neg x
/-- Ring homomorphisms preserve subtraction. -/
@[simp] theorem map_sub {α β} [ring α] [ring β] (f : α →+* β) (x y : α) :
f (x - y) = (f x) - (f y) := (f : α →+ β).map_sub x y
/-- A ring homomorphism is injective iff its kernel is trivial. -/
theorem injective_iff {α β} [ring α] [non_assoc_semiring β] (f : α →+* β) :
function.injective f ↔ (∀ a, f a = 0 → a = 0) :=
(f : α →+ β).injective_iff
/-- A ring homomorphism is injective iff its kernel is trivial. -/
theorem injective_iff' {α β} [ring α] [non_assoc_semiring β] (f : α →+* β) :
function.injective f ↔ (∀ a, f a = 0 ↔ a = 0) :=
(f : α →+ β).injective_iff'
/-- Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. -/
def mk' {γ} [non_assoc_semiring α] [ring γ] (f : α →* γ)
(map_add : ∀ a b : α, f (a + b) = f a + f b) :
α →+* γ :=
{ to_fun := f,
.. add_monoid_hom.mk' f map_add, .. f }
end ring_hom
/-- A commutative ring is a `ring` with commutative multiplication. -/
@[protect_proj, ancestor ring comm_semigroup]
class comm_ring (α : Type u) extends ring α, comm_monoid α
@[priority 100] -- see Note [lower instance priority]
instance comm_ring.to_comm_semiring [s : comm_ring α] : comm_semiring α :=
{ mul_zero := mul_zero, zero_mul := zero_mul, ..s }
section ring
variables [ring α] {a b c : α}
theorem dvd_neg_of_dvd (h : a ∣ b) : (a ∣ -b) :=
dvd.elim h
(assume c, assume : b = a * c,
dvd.intro (-c) (by simp [this]))
theorem dvd_of_dvd_neg (h : a ∣ -b) : (a ∣ b) :=
let t := dvd_neg_of_dvd h in by rwa neg_neg at t
/-- An element a of a ring divides the additive inverse of an element b iff a divides b. -/
@[simp] lemma dvd_neg (a b : α) : (a ∣ -b) ↔ (a ∣ b) :=
⟨dvd_of_dvd_neg, dvd_neg_of_dvd⟩
theorem neg_dvd_of_dvd (h : a ∣ b) : -a ∣ b :=
dvd.elim h
(assume c, assume : b = a * c,
dvd.intro (-c) (by simp [this]))
theorem dvd_of_neg_dvd (h : -a ∣ b) : a ∣ b :=
let t := neg_dvd_of_dvd h in by rwa neg_neg at t
/-- The additive inverse of an element a of a ring divides another element b iff a divides b. -/
@[simp] lemma neg_dvd (a b : α) : (-a ∣ b) ↔ (a ∣ b) :=
⟨dvd_of_neg_dvd, neg_dvd_of_dvd⟩
theorem dvd_sub (h₁ : a ∣ b) (h₂ : a ∣ c) : a ∣ b - c :=
by { rw sub_eq_add_neg, exact dvd_add h₁ (dvd_neg_of_dvd h₂) }
theorem dvd_add_iff_left (h : a ∣ c) : a ∣ b ↔ a ∣ b + c :=
⟨λh₂, dvd_add h₂ h, λH, by have t := dvd_sub H h; rwa add_sub_cancel at t⟩
theorem dvd_add_iff_right (h : a ∣ b) : a ∣ c ↔ a ∣ b + c :=
by rw add_comm; exact dvd_add_iff_left h
theorem two_dvd_bit1 : 2 ∣ bit1 a ↔ (2 : α) ∣ 1 := (dvd_add_iff_right (@two_dvd_bit0 _ _ a)).symm
/-- If an element a divides another element c in a commutative ring, a divides the sum of another
element b with c iff a divides b. -/
theorem dvd_add_left (h : a ∣ c) : a ∣ b + c ↔ a ∣ b :=
(dvd_add_iff_left h).symm
/-- If an element a divides another element b in a commutative ring, a divides the sum of b and
another element c iff a divides c. -/
theorem dvd_add_right (h : a ∣ b) : a ∣ b + c ↔ a ∣ c :=
(dvd_add_iff_right h).symm
/-- An element a divides the sum a + b if and only if a divides b.-/
@[simp] lemma dvd_add_self_left {a b : α} : a ∣ a + b ↔ a ∣ b :=
dvd_add_right (dvd_refl a)
/-- An element a divides the sum b + a if and only if a divides b.-/
@[simp] lemma dvd_add_self_right {a b : α} : a ∣ b + a ↔ a ∣ b :=
dvd_add_left (dvd_refl a)
lemma dvd_iff_dvd_of_dvd_sub {a b c : α} (h : a ∣ (b - c)) : (a ∣ b ↔ a ∣ c) :=
begin
split,
{ intro h',
convert dvd_sub h' h,
exact eq.symm (sub_sub_self b c) },
{ intro h',
convert dvd_add h h',
exact eq_add_of_sub_eq rfl }
end
@[simp] theorem even_neg (a : α) : even (-a) ↔ even a :=
dvd_neg _ _
end ring
section comm_ring
variables [comm_ring α] {a b c : α}
/-- Pullback a `comm_ring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.comm_ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
comm_ring β :=
{ .. hf.ring f zero one add mul neg sub, .. hf.comm_semigroup f mul }
/-- Pullback a `comm_ring` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.surjective.comm_ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
comm_ring β :=
{ .. hf.ring f zero one add mul neg sub, .. hf.comm_semigroup f mul }
local attribute [simp] add_assoc add_comm add_left_comm mul_comm
/-- Representation of a difference of two squares in a commutative ring as a product. -/
theorem mul_self_sub_mul_self (a b : α) : a * a - b * b = (a + b) * (a - b) :=
by rw [add_mul, mul_sub, mul_sub, mul_comm a b, sub_add_sub_cancel]
lemma mul_self_sub_one (a : α) : a * a - 1 = (a + 1) * (a - 1) :=
by rw [← mul_self_sub_mul_self, mul_one]
/-- Vieta's formula for a quadratic equation, relating the coefficients of the polynomial with
its roots. This particular version states that if we have a root `x` of a monic quadratic
polynomial, then there is another root `y` such that `x + y` is negative the `a_1` coefficient
and `x * y` is the `a_0` coefficient. -/
lemma Vieta_formula_quadratic {b c x : α} (h : x * x - b * x + c = 0) :
∃ y : α, y * y - b * y + c = 0 ∧ x + y = b ∧ x * y = c :=
begin
have : c = -(x * x - b * x) := (neg_eq_of_add_eq_zero h).symm,
have : c = x * (b - x), by subst this; simp [mul_sub, mul_comm],
refine ⟨b - x, _, by simp, by rw this⟩,
rw [this, sub_add, ← sub_mul, sub_self]
end
lemma dvd_mul_sub_mul {k a b x y : α} (hab : k ∣ a - b) (hxy : k ∣ x - y) :
k ∣ a * x - b * y :=
begin
convert dvd_add (dvd_mul_of_dvd_right hxy a) (dvd_mul_of_dvd_left hab y),
rw [mul_sub_left_distrib, mul_sub_right_distrib],
simp only [sub_eq_add_neg, add_assoc, neg_add_cancel_left],
end
end comm_ring
lemma succ_ne_self [ring α] [nontrivial α] (a : α) : a + 1 ≠ a :=
λ h, one_ne_zero ((add_right_inj a).mp (by simp [h]))
lemma pred_ne_self [ring α] [nontrivial α] (a : α) : a - 1 ≠ a :=
λ h, one_ne_zero (neg_injective ((add_right_inj a).mp (by simpa [sub_eq_add_neg] using h)))
/-- A domain is a ring with no zero divisors, i.e. satisfying
the condition `a * b = 0 ↔ a = 0 ∨ b = 0`. Alternatively, a domain
is an integral domain without assuming commutativity of multiplication. -/
@[protect_proj] class domain (α : Type u) extends ring α, nontrivial α :=
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀ a b : α, a * b = 0 → a = 0 ∨ b = 0)
section domain
variable [domain α]
@[priority 100] -- see Note [lower instance priority]
instance domain.to_no_zero_divisors : no_zero_divisors α :=
⟨domain.eq_zero_or_eq_zero_of_mul_eq_zero⟩
@[priority 100] -- see Note [lower instance priority]
instance domain.to_cancel_monoid_with_zero : cancel_monoid_with_zero α :=
{ mul_left_cancel_of_ne_zero := λ a b c ha,
by { rw [← sub_eq_zero, ← mul_sub], simp [ha, sub_eq_zero] },
mul_right_cancel_of_ne_zero := λ a b c hb,
by { rw [← sub_eq_zero, ← sub_mul], simp [hb, sub_eq_zero] },
.. (infer_instance : semiring α) }
/-- Pullback a `domain` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.domain [has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β]
[has_sub β] (f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
domain β :=
{ .. hf.ring f zero one add mul neg sub, .. pullback_nonzero f zero one,
.. hf.no_zero_divisors f zero mul }
end domain
/-!
### Integral domains
-/
/-- An integral domain is a commutative ring with no zero divisors, i.e. satisfying the condition
`a * b = 0 ↔ a = 0 ∨ b = 0`. Alternatively, an integral domain is a domain with commutative
multiplication. -/
@[protect_proj, ancestor comm_ring domain]
class integral_domain (α : Type u) extends comm_ring α, domain α
section integral_domain
variables [integral_domain α] {a b c d e : α}
@[priority 100] -- see Note [lower instance priority]
instance integral_domain.to_comm_cancel_monoid_with_zero : comm_cancel_monoid_with_zero α :=
{ ..comm_semiring.to_comm_monoid_with_zero, ..domain.to_cancel_monoid_with_zero }
/-- Pullback an `integral_domain` instance along an injective function.
See note [reducible non-instances]. -/
@[reducible]
protected def function.injective.integral_domain [has_zero β] [has_one β] [has_add β] [has_mul β]
[has_neg β] [has_sub β] (f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
integral_domain β :=
{ .. hf.comm_ring f zero one add mul neg sub, .. hf.domain f zero one add mul neg sub }
lemma mul_self_eq_mul_self_iff {a b : α} : a * a = b * b ↔ a = b ∨ a = -b :=
by rw [← sub_eq_zero, mul_self_sub_mul_self, mul_eq_zero, or_comm, sub_eq_zero,
add_eq_zero_iff_eq_neg]
lemma mul_self_eq_one_iff {a : α} : a * a = 1 ↔ a = 1 ∨ a = -1 :=
by rw [← mul_self_eq_mul_self_iff, one_mul]
/-- In the unit group of an integral domain, a unit is its own inverse iff the unit is one or
one's additive inverse. -/
lemma units.inv_eq_self_iff (u : units α) : u⁻¹ = u ↔ u = 1 ∨ u = -1 :=
by { rw inv_eq_iff_mul_eq_one, simp only [units.ext_iff], push_cast, exact mul_self_eq_one_iff }
/--
Makes a ring homomorphism from an additive group homomorphism from a commutative ring to an integral
domain that commutes with self multiplication, assumes that two is nonzero and one is sent to one.
-/
def add_monoid_hom.mk_ring_hom_of_mul_self_of_two_ne_zero [comm_ring β] (f : β →+ α)
(h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0) (h_one : f 1 = 1) : β →+* α :=
{ map_one' := h_one,
map_mul' := begin
intros x y,
have hxy := h (x + y),
rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul,
mul_add, mul_add, ← sub_eq_zero, add_comm, ← sub_sub, ← sub_sub, ← sub_sub,
mul_comm y x, mul_comm (f y) (f x)] at hxy,
simp only [add_assoc, add_sub_assoc, add_sub_cancel'_right] at hxy,
rw [sub_sub, ← two_mul, ← add_sub_assoc, ← two_mul, ← mul_sub, mul_eq_zero, sub_eq_zero,
or_iff_not_imp_left] at hxy,
exact hxy h_two,
end,
..f }
@[simp]
lemma add_monoid_hom.coe_fn_mk_ring_hom_of_mul_self_of_two_ne_zero [comm_ring β] (f : β →+ α)
(h h_two h_one) :
(f.mk_ring_hom_of_mul_self_of_two_ne_zero h h_two h_one : β → α) = f := rfl
@[simp]
lemma add_monoid_hom.coe_add_monoid_hom_mk_ring_hom_of_mul_self_of_two_ne_zero [comm_ring β]
(f : β →+ α) (h h_two h_one) :
(f.mk_ring_hom_of_mul_self_of_two_ne_zero h h_two h_one : β →+ α) = f := by {ext, simp}
end integral_domain
namespace ring
variables {M₀ : Type*} [monoid_with_zero M₀]
open_locale classical
/-- Introduce a function `inverse` on a monoid with zero `M₀`, which sends `x` to `x⁻¹` if `x` is
invertible and to `0` otherwise. This definition is somewhat ad hoc, but one needs a fully (rather
than partially) defined inverse function for some purposes, including for calculus.
Note that while this is in the `ring` namespace for brevity, it requires the weaker assumption
`monoid_with_zero M₀` instead of `ring M₀`. -/
noncomputable def inverse : M₀ → M₀ :=
λ x, if h : is_unit x then (((classical.some h)⁻¹ : units M₀) : M₀) else 0
/-- By definition, if `x` is invertible then `inverse x = x⁻¹`. -/
@[simp] lemma inverse_unit (u : units M₀) : inverse (u : M₀) = (u⁻¹ : units M₀) :=
begin
simp only [units.is_unit, inverse, dif_pos],
exact units.inv_unique (classical.some_spec u.is_unit)
end
/-- By definition, if `x` is not invertible then `inverse x = 0`. -/
@[simp] lemma inverse_non_unit (x : M₀) (h : ¬(is_unit x)) : inverse x = 0 := dif_neg h
end ring
/-- A predicate to express that a ring is an integral domain.
This is mainly useful because such a predicate does not contain data,
and can therefore be easily transported along ring isomorphisms. -/
structure is_integral_domain (R : Type u) [ring R] extends nontrivial R : Prop :=
(mul_comm : ∀ (x y : R), x * y = y * x)
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀ x y : R, x * y = 0 → x = 0 ∨ y = 0)
-- The linter does not recognize that is_integral_domain.to_nontrivial is a structure
-- projection, disable it
attribute [nolint def_lemma doc_blame] is_integral_domain.to_nontrivial
/-- Every integral domain satisfies the predicate for integral domains. -/
lemma integral_domain.to_is_integral_domain (R : Type u) [integral_domain R] :
is_integral_domain R :=
{ .. (‹_› : integral_domain R) }
/-- If a ring satisfies the predicate for integral domains,
then it can be endowed with an `integral_domain` instance
whose data is definitionally equal to the existing data. -/
def is_integral_domain.to_integral_domain (R : Type u) [ring R] (h : is_integral_domain R) :
integral_domain R :=
{ .. (‹_› : ring R), .. (‹_› : is_integral_domain R) }
namespace semiconj_by
@[simp] lemma add_right [distrib R] {a x y x' y' : R}
(h : semiconj_by a x y) (h' : semiconj_by a x' y') :
semiconj_by a (x + x') (y + y') :=
by simp only [semiconj_by, left_distrib, right_distrib, h.eq, h'.eq]
@[simp] lemma add_left [distrib R] {a b x y : R}
(ha : semiconj_by a x y) (hb : semiconj_by b x y) :
semiconj_by (a + b) x y :=
by simp only [semiconj_by, left_distrib, right_distrib, ha.eq, hb.eq]
variables [ring R] {a b x y x' y' : R}
lemma neg_right (h : semiconj_by a x y) : semiconj_by a (-x) (-y) :=
by simp only [semiconj_by, h.eq, neg_mul_eq_neg_mul_symm, mul_neg_eq_neg_mul_symm]
@[simp] lemma neg_right_iff : semiconj_by a (-x) (-y) ↔ semiconj_by a x y :=
⟨λ h, neg_neg x ▸ neg_neg y ▸ h.neg_right, semiconj_by.neg_right⟩
lemma neg_left (h : semiconj_by a x y) : semiconj_by (-a) x y :=
by simp only [semiconj_by, h.eq, neg_mul_eq_neg_mul_symm, mul_neg_eq_neg_mul_symm]
@[simp] lemma neg_left_iff : semiconj_by (-a) x y ↔ semiconj_by a x y :=
⟨λ h, neg_neg a ▸ h.neg_left, semiconj_by.neg_left⟩
@[simp] lemma neg_one_right (a : R) : semiconj_by a (-1) (-1) :=
(one_right a).neg_right
@[simp] lemma neg_one_left (x : R) : semiconj_by (-1) x x :=
(semiconj_by.one_left x).neg_left
@[simp] lemma sub_right (h : semiconj_by a x y) (h' : semiconj_by a x' y') :
semiconj_by a (x - x') (y - y') :=
by simpa only [sub_eq_add_neg] using h.add_right h'.neg_right
@[simp] lemma sub_left (ha : semiconj_by a x y) (hb : semiconj_by b x y) :
semiconj_by (a - b) x y :=
by simpa only [sub_eq_add_neg] using ha.add_left hb.neg_left
end semiconj_by
namespace commute
@[simp] theorem add_right [distrib R] {a b c : R} :
commute a b → commute a c → commute a (b + c) :=
semiconj_by.add_right
@[simp] theorem add_left [distrib R] {a b c : R} :
commute a c → commute b c → commute (a + b) c :=
semiconj_by.add_left
lemma bit0_right [distrib R] {x y : R} (h : commute x y) : commute x (bit0 y) :=
h.add_right h
lemma bit0_left [distrib R] {x y : R} (h : commute x y) : commute (bit0 x) y :=
h.add_left h
lemma bit1_right [semiring R] {x y : R} (h : commute x y) : commute x (bit1 y) :=
h.bit0_right.add_right (commute.one_right x)
lemma bit1_left [semiring R] {x y : R} (h : commute x y) : commute (bit1 x) y :=
h.bit0_left.add_left (commute.one_left y)
variables [ring R] {a b c : R}
theorem neg_right : commute a b → commute a (- b) := semiconj_by.neg_right
@[simp] theorem neg_right_iff : commute a (-b) ↔ commute a b := semiconj_by.neg_right_iff
theorem neg_left : commute a b → commute (- a) b := semiconj_by.neg_left
@[simp] theorem neg_left_iff : commute (-a) b ↔ commute a b := semiconj_by.neg_left_iff
@[simp] theorem neg_one_right (a : R) : commute a (-1) := semiconj_by.neg_one_right a
@[simp] theorem neg_one_left (a : R): commute (-1) a := semiconj_by.neg_one_left a
@[simp] theorem sub_right : commute a b → commute a c → commute a (b - c) := semiconj_by.sub_right
@[simp] theorem sub_left : commute a c → commute b c → commute (a - b) c := semiconj_by.sub_left
end commute
|
3202f37654976bc35810977ed1dbcaa7e82f9699 | 9be442d9ec2fcf442516ed6e9e1660aa9071b7bd | /tests/lean/ctorUnivTooBig.lean | c55ec46566eecfa38333cf7bacbd24c1f80e36ae | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | EdAyers/lean4 | 57ac632d6b0789cb91fab2170e8c9e40441221bd | 37ba0df5841bde51dbc2329da81ac23d4f6a4de4 | refs/heads/master | 1,676,463,245,298 | 1,660,619,433,000 | 1,660,619,433,000 | 183,433,437 | 1 | 0 | Apache-2.0 | 1,657,612,672,000 | 1,556,196,574,000 | Lean | UTF-8 | Lean | false | false | 1,146 | lean | inductive Bla : Type u where
| nil : Bla
| cons : (α : Type u) → (a : α) → Bla → Bla -- Error
inductive Foo : Type where
| leaf
| mk (α : Type) (a : α) -- Error
| mk₂
inductive Foo' : Type u where
| leaf
| mk : Sort (max u v) → a → Foo' -- Error
| mk₂
inductive Boo : Type u where
| nil : Boo
| cons : (α : Sort u) → (a : α) → Boo → Boo
namespace Ex1
inductive Member : α → List α → Type u
| head : Member a (a::as)
| tail : Member a bs → Member a (b::bs)
#check @Member.head
end Ex1
namespace Ex2
inductive Member : α → List α → Type
| head : Member a (a::as)
| tail : Member a bs → Member a (b::bs)
end Ex2
namespace Ex3
inductive Member : α → List α → Type (max u v)
| head : Member a (a::as) -- Error
| tail : Member a bs → Member a (b::bs)
end Ex3
namespace Ex4
inductive Member : α → List α → Type (u+1)
| head : Member a (a::as) -- Error
| tail : Member a bs → Member a (b::bs)
end Ex4
namespace Ex5
inductive Member : α → List α → Type 1
| head : Member a (a::as) -- Error
| tail : Member a bs → Member a (b::bs)
end Ex5
|
e8143cfc1f20e8aaa7b4b08dc92637a9846760b4 | 02005f45e00c7ecf2c8ca5db60251bd1e9c860b5 | /src/control/fold.lean | ddddae622dd14518deeb575cbb4c21742770eb9c | [
"Apache-2.0"
] | permissive | anthony2698/mathlib | 03cd69fe5c280b0916f6df2d07c614c8e1efe890 | 407615e05814e98b24b2ff322b14e8e3eb5e5d67 | refs/heads/master | 1,678,792,774,873 | 1,614,371,563,000 | 1,614,371,563,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 14,745 | lean | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import algebra.free_monoid
import algebra.opposites
import control.traversable.instances
import control.traversable.lemmas
import category_theory.category
import category_theory.endomorphism
import category_theory.types
import category_theory.category.Kleisli
/-!
# List folds generalized to `traversable`
Informally, we can think of `foldl` as a special case of `traverse` where we do not care about the
reconstructed data structure and, in a state monad, we care about the final state.
The obvious way to define `foldl` would be to use the state monad but it
is nicer to reason about a more abstract interface with `fold_map` as a
primitive and `fold_map_hom` as a defining property.
```
def fold_map {α ω} [has_one ω] [has_mul ω] (f : α → ω) : t α → ω := ...
lemma fold_map_hom (α β)
[monoid α] [monoid β] (f : α → β) [is_monoid_hom f]
(g : γ → α) (x : t γ) :
f (fold_map g x) = fold_map (f ∘ g) x :=
...
```
`fold_map` uses a monoid ω to accumulate a value for every element of
a data structure and `fold_map_hom` uses a monoid homomorphism to
substitute the monoid used by `fold_map`. The two are sufficient to
define `foldl`, `foldr` and `to_list`. `to_list` permits the
formulation of specifications in terms of operations on lists.
Each fold function can be defined using a specialized
monoid. `to_list` uses a free monoid represented as a list with
concatenation while `foldl` uses endofunctions together with function
composition.
The definition through monoids uses `traverse` together with the
applicative functor `const m` (where `m` is the monoid). As an
implementation, `const` guarantees that no resource is spent on
reconstructing the structure during traversal.
A special class could be defined for `foldable`, similarly to Haskell,
but the author cannot think of instances of `foldable` that are not also
`traversable`.
-/
universes u v
open ulift category_theory opposite
namespace monoid
variables {m : Type u → Type u} [monad m]
variables {α β : Type u}
/--
For a list, foldl f x [y₀,y₁] reduces as follows:
```
calc foldl f x [y₀,y₁]
= foldl f (f x y₀) [y₁] : rfl
... = foldl f (f (f x y₀) y₁) [] : rfl
... = f (f x y₀) y₁ : rfl
```
with
```
f : α → β → α
x : α
[y₀,y₁] : list β
```
We can view the above as a composition of functions:
```
... = f (f x y₀) y₁ : rfl
... = flip f y₁ (flip f y₀ x) : rfl
... = (flip f y₁ ∘ flip f y₀) x : rfl
```
We can use traverse and const to construct this composition:
```
calc const.run (traverse (λ y, const.mk' (flip f y)) [y₀,y₁]) x
= const.run ((::) <$> const.mk' (flip f y₀) <*> traverse (λ y, const.mk' (flip f y)) [y₁]) x
... = const.run ((::) <$> const.mk' (flip f y₀) <*>
( (::) <$> const.mk' (flip f y₁) <*> traverse (λ y, const.mk' (flip f y)) [] )) x
... = const.run ((::) <$> const.mk' (flip f y₀) <*>
( (::) <$> const.mk' (flip f y₁) <*> pure [] )) x
... = const.run ( ((::) <$> const.mk' (flip f y₁) <*> pure []) ∘
((::) <$> const.mk' (flip f y₀)) ) x
... = const.run ( const.mk' (flip f y₁) ∘ const.mk' (flip f y₀) ) x
... = const.run ( flip f y₁ ∘ flip f y₀ ) x
... = f (f x y₀) y₁
```
And this is how `const` turns a monoid into an applicative functor and
how the monoid of endofunctions define `foldl`.
-/
@[reducible] def foldl (α : Type u) : Type u := (End α)ᵒᵖ
def foldl.mk (f : α → α) : foldl α := op f
def foldl.get (x : foldl α) : α → α := unop x
def foldl.of_free_monoid (f : β → α → β) (xs : free_monoid α) : monoid.foldl β :=
op $ flip (list.foldl f) xs
@[reducible] def foldr (α : Type u) : Type u := End α
def foldr.mk (f : α → α) : foldr α := f
def foldr.get (x : foldr α) : α → α := x
def foldr.of_free_monoid (f : α → β → β) (xs : free_monoid α) : monoid.foldr β :=
flip (list.foldr f) xs
@[reducible] def mfoldl (m : Type u → Type u) [monad m] (α : Type u) : Type u :=
opposite $ End $ Kleisli.mk m α
def mfoldl.mk (f : α → m α) : mfoldl m α := op f
def mfoldl.get (x : mfoldl m α) : α → m α := unop x
def mfoldl.of_free_monoid (f : β → α → m β) (xs : free_monoid α) : monoid.mfoldl m β :=
op $ flip (list.mfoldl f) xs
@[reducible] def mfoldr (m : Type u → Type u) [monad m] (α : Type u) : Type u :=
End $ Kleisli.mk m α
def mfoldr.mk (f : α → m α) : mfoldr m α := f
def mfoldr.get (x : mfoldr m α) : α → m α := x
def mfoldr.of_free_monoid (f : α → β → m β) (xs : free_monoid α) : monoid.mfoldr m β :=
flip (list.mfoldr f) xs
end monoid
namespace traversable
open monoid functor
section defs
variables {α β : Type u} {t : Type u → Type u} [traversable t]
def fold_map {α ω} [has_one ω] [has_mul ω] (f : α → ω) : t α → ω :=
traverse (const.mk' ∘ f)
def foldl (f : α → β → α) (x : α) (xs : t β) : α :=
(fold_map (foldl.mk ∘ flip f) xs).get x
def foldr (f : α → β → β) (x : β) (xs : t α) : β :=
(fold_map (foldr.mk ∘ f) xs).get x
/--
Conceptually, `to_list` collects all the elements of a collection
in a list. This idea is formalized by
`lemma to_list_spec (x : t α) : to_list x = fold_map free_monoid.mk x`.
The definition of `to_list` is based on `foldl` and `list.cons` for
speed. It is faster than using `fold_map free_monoid.mk` because, by
using `foldl` and `list.cons`, each insertion is done in constant
time. As a consequence, `to_list` performs in linear.
On the other hand, `fold_map free_monoid.mk` creates a singleton list
around each element and concatenates all the resulting lists. In
`xs ++ ys`, concatenation takes a time proportional to `length xs`. Since
the order in which concatenation is evaluated is unspecified, nothing
prevents each element of the traversable to be appended at the end
`xs ++ [x]` which would yield a `O(n²)` run time. -/
def to_list : t α → list α :=
list.reverse ∘ foldl (flip list.cons) []
def length (xs : t α) : ℕ :=
down $ foldl (λ l _, up $ l.down + 1) (up 0) xs
variables {m : Type u → Type u} [monad m]
def mfoldl (f : α → β → m α) (x : α) (xs : t β) : m α :=
(fold_map (mfoldl.mk ∘ flip f) xs).get x
def mfoldr (f : α → β → m β) (x : β) (xs : t α) : m β :=
(fold_map (mfoldr.mk ∘ f) xs).get x
end defs
section applicative_transformation
variables {α β γ : Type u}
open function (hiding const) is_monoid_hom
def map_fold [monoid α] [monoid β] (f : α → β) [is_monoid_hom f] :
applicative_transformation (const α) (const β) :=
{ app := λ x, f,
preserves_seq' := by { intros, simp only [map_mul f, (<*>)], },
preserves_pure' := by { intros, simp only [map_one f, pure] } }
def free.mk : α → free_monoid α := list.ret
def free.map (f : α → β) : free_monoid α → free_monoid β := list.map f
lemma free.map_eq_map (f : α → β) (xs : list α) :
f <$> xs = free.map f xs := rfl
instance (f : α → β) : is_monoid_hom (free.map f) :=
{ map_mul := λ x y, by simp only [free.map, free_monoid.mul_def, list.map_append, free_add_monoid.add_def],
map_one := by simp only [free.map, free_monoid.one_def, list.map, free_add_monoid.zero_def] }
instance fold_foldl (f : β → α → β) :
is_monoid_hom (foldl.of_free_monoid f) :=
{ map_one := rfl,
map_mul := by intros; simp only [free_monoid.mul_def, foldl.of_free_monoid, flip, unop_op,
list.foldl_append, op_inj_iff]; refl }
lemma foldl.unop_of_free_monoid (f : β → α → β) (xs : free_monoid α) (a : β) :
unop (foldl.of_free_monoid f xs) a = list.foldl f a xs := rfl
instance fold_foldr (f : α → β → β) :
is_monoid_hom (foldr.of_free_monoid f) :=
{ map_one := rfl,
map_mul := by intros; simp only [free_monoid.mul_def, foldr.of_free_monoid, list.foldr_append, flip]; refl }
variables (m : Type u → Type u) [monad m] [is_lawful_monad m]
@[simp]
lemma mfoldl.unop_of_free_monoid (f : β → α → m β) (xs : free_monoid α) (a : β) :
unop (mfoldl.of_free_monoid f xs) a = list.mfoldl f a xs := rfl
instance fold_mfoldl (f : β → α → m β) :
is_monoid_hom (mfoldl.of_free_monoid f) :=
{ map_one := rfl,
map_mul := by intros; apply unop_injective; ext; apply list.mfoldl_append }
instance fold_mfoldr (f : α → β → m β) :
is_monoid_hom (mfoldr.of_free_monoid f) :=
{ map_one := rfl,
map_mul := by intros; ext; apply list.mfoldr_append }
variables {t : Type u → Type u} [traversable t] [is_lawful_traversable t]
open is_lawful_traversable
lemma fold_map_hom
[monoid α] [monoid β] (f : α → β) [is_monoid_hom f]
(g : γ → α) (x : t γ) :
f (fold_map g x) = fold_map (f ∘ g) x :=
calc f (fold_map g x)
= f (traverse (const.mk' ∘ g) x) : rfl
... = (map_fold f).app _ (traverse (const.mk' ∘ g) x) : rfl
... = traverse ((map_fold f).app _ ∘ (const.mk' ∘ g)) x : naturality (map_fold f) _ _
... = fold_map (f ∘ g) x : rfl
lemma fold_map_hom_free
[monoid β] (f : free_monoid α → β) [is_monoid_hom f] (x : t α) :
f (fold_map free.mk x) = fold_map (f ∘ free.mk) x :=
fold_map_hom _ _ x
variable {m}
lemma fold_mfoldl_cons (f : α → β → m α) (x : β) (y : α) :
list.mfoldl f y (free.mk x) = f y x :=
by simp only [free.mk, list.ret, list.mfoldl, bind_pure]
lemma fold_mfoldr_cons (f : β → α → m α) (x : β) (y : α) :
list.mfoldr f y (free.mk x) = f x y :=
by simp only [free.mk, list.ret, list.mfoldr, pure_bind]
end applicative_transformation
section equalities
open is_lawful_traversable list (cons)
variables {α β γ : Type u}
variables {t : Type u → Type u} [traversable t] [is_lawful_traversable t]
@[simp]
lemma foldl.of_free_monoid_comp_free_mk (f : α → β → α) :
foldl.of_free_monoid f ∘ free.mk = foldl.mk ∘ flip f := rfl
@[simp]
lemma foldr.of_free_monoid_comp_free_mk (f : β → α → α) :
foldr.of_free_monoid f ∘ free.mk = foldr.mk ∘ f := rfl
@[simp]
lemma mfoldl.of_free_monoid_comp_free_mk {m} [monad m] [is_lawful_monad m] (f : α → β → m α) :
mfoldl.of_free_monoid f ∘ free.mk = mfoldl.mk ∘ flip f :=
by ext; simp only [(∘), mfoldl.of_free_monoid, mfoldl.mk, flip, fold_mfoldl_cons]; refl
@[simp]
lemma mfoldr.of_free_monoid_comp_free_mk {m} [monad m] [is_lawful_monad m] (f : β → α → m α) :
mfoldr.of_free_monoid f ∘ free.mk = mfoldr.mk ∘ f :=
by { ext, simp only [(∘), mfoldr.of_free_monoid, mfoldr.mk, flip, fold_mfoldr_cons] }
lemma to_list_spec (xs : t α) :
to_list xs = (fold_map free.mk xs : free_monoid _) :=
eq.symm $
calc fold_map free.mk xs
= (fold_map free.mk xs).reverse.reverse : by simp only [list.reverse_reverse]
... = (list.foldr cons [] (fold_map free.mk xs).reverse).reverse
: by simp only [list.foldr_eta]
... = (unop (foldl.of_free_monoid (flip cons) (fold_map free.mk xs)) []).reverse
: by simp only [flip,list.foldr_reverse,foldl.of_free_monoid, unop_op]
... = to_list xs : begin
rw fold_map_hom_free (foldl.of_free_monoid (flip cons)),
simp only [to_list, foldl, list.reverse_inj, foldl.get,
foldl.of_free_monoid_comp_free_mk],
all_goals { apply_instance }
end
lemma fold_map_map [monoid γ] (f : α → β) (g : β → γ) (xs : t α) :
fold_map g (f <$> xs) = fold_map (g ∘ f) xs :=
by simp only [fold_map,traverse_map]
lemma foldl_to_list (f : α → β → α) (xs : t β) (x : α) :
foldl f x xs = list.foldl f x (to_list xs) :=
begin
rw ← foldl.unop_of_free_monoid,
simp only [foldl, to_list_spec, fold_map_hom_free (foldl.of_free_monoid f),
foldl.of_free_monoid_comp_free_mk, foldl.get]
end
lemma foldr_to_list (f : α → β → β) (xs : t α) (x : β) :
foldr f x xs = list.foldr f x (to_list xs) :=
begin
change _ = foldr.of_free_monoid _ _ _,
simp only [foldr, to_list_spec, fold_map_hom_free (foldr.of_free_monoid f),
foldr.of_free_monoid_comp_free_mk, foldr.get]
end
lemma to_list_map (f : α → β) (xs : t α) :
to_list (f <$> xs) = f <$> to_list xs :=
by simp only [to_list_spec,free.map_eq_map,fold_map_hom (free.map f), fold_map_map]; refl
@[simp] theorem foldl_map (g : β → γ) (f : α → γ → α) (a : α) (l : t β) :
foldl f a (g <$> l) = foldl (λ x y, f x (g y)) a l :=
by simp only [foldl, fold_map_map, (∘), flip]
@[simp] theorem foldr_map (g : β → γ) (f : γ → α → α) (a : α) (l : t β) :
foldr f a (g <$> l) = foldr (f ∘ g) a l :=
by simp only [foldr, fold_map_map, (∘), flip]
@[simp] theorem to_list_eq_self {xs : list α} : to_list xs = xs :=
begin
simp only [to_list_spec, fold_map, traverse],
induction xs,
case list.nil { refl },
case list.cons : _ _ ih { unfold list.traverse list.ret, rw ih, refl }
end
theorem length_to_list {xs : t α} : length xs = list.length (to_list xs) :=
begin
unfold length,
rw foldl_to_list,
generalize : to_list xs = ys,
let f := λ (n : ℕ) (a : α), n + 1,
transitivity list.foldl f 0 ys,
{ generalize : 0 = n,
induction ys with _ _ ih generalizing n,
{ simp only [list.foldl_nil] },
{ simp only [list.foldl, ih (n+1)] } },
{ induction ys with _ tl ih,
{ simp only [list.length, list.foldl_nil] },
{ simp only [list.foldl, list.length],
rw [← ih],
exact tl.foldl_hom (λx, x+1) f f 0 (λ n x, rfl) } }
end
variables {m : Type u → Type u} [monad m] [is_lawful_monad m]
section
local attribute [semireducible] opposite
lemma mfoldl_to_list {f : α → β → m α} {x : α} {xs : t β} :
mfoldl f x xs = list.mfoldl f x (to_list xs) :=
begin
change _ = unop (mfoldl.of_free_monoid f (to_list xs)) x,
simp only [mfoldl, to_list_spec, fold_map_hom_free (mfoldl.of_free_monoid f),
mfoldl.of_free_monoid_comp_free_mk, mfoldl.get]
end
end
lemma mfoldr_to_list (f : α → β → m β) (x : β) (xs : t α) :
mfoldr f x xs = list.mfoldr f x (to_list xs) :=
begin
change _ = mfoldr.of_free_monoid f (to_list xs) x,
simp only [mfoldr, to_list_spec, fold_map_hom_free (mfoldr.of_free_monoid f),
mfoldr.of_free_monoid_comp_free_mk, mfoldr.get]
end
@[simp] theorem mfoldl_map (g : β → γ) (f : α → γ → m α) (a : α) (l : t β) :
mfoldl f a (g <$> l) = mfoldl (λ x y, f x (g y)) a l :=
by simp only [mfoldl, fold_map_map, (∘), flip]
@[simp] theorem mfoldr_map (g : β → γ) (f : γ → α → m α) (a : α) (l : t β) :
mfoldr f a (g <$> l) = mfoldr (f ∘ g) a l :=
by simp only [mfoldr, fold_map_map, (∘), flip]
end equalities
end traversable
|
09f4aa5b58bc6688fdf7c3a5ac3b6fb8ed363605 | a4673261e60b025e2c8c825dfa4ab9108246c32e | /tests/lean/Process.lean | 425e86fe398a769967c682c27b0fcfe9f50a96b8 | [
"Apache-2.0"
] | permissive | jcommelin/lean4 | c02dec0cc32c4bccab009285475f265f17d73228 | 2909313475588cc20ac0436e55548a4502050d0a | refs/heads/master | 1,674,129,550,893 | 1,606,415,348,000 | 1,606,415,348,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,050 | lean |
open IO.Process
def usingIO {α} (x : IO α) : IO α := x
#eval usingIO do
let child ← spawn { cmd := "sh", args := #["-c", "exit 1"] };
child.wait
#eval usingIO do
let child ← spawn { cmd := "sh", args := #["-c", "echo hi!"] };
child.wait
#eval usingIO do
let child ← spawn { cmd := "sh", args := #["-c", "echo ho!"], stdout := Stdio.piped };
child.wait;
child.stdout.readToEnd
#eval usingIO do
let child ← spawn { cmd := "head", args := #["-n1"], stdin := Stdio.piped, stdout := Stdio.piped };
child.stdin.putStrLn "hu!";
child.stdin.flush;
child.wait;
child.stdout.readToEnd
#eval usingIO do
let child ← spawn { cmd := "true", stdin := Stdio.piped };
child.wait;
child.stdin.putStrLn "ha!";
child.stdin.flush <|> IO.println "flush of broken pipe failed"
#eval usingIO do
-- produce enough output to fill both pipes on all platforms
let out ← output { cmd := "sh", args := #["-c", "printf '%100000s' >& 2; printf '%100001s'"] };
IO.println out.stdout.length;
IO.println out.stderr.length
|
baf2d87f7889bffab95a4d80669fa058ab6c136d | 649957717d58c43b5d8d200da34bf374293fe739 | /src/data/padics/padic_integers.lean | 394326207d71b6f7c3af98e352ace78543767f62 | [
"Apache-2.0"
] | permissive | Vtec234/mathlib | b50c7b21edea438df7497e5ed6a45f61527f0370 | fb1848bbbfce46152f58e219dc0712f3289d2b20 | refs/heads/master | 1,592,463,095,113 | 1,562,737,749,000 | 1,562,737,749,000 | 196,202,858 | 0 | 0 | Apache-2.0 | 1,562,762,338,000 | 1,562,762,337,000 | null | UTF-8 | Lean | false | false | 9,392 | lean | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro
Define the p-adic integers ℤ_p as a subtype of ℚ_p. Construct algebraic structures on ℤ_p.
-/
import data.padics.padic_numbers ring_theory.ideals data.int.modeq
import tactic.linarith
open nat padic metric
noncomputable theory
local attribute [instance] classical.prop_decidable
def padic_int (p : ℕ) [p.prime] := {x : ℚ_[p] // ∥x∥ ≤ 1}
notation `ℤ_[`p`]` := padic_int p
namespace padic_int
variables {p : ℕ} [nat.prime p]
def add : ℤ_[p] → ℤ_[p] → ℤ_[p]
| ⟨x, hx⟩ ⟨y, hy⟩ := ⟨x+y,
le_trans (padic_norm_e.nonarchimedean _ _) (max_le_iff.2 ⟨hx,hy⟩)⟩
def mul : ℤ_[p] → ℤ_[p] → ℤ_[p]
| ⟨x, hx⟩ ⟨y, hy⟩ := ⟨x*y,
begin rw padic_norm_e.mul, apply mul_le_one; {assumption <|> apply norm_nonneg} end⟩
def neg : ℤ_[p] → ℤ_[p]
| ⟨x, hx⟩ := ⟨-x, by simpa⟩
instance : ring ℤ_[p] :=
begin
refine { add := add,
mul := mul,
neg := neg,
zero := ⟨0, by simp [zero_le_one]⟩,
one := ⟨1, by simp⟩,
.. };
{repeat {rintro ⟨_, _⟩}, simp [mul_assoc, left_distrib, right_distrib, add, mul, neg]}
end
lemma zero_def : ∀ x : ℤ_[p], x = 0 ↔ x.val = 0
| ⟨x, _⟩ := ⟨subtype.mk.inj, λ h, by simp at h; simp only [h]; refl⟩
@[simp] lemma add_def : ∀ (x y : ℤ_[p]), (x+y).val = x.val + y.val
| ⟨x, hx⟩ ⟨y, hy⟩ := rfl
@[simp] lemma mul_def : ∀ (x y : ℤ_[p]), (x*y).val = x.val * y.val
| ⟨x, hx⟩ ⟨y, hy⟩ := rfl
@[simp] lemma mk_zero {h} : (⟨0, h⟩ : ℤ_[p]) = (0 : ℤ_[p]) := rfl
instance : has_coe ℤ_[p] ℚ_[p] := ⟨subtype.val⟩
@[simp] lemma val_eq_coe (z : ℤ_[p]) : z.val = ↑z := rfl
@[simp, move_cast] lemma coe_add : ∀ (z1 z2 : ℤ_[p]), (↑(z1 + z2) : ℚ_[p]) = ↑z1 + ↑z2
| ⟨_, _⟩ ⟨_, _⟩ := rfl
@[simp, move_cast] lemma coe_mul : ∀ (z1 z2 : ℤ_[p]), (↑(z1 * z2) : ℚ_[p]) = ↑z1 * ↑z2
| ⟨_, _⟩ ⟨_, _⟩ := rfl
@[simp, move_cast] lemma coe_neg : ∀ (z1 : ℤ_[p]), (↑(-z1) : ℚ_[p]) = -↑z1
| ⟨_, _⟩ := rfl
@[simp, move_cast] lemma coe_sub : ∀ (z1 z2 : ℤ_[p]), (↑(z1 - z2) : ℚ_[p]) = ↑z1 - ↑z2
| ⟨_, _⟩ ⟨_, _⟩ := rfl
@[simp, squash_cast] lemma coe_one : (↑(1 : ℤ_[p]) : ℚ_[p]) = 1 := rfl
@[simp, squash_cast] lemma coe_coe : ∀ n : ℕ, (↑(↑n : ℤ_[p]) : ℚ_[p]) = (↑n : ℚ_[p])
| 0 := rfl
| (k+1) := by simp [coe_coe]
@[simp, squash_cast] lemma coe_zero : (↑(0 : ℤ_[p]) : ℚ_[p]) = 0 := rfl
@[simp, move_cast] lemma cast_pow (x : ℤ_[p]) : ∀ (n : ℕ), (↑(x^n) : ℚ_[p]) = (↑x : ℚ_[p])^n
| 0 := by simp
| (k+1) := by simp [monoid.pow, pow]; congr; apply cast_pow
lemma mk_coe : ∀ (k : ℤ_[p]), (⟨↑k, k.2⟩ : ℤ_[p]) = k
| ⟨_, _⟩ := rfl
def inv : ℤ_[p] → ℤ_[p]
| ⟨k, _⟩ := if h : ∥k∥ = 1 then ⟨1/k, by simp [h]⟩ else 0
end padic_int
section instances
variables {p : ℕ} [nat.prime p]
@[reducible] def padic_norm_z (z : ℤ_[p]) : ℝ := ∥z.val∥
instance : metric_space ℤ_[p] := subtype.metric_space
instance : has_norm ℤ_[p] := ⟨padic_norm_z⟩
instance : normed_ring ℤ_[p] :=
{ dist_eq := λ ⟨_, _⟩ ⟨_, _⟩, rfl,
norm_mul := λ ⟨_, _⟩ ⟨_, _⟩, norm_mul_le _ _ }
instance padic_norm_z.is_absolute_value : is_absolute_value (λ z : ℤ_[p], ∥z∥) :=
{ abv_nonneg := norm_nonneg,
abv_eq_zero := λ ⟨_, _⟩, by simp [norm_eq_zero, padic_int.zero_def],
abv_add := λ ⟨_,_⟩ ⟨_, _⟩, norm_triangle _ _,
abv_mul := λ _ _, by unfold norm; simp [padic_norm_z] }
protected lemma padic_int.pmul_comm : ∀ z1 z2 : ℤ_[p], z1*z2 = z2*z1
| ⟨q1, h1⟩ ⟨q2, h2⟩ := show (⟨q1*q2, _⟩ : ℤ_[p]) = ⟨q2*q1, _⟩, by simp [mul_comm]
instance : comm_ring ℤ_[p] :=
{ mul_comm := padic_int.pmul_comm,
..padic_int.ring }
protected lemma padic_int.zero_ne_one : (0 : ℤ_[p]) ≠ 1 :=
show (⟨(0 : ℚ_[p]), _⟩ : ℤ_[p]) ≠ ⟨(1 : ℚ_[p]), _⟩, from mt subtype.ext.1 zero_ne_one
protected lemma padic_int.eq_zero_or_eq_zero_of_mul_eq_zero :
∀ (a b : ℤ_[p]), a * b = 0 → a = 0 ∨ b = 0
| ⟨a, ha⟩ ⟨b, hb⟩ := λ h : (⟨a * b, _⟩ : ℤ_[p]) = ⟨0, _⟩,
have a * b = 0, from subtype.ext.1 h,
(mul_eq_zero_iff_eq_zero_or_eq_zero.1 this).elim
(λ h1, or.inl (by simp [h1]; refl))
(λ h2, or.inr (by simp [h2]; refl))
instance : integral_domain ℤ_[p] :=
{ eq_zero_or_eq_zero_of_mul_eq_zero := padic_int.eq_zero_or_eq_zero_of_mul_eq_zero,
zero_ne_one := padic_int.zero_ne_one,
..padic_int.comm_ring }
end instances
namespace padic_norm_z
variables {p : ℕ} [nat.prime p]
lemma le_one : ∀ z : ℤ_[p], ∥z∥ ≤ 1
| ⟨_, h⟩ := h
@[simp] lemma one : ∥(1 : ℤ_[p])∥ = 1 := by simp [norm, padic_norm_z]
@[simp] lemma mul (z1 z2 : ℤ_[p]) : ∥z1 * z2∥ = ∥z1∥ * ∥z2∥ :=
by unfold norm; simp [padic_norm_z]
@[simp] lemma pow (z : ℤ_[p]) : ∀ n : ℕ, ∥z^n∥ = ∥z∥^n
| 0 := by simp
| (k+1) := show ∥z*z^k∥ = ∥z∥*∥z∥^k, by {rw mul, congr, apply pow}
theorem nonarchimedean : ∀ (q r : ℤ_[p]), ∥q + r∥ ≤ max (∥q∥) (∥r∥)
| ⟨_, _⟩ ⟨_, _⟩ := padic_norm_e.nonarchimedean _ _
theorem add_eq_max_of_ne : ∀ {q r : ℤ_[p]}, ∥q∥ ≠ ∥r∥ → ∥q+r∥ = max (∥q∥) (∥r∥)
| ⟨_, _⟩ ⟨_, _⟩ := padic_norm_e.add_eq_max_of_ne
@[simp] lemma norm_one : ∥(1 : ℤ_[p])∥ = 1 := norm_one
lemma eq_of_norm_add_lt_right {z1 z2 : ℤ_[p]}
(h : ∥z1 + z2∥ < ∥z2∥) : ∥z1∥ = ∥z2∥ :=
by_contradiction $ λ hne,
not_lt_of_ge (by rw padic_norm_z.add_eq_max_of_ne hne; apply le_max_right) h
lemma eq_of_norm_add_lt_left {z1 z2 : ℤ_[p]}
(h : ∥z1 + z2∥ < ∥z1∥) : ∥z1∥ = ∥z2∥ :=
by_contradiction $ λ hne,
not_lt_of_ge (by rw padic_norm_z.add_eq_max_of_ne hne; apply le_max_left) h
@[simp] lemma padic_norm_e_of_padic_int (z : ℤ_[p]) : ∥(↑z : ℚ_[p])∥ = ∥z∥ :=
by simp [norm, padic_norm_z]
@[simp] lemma padic_norm_z_eq_padic_norm_e {q : ℚ_[p]} (hq : ∥q∥ ≤ 1) :
@norm ℤ_[p] _ ⟨q, hq⟩ = ∥q∥ := rfl
end padic_norm_z
private lemma mul_lt_one {α} [decidable_linear_ordered_comm_ring α] {a b : α} (hbz : 0 < b)
(ha : a < 1) (hb : b < 1) : a * b < 1 :=
suffices a*b < 1*1, by simpa,
mul_lt_mul ha (le_of_lt hb) hbz zero_le_one
private lemma mul_lt_one_of_le_of_lt {α} [decidable_linear_ordered_comm_ring α] {a b : α} (ha : a ≤ 1)
(hbz : 0 ≤ b) (hb : b < 1) : a * b < 1 :=
if hb' : b = 0 then by simpa [hb'] using zero_lt_one
else if ha' : a = 1 then by simpa [ha']
else mul_lt_one (lt_of_le_of_ne hbz (ne.symm hb')) (lt_of_le_of_ne ha ha') hb
namespace padic_int
variables {p : ℕ} [nat.prime p]
local attribute [reducible] padic_int
lemma mul_inv : ∀ {z : ℤ_[p]}, ∥z∥ = 1 → z * z.inv = 1
| ⟨k, _⟩ h :=
begin
have hk : k ≠ 0, from λ h', @zero_ne_one ℚ_[p] _ (by simpa [h'] using h),
unfold padic_int.inv, split_ifs,
{ change (⟨k * (1/k), _⟩ : ℤ_[p]) = 1,
simp [hk], refl },
{ apply subtype.ext.2, simp [mul_inv_cancel hk] }
end
lemma inv_mul {z : ℤ_[p]} (hz : ∥z∥ = 1) : z.inv * z = 1 :=
by rw [mul_comm, mul_inv hz]
lemma is_unit_iff {z : ℤ_[p]} : is_unit z ↔ ∥z∥ = 1 :=
⟨λ h, begin
rcases is_unit_iff_dvd_one.1 h with ⟨w, eq⟩,
refine le_antisymm (padic_norm_z.le_one _) _,
have := mul_le_mul_of_nonneg_left (padic_norm_z.le_one w) (norm_nonneg z),
rwa [mul_one, ← padic_norm_z.mul, ← eq, padic_norm_z.one] at this
end, λ h, ⟨⟨z, z.inv, mul_inv h, inv_mul h⟩, rfl⟩⟩
lemma norm_lt_one_add {z1 z2 : ℤ_[p]} (hz1 : ∥z1∥ < 1) (hz2 : ∥z2∥ < 1) : ∥z1 + z2∥ < 1 :=
lt_of_le_of_lt (padic_norm_z.nonarchimedean _ _) (max_lt hz1 hz2)
lemma norm_lt_one_mul {z1 z2 : ℤ_[p]} (hz2 : ∥z2∥ < 1) : ∥z1 * z2∥ < 1 :=
calc ∥z1 * z2∥ = ∥z1∥ * ∥z2∥ : by simp
... < 1 : mul_lt_one_of_le_of_lt (padic_norm_z.le_one _) (norm_nonneg _) hz2
@[simp] lemma mem_nonunits {z : ℤ_[p]} : z ∈ nonunits ℤ_[p] ↔ ∥z∥ < 1 :=
by rw lt_iff_le_and_ne; simp [padic_norm_z.le_one z, nonunits, is_unit_iff]
instance : local_ring ℤ_[p] :=
local_of_nonunits_ideal zero_ne_one $ λ x y, by simp; exact norm_lt_one_add
private def cau_seq_to_rat_cau_seq (f : cau_seq ℤ_[p] norm) :
cau_seq ℚ_[p] (λ a, ∥a∥) :=
⟨ λ n, f n,
λ _ hε, by simpa [norm, padic_norm_z] using f.cauchy hε ⟩
instance complete : cau_seq.is_complete ℤ_[p] norm :=
⟨ λ f,
have hqn : ∥cau_seq.lim (cau_seq_to_rat_cau_seq f)∥ ≤ 1,
from padic_norm_e_lim_le zero_lt_one (λ _, padic_norm_z.le_one _),
⟨ ⟨_, hqn⟩,
λ ε, by simpa [norm, padic_norm_z] using cau_seq.equiv_lim (cau_seq_to_rat_cau_seq f) ε⟩⟩
end padic_int
namespace padic_norm_z
variables {p : ℕ} [nat.prime p]
lemma padic_val_of_cong_pow_p {z1 z2 : ℤ} {n : ℕ} (hz : z1 ≡ z2 [ZMOD ↑(p^n)]) :
∥(z1 - z2 : ℚ_[p])∥ ≤ ↑(↑p ^ (-n : ℤ) : ℚ) :=
have hdvd : ↑(p^n) ∣ z2 - z1, from int.modeq.modeq_iff_dvd.1 hz,
have (z2 - z1 : ℚ_[p]) = ↑(↑(z2 - z1) : ℚ), by norm_cast,
begin
rw [norm_sub_rev, this, padic_norm_e.eq_padic_norm],
exact_mod_cast padic_norm.le_of_dvd p hdvd
end
end padic_norm_z
|
71e5c27d399b05638600b1b5fdcc58f684576793 | 947fa6c38e48771ae886239b4edce6db6e18d0fb | /src/logic/equiv/fin.lean | 348637dd9098ce76bb11ced4b94f66b8e8500225 | [
"Apache-2.0"
] | permissive | ramonfmir/mathlib | c5dc8b33155473fab97c38bd3aa6723dc289beaa | 14c52e990c17f5a00c0cc9e09847af16fabbed25 | refs/heads/master | 1,661,979,343,526 | 1,660,830,384,000 | 1,660,830,384,000 | 182,072,989 | 0 | 0 | null | 1,555,585,876,000 | 1,555,585,876,000 | null | UTF-8 | Lean | false | false | 15,278 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import data.fin.vec_notation
import logic.equiv.basic
/-!
# Equivalences for `fin n`
-/
universes u
variables {m n : ℕ}
/-- Equivalence between `fin 0` and `empty`. -/
def fin_zero_equiv : fin 0 ≃ empty :=
equiv.equiv_empty _
/-- Equivalence between `fin 0` and `pempty`. -/
def fin_zero_equiv' : fin 0 ≃ pempty.{u} :=
equiv.equiv_pempty _
/-- Equivalence between `fin 1` and `unit`. -/
def fin_one_equiv : fin 1 ≃ unit :=
equiv.equiv_punit _
/-- Equivalence between `fin 2` and `bool`. -/
def fin_two_equiv : fin 2 ≃ bool :=
{ to_fun := ![ff, tt],
inv_fun := λ b, cond b 1 0,
left_inv := fin.forall_fin_two.2 $ by simp,
right_inv := bool.forall_bool.2 $ by simp }
/-- `Π i : fin 2, α i` is equivalent to `α 0 × α 1`. See also `fin_two_arrow_equiv` for a
non-dependent version and `prod_equiv_pi_fin_two` for a version with inputs `α β : Type u`. -/
@[simps {fully_applied := ff}] def pi_fin_two_equiv (α : fin 2 → Type u) : (Π i, α i) ≃ α 0 × α 1 :=
{ to_fun := λ f, (f 0, f 1),
inv_fun := λ p, fin.cons p.1 $ fin.cons p.2 fin_zero_elim,
left_inv := λ f, funext $ fin.forall_fin_two.2 ⟨rfl, rfl⟩,
right_inv := λ ⟨x, y⟩, rfl }
lemma fin.preimage_apply_01_prod {α : fin 2 → Type u} (s : set (α 0)) (t : set (α 1)) :
(λ f : Π i, α i, (f 0, f 1)) ⁻¹' s ×ˢ t =
set.pi set.univ (fin.cons s $ fin.cons t fin.elim0) :=
begin
ext f,
have : (fin.cons s (fin.cons t fin.elim0) : Π i, set (α i)) 1 = t := rfl,
simp [fin.forall_fin_two, this]
end
lemma fin.preimage_apply_01_prod' {α : Type u} (s t : set α) :
(λ f : fin 2 → α, (f 0, f 1)) ⁻¹' s ×ˢ t = set.pi set.univ ![s, t] :=
fin.preimage_apply_01_prod s t
/-- A product space `α × β` is equivalent to the space `Π i : fin 2, γ i`, where
`γ = fin.cons α (fin.cons β fin_zero_elim)`. See also `pi_fin_two_equiv` and
`fin_two_arrow_equiv`. -/
@[simps {fully_applied := ff }] def prod_equiv_pi_fin_two (α β : Type u) :
α × β ≃ Π i : fin 2, ![α, β] i :=
(pi_fin_two_equiv (fin.cons α (fin.cons β fin_zero_elim))).symm
/-- The space of functions `fin 2 → α` is equivalent to `α × α`. See also `pi_fin_two_equiv` and
`prod_equiv_pi_fin_two`. -/
@[simps { fully_applied := ff }] def fin_two_arrow_equiv (α : Type*) : (fin 2 → α) ≃ α × α :=
{ inv_fun := λ x, ![x.1, x.2],
.. pi_fin_two_equiv (λ _, α) }
/-- `Π i : fin 2, α i` is order equivalent to `α 0 × α 1`. See also `order_iso.fin_two_arrow_equiv`
for a non-dependent version. -/
def order_iso.pi_fin_two_iso (α : fin 2 → Type u) [Π i, preorder (α i)] :
(Π i, α i) ≃o α 0 × α 1 :=
{ to_equiv := pi_fin_two_equiv α,
map_rel_iff' := λ f g, iff.symm fin.forall_fin_two }
/-- The space of functions `fin 2 → α` is order equivalent to `α × α`. See also
`order_iso.pi_fin_two_iso`. -/
def order_iso.fin_two_arrow_iso (α : Type*) [preorder α] : (fin 2 → α) ≃o α × α :=
{ to_equiv := fin_two_arrow_equiv α, .. order_iso.pi_fin_two_iso (λ _, α) }
/-- The 'identity' equivalence between `fin n` and `fin m` when `n = m`. -/
def fin_congr {n m : ℕ} (h : n = m) : fin n ≃ fin m :=
(fin.cast h).to_equiv
@[simp] lemma fin_congr_apply_mk {n m : ℕ} (h : n = m) (k : ℕ) (w : k < n) :
fin_congr h ⟨k, w⟩ = ⟨k, by { subst h, exact w }⟩ :=
rfl
@[simp] lemma fin_congr_symm {n m : ℕ} (h : n = m) :
(fin_congr h).symm = fin_congr h.symm := rfl
@[simp] lemma fin_congr_apply_coe {n m : ℕ} (h : n = m) (k : fin n) :
(fin_congr h k : ℕ) = k :=
by { cases k, refl, }
lemma fin_congr_symm_apply_coe {n m : ℕ} (h : n = m) (k : fin m) :
((fin_congr h).symm k : ℕ) = k :=
by { cases k, refl, }
/-- An equivalence that removes `i` and maps it to `none`.
This is a version of `fin.pred_above` that produces `option (fin n)` instead of
mapping both `i.cast_succ` and `i.succ` to `i`. -/
def fin_succ_equiv' {n : ℕ} (i : fin (n + 1)) :
fin (n + 1) ≃ option (fin n) :=
{ to_fun := i.insert_nth none some,
inv_fun := λ x, x.cases_on' i (fin.succ_above i),
left_inv := λ x, fin.succ_above_cases i (by simp) (λ j, by simp) x,
right_inv := λ x, by cases x; dsimp; simp }
@[simp] lemma fin_succ_equiv'_at {n : ℕ} (i : fin (n + 1)) :
(fin_succ_equiv' i) i = none := by simp [fin_succ_equiv']
@[simp] lemma fin_succ_equiv'_succ_above {n : ℕ} (i : fin (n + 1)) (j : fin n) :
fin_succ_equiv' i (i.succ_above j) = some j :=
@fin.insert_nth_apply_succ_above n (λ _, option (fin n)) i _ _ _
lemma fin_succ_equiv'_below {n : ℕ} {i : fin (n + 1)} {m : fin n} (h : m.cast_succ < i) :
(fin_succ_equiv' i) m.cast_succ = some m :=
by rw [← fin.succ_above_below _ _ h, fin_succ_equiv'_succ_above]
lemma fin_succ_equiv'_above {n : ℕ} {i : fin (n + 1)} {m : fin n} (h : i ≤ m.cast_succ) :
(fin_succ_equiv' i) m.succ = some m :=
by rw [← fin.succ_above_above _ _ h, fin_succ_equiv'_succ_above]
@[simp] lemma fin_succ_equiv'_symm_none {n : ℕ} (i : fin (n + 1)) :
(fin_succ_equiv' i).symm none = i := rfl
@[simp] lemma fin_succ_equiv'_symm_some {n : ℕ} (i : fin (n + 1)) (j : fin n) :
(fin_succ_equiv' i).symm (some j) = i.succ_above j :=
rfl
lemma fin_succ_equiv'_symm_some_below {n : ℕ} {i : fin (n + 1)} {m : fin n} (h : m.cast_succ < i) :
(fin_succ_equiv' i).symm (some m) = m.cast_succ :=
fin.succ_above_below i m h
lemma fin_succ_equiv'_symm_some_above {n : ℕ} {i : fin (n + 1)} {m : fin n} (h : i ≤ m.cast_succ) :
(fin_succ_equiv' i).symm (some m) = m.succ :=
fin.succ_above_above i m h
lemma fin_succ_equiv'_symm_coe_below {n : ℕ} {i : fin (n + 1)} {m : fin n} (h : m.cast_succ < i) :
(fin_succ_equiv' i).symm m = m.cast_succ :=
fin_succ_equiv'_symm_some_below h
lemma fin_succ_equiv'_symm_coe_above {n : ℕ} {i : fin (n + 1)} {m : fin n} (h : i ≤ m.cast_succ) :
(fin_succ_equiv' i).symm m = m.succ :=
fin_succ_equiv'_symm_some_above h
/-- Equivalence between `fin (n + 1)` and `option (fin n)`.
This is a version of `fin.pred` that produces `option (fin n)` instead of
requiring a proof that the input is not `0`. -/
def fin_succ_equiv (n : ℕ) : fin (n + 1) ≃ option (fin n) :=
fin_succ_equiv' 0
@[simp] lemma fin_succ_equiv_zero {n : ℕ} :
(fin_succ_equiv n) 0 = none :=
rfl
@[simp] lemma fin_succ_equiv_succ {n : ℕ} (m : fin n):
(fin_succ_equiv n) m.succ = some m :=
fin_succ_equiv'_above (fin.zero_le _)
@[simp] lemma fin_succ_equiv_symm_none {n : ℕ} :
(fin_succ_equiv n).symm none = 0 :=
fin_succ_equiv'_symm_none _
@[simp] lemma fin_succ_equiv_symm_some {n : ℕ} (m : fin n) :
(fin_succ_equiv n).symm (some m) = m.succ :=
congr_fun fin.succ_above_zero m
@[simp] lemma fin_succ_equiv_symm_coe {n : ℕ} (m : fin n) :
(fin_succ_equiv n).symm m = m.succ :=
fin_succ_equiv_symm_some m
/-- The equiv version of `fin.pred_above_zero`. -/
lemma fin_succ_equiv'_zero {n : ℕ} :
fin_succ_equiv' (0 : fin (n + 1)) = fin_succ_equiv n := rfl
/-- `equiv` between `fin (n + 1)` and `option (fin n)` sending `fin.last n` to `none` -/
def fin_succ_equiv_last {n : ℕ} : fin (n + 1) ≃ option (fin n) :=
fin_succ_equiv' (fin.last n)
@[simp] lemma fin_succ_equiv_last_cast_succ {n : ℕ} (i : fin n) :
fin_succ_equiv_last i.cast_succ = some i :=
fin_succ_equiv'_below i.2
@[simp] lemma fin_succ_equiv_last_last {n : ℕ} :
fin_succ_equiv_last (fin.last n) = none :=
by simp [fin_succ_equiv_last]
@[simp] lemma fin_succ_equiv_last_symm_some {n : ℕ} (i : fin n) :
fin_succ_equiv_last.symm (some i) = i.cast_succ :=
fin_succ_equiv'_symm_some_below i.2
@[simp] lemma fin_succ_equiv_last_symm_coe {n : ℕ} (i : fin n) :
fin_succ_equiv_last.symm ↑i = i.cast_succ :=
fin_succ_equiv'_symm_some_below i.2
@[simp] lemma fin_succ_equiv_last_symm_none {n : ℕ} :
fin_succ_equiv_last.symm none = fin.last n :=
fin_succ_equiv'_symm_none _
/-- Equivalence between `Π j : fin (n + 1), α j` and `α i × Π j : fin n, α (fin.succ_above i j)`. -/
@[simps { fully_applied := ff}]
def equiv.pi_fin_succ_above_equiv {n : ℕ} (α : fin (n + 1) → Type u) (i : fin (n + 1)) :
(Π j, α j) ≃ α i × (Π j, α (i.succ_above j)) :=
{ to_fun := λ f, (f i, λ j, f (i.succ_above j)),
inv_fun := λ f, i.insert_nth f.1 f.2,
left_inv := λ f, by simp [fin.insert_nth_eq_iff],
right_inv := λ f, by simp }
/-- Order isomorphism between `Π j : fin (n + 1), α j` and
`α i × Π j : fin n, α (fin.succ_above i j)`. -/
def order_iso.pi_fin_succ_above_iso {n : ℕ} (α : fin (n + 1) → Type u) [Π i, has_le (α i)]
(i : fin (n + 1)) :
(Π j, α j) ≃o α i × (Π j, α (i.succ_above j)) :=
{ to_equiv := equiv.pi_fin_succ_above_equiv α i,
map_rel_iff' := λ f g, i.forall_iff_succ_above.symm }
/-- Equivalence between `fin (n + 1) → β` and `β × (fin n → β)`. -/
@[simps { fully_applied := ff}]
def equiv.pi_fin_succ (n : ℕ) (β : Type u) :
(fin (n+1) → β) ≃ β × (fin n → β) :=
equiv.pi_fin_succ_above_equiv (λ _, β) 0
/-- Equivalence between `fin m ⊕ fin n` and `fin (m + n)` -/
def fin_sum_fin_equiv : fin m ⊕ fin n ≃ fin (m + n) :=
{ to_fun := sum.elim (fin.cast_add n) (fin.nat_add m),
inv_fun := λ i, @fin.add_cases m n (λ _, fin m ⊕ fin n) sum.inl sum.inr i,
left_inv := λ x, by { cases x with y y; dsimp; simp },
right_inv := λ x, by refine fin.add_cases (λ i, _) (λ i, _) x; simp }
@[simp] lemma fin_sum_fin_equiv_apply_left (i : fin m) :
(fin_sum_fin_equiv (sum.inl i) : fin (m + n)) = fin.cast_add n i := rfl
@[simp] lemma fin_sum_fin_equiv_apply_right (i : fin n) :
(fin_sum_fin_equiv (sum.inr i) : fin (m + n)) = fin.nat_add m i := rfl
@[simp] lemma fin_sum_fin_equiv_symm_apply_cast_add (x : fin m) :
fin_sum_fin_equiv.symm (fin.cast_add n x) = sum.inl x :=
fin_sum_fin_equiv.symm_apply_apply (sum.inl x)
@[simp] lemma fin_sum_fin_equiv_symm_apply_nat_add (x : fin n) :
fin_sum_fin_equiv.symm (fin.nat_add m x) = sum.inr x :=
fin_sum_fin_equiv.symm_apply_apply (sum.inr x)
@[simp] lemma fin_sum_fin_equiv_symm_last :
fin_sum_fin_equiv.symm (fin.last n) = sum.inr 0 :=
fin_sum_fin_equiv_symm_apply_nat_add 0
/-- The equivalence between `fin (m + n)` and `fin (n + m)` which rotates by `n`. -/
def fin_add_flip : fin (m + n) ≃ fin (n + m) :=
(fin_sum_fin_equiv.symm.trans (equiv.sum_comm _ _)).trans fin_sum_fin_equiv
@[simp] lemma fin_add_flip_apply_cast_add (k : fin m) (n : ℕ) :
fin_add_flip (fin.cast_add n k) = fin.nat_add n k :=
by simp [fin_add_flip]
@[simp] lemma fin_add_flip_apply_nat_add (k : fin n) (m : ℕ) :
fin_add_flip (fin.nat_add m k) = fin.cast_add m k :=
by simp [fin_add_flip]
@[simp] lemma fin_add_flip_apply_mk_left {k : ℕ} (h : k < m)
(hk : k < m + n := nat.lt_add_right k m n h)
(hnk : n + k < n + m := add_lt_add_left h n) :
fin_add_flip (⟨k, hk⟩ : fin (m + n)) = ⟨n + k, hnk⟩ :=
by convert fin_add_flip_apply_cast_add ⟨k, h⟩ n
@[simp] lemma fin_add_flip_apply_mk_right {k : ℕ} (h₁ : m ≤ k) (h₂ : k < m + n) :
fin_add_flip (⟨k, h₂⟩ : fin (m + n)) = ⟨k - m, tsub_le_self.trans_lt $ add_comm m n ▸ h₂⟩ :=
begin
convert fin_add_flip_apply_nat_add ⟨k - m, (tsub_lt_iff_right h₁).2 _⟩ m,
{ simp [add_tsub_cancel_of_le h₁] },
{ rwa add_comm }
end
/-- Rotate `fin n` one step to the right. -/
def fin_rotate : Π n, equiv.perm (fin n)
| 0 := equiv.refl _
| (n+1) := fin_add_flip.trans (fin_congr (add_comm _ _))
lemma fin_rotate_of_lt {k : ℕ} (h : k < n) :
fin_rotate (n+1) ⟨k, lt_of_lt_of_le h (nat.le_succ _)⟩ = ⟨k + 1, nat.succ_lt_succ h⟩ :=
begin
dsimp [fin_rotate],
simp [h, add_comm],
end
lemma fin_rotate_last' : fin_rotate (n+1) ⟨n, lt_add_one _⟩ = ⟨0, nat.zero_lt_succ _⟩ :=
begin
dsimp [fin_rotate],
rw fin_add_flip_apply_mk_right,
simp,
end
lemma fin_rotate_last : fin_rotate (n+1) (fin.last _) = 0 :=
fin_rotate_last'
lemma fin.snoc_eq_cons_rotate {α : Type*} (v : fin n → α) (a : α) :
@fin.snoc _ (λ _, α) v a = (λ i, @fin.cons _ (λ _, α) a v (fin_rotate _ i)) :=
begin
ext ⟨i, h⟩,
by_cases h' : i < n,
{ rw [fin_rotate_of_lt h', fin.snoc, fin.cons, dif_pos h'],
refl, },
{ have h'' : n = i,
{ simp only [not_lt] at h', exact (nat.eq_of_le_of_lt_succ h' h).symm, },
subst h'',
rw [fin_rotate_last', fin.snoc, fin.cons, dif_neg (lt_irrefl _)],
refl, }
end
@[simp] lemma fin_rotate_zero : fin_rotate 0 = equiv.refl _ := rfl
@[simp] lemma fin_rotate_one : fin_rotate 1 = equiv.refl _ :=
subsingleton.elim _ _
@[simp] lemma fin_rotate_succ_apply {n : ℕ} (i : fin n.succ) :
fin_rotate n.succ i = i + 1 :=
begin
cases n,
{ simp },
rcases i.le_last.eq_or_lt with rfl|h,
{ simp [fin_rotate_last] },
{ cases i,
simp only [fin.lt_iff_coe_lt_coe, fin.coe_last, fin.coe_mk] at h,
simp [fin_rotate_of_lt h, fin.eq_iff_veq, fin.add_def, nat.mod_eq_of_lt (nat.succ_lt_succ h)] },
end
@[simp] lemma fin_rotate_apply_zero {n : ℕ} : fin_rotate n.succ 0 = 1 :=
by rw [fin_rotate_succ_apply, zero_add]
lemma coe_fin_rotate_of_ne_last {n : ℕ} {i : fin n.succ} (h : i ≠ fin.last n) :
(fin_rotate n.succ i : ℕ) = i + 1 :=
begin
rw fin_rotate_succ_apply,
have : (i : ℕ) < n := lt_of_le_of_ne (nat.succ_le_succ_iff.mp i.2) (fin.coe_injective.ne h),
exact fin.coe_add_one_of_lt this
end
lemma coe_fin_rotate {n : ℕ} (i : fin n.succ) :
(fin_rotate n.succ i : ℕ) = if i = fin.last n then 0 else i + 1 :=
by rw [fin_rotate_succ_apply, fin.coe_add_one i]
/-- Equivalence between `fin m × fin n` and `fin (m * n)` -/
@[simps]
def fin_prod_fin_equiv : fin m × fin n ≃ fin (m * n) :=
{ to_fun := λ x, ⟨x.2 + n * x.1,
calc x.2.1 + n * x.1.1 + 1
= x.1.1 * n + x.2.1 + 1 : by ac_refl
... ≤ x.1.1 * n + n : nat.add_le_add_left x.2.2 _
... = (x.1.1 + 1) * n : eq.symm $ nat.succ_mul _ _
... ≤ m * n : nat.mul_le_mul_right _ x.1.2⟩,
inv_fun := λ x, (x.div_nat, x.mod_nat),
left_inv := λ ⟨x, y⟩,
have H : 0 < n, from nat.pos_of_ne_zero $ λ H, nat.not_lt_zero y.1 $ H ▸ y.2,
prod.ext
(fin.eq_of_veq $ calc
(y.1 + n * x.1) / n
= y.1 / n + x.1 : nat.add_mul_div_left _ _ H
... = 0 + x.1 : by rw nat.div_eq_of_lt y.2
... = x.1 : nat.zero_add x.1)
(fin.eq_of_veq $ calc
(y.1 + n * x.1) % n
= y.1 % n : nat.add_mul_mod_self_left _ _ _
... = y.1 : nat.mod_eq_of_lt y.2),
right_inv := λ x, fin.eq_of_veq $ nat.mod_add_div _ _ }
/-- Promote a `fin n` into a larger `fin m`, as a subtype where the underlying
values are retained. This is the `order_iso` version of `fin.cast_le`. -/
@[simps apply symm_apply]
def fin.cast_le_order_iso {n m : ℕ} (h : n ≤ m) : fin n ≃o {i : fin m // (i : ℕ) < n} :=
{ to_fun := λ i, ⟨fin.cast_le h i, by simpa using i.is_lt⟩,
inv_fun := λ i, ⟨i, i.prop⟩,
left_inv := λ _, by simp,
right_inv := λ _, by simp,
map_rel_iff' := λ _ _, by simp }
/-- `fin 0` is a subsingleton. -/
instance subsingleton_fin_zero : subsingleton (fin 0) :=
fin_zero_equiv.subsingleton
/-- `fin 1` is a subsingleton. -/
instance subsingleton_fin_one : subsingleton (fin 1) :=
fin_one_equiv.subsingleton
|
3d5302c99c473661fdad4d908094179b1faf1363 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/run/2018.lean | 494143bb9bdf12d1d75dd2cce68c70f95242c14f | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 524 | lean | def works (l : List α) : List α :=
match l with
| [] => []
| _::tail =>
works tail
decreasing_by decreasing_tactic -- to force well-founded recursion
-- Both of these appear in mathlib
@[simp]
theorem add_zero (n : Nat) : n + 0 = n := by sorry
@[simp]
theorem lt_add_iff_pos_left (a : Nat) {b : Nat} :
a < b + a ↔ 0 < b
:= by sorry
-- Breaking this:
def should_still_work (l : List α) : List α :=
match l with
| [] => []
| _::tail =>
should_still_work tail
decreasing_by decreasing_tactic
|
0ceaf90a8002040d6e99a5545aebcacc0f8b4c58 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/topology/metric_space/closeds.lean | e08a06bb4a4bdd8df014548598566e2568b1f10c | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 21,237 | lean | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import analysis.specific_limits.basic
import topology.metric_space.hausdorff_distance
import topology.sets.compacts
/-!
# Closed subsets
This file defines the metric and emetric space structure on the types of closed subsets and nonempty
compact subsets of a metric or emetric space.
The Hausdorff distance induces an emetric space structure on the type of closed subsets
of an emetric space, called `closeds`. Its completeness, resp. compactness, resp.
second-countability, follow from the corresponding properties of the original space.
In a metric space, the type of nonempty compact subsets (called `nonempty_compacts`) also
inherits a metric space structure from the Hausdorff distance, as the Hausdorff edistance is
always finite in this context.
-/
noncomputable theory
open_locale classical topological_space ennreal
universe u
open classical set function topological_space filter
namespace emetric
section
variables {α : Type u} [emetric_space α] {s : set α}
/-- In emetric spaces, the Hausdorff edistance defines an emetric space structure
on the type of closed subsets -/
instance closeds.emetric_space : emetric_space (closeds α) :=
{ edist := λs t, Hausdorff_edist (s : set α) t,
edist_self := λs, Hausdorff_edist_self,
edist_comm := λs t, Hausdorff_edist_comm,
edist_triangle := λs t u, Hausdorff_edist_triangle,
eq_of_edist_eq_zero :=
λ s t h, closeds.ext $ (Hausdorff_edist_zero_iff_eq_of_closed s.closed t.closed).1 h }
/-- The edistance to a closed set depends continuously on the point and the set -/
lemma continuous_inf_edist_Hausdorff_edist :
continuous (λ p : α × (closeds α), inf_edist p.1 p.2) :=
begin
refine continuous_of_le_add_edist 2 (by simp) _,
rintros ⟨x, s⟩ ⟨y, t⟩,
calc inf_edist x s ≤ inf_edist x t + Hausdorff_edist (t : set α) s :
inf_edist_le_inf_edist_add_Hausdorff_edist
... ≤ inf_edist y t + edist x y + Hausdorff_edist (t : set α) s :
add_le_add_right inf_edist_le_inf_edist_add_edist _
... = inf_edist y t + (edist x y + Hausdorff_edist (s : set α) t)
: by rw [add_assoc, Hausdorff_edist_comm]
... ≤ inf_edist y t + (edist (x, s) (y, t) + edist (x, s) (y, t)) :
add_le_add_left (add_le_add (le_max_left _ _) (le_max_right _ _)) _
... = inf_edist y t + 2 * edist (x, s) (y, t) :
by rw [← mul_two, mul_comm]
end
/-- Subsets of a given closed subset form a closed set -/
lemma is_closed_subsets_of_is_closed (hs : is_closed s) :
is_closed {t : closeds α | (t : set α) ⊆ s} :=
begin
refine is_closed_of_closure_subset (λt ht x hx, _),
-- t : closeds α, ht : t ∈ closure {t : closeds α | t ⊆ s},
-- x : α, hx : x ∈ t
-- goal : x ∈ s
have : x ∈ closure s,
{ refine mem_closure_iff.2 (λε εpos, _),
rcases mem_closure_iff.1 ht ε εpos with ⟨u, hu, Dtu⟩,
-- u : closeds α, hu : u ∈ {t : closeds α | t ⊆ s}, hu' : edist t u < ε
rcases exists_edist_lt_of_Hausdorff_edist_lt hx Dtu with ⟨y, hy, Dxy⟩,
-- y : α, hy : y ∈ u, Dxy : edist x y < ε
exact ⟨y, hu hy, Dxy⟩ },
rwa hs.closure_eq at this,
end
/-- By definition, the edistance on `closeds α` is given by the Hausdorff edistance -/
lemma closeds.edist_eq {s t : closeds α} : edist s t = Hausdorff_edist (s : set α) t := rfl
/-- In a complete space, the type of closed subsets is complete for the
Hausdorff edistance. -/
instance closeds.complete_space [complete_space α] : complete_space (closeds α) :=
begin
/- We will show that, if a sequence of sets `s n` satisfies
`edist (s n) (s (n+1)) < 2^{-n}`, then it converges. This is enough to guarantee
completeness, by a standard completeness criterion.
We use the shorthand `B n = 2^{-n}` in ennreal. -/
let B : ℕ → ℝ≥0∞ := λ n, (2⁻¹)^n,
have B_pos : ∀ n, (0:ℝ≥0∞) < B n,
by simp [B, ennreal.pow_pos],
have B_ne_top : ∀ n, B n ≠ ⊤,
by simp [B, ennreal.pow_ne_top],
/- Consider a sequence of closed sets `s n` with `edist (s n) (s (n+1)) < B n`.
We will show that it converges. The limit set is t0 = ⋂n, closure (⋃m≥n, s m).
We will have to show that a point in `s n` is close to a point in `t0`, and a point
in `t0` is close to a point in `s n`. The completeness then follows from a
standard criterion. -/
refine complete_of_convergent_controlled_sequences B B_pos (λs hs, _),
let t0 := ⋂ n, closure (⋃ m ≥ n, s m : set α),
let t : closeds α := ⟨t0, is_closed_Inter (λ_, is_closed_closure)⟩,
use t,
-- The inequality is written this way to agree with `edist_le_of_edist_le_geometric_of_tendsto₀`
have I1 : ∀ n, ∀ x ∈ s n, ∃ y ∈ t0, edist x y ≤ 2 * B n,
{ /- This is the main difficulty of the proof. Starting from `x ∈ s n`, we want
to find a point in `t0` which is close to `x`. Define inductively a sequence of
points `z m` with `z n = x` and `z m ∈ s m` and `edist (z m) (z (m+1)) ≤ B m`. This is
possible since the Hausdorff distance between `s m` and `s (m+1)` is at most `B m`.
This sequence is a Cauchy sequence, therefore converging as the space is complete, to
a limit which satisfies the required properties. -/
assume n x hx,
obtain ⟨z, hz₀, hz⟩ : ∃ z : Π l, s (n + l), (z 0 : α) = x ∧
∀ k, edist (z k:α) (z (k+1):α) ≤ B n / 2^k,
{ -- We prove existence of the sequence by induction.
have : ∀ l (z : s (n + l)), ∃ z' : s (n + l + 1), edist (z : α) z' ≤ B n / 2^l,
{ assume l z,
obtain ⟨z', z'_mem, hz'⟩ : ∃ z' ∈ s (n + l + 1), edist (z : α) z' < B n / 2^l,
{ refine exists_edist_lt_of_Hausdorff_edist_lt _ _,
{ exact s (n + l) },
{ exact z.2 },
simp only [B, ennreal.inv_pow, div_eq_mul_inv],
rw [← pow_add],
apply hs; simp },
exact ⟨⟨z', z'_mem⟩, le_of_lt hz'⟩ },
use [λ k, nat.rec_on k ⟨x, hx⟩ (λl z, some (this l z)), rfl],
exact λ k, some_spec (this k _) },
-- it follows from the previous bound that `z` is a Cauchy sequence
have : cauchy_seq (λ k, ((z k):α)),
from cauchy_seq_of_edist_le_geometric_two (B n) (B_ne_top n) hz,
-- therefore, it converges
rcases cauchy_seq_tendsto_of_complete this with ⟨y, y_lim⟩,
use y,
-- the limit point `y` will be the desired point, in `t0` and close to our initial point `x`.
-- First, we check it belongs to `t0`.
have : y ∈ t0 := mem_Inter.2 (λk, mem_closure_of_tendsto y_lim
begin
simp only [exists_prop, set.mem_Union, filter.eventually_at_top, set.mem_preimage,
set.preimage_Union],
exact ⟨k, λ m hm, ⟨n+m, zero_add k ▸ add_le_add (zero_le n) hm, (z m).2⟩⟩
end),
use this,
-- Then, we check that `y` is close to `x = z n`. This follows from the fact that `y`
-- is the limit of `z k`, and the distance between `z n` and `z k` has already been estimated.
rw [← hz₀],
exact edist_le_of_edist_le_geometric_two_of_tendsto₀ (B n) hz y_lim },
have I2 : ∀ n, ∀ x ∈ t0, ∃ y ∈ s n, edist x y ≤ 2 * B n,
{ /- For the (much easier) reverse inequality, we start from a point `x ∈ t0` and we want
to find a point `y ∈ s n` which is close to `x`.
`x` belongs to `t0`, the intersection of the closures. In particular, it is well
approximated by a point `z` in `⋃m≥n, s m`, say in `s m`. Since `s m` and
`s n` are close, this point is itself well approximated by a point `y` in `s n`,
as required. -/
assume n x xt0,
have : x ∈ closure (⋃ m ≥ n, s m : set α), by apply mem_Inter.1 xt0 n,
rcases mem_closure_iff.1 this (B n) (B_pos n) with ⟨z, hz, Dxz⟩,
-- z : α, Dxz : edist x z < B n,
simp only [exists_prop, set.mem_Union] at hz,
rcases hz with ⟨m, ⟨m_ge_n, hm⟩⟩,
-- m : ℕ, m_ge_n : m ≥ n, hm : z ∈ s m
have : Hausdorff_edist (s m : set α) (s n) < B n := hs n m n m_ge_n (le_refl n),
rcases exists_edist_lt_of_Hausdorff_edist_lt hm this with ⟨y, hy, Dzy⟩,
-- y : α, hy : y ∈ s n, Dzy : edist z y < B n
exact ⟨y, hy, calc
edist x y ≤ edist x z + edist z y : edist_triangle _ _ _
... ≤ B n + B n : add_le_add (le_of_lt Dxz) (le_of_lt Dzy)
... = 2 * B n : (two_mul _).symm ⟩ },
-- Deduce from the above inequalities that the distance between `s n` and `t0` is at most `2 B n`.
have main : ∀n:ℕ, edist (s n) t ≤ 2 * B n := λn, Hausdorff_edist_le_of_mem_edist (I1 n) (I2 n),
-- from this, the convergence of `s n` to `t0` follows.
refine tendsto_at_top.2 (λε εpos, _),
have : tendsto (λn, 2 * B n) at_top (𝓝 (2 * 0)),
from ennreal.tendsto.const_mul
(ennreal.tendsto_pow_at_top_nhds_0_of_lt_1 $ by simp [ennreal.one_lt_two])
(or.inr $ by simp),
rw mul_zero at this,
obtain ⟨N, hN⟩ : ∃ N, ∀ b ≥ N, ε > 2 * B b,
from ((tendsto_order.1 this).2 ε εpos).exists_forall_of_at_top,
exact ⟨N, λn hn, lt_of_le_of_lt (main n) (hN n hn)⟩
end
/-- In a compact space, the type of closed subsets is compact. -/
instance closeds.compact_space [compact_space α] : compact_space (closeds α) :=
⟨begin
/- by completeness, it suffices to show that it is totally bounded,
i.e., for all ε>0, there is a finite set which is ε-dense.
start from a set `s` which is ε-dense in α. Then the subsets of `s`
are finitely many, and ε-dense for the Hausdorff distance. -/
refine is_compact_of_totally_bounded_is_closed (emetric.totally_bounded_iff.2 (λε εpos, _))
is_closed_univ,
rcases exists_between εpos with ⟨δ, δpos, δlt⟩,
rcases emetric.totally_bounded_iff.1
(is_compact_iff_totally_bounded_is_complete.1 (@is_compact_univ α _ _)).1 δ δpos
with ⟨s, fs, hs⟩,
-- s : set α, fs : s.finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
-- we first show that any set is well approximated by a subset of `s`.
have main : ∀ u : set α, ∃v ⊆ s, Hausdorff_edist u v ≤ δ,
{ assume u,
let v := {x : α | x ∈ s ∧ ∃y∈u, edist x y < δ},
existsi [v, ((λx hx, hx.1) : v ⊆ s)],
refine Hausdorff_edist_le_of_mem_edist _ _,
{ assume x hx,
have : x ∈ ⋃y ∈ s, ball y δ := hs (by simp),
rcases mem_Union₂.1 this with ⟨y, ys, dy⟩,
have : edist y x < δ := by simp at dy; rwa [edist_comm] at dy,
exact ⟨y, ⟨ys, ⟨x, hx, this⟩⟩, le_of_lt dy⟩ },
{ rintros x ⟨hx1, ⟨y, yu, hy⟩⟩,
exact ⟨y, yu, le_of_lt hy⟩ }},
-- introduce the set F of all subsets of `s` (seen as members of `closeds α`).
let F := {f : closeds α | (f : set α) ⊆ s},
refine ⟨F, _, λ u _, _⟩,
-- `F` is finite
{ apply @finite.of_finite_image _ _ F coe,
{ apply fs.finite_subsets.subset (λb, _),
simp only [and_imp, set.mem_image, set.mem_set_of_eq, exists_imp_distrib],
assume x hx hx',
rwa hx' at hx },
{ exact set_like.coe_injective.inj_on F } },
-- `F` is ε-dense
{ obtain ⟨t0, t0s, Dut0⟩ := main u,
have : is_closed t0 := (fs.subset t0s).is_compact.is_closed,
let t : closeds α := ⟨t0, this⟩,
have : t ∈ F := t0s,
have : edist u t < ε := lt_of_le_of_lt Dut0 δlt,
apply mem_Union₂.2,
exact ⟨t, ‹t ∈ F›, this⟩ }
end⟩
/-- In an emetric space, the type of non-empty compact subsets is an emetric space,
where the edistance is the Hausdorff edistance -/
instance nonempty_compacts.emetric_space : emetric_space (nonempty_compacts α) :=
{ edist := λ s t, Hausdorff_edist (s : set α) t,
edist_self := λs, Hausdorff_edist_self,
edist_comm := λs t, Hausdorff_edist_comm,
edist_triangle := λs t u, Hausdorff_edist_triangle,
eq_of_edist_eq_zero := λ s t h, nonempty_compacts.ext $ begin
have : closure (s : set α) = closure t := Hausdorff_edist_zero_iff_closure_eq_closure.1 h,
rwa [s.is_compact.is_closed.closure_eq, t.is_compact.is_closed.closure_eq] at this,
end }
/-- `nonempty_compacts.to_closeds` is a uniform embedding (as it is an isometry) -/
lemma nonempty_compacts.to_closeds.uniform_embedding :
uniform_embedding (@nonempty_compacts.to_closeds α _ _) :=
isometry.uniform_embedding $ λx y, rfl
/-- The range of `nonempty_compacts.to_closeds` is closed in a complete space -/
lemma nonempty_compacts.is_closed_in_closeds [complete_space α] :
is_closed (range $ @nonempty_compacts.to_closeds α _ _) :=
begin
have : range nonempty_compacts.to_closeds =
{s : closeds α | (s : set α).nonempty ∧ is_compact (s : set α) },
{ ext s,
refine ⟨_, λ h, ⟨⟨⟨s, h.2⟩, h.1⟩, closeds.ext rfl⟩⟩,
rintro ⟨s, hs, rfl⟩,
exact ⟨s.nonempty, s.is_compact⟩ },
rw this,
refine is_closed_of_closure_subset (λs hs, ⟨_, _⟩),
{ -- take a set set t which is nonempty and at a finite distance of s
rcases mem_closure_iff.1 hs ⊤ ennreal.coe_lt_top with ⟨t, ht, Dst⟩,
rw edist_comm at Dst,
-- since `t` is nonempty, so is `s`
exact nonempty_of_Hausdorff_edist_ne_top ht.1 (ne_of_lt Dst) },
{ refine is_compact_iff_totally_bounded_is_complete.2 ⟨_, s.closed.is_complete⟩,
refine totally_bounded_iff.2 (λε (εpos : 0 < ε), _),
-- we have to show that s is covered by finitely many eballs of radius ε
-- pick a nonempty compact set t at distance at most ε/2 of s
rcases mem_closure_iff.1 hs (ε/2) (ennreal.half_pos εpos.ne') with ⟨t, ht, Dst⟩,
-- cover this space with finitely many balls of radius ε/2
rcases totally_bounded_iff.1 (is_compact_iff_totally_bounded_is_complete.1 ht.2).1 (ε/2)
(ennreal.half_pos εpos.ne') with ⟨u, fu, ut⟩,
refine ⟨u, ⟨fu, λx hx, _⟩⟩,
-- u : set α, fu : u.finite, ut : t ⊆ ⋃ (y : α) (H : y ∈ u), eball y (ε / 2)
-- then s is covered by the union of the balls centered at u of radius ε
rcases exists_edist_lt_of_Hausdorff_edist_lt hx Dst with ⟨z, hz, Dxz⟩,
rcases mem_Union₂.1 (ut hz) with ⟨y, hy, Dzy⟩,
have : edist x y < ε := calc
edist x y ≤ edist x z + edist z y : edist_triangle _ _ _
... < ε/2 + ε/2 : ennreal.add_lt_add Dxz Dzy
... = ε : ennreal.add_halves _,
exact mem_bUnion hy this },
end
/-- In a complete space, the type of nonempty compact subsets is complete. This follows
from the same statement for closed subsets -/
instance nonempty_compacts.complete_space [complete_space α] :
complete_space (nonempty_compacts α) :=
(complete_space_iff_is_complete_range
nonempty_compacts.to_closeds.uniform_embedding.to_uniform_inducing).2 $
nonempty_compacts.is_closed_in_closeds.is_complete
/-- In a compact space, the type of nonempty compact subsets is compact. This follows from
the same statement for closed subsets -/
instance nonempty_compacts.compact_space [compact_space α] : compact_space (nonempty_compacts α) :=
⟨begin
rw nonempty_compacts.to_closeds.uniform_embedding.embedding.is_compact_iff_is_compact_image,
rw [image_univ],
exact nonempty_compacts.is_closed_in_closeds.is_compact
end⟩
/-- In a second countable space, the type of nonempty compact subsets is second countable -/
instance nonempty_compacts.second_countable_topology [second_countable_topology α] :
second_countable_topology (nonempty_compacts α) :=
begin
haveI : separable_space (nonempty_compacts α) :=
begin
/- To obtain a countable dense subset of `nonempty_compacts α`, start from
a countable dense subset `s` of α, and then consider all its finite nonempty subsets.
This set is countable and made of nonempty compact sets. It turns out to be dense:
by total boundedness, any compact set `t` can be covered by finitely many small balls, and
approximations in `s` of the centers of these balls give the required finite approximation
of `t`. -/
rcases exists_countable_dense α with ⟨s, cs, s_dense⟩,
let v0 := {t : set α | t.finite ∧ t ⊆ s},
let v : set (nonempty_compacts α) := {t : nonempty_compacts α | (t : set α) ∈ v0},
refine ⟨⟨v, _, _⟩⟩,
{ have : v0.countable, from countable_set_of_finite_subset cs,
exact this.preimage set_like.coe_injective },
{ refine λt, mem_closure_iff.2 (λε εpos, _),
-- t is a compact nonempty set, that we have to approximate uniformly by a a set in `v`.
rcases exists_between εpos with ⟨δ, δpos, δlt⟩,
have δpos' : 0 < δ / 2, from ennreal.half_pos δpos.ne',
-- construct a map F associating to a point in α an approximating point in s, up to δ/2.
have Exy : ∀x, ∃y, y ∈ s ∧ edist x y < δ/2,
{ assume x,
rcases mem_closure_iff.1 (s_dense x) (δ/2) δpos' with ⟨y, ys, hy⟩,
exact ⟨y, ⟨ys, hy⟩⟩ },
let F := λx, some (Exy x),
have Fspec : ∀x, F x ∈ s ∧ edist x (F x) < δ/2 := λx, some_spec (Exy x),
-- cover `t` with finitely many balls. Their centers form a set `a`
have : totally_bounded (t : set α) := t.is_compact.totally_bounded,
rcases totally_bounded_iff.1 this (δ/2) δpos' with ⟨a, af, ta⟩,
-- a : set α, af : a.finite, ta : t ⊆ ⋃ (y : α) (H : y ∈ a), eball y (δ / 2)
-- replace each center by a nearby approximation in `s`, giving a new set `b`
let b := F '' a,
have : b.finite := af.image _,
have tb : ∀ x ∈ t, ∃ y ∈ b, edist x y < δ,
{ assume x hx,
rcases mem_Union₂.1 (ta hx) with ⟨z, za, Dxz⟩,
existsi [F z, mem_image_of_mem _ za],
calc edist x (F z) ≤ edist x z + edist z (F z) : edist_triangle _ _ _
... < δ/2 + δ/2 : ennreal.add_lt_add Dxz (Fspec z).2
... = δ : ennreal.add_halves _ },
-- keep only the points in `b` that are close to point in `t`, yielding a new set `c`
let c := {y ∈ b | ∃ x ∈ t, edist x y < δ},
have : c.finite := ‹b.finite›.subset (λx hx, hx.1),
-- points in `t` are well approximated by points in `c`
have tc : ∀ x ∈ t, ∃ y ∈ c, edist x y ≤ δ,
{ assume x hx,
rcases tb x hx with ⟨y, yv, Dxy⟩,
have : y ∈ c := by simp [c, -mem_image]; exact ⟨yv, ⟨x, hx, Dxy⟩⟩,
exact ⟨y, this, le_of_lt Dxy⟩ },
-- points in `c` are well approximated by points in `t`
have ct : ∀ y ∈ c, ∃ x ∈ t, edist y x ≤ δ,
{ rintro y ⟨hy1, x, xt, Dyx⟩,
have : edist y x ≤ δ := calc
edist y x = edist x y : edist_comm _ _
... ≤ δ : le_of_lt Dyx,
exact ⟨x, xt, this⟩ },
-- it follows that their Hausdorff distance is small
have : Hausdorff_edist (t :set α) c ≤ δ :=
Hausdorff_edist_le_of_mem_edist tc ct,
have Dtc : Hausdorff_edist (t : set α) c < ε := this.trans_lt δlt,
-- the set `c` is not empty, as it is well approximated by a nonempty set
have hc : c.nonempty,
from nonempty_of_Hausdorff_edist_ne_top t.nonempty (ne_top_of_lt Dtc),
-- let `d` be the version of `c` in the type `nonempty_compacts α`
let d : nonempty_compacts α := ⟨⟨c, ‹c.finite›.is_compact⟩, hc⟩,
have : c ⊆ s,
{ assume x hx,
rcases (mem_image _ _ _).1 hx.1 with ⟨y, ⟨ya, yx⟩⟩,
rw ← yx,
exact (Fspec y).1 },
have : d ∈ v := ⟨‹c.finite›, this⟩,
-- we have proved that `d` is a good approximation of `t` as requested
exact ⟨d, ‹d ∈ v›, Dtc⟩ },
end,
exact uniform_space.second_countable_of_separable (nonempty_compacts α),
end
end --section
end emetric --namespace
namespace metric
section
variables {α : Type u} [metric_space α]
/-- `nonempty_compacts α` inherits a metric space structure, as the Hausdorff
edistance between two such sets is finite. -/
instance nonempty_compacts.metric_space : metric_space (nonempty_compacts α) :=
emetric_space.to_metric_space $ λ x y, Hausdorff_edist_ne_top_of_nonempty_of_bounded
x.nonempty y.nonempty x.is_compact.bounded y.is_compact.bounded
/-- The distance on `nonempty_compacts α` is the Hausdorff distance, by construction -/
lemma nonempty_compacts.dist_eq {x y : nonempty_compacts α} :
dist x y = Hausdorff_dist (x : set α) y := rfl
lemma lipschitz_inf_dist_set (x : α) : lipschitz_with 1 (λ s : nonempty_compacts α, inf_dist x s) :=
lipschitz_with.of_le_add $ assume s t,
by { rw dist_comm,
exact inf_dist_le_inf_dist_add_Hausdorff_dist (edist_ne_top t s) }
lemma lipschitz_inf_dist : lipschitz_with 2 (λ p : α × (nonempty_compacts α), inf_dist p.1 p.2) :=
@lipschitz_with.uncurry _ _ _ _ _ _ (λ (x : α) (s : nonempty_compacts α), inf_dist x s) 1 1
(λ s, lipschitz_inf_dist_pt s) lipschitz_inf_dist_set
lemma uniform_continuous_inf_dist_Hausdorff_dist :
uniform_continuous (λ p : α × (nonempty_compacts α), inf_dist p.1 p.2) :=
lipschitz_inf_dist.uniform_continuous
end --section
end metric --namespace
|
4ce9dc59d07ac3304b9e0dab88b7bf6efff13e84 | e00ea76a720126cf9f6d732ad6216b5b824d20a7 | /src/category_theory/adjunction/basic.lean | b7c3301e955ecabc72d35f7d4f82afd14c1dc277 | [
"Apache-2.0"
] | permissive | vaibhavkarve/mathlib | a574aaf68c0a431a47fa82ce0637f0f769826bfe | 17f8340912468f49bdc30acdb9a9fa02eeb0473a | refs/heads/master | 1,621,263,802,637 | 1,585,399,588,000 | 1,585,399,588,000 | 250,833,447 | 0 | 0 | Apache-2.0 | 1,585,410,341,000 | 1,585,410,341,000 | null | UTF-8 | Lean | false | false | 10,550 | lean | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Johan Commelin
-/
import category_theory.equivalence
import data.equiv.basic
namespace category_theory
open category
universes v₁ v₂ v₃ u₁ u₂ u₃ -- declare the `v`'s first; see `category_theory.category` for an explanation
local attribute [elab_simple] whisker_left whisker_right
variables {C : Type u₁} [𝒞 : category.{v₁} C] {D : Type u₂} [𝒟 : category.{v₂} D]
include 𝒞 𝒟
/--
`F ⊣ G` represents the data of an adjunction between two functors
`F : C ⥤ D` and `G : D ⥤ C`. `F` is the left adjoint and `G` is the right adjoint.
-/
structure adjunction (F : C ⥤ D) (G : D ⥤ C) :=
(hom_equiv : Π (X Y), (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y))
(unit : 𝟭 C ⟶ F.comp G)
(counit : G.comp F ⟶ 𝟭 D)
(hom_equiv_unit' : Π {X Y f}, (hom_equiv X Y) f = (unit : _ ⟶ _).app X ≫ G.map f . obviously)
(hom_equiv_counit' : Π {X Y g}, (hom_equiv X Y).symm g = F.map g ≫ counit.app Y . obviously)
infix ` ⊣ `:15 := adjunction
class is_left_adjoint (left : C ⥤ D) :=
(right : D ⥤ C)
(adj : left ⊣ right)
class is_right_adjoint (right : D ⥤ C) :=
(left : C ⥤ D)
(adj : left ⊣ right)
def left_adjoint (R : D ⥤ C) [is_right_adjoint R] : C ⥤ D :=
is_right_adjoint.left R
def right_adjoint (L : C ⥤ D) [is_left_adjoint L] : D ⥤ C :=
is_left_adjoint.right L
namespace adjunction
restate_axiom hom_equiv_unit'
restate_axiom hom_equiv_counit'
attribute [simp, priority 10] hom_equiv_unit hom_equiv_counit
section
variables {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) {X' X : C} {Y Y' : D}
@[simp, priority 10] lemma hom_equiv_naturality_left_symm (f : X' ⟶ X) (g : X ⟶ G.obj Y) :
(adj.hom_equiv X' Y).symm (f ≫ g) = F.map f ≫ (adj.hom_equiv X Y).symm g :=
by rw [hom_equiv_counit, F.map_comp, assoc, adj.hom_equiv_counit.symm]
@[simp] lemma hom_equiv_naturality_left (f : X' ⟶ X) (g : F.obj X ⟶ Y) :
(adj.hom_equiv X' Y) (F.map f ≫ g) = f ≫ (adj.hom_equiv X Y) g :=
by rw [← equiv.eq_symm_apply]; simp [-hom_equiv_unit]
@[simp, priority 10] lemma hom_equiv_naturality_right (f : F.obj X ⟶ Y) (g : Y ⟶ Y') :
(adj.hom_equiv X Y') (f ≫ g) = (adj.hom_equiv X Y) f ≫ G.map g :=
by rw [hom_equiv_unit, G.map_comp, ← assoc, ←hom_equiv_unit]
@[simp] lemma hom_equiv_naturality_right_symm (f : X ⟶ G.obj Y) (g : Y ⟶ Y') :
(adj.hom_equiv X Y').symm (f ≫ G.map g) = (adj.hom_equiv X Y).symm f ≫ g :=
by rw [equiv.symm_apply_eq]; simp [-hom_equiv_counit]
@[simp] lemma left_triangle :
(whisker_right adj.unit F) ≫ (whisker_left F adj.counit) = nat_trans.id _ :=
begin
ext, dsimp,
erw [← adj.hom_equiv_counit, equiv.symm_apply_eq, adj.hom_equiv_unit],
simp
end
@[simp] lemma right_triangle :
(whisker_left G adj.unit) ≫ (whisker_right adj.counit G) = nat_trans.id _ :=
begin
ext, dsimp,
erw [← adj.hom_equiv_unit, ← equiv.eq_symm_apply, adj.hom_equiv_counit],
simp
end
@[simp, reassoc] lemma left_triangle_components :
F.map (adj.unit.app X) ≫ adj.counit.app (F.obj X) = 𝟙 (F.obj X) :=
congr_arg (λ (t : nat_trans _ (𝟭 C ⋙ F)), t.app X) adj.left_triangle
@[simp, reassoc] lemma right_triangle_components {Y : D} :
adj.unit.app (G.obj Y) ≫ G.map (adj.counit.app Y) = 𝟙 (G.obj Y) :=
congr_arg (λ (t : nat_trans _ (G ⋙ 𝟭 C)), t.app Y) adj.right_triangle
@[simp, reassoc] lemma counit_naturality {X Y : D} (f : X ⟶ Y) :
F.map (G.map f) ≫ (adj.counit).app Y = (adj.counit).app X ≫ f :=
adj.counit.naturality f
@[simp, reassoc] lemma unit_naturality {X Y : C} (f : X ⟶ Y) :
(adj.unit).app X ≫ G.map (F.map f) = f ≫ (adj.unit).app Y :=
(adj.unit.naturality f).symm
end
end adjunction
namespace adjunction
structure core_hom_equiv (F : C ⥤ D) (G : D ⥤ C) :=
(hom_equiv : Π (X Y), (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y))
(hom_equiv_naturality_left_symm' : Π {X' X Y} (f : X' ⟶ X) (g : X ⟶ G.obj Y),
(hom_equiv X' Y).symm (f ≫ g) = F.map f ≫ (hom_equiv X Y).symm g . obviously)
(hom_equiv_naturality_right' : Π {X Y Y'} (f : F.obj X ⟶ Y) (g : Y ⟶ Y'),
(hom_equiv X Y') (f ≫ g) = (hom_equiv X Y) f ≫ G.map g . obviously)
namespace core_hom_equiv
restate_axiom hom_equiv_naturality_left_symm'
restate_axiom hom_equiv_naturality_right'
attribute [simp, priority 10] hom_equiv_naturality_left_symm hom_equiv_naturality_right
variables {F : C ⥤ D} {G : D ⥤ C} (adj : core_hom_equiv F G) {X' X : C} {Y Y' : D}
@[simp] lemma hom_equiv_naturality_left (f : X' ⟶ X) (g : F.obj X ⟶ Y) :
(adj.hom_equiv X' Y) (F.map f ≫ g) = f ≫ (adj.hom_equiv X Y) g :=
by rw [← equiv.eq_symm_apply]; simp
@[simp] lemma hom_equiv_naturality_right_symm (f : X ⟶ G.obj Y) (g : Y ⟶ Y') :
(adj.hom_equiv X Y').symm (f ≫ G.map g) = (adj.hom_equiv X Y).symm f ≫ g :=
by rw [equiv.symm_apply_eq]; simp
end core_hom_equiv
structure core_unit_counit (F : C ⥤ D) (G : D ⥤ C) :=
(unit : 𝟭 C ⟶ F.comp G)
(counit : G.comp F ⟶ 𝟭 D)
(left_triangle' : whisker_right unit F ≫ (functor.associator F G F).hom ≫ whisker_left F counit = nat_trans.id (𝟭 C ⋙ F) . obviously)
(right_triangle' : whisker_left G unit ≫ (functor.associator G F G).inv ≫ whisker_right counit G = nat_trans.id (G ⋙ 𝟭 C) . obviously)
namespace core_unit_counit
restate_axiom left_triangle'
restate_axiom right_triangle'
attribute [simp] left_triangle right_triangle
end core_unit_counit
variables {F : C ⥤ D} {G : D ⥤ C}
def mk_of_hom_equiv (adj : core_hom_equiv F G) : F ⊣ G :=
{ unit :=
{ app := λ X, (adj.hom_equiv X (F.obj X)) (𝟙 (F.obj X)),
naturality' :=
begin
intros,
erw [← adj.hom_equiv_naturality_left, ← adj.hom_equiv_naturality_right],
dsimp, simp
end },
counit :=
{ app := λ Y, (adj.hom_equiv _ _).inv_fun (𝟙 (G.obj Y)),
naturality' :=
begin
intros,
erw [← adj.hom_equiv_naturality_left_symm, ← adj.hom_equiv_naturality_right_symm],
dsimp, simp
end },
hom_equiv_unit' := λ X Y f, by erw [← adj.hom_equiv_naturality_right]; simp,
hom_equiv_counit' := λ X Y f, by erw [← adj.hom_equiv_naturality_left_symm]; simp,
.. adj }
def mk_of_unit_counit (adj : core_unit_counit F G) : F ⊣ G :=
{ hom_equiv := λ X Y,
{ to_fun := λ f, adj.unit.app X ≫ G.map f,
inv_fun := λ g, F.map g ≫ adj.counit.app Y,
left_inv := λ f, begin
change F.map (_ ≫ _) ≫ _ = _,
rw [F.map_comp, assoc, ←functor.comp_map, adj.counit.naturality, ←assoc],
convert id_comp _ f,
have t := congr_arg (λ t : nat_trans _ _, t.app _) adj.left_triangle,
dsimp at t,
simp only [id_comp] at t,
exact t,
end,
right_inv := λ g, begin
change _ ≫ G.map (_ ≫ _) = _,
rw [G.map_comp, ←assoc, ←functor.comp_map, ←adj.unit.naturality, assoc],
convert comp_id _ g,
have t := congr_arg (λ t : nat_trans _ _, t.app _) adj.right_triangle,
dsimp at t,
simp only [id_comp] at t,
exact t,
end },
.. adj }
section
omit 𝒟
def id : 𝟭 C ⊣ 𝟭 C :=
{ hom_equiv := λ X Y, equiv.refl _,
unit := 𝟙 _,
counit := 𝟙 _ }
end
section
variables {E : Type u₃} [ℰ : category.{v₃} E] (H : D ⥤ E) (I : E ⥤ D)
def comp (adj₁ : F ⊣ G) (adj₂ : H ⊣ I) : F ⋙ H ⊣ I ⋙ G :=
{ hom_equiv := λ X Z, equiv.trans (adj₂.hom_equiv _ _) (adj₁.hom_equiv _ _),
unit := adj₁.unit ≫
(whisker_left F $ whisker_right adj₂.unit G) ≫ (functor.associator _ _ _).inv,
counit := (functor.associator _ _ _).hom ≫
(whisker_left I $ whisker_right adj₁.counit H) ≫ adj₂.counit }
end
section construct_left
-- Construction of a left adjoint. In order to construct a left
-- adjoint to a functor G : D → C, it suffices to give the object part
-- of a functor F : C → D together with isomorphisms Hom(FX, Y) ≃
-- Hom(X, GY) natural in Y. The action of F on morphisms can be
-- constructed from this data.
variables {F_obj : C → D} {G}
variables (e : Π X Y, (F_obj X ⟶ Y) ≃ (X ⟶ G.obj Y))
variables (he : Π X Y Y' g h, e X Y' (h ≫ g) = e X Y h ≫ G.map g)
include he
private lemma he' {X Y Y'} (f g) : (e X Y').symm (f ≫ G.map g) = (e X Y).symm f ≫ g :=
by intros; rw [equiv.symm_apply_eq, he]; simp
def left_adjoint_of_equiv : C ⥤ D :=
{ obj := F_obj,
map := λ X X' f, (e X (F_obj X')).symm (f ≫ e X' (F_obj X') (𝟙 _)),
map_comp' := λ X X' X'' f f', begin
rw [equiv.symm_apply_eq, he, equiv.apply_symm_apply],
conv { to_rhs, rw [assoc, ←he, id_comp, equiv.apply_symm_apply] },
simp
end }
def adjunction_of_equiv_left : left_adjoint_of_equiv e he ⊣ G :=
mk_of_hom_equiv
{ hom_equiv := e,
hom_equiv_naturality_left_symm' :=
begin
intros,
erw [← he' e he, ← equiv.apply_eq_iff_eq],
simp [(he _ _ _ _ _).symm]
end }
end construct_left
section construct_right
-- Construction of a right adjoint, analogous to the above.
variables {F} {G_obj : D → C}
variables (e : Π X Y, (F.obj X ⟶ Y) ≃ (X ⟶ G_obj Y))
variables (he : Π X' X Y f g, e X' Y (F.map f ≫ g) = f ≫ e X Y g)
include he
private lemma he' {X' X Y} (f g) : F.map f ≫ (e X Y).symm g = (e X' Y).symm (f ≫ g) :=
by intros; rw [equiv.eq_symm_apply, he]; simp
def right_adjoint_of_equiv : D ⥤ C :=
{ obj := G_obj,
map := λ Y Y' g, (e (G_obj Y) Y') ((e (G_obj Y) Y).symm (𝟙 _) ≫ g),
map_comp' := λ Y Y' Y'' g g', begin
rw [← equiv.eq_symm_apply, ← he' e he, equiv.symm_apply_apply],
conv { to_rhs, rw [← assoc, he' e he, comp_id, equiv.symm_apply_apply] },
simp
end }
def adjunction_of_equiv_right : F ⊣ right_adjoint_of_equiv e he :=
mk_of_hom_equiv
{ hom_equiv := e,
hom_equiv_naturality_left_symm' := by intros; rw [equiv.symm_apply_eq, he]; simp,
hom_equiv_naturality_right' :=
begin
intros X Y Y' g h,
erw [←he, equiv.apply_eq_iff_eq, ←assoc, he' e he, comp_id, equiv.symm_apply_apply]
end }
end construct_right
end adjunction
open adjunction
namespace equivalence
def to_adjunction (e : C ≌ D) : e.functor ⊣ e.inverse :=
mk_of_unit_counit ⟨e.unit, e.counit,
by { ext, dsimp, simp only [id_comp], exact e.functor_unit_comp _, },
by { ext, dsimp, simp only [id_comp], exact e.unit_inverse_comp _, }⟩
end equivalence
namespace functor
def adjunction (E : C ⥤ D) [is_equivalence E] : E ⊣ E.inv :=
(E.as_equivalence).to_adjunction
end functor
end category_theory
|
449b11a94b70e0fcb2461ccbc622ec005762c324 | c3f2fcd060adfa2ca29f924839d2d925e8f2c685 | /library/algebra/order.lean | 95f7b9ad98e8e30877e5c31ba8c7bebcb34a9944 | [
"Apache-2.0"
] | permissive | respu/lean | 6582d19a2f2838a28ecd2b3c6f81c32d07b5341d | 8c76419c60b63d0d9f7bc04ebb0b99812d0ec654 | refs/heads/master | 1,610,882,451,231 | 1,427,747,084,000 | 1,427,747,429,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 13,819 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: algebra.order
Author: Jeremy Avigad
Various types of orders. We develop weak orders "≤" and strict orders "<" separately. We also
consider structures with both, where the two are related by
x < y ↔ (x ≤ y ∧ x ≠ y) (order_pair)
x ≤ y ↔ (x < y ∨ x = y) (strong_order_pair)
These might not hold constructively in some applications, but we can define additional structures
with both < and ≤ as needed.
-/
import logic.eq logic.connectives
open eq eq.ops
namespace algebra
variable {A : Type}
/- overloaded symbols -/
structure has_le [class] (A : Type) :=
(le : A → A → Prop)
structure has_lt [class] (A : Type) :=
(lt : A → A → Prop)
infixl `<=` := has_le.le
infixl `≤` := has_le.le
infixl `<` := has_lt.lt
definition has_le.ge [reducible] {A : Type} [s : has_le A] (a b : A) := b ≤ a
notation a ≥ b := has_le.ge a b
notation a >= b := has_le.ge a b
definition has_lt.gt [reducible] {A : Type} [s : has_lt A] (a b : A) := b < a
notation a > b := has_lt.gt a b
theorem le_of_eq_of_le {A : Type} [s : has_le A] {a b c : A} (H1 : a = b) (H2 : b ≤ c) : a ≤ c :=
H1⁻¹ ▸ H2
theorem le_of_le_of_eq {A : Type} [s : has_le A] {a b c : A} (H1 : a ≤ b) (H2 : b = c) : a ≤ c :=
H2 ▸ H1
theorem lt_of_eq_of_lt {A : Type} [s : has_lt A] {a b c : A} (H1 : a = b) (H2 : b < c) : a < c :=
H1⁻¹ ▸ H2
theorem lt_of_lt_of_eq {A : Type} [s : has_lt A] {a b c : A} (H1 : a < b) (H2 : b = c) : a < c :=
H2 ▸ H1
theorem ge_of_eq_of_ge {A : Type} [s : has_le A] {a b c : A} (H1 : a = b) (H2 : b ≥ c) : a ≥ c :=
H1⁻¹ ▸ H2
theorem ge_of_ge_of_eq {A : Type} [s : has_le A] {a b c : A} (H1 : a ≥ b) (H2 : b = c) : a ≥ c :=
H2 ▸ H1
theorem gt_of_eq_of_gt {A : Type} [s : has_lt A] {a b c : A} (H1 : a = b) (H2 : b > c) : a > c :=
H1⁻¹ ▸ H2
theorem gt_of_gt_of_eq {A : Type} [s : has_lt A] {a b c : A} (H1 : a > b) (H2 : b = c) : a > c :=
H2 ▸ H1
calc_trans le_of_eq_of_le
calc_trans le_of_le_of_eq
calc_trans lt_of_eq_of_lt
calc_trans lt_of_lt_of_eq
calc_trans ge_of_eq_of_ge
calc_trans ge_of_ge_of_eq
calc_trans gt_of_eq_of_gt
calc_trans gt_of_gt_of_eq
/- weak orders -/
structure weak_order [class] (A : Type) extends has_le A :=
(le_refl : ∀a, le a a)
(le_trans : ∀a b c, le a b → le b c → le a c)
(le_antisymm : ∀a b, le a b → le b a → a = b)
section
variable [s : weak_order A]
include s
theorem le.refl (a : A) : a ≤ a := !weak_order.le_refl
theorem le.trans {a b c : A} : a ≤ b → b ≤ c → a ≤ c := !weak_order.le_trans
calc_trans le.trans
theorem ge.trans {a b c : A} (H1 : a ≥ b) (H2: b ≥ c) : a ≥ c := le.trans H2 H1
calc_trans ge.trans
theorem le.antisymm {a b : A} : a ≤ b → b ≤ a → a = b := !weak_order.le_antisymm
end
structure linear_weak_order [class] (A : Type) extends weak_order A :=
(le_total : ∀a b, le a b ∨ le b a)
theorem le.total [s : linear_weak_order A] (a b : A) : a ≤ b ∨ b ≤ a :=
!linear_weak_order.le_total
/- strict orders -/
structure strict_order [class] (A : Type) extends has_lt A :=
(lt_irrefl : ∀a, ¬ lt a a)
(lt_trans : ∀a b c, lt a b → lt b c → lt a c)
section
variable [s : strict_order A]
include s
theorem lt.irrefl (a : A) : ¬ a < a := !strict_order.lt_irrefl
theorem lt.trans {a b c : A} : a < b → b < c → a < c := !strict_order.lt_trans
calc_trans lt.trans
theorem gt.trans {a b c : A} (H1 : a > b) (H2: b > c) : a > c := lt.trans H2 H1
calc_trans gt.trans
theorem ne_of_lt {a b : A} : a < b → a ≠ b :=
assume lt_ab : a < b, assume eq_ab : a = b,
show false, from lt.irrefl b (eq_ab ▸ lt_ab)
theorem lt.asymm {a b : A} (H : a < b) : ¬ b < a :=
assume H1 : b < a, lt.irrefl _ (lt.trans H H1)
end
/- well-founded orders -/
-- TODO: do these duplicate what Leo has done? if so, eliminate
structure wf_strict_order [class] (A : Type) extends strict_order A :=
(wf_rec : ∀P : A → Type, (∀x, (∀y, lt y x → P y) → P x) → ∀x, P x)
definition wf.rec_on {A : Type} [s : wf_strict_order A] {P : A → Type}
(x : A) (H : ∀x, (∀y, wf_strict_order.lt y x → P y) → P x) : P x :=
wf_strict_order.wf_rec P H x
theorem wf.ind_on.{u v} {A : Type.{u}} [s : wf_strict_order.{u 0} A] {P : A → Prop}
(x : A) (H : ∀x, (∀y, wf_strict_order.lt y x → P y) → P x) : P x :=
wf.rec_on x H
/- structures with a weak and a strict order -/
structure order_pair [class] (A : Type) extends weak_order A, has_lt A :=
(lt_iff_le_ne : ∀a b, lt a b ↔ (le a b ∧ a ≠ b))
section
variable [s : order_pair A]
variables {a b c : A}
include s
theorem lt_iff_le_and_ne : a < b ↔ (a ≤ b ∧ a ≠ b) :=
!order_pair.lt_iff_le_ne
theorem le_of_lt (H : a < b) : a ≤ b :=
and.elim_left (iff.mp lt_iff_le_and_ne H)
theorem lt_of_le_of_ne (H1 : a ≤ b) (H2 : a ≠ b) : a < b :=
iff.mp (iff.symm lt_iff_le_and_ne) (and.intro H1 H2)
private theorem lt_irrefl (s' : order_pair A) (a : A) : ¬ a < a :=
assume H : a < a,
have H1 : a ≠ a, from and.elim_right (iff.mp !lt_iff_le_and_ne H),
H1 rfl
private theorem lt_trans (s' : order_pair A) (a b c: A) (lt_ab : a < b) (lt_bc : b < c) : a < c :=
have le_ab : a ≤ b, from le_of_lt lt_ab,
have le_bc : b ≤ c, from le_of_lt lt_bc,
have le_ac : a ≤ c, from le.trans le_ab le_bc,
have ne_ac : a ≠ c, from
assume eq_ac : a = c,
have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
have eq_ab : a = b, from le.antisymm le_ab le_ba,
have ne_ab : a ≠ b, from and.elim_right (iff.mp lt_iff_le_and_ne lt_ab),
ne_ab eq_ab,
show a < c, from lt_of_le_of_ne le_ac ne_ac
definition order_pair.to_strict_order [instance] [coercion] [reducible] : strict_order A :=
⦃ strict_order, s, lt_irrefl := lt_irrefl s, lt_trans := lt_trans s ⦄
theorem lt_of_lt_of_le : a < b → b ≤ c → a < c :=
assume lt_ab : a < b,
assume le_bc : b ≤ c,
have le_ac : a ≤ c, from le.trans (le_of_lt lt_ab) le_bc,
have ne_ac : a ≠ c, from
assume eq_ac : a = c,
have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
have eq_ab : a = b, from le.antisymm (le_of_lt lt_ab) le_ba,
show false, from ne_of_lt lt_ab eq_ab,
show a < c, from lt_of_le_of_ne le_ac ne_ac
theorem lt_of_le_of_lt : a ≤ b → b < c → a < c :=
assume le_ab : a ≤ b,
assume lt_bc : b < c,
have le_ac : a ≤ c, from le.trans le_ab (le_of_lt lt_bc),
have ne_ac : a ≠ c, from
assume eq_ac : a = c,
have le_cb : c ≤ b, from eq_ac ▸ le_ab,
have eq_bc : b = c, from le.antisymm (le_of_lt lt_bc) le_cb,
show false, from ne_of_lt lt_bc eq_bc,
show a < c, from lt_of_le_of_ne le_ac ne_ac
theorem gt_of_gt_of_ge (H1 : a > b) (H2 : b ≥ c) : a > c := lt_of_le_of_lt H2 H1
theorem gt_of_ge_of_gt (H1 : a ≥ b) (H2 : b > c) : a > c := lt_of_lt_of_le H2 H1
calc_trans lt_of_lt_of_le
calc_trans lt_of_le_of_lt
calc_trans gt_of_gt_of_ge
calc_trans gt_of_ge_of_gt
theorem not_le_of_lt (H : a < b) : ¬ b ≤ a :=
assume H1 : b ≤ a,
lt.irrefl _ (lt_of_lt_of_le H H1)
theorem not_lt_of_le (H : a ≤ b) : ¬ b < a :=
assume H1 : b < a,
lt.irrefl _ (lt_of_le_of_lt H H1)
end
structure strong_order_pair [class] (A : Type) extends order_pair A :=
(le_iff_lt_or_eq : ∀a b, le a b ↔ lt a b ∨ a = b)
theorem le_iff_lt_or_eq [s : strong_order_pair A] {a b : A} : a ≤ b ↔ a < b ∨ a = b :=
!strong_order_pair.le_iff_lt_or_eq
theorem lt_or_eq_of_le [s : strong_order_pair A] {a b : A} (le_ab : a ≤ b) : a < b ∨ a = b :=
iff.mp le_iff_lt_or_eq le_ab
-- We can also construct a strong order pair by defining a strict order, and then defining
-- x ≤ y ↔ x < y ∨ x = y
structure strict_order_with_le [class] (A : Type) extends strict_order A, has_le A :=
(le_iff_lt_or_eq : ∀a b, le a b ↔ lt a b ∨ a = b)
private theorem le_refl (s : strict_order_with_le A) (a : A) : a ≤ a :=
iff.mp (iff.symm !strict_order_with_le.le_iff_lt_or_eq) (or.intro_right _ rfl)
private theorem le_trans (s : strict_order_with_le A) (a b c : A) (le_ab : a ≤ b) (le_bc : b ≤ c) : a ≤ c :=
or.elim (iff.mp !strict_order_with_le.le_iff_lt_or_eq le_ab)
(assume lt_ab : a < b,
or.elim (iff.mp !strict_order_with_le.le_iff_lt_or_eq le_bc)
(assume lt_bc : b < c,
iff.elim_right
!strict_order_with_le.le_iff_lt_or_eq (or.intro_left _ (lt.trans lt_ab lt_bc)))
(assume eq_bc : b = c, eq_bc ▸ le_ab))
(assume eq_ab : a = b,
eq_ab⁻¹ ▸ le_bc)
private theorem le_antisymm (s : strict_order_with_le A) (a b : A) (le_ab : a ≤ b) (le_ba : b ≤ a) : a = b :=
or.elim (iff.mp !strict_order_with_le.le_iff_lt_or_eq le_ab)
(assume lt_ab : a < b,
or.elim (iff.mp !strict_order_with_le.le_iff_lt_or_eq le_ba)
(assume lt_ba : b < a, absurd (lt.trans lt_ab lt_ba) (lt.irrefl a))
(assume eq_ba : b = a, eq_ba⁻¹))
(assume eq_ab : a = b, eq_ab)
private theorem lt_iff_le_ne (s : strict_order_with_le A) (a b : A) : a < b ↔ a ≤ b ∧ a ≠ b :=
iff.intro
(assume lt_ab : a < b,
have le_ab : a ≤ b, from
iff.elim_right !strict_order_with_le.le_iff_lt_or_eq (or.intro_left _ lt_ab),
show a ≤ b ∧ a ≠ b, from and.intro le_ab (ne_of_lt lt_ab))
(assume H : a ≤ b ∧ a ≠ b,
have H1 : a < b ∨ a = b, from
iff.mp !strict_order_with_le.le_iff_lt_or_eq (and.elim_left H),
show a < b, from or_resolve_left H1 (and.elim_right H))
definition strict_order_with_le.to_order_pair [instance] [coercion] [reducible] [s : strict_order_with_le A] :
strong_order_pair A :=
⦃ strong_order_pair, s,
le_refl := le_refl s,
le_trans := le_trans s,
le_antisymm := le_antisymm s,
lt_iff_le_ne := lt_iff_le_ne s ⦄
/- linear orders -/
structure linear_order_pair [class] (A : Type) extends order_pair A, linear_weak_order A
structure linear_strong_order_pair [class] (A : Type) extends strong_order_pair A,
linear_weak_order A
section
variable [s : linear_strong_order_pair A]
variables (a b c : A)
include s
theorem lt.trichotomy : a < b ∨ a = b ∨ b < a :=
or.elim (le.total a b)
(assume H : a ≤ b,
or.elim (iff.mp !le_iff_lt_or_eq H) (assume H1, or.inl H1) (assume H1, or.inr (or.inl H1)))
(assume H : b ≤ a,
or.elim (iff.mp !le_iff_lt_or_eq H)
(assume H1, or.inr (or.inr H1))
(assume H1, or.inr (or.inl (H1⁻¹))))
theorem lt.by_cases {a b : A} {P : Prop}
(H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
or.elim !lt.trichotomy
(assume H, H1 H)
(assume H, or.elim H (assume H', H2 H') (assume H', H3 H'))
definition linear_strong_order_pair.to_linear_order_pair [instance] [coercion] [reducible]
: linear_order_pair A :=
⦃ linear_order_pair, s ⦄
theorem le_of_not_lt {a b : A} (H : ¬ a < b) : b ≤ a :=
lt.by_cases (assume H', absurd H' H) (assume H', H' ▸ !le.refl) (assume H', le_of_lt H')
theorem lt_of_not_le {a b : A} (H : ¬ a ≤ b) : b < a :=
lt.by_cases
(assume H', absurd (le_of_lt H') H)
(assume H', absurd (H' ▸ !le.refl) H)
(assume H', H')
theorem lt_or_ge : a < b ∨ a ≥ b :=
lt.by_cases
(assume H1 : a < b, or.inl H1)
(assume H1 : a = b, or.inr (H1 ▸ le.refl a))
(assume H1 : a > b, or.inr (le_of_lt H1))
theorem le_or_gt : a ≤ b ∨ a > b :=
!or.swap (lt_or_ge b a)
theorem lt_or_gt_of_ne {a b : A} (H : a ≠ b) : a < b ∨ a > b :=
lt.by_cases (assume H1, or.inl H1) (assume H1, absurd H1 H) (assume H1, or.inr H1)
end
structure decidable_linear_order [class] (A : Type) extends linear_strong_order_pair A :=
(decidable_lt : decidable_rel lt)
section
variable [s : decidable_linear_order A]
variables {a b c d : A}
include s
open decidable
definition decidable_lt [instance] : decidable (a < b) :=
@decidable_linear_order.decidable_lt _ _ _ _
definition decidable_le [instance] : decidable (a ≤ b) :=
by_cases
(assume H : a < b, inl (le_of_lt H))
(assume H : ¬ a < b,
have H1 : b ≤ a, from le_of_not_lt H,
by_cases
(assume H2 : b < a, inr (not_le_of_lt H2))
(assume H2 : ¬ b < a, inl (le_of_not_lt H2)))
definition decidable_eq [instance] : decidable (a = b) :=
by_cases
(assume H : a ≤ b,
by_cases
(assume H1 : b ≤ a, inl (le.antisymm H H1))
(assume H1 : ¬ b ≤ a, inr (assume H2 : a = b, H1 (H2 ▸ le.refl a))))
(assume H : ¬ a ≤ b,
(inr (assume H1 : a = b, H (H1 ▸ !le.refl))))
-- testing equality first may result in more definitional equalities
definition lt.cases {B : Type} (a b : A) (t_lt t_eq t_gt : B) : B :=
if a = b then t_eq else (if a < b then t_lt else t_gt)
theorem lt.cases_of_eq {B : Type} {a b : A} {t_lt t_eq t_gt : B} (H : a = b) :
lt.cases a b t_lt t_eq t_gt = t_eq := if_pos H
theorem lt.cases_of_lt {B : Type} {a b : A} {t_lt t_eq t_gt : B} (H : a < b) :
lt.cases a b t_lt t_eq t_gt = t_lt :=
if_neg (ne_of_lt H) ⬝ if_pos H
theorem lt.cases_of_gt {B : Type} {a b : A} {t_lt t_eq t_gt : B} (H : a > b) :
lt.cases a b t_lt t_eq t_gt = t_gt :=
if_neg (ne.symm (ne_of_lt H)) ⬝ if_neg (lt.asymm H)
end
end algebra
/-
For reference, these are all the transitivity rules defined in this file:
calc_trans le_of_eq_of_le
calc_trans le_of_le_of_eq
calc_trans lt_of_eq_of_lt
calc_trans lt_of_lt_of_eq
calc_trans le.trans
calc_trans lt.trans
calc_trans lt_of_lt_of_le
calc_trans lt_of_le_of_lt
calc_trans ge_of_eq_of_ge
calc_trans ge_of_ge_of_eq
calc_trans gt_of_eq_of_gt
calc_trans gt_of_gt_of_eq
calc_trans ge.trans
calc_trans gt.trans
calc_trans gt_of_gt_of_ge
calc_trans gt_of_ge_of_gt
-/
|
fa6b282aa728da304ca3d8870469369333eeeaca | a0e23cfdd129a671bf3154ee1a8a3a72bf4c7940 | /stage0/src/Lean/Elab/InfoTree.lean | eea75395cb6a1f941e364537e7d5bb62ae0c7168 | [
"Apache-2.0"
] | permissive | WojciechKarpiel/lean4 | 7f89706b8e3c1f942b83a2c91a3a00b05da0e65b | f6e1314fa08293dea66a329e05b6c196a0189163 | refs/heads/master | 1,686,633,402,214 | 1,625,821,189,000 | 1,625,821,258,000 | 384,640,886 | 0 | 0 | Apache-2.0 | 1,625,903,617,000 | 1,625,903,026,000 | null | UTF-8 | Lean | false | false | 15,120 | lean | /-
Copyright (c) 2020 Wojciech Nawrocki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wojciech Nawrocki, Leonardo de Moura, Sebastian Ullrich
-/
import Lean.Data.Position
import Lean.Expr
import Lean.Message
import Lean.Data.Json
import Lean.Meta.Basic
import Lean.Meta.PPGoal
namespace Lean.Elab
open Std (PersistentArray PersistentArray.empty PersistentHashMap)
/- Context after executing `liftTermElabM`.
Note that the term information collected during elaboration may contain metavariables, and their
assignments are stored at `mctx`. -/
structure ContextInfo where
env : Environment
fileMap : FileMap
mctx : MetavarContext := {}
options : Options := {}
currNamespace : Name := Name.anonymous
openDecls : List OpenDecl := []
deriving Inhabited
/-- An elaboration step -/
structure ElabInfo where
elaborator : Name
stx : Syntax
deriving Inhabited
structure TermInfo extends ElabInfo where
lctx : LocalContext -- The local context when the term was elaborated.
expectedType? : Option Expr
expr : Expr
deriving Inhabited
structure CommandInfo extends ElabInfo where
deriving Inhabited
inductive CompletionInfo where
| dot (termInfo : TermInfo) (field? : Option Syntax) (expectedType? : Option Expr)
| id (stx : Syntax) (id : Name) (danglingDot : Bool) (lctx : LocalContext) (expectedType? : Option Expr)
| namespaceId (stx : Syntax)
| option (stx : Syntax)
| endSection (stx : Syntax) (scopeNames : List String)
| tactic (stx : Syntax) (goals : List MVarId)
-- TODO `import`
def CompletionInfo.stx : CompletionInfo → Syntax
| dot i .. => i.stx
| id stx .. => stx
| namespaceId stx => stx
| option stx => stx
| endSection stx .. => stx
| tactic stx .. => stx
structure FieldInfo where
name : Name
lctx : LocalContext
val : Expr
stx : Syntax
deriving Inhabited
/- We store the list of goals before and after the execution of a tactic.
We also store the metavariable context at each time since, we want to unassigned metavariables
at tactic execution time to be displayed as `?m...`. -/
structure TacticInfo extends ElabInfo where
mctxBefore : MetavarContext
goalsBefore : List MVarId
mctxAfter : MetavarContext
goalsAfter : List MVarId
deriving Inhabited
structure MacroExpansionInfo where
lctx : LocalContext -- The local context when the macro was expanded.
stx : Syntax
output : Syntax
deriving Inhabited
inductive Info where
| ofTacticInfo (i : TacticInfo)
| ofTermInfo (i : TermInfo)
| ofCommandInfo (i : CommandInfo)
| ofMacroExpansionInfo (i : MacroExpansionInfo)
| ofFieldInfo (i : FieldInfo)
| ofCompletionInfo (i : CompletionInfo)
deriving Inhabited
inductive InfoTree where
| context (i : ContextInfo) (t : InfoTree) -- The context object is created by `liftTermElabM` at `Command.lean`
| node (i : Info) (children : PersistentArray InfoTree) -- The children contains information for nested term elaboration and tactic evaluation
| ofJson (j : Json) -- For user data
| hole (mvarId : MVarId) -- The elaborator creates holes (aka metavariables) for tactics and postponed terms
deriving Inhabited
partial def InfoTree.findInfo? (p : Info → Bool) (t : InfoTree) : Option InfoTree :=
match t with
| context _ t => findInfo? p t
| node i ts =>
if p i then
some t
else
ts.findSome? (findInfo? p)
| _ => none
structure InfoState where
enabled : Bool := false
assignment : PersistentHashMap MVarId InfoTree := {} -- map from holeId to InfoTree
trees : PersistentArray InfoTree := {}
deriving Inhabited
class MonadInfoTree (m : Type → Type) where
getInfoState : m InfoState
modifyInfoState : (InfoState → InfoState) → m Unit
export MonadInfoTree (getInfoState modifyInfoState)
instance [MonadLift m n] [MonadInfoTree m] : MonadInfoTree n where
getInfoState := liftM (getInfoState : m _)
modifyInfoState f := liftM (modifyInfoState f : m _)
partial def InfoTree.substitute (tree : InfoTree) (assignment : PersistentHashMap MVarId InfoTree) : InfoTree :=
match tree with
| node i c => node i <| c.map (substitute · assignment)
| context i t => context i (substitute t assignment)
| ofJson j => ofJson j
| hole id => match assignment.find? id with
| none => hole id
| some tree => substitute tree assignment
def ContextInfo.runMetaM (info : ContextInfo) (lctx : LocalContext) (x : MetaM α) : IO α := do
let x := x.run { lctx := lctx } { mctx := info.mctx }
let ((a, _), _) ← x.toIO { options := info.options, currNamespace := info.currNamespace, openDecls := info.openDecls } { env := info.env }
return a
def ContextInfo.toPPContext (info : ContextInfo) (lctx : LocalContext) : PPContext :=
{ env := info.env, mctx := info.mctx, lctx := lctx,
opts := info.options, currNamespace := info.currNamespace, openDecls := info.openDecls }
def ContextInfo.ppSyntax (info : ContextInfo) (lctx : LocalContext) (stx : Syntax) : IO Format := do
ppTerm (info.toPPContext lctx) stx
private def formatStxRange (ctx : ContextInfo) (stx : Syntax) : Format := do
let pos := stx.getPos?.getD 0
let endPos := stx.getTailPos?.getD pos
return f!"{fmtPos pos stx.getHeadInfo}-{fmtPos endPos stx.getTailInfo}"
where fmtPos pos info :=
let pos := format <| ctx.fileMap.toPosition pos
match info with
| SourceInfo.original .. => pos
| _ => f!"{pos}†"
private def formatElabInfo (ctx : ContextInfo) (info : ElabInfo) : Format :=
if info.elaborator.isAnonymous then
formatStxRange ctx info.stx
else
f!"{formatStxRange ctx info.stx} @ {info.elaborator}"
def TermInfo.runMetaM (info : TermInfo) (ctx : ContextInfo) (x : MetaM α) : IO α :=
ctx.runMetaM info.lctx x
def TermInfo.format (ctx : ContextInfo) (info : TermInfo) : IO Format := do
info.runMetaM ctx do
try
return f!"{← Meta.ppExpr info.expr} : {← Meta.ppExpr (← Meta.inferType info.expr)} @ {formatElabInfo ctx info.toElabInfo}"
catch _ =>
return f!"{← Meta.ppExpr info.expr} : <failed-to-infer-type> @ {formatElabInfo ctx info.toElabInfo}"
def CompletionInfo.format (ctx : ContextInfo) (info : CompletionInfo) : IO Format :=
match info with
| CompletionInfo.dot i (expectedType? := expectedType?) .. => return f!"[.] {← i.format ctx} : {expectedType?}"
| CompletionInfo.id stx _ _ lctx expectedType? => ctx.runMetaM lctx do return f!"[.] {stx} : {expectedType?} @ {formatStxRange ctx info.stx}"
| _ => return f!"[.] {info.stx} @ {formatStxRange ctx info.stx}"
def CommandInfo.format (ctx : ContextInfo) (info : CommandInfo) : IO Format := do
return f!"command @ {formatElabInfo ctx info.toElabInfo}"
def FieldInfo.format (ctx : ContextInfo) (info : FieldInfo) : IO Format := do
ctx.runMetaM info.lctx do
return f!"{info.name} : {← Meta.ppExpr (← Meta.inferType info.val)} := {← Meta.ppExpr info.val} @ {formatStxRange ctx info.stx}"
def ContextInfo.ppGoals (ctx : ContextInfo) (goals : List MVarId) : IO Format :=
if goals.isEmpty then
return "no goals"
else
ctx.runMetaM {} (return Std.Format.prefixJoin "\n" (← goals.mapM Meta.ppGoal))
def TacticInfo.format (ctx : ContextInfo) (info : TacticInfo) : IO Format := do
let ctxB := { ctx with mctx := info.mctxBefore }
let ctxA := { ctx with mctx := info.mctxAfter }
let goalsBefore ← ctxB.ppGoals info.goalsBefore
let goalsAfter ← ctxA.ppGoals info.goalsAfter
return f!"Tactic @ {formatElabInfo ctx info.toElabInfo}\n{info.stx}\nbefore {goalsBefore}\nafter {goalsAfter}"
def MacroExpansionInfo.format (ctx : ContextInfo) (info : MacroExpansionInfo) : IO Format := do
let stx ← ctx.ppSyntax info.lctx info.stx
let output ← ctx.ppSyntax info.lctx info.output
return f!"Macro expansion\n{stx}\n===>\n{output}"
def Info.format (ctx : ContextInfo) : Info → IO Format
| ofTacticInfo i => i.format ctx
| ofTermInfo i => i.format ctx
| ofCommandInfo i => i.format ctx
| ofMacroExpansionInfo i => i.format ctx
| ofFieldInfo i => i.format ctx
| ofCompletionInfo i => i.format ctx
def Info.toElabInfo? : Info → Option ElabInfo
| ofTacticInfo i => some i.toElabInfo
| ofTermInfo i => some i.toElabInfo
| ofCommandInfo i => some i.toElabInfo
| ofMacroExpansionInfo i => none
| ofFieldInfo i => none
| ofCompletionInfo i => none
/--
Helper function for propagating the tactic metavariable context to its children nodes.
We need this function because we preserve `TacticInfo` nodes during backtracking *and* their
children. Moreover, we backtrack the metavariable context to undo metavariable assignments.
`TacticInfo` nodes save the metavariable context before/after the tactic application, and
can be pretty printed without any extra information. This is not the case for `TermInfo` nodes.
Without this function, the formatting method would often fail when processing `TermInfo` nodes
that are children of `TacticInfo` nodes that have been preserved during backtracking.
Saving the metavariable context at `TermInfo` nodes is also not a good option because
at `TermInfo` creation time, the metavariable context often miss information, e.g.,
a TC problem has not been resolved, a postponed subterm has not been elaborated, etc.
See `Term.SavedState.restore`.
-/
def Info.updateContext? : Option ContextInfo → Info → Option ContextInfo
| some ctx, ofTacticInfo i => some { ctx with mctx := i.mctxAfter }
| ctx?, _ => ctx?
partial def InfoTree.format (tree : InfoTree) (ctx? : Option ContextInfo := none) : IO Format := do
match tree with
| ofJson j => return toString j
| hole id => return toString id
| context i t => format t i
| node i cs => match ctx? with
| none => return "<context-not-available>"
| some ctx =>
let fmt ← i.format ctx
if cs.size == 0 then
return fmt
else
let ctx? := i.updateContext? ctx?
return f!"{fmt}{Std.Format.nestD <| Std.Format.prefixJoin "\n" (← cs.toList.mapM fun c => format c ctx?)}"
section
variable [Monad m] [MonadInfoTree m]
@[inline] private def modifyInfoTrees (f : PersistentArray InfoTree → PersistentArray InfoTree) : m Unit :=
modifyInfoState fun s => { s with trees := f s.trees }
def getResetInfoTrees : m (PersistentArray InfoTree) := do
let trees := (← getInfoState).trees
modifyInfoTrees fun _ => {}
return trees
def pushInfoTree (t : InfoTree) : m Unit := do
if (← getInfoState).enabled then
modifyInfoTrees fun ts => ts.push t
def pushInfoLeaf (t : Info) : m Unit := do
if (← getInfoState).enabled then
pushInfoTree <| InfoTree.node (children := {}) t
def addCompletionInfo (info : CompletionInfo) : m Unit := do
pushInfoLeaf <| Info.ofCompletionInfo info
def resolveGlobalConstNoOverloadWithInfo [MonadResolveName m] [MonadEnv m] [MonadError m] (stx : Syntax) (id := stx.getId) (expectedType? : Option Expr := none) : m Name := do
let n ← resolveGlobalConstNoOverload id
if (← getInfoState).enabled then
-- we do not store a specific elaborator since identifiers are special-cased by the server anyway
pushInfoLeaf <| Info.ofTermInfo { elaborator := Name.anonymous, lctx := LocalContext.empty, expr := (← mkConstWithLevelParams n), stx, expectedType? }
return n
def resolveGlobalConstWithInfos [MonadResolveName m] [MonadEnv m] [MonadError m] (stx : Syntax) (id := stx.getId) (expectedType? : Option Expr := none) : m (List Name) := do
let ns ← resolveGlobalConst id
if (← getInfoState).enabled then
for n in ns do
pushInfoLeaf <| Info.ofTermInfo { elaborator := Name.anonymous, lctx := LocalContext.empty, expr := (← mkConstWithLevelParams n), stx, expectedType? }
return ns
def withInfoContext' [MonadFinally m] (x : m α) (mkInfo : α → m (Sum Info MVarId)) : m α := do
if (← getInfoState).enabled then
let treesSaved ← getResetInfoTrees
Prod.fst <$> MonadFinally.tryFinally' x fun a? => do
match a? with
| none => modifyInfoTrees fun _ => treesSaved
| some a =>
let info ← mkInfo a
modifyInfoTrees fun trees =>
match info with
| Sum.inl info => treesSaved.push <| InfoTree.node info trees
| Sum.inr mvaId => treesSaved.push <| InfoTree.hole mvaId
else
x
def withInfoTreeContext [MonadFinally m] (x : m α) (mkInfoTree : PersistentArray InfoTree → m InfoTree) : m α := do
if (← getInfoState).enabled then
let treesSaved ← getResetInfoTrees
Prod.fst <$> MonadFinally.tryFinally' x fun _ => do
let st ← getInfoState
let tree ← mkInfoTree st.trees
modifyInfoTrees fun _ => treesSaved.push tree
else
x
@[inline] def withInfoContext [MonadFinally m] (x : m α) (mkInfo : m Info) : m α := do
withInfoTreeContext x (fun trees => do return InfoTree.node (← mkInfo) trees)
def withSaveInfoContext [MonadFinally m] [MonadEnv m] [MonadOptions m] [MonadMCtx m] [MonadResolveName m] [MonadFileMap m] (x : m α) : m α := do
if (← getInfoState).enabled then
let treesSaved ← getResetInfoTrees
Prod.fst <$> MonadFinally.tryFinally' x fun _ => do
let st ← getInfoState
let trees ← st.trees.mapM fun tree => do
let tree := tree.substitute st.assignment
InfoTree.context {
env := (← getEnv), fileMap := (← getFileMap), mctx := (← getMCtx), currNamespace := (← getCurrNamespace), openDecls := (← getOpenDecls), options := (← getOptions)
} tree
modifyInfoTrees fun _ => treesSaved ++ trees
else
x
def getInfoHoleIdAssignment? (mvarId : MVarId) : m (Option InfoTree) :=
return (← getInfoState).assignment[mvarId]
def assignInfoHoleId (mvarId : MVarId) (infoTree : InfoTree) : m Unit := do
assert! (← getInfoHoleIdAssignment? mvarId).isNone
modifyInfoState fun s => { s with assignment := s.assignment.insert mvarId infoTree }
end
def withMacroExpansionInfo [MonadFinally m] [Monad m] [MonadInfoTree m] [MonadLCtx m] (stx output : Syntax) (x : m α) : m α :=
let mkInfo : m Info := do
return Info.ofMacroExpansionInfo {
lctx := (← getLCtx)
stx, output
}
withInfoContext x mkInfo
@[inline] def withInfoHole [MonadFinally m] [Monad m] [MonadInfoTree m] (mvarId : MVarId) (x : m α) : m α := do
if (← getInfoState).enabled then
let treesSaved ← getResetInfoTrees
Prod.fst <$> MonadFinally.tryFinally' x fun a? => modifyInfoState fun s =>
if s.trees.size > 0 then
{ s with trees := treesSaved, assignment := s.assignment.insert mvarId s.trees[s.trees.size - 1] }
else
{ s with trees := treesSaved }
else
x
def enableInfoTree [MonadInfoTree m] (flag := true) : m Unit :=
modifyInfoState fun s => { s with enabled := flag }
def getInfoTrees [MonadInfoTree m] [Monad m] : m (PersistentArray InfoTree) :=
return (← getInfoState).trees
end Lean.Elab
|
fda76a5e138d5ee0dcc6cd19fc2e4029f5baae2a | 4d2583807a5ac6caaffd3d7a5f646d61ca85d532 | /src/ring_theory/perfection.lean | 6b35acc55bc5a52bc2ea8fbb36413af5bd42736c | [
"Apache-2.0"
] | permissive | AntoineChambert-Loir/mathlib | 64aabb896129885f12296a799818061bc90da1ff | 07be904260ab6e36a5769680b6012f03a4727134 | refs/heads/master | 1,693,187,631,771 | 1,636,719,886,000 | 1,636,719,886,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 25,338 | lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import algebra.char_p
import algebra.ring.pi
import analysis.special_functions.pow
import field_theory.perfect_closure
import ring_theory.localization
import ring_theory.subring
import ring_theory.valuation.integers
/-!
# Ring Perfection and Tilt
In this file we define the perfection of a ring of characteristic p, and the tilt of a field
given a valuation to `ℝ≥0`.
## TODO
Define the valuation on the tilt, and define a characteristic predicate for the tilt.
-/
universes u₁ u₂ u₃ u₄
open_locale nnreal
/-- The perfection of a monoid `M`, defined to be the projective limit of `M`
using the `p`-th power maps `M → M` indexed by the natural numbers, implemented as
`{ f : ℕ → M | ∀ n, f (n + 1) ^ p = f n }`. -/
def monoid.perfection (M : Type u₁) [comm_monoid M] (p : ℕ) : submonoid (ℕ → M) :=
{ carrier := { f | ∀ n, f (n + 1) ^ p = f n },
one_mem' := λ n, one_pow _,
mul_mem' := λ f g hf hg n, (mul_pow _ _ _).trans $ congr_arg2 _ (hf n) (hg n) }
/-- The perfection of a ring `R` with characteristic `p`, as a subsemiring,
defined to be the projective limit of `R` using the Frobenius maps `R → R`
indexed by the natural numbers, implemented as `{ f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }`. -/
def ring.perfection_subsemiring (R : Type u₁) [comm_semiring R]
(p : ℕ) [hp : fact p.prime] [char_p R p] :
subsemiring (ℕ → R) :=
{ zero_mem' := λ n, zero_pow $ hp.1.pos,
add_mem' := λ f g hf hg n, (frobenius_add R p _ _).trans $ congr_arg2 _ (hf n) (hg n),
.. monoid.perfection R p }
/-- The perfection of a ring `R` with characteristic `p`, as a subring,
defined to be the projective limit of `R` using the Frobenius maps `R → R`
indexed by the natural numbers, implemented as `{ f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }`. -/
def ring.perfection_subring (R : Type u₁) [comm_ring R]
(p : ℕ) [hp : fact p.prime] [char_p R p] :
subring (ℕ → R) :=
(ring.perfection_subsemiring R p).to_subring $ λ n, by simp_rw [← frobenius_def, pi.neg_apply,
pi.one_apply, ring_hom.map_neg, ring_hom.map_one]
/-- The perfection of a ring `R` with characteristic `p`,
defined to be the projective limit of `R` using the Frobenius maps `R → R`
indexed by the natural numbers, implemented as `{f : ℕ → R // ∀ n, f (n + 1) ^ p = f n}`. -/
def ring.perfection (R : Type u₁) [comm_semiring R] (p : ℕ) : Type u₁ :=
{f // ∀ (n : ℕ), (f : ℕ → R) (n + 1) ^ p = f n}
namespace perfection
variables (R : Type u₁) [comm_semiring R] (p : ℕ) [hp : fact p.prime] [char_p R p]
include hp
instance : comm_semiring (ring.perfection R p) :=
(ring.perfection_subsemiring R p).to_comm_semiring
instance : char_p (ring.perfection R p) p :=
char_p.subsemiring (ℕ → R) p (ring.perfection_subsemiring R p)
instance ring (R : Type u₁) [comm_ring R] [char_p R p] : ring (ring.perfection R p) :=
(ring.perfection_subring R p).to_ring
instance comm_ring (R : Type u₁) [comm_ring R] [char_p R p] : comm_ring (ring.perfection R p) :=
(ring.perfection_subring R p).to_comm_ring
instance : inhabited (ring.perfection R p) := ⟨0⟩
/-- The `n`-th coefficient of an element of the perfection. -/
def coeff (n : ℕ) : ring.perfection R p →+* R :=
{ to_fun := λ f, f.1 n,
map_one' := rfl,
map_mul' := λ f g, rfl,
map_zero' := rfl,
map_add' := λ f g, rfl }
variables {R p}
@[ext] lemma ext {f g : ring.perfection R p} (h : ∀ n, coeff R p n f = coeff R p n g) : f = g :=
subtype.eq $ funext h
variables (R p)
/-- The `p`-th root of an element of the perfection. -/
def pth_root : ring.perfection R p →+* ring.perfection R p :=
{ to_fun := λ f, ⟨λ n, coeff R p (n + 1) f, λ n, f.2 _⟩,
map_one' := rfl,
map_mul' := λ f g, rfl,
map_zero' := rfl,
map_add' := λ f g, rfl }
variables {R p}
@[simp] lemma coeff_mk (f : ℕ → R) (hf) (n : ℕ) : coeff R p n ⟨f, hf⟩ = f n := rfl
lemma coeff_pth_root (f : ring.perfection R p) (n : ℕ) :
coeff R p n (pth_root R p f) = coeff R p (n + 1) f :=
rfl
lemma coeff_pow_p (f : ring.perfection R p) (n : ℕ) :
coeff R p (n + 1) (f ^ p) = coeff R p n f :=
by { rw ring_hom.map_pow, exact f.2 n }
lemma coeff_pow_p' (f : ring.perfection R p) (n : ℕ) :
coeff R p (n + 1) f ^ p = coeff R p n f :=
f.2 n
lemma coeff_frobenius (f : ring.perfection R p) (n : ℕ) :
coeff R p (n + 1) (frobenius _ p f) = coeff R p n f :=
by apply coeff_pow_p f n -- `coeff_pow_p f n` also works but is slow!
lemma coeff_iterate_frobenius (f : ring.perfection R p) (n m : ℕ) :
coeff R p (n + m) (frobenius _ p ^[m] f) = coeff R p n f :=
nat.rec_on m rfl $ λ m ih, by erw [function.iterate_succ_apply', coeff_frobenius, ih]
lemma coeff_iterate_frobenius' (f : ring.perfection R p) (n m : ℕ) (hmn : m ≤ n) :
coeff R p n (frobenius _ p ^[m] f) = coeff R p (n - m) f :=
eq.symm $ (coeff_iterate_frobenius _ _ m).symm.trans $ (tsub_add_cancel_of_le hmn).symm ▸ rfl
lemma pth_root_frobenius : (pth_root R p).comp (frobenius _ p) = ring_hom.id _ :=
ring_hom.ext $ λ x, ext $ λ n,
by rw [ring_hom.comp_apply, ring_hom.id_apply, coeff_pth_root, coeff_frobenius]
lemma frobenius_pth_root : (frobenius _ p).comp (pth_root R p) = ring_hom.id _ :=
ring_hom.ext $ λ x, ext $ λ n,
by rw [ring_hom.comp_apply, ring_hom.id_apply, ring_hom.map_frobenius, coeff_pth_root,
← ring_hom.map_frobenius, coeff_frobenius]
lemma coeff_add_ne_zero {f : ring.perfection R p} {n : ℕ} (hfn : coeff R p n f ≠ 0) (k : ℕ) :
coeff R p (n + k) f ≠ 0 :=
nat.rec_on k hfn $ λ k ih h, ih $ by erw [← coeff_pow_p, ring_hom.map_pow, h, zero_pow hp.1.pos]
lemma coeff_ne_zero_of_le {f : ring.perfection R p} {m n : ℕ} (hfm : coeff R p m f ≠ 0)
(hmn : m ≤ n) : coeff R p n f ≠ 0 :=
let ⟨k, hk⟩ := nat.exists_eq_add_of_le hmn in hk.symm ▸ coeff_add_ne_zero hfm k
variables (R p)
instance perfect_ring : perfect_ring (ring.perfection R p) p :=
{ pth_root' := pth_root R p,
frobenius_pth_root' := congr_fun $ congr_arg ring_hom.to_fun $ @frobenius_pth_root R _ p _ _,
pth_root_frobenius' := congr_fun $ congr_arg ring_hom.to_fun $ @pth_root_frobenius R _ p _ _ }
/-- Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* perfection S p`. -/
@[simps] def lift (R : Type u₁) [comm_semiring R] [char_p R p] [perfect_ring R p]
(S : Type u₂) [comm_semiring S] [char_p S p] :
(R →+* S) ≃ (R →+* ring.perfection S p) :=
{ to_fun := λ f,
{ to_fun := λ r, ⟨λ n, f $ _root_.pth_root R p ^[n] r,
λ n, by rw [← f.map_pow, function.iterate_succ_apply', pth_root_pow_p]⟩,
map_one' := ext $ λ n, (congr_arg f $ ring_hom.iterate_map_one _ _).trans f.map_one,
map_mul' := λ x y, ext $ λ n, (congr_arg f $ ring_hom.iterate_map_mul _ _ _ _).trans $
f.map_mul _ _,
map_zero' := ext $ λ n, (congr_arg f $ ring_hom.iterate_map_zero _ _).trans f.map_zero,
map_add' := λ x y, ext $ λ n, (congr_arg f $ ring_hom.iterate_map_add _ _ _ _).trans $
f.map_add _ _ },
inv_fun := ring_hom.comp $ coeff S p 0,
left_inv := λ f, ring_hom.ext $ λ r, rfl,
right_inv := λ f, ring_hom.ext $ λ r, ext $ λ n,
show coeff S p 0 (f (_root_.pth_root R p ^[n] r)) = coeff S p n (f r),
by rw [← coeff_iterate_frobenius _ 0 n, zero_add, ← ring_hom.map_iterate_frobenius,
right_inverse_pth_root_frobenius.iterate] }
lemma hom_ext {R : Type u₁} [comm_semiring R] [char_p R p] [perfect_ring R p]
{S : Type u₂} [comm_semiring S] [char_p S p] {f g : R →+* ring.perfection S p}
(hfg : ∀ x, coeff S p 0 (f x) = coeff S p 0 (g x)) : f = g :=
(lift p R S).symm.injective $ ring_hom.ext hfg
variables {R} {S : Type u₂} [comm_semiring S] [char_p S p]
/-- A ring homomorphism `R →+* S` induces `perfection R p →+* perfection S p` -/
@[simps] def map (φ : R →+* S) : ring.perfection R p →+* ring.perfection S p :=
{ to_fun := λ f, ⟨λ n, φ (coeff R p n f), λ n, by rw [← φ.map_pow, coeff_pow_p']⟩,
map_one' := subtype.eq $ funext $ λ n, φ.map_one,
map_mul' := λ f g, subtype.eq $ funext $ λ n, φ.map_mul _ _,
map_zero' := subtype.eq $ funext $ λ n, φ.map_zero,
map_add' := λ f g, subtype.eq $ funext $ λ n, φ.map_add _ _ }
lemma coeff_map (φ : R →+* S) (f : ring.perfection R p) (n : ℕ) :
coeff S p n (map p φ f) = φ (coeff R p n f) :=
rfl
end perfection
/-- A perfection map to a ring of characteristic `p` is a map that is isomorphic
to its perfection. -/
@[nolint has_inhabited_instance] structure perfection_map (p : ℕ) [fact p.prime]
{R : Type u₁} [comm_semiring R] [char_p R p]
{P : Type u₂} [comm_semiring P] [char_p P p] [perfect_ring P p] (π : P →+* R) : Prop :=
(injective : ∀ ⦃x y : P⦄, (∀ n, π (pth_root P p ^[n] x) = π (pth_root P p ^[n] y)) → x = y)
(surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) →
∃ x : P, ∀ n, π (pth_root P p ^[n] x) = f n)
namespace perfection_map
variables {p : ℕ} [fact p.prime]
variables {R : Type u₁} [comm_semiring R] [char_p R p]
variables {P : Type u₃} [comm_semiring P] [char_p P p] [perfect_ring P p]
/-- Create a `perfection_map` from an isomorphism to the perfection. -/
@[simps] lemma mk' {f : P →+* R} (g : P ≃+* ring.perfection R p)
(hfg : perfection.lift p P R f = g) :
perfection_map p f :=
{ injective := λ x y hxy, g.injective $ (ring_hom.ext_iff.1 hfg x).symm.trans $
eq.symm $ (ring_hom.ext_iff.1 hfg y).symm.trans $ perfection.ext $ λ n, (hxy n).symm,
surjective := λ y hy, let ⟨x, hx⟩ := g.surjective ⟨y, hy⟩ in
⟨x, λ n, show perfection.coeff R p n (perfection.lift p P R f x) =
perfection.coeff R p n ⟨y, hy⟩,
by rw [hfg, ← coe_fn_coe_base, hx]⟩ }
variables (p R P)
/-- The canonical perfection map from the perfection of a ring. -/
lemma of : perfection_map p (perfection.coeff R p 0) :=
mk' (ring_equiv.refl _) $ (equiv.apply_eq_iff_eq_symm_apply _).2 rfl
/-- For a perfect ring, it itself is the perfection. -/
lemma id [perfect_ring R p] : perfection_map p (ring_hom.id R) :=
{ injective := λ x y hxy, hxy 0,
surjective := λ f hf, ⟨f 0, λ n, show pth_root R p ^[n] (f 0) = f n,
from nat.rec_on n rfl $ λ n ih, injective_pow_p p $
by rw [function.iterate_succ_apply', pth_root_pow_p _, ih, hf]⟩ }
variables {p R P}
/-- A perfection map induces an isomorphism to the prefection. -/
noncomputable def equiv {π : P →+* R} (m : perfection_map p π) : P ≃+* ring.perfection R p :=
ring_equiv.of_bijective (perfection.lift p P R π)
⟨λ x y hxy, m.injective $ λ n, (congr_arg (perfection.coeff R p n) hxy : _),
λ f, let ⟨x, hx⟩ := m.surjective f.1 f.2 in ⟨x, perfection.ext $ hx⟩⟩
lemma equiv_apply {π : P →+* R} (m : perfection_map p π) (x : P) :
m.equiv x = perfection.lift p P R π x :=
rfl
lemma comp_equiv {π : P →+* R} (m : perfection_map p π) (x : P) :
perfection.coeff R p 0 (m.equiv x) = π x :=
rfl
lemma comp_equiv' {π : P →+* R} (m : perfection_map p π) :
(perfection.coeff R p 0).comp ↑m.equiv = π :=
ring_hom.ext $ λ x, rfl
lemma comp_symm_equiv {π : P →+* R} (m : perfection_map p π) (f : ring.perfection R p) :
π (m.equiv.symm f) = perfection.coeff R p 0 f :=
(m.comp_equiv _).symm.trans $ congr_arg _ $ m.equiv.apply_symm_apply f
lemma comp_symm_equiv' {π : P →+* R} (m : perfection_map p π) :
π.comp ↑m.equiv.symm = perfection.coeff R p 0 :=
ring_hom.ext m.comp_symm_equiv
variables (p R P)
/-- Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* P`,
where `P` is any perfection of `S`. -/
@[simps] noncomputable def lift [perfect_ring R p] (S : Type u₂) [comm_semiring S] [char_p S p]
(P : Type u₃) [comm_semiring P] [char_p P p] [perfect_ring P p]
(π : P →+* S) (m : perfection_map p π) :
(R →+* S) ≃ (R →+* P) :=
{ to_fun := λ f, ring_hom.comp ↑m.equiv.symm $ perfection.lift p R S f,
inv_fun := λ f, π.comp f,
left_inv := λ f, by { simp_rw [← ring_hom.comp_assoc, comp_symm_equiv'],
exact (perfection.lift p R S).symm_apply_apply f },
right_inv := λ f, ring_hom.ext $ λ x, m.equiv.injective $ (m.equiv.apply_symm_apply _).trans $
show perfection.lift p R S (π.comp f) x = ring_hom.comp ↑m.equiv f x,
from ring_hom.ext_iff.1 ((perfection.lift p R S).apply_eq_iff_eq_symm_apply.2 rfl) _ }
variables {R p}
lemma hom_ext [perfect_ring R p] {S : Type u₂} [comm_semiring S] [char_p S p]
{P : Type u₃} [comm_semiring P] [char_p P p] [perfect_ring P p]
(π : P →+* S) (m : perfection_map p π) {f g : R →+* P}
(hfg : ∀ x, π (f x) = π (g x)) : f = g :=
(lift p R S P π m).symm.injective $ ring_hom.ext hfg
variables {R P} (p) {S : Type u₂} [comm_semiring S] [char_p S p]
variables {Q : Type u₄} [comm_semiring Q] [char_p Q p] [perfect_ring Q p]
/-- A ring homomorphism `R →+* S` induces `P →+* Q`, a map of the respective perfections. -/
@[nolint unused_arguments]
noncomputable def map {π : P →+* R} (m : perfection_map p π) {σ : Q →+* S} (n : perfection_map p σ)
(φ : R →+* S) : P →+* Q :=
lift p P S Q σ n $ φ.comp π
lemma comp_map {π : P →+* R} (m : perfection_map p π) {σ : Q →+* S} (n : perfection_map p σ)
(φ : R →+* S) : σ.comp (map p m n φ) = φ.comp π :=
(lift p P S Q σ n).symm_apply_apply _
lemma map_map {π : P →+* R} (m : perfection_map p π) {σ : Q →+* S} (n : perfection_map p σ)
(φ : R →+* S) (x : P) : σ (map p m n φ x) = φ (π x) :=
ring_hom.ext_iff.1 (comp_map p m n φ) x
-- Why is this slow?
lemma map_eq_map (φ : R →+* S) :
@map p _ R _ _ _ _ _ _ S _ _ _ _ _ _ _ (of p R) _ (of p S) φ = perfection.map p φ :=
hom_ext _ (of p S) $ λ f, by rw [map_map, perfection.coeff_map]
end perfection_map
section perfectoid
variables (K : Type u₁) [field K] (v : valuation K ℝ≥0)
variables (O : Type u₂) [comm_ring O] [algebra O K] (hv : v.integers O)
variables (p : ℕ)
include hv
/-- `O/(p)` for `O`, ring of integers of `K`. -/
@[nolint unused_arguments has_inhabited_instance] def mod_p :=
(ideal.span {p} : ideal O).quotient
variables [hp : fact p.prime] [hvp : fact (v p ≠ 1)]
namespace mod_p
instance : comm_ring (mod_p K v O hv p) :=
ideal.quotient.comm_ring _
include hp hvp
instance : char_p (mod_p K v O hv p) p :=
char_p.quotient O p $ mt hv.one_of_is_unit $ ((algebra_map O K).map_nat_cast p).symm ▸ hvp.1
instance : nontrivial (mod_p K v O hv p) :=
char_p.nontrivial_of_char_ne_one hp.1.ne_one
section classical
local attribute [instance] classical.dec
omit hp hvp
/-- For a field `K` with valuation `v : K → ℝ≥0` and ring of integers `O`,
a function `O/(p) → ℝ≥0` that sends `0` to `0` and `x + (p)` to `v(x)` as long as `x ∉ (p)`. -/
noncomputable def pre_val (x : mod_p K v O hv p) : ℝ≥0 :=
if x = 0 then 0 else v (algebra_map O K x.out')
variables {K v O hv p}
lemma pre_val_mk {x : O} (hx : (ideal.quotient.mk _ x : mod_p K v O hv p) ≠ 0) :
pre_val K v O hv p (ideal.quotient.mk _ x) = v (algebra_map O K x) :=
begin
obtain ⟨r, hr⟩ := ideal.mem_span_singleton'.1 (ideal.quotient.eq.1 $ quotient.sound' $
@quotient.mk_out' O (ideal.span {p} : ideal O).quotient_rel x),
refine (if_neg hx).trans (v.map_eq_of_sub_lt $ lt_of_not_ge' _),
erw [← ring_hom.map_sub, ← hr, hv.le_iff_dvd],
exact λ hprx, hx (ideal.quotient.eq_zero_iff_mem.2 $ ideal.mem_span_singleton.2 $
dvd_of_mul_left_dvd hprx),
end
lemma pre_val_zero : pre_val K v O hv p 0 = 0 :=
if_pos rfl
lemma pre_val_mul {x y : mod_p K v O hv p} (hxy0 : x * y ≠ 0) :
pre_val K v O hv p (x * y) = pre_val K v O hv p x * pre_val K v O hv p y :=
begin
have hx0 : x ≠ 0 := mt (by { rintro rfl, rw zero_mul }) hxy0,
have hy0 : y ≠ 0 := mt (by { rintro rfl, rw mul_zero }) hxy0,
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surjective y,
rw ← ring_hom.map_mul at hxy0 ⊢,
rw [pre_val_mk hx0, pre_val_mk hy0, pre_val_mk hxy0, ring_hom.map_mul, v.map_mul]
end
lemma pre_val_add (x y : mod_p K v O hv p) :
pre_val K v O hv p (x + y) ≤ max (pre_val K v O hv p x) (pre_val K v O hv p y) :=
begin
by_cases hx0 : x = 0, { rw [hx0, zero_add], exact le_max_right _ _ },
by_cases hy0 : y = 0, { rw [hy0, add_zero], exact le_max_left _ _ },
by_cases hxy0 : x + y = 0, { rw [hxy0, pre_val_zero], exact zero_le _ },
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surjective y,
rw ← ring_hom.map_add at hxy0 ⊢,
rw [pre_val_mk hx0, pre_val_mk hy0, pre_val_mk hxy0, ring_hom.map_add], exact v.map_add _ _
end
lemma v_p_lt_pre_val {x : mod_p K v O hv p} : v p < pre_val K v O hv p x ↔ x ≠ 0 :=
begin
refine ⟨λ h hx, by { rw [hx, pre_val_zero] at h, exact not_lt_zero' h },
λ h, lt_of_not_ge' $ λ hp, h _⟩,
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
rw [pre_val_mk h, ← (algebra_map O K).map_nat_cast p, hv.le_iff_dvd] at hp,
rw [ideal.quotient.eq_zero_iff_mem, ideal.mem_span_singleton], exact hp
end
lemma pre_val_eq_zero {x : mod_p K v O hv p} : pre_val K v O hv p x = 0 ↔ x = 0 :=
⟨λ hvx, classical.by_contradiction $ λ hx0 : x ≠ 0,
by { rw [← v_p_lt_pre_val, hvx] at hx0, exact not_lt_zero' hx0 },
λ hx, hx.symm ▸ pre_val_zero⟩
variables (hv hvp)
lemma v_p_lt_val {x : O} :
v p < v (algebra_map O K x) ↔ (ideal.quotient.mk _ x : mod_p K v O hv p) ≠ 0 :=
by rw [lt_iff_not_ge', not_iff_not, ← (algebra_map O K).map_nat_cast p, hv.le_iff_dvd,
ideal.quotient.eq_zero_iff_mem, ideal.mem_span_singleton]
open nnreal
variables {hv} [hvp]
include hp
lemma mul_ne_zero_of_pow_p_ne_zero {x y : mod_p K v O hv p} (hx : x ^ p ≠ 0) (hy : y ^ p ≠ 0) :
x * y ≠ 0 :=
begin
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surjective y,
have h1p : (0 : ℝ) < 1 / p := one_div_pos.2 (nat.cast_pos.2 hp.1.pos),
rw ← ring_hom.map_mul, rw ← ring_hom.map_pow at hx hy,
rw ← v_p_lt_val hv at hx hy ⊢,
rw [ring_hom.map_pow, v.map_pow, ← rpow_lt_rpow_iff h1p, ← rpow_nat_cast, ← rpow_mul,
mul_one_div_cancel (nat.cast_ne_zero.2 hp.1.ne_zero : (p : ℝ) ≠ 0), rpow_one] at hx hy,
rw [ring_hom.map_mul, v.map_mul], refine lt_of_le_of_lt _ (mul_lt_mul₀ hx hy),
by_cases hvp : v p = 0, { rw hvp, exact zero_le _ }, replace hvp := zero_lt_iff.2 hvp,
conv_lhs { rw ← rpow_one (v p) }, rw ← rpow_add (ne_of_gt hvp),
refine rpow_le_rpow_of_exponent_ge hvp ((algebra_map O K).map_nat_cast p ▸ hv.2 _) _,
rw [← add_div, div_le_one (nat.cast_pos.2 hp.1.pos : 0 < (p : ℝ))], exact_mod_cast hp.1.two_le
end
end classical
end mod_p
/-- Perfection of `O/(p)` where `O` is the ring of integers of `K`. -/
@[nolint has_inhabited_instance] def pre_tilt :=
ring.perfection (mod_p K v O hv p) p
include hp hvp
namespace pre_tilt
instance : comm_ring (pre_tilt K v O hv p) :=
perfection.comm_ring p _
instance : char_p (pre_tilt K v O hv p) p :=
perfection.char_p (mod_p K v O hv p) p
section classical
open_locale classical
open perfection
/-- The valuation `Perfection(O/(p)) → ℝ≥0` as a function.
Given `f ∈ Perfection(O/(p))`, if `f = 0` then output `0`;
otherwise output `pre_val(f(n))^(p^n)` for any `n` such that `f(n) ≠ 0`. -/
noncomputable def val_aux (f : pre_tilt K v O hv p) : ℝ≥0 :=
if h : ∃ n, coeff _ _ n f ≠ 0
then mod_p.pre_val K v O hv p (coeff _ _ (nat.find h) f) ^ (p ^ nat.find h)
else 0
variables {K v O hv p}
lemma coeff_nat_find_add_ne_zero {f : pre_tilt K v O hv p} {h : ∃ n, coeff _ _ n f ≠ 0} (k : ℕ) :
coeff _ _ (nat.find h + k) f ≠ 0 :=
coeff_add_ne_zero (nat.find_spec h) k
lemma val_aux_eq {f : pre_tilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0) :
val_aux K v O hv p f = mod_p.pre_val K v O hv p (coeff _ _ n f) ^ (p ^ n) :=
begin
have h : ∃ n, coeff _ _ n f ≠ 0 := ⟨n, hfn⟩,
rw [val_aux, dif_pos h],
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le (nat.find_min' h hfn),
induction k with k ih, { refl },
obtain ⟨x, hx⟩ := ideal.quotient.mk_surjective (coeff _ _ (nat.find h + k + 1) f),
have h1 : (ideal.quotient.mk _ x : mod_p K v O hv p) ≠ 0 := hx.symm ▸ hfn,
have h2 : (ideal.quotient.mk _ (x ^ p) : mod_p K v O hv p) ≠ 0,
by { erw [ring_hom.map_pow, hx, ← ring_hom.map_pow, coeff_pow_p],
exact coeff_nat_find_add_ne_zero k },
erw [ih (coeff_nat_find_add_ne_zero k), ← hx, ← coeff_pow_p, ring_hom.map_pow, ← hx,
← ring_hom.map_pow, mod_p.pre_val_mk h1, mod_p.pre_val_mk h2,
ring_hom.map_pow, v.map_pow, ← pow_mul, pow_succ], refl
end
lemma val_aux_zero : val_aux K v O hv p 0 = 0 :=
dif_neg $ λ ⟨n, hn⟩, hn rfl
lemma val_aux_one : val_aux K v O hv p 1 = 1 :=
(val_aux_eq $ show coeff (mod_p K v O hv p) p 0 1 ≠ 0, from one_ne_zero).trans $
by { rw [pow_zero, pow_one, ring_hom.map_one, ← (ideal.quotient.mk _).map_one, mod_p.pre_val_mk,
ring_hom.map_one, v.map_one], exact @one_ne_zero (mod_p K v O hv p) _ _ }
lemma val_aux_mul (f g : pre_tilt K v O hv p) :
val_aux K v O hv p (f * g) = val_aux K v O hv p f * val_aux K v O hv p g :=
begin
by_cases hf : f = 0, { rw [hf, zero_mul, val_aux_zero, zero_mul] },
by_cases hg : g = 0, { rw [hg, mul_zero, val_aux_zero, mul_zero] },
obtain ⟨m, hm⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 (λ h, hf $ perfection.ext h),
obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 (λ h, hg $ perfection.ext h),
replace hm := coeff_ne_zero_of_le hm (le_max_left m n),
replace hn := coeff_ne_zero_of_le hn (le_max_right m n),
have hfg : coeff _ _ (max m n + 1) (f * g) ≠ 0,
{ rw ring_hom.map_mul,
refine mod_p.mul_ne_zero_of_pow_p_ne_zero _ _,
{ rw [← ring_hom.map_pow, coeff_pow_p f], assumption },
{ rw [← ring_hom.map_pow, coeff_pow_p g], assumption } },
rw [val_aux_eq (coeff_add_ne_zero hm 1), val_aux_eq (coeff_add_ne_zero hn 1), val_aux_eq hfg],
rw ring_hom.map_mul at hfg ⊢, rw [mod_p.pre_val_mul hfg, mul_pow]
end
lemma val_aux_add (f g : pre_tilt K v O hv p) :
val_aux K v O hv p (f + g) ≤ max (val_aux K v O hv p f) (val_aux K v O hv p g) :=
begin
by_cases hf : f = 0, { rw [hf, zero_add, val_aux_zero, max_eq_right], exact zero_le _ },
by_cases hg : g = 0, { rw [hg, add_zero, val_aux_zero, max_eq_left], exact zero_le _ },
by_cases hfg : f + g = 0, { rw [hfg, val_aux_zero], exact zero_le _ },
replace hf : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 (λ h, hf $ perfection.ext h),
replace hg : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 (λ h, hg $ perfection.ext h),
replace hfg : ∃ n, coeff _ _ n (f + g) ≠ 0 := not_forall.1 (λ h, hfg $ perfection.ext h),
obtain ⟨m, hm⟩ := hf, obtain ⟨n, hn⟩ := hg, obtain ⟨k, hk⟩ := hfg,
replace hm := coeff_ne_zero_of_le hm (le_trans (le_max_left m n) (le_max_left _ k)),
replace hn := coeff_ne_zero_of_le hn (le_trans (le_max_right m n) (le_max_left _ k)),
replace hk := coeff_ne_zero_of_le hk (le_max_right (max m n) k),
rw [val_aux_eq hm, val_aux_eq hn, val_aux_eq hk, ring_hom.map_add],
cases le_max_iff.1
(mod_p.pre_val_add (coeff _ _ (max (max m n) k) f) (coeff _ _ (max (max m n) k) g)) with h h,
{ exact le_max_of_le_left (pow_le_pow_of_le_left' h _) },
{ exact le_max_of_le_right (pow_le_pow_of_le_left' h _) }
end
variables (K v O hv p)
/-- The valuation `Perfection(O/(p)) → ℝ≥0`.
Given `f ∈ Perfection(O/(p))`, if `f = 0` then output `0`;
otherwise output `pre_val(f(n))^(p^n)` for any `n` such that `f(n) ≠ 0`. -/
noncomputable def val : valuation (pre_tilt K v O hv p) ℝ≥0 :=
{ to_fun := val_aux K v O hv p,
map_one' := val_aux_one,
map_mul' := val_aux_mul,
map_zero' := val_aux_zero,
map_add' := val_aux_add }
variables {K v O hv p}
lemma map_eq_zero {f : pre_tilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 :=
begin
by_cases hf0 : f = 0, { rw hf0, exact iff_of_true (valuation.map_zero _) rfl },
obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 (λ h, hf0 $ perfection.ext h),
show val_aux K v O hv p f = 0 ↔ f = 0, refine iff_of_false (λ hvf, hn _) hf0,
rw val_aux_eq hn at hvf, replace hvf := pow_eq_zero hvf, rwa mod_p.pre_val_eq_zero at hvf
end
end classical
instance : is_domain (pre_tilt K v O hv p) :=
{ exists_pair_ne := (char_p.nontrivial_of_char_ne_one hp.1.ne_one).1,
eq_zero_or_eq_zero_of_mul_eq_zero := λ f g hfg,
by { simp_rw ← map_eq_zero at hfg ⊢, contrapose! hfg, rw valuation.map_mul,
exact mul_ne_zero hfg.1 hfg.2 },
.. (infer_instance : comm_ring (pre_tilt K v O hv p)) }
end pre_tilt
/-- The tilt of a field, as defined in Perfectoid Spaces by Peter Scholze, as in
[scholze2011perfectoid]. Given a field `K` with valuation `K → ℝ≥0` and ring of integers `O`,
this is implemented as the fraction field of the perfection of `O/(p)`. -/
@[nolint has_inhabited_instance] def tilt :=
fraction_ring (pre_tilt K v O hv p)
namespace tilt
noncomputable instance : field (tilt K v O hv p) :=
fraction_ring.field
end tilt
end perfectoid
|
58e2ba3c6593a2f94bf92fef9d728e8529fb88ae | ce6917c5bacabee346655160b74a307b4a5ab620 | /src/ch5/ex0408.lean | 8751e72066b0bbd5e17c304050d03ec031c8afde | [] | no_license | Ailrun/Theorem_Proving_in_Lean | ae6a23f3c54d62d401314d6a771e8ff8b4132db2 | 2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68 | refs/heads/master | 1,609,838,270,467 | 1,586,846,743,000 | 1,586,846,743,000 | 240,967,761 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 312 | lean | example (p q r : Prop) : p ∧ (q ∨ r) → (p ∧ q) ∨ (p ∧ r) :=
begin
intro h,
cases h with hp hqr,
show (p ∧ q) ∨ (p ∧ r),
cases hqr with hq hr,
have hpq : p ∧ q,
split; assumption,
left, exact hpq,
have hpr : p ∧ r,
split; assumption,
right, exact hpr
end
|
ab6f3b1afe1903abb5416fa69bc9fa0b6335e7ac | 491068d2ad28831e7dade8d6dff871c3e49d9431 | /library/data/nat/parity.lean | 663dd9913670d813279f6002ef8a132c5431fe1e | [
"Apache-2.0"
] | permissive | davidmueller13/lean | 65a3ed141b4088cd0a268e4de80eb6778b21a0e9 | c626e2e3c6f3771e07c32e82ee5b9e030de5b050 | refs/heads/master | 1,611,278,313,401 | 1,444,021,177,000 | 1,444,021,177,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 9,910 | lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
Parity
-/
import data.nat.power logic.identities
namespace nat
open decidable
definition even (n : nat) := n mod 2 = 0
definition decidable_even [instance] : ∀ n, decidable (even n) :=
take n, !nat.has_decidable_eq
definition odd (n : nat) := ¬even n
definition decidable_odd [instance] : ∀ n, decidable (odd n) :=
take n, decidable_not
lemma even_of_dvd {n} : 2 ∣ n → even n :=
mod_eq_zero_of_dvd
lemma dvd_of_even {n} : even n → 2 ∣ n :=
dvd_of_mod_eq_zero
lemma not_odd_zero : ¬ odd 0 :=
dec_trivial
lemma even_zero : even 0 :=
dec_trivial
lemma odd_one : odd 1 :=
dec_trivial
lemma not_even_one : ¬ even 1 :=
dec_trivial
lemma odd_eq_not_even (n : nat) : odd n = ¬ even n :=
rfl
lemma odd_iff_not_even (n : nat) : odd n ↔ ¬ even n :=
!iff.refl
lemma odd_of_not_even {n} : ¬ even n → odd n :=
suppose ¬ even n,
iff.mpr !odd_iff_not_even this
lemma even_of_not_odd {n} : ¬ odd n → even n :=
suppose ¬ odd n,
not_not_elim (iff.mp (not_iff_not_of_iff !odd_iff_not_even) this)
lemma not_odd_of_even {n} : even n → ¬ odd n :=
suppose even n,
iff.mpr (not_iff_not_of_iff !odd_iff_not_even) (not_not_intro this)
lemma not_even_of_odd {n} : odd n → ¬ even n :=
suppose odd n,
iff.mp !odd_iff_not_even this
lemma odd_succ_of_even {n} : even n → odd (succ n) :=
suppose even n,
by_contradiction (suppose ¬ odd (succ n),
assert 0 = 1, from calc
0 = (n+1) mod 2 : even_of_not_odd this
... = 1 mod 2 : add_mod_eq_add_mod_right 1 `even n`,
by contradiction)
lemma eq_1_of_ne_0_lt_2 : ∀ {n : nat}, n ≠ 0 → n < 2 → n = 1
| 0 h₁ h₂ := absurd rfl h₁
| 1 h₁ h₂ := rfl
| (n+2) h₁ h₂ := absurd (lt_of_succ_lt_succ (lt_of_succ_lt_succ h₂)) !not_lt_zero
lemma mod_eq_of_odd {n} : odd n → n mod 2 = 1 :=
suppose odd n,
have ¬ n mod 2 = 0, from this,
have n mod 2 < 2, from mod_lt n dec_trivial,
eq_1_of_ne_0_lt_2 `¬ n mod 2 = 0` `n mod 2 < 2`
lemma odd_of_mod_eq {n} : n mod 2 = 1 → odd n :=
suppose n mod 2 = 1,
by_contradiction (suppose ¬ odd n,
assert n mod 2 = 0, from even_of_not_odd this,
by rewrite this at *; contradiction)
lemma even_succ_of_odd {n} : odd n → even (succ n) :=
suppose odd n,
assert n mod 2 = 1 mod 2, from mod_eq_of_odd this,
assert (n+1) mod 2 = 2 mod 2, from add_mod_eq_add_mod_right 1 this,
by rewrite mod_self at this; exact this
lemma odd_succ_succ_of_odd {n} : odd n → odd (succ (succ n)) :=
suppose odd n,
odd_succ_of_even (even_succ_of_odd this)
lemma even_succ_succ_of_even {n} : even n → even (succ (succ n)) :=
suppose even n,
even_succ_of_odd (odd_succ_of_even this)
lemma even_of_odd_succ {n} : odd (succ n) → even n :=
suppose odd (succ n),
by_contradiction (suppose ¬ even n,
have odd n, from odd_of_not_even this,
have even (succ n), from even_succ_of_odd this,
absurd this (not_even_of_odd `odd (succ n)`))
lemma odd_of_even_succ {n} : even (succ n) → odd n :=
suppose even (succ n),
by_contradiction (suppose ¬ odd n,
have even n, from even_of_not_odd this,
have odd (succ n), from odd_succ_of_even this,
absurd `even (succ n)` (not_even_of_odd this))
lemma even_of_even_succ_succ {n} : even (succ (succ n)) → even n :=
suppose even (n+2),
even_of_odd_succ (odd_of_even_succ this)
lemma odd_of_odd_succ_succ {n} : odd (succ (succ n)) → odd n :=
suppose odd (n+2),
odd_of_even_succ (even_of_odd_succ this)
lemma dvd_of_odd {n} : odd n → 2 ∣ n+1 :=
suppose odd n,
dvd_of_even (even_succ_of_odd this)
lemma odd_of_dvd {n} : 2 ∣ n+1 → odd n :=
suppose 2 ∣ n+1,
odd_of_even_succ (even_of_dvd this)
lemma even_two_mul : ∀ n, even (2 * n) :=
take n, even_of_dvd (dvd_mul_right 2 n)
lemma odd_two_mul_plus_one : ∀ n, odd (2 * n + 1) :=
take n, odd_succ_of_even (even_two_mul n)
lemma not_even_two_mul_plus_one : ∀ n, ¬ even (2 * n + 1) :=
take n, not_even_of_odd (odd_two_mul_plus_one n)
lemma not_odd_two_mul : ∀ n, ¬ odd (2 * n) :=
take n, not_odd_of_even (even_two_mul n)
lemma even_pred_of_odd : ∀ {n}, odd n → even (pred n)
| 0 h := absurd h not_odd_zero
| (n+1) h := even_of_odd_succ h
lemma even_or_odd : ∀ n, even n ∨ odd n :=
λ n, by_cases
(λ h : even n, or.inl h)
(λ h : ¬ even n, or.inr (odd_of_not_even h))
lemma exists_of_even {n} : even n → ∃ k, n = 2*k :=
λ h, exists_eq_mul_right_of_dvd (dvd_of_even h)
lemma exists_of_odd : ∀ {n}, odd n → ∃ k, n = 2*k + 1
| 0 h := absurd h not_odd_zero
| (n+1) h :=
obtain k (hk : n = 2*k), from exists_of_even (even_of_odd_succ h),
exists.intro k (by subst n)
lemma even_of_exists {n} : (∃ k, n = 2 * k) → even n :=
suppose ∃ k, n = 2 * k,
obtain k (hk : n = 2 * k), from this,
have 2 ∣ n, by subst n; apply dvd_mul_right,
even_of_dvd this
lemma odd_of_exists {n} : (∃ k, n = 2 * k + 1) → odd n :=
assume h, by_contradiction (λ hn,
have even n, from even_of_not_odd hn,
have ∃ k, n = 2 * k, from exists_of_even this,
obtain k₁ (hk₁ : n = 2 * k₁ + 1), from h,
obtain k₂ (hk₂ : n = 2 * k₂), from this,
assert (2 * k₁ + 1) mod 2 = (2 * k₂) mod 2, by rewrite [-hk₁, -hk₂],
begin
rewrite [mul_mod_right at this, add.comm at this, add_mul_mod_self_left at this],
contradiction
end)
lemma even_add_of_even_of_even {n m} : even n → even m → even (n+m) :=
suppose even n, suppose even m,
obtain k₁ (hk₁ : n = 2 * k₁), from exists_of_even `even n`,
obtain k₂ (hk₂ : m = 2 * k₂), from exists_of_even `even m`,
even_of_exists (exists.intro (k₁+k₂) (by rewrite [hk₁, hk₂, mul.left_distrib]))
lemma even_add_of_odd_of_odd {n m} : odd n → odd m → even (n+m) :=
suppose odd n, suppose odd m,
assert even (succ n + succ m), from even_add_of_even_of_even (even_succ_of_odd `odd n`) (even_succ_of_odd `odd m`),
have even(succ (succ (n + m))), by rewrite [add_succ at this, succ_add at this]; exact this,
even_of_even_succ_succ this
lemma odd_add_of_even_of_odd {n m} : even n → odd m → odd (n+m) :=
suppose even n, suppose odd m,
assert even (n + succ m), from even_add_of_even_of_even `even n` (even_succ_of_odd `odd m`),
odd_of_even_succ this
lemma odd_add_of_odd_of_even {n m} : odd n → even m → odd (n+m) :=
suppose odd n, suppose even m,
assert odd (m+n), from odd_add_of_even_of_odd `even m` `odd n`,
by rewrite add.comm at this; exact this
lemma even_mul_of_even_left {n} (m) : even n → even (n*m) :=
suppose even n,
obtain k (hk : n = 2*k), from exists_of_even this,
even_of_exists (exists.intro (k*m) (by rewrite [hk, mul.assoc]))
lemma even_mul_of_even_right {n} (m) : even n → even (m*n) :=
suppose even n,
assert even (n*m), from even_mul_of_even_left _ this,
by rewrite mul.comm at this; exact this
lemma odd_mul_of_odd_of_odd {n m} : odd n → odd m → odd (n*m) :=
suppose odd n, suppose odd m,
assert even (n * succ m), from even_mul_of_even_right _ (even_succ_of_odd `odd m`),
assert even (n * m + n), by rewrite mul_succ at this; exact this,
by_contradiction (suppose ¬ odd (n*m),
assert even (n*m), from even_of_not_odd this,
absurd `even (n * m + n)` (not_even_of_odd (odd_add_of_even_of_odd this `odd n`)))
lemma even_of_even_mul_self {n} : even (n * n) → even n :=
suppose even (n * n),
by_contradiction (suppose odd n,
have odd (n * n), from odd_mul_of_odd_of_odd this this,
show false, from this `even (n * n)`)
lemma odd_of_odd_mul_self {n} : odd (n * n) → odd n :=
suppose odd (n * n),
suppose even n,
have even (n * n), from !even_mul_of_even_left this,
show false, from `odd (n * n)` this
lemma odd_pow {n m} (h : odd n) : odd (n^m) :=
nat.induction_on m
(show odd (n^0), from dec_trivial)
(take m, suppose odd (n^m),
show odd (n^(m+1)), from odd_mul_of_odd_of_odd h this)
lemma even_pow {n m} (mpos : m > 0) (h : even n) : even (n^m) :=
have h₁ : ∀ m, even (n^succ m),
from take m, nat.induction_on m
(show even (n^1), by rewrite pow_one; apply h)
(take m, suppose even (n^succ m),
show even (n^(succ (succ m))), from !even_mul_of_even_left h),
obtain m' (h₂ : m = succ m'), from exists_eq_succ_of_pos mpos,
show even (n^m), by rewrite h₂; apply h₁
lemma odd_of_odd_pow {n m} (mpos : m > 0) (h : odd (n^m)) : odd n :=
suppose even n,
have even (n^m), from even_pow mpos this,
show false, from `odd (n^m)` this
lemma even_of_even_pow {n m} (h : even (n^m)) : even n :=
by_contradiction
(suppose odd n,
have odd (n^m), from odd_pow this,
show false, from this `even (n^m)`)
lemma eq_of_div2_of_even {n m : nat} : n div 2 = m div 2 → (even n ↔ even m) → n = m :=
assume h₁ h₂,
or.elim (em (even n))
(suppose even n, or.elim (em (even m))
(suppose even m,
obtain w₁ (hw₁ : n = 2*w₁), from exists_of_even `even n`,
obtain w₂ (hw₂ : m = 2*w₂), from exists_of_even `even m`,
begin
substvars, rewrite [mul.comm 2 w₁ at h₁, mul.comm 2 w₂ at h₁,
*mul_div_cancel _ (dec_trivial : 2 > 0) at h₁, h₁]
end)
(suppose odd m, absurd `odd m` (not_odd_of_even (iff.mp h₂ `even n`))))
(suppose odd n, or.elim (em (even m))
(suppose even m, absurd `odd n` (not_odd_of_even (iff.mpr h₂ `even m`)))
(suppose odd m,
assert d : 1 div 2 = 0, from dec_trivial,
obtain w₁ (hw₁ : n = 2*w₁ + 1), from exists_of_odd `odd n`,
obtain w₂ (hw₂ : m = 2*w₂ + 1), from exists_of_odd `odd m`,
begin
substvars,
rewrite [add.comm at h₁, add_mul_div_self_left _ _ (dec_trivial : 2 > 0) at h₁, d at h₁, zero_add at h₁],
rewrite [add.comm at h₁, add_mul_div_self_left _ _ (dec_trivial : 2 > 0) at h₁, d at h₁, zero_add at h₁],
rewrite h₁
end))
end nat
|
40fe07f61f8cd7c9a51d28162875d732161a8145 | 947fa6c38e48771ae886239b4edce6db6e18d0fb | /src/ring_theory/multiplicity.lean | d138e40faf23664e72257b734adf533e43943c88 | [
"Apache-2.0"
] | permissive | ramonfmir/mathlib | c5dc8b33155473fab97c38bd3aa6723dc289beaa | 14c52e990c17f5a00c0cc9e09847af16fabbed25 | refs/heads/master | 1,661,979,343,526 | 1,660,830,384,000 | 1,660,830,384,000 | 182,072,989 | 0 | 0 | null | 1,555,585,876,000 | 1,555,585,876,000 | null | UTF-8 | Lean | false | false | 21,889 | lean | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import algebra.associated
import algebra.big_operators.basic
import ring_theory.valuation.basic
/-!
# Multiplicity of a divisor
For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves
several basic results on it.
## Main definitions
* `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest
number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers
`n`.
* `multiplicity.finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite.
-/
variables {α : Type*}
open nat part
open_locale big_operators
/-- `multiplicity a b` returns the largest natural number `n` such that
`a ^ n ∣ b`, as an `part_enat` or natural with infinity. If `∀ n, a ^ n ∣ b`,
then it returns `⊤`-/
def multiplicity [comm_monoid α] [decidable_rel ((∣) : α → α → Prop)] (a b : α) : part_enat :=
part_enat.find $ λ n, ¬a ^ (n + 1) ∣ b
namespace multiplicity
section comm_monoid
variables [comm_monoid α]
/-- `multiplicity.finite a b` indicates that the multiplicity of `a` in `b` is finite. -/
@[reducible] def finite (a b : α) : Prop := ∃ n : ℕ, ¬a ^ (n + 1) ∣ b
lemma finite_iff_dom [decidable_rel ((∣) : α → α → Prop)] {a b : α} :
finite a b ↔ (multiplicity a b).dom := iff.rfl
lemma finite_def {a b : α} : finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b := iff.rfl
@[norm_cast]
theorem int.coe_nat_multiplicity (a b : ℕ) :
multiplicity (a : ℤ) (b : ℤ) = multiplicity a b :=
begin
apply part.ext',
{ repeat { rw [← finite_iff_dom, finite_def] },
norm_cast },
{ intros h1 h2,
apply _root_.le_antisymm; { apply nat.find_mono, norm_cast, simp } }
end
lemma not_finite_iff_forall {a b : α} : (¬ finite a b) ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨λ h n, nat.cases_on n (by { rw pow_zero, exact one_dvd _ }) (by simpa [finite, not_not] using h),
by simp [finite, multiplicity, not_not]; tauto⟩
lemma not_unit_of_finite {a b : α} (h : finite a b) : ¬is_unit a :=
let ⟨n, hn⟩ := h in mt (is_unit_iff_forall_dvd.1 ∘ is_unit.pow (n + 1)) $
λ h, hn (h b)
lemma finite_of_finite_mul_left {a b c : α} : finite a (b * c) → finite a c :=
λ ⟨n, hn⟩, ⟨n, λ h, hn (h.trans (by simp [mul_pow]))⟩
lemma finite_of_finite_mul_right {a b c : α} : finite a (b * c) → finite a b :=
by rw mul_comm; exact finite_of_finite_mul_left
variable [decidable_rel ((∣) : α → α → Prop)]
lemma pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : part_enat) ≤ multiplicity a b → a ^ k ∣ b :=
by { rw ← part_enat.some_eq_coe, exact
nat.cases_on k (λ _, by { rw pow_zero, exact one_dvd _ })
(λ k ⟨h₁, h₂⟩, by_contradiction (λ hk, (nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk))) }
lemma pow_multiplicity_dvd {a b : α} (h : finite a b) : a ^ get (multiplicity a b) h ∣ b :=
pow_dvd_of_le_multiplicity (by rw part_enat.coe_get)
lemma is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b :=
λ h, by rw [part_enat.lt_coe_iff] at hm; exact nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h)
lemma is_greatest' {a b : α} {m : ℕ} (h : finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b :=
is_greatest (by rwa [← part_enat.coe_lt_coe, part_enat.coe_get] at hm)
lemma pos_of_dvd {a b : α} (hfin : finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin :=
begin
refine zero_lt_iff.2 (λ h, _),
simpa [hdiv] using (is_greatest' hfin (lt_one_iff.mpr h)),
end
lemma unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : part_enat) = multiplicity a b :=
le_antisymm (le_of_not_gt (λ hk', is_greatest hk' hk)) $
have finite a b, from ⟨k, hsucc⟩,
by { rw [part_enat.le_coe_iff], exact ⟨this, nat.find_min' _ hsucc⟩ }
lemma unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬ a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ :=
by rw [← part_enat.coe_inj, part_enat.coe_get, unique hk hsucc]
lemma le_multiplicity_of_pow_dvd {a b : α}
{k : ℕ} (hk : a ^ k ∣ b) : (k : part_enat) ≤ multiplicity a b :=
le_of_not_gt $ λ hk', is_greatest hk' hk
lemma pow_dvd_iff_le_multiplicity {a b : α}
{k : ℕ} : a ^ k ∣ b ↔ (k : part_enat) ≤ multiplicity a b :=
⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩
lemma multiplicity_lt_iff_neg_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : part_enat) ↔ ¬ a ^ k ∣ b :=
by { rw [pow_dvd_iff_le_multiplicity, not_le] }
lemma eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : part_enat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b :=
begin
rw [← part_enat.some_eq_coe],
exact ⟨λ h, let ⟨h₁, h₂⟩ := eq_some_iff.1 h in
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest
(by { rw [part_enat.lt_coe_iff], exact ⟨h₁, lt_succ_self _⟩ })⟩,
λ h, eq_some_iff.2 ⟨⟨n, h.2⟩, eq.symm $ unique' h.1 h.2⟩⟩
end
lemma eq_top_iff {a b : α} :
multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(part_enat.find_eq_top_iff _).trans $
by { simp only [not_not],
exact ⟨λ h n, nat.cases_on n (by { rw pow_zero, exact one_dvd _}) (λ n, h _), λ h n, h _⟩ }
@[simp] lemma is_unit_left {a : α} (b : α) (ha : is_unit a) : multiplicity a b = ⊤ :=
eq_top_iff.2 (λ _, is_unit_iff_forall_dvd.1 (ha.pow _) _)
lemma is_unit_right {a b : α} (ha : ¬is_unit a) (hb : is_unit b) :
multiplicity a b = 0 :=
eq_coe_iff.2 ⟨show a ^ 0 ∣ b, by simp only [pow_zero, one_dvd],
by { rw pow_one, exact λ h, mt (is_unit_of_dvd_unit h) ha hb }⟩
@[simp] lemma one_left (b : α) : multiplicity 1 b = ⊤ := is_unit_left b is_unit_one
lemma one_right {a : α} (ha : ¬is_unit a) : multiplicity a 1 = 0 := is_unit_right ha is_unit_one
@[simp] lemma get_one_right {a : α} (ha : finite a 1) : get (multiplicity a 1) ha = 0 :=
begin
rw [part_enat.get_eq_iff_eq_coe, eq_coe_iff, pow_zero],
simpa [is_unit_iff_dvd_one.symm] using not_unit_of_finite ha,
end
@[simp] lemma unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ :=
is_unit_left a u.is_unit
lemma unit_right {a : α} (ha : ¬is_unit a) (u : αˣ) : multiplicity a u = 0 :=
is_unit_right ha u.is_unit
lemma multiplicity_eq_zero_of_not_dvd {a b : α} (ha : ¬a ∣ b) : multiplicity a b = 0 :=
by { rw [← nat.cast_zero, eq_coe_iff], simpa }
lemma eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬ finite a b :=
part.eq_none_iff'
lemma ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ finite a b :=
by rw [ne.def, eq_top_iff_not_finite, not_not]
lemma lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ finite a b :=
by rw [lt_top_iff_ne_top, ne_top_iff_finite]
lemma exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : finite a b) :
∃ (c : α), b = a ^ ((multiplicity a b).get hfin) * c ∧ ¬ a ∣ c :=
begin
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin,
refine ⟨c, hc, _⟩,
rintro ⟨k, hk⟩,
rw [hk, ← mul_assoc, ← pow_succ'] at hc,
have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩,
exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁,
end
open_locale classical
lemma multiplicity_le_multiplicity_iff {a b c d : α} : multiplicity a b ≤ multiplicity c d ↔
(∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d) :=
⟨λ h n hab, (pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h)),
λ h, if hab : finite a b
then by rw [← part_enat.coe_get (finite_iff_dom.1 hab)];
exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _))
else
have ∀ n : ℕ, c ^ n ∣ d, from λ n, h n (not_finite_iff_forall.1 hab _),
by rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2
(not_finite_iff_forall.2 this)]⟩
lemma multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
multiplicity b c ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 $ λ n h, (pow_dvd_pow_of_dvd hdvd n).trans h
lemma eq_of_associated_left {a b c : α} (h : associated a b) :
multiplicity b c = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd)
(multiplicity_le_multiplicity_of_dvd_left h.symm.dvd)
lemma multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) :
multiplicity a b ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 $ λ n hb, hb.trans h
lemma eq_of_associated_right {a b c : α} (h : associated b c) :
multiplicity a b = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd)
(multiplicity_le_multiplicity_of_dvd_right h.symm.dvd)
lemma dvd_of_multiplicity_pos {a b : α} (h : (0 : part_enat) < multiplicity a b) : a ∣ b :=
begin
rw ← pow_one a,
apply pow_dvd_of_le_multiplicity,
simpa only [nat.cast_one, part_enat.pos_iff_one_le] using h
end
lemma dvd_iff_multiplicity_pos {a b : α} : (0 : part_enat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos,
λ hdvd, lt_of_le_of_ne (zero_le _) (λ heq, is_greatest
(show multiplicity a b < ↑1,
by simpa only [heq, nat.cast_zero] using part_enat.coe_lt_coe.mpr zero_lt_one)
(by rwa pow_one a))⟩
lemma finite_nat_iff {a b : ℕ} : finite a b ↔ (a ≠ 1 ∧ 0 < b) :=
begin
rw [← not_iff_not, not_finite_iff_forall, not_and_distrib, ne.def,
not_not, not_lt, nat.le_zero_iff],
exact ⟨λ h, or_iff_not_imp_right.2 (λ hb,
have ha : a ≠ 0, from λ ha, by simpa [ha] using h 1,
by_contradiction (λ ha1 : a ≠ 1,
have ha_gt_one : 1 < a, from
lt_of_not_ge (λ ha', by { clear h, revert ha ha1, dec_trivial! }),
not_lt_of_ge (le_of_dvd (nat.pos_of_ne_zero hb) (h b))
(lt_pow_self ha_gt_one b))),
λ h, by cases h; simp *⟩
end
alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos
end comm_monoid
section comm_monoid_with_zero
variable [comm_monoid_with_zero α]
lemma ne_zero_of_finite {a b : α} (h : finite a b) : b ≠ 0 :=
let ⟨n, hn⟩ := h in λ hb, by simpa [hb] using hn
variable [decidable_rel ((∣) : α → α → Prop)]
@[simp] protected lemma zero (a : α) : multiplicity a 0 = ⊤ :=
part.eq_none_iff.2 (λ n ⟨⟨k, hk⟩, _⟩, hk (dvd_zero _))
@[simp] lemma multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 :=
begin
apply multiplicity.multiplicity_eq_zero_of_not_dvd,
rwa zero_dvd_iff,
end
lemma multiplicity_mk_eq_multiplicity [decidable_rel ((∣) : associates α → associates α → Prop)]
{a b : α} : multiplicity (associates.mk a) (associates.mk b) = multiplicity a b :=
begin
by_cases h : finite a b,
{ rw ← part_enat.coe_get (finite_iff_dom.mp h),
refine (multiplicity.unique
(show (associates.mk a)^(((multiplicity a b).get h)) ∣ associates.mk b, from _) _).symm ;
rw [← associates.mk_pow, associates.mk_dvd_mk],
{ exact pow_multiplicity_dvd h },
{ exact is_greatest ((part_enat.lt_coe_iff _ _).mpr (exists.intro
(finite_iff_dom.mp h) (nat.lt_succ_self _))) } },
{ suffices : ¬ (finite (associates.mk a) (associates.mk b)),
{ rw [finite_iff_dom, part_enat.not_dom_iff_eq_top] at h this,
rw [h, this] },
refine not_finite_iff_forall.mpr (λ n, by {rw [← associates.mk_pow, associates.mk_dvd_mk],
exact not_finite_iff_forall.mp h n }) }
end
end comm_monoid_with_zero
section comm_semiring
variables [comm_semiring α] [decidable_rel ((∣) : α → α → Prop)]
lemma min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(λ h, by rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact λ n hn, dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn))
(λ h, by rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact λ n hn, dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn)
end comm_semiring
section comm_ring
variables [comm_ring α] [decidable_rel ((∣) : α → α → Prop)]
open_locale classical
@[simp] protected lemma neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
part.ext' (by simp only [multiplicity, part_enat.find, dvd_neg])
(λ h₁ h₂, part_enat.coe_inj.1 (by rw [part_enat.coe_get]; exact
eq.symm (unique ((dvd_neg _ _).2 (pow_multiplicity_dvd _))
(mt (dvd_neg _ _).1 (is_greatest' _ (lt_succ_self _))))))
theorem int.nat_abs (a : ℕ) (b : ℤ) : multiplicity a b.nat_abs = multiplicity (a : ℤ) b :=
begin
cases int.nat_abs_eq b with h h; conv_rhs { rw h },
{ rw [int.coe_nat_multiplicity], },
{ rw [multiplicity.neg, int.coe_nat_multiplicity], },
end
lemma multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b :=
begin
apply le_antisymm,
{ apply part_enat.le_of_lt_add_one,
cases part_enat.ne_top_iff.mp (part_enat.ne_top_of_lt h) with k hk,
rw [hk], rw_mod_cast [multiplicity_lt_iff_neg_dvd], intro h_dvd,
rw [← dvd_add_iff_right] at h_dvd,
apply multiplicity.is_greatest _ h_dvd, rw [hk], apply_mod_cast nat.lt_succ_self,
rw [pow_dvd_iff_le_multiplicity, nat.cast_add, ← hk, nat.cast_one],
exact part_enat.add_one_le_of_lt h },
{ convert min_le_multiplicity_add, rw [min_eq_right (le_of_lt h)] }
end
lemma multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b :=
by { rw [sub_eq_add_neg, multiplicity_add_of_gt]; rwa [multiplicity.neg] }
lemma multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) :=
begin
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with hab|hab|hab,
{ rw [add_comm, multiplicity_add_of_gt hab, min_eq_left], exact le_of_lt hab },
{ contradiction },
{ rw [multiplicity_add_of_gt hab, min_eq_right], exact le_of_lt hab},
end
end comm_ring
section cancel_comm_monoid_with_zero
variables [cancel_comm_monoid_with_zero α]
lemma finite_mul_aux {p : α} (hp : prime p) : ∀ {n m : ℕ} {a b : α},
¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n m := λ a b ha hb ⟨s, hs⟩,
have p ∣ a * b, from ⟨p ^ (n + m) * s,
by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩,
(hp.2.2 a b this).elim
(λ ⟨x, hx⟩, have hn0 : 0 < n,
from nat.pos_of_ne_zero (λ hn0, by clear _fun_match _fun_match; simpa [hx, hn0] using ha),
have wf : (n - 1) < n, from tsub_lt_self hn0 dec_trivial,
have hpx : ¬ p ^ (n - 1 + 1) ∣ x,
from λ ⟨y, hy⟩, ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1
$ by rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩),
have 1 ≤ n + m, from le_trans hn0 (nat.le_add_right n m),
finite_mul_aux hpx hb ⟨s, mul_right_cancel₀ hp.1 begin
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this],
clear _fun_match _fun_match finite_mul_aux,
simp [*, mul_comm, mul_assoc, mul_left_comm, pow_add] at *
end⟩)
(λ ⟨x, hx⟩, have hm0 : 0 < m,
from nat.pos_of_ne_zero (λ hm0, by clear _fun_match _fun_match; simpa [hx, hm0] using hb),
have wf : (m - 1) < m, from tsub_lt_self hm0 dec_trivial,
have hpx : ¬ p ^ (m - 1 + 1) ∣ x,
from λ ⟨y, hy⟩, hb (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1
$ by rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩),
finite_mul_aux ha hpx ⟨s, mul_right_cancel₀ hp.1 begin
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)],
clear _fun_match _fun_match finite_mul_aux,
simp [*, mul_comm, mul_assoc, mul_left_comm, pow_add] at *
end⟩)
lemma finite_mul {p a b : α} (hp : prime p) : finite p a → finite p b → finite p (a * b) :=
λ ⟨n, hn⟩ ⟨m, hm⟩, ⟨n + m, finite_mul_aux hp hn hm⟩
lemma finite_mul_iff {p a b : α} (hp : prime p) : finite p (a * b) ↔ finite p a ∧ finite p b :=
⟨λ h, ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩,
λ h, finite_mul hp h.1 h.2⟩
lemma finite_pow {p a : α} (hp : prime p) : Π {k : ℕ} (ha : finite p a), finite p (a ^ k)
| 0 ha := ⟨0, by simp [mt is_unit_iff_dvd_one.2 hp.2.1]⟩
| (k+1) ha := by rw [pow_succ]; exact finite_mul hp ha (finite_pow ha)
variable [decidable_rel ((∣) : α → α → Prop)]
@[simp] lemma multiplicity_self {a : α} (ha : ¬is_unit a) (ha0 : a ≠ 0) :
multiplicity a a = 1 :=
by { rw ← nat.cast_one, exact
eq_coe_iff.2 ⟨by simp, λ ⟨b, hb⟩, ha (is_unit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 $ by { clear _fun_match,
simpa [pow_succ, mul_assoc] using hb }⟩)⟩ }
@[simp] lemma get_multiplicity_self {a : α} (ha : finite a a) :
get (multiplicity a a) ha = 1 :=
part_enat.get_eq_iff_eq_coe.2 (eq_coe_iff.2
⟨by simp, λ ⟨b, hb⟩,
by rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact mt is_unit_iff_dvd_one.2 (not_unit_of_finite ha)
⟨b, by clear _fun_match; simp * at *⟩⟩)
protected lemma mul' {p a b : α} (hp : prime p)
(h : (multiplicity p (a * b)).dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a,
from pow_multiplicity_dvd _,
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b,
from pow_multiplicity_dvd _,
have hpoweq : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2,
by simp [pow_add],
have hdiv : p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣ a * b,
by rw [hpoweq]; apply mul_dvd_mul; assumption,
have hsucc : ¬p ^ ((get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) + 1) ∣ a * b,
from λ h, by exact
not_or (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
(_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h),
by rw [← part_enat.coe_inj, part_enat.coe_get, eq_coe_iff];
exact ⟨hdiv, hsucc⟩
open_locale classical
protected lemma mul {p a b : α} (hp : prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : finite p a ∧ finite p b then
by rw [← part_enat.coe_get (finite_iff_dom.1 h.1), ← part_enat.coe_get (finite_iff_dom.1 h.2),
← part_enat.coe_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)),
← nat.cast_add, part_enat.coe_inj, multiplicity.mul' hp]; refl
else begin
rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)],
cases not_and_distrib.1 h with h h;
simp [eq_top_iff_not_finite.2 h]
end
lemma finset.prod {β : Type*} {p : α} (hp : prime p) (s : finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) :=
begin
classical,
induction s using finset.induction with a s has ih h,
{ simp only [finset.sum_empty, finset.prod_empty],
convert one_right hp.not_unit },
{ simp [has, ← ih],
convert multiplicity.mul hp }
end
protected lemma pow' {p a : α} (hp : prime p) (ha : finite p a) : ∀ {k : ℕ},
get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha
| 0 := by simp [one_right hp.not_unit]
| (k+1) := have multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k), by rw pow_succ,
by rw [get_eq_get_of_eq _ _ this, multiplicity.mul' hp, pow', add_mul, one_mul, add_comm]
lemma pow {p a : α} (hp : prime p) : ∀ {k : ℕ},
multiplicity p (a ^ k) = k • (multiplicity p a)
| 0 := by simp [one_right hp.not_unit]
| (succ k) := by simp [pow_succ, succ_nsmul, pow, multiplicity.mul hp]
lemma multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬ is_unit p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
by { rw [eq_coe_iff], use dvd_rfl, rw [pow_dvd_pow_iff h0 hu], apply nat.not_succ_le_self }
lemma multiplicity_pow_self_of_prime {p : α} (hp : prime p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
multiplicity_pow_self hp.ne_zero hp.not_unit n
end cancel_comm_monoid_with_zero
section valuation
variables {R : Type*} [comm_ring R] [is_domain R] {p : R}
[decidable_rel (has_dvd.dvd : R → R → Prop)]
/-- `multiplicity` of a prime inan integral domain as an additive valuation to `part_enat`. -/
noncomputable def add_valuation (hp : prime p) : add_valuation R part_enat :=
add_valuation.of (multiplicity p) (multiplicity.zero _) (one_right hp.not_unit)
(λ _ _, min_le_multiplicity_add) (λ a b, multiplicity.mul hp)
@[simp]
lemma add_valuation_apply {hp : prime p} {r : R} : add_valuation hp r = multiplicity p r := rfl
end valuation
end multiplicity
section nat
open multiplicity
lemma multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b)
(hab : nat.coprime a b) : multiplicity p a = 0 :=
begin
rw [multiplicity_le_multiplicity_iff] at hle,
rw [← nonpos_iff_eq_zero, ← not_lt, part_enat.pos_iff_one_le, ← nat.cast_one,
← pow_dvd_iff_le_multiplicity],
assume h,
have := nat.dvd_gcd h (hle _ h),
rw [coprime.gcd_eq_one hab, nat.dvd_one, pow_one] at this,
exact hp this
end
end nat
|
c1559f1d4f082ac98561321b02d33436e0bee4aa | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/linear_algebra/std_basis.lean | 7c3fc0d352f2e7efdf925b5f19aad4ec97cbea3b | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 9,573 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import linear_algebra.basic
import linear_algebra.basis
import linear_algebra.pi
/-!
# The standard basis
This file defines the standard basis `std_basis R φ i b j`, which is `b` where `i = j` and `0`
elsewhere.
To give a concrete example, `std_basis R (λ (i : fin 3), R) i 1` gives the `i`th unit basis vector
in `R³`, and `pi.is_basis_fun` proves this is a basis over `fin 3 → R`.
## Main definitions
- `linear_map.std_basis R ϕ i b`: the `i`'th standard `R`-basis vector on `Π i, ϕ i`,
scaled by `b`.
## Main results
- `pi.is_basis_std_basis`: `std_basis` turns a component-wise basis into a basis on the product
type.
- `pi.is_basis_fun`: `std_basis R (λ _, R) i 1` is a basis for `n → R`.
-/
open function submodule
open_locale big_operators
open_locale big_operators
namespace linear_map
variables (R : Type*) {ι : Type*} [semiring R] (φ : ι → Type*)
[Π i, add_comm_monoid (φ i)] [Π i, module R (φ i)] [decidable_eq ι]
/-- The standard basis of the product of `φ`. -/
def std_basis : Π (i : ι), φ i →ₗ[R] (Πi, φ i) := single
lemma std_basis_apply (i : ι) (b : φ i) : std_basis R φ i b = update 0 i b :=
rfl
lemma coe_std_basis (i : ι) : ⇑(std_basis R φ i) = pi.single i :=
funext $ std_basis_apply R φ i
@[simp] lemma std_basis_same (i : ι) (b : φ i) : std_basis R φ i b i = b :=
by rw [std_basis_apply, update_same]
lemma std_basis_ne (i j : ι) (h : j ≠ i) (b : φ i) : std_basis R φ i b j = 0 :=
by rw [std_basis_apply, update_noteq h]; refl
lemma std_basis_eq_pi_diag (i : ι) : std_basis R φ i = pi (diag i) :=
begin
ext x j,
convert (update_apply 0 x i j _).symm,
refl,
end
lemma ker_std_basis (i : ι) : ker (std_basis R φ i) = ⊥ :=
ker_eq_bot_of_injective $ assume f g hfg,
have std_basis R φ i f i = std_basis R φ i g i := hfg ▸ rfl,
by simpa only [std_basis_same]
lemma proj_comp_std_basis (i j : ι) : (proj i).comp (std_basis R φ j) = diag j i :=
by rw [std_basis_eq_pi_diag, proj_pi]
lemma proj_std_basis_same (i : ι) : (proj i).comp (std_basis R φ i) = id :=
by ext b; simp
lemma proj_std_basis_ne (i j : ι) (h : i ≠ j) : (proj i).comp (std_basis R φ j) = 0 :=
by ext b; simp [std_basis_ne R φ _ _ h]
lemma supr_range_std_basis_le_infi_ker_proj (I J : set ι) (h : disjoint I J) :
(⨆i∈I, range (std_basis R φ i)) ≤ (⨅i∈J, ker (proj i)) :=
begin
refine (supr_le $ assume i, supr_le $ assume hi, range_le_iff_comap.2 _),
simp only [(ker_comp _ _).symm, eq_top_iff, set_like.le_def, mem_ker, comap_infi, mem_infi],
assume b hb j hj,
have : i ≠ j := assume eq, h ⟨hi, eq.symm ▸ hj⟩,
rw [proj_std_basis_ne R φ j i this.symm, zero_apply]
end
lemma infi_ker_proj_le_supr_range_std_basis {I : finset ι} {J : set ι} (hu : set.univ ⊆ ↑I ∪ J) :
(⨅ i∈J, ker (proj i)) ≤ (⨆i∈I, range (std_basis R φ i)) :=
set_like.le_def.2
begin
assume b hb,
simp only [mem_infi, mem_ker, proj_apply] at hb,
rw ← show ∑ i in I, std_basis R φ i (b i) = b,
{ ext i,
rw [finset.sum_apply, ← std_basis_same R φ i (b i)],
refine finset.sum_eq_single i (assume j hjI ne, std_basis_ne _ _ _ _ ne.symm _) _,
assume hiI,
rw [std_basis_same],
exact hb _ ((hu trivial).resolve_left hiI) },
exact sum_mem _ (assume i hiI, mem_supr_of_mem i $ mem_supr_of_mem hiI $
(std_basis R φ i).mem_range_self (b i))
end
lemma supr_range_std_basis_eq_infi_ker_proj {I J : set ι}
(hd : disjoint I J) (hu : set.univ ⊆ I ∪ J) (hI : set.finite I) :
(⨆i∈I, range (std_basis R φ i)) = (⨅i∈J, ker (proj i)) :=
begin
refine le_antisymm (supr_range_std_basis_le_infi_ker_proj _ _ _ _ hd) _,
have : set.univ ⊆ ↑hI.to_finset ∪ J, { rwa [hI.coe_to_finset] },
refine le_trans (infi_ker_proj_le_supr_range_std_basis R φ this) (supr_le_supr $ assume i, _),
rw [set.finite.mem_to_finset],
exact le_refl _
end
lemma supr_range_std_basis [fintype ι] : (⨆i:ι, range (std_basis R φ i)) = ⊤ :=
have (set.univ : set ι) ⊆ ↑(finset.univ : finset ι) ∪ ∅ := by rw [finset.coe_univ, set.union_empty],
begin
apply top_unique,
convert (infi_ker_proj_le_supr_range_std_basis R φ this),
exact infi_emptyset.symm,
exact (funext $ λi, (@supr_pos _ _ _ (λh, range (std_basis R φ i)) $ finset.mem_univ i).symm)
end
lemma disjoint_std_basis_std_basis (I J : set ι) (h : disjoint I J) :
disjoint (⨆i∈I, range (std_basis R φ i)) (⨆i∈J, range (std_basis R φ i)) :=
begin
refine disjoint.mono
(supr_range_std_basis_le_infi_ker_proj _ _ _ _ $ disjoint_compl_right)
(supr_range_std_basis_le_infi_ker_proj _ _ _ _ $ disjoint_compl_right) _,
simp only [disjoint, set_like.le_def, mem_infi, mem_inf, mem_ker, mem_bot, proj_apply,
funext_iff],
rintros b ⟨hI, hJ⟩ i,
classical,
by_cases hiI : i ∈ I,
{ by_cases hiJ : i ∈ J,
{ exact (h ⟨hiI, hiJ⟩).elim },
{ exact hJ i hiJ } },
{ exact hI i hiI }
end
lemma std_basis_eq_single {a : R} :
(λ (i : ι), (std_basis R (λ _ : ι, R) i) a) = λ (i : ι), (finsupp.single i a) :=
begin
ext i j,
rw [std_basis_apply, finsupp.single_apply],
split_ifs,
{ rw [h, function.update_same] },
{ rw [function.update_noteq (ne.symm h)], refl },
end
end linear_map
namespace pi
open linear_map
open set
variables {R : Type*}
section module
variables {η : Type*} {ιs : η → Type*} {Ms : η → Type*}
lemma linear_independent_std_basis [ring R] [∀i, add_comm_group (Ms i)] [∀i, module R (Ms i)]
[decidable_eq η] (v : Πj, ιs j → (Ms j)) (hs : ∀i, linear_independent R (v i)) :
linear_independent R (λ (ji : Σ j, ιs j), std_basis R Ms ji.1 (v ji.1 ji.2)) :=
begin
have hs' : ∀j : η, linear_independent R (λ i : ιs j, std_basis R Ms j (v j i)),
{ intro j,
exact (hs j).map' _ (ker_std_basis _ _ _) },
apply linear_independent_Union_finite hs',
{ assume j J _ hiJ,
simp [(set.Union.equations._eqn_1 _).symm, submodule.span_image, submodule.span_Union],
have h₀ : ∀ j, span R (range (λ (i : ιs j), std_basis R Ms j (v j i)))
≤ range (std_basis R Ms j),
{ intro j,
rw [span_le, linear_map.range_coe],
apply range_comp_subset_range },
have h₁ : span R (range (λ (i : ιs j), std_basis R Ms j (v j i)))
≤ ⨆ i ∈ {j}, range (std_basis R Ms i),
{ rw @supr_singleton _ _ _ (λ i, linear_map.range (std_basis R (λ (j : η), Ms j) i)),
apply h₀ },
have h₂ : (⨆ j ∈ J, span R (range (λ (i : ιs j), std_basis R Ms j (v j i)))) ≤
⨆ j ∈ J, range (std_basis R (λ (j : η), Ms j) j) :=
supr_le_supr (λ i, supr_le_supr (λ H, h₀ i)),
have h₃ : disjoint (λ (i : η), i ∈ {j}) J,
{ convert set.disjoint_singleton_left.2 hiJ using 0 },
exact (disjoint_std_basis_std_basis _ _ _ _ h₃).mono h₁ h₂ }
end
variables [semiring R] [∀i, add_comm_monoid (Ms i)] [∀i, module R (Ms i)]
variable [fintype η]
section
open linear_equiv
/-- `pi.basis (s : ∀ j, basis (ιs j) R (Ms j))` is the `Σ j, ιs j`-indexed basis on `Π j, Ms j`
given by `s j` on each component. -/
protected noncomputable def basis (s : ∀ j, basis (ιs j) R (Ms j)) :
basis (Σ j, ιs j) R (Π j, Ms j) :=
-- The `add_comm_monoid (Π j, Ms j)` instance was hard to find.
-- Defining this in tactic mode seems to shake up instance search enough that it works by itself.
by { refine basis.of_repr (linear_equiv.trans _ (finsupp.sigma_finsupp_lequiv_pi_finsupp R).symm),
exact linear_equiv.Pi_congr_right (λ j, (s j).repr) }
@[simp] lemma basis_repr_std_basis [decidable_eq η] (s : ∀ j, basis (ιs j) R (Ms j)) (j i) :
(pi.basis s).repr (std_basis R _ j (s j i)) = finsupp.single ⟨j, i⟩ 1 :=
begin
ext ⟨j', i'⟩,
by_cases hj : j = j',
{ subst hj,
simp only [pi.basis, linear_equiv.trans_apply, basis.repr_self, std_basis_same,
linear_equiv.Pi_congr_right_apply, finsupp.sigma_finsupp_lequiv_pi_finsupp_symm_apply],
symmetry,
exact basis.finsupp.single_apply_left
(λ i i' (h : (⟨j, i⟩ : Σ j, ιs j) = ⟨j, i'⟩), eq_of_heq (sigma.mk.inj h).2) _ _ _ },
simp only [pi.basis, linear_equiv.trans_apply, finsupp.sigma_finsupp_lequiv_pi_finsupp_symm_apply,
linear_equiv.Pi_congr_right_apply],
dsimp,
rw [std_basis_ne _ _ _ _ (ne.symm hj), linear_equiv.map_zero, finsupp.zero_apply,
finsupp.single_eq_of_ne],
rintros ⟨⟩,
contradiction
end
@[simp] lemma basis_apply [decidable_eq η] (s : ∀ j, basis (ιs j) R (Ms j)) (ji) :
pi.basis s ji = std_basis R _ ji.1 (s ji.1 ji.2) :=
basis.apply_eq_iff.mpr (by simp)
@[simp] lemma basis_repr (s : ∀ j, basis (ιs j) R (Ms j)) (x) (ji) :
(pi.basis s).repr x ji = (s ji.1).repr (x ji.1) ji.2 :=
rfl
end
section
variables (R η)
/-- The basis on `η → R` where the `i`th basis vector is `function.update 0 i 1`. -/
noncomputable def basis_fun : basis η R (Π (j : η), R) :=
basis.of_equiv_fun (linear_equiv.refl _ _)
@[simp] lemma basis_fun_apply [decidable_eq η] (i) :
basis_fun R η i = std_basis R (λ (i : η), R) i 1 :=
by { simp only [basis_fun, basis.coe_of_equiv_fun, linear_equiv.refl_symm,
linear_equiv.refl_apply, std_basis_apply],
congr /- Get rid of a `decidable_eq` mismatch. -/ }
@[simp] lemma basis_fun_repr (x : η → R) (i : η) :
(pi.basis_fun R η).repr x i = x i :=
by simp [basis_fun]
end
end module
end pi
|
8a9e90e90b5e7f18cc81b45c0ac06344777eaca5 | bb31430994044506fa42fd667e2d556327e18dfe | /src/ring_theory/derivation.lean | 5b4ad1574a5df003baaacbdec83f66fbfe68f9f7 | [
"Apache-2.0"
] | permissive | sgouezel/mathlib | 0cb4e5335a2ba189fa7af96d83a377f83270e503 | 00638177efd1b2534fc5269363ebf42a7871df9a | refs/heads/master | 1,674,527,483,042 | 1,673,665,568,000 | 1,673,665,568,000 | 119,598,202 | 0 | 0 | null | 1,517,348,647,000 | 1,517,348,646,000 | null | UTF-8 | Lean | false | false | 47,780 | lean | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import ring_theory.adjoin.basic
import algebra.lie.of_associative
import ring_theory.ideal.cotangent
import ring_theory.is_tensor_product
/-!
# Derivations
This file defines derivation. A derivation `D` from the `R`-algebra `A` to the `A`-module `M` is an
`R`-linear map that satisfy the Leibniz rule `D (a * b) = a * D b + D a * b`.
## Main results
- `derivation`: The type of `R`-derivations from `A` to `M`. This has an `A`-module structure.
- `derivation.llcomp`: We may compose linear maps and derivations to obtain a derivation,
and the composition is bilinear.
- `derivation.lie_algebra`: The `R`-derivations from `A` to `A` form an lie algebra over `R`.
- `derivation_to_square_zero_equiv_lift`: The `R`-derivations from `A` into a square-zero ideal `I`
of `B` corresponds to the lifts `A →ₐ[R] B` of the map `A →ₐ[R] B ⧸ I`.
- `kaehler_differential`: The module of kaehler differentials. For an `R`-algebra `S`, we provide
the notation `Ω[S⁄R]` for `kaehler_differential R S`.
Note that the slash is `\textfractionsolidus`.
- `kaehler_differential.D`: The derivation into the module of kaehler differentials.
- `kaehler_differential.span_range_derivation`: The image of `D` spans `Ω[S⁄R]` as an `S`-module.
- `kaehler_differential.linear_map_equiv_derivation`:
The isomorphism `Hom_R(Ω[S⁄R], M) ≃ₗ[S] Der_R(S, M)`.
- `kaehler_differential.quot_ker_total_equiv`: An alternative description of `Ω[S⁄R]` as `S` copies
of `S` with kernel (`kaehler_differential.ker_total`) generated by the relations:
1. `dx + dy = d(x + y)`
2. `x dy + y dx = d(x * y)`
3. `dr = 0` for `r ∈ R`
- `kaehler_differential.map`: Given a map between the arrows `R → A` and `S → B`, we have an
`A`-linear map `Ω[A⁄R] → Ω[B⁄S]`.
## Future project
- Generalize derivations into bimodules.
- Define a `is_kaehler_differential` predicate.
-/
open algebra
open_locale big_operators
/-- `D : derivation R A M` is an `R`-linear map from `A` to `M` that satisfies the `leibniz`
equality. We also require that `D 1 = 0`. See `derivation.mk'` for a constructor that deduces this
assumption from the Leibniz rule when `M` is cancellative.
TODO: update this when bimodules are defined. -/
@[protect_proj]
structure derivation (R : Type*) (A : Type*) [comm_semiring R] [comm_semiring A]
[algebra R A] (M : Type*) [add_comm_monoid M] [module A M] [module R M]
extends A →ₗ[R] M :=
(map_one_eq_zero' : to_linear_map 1 = 0)
(leibniz' (a b : A) : to_linear_map (a * b) = a • to_linear_map b + b • to_linear_map a)
/-- The `linear_map` underlying a `derivation`. -/
add_decl_doc derivation.to_linear_map
namespace derivation
section
variables {R : Type*} [comm_semiring R]
variables {A : Type*} [comm_semiring A] [algebra R A]
variables {M : Type*} [add_comm_monoid M] [module A M] [module R M]
variables (D : derivation R A M) {D1 D2 : derivation R A M} (r : R) (a b : A)
instance : add_monoid_hom_class (derivation R A M) A M :=
{ coe := λ D, D.to_fun,
coe_injective' := λ D1 D2 h, by { cases D1, cases D2, congr, exact fun_like.coe_injective h },
map_add := λ D, D.to_linear_map.map_add',
map_zero := λ D, D.to_linear_map.map_zero }
/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`
directly. -/
instance : has_coe_to_fun (derivation R A M) (λ _, A → M) := ⟨λ D, D.to_linear_map.to_fun⟩
-- Not a simp lemma because it can be proved via `coe_fn_coe` + `to_linear_map_eq_coe`
lemma to_fun_eq_coe : D.to_fun = ⇑D := rfl
instance has_coe_to_linear_map : has_coe (derivation R A M) (A →ₗ[R] M) :=
⟨λ D, D.to_linear_map⟩
@[simp] lemma to_linear_map_eq_coe : D.to_linear_map = D := rfl
@[simp] lemma mk_coe (f : A →ₗ[R] M) (h₁ h₂) :
((⟨f, h₁, h₂⟩ : derivation R A M) : A → M) = f := rfl
@[simp, norm_cast]
lemma coe_fn_coe (f : derivation R A M) : ⇑(f : A →ₗ[R] M) = f := rfl
lemma coe_injective : @function.injective (derivation R A M) (A → M) coe_fn :=
fun_like.coe_injective
@[ext] theorem ext (H : ∀ a, D1 a = D2 a) : D1 = D2 :=
fun_like.ext _ _ H
lemma congr_fun (h : D1 = D2) (a : A) : D1 a = D2 a := fun_like.congr_fun h a
protected lemma map_add : D (a + b) = D a + D b := map_add D a b
protected lemma map_zero : D 0 = 0 := map_zero D
@[simp] lemma map_smul : D (r • a) = r • D a := D.to_linear_map.map_smul r a
@[simp] lemma leibniz : D (a * b) = a • D b + b • D a := D.leibniz' _ _
lemma map_sum {ι : Type*} (s : finset ι) (f : ι → A) : D (∑ i in s, f i) = ∑ i in s, D (f i) :=
D.to_linear_map.map_sum
@[simp, priority 900] lemma map_smul_of_tower {S : Type*} [has_smul S A] [has_smul S M]
[linear_map.compatible_smul A M S R] (D : derivation R A M) (r : S) (a : A) :
D (r • a) = r • D a :=
D.to_linear_map.map_smul_of_tower r a
@[simp] lemma map_one_eq_zero : D 1 = 0 := D.map_one_eq_zero'
@[simp] lemma map_algebra_map : D (algebra_map R A r) = 0 :=
by rw [←mul_one r, ring_hom.map_mul, ring_hom.map_one, ←smul_def, map_smul, map_one_eq_zero,
smul_zero]
@[simp] lemma map_coe_nat (n : ℕ) : D (n : A) = 0 :=
by rw [← nsmul_one, D.map_smul_of_tower n, map_one_eq_zero, smul_zero]
@[simp] lemma leibniz_pow (n : ℕ) : D (a ^ n) = n • a ^ (n - 1) • D a :=
begin
induction n with n ihn,
{ rw [pow_zero, map_one_eq_zero, zero_smul] },
{ rcases (zero_le n).eq_or_lt with (rfl|hpos),
{ rw [pow_one, one_smul, pow_zero, one_smul] },
{ have : a * a ^ (n - 1) = a ^ n, by rw [← pow_succ, nat.sub_add_cancel hpos],
simp only [pow_succ, leibniz, ihn, smul_comm a n, smul_smul a, add_smul, this,
nat.succ_eq_add_one, nat.add_succ_sub_one, add_zero, one_nsmul] } }
end
lemma eq_on_adjoin {s : set A} (h : set.eq_on D1 D2 s) : set.eq_on D1 D2 (adjoin R s) :=
λ x hx, algebra.adjoin_induction hx h
(λ r, (D1.map_algebra_map r).trans (D2.map_algebra_map r).symm)
(λ x y hx hy, by simp only [map_add, *])
(λ x y hx hy, by simp only [leibniz, *])
/-- If adjoin of a set is the whole algebra, then any two derivations equal on this set are equal
on the whole algebra. -/
lemma ext_of_adjoin_eq_top (s : set A) (hs : adjoin R s = ⊤) (h : set.eq_on D1 D2 s) : D1 = D2 :=
ext $ λ a, eq_on_adjoin h $ hs.symm ▸ trivial
/- Data typeclasses -/
instance : has_zero (derivation R A M) :=
⟨{ to_linear_map := 0,
map_one_eq_zero' := rfl,
leibniz' := λ a b, by simp only [add_zero, linear_map.zero_apply, smul_zero] }⟩
@[simp] lemma coe_zero : ⇑(0 : derivation R A M) = 0 := rfl
@[simp] lemma coe_zero_linear_map : ↑(0 : derivation R A M) = (0 : A →ₗ[R] M) := rfl
lemma zero_apply (a : A) : (0 : derivation R A M) a = 0 := rfl
instance : has_add (derivation R A M) :=
⟨λ D1 D2,
{ to_linear_map := D1 + D2,
map_one_eq_zero' := by simp,
leibniz' := λ a b, by simp only [leibniz, linear_map.add_apply,
coe_fn_coe, smul_add, add_add_add_comm] }⟩
@[simp] lemma coe_add (D1 D2 : derivation R A M) : ⇑(D1 + D2) = D1 + D2 := rfl
@[simp] lemma coe_add_linear_map (D1 D2 : derivation R A M) : ↑(D1 + D2) = (D1 + D2 : A →ₗ[R] M) :=
rfl
lemma add_apply : (D1 + D2) a = D1 a + D2 a := rfl
instance : inhabited (derivation R A M) := ⟨0⟩
section scalar
variables {S : Type*} [monoid S] [distrib_mul_action S M] [smul_comm_class R S M]
[smul_comm_class S A M]
@[priority 100]
instance : has_smul S (derivation R A M) :=
⟨λ r D,
{ to_linear_map := r • D,
map_one_eq_zero' := by rw [linear_map.smul_apply, coe_fn_coe, D.map_one_eq_zero, smul_zero],
leibniz' := λ a b, by simp only [linear_map.smul_apply, coe_fn_coe, leibniz, smul_add,
smul_comm r] }⟩
@[simp] lemma coe_smul (r : S) (D : derivation R A M) : ⇑(r • D) = r • D := rfl
@[simp] lemma coe_smul_linear_map (r : S) (D : derivation R A M) :
↑(r • D) = (r • D : A →ₗ[R] M) := rfl
lemma smul_apply (r : S) (D : derivation R A M) : (r • D) a = r • D a := rfl
instance : add_comm_monoid (derivation R A M) :=
coe_injective.add_comm_monoid _ coe_zero coe_add (λ _ _, rfl)
/-- `coe_fn` as an `add_monoid_hom`. -/
def coe_fn_add_monoid_hom : derivation R A M →+ (A → M) :=
{ to_fun := coe_fn, map_zero' := coe_zero, map_add' := coe_add }
@[priority 100]
instance : distrib_mul_action S (derivation R A M) :=
function.injective.distrib_mul_action coe_fn_add_monoid_hom coe_injective coe_smul
instance [distrib_mul_action Sᵐᵒᵖ M] [is_central_scalar S M] :
is_central_scalar S (derivation R A M) :=
{ op_smul_eq_smul := λ _ _, ext $ λ _, op_smul_eq_smul _ _}
end scalar
@[priority 100]
instance {S : Type*} [semiring S] [module S M] [smul_comm_class R S M] [smul_comm_class S A M] :
module S (derivation R A M) :=
function.injective.module S coe_fn_add_monoid_hom coe_injective coe_smul
instance [is_scalar_tower R A M] : is_scalar_tower R A (derivation R A M) :=
⟨λ x y z, ext (λ a, smul_assoc _ _ _)⟩
section push_forward
variables {N : Type*} [add_comm_monoid N] [module A N] [module R N] [is_scalar_tower R A M]
[is_scalar_tower R A N]
variables (f : M →ₗ[A] N) (e : M ≃ₗ[A] N)
/-- We can push forward derivations using linear maps, i.e., the composition of a derivation with a
linear map is a derivation. Furthermore, this operation is linear on the spaces of derivations. -/
def _root_.linear_map.comp_der : derivation R A M →ₗ[R] derivation R A N :=
{ to_fun := λ D,
{ to_linear_map := (f : M →ₗ[R] N).comp (D : A →ₗ[R] M),
map_one_eq_zero' := by simp only [linear_map.comp_apply, coe_fn_coe, map_one_eq_zero, map_zero],
leibniz' := λ a b, by simp only [coe_fn_coe, linear_map.comp_apply, linear_map.map_add,
leibniz, linear_map.coe_coe_is_scalar_tower, linear_map.map_smul] },
map_add' := λ D₁ D₂, by { ext, exact linear_map.map_add _ _ _, },
map_smul' := λ r D, by { ext, exact linear_map.map_smul _ _ _, }, }
@[simp] lemma coe_to_linear_map_comp :
(f.comp_der D : A →ₗ[R] N) = (f : M →ₗ[R] N).comp (D : A →ₗ[R] M) :=
rfl
@[simp] lemma coe_comp :
(f.comp_der D : A → N) = (f : M →ₗ[R] N).comp (D : A →ₗ[R] M) :=
rfl
/-- The composition of a derivation with a linear map as a bilinear map -/
@[simps]
def llcomp : (M →ₗ[A] N) →ₗ[A] derivation R A M →ₗ[R] derivation R A N :=
{ to_fun := λ f, f.comp_der,
map_add' := λ f₁ f₂, by { ext, refl },
map_smul' := λ r D, by { ext, refl } }
/-- Pushing a derivation foward through a linear equivalence is an equivalence. -/
def _root_.linear_equiv.comp_der : derivation R A M ≃ₗ[R] derivation R A N :=
{ inv_fun := e.symm.to_linear_map.comp_der,
left_inv := λ D, by { ext a, exact e.symm_apply_apply (D a) },
right_inv := λ D, by { ext a, exact e.apply_symm_apply (D a) },
..e.to_linear_map.comp_der }
end push_forward
section restrict_scalars
variables {S : Type*} [comm_semiring S]
variables [algebra S A] [module S M] [linear_map.compatible_smul A M R S]
variables (R)
/-- If `A` is both an `R`-algebra and an `S`-algebra; `M` is both an `R`-module and an `S`-module,
then an `S`-derivation `A → M` is also an `R`-derivation if it is also `R`-linear. -/
protected
def restrict_scalars (d : derivation S A M) : derivation R A M :=
{ map_one_eq_zero' := d.map_one_eq_zero,
leibniz' := d.leibniz,
to_linear_map := d.to_linear_map.restrict_scalars R }
end restrict_scalars
end
section cancel
variables {R : Type*} [comm_semiring R] {A : Type*} [comm_semiring A] [algebra R A]
{M : Type*} [add_cancel_comm_monoid M] [module R M] [module A M]
/-- Define `derivation R A M` from a linear map when `M` is cancellative by verifying the Leibniz
rule. -/
def mk' (D : A →ₗ[R] M) (h : ∀ a b, D (a * b) = a • D b + b • D a) : derivation R A M :=
{ to_linear_map := D,
map_one_eq_zero' := add_right_eq_self.1 $ by simpa only [one_smul, one_mul] using (h 1 1).symm,
leibniz' := h }
@[simp] lemma coe_mk' (D : A →ₗ[R] M) (h) : ⇑(mk' D h) = D := rfl
@[simp] lemma coe_mk'_linear_map (D : A →ₗ[R] M) (h) : (mk' D h : A →ₗ[R] M) = D := rfl
end cancel
section
variables {R : Type*} [comm_ring R]
variables {A : Type*} [comm_ring A] [algebra R A]
section
variables {M : Type*} [add_comm_group M] [module A M] [module R M]
variables (D : derivation R A M) {D1 D2 : derivation R A M} (r : R) (a b : A)
protected lemma map_neg : D (-a) = -D a := map_neg D a
protected lemma map_sub : D (a - b) = D a - D b := map_sub D a b
@[simp] lemma map_coe_int (n : ℤ) : D (n : A) = 0 :=
by rw [← zsmul_one, D.map_smul_of_tower n, map_one_eq_zero, smul_zero]
lemma leibniz_of_mul_eq_one {a b : A} (h : a * b = 1) : D a = -a^2 • D b :=
begin
rw neg_smul,
refine eq_neg_of_add_eq_zero_left _,
calc D a + a ^ 2 • D b = a • b • D a + a • a • D b : by simp only [smul_smul, h, one_smul, sq]
... = a • D (a * b) : by rw [leibniz, smul_add, add_comm]
... = 0 : by rw [h, map_one_eq_zero, smul_zero]
end
lemma leibniz_inv_of [invertible a] : D (⅟a) = -⅟a^2 • D a :=
D.leibniz_of_mul_eq_one $ inv_of_mul_self a
lemma leibniz_inv {K : Type*} [field K] [module K M] [algebra R K] (D : derivation R K M) (a : K) :
D (a⁻¹) = -a⁻¹ ^ 2 • D a :=
begin
rcases eq_or_ne a 0 with (rfl|ha),
{ simp },
{ exact D.leibniz_of_mul_eq_one (inv_mul_cancel ha) }
end
instance : has_neg (derivation R A M) :=
⟨λ D, mk' (-D) $ λ a b,
by simp only [linear_map.neg_apply, smul_neg, neg_add_rev, leibniz, coe_fn_coe, add_comm]⟩
@[simp] lemma coe_neg (D : derivation R A M) : ⇑(-D) = -D := rfl
@[simp] lemma coe_neg_linear_map (D : derivation R A M) : ↑(-D) = (-D : A →ₗ[R] M) :=
rfl
lemma neg_apply : (-D) a = -D a := rfl
instance : has_sub (derivation R A M) :=
⟨λ D1 D2, mk' (D1 - D2 : A →ₗ[R] M) $ λ a b,
by simp only [linear_map.sub_apply, leibniz, coe_fn_coe, smul_sub, add_sub_add_comm]⟩
@[simp] lemma coe_sub (D1 D2 : derivation R A M) : ⇑(D1 - D2) = D1 - D2 := rfl
@[simp] lemma coe_sub_linear_map (D1 D2 : derivation R A M) : ↑(D1 - D2) = (D1 - D2 : A →ₗ[R] M) :=
rfl
lemma sub_apply : (D1 - D2) a = D1 a - D2 a := rfl
instance : add_comm_group (derivation R A M) :=
coe_injective.add_comm_group _ coe_zero coe_add coe_neg coe_sub (λ _ _, rfl) (λ _ _, rfl)
end
section lie_structures
/-! # Lie structures -/
variables (D : derivation R A A) {D1 D2 : derivation R A A} (r : R) (a b : A)
/-- The commutator of derivations is again a derivation. -/
instance : has_bracket (derivation R A A) (derivation R A A) :=
⟨λ D1 D2, mk' (⁅(D1 : module.End R A), (D2 : module.End R A)⁆) $ λ a b,
by { simp only [ring.lie_def, map_add, id.smul_eq_mul, linear_map.mul_apply, leibniz, coe_fn_coe,
linear_map.sub_apply], ring, }⟩
@[simp] lemma commutator_coe_linear_map :
↑⁅D1, D2⁆ = ⁅(D1 : module.End R A), (D2 : module.End R A)⁆ := rfl
lemma commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) := rfl
instance : lie_ring (derivation R A A) :=
{ add_lie := λ d e f, by { ext a, simp only [commutator_apply, add_apply, map_add], ring, },
lie_add := λ d e f, by { ext a, simp only [commutator_apply, add_apply, map_add], ring, },
lie_self := λ d, by { ext a, simp only [commutator_apply, add_apply, map_add], ring_nf, },
leibniz_lie := λ d e f,
by { ext a, simp only [commutator_apply, add_apply, sub_apply, map_sub], ring, } }
instance : lie_algebra R (derivation R A A) :=
{ lie_smul := λ r d e, by { ext a, simp only [commutator_apply, map_smul, smul_sub, smul_apply]},
..derivation.module }
end lie_structures
end
end derivation
section to_square_zero
universes u v w
variables {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [comm_semiring A] [comm_ring B]
variables [algebra R A] [algebra R B] (I : ideal B) (hI : I ^ 2 = ⊥)
/-- If `f₁ f₂ : A →ₐ[R] B` are two lifts of the same `A →ₐ[R] B ⧸ I`,
we may define a map `f₁ - f₂ : A →ₗ[R] I`. -/
def diff_to_ideal_of_quotient_comp_eq (f₁ f₂ : A →ₐ[R] B)
(e : (ideal.quotient.mkₐ R I).comp f₁ = (ideal.quotient.mkₐ R I).comp f₂) :
A →ₗ[R] I :=
linear_map.cod_restrict (I.restrict_scalars _) (f₁.to_linear_map - f₂.to_linear_map)
begin
intro x,
change f₁ x - f₂ x ∈ I,
rw [← ideal.quotient.eq, ← ideal.quotient.mkₐ_eq_mk R, ← alg_hom.comp_apply, e],
refl,
end
@[simp]
lemma diff_to_ideal_of_quotient_comp_eq_apply (f₁ f₂ : A →ₐ[R] B)
(e : (ideal.quotient.mkₐ R I).comp f₁ = (ideal.quotient.mkₐ R I).comp f₂) (x : A) :
((diff_to_ideal_of_quotient_comp_eq I f₁ f₂ e) x : B) = f₁ x - f₂ x :=
rfl
variables [algebra A B] [is_scalar_tower R A B]
include hI
/-- Given a tower of algebras `R → A → B`, and a square-zero `I : ideal B`, each lift `A →ₐ[R] B`
of the canonical map `A →ₐ[R] B ⧸ I` corresponds to a `R`-derivation from `A` to `I`. -/
def derivation_to_square_zero_of_lift
(f : A →ₐ[R] B) (e : (ideal.quotient.mkₐ R I).comp f = is_scalar_tower.to_alg_hom R A (B ⧸ I)) :
derivation R A I :=
begin
refine
{ map_one_eq_zero' := _,
leibniz' := _,
..(diff_to_ideal_of_quotient_comp_eq I f (is_scalar_tower.to_alg_hom R A B) _) },
{ rw e, ext, refl },
{ ext, change f 1 - algebra_map A B 1 = 0, rw [map_one, map_one, sub_self] },
{ intros x y,
let F := diff_to_ideal_of_quotient_comp_eq I f (is_scalar_tower.to_alg_hom R A B)
(by { rw e, ext, refl }),
have : (f x - algebra_map A B x) * (f y - algebra_map A B y) = 0,
{ rw [← ideal.mem_bot, ← hI, pow_two],
convert (ideal.mul_mem_mul (F x).2 (F y).2) using 1 },
ext,
dsimp only [submodule.coe_add, submodule.coe_mk, linear_map.coe_mk,
diff_to_ideal_of_quotient_comp_eq_apply, submodule.coe_smul_of_tower,
is_scalar_tower.coe_to_alg_hom', linear_map.to_fun_eq_coe],
simp only [map_mul, sub_mul, mul_sub, algebra.smul_def] at ⊢ this,
rw [sub_eq_iff_eq_add, sub_eq_iff_eq_add] at this,
rw this,
ring }
end
lemma derivation_to_square_zero_of_lift_apply (f : A →ₐ[R] B)
(e : (ideal.quotient.mkₐ R I).comp f = is_scalar_tower.to_alg_hom R A (B ⧸ I))
(x : A) : (derivation_to_square_zero_of_lift I hI f e x : B) = f x - algebra_map A B x := rfl
/-- Given a tower of algebras `R → A → B`, and a square-zero `I : ideal B`, each `R`-derivation
from `A` to `I` corresponds to a lift `A →ₐ[R] B` of the canonical map `A →ₐ[R] B ⧸ I`. -/
def lift_of_derivation_to_square_zero (f : derivation R A I) :
A →ₐ[R] B :=
{ map_one' := show (f 1 : B) + algebra_map A B 1 = 1,
by rw [map_one, f.map_one_eq_zero, submodule.coe_zero, zero_add],
map_mul' := λ x y, begin
have : (f x : B) * (f y) = 0,
{ rw [← ideal.mem_bot, ← hI, pow_two], convert (ideal.mul_mem_mul (f x).2 (f y).2) using 1 },
dsimp,
simp only [map_mul, f.leibniz, add_mul, mul_add, submodule.coe_add,
submodule.coe_smul_of_tower, algebra.smul_def, this],
ring
end,
commutes' := λ r, by { dsimp, simp [← is_scalar_tower.algebra_map_apply R A B r] },
map_zero' := ((I.restrict_scalars R).subtype.comp f.to_linear_map +
(is_scalar_tower.to_alg_hom R A B).to_linear_map).map_zero,
..((I.restrict_scalars R).subtype.comp f.to_linear_map +
(is_scalar_tower.to_alg_hom R A B).to_linear_map) }
lemma lift_of_derivation_to_square_zero_apply (f : derivation R A I) (x : A) :
lift_of_derivation_to_square_zero I hI f x = f x + algebra_map A B x := rfl
@[simp] lemma lift_of_derivation_to_square_zero_mk_apply (d : derivation R A I) (x : A) :
ideal.quotient.mk I (lift_of_derivation_to_square_zero I hI d x) = algebra_map A (B ⧸ I) x :=
by { rw [lift_of_derivation_to_square_zero_apply, map_add,
ideal.quotient.eq_zero_iff_mem.mpr (d x).prop, zero_add], refl }
/-- Given a tower of algebras `R → A → B`, and a square-zero `I : ideal B`,
there is a 1-1 correspondance between `R`-derivations from `A` to `I` and
lifts `A →ₐ[R] B` of the canonical map `A →ₐ[R] B ⧸ I`. -/
@[simps]
def derivation_to_square_zero_equiv_lift :
derivation R A I ≃
{ f : A →ₐ[R] B // (ideal.quotient.mkₐ R I).comp f = is_scalar_tower.to_alg_hom R A (B ⧸ I) } :=
begin
refine ⟨λ d, ⟨lift_of_derivation_to_square_zero I hI d, _⟩, λ f,
(derivation_to_square_zero_of_lift I hI f.1 f.2 : _), _, _⟩,
{ ext x, exact lift_of_derivation_to_square_zero_mk_apply I hI d x },
{ intro d, ext x, exact add_sub_cancel (d x : B) (algebra_map A B x) },
{ rintro ⟨f, hf⟩, ext x, exact sub_add_cancel (f x) (algebra_map A B x) }
end
end to_square_zero
section kaehler_differential
open_locale tensor_product
variables (R S : Type*) [comm_ring R] [comm_ring S] [algebra R S]
/-- The kernel of the multiplication map `S ⊗[R] S →ₐ[R] S`. -/
abbreviation kaehler_differential.ideal : ideal (S ⊗[R] S) :=
ring_hom.ker (tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S)
variable {S}
lemma kaehler_differential.one_smul_sub_smul_one_mem_ideal (a : S) :
(1 : S) ⊗ₜ[R] a - a ⊗ₜ[R] (1 : S) ∈ kaehler_differential.ideal R S :=
by simp [ring_hom.mem_ker]
variables {R}
variables {M : Type*} [add_comm_group M] [module R M] [module S M] [is_scalar_tower R S M]
/-- For a `R`-derivation `S → M`, this is the map `S ⊗[R] S →ₗ[S] M` sending `s ⊗ₜ t ↦ s • D t`. -/
def derivation.tensor_product_to (D : derivation R S M) : S ⊗[R] S →ₗ[S] M :=
tensor_product.algebra_tensor_module.lift ((linear_map.lsmul S (S →ₗ[R] M)).flip D.to_linear_map)
lemma derivation.tensor_product_to_tmul (D : derivation R S M) (s t : S) :
D.tensor_product_to (s ⊗ₜ t) = s • D t :=
rfl
lemma derivation.tensor_product_to_mul (D : derivation R S M) (x y : S ⊗[R] S) :
D.tensor_product_to (x * y) = tensor_product.lmul' R x • D.tensor_product_to y +
tensor_product.lmul' R y • D.tensor_product_to x :=
begin
apply tensor_product.induction_on x,
{ rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero] },
swap, { rintros, simp only [add_mul, map_add, add_smul, *, smul_add], rw add_add_add_comm },
intros x₁ x₂,
apply tensor_product.induction_on y,
{ rw [mul_zero, map_zero, map_zero, zero_smul, smul_zero, add_zero] },
swap, { rintros, simp only [mul_add, map_add, add_smul, *, smul_add], rw add_add_add_comm },
intros x y,
simp only [tensor_product.tmul_mul_tmul, derivation.tensor_product_to,
tensor_product.algebra_tensor_module.lift_apply, tensor_product.lift.tmul',
tensor_product.lmul'_apply_tmul],
dsimp,
rw D.leibniz,
simp only [smul_smul, smul_add, mul_comm (x * y) x₁, mul_right_comm x₁ x₂, ← mul_assoc],
end
variables (R S)
/-- The kernel of `S ⊗[R] S →ₐ[R] S` is generated by `1 ⊗ s - s ⊗ 1` as a `S`-module. -/
lemma kaehler_differential.submodule_span_range_eq_ideal :
submodule.span S (set.range $ λ s : S, (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
(kaehler_differential.ideal R S).restrict_scalars S :=
begin
apply le_antisymm,
{ rw submodule.span_le,
rintros _ ⟨s, rfl⟩,
exact kaehler_differential.one_smul_sub_smul_one_mem_ideal _ _ },
{ rintros x (hx : _ = _),
have : x - (tensor_product.lmul' R x) ⊗ₜ[R] (1 : S) = x,
{ rw [hx, tensor_product.zero_tmul, sub_zero] },
rw ← this,
clear this hx,
apply tensor_product.induction_on x; clear x,
{ rw [map_zero, tensor_product.zero_tmul, sub_zero], exact zero_mem _ },
{ intros x y,
convert_to x • (1 ⊗ₜ y - y ⊗ₜ 1) ∈ _,
{ rw [tensor_product.lmul'_apply_tmul, smul_sub, tensor_product.smul_tmul',
tensor_product.smul_tmul', smul_eq_mul, smul_eq_mul, mul_one] },
{ refine submodule.smul_mem _ x _,
apply submodule.subset_span,
exact set.mem_range_self y } },
{ intros x y hx hy,
rw [map_add, tensor_product.add_tmul, ← sub_add_sub_comm],
exact add_mem hx hy } }
end
lemma kaehler_differential.span_range_eq_ideal :
ideal.span (set.range $ λ s : S, (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
kaehler_differential.ideal R S :=
begin
apply le_antisymm,
{ rw ideal.span_le,
rintros _ ⟨s, rfl⟩,
exact kaehler_differential.one_smul_sub_smul_one_mem_ideal _ _ },
{ change (kaehler_differential.ideal R S).restrict_scalars S ≤ (ideal.span _).restrict_scalars S,
rw [← kaehler_differential.submodule_span_range_eq_ideal, ideal.span],
conv_rhs { rw ← submodule.span_span_of_tower S },
exact submodule.subset_span }
end
/--
The module of Kähler differentials (Kahler differentials, Kaehler differentials).
This is implemented as `I / I ^ 2` with `I` the kernel of the multiplication map `S ⊗[R] S →ₐ[R] S`.
To view elements as a linear combination of the form `s • D s'`, use
`kaehler_differential.tensor_product_to_surjective` and `derivation.tensor_product_to_tmul`.
We also provide the notation `Ω[S⁄R]` for `kaehler_differential R S`.
Note that the slash is `\textfractionsolidus`.
-/
@[derive [add_comm_group, module (S ⊗[R] S)]]
def kaehler_differential : Type* := (kaehler_differential.ideal R S).cotangent
notation `Ω[`:100 S `⁄`:0 R `]`:0 := kaehler_differential R S
instance : nonempty Ω[S⁄R] := ⟨0⟩
instance kaehler_differential.module' {R' : Type*} [comm_ring R'] [algebra R' S] :
module R' Ω[S⁄R] :=
(module.comp_hom (kaehler_differential.ideal R S).cotangent (algebra_map R' S) : _)
instance : is_scalar_tower S (S ⊗[R] S) Ω[S⁄R] :=
ideal.cotangent.is_scalar_tower _
instance kaehler_differential.is_scalar_tower_of_tower {R₁ R₂ : Type*} [comm_ring R₁] [comm_ring R₂]
[algebra R₁ S] [algebra R₂ S] [algebra R₁ R₂] [is_scalar_tower R₁ R₂ S] :
is_scalar_tower R₁ R₂ Ω[S⁄R] :=
begin
convert restrict_scalars.is_scalar_tower R₁ R₂ Ω[S⁄R] using 1,
ext x m,
show algebra_map R₁ S x • m = algebra_map R₂ S (algebra_map R₁ R₂ x) • m,
rw ← is_scalar_tower.algebra_map_apply,
end
instance kaehler_differential.is_scalar_tower' :
is_scalar_tower R (S ⊗[R] S) Ω[S⁄R] :=
begin
convert restrict_scalars.is_scalar_tower R (S ⊗[R] S) Ω[S⁄R] using 1,
ext x m,
show algebra_map R S x • m = algebra_map R (S ⊗[R] S) x • m,
simp_rw [is_scalar_tower.algebra_map_apply R S (S ⊗[R] S), is_scalar_tower.algebra_map_smul]
end
/-- The quotient map `I → Ω[S⁄R]` with `I` being the kernel of `S ⊗[R] S → S`. -/
def kaehler_differential.from_ideal : kaehler_differential.ideal R S →ₗ[S ⊗[R] S] Ω[S⁄R] :=
(kaehler_differential.ideal R S).to_cotangent
/-- (Implementation) The underlying linear map of the derivation into `Ω[S⁄R]`. -/
def kaehler_differential.D_linear_map : S →ₗ[R] Ω[S⁄R] :=
((kaehler_differential.from_ideal R S).restrict_scalars R).comp
((tensor_product.include_right.to_linear_map - tensor_product.include_left.to_linear_map :
S →ₗ[R] S ⊗[R] S).cod_restrict ((kaehler_differential.ideal R S).restrict_scalars R)
(kaehler_differential.one_smul_sub_smul_one_mem_ideal R) : _ →ₗ[R] _)
lemma kaehler_differential.D_linear_map_apply (s : S) :
kaehler_differential.D_linear_map R S s = (kaehler_differential.ideal R S).to_cotangent
⟨1 ⊗ₜ s - s ⊗ₜ 1, kaehler_differential.one_smul_sub_smul_one_mem_ideal R s⟩ :=
rfl
/-- The universal derivation into `Ω[S⁄R]`. -/
def kaehler_differential.D : derivation R S Ω[S⁄R] :=
{ map_one_eq_zero' := begin
dsimp [kaehler_differential.D_linear_map_apply],
rw [ideal.to_cotangent_eq_zero, subtype.coe_mk, sub_self],
exact zero_mem _
end,
leibniz' := λ a b, begin
dsimp [kaehler_differential.D_linear_map_apply],
rw [← linear_map.map_smul_of_tower, ← linear_map.map_smul_of_tower, ← map_add,
ideal.to_cotangent_eq, pow_two],
convert submodule.mul_mem_mul (kaehler_differential.one_smul_sub_smul_one_mem_ideal R a : _)
(kaehler_differential.one_smul_sub_smul_one_mem_ideal R b : _) using 1,
simp only [add_subgroup_class.coe_sub, submodule.coe_add, submodule.coe_mk,
tensor_product.tmul_mul_tmul, mul_sub, sub_mul, mul_comm b,
submodule.coe_smul_of_tower, smul_sub, tensor_product.smul_tmul', smul_eq_mul, mul_one],
ring_nf,
end,
..(kaehler_differential.D_linear_map R S) }
lemma kaehler_differential.D_apply (s : S) :
kaehler_differential.D R S s = (kaehler_differential.ideal R S).to_cotangent
⟨1 ⊗ₜ s - s ⊗ₜ 1, kaehler_differential.one_smul_sub_smul_one_mem_ideal R s⟩ :=
rfl
lemma kaehler_differential.span_range_derivation :
submodule.span S (set.range $ kaehler_differential.D R S) = ⊤ :=
begin
rw _root_.eq_top_iff,
rintros x -,
obtain ⟨⟨x, hx⟩, rfl⟩ := ideal.to_cotangent_surjective _ x,
have : x ∈ (kaehler_differential.ideal R S).restrict_scalars S := hx,
rw ← kaehler_differential.submodule_span_range_eq_ideal at this,
suffices : ∃ hx, (kaehler_differential.ideal R S).to_cotangent ⟨x, hx⟩ ∈
submodule.span S (set.range $ kaehler_differential.D R S),
{ exact this.some_spec },
apply submodule.span_induction this,
{ rintros _ ⟨x, rfl⟩,
refine ⟨kaehler_differential.one_smul_sub_smul_one_mem_ideal R x, _⟩,
apply submodule.subset_span,
exact ⟨x, kaehler_differential.D_linear_map_apply R S x⟩ },
{ exact ⟨zero_mem _, zero_mem _⟩ },
{ rintros x y ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩, exact ⟨add_mem hx₁ hy₁, add_mem hx₂ hy₂⟩ },
{ rintros r x ⟨hx₁, hx₂⟩, exact ⟨((kaehler_differential.ideal R S).restrict_scalars S).smul_mem
r hx₁, submodule.smul_mem _ r hx₂⟩ }
end
variables {R S}
/-- The linear map from `Ω[S⁄R]`, associated with a derivation. -/
def derivation.lift_kaehler_differential (D : derivation R S M) : Ω[S⁄R] →ₗ[S] M :=
begin
refine ((kaehler_differential.ideal R S • ⊤ :
submodule (S ⊗[R] S) (kaehler_differential.ideal R S)).restrict_scalars S).liftq _ _,
{ exact D.tensor_product_to.comp ((kaehler_differential.ideal R S).subtype.restrict_scalars S) },
{ intros x hx,
change _ = _,
apply submodule.smul_induction_on hx; clear hx x,
{ rintros x (hx : _ = _) ⟨y, hy : _ = _⟩ -,
dsimp,
rw [derivation.tensor_product_to_mul, hx, hy, zero_smul, zero_smul, zero_add] },
{ intros x y ex ey, rw [map_add, ex, ey, zero_add] } }
end
lemma derivation.lift_kaehler_differential_apply (D : derivation R S M) (x) :
D.lift_kaehler_differential
((kaehler_differential.ideal R S).to_cotangent x) = D.tensor_product_to x :=
rfl
lemma derivation.lift_kaehler_differential_comp (D : derivation R S M) :
D.lift_kaehler_differential.comp_der (kaehler_differential.D R S) = D :=
begin
ext a,
dsimp [kaehler_differential.D_apply],
refine (D.lift_kaehler_differential_apply _).trans _,
rw [subtype.coe_mk, map_sub, derivation.tensor_product_to_tmul,
derivation.tensor_product_to_tmul, one_smul, D.map_one_eq_zero, smul_zero, sub_zero],
end
@[simp] lemma derivation.lift_kaehler_differential_comp_D (D' : derivation R S M) (x : S) :
D'.lift_kaehler_differential (kaehler_differential.D R S x) = D' x :=
derivation.congr_fun D'.lift_kaehler_differential_comp x
@[ext]
lemma derivation.lift_kaehler_differential_unique
(f f' : Ω[S⁄R] →ₗ[S] M)
(hf : f.comp_der (kaehler_differential.D R S) =
f'.comp_der (kaehler_differential.D R S)) :
f = f' :=
begin
apply linear_map.ext,
intro x,
have : x ∈ submodule.span S (set.range $ kaehler_differential.D R S),
{ rw kaehler_differential.span_range_derivation, trivial },
apply submodule.span_induction this,
{ rintros _ ⟨x, rfl⟩, exact congr_arg (λ D : derivation R S M, D x) hf },
{ rw [map_zero, map_zero] },
{ intros x y hx hy, rw [map_add, map_add, hx, hy] },
{ intros a x e, rw [map_smul, map_smul, e] }
end
variables (R S)
lemma derivation.lift_kaehler_differential_D :
(kaehler_differential.D R S).lift_kaehler_differential = linear_map.id :=
derivation.lift_kaehler_differential_unique _ _
(kaehler_differential.D R S).lift_kaehler_differential_comp
variables {R S}
lemma kaehler_differential.D_tensor_product_to (x : kaehler_differential.ideal R S) :
(kaehler_differential.D R S).tensor_product_to x =
(kaehler_differential.ideal R S).to_cotangent x :=
begin
rw [← derivation.lift_kaehler_differential_apply, derivation.lift_kaehler_differential_D],
refl,
end
variables (R S)
lemma kaehler_differential.tensor_product_to_surjective :
function.surjective (kaehler_differential.D R S).tensor_product_to :=
begin
intro x, obtain ⟨x, rfl⟩ := (kaehler_differential.ideal R S).to_cotangent_surjective x,
exact ⟨x, kaehler_differential.D_tensor_product_to x⟩
end
/-- The `S`-linear maps from `Ω[S⁄R]` to `M` are (`S`-linearly) equivalent to `R`-derivations
from `S` to `M`. -/
def kaehler_differential.linear_map_equiv_derivation : (Ω[S⁄R] →ₗ[S] M) ≃ₗ[S] derivation R S M :=
{ inv_fun := derivation.lift_kaehler_differential,
left_inv := λ f, derivation.lift_kaehler_differential_unique _ _
(derivation.lift_kaehler_differential_comp _),
right_inv := derivation.lift_kaehler_differential_comp,
..(derivation.llcomp.flip $ kaehler_differential.D R S) }
/-- The quotient ring of `S ⊗ S ⧸ J ^ 2` by `Ω[S⁄R]` is isomorphic to `S`. -/
def kaehler_differential.quotient_cotangent_ideal_ring_equiv :
(S ⊗ S ⧸ kaehler_differential.ideal R S ^ 2) ⧸
(kaehler_differential.ideal R S).cotangent_ideal ≃+* S :=
begin
have : function.right_inverse tensor_product.include_left
(↑(tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S) : S ⊗[R] S →+* S),
{ intro x, rw [alg_hom.coe_to_ring_hom, ← alg_hom.comp_apply,
tensor_product.lmul'_comp_include_left], refl },
refine (ideal.quot_cotangent _).trans _,
refine (ideal.quot_equiv_of_eq _).trans (ring_hom.quotient_ker_equiv_of_right_inverse this),
ext, refl,
end
/-- The quotient ring of `S ⊗ S ⧸ J ^ 2` by `Ω[S⁄R]` is isomorphic to `S` as an `S`-algebra. -/
def kaehler_differential.quotient_cotangent_ideal :
((S ⊗ S ⧸ kaehler_differential.ideal R S ^ 2) ⧸
(kaehler_differential.ideal R S).cotangent_ideal) ≃ₐ[S] S :=
{ commutes' := (kaehler_differential.quotient_cotangent_ideal_ring_equiv R S).apply_symm_apply,
..kaehler_differential.quotient_cotangent_ideal_ring_equiv R S }
lemma kaehler_differential.End_equiv_aux (f : S →ₐ[R] S ⊗ S ⧸ kaehler_differential.ideal R S ^ 2) :
(ideal.quotient.mkₐ R (kaehler_differential.ideal R S).cotangent_ideal).comp f =
is_scalar_tower.to_alg_hom R S _ ↔
(tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S).ker_square_lift.comp f = alg_hom.id R S :=
begin
rw [alg_hom.ext_iff, alg_hom.ext_iff],
apply forall_congr,
intro x,
have e₁ : (tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S).ker_square_lift (f x) =
kaehler_differential.quotient_cotangent_ideal_ring_equiv R S
(ideal.quotient.mk (kaehler_differential.ideal R S).cotangent_ideal $ f x),
{ generalize : f x = y, obtain ⟨y, rfl⟩ := ideal.quotient.mk_surjective y, refl },
have e₂ : x = kaehler_differential.quotient_cotangent_ideal_ring_equiv
R S (is_scalar_tower.to_alg_hom R S _ x),
{ exact (mul_one x).symm },
split,
{ intro e,
exact (e₁.trans (@ring_equiv.congr_arg _ _ _ _ _ _
(kaehler_differential.quotient_cotangent_ideal_ring_equiv R S) _ _ e)).trans e₂.symm },
{ intro e, apply (kaehler_differential.quotient_cotangent_ideal_ring_equiv R S).injective,
exact e₁.symm.trans (e.trans e₂) }
end
/-- Derivations into `Ω[S⁄R]` is equivalent to derivations
into `(kaehler_differential.ideal R S).cotangent_ideal` -/
-- This has type
-- `derivation R S Ω[ S / R ] ≃ₗ[R] derivation R S (kaehler_differential.ideal R S).cotangent_ideal`
-- But lean times-out if this is given explicitly.
noncomputable
def kaehler_differential.End_equiv_derivation' :=
@linear_equiv.comp_der R _ _ _ _ Ω[S⁄R] _ _ _ _ _ _ _ _ _
((kaehler_differential.ideal R S).cotangent_equiv_ideal.restrict_scalars S)
/-- (Implementation) An `equiv` version of `kaehler_differential.End_equiv_aux`.
Used in `kaehler_differential.End_equiv`. -/
def kaehler_differential.End_equiv_aux_equiv :
{f // (ideal.quotient.mkₐ R (kaehler_differential.ideal R S).cotangent_ideal).comp f =
is_scalar_tower.to_alg_hom R S _ } ≃
{ f // (tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S).ker_square_lift.comp f = alg_hom.id R S } :=
(equiv.refl _).subtype_equiv (kaehler_differential.End_equiv_aux R S)
/--
The endomorphisms of `Ω[S⁄R]` corresponds to sections of the surjection `S ⊗[R] S ⧸ J ^ 2 →ₐ[R] S`,
with `J` being the kernel of the multiplication map `S ⊗[R] S →ₐ[R] S`.
-/
noncomputable
def kaehler_differential.End_equiv :
module.End S Ω[S⁄R] ≃
{ f // (tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S).ker_square_lift.comp f = alg_hom.id R S } :=
(kaehler_differential.linear_map_equiv_derivation R S).to_equiv.trans $
(kaehler_differential.End_equiv_derivation' R S).to_equiv.trans $
(derivation_to_square_zero_equiv_lift
(kaehler_differential.ideal R S).cotangent_ideal
(kaehler_differential.ideal R S).cotangent_ideal_square).trans $
kaehler_differential.End_equiv_aux_equiv R S
section presentation
open kaehler_differential (D)
open finsupp (single)
/-- The `S`-submodule of `S →₀ S` (the direct sum of copies of `S` indexed by `S`) generated by
the relations:
1. `dx + dy = d(x + y)`
2. `x dy + y dx = d(x * y)`
3. `dr = 0` for `r ∈ R`
where `db` is the unit in the copy of `S` with index `b`.
This is the kernel of the surjection `finsupp.total S Ω[S⁄R] S (kaehler_differential.D R S)`.
See `kaehler_differential.ker_total_eq` and `kaehler_differential.total_surjective`.
-/
noncomputable
def kaehler_differential.ker_total : submodule S (S →₀ S) :=
submodule.span S
((set.range (λ (x : S × S), single x.1 1 + single x.2 1 - single (x.1 + x.2) 1)) ∪
(set.range (λ (x : S × S), single x.2 x.1 + single x.1 x.2 - single (x.1 * x.2) 1)) ∪
(set.range (λ x : R, single (algebra_map R S x) 1)))
local notation x `𝖣` y := (kaehler_differential.ker_total R S).mkq (single y x)
lemma kaehler_differential.ker_total_mkq_single_add (x y z) :
(z 𝖣 (x + y)) = (z 𝖣 x) + (z 𝖣 y) :=
begin
rw [← map_add, eq_comm, ← sub_eq_zero, ← map_sub, submodule.mkq_apply,
submodule.quotient.mk_eq_zero],
simp_rw [← finsupp.smul_single_one _ z, ← smul_add, ← smul_sub],
exact submodule.smul_mem _ _ (submodule.subset_span (or.inl $ or.inl $ ⟨⟨_, _⟩, rfl⟩)),
end
lemma kaehler_differential.ker_total_mkq_single_mul (x y z) :
(z 𝖣 (x * y)) = ((z * x) 𝖣 y) + ((z * y) 𝖣 x) :=
begin
rw [← map_add, eq_comm, ← sub_eq_zero, ← map_sub, submodule.mkq_apply,
submodule.quotient.mk_eq_zero],
simp_rw [← finsupp.smul_single_one _ z, ← @smul_eq_mul _ _ z,
← finsupp.smul_single, ← smul_add, ← smul_sub],
exact submodule.smul_mem _ _ (submodule.subset_span (or.inl $ or.inr $ ⟨⟨_, _⟩, rfl⟩)),
end
lemma kaehler_differential.ker_total_mkq_single_algebra_map (x y) :
(y 𝖣 (algebra_map R S x)) = 0 :=
begin
rw [submodule.mkq_apply, submodule.quotient.mk_eq_zero, ← finsupp.smul_single_one _ y],
exact submodule.smul_mem _ _ (submodule.subset_span (or.inr $ ⟨_, rfl⟩)),
end
lemma kaehler_differential.ker_total_mkq_single_algebra_map_one (x) :
(x 𝖣 1) = 0 :=
begin
rw [← (algebra_map R S).map_one, kaehler_differential.ker_total_mkq_single_algebra_map],
end
lemma kaehler_differential.ker_total_mkq_single_smul (r : R) (x y) :
(y 𝖣 (r • x)) = r • (y 𝖣 x) :=
begin
rw [algebra.smul_def, kaehler_differential.ker_total_mkq_single_mul,
kaehler_differential.ker_total_mkq_single_algebra_map, add_zero,
← linear_map.map_smul_of_tower, finsupp.smul_single, mul_comm, algebra.smul_def],
end
/-- The (universal) derivation into `(S →₀ S) ⧸ kaehler_differential.ker_total R S`. -/
noncomputable
def kaehler_differential.derivation_quot_ker_total :
derivation R S ((S →₀ S) ⧸ kaehler_differential.ker_total R S) :=
{ to_fun := λ x, 1 𝖣 x,
map_add' := λ x y, kaehler_differential.ker_total_mkq_single_add _ _ _ _ _,
map_smul' := λ r s, kaehler_differential.ker_total_mkq_single_smul _ _ _ _ _,
map_one_eq_zero' := kaehler_differential.ker_total_mkq_single_algebra_map_one _ _ _,
leibniz' := λ a b, (kaehler_differential.ker_total_mkq_single_mul _ _ _ _ _).trans
(by { simp_rw [← finsupp.smul_single_one _ (1 * _ : S)], dsimp, simp }) }
lemma kaehler_differential.derivation_quot_ker_total_apply (x) :
kaehler_differential.derivation_quot_ker_total R S x = (1 𝖣 x) := rfl
lemma kaehler_differential.derivation_quot_ker_total_lift_comp_total :
(kaehler_differential.derivation_quot_ker_total R S).lift_kaehler_differential.comp
(finsupp.total S Ω[S⁄R] S (kaehler_differential.D R S)) = submodule.mkq _ :=
begin
apply finsupp.lhom_ext,
intros a b,
conv_rhs { rw [← finsupp.smul_single_one a b, linear_map.map_smul] },
simp [kaehler_differential.derivation_quot_ker_total_apply],
end
lemma kaehler_differential.ker_total_eq :
(finsupp.total S Ω[S⁄R] S (kaehler_differential.D R S)).ker =
kaehler_differential.ker_total R S :=
begin
apply le_antisymm,
{ conv_rhs { rw ← (kaehler_differential.ker_total R S).ker_mkq },
rw ← kaehler_differential.derivation_quot_ker_total_lift_comp_total,
exact linear_map.ker_le_ker_comp _ _ },
{ rw [kaehler_differential.ker_total, submodule.span_le],
rintros _ ((⟨⟨x, y⟩, rfl⟩|⟨⟨x, y⟩, rfl⟩)|⟨x, rfl⟩); dsimp; simp [linear_map.mem_ker] },
end
lemma kaehler_differential.total_surjective :
function.surjective (finsupp.total S Ω[S⁄R] S (kaehler_differential.D R S)) :=
begin
rw [← linear_map.range_eq_top, finsupp.range_total, kaehler_differential.span_range_derivation],
end
/-- `Ω[S⁄R]` is isomorphic to `S` copies of `S` with kernel `kaehler_differential.ker_total`. -/
@[simps] noncomputable
def kaehler_differential.quot_ker_total_equiv :
((S →₀ S) ⧸ kaehler_differential.ker_total R S) ≃ₗ[S] Ω[S⁄R] :=
{ inv_fun := (kaehler_differential.derivation_quot_ker_total R S).lift_kaehler_differential,
left_inv := begin
intro x,
obtain ⟨x, rfl⟩ := submodule.mkq_surjective _ x,
exact linear_map.congr_fun
(kaehler_differential.derivation_quot_ker_total_lift_comp_total R S : _) x,
end,
right_inv := begin
intro x,
obtain ⟨x, rfl⟩ := kaehler_differential.total_surjective R S x,
erw linear_map.congr_fun
(kaehler_differential.derivation_quot_ker_total_lift_comp_total R S : _) x,
refl
end,
..(kaehler_differential.ker_total R S).liftq
(finsupp.total S Ω[S⁄R] S (kaehler_differential.D R S))
(kaehler_differential.ker_total_eq R S).ge }
lemma kaehler_differential.quot_ker_total_equiv_symm_comp_D :
(kaehler_differential.quot_ker_total_equiv R S).symm.to_linear_map.comp_der
(kaehler_differential.D R S) = kaehler_differential.derivation_quot_ker_total R S :=
by convert
(kaehler_differential.derivation_quot_ker_total R S).lift_kaehler_differential_comp using 0
variables (A B : Type*) [comm_ring A] [comm_ring B] [algebra R A] [algebra S B] [algebra R B]
variables [algebra A B] [is_scalar_tower R S B] [is_scalar_tower R A B]
-- The map `(A →₀ A) →ₗ[A] (B →₀ B)`
local notation `finsupp_map` := ((finsupp.map_range.linear_map (algebra.of_id A B).to_linear_map)
.comp (finsupp.lmap_domain A A (algebra_map A B)))
lemma kaehler_differential.ker_total_map (h : function.surjective (algebra_map A B)) :
(kaehler_differential.ker_total R A).map finsupp_map ⊔
submodule.span A (set.range (λ x : S, single (algebra_map S B x) (1 : B))) =
(kaehler_differential.ker_total S B).restrict_scalars _ :=
begin
rw [kaehler_differential.ker_total, submodule.map_span, kaehler_differential.ker_total,
← submodule.span_eq_restrict_scalars _ _ _ _ h],
simp_rw [set.image_union, submodule.span_union, ← set.image_univ, set.image_image,
set.image_univ, map_sub, map_add],
simp only [linear_map.comp_apply, finsupp.map_range.linear_map_apply, finsupp.map_range_single,
finsupp.lmap_domain_apply, finsupp.map_domain_single, alg_hom.to_linear_map_apply,
algebra.of_id_apply, ← is_scalar_tower.algebra_map_apply, map_one, map_add, map_mul],
simp_rw [sup_assoc, ← (h.prod_map h).range_comp],
congr' 3,
rw [sup_eq_right],
apply submodule.span_mono,
simp_rw is_scalar_tower.algebra_map_apply R S B,
exact set.range_comp_subset_range (algebra_map R S) (λ x, single (algebra_map S B x) (1 : B))
end
end presentation
section exact_sequence
local attribute [irreducible] kaehler_differential
/- We have the commutative diagram
A --→ B
↑ ↑
| |
R --→ S -/
variables (A B : Type*) [comm_ring A] [comm_ring B] [algebra R A] [algebra R B]
variables [algebra A B] [algebra S B] [is_scalar_tower R A B] [is_scalar_tower R S B]
variables {R B}
/-- For a tower `R → A → B` and an `R`-derivation `B → M`, we may compose with `A → B` to obtain an
`R`-derivation `A → M`. -/
def derivation.comp_algebra_map [module A M] [module B M] [is_scalar_tower A B M]
(d : derivation R B M) : derivation R A M :=
{ map_one_eq_zero' := by simp,
leibniz' := λ a b, by simp,
to_linear_map := d.to_linear_map.comp (is_scalar_tower.to_alg_hom R A B).to_linear_map }
variables (R B)
/-- The map `Ω[A⁄R] →ₗ[A] Ω[B⁄R]` given a square
A --→ B
↑ ↑
| |
R --→ S -/
def kaehler_differential.map : Ω[A⁄R] →ₗ[A] Ω[B⁄S] :=
derivation.lift_kaehler_differential
(((kaehler_differential.D S B).restrict_scalars R).comp_algebra_map A)
lemma kaehler_differential.map_comp_der :
(kaehler_differential.map R S A B).comp_der (kaehler_differential.D R A) =
(((kaehler_differential.D S B).restrict_scalars R).comp_algebra_map A) :=
derivation.lift_kaehler_differential_comp _
lemma kaehler_differential.map_D (x : A) :
kaehler_differential.map R S A B (kaehler_differential.D R A x) =
kaehler_differential.D S B (algebra_map A B x) :=
derivation.congr_fun (kaehler_differential.map_comp_der R S A B) x
open is_scalar_tower (to_alg_hom)
lemma kaehler_differential.map_surjective_of_surjective
(h : function.surjective (algebra_map A B)) :
function.surjective (kaehler_differential.map R S A B) :=
begin
rw [← linear_map.range_eq_top, _root_.eq_top_iff, ← @submodule.restrict_scalars_top B A,
← kaehler_differential.span_range_derivation, ← submodule.span_eq_restrict_scalars _ _ _ _ h,
submodule.span_le],
rintros _ ⟨x, rfl⟩,
obtain ⟨y, rfl⟩ := h x,
rw ← kaehler_differential.map_D R S A B,
exact ⟨_, rfl⟩,
end
/-- The lift of the map `Ω[A⁄R] →ₗ[A] Ω[B⁄R]` to the base change along `A → B`.
This is the first map in the exact sequence `B ⊗[A] Ω[A⁄R] → Ω[B⁄R] → Ω[B⁄A] → 0`. -/
noncomputable
def kaehler_differential.map_base_change : B ⊗[A] Ω[A⁄R] →ₗ[B] Ω[B⁄R] :=
(tensor_product.is_base_change A Ω[A⁄R] B).lift (kaehler_differential.map R R A B)
@[simp]
lemma kaehler_differential.map_base_change_tmul (x : B) (y : Ω[A⁄R]) :
kaehler_differential.map_base_change R A B (x ⊗ₜ y) =
x • kaehler_differential.map R R A B y :=
begin
conv_lhs { rw [← mul_one x, ← smul_eq_mul, ← tensor_product.smul_tmul', linear_map.map_smul] },
congr' 1,
exact is_base_change.lift_eq _ _ _
end
end exact_sequence
end kaehler_differential
|
24258a917d5b17ea1f2563b6aff2dbed4d889c03 | 947fa6c38e48771ae886239b4edce6db6e18d0fb | /src/group_theory/sylow.lean | 304130b9e311361424e735f6b62e6e5e6a35bf32 | [
"Apache-2.0"
] | permissive | ramonfmir/mathlib | c5dc8b33155473fab97c38bd3aa6723dc289beaa | 14c52e990c17f5a00c0cc9e09847af16fabbed25 | refs/heads/master | 1,661,979,343,526 | 1,660,830,384,000 | 1,660,830,384,000 | 182,072,989 | 0 | 0 | null | 1,555,585,876,000 | 1,555,585,876,000 | null | UTF-8 | Lean | false | false | 33,603 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import data.nat.factorization.basic
import data.set_like.fintype
import group_theory.group_action.conj_act
import group_theory.p_group
import group_theory.noncomm_pi_coprod
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `is_p_group.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modeq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open fintype mul_action subgroup
section infinite_sylow
variables (p : ℕ) (G : Type*) [group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure sylow extends subgroup G :=
(is_p_group' : is_p_group p to_subgroup)
(is_maximal' : ∀ {Q : subgroup G}, is_p_group p Q → to_subgroup ≤ Q → Q = to_subgroup)
variables {p} {G}
namespace sylow
instance : has_coe (sylow p G) (subgroup G) := ⟨sylow.to_subgroup⟩
@[simp] lemma to_subgroup_eq_coe {P : sylow p G} : P.to_subgroup = ↑P := rfl
@[ext] lemma ext {P Q : sylow p G} (h : (P : subgroup G) = Q) : P = Q :=
by cases P; cases Q; congr'
lemma ext_iff {P Q : sylow p G} : P = Q ↔ (P : subgroup G) = Q :=
⟨congr_arg coe, ext⟩
instance : set_like (sylow p G) G :=
{ coe := coe,
coe_injective' := λ P Q h, ext (set_like.coe_injective h) }
instance : subgroup_class (sylow p G) G :=
{ mul_mem := λ s, s.mul_mem',
one_mem := λ s, s.one_mem',
inv_mem := λ s, s.inv_mem' }
variables (P : sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mul_action_left {α : Type*} [mul_action G α] : mul_action P α :=
subgroup.mul_action ↑P
variables {K : Type*} [group K] (ϕ : K →* G) {N : subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comap_of_ker_is_p_group (hϕ : is_p_group p ϕ.ker) (h : ↑P ≤ ϕ.range) : sylow p K :=
{ P.1.comap ϕ with
is_p_group' := P.2.comap_of_ker_is_p_group ϕ hϕ,
is_maximal' := λ Q hQ hle, by
{ rw ← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle)),
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }, }
@[simp] lemma coe_comap_of_ker_is_p_group (hϕ : is_p_group p ϕ.ker) (h : ↑P ≤ ϕ.range) :
↑(P.comap_of_ker_is_p_group ϕ hϕ h) = subgroup.comap ϕ ↑P := rfl
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comap_of_injective (hϕ : function.injective ϕ) (h : ↑P ≤ ϕ.range) : sylow p K :=
P.comap_of_ker_is_p_group ϕ (is_p_group.ker_is_p_group_of_injective hϕ) h
@[simp] lemma coe_comap_of_injective (hϕ : function.injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comap_of_injective ϕ hϕ h) = subgroup.comap ϕ ↑P := rfl
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
def subtype (h : ↑P ≤ N) : sylow p N :=
P.comap_of_injective N.subtype subtype.coe_injective (by simp [h])
@[simp] lemma coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroup.comap N.subtype ↑P := rfl
end sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
lemma is_p_group.exists_le_sylow {P : subgroup G} (hP : is_p_group p P) :
∃ Q : sylow p G, P ≤ Q :=
exists.elim (zorn_nonempty_partial_order₀ {Q : subgroup G | is_p_group p Q} (λ c hc1 hc2 Q hQ,
⟨ { carrier := ⋃ (R : c), R,
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩,
inv_mem' := λ g ⟨_, ⟨R, rfl⟩, hg⟩, ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩,
mul_mem' := λ g h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩, (hc2.total R.2 S.2).elim
(λ T, ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) (λ T, ⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩) },
λ ⟨g, _, ⟨S, rfl⟩, hg⟩, by
{ refine exists_imp_exists (λ k hk, _) (hc1 S.2 ⟨g, hg⟩),
rwa [subtype.ext_iff, coe_pow] at hk ⊢ },
λ M hM g hg, ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩) P hP) (λ Q ⟨hQ1, hQ2, hQ3⟩, ⟨⟨Q, hQ1, hQ3⟩, hQ2⟩)
instance sylow.nonempty : nonempty (sylow p G) :=
nonempty_of_exists is_p_group.of_bot.exists_le_sylow
noncomputable instance sylow.inhabited : inhabited (sylow p G) :=
classical.inhabited_of_nonempty sylow.nonempty
lemma sylow.exists_comap_eq_of_ker_is_p_group {H : Type*} [group H] (P : sylow p H)
{f : H →* G} (hf : is_p_group p f.ker) : ∃ Q : sylow p G, (Q : subgroup G).comap f = P :=
exists_imp_exists (λ Q hQ, P.3 (Q.2.comap_of_ker_is_p_group f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
lemma sylow.exists_comap_eq_of_injective {H : Type*} [group H] (P : sylow p H)
{f : H →* G} (hf : function.injective f) : ∃ Q : sylow p G, (Q : subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_is_p_group (is_p_group.ker_is_p_group_of_injective hf)
lemma sylow.exists_comap_subtype_eq {H : subgroup G} (P : sylow p H) :
∃ Q : sylow p G, (Q : subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective subtype.coe_injective
/-- If the kernel of `f : H →* G` is a `p`-group,
then `fintype (sylow p G)` implies `fintype (sylow p H)`. -/
noncomputable def sylow.fintype_of_ker_is_p_group {H : Type*} [group H]
{f : H →* G} (hf : is_p_group p f.ker) [fintype (sylow p G)] : fintype (sylow p H) :=
let h_exists := λ P : sylow p H, P.exists_comap_eq_of_ker_is_p_group hf,
g : sylow p H → sylow p G := λ P, classical.some (h_exists P),
hg : ∀ P : sylow p H, (g P).1.comap f = P := λ P, classical.some_spec (h_exists P) in
fintype.of_injective g (λ P Q h, sylow.ext (by simp only [←hg, h]))
/-- If `f : H →* G` is injective, then `fintype (sylow p G)` implies `fintype (sylow p H)`. -/
noncomputable def sylow.fintype_of_injective {H : Type*} [group H]
{f : H →* G} (hf : function.injective f) [fintype (sylow p G)] : fintype (sylow p H) :=
sylow.fintype_of_ker_is_p_group (is_p_group.ker_is_p_group_of_injective hf)
/-- If `H` is a subgroup of `G`, then `fintype (sylow p G)` implies `fintype (sylow p H)`. -/
noncomputable instance (H : subgroup G) [fintype (sylow p G)] : fintype (sylow p H) :=
sylow.fintype_of_injective (show function.injective H.subtype, from subtype.coe_injective)
/-- If `H` is a subgroup of `G`, then `finite (sylow p G)` implies `finite (sylow p H)`. -/
instance (H : subgroup G) [finite (sylow p G)] : finite (sylow p H) :=
by { casesI nonempty_fintype (sylow p G), apply_instance }
open_locale pointwise
/-- `subgroup.pointwise_mul_action` preserves Sylow subgroups. -/
instance sylow.pointwise_mul_action {α : Type*} [group α] [mul_distrib_mul_action α G] :
mul_action α (sylow p G) :=
{ smul := λ g P, ⟨g • P, P.2.map _, λ Q hQ hS, inv_smul_eq_iff.mp (P.3 (hQ.map _)
(λ s hs, (congr_arg (∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs)))))⟩,
one_smul := λ P, sylow.ext (one_smul α P),
mul_smul := λ g h P, sylow.ext (mul_smul g h P) }
lemma sylow.pointwise_smul_def {α : Type*} [group α] [mul_distrib_mul_action α G]
{g : α} {P : sylow p G} : ↑(g • P) = g • (P : subgroup G) := rfl
instance sylow.mul_action : mul_action G (sylow p G) :=
comp_hom _ mul_aut.conj
lemma sylow.smul_def {g : G} {P : sylow p G} : g • P = mul_aut.conj g • P := rfl
lemma sylow.coe_subgroup_smul {g : G} {P : sylow p G} :
↑(g • P) = mul_aut.conj g • (P : subgroup G) := rfl
lemma sylow.coe_smul {g : G} {P : sylow p G} :
↑(g • P) = mul_aut.conj g • (P : set G) := rfl
lemma sylow.smul_le {P : sylow p G} {H : subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
subgroup.conj_smul_le_of_le hP h
lemma sylow.smul_subtype {P : sylow p G} {H : subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (sylow.smul_le hP h) :=
sylow.ext (subgroup.conj_smul_subgroup_of hP h)
lemma sylow.smul_eq_iff_mem_normalizer {g : G} {P : sylow p G} :
g • P = P ↔ g ∈ (P : subgroup G).normalizer :=
begin
rw [eq_comm, set_like.ext_iff, ←inv_mem_iff, mem_normalizer_iff, inv_inv],
exact forall_congr (λ h, iff_congr iff.rfl ⟨λ ⟨a, b, c⟩, (congr_arg _ c).mp
((congr_arg (∈ P.1) (mul_aut.inv_apply_self G (mul_aut.conj g) a)).mpr b),
λ hh, ⟨(mul_aut.conj g)⁻¹ h, hh, mul_aut.apply_inv_self G (mul_aut.conj g) h⟩⟩),
end
lemma sylow.smul_eq_of_normal {g : G} {P : sylow p G} [h : (P : subgroup G).normal] :
g • P = P :=
by simp only [sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
lemma subgroup.sylow_mem_fixed_points_iff (H : subgroup G) {P : sylow p G} :
P ∈ fixed_points H (sylow p G) ↔ H ≤ (P : subgroup G).normalizer :=
by simp_rw [set_like.le_def, ←sylow.smul_eq_iff_mem_normalizer]; exact subtype.forall
lemma is_p_group.inf_normalizer_sylow {P : subgroup G} (hP : is_p_group p P) (Q : sylow p G) :
P ⊓ (Q : subgroup G).normalizer = P ⊓ Q :=
le_antisymm (le_inf inf_le_left (sup_eq_right.mp (Q.3 (hP.to_inf_left.to_sup_of_normal_right'
Q.2 inf_le_right) le_sup_right))) (inf_le_inf_left P le_normalizer)
lemma is_p_group.sylow_mem_fixed_points_iff
{P : subgroup G} (hP : is_p_group p P) {Q : sylow p G} :
Q ∈ fixed_points P (sylow p G) ↔ P ≤ Q :=
by rw [P.sylow_mem_fixed_points_iff, ←inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : fact p.prime] [finite (sylow p G)] : is_pretransitive G (sylow p G) :=
⟨λ P Q, by
{ classical,
casesI nonempty_fintype (sylow p G),
have H := λ {R : sylow p G} {S : orbit G P},
calc S ∈ fixed_points R (orbit G P)
↔ S.1 ∈ fixed_points R (sylow p G) : forall_congr (λ a, subtype.ext_iff)
... ↔ R.1 ≤ S : R.2.sylow_mem_fixed_points_iff
... ↔ S.1.1 = R : ⟨λ h, R.3 S.1.2 h, ge_of_eq⟩,
suffices : set.nonempty (fixed_points Q (orbit G P)),
{ exact exists.elim this (λ R hR, (congr_arg _ (sylow.ext (H.mp hR))).mp R.2) },
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card,
refine λ h, hp.out.not_dvd_one (nat.modeq_zero_iff_dvd.mp _),
calc 1 = card (fixed_points P (orbit G P)) : _
... ≡ card (orbit G P) [MOD p] : (P.2.card_modeq_card_fixed_points (orbit G P)).symm
... ≡ 0 [MOD p] : nat.modeq_zero_iff_dvd.mpr h,
rw ← set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P),
refine card_congr' (congr_arg _ (eq.symm _)),
rw set.eq_singleton_iff_unique_mem,
exact ⟨H.mpr rfl, λ R h, subtype.ext (sylow.ext (H.mp h))⟩ }⟩
variables (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
lemma card_sylow_modeq_one [fact p.prime] [fintype (sylow p G)] : card (sylow p G) ≡ 1 [MOD p] :=
begin
refine sylow.nonempty.elim (λ P : sylow p G, _),
have : fixed_points P.1 (sylow p G) = {P} :=
set.ext (λ Q : sylow p G, calc Q ∈ fixed_points P (sylow p G)
↔ P.1 ≤ Q : P.2.sylow_mem_fixed_points_iff
... ↔ Q.1 = P.1 : ⟨P.3 Q.2, ge_of_eq⟩
... ↔ Q ∈ {P} : sylow.ext_iff.symm.trans set.mem_singleton_iff.symm),
haveI : fintype (fixed_points P.1 (sylow p G)), { rw this, apply_instance },
have : card (fixed_points P.1 (sylow p G)) = 1, { simp [this] },
exact (P.2.card_modeq_card_fixed_points (sylow p G)).trans (by rw this),
end
lemma not_dvd_card_sylow [hp : fact p.prime] [fintype (sylow p G)] : ¬ p ∣ card (sylow p G) :=
λ h, hp.1.ne_one (nat.dvd_one.mp ((nat.modeq_iff_dvd' zero_le_one).mp
((nat.modeq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modeq_one p G))))
variables {p} {G}
/-- Sylow subgroups are isomorphic -/
def sylow.equiv_smul (P : sylow p G) (g : G) : P ≃* (g • P : sylow p G) :=
equiv_smul (mul_aut.conj g) ↑P
/-- Sylow subgroups are isomorphic -/
noncomputable def sylow.equiv [fact p.prime] [finite (sylow p G)] (P Q : sylow p G) :
P ≃* Q :=
begin
rw ← classical.some_spec (exists_smul_eq G P Q),
exact P.equiv_smul (classical.some (exists_smul_eq G P Q)),
end
@[simp] lemma sylow.orbit_eq_top [fact p.prime] [finite (sylow p G)] (P : sylow p G) :
orbit G P = ⊤ :=
top_le_iff.mp (λ Q hQ, exists_smul_eq G P Q)
lemma sylow.stabilizer_eq_normalizer (P : sylow p G) :
stabilizer G P = (P : subgroup G).normalizer :=
ext (λ g, sylow.smul_eq_iff_mem_normalizer)
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def sylow.equiv_quotient_normalizer [fact p.prime] [fintype (sylow p G)]
(P : sylow p G) : sylow p G ≃ G ⧸ (P : subgroup G).normalizer :=
calc sylow p G ≃ (⊤ : set (sylow p G)) : (equiv.set.univ (sylow p G)).symm
... ≃ orbit G P : by rw P.orbit_eq_top
... ≃ G ⧸ (stabilizer G P) : orbit_equiv_quotient_stabilizer G P
... ≃ G ⧸ (P : subgroup G).normalizer : by rw P.stabilizer_eq_normalizer
noncomputable instance [fact p.prime] [fintype (sylow p G)] (P : sylow p G) :
fintype (G ⧸ (P : subgroup G).normalizer) :=
of_equiv (sylow p G) P.equiv_quotient_normalizer
lemma card_sylow_eq_card_quotient_normalizer [fact p.prime] [fintype (sylow p G)] (P : sylow p G) :
card (sylow p G) = card (G ⧸ (P : subgroup G).normalizer) :=
card_congr P.equiv_quotient_normalizer
lemma card_sylow_eq_index_normalizer [fact p.prime] [fintype (sylow p G)] (P : sylow p G) :
card (sylow p G) = (P : subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : subgroup G).normalizer.index_eq_card.symm
lemma card_sylow_dvd_index [fact p.prime] [fintype (sylow p G)] (P : sylow p G) :
card (sylow p G) ∣ (P : subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans (index_dvd_of_le le_normalizer)
lemma not_dvd_index_sylow' [hp : fact p.prime] (P : sylow p G) [(P : subgroup G).normal]
(hP : (P : subgroup G).index ≠ 0) : ¬ p ∣ (P : subgroup G).index :=
begin
intro h,
haveI : fintype (G ⧸ (P : subgroup G)) := fintype_of_index_ne_zero hP,
rw index_eq_card at h,
obtain ⟨x, hx⟩ := exists_prime_order_of_dvd_card p h,
have h := is_p_group.of_card ((order_eq_card_zpowers.symm.trans hx).trans (pow_one p).symm),
let Q := (zpowers x).comap (quotient_group.mk' (P : subgroup G)),
have hQ : is_p_group p Q,
{ apply h.comap_of_ker_is_p_group,
rw [quotient_group.ker_mk],
exact P.2 },
replace hp := mt order_of_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one),
rw [←zpowers_eq_bot, ←ne, ←bot_lt_iff_ne_bot, ←comap_lt_comap_of_surjective
(quotient_group.mk'_surjective _), monoid_hom.comap_bot, quotient_group.ker_mk] at hp,
exact hp.ne' (P.3 hQ hp.le),
end
lemma not_dvd_index_sylow [hp : fact p.prime] [finite (sylow p G)] (P : sylow p G)
(hP : relindex ↑P (P : subgroup G).normalizer ≠ 0) : ¬ p ∣ (P : subgroup G).index :=
begin
casesI nonempty_fintype (sylow p G),
rw [←relindex_mul_index le_normalizer, ←card_sylow_eq_index_normalizer],
haveI : (P.subtype le_normalizer : subgroup (P : subgroup G).normalizer).normal :=
subgroup.normal_in_normalizer,
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer) hP,
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G),
end
/-- Frattini's Argument: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
lemma sylow.normalizer_sup_eq_top {p : ℕ} [fact p.prime] {N : subgroup G} [N.normal]
[finite (sylow p N)] (P : sylow p N) : ((↑P : subgroup N).map N.subtype).normalizer ⊔ N = ⊤ :=
begin
refine top_le_iff.mp (λ g hg, _),
obtain ⟨n, hn⟩ := exists_smul_eq N ((mul_aut.conj_normal g : mul_aut N) • P) P,
rw [←inv_mul_cancel_left ↑n g, sup_comm],
apply mul_mem_sup (N.inv_mem n.2),
rw [sylow.smul_def, ←mul_smul, ←mul_aut.conj_normal_coe, ←mul_aut.conj_normal.map_mul,
sylow.ext_iff, sylow.pointwise_smul_def, pointwise_smul_def] at hn,
refine λ x, (mem_map_iff_mem (show function.injective (mul_aut.conj (↑n * g)).to_monoid_hom,
from (mul_aut.conj (↑n * g)).injective)).symm.trans _,
rw [map_map, ←(congr_arg (map N.subtype) hn), map_map],
refl,
end
end infinite_sylow
open equiv equiv.perm finset function list quotient_group
open_locale big_operators
universes u v w
variables {G : Type u} {α : Type v} {β : Type w} [group G]
local attribute [instance, priority 10] subtype.fintype set_fintype classical.prop_decidable
lemma quotient_group.card_preimage_mk [fintype G] (s : subgroup G)
(t : set (G ⧸ s)) : fintype.card (quotient_group.mk ⁻¹' t) =
fintype.card s * fintype.card t :=
by rw [← fintype.card_prod, fintype.card_congr
(preimage_mk_equiv_subgroup_times_set _ _)]
namespace sylow
open subgroup submonoid mul_action
lemma mem_fixed_points_mul_left_cosets_iff_mem_normalizer {H : subgroup G} [finite ↥(H : set G)]
{x : G} :
(x : G ⧸ H) ∈ fixed_points H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨λ hx, have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x,
from λ _, ((mem_fixed_points' _).1 hx _),
inv_mem_iff.1 (mem_normalizer_fintype (λ n (hn : n ∈ H),
have (n⁻¹ * x)⁻¹ * x ∈ H := quotient_group.eq.1 (ha (mem_orbit _ ⟨n⁻¹, H.inv_mem hn⟩)),
show _ ∈ H, by {rw [mul_inv_rev, inv_inv] at this, convert this, rw inv_inv}
)),
λ (hx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H),
(mem_fixed_points' _).2 $ λ y, quotient.induction_on' y $ λ y hy, quotient_group.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy in
have hb₂ : (b * x)⁻¹ * y ∈ H := quotient_group.eq.1 hb₂,
inv_mem_iff.1 $ (hx _).2 $ (mul_mem_cancel_left (inv_mem hb₁)).1
$ by rw hx at hb₂;
simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixed_points_mul_left_cosets_equiv_quotient (H : subgroup G) [finite (H : set G)] :
mul_action.fixed_points H (G ⧸ H) ≃
normalizer H ⧸ (subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
@subtype_quotient_equiv_quotient_subtype G (normalizer H : set G) (id _) (id _) (fixed_points _ _)
(λ a, (@mem_fixed_points_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by { intros, rw setoid_has_equiv, simp only [left_rel_apply], refl })
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
lemma card_quotient_normalizer_modeq_card_quotient [fintype G] {p : ℕ} {n : ℕ} [hp : fact p.prime]
{H : subgroup G} (hH : fintype.card H = p ^ n) :
card (normalizer H ⧸ (subgroup.comap ((normalizer H).subtype : normalizer H →* G) H))
≡ card (G ⧸ H) [MOD p] :=
begin
rw [← fintype.card_congr (fixed_points_mul_left_cosets_equiv_quotient H)],
exact ((is_p_group.of_card hH).card_modeq_card_fixed_points _).symm
end
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
lemma card_normalizer_modeq_card [fintype G] {p : ℕ} {n : ℕ} [hp : fact p.prime]
{H : subgroup G} (hH : fintype.card H = p ^ n) :
card (normalizer H) ≡ card G [MOD p ^ (n + 1)] :=
have subgroup.comap ((normalizer H).subtype : normalizer H →* G) H ≃ H,
from set.bij_on.equiv (normalizer H).subtype
⟨λ _, id, λ _ _ _ _ h, subtype.val_injective h,
λ x hx, ⟨⟨x, le_normalizer hx⟩, hx, rfl⟩⟩,
begin
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup
(subgroup.comap ((normalizer H).subtype : normalizer H →* G) H),
fintype.card_congr this, hH, pow_succ],
exact (card_quotient_normalizer_modeq_card_quotient hH).mul_right' _
end
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
lemma prime_dvd_card_quotient_normalizer [fintype G] {p : ℕ} {n : ℕ} [hp : fact p.prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : subgroup G} (hH : fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ (subgroup.comap ((normalizer H).subtype : normalizer H →* G) H)) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd in
have hcard : card (G ⧸ H) = s * p :=
(nat.mul_left_inj (show card H > 0, from fintype.card_pos_iff.2
⟨⟨1, H.one_mem⟩⟩)).1
(by rwa [← card_eq_card_quotient_mul_card_subgroup H, hH, hs,
pow_succ', mul_assoc, mul_comm p]),
have hm : s * p % p =
card (normalizer H ⧸ (subgroup.comap ((normalizer H).subtype : normalizer H →* G) H)) % p :=
hcard ▸ (card_quotient_normalizer_modeq_card_quotient hH).symm,
nat.dvd_of_mod_eq_zero
(by rwa [nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
lemma prime_pow_dvd_card_normalizer [fintype G] {p : ℕ} {n : ℕ} [hp : fact p.prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : subgroup G} (hH : fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
nat.modeq_zero_iff_dvd.1 ((card_normalizer_modeq_card hH).trans
hdvd.modeq_zero_nat)
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [fintype G] {p : ℕ} {n : ℕ} [hp : fact p.prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : subgroup G} (hH : fintype.card H = p ^ n) :
∃ K : subgroup G, fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd in
have hcard : card (G ⧸ H) = s * p :=
(nat.mul_left_inj (show card H > 0, from fintype.card_pos_iff.2
⟨⟨1, H.one_mem⟩⟩)).1
(by rwa [← card_eq_card_quotient_mul_card_subgroup H, hH, hs,
pow_succ', mul_assoc, mul_comm p]),
have hm : s * p % p =
card (normalizer H ⧸ (subgroup.comap (normalizer H).subtype H)) % p :=
card_congr (fixed_points_mul_left_cosets_equiv_quotient H) ▸ hcard ▸
(is_p_group.of_card hH).card_modeq_card_fixed_points _,
have hm' : p ∣ card (normalizer H ⧸ (subgroup.comap (normalizer H).subtype H)) :=
nat.dvd_of_mod_eq_zero
(by rwa [nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm),
let ⟨x, hx⟩ := @exists_prime_order_of_dvd_card _ (quotient_group.quotient.group _) _ _ hp hm' in
have hequiv : H ≃ (subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
⟨λ a, ⟨⟨a.1, le_normalizer a.2⟩, a.2⟩, λ a, ⟨a.1.1, a.2⟩,
λ ⟨_, _⟩, rfl, λ ⟨⟨_, _⟩, _⟩, rfl⟩,
⟨subgroup.map ((normalizer H).subtype) (subgroup.comap
(quotient_group.mk' (comap H.normalizer.subtype H)) (zpowers x)),
begin
show card ↥(map H.normalizer.subtype
(comap (mk' (comap H.normalizer.subtype H)) (subgroup.zpowers x))) = p ^ (n + 1),
suffices : card ↥(subtype.val '' ((subgroup.comap (mk' (comap H.normalizer.subtype H))
(zpowers x)) : set (↥(H.normalizer)))) = p^(n+1),
{ convert this using 2 },
rw [set.card_image_of_injective
(subgroup.comap (mk' (comap H.normalizer.subtype H)) (zpowers x) : set (H.normalizer))
subtype.val_injective,
pow_succ', ← hH, fintype.card_congr hequiv, ← hx, order_eq_card_zpowers,
← fintype.card_prod],
exact @fintype.card_congr _ _ (id _) (id _) (preimage_mk_equiv_subgroup_times_set _ _)
end,
begin
assume y hy,
simp only [exists_prop, subgroup.coe_subtype, mk'_apply, subgroup.mem_map, subgroup.mem_comap],
refine ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩,
rw [zpow_zero, eq_comm, quotient_group.eq_one_iff],
simpa using hy
end⟩
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [fintype G] (p : ℕ) : ∀ {n m : ℕ} [hp : fact p.prime]
(hdvd : p ^ m ∣ card G) (H : subgroup G) (hH : card H = p ^ n) (hnm : n ≤ m),
∃ K : subgroup G, card K = p ^ m ∧ H ≤ K
| n m := λ hp hdvd H hH hnm,
(lt_or_eq_of_le hnm).elim
(λ hnm : n < m,
have h0m : 0 < m, from (lt_of_le_of_lt n.zero_le hnm),
have wf : m - 1 < m, from nat.sub_lt h0m zero_lt_one,
have hnm1 : n ≤ m - 1, from le_tsub_of_add_le_right hnm,
let ⟨K, hK⟩ := @exists_subgroup_card_pow_prime_le n (m - 1) hp
(nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1 in
have hdvd' : p ^ ((m - 1) + 1) ∣ card G, by rwa [tsub_add_cancel_of_le h0m.nat_succ_le],
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ hp hdvd' K hK.1 in
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
(λ hnm : n = m, ⟨H, by simp [hH, hnm]⟩)
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [fintype G] (p : ℕ) {n : ℕ} [fact p.prime]
(hdvd : p ^ n ∣ card G) : ∃ K : subgroup G, fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (by simp) n.zero_le in
⟨K, hK.1⟩
lemma pow_dvd_card_of_pow_dvd_card [fintype G] {p n : ℕ} [hp : fact p.prime] (P : sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P
index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left ((index_mul_card P.1).symm ▸ hdvd)
lemma dvd_card_of_dvd_card [fintype G] {p : ℕ} [fact p.prime] (P : sylow p G)
(hdvd : p ∣ card G) : p ∣ card P :=
begin
rw ← pow_one p at hdvd,
have key := P.pow_dvd_card_of_pow_dvd_card hdvd,
rwa pow_one at key,
end
/-- Sylow subgroups are Hall subgroups. -/
lemma card_coprime_index [fintype G] {p : ℕ} [hp : fact p.prime] (P : sylow p G) :
(card P).coprime (index (P : subgroup G)) :=
let ⟨n, hn⟩ := is_p_group.iff_card.mp P.2 in
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
lemma ne_bot_of_dvd_card [fintype G] {p : ℕ} [hp : fact p.prime] (P : sylow p G)
(hdvd : p ∣ card G) : (P : subgroup G) ≠ ⊥ :=
begin
refine λ h, hp.out.not_dvd_one _,
have key : p ∣ card (P : subgroup G) := P.dvd_card_of_dvd_card hdvd,
rwa [h, card_bot] at key,
end
/-- The cardinality of a Sylow group is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
lemma card_eq_multiplicity [fintype G] {p : ℕ} [hp : fact p.prime] (P : sylow p G) :
card P = p ^ nat.factorization (card G) p :=
begin
obtain ⟨n, heq : card P = _⟩ := is_p_group.iff_card.mp (P.is_p_group'),
refine nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (nat.ord_proj_dvd _ p)),
rw [heq, ←hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0, from card_ne_zero), ←heq],
exact P.1.card_subgroup_dvd_card,
end
lemma subsingleton_of_normal {p : ℕ} [fact p.prime] [finite (sylow p G)] (P : sylow p G)
(h : (P : subgroup G).normal) : subsingleton (sylow p G) :=
begin
apply subsingleton.intro,
intros Q R,
obtain ⟨x, h1⟩ := exists_smul_eq G P Q,
obtain ⟨x, h2⟩ := exists_smul_eq G P R,
rw sylow.smul_eq_of_normal at h1 h2,
rw [← h1, ← h2],
end
section pointwise
open_locale pointwise
lemma characteristic_of_normal {p : ℕ} [fact p.prime] [finite (sylow p G)] (P : sylow p G)
(h : (P : subgroup G).normal) :
(P : subgroup G).characteristic :=
begin
haveI := sylow.subsingleton_of_normal P h,
rw characteristic_iff_map_eq,
intros Φ,
show (Φ • P).to_subgroup = P.to_subgroup,
congr,
end
end pointwise
lemma normal_of_normalizer_normal {p : ℕ} [fact p.prime] [finite (sylow p G)]
(P : sylow p G) (hn : (↑P : subgroup G).normalizer.normal) :
(↑P : subgroup G).normal :=
by rw [←normalizer_eq_top, ←normalizer_sup_eq_top (P.subtype le_normalizer), coe_subtype,
map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _))), sup_idem]
@[simp] lemma normalizer_normalizer {p : ℕ} [fact p.prime] [finite (sylow p G)] (P : sylow p G) :
(↑P : subgroup G).normalizer.normalizer = (↑P : subgroup G).normalizer :=
begin
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer)),
simp_rw [←normalizer_eq_top, coe_subtype, ←comap_subtype_normalizer_eq le_normalizer,
←comap_subtype_normalizer_eq le_rfl, comap_subtype_self_eq_top] at this,
rw [←subtype_range (P : subgroup G).normalizer.normalizer, monoid_hom.range_eq_map, ←this rfl],
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _))),
end
lemma normal_of_all_max_subgroups_normal [finite G]
(hnc : ∀ (H : subgroup G), is_coatom H → H.normal)
{p : ℕ} [fact p.prime] [finite (sylow p G)] (P : sylow p G) :
(↑P : subgroup G).normal :=
normalizer_eq_top.mp begin
rcases eq_top_or_exists_le_coatom ((↑P : subgroup G).normalizer) with heq | ⟨K, hK, hNK⟩,
{ exact heq },
{ haveI := hnc _ hK,
have hPK := le_trans le_normalizer hNK,
let P' := P.subtype hPK,
exfalso,
apply hK.1,
calc K = (↑P : subgroup G).normalizer ⊔ K : by { rw sup_eq_right.mpr, exact hNK }
... = (map K.subtype (↑P' : subgroup K)).normalizer ⊔ K : by simp [map_comap_eq_self, hPK]
... = ⊤ : normalizer_sup_eq_top P' },
end
lemma normal_of_normalizer_condition (hnc : normalizer_condition G)
{p : ℕ} [fact p.prime] [finite (sylow p G)] (P : sylow p G) :
(↑P : subgroup G).normal :=
normalizer_eq_top.mp $ normalizer_condition_iff_only_full_group_self_normalizing.mp hnc _ $
normalizer_normalizer _
open_locale big_operators
/-- If all its sylow groups are normal, then a finite group is isomorphic to the direct product
of these sylow groups.
-/
noncomputable
def direct_product_of_normal [fintype G]
(hn : ∀ {p : ℕ} [fact p.prime] (P : sylow p G), (↑P : subgroup G).normal) :
(Π p : (card G).factorization.support, Π P : sylow p G, (↑P : subgroup G)) ≃* G :=
begin
set ps := (fintype.card G).factorization.support,
-- “The” sylow group for p
let P : Π p, sylow p G := default,
have hcomm : pairwise (λ (p₁ p₂ : ps), ∀ (x y : G), x ∈ P p₁ → y ∈ P p₂ → commute x y),
{ rintros ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne,
haveI hp₁' := fact.mk (nat.prime_of_mem_factorization hp₁),
haveI hp₂' := fact.mk (nat.prime_of_mem_factorization hp₂),
have hne' : p₁ ≠ p₂, by simpa using hne,
apply subgroup.commute_of_normal_of_disjoint _ _ (hn (P p₁)) (hn (P p₂)),
apply is_p_group.disjoint_of_ne p₁ p₂ hne' _ _ (P p₁).is_p_group' (P p₂).is_p_group', },
refine mul_equiv.trans _ _,
-- There is only one sylow group for each p, so the inner product is trivial
show (Π p : ps, Π P : sylow p G, P) ≃* (Π p : ps, P p),
{ -- here we need to help the elaborator with an explicit instantiation
apply @mul_equiv.Pi_congr_right ps (λ p, (Π P : sylow p G, P)) (λ p, P p) _ _ ,
rintro ⟨p, hp⟩,
haveI hp' := fact.mk (nat.prime_of_mem_factorization hp),
haveI := subsingleton_of_normal _ (hn (P p)),
change (Π (P : sylow p G), P) ≃* P p,
exact mul_equiv.Pi_subsingleton _ _, },
show (Π p : ps, P p) ≃* G,
apply mul_equiv.of_bijective (subgroup.noncomm_pi_coprod hcomm),
apply (bijective_iff_injective_and_card _).mpr,
split,
show injective _,
{ apply subgroup.injective_noncomm_pi_coprod_of_independent,
apply independent_of_coprime_order hcomm,
rintros ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne,
haveI hp₁' := fact.mk (nat.prime_of_mem_factorization hp₁),
haveI hp₂' := fact.mk (nat.prime_of_mem_factorization hp₂),
have hne' : p₁ ≠ p₂, by simpa using hne,
apply is_p_group.coprime_card_of_ne p₁ p₂ hne' _ _ (P p₁).is_p_group' (P p₂).is_p_group', },
show card (Π (p : ps), P p) = card G,
{ calc card (Π (p : ps), P p)
= ∏ (p : ps), card ↥(P p) : fintype.card_pi
... = ∏ (p : ps), p.1 ^ (card G).factorization p.1 :
begin
congr' 1 with ⟨p, hp⟩,
exact @card_eq_multiplicity _ _ _ p ⟨nat.prime_of_mem_factorization hp⟩ (P p)
end
... = ∏ p in ps, p ^ (card G).factorization p :
finset.prod_finset_coe (λ p, p ^ (card G).factorization p) _
... = (card G).factorization.prod pow : rfl
... = card G : nat.factorization_prod_pow_eq_self fintype.card_ne_zero }
end
end sylow
|
dc02419cb330190e76b41c30f6769a317e1e225e | 302b541ac2e998a523ae04da7673fd0932ded126 | /tests/simple/ctor.lean | e320418a37cfdc0d4a13e4c21fe9978d9cb1e49c | [] | no_license | mattweingarten/lambdapure | 4aeff69e8e3b8e78ea3c0a2b9b61770ef5a689b1 | f920a4ad78e6b1e3651f30bf8445c9105dfa03a8 | refs/heads/master | 1,680,665,168,790 | 1,618,420,180,000 | 1,618,420,180,000 | 310,816,264 | 2 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 167 | lean | set_option trace.compiler.ir.init true
inductive Tree
| Nil
| Node (l r : Tree) : Tree
open Tree
def mkTree : Nat -> Tree
| 0 => Nil
| x => Node Nil Nil
|
3cd3d1a30cc7d2c8aa7b3b2d0567480c3931ac7f | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/data/pfunctor/multivariate/basic.lean | e41829687d78257469405804e3777f7f5e3f9365 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 8,061 | lean | /-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad, Simon Hudon
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.control.functor.multivariate
import Mathlib.data.pfunctor.univariate.default
import Mathlib.data.sigma.default
import Mathlib.PostPort
universes u l u_1
namespace Mathlib
/-!
# Multivariate polynomial functors.
Multivariate polynomial functors are used for defining M-types and W-types.
They map a type vector `α` to the type `Σ a : A, B a ⟹ α`, with `A : Type` and
`B : A → typevec n`. They interact well with Lean's inductive definitions because
they guarantee that occurrences of `α` are positive.
-/
/--
multivariate polynomial functors
-/
structure mvpfunctor (n : ℕ)
where
A : Type u
B : A → typevec n
namespace mvpfunctor
/-- Applying `P` to an object of `Type` -/
def obj {n : ℕ} (P : mvpfunctor n) (α : typevec n) :=
sigma fun (a : A P) => typevec.arrow (B P a) α
/-- Applying `P` to a morphism of `Type` -/
def map {n : ℕ} (P : mvpfunctor n) {α : typevec n} {β : typevec n} (f : typevec.arrow α β) : obj P α → obj P β :=
fun (_x : obj P α) => sorry
protected instance inhabited {n : ℕ} : Inhabited (mvpfunctor n) :=
{ default := mk Inhabited.default fun (_x : Inhabited.default) => Inhabited.default }
protected instance obj.inhabited {n : ℕ} (P : mvpfunctor n) {α : typevec n} [Inhabited (A P)] [(i : fin2 n) → Inhabited (α i)] : Inhabited (obj P α) :=
{ default := sigma.mk Inhabited.default fun (_x : fin2 n) (_x_1 : B P Inhabited.default _x) => Inhabited.default }
protected instance obj.mvfunctor {n : ℕ} (P : mvpfunctor n) : mvfunctor (obj P) :=
mvfunctor.mk (map P)
theorem map_eq {n : ℕ} (P : mvpfunctor n) {α : typevec n} {β : typevec n} (g : typevec.arrow α β) (a : A P) (f : typevec.arrow (B P a) α) : mvfunctor.map g (sigma.mk a f) = sigma.mk a (typevec.comp g f) :=
rfl
theorem id_map {n : ℕ} (P : mvpfunctor n) {α : typevec n} (x : obj P α) : mvfunctor.map typevec.id x = x :=
sigma.cases_on x
fun (x_fst : A P) (x_snd : typevec.arrow (B P x_fst) α) =>
idRhs (mvfunctor.map typevec.id (sigma.mk x_fst x_snd) = mvfunctor.map typevec.id (sigma.mk x_fst x_snd)) rfl
theorem comp_map {n : ℕ} (P : mvpfunctor n) {α : typevec n} {β : typevec n} {γ : typevec n} (f : typevec.arrow α β) (g : typevec.arrow β γ) (x : obj P α) : mvfunctor.map (typevec.comp g f) x = mvfunctor.map g (mvfunctor.map f x) := sorry
protected instance obj.is_lawful_mvfunctor {n : ℕ} (P : mvpfunctor n) : is_lawful_mvfunctor (obj P) :=
is_lawful_mvfunctor.mk (id_map P) (comp_map P)
/-- Constant functor where the input object does not affect the output -/
def const (n : ℕ) (A : Type u) : mvpfunctor n :=
mk A fun (a : A) (i : fin2 n) => pempty
/-- Constructor for the constant functor -/
def const.mk (n : ℕ) {A : Type u} (x : A) {α : typevec n} : obj (const n A) α :=
sigma.mk x fun (i : fin2 n) (a : B (const n A) x i) => pempty.elim a
/-- Destructor for the constant functor -/
def const.get {n : ℕ} {A : Type u} {α : typevec n} (x : obj (const n A) α) : A :=
sigma.fst x
@[simp] theorem const.get_map {n : ℕ} {A : Type u} {α : typevec n} {β : typevec n} (f : typevec.arrow α β) (x : obj (const n A) α) : const.get (mvfunctor.map f x) = const.get x :=
sigma.cases_on x
fun (x_fst : A (const n A)) (x_snd : typevec.arrow (B (const n A) x_fst) α) =>
Eq.refl (const.get (mvfunctor.map f (sigma.mk x_fst x_snd)))
@[simp] theorem const.get_mk {n : ℕ} {A : Type u} {α : typevec n} (x : A) : const.get (const.mk n x) = x :=
Eq.refl (const.get (const.mk n x))
@[simp] theorem const.mk_get {n : ℕ} {A : Type u} {α : typevec n} (x : obj (const n A) α) : const.mk n (const.get x) = x := sorry
/-- Functor composition on polynomial functors -/
def comp {n : ℕ} {m : ℕ} (P : mvpfunctor n) (Q : fin2 n → mvpfunctor m) : mvpfunctor m :=
mk (sigma fun (a₂ : A P) => (i : fin2 n) → B P a₂ i → A (Q i))
fun (a : sigma fun (a₂ : A P) => (i : fin2 n) → B P a₂ i → A (Q i)) (i : fin2 m) =>
sigma fun (j : fin2 n) => sigma fun (b : B P (sigma.fst a) j) => B (Q j) (sigma.snd a j b) i
/-- Constructor for functor composition -/
def comp.mk {n : ℕ} {m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α : typevec m} (x : obj P fun (i : fin2 n) => obj (Q i) α) : obj (comp P Q) α :=
sigma.mk (sigma.mk (sigma.fst x) fun (i : fin2 n) (a : B P (sigma.fst x) i) => sigma.fst (sigma.snd x i a))
fun (i : fin2 m)
(a :
B (comp P Q) (sigma.mk (sigma.fst x) fun (i : fin2 n) (a : B P (sigma.fst x) i) => sigma.fst (sigma.snd x i a))
i) =>
sigma.snd (sigma.snd x (sigma.fst a) (sigma.fst (sigma.snd a))) i (sigma.snd (sigma.snd a))
/-- Destructor for functor composition -/
def comp.get {n : ℕ} {m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α : typevec m} (x : obj (comp P Q) α) : obj P fun (i : fin2 n) => obj (Q i) α :=
sigma.mk (sigma.fst (sigma.fst x))
fun (i : fin2 n) (a : B P (sigma.fst (sigma.fst x)) i) =>
sigma.mk (sigma.snd (sigma.fst x) i a)
fun (j : fin2 m) (b : B (Q i) (sigma.snd (sigma.fst x) i a) j) => sigma.snd x j (sigma.mk i (sigma.mk a b))
theorem comp.get_map {n : ℕ} {m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α : typevec m} {β : typevec m} (f : typevec.arrow α β) (x : obj (comp P Q) α) : comp.get (mvfunctor.map f x) = mvfunctor.map (fun (i : fin2 n) (x : obj (Q i) α) => mvfunctor.map f x) (comp.get x) :=
sigma.cases_on x
fun (x_fst : A (comp P Q)) (x_snd : typevec.arrow (B (comp P Q) x_fst) α) =>
Eq.refl (comp.get (mvfunctor.map f (sigma.mk x_fst x_snd)))
@[simp] theorem comp.get_mk {n : ℕ} {m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α : typevec m} (x : obj P fun (i : fin2 n) => obj (Q i) α) : comp.get (comp.mk x) = x := sorry
@[simp] theorem comp.mk_get {n : ℕ} {m : ℕ} {P : mvpfunctor n} {Q : fin2 n → mvpfunctor m} {α : typevec m} (x : obj (comp P Q) α) : comp.mk (comp.get x) = x := sorry
/-
lifting predicates and relations
-/
theorem liftp_iff {n : ℕ} {P : mvpfunctor n} {α : typevec n} (p : {i : fin2 n} → α i → Prop) (x : obj P α) : mvfunctor.liftp p x ↔
∃ (a : A P), ∃ (f : typevec.arrow (B P a) α), x = sigma.mk a f ∧ ∀ (i : fin2 n) (j : B P a i), p (f i j) := sorry
theorem liftp_iff' {n : ℕ} {P : mvpfunctor n} {α : typevec n} (p : {i : fin2 n} → α i → Prop) (a : A P) (f : typevec.arrow (B P a) α) : mvfunctor.liftp p (sigma.mk a f) ↔ ∀ (i : fin2 n) (x : B P a i), p (f i x) := sorry
theorem liftr_iff {n : ℕ} {P : mvpfunctor n} {α : typevec n} (r : {i : fin2 n} → α i → α i → Prop) (x : obj P α) (y : obj P α) : mvfunctor.liftr r x y ↔
∃ (a : A P),
∃ (f₀ : typevec.arrow (B P a) α),
∃ (f₁ : typevec.arrow (B P a) α),
x = sigma.mk a f₀ ∧ y = sigma.mk a f₁ ∧ ∀ (i : fin2 n) (j : B P a i), r (f₀ i j) (f₁ i j) := sorry
theorem supp_eq {n : ℕ} {P : mvpfunctor n} {α : typevec n} (a : A P) (f : typevec.arrow (B P a) α) (i : fin2 n) : mvfunctor.supp (sigma.mk a f) i = f i '' set.univ := sorry
end mvpfunctor
/-
Decomposing an n+1-ary pfunctor.
-/
namespace mvpfunctor
/-- Split polynomial functor, get a n-ary functor
from a `n+1`-ary functor -/
def drop {n : ℕ} (P : mvpfunctor (n + 1)) : mvpfunctor n :=
mk (A P) fun (a : A P) => typevec.drop (B P a)
/-- Split polynomial functor, get a univariate functor
from a `n+1`-ary functor -/
def last {n : ℕ} (P : mvpfunctor (n + 1)) : pfunctor :=
pfunctor.mk (A P) fun (a : A P) => typevec.last (B P a)
/-- append arrows of a polynomial functor application -/
def append_contents {n : ℕ} (P : mvpfunctor (n + 1)) {α : typevec n} {β : Type u_1} {a : A P} (f' : typevec.arrow (B (drop P) a) α) (f : pfunctor.B (last P) a → β) : typevec.arrow (B P a) (α ::: β) :=
typevec.split_fun f' f
|
552685e4c4929124eec38aed5a979b8ff382447a | 4727251e0cd73359b15b664c3170e5d754078599 | /src/analysis/analytic/inverse.lean | 06ecf81952709d1a41a6d91f11c5c2307df3fc80 | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 27,104 | lean | /-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import analysis.analytic.composition
/-!
# Inverse of analytic functions
We construct the left and right inverse of a formal multilinear series with invertible linear term,
we prove that they coincide and study their properties (notably convergence).
## Main statements
* `p.left_inv i`: the formal left inverse of the formal multilinear series `p`,
for `i : E ≃L[𝕜] F` which coincides with `p₁`.
* `p.right_inv i`: the formal right inverse of the formal multilinear series `p`,
for `i : E ≃L[𝕜] F` which coincides with `p₁`.
* `p.left_inv_comp` says that `p.left_inv i` is indeed a left inverse to `p` when `p₁ = i`.
* `p.right_inv_comp` says that `p.right_inv i` is indeed a right inverse to `p` when `p₁ = i`.
* `p.left_inv_eq_right_inv`: the two inverses coincide.
* `p.radius_right_inv_pos_of_radius_pos`: if a power series has a positive radius of convergence,
then so does its inverse.
-/
open_locale big_operators classical topological_space
open finset filter
namespace formal_multilinear_series
variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜]
{E : Type*} [normed_group E] [normed_space 𝕜 E]
{F : Type*} [normed_group F] [normed_space 𝕜 F]
/-! ### The left inverse of a formal multilinear series -/
/-- The left inverse of a formal multilinear series, where the `n`-th term is defined inductively
in terms of the previous ones to make sure that `(left_inv p i) ∘ p = id`. For this, the linear term
`p₁` in `p` should be invertible. In the definition, `i` is a linear isomorphism that should
coincide with `p₁`, so that one can use its inverse in the construction. The definition does not
use that `i = p₁`, but proofs that the definition is well-behaved do.
The `n`-th term in `q ∘ p` is `∑ qₖ (p_{j₁}, ..., p_{jₖ})` over `j₁ + ... + jₖ = n`. In this
expression, `qₙ` appears only once, in `qₙ (p₁, ..., p₁)`. We adjust the definition so that this
term compensates the rest of the sum, using `i⁻¹` as an inverse to `p₁`.
These formulas only make sense when the constant term `p₀` vanishes. The definition we give is
general, but it ignores the value of `p₀`.
-/
noncomputable def left_inv (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
formal_multilinear_series 𝕜 F E
| 0 := 0
| 1 := (continuous_multilinear_curry_fin1 𝕜 F E).symm i.symm
| (n+2) := - ∑ c : {c : composition (n+2) // c.length < n + 2},
have (c : composition (n+2)).length < n+2 := c.2,
(left_inv (c : composition (n+2)).length).comp_along_composition
(p.comp_continuous_linear_map i.symm) c
@[simp] lemma left_inv_coeff_zero (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.left_inv i 0 = 0 := by rw left_inv
@[simp] lemma left_inv_coeff_one (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.left_inv i 1 = (continuous_multilinear_curry_fin1 𝕜 F E).symm i.symm := by rw left_inv
/-- The left inverse does not depend on the zeroth coefficient of a formal multilinear
series. -/
lemma left_inv_remove_zero (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.remove_zero.left_inv i = p.left_inv i :=
begin
ext1 n,
induction n using nat.strong_rec' with n IH,
cases n, { simp }, -- if one replaces `simp` with `refl`, the proof times out in the kernel.
cases n, { simp }, -- TODO: why?
simp only [left_inv, neg_inj],
refine finset.sum_congr rfl (λ c cuniv, _),
rcases c with ⟨c, hc⟩,
ext v,
dsimp,
simp [IH _ hc],
end
/-- The left inverse to a formal multilinear series is indeed a left inverse, provided its linear
term is invertible. -/
lemma left_inv_comp (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuous_multilinear_curry_fin1 𝕜 E F).symm i) :
(left_inv p i).comp p = id 𝕜 E :=
begin
ext n v,
cases n,
{ simp only [left_inv, continuous_multilinear_map.zero_apply, id_apply_ne_one, ne.def,
not_false_iff, zero_ne_one, comp_coeff_zero']},
cases n,
{ simp only [left_inv, comp_coeff_one, h, id_apply_one, continuous_linear_equiv.coe_apply,
continuous_linear_equiv.symm_apply_apply, continuous_multilinear_curry_fin1_symm_apply] },
have A : (finset.univ : finset (composition (n+2)))
= {c | composition.length c < n + 2}.to_finset ∪ {composition.ones (n+2)},
{ refine subset.antisymm (λ c hc, _) (subset_univ _),
by_cases h : c.length < n + 2,
{ simp [h] },
{ simp [composition.eq_ones_iff_le_length.2 (not_lt.1 h)] } },
have B : disjoint ({c | composition.length c < n + 2} : set (composition (n + 2))).to_finset
{composition.ones (n+2)}, by simp,
have C : (p.left_inv i (composition.ones (n + 2)).length)
(λ (j : fin (composition.ones n.succ.succ).length), p 1 (λ k,
v ((fin.cast_le (composition.length_le _)) j)))
= p.left_inv i (n+2) (λ (j : fin (n+2)), p 1 (λ k, v j)),
{ apply formal_multilinear_series.congr _ (composition.ones_length _) (λ j hj1 hj2, _),
exact formal_multilinear_series.congr _ rfl (λ k hk1 hk2, by congr) },
have D : p.left_inv i (n+2) (λ (j : fin (n+2)), p 1 (λ k, v j)) =
- ∑ (c : composition (n + 2)) in {c : composition (n + 2) | c.length < n + 2}.to_finset,
(p.left_inv i c.length) (p.apply_composition c v),
{ simp only [left_inv, continuous_multilinear_map.neg_apply, neg_inj,
continuous_multilinear_map.sum_apply],
convert (sum_to_finset_eq_subtype (λ (c : composition (n+2)), c.length < n+2)
(λ (c : composition (n+2)), (continuous_multilinear_map.comp_along_composition
(p.comp_continuous_linear_map ↑(i.symm)) c (p.left_inv i c.length))
(λ (j : fin (n + 2)), p 1 (λ (k : fin 1), v j)))).symm.trans _,
simp only [comp_continuous_linear_map_apply_composition,
continuous_multilinear_map.comp_along_composition_apply],
congr,
ext c,
congr,
ext k,
simp [h] },
simp [formal_multilinear_series.comp, show n + 2 ≠ 1, by dec_trivial, A, finset.sum_union B,
apply_composition_ones, C, D],
end
/-! ### The right inverse of a formal multilinear series -/
/-- The right inverse of a formal multilinear series, where the `n`-th term is defined inductively
in terms of the previous ones to make sure that `p ∘ (right_inv p i) = id`. For this, the linear
term `p₁` in `p` should be invertible. In the definition, `i` is a linear isomorphism that should
coincide with `p₁`, so that one can use its inverse in the construction. The definition does not
use that `i = p₁`, but proofs that the definition is well-behaved do.
The `n`-th term in `p ∘ q` is `∑ pₖ (q_{j₁}, ..., q_{jₖ})` over `j₁ + ... + jₖ = n`. In this
expression, `qₙ` appears only once, in `p₁ (qₙ)`. We adjust the definition of `qₙ` so that this
term compensates the rest of the sum, using `i⁻¹` as an inverse to `p₁`.
These formulas only make sense when the constant term `p₀` vanishes. The definition we give is
general, but it ignores the value of `p₀`.
-/
noncomputable def right_inv (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
formal_multilinear_series 𝕜 F E
| 0 := 0
| 1 := (continuous_multilinear_curry_fin1 𝕜 F E).symm i.symm
| (n+2) :=
let q : formal_multilinear_series 𝕜 F E := λ k, if h : k < n + 2 then right_inv k else 0 in
- (i.symm : F →L[𝕜] E).comp_continuous_multilinear_map ((p.comp q) (n+2))
@[simp] lemma right_inv_coeff_zero (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.right_inv i 0 = 0 := by rw right_inv
@[simp] lemma right_inv_coeff_one (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.right_inv i 1 = (continuous_multilinear_curry_fin1 𝕜 F E).symm i.symm := by rw right_inv
/-- The right inverse does not depend on the zeroth coefficient of a formal multilinear
series. -/
lemma right_inv_remove_zero (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) :
p.remove_zero.right_inv i = p.right_inv i :=
begin
ext1 n,
induction n using nat.strong_rec' with n IH,
cases n, { simp },
cases n, { simp },
simp only [right_inv, neg_inj],
unfold_coes,
congr' 1,
rw remove_zero_comp_of_pos _ _ (show 0 < n+2, by dec_trivial),
congr' 1,
ext k,
by_cases hk : k < n+2; simp [hk, IH]
end
lemma comp_right_inv_aux1 {n : ℕ} (hn : 0 < n)
(p : formal_multilinear_series 𝕜 E F) (q : formal_multilinear_series 𝕜 F E) (v : fin n → F) :
p.comp q n v =
(∑ (c : composition n) in {c : composition n | 1 < c.length}.to_finset,
p c.length (q.apply_composition c v)) + p 1 (λ i, q n v) :=
begin
have A : (finset.univ : finset (composition n))
= {c | 1 < composition.length c}.to_finset ∪ {composition.single n hn},
{ refine subset.antisymm (λ c hc, _) (subset_univ _),
by_cases h : 1 < c.length,
{ simp [h] },
{ have : c.length = 1,
by { refine (eq_iff_le_not_lt.2 ⟨ _, h⟩).symm, exact c.length_pos_of_pos hn },
rw ← composition.eq_single_iff_length hn at this,
simp [this] } },
have B : disjoint ({c | 1 < composition.length c} : set (composition n)).to_finset
{composition.single n hn}, by simp,
have C : p (composition.single n hn).length
(q.apply_composition (composition.single n hn) v)
= p 1 (λ (i : fin 1), q n v),
{ apply p.congr (composition.single_length hn) (λ j hj1 hj2, _),
simp [apply_composition_single] },
simp [formal_multilinear_series.comp, A, finset.sum_union B, C],
end
lemma comp_right_inv_aux2
(p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (v : fin (n + 2) → F) :
∑ (c : composition (n + 2)) in {c : composition (n + 2) | 1 < c.length}.to_finset,
p c.length (apply_composition (λ (k : ℕ), ite (k < n + 2) (p.right_inv i k) 0) c v) =
∑ (c : composition (n + 2)) in {c : composition (n + 2) | 1 < c.length}.to_finset,
p c.length ((p.right_inv i).apply_composition c v) :=
begin
have N : 0 < n + 2, by dec_trivial,
refine sum_congr rfl (λ c hc, p.congr rfl (λ j hj1 hj2, _)),
have : ∀ k, c.blocks_fun k < n + 2,
{ simp only [set.mem_to_finset, set.mem_set_of_eq] at hc,
simp [← composition.ne_single_iff N, composition.eq_single_iff_length, ne_of_gt hc] },
simp [apply_composition, this],
end
/-- The right inverse to a formal multilinear series is indeed a right inverse, provided its linear
term is invertible and its constant term vanishes. -/
lemma comp_right_inv (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuous_multilinear_curry_fin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
p.comp (right_inv p i) = id 𝕜 F :=
begin
ext n v,
cases n,
{ simp only [h0, continuous_multilinear_map.zero_apply, id_apply_ne_one, ne.def, not_false_iff,
zero_ne_one, comp_coeff_zero']},
cases n,
{ simp only [comp_coeff_one, h, right_inv, continuous_linear_equiv.apply_symm_apply, id_apply_one,
continuous_linear_equiv.coe_apply, continuous_multilinear_curry_fin1_symm_apply] },
have N : 0 < n+2, by dec_trivial,
simp [comp_right_inv_aux1 N, h, right_inv, lt_irrefl n, show n + 2 ≠ 1, by dec_trivial,
← sub_eq_add_neg, sub_eq_zero, comp_right_inv_aux2],
end
lemma right_inv_coeff (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (hn : 2 ≤ n) :
p.right_inv i n = - (i.symm : F →L[𝕜] E).comp_continuous_multilinear_map
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition n)),
p.comp_along_composition (p.right_inv i) c) :=
begin
cases n, { exact false.elim (zero_lt_two.not_le hn) },
cases n, { exact false.elim (one_lt_two.not_le hn) },
simp only [right_inv, neg_inj],
congr' 1,
ext v,
have N : 0 < n + 2, by dec_trivial,
have : (p 1) (λ (i : fin 1), 0) = 0 := continuous_multilinear_map.map_zero _,
simp [comp_right_inv_aux1 N, lt_irrefl n, this, comp_right_inv_aux2]
end
/-! ### Coincidence of the left and the right inverse -/
private lemma left_inv_eq_right_inv_aux (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuous_multilinear_curry_fin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
left_inv p i = right_inv p i := calc
left_inv p i = (left_inv p i).comp (id 𝕜 F) : by simp
... = (left_inv p i).comp (p.comp (right_inv p i)) : by rw comp_right_inv p i h h0
... = ((left_inv p i).comp p).comp (right_inv p i) : by rw comp_assoc
... = (id 𝕜 E).comp (right_inv p i) : by rw left_inv_comp p i h
... = right_inv p i : by simp
/-- The left inverse and the right inverse of a formal multilinear series coincide. This is not at
all obvious from their definition, but it follows from uniqueness of inverses (which comes from the
fact that composition is associative on formal multilinear series). -/
theorem left_inv_eq_right_inv (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuous_multilinear_curry_fin1 𝕜 E F).symm i) :
left_inv p i = right_inv p i := calc
left_inv p i = left_inv p.remove_zero i : by rw left_inv_remove_zero
... = right_inv p.remove_zero i : by { apply left_inv_eq_right_inv_aux; simp [h] }
... = right_inv p i : by rw right_inv_remove_zero
/-!
### Convergence of the inverse of a power series
Assume that `p` is a convergent multilinear series, and let `q` be its (left or right) inverse.
Using the left-inverse formula gives
$$
q_n = - (p_1)^{-n} \sum_{k=0}^{n-1} \sum_{i_1 + \dotsc + i_k = n} q_k (p_{i_1}, \dotsc, p_{i_k}).
$$
Assume for simplicity that we are in dimension `1` and `p₁ = 1`. In the formula for `qₙ`, the term
`q_{n-1}` appears with a multiplicity of `n-1` (choosing the index `i_j` for which `i_j = 2` while
all the other indices are equal to `1`), which indicates that `qₙ` might grow like `n!`. This is
bad for summability properties.
It turns out that the right-inverse formula is better behaved, and should instead be used for this
kind of estimate. It reads
$$
q_n = - (p_1)^{-1} \sum_{k=2}^n \sum_{i_1 + \dotsc + i_k = n} p_k (q_{i_1}, \dotsc, q_{i_k}).
$$
Here, `q_{n-1}` can only appear in the term with `k = 2`, and it only appears twice, so there is
hope this formula can lead to an at most geometric behavior.
Let `Qₙ = ∥qₙ∥`. Bounding `∥pₖ∥` with `C r^k` gives an inequality
$$
Q_n ≤ C' \sum_{k=2}^n r^k \sum_{i_1 + \dotsc + i_k = n} Q_{i_1} \dotsm Q_{i_k}.
$$
This formula is not enough to prove by naive induction on `n` a bound of the form `Qₙ ≤ D R^n`.
However, assuming that the inequality above were an equality, one could get a formula for the
generating series of the `Qₙ`:
$$
\begin{align}
Q(z) & := \sum Q_n z^n = Q_1 z + C' \sum_{2 \leq k \leq n} \sum_{i_1 + \dotsc + i_k = n}
(r z^{i_1} Q_{i_1}) \dotsm (r z^{i_k} Q_{i_k})
\\ & = Q_1 z + C' \sum_{k = 2}^\infty (\sum_{i_1 \geq 1} r z^{i_1} Q_{i_1})
\dotsm (\sum_{i_k \geq 1} r z^{i_k} Q_{i_k})
\\ & = Q_1 z + C' \sum_{k = 2}^\infty (r Q(z))^k
= Q_1 z + C' (r Q(z))^2 / (1 - r Q(z)).
\end{align}
$$
One can solve this formula explicitly. The solution is analytic in a neighborhood of `0` in `ℂ`,
hence its coefficients grow at most geometrically (by a contour integral argument), and therefore
the original `Qₙ`, which are bounded by these ones, are also at most geometric.
This classical argument is not really satisfactory, as it requires an a priori bound on a complex
analytic function. Another option would be to compute explicitly its terms (with binomial
coefficients) to obtain an explicit geometric bound, but this would be very painful.
Instead, we will use the above intuition, but in a slightly different form, with finite sums and an
induction. I learnt this trick in [pöschel2017siegelsternberg]. Let
$S_n = \sum_{k=1}^n Q_k a^k$ (where `a` is a positive real parameter to be chosen suitably small).
The above computation but with finite sums shows that
$$
S_n \leq Q_1 a + C' \sum_{k=2}^n (r S_{n-1})^k.
$$
In particular, $S_n \leq Q_1 a + C' (r S_{n-1})^2 / (1- r S_{n-1})$.
Assume that $S_{n-1} \leq K a$, where `K > Q₁` is fixed and `a` is small enough so that
`r K a ≤ 1/2` (to control the denominator). Then this equation gives a bound
$S_n \leq Q_1 a + 2 C' r^2 K^2 a^2$.
If `a` is small enough, this is bounded by `K a` as the second term is quadratic in `a`, and
therefore negligible.
By induction, we deduce `Sₙ ≤ K a` for all `n`, which gives in particular the fact that `aⁿ Qₙ`
remains bounded.
-/
/-- First technical lemma to control the growth of coefficients of the inverse. Bound the explicit
expression for `∑_{k<n+1} aᵏ Qₖ` in terms of a sum of powers of the same sum one step before,
in a general abstract setup. -/
lemma radius_right_inv_pos_of_radius_pos_aux1
(n : ℕ) (p : ℕ → ℝ) (hp : ∀ k, 0 ≤ p k) {r a : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a) :
∑ k in Ico 2 (n + 1), a ^ k *
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition k)),
r ^ c.length * ∏ j, p (c.blocks_fun j))
≤ ∑ j in Ico 2 (n + 1), r ^ j * (∑ k in Ico 1 n, a ^ k * p k) ^ j :=
calc
∑ k in Ico 2 (n + 1), a ^ k *
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition k)),
r ^ c.length * ∏ j, p (c.blocks_fun j))
= ∑ k in Ico 2 (n + 1),
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition k)),
∏ j, r * (a ^ (c.blocks_fun j) * p (c.blocks_fun j))) :
begin
simp_rw [mul_sum],
apply sum_congr rfl (λ k hk, _),
apply sum_congr rfl (λ c hc, _),
rw [prod_mul_distrib, prod_mul_distrib, prod_pow_eq_pow_sum, composition.sum_blocks_fun,
prod_const, card_fin],
ring,
end
... ≤ ∑ d in comp_partial_sum_target 2 (n + 1) n,
∏ (j : fin d.2.length), r * (a ^ d.2.blocks_fun j * p (d.2.blocks_fun j)) :
begin
rw sum_sigma',
refine sum_le_sum_of_subset_of_nonneg _ (λ x hx1 hx2,
prod_nonneg (λ j hj, mul_nonneg hr (mul_nonneg (pow_nonneg ha _) (hp _)))),
rintros ⟨k, c⟩ hd,
simp only [set.mem_to_finset, mem_Ico, mem_sigma, set.mem_set_of_eq] at hd,
simp only [mem_comp_partial_sum_target_iff],
refine ⟨hd.2, c.length_le.trans_lt hd.1.2, λ j, _⟩,
have : c ≠ composition.single k (zero_lt_two.trans_le hd.1.1),
by simp [composition.eq_single_iff_length, ne_of_gt hd.2],
rw composition.ne_single_iff at this,
exact (this j).trans_le (nat.lt_succ_iff.mp hd.1.2)
end
... = ∑ e in comp_partial_sum_source 2 (n+1) n, ∏ (j : fin e.1), r * (a ^ e.2 j * p (e.2 j)) :
begin
symmetry,
apply comp_change_of_variables_sum,
rintros ⟨k, blocks_fun⟩ H,
have K : (comp_change_of_variables 2 (n + 1) n ⟨k, blocks_fun⟩ H).snd.length = k, by simp,
congr' 2; try { rw K },
rw fin.heq_fun_iff K.symm,
assume j,
rw comp_change_of_variables_blocks_fun,
end
... = ∑ j in Ico 2 (n+1), r ^ j * (∑ k in Ico 1 n, a ^ k * p k) ^ j :
begin
rw [comp_partial_sum_source, ← sum_sigma' (Ico 2 (n + 1))
(λ (k : ℕ), (fintype.pi_finset (λ (i : fin k), Ico 1 n) : finset (fin k → ℕ)))
(λ n e, ∏ (j : fin n), r * (a ^ e j * p (e j)))],
apply sum_congr rfl (λ j hj, _),
simp only [← @multilinear_map.mk_pi_algebra_apply ℝ (fin j) _ _ ℝ],
simp only [← multilinear_map.map_sum_finset (multilinear_map.mk_pi_algebra ℝ (fin j) ℝ)
(λ k (m : ℕ), r * (a ^ m * p m))],
simp only [multilinear_map.mk_pi_algebra_apply],
dsimp,
simp [prod_const, ← mul_sum, mul_pow],
end
/-- Second technical lemma to control the growth of coefficients of the inverse. Bound the explicit
expression for `∑_{k<n+1} aᵏ Qₖ` in terms of a sum of powers of the same sum one step before,
in the specific setup we are interesting in, by reducing to the general bound in
`radius_right_inv_pos_of_radius_pos_aux1`. -/
lemma radius_right_inv_pos_of_radius_pos_aux2
{n : ℕ} (hn : 2 ≤ n + 1) (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
{r a C : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a) (hC : 0 ≤ C) (hp : ∀ n, ∥p n∥ ≤ C * r ^ n) :
(∑ k in Ico 1 (n + 1), a ^ k * ∥p.right_inv i k∥) ≤
∥(i.symm : F →L[𝕜] E)∥ * a + ∥(i.symm : F →L[𝕜] E)∥ * C * ∑ k in Ico 2 (n + 1),
(r * ((∑ j in Ico 1 n, a ^ j * ∥p.right_inv i j∥))) ^ k :=
let I := ∥(i.symm : F →L[𝕜] E)∥ in calc
∑ k in Ico 1 (n + 1), a ^ k * ∥p.right_inv i k∥
= a * I + ∑ k in Ico 2 (n + 1), a ^ k * ∥p.right_inv i k∥ :
by simp only [linear_isometry_equiv.norm_map, pow_one, right_inv_coeff_one,
nat.Ico_succ_singleton, sum_singleton, ← sum_Ico_consecutive _ one_le_two hn]
... = a * I + ∑ k in Ico 2 (n + 1), a ^ k *
∥(i.symm : F →L[𝕜] E).comp_continuous_multilinear_map
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition k)),
p.comp_along_composition (p.right_inv i) c)∥ :
begin
congr' 1,
apply sum_congr rfl (λ j hj, _),
rw [right_inv_coeff _ _ _ (mem_Ico.1 hj).1, norm_neg],
end
... ≤ a * ∥(i.symm : F →L[𝕜] E)∥ + ∑ k in Ico 2 (n + 1), a ^ k * (I *
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition k)),
C * r ^ c.length * ∏ j, ∥p.right_inv i (c.blocks_fun j)∥)) :
begin
apply_rules [add_le_add, le_refl, sum_le_sum (λ j hj, _), mul_le_mul_of_nonneg_left,
pow_nonneg, ha],
apply (continuous_linear_map.norm_comp_continuous_multilinear_map_le _ _).trans,
apply mul_le_mul_of_nonneg_left _ (norm_nonneg _),
apply (norm_sum_le _ _).trans,
apply sum_le_sum (λ c hc, _),
apply (comp_along_composition_norm _ _ _).trans,
apply mul_le_mul_of_nonneg_right (hp _),
exact prod_nonneg (λ j hj, norm_nonneg _),
end
... = I * a + I * C * ∑ k in Ico 2 (n + 1), a ^ k *
(∑ c in ({c | 1 < composition.length c}.to_finset : finset (composition k)),
r ^ c.length * ∏ j, ∥p.right_inv i (c.blocks_fun j)∥) :
begin
simp_rw [mul_assoc C, ← mul_sum, ← mul_assoc, mul_comm _ (∥↑i.symm∥), mul_assoc, ← mul_sum,
← mul_assoc, mul_comm _ C, mul_assoc, ← mul_sum],
ring,
end
... ≤ I * a + I * C * ∑ k in Ico 2 (n+1), (r * ((∑ j in Ico 1 n, a ^ j * ∥p.right_inv i j∥))) ^ k :
begin
apply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, norm_nonneg, hC, mul_nonneg],
simp_rw [mul_pow],
apply radius_right_inv_pos_of_radius_pos_aux1 n (λ k, ∥p.right_inv i k∥)
(λ k, norm_nonneg _) hr ha,
end
/-- If a a formal multilinear series has a positive radius of convergence, then its right inverse
also has a positive radius of convergence. -/
theorem radius_right_inv_pos_of_radius_pos (p : formal_multilinear_series 𝕜 E F) (i : E ≃L[𝕜] F)
(hp : 0 < p.radius) : 0 < (p.right_inv i).radius :=
begin
obtain ⟨C, r, Cpos, rpos, ple⟩ : ∃ C r (hC : 0 < C) (hr : 0 < r), ∀ (n : ℕ), ∥p n∥ ≤ C * r ^ n :=
le_mul_pow_of_radius_pos p hp,
let I := ∥(i.symm : F →L[𝕜] E)∥,
-- choose `a` small enough to make sure that `∑_{k ≤ n} aᵏ Qₖ` will be controllable by
-- induction
obtain ⟨a, apos, ha1, ha2⟩ : ∃ a (apos : 0 < a),
(2 * I * C * r^2 * (I + 1) ^ 2 * a ≤ 1) ∧ (r * (I + 1) * a ≤ 1/2),
{ have : tendsto (λ a, 2 * I * C * r^2 * (I + 1) ^ 2 * a) (𝓝 0)
(𝓝 (2 * I * C * r^2 * (I + 1) ^ 2 * 0)) := tendsto_const_nhds.mul tendsto_id,
have A : ∀ᶠ a in 𝓝 0, 2 * I * C * r^2 * (I + 1) ^ 2 * a < 1,
by { apply (tendsto_order.1 this).2, simp [zero_lt_one] },
have : tendsto (λ a, r * (I + 1) * a) (𝓝 0)
(𝓝 (r * (I + 1) * 0)) := tendsto_const_nhds.mul tendsto_id,
have B : ∀ᶠ a in 𝓝 0, r * (I + 1) * a < 1/2,
by { apply (tendsto_order.1 this).2, simp [zero_lt_one] },
have C : ∀ᶠ a in 𝓝[>] (0 : ℝ), (0 : ℝ) < a,
by { filter_upwards [self_mem_nhds_within] with _ ha using ha },
rcases (C.and ((A.and B).filter_mono inf_le_left)).exists with ⟨a, ha⟩,
exact ⟨a, ha.1, ha.2.1.le, ha.2.2.le⟩ },
-- check by induction that the partial sums are suitably bounded, using the choice of `a` and the
-- inductive control from Lemma `radius_right_inv_pos_of_radius_pos_aux2`.
let S := λ n, ∑ k in Ico 1 n, a ^ k * ∥p.right_inv i k∥,
have IRec : ∀ n, 1 ≤ n → S n ≤ (I + 1) * a,
{ apply nat.le_induction,
{ simp only [S],
rw [Ico_eq_empty_of_le (le_refl 1), sum_empty],
exact mul_nonneg (add_nonneg (norm_nonneg _) zero_le_one) apos.le },
{ assume n one_le_n hn,
have In : 2 ≤ n + 1, by linarith,
have Snonneg : 0 ≤ S n :=
sum_nonneg (λ x hx, mul_nonneg (pow_nonneg apos.le _) (norm_nonneg _)),
have rSn : r * S n ≤ 1/2 := calc
r * S n ≤ r * ((I+1) * a) : mul_le_mul_of_nonneg_left hn rpos.le
... ≤ 1/2 : by rwa [← mul_assoc],
calc S (n + 1) ≤ I * a + I * C * ∑ k in Ico 2 (n + 1), (r * S n)^k :
radius_right_inv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple
... = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) :
by { rw geom_sum_Ico' _ In, exact ne_of_lt (rSn.trans_lt (by norm_num)) }
... ≤ I * a + I * C * ((r * S n) ^ 2 / (1/2)) :
begin
apply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, mul_nonneg, norm_nonneg,
Cpos.le],
refine div_le_div (sq_nonneg _) _ (by norm_num) (by linarith),
simp only [sub_le_self_iff],
apply pow_nonneg (mul_nonneg rpos.le Snonneg),
end
... = I * a + 2 * I * C * (r * S n) ^ 2 : by ring
... ≤ I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 :
by apply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, mul_nonneg, norm_nonneg,
Cpos.le, zero_le_two, pow_le_pow_of_le_left, rpos.le]
... = (I + 2 * I * C * r^2 * (I + 1) ^ 2 * a) * a : by ring
... ≤ (I + 1) * a :
by apply_rules [mul_le_mul_of_nonneg_right, apos.le, add_le_add, le_refl] } },
-- conclude that all coefficients satisfy `aⁿ Qₙ ≤ (I + 1) a`.
let a' : nnreal := ⟨a, apos.le⟩,
suffices H : (a' : ennreal) ≤ (p.right_inv i).radius,
by { apply lt_of_lt_of_le _ H, exact_mod_cast apos },
apply le_radius_of_bound _ ((I + 1) * a) (λ n, _),
by_cases hn : n = 0,
{ have : ∥p.right_inv i n∥ = ∥p.right_inv i 0∥, by congr; try { rw hn },
simp only [this, norm_zero, zero_mul, right_inv_coeff_zero],
apply_rules [mul_nonneg, add_nonneg, norm_nonneg, zero_le_one, apos.le] },
{ have one_le_n : 1 ≤ n := bot_lt_iff_ne_bot.2 hn,
calc ∥p.right_inv i n∥ * ↑a' ^ n = a ^ n * ∥p.right_inv i n∥ : mul_comm _ _
... ≤ ∑ k in Ico 1 (n + 1), a ^ k * ∥p.right_inv i k∥ :
begin
have : ∀ k ∈ Ico 1 (n + 1), 0 ≤ a ^ k * ∥p.right_inv i k∥ :=
λ k hk, mul_nonneg (pow_nonneg apos.le _) (norm_nonneg _),
exact single_le_sum this (by simp [one_le_n]),
end
... ≤ (I + 1) * a : IRec (n + 1) (by dec_trivial) }
end
end formal_multilinear_series
|
4b0bcdda9381bdb6733a6ca56854c9588c2f443f | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/algebra/algebra/tower.lean | 5f0db19a076bdb964e37aa8663d679e8645d5d9e | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 11,613 | lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import algebra.algebra.subalgebra
/-!
# Towers of algebras
In this file we prove basic facts about towers of algebra.
An algebra tower A/S/R is expressed by having instances of `algebra A S`,
`algebra R S`, `algebra R A` and `is_scalar_tower R S A`, the later asserting the
compatibility condition `(r • s) • a = r • (s • a)`.
An important definition is `to_alg_hom R S A`, the canonical `R`-algebra homomorphism `S →ₐ[R] A`.
-/
universes u v w u₁ v₁
variables (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace algebra
variables [comm_semiring R] [semiring A] [algebra R A]
variables [add_comm_monoid M] [module R M] [module A M] [is_scalar_tower R A M]
variables {A}
/-- The `R`-algebra morphism `A → End (M)` corresponding to the representation of the algebra `A`
on the `R`-module `M`. -/
def lsmul : A →ₐ[R] module.End R M :=
{ map_one' := by { ext m, exact one_smul A m },
map_mul' := by { intros a b, ext c, exact smul_assoc a b c },
map_zero' := by { ext m, exact zero_smul A m },
commutes' := by { intro r, ext m, exact algebra_map_smul A r m },
.. (show A →ₗ[R] M →ₗ[R] M, from linear_map.mk₂ R (•)
(λ x y z, add_smul x y z)
(λ c x y, smul_assoc c x y)
(λ x y z, smul_add x y z)
(λ c x y, smul_algebra_smul_comm c x y)) }
@[simp] lemma lsmul_coe (a : A) : (lsmul R M a : M → M) = (•) a := rfl
@[simp] lemma lmul_algebra_map (x : R) :
lmul R A (algebra_map R A x) = algebra.lsmul R A x :=
eq.symm $ linear_map.ext $ smul_def'' x
end algebra
namespace is_scalar_tower
section module
variables [comm_semiring R] [semiring A] [algebra R A]
variables [add_comm_monoid M] [module R M] [module A M] [is_scalar_tower R A M]
variables {R} (A) {M}
theorem algebra_map_smul (r : R) (x : M) : algebra_map R A r • x = r • x :=
by rw [algebra.algebra_map_eq_smul_one, smul_assoc, one_smul]
end module
section semiring
variables [comm_semiring R] [comm_semiring S] [semiring A] [semiring B]
variables [algebra R S] [algebra S A] [algebra S B]
variables {R S A}
theorem of_algebra_map_eq [algebra R A]
(h : ∀ x, algebra_map R A x = algebra_map S A (algebra_map R S x)) :
is_scalar_tower R S A :=
⟨λ x y z, by simp_rw [algebra.smul_def, ring_hom.map_mul, mul_assoc, h]⟩
/-- See note [partially-applied ext lemmas]. -/
theorem of_algebra_map_eq' [algebra R A]
(h : algebra_map R A = (algebra_map S A).comp (algebra_map R S)) :
is_scalar_tower R S A :=
of_algebra_map_eq $ ring_hom.ext_iff.1 h
variables (R S A)
instance subalgebra (S₀ : subalgebra R S) : is_scalar_tower S₀ S A :=
of_algebra_map_eq $ λ x, rfl
variables [algebra R A] [algebra R B]
variables [is_scalar_tower R S A] [is_scalar_tower R S B]
theorem algebra_map_eq :
algebra_map R A = (algebra_map S A).comp (algebra_map R S) :=
ring_hom.ext $ λ x, by simp_rw [ring_hom.comp_apply, algebra.algebra_map_eq_smul_one,
smul_assoc, one_smul]
theorem algebra_map_apply (x : R) : algebra_map R A x = algebra_map S A (algebra_map R S x) :=
by rw [algebra_map_eq R S A, ring_hom.comp_apply]
instance subalgebra' (S₀ : subalgebra R S) : is_scalar_tower R S₀ A :=
@is_scalar_tower.of_algebra_map_eq R S₀ A _ _ _ _ _ _ $ λ _,
(is_scalar_tower.algebra_map_apply R S A _ : _)
@[ext] lemma algebra.ext {S : Type u} {A : Type v} [comm_semiring S] [semiring A]
(h1 h2 : algebra S A) (h : ∀ {r : S} {x : A}, (by haveI := h1; exact r • x) = r • x) : h1 = h2 :=
begin
unfreezingI { cases h1 with f1 g1 h11 h12, cases h2 with f2 g2 h21 h22,
cases f1, cases f2, congr', { ext r x, exact h },
ext r, erw [← mul_one (g1 r), ← h12, ← mul_one (g2 r), ← h22, h], refl }
end
/-- In a tower, the canonical map from the middle element to the top element is an
algebra homomorphism over the bottom element. -/
def to_alg_hom : S →ₐ[R] A :=
{ commutes' := λ _, (algebra_map_apply _ _ _ _).symm,
.. algebra_map S A }
lemma to_alg_hom_apply (y : S) : to_alg_hom R S A y = algebra_map S A y := rfl
@[simp] lemma coe_to_alg_hom : ↑(to_alg_hom R S A) = algebra_map S A :=
ring_hom.ext $ λ _, rfl
@[simp] lemma coe_to_alg_hom' : (to_alg_hom R S A : S → A) = algebra_map S A :=
rfl
variables (R) {S A B}
-- conflicts with is_scalar_tower.subalgebra
@[priority 999] instance subsemiring (U : subsemiring S) : is_scalar_tower U S A :=
of_algebra_map_eq $ λ x, rfl
section
local attribute [instance] algebra.of_is_subring subset.comm_ring
-- conflicts with is_scalar_tower.subalgebra
@[priority 999] instance subring {S A : Type*} [comm_ring S] [ring A] [algebra S A]
(U : set S) [is_subring U] : is_scalar_tower U S A :=
of_algebra_map_eq $ λ x, rfl
end
@[nolint instance_priority]
instance of_ring_hom {R A B : Type*} [comm_semiring R] [comm_semiring A] [comm_semiring B]
[algebra R A] [algebra R B] (f : A →ₐ[R] B) :
@is_scalar_tower R A B _ (f.to_ring_hom.to_algebra.to_has_scalar) _ :=
by { letI := (f : A →+* B).to_algebra, exact of_algebra_map_eq (λ x, (f.commutes x).symm) }
end semiring
section division_ring
variables [field R] [division_ring S] [algebra R S] [char_zero R] [char_zero S]
instance rat : is_scalar_tower ℚ R S :=
of_algebra_map_eq $ λ x, ((algebra_map R S).map_rat_cast x).symm
end division_ring
end is_scalar_tower
section homs
variables [comm_semiring R] [comm_semiring S] [semiring A] [semiring B]
variables [algebra R S] [algebra S A] [algebra S B]
variables [algebra R A] [algebra R B]
variables [is_scalar_tower R S A] [is_scalar_tower R S B]
variables (R) {A S B}
open is_scalar_tower
/-- Given a scalar tower `R`, `S`, `A` of algebras, reinterpret an `S`-subalgebra of `A` an as an
`R`-subalgebra. -/
def subalgebra.restrict_scalars (iSB : subalgebra S A) :
subalgebra R A :=
{ one_mem' := iSB.one_mem,
mul_mem' := λ _ _, iSB.mul_mem,
algebra_map_mem' := λ r, begin
rw is_scalar_tower.algebra_map_eq R S,
exact iSB.algebra_map_mem' _,
end,
.. iSB.to_submodule.restrict_scalars R }
namespace alg_hom
/-- R ⟶ S induces S-Alg ⥤ R-Alg -/
def restrict_scalars (f : A →ₐ[S] B) : A →ₐ[R] B :=
{ commutes' := λ r, by { rw [algebra_map_apply R S A, algebra_map_apply R S B],
exact f.commutes (algebra_map R S r) },
.. (f : A →+* B) }
lemma restrict_scalars_apply (f : A →ₐ[S] B) (x : A) : f.restrict_scalars R x = f x := rfl
@[simp] lemma coe_restrict_scalars (f : A →ₐ[S] B) : (f.restrict_scalars R : A →+* B) = f := rfl
@[simp] lemma coe_restrict_scalars' (f : A →ₐ[S] B) : (restrict_scalars R f : A → B) = f := rfl
end alg_hom
namespace alg_equiv
/-- R ⟶ S induces S-Alg ⥤ R-Alg -/
def restrict_scalars (f : A ≃ₐ[S] B) : A ≃ₐ[R] B :=
{ commutes' := λ r, by { rw [algebra_map_apply R S A, algebra_map_apply R S B],
exact f.commutes (algebra_map R S r) },
.. (f : A ≃+* B) }
lemma restrict_scalars_apply (f : A ≃ₐ[S] B) (x : A) : f.restrict_scalars R x = f x := rfl
@[simp] lemma coe_restrict_scalars (f : A ≃ₐ[S] B) : (f.restrict_scalars R : A ≃+* B) = f := rfl
@[simp] lemma coe_restrict_scalars' (f : A ≃ₐ[S] B) : (restrict_scalars R f : A → B) = f := rfl
end alg_equiv
end homs
namespace subalgebra
open is_scalar_tower
section semiring
variables (R) {S A} [comm_semiring R] [comm_semiring S] [semiring A]
variables [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A]
/-- If A/S/R is a tower of algebras then the `res`triction of a S-subalgebra of A is
an R-subalgebra of A. -/
def res (U : subalgebra S A) : subalgebra R A :=
{ algebra_map_mem' := λ x, by { rw algebra_map_apply R S A, exact U.algebra_map_mem _ },
.. U }
@[simp] lemma res_top : res R (⊤ : subalgebra S A) = ⊤ :=
algebra.eq_top_iff.2 $ λ _, show _ ∈ (⊤ : subalgebra S A), from algebra.mem_top
@[simp] lemma mem_res {U : subalgebra S A} {x : A} : x ∈ res R U ↔ x ∈ U := iff.rfl
lemma res_inj {U V : subalgebra S A} (H : res R U = res R V) : U = V :=
ext $ λ x, by rw [← mem_res R, H, mem_res]
/-- Produces a map from `subalgebra.under`. -/
def of_under {R A B : Type*} [comm_semiring R] [comm_semiring A] [semiring B]
[algebra R A] [algebra R B] (S : subalgebra R A) (U : subalgebra S A)
[algebra S B] [is_scalar_tower R S B] (f : U →ₐ[S] B) : S.under U →ₐ[R] B :=
{ commutes' := λ r, (f.commutes (algebra_map R S r)).trans (algebra_map_apply R S B r).symm,
.. f }
end semiring
end subalgebra
namespace is_scalar_tower
open subalgebra
variables [comm_semiring R] [comm_semiring S] [comm_semiring A]
variables [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A]
theorem range_under_adjoin (t : set A) :
(to_alg_hom R S A).range.under (algebra.adjoin _ t) = res R (algebra.adjoin S t) :=
subalgebra.ext $ λ z,
show z ∈ subsemiring.closure (set.range (algebra_map (to_alg_hom R S A).range A) ∪ t : set A) ↔
z ∈ subsemiring.closure (set.range (algebra_map S A) ∪ t : set A),
from suffices set.range (algebra_map (to_alg_hom R S A).range A) = set.range (algebra_map S A),
by rw this,
by { ext z, exact ⟨λ ⟨⟨x, y, h1⟩, h2⟩, ⟨y, h2 ▸ h1⟩, λ ⟨y, hy⟩, ⟨⟨z, y, hy⟩, rfl⟩⟩ }
end is_scalar_tower
section semiring
variables {R S A}
variables [comm_semiring R] [semiring S] [add_comm_monoid A]
variables [algebra R S] [module S A] [module R A] [is_scalar_tower R S A]
namespace submodule
open is_scalar_tower
theorem smul_mem_span_smul_of_mem {s : set S} {t : set A} {k : S} (hks : k ∈ span R s)
{x : A} (hx : x ∈ t) : k • x ∈ span R (s • t) :=
span_induction hks (λ c hc, subset_span $ set.mem_smul.2 ⟨c, x, hc, hx, rfl⟩)
(by { rw zero_smul, exact zero_mem _ })
(λ c₁ c₂ ih₁ ih₂, by { rw add_smul, exact add_mem _ ih₁ ih₂ })
(λ b c hc, by { rw is_scalar_tower.smul_assoc, exact smul_mem _ _ hc })
theorem smul_mem_span_smul {s : set S} (hs : span R s = ⊤) {t : set A} {k : S}
{x : A} (hx : x ∈ span R t) :
k • x ∈ span R (s • t) :=
span_induction hx (λ x hx, smul_mem_span_smul_of_mem (hs.symm ▸ mem_top) hx)
(by { rw smul_zero, exact zero_mem _ })
(λ x y ihx ihy, by { rw smul_add, exact add_mem _ ihx ihy })
(λ c x hx, smul_comm c k x ▸ smul_mem _ _ hx)
theorem smul_mem_span_smul' {s : set S} (hs : span R s = ⊤) {t : set A} {k : S}
{x : A} (hx : x ∈ span R (s • t)) :
k • x ∈ span R (s • t) :=
span_induction hx (λ x hx, let ⟨p, q, hp, hq, hpq⟩ := set.mem_smul.1 hx in
by { rw [← hpq, smul_smul], exact smul_mem_span_smul_of_mem (hs.symm ▸ mem_top) hq })
(by { rw smul_zero, exact zero_mem _ })
(λ x y ihx ihy, by { rw smul_add, exact add_mem _ ihx ihy })
(λ c x hx, smul_comm c k x ▸ smul_mem _ _ hx)
theorem span_smul {s : set S} (hs : span R s = ⊤) (t : set A) :
span R (s • t) = (span S t).restrict_scalars R :=
le_antisymm (span_le.2 $ λ x hx, let ⟨p, q, hps, hqt, hpqx⟩ := set.mem_smul.1 hx in
hpqx ▸ (span S t).smul_mem p (subset_span hqt)) $
λ p hp, span_induction hp (λ x hx, one_smul S x ▸ smul_mem_span_smul hs (subset_span hx))
(zero_mem _)
(λ _ _, add_mem _)
(λ k x hx, smul_mem_span_smul' hs hx)
end submodule
end semiring
section ring
namespace algebra
variables [comm_semiring R] [ring A] [algebra R A]
variables [add_comm_group M] [module A M] [module R M] [is_scalar_tower R A M]
lemma lsmul_injective [no_zero_smul_divisors A M] {x : A} (hx : x ≠ 0) :
function.injective (lsmul R M x) :=
smul_left_injective _ hx
end algebra
end ring
|
f5c49057e4b6d37cf8fa44773922912ebd1a01f5 | e00ea76a720126cf9f6d732ad6216b5b824d20a7 | /src/tactic/linarith.lean | a6757a686bd052f18d116f97ab9b706458743641 | [
"Apache-2.0"
] | permissive | vaibhavkarve/mathlib | a574aaf68c0a431a47fa82ce0637f0f769826bfe | 17f8340912468f49bdc30acdb9a9fa02eeb0473a | refs/heads/master | 1,621,263,802,637 | 1,585,399,588,000 | 1,585,399,588,000 | 250,833,447 | 0 | 0 | Apache-2.0 | 1,585,410,341,000 | 1,585,410,341,000 | null | UTF-8 | Lean | false | false | 35,244 | lean | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Robert Y. Lewis
-/
import tactic.ring data.nat.gcd data.list.defs meta.rb_map data.tree
/-!
# `linarith`
A tactic for discharging linear arithmetic goals using Fourier-Motzkin elimination.
`linarith` is (in principle) complete for ℚ and ℝ. It is not complete for non-dense orders, i.e. ℤ.
- @TODO: investigate storing comparisons in a list instead of a set, for possible efficiency gains
- @TODO: delay proofs of denominator normalization and nat casting until after contradiction is
found
-/
meta def nat.to_pexpr : ℕ → pexpr
| 0 := ``(0)
| 1 := ``(1)
| n := if n % 2 = 0 then ``(bit0 %%(nat.to_pexpr (n/2))) else ``(bit1 %%(nat.to_pexpr (n/2)))
open native
namespace linarith
section lemmas
lemma int.coe_nat_bit0 (n : ℕ) : (↑(bit0 n : ℕ) : ℤ) = bit0 (↑n : ℤ) := by simp [bit0]
lemma int.coe_nat_bit1 (n : ℕ) : (↑(bit1 n : ℕ) : ℤ) = bit1 (↑n : ℤ) := by simp [bit1, bit0]
lemma int.coe_nat_bit0_mul (n : ℕ) (x : ℕ) : (↑(bit0 n * x) : ℤ) = (↑(bit0 n) : ℤ) * (↑x : ℤ) := by simp
lemma int.coe_nat_bit1_mul (n : ℕ) (x : ℕ) : (↑(bit1 n * x) : ℤ) = (↑(bit1 n) : ℤ) * (↑x : ℤ) := by simp
lemma int.coe_nat_one_mul (x : ℕ) : (↑(1 * x) : ℤ) = 1 * (↑x : ℤ) := by simp
lemma int.coe_nat_zero_mul (x : ℕ) : (↑(0 * x) : ℤ) = 0 * (↑x : ℤ) := by simp
lemma int.coe_nat_mul_bit0 (n : ℕ) (x : ℕ) : (↑(x * bit0 n) : ℤ) = (↑x : ℤ) * (↑(bit0 n) : ℤ) := by simp
lemma int.coe_nat_mul_bit1 (n : ℕ) (x : ℕ) : (↑(x * bit1 n) : ℤ) = (↑x : ℤ) * (↑(bit1 n) : ℤ) := by simp
lemma int.coe_nat_mul_one (x : ℕ) : (↑(x * 1) : ℤ) = (↑x : ℤ) * 1 := by simp
lemma int.coe_nat_mul_zero (x : ℕ) : (↑(x * 0) : ℤ) = (↑x : ℤ) * 0 := by simp
lemma nat_eq_subst {n1 n2 : ℕ} {z1 z2 : ℤ} (hn : n1 = n2) (h1 : ↑n1 = z1) (h2 : ↑n2 = z2) : z1 = z2 :=
by simpa [eq.symm h1, eq.symm h2, int.coe_nat_eq_coe_nat_iff]
lemma nat_le_subst {n1 n2 : ℕ} {z1 z2 : ℤ} (hn : n1 ≤ n2) (h1 : ↑n1 = z1) (h2 : ↑n2 = z2) : z1 ≤ z2 :=
by simpa [eq.symm h1, eq.symm h2, int.coe_nat_le]
lemma nat_lt_subst {n1 n2 : ℕ} {z1 z2 : ℤ} (hn : n1 < n2) (h1 : ↑n1 = z1) (h2 : ↑n2 = z2) : z1 < z2 :=
by simpa [eq.symm h1, eq.symm h2, int.coe_nat_lt]
lemma eq_of_eq_of_eq {α} [ordered_semiring α] {a b : α} (ha : a = 0) (hb : b = 0) : a + b = 0 :=
by simp *
lemma le_of_eq_of_le {α} [ordered_semiring α] {a b : α} (ha : a = 0) (hb : b ≤ 0) : a + b ≤ 0 :=
by simp *
lemma lt_of_eq_of_lt {α} [ordered_semiring α] {a b : α} (ha : a = 0) (hb : b < 0) : a + b < 0 :=
by simp *
lemma le_of_le_of_eq {α} [ordered_semiring α] {a b : α} (ha : a ≤ 0) (hb : b = 0) : a + b ≤ 0 :=
by simp *
lemma lt_of_lt_of_eq {α} [ordered_semiring α] {a b : α} (ha : a < 0) (hb : b = 0) : a + b < 0 :=
by simp *
lemma mul_neg {α} [ordered_ring α] {a b : α} (ha : a < 0) (hb : b > 0) : b * a < 0 :=
have (-b)*a > 0, from mul_pos_of_neg_of_neg (neg_neg_of_pos hb) ha,
neg_of_neg_pos (by simpa)
lemma mul_nonpos {α} [ordered_ring α] {a b : α} (ha : a ≤ 0) (hb : b > 0) : b * a ≤ 0 :=
have (-b)*a ≥ 0, from mul_nonneg_of_nonpos_of_nonpos (le_of_lt (neg_neg_of_pos hb)) ha,
(by simpa)
lemma mul_eq {α} [ordered_semiring α] {a b : α} (ha : a = 0) (hb : b > 0) : b * a = 0 :=
by simp *
lemma eq_of_not_lt_of_not_gt {α} [linear_order α] (a b : α) (h1 : ¬ a < b) (h2 : ¬ b < a) : a = b :=
le_antisymm (le_of_not_gt h2) (le_of_not_gt h1)
lemma add_subst {α} [ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
n * (e1 + e2) = t1 + t2 := by simp [left_distrib, *]
lemma sub_subst {α} [ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
n * (e1 - e2) = t1 - t2 := by simp [left_distrib, *, sub_eq_add_neg]
lemma neg_subst {α} [ring α] {n e t : α} (h1 : n * e = t) : n * (-e) = -t := by simp *
private meta def apnn : tactic unit := `[norm_num]
lemma mul_subst {α} [comm_ring α] {n1 n2 k e1 e2 t1 t2 : α} (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2)
(h3 : n1*n2 = k . apnn) : k * (e1 * e2) = t1 * t2 :=
have h3 : n1 * n2 = k, from h3,
by rw [←h3, mul_comm n1, mul_assoc n2, ←mul_assoc n1, h1, ←mul_assoc n2, mul_comm n2, mul_assoc, h2] -- OUCH
lemma div_subst {α} [field α] {n1 n2 k e1 e2 t1 : α} (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1*n2 = k) :
k * (e1 / e2) = t1 :=
by rw [←h3, mul_assoc, mul_div_comm, h2, ←mul_assoc, h1, mul_comm, one_mul]
end lemmas
section datatypes
@[derive decidable_eq, derive inhabited]
inductive ineq
| eq | le | lt
open ineq
def ineq.max : ineq → ineq → ineq
| eq a := a
| le a := a
| lt a := lt
def ineq.is_lt : ineq → ineq → bool
| eq le := tt
| eq lt := tt
| le lt := tt
| _ _ := ff
def ineq.to_string : ineq → string
| eq := "="
| le := "≤"
| lt := "<"
instance : has_to_string ineq := ⟨ineq.to_string⟩
/--
The main datatype for FM elimination.
Variables are represented by natural numbers, each of which has an integer coefficient.
Index 0 is reserved for constants, i.e. `coeffs.find 0` is the coefficient of 1.
The represented term is `coeffs.keys.sum (λ i, coeffs.find i * Var[i])`.
str determines the direction of the comparison -- is it < 0, ≤ 0, or = 0?
-/
@[derive _root_.inhabited]
meta structure comp :=
(str : ineq)
(coeffs : rb_map ℕ int)
meta inductive comp_source
| assump : ℕ → comp_source
| add : comp_source → comp_source → comp_source
| scale : ℕ → comp_source → comp_source
meta def comp_source.flatten : comp_source → rb_map ℕ ℕ
| (comp_source.assump n) := mk_rb_map.insert n 1
| (comp_source.add c1 c2) := (comp_source.flatten c1).add (comp_source.flatten c2)
| (comp_source.scale n c) := (comp_source.flatten c).map (λ v, v * n)
meta def comp_source.to_string : comp_source → string
| (comp_source.assump e) := to_string e
| (comp_source.add c1 c2) := comp_source.to_string c1 ++ " + " ++ comp_source.to_string c2
| (comp_source.scale n c) := to_string n ++ " * " ++ comp_source.to_string c
meta instance comp_source.has_to_format : has_to_format comp_source :=
⟨λ a, comp_source.to_string a⟩
meta structure pcomp :=
(c : comp)
(src : comp_source)
meta def map_lt (m1 m2 : rb_map ℕ int) : bool :=
list.lex (prod.lex (<) (<)) m1.to_list m2.to_list
-- make more efficient
meta def comp.lt (c1 c2 : comp) : bool :=
(c1.str.is_lt c2.str) || (c1.str = c2.str) && map_lt c1.coeffs c2.coeffs
meta instance comp.has_lt : has_lt comp := ⟨λ a b, comp.lt a b⟩
meta instance pcomp.has_lt : has_lt pcomp := ⟨λ p1 p2, p1.c < p2.c⟩
-- short-circuit type class inference
meta instance pcomp.has_lt_dec : decidable_rel ((<) : pcomp → pcomp → Prop) := by apply_instance
meta def comp.coeff_of (c : comp) (a : ℕ) : ℤ :=
c.coeffs.zfind a
meta def comp.scale (c : comp) (n : ℕ) : comp :=
{ c with coeffs := c.coeffs.map ((*) (n : ℤ)) }
meta def comp.add (c1 c2 : comp) : comp :=
⟨c1.str.max c2.str, c1.coeffs.add c2.coeffs⟩
meta def pcomp.scale (c : pcomp) (n : ℕ) : pcomp :=
⟨c.c.scale n, comp_source.scale n c.src⟩
meta def pcomp.add (c1 c2 : pcomp) : pcomp :=
⟨c1.c.add c2.c, comp_source.add c1.src c2.src⟩
meta instance pcomp.to_format : has_to_format pcomp :=
⟨λ p, to_fmt p.c.coeffs ++ to_string p.c.str ++ "0"⟩
meta instance comp.to_format : has_to_format comp :=
⟨λ p, to_fmt p.coeffs⟩
end datatypes
section fm_elim
/-- If `c1` and `c2` both contain variable `a` with opposite coefficients,
produces `v1`, `v2`, and `c` such that `a` has been cancelled in `c := v1*c1 + v2*c2`. -/
meta def elim_var (c1 c2 : comp) (a : ℕ) : option (ℕ × ℕ × comp) :=
let v1 := c1.coeff_of a,
v2 := c2.coeff_of a in
if v1 * v2 < 0 then
let vlcm := nat.lcm v1.nat_abs v2.nat_abs,
v1' := vlcm / v1.nat_abs,
v2' := vlcm / v2.nat_abs in
some ⟨v1', v2', comp.add (c1.scale v1') (c2.scale v2')⟩
else none
meta def pelim_var (p1 p2 : pcomp) (a : ℕ) : option pcomp :=
do (n1, n2, c) ← elim_var p1.c p2.c a,
return ⟨c, comp_source.add (p1.src.scale n1) (p2.src.scale n2)⟩
meta def comp.is_contr (c : comp) : bool := c.coeffs.empty ∧ c.str = ineq.lt
meta def pcomp.is_contr (p : pcomp) : bool := p.c.is_contr
meta def elim_with_set (a : ℕ) (p : pcomp) (comps : rb_set pcomp) : rb_set pcomp :=
if ¬ p.c.coeffs.contains a then mk_rb_set.insert p else
comps.fold mk_rb_set $ λ pc s,
match pelim_var p pc a with
| some pc := s.insert pc
| none := s
end
/--
The state for the elimination monad.
* `vars`: the set of variables present in `comps`
* `comps`: a set of comparisons
* `inputs`: a set of pairs of exprs `(t, pf)`, where `t` is a term and `pf` is a proof that
`t {<, ≤, =} 0`, indexed by `ℕ`.
* `has_false`: stores a `pcomp` of `0 < 0` if one has been found
TODO: is it more efficient to store comps as a list, to avoid comparisons?
-/
meta structure linarith_structure :=
(vars : rb_set ℕ)
(comps : rb_set pcomp)
@[reducible] meta def linarith_monad :=
state_t linarith_structure (except_t pcomp id)
meta instance : monad linarith_monad := state_t.monad
meta instance : monad_except pcomp linarith_monad :=
state_t.monad_except pcomp
meta def get_vars : linarith_monad (rb_set ℕ) :=
linarith_structure.vars <$> get
meta def get_var_list : linarith_monad (list ℕ) :=
rb_set.to_list <$> get_vars
meta def get_comps : linarith_monad (rb_set pcomp) :=
linarith_structure.comps <$> get
meta def validate : linarith_monad unit :=
do ⟨_, comps⟩ ← get,
match comps.to_list.find (λ p : pcomp, p.is_contr) with
| none := return ()
| some c := throw c
end
meta def update (vars : rb_set ℕ) (comps : rb_set pcomp) : linarith_monad unit :=
state_t.put ⟨vars, comps⟩ >> validate
meta def monad.elim_var (a : ℕ) : linarith_monad unit :=
do vs ← get_vars,
when (vs.contains a) $
do comps ← get_comps,
let cs' := comps.fold mk_rb_set (λ p s, s.union (elim_with_set a p comps)),
update (vs.erase a) cs'
meta def elim_all_vars : linarith_monad unit :=
get_var_list >>= list.mmap' monad.elim_var
end fm_elim
section parse
open ineq tactic
meta def map_of_expr_mul_aux (c1 c2 : rb_map ℕ ℤ) : option (rb_map ℕ ℤ) :=
match c1.keys, c2.keys with
| [0], _ := some $ c2.scale (c1.zfind 0)
| _, [0] := some $ c1.scale (c2.zfind 0)
| [], _ := some mk_rb_map
| _, [] := some mk_rb_map
| _, _ := none
end
meta def list.mfind {α} (tac : α → tactic unit) : list α → tactic α
| [] := failed
| (h::t) := tac h >> return h <|> list.mfind t
meta def rb_map.find_defeq (red : transparency) {v} (m : expr_map v) (e : expr) : tactic v :=
prod.snd <$> list.mfind (λ p, is_def_eq e p.1 red) m.to_list
/--
Turns an expression into a map from `ℕ` to `ℤ`, for use in a `comp` object.
The `expr_map` `ℕ` argument identifies which expressions have already been assigned numbers.
Returns a new map.
-/
meta def map_of_expr (red : transparency) : expr_map ℕ → expr → tactic (expr_map ℕ × rb_map ℕ ℤ)
| m e@`(%%e1 * %%e2) :=
(do (m', comp1) ← map_of_expr m e1,
(m', comp2) ← map_of_expr m' e2,
mp ← map_of_expr_mul_aux comp1 comp2,
return (m', mp)) <|>
(do k ← rb_map.find_defeq red m e, return (m, mk_rb_map.insert k 1)) <|>
(let n := m.size + 1 in return (m.insert e n, mk_rb_map.insert n 1))
| m `(%%e1 + %%e2) :=
do (m', comp1) ← map_of_expr m e1,
(m', comp2) ← map_of_expr m' e2,
return (m', comp1.add comp2)
| m `(%%e1 - %%e2) :=
do (m', comp1) ← map_of_expr m e1,
(m', comp2) ← map_of_expr m' e2,
return (m', comp1.add (comp2.scale (-1)))
| m `(-%%e) := do (m', comp) ← map_of_expr m e, return (m', comp.scale (-1))
| m e :=
match e.to_int with
| some 0 := return ⟨m, mk_rb_map⟩
| some z := return ⟨m, mk_rb_map.insert 0 z⟩
| none :=
(do k ← rb_map.find_defeq red m e, return (m, mk_rb_map.insert k 1)) <|>
(let n := m.size + 1 in return (m.insert e n, mk_rb_map.insert n 1))
end
meta def parse_into_comp_and_expr : expr → option (ineq × expr)
| `(%%e < 0) := (ineq.lt, e)
| `(%%e ≤ 0) := (ineq.le, e)
| `(%%e = 0) := (ineq.eq, e)
| _ := none
meta def to_comp (red : transparency) (e : expr) (m : expr_map ℕ) : tactic (comp × expr_map ℕ) :=
do (iq, e) ← parse_into_comp_and_expr e,
(m', comp') ← map_of_expr red m e,
return ⟨⟨iq, comp'⟩, m'⟩
meta def to_comp_fold (red : transparency) : expr_map ℕ → list expr →
tactic (list (option comp) × expr_map ℕ)
| m [] := return ([], m)
| m (h::t) :=
(do (c, m') ← to_comp red h m,
(l, mp) ← to_comp_fold m' t,
return (c::l, mp)) <|>
(do (l, mp) ← to_comp_fold m t,
return (none::l, mp))
/--
Takes a list of proofs of props of the form `t {<, ≤, =} 0`, and creates a
`linarith_structure`.
-/
meta def mk_linarith_structure (red : transparency) (l : list expr) :
tactic (linarith_structure × rb_map ℕ (expr × expr)) :=
do pftps ← l.mmap infer_type,
(l', map) ← to_comp_fold red mk_rb_map pftps,
let lz := list.enum $ ((l.zip pftps).zip l').filter_map (λ ⟨a, b⟩, prod.mk a <$> b),
let prmap := rb_map.of_list $ lz.map (λ ⟨n, x⟩, (n, x.1)),
let vars : rb_set ℕ := rb_map.set_of_list $ list.range map.size.succ,
let pc : rb_set pcomp := rb_map.set_of_list $
lz.map (λ ⟨n, x⟩, ⟨x.2, comp_source.assump n⟩),
return (⟨vars, pc⟩, prmap)
meta def linarith_monad.run (red : transparency) {α} (tac : linarith_monad α) (l : list expr) :
tactic ((pcomp ⊕ α) × rb_map ℕ (expr × expr)) :=
do (struct, inputs) ← mk_linarith_structure red l,
match (state_t.run (validate >> tac) struct).run with
| (except.ok (a, _)) := return (sum.inr a, inputs)
| (except.error contr) := return (sum.inl contr, inputs)
end
end parse
section prove
open ineq tactic
meta def get_rel_sides : expr → tactic (expr × expr)
| `(%%a < %%b) := return (a, b)
| `(%%a ≤ %%b) := return (a, b)
| `(%%a = %%b) := return (a, b)
| `(%%a ≥ %%b) := return (a, b)
| `(%%a > %%b) := return (a, b)
| _ := failed
meta def mul_expr (n : ℕ) (e : expr) : pexpr :=
if n = 1 then ``(%%e) else
``(%%(nat.to_pexpr n) * %%e)
meta def add_exprs_aux : pexpr → list pexpr → pexpr
| p [] := p
| p [a] := ``(%%p + %%a)
| p (h::t) := add_exprs_aux ``(%%p + %%h) t
meta def add_exprs : list pexpr → pexpr
| [] := ``(0)
| (h::t) := add_exprs_aux h t
meta def find_contr (m : rb_set pcomp) : option pcomp :=
m.keys.find (λ p, p.c.is_contr)
meta def ineq_const_mul_nm : ineq → name
| lt := ``mul_neg
| le := ``mul_nonpos
| eq := ``mul_eq
meta def ineq_const_nm : ineq → ineq → (name × ineq)
| eq eq := (``eq_of_eq_of_eq, eq)
| eq le := (``le_of_eq_of_le, le)
| eq lt := (``lt_of_eq_of_lt, lt)
| le eq := (``le_of_le_of_eq, le)
| le le := (`add_nonpos, le)
| le lt := (`add_neg_of_nonpos_of_neg, lt)
| lt eq := (``lt_of_lt_of_eq, lt)
| lt le := (`add_neg_of_neg_of_nonpos, lt)
| lt lt := (`add_neg, lt)
meta def mk_single_comp_zero_pf (c : ℕ) (h : expr) : tactic (ineq × expr) :=
do tp ← infer_type h,
some (iq, e) ← return $ parse_into_comp_and_expr tp,
if c = 0 then
do e' ← mk_app ``zero_mul [e], return (eq, e')
else if c = 1 then return (iq, h)
else
do nm ← resolve_name (ineq_const_mul_nm iq),
tp ← (prod.snd <$> (infer_type h >>= get_rel_sides)) >>= infer_type,
cpos ← to_expr ``((%%c.to_pexpr : %%tp) > 0),
(_, ex) ← solve_aux cpos `[norm_num, done],
-- e' ← mk_app (ineq_const_mul_nm iq) [h, ex], -- this takes many seconds longer in some examples! why?
e' ← to_expr ``(%%nm %%h %%ex) ff,
return (iq, e')
meta def mk_lt_zero_pf_aux (c : ineq) (pf npf : expr) (coeff : ℕ) : tactic (ineq × expr) :=
do (iq, h') ← mk_single_comp_zero_pf coeff npf,
let (nm, niq) := ineq_const_nm c iq,
n ← resolve_name nm,
e' ← to_expr ``(%%n %%pf %%h'),
return (niq, e')
/--
Takes a list of coefficients `[c]` and list of expressions, of equal length.
Each expression is a proof of a prop of the form `t {<, ≤, =} 0`.
Produces a proof that the sum of `(c*t) {<, ≤, =} 0`,
where the `comp` is as strong as possible.
-/
meta def mk_lt_zero_pf : list ℕ → list expr → tactic expr
| _ [] := fail "no linear hypotheses found"
| [c] [h] := prod.snd <$> mk_single_comp_zero_pf c h
| (c::ct) (h::t) :=
do (iq, h') ← mk_single_comp_zero_pf c h,
prod.snd <$> (ct.zip t).mfoldl (λ pr ce, mk_lt_zero_pf_aux pr.1 pr.2 ce.2 ce.1) (iq, h')
| _ _ := fail "not enough args to mk_lt_zero_pf"
meta def term_of_ineq_prf (prf : expr) : tactic expr :=
do (lhs, _) ← infer_type prf >>= get_rel_sides,
return lhs
meta structure linarith_config :=
(discharger : tactic unit := `[ring])
(restrict_type : option Type := none)
(restrict_type_reflect : reflected restrict_type . apply_instance)
(exfalso : bool := tt)
(transparency : transparency := reducible)
meta def ineq_pf_tp (pf : expr) : tactic expr :=
do (_, z) ← infer_type pf >>= get_rel_sides,
infer_type z
meta def mk_neg_one_lt_zero_pf (tp : expr) : tactic expr :=
to_expr ``((neg_neg_of_pos zero_lt_one : -1 < (0 : %%tp)))
/--
Assumes `e` is a proof that `t = 0`. Creates a proof that `-t = 0`.
-/
meta def mk_neg_eq_zero_pf (e : expr) : tactic expr :=
to_expr ``(neg_eq_zero.mpr %%e)
meta def add_neg_eq_pfs : list expr → tactic (list expr)
| [] := return []
| (h::t) :=
do some (iq, tp) ← parse_into_comp_and_expr <$> infer_type h,
match iq with
| ineq.eq := do nep ← mk_neg_eq_zero_pf h, tl ← add_neg_eq_pfs t, return $ h::nep::tl
| _ := list.cons h <$> add_neg_eq_pfs t
end
/--
Takes a list of proofs of propositions of the form `t {<, ≤, =} 0`,
and tries to prove the goal `false`.
-/
meta def prove_false_by_linarith1 (cfg : linarith_config) : list expr → tactic unit
| [] := fail "no args to linarith"
| l@(h::t) :=
do l' ← add_neg_eq_pfs l,
hz ← ineq_pf_tp h >>= mk_neg_one_lt_zero_pf,
(sum.inl contr, inputs) ← elim_all_vars.run cfg.transparency (hz::l')
| fail "linarith failed to find a contradiction",
let coeffs := inputs.keys.map (λ k, (contr.src.flatten.ifind k)),
let pfs : list expr := inputs.keys.map (λ k, (inputs.ifind k).1),
let zip := (coeffs.zip pfs).filter (λ pr, pr.1 ≠ 0),
let (coeffs, pfs) := zip.unzip,
mls ← zip.mmap (λ pr, do e ← term_of_ineq_prf pr.2, return (mul_expr pr.1 e)),
sm ← to_expr $ add_exprs mls,
tgt ← to_expr ``(%%sm = 0),
(a, b) ← solve_aux tgt (cfg.discharger >> done),
pf ← mk_lt_zero_pf coeffs pfs,
pftp ← infer_type pf,
(_, nep, _) ← rewrite_core b pftp,
pf' ← mk_eq_mp nep pf,
mk_app `lt_irrefl [pf'] >>= exact
end prove
section normalize
open tactic
set_option eqn_compiler.max_steps 50000
meta def rem_neg (prf : expr) : expr → tactic expr
| `(_ ≤ _) := to_expr ``(lt_of_not_ge %%prf)
| `(_ < _) := to_expr ``(le_of_not_gt %%prf)
| `(_ > _) := to_expr ``(le_of_not_gt %%prf)
| `(_ ≥ _) := to_expr ``(lt_of_not_ge %%prf)
| e := failed
meta def rearr_comp : expr → expr → tactic expr
| prf `(%%a ≤ 0) := return prf
| prf `(%%a < 0) := return prf
| prf `(%%a = 0) := return prf
| prf `(%%a ≥ 0) := to_expr ``(neg_nonpos.mpr %%prf)
| prf `(%%a > 0) := to_expr ``(neg_neg_of_pos %%prf)
| prf `(0 ≥ %%a) := to_expr ``(show %%a ≤ 0, from %%prf)
| prf `(0 > %%a) := to_expr ``(show %%a < 0, from %%prf)
| prf `(0 = %%a) := to_expr ``(eq.symm %%prf)
| prf `(0 ≤ %%a) := to_expr ``(neg_nonpos.mpr %%prf)
| prf `(0 < %%a) := to_expr ``(neg_neg_of_pos %%prf)
| prf `(%%a ≤ %%b) := to_expr ``(sub_nonpos.mpr %%prf)
| prf `(%%a < %%b) := to_expr ``(sub_neg_of_lt %%prf)
| prf `(%%a = %%b) := to_expr ``(sub_eq_zero.mpr %%prf)
| prf `(%%a > %%b) := to_expr ``(sub_neg_of_lt %%prf)
| prf `(%%a ≥ %%b) := to_expr ``(sub_nonpos.mpr %%prf)
| prf `(¬ %%t) := do nprf ← rem_neg prf t, tp ← infer_type nprf, rearr_comp nprf tp
| prf _ := fail "couldn't rearrange comp"
meta def is_numeric : expr → option ℚ
| `(%%e1 + %%e2) := (+) <$> is_numeric e1 <*> is_numeric e2
| `(%%e1 - %%e2) := has_sub.sub <$> is_numeric e1 <*> is_numeric e2
| `(%%e1 * %%e2) := (*) <$> is_numeric e1 <*> is_numeric e2
| `(%%e1 / %%e2) := (/) <$> is_numeric e1 <*> is_numeric e2
| `(-%%e) := rat.neg <$> is_numeric e
| e := e.to_rat
meta def find_cancel_factor : expr → ℕ × tree ℕ
| `(%%e1 + %%e2) :=
let (v1, t1) := find_cancel_factor e1, (v2, t2) := find_cancel_factor e2, lcm := v1.lcm v2 in
(lcm, tree.node lcm t1 t2)
| `(%%e1 - %%e2) :=
let (v1, t1) := find_cancel_factor e1, (v2, t2) := find_cancel_factor e2, lcm := v1.lcm v2 in
(lcm, tree.node lcm t1 t2)
| `(%%e1 * %%e2) :=
match is_numeric e1, is_numeric e2 with
| none, none := (1, tree.node 1 tree.nil tree.nil)
| _, _ :=
let (v1, t1) := find_cancel_factor e1, (v2, t2) := find_cancel_factor e2, pd := v1*v2 in
(pd, tree.node pd t1 t2)
end
| `(%%e1 / %%e2) :=
match is_numeric e2 with
| some q := let (v1, t1) := find_cancel_factor e1, n := v1.lcm q.num.nat_abs in
(n, tree.node n t1 (tree.node q.num.nat_abs tree.nil tree.nil))
| none := (1, tree.node 1 tree.nil tree.nil)
end
| `(-%%e) := find_cancel_factor e
| _ := (1, tree.node 1 tree.nil tree.nil)
open tree
meta def mk_prod_prf : ℕ → tree ℕ → expr → tactic expr
| v (node _ lhs rhs) `(%%e1 + %%e2) :=
do v1 ← mk_prod_prf v lhs e1, v2 ← mk_prod_prf v rhs e2, mk_app ``add_subst [v1, v2]
| v (node _ lhs rhs) `(%%e1 - %%e2) :=
do v1 ← mk_prod_prf v lhs e1, v2 ← mk_prod_prf v rhs e2, mk_app ``sub_subst [v1, v2]
| v (node n lhs@(node ln _ _) rhs) `(%%e1 * %%e2) :=
do tp ← infer_type e1, v1 ← mk_prod_prf ln lhs e1, v2 ← mk_prod_prf (v/ln) rhs e2,
ln' ← tp.of_nat ln, vln' ← tp.of_nat (v/ln), v' ← tp.of_nat v,
ntp ← to_expr ``(%%ln' * %%vln' = %%v'),
(_, npf) ← solve_aux ntp `[norm_num, done],
mk_app ``mul_subst [v1, v2, npf]
| v (node n lhs rhs@(node rn _ _)) `(%%e1 / %%e2) :=
do tp ← infer_type e1, v1 ← mk_prod_prf (v/rn) lhs e1,
rn' ← tp.of_nat rn, vrn' ← tp.of_nat (v/rn), n' ← tp.of_nat n, v' ← tp.of_nat v,
ntp ← to_expr ``(%%rn' / %%e2 = 1),
(_, npf) ← solve_aux ntp `[norm_num, done],
ntp2 ← to_expr ``(%%vrn' * %%n' = %%v'),
(_, npf2) ← solve_aux ntp2 `[norm_num, done],
mk_app ``div_subst [v1, npf, npf2]
| v t `(-%%e) := do v' ← mk_prod_prf v t e, mk_app ``neg_subst [v']
| v _ e :=
do tp ← infer_type e,
v' ← tp.of_nat v,
e' ← to_expr ``(%%v' * %%e),
mk_app `eq.refl [e']
/--
Given `e`, a term with rational division, produces a natural number `n` and a proof of `n*e = e'`,
where `e'` has no division.
-/
meta def kill_factors (e : expr) : tactic (ℕ × expr) :=
let (n, t) := find_cancel_factor e in
do e' ← mk_prod_prf n t e, return (n, e')
open expr
meta def expr_contains (n : name) : expr → bool
| (const nm _) := nm = n
| (lam _ _ _ bd) := expr_contains bd
| (pi _ _ _ bd) := expr_contains bd
| (app e1 e2) := expr_contains e1 || expr_contains e2
| _ := ff
lemma sub_into_lt {α} [ordered_semiring α] {a b : α} (he : a = b) (hl : a ≤ 0) : b ≤ 0 :=
by rwa he at hl
meta def norm_hyp_aux (h' lhs : expr) : tactic expr :=
do (v, lhs') ← kill_factors lhs,
if v = 1 then return h' else do
(ih, h'') ← mk_single_comp_zero_pf v h',
(_, nep, _) ← infer_type h'' >>= rewrite_core lhs',
mk_eq_mp nep h''
meta def norm_hyp (h : expr) : tactic expr :=
do htp ← infer_type h,
h' ← rearr_comp h htp,
some (c, lhs) ← parse_into_comp_and_expr <$> infer_type h',
if expr_contains `has_div.div lhs then
norm_hyp_aux h' lhs
else return h'
meta def get_contr_lemma_name : expr → option name
| `(%%a < %%b) := return `lt_of_not_ge
| `(%%a ≤ %%b) := return `le_of_not_gt
| `(%%a = %%b) := return ``eq_of_not_lt_of_not_gt
| `(%%a ≠ %%b) := return `not.intro
| `(%%a ≥ %%b) := return `le_of_not_gt
| `(%%a > %%b) := return `lt_of_not_ge
| `(¬ %%a < %%b) := return `not.intro
| `(¬ %%a ≤ %%b) := return `not.intro
| `(¬ %%a = %%b) := return `not.intro
| `(¬ %%a ≥ %%b) := return `not.intro
| `(¬ %%a > %%b) := return `not.intro
| _ := none
/-- Assumes the input `t` is of type `ℕ`. Produces `t'` of type `ℤ` such that `↑t = t'` and
a proof of equality. -/
meta def cast_expr (e : expr) : tactic (expr × expr) :=
do s ← [`int.coe_nat_add, `int.coe_nat_zero, `int.coe_nat_one,
``int.coe_nat_bit0_mul, ``int.coe_nat_bit1_mul, ``int.coe_nat_zero_mul, ``int.coe_nat_one_mul,
``int.coe_nat_mul_bit0, ``int.coe_nat_mul_bit1, ``int.coe_nat_mul_zero, ``int.coe_nat_mul_one,
``int.coe_nat_bit0, ``int.coe_nat_bit1].mfoldl simp_lemmas.add_simp simp_lemmas.mk,
ce ← to_expr ``(↑%%e : ℤ),
simplify s [] ce {fail_if_unchanged := ff}
meta def is_nat_int_coe : expr → option expr
| `((↑(%%n : ℕ) : ℤ)) := some n
| _ := none
meta def mk_coe_nat_nonneg_prf (e : expr) : tactic expr :=
mk_app `int.coe_nat_nonneg [e]
meta def get_nat_comps : expr → list expr
| `(%%a + %%b) := (get_nat_comps a).append (get_nat_comps b)
| `(%%a * %%b) := (get_nat_comps a).append (get_nat_comps b)
| e := match is_nat_int_coe e with
| some e' := [e']
| none := []
end
meta def mk_coe_nat_nonneg_prfs (e : expr) : tactic (list expr) :=
(get_nat_comps e).mmap mk_coe_nat_nonneg_prf
meta def mk_cast_eq_and_nonneg_prfs (pf a b : expr) (ln : name) : tactic (list expr) :=
do (a', prfa) ← cast_expr a,
(b', prfb) ← cast_expr b,
la ← mk_coe_nat_nonneg_prfs a',
lb ← mk_coe_nat_nonneg_prfs b',
pf' ← mk_app ln [pf, prfa, prfb],
return $ pf'::(la.append lb)
meta def mk_int_pfs_of_nat_pf (pf : expr) : tactic (list expr) :=
do tp ← infer_type pf,
match tp with
| `(%%a = %%b) := mk_cast_eq_and_nonneg_prfs pf a b ``nat_eq_subst
| `(%%a ≤ %%b) := mk_cast_eq_and_nonneg_prfs pf a b ``nat_le_subst
| `(%%a < %%b) := mk_cast_eq_and_nonneg_prfs pf a b ``nat_lt_subst
| `(%%a ≥ %%b) := mk_cast_eq_and_nonneg_prfs pf b a ``nat_le_subst
| `(%%a > %%b) := mk_cast_eq_and_nonneg_prfs pf b a ``nat_lt_subst
| `(¬ %%a ≤ %%b) := do pf' ← mk_app ``lt_of_not_ge [pf], mk_cast_eq_and_nonneg_prfs pf' b a ``nat_lt_subst
| `(¬ %%a < %%b) := do pf' ← mk_app ``le_of_not_gt [pf], mk_cast_eq_and_nonneg_prfs pf' b a ``nat_le_subst
| `(¬ %%a ≥ %%b) := do pf' ← mk_app ``lt_of_not_ge [pf], mk_cast_eq_and_nonneg_prfs pf' a b ``nat_lt_subst
| `(¬ %%a > %%b) := do pf' ← mk_app ``le_of_not_gt [pf], mk_cast_eq_and_nonneg_prfs pf' a b ``nat_le_subst
| _ := fail "mk_int_pfs_of_nat_pf failed: proof is not an inequality"
end
meta def mk_non_strict_int_pf_of_strict_int_pf (pf : expr) : tactic expr :=
do tp ← infer_type pf,
match tp with
| `(%%a < %%b) := to_expr ``(@cast (%%a < %%b) (%%a + 1 ≤ %%b) (by refl) %%pf)
| `(%%a > %%b) := to_expr ``(@cast (%%a > %%b) (%%a ≥ %%b + 1) (by refl) %%pf)
| `(¬ %%a ≤ %%b) := to_expr ``(@cast (%%a > %%b) (%%a ≥ %%b + 1) (by refl) (lt_of_not_ge %%pf))
| `(¬ %%a ≥ %%b) := to_expr ``(@cast (%%a < %%b) (%%a + 1 ≤ %%b) (by refl) (lt_of_not_ge %%pf))
| _ := fail "mk_non_strict_int_pf_of_strict_int_pf failed: proof is not an inequality"
end
meta def guard_is_nat_prop : expr → tactic unit
| `(%%a = _) := infer_type a >>= unify `(ℕ)
| `(%%a ≤ _) := infer_type a >>= unify `(ℕ)
| `(%%a < _) := infer_type a >>= unify `(ℕ)
| `(%%a ≥ _) := infer_type a >>= unify `(ℕ)
| `(%%a > _) := infer_type a >>= unify `(ℕ)
| `(¬ %%p) := guard_is_nat_prop p
| _ := failed
meta def guard_is_strict_int_prop : expr → tactic unit
| `(%%a < _) := infer_type a >>= unify `(ℤ)
| `(%%a > _) := infer_type a >>= unify `(ℤ)
| `(¬ %%a ≤ _) := infer_type a >>= unify `(ℤ)
| `(¬ %%a ≥ _) := infer_type a >>= unify `(ℤ)
| _ := failed
meta def replace_nat_pfs : list expr → tactic (list expr)
| [] := return []
| (h::t) :=
(do infer_type h >>= guard_is_nat_prop,
ls ← mk_int_pfs_of_nat_pf h,
list.append ls <$> replace_nat_pfs t) <|> list.cons h <$> replace_nat_pfs t
meta def replace_strict_int_pfs : list expr → tactic (list expr)
| [] := return []
| (h::t) :=
(do infer_type h >>= guard_is_strict_int_prop,
l ← mk_non_strict_int_pf_of_strict_int_pf h,
list.cons l <$> replace_strict_int_pfs t) <|> list.cons h <$> replace_strict_int_pfs t
meta def partition_by_type_aux : rb_lmap expr expr → list expr → tactic (rb_lmap expr expr)
| m [] := return m
| m (h::t) := do tp ← ineq_pf_tp h, partition_by_type_aux (m.insert tp h) t
meta def partition_by_type (l : list expr) : tactic (rb_lmap expr expr) :=
partition_by_type_aux mk_rb_map l
private meta def try_linarith_on_lists (cfg : linarith_config) (ls : list (list expr)) : tactic unit :=
(first $ ls.map $ prove_false_by_linarith1 cfg) <|> fail "linarith failed"
/--
Takes a list of proofs of propositions.
Filters out the proofs of linear (in)equalities,
and tries to use them to prove `false`.
If `pref_type` is given, starts by working over this type.
-/
meta def prove_false_by_linarith (cfg : linarith_config) (pref_type : option expr) (l : list expr) : tactic unit :=
do l' ← replace_nat_pfs l,
l'' ← replace_strict_int_pfs l',
ls ← list.reduce_option <$> l''.mmap (λ h, (do s ← norm_hyp h, return (some s)) <|> return none)
>>= partition_by_type,
pref_type ← (unify pref_type.iget `(ℕ) >> return (some `(ℤ) : option expr)) <|> return pref_type,
match cfg.restrict_type, rb_map.values ls, pref_type with
| some rtp, _, _ :=
do m ← mk_mvar, unify `(some %%m : option Type) cfg.restrict_type_reflect, m ← instantiate_mvars m,
prove_false_by_linarith1 cfg (ls.ifind m)
| none, [ls'], _ := prove_false_by_linarith1 cfg ls'
| none, ls', none := try_linarith_on_lists cfg ls'
| none, _, (some t) := prove_false_by_linarith1 cfg (ls.ifind t) <|>
try_linarith_on_lists cfg (rb_map.values (ls.erase t))
end
end normalize
end linarith
section
open tactic linarith
open lean lean.parser interactive tactic interactive.types
local postfix `?`:9001 := optional
local postfix *:9001 := many
meta def linarith.elab_arg_list : option (list pexpr) → tactic (list expr)
| none := return []
| (some l) := l.mmap i_to_expr
meta def linarith.preferred_type_of_goal : option expr → tactic (option expr)
| none := return none
| (some e) := some <$> ineq_pf_tp e
/--
`linarith.interactive_aux cfg o_goal restrict_hyps args`:
* `cfg` is a `linarith_config` object
* `o_goal : option expr` is the local constant corresponding to the former goal, if there was one
* `restrict_hyps : bool` is `tt` if `linarith only [...]` was used
* `args : option (list pexpr)` is the optional list of arguments in `linarith [...]`
-/
meta def linarith.interactive_aux (cfg : linarith_config) :
option expr → bool → option (list pexpr) → tactic unit
| none tt none := fail "linarith only called with no arguments"
| none tt (some l) := l.mmap i_to_expr >>= prove_false_by_linarith cfg none
| (some e) tt l :=
do tp ← ineq_pf_tp e,
list.cons e <$> linarith.elab_arg_list l >>= prove_false_by_linarith cfg (some tp)
| oe ff l :=
do otp ← linarith.preferred_type_of_goal oe,
list.append <$> local_context <*>
(list.filter (λ a, bnot $ expr.is_local_constant a) <$> linarith.elab_arg_list l) >>=
prove_false_by_linarith cfg otp
/--
Tries to prove a goal of `false` by linear arithmetic on hypotheses.
If the goal is a linear (in)equality, tries to prove it by contradiction.
If the goal is not `false` or an inequality, applies `exfalso` and tries linarith on the
hypotheses.
* `linarith` will use all relevant hypotheses in the local context.
* `linarith [t1, t2, t3]` will add proof terms t1, t2, t3 to the local context.
* `linarith only [h1, h2, h3, t1, t2, t3]` will use only the goal (if relevant), local hypotheses
`h1`, `h2`, `h3`, and proofs `t1`, `t2`, `t3`. It will ignore the rest of the local context.
* `linarith!` will use a stronger reducibility setting to identify atoms.
Config options:
* `linarith {exfalso := ff}` will fail on a goal that is neither an inequality nor `false`
* `linarith {restrict_type := T}` will run only on hypotheses that are inequalities over `T`
* `linarith {discharger := tac}` will use `tac` instead of `ring` for normalization.
Options: `ring2`, `ring SOP`, `simp`
---
`linarith` attempts to find a contradiction between hypotheses that are linear (in)equalities.
Equivalently, it can prove a linear inequality by assuming its negation and proving `false`.
In theory, `linarith` should prove any goal that is true in the theory of linear arithmetic over
the rationals. While there is some special handling for non-dense orders like `nat` and `int`,
this tactic is not complete for these theories and will not prove every true goal.
An example:
```lean
example (x y z : ℚ) (h1 : 2*x < 3*y) (h2 : -4*x + 2*z < 0)
(h3 : 12*y - 4* z < 0) : false :=
by linarith
```
`linarith` will use all appropriate hypotheses and the negation of the goal, if applicable.
`linarith [t1, t2, t3]` will additionally use proof terms `t1, t2, t3`.
`linarith only [h1, h2, h3, t1, t2, t3]` will use only the goal (if relevant), local hypotheses
`h1`, `h2`, `h3`, and proofs `t1`, `t2`, `t3`. It will ignore the rest of the local context.
`linarith!` will use a stronger reducibility setting to try to identify atoms. For example,
```lean
example (x : ℚ) : id x ≥ x :=
by linarith
```
will fail, because `linarith` will not identify `x` and `id x`. `linarith!` will.
This can sometimes be expensive.
`linarith {discharger := tac, restrict_type := tp, exfalso := ff}` takes a config object with three
optional arguments:
* `discharger` specifies a tactic to be used for reducing an algebraic equation in the
proof stage. The default is `ring`. Other options currently include `ring SOP` or `simp` for basic
problems.
* `restrict_type` will only use hypotheses that are inequalities over `tp`. This is useful
if you have e.g. both integer and rational valued inequalities in the local context, which can
sometimes confuse the tactic.
* If `exfalso` is false, `linarith` will fail when the goal is neither an inequality nor `false`.
(True by default.)
-/
meta def tactic.interactive.linarith (red : parse ((tk "!")?))
(restr : parse ((tk "only")?)) (hyps : parse pexpr_list?)
(cfg : linarith_config := {}) : tactic unit :=
let cfg :=
if red.is_some then {cfg with transparency := semireducible, discharger := `[ring!]}
else cfg in
do t ← target,
match get_contr_lemma_name t with
| some nm := seq (applyc nm) $
do t ← intro1, linarith.interactive_aux cfg (some t) restr.is_some hyps
| none := if cfg.exfalso then exfalso >> linarith.interactive_aux cfg none restr.is_some hyps
else fail "linarith failed: target type is not an inequality."
end
add_hint_tactic "linarith"
add_tactic_doc
{ name := "linarith",
category := doc_category.tactic,
decl_names := [`tactic.interactive.linarith],
tags := ["arithmetic", "decision procedure", "finishing"] }
end
|
b231c8baa81ee466fb4474e55e876c854fd43380 | d29d82a0af640c937e499f6be79fc552eae0aa13 | /src/topology/algebra/group.lean | 3413057de13aba0af7d323978fd38777871292cf | [
"Apache-2.0"
] | permissive | AbdulMajeedkhurasani/mathlib | 835f8a5c5cf3075b250b3737172043ab4fa1edf6 | 79bc7323b164aebd000524ebafd198eb0e17f956 | refs/heads/master | 1,688,003,895,660 | 1,627,788,521,000 | 1,627,788,521,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 29,428 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import order.filter.pointwise
import group_theory.quotient_group
import topology.algebra.monoid
import topology.homeomorph
/-!
# Theory of topological groups
This file defines the following typeclasses:
* `topological_group`, `topological_add_group`: multiplicative and additive topological groups,
i.e., groups with continuous `(*)` and `(⁻¹)` / `(+)` and `(-)`;
* `has_continuous_sub G` means that `G` has a continuous subtraction operation.
There is an instance deducing `has_continuous_sub` from `topological_group` but we use a separate
typeclass because, e.g., `ℕ` and `ℝ≥0` have continuous subtraction but are not additive groups.
We also define `homeomorph` versions of several `equiv`s: `homeomorph.mul_left`,
`homeomorph.mul_right`, `homeomorph.inv`, and prove a few facts about neighbourhood filters in
groups.
## Tags
topological space, group, topological group
-/
open classical set filter topological_space function
open_locale classical topological_space filter
universes u v w x
variables {α : Type u} {β : Type v} {G : Type w} {H : Type x}
section continuous_mul_group
/-!
### Groups with continuous multiplication
In this section we prove a few statements about groups with continuous `(*)`.
-/
variables [topological_space G] [group G] [has_continuous_mul G]
/-- Multiplication from the left in a topological group as a homeomorphism. -/
@[to_additive "Addition from the left in a topological additive group as a homeomorphism."]
protected def homeomorph.mul_left (a : G) : G ≃ₜ G :=
{ continuous_to_fun := continuous_const.mul continuous_id,
continuous_inv_fun := continuous_const.mul continuous_id,
.. equiv.mul_left a }
@[simp, to_additive]
lemma homeomorph.coe_mul_left (a : G) : ⇑(homeomorph.mul_left a) = (*) a := rfl
@[to_additive]
lemma homeomorph.mul_left_symm (a : G) : (homeomorph.mul_left a).symm = homeomorph.mul_left a⁻¹ :=
by { ext, refl }
@[to_additive]
lemma is_open_map_mul_left (a : G) : is_open_map (λ x, a * x) :=
(homeomorph.mul_left a).is_open_map
@[to_additive]
lemma is_closed_map_mul_left (a : G) : is_closed_map (λ x, a * x) :=
(homeomorph.mul_left a).is_closed_map
/-- Multiplication from the right in a topological group as a homeomorphism. -/
@[to_additive "Addition from the right in a topological additive group as a homeomorphism."]
protected def homeomorph.mul_right (a : G) :
G ≃ₜ G :=
{ continuous_to_fun := continuous_id.mul continuous_const,
continuous_inv_fun := continuous_id.mul continuous_const,
.. equiv.mul_right a }
@[to_additive]
lemma is_open_map_mul_right (a : G) : is_open_map (λ x, x * a) :=
(homeomorph.mul_right a).is_open_map
@[to_additive]
lemma is_closed_map_mul_right (a : G) : is_closed_map (λ x, x * a) :=
(homeomorph.mul_right a).is_closed_map
@[to_additive]
lemma is_open_map_div_right (a : G) : is_open_map (λ x, x / a) :=
by simpa only [div_eq_mul_inv] using is_open_map_mul_right (a⁻¹)
@[to_additive]
lemma is_closed_map_div_right (a : G) : is_closed_map (λ x, x / a) :=
by simpa only [div_eq_mul_inv] using is_closed_map_mul_right (a⁻¹)
@[to_additive]
lemma discrete_topology_of_open_singleton_one (h : is_open ({1} : set G)) : discrete_topology G :=
begin
rw ← singletons_open_iff_discrete,
intro g,
suffices : {g} = (λ (x : G), g⁻¹ * x) ⁻¹' {1},
{ rw this, exact (continuous_mul_left (g⁻¹)).is_open_preimage _ h, },
simp only [mul_one, set.preimage_mul_left_singleton, eq_self_iff_true,
inv_inv, set.singleton_eq_singleton_iff],
end
end continuous_mul_group
section topological_group
/-!
### Topological groups
A topological group is a group in which the multiplication and inversion operations are
continuous. Topological additive groups are defined in the same way. Equivalently, we can require
that the division operation `λ x y, x * y⁻¹` (resp., subtraction) is continuous.
-/
/-- A topological (additive) group is a group in which the addition and negation operations are
continuous. -/
class topological_add_group (G : Type u) [topological_space G] [add_group G]
extends has_continuous_add G : Prop :=
(continuous_neg : continuous (λa:G, -a))
/-- A topological group is a group in which the multiplication and inversion operations are
continuous. -/
@[to_additive]
class topological_group (G : Type*) [topological_space G] [group G]
extends has_continuous_mul G : Prop :=
(continuous_inv : continuous (has_inv.inv : G → G))
variables [topological_space G] [group G] [topological_group G]
export topological_group (continuous_inv)
export topological_add_group (continuous_neg)
@[to_additive]
lemma continuous_on_inv {s : set G} : continuous_on has_inv.inv s :=
continuous_inv.continuous_on
@[to_additive]
lemma continuous_within_at_inv {s : set G} {x : G} : continuous_within_at has_inv.inv s x :=
continuous_inv.continuous_within_at
@[to_additive]
lemma continuous_at_inv {x : G} : continuous_at has_inv.inv x :=
continuous_inv.continuous_at
@[to_additive]
lemma tendsto_inv (a : G) : tendsto has_inv.inv (𝓝 a) (𝓝 (a⁻¹)) :=
continuous_at_inv
/-- If a function converges to a value in a multiplicative topological group, then its inverse
converges to the inverse of this value. For the version in normed fields assuming additionally
that the limit is nonzero, use `tendsto.inv'`. -/
@[to_additive]
lemma filter.tendsto.inv {f : α → G} {l : filter α} {y : G} (h : tendsto f l (𝓝 y)) :
tendsto (λ x, (f x)⁻¹) l (𝓝 y⁻¹) :=
(continuous_inv.tendsto y).comp h
variables [topological_space α] {f : α → G} {s : set α} {x : α}
@[continuity, to_additive]
lemma continuous.inv (hf : continuous f) : continuous (λx, (f x)⁻¹) :=
continuous_inv.comp hf
attribute [continuity] continuous.neg -- TODO
@[to_additive]
lemma continuous_on.inv (hf : continuous_on f s) : continuous_on (λx, (f x)⁻¹) s :=
continuous_inv.comp_continuous_on hf
@[to_additive]
lemma continuous_within_at.inv (hf : continuous_within_at f s x) :
continuous_within_at (λ x, (f x)⁻¹) s x :=
hf.inv
@[instance, to_additive]
instance [topological_space H] [group H] [topological_group H] :
topological_group (G × H) :=
{ continuous_inv := continuous_inv.prod_map continuous_inv }
@[to_additive]
instance pi.topological_group {C : β → Type*} [∀ b, topological_space (C b)]
[∀ b, group (C b)] [∀ b, topological_group (C b)] : topological_group (Π b, C b) :=
{ continuous_inv := continuous_pi (λ i, (continuous_apply i).inv) }
variable (G)
/-- Inversion in a topological group as a homeomorphism. -/
@[to_additive "Negation in a topological group as a homeomorphism."]
protected def homeomorph.inv : G ≃ₜ G :=
{ continuous_to_fun := continuous_inv,
continuous_inv_fun := continuous_inv,
.. equiv.inv G }
@[to_additive]
lemma nhds_one_symm : comap has_inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) :=
((homeomorph.inv G).comap_nhds_eq _).trans (congr_arg nhds one_inv)
/-- The map `(x, y) ↦ (x, xy)` as a homeomorphism. This is a shear mapping. -/
@[to_additive "The map `(x, y) ↦ (x, x + y)` as a homeomorphism.
This is a shear mapping."]
protected def homeomorph.shear_mul_right : G × G ≃ₜ G × G :=
{ continuous_to_fun := continuous_fst.prod_mk continuous_mul,
continuous_inv_fun := continuous_fst.prod_mk $ continuous_fst.inv.mul continuous_snd,
.. equiv.prod_shear (equiv.refl _) equiv.mul_left }
@[simp, to_additive]
lemma homeomorph.shear_mul_right_coe :
⇑(homeomorph.shear_mul_right G) = λ z : G × G, (z.1, z.1 * z.2) :=
rfl
@[simp, to_additive]
lemma homeomorph.shear_mul_right_symm_coe :
⇑(homeomorph.shear_mul_right G).symm = λ z : G × G, (z.1, z.1⁻¹ * z.2) :=
rfl
variable {G}
@[to_additive]
lemma inv_closure (s : set G) : (closure s)⁻¹ = closure s⁻¹ :=
(homeomorph.inv G).preimage_closure s
/-- The (topological-space) closure of a subgroup of a space `M` with `has_continuous_mul` is
itself a subgroup. -/
@[to_additive "The (topological-space) closure of an additive subgroup of a space `M` with
`has_continuous_add` is itself an additive subgroup."]
def subgroup.topological_closure (s : subgroup G) : subgroup G :=
{ carrier := closure (s : set G),
inv_mem' := λ g m, by simpa [←mem_inv, inv_closure] using m,
..s.to_submonoid.topological_closure }
@[simp, to_additive] lemma subgroup.topological_closure_coe {s : subgroup G} :
(s.topological_closure : set G) = closure s :=
rfl
@[to_additive]
instance subgroup.topological_closure_topological_group (s : subgroup G) :
topological_group (s.topological_closure) :=
{ continuous_inv :=
begin
apply continuous_induced_rng,
change continuous (λ p : s.topological_closure, (p : G)⁻¹),
continuity,
end
..s.to_submonoid.topological_closure_has_continuous_mul}
@[to_additive] lemma subgroup.subgroup_topological_closure (s : subgroup G) :
s ≤ s.topological_closure :=
subset_closure
@[to_additive] lemma subgroup.is_closed_topological_closure (s : subgroup G) :
is_closed (s.topological_closure : set G) :=
by convert is_closed_closure
@[to_additive] lemma subgroup.topological_closure_minimal
(s : subgroup G) {t : subgroup G} (h : s ≤ t) (ht : is_closed (t : set G)) :
s.topological_closure ≤ t :=
closure_minimal h ht
@[to_additive] lemma dense_range.topological_closure_map_subgroup [group H] [topological_space H]
[topological_group H] {f : G →* H} (hf : continuous f) (hf' : dense_range f) {s : subgroup G}
(hs : s.topological_closure = ⊤) :
(s.map f).topological_closure = ⊤ :=
begin
rw set_like.ext'_iff at hs ⊢,
simp only [subgroup.topological_closure_coe, subgroup.coe_top, ← dense_iff_closure_eq] at hs ⊢,
exact hf'.dense_image hf hs
end
@[to_additive exists_nhds_half_neg]
lemma exists_nhds_split_inv {s : set G} (hs : s ∈ 𝓝 (1 : G)) :
∃ V ∈ 𝓝 (1 : G), ∀ (v ∈ V) (w ∈ V), v / w ∈ s :=
have ((λp : G × G, p.1 * p.2⁻¹) ⁻¹' s) ∈ 𝓝 ((1, 1) : G × G),
from continuous_at_fst.mul continuous_at_snd.inv (by simpa),
by simpa only [div_eq_mul_inv, nhds_prod_eq, mem_prod_self_iff, prod_subset_iff, mem_preimage]
using this
@[to_additive]
lemma nhds_translation_mul_inv (x : G) : comap (λ y : G, y * x⁻¹) (𝓝 1) = 𝓝 x :=
((homeomorph.mul_right x⁻¹).comap_nhds_eq 1).trans $ show 𝓝 (1 * x⁻¹⁻¹) = 𝓝 x, by simp
@[simp, to_additive] lemma map_mul_left_nhds (x y : G) : map ((*) x) (𝓝 y) = 𝓝 (x * y) :=
(homeomorph.mul_left x).map_nhds_eq y
@[to_additive] lemma map_mul_left_nhds_one (x : G) : map ((*) x) (𝓝 1) = 𝓝 x := by simp
@[to_additive]
lemma topological_group.ext {G : Type*} [group G] {t t' : topological_space G}
(tg : @topological_group G t _) (tg' : @topological_group G t' _)
(h : @nhds G t 1 = @nhds G t' 1) : t = t' :=
eq_of_nhds_eq_nhds $ λ x, by
rw [← @nhds_translation_mul_inv G t _ _ x , ← @nhds_translation_mul_inv G t' _ _ x , ← h]
@[to_additive]
lemma topological_group.of_nhds_aux {G : Type*} [group G] [topological_space G]
(hinv : tendsto (λ (x : G), x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ (x₀ : G), 𝓝 x₀ = map (λ (x : G), x₀ * x) (𝓝 1))
(hconj : ∀ (x₀ : G), map (λ (x : G), x₀ * x * x₀⁻¹) (𝓝 1) ≤ 𝓝 1) : continuous (λ x : G, x⁻¹) :=
begin
rw continuous_iff_continuous_at,
rintros x₀,
have key : (λ x, (x₀*x)⁻¹) = (λ x, x₀⁻¹*x) ∘ (λ x, x₀*x*x₀⁻¹) ∘ (λ x, x⁻¹),
by {ext ; simp[mul_assoc] },
calc map (λ x, x⁻¹) (𝓝 x₀)
= map (λ x, x⁻¹) (map (λ x, x₀*x) $ 𝓝 1) : by rw hleft
... = map (λ x, (x₀*x)⁻¹) (𝓝 1) : by rw filter.map_map
... = map (((λ x, x₀⁻¹*x) ∘ (λ x, x₀*x*x₀⁻¹)) ∘ (λ x, x⁻¹)) (𝓝 1) : by rw key
... = map ((λ x, x₀⁻¹*x) ∘ (λ x, x₀*x*x₀⁻¹)) _ : by rw ← filter.map_map
... ≤ map ((λ x, x₀⁻¹ * x) ∘ λ x, x₀ * x * x₀⁻¹) (𝓝 1) : map_mono hinv
... = map (λ x, x₀⁻¹ * x) (map (λ x, x₀ * x * x₀⁻¹) (𝓝 1)) : filter.map_map
... ≤ map (λ x, x₀⁻¹ * x) (𝓝 1) : map_mono (hconj x₀)
... = 𝓝 x₀⁻¹ : (hleft _).symm
end
@[to_additive]
lemma topological_group.of_nhds_one' {G : Type*} [group G] [topological_space G]
(hmul : tendsto (uncurry ((*) : G → G → G)) ((𝓝 1) ×ᶠ 𝓝 1) (𝓝 1))
(hinv : tendsto (λ x : G, x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1))
(hright : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x*x₀) (𝓝 1)) : topological_group G :=
begin
refine { continuous_mul := (has_continuous_mul.of_nhds_one hmul hleft hright).continuous_mul,
continuous_inv := topological_group.of_nhds_aux hinv hleft _ },
intros x₀,
suffices : map (λ (x : G), x₀ * x * x₀⁻¹) (𝓝 1) = 𝓝 1, by simp [this, le_refl],
rw [show (λ x, x₀ * x * x₀⁻¹) = (λ x, x₀ * x) ∘ λ x, x*x₀⁻¹, by {ext, simp [mul_assoc] },
← filter.map_map, ← hright, hleft x₀⁻¹, filter.map_map],
convert map_id,
ext,
simp
end
@[to_additive]
lemma topological_group.of_nhds_one {G : Type u} [group G] [topological_space G]
(hmul : tendsto (uncurry ((*) : G → G → G)) ((𝓝 1) ×ᶠ 𝓝 1) (𝓝 1))
(hinv : tendsto (λ x : G, x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1))
(hconj : ∀ x₀ : G, tendsto (λ x, x₀*x*x₀⁻¹) (𝓝 1) (𝓝 1)) : topological_group G :=
{ continuous_mul := begin
rw continuous_iff_continuous_at,
rintros ⟨x₀, y₀⟩,
have key : (λ (p : G × G), x₀ * p.1 * (y₀ * p.2)) =
((λ x, x₀*y₀*x) ∘ (uncurry (*)) ∘ (prod.map (λ x, y₀⁻¹*x*y₀) id)),
by { ext, simp [uncurry, prod.map, mul_assoc] },
specialize hconj y₀⁻¹, rw inv_inv at hconj,
calc map (λ (p : G × G), p.1 * p.2) (𝓝 (x₀, y₀))
= map (λ (p : G × G), p.1 * p.2) ((𝓝 x₀) ×ᶠ 𝓝 y₀)
: by rw nhds_prod_eq
... = map (λ (p : G × G), x₀ * p.1 * (y₀ * p.2)) ((𝓝 1) ×ᶠ (𝓝 1))
: by rw [hleft x₀, hleft y₀, prod_map_map_eq, filter.map_map]
... = map (((λ x, x₀*y₀*x) ∘ (uncurry (*))) ∘ (prod.map (λ x, y₀⁻¹*x*y₀) id))((𝓝 1) ×ᶠ (𝓝 1))
: by rw key
... = map ((λ x, x₀*y₀*x) ∘ (uncurry (*))) ((map (λ x, y₀⁻¹*x*y₀) $ 𝓝 1) ×ᶠ (𝓝 1))
: by rw [← filter.map_map, ← prod_map_map_eq', map_id]
... ≤ map ((λ x, x₀*y₀*x) ∘ (uncurry (*))) ((𝓝 1) ×ᶠ (𝓝 1))
: map_mono (filter.prod_mono hconj $ le_refl _)
... = map (λ x, x₀*y₀*x) (map (uncurry (*)) ((𝓝 1) ×ᶠ (𝓝 1))) : by rw filter.map_map
... ≤ map (λ x, x₀*y₀*x) (𝓝 1) : map_mono hmul
... = 𝓝 (x₀*y₀) : (hleft _).symm
end,
continuous_inv := topological_group.of_nhds_aux hinv hleft hconj}
@[to_additive]
lemma topological_group.of_comm_of_nhds_one {G : Type*} [comm_group G] [topological_space G]
(hmul : tendsto (uncurry ((*) : G → G → G)) ((𝓝 1) ×ᶠ 𝓝 1) (𝓝 1))
(hinv : tendsto (λ x : G, x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (λ x, x₀*x) (𝓝 1)) : topological_group G :=
topological_group.of_nhds_one hmul hinv hleft (by simpa using tendsto_id)
end topological_group
section quotient_topological_group
variables [topological_space G] [group G] [topological_group G] (N : subgroup G) (n : N.normal)
@[to_additive]
instance {G : Type*} [group G] [topological_space G] (N : subgroup G) :
topological_space (quotient_group.quotient N) :=
quotient.topological_space
open quotient_group
@[to_additive]
lemma quotient_group.is_open_map_coe : is_open_map (coe : G → quotient N) :=
begin
intros s s_op,
change is_open ((coe : G → quotient N) ⁻¹' (coe '' s)),
rw quotient_group.preimage_image_coe N s,
exact is_open_Union (λ n, (continuous_mul_right _).is_open_preimage s s_op)
end
@[to_additive]
instance topological_group_quotient [N.normal] : topological_group (quotient N) :=
{ continuous_mul := begin
have cont : continuous ((coe : G → quotient N) ∘ (λ (p : G × G), p.fst * p.snd)) :=
continuous_quot_mk.comp continuous_mul,
have quot : quotient_map (λ p : G × G, ((p.1:quotient N), (p.2:quotient N))),
{ apply is_open_map.to_quotient_map,
{ exact (quotient_group.is_open_map_coe N).prod (quotient_group.is_open_map_coe N) },
{ exact continuous_quot_mk.prod_map continuous_quot_mk },
{ exact (surjective_quot_mk _).prod_map (surjective_quot_mk _) } },
exact (quotient_map.continuous_iff quot).2 cont,
end,
continuous_inv := begin
have : continuous ((coe : G → quotient N) ∘ (λ (a : G), a⁻¹)) :=
continuous_quot_mk.comp continuous_inv,
convert continuous_quotient_lift _ this,
end }
attribute [instance] topological_add_group_quotient
end quotient_topological_group
/-- A typeclass saying that `λ p : G × G, p.1 - p.2` is a continuous function. This property
automatically holds for topological additive groups but it also holds, e.g., for `ℝ≥0`. -/
class has_continuous_sub (G : Type*) [topological_space G] [has_sub G] : Prop :=
(continuous_sub : continuous (λ p : G × G, p.1 - p.2))
@[priority 100] -- see Note [lower instance priority]
instance topological_add_group.to_has_continuous_sub [topological_space G] [add_group G]
[topological_add_group G] :
has_continuous_sub G :=
⟨by { simp only [sub_eq_add_neg], exact continuous_fst.add continuous_snd.neg }⟩
export has_continuous_sub (continuous_sub)
section has_continuous_sub
variables [topological_space G] [has_sub G] [has_continuous_sub G]
lemma filter.tendsto.sub {f g : α → G} {l : filter α} {a b : G} (hf : tendsto f l (𝓝 a))
(hg : tendsto g l (𝓝 b)) :
tendsto (λx, f x - g x) l (𝓝 (a - b)) :=
(continuous_sub.tendsto (a, b)).comp (hf.prod_mk_nhds hg)
variables [topological_space α] {f g : α → G} {s : set α} {x : α}
@[continuity] lemma continuous.sub (hf : continuous f) (hg : continuous g) :
continuous (λ x, f x - g x) :=
continuous_sub.comp (hf.prod_mk hg : _)
lemma continuous_within_at.sub (hf : continuous_within_at f s x) (hg : continuous_within_at g s x) :
continuous_within_at (λ x, f x - g x) s x :=
hf.sub hg
lemma continuous_on.sub (hf : continuous_on f s) (hg : continuous_on g s) :
continuous_on (λx, f x - g x) s :=
λ x hx, (hf x hx).sub (hg x hx)
end has_continuous_sub
lemma nhds_translation [topological_space G] [add_group G] [topological_add_group G] (x : G) :
comap (λy:G, y - x) (𝓝 0) = 𝓝 x :=
by simpa only [sub_eq_add_neg] using nhds_translation_add_neg x
/-- additive group with a neighbourhood around 0.
Only used to construct a topology and uniform space.
This is currently only available for commutative groups, but it can be extended to
non-commutative groups too.
-/
class add_group_with_zero_nhd (G : Type u) extends add_comm_group G :=
(Z [] : filter G)
(zero_Z : pure 0 ≤ Z)
(sub_Z : tendsto (λp:G×G, p.1 - p.2) (Z ×ᶠ Z) Z)
namespace add_group_with_zero_nhd
variables (G) [add_group_with_zero_nhd G]
local notation `Z` := add_group_with_zero_nhd.Z
@[priority 100] -- see Note [lower instance priority]
instance : topological_space G :=
topological_space.mk_of_nhds $ λa, map (λx, x + a) (Z G)
variables {G}
lemma neg_Z : tendsto (λa:G, - a) (Z G) (Z G) :=
have tendsto (λa, (0:G)) (Z G) (Z G),
by refine le_trans (assume h, _) zero_Z; simp [univ_mem_sets'] {contextual := tt},
have tendsto (λa:G, 0 - a) (Z G) (Z G), from
sub_Z.comp (tendsto.prod_mk this tendsto_id),
by simpa
lemma add_Z : tendsto (λp:G×G, p.1 + p.2) (Z G ×ᶠ Z G) (Z G) :=
suffices tendsto (λp:G×G, p.1 - -p.2) (Z G ×ᶠ Z G) (Z G),
by simpa [sub_eq_add_neg],
sub_Z.comp (tendsto.prod_mk tendsto_fst (neg_Z.comp tendsto_snd))
lemma exists_Z_half {s : set G} (hs : s ∈ Z G) : ∃ V ∈ Z G, ∀ (v ∈ V) (w ∈ V), v + w ∈ s :=
begin
have : ((λa:G×G, a.1 + a.2) ⁻¹' s) ∈ Z G ×ᶠ Z G := add_Z (by simpa using hs),
rcases mem_prod_self_iff.1 this with ⟨V, H, H'⟩,
exact ⟨V, H, prod_subset_iff.1 H'⟩
end
lemma nhds_eq (a : G) : 𝓝 a = map (λx, x + a) (Z G) :=
topological_space.nhds_mk_of_nhds _ _
(assume a, calc pure a = map (λx, x + a) (pure 0) : by simp
... ≤ _ : map_mono zero_Z)
(assume b s hs,
let ⟨t, ht, eqt⟩ := exists_Z_half hs in
have t0 : (0:G) ∈ t, by simpa using zero_Z ht,
begin
refine ⟨(λx:G, x + b) '' t, image_mem_map ht, _, _⟩,
{ refine set.image_subset_iff.2 (assume b hbt, _),
simpa using eqt 0 t0 b hbt },
{ rintros _ ⟨c, hb, rfl⟩,
refine (Z G).sets_of_superset ht (assume x hxt, _),
simpa [add_assoc] using eqt _ hxt _ hb }
end)
lemma nhds_zero_eq_Z : 𝓝 0 = Z G := by simp [nhds_eq]; exact filter.map_id
@[priority 100] -- see Note [lower instance priority]
instance : has_continuous_add G :=
⟨ continuous_iff_continuous_at.2 $ assume ⟨a, b⟩,
begin
rw [continuous_at, nhds_prod_eq, nhds_eq, nhds_eq, nhds_eq, filter.prod_map_map_eq,
tendsto_map'_iff],
suffices : tendsto ((λx:G, (a + b) + x) ∘ (λp:G×G,p.1 + p.2)) (Z G ×ᶠ Z G)
(map (λx:G, (a + b) + x) (Z G)),
{ simpa [(∘), add_comm, add_left_comm] },
exact tendsto_map.comp add_Z
end ⟩
@[priority 100] -- see Note [lower instance priority]
instance : topological_add_group G :=
⟨continuous_iff_continuous_at.2 $ assume a,
begin
rw [continuous_at, nhds_eq, nhds_eq, tendsto_map'_iff],
suffices : tendsto ((λx:G, x - a) ∘ (λx:G, -x)) (Z G) (map (λx:G, x - a) (Z G)),
{ simpa [(∘), add_comm, sub_eq_add_neg] using this },
exact tendsto_map.comp neg_Z
end⟩
end add_group_with_zero_nhd
section filter_mul
section
variables [topological_space G] [group G] [topological_group G]
@[to_additive]
lemma is_open.mul_left {s t : set G} : is_open t → is_open (s * t) := λ ht,
begin
have : ∀a, is_open ((λ (x : G), a * x) '' t) :=
assume a, is_open_map_mul_left a t ht,
rw ← Union_mul_left_image,
exact is_open_Union (λa, is_open_Union $ λha, this _),
end
@[to_additive]
lemma is_open.mul_right {s t : set G} : is_open s → is_open (s * t) := λ hs,
begin
have : ∀a, is_open ((λ (x : G), x * a) '' s),
assume a, apply is_open_map_mul_right, exact hs,
rw ← Union_mul_right_image,
exact is_open_Union (λa, is_open_Union $ λha, this _),
end
variables (G)
lemma topological_group.t1_space (h : @is_closed G _ {1}) : t1_space G :=
⟨assume x, by { convert is_closed_map_mul_right x _ h, simp }⟩
lemma topological_group.regular_space [t1_space G] : regular_space G :=
⟨assume s a hs ha,
let f := λ p : G × G, p.1 * (p.2)⁻¹ in
have hf : continuous f := continuous_fst.mul continuous_snd.inv,
-- a ∈ -s implies f (a, 1) ∈ -s, and so (a, 1) ∈ f⁻¹' (-s);
-- and so can find t₁ t₂ open such that a ∈ t₁ × t₂ ⊆ f⁻¹' (-s)
let ⟨t₁, t₂, ht₁, ht₂, a_mem_t₁, one_mem_t₂, t_subset⟩ :=
is_open_prod_iff.1 ((is_open_compl_iff.2 hs).preimage hf) a (1:G) (by simpa [f]) in
begin
use [s * t₂, ht₂.mul_left, λ x hx, ⟨x, 1, hx, one_mem_t₂, mul_one _⟩],
rw [nhds_within, inf_principal_eq_bot, mem_nhds_iff],
refine ⟨t₁, _, ht₁, a_mem_t₁⟩,
rintros x hx ⟨y, z, hy, hz, yz⟩,
have : x * z⁻¹ ∈ sᶜ := (prod_subset_iff.1 t_subset) x hx z hz,
have : x * z⁻¹ ∈ s, rw ← yz, simpa,
contradiction
end⟩
local attribute [instance] topological_group.regular_space
lemma topological_group.t2_space [t1_space G] : t2_space G := regular_space.t2_space G
end
section
/-! Some results about an open set containing the product of two sets in a topological group. -/
variables [topological_space G] [group G] [topological_group G]
/-- Given a compact set `K` inside an open set `U`, there is a open neighborhood `V` of `1`
such that `KV ⊆ U`. -/
@[to_additive "Given a compact set `K` inside an open set `U`, there is a open neighborhood `V` of
`0` such that `K + V ⊆ U`."]
lemma compact_open_separated_mul {K U : set G} (hK : is_compact K) (hU : is_open U) (hKU : K ⊆ U) :
∃ V : set G, is_open V ∧ (1 : G) ∈ V ∧ K * V ⊆ U :=
begin
let W : G → set G := λ x, (λ y, x * y) ⁻¹' U,
have h1W : ∀ x, is_open (W x) := λ x, hU.preimage (continuous_mul_left x),
have h2W : ∀ x ∈ K, (1 : G) ∈ W x := λ x hx, by simp only [mem_preimage, mul_one, hKU hx],
choose V hV using λ x : K, exists_open_nhds_one_mul_subset ((h1W x).mem_nhds (h2W x.1 x.2)),
let X : K → set G := λ x, (λ y, (x : G)⁻¹ * y) ⁻¹' (V x),
obtain ⟨t, ht⟩ : ∃ t : finset ↥K, K ⊆ ⋃ i ∈ t, X i,
{ refine hK.elim_finite_subcover X (λ x, (hV x).1.preimage (continuous_mul_left x⁻¹)) _,
intros x hx, rw [mem_Union], use ⟨x, hx⟩, rw [mem_preimage], convert (hV _).2.1,
simp only [mul_left_inv, subtype.coe_mk] },
refine ⟨⋂ x ∈ t, V x, is_open_bInter (finite_mem_finset _) (λ x hx, (hV x).1), _, _⟩,
{ simp only [mem_Inter], intros x hx, exact (hV x).2.1 },
rintro _ ⟨x, y, hx, hy, rfl⟩, simp only [mem_Inter] at hy,
have := ht hx, simp only [mem_Union, mem_preimage] at this, rcases this with ⟨z, h1z, h2z⟩,
have : (z : G)⁻¹ * x * y ∈ W z := (hV z).2.2 (mul_mem_mul h2z (hy z h1z)),
rw [mem_preimage] at this, convert this using 1, simp only [mul_assoc, mul_inv_cancel_left]
end
/-- A compact set is covered by finitely many left multiplicative translates of a set
with non-empty interior. -/
@[to_additive "A compact set is covered by finitely many left additive translates of a set
with non-empty interior."]
lemma compact_covered_by_mul_left_translates {K V : set G} (hK : is_compact K)
(hV : (interior V).nonempty) : ∃ t : finset G, K ⊆ ⋃ g ∈ t, (λ h, g * h) ⁻¹' V :=
begin
obtain ⟨t, ht⟩ : ∃ t : finset G, K ⊆ ⋃ x ∈ t, interior (((*) x) ⁻¹' V),
{ refine hK.elim_finite_subcover (λ x, interior $ ((*) x) ⁻¹' V) (λ x, is_open_interior) _,
cases hV with g₀ hg₀,
refine λ g hg, mem_Union.2 ⟨g₀ * g⁻¹, _⟩,
refine preimage_interior_subset_interior_preimage (continuous_const.mul continuous_id) _,
rwa [mem_preimage, inv_mul_cancel_right] },
exact ⟨t, subset.trans ht $ bUnion_subset_bUnion_right $ λ g hg, interior_subset⟩
end
/-- Every locally compact separable topological group is σ-compact.
Note: this is not true if we drop the topological group hypothesis. -/
@[priority 100, to_additive separable_locally_compact_add_group.sigma_compact_space]
instance separable_locally_compact_group.sigma_compact_space
[separable_space G] [locally_compact_space G] : sigma_compact_space G :=
begin
obtain ⟨L, hLc, hL1⟩ := exists_compact_mem_nhds (1 : G),
refine ⟨⟨λ n, (λ x, x * dense_seq G n) ⁻¹' L, _, _⟩⟩,
{ intro n, exact (homeomorph.mul_right _).compact_preimage.mpr hLc },
{ refine Union_eq_univ_iff.2 (λ x, _),
obtain ⟨_, ⟨n, rfl⟩, hn⟩ : (range (dense_seq G) ∩ (λ y, x * y) ⁻¹' L).nonempty,
{ rw [← (homeomorph.mul_left x).apply_symm_apply 1] at hL1,
exact (dense_range_dense_seq G).inter_nhds_nonempty
((homeomorph.mul_left x).continuous.continuous_at $ hL1) },
exact ⟨n, hn⟩ }
end
end
section
variables [topological_space G] [comm_group G] [topological_group G]
@[to_additive]
lemma nhds_mul (x y : G) : 𝓝 (x * y) = 𝓝 x * 𝓝 y :=
filter_eq $ set.ext $ assume s,
begin
rw [← nhds_translation_mul_inv x, ← nhds_translation_mul_inv y, ← nhds_translation_mul_inv (x*y)],
split,
{ rintros ⟨t, ht, ts⟩,
rcases exists_nhds_one_split ht with ⟨V, V1, h⟩,
refine ⟨(λa, a * x⁻¹) ⁻¹' V, (λa, a * y⁻¹) ⁻¹' V,
⟨V, V1, subset.refl _⟩, ⟨V, V1, subset.refl _⟩, _⟩,
rintros a ⟨v, w, v_mem, w_mem, rfl⟩,
apply ts,
simpa [mul_comm, mul_assoc, mul_left_comm] using h (v * x⁻¹) v_mem (w * y⁻¹) w_mem },
{ rintros ⟨a, c, ⟨b, hb, ba⟩, ⟨d, hd, dc⟩, ac⟩,
refine ⟨b ∩ d, inter_mem_sets hb hd, assume v, _⟩,
simp only [preimage_subset_iff, mul_inv_rev, mem_preimage] at *,
rintros ⟨vb, vd⟩,
refine ac ⟨v * y⁻¹, y, _, _, _⟩,
{ rw ← mul_assoc _ _ _ at vb, exact ba _ vb },
{ apply dc y, rw mul_right_inv, exact mem_of_mem_nhds hd },
{ simp only [inv_mul_cancel_right] } }
end
@[to_additive]
lemma nhds_is_mul_hom : is_mul_hom (λx:G, 𝓝 x) := ⟨λ_ _, nhds_mul _ _⟩
end
end filter_mul
instance additive.topological_add_group {G} [h : topological_space G]
[group G] [topological_group G] : @topological_add_group (additive G) h _ :=
{ continuous_neg := @continuous_inv G _ _ _ }
instance multiplicative.topological_group {G} [h : topological_space G]
[add_group G] [topological_add_group G] : @topological_group (multiplicative G) h _ :=
{ continuous_inv := @continuous_neg G _ _ _ }
namespace units
variables [monoid α] [topological_space α] [has_continuous_mul α]
instance : topological_group (units α) :=
{ continuous_inv := continuous_induced_rng ((continuous_unop.comp (continuous_snd.comp
(@continuous_embed_product α _ _))).prod_mk (continuous_op.comp continuous_coe)) }
end units
|
6c946207a800c64f74891705fbfe1e48bbafe879 | fb4c13c9b8a3d8624041a268d2d3348d53c1aecb | /library/standard/list2.lean | ed0036bf4ccbe9b6bcd376d98ac88cbe8895b8de | [
"Apache-2.0"
] | permissive | avigad/libraries | 90f149eedb7f621b7af5bd469f0b4869f1bc35c5 | 1a21725f97d83ae7531eaeb38d175bad8c0f1d89 | refs/heads/master | 1,610,958,129,193 | 1,405,610,493,000 | 1,405,610,493,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 5,010 | lean | ----------------------------------------------------------------------------------------------------
--- Copyright (c) 2014 Parikshit Khanna. All rights reserved.
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Parikshit Khanna, Jeremy Avigad
----------------------------------------------------------------------------------------------------
-- Theory list2
-- ============
--
-- More basic properties of lists.
import macros
import tactic
import nat
import list
using nat
unary_nat
namespace list
axiom sorry {P : Bool} : P
theorem insert_cons_ge (n m : ℕ) (H : m ≤ n) (l : list ℕ) : insert n (cons m l) = cons m (insert n l)
:= trans (insert_cons n m l) (imp_if_eq H _ _)
theorem insert_cons_lt (n m : ℕ) (H : n < m) (l : list ℕ) : insert n (cons m l) = cons n (cons m l)
:= trans (insert_cons n m l) (not_imp_if_eq (lt_imp_not_ge H) _ _)
-- temporary, until Lean 2.0
theorem ge_iff_le (n m : ℕ) : (n ≥ m) = (m ≤ n) := refl _
theorem gt_iff_lt (n m : ℕ) : (n > m) = (m < n) := refl _
add_rewrite insert_cons_ge insert_cons_lt ge_iff_le gt_iff_lt
-- (if n ≥ m then cons m (insert n l) else cons n (cons m l))
theorem insert_comm_aux (x y : ℕ) (H : x < y) (l : list ℕ) : insert x (insert y l) = insert y (insert x l)
:=
list_induction_on l
(show insert x (insert y nil) = insert y (insert x nil), from (by simp) (lt_imp_le H))
(take z l,
assume IH : insert x (insert y l) = insert y (insert x l),
show insert x (insert y (cons z l)) = insert y (insert x (cons z l)), from
or_elim (le_or_gt z y)
(assume H1 : z ≤ y,
or_elim (le_or_gt z x)
(assume H2 : z ≤ x, by simp)
(assume H2 : x < z,
have H3 : x ≤ y, from lt_imp_le H,
by simp))
(assume H1 : y < z,
have H2 : x < z, from lt_trans H H1,
have H3 : x ≤ y, from lt_imp_le H,
by simp))
theorem insert_comm (x y : ℕ) (l : list ℕ) : insert x (insert y l) = insert y (insert x l)
:=
or_elim (trichotomy x y)
(assume H, insert_comm_aux x y H l)
(assume H, or_elim H
(assume H1, subst (refl _) H1)
(assume H1, symm (insert_comm_aux y x H1 l)))
-- definition map {T : Type} {S : Type} (P : T → S) (x0 : S) : list T → list S
-- := list_rec (cons x0 nil) (fun x l b, cons (P x) b)
-- definition foldr {T : Type} {S : Type} (f : T → T → T) (x0 : T) : list T → T
-- := list_rec x0 (fun x l b,f x b)
-- -- l = x [a b c d]
-- -- b = f(c f( b (f a d)))) f( a (f b (f c d))) f (f (f a b) c) d
-- -- r = f (c f(b f(a (f x d)))) f a f b f c d f ( f (f ( f x a) b) c) d
-- definition last {T : Type} (x0 : T) (l : list T) : T
-- := head x0 (reverse l)
-- theorem find_insert (x y : ℕ) (l : list ℕ) : find x (insert y (asort l)) = (if y ≥ x then find x (asort l) else succ (find x (asort l)))
-- :=
-- by_cases _
-- (assume H : y ≥ x,
-- list_induction_on l
-- ((by simp) (asort_nil) (insert_nil y) (find_cons x y nil)
-- (find_nil x))
-- (take z l,
-- assume IH : find x (insert y (asort l)) = (if y ≥ x then find x l else succ (find x l)),
-- by_cases _
-- (assume H1 : y ≥ z,
-- by_cases _
-- (assume H2 : z ≥ x,
-- (by simp) (asort_cons z l) (insert_cons _ _ l) (find_cons _ _ l))
-- (assume H2 : ¬ z ≥ x,
-- (by simp) (insert_cons _ _ l) (find_cons _ _ l)))
-- (assume H1 : ¬ y ≥ z,
-- by_cases _
-- (assume H2 : z ≥ x,
-- (by simp) (insert_cons _ _ l) (find_cons _ _ l))
-- (assume H2 : ¬ z ≥ x,
-- (by simp) (insert_cons _ _ l) (find_cons _ _ l))))
-- (assume H : ¬ y ≥ x,
-- list_induction_on l
-- ((by simp) (asort_nil) (insert_nil y) (find_cons x y nil)
-- (find_nil x))
-- (take z l,
-- assume IH : find x (insert y (asort l)) = (if y ≥ x then find x l else succ (find x l)),
-- by_cases _
-- (assume H1 : y ≥ z,
-- by_cases _
-- (assume H2 : z ≥ x,
-- (by simp) (asort_cons z l) (insert_cons _ _ l) (find_cons _ _ l))
-- (assume H2 : ¬ z ≥ x,
-- (by simp) (insert_cons _ _ l) (find_cons _ _ l)))
-- (assume H1 : ¬ y ≥ z,
-- by_cases _
-- (assume H2 : z ≥ x,
-- (by simp) (insert_cons _ _ l) (find_cons _ _ l))
-- (assume H2 : ¬ z ≥ x,
-- (by simp) (insert_cons _ _ l) (find_cons _ _ l))))))
theorem find_insert (x y : ℕ) (l : list ℕ) :
mem x l → (find x (insert y (asort l)) = (if y ≥ x then find x (asort l) else succ (find x (asort l))))
:= list_induction_on l
(assume H1 : mem x nil,
show find x (insert y (asort nil)) = (if y ≥ x then find x (asort nil) else succ (find x (asort nil))),
from false_elim _ ((iff_elim_left (mem_nil x)) H1)
) (take z l,
sorry)
|
c4bd42852944a255758d937769e521d246f3c096 | 6dc0c8ce7a76229dd81e73ed4474f15f88a9e294 | /src/Init/Data/Char/Basic.lean | 58a4de6774c08623460ef47fe6dd54f44345a2af | [
"Apache-2.0"
] | permissive | williamdemeo/lean4 | 72161c58fe65c3ad955d6a3050bb7d37c04c0d54 | 6d00fcf1d6d873e195f9220c668ef9c58e9c4a35 | refs/heads/master | 1,678,305,356,877 | 1,614,708,995,000 | 1,614,708,995,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,050 | lean | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
prelude
import Init.Data.UInt
@[inline, reducible] def isValidChar (n : UInt32) : Prop :=
n < 0xd800 ∨ (0xdfff < n ∧ n < 0x110000)
namespace Char
protected def Less (a b : Char) : Prop := a.val < b.val
protected def LessEq (a b : Char) : Prop := a.val ≤ b.val
instance : HasLess Char := ⟨Char.Less⟩
instance : HasLessEq Char := ⟨Char.LessEq⟩
protected def lt (a b : Char) : Bool := a.val < b.val
instance (a b : Char) : Decidable (a < b) :=
UInt32.decLt _ _
instance (a b : Char) : Decidable (a ≤ b) :=
UInt32.decLe _ _
abbrev isValidCharNat (n : Nat) : Prop :=
n < 0xd800 ∨ (0xdfff < n ∧ n < 0x110000)
theorem isValidUInt32 (n : Nat) (h : isValidCharNat n) : n < UInt32.size := by
match h with
| Or.inl h =>
apply Nat.ltTrans h
exact decide!
| Or.inr ⟨h₁, h₂⟩ =>
apply Nat.ltTrans h₂
exact decide!
theorem isValidCharOfValidNat (n : Nat) (h : isValidCharNat n) : isValidChar (UInt32.ofNat' n (isValidUInt32 n h)) :=
match h with
| Or.inl h => Or.inl h
| Or.inr ⟨h₁, h₂⟩ => Or.inr ⟨h₁, h₂⟩
theorem isValidChar0 : isValidChar 0 :=
Or.inl decide!
@[inline] def toNat (c : Char) : Nat :=
c.val.toNat
instance : Inhabited Char where
default := 'A'
def isWhitespace (c : Char) : Bool :=
c = ' ' || c = '\t' || c = '\r' || c = '\n'
def isUpper (c : Char) : Bool :=
c.val ≥ 65 && c.val ≤ 90
def isLower (c : Char) : Bool :=
c.val ≥ 97 && c.val ≤ 122
def isAlpha (c : Char) : Bool :=
c.isUpper || c.isLower
def isDigit (c : Char) : Bool :=
c.val ≥ 48 && c.val ≤ 57
def isAlphanum (c : Char) : Bool :=
c.isAlpha || c.isDigit
def toLower (c : Char) : Char :=
let n := toNat c;
if n >= 65 ∧ n <= 90 then ofNat (n + 32) else c
def toUpper (c : Char) : Char :=
let n := toNat c;
if n >= 97 ∧ n <= 122 then ofNat (n - 32) else c
end Char
|
6cebf2d9af66a8aa7c6229d2ce6f0fd1da7185f2 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/group_theory/double_coset.lean | 67b06f2a43f6286225b31bd4179ad561c34681ab | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 7,515 | lean | /-
Copyright (c) 2021 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import data.setoid.basic
import group_theory.subgroup.basic
import group_theory.coset
import group_theory.subgroup.pointwise
import data.set.basic
import tactic.group
/-!
# Double cosets
This file defines double cosets for two subgroups `H K` of a group `G` and the quotient of `G` by
the double coset relation, i.e. `H \ G / K`. We also prove that `G` can be writen as a disjoint
union of the double cosets and that if one of `H` or `K` is the trivial group (i.e. `⊥` ) then
this is the usual left or right quotient of a group by a subgroup.
## Main definitions
* `rel`: The double coset relation defined by two subgroups `H K` of `G`.
* `double_coset.quotient`: The quotient of `G` by the double coset relation, i.e, ``H \ G / K`.
-/
variables {G : Type*} [group G] {α : Type*} [has_mul α] (J: subgroup G) (g : G)
namespace doset
open_locale pointwise
/--The double_coset as an element of `set α` corresponding to `s a t` -/
def _root_.doset (a : α) (s t : set α) : set α := s * {a} * t
lemma mem_doset {s t : set α} {a b : α} : b ∈ doset a s t ↔ ∃ (x ∈ s) (y ∈ t), b = x * a * y :=
⟨λ ⟨_, y, ⟨x, _, hx, rfl, rfl⟩, hy, h⟩, ⟨x, hx, y, hy, h.symm⟩,
λ ⟨x, hx, y, hy, h⟩, ⟨x * a, y, ⟨x, a, hx, rfl, rfl⟩, hy, h.symm⟩⟩
lemma mem_doset_self (H K : subgroup G) (a : G) : a ∈ doset a H K :=
mem_doset.mpr ⟨1, H.one_mem, 1, K.one_mem, (one_mul a).symm.trans (mul_one (1 * a)).symm⟩
lemma doset_eq_of_mem {H K : subgroup G} {a b : G} (hb : b ∈ doset a H K) :
doset b H K = doset a H K :=
begin
obtain ⟨_, k, ⟨h, a, hh, (rfl : _ = _), rfl⟩, hk, rfl⟩ := hb,
rw [doset, doset, ←set.singleton_mul_singleton, ←set.singleton_mul_singleton, mul_assoc,
mul_assoc, subgroup.singleton_mul_subgroup hk, ←mul_assoc, ←mul_assoc,
subgroup.subgroup_mul_singleton hh],
end
lemma mem_doset_of_not_disjoint {H K : subgroup G} {a b : G}
(h : ¬ disjoint (doset a H K) (doset b H K)) : b ∈ doset a H K :=
begin
rw set.not_disjoint_iff at h,
simp only [mem_doset] at *,
obtain ⟨x, ⟨l, hl, r, hr, hrx⟩, y, hy, ⟨r', hr', rfl⟩⟩ := h,
refine ⟨y⁻¹ * l, H.mul_mem (H.inv_mem hy) (hl), r * r'⁻¹, K.mul_mem hr (K.inv_mem hr'), _⟩,
rwa [mul_assoc, mul_assoc, eq_inv_mul_iff_mul_eq, ←mul_assoc, ←mul_assoc, eq_mul_inv_iff_mul_eq],
end
lemma eq_of_not_disjoint {H K : subgroup G} {a b : G} (h: ¬ disjoint (doset a H K) (doset b H K)) :
doset a H K = doset b H K :=
begin
rw disjoint.comm at h,
have ha : a ∈ doset b H K := mem_doset_of_not_disjoint h,
apply doset_eq_of_mem ha,
end
/-- The setoid defined by the double_coset relation -/
def setoid (H K : set G) : setoid G :=
setoid.ker (λ x, doset x H K)
/-- Quotient of `G` by the double coset relation, i.e. `H \ G / K` -/
def quotient (H K : set G) : Type* :=
quotient (setoid H K)
lemma rel_iff {H K : subgroup G} {x y : G} :
(setoid ↑H ↑K).rel x y ↔ ∃ (a ∈ H) (b ∈ K), y = a * x * b :=
iff.trans ⟨λ hxy, (congr_arg _ hxy).mpr (mem_doset_self H K y),
λ hxy, (doset_eq_of_mem hxy).symm⟩ mem_doset
lemma bot_rel_eq_left_rel (H : subgroup G) :
(setoid ↑(⊥ : subgroup G) ↑H).rel = (quotient_group.left_rel H).rel :=
begin
ext a b,
rw rel_iff,
split,
{ rintros ⟨a, (rfl : a = 1), b, hb, rfl⟩,
change a⁻¹ * (1 * a * b) ∈ H,
rwa [one_mul, inv_mul_cancel_left] },
{ rintro (h : a⁻¹ * b ∈ H),
exact ⟨1, rfl, a⁻¹ * b, h, by rw [one_mul, mul_inv_cancel_left]⟩ },
end
lemma rel_bot_eq_right_group_rel (H : subgroup G) :
(setoid ↑H ↑(⊥ : subgroup G)).rel = (quotient_group.right_rel H).rel :=
begin
ext a b,
rw rel_iff,
split,
{ rintros ⟨b, hb, a, (rfl : a = 1), rfl⟩,
change b * a * 1 * a⁻¹ ∈ H,
rwa [mul_one, mul_inv_cancel_right] },
{ rintro (h : b * a⁻¹ ∈ H),
exact ⟨b * a⁻¹, h, 1, rfl, by rw [mul_one, inv_mul_cancel_right]⟩ },
end
/--Create a doset out of an element of `H \ G / K`-/
def quot_to_doset (H K : subgroup G) (q : quotient ↑H ↑K) : set G := (doset q.out' H K)
/--Map from `G` to `H \ G / K`-/
abbreviation mk (H K : subgroup G) (a : G) : quotient ↑H ↑K :=
quotient.mk' a
instance (H K : subgroup G) : inhabited (quotient ↑H ↑K) := ⟨mk H K (1 : G)⟩
lemma eq (H K : subgroup G) (a b : G) : mk H K a = mk H K b ↔ ∃ (h ∈ H) (k ∈ K), b = h * a * k :=
by { rw quotient.eq', apply rel_iff, }
lemma out_eq' (H K : subgroup G) (q : quotient ↑H ↑K) : mk H K q.out' = q :=
quotient.out_eq' q
lemma mk_out'_eq_mul (H K : subgroup G) (g : G) :
∃ (h k : G), (h ∈ H) ∧ (k ∈ K) ∧ (mk H K g : quotient ↑H ↑K).out' = h * g * k :=
begin
have := eq H K (mk H K g : quotient ↑H ↑K).out' g,
rw out_eq' at this,
obtain ⟨h, h_h, k, hk, T⟩ := this.1 rfl,
refine ⟨h⁻¹, k⁻¹, (H.inv_mem h_h), K.inv_mem hk, eq_mul_inv_of_mul_eq (eq_inv_mul_of_mul_eq _)⟩,
rw [← mul_assoc, ← T]
end
lemma mk_eq_of_doset_eq {H K : subgroup G} {a b : G} (h : doset a H K = doset b H K) :
mk H K a = mk H K b :=
begin
rw eq,
exact mem_doset.mp (h.symm ▸ mem_doset_self H K b)
end
lemma disjoint_out' {H K : subgroup G} {a b : quotient H.1 K} :
a ≠ b → disjoint (doset a.out' H K) (doset b.out' H K) :=
begin
contrapose!,
intro h,
simpa [out_eq'] using mk_eq_of_doset_eq (eq_of_not_disjoint h),
end
lemma union_quot_to_doset (H K : subgroup G) : (⋃ q, quot_to_doset H K q) = set.univ :=
begin
ext x,
simp only [set.mem_Union, quot_to_doset, mem_doset, set_like.mem_coe, exists_prop,
set.mem_univ, iff_true],
use mk H K x,
obtain ⟨h, k, h3, h4, h5⟩ := mk_out'_eq_mul H K x,
refine ⟨h⁻¹, H.inv_mem h3, k⁻¹, K.inv_mem h4, _⟩,
simp only [h5, subgroup.coe_mk, ←mul_assoc, one_mul, mul_left_inv, mul_inv_cancel_right],
end
lemma doset_union_right_coset (H K : subgroup G) (a : G) :
(⋃ (k : K), right_coset ↑H (a * k)) = doset a H K :=
begin
ext x,
simp only [mem_right_coset_iff, exists_prop, mul_inv_rev, set.mem_Union, mem_doset,
subgroup.mem_carrier, set_like.mem_coe],
split,
{rintro ⟨y, h_h⟩,
refine ⟨x * (y⁻¹ * a⁻¹), h_h, y, y.2, _⟩,
simp only [← mul_assoc, subgroup.coe_mk, inv_mul_cancel_right]},
{rintros ⟨x, hx, y, hy, hxy⟩,
refine ⟨⟨y,hy⟩,_⟩,
simp only [hxy, ←mul_assoc, hx, mul_inv_cancel_right, subgroup.coe_mk]},
end
lemma doset_union_left_coset (H K : subgroup G) (a : G) :
(⋃ (h : H), left_coset (h * a : G) K) = doset a H K :=
begin
ext x,
simp only [mem_left_coset_iff, mul_inv_rev, set.mem_Union, mem_doset],
split,
{ rintro ⟨y, h_h⟩,
refine ⟨y, y.2, a⁻¹ * y⁻¹ * x, h_h, _⟩,
simp only [←mul_assoc, one_mul, mul_right_inv, mul_inv_cancel_right]},
{ rintros ⟨x, hx, y, hy, hxy⟩,
refine ⟨⟨x, hx⟩, _⟩,
simp only [hxy, ←mul_assoc, hy, one_mul, mul_left_inv, subgroup.coe_mk, inv_mul_cancel_right]},
end
lemma left_bot_eq_left_quot (H : subgroup G) :
quotient (⊥ : subgroup G).1 H = (G ⧸ H) :=
begin
unfold quotient,
congr,
ext,
simp_rw ← bot_rel_eq_left_rel H,
refl,
end
lemma right_bot_eq_right_quot (H : subgroup G) :
quotient H.1 (⊥ : subgroup G) = _root_.quotient (quotient_group.right_rel H) :=
begin
unfold quotient,
congr,
ext,
simp_rw ← rel_bot_eq_right_group_rel H,
refl,
end
end doset
|
7d3da54dcc7d543e2d97b35556ffee3236c11432 | 2a70b774d16dbdf5a533432ee0ebab6838df0948 | /_target/deps/mathlib/src/order/basic.lean | c9e964e1e482c63423ad012c6766bc1035d0e120 | [
"Apache-2.0"
] | permissive | hjvromen/lewis | 40b035973df7c77ebf927afab7878c76d05ff758 | 105b675f73630f028ad5d890897a51b3c1146fb0 | refs/heads/master | 1,677,944,636,343 | 1,676,555,301,000 | 1,676,555,301,000 | 327,553,599 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 23,362 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import data.subtype
import data.prod
open function
/-!
# Basic definitions about `≤` and `<`
## Definitions
### Predicates on functions
- `monotone f`: a function between two types equipped with `≤` is monotone
if `a ≤ b` implies `f a ≤ f b`.
- `strict_mono f` : a function between two types equipped with `<` is strictly monotone
if `a < b` implies `f a < f b`.
- `order_dual α` : a type tag reversing the meaning of all inequalities.
### Transfering orders
- `order.preimage`, `preorder.lift`: transfer a (pre)order on `β` to an order on `α`
using a function `f : α → β`.
- `partial_order.lift`, `linear_order.lift`: transfer a partial (resp., linear) order on `β` to a
partial (resp., linear) order on `α` using an injective function `f`.
### Extra classes
- `no_top_order`, `no_bot_order`: an order without a maximal/minimal element.
- `densely_ordered`: an order with no gaps, i.e. for any two elements `a<b` there exists
`c`, `a<c<b`.
## Main theorems
- `monotone_of_monotone_nat`: if `f : ℕ → α` and `f n ≤ f (n + 1)` for all `n`, then
`f` is monotone;
- `strict_mono.nat`: if `f : ℕ → α` and `f n < f (n + 1)` for all `n`, then f is strictly monotone.
## TODO
- expand module docs
- automatic construction of dual definitions / theorems
## See also
- `algebra.order` for basic lemmas about orders, and projection notation for orders
## Tags
preorder, order, partial order, linear order, monotone, strictly monotone
-/
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w} {r : α → α → Prop}
theorem preorder.ext {α} {A B : preorder α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
begin
casesI A, casesI B, congr,
{ funext x y, exact propext (H x y) },
{ funext x y,
dsimp [(≤)] at A_lt_iff_le_not_le B_lt_iff_le_not_le H,
simp [A_lt_iff_le_not_le, B_lt_iff_le_not_le, H] },
end
theorem partial_order.ext {α} {A B : partial_order α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
by { haveI this := preorder.ext H,
casesI A, casesI B, injection this, congr' }
theorem linear_order.ext {α} {A B : linear_order α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
by { haveI this := partial_order.ext H,
casesI A, casesI B, injection this, congr' }
/-- Given a relation `R` on `β` and a function `f : α → β`,
the preimage relation on `α` is defined by `x ≤ y ↔ f x ≤ f y`.
It is the unique relation on `α` making `f` a `rel_embedding`
(assuming `f` is injective). -/
@[simp] def order.preimage {α β} (f : α → β) (s : β → β → Prop) (x y : α) := s (f x) (f y)
infix ` ⁻¹'o `:80 := order.preimage
/-- The preimage of a decidable order is decidable. -/
instance order.preimage.decidable {α β} (f : α → β) (s : β → β → Prop) [H : decidable_rel s] :
decidable_rel (f ⁻¹'o s) :=
λ x y, H _ _
section monotone
variables [preorder α] [preorder β] [preorder γ]
/-- A function between preorders is monotone if
`a ≤ b` implies `f a ≤ f b`. -/
def monotone (f : α → β) := ∀⦃a b⦄, a ≤ b → f a ≤ f b
theorem monotone_id : @monotone α α _ _ id := assume x y h, h
theorem monotone_const {b : β} : monotone (λ(a:α), b) := assume x y h, le_refl b
protected theorem monotone.comp {g : β → γ} {f : α → β} (m_g : monotone g) (m_f : monotone f) :
monotone (g ∘ f) :=
assume a b h, m_g (m_f h)
protected theorem monotone.iterate {f : α → α} (hf : monotone f) (n : ℕ) : monotone (f^[n]) :=
nat.rec_on n monotone_id (λ n ihn, ihn.comp hf)
lemma monotone_of_monotone_nat {f : ℕ → α} (hf : ∀n, f n ≤ f (n + 1)) :
monotone f | n m h :=
begin
induction h,
{ refl },
{ transitivity, assumption, exact hf _ }
end
lemma monotone.reflect_lt {α β} [linear_order α] [preorder β] {f : α → β} (hf : monotone f)
{x x' : α} (h : f x < f x') : x < x' :=
by { rw [← not_le], intro h', apply not_le_of_lt h, exact hf h' }
/-- If `f` is a monotone function from `ℕ` to a preorder such that `y` lies between `f x` and
`f (x + 1)`, then `y` doesn't lie in the range of `f`. -/
lemma monotone.ne_of_lt_of_lt_nat {α} [preorder α] {f : ℕ → α} (hf : monotone f)
(x x' : ℕ) {y : α} (h1 : f x < y) (h2 : y < f (x + 1)) : f x' ≠ y :=
by { rintro rfl, apply (hf.reflect_lt h1).not_le, exact nat.le_of_lt_succ (hf.reflect_lt h2) }
/-- If `f` is a monotone function from `ℤ` to a preorder such that `y` lies between `f x` and
`f (x + 1)`, then `y` doesn't lie in the range of `f`. -/
lemma monotone.ne_of_lt_of_lt_int {α} [preorder α] {f : ℤ → α} (hf : monotone f)
(x x' : ℤ) {y : α} (h1 : f x < y) (h2 : y < f (x + 1)) : f x' ≠ y :=
by { rintro rfl, apply (hf.reflect_lt h1).not_le, exact int.le_of_lt_add_one (hf.reflect_lt h2) }
end monotone
/-- A function `f` is strictly monotone if `a < b` implies `f a < f b`. -/
def strict_mono [has_lt α] [has_lt β] (f : α → β) : Prop :=
∀ ⦃a b⦄, a < b → f a < f b
lemma strict_mono_id [has_lt α] : strict_mono (id : α → α) := λ a b, id
/-- A function `f` is strictly monotone increasing on `t` if `x < y` for `x,y ∈ t` implies
`f x < f y`. -/
def strict_mono_incr_on [has_lt α] [has_lt β] (f : α → β) (t : set α) : Prop :=
∀ ⦃x⦄ (hx : x ∈ t) ⦃y⦄ (hy : y ∈ t), x < y → f x < f y
/-- A function `f` is strictly monotone decreasing on `t` if `x < y` for `x,y ∈ t` implies
`f y < f x`. -/
def strict_mono_decr_on [has_lt α] [has_lt β] (f : α → β) (t : set α) : Prop :=
∀ ⦃x⦄ (hx : x ∈ t) ⦃y⦄ (hy : y ∈ t), x < y → f y < f x
/-- Type tag for a set with dual order: `≤` means `≥` and `<` means `>`. -/
def order_dual (α : Type*) := α
namespace order_dual
instance (α : Type*) [h : nonempty α] : nonempty (order_dual α) := h
instance (α : Type*) [h : subsingleton α] : subsingleton (order_dual α) := h
instance (α : Type*) [has_le α] : has_le (order_dual α) := ⟨λx y:α, y ≤ x⟩
instance (α : Type*) [has_lt α] : has_lt (order_dual α) := ⟨λx y:α, y < x⟩
-- `dual_le` and `dual_lt` should not be simp lemmas:
-- they cause a loop since `α` and `order_dual α` are definitionally equal
lemma dual_le [has_le α] {a b : α} :
@has_le.le (order_dual α) _ a b ↔ @has_le.le α _ b a := iff.rfl
lemma dual_lt [has_lt α] {a b : α} :
@has_lt.lt (order_dual α) _ a b ↔ @has_lt.lt α _ b a := iff.rfl
lemma dual_compares [has_lt α] {a b : α} {o : ordering} :
@ordering.compares (order_dual α) _ o a b ↔ @ordering.compares α _ o b a :=
by { cases o, exacts [iff.rfl, eq_comm, iff.rfl] }
instance (α : Type*) [preorder α] : preorder (order_dual α) :=
{ le_refl := le_refl,
le_trans := assume a b c hab hbc, hbc.trans hab,
lt_iff_le_not_le := λ _ _, lt_iff_le_not_le,
.. order_dual.has_le α,
.. order_dual.has_lt α }
instance (α : Type*) [partial_order α] : partial_order (order_dual α) :=
{ le_antisymm := assume a b hab hba, @le_antisymm α _ a b hba hab, .. order_dual.preorder α }
instance (α : Type*) [linear_order α] : linear_order (order_dual α) :=
{ le_total := assume a b:α, le_total b a,
decidable_le := show decidable_rel (λa b:α, b ≤ a), by apply_instance,
decidable_lt := show decidable_rel (λa b:α, b < a), by apply_instance,
.. order_dual.partial_order α }
instance : Π [inhabited α], inhabited (order_dual α) := id
theorem preorder.dual_dual (α : Type*) [H : preorder α] :
order_dual.preorder (order_dual α) = H :=
preorder.ext $ λ _ _, iff.rfl
theorem partial_order.dual_dual (α : Type*) [H : partial_order α] :
order_dual.partial_order (order_dual α) = H :=
partial_order.ext $ λ _ _, iff.rfl
theorem linear_order.dual_dual (α : Type*) [H : linear_order α] :
order_dual.linear_order (order_dual α) = H :=
linear_order.ext $ λ _ _, iff.rfl
theorem cmp_le_flip {α} [has_le α] [@decidable_rel α (≤)] (x y : α) :
@cmp_le (order_dual α) _ _ x y = cmp_le y x := rfl
end order_dual
namespace strict_mono_incr_on
section dual
variables [preorder α] [preorder β] {f : α → β} {s : set α}
protected lemma dual (H : strict_mono_incr_on f s) :
@strict_mono_incr_on (order_dual α) (order_dual β) _ _ f s :=
λ x hx y hy, H hy hx
protected lemma dual_right (H : strict_mono_incr_on f s) :
@strict_mono_decr_on α (order_dual β) _ _ f s :=
H
end dual
variables [linear_order α] [preorder β] {f : α → β} {s : set α} {x y : α}
lemma le_iff_le (H : strict_mono_incr_on f s) (hx : x ∈ s) (hy : y ∈ s) :
f x ≤ f y ↔ x ≤ y :=
⟨λ h, le_of_not_gt $ λ h', not_le_of_lt (H hy hx h') h,
λ h, (lt_or_eq_of_le h).elim (λ h', le_of_lt (H hx hy h')) (λ h', h' ▸ le_refl _)⟩
lemma lt_iff_lt (H : strict_mono_incr_on f s) (hx : x ∈ s) (hy : y ∈ s) :
f x < f y ↔ x < y :=
by simp only [H.le_iff_le, hx, hy, lt_iff_le_not_le]
protected theorem compares (H : strict_mono_incr_on f s) (hx : x ∈ s) (hy : y ∈ s) :
∀ {o}, ordering.compares o (f x) (f y) ↔ ordering.compares o x y
| ordering.lt := H.lt_iff_lt hx hy
| ordering.eq := ⟨λ h, le_antisymm ((H.le_iff_le hx hy).1 h.le) ((H.le_iff_le hy hx).1 h.symm.le),
congr_arg _⟩
| ordering.gt := H.lt_iff_lt hy hx
end strict_mono_incr_on
namespace strict_mono_decr_on
section dual
variables [preorder α] [preorder β] {f : α → β} {s : set α}
protected lemma dual (H : strict_mono_decr_on f s) :
@strict_mono_decr_on (order_dual α) (order_dual β) _ _ f s :=
λ x hx y hy, H hy hx
protected lemma dual_right (H : strict_mono_decr_on f s) :
@strict_mono_incr_on α (order_dual β) _ _ f s :=
H
end dual
variables [linear_order α] [preorder β] {f : α → β} {s : set α} {x y : α}
lemma le_iff_le (H : strict_mono_decr_on f s) (hx : x ∈ s) (hy : y ∈ s) :
f x ≤ f y ↔ y ≤ x :=
H.dual_right.le_iff_le hy hx
lemma lt_iff_lt (H : strict_mono_decr_on f s) (hx : x ∈ s) (hy : y ∈ s) :
f x < f y ↔ y < x :=
H.dual_right.lt_iff_lt hy hx
protected theorem compares (H : strict_mono_decr_on f s) (hx : x ∈ s) (hy : y ∈ s) {o : ordering} :
ordering.compares o (f x) (f y) ↔ ordering.compares o y x :=
order_dual.dual_compares.trans $ H.dual_right.compares hy hx
end strict_mono_decr_on
namespace strict_mono
open ordering function
protected lemma strict_mono_incr_on [has_lt α] [has_lt β] {f : α → β} (hf : strict_mono f)
(s : set α) :
strict_mono_incr_on f s :=
λ x hx y hy hxy, hf hxy
lemma comp [has_lt α] [has_lt β] [has_lt γ] {g : β → γ} {f : α → β}
(hg : strict_mono g) (hf : strict_mono f) :
strict_mono (g ∘ f) :=
λ a b h, hg (hf h)
protected theorem iterate [has_lt α] {f : α → α} (hf : strict_mono f) (n : ℕ) :
strict_mono (f^[n]) :=
nat.rec_on n strict_mono_id (λ n ihn, ihn.comp hf)
lemma id_le {φ : ℕ → ℕ} (h : strict_mono φ) : ∀ n, n ≤ φ n :=
λ n, nat.rec_on n (nat.zero_le _)
(λ n hn, nat.succ_le_of_lt (lt_of_le_of_lt hn $ h $ nat.lt_succ_self n))
protected lemma ite' [preorder α] [has_lt β] {f g : α → β} (hf : strict_mono f) (hg : strict_mono g)
{p : α → Prop} [decidable_pred p] (hp : ∀ ⦃x y⦄, x < y → p y → p x)
(hfg : ∀ ⦃x y⦄, p x → ¬p y → x < y → f x < g y) :
strict_mono (λ x, if p x then f x else g x) :=
begin
intros x y h,
by_cases hy : p y,
{ have hx : p x := hp h hy,
simpa [hx, hy] using hf h },
{ by_cases hx : p x,
{ simpa [hx, hy] using hfg hx hy h },
{ simpa [hx, hy] using hg h} }
end
protected lemma ite [preorder α] [preorder β] {f g : α → β} (hf : strict_mono f)
(hg : strict_mono g) {p : α → Prop} [decidable_pred p] (hp : ∀ ⦃x y⦄, x < y → p y → p x)
(hfg : ∀ x, f x ≤ g x) :
strict_mono (λ x, if p x then f x else g x) :=
hf.ite' hg hp $ λ x y hx hy h, (hf h).trans_le (hfg y)
section
variables [linear_order α] [preorder β] {f : α → β}
lemma lt_iff_lt (H : strict_mono f) {a b} : f a < f b ↔ a < b :=
(H.strict_mono_incr_on set.univ).lt_iff_lt trivial trivial
protected theorem compares (H : strict_mono f) {a b} {o} :
compares o (f a) (f b) ↔ compares o a b :=
(H.strict_mono_incr_on set.univ).compares trivial trivial
lemma injective (H : strict_mono f) : injective f :=
λ x y h, show compares eq x y, from H.compares.1 h
lemma le_iff_le (H : strict_mono f) {a b} : f a ≤ f b ↔ a ≤ b :=
(H.strict_mono_incr_on set.univ).le_iff_le trivial trivial
lemma top_preimage_top (H : strict_mono f) {a} (h_top : ∀ p, p ≤ f a) (x : α) : x ≤ a :=
H.le_iff_le.mp (h_top (f x))
lemma bot_preimage_bot (H : strict_mono f) {a} (h_bot : ∀ p, f a ≤ p) (x : α) : a ≤ x :=
H.le_iff_le.mp (h_bot (f x))
end
protected lemma nat {β} [preorder β] {f : ℕ → β} (h : ∀n, f n < f (n+1)) : strict_mono f :=
by { intros n m hnm, induction hnm with m' hnm' ih, apply h, exact ih.trans (h _) }
-- `preorder α` isn't strong enough: if the preorder on α is an equivalence relation,
-- then `strict_mono f` is vacuously true.
lemma monotone [partial_order α] [preorder β] {f : α → β} (H : strict_mono f) : monotone f :=
λ a b h, (lt_or_eq_of_le h).rec (le_of_lt ∘ (@H _ _)) (by rintro rfl; refl)
end strict_mono
section
open function
lemma injective_of_lt_imp_ne [linear_order α] {f : α → β} (h : ∀ x y, x < y → f x ≠ f y) :
injective f :=
begin
intros x y k,
contrapose k,
rw [←ne.def, ne_iff_lt_or_gt] at k,
cases k,
{ apply h _ _ k },
{ rw eq_comm,
apply h _ _ k }
end
lemma strict_mono_of_monotone_of_injective [partial_order α] [partial_order β] {f : α → β}
(h₁ : monotone f) (h₂ : injective f) : strict_mono f :=
λ a b h,
begin
rw lt_iff_le_and_ne at ⊢ h,
exact ⟨h₁ h.1, λ e, h.2 (h₂ e)⟩
end
lemma monotone.strict_mono_iff_injective [linear_order α] [partial_order β] {f : α → β}
(h : monotone f) : strict_mono f ↔ injective f :=
⟨λ h, h.injective, strict_mono_of_monotone_of_injective h⟩
lemma strict_mono_of_le_iff_le [preorder α] [preorder β] {f : α → β}
(h : ∀ x y, x ≤ y ↔ f x ≤ f y) : strict_mono f :=
λ a b, by simp [lt_iff_le_not_le, h] {contextual := tt}
end
/-! ### Order instances on the function space -/
instance pi.preorder {ι : Type u} {α : ι → Type v} [∀i, preorder (α i)] : preorder (Πi, α i) :=
{ le := λx y, ∀i, x i ≤ y i,
le_refl := assume a i, le_refl (a i),
le_trans := assume a b c h₁ h₂ i, le_trans (h₁ i) (h₂ i) }
lemma pi.le_def {ι : Type u} {α : ι → Type v} [∀i, preorder (α i)] {x y : Π i, α i} :
x ≤ y ↔ ∀ i, x i ≤ y i :=
iff.rfl
lemma le_update_iff {ι : Type u} {α : ι → Type v} [∀i, preorder (α i)] [decidable_eq ι]
{x y : Π i, α i} {i : ι} {a : α i} :
x ≤ function.update y i a ↔ x i ≤ a ∧ ∀ j ≠ i, x j ≤ y j :=
function.forall_update_iff _ (λ j z, x j ≤ z)
lemma update_le_iff {ι : Type u} {α : ι → Type v} [∀i, preorder (α i)] [decidable_eq ι]
{x y : Π i, α i} {i : ι} {a : α i} :
function.update x i a ≤ y ↔ a ≤ y i ∧ ∀ j ≠ i, x j ≤ y j :=
function.forall_update_iff _ (λ j z, z ≤ y j)
instance pi.partial_order {ι : Type u} {α : ι → Type v} [∀i, partial_order (α i)] :
partial_order (Πi, α i) :=
{ le_antisymm := λf g h1 h2, funext (λb, le_antisymm (h1 b) (h2 b)),
..pi.preorder }
theorem comp_le_comp_left_of_monotone [preorder α] [preorder β]
{f : β → α} {g h : γ → β} (m_f : monotone f) (le_gh : g ≤ h) :
has_le.le.{max w u} (f ∘ g) (f ∘ h) :=
assume x, m_f (le_gh x)
section monotone
variables [preorder α] [preorder γ]
protected theorem monotone.order_dual {f : α → γ} (hf : monotone f) :
@monotone (order_dual α) (order_dual γ) _ _ f :=
λ x y hxy, hf hxy
theorem monotone_lam {f : α → β → γ} (m : ∀b, monotone (λa, f a b)) : monotone f :=
assume a a' h b, m b h
theorem monotone_app (f : β → α → γ) (b : β) (m : monotone (λa b, f b a)) : monotone (f b) :=
assume a a' h, m h b
end monotone
theorem strict_mono.order_dual [has_lt α] [has_lt β] {f : α → β} (hf : strict_mono f) :
@strict_mono (order_dual α) (order_dual β) _ _ f :=
λ x y hxy, hf hxy
/-- Transfer a `preorder` on `β` to a `preorder` on `α` using a function `f : α → β`. -/
def preorder.lift {α β} [preorder β] (f : α → β) : preorder α :=
{ le := λx y, f x ≤ f y,
le_refl := λ a, le_refl _,
le_trans := λ a b c, le_trans,
lt := λx y, f x < f y,
lt_iff_le_not_le := λ a b, lt_iff_le_not_le }
/-- Transfer a `partial_order` on `β` to a `partial_order` on `α` using an injective
function `f : α → β`. -/
def partial_order.lift {α β} [partial_order β] (f : α → β) (inj : injective f) :
partial_order α :=
{ le_antisymm := λ a b h₁ h₂, inj (le_antisymm h₁ h₂), .. preorder.lift f }
/-- Transfer a `linear_order` on `β` to a `linear_order` on `α` using an injective
function `f : α → β`. -/
def linear_order.lift {α β} [linear_order β] (f : α → β) (inj : injective f) :
linear_order α :=
{ le_total := λx y, le_total (f x) (f y),
decidable_le := λ x y, (infer_instance : decidable (f x ≤ f y)),
decidable_lt := λ x y, (infer_instance : decidable (f x < f y)),
decidable_eq := λ x y, decidable_of_iff _ inj.eq_iff,
.. partial_order.lift f inj }
instance subtype.preorder {α} [preorder α] (p : α → Prop) : preorder (subtype p) :=
preorder.lift subtype.val
@[simp] lemma subtype.mk_le_mk {α} [preorder α] {p : α → Prop} {x y : α} {hx : p x} {hy : p y} :
(⟨x, hx⟩ : subtype p) ≤ ⟨y, hy⟩ ↔ x ≤ y :=
iff.rfl
@[simp] lemma subtype.mk_lt_mk {α} [preorder α] {p : α → Prop} {x y : α} {hx : p x} {hy : p y} :
(⟨x, hx⟩ : subtype p) < ⟨y, hy⟩ ↔ x < y :=
iff.rfl
@[simp, norm_cast] lemma subtype.coe_le_coe {α} [preorder α] {p : α → Prop} {x y : subtype p} :
(x : α) ≤ y ↔ x ≤ y :=
iff.rfl
@[simp, norm_cast] lemma subtype.coe_lt_coe {α} [preorder α] {p : α → Prop} {x y : subtype p} :
(x : α) < y ↔ x < y :=
iff.rfl
instance subtype.partial_order {α} [partial_order α] (p : α → Prop) :
partial_order (subtype p) :=
partial_order.lift subtype.val subtype.val_injective
instance subtype.linear_order {α} [linear_order α] (p : α → Prop) : linear_order (subtype p) :=
linear_order.lift subtype.val subtype.val_injective
lemma subtype.mono_coe [preorder α] (t : set α) : monotone (coe : (subtype t) → α) :=
λ x y, id
lemma subtype.strict_mono_coe [preorder α] (t : set α) : strict_mono (coe : (subtype t) → α) :=
λ x y, id
instance prod.has_le (α : Type u) (β : Type v) [has_le α] [has_le β] : has_le (α × β) :=
⟨λp q, p.1 ≤ q.1 ∧ p.2 ≤ q.2⟩
instance prod.preorder (α : Type u) (β : Type v) [preorder α] [preorder β] : preorder (α × β) :=
{ le_refl := assume ⟨a, b⟩, ⟨le_refl a, le_refl b⟩,
le_trans := assume ⟨a, b⟩ ⟨c, d⟩ ⟨e, f⟩ ⟨hac, hbd⟩ ⟨hce, hdf⟩,
⟨le_trans hac hce, le_trans hbd hdf⟩,
.. prod.has_le α β }
/-- The pointwise partial order on a product.
(The lexicographic ordering is defined in order/lexicographic.lean, and the instances are
available via the type synonym `lex α β = α × β`.) -/
instance prod.partial_order (α : Type u) (β : Type v) [partial_order α] [partial_order β] :
partial_order (α × β) :=
{ le_antisymm := assume ⟨a, b⟩ ⟨c, d⟩ ⟨hac, hbd⟩ ⟨hca, hdb⟩,
prod.ext (le_antisymm hac hca) (le_antisymm hbd hdb),
.. prod.preorder α β }
/-!
### Additional order classes
-/
/-- order without a top element; somtimes called cofinal -/
class no_top_order (α : Type u) [preorder α] : Prop :=
(no_top : ∀a:α, ∃a', a < a')
lemma no_top [preorder α] [no_top_order α] : ∀a:α, ∃a', a < a' :=
no_top_order.no_top
instance nonempty_gt {α : Type u} [preorder α] [no_top_order α] (a : α) :
nonempty {x // a < x} :=
nonempty_subtype.2 (no_top a)
/-- order without a bottom element; somtimes called coinitial or dense -/
class no_bot_order (α : Type u) [preorder α] : Prop :=
(no_bot : ∀a:α, ∃a', a' < a)
lemma no_bot [preorder α] [no_bot_order α] : ∀a:α, ∃a', a' < a :=
no_bot_order.no_bot
instance order_dual.no_top_order (α : Type u) [preorder α] [no_bot_order α] :
no_top_order (order_dual α) :=
⟨λ a, @no_bot α _ _ a⟩
instance order_dual.no_bot_order (α : Type u) [preorder α] [no_top_order α] :
no_bot_order (order_dual α) :=
⟨λ a, @no_top α _ _ a⟩
instance nonempty_lt {α : Type u} [preorder α] [no_bot_order α] (a : α) :
nonempty {x // x < a} :=
nonempty_subtype.2 (no_bot a)
/-- An order is dense if there is an element between any pair of distinct elements. -/
class densely_ordered (α : Type u) [preorder α] : Prop :=
(dense : ∀a₁ a₂:α, a₁ < a₂ → ∃a, a₁ < a ∧ a < a₂)
lemma exists_between [preorder α] [densely_ordered α] : ∀{a₁ a₂:α}, a₁ < a₂ → ∃a, a₁ < a ∧ a < a₂ :=
densely_ordered.dense
instance order_dual.densely_ordered (α : Type u) [preorder α] [densely_ordered α] :
densely_ordered (order_dual α) :=
⟨λ a₁ a₂ ha, (@exists_between α _ _ _ _ ha).imp $ λ a, and.symm⟩
lemma le_of_forall_le_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
(h : ∀a₃>a₂, a₁ ≤ a₃) :
a₁ ≤ a₂ :=
le_of_not_gt $ assume ha,
let ⟨a, ha₁, ha₂⟩ := exists_between ha in
lt_irrefl a $ lt_of_lt_of_le ‹a < a₁› (h _ ‹a₂ < a›)
lemma eq_of_le_of_forall_le_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
(h₁ : a₂ ≤ a₁) (h₂ : ∀a₃>a₂, a₁ ≤ a₃) : a₁ = a₂ :=
le_antisymm (le_of_forall_le_of_dense h₂) h₁
lemma le_of_forall_ge_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
(h : ∀a₃<a₁, a₃ ≤ a₂) :
a₁ ≤ a₂ :=
le_of_not_gt $ assume ha,
let ⟨a, ha₁, ha₂⟩ := exists_between ha in
lt_irrefl a $ lt_of_le_of_lt (h _ ‹a < a₁›) ‹a₂ < a›
lemma eq_of_le_of_forall_ge_of_dense [linear_order α] [densely_ordered α] {a₁ a₂ : α}
(h₁ : a₂ ≤ a₁) (h₂ : ∀a₃<a₁, a₃ ≤ a₂) : a₁ = a₂ :=
le_antisymm (le_of_forall_ge_of_dense h₂) h₁
lemma dense_or_discrete [linear_order α] (a₁ a₂ : α) :
(∃a, a₁ < a ∧ a < a₂) ∨ ((∀a>a₁, a₂ ≤ a) ∧ (∀a<a₂, a ≤ a₁)) :=
or_iff_not_imp_left.2 $ assume h,
⟨assume a ha₁, le_of_not_gt $ assume ha₂, h ⟨a, ha₁, ha₂⟩,
assume a ha₂, le_of_not_gt $ assume ha₁, h ⟨a, ha₁, ha₂⟩⟩
variables {s : β → β → Prop} {t : γ → γ → Prop}
/-- Type synonym to create an instance of `linear_order` from a
`partial_order` and `[is_total α (≤)]` -/
def as_linear_order (α : Type u) := α
instance {α} [inhabited α] : inhabited (as_linear_order α) :=
⟨ (default α : α) ⟩
noncomputable instance as_linear_order.linear_order {α} [partial_order α] [is_total α (≤)] :
linear_order (as_linear_order α) :=
{ le_total := @total_of α (≤) _,
decidable_le := classical.dec_rel _,
.. (_ : partial_order α) }
|
d8328a4c6e2afa8335a83dc47177271da10c2e42 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/measure_theory/integral/bochner.lean | 5fcec00ea6050a741dfb47516bc131003dbf79ab | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 80,483 | lean | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import measure_theory.integral.set_to_l1
/-!
# Bochner integral
The Bochner integral extends the definition of the Lebesgue integral to functions that map from a
measure space into a Banach space (complete normed vector space). It is constructed here by
extending the integral on simple functions.
## Main definitions
The Bochner integral is defined through the extension process described in the file `set_to_L1`,
which follows these steps:
1. Define the integral of the indicator of a set. This is `weighted_smul μ s x = (μ s).to_real * x`.
`weighted_smul μ` is shown to be linear in the value `x` and `dominated_fin_meas_additive`
(defined in the file `set_to_L1`) with respect to the set `s`.
2. Define the integral on simple functions of the type `simple_func α E` (notation : `α →ₛ E`)
where `E` is a real normed space. (See `simple_func.integral` for details.)
3. Transfer this definition to define the integral on `L1.simple_func α E` (notation :
`α →₁ₛ[μ] E`), see `L1.simple_func.integral`. Show that this integral is a continuous linear
map from `α →₁ₛ[μ] E` to `E`.
4. Define the Bochner integral on L1 functions by extending the integral on integrable simple
functions `α →₁ₛ[μ] E` using `continuous_linear_map.extend` and the fact that the embedding of
`α →₁ₛ[μ] E` into `α →₁[μ] E` is dense.
5. Define the Bochner integral on functions as the Bochner integral of its equivalence class in L1
space, if it is in L1, and 0 otherwise.
The result of that construction is `∫ a, f a ∂μ`, which is definitionally equal to
`set_to_fun (dominated_fin_meas_additive_weighted_smul μ) f`. Some basic properties of the integral
(like linearity) are particular cases of the properties of `set_to_fun` (which are described in the
file `set_to_L1`).
## Main statements
1. Basic properties of the Bochner integral on functions of type `α → E`, where `α` is a measure
space and `E` is a real normed space.
* `integral_zero` : `∫ 0 ∂μ = 0`
* `integral_add` : `∫ x, f x + g x ∂μ = ∫ x, f ∂μ + ∫ x, g x ∂μ`
* `integral_neg` : `∫ x, - f x ∂μ = - ∫ x, f x ∂μ`
* `integral_sub` : `∫ x, f x - g x ∂μ = ∫ x, f x ∂μ - ∫ x, g x ∂μ`
* `integral_smul` : `∫ x, r • f x ∂μ = r • ∫ x, f x ∂μ`
* `integral_congr_ae` : `f =ᵐ[μ] g → ∫ x, f x ∂μ = ∫ x, g x ∂μ`
* `norm_integral_le_integral_norm` : `‖∫ x, f x ∂μ‖ ≤ ∫ x, ‖f x‖ ∂μ`
2. Basic properties of the Bochner integral on functions of type `α → ℝ`, where `α` is a measure
space.
* `integral_nonneg_of_ae` : `0 ≤ᵐ[μ] f → 0 ≤ ∫ x, f x ∂μ`
* `integral_nonpos_of_ae` : `f ≤ᵐ[μ] 0 → ∫ x, f x ∂μ ≤ 0`
* `integral_mono_ae` : `f ≤ᵐ[μ] g → ∫ x, f x ∂μ ≤ ∫ x, g x ∂μ`
* `integral_nonneg` : `0 ≤ f → 0 ≤ ∫ x, f x ∂μ`
* `integral_nonpos` : `f ≤ 0 → ∫ x, f x ∂μ ≤ 0`
* `integral_mono` : `f ≤ᵐ[μ] g → ∫ x, f x ∂μ ≤ ∫ x, g x ∂μ`
3. Propositions connecting the Bochner integral with the integral on `ℝ≥0∞`-valued functions,
which is called `lintegral` and has the notation `∫⁻`.
* `integral_eq_lintegral_max_sub_lintegral_min` : `∫ x, f x ∂μ = ∫⁻ x, f⁺ x ∂μ - ∫⁻ x, f⁻ x ∂μ`,
where `f⁺` is the positive part of `f` and `f⁻` is the negative part of `f`.
* `integral_eq_lintegral_of_nonneg_ae` : `0 ≤ᵐ[μ] f → ∫ x, f x ∂μ = ∫⁻ x, f x ∂μ`
4. `tendsto_integral_of_dominated_convergence` : the Lebesgue dominated convergence theorem
5. (In the file `set_integral`) integration commutes with continuous linear maps.
* `continuous_linear_map.integral_comp_comm`
* `linear_isometry.integral_comp_comm`
## Notes
Some tips on how to prove a proposition if the API for the Bochner integral is not enough so that
you need to unfold the definition of the Bochner integral and go back to simple functions.
One method is to use the theorem `integrable.induction` in the file `simple_func_dense_lp` (or one
of the related results, like `Lp.induction` for functions in `Lp`), which allows you to prove
something for an arbitrary integrable function.
Another method is using the following steps.
See `integral_eq_lintegral_max_sub_lintegral_min` for a complicated example, which proves that
`∫ f = ∫⁻ f⁺ - ∫⁻ f⁻`, with the first integral sign being the Bochner integral of a real-valued
function `f : α → ℝ`, and second and third integral sign being the integral on `ℝ≥0∞`-valued
functions (called `lintegral`). The proof of `integral_eq_lintegral_max_sub_lintegral_min` is
scattered in sections with the name `pos_part`.
Here are the usual steps of proving that a property `p`, say `∫ f = ∫⁻ f⁺ - ∫⁻ f⁻`, holds for all
functions :
1. First go to the `L¹` space.
For example, if you see `ennreal.to_real (∫⁻ a, ennreal.of_real $ ‖f a‖)`, that is the norm of
`f` in `L¹` space. Rewrite using `L1.norm_of_fun_eq_lintegral_norm`.
2. Show that the set `{f ∈ L¹ | ∫ f = ∫⁻ f⁺ - ∫⁻ f⁻}` is closed in `L¹` using `is_closed_eq`.
3. Show that the property holds for all simple functions `s` in `L¹` space.
Typically, you need to convert various notions to their `simple_func` counterpart, using lemmas
like `L1.integral_coe_eq_integral`.
4. Since simple functions are dense in `L¹`,
```
univ = closure {s simple}
= closure {s simple | ∫ s = ∫⁻ s⁺ - ∫⁻ s⁻} : the property holds for all simple functions
⊆ closure {f | ∫ f = ∫⁻ f⁺ - ∫⁻ f⁻}
= {f | ∫ f = ∫⁻ f⁺ - ∫⁻ f⁻} : closure of a closed set is itself
```
Use `is_closed_property` or `dense_range.induction_on` for this argument.
## Notations
* `α →ₛ E` : simple functions (defined in `measure_theory/integration`)
* `α →₁[μ] E` : functions in L1 space, i.e., equivalence classes of integrable functions (defined in
`measure_theory/lp_space`)
* `α →₁ₛ[μ] E` : simple functions in L1 space, i.e., equivalence classes of integrable simple
functions (defined in `measure_theory/simple_func_dense`)
* `∫ a, f a ∂μ` : integral of `f` with respect to a measure `μ`
* `∫ a, f a` : integral of `f` with respect to `volume`, the default measure on the ambient type
We also define notations for integral on a set, which are described in the file
`measure_theory/set_integral`.
Note : `ₛ` is typed using `\_s`. Sometimes it shows as a box if the font is missing.
## Tags
Bochner integral, simple function, function space, Lebesgue dominated convergence theorem
-/
noncomputable theory
open_locale topological_space big_operators nnreal ennreal measure_theory
open set filter topological_space ennreal emetric
namespace measure_theory
variables {α E F 𝕜 : Type*}
section weighted_smul
open continuous_linear_map
variables [normed_add_comm_group F] [normed_space ℝ F] {m : measurable_space α} {μ : measure α}
/-- Given a set `s`, return the continuous linear map `λ x, (μ s).to_real • x`. The extension of
that set function through `set_to_L1` gives the Bochner integral of L1 functions. -/
def weighted_smul {m : measurable_space α} (μ : measure α) (s : set α) : F →L[ℝ] F :=
(μ s).to_real • (continuous_linear_map.id ℝ F)
lemma weighted_smul_apply {m : measurable_space α} (μ : measure α) (s : set α) (x : F) :
weighted_smul μ s x = (μ s).to_real • x :=
by simp [weighted_smul]
@[simp] lemma weighted_smul_zero_measure {m : measurable_space α} :
weighted_smul (0 : measure α) = (0 : set α → F →L[ℝ] F) :=
by { ext1, simp [weighted_smul], }
@[simp] lemma weighted_smul_empty {m : measurable_space α} (μ : measure α) :
weighted_smul μ ∅ = (0 : F →L[ℝ] F) :=
by { ext1 x, rw [weighted_smul_apply], simp, }
lemma weighted_smul_add_measure {m : measurable_space α} (μ ν : measure α) {s : set α}
(hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
(weighted_smul (μ + ν) s : F →L[ℝ] F) = weighted_smul μ s + weighted_smul ν s :=
begin
ext1 x,
push_cast,
simp_rw [pi.add_apply, weighted_smul_apply],
push_cast,
rw [pi.add_apply, ennreal.to_real_add hμs hνs, add_smul],
end
lemma weighted_smul_smul_measure {m : measurable_space α} (μ : measure α) (c : ℝ≥0∞) {s : set α} :
(weighted_smul (c • μ) s : F →L[ℝ] F) = c.to_real • weighted_smul μ s :=
begin
ext1 x,
push_cast,
simp_rw [pi.smul_apply, weighted_smul_apply],
push_cast,
simp_rw [pi.smul_apply, smul_eq_mul, to_real_mul, smul_smul],
end
lemma weighted_smul_congr (s t : set α) (hst : μ s = μ t) :
(weighted_smul μ s : F →L[ℝ] F) = weighted_smul μ t :=
by { ext1 x, simp_rw weighted_smul_apply, congr' 2, }
lemma weighted_smul_null {s : set α} (h_zero : μ s = 0) : (weighted_smul μ s : F →L[ℝ] F) = 0 :=
by { ext1 x, rw [weighted_smul_apply, h_zero], simp, }
lemma weighted_smul_union' (s t : set α) (ht : measurable_set t)
(hs_finite : μ s ≠ ∞) (ht_finite : μ t ≠ ∞) (h_inter : s ∩ t = ∅) :
(weighted_smul μ (s ∪ t) : F →L[ℝ] F) = weighted_smul μ s + weighted_smul μ t :=
begin
ext1 x,
simp_rw [add_apply, weighted_smul_apply,
measure_union (set.disjoint_iff_inter_eq_empty.mpr h_inter) ht,
ennreal.to_real_add hs_finite ht_finite, add_smul],
end
@[nolint unused_arguments]
lemma weighted_smul_union (s t : set α) (hs : measurable_set s) (ht : measurable_set t)
(hs_finite : μ s ≠ ∞) (ht_finite : μ t ≠ ∞) (h_inter : s ∩ t = ∅) :
(weighted_smul μ (s ∪ t) : F →L[ℝ] F) = weighted_smul μ s + weighted_smul μ t :=
weighted_smul_union' s t ht hs_finite ht_finite h_inter
lemma weighted_smul_smul [normed_field 𝕜] [normed_space 𝕜 F] [smul_comm_class ℝ 𝕜 F]
(c : 𝕜) (s : set α) (x : F) :
weighted_smul μ s (c • x) = c • weighted_smul μ s x :=
by { simp_rw [weighted_smul_apply, smul_comm], }
lemma norm_weighted_smul_le (s : set α) : ‖(weighted_smul μ s : F →L[ℝ] F)‖ ≤ (μ s).to_real :=
calc ‖(weighted_smul μ s : F →L[ℝ] F)‖ = ‖(μ s).to_real‖ * ‖continuous_linear_map.id ℝ F‖ :
norm_smul _ _
... ≤ ‖(μ s).to_real‖ : (mul_le_mul_of_nonneg_left norm_id_le (norm_nonneg _)).trans (mul_one _).le
... = abs (μ s).to_real : real.norm_eq_abs _
... = (μ s).to_real : abs_eq_self.mpr ennreal.to_real_nonneg
lemma dominated_fin_meas_additive_weighted_smul {m : measurable_space α} (μ : measure α) :
dominated_fin_meas_additive μ (weighted_smul μ : set α → F →L[ℝ] F) 1 :=
⟨weighted_smul_union, λ s _ _, (norm_weighted_smul_le s).trans (one_mul _).symm.le⟩
lemma weighted_smul_nonneg (s : set α) (x : ℝ) (hx : 0 ≤ x) : 0 ≤ weighted_smul μ s x :=
begin
simp only [weighted_smul, algebra.id.smul_eq_mul, coe_smul', id.def, coe_id', pi.smul_apply],
exact mul_nonneg to_real_nonneg hx,
end
end weighted_smul
local infixr ` →ₛ `:25 := simple_func
namespace simple_func
section pos_part
variables [linear_order E] [has_zero E] [measurable_space α]
/-- Positive part of a simple function. -/
def pos_part (f : α →ₛ E) : α →ₛ E := f.map (λ b, max b 0)
/-- Negative part of a simple function. -/
def neg_part [has_neg E] (f : α →ₛ E) : α →ₛ E := pos_part (-f)
lemma pos_part_map_norm (f : α →ₛ ℝ) : (pos_part f).map norm = pos_part f :=
by { ext, rw [map_apply, real.norm_eq_abs, abs_of_nonneg], exact le_max_right _ _ }
lemma neg_part_map_norm (f : α →ₛ ℝ) : (neg_part f).map norm = neg_part f :=
by { rw neg_part, exact pos_part_map_norm _ }
lemma pos_part_sub_neg_part (f : α →ₛ ℝ) : f.pos_part - f.neg_part = f :=
begin
simp only [pos_part, neg_part],
ext a,
rw coe_sub,
exact max_zero_sub_eq_self (f a)
end
end pos_part
section integral
/-!
### The Bochner integral of simple functions
Define the Bochner integral of simple functions of the type `α →ₛ β` where `β` is a normed group,
and prove basic property of this integral.
-/
open finset
variables [normed_add_comm_group E] [normed_add_comm_group F] [normed_space ℝ F] {p : ℝ≥0∞}
{G F' : Type*} [normed_add_comm_group G] [normed_add_comm_group F'] [normed_space ℝ F']
{m : measurable_space α} {μ : measure α}
/-- Bochner integral of simple functions whose codomain is a real `normed_space`.
This is equal to `∑ x in f.range, (μ (f ⁻¹' {x})).to_real • x` (see `integral_eq`). -/
def integral {m : measurable_space α} (μ : measure α) (f : α →ₛ F) : F :=
f.set_to_simple_func (weighted_smul μ)
lemma integral_def {m : measurable_space α} (μ : measure α) (f : α →ₛ F) :
f.integral μ = f.set_to_simple_func (weighted_smul μ) := rfl
lemma integral_eq {m : measurable_space α} (μ : measure α) (f : α →ₛ F) :
f.integral μ = ∑ x in f.range, (μ (f ⁻¹' {x})).to_real • x :=
by simp [integral, set_to_simple_func, weighted_smul_apply]
lemma integral_eq_sum_filter [decidable_pred (λ x : F, x ≠ 0)] {m : measurable_space α} (f : α →ₛ F)
(μ : measure α) :
f.integral μ = ∑ x in f.range.filter (λ x, x ≠ 0), (μ (f ⁻¹' {x})).to_real • x :=
by { rw [integral_def, set_to_simple_func_eq_sum_filter], simp_rw weighted_smul_apply, congr }
/-- The Bochner integral is equal to a sum over any set that includes `f.range` (except `0`). -/
lemma integral_eq_sum_of_subset [decidable_pred (λ x : F, x ≠ 0)] {f : α →ₛ F} {s : finset F}
(hs : f.range.filter (λ x, x ≠ 0) ⊆ s) : f.integral μ = ∑ x in s, (μ (f ⁻¹' {x})).to_real • x :=
begin
rw [simple_func.integral_eq_sum_filter, finset.sum_subset hs],
rintro x - hx, rw [finset.mem_filter, not_and_distrib, ne.def, not_not] at hx,
rcases hx with hx|rfl; [skip, simp],
rw [simple_func.mem_range] at hx, rw [preimage_eq_empty]; simp [set.disjoint_singleton_left, hx]
end
@[simp] lemma integral_const {m : measurable_space α} (μ : measure α) (y : F) :
(const α y).integral μ = (μ univ).to_real • y :=
by classical; calc (const α y).integral μ = ∑ z in {y}, (μ ((const α y) ⁻¹' {z})).to_real • z :
integral_eq_sum_of_subset $ (filter_subset _ _).trans (range_const_subset _ _)
... = (μ univ).to_real • y : by simp
@[simp] lemma integral_piecewise_zero {m : measurable_space α} (f : α →ₛ F) (μ : measure α)
{s : set α} (hs : measurable_set s) :
(piecewise s hs f 0).integral μ = f.integral (μ.restrict s) :=
begin
classical,
refine (integral_eq_sum_of_subset _).trans
((sum_congr rfl $ λ y hy, _).trans (integral_eq_sum_filter _ _).symm),
{ intros y hy,
simp only [mem_filter, mem_range, coe_piecewise, coe_zero, piecewise_eq_indicator,
mem_range_indicator] at *,
rcases hy with ⟨⟨rfl, -⟩|⟨x, hxs, rfl⟩, h₀⟩,
exacts [(h₀ rfl).elim, ⟨set.mem_range_self _, h₀⟩] },
{ dsimp,
rw [set.piecewise_eq_indicator, indicator_preimage_of_not_mem,
measure.restrict_apply (f.measurable_set_preimage _)],
exact λ h₀, (mem_filter.1 hy).2 (eq.symm h₀) }
end
/-- Calculate the integral of `g ∘ f : α →ₛ F`, where `f` is an integrable function from `α` to `E`
and `g` is a function from `E` to `F`. We require `g 0 = 0` so that `g ∘ f` is integrable. -/
lemma map_integral (f : α →ₛ E) (g : E → F) (hf : integrable f μ) (hg : g 0 = 0) :
(f.map g).integral μ = ∑ x in f.range, (ennreal.to_real (μ (f ⁻¹' {x}))) • (g x) :=
map_set_to_simple_func _ weighted_smul_union hf hg
/-- `simple_func.integral` and `simple_func.lintegral` agree when the integrand has type
`α →ₛ ℝ≥0∞`. But since `ℝ≥0∞` is not a `normed_space`, we need some form of coercion.
See `integral_eq_lintegral` for a simpler version. -/
lemma integral_eq_lintegral' {f : α →ₛ E} {g : E → ℝ≥0∞} (hf : integrable f μ) (hg0 : g 0 = 0)
(ht : ∀ b, g b ≠ ∞) :
(f.map (ennreal.to_real ∘ g)).integral μ = ennreal.to_real (∫⁻ a, g (f a) ∂μ) :=
begin
have hf' : f.fin_meas_supp μ := integrable_iff_fin_meas_supp.1 hf,
simp only [← map_apply g f, lintegral_eq_lintegral],
rw [map_integral f _ hf, map_lintegral, ennreal.to_real_sum],
{ refine finset.sum_congr rfl (λb hb, _),
rw [smul_eq_mul, to_real_mul, mul_comm] },
{ assume a ha,
by_cases a0 : a = 0,
{ rw [a0, hg0, zero_mul], exact with_top.zero_ne_top },
{ apply mul_ne_top (ht a) (hf'.meas_preimage_singleton_ne_zero a0).ne } },
{ simp [hg0] }
end
variables [normed_field 𝕜] [normed_space 𝕜 E] [normed_space ℝ E] [smul_comm_class ℝ 𝕜 E]
lemma integral_congr {f g : α →ₛ E} (hf : integrable f μ) (h : f =ᵐ[μ] g) :
f.integral μ = g.integral μ :=
set_to_simple_func_congr (weighted_smul μ) (λ s hs, weighted_smul_null) weighted_smul_union hf h
/-- `simple_func.bintegral` and `simple_func.integral` agree when the integrand has type
`α →ₛ ℝ≥0∞`. But since `ℝ≥0∞` is not a `normed_space`, we need some form of coercion. -/
lemma integral_eq_lintegral {f : α →ₛ ℝ} (hf : integrable f μ) (h_pos : 0 ≤ᵐ[μ] f) :
f.integral μ = ennreal.to_real (∫⁻ a, ennreal.of_real (f a) ∂μ) :=
begin
have : f =ᵐ[μ] f.map (ennreal.to_real ∘ ennreal.of_real) :=
h_pos.mono (λ a h, (ennreal.to_real_of_real h).symm),
rw [← integral_eq_lintegral' hf],
exacts [integral_congr hf this, ennreal.of_real_zero, λ b, ennreal.of_real_ne_top]
end
lemma integral_add {f g : α →ₛ E} (hf : integrable f μ) (hg : integrable g μ) :
integral μ (f + g) = integral μ f + integral μ g :=
set_to_simple_func_add _ weighted_smul_union hf hg
lemma integral_neg {f : α →ₛ E} (hf : integrable f μ) : integral μ (-f) = - integral μ f :=
set_to_simple_func_neg _ weighted_smul_union hf
lemma integral_sub {f g : α →ₛ E} (hf : integrable f μ) (hg : integrable g μ) :
integral μ (f - g) = integral μ f - integral μ g :=
set_to_simple_func_sub _ weighted_smul_union hf hg
lemma integral_smul (c : 𝕜) {f : α →ₛ E} (hf : integrable f μ) :
integral μ (c • f) = c • integral μ f :=
set_to_simple_func_smul _ weighted_smul_union weighted_smul_smul c hf
lemma norm_set_to_simple_func_le_integral_norm (T : set α → E →L[ℝ] F) {C : ℝ}
(hT_norm : ∀ s, measurable_set s → μ s < ∞ → ‖T s‖ ≤ C * (μ s).to_real) {f : α →ₛ E}
(hf : integrable f μ) :
‖f.set_to_simple_func T‖ ≤ C * (f.map norm).integral μ :=
calc ‖f.set_to_simple_func T‖
≤ C * ∑ x in f.range, ennreal.to_real (μ (f ⁻¹' {x})) * ‖x‖ :
norm_set_to_simple_func_le_sum_mul_norm_of_integrable T hT_norm f hf
... = C * (f.map norm).integral μ : by { rw map_integral f norm hf norm_zero, simp_rw smul_eq_mul, }
lemma norm_integral_le_integral_norm (f : α →ₛ E) (hf : integrable f μ) :
‖f.integral μ‖ ≤ (f.map norm).integral μ :=
begin
refine (norm_set_to_simple_func_le_integral_norm _ (λ s _ _, _) hf).trans (one_mul _).le,
exact (norm_weighted_smul_le s).trans (one_mul _).symm.le,
end
lemma integral_add_measure {ν} (f : α →ₛ E) (hf : integrable f (μ + ν)) :
f.integral (μ + ν) = f.integral μ + f.integral ν :=
begin
simp_rw [integral_def],
refine set_to_simple_func_add_left' (weighted_smul μ) (weighted_smul ν) (weighted_smul (μ + ν))
(λ s hs hμνs, _) hf,
rw [lt_top_iff_ne_top, measure.coe_add, pi.add_apply, ennreal.add_ne_top] at hμνs,
rw weighted_smul_add_measure _ _ hμνs.1 hμνs.2,
end
end integral
end simple_func
namespace L1
open ae_eq_fun Lp.simple_func Lp
variables [normed_add_comm_group E] [normed_add_comm_group F] {m : measurable_space α}
{μ : measure α}
variables {α E μ}
namespace simple_func
lemma norm_eq_integral (f : α →₁ₛ[μ] E) : ‖f‖ = ((to_simple_func f).map norm).integral μ :=
begin
rw [norm_eq_sum_mul f, (to_simple_func f).map_integral norm (simple_func.integrable f) norm_zero],
simp_rw smul_eq_mul,
end
section pos_part
/-- Positive part of a simple function in L1 space. -/
def pos_part (f : α →₁ₛ[μ] ℝ) : α →₁ₛ[μ] ℝ := ⟨Lp.pos_part (f : α →₁[μ] ℝ),
begin
rcases f with ⟨f, s, hsf⟩,
use s.pos_part,
simp only [subtype.coe_mk, Lp.coe_pos_part, ← hsf, ae_eq_fun.pos_part_mk, simple_func.pos_part,
simple_func.coe_map, mk_eq_mk],
end ⟩
/-- Negative part of a simple function in L1 space. -/
def neg_part (f : α →₁ₛ[μ] ℝ) : α →₁ₛ[μ] ℝ := pos_part (-f)
@[norm_cast]
lemma coe_pos_part (f : α →₁ₛ[μ] ℝ) : (pos_part f : α →₁[μ] ℝ) = Lp.pos_part (f : α →₁[μ] ℝ) := rfl
@[norm_cast]
lemma coe_neg_part (f : α →₁ₛ[μ] ℝ) : (neg_part f : α →₁[μ] ℝ) = Lp.neg_part (f : α →₁[μ] ℝ) := rfl
end pos_part
section simple_func_integral
/-!
### The Bochner integral of `L1`
Define the Bochner integral on `α →₁ₛ[μ] E` by extension from the simple functions `α →₁ₛ[μ] E`,
and prove basic properties of this integral. -/
variables [normed_field 𝕜] [normed_space 𝕜 E] [normed_space ℝ E] [smul_comm_class ℝ 𝕜 E]
{F' : Type*} [normed_add_comm_group F'] [normed_space ℝ F']
local attribute [instance] simple_func.normed_space
/-- The Bochner integral over simple functions in L1 space. -/
def integral (f : α →₁ₛ[μ] E) : E := ((to_simple_func f)).integral μ
lemma integral_eq_integral (f : α →₁ₛ[μ] E) : integral f = ((to_simple_func f)).integral μ := rfl
lemma integral_eq_lintegral {f : α →₁ₛ[μ] ℝ} (h_pos : 0 ≤ᵐ[μ] (to_simple_func f)) :
integral f = ennreal.to_real (∫⁻ a, ennreal.of_real ((to_simple_func f) a) ∂μ) :=
by rw [integral, simple_func.integral_eq_lintegral (simple_func.integrable f) h_pos]
lemma integral_eq_set_to_L1s (f : α →₁ₛ[μ] E) : integral f = set_to_L1s (weighted_smul μ) f := rfl
lemma integral_congr {f g : α →₁ₛ[μ] E} (h : to_simple_func f =ᵐ[μ] to_simple_func g) :
integral f = integral g :=
simple_func.integral_congr (simple_func.integrable f) h
lemma integral_add (f g : α →₁ₛ[μ] E) : integral (f + g) = integral f + integral g :=
set_to_L1s_add _ (λ _ _, weighted_smul_null) weighted_smul_union _ _
lemma integral_smul (c : 𝕜) (f : α →₁ₛ[μ] E) :
integral (c • f) = c • integral f :=
set_to_L1s_smul _ (λ _ _, weighted_smul_null) weighted_smul_union weighted_smul_smul c f
lemma norm_integral_le_norm (f : α →₁ₛ[μ] E) : ‖integral f‖ ≤ ‖f‖ :=
begin
rw [integral, norm_eq_integral],
exact (to_simple_func f).norm_integral_le_integral_norm (simple_func.integrable f)
end
variables {E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E'] [normed_space 𝕜 E']
variables (α E μ 𝕜)
/-- The Bochner integral over simple functions in L1 space as a continuous linear map. -/
def integral_clm' : (α →₁ₛ[μ] E) →L[𝕜] E :=
linear_map.mk_continuous ⟨integral, integral_add, integral_smul⟩
1 (λf, le_trans (norm_integral_le_norm _) $ by rw one_mul)
/-- The Bochner integral over simple functions in L1 space as a continuous linear map over ℝ. -/
def integral_clm : (α →₁ₛ[μ] E) →L[ℝ] E := integral_clm' α E ℝ μ
variables {α E μ 𝕜}
local notation (name := simple_func.integral_clm) `Integral` := integral_clm α E μ
open continuous_linear_map
lemma norm_Integral_le_one : ‖Integral‖ ≤ 1 :=
linear_map.mk_continuous_norm_le _ (zero_le_one) _
section pos_part
lemma pos_part_to_simple_func (f : α →₁ₛ[μ] ℝ) :
to_simple_func (pos_part f) =ᵐ[μ] (to_simple_func f).pos_part :=
begin
have eq : ∀ a, (to_simple_func f).pos_part a = max ((to_simple_func f) a) 0 := λa, rfl,
have ae_eq : ∀ᵐ a ∂μ, to_simple_func (pos_part f) a = max ((to_simple_func f) a) 0,
{ filter_upwards [to_simple_func_eq_to_fun (pos_part f), Lp.coe_fn_pos_part (f : α →₁[μ] ℝ),
to_simple_func_eq_to_fun f] with _ _ h₂ _,
convert h₂, },
refine ae_eq.mono (assume a h, _),
rw [h, eq],
end
lemma neg_part_to_simple_func (f : α →₁ₛ[μ] ℝ) :
to_simple_func (neg_part f) =ᵐ[μ] (to_simple_func f).neg_part :=
begin
rw [simple_func.neg_part, measure_theory.simple_func.neg_part],
filter_upwards [pos_part_to_simple_func (-f), neg_to_simple_func f],
assume a h₁ h₂,
rw h₁,
show max _ _ = max _ _,
rw h₂,
refl
end
lemma integral_eq_norm_pos_part_sub (f : α →₁ₛ[μ] ℝ) :
integral f = ‖pos_part f‖ - ‖neg_part f‖ :=
begin
-- Convert things in `L¹` to their `simple_func` counterpart
have ae_eq₁ : (to_simple_func f).pos_part =ᵐ[μ] (to_simple_func (pos_part f)).map norm,
{ filter_upwards [pos_part_to_simple_func f] with _ h,
rw [simple_func.map_apply, h],
conv_lhs { rw [← simple_func.pos_part_map_norm, simple_func.map_apply] } },
-- Convert things in `L¹` to their `simple_func` counterpart
have ae_eq₂ : (to_simple_func f).neg_part =ᵐ[μ] (to_simple_func (neg_part f)).map norm,
{ filter_upwards [neg_part_to_simple_func f] with _ h,
rw [simple_func.map_apply, h],
conv_lhs { rw [← simple_func.neg_part_map_norm, simple_func.map_apply], }, },
-- Convert things in `L¹` to their `simple_func` counterpart
have ae_eq : ∀ᵐ a ∂μ, (to_simple_func f).pos_part a - (to_simple_func f).neg_part a =
(to_simple_func (pos_part f)).map norm a - (to_simple_func (neg_part f)).map norm a,
{ filter_upwards [ae_eq₁, ae_eq₂] with _ h₁ h₂,
rw [h₁, h₂], },
rw [integral, norm_eq_integral, norm_eq_integral, ← simple_func.integral_sub],
{ show (to_simple_func f).integral μ =
((to_simple_func (pos_part f)).map norm - (to_simple_func (neg_part f)).map norm).integral μ,
apply measure_theory.simple_func.integral_congr (simple_func.integrable f),
filter_upwards [ae_eq₁, ae_eq₂] with _ h₁ h₂,
show _ = _ - _,
rw [← h₁, ← h₂],
have := (to_simple_func f).pos_part_sub_neg_part,
conv_lhs {rw ← this},
refl, },
{ exact (simple_func.integrable f).pos_part.congr ae_eq₁ },
{ exact (simple_func.integrable f).neg_part.congr ae_eq₂ }
end
end pos_part
end simple_func_integral
end simple_func
open simple_func
local notation (name := simple_func.integral_clm) `Integral` := @integral_clm α E _ _ _ _ _ μ _
variables [normed_space ℝ E] [nontrivially_normed_field 𝕜] [normed_space 𝕜 E]
[smul_comm_class ℝ 𝕜 E] [normed_space ℝ F] [complete_space E]
section integration_in_L1
local attribute [instance] simple_func.normed_space
open continuous_linear_map
variables (𝕜)
/-- The Bochner integral in L1 space as a continuous linear map. -/
def integral_clm' : (α →₁[μ] E) →L[𝕜] E :=
(integral_clm' α E 𝕜 μ).extend
(coe_to_Lp α E 𝕜) (simple_func.dense_range one_ne_top) simple_func.uniform_inducing
variables {𝕜}
/-- The Bochner integral in L1 space as a continuous linear map over ℝ. -/
def integral_clm : (α →₁[μ] E) →L[ℝ] E := integral_clm' ℝ
/-- The Bochner integral in L1 space -/
def integral (f : α →₁[μ] E) : E := integral_clm f
lemma integral_eq (f : α →₁[μ] E) : integral f = integral_clm f := rfl
lemma integral_eq_set_to_L1 (f : α →₁[μ] E) :
integral f = set_to_L1 (dominated_fin_meas_additive_weighted_smul μ) f :=
rfl
@[norm_cast] lemma simple_func.integral_L1_eq_integral (f : α →₁ₛ[μ] E) :
integral (f : α →₁[μ] E) = (simple_func.integral f) :=
set_to_L1_eq_set_to_L1s_clm (dominated_fin_meas_additive_weighted_smul μ) f
variables (α E)
@[simp] lemma integral_zero : integral (0 : α →₁[μ] E) = 0 :=
map_zero integral_clm
variables {α E}
lemma integral_add (f g : α →₁[μ] E) : integral (f + g) = integral f + integral g :=
map_add integral_clm f g
lemma integral_neg (f : α →₁[μ] E) : integral (-f) = - integral f :=
map_neg integral_clm f
lemma integral_sub (f g : α →₁[μ] E) : integral (f - g) = integral f - integral g :=
map_sub integral_clm f g
lemma integral_smul (c : 𝕜) (f : α →₁[μ] E) : integral (c • f) = c • integral f :=
show (integral_clm' 𝕜) (c • f) = c • (integral_clm' 𝕜) f, from map_smul (integral_clm' 𝕜) c f
local notation (name := integral_clm) `Integral` := @integral_clm α E _ _ μ _ _
local notation (name := simple_func.integral_clm') `sIntegral` :=
@simple_func.integral_clm α E _ _ μ _
lemma norm_Integral_le_one : ‖Integral‖ ≤ 1 :=
norm_set_to_L1_le (dominated_fin_meas_additive_weighted_smul μ) zero_le_one
lemma norm_integral_le (f : α →₁[μ] E) : ‖integral f‖ ≤ ‖f‖ :=
calc ‖integral f‖ = ‖Integral f‖ : rfl
... ≤ ‖Integral‖ * ‖f‖ : le_op_norm _ _
... ≤ 1 * ‖f‖ : mul_le_mul_of_nonneg_right norm_Integral_le_one $ norm_nonneg _
... = ‖f‖ : one_mul _
@[continuity]
lemma continuous_integral : continuous (λ (f : α →₁[μ] E), integral f) :=
L1.integral_clm.continuous
section pos_part
lemma integral_eq_norm_pos_part_sub (f : α →₁[μ] ℝ) :
integral f = ‖Lp.pos_part f‖ - ‖Lp.neg_part f‖ :=
begin
-- Use `is_closed_property` and `is_closed_eq`
refine @is_closed_property _ _ _ (coe : (α →₁ₛ[μ] ℝ) → (α →₁[μ] ℝ))
(λ f : α →₁[μ] ℝ, integral f = ‖Lp.pos_part f‖ - ‖Lp.neg_part f‖)
(simple_func.dense_range one_ne_top) (is_closed_eq _ _) _ f,
{ exact cont _ },
{ refine continuous.sub (continuous_norm.comp Lp.continuous_pos_part)
(continuous_norm.comp Lp.continuous_neg_part) },
-- Show that the property holds for all simple functions in the `L¹` space.
{ assume s,
norm_cast,
exact simple_func.integral_eq_norm_pos_part_sub _ }
end
end pos_part
end integration_in_L1
end L1
/-!
## The Bochner integral on functions
Define the Bochner integral on functions generally to be the `L1` Bochner integral, for integrable
functions, and 0 otherwise; prove its basic properties.
-/
variables [normed_add_comm_group E] [normed_space ℝ E] [complete_space E]
[nontrivially_normed_field 𝕜] [normed_space 𝕜 E] [smul_comm_class ℝ 𝕜 E]
[normed_add_comm_group F] [normed_space ℝ F] [complete_space F]
section
open_locale classical
/-- The Bochner integral -/
def integral {m : measurable_space α} (μ : measure α) (f : α → E) : E :=
if hf : integrable f μ then L1.integral (hf.to_L1 f) else 0
end
/-! In the notation for integrals, an expression like `∫ x, g ‖x‖ ∂μ` will not be parsed correctly,
and needs parentheses. We do not set the binding power of `r` to `0`, because then
`∫ x, f x = 0` will be parsed incorrectly. -/
notation `∫` binders `, ` r:(scoped:60 f, f) ` ∂` μ:70 := integral μ r
notation `∫` binders `, ` r:(scoped:60 f, integral volume f) := r
notation `∫` binders ` in ` s `, ` r:(scoped:60 f, f) ` ∂` μ:70 := integral (measure.restrict μ s) r
notation `∫` binders ` in ` s `, ` r:(scoped:60 f, integral (measure.restrict volume s) f) := r
section properties
open continuous_linear_map measure_theory.simple_func
variables {f g : α → E} {m : measurable_space α} {μ : measure α}
lemma integral_eq (f : α → E) (hf : integrable f μ) :
∫ a, f a ∂μ = L1.integral (hf.to_L1 f) :=
@dif_pos _ (id _) hf _ _ _
lemma integral_eq_set_to_fun (f : α → E) :
∫ a, f a ∂μ = set_to_fun μ (weighted_smul μ) (dominated_fin_meas_additive_weighted_smul μ) f :=
rfl
lemma L1.integral_eq_integral (f : α →₁[μ] E) : L1.integral f = ∫ a, f a ∂μ :=
(L1.set_to_fun_eq_set_to_L1 (dominated_fin_meas_additive_weighted_smul μ) f).symm
lemma integral_undef (h : ¬ integrable f μ) : ∫ a, f a ∂μ = 0 :=
@dif_neg _ (id _) h _ _ _
lemma integral_non_ae_strongly_measurable (h : ¬ ae_strongly_measurable f μ) : ∫ a, f a ∂μ = 0 :=
integral_undef $ not_and_of_not_left _ h
variables (α E)
lemma integral_zero : ∫ a : α, (0:E) ∂μ = 0 :=
set_to_fun_zero (dominated_fin_meas_additive_weighted_smul μ)
@[simp] lemma integral_zero' : integral μ (0 : α → E) = 0 :=
integral_zero α E
variables {α E}
lemma integral_add (hf : integrable f μ) (hg : integrable g μ) :
∫ a, f a + g a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ :=
set_to_fun_add (dominated_fin_meas_additive_weighted_smul μ) hf hg
lemma integral_add' (hf : integrable f μ) (hg : integrable g μ) :
∫ a, (f + g) a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ :=
integral_add hf hg
lemma integral_finset_sum {ι} (s : finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, integrable (f i) μ) :
∫ a, ∑ i in s, f i a ∂μ = ∑ i in s, ∫ a, f i a ∂μ :=
set_to_fun_finset_sum (dominated_fin_meas_additive_weighted_smul _) s hf
lemma integral_neg (f : α → E) : ∫ a, -f a ∂μ = - ∫ a, f a ∂μ :=
set_to_fun_neg (dominated_fin_meas_additive_weighted_smul μ) f
lemma integral_neg' (f : α → E) : ∫ a, (-f) a ∂μ = - ∫ a, f a ∂μ :=
integral_neg f
lemma integral_sub (hf : integrable f μ) (hg : integrable g μ) :
∫ a, f a - g a ∂μ = ∫ a, f a ∂μ - ∫ a, g a ∂μ :=
set_to_fun_sub (dominated_fin_meas_additive_weighted_smul μ) hf hg
lemma integral_sub' (hf : integrable f μ) (hg : integrable g μ) :
∫ a, (f - g) a ∂μ = ∫ a, f a ∂μ - ∫ a, g a ∂μ :=
integral_sub hf hg
lemma integral_smul (c : 𝕜) (f : α → E) :
∫ a, c • (f a) ∂μ = c • ∫ a, f a ∂μ :=
set_to_fun_smul (dominated_fin_meas_additive_weighted_smul μ) weighted_smul_smul c f
lemma integral_mul_left (r : ℝ) (f : α → ℝ) : ∫ a, r * (f a) ∂μ = r * ∫ a, f a ∂μ :=
integral_smul r f
lemma integral_mul_right (r : ℝ) (f : α → ℝ) : ∫ a, (f a) * r ∂μ = ∫ a, f a ∂μ * r :=
by { simp only [mul_comm], exact integral_mul_left r f }
lemma integral_div (r : ℝ) (f : α → ℝ) : ∫ a, (f a) / r ∂μ = ∫ a, f a ∂μ / r :=
integral_mul_right r⁻¹ f
lemma integral_congr_ae (h : f =ᵐ[μ] g) : ∫ a, f a ∂μ = ∫ a, g a ∂μ :=
set_to_fun_congr_ae (dominated_fin_meas_additive_weighted_smul μ) h
@[simp] lemma L1.integral_of_fun_eq_integral {f : α → E} (hf : integrable f μ) :
∫ a, (hf.to_L1 f) a ∂μ = ∫ a, f a ∂μ :=
set_to_fun_to_L1 (dominated_fin_meas_additive_weighted_smul μ) hf
@[continuity]
lemma continuous_integral : continuous (λ (f : α →₁[μ] E), ∫ a, f a ∂μ) :=
continuous_set_to_fun (dominated_fin_meas_additive_weighted_smul μ)
lemma norm_integral_le_lintegral_norm (f : α → E) :
‖∫ a, f a ∂μ‖ ≤ ennreal.to_real (∫⁻ a, (ennreal.of_real ‖f a‖) ∂μ) :=
begin
by_cases hf : integrable f μ,
{ rw [integral_eq f hf, ← integrable.norm_to_L1_eq_lintegral_norm f hf],
exact L1.norm_integral_le _ },
{ rw [integral_undef hf, norm_zero], exact to_real_nonneg }
end
lemma ennnorm_integral_le_lintegral_ennnorm (f : α → E) :
(‖∫ a, f a ∂μ‖₊ : ℝ≥0∞) ≤ ∫⁻ a, ‖f a‖₊ ∂μ :=
by { simp_rw [← of_real_norm_eq_coe_nnnorm], apply ennreal.of_real_le_of_le_to_real,
exact norm_integral_le_lintegral_norm f }
lemma integral_eq_zero_of_ae {f : α → E} (hf : f =ᵐ[μ] 0) : ∫ a, f a ∂μ = 0 :=
by simp [integral_congr_ae hf, integral_zero]
/-- If `f` has finite integral, then `∫ x in s, f x ∂μ` is absolutely continuous in `s`: it tends
to zero as `μ s` tends to zero. -/
lemma has_finite_integral.tendsto_set_integral_nhds_zero {ι} {f : α → E}
(hf : has_finite_integral f μ) {l : filter ι} {s : ι → set α} (hs : tendsto (μ ∘ s) l (𝓝 0)) :
tendsto (λ i, ∫ x in s i, f x ∂μ) l (𝓝 0) :=
begin
rw [tendsto_zero_iff_norm_tendsto_zero],
simp_rw [← coe_nnnorm, ← nnreal.coe_zero, nnreal.tendsto_coe, ← ennreal.tendsto_coe,
ennreal.coe_zero],
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
(tendsto_set_lintegral_zero (ne_of_lt hf) hs) (λ i, zero_le _)
(λ i, ennnorm_integral_le_lintegral_ennnorm _)
end
/-- If `f` is integrable, then `∫ x in s, f x ∂μ` is absolutely continuous in `s`: it tends
to zero as `μ s` tends to zero. -/
lemma integrable.tendsto_set_integral_nhds_zero {ι} {f : α → E}
(hf : integrable f μ) {l : filter ι} {s : ι → set α} (hs : tendsto (μ ∘ s) l (𝓝 0)) :
tendsto (λ i, ∫ x in s i, f x ∂μ) l (𝓝 0) :=
hf.2.tendsto_set_integral_nhds_zero hs
/-- If `F i → f` in `L1`, then `∫ x, F i x ∂μ → ∫ x, f x ∂μ`. -/
lemma tendsto_integral_of_L1 {ι} (f : α → E) (hfi : integrable f μ)
{F : ι → α → E} {l : filter ι} (hFi : ∀ᶠ i in l, integrable (F i) μ)
(hF : tendsto (λ i, ∫⁻ x, ‖F i x - f x‖₊ ∂μ) l (𝓝 0)) :
tendsto (λ i, ∫ x, F i x ∂μ) l (𝓝 $ ∫ x, f x ∂μ) :=
tendsto_set_to_fun_of_L1 (dominated_fin_meas_additive_weighted_smul μ) f hfi hFi hF
/-- Lebesgue dominated convergence theorem provides sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their integrals.
We could weaken the condition `bound_integrable` to require `has_finite_integral bound μ` instead
(i.e. not requiring that `bound` is measurable), but in all applications proving integrability
is easier. -/
theorem tendsto_integral_of_dominated_convergence {F : ℕ → α → E} {f : α → E} (bound : α → ℝ)
(F_measurable : ∀ n, ae_strongly_measurable (F n) μ)
(bound_integrable : integrable bound μ)
(h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
(h_lim : ∀ᵐ a ∂μ, tendsto (λ n, F n a) at_top (𝓝 (f a))) :
tendsto (λn, ∫ a, F n a ∂μ) at_top (𝓝 $ ∫ a, f a ∂μ) :=
tendsto_set_to_fun_of_dominated_convergence (dominated_fin_meas_additive_weighted_smul μ) bound
F_measurable bound_integrable h_bound h_lim
/-- Lebesgue dominated convergence theorem for filters with a countable basis -/
lemma tendsto_integral_filter_of_dominated_convergence {ι} {l : filter ι}
[l.is_countably_generated]
{F : ι → α → E} {f : α → E} (bound : α → ℝ)
(hF_meas : ∀ᶠ n in l, ae_strongly_measurable (F n) μ)
(h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
(bound_integrable : integrable bound μ)
(h_lim : ∀ᵐ a ∂μ, tendsto (λ n, F n a) l (𝓝 (f a))) :
tendsto (λn, ∫ a, F n a ∂μ) l (𝓝 $ ∫ a, f a ∂μ) :=
tendsto_set_to_fun_filter_of_dominated_convergence (dominated_fin_meas_additive_weighted_smul μ)
bound hF_meas h_bound bound_integrable h_lim
/-- Lebesgue dominated convergence theorem for series. -/
lemma has_sum_integral_of_dominated_convergence {ι} [countable ι]
{F : ι → α → E} {f : α → E} (bound : ι → α → ℝ)
(hF_meas : ∀ n, ae_strongly_measurable (F n) μ)
(h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound n a)
(bound_summable : ∀ᵐ a ∂μ, summable (λ n, bound n a))
(bound_integrable : integrable (λ a, ∑' n, bound n a) μ)
(h_lim : ∀ᵐ a ∂μ, has_sum (λ n, F n a) (f a)) :
has_sum (λn, ∫ a, F n a ∂μ) (∫ a, f a ∂μ) :=
begin
have hb_nonneg : ∀ᵐ a ∂μ, ∀ n, 0 ≤ bound n a :=
eventually_countable_forall.2 (λ n, (h_bound n).mono $ λ a, (norm_nonneg _).trans),
have hb_le_tsum : ∀ n, bound n ≤ᵐ[μ] (λ a, ∑' n, bound n a),
{ intro n,
filter_upwards [hb_nonneg, bound_summable] with _ ha0 ha_sum
using le_tsum ha_sum _ (λ i _, ha0 i) },
have hF_integrable : ∀ n, integrable (F n) μ,
{ refine λ n, bound_integrable.mono' (hF_meas n) _,
exact eventually_le.trans (h_bound n) (hb_le_tsum n) },
simp only [has_sum, ← integral_finset_sum _ (λ n _, hF_integrable n)],
refine tendsto_integral_filter_of_dominated_convergence (λ a, ∑' n, bound n a) _ _
bound_integrable h_lim,
{ exact eventually_of_forall (λ s, s.ae_strongly_measurable_sum $ λ n hn, hF_meas n) },
{ refine eventually_of_forall (λ s, _),
filter_upwards [eventually_countable_forall.2 h_bound, hb_nonneg, bound_summable]
with a hFa ha0 has,
calc ‖∑ n in s, F n a‖ ≤ ∑ n in s, bound n a : norm_sum_le_of_le _ (λ n hn, hFa n)
... ≤ ∑' n, bound n a : sum_le_tsum _ (λ n hn, ha0 n) has },
end
variables {X : Type*} [topological_space X] [first_countable_topology X]
lemma continuous_within_at_of_dominated {F : X → α → E} {x₀ : X} {bound : α → ℝ} {s : set X}
(hF_meas : ∀ᶠ x in 𝓝[s] x₀, ae_strongly_measurable (F x) μ)
(h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a)
(bound_integrable : integrable bound μ)
(h_cont : ∀ᵐ a ∂μ, continuous_within_at (λ x, F x a) s x₀) :
continuous_within_at (λ x, ∫ a, F x a ∂μ) s x₀ :=
continuous_within_at_set_to_fun_of_dominated (dominated_fin_meas_additive_weighted_smul μ) hF_meas
h_bound bound_integrable h_cont
lemma continuous_at_of_dominated {F : X → α → E} {x₀ : X} {bound : α → ℝ}
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) μ)
(h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a)
(bound_integrable : integrable bound μ) (h_cont : ∀ᵐ a ∂μ, continuous_at (λ x, F x a) x₀) :
continuous_at (λ x, ∫ a, F x a ∂μ) x₀ :=
continuous_at_set_to_fun_of_dominated (dominated_fin_meas_additive_weighted_smul μ) hF_meas h_bound
bound_integrable h_cont
lemma continuous_on_of_dominated {F : X → α → E} {bound : α → ℝ} {s : set X}
(hF_meas : ∀ x ∈ s, ae_strongly_measurable (F x) μ)
(h_bound : ∀ x ∈ s, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a)
(bound_integrable : integrable bound μ)
(h_cont : ∀ᵐ a ∂μ, continuous_on (λ x, F x a) s) :
continuous_on (λ x, ∫ a, F x a ∂μ) s :=
continuous_on_set_to_fun_of_dominated (dominated_fin_meas_additive_weighted_smul μ) hF_meas
h_bound bound_integrable h_cont
lemma continuous_of_dominated {F : X → α → E} {bound : α → ℝ}
(hF_meas : ∀ x, ae_strongly_measurable (F x) μ) (h_bound : ∀ x, ∀ᵐ a ∂μ, ‖F x a‖ ≤ bound a)
(bound_integrable : integrable bound μ) (h_cont : ∀ᵐ a ∂μ, continuous (λ x, F x a)) :
continuous (λ x, ∫ a, F x a ∂μ) :=
continuous_set_to_fun_of_dominated (dominated_fin_meas_additive_weighted_smul μ) hF_meas h_bound
bound_integrable h_cont
/-- The Bochner integral of a real-valued function `f : α → ℝ` is the difference between the
integral of the positive part of `f` and the integral of the negative part of `f`. -/
lemma integral_eq_lintegral_pos_part_sub_lintegral_neg_part {f : α → ℝ} (hf : integrable f μ) :
∫ a, f a ∂μ =
ennreal.to_real (∫⁻ a, (ennreal.of_real $ f a) ∂μ) -
ennreal.to_real (∫⁻ a, (ennreal.of_real $ - f a) ∂μ) :=
let f₁ := hf.to_L1 f in
-- Go to the `L¹` space
have eq₁ : ennreal.to_real (∫⁻ a, (ennreal.of_real $ f a) ∂μ) = ‖Lp.pos_part f₁‖ :=
begin
rw L1.norm_def,
congr' 1,
apply lintegral_congr_ae,
filter_upwards [Lp.coe_fn_pos_part f₁, hf.coe_fn_to_L1] with _ h₁ h₂,
rw [h₁, h₂, ennreal.of_real],
congr' 1,
apply nnreal.eq,
rw real.nnnorm_of_nonneg (le_max_right _ _),
simp only [real.coe_to_nnreal', subtype.coe_mk],
end,
-- Go to the `L¹` space
have eq₂ : ennreal.to_real (∫⁻ a, (ennreal.of_real $ - f a) ∂μ) = ‖Lp.neg_part f₁‖ :=
begin
rw L1.norm_def,
congr' 1,
apply lintegral_congr_ae,
filter_upwards [Lp.coe_fn_neg_part f₁, hf.coe_fn_to_L1] with _ h₁ h₂,
rw [h₁, h₂, ennreal.of_real],
congr' 1,
apply nnreal.eq,
simp only [real.coe_to_nnreal', coe_nnnorm, nnnorm_neg],
rw [real.norm_of_nonpos (min_le_right _ _), ← max_neg_neg, neg_zero],
end,
begin
rw [eq₁, eq₂, integral, dif_pos],
exact L1.integral_eq_norm_pos_part_sub _
end
lemma integral_eq_lintegral_of_nonneg_ae
{f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfm : ae_strongly_measurable f μ) :
∫ a, f a ∂μ = ennreal.to_real (∫⁻ a, (ennreal.of_real $ f a) ∂μ) :=
begin
by_cases hfi : integrable f μ,
{ rw integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfi,
have h_min : ∫⁻ a, ennreal.of_real (-f a) ∂μ = 0,
{ rw lintegral_eq_zero_iff',
{ refine hf.mono _,
simp only [pi.zero_apply],
assume a h,
simp only [h, neg_nonpos, of_real_eq_zero], },
{ exact measurable_of_real.comp_ae_measurable hfm.ae_measurable.neg } },
rw [h_min, zero_to_real, _root_.sub_zero] },
{ rw integral_undef hfi,
simp_rw [integrable, hfm, has_finite_integral_iff_norm, lt_top_iff_ne_top, ne.def, true_and,
not_not] at hfi,
have : ∫⁻ (a : α), ennreal.of_real (f a) ∂μ = ∫⁻ a, (ennreal.of_real ‖f a‖) ∂μ,
{ refine lintegral_congr_ae (hf.mono $ assume a h, _),
rw [real.norm_eq_abs, abs_of_nonneg h] },
rw [this, hfi], refl }
end
lemma integral_norm_eq_lintegral_nnnorm {G} [normed_add_comm_group G]
{f : α → G} (hf : ae_strongly_measurable f μ) :
∫ x, ‖f x‖ ∂μ = ennreal.to_real ∫⁻ x, ‖f x‖₊ ∂μ :=
begin
rw integral_eq_lintegral_of_nonneg_ae _ hf.norm,
{ simp_rw [of_real_norm_eq_coe_nnnorm], },
{ refine ae_of_all _ _, simp_rw [pi.zero_apply, norm_nonneg, imp_true_iff] },
end
lemma of_real_integral_norm_eq_lintegral_nnnorm {G} [normed_add_comm_group G] {f : α → G}
(hf : integrable f μ) :
ennreal.of_real ∫ x, ‖f x‖ ∂μ = ∫⁻ x, ‖f x‖₊ ∂μ :=
by rw [integral_norm_eq_lintegral_nnnorm hf.ae_strongly_measurable,
ennreal.of_real_to_real (lt_top_iff_ne_top.mp hf.2)]
lemma integral_eq_integral_pos_part_sub_integral_neg_part {f : α → ℝ} (hf : integrable f μ) :
∫ a, f a ∂μ = (∫ a, real.to_nnreal (f a) ∂μ) - (∫ a, real.to_nnreal (-f a) ∂μ) :=
begin
rw [← integral_sub hf.real_to_nnreal],
{ simp },
{ exact hf.neg.real_to_nnreal }
end
lemma integral_nonneg_of_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ ∫ a, f a ∂μ :=
set_to_fun_nonneg (dominated_fin_meas_additive_weighted_smul μ)
(λ s _ _, weighted_smul_nonneg s) hf
lemma lintegral_coe_eq_integral (f : α → ℝ≥0) (hfi : integrable (λ x, (f x : ℝ)) μ) :
∫⁻ a, f a ∂μ = ennreal.of_real ∫ a, f a ∂μ :=
begin
simp_rw [integral_eq_lintegral_of_nonneg_ae (eventually_of_forall (λ x, (f x).coe_nonneg))
hfi.ae_strongly_measurable, ← ennreal.coe_nnreal_eq], rw [ennreal.of_real_to_real],
rw [← lt_top_iff_ne_top], convert hfi.has_finite_integral, ext1 x, rw [nnreal.nnnorm_eq]
end
lemma of_real_integral_eq_lintegral_of_real {f : α → ℝ} (hfi : integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
ennreal.of_real (∫ x, f x ∂μ) = ∫⁻ x, ennreal.of_real (f x) ∂μ :=
begin
simp_rw [integral_congr_ae (show f =ᵐ[μ] λ x, ‖f x‖,
by { filter_upwards [f_nn] with x hx,
rw [real.norm_eq_abs, abs_eq_self.mpr hx], }),
of_real_integral_norm_eq_lintegral_nnnorm hfi, ←of_real_norm_eq_coe_nnnorm],
apply lintegral_congr_ae,
filter_upwards [f_nn] with x hx,
exact congr_arg ennreal.of_real (by rw [real.norm_eq_abs, abs_eq_self.mpr hx]),
end
lemma integral_to_real {f : α → ℝ≥0∞} (hfm : ae_measurable f μ) (hf : ∀ᵐ x ∂μ, f x < ∞) :
∫ a, (f a).to_real ∂μ = (∫⁻ a, f a ∂μ).to_real :=
begin
rw [integral_eq_lintegral_of_nonneg_ae _ hfm.ennreal_to_real.ae_strongly_measurable],
{ rw lintegral_congr_ae, refine hf.mp (eventually_of_forall _),
intros x hx, rw [lt_top_iff_ne_top] at hx, simp [hx] },
{ exact (eventually_of_forall $ λ x, ennreal.to_real_nonneg) }
end
lemma lintegral_coe_le_coe_iff_integral_le {f : α → ℝ≥0} (hfi : integrable (λ x, (f x : ℝ)) μ)
{b : ℝ≥0} :
∫⁻ a, f a ∂μ ≤ b ↔ ∫ a, (f a : ℝ) ∂μ ≤ b :=
by rw [lintegral_coe_eq_integral f hfi, ennreal.of_real, ennreal.coe_le_coe,
real.to_nnreal_le_iff_le_coe]
lemma integral_coe_le_of_lintegral_coe_le {f : α → ℝ≥0} {b : ℝ≥0} (h : ∫⁻ a, f a ∂μ ≤ b) :
∫ a, (f a : ℝ) ∂μ ≤ b :=
begin
by_cases hf : integrable (λ a, (f a : ℝ)) μ,
{ exact (lintegral_coe_le_coe_iff_integral_le hf).1 h },
{ rw integral_undef hf, exact b.2 }
end
lemma integral_nonneg {f : α → ℝ} (hf : 0 ≤ f) : 0 ≤ ∫ a, f a ∂μ :=
integral_nonneg_of_ae $ eventually_of_forall hf
lemma integral_nonpos_of_ae {f : α → ℝ} (hf : f ≤ᵐ[μ] 0) : ∫ a, f a ∂μ ≤ 0 :=
begin
have hf : 0 ≤ᵐ[μ] (-f) := hf.mono (assume a h, by rwa [pi.neg_apply, pi.zero_apply, neg_nonneg]),
have : 0 ≤ ∫ a, -f a ∂μ := integral_nonneg_of_ae hf,
rwa [integral_neg, neg_nonneg] at this,
end
lemma integral_nonpos {f : α → ℝ} (hf : f ≤ 0) : ∫ a, f a ∂μ ≤ 0 :=
integral_nonpos_of_ae $ eventually_of_forall hf
lemma integral_eq_zero_iff_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : integrable f μ) :
∫ x, f x ∂μ = 0 ↔ f =ᵐ[μ] 0 :=
by simp_rw [integral_eq_lintegral_of_nonneg_ae hf hfi.1, ennreal.to_real_eq_zero_iff,
lintegral_eq_zero_iff' (ennreal.measurable_of_real.comp_ae_measurable hfi.1.ae_measurable),
← ennreal.not_lt_top, ← has_finite_integral_iff_of_real hf, hfi.2, not_true, or_false,
← hf.le_iff_eq, filter.eventually_eq, filter.eventually_le, (∘), pi.zero_apply,
ennreal.of_real_eq_zero]
lemma integral_eq_zero_iff_of_nonneg {f : α → ℝ} (hf : 0 ≤ f) (hfi : integrable f μ) :
∫ x, f x ∂μ = 0 ↔ f =ᵐ[μ] 0 :=
integral_eq_zero_iff_of_nonneg_ae (eventually_of_forall hf) hfi
lemma integral_pos_iff_support_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : integrable f μ) :
(0 < ∫ x, f x ∂μ) ↔ 0 < μ (function.support f) :=
by simp_rw [(integral_nonneg_of_ae hf).lt_iff_ne, pos_iff_ne_zero, ne.def, @eq_comm ℝ 0,
integral_eq_zero_iff_of_nonneg_ae hf hfi, filter.eventually_eq, ae_iff, pi.zero_apply,
function.support]
lemma integral_pos_iff_support_of_nonneg {f : α → ℝ} (hf : 0 ≤ f) (hfi : integrable f μ) :
(0 < ∫ x, f x ∂μ) ↔ 0 < μ (function.support f) :=
integral_pos_iff_support_of_nonneg_ae (eventually_of_forall hf) hfi
section normed_add_comm_group
variables {H : Type*} [normed_add_comm_group H]
lemma L1.norm_eq_integral_norm (f : α →₁[μ] H) : ‖f‖ = ∫ a, ‖f a‖ ∂μ :=
begin
simp only [snorm, snorm', ennreal.one_to_real, ennreal.rpow_one, Lp.norm_def,
if_false, ennreal.one_ne_top, one_ne_zero, _root_.div_one],
rw integral_eq_lintegral_of_nonneg_ae (eventually_of_forall (by simp [norm_nonneg]))
(Lp.ae_strongly_measurable f).norm,
simp [of_real_norm_eq_coe_nnnorm]
end
lemma L1.norm_of_fun_eq_integral_norm {f : α → H} (hf : integrable f μ) :
‖hf.to_L1 f‖ = ∫ a, ‖f a‖ ∂μ :=
begin
rw L1.norm_eq_integral_norm,
refine integral_congr_ae _,
apply hf.coe_fn_to_L1.mono,
intros a ha,
simp [ha]
end
lemma mem_ℒp.snorm_eq_integral_rpow_norm {f : α → H} {p : ℝ≥0∞} (hp1 : p ≠ 0) (hp2 : p ≠ ∞)
(hf : mem_ℒp f p μ) :
snorm f p μ = ennreal.of_real ((∫ a, ‖f a‖ ^ p.to_real ∂μ) ^ (p.to_real ⁻¹)) :=
begin
have A : ∫⁻ (a : α), ennreal.of_real (‖f a‖ ^ p.to_real) ∂μ = ∫⁻ (a : α), ‖f a‖₊ ^ p.to_real ∂μ,
{ apply lintegral_congr (λ x, _),
rw [← of_real_rpow_of_nonneg (norm_nonneg _) to_real_nonneg, of_real_norm_eq_coe_nnnorm] },
simp only [snorm_eq_lintegral_rpow_nnnorm hp1 hp2, one_div],
rw integral_eq_lintegral_of_nonneg_ae, rotate,
{ exact eventually_of_forall (λ x, real.rpow_nonneg_of_nonneg (norm_nonneg _) _) },
{ exact (hf.ae_strongly_measurable.norm.ae_measurable.pow_const _).ae_strongly_measurable },
rw [A, ← of_real_rpow_of_nonneg to_real_nonneg (inv_nonneg.2 to_real_nonneg), of_real_to_real],
exact (lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top hp1 hp2 hf.2).ne
end
end normed_add_comm_group
lemma integral_mono_ae {f g : α → ℝ} (hf : integrable f μ) (hg : integrable g μ) (h : f ≤ᵐ[μ] g) :
∫ a, f a ∂μ ≤ ∫ a, g a ∂μ :=
set_to_fun_mono (dominated_fin_meas_additive_weighted_smul μ) (λ s _ _, weighted_smul_nonneg s)
hf hg h
@[mono] lemma integral_mono {f g : α → ℝ} (hf : integrable f μ) (hg : integrable g μ) (h : f ≤ g) :
∫ a, f a ∂μ ≤ ∫ a, g a ∂μ :=
integral_mono_ae hf hg $ eventually_of_forall h
lemma integral_mono_of_nonneg {f g : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hgi : integrable g μ)
(h : f ≤ᵐ[μ] g) : ∫ a, f a ∂μ ≤ ∫ a, g a ∂μ :=
begin
by_cases hfm : ae_strongly_measurable f μ,
{ refine integral_mono_ae ⟨hfm, _⟩ hgi h,
refine (hgi.has_finite_integral.mono $ h.mp $ hf.mono $ λ x hf hfg, _),
simpa [abs_of_nonneg hf, abs_of_nonneg (le_trans hf hfg)] },
{ rw [integral_non_ae_strongly_measurable hfm],
exact integral_nonneg_of_ae (hf.trans h) }
end
lemma integral_mono_measure {f : α → ℝ} {ν} (hle : μ ≤ ν) (hf : 0 ≤ᵐ[ν] f) (hfi : integrable f ν) :
∫ a, f a ∂μ ≤ ∫ a, f a ∂ν :=
begin
have hfi' : integrable f μ := hfi.mono_measure hle,
have hf' : 0 ≤ᵐ[μ] f := hle.absolutely_continuous hf,
rw [integral_eq_lintegral_of_nonneg_ae hf' hfi'.1, integral_eq_lintegral_of_nonneg_ae hf hfi.1,
ennreal.to_real_le_to_real],
exacts [lintegral_mono' hle le_rfl, ((has_finite_integral_iff_of_real hf').1 hfi'.2).ne,
((has_finite_integral_iff_of_real hf).1 hfi.2).ne]
end
lemma norm_integral_le_integral_norm (f : α → E) : ‖(∫ a, f a ∂μ)‖ ≤ ∫ a, ‖f a‖ ∂μ :=
have le_ae : ∀ᵐ a ∂μ, 0 ≤ ‖f a‖ := eventually_of_forall (λa, norm_nonneg _),
classical.by_cases
( λh : ae_strongly_measurable f μ,
calc ‖∫ a, f a ∂μ‖ ≤ ennreal.to_real (∫⁻ a, (ennreal.of_real ‖f a‖) ∂μ) :
norm_integral_le_lintegral_norm _
... = ∫ a, ‖f a‖ ∂μ : (integral_eq_lintegral_of_nonneg_ae le_ae $ h.norm).symm )
( λh : ¬ae_strongly_measurable f μ,
begin
rw [integral_non_ae_strongly_measurable h, norm_zero],
exact integral_nonneg_of_ae le_ae
end )
lemma norm_integral_le_of_norm_le {f : α → E} {g : α → ℝ} (hg : integrable g μ)
(h : ∀ᵐ x ∂μ, ‖f x‖ ≤ g x) : ‖∫ x, f x ∂μ‖ ≤ ∫ x, g x ∂μ :=
calc ‖∫ x, f x ∂μ‖ ≤ ∫ x, ‖f x‖ ∂μ : norm_integral_le_integral_norm f
... ≤ ∫ x, g x ∂μ :
integral_mono_of_nonneg (eventually_of_forall $ λ x, norm_nonneg _) hg h
lemma simple_func.integral_eq_integral (f : α →ₛ E) (hfi : integrable f μ) :
f.integral μ = ∫ x, f x ∂μ :=
begin
rw [integral_eq f hfi, ← L1.simple_func.to_Lp_one_eq_to_L1,
L1.simple_func.integral_L1_eq_integral, L1.simple_func.integral_eq_integral],
exact simple_func.integral_congr hfi (Lp.simple_func.to_simple_func_to_Lp _ _).symm
end
lemma simple_func.integral_eq_sum (f : α →ₛ E) (hfi : integrable f μ) :
∫ x, f x ∂μ = ∑ x in f.range, (ennreal.to_real (μ (f ⁻¹' {x}))) • x :=
by { rw [← f.integral_eq_integral hfi, simple_func.integral, ← simple_func.integral_eq], refl, }
@[simp] lemma integral_const (c : E) : ∫ x : α, c ∂μ = (μ univ).to_real • c :=
begin
cases (@le_top _ _ _ (μ univ)).lt_or_eq with hμ hμ,
{ haveI : is_finite_measure μ := ⟨hμ⟩,
exact set_to_fun_const (dominated_fin_meas_additive_weighted_smul _) _, },
{ by_cases hc : c = 0,
{ simp [hc, integral_zero] },
{ have : ¬integrable (λ x : α, c) μ,
{ simp only [integrable_const_iff, not_or_distrib],
exact ⟨hc, hμ.not_lt⟩ },
simp [integral_undef, *] } }
end
lemma norm_integral_le_of_norm_le_const [is_finite_measure μ] {f : α → E} {C : ℝ}
(h : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
‖∫ x, f x ∂μ‖ ≤ C * (μ univ).to_real :=
calc ‖∫ x, f x ∂μ‖ ≤ ∫ x, C ∂μ : norm_integral_le_of_norm_le (integrable_const C) h
... = C * (μ univ).to_real : by rw [integral_const, smul_eq_mul, mul_comm]
lemma tendsto_integral_approx_on_of_measurable
[measurable_space E] [borel_space E]
{f : α → E} {s : set E} [separable_space s] (hfi : integrable f μ)
(hfm : measurable f) (hs : ∀ᵐ x ∂μ, f x ∈ closure s) {y₀ : E} (h₀ : y₀ ∈ s)
(h₀i : integrable (λ x, y₀) μ) :
tendsto (λ n, (simple_func.approx_on f hfm s y₀ h₀ n).integral μ) at_top (𝓝 $ ∫ x, f x ∂μ) :=
begin
have hfi' := simple_func.integrable_approx_on hfm hfi h₀ h₀i,
simp only [simple_func.integral_eq_integral _ (hfi' _)],
exact tendsto_set_to_fun_approx_on_of_measurable (dominated_fin_meas_additive_weighted_smul μ)
hfi hfm hs h₀ h₀i,
end
lemma tendsto_integral_approx_on_of_measurable_of_range_subset
[measurable_space E] [borel_space E] {f : α → E}
(fmeas : measurable f) (hf : integrable f μ) (s : set E) [separable_space s]
(hs : range f ∪ {0} ⊆ s) :
tendsto (λ n, (simple_func.approx_on f fmeas s 0 (hs $ by simp) n).integral μ) at_top
(𝓝 $ ∫ x, f x ∂μ) :=
begin
apply tendsto_integral_approx_on_of_measurable hf fmeas _ _ (integrable_zero _ _ _),
exact eventually_of_forall (λ x, subset_closure (hs (set.mem_union_left _ (mem_range_self _)))),
end
variable {ν : measure α}
lemma integral_add_measure {f : α → E} (hμ : integrable f μ) (hν : integrable f ν) :
∫ x, f x ∂(μ + ν) = ∫ x, f x ∂μ + ∫ x, f x ∂ν :=
begin
have hfi := hμ.add_measure hν,
simp_rw [integral_eq_set_to_fun],
have hμ_dfma : dominated_fin_meas_additive (μ + ν) (weighted_smul μ : set α → E →L[ℝ] E) 1,
from dominated_fin_meas_additive.add_measure_right μ ν
(dominated_fin_meas_additive_weighted_smul μ) zero_le_one,
have hν_dfma : dominated_fin_meas_additive (μ + ν) (weighted_smul ν : set α → E →L[ℝ] E) 1,
from dominated_fin_meas_additive.add_measure_left μ ν
(dominated_fin_meas_additive_weighted_smul ν) zero_le_one,
rw [← set_to_fun_congr_measure_of_add_right hμ_dfma (dominated_fin_meas_additive_weighted_smul μ)
f hfi,
← set_to_fun_congr_measure_of_add_left hν_dfma (dominated_fin_meas_additive_weighted_smul ν)
f hfi],
refine set_to_fun_add_left' _ _ _ (λ s hs hμνs, _) f,
rw [measure.coe_add, pi.add_apply, add_lt_top] at hμνs,
rw [weighted_smul, weighted_smul, weighted_smul, ← add_smul, measure.coe_add, pi.add_apply,
to_real_add hμνs.1.ne hμνs.2.ne],
end
@[simp] lemma integral_zero_measure {m : measurable_space α} (f : α → E) :
∫ x, f x ∂(0 : measure α) = 0 :=
set_to_fun_measure_zero (dominated_fin_meas_additive_weighted_smul _) rfl
theorem integral_finset_sum_measure {ι} {m : measurable_space α} {f : α → E}
{μ : ι → measure α} {s : finset ι} (hf : ∀ i ∈ s, integrable f (μ i)) :
∫ a, f a ∂(∑ i in s, μ i) = ∑ i in s, ∫ a, f a ∂μ i :=
begin
classical,
refine finset.induction_on' s _ _, -- `induction s using finset.induction_on'` fails
{ simp },
{ intros i t hi ht hit iht,
simp only [finset.sum_insert hit, ← iht],
exact integral_add_measure (hf _ hi) (integrable_finset_sum_measure.2 $ λ j hj, hf j (ht hj)) }
end
lemma nndist_integral_add_measure_le_lintegral (h₁ : integrable f μ) (h₂ : integrable f ν) :
(nndist (∫ x, f x ∂μ) (∫ x, f x ∂(μ + ν)) : ℝ≥0∞) ≤ ∫⁻ x, ‖f x‖₊ ∂ν :=
begin
rw [integral_add_measure h₁ h₂, nndist_comm, nndist_eq_nnnorm, add_sub_cancel'],
exact ennnorm_integral_le_lintegral_ennnorm _
end
theorem has_sum_integral_measure {ι} {m : measurable_space α} {f : α → E} {μ : ι → measure α}
(hf : integrable f (measure.sum μ)) :
has_sum (λ i, ∫ a, f a ∂μ i) (∫ a, f a ∂measure.sum μ) :=
begin
have hfi : ∀ i, integrable f (μ i) := λ i, hf.mono_measure (measure.le_sum _ _),
simp only [has_sum, ← integral_finset_sum_measure (λ i _, hfi i)],
refine metric.nhds_basis_ball.tendsto_right_iff.mpr (λ ε ε0, _),
lift ε to ℝ≥0 using ε0.le,
have hf_lt : ∫⁻ x, ‖f x‖₊ ∂(measure.sum μ) < ∞ := hf.2,
have hmem : ∀ᶠ y in 𝓝 ∫⁻ x, ‖f x‖₊ ∂(measure.sum μ), ∫⁻ x, ‖f x‖₊ ∂(measure.sum μ) < y + ε,
{ refine tendsto_id.add tendsto_const_nhds (lt_mem_nhds $ ennreal.lt_add_right _ _),
exacts [hf_lt.ne, ennreal.coe_ne_zero.2 (nnreal.coe_ne_zero.1 ε0.ne')] },
refine ((has_sum_lintegral_measure (λ x, ‖f x‖₊) μ).eventually hmem).mono (λ s hs, _),
obtain ⟨ν, hν⟩ : ∃ ν, (∑ i in s, μ i) + ν = measure.sum μ,
{ refine ⟨measure.sum (λ i : ↥(sᶜ : set ι), μ i), _⟩,
simpa only [← measure.sum_coe_finset] using measure.sum_add_sum_compl (s : set ι) μ },
rw [metric.mem_ball, ← coe_nndist, nnreal.coe_lt_coe, ← ennreal.coe_lt_coe, ← hν],
rw [← hν, integrable_add_measure] at hf,
refine (nndist_integral_add_measure_le_lintegral hf.1 hf.2).trans_lt _,
rw [← hν, lintegral_add_measure, lintegral_finset_sum_measure] at hs,
exact lt_of_add_lt_add_left hs
end
theorem integral_sum_measure {ι} {m : measurable_space α} {f : α → E} {μ : ι → measure α}
(hf : integrable f (measure.sum μ)) :
∫ a, f a ∂measure.sum μ = ∑' i, ∫ a, f a ∂μ i :=
(has_sum_integral_measure hf).tsum_eq.symm
@[simp] lemma integral_smul_measure (f : α → E) (c : ℝ≥0∞) :
∫ x, f x ∂(c • μ) = c.to_real • ∫ x, f x ∂μ :=
begin
-- First we consider the “degenerate” case `c = ∞`
rcases eq_or_ne c ∞ with rfl|hc,
{ rw [ennreal.top_to_real, zero_smul, integral_eq_set_to_fun, set_to_fun_top_smul_measure], },
-- Main case: `c ≠ ∞`
simp_rw [integral_eq_set_to_fun, ← set_to_fun_smul_left],
have hdfma :
dominated_fin_meas_additive μ (weighted_smul (c • μ) : set α → E →L[ℝ] E) c.to_real,
from mul_one c.to_real
▸ (dominated_fin_meas_additive_weighted_smul (c • μ)).of_smul_measure c hc,
have hdfma_smul := (dominated_fin_meas_additive_weighted_smul (c • μ)),
rw ← set_to_fun_congr_smul_measure c hc hdfma hdfma_smul f,
exact set_to_fun_congr_left' _ _ (λ s hs hμs, weighted_smul_smul_measure μ c) f,
end
lemma integral_map_of_strongly_measurable {β} [measurable_space β] {φ : α → β} (hφ : measurable φ)
{f : β → E} (hfm : strongly_measurable f) :
∫ y, f y ∂(measure.map φ μ) = ∫ x, f (φ x) ∂μ :=
begin
by_cases hfi : integrable f (measure.map φ μ), swap,
{ rw [integral_undef hfi, integral_undef],
rwa [← integrable_map_measure hfm.ae_strongly_measurable hφ.ae_measurable] },
borelize E,
haveI : separable_space (range f ∪ {0} : set E) := hfm.separable_space_range_union_singleton,
refine tendsto_nhds_unique
(tendsto_integral_approx_on_of_measurable_of_range_subset hfm.measurable hfi _ subset.rfl) _,
convert tendsto_integral_approx_on_of_measurable_of_range_subset (hfm.measurable.comp hφ)
((integrable_map_measure hfm.ae_strongly_measurable hφ.ae_measurable).1 hfi) (range f ∪ {0})
(by simp [insert_subset_insert, set.range_comp_subset_range]) using 1,
ext1 i,
simp only [simple_func.approx_on_comp, simple_func.integral_eq, measure.map_apply, hφ,
simple_func.measurable_set_preimage, ← preimage_comp, simple_func.coe_comp],
refine (finset.sum_subset (simple_func.range_comp_subset_range _ hφ) (λ y _ hy, _)).symm,
rw [simple_func.mem_range, ← set.preimage_singleton_eq_empty, simple_func.coe_comp] at hy,
rw [hy],
simp,
end
lemma integral_map {β} [measurable_space β] {φ : α → β} (hφ : ae_measurable φ μ)
{f : β → E} (hfm : ae_strongly_measurable f (measure.map φ μ)) :
∫ y, f y ∂(measure.map φ μ) = ∫ x, f (φ x) ∂μ :=
let g := hfm.mk f in calc
∫ y, f y ∂(measure.map φ μ) = ∫ y, g y ∂(measure.map φ μ) : integral_congr_ae hfm.ae_eq_mk
... = ∫ y, g y ∂(measure.map (hφ.mk φ) μ) :
by { congr' 1, exact measure.map_congr hφ.ae_eq_mk }
... = ∫ x, g (hφ.mk φ x) ∂μ :
integral_map_of_strongly_measurable hφ.measurable_mk hfm.strongly_measurable_mk
... = ∫ x, g (φ x) ∂μ : integral_congr_ae (hφ.ae_eq_mk.symm.fun_comp _)
... = ∫ x, f (φ x) ∂μ : integral_congr_ae $ ae_eq_comp hφ (hfm.ae_eq_mk).symm
lemma _root_.measurable_embedding.integral_map {β} {_ : measurable_space β} {f : α → β}
(hf : measurable_embedding f) (g : β → E) :
∫ y, g y ∂(measure.map f μ) = ∫ x, g (f x) ∂μ :=
begin
by_cases hgm : ae_strongly_measurable g (measure.map f μ),
{ exact integral_map hf.measurable.ae_measurable hgm },
{ rw [integral_non_ae_strongly_measurable hgm, integral_non_ae_strongly_measurable],
rwa ← hf.ae_strongly_measurable_map_iff }
end
lemma _root_.closed_embedding.integral_map {β} [topological_space α] [borel_space α]
[topological_space β] [measurable_space β] [borel_space β]
{φ : α → β} (hφ : closed_embedding φ) (f : β → E) :
∫ y, f y ∂(measure.map φ μ) = ∫ x, f (φ x) ∂μ :=
hφ.measurable_embedding.integral_map _
lemma integral_map_equiv {β} [measurable_space β] (e : α ≃ᵐ β) (f : β → E) :
∫ y, f y ∂(measure.map e μ) = ∫ x, f (e x) ∂μ :=
e.measurable_embedding.integral_map f
lemma measure_preserving.integral_comp {β} {_ : measurable_space β} {f : α → β} {ν}
(h₁ : measure_preserving f μ ν) (h₂ : measurable_embedding f) (g : β → E) :
∫ x, g (f x) ∂μ = ∫ y, g y ∂ν :=
h₁.map_eq ▸ (h₂.integral_map g).symm
lemma set_integral_eq_subtype {α} [measure_space α] {s : set α} (hs : measurable_set s)
(f : α → E) :
∫ x in s, f x = ∫ x : s, f x :=
by { rw ← map_comap_subtype_coe hs, exact (measurable_embedding.subtype_coe hs).integral_map _ }
@[simp] lemma integral_dirac' [measurable_space α] (f : α → E) (a : α)
(hfm : strongly_measurable f) :
∫ x, f x ∂(measure.dirac a) = f a :=
begin
borelize E,
calc ∫ x, f x ∂(measure.dirac a) = ∫ x, f a ∂(measure.dirac a) :
integral_congr_ae $ ae_eq_dirac' hfm.measurable
... = f a : by simp [measure.dirac_apply_of_mem]
end
@[simp] lemma integral_dirac [measurable_space α] [measurable_singleton_class α]
(f : α → E) (a : α) : ∫ x, f x ∂(measure.dirac a) = f a :=
calc ∫ x, f x ∂(measure.dirac a) = ∫ x, f a ∂(measure.dirac a) :
integral_congr_ae $ ae_eq_dirac f
... = f a : by simp [measure.dirac_apply_of_mem]
lemma mul_meas_ge_le_integral_of_nonneg [is_finite_measure μ] {f : α → ℝ} (hf_nonneg : 0 ≤ f)
(hf_int : integrable f μ) (ε : ℝ) :
ε * (μ {x | ε ≤ f x}).to_real ≤ ∫ x, f x ∂μ :=
begin
cases lt_or_le ε 0 with hε hε,
{ exact (mul_nonpos_of_nonpos_of_nonneg hε.le ennreal.to_real_nonneg).trans
(integral_nonneg hf_nonneg), },
rw [integral_eq_lintegral_of_nonneg_ae (eventually_of_forall (λ x, hf_nonneg x))
hf_int.ae_strongly_measurable, ← ennreal.to_real_of_real hε, ← ennreal.to_real_mul],
have : {x : α | (ennreal.of_real ε).to_real ≤ f x}
= {x : α | ennreal.of_real ε ≤ (λ x, ennreal.of_real (f x)) x},
{ ext1 x,
rw [set.mem_set_of_eq, set.mem_set_of_eq, ← ennreal.to_real_of_real (hf_nonneg x)],
exact ennreal.to_real_le_to_real ennreal.of_real_ne_top ennreal.of_real_ne_top, },
rw this,
have h_meas : ae_measurable (λ x, ennreal.of_real (f x)) μ,
from measurable_id'.ennreal_of_real.comp_ae_measurable hf_int.ae_measurable,
have h_mul_meas_le := @mul_meas_ge_le_lintegral₀ _ _ μ _ h_meas (ennreal.of_real ε),
rw ennreal.to_real_le_to_real _ _,
{ exact h_mul_meas_le, },
{ simp only [ne.def, with_top.mul_eq_top_iff, ennreal.of_real_eq_zero, not_le,
ennreal.of_real_ne_top, false_and, or_false, not_and],
exact λ _, measure_ne_top _ _, },
{ have h_lt_top : ∫⁻ a, ‖f a‖₊ ∂μ < ∞ := hf_int.has_finite_integral,
simp_rw [← of_real_norm_eq_coe_nnnorm, real.norm_eq_abs] at h_lt_top,
convert h_lt_top.ne,
ext1 x,
rw abs_of_nonneg (hf_nonneg x), },
end
/-- Hölder's inequality for the integral of a product of norms. The integral of the product of two
norms of functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are
conjugate exponents. -/
theorem integral_mul_norm_le_Lp_mul_Lq {E} [normed_add_comm_group E] {f g : α → E}
{p q : ℝ} (hpq : p.is_conjugate_exponent q)
(hf : mem_ℒp f (ennreal.of_real p) μ) (hg : mem_ℒp g (ennreal.of_real q) μ) :
∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1/p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1/q) :=
begin
-- translate the Bochner integrals into Lebesgue integrals.
rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae,
integral_eq_lintegral_of_nonneg_ae],
rotate 1,
{ exact eventually_of_forall (λ x, real.rpow_nonneg_of_nonneg (norm_nonneg _) _), },
{ exact (hg.1.norm.ae_measurable.pow ae_measurable_const).ae_strongly_measurable, },
{ exact eventually_of_forall (λ x, real.rpow_nonneg_of_nonneg (norm_nonneg _) _),},
{ exact (hf.1.norm.ae_measurable.pow ae_measurable_const).ae_strongly_measurable, },
{ exact eventually_of_forall (λ x, mul_nonneg (norm_nonneg _) (norm_nonneg _)), },
{ exact hf.1.norm.mul hg.1.norm, },
rw [ennreal.to_real_rpow, ennreal.to_real_rpow, ← ennreal.to_real_mul],
-- replace norms by nnnorm
have h_left : ∫⁻ a, ennreal.of_real (‖f a‖ * ‖g a‖) ∂μ
= ∫⁻ a, ((λ x, (‖f x‖₊ : ℝ≥0∞)) * (λ x, ‖g x‖₊)) a ∂μ,
{ simp_rw [pi.mul_apply, ← of_real_norm_eq_coe_nnnorm, ennreal.of_real_mul (norm_nonneg _)], },
have h_right_f : ∫⁻ a, ennreal.of_real (‖f a‖ ^ p) ∂μ = ∫⁻ a, ‖f a‖₊ ^ p ∂μ,
{ refine lintegral_congr (λ x, _),
rw [← of_real_norm_eq_coe_nnnorm, ennreal.of_real_rpow_of_nonneg (norm_nonneg _) hpq.nonneg], },
have h_right_g : ∫⁻ a, ennreal.of_real (‖g a‖ ^ q) ∂μ = ∫⁻ a, ‖g a‖₊ ^ q ∂μ,
{ refine lintegral_congr (λ x, _),
rw [← of_real_norm_eq_coe_nnnorm,
ennreal.of_real_rpow_of_nonneg (norm_nonneg _) hpq.symm.nonneg], },
rw [h_left, h_right_f, h_right_g],
-- we can now apply `ennreal.lintegral_mul_le_Lp_mul_Lq` (up to the `to_real` application)
refine ennreal.to_real_mono _ _,
{ refine ennreal.mul_ne_top _ _,
{ convert hf.snorm_ne_top,
rw snorm_eq_lintegral_rpow_nnnorm,
{ rw ennreal.to_real_of_real hpq.nonneg, },
{ rw [ne.def, ennreal.of_real_eq_zero, not_le],
exact hpq.pos, },
{ exact ennreal.coe_ne_top, }, },
{ convert hg.snorm_ne_top,
rw snorm_eq_lintegral_rpow_nnnorm,
{ rw ennreal.to_real_of_real hpq.symm.nonneg, },
{ rw [ne.def, ennreal.of_real_eq_zero, not_le],
exact hpq.symm.pos, },
{ exact ennreal.coe_ne_top, }, }, },
{ exact ennreal.lintegral_mul_le_Lp_mul_Lq μ hpq hf.1.nnnorm.ae_measurable.coe_nnreal_ennreal
hg.1.nnnorm.ae_measurable.coe_nnreal_ennreal, },
end
/-- Hölder's inequality for functions `α → ℝ`. The integral of the product of two nonnegative
functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are conjugate
exponents. -/
theorem integral_mul_le_Lp_mul_Lq_of_nonneg {p q : ℝ}
(hpq : p.is_conjugate_exponent q) {f g : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f) (hg_nonneg : 0 ≤ᵐ[μ] g)
(hf : mem_ℒp f (ennreal.of_real p) μ) (hg : mem_ℒp g (ennreal.of_real q) μ) :
∫ a, f a * g a ∂μ ≤ (∫ a, (f a) ^ p ∂μ) ^ (1/p) * (∫ a, (g a) ^ q ∂μ) ^ (1/q) :=
begin
have h_left : ∫ a, f a * g a ∂μ = ∫ a, ‖f a‖ * ‖g a‖ ∂μ,
{ refine integral_congr_ae _,
filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg,
rw [real.norm_of_nonneg hxf, real.norm_of_nonneg hxg], },
have h_right_f : ∫ a, (f a) ^ p ∂μ = ∫ a, ‖f a‖ ^ p ∂μ,
{ refine integral_congr_ae _,
filter_upwards [hf_nonneg] with x hxf,
rw real.norm_of_nonneg hxf, },
have h_right_g : ∫ a, (g a) ^ q ∂μ = ∫ a, ‖g a‖ ^ q ∂μ,
{ refine integral_congr_ae _,
filter_upwards [hg_nonneg] with x hxg,
rw real.norm_of_nonneg hxg, },
rw [h_left, h_right_f, h_right_g],
exact integral_mul_norm_le_Lp_mul_Lq hpq hf hg,
end
end properties
mk_simp_attribute integral_simps "Simp set for integral rules."
attribute [integral_simps] integral_neg integral_smul L1.integral_add L1.integral_sub
L1.integral_smul L1.integral_neg
attribute [irreducible] integral L1.integral
section integral_trim
variables {H β γ : Type*} [normed_add_comm_group H]
{m m0 : measurable_space β} {μ : measure β}
/-- Simple function seen as simple function of a larger `measurable_space`. -/
def simple_func.to_larger_space (hm : m ≤ m0) (f : @simple_func β m γ) : simple_func β γ :=
⟨@simple_func.to_fun β m γ f, λ x, hm _ (@simple_func.measurable_set_fiber β γ m f x),
@simple_func.finite_range β γ m f⟩
lemma simple_func.coe_to_larger_space_eq (hm : m ≤ m0) (f : @simple_func β m γ) :
⇑(f.to_larger_space hm) = f :=
rfl
lemma integral_simple_func_larger_space (hm : m ≤ m0) (f : @simple_func β m F)
(hf_int : integrable f μ) :
∫ x, f x ∂μ = ∑ x in (@simple_func.range β F m f), (ennreal.to_real (μ (f ⁻¹' {x}))) • x :=
begin
simp_rw ← f.coe_to_larger_space_eq hm,
have hf_int : integrable (f.to_larger_space hm) μ, by rwa simple_func.coe_to_larger_space_eq,
rw simple_func.integral_eq_sum _ hf_int,
congr,
end
lemma integral_trim_simple_func (hm : m ≤ m0) (f : @simple_func β m F) (hf_int : integrable f μ) :
∫ x, f x ∂μ = ∫ x, f x ∂(μ.trim hm) :=
begin
have hf : strongly_measurable[m] f, from @simple_func.strongly_measurable β F m _ f,
have hf_int_m := hf_int.trim hm hf,
rw [integral_simple_func_larger_space (le_refl m) f hf_int_m,
integral_simple_func_larger_space hm f hf_int],
congr' with x,
congr,
exact (trim_measurable_set_eq hm (@simple_func.measurable_set_fiber β F m f x)).symm,
end
lemma integral_trim (hm : m ≤ m0) {f : β → F} (hf : strongly_measurable[m] f) :
∫ x, f x ∂μ = ∫ x, f x ∂(μ.trim hm) :=
begin
borelize F,
by_cases hf_int : integrable f μ,
swap,
{ have hf_int_m : ¬ integrable f (μ.trim hm),
from λ hf_int_m, hf_int (integrable_of_integrable_trim hm hf_int_m),
rw [integral_undef hf_int, integral_undef hf_int_m], },
haveI : separable_space (range f ∪ {0} : set F) := hf.separable_space_range_union_singleton,
let f_seq := @simple_func.approx_on F β _ _ _ m _ hf.measurable (range f ∪ {0}) 0 (by simp) _,
have hf_seq_meas : ∀ n, strongly_measurable[m] (f_seq n),
from λ n, @simple_func.strongly_measurable β F m _ (f_seq n),
have hf_seq_int : ∀ n, integrable (f_seq n) μ,
from simple_func.integrable_approx_on_range (hf.mono hm).measurable hf_int,
have hf_seq_int_m : ∀ n, integrable (f_seq n) (μ.trim hm),
from λ n, (hf_seq_int n).trim hm (hf_seq_meas n) ,
have hf_seq_eq : ∀ n, ∫ x, f_seq n x ∂μ = ∫ x, f_seq n x ∂(μ.trim hm),
from λ n, integral_trim_simple_func hm (f_seq n) (hf_seq_int n),
have h_lim_1 : at_top.tendsto (λ n, ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂μ)),
{ refine tendsto_integral_of_L1 f hf_int (eventually_of_forall hf_seq_int) _,
exact simple_func.tendsto_approx_on_range_L1_nnnorm (hf.mono hm).measurable hf_int, },
have h_lim_2 : at_top.tendsto (λ n, ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂(μ.trim hm))),
{ simp_rw hf_seq_eq,
refine @tendsto_integral_of_L1 β F _ _ _ m (μ.trim hm) _ f
(hf_int.trim hm hf) _ _ (eventually_of_forall hf_seq_int_m) _,
exact @simple_func.tendsto_approx_on_range_L1_nnnorm β F m _ _ _ f _ _
hf.measurable (hf_int.trim hm hf), },
exact tendsto_nhds_unique h_lim_1 h_lim_2,
end
lemma integral_trim_ae (hm : m ≤ m0) {f : β → F} (hf : ae_strongly_measurable f (μ.trim hm)) :
∫ x, f x ∂μ = ∫ x, f x ∂(μ.trim hm) :=
begin
rw [integral_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk), integral_congr_ae hf.ae_eq_mk],
exact integral_trim hm hf.strongly_measurable_mk,
end
lemma ae_eq_trim_of_strongly_measurable
[topological_space γ] [metrizable_space γ] (hm : m ≤ m0) {f g : β → γ}
(hf : strongly_measurable[m] f) (hg : strongly_measurable[m] g) (hfg : f =ᵐ[μ] g) :
f =ᵐ[μ.trim hm] g :=
begin
rwa [eventually_eq, ae_iff, trim_measurable_set_eq hm _],
exact (hf.measurable_set_eq_fun hg).compl
end
lemma ae_eq_trim_iff [topological_space γ] [metrizable_space γ]
(hm : m ≤ m0) {f g : β → γ} (hf : strongly_measurable[m] f) (hg : strongly_measurable[m] g) :
f =ᵐ[μ.trim hm] g ↔ f =ᵐ[μ] g :=
⟨ae_eq_of_ae_eq_trim, ae_eq_trim_of_strongly_measurable hm hf hg⟩
lemma ae_le_trim_of_strongly_measurable
[linear_order γ] [topological_space γ] [order_closed_topology γ] [pseudo_metrizable_space γ]
(hm : m ≤ m0) {f g : β → γ} (hf : strongly_measurable[m] f) (hg : strongly_measurable[m] g)
(hfg : f ≤ᵐ[μ] g) :
f ≤ᵐ[μ.trim hm] g :=
begin
rwa [eventually_le, ae_iff, trim_measurable_set_eq hm _],
exact (hf.measurable_set_le hg).compl,
end
lemma ae_le_trim_iff
[linear_order γ] [topological_space γ] [order_closed_topology γ] [pseudo_metrizable_space γ]
(hm : m ≤ m0) {f g : β → γ} (hf : strongly_measurable[m] f) (hg : strongly_measurable[m] g) :
f ≤ᵐ[μ.trim hm] g ↔ f ≤ᵐ[μ] g :=
⟨ae_le_of_ae_le_trim, ae_le_trim_of_strongly_measurable hm hf hg⟩
end integral_trim
section snorm_bound
variables {m0 : measurable_space α} {μ : measure α}
lemma snorm_one_le_of_le {r : ℝ≥0} {f : α → ℝ}
(hfint : integrable f μ) (hfint' : 0 ≤ ∫ x, f x ∂μ) (hf : ∀ᵐ ω ∂μ, f ω ≤ r) :
snorm f 1 μ ≤ 2 * μ set.univ * r :=
begin
by_cases hr : r = 0,
{ suffices : f =ᵐ[μ] 0,
{ rw [snorm_congr_ae this, snorm_zero, hr, ennreal.coe_zero, mul_zero],
exact le_rfl },
rw [hr, nonneg.coe_zero] at hf,
have hnegf : ∫ x, -f x ∂μ = 0,
{ rw [integral_neg, neg_eq_zero],
exact le_antisymm (integral_nonpos_of_ae hf) hfint' },
have := (integral_eq_zero_iff_of_nonneg_ae _ hfint.neg).1 hnegf,
{ filter_upwards [this] with ω hω,
rwa [pi.neg_apply, pi.zero_apply, neg_eq_zero] at hω },
{ filter_upwards [hf] with ω hω,
rwa [pi.zero_apply, pi.neg_apply, right.nonneg_neg_iff] } },
by_cases hμ : is_finite_measure μ,
swap,
{ have : μ set.univ = ∞,
{ by_contra hμ',
exact hμ (is_finite_measure.mk $ lt_top_iff_ne_top.2 hμ') },
rw [this, ennreal.mul_top, if_neg, ennreal.top_mul, if_neg],
{ exact le_top },
{ simp [hr] },
{ norm_num } },
haveI := hμ,
rw [integral_eq_integral_pos_part_sub_integral_neg_part hfint, sub_nonneg] at hfint',
have hposbdd : ∫ ω, max (f ω) 0 ∂μ ≤ (μ set.univ).to_real • r,
{ rw ← integral_const,
refine integral_mono_ae hfint.real_to_nnreal (integrable_const r) _,
filter_upwards [hf] with ω hω using real.to_nnreal_le_iff_le_coe.2 hω },
rw [mem_ℒp.snorm_eq_integral_rpow_norm one_ne_zero ennreal.one_ne_top
(mem_ℒp_one_iff_integrable.2 hfint),
ennreal.of_real_le_iff_le_to_real (ennreal.mul_ne_top
(ennreal.mul_ne_top ennreal.two_ne_top $ @measure_ne_top _ _ _ hμ _) ennreal.coe_ne_top)],
simp_rw [ennreal.one_to_real, _root_.inv_one, real.rpow_one, real.norm_eq_abs,
← max_zero_add_max_neg_zero_eq_abs_self, ← real.coe_to_nnreal'],
rw integral_add hfint.real_to_nnreal,
{ simp only [real.coe_to_nnreal', ennreal.to_real_mul, ennreal.to_real_bit0,
ennreal.one_to_real, ennreal.coe_to_real] at hfint' ⊢,
refine (add_le_add_left hfint' _).trans _,
rwa [← two_mul, mul_assoc, mul_le_mul_left (two_pos : (0 : ℝ) < 2)] },
{ exact hfint.neg.sup (integrable_zero _ _ μ) }
end
lemma snorm_one_le_of_le' {r : ℝ} {f : α → ℝ}
(hfint : integrable f μ) (hfint' : 0 ≤ ∫ x, f x ∂μ) (hf : ∀ᵐ ω ∂μ, f ω ≤ r) :
snorm f 1 μ ≤ 2 * μ set.univ * ennreal.of_real r :=
begin
refine snorm_one_le_of_le hfint hfint' _,
simp only [real.coe_to_nnreal', le_max_iff],
filter_upwards [hf] with ω hω using or.inl hω,
end
end snorm_bound
end measure_theory
|
cd613ad6ba24f16b85faf28e572c8a1066afa642 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /src/Lean/Meta/Tactic/Simp/SimpTheorems.lean | 87cea99d203cacfe3d224953ad1c6fdd1dfa42c9 | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 20,504 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.ScopedEnvExtension
import Lean.Util.Recognizers
import Lean.Meta.DiscrTree
import Lean.Meta.AppBuilder
import Lean.Meta.Eqns
import Lean.Meta.Tactic.AuxLemma
import Lean.DocString
namespace Lean.Meta
/--
An `Origin` is an identifier for simp theorems which indicates roughly
what action the user took which lead to this theorem existing in the simp set.
-/
inductive Origin where
/-- A global declaration in the environment. -/
| decl (declName : Name)
/--
A local hypothesis.
When `contextual := true` is enabled, this fvar may exist in an extension
of the current local context; it will not be used for rewriting by simp once
it is out of scope but it may end up in the `usedSimps` trace.
-/
| fvar (fvarId : FVarId)
/--
A proof term provided directly to a call to `simp [ref, ...]` where `ref`
is the provided simp argument (of kind `Parser.Tactic.simpLemma`).
The `id` is a unique identifier for the call.
-/
| stx (id : Name) (ref : Syntax)
/--
Some other origin. `name` should not collide with the other types
for erasure to work correctly, and simp trace will ignore this lemma.
The other origins should be preferred if possible.
-/
| other (name : Name)
deriving Inhabited, Repr
/-- A unique identifier corresponding to the origin. -/
def Origin.key : Origin → Name
| .decl declName => declName
| .fvar fvarId => fvarId.name
| .stx id _ => id
| .other name => name
instance : BEq Origin := ⟨(·.key == ·.key)⟩
instance : Hashable Origin := ⟨(hash ·.key)⟩
/-
Note: we want to use iota reduction when indexing instaces. Otherwise,
we cannot use simp theorems such as
```
@[simp] theorem liftOn_mk (a : α) (f : α → γ) (h : ∀ a₁ a₂, r a₁ a₂ → f a₁ = f a₂) :
Quot.liftOn (Quot.mk r a) f h = f a := rfl
```
If we use `iota`, then the lhs is reduced to `f a`.
See comment at `DiscrTree`.
-/
abbrev SimpTheoremKey := DiscrTree.Key (simpleReduce := true)
/--
The fields `levelParams` and `proof` are used to encode the proof of the simp theorem.
If the `proof` is a global declaration `c`, we store `Expr.const c []` at `proof` without the universe levels, and `levelParams` is set to `#[]`
When using the lemma, we create fresh universe metavariables.
Motivation: most simp theorems are global declarations, and this approach is faster and saves memory.
The field `levelParams` is not empty only when we elaborate an expression provided by the user, and it contains universe metavariables.
Then, we use `abstractMVars` to abstract the universe metavariables and create new fresh universe parameters that are stored at the field `levelParams`.
-/
structure SimpTheorem where
keys : Array SimpTheoremKey := #[]
/--
It stores universe parameter names for universe polymorphic proofs.
Recall that it is non-empty only when we elaborate an expression provided by the user.
When `proof` is just a constant, we can use the universe parameter names stored in the declaration.
-/
levelParams : Array Name := #[]
proof : Expr
priority : Nat := eval_prio default
post : Bool := true
/-- `perm` is true if lhs and rhs are identical modulo permutation of variables. -/
perm : Bool := false
/--
`origin` is mainly relevant for producing trace messages.
It is also viewed an `id` used to "erase" `simp` theorems from `SimpTheorems`.
-/
origin : Origin
/-- `rfl` is true if `proof` is by `Eq.refl` or `rfl`. -/
rfl : Bool
deriving Inhabited
mutual
partial def isRflProofCore (type : Expr) (proof : Expr) : CoreM Bool := do
match type with
| .forallE _ _ type _ =>
if let .lam _ _ proof _ := proof then
isRflProofCore type proof
else
return false
| _ =>
if type.isAppOfArity ``Eq 3 then
if proof.isAppOfArity ``Eq.refl 2 || proof.isAppOfArity ``rfl 2 then
return true
else if proof.isAppOfArity ``Eq.symm 4 then
-- `Eq.symm` of rfl theorem is a rfl theorem
isRflProofCore type proof.appArg! -- small hack: we don't need to set the exact type
else if proof.isApp && proof.getAppFn.isConst then
-- The application of a `rfl` theorem is a `rfl` theorem
isRflTheorem proof.getAppFn.constName!
else
return false
else
return false
partial def isRflTheorem (declName : Name) : CoreM Bool := do
let .thmInfo info ← getConstInfo declName | return false
isRflProofCore info.type info.value
end
def isRflProof (proof : Expr) : MetaM Bool := do
if let .const declName .. := proof then
isRflTheorem declName
else
isRflProofCore (← inferType proof) proof
instance : ToFormat SimpTheorem where
format s :=
let perm := if s.perm then ":perm" else ""
let name := format s.origin.key
let prio := f!":{s.priority}"
name ++ prio ++ perm
def ppOrigin [Monad m] [MonadEnv m] [MonadError m] : Origin → m MessageData
| .decl n => mkConstWithLevelParams n
| .fvar n => return mkFVar n
| .stx _ ref => return ref
| .other n => return n
def ppSimpTheorem [Monad m] [MonadLiftT IO m] [MonadEnv m] [MonadError m] (s : SimpTheorem) : m MessageData := do
let perm := if s.perm then ":perm" else ""
let name ← ppOrigin s.origin
let prio := m!":{s.priority}"
return name ++ prio ++ perm
instance : BEq SimpTheorem where
beq e₁ e₂ := e₁.proof == e₂.proof
abbrev SimpTheoremTree := DiscrTree SimpTheorem (simpleReduce := true)
structure SimpTheorems where
pre : SimpTheoremTree := DiscrTree.empty
post : SimpTheoremTree := DiscrTree.empty
lemmaNames : PHashSet Origin := {}
toUnfold : PHashSet Name := {}
erased : PHashSet Origin := {}
toUnfoldThms : PHashMap Name (Array Name) := {}
deriving Inhabited
def addSimpTheoremEntry (d : SimpTheorems) (e : SimpTheorem) : SimpTheorems :=
if e.post then
{ d with post := d.post.insertCore e.keys e, lemmaNames := updateLemmaNames d.lemmaNames }
else
{ d with pre := d.pre.insertCore e.keys e, lemmaNames := updateLemmaNames d.lemmaNames }
where
updateLemmaNames (s : PHashSet Origin) : PHashSet Origin :=
s.insert e.origin
def SimpTheorems.addDeclToUnfoldCore (d : SimpTheorems) (declName : Name) : SimpTheorems :=
{ d with toUnfold := d.toUnfold.insert declName }
/-- Return `true` if `declName` is tagged to be unfolded using `unfoldDefinition?` (i.e., without using equational theorems). -/
def SimpTheorems.isDeclToUnfold (d : SimpTheorems) (declName : Name) : Bool :=
d.toUnfold.contains declName
def SimpTheorems.isLemma (d : SimpTheorems) (thmId : Origin) : Bool :=
d.lemmaNames.contains thmId
/-- Register the equational theorems for the given definition. -/
def SimpTheorems.registerDeclToUnfoldThms (d : SimpTheorems) (declName : Name) (eqThms : Array Name) : SimpTheorems :=
{ d with toUnfoldThms := d.toUnfoldThms.insert declName eqThms }
partial def SimpTheorems.eraseCore (d : SimpTheorems) (thmId : Origin) : SimpTheorems :=
let d := { d with erased := d.erased.insert thmId, lemmaNames := d.lemmaNames.erase thmId }
if let .decl declName := thmId then
let d := { d with toUnfold := d.toUnfold.erase declName }
if let some thms := d.toUnfoldThms.find? declName then
thms.foldl (init := d) (eraseCore · <| .decl ·)
else
d
else
d
def SimpTheorems.erase [Monad m] [MonadError m] (d : SimpTheorems) (thmId : Origin) : m SimpTheorems := do
unless d.isLemma thmId ||
match thmId with
| .decl declName => d.isDeclToUnfold declName || d.toUnfoldThms.contains declName
| _ => false
do
throwError "'{thmId.key}' does not have [simp] attribute"
return d.eraseCore thmId
private partial def isPerm : Expr → Expr → MetaM Bool
| Expr.app f₁ a₁, Expr.app f₂ a₂ => isPerm f₁ f₂ <&&> isPerm a₁ a₂
| Expr.mdata _ s, t => isPerm s t
| s, Expr.mdata _ t => isPerm s t
| s@(Expr.mvar ..), t@(Expr.mvar ..) => isDefEq s t
| Expr.forallE n₁ d₁ b₁ _, Expr.forallE _ d₂ b₂ _ => isPerm d₁ d₂ <&&> withLocalDeclD n₁ d₁ fun x => isPerm (b₁.instantiate1 x) (b₂.instantiate1 x)
| Expr.lam n₁ d₁ b₁ _, Expr.lam _ d₂ b₂ _ => isPerm d₁ d₂ <&&> withLocalDeclD n₁ d₁ fun x => isPerm (b₁.instantiate1 x) (b₂.instantiate1 x)
| Expr.letE n₁ t₁ v₁ b₁ _, Expr.letE _ t₂ v₂ b₂ _ =>
isPerm t₁ t₂ <&&> isPerm v₁ v₂ <&&> withLetDecl n₁ t₁ v₁ fun x => isPerm (b₁.instantiate1 x) (b₂.instantiate1 x)
| Expr.proj _ i₁ b₁, Expr.proj _ i₂ b₂ => pure (i₁ == i₂) <&&> isPerm b₁ b₂
| s, t => return s == t
private def checkBadRewrite (lhs rhs : Expr) : MetaM Unit := do
let lhs ← DiscrTree.reduceDT lhs (root := true) (simpleReduce := true)
if lhs == rhs && lhs.isFVar then
throwError "invalid `simp` theorem, equation is equivalent to{indentExpr (← mkEq lhs rhs)}"
private partial def shouldPreprocess (type : Expr) : MetaM Bool :=
forallTelescopeReducing type fun _ result => do
if let some (_, lhs, rhs) := result.eq? then
checkBadRewrite lhs rhs
return false
else
return true
private partial def preprocess (e type : Expr) (inv : Bool) (isGlobal : Bool) : MetaM (List (Expr × Expr)) :=
go e type
where
go (e type : Expr) : MetaM (List (Expr × Expr)) := do
let type ← whnf type
if type.isForall then
forallTelescopeReducing type fun xs type => do
let e := mkAppN e xs
let ps ← go e type
ps.mapM fun (e, type) =>
return (← mkLambdaFVars xs e, ← mkForallFVars xs type)
else if let some (_, lhs, rhs) := type.eq? then
if isGlobal then
checkBadRewrite lhs rhs
if inv then
let type ← mkEq rhs lhs
let e ← mkEqSymm e
return [(e, type)]
else
return [(e, type)]
else if let some (lhs, rhs) := type.iff? then
if isGlobal then
checkBadRewrite lhs rhs
if inv then
let type ← mkEq rhs lhs
let e ← mkEqSymm (← mkPropExt e)
return [(e, type)]
else
let type ← mkEq lhs rhs
let e ← mkPropExt e
return [(e, type)]
else if let some (_, lhs, rhs) := type.ne? then
if inv then
throwError "invalid '←' modifier in rewrite rule to 'False'"
if rhs.isConstOf ``Bool.true then
return [(← mkAppM ``Bool.of_not_eq_true #[e], ← mkEq lhs (mkConst ``Bool.false))]
else if rhs.isConstOf ``Bool.false then
return [(← mkAppM ``Bool.of_not_eq_false #[e], ← mkEq lhs (mkConst ``Bool.true))]
let type ← mkEq (← mkEq lhs rhs) (mkConst ``False)
let e ← mkEqFalse e
return [(e, type)]
else if let some p := type.not? then
if inv then
throwError "invalid '←' modifier in rewrite rule to 'False'"
if let some (_, lhs, rhs) := p.eq? then
if rhs.isConstOf ``Bool.true then
return [(← mkAppM ``Bool.of_not_eq_true #[e], ← mkEq lhs (mkConst ``Bool.false))]
else if rhs.isConstOf ``Bool.false then
return [(← mkAppM ``Bool.of_not_eq_false #[e], ← mkEq lhs (mkConst ``Bool.true))]
let type ← mkEq p (mkConst ``False)
let e ← mkEqFalse e
return [(e, type)]
else if let some (type₁, type₂) := type.and? then
let e₁ := mkProj ``And 0 e
let e₂ := mkProj ``And 1 e
return (← go e₁ type₁) ++ (← go e₂ type₂)
else
if inv then
throwError "invalid '←' modifier in rewrite rule to 'True'"
let type ← mkEq type (mkConst ``True)
let e ← mkEqTrue e
return [(e, type)]
private def checkTypeIsProp (type : Expr) : MetaM Unit :=
unless (← isProp type) do
throwError "invalid 'simp', proposition expected{indentExpr type}"
private def mkSimpTheoremCore (origin : Origin) (e : Expr) (levelParams : Array Name) (proof : Expr) (post : Bool) (prio : Nat) : MetaM SimpTheorem := do
assert! origin != .fvar ⟨.anonymous⟩
let type ← instantiateMVars (← inferType e)
withNewMCtxDepth do
let (_, _, type) ← withReducible <| forallMetaTelescopeReducing type
let type ← whnfR type
let (keys, perm) ←
match type.eq? with
| some (_, lhs, rhs) => pure (← DiscrTree.mkPath lhs, ← isPerm lhs rhs)
| none => throwError "unexpected kind of 'simp' theorem{indentExpr type}"
return { origin, keys, perm, post, levelParams, proof, priority := prio, rfl := (← isRflProof proof) }
private def mkSimpTheoremsFromConst (declName : Name) (post : Bool) (inv : Bool) (prio : Nat) : MetaM (Array SimpTheorem) := do
let cinfo ← getConstInfo declName
let val := mkConst declName (cinfo.levelParams.map mkLevelParam)
withReducible do
let type ← inferType val
checkTypeIsProp type
if inv || (← shouldPreprocess type) then
let mut r := #[]
for (val, type) in (← preprocess val type inv (isGlobal := true)) do
let auxName ← mkAuxLemma cinfo.levelParams type val
r := r.push <| (← mkSimpTheoremCore (.decl declName) (mkConst auxName (cinfo.levelParams.map mkLevelParam)) #[] (mkConst auxName) post prio)
return r
else
return #[← mkSimpTheoremCore (.decl declName) (mkConst declName (cinfo.levelParams.map mkLevelParam)) #[] (mkConst declName) post prio]
inductive SimpEntry where
| thm : SimpTheorem → SimpEntry
| toUnfold : Name → SimpEntry
| toUnfoldThms : Name → Array Name → SimpEntry
deriving Inhabited
abbrev SimpExtension := SimpleScopedEnvExtension SimpEntry SimpTheorems
def SimpExtension.getTheorems (ext : SimpExtension) : CoreM SimpTheorems :=
return ext.getState (← getEnv)
def addSimpTheorem (ext : SimpExtension) (declName : Name) (post : Bool) (inv : Bool) (attrKind : AttributeKind) (prio : Nat) : MetaM Unit := do
let simpThms ← mkSimpTheoremsFromConst declName post inv prio
for simpThm in simpThms do
ext.add (SimpEntry.thm simpThm) attrKind
def mkSimpAttr (attrName : Name) (attrDescr : String) (ext : SimpExtension)
(ref : Name := by exact decl_name%) : IO Unit :=
registerBuiltinAttribute {
ref := ref
name := attrName
descr := attrDescr
applicationTime := AttributeApplicationTime.afterCompilation
add := fun declName stx attrKind =>
let go : MetaM Unit := do
let info ← getConstInfo declName
let post := if stx[1].isNone then true else stx[1][0].getKind == ``Lean.Parser.Tactic.simpPost
let prio ← getAttrParamOptPrio stx[2]
if (← isProp info.type) then
addSimpTheorem ext declName post (inv := false) attrKind prio
else if info.hasValue then
if let some eqns ← getEqnsFor? declName then
for eqn in eqns do
addSimpTheorem ext eqn post (inv := false) attrKind prio
ext.add (SimpEntry.toUnfoldThms declName eqns) attrKind
if hasSmartUnfoldingDecl (← getEnv) declName then
ext.add (SimpEntry.toUnfold declName) attrKind
else
ext.add (SimpEntry.toUnfold declName) attrKind
else
throwError "invalid 'simp', it is not a proposition nor a definition (to unfold)"
discard <| go.run {} {}
erase := fun declName => do
let s := ext.getState (← getEnv)
let s ← s.erase (.decl declName)
modifyEnv fun env => ext.modifyState env fun _ => s
}
def mkSimpExt (name : Name := by exact decl_name%) : IO SimpExtension :=
registerSimpleScopedEnvExtension {
name := name
initial := {}
addEntry := fun d e =>
match e with
| SimpEntry.thm e => addSimpTheoremEntry d e
| SimpEntry.toUnfold n => d.addDeclToUnfoldCore n
| SimpEntry.toUnfoldThms n thms => d.registerDeclToUnfoldThms n thms
}
abbrev SimpExtensionMap := HashMap Name SimpExtension
builtin_initialize simpExtensionMapRef : IO.Ref SimpExtensionMap ← IO.mkRef {}
def registerSimpAttr (attrName : Name) (attrDescr : String)
(ref : Name := by exact decl_name%) : IO SimpExtension := do
let ext ← mkSimpExt ref
mkSimpAttr attrName attrDescr ext ref -- Remark: it will fail if it is not performed during initialization
simpExtensionMapRef.modify fun map => map.insert attrName ext
return ext
builtin_initialize simpExtension : SimpExtension ← registerSimpAttr `simp "simplification theorem"
def getSimpExtension? (attrName : Name) : IO (Option SimpExtension) :=
return (← simpExtensionMapRef.get).find? attrName
def getSimpTheorems : CoreM SimpTheorems :=
simpExtension.getTheorems
/-- Auxiliary method for adding a global declaration to a `SimpTheorems` datastructure. -/
def SimpTheorems.addConst (s : SimpTheorems) (declName : Name) (post := true) (inv := false) (prio : Nat := eval_prio default) : MetaM SimpTheorems := do
let s := { s with erased := s.erased.erase (.decl declName) }
let simpThms ← mkSimpTheoremsFromConst declName post inv prio
return simpThms.foldl addSimpTheoremEntry s
def SimpTheorem.getValue (simpThm : SimpTheorem) : MetaM Expr := do
if simpThm.proof.isConst && simpThm.levelParams.isEmpty then
let info ← getConstInfo simpThm.proof.constName!
if info.levelParams.isEmpty then
return simpThm.proof
else
return simpThm.proof.updateConst! (← info.levelParams.mapM (fun _ => mkFreshLevelMVar))
else
let us ← simpThm.levelParams.mapM fun _ => mkFreshLevelMVar
return simpThm.proof.instantiateLevelParamsArray simpThm.levelParams us
private def preprocessProof (val : Expr) (inv : Bool) : MetaM (Array Expr) := do
let type ← inferType val
checkTypeIsProp type
let ps ← preprocess val type inv (isGlobal := false)
return ps.toArray.map fun (val, _) => val
/-- Auxiliary method for creating simp theorems from a proof term `val`. -/
def mkSimpTheorems (id : Origin) (levelParams : Array Name) (proof : Expr) (post := true) (inv := false) (prio : Nat := eval_prio default) : MetaM (Array SimpTheorem) :=
withReducible do
(← preprocessProof proof inv).mapM fun val => mkSimpTheoremCore id val levelParams val post prio
/-- Auxiliary method for adding a local simp theorem to a `SimpTheorems` datastructure. -/
def SimpTheorems.add (s : SimpTheorems) (id : Origin) (levelParams : Array Name) (proof : Expr) (inv := false) (post := true) (prio : Nat := eval_prio default) : MetaM SimpTheorems := do
if proof.isConst then
s.addConst proof.constName! post inv prio
else
let simpThms ← mkSimpTheorems id levelParams proof post inv prio
return simpThms.foldl addSimpTheoremEntry s
def SimpTheorems.addDeclToUnfold (d : SimpTheorems) (declName : Name) : MetaM SimpTheorems := do
if let some eqns ← getEqnsFor? declName then
let mut d := d
for eqn in eqns do
d ← SimpTheorems.addConst d eqn
if hasSmartUnfoldingDecl (← getEnv) declName then
d := d.addDeclToUnfoldCore declName
return d
else
return d.addDeclToUnfoldCore declName
abbrev SimpTheoremsArray := Array SimpTheorems
def SimpTheoremsArray.addTheorem (thmsArray : SimpTheoremsArray) (id : Origin) (h : Expr) : MetaM SimpTheoremsArray :=
if thmsArray.isEmpty then
let thms : SimpTheorems := {}
return #[ (← thms.add id #[] h) ]
else
thmsArray.modifyM 0 fun thms => thms.add id #[] h
def SimpTheoremsArray.eraseTheorem (thmsArray : SimpTheoremsArray) (thmId : Origin) : SimpTheoremsArray :=
thmsArray.map fun thms => thms.eraseCore thmId
def SimpTheoremsArray.isErased (thmsArray : SimpTheoremsArray) (thmId : Origin) : Bool :=
thmsArray.any fun thms => thms.erased.contains thmId
def SimpTheoremsArray.isDeclToUnfold (thmsArray : SimpTheoremsArray) (declName : Name) : Bool :=
thmsArray.any fun thms => thms.isDeclToUnfold declName
macro (name := _root_.Lean.Parser.Command.registerSimpAttr) doc:(docComment)?
"register_simp_attr" id:ident : command => do
let str := id.getId.toString
let idParser := mkIdentFrom id (`Parser.Attr ++ id.getId)
let descr := quote (removeLeadingSpaces (doc.map (·.getDocString) |>.getD s!"simp set for {id.getId.toString}"))
`($[$doc:docComment]? initialize ext : SimpExtension ← registerSimpAttr $(quote id.getId) $descr $(quote id.getId)
$[$doc:docComment]? syntax (name := $idParser:ident) $(quote str):str (Parser.Tactic.simpPre <|> Parser.Tactic.simpPost)? (prio)? : attr)
end Meta
end Lean
|
fcb71c262a61c93e18913d50be1058266f48f7f7 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/group_theory/group_action/sum.lean | 253eb5990aa13cc53c3f9d6cbc9cf9ec5864455c | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 2,694 | lean | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import group_theory.group_action.defs
/-!
# Sum instances for additive and multiplicative actions
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file defines instances for additive and multiplicative actions on the binary `sum` type.
## See also
* `group_theory.group_action.option`
* `group_theory.group_action.pi`
* `group_theory.group_action.prod`
* `group_theory.group_action.sigma`
-/
variables {M N P α β γ : Type*}
namespace sum
section has_smul
variables [has_smul M α] [has_smul M β] [has_smul N α] [has_smul N β] (a : M) (b : α)
(c : β) (x : α ⊕ β)
@[to_additive sum.has_vadd] instance : has_smul M (α ⊕ β) := ⟨λ a, sum.map ((•) a) ((•) a)⟩
@[to_additive] lemma smul_def : a • x = x.map ((•) a) ((•) a) := rfl
@[simp, to_additive] lemma smul_inl : a • (inl b : α ⊕ β) = inl (a • b) := rfl
@[simp, to_additive] lemma smul_inr : a • (inr c : α ⊕ β) = inr (a • c) := rfl
@[simp, to_additive] lemma smul_swap : (a • x).swap = a • x.swap := by cases x; refl
instance [has_smul M N] [is_scalar_tower M N α] [is_scalar_tower M N β] :
is_scalar_tower M N (α ⊕ β) :=
⟨λ a b x,
by { cases x, exacts [congr_arg inl (smul_assoc _ _ _), congr_arg inr (smul_assoc _ _ _)] }⟩
@[to_additive] instance [smul_comm_class M N α] [smul_comm_class M N β] :
smul_comm_class M N (α ⊕ β) :=
⟨λ a b x,
by { cases x, exacts [congr_arg inl (smul_comm _ _ _), congr_arg inr (smul_comm _ _ _)] }⟩
@[to_additive]
instance [has_smul Mᵐᵒᵖ α] [has_smul Mᵐᵒᵖ β] [is_central_scalar M α] [is_central_scalar M β] :
is_central_scalar M (α ⊕ β) :=
⟨λ a x,
by { cases x, exacts [congr_arg inl (op_smul_eq_smul _ _), congr_arg inr (op_smul_eq_smul _ _)] }⟩
@[to_additive] instance has_faithful_smul_left [has_faithful_smul M α] :
has_faithful_smul M (α ⊕ β) :=
⟨λ x y h, eq_of_smul_eq_smul $ λ a : α, by injection h (inl a)⟩
@[to_additive] instance has_faithful_smul_right [has_faithful_smul M β] :
has_faithful_smul M (α ⊕ β) :=
⟨λ x y h, eq_of_smul_eq_smul $ λ b : β, by injection h (inr b)⟩
end has_smul
@[to_additive] instance {m : monoid M} [mul_action M α] [mul_action M β] : mul_action M (α ⊕ β) :=
{ mul_smul := λ a b x,
by { cases x, exacts [congr_arg inl (mul_smul _ _ _), congr_arg inr (mul_smul _ _ _)] },
one_smul := λ x,
by { cases x, exacts [congr_arg inl (one_smul _ _), congr_arg inr (one_smul _ _)] } }
end sum
|
25c3a1d4606e73e2fd422af9972f3c7c2bd012bf | 367134ba5a65885e863bdc4507601606690974c1 | /src/data/polynomial/iterated_deriv.lean | 9eb1378bcd67f51f483cd966d697ac5e1e79d07f | [
"Apache-2.0"
] | permissive | kodyvajjha/mathlib | 9bead00e90f68269a313f45f5561766cfd8d5cad | b98af5dd79e13a38d84438b850a2e8858ec21284 | refs/heads/master | 1,624,350,366,310 | 1,615,563,062,000 | 1,615,563,062,000 | 162,666,963 | 0 | 0 | Apache-2.0 | 1,545,367,651,000 | 1,545,367,651,000 | null | UTF-8 | Lean | false | false | 8,108 | lean | /-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import data.polynomial.derivative
import logic.function.iterate
import data.finset.intervals
import tactic.ring
import tactic.linarith
/-!
# Theory of iterated derivative
We define and prove some lemmas about iterated (formal) derivative for polynomials over a semiring.
-/
noncomputable theory
open finset nat polynomial
open_locale big_operators
namespace polynomial
universes u
variable {R : Type u}
section semiring
variables [semiring R] (r : R) (f p q : polynomial R) (n k : ℕ)
/-- `iterated_deriv f n` is the `n`-th formal derivative of the polynomial `f` -/
def iterated_deriv : polynomial R := derivative ^[n] f
@[simp] lemma iterated_deriv_zero_right : iterated_deriv f 0 = f := rfl
lemma iterated_deriv_succ : iterated_deriv f (n + 1) = (iterated_deriv f n).derivative :=
by rw [iterated_deriv, iterated_deriv, function.iterate_succ']
@[simp] lemma iterated_deriv_zero_left : iterated_deriv (0 : polynomial R) n = 0 :=
begin
induction n with n hn,
{ exact iterated_deriv_zero_right _ },
{ rw [iterated_deriv_succ, hn, derivative_zero] },
end
@[simp] lemma iterated_deriv_add :
iterated_deriv (p + q) n = iterated_deriv p n + iterated_deriv q n :=
begin
induction n with n ih,
{ simp only [iterated_deriv_zero_right], },
{ simp only [iterated_deriv_succ, ih, derivative_add] }
end
@[simp] lemma iterated_deriv_smul : iterated_deriv (r • p) n = r • iterated_deriv p n :=
begin
induction n with n ih,
{ simp only [iterated_deriv_zero_right] },
{ simp only [iterated_deriv_succ, ih, derivative_smul] }
end
@[simp] lemma iterated_deriv_X_zero : iterated_deriv (X : polynomial R) 0 = X :=
by simp only [iterated_deriv_zero_right]
@[simp] lemma iterated_deriv_X_one : iterated_deriv (X : polynomial R) 1 = 1 :=
by simp only [iterated_deriv, derivative_X, function.iterate_one]
@[simp] lemma iterated_deriv_X (h : 1 < n) : iterated_deriv (X : polynomial R) n = 0 :=
begin
induction n with n ih,
{ exfalso, exact not_lt_zero 1 h},
{ simp only [iterated_deriv_succ],
by_cases H : n = 1,
{ rw H, simp only [iterated_deriv_X_one, derivative_one] },
{ replace h : 1 < n := array.push_back_idx h (ne.symm H),
rw ih h, simp only [derivative_zero] } }
end
@[simp] lemma iterated_deriv_C_zero : iterated_deriv (C r) 0 = C r :=
by simp only [iterated_deriv_zero_right]
@[simp] lemma iterated_deriv_C (h : 0 < n) : iterated_deriv (C r) n = 0 :=
begin
induction n with n ih,
{ exfalso, exact nat.lt_asymm h h },
{ by_cases H : n = 0,
{ rw [iterated_deriv_succ, H], simp only [iterated_deriv_C_zero, derivative_C]},
{ replace h : 0 < n := nat.pos_of_ne_zero H,
rw [iterated_deriv_succ, ih h], simp only [derivative_zero] } }
end
@[simp] lemma iterated_deriv_one_zero : iterated_deriv (1 : polynomial R) 0 = 1 :=
by simp only [iterated_deriv_zero_right]
@[simp] lemma iterated_deriv_one : 0 < n → iterated_deriv (1 : polynomial R) n = 0 := λ h,
begin
have eq1 : (1 : polynomial R) = C 1 := by simp only [ring_hom.map_one],
rw eq1, exact iterated_deriv_C _ _ h,
end
end semiring
section ring
variables [ring R] (p q : polynomial R) (n : ℕ)
@[simp] lemma iterated_deriv_neg : iterated_deriv (-p) n = - iterated_deriv p n :=
begin
induction n with n ih,
{ simp only [iterated_deriv_zero_right] },
{ simp only [iterated_deriv_succ, ih, derivative_neg] }
end
@[simp] lemma iterated_deriv_sub :
iterated_deriv (p - q) n = iterated_deriv p n - iterated_deriv q n :=
by rw [sub_eq_add_neg, iterated_deriv_add, iterated_deriv_neg, ←sub_eq_add_neg]
end ring
section comm_semiring
variable [comm_semiring R]
variables (f p q : polynomial R) (n k : ℕ)
lemma coeff_iterated_deriv_as_prod_Ico :
∀ m : ℕ, (iterated_deriv f k).coeff m = (∏ i in Ico m.succ (m + k.succ), i) * (f.coeff (m+k)) :=
begin
induction k with k ih,
{ simp only [add_zero, forall_const, one_mul, Ico.self_eq_empty, eq_self_iff_true,
iterated_deriv_zero_right, prod_empty] },
{ intro m, rw [iterated_deriv_succ, coeff_derivative, ih (m+1), mul_right_comm],
apply congr_arg2,
{ have set_eq : (Ico m.succ (m + k.succ.succ)) = (Ico (m + 1).succ (m + 1 + k.succ)) ∪ {m+1},
{ rw [union_comm, ←insert_eq, Ico.insert_succ_bot, add_succ, add_succ, add_succ _ k,
←succ_eq_add_one, succ_add],
rw succ_eq_add_one,
linarith },
rw [set_eq, prod_union],
apply congr_arg2,
{ refl },
{ simp only [prod_singleton], norm_cast },
{ simp only [succ_pos', disjoint_singleton, and_true, lt_add_iff_pos_right, not_le, Ico.mem],
exact lt_add_one (m + 1) } },
{ exact congr_arg _ (succ_add m k) } },
end
lemma coeff_iterated_deriv_as_prod_range :
∀ m : ℕ, (iterated_deriv f k).coeff m = f.coeff (m + k) * (∏ i in finset.range k, ↑(m + k - i)) :=
begin
induction k with k ih,
{ simp },
intro m,
calc (f.iterated_deriv k.succ).coeff m
= f.coeff (m + k.succ) * (∏ i in finset.range k, ↑(m + k.succ - i)) * (m + 1) :
by rw [iterated_deriv_succ, coeff_derivative, ih m.succ, succ_add, add_succ]
... = f.coeff (m + k.succ) * (↑(m + 1) * (∏ (i : ℕ) in range k, ↑(m + k.succ - i))) :
by { push_cast, ring }
... = f.coeff (m + k.succ) * (∏ (i : ℕ) in range k.succ, ↑(m + k.succ - i)) :
by { rw [prod_range_succ, nat.add_sub_assoc (le_succ k), nat.succ_sub le_rfl, nat.sub_self] }
end
lemma iterated_deriv_eq_zero_of_nat_degree_lt (h : f.nat_degree < n) : iterated_deriv f n = 0 :=
begin
ext m,
rw [coeff_iterated_deriv_as_prod_range, coeff_zero, coeff_eq_zero_of_nat_degree_lt, zero_mul],
linarith
end
lemma iterated_deriv_mul :
iterated_deriv (p * q) n =
∑ k in range n.succ,
(C (n.choose k : R)) * iterated_deriv p (n - k) * iterated_deriv q k :=
begin
induction n with n IH,
{ simp },
calc (p * q).iterated_deriv n.succ
= (∑ (k : ℕ) in range n.succ,
C ↑(n.choose k) * p.iterated_deriv (n - k) * q.iterated_deriv k).derivative :
by rw [iterated_deriv_succ, IH]
... = ∑ (k : ℕ) in range n.succ,
C ↑(n.choose k) * p.iterated_deriv (n - k + 1) * q.iterated_deriv k +
∑ (k : ℕ) in range n.succ,
C ↑(n.choose k) * p.iterated_deriv (n - k) * q.iterated_deriv (k + 1) :
by simp_rw [derivative_sum, derivative_mul, derivative_C, zero_mul, zero_add,
iterated_deriv_succ, sum_add_distrib]
... = (∑ (k : ℕ) in range n.succ,
C ↑(n.choose k.succ) * p.iterated_deriv (n - k) * q.iterated_deriv (k + 1) +
C ↑1 * p.iterated_deriv n.succ * q.iterated_deriv 0) +
∑ (k : ℕ) in range n.succ,
C ↑(n.choose k) * p.iterated_deriv (n - k) * q.iterated_deriv (k + 1) : _
... = ∑ (k : ℕ) in range n.succ,
C ↑(n.choose k) * p.iterated_deriv (n - k) * q.iterated_deriv (k + 1) +
∑ (k : ℕ) in range n.succ,
C ↑(n.choose k.succ) * p.iterated_deriv (n - k) * q.iterated_deriv (k + 1) +
C ↑1 * p.iterated_deriv n.succ * q.iterated_deriv 0 :
by ring
... = ∑ (i : ℕ) in range n.succ,
C ↑(n.succ.choose (i + 1)) * p.iterated_deriv (n + 1 - (i + 1)) * q.iterated_deriv (i + 1) +
C ↑1 * p.iterated_deriv n.succ * q.iterated_deriv 0 :
by simp_rw [choose_succ_succ, succ_sub_succ, cast_add, C.map_add, add_mul, sum_add_distrib]
... = ∑ (k : ℕ) in range n.succ.succ,
C ↑(n.succ.choose k) * p.iterated_deriv (n.succ - k) * q.iterated_deriv k :
by rw [sum_range_succ' _ n.succ, choose_zero_right, nat.sub_zero],
congr,
refine (sum_range_succ' _ _).trans (congr_arg2 (+) _ _),
{ rw [sum_range_succ, nat.choose_succ_self, cast_zero, C.map_zero, zero_mul, zero_mul, zero_add],
refine sum_congr rfl (λ k hk, _),
rw mem_range at hk,
congr,
rw [← nat.sub_add_comm (nat.succ_le_of_lt hk), nat.succ_sub_succ] },
{ rw [choose_zero_right, nat.sub_zero] },
end
end comm_semiring
end polynomial
|
bbe61856b0887a8a655bf6164d29243094ed337f | 4727251e0cd73359b15b664c3170e5d754078599 | /src/tactic/local_cache.lean | aa6f053801c64127a83df83dd5f7436f73c67088 | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 7,335 | lean | /-
Copyright (c) 2019 Keeley Hoek. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Keeley Hoek
-/
import tactic.norm_num
namespace tactic
namespace local_cache
namespace internal
variables {α : Type} [reflected α] [has_reflect α]
meta def mk_full_namespace (ns : name) : name := `local_cache ++ ns
meta def save_data (dn : name) (a : α) [reflected a] : tactic unit :=
tactic.add_decl $ mk_definition dn [] (reflect α) (reflect a)
meta def load_data (dn : name) : tactic α :=
do e ← tactic.get_env,
d ← e.get dn,
tactic.eval_expr α d.value
meta def poke_data (dn : name) : tactic bool :=
do e ← tactic.get_env,
return (e.get dn).to_bool
meta def run_once_under_name {α : Type} [reflected α] [has_reflect α] (t : tactic α)
(cache_name : name) : tactic α :=
do load_data cache_name <|>
do
{ a ← t,
save_data cache_name a,
return a }
-- We maintain two separate caches with different scopes:
-- one local to `begin ... end` or `by` blocks, and another
-- for entire `def`/`lemma`s.
meta structure cache_scope :=
-- Returns the name of the def used to store the contents of is cache,
-- making a new one and recording this in private state if neccesary.
(get_name : name → tactic name)
-- Same as above but fails instead of making a new name, and never
-- mutates state.
(try_get_name : name → tactic name)
-- Asks whether the namespace `ns` currently has a value-in-cache
(present : name → tactic bool)
-- Clear cache associated to namespace `ns`
(clear : name → tactic unit)
namespace block_local
-- `mk_new` gives a way to generate a new name if no current one
-- exists.
private meta def get_name_aux (ns : name) (mk_new : options → name → tactic name) : tactic name :=
do o ← tactic.get_options,
let opt := mk_full_namespace ns,
match o.get_string opt "" with
| "" := mk_new o opt
| s := return $ name.from_components $ s.split (= '.')
end
meta def get_name (ns : name) : tactic name :=
get_name_aux ns $ λ o opt,
do n ← mk_user_fresh_name,
tactic.set_options $ o.set_string opt n.to_string,
return n
-- Like `get_name`, but fail if `ns` does not have a cached
-- decl name (we create a new one above).
meta def try_get_name (ns : name) : tactic name :=
get_name_aux ns $ λ o opt, fail format!"no cache for \"{ns}\""
meta def present (ns : name) : tactic bool :=
do o ← tactic.get_options,
match o.get_string (mk_full_namespace ns) "" with
| "" := return ff
| s := return tt
end
meta def clear (ns : name) : tactic unit :=
do o ← tactic.get_options,
set_options $ o.set_string (mk_full_namespace ns) ""
end block_local
namespace def_local
-- Fowler-Noll-Vo hash function (FNV-1a)
section fnv_a1
def FNV_OFFSET_BASIS := 0xcbf29ce484222325
def FNV_PRIME := 0x100000001b3
def RADIX := by apply_normed 2^64
def hash_byte (seed : ℕ) (c : char) : ℕ :=
let n : ℕ := c.to_nat in ((seed.lxor n) * FNV_PRIME) % RADIX
def hash_string (s : string) : ℕ :=
s.to_list.foldl hash_byte FNV_OFFSET_BASIS
end fnv_a1
meta def hash_context : tactic string :=
do ns ← open_namespaces,
dn ← decl_name,
let flat := ((list.cons dn ns).map to_string).foldl string.append "",
return $ (to_string dn) ++ (to_string (hash_string flat))
meta def get_root_name (ns : name) : tactic name :=
do hc ← hash_context,
return $ mk_full_namespace $ hc ++ ns
meta def apply_tag (n : name) (tag : ℕ) : name := n ++ to_string format!"t{tag}"
meta def mk_dead_name (n : name) : name := n ++ `dead
meta def kill_name (n : name) : tactic unit :=
save_data (mk_dead_name n) ()
meta def is_name_dead (n : name) : tactic bool :=
do { witness : unit ← load_data $ mk_dead_name n, return true }
<|> return false
-- `get_with_status_tag_aux rn n` fails exactly when `rn ++ to_string n` does
-- not exist.
private meta def get_with_status_tag_aux (rn : name) : ℕ → tactic (ℕ × bool)
| tag := do let n := apply_tag rn tag,
present ← poke_data n,
if ¬present then fail format!"{rn} never seen in cache!"
else do is_dead ← is_name_dead n,
if is_dead then get_with_status_tag_aux (tag + 1)
<|> return (tag, false)
else return (tag, true)
-- Find the latest tag for the name `rn` and report whether it is alive.
meta def get_tag_with_status (rn : name) : tactic (ℕ × bool) :=
get_with_status_tag_aux rn 0
meta def get_name (ns : name) : tactic name :=
do rn ← get_root_name ns,
(tag, alive) ← get_tag_with_status rn <|> return (0, true),
return $ apply_tag rn $ if alive then tag
else tag + 1
meta def try_get_name (ns : name) : tactic name :=
do rn ← get_root_name ns,
(tag, alive) ← get_tag_with_status rn,
if alive then return $ apply_tag rn tag
else fail format!"no cache for \"{ns}\""
meta def present (ns : name) : tactic bool :=
do rn ← get_root_name ns,
(prod.snd <$> get_tag_with_status rn) <|> return false
meta def clear (ns : name) : tactic unit :=
do { n ← try_get_name ns, kill_name n }
<|> skip
end def_local
end internal
open internal
/-- This scope propogates the cache within a `begin ... end` or `by` block
and its decendants. -/
meta def cache_scope.block_local : cache_scope :=
⟨ block_local.get_name,
block_local.try_get_name,
block_local.present,
block_local.clear ⟩
/-- This scope propogates the cache within an entire `def`/`lemma`. -/
meta def cache_scope.def_local : cache_scope :=
⟨ def_local.get_name,
def_local.try_get_name,
def_local.present,
def_local.clear ⟩
open cache_scope
/-- Asks whether the namespace `ns` currently has a value-in-cache. -/
meta def present (ns : name) (s : cache_scope := block_local) : tactic bool :=
s.present ns
/-- Clear cache associated to namespace `ns`. -/
meta def clear (ns : name) (s : cache_scope := block_local) : tactic unit :=
s.clear ns
/-- Gets the (optionally present) value-in-cache for `ns`. -/
meta def get (ns : name) (α : Type) [reflected α] [has_reflect α] (s : cache_scope := block_local) :
tactic (option α) :=
do dn ← some <$> s.try_get_name ns <|> return none,
match dn with
| none := return none
| some dn := some <$> load_data dn
end
-- Note: we can't just use `<|>` on `load_data` since it will fail
-- when a cached value is not present *as well as* when the type of
-- `α` is just wrong.
end local_cache
open local_cache local_cache.internal
/-- Using the namespace `ns` as its key, when called for the first
time `run_once ns t` runs `t`, then saves and returns the result.
Upon subsequent invocations in the same tactic block, with the scope
of the caching being inherited by child tactic blocks) we return the
cached result directly.
You can configure the cached scope to be entire `def`/`lemma`s changing
the optional cache_scope argument to `cache_scope.def_local`.
Note: the caches backing each scope are different.
If `α` is just `unit`, this means we just run `t` once each tactic
block. -/
meta def run_once {α : Type} [reflected α] [has_reflect α] (ns : name) (t : tactic α)
(s : cache_scope := cache_scope.block_local) : tactic α :=
s.get_name ns >>= run_once_under_name t
end tactic
|
8c8ff7da0ba38bfe9272189c6374300a057d732c | d450724ba99f5b50b57d244eb41fef9f6789db81 | /src/Submissions/exam1.lean | 08943b5319944253a6215b7fde492ce94ab76842 | [] | no_license | jakekauff/CS2120F21 | 4f009adeb4ce4a148442b562196d66cc6c04530c | e69529ec6f5d47a554291c4241a3d8ec4fe8f5ad | refs/heads/main | 1,693,841,880,030 | 1,637,604,848,000 | 1,637,604,848,000 | 399,946,698 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 11,250 | lean | /-
*******************************
PART 1 of 2: AXIOMS [50 points]
*******************************
-/
/-
Explain the inference (introduction and
elimination) rules of predicate logic as
directed below. To give you guidance on
how to answer these questions we include
answers for some questions.
-/
-- -------------------------------------
/- #1: → implies [5 points]
INTRODUCTION
Explain the introduction rule for →.
[We give you the answer here.]
In English: Given propositions, P and Q,
a derivation (computation) a proof of Q
from any proof of P, is a proof of P → Q.
In constructive logic, a derivation is a
given as a function definition. A proof
of P → Q literally is a function, that,
when given any proof of P as an argument
returns a proof of Q. If you define such
a function, you have proved P → Q.
ELIMINATION
Complete the definition of the elimination
rule for →.
(P Q : Prop) (p2q : P → Q) (p : P)
----------------------------------
[q : Q]
-/
-- Give a formal proof of the following
example : ∀ (P Q : Prop) (p2q : P → Q) (p : P), Q :=
begin
assume P Q,
assume pq,
assume p,
apply pq p,
end
-- Extra credit [2 points]. Who invented this principle? Modus Ponens
-- -------------------------------------
/- #2: true [5 points]
INTRODUCTION
Give the introduction rule for true using
inference rule notation.
[Here's our answer.]
---------- intro
(pf: true)
Give a brief English language explanation of
the introduction rule for true.
-- Because it is inherently true, there exists a proof of true.
ELIMINATION
Give an English language description of the
eliimination rule for true.
[Our answer]
A proof of true carries no information so
there's no use for an elimination rule.
-/
-- Give a formal proof of the following:
example : true :=
begin
apply true.intro,
end
-- -------------------------------------
/- #3: ∧ - and [10 points]
INTRODUCTION
Using inference rule notation, give the
introduction rule for ∧.
[Our answer]
(P Q : Prop) (p : P) (q : Q)
---------------------------- intro
(pq : P ∧ Q)
Given an English language description of
this inference rule. What does it really
say, in plain simple English.
-- Given an arbitrary proposition P and an arbitrary proposition Q,
assuming that P is true and Q is true, it follows that P and Q is true.
ELIMINATION
Given the elimination rules for ∧ in both
inference rule and English language forms.
(P Q : Prop) (pq : P ∧ Q)
------------------------- ∧ elim left
[p : P]
(P Q : Prop) (pq : P ∧ Q)
------------------------- ∧ elim right
[q : Q]
English: Given arbitrary propositions P and Q, a proof of
P and a proof of Q can be derived from a proof of P and Q.
-/
/-
Formally state and prove the theorem that,
for any propositions P and Q, Q ∧ P → P.
-/
example : ∀(P Q : Prop), Q ∧ P → P :=
begin
intros P Q qnp,
exact qnp.right,
end
-- -------------------------------------
/- #4: ∀ - for all [10 points]
INTRODUCTION
Explain in English the introduction rule for ∀. If
T is any type (such as nat) and Q is any proposition
(often in the form of a predicate on values of the
given type), how do you prove ∀ (t : T), Q? What is
the introduction rule for ∀?
-- If T is any type and Q is any proposition,
you prove ∀ (t : T), Q by first assuming that t is
of an arbitrary but specific type and then proving that Q is true
for t. It then follows that Q is true for all of this type.
The introduction rule for ∀ is as follows: we assume an
arbitrary premise and then show that the premise holds.
ELIMINATION
Here's an inference rule representation of the elim
rule for ∀. First, complete the inference rule by
filling in the bottom half, then Explain in English
what it says.
(T : Type) (Q : Prop), (pf : ∀ (t : T), Q) (t : T)
-------------------------------------------------- elim
[pf : Q]
-- Given an arbitrary type T and an arbitrary proposition Q,
one can apply a proof that Q holds for all t of type T to
a value t of type T to obtain a proof of Q.
Given a proof, (pf : ∀ (t : T), Q), and a value, (t : T),
briefly explain in English how you *use* pf to derive a
proof of Q.
-- Given a proof, (pf : ∀ (t : T), Q), and a value, (t : T),
you use pf as a sort of function that takes t as an argument
to produce a proof of Q.
-/
/-
Consider the following assumptions, and then in the
context of these assumptions, given answers to the
challenge problems.
-/
axioms
(Person : Type)
(KnowsLogic BetterComputerScientist : Person → Prop)
(LogicMakesYouBetterAtCS:
∀ (p : Person), KnowsLogic p → BetterComputerScientist p)
-- formalizee the following assumptions here
-- (1) Lynn is a person
-- (2) Lynn knows logic
(Lynn : Person)
(LynnKnowsLogic :
∀ (Lynn : Person), KnowsLogic Lynn)
-- Knowing logic makes you a better computer scientist.
/-
Now, formally state and prove the proposition that
Lynn is a better computer scientist
-/
example : ∀(Lynn : Person), BetterComputerScientist Lynn :=
begin
assume Lynn,
apply LogicMakesYouBetterAtCS,
apply LynnKnowsLogic,
end
-- -------------------------------------
/- #5: ¬ - not [10 points]
The ¬ connective in Lean is short for *not*. Give
the short formal definition of ¬ in Lean. Note that
surround the place where you give your answer with
a namespace. This is just to avoid conflicting with
Lean's definition of not.
-/
namespace hidden
def not (P : Prop) := P → false -- fill in the placeholder
end hidden
/-
Explain precisely in English the "proof strategy"
of "proof by negation." Explain how one uses this
strategy to prove a proposition, ¬P.
-/
-- The proof strategy of proof by negation involves
-- assuming a proposition to be true, then showing
-- a contradiction to prove the proposition false.
-- Given an arbitrary proposition P, to derive a
-- proof of ¬P using proof by negation, we first
-- assume P → True. Then we show that in this context there is
-- a contradiction, so P → false. By the definition of
-- ¬P, we can use the proof of P → false to derive a
-- proof of ¬P.
/-
Explain precisely in English the "proof strategy"
of "proof by contradiction." Explain how one uses
this strategy to prove a proposition, P (notice
the lack of a ¬ in front of the P).
Fill in the blanks the following partial answer:
To prove P, assume __¬P__ and show that __this assumption yields a contradiction__.
From this derivation you can conclude __¬¬P__.
Then you apply the rule of negation __elimination__
to that result to arrive a a proof of P. We have
seen that the inference rule you apply in the
last step is not constructively valid but that it
is __classically__ valid, and that accepting the axiom
of the __excluded middle__ suffices to establish negation
__elimination__ (better called double __negation__ __elimination__)
as a theorem.
-/
-- -------------------------------------
/-
#6: ↔ - if and only if, equivalent to [10 points]
-/
/-
ELIMINATION
As with ∧, ↔ has two elimination rules. You can
see their statements here.
-/
#check @iff.elim_left
#check @iff.elim_right
/-
Formally state and prove the theorem that
biimplication, ↔, is *commutative*. Note:
put parentheses around each ↔ proposition,
as → has higher precedence than ↔. Recall
that iff has both elim_left and elim_right
rules, just like ∧.
-/
example : ∀(P Q : Prop), (P ↔ Q) → (Q ↔ P) :=
begin
intros P Q,
assume PiffQ,
apply iff.intro,
apply iff.elim_right PiffQ,
apply iff.elim_left PiffQ,
end
/-
************************************************
PART 2 of 3: PROPOSITIONS AND PROOFS [50 points]
************************************************
-/
/- #7 [20 points]
First, give a fluent English rendition of
the following proposition. Note that the
identifier we use, elantp, is short for
"everyone likes a nice, talented person."
Then give a formal proof. Hint: try the
"intros" tactic by itself where you'd
previously have used assume followed by
a list of identifiers. Think about what
each expression means.
-/
def ELJL : Prop :=
∀ (Person : Type)
(Nice : Person → Prop)
(Talented : Person → Prop)
(Likes : Person → Person → Prop)
(elantp : ∀ (p : Person),
Nice p → Talented p →
∀ (q : Person), Likes q p)
(JohnLennon : Person)
(JLNT : Nice JohnLennon ∧ Talented JohnLennon),
(∀ (p : Person), Likes p JohnLennon)
-- English Rendition:
/-
Given any person P, and that P is a nice and talented person, then for
any person Q, Q likes P. John Lennon is a person who is nice and talented.
For any person p, p likes John Lennon.
-/
example : ELJL :=
begin
intros Person Nice Talented Likes elantp JohnLennon JLNT p,
apply elantp JohnLennon,
apply and.elim_left JLNT,
apply and.elim_right JLNT,
end
/- #8 [10 points]
If every car is either heavy or light, and red or
blue, and we want a prove by cases that every car
is rad, then:
-- how many cases will need to be considered? __4__
-- list the cases (informaly)
-- heavy and blue
-- heavy and red
-- light and blue
-- light and red
-/
/-
*********
RELATIONS
*********
-/
/- #9 [20 points]
Equality can be understood as a binary relation
on objects of a given type. There is a binary
equality relation on natural numbers, rational
numbers, on strings, on Booleans, and so forth.
We also saw that from the two axioms (inference
rules) for equality, we could prove that it is
not only reflexive, but also both symmetric and
transitive.
Complete the following definitions to express
the propositions that equality is respectively
symmetric and transitive. (Don't worry about
proving these propositions. We just want you
to write them formally, to show that you what
the terms means.)
-/
def eq_is_symmetric : Prop :=
∀ (T : Type) (x y : T), x = y → y = x
def eq_is_transitive : Prop :=
∀ (T : Type) (x y z : T), x = y → y = z → x = z
/-
************
EXTRA CREDIT
************
-/
/-
EXTRA CREDIT: State and prove the proposition
that (double) negation elimination is equivalent
to excluded middle. You get 1 point for writing
the correct proposition, 2 points for proving it
in one direction and five points for proving it
both directions.
-/
def negelim_equiv_exmid :
∀ (P : Prop), (¬¬P → P) ↔ (P ∨ ¬P) :=
begin
assume P,
apply iff.intro _ _,
--backward
show (P ∨ ¬P) → (¬¬P → P),
assume pornp,
assume nnp,
cases pornp with p np,
exact p,
apply false.elim,
have f := nnp np,
exact f,
-- forward
assume nnp_p,
end
/-
EXTRA CREDIT: Formally express and prove the
proposition that if there is someone everyone
loves, and loves is a symmetric relation, then
thre is someone who loves everyone. [5 points]
-/
axiom Loves : Person → Person → Prop
axiom EveryoneLoves : ∀ (P : Person), ∃ (R : Person), Loves R P
axiom LovesIsSymmetric : ∀ (P R : Person), (Loves R P) ↔ (Loves P R)
example : (∀(P:Person), ∃ (R : Person), Loves P R → true) :=
begin
intros,
have idk :=
end |
1e707d783fbec4f9b3c017482e6322ec8241fe96 | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/set_theory/ordinal_arithmetic.lean | b2663c470c0736fa609ae6aff3710a6c62259dcc | [
"Apache-2.0"
] | permissive | dexmagic/mathlib | ff48eefc56e2412429b31d4fddd41a976eb287ce | 7a5d15a955a92a90e1d398b2281916b9c41270b2 | refs/heads/master | 1,693,481,322,046 | 1,633,360,193,000 | 1,633,360,193,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 70,429 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import set_theory.ordinal
/-!
# Ordinal arithmetic
Ordinals have an addition (corresponding to disjoint union) that turns them into an additive
monoid, and a multiplication (corresponding to the lexicographic order on the product) that turns
them into a monoid. One can also define correspondingly a subtraction, a division, a successor
function, a power function and a logarithm function.
We also define limit ordinals and prove the basic induction principle on ordinals separating
successor ordinals and limit ordinals, in `limit_rec_on`.
## Main definitions and results
* `o₁ + o₂` is the order on the disjoint union of `o₁` and `o₂` obtained by declaring that
every element of `o₁` is smaller than every element of `o₂`.
* `o₁ - o₂` is the unique ordinal `o` such that `o₂ + o = o₁`, when `o₂ ≤ o₁`.
* `o₁ * o₂` is the lexicographic order on `o₂ × o₁`.
* `o₁ / o₂` is the ordinal `o` such that `o₁ = o₂ * o + o'` with `o' < o₂`. We also define the
divisibility predicate, and a modulo operation.
* `succ o = o + 1` is the successor of `o`.
* `pred o` if the predecessor of `o`. If `o` is not a successor, we set `pred o = o`.
We also define the power function and the logarithm function on ordinals, and discuss the properties
of casts of natural numbers of and of `omega` with respect to these operations.
Some properties of the operations are also used to discuss general tools on ordinals:
* `is_limit o`: an ordinal is a limit ordinal if it is neither `0` nor a successor.
* `limit_rec_on` is the main induction principle of ordinals: if one can prove a property by
induction at successor ordinals and at limit ordinals, then it holds for all ordinals.
* `is_normal`: a function `f : ordinal → ordinal` satisfies `is_normal` if it is strictly increasing
and order-continuous, i.e., the image `f o` of a limit ordinal `o` is the sup of `f a` for
`a < o`.
* `nfp f a`: the next fixed point of a function `f` on ordinals, above `a`. It behaves well
for normal functions.
* `CNF b o` is the Cantor normal form of the ordinal `o` in base `b`.
* `sup`: the supremum of an indexed family of ordinals in `Type u`, as an ordinal in `Type u`.
* `bsup`: the supremum of a set of ordinals indexed by ordinals less than a given ordinal `o`.
-/
noncomputable theory
open function cardinal set equiv
open_locale classical cardinal
universes u v w
variables {α : Type*} {β : Type*} {γ : Type*}
{r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop}
namespace ordinal
/-! ### Further properties of addition on ordinals -/
@[simp] theorem lift_add (a b) : lift (a + b) = lift a + lift b :=
quotient.induction_on₂ a b $ λ ⟨α, r, _⟩ ⟨β, s, _⟩,
quotient.sound ⟨(rel_iso.preimage equiv.ulift _).trans
(rel_iso.sum_lex_congr (rel_iso.preimage equiv.ulift _)
(rel_iso.preimage equiv.ulift _)).symm⟩
@[simp] theorem lift_succ (a) : lift (succ a) = succ (lift a) :=
by unfold succ; simp only [lift_add, lift_one]
theorem add_le_add_iff_left (a) {b c : ordinal} : a + b ≤ a + c ↔ b ≤ c :=
⟨induction_on a $ λ α r hr, induction_on b $ λ β₁ s₁ hs₁, induction_on c $ λ β₂ s₂ hs₂ ⟨f⟩, ⟨
have fl : ∀ a, f (sum.inl a) = sum.inl a := λ a,
by simpa only [initial_seg.trans_apply, initial_seg.le_add_apply]
using @initial_seg.eq _ _ _ _ (@sum.lex.is_well_order _ _ _ _ hr hs₂)
((initial_seg.le_add r s₁).trans f) (initial_seg.le_add r s₂) a,
have ∀ b, {b' // f (sum.inr b) = sum.inr b'}, begin
intro b, cases e : f (sum.inr b),
{ rw ← fl at e, have := f.inj' e, contradiction },
{ exact ⟨_, rfl⟩ }
end,
let g (b) := (this b).1 in
have fr : ∀ b, f (sum.inr b) = sum.inr (g b), from λ b, (this b).2,
⟨⟨⟨g, λ x y h, by injection f.inj'
(by rw [fr, fr, h] : f (sum.inr x) = f (sum.inr y))⟩,
λ a b, by simpa only [sum.lex_inr_inr, fr, rel_embedding.coe_fn_to_embedding,
initial_seg.coe_fn_to_rel_embedding, function.embedding.coe_fn_mk]
using @rel_embedding.map_rel_iff _ _ _ _ f.to_rel_embedding (sum.inr a) (sum.inr b)⟩,
λ a b H, begin
rcases f.init' (by rw fr; exact sum.lex_inr_inr.2 H) with ⟨a'|a', h⟩,
{ rw fl at h, cases h },
{ rw fr at h, exact ⟨a', sum.inr.inj h⟩ }
end⟩⟩,
λ h, add_le_add_left h _⟩
theorem add_succ (o₁ o₂ : ordinal) : o₁ + succ o₂ = succ (o₁ + o₂) :=
(add_assoc _ _ _).symm
@[simp] theorem succ_zero : succ 0 = 1 := zero_add _
theorem one_le_iff_pos {o : ordinal} : 1 ≤ o ↔ 0 < o :=
by rw [← succ_zero, succ_le]
theorem one_le_iff_ne_zero {o : ordinal} : 1 ≤ o ↔ o ≠ 0 :=
by rw [one_le_iff_pos, ordinal.pos_iff_ne_zero]
theorem succ_pos (o : ordinal) : 0 < succ o :=
lt_of_le_of_lt (ordinal.zero_le _) (lt_succ_self _)
theorem succ_ne_zero (o : ordinal) : succ o ≠ 0 :=
ne_of_gt $ succ_pos o
@[simp] theorem card_succ (o : ordinal) : card (succ o) = card o + 1 :=
by simp only [succ, card_add, card_one]
theorem nat_cast_succ (n : ℕ) : (succ n : ordinal) = n.succ := rfl
theorem add_left_cancel (a) {b c : ordinal} : a + b = a + c ↔ b = c :=
by simp only [le_antisymm_iff, add_le_add_iff_left]
theorem lt_succ {a b : ordinal} : a < succ b ↔ a ≤ b :=
by rw [← not_le, succ_le, not_lt]
theorem add_lt_add_iff_left (a) {b c : ordinal} : a + b < a + c ↔ b < c :=
by rw [← not_le, ← not_le, add_le_add_iff_left]
theorem lt_of_add_lt_add_right {a b c : ordinal} : a + b < c + b → a < c :=
lt_imp_lt_of_le_imp_le (λ h, add_le_add_right h _)
@[simp] theorem succ_lt_succ {a b : ordinal} : succ a < succ b ↔ a < b :=
by rw [lt_succ, succ_le]
@[simp] theorem succ_le_succ {a b : ordinal} : succ a ≤ succ b ↔ a ≤ b :=
le_iff_le_iff_lt_iff_lt.2 succ_lt_succ
theorem succ_inj {a b : ordinal} : succ a = succ b ↔ a = b :=
by simp only [le_antisymm_iff, succ_le_succ]
theorem add_le_add_iff_right {a b : ordinal} (n : ℕ) : a + n ≤ b + n ↔ a ≤ b :=
by induction n with n ih; [rw [nat.cast_zero, add_zero, add_zero],
rw [← nat_cast_succ, add_succ, add_succ, succ_le_succ, ih]]
theorem add_right_cancel {a b : ordinal} (n : ℕ) : a + n = b + n ↔ a = b :=
by simp only [le_antisymm_iff, add_le_add_iff_right]
/-! ### The zero ordinal -/
@[simp] theorem card_eq_zero {o} : card o = 0 ↔ o = 0 :=
⟨induction_on o $ λ α r _ h, begin
refine le_antisymm (le_of_not_lt $
λ hn, ne_zero_iff_nonempty.2 _ h) (ordinal.zero_le _),
rw [← succ_le, succ_zero] at hn, cases hn with f,
exact ⟨f punit.star⟩
end, λ e, by simp only [e, card_zero]⟩
@[simp] theorem type_eq_zero_of_empty [is_well_order α r] [is_empty α] : type r = 0 :=
card_eq_zero.symm.mpr eq_zero_of_is_empty
@[simp] theorem type_eq_zero_iff_is_empty [is_well_order α r] : type r = 0 ↔ is_empty α :=
(@card_eq_zero (type r)).symm.trans eq_zero_iff_is_empty
theorem type_ne_zero_iff_nonempty [is_well_order α r] : type r ≠ 0 ↔ nonempty α :=
(not_congr (@card_eq_zero (type r))).symm.trans ne_zero_iff_nonempty
protected lemma one_ne_zero : (1 : ordinal) ≠ 0 :=
type_ne_zero_iff_nonempty.2 ⟨punit.star⟩
instance : nontrivial ordinal.{u} :=
⟨⟨1, 0, ordinal.one_ne_zero⟩⟩
theorem zero_lt_one : (0 : ordinal) < 1 :=
lt_iff_le_and_ne.2 ⟨ordinal.zero_le _, ne.symm $ ordinal.one_ne_zero⟩
/-! ### The predecessor of an ordinal -/
/-- The ordinal predecessor of `o` is `o'` if `o = succ o'`,
and `o` otherwise. -/
def pred (o : ordinal.{u}) : ordinal.{u} :=
if h : ∃ a, o = succ a then classical.some h else o
@[simp] theorem pred_succ (o) : pred (succ o) = o :=
by have h : ∃ a, succ o = succ a := ⟨_, rfl⟩;
simpa only [pred, dif_pos h] using (succ_inj.1 $ classical.some_spec h).symm
theorem pred_le_self (o) : pred o ≤ o :=
if h : ∃ a, o = succ a then let ⟨a, e⟩ := h in
by rw [e, pred_succ]; exact le_of_lt (lt_succ_self _)
else by rw [pred, dif_neg h]
theorem pred_eq_iff_not_succ {o} : pred o = o ↔ ¬ ∃ a, o = succ a :=
⟨λ e ⟨a, e'⟩, by rw [e', pred_succ] at e; exact ne_of_lt (lt_succ_self _) e,
λ h, dif_neg h⟩
theorem pred_lt_iff_is_succ {o} : pred o < o ↔ ∃ a, o = succ a :=
iff.trans (by simp only [le_antisymm_iff, pred_le_self, true_and, not_le])
(iff_not_comm.1 pred_eq_iff_not_succ).symm
theorem succ_pred_iff_is_succ {o} : succ (pred o) = o ↔ ∃ a, o = succ a :=
⟨λ e, ⟨_, e.symm⟩, λ ⟨a, e⟩, by simp only [e, pred_succ]⟩
theorem succ_lt_of_not_succ {o} (h : ¬ ∃ a, o = succ a) {b} : succ b < o ↔ b < o :=
⟨lt_trans (lt_succ_self _), λ l,
lt_of_le_of_ne (succ_le.2 l) (λ e, h ⟨_, e.symm⟩)⟩
theorem lt_pred {a b} : a < pred b ↔ succ a < b :=
if h : ∃ a, b = succ a then let ⟨c, e⟩ := h in
by rw [e, pred_succ, succ_lt_succ]
else by simp only [pred, dif_neg h, succ_lt_of_not_succ h]
theorem pred_le {a b} : pred a ≤ b ↔ a ≤ succ b :=
le_iff_le_iff_lt_iff_lt.2 lt_pred
@[simp] theorem lift_is_succ {o} : (∃ a, lift o = succ a) ↔ (∃ a, o = succ a) :=
⟨λ ⟨a, h⟩,
let ⟨b, e⟩ := lift_down $ show a ≤ lift o, from le_of_lt $
h.symm ▸ lt_succ_self _ in
⟨b, lift_inj.1 $ by rw [h, ← e, lift_succ]⟩,
λ ⟨a, h⟩, ⟨lift a, by simp only [h, lift_succ]⟩⟩
@[simp] theorem lift_pred (o) : lift (pred o) = pred (lift o) :=
if h : ∃ a, o = succ a then
by cases h with a e; simp only [e, pred_succ, lift_succ]
else by rw [pred_eq_iff_not_succ.2 h,
pred_eq_iff_not_succ.2 (mt lift_is_succ.1 h)]
/-! ### Limit ordinals -/
/-- A limit ordinal is an ordinal which is not zero and not a successor. -/
def is_limit (o : ordinal) : Prop := o ≠ 0 ∧ ∀ a < o, succ a < o
theorem not_zero_is_limit : ¬ is_limit 0
| ⟨h, _⟩ := h rfl
theorem not_succ_is_limit (o) : ¬ is_limit (succ o)
| ⟨_, h⟩ := lt_irrefl _ (h _ (lt_succ_self _))
theorem not_succ_of_is_limit {o} (h : is_limit o) : ¬ ∃ a, o = succ a
| ⟨a, e⟩ := not_succ_is_limit a (e ▸ h)
theorem succ_lt_of_is_limit {o} (h : is_limit o) {a} : succ a < o ↔ a < o :=
⟨lt_trans (lt_succ_self _), h.2 _⟩
theorem le_succ_of_is_limit {o} (h : is_limit o) {a} : o ≤ succ a ↔ o ≤ a :=
le_iff_le_iff_lt_iff_lt.2 $ succ_lt_of_is_limit h
theorem limit_le {o} (h : is_limit o) {a} : o ≤ a ↔ ∀ x < o, x ≤ a :=
⟨λ h x l, le_trans (le_of_lt l) h,
λ H, (le_succ_of_is_limit h).1 $ le_of_not_lt $ λ hn,
not_lt_of_le (H _ hn) (lt_succ_self _)⟩
theorem lt_limit {o} (h : is_limit o) {a} : a < o ↔ ∃ x < o, a < x :=
by simpa only [not_ball, not_le] using not_congr (@limit_le _ h a)
@[simp] theorem lift_is_limit (o) : is_limit (lift o) ↔ is_limit o :=
and_congr (not_congr $ by simpa only [lift_zero] using @lift_inj o 0)
⟨λ H a h, lift_lt.1 $ by simpa only [lift_succ] using H _ (lift_lt.2 h),
λ H a h, let ⟨a', e⟩ := lift_down (le_of_lt h) in
by rw [← e, ← lift_succ, lift_lt];
rw [← e, lift_lt] at h; exact H a' h⟩
theorem is_limit.pos {o : ordinal} (h : is_limit o) : 0 < o :=
lt_of_le_of_ne (ordinal.zero_le _) h.1.symm
theorem is_limit.one_lt {o : ordinal} (h : is_limit o) : 1 < o :=
by simpa only [succ_zero] using h.2 _ h.pos
theorem is_limit.nat_lt {o : ordinal} (h : is_limit o) : ∀ n : ℕ, (n : ordinal) < o
| 0 := h.pos
| (n+1) := h.2 _ (is_limit.nat_lt n)
theorem zero_or_succ_or_limit (o : ordinal) :
o = 0 ∨ (∃ a, o = succ a) ∨ is_limit o :=
if o0 : o = 0 then or.inl o0 else
if h : ∃ a, o = succ a then or.inr (or.inl h) else
or.inr $ or.inr ⟨o0, λ a, (succ_lt_of_not_succ h).2⟩
/-- Main induction principle of ordinals: if one can prove a property by
induction at successor ordinals and at limit ordinals, then it holds for all ordinals. -/
@[elab_as_eliminator] def limit_rec_on {C : ordinal → Sort*}
(o : ordinal) (H₁ : C 0) (H₂ : ∀ o, C o → C (succ o))
(H₃ : ∀ o, is_limit o → (∀ o' < o, C o') → C o) : C o :=
wf.fix (λ o IH,
if o0 : o = 0 then by rw o0; exact H₁ else
if h : ∃ a, o = succ a then
by rw ← succ_pred_iff_is_succ.2 h; exact
H₂ _ (IH _ $ pred_lt_iff_is_succ.2 h)
else H₃ _ ⟨o0, λ a, (succ_lt_of_not_succ h).2⟩ IH) o
@[simp] theorem limit_rec_on_zero {C} (H₁ H₂ H₃) : @limit_rec_on C 0 H₁ H₂ H₃ = H₁ :=
by rw [limit_rec_on, well_founded.fix_eq, dif_pos rfl]; refl
@[simp] theorem limit_rec_on_succ {C} (o H₁ H₂ H₃) :
@limit_rec_on C (succ o) H₁ H₂ H₃ = H₂ o (@limit_rec_on C o H₁ H₂ H₃) :=
begin
have h : ∃ a, succ o = succ a := ⟨_, rfl⟩,
rw [limit_rec_on, well_founded.fix_eq,
dif_neg (succ_ne_zero o), dif_pos h],
generalize : limit_rec_on._proof_2 (succ o) h = h₂,
generalize : limit_rec_on._proof_3 (succ o) h = h₃,
revert h₂ h₃, generalize e : pred (succ o) = o', intros,
rw pred_succ at e, subst o', refl
end
@[simp] theorem limit_rec_on_limit {C} (o H₁ H₂ H₃ h) :
@limit_rec_on C o H₁ H₂ H₃ = H₃ o h (λ x h, @limit_rec_on C x H₁ H₂ H₃) :=
by rw [limit_rec_on, well_founded.fix_eq,
dif_neg h.1, dif_neg (not_succ_of_is_limit h)]; refl
lemma has_succ_of_is_limit {α} {r : α → α → Prop} [wo : is_well_order α r]
(h : (type r).is_limit) (x : α) : ∃y, r x y :=
begin
use enum r (typein r x).succ (h.2 _ (typein_lt_type r x)),
convert (enum_lt (typein_lt_type r x) _).mpr (lt_succ_self _), rw [enum_typein]
end
lemma type_subrel_lt (o : ordinal.{u}) :
type (subrel (<) {o' : ordinal | o' < o}) = ordinal.lift.{u+1} o :=
begin
refine quotient.induction_on o _,
rintro ⟨α, r, wo⟩, resetI, apply quotient.sound,
constructor, symmetry, refine (rel_iso.preimage equiv.ulift r).trans (typein_iso r)
end
lemma mk_initial_seg (o : ordinal.{u}) :
#{o' : ordinal | o' < o} = cardinal.lift.{u+1} o.card :=
by rw [lift_card, ←type_subrel_lt, card_type]
/-! ### Normal ordinal functions -/
/-- A normal ordinal function is a strictly increasing function which is
order-continuous, i.e., the image `f o` of a limit ordinal `o` is the sup of `f a` for
`a < o`. -/
def is_normal (f : ordinal → ordinal) : Prop :=
(∀ o, f o < f (succ o)) ∧ ∀ o, is_limit o → ∀ a, f o ≤ a ↔ ∀ b < o, f b ≤ a
theorem is_normal.limit_le {f} (H : is_normal f) : ∀ {o}, is_limit o →
∀ {a}, f o ≤ a ↔ ∀ b < o, f b ≤ a := H.2
theorem is_normal.limit_lt {f} (H : is_normal f) {o} (h : is_limit o) {a} :
a < f o ↔ ∃ b < o, a < f b :=
not_iff_not.1 $ by simpa only [exists_prop, not_exists, not_and, not_lt] using H.2 _ h a
theorem is_normal.lt_iff {f} (H : is_normal f) {a b} : f a < f b ↔ a < b :=
strict_mono.lt_iff_lt $ λ a b,
limit_rec_on b (not.elim (not_lt_of_le $ ordinal.zero_le _))
(λ b IH h, (lt_or_eq_of_le (lt_succ.1 h)).elim
(λ h, lt_trans (IH h) (H.1 _))
(λ e, e ▸ H.1 _))
(λ b l IH h, lt_of_lt_of_le (H.1 a)
((H.2 _ l _).1 (le_refl _) _ (l.2 _ h)))
theorem is_normal.le_iff {f} (H : is_normal f) {a b} : f a ≤ f b ↔ a ≤ b :=
le_iff_le_iff_lt_iff_lt.2 H.lt_iff
theorem is_normal.inj {f} (H : is_normal f) {a b} : f a = f b ↔ a = b :=
by simp only [le_antisymm_iff, H.le_iff]
theorem is_normal.le_self {f} (H : is_normal f) (a) : a ≤ f a :=
limit_rec_on a (ordinal.zero_le _)
(λ a IH, succ_le.2 $ lt_of_le_of_lt IH (H.1 _))
(λ a l IH, (limit_le l).2 $ λ b h,
le_trans (IH b h) $ H.le_iff.2 $ le_of_lt h)
theorem is_normal.le_set {f} (H : is_normal f) (p : ordinal → Prop)
(p0 : ∃ x, p x) (S)
(H₂ : ∀ o, S ≤ o ↔ ∀ a, p a → a ≤ o) {o} :
f S ≤ o ↔ ∀ a, p a → f a ≤ o :=
⟨λ h a pa, le_trans (H.le_iff.2 ((H₂ _).1 (le_refl _) _ pa)) h,
λ h, begin
revert H₂, apply limit_rec_on S,
{ intro H₂,
cases p0 with x px,
have := ordinal.le_zero.1 ((H₂ _).1 (ordinal.zero_le _) _ px),
rw this at px, exact h _ px },
{ intros S _ H₂,
rcases not_ball.1 (mt (H₂ S).2 $ not_le_of_lt $ lt_succ_self _) with ⟨a, h₁, h₂⟩,
exact le_trans (H.le_iff.2 $ succ_le.2 $ not_le.1 h₂) (h _ h₁) },
{ intros S L _ H₂, apply (H.2 _ L _).2, intros a h',
rcases not_ball.1 (mt (H₂ a).2 (not_le.2 h')) with ⟨b, h₁, h₂⟩,
exact le_trans (H.le_iff.2 $ le_of_lt $ not_le.1 h₂) (h _ h₁) }
end⟩
theorem is_normal.le_set' {f} (H : is_normal f) (p : α → Prop) (g : α → ordinal)
(p0 : ∃ x, p x) (S)
(H₂ : ∀ o, S ≤ o ↔ ∀ a, p a → g a ≤ o) {o} :
f S ≤ o ↔ ∀ a, p a → f (g a) ≤ o :=
(H.le_set (λ x, ∃ y, p y ∧ x = g y)
(let ⟨x, px⟩ := p0 in ⟨_, _, px, rfl⟩) _
(λ o, (H₂ o).trans ⟨λ H a ⟨y, h1, h2⟩, h2.symm ▸ H y h1,
λ H a h1, H (g a) ⟨a, h1, rfl⟩⟩)).trans
⟨λ H a h, H (g a) ⟨a, h, rfl⟩, λ H a ⟨y, h1, h2⟩, h2.symm ▸ H y h1⟩
theorem is_normal.refl : is_normal id :=
⟨λ x, lt_succ_self _, λ o l a, limit_le l⟩
theorem is_normal.trans {f g} (H₁ : is_normal f) (H₂ : is_normal g) :
is_normal (λ x, f (g x)) :=
⟨λ x, H₁.lt_iff.2 (H₂.1 _),
λ o l a, H₁.le_set' (< o) g ⟨_, l.pos⟩ _ (λ c, H₂.2 _ l _)⟩
theorem is_normal.is_limit {f} (H : is_normal f) {o} (l : is_limit o) :
is_limit (f o) :=
⟨ne_of_gt $ lt_of_le_of_lt (ordinal.zero_le _) $ H.lt_iff.2 l.pos,
λ a h, let ⟨b, h₁, h₂⟩ := (H.limit_lt l).1 h in
lt_of_le_of_lt (succ_le.2 h₂) (H.lt_iff.2 h₁)⟩
theorem add_le_of_limit {a b c : ordinal.{u}}
(h : is_limit b) : a + b ≤ c ↔ ∀ b' < b, a + b' ≤ c :=
⟨λ h b' l, le_trans (add_le_add_left (le_of_lt l) _) h,
λ H, le_of_not_lt $
induction_on a (λ α r _, induction_on b $ λ β s _ h H l, begin
resetI,
suffices : ∀ x : β, sum.lex r s (sum.inr x) (enum _ _ l),
{ cases enum _ _ l with x x,
{ cases this (enum s 0 h.pos) },
{ exact irrefl _ (this _) } },
intros x,
rw [← typein_lt_typein (sum.lex r s), typein_enum],
have := H _ (h.2 _ (typein_lt_type s x)),
rw [add_succ, succ_le] at this,
refine lt_of_le_of_lt (type_le'.2
⟨rel_embedding.of_monotone (λ a, _) (λ a b, _)⟩) this,
{ rcases a with ⟨a | b, h⟩,
{ exact sum.inl a },
{ exact sum.inr ⟨b, by cases h; assumption⟩ } },
{ rcases a with ⟨a | a, h₁⟩; rcases b with ⟨b | b, h₂⟩; cases h₁; cases h₂;
rintro ⟨⟩; constructor; assumption }
end) h H⟩
theorem add_is_normal (a : ordinal) : is_normal ((+) a) :=
⟨λ b, (add_lt_add_iff_left a).2 (lt_succ_self _),
λ b l c, add_le_of_limit l⟩
theorem add_is_limit (a) {b} : is_limit b → is_limit (a + b) :=
(add_is_normal a).is_limit
/-! ### Subtraction on ordinals-/
/-- `a - b` is the unique ordinal satisfying
`b + (a - b) = a` when `b ≤ a`. -/
def sub (a b : ordinal.{u}) : ordinal.{u} :=
omin {o | a ≤ b+o} ⟨a, le_add_left _ _⟩
instance : has_sub ordinal := ⟨sub⟩
theorem le_add_sub (a b : ordinal) : a ≤ b + (a - b) :=
omin_mem {o | a ≤ b+o} _
theorem sub_le {a b c : ordinal} : a - b ≤ c ↔ a ≤ b + c :=
⟨λ h, le_trans (le_add_sub a b) (add_le_add_left h _),
λ h, omin_le h⟩
theorem lt_sub {a b c : ordinal} : a < b - c ↔ c + a < b :=
lt_iff_lt_of_le_iff_le sub_le
theorem add_sub_cancel (a b : ordinal) : a + b - a = b :=
le_antisymm (sub_le.2 $ le_refl _)
((add_le_add_iff_left a).1 $ le_add_sub _ _)
theorem sub_eq_of_add_eq {a b c : ordinal} (h : a + b = c) : c - a = b :=
h ▸ add_sub_cancel _ _
theorem sub_le_self (a b : ordinal) : a - b ≤ a :=
sub_le.2 $ le_add_left _ _
theorem add_sub_cancel_of_le {a b : ordinal} (h : b ≤ a) : b + (a - b) = a :=
le_antisymm begin
rcases zero_or_succ_or_limit (a-b) with e|⟨c,e⟩|l,
{ simp only [e, add_zero, h] },
{ rw [e, add_succ, succ_le, ← lt_sub, e], apply lt_succ_self },
{ exact (add_le_of_limit l).2 (λ c l, le_of_lt (lt_sub.1 l)) }
end (le_add_sub _ _)
@[simp] theorem sub_zero (a : ordinal) : a - 0 = a :=
by simpa only [zero_add] using add_sub_cancel 0 a
@[simp] theorem zero_sub (a : ordinal) : 0 - a = 0 :=
by rw ← ordinal.le_zero; apply sub_le_self
@[simp] theorem sub_self (a : ordinal) : a - a = 0 :=
by simpa only [add_zero] using add_sub_cancel a 0
protected theorem sub_eq_zero_iff_le {a b : ordinal} : a - b = 0 ↔ a ≤ b :=
⟨λ h, by simpa only [h, add_zero] using le_add_sub a b,
λ h, by rwa [← ordinal.le_zero, sub_le, add_zero]⟩
theorem sub_sub (a b c : ordinal) : a - b - c = a - (b + c) :=
eq_of_forall_ge_iff $ λ d, by rw [sub_le, sub_le, sub_le, add_assoc]
theorem add_sub_add_cancel (a b c : ordinal) : a + b - (a + c) = b - c :=
by rw [← sub_sub, add_sub_cancel]
theorem sub_is_limit {a b} (l : is_limit a) (h : b < a) : is_limit (a - b) :=
⟨ne_of_gt $ lt_sub.2 $ by rwa add_zero,
λ c h, by rw [lt_sub, add_succ]; exact l.2 _ (lt_sub.1 h)⟩
@[simp] theorem one_add_omega : 1 + omega.{u} = omega :=
begin
refine le_antisymm _ (le_add_left _ _),
rw [omega, one_eq_lift_type_unit, ← lift_add, lift_le, type_add],
have : is_well_order unit empty_relation := by apply_instance,
refine ⟨rel_embedding.collapse (rel_embedding.of_monotone _ _)⟩,
{ apply sum.rec, exact λ _, 0, exact nat.succ },
{ intros a b, cases a; cases b; intro H; cases H with _ _ H _ _ H;
[cases H, exact nat.succ_pos _, exact nat.succ_lt_succ H] }
end
@[simp, priority 990]
theorem one_add_of_omega_le {o} (h : omega ≤ o) : 1 + o = o :=
by rw [← add_sub_cancel_of_le h, ← add_assoc, one_add_omega]
/-! ### Multiplication of ordinals-/
/-- The multiplication of ordinals `o₁` and `o₂` is the (well founded) lexicographic order on
`o₂ × o₁`. -/
instance : monoid ordinal.{u} :=
{ mul := λ a b, quotient.lift_on₂ a b
(λ ⟨α, r, wo⟩ ⟨β, s, wo'⟩, ⟦⟨β × α, prod.lex s r, by exactI prod.lex.is_well_order⟩⟧
: Well_order → Well_order → ordinal) $
λ ⟨α₁, r₁, o₁⟩ ⟨α₂, r₂, o₂⟩ ⟨β₁, s₁, p₁⟩ ⟨β₂, s₂, p₂⟩ ⟨f⟩ ⟨g⟩,
quot.sound ⟨rel_iso.prod_lex_congr g f⟩,
one := 1,
mul_assoc := λ a b c, quotient.induction_on₃ a b c $ λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩,
eq.symm $ quotient.sound ⟨⟨prod_assoc _ _ _, λ a b, begin
rcases a with ⟨⟨a₁, a₂⟩, a₃⟩,
rcases b with ⟨⟨b₁, b₂⟩, b₃⟩,
simp [prod.lex_def, and_or_distrib_left, or_assoc, and_assoc]
end⟩⟩,
mul_one := λ a, induction_on a $ λ α r _, quotient.sound
⟨⟨punit_prod _, λ a b, by rcases a with ⟨⟨⟨⟩⟩, a⟩; rcases b with ⟨⟨⟨⟩⟩, b⟩;
simp only [prod.lex_def, empty_relation, false_or];
simp only [eq_self_iff_true, true_and]; refl⟩⟩,
one_mul := λ a, induction_on a $ λ α r _, quotient.sound
⟨⟨prod_punit _, λ a b, by rcases a with ⟨a, ⟨⟨⟩⟩⟩; rcases b with ⟨b, ⟨⟨⟩⟩⟩;
simp only [prod.lex_def, empty_relation, and_false, or_false]; refl⟩⟩ }
@[simp] theorem type_mul {α β : Type u} (r : α → α → Prop) (s : β → β → Prop)
[is_well_order α r] [is_well_order β s] : type r * type s = type (prod.lex s r) := rfl
@[simp] theorem lift_mul (a b) : lift (a * b) = lift a * lift b :=
quotient.induction_on₂ a b $ λ ⟨α, r, _⟩ ⟨β, s, _⟩,
quotient.sound ⟨(rel_iso.preimage equiv.ulift _).trans
(rel_iso.prod_lex_congr (rel_iso.preimage equiv.ulift _)
(rel_iso.preimage equiv.ulift _)).symm⟩
@[simp] theorem card_mul (a b) : card (a * b) = card a * card b :=
quotient.induction_on₂ a b $ λ ⟨α, r, _⟩ ⟨β, s, _⟩,
mul_comm (mk β) (mk α)
@[simp] theorem mul_zero (a : ordinal) : a * 0 = 0 :=
induction_on a $ λ α _ _, by exactI type_eq_zero_of_empty
@[simp] theorem zero_mul (a : ordinal) : 0 * a = 0 :=
induction_on a $ λ α _ _, by exactI type_eq_zero_of_empty
theorem mul_add (a b c : ordinal) : a * (b + c) = a * b + a * c :=
quotient.induction_on₃ a b c $ λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩,
quotient.sound ⟨⟨sum_prod_distrib _ _ _, begin
rintro ⟨a₁|a₁, a₂⟩ ⟨b₁|b₁, b₂⟩; simp only [prod.lex_def,
sum.lex_inl_inl, sum.lex.sep, sum.lex_inr_inl, sum.lex_inr_inr,
sum_prod_distrib_apply_left, sum_prod_distrib_apply_right];
simp only [sum.inl.inj_iff, true_or, false_and, false_or]
end⟩⟩
@[simp] theorem mul_add_one (a b : ordinal) : a * (b + 1) = a * b + a :=
by simp only [mul_add, mul_one]
@[simp] theorem mul_succ (a b : ordinal) : a * succ b = a * b + a := mul_add_one _ _
theorem mul_le_mul_left {a b} (c : ordinal) : a ≤ b → c * a ≤ c * b :=
quotient.induction_on₃ a b c $ λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨f⟩, begin
resetI,
refine type_le'.2 ⟨rel_embedding.of_monotone
(λ a, (f a.1, a.2))
(λ a b h, _)⟩, clear_,
cases h with a₁ b₁ a₂ b₂ h' a b₁ b₂ h',
{ exact prod.lex.left _ _ (f.to_rel_embedding.map_rel_iff.2 h') },
{ exact prod.lex.right _ h' }
end
theorem mul_le_mul_right {a b} (c : ordinal) : a ≤ b → a * c ≤ b * c :=
quotient.induction_on₃ a b c $ λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨f⟩, begin
resetI,
refine type_le'.2 ⟨rel_embedding.of_monotone
(λ a, (a.1, f a.2))
(λ a b h, _)⟩,
cases h with a₁ b₁ a₂ b₂ h' a b₁ b₂ h',
{ exact prod.lex.left _ _ h' },
{ exact prod.lex.right _ (f.to_rel_embedding.map_rel_iff.2 h') }
end
theorem mul_le_mul {a b c d : ordinal} (h₁ : a ≤ c) (h₂ : b ≤ d) : a * b ≤ c * d :=
le_trans (mul_le_mul_left _ h₂) (mul_le_mul_right _ h₁)
private lemma mul_le_of_limit_aux {α β r s} [is_well_order α r] [is_well_order β s]
{c} (h : is_limit (type s)) (H : ∀ b' < type s, type r * b' ≤ c)
(l : c < type r * type s) : false :=
begin
suffices : ∀ a b, prod.lex s r (b, a) (enum _ _ l),
{ cases enum _ _ l with b a, exact irrefl _ (this _ _) },
intros a b,
rw [← typein_lt_typein (prod.lex s r), typein_enum],
have := H _ (h.2 _ (typein_lt_type s b)),
rw [mul_succ] at this,
have := lt_of_lt_of_le ((add_lt_add_iff_left _).2
(typein_lt_type _ a)) this,
refine lt_of_le_of_lt _ this,
refine (type_le'.2 _),
constructor,
refine rel_embedding.of_monotone (λ a, _) (λ a b, _),
{ rcases a with ⟨⟨b', a'⟩, h⟩,
by_cases e : b = b',
{ refine sum.inr ⟨a', _⟩,
subst e, cases h with _ _ _ _ h _ _ _ h,
{ exact (irrefl _ h).elim },
{ exact h } },
{ refine sum.inl (⟨b', _⟩, a'),
cases h with _ _ _ _ h _ _ _ h,
{ exact h }, { exact (e rfl).elim } } },
{ rcases a with ⟨⟨b₁, a₁⟩, h₁⟩,
rcases b with ⟨⟨b₂, a₂⟩, h₂⟩,
intro h, by_cases e₁ : b = b₁; by_cases e₂ : b = b₂,
{ substs b₁ b₂,
simpa only [subrel_val, prod.lex_def, @irrefl _ s _ b, true_and, false_or, eq_self_iff_true,
dif_pos, sum.lex_inr_inr] using h },
{ subst b₁,
simp only [subrel_val, prod.lex_def, e₂, prod.lex_def, dif_pos, subrel_val, eq_self_iff_true,
or_false, dif_neg, not_false_iff, sum.lex_inr_inl, false_and] at h ⊢,
cases h₂; [exact asymm h h₂_h, exact e₂ rfl] },
{ simp only [e₂, dif_pos, eq_self_iff_true, dif_neg e₁, not_false_iff, sum.lex.sep] },
{ simpa only [dif_neg e₁, dif_neg e₂, prod.lex_def, subrel_val, subtype.mk_eq_mk,
sum.lex_inl_inl] using h } }
end
theorem mul_le_of_limit {a b c : ordinal.{u}}
(h : is_limit b) : a * b ≤ c ↔ ∀ b' < b, a * b' ≤ c :=
⟨λ h b' l, le_trans (mul_le_mul_left _ (le_of_lt l)) h,
λ H, le_of_not_lt $ induction_on a (λ α r _, induction_on b $ λ β s _,
by exactI mul_le_of_limit_aux) h H⟩
theorem mul_is_normal {a : ordinal} (h : 0 < a) : is_normal ((*) a) :=
⟨λ b, by rw mul_succ; simpa only [add_zero] using (add_lt_add_iff_left (a*b)).2 h,
λ b l c, mul_le_of_limit l⟩
theorem lt_mul_of_limit {a b c : ordinal.{u}}
(h : is_limit c) : a < b * c ↔ ∃ c' < c, a < b * c' :=
by simpa only [not_ball, not_le] using not_congr (@mul_le_of_limit b c a h)
theorem mul_lt_mul_iff_left {a b c : ordinal} (a0 : 0 < a) : a * b < a * c ↔ b < c :=
(mul_is_normal a0).lt_iff
theorem mul_le_mul_iff_left {a b c : ordinal} (a0 : 0 < a) : a * b ≤ a * c ↔ b ≤ c :=
(mul_is_normal a0).le_iff
theorem mul_lt_mul_of_pos_left {a b c : ordinal}
(h : a < b) (c0 : 0 < c) : c * a < c * b :=
(mul_lt_mul_iff_left c0).2 h
theorem mul_pos {a b : ordinal} (h₁ : 0 < a) (h₂ : 0 < b) : 0 < a * b :=
by simpa only [mul_zero] using mul_lt_mul_of_pos_left h₂ h₁
theorem mul_ne_zero {a b : ordinal} : a ≠ 0 → b ≠ 0 → a * b ≠ 0 :=
by simpa only [ordinal.pos_iff_ne_zero] using mul_pos
theorem le_of_mul_le_mul_left {a b c : ordinal}
(h : c * a ≤ c * b) (h0 : 0 < c) : a ≤ b :=
le_imp_le_of_lt_imp_lt (λ h', mul_lt_mul_of_pos_left h' h0) h
theorem mul_right_inj {a b c : ordinal} (a0 : 0 < a) : a * b = a * c ↔ b = c :=
(mul_is_normal a0).inj
theorem mul_is_limit {a b : ordinal}
(a0 : 0 < a) : is_limit b → is_limit (a * b) :=
(mul_is_normal a0).is_limit
theorem mul_is_limit_left {a b : ordinal}
(l : is_limit a) (b0 : 0 < b) : is_limit (a * b) :=
begin
rcases zero_or_succ_or_limit b with rfl|⟨b,rfl⟩|lb,
{ exact (lt_irrefl _).elim b0 },
{ rw mul_succ, exact add_is_limit _ l },
{ exact mul_is_limit l.pos lb }
end
/-! ### Division on ordinals -/
protected lemma div_aux (a b : ordinal.{u}) (h : b ≠ 0) : set.nonempty {o | a < b * succ o} :=
⟨a, succ_le.1 $
by simpa only [succ_zero, one_mul]
using mul_le_mul_right (succ a) (succ_le.2 (ordinal.pos_iff_ne_zero.2 h))⟩
/-- `a / b` is the unique ordinal `o` satisfying
`a = b * o + o'` with `o' < b`. -/
protected def div (a b : ordinal.{u}) : ordinal.{u} :=
if h : b = 0 then 0 else omin {o | a < b * succ o} (ordinal.div_aux a b h)
instance : has_div ordinal := ⟨ordinal.div⟩
@[simp] theorem div_zero (a : ordinal) : a / 0 = 0 := dif_pos rfl
lemma div_def (a) {b : ordinal} (h : b ≠ 0) :
a / b = omin {o | a < b * succ o} (ordinal.div_aux a b h) := dif_neg h
theorem lt_mul_succ_div (a) {b : ordinal} (h : b ≠ 0) : a < b * succ (a / b) :=
by rw div_def a h; exact omin_mem {o | a < b * succ o} _
theorem lt_mul_div_add (a) {b : ordinal} (h : b ≠ 0) : a < b * (a / b) + b :=
by simpa only [mul_succ] using lt_mul_succ_div a h
theorem div_le {a b c : ordinal} (b0 : b ≠ 0) : a / b ≤ c ↔ a < b * succ c :=
⟨λ h, lt_of_lt_of_le (lt_mul_succ_div a b0) (mul_le_mul_left _ $ succ_le_succ.2 h),
λ h, by rw div_def a b0; exact omin_le h⟩
theorem lt_div {a b c : ordinal} (c0 : c ≠ 0) : a < b / c ↔ c * succ a ≤ b :=
by rw [← not_le, div_le c0, not_lt]
theorem le_div {a b c : ordinal} (c0 : c ≠ 0) :
a ≤ b / c ↔ c * a ≤ b :=
begin
apply limit_rec_on a,
{ simp only [mul_zero, ordinal.zero_le] },
{ intros, rw [succ_le, lt_div c0] },
{ simp only [mul_le_of_limit, limit_le, iff_self, forall_true_iff] {contextual := tt} }
end
theorem div_lt {a b c : ordinal} (b0 : b ≠ 0) :
a / b < c ↔ a < b * c :=
lt_iff_lt_of_le_iff_le $ le_div b0
theorem div_le_of_le_mul {a b c : ordinal} (h : a ≤ b * c) : a / b ≤ c :=
if b0 : b = 0 then by simp only [b0, div_zero, ordinal.zero_le] else
(div_le b0).2 $ lt_of_le_of_lt h $
mul_lt_mul_of_pos_left (lt_succ_self _) (ordinal.pos_iff_ne_zero.2 b0)
theorem mul_lt_of_lt_div {a b c : ordinal} : a < b / c → c * a < b :=
lt_imp_lt_of_le_imp_le div_le_of_le_mul
@[simp] theorem zero_div (a : ordinal) : 0 / a = 0 :=
ordinal.le_zero.1 $ div_le_of_le_mul $ ordinal.zero_le _
theorem mul_div_le (a b : ordinal) : b * (a / b) ≤ a :=
if b0 : b = 0 then by simp only [b0, zero_mul, ordinal.zero_le] else (le_div b0).1 (le_refl _)
theorem mul_add_div (a) {b : ordinal} (b0 : b ≠ 0) (c) : (b * a + c) / b = a + c / b :=
begin
apply le_antisymm,
{ apply (div_le b0).2,
rw [mul_succ, mul_add, add_assoc, add_lt_add_iff_left],
apply lt_mul_div_add _ b0 },
{ rw [le_div b0, mul_add, add_le_add_iff_left],
apply mul_div_le }
end
theorem div_eq_zero_of_lt {a b : ordinal} (h : a < b) : a / b = 0 :=
begin
rw [← ordinal.le_zero, div_le $ ordinal.pos_iff_ne_zero.1 $ lt_of_le_of_lt (ordinal.zero_le _) h],
simpa only [succ_zero, mul_one] using h
end
@[simp] theorem mul_div_cancel (a) {b : ordinal} (b0 : b ≠ 0) : b * a / b = a :=
by simpa only [add_zero, zero_div] using mul_add_div a b0 0
@[simp] theorem div_one (a : ordinal) : a / 1 = a :=
by simpa only [one_mul] using mul_div_cancel a ordinal.one_ne_zero
@[simp] theorem div_self {a : ordinal} (h : a ≠ 0) : a / a = 1 :=
by simpa only [mul_one] using mul_div_cancel 1 h
theorem mul_sub (a b c : ordinal) : a * (b - c) = a * b - a * c :=
if a0 : a = 0 then by simp only [a0, zero_mul, sub_self] else
eq_of_forall_ge_iff $ λ d,
by rw [sub_le, ← le_div a0, sub_le, ← le_div a0, mul_add_div _ a0]
theorem is_limit_add_iff {a b} : is_limit (a + b) ↔ is_limit b ∨ (b = 0 ∧ is_limit a) :=
begin
split; intro h,
{ by_cases h' : b = 0,
{ rw [h', add_zero] at h, right, exact ⟨h', h⟩ },
left, rw [←add_sub_cancel a b], apply sub_is_limit h,
suffices : a + 0 < a + b, simpa only [add_zero],
rwa [add_lt_add_iff_left, ordinal.pos_iff_ne_zero] },
rcases h with h|⟨rfl, h⟩, exact add_is_limit a h, simpa only [add_zero]
end
theorem dvd_add_iff : ∀ {a b c : ordinal}, a ∣ b → (a ∣ b + c ↔ a ∣ c)
| a _ c ⟨b, rfl⟩ :=
⟨λ ⟨d, e⟩, ⟨d - b, by rw [mul_sub, ← e, add_sub_cancel]⟩,
λ ⟨d, e⟩, by { rw [e, ← mul_add], apply dvd_mul_right }⟩
theorem dvd_add {a b c : ordinal} (h₁ : a ∣ b) : a ∣ c → a ∣ b + c :=
(dvd_add_iff h₁).2
theorem dvd_zero (a : ordinal) : a ∣ 0 := ⟨_, (mul_zero _).symm⟩
theorem zero_dvd {a : ordinal} : 0 ∣ a ↔ a = 0 :=
⟨λ ⟨h, e⟩, by simp only [e, zero_mul], λ e, e.symm ▸ dvd_zero _⟩
theorem one_dvd (a : ordinal) : 1 ∣ a := ⟨a, (one_mul _).symm⟩
theorem div_mul_cancel : ∀ {a b : ordinal}, a ≠ 0 → a ∣ b → a * (b / a) = b
| a _ a0 ⟨b, rfl⟩ := by rw [mul_div_cancel _ a0]
theorem le_of_dvd : ∀ {a b : ordinal}, b ≠ 0 → a ∣ b → a ≤ b
| a _ b0 ⟨b, rfl⟩ := by simpa only [mul_one] using mul_le_mul_left a
(one_le_iff_ne_zero.2 (λ h : b = 0, by simpa only [h, mul_zero] using b0))
theorem dvd_antisymm {a b : ordinal} (h₁ : a ∣ b) (h₂ : b ∣ a) : a = b :=
if a0 : a = 0 then by subst a; exact (zero_dvd.1 h₁).symm else
if b0 : b = 0 then by subst b; exact zero_dvd.1 h₂ else
le_antisymm (le_of_dvd b0 h₁) (le_of_dvd a0 h₂)
/-- `a % b` is the unique ordinal `o'` satisfying
`a = b * o + o'` with `o' < b`. -/
instance : has_mod ordinal := ⟨λ a b, a - b * (a / b)⟩
theorem mod_def (a b : ordinal) : a % b = a - b * (a / b) := rfl
@[simp] theorem mod_zero (a : ordinal) : a % 0 = a :=
by simp only [mod_def, div_zero, zero_mul, sub_zero]
theorem mod_eq_of_lt {a b : ordinal} (h : a < b) : a % b = a :=
by simp only [mod_def, div_eq_zero_of_lt h, mul_zero, sub_zero]
@[simp] theorem zero_mod (b : ordinal) : 0 % b = 0 :=
by simp only [mod_def, zero_div, mul_zero, sub_self]
theorem div_add_mod (a b : ordinal) : b * (a / b) + a % b = a :=
add_sub_cancel_of_le $ mul_div_le _ _
theorem mod_lt (a) {b : ordinal} (h : b ≠ 0) : a % b < b :=
(add_lt_add_iff_left (b * (a / b))).1 $
by rw div_add_mod; exact lt_mul_div_add a h
@[simp] theorem mod_self (a : ordinal) : a % a = 0 :=
if a0 : a = 0 then by simp only [a0, zero_mod] else
by simp only [mod_def, div_self a0, mul_one, sub_self]
@[simp] theorem mod_one (a : ordinal) : a % 1 = 0 :=
by simp only [mod_def, div_one, one_mul, sub_self]
/-! ### Supremum of a family of ordinals -/
/-- The supremum of a family of ordinals -/
def sup {ι} (f : ι → ordinal) : ordinal :=
omin {c | ∀ i, f i ≤ c}
⟨(sup (cardinal.succ ∘ card ∘ f)).ord, λ i, le_of_lt $
cardinal.lt_ord.2 (lt_of_lt_of_le (cardinal.lt_succ_self _) (le_sup _ _))⟩
theorem le_sup {ι} (f : ι → ordinal) : ∀ i, f i ≤ sup f :=
omin_mem {c | ∀ i, f i ≤ c} _
theorem sup_le {ι} {f : ι → ordinal} {a} : sup f ≤ a ↔ ∀ i, f i ≤ a :=
⟨λ h i, le_trans (le_sup _ _) h, λ h, omin_le h⟩
theorem lt_sup {ι} {f : ι → ordinal} {a} : a < sup f ↔ ∃ i, a < f i :=
by simpa only [not_forall, not_le] using not_congr (@sup_le _ f a)
theorem is_normal.sup {f} (H : is_normal f)
{ι} {g : ι → ordinal} (h : nonempty ι) : f (sup g) = sup (f ∘ g) :=
eq_of_forall_ge_iff $ λ a,
by rw [sup_le, comp, H.le_set' (λ_:ι, true) g (let ⟨i⟩ := h in ⟨i, ⟨⟩⟩)];
intros; simp only [sup_le, true_implies_iff]
theorem sup_ord {ι} (f : ι → cardinal) : sup (λ i, (f i).ord) = (cardinal.sup f).ord :=
eq_of_forall_ge_iff $ λ a, by simp only [sup_le, cardinal.ord_le, cardinal.sup_le]
lemma sup_succ {ι} (f : ι → ordinal) : sup (λ i, succ (f i)) ≤ succ (sup f) :=
by { rw [ordinal.sup_le], intro i, rw ordinal.succ_le_succ, apply ordinal.le_sup }
lemma unbounded_range_of_sup_ge {α β : Type u} (r : α → α → Prop) [is_well_order α r] (f : β → α)
(h : type r ≤ sup.{u u} (typein r ∘ f)) : unbounded r (range f) :=
begin
apply (not_bounded_iff _).mp, rintro ⟨x, hx⟩, apply not_lt_of_ge h,
refine lt_of_le_of_lt _ (typein_lt_type r x), rw [sup_le], intro y,
apply le_of_lt, rw typein_lt_typein, apply hx, apply mem_range_self
end
/-- The supremum of a family of ordinals indexed by the set
of ordinals less than some `o : ordinal.{u}`.
(This is not a special case of `sup` over the subtype,
because `{a // a < o} : Type (u+1)` and `sup` only works over
families in `Type u`.) -/
def bsup (o : ordinal.{u}) : (Π a < o, ordinal.{max u v}) → ordinal.{max u v} :=
match o, o.out, o.out_eq with
| _, ⟨α, r, _⟩, rfl, f := by exactI sup (λ a, f (typein r a) (typein_lt_type _ _))
end
theorem bsup_le {o f a} : bsup.{u v} o f ≤ a ↔ ∀ i h, f i h ≤ a :=
match o, o.out, o.out_eq, f :
∀ o w (e : ⟦w⟧ = o) (f : Π (a : ordinal.{u}), a < o → ordinal.{(max u v)}),
bsup._match_1 o w e f ≤ a ↔ ∀ i h, f i h ≤ a with
| _, ⟨α, r, _⟩, rfl, f := by rw [bsup._match_1, sup_le]; exactI
⟨λ H i h, by simpa only [typein_enum] using H (enum r i h), λ H b, H _ _⟩
end
theorem bsup_type (r : α → α → Prop) [is_well_order α r] (f) :
bsup (type r) f = sup (λ a, f (typein r a) (typein_lt_type _ _)) :=
eq_of_forall_ge_iff $ λ o,
by rw [bsup_le, sup_le]; exact
⟨λ H b, H _ _, λ H i h, by simpa only [typein_enum] using H (enum r i h)⟩
theorem le_bsup {o} (f : Π a < o, ordinal) (i h) : f i h ≤ bsup o f :=
bsup_le.1 (le_refl _) _ _
theorem lt_bsup {o : ordinal} {f : Π a < o, ordinal}
(hf : ∀{a a'} (ha : a < o) (ha' : a' < o), a < a' → f a ha < f a' ha')
(ho : o.is_limit) (i h) : f i h < bsup o f :=
lt_of_lt_of_le (hf _ _ $ lt_succ_self i) (le_bsup f i.succ $ ho.2 _ h)
theorem bsup_id {o} (ho : is_limit o) : bsup.{u u} o (λ x _, x) = o :=
begin
apply le_antisymm, rw [bsup_le], intro i, apply le_of_lt,
rw [←not_lt], intro h, apply lt_irrefl (bsup.{u u} o (λ x _, x)),
apply lt_of_le_of_lt _ (lt_bsup _ ho _ h), refl, intros, assumption
end
theorem is_normal.bsup {f} (H : is_normal f)
{o : ordinal} : ∀ (g : Π a < o, ordinal) (h : o ≠ 0),
f (bsup o g) = bsup o (λ a h, f (g a h)) :=
induction_on o $ λ α r _ g h,
by resetI; rw [bsup_type,
H.sup (type_ne_zero_iff_nonempty.1 h), bsup_type]
theorem is_normal.bsup_eq {f} (H : is_normal f) {o : ordinal} (h : is_limit o) :
bsup.{u} o (λx _, f x) = f o :=
by { rw [←is_normal.bsup.{u u} H (λ x _, x) h.1, bsup_id h] }
/-! ### Ordinal exponential -/
/-- The ordinal exponential, defined by transfinite recursion. -/
def power (a b : ordinal) : ordinal :=
if a = 0 then 1 - b else
limit_rec_on b 1 (λ _ IH, IH * a) (λ b _, bsup.{u u} b)
instance : has_pow ordinal ordinal := ⟨power⟩
local infixr ^ := @pow ordinal ordinal ordinal.has_pow
theorem zero_power' (a : ordinal) : 0 ^ a = 1 - a :=
by simp only [pow, power, if_pos rfl]
@[simp] theorem zero_power {a : ordinal} (a0 : a ≠ 0) : 0 ^ a = 0 :=
by rwa [zero_power', ordinal.sub_eq_zero_iff_le, one_le_iff_ne_zero]
@[simp] theorem power_zero (a : ordinal) : a ^ 0 = 1 :=
by by_cases a = 0; [simp only [pow, power, if_pos h, sub_zero],
simp only [pow, power, if_neg h, limit_rec_on_zero]]
@[simp] theorem power_succ (a b : ordinal) : a ^ succ b = a ^ b * a :=
if h : a = 0 then by subst a; simp only [zero_power (succ_ne_zero _), mul_zero]
else by simp only [pow, power, limit_rec_on_succ, if_neg h]
theorem power_limit {a b : ordinal} (a0 : a ≠ 0) (h : is_limit b) :
a ^ b = bsup.{u u} b (λ c _, a ^ c) :=
by simp only [pow, power, if_neg a0]; rw limit_rec_on_limit _ _ _ _ h; refl
theorem power_le_of_limit {a b c : ordinal} (a0 : a ≠ 0) (h : is_limit b) :
a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c :=
by rw [power_limit a0 h, bsup_le]
theorem lt_power_of_limit {a b c : ordinal} (b0 : b ≠ 0) (h : is_limit c) :
a < b ^ c ↔ ∃ c' < c, a < b ^ c' :=
by rw [← not_iff_not, not_exists]; simp only [not_lt, power_le_of_limit b0 h, exists_prop, not_and]
@[simp] theorem power_one (a : ordinal) : a ^ 1 = a :=
by rw [← succ_zero, power_succ]; simp only [power_zero, one_mul]
@[simp] theorem one_power (a : ordinal) : 1 ^ a = 1 :=
begin
apply limit_rec_on a,
{ simp only [power_zero] },
{ intros _ ih, simp only [power_succ, ih, mul_one] },
refine λ b l IH, eq_of_forall_ge_iff (λ c, _),
rw [power_le_of_limit ordinal.one_ne_zero l],
exact ⟨λ H, by simpa only [power_zero] using H 0 l.pos,
λ H b' h, by rwa IH _ h⟩,
end
theorem power_pos {a : ordinal} (b)
(a0 : 0 < a) : 0 < a ^ b :=
begin
have h0 : 0 < a ^ 0, {simp only [power_zero, zero_lt_one]},
apply limit_rec_on b,
{ exact h0 },
{ intros b IH, rw [power_succ],
exact mul_pos IH a0 },
{ exact λ b l _, (lt_power_of_limit (ordinal.pos_iff_ne_zero.1 a0) l).2
⟨0, l.pos, h0⟩ },
end
theorem power_ne_zero {a : ordinal} (b)
(a0 : a ≠ 0) : a ^ b ≠ 0 :=
ordinal.pos_iff_ne_zero.1 $ power_pos b $ ordinal.pos_iff_ne_zero.2 a0
theorem power_is_normal {a : ordinal} (h : 1 < a) : is_normal ((^) a) :=
have a0 : 0 < a, from lt_trans zero_lt_one h,
⟨λ b, by simpa only [mul_one, power_succ] using
(mul_lt_mul_iff_left (power_pos b a0)).2 h,
λ b l c, power_le_of_limit (ne_of_gt a0) l⟩
theorem power_lt_power_iff_right {a b c : ordinal}
(a1 : 1 < a) : a ^ b < a ^ c ↔ b < c :=
(power_is_normal a1).lt_iff
theorem power_le_power_iff_right {a b c : ordinal}
(a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c :=
(power_is_normal a1).le_iff
theorem power_right_inj {a b c : ordinal}
(a1 : 1 < a) : a ^ b = a ^ c ↔ b = c :=
(power_is_normal a1).inj
theorem power_is_limit {a b : ordinal}
(a1 : 1 < a) : is_limit b → is_limit (a ^ b) :=
(power_is_normal a1).is_limit
theorem power_is_limit_left {a b : ordinal}
(l : is_limit a) (hb : b ≠ 0) : is_limit (a ^ b) :=
begin
rcases zero_or_succ_or_limit b with e|⟨b,rfl⟩|l',
{ exact absurd e hb },
{ rw power_succ,
exact mul_is_limit (power_pos _ l.pos) l },
{ exact power_is_limit l.one_lt l' }
end
theorem power_le_power_right {a b c : ordinal}
(h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c :=
begin
cases lt_or_eq_of_le (one_le_iff_pos.2 h₁) with h₁ h₁,
{ exact (power_le_power_iff_right h₁).2 h₂ },
{ subst a, simp only [one_power] }
end
theorem power_le_power_left {a b : ordinal} (c)
(ab : a ≤ b) : a ^ c ≤ b ^ c :=
begin
by_cases a0 : a = 0,
{ subst a, by_cases c0 : c = 0,
{ subst c, simp only [power_zero] },
{ simp only [zero_power c0, ordinal.zero_le] } },
{ apply limit_rec_on c,
{ simp only [power_zero] },
{ intros c IH, simpa only [power_succ] using mul_le_mul IH ab },
{ exact λ c l IH, (power_le_of_limit a0 l).2
(λ b' h, le_trans (IH _ h) (power_le_power_right
(lt_of_lt_of_le (ordinal.pos_iff_ne_zero.2 a0) ab) (le_of_lt h))) } }
end
theorem le_power_self {a : ordinal} (b) (a1 : 1 < a) : b ≤ a ^ b :=
(power_is_normal a1).le_self _
theorem power_lt_power_left_of_succ {a b c : ordinal}
(ab : a < b) : a ^ succ c < b ^ succ c :=
by rw [power_succ, power_succ]; exact
lt_of_le_of_lt
(mul_le_mul_right _ $ power_le_power_left _ $ le_of_lt ab)
(mul_lt_mul_of_pos_left ab (power_pos _ (lt_of_le_of_lt (ordinal.zero_le _) ab)))
theorem power_add (a b c : ordinal) : a ^ (b + c) = a ^ b * a ^ c :=
begin
by_cases a0 : a = 0,
{ subst a,
by_cases c0 : c = 0, {simp only [c0, add_zero, power_zero, mul_one]},
have : b+c ≠ 0 := ne_of_gt (lt_of_lt_of_le
(ordinal.pos_iff_ne_zero.2 c0) (le_add_left _ _)),
simp only [zero_power c0, zero_power this, mul_zero] },
cases eq_or_lt_of_le (one_le_iff_ne_zero.2 a0) with a1 a1,
{ subst a1, simp only [one_power, mul_one] },
apply limit_rec_on c,
{ simp only [add_zero, power_zero, mul_one] },
{ intros c IH,
rw [add_succ, power_succ, IH, power_succ, mul_assoc] },
{ intros c l IH,
refine eq_of_forall_ge_iff (λ d, (((power_is_normal a1).trans
(add_is_normal b)).limit_le l).trans _),
simp only [IH] {contextual := tt},
exact (((mul_is_normal $ power_pos b (ordinal.pos_iff_ne_zero.2 a0)).trans
(power_is_normal a1)).limit_le l).symm }
end
theorem power_dvd_power (a) {b c : ordinal}
(h : b ≤ c) : a ^ b ∣ a ^ c :=
by { rw [← add_sub_cancel_of_le h, power_add], apply dvd_mul_right }
theorem power_dvd_power_iff {a b c : ordinal}
(a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c :=
⟨λ h, le_of_not_lt $ λ hn,
not_le_of_lt ((power_lt_power_iff_right a1).2 hn) $
le_of_dvd (power_ne_zero _ $ one_le_iff_ne_zero.1 $ le_of_lt a1) h,
power_dvd_power _⟩
theorem power_mul (a b c : ordinal) : a ^ (b * c) = (a ^ b) ^ c :=
begin
by_cases b0 : b = 0, {simp only [b0, zero_mul, power_zero, one_power]},
by_cases a0 : a = 0,
{ subst a,
by_cases c0 : c = 0, {simp only [c0, mul_zero, power_zero]},
simp only [zero_power b0, zero_power c0, zero_power (mul_ne_zero b0 c0)] },
cases eq_or_lt_of_le (one_le_iff_ne_zero.2 a0) with a1 a1,
{ subst a1, simp only [one_power] },
apply limit_rec_on c,
{ simp only [mul_zero, power_zero] },
{ intros c IH,
rw [mul_succ, power_add, IH, power_succ] },
{ intros c l IH,
refine eq_of_forall_ge_iff (λ d, (((power_is_normal a1).trans
(mul_is_normal (ordinal.pos_iff_ne_zero.2 b0))).limit_le l).trans _),
simp only [IH] {contextual := tt},
exact (power_le_of_limit (power_ne_zero _ a0) l).symm }
end
/-! ### Ordinal logarithm -/
/-- The ordinal logarithm is the solution `u` to the equation
`x = b ^ u * v + w` where `v < b` and `w < b`. -/
def log (b : ordinal) (x : ordinal) : ordinal :=
if h : 1 < b then pred $
omin {o | x < b^o} ⟨succ x, succ_le.1 (le_power_self _ h)⟩
else 0
@[simp] theorem log_not_one_lt {b : ordinal} (b1 : ¬ 1 < b) (x : ordinal) : log b x = 0 :=
by simp only [log, dif_neg b1]
theorem log_def {b : ordinal} (b1 : 1 < b) (x : ordinal) : log b x =
pred (omin {o | x < b^o} (log._proof_1 b x b1)) :=
by simp only [log, dif_pos b1]
@[simp] theorem log_zero (b : ordinal) : log b 0 = 0 :=
if b1 : 1 < b then
by rw [log_def b1, ← ordinal.le_zero, pred_le];
apply omin_le; change 0<b^succ 0;
rw [succ_zero, power_one];
exact lt_trans zero_lt_one b1
else by simp only [log_not_one_lt b1]
theorem succ_log_def {b x : ordinal} (b1 : 1 < b) (x0 : 0 < x) : succ (log b x) =
omin {o | x < b^o} (log._proof_1 b x b1) :=
begin
let t := omin {o | x < b^o} (log._proof_1 b x b1),
have : x < b ^ t := omin_mem {o | x < b^o} _,
rcases zero_or_succ_or_limit t with h|h|h,
{ refine (not_lt_of_le (one_le_iff_pos.2 x0) _).elim,
simpa only [h, power_zero] },
{ rw [show log b x = pred t, from log_def b1 x,
succ_pred_iff_is_succ.2 h] },
{ rcases (lt_power_of_limit (ne_of_gt $ lt_trans zero_lt_one b1) h).1 this with ⟨a, h₁, h₂⟩,
exact (not_le_of_lt h₁).elim (le_omin.1 (le_refl t) a h₂) }
end
theorem lt_power_succ_log {b : ordinal} (b1 : 1 < b) (x : ordinal) :
x < b ^ succ (log b x) :=
begin
cases lt_or_eq_of_le (ordinal.zero_le x) with x0 x0,
{ rw [succ_log_def b1 x0], exact omin_mem {o | x < b^o} _ },
{ subst x, apply power_pos _ (lt_trans zero_lt_one b1) }
end
theorem power_log_le (b) {x : ordinal} (x0 : 0 < x) :
b ^ log b x ≤ x :=
begin
by_cases b0 : b = 0,
{ rw [b0, zero_power'],
refine le_trans (sub_le_self _ _) (one_le_iff_pos.2 x0) },
cases lt_or_eq_of_le (one_le_iff_ne_zero.2 b0) with b1 b1,
{ refine le_of_not_lt (λ h, not_le_of_lt (lt_succ_self (log b x)) _),
have := @omin_le {o | x < b^o} _ _ h,
rwa ← succ_log_def b1 x0 at this },
{ rw [← b1, one_power], exact one_le_iff_pos.2 x0 }
end
theorem le_log {b x c : ordinal} (b1 : 1 < b) (x0 : 0 < x) :
c ≤ log b x ↔ b ^ c ≤ x :=
⟨λ h, le_trans ((power_le_power_iff_right b1).2 h) (power_log_le b x0),
λ h, le_of_not_lt $ λ hn,
not_le_of_lt (lt_power_succ_log b1 x) $
le_trans ((power_le_power_iff_right b1).2 (succ_le.2 hn)) h⟩
theorem log_lt {b x c : ordinal} (b1 : 1 < b) (x0 : 0 < x) :
log b x < c ↔ x < b ^ c :=
lt_iff_lt_of_le_iff_le (le_log b1 x0)
theorem log_le_log (b) {x y : ordinal} (xy : x ≤ y) :
log b x ≤ log b y :=
if x0 : x = 0 then by simp only [x0, log_zero, ordinal.zero_le] else
have x0 : 0 < x, from ordinal.pos_iff_ne_zero.2 x0,
if b1 : 1 < b then
(le_log b1 (lt_of_lt_of_le x0 xy)).2 $ le_trans (power_log_le _ x0) xy
else by simp only [log_not_one_lt b1, ordinal.zero_le]
theorem log_le_self (b x : ordinal) : log b x ≤ x :=
if x0 : x = 0 then by simp only [x0, log_zero, ordinal.zero_le] else
if b1 : 1 < b then
le_trans (le_power_self _ b1) (power_log_le b (ordinal.pos_iff_ne_zero.2 x0))
else by simp only [log_not_one_lt b1, ordinal.zero_le]
/-! ### The Cantor normal form -/
theorem CNF_aux {b o : ordinal} (b0 : b ≠ 0) (o0 : o ≠ 0) :
o % b ^ log b o < o :=
lt_of_lt_of_le
(mod_lt _ $ power_ne_zero _ b0)
(power_log_le _ $ ordinal.pos_iff_ne_zero.2 o0)
/-- Proving properties of ordinals by induction over their Cantor normal form. -/
@[elab_as_eliminator] noncomputable def CNF_rec {b : ordinal} (b0 : b ≠ 0)
{C : ordinal → Sort*}
(H0 : C 0)
(H : ∀ o, o ≠ 0 → o % b ^ log b o < o → C (o % b ^ log b o) → C o)
: ∀ o, C o
| o :=
if o0 : o = 0 then by rw o0; exact H0 else
have _, from CNF_aux b0 o0,
H o o0 this (CNF_rec (o % b ^ log b o))
using_well_founded {dec_tac := `[assumption]}
@[simp] theorem CNF_rec_zero {b} (b0) {C H0 H} : @CNF_rec b b0 C H0 H 0 = H0 :=
by rw [CNF_rec, dif_pos rfl]; refl
@[simp] theorem CNF_rec_ne_zero {b} (b0) {C H0 H o} (o0) :
@CNF_rec b b0 C H0 H o = H o o0 (CNF_aux b0 o0) (@CNF_rec b b0 C H0 H _) :=
by rw [CNF_rec, dif_neg o0]
/-- The Cantor normal form of an ordinal is the list of coefficients
in the base-`b` expansion of `o`.
CNF b (b ^ u₁ * v₁ + b ^ u₂ * v₂) = [(u₁, v₁), (u₂, v₂)] -/
noncomputable def CNF (b := omega) (o : ordinal) : list (ordinal × ordinal) :=
if b0 : b = 0 then [] else
CNF_rec b0 [] (λ o o0 h IH, (log b o, o / b ^ log b o) :: IH) o
@[simp] theorem zero_CNF (o) : CNF 0 o = [] :=
dif_pos rfl
@[simp] theorem CNF_zero (b) : CNF b 0 = [] :=
if b0 : b = 0 then dif_pos b0 else
(dif_neg b0).trans $ CNF_rec_zero _
theorem CNF_ne_zero {b o : ordinal} (b0 : b ≠ 0) (o0 : o ≠ 0) :
CNF b o = (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) :=
by unfold CNF; rw [dif_neg b0, dif_neg b0, CNF_rec_ne_zero b0 o0]
theorem one_CNF {o : ordinal} (o0 : o ≠ 0) :
CNF 1 o = [(0, o)] :=
by rw [CNF_ne_zero ordinal.one_ne_zero o0, log_not_one_lt (lt_irrefl _), power_zero, mod_one,
CNF_zero, div_one]
theorem CNF_foldr {b : ordinal} (b0 : b ≠ 0) (o) :
(CNF b o).foldr (λ p r, b ^ p.1 * p.2 + r) 0 = o :=
CNF_rec b0 (by rw CNF_zero; refl)
(λ o o0 h IH, by rw [CNF_ne_zero b0 o0, list.foldr_cons, IH, div_add_mod]) o
theorem CNF_pairwise_aux (b := omega) (o) :
(∀ p ∈ CNF b o, prod.fst p ≤ log b o) ∧
(CNF b o).pairwise (λ p q, q.1 < p.1) :=
begin
by_cases b0 : b = 0,
{ simp only [b0, zero_CNF, list.pairwise.nil, and_true], exact λ _, false.elim },
cases lt_or_eq_of_le (one_le_iff_ne_zero.2 b0) with b1 b1,
{ refine CNF_rec b0 _ _ o,
{ simp only [CNF_zero, list.pairwise.nil, and_true], exact λ _, false.elim },
intros o o0 H IH, cases IH with IH₁ IH₂,
simp only [CNF_ne_zero b0 o0, list.forall_mem_cons, list.pairwise_cons, IH₂, and_true],
refine ⟨⟨le_refl _, λ p m, _⟩, λ p m, _⟩,
{ exact le_trans (IH₁ p m) (log_le_log _ $ le_of_lt H) },
{ refine lt_of_le_of_lt (IH₁ p m) ((log_lt b1 _).2 _),
{ rw ordinal.pos_iff_ne_zero, intro e,
rw e at m, simpa only [CNF_zero] using m },
{ exact mod_lt _ (power_ne_zero _ b0) } } },
{ by_cases o0 : o = 0,
{ simp only [o0, CNF_zero, list.pairwise.nil, and_true], exact λ _, false.elim },
rw [← b1, one_CNF o0],
simp only [list.mem_singleton, log_not_one_lt (lt_irrefl _), forall_eq, le_refl, true_and,
list.pairwise_singleton] }
end
theorem CNF_pairwise (b := omega) (o) :
(CNF b o).pairwise (λ p q, prod.fst q < p.1) :=
(CNF_pairwise_aux _ _).2
theorem CNF_fst_le_log (b := omega) (o) :
∀ p ∈ CNF b o, prod.fst p ≤ log b o :=
(CNF_pairwise_aux _ _).1
theorem CNF_fst_le (b := omega) (o) (p ∈ CNF b o) : prod.fst p ≤ o :=
le_trans (CNF_fst_le_log _ _ p H) (log_le_self _ _)
theorem CNF_snd_lt {b : ordinal} (b1 : 1 < b) (o) :
∀ p ∈ CNF b o, prod.snd p < b :=
begin
have b0 := ne_of_gt (lt_trans zero_lt_one b1),
refine CNF_rec b0 (λ _, by rw [CNF_zero]; exact false.elim) _ o,
intros o o0 H IH,
simp only [CNF_ne_zero b0 o0, list.mem_cons_iff, forall_eq_or_imp, iff_true_intro IH, and_true],
rw [div_lt (power_ne_zero _ b0), ← power_succ],
exact lt_power_succ_log b1 _,
end
theorem CNF_sorted (b := omega) (o) :
((CNF b o).map prod.fst).sorted (>) :=
by rw [list.sorted, list.pairwise_map]; exact CNF_pairwise b o
/-! ### Casting naturals into ordinals, compatibility with operations -/
@[simp] theorem nat_cast_mul {m n : ℕ} : ((m * n : ℕ) : ordinal) = m * n :=
by induction n with n IH; [simp only [nat.cast_zero, nat.mul_zero, mul_zero],
rw [nat.mul_succ, nat.cast_add, IH, nat.cast_succ, mul_add_one]]
@[simp] theorem nat_cast_power {m n : ℕ} : ((pow m n : ℕ) : ordinal) = m ^ n :=
by induction n with n IH; [simp only [pow_zero, nat.cast_zero, power_zero, nat.cast_one],
rw [pow_succ', nat_cast_mul, IH, nat.cast_succ, ← succ_eq_add_one, power_succ]]
@[simp] theorem nat_cast_le {m n : ℕ} : (m : ordinal) ≤ n ↔ m ≤ n :=
by rw [← cardinal.ord_nat, ← cardinal.ord_nat,
cardinal.ord_le_ord, cardinal.nat_cast_le]
@[simp] theorem nat_cast_lt {m n : ℕ} : (m : ordinal) < n ↔ m < n :=
by simp only [lt_iff_le_not_le, nat_cast_le]
@[simp] theorem nat_cast_inj {m n : ℕ} : (m : ordinal) = n ↔ m = n :=
by simp only [le_antisymm_iff, nat_cast_le]
@[simp] theorem nat_cast_eq_zero {n : ℕ} : (n : ordinal) = 0 ↔ n = 0 :=
@nat_cast_inj n 0
theorem nat_cast_ne_zero {n : ℕ} : (n : ordinal) ≠ 0 ↔ n ≠ 0 :=
not_congr nat_cast_eq_zero
@[simp] theorem nat_cast_pos {n : ℕ} : (0 : ordinal) < n ↔ 0 < n :=
@nat_cast_lt 0 n
@[simp] theorem nat_cast_sub {m n : ℕ} : ((m - n : ℕ) : ordinal) = m - n :=
(_root_.le_total m n).elim
(λ h, by rw [nat.sub_eq_zero_iff_le.2 h, ordinal.sub_eq_zero_iff_le.2 (nat_cast_le.2 h)]; refl)
(λ h, (add_left_cancel n).1 $ by rw [← nat.cast_add,
nat.add_sub_cancel' h, add_sub_cancel_of_le (nat_cast_le.2 h)])
@[simp] theorem nat_cast_div {m n : ℕ} : ((m / n : ℕ) : ordinal) = m / n :=
if n0 : n = 0 then by simp only [n0, nat.div_zero, nat.cast_zero, div_zero] else
have n0':_, from nat_cast_ne_zero.2 n0,
le_antisymm
(by rw [le_div n0', ← nat_cast_mul, nat_cast_le, mul_comm];
apply nat.div_mul_le_self)
(by rw [div_le n0', succ, ← nat.cast_succ, ← nat_cast_mul,
nat_cast_lt, mul_comm, ← nat.div_lt_iff_lt_mul _ _ (nat.pos_of_ne_zero n0)];
apply nat.lt_succ_self)
@[simp] theorem nat_cast_mod {m n : ℕ} : ((m % n : ℕ) : ordinal) = m % n :=
by rw [← add_left_cancel (n*(m/n)), div_add_mod, ← nat_cast_div, ← nat_cast_mul, ← nat.cast_add,
nat.div_add_mod]
@[simp] theorem nat_le_card {o} {n : ℕ} : (n : cardinal) ≤ card o ↔ (n : ordinal) ≤ o :=
⟨λ h, by rwa [← cardinal.ord_le, cardinal.ord_nat] at h,
λ h, card_nat n ▸ card_le_card h⟩
@[simp] theorem nat_lt_card {o} {n : ℕ} : (n : cardinal) < card o ↔ (n : ordinal) < o :=
by rw [← succ_le, ← cardinal.succ_le, ← cardinal.nat_succ, nat_le_card]; refl
@[simp] theorem card_lt_nat {o} {n : ℕ} : card o < n ↔ o < n :=
lt_iff_lt_of_le_iff_le nat_le_card
@[simp] theorem card_le_nat {o} {n : ℕ} : card o ≤ n ↔ o ≤ n :=
le_iff_le_iff_lt_iff_lt.2 nat_lt_card
@[simp] theorem card_eq_nat {o} {n : ℕ} : card o = n ↔ o = n :=
by simp only [le_antisymm_iff, card_le_nat, nat_le_card]
@[simp] theorem type_fin (n : ℕ) : @type (fin n) (<) _ = n :=
by rw [← card_eq_nat, card_type, mk_fin]
@[simp] theorem lift_nat_cast (n : ℕ) : lift n = n :=
by induction n with n ih; [simp only [nat.cast_zero, lift_zero],
simp only [nat.cast_succ, lift_add, ih, lift_one]]
theorem lift_type_fin (n : ℕ) : lift (@type (fin n) (<) _) = n :=
by simp only [type_fin, lift_nat_cast]
theorem fintype_card (r : α → α → Prop) [is_well_order α r] [fintype α] : type r = fintype.card α :=
by rw [← card_eq_nat, card_type, fintype_card]
end ordinal
/-! ### Properties of `omega` -/
namespace cardinal
open ordinal
@[simp] theorem ord_omega : ord.{u} omega = ordinal.omega :=
le_antisymm (ord_le.2 $ le_refl _) $
le_of_forall_lt $ λ o h, begin
rcases ordinal.lt_lift_iff.1 h with ⟨o, rfl, h'⟩,
rw [lt_ord, ← lift_card, ← lift_omega.{0 u},
lift_lt, ← typein_enum (<) h'],
exact lt_omega_iff_fintype.2 ⟨set.fintype_lt_nat _⟩
end
@[simp] theorem add_one_of_omega_le {c} (h : omega ≤ c) : c + 1 = c :=
by rw [add_comm, ← card_ord c, ← card_one,
← card_add, one_add_of_omega_le];
rwa [← ord_omega, ord_le_ord]
end cardinal
namespace ordinal
theorem lt_omega {o : ordinal.{u}} : o < omega ↔ ∃ n : ℕ, o = n :=
by rw [← cardinal.ord_omega, cardinal.lt_ord, lt_omega]; simp only [card_eq_nat]
theorem nat_lt_omega (n : ℕ) : (n : ordinal) < omega :=
lt_omega.2 ⟨_, rfl⟩
theorem omega_pos : 0 < omega := nat_lt_omega 0
theorem omega_ne_zero : omega ≠ 0 := ne_of_gt omega_pos
theorem one_lt_omega : 1 < omega := by simpa only [nat.cast_one] using nat_lt_omega 1
theorem omega_is_limit : is_limit omega :=
⟨omega_ne_zero, λ o h,
let ⟨n, e⟩ := lt_omega.1 h in
by rw [e]; exact nat_lt_omega (n+1)⟩
theorem omega_le {o : ordinal.{u}} : omega ≤ o ↔ ∀ n : ℕ, (n : ordinal) ≤ o :=
⟨λ h n, le_trans (le_of_lt (nat_lt_omega _)) h,
λ H, le_of_forall_lt $ λ a h,
let ⟨n, e⟩ := lt_omega.1 h in
by rw [e, ← succ_le]; exact H (n+1)⟩
theorem nat_lt_limit {o} (h : is_limit o) : ∀ n : ℕ, (n : ordinal) < o
| 0 := lt_of_le_of_ne (ordinal.zero_le o) h.1.symm
| (n+1) := h.2 _ (nat_lt_limit n)
theorem omega_le_of_is_limit {o} (h : is_limit o) : omega ≤ o :=
omega_le.2 $ λ n, le_of_lt $ nat_lt_limit h n
theorem add_omega {a : ordinal} (h : a < omega) : a + omega = omega :=
begin
rcases lt_omega.1 h with ⟨n, rfl⟩,
clear h, induction n with n IH,
{ rw [nat.cast_zero, zero_add] },
{ rw [nat.cast_succ, add_assoc, one_add_of_omega_le (le_refl _), IH] }
end
theorem add_lt_omega {a b : ordinal} (ha : a < omega) (hb : b < omega) : a + b < omega :=
match a, b, lt_omega.1 ha, lt_omega.1 hb with
| _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ := by rw [← nat.cast_add]; apply nat_lt_omega
end
theorem mul_lt_omega {a b : ordinal} (ha : a < omega) (hb : b < omega) : a * b < omega :=
match a, b, lt_omega.1 ha, lt_omega.1 hb with
| _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ := by rw [← nat_cast_mul]; apply nat_lt_omega
end
theorem is_limit_iff_omega_dvd {a : ordinal} : is_limit a ↔ a ≠ 0 ∧ omega ∣ a :=
begin
refine ⟨λ l, ⟨l.1, ⟨a / omega, le_antisymm _ (mul_div_le _ _)⟩⟩, λ h, _⟩,
{ refine (limit_le l).2 (λ x hx, le_of_lt _),
rw [← div_lt omega_ne_zero, ← succ_le, le_div omega_ne_zero,
mul_succ, add_le_of_limit omega_is_limit],
intros b hb,
rcases lt_omega.1 hb with ⟨n, rfl⟩,
exact le_trans (add_le_add_right (mul_div_le _ _) _)
(le_of_lt $ lt_sub.1 $ nat_lt_limit (sub_is_limit l hx) _) },
{ rcases h with ⟨a0, b, rfl⟩,
refine mul_is_limit_left omega_is_limit
(ordinal.pos_iff_ne_zero.2 $ mt _ a0),
intro e, simp only [e, mul_zero] }
end
local infixr ^ := @pow ordinal ordinal ordinal.has_pow
theorem power_lt_omega {a b : ordinal} (ha : a < omega) (hb : b < omega) : a ^ b < omega :=
match a, b, lt_omega.1 ha, lt_omega.1 hb with
| _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ := by rw [← nat_cast_power]; apply nat_lt_omega
end
theorem add_omega_power {a b : ordinal} (h : a < omega ^ b) : a + omega ^ b = omega ^ b :=
begin
refine le_antisymm _ (le_add_left _ _),
revert h, apply limit_rec_on b,
{ intro h, rw [power_zero, ← succ_zero, lt_succ, ordinal.le_zero] at h,
rw [h, zero_add] },
{ intros b _ h, rw [power_succ] at h,
rcases (lt_mul_of_limit omega_is_limit).1 h with ⟨x, xo, ax⟩,
refine le_trans (add_le_add_right (le_of_lt ax) _) _,
rw [power_succ, ← mul_add, add_omega xo] },
{ intros b l IH h, rcases (lt_power_of_limit omega_ne_zero l).1 h with ⟨x, xb, ax⟩,
refine (((add_is_normal a).trans (power_is_normal one_lt_omega))
.limit_le l).2 (λ y yb, _),
let z := max x y,
have := IH z (max_lt xb yb)
(lt_of_lt_of_le ax $ power_le_power_right omega_pos (le_max_left _ _)),
exact le_trans (add_le_add_left (power_le_power_right omega_pos (le_max_right _ _)) _)
(le_trans this (power_le_power_right omega_pos $ le_of_lt $ max_lt xb yb)) }
end
theorem add_lt_omega_power {a b c : ordinal} (h₁ : a < omega ^ c) (h₂ : b < omega ^ c) :
a + b < omega ^ c :=
by rwa [← add_omega_power h₁, add_lt_add_iff_left]
theorem add_absorp {a b c : ordinal} (h₁ : a < omega ^ b) (h₂ : omega ^ b ≤ c) : a + c = c :=
by rw [← add_sub_cancel_of_le h₂, ← add_assoc, add_omega_power h₁]
theorem add_absorp_iff {o : ordinal} (o0 : 0 < o) : (∀ a < o, a + o = o) ↔ ∃ a, o = omega ^ a :=
⟨λ H, ⟨log omega o, begin
refine ((lt_or_eq_of_le (power_log_le _ o0))
.resolve_left $ λ h, _).symm,
have := H _ h,
have := lt_power_succ_log one_lt_omega o,
rw [power_succ, lt_mul_of_limit omega_is_limit] at this,
rcases this with ⟨a, ao, h'⟩,
rcases lt_omega.1 ao with ⟨n, rfl⟩, clear ao,
revert h', apply not_lt_of_le,
suffices e : omega ^ log omega o * ↑n + o = o,
{ simpa only [e] using le_add_right (omega ^ log omega o * ↑n) o },
induction n with n IH, {simp only [nat.cast_zero, mul_zero, zero_add]},
simp only [nat.cast_succ, mul_add_one, add_assoc, this, IH]
end⟩,
λ ⟨b, e⟩, e.symm ▸ λ a, add_omega_power⟩
theorem add_mul_limit_aux {a b c : ordinal} (ba : b + a = a)
(l : is_limit c)
(IH : ∀ c' < c, (a + b) * succ c' = a * succ c' + b) :
(a + b) * c = a * c :=
le_antisymm
((mul_le_of_limit l).2 $ λ c' h, begin
apply le_trans (mul_le_mul_left _ (le_of_lt $ lt_succ_self _)),
rw IH _ h,
apply le_trans (add_le_add_left _ _),
{ rw ← mul_succ, exact mul_le_mul_left _ (succ_le.2 $ l.2 _ h) },
{ rw ← ba, exact le_add_right _ _ }
end)
(mul_le_mul_right _ (le_add_right _ _))
theorem add_mul_succ {a b : ordinal} (c) (ba : b + a = a) :
(a + b) * succ c = a * succ c + b :=
begin
apply limit_rec_on c,
{ simp only [succ_zero, mul_one] },
{ intros c IH,
rw [mul_succ, IH, ← add_assoc, add_assoc _ b, ba, ← mul_succ] },
{ intros c l IH,
have := add_mul_limit_aux ba l IH,
rw [mul_succ, add_mul_limit_aux ba l IH, mul_succ, add_assoc] }
end
theorem add_mul_limit {a b c : ordinal} (ba : b + a = a)
(l : is_limit c) : (a + b) * c = a * c :=
add_mul_limit_aux ba l (λ c' _, add_mul_succ c' ba)
theorem mul_omega {a : ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega :=
le_antisymm
((mul_le_of_limit omega_is_limit).2 $ λ b hb, le_of_lt (mul_lt_omega ha hb))
(by simpa only [one_mul] using mul_le_mul_right omega (one_le_iff_pos.2 a0))
theorem mul_lt_omega_power {a b c : ordinal}
(c0 : 0 < c) (ha : a < omega ^ c) (hb : b < omega) : a * b < omega ^ c :=
if b0 : b = 0 then by simp only [b0, mul_zero, power_pos _ omega_pos] else begin
rcases zero_or_succ_or_limit c with rfl|⟨c,rfl⟩|l,
{ exact (lt_irrefl _).elim c0 },
{ rw power_succ at ha,
rcases ((mul_is_normal $ power_pos _ omega_pos).limit_lt
omega_is_limit).1 ha with ⟨n, hn, an⟩,
refine lt_of_le_of_lt (mul_le_mul_right _ (le_of_lt an)) _,
rw [power_succ, mul_assoc, mul_lt_mul_iff_left (power_pos _ omega_pos)],
exact mul_lt_omega hn hb },
{ rcases ((power_is_normal one_lt_omega).limit_lt l).1 ha with ⟨x, hx, ax⟩,
refine lt_of_le_of_lt (mul_le_mul (le_of_lt ax) (le_of_lt hb)) _,
rw [← power_succ, power_lt_power_iff_right one_lt_omega],
exact l.2 _ hx }
end
theorem mul_omega_dvd {a : ordinal}
(a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b
| _ ⟨b, rfl⟩ := by rw [← mul_assoc, mul_omega a0 ha]
theorem mul_omega_power_power {a b : ordinal} (a0 : 0 < a) (h : a < omega ^ omega ^ b) :
a * omega ^ omega ^ b = omega ^ omega ^ b :=
begin
by_cases b0 : b = 0, {rw [b0, power_zero, power_one] at h ⊢, exact mul_omega a0 h},
refine le_antisymm _
(by simpa only [one_mul] using mul_le_mul_right (omega^omega^b) (one_le_iff_pos.2 a0)),
rcases (lt_power_of_limit omega_ne_zero (power_is_limit_left omega_is_limit b0)).1 h
with ⟨x, xb, ax⟩,
refine le_trans (mul_le_mul_right _ (le_of_lt ax)) _,
rw [← power_add, add_omega_power xb]
end
theorem power_omega {a : ordinal} (a1 : 1 < a) (h : a < omega) : a ^ omega = omega :=
le_antisymm
((power_le_of_limit (one_le_iff_ne_zero.1 $ le_of_lt a1) omega_is_limit).2
(λ b hb, le_of_lt (power_lt_omega h hb)))
(le_power_self _ a1)
/-! ### Fixed points of normal functions -/
/-- The next fixed point function, the least fixed point of the
normal function `f` above `a`. -/
def nfp (f : ordinal → ordinal) (a : ordinal) :=
sup (λ n : ℕ, f^[n] a)
theorem iterate_le_nfp (f a n) : f^[n] a ≤ nfp f a :=
le_sup _ n
theorem le_nfp_self (f a) : a ≤ nfp f a :=
iterate_le_nfp f a 0
theorem is_normal.lt_nfp {f} (H : is_normal f) {a b} :
f b < nfp f a ↔ b < nfp f a :=
lt_sup.trans $ iff.trans
(by exact
⟨λ ⟨n, h⟩, ⟨n, lt_of_le_of_lt (H.le_self _) h⟩,
λ ⟨n, h⟩, ⟨n+1, by rw iterate_succ'; exact H.lt_iff.2 h⟩⟩)
lt_sup.symm
theorem is_normal.nfp_le {f} (H : is_normal f) {a b} :
nfp f a ≤ f b ↔ nfp f a ≤ b :=
le_iff_le_iff_lt_iff_lt.2 H.lt_nfp
theorem is_normal.nfp_le_fp {f} (H : is_normal f) {a b}
(ab : a ≤ b) (h : f b ≤ b) : nfp f a ≤ b :=
sup_le.2 $ λ i, begin
induction i with i IH generalizing a, {exact ab},
exact IH (le_trans (H.le_iff.2 ab) h),
end
theorem is_normal.nfp_fp {f} (H : is_normal f) (a) : f (nfp f a) = nfp f a :=
begin
refine le_antisymm _ (H.le_self _),
cases le_or_lt (f a) a with aa aa,
{ rwa le_antisymm (H.nfp_le_fp (le_refl _) aa) (le_nfp_self _ _) },
rcases zero_or_succ_or_limit (nfp f a) with e|⟨b, e⟩|l,
{ refine @le_trans _ _ _ (f a) _ (H.le_iff.2 _) (iterate_le_nfp f a 1),
simp only [e, ordinal.zero_le] },
{ have : f b < nfp f a := H.lt_nfp.2 (by simp only [e, lt_succ_self]),
rw [e, lt_succ] at this,
have ab : a ≤ b,
{ rw [← lt_succ, ← e],
exact lt_of_lt_of_le aa (iterate_le_nfp f a 1) },
refine le_trans (H.le_iff.2 (H.nfp_le_fp ab this))
(le_trans this (le_of_lt _)),
simp only [e, lt_succ_self] },
{ exact (H.2 _ l _).2 (λ b h, le_of_lt (H.lt_nfp.2 h)) }
end
theorem is_normal.le_nfp {f} (H : is_normal f) {a b} :
f b ≤ nfp f a ↔ b ≤ nfp f a :=
⟨le_trans (H.le_self _), λ h,
by simpa only [H.nfp_fp] using H.le_iff.2 h⟩
theorem nfp_eq_self {f : ordinal → ordinal} {a} (h : f a = a) : nfp f a = a :=
le_antisymm (sup_le.mpr $ λ i, by rw [iterate_fixed h]) (le_nfp_self f a)
/-- The derivative of a normal function `f` is
the sequence of fixed points of `f`. -/
def deriv (f : ordinal → ordinal) (o : ordinal) : ordinal :=
limit_rec_on o (nfp f 0)
(λ a IH, nfp f (succ IH))
(λ a l, bsup.{u u} a)
@[simp] theorem deriv_zero (f) : deriv f 0 = nfp f 0 := limit_rec_on_zero _ _ _
@[simp] theorem deriv_succ (f o) : deriv f (succ o) = nfp f (succ (deriv f o)) :=
limit_rec_on_succ _ _ _ _
theorem deriv_limit (f) {o} : is_limit o →
deriv f o = bsup.{u u} o (λ a _, deriv f a) :=
limit_rec_on_limit _ _ _ _
theorem deriv_is_normal (f) : is_normal (deriv f) :=
⟨λ o, by rw [deriv_succ, ← succ_le]; apply le_nfp_self,
λ o l a, by rw [deriv_limit _ l, bsup_le]⟩
theorem is_normal.deriv_fp {f} (H : is_normal f) (o) : f (deriv.{u} f o) = deriv f o :=
begin
apply limit_rec_on o,
{ rw [deriv_zero, H.nfp_fp] },
{ intros o ih, rw [deriv_succ, H.nfp_fp] },
intros o l IH,
rw [deriv_limit _ l, is_normal.bsup.{u u u} H _ l.1],
refine eq_of_forall_ge_iff (λ c, _),
simp only [bsup_le, IH] {contextual:=tt}
end
theorem is_normal.fp_iff_deriv {f} (H : is_normal f)
{a} : f a ≤ a ↔ ∃ o, a = deriv f o :=
⟨λ ha, begin
suffices : ∀ o (_:a ≤ deriv f o), ∃ o, a = deriv f o,
from this a ((deriv_is_normal _).le_self _),
intro o, apply limit_rec_on o,
{ intros h₁,
refine ⟨0, le_antisymm h₁ _⟩,
rw deriv_zero,
exact H.nfp_le_fp (ordinal.zero_le _) ha },
{ intros o IH h₁,
cases le_or_lt a (deriv f o), {exact IH h},
refine ⟨succ o, le_antisymm h₁ _⟩,
rw deriv_succ,
exact H.nfp_le_fp (succ_le.2 h) ha },
{ intros o l IH h₁,
cases eq_or_lt_of_le h₁, {exact ⟨_, h⟩},
rw [deriv_limit _ l, ← not_le, bsup_le, not_ball] at h,
exact let ⟨o', h, hl⟩ := h in IH o' h (le_of_not_le hl) }
end, λ ⟨o, e⟩, e.symm ▸ le_of_eq (H.deriv_fp _)⟩
end ordinal
|
cdbe7ebe4f8f45d268cf625e61dd8074eafc2c08 | d1a52c3f208fa42c41df8278c3d280f075eb020c | /tests/lean/run/coeIssue2.lean | 87f9f73c55bdcc4908c9fdf589a3a01abb93b725 | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | cipher1024/lean4 | 6e1f98bb58e7a92b28f5364eb38a14c8d0aae393 | 69114d3b50806264ef35b57394391c3e738a9822 | refs/heads/master | 1,642,227,983,603 | 1,642,011,696,000 | 1,642,011,696,000 | 228,607,691 | 0 | 0 | Apache-2.0 | 1,576,584,269,000 | 1,576,584,268,000 | null | UTF-8 | Lean | false | false | 838 | lean |
structure A :=
(a : Nat)
structure B :=
(a : Nat)
structure C :=
(a : Nat)
instance : Coe A B := ⟨fun s => ⟨s.1⟩⟩
instance : Coe A C := ⟨fun s => ⟨s.1⟩⟩
def f {α} (a b : α) := a
def forceB {α} (b : B) (a : α) := a
def forceC {α} (c : C) (a : α) := a
def forceA {α} (a : A) (o : α) := o
def f1 (x : _) (y : _) (z : _) :=
let a1 := f x y;
let a2 := f y z;
forceB x a1 -- works
def f2 (x : _) (y : _) (z : _) :=
let a1 := f x (coe y);
let a2 := f (coe y) z;
forceB x (forceC z (forceA y (a1, a2))) -- works because we manually added the coercions above
#exit
def f3 (x : _) (y : _) (z : _) :=
let a1 := f x y;
let a2 := f y z;
/- Fails because we "missed" the opportunity to insert coercions around `y`.
I think we should **not** support this kind of example. -/
forceB x (forceC z (forceA y (a1, a2)))
|
e225fff740f5dd50ba980edd27199c8f67b3dfa1 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/analysis/special_functions/exp_deriv.lean | 4c88a9c96fef7c61e9993df41b4364c7181b792f | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 10,166 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import analysis.calculus.inverse
import analysis.complex.real_deriv
import analysis.special_functions.exp
/-!
# Complex and real exponential
In this file we prove that `complex.exp` and `real.exp` are infinitely smooth functions.
## Tags
exp, derivative
-/
noncomputable theory
open filter asymptotics set function
open_locale classical topological_space
namespace complex
variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] [normed_algebra 𝕜 ℂ]
/-- The complex exponential is everywhere differentiable, with the derivative `exp x`. -/
lemma has_deriv_at_exp (x : ℂ) : has_deriv_at exp (exp x) x :=
begin
rw has_deriv_at_iff_is_o_nhds_zero,
have : (1 : ℕ) < 2 := by norm_num,
refine (is_O.of_bound (∥exp x∥) _).trans_is_o (is_o_pow_id this),
filter_upwards [metric.ball_mem_nhds (0 : ℂ) zero_lt_one],
simp only [metric.mem_ball, dist_zero_right, norm_pow],
exact λ z hz, exp_bound_sq x z hz.le,
end
lemma differentiable_exp : differentiable 𝕜 exp :=
λ x, (has_deriv_at_exp x).differentiable_at.restrict_scalars 𝕜
lemma differentiable_at_exp {x : ℂ} : differentiable_at 𝕜 exp x :=
differentiable_exp x
@[simp] lemma deriv_exp : deriv exp = exp :=
funext $ λ x, (has_deriv_at_exp x).deriv
@[simp] lemma iter_deriv_exp : ∀ n : ℕ, (deriv^[n] exp) = exp
| 0 := rfl
| (n+1) := by rw [iterate_succ_apply, deriv_exp, iter_deriv_exp n]
lemma cont_diff_exp : ∀ {n}, cont_diff 𝕜 n exp :=
begin
refine cont_diff_all_iff_nat.2 (λ n, _),
have : cont_diff ℂ ↑n exp,
{ induction n with n ihn,
{ exact cont_diff_zero.2 continuous_exp },
{ rw cont_diff_succ_iff_deriv,
use differentiable_exp,
rwa deriv_exp }, },
exact this.restrict_scalars 𝕜
end
lemma has_strict_deriv_at_exp (x : ℂ) : has_strict_deriv_at exp (exp x) x :=
cont_diff_exp.cont_diff_at.has_strict_deriv_at' (has_deriv_at_exp x) le_rfl
lemma has_strict_fderiv_at_exp_real (x : ℂ) :
has_strict_fderiv_at exp (exp x • (1 : ℂ →L[ℝ] ℂ)) x :=
(has_strict_deriv_at_exp x).complex_to_real_fderiv
lemma is_open_map_exp : is_open_map exp :=
open_map_of_strict_deriv has_strict_deriv_at_exp exp_ne_zero
end complex
section
variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] [normed_algebra 𝕜 ℂ]
{f : 𝕜 → ℂ} {f' : ℂ} {x : 𝕜} {s : set 𝕜}
lemma has_strict_deriv_at.cexp (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, complex.exp (f x)) (complex.exp (f x) * f') x :=
(complex.has_strict_deriv_at_exp (f x)).comp x hf
lemma has_deriv_at.cexp (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, complex.exp (f x)) (complex.exp (f x) * f') x :=
(complex.has_deriv_at_exp (f x)).comp x hf
lemma has_deriv_within_at.cexp (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, complex.exp (f x)) (complex.exp (f x) * f') s x :=
(complex.has_deriv_at_exp (f x)).comp_has_deriv_within_at x hf
lemma deriv_within_cexp (hf : differentiable_within_at 𝕜 f s x)
(hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λ x, complex.exp (f x)) s x = complex.exp (f x) * deriv_within f s x :=
hf.has_deriv_within_at.cexp.deriv_within hxs
@[simp] lemma deriv_cexp (hc : differentiable_at 𝕜 f x) :
deriv (λ x, complex.exp (f x)) x = complex.exp (f x) * deriv f x :=
hc.has_deriv_at.cexp.deriv
end
section
variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] [normed_algebra 𝕜 ℂ]
{E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] {f : E → ℂ} {f' : E →L[𝕜] ℂ}
{x : E} {s : set E}
lemma has_strict_fderiv_at.cexp (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, complex.exp (f x)) (complex.exp (f x) • f') x :=
(complex.has_strict_deriv_at_exp (f x)).comp_has_strict_fderiv_at x hf
lemma has_fderiv_within_at.cexp (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, complex.exp (f x)) (complex.exp (f x) • f') s x :=
(complex.has_deriv_at_exp (f x)).comp_has_fderiv_within_at x hf
lemma has_fderiv_at.cexp (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, complex.exp (f x)) (complex.exp (f x) • f') x :=
has_fderiv_within_at_univ.1 $ hf.has_fderiv_within_at.cexp
lemma differentiable_within_at.cexp (hf : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (λ x, complex.exp (f x)) s x :=
hf.has_fderiv_within_at.cexp.differentiable_within_at
@[simp] lemma differentiable_at.cexp (hc : differentiable_at 𝕜 f x) :
differentiable_at 𝕜 (λ x, complex.exp (f x)) x :=
hc.has_fderiv_at.cexp.differentiable_at
lemma differentiable_on.cexp (hc : differentiable_on 𝕜 f s) :
differentiable_on 𝕜 (λ x, complex.exp (f x)) s :=
λ x h, (hc x h).cexp
@[simp] lemma differentiable.cexp (hc : differentiable 𝕜 f) :
differentiable 𝕜 (λ x, complex.exp (f x)) :=
λ x, (hc x).cexp
lemma cont_diff.cexp {n} (h : cont_diff 𝕜 n f) :
cont_diff 𝕜 n (λ x, complex.exp (f x)) :=
complex.cont_diff_exp.comp h
lemma cont_diff_at.cexp {n} (hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ x, complex.exp (f x)) x :=
complex.cont_diff_exp.cont_diff_at.comp x hf
lemma cont_diff_on.cexp {n} (hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ x, complex.exp (f x)) s :=
complex.cont_diff_exp.comp_cont_diff_on hf
lemma cont_diff_within_at.cexp {n} (hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (λ x, complex.exp (f x)) s x :=
complex.cont_diff_exp.cont_diff_at.comp_cont_diff_within_at x hf
end
namespace real
variables {x y z : ℝ}
lemma has_strict_deriv_at_exp (x : ℝ) : has_strict_deriv_at exp (exp x) x :=
(complex.has_strict_deriv_at_exp x).real_of_complex
lemma has_deriv_at_exp (x : ℝ) : has_deriv_at exp (exp x) x :=
(complex.has_deriv_at_exp x).real_of_complex
lemma cont_diff_exp {n} : cont_diff ℝ n exp :=
complex.cont_diff_exp.real_of_complex
lemma differentiable_exp : differentiable ℝ exp :=
λx, (has_deriv_at_exp x).differentiable_at
lemma differentiable_at_exp : differentiable_at ℝ exp x :=
differentiable_exp x
@[simp] lemma deriv_exp : deriv exp = exp :=
funext $ λ x, (has_deriv_at_exp x).deriv
@[simp] lemma iter_deriv_exp : ∀ n : ℕ, (deriv^[n] exp) = exp
| 0 := rfl
| (n+1) := by rw [iterate_succ_apply, deriv_exp, iter_deriv_exp n]
end real
section
/-! Register lemmas for the derivatives of the composition of `real.exp` with a differentiable
function, for standalone use and use with `simp`. -/
variables {f : ℝ → ℝ} {f' x : ℝ} {s : set ℝ}
lemma has_strict_deriv_at.exp (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, real.exp (f x)) (real.exp (f x) * f') x :=
(real.has_strict_deriv_at_exp (f x)).comp x hf
lemma has_deriv_at.exp (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, real.exp (f x)) (real.exp (f x) * f') x :=
(real.has_deriv_at_exp (f x)).comp x hf
lemma has_deriv_within_at.exp (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, real.exp (f x)) (real.exp (f x) * f') s x :=
(real.has_deriv_at_exp (f x)).comp_has_deriv_within_at x hf
lemma deriv_within_exp (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, real.exp (f x)) s x = real.exp (f x) * (deriv_within f s x) :=
hf.has_deriv_within_at.exp.deriv_within hxs
@[simp] lemma deriv_exp (hc : differentiable_at ℝ f x) :
deriv (λx, real.exp (f x)) x = real.exp (f x) * (deriv f x) :=
hc.has_deriv_at.exp.deriv
end
section
/-! Register lemmas for the derivatives of the composition of `real.exp` with a differentiable
function, for standalone use and use with `simp`. -/
variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ}
{x : E} {s : set E}
lemma cont_diff.exp {n} (hf : cont_diff ℝ n f) :
cont_diff ℝ n (λ x, real.exp (f x)) :=
real.cont_diff_exp.comp hf
lemma cont_diff_at.exp {n} (hf : cont_diff_at ℝ n f x) :
cont_diff_at ℝ n (λ x, real.exp (f x)) x :=
real.cont_diff_exp.cont_diff_at.comp x hf
lemma cont_diff_on.exp {n} (hf : cont_diff_on ℝ n f s) :
cont_diff_on ℝ n (λ x, real.exp (f x)) s :=
real.cont_diff_exp.comp_cont_diff_on hf
lemma cont_diff_within_at.exp {n} (hf : cont_diff_within_at ℝ n f s x) :
cont_diff_within_at ℝ n (λ x, real.exp (f x)) s x :=
real.cont_diff_exp.cont_diff_at.comp_cont_diff_within_at x hf
lemma has_fderiv_within_at.exp (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, real.exp (f x)) (real.exp (f x) • f') s x :=
(real.has_deriv_at_exp (f x)).comp_has_fderiv_within_at x hf
lemma has_fderiv_at.exp (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, real.exp (f x)) (real.exp (f x) • f') x :=
(real.has_deriv_at_exp (f x)).comp_has_fderiv_at x hf
lemma has_strict_fderiv_at.exp (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, real.exp (f x)) (real.exp (f x) • f') x :=
(real.has_strict_deriv_at_exp (f x)).comp_has_strict_fderiv_at x hf
lemma differentiable_within_at.exp (hf : differentiable_within_at ℝ f s x) :
differentiable_within_at ℝ (λ x, real.exp (f x)) s x :=
hf.has_fderiv_within_at.exp.differentiable_within_at
@[simp] lemma differentiable_at.exp (hc : differentiable_at ℝ f x) :
differentiable_at ℝ (λx, real.exp (f x)) x :=
hc.has_fderiv_at.exp.differentiable_at
lemma differentiable_on.exp (hc : differentiable_on ℝ f s) :
differentiable_on ℝ (λx, real.exp (f x)) s :=
λ x h, (hc x h).exp
@[simp] lemma differentiable.exp (hc : differentiable ℝ f) :
differentiable ℝ (λx, real.exp (f x)) :=
λ x, (hc x).exp
lemma fderiv_within_exp (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λx, real.exp (f x)) s x = real.exp (f x) • (fderiv_within ℝ f s x) :=
hf.has_fderiv_within_at.exp.fderiv_within hxs
@[simp] lemma fderiv_exp (hc : differentiable_at ℝ f x) :
fderiv ℝ (λx, real.exp (f x)) x = real.exp (f x) • (fderiv ℝ f x) :=
hc.has_fderiv_at.exp.fderiv
end
|
167d2ecd25466cf859d333ae4ed27b2c5754340c | 01f6b345a06ece970e589d4bbc68ee8b9b2cf58a | /src/galois_action.lean | 06a7b30f4047bf3fe70d1af7077e687417eb1e91 | [] | no_license | mariainesdff/norm_extensions_journal_submission | 6077acb98a7200de4553e653d81d54fb5d2314c8 | d396130660935464fbc683f9aaf37fff8a890baa | refs/heads/master | 1,686,685,693,347 | 1,684,065,115,000 | 1,684,065,115,000 | 603,823,641 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,393 | lean | /-
Copyright (c) 2023 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Cp_def
/-!
# The Galois action on the `p`-adic complex numbers.
In this file, we show that the action of the absolute Galois group of `ℚ_[p]` on `Q_p_alg p`
extends to an action on `ℂ_[p]`.
## Tags
p-adic, p adic, padic, galois action, galois
-/
noncomputable theory
variables {p : ℕ} [fact (nat.prime p)]
/-- The elements of the Galois group Gal((Q_p_alg p)/ℚ_[p]) are isometries. -/
lemma galois_map_isometry (σ : (Q_p_alg p) ≃ₐ[ℚ_[p]] (Q_p_alg p)) : isometry σ :=
begin
rw isometry_iff_dist_eq,
intros x y,
simp only [dist_eq_norm, ← map_sub, Q_p_alg.norm_def,
spectral_norm_aut_isom (Q_p_alg.is_algebraic p) σ (x - y)],
end
/-- Any `σ` in `Q_p_alg p ≃ₐ[ℚ_[p]] Q_p_alg p` acts uniformly continuously on `Q_p_alg p`. -/
instance : has_uniform_continuous_const_smul (Q_p_alg p ≃ₐ[ℚ_[p]] Q_p_alg p) (Q_p_alg p) :=
⟨λ σ, (galois_map_isometry σ).uniform_continuous⟩
/-- The action of the absolute galois of `ℚ_[p]` on `Q_p_alg p` extends to an action on `ℂ_[p]`.-/
instance : mul_action ((Q_p_alg p) ≃ₐ[ℚ_[p]] (Q_p_alg p)) (ℂ_[p]) :=
uniform_space.completion.mul_action ((Q_p_alg p) ≃ₐ[ℚ_[p]] (Q_p_alg p)) (Q_p_alg p) |
1164ec0567ebc9fcb934ce124e7a16fe57c36644 | e0f9ba56b7fedc16ef8697f6caeef5898b435143 | /src/data/complex/basic.lean | 75db0333b26a5a4b45194d004c8fb446ae8b21f2 | [
"Apache-2.0"
] | permissive | anrddh/mathlib | 6a374da53c7e3a35cb0298b0cd67824efef362b4 | a4266a01d2dcb10de19369307c986d038c7bb6a6 | refs/heads/master | 1,656,710,827,909 | 1,589,560,456,000 | 1,589,560,456,000 | 264,271,800 | 0 | 0 | Apache-2.0 | 1,589,568,062,000 | 1,589,568,061,000 | null | UTF-8 | Lean | false | false | 19,437 | lean | /-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
The complex numbers, modelled as R^2 in the obvious way.
-/
import data.real.basic
structure complex : Type :=
(re : ℝ) (im : ℝ)
notation `ℂ` := complex
namespace complex
@[simp] theorem eta : ∀ z : ℂ, complex.mk z.re z.im = z
| ⟨a, b⟩ := rfl
@[ext]
theorem ext : ∀ {z w : ℂ}, z.re = w.re → z.im = w.im → z = w
| ⟨zr, zi⟩ ⟨_, _⟩ rfl rfl := rfl
theorem ext_iff {z w : ℂ} : z = w ↔ z.re = w.re ∧ z.im = w.im :=
⟨λ H, by simp [H], and.rec ext⟩
instance : has_coe ℝ ℂ := ⟨λ r, ⟨r, 0⟩⟩
@[simp, norm_cast] lemma of_real_re (r : ℝ) : (r : ℂ).re = r := rfl
@[simp, norm_cast] lemma of_real_im (r : ℝ) : (r : ℂ).im = 0 := rfl
@[simp, norm_cast] theorem of_real_inj {z w : ℝ} : (z : ℂ) = w ↔ z = w :=
⟨congr_arg re, congr_arg _⟩
instance : has_zero ℂ := ⟨(0 : ℝ)⟩
instance : inhabited ℂ := ⟨0⟩
@[simp] lemma zero_re : (0 : ℂ).re = 0 := rfl
@[simp] lemma zero_im : (0 : ℂ).im = 0 := rfl
@[simp, norm_cast] lemma of_real_zero : ((0 : ℝ) : ℂ) = 0 := rfl
@[simp] theorem of_real_eq_zero {z : ℝ} : (z : ℂ) = 0 ↔ z = 0 := of_real_inj
theorem of_real_ne_zero {z : ℝ} : (z : ℂ) ≠ 0 ↔ z ≠ 0 := not_congr of_real_eq_zero
instance : has_one ℂ := ⟨(1 : ℝ)⟩
@[simp] lemma one_re : (1 : ℂ).re = 1 := rfl
@[simp] lemma one_im : (1 : ℂ).im = 0 := rfl
@[simp, norm_cast] lemma of_real_one : ((1 : ℝ) : ℂ) = 1 := rfl
def I : ℂ := ⟨0, 1⟩
@[simp] lemma I_re : I.re = 0 := rfl
@[simp] lemma I_im : I.im = 1 := rfl
instance : has_add ℂ := ⟨λ z w, ⟨z.re + w.re, z.im + w.im⟩⟩
@[simp] lemma add_re (z w : ℂ) : (z + w).re = z.re + w.re := rfl
@[simp] lemma add_im (z w : ℂ) : (z + w).im = z.im + w.im := rfl
@[simp, norm_cast] lemma of_real_add (r s : ℝ) : ((r + s : ℝ) : ℂ) = r + s := ext_iff.2 $ by simp
@[simp, norm_cast] lemma of_real_bit0 (r : ℝ) : ((bit0 r : ℝ) : ℂ) = bit0 r := ext_iff.2 $ by simp [bit0]
@[simp, norm_cast] lemma of_real_bit1 (r : ℝ) : ((bit1 r : ℝ) : ℂ) = bit1 r := ext_iff.2 $ by simp [bit1]
instance : has_neg ℂ := ⟨λ z, ⟨-z.re, -z.im⟩⟩
@[simp] lemma neg_re (z : ℂ) : (-z).re = -z.re := rfl
@[simp] lemma neg_im (z : ℂ) : (-z).im = -z.im := rfl
@[simp, norm_cast] lemma of_real_neg (r : ℝ) : ((-r : ℝ) : ℂ) = -r := ext_iff.2 $ by simp
instance : has_mul ℂ := ⟨λ z w, ⟨z.re * w.re - z.im * w.im, z.re * w.im + z.im * w.re⟩⟩
@[simp] lemma mul_re (z w : ℂ) : (z * w).re = z.re * w.re - z.im * w.im := rfl
@[simp] lemma mul_im (z w : ℂ) : (z * w).im = z.re * w.im + z.im * w.re := rfl
@[simp, norm_cast] lemma of_real_mul (r s : ℝ) : ((r * s : ℝ) : ℂ) = r * s := ext_iff.2 $ by simp
lemma smul_re (r : ℝ) (z : ℂ) : (↑r * z).re = r * z.re := by simp
lemma smul_im (r : ℝ) (z : ℂ) : (↑r * z).im = r * z.im := by simp
@[simp] lemma I_mul_I : I * I = -1 := ext_iff.2 $ by simp
lemma I_ne_zero : (I : ℂ) ≠ 0 := mt (congr_arg im) zero_ne_one.symm
lemma mk_eq_add_mul_I (a b : ℝ) : complex.mk a b = a + b * I :=
ext_iff.2 $ by simp
@[simp] lemma re_add_im (z : ℂ) : (z.re : ℂ) + z.im * I = z :=
ext_iff.2 $ by simp
def real_prod_equiv : ℂ ≃ (ℝ × ℝ) :=
{ to_fun := λ z, ⟨z.re, z.im⟩,
inv_fun := λ p, ⟨p.1, p.2⟩,
left_inv := λ ⟨x, y⟩, rfl,
right_inv := λ ⟨x, y⟩, rfl }
@[simp] theorem real_prod_equiv_apply (z : ℂ) : real_prod_equiv z = (z.re, z.im) := rfl
theorem real_prod_equiv_symm_re (x y : ℝ) : (real_prod_equiv.symm (x, y)).re = x := rfl
theorem real_prod_equiv_symm_im (x y : ℝ) : (real_prod_equiv.symm (x, y)).im = y := rfl
def conj (z : ℂ) : ℂ := ⟨z.re, -z.im⟩
@[simp] lemma conj_re (z : ℂ) : (conj z).re = z.re := rfl
@[simp] lemma conj_im (z : ℂ) : (conj z).im = -z.im := rfl
@[simp] lemma conj_of_real (r : ℝ) : conj r = r := ext_iff.2 $ by simp [conj]
@[simp] lemma conj_zero : conj 0 = 0 := ext_iff.2 $ by simp [conj]
@[simp] lemma conj_one : conj 1 = 1 := ext_iff.2 $ by simp
@[simp] lemma conj_I : conj I = -I := ext_iff.2 $ by simp
@[simp] lemma conj_add (z w : ℂ) : conj (z + w) = conj z + conj w :=
ext_iff.2 $ by simp [add_comm]
@[simp] lemma conj_neg (z : ℂ) : conj (-z) = -conj z := rfl
@[simp] lemma conj_neg_I : conj (-I) = I := ext_iff.2 $ by simp
@[simp] lemma conj_mul (z w : ℂ) : conj (z * w) = conj z * conj w :=
ext_iff.2 $ by simp [add_comm]
@[simp] lemma conj_conj (z : ℂ) : conj (conj z) = z :=
ext_iff.2 $ by simp
lemma conj_involutive : function.involutive conj := conj_conj
lemma conj_bijective : function.bijective conj := conj_involutive.bijective
lemma conj_inj {z w : ℂ} : conj z = conj w ↔ z = w :=
conj_bijective.1.eq_iff
@[simp] lemma conj_eq_zero {z : ℂ} : conj z = 0 ↔ z = 0 :=
by simpa using @conj_inj z 0
lemma eq_conj_iff_real {z : ℂ} : conj z = z ↔ ∃ r : ℝ, z = r :=
⟨λ h, ⟨z.re, ext rfl $ eq_zero_of_neg_eq (congr_arg im h)⟩,
λ ⟨h, e⟩, by rw [e, conj_of_real]⟩
lemma eq_conj_iff_re {z : ℂ} : conj z = z ↔ (z.re : ℂ) = z :=
eq_conj_iff_real.trans ⟨by rintro ⟨r, rfl⟩; simp, λ h, ⟨_, h.symm⟩⟩
def norm_sq (z : ℂ) : ℝ := z.re * z.re + z.im * z.im
@[simp] lemma norm_sq_of_real (r : ℝ) : norm_sq r = r * r :=
by simp [norm_sq]
@[simp] lemma norm_sq_zero : norm_sq 0 = 0 := by simp [norm_sq]
@[simp] lemma norm_sq_one : norm_sq 1 = 1 := by simp [norm_sq]
@[simp] lemma norm_sq_I : norm_sq I = 1 := by simp [norm_sq]
lemma norm_sq_nonneg (z : ℂ) : 0 ≤ norm_sq z :=
add_nonneg (mul_self_nonneg _) (mul_self_nonneg _)
@[simp] lemma norm_sq_eq_zero {z : ℂ} : norm_sq z = 0 ↔ z = 0 :=
⟨λ h, ext
(eq_zero_of_mul_self_add_mul_self_eq_zero h)
(eq_zero_of_mul_self_add_mul_self_eq_zero $ (add_comm _ _).trans h),
λ h, h.symm ▸ norm_sq_zero⟩
@[simp] lemma norm_sq_pos {z : ℂ} : 0 < norm_sq z ↔ z ≠ 0 :=
by rw [lt_iff_le_and_ne, ne, eq_comm]; simp [norm_sq_nonneg]
@[simp] lemma norm_sq_neg (z : ℂ) : norm_sq (-z) = norm_sq z :=
by simp [norm_sq]
@[simp] lemma norm_sq_conj (z : ℂ) : norm_sq (conj z) = norm_sq z :=
by simp [norm_sq]
@[simp] lemma norm_sq_mul (z w : ℂ) : norm_sq (z * w) = norm_sq z * norm_sq w :=
by dsimp [norm_sq]; ring
lemma norm_sq_add (z w : ℂ) : norm_sq (z + w) =
norm_sq z + norm_sq w + 2 * (z * conj w).re :=
by dsimp [norm_sq]; ring
lemma re_sq_le_norm_sq (z : ℂ) : z.re * z.re ≤ norm_sq z :=
le_add_of_nonneg_right (mul_self_nonneg _)
lemma im_sq_le_norm_sq (z : ℂ) : z.im * z.im ≤ norm_sq z :=
le_add_of_nonneg_left (mul_self_nonneg _)
theorem mul_conj (z : ℂ) : z * conj z = norm_sq z :=
ext_iff.2 $ by simp [norm_sq, mul_comm, sub_eq_neg_add, add_comm]
theorem add_conj (z : ℂ) : z + conj z = (2 * z.re : ℝ) :=
ext_iff.2 $ by simp [two_mul]
instance : comm_ring ℂ :=
by refine { zero := 0, add := (+), neg := has_neg.neg, one := 1, mul := (*), ..};
{ intros, apply ext_iff.2; split; simp; ring }
/-- Coercion `ℝ → ℂ` as a `ring_hom`. -/
def of_real : ℝ →+* ℂ := ⟨coe, of_real_one, of_real_mul, of_real_zero, of_real_add⟩
@[simp] lemma of_real_eq_coe (r : ℝ) : of_real r = r := rfl
@[simp] lemma I_sq : I ^ 2 = -1 := by rw [pow_two, I_mul_I]
@[simp] lemma bit0_re (z : ℂ) : (bit0 z).re = bit0 z.re := rfl
@[simp] lemma bit1_re (z : ℂ) : (bit1 z).re = bit1 z.re := rfl
@[simp] lemma bit0_im (z : ℂ) : (bit0 z).im = bit0 z.im := eq.refl _
@[simp] lemma bit1_im (z : ℂ) : (bit1 z).im = bit0 z.im := add_zero _
@[simp] lemma sub_re (z w : ℂ) : (z - w).re = z.re - w.re := rfl
@[simp] lemma sub_im (z w : ℂ) : (z - w).im = z.im - w.im := rfl
@[simp, norm_cast] lemma of_real_sub (r s : ℝ) : ((r - s : ℝ) : ℂ) = r - s := ext_iff.2 $ by simp
@[simp, norm_cast] lemma of_real_pow (r : ℝ) (n : ℕ) : ((r ^ n : ℝ) : ℂ) = r ^ n :=
by induction n; simp [*, of_real_mul, pow_succ]
theorem sub_conj (z : ℂ) : z - conj z = (2 * z.im : ℝ) * I :=
ext_iff.2 $ by simp [two_mul, sub_eq_add_neg]
lemma conj_pow (z : ℂ) (n : ℕ) : conj (z ^ n) = conj z ^ n :=
by induction n; simp [*, conj_mul, pow_succ]
@[simp] lemma conj_two : conj (2 : ℂ) = 2 := by apply complex.ext; simp
lemma norm_sq_sub (z w : ℂ) : norm_sq (z - w) =
norm_sq z + norm_sq w - 2 * (z * conj w).re :=
by rw [sub_eq_add_neg, norm_sq_add]; simp [-mul_re, add_comm, add_left_comm, sub_eq_add_neg]
noncomputable instance : has_inv ℂ := ⟨λ z, conj z * ((norm_sq z)⁻¹:ℝ)⟩
theorem inv_def (z : ℂ) : z⁻¹ = conj z * ((norm_sq z)⁻¹:ℝ) := rfl
@[simp] lemma inv_re (z : ℂ) : (z⁻¹).re = z.re / norm_sq z := by simp [inv_def, division_def]
@[simp] lemma inv_im (z : ℂ) : (z⁻¹).im = -z.im / norm_sq z := by simp [inv_def, division_def]
@[simp, norm_cast] lemma of_real_inv (r : ℝ) : ((r⁻¹ : ℝ) : ℂ) = r⁻¹ :=
ext_iff.2 $ begin
simp,
by_cases r = 0, {simp [h]},
rw [← div_div_eq_div_mul, div_self h, one_div_eq_inv]
end
protected lemma inv_zero : (0⁻¹ : ℂ) = 0 :=
by rw [← of_real_zero, ← of_real_inv, inv_zero]
protected theorem mul_inv_cancel {z : ℂ} (h : z ≠ 0) : z * z⁻¹ = 1 :=
by rw [inv_def, ← mul_assoc, mul_conj, ← of_real_mul,
mul_inv_cancel (mt norm_sq_eq_zero.1 h), of_real_one]
noncomputable instance : field ℂ :=
{ inv := has_inv.inv,
zero_ne_one := mt (congr_arg re) zero_ne_one,
mul_inv_cancel := @complex.mul_inv_cancel,
inv_zero := complex.inv_zero,
..complex.comm_ring }
instance re.is_add_group_hom : is_add_group_hom complex.re :=
{ map_add := complex.add_re }
instance im.is_add_group_hom : is_add_group_hom complex.im :=
{ map_add := complex.add_im }
instance : is_ring_hom conj :=
by refine_struct {..}; simp
instance of_real.is_ring_hom : is_ring_hom (coe : ℝ → ℂ) :=
by refine_struct {..}; simp
lemma div_re (z w : ℂ) : (z / w).re = z.re * w.re / norm_sq w + z.im * w.im / norm_sq w :=
by simp [div_eq_mul_inv, mul_assoc, sub_eq_add_neg, add_comm, add_left_comm]
lemma div_im (z w : ℂ) : (z / w).im = z.im * w.re / norm_sq w - z.re * w.im / norm_sq w :=
by simp [div_eq_mul_inv, mul_assoc, sub_eq_add_neg, add_comm, add_left_comm]
@[simp, norm_cast] lemma of_real_div (r s : ℝ) : ((r / s : ℝ) : ℂ) = r / s :=
is_ring_hom.map_div coe
@[simp, norm_cast] lemma of_real_fpow (r : ℝ) (n : ℤ) : ((r ^ n : ℝ) : ℂ) = (r : ℂ) ^ n :=
is_ring_hom.map_fpow of_real r n
@[simp, norm_cast] theorem of_real_int_cast : ∀ n : ℤ, ((n : ℝ) : ℂ) = n :=
of_real.map_int_cast
@[simp, norm_cast] theorem of_real_nat_cast (n : ℕ) : ((n : ℝ) : ℂ) = n :=
of_real.map_nat_cast n
@[simp] lemma conj_sub (z w : ℂ) : conj (z - w) = conj z - conj w :=
by simp [sub_eq_add_neg]
@[simp] lemma conj_inv (z : ℂ) : conj z⁻¹ = (conj z)⁻¹ :=
by ext; simp [neg_div]
@[simp] lemma conj_div (z w : ℂ) : conj (z / w) = conj z / conj w :=
by rw [division_def, conj_mul, conj_inv]; refl
@[simp] lemma div_I (z : ℂ) : z / I = -(z * I) :=
(div_eq_iff_mul_eq I_ne_zero).2 $ by simp [mul_assoc]
@[simp] lemma inv_I : I⁻¹ = -I :=
by simp [inv_eq_one_div]
@[simp] lemma norm_sq_inv (z : ℂ) : norm_sq z⁻¹ = (norm_sq z)⁻¹ :=
by classical; exact
if h : z = 0 then by simp [h] else
(domain.mul_left_inj (mt norm_sq_eq_zero.1 h)).1 $
by rw [← norm_sq_mul]; simp [h, -norm_sq_mul]
@[simp] lemma norm_sq_div (z w : ℂ) : norm_sq (z / w) = norm_sq z / norm_sq w :=
by rw [division_def, norm_sq_mul, norm_sq_inv]; refl
instance char_zero_complex : char_zero ℂ :=
add_group.char_zero_of_inj_zero $ λ n h,
by rwa [← of_real_nat_cast, of_real_eq_zero, nat.cast_eq_zero] at h
@[simp, norm_cast] theorem of_real_rat_cast : ∀ n : ℚ, ((n : ℝ) : ℂ) = n :=
of_real.map_rat_cast
theorem re_eq_add_conj (z : ℂ) : (z.re : ℂ) = (z + conj z) / 2 :=
by rw [add_conj]; simp; rw [mul_div_cancel_left (z.re:ℂ) two_ne_zero']
@[simp, norm_cast] lemma nat_cast_re (n : ℕ) : (n : ℂ).re = n :=
by rw [← of_real_nat_cast, of_real_re]
@[simp, norm_cast] lemma nat_cast_im (n : ℕ) : (n : ℂ).im = 0 :=
by rw [← of_real_nat_cast, of_real_im]
@[simp, norm_cast] lemma int_cast_re (n : ℤ) : (n : ℂ).re = n :=
by rw [← of_real_int_cast, of_real_re]
@[simp, norm_cast] lemma int_cast_im (n : ℤ) : (n : ℂ).im = 0 :=
by rw [← of_real_int_cast, of_real_im]
@[simp, norm_cast] lemma rat_cast_re (q : ℚ) : (q : ℂ).re = q :=
by rw [← of_real_rat_cast, of_real_re]
@[simp, norm_cast] lemma rat_cast_im (q : ℚ) : (q : ℂ).im = 0 :=
by rw [← of_real_rat_cast, of_real_im]
noncomputable def abs (z : ℂ) : ℝ := (norm_sq z).sqrt
local notation `abs'` := _root_.abs
@[simp] lemma abs_of_real (r : ℝ) : abs r = abs' r :=
by simp [abs, norm_sq_of_real, real.sqrt_mul_self_eq_abs]
lemma abs_of_nonneg {r : ℝ} (h : 0 ≤ r) : abs r = r :=
(abs_of_real _).trans (abs_of_nonneg h)
lemma abs_of_nat (n : ℕ) : complex.abs n = n :=
calc complex.abs n = complex.abs (n:ℝ) : by rw [of_real_nat_cast]
... = _ : abs_of_nonneg (nat.cast_nonneg n)
lemma mul_self_abs (z : ℂ) : abs z * abs z = norm_sq z :=
real.mul_self_sqrt (norm_sq_nonneg _)
@[simp] lemma abs_zero : abs 0 = 0 := by simp [abs]
@[simp] lemma abs_one : abs 1 = 1 := by simp [abs]
@[simp] lemma abs_I : abs I = 1 := by simp [abs]
@[simp] lemma abs_two : abs 2 = 2 :=
calc abs 2 = abs (2 : ℝ) : by rw [of_real_bit0, of_real_one]
... = (2 : ℝ) : abs_of_nonneg (by norm_num)
lemma abs_nonneg (z : ℂ) : 0 ≤ abs z :=
real.sqrt_nonneg _
@[simp] lemma abs_eq_zero {z : ℂ} : abs z = 0 ↔ z = 0 :=
(real.sqrt_eq_zero $ norm_sq_nonneg _).trans norm_sq_eq_zero
@[simp] lemma abs_conj (z : ℂ) : abs (conj z) = abs z :=
by simp [abs]
@[simp] lemma abs_mul (z w : ℂ) : abs (z * w) = abs z * abs w :=
by rw [abs, norm_sq_mul, real.sqrt_mul (norm_sq_nonneg _)]; refl
lemma abs_re_le_abs (z : ℂ) : abs' z.re ≤ abs z :=
by rw [mul_self_le_mul_self_iff (_root_.abs_nonneg z.re) (abs_nonneg _),
abs_mul_abs_self, mul_self_abs];
apply re_sq_le_norm_sq
lemma abs_im_le_abs (z : ℂ) : abs' z.im ≤ abs z :=
by rw [mul_self_le_mul_self_iff (_root_.abs_nonneg z.im) (abs_nonneg _),
abs_mul_abs_self, mul_self_abs];
apply im_sq_le_norm_sq
lemma re_le_abs (z : ℂ) : z.re ≤ abs z :=
(abs_le.1 (abs_re_le_abs _)).2
lemma im_le_abs (z : ℂ) : z.im ≤ abs z :=
(abs_le.1 (abs_im_le_abs _)).2
lemma abs_add (z w : ℂ) : abs (z + w) ≤ abs z + abs w :=
(mul_self_le_mul_self_iff (abs_nonneg _)
(add_nonneg (abs_nonneg _) (abs_nonneg _))).2 $
begin
rw [mul_self_abs, add_mul_self_eq, mul_self_abs, mul_self_abs,
add_right_comm, norm_sq_add, add_le_add_iff_left,
mul_assoc, mul_le_mul_left (@two_pos ℝ _)],
simpa [-mul_re] using re_le_abs (z * conj w)
end
instance : is_absolute_value abs :=
{ abv_nonneg := abs_nonneg,
abv_eq_zero := λ _, abs_eq_zero,
abv_add := abs_add,
abv_mul := abs_mul }
open is_absolute_value
@[simp] lemma abs_abs (z : ℂ) : abs' (abs z) = abs z :=
_root_.abs_of_nonneg (abs_nonneg _)
@[simp] lemma abs_pos {z : ℂ} : 0 < abs z ↔ z ≠ 0 := abv_pos abs
@[simp] lemma abs_neg : ∀ z, abs (-z) = abs z := abv_neg abs
lemma abs_sub : ∀ z w, abs (z - w) = abs (w - z) := abv_sub abs
lemma abs_sub_le : ∀ a b c, abs (a - c) ≤ abs (a - b) + abs (b - c) := abv_sub_le abs
@[simp] theorem abs_inv : ∀ z, abs z⁻¹ = (abs z)⁻¹ := abv_inv abs
@[simp] theorem abs_div : ∀ z w, abs (z / w) = abs z / abs w := abv_div abs
lemma abs_abs_sub_le_abs_sub : ∀ z w, abs' (abs z - abs w) ≤ abs (z - w) := abs_abv_sub_le_abv_sub abs
lemma abs_le_abs_re_add_abs_im (z : ℂ) : abs z ≤ abs' z.re + abs' z.im :=
by simpa [re_add_im] using abs_add z.re (z.im * I)
lemma abs_re_div_abs_le_one (z : ℂ) : abs' (z.re / z.abs) ≤ 1 :=
by classical; exact
if hz : z = 0 then by simp [hz, zero_le_one]
else by rw [_root_.abs_div, abs_abs]; exact
div_le_of_le_mul (abs_pos.2 hz) (by rw mul_one; exact abs_re_le_abs _)
lemma abs_im_div_abs_le_one (z : ℂ) : abs' (z.im / z.abs) ≤ 1 :=
by classical; exact
if hz : z = 0 then by simp [hz, zero_le_one]
else by rw [_root_.abs_div, abs_abs]; exact
div_le_of_le_mul (abs_pos.2 hz) (by rw mul_one; exact abs_im_le_abs _)
@[simp, norm_cast] lemma abs_cast_nat (n : ℕ) : abs (n : ℂ) = n :=
by rw [← of_real_nat_cast, abs_of_nonneg (nat.cast_nonneg n)]
lemma norm_sq_eq_abs (x : ℂ) : norm_sq x = abs x ^ 2 :=
by rw [abs, pow_two, real.mul_self_sqrt (norm_sq_nonneg _)]
theorem is_cau_seq_re (f : cau_seq ℂ abs) : is_cau_seq abs' (λ n, (f n).re) :=
λ ε ε0, (f.cauchy ε0).imp $ λ i H j ij,
lt_of_le_of_lt (by simpa using abs_re_le_abs (f j - f i)) (H _ ij)
theorem is_cau_seq_im (f : cau_seq ℂ abs) : is_cau_seq abs' (λ n, (f n).im) :=
λ ε ε0, (f.cauchy ε0).imp $ λ i H j ij,
lt_of_le_of_lt (by simpa using abs_im_le_abs (f j - f i)) (H _ ij)
noncomputable def cau_seq_re (f : cau_seq ℂ abs) : cau_seq ℝ abs' :=
⟨_, is_cau_seq_re f⟩
noncomputable def cau_seq_im (f : cau_seq ℂ abs) : cau_seq ℝ abs' :=
⟨_, is_cau_seq_im f⟩
lemma is_cau_seq_abs {f : ℕ → ℂ} (hf : is_cau_seq abs f) :
is_cau_seq abs' (abs ∘ f) :=
λ ε ε0, let ⟨i, hi⟩ := hf ε ε0 in
⟨i, λ j hj, lt_of_le_of_lt (abs_abs_sub_le_abs_sub _ _) (hi j hj)⟩
noncomputable def lim_aux (f : cau_seq ℂ abs) : ℂ :=
⟨cau_seq.lim (cau_seq_re f), cau_seq.lim (cau_seq_im f)⟩
theorem equiv_lim_aux (f : cau_seq ℂ abs) : f ≈ cau_seq.const abs (lim_aux f) :=
λ ε ε0, (exists_forall_ge_and
(cau_seq.equiv_lim ⟨_, is_cau_seq_re f⟩ _ (half_pos ε0))
(cau_seq.equiv_lim ⟨_, is_cau_seq_im f⟩ _ (half_pos ε0))).imp $
λ i H j ij, begin
cases H _ ij with H₁ H₂,
apply lt_of_le_of_lt (abs_le_abs_re_add_abs_im _),
dsimp [lim_aux] at *,
have := add_lt_add H₁ H₂,
rwa add_halves at this,
end
noncomputable instance : cau_seq.is_complete ℂ abs :=
⟨λ f, ⟨lim_aux f, equiv_lim_aux f⟩⟩
open cau_seq
lemma lim_eq_lim_im_add_lim_re (f : cau_seq ℂ abs) : lim f =
↑(lim (cau_seq_re f)) + ↑(lim (cau_seq_im f)) * I :=
lim_eq_of_equiv_const $
calc f ≈ _ : equiv_lim_aux f
... = cau_seq.const abs (↑(lim (cau_seq_re f)) + ↑(lim (cau_seq_im f)) * I) :
cau_seq.ext (λ _, complex.ext (by simp [lim_aux, cau_seq_re]) (by simp [lim_aux, cau_seq_im]))
lemma lim_re (f : cau_seq ℂ abs) : lim (cau_seq_re f) = (lim f).re :=
by rw [lim_eq_lim_im_add_lim_re]; simp
lemma lim_im (f : cau_seq ℂ abs) : lim (cau_seq_im f) = (lim f).im :=
by rw [lim_eq_lim_im_add_lim_re]; simp
lemma is_cau_seq_conj (f : cau_seq ℂ abs) : is_cau_seq abs (λ n, conj (f n)) :=
λ ε ε0, let ⟨i, hi⟩ := f.2 ε ε0 in
⟨i, λ j hj, by rw [← conj_sub, abs_conj]; exact hi j hj⟩
noncomputable def cau_seq_conj (f : cau_seq ℂ abs) : cau_seq ℂ abs := ⟨_, is_cau_seq_conj f⟩
lemma lim_conj (f : cau_seq ℂ abs) : lim (cau_seq_conj f) = conj (lim f) :=
complex.ext (by simp [cau_seq_conj, (lim_re _).symm, cau_seq_re])
(by simp [cau_seq_conj, (lim_im _).symm, cau_seq_im, (lim_neg _).symm]; refl)
noncomputable def cau_seq_abs (f : cau_seq ℂ abs) : cau_seq ℝ abs' :=
⟨_, is_cau_seq_abs f.2⟩
lemma lim_abs (f : cau_seq ℂ abs) : lim (cau_seq_abs f) = abs (lim f) :=
lim_eq_of_equiv_const (λ ε ε0,
let ⟨i, hi⟩ := equiv_lim f ε ε0 in
⟨i, λ j hj, lt_of_le_of_lt (abs_abs_sub_le_abs_sub _ _) (hi j hj)⟩)
end complex
|
3e0277d793f1ff3e5cbf7d44922c1a02f2bbe1ad | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/category_theory/endomorphism.lean | 958b134183dad5b7de15349666712e4fb3b6b3d5 | [
"Apache-2.0"
] | permissive | robertylewis/mathlib | 3d16e3e6daf5ddde182473e03a1b601d2810952c | 1d13f5b932f5e40a8308e3840f96fc882fae01f0 | refs/heads/master | 1,651,379,945,369 | 1,644,276,960,000 | 1,644,276,960,000 | 98,875,504 | 0 | 0 | Apache-2.0 | 1,644,253,514,000 | 1,501,495,700,000 | Lean | UTF-8 | Lean | false | false | 4,648 | lean | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Scott Morrison, Simon Hudon
-/
import category_theory.groupoid
import category_theory.opposites
import data.equiv.mul_add
import group_theory.group_action.defs
/-!
# Endomorphisms
Definition and basic properties of endomorphisms and automorphisms of an object in a category.
For each `X : C`, we provide `End X := X ⟶ X` with a monoid structure,
and `Aut X := X ≅ X ` with a group structure.
-/
universes v v' u u'
namespace category_theory
/-- Endomorphisms of an object in a category. Arguments order in multiplication agrees with
`function.comp`, not with `category.comp`. -/
def End {C : Type u} [category_struct.{v} C] (X : C) := X ⟶ X
namespace End
section struct
variables {C : Type u} [category_struct.{v} C] (X : C)
instance has_one : has_one (End X) := ⟨𝟙 X⟩
instance inhabited : inhabited (End X) := ⟨𝟙 X⟩
/-- Multiplication of endomorphisms agrees with `function.comp`, not `category_struct.comp`. -/
instance has_mul : has_mul (End X) := ⟨λ x y, y ≫ x⟩
variable {X}
/-- Assist the typechecker by expressing a morphism `X ⟶ X` as a term of `End X`. -/
def of (f : X ⟶ X) : End X := f
/-- Assist the typechecker by expressing an endomorphism `f : End X` as a term of `X ⟶ X`. -/
def as_hom (f : End X) : X ⟶ X := f
@[simp] lemma one_def : (1 : End X) = 𝟙 X := rfl
@[simp] lemma mul_def (xs ys : End X) : xs * ys = ys ≫ xs := rfl
end struct
/-- Endomorphisms of an object form a monoid -/
instance monoid {C : Type u} [category.{v} C] {X : C} : monoid (End X) :=
{ mul_one := category.id_comp,
one_mul := category.comp_id,
mul_assoc := λ x y z, (category.assoc z y x).symm,
..End.has_mul X, ..End.has_one X }
section mul_action
variables {C : Type u} [category.{v} C]
open opposite
instance mul_action_right {X Y : C} : mul_action (End Y) (X ⟶ Y) :=
{ smul := λ r f, f ≫ r,
one_smul := category.comp_id,
mul_smul := λ r s f, eq.symm $ category.assoc _ _ _ }
instance mul_action_left {X : Cᵒᵖ} {Y : C} : mul_action (End X) (unop X ⟶ Y) :=
{ smul := λ r f, r.unop ≫ f,
one_smul := category.id_comp,
mul_smul := λ r s f, category.assoc _ _ _ }
lemma smul_right {X Y : C} {r : End Y} {f : X ⟶ Y} : r • f = f ≫ r := rfl
lemma smul_left {X : Cᵒᵖ} {Y : C} {r : (End X)} {f : unop X ⟶ Y} : r • f = r.unop ≫ f := rfl
end mul_action
/-- In a groupoid, endomorphisms form a group -/
instance group {C : Type u} [groupoid.{v} C] (X : C) : group (End X) :=
{ mul_left_inv := groupoid.comp_inv, inv := groupoid.inv, ..End.monoid }
end End
lemma is_unit_iff_is_iso {C : Type u} [category.{v} C] {X : C} (f : End X) :
is_unit (f : End X) ↔ is_iso f :=
⟨λ h, { out := ⟨h.unit.inv, ⟨h.unit.inv_val, h.unit.val_inv⟩⟩ },
λ h, by exactI ⟨⟨f, inv f, by simp, by simp⟩, rfl⟩⟩
variables {C : Type u} [category.{v} C] (X : C)
/--
Automorphisms of an object in a category.
The order of arguments in multiplication agrees with
`function.comp`, not with `category.comp`.
-/
def Aut (X : C) := X ≅ X
attribute [ext Aut] iso.ext
namespace Aut
instance inhabited : inhabited (Aut X) := ⟨iso.refl X⟩
instance : group (Aut X) :=
by refine_struct
{ one := iso.refl X,
inv := iso.symm,
mul := flip iso.trans,
div := _,
npow := @npow_rec (Aut X) ⟨iso.refl X⟩ ⟨flip iso.trans⟩,
zpow := @zpow_rec (Aut X) ⟨iso.refl X⟩ ⟨flip iso.trans⟩ ⟨iso.symm⟩ };
intros; try { refl }; ext;
simp [flip, (*), monoid.mul, mul_one_class.mul, mul_one_class.one, has_one.one, monoid.one,
has_inv.inv]
/--
Units in the monoid of endomorphisms of an object
are (multiplicatively) equivalent to automorphisms of that object.
-/
def units_End_equiv_Aut : (End X)ˣ ≃* Aut X :=
{ to_fun := λ f, ⟨f.1, f.2, f.4, f.3⟩,
inv_fun := λ f, ⟨f.1, f.2, f.4, f.3⟩,
left_inv := λ ⟨f₁, f₂, f₃, f₄⟩, rfl,
right_inv := λ ⟨f₁, f₂, f₃, f₄⟩, rfl,
map_mul' := λ f g, by rcases f; rcases g; refl }
end Aut
namespace functor
variables {D : Type u'} [category.{v'} D] (f : C ⥤ D) (X)
/-- `f.map` as a monoid hom between endomorphism monoids. -/
@[simps] def map_End : End X →* End (f.obj X) :=
{ to_fun := functor.map f,
map_mul' := λ x y, f.map_comp y x,
map_one' := f.map_id X }
/-- `f.map_iso` as a group hom between automorphism groups. -/
def map_Aut : Aut X →* Aut (f.obj X) :=
{ to_fun := f.map_iso,
map_mul' := λ x y, f.map_iso_trans y x,
map_one' := f.map_iso_refl X }
end functor
end category_theory
|
bccb18c1be9ea771d299c70d08f6ac52045b5160 | d7189ea2ef694124821b033e533f18905b5e87ef | /galois/list/init.lean | 84d46813559764e4465c12c17b2e1fbf56596392 | [
"Apache-2.0"
] | permissive | digama0/lean-protocol-support | eaa7e6f8b8e0d5bbfff1f7f52bfb79a3b11b0f59 | cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda | refs/heads/master | 1,625,421,450,627 | 1,506,035,462,000 | 1,506,035,462,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 972 | lean | /- Defines theorems about init and supporting function -/
import galois.list.simplify_eq
universe variable u
namespace list
variable {α : Type u}
theorem length_init : ∀ (l : list α), length (init l) = length l - 1
| nil := by simp [init]
| (x::l) :=
begin
have p := length_init l,
cases l with b r,
-- Case where l = nil
{ simp [init] },
-- Case where l = b :: r
{ simp [init, p, nat.add_sub_cancel, nat.add_sub_cancel_left], }
end
-- A list with length 0 must be nil nil
theorem length_0_implies_nil {l : list α} (h : length l = 0) : l = nil :=
begin
cases l,
simp,
contradiction
end
-- Maps a non-empty list to the initial part and last part.
theorem init_append_last_self (l : list α) (pr : l ≠ nil) : init l ++ [last l pr] = l :=
begin
revert pr,
induction l with x l ind,
{ contradiction },
{ intros pr,
cases l with y l,
{ simp [init] },
{ simp [init, last, ind (last._main._proof_2 y l)] }
},
end
end list
|
455e42d044654bc47327d8ad6df2c5d6771fac5c | 4727251e0cd73359b15b664c3170e5d754078599 | /src/testing/slim_check/testable.lean | 9888738ef57d3eb5d86faff40c0fc82e3eaf415a | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 28,298 | lean | /-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import testing.slim_check.sampleable
/-!
# `testable` Class
Testable propositions have a procedure that can generate counter-examples
together with a proof that they invalidate the proposition.
This is a port of the Haskell QuickCheck library.
## Creating Customized Instances
The type classes `testable` and `sampleable` are the means by which
`slim_check` creates samples and tests them. For instance, the proposition
`∀ i j : ℕ, i ≤ j` has a `testable` instance because `ℕ` is sampleable
and `i ≤ j` is decidable. Once `slim_check` finds the `testable`
instance, it can start using the instance to repeatedly creating samples
and checking whether they satisfy the property. This allows the
user to create new instances and apply `slim_check` to new situations.
### Polymorphism
The property `testable.check (∀ (α : Type) (xs ys : list α), xs ++ ys
= ys ++ xs)` shows us that type-polymorphic properties can be
tested. `α` is instantiated with `ℤ` first and then tested as normal
monomorphic properties.
The monomorphisation limits the applicability of `slim_check` to
polymorphic properties that can be stated about integers. The
limitation may be lifted in the future but, for now, if
one wishes to use a different type than `ℤ`, one has to refer to
the desired type.
### What do I do if I'm testing a property about my newly defined type?
Let us consider a type made for a new formalization:
```lean
structure my_type :=
(x y : ℕ)
(h : x ≤ y)
```
How do we test a property about `my_type`? For instance, let us consider
`testable.check $ ∀ a b : my_type, a.y ≤ b.x → a.x ≤ b.y`. Writing this
property as is will give us an error because we do not have an instance
of `sampleable my_type`. We can define one as follows:
```lean
instance : sampleable my_type :=
{ sample := do
x ← sample ℕ,
xy_diff ← sample ℕ,
return { x := x, y := x + xy_diff, h := /- some proof -/ } }
```
We can see that the instance is very simple because our type is built
up from other type that have `sampleable` instances. `sampleable` also
has a `shrink` method but it is optional. We may want to implement one
for ease of testing as:
```lean
/- ... -/
shrink := λ ⟨x,y,h⟩, (λ ⟨x,y⟩, { x := x, y := x + y, h := /- proof -/}) <$> shrink (x, y - x) }
```
Again, we take advantage of the fact that other types have useful
`shrink` implementations, in this case `prod`.
### Optimizing the sampling
Some properties are guarded by a proposition. For instance, recall this
example:
```lean
#eval testable.check (∀ x : ℕ, 2 ∣ x → x < 100)
```
When testing the above example, we generate a natural number, we check
that it is even and test it if it is even or throw it away and start
over otherwise. Statistically, we can expect half of our samples to be
thrown away by such a filter. Sometimes, the filter is more
restrictive. For instance we might need `x` to be a `prime`
number. This would cause most of our samples to be discarded.
We can help `slim_check` find good samples by providing specialized
sampleable instances. Below, we show an instance for the subtype
of even natural numbers. This means that, when producing
a sample, it is forced to produce a proof that it is even.
```lean
instance {k : ℕ} [fact (0 < k)] : sampleable { x : ℕ // k ∣ x } :=
{ sample := do { n ← sample ℕ, pure ⟨k*n, dvd_mul_right _ _⟩ },
shrink := λ ⟨x,h⟩, (λ y, ⟨k*y, dvd_mul_right _ _⟩) <$> shrink x }
```
Such instance will be preferred when testing a proposition of the shape
`∀ x : T, p x → q`
We can observe the effect by enabling tracing:
```lean
/- no specialized sampling -/
#eval testable.check (∀ x : ℕ, 2 ∣ x → x < 100) { trace_discarded := tt }
-- discard
-- x := 1
-- discard
-- x := 41
-- discard
-- x := 3
-- discard
-- x := 5
-- discard
-- x := 5
-- discard
-- x := 197
-- discard
-- x := 469
-- discard
-- x := 9
-- discard
-- ===================
-- Found problems!
-- x := 552
-- -------------------
/- let us define a specialized sampling instance -/
instance {k : ℕ} : sampleable { x : ℕ // k ∣ x } :=
{ sample := do { n ← sample ℕ, pure ⟨k*n, dvd_mul_right _ _⟩ },
shrink := λ ⟨x,h⟩, (λ y, ⟨k*y, dvd_mul_right _ _⟩) <$> shrink x }
#eval testable.check (∀ x : ℕ, 2 ∣ x → x < 100) { enable_tracing := tt }
-- ===================
-- Found problems!
-- x := 358
-- -------------------
```
Similarly, it is common to write properties of the form: `∀ i j, i ≤ j → ...`
as the following example show:
```lean
#eval check (∀ i j k : ℕ, j < k → i - k < i - j)
```
Without subtype instances, the above property discards many samples
because `j < k` does not hold. Fortunately, we have appropriate
instance to choose `k` intelligently.
## Main definitions
* `testable` class
* `testable.check`: a way to test a proposition using random examples
## Tags
random testing
## References
* https://hackage.haskell.org/package/QuickCheck
-/
universes u v
variables var var' : string
variable α : Type u
variable β : α → Prop
variable f : Type → Prop
namespace slim_check
/-- Result of trying to disprove `p`
The constructors are:
* `success : (psum unit p) → test_result`
succeed when we find another example satisfying `p`
In `success h`, `h` is an optional proof of the proposition.
Without the proof, all we know is that we found one example
where `p` holds. With a proof, the one test was sufficient to
prove that `p` holds and we do not need to keep finding examples.
* `gave_up {} : ℕ → test_result`
give up when a well-formed example cannot be generated.
`gave_up n` tells us that `n` invalid examples were tried.
Above 100, we give up on the proposition and report that we
did not find a way to properly test it.
* `failure : ¬ p → (list string) → ℕ → test_result`
a counter-example to `p`; the strings specify values for the relevant variables.
`failure h vs n` also carries a proof that `p` does not hold. This way, we can
guarantee that there will be no false positive. The last component, `n`,
is the number of times that the counter-example was shrunk.
-/
@[derive inhabited]
inductive test_result (p : Prop)
| success : (psum unit p) → test_result
| gave_up {} : ℕ → test_result
| failure : ¬ p → (list string) → ℕ → test_result
/-- format a `test_result` as a string. -/
protected def test_result.to_string {p} : test_result p → string
| (test_result.success (psum.inl ())) := "success (without proof)"
| (test_result.success (psum.inr h)) := "success (with proof)"
| (test_result.gave_up n) := sformat!"gave up {n} times"
| (test_result.failure a vs _) := sformat!"failed {vs}"
/-- configuration for testing a property -/
@[derive [has_reflect, inhabited]]
structure slim_check_cfg :=
(num_inst : ℕ := 100) -- number of examples
(max_size : ℕ := 100) -- final size argument
(trace_discarded : bool := ff) -- enable the printing out of discarded samples
(trace_success : bool := ff) -- enable the printing out of successful tests
(trace_shrink : bool := ff) -- enable the printing out of shrinking steps
(trace_shrink_candidates : bool := ff) -- enable the printing out of shrinking candidates
(random_seed : option ℕ := none) -- specify a seed to the random number generator to
-- obtain a deterministic behavior
(quiet : bool := ff) -- suppress success message when running `slim_check`
instance {p} : has_to_string (test_result p) := ⟨ test_result.to_string ⟩
/--
`printable_prop p` allows one to print a proposition so that
`slim_check` can indicate how values relate to each other.
-/
class printable_prop (p : Prop) :=
(print_prop : option string)
@[priority 100] -- see [note priority]
instance default_printable_prop {p} : printable_prop p :=
⟨ none ⟩
/-- `testable p` uses random examples to try to disprove `p`. -/
class testable (p : Prop) :=
(run [] (cfg : slim_check_cfg) (minimize : bool) : gen (test_result p))
open _root_.list
open test_result
/-- applicative combinator proof carrying test results -/
def combine {p q : Prop} : psum unit (p → q) → psum unit p → psum unit q
| (psum.inr f) (psum.inr x) := psum.inr (f x)
| _ _ := psum.inl ()
/-- Combine the test result for properties `p` and `q` to create a test for their conjunction. -/
def and_counter_example {p q : Prop} :
test_result p →
test_result q →
test_result (p ∧ q)
| (failure Hce xs n) _ := failure (λ h, Hce h.1) xs n
| _ (failure Hce xs n) := failure (λ h, Hce h.2) xs n
| (success xs) (success ys) := success $ combine (combine (psum.inr and.intro) xs) ys
| (gave_up n) (gave_up m) := gave_up $ n + m
| (gave_up n) _ := gave_up n
| _ (gave_up n) := gave_up n
/-- Combine the test result for properties `p` and `q` to create a test for their disjunction -/
def or_counter_example {p q : Prop} :
test_result p →
test_result q →
test_result (p ∨ q)
| (failure Hce xs n) (failure Hce' ys n') := failure (λ h, or_iff_not_and_not.1 h ⟨Hce, Hce'⟩)
(xs ++ ys) (n + n')
| (success xs) _ := success $ combine (psum.inr or.inl) xs
| _ (success ys) := success $ combine (psum.inr or.inr) ys
| (gave_up n) (gave_up m) := gave_up $ n + m
| (gave_up n) _ := gave_up n
| _ (gave_up n) := gave_up n
/-- If `q → p`, then `¬ p → ¬ q` which means that testing `p` can allow us
to find counter-examples to `q`. -/
def convert_counter_example {p q : Prop}
(h : q → p) :
test_result p →
opt_param (psum unit (p → q)) (psum.inl ()) →
test_result q
| (failure Hce xs n) _ := failure (mt h Hce) xs n
| (success Hp) Hpq := success (combine Hpq Hp)
| (gave_up n) _ := gave_up n
/-- Test `q` by testing `p` and proving the equivalence between the two. -/
def convert_counter_example' {p q : Prop}
(h : p ↔ q) (r : test_result p) :
test_result q :=
convert_counter_example h.2 r (psum.inr h.1)
/-- When we assign a value to a universally quantified variable,
we record that value using this function so that our counter-examples
can be informative. -/
def add_to_counter_example (x : string) {p q : Prop}
(h : q → p) :
test_result p →
opt_param (psum unit (p → q)) (psum.inl ()) →
test_result q
| (failure Hce xs n) _ := failure (mt h Hce) (x :: xs) n
| r hpq := convert_counter_example h r hpq
/-- Add some formatting to the information recorded by `add_to_counter_example`. -/
def add_var_to_counter_example {γ : Type v} [has_repr γ]
(var : string) (x : γ) {p q : Prop}
(h : q → p) : test_result p →
opt_param (psum unit (p → q)) (psum.inl ()) →
test_result q :=
@add_to_counter_example (var ++ " := " ++ repr x) _ _ h
/-- Gadget used to introspect the name of bound variables.
It is used with the `testable` typeclass so that
`testable (named_binder "x" (∀ x, p x))` can use the variable name
of `x` in error messages displayed to the user. If we find that instantiating
the above quantifier with 3 falsifies it, we can print:
```
==============
Problem found!
==============
x := 3
```
-/
@[simp, nolint unused_arguments]
def named_binder (n : string) (p : Prop) : Prop := p
/-- Is the given test result a failure? -/
def is_failure {p} : test_result p → bool
| (test_result.failure _ _ _) := tt
| _ := ff
instance and_testable (p q : Prop) [testable p] [testable q] :
testable (p ∧ q) :=
⟨ λ cfg min, do
xp ← testable.run p cfg min,
xq ← testable.run q cfg min,
pure $ and_counter_example xp xq ⟩
instance or_testable (p q : Prop) [testable p] [testable q] :
testable (p ∨ q) :=
⟨ λ cfg min, do
xp ← testable.run p cfg min,
match xp with
| success (psum.inl h) := pure $ success (psum.inl h)
| success (psum.inr h) := pure $ success (psum.inr $ or.inl h)
| _ := do
xq ← testable.run q cfg min,
pure $ or_counter_example xp xq
end ⟩
instance iff_testable (p q : Prop) [testable ((p ∧ q) ∨ (¬ p ∧ ¬ q))] :
testable (p ↔ q) :=
⟨ λ cfg min, do
xp ← testable.run ((p ∧ q) ∨ (¬ p ∧ ¬ q)) cfg min,
return $ convert_counter_example' (by tauto!) xp ⟩
open printable_prop
@[priority 1000]
instance dec_guard_testable (p : Prop) [printable_prop p] [decidable p] (β : p → Prop)
[∀ h, testable (β h)] : testable (named_binder var $ Π h, β h) :=
⟨ λ cfg min, do
if h : p
then
match print_prop p with
| none := (λ r, convert_counter_example ($ h) r (psum.inr $ λ q _, q)) <$>
testable.run (β h) cfg min
| some str := (λ r, add_to_counter_example (sformat!"guard: {str}") ($ h) r
(psum.inr $ λ q _, q)) <$> testable.run (β h) cfg min
end
else if cfg.trace_discarded ∨ cfg.trace_success then
match print_prop p with
| none := trace "discard" $ return $ gave_up 1
| some str := trace sformat!"discard: {str} does not hold" $ return $ gave_up 1
end
else return $ gave_up 1 ⟩
/-- Type tag that replaces a type's `has_repr` instance with its `has_to_string` instance. -/
def use_has_to_string (α : Type*) := α
instance use_has_to_string.inhabited [I : inhabited α] : inhabited (use_has_to_string α) := I
/-- Add the type tag `use_has_to_string` to an expression's type. -/
def use_has_to_string.mk {α} (x : α) : use_has_to_string α := x
instance [has_to_string α] : has_repr (use_has_to_string α) :=
⟨ @to_string α _ ⟩
@[priority 2000]
instance all_types_testable [testable (f ℤ)] : testable (named_binder var $ Π x, f x) :=
⟨ λ cfg min, do
r ← testable.run (f ℤ) cfg min,
return $ add_var_to_counter_example var (use_has_to_string.mk "ℤ") ($ ℤ) r ⟩
/-- Trace the value of sampled variables if the sample is discarded. -/
def trace_if_giveup {p α β} [has_repr α] (tracing_enabled : bool) (var : string) (val : α) :
test_result p → thunk β → β
| (test_result.gave_up _) :=
if tracing_enabled then trace (sformat!" {var} := {repr val}")
else ($ ())
| _ := ($ ())
/-- testable instance for a property iterating over the element of a list -/
@[priority 5000]
instance test_forall_in_list
[∀ x, testable (β x)] [has_repr α] :
Π xs : list α, testable (named_binder var $ ∀ x, named_binder var' $ x ∈ xs → β x)
| [] := ⟨ λ tracing min, return $ success $ psum.inr (by { introv x h, cases h} ) ⟩
| (x :: xs) :=
⟨ λ cfg min, do
r ← testable.run (β x) cfg min,
trace_if_giveup cfg.trace_discarded var x r $
match r with
| failure _ _ _ := return $ add_var_to_counter_example var x
(by { intro h, apply h, left, refl }) r
| success hp := do
rs ← @testable.run _ (test_forall_in_list xs) cfg min,
return $ convert_counter_example
(by { intros h i h',
apply h,
right, apply h' })
rs
(combine (psum.inr
$ by { intros j h, simp only [ball_cons, named_binder],
split ; assumption, } ) hp)
| gave_up n := do
rs ← @testable.run _ (test_forall_in_list xs) cfg min,
match rs with
| (success _) := return $ gave_up n
| (failure Hce xs n) := return $ failure
(by { simp only [ball_cons, named_binder],
apply not_and_of_not_right _ Hce, }) xs n
| (gave_up n') := return $ gave_up (n + n')
end
end ⟩
/-- Test proposition `p` by randomly selecting one of the provided
testable instances. -/
def combine_testable (p : Prop)
(t : list $ testable p) (h : 0 < t.length) : testable p :=
⟨ λ cfg min, have 0 < length (map (λ t, @testable.run _ t cfg min) t),
by { rw [length_map], apply h },
gen.one_of (list.map (λ t, @testable.run _ t cfg min) t) this ⟩
open sampleable_ext
/--
Format the counter-examples found in a test failure.
-/
def format_failure (s : string) (xs : list string) (n : ℕ) : string :=
let counter_ex := string.intercalate "\n" xs in
sformat!"
===================
{s}
{counter_ex}
({n} shrinks)
-------------------
"
/--
Format the counter-examples found in a test failure.
-/
def format_failure' (s : string) {p} : test_result p → string
| (success a) := ""
| (gave_up a) := ""
| (test_result.failure _ xs n) := format_failure s xs n
/--
Increase the number of shrinking steps in a test result.
-/
def add_shrinks {p} (n : ℕ) : test_result p → test_result p
| r@(success a) := r
| r@(gave_up a) := r
| (test_result.failure h vs n') := test_result.failure h vs $ n + n'
/-- Shrink a counter-example `x` by using `shrink x`, picking the first
candidate that falsifies a property and recursively shrinking that one.
The process is guaranteed to terminate because `shrink x` produces
a proof that all the values it produces are smaller (according to `sizeof`)
than `x`. -/
def minimize_aux [sampleable_ext α] [∀ x, testable (β x)] (cfg : slim_check_cfg) (var : string) :
proxy_repr α → ℕ → option_t gen (Σ x, test_result (β (interp α x))) :=
well_founded.fix has_well_founded.wf $ λ x f_rec n, do
if cfg.trace_shrink_candidates
then return $ trace sformat!
"candidates for {var} :=\n{repr (sampleable_ext.shrink x).to_list}\n" ()
else pure (),
⟨y,r,⟨h₁⟩⟩ ← (sampleable_ext.shrink x).mfirst (λ ⟨a,h⟩, do
⟨r⟩ ← monad_lift (uliftable.up $ testable.run (β (interp α a)) cfg tt
: gen (ulift $ test_result $ β $ interp α a)),
if is_failure r
then pure (⟨a, r, ⟨h⟩⟩ : (Σ a, test_result (β (interp α a)) × plift (sizeof_lt a x)))
else failure),
if cfg.trace_shrink then return $
trace (sformat!"{var} := {repr y}" ++ format_failure' "Shrink counter-example:" r) ()
else pure (),
f_rec y h₁ (n+1) <|> pure ⟨y,add_shrinks (n+1) r⟩
/-- Once a property fails to hold on an example, look for smaller counter-examples
to show the user. -/
def minimize [sampleable_ext α] [∀ x, testable (β x)] (cfg : slim_check_cfg) (var : string)
(x : proxy_repr α) (r : test_result (β (interp α x))) :
gen (Σ x, test_result (β (interp α x))) := do
if cfg.trace_shrink then return $
trace (sformat!"{var} := {repr x}" ++ format_failure' "Shrink counter-example:" r) ()
else pure (),
x' ← option_t.run $ minimize_aux α _ cfg var x 0,
pure $ x'.get_or_else ⟨x, r⟩
@[priority 2000]
instance exists_testable (p : Prop)
[testable (named_binder var (∀ x, named_binder var' $ β x → p))] :
testable (named_binder var' (named_binder var (∃ x, β x) → p)) :=
⟨ λ cfg min, do
x ← testable.run (named_binder var (∀ x, named_binder var' $ β x → p)) cfg min,
pure $ convert_counter_example' exists_imp_distrib.symm x ⟩
/-- Test a universal property by creating a sample of the right type and instantiating the
bound variable with it -/
instance var_testable [sampleable_ext α] [∀ x, testable (β x)] :
testable (named_binder var $ Π x : α, β x) :=
⟨ λ cfg min, do
uliftable.adapt_down (sampleable_ext.sample α) $
λ x, do
r ← testable.run (β (sampleable_ext.interp α x)) cfg ff,
uliftable.adapt_down (if is_failure r ∧ min
then minimize _ _ cfg var x r
else if cfg.trace_success
then trace (sformat!" {var} := {repr x}") $ pure ⟨x,r⟩
else pure ⟨x,r⟩) $
λ ⟨x,r⟩, return $ trace_if_giveup cfg.trace_discarded var x r (add_var_to_counter_example var x
($ sampleable_ext.interp α x) r) ⟩
/-- Test a universal property about propositions -/
instance prop_var_testable (β : Prop → Prop) [I : ∀ b : bool, testable (β b)] :
testable (named_binder var $ Π p : Prop, β p) :=
⟨λ cfg min, do
convert_counter_example (λ h (b : bool), h b) <$> @testable.run
(named_binder var $ Π b : bool, β b) _ cfg min⟩
@[priority 3000]
instance unused_var_testable (β) [inhabited α] [testable β] :
testable (named_binder var $ Π x : α, β) :=
⟨ λ cfg min, do
r ← testable.run β cfg min,
pure $ convert_counter_example ($ default) r (psum.inr $ λ x _, x) ⟩
@[priority 2000]
instance subtype_var_testable {p : α → Prop}
[∀ x, printable_prop (p x)]
[∀ x, testable (β x)]
[I : sampleable_ext (subtype p)] :
testable (named_binder var $ Π x : α, named_binder var' $ p x → β x) :=
⟨ λ cfg min,
do let test (x : subtype p) : testable (β x) :=
⟨ λ cfg min, do
r ← testable.run (β x.val) cfg min,
match print_prop (p x) with
| none := pure r
| some str := pure $ add_to_counter_example sformat!"guard: {str} (by construction)"
id r (psum.inr id)
end ⟩,
r ← @testable.run (∀ x : subtype p, β x.val) (@slim_check.var_testable var _ _ I test) cfg
min,
pure $ convert_counter_example'
⟨λ (h : ∀ x : subtype p, β x) x h', h ⟨x,h'⟩,
λ h ⟨x,h'⟩, h x h'⟩
r ⟩
@[priority 100]
instance decidable_testable (p : Prop) [printable_prop p] [decidable p] : testable p :=
⟨ λ cfg min, return $
if h : p then success (psum.inr h)
else
match print_prop p with
| none := failure h [] 0
| some str := failure h [sformat!"issue: {str} does not hold"] 0
end ⟩
instance eq.printable_prop {α} [has_repr α] (x y : α) : printable_prop (x = y) :=
⟨ some sformat!"{repr x} = {repr y}" ⟩
instance ne.printable_prop {α} [has_repr α] (x y : α) : printable_prop (x ≠ y) :=
⟨ some sformat!"{repr x} ≠ {repr y}" ⟩
instance le.printable_prop {α} [has_le α] [has_repr α] (x y : α) : printable_prop (x ≤ y) :=
⟨ some sformat!"{repr x} ≤ {repr y}" ⟩
instance lt.printable_prop {α} [has_lt α] [has_repr α] (x y : α) : printable_prop (x < y) :=
⟨ some sformat!"{repr x} < {repr y}" ⟩
instance perm.printable_prop {α} [has_repr α] (xs ys : list α) : printable_prop (xs ~ ys) :=
⟨ some sformat!"{repr xs} ~ {repr ys}" ⟩
instance and.printable_prop (x y : Prop) [printable_prop x] [printable_prop y] :
printable_prop (x ∧ y) :=
⟨ do x' ← print_prop x,
y' ← print_prop y,
some sformat!"({x'} ∧ {y'})" ⟩
instance or.printable_prop (x y : Prop) [printable_prop x] [printable_prop y] :
printable_prop (x ∨ y) :=
⟨ do x' ← print_prop x,
y' ← print_prop y,
some sformat!"({x'} ∨ {y'})" ⟩
instance iff.printable_prop (x y : Prop) [printable_prop x] [printable_prop y] :
printable_prop (x ↔ y) :=
⟨ do x' ← print_prop x,
y' ← print_prop y,
some sformat!"({x'} ↔ {y'})" ⟩
instance imp.printable_prop (x y : Prop) [printable_prop x] [printable_prop y] :
printable_prop (x → y) :=
⟨ do x' ← print_prop x,
y' ← print_prop y,
some sformat!"({x'} → {y'})" ⟩
instance not.printable_prop (x : Prop) [printable_prop x] : printable_prop (¬ x) :=
⟨ do x' ← print_prop x,
some sformat!"¬ {x'}" ⟩
instance true.printable_prop : printable_prop true :=
⟨ some "true" ⟩
instance false.printable_prop : printable_prop false :=
⟨ some "false" ⟩
instance bool.printable_prop (b : bool) : printable_prop b :=
⟨ some $ if b then "true" else "false" ⟩
section io
open _root_.nat
variable {p : Prop}
/-- Execute `cmd` and repeat every time the result is `gave_up` (at most
`n` times). -/
def retry (cmd : rand (test_result p)) : ℕ → rand (test_result p)
| 0 := return $ gave_up 1
| (succ n) := do
r ← cmd,
match r with
| success hp := return $ success hp
| (failure Hce xs n) := return (failure Hce xs n)
| (gave_up _) := retry n
end
/-- Count the number of times the test procedure gave up. -/
def give_up (x : ℕ) : test_result p → test_result p
| (success (psum.inl ())) := gave_up x
| (success (psum.inr p)) := success (psum.inr p)
| (gave_up n) := gave_up (n+x)
| (failure Hce xs n) := failure Hce xs n
variable (p)
variable [testable p]
/-- Try `n` times to find a counter-example for `p`. -/
def testable.run_suite_aux (cfg : slim_check_cfg) : test_result p → ℕ → rand (test_result p)
| r 0 := return r
| r (succ n) :=
do let size := (cfg.num_inst - n - 1) * cfg.max_size / cfg.num_inst,
when cfg.trace_success $ return $ trace sformat!"[slim_check: sample]" (),
x ← retry ( (testable.run p cfg tt).run ⟨ size ⟩) 10,
match x with
| (success (psum.inl ())) := testable.run_suite_aux r n
| (success (psum.inr Hp)) := return $ success (psum.inr Hp)
| (failure Hce xs n) := return (failure Hce xs n)
| (gave_up g) := testable.run_suite_aux (give_up g r) n
end
/-- Try to find a counter-example of `p`. -/
def testable.run_suite (cfg : slim_check_cfg := {}) : rand (test_result p) :=
testable.run_suite_aux p cfg (success $ psum.inl ()) cfg.num_inst
/-- Run a test suite for `p` in `io`. -/
def testable.check' (cfg : slim_check_cfg := {}) : io (test_result p) :=
match cfg.random_seed with
| some seed := io.run_rand_with seed (testable.run_suite p cfg)
| none := io.run_rand (testable.run_suite p cfg)
end
namespace tactic
open _root_.tactic expr
/-!
## Decorations
Instances of `testable` use `named_binder` as a decoration on
propositions in order to access the name of bound variables, as in
`named_binder "x" (forall x, x < y)`. This helps the
`testable` instances create useful error messages where variables
are matched with values that falsify a given proposition.
The following functions help support the gadget so that the user does
not have to put them in themselves.
-/
/-- `add_existential_decorations p` adds `a `named_binder` annotation at the
root of `p` if `p` is an existential quantification. -/
meta def add_existential_decorations : expr → expr
| e@`(@Exists %%α %%(lam n bi d b)) :=
let n := to_string n in
const ``named_binder [] (`(n) : expr) e
| e := e
/-- Traverse the syntax of a proposition to find universal quantifiers
and existential quantifiers and add `named_binder` annotations next to
them. -/
meta def add_decorations : expr → expr | e :=
e.replace $ λ e _,
match e with
| (pi n bi d b) :=
let n := to_string n in
some $ const ``named_binder [] (`(n) : expr)
(pi n bi (add_existential_decorations d) (add_decorations b))
| e := none
end
/-- `decorations_of p` is used as a hint to `mk_decorations` to specify
that the goal should be satisfied with a proposition equivalent to `p`
with added annotations. -/
@[reducible, nolint unused_arguments]
def decorations_of (p : Prop) := Prop
/-- In a goal of the shape `⊢ tactic.decorations_of p`, `mk_decoration` examines
the syntax of `p` and add `named_binder` around universal quantifications and
existential quantifications to improve error messages.
This tool can be used in the declaration of a function as follows:
```lean
def foo (p : Prop) (p' : tactic.decorations_of p . mk_decorations) [testable p'] : ...
```
`p` is the parameter given by the user, `p'` is an equivalent proposition where
the quantifiers are annotated with `named_binder`.
-/
meta def mk_decorations : tactic unit := do
`(tactic.decorations_of %%p) ← target,
exact $ add_decorations p
end tactic
/-- Run a test suite for `p` and return true or false: should we believe that `p` holds? -/
def testable.check (p : Prop) (cfg : slim_check_cfg := {})
(p' : tactic.decorations_of p . tactic.mk_decorations) [testable p'] : io punit := do
x ← match cfg.random_seed with
| some seed := io.run_rand_with seed (testable.run_suite p' cfg)
| none := io.run_rand (testable.run_suite p' cfg)
end,
match x with
| (success _) := when (¬ cfg.quiet) $ io.put_str_ln "Success"
| (gave_up n) := io.fail sformat!"Gave up {repr n} times"
| (failure _ xs n) := do
io.fail $ format_failure "Found problems!" xs n
end
end io
end slim_check
|
359ed245a413e47ca399ef9e14745f3a9c0941d1 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/topology/metric_space/metrizable.lean | d158d8e7691aa4373af44f915b04acefee1c3255 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 11,636 | lean | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import topology.urysohns_lemma
import topology.continuous_function.bounded
import topology.uniform_space.cauchy
/-!
# Metrizability of a T₃ topological space with second countable topology
In this file we define metrizable topological spaces, i.e., topological spaces for which there
exists a metric space structure that generates the same topology.
We also show that a T₃ topological space with second countable topology `X` is metrizable.
First we prove that `X` can be embedded into `l^∞`, then use this embedding to pull back the metric
space structure.
-/
open set filter metric
open_locale bounded_continuous_function filter topological_space
namespace topological_space
variables {ι X Y : Type*} {π : ι → Type*} [topological_space X] [topological_space Y]
[finite ι] [Π i, topological_space (π i)]
/-- A topological space is *pseudo metrizable* if there exists a pseudo metric space structure
compatible with the topology. To endow such a space with a compatible distance, use
`letI : pseudo_metric_space X := topological_space.pseudo_metrizable_space_pseudo_metric X`. -/
class pseudo_metrizable_space (X : Type*) [t : topological_space X] : Prop :=
(exists_pseudo_metric : ∃ (m : pseudo_metric_space X), m.to_uniform_space.to_topological_space = t)
@[priority 100]
instance _root_.pseudo_metric_space.to_pseudo_metrizable_space {X : Type*}
[m : pseudo_metric_space X] :
pseudo_metrizable_space X :=
⟨⟨m, rfl⟩⟩
/-- Construct on a metrizable space a metric compatible with the topology. -/
noncomputable def pseudo_metrizable_space_pseudo_metric
(X : Type*) [topological_space X] [h : pseudo_metrizable_space X] :
pseudo_metric_space X :=
h.exists_pseudo_metric.some.replace_topology h.exists_pseudo_metric.some_spec.symm
instance pseudo_metrizable_space_prod [pseudo_metrizable_space X] [pseudo_metrizable_space Y] :
pseudo_metrizable_space (X × Y) :=
begin
letI : pseudo_metric_space X := pseudo_metrizable_space_pseudo_metric X,
letI : pseudo_metric_space Y := pseudo_metrizable_space_pseudo_metric Y,
apply_instance
end
/-- Given an inducing map of a topological space into a pseudo metrizable space, the source space
is also pseudo metrizable. -/
lemma _root_.inducing.pseudo_metrizable_space [pseudo_metrizable_space Y] {f : X → Y}
(hf : inducing f) :
pseudo_metrizable_space X :=
begin
letI : pseudo_metric_space Y := pseudo_metrizable_space_pseudo_metric Y,
exact ⟨⟨hf.comap_pseudo_metric_space, rfl⟩⟩
end
/-- Every pseudo-metrizable space is first countable. -/
@[priority 100]
instance pseudo_metrizable_space.first_countable_topology [h : pseudo_metrizable_space X] :
topological_space.first_countable_topology X :=
begin
unfreezingI { rcases h with ⟨_, hm⟩, rw ←hm },
exact @uniform_space.first_countable_topology X pseudo_metric_space.to_uniform_space
emetric.uniformity.filter.is_countably_generated,
end
instance pseudo_metrizable_space.subtype [pseudo_metrizable_space X]
(s : set X) : pseudo_metrizable_space s :=
inducing_coe.pseudo_metrizable_space
instance pseudo_metrizable_space_pi [Π i, pseudo_metrizable_space (π i)] :
pseudo_metrizable_space (Π i, π i) :=
by { casesI nonempty_fintype ι, letI := λ i, pseudo_metrizable_space_pseudo_metric (π i),
apply_instance }
/-- A topological space is metrizable if there exists a metric space structure compatible with the
topology. To endow such a space with a compatible distance, use
`letI : metric_space X := topological_space.metrizable_space_metric X` -/
class metrizable_space (X : Type*) [t : topological_space X] : Prop :=
(exists_metric : ∃ (m : metric_space X), m.to_uniform_space.to_topological_space = t)
@[priority 100]
instance _root_.metric_space.to_metrizable_space {X : Type*} [m : metric_space X] :
metrizable_space X :=
⟨⟨m, rfl⟩⟩
@[priority 100]
instance metrizable_space.to_pseudo_metrizable_space [h : metrizable_space X] :
pseudo_metrizable_space X :=
⟨let ⟨m, hm⟩ := h.1 in ⟨m.to_pseudo_metric_space, hm⟩⟩
/-- Construct on a metrizable space a metric compatible with the topology. -/
noncomputable def metrizable_space_metric (X : Type*) [topological_space X]
[h : metrizable_space X] :
metric_space X :=
h.exists_metric.some.replace_topology h.exists_metric.some_spec.symm
@[priority 100]
instance t2_space_of_metrizable_space [metrizable_space X] : t2_space X :=
by { letI : metric_space X := metrizable_space_metric X, apply_instance }
instance metrizable_space_prod [metrizable_space X] [metrizable_space Y] :
metrizable_space (X × Y) :=
begin
letI : metric_space X := metrizable_space_metric X,
letI : metric_space Y := metrizable_space_metric Y,
apply_instance
end
/-- Given an embedding of a topological space into a metrizable space, the source space is also
metrizable. -/
lemma _root_.embedding.metrizable_space [metrizable_space Y] {f : X → Y} (hf : embedding f) :
metrizable_space X :=
begin
letI : metric_space Y := metrizable_space_metric Y,
exact ⟨⟨hf.comap_metric_space f, rfl⟩⟩
end
instance metrizable_space.subtype [metrizable_space X] (s : set X) : metrizable_space s :=
embedding_subtype_coe.metrizable_space
instance metrizable_space_pi [Π i, metrizable_space (π i)] : metrizable_space (Π i, π i) :=
by { casesI nonempty_fintype ι, letI := λ i, metrizable_space_metric (π i), apply_instance }
variables (X) [t3_space X] [second_countable_topology X]
/-- A T₃ topological space with second countable topology can be embedded into `l^∞ = ℕ →ᵇ ℝ`.
-/
lemma exists_embedding_l_infty : ∃ f : X → (ℕ →ᵇ ℝ), embedding f :=
begin
haveI : normal_space X := normal_space_of_t3_second_countable X,
-- Choose a countable basis, and consider the set `s` of pairs of set `(U, V)` such that `U ∈ B`,
-- `V ∈ B`, and `closure U ⊆ V`.
rcases exists_countable_basis X with ⟨B, hBc, -, hB⟩,
set s : set (set X × set X) := {UV ∈ B ×ˢ B| closure UV.1 ⊆ UV.2},
-- `s` is a countable set.
haveI : encodable s := ((hBc.prod hBc).mono (inter_subset_left _ _)).to_encodable,
-- We don't have the space of bounded (possibly discontinuous) functions, so we equip `s`
-- with the discrete topology and deal with `s →ᵇ ℝ` instead.
letI : topological_space s := ⊥, haveI : discrete_topology s := ⟨rfl⟩,
rsuffices ⟨f, hf⟩ : ∃ f : X → (s →ᵇ ℝ), embedding f,
{ exact ⟨λ x, (f x).extend (encodable.encode' s) 0, (bounded_continuous_function.isometry_extend
(encodable.encode' s) (0 : ℕ →ᵇ ℝ)).embedding.comp hf⟩ },
have hd : ∀ UV : s, disjoint (closure UV.1.1) (UV.1.2ᶜ) :=
λ UV, disjoint_compl_right.mono_right (compl_subset_compl.2 UV.2.2),
-- Choose a sequence of `εₙ > 0`, `n : s`, that is bounded above by `1` and tends to zero
-- along the `cofinite` filter.
obtain ⟨ε, ε01, hε⟩ : ∃ ε : s → ℝ, (∀ UV, ε UV ∈ Ioc (0 : ℝ) 1) ∧ tendsto ε cofinite (𝓝 0),
{ rcases pos_sum_of_encodable zero_lt_one s with ⟨ε, ε0, c, hεc, hc1⟩,
refine ⟨ε, λ UV, ⟨ε0 UV, _⟩, hεc.summable.tendsto_cofinite_zero⟩,
exact (le_has_sum hεc UV $ λ _ _, (ε0 _).le).trans hc1 },
/- For each `UV = (U, V) ∈ s` we use Urysohn's lemma to choose a function `f UV` that is equal to
zero on `U` and is equal to `ε UV` on the complement to `V`. -/
have : ∀ UV : s, ∃ f : C(X, ℝ), eq_on f 0 UV.1.1 ∧ eq_on f (λ _, ε UV) UV.1.2ᶜ ∧
∀ x, f x ∈ Icc 0 (ε UV),
{ intro UV,
rcases exists_continuous_zero_one_of_closed is_closed_closure
(hB.is_open UV.2.1.2).is_closed_compl (hd UV) with ⟨f, hf₀, hf₁, hf01⟩,
exact ⟨ε UV • f, λ x hx, by simp [hf₀ (subset_closure hx)], λ x hx, by simp [hf₁ hx],
λ x, ⟨mul_nonneg (ε01 _).1.le (hf01 _).1, mul_le_of_le_one_right (ε01 _).1.le (hf01 _).2⟩⟩ },
choose f hf0 hfε hf0ε,
have hf01 : ∀ UV x, f UV x ∈ Icc (0 : ℝ) 1,
from λ UV x, Icc_subset_Icc_right (ε01 _).2 (hf0ε _ _),
/- The embedding is given by `F x UV = f UV x`. -/
set F : X → s →ᵇ ℝ := λ x, ⟨⟨λ UV, f UV x, continuous_of_discrete_topology⟩, 1, λ UV₁ UV₂,
real.dist_le_of_mem_Icc_01 (hf01 _ _) (hf01 _ _)⟩,
have hF : ∀ x UV, F x UV = f UV x := λ _ _, rfl,
refine ⟨F, embedding.mk' _ (λ x y hxy, _) (λ x, le_antisymm _ _)⟩,
{ /- First we prove that `F` is injective. Indeed, if `F x = F y` and `x ≠ y`, then we can find
`(U, V) ∈ s` such that `x ∈ U` and `y ∉ V`, hence `F x UV = 0 ≠ ε UV = F y UV`. -/
refine not_not.1 (λ Hne, _), -- `by_contra Hne` timeouts
rcases hB.mem_nhds_iff.1 (is_open_ne.mem_nhds Hne) with ⟨V, hVB, hxV, hVy⟩,
rcases hB.exists_closure_subset (hB.mem_nhds hVB hxV) with ⟨U, hUB, hxU, hUV⟩,
set UV : ↥s := ⟨(U, V), ⟨hUB, hVB⟩, hUV⟩,
apply (ε01 UV).1.ne,
calc (0 : ℝ) = F x UV : (hf0 UV hxU).symm
... = F y UV : by rw hxy
... = ε UV : hfε UV (λ h : y ∈ V, hVy h rfl) },
{ /- Now we prove that each neighborhood `V` of `x : X` include a preimage of a neighborhood of
`F x` under `F`. Without loss of generality, `V` belongs to `B`. Choose `U ∈ B` such that
`x ∈ V` and `closure V ⊆ U`. Then the preimage of the `(ε (U, V))`-neighborhood of `F x`
is included by `V`. -/
refine ((nhds_basis_ball.comap _).le_basis_iff hB.nhds_has_basis).2 _,
rintro V ⟨hVB, hxV⟩,
rcases hB.exists_closure_subset (hB.mem_nhds hVB hxV) with ⟨U, hUB, hxU, hUV⟩,
set UV : ↥s := ⟨(U, V), ⟨hUB, hVB⟩, hUV⟩,
refine ⟨ε UV, (ε01 UV).1, λ y (hy : dist (F y) (F x) < ε UV), _⟩,
replace hy : dist (F y UV) (F x UV) < ε UV,
from (bounded_continuous_function.dist_coe_le_dist _).trans_lt hy,
contrapose! hy,
rw [hF, hF, hfε UV hy, hf0 UV hxU, pi.zero_apply, dist_zero_right],
exact le_abs_self _ },
{ /- Finally, we prove that `F` is continuous. Given `δ > 0`, consider the set `T` of `(U, V) ∈ s`
such that `ε (U, V) ≥ δ`. Since `ε` tends to zero, `T` is finite. Since each `f` is continuous,
we can choose a neighborhood such that `dist (F y (U, V)) (F x (U, V)) ≤ δ` for any
`(U, V) ∈ T`. For `(U, V) ∉ T`, the same inequality is true because both `F y (U, V)` and
`F x (U, V)` belong to the interval `[0, ε (U, V)]`. -/
refine (nhds_basis_closed_ball.comap _).ge_iff.2 (λ δ δ0, _),
have h_fin : {UV : s | δ ≤ ε UV}.finite, by simpa only [← not_lt] using hε (gt_mem_nhds δ0),
have : ∀ᶠ y in 𝓝 x, ∀ UV, δ ≤ ε UV → dist (F y UV) (F x UV) ≤ δ,
{ refine (eventually_all_finite h_fin).2 (λ UV hUV, _),
exact (f UV).continuous.tendsto x (closed_ball_mem_nhds _ δ0) },
refine this.mono (λ y hy, (bounded_continuous_function.dist_le δ0.le).2 $ λ UV, _),
cases le_total δ (ε UV) with hle hle,
exacts [hy _ hle, (real.dist_le_of_mem_Icc (hf0ε _ _) (hf0ε _ _)).trans (by rwa sub_zero)] }
end
/-- *Urysohn's metrization theorem* (Tychonoff's version): a T₃ topological space with second
countable topology `X` is metrizable, i.e., there exists a metric space structure that generates the
same topology. -/
lemma metrizable_space_of_t3_second_countable : metrizable_space X :=
let ⟨f, hf⟩ := exists_embedding_l_infty X in hf.metrizable_space
instance : metrizable_space ennreal := metrizable_space_of_t3_second_countable ennreal
end topological_space
|
825861048ead3ca063ccb23868993d421d70f4b5 | 75db7e3219bba2fbf41bf5b905f34fcb3c6ca3f2 | /tests/lean/let1.lean | 5741efe453d6cf60fd6fc192029b42e4f882a910 | [
"Apache-2.0"
] | permissive | jroesch/lean | 30ef0860fa905d35b9ad6f76de1a4f65c9af6871 | 3de4ec1a6ce9a960feb2a48eeea8b53246fa34f2 | refs/heads/master | 1,586,090,835,348 | 1,455,142,203,000 | 1,455,142,277,000 | 51,536,958 | 1 | 0 | null | 1,455,215,811,000 | 1,455,215,811,000 | null | UTF-8 | Lean | false | false | 1,163 | lean | prelude -- Correct version
check let bool := Type.{0},
and (p q : bool) := ∀ c : bool, (p → q → c) → c,
infixl `∧`:25 := and,
and_intro (p q : bool) (H1 : p) (H2 : q) : p ∧ q
:= λ (c : bool) (H : p → q → c), H H1 H2,
and_elim_left (p q : bool) (H : p ∧ q) : p
:= H p (λ (H1 : p) (H2 : q), H1),
and_elim_right (p q : bool) (H : p ∧ q) : q
:= H q (λ (H1 : p) (H2 : q), H2)
in and_intro
-- TODO(Leo): fix expected output as soon as elaborator starts checking let-expression type again
check let bool := Type.{0},
and (p q : bool) := ∀ c : bool, (p → q → c) → c,
infixl `∧`:25 := and,
and_intro [visible] (p q : bool) (H1 : p) (H2 : q) : q ∧ p
:= λ (c : bool) (H : p → q → c), H H1 H2,
and_elim_left (p q : bool) (H : p ∧ q) : p
:= H p (λ (H1 : p) (H2 : q), H1),
and_elim_right (p q : bool) (H : p ∧ q) : q
:= H q (λ (H1 : p) (H2 : q), H2)
in and_intro
|
f83afb965328108f810f668a60ffef34c660ccea | a721fe7446524f18ba361625fc01033d9c8b7a78 | /src/principia/statements.lean | bd9e0863f15da35c9d8fe6709e36010c5b688d02 | [] | no_license | Sterrs/leaning | 8fd80d1f0a6117a220bb2e57ece639b9a63deadc | 3901cc953694b33adda86cb88ca30ba99594db31 | refs/heads/master | 1,627,023,822,744 | 1,616,515,221,000 | 1,616,515,221,000 | 245,512,190 | 2 | 0 | null | 1,616,429,050,000 | 1,583,527,118,000 | Lean | UTF-8 | Lean | false | false | 1,438 | lean | -- vim: ts=2 sw=0 sts=-1 et ai tw=70
import .mynat.parity
import .mynat.sum
-- This file is specifically for statements of famous theorems.
-- It is a useful and fun thing to state these, but they're better off
-- in a different file because they tend to use sorry
open hidden.mynat
open hidden
-- Landau's Problems about prime numbers
-- https://en.wikipedia.org/wiki/Landau%27s_problems
theorem goldbachs_conjecture {m : mynat} :
even m → ∃ p q: mynat, prime p → prime q → p + q = m := sorry
theorem twin_prime_conjecture:
infinitely_many (λ p, prime p ∧ prime (p+2)) := sorry
theorem legendres_conjecture {m : mynat} :
∃ p, m*m ≤ p ∧ p ≤ (m+1)*(m+1) ∧ prime p := sorry
theorem landau4:
infinitely_many (λ n, prime (n*n + 1)) := sorry
-- instructive exercise in disproof by counterexample
theorem fermats_penultimate_theorem:
¬ ∀ a b c n: mynat, a^n + b^n = c^n → n = 2 :=
begin
assume hflt,
have h := hflt 1 1 2 1 rfl,
cases h,
end
theorem fermats_last_theorem (a b c n: mynat):
a ≠ 0 → b ≠ 0 → a ^ n + b ^ n = c ^ n → n = 1 ∨ n = 2 := sorry
def collatz_sequence : mynat → sequence mynat
| m zero := m
| m (succ n) :=
begin
have prev := collatz_sequence m n,
exact
if even prev then
sorry -- Need division
else
3 * prev + 1,
end
-- "Mathematics may not be ready for such problems"
theorem collatz_conjecture:
∀ m, ∃ n, collatz_sequence m n = 1 := sorry
|
53fb99908eef72201ff17f51810dadc41a983a8a | e2fc96178628c7451e998a0db2b73877d0648be5 | /src/utilities/written_by_others/trim_assoc.lean | ac0bce6b63bdad1ff13fe363b232317ffdb01c54 | [
"BSD-2-Clause"
] | permissive | madvorak/grammars | cd324ae19b28f7b8be9c3ad010ef7bf0fabe5df2 | 1447343a45fcb7821070f1e20b57288d437323a6 | refs/heads/main | 1,692,383,644,884 | 1,692,032,429,000 | 1,692,032,429,000 | 453,948,141 | 7 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,047 | lean | -- Written by Damiano Testa. BSD license applies.
import tactic
private def in_match {α : Type*} [decidable_eq α] : list α → list α → list α
| [] _ := []
| _ [] := []
| (a::as) (b::bs) := if a = b then (a :: in_match as bs) else []
private def split3 {α : Type*} [decidable_eq α] (l r : list α) : list α × (list α × list α) × list α :=
let ini := in_match l r in
let lx_tail := l.drop ini.length in
let rx_tail := r.drop ini.length in
let lx_tail_rev := lx_tail.reverse in
let rx_tail_rev := rx_tail.reverse in
let fin := (in_match lx_tail_rev rx_tail_rev).reverse in
(ini,
((lx_tail_rev.drop fin.length).reverse,
(rx_tail_rev.drop fin.length).reverse),
fin)
private meta def sum_up_with_default {α : Type*} (e : α) (op : α → α → α) : list α → α
| [] := e
| [a] := a
| (a::as) := as.foldl (λ x y, op x y) a
namespace tactic
private meta def list_binary_operands (f : expr) : expr → tactic (list expr)
| x@(expr.app (expr.app g a) b) := do
some _ ← try_core (unify f g) | pure [x],
as ← list_binary_operands a,
bs ← list_binary_operands b,
pure (as ++ bs)
| a := pure [a]
private meta def assoc_unit (typ : expr) : name → tactic expr
| `has_append.append := to_expr ``([] : %%typ) tt ff
| `has_mul.mul := to_expr ``(1 : %%typ) tt ff
| `has_add.add := to_expr ``(0 : %%typ) tt ff
| _ := fail "the tactic does not support this operation"
private meta def assoc_tac : name → tactic unit
| `has_append.append := `[{ simp only [list.append_assoc, list.nil_append, list.append_nil] }]
| `has_mul.mul := `[{ simp only [mul_assoc, mul_one, one_mul] }]
| `has_add.add := `[{ simp only [add_assoc, add_zero, zero_add] }]
| _ := fail "the tactic does not support this operation"
meta def interactive.trim : tactic unit := do
`(%%lhs = %%rhs) ← target >>= instantiate_mvars <|> fail "goal is not an equality",
et ← infer_type lhs,
oper ← match lhs with
| (expr.app (expr.app f _) _) := pure f
| _ := match rhs with
| (expr.app (expr.app f _) _) := pure f
| _ := fail "no operation found"
end
end,
opl ← list_binary_operands oper lhs,
opr ← list_binary_operands oper rhs,
let (ini, (ldiff, rdiff), fin) := split3 opl opr,
un ← assoc_unit et oper.get_app_fn.const_name <|> pure lhs,
let rec_l := [ini, ldiff, fin].map $ sum_up_with_default un (λ x y, oper.mk_app [x, y]),
let rec_r := [ini, rdiff, fin].map $ sum_up_with_default un (λ x y, oper.mk_app [x, y]),
let nleft := sum_up_with_default un (λ x y, oper.mk_app [x, y]) rec_l,
let nright := sum_up_with_default un (λ x y, oper.mk_app [x, y]) rec_r,
l_eq ← mk_app `eq [lhs, nleft],
r_eq ← mk_app `eq [nright, rhs],
(_, pr_left) ← solve_aux l_eq (assoc_tac oper.get_app_fn.const_name),
(_, pr_right) ← solve_aux r_eq (assoc_tac oper.get_app_fn.const_name),
refine ``(eq.trans %%pr_left (eq.trans _ %%pr_right)),
tactic.congr' (some 1),
try $ tactic.congr' (some 1)
add_hint_tactic "trim"
end tactic
|
95dd8bef871e4a45418567aace0da5b443cee2d0 | 47181b4ef986292573c77e09fcb116584d37ea8a | /src/for_mathlib/exp_log.lean | 4ef0ca75bc48fd66f4cf98533e40a9fb2f167f69 | [
"MIT"
] | permissive | RaitoBezarius/berkovich-spaces | 87662a2bdb0ac0beed26e3338b221e3f12107b78 | 0a49f75a599bcb20333ec86b301f84411f04f7cf | refs/heads/main | 1,690,520,666,912 | 1,629,328,012,000 | 1,629,328,012,000 | 332,238,095 | 4 | 0 | MIT | 1,629,312,085,000 | 1,611,414,506,000 | Lean | UTF-8 | Lean | false | false | 506 | lean | import analysis.special_functions.exp_log
lemma real.log_ne_zero_of_ne_one (x: ℝ) (hx_pos: 0 < x) (hx: x ≠ 1): real.log x ≠ 0 :=
begin
by_cases (1 < x),
exact ne_of_gt (real.log_pos h),
push_neg at h,
exact ne_of_lt (real.log_neg hx_pos (lt_of_le_of_ne h hx)),
end
lemma log_inj_pos {x y: ℝ} (x_pos: 0 < x) (y_pos: 0 < y):
real.log x = real.log y → x = y :=
λ h, le_antisymm
((real.log_le_log x_pos y_pos).1 $ le_of_eq h)
((real.log_le_log y_pos x_pos).1 $ le_of_eq $ eq.symm h)
|
f52f2bd03edc0747a1a26fc8dd3a935cbabdf09c | f1b175e38ffc5cc1c7c5551a72d0dbaf70786f83 | /analysis/topology/topological_space.lean | 22dd30c8427a1ab12aa637a917ca5b215e314ec1 | [
"Apache-2.0"
] | permissive | mjendrusch/mathlib | df3ae884dd5ce38c7edf452bcbfd3baf4e3a6214 | 5c209edb7eb616a26f64efe3500f2b1ba95b8d55 | refs/heads/master | 1,585,663,284,800 | 1,539,062,055,000 | 1,539,062,055,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 62,954 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
Theory of topological spaces.
Parts of the formalization is based on the books:
N. Bourbaki: General Topology
I. M. James: Topologies and Uniformities
A major difference is that this formalization is heavily based on the filter library.
-/
import order.filter data.set.countable tactic
open set filter lattice classical
local attribute [instance] prop_decidable
universes u v w
structure topological_space (α : Type u) :=
(is_open : set α → Prop)
(is_open_univ : is_open univ)
(is_open_inter : ∀s t, is_open s → is_open t → is_open (s ∩ t))
(is_open_sUnion : ∀s, (∀t∈s, is_open t) → is_open (⋃₀ s))
attribute [class] topological_space
section topological_space
variables {α : Type u} {β : Type v} {ι : Sort w} {a a₁ a₂ : α} {s s₁ s₂ : set α} {p p₁ p₂ : α → Prop}
lemma topological_space_eq : ∀ {f g : topological_space α}, f.is_open = g.is_open → f = g
| ⟨a, _, _, _⟩ ⟨b, _, _, _⟩ rfl := rfl
section
variables [t : topological_space α]
include t
/-- `is_open s` means that `s` is open in the ambient topological space on `α` -/
def is_open (s : set α) : Prop := topological_space.is_open t s
@[simp]
lemma is_open_univ : is_open (univ : set α) := topological_space.is_open_univ t
lemma is_open_inter (h₁ : is_open s₁) (h₂ : is_open s₂) : is_open (s₁ ∩ s₂) :=
topological_space.is_open_inter t s₁ s₂ h₁ h₂
lemma is_open_sUnion {s : set (set α)} (h : ∀t ∈ s, is_open t) : is_open (⋃₀ s) :=
topological_space.is_open_sUnion t s h
end
lemma is_open_fold {s : set α} {t : topological_space α} : t.is_open s = @is_open α t s :=
rfl
variables [topological_space α]
lemma is_open_Union {f : ι → set α} (h : ∀i, is_open (f i)) : is_open (⋃i, f i) :=
is_open_sUnion $ by rintro _ ⟨i, rfl⟩; exact h i
lemma is_open_bUnion {s : set β} {f : β → set α} (h : ∀i∈s, is_open (f i)) :
is_open (⋃i∈s, f i) :=
is_open_Union $ assume i, is_open_Union $ assume hi, h i hi
lemma is_open_union (h₁ : is_open s₁) (h₂ : is_open s₂) : is_open (s₁ ∪ s₂) :=
by rw union_eq_Union; exact is_open_Union (bool.forall_bool.2 ⟨h₂, h₁⟩)
@[simp] lemma is_open_empty : is_open (∅ : set α) :=
by rw ← sUnion_empty; exact is_open_sUnion (assume a, false.elim)
lemma is_open_sInter {s : set (set α)} (hs : finite s) : (∀t ∈ s, is_open t) → is_open (⋂₀ s) :=
finite.induction_on hs (λ _, by rw sInter_empty; exact is_open_univ) $
λ a s has hs ih h, by rw sInter_insert; exact
is_open_inter (h _ $ mem_insert _ _) (ih $ λ t, h t ∘ mem_insert_of_mem _)
lemma is_open_bInter {s : set β} {f : β → set α} (hs : finite s) :
(∀i∈s, is_open (f i)) → is_open (⋂i∈s, f i) :=
finite.induction_on hs
(λ _, by rw bInter_empty; exact is_open_univ)
(λ a s has hs ih h, by rw bInter_insert; exact
is_open_inter (h a (mem_insert _ _)) (ih (λ i hi, h i (mem_insert_of_mem _ hi))))
lemma is_open_const {p : Prop} : is_open {a : α | p} :=
by_cases
(assume : p, begin simp only [this]; exact is_open_univ end)
(assume : ¬ p, begin simp only [this]; exact is_open_empty end)
lemma is_open_and : is_open {a | p₁ a} → is_open {a | p₂ a} → is_open {a | p₁ a ∧ p₂ a} :=
is_open_inter
/-- A set is closed if its complement is open -/
def is_closed (s : set α) : Prop := is_open (-s)
@[simp] lemma is_closed_empty : is_closed (∅ : set α) :=
by unfold is_closed; rw compl_empty; exact is_open_univ
@[simp] lemma is_closed_univ : is_closed (univ : set α) :=
by unfold is_closed; rw compl_univ; exact is_open_empty
lemma is_closed_union : is_closed s₁ → is_closed s₂ → is_closed (s₁ ∪ s₂) :=
λ h₁ h₂, by unfold is_closed; rw compl_union; exact is_open_inter h₁ h₂
lemma is_closed_sInter {s : set (set α)} : (∀t ∈ s, is_closed t) → is_closed (⋂₀ s) :=
by simp only [is_closed, compl_sInter, sUnion_image]; exact assume h, is_open_Union $ assume t, is_open_Union $ assume ht, h t ht
lemma is_closed_Inter {f : ι → set α} (h : ∀i, is_closed (f i)) : is_closed (⋂i, f i ) :=
is_closed_sInter $ assume t ⟨i, (heq : t = f i)⟩, heq.symm ▸ h i
@[simp] lemma is_open_compl_iff {s : set α} : is_open (-s) ↔ is_closed s := iff.rfl
@[simp] lemma is_closed_compl_iff {s : set α} : is_closed (-s) ↔ is_open s :=
by rw [←is_open_compl_iff, compl_compl]
lemma is_open_diff {s t : set α} (h₁ : is_open s) (h₂ : is_closed t) : is_open (s \ t) :=
is_open_inter h₁ $ is_open_compl_iff.mpr h₂
lemma is_closed_inter (h₁ : is_closed s₁) (h₂ : is_closed s₂) : is_closed (s₁ ∩ s₂) :=
by rw [is_closed, compl_inter]; exact is_open_union h₁ h₂
lemma is_closed_Union {s : set β} {f : β → set α} (hs : finite s) :
(∀i∈s, is_closed (f i)) → is_closed (⋃i∈s, f i) :=
finite.induction_on hs
(λ _, by rw bUnion_empty; exact is_closed_empty)
(λ a s has hs ih h, by rw bUnion_insert; exact
is_closed_union (h a (mem_insert _ _)) (ih (λ i hi, h i (mem_insert_of_mem _ hi))))
lemma is_closed_imp [topological_space α] {p q : α → Prop}
(hp : is_open {x | p x}) (hq : is_closed {x | q x}) : is_closed {x | p x → q x} :=
have {x | p x → q x} = (- {x | p x}) ∪ {x | q x}, from set.ext $ λ x, imp_iff_not_or,
by rw [this]; exact is_closed_union (is_closed_compl_iff.mpr hp) hq
lemma is_open_neg : is_closed {a | p a} → is_open {a | ¬ p a} :=
is_open_compl_iff.mpr
/-- The interior of a set `s` is the largest open subset of `s`. -/
def interior (s : set α) : set α := ⋃₀ {t | is_open t ∧ t ⊆ s}
lemma mem_interior {s : set α} {x : α} :
x ∈ interior s ↔ ∃ t ⊆ s, is_open t ∧ x ∈ t :=
by simp only [interior, mem_set_of_eq, exists_prop, and_assoc, and.left_comm]
@[simp] lemma is_open_interior {s : set α} : is_open (interior s) :=
is_open_sUnion $ assume t ⟨h₁, h₂⟩, h₁
lemma interior_subset {s : set α} : interior s ⊆ s :=
sUnion_subset $ assume t ⟨h₁, h₂⟩, h₂
lemma interior_maximal {s t : set α} (h₁ : t ⊆ s) (h₂ : is_open t) : t ⊆ interior s :=
subset_sUnion_of_mem ⟨h₂, h₁⟩
lemma interior_eq_of_open {s : set α} (h : is_open s) : interior s = s :=
subset.antisymm interior_subset (interior_maximal (subset.refl s) h)
lemma interior_eq_iff_open {s : set α} : interior s = s ↔ is_open s :=
⟨assume h, h ▸ is_open_interior, interior_eq_of_open⟩
lemma subset_interior_iff_open {s : set α} : s ⊆ interior s ↔ is_open s :=
by simp only [interior_eq_iff_open.symm, subset.antisymm_iff, interior_subset, true_and]
lemma subset_interior_iff_subset_of_open {s t : set α} (h₁ : is_open s) :
s ⊆ interior t ↔ s ⊆ t :=
⟨assume h, subset.trans h interior_subset, assume h₂, interior_maximal h₂ h₁⟩
lemma interior_mono {s t : set α} (h : s ⊆ t) : interior s ⊆ interior t :=
interior_maximal (subset.trans interior_subset h) is_open_interior
@[simp] lemma interior_empty : interior (∅ : set α) = ∅ :=
interior_eq_of_open is_open_empty
@[simp] lemma interior_univ : interior (univ : set α) = univ :=
interior_eq_of_open is_open_univ
@[simp] lemma interior_interior {s : set α} : interior (interior s) = interior s :=
interior_eq_of_open is_open_interior
@[simp] lemma interior_inter {s t : set α} : interior (s ∩ t) = interior s ∩ interior t :=
subset.antisymm
(subset_inter (interior_mono $ inter_subset_left s t) (interior_mono $ inter_subset_right s t))
(interior_maximal (inter_subset_inter interior_subset interior_subset) $ is_open_inter is_open_interior is_open_interior)
lemma interior_union_is_closed_of_interior_empty {s t : set α} (h₁ : is_closed s) (h₂ : interior t = ∅) :
interior (s ∪ t) = interior s :=
have interior (s ∪ t) ⊆ s, from
assume x ⟨u, ⟨(hu₁ : is_open u), (hu₂ : u ⊆ s ∪ t)⟩, (hx₁ : x ∈ u)⟩,
classical.by_contradiction $ assume hx₂ : x ∉ s,
have u \ s ⊆ t,
from assume x ⟨h₁, h₂⟩, or.resolve_left (hu₂ h₁) h₂,
have u \ s ⊆ interior t,
by rwa subset_interior_iff_subset_of_open (is_open_diff hu₁ h₁),
have u \ s ⊆ ∅,
by rwa h₂ at this,
this ⟨hx₁, hx₂⟩,
subset.antisymm
(interior_maximal this is_open_interior)
(interior_mono $ subset_union_left _ _)
lemma is_open_iff_forall_mem_open : is_open s ↔ ∀ x ∈ s, ∃ t ⊆ s, is_open t ∧ x ∈ t :=
by rw ← subset_interior_iff_open; simp only [subset_def, mem_interior]
/-- The closure of `s` is the smallest closed set containing `s`. -/
def closure (s : set α) : set α := ⋂₀ {t | is_closed t ∧ s ⊆ t}
@[simp] lemma is_closed_closure {s : set α} : is_closed (closure s) :=
is_closed_sInter $ assume t ⟨h₁, h₂⟩, h₁
lemma subset_closure {s : set α} : s ⊆ closure s :=
subset_sInter $ assume t ⟨h₁, h₂⟩, h₂
lemma closure_minimal {s t : set α} (h₁ : s ⊆ t) (h₂ : is_closed t) : closure s ⊆ t :=
sInter_subset_of_mem ⟨h₂, h₁⟩
lemma closure_eq_of_is_closed {s : set α} (h : is_closed s) : closure s = s :=
subset.antisymm (closure_minimal (subset.refl s) h) subset_closure
lemma closure_eq_iff_is_closed {s : set α} : closure s = s ↔ is_closed s :=
⟨assume h, h ▸ is_closed_closure, closure_eq_of_is_closed⟩
lemma closure_subset_iff_subset_of_is_closed {s t : set α} (h₁ : is_closed t) :
closure s ⊆ t ↔ s ⊆ t :=
⟨subset.trans subset_closure, assume h, closure_minimal h h₁⟩
lemma closure_mono {s t : set α} (h : s ⊆ t) : closure s ⊆ closure t :=
closure_minimal (subset.trans h subset_closure) is_closed_closure
@[simp] lemma closure_empty : closure (∅ : set α) = ∅ :=
closure_eq_of_is_closed is_closed_empty
lemma closure_empty_iff (s : set α) : closure s = ∅ ↔ s = ∅ :=
begin
split; intro h,
{ rw set.eq_empty_iff_forall_not_mem,
intros x H,
simpa only [h] using subset_closure H },
{ exact (eq.symm h) ▸ closure_empty },
end
@[simp] lemma closure_univ : closure (univ : set α) = univ :=
closure_eq_of_is_closed is_closed_univ
@[simp] lemma closure_closure {s : set α} : closure (closure s) = closure s :=
closure_eq_of_is_closed is_closed_closure
@[simp] lemma closure_union {s t : set α} : closure (s ∪ t) = closure s ∪ closure t :=
subset.antisymm
(closure_minimal (union_subset_union subset_closure subset_closure) $ is_closed_union is_closed_closure is_closed_closure)
(union_subset (closure_mono $ subset_union_left _ _) (closure_mono $ subset_union_right _ _))
lemma interior_subset_closure {s : set α} : interior s ⊆ closure s :=
subset.trans interior_subset subset_closure
lemma closure_eq_compl_interior_compl {s : set α} : closure s = - interior (- s) :=
begin
unfold interior closure is_closed,
rw [compl_sUnion, compl_image_set_of],
simp only [compl_subset_compl]
end
@[simp] lemma interior_compl {s : set α} : interior (- s) = - closure s :=
by simp [closure_eq_compl_interior_compl]
@[simp] lemma closure_compl {s : set α} : closure (- s) = - interior s :=
by simp [closure_eq_compl_interior_compl]
theorem mem_closure_iff {s : set α} {a : α} : a ∈ closure s ↔ ∀ o, is_open o → a ∈ o → o ∩ s ≠ ∅ :=
⟨λ h o oo ao os,
have s ⊆ -o, from λ x xs xo, @ne_empty_of_mem α (o∩s) x ⟨xo, xs⟩ os,
closure_minimal this (is_closed_compl_iff.2 oo) h ao,
λ H c ⟨h₁, h₂⟩, classical.by_contradiction $ λ nc,
let ⟨x, hc, hs⟩ := exists_mem_of_ne_empty (H _ h₁ nc) in hc (h₂ hs)⟩
lemma dense_iff_inter_open {s : set α} : closure s = univ ↔ ∀ U, is_open U → U ≠ ∅ → U ∩ s ≠ ∅ :=
begin
split ; intro h,
{ intros U U_op U_ne,
cases exists_mem_of_ne_empty U_ne with x x_in,
exact mem_closure_iff.1 (by simp only [h]) U U_op x_in },
{ apply eq_univ_of_forall, intro x,
rw mem_closure_iff,
intros U U_op x_in,
exact h U U_op (ne_empty_of_mem x_in) },
end
/-- The frontier of a set is the set of points between the closure and interior. -/
def frontier (s : set α) : set α := closure s \ interior s
lemma frontier_eq_closure_inter_closure {s : set α} :
frontier s = closure s ∩ closure (- s) :=
by rw [closure_compl, frontier, diff_eq]
@[simp] lemma frontier_compl (s : set α) : frontier (-s) = frontier s :=
by simp only [frontier_eq_closure_inter_closure, lattice.neg_neg, inter_comm]
/-- neighbourhood filter -/
def nhds (a : α) : filter α := (⨅ s ∈ {s : set α | a ∈ s ∧ is_open s}, principal s)
lemma tendsto_nhds {m : β → α} {f : filter β} (h : ∀s, a ∈ s → is_open s → m ⁻¹' s ∈ f.sets) :
tendsto m f (nhds a) :=
show map m f ≤ (⨅ s ∈ {s : set α | a ∈ s ∧ is_open s}, principal s),
from le_infi $ assume s, le_infi $ assume ⟨ha, hs⟩, le_principal_iff.mpr $ h s ha hs
lemma tendsto_const_nhds {a : α} {f : filter β} : tendsto (λb:β, a) f (nhds a) :=
tendsto_nhds $ assume s ha hs, univ_mem_sets' $ assume _, ha
lemma nhds_sets {a : α} : (nhds a).sets = {s | ∃t⊆s, is_open t ∧ a ∈ t} :=
calc (nhds a).sets = (⋃s∈{s : set α| a ∈ s ∧ is_open s}, (principal s).sets) : infi_sets_eq'
(assume x ⟨hx₁, hx₂⟩ y ⟨hy₁, hy₂⟩,
⟨x ∩ y, ⟨⟨hx₁, hy₁⟩, is_open_inter hx₂ hy₂⟩,
le_principal_iff.2 (inter_subset_left _ _),
le_principal_iff.2 (inter_subset_right _ _)⟩)
⟨univ, mem_univ _, is_open_univ⟩
... = {s | ∃t⊆s, is_open t ∧ a ∈ t} :
le_antisymm
(supr_le $ assume i, supr_le $ assume ⟨hi₁, hi₂⟩ t ht, ⟨i, ht, hi₂, hi₁⟩)
(assume t ⟨i, hi₁, hi₂, hi₃⟩, mem_Union.2 ⟨i, mem_Union.2 ⟨⟨hi₃, hi₂⟩, hi₁⟩⟩)
lemma map_nhds {a : α} {f : α → β} :
map f (nhds a) = (⨅ s ∈ {s : set α | a ∈ s ∧ is_open s}, principal (image f s)) :=
calc map f (nhds a) = (⨅ s ∈ {s : set α | a ∈ s ∧ is_open s}, map f (principal s)) :
map_binfi_eq
(assume x ⟨hx₁, hx₂⟩ y ⟨hy₁, hy₂⟩,
⟨x ∩ y, ⟨⟨hx₁, hy₁⟩, is_open_inter hx₂ hy₂⟩,
le_principal_iff.2 (inter_subset_left _ _),
le_principal_iff.2 (inter_subset_right _ _)⟩)
⟨univ, mem_univ _, is_open_univ⟩
... = _ : by simp only [map_principal]
lemma mem_nhds_sets_iff {a : α} {s : set α} :
s ∈ (nhds a).sets ↔ ∃t⊆s, is_open t ∧ a ∈ t :=
by simp only [nhds_sets, mem_set_of_eq, exists_prop]
lemma mem_of_nhds {a : α} {s : set α} : s ∈ (nhds a).sets → a ∈ s :=
λ H, let ⟨t, ht, _, hs⟩ := mem_nhds_sets_iff.1 H in ht hs
lemma mem_nhds_sets {a : α} {s : set α} (hs : is_open s) (ha : a ∈ s) :
s ∈ (nhds a).sets :=
mem_nhds_sets_iff.2 ⟨s, subset.refl _, hs, ha⟩
lemma pure_le_nhds : pure ≤ (nhds : α → filter α) :=
assume a, le_infi $ assume s, le_infi $ assume ⟨h₁, _⟩, principal_mono.mpr $
singleton_subset_iff.2 h₁
@[simp] lemma nhds_neq_bot {a : α} : nhds a ≠ ⊥ :=
assume : nhds a = ⊥,
have pure a = (⊥ : filter α),
from lattice.bot_unique $ this ▸ pure_le_nhds a,
pure_neq_bot this
lemma interior_eq_nhds {s : set α} : interior s = {a | nhds a ≤ principal s} :=
set.ext $ λ x, by simp only [mem_interior, le_principal_iff, mem_nhds_sets_iff]; refl
lemma mem_interior_iff_mem_nhds {s : set α} {a : α} :
a ∈ interior s ↔ s ∈ (nhds a).sets :=
by simp only [interior_eq_nhds, le_principal_iff]; refl
lemma is_open_iff_nhds {s : set α} : is_open s ↔ ∀a∈s, nhds a ≤ principal s :=
calc is_open s ↔ s ⊆ interior s : subset_interior_iff_open.symm
... ↔ (∀a∈s, nhds a ≤ principal s) : by rw [interior_eq_nhds]; refl
lemma is_open_iff_mem_nhds {s : set α} : is_open s ↔ ∀a∈s, s ∈ (nhds a).sets :=
is_open_iff_nhds.trans $ forall_congr $ λ _, imp_congr_right $ λ _, le_principal_iff
lemma closure_eq_nhds {s : set α} : closure s = {a | nhds a ⊓ principal s ≠ ⊥} :=
calc closure s = - interior (- s) : closure_eq_compl_interior_compl
... = {a | ¬ nhds a ≤ principal (-s)} : by rw [interior_eq_nhds]; refl
... = {a | nhds a ⊓ principal s ≠ ⊥} : set.ext $ assume a, not_congr
(inf_eq_bot_iff_le_compl
(show principal s ⊔ principal (-s) = ⊤, by simp only [sup_principal, union_compl_self, principal_univ])
(by simp only [inf_principal, inter_compl_self, principal_empty])).symm
theorem mem_closure_iff_nhds {s : set α} {a : α} : a ∈ closure s ↔ ∀ t ∈ (nhds a).sets, t ∩ s ≠ ∅ :=
mem_closure_iff.trans
⟨λ H t ht, subset_ne_empty
(inter_subset_inter_left _ interior_subset)
(H _ is_open_interior (mem_interior_iff_mem_nhds.2 ht)),
λ H o oo ao, H _ (mem_nhds_sets oo ao)⟩
lemma is_closed_iff_nhds {s : set α} : is_closed s ↔ ∀a, nhds a ⊓ principal s ≠ ⊥ → a ∈ s :=
calc is_closed s ↔ closure s = s : by rw [closure_eq_iff_is_closed]
... ↔ closure s ⊆ s : ⟨assume h, by rw h, assume h, subset.antisymm h subset_closure⟩
... ↔ (∀a, nhds a ⊓ principal s ≠ ⊥ → a ∈ s) : by rw [closure_eq_nhds]; refl
lemma closure_inter_open {s t : set α} (h : is_open s) : s ∩ closure t ⊆ closure (s ∩ t) :=
assume a ⟨hs, ht⟩,
have s ∈ (nhds a).sets, from mem_nhds_sets h hs,
have nhds a ⊓ principal s = nhds a, from inf_of_le_left $ by rwa le_principal_iff,
have nhds a ⊓ principal (s ∩ t) ≠ ⊥,
from calc nhds a ⊓ principal (s ∩ t) = nhds a ⊓ (principal s ⊓ principal t) : by rw inf_principal
... = nhds a ⊓ principal t : by rw [←inf_assoc, this]
... ≠ ⊥ : by rw [closure_eq_nhds] at ht; assumption,
by rw [closure_eq_nhds]; assumption
lemma closure_diff {s t : set α} : closure s - closure t ⊆ closure (s - t) :=
calc closure s \ closure t = (- closure t) ∩ closure s : by simp only [diff_eq, inter_comm]
... ⊆ closure (- closure t ∩ s) : closure_inter_open $ is_open_compl_iff.mpr $ is_closed_closure
... = closure (s \ closure t) : by simp only [diff_eq, inter_comm]
... ⊆ closure (s \ t) : closure_mono $ diff_subset_diff (subset.refl s) subset_closure
lemma mem_of_closed_of_tendsto {f : β → α} {b : filter β} {a : α} {s : set α}
(hb : b ≠ ⊥) (hf : tendsto f b (nhds a)) (hs : is_closed s) (h : f ⁻¹' s ∈ b.sets) : a ∈ s :=
have b.map f ≤ nhds a ⊓ principal s,
from le_trans (le_inf (le_refl _) (le_principal_iff.mpr h)) (inf_le_inf hf (le_refl _)),
is_closed_iff_nhds.mp hs a $ neq_bot_of_le_neq_bot (map_ne_bot hb) this
lemma mem_closure_of_tendsto {f : β → α} {x : filter β} {a : α} {s : set α}
(hf : tendsto f x (nhds a)) (hs : is_closed s) (h : x ⊓ principal (f ⁻¹' s) ≠ ⊥) : a ∈ s :=
is_closed_iff_nhds.mp hs _ $ neq_bot_of_le_neq_bot (@map_ne_bot _ _ _ f h) $
le_inf (le_trans (map_mono $ inf_le_left) hf) $
le_trans (map_mono $ inf_le_right_of_le $ by simp only [comap_principal, le_principal_iff]; exact subset.refl _) (@map_comap_le _ _ _ f)
/- locally finite family [General Topology (Bourbaki, 1995)] -/
section locally_finite
/-- A family of sets in `set α` is locally finite if at every point `x:α`,
there is a neighborhood of `x` which meets only finitely many sets in the family -/
def locally_finite (f : β → set α) :=
∀x:α, ∃t∈(nhds x).sets, finite {i | f i ∩ t ≠ ∅ }
lemma locally_finite_of_finite {f : β → set α} (h : finite (univ : set β)) : locally_finite f :=
assume x, ⟨univ, univ_mem_sets, finite_subset h $ subset_univ _⟩
lemma locally_finite_subset
{f₁ f₂ : β → set α} (hf₂ : locally_finite f₂) (hf : ∀b, f₁ b ⊆ f₂ b) : locally_finite f₁ :=
assume a,
let ⟨t, ht₁, ht₂⟩ := hf₂ a in
⟨t, ht₁, finite_subset ht₂ $ assume i hi,
neq_bot_of_le_neq_bot hi $ inter_subset_inter (hf i) $ subset.refl _⟩
lemma is_closed_Union_of_locally_finite {f : β → set α}
(h₁ : locally_finite f) (h₂ : ∀i, is_closed (f i)) : is_closed (⋃i, f i) :=
is_open_iff_nhds.mpr $ assume a, assume h : a ∉ (⋃i, f i),
have ∀i, a ∈ -f i,
from assume i hi, h $ mem_Union.2 ⟨i, hi⟩,
have ∀i, - f i ∈ (nhds a).sets,
by rw [nhds_sets]; exact assume i, ⟨- f i, subset.refl _, h₂ i, this i⟩,
let ⟨t, h_sets, (h_fin : finite {i | f i ∩ t ≠ ∅ })⟩ := h₁ a in
calc nhds a ≤ principal (t ∩ (⋂ i∈{i | f i ∩ t ≠ ∅ }, - f i)) :
begin
rw [le_principal_iff],
apply @filter.inter_mem_sets _ (nhds a) _ _ h_sets,
apply @filter.Inter_mem_sets _ (nhds a) _ _ _ h_fin,
exact assume i h, this i
end
... ≤ principal (- ⋃i, f i) :
begin
simp only [principal_mono, subset_def, mem_compl_eq, mem_inter_eq,
mem_Inter, mem_set_of_eq, mem_Union, and_imp, not_exists,
not_eq_empty_iff_exists, exists_imp_distrib, (≠)],
exact assume x xt ht i xfi, ht i x xfi xt xfi
end
end locally_finite
/- compact sets -/
section compact
/-- A set `s` is compact if for every filter `f` that contains `s`,
every set of `f` also meets every neighborhood of some `a ∈ s`. -/
def compact (s : set α) := ∀f, f ≠ ⊥ → f ≤ principal s → ∃a∈s, f ⊓ nhds a ≠ ⊥
lemma compact_inter {s t : set α} (hs : compact s) (ht : is_closed t) : compact (s ∩ t) :=
assume f hnf hstf,
let ⟨a, hsa, (ha : f ⊓ nhds a ≠ ⊥)⟩ := hs f hnf (le_trans hstf (le_principal_iff.2 (inter_subset_left _ _))) in
have ∀a, principal t ⊓ nhds a ≠ ⊥ → a ∈ t,
by intro a; rw [inf_comm]; rw [is_closed_iff_nhds] at ht; exact ht a,
have a ∈ t,
from this a $ neq_bot_of_le_neq_bot ha $ inf_le_inf (le_trans hstf (le_principal_iff.2 (inter_subset_right _ _))) (le_refl _),
⟨a, ⟨hsa, this⟩, ha⟩
lemma compact_diff {s t : set α} (hs : compact s) (ht : is_open t) : compact (s \ t) :=
compact_inter hs (is_closed_compl_iff.mpr ht)
lemma compact_of_is_closed_subset {s t : set α}
(hs : compact s) (ht : is_closed t) (h : t ⊆ s) : compact t :=
by convert ← compact_inter hs ht; exact inter_eq_self_of_subset_right h
lemma compact_adherence_nhdset {s t : set α} {f : filter α}
(hs : compact s) (hf₂ : f ≤ principal s) (ht₁ : is_open t) (ht₂ : ∀a∈s, nhds a ⊓ f ≠ ⊥ → a ∈ t) :
t ∈ f.sets :=
classical.by_cases mem_sets_of_neq_bot $
assume : f ⊓ principal (- t) ≠ ⊥,
let ⟨a, ha, (hfa : f ⊓ principal (-t) ⊓ nhds a ≠ ⊥)⟩ := hs _ this $ inf_le_left_of_le hf₂ in
have a ∈ t,
from ht₂ a ha $ neq_bot_of_le_neq_bot hfa $ le_inf inf_le_right $ inf_le_left_of_le inf_le_left,
have nhds a ⊓ principal (-t) ≠ ⊥,
from neq_bot_of_le_neq_bot hfa $ le_inf inf_le_right $ inf_le_left_of_le inf_le_right,
have ∀s∈(nhds a ⊓ principal (-t)).sets, s ≠ ∅,
from forall_sets_neq_empty_iff_neq_bot.mpr this,
have false,
from this _ ⟨t, mem_nhds_sets ht₁ ‹a ∈ t›, -t, subset.refl _, subset.refl _⟩ (inter_compl_self _),
by contradiction
lemma compact_iff_ultrafilter_le_nhds {s : set α} :
compact s ↔ (∀f, ultrafilter f → f ≤ principal s → ∃a∈s, f ≤ nhds a) :=
⟨assume hs : compact s, assume f hf hfs,
let ⟨a, ha, h⟩ := hs _ hf.left hfs in
⟨a, ha, le_of_ultrafilter hf h⟩,
assume hs : (∀f, ultrafilter f → f ≤ principal s → ∃a∈s, f ≤ nhds a),
assume f hf hfs,
let ⟨a, ha, (h : ultrafilter_of f ≤ nhds a)⟩ :=
hs (ultrafilter_of f) (ultrafilter_ultrafilter_of hf) (le_trans ultrafilter_of_le hfs) in
have ultrafilter_of f ⊓ nhds a ≠ ⊥,
by simp only [inf_of_le_left, h]; exact (ultrafilter_ultrafilter_of hf).left,
⟨a, ha, neq_bot_of_le_neq_bot this (inf_le_inf ultrafilter_of_le (le_refl _))⟩⟩
lemma compact_elim_finite_subcover {s : set α} {c : set (set α)}
(hs : compact s) (hc₁ : ∀t∈c, is_open t) (hc₂ : s ⊆ ⋃₀ c) : ∃c'⊆c, finite c' ∧ s ⊆ ⋃₀ c' :=
classical.by_contradiction $ assume h,
have h : ∀{c'}, c' ⊆ c → finite c' → ¬ s ⊆ ⋃₀ c',
from assume c' h₁ h₂ h₃, h ⟨c', h₁, h₂, h₃⟩,
let
f : filter α := (⨅c':{c' : set (set α) // c' ⊆ c ∧ finite c'}, principal (s - ⋃₀ c')),
⟨a, ha⟩ := @exists_mem_of_ne_empty α s
(assume h', h (empty_subset _) finite_empty $ h'.symm ▸ empty_subset _)
in
have f ≠ ⊥, from infi_neq_bot_of_directed ⟨a⟩
(assume ⟨c₁, hc₁, hc'₁⟩ ⟨c₂, hc₂, hc'₂⟩, ⟨⟨c₁ ∪ c₂, union_subset hc₁ hc₂, finite_union hc'₁ hc'₂⟩,
principal_mono.mpr $ diff_subset_diff_right $ sUnion_mono $ subset_union_left _ _,
principal_mono.mpr $ diff_subset_diff_right $ sUnion_mono $ subset_union_right _ _⟩)
(assume ⟨c', hc'₁, hc'₂⟩, show principal (s \ _) ≠ ⊥, by simp only [ne.def, principal_eq_bot_iff, diff_eq_empty]; exact h hc'₁ hc'₂),
have f ≤ principal s, from infi_le_of_le ⟨∅, empty_subset _, finite_empty⟩ $
show principal (s \ ⋃₀∅) ≤ principal s, from le_principal_iff.2 (diff_subset _ _),
let
⟨a, ha, (h : f ⊓ nhds a ≠ ⊥)⟩ := hs f ‹f ≠ ⊥› this,
⟨t, ht₁, (ht₂ : a ∈ t)⟩ := hc₂ ha
in
have f ≤ principal (-t),
from infi_le_of_le ⟨{t}, by rwa singleton_subset_iff, finite_insert _ finite_empty⟩ $
principal_mono.mpr $
show s - ⋃₀{t} ⊆ - t, begin rw sUnion_singleton; exact assume x ⟨_, hnt⟩, hnt end,
have is_closed (- t), from is_open_compl_iff.mp $ by rw lattice.neg_neg; exact hc₁ t ht₁,
have a ∈ - t, from is_closed_iff_nhds.mp this _ $ neq_bot_of_le_neq_bot h $
le_inf inf_le_right (inf_le_left_of_le ‹f ≤ principal (- t)›),
this ‹a ∈ t›
lemma compact_elim_finite_subcover_image {s : set α} {b : set β} {c : β → set α}
(hs : compact s) (hc₁ : ∀i∈b, is_open (c i)) (hc₂ : s ⊆ ⋃i∈b, c i) :
∃b'⊆b, finite b' ∧ s ⊆ ⋃i∈b', c i :=
if h : b = ∅ then ⟨∅, empty_subset _, finite_empty, h ▸ hc₂⟩ else
let ⟨i, hi⟩ := exists_mem_of_ne_empty h in
have hc'₁ : ∀i∈c '' b, is_open i, from assume i ⟨j, hj, h⟩, h ▸ hc₁ _ hj,
have hc'₂ : s ⊆ ⋃₀ (c '' b), by rwa set.sUnion_image,
let ⟨d, hd₁, hd₂, hd₃⟩ := compact_elim_finite_subcover hs hc'₁ hc'₂ in
have ∀x : d, ∃i, i ∈ b ∧ c i = x, from assume ⟨x, hx⟩, hd₁ hx,
let ⟨f', hf⟩ := axiom_of_choice this,
f := λx:set α, (if h : x ∈ d then f' ⟨x, h⟩ else i : β) in
have ∀(x : α) (i : set α), i ∈ d → x ∈ i → (∃ (i : β), i ∈ f '' d ∧ x ∈ c i),
from assume x i hid hxi, ⟨f i, mem_image_of_mem f hid,
by simpa only [f, dif_pos hid, (hf ⟨_, hid⟩).2] using hxi⟩,
⟨f '' d,
assume i ⟨j, hj, h⟩,
h ▸ by simpa only [f, dif_pos hj] using (hf ⟨_, hj⟩).1,
finite_image f hd₂,
subset.trans hd₃ $ by simpa only [subset_def, mem_sUnion, exists_imp_distrib, mem_Union, exists_prop, and_imp]⟩
lemma compact_of_finite_subcover {s : set α}
(h : ∀c, (∀t∈c, is_open t) → s ⊆ ⋃₀ c → ∃c'⊆c, finite c' ∧ s ⊆ ⋃₀ c') : compact s :=
assume f hfn hfs, classical.by_contradiction $ assume : ¬ (∃x∈s, f ⊓ nhds x ≠ ⊥),
have hf : ∀x∈s, nhds x ⊓ f = ⊥,
by simpa only [not_exists, not_not, inf_comm],
have ¬ ∃x∈s, ∀t∈f.sets, x ∈ closure t,
from assume ⟨x, hxs, hx⟩,
have ∅ ∈ (nhds x ⊓ f).sets, by rw [empty_in_sets_eq_bot, hf x hxs],
let ⟨t₁, ht₁, t₂, ht₂, ht⟩ := by rw [mem_inf_sets] at this; exact this in
have ∅ ∈ (nhds x ⊓ principal t₂).sets,
from (nhds x ⊓ principal t₂).sets_of_superset (inter_mem_inf_sets ht₁ (subset.refl t₂)) ht,
have nhds x ⊓ principal t₂ = ⊥,
by rwa [empty_in_sets_eq_bot] at this,
by simp only [closure_eq_nhds] at hx; exact hx t₂ ht₂ this,
have ∀x∈s, ∃t∈f.sets, x ∉ closure t, by simpa only [not_exists, not_forall],
let c := (λt, - closure t) '' f.sets, ⟨c', hcc', hcf, hsc'⟩ := h c
(assume t ⟨s, hs, h⟩, h ▸ is_closed_closure) (by simpa only [subset_def, sUnion_image, mem_Union]) in
let ⟨b, hb⟩ := axiom_of_choice $
show ∀s:c', ∃t, t ∈ f.sets ∧ - closure t = s,
from assume ⟨x, hx⟩, hcc' hx in
have (⋂s∈c', if h : s ∈ c' then b ⟨s, h⟩ else univ) ∈ f.sets,
from Inter_mem_sets hcf $ assume t ht, by rw [dif_pos ht]; exact (hb ⟨t, ht⟩).left,
have s ∩ (⋂s∈c', if h : s ∈ c' then b ⟨s, h⟩ else univ) ∈ f.sets,
from inter_mem_sets (le_principal_iff.1 hfs) this,
have ∅ ∈ f.sets,
from mem_sets_of_superset this $ assume x ⟨hxs, hxi⟩,
let ⟨t, htc', hxt⟩ := (show ∃t ∈ c', x ∈ t, from hsc' hxs) in
have -closure (b ⟨t, htc'⟩) = t, from (hb _).right,
have x ∈ - t,
from this ▸ (calc x ∈ b ⟨t, htc'⟩ : by rw mem_bInter_iff at hxi; have h := hxi t htc'; rwa [dif_pos htc'] at h
... ⊆ closure (b ⟨t, htc'⟩) : subset_closure
... ⊆ - - closure (b ⟨t, htc'⟩) : by rw lattice.neg_neg; exact subset.refl _),
show false, from this hxt,
hfn $ by rwa [empty_in_sets_eq_bot] at this
lemma compact_iff_finite_subcover {s : set α} :
compact s ↔ (∀c, (∀t∈c, is_open t) → s ⊆ ⋃₀ c → ∃c'⊆c, finite c' ∧ s ⊆ ⋃₀ c') :=
⟨assume hc c, compact_elim_finite_subcover hc, compact_of_finite_subcover⟩
lemma compact_empty : compact (∅ : set α) :=
assume f hnf hsf, not.elim hnf $
empty_in_sets_eq_bot.1 $ le_principal_iff.1 hsf
lemma compact_singleton {a : α} : compact ({a} : set α) :=
compact_of_finite_subcover $ assume c hc₁ hc₂,
let ⟨i, hic, hai⟩ := (show ∃i ∈ c, a ∈ i, from mem_sUnion.1 $ singleton_subset_iff.1 hc₂) in
⟨{i}, singleton_subset_iff.2 hic, finite_singleton _, by rwa [sUnion_singleton, singleton_subset_iff]⟩
lemma compact_bUnion_of_compact {s : set β} {f : β → set α} (hs : finite s) :
(∀i ∈ s, compact (f i)) → compact (⋃i ∈ s, f i) :=
assume hf, compact_of_finite_subcover $ assume c c_open c_cover,
have ∀i : subtype s, ∃c' ⊆ c, finite c' ∧ f i ⊆ ⋃₀ c', from
assume ⟨i, hi⟩, compact_elim_finite_subcover (hf i hi) c_open
(calc f i ⊆ ⋃i ∈ s, f i : subset_bUnion_of_mem hi
... ⊆ ⋃₀ c : c_cover),
let ⟨finite_subcovers, h⟩ := axiom_of_choice this in
let c' := ⋃i, finite_subcovers i in
have c' ⊆ c, from Union_subset (λi, (h i).fst),
have finite c', from @finite_Union _ _ hs.fintype _ (λi, (h i).snd.1),
have (⋃i ∈ s, f i) ⊆ ⋃₀ c', from bUnion_subset $ λi hi, calc
f i ⊆ ⋃₀ finite_subcovers ⟨i,hi⟩ : (h ⟨i,hi⟩).snd.2
... ⊆ ⋃₀ c' : sUnion_mono (subset_Union _ _),
⟨c', ‹c' ⊆ c›, ‹finite c'›, this⟩
lemma compact_of_finite {s : set α} (hs : finite s) : compact s :=
let s' : set α := ⋃i ∈ s, {i} in
have e : s' = s, from ext $ λi, by simp only [mem_bUnion_iff, mem_singleton_iff, exists_eq_right'],
have compact s', from compact_bUnion_of_compact hs (λ_ _, compact_singleton),
e ▸ this
end compact
/- separation axioms -/
section separation
/-- A T₁ space, also known as a Fréchet space, is a topological space
where for every pair `x ≠ y`, there is an open set containing `x` and not `y`.
Equivalently, every singleton set is closed. -/
class t1_space (α : Type u) [topological_space α] :=
(t1 : ∀x, is_closed ({x} : set α))
lemma is_closed_singleton [t1_space α] {x : α} : is_closed ({x} : set α) :=
t1_space.t1 x
lemma compl_singleton_mem_nhds [t1_space α] {x y : α} (h : y ≠ x) : - {x} ∈ (nhds y).sets :=
mem_nhds_sets is_closed_singleton $ by rwa [mem_compl_eq, mem_singleton_iff]
@[simp] lemma closure_singleton [topological_space α] [t1_space α] {a : α} :
closure ({a} : set α) = {a} :=
closure_eq_of_is_closed is_closed_singleton
/-- A T₂ space, also known as a Hausdorff space, is one in which for every
`x ≠ y` there exists disjoint open sets around `x` and `y`. This is
the most widely used of the separation axioms. -/
class t2_space (α : Type u) [topological_space α] :=
(t2 : ∀x y, x ≠ y → ∃u v : set α, is_open u ∧ is_open v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅)
lemma t2_separation [t2_space α] {x y : α} (h : x ≠ y) :
∃u v : set α, is_open u ∧ is_open v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅ :=
t2_space.t2 x y h
instance t2_space.t1_space [topological_space α] [t2_space α] : t1_space α :=
⟨assume x,
have ∀y, y ≠ x ↔ ∃ (i : set α), (x ∉ i ∧ is_open i) ∧ y ∈ i,
from assume y, ⟨assume h',
let ⟨u, v, hu, hv, hy, hx, h⟩ := t2_separation h' in
have x ∉ u,
from assume : x ∈ u,
have x ∈ u ∩ v, from ⟨this, hx⟩,
by rwa [h] at this,
⟨u, ⟨this, hu⟩, hy⟩,
assume ⟨s, ⟨hx, hs⟩, hy⟩ h, hx $ h ▸ hy⟩,
have (-{x} : set α) = (⋃s∈{s : set α | x ∉ s ∧ is_open s}, s),
by apply set.ext; simpa only [mem_compl_eq, mem_singleton_iff, mem_bUnion_iff],
show is_open (- {x}),
by rw [this]; exact (is_open_Union $ assume s, is_open_Union $ assume ⟨_, hs⟩, hs)⟩
lemma eq_of_nhds_neq_bot [ht : t2_space α] {x y : α} (h : nhds x ⊓ nhds y ≠ ⊥) : x = y :=
classical.by_contradiction $ assume : x ≠ y,
let ⟨u, v, hu, hv, hx, hy, huv⟩ := t2_space.t2 x y this in
have u ∩ v ∈ (nhds x ⊓ nhds y).sets,
from inter_mem_inf_sets (mem_nhds_sets hu hx) (mem_nhds_sets hv hy),
h $ empty_in_sets_eq_bot.mp $ huv ▸ this
@[simp] lemma nhds_eq_nhds_iff {a b : α} [t2_space α] : nhds a = nhds b ↔ a = b :=
⟨assume h, eq_of_nhds_neq_bot $ by rw [h, inf_idem]; exact nhds_neq_bot, assume h, h ▸ rfl⟩
@[simp] lemma nhds_le_nhds_iff {a b : α} [t2_space α] : nhds a ≤ nhds b ↔ a = b :=
⟨assume h, eq_of_nhds_neq_bot $ by rw [inf_of_le_left h]; exact nhds_neq_bot, assume h, h ▸ le_refl _⟩
lemma tendsto_nhds_unique [t2_space α] {f : β → α} {l : filter β} {a b : α}
(hl : l ≠ ⊥) (ha : tendsto f l (nhds a)) (hb : tendsto f l (nhds b)) : a = b :=
eq_of_nhds_neq_bot $ neq_bot_of_le_neq_bot (map_ne_bot hl) $ le_inf ha hb
end separation
section regularity
/-- A T₃ space, also known as a regular space (although this condition sometimes
omits T₂), is one in which for every closed `C` and `x ∉ C`, there exist
disjoint open sets containing `x` and `C` respectively. -/
class regular_space (α : Type u) [topological_space α] extends t2_space α :=
(regular : ∀{s:set α} {a}, is_closed s → a ∉ s → ∃t, is_open t ∧ s ⊆ t ∧ nhds a ⊓ principal t = ⊥)
lemma nhds_is_closed [regular_space α] {a : α} {s : set α} (h : s ∈ (nhds a).sets) :
∃t∈(nhds a).sets, t ⊆ s ∧ is_closed t :=
let ⟨s', h₁, h₂, h₃⟩ := mem_nhds_sets_iff.mp h in
have ∃t, is_open t ∧ -s' ⊆ t ∧ nhds a ⊓ principal t = ⊥,
from regular_space.regular (is_closed_compl_iff.mpr h₂) (not_not_intro h₃),
let ⟨t, ht₁, ht₂, ht₃⟩ := this in
⟨-t,
mem_sets_of_neq_bot $ by rwa [lattice.neg_neg],
subset.trans (compl_subset_comm.1 ht₂) h₁,
is_closed_compl_iff.mpr ht₁⟩
end regularity
/- generating sets -/
end topological_space
namespace topological_space
variables {α : Type u}
/-- The least topology containing a collection of basic sets. -/
inductive generate_open (g : set (set α)) : set α → Prop
| basic : ∀s∈g, generate_open s
| univ : generate_open univ
| inter : ∀s t, generate_open s → generate_open t → generate_open (s ∩ t)
| sUnion : ∀k, (∀s∈k, generate_open s) → generate_open (⋃₀ k)
/-- The smallest topological space containing the collection `g` of basic sets -/
def generate_from (g : set (set α)) : topological_space α :=
{ is_open := generate_open g,
is_open_univ := generate_open.univ g,
is_open_inter := generate_open.inter,
is_open_sUnion := generate_open.sUnion }
lemma nhds_generate_from {g : set (set α)} {a : α} :
@nhds α (generate_from g) a = (⨅s∈{s | a ∈ s ∧ s ∈ g}, principal s) :=
le_antisymm
(infi_le_infi $ assume s, infi_le_infi_const $ assume ⟨as, sg⟩, ⟨as, generate_open.basic _ sg⟩)
(le_infi $ assume s, le_infi $ assume ⟨as, hs⟩,
have ∀s, generate_open g s → a ∈ s → (⨅s∈{s | a ∈ s ∧ s ∈ g}, principal s) ≤ principal s,
begin
intros s hs,
induction hs,
case generate_open.basic : s hs
{ exact assume as, infi_le_of_le s $ infi_le _ ⟨as, hs⟩ },
case generate_open.univ
{ rw [principal_univ],
exact assume _, le_top },
case generate_open.inter : s t hs' ht' hs ht
{ exact assume ⟨has, hat⟩, calc _ ≤ principal s ⊓ principal t : le_inf (hs has) (ht hat)
... = _ : inf_principal },
case generate_open.sUnion : k hk' hk
{ exact λ ⟨t, htk, hat⟩, calc _ ≤ principal t : hk t htk hat
... ≤ _ : le_principal_iff.2 $ subset_sUnion_of_mem htk }
end,
this s hs as)
lemma tendsto_nhds_generate_from {β : Type*} {m : α → β} {f : filter α} {g : set (set β)} {b : β}
(h : ∀s∈g, b ∈ s → m ⁻¹' s ∈ f.sets) : tendsto m f (@nhds β (generate_from g) b) :=
by rw [nhds_generate_from]; exact
(tendsto_infi.2 $ assume s, tendsto_infi.2 $ assume ⟨hbs, hsg⟩, tendsto_principal.2 $ h s hsg hbs)
protected def mk_of_nhds (n : α → filter α) : topological_space α :=
{ is_open := λs, ∀a∈s, s ∈ (n a).sets,
is_open_univ := assume x h, univ_mem_sets,
is_open_inter := assume s t hs ht x ⟨hxs, hxt⟩, inter_mem_sets (hs x hxs) (ht x hxt),
is_open_sUnion := assume s hs a ⟨x, hx, hxa⟩, mem_sets_of_superset (hs x hx _ hxa) (set.subset_sUnion_of_mem hx) }
lemma nhds_mk_of_nhds (n : α → filter α) (a : α)
(h₀ : pure ≤ n) (h₁ : ∀{a s}, s ∈ (n a).sets → ∃t∈(n a).sets, t ⊆ s ∧ ∀a'∈t, t ∈ (n a').sets) :
@nhds α (topological_space.mk_of_nhds n) a = n a :=
by letI := topological_space.mk_of_nhds n; from
(le_antisymm
(assume s hs, let ⟨t, ht, hst, h⟩ := h₁ hs in
have t ∈ (nhds a).sets, from mem_nhds_sets h (mem_pure_sets.1 $ h₀ a ht),
(nhds a).sets_of_superset this hst)
(assume s hs,
let ⟨t, hts, ht, hat⟩ := (@mem_nhds_sets_iff α (topological_space.mk_of_nhds n) _ _).1 hs in
(n a).sets_of_superset (ht _ hat) hts))
end topological_space
section lattice
variables {α : Type u} {β : Type v}
instance : partial_order (topological_space α) :=
{ le := λt s, t.is_open ≤ s.is_open,
le_antisymm := assume t s h₁ h₂, topological_space_eq $ le_antisymm h₁ h₂,
le_refl := assume t, le_refl t.is_open,
le_trans := assume a b c h₁ h₂, @le_trans _ _ a.is_open b.is_open c.is_open h₁ h₂ }
lemma generate_from_le_iff_subset_is_open {g : set (set α)} {t : topological_space α} :
topological_space.generate_from g ≤ t ↔ g ⊆ {s | t.is_open s} :=
iff.intro
(assume ht s hs, ht _ $ topological_space.generate_open.basic s hs)
(assume hg s hs, hs.rec_on (assume v hv, hg hv)
t.is_open_univ (assume u v _ _, t.is_open_inter u v) (assume k _, t.is_open_sUnion k))
protected def mk_of_closure (s : set (set α))
(hs : {u | (topological_space.generate_from s).is_open u} = s) : topological_space α :=
{ is_open := λu, u ∈ s,
is_open_univ := hs ▸ topological_space.generate_open.univ _,
is_open_inter := hs ▸ topological_space.generate_open.inter,
is_open_sUnion := hs ▸ topological_space.generate_open.sUnion }
lemma mk_of_closure_sets {s : set (set α)}
{hs : {u | (topological_space.generate_from s).is_open u} = s} :
mk_of_closure s hs = topological_space.generate_from s :=
topological_space_eq hs.symm
def gi_generate_from (α : Type*) :
galois_insertion topological_space.generate_from (λt:topological_space α, {s | t.is_open s}) :=
{ gc := assume g t, generate_from_le_iff_subset_is_open,
le_l_u := assume ts s hs, topological_space.generate_open.basic s hs,
choice := λg hg, mk_of_closure g
(subset.antisymm hg $ generate_from_le_iff_subset_is_open.1 $ le_refl _),
choice_eq := assume s hs, mk_of_closure_sets }
lemma generate_from_mono {α} {g₁ g₂ : set (set α)} (h : g₁ ⊆ g₂) :
topological_space.generate_from g₁ ≤ topological_space.generate_from g₂ :=
(gi_generate_from _).gc.monotone_l h
instance {α : Type u} : complete_lattice (topological_space α) :=
(gi_generate_from α).lift_complete_lattice
@[simp] lemma is_open_top {s : set α} : @is_open α ⊤ s := trivial
lemma le_of_nhds_le_nhds {t₁ t₂ : topological_space α} (h : ∀x, @nhds α t₂ x ≤ @nhds α t₁ x) :
t₁ ≤ t₂ :=
assume s, show @is_open α t₁ s → @is_open α t₂ s,
begin simp only [is_open_iff_nhds, le_principal_iff];
exact assume hs a ha, h _ $ hs _ ha end
lemma eq_of_nhds_eq_nhds {t₁ t₂ : topological_space α} (h : ∀x, @nhds α t₂ x = @nhds α t₁ x) :
t₁ = t₂ :=
le_antisymm
(le_of_nhds_le_nhds $ assume x, le_of_eq $ h x)
(le_of_nhds_le_nhds $ assume x, le_of_eq $ (h x).symm)
end lattice
section galois_connection
variables {α : Type*} {β : Type*} {γ : Type*}
/-- Given `f : α → β` and a topology on `β`, the induced topology on `α` is the collection of
sets that are preimages of some open set in `β`. This is the coarsest topology that
makes `f` continuous. -/
def topological_space.induced {α : Type u} {β : Type v} (f : α → β) (t : topological_space β) :
topological_space α :=
{ is_open := λs, ∃s', t.is_open s' ∧ s = f ⁻¹' s',
is_open_univ := ⟨univ, t.is_open_univ, preimage_univ.symm⟩,
is_open_inter := by rintro s₁ s₂ ⟨s'₁, hs₁, rfl⟩ ⟨s'₂, hs₂, rfl⟩;
exact ⟨s'₁ ∩ s'₂, t.is_open_inter _ _ hs₁ hs₂, preimage_inter⟩,
is_open_sUnion := assume s h,
begin
simp only [classical.skolem] at h,
cases h with f hf,
apply exists.intro (⋃(x : set α) (h : x ∈ s), f x h),
simp only [sUnion_eq_bUnion, preimage_Union, (λx h, (hf x h).right.symm)], refine ⟨_, rfl⟩,
exact (@is_open_Union β _ t _ $ assume i,
show is_open (⋃h, f i h), from @is_open_Union β _ t _ $ assume h, (hf i h).left)
end }
lemma is_closed_induced_iff [t : topological_space β] {s : set α} {f : α → β} :
@is_closed α (t.induced f) s ↔ (∃t, is_closed t ∧ s = f ⁻¹' t) :=
⟨assume ⟨t, ht, heq⟩, ⟨-t, is_closed_compl_iff.2 ht, by simp only [preimage_compl, heq.symm, lattice.neg_neg]⟩,
assume ⟨t, ht, heq⟩, ⟨-t, ht, by simp only [preimage_compl, heq.symm]⟩⟩
/-- Given `f : α → β` and a topology on `α`, the coinduced topology on `β` is defined
such that `s:set β` is open if the preimage of `s` is open. This is the finest topology that
makes `f` continuous. -/
def topological_space.coinduced {α : Type u} {β : Type v} (f : α → β) (t : topological_space α) :
topological_space β :=
{ is_open := λs, t.is_open (f ⁻¹' s),
is_open_univ := by rw preimage_univ; exact t.is_open_univ,
is_open_inter := assume s₁ s₂ h₁ h₂, by rw preimage_inter; exact t.is_open_inter _ _ h₁ h₂,
is_open_sUnion := assume s h, by rw [preimage_sUnion]; exact (@is_open_Union _ _ t _ $ assume i,
show is_open (⋃ (H : i ∈ s), f ⁻¹' i), from
@is_open_Union _ _ t _ $ assume hi, h i hi) }
variables {t t₁ t₂ : topological_space α} {t' : topological_space β} {f : α → β} {g : β → α}
lemma induced_le_iff_le_coinduced {f : α → β } {tα : topological_space α} {tβ : topological_space β} :
tβ.induced f ≤ tα ↔ tβ ≤ tα.coinduced f :=
iff.intro
(assume h s hs, show tα.is_open (f ⁻¹' s), from h _ ⟨s, hs, rfl⟩)
(assume h s ⟨t, ht, hst⟩, hst.symm ▸ h _ ht)
lemma gc_induced_coinduced (f : α → β) :
galois_connection (topological_space.induced f) (topological_space.coinduced f) :=
assume f g, induced_le_iff_le_coinduced
lemma induced_mono (h : t₁ ≤ t₂) : t₁.induced g ≤ t₂.induced g :=
(gc_induced_coinduced g).monotone_l h
lemma coinduced_mono (h : t₁ ≤ t₂) : t₁.coinduced f ≤ t₂.coinduced f :=
(gc_induced_coinduced f).monotone_u h
@[simp] lemma induced_bot : (⊥ : topological_space α).induced g = ⊥ :=
(gc_induced_coinduced g).l_bot
@[simp] lemma induced_sup : (t₁ ⊔ t₂).induced g = t₁.induced g ⊔ t₂.induced g :=
(gc_induced_coinduced g).l_sup
@[simp] lemma induced_supr {ι : Sort w} {t : ι → topological_space α} :
(⨆i, t i).induced g = (⨆i, (t i).induced g) :=
(gc_induced_coinduced g).l_supr
@[simp] lemma coinduced_top : (⊤ : topological_space α).coinduced f = ⊤ :=
(gc_induced_coinduced f).u_top
@[simp] lemma coinduced_inf : (t₁ ⊓ t₂).coinduced f = t₁.coinduced f ⊓ t₂.coinduced f :=
(gc_induced_coinduced f).u_inf
@[simp] lemma coinduced_infi {ι : Sort w} {t : ι → topological_space α} :
(⨅i, t i).coinduced f = (⨅i, (t i).coinduced f) :=
(gc_induced_coinduced f).u_infi
lemma induced_id [t : topological_space α] : t.induced id = t :=
topological_space_eq $ funext $ assume s, propext $
⟨assume ⟨s', hs, h⟩, h.symm ▸ hs, assume hs, ⟨s, hs, rfl⟩⟩
lemma induced_compose [tβ : topological_space β] [tγ : topological_space γ]
{f : α → β} {g : β → γ} : (tγ.induced g).induced f = tγ.induced (g ∘ f) :=
topological_space_eq $ funext $ assume s, propext $
⟨assume ⟨s', ⟨s, hs, h₂⟩, h₁⟩, h₁.symm ▸ h₂.symm ▸ ⟨s, hs, rfl⟩,
assume ⟨s, hs, h⟩, ⟨preimage g s, ⟨s, hs, rfl⟩, h ▸ rfl⟩⟩
lemma coinduced_id [t : topological_space α] : t.coinduced id = t :=
topological_space_eq rfl
lemma coinduced_compose [tα : topological_space α]
{f : α → β} {g : β → γ} : (tα.coinduced f).coinduced g = tα.coinduced (g ∘ f) :=
topological_space_eq rfl
end galois_connection
/- constructions using the complete lattice structure -/
section constructions
open topological_space
variables {α : Type u} {β : Type v}
instance inhabited_topological_space {α : Type u} : inhabited (topological_space α) :=
⟨⊤⟩
lemma t2_space_top : @t2_space α ⊤ :=
{ t2 := assume x y hxy, ⟨{x}, {y}, trivial, trivial, mem_insert _ _, mem_insert _ _,
eq_empty_iff_forall_not_mem.2 $ by intros z hz;
cases eq_of_mem_singleton hz.1; cases eq_of_mem_singleton hz.2; cc⟩ }
instance : topological_space empty := ⊤
instance : topological_space unit := ⊤
instance : topological_space bool := ⊤
instance : topological_space ℕ := ⊤
instance : topological_space ℤ := ⊤
instance sierpinski_space : topological_space Prop :=
generate_from {{true}}
instance {p : α → Prop} [t : topological_space α] : topological_space (subtype p) :=
induced subtype.val t
instance {r : α → α → Prop} [t : topological_space α] : topological_space (quot r) :=
coinduced (quot.mk r) t
instance {s : setoid α} [t : topological_space α] : topological_space (quotient s) :=
coinduced quotient.mk t
instance [t₁ : topological_space α] [t₂ : topological_space β] : topological_space (α × β) :=
induced prod.fst t₁ ⊔ induced prod.snd t₂
instance [t₁ : topological_space α] [t₂ : topological_space β] : topological_space (α ⊕ β) :=
coinduced sum.inl t₁ ⊓ coinduced sum.inr t₂
instance {β : α → Type v} [t₂ : Πa, topological_space (β a)] : topological_space (sigma β) :=
⨅a, coinduced (sigma.mk a) (t₂ a)
instance Pi.topological_space {β : α → Type v} [t₂ : Πa, topological_space (β a)] : topological_space (Πa, β a) :=
⨆a, induced (λf, f a) (t₂ a)
instance [topological_space α] : topological_space (list α) :=
topological_space.mk_of_nhds (traverse nhds)
lemma nhds_list [topological_space α] (as : list α) : nhds as = traverse nhds as :=
begin
refine nhds_mk_of_nhds _ _ _ _,
{ assume l, induction l,
case list.nil { exact le_refl _ },
case list.cons : a l ih {
suffices : list.cons <$> pure a <*> pure l ≤ list.cons <$> nhds a <*> traverse nhds l,
{ simpa only [-filter.pure_def] with functor_norm using this },
exact filter.seq_mono (filter.map_mono $ pure_le_nhds a) ih } },
{ assume l s hs,
rcases (mem_traverse_sets_iff _ _).1 hs with ⟨u, hu, hus⟩, clear as hs,
have : ∃v:list (set α), l.forall₂ (λa s, is_open s ∧ a ∈ s) v ∧ sequence v ⊆ s,
{ induction hu generalizing s,
case list.forall₂.nil : hs this { existsi [], simpa only [list.forall₂_nil_left_iff, exists_eq_left] },
case list.forall₂.cons : a s as ss ht h ih t hts {
rcases mem_nhds_sets_iff.1 ht with ⟨u, hut, hu⟩,
rcases ih (subset.refl _) with ⟨v, hv, hvss⟩,
exact ⟨u::v, list.forall₂.cons hu hv,
subset.trans (set.seq_mono (set.image_subset _ hut) hvss) hts⟩ } },
rcases this with ⟨v, hv, hvs⟩,
refine ⟨sequence v, mem_traverse_sets _ _ _, hvs, _⟩,
{ exact hv.imp (assume a s ⟨hs, ha⟩, mem_nhds_sets hs ha) },
{ assume u hu,
have hu := (list.mem_traverse _ _).1 hu,
have : list.forall₂ (λa s, is_open s ∧ a ∈ s) u v,
{ refine list.forall₂.flip _,
replace hv := hv.flip,
simp only [list.forall₂_and_left, flip] at ⊢ hv,
exact ⟨hv.1, hu.flip⟩ },
exact mem_traverse_sets _ _ (this.imp $ assume a s ⟨hs, ha⟩, mem_nhds_sets hs ha) } }
end
lemma quotient_dense_of_dense [setoid α] [topological_space α] {s : set α} (H : ∀ x, x ∈ closure s) :
closure (quotient.mk '' s) = univ :=
eq_univ_of_forall $ λ x, begin
rw mem_closure_iff,
intros U U_op x_in_U,
let V := quotient.mk ⁻¹' U,
cases quotient.exists_rep x with y y_x,
have y_in_V : y ∈ V, by simp only [mem_preimage_eq, y_x, x_in_U],
have V_op : is_open V := U_op,
have : V ∩ s ≠ ∅ := mem_closure_iff.1 (H y) V V_op y_in_V,
rcases exists_mem_of_ne_empty this with ⟨w, w_in_V, w_in_range⟩,
exact ne_empty_of_mem ⟨w_in_V, mem_image_of_mem quotient.mk w_in_range⟩
end
lemma generate_from_le {t : topological_space α} { g : set (set α) } (h : ∀s∈g, is_open s) :
generate_from g ≤ t :=
generate_from_le_iff_subset_is_open.2 h
protected def topological_space.nhds_adjoint (a : α) (f : filter α) : topological_space α :=
{ is_open := λs, a ∈ s → s ∈ f.sets,
is_open_univ := assume s, univ_mem_sets,
is_open_inter := assume s t hs ht ⟨has, hat⟩, inter_mem_sets (hs has) (ht hat),
is_open_sUnion := assume k hk ⟨u, hu, hau⟩, mem_sets_of_superset (hk u hu hau) (subset_sUnion_of_mem hu) }
lemma gc_nhds (a : α) :
@galois_connection _ (order_dual (filter α)) _ _ (λt, @nhds α t a) (topological_space.nhds_adjoint a) :=
assume t (f : filter α), show f ≤ @nhds α t a ↔ _, from iff.intro
(assume h s hs has, h $ @mem_nhds_sets α t a s hs has)
(assume h, le_infi $ assume u, le_infi $ assume ⟨hau, hu⟩, le_principal_iff.2 $ h _ hu hau)
lemma nhds_mono {t₁ t₂ : topological_space α} {a : α} (h : t₁ ≤ t₂) :
@nhds α t₂ a ≤ @nhds α t₁ a := (gc_nhds a).monotone_l h
lemma nhds_supr {ι : Sort*} {t : ι → topological_space α} {a : α} :
@nhds α (supr t) a = (⨅i, @nhds α (t i) a) := (gc_nhds a).l_supr
lemma nhds_Sup {s : set (topological_space α)} {a : α} :
@nhds α (Sup s) a = (⨅t∈s, @nhds α t a) := (gc_nhds a).l_Sup
lemma nhds_sup {t₁ t₂ : topological_space α} {a : α} :
@nhds α (t₁ ⊔ t₂) a = @nhds α t₁ a ⊓ @nhds α t₂ a := (gc_nhds a).l_sup
lemma nhds_bot {a : α} : @nhds α ⊥ a = ⊤ := (gc_nhds a).l_bot
private lemma separated_by_f
[tα : topological_space α] [tβ : topological_space β] [t2_space β]
(f : α → β) (hf : induced f tβ ≤ tα) {x y : α} (h : f x ≠ f y) :
∃u v : set α, is_open u ∧ is_open v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅ :=
let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h in
⟨f ⁻¹' u, f ⁻¹' v, hf _ ⟨u, uo, rfl⟩, hf _ ⟨v, vo, rfl⟩, xu, yv,
by rw [←preimage_inter, uv, preimage_empty]⟩
instance {p : α → Prop} [t : topological_space α] [t2_space α] : t2_space (subtype p) :=
⟨assume x y h,
separated_by_f subtype.val (le_refl _) (mt subtype.eq h)⟩
instance [t₁ : topological_space α] [t2_space α] [t₂ : topological_space β] [t2_space β] :
t2_space (α × β) :=
⟨assume ⟨x₁,x₂⟩ ⟨y₁,y₂⟩ h,
or.elim (not_and_distrib.mp (mt prod.ext_iff.mpr h))
(λ h₁, separated_by_f prod.fst le_sup_left h₁)
(λ h₂, separated_by_f prod.snd le_sup_right h₂)⟩
instance Pi.t2_space {β : α → Type v} [t₂ : Πa, topological_space (β a)] [Πa, t2_space (β a)] :
t2_space (Πa, β a) :=
⟨assume x y h,
let ⟨i, hi⟩ := not_forall.mp (mt funext h) in
separated_by_f (λz, z i) (le_supr _ i) hi⟩
end constructions
namespace topological_space
/- countability axioms
For our applications we are interested that there exists a countable basis, but we do not need the
concrete basis itself. This allows us to declare these type classes as `Prop` to use them as mixins.
-/
variables {α : Type u} [t : topological_space α]
include t
/-- A topological basis is one that satisfies the necessary conditions so that
it suffices to take unions of the basis sets to get a topology (without taking
finite intersections as well). -/
def is_topological_basis (s : set (set α)) : Prop :=
(∀t₁∈s, ∀t₂∈s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃∈s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂) ∧
(⋃₀ s) = univ ∧
t = generate_from s
lemma is_topological_basis_of_subbasis {s : set (set α)} (hs : t = generate_from s) :
is_topological_basis ((λf, ⋂₀ f) '' {f:set (set α) | finite f ∧ f ⊆ s ∧ ⋂₀ f ≠ ∅}) :=
let b' := (λf, ⋂₀ f) '' {f:set (set α) | finite f ∧ f ⊆ s ∧ ⋂₀ f ≠ ∅} in
⟨assume s₁ ⟨t₁, ⟨hft₁, ht₁b, ht₁⟩, eq₁⟩ s₂ ⟨t₂, ⟨hft₂, ht₂b, ht₂⟩, eq₂⟩,
have ie : ⋂₀(t₁ ∪ t₂) = ⋂₀ t₁ ∩ ⋂₀ t₂, from Inf_union,
eq₁ ▸ eq₂ ▸ assume x h,
⟨_, ⟨t₁ ∪ t₂, ⟨finite_union hft₁ hft₂, union_subset ht₁b ht₂b,
by simpa only [ie] using ne_empty_of_mem h⟩, ie⟩, h, subset.refl _⟩,
eq_univ_iff_forall.2 $ assume a, ⟨univ, ⟨∅, ⟨finite_empty, empty_subset _,
by rw sInter_empty; exact nonempty_iff_univ_ne_empty.1 ⟨a⟩⟩, sInter_empty⟩, mem_univ _⟩,
have generate_from s = generate_from b',
from le_antisymm
(generate_from_le $ assume s hs,
by_cases
(assume : s = ∅, by rw [this]; apply @is_open_empty _ _)
(assume : s ≠ ∅, generate_open.basic _ ⟨{s}, ⟨finite_singleton s, singleton_subset_iff.2 hs,
by rwa [sInter_singleton]⟩, sInter_singleton s⟩))
(generate_from_le $ assume u ⟨t, ⟨hft, htb, ne⟩, eq⟩,
eq ▸ @is_open_sInter _ (generate_from s) _ hft (assume s hs, generate_open.basic _ $ htb hs)),
this ▸ hs⟩
lemma is_topological_basis_of_open_of_nhds {s : set (set α)}
(h_open : ∀ u ∈ s, _root_.is_open u)
(h_nhds : ∀(a:α) (u : set α), a ∈ u → _root_.is_open u → ∃v ∈ s, a ∈ v ∧ v ⊆ u) :
is_topological_basis s :=
⟨assume t₁ ht₁ t₂ ht₂ x ⟨xt₁, xt₂⟩,
h_nhds x (t₁ ∩ t₂) ⟨xt₁, xt₂⟩
(is_open_inter _ _ _ (h_open _ ht₁) (h_open _ ht₂)),
eq_univ_iff_forall.2 $ assume a,
let ⟨u, h₁, h₂, _⟩ := h_nhds a univ trivial (is_open_univ _) in
⟨u, h₁, h₂⟩,
le_antisymm
(assume u hu,
(@is_open_iff_nhds α (generate_from _) _).mpr $ assume a hau,
let ⟨v, hvs, hav, hvu⟩ := h_nhds a u hau hu in
by rw nhds_generate_from; exact infi_le_of_le v (infi_le_of_le ⟨hav, hvs⟩ $ le_principal_iff.2 hvu))
(generate_from_le h_open)⟩
lemma mem_nhds_of_is_topological_basis {a : α} {s : set α} {b : set (set α)}
(hb : is_topological_basis b) : s ∈ (nhds a).sets ↔ ∃t∈b, a ∈ t ∧ t ⊆ s :=
begin
rw [hb.2.2, nhds_generate_from, infi_sets_eq'],
{ simp only [mem_bUnion_iff, exists_prop, mem_set_of_eq, and_assoc, and.left_comm]; refl },
{ exact assume s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩,
have a ∈ s ∩ t, from ⟨hs₁, ht₁⟩,
let ⟨u, hu₁, hu₂, hu₃⟩ := hb.1 _ hs₂ _ ht₂ _ this in
⟨u, ⟨hu₂, hu₁⟩, le_principal_iff.2 (subset.trans hu₃ (inter_subset_left _ _)),
le_principal_iff.2 (subset.trans hu₃ (inter_subset_right _ _))⟩ },
{ rcases eq_univ_iff_forall.1 hb.2.1 a with ⟨i, h1, h2⟩,
exact ⟨i, h2, h1⟩ }
end
lemma is_open_of_is_topological_basis {s : set α} {b : set (set α)}
(hb : is_topological_basis b) (hs : s ∈ b) : _root_.is_open s :=
is_open_iff_mem_nhds.2 $ λ a as,
(mem_nhds_of_is_topological_basis hb).2 ⟨s, hs, as, subset.refl _⟩
lemma mem_basis_subset_of_mem_open {b : set (set α)}
(hb : is_topological_basis b) {a:α} {u : set α} (au : a ∈ u)
(ou : _root_.is_open u) : ∃v ∈ b, a ∈ v ∧ v ⊆ u :=
(mem_nhds_of_is_topological_basis hb).1 $ mem_nhds_sets ou au
lemma sUnion_basis_of_is_open {B : set (set α)}
(hB : is_topological_basis B) {u : set α} (ou : _root_.is_open u) :
∃ S ⊆ B, u = ⋃₀ S :=
⟨{s ∈ B | s ⊆ u}, λ s h, h.1, set.ext $ λ a,
⟨λ ha, let ⟨b, hb, ab, bu⟩ := mem_basis_subset_of_mem_open hB ha ou in
⟨b, ⟨hb, bu⟩, ab⟩,
λ ⟨b, ⟨hb, bu⟩, ab⟩, bu ab⟩⟩
lemma Union_basis_of_is_open {B : set (set α)}
(hB : is_topological_basis B) {u : set α} (ou : _root_.is_open u) :
∃ (β : Type u) (f : β → set α), u = (⋃ i, f i) ∧ ∀ i, f i ∈ B :=
let ⟨S, sb, su⟩ := sUnion_basis_of_is_open hB ou in
⟨S, subtype.val, su.trans set.sUnion_eq_Union, λ ⟨b, h⟩, sb h⟩
variables (α)
/-- A separable space is one with a countable dense subset. -/
class separable_space : Prop :=
(exists_countable_closure_eq_univ : ∃s:set α, countable s ∧ closure s = univ)
/-- A first-countable space is one in which every point has a
countable neighborhood basis. -/
class first_countable_topology : Prop :=
(nhds_generated_countable : ∀a:α, ∃s:set (set α), countable s ∧ nhds a = (⨅t∈s, principal t))
/-- A second-countable space is one with a countable basis. -/
class second_countable_topology : Prop :=
(is_open_generated_countable : ∃b:set (set α), countable b ∧ t = topological_space.generate_from b)
instance second_countable_topology.to_first_countable_topology
[second_countable_topology α] : first_countable_topology α :=
let ⟨b, hb, eq⟩ := second_countable_topology.is_open_generated_countable α in
⟨assume a, ⟨{s | a ∈ s ∧ s ∈ b},
countable_subset (assume x ⟨_, hx⟩, hx) hb, by rw [eq, nhds_generate_from]⟩⟩
lemma is_open_generated_countable_inter [second_countable_topology α] :
∃b:set (set α), countable b ∧ ∅ ∉ b ∧ is_topological_basis b :=
let ⟨b, hb₁, hb₂⟩ := second_countable_topology.is_open_generated_countable α in
let b' := (λs, ⋂₀ s) '' {s:set (set α) | finite s ∧ s ⊆ b ∧ ⋂₀ s ≠ ∅} in
⟨b',
countable_image _ $ countable_subset
(by simp only [(and_assoc _ _).symm]; exact inter_subset_left _ _)
(countable_set_of_finite_subset hb₁),
assume ⟨s, ⟨_, _, hn⟩, hp⟩, hn hp,
is_topological_basis_of_subbasis hb₂⟩
instance second_countable_topology.to_separable_space
[second_countable_topology α] : separable_space α :=
let ⟨b, hb₁, hb₂, hb₃, hb₄, eq⟩ := is_open_generated_countable_inter α in
have nhds_eq : ∀a, nhds a = (⨅ s : {s : set α // a ∈ s ∧ s ∈ b}, principal s.val),
by intro a; rw [eq, nhds_generate_from, infi_subtype]; refl,
have ∀s∈b, ∃a, a ∈ s, from assume s hs, exists_mem_of_ne_empty $ assume eq, hb₂ $ eq ▸ hs,
have ∃f:∀s∈b, α, ∀s h, f s h ∈ s, by simp only [skolem] at this; exact this,
let ⟨f, hf⟩ := this in
⟨⟨(⋃s∈b, ⋃h:s∈b, {f s h}),
countable_bUnion hb₁ (λ _ _, countable_Union_Prop $ λ _, countable_singleton _),
set.ext $ assume a,
have a ∈ (⋃₀ b), by rw [hb₄]; exact trivial,
let ⟨t, ht₁, ht₂⟩ := this in
have w : {s : set α // a ∈ s ∧ s ∈ b}, from ⟨t, ht₂, ht₁⟩,
suffices (⨅ (x : {s // a ∈ s ∧ s ∈ b}), principal (x.val ∩ ⋃s (h₁ h₂ : s ∈ b), {f s h₂})) ≠ ⊥,
by simpa only [closure_eq_nhds, nhds_eq, infi_inf w, inf_principal, mem_set_of_eq, mem_univ, iff_true],
infi_neq_bot_of_directed ⟨a⟩
(assume ⟨s₁, has₁, hs₁⟩ ⟨s₂, has₂, hs₂⟩,
have a ∈ s₁ ∩ s₂, from ⟨has₁, has₂⟩,
let ⟨s₃, hs₃, has₃, hs⟩ := hb₃ _ hs₁ _ hs₂ _ this in
⟨⟨s₃, has₃, hs₃⟩, begin
simp only [le_principal_iff, mem_principal_sets, (≥)],
simp only [subset_inter_iff] at hs, split;
apply inter_subset_inter_left; simp only [hs]
end⟩)
(assume ⟨s, has, hs⟩,
have s ∩ (⋃ (s : set α) (H h : s ∈ b), {f s h}) ≠ ∅,
from ne_empty_of_mem ⟨hf _ hs, mem_bUnion hs $ mem_Union.mpr ⟨hs, mem_singleton _⟩⟩,
mt principal_eq_bot_iff.1 this) ⟩⟩
lemma is_open_sUnion_countable [second_countable_topology α]
(S : set (set α)) (H : ∀ s ∈ S, _root_.is_open s) :
∃ T : set (set α), countable T ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
let ⟨B, cB, _, bB⟩ := is_open_generated_countable_inter α in
begin
let B' := {b ∈ B | ∃ s ∈ S, b ⊆ s},
choose f hf using assume b:B', b.2.2,
change B' → set α at f,
haveI : encodable B' := (countable_subset (sep_subset _ _) cB).to_encodable,
have : range f ⊆ S := range_subset_iff.2 (λ x, (hf x).fst),
exact ⟨_, countable_range f, this,
subset.antisymm (sUnion_subset_sUnion this) $
sUnion_subset $ λ s hs x xs,
let ⟨b, hb, xb, bs⟩ := mem_basis_subset_of_mem_open bB xs (H _ hs) in
⟨_, ⟨⟨_, hb, _, hs, bs⟩, rfl⟩, (hf _).snd xb⟩⟩
end
end topological_space
section limit
variables {α : Type u} [inhabited α] [topological_space α]
open classical
/-- If `f` is a filter, then `lim f` is a limit of the filter, if it exists. -/
noncomputable def lim (f : filter α) : α := epsilon $ λa, f ≤ nhds a
lemma lim_spec {f : filter α} (h : ∃a, f ≤ nhds a) : f ≤ nhds (lim f) := epsilon_spec h
variables [t2_space α] {f : filter α}
lemma lim_eq {a : α} (hf : f ≠ ⊥) (h : f ≤ nhds a) : lim f = a :=
eq_of_nhds_neq_bot $ neq_bot_of_le_neq_bot hf $ le_inf (lim_spec ⟨_, h⟩) h
@[simp] lemma lim_nhds_eq {a : α} : lim (nhds a) = a :=
lim_eq nhds_neq_bot (le_refl _)
@[simp] lemma lim_nhds_eq_of_closure {a : α} {s : set α} (h : a ∈ closure s) :
lim (nhds a ⊓ principal s) = a :=
lim_eq begin rw [closure_eq_nhds] at h, exact h end inf_le_left
end limit
|
4537bdff510121559d6a09b91047f65a3aff8828 | a7eef317ddec01b9fc6cfbb876fe7ac00f205ac7 | /src/category_theory/limits/types.lean | 5bb6859925d9ef86871b3783db9a7e0681da1747 | [
"Apache-2.0"
] | permissive | kmill/mathlib | ea5a007b67ae4e9e18dd50d31d8aa60f650425ee | 1a419a9fea7b959317eddd556e1bb9639f4dcc05 | refs/heads/master | 1,668,578,197,719 | 1,593,629,163,000 | 1,593,629,163,000 | 276,482,939 | 0 | 0 | null | 1,593,637,960,000 | 1,593,637,959,000 | null | UTF-8 | Lean | false | false | 5,305 | lean | /-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Reid Barton
-/
import category_theory.limits.shapes.images
universes v u -- declare the `v`'s first; see `category_theory.category` for an explanation
open category_theory
open category_theory.limits
namespace category_theory.limits.types
variables {J : Type u} [small_category J]
/-- (internal implementation) the limit cone of a functor, implemented as flat sections of a pi type -/
def limit_ (F : J ⥤ Type u) : cone F :=
{ X := F.sections,
π := { app := λ j u, u.val j } }
local attribute [elab_simple] congr_fun
/-- (internal implementation) the fact that the proposed limit cone is the limit -/
def limit_is_limit_ (F : J ⥤ Type u) : is_limit (limit_ F) :=
{ lift := λ s v, ⟨λ j, s.π.app j v, λ j j' f, congr_fun (cone.w s f) _⟩,
uniq' := by { intros, ext x j, exact congr_fun (w j) x } }
instance : has_limits (Type u) :=
{ has_limits_of_shape := λ J 𝒥,
{ has_limit := λ F, by exactI { cone := limit_ F, is_limit := limit_is_limit_ F } } }
@[simp] lemma types_limit (F : J ⥤ Type u) :
limits.limit F = {u : Π j, F.obj j // ∀ {j j'} f, F.map f (u j) = u j'} := rfl
@[simp] lemma types_limit_π (F : J ⥤ Type u) (j : J) (g : limit F) :
limit.π F j g = g.val j := rfl
@[simp] lemma types_limit_pre
(F : J ⥤ Type u) {K : Type u} [𝒦 : small_category K] (E : K ⥤ J) (g : limit F) :
limit.pre F E g = (⟨λ k, g.val (E.obj k), by obviously⟩ : limit (E ⋙ F)) := rfl
@[simp] lemma types_limit_map {F G : J ⥤ Type u} (α : F ⟶ G) (g : limit F) :
(lim.map α : limit F → limit G) g =
(⟨λ j, (α.app j) (g.val j), λ j j' f,
by {rw ←functor_to_types.naturality, dsimp, rw ←(g.prop f)}⟩ : limit G) := rfl
@[simp] lemma types_limit_lift (F : J ⥤ Type u) (c : cone F) (x : c.X) :
limit.lift F c x = (⟨λ j, c.π.app j x, λ j j' f, congr_fun (cone.w c f) x⟩ : limit F) :=
rfl
/-- (internal implementation) the limit cone of a functor, implemented as a quotient of a sigma type -/
def colimit_ (F : J ⥤ Type u) : cocone F :=
{ X := @quot (Σ j, F.obj j) (λ p p', ∃ f : p.1 ⟶ p'.1, p'.2 = F.map f p.2),
ι :=
{ app := λ j x, quot.mk _ ⟨j, x⟩,
naturality' := λ j j' f, funext $ λ x, eq.symm (quot.sound ⟨f, rfl⟩) } }
local attribute [elab_with_expected_type] quot.lift
/-- (internal implementation) the fact that the proposed colimit cocone is the colimit -/
def colimit_is_colimit_ (F : J ⥤ Type u) : is_colimit (colimit_ F) :=
{ desc := λ s, quot.lift (λ (p : Σ j, F.obj j), s.ι.app p.1 p.2)
(assume ⟨j, x⟩ ⟨j', x'⟩ ⟨f, hf⟩, by rw hf; exact (congr_fun (cocone.w s f) x).symm) }
instance : has_colimits (Type u) :=
{ has_colimits_of_shape := λ J 𝒥,
{ has_colimit := λ F, by exactI { cocone := colimit_ F, is_colimit := colimit_is_colimit_ F } } }
@[simp] lemma types_colimit (F : J ⥤ Type u) :
limits.colimit F = @quot (Σ j, F.obj j) (λ p p', ∃ f : p.1 ⟶ p'.1, p'.2 = F.map f p.2) := rfl
@[simp] lemma types_colimit_ι (F : J ⥤ Type u) (j : J) :
colimit.ι F j = λ x, quot.mk _ ⟨j, x⟩ := rfl
@[simp] lemma types_colimit_pre
(F : J ⥤ Type u) {K : Type u} [𝒦 : small_category K] (E : K ⥤ J) :
colimit.pre F E =
quot.lift (λ p, quot.mk _ ⟨E.obj p.1, p.2⟩) (λ p p' ⟨f, h⟩, quot.sound ⟨E.map f, h⟩) := rfl
@[simp] lemma types_colimit_map {F G : J ⥤ Type u} (α : F ⟶ G) :
(colim.map α : colimit F → colimit G) =
quot.lift
(λ p, quot.mk _ ⟨p.1, (α.app p.1) p.2⟩)
(λ p p' ⟨f, h⟩, quot.sound ⟨f, by rw h; exact functor_to_types.naturality _ _ α f _⟩) := rfl
@[simp] lemma types_colimit_desc (F : J ⥤ Type u) (c : cocone F) :
colimit.desc F c =
quot.lift
(λ p, c.ι.app p.1 p.2)
(λ p p' ⟨f, h⟩, by rw h; exact (functor_to_types.naturality _ _ c.ι f _).symm) := rfl
variables {α β : Type u} (f : α ⟶ β)
section -- implementation of `has_image`
/-- the image of a morphism in Type is just `set.range f` -/
def image : Type u := set.range f
instance [inhabited α] : inhabited (image f) :=
{ default := ⟨f (default α), ⟨_, rfl⟩⟩ }
/-- the inclusion of `image f` into the target -/
def image.ι : image f ⟶ β := subtype.val
instance : mono (image.ι f) :=
(mono_iff_injective _).2 subtype.val_injective
variables {f}
/-- the universal property for the image factorisation -/
noncomputable def image.lift (F' : mono_factorisation f) : image f ⟶ F'.I :=
(λ x, F'.e (classical.indefinite_description _ x.2).1 : image f → F'.I)
lemma image.lift_fac (F' : mono_factorisation f) : image.lift F' ≫ F'.m = image.ι f :=
begin
ext x,
change (F'.e ≫ F'.m) _ = _,
rw [F'.fac, (classical.indefinite_description _ x.2).2],
refl,
end
end
/-- the factorisation of any morphism in AddCommGroup through a mono. -/
def mono_factorisation : mono_factorisation f :=
{ I := image f,
m := image.ι f,
e := set.range_factorization f }
noncomputable instance : has_image f :=
{ F := mono_factorisation f,
is_image :=
{ lift := image.lift,
lift_fac' := image.lift_fac } }
noncomputable instance : has_images (Type u) :=
{ has_image := infer_instance }
end category_theory.limits.types
|
13e95fc05df09b936c22fd1c5f4a465f8a687f6d | ea11767c9c6a467c4b7710ec6f371c95cfc023fd | /src/monoidal_categories/tensor_with_object.lean | 95dfd9cf6e7565db83d4c8ccd81fefbdde761c3f | [] | no_license | RitaAhmadi/lean-monoidal-categories | 68a23f513e902038e44681336b87f659bbc281e0 | 81f43e1e0d623a96695aa8938951d7422d6d7ba6 | refs/heads/master | 1,651,458,686,519 | 1,529,824,613,000 | 1,529,824,613,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 718 | lean | -- Copyright (c) 2017 Scott Morrison. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Stephen Morgan, Scott Morrison
import .monoidal_category
open categories
open categories.functor
open categories.products
open categories.natural_transformation
namespace categories.monoidal_category
universe variables u v
variables {C : Type u} [𝒞 : monoidal_category.{u v} C]
include 𝒞
definition tensor_on_left (Z : C) : C ↝ C :=
{ onObjects := λ X, Z ⊗ X,
onMorphisms := λ X Y f, (𝟙 Z) ⊗ f }
definition tensor_on_right (Z : C) : C ↝ C :=
{ onObjects := λ X, X ⊗ Z,
onMorphisms := λ X Y f, f ⊗ (𝟙 Z) }
end categories.monoidal_category |
627d48d793321b4dfcdb5175fcf331d4258ef74d | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/control/equiv_functor/instances.lean | 793d7e910ae041ae9468dd8d9f35b92b6e8d5ec9 | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 1,037 | lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import data.fintype.basic
import control.equiv_functor
/-!
# `equiv_functor` instances
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
We derive some `equiv_functor` instances, to enable `equiv_rw` to rewrite under these functions.
-/
open equiv
instance equiv_functor_unique : equiv_functor unique :=
{ map := λ α β e, equiv.unique_congr e, }
instance equiv_functor_perm : equiv_functor perm :=
{ map := λ α β e p, (e.symm.trans p).trans e }
-- There is a classical instance of `is_lawful_functor finset` available,
-- but we provide this computable alternative separately.
instance equiv_functor_finset : equiv_functor finset :=
{ map := λ α β e s, s.map e.to_embedding, }
instance equiv_functor_fintype : equiv_functor fintype :=
{ map := λ α β e s, by exactI fintype.of_bijective e e.bijective, }
|
146426c019316ee27f6843c48acf5df8da230adf | 3dc4623269159d02a444fe898d33e8c7e7e9461b | /.github/workflows/geo/src/adjunction.lean | 11d5238506fe0d5e255888888b9c13a0418a5317 | [] | no_license | Or7ando/lean | cc003e6c41048eae7c34aa6bada51c9e9add9e66 | d41169cf4e416a0d42092fb6bdc14131cee9dd15 | refs/heads/master | 1,650,600,589,722 | 1,587,262,906,000 | 1,587,262,906,000 | 255,387,160 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 4,758 | lean | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import category_theory.equivalence
import category_theory.adjunction
import category_theory.yoneda
import category_theory.natural_isomorphism
import category_theory.fully_faithful
namespace category_theory
open category
universes v₁ v₂ v₃ u₁ u₂ u₃ -- declare the `v`'s first; see `category_theory.category` for an explanation
local attribute [elab_simple] whisker_left whisker_right
section
variables {C : Type u₁} [𝒞 : category.{v₁} C] {D : Type u₂} [𝒟 : category.{v₂} D]
include 𝒞 𝒟
-- Some basic adjunction properties
@[reducible]
def equiv_homset_left_of_nat_iso {C : Type u₁} [𝒞 : category.{v₁} C] {D : Type u₂} [𝒟 : category.{v₂} D]
{F G : C ⥤ D} (iso : F ≅ G) {X : C} {Y : D} :
(F.obj X ⟶ Y) ≃ (G.obj X ⟶ Y) :=
⟨λ f, (iso.app _).inv ≫ f, λ g, (iso.app _).hom ≫ g, λ f, begin dsimp, rw ← assoc, simp end, λ g, begin dsimp, rw ← assoc, simp end⟩
@[reducible]
def equiv_homset_right_of_nat_iso {C : Type u₁} [𝒞 : category.{v₁} C] {D : Type u₂} [𝒟 : category.{v₂} D]
{G H : D ⥤ C} (iso : G ≅ H) {X : C} {Y : D} :
(X ⟶ G.obj Y) ≃ (X ⟶ H.obj Y) :=
⟨λ f, f ≫ (iso.app _).hom, λ g, g ≫ (iso.app _).inv, λ f, by simp, λ g, by simp⟩
def adjunction_of_nat_iso_left {C : Type u₁} [𝒞 : category.{v₁} C] {D : Type u₂} [𝒟 : category.{v₂} D]
{F G : C ⥤ D} {H : D ⥤ C} (adj : F ⊣ H) (iso : F ≅ G) :
G ⊣ H :=
adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y, equiv.trans (equiv_homset_left_of_nat_iso iso.symm) (adj.hom_equiv X Y),
hom_equiv_naturality_left_symm' := begin intros, simp, rw ← assoc, rw ← assoc, rw ← assoc, rw ← assoc, congr' 2, simp end,
hom_equiv_naturality_right' := λ X Y Y' f g, by simp}
def adjunction_of_nat_iso_right {C : Type u₁} [𝒞 : category.{v₁} C] {D : Type u₂} [𝒟 : category.{v₂} D]
{F : C ⥤ D} {G H : D ⥤ C} (adj : F ⊣ G) (iso : G ≅ H) :
F ⊣ H :=
adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y, equiv.trans (adj.hom_equiv X Y) (equiv_homset_right_of_nat_iso iso),
hom_equiv_naturality_left_symm' := λ X X' Y f g, by simp,
hom_equiv_naturality_right' := λ X Y Y' f g, begin simp, congr' 1, rw ← assoc, rw ← assoc, congr' 1, rw [nat_trans.naturality] end}
end
section
variables {C : Type u₁} [𝒞 : category.{v₁} C]
{D : Type u₂} [𝒟 : category.{v₂} D]
{E : Type u₃} [ℰ : category.{v₃} E]
include 𝒞 𝒟 ℰ
def faithful_functor_right_cancel {F G : C ⥤ D} {H : D ⥤ E}
[full H] [faithful H] (comp_iso: F ⋙ H ≅ G ⋙ H) : F ≅ G :=
begin
refine nat_iso.of_components (λX, preimage_iso (comp_iso.app X)) _,
intros X Y f,
apply functor.injectivity H,
simp only [preimage_iso_hom, iso.app, functor.image_preimage, functor.map_comp],
exact comp_iso.hom.naturality f,
end
end
section
variables {C : Type u₁} [𝒞 : category.{v₁} C]
{D : Type u₂} [𝒟 : category.{v₂} D]
include 𝒞 𝒟
def left_adjoints_coyoneda_equiv {F F' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G):
F.op ⋙ coyoneda ≅ F'.op ⋙ coyoneda :=
begin
refine nat_iso.of_components _ _,
{ intro X,
refine nat_iso.of_components _ _,
{ intro Y,
exact equiv.to_iso ((adj1.hom_equiv X.unop Y).trans ((adj2.hom_equiv X.unop Y).symm)) },
tidy },
tidy
end
def left_adjoint_uniq {F F' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G) : F ≅ F' :=
nat_iso.unop
(faithful_functor_right_cancel
(left_adjoints_coyoneda_equiv adj2 adj1))
def adjunction_op {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) : G.op ⊣ F.op :=
adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y,
{ to_fun := λ f, ((adj.hom_equiv (Y.unop) (X.unop)).inv_fun f.unop).op,
inv_fun := λ g, ((adj.hom_equiv (Y.unop) (X.unop)).to_fun g.unop).op,
left_inv := λ f, by { dsimp, rw (adj.hom_equiv _ _).right_inv, refl },
right_inv := λ f, by { dsimp, rw (adj.hom_equiv _ _).left_inv, refl } },
hom_equiv_naturality_left_symm' := λ X' X Y f g,
begin
dsimp,
apply has_hom.hom.unop_inj,
rw [unop_comp, has_hom.hom.unop_op],
apply adj.hom_equiv_naturality_right
end,
hom_equiv_naturality_right' := λ Y' Y X f g,
begin
dsimp,
apply has_hom.hom.unop_inj,
rw [unop_comp, has_hom.hom.unop_op],
apply adj.hom_equiv_naturality_left_symm
end
}
def right_adjoint_uniq {F : C ⥤ D} {G G' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F ⊣ G') : G ≅ G' :=
nat_iso.unop (left_adjoint_uniq (adjunction_op adj2) (adjunction_op adj1))
end
end category_theory |
f1061b123ed43effb722c89d2df65a38b9ac70c7 | 626e312b5c1cb2d88fca108f5933076012633192 | /src/ring_theory/unique_factorization_domain.lean | 15394d8e64806c6dfa30b688c685cedb0b3159bd | [
"Apache-2.0"
] | permissive | Bioye97/mathlib | 9db2f9ee54418d29dd06996279ba9dc874fd6beb | 782a20a27ee83b523f801ff34efb1a9557085019 | refs/heads/master | 1,690,305,956,488 | 1,631,067,774,000 | 1,631,067,774,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 51,819 | lean | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import algebra.gcd_monoid.basic
import ring_theory.integral_domain
import ring_theory.noetherian
/-!
# Unique factorization
## Main Definitions
* `wf_dvd_monoid` holds for `monoid`s for which a strict divisibility relation is
well-founded.
* `unique_factorization_monoid` holds for `wf_dvd_monoid`s where
`irreducible` is equivalent to `prime`
## To do
* set up the complete lattice structure on `factor_set`.
-/
variables {α : Type*}
local infix ` ~ᵤ ` : 50 := associated
/-- Well-foundedness of the strict version of |, which is equivalent to the descending chain
condition on divisibility and to the ascending chain condition on
principal ideals in an integral domain.
-/
class wf_dvd_monoid (α : Type*) [comm_monoid_with_zero α] : Prop :=
(well_founded_dvd_not_unit : well_founded (@dvd_not_unit α _))
export wf_dvd_monoid (well_founded_dvd_not_unit)
@[priority 100] -- see Note [lower instance priority]
instance is_noetherian_ring.wf_dvd_monoid [integral_domain α] [is_noetherian_ring α] :
wf_dvd_monoid α :=
⟨by { convert inv_image.wf (λ a, ideal.span ({a} : set α)) (well_founded_submodule_gt _ _),
ext,
exact ideal.span_singleton_lt_span_singleton.symm }⟩
namespace wf_dvd_monoid
variables [comm_monoid_with_zero α]
open associates nat
theorem of_wf_dvd_monoid_associates (h : wf_dvd_monoid (associates α)): wf_dvd_monoid α :=
⟨begin
haveI := h,
refine (surjective.well_founded_iff mk_surjective _).2 wf_dvd_monoid.well_founded_dvd_not_unit,
intros, rw mk_dvd_not_unit_mk_iff
end⟩
variables [wf_dvd_monoid α]
instance wf_dvd_monoid_associates : wf_dvd_monoid (associates α) :=
⟨begin
refine (surjective.well_founded_iff mk_surjective _).1 wf_dvd_monoid.well_founded_dvd_not_unit,
intros, rw mk_dvd_not_unit_mk_iff
end⟩
theorem well_founded_associates : well_founded ((<) : associates α → associates α → Prop) :=
subrelation.wf (λ x y, dvd_not_unit_of_lt) wf_dvd_monoid.well_founded_dvd_not_unit
local attribute [elab_as_eliminator] well_founded.fix
lemma exists_irreducible_factor {a : α} (ha : ¬ is_unit a) (ha0 : a ≠ 0) :
∃ i, irreducible i ∧ i ∣ a :=
(irreducible_or_factor a ha).elim (λ hai, ⟨a, hai, dvd_rfl⟩)
(well_founded.fix
wf_dvd_monoid.well_founded_dvd_not_unit
(λ a ih ha ha0 ⟨x, y, hx, hy, hxy⟩,
have hx0 : x ≠ 0, from λ hx0, ha0 (by rw [← hxy, hx0, zero_mul]),
(irreducible_or_factor x hx).elim
(λ hxi, ⟨x, hxi, hxy ▸ by simp⟩)
(λ hxf, let ⟨i, hi⟩ := ih x ⟨hx0, y, hy, hxy.symm⟩ hx hx0 hxf in
⟨i, hi.1, hi.2.trans (hxy ▸ by simp)⟩)) a ha ha0)
@[elab_as_eliminator] lemma induction_on_irreducible {P : α → Prop} (a : α)
(h0 : P 0) (hu : ∀ u : α, is_unit u → P u)
(hi : ∀ a i : α, a ≠ 0 → irreducible i → P a → P (i * a)) :
P a :=
by haveI := classical.dec; exact
well_founded.fix wf_dvd_monoid.well_founded_dvd_not_unit
(λ a ih, if ha0 : a = 0 then ha0.symm ▸ h0
else if hau : is_unit a then hu a hau
else let ⟨i, hii, ⟨b, hb⟩⟩ := exists_irreducible_factor hau ha0 in
have hb0 : b ≠ 0, from λ hb0, by simp * at *,
hb.symm ▸ hi _ _ hb0 hii (ih _ ⟨hb0, i,
hii.1, by rw [hb, mul_comm]⟩))
a
lemma exists_factors (a : α) : a ≠ 0 →
∃f : multiset α, (∀b ∈ f, irreducible b) ∧ associated f.prod a :=
wf_dvd_monoid.induction_on_irreducible a
(λ h, (h rfl).elim)
(λ u hu _, ⟨0, ⟨by simp [hu], associated.symm (by simp [hu, associated_one_iff_is_unit])⟩⟩)
(λ a i ha0 hii ih hia0,
let ⟨s, hs⟩ := ih ha0 in
⟨i ::ₘ s, ⟨by clear _let_match; finish,
by { rw multiset.prod_cons,
exact hs.2.mul_left _ }⟩⟩)
end wf_dvd_monoid
theorem wf_dvd_monoid.of_well_founded_associates [comm_cancel_monoid_with_zero α]
(h : well_founded ((<) : associates α → associates α → Prop)) : wf_dvd_monoid α :=
wf_dvd_monoid.of_wf_dvd_monoid_associates
⟨by { convert h, ext, exact associates.dvd_not_unit_iff_lt }⟩
theorem wf_dvd_monoid.iff_well_founded_associates [comm_cancel_monoid_with_zero α] :
wf_dvd_monoid α ↔ well_founded ((<) : associates α → associates α → Prop) :=
⟨by apply wf_dvd_monoid.well_founded_associates, wf_dvd_monoid.of_well_founded_associates⟩
section prio
set_option default_priority 100 -- see Note [default priority]
/-- unique factorization monoids.
These are defined as `comm_cancel_monoid_with_zero`s with well-founded strict divisibility
relations, but this is equivalent to more familiar definitions:
Each element (except zero) is uniquely represented as a multiset of irreducible factors.
Uniqueness is only up to associated elements.
Each element (except zero) is non-uniquely represented as a multiset
of prime factors.
To define a UFD using the definition in terms of multisets
of irreducible factors, use the definition `of_exists_unique_irreducible_factors`
To define a UFD using the definition in terms of multisets
of prime factors, use the definition `of_exists_prime_factors`
-/
class unique_factorization_monoid (α : Type*) [comm_cancel_monoid_with_zero α]
extends wf_dvd_monoid α : Prop :=
(irreducible_iff_prime : ∀ {a : α}, irreducible a ↔ prime a)
instance ufm_of_gcd_of_wf_dvd_monoid [nontrivial α] [comm_cancel_monoid_with_zero α]
[wf_dvd_monoid α] [gcd_monoid α] : unique_factorization_monoid α :=
{ irreducible_iff_prime := λ _, gcd_monoid.irreducible_iff_prime
.. ‹wf_dvd_monoid α› }
instance associates.ufm [comm_cancel_monoid_with_zero α]
[unique_factorization_monoid α] : unique_factorization_monoid (associates α) :=
{ irreducible_iff_prime := by { rw ← associates.irreducible_iff_prime_iff,
apply unique_factorization_monoid.irreducible_iff_prime, }
.. (wf_dvd_monoid.wf_dvd_monoid_associates : wf_dvd_monoid (associates α)) }
end prio
namespace unique_factorization_monoid
variables [comm_cancel_monoid_with_zero α] [unique_factorization_monoid α]
theorem exists_prime_factors (a : α) : a ≠ 0 →
∃ f : multiset α, (∀b ∈ f, prime b) ∧ f.prod ~ᵤ a :=
by { simp_rw ← unique_factorization_monoid.irreducible_iff_prime,
apply wf_dvd_monoid.exists_factors a }
@[elab_as_eliminator] lemma induction_on_prime {P : α → Prop}
(a : α) (h₁ : P 0) (h₂ : ∀ x : α, is_unit x → P x)
(h₃ : ∀ a p : α, a ≠ 0 → prime p → P a → P (p * a)) : P a :=
begin
simp_rw ← unique_factorization_monoid.irreducible_iff_prime at h₃,
exact wf_dvd_monoid.induction_on_irreducible a h₁ h₂ h₃,
end
lemma factors_unique : ∀{f g : multiset α},
(∀x∈f, irreducible x) → (∀x∈g, irreducible x) → f.prod ~ᵤ g.prod →
multiset.rel associated f g :=
by haveI := classical.dec_eq α; exact
λ f, multiset.induction_on f
(λ g _ hg h,
multiset.rel_zero_left.2 $
multiset.eq_zero_of_forall_not_mem (λ x hx,
have is_unit g.prod, by simpa [associated_one_iff_is_unit] using h.symm,
(hg x hx).not_unit (is_unit_iff_dvd_one.2 ((multiset.dvd_prod hx).trans
(is_unit_iff_dvd_one.1 this)))))
(λ p f ih g hf hg hfg,
let ⟨b, hbg, hb⟩ := exists_associated_mem_of_dvd_prod
(irreducible_iff_prime.1 (hf p (by simp)))
(λ q hq, irreducible_iff_prime.1 (hg _ hq)) $
hfg.dvd_iff_dvd_right.1
(show p ∣ (p ::ₘ f).prod, by simp) in
begin
rw ← multiset.cons_erase hbg,
exact multiset.rel.cons hb (ih (λ q hq, hf _ (by simp [hq]))
(λ q (hq : q ∈ g.erase b), hg q (multiset.mem_of_mem_erase hq))
(associated.of_mul_left
(by rwa [← multiset.prod_cons, ← multiset.prod_cons, multiset.cons_erase hbg]) hb
(hf p (by simp)).ne_zero))
end)
end unique_factorization_monoid
lemma prime_factors_unique [comm_cancel_monoid_with_zero α] : ∀ {f g : multiset α},
(∀ x ∈ f, prime x) → (∀ x ∈ g, prime x) → f.prod ~ᵤ g.prod →
multiset.rel associated f g :=
by haveI := classical.dec_eq α; exact
λ f, multiset.induction_on f
(λ g _ hg h,
multiset.rel_zero_left.2 $
multiset.eq_zero_of_forall_not_mem $ λ x hx,
have is_unit g.prod, by simpa [associated_one_iff_is_unit] using h.symm,
(hg x hx).not_unit $ is_unit_iff_dvd_one.2 $
(multiset.dvd_prod hx).trans (is_unit_iff_dvd_one.1 this))
(λ p f ih g hf hg hfg,
let ⟨b, hbg, hb⟩ := exists_associated_mem_of_dvd_prod
(hf p (by simp)) (λ q hq, hg _ hq) $
hfg.dvd_iff_dvd_right.1
(show p ∣ (p ::ₘ f).prod, by simp) in
begin
rw ← multiset.cons_erase hbg,
exact multiset.rel.cons hb (ih (λ q hq, hf _ (by simp [hq]))
(λ q (hq : q ∈ g.erase b), hg q (multiset.mem_of_mem_erase hq))
(associated.of_mul_left
(by rwa [← multiset.prod_cons, ← multiset.prod_cons, multiset.cons_erase hbg]) hb
(hf p (by simp)).ne_zero)),
end)
/-- If an irreducible has a prime factorization,
then it is an associate of one of its prime factors. -/
lemma prime_factors_irreducible [comm_cancel_monoid_with_zero α] {a : α} {f : multiset α}
(ha : irreducible a) (pfa : (∀b ∈ f, prime b) ∧ f.prod ~ᵤ a) :
∃ p, a ~ᵤ p ∧ f = {p} :=
begin
haveI := classical.dec_eq α,
refine multiset.induction_on f (λ h, (ha.not_unit
(associated_one_iff_is_unit.1 (associated.symm h))).elim) _ pfa.2 pfa.1,
rintros p s _ ⟨u, hu⟩ hs,
use p,
have hs0 : s = 0,
{ by_contra hs0,
obtain ⟨q, hq⟩ := multiset.exists_mem_of_ne_zero hs0,
apply (hs q (by simp [hq])).2.1,
refine (ha.is_unit_or_is_unit (_ : _ = ((p * ↑u) * (s.erase q).prod) * _)).resolve_left _,
{ rw [mul_right_comm _ _ q, mul_assoc, ← multiset.prod_cons, multiset.cons_erase hq, ← hu,
mul_comm, mul_comm p _, mul_assoc],
simp, },
apply mt is_unit_of_mul_is_unit_left (mt is_unit_of_mul_is_unit_left _),
apply (hs p (multiset.mem_cons_self _ _)).2.1 },
simp only [mul_one, multiset.prod_cons, multiset.prod_zero, hs0] at *,
exact ⟨associated.symm ⟨u, hu⟩, rfl⟩,
end
section exists_prime_factors
variables [comm_cancel_monoid_with_zero α]
variables (pf : ∀ (a : α), a ≠ 0 → ∃ f : multiset α, (∀b ∈ f, prime b) ∧ f.prod ~ᵤ a)
include pf
lemma wf_dvd_monoid.of_exists_prime_factors : wf_dvd_monoid α :=
⟨begin
classical,
apply rel_hom.well_founded (rel_hom.mk _ _) (with_top.well_founded_lt nat.lt_wf),
{ intro a,
by_cases h : a = 0, { exact ⊤ },
exact (classical.some (pf a h)).card },
rintros a b ⟨ane0, ⟨c, hc, b_eq⟩⟩,
rw dif_neg ane0,
by_cases h : b = 0, { simp [h, lt_top_iff_ne_top] },
rw [dif_neg h, with_top.coe_lt_coe],
have cne0 : c ≠ 0, { refine mt (λ con, _) h, rw [b_eq, con, mul_zero] },
calc multiset.card (classical.some (pf a ane0))
< _ + multiset.card (classical.some (pf c cne0)) :
lt_add_of_pos_right _ (multiset.card_pos.mpr (λ con, hc (associated_one_iff_is_unit.mp _)))
... = multiset.card (classical.some (pf a ane0) + classical.some (pf c cne0)) :
(multiset.card_add _ _).symm
... = multiset.card (classical.some (pf b h)) :
multiset.card_eq_card_of_rel (prime_factors_unique _ (classical.some_spec (pf _ h)).1 _),
{ convert (classical.some_spec (pf c cne0)).2.symm,
rw [con, multiset.prod_zero] },
{ intros x hadd,
rw multiset.mem_add at hadd,
cases hadd; apply (classical.some_spec (pf _ _)).1 _ hadd },
{ rw multiset.prod_add,
transitivity a * c,
{ apply associated.mul_mul; apply (classical.some_spec (pf _ _)).2 },
{ rw ← b_eq,
apply (classical.some_spec (pf _ _)).2.symm, } }
end⟩
lemma irreducible_iff_prime_of_exists_prime_factors {p : α} : irreducible p ↔ prime p :=
begin
by_cases hp0 : p = 0,
{ simp [hp0] },
refine ⟨λ h, _, prime.irreducible⟩,
obtain ⟨f, hf⟩ := pf p hp0,
obtain ⟨q, hq, rfl⟩ := prime_factors_irreducible h hf,
rw hq.prime_iff,
exact hf.1 q (multiset.mem_singleton_self _)
end
theorem unique_factorization_monoid.of_exists_prime_factors :
unique_factorization_monoid α :=
{ irreducible_iff_prime := λ _, irreducible_iff_prime_of_exists_prime_factors pf,
.. wf_dvd_monoid.of_exists_prime_factors pf }
end exists_prime_factors
theorem unique_factorization_monoid.iff_exists_prime_factors [comm_cancel_monoid_with_zero α] :
unique_factorization_monoid α ↔
(∀ (a : α), a ≠ 0 → ∃ f : multiset α, (∀b ∈ f, prime b) ∧ f.prod ~ᵤ a) :=
⟨λ h, @unique_factorization_monoid.exists_prime_factors _ _ h,
unique_factorization_monoid.of_exists_prime_factors⟩
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [comm_cancel_monoid_with_zero α]
(eif : ∀ (a : α), a ≠ 0 → ∃ f : multiset α, (∀b ∈ f, irreducible b) ∧ f.prod ~ᵤ a)
(uif : ∀ (f g : multiset α),
(∀ x ∈ f, irreducible x) → (∀ x ∈ g, irreducible x) → f.prod ~ᵤ g.prod →
multiset.rel associated f g)
(p : α) : irreducible p ↔ prime p :=
⟨by letI := classical.dec_eq α; exact λ hpi,
⟨hpi.ne_zero, hpi.1,
λ a b ⟨x, hx⟩,
if hab0 : a * b = 0
then (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim
(λ ha0, by simp [ha0])
(λ hb0, by simp [hb0])
else
have hx0 : x ≠ 0, from λ hx0, by simp * at *,
have ha0 : a ≠ 0, from left_ne_zero_of_mul hab0,
have hb0 : b ≠ 0, from right_ne_zero_of_mul hab0,
begin
cases eif x hx0 with fx hfx,
cases eif a ha0 with fa hfa,
cases eif b hb0 with fb hfb,
have h : multiset.rel associated (p ::ₘ fx) (fa + fb),
{ apply uif,
{ exact λ i hi, (multiset.mem_cons.1 hi).elim (λ hip, hip.symm ▸ hpi) (hfx.1 _), },
{ exact λ i hi, (multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _), },
calc multiset.prod (p ::ₘ fx)
~ᵤ a * b : by rw [hx, multiset.prod_cons];
exact hfx.2.mul_left _
... ~ᵤ (fa).prod * (fb).prod :
hfa.2.symm.mul_mul hfb.2.symm
... = _ : by rw multiset.prod_add, },
exact let ⟨q, hqf, hq⟩ := multiset.exists_mem_of_rel_of_mem h
(multiset.mem_cons_self p _) in
(multiset.mem_add.1 hqf).elim
(λ hqa, or.inl $ hq.dvd_iff_dvd_left.2 $
hfa.2.dvd_iff_dvd_right.1
(multiset.dvd_prod hqa))
(λ hqb, or.inr $ hq.dvd_iff_dvd_left.2 $
hfb.2.dvd_iff_dvd_right.1
(multiset.dvd_prod hqb))
end⟩, prime.irreducible⟩
theorem unique_factorization_monoid.of_exists_unique_irreducible_factors
[comm_cancel_monoid_with_zero α]
(eif : ∀ (a : α), a ≠ 0 → ∃ f : multiset α, (∀b ∈ f, irreducible b) ∧ f.prod ~ᵤ a)
(uif : ∀ (f g : multiset α),
(∀ x ∈ f, irreducible x) → (∀ x ∈ g, irreducible x) → f.prod ~ᵤ g.prod →
multiset.rel associated f g) :
unique_factorization_monoid α :=
unique_factorization_monoid.of_exists_prime_factors (by
{ convert eif,
simp_rw irreducible_iff_prime_of_exists_unique_irreducible_factors eif uif })
namespace unique_factorization_monoid
variables [comm_cancel_monoid_with_zero α] [decidable_eq α] [nontrivial α] [normalization_monoid α]
variables [unique_factorization_monoid α]
/-- Noncomputably determines the multiset of prime factors. -/
noncomputable def factors (a : α) : multiset α := if h : a = 0 then 0 else
multiset.map normalize $ classical.some (unique_factorization_monoid.exists_prime_factors a h)
theorem factors_prod {a : α} (ane0 : a ≠ 0) : associated (factors a).prod a :=
begin
rw [factors, dif_neg ane0],
refine associated.trans _ (classical.some_spec (exists_prime_factors a ane0)).2,
rw [← associates.mk_eq_mk_iff_associated, ← associates.prod_mk, ← associates.prod_mk,
multiset.map_map],
congr' 2,
ext,
rw [function.comp_apply, associates.mk_normalize],
end
theorem prime_of_factor {a : α} : ∀ (x : α), x ∈ factors a → prime x :=
begin
rw [factors],
split_ifs with ane0, { simp },
intros x hx, rcases multiset.mem_map.1 hx with ⟨y, ⟨hy, rfl⟩⟩,
rw (normalize_associated _).prime_iff,
exact (classical.some_spec (unique_factorization_monoid.exists_prime_factors a ane0)).1 y hy,
end
theorem irreducible_of_factor {a : α} : ∀ (x : α), x ∈ factors a → irreducible x :=
λ x h, (prime_of_factor x h).irreducible
theorem normalize_factor {a : α} : ∀ (x : α), x ∈ factors a → normalize x = x :=
begin
rw factors,
split_ifs with h, { simp },
intros x hx,
obtain ⟨y, hy, rfl⟩ := multiset.mem_map.1 hx,
apply normalize_idem
end
lemma factors_irreducible {a : α} (ha : irreducible a) :
factors a = {normalize a} :=
begin
obtain ⟨p, a_assoc, hp⟩ := prime_factors_irreducible ha
⟨prime_of_factor, factors_prod ha.ne_zero⟩,
have p_mem : p ∈ factors a,
{ rw hp, exact multiset.mem_singleton_self _ },
convert hp,
rwa [← normalize_factor p p_mem, normalize_eq_normalize_iff, dvd_dvd_iff_associated]
end
lemma exists_mem_factors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : irreducible p) : p ∣ a →
∃ q ∈ factors a, p ~ᵤ q :=
λ ⟨b, hb⟩,
have hb0 : b ≠ 0, from λ hb0, by simp * at *,
have multiset.rel associated (p ::ₘ factors b) (factors a),
from factors_unique
(λ x hx, (multiset.mem_cons.1 hx).elim (λ h, h.symm ▸ hp)
(irreducible_of_factor _))
irreducible_of_factor
(associated.symm $ calc multiset.prod (factors a) ~ᵤ a : factors_prod ha0
... = p * b : hb
... ~ᵤ multiset.prod (p ::ₘ factors b) :
by rw multiset.prod_cons; exact (factors_prod hb0).symm.mul_left _),
multiset.exists_mem_of_rel_of_mem this (by simp)
@[simp] lemma factors_zero : factors (0 : α) = 0 := dif_pos rfl
@[simp] lemma factors_one : factors (1 : α) = 0 :=
begin
rw ← multiset.rel_zero_right,
apply factors_unique irreducible_of_factor,
{ intros x hx,
exfalso,
apply multiset.not_mem_zero x hx },
{ simp [factors_prod (@one_ne_zero α _ _)] },
apply_instance
end
@[simp] lemma factors_mul {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
factors (x * y) = factors x + factors y :=
begin
have h : (normalize : α → α) = associates.out ∘ associates.mk,
{ ext, rw [function.comp_apply, associates.out_mk], },
rw [← multiset.map_id' (factors (x * y)), ← multiset.map_id' (factors x),
← multiset.map_id' (factors y), ← multiset.map_congr normalize_factor,
← multiset.map_congr normalize_factor, ← multiset.map_congr normalize_factor,
← multiset.map_add, h, ← multiset.map_map associates.out, eq_comm,
← multiset.map_map associates.out],
refine congr rfl _,
apply multiset.map_mk_eq_map_mk_of_rel,
apply factors_unique,
{ intros x hx,
rcases multiset.mem_add.1 hx with hx | hx;
exact irreducible_of_factor x hx },
{ exact irreducible_of_factor },
{ rw multiset.prod_add,
exact ((factors_prod hx).mul_mul (factors_prod hy)).trans
(factors_prod (mul_ne_zero hx hy)).symm }
end
@[simp] lemma factors_pow {x : α} (n : ℕ) :
factors (x ^ n) = n • factors x :=
begin
induction n with n ih,
{ simp },
by_cases h0 : x = 0,
{ simp [h0, zero_pow n.succ_pos, smul_zero] },
rw [pow_succ, succ_nsmul, factors_mul h0 (pow_ne_zero _ h0), ih],
end
lemma dvd_iff_factors_le_factors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
x ∣ y ↔ factors x ≤ factors y :=
begin
split,
{ rintro ⟨c, rfl⟩,
simp [hx, right_ne_zero_of_mul hy] },
{ rw [← (factors_prod hx).dvd_iff_dvd_left, ← (factors_prod hy).dvd_iff_dvd_right],
apply multiset.prod_dvd_prod }
end
end unique_factorization_monoid
namespace unique_factorization_monoid
open_locale classical
open multiset associates
noncomputable theory
variables [comm_cancel_monoid_with_zero α] [nontrivial α] [unique_factorization_monoid α]
/-- Noncomputably defines a `normalization_monoid` structure on a `unique_factorization_monoid`. -/
protected def normalization_monoid : normalization_monoid α :=
normalization_monoid_of_monoid_hom_right_inverse {
to_fun := λ a : associates α, if a = 0 then 0 else ((factors a).map
(classical.some mk_surjective.has_right_inverse : associates α → α)).prod,
map_one' := by simp,
map_mul' := λ x y, by {
by_cases hx : x = 0, { simp [hx] },
by_cases hy : y = 0, { simp [hy] },
simp [hx, hy] } } begin
intro x,
dsimp,
by_cases hx : x = 0, { simp [hx] },
have h : associates.mk_monoid_hom ∘ (classical.some mk_surjective.has_right_inverse) =
(id : associates α → associates α),
{ ext x,
rw [function.comp_apply, mk_monoid_hom_apply,
classical.some_spec mk_surjective.has_right_inverse x],
refl },
rw [if_neg hx, ← mk_monoid_hom_apply, monoid_hom.map_multiset_prod, map_map, h, map_id,
← associated_iff_eq],
apply factors_prod hx
end
instance : inhabited (normalization_monoid α) := ⟨unique_factorization_monoid.normalization_monoid⟩
end unique_factorization_monoid
namespace unique_factorization_monoid
variables {R : Type*} [comm_cancel_monoid_with_zero R] [unique_factorization_monoid R]
lemma no_factors_of_no_prime_factors {a b : R} (ha : a ≠ 0)
(h : (∀ {d}, d ∣ a → d ∣ b → ¬ prime d)) : ∀ {d}, d ∣ a → d ∣ b → is_unit d :=
λ d, induction_on_prime d
(by { simp only [zero_dvd_iff], intros, contradiction })
(λ x hx _ _, hx)
(λ d q hp hq ih dvd_a dvd_b,
absurd hq (h (dvd_of_mul_right_dvd dvd_a) (dvd_of_mul_right_dvd dvd_b)))
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `is_coprime.dvd_of_dvd_mul_left`. -/
lemma dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
(∀ {d}, d ∣ a → d ∣ c → ¬ prime d) → a ∣ b * c → a ∣ b :=
begin
refine induction_on_prime c _ _ _,
{ intro no_factors,
simp only [dvd_zero, mul_zero, forall_prop_of_true],
haveI := classical.prop_decidable,
exact is_unit_iff_forall_dvd.mp
(no_factors_of_no_prime_factors ha @no_factors (dvd_refl a) (dvd_zero a)) _ },
{ rintros _ ⟨x, rfl⟩ _ a_dvd_bx,
apply units.dvd_mul_right.mp a_dvd_bx },
{ intros c p hc hp ih no_factors a_dvd_bpc,
apply ih (λ q dvd_a dvd_c hq, no_factors dvd_a (dvd_c.mul_left _) hq),
rw mul_left_comm at a_dvd_bpc,
refine or.resolve_left (hp.left_dvd_or_dvd_right_of_dvd_mul a_dvd_bpc) (λ h, _),
exact no_factors h (dvd_mul_right p c) hp }
end
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `b` have no common prime factors, `a ∣ c`.
Compare `is_coprime.dvd_of_dvd_mul_right`. -/
lemma dvd_of_dvd_mul_right_of_no_prime_factors {a b c : R} (ha : a ≠ 0)
(no_factors : ∀ {d}, d ∣ a → d ∣ b → ¬ prime d) : a ∣ b * c → a ∣ c :=
by simpa [mul_comm b c] using dvd_of_dvd_mul_left_of_no_prime_factors ha @no_factors
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
lemma exists_reduced_factors : ∀ (a ≠ (0 : R)) b,
∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → is_unit d) ∧ c' * a' = a ∧ c' * b' = b :=
begin
haveI := classical.prop_decidable,
intros a,
refine induction_on_prime a _ _ _,
{ intros, contradiction },
{ intros a a_unit a_ne_zero b,
use [a, b, 1],
split,
{ intros p p_dvd_a _,
exact is_unit_of_dvd_unit p_dvd_a a_unit },
{ simp } },
{ intros a p a_ne_zero p_prime ih_a pa_ne_zero b,
by_cases p ∣ b,
{ rcases h with ⟨b, rfl⟩,
obtain ⟨a', b', c', no_factor, ha', hb'⟩ := ih_a a_ne_zero b,
refine ⟨a', b', p * c', @no_factor, _, _⟩,
{ rw [mul_assoc, ha'] },
{ rw [mul_assoc, hb'] } },
{ obtain ⟨a', b', c', coprime, rfl, rfl⟩ := ih_a a_ne_zero b,
refine ⟨p * a', b', c', _, mul_left_comm _ _ _, rfl⟩,
intros q q_dvd_pa' q_dvd_b',
cases p_prime.left_dvd_or_dvd_right_of_dvd_mul q_dvd_pa' with p_dvd_q q_dvd_a',
{ have : p ∣ c' * b' := dvd_mul_of_dvd_right (p_dvd_q.trans q_dvd_b') _,
contradiction },
exact coprime q_dvd_a' q_dvd_b' } }
end
lemma exists_reduced_factors' (a b : R) (hb : b ≠ 0) :
∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → is_unit d) ∧ c' * a' = a ∧ c' * b' = b :=
let ⟨b', a', c', no_factor, hb, ha⟩ := exists_reduced_factors b hb a
in ⟨a', b', c', λ _ hpb hpa, no_factor hpa hpb, ha, hb⟩
section multiplicity
variables [nontrivial R] [normalization_monoid R] [decidable_eq R]
variables [decidable_rel (has_dvd.dvd : R → R → Prop)]
open multiplicity multiset
lemma le_multiplicity_iff_repeat_le_factors {a b : R} {n : ℕ} (ha : irreducible a) (hb : b ≠ 0) :
↑n ≤ multiplicity a b ↔ repeat (normalize a) n ≤ factors b :=
begin
rw ← pow_dvd_iff_le_multiplicity,
revert b,
induction n with n ih, { simp },
intros b hb,
split,
{ rintro ⟨c, rfl⟩,
rw [ne.def, pow_succ, mul_assoc, mul_eq_zero, decidable.not_or_iff_and_not] at hb,
rw [pow_succ, mul_assoc, factors_mul hb.1 hb.2, repeat_succ, factors_irreducible ha,
singleton_add, cons_le_cons_iff, ← ih hb.2],
apply dvd.intro _ rfl },
{ rw [multiset.le_iff_exists_add],
rintro ⟨u, hu⟩,
rw [← (factors_prod hb).dvd_iff_dvd_right, hu, prod_add, prod_repeat],
exact (associated.pow_pow $ associated_normalize a).dvd.trans (dvd.intro u.prod rfl) }
end
lemma multiplicity_eq_count_factors {a b : R} (ha : irreducible a) (hb : b ≠ 0) :
multiplicity a b = (factors b).count (normalize a) :=
begin
apply le_antisymm,
{ apply enat.le_of_lt_add_one,
rw [← enat.coe_one, ← enat.coe_add, lt_iff_not_ge, ge_iff_le,
le_multiplicity_iff_repeat_le_factors ha hb, ← le_count_iff_repeat_le],
simp },
rw [le_multiplicity_iff_repeat_le_factors ha hb, ← le_count_iff_repeat_le],
end
end multiplicity
end unique_factorization_monoid
namespace associates
open unique_factorization_monoid associated multiset
variables [comm_cancel_monoid_with_zero α]
/-- `factor_set α` representation elements of unique factorization domain as multisets.
`multiset α` produced by `factors` are only unique up to associated elements, while the multisets in
`factor_set α` are unqiue by equality and restricted to irreducible elements. This gives us a
representation of each element as a unique multisets (or the added ⊤ for 0), which has a complete
lattice struture. Infimum is the greatest common divisor and supremum is the least common multiple.
-/
@[reducible] def {u} factor_set (α : Type u) [comm_cancel_monoid_with_zero α] :
Type u :=
with_top (multiset { a : associates α // irreducible a })
local attribute [instance] associated.setoid
theorem factor_set.coe_add {a b : multiset { a : associates α // irreducible a }} :
(↑(a + b) : factor_set α) = a + b :=
by norm_cast
lemma factor_set.sup_add_inf_eq_add [decidable_eq (associates α)] :
∀(a b : factor_set α), a ⊔ b + a ⊓ b = a + b
| none b := show ⊤ ⊔ b + ⊤ ⊓ b = ⊤ + b, by simp
| a none := show a ⊔ ⊤ + a ⊓ ⊤ = a + ⊤, by simp
| (some a) (some b) := show (a : factor_set α) ⊔ b + a ⊓ b = a + b, from
begin
rw [← with_top.coe_sup, ← with_top.coe_inf, ← with_top.coe_add, ← with_top.coe_add,
with_top.coe_eq_coe],
exact multiset.union_add_inter _ _
end
/-- Evaluates the product of a `factor_set` to be the product of the corresponding multiset,
or `0` if there is none. -/
def factor_set.prod : factor_set α → associates α
| none := 0
| (some s) := (s.map coe).prod
@[simp] theorem prod_top : (⊤ : factor_set α).prod = 0 := rfl
@[simp] theorem prod_coe {s : multiset { a : associates α // irreducible a }} :
(s : factor_set α).prod = (s.map coe).prod :=
rfl
@[simp] theorem prod_add : ∀(a b : factor_set α), (a + b).prod = a.prod * b.prod
| none b := show (⊤ + b).prod = (⊤:factor_set α).prod * b.prod, by simp
| a none := show (a + ⊤).prod = a.prod * (⊤:factor_set α).prod, by simp
| (some a) (some b) :=
show (↑a + ↑b:factor_set α).prod = (↑a:factor_set α).prod * (↑b:factor_set α).prod,
by rw [← factor_set.coe_add, prod_coe, prod_coe, prod_coe, multiset.map_add, multiset.prod_add]
theorem prod_mono : ∀{a b : factor_set α}, a ≤ b → a.prod ≤ b.prod
| none b h := have b = ⊤, from top_unique h, by rw [this, prod_top]; exact le_refl _
| a none h := show a.prod ≤ (⊤ : factor_set α).prod, by simp; exact le_top
| (some a) (some b) h := prod_le_prod $ multiset.map_le_map $ with_top.coe_le_coe.1 $ h
/-- `bcount p s` is the multiplicity of `p` in the factor_set `s` (with bundled `p`)-/
def bcount [decidable_eq (associates α)] (p : {a : associates α // irreducible a}) :
factor_set α → ℕ
| none := 0
| (some s) := s.count p
variables [dec_irr : Π (p : associates α), decidable (irreducible p)]
include dec_irr
/-- `count p s` is the multiplicity of the irreducible `p` in the factor_set `s`.
If `p` is not irreducible, `count p s` is defined to be `0`. -/
def count [decidable_eq (associates α)] (p : associates α) :
factor_set α → ℕ :=
if hp : irreducible p then bcount ⟨p, hp⟩ else 0
@[simp] lemma count_some [decidable_eq (associates α)] {p : associates α} (hp : irreducible p)
(s : multiset _) : count p (some s) = s.count ⟨p, hp⟩:=
by { dunfold count, split_ifs, refl }
@[simp] lemma count_zero [decidable_eq (associates α)] {p : associates α} (hp : irreducible p) :
count p (0 : factor_set α) = 0 :=
by { dunfold count, split_ifs, refl }
lemma count_reducible [decidable_eq (associates α)] {p : associates α} (hp : ¬ irreducible p) :
count p = 0 := dif_neg hp
omit dec_irr
/-- membership in a factor_set (bundled version) -/
def bfactor_set_mem : {a : associates α // irreducible a} → (factor_set α) → Prop
| _ ⊤ := true
| p (some l) := p ∈ l
include dec_irr
/-- `factor_set_mem p s` is the predicate that the irreducible `p` is a member of
`s : factor_set α`.
If `p` is not irreducible, `p` is not a member of any `factor_set`. -/
def factor_set_mem (p : associates α) (s : factor_set α) : Prop :=
if hp : irreducible p then bfactor_set_mem ⟨p, hp⟩ s else false
instance : has_mem (associates α) (factor_set α) := ⟨factor_set_mem⟩
@[simp] lemma factor_set_mem_eq_mem (p : associates α) (s : factor_set α) :
factor_set_mem p s = (p ∈ s) := rfl
lemma mem_factor_set_top {p : associates α} {hp : irreducible p} :
p ∈ (⊤ : factor_set α) :=
begin
dunfold has_mem.mem, dunfold factor_set_mem, split_ifs, exact trivial
end
lemma mem_factor_set_some {p : associates α} {hp : irreducible p}
{l : multiset {a : associates α // irreducible a }} :
p ∈ (l : factor_set α) ↔ subtype.mk p hp ∈ l :=
begin
dunfold has_mem.mem, dunfold factor_set_mem, split_ifs, refl
end
lemma reducible_not_mem_factor_set {p : associates α} (hp : ¬ irreducible p)
(s : factor_set α) : ¬ p ∈ s :=
λ (h : if hp : irreducible p then bfactor_set_mem ⟨p, hp⟩ s else false),
by rwa [dif_neg hp] at h
omit dec_irr
variable [unique_factorization_monoid α]
theorem unique' {p q : multiset (associates α)} :
(∀a∈p, irreducible a) → (∀a∈q, irreducible a) → p.prod = q.prod → p = q :=
begin
apply multiset.induction_on_multiset_quot p,
apply multiset.induction_on_multiset_quot q,
assume s t hs ht eq,
refine multiset.map_mk_eq_map_mk_of_rel (unique_factorization_monoid.factors_unique _ _ _),
{ exact assume a ha, ((irreducible_mk _).1 $ hs _ $ multiset.mem_map_of_mem _ ha) },
{ exact assume a ha, ((irreducible_mk _).1 $ ht _ $ multiset.mem_map_of_mem _ ha) },
simpa [quot_mk_eq_mk, prod_mk, mk_eq_mk_iff_associated] using eq
end
variables [nontrivial α] [normalization_monoid α]
private theorem forall_map_mk_factors_irreducible [decidable_eq α] (x : α) (hx : x ≠ 0) :
∀(a : associates α), a ∈ multiset.map associates.mk (factors x) → irreducible a :=
begin
assume a ha,
rcases multiset.mem_map.1 ha with ⟨c, hc, rfl⟩,
exact (irreducible_mk c).2 (irreducible_of_factor _ hc)
end
theorem prod_le_prod_iff_le {p q : multiset (associates α)}
(hp : ∀a∈p, irreducible a) (hq : ∀a∈q, irreducible a) :
p.prod ≤ q.prod ↔ p ≤ q :=
iff.intro
begin
classical,
rintros ⟨⟨c⟩, eqc⟩,
have : c ≠ 0, from (mt mk_eq_zero.2 $
assume (hc : quot.mk setoid.r c = 0),
have (0 : associates α) ∈ q, from prod_eq_zero_iff.1 $ eqc.symm ▸ hc.symm ▸ mul_zero _,
not_irreducible_zero ((irreducible_mk 0).1 $ hq _ this)),
have : associates.mk (factors c).prod = quot.mk setoid.r c,
from mk_eq_mk_iff_associated.2 (factors_prod this),
refine multiset.le_iff_exists_add.2 ⟨(factors c).map associates.mk, unique' hq _ _⟩,
{ assume x hx,
rcases multiset.mem_add.1 hx with h | h,
exact hp x h,
exact forall_map_mk_factors_irreducible c ‹c ≠ 0› _ h },
{ simp [multiset.prod_add, prod_mk, *] at * }
end
prod_le_prod
variables [dec : decidable_eq α] [dec' : decidable_eq (associates α)]
include dec
/-- This returns the multiset of irreducible factors as a `factor_set`,
a multiset of irreducible associates `with_top`. -/
noncomputable def factors' (a : α) :
multiset { a : associates α // irreducible a } :=
(factors a).pmap (λa ha, ⟨associates.mk a, (irreducible_mk _).2 ha⟩)
(irreducible_of_factor)
@[simp] theorem map_subtype_coe_factors' {a : α} :
(factors' a).map coe = (factors a).map associates.mk :=
by simp [factors', multiset.map_pmap, multiset.pmap_eq_map]
theorem factors'_cong {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) (h : a ~ᵤ b) :
factors' a = factors' b :=
have multiset.rel associated (factors a) (factors b), from
factors_unique irreducible_of_factor irreducible_of_factor
((factors_prod ha).trans $ h.trans $ (factors_prod hb).symm),
by simpa [(multiset.map_eq_map subtype.coe_injective).symm, rel_associated_iff_map_eq_map.symm]
include dec'
/-- This returns the multiset of irreducible factors of an associate as a `factor_set`,
a multiset of irreducible associates `with_top`. -/
noncomputable def factors (a : associates α) :
factor_set α :=
begin
refine (if h : a = 0 then ⊤ else
quotient.hrec_on a (λx h, some $ factors' x) _ h),
assume a b hab,
apply function.hfunext,
{ have : a ~ᵤ 0 ↔ b ~ᵤ 0, from
iff.intro (assume ha0, hab.symm.trans ha0) (assume hb0, hab.trans hb0),
simp only [associated_zero_iff_eq_zero] at this,
simp only [quotient_mk_eq_mk, this, mk_eq_zero] },
exact (assume ha hb eq, heq_of_eq $ congr_arg some $ factors'_cong
(λ c, ha (mk_eq_zero.2 c)) (λ c, hb (mk_eq_zero.2 c)) hab)
end
@[simp] theorem factors_0 : (0 : associates α).factors = ⊤ :=
dif_pos rfl
@[simp] theorem factors_mk (a : α) (h : a ≠ 0) :
(associates.mk a).factors = factors' a :=
by { classical, apply dif_neg, apply (mt mk_eq_zero.1 h) }
theorem prod_factors : ∀(s : factor_set α), s.prod.factors = s
| none := by simp [factor_set.prod]; refl
| (some s) :=
begin
unfold factor_set.prod,
generalize eq_a : (s.map coe).prod = a,
rcases a with ⟨a⟩,
rw quot_mk_eq_mk at *,
have : (s.map (coe : _ → associates α)).prod ≠ 0, from assume ha,
let ⟨⟨a, ha⟩, h, eq⟩ := multiset.mem_map.1 (prod_eq_zero_iff.1 ha) in
have irreducible (0 : associates α), from eq ▸ ha,
not_irreducible_zero ((irreducible_mk _).1 this),
have ha : a ≠ 0, by simp [*] at *,
suffices : (unique_factorization_monoid.factors a).map associates.mk = s.map coe,
{ rw [factors_mk a ha],
apply congr_arg some _,
simpa [(multiset.map_eq_map subtype.coe_injective).symm] },
refine unique'
(forall_map_mk_factors_irreducible _ ha)
(assume a ha, let ⟨⟨x, hx⟩, ha, eq⟩ := multiset.mem_map.1 ha in eq ▸ hx)
_,
rw [prod_mk, eq_a, mk_eq_mk_iff_associated],
exact factors_prod ha
end
@[simp]
theorem factors_prod (a : associates α) : a.factors.prod = a :=
quotient.induction_on a $ assume a, decidable.by_cases
(assume : associates.mk a = 0, by simp [quotient_mk_eq_mk, this])
(assume : associates.mk a ≠ 0,
have a ≠ 0, by simp * at *,
by simp [this, quotient_mk_eq_mk, prod_mk, mk_eq_mk_iff_associated.2 (factors_prod this)])
theorem eq_of_factors_eq_factors {a b : associates α} (h : a.factors = b.factors) : a = b :=
have a.factors.prod = b.factors.prod, by rw h,
by rwa [factors_prod, factors_prod] at this
omit dec dec'
theorem eq_of_prod_eq_prod {a b : factor_set α} (h : a.prod = b.prod) : a = b :=
begin
classical,
have : a.prod.factors = b.prod.factors, by rw h,
rwa [prod_factors, prod_factors] at this
end
include dec dec'
@[simp] theorem factors_mul (a b : associates α) : (a * b).factors = a.factors + b.factors :=
eq_of_prod_eq_prod $ eq_of_factors_eq_factors $
by rw [prod_add, factors_prod, factors_prod, factors_prod]
theorem factors_mono : ∀{a b : associates α}, a ≤ b → a.factors ≤ b.factors
| s t ⟨d, rfl⟩ := by rw [factors_mul] ; exact le_add_of_nonneg_right bot_le
theorem factors_le {a b : associates α} : a.factors ≤ b.factors ↔ a ≤ b :=
iff.intro
(assume h, have a.factors.prod ≤ b.factors.prod, from prod_mono h,
by rwa [factors_prod, factors_prod] at this)
factors_mono
omit dec dec'
theorem prod_le {a b : factor_set α} : a.prod ≤ b.prod ↔ a ≤ b :=
begin
classical,
exact iff.intro
(assume h, have a.prod.factors ≤ b.prod.factors, from factors_mono h,
by rwa [prod_factors, prod_factors] at this)
prod_mono
end
include dec dec'
noncomputable instance : has_sup (associates α) := ⟨λa b, (a.factors ⊔ b.factors).prod⟩
noncomputable instance : has_inf (associates α) := ⟨λa b, (a.factors ⊓ b.factors).prod⟩
noncomputable instance : bounded_lattice (associates α) :=
{ sup := (⊔),
inf := (⊓),
sup_le :=
assume a b c hac hbc, factors_prod c ▸ prod_mono (sup_le (factors_mono hac) (factors_mono hbc)),
le_sup_left := assume a b,
le_trans (le_of_eq (factors_prod a).symm) $ prod_mono $ le_sup_left,
le_sup_right := assume a b,
le_trans (le_of_eq (factors_prod b).symm) $ prod_mono $ le_sup_right,
le_inf :=
assume a b c hac hbc, factors_prod a ▸ prod_mono (le_inf (factors_mono hac) (factors_mono hbc)),
inf_le_left := assume a b,
le_trans (prod_mono inf_le_left) (le_of_eq (factors_prod a)),
inf_le_right := assume a b,
le_trans (prod_mono inf_le_right) (le_of_eq (factors_prod b)),
.. associates.partial_order,
.. associates.order_top,
.. associates.order_bot }
lemma sup_mul_inf (a b : associates α) : (a ⊔ b) * (a ⊓ b) = a * b :=
show (a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod = a * b,
begin
refine eq_of_factors_eq_factors _,
rw [← prod_add, prod_factors, factors_mul, factor_set.sup_add_inf_eq_add]
end
include dec_irr
lemma dvd_of_mem_factors {a p : associates α} {hp : irreducible p}
(hm : p ∈ factors a) : p ∣ a :=
begin
by_cases ha0 : a = 0, { rw ha0, exact dvd_zero p },
obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0,
rw [← associates.factors_prod a],
rw [← ha', factors_mk a0 nza] at hm ⊢,
erw prod_coe,
apply multiset.dvd_prod, apply multiset.mem_map.mpr,
exact ⟨⟨p, hp⟩, mem_factor_set_some.mp hm, rfl⟩
end
omit dec'
lemma dvd_of_mem_factors' {a : α} {p : associates α} {hp : irreducible p} {hz : a ≠ 0}
(h_mem : subtype.mk p hp ∈ factors' a) : p ∣ associates.mk a :=
by { haveI := classical.dec_eq (associates α),
apply @dvd_of_mem_factors _ _ _ _ _ _ _ _ _ _ hp,
rw factors_mk _ hz,
apply mem_factor_set_some.2 h_mem }
omit dec_irr
lemma mem_factors'_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : irreducible p) (hd : p ∣ a) :
subtype.mk (associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a :=
begin
obtain ⟨q, hq, hpq⟩ := exists_mem_factors_of_dvd ha0 hp hd,
apply multiset.mem_pmap.mpr, use q, use hq,
exact subtype.eq (eq.symm (mk_eq_mk_iff_associated.mpr hpq))
end
include dec_irr
lemma mem_factors'_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : irreducible p) :
subtype.mk (associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a ↔ p ∣ a :=
begin
split,
{ rw ← mk_dvd_mk, apply dvd_of_mem_factors', apply ha0 },
{ apply mem_factors'_of_dvd ha0 }
end
include dec'
lemma mem_factors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : irreducible p) (hd : p ∣ a) :
(associates.mk p) ∈ factors (associates.mk a) :=
begin
rw factors_mk _ ha0, exact mem_factor_set_some.mpr (mem_factors'_of_dvd ha0 hp hd)
end
lemma mem_factors_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : irreducible p) :
(associates.mk p) ∈ factors (associates.mk a) ↔ p ∣ a :=
begin
split,
{ rw ← mk_dvd_mk, apply dvd_of_mem_factors, exact (irreducible_mk p).mpr hp },
{ apply mem_factors_of_dvd ha0 hp }
end
lemma exists_prime_dvd_of_not_inf_one {a b : α}
(ha : a ≠ 0) (hb : b ≠ 0) (h : (associates.mk a) ⊓ (associates.mk b) ≠ 1) :
∃ (p : α), prime p ∧ p ∣ a ∧ p ∣ b :=
begin
have hz : (factors (associates.mk a)) ⊓ (factors (associates.mk b)) ≠ 0,
{ contrapose! h with hf,
change ((factors (associates.mk a)) ⊓ (factors (associates.mk b))).prod = 1,
rw hf,
exact multiset.prod_zero },
rw [factors_mk a ha, factors_mk b hb, ← with_top.coe_inf] at hz,
obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := multiset.exists_mem_of_ne_zero ((mt with_top.coe_eq_coe.mpr) hz),
rw multiset.inf_eq_inter at p0_mem,
obtain ⟨p, rfl⟩ : ∃ p, associates.mk p = p0 := quot.exists_rep p0,
refine ⟨p, _, _, _⟩,
{ rw [← irreducible_iff_prime, ← irreducible_mk],
exact p0_irr },
{ apply dvd_of_mk_le_mk,
apply dvd_of_mem_factors' (multiset.mem_inter.mp p0_mem).left,
apply ha, },
{ apply dvd_of_mk_le_mk,
apply dvd_of_mem_factors' (multiset.mem_inter.mp p0_mem).right,
apply hb }
end
theorem coprime_iff_inf_one {a b : α} (ha0 : a ≠ 0) (hb0 : b ≠ 0) :
(associates.mk a) ⊓ (associates.mk b) = 1 ↔ ∀ {d : α}, d ∣ a → d ∣ b → ¬ prime d :=
begin
split,
{ intros hg p ha hb hp,
refine ((associates.prime_mk _).mpr hp).not_unit (is_unit_of_dvd_one _ _),
rw ← hg,
exact le_inf (mk_le_mk_of_dvd ha) (mk_le_mk_of_dvd hb) },
{ contrapose,
intros hg hc,
obtain ⟨p, hp, hpa, hpb⟩ := exists_prime_dvd_of_not_inf_one ha0 hb0 hg,
exact hc hpa hpb hp }
end
omit dec_irr
theorem factors_prime_pow {p : associates α} (hp : irreducible p)
(k : ℕ) : factors (p ^ k) = some (multiset.repeat ⟨p, hp⟩ k) :=
eq_of_prod_eq_prod (by rw [associates.factors_prod, factor_set.prod, multiset.map_repeat,
multiset.prod_repeat, subtype.coe_mk])
include dec_irr
theorem prime_pow_dvd_iff_le {m p : associates α} (h₁ : m ≠ 0)
(h₂ : irreducible p) {k : ℕ} : p ^ k ≤ m ↔ k ≤ count p m.factors :=
begin
obtain ⟨a, nz, rfl⟩ := associates.exists_non_zero_rep h₁,
rw [factors_mk _ nz, ← with_top.some_eq_coe, count_some, multiset.le_count_iff_repeat_le,
← factors_le, factors_prime_pow h₂, factors_mk _ nz],
exact with_top.coe_le_coe
end
theorem le_of_count_ne_zero {m p : associates α} (h0 : m ≠ 0)
(hp : irreducible p) : count p m.factors ≠ 0 → p ≤ m :=
begin
rw [← pos_iff_ne_zero],
intro h,
rw [← pow_one p],
apply (prime_pow_dvd_iff_le h0 hp).2,
simpa only
end
theorem count_mul {a : associates α} (ha : a ≠ 0) {b : associates α} (hb : b ≠ 0)
{p : associates α} (hp : irreducible p) :
count p (factors (a * b)) = count p a.factors + count p b.factors :=
begin
obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha,
obtain ⟨b0, nzb, hb'⟩ := exists_non_zero_rep hb,
rw [factors_mul, ← ha', ← hb', factors_mk a0 nza, factors_mk b0 nzb, ← factor_set.coe_add,
← with_top.some_eq_coe, ← with_top.some_eq_coe, ← with_top.some_eq_coe, count_some hp,
multiset.count_add, count_some hp, count_some hp]
end
theorem count_of_coprime {a : associates α} (ha : a ≠ 0) {b : associates α} (hb : b ≠ 0)
(hab : ∀ d, d ∣ a → d ∣ b → ¬ prime d) {p : associates α} (hp : irreducible p) :
count p a.factors = 0 ∨ count p b.factors = 0 :=
begin
rw [or_iff_not_imp_left, ← ne.def],
intro hca,
contrapose! hab with hcb,
exact ⟨p, le_of_count_ne_zero ha hp hca, le_of_count_ne_zero hb hp hcb,
(irreducible_iff_prime.mp hp)⟩,
end
theorem count_mul_of_coprime {a : associates α} (ha : a ≠ 0) {b : associates α} (hb : b ≠ 0)
{p : associates α} (hp : irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬ prime d) :
count p a.factors = 0 ∨ count p a.factors = count p (a * b).factors :=
begin
cases count_of_coprime ha hb hab hp with hz hb0, { tauto },
apply or.intro_right,
rw [count_mul ha hb hp, hb0, add_zero]
end
theorem count_mul_of_coprime' {a : associates α} (ha : a ≠ 0) {b : associates α} (hb : b ≠ 0)
{p : associates α} (hp : irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬ prime d) :
count p (a * b).factors = count p a.factors
∨ count p (a * b).factors = count p b.factors :=
begin
rw [count_mul ha hb hp],
cases count_of_coprime ha hb hab hp with ha0 hb0,
{ apply or.intro_right, rw [ha0, zero_add] },
{ apply or.intro_left, rw [hb0, add_zero] }
end
theorem dvd_count_of_dvd_count_mul {a b : associates α} (ha : a ≠ 0) (hb : b ≠ 0)
{p : associates α} (hp : irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬ prime d)
{k : ℕ} (habk : k ∣ count p (a * b).factors) : k ∣ count p a.factors :=
begin
cases count_of_coprime ha hb hab hp with hz h,
{ rw hz, exact dvd_zero k },
{ rw [count_mul ha hb hp, h] at habk, exact habk }
end
omit dec_irr
@[simp] lemma factors_one : factors (1 : associates α) = 0 :=
begin
apply eq_of_prod_eq_prod,
rw associates.factors_prod,
exact multiset.prod_zero,
end
@[simp] theorem pow_factors {a : associates α} {k : ℕ} : (a ^ k).factors = k • a.factors :=
begin
induction k with n h,
{ rw [zero_nsmul, pow_zero], exact factors_one },
{ rw [pow_succ, succ_nsmul, factors_mul, h] }
end
include dec_irr
lemma count_pow {a : associates α} (ha : a ≠ 0) {p : associates α} (hp : irreducible p)
(k : ℕ) : count p (a ^ k).factors = k * count p a.factors :=
begin
induction k with n h,
{ rw [pow_zero, factors_one, zero_mul, count_zero hp] },
{ rw [pow_succ, count_mul ha (pow_ne_zero _ ha) hp, h, nat.succ_eq_add_one], ring }
end
theorem dvd_count_pow {a : associates α} (ha : a ≠ 0) {p : associates α} (hp : irreducible p)
(k : ℕ) : k ∣ count p (a ^ k).factors := by { rw count_pow ha hp, apply dvd_mul_right }
theorem is_pow_of_dvd_count {a : associates α} (ha : a ≠ 0) {k : ℕ}
(hk : ∀ (p : associates α) (hp : irreducible p), k ∣ count p a.factors) :
∃ (b : associates α), a = b ^ k :=
begin
obtain ⟨a0, hz, rfl⟩ := exists_non_zero_rep ha,
rw [factors_mk a0 hz] at hk,
have hk' : ∀ p, p ∈ (factors' a0) → k ∣ (factors' a0).count p,
{ rintros p -,
have pp : p = ⟨p.val, p.2⟩, { simp only [subtype.coe_eta, subtype.val_eq_coe] },
rw [pp, ← count_some p.2], exact hk p.val p.2 },
obtain ⟨u, hu⟩ := multiset.exists_smul_of_dvd_count _ hk',
use (u : factor_set α).prod,
apply eq_of_factors_eq_factors,
rw [pow_factors, prod_factors, factors_mk a0 hz, ← with_top.some_eq_coe, hu],
exact with_bot.coe_nsmul u k
end
omit dec
omit dec_irr
omit dec'
theorem eq_pow_of_mul_eq_pow {a b c : associates α} (ha : a ≠ 0) (hb : b ≠ 0)
(hab : ∀ d, d ∣ a → d ∣ b → ¬ prime d) {k : ℕ} (h : a * b = c ^ k) :
∃ (d : associates α), a = d ^ k :=
begin
classical,
by_cases hk0 : k = 0,
{ use 1,
rw [hk0, pow_zero] at h ⊢,
apply (mul_eq_one_iff.1 h).1 },
{ refine is_pow_of_dvd_count ha _,
intros p hp,
apply dvd_count_of_dvd_count_mul ha hb hp hab,
rw h,
apply dvd_count_pow _ hp,
rintros rfl,
rw zero_pow' _ hk0 at h,
cases mul_eq_zero.mp h; contradiction }
end
end associates
section
open associates unique_factorization_monoid
/-- `to_gcd_monoid` constructs a GCD monoid out of a normalization on a
unique factorization domain. -/
noncomputable def unique_factorization_monoid.to_gcd_monoid
(α : Type*) [comm_cancel_monoid_with_zero α] [nontrivial α] [unique_factorization_monoid α]
[normalization_monoid α] [decidable_eq (associates α)] [decidable_eq α] : gcd_monoid α :=
{ gcd := λa b, (associates.mk a ⊓ associates.mk b).out,
lcm := λa b, (associates.mk a ⊔ associates.mk b).out,
gcd_dvd_left := assume a b, (out_dvd_iff a (associates.mk a ⊓ associates.mk b)).2 $ inf_le_left,
gcd_dvd_right := assume a b, (out_dvd_iff b (associates.mk a ⊓ associates.mk b)).2 $ inf_le_right,
dvd_gcd := assume a b c hac hab, show a ∣ (associates.mk c ⊓ associates.mk b).out,
by rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff]; exact ⟨hac, hab⟩,
lcm_zero_left := assume a, show (⊤ ⊔ associates.mk a).out = 0, by simp,
lcm_zero_right := assume a, show (associates.mk a ⊔ ⊤).out = 0, by simp,
gcd_mul_lcm := assume a b,
show (associates.mk a ⊓ associates.mk b).out * (associates.mk a ⊔ associates.mk b).out =
normalize (a * b),
by rw [← out_mk, ← out_mul, mul_comm, sup_mul_inf]; refl,
normalize_gcd := assume a b, by convert normalize_out _,
.. ‹normalization_monoid α› }
end
namespace unique_factorization_monoid
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintype_subtype_dvd {M : Type*} [comm_cancel_monoid_with_zero M]
[unique_factorization_monoid M] [fintype (units M)]
(y : M) (hy : y ≠ 0) :
fintype {x // x ∣ y} :=
begin
haveI : nontrivial M := ⟨⟨y, 0, hy⟩⟩,
haveI : normalization_monoid M := unique_factorization_monoid.normalization_monoid,
haveI := classical.dec_eq M,
haveI := classical.dec_eq (associates M),
-- We'll show `λ (u : units M) (f ⊆ factors y) → u * Π f` is injective
-- and has image exactly the divisors of `y`.
refine fintype.of_finset
(((factors y).powerset.to_finset.product (finset.univ : finset (units M))).image
(λ s, (s.snd : M) * s.fst.prod))
(λ x, _),
simp only [exists_prop, finset.mem_image, finset.mem_product, finset.mem_univ, and_true,
multiset.mem_to_finset, multiset.mem_powerset, exists_eq_right, multiset.mem_map],
split,
{ rintros ⟨s, hs, rfl⟩,
have prod_s_ne : s.fst.prod ≠ 0,
{ intro hz,
apply hy (eq_zero_of_zero_dvd _),
have hz := (@multiset.prod_eq_zero_iff M _ _ _ s.fst).mp hz,
rw ← (factors_prod hy).dvd_iff_dvd_right,
exact multiset.dvd_prod (multiset.mem_of_le hs hz) },
show (s.snd : M) * s.fst.prod ∣ y,
rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
← (factors_prod hy).dvd_iff_dvd_right],
exact multiset.prod_dvd_prod hs },
{ rintro (h : x ∣ y),
have hx : x ≠ 0, { refine mt (λ hx, _) hy, rwa [hx, zero_dvd_iff] at h },
obtain ⟨u, hu⟩ := factors_prod hx,
refine ⟨⟨factors x, u⟩, _, (mul_comm _ _).trans hu⟩,
exact (dvd_iff_factors_le_factors hx hy).mp h }
end
end unique_factorization_monoid
|
0ea1d0c84ed801dabb001bf656f2d5771e69b638 | 94e33a31faa76775069b071adea97e86e218a8ee | /src/data/finsupp/multiset.lean | 7b000b7ca7a2c3c62bf0eae5e4585409a64eb474 | [
"Apache-2.0"
] | permissive | urkud/mathlib | eab80095e1b9f1513bfb7f25b4fa82fa4fd02989 | 6379d39e6b5b279df9715f8011369a301b634e41 | refs/heads/master | 1,658,425,342,662 | 1,658,078,703,000 | 1,658,078,703,000 | 186,910,338 | 0 | 0 | Apache-2.0 | 1,568,512,083,000 | 1,557,958,709,000 | Lean | UTF-8 | Lean | false | false | 7,260 | lean | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import data.finsupp.order
/-!
# Equivalence between `multiset` and `ℕ`-valued finitely supported functions
This defines `finsupp.to_multiset` the equivalence between `α →₀ ℕ` and `multiset α`, along
with `multiset.to_finsupp` the reverse equivalence and `finsupp.order_iso_multiset` the equivalence
promoted to an order isomorphism.
-/
open finset
open_locale big_operators classical
noncomputable theory
variables {α β ι : Type*}
namespace finsupp
/-- Given `f : α →₀ ℕ`, `f.to_multiset` is the multiset with multiplicities given by the values of
`f` on the elements of `α`. We define this function as an `add_equiv`. -/
def to_multiset : (α →₀ ℕ) ≃+ multiset α :=
{ to_fun := λ f, f.sum (λa n, n • {a}),
inv_fun := λ s, ⟨s.to_finset, λ a, s.count a, λ a, by simp⟩,
left_inv := λ f, ext $ λ a, by
{ simp only [sum, multiset.count_sum', multiset.count_singleton, mul_boole, coe_mk,
mem_support_iff, multiset.count_nsmul, finset.sum_ite_eq, ite_not, ite_eq_right_iff],
exact eq.symm },
right_inv := λ s, by simp only [sum, coe_mk, multiset.to_finset_sum_count_nsmul_eq],
map_add' := λ f g, sum_add_index' (λ a, zero_nsmul _) (λ a, add_nsmul _) }
lemma to_multiset_zero : (0 : α →₀ ℕ).to_multiset = 0 := rfl
lemma to_multiset_add (m n : α →₀ ℕ) : (m + n).to_multiset = m.to_multiset + n.to_multiset :=
to_multiset.map_add m n
lemma to_multiset_apply (f : α →₀ ℕ) : f.to_multiset = f.sum (λ a n, n • {a}) := rfl
@[simp]
lemma to_multiset_symm_apply [decidable_eq α] (s : multiset α) (x : α) :
finsupp.to_multiset.symm s x = s.count x :=
by convert rfl
@[simp] lemma to_multiset_single (a : α) (n : ℕ) : to_multiset (single a n) = n • {a} :=
by rw [to_multiset_apply, sum_single_index]; apply zero_nsmul
lemma to_multiset_sum {f : ι → α →₀ ℕ} (s : finset ι) :
finsupp.to_multiset (∑ i in s, f i) = ∑ i in s, finsupp.to_multiset (f i) :=
add_equiv.map_sum _ _ _
lemma to_multiset_sum_single (s : finset ι) (n : ℕ) :
finsupp.to_multiset (∑ i in s, single i n) = n • s.val :=
by simp_rw [to_multiset_sum, finsupp.to_multiset_single, sum_nsmul, sum_multiset_singleton]
lemma card_to_multiset (f : α →₀ ℕ) : f.to_multiset.card = f.sum (λ a, id) :=
by simp [to_multiset_apply, add_monoid_hom.map_finsupp_sum, function.id_def]
lemma to_multiset_map (f : α →₀ ℕ) (g : α → β) :
f.to_multiset.map g = (f.map_domain g).to_multiset :=
begin
refine f.induction _ _,
{ rw [to_multiset_zero, multiset.map_zero, map_domain_zero, to_multiset_zero] },
{ assume a n f _ _ ih,
rw [to_multiset_add, multiset.map_add, ih, map_domain_add, map_domain_single,
to_multiset_single, to_multiset_add, to_multiset_single,
← multiset.coe_map_add_monoid_hom, (multiset.map_add_monoid_hom g).map_nsmul],
refl }
end
@[simp] lemma prod_to_multiset [comm_monoid α] (f : α →₀ ℕ) :
f.to_multiset.prod = f.prod (λ a n, a ^ n) :=
begin
refine f.induction _ _,
{ rw [to_multiset_zero, multiset.prod_zero, finsupp.prod_zero_index] },
{ assume a n f _ _ ih,
rw [to_multiset_add, multiset.prod_add, ih, to_multiset_single, multiset.prod_nsmul,
finsupp.prod_add_index' pow_zero pow_add, finsupp.prod_single_index, multiset.prod_singleton],
{ exact pow_zero a } }
end
@[simp] lemma to_finset_to_multiset [decidable_eq α] (f : α →₀ ℕ) :
f.to_multiset.to_finset = f.support :=
begin
refine f.induction _ _,
{ rw [to_multiset_zero, multiset.to_finset_zero, support_zero] },
{ assume a n f ha hn ih,
rw [to_multiset_add, multiset.to_finset_add, ih, to_multiset_single, support_add_eq,
support_single_ne_zero _ hn, multiset.to_finset_nsmul _ _ hn, multiset.to_finset_singleton],
refine disjoint.mono_left support_single_subset _,
rwa [finset.disjoint_singleton_left] }
end
@[simp] lemma count_to_multiset [decidable_eq α] (f : α →₀ ℕ) (a : α) :
f.to_multiset.count a = f a :=
calc f.to_multiset.count a = f.sum (λx n, (n • {x} : multiset α).count a) :
(multiset.count_add_monoid_hom a).map_sum _ f.support
... = f.sum (λx n, n * ({x} : multiset α).count a) : by simp only [multiset.count_nsmul]
... = f a * ({a} : multiset α).count a : sum_eq_single _
(λ a' _ H, by simp only [multiset.count_singleton, if_false, H.symm, mul_zero])
(λ H, by simp only [not_mem_support_iff.1 H, zero_mul])
... = f a : by rw [multiset.count_singleton_self, mul_one]
@[simp] lemma mem_to_multiset (f : α →₀ ℕ) (i : α) : i ∈ f.to_multiset ↔ i ∈ f.support :=
by rw [←multiset.count_ne_zero, finsupp.count_to_multiset, finsupp.mem_support_iff]
end finsupp
namespace multiset
/-- Given a multiset `s`, `s.to_finsupp` returns the finitely supported function on `ℕ` given by
the multiplicities of the elements of `s`. -/
def to_finsupp : multiset α ≃+ (α →₀ ℕ) := finsupp.to_multiset.symm
@[simp] lemma to_finsupp_support [decidable_eq α] (s : multiset α) :
s.to_finsupp.support = s.to_finset :=
by convert rfl
@[simp] lemma to_finsupp_apply [decidable_eq α] (s : multiset α) (a : α) :
to_finsupp s a = s.count a :=
by convert rfl
lemma to_finsupp_zero : to_finsupp (0 : multiset α) = 0 := add_equiv.map_zero _
lemma to_finsupp_add (s t : multiset α) : to_finsupp (s + t) = to_finsupp s + to_finsupp t :=
to_finsupp.map_add s t
@[simp] lemma to_finsupp_singleton (a : α) : to_finsupp ({a} : multiset α) = finsupp.single a 1 :=
finsupp.to_multiset.symm_apply_eq.2 $ by simp
@[simp] lemma to_finsupp_to_multiset (s : multiset α) : s.to_finsupp.to_multiset = s :=
finsupp.to_multiset.apply_symm_apply s
lemma to_finsupp_eq_iff {s : multiset α} {f : α →₀ ℕ} : s.to_finsupp = f ↔ s = f.to_multiset :=
finsupp.to_multiset.symm_apply_eq
end multiset
@[simp] lemma finsupp.to_multiset_to_finsupp (f : α →₀ ℕ) : f.to_multiset.to_finsupp = f :=
finsupp.to_multiset.symm_apply_apply f
/-! ### As an order isomorphism -/
namespace finsupp
/-- `finsupp.to_multiset` as an order isomorphism. -/
def order_iso_multiset : (ι →₀ ℕ) ≃o multiset ι :=
{ to_equiv := to_multiset.to_equiv,
map_rel_iff' := λ f g, by simp [multiset.le_iff_count, le_def] }
@[simp] lemma coe_order_iso_multiset : ⇑(@order_iso_multiset ι) = to_multiset := rfl
@[simp] lemma coe_order_iso_multiset_symm : ⇑(@order_iso_multiset ι).symm = multiset.to_finsupp :=
rfl
lemma to_multiset_strict_mono : strict_mono (@to_multiset ι) := (@order_iso_multiset ι).strict_mono
lemma sum_id_lt_of_lt (m n : ι →₀ ℕ) (h : m < n) : m.sum (λ _, id) < n.sum (λ _, id) :=
begin
rw [←card_to_multiset, ←card_to_multiset],
apply multiset.card_lt_of_lt,
exact to_multiset_strict_mono h
end
variable (ι)
/-- The order on `ι →₀ ℕ` is well-founded. -/
lemma lt_wf : well_founded (@has_lt.lt (ι →₀ ℕ) _) :=
subrelation.wf (sum_id_lt_of_lt) $ inv_image.wf _ nat.lt_wf
end finsupp
lemma multiset.to_finsupp_strict_mono : strict_mono (@multiset.to_finsupp ι) :=
(@finsupp.order_iso_multiset ι).symm.strict_mono
|
b278a1c5e72620940c6509148ab8718502be6b56 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/number_theory/class_number/function_field.lean | b56a75a38d58a94003ecb7012d6b2d7010dd9b75 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 1,748 | lean | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import number_theory.class_number.admissible_card_pow_degree
import number_theory.class_number.finite
import number_theory.function_field
/-!
# Class numbers of function fields
This file defines the class number of a function field as the (finite) cardinality of
the class group of its ring of integers. It also proves some elementary results
on the class number.
## Main definitions
- `function_field.class_number`: the class number of a function field is the (finite)
cardinality of the class group of its ring of integers
-/
namespace function_field
open_locale polynomial
variables (Fq F : Type) [field Fq] [fintype Fq] [field F]
variables [algebra Fq[X] F] [algebra (ratfunc Fq) F]
variables [is_scalar_tower Fq[X] (ratfunc Fq) F]
variables [function_field Fq F] [is_separable (ratfunc Fq) F]
open_locale classical
namespace ring_of_integers
open function_field
noncomputable instance : fintype (class_group (ring_of_integers Fq F)) :=
class_group.fintype_of_admissible_of_finite (ratfunc Fq) F
(polynomial.card_pow_degree_is_admissible : absolute_value.is_admissible
(polynomial.card_pow_degree : absolute_value Fq[X] ℤ))
end ring_of_integers
/-- The class number in a function field is the (finite) cardinality of the class group. -/
noncomputable def class_number : ℕ := fintype.card (class_group (ring_of_integers Fq F))
/-- The class number of a function field is `1` iff the ring of integers is a PID. -/
theorem class_number_eq_one_iff :
class_number Fq F = 1 ↔ is_principal_ideal_ring (ring_of_integers Fq F) :=
card_class_group_eq_one_iff
end function_field
|
4972fcccac0aa58a57595d0be92a5ca9026012dc | bbecf0f1968d1fba4124103e4f6b55251d08e9c4 | /src/measure_theory/constructions/prod.lean | 6dea8adee06a5b09b8faf10d2cc480600f194cb3 | [
"Apache-2.0"
] | permissive | waynemunro/mathlib | e3fd4ff49f4cb43d4a8ded59d17be407bc5ee552 | 065a70810b5480d584033f7bbf8e0409480c2118 | refs/heads/master | 1,693,417,182,397 | 1,634,644,781,000 | 1,634,644,781,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 47,115 | lean | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import measure_theory.measure.giry_monad
import measure_theory.integral.set_integral
/-!
# The product measure
In this file we define and prove properties about the binary product measure. If `α` and `β` have
σ-finite measures `μ` resp. `ν` then `α × β` can be equipped with a σ-finite measure `μ.prod ν` that
satisfies `(μ.prod ν) s = ∫⁻ x, ν {y | (x, y) ∈ s} ∂μ`.
We also have `(μ.prod ν) (s.prod t) = μ s * ν t`, i.e. the measure of a rectangle is the product of
the measures of the sides.
We also prove Tonelli's theorem and Fubini's theorem.
## Main definition
* `measure_theory.measure.prod`: The product of two measures.
## Main results
* `measure_theory.measure.prod_apply` states `μ.prod ν s = ∫⁻ x, ν {y | (x, y) ∈ s} ∂μ`
for measurable `s`. `measure_theory.measure.prod_apply_symm` is the reversed version.
* `measure_theory.measure.prod_prod` states `μ.prod ν (s.prod t) = μ s * ν t` for measurable sets
`s` and `t`.
* `measure_theory.lintegral_prod`: Tonelli's theorem. It states that for a measurable function
`α × β → ℝ≥0∞` we have `∫⁻ z, f z ∂(μ.prod ν) = ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ`. The version
for functions `α → β → ℝ≥0∞` is reversed, and called `lintegral_lintegral`. Both versions have
a variant with `_symm` appended, where the order of integration is reversed.
The lemma `measurable.lintegral_prod_right'` states that the inner integral of the right-hand side
is measurable.
* `measure_theory.integrable_prod_iff` states that a binary function is integrable iff both
* `y ↦ f (x, y)` is integrable for almost every `x`, and
* the function `x ↦ ∫ ∥f (x, y)∥ dy` is integrable.
* `measure_theory.integral_prod`: Fubini's theorem. It states that for a integrable function
`α × β → E` (where `E` is a second countable Banach space) we have
`∫ z, f z ∂(μ.prod ν) = ∫ x, ∫ y, f (x, y) ∂ν ∂μ`. This theorem has the same variants as
Tonelli's theorem. The lemma `measure_theory.integrable.integral_prod_right` states that the
inner integral of the right-hand side is integrable.
## Implementation Notes
Many results are proven twice, once for functions in curried form (`α → β → γ`) and one for
functions in uncurried form (`α × β → γ`). The former often has an assumption
`measurable (uncurry f)`, which could be inconvenient to discharge, but for the latter it is more
common that the function has to be given explicitly, since Lean cannot synthesize the function by
itself. We name the lemmas about the uncurried form with a prime.
Tonelli's theorem and Fubini's theorem have a different naming scheme, since the version for the
uncurried version is reversed.
## Tags
product measure, Fubini's theorem, Tonelli's theorem, Fubini-Tonelli theorem
-/
noncomputable theory
open_locale classical topological_space ennreal measure_theory
open set function real ennreal
open measure_theory measurable_space measure_theory.measure
open topological_space (hiding generate_from)
open filter (hiding prod_eq map)
variables {α α' β β' γ E : Type*}
/-- Rectangles formed by π-systems form a π-system. -/
lemma is_pi_system.prod {C : set (set α)} {D : set (set β)} (hC : is_pi_system C)
(hD : is_pi_system D) : is_pi_system (image2 set.prod C D) :=
begin
rintro _ _ ⟨s₁, t₁, hs₁, ht₁, rfl⟩ ⟨s₂, t₂, hs₂, ht₂, rfl⟩ hst,
rw [prod_inter_prod] at hst ⊢, rw [prod_nonempty_iff] at hst,
exact mem_image2_of_mem (hC _ _ hs₁ hs₂ hst.1) (hD _ _ ht₁ ht₂ hst.2)
end
/-- Rectangles of countably spanning sets are countably spanning. -/
lemma is_countably_spanning.prod {C : set (set α)} {D : set (set β)}
(hC : is_countably_spanning C) (hD : is_countably_spanning D) :
is_countably_spanning (image2 set.prod C D) :=
begin
rcases ⟨hC, hD⟩ with ⟨⟨s, h1s, h2s⟩, t, h1t, h2t⟩,
refine ⟨λ n, (s n.unpair.1).prod (t n.unpair.2), λ n, mem_image2_of_mem (h1s _) (h1t _), _⟩,
rw [Union_unpair_prod, h2s, h2t, univ_prod_univ]
end
variables [measurable_space α] [measurable_space α'] [measurable_space β] [measurable_space β']
variables [measurable_space γ]
variables {μ : measure α} {ν : measure β} {τ : measure γ}
variables [normed_group E] [measurable_space E]
/-! ### Measurability
Before we define the product measure, we can talk about the measurability of operations on binary
functions. We show that if `f` is a binary measurable function, then the function that integrates
along one of the variables (using either the Lebesgue or Bochner integral) is measurable.
-/
/-- The product of generated σ-algebras is the one generated by rectangles, if both generating sets
are countably spanning. -/
lemma generate_from_prod_eq {α β} {C : set (set α)} {D : set (set β)}
(hC : is_countably_spanning C) (hD : is_countably_spanning D) :
@prod.measurable_space _ _ (generate_from C) (generate_from D) =
generate_from (image2 set.prod C D) :=
begin
apply le_antisymm,
{ refine sup_le _ _; rw [comap_generate_from];
apply generate_from_le; rintro _ ⟨s, hs, rfl⟩,
{ rcases hD with ⟨t, h1t, h2t⟩,
rw [← prod_univ, ← h2t, prod_Union],
apply measurable_set.Union,
intro n, apply measurable_set_generate_from,
exact ⟨s, t n, hs, h1t n, rfl⟩ },
{ rcases hC with ⟨t, h1t, h2t⟩,
rw [← univ_prod, ← h2t, Union_prod_const],
apply measurable_set.Union,
rintro n, apply measurable_set_generate_from,
exact mem_image2_of_mem (h1t n) hs } },
{ apply generate_from_le, rintro _ ⟨s, t, hs, ht, rfl⟩, rw [prod_eq],
apply (measurable_fst _).inter (measurable_snd _),
{ exact measurable_set_generate_from hs },
{ exact measurable_set_generate_from ht } }
end
/-- If `C` and `D` generate the σ-algebras on `α` resp. `β`, then rectangles formed by `C` and `D`
generate the σ-algebra on `α × β`. -/
lemma generate_from_eq_prod {C : set (set α)} {D : set (set β)} (hC : generate_from C = ‹_›)
(hD : generate_from D = ‹_›) (h2C : is_countably_spanning C) (h2D : is_countably_spanning D) :
generate_from (image2 set.prod C D) = prod.measurable_space :=
by rw [← hC, ← hD, generate_from_prod_eq h2C h2D]
/-- The product σ-algebra is generated from boxes, i.e. `s.prod t` for sets `s : set α` and
`t : set β`. -/
lemma generate_from_prod :
generate_from (image2 set.prod {s : set α | measurable_set s} {t : set β | measurable_set t}) =
prod.measurable_space :=
generate_from_eq_prod generate_from_measurable_set generate_from_measurable_set
is_countably_spanning_measurable_set is_countably_spanning_measurable_set
/-- Rectangles form a π-system. -/
lemma is_pi_system_prod :
is_pi_system (image2 set.prod {s : set α | measurable_set s} {t : set β | measurable_set t}) :=
is_pi_system_measurable_set.prod is_pi_system_measurable_set
/-- If `ν` is a finite measure, and `s ⊆ α × β` is measurable, then `x ↦ ν { y | (x, y) ∈ s }` is
a measurable function. `measurable_measure_prod_mk_left` is strictly more general. -/
lemma measurable_measure_prod_mk_left_finite [is_finite_measure ν] {s : set (α × β)}
(hs : measurable_set s) : measurable (λ x, ν (prod.mk x ⁻¹' s)) :=
begin
refine induction_on_inter generate_from_prod.symm is_pi_system_prod _ _ _ _ hs,
{ simp [measurable_zero, const_def] },
{ rintro _ ⟨s, t, hs, ht, rfl⟩, simp only [mk_preimage_prod_right_eq_if, measure_if],
exact measurable_const.indicator hs },
{ intros t ht h2t,
simp_rw [preimage_compl, measure_compl (measurable_prod_mk_left ht) (measure_ne_top ν _)],
exact h2t.const_sub _ },
{ intros f h1f h2f h3f, simp_rw [preimage_Union],
have : ∀ b, ν (⋃ i, prod.mk b ⁻¹' f i) = ∑' i, ν (prod.mk b ⁻¹' f i) :=
λ b, measure_Union (λ i j hij, disjoint.preimage _ (h1f i j hij))
(λ i, measurable_prod_mk_left (h2f i)),
simp_rw [this], apply measurable.ennreal_tsum h3f },
end
/-- If `ν` is a σ-finite measure, and `s ⊆ α × β` is measurable, then `x ↦ ν { y | (x, y) ∈ s }` is
a measurable function. -/
lemma measurable_measure_prod_mk_left [sigma_finite ν] {s : set (α × β)}
(hs : measurable_set s) : measurable (λ x, ν (prod.mk x ⁻¹' s)) :=
begin
have : ∀ x, measurable_set (prod.mk x ⁻¹' s) := λ x, measurable_prod_mk_left hs,
simp only [← @supr_restrict_spanning_sets _ _ ν, this],
apply measurable_supr, intro i,
haveI := fact.mk (measure_spanning_sets_lt_top ν i),
exact measurable_measure_prod_mk_left_finite hs
end
/-- If `μ` is a σ-finite measure, and `s ⊆ α × β` is measurable, then `y ↦ μ { x | (x, y) ∈ s }` is
a measurable function. -/
lemma measurable_measure_prod_mk_right {μ : measure α} [sigma_finite μ] {s : set (α × β)}
(hs : measurable_set s) : measurable (λ y, μ ((λ x, (x, y)) ⁻¹' s)) :=
measurable_measure_prod_mk_left (measurable_set_swap_iff.mpr hs)
lemma measurable.map_prod_mk_left [sigma_finite ν] : measurable (λ x : α, map (prod.mk x) ν) :=
begin
apply measurable_of_measurable_coe, intros s hs,
simp_rw [map_apply measurable_prod_mk_left hs],
exact measurable_measure_prod_mk_left hs
end
lemma measurable.map_prod_mk_right {μ : measure α} [sigma_finite μ] :
measurable (λ y : β, map (λ x : α, (x, y)) μ) :=
begin
apply measurable_of_measurable_coe, intros s hs,
simp_rw [map_apply measurable_prod_mk_right hs],
exact measurable_measure_prod_mk_right hs
end
/-- The Lebesgue integral is measurable. This shows that the integrand of (the right-hand-side of)
Tonelli's theorem is measurable. -/
lemma measurable.lintegral_prod_right' [sigma_finite ν] :
∀ {f : α × β → ℝ≥0∞} (hf : measurable f), measurable (λ x, ∫⁻ y, f (x, y) ∂ν) :=
begin
have m := @measurable_prod_mk_left,
refine measurable.ennreal_induction _ _ _,
{ intros c s hs, simp only [← indicator_comp_right],
suffices : measurable (λ x, c * ν (prod.mk x ⁻¹' s)),
{ simpa [lintegral_indicator _ (m hs)] },
exact (measurable_measure_prod_mk_left hs).const_mul _ },
{ rintro f g - hf hg h2f h2g, simp_rw [pi.add_apply, lintegral_add (hf.comp m) (hg.comp m)],
exact h2f.add h2g },
{ intros f hf h2f h3f,
have := measurable_supr h3f,
have : ∀ x, monotone (λ n y, f n (x, y)) := λ x i j hij y, h2f hij (x, y),
simpa [lintegral_supr (λ n, (hf n).comp m), this] }
end
/-- The Lebesgue integral is measurable. This shows that the integrand of (the right-hand-side of)
Tonelli's theorem is measurable.
This version has the argument `f` in curried form. -/
lemma measurable.lintegral_prod_right [sigma_finite ν] {f : α → β → ℝ≥0∞}
(hf : measurable (uncurry f)) : measurable (λ x, ∫⁻ y, f x y ∂ν) :=
hf.lintegral_prod_right'
/-- The Lebesgue integral is measurable. This shows that the integrand of (the right-hand-side of)
the symmetric version of Tonelli's theorem is measurable. -/
lemma measurable.lintegral_prod_left' [sigma_finite μ] {f : α × β → ℝ≥0∞}
(hf : measurable f) : measurable (λ y, ∫⁻ x, f (x, y) ∂μ) :=
(measurable_swap_iff.mpr hf).lintegral_prod_right'
/-- The Lebesgue integral is measurable. This shows that the integrand of (the right-hand-side of)
the symmetric version of Tonelli's theorem is measurable.
This version has the argument `f` in curried form. -/
lemma measurable.lintegral_prod_left [sigma_finite μ] {f : α → β → ℝ≥0∞}
(hf : measurable (uncurry f)) : measurable (λ y, ∫⁻ x, f x y ∂μ) :=
hf.lintegral_prod_left'
lemma measurable_set_integrable [sigma_finite ν] [opens_measurable_space E] ⦃f : α → β → E⦄
(hf : measurable (uncurry f)) : measurable_set { x | integrable (f x) ν } :=
begin
simp_rw [integrable, hf.of_uncurry_left.ae_measurable, true_and],
exact measurable_set_lt (measurable.lintegral_prod_right hf.ennnorm) measurable_const
end
section
variables [second_countable_topology E] [normed_space ℝ E]
[complete_space E] [borel_space E]
/-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable.
This version has `f` in curried form. -/
lemma measurable.integral_prod_right [sigma_finite ν] ⦃f : α → β → E⦄
(hf : measurable (uncurry f)) : measurable (λ x, ∫ y, f x y ∂ν) :=
begin
let s : ℕ → simple_func (α × β) E := simple_func.approx_on _ hf univ _ (mem_univ 0),
let s' : ℕ → α → simple_func β E := λ n x, (s n).comp (prod.mk x) measurable_prod_mk_left,
let f' : ℕ → α → E := λ n, {x | integrable (f x) ν}.indicator
(λ x, (s' n x).integral ν),
have hf' : ∀ n, measurable (f' n),
{ intro n, refine measurable.indicator _ (measurable_set_integrable hf),
have : ∀ x, (s' n x).range.filter (λ x, x ≠ 0) ⊆ (s n).range,
{ intros x, refine finset.subset.trans (finset.filter_subset _ _) _, intro y,
simp_rw [simple_func.mem_range], rintro ⟨z, rfl⟩, exact ⟨(x, z), rfl⟩ },
simp only [simple_func.integral_eq_sum_of_subset (this _)],
refine finset.measurable_sum _ (λ x _, _),
refine (measurable.ennreal_to_real _).smul_const _,
simp only [simple_func.coe_comp, preimage_comp] {single_pass := tt},
apply measurable_measure_prod_mk_left,
exact (s n).measurable_set_fiber x },
have h2f' : tendsto f' at_top (𝓝 (λ (x : α), ∫ (y : β), f x y ∂ν)),
{ rw [tendsto_pi], intro x,
by_cases hfx : integrable (f x) ν,
{ have : ∀ n, integrable (s' n x) ν,
{ intro n, apply (hfx.norm.add hfx.norm).mono' (s' n x).measurable.ae_measurable,
apply eventually_of_forall, intro y,
simp_rw [s', simple_func.coe_comp], exact simple_func.norm_approx_on_zero_le _ _ (x, y) n },
simp only [f', hfx, simple_func.integral_eq_integral _ (this _), indicator_of_mem,
mem_set_of_eq],
refine tendsto_integral_of_dominated_convergence (λ y, ∥f x y∥ + ∥f x y∥)
(λ n, (s' n x).ae_measurable) hf.of_uncurry_left.ae_measurable (hfx.norm.add hfx.norm) _ _,
{ exact λ n, eventually_of_forall (λ y, simple_func.norm_approx_on_zero_le _ _ (x, y) n) },
{ exact eventually_of_forall (λ y, simple_func.tendsto_approx_on _ _ (by simp)) } },
{ simpa [f', hfx, integral_undef] using @tendsto_const_nhds _ _ _ (0 : E) _, } },
exact measurable_of_tendsto_metric hf' h2f'
end
/-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
Fubini's theorem is measurable. -/
lemma measurable.integral_prod_right' [sigma_finite ν] ⦃f : α × β → E⦄
(hf : measurable f) : measurable (λ x, ∫ y, f (x, y) ∂ν) :=
by { rw [← uncurry_curry f] at hf, exact hf.integral_prod_right }
/-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
the symmetric version of Fubini's theorem is measurable.
This version has `f` in curried form. -/
lemma measurable.integral_prod_left [sigma_finite μ] ⦃f : α → β → E⦄
(hf : measurable (uncurry f)) : measurable (λ y, ∫ x, f x y ∂μ) :=
(hf.comp measurable_swap).integral_prod_right'
/-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
the symmetric version of Fubini's theorem is measurable. -/
lemma measurable.integral_prod_left' [sigma_finite μ] ⦃f : α × β → E⦄
(hf : measurable f) : measurable (λ y, ∫ x, f (x, y) ∂μ) :=
(hf.comp measurable_swap).integral_prod_right'
end
/-! ### The product measure -/
namespace measure_theory
namespace measure
/-- The binary product of measures. They are defined for arbitrary measures, but we basically
prove all properties under the assumption that at least one of them is σ-finite. -/
@[irreducible] protected def prod (μ : measure α) (ν : measure β) : measure (α × β) :=
bind μ $ λ x : α, map (prod.mk x) ν
instance prod.measure_space {α β} [measure_space α] [measure_space β] : measure_space (α × β) :=
{ volume := volume.prod volume }
variables {μ ν} [sigma_finite ν]
lemma volume_eq_prod (α β) [measure_space α] [measure_space β] :
(volume : measure (α × β)) = (volume : measure α).prod (volume : measure β) :=
rfl
lemma prod_apply {s : set (α × β)} (hs : measurable_set s) :
μ.prod ν s = ∫⁻ x, ν (prod.mk x ⁻¹' s) ∂μ :=
by simp_rw [measure.prod, bind_apply hs measurable.map_prod_mk_left,
map_apply measurable_prod_mk_left hs]
@[simp] lemma prod_prod {s : set α} {t : set β}
(hs : measurable_set s) (ht : measurable_set t) : μ.prod ν (s.prod t) = μ s * ν t :=
by simp_rw [prod_apply (hs.prod ht), mk_preimage_prod_right_eq_if, measure_if,
lintegral_indicator _ hs, lintegral_const, restrict_apply measurable_set.univ,
univ_inter, mul_comm]
local attribute [instance] nonempty_measurable_superset
/-- If we don't assume measurability of `s` and `t`, we can bound the measure of their product. -/
lemma prod_prod_le (s : set α) (t : set β) : μ.prod ν (s.prod t) ≤ μ s * ν t :=
begin
by_cases hs0 : μ s = 0,
{ rcases (exists_measurable_superset_of_null hs0) with ⟨s', hs', h2s', h3s'⟩,
convert measure_mono (prod_mono hs' (subset_univ _)),
simp_rw [hs0, prod_prod h2s' measurable_set.univ, h3s', zero_mul] },
by_cases hti : ν t = ∞,
{ convert le_top, simp_rw [hti, ennreal.mul_top, hs0, if_false] },
rw [measure_eq_infi' μ],
simp_rw [ennreal.infi_mul hti],
refine le_infi _,
rintro ⟨s', h1s', h2s'⟩,
rw [subtype.coe_mk],
by_cases ht0 : ν t = 0,
{ rcases (exists_measurable_superset_of_null ht0) with ⟨t', ht', h2t', h3t'⟩,
convert measure_mono (prod_mono (subset_univ _) ht'),
simp_rw [ht0, prod_prod measurable_set.univ h2t', h3t', mul_zero] },
by_cases hsi : μ s' = ∞,
{ convert le_top, simp_rw [hsi, ennreal.top_mul, ht0, if_false] },
rw [measure_eq_infi' ν],
simp_rw [ennreal.mul_infi hsi],
refine le_infi _,
rintro ⟨t', h1t', h2t'⟩,
convert measure_mono (prod_mono h1s' h1t'),
simp [prod_prod h2s' h2t'],
end
lemma ae_measure_lt_top {s : set (α × β)} (hs : measurable_set s)
(h2s : (μ.prod ν) s ≠ ∞) : ∀ᵐ x ∂μ, ν (prod.mk x ⁻¹' s) < ∞ :=
by { simp_rw [prod_apply hs] at h2s, refine ae_lt_top (measurable_measure_prod_mk_left hs) h2s }
lemma integrable_measure_prod_mk_left {s : set (α × β)}
(hs : measurable_set s) (h2s : (μ.prod ν) s ≠ ∞) :
integrable (λ x, (ν (prod.mk x ⁻¹' s)).to_real) μ :=
begin
refine ⟨(measurable_measure_prod_mk_left hs).ennreal_to_real.ae_measurable, _⟩,
simp_rw [has_finite_integral, ennnorm_eq_of_real to_real_nonneg],
convert h2s.lt_top using 1, simp_rw [prod_apply hs], apply lintegral_congr_ae,
refine (ae_measure_lt_top hs h2s).mp _, apply eventually_of_forall, intros x hx,
rw [lt_top_iff_ne_top] at hx, simp [of_real_to_real, hx],
end
/-- Note: the assumption `hs` cannot be dropped. For a counterexample, see
Walter Rudin *Real and Complex Analysis*, example (c) in section 8.9. -/
lemma measure_prod_null {s : set (α × β)}
(hs : measurable_set s) : μ.prod ν s = 0 ↔ (λ x, ν (prod.mk x ⁻¹' s)) =ᵐ[μ] 0 :=
by simp_rw [prod_apply hs, lintegral_eq_zero_iff (measurable_measure_prod_mk_left hs)]
/-- Note: the converse is not true without assuming that `s` is measurable. For a counterexample,
see Walter Rudin *Real and Complex Analysis*, example (c) in section 8.9. -/
lemma measure_ae_null_of_prod_null {s : set (α × β)}
(h : μ.prod ν s = 0) : (λ x, ν (prod.mk x ⁻¹' s)) =ᵐ[μ] 0 :=
begin
obtain ⟨t, hst, mt, ht⟩ := exists_measurable_superset_of_null h,
simp_rw [measure_prod_null mt] at ht,
rw [eventually_le_antisymm_iff],
exact ⟨eventually_le.trans_eq
(eventually_of_forall $ λ x, (measure_mono (preimage_mono hst) : _)) ht,
eventually_of_forall $ λ x, zero_le _⟩
end
/-- Note: the converse is not true. For a counterexample, see
Walter Rudin *Real and Complex Analysis*, example (c) in section 8.9. -/
lemma ae_ae_of_ae_prod {p : α × β → Prop} (h : ∀ᵐ z ∂μ.prod ν, p z) :
∀ᵐ x ∂ μ, ∀ᵐ y ∂ ν, p (x, y) :=
measure_ae_null_of_prod_null h
/-- `μ.prod ν` has finite spanning sets in rectangles of finite spanning sets. -/
def finite_spanning_sets_in.prod {ν : measure β} {C : set (set α)} {D : set (set β)}
(hμ : μ.finite_spanning_sets_in C) (hν : ν.finite_spanning_sets_in D)
(hC : ∀ s ∈ C, measurable_set s) (hD : ∀ t ∈ D, measurable_set t) :
(μ.prod ν).finite_spanning_sets_in (image2 set.prod C D) :=
begin
haveI := hν.sigma_finite,
refine ⟨λ n, (hμ.set n.unpair.1).prod (hν.set n.unpair.2),
λ n, mem_image2_of_mem (hμ.set_mem _) (hν.set_mem _), λ n, _, _⟩,
{ simp_rw [prod_prod (hC _ (hμ.set_mem _)) (hD _ (hν.set_mem _))],
exact mul_lt_top (hμ.finite _).ne (hν.finite _).ne },
{ simp_rw [Union_unpair_prod, hμ.spanning, hν.spanning, univ_prod_univ] }
end
lemma prod_fst_absolutely_continuous : map prod.fst (μ.prod ν) ≪ μ :=
begin
refine absolutely_continuous.mk (λ s hs h2s, _),
simp_rw [map_apply measurable_fst hs, ← prod_univ, prod_prod hs measurable_set.univ],
rw [h2s, zero_mul] -- for some reason `simp_rw [h2s]` doesn't work
end
lemma prod_snd_absolutely_continuous : map prod.snd (μ.prod ν) ≪ ν :=
begin
refine absolutely_continuous.mk (λ s hs h2s, _),
simp_rw [map_apply measurable_snd hs, ← univ_prod, prod_prod measurable_set.univ hs],
rw [h2s, mul_zero] -- for some reason `simp_rw [h2s]` doesn't work
end
variables [sigma_finite μ]
instance prod.sigma_finite : sigma_finite (μ.prod ν) :=
(μ.to_finite_spanning_sets_in.prod ν.to_finite_spanning_sets_in (λ _, id) (λ _, id)).sigma_finite
/-- A measure on a product space equals the product measure if they are equal on rectangles
with as sides sets that generate the corresponding σ-algebras. -/
lemma prod_eq_generate_from {μ : measure α} {ν : measure β} {C : set (set α)}
{D : set (set β)} (hC : generate_from C = ‹_›)
(hD : generate_from D = ‹_›) (h2C : is_pi_system C) (h2D : is_pi_system D)
(h3C : μ.finite_spanning_sets_in C) (h3D : ν.finite_spanning_sets_in D)
{μν : measure (α × β)}
(h₁ : ∀ (s ∈ C) (t ∈ D), μν (set.prod s t) = μ s * ν t) : μ.prod ν = μν :=
begin
have h4C : ∀ (s : set α), s ∈ C → measurable_set s,
{ intros s hs, rw [← hC], exact measurable_set_generate_from hs },
have h4D : ∀ (t : set β), t ∈ D → measurable_set t,
{ intros t ht, rw [← hD], exact measurable_set_generate_from ht },
refine (h3C.prod h3D h4C h4D).ext
(generate_from_eq_prod hC hD h3C.is_countably_spanning h3D.is_countably_spanning).symm
(h2C.prod h2D) _,
{ rintro _ ⟨s, t, hs, ht, rfl⟩, haveI := h3D.sigma_finite,
simp_rw [h₁ s hs t ht, prod_prod (h4C s hs) (h4D t ht)] }
end
/-- A measure on a product space equals the product measure if they are equal on rectangles. -/
lemma prod_eq {μν : measure (α × β)}
(h : ∀ s t, measurable_set s → measurable_set t → μν (s.prod t) = μ s * ν t) : μ.prod ν = μν :=
prod_eq_generate_from generate_from_measurable_set generate_from_measurable_set
is_pi_system_measurable_set is_pi_system_measurable_set
μ.to_finite_spanning_sets_in ν.to_finite_spanning_sets_in (λ s hs t ht, h s t hs ht)
lemma prod_swap : map prod.swap (μ.prod ν) = ν.prod μ :=
begin
refine (prod_eq _).symm,
intros s t hs ht,
simp_rw [map_apply measurable_swap (hs.prod ht), preimage_swap_prod, prod_prod ht hs, mul_comm]
end
lemma prod_apply_symm {s : set (α × β)} (hs : measurable_set s) :
μ.prod ν s = ∫⁻ y, μ ((λ x, (x, y)) ⁻¹' s) ∂ν :=
by { rw [← prod_swap, map_apply measurable_swap hs],
simp only [prod_apply (measurable_swap hs)], refl }
lemma prod_assoc_prod [sigma_finite τ] :
map measurable_equiv.prod_assoc ((μ.prod ν).prod τ) = μ.prod (ν.prod τ) :=
begin
refine (prod_eq_generate_from generate_from_measurable_set generate_from_prod
is_pi_system_measurable_set is_pi_system_prod μ.to_finite_spanning_sets_in
(ν.to_finite_spanning_sets_in.prod τ.to_finite_spanning_sets_in (λ _, id) (λ _, id)) _).symm,
rintro s hs _ ⟨t, u, ht, hu, rfl⟩, rw [mem_set_of_eq] at hs ht hu,
simp_rw [map_apply (measurable_equiv.measurable _) (hs.prod (ht.prod hu)), prod_prod ht hu,
measurable_equiv.prod_assoc, measurable_equiv.coe_mk, equiv.prod_assoc_preimage,
prod_prod (hs.prod ht) hu, prod_prod hs ht, mul_assoc]
end
/-! ### The product of specific measures -/
lemma prod_restrict {s : set α} {t : set β} (hs : measurable_set s) (ht : measurable_set t) :
(μ.restrict s).prod (ν.restrict t) = (μ.prod ν).restrict (s.prod t) :=
begin
refine prod_eq (λ s' t' hs' ht', _),
simp_rw [restrict_apply (hs'.prod ht'), prod_inter_prod, prod_prod (hs'.inter hs) (ht'.inter ht),
restrict_apply hs', restrict_apply ht']
end
lemma restrict_prod_eq_prod_univ {s : set α} (hs : measurable_set s) :
(μ.restrict s).prod ν = (μ.prod ν).restrict (s.prod univ) :=
begin
have : ν = ν.restrict set.univ := measure.restrict_univ.symm,
rwa [this, measure.prod_restrict, ← this],
exact measurable_set.univ,
end
lemma prod_dirac (y : β) : μ.prod (dirac y) = map (λ x, (x, y)) μ :=
begin
refine prod_eq (λ s t hs ht, _),
simp_rw [map_apply measurable_prod_mk_right (hs.prod ht), mk_preimage_prod_left_eq_if, measure_if,
dirac_apply' _ ht, ← indicator_mul_right _ (λ x, μ s), pi.one_apply, mul_one]
end
lemma dirac_prod (x : α) : (dirac x).prod ν = map (prod.mk x) ν :=
begin
refine prod_eq (λ s t hs ht, _),
simp_rw [map_apply measurable_prod_mk_left (hs.prod ht), mk_preimage_prod_right_eq_if, measure_if,
dirac_apply' _ hs, ← indicator_mul_left _ _ (λ x, ν t), pi.one_apply, one_mul]
end
lemma dirac_prod_dirac {x : α} {y : β} : (dirac x).prod (dirac y) = dirac (x, y) :=
by rw [prod_dirac, map_dirac measurable_prod_mk_right]
lemma prod_sum {ι : Type*} [fintype ι] (ν : ι → measure β) [∀ i, sigma_finite (ν i)] :
μ.prod (sum ν) = sum (λ i, μ.prod (ν i)) :=
begin
refine prod_eq (λ s t hs ht, _),
simp_rw [sum_apply _ (hs.prod ht), sum_apply _ ht, prod_prod hs ht, tsum_fintype, finset.mul_sum]
end
lemma sum_prod {ι : Type*} [fintype ι] (μ : ι → measure α) [∀ i, sigma_finite (μ i)] :
(sum μ).prod ν = sum (λ i, (μ i).prod ν) :=
begin
refine prod_eq (λ s t hs ht, _),
simp_rw [sum_apply _ (hs.prod ht), sum_apply _ hs, prod_prod hs ht, tsum_fintype, finset.sum_mul]
end
lemma prod_add (ν' : measure β) [sigma_finite ν'] : μ.prod (ν + ν') = μ.prod ν + μ.prod ν' :=
by { refine prod_eq (λ s t hs ht, _), simp_rw [add_apply, prod_prod hs ht, left_distrib] }
lemma add_prod (μ' : measure α) [sigma_finite μ'] : (μ + μ').prod ν = μ.prod ν + μ'.prod ν :=
by { refine prod_eq (λ s t hs ht, _), simp_rw [add_apply, prod_prod hs ht, right_distrib] }
@[simp] lemma zero_prod (ν : measure β) : (0 : measure α).prod ν = 0 :=
by { rw measure.prod, exact bind_zero_left _ }
@[simp] lemma prod_zero (μ : measure α) : μ.prod (0 : measure β) = 0 :=
by simp [measure.prod]
lemma map_prod_map {δ} [measurable_space δ] {f : α → β} {g : γ → δ}
{μa : measure α} {μc : measure γ} (hfa : sigma_finite (map f μa))
(hgc : sigma_finite (map g μc)) (hf : measurable f) (hg : measurable g) :
(map f μa).prod (map g μc) = map (prod.map f g) (μa.prod μc) :=
begin
haveI := hgc.of_map μc hg,
refine prod_eq (λ s t hs ht, _),
rw [map_apply (hf.prod_map hg) (hs.prod ht), map_apply hf hs, map_apply hg ht],
exact prod_prod (hf hs) (hg ht)
end
end measure
end measure_theory
open measure_theory.measure
section
lemma ae_measurable.prod_swap [sigma_finite μ] [sigma_finite ν] {f : β × α → γ}
(hf : ae_measurable f (ν.prod μ)) : ae_measurable (λ (z : α × β), f z.swap) (μ.prod ν) :=
by { rw ← prod_swap at hf, exact hf.comp_measurable measurable_swap }
lemma ae_measurable.fst [sigma_finite ν] {f : α → γ}
(hf : ae_measurable f μ) : ae_measurable (λ (z : α × β), f z.1) (μ.prod ν) :=
hf.comp_measurable' measurable_fst prod_fst_absolutely_continuous
lemma ae_measurable.snd [sigma_finite ν] {f : β → γ}
(hf : ae_measurable f ν) : ae_measurable (λ (z : α × β), f z.2) (μ.prod ν) :=
hf.comp_measurable' measurable_snd prod_snd_absolutely_continuous
/-- The Bochner integral is a.e.-measurable.
This shows that the integrand of (the right-hand-side of) Fubini's theorem is a.e.-measurable. -/
lemma ae_measurable.integral_prod_right' [sigma_finite ν]
[second_countable_topology E] [normed_space ℝ E] [borel_space E] [complete_space E]
⦃f : α × β → E⦄ (hf : ae_measurable f (μ.prod ν)) : ae_measurable (λ x, ∫ y, f (x, y) ∂ν) μ :=
⟨λ x, ∫ y, hf.mk f (x, y) ∂ν, hf.measurable_mk.integral_prod_right', begin
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk],
assume x hx,
exact integral_congr_ae hx
end⟩
lemma ae_measurable.prod_mk_left [sigma_finite ν] {f : α × β → γ}
(hf : ae_measurable f (μ.prod ν)) : ∀ᵐ x ∂μ, ae_measurable (λ y, f (x, y)) ν :=
begin
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk],
intros x hx,
exact ⟨λ y, hf.mk f (x, y), hf.measurable_mk.comp measurable_prod_mk_left, hx⟩
end
end
namespace measure_theory
/-! ### The Lebesgue integral on a product -/
variables [sigma_finite ν]
lemma lintegral_prod_swap [sigma_finite μ] (f : α × β → ℝ≥0∞)
(hf : ae_measurable f (μ.prod ν)) : ∫⁻ z, f z.swap ∂(ν.prod μ) = ∫⁻ z, f z ∂(μ.prod ν) :=
by { rw ← prod_swap at hf, rw [← lintegral_map' hf measurable_swap, prod_swap] }
/-- **Tonelli's Theorem**: For `ℝ≥0∞`-valued measurable functions on `α × β`,
the integral of `f` is equal to the iterated integral. -/
lemma lintegral_prod_of_measurable :
∀ (f : α × β → ℝ≥0∞) (hf : measurable f), ∫⁻ z, f z ∂(μ.prod ν) = ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ :=
begin
have m := @measurable_prod_mk_left,
refine measurable.ennreal_induction _ _ _,
{ intros c s hs, simp only [← indicator_comp_right],
simp [lintegral_indicator, m hs, hs, lintegral_const_mul, measurable_measure_prod_mk_left hs,
prod_apply] },
{ rintro f g - hf hg h2f h2g,
simp [lintegral_add, measurable.lintegral_prod_right', hf.comp m, hg.comp m,
hf, hg, h2f, h2g] },
{ intros f hf h2f h3f,
have kf : ∀ x n, measurable (λ y, f n (x, y)) := λ x n, (hf n).comp m,
have k2f : ∀ x, monotone (λ n y, f n (x, y)) := λ x i j hij y, h2f hij (x, y),
have lf : ∀ n, measurable (λ x, ∫⁻ y, f n (x, y) ∂ν) := λ n, (hf n).lintegral_prod_right',
have l2f : monotone (λ n x, ∫⁻ y, f n (x, y) ∂ν) := λ i j hij x, lintegral_mono (k2f x hij),
simp only [lintegral_supr hf h2f, lintegral_supr (kf _), k2f, lintegral_supr lf l2f, h3f] },
end
/-- **Tonelli's Theorem**: For `ℝ≥0∞`-valued almost everywhere measurable functions on `α × β`,
the integral of `f` is equal to the iterated integral. -/
lemma lintegral_prod (f : α × β → ℝ≥0∞) (hf : ae_measurable f (μ.prod ν)) :
∫⁻ z, f z ∂(μ.prod ν) = ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ :=
begin
have A : ∫⁻ z, f z ∂(μ.prod ν) = ∫⁻ z, hf.mk f z ∂(μ.prod ν) :=
lintegral_congr_ae hf.ae_eq_mk,
have B : ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ = ∫⁻ x, ∫⁻ y, hf.mk f (x, y) ∂ν ∂μ,
{ apply lintegral_congr_ae,
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk],
assume a ha,
exact lintegral_congr_ae ha },
rw [A, B, lintegral_prod_of_measurable _ hf.measurable_mk],
apply_instance
end
/-- The symmetric verion of Tonelli's Theorem: For `ℝ≥0∞`-valued almost everywhere measurable
functions on `α × β`, the integral of `f` is equal to the iterated integral, in reverse order. -/
lemma lintegral_prod_symm [sigma_finite μ] (f : α × β → ℝ≥0∞)
(hf : ae_measurable f (μ.prod ν)) : ∫⁻ z, f z ∂(μ.prod ν) = ∫⁻ y, ∫⁻ x, f (x, y) ∂μ ∂ν :=
by { simp_rw [← lintegral_prod_swap f hf], exact lintegral_prod _ hf.prod_swap }
/-- The symmetric verion of Tonelli's Theorem: For `ℝ≥0∞`-valued measurable
functions on `α × β`, the integral of `f` is equal to the iterated integral, in reverse order. -/
lemma lintegral_prod_symm' [sigma_finite μ] (f : α × β → ℝ≥0∞)
(hf : measurable f) : ∫⁻ z, f z ∂(μ.prod ν) = ∫⁻ y, ∫⁻ x, f (x, y) ∂μ ∂ν :=
lintegral_prod_symm f hf.ae_measurable
/-- The reversed version of **Tonelli's Theorem**. In this version `f` is in curried form, which
makes it easier for the elaborator to figure out `f` automatically. -/
lemma lintegral_lintegral ⦃f : α → β → ℝ≥0∞⦄
(hf : ae_measurable (uncurry f) (μ.prod ν)) :
∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ = ∫⁻ z, f z.1 z.2 ∂(μ.prod ν) :=
(lintegral_prod _ hf).symm
/-- The reversed version of **Tonelli's Theorem** (symmetric version). In this version `f` is in
curried form, which makes it easier for the elaborator to figure out `f` automatically. -/
lemma lintegral_lintegral_symm [sigma_finite μ] ⦃f : α → β → ℝ≥0∞⦄
(hf : ae_measurable (uncurry f) (μ.prod ν)) :
∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ = ∫⁻ z, f z.2 z.1 ∂(ν.prod μ) :=
(lintegral_prod_symm _ hf.prod_swap).symm
/-- Change the order of Lebesgue integration. -/
lemma lintegral_lintegral_swap [sigma_finite μ] ⦃f : α → β → ℝ≥0∞⦄
(hf : ae_measurable (uncurry f) (μ.prod ν)) :
∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ = ∫⁻ y, ∫⁻ x, f x y ∂μ ∂ν :=
(lintegral_lintegral hf).trans (lintegral_prod_symm _ hf)
lemma lintegral_prod_mul {f : α → ℝ≥0∞} {g : β → ℝ≥0∞}
(hf : ae_measurable f μ) (hg : ae_measurable g ν) :
∫⁻ z, f z.1 * g z.2 ∂(μ.prod ν) = ∫⁻ x, f x ∂μ * ∫⁻ y, g y ∂ν :=
by simp [lintegral_prod _ (hf.fst.mul hg.snd), lintegral_lintegral_mul hf hg]
/-! ### Integrability on a product -/
section
variables [opens_measurable_space E]
lemma integrable.swap [sigma_finite μ] ⦃f : α × β → E⦄
(hf : integrable f (μ.prod ν)) : integrable (f ∘ prod.swap) (ν.prod μ) :=
⟨hf.ae_measurable.prod_swap,
(lintegral_prod_swap _ hf.ae_measurable.ennnorm : _).le.trans_lt hf.has_finite_integral⟩
lemma integrable_swap_iff [sigma_finite μ] ⦃f : α × β → E⦄ :
integrable (f ∘ prod.swap) (ν.prod μ) ↔ integrable f (μ.prod ν) :=
⟨λ hf, by { convert hf.swap, ext ⟨x, y⟩, refl }, λ hf, hf.swap⟩
lemma has_finite_integral_prod_iff ⦃f : α × β → E⦄ (h1f : measurable f) :
has_finite_integral f (μ.prod ν) ↔ (∀ᵐ x ∂ μ, has_finite_integral (λ y, f (x, y)) ν) ∧
has_finite_integral (λ x, ∫ y, ∥f (x, y)∥ ∂ν) μ :=
begin
simp only [has_finite_integral, lintegral_prod_of_measurable _ h1f.ennnorm],
have : ∀ x, ∀ᵐ y ∂ν, 0 ≤ ∥f (x, y)∥ := λ x, eventually_of_forall (λ y, norm_nonneg _),
simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
(h1f.norm.comp measurable_prod_mk_left).ae_measurable,
ennnorm_eq_of_real to_real_nonneg, of_real_norm_eq_coe_nnnorm],
-- this fact is probably too specialized to be its own lemma
have : ∀ {p q r : Prop} (h1 : r → p), (r ↔ p ∧ q) ↔ (p → (r ↔ q)) :=
λ p q r h1, by rw [← and.congr_right_iff, and_iff_right_of_imp h1],
rw [this],
{ intro h2f, rw lintegral_congr_ae,
refine h2f.mp _, apply eventually_of_forall, intros x hx, dsimp only,
rw [of_real_to_real], rw [← lt_top_iff_ne_top], exact hx },
{ intro h2f, refine ae_lt_top _ h2f.ne, exact h1f.ennnorm.lintegral_prod_right' },
end
lemma has_finite_integral_prod_iff' ⦃f : α × β → E⦄ (h1f : ae_measurable f (μ.prod ν)) :
has_finite_integral f (μ.prod ν) ↔ (∀ᵐ x ∂ μ, has_finite_integral (λ y, f (x, y)) ν) ∧
has_finite_integral (λ x, ∫ y, ∥f (x, y)∥ ∂ν) μ :=
begin
rw [has_finite_integral_congr h1f.ae_eq_mk, has_finite_integral_prod_iff h1f.measurable_mk],
apply and_congr,
{ apply eventually_congr,
filter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm],
assume x hx,
exact has_finite_integral_congr hx },
{ apply has_finite_integral_congr,
filter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm],
assume x hx,
exact integral_congr_ae (eventually_eq.fun_comp hx _) },
{ apply_instance }
end
/-- A binary function is integrable if the function `y ↦ f (x, y)` is integrable for almost every
`x` and the function `x ↦ ∫ ∥f (x, y)∥ dy` is integrable. -/
lemma integrable_prod_iff ⦃f : α × β → E⦄ (h1f : ae_measurable f (μ.prod ν)) :
integrable f (μ.prod ν) ↔
(∀ᵐ x ∂ μ, integrable (λ y, f (x, y)) ν) ∧ integrable (λ x, ∫ y, ∥f (x, y)∥ ∂ν) μ :=
by simp [integrable, h1f, has_finite_integral_prod_iff', h1f.norm.integral_prod_right',
h1f.prod_mk_left]
/-- A binary function is integrable if the function `x ↦ f (x, y)` is integrable for almost every
`y` and the function `y ↦ ∫ ∥f (x, y)∥ dx` is integrable. -/
lemma integrable_prod_iff' [sigma_finite μ] ⦃f : α × β → E⦄ (h1f : ae_measurable f (μ.prod ν)) :
integrable f (μ.prod ν) ↔
(∀ᵐ y ∂ ν, integrable (λ x, f (x, y)) μ) ∧ integrable (λ y, ∫ x, ∥f (x, y)∥ ∂μ) ν :=
by { convert integrable_prod_iff (h1f.prod_swap) using 1, rw [integrable_swap_iff] }
lemma integrable.prod_left_ae [sigma_finite μ] ⦃f : α × β → E⦄
(hf : integrable f (μ.prod ν)) : ∀ᵐ y ∂ ν, integrable (λ x, f (x, y)) μ :=
((integrable_prod_iff' hf.ae_measurable).mp hf).1
lemma integrable.prod_right_ae [sigma_finite μ] ⦃f : α × β → E⦄
(hf : integrable f (μ.prod ν)) : ∀ᵐ x ∂ μ, integrable (λ y, f (x, y)) ν :=
hf.swap.prod_left_ae
lemma integrable.integral_norm_prod_left ⦃f : α × β → E⦄
(hf : integrable f (μ.prod ν)) : integrable (λ x, ∫ y, ∥f (x, y)∥ ∂ν) μ :=
((integrable_prod_iff hf.ae_measurable).mp hf).2
lemma integrable.integral_norm_prod_right [sigma_finite μ] ⦃f : α × β → E⦄
(hf : integrable f (μ.prod ν)) : integrable (λ y, ∫ x, ∥f (x, y)∥ ∂μ) ν :=
hf.swap.integral_norm_prod_left
end
variables [second_countable_topology E] [normed_space ℝ E]
[complete_space E] [borel_space E]
lemma integrable.integral_prod_left ⦃f : α × β → E⦄
(hf : integrable f (μ.prod ν)) : integrable (λ x, ∫ y, f (x, y) ∂ν) μ :=
integrable.mono hf.integral_norm_prod_left hf.ae_measurable.integral_prod_right' $
eventually_of_forall $ λ x, (norm_integral_le_integral_norm _).trans_eq $
(norm_of_nonneg $ integral_nonneg_of_ae $ eventually_of_forall $
λ y, (norm_nonneg (f (x, y)) : _)).symm
lemma integrable.integral_prod_right [sigma_finite μ] ⦃f : α × β → E⦄
(hf : integrable f (μ.prod ν)) : integrable (λ y, ∫ x, f (x, y) ∂μ) ν :=
hf.swap.integral_prod_left
/-! ### The Bochner integral on a product -/
variables [sigma_finite μ]
lemma integral_prod_swap (f : α × β → E)
(hf : ae_measurable f (μ.prod ν)) : ∫ z, f z.swap ∂(ν.prod μ) = ∫ z, f z ∂(μ.prod ν) :=
begin
rw ← prod_swap at hf,
rw [← integral_map measurable_swap hf, prod_swap]
end
variables {E' : Type*} [measurable_space E'] [normed_group E'] [borel_space E'] [complete_space E']
[normed_space ℝ E'] [second_countable_topology E']
/-! Some rules about the sum/difference of double integrals. They follow from `integral_add`, but
we separate them out as separate lemmas, because they involve quite some steps. -/
/-- Integrals commute with addition inside another integral. `F` can be any function. -/
lemma integral_fn_integral_add ⦃f g : α × β → E⦄ (F : E → E')
(hf : integrable f (μ.prod ν)) (hg : integrable g (μ.prod ν)) :
∫ x, F (∫ y, f (x, y) + g (x, y) ∂ν) ∂μ = ∫ x, F (∫ y, f (x, y) ∂ν + ∫ y, g (x, y) ∂ν) ∂μ :=
begin
refine integral_congr_ae _,
filter_upwards [hf.prod_right_ae, hg.prod_right_ae],
intros x h2f h2g, simp [integral_add h2f h2g],
end
/-- Integrals commute with subtraction inside another integral.
`F` can be any measurable function. -/
lemma integral_fn_integral_sub ⦃f g : α × β → E⦄ (F : E → E')
(hf : integrable f (μ.prod ν)) (hg : integrable g (μ.prod ν)) :
∫ x, F (∫ y, f (x, y) - g (x, y) ∂ν) ∂μ = ∫ x, F (∫ y, f (x, y) ∂ν - ∫ y, g (x, y) ∂ν) ∂μ :=
begin
refine integral_congr_ae _,
filter_upwards [hf.prod_right_ae, hg.prod_right_ae],
intros x h2f h2g, simp [integral_sub h2f h2g]
end
/-- Integrals commute with subtraction inside a lower Lebesgue integral.
`F` can be any function. -/
lemma lintegral_fn_integral_sub ⦃f g : α × β → E⦄
(F : E → ℝ≥0∞) (hf : integrable f (μ.prod ν)) (hg : integrable g (μ.prod ν)) :
∫⁻ x, F (∫ y, f (x, y) - g (x, y) ∂ν) ∂μ = ∫⁻ x, F (∫ y, f (x, y) ∂ν - ∫ y, g (x, y) ∂ν) ∂μ :=
begin
refine lintegral_congr_ae _,
filter_upwards [hf.prod_right_ae, hg.prod_right_ae],
intros x h2f h2g, simp [integral_sub h2f h2g]
end
/-- Double integrals commute with addition. -/
lemma integral_integral_add ⦃f g : α × β → E⦄
(hf : integrable f (μ.prod ν)) (hg : integrable g (μ.prod ν)) :
∫ x, ∫ y, f (x, y) + g (x, y) ∂ν ∂μ = ∫ x, ∫ y, f (x, y) ∂ν ∂μ + ∫ x, ∫ y, g (x, y) ∂ν ∂μ :=
(integral_fn_integral_add id hf hg).trans $
integral_add hf.integral_prod_left hg.integral_prod_left
/-- Double integrals commute with addition. This is the version with `(f + g) (x, y)`
(instead of `f (x, y) + g (x, y)`) in the LHS. -/
lemma integral_integral_add' ⦃f g : α × β → E⦄
(hf : integrable f (μ.prod ν)) (hg : integrable g (μ.prod ν)) :
∫ x, ∫ y, (f + g) (x, y) ∂ν ∂μ = ∫ x, ∫ y, f (x, y) ∂ν ∂μ + ∫ x, ∫ y, g (x, y) ∂ν ∂μ :=
integral_integral_add hf hg
/-- Double integrals commute with subtraction. -/
lemma integral_integral_sub ⦃f g : α × β → E⦄
(hf : integrable f (μ.prod ν)) (hg : integrable g (μ.prod ν)) :
∫ x, ∫ y, f (x, y) - g (x, y) ∂ν ∂μ = ∫ x, ∫ y, f (x, y) ∂ν ∂μ - ∫ x, ∫ y, g (x, y) ∂ν ∂μ :=
(integral_fn_integral_sub id hf hg).trans $
integral_sub hf.integral_prod_left hg.integral_prod_left
/-- Double integrals commute with subtraction. This is the version with `(f - g) (x, y)`
(instead of `f (x, y) - g (x, y)`) in the LHS. -/
lemma integral_integral_sub' ⦃f g : α × β → E⦄
(hf : integrable f (μ.prod ν)) (hg : integrable g (μ.prod ν)) :
∫ x, ∫ y, (f - g) (x, y) ∂ν ∂μ = ∫ x, ∫ y, f (x, y) ∂ν ∂μ - ∫ x, ∫ y, g (x, y) ∂ν ∂μ :=
integral_integral_sub hf hg
/-- The map that sends an L¹-function `f : α × β → E` to `∫∫f` is continuous. -/
lemma continuous_integral_integral :
continuous (λ (f : α × β →₁[μ.prod ν] E), ∫ x, ∫ y, f (x, y) ∂ν ∂μ) :=
begin
rw [continuous_iff_continuous_at], intro g,
refine tendsto_integral_of_L1 _ (L1.integrable_coe_fn g).integral_prod_left
(eventually_of_forall $ λ h, (L1.integrable_coe_fn h).integral_prod_left) _,
simp_rw [← lintegral_fn_integral_sub (λ x, (nnnorm x : ℝ≥0∞)) (L1.integrable_coe_fn _)
(L1.integrable_coe_fn g)],
refine tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (λ i, zero_le _) _,
{ exact λ i, ∫⁻ x, ∫⁻ y, nnnorm (i (x, y) - g (x, y)) ∂ν ∂μ },
swap, { exact λ i, lintegral_mono (λ x, ennnorm_integral_le_lintegral_ennnorm _) },
show tendsto (λ (i : α × β →₁[μ.prod ν] E),
∫⁻ x, ∫⁻ (y : β), nnnorm (i (x, y) - g (x, y)) ∂ν ∂μ) (𝓝 g) (𝓝 0),
have : ∀ (i : α × β →₁[μ.prod ν] E), measurable (λ z, (nnnorm (i z - g z) : ℝ≥0∞)) :=
λ i, ((Lp.measurable i).sub (Lp.measurable g)).ennnorm,
simp_rw [← lintegral_prod_of_measurable _ (this _), ← L1.of_real_norm_sub_eq_lintegral,
← of_real_zero],
refine (continuous_of_real.tendsto 0).comp _,
rw [← tendsto_iff_norm_tendsto_zero], exact tendsto_id
end
/-- **Fubini's Theorem**: For integrable functions on `α × β`,
the Bochner integral of `f` is equal to the iterated Bochner integral.
`integrable_prod_iff` can be useful to show that the function in question in integrable.
`measure_theory.integrable.integral_prod_right` is useful to show that the inner integral
of the right-hand side is integrable. -/
lemma integral_prod : ∀ (f : α × β → E) (hf : integrable f (μ.prod ν)),
∫ z, f z ∂(μ.prod ν) = ∫ x, ∫ y, f (x, y) ∂ν ∂μ :=
begin
apply integrable.induction,
{ intros c s hs h2s,
simp_rw [integral_indicator hs, ← indicator_comp_right,
function.comp, integral_indicator (measurable_prod_mk_left hs),
set_integral_const, integral_smul_const,
integral_to_real (measurable_measure_prod_mk_left hs).ae_measurable
(ae_measure_lt_top hs h2s.ne), prod_apply hs] },
{ intros f g hfg i_f i_g hf hg,
simp_rw [integral_add' i_f i_g, integral_integral_add' i_f i_g, hf, hg] },
{ exact is_closed_eq continuous_integral continuous_integral_integral },
{ intros f g hfg i_f hf, convert hf using 1,
{ exact integral_congr_ae hfg.symm },
{ refine integral_congr_ae _,
refine (ae_ae_of_ae_prod hfg).mp _,
apply eventually_of_forall, intros x hfgx,
exact integral_congr_ae (ae_eq_symm hfgx) } }
end
/-- Symmetric version of **Fubini's Theorem**: For integrable functions on `α × β`,
the Bochner integral of `f` is equal to the iterated Bochner integral.
This version has the integrals on the right-hand side in the other order. -/
lemma integral_prod_symm (f : α × β → E) (hf : integrable f (μ.prod ν)) :
∫ z, f z ∂(μ.prod ν) = ∫ y, ∫ x, f (x, y) ∂μ ∂ν :=
by { simp_rw [← integral_prod_swap f hf.ae_measurable], exact integral_prod _ hf.swap }
/-- Reversed version of **Fubini's Theorem**. -/
lemma integral_integral {f : α → β → E} (hf : integrable (uncurry f) (μ.prod ν)) :
∫ x, ∫ y, f x y ∂ν ∂μ = ∫ z, f z.1 z.2 ∂(μ.prod ν) :=
(integral_prod _ hf).symm
/-- Reversed version of **Fubini's Theorem** (symmetric version). -/
lemma integral_integral_symm {f : α → β → E} (hf : integrable (uncurry f) (μ.prod ν)) :
∫ x, ∫ y, f x y ∂ν ∂μ = ∫ z, f z.2 z.1 ∂(ν.prod μ) :=
(integral_prod_symm _ hf.swap).symm
/-- Change the order of Bochner integration. -/
lemma integral_integral_swap ⦃f : α → β → E⦄ (hf : integrable (uncurry f) (μ.prod ν)) :
∫ x, ∫ y, f x y ∂ν ∂μ = ∫ y, ∫ x, f x y ∂μ ∂ν :=
(integral_integral hf).trans (integral_prod_symm _ hf)
end measure_theory
|
2b36af95602a27bb295c668f2d77e6281a16af2c | c777c32c8e484e195053731103c5e52af26a25d1 | /src/number_theory/well_approximable.lean | 93e703e48f198ad4508676e79fb92e61914de515 | [
"Apache-2.0"
] | permissive | kbuzzard/mathlib | 2ff9e85dfe2a46f4b291927f983afec17e946eb8 | 58537299e922f9c77df76cb613910914a479c1f7 | refs/heads/master | 1,685,313,702,744 | 1,683,974,212,000 | 1,683,974,212,000 | 128,185,277 | 1 | 0 | null | 1,522,920,600,000 | 1,522,920,600,000 | null | UTF-8 | Lean | false | false | 16,717 | lean | /-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import dynamics.ergodic.add_circle
import measure_theory.covering.liminf_limsup
/-!
# Well-approximable numbers and Gallagher's ergodic theorem
Gallagher's ergodic theorem is a result in metric number theory. It thus belongs to that branch of
mathematics concerning arithmetic properties of real numbers which hold almost eveywhere with
respect to the Lebesgue measure.
Gallagher's theorem concerns the approximation of real numbers by rational numbers. The input is a
sequence of distances `δ₁, δ₂, ...`, and the theorem concerns the set of real numbers `x` for which
there is an infinity of solutions to:
$$
|x - m/n| < δₙ,
$$
where the rational number `m/n` is in lowest terms. The result is that for any `δ`, this set is
either almost all `x` or almost no `x`.
This result was proved by Gallagher in 1959
[P. Gallagher, *Approximation by reduced fractions*](Gallagher1961). It is formalised here as
`add_circle.add_well_approximable_ae_empty_or_univ` except with `x` belonging to the circle `ℝ ⧸ ℤ`
since this turns out to be more natural.
Given a particular `δ`, the Duffin-Schaeffer conjecture (now a theorem) gives a criterion for
deciding which of the two cases in the conclusion of Gallagher's theorem actually occurs. It was
proved by Koukoulopoulos and Maynard in 2019
[D. Koukoulopoulos, J. Maynard, *On the Duffin-Schaeffer conjecture*](KoukoulopoulosMaynard2020).
We do *not* include a formalisation of the Koukoulopoulos-Maynard result here.
## Main definitions and results:
* `approx_order_of`: in a seminormed group `A`, given `n : ℕ` and `δ : ℝ`, `approx_order_of A n δ`
is the set of elements within a distance `δ` of a point of order `n`.
* `well_approximable`: in a seminormed group `A`, given a sequence of distances `δ₁, δ₂, ...`,
`well_approximable A δ` is the limsup as `n → ∞` of the sets `approx_order_of A n δₙ`. Thus, it
is the set of points that lie in infinitely many of the sets `approx_order_of A n δₙ`.
* `add_circle.add_well_approximable_ae_empty_or_univ`: *Gallagher's ergodic theorem* says that for
for the (additive) circle `𝕊`, for any sequence of distances `δ`, the set
`add_well_approximable 𝕊 δ` is almost empty or almost full.
## TODO:
The hypothesis `hδ` in `add_circle.add_well_approximable_ae_empty_or_univ` can be dropped.
An elementary (non-measure-theoretic) argument shows that if `¬ hδ` holds then
`add_well_approximable 𝕊 δ = univ` (provided `δ` is non-negative).
-/
open set filter function metric measure_theory
open_locale measure_theory topology pointwise
/-- In a seminormed group `A`, given `n : ℕ` and `δ : ℝ`, `approx_order_of A n δ` is the set of
elements within a distance `δ` of a point of order `n`. -/
@[to_additive approx_add_order_of "In a seminormed additive group `A`, given `n : ℕ` and `δ : ℝ`,
`approx_add_order_of A n δ` is the set of elements within a distance `δ` of a point of order `n`."]
def approx_order_of (A : Type*) [seminormed_group A] (n : ℕ) (δ : ℝ) : set A :=
thickening δ {y | order_of y = n}
@[to_additive mem_approx_add_order_of_iff]
lemma mem_approx_order_of_iff {A : Type*} [seminormed_group A] {n : ℕ} {δ : ℝ} {a : A} :
a ∈ approx_order_of A n δ ↔ ∃ (b : A), order_of b = n ∧ a ∈ ball b δ :=
by simp only [approx_order_of, thickening_eq_bUnion_ball, mem_Union₂, mem_set_of_eq, exists_prop]
/-- In a seminormed group `A`, given a sequence of distances `δ₁, δ₂, ...`, `well_approximable A δ`
is the limsup as `n → ∞` of the sets `approx_order_of A n δₙ`. Thus, it is the set of points that
lie in infinitely many of the sets `approx_order_of A n δₙ`. -/
@[to_additive add_well_approximable "In a seminormed additive group `A`, given a sequence of
distances `δ₁, δ₂, ...`, `add_well_approximable A δ` is the limsup as `n → ∞` of the sets
`approx_add_order_of A n δₙ`. Thus, it is the set of points that lie in infinitely many of the sets
`approx_add_order_of A n δₙ`."]
def well_approximable (A : Type*) [seminormed_group A] (δ : ℕ → ℝ) : set A :=
blimsup (λ n, approx_order_of A n (δ n)) at_top (λ n, 0 < n)
@[to_additive mem_add_well_approximable_iff]
lemma mem_well_approximable_iff {A : Type*} [seminormed_group A] {δ : ℕ → ℝ} {a : A} :
a ∈ well_approximable A δ ↔ a ∈ blimsup (λ n, approx_order_of A n (δ n)) at_top (λ n, 0 < n) :=
iff.rfl
namespace approx_order_of
variables {A : Type*} [seminormed_comm_group A] {a : A} {m n : ℕ} (δ : ℝ)
@[to_additive]
lemma image_pow_subset_of_coprime (hm : 0 < m) (hmn : n.coprime m) :
(λ y, y^m) '' (approx_order_of A n δ) ⊆ approx_order_of A n (m * δ) :=
begin
rintros - ⟨a, ha, rfl⟩,
obtain ⟨b, hb, hab⟩ := mem_approx_order_of_iff.mp ha,
replace hb : b^m ∈ {u : A | order_of u = n}, { rw ← hb at hmn ⊢, exact order_of_pow_coprime hmn },
apply ball_subset_thickening hb ((m : ℝ) • δ),
convert pow_mem_ball hm hab using 1,
simp only [nsmul_eq_mul, algebra.id.smul_eq_mul],
end
@[to_additive]
lemma image_pow_subset (n : ℕ) (hm : 0 < m) :
(λ y, y^m) '' (approx_order_of A (n * m) δ) ⊆ approx_order_of A n (m * δ) :=
begin
rintros - ⟨a, ha, rfl⟩,
obtain ⟨b, hb : order_of b = n * m, hab : a ∈ ball b δ⟩ := mem_approx_order_of_iff.mp ha,
replace hb : b^m ∈ {y : A | order_of y = n},
{ rw [mem_set_of_eq, order_of_pow' b hm.ne', hb, nat.gcd_mul_left_left, n.mul_div_cancel hm], },
apply ball_subset_thickening hb (m * δ),
convert pow_mem_ball hm hab,
simp only [nsmul_eq_mul],
end
@[to_additive]
lemma smul_subset_of_coprime (han : (order_of a).coprime n) :
a • approx_order_of A n δ ⊆ approx_order_of A ((order_of a) * n) δ :=
begin
simp_rw [approx_order_of, thickening_eq_bUnion_ball, ← image_smul, image_Union₂,
image_smul, smul_ball'', smul_eq_mul, mem_set_of_eq],
refine Union₂_subset_iff.mpr (λ b hb c hc, _),
simp only [mem_Union, exists_prop],
refine ⟨a * b, _, hc⟩,
rw ← hb at ⊢ han,
exact (commute.all a b).order_of_mul_eq_mul_order_of_of_coprime han,
end
@[to_additive vadd_eq_of_mul_dvd]
lemma smul_eq_of_mul_dvd (hn : 0 < n) (han : (order_of a)^2 ∣ n) :
a • approx_order_of A n δ = approx_order_of A n δ :=
begin
simp_rw [approx_order_of, thickening_eq_bUnion_ball, ← image_smul, image_Union₂,
image_smul, smul_ball'', smul_eq_mul, mem_set_of_eq],
replace han : ∀ {b : A}, order_of b = n → order_of (a * b) = n,
{ intros b hb,
rw ← hb at han hn,
rw sq at han,
rwa [(commute.all a b).order_of_mul_eq_right_of_forall_prime_mul_dvd (order_of_pos_iff.mp hn)
(λ p hp hp', dvd_trans (mul_dvd_mul_right hp' $ order_of a) han)], },
let f : {b : A | order_of b = n} → {b : A | order_of b = n} := λ b, ⟨a * b, han b.property⟩,
have hf : surjective f,
{ rintros ⟨b, hb⟩,
refine ⟨⟨a⁻¹ * b, _⟩, _⟩,
{ rw [mem_set_of_eq, ← order_of_inv, mul_inv_rev, inv_inv, mul_comm],
apply han,
simpa, },
{ simp only [subtype.mk_eq_mk, subtype.coe_mk, mul_inv_cancel_left], }, },
simpa only [f, mem_set_of_eq, subtype.coe_mk, Union_coe_set] using
hf.Union_comp (λ b, ball (b : A) δ),
end
end approx_order_of
namespace unit_add_circle
lemma mem_approx_add_order_of_iff {δ : ℝ} {x : unit_add_circle} {n : ℕ} (hn : 0 < n) :
x ∈ approx_add_order_of unit_add_circle n δ ↔
∃ m < n, gcd m n = 1 ∧ ‖x - ↑((m : ℝ) / n)‖ < δ :=
begin
haveI := real.fact_zero_lt_one,
simp only [mem_approx_add_order_of_iff, mem_set_of_eq, ball, exists_prop, dist_eq_norm,
add_circle.add_order_of_eq_pos_iff hn, mul_one],
split,
{ rintros ⟨y, ⟨m, hm₁, hm₂, rfl⟩, hx⟩, exact ⟨m, hm₁, hm₂, hx⟩, },
{ rintros ⟨m, hm₁, hm₂, hx⟩, exact ⟨↑((m : ℝ) / n), ⟨m, hm₁, hm₂, rfl⟩, hx⟩, },
end
lemma mem_add_well_approximable_iff (δ : ℕ → ℝ) (x : unit_add_circle) :
x ∈ add_well_approximable unit_add_circle δ ↔
{n : ℕ | ∃ m < n, gcd m n = 1 ∧ ‖x - ↑((m : ℝ) / n)‖ < δ n}.infinite :=
begin
simp only [mem_add_well_approximable_iff, ← nat.cofinite_eq_at_top, cofinite.blimsup_set_eq,
mem_set_of_eq],
refine iff_of_eq (congr_arg set.infinite $ ext (λ n, ⟨λ hn, _, λ hn, _⟩)),
{ exact (mem_approx_add_order_of_iff hn.1).mp hn.2, },
{ have h : 0 < n := by { obtain ⟨m, hm₁, hm₂, hm₃⟩ := hn, exact pos_of_gt hm₁, },
exact ⟨h, (mem_approx_add_order_of_iff h).mpr hn⟩, },
end
end unit_add_circle
namespace add_circle
variables {T : ℝ} [hT : fact (0 < T)]
include hT
local notation a `∤` b := ¬ a ∣ b
local notation a `∣∣` b := (a ∣ b) ∧ (a*a ∤ b)
local notation `𝕊` := add_circle T
/-- *Gallagher's ergodic theorem* on Diophantine approximation. -/
theorem add_well_approximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : tendsto δ at_top (𝓝 0)) :
(∀ᵐ x, ¬ add_well_approximable 𝕊 δ x) ∨ ∀ᵐ x, add_well_approximable 𝕊 δ x :=
begin
/- Sketch of proof:
Let `E := add_well_approximable 𝕊 δ`. For each prime `p : ℕ`, we can partition `E` into three
pieces `E = (A p) ∪ (B p) ∪ (C p)` where:
`A p = blimsup (approx_add_order_of 𝕊 n (δ n)) at_top (λ n, 0 < n ∧ (p ∤ n))`
`B p = blimsup (approx_add_order_of 𝕊 n (δ n)) at_top (λ n, 0 < n ∧ (p ∣∣ n))`
`C p = blimsup (approx_add_order_of 𝕊 n (δ n)) at_top (λ n, 0 < n ∧ (p*p ∣ n))`.
(In other words, `A p` is the set of points `x` for which there exist infinitely-many `n` such
that `x` is within a distance `δ n` of a point of order `n` and `p ∤ n`. Similarly for `B`, `C`.)
These sets have the following key properties:
1. `A p` is almost invariant under the ergodic map `y ↦ p • y`
2. `B p` is almost invariant under the ergodic map `y ↦ p • y + 1/p`
3. `C p` is invariant under the map `y ↦ y + 1/p`
To prove 1 and 2 we need the key result `blimsup_thickening_mul_ae_eq` but 3 is elementary.
It follows from `add_circle.ergodic_nsmul_add` and `ergodic.ae_empty_or_univ_of_image_ae_le` that
if either `A p` or `B p` is not almost empty for any `p`, then it is almost full and thus so is
`E`. We may therefore assume that both `A p` and `B p` are almost empty for all `p`. We thus have
`E` is almost equal to `C p` for every prime. Combining this with 3 we find that `E` is almost
invariant under the map `y ↦ y + 1/p` for every prime `p`. The required result then follows from
`add_circle.ae_empty_or_univ_of_forall_vadd_ae_eq_self`. -/
letI : semilattice_sup nat.primes := nat.subtype.semilattice_sup _,
set μ : measure 𝕊 := volume,
set u : nat.primes → 𝕊 := λ p, ↑(((↑(1 : ℕ) : ℝ) / p) * T),
have hu₀ : ∀ (p : nat.primes), add_order_of (u p) = (p : ℕ),
{ rintros ⟨p, hp⟩, exact add_order_of_div_of_gcd_eq_one hp.pos (gcd_one_left p), },
have hu : tendsto (add_order_of ∘ u) at_top at_top,
{ rw (funext hu₀ : add_order_of ∘ u = coe),
have h_mono : monotone (coe : nat.primes → ℕ) := λ p q hpq, hpq,
refine h_mono.tendsto_at_top_at_top (λ n, _),
obtain ⟨p, hp, hp'⟩ := n.exists_infinite_primes,
exact ⟨⟨p, hp'⟩, hp⟩, },
set E := add_well_approximable 𝕊 δ,
set X : ℕ → set 𝕊 := λ n, approx_add_order_of 𝕊 n (δ n),
set A : ℕ → set 𝕊 := λ p, blimsup X at_top (λ n, 0 < n ∧ (p ∤ n)),
set B : ℕ → set 𝕊 := λ p, blimsup X at_top (λ n, 0 < n ∧ (p ∣∣ n)),
set C : ℕ → set 𝕊 := λ p, blimsup X at_top (λ n, 0 < n ∧ (p^2 ∣ n)),
have hA₀ : ∀ p, measurable_set (A p) :=
λ p, measurable_set.measurable_set_blimsup (λ n hn, is_open_thickening.measurable_set),
have hB₀ : ∀ p, measurable_set (B p) :=
λ p, measurable_set.measurable_set_blimsup (λ n hn, is_open_thickening.measurable_set),
have hE₀ : null_measurable_set E μ,
{ refine (measurable_set.measurable_set_blimsup
(λ n hn, is_open.measurable_set _)).null_measurable_set,
exact is_open_thickening, },
have hE₁ : ∀ p, E = (A p) ∪ (B p) ∪ (C p),
{ intros p,
simp only [E, add_well_approximable, ← blimsup_or_eq_sup, ← and_or_distrib_left, ← sup_eq_union,
sq],
congr,
refine funext (λ n, propext $ iff_self_and.mpr (λ hn, _)),
-- `tauto` can finish from here but unfortunately it's very slow.
simp only [(em (p ∣ n)).symm, (em (p*p ∣ n)).symm, or_and_distrib_left, or_true, true_and,
or_assoc], },
have hE₂ : ∀ (p : nat.primes), A p =ᵐ[μ] (∅ : set 𝕊) ∧ B p =ᵐ[μ] (∅ : set 𝕊) → E =ᵐ[μ] C p,
{ rintros p ⟨hA, hB⟩,
rw hE₁ p,
exact union_ae_eq_right_of_ae_eq_empty ((union_ae_eq_right_of_ae_eq_empty hA).trans hB), },
have hA : ∀ (p : nat.primes), A p =ᵐ[μ] (∅ : set 𝕊) ∨ A p =ᵐ[μ] univ,
{ rintros ⟨p, hp⟩,
let f : 𝕊 → 𝕊 := λ y, (p : ℕ) • y,
suffices : f '' (A p) ⊆
blimsup (λ n, approx_add_order_of 𝕊 n (p * δ n)) at_top (λ n, 0 < n ∧ (p ∤ n)),
{ apply (ergodic_nsmul hp.one_lt).ae_empty_or_univ_of_image_ae_le (hA₀ p),
apply (has_subset.subset.eventually_le this).congr eventually_eq.rfl,
exact blimsup_thickening_mul_ae_eq μ
(λ n, 0 < n ∧ (p ∤ n)) (λ n, {y | add_order_of y = n}) (nat.cast_pos.mpr hp.pos) _ hδ, },
refine (Sup_hom.set_image f).apply_blimsup_le.trans (mono_blimsup $ λ n hn, _),
replace hn := nat.coprime_comm.mp (hp.coprime_iff_not_dvd.2 hn.2),
exact approx_add_order_of.image_nsmul_subset_of_coprime (δ n) hp.pos hn, },
have hB : ∀ (p : nat.primes), B p =ᵐ[μ] (∅ : set 𝕊) ∨ B p =ᵐ[μ] univ,
{ rintros ⟨p, hp⟩,
let x := u ⟨p, hp⟩,
let f : 𝕊 → 𝕊 := λ y, p • y + x,
suffices : f '' (B p) ⊆
blimsup (λ n, approx_add_order_of 𝕊 n (p * δ n)) at_top (λ n, 0 < n ∧ (p ∣∣ n)),
{ apply (ergodic_nsmul_add x hp.one_lt).ae_empty_or_univ_of_image_ae_le (hB₀ p),
apply (has_subset.subset.eventually_le this).congr eventually_eq.rfl,
exact blimsup_thickening_mul_ae_eq μ
(λ n, 0 < n ∧ (p ∣∣ n)) (λ n, {y | add_order_of y = n}) (nat.cast_pos.mpr hp.pos) _ hδ, },
refine (Sup_hom.set_image f).apply_blimsup_le.trans (mono_blimsup _),
rintros n ⟨hn, h_div, h_ndiv⟩,
have h_cop : (add_order_of x).coprime (n/p),
{ obtain ⟨q, rfl⟩ := h_div,
rw [hu₀, subtype.coe_mk, hp.coprime_iff_not_dvd, q.mul_div_cancel_left hp.pos],
exact λ contra, h_ndiv (mul_dvd_mul_left p contra), },
replace h_div : n / p * p = n := nat.div_mul_cancel h_div,
have hf : f = (λ y, x + y) ∘ (λ y, p • y), { ext, simp [add_comm x], },
simp_rw [comp_app],
rw [le_eq_subset, Sup_hom.set_image_to_fun, hf, image_comp],
have := @monotone_image 𝕊 𝕊 (λ y, x + y),
specialize this (approx_add_order_of.image_nsmul_subset (δ n) (n/p) hp.pos),
simp only [h_div] at this ⊢,
refine this.trans _,
convert approx_add_order_of.vadd_subset_of_coprime (p * δ n) h_cop,
simp only [hu₀, subtype.coe_mk, h_div, mul_comm p], },
change (∀ᵐ x, x ∉ E) ∨ E ∈ volume.ae,
rw [← eventually_eq_empty, ← eventually_eq_univ],
have hC : ∀ (p : nat.primes), (u p) +ᵥ C p = C p,
{ intros p,
let e := (add_action.to_perm (u p) : equiv.perm 𝕊).to_order_iso_set,
change e (C p) = C p,
rw [e.apply_blimsup, ← hu₀ p],
exact blimsup_congr (eventually_of_forall $ λ n hn,
approx_add_order_of.vadd_eq_of_mul_dvd (δ n) hn.1 hn.2), },
by_cases h : ∀ (p : nat.primes), A p =ᵐ[μ] (∅ : set 𝕊) ∧ B p =ᵐ[μ] (∅ : set 𝕊),
{ replace h : ∀ (p : nat.primes), ((u p) +ᵥ E : set _) =ᵐ[μ] E,
{ intros p,
replace hE₂ : E =ᵐ[μ] C p := hE₂ p (h p),
have h_qmp : measure_theory.measure.quasi_measure_preserving ((+ᵥ) (-u p)) μ μ :=
(measure_preserving_vadd _ μ).quasi_measure_preserving,
refine (h_qmp.vadd_ae_eq_of_ae_eq (u p) hE₂).trans (ae_eq_trans _ hE₂.symm),
rw hC, },
exact ae_empty_or_univ_of_forall_vadd_ae_eq_self hE₀ h hu, },
{ right,
simp only [not_forall, not_and_distrib] at h,
obtain ⟨p, hp⟩ := h,
rw hE₁ p,
cases hp,
{ cases hA p, { contradiction, },
simp only [h, union_ae_eq_univ_of_ae_eq_univ_left], },
{ cases hB p, { contradiction, },
simp only [h, union_ae_eq_univ_of_ae_eq_univ_left, union_ae_eq_univ_of_ae_eq_univ_right], } },
end
end add_circle
|
6ba294f294d67a1000db70a19cc8b36577d2a605 | 4efff1f47634ff19e2f786deadd394270a59ecd2 | /src/ring_theory/integral_domain.lean | 273ea66c82f0b389d2fe75d4a4a0a003f6cf6475 | [
"Apache-2.0"
] | permissive | agjftucker/mathlib | d634cd0d5256b6325e3c55bb7fb2403548371707 | 87fe50de17b00af533f72a102d0adefe4a2285e8 | refs/heads/master | 1,625,378,131,941 | 1,599,166,526,000 | 1,599,166,526,000 | 160,748,509 | 0 | 0 | Apache-2.0 | 1,544,141,789,000 | 1,544,141,789,000 | null | UTF-8 | Lean | false | false | 6,774 | lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes
-/
import data.fintype.card
import data.polynomial.ring_division
import group_theory.order_of_element
import algebra.geom_sum
/-!
# Integral domains
Assorted theorems about integral domains.
## Main theorems
* `is_cyclic_of_subgroup_integral_domain` : A finite subgroup of the units of an integral domain
is cyclic.
* `field_of_integral_domain` : A finite integral domain is a field.
## Tags
integral domain, finite integral domain, finite field
-/
section
open finset polynomial function
open_locale big_operators nat
variables {R : Type*} {G : Type*} [integral_domain R] [group G] [fintype G]
lemma card_nth_roots_subgroup_units (f : G →* R) (hf : injective f) {n : ℕ} (hn : 0 < n) (g₀ : G) :
({g ∈ univ | g ^ n = g₀} : finset G).card ≤ (nth_roots n (f g₀)).card :=
begin
apply card_le_card_of_inj_on f,
{ intros g hg, rw [sep_def, mem_filter] at hg, rw [mem_nth_roots hn, ← f.map_pow, hg.2] },
{ intros, apply hf, assumption }
end
/-- A finite subgroup of the unit group of an integral domain is cyclic. -/
lemma is_cyclic_of_subgroup_integral_domain (f : G →* R) (hf : injective f) : is_cyclic G :=
begin
haveI := classical.dec_eq G,
apply is_cyclic_of_card_pow_eq_one_le,
intros n hn,
convert (le_trans (card_nth_roots_subgroup_units f hf hn 1) (card_nth_roots n (f 1)))
end
/-- The unit group of a finite integral domain is cyclic. -/
instance [fintype R] : is_cyclic (units R) :=
is_cyclic_of_subgroup_integral_domain (units.coe_hom R) $ units.ext
/-- Every finite integral domain is a field. -/
def field_of_integral_domain [decidable_eq R] [fintype R] : field R :=
{ inv := λ a, if h : a = 0 then 0
else fintype.bij_inv (show function.bijective (* a),
from fintype.injective_iff_bijective.1 $ λ _ _, mul_right_cancel' h) 1,
mul_inv_cancel := λ a ha, show a * dite _ _ _ = _, by rw [dif_neg ha, mul_comm];
exact fintype.right_inverse_bij_inv (show function.bijective (* a), from _) 1,
inv_zero := dif_pos rfl,
..show integral_domain R, by apply_instance }
section
variables (S : set (units R)) [is_subgroup S] [fintype S]
/-- A finite subgroup of the units of an integral domain is cyclic. -/
instance subgroup_units_cyclic : is_cyclic S :=
begin
refine is_cyclic_of_subgroup_integral_domain ⟨(coe : S → R), _, _⟩
(units.ext.comp subtype.val_injective),
{ simp only [is_submonoid.coe_one, units.coe_one, coe_coe] },
{ intros, simp only [is_submonoid.coe_mul, units.coe_mul, coe_coe] },
end
end
lemma card_fiber_eq_of_mem_range {H : Type*} [group H] [decidable_eq H]
(f : G →* H) {x y : H} (hx : x ∈ set.range f) (hy : y ∈ set.range f) :
(univ.filter $ λ g, f g = x).card = (univ.filter $ λ g, f g = y).card :=
begin
rcases hx with ⟨x, rfl⟩,
rcases hy with ⟨y, rfl⟩,
refine card_congr (λ g _, g * x⁻¹ * y) _ _ (λ g hg, ⟨g * y⁻¹ * x, _⟩),
{ simp only [mem_filter, one_mul, monoid_hom.map_mul, mem_univ, mul_right_inv,
eq_self_iff_true, monoid_hom.map_mul_inv, and_self, forall_true_iff] {contextual := tt} },
{ simp only [mul_left_inj, imp_self, forall_2_true_iff], },
{ simp only [true_and, mem_filter, mem_univ] at hg,
simp only [hg, mem_filter, one_mul, monoid_hom.map_mul, mem_univ, mul_right_inv,
eq_self_iff_true, exists_prop_of_true, monoid_hom.map_mul_inv, and_self,
mul_inv_cancel_right, inv_mul_cancel_right], }
end
/-- In an integral domain, a sum indexed by a nontrivial homomorphism from a finite group is zero. -/
lemma sum_hom_units_eq_zero (f : G →* R) (hf : f ≠ 1) : ∑ g : G, f g = 0 :=
begin
classical,
obtain ⟨x, hx⟩ : ∃ x : set.range f.to_hom_units, ∀ y : set.range f.to_hom_units, y ∈ powers x,
from is_cyclic.exists_monoid_generator (set.range (f.to_hom_units)),
have hx1 : x ≠ 1,
{ rintro rfl,
apply hf,
ext g,
rw [monoid_hom.one_apply],
cases hx ⟨f.to_hom_units g, g, rfl⟩ with n hn,
rwa [subtype.ext_iff, units.ext_iff, subtype.coe_mk, monoid_hom.coe_to_hom_units,
is_submonoid.coe_pow, units.coe_pow, is_submonoid.coe_one, units.coe_one,
_root_.one_pow, eq_comm] at hn, },
replace hx1 : (x : R) - 1 ≠ 0,
from λ h, hx1 (subtype.eq (units.ext (sub_eq_zero.1 h))),
let c := (univ.filter (λ g, f.to_hom_units g = 1)).card,
calc ∑ g : G, f g
= ∑ g : G, f.to_hom_units g : rfl
... = ∑ u : units R in univ.image f.to_hom_units, (univ.filter (λ g, f.to_hom_units g = u)).card • u :
sum_comp (coe : units R → R) f.to_hom_units
... = ∑ u : units R in univ.image f.to_hom_units, c • u :
sum_congr rfl (λ u hu, congr_arg2 _ _ rfl) -- remaining goal 1, proven below
... = ∑ b : set.range f.to_hom_units, c • ↑b : finset.sum_subtype
(by simp only [mem_image, set.mem_range, forall_const, iff_self, mem_univ, exists_prop_of_true]) _
... = c • ∑ b : set.range f.to_hom_units, (b : R) : smul_sum.symm
... = c • 0 : congr_arg2 _ rfl _ -- remaining goal 2, proven below
... = 0 : smul_zero _,
{ -- remaining goal 1
show (univ.filter (λ (g : G), f.to_hom_units g = u)).card = c,
apply card_fiber_eq_of_mem_range f.to_hom_units,
{ simpa only [mem_image, mem_univ, exists_prop_of_true, set.mem_range] using hu, },
{ exact ⟨1, f.to_hom_units.map_one⟩ } },
-- remaining goal 2
show ∑ b : set.range f.to_hom_units, (b : R) = 0,
calc ∑ b : set.range f.to_hom_units, (b : R)
= ∑ n in range (order_of x), x ^ n :
eq.symm $ sum_bij (λ n _, x ^ n)
(by simp only [mem_univ, forall_true_iff])
(by simp only [is_submonoid.coe_pow, eq_self_iff_true, units.coe_pow, coe_coe, forall_true_iff])
(λ m n hm hn, pow_injective_of_lt_order_of _
(by simpa only [mem_range] using hm)
(by simpa only [mem_range] using hn))
(λ b hb, let ⟨n, hn⟩ := hx b in ⟨n % order_of x, mem_range.2 (nat.mod_lt _ (order_of_pos _)),
by rw [← pow_eq_mod_order_of, hn]⟩)
... = 0 : _,
rw [← mul_left_inj' hx1, zero_mul, ← geom_series, geom_sum_mul, coe_coe],
norm_cast,
rw [pow_order_of_eq_one, is_submonoid.coe_one, units.coe_one, sub_self],
end
/-- In an integral domain, a sum indexed by a homomorphism from a finite group is zero,
unless the homomorphism is trivial, in which case the sum is equal to the cardinality of the group. -/
lemma sum_hom_units (f : G →* R) [decidable (f = 1)] :
∑ g : G, f g = if f = 1 then fintype.card G else 0 :=
begin
split_ifs with h h,
{ simp [h, card_univ] },
{ exact sum_hom_units_eq_zero f h }
end
end
|
24e2af0e0120d3dee095194a935e8331717ddd9f | bbecf0f1968d1fba4124103e4f6b55251d08e9c4 | /src/measure_theory/measurable_space.lean | 4adf936993bf2884a7200d29481b2a3a253d22dc | [
"Apache-2.0"
] | permissive | waynemunro/mathlib | e3fd4ff49f4cb43d4a8ded59d17be407bc5ee552 | 065a70810b5480d584033f7bbf8e0409480c2118 | refs/heads/master | 1,693,417,182,397 | 1,634,644,781,000 | 1,634,644,781,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 48,701 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import measure_theory.measurable_space_def
import measure_theory.tactic
import data.tprod
import data.equiv.fin
/-!
# Measurable spaces and measurable functions
This file provides properties of measurable spaces and the functions and isomorphisms
between them. The definition of a measurable space is in `measure_theory.measurable_space_def`.
A measurable space is a set equipped with a σ-algebra, a collection of
subsets closed under complementation and countable union. A function
between measurable spaces is measurable if the preimage of each
measurable subset is measurable.
σ-algebras on a fixed set `α` form a complete lattice. Here we order
σ-algebras by writing `m₁ ≤ m₂` if every set which is `m₁`-measurable is
also `m₂`-measurable (that is, `m₁` is a subset of `m₂`). In particular, any
collection of subsets of `α` generates a smallest σ-algebra which
contains all of them. A function `f : α → β` induces a Galois connection
between the lattices of σ-algebras on `α` and `β`.
A measurable equivalence between measurable spaces is an equivalence
which respects the σ-algebras, that is, for which both directions of
the equivalence are measurable functions.
We say that a filter `f` is measurably generated if every set `s ∈ f` includes a measurable
set `t ∈ f`. This property is useful, e.g., to extract a measurable witness of `filter.eventually`.
## Notation
* We write `α ≃ᵐ β` for measurable equivalences between the measurable spaces `α` and `β`.
This should not be confused with `≃ₘ` which is used for diffeomorphisms between manifolds.
## Implementation notes
Measurability of a function `f : α → β` between measurable spaces is
defined in terms of the Galois connection induced by f.
## References
* <https://en.wikipedia.org/wiki/Measurable_space>
* <https://en.wikipedia.org/wiki/Sigma-algebra>
* <https://en.wikipedia.org/wiki/Dynkin_system>
## Tags
measurable space, σ-algebra, measurable function, measurable equivalence, dynkin system,
π-λ theorem, π-system
-/
open set encodable function equiv
open_locale classical filter
variables {α β γ δ δ' : Type*} {ι : Sort*} {s t u : set α}
namespace measurable_space
section functors
variables {m m₁ m₂ : measurable_space α} {m' : measurable_space β} {f : α → β} {g : β → α}
/-- The forward image of a measurable space under a function. `map f m` contains the sets
`s : set β` whose preimage under `f` is measurable. -/
protected def map (f : α → β) (m : measurable_space α) : measurable_space β :=
{ measurable_set' := λ s, m.measurable_set' $ f ⁻¹' s,
measurable_set_empty := m.measurable_set_empty,
measurable_set_compl := assume s hs, m.measurable_set_compl _ hs,
measurable_set_Union := assume f hf, by { rw preimage_Union, exact m.measurable_set_Union _ hf }}
@[simp] lemma map_id : m.map id = m :=
measurable_space.ext $ assume s, iff.rfl
@[simp] lemma map_comp {f : α → β} {g : β → γ} : (m.map f).map g = m.map (g ∘ f) :=
measurable_space.ext $ assume s, iff.rfl
/-- The reverse image of a measurable space under a function. `comap f m` contains the sets
`s : set α` such that `s` is the `f`-preimage of a measurable set in `β`. -/
protected def comap (f : α → β) (m : measurable_space β) : measurable_space α :=
{ measurable_set' := λ s, ∃s', m.measurable_set' s' ∧ f ⁻¹' s' = s,
measurable_set_empty := ⟨∅, m.measurable_set_empty, rfl⟩,
measurable_set_compl := assume s ⟨s', h₁, h₂⟩, ⟨s'ᶜ, m.measurable_set_compl _ h₁, h₂ ▸ rfl⟩,
measurable_set_Union := assume s hs,
let ⟨s', hs'⟩ := classical.axiom_of_choice hs in
⟨⋃ i, s' i, m.measurable_set_Union _ (λ i, (hs' i).left), by simp [hs'] ⟩ }
@[simp] lemma comap_id : m.comap id = m :=
measurable_space.ext $ assume s, ⟨assume ⟨s', hs', h⟩, h ▸ hs', assume h, ⟨s, h, rfl⟩⟩
@[simp] lemma comap_comp {f : β → α} {g : γ → β} : (m.comap f).comap g = m.comap (f ∘ g) :=
measurable_space.ext $ assume s,
⟨assume ⟨t, ⟨u, h, hu⟩, ht⟩, ⟨u, h, ht ▸ hu ▸ rfl⟩, assume ⟨t, h, ht⟩, ⟨f ⁻¹' t, ⟨_, h, rfl⟩, ht⟩⟩
lemma comap_le_iff_le_map {f : α → β} : m'.comap f ≤ m ↔ m' ≤ m.map f :=
⟨assume h s hs, h _ ⟨_, hs, rfl⟩, assume h s ⟨t, ht, heq⟩, heq ▸ h _ ht⟩
lemma gc_comap_map (f : α → β) :
galois_connection (measurable_space.comap f) (measurable_space.map f) :=
assume f g, comap_le_iff_le_map
lemma map_mono (h : m₁ ≤ m₂) : m₁.map f ≤ m₂.map f := (gc_comap_map f).monotone_u h
lemma monotone_map : monotone (measurable_space.map f) := assume a b h, map_mono h
lemma comap_mono (h : m₁ ≤ m₂) : m₁.comap g ≤ m₂.comap g := (gc_comap_map g).monotone_l h
lemma monotone_comap : monotone (measurable_space.comap g) := assume a b h, comap_mono h
@[simp] lemma comap_bot : (⊥ : measurable_space α).comap g = ⊥ := (gc_comap_map g).l_bot
@[simp] lemma comap_sup : (m₁ ⊔ m₂).comap g = m₁.comap g ⊔ m₂.comap g := (gc_comap_map g).l_sup
@[simp] lemma comap_supr {m : ι → measurable_space α} : (⨆i, m i).comap g = (⨆i, (m i).comap g) :=
(gc_comap_map g).l_supr
@[simp] lemma map_top : (⊤ : measurable_space α).map f = ⊤ := (gc_comap_map f).u_top
@[simp] lemma map_inf : (m₁ ⊓ m₂).map f = m₁.map f ⊓ m₂.map f := (gc_comap_map f).u_inf
@[simp] lemma map_infi {m : ι → measurable_space α} : (⨅i, m i).map f = (⨅i, (m i).map f) :=
(gc_comap_map f).u_infi
lemma comap_map_le : (m.map f).comap f ≤ m := (gc_comap_map f).l_u_le _
lemma le_map_comap : m ≤ (m.comap g).map g := (gc_comap_map g).le_u_l _
end functors
lemma generate_from_le_generate_from {s t : set (set α)} (h : s ⊆ t) :
generate_from s ≤ generate_from t :=
gi_generate_from.gc.monotone_l h
lemma generate_from_sup_generate_from {s t : set (set α)} :
generate_from s ⊔ generate_from t = generate_from (s ∪ t) :=
(@gi_generate_from α).gc.l_sup.symm
lemma comap_generate_from {f : α → β} {s : set (set β)} :
(generate_from s).comap f = generate_from (preimage f '' s) :=
le_antisymm
(comap_le_iff_le_map.2 $ generate_from_le $ assume t hts,
generate_measurable.basic _ $ mem_image_of_mem _ $ hts)
(generate_from_le $ assume t ⟨u, hu, eq⟩, eq ▸ ⟨u, generate_measurable.basic _ hu, rfl⟩)
end measurable_space
section measurable_functions
open measurable_space
lemma measurable_iff_le_map {m₁ : measurable_space α} {m₂ : measurable_space β} {f : α → β} :
measurable f ↔ m₂ ≤ m₁.map f :=
iff.rfl
alias measurable_iff_le_map ↔ measurable.le_map measurable.of_le_map
lemma measurable_iff_comap_le {m₁ : measurable_space α} {m₂ : measurable_space β} {f : α → β} :
measurable f ↔ m₂.comap f ≤ m₁ :=
comap_le_iff_le_map.symm
alias measurable_iff_comap_le ↔ measurable.comap_le measurable.of_comap_le
lemma measurable.mono {ma ma' : measurable_space α} {mb mb' : measurable_space β} {f : α → β}
(hf : @measurable α β ma mb f) (ha : ma ≤ ma') (hb : mb' ≤ mb) :
@measurable α β ma' mb' f :=
λ t ht, ha _ $ hf $ hb _ ht
@[measurability]
lemma measurable_from_top [measurable_space β] {f : α → β} : @measurable _ _ ⊤ _ f :=
λ s hs, trivial
lemma measurable_generate_from [measurable_space α] {s : set (set β)} {f : α → β}
(h : ∀ t ∈ s, measurable_set (f ⁻¹' t)) : @measurable _ _ _ (generate_from s) f :=
measurable.of_le_map $ generate_from_le h
variables [measurable_space α] [measurable_space β] [measurable_space γ]
@[measurability]
lemma measurable_set_preimage {f : α → β} {t : set β} (hf : measurable f) (ht : measurable_set t) :
measurable_set (f ⁻¹' t) :=
hf ht
@[measurability] lemma measurable.iterate {f : α → α} (hf : measurable f) : ∀ n, measurable (f^[n])
| 0 := measurable_id
| (n+1) := (measurable.iterate n).comp hf
@[nontriviality, measurability]
lemma subsingleton.measurable [subsingleton α] {f : α → β} : measurable f :=
λ s hs, @subsingleton.measurable_set α _ _ _
@[measurability]
lemma measurable.piecewise {s : set α} {_ : decidable_pred (∈ s)} {f g : α → β}
(hs : measurable_set s) (hf : measurable f) (hg : measurable g) :
measurable (piecewise s f g) :=
begin
intros t ht,
rw piecewise_preimage,
exact hs.ite (hf ht) (hg ht)
end
/-- this is slightly different from `measurable.piecewise`. It can be used to show
`measurable (ite (x=0) 0 1)` by
`exact measurable.ite (measurable_set_singleton 0) measurable_const measurable_const`,
but replacing `measurable.ite` by `measurable.piecewise` in that example proof does not work. -/
lemma measurable.ite {p : α → Prop} {_ : decidable_pred p} {f g : α → β}
(hp : measurable_set {a : α | p a}) (hf : measurable f) (hg : measurable g) :
measurable (λ x, ite (p x) (f x) (g x)) :=
measurable.piecewise hp hf hg
@[measurability]
lemma measurable.indicator [has_zero β] {s : set α} {f : α → β}
(hf : measurable f) (hs : measurable_set s) : measurable (s.indicator f) :=
hf.piecewise hs measurable_const
@[to_additive]
lemma measurable_one [has_one α] : measurable (1 : β → α) := @measurable_const _ _ _ _ 1
lemma measurable_of_empty [is_empty α] (f : α → β) : measurable f :=
begin
assume s hs,
convert measurable_set.empty,
exact eq_empty_of_is_empty _,
end
lemma measurable_of_empty_codomain [is_empty β] (f : α → β) : measurable f :=
by { haveI := function.is_empty f, exact measurable_of_empty f }
/-- A version of `measurable_const` that assumes `f x = f y` for all `x, y`. This version works
for functions between empty types. -/
lemma measurable_const' {f : β → α} (hf : ∀ x y, f x = f y) : measurable f :=
begin
casesI is_empty_or_nonempty β,
{ exact measurable_of_empty f },
{ convert measurable_const, exact funext (λ x, hf x h.some) }
end
@[to_additive, measurability] lemma measurable_set_mul_support [has_one β]
[measurable_singleton_class β] {f : α → β} (hf : measurable f) :
measurable_set (mul_support f) :=
hf (measurable_set_singleton 1).compl
attribute [measurability] measurable_set_support
/-- If a function coincides with a measurable function outside of a countable set, it is
measurable. -/
lemma measurable.measurable_of_countable_ne [measurable_singleton_class α]
{f g : α → β} (hf : measurable f) (h : countable {x | f x ≠ g x}) : measurable g :=
begin
assume t ht,
have : g ⁻¹' t = (g ⁻¹' t ∩ {x | f x = g x}ᶜ) ∪ (g ⁻¹' t ∩ {x | f x = g x}),
by simp [← inter_union_distrib_left],
rw this,
apply measurable_set.union (h.mono (inter_subset_right _ _)).measurable_set,
have : g ⁻¹' t ∩ {x : α | f x = g x} = f ⁻¹' t ∩ {x : α | f x = g x},
by { ext x, simp {contextual := tt} },
rw this,
exact (hf ht).inter h.measurable_set.of_compl,
end
lemma measurable_of_fintype [fintype α] [measurable_singleton_class α] (f : α → β) :
measurable f :=
λ s hs, (finite.of_fintype (f ⁻¹' s)).measurable_set
end measurable_functions
section constructions
variables [measurable_space α] [measurable_space β] [measurable_space γ]
instance : measurable_space empty := ⊤
instance : measurable_space punit := ⊤ -- this also works for `unit`
instance : measurable_space bool := ⊤
instance : measurable_space ℕ := ⊤
instance : measurable_space ℤ := ⊤
instance : measurable_space ℚ := ⊤
lemma measurable_to_encodable [encodable α] {f : β → α} (h : ∀ y, measurable_set (f ⁻¹' {f y})) :
measurable f :=
begin
assume s hs,
rw [← bUnion_preimage_singleton],
refine measurable_set.Union (λ y, measurable_set.Union_Prop $ λ hy, _),
by_cases hyf : y ∈ range f,
{ rcases hyf with ⟨y, rfl⟩,
apply h },
{ simp only [preimage_singleton_eq_empty.2 hyf, measurable_set.empty] }
end
@[measurability] lemma measurable_unit (f : unit → α) : measurable f :=
measurable_from_top
section nat
@[measurability] lemma measurable_from_nat {f : ℕ → α} : measurable f :=
measurable_from_top
lemma measurable_to_nat {f : α → ℕ} : (∀ y, measurable_set (f ⁻¹' {f y})) → measurable f :=
measurable_to_encodable
lemma measurable_find_greatest' {p : α → ℕ → Prop}
{N} (hN : ∀ k ≤ N, measurable_set {x | nat.find_greatest (p x) N = k}) :
measurable (λ x, nat.find_greatest (p x) N) :=
measurable_to_nat $ λ x, hN _ nat.find_greatest_le
lemma measurable_find_greatest {p : α → ℕ → Prop} {N} (hN : ∀ k ≤ N, measurable_set {x | p x k}) :
measurable (λ x, nat.find_greatest (p x) N) :=
begin
refine measurable_find_greatest' (λ k hk, _),
simp only [nat.find_greatest_eq_iff, set_of_and, set_of_forall, ← compl_set_of],
repeat { apply_rules [measurable_set.inter, measurable_set.const, measurable_set.Inter,
measurable_set.Inter_Prop, measurable_set.compl, hN]; try { intros } }
end
lemma measurable_find {p : α → ℕ → Prop} (hp : ∀ x, ∃ N, p x N)
(hm : ∀ k, measurable_set {x | p x k}) :
measurable (λ x, nat.find (hp x)) :=
begin
refine measurable_to_nat (λ x, _),
rw [preimage_find_eq_disjointed],
exact measurable_set.disjointed hm _
end
end nat
section subtype
instance {α} {p : α → Prop} [m : measurable_space α] : measurable_space (subtype p) :=
m.comap (coe : _ → α)
@[measurability] lemma measurable_subtype_coe {p : α → Prop} : measurable (coe : subtype p → α) :=
measurable_space.le_map_comap
@[measurability] lemma measurable.subtype_coe {p : β → Prop} {f : α → subtype p}
(hf : measurable f) :
measurable (λ a : α, (f a : β)) :=
measurable_subtype_coe.comp hf
@[measurability]
lemma measurable.subtype_mk {p : β → Prop} {f : α → β} (hf : measurable f) {h : ∀ x, p (f x)} :
measurable (λ x, (⟨f x, h x⟩ : subtype p)) :=
λ t ⟨s, hs⟩, hs.2 ▸ by simp only [← preimage_comp, (∘), subtype.coe_mk, hf hs.1]
lemma measurable_set.subtype_image {s : set α} {t : set s}
(hs : measurable_set s) : measurable_set t → measurable_set ((coe : s → α) '' t)
| ⟨u, (hu : measurable_set u), (eq : coe ⁻¹' u = t)⟩ :=
begin
rw [← eq, subtype.image_preimage_coe],
exact hu.inter hs
end
lemma measurable_of_measurable_union_cover
{f : α → β} (s t : set α) (hs : measurable_set s) (ht : measurable_set t) (h : univ ⊆ s ∪ t)
(hc : measurable (λ a : s, f a)) (hd : measurable (λ a : t, f a)) :
measurable f :=
begin
intros u hu,
convert (hs.subtype_image (hc hu)).union (ht.subtype_image (hd hu)),
change f ⁻¹' u = coe '' (coe ⁻¹' (f ⁻¹' u) : set s) ∪ coe '' (coe ⁻¹' (f ⁻¹' u) : set t),
rw [image_preimage_eq_inter_range, image_preimage_eq_inter_range, subtype.range_coe,
subtype.range_coe, ← inter_distrib_left, univ_subset_iff.1 h, inter_univ],
end
instance {α} {p : α → Prop} [measurable_space α] [measurable_singleton_class α] :
measurable_singleton_class (subtype p) :=
{ measurable_set_singleton := λ x,
begin
have : measurable_set {(x : α)} := measurable_set_singleton _,
convert @measurable_subtype_coe α _ p _ this,
ext y,
simp [subtype.ext_iff],
end }
lemma measurable_of_measurable_on_compl_finite [measurable_singleton_class α]
{f : α → β} (s : set α) (hs : finite s) (hf : measurable (set.restrict f sᶜ)) :
measurable f :=
begin
letI : fintype s := finite.fintype hs,
exact measurable_of_measurable_union_cover s sᶜ hs.measurable_set hs.measurable_set.compl
(by simp only [union_compl_self]) (measurable_of_fintype _) hf
end
lemma measurable_of_measurable_on_compl_singleton [measurable_singleton_class α]
{f : α → β} (a : α) (hf : measurable (set.restrict f {x | x ≠ a})) :
measurable f :=
measurable_of_measurable_on_compl_finite {a} (finite_singleton a) hf
end subtype
section prod
instance {α β} [m₁ : measurable_space α] [m₂ : measurable_space β] : measurable_space (α × β) :=
m₁.comap prod.fst ⊔ m₂.comap prod.snd
@[measurability] lemma measurable_fst : measurable (prod.fst : α × β → α) :=
measurable.of_comap_le le_sup_left
lemma measurable.fst {f : α → β × γ} (hf : measurable f) :
measurable (λ a : α, (f a).1) :=
measurable_fst.comp hf
@[measurability] lemma measurable_snd : measurable (prod.snd : α × β → β) :=
measurable.of_comap_le le_sup_right
lemma measurable.snd {f : α → β × γ} (hf : measurable f) :
measurable (λ a : α, (f a).2) :=
measurable_snd.comp hf
@[measurability] lemma measurable.prod {f : α → β × γ}
(hf₁ : measurable (λ a, (f a).1)) (hf₂ : measurable (λ a, (f a).2)) : measurable f :=
measurable.of_le_map $ sup_le
(by { rw [measurable_space.comap_le_iff_le_map, measurable_space.map_comp], exact hf₁ })
(by { rw [measurable_space.comap_le_iff_le_map, measurable_space.map_comp], exact hf₂ })
lemma measurable_prod {f : α → β × γ} : measurable f ↔
measurable (λ a, (f a).1) ∧ measurable (λ a, (f a).2) :=
⟨λ hf, ⟨measurable_fst.comp hf, measurable_snd.comp hf⟩, λ h, measurable.prod h.1 h.2⟩
lemma measurable.prod_mk {f : α → β} {g : α → γ} (hf : measurable f) (hg : measurable g) :
measurable (λ a : α, (f a, g a)) :=
measurable.prod hf hg
lemma measurable_prod_mk_left {x : α} : measurable (@prod.mk _ β x) :=
measurable_const.prod_mk measurable_id
lemma measurable_prod_mk_right {y : β} : measurable (λ x : α, (x, y)) :=
measurable_id.prod_mk measurable_const
lemma measurable.prod_map [measurable_space δ] {f : α → β} {g : γ → δ} (hf : measurable f)
(hg : measurable g) : measurable (prod.map f g) :=
(hf.comp measurable_fst).prod_mk (hg.comp measurable_snd)
lemma measurable.of_uncurry_left {f : α → β → γ} (hf : measurable (uncurry f)) {x : α} :
measurable (f x) :=
hf.comp measurable_prod_mk_left
lemma measurable.of_uncurry_right {f : α → β → γ} (hf : measurable (uncurry f)) {y : β} :
measurable (λ x, f x y) :=
hf.comp measurable_prod_mk_right
@[measurability] lemma measurable_swap : measurable (prod.swap : α × β → β × α) :=
measurable.prod measurable_snd measurable_fst
lemma measurable_swap_iff {f : α × β → γ} : measurable (f ∘ prod.swap) ↔ measurable f :=
⟨λ hf, by { convert hf.comp measurable_swap, ext ⟨x, y⟩, refl }, λ hf, hf.comp measurable_swap⟩
@[measurability]
lemma measurable_set.prod {s : set α} {t : set β} (hs : measurable_set s) (ht : measurable_set t) :
measurable_set (s.prod t) :=
measurable_set.inter (measurable_fst hs) (measurable_snd ht)
lemma measurable_set_prod_of_nonempty {s : set α} {t : set β} (h : (s.prod t).nonempty) :
measurable_set (s.prod t) ↔ measurable_set s ∧ measurable_set t :=
begin
rcases h with ⟨⟨x, y⟩, hx, hy⟩,
refine ⟨λ hst, _, λ h, h.1.prod h.2⟩,
have : measurable_set ((λ x, (x, y)) ⁻¹' s.prod t) := measurable_id.prod_mk measurable_const hst,
have : measurable_set (prod.mk x ⁻¹' s.prod t) := measurable_const.prod_mk measurable_id hst,
simp * at *
end
lemma measurable_set_prod {s : set α} {t : set β} :
measurable_set (s.prod t) ↔ (measurable_set s ∧ measurable_set t) ∨ s = ∅ ∨ t = ∅ :=
begin
cases (s.prod t).eq_empty_or_nonempty with h h,
{ simp [h, prod_eq_empty_iff.mp h] },
{ simp [←not_nonempty_iff_eq_empty, prod_nonempty_iff.mp h, measurable_set_prod_of_nonempty h] }
end
lemma measurable_set_swap_iff {s : set (α × β)} :
measurable_set (prod.swap ⁻¹' s) ↔ measurable_set s :=
⟨λ hs, by { convert measurable_swap hs, ext ⟨x, y⟩, refl }, λ hs, measurable_swap hs⟩
lemma measurable_from_prod_encodable [encodable β] [measurable_singleton_class β]
{f : α × β → γ} (hf : ∀ y, measurable (λ x, f (x, y))) :
measurable f :=
begin
intros s hs,
have : f ⁻¹' s = ⋃ y, ((λ x, f (x, y)) ⁻¹' s).prod {y},
{ ext1 ⟨x, y⟩,
simp [and_assoc, and.left_comm] },
rw this,
exact measurable_set.Union (λ y, (hf y hs).prod (measurable_set_singleton y))
end
end prod
section pi
variables {π : δ → Type*}
instance measurable_space.pi [m : Π a, measurable_space (π a)] : measurable_space (Π a, π a) :=
⨆ a, (m a).comap (λ b, b a)
variables [Π a, measurable_space (π a)] [measurable_space γ]
lemma measurable_pi_iff {g : α → Π a, π a} :
measurable g ↔ ∀ a, measurable (λ x, g x a) :=
by simp_rw [measurable_iff_comap_le, measurable_space.pi, measurable_space.comap_supr,
measurable_space.comap_comp, function.comp, supr_le_iff]
@[measurability]
lemma measurable_pi_apply (a : δ) : measurable (λ f : Π a, π a, f a) :=
measurable.of_comap_le $ le_supr _ a
@[measurability]
lemma measurable.eval {a : δ} {g : α → Π a, π a}
(hg : measurable g) : measurable (λ x, g x a) :=
(measurable_pi_apply a).comp hg
@[measurability]
lemma measurable_pi_lambda (f : α → Π a, π a) (hf : ∀ a, measurable (λ c, f c a)) :
measurable f :=
measurable_pi_iff.mpr hf
/-- The function `update f a : π a → Π a, π a` is always measurable.
This doesn't require `f` to be measurable.
This should not be confused with the statement that `update f a x` is measurable. -/
@[measurability]
lemma measurable_update (f : Π (a : δ), π a) {a : δ} : measurable (update f a) :=
begin
apply measurable_pi_lambda,
intro x, by_cases hx : x = a,
{ cases hx, convert measurable_id, ext, simp },
simp_rw [update_noteq hx], apply measurable_const,
end
/- Even though we cannot use projection notation, we still keep a dot to be consistent with similar
lemmas, like `measurable_set.prod`. -/
@[measurability]
lemma measurable_set.pi {s : set δ} {t : Π i : δ, set (π i)} (hs : countable s)
(ht : ∀ i ∈ s, measurable_set (t i)) :
measurable_set (s.pi t) :=
by { rw [pi_def], exact measurable_set.bInter hs (λ i hi, measurable_pi_apply _ (ht i hi)) }
lemma measurable_set.univ_pi [encodable δ] {t : Π i : δ, set (π i)}
(ht : ∀ i, measurable_set (t i)) : measurable_set (pi univ t) :=
measurable_set.pi (countable_encodable _) (λ i _, ht i)
lemma measurable_set_pi_of_nonempty {s : set δ} {t : Π i, set (π i)} (hs : countable s)
(h : (pi s t).nonempty) : measurable_set (pi s t) ↔ ∀ i ∈ s, measurable_set (t i) :=
begin
rcases h with ⟨f, hf⟩, refine ⟨λ hst i hi, _, measurable_set.pi hs⟩,
convert measurable_update f hst, rw [update_preimage_pi hi], exact λ j hj _, hf j hj
end
lemma measurable_set_pi {s : set δ} {t : Π i, set (π i)} (hs : countable s) :
measurable_set (pi s t) ↔ (∀ i ∈ s, measurable_set (t i)) ∨ pi s t = ∅ :=
begin
cases (pi s t).eq_empty_or_nonempty with h h,
{ simp [h] },
{ simp [measurable_set_pi_of_nonempty hs, h, ← not_nonempty_iff_eq_empty] }
end
section
variable (π)
@[measurability]
lemma measurable_pi_equiv_pi_subtype_prod_symm (p : δ → Prop) [decidable_pred p] :
measurable (equiv.pi_equiv_pi_subtype_prod p π).symm :=
begin
apply measurable_pi_iff.2 (λ j, _),
by_cases hj : p j,
{ simp only [hj, dif_pos, equiv.pi_equiv_pi_subtype_prod_symm_apply],
have : measurable (λ (f : (Π (i : {x // p x}), π ↑i)), f ⟨j, hj⟩) :=
measurable_pi_apply ⟨j, hj⟩,
exact measurable.comp this measurable_fst },
{ simp only [hj, equiv.pi_equiv_pi_subtype_prod_symm_apply, dif_neg, not_false_iff],
have : measurable (λ (f : (Π (i : {x // ¬ p x}), π ↑i)), f ⟨j, hj⟩) :=
measurable_pi_apply ⟨j, hj⟩,
exact measurable.comp this measurable_snd }
end
@[measurability]
lemma measurable_pi_equiv_pi_subtype_prod (p : δ → Prop) [decidable_pred p] :
measurable (equiv.pi_equiv_pi_subtype_prod p π) :=
begin
refine measurable_prod.2 _,
split;
{ apply measurable_pi_iff.2 (λ j, _),
simp only [pi_equiv_pi_subtype_prod_apply, measurable_pi_apply] }
end
end
section fintype
local attribute [instance] fintype.encodable
lemma measurable_set.pi_fintype [fintype δ] {s : set δ} {t : Π i, set (π i)}
(ht : ∀ i ∈ s, measurable_set (t i)) : measurable_set (pi s t) :=
measurable_set.pi (countable_encodable _) ht
lemma measurable_set.univ_pi_fintype [fintype δ] {t : Π i, set (π i)}
(ht : ∀ i, measurable_set (t i)) : measurable_set (pi univ t) :=
measurable_set.pi_fintype (λ i _, ht i)
end fintype
end pi
instance tprod.measurable_space (π : δ → Type*) [∀ x, measurable_space (π x)] :
∀ (l : list δ), measurable_space (list.tprod π l)
| [] := punit.measurable_space
| (i :: is) := @prod.measurable_space _ _ _ (tprod.measurable_space is)
section tprod
open list
variables {π : δ → Type*} [∀ x, measurable_space (π x)]
lemma measurable_tprod_mk (l : list δ) : measurable (@tprod.mk δ π l) :=
begin
induction l with i l ih,
{ exact measurable_const },
{ exact (measurable_pi_apply i).prod_mk ih }
end
lemma measurable_tprod_elim : ∀ {l : list δ} {i : δ} (hi : i ∈ l),
measurable (λ (v : tprod π l), v.elim hi)
| (i :: is) j hj := begin
by_cases hji : j = i,
{ subst hji, simp [measurable_fst] },
{ rw [funext $ tprod.elim_of_ne _ hji],
exact (measurable_tprod_elim (hj.resolve_left hji)).comp measurable_snd }
end
lemma measurable_tprod_elim' {l : list δ} (h : ∀ i, i ∈ l) :
measurable (tprod.elim' h : tprod π l → Π i, π i) :=
measurable_pi_lambda _ (λ i, measurable_tprod_elim (h i))
lemma measurable_set.tprod (l : list δ) {s : ∀ i, set (π i)} (hs : ∀ i, measurable_set (s i)) :
measurable_set (set.tprod l s) :=
by { induction l with i l ih, exact measurable_set.univ, exact (hs i).prod ih }
end tprod
instance {α β} [m₁ : measurable_space α] [m₂ : measurable_space β] : measurable_space (α ⊕ β) :=
m₁.map sum.inl ⊓ m₂.map sum.inr
section sum
@[measurability] lemma measurable_inl : measurable (@sum.inl α β) :=
measurable.of_le_map inf_le_left
@[measurability] lemma measurable_inr : measurable (@sum.inr α β) :=
measurable.of_le_map inf_le_right
lemma measurable_sum {f : α ⊕ β → γ}
(hl : measurable (f ∘ sum.inl)) (hr : measurable (f ∘ sum.inr)) : measurable f :=
measurable.of_comap_le $ le_inf
(measurable_space.comap_le_iff_le_map.2 $ hl)
(measurable_space.comap_le_iff_le_map.2 $ hr)
@[measurability]
lemma measurable.sum_elim {f : α → γ} {g : β → γ} (hf : measurable f) (hg : measurable g) :
measurable (sum.elim f g) :=
measurable_sum hf hg
lemma measurable_set.inl_image {s : set α} (hs : measurable_set s) :
measurable_set (sum.inl '' s : set (α ⊕ β)) :=
⟨show measurable_set (sum.inl ⁻¹' _), by { rwa [preimage_image_eq], exact (λ a b, sum.inl.inj) },
have sum.inr ⁻¹' (sum.inl '' s : set (α ⊕ β)) = ∅ :=
eq_empty_of_subset_empty $ assume x ⟨y, hy, eq⟩, by contradiction,
show measurable_set (sum.inr ⁻¹' _), by { rw [this], exact measurable_set.empty }⟩
lemma measurable_set_range_inl : measurable_set (range sum.inl : set (α ⊕ β)) :=
by { rw [← image_univ], exact measurable_set.univ.inl_image }
lemma measurable_set_inr_image {s : set β} (hs : measurable_set s) :
measurable_set (sum.inr '' s : set (α ⊕ β)) :=
⟨ have sum.inl ⁻¹' (sum.inr '' s : set (α ⊕ β)) = ∅ :=
eq_empty_of_subset_empty $ assume x ⟨y, hy, eq⟩, by contradiction,
show measurable_set (sum.inl ⁻¹' _), by { rw [this], exact measurable_set.empty },
show measurable_set (sum.inr ⁻¹' _), by { rwa [preimage_image_eq], exact λ a b, sum.inr.inj }⟩
lemma measurable_set_range_inr : measurable_set (range sum.inr : set (α ⊕ β)) :=
by { rw [← image_univ], exact measurable_set_inr_image measurable_set.univ }
end sum
instance {α} {β : α → Type*} [m : Πa, measurable_space (β a)] : measurable_space (sigma β) :=
⨅a, (m a).map (sigma.mk a)
end constructions
/-- Equivalences between measurable spaces. Main application is the simplification of measurability
statements along measurable equivalences. -/
structure measurable_equiv (α β : Type*) [measurable_space α] [measurable_space β] extends α ≃ β :=
(measurable_to_fun : measurable to_equiv)
(measurable_inv_fun : measurable to_equiv.symm)
infix ` ≃ᵐ `:25 := measurable_equiv
namespace measurable_equiv
variables (α β) [measurable_space α] [measurable_space β] [measurable_space γ] [measurable_space δ]
instance : has_coe_to_fun (α ≃ᵐ β) :=
⟨λ _, α → β, λ e, e.to_equiv⟩
variables {α β}
@[simp] lemma coe_to_equiv (e : α ≃ᵐ β) : (e.to_equiv : α → β) = e := rfl
@[measurability]
protected lemma measurable (e : α ≃ᵐ β) : measurable (e : α → β) :=
e.measurable_to_fun
@[simp] lemma coe_mk (e : α ≃ β) (h1 : measurable e) (h2 : measurable e.symm) :
((⟨e, h1, h2⟩ : α ≃ᵐ β) : α → β) = e := rfl
/-- Any measurable space is equivalent to itself. -/
def refl (α : Type*) [measurable_space α] : α ≃ᵐ α :=
{ to_equiv := equiv.refl α,
measurable_to_fun := measurable_id, measurable_inv_fun := measurable_id }
instance : inhabited (α ≃ᵐ α) := ⟨refl α⟩
/-- The composition of equivalences between measurable spaces. -/
def trans (ab : α ≃ᵐ β) (bc : β ≃ᵐ γ) :
α ≃ᵐ γ :=
{ to_equiv := ab.to_equiv.trans bc.to_equiv,
measurable_to_fun := bc.measurable_to_fun.comp ab.measurable_to_fun,
measurable_inv_fun := ab.measurable_inv_fun.comp bc.measurable_inv_fun }
/-- The inverse of an equivalence between measurable spaces. -/
def symm (ab : α ≃ᵐ β) : β ≃ᵐ α :=
{ to_equiv := ab.to_equiv.symm,
measurable_to_fun := ab.measurable_inv_fun,
measurable_inv_fun := ab.measurable_to_fun }
@[simp] lemma coe_to_equiv_symm (e : α ≃ᵐ β) : (e.to_equiv.symm : β → α) = e.symm := rfl
/-- See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. -/
def simps.apply (h : α ≃ᵐ β) : α → β := h
/-- See Note [custom simps projection] -/
def simps.symm_apply (h : α ≃ᵐ β) : β → α := h.symm
initialize_simps_projections measurable_equiv
(to_equiv_to_fun → apply, to_equiv_inv_fun → symm_apply)
lemma to_equiv_injective : injective (to_equiv : (α ≃ᵐ β) → (α ≃ β)) :=
by { rintro ⟨e₁, _, _⟩ ⟨e₂, _, _⟩ (rfl : e₁ = e₂), refl }
@[ext] lemma ext {e₁ e₂ : α ≃ᵐ β} (h : (e₁ : α → β) = e₂) : e₁ = e₂ :=
to_equiv_injective $ equiv.coe_fn_injective h
@[simp] lemma symm_mk (e : α ≃ β) (h1 : measurable e) (h2 : measurable e.symm) :
(⟨e, h1, h2⟩ : α ≃ᵐ β).symm = ⟨e.symm, h2, h1⟩ := rfl
attribute [simps apply to_equiv] trans refl
@[simp] lemma symm_refl (α : Type*) [measurable_space α] : (refl α).symm = refl α := rfl
@[simp] theorem symm_comp_self (e : α ≃ᵐ β) : e.symm ∘ e = id := funext e.left_inv
@[simp] theorem self_comp_symm (e : α ≃ᵐ β) : e ∘ e.symm = id := funext e.right_inv
@[simp] theorem apply_symm_apply (e : α ≃ᵐ β) (y : β) : e (e.symm y) = y := e.right_inv y
@[simp] theorem symm_apply_apply (e : α ≃ᵐ β) (x : α) : e.symm (e x) = x := e.left_inv x
@[simp] theorem symm_trans_self (e : α ≃ᵐ β) : e.symm.trans e = refl β :=
ext e.self_comp_symm
@[simp] theorem self_trans_symm (e : α ≃ᵐ β) : e.trans e.symm = refl α :=
ext e.symm_comp_self
protected theorem surjective (e : α ≃ᵐ β) : surjective e := e.to_equiv.surjective
protected theorem bijective (e : α ≃ᵐ β) : bijective e := e.to_equiv.bijective
protected theorem injective (e : α ≃ᵐ β) : injective e := e.to_equiv.injective
/-- Equal measurable spaces are equivalent. -/
protected def cast {α β} [i₁ : measurable_space α] [i₂ : measurable_space β]
(h : α = β) (hi : i₁ == i₂) : α ≃ᵐ β :=
{ to_equiv := equiv.cast h,
measurable_to_fun := by { substI h, substI hi, exact measurable_id },
measurable_inv_fun := by { substI h, substI hi, exact measurable_id }}
protected lemma measurable_coe_iff {f : β → γ} (e : α ≃ᵐ β) :
measurable (f ∘ e) ↔ measurable f :=
iff.intro
(assume hfe,
have measurable (f ∘ (e.symm.trans e).to_equiv) := hfe.comp e.symm.measurable,
by rwa [coe_to_equiv, symm_trans_self] at this)
(λ h, h.comp e.measurable)
/-- Products of equivalent measurable spaces are equivalent. -/
def prod_congr (ab : α ≃ᵐ β) (cd : γ ≃ᵐ δ) : α × γ ≃ᵐ β × δ :=
{ to_equiv := prod_congr ab.to_equiv cd.to_equiv,
measurable_to_fun := (ab.measurable_to_fun.comp measurable_id.fst).prod_mk
(cd.measurable_to_fun.comp measurable_id.snd),
measurable_inv_fun := (ab.measurable_inv_fun.comp measurable_id.fst).prod_mk
(cd.measurable_inv_fun.comp measurable_id.snd) }
/-- Products of measurable spaces are symmetric. -/
def prod_comm : α × β ≃ᵐ β × α :=
{ to_equiv := prod_comm α β,
measurable_to_fun := measurable_id.snd.prod_mk measurable_id.fst,
measurable_inv_fun := measurable_id.snd.prod_mk measurable_id.fst }
/-- Products of measurable spaces are associative. -/
def prod_assoc : (α × β) × γ ≃ᵐ α × (β × γ) :=
{ to_equiv := prod_assoc α β γ,
measurable_to_fun := measurable_fst.fst.prod_mk $ measurable_fst.snd.prod_mk measurable_snd,
measurable_inv_fun := (measurable_fst.prod_mk measurable_snd.fst).prod_mk measurable_snd.snd }
/-- Sums of measurable spaces are symmetric. -/
def sum_congr (ab : α ≃ᵐ β) (cd : γ ≃ᵐ δ) : α ⊕ γ ≃ᵐ β ⊕ δ :=
{ to_equiv := sum_congr ab.to_equiv cd.to_equiv,
measurable_to_fun :=
begin
cases ab with ab' abm, cases ab', cases cd with cd' cdm, cases cd',
refine measurable_sum (measurable_inl.comp abm) (measurable_inr.comp cdm)
end,
measurable_inv_fun :=
begin
cases ab with ab' _ abm, cases ab', cases cd with cd' _ cdm, cases cd',
refine measurable_sum (measurable_inl.comp abm) (measurable_inr.comp cdm)
end }
/-- `set.prod s t ≃ (s × t)` as measurable spaces. -/
def set.prod (s : set α) (t : set β) : s.prod t ≃ᵐ s × t :=
{ to_equiv := equiv.set.prod s t,
measurable_to_fun := measurable_id.subtype_coe.fst.subtype_mk.prod_mk
measurable_id.subtype_coe.snd.subtype_mk,
measurable_inv_fun := measurable.subtype_mk $ measurable_id.fst.subtype_coe.prod_mk
measurable_id.snd.subtype_coe }
/-- `univ α ≃ α` as measurable spaces. -/
def set.univ (α : Type*) [measurable_space α] : (univ : set α) ≃ᵐ α :=
{ to_equiv := equiv.set.univ α,
measurable_to_fun := measurable_id.subtype_coe,
measurable_inv_fun := measurable_id.subtype_mk }
/-- `{a} ≃ unit` as measurable spaces. -/
def set.singleton (a : α) : ({a} : set α) ≃ᵐ unit :=
{ to_equiv := equiv.set.singleton a,
measurable_to_fun := measurable_const,
measurable_inv_fun := measurable_const }
/-- A set is equivalent to its image under a function `f` as measurable spaces,
if `f` is an injective measurable function that sends measurable sets to measurable sets. -/
noncomputable def set.image (f : α → β) (s : set α) (hf : injective f)
(hfm : measurable f) (hfi : ∀ s, measurable_set s → measurable_set (f '' s)) : s ≃ᵐ (f '' s) :=
{ to_equiv := equiv.set.image f s hf,
measurable_to_fun := (hfm.comp measurable_id.subtype_coe).subtype_mk,
measurable_inv_fun :=
begin
rintro t ⟨u, hu, rfl⟩, simp [preimage_preimage, set.image_symm_preimage hf],
exact measurable_subtype_coe (hfi u hu)
end }
/-- The domain of `f` is equivalent to its range as measurable spaces,
if `f` is an injective measurable function that sends measurable sets to measurable sets. -/
noncomputable def set.range (f : α → β) (hf : injective f) (hfm : measurable f)
(hfi : ∀ s, measurable_set s → measurable_set (f '' s)) :
α ≃ᵐ (range f) :=
(measurable_equiv.set.univ _).symm.trans $
(measurable_equiv.set.image f univ hf hfm hfi).trans $
measurable_equiv.cast (by rw image_univ) (by rw image_univ)
/-- `α` is equivalent to its image in `α ⊕ β` as measurable spaces. -/
def set.range_inl : (range sum.inl : set (α ⊕ β)) ≃ᵐ α :=
{ to_fun := λ ab, match ab with
| ⟨sum.inl a, _⟩ := a
| ⟨sum.inr b, p⟩ := have false, by { cases p, contradiction }, this.elim
end,
inv_fun := λ a, ⟨sum.inl a, a, rfl⟩,
left_inv := by { rintro ⟨ab, a, rfl⟩, refl },
right_inv := assume a, rfl,
measurable_to_fun := assume s (hs : measurable_set s),
begin
refine ⟨_, hs.inl_image, set.ext _⟩,
rintros ⟨ab, a, rfl⟩,
simp [set.range_inl._match_1]
end,
measurable_inv_fun := measurable.subtype_mk measurable_inl }
/-- `β` is equivalent to its image in `α ⊕ β` as measurable spaces. -/
def set.range_inr : (range sum.inr : set (α ⊕ β)) ≃ᵐ β :=
{ to_fun := λ ab, match ab with
| ⟨sum.inr b, _⟩ := b
| ⟨sum.inl a, p⟩ := have false, by { cases p, contradiction }, this.elim
end,
inv_fun := λ b, ⟨sum.inr b, b, rfl⟩,
left_inv := by { rintro ⟨ab, b, rfl⟩, refl },
right_inv := assume b, rfl,
measurable_to_fun := assume s (hs : measurable_set s),
begin
refine ⟨_, measurable_set_inr_image hs, set.ext _⟩,
rintros ⟨ab, b, rfl⟩,
simp [set.range_inr._match_1]
end,
measurable_inv_fun := measurable.subtype_mk measurable_inr }
/-- Products distribute over sums (on the right) as measurable spaces. -/
def sum_prod_distrib (α β γ) [measurable_space α] [measurable_space β] [measurable_space γ] :
(α ⊕ β) × γ ≃ᵐ (α × γ) ⊕ (β × γ) :=
{ to_equiv := sum_prod_distrib α β γ,
measurable_to_fun :=
begin
refine measurable_of_measurable_union_cover
((range sum.inl).prod univ)
((range sum.inr).prod univ)
(measurable_set_range_inl.prod measurable_set.univ)
(measurable_set_range_inr.prod measurable_set.univ)
(by { rintro ⟨a|b, c⟩; simp [set.prod_eq] })
_
_,
{ refine (set.prod (range sum.inl) univ).symm.measurable_coe_iff.1 _,
refine (prod_congr set.range_inl (set.univ _)).symm.measurable_coe_iff.1 _,
dsimp [(∘)],
convert measurable_inl,
ext ⟨a, c⟩, refl },
{ refine (set.prod (range sum.inr) univ).symm.measurable_coe_iff.1 _,
refine (prod_congr set.range_inr (set.univ _)).symm.measurable_coe_iff.1 _,
dsimp [(∘)],
convert measurable_inr,
ext ⟨b, c⟩, refl }
end,
measurable_inv_fun :=
measurable_sum
((measurable_inl.comp measurable_fst).prod_mk measurable_snd)
((measurable_inr.comp measurable_fst).prod_mk measurable_snd) }
/-- Products distribute over sums (on the left) as measurable spaces. -/
def prod_sum_distrib (α β γ) [measurable_space α] [measurable_space β] [measurable_space γ] :
α × (β ⊕ γ) ≃ᵐ (α × β) ⊕ (α × γ) :=
prod_comm.trans $ (sum_prod_distrib _ _ _).trans $ sum_congr prod_comm prod_comm
/-- Products distribute over sums as measurable spaces. -/
def sum_prod_sum (α β γ δ)
[measurable_space α] [measurable_space β] [measurable_space γ] [measurable_space δ] :
(α ⊕ β) × (γ ⊕ δ) ≃ᵐ ((α × γ) ⊕ (α × δ)) ⊕ ((β × γ) ⊕ (β × δ)) :=
(sum_prod_distrib _ _ _).trans $ sum_congr (prod_sum_distrib _ _ _) (prod_sum_distrib _ _ _)
variables {π π' : δ' → Type*} [∀ x, measurable_space (π x)] [∀ x, measurable_space (π' x)]
/-- A family of measurable equivalences `Π a, β₁ a ≃ᵐ β₂ a` generates a measurable equivalence
between `Π a, β₁ a` and `Π a, β₂ a`. -/
def Pi_congr_right (e : Π a, π a ≃ᵐ π' a) : (Π a, π a) ≃ᵐ (Π a, π' a) :=
{ to_equiv := Pi_congr_right (λ a, (e a).to_equiv),
measurable_to_fun :=
measurable_pi_lambda _ (λ i, (e i).measurable_to_fun.comp (measurable_pi_apply i)),
measurable_inv_fun :=
measurable_pi_lambda _ (λ i, (e i).measurable_inv_fun.comp (measurable_pi_apply i)) }
/-- Pi-types are measurably equivalent to iterated products. -/
@[simps {fully_applied := ff}]
noncomputable def pi_measurable_equiv_tprod {l : list δ'} (hnd : l.nodup) (h : ∀ i, i ∈ l) :
(Π i, π i) ≃ᵐ list.tprod π l :=
{ to_equiv := list.tprod.pi_equiv_tprod hnd h,
measurable_to_fun := measurable_tprod_mk l,
measurable_inv_fun := measurable_tprod_elim' h }
/-- If `α` has a unique term, then the type of function `α → β` is measurably equivalent to `β`. -/
@[simps {fully_applied := ff}] def fun_unique (α β : Type*) [unique α] [measurable_space β] :
(α → β) ≃ᵐ β :=
{ to_equiv := equiv.fun_unique α β,
measurable_to_fun := measurable_pi_apply _,
measurable_inv_fun := measurable_pi_iff.2 $ λ b, measurable_id }
/-- The space `Π i : fin 2, α i` is measurably equivalent to `α 0 × α 1`. -/
@[simps {fully_applied := ff}] def pi_fin_two (α : fin 2 → Type*) [∀ i, measurable_space (α i)] :
(Π i, α i) ≃ᵐ α 0 × α 1 :=
{ to_equiv := pi_fin_two_equiv α,
measurable_to_fun := measurable.prod (measurable_pi_apply _) (measurable_pi_apply _),
measurable_inv_fun := measurable_pi_iff.2 $
fin.forall_fin_two.2 ⟨measurable_fst, measurable_snd⟩ }
/-- The space `fin 2 → α` is measurably equivalent to `α × α`. -/
@[simps {fully_applied := ff}] def fin_two_arrow : (fin 2 → α) ≃ᵐ α × α := pi_fin_two (λ _, α)
end measurable_equiv
namespace filter
variables [measurable_space α]
/-- A filter `f` is measurably generates if each `s ∈ f` includes a measurable `t ∈ f`. -/
class is_measurably_generated (f : filter α) : Prop :=
(exists_measurable_subset : ∀ ⦃s⦄, s ∈ f → ∃ t ∈ f, measurable_set t ∧ t ⊆ s)
instance is_measurably_generated_bot : is_measurably_generated (⊥ : filter α) :=
⟨λ _ _, ⟨∅, mem_bot, measurable_set.empty, empty_subset _⟩⟩
instance is_measurably_generated_top : is_measurably_generated (⊤ : filter α) :=
⟨λ s hs, ⟨univ, univ_mem, measurable_set.univ, λ x _, hs x⟩⟩
lemma eventually.exists_measurable_mem {f : filter α} [is_measurably_generated f]
{p : α → Prop} (h : ∀ᶠ x in f, p x) :
∃ s ∈ f, measurable_set s ∧ ∀ x ∈ s, p x :=
is_measurably_generated.exists_measurable_subset h
lemma eventually.exists_measurable_mem_of_lift' {f : filter α} [is_measurably_generated f]
{p : set α → Prop} (h : ∀ᶠ s in f.lift' powerset, p s) :
∃ s ∈ f, measurable_set s ∧ p s :=
let ⟨s, hsf, hs⟩ := eventually_lift'_powerset.1 h,
⟨t, htf, htm, hts⟩ := is_measurably_generated.exists_measurable_subset hsf
in ⟨t, htf, htm, hs t hts⟩
instance inf_is_measurably_generated (f g : filter α) [is_measurably_generated f]
[is_measurably_generated g] :
is_measurably_generated (f ⊓ g) :=
begin
refine ⟨_⟩,
rintros t ⟨sf, hsf, sg, hsg, rfl⟩,
rcases is_measurably_generated.exists_measurable_subset hsf with ⟨s'f, hs'f, hmf, hs'sf⟩,
rcases is_measurably_generated.exists_measurable_subset hsg with ⟨s'g, hs'g, hmg, hs'sg⟩,
refine ⟨s'f ∩ s'g, inter_mem_inf hs'f hs'g, hmf.inter hmg, _⟩,
exact inter_subset_inter hs'sf hs'sg
end
lemma principal_is_measurably_generated_iff {s : set α} :
is_measurably_generated (𝓟 s) ↔ measurable_set s :=
begin
refine ⟨_, λ hs, ⟨λ t ht, ⟨s, mem_principal_self s, hs, ht⟩⟩⟩,
rintros ⟨hs⟩,
rcases hs (mem_principal_self s) with ⟨t, ht, htm, hts⟩,
have : t = s := subset.antisymm hts ht,
rwa ← this
end
alias principal_is_measurably_generated_iff ↔
_ measurable_set.principal_is_measurably_generated
instance infi_is_measurably_generated {f : ι → filter α} [∀ i, is_measurably_generated (f i)] :
is_measurably_generated (⨅ i, f i) :=
begin
refine ⟨λ s hs, _⟩,
rw [← equiv.plift.surjective.infi_comp, mem_infi] at hs,
rcases hs with ⟨t, ht, ⟨V, hVf, rfl⟩⟩,
choose U hUf hU using λ i, is_measurably_generated.exists_measurable_subset (hVf i),
refine ⟨⋂ i : t, U i, _, _, _⟩,
{ rw [← equiv.plift.surjective.infi_comp, mem_infi],
refine ⟨t, ht, U, hUf, rfl⟩ },
{ haveI := ht.countable.to_encodable,
refine measurable_set.Inter (λ i, (hU i).1) },
{ exact Inter_subset_Inter (λ i, (hU i).2) }
end
end filter
/-- We say that a collection of sets is countably spanning if a countable subset spans the
whole type. This is a useful condition in various parts of measure theory. For example, it is
a needed condition to show that the product of two collections generate the product sigma algebra,
see `generate_from_prod_eq`. -/
def is_countably_spanning (C : set (set α)) : Prop :=
∃ (s : ℕ → set α), (∀ n, s n ∈ C) ∧ (⋃ n, s n) = univ
lemma is_countably_spanning_measurable_set [measurable_space α] :
is_countably_spanning {s : set α | measurable_set s} :=
⟨λ _, univ, λ _, measurable_set.univ, Union_const _⟩
namespace measurable_set
/-!
### Typeclasses on `subtype measurable_set`
-/
variables [measurable_space α]
instance : has_mem α (subtype (measurable_set : set α → Prop)) :=
⟨λ a s, a ∈ (s : set α)⟩
@[simp] lemma mem_coe (a : α) (s : subtype (measurable_set : set α → Prop)) :
a ∈ (s : set α) ↔ a ∈ s := iff.rfl
instance : has_emptyc (subtype (measurable_set : set α → Prop)) :=
⟨⟨∅, measurable_set.empty⟩⟩
@[simp] lemma coe_empty : ↑(∅ : subtype (measurable_set : set α → Prop)) = (∅ : set α) := rfl
instance [measurable_singleton_class α] : has_insert α (subtype (measurable_set : set α → Prop)) :=
⟨λ a s, ⟨has_insert.insert a s, s.prop.insert a⟩⟩
@[simp] lemma coe_insert [measurable_singleton_class α] (a : α)
(s : subtype (measurable_set : set α → Prop)) :
↑(has_insert.insert a s) = (has_insert.insert a s : set α) := rfl
instance : has_compl (subtype (measurable_set : set α → Prop)) :=
⟨λ x, ⟨xᶜ, x.prop.compl⟩⟩
@[simp] lemma coe_compl (s : subtype (measurable_set : set α → Prop)) : ↑(sᶜ) = (sᶜ : set α) := rfl
instance : has_union (subtype (measurable_set : set α → Prop)) :=
⟨λ x y, ⟨x ∪ y, x.prop.union y.prop⟩⟩
@[simp] lemma coe_union (s t : subtype (measurable_set : set α → Prop)) :
↑(s ∪ t) = (s ∪ t : set α) := rfl
instance : has_inter (subtype (measurable_set : set α → Prop)) :=
⟨λ x y, ⟨x ∩ y, x.prop.inter y.prop⟩⟩
@[simp] lemma coe_inter (s t : subtype (measurable_set : set α → Prop)) :
↑(s ∩ t) = (s ∩ t : set α) := rfl
instance : has_sdiff (subtype (measurable_set : set α → Prop)) :=
⟨λ x y, ⟨x \ y, x.prop.diff y.prop⟩⟩
@[simp] lemma coe_sdiff (s t : subtype (measurable_set : set α → Prop)) :
↑(s \ t) = (s \ t : set α) := rfl
instance : has_bot (subtype (measurable_set : set α → Prop)) :=
⟨⟨⊥, measurable_set.empty⟩⟩
@[simp] lemma coe_bot : ↑(⊥ : subtype (measurable_set : set α → Prop)) = (⊥ : set α) := rfl
instance : has_top (subtype (measurable_set : set α → Prop)) :=
⟨⟨⊤, measurable_set.univ⟩⟩
@[simp] lemma coe_top : ↑(⊤ : subtype (measurable_set : set α → Prop)) = (⊤ : set α) := rfl
instance : partial_order (subtype (measurable_set : set α → Prop)) :=
partial_order.lift _ subtype.coe_injective
instance : bounded_distrib_lattice (subtype (measurable_set : set α → Prop)) :=
{ sup := (∪),
le_sup_left := λ a b, show (a : set α) ≤ a ⊔ b, from le_sup_left,
le_sup_right := λ a b, show (b : set α) ≤ a ⊔ b, from le_sup_right,
sup_le := λ a b c ha hb, show (a ⊔ b : set α) ≤ c, from sup_le ha hb,
inf := (∩),
inf_le_left := λ a b, show (a ⊓ b : set α) ≤ a, from inf_le_left,
inf_le_right := λ a b, show (a ⊓ b : set α) ≤ b, from inf_le_right,
le_inf := λ a b c ha hb, show (a : set α) ≤ b ⊓ c, from le_inf ha hb,
top := ⊤,
le_top := λ a, show (a : set α) ≤ ⊤, from le_top,
bot := ⊥,
bot_le := λ a, show (⊥ : set α) ≤ a, from bot_le,
le_sup_inf := λ x y z, show ((x ⊔ y) ⊓ (x ⊔ z) : set α) ≤ x ⊔ y ⊓ z, from le_sup_inf,
.. measurable_set.subtype.partial_order }
instance : boolean_algebra (subtype (measurable_set : set α → Prop)) :=
{ sdiff := (\),
sup_inf_sdiff := λ a b, subtype.eq $ sup_inf_sdiff a b,
inf_inf_sdiff := λ a b, subtype.eq $ inf_inf_sdiff a b,
compl := has_compl.compl,
inf_compl_le_bot := λ a, boolean_algebra.inf_compl_le_bot (a : set α),
top_le_sup_compl := λ a, boolean_algebra.top_le_sup_compl (a : set α),
sdiff_eq := λ a b, subtype.eq $ sdiff_eq,
.. measurable_set.subtype.bounded_distrib_lattice }
end measurable_set
|
57aa610adebb2c30181c3c10734c405fc5f72e39 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/linear_algebra/finsupp_vector_space.lean | 2111ed5953326dfb005f9c57daf9a8cd682e4888 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 4,637 | lean | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Johannes Hölzl
Linear structures on function with finite support `ι →₀ β`.
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.mv_polynomial.default
import Mathlib.linear_algebra.dimension
import Mathlib.linear_algebra.direct_sum.finsupp
import Mathlib.PostPort
universes u_1 u_2 u_3 u_4 u_5 u v w
namespace Mathlib
namespace finsupp
theorem linear_independent_single {R : Type u_1} {M : Type u_2} {ι : Type u_3} [ring R] [add_comm_group M] [module R M] {φ : ι → Type u_4} {f : (ι : ι) → φ ι → M} (hf : ∀ (i : ι), linear_independent R (f i)) : linear_independent R fun (ix : sigma fun (i : ι) => φ i) => single (sigma.fst ix) (f (sigma.fst ix) (sigma.snd ix)) := sorry
theorem is_basis_single {R : Type u_1} {M : Type u_2} {ι : Type u_3} [ring R] [add_comm_group M] [module R M] {φ : ι → Type u_4} (f : (ι : ι) → φ ι → M) (hf : ∀ (i : ι), is_basis R (f i)) : is_basis R fun (ix : sigma fun (i : ι) => φ i) => single (sigma.fst ix) (f (sigma.fst ix) (sigma.snd ix)) := sorry
theorem is_basis_single_one {R : Type u_1} {ι : Type u_3} [ring R] : is_basis R fun (i : ι) => single i 1 := sorry
/-- If b : ι → M and c : κ → N are bases then so is λ i, b i.1 ⊗ₜ c i.2 : ι × κ → M ⊗ N. -/
theorem is_basis.tensor_product {R : Type u_1} {M : Type u_2} {N : Type u_3} {ι : Type u_4} {κ : Type u_5} [comm_ring R] [add_comm_group M] [module R M] [add_comm_group N] [module R N] {b : ι → M} (hb : is_basis R b) {c : κ → N} (hc : is_basis R c) : is_basis R fun (i : ι × κ) => tensor_product.tmul R (b (prod.fst i)) (c (prod.snd i)) := sorry
theorem dim_eq {K : Type u} {V : Type v} {ι : Type v} [field K] [add_comm_group V] [vector_space K V] : vector_space.dim K (ι →₀ V) = cardinal.mk ι * vector_space.dim K V := sorry
end finsupp
/- We use `universe variables` instead of `universes` here because universes introduced by the
`universes` keyword do not get replaced by metavariables once a lemma has been proven. So if you
prove a lemma using universe `u`, you can only apply it to universe `u` in other lemmas of the
same section. -/
theorem equiv_of_dim_eq_lift_dim {K : Type u} {V : Type v} {V' : Type w} [field K] [add_comm_group V] [vector_space K V] [add_comm_group V'] [vector_space K V'] (h : cardinal.lift (vector_space.dim K V) = cardinal.lift (vector_space.dim K V')) : Nonempty (linear_equiv K V V') := sorry
/-- Two `K`-vector spaces are equivalent if their dimension is the same. -/
def equiv_of_dim_eq_dim {K : Type u} {V₁ : Type v} {V₂ : Type v} [field K] [add_comm_group V₁] [vector_space K V₁] [add_comm_group V₂] [vector_space K V₂] (h : vector_space.dim K V₁ = vector_space.dim K V₂) : linear_equiv K V₁ V₂ :=
Classical.choice sorry
/-- An `n`-dimensional `K`-vector space is equivalent to `fin n → K`. -/
def fin_dim_vectorspace_equiv {K : Type u} {V : Type v} [field K] [add_comm_group V] [vector_space K V] (n : ℕ) (hn : vector_space.dim K V = ↑n) : linear_equiv K V (fin n → K) :=
Classical.choice sorry
theorem eq_bot_iff_dim_eq_zero {K : Type u} {V : Type v} [field K] [add_comm_group V] [vector_space K V] (p : submodule K V) (h : vector_space.dim K ↥p = 0) : p = ⊥ :=
let e : linear_equiv K ↥p ↥⊥ :=
equiv_of_dim_eq_dim (eq.mpr (id (Eq._oldrec (Eq.refl (vector_space.dim K ↥p = vector_space.dim K ↥⊥)) dim_bot)) h);
linear_equiv.eq_bot_of_equiv p e
theorem injective_of_surjective {K : Type u} {V₁ : Type v} {V₂ : Type v} [field K] [add_comm_group V₁] [vector_space K V₁] [add_comm_group V₂] [vector_space K V₂] (f : linear_map K V₁ V₂) (hV₁ : vector_space.dim K V₁ < cardinal.omega) (heq : vector_space.dim K V₂ = vector_space.dim K V₁) (hf : linear_map.range f = ⊤) : linear_map.ker f = ⊥ := sorry
theorem cardinal_mk_eq_cardinal_mk_field_pow_dim {K : Type u} {V : Type u} [field K] [add_comm_group V] [vector_space K V] (h : vector_space.dim K V < cardinal.omega) : cardinal.mk V = cardinal.mk K ^ vector_space.dim K V := sorry
theorem cardinal_lt_omega_of_dim_lt_omega {K : Type u} {V : Type u} [field K] [add_comm_group V] [vector_space K V] [fintype K] (h : vector_space.dim K V < cardinal.omega) : cardinal.mk V < cardinal.omega :=
eq.mpr (id (Eq._oldrec (Eq.refl (cardinal.mk V < cardinal.omega)) (cardinal_mk_eq_cardinal_mk_field_pow_dim h)))
(cardinal.power_lt_omega (iff.mpr cardinal.lt_omega_iff_fintype (Nonempty.intro infer_instance)) h)
|
0a3e8d617b2b878c0165169c3c970668175bf373 | 432d948a4d3d242fdfb44b81c9e1b1baacd58617 | /src/measure_theory/set_integral.lean | 0ff1b93f8a57596ad9fd16d3299c6a8f234618eb | [
"Apache-2.0"
] | permissive | JLimperg/aesop3 | 306cc6570c556568897ed2e508c8869667252e8a | a4a116f650cc7403428e72bd2e2c4cda300fe03f | refs/heads/master | 1,682,884,916,368 | 1,620,320,033,000 | 1,620,320,033,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 46,992 | lean | /-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import measure_theory.bochner_integration
import analysis.normed_space.indicator_function
/-!
# Set integral
In this file we prove some properties of `∫ x in s, f x ∂μ`. Recall that this notation
is defined as `∫ x, f x ∂(μ.restrict s)`. In `integral_indicator` we prove that for a measurable
function `f` and a measurable set `s` this definition coincides with another natural definition:
`∫ x, indicator s f x ∂μ = ∫ x in s, f x ∂μ`, where `indicator s f x` is equal to `f x` for `x ∈ s`
and is zero otherwise.
Since `∫ x in s, f x ∂μ` is a notation, one can rewrite or apply any theorem about `∫ x, f x ∂μ`
directly. In this file we prove some theorems about dependence of `∫ x in s, f x ∂μ` on `s`, e.g.
`integral_union`, `integral_empty`, `integral_univ`.
We also define `integrable_on f s μ := integrable f (μ.restrict s)` and prove theorems like
`integrable_on_union : integrable_on f (s ∪ t) μ ↔ integrable_on f s μ ∧ integrable_on f t μ`.
Next we define a predicate `integrable_at_filter (f : α → E) (l : filter α) (μ : measure α)`
saying that `f` is integrable at some set `s ∈ l` and prove that a measurable function is integrable
at `l` with respect to `μ` provided that `f` is bounded above at `l ⊓ μ.ae` and `μ` is finite
at `l`.
Finally, we prove a version of the
[Fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus)
for set integral, see `filter.tendsto.integral_sub_linear_is_o_ae` and its corollaries.
Namely, consider a measurably generated filter `l`, a measure `μ` finite at this filter, and
a function `f` that has a finite limit `c` at `l ⊓ μ.ae`. Then `∫ x in s, f x ∂μ = μ s • c + o(μ s)`
as `s` tends to `l.lift' powerset`, i.e. for any `ε>0` there exists `t ∈ l` such that
`∥∫ x in s, f x ∂μ - μ s • c∥ ≤ ε * μ s` whenever `s ⊆ t`. We also formulate a version of this
theorem for a locally finite measure `μ` and a function `f` continuous at a point `a`.
## Notation
We provide the following notations for expressing the integral of a function on a set :
* `∫ a in s, f a ∂μ` is `measure_theory.integral (μ.restrict s) f`
* `∫ a in s, f a` is `∫ a in s, f a ∂volume`
Note that the set notations are defined in the file `measure_theory/bochner_integration`,
but we reference them here because all theorems about set integrals are in this file.
## TODO
The file ends with over a hundred lines of commented out code. This is the old contents of this file
using the `indicator` approach to the definition of `∫ x in s, f x ∂μ`. This code should be
migrated to the new definition.
-/
noncomputable theory
open set filter topological_space measure_theory function
open_locale classical topological_space interval big_operators filter ennreal measure_theory
variables {α β E F : Type*} [measurable_space α]
section piecewise
variables {μ : measure α} {s : set α} {f g : α → β}
lemma piecewise_ae_eq_restrict (hs : measurable_set s) : piecewise s f g =ᵐ[μ.restrict s] f :=
begin
rw [ae_restrict_eq hs],
exact (piecewise_eq_on s f g).eventually_eq.filter_mono inf_le_right
end
lemma piecewise_ae_eq_restrict_compl (hs : measurable_set s) :
piecewise s f g =ᵐ[μ.restrict sᶜ] g :=
begin
rw [ae_restrict_eq hs.compl],
exact (piecewise_eq_on_compl s f g).eventually_eq.filter_mono inf_le_right
end
end piecewise
section indicator_function
variables [has_zero β] {μ : measure α} {s : set α} {f : α → β}
lemma indicator_ae_eq_restrict (hs : measurable_set s) : indicator s f =ᵐ[μ.restrict s] f :=
piecewise_ae_eq_restrict hs
lemma indicator_ae_eq_restrict_compl (hs : measurable_set s) : indicator s f =ᵐ[μ.restrict sᶜ] 0 :=
piecewise_ae_eq_restrict_compl hs
end indicator_function
section
variables [measurable_space β] {l l' : filter α} {f g : α → β} {μ ν : measure α}
/-- A function `f` is measurable at filter `l` w.r.t. a measure `μ` if it is ae-measurable
w.r.t. `μ.restrict s` for some `s ∈ l`. -/
def measurable_at_filter (f : α → β) (l : filter α) (μ : measure α . volume_tac) :=
∃ s ∈ l, ae_measurable f (μ.restrict s)
@[simp] lemma measurable_at_bot {f : α → β} : measurable_at_filter f ⊥ μ :=
⟨∅, mem_bot_sets, by simp⟩
protected lemma measurable_at_filter.eventually (h : measurable_at_filter f l μ) :
∀ᶠ s in l.lift' powerset, ae_measurable f (μ.restrict s) :=
(eventually_lift'_powerset' $ λ s t, ae_measurable.mono_set).2 h
protected lemma measurable_at_filter.filter_mono (h : measurable_at_filter f l μ) (h' : l' ≤ l) :
measurable_at_filter f l' μ :=
let ⟨s, hsl, hs⟩ := h in ⟨s, h' hsl, hs⟩
protected lemma ae_measurable.measurable_at_filter (h : ae_measurable f μ) :
measurable_at_filter f l μ :=
⟨univ, univ_mem_sets, by rwa measure.restrict_univ⟩
lemma ae_measurable.measurable_at_filter_of_mem {s} (h : ae_measurable f (μ.restrict s))
(hl : s ∈ l):
measurable_at_filter f l μ :=
⟨s, hl, h⟩
protected lemma measurable.measurable_at_filter (h : measurable f) :
measurable_at_filter f l μ :=
h.ae_measurable.measurable_at_filter
end
namespace measure_theory
section normed_group
lemma has_finite_integral_restrict_of_bounded [normed_group E] {f : α → E} {s : set α}
{μ : measure α} {C} (hs : μ s < ∞) (hf : ∀ᵐ x ∂(μ.restrict s), ∥f x∥ ≤ C) :
has_finite_integral f (μ.restrict s) :=
by haveI : finite_measure (μ.restrict s) := ⟨by rwa [measure.restrict_apply_univ]⟩;
exact has_finite_integral_of_bounded hf
variables [normed_group E] [measurable_space E] {f g : α → E} {s t : set α} {μ ν : measure α}
/-- A function is `integrable_on` a set `s` if it is a measurable function and if the integral of
its pointwise norm over `s` is less than infinity. -/
def integrable_on (f : α → E) (s : set α) (μ : measure α . volume_tac) : Prop :=
integrable f (μ.restrict s)
lemma integrable_on.integrable (h : integrable_on f s μ) :
integrable f (μ.restrict s) :=
h
@[simp] lemma integrable_on_empty : integrable_on f ∅ μ :=
by simp [integrable_on, integrable_zero_measure]
@[simp] lemma integrable_on_univ : integrable_on f univ μ ↔ integrable f μ :=
by rw [integrable_on, measure.restrict_univ]
lemma integrable_on_zero : integrable_on (λ _, (0:E)) s μ := integrable_zero _ _ _
lemma integrable_on_const {C : E} : integrable_on (λ _, C) s μ ↔ C = 0 ∨ μ s < ∞ :=
integrable_const_iff.trans $ by rw [measure.restrict_apply_univ]
lemma integrable_on.mono (h : integrable_on f t ν) (hs : s ⊆ t) (hμ : μ ≤ ν) :
integrable_on f s μ :=
h.mono_measure $ measure.restrict_mono hs hμ
lemma integrable_on.mono_set (h : integrable_on f t μ) (hst : s ⊆ t) :
integrable_on f s μ :=
h.mono hst (le_refl _)
lemma integrable_on.mono_measure (h : integrable_on f s ν) (hμ : μ ≤ ν) :
integrable_on f s μ :=
h.mono (subset.refl _) hμ
lemma integrable_on.mono_set_ae (h : integrable_on f t μ) (hst : s ≤ᵐ[μ] t) :
integrable_on f s μ :=
h.integrable.mono_measure $ restrict_mono_ae hst
lemma integrable.integrable_on (h : integrable f μ) : integrable_on f s μ :=
h.mono_measure $ measure.restrict_le_self
lemma integrable.integrable_on' (h : integrable f (μ.restrict s)) : integrable_on f s μ :=
h
lemma integrable_on.left_of_union (h : integrable_on f (s ∪ t) μ) : integrable_on f s μ :=
h.mono_set $ subset_union_left _ _
lemma integrable_on.right_of_union (h : integrable_on f (s ∪ t) μ) : integrable_on f t μ :=
h.mono_set $ subset_union_right _ _
lemma integrable_on.union (hs : integrable_on f s μ) (ht : integrable_on f t μ) :
integrable_on f (s ∪ t) μ :=
(hs.add_measure ht).mono_measure $ measure.restrict_union_le _ _
@[simp] lemma integrable_on_union :
integrable_on f (s ∪ t) μ ↔ integrable_on f s μ ∧ integrable_on f t μ :=
⟨λ h, ⟨h.left_of_union, h.right_of_union⟩, λ h, h.1.union h.2⟩
@[simp] lemma integrable_on_finite_union {s : set β} (hs : finite s)
{t : β → set α} : integrable_on f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, integrable_on f (t i) μ :=
begin
apply hs.induction_on,
{ simp },
{ intros a s ha hs hf, simp [hf, or_imp_distrib, forall_and_distrib] }
end
@[simp] lemma integrable_on_finset_union {s : finset β} {t : β → set α} :
integrable_on f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, integrable_on f (t i) μ :=
integrable_on_finite_union s.finite_to_set
lemma integrable_on.add_measure (hμ : integrable_on f s μ) (hν : integrable_on f s ν) :
integrable_on f s (μ + ν) :=
by { delta integrable_on, rw measure.restrict_add, exact hμ.integrable.add_measure hν }
@[simp] lemma integrable_on_add_measure :
integrable_on f s (μ + ν) ↔ integrable_on f s μ ∧ integrable_on f s ν :=
⟨λ h, ⟨h.mono_measure (measure.le_add_right (le_refl _)),
h.mono_measure (measure.le_add_left (le_refl _))⟩,
λ h, h.1.add_measure h.2⟩
lemma ae_measurable_indicator_iff (hs : measurable_set s) :
ae_measurable f (μ.restrict s) ↔ ae_measurable (indicator s f) μ :=
begin
split,
{ assume h,
refine ⟨indicator s (h.mk f), h.measurable_mk.indicator hs, _⟩,
have A : s.indicator f =ᵐ[μ.restrict s] s.indicator (ae_measurable.mk f h) :=
(indicator_ae_eq_restrict hs).trans (h.ae_eq_mk.trans $ (indicator_ae_eq_restrict hs).symm),
have B : s.indicator f =ᵐ[μ.restrict sᶜ] s.indicator (ae_measurable.mk f h) :=
(indicator_ae_eq_restrict_compl hs).trans (indicator_ae_eq_restrict_compl hs).symm,
have : s.indicator f =ᵐ[μ.restrict s + μ.restrict sᶜ] s.indicator (ae_measurable.mk f h) :=
ae_add_measure_iff.2 ⟨A, B⟩,
simpa only [hs, measure.restrict_add_restrict_compl] using this },
{ assume h,
exact (h.mono_measure measure.restrict_le_self).congr (indicator_ae_eq_restrict hs) }
end
lemma integrable_indicator_iff (hs : measurable_set s) :
integrable (indicator s f) μ ↔ integrable_on f s μ :=
by simp [integrable_on, integrable, has_finite_integral, nnnorm_indicator_eq_indicator_nnnorm,
ennreal.coe_indicator, lintegral_indicator _ hs, ae_measurable_indicator_iff hs]
lemma integrable_on.indicator (h : integrable_on f s μ) (hs : measurable_set s) :
integrable (indicator s f) μ :=
(integrable_indicator_iff hs).2 h
lemma integrable.indicator (h : integrable f μ) (hs : measurable_set s) :
integrable (indicator s f) μ :=
h.integrable_on.indicator hs
/-- We say that a function `f` is *integrable at filter* `l` if it is integrable on some
set `s ∈ l`. Equivalently, it is eventually integrable on `s` in `l.lift' powerset`. -/
def integrable_at_filter (f : α → E) (l : filter α) (μ : measure α . volume_tac) :=
∃ s ∈ l, integrable_on f s μ
variables {l l' : filter α}
protected lemma integrable_at_filter.eventually (h : integrable_at_filter f l μ) :
∀ᶠ s in l.lift' powerset, integrable_on f s μ :=
by { refine (eventually_lift'_powerset' $ λ s t hst ht, _).2 h, exact ht.mono_set hst }
lemma integrable_at_filter.filter_mono (hl : l ≤ l') (hl' : integrable_at_filter f l' μ) :
integrable_at_filter f l μ :=
let ⟨s, hs, hsf⟩ := hl' in ⟨s, hl hs, hsf⟩
lemma integrable_at_filter.inf_of_left (hl : integrable_at_filter f l μ) :
integrable_at_filter f (l ⊓ l') μ :=
hl.filter_mono inf_le_left
lemma integrable_at_filter.inf_of_right (hl : integrable_at_filter f l μ) :
integrable_at_filter f (l' ⊓ l) μ :=
hl.filter_mono inf_le_right
@[simp] lemma integrable_at_filter.inf_ae_iff {l : filter α} :
integrable_at_filter f (l ⊓ μ.ae) μ ↔ integrable_at_filter f l μ :=
begin
refine ⟨_, λ h, h.filter_mono inf_le_left⟩,
rintros ⟨s, ⟨t, ht, u, hu, hs⟩, hf⟩,
refine ⟨t, ht, _⟩,
refine hf.integrable.mono_measure (λ v hv, _),
simp only [measure.restrict_apply hv],
refine measure_mono_ae (mem_sets_of_superset hu $ λ x hx, _),
exact λ ⟨hv, ht⟩, ⟨hv, hs ⟨ht, hx⟩⟩
end
alias integrable_at_filter.inf_ae_iff ↔ measure_theory.integrable_at_filter.of_inf_ae _
/-- If `μ` is a measure finite at filter `l` and `f` is a function such that its norm is bounded
above at `l`, then `f` is integrable at `l`. -/
lemma measure.finite_at_filter.integrable_at_filter {l : filter α} [is_measurably_generated l]
(hfm : measurable_at_filter f l μ) (hμ : μ.finite_at_filter l)
(hf : l.is_bounded_under (≤) (norm ∘ f)) :
integrable_at_filter f l μ :=
begin
obtain ⟨C, hC⟩ : ∃ C, ∀ᶠ s in (l.lift' powerset), ∀ x ∈ s, ∥f x∥ ≤ C,
from hf.imp (λ C hC, eventually_lift'_powerset.2 ⟨_, hC, λ t, id⟩),
rcases (hfm.eventually.and (hμ.eventually.and hC)).exists_measurable_mem_of_lift'
with ⟨s, hsl, hsm, hfm, hμ, hC⟩,
refine ⟨s, hsl, ⟨hfm, has_finite_integral_restrict_of_bounded hμ _⟩⟩,
exact C,
rw [ae_restrict_eq hsm, eventually_inf_principal],
exact eventually_of_forall hC
end
lemma measure.finite_at_filter.integrable_at_filter_of_tendsto_ae
{l : filter α} [is_measurably_generated l] (hfm : measurable_at_filter f l μ)
(hμ : μ.finite_at_filter l) {b} (hf : tendsto f (l ⊓ μ.ae) (𝓝 b)) :
integrable_at_filter f l μ :=
(hμ.inf_of_left.integrable_at_filter (hfm.filter_mono inf_le_left)
hf.norm.is_bounded_under_le).of_inf_ae
alias measure.finite_at_filter.integrable_at_filter_of_tendsto_ae ←
filter.tendsto.integrable_at_filter_ae
lemma measure.finite_at_filter.integrable_at_filter_of_tendsto {l : filter α}
[is_measurably_generated l] (hfm : measurable_at_filter f l μ) (hμ : μ.finite_at_filter l)
{b} (hf : tendsto f l (𝓝 b)) :
integrable_at_filter f l μ :=
hμ.integrable_at_filter hfm hf.norm.is_bounded_under_le
alias measure.finite_at_filter.integrable_at_filter_of_tendsto ← filter.tendsto.integrable_at_filter
variables [borel_space E] [second_countable_topology E]
lemma integrable_add [opens_measurable_space E] {f g : α → E}
(h : disjoint (support f) (support g)) (hf : measurable f) (hg : measurable g) :
integrable (f + g) μ ↔ integrable f μ ∧ integrable g μ :=
begin
refine ⟨λ hfg, ⟨_, _⟩, λ h, h.1.add h.2⟩,
{ rw ← indicator_add_eq_left h, exact hfg.indicator (measurable_set_support hf) },
{ rw ← indicator_add_eq_right h, exact hfg.indicator (measurable_set_support hg) }
end
/-- To prove something for an arbitrary integrable function in a second countable
Borel normed group, it suffices to show that
* the property holds for (multiples of) characteristic functions;
* is closed under addition;
* the set of functions in the `L¹` space for which the property holds is closed.
* the property is closed under the almost-everywhere equal relation.
It is possible to make the hypotheses in the induction steps a bit stronger, and such conditions
can be added once we need them (for example in `h_add` it is only necessary to consider the sum of
a simple function with a multiple of a characteristic function and that the intersection
of their images is a subset of `{0}`).
-/
@[elab_as_eliminator]
lemma integrable.induction (P : (α → E) → Prop)
(h_ind : ∀ (c : E) ⦃s⦄, measurable_set s → μ s < ∞ → P (s.indicator (λ _, c)))
(h_add : ∀ ⦃f g : α → E⦄, disjoint (support f) (support g) → integrable f μ → integrable g μ →
P f → P g → P (f + g))
(h_closed : is_closed {f : α →₁[μ] E | P f} )
(h_ae : ∀ ⦃f g⦄, f =ᵐ[μ] g → integrable f μ → P f → P g) :
∀ ⦃f : α → E⦄ (hf : integrable f μ), P f :=
begin
have : ∀ (f : simple_func α E), integrable f μ → P f,
{ refine simple_func.induction _ _,
{ intros c s hs h, dsimp only [simple_func.coe_const, simple_func.const_zero,
piecewise_eq_indicator, simple_func.coe_zero, simple_func.coe_piecewise] at h ⊢,
by_cases hc : c = 0,
{ subst hc, convert h_ind 0 measurable_set.empty (by simp) using 1, simp [const] },
apply h_ind c hs,
have : (nnnorm c : ℝ≥0∞) * μ s < ∞,
{ have := @comp_indicator _ _ _ _ (λ x : E, (nnnorm x : ℝ≥0∞)) (const α c) s,
dsimp only at this,
have h' := h.has_finite_integral,
simpa [has_finite_integral, this, lintegral_indicator, hs] using h' },
exact ennreal.lt_top_of_mul_lt_top_right this (by simp [hc]) },
{ intros f g hfg hf hg int_fg,
rw [simple_func.coe_add, integrable_add hfg f.measurable g.measurable] at int_fg,
refine h_add hfg int_fg.1 int_fg.2 (hf int_fg.1) (hg int_fg.2) } },
have : ∀ (f : α →₁ₛ[μ] E), P f,
{ intro f,
exact h_ae (L1.simple_func.to_simple_func_eq_to_fun f) (L1.simple_func.integrable f)
(this (L1.simple_func.to_simple_func f) (L1.simple_func.integrable f)) },
have : ∀ (f : α →₁[μ] E), P f :=
λ f, L1.simple_func.dense_range.induction_on f h_closed this,
exact λ f hf, h_ae hf.coe_fn_to_L1 (L1.integrable_coe_fn _) (this (hf.to_L1 f)),
end
variables [complete_space E] [normed_space ℝ E]
lemma set_integral_congr_ae (hs : measurable_set s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ :=
integral_congr_ae ((ae_restrict_iff' hs).2 h)
lemma set_integral_congr (hs : measurable_set s) (h : eq_on f g s) :
∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ :=
set_integral_congr_ae hs $ eventually_of_forall h
lemma integral_union (hst : disjoint s t) (hs : measurable_set s) (ht : measurable_set t)
(hfs : integrable_on f s μ) (hft : integrable_on f t μ) :
∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ + ∫ x in t, f x ∂μ :=
by simp only [integrable_on, measure.restrict_union hst hs ht, integral_add_measure hfs hft]
lemma integral_empty : ∫ x in ∅, f x ∂μ = 0 := by rw [measure.restrict_empty, integral_zero_measure]
lemma integral_univ : ∫ x in univ, f x ∂μ = ∫ x, f x ∂μ := by rw [measure.restrict_univ]
lemma integral_add_compl (hs : measurable_set s) (hfi : integrable f μ) :
∫ x in s, f x ∂μ + ∫ x in sᶜ, f x ∂μ = ∫ x, f x ∂μ :=
by rw [← integral_union (@disjoint_compl_right (set α) _ _) hs hs.compl
hfi.integrable_on hfi.integrable_on, union_compl_self, integral_univ]
/-- For a function `f` and a measurable set `s`, the integral of `indicator s f`
over the whole space is equal to `∫ x in s, f x ∂μ` defined as `∫ x, f x ∂(μ.restrict s)`. -/
lemma integral_indicator (hs : measurable_set s) :
∫ x, indicator s f x ∂μ = ∫ x in s, f x ∂μ :=
begin
by_cases hf : ae_measurable f (μ.restrict s), swap,
{ rw integral_non_ae_measurable hf,
rw [ae_measurable_indicator_iff hs] at hf,
exact integral_non_ae_measurable hf },
by_cases hfi : integrable_on f s μ, swap,
{ rwa [integral_undef, integral_undef],
rwa integrable_indicator_iff hs },
calc ∫ x, indicator s f x ∂μ = ∫ x in s, indicator s f x ∂μ + ∫ x in sᶜ, indicator s f x ∂μ :
(integral_add_compl hs (hfi.indicator hs)).symm
... = ∫ x in s, f x ∂μ + ∫ x in sᶜ, 0 ∂μ :
congr_arg2 (+) (integral_congr_ae (indicator_ae_eq_restrict hs))
(integral_congr_ae (indicator_ae_eq_restrict_compl hs))
... = ∫ x in s, f x ∂μ : by simp
end
lemma set_integral_const (c : E) : ∫ x in s, c ∂μ = (μ s).to_real • c :=
by rw [integral_const, measure.restrict_apply_univ]
@[simp]
lemma integral_indicator_const (e : E) ⦃s : set α⦄ (s_meas : measurable_set s) :
∫ (a : α), s.indicator (λ (x : α), e) a ∂μ = (μ s).to_real • e :=
by rw [integral_indicator s_meas, ← set_integral_const]
lemma set_integral_map {β} [measurable_space β] {g : α → β} {f : β → E} {s : set β}
(hs : measurable_set s) (hf : ae_measurable f (measure.map g μ)) (hg : measurable g) :
∫ y in s, f y ∂(measure.map g μ) = ∫ x in g ⁻¹' s, f (g x) ∂μ :=
begin
rw [measure.restrict_map hg hs, integral_map hg (hf.mono_measure _)],
exact measure.map_mono g measure.restrict_le_self
end
lemma set_integral_map_of_closed_embedding [topological_space α] [borel_space α]
{β} [measurable_space β] [topological_space β] [borel_space β]
{g : α → β} {f : β → E} {s : set β} (hs : measurable_set s) (hg : closed_embedding g) :
∫ y in s, f y ∂(measure.map g μ) = ∫ x in g ⁻¹' s, f (g x) ∂μ :=
by rw [measure.restrict_map hg.measurable hs, integral_map_of_closed_embedding hg]
lemma norm_set_integral_le_of_norm_le_const_ae {C : ℝ} (hs : μ s < ∞)
(hC : ∀ᵐ x ∂μ.restrict s, ∥f x∥ ≤ C) :
∥∫ x in s, f x ∂μ∥ ≤ C * (μ s).to_real :=
begin
rw ← measure.restrict_apply_univ at *,
haveI : finite_measure (μ.restrict s) := ⟨‹_›⟩,
exact norm_integral_le_of_norm_le_const hC
end
lemma norm_set_integral_le_of_norm_le_const_ae' {C : ℝ} (hs : μ s < ∞)
(hC : ∀ᵐ x ∂μ, x ∈ s → ∥f x∥ ≤ C) (hfm : ae_measurable f (μ.restrict s)) :
∥∫ x in s, f x ∂μ∥ ≤ C * (μ s).to_real :=
begin
apply norm_set_integral_le_of_norm_le_const_ae hs,
have A : ∀ᵐ (x : α) ∂μ, x ∈ s → ∥ae_measurable.mk f hfm x∥ ≤ C,
{ filter_upwards [hC, hfm.ae_mem_imp_eq_mk],
assume a h1 h2 h3,
rw [← h2 h3],
exact h1 h3 },
have B : measurable_set {x | ∥(hfm.mk f) x∥ ≤ C} := hfm.measurable_mk.norm measurable_set_Iic,
filter_upwards [hfm.ae_eq_mk, (ae_restrict_iff B).2 A],
assume a h1 h2,
rwa h1
end
lemma norm_set_integral_le_of_norm_le_const_ae'' {C : ℝ} (hs : μ s < ∞) (hsm : measurable_set s)
(hC : ∀ᵐ x ∂μ, x ∈ s → ∥f x∥ ≤ C) :
∥∫ x in s, f x ∂μ∥ ≤ C * (μ s).to_real :=
norm_set_integral_le_of_norm_le_const_ae hs $ by rwa [ae_restrict_eq hsm, eventually_inf_principal]
lemma norm_set_integral_le_of_norm_le_const {C : ℝ} (hs : μ s < ∞)
(hC : ∀ x ∈ s, ∥f x∥ ≤ C) (hfm : ae_measurable f (μ.restrict s)) :
∥∫ x in s, f x ∂μ∥ ≤ C * (μ s).to_real :=
norm_set_integral_le_of_norm_le_const_ae' hs (eventually_of_forall hC) hfm
lemma norm_set_integral_le_of_norm_le_const' {C : ℝ} (hs : μ s < ∞) (hsm : measurable_set s)
(hC : ∀ x ∈ s, ∥f x∥ ≤ C) :
∥∫ x in s, f x ∂μ∥ ≤ C * (μ s).to_real :=
norm_set_integral_le_of_norm_le_const_ae'' hs hsm $ eventually_of_forall hC
lemma set_integral_eq_zero_iff_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ.restrict s] f)
(hfi : integrable_on f s μ) :
∫ x in s, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict s] 0 :=
integral_eq_zero_iff_of_nonneg_ae hf hfi
lemma set_integral_pos_iff_support_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ.restrict s] f)
(hfi : integrable_on f s μ) :
0 < ∫ x in s, f x ∂μ ↔ 0 < μ (support f ∩ s) :=
begin
rw [integral_pos_iff_support_of_nonneg_ae hf hfi, restrict_apply_of_null_measurable_set],
exact hfi.ae_measurable.null_measurable_set (measurable_set_singleton 0).compl
end
end normed_group
section mono
variables {μ : measure α} {f g : α → ℝ} {s : set α}
(hf : integrable_on f s μ) (hg : integrable_on g s μ)
lemma set_integral_mono_ae_restrict (h : f ≤ᵐ[μ.restrict s] g) :
∫ a in s, f a ∂μ ≤ ∫ a in s, g a ∂μ :=
integral_mono_ae hf hg h
lemma set_integral_mono_ae (h : f ≤ᵐ[μ] g) :
∫ a in s, f a ∂μ ≤ ∫ a in s, g a ∂μ :=
set_integral_mono_ae_restrict hf hg (ae_restrict_of_ae h)
lemma set_integral_mono_on (hs : measurable_set s) (h : ∀ x ∈ s, f x ≤ g x) :
∫ a in s, f a ∂μ ≤ ∫ a in s, g a ∂μ :=
set_integral_mono_ae_restrict hf hg
(by simp [hs, eventually_le, eventually_inf_principal, ae_of_all _ h])
lemma set_integral_mono (h : f ≤ g) :
∫ a in s, f a ∂μ ≤ ∫ a in s, g a ∂μ :=
integral_mono hf hg h
end mono
section nonneg
variables {μ : measure α} {f : α → ℝ} {s : set α}
lemma set_integral_nonneg_of_ae_restrict (hf : 0 ≤ᵐ[μ.restrict s] f) :
(0:ℝ) ≤ (∫ a in s, f a ∂μ) :=
integral_nonneg_of_ae hf
lemma set_integral_nonneg_of_ae (hf : 0 ≤ᵐ[μ] f) : (0:ℝ) ≤ (∫ a in s, f a ∂μ) :=
set_integral_nonneg_of_ae_restrict (ae_restrict_of_ae hf)
lemma set_integral_nonneg (hs : measurable_set s) (hf : ∀ a, a ∈ s → 0 ≤ f a) :
(0:ℝ) ≤ (∫ a in s, f a ∂μ) :=
set_integral_nonneg_of_ae_restrict ((ae_restrict_iff' hs).mpr (ae_of_all μ hf))
end nonneg
end measure_theory
open measure_theory asymptotics metric
variables {ι : Type*} [measurable_space E] [normed_group E]
/-- Fundamental theorem of calculus for set integrals: if `μ` is a measure that is finite at a
filter `l` and `f` is a measurable function that has a finite limit `b` at `l ⊓ μ.ae`, then `∫ x in
s i, f x ∂μ = μ (s i) • b + o(μ (s i))` at a filter `li` provided that `s i` tends to `l.lift'
powerset` along `li`. Since `μ (s i)` is an `ℝ≥0∞` number, we use `(μ (s i)).to_real` in the
actual statement.
Often there is a good formula for `(μ (s i)).to_real`, so the formalization can take an optional
argument `m` with this formula and a proof `of `(λ i, (μ (s i)).to_real) =ᶠ[li] m`. Without these
arguments, `m i = (μ (s i)).to_real` is used in the output. -/
lemma filter.tendsto.integral_sub_linear_is_o_ae
[normed_space ℝ E] [second_countable_topology E] [complete_space E] [borel_space E]
{μ : measure α} {l : filter α} [l.is_measurably_generated]
{f : α → E} {b : E} (h : tendsto f (l ⊓ μ.ae) (𝓝 b))
(hfm : measurable_at_filter f l μ) (hμ : μ.finite_at_filter l)
{s : ι → set α} {li : filter ι} (hs : tendsto s li (l.lift' powerset))
(m : ι → ℝ := λ i, (μ (s i)).to_real)
(hsμ : (λ i, (μ (s i)).to_real) =ᶠ[li] m . tactic.interactive.refl) :
is_o (λ i, ∫ x in s i, f x ∂μ - m i • b) m li :=
begin
suffices : is_o (λ s, ∫ x in s, f x ∂μ - (μ s).to_real • b) (λ s, (μ s).to_real)
(l.lift' powerset),
from (this.comp_tendsto hs).congr' (hsμ.mono $ λ a ha, ha ▸ rfl) hsμ,
refine is_o_iff.2 (λ ε ε₀, _),
have : ∀ᶠ s in l.lift' powerset, ∀ᶠ x in μ.ae, x ∈ s → f x ∈ closed_ball b ε :=
eventually_lift'_powerset_eventually.2 (h.eventually $ closed_ball_mem_nhds _ ε₀),
filter_upwards [hμ.eventually, (hμ.integrable_at_filter_of_tendsto_ae hfm h).eventually,
hfm.eventually, this],
simp only [mem_closed_ball, dist_eq_norm],
intros s hμs h_integrable hfm h_norm,
rw [← set_integral_const, ← integral_sub h_integrable (integrable_on_const.2 $ or.inr hμs),
real.norm_eq_abs, abs_of_nonneg ennreal.to_real_nonneg],
exact norm_set_integral_le_of_norm_le_const_ae' hμs h_norm (hfm.sub ae_measurable_const)
end
/-- Fundamental theorem of calculus for set integrals, `nhds_within` version: if `μ` is a locally
finite measure and `f` is an almost everywhere measurable function that is continuous at a point `a`
within a measurable set `t`, then `∫ x in s i, f x ∂μ = μ (s i) • f a + o(μ (s i))` at a filter `li`
provided that `s i` tends to `(𝓝[t] a).lift' powerset` along `li`. Since `μ (s i)` is an `ℝ≥0∞`
number, we use `(μ (s i)).to_real` in the actual statement.
Often there is a good formula for `(μ (s i)).to_real`, so the formalization can take an optional
argument `m` with this formula and a proof `of `(λ i, (μ (s i)).to_real) =ᶠ[li] m`. Without these
arguments, `m i = (μ (s i)).to_real` is used in the output. -/
lemma continuous_within_at.integral_sub_linear_is_o_ae
[topological_space α] [opens_measurable_space α]
[normed_space ℝ E] [second_countable_topology E] [complete_space E] [borel_space E]
{μ : measure α} [locally_finite_measure μ] {a : α} {t : set α}
{f : α → E} (ha : continuous_within_at f t a) (ht : measurable_set t)
(hfm : measurable_at_filter f (𝓝[t] a) μ)
{s : ι → set α} {li : filter ι} (hs : tendsto s li ((𝓝[t] a).lift' powerset))
(m : ι → ℝ := λ i, (μ (s i)).to_real)
(hsμ : (λ i, (μ (s i)).to_real) =ᶠ[li] m . tactic.interactive.refl) :
is_o (λ i, ∫ x in s i, f x ∂μ - m i • f a) m li :=
by haveI : (𝓝[t] a).is_measurably_generated := ht.nhds_within_is_measurably_generated _;
exact (ha.mono_left inf_le_left).integral_sub_linear_is_o_ae
hfm (μ.finite_at_nhds_within a t) hs m hsμ
/-- Fundamental theorem of calculus for set integrals, `nhds` version: if `μ` is a locally finite
measure and `f` is an almost everywhere measurable function that is continuous at a point `a`, then
`∫ x in s i, f x ∂μ = μ (s i) • f a + o(μ (s i))` at `li` provided that `s` tends to `(𝓝 a).lift'
powerset` along `li. Since `μ (s i)` is an `ℝ≥0∞` number, we use `(μ (s i)).to_real` in the
actual statement.
Often there is a good formula for `(μ (s i)).to_real`, so the formalization can take an optional
argument `m` with this formula and a proof `of `(λ i, (μ (s i)).to_real) =ᶠ[li] m`. Without these
arguments, `m i = (μ (s i)).to_real` is used in the output. -/
lemma continuous_at.integral_sub_linear_is_o_ae
[topological_space α] [opens_measurable_space α]
[normed_space ℝ E] [second_countable_topology E] [complete_space E] [borel_space E]
{μ : measure α} [locally_finite_measure μ] {a : α}
{f : α → E} (ha : continuous_at f a) (hfm : measurable_at_filter f (𝓝 a) μ)
{s : ι → set α} {li : filter ι} (hs : tendsto s li ((𝓝 a).lift' powerset))
(m : ι → ℝ := λ i, (μ (s i)).to_real)
(hsμ : (λ i, (μ (s i)).to_real) =ᶠ[li] m . tactic.interactive.refl) :
is_o (λ i, ∫ x in s i, f x ∂μ - m i • f a) m li :=
(ha.mono_left inf_le_left).integral_sub_linear_is_o_ae hfm (μ.finite_at_nhds a) hs m hsμ
/-- If a function is integrable at `𝓝[s] x` for each point `x` of a compact set `s`, then it is
integrable on `s`. -/
lemma is_compact.integrable_on_of_nhds_within [topological_space α] {μ : measure α} {s : set α}
(hs : is_compact s) {f : α → E} (hf : ∀ x ∈ s, integrable_at_filter f (𝓝[s] x) μ) :
integrable_on f s μ :=
is_compact.induction_on hs integrable_on_empty (λ s t hst ht, ht.mono_set hst)
(λ s t hs ht, hs.union ht) hf
/-- A function which is continuous on a set `s` is almost everywhere measurable with respect to
`μ.restrict s`. -/
lemma continuous_on.ae_measurable [topological_space α] [opens_measurable_space α] [borel_space E]
{f : α → E} {s : set α} {μ : measure α} (hf : continuous_on f s) (hs : measurable_set s) :
ae_measurable f (μ.restrict s) :=
begin
refine ⟨indicator s f, _, (indicator_ae_eq_restrict hs).symm⟩,
apply measurable_of_is_open,
assume t ht,
obtain ⟨u, u_open, hu⟩ : ∃ (u : set α), is_open u ∧ f ⁻¹' t ∩ s = u ∩ s :=
_root_.continuous_on_iff'.1 hf t ht,
rw [indicator_preimage, set.ite, hu],
exact (u_open.measurable_set.inter hs).union ((measurable_zero ht.measurable_set).diff hs)
end
lemma continuous_on.integrable_at_nhds_within
[topological_space α] [opens_measurable_space α] [borel_space E]
{μ : measure α} [locally_finite_measure μ] {a : α} {t : set α} {f : α → E}
(hft : continuous_on f t) (ht : measurable_set t) (ha : a ∈ t) :
integrable_at_filter f (𝓝[t] a) μ :=
by haveI : (𝓝[t] a).is_measurably_generated := ht.nhds_within_is_measurably_generated _;
exact (hft a ha).integrable_at_filter ⟨_, self_mem_nhds_within, hft.ae_measurable ht⟩
(μ.finite_at_nhds_within _ _)
/-- Fundamental theorem of calculus for set integrals, `nhds_within` version: if `μ` is a locally
finite measure, `f` is continuous on a measurable set `t`, and `a ∈ t`, then `∫ x in (s i), f x ∂μ =
μ (s i) • f a + o(μ (s i))` at `li` provided that `s i` tends to `(𝓝[t] a).lift' powerset` along
`li`. Since `μ (s i)` is an `ℝ≥0∞` number, we use `(μ (s i)).to_real` in the actual statement.
Often there is a good formula for `(μ (s i)).to_real`, so the formalization can take an optional
argument `m` with this formula and a proof `of `(λ i, (μ (s i)).to_real) =ᶠ[li] m`. Without these
arguments, `m i = (μ (s i)).to_real` is used in the output. -/
lemma continuous_on.integral_sub_linear_is_o_ae
[topological_space α] [opens_measurable_space α]
[normed_space ℝ E] [second_countable_topology E] [complete_space E] [borel_space E]
{μ : measure α} [locally_finite_measure μ] {a : α} {t : set α}
{f : α → E} (hft : continuous_on f t) (ha : a ∈ t) (ht : measurable_set t)
{s : ι → set α} {li : filter ι} (hs : tendsto s li ((𝓝[t] a).lift' powerset))
(m : ι → ℝ := λ i, (μ (s i)).to_real)
(hsμ : (λ i, (μ (s i)).to_real) =ᶠ[li] m . tactic.interactive.refl) :
is_o (λ i, ∫ x in s i, f x ∂μ - m i • f a) m li :=
(hft a ha).integral_sub_linear_is_o_ae ht ⟨t, self_mem_nhds_within, hft.ae_measurable ht⟩ hs m hsμ
/-- A function `f` continuous on a compact set `s` is integrable on this set with respect to any
locally finite measure. -/
lemma continuous_on.integrable_on_compact
[topological_space α] [opens_measurable_space α] [borel_space E]
[t2_space α] {μ : measure α} [locally_finite_measure μ]
{s : set α} (hs : is_compact s) {f : α → E} (hf : continuous_on f s) :
integrable_on f s μ :=
hs.integrable_on_of_nhds_within $ λ x hx, hf.integrable_at_nhds_within hs.measurable_set hx
/-- A continuous function `f` is integrable on any compact set with respect to any locally finite
measure. -/
lemma continuous.integrable_on_compact
[topological_space α] [opens_measurable_space α] [t2_space α]
[borel_space E] {μ : measure α} [locally_finite_measure μ] {s : set α}
(hs : is_compact s) {f : α → E} (hf : continuous f) :
integrable_on f s μ :=
hf.continuous_on.integrable_on_compact hs
/-- A continuous function with compact closure of the support is integrable on the whole space. -/
lemma continuous.integrable_of_compact_closure_support
[topological_space α] [opens_measurable_space α] [t2_space α] [borel_space E]
{μ : measure α} [locally_finite_measure μ] {f : α → E} (hf : continuous f)
(hfc : is_compact (closure $ support f)) :
integrable f μ :=
begin
rw [← indicator_eq_self.2 (@subset_closure _ _ (support f)),
integrable_indicator_iff is_closed_closure.measurable_set],
{ exact hf.integrable_on_compact hfc },
{ apply_instance }
end
section
/-! ### Continuous linear maps composed with integration
The goal of this section is to prove that integration commutes with continuous linear maps.
This holds for simple functions. The general result follows from the continuity of all involved
operations on the space `L¹`. Note that composition by a continuous linear map on `L¹` is not just
the composition, as we are dealing with classes of functions, but it has already been defined
as `continuous_linear_map.comp_Lp`. We take advantage of this construction here.
-/
variables {μ : measure α} [normed_space ℝ E]
variables [normed_group F] [normed_space ℝ F]
variables {p : ennreal}
local attribute [instance] fact_one_le_one_ennreal
namespace continuous_linear_map
variables [measurable_space F] [borel_space F]
lemma integrable_comp [opens_measurable_space E] {φ : α → E} (L : E →L[ℝ] F)
(φ_int : integrable φ μ) : integrable (λ (a : α), L (φ a)) μ :=
((integrable.norm φ_int).const_mul ∥L∥).mono' (L.measurable.comp_ae_measurable φ_int.ae_measurable)
(eventually_of_forall $ λ a, L.le_op_norm (φ a))
variables [second_countable_topology F] [complete_space F]
[borel_space E] [second_countable_topology E]
lemma integral_comp_Lp (L : E →L[ℝ] F) (φ : Lp E p μ) :
∫ a, (L.comp_Lp φ) a ∂μ = ∫ a, L (φ a) ∂μ :=
integral_congr_ae $ coe_fn_comp_Lp _ _
lemma continuous_integral_comp_L1 (L : E →L[ℝ] F) :
continuous (λ (φ : α →₁[μ] E), ∫ (a : α), L (φ a) ∂μ) :=
begin
rw ← funext L.integral_comp_Lp,
exact continuous_integral.comp (L.comp_LpL 1 μ).continuous
end
variables [complete_space E]
lemma integral_comp_comm (L : E →L[ℝ] F) {φ : α → E} (φ_int : integrable φ μ) :
∫ a, L (φ a) ∂μ = L (∫ a, φ a ∂μ) :=
begin
apply integrable.induction (λ φ, ∫ a, L (φ a) ∂μ = L (∫ a, φ a ∂μ)),
{ intros e s s_meas s_finite,
rw [integral_indicator_const e s_meas, continuous_linear_map.map_smul,
← integral_indicator_const (L e) s_meas],
congr' 1 with a,
rw set.indicator_comp_of_zero L.map_zero },
{ intros f g H f_int g_int hf hg,
simp [L.map_add, integral_add f_int g_int,
integral_add (L.integrable_comp f_int) (L.integrable_comp g_int), hf, hg] },
{ exact is_closed_eq L.continuous_integral_comp_L1 (L.continuous.comp continuous_integral) },
{ intros f g hfg f_int hf,
convert hf using 1 ; clear hf,
{ exact integral_congr_ae (hfg.fun_comp L).symm },
{ rw integral_congr_ae hfg.symm } },
all_goals { assumption }
end
lemma integral_apply {H : Type*} [normed_group H] [normed_space ℝ H]
[second_countable_topology $ H →L[ℝ] E] {φ : α → H →L[ℝ] E} (φ_int : integrable φ μ) (v : H) :
(∫ a, φ a ∂μ) v = ∫ a, φ a v ∂μ :=
((continuous_linear_map.apply ℝ E v).integral_comp_comm φ_int).symm
lemma integral_comp_comm' (L : E →L[ℝ] F) {K} (hL : antilipschitz_with K L) (φ : α → E) :
∫ a, L (φ a) ∂μ = L (∫ a, φ a ∂μ) :=
begin
by_cases h : integrable φ μ,
{ exact integral_comp_comm L h },
have : ¬ (integrable (L ∘ φ) μ),
by rwa lipschitz_with.integrable_comp_iff_of_antilipschitz L.lipschitz hL (L.map_zero),
simp [integral_undef, h, this]
end
lemma integral_comp_L1_comm (L : E →L[ℝ] F) (φ : α →₁[μ] E) : ∫ a, L (φ a) ∂μ = L (∫ a, φ a ∂μ) :=
L.integral_comp_comm (L1.integrable_coe_fn φ)
end continuous_linear_map
namespace linear_isometry
variables [measurable_space F] [borel_space F] [complete_space E]
[second_countable_topology F] [complete_space F]
[borel_space E] [second_countable_topology E]
lemma integral_comp_comm (L : E →ₗᵢ[ℝ] F) (φ : α → E) :
∫ a, L (φ a) ∂μ = L (∫ a, φ a ∂μ) :=
L.to_continuous_linear_map.integral_comp_comm' L.antilipschitz _
end linear_isometry
variables [borel_space E] [second_countable_topology E] [complete_space E]
[measurable_space F] [borel_space F] [second_countable_topology F] [complete_space F]
@[norm_cast] lemma integral_of_real {𝕜 : Type*} [is_R_or_C 𝕜] [measurable_space 𝕜] [borel_space 𝕜]
{f : α → ℝ} :
∫ a, (f a : 𝕜) ∂μ = ↑∫ a, f a ∂μ :=
linear_isometry.integral_comp_comm (@is_R_or_C.of_real_li 𝕜 _) f
lemma integral_conj {𝕜 : Type*} [is_R_or_C 𝕜] [measurable_space 𝕜] [borel_space 𝕜] {f : α → 𝕜} :
∫ a, is_R_or_C.conj (f a) ∂μ = is_R_or_C.conj ∫ a, f a ∂μ :=
linear_isometry.integral_comp_comm (@is_R_or_C.conj_li 𝕜 _) f
lemma fst_integral {f : α → E × F} (hf : integrable f μ) :
(∫ x, f x ∂μ).1 = ∫ x, (f x).1 ∂μ :=
((continuous_linear_map.fst ℝ E F).integral_comp_comm hf).symm
lemma snd_integral {f : α → E × F} (hf : integrable f μ) :
(∫ x, f x ∂μ).2 = ∫ x, (f x).2 ∂μ :=
((continuous_linear_map.snd ℝ E F).integral_comp_comm hf).symm
lemma integral_pair {f : α → E} {g : α → F} (hf : integrable f μ) (hg : integrable g μ) :
∫ x, (f x, g x) ∂μ = (∫ x, f x ∂μ, ∫ x, g x ∂μ) :=
have _ := hf.prod_mk hg, prod.ext (fst_integral this) (snd_integral this)
lemma integral_smul_const (f : α → ℝ) (c : E) :
∫ x, f x • c ∂μ = (∫ x, f x ∂μ) • c :=
begin
by_cases hf : integrable f μ,
{ exact ((continuous_linear_map.id ℝ ℝ).smul_right c).integral_comp_comm hf },
{ by_cases hc : c = 0,
{ simp only [hc, integral_zero, smul_zero] },
rw [integral_undef hf, integral_undef, zero_smul],
simp_rw [integrable_smul_const hc, hf, not_false_iff] }
end
end
/-
namespace integrable
variables [measurable_space α] [measurable_space β] [normed_group E]
protected lemma measure_mono
end integrable
end measure_theory
section integral_on
variables [measurable_space α]
[normed_group β] [second_countable_topology β] [normed_space ℝ β] [complete_space β]
[measurable_space β] [borel_space β]
{s t : set α} {f g : α → β} {μ : measure α}
open set
lemma integral_on_congr (hf : measurable f) (hg : measurable g) (hs : measurable_set s)
(h : ∀ᵐ a ∂μ, a ∈ s → f a = g a) : ∫ a in s, f a ∂μ = ∫ a in s, g a ∂μ :=
integral_congr_ae hf hg $ _
lemma integral_on_congr_of_set (hsm : measurable_on s f) (htm : measurable_on t f)
(h : ∀ᵐ a, a ∈ s ↔ a ∈ t) : (∫ a in s, f a) = (∫ a in t, f a) :=
integral_congr_ae hsm htm $ indicator_congr_of_set h
lemma integral_on_add {s : set α} (hfm : measurable_on s f) (hfi : integrable_on s f)
(hgm : measurable_on s g) (hgi : integrable_on s g) :
(∫ a in s, f a + g a) = (∫ a in s, f a) + (∫ a in s, g a) :=
by { simp only [indicator_add], exact integral_add hfm hfi hgm hgi }
lemma integral_on_sub (hfm : measurable_on s f) (hfi : integrable_on s f) (hgm : measurable_on s g)
(hgi : integrable_on s g) : (∫ a in s, f a - g a) = (∫ a in s, f a) - (∫ a in s, g a) :=
by { simp only [indicator_sub], exact integral_sub hfm hfi hgm hgi }
lemma integral_on_le_integral_on_ae {f g : α → ℝ} (hfm : measurable_on s f)
(hfi : integrable_on s f) (hgm : measurable_on s g) (hgi : integrable_on s g)
(h : ∀ᵐ a, a ∈ s → f a ≤ g a) :
(∫ a in s, f a) ≤ (∫ a in s, g a) :=
begin
apply integral_le_integral_ae hfm hfi hgm hgi,
apply indicator_le_indicator_ae,
exact h
end
lemma integral_on_le_integral_on {f g : α → ℝ} (hfm : measurable_on s f) (hfi : integrable_on s f)
(hgm : measurable_on s g) (hgi : integrable_on s g) (h : ∀ a, a ∈ s → f a ≤ g a) :
(∫ a in s, f a) ≤ (∫ a in s, g a) :=
integral_on_le_integral_on_ae hfm hfi hgm hgi $ by filter_upwards [] h
lemma integral_on_union (hsm : measurable_on s f) (hsi : integrable_on s f)
(htm : measurable_on t f) (hti : integrable_on t f) (h : disjoint s t) :
(∫ a in (s ∪ t), f a) = (∫ a in s, f a) + (∫ a in t, f a) :=
by { rw [indicator_union_of_disjoint h, integral_add hsm hsi htm hti] }
lemma integral_on_union_ae (hs : measurable_set s) (ht : measurable_set t) (hsm : measurable_on s f)
(hsi : integrable_on s f) (htm : measurable_on t f) (hti : integrable_on t f)
(h : ∀ᵐ a, a ∉ s ∩ t) :
(∫ a in (s ∪ t), f a) = (∫ a in s, f a) + (∫ a in t, f a) :=
begin
have := integral_congr_ae _ _ (indicator_union_ae h f),
rw [this, integral_add hsm hsi htm hti],
{ exact hsm.union hs ht htm },
{ exact measurable.add hsm htm }
end
lemma integral_on_nonneg_of_ae {f : α → ℝ} (hf : ∀ᵐ a, a ∈ s → 0 ≤ f a) : (0:ℝ) ≤ (∫ a in s, f a) :=
integral_nonneg_of_ae $ by { filter_upwards [hf] λ a h, indicator_nonneg' h }
lemma integral_on_nonneg {f : α → ℝ} (hf : ∀ a, a ∈ s → 0 ≤ f a) : (0:ℝ) ≤ (∫ a in s, f a) :=
integral_on_nonneg_of_ae $ univ_mem_sets' hf
lemma integral_on_nonpos_of_ae {f : α → ℝ} (hf : ∀ᵐ a, a ∈ s → f a ≤ 0) : (∫ a in s, f a) ≤ 0 :=
integral_nonpos_of_nonpos_ae $ by { filter_upwards [hf] λ a h, indicator_nonpos' h }
lemma integral_on_nonpos {f : α → ℝ} (hf : ∀ a, a ∈ s → f a ≤ 0) : (∫ a in s, f a) ≤ 0 :=
integral_on_nonpos_of_ae $ univ_mem_sets' hf
lemma tendsto_integral_on_of_monotone {s : ℕ → set α} {f : α → β} (hsm : ∀i, measurable_set (s i))
(h_mono : monotone s) (hfm : measurable_on (Union s) f) (hfi : integrable_on (Union s) f) :
tendsto (λi, ∫ a in (s i), f a) at_top (nhds (∫ a in (Union s), f a)) :=
let bound : α → ℝ := indicator (Union s) (λa, ∥f a∥) in
begin
apply tendsto_integral_of_dominated_convergence,
{ assume i, exact hfm.subset (hsm i) (subset_Union _ _) },
{ assumption },
{ show integrable_on (Union s) (λa, ∥f a∥), rwa integrable_on_norm_iff },
{ assume i, apply ae_of_all,
assume a,
rw [norm_indicator_eq_indicator_norm],
exact indicator_le_indicator_of_subset (subset_Union _ _) (λa, norm_nonneg _) _ },
{ filter_upwards [] λa, le_trans (tendsto_indicator_of_monotone _ h_mono _ _) (pure_le_nhds _) }
end
lemma tendsto_integral_on_of_antimono (s : ℕ → set α) (f : α → β) (hsm : ∀i, measurable_set (s i))
(h_mono : ∀i j, i ≤ j → s j ⊆ s i) (hfm : measurable_on (s 0) f) (hfi : integrable_on (s 0) f) :
tendsto (λi, ∫ a in (s i), f a) at_top (nhds (∫ a in (Inter s), f a)) :=
let bound : α → ℝ := indicator (s 0) (λa, ∥f a∥) in
begin
apply tendsto_integral_of_dominated_convergence,
{ assume i, refine hfm.subset (hsm i) (h_mono _ _ (zero_le _)) },
{ exact hfm.subset (measurable_set.Inter hsm) (Inter_subset _ _) },
{ show integrable_on (s 0) (λa, ∥f a∥), rwa integrable_on_norm_iff },
{ assume i, apply ae_of_all,
assume a,
rw [norm_indicator_eq_indicator_norm],
refine indicator_le_indicator_of_subset (h_mono _ _ (zero_le _)) (λa, norm_nonneg _) _ },
{ filter_upwards [] λa, le_trans (tendsto_indicator_of_antimono _ h_mono _ _) (pure_le_nhds _) }
end
-- TODO : prove this for an encodable type
-- by proving an encodable version of `filter.is_countably_generated_at_top_finset_nat `
lemma integral_on_Union (s : ℕ → set α) (f : α → β) (hm : ∀i, measurable_set (s i))
(hd : ∀ i j, i ≠ j → s i ∩ s j = ∅) (hfm : measurable_on (Union s) f)
(hfi : integrable_on (Union s) f) :
(∫ a in (Union s), f a) = ∑'i, ∫ a in s i, f a :=
suffices h : tendsto (λn:finset ℕ, ∑ i in n, ∫ a in s i, f a) at_top (𝓝 $ (∫ a in (Union s), f a)),
by { rwa has_sum.tsum_eq },
begin
have : (λn:finset ℕ, ∑ i in n, ∫ a in s i, f a) = λn:finset ℕ, ∫ a in (⋃i∈n, s i), f a,
{ funext,
rw [← integral_finset_sum, indicator_finset_bUnion],
{ assume i hi j hj hij, exact hd i j hij },
{ assume i, refine hfm.subset (hm _) (subset_Union _ _) },
{ assume i, refine hfi.subset (subset_Union _ _) } },
rw this,
refine tendsto_integral_filter_of_dominated_convergence _ _ _ _ _ _ _,
{ exact indicator (Union s) (λ a, ∥f a∥) },
{ exact is_countably_generated_at_top_finset_nat },
{ refine univ_mem_sets' (λ n, _),
simp only [mem_set_of_eq],
refine hfm.subset (measurable_set.Union (λ i, measurable_set.Union_Prop (λh, hm _)))
(bUnion_subset_Union _ _), },
{ assumption },
{ refine univ_mem_sets' (λ n, univ_mem_sets' $ _),
simp only [mem_set_of_eq],
assume a,
rw ← norm_indicator_eq_indicator_norm,
refine norm_indicator_le_of_subset (bUnion_subset_Union _ _) _ _ },
{ rw [← integrable_on, integrable_on_norm_iff], assumption },
{ filter_upwards [] λa, le_trans (tendsto_indicator_bUnion_finset _ _ _) (pure_le_nhds _) }
end
end integral_on
-/
|
6eafe7e6208c96e1626ce7756650d459fce7ef3d | 206422fb9edabf63def0ed2aa3f489150fb09ccb | /src/topology/algebra/ordered.lean | 2e3e2b0dadde8d2eec04638536a193cfd5e0d063 | [
"Apache-2.0"
] | permissive | hamdysalah1/mathlib | b915f86b2503feeae268de369f1b16932321f097 | 95454452f6b3569bf967d35aab8d852b1ddf8017 | refs/heads/master | 1,677,154,116,545 | 1,611,797,994,000 | 1,611,797,994,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 150,857 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import tactic.linarith
import tactic.tfae
import algebra.archimedean
import algebra.group.pi
import algebra.ordered_ring
import order.liminf_limsup
import data.set.intervals.image_preimage
import data.set.intervals.ord_connected
import data.set.intervals.surj_on
import data.set.intervals.pi
import topology.algebra.group
import topology.extend_from_subset
import order.filter.interval
/-!
# Theory of topology on ordered spaces
## Main definitions
The order topology on an ordered space is the topology generated by all open intervals (or
equivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `preorder.topology α`.
However, we do *not* register it as an instance (as many existing ordered types already have
topologies, which would be equal but not definitionally equal to `preorder.topology α`). Instead,
we introduce a class `order_topology α`(which is a `Prop`, also known as a mixin) saying that on
the type `α` having already a topological space structure and a preorder structure, the topological
structure is equal to the order topology.
We also introduce another (mixin) class `order_closed_topology α` saying that the set of points
`(x, y)` with `x ≤ y` is closed in the product space. This is automatically satisfied on a linear
order with the order topology.
We prove many basic properties of such topologies.
## Main statements
This file contains the proofs of the following facts. For exact requirements
(`order_closed_topology` vs `order_topology`, `preorder` vs `partial_order` vs `linear_order` etc)
see their statements.
### Open / closed sets
* `is_open_lt` : if `f` and `g` are continuous functions, then `{x | f x < g x}` is open;
* `is_open_Iio`, `is_open_Ioi`, `is_open_Ioo` : open intervals are open;
* `is_closed_le` : if `f` and `g` are continuous functions, then `{x | f x ≤ g x}` is closed;
* `is_closed_Iic`, `is_closed_Ici`, `is_closed_Icc` : closed intervals are closed;
* `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x | f x ≤ g x}`
and `{x | f x < g x}` are included by `{x | f x = g x}`;
* `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any
neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood
of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`.
### Convergence and inequalities
* `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually
`f x ≤ g x`, then `a ≤ b`
* `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b`
(resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a); we also provide primed versions
that assume the inequalities to hold for all `x`.
### Min, max, `Sup` and `Inf`
* `continuous.min`, `continuous.max`: pointwise `min`/`max` of two continuous functions is
continuous.
* `tendsto.min`, `tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise
`min`/`max` tend to `min a b` and `max a b`, respectively.
* `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem,
sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h`
both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`.
### Connected sets and Intermediate Value Theorem
* `is_preconnected_I??` : all intervals `I??` are preconnected,
* `is_preconnected.intermediate_value`, `intermediate_value_univ` : Intermediate Value Theorem for
connected sets and connected spaces, respectively;
* `intermediate_value_Icc`, `intermediate_value_Icc'`: Intermediate Value Theorem for functions
on closed intervals.
### Miscellaneous facts
* `is_compact.exists_forall_le`, `is_compact.exists_forall_ge` : extreme value theorem, a continuous
function on a compact set takes its minimum and maximum values.
* `is_closed.Icc_subset_of_forall_mem_nhds_within` : “Continuous induction” principle;
if `s ∩ [a, b]` is closed, `a ∈ s`, and for each `x ∈ [a, b) ∩ s` some of its right neighborhoods
is included `s`, then `[a, b] ⊆ s`.
* `is_closed.Icc_subset_of_forall_exists_gt`, `is_closed.mem_of_ge_of_forall_exists_gt` : two
other versions of the “continuous induction” principle.
## Implementation
We do _not_ register the order topology as an instance on a preorder (or even on a linear order).
Indeed, on many such spaces, a topology has already been constructed in a different way (think
of the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`),
and is in general not defeq to the one generated by the intervals. We make it available as a
definition `preorder.topology α` though, that can be registered as an instance when necessary, or
for specific types.
-/
open classical set filter topological_space
open function (curry uncurry)
open_locale topological_space classical filter
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w}
/-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the
set of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin.
This property is satisfied for the order topology on a linear order, but it can be satisfied more
generally, and suffices to derive many interesting properties relating order and topology. -/
class order_closed_topology (α : Type*) [topological_space α] [preorder α] : Prop :=
(is_closed_le' : is_closed {p:α×α | p.1 ≤ p.2})
instance : Π [topological_space α], topological_space (order_dual α) := id
section order_closed_topology
section preorder
variables [topological_space α] [preorder α] [t : order_closed_topology α]
include t
lemma is_closed_le_prod : is_closed {p : α × α | p.1 ≤ p.2} :=
t.is_closed_le'
lemma is_closed_le [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :
is_closed {b | f b ≤ g b} :=
continuous_iff_is_closed.mp (hf.prod_mk hg) _ is_closed_le_prod
lemma is_closed_le' (a : α) : is_closed {b | b ≤ a} :=
is_closed_le continuous_id continuous_const
lemma is_closed_Iic {a : α} : is_closed (Iic a) :=
is_closed_le' a
lemma is_closed_ge' (a : α) : is_closed {b | a ≤ b} :=
is_closed_le continuous_const continuous_id
lemma is_closed_Ici {a : α} : is_closed (Ici a) :=
is_closed_ge' a
instance : order_closed_topology (order_dual α) :=
⟨(@order_closed_topology.is_closed_le' α _ _ _).preimage continuous_swap⟩
lemma is_closed_Icc {a b : α} : is_closed (Icc a b) :=
is_closed_inter is_closed_Ici is_closed_Iic
@[simp] lemma closure_Icc (a b : α) : closure (Icc a b) = Icc a b :=
is_closed_Icc.closure_eq
@[simp] lemma closure_Iic (a : α) : closure (Iic a) = Iic a :=
is_closed_Iic.closure_eq
@[simp] lemma closure_Ici (a : α) : closure (Ici a) = Ici a :=
is_closed_Ici.closure_eq
lemma le_of_tendsto_of_tendsto {f g : β → α} {b : filter β} {a₁ a₂ : α} [ne_bot b]
(hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) :
a₁ ≤ a₂ :=
have tendsto (λb, (f b, g b)) b (𝓝 (a₁, a₂)),
by rw [nhds_prod_eq]; exact hf.prod_mk hg,
show (a₁, a₂) ∈ {p:α×α | p.1 ≤ p.2},
from t.is_closed_le'.mem_of_tendsto this h
lemma le_of_tendsto_of_tendsto' {f g : β → α} {b : filter β} {a₁ a₂ : α} [ne_bot b]
(hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) :
a₁ ≤ a₂ :=
le_of_tendsto_of_tendsto hf hg (eventually_of_forall h)
lemma le_of_tendsto {f : β → α} {a b : α} {x : filter β}
[ne_bot x] (lim : tendsto f x (𝓝 a)) (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b :=
le_of_tendsto_of_tendsto lim tendsto_const_nhds h
lemma le_of_tendsto' {f : β → α} {a b : α} {x : filter β}
[ne_bot x] (lim : tendsto f x (𝓝 a)) (h : ∀ c, f c ≤ b) : a ≤ b :=
le_of_tendsto lim (eventually_of_forall h)
lemma ge_of_tendsto {f : β → α} {a b : α} {x : filter β} [ne_bot x]
(lim : tendsto f x (𝓝 a)) (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a :=
le_of_tendsto_of_tendsto tendsto_const_nhds lim h
lemma ge_of_tendsto' {f : β → α} {a b : α} {x : filter β} [ne_bot x]
(lim : tendsto f x (𝓝 a)) (h : ∀ c, b ≤ f c) : b ≤ a :=
ge_of_tendsto lim (eventually_of_forall h)
@[simp]
lemma closure_le_eq [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :
closure {b | f b ≤ g b} = {b | f b ≤ g b} :=
(is_closed_le hf hg).closure_eq
lemma closure_lt_subset_le [topological_space β] {f g : β → α} (hf : continuous f)
(hg : continuous g) :
closure {b | f b < g b} ⊆ {b | f b ≤ g b} :=
by { rw [←closure_le_eq hf hg], exact closure_mono (λ b, le_of_lt) }
lemma continuous_within_at.closure_le [topological_space β]
{f g : β → α} {s : set β} {x : β} (hx : x ∈ closure s)
(hf : continuous_within_at f s x)
(hg : continuous_within_at g s x)
(h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x :=
show (f x, g x) ∈ {p : α × α | p.1 ≤ p.2},
from order_closed_topology.is_closed_le'.closure_subset ((hf.prod hg).mem_closure hx h)
/-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`,
then the set `{x ∈ s | f x ≤ g x}` is a closed set. -/
lemma is_closed.is_closed_le [topological_space β] {f g : β → α} {s : set β} (hs : is_closed s)
(hf : continuous_on f s) (hg : continuous_on g s) :
is_closed {x ∈ s | f x ≤ g x} :=
(hf.prod hg).preimage_closed_of_closed hs order_closed_topology.is_closed_le'
omit t
lemma nhds_within_Ici_ne_bot {a b : α} (H₂ : a ≤ b) :
ne_bot (𝓝[Ici a] b) :=
nhds_within_ne_bot_of_mem H₂
@[instance] lemma nhds_within_Ici_self_ne_bot (a : α) :
ne_bot (𝓝[Ici a] a) :=
nhds_within_Ici_ne_bot (le_refl a)
lemma nhds_within_Iic_ne_bot {a b : α} (H : a ≤ b) :
ne_bot (𝓝[Iic b] a) :=
nhds_within_ne_bot_of_mem H
@[instance] lemma nhds_within_Iic_self_ne_bot (a : α) :
ne_bot (𝓝[Iic a] a) :=
nhds_within_Iic_ne_bot (le_refl a)
end preorder
section partial_order
variables [topological_space α] [partial_order α] [t : order_closed_topology α]
include t
private lemma is_closed_eq : is_closed {p : α × α | p.1 = p.2} :=
by simp only [le_antisymm_iff];
exact is_closed_inter t.is_closed_le' (is_closed_le continuous_snd continuous_fst)
@[priority 90] -- see Note [lower instance priority]
instance order_closed_topology.to_t2_space : t2_space α :=
{ t2 :=
have is_open {p : α × α | p.1 ≠ p.2}, from is_closed_eq,
assume a b h,
let ⟨u, v, hu, hv, ha, hb, h⟩ := is_open_prod_iff.mp this a b h in
⟨u, v, hu, hv, ha, hb,
set.eq_empty_iff_forall_not_mem.2 $ assume a ⟨h₁, h₂⟩,
have a ≠ a, from @h (a, a) ⟨h₁, h₂⟩,
this rfl⟩ }
end partial_order
section linear_order
variables [topological_space α] [linear_order α] [order_closed_topology α]
lemma is_open_lt_prod : is_open {p : α × α | p.1 < p.2} :=
by { simp_rw [← is_closed_compl_iff, compl_set_of, not_lt],
exact is_closed_le continuous_snd continuous_fst }
lemma is_open_lt [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :
is_open {b | f b < g b} :=
by simp [lt_iff_not_ge, -not_le]; exact is_closed_le hg hf
variables {a b : α}
lemma is_open_Iio : is_open (Iio a) :=
is_open_lt continuous_id continuous_const
lemma is_open_Ioi : is_open (Ioi a) :=
is_open_lt continuous_const continuous_id
lemma is_open_Ioo : is_open (Ioo a b) :=
is_open_inter is_open_Ioi is_open_Iio
@[simp] lemma interior_Ioi : interior (Ioi a) = Ioi a :=
is_open_Ioi.interior_eq
@[simp] lemma interior_Iio : interior (Iio a) = Iio a :=
is_open_Iio.interior_eq
@[simp] lemma interior_Ioo : interior (Ioo a b) = Ioo a b :=
is_open_Ioo.interior_eq
variables [topological_space γ]
/-- Intermediate value theorem for two functions: if `f` and `g` are two continuous functions
on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`. -/
lemma intermediate_value_univ₂ [preconnected_space γ] {a b : γ} {f g : γ → α} (hf : continuous f)
(hg : continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b) :
∃ x, f x = g x :=
begin
obtain ⟨x, h, hfg, hgf⟩ : (univ ∩ {x | f x ≤ g x ∧ g x ≤ f x}).nonempty,
from is_preconnected_closed_iff.1 preconnected_space.is_preconnected_univ _ _
(is_closed_le hf hg) (is_closed_le hg hf) (λ x hx, le_total _ _) ⟨a, trivial, ha⟩
⟨b, trivial, hb⟩,
exact ⟨x, le_antisymm hfg hgf⟩
end
/-- Intermediate value theorem for two functions: if `f` and `g` are two functions continuous
on a preconnected set `s` and for some `a b ∈ s` we have `f a ≤ g a` and `g b ≤ f b`,
then for some `x ∈ s` we have `f x = g x`. -/
lemma is_preconnected.intermediate_value₂ {s : set γ} (hs : is_preconnected s)
{a b : γ} (ha : a ∈ s) (hb : b ∈ s) {f g : γ → α}
(hf : continuous_on f s) (hg : continuous_on g s) (ha' : f a ≤ g a) (hb' : g b ≤ f b) :
∃ x ∈ s, f x = g x :=
let ⟨x, hx⟩ := @intermediate_value_univ₂ α s _ _ _ _ (subtype.preconnected_space hs) ⟨a, ha⟩ ⟨b, hb⟩
_ _ (continuous_on_iff_continuous_restrict.1 hf) (continuous_on_iff_continuous_restrict.1 hg)
ha' hb'
in ⟨x, x.2, hx⟩
/-- Intermediate Value Theorem for continuous functions on connected sets. -/
lemma is_preconnected.intermediate_value {s : set γ} (hs : is_preconnected s)
{a b : γ} (ha : a ∈ s) (hb : b ∈ s) {f : γ → α} (hf : continuous_on f s) :
Icc (f a) (f b) ⊆ f '' s :=
λ x hx, mem_image_iff_bex.2 $ hs.intermediate_value₂ ha hb hf continuous_on_const hx.1 hx.2
/-- Intermediate Value Theorem for continuous functions on connected spaces. -/
lemma intermediate_value_univ [preconnected_space γ] (a b : γ) {f : γ → α} (hf : continuous f) :
Icc (f a) (f b) ⊆ range f :=
λ x hx, intermediate_value_univ₂ hf continuous_const hx.1 hx.2
/-- Intermediate Value Theorem for continuous functions on connected spaces. -/
lemma mem_range_of_exists_le_of_exists_ge [preconnected_space γ] {c : α} {f : γ → α}
(hf : continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b) :
c ∈ range f :=
let ⟨a, ha⟩ := h₁, ⟨b, hb⟩ := h₂ in intermediate_value_univ a b hf ⟨ha, hb⟩
/-- If a preconnected set contains endpoints of an interval, then it includes the whole interval. -/
lemma is_preconnected.Icc_subset {s : set α} (hs : is_preconnected s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) :
Icc a b ⊆ s :=
by simpa only [image_id] using hs.intermediate_value ha hb continuous_on_id
/-- If a preconnected set contains endpoints of an interval, then it includes the whole interval. -/
lemma is_connected.Icc_subset {s : set α} (hs : is_connected s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) :
Icc a b ⊆ s :=
hs.2.Icc_subset ha hb
/-- If preconnected set in a linear order space is unbounded below and above, then it is the whole
space. -/
lemma is_preconnected.eq_univ_of_unbounded {s : set α} (hs : is_preconnected s) (hb : ¬bdd_below s)
(ha : ¬bdd_above s) :
s = univ :=
begin
refine eq_univ_of_forall (λ x, _),
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := not_bdd_below_iff.1 hb x,
obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bdd_above_iff.1 ha x,
exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩
end
/-!
### Neighborhoods to the left and to the right on an `order_closed_topology`
Limits to the left and to the right of real functions are defined in terms of neighborhoods to
the left and to the right, either open or closed, i.e., members of `𝓝[Ioi a] a` and
`𝓝[Ici a] a` on the right, and similarly on the left. Here we simply prove that all
right-neighborhoods of a point are equal, and we'll prove later other useful characterizations which
require the stronger hypothesis `order_topology α` -/
/-!
#### Right neighborhoods, point excluded
-/
lemma Ioo_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Ioo a c ∈ 𝓝[Ioi b] b :=
mem_nhds_within.2 ⟨Iio c, is_open_Iio, H.2,
by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩
lemma Ioc_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Ioc a c ∈ 𝓝[Ioi b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Ioi H) Ioo_subset_Ioc_self
lemma Ico_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Ico a c ∈ 𝓝[Ioi b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Ioi H) Ioo_subset_Ico_self
lemma Icc_mem_nhds_within_Ioi {a b c : α} (H : b ∈ Ico a c) :
Icc a c ∈ 𝓝[Ioi b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Ioi H) Ioo_subset_Icc_self
@[simp] lemma nhds_within_Ioc_eq_nhds_within_Ioi {a b : α} (h : a < b) :
𝓝[Ioc a b] a = 𝓝[Ioi a] a :=
le_antisymm (nhds_within_mono _ Ioc_subset_Ioi_self) $
nhds_within_le_of_mem $ Ioc_mem_nhds_within_Ioi $ left_mem_Ico.2 h
@[simp] lemma nhds_within_Ioo_eq_nhds_within_Ioi {a b : α} (h : a < b) :
𝓝[Ioo a b] a = 𝓝[Ioi a] a :=
le_antisymm (nhds_within_mono _ Ioo_subset_Ioi_self) $
nhds_within_le_of_mem $ Ioo_mem_nhds_within_Ioi $ left_mem_Ico.2 h
@[simp]
lemma continuous_within_at_Ioc_iff_Ioi [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioc a b) a ↔ continuous_within_at f (Ioi a) a :=
by simp only [continuous_within_at, nhds_within_Ioc_eq_nhds_within_Ioi h]
@[simp]
lemma continuous_within_at_Ioo_iff_Ioi [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioo a b) a ↔ continuous_within_at f (Ioi a) a :=
by simp only [continuous_within_at, nhds_within_Ioo_eq_nhds_within_Ioi h]
/-!
#### Left neighborhoods, point excluded
-/
lemma Ioo_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Ioo a c ∈ 𝓝[Iio b] b :=
by simpa only [dual_Ioo] using @Ioo_mem_nhds_within_Ioi (order_dual α) _ _ _ _ _ _ ⟨H.2, H.1⟩
lemma Ico_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Ico a c ∈ 𝓝[Iio b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Iio H) Ioo_subset_Ico_self
lemma Ioc_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Ioc a c ∈ 𝓝[Iio b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Iio H) Ioo_subset_Ioc_self
lemma Icc_mem_nhds_within_Iio {a b c : α} (H : b ∈ Ioc a c) :
Icc a c ∈ 𝓝[Iio b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Iio H) Ioo_subset_Icc_self
@[simp] lemma nhds_within_Ico_eq_nhds_within_Iio {a b : α} (h : a < b) :
𝓝[Ico a b] b = 𝓝[Iio b] b :=
by simpa only [dual_Ioc] using @nhds_within_Ioc_eq_nhds_within_Ioi (order_dual α) _ _ _ _ _ h
@[simp] lemma nhds_within_Ioo_eq_nhds_within_Iio {a b : α} (h : a < b) :
𝓝[Ioo a b] b = 𝓝[Iio b] b :=
by simpa only [dual_Ioo] using @nhds_within_Ioo_eq_nhds_within_Ioi (order_dual α) _ _ _ _ _ h
@[simp] lemma continuous_within_at_Ico_iff_Iio {a b : α} {f : α → γ} (h : a < b) :
continuous_within_at f (Ico a b) b ↔ continuous_within_at f (Iio b) b :=
by simp only [continuous_within_at, nhds_within_Ico_eq_nhds_within_Iio h]
@[simp] lemma continuous_within_at_Ioo_iff_Iio {a b : α} {f : α → γ} (h : a < b) :
continuous_within_at f (Ioo a b) b ↔ continuous_within_at f (Iio b) b :=
by simp only [continuous_within_at, nhds_within_Ioo_eq_nhds_within_Iio h]
/-!
#### Right neighborhoods, point included
-/
lemma Ioo_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ioo a c) :
Ioo a c ∈ 𝓝[Ici b] b :=
mem_nhds_within_of_mem_nhds $ mem_nhds_sets is_open_Ioo H
lemma Ioc_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ioo a c) :
Ioc a c ∈ 𝓝[Ici b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Ici H) Ioo_subset_Ioc_self
lemma Ico_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ico a c) :
Ico a c ∈ 𝓝[Ici b] b :=
mem_nhds_within.2 ⟨Iio c, is_open_Iio, H.2,
by simp only [inter_comm, Ici_inter_Iio, Ico_subset_Ico_left H.1]⟩
lemma Icc_mem_nhds_within_Ici {a b c : α} (H : b ∈ Ico a c) :
Icc a c ∈ 𝓝[Ici b] b :=
mem_sets_of_superset (Ico_mem_nhds_within_Ici H) Ico_subset_Icc_self
@[simp] lemma nhds_within_Icc_eq_nhds_within_Ici {a b : α} (h : a < b) :
𝓝[Icc a b] a = 𝓝[Ici a] a :=
le_antisymm (nhds_within_mono _ Icc_subset_Ici_self) $
nhds_within_le_of_mem $ Icc_mem_nhds_within_Ici $ left_mem_Ico.2 h
@[simp] lemma nhds_within_Ico_eq_nhds_within_Ici {a b : α} (h : a < b) :
𝓝[Ico a b] a = 𝓝[Ici a] a :=
le_antisymm (nhds_within_mono _ (λ x, and.left)) $
nhds_within_le_of_mem $ Ico_mem_nhds_within_Ici $ left_mem_Ico.2 h
@[simp]
lemma continuous_within_at_Icc_iff_Ici [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Icc a b) a ↔ continuous_within_at f (Ici a) a :=
by simp only [continuous_within_at, nhds_within_Icc_eq_nhds_within_Ici h]
@[simp]
lemma continuous_within_at_Ico_iff_Ici [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ico a b) a ↔ continuous_within_at f (Ici a) a :=
by simp only [continuous_within_at, nhds_within_Ico_eq_nhds_within_Ici h]
/-!
#### Left neighborhoods, point included
-/
lemma Ioo_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioo a c) :
Ioo a c ∈ 𝓝[Iic b] b :=
mem_nhds_within_of_mem_nhds $ mem_nhds_sets is_open_Ioo H
lemma Ico_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioo a c) :
Ico a c ∈ 𝓝[Iic b] b :=
mem_sets_of_superset (Ioo_mem_nhds_within_Iic H) Ioo_subset_Ico_self
lemma Ioc_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioc a c) :
Ioc a c ∈ 𝓝[Iic b] b :=
by simpa only [dual_Ico] using @Ico_mem_nhds_within_Ici (order_dual α) _ _ _ _ _ _ ⟨H.2, H.1⟩
lemma Icc_mem_nhds_within_Iic {a b c : α} (H : b ∈ Ioc a c) :
Icc a c ∈ 𝓝[Iic b] b :=
mem_sets_of_superset (Ioc_mem_nhds_within_Iic H) Ioc_subset_Icc_self
@[simp] lemma nhds_within_Icc_eq_nhds_within_Iic {a b : α} (h : a < b) :
𝓝[Icc a b] b = 𝓝[Iic b] b :=
by simpa only [dual_Icc] using @nhds_within_Icc_eq_nhds_within_Ici (order_dual α) _ _ _ _ _ h
@[simp] lemma nhds_within_Ioc_eq_nhds_within_Iic {a b : α} (h : a < b) :
𝓝[Ioc a b] b = 𝓝[Iic b] b :=
by simpa only [dual_Ico] using @nhds_within_Ico_eq_nhds_within_Ici (order_dual α) _ _ _ _ _ h
@[simp]
lemma continuous_within_at_Icc_iff_Iic [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Icc a b) b ↔ continuous_within_at f (Iic b) b :=
by simp only [continuous_within_at, nhds_within_Icc_eq_nhds_within_Iic h]
@[simp]
lemma continuous_within_at_Ioc_iff_Iic [topological_space β] {a b : α} {f : α → β} (h : a < b) :
continuous_within_at f (Ioc a b) b ↔ continuous_within_at f (Iic b) b :=
by simp only [continuous_within_at, nhds_within_Ioc_eq_nhds_within_Iic h]
end linear_order
section linear_order
variables [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α}
section
variables [topological_space β]
lemma frontier_le_subset_eq (hf : continuous f) (hg : continuous g) :
frontier {b | f b ≤ g b} ⊆ {b | f b = g b} :=
begin
rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg],
rintros b ⟨hb₁, hb₂⟩,
refine le_antisymm hb₁ (closure_lt_subset_le hg hf _),
convert hb₂ using 2, simp only [not_le.symm], refl
end
lemma frontier_lt_subset_eq (hf : continuous f) (hg : continuous g) :
frontier {b | f b < g b} ⊆ {b | f b = g b} :=
by rw ← frontier_compl;
convert frontier_le_subset_eq hg hf; simp [ext_iff, eq_comm]
@[continuity] lemma continuous.min (hf : continuous f) (hg : continuous g) :
continuous (λb, min (f b) (g b)) :=
have ∀b∈frontier {b | f b ≤ g b}, f b = g b, from assume b hb, frontier_le_subset_eq hf hg hb,
continuous_if this hf hg
@[continuity] lemma continuous.max (hf : continuous f) (hg : continuous g) :
continuous (λb, max (f b) (g b)) :=
@continuous.min (order_dual α) _ _ _ _ _ _ _ hf hg
end
lemma continuous_min : continuous (λ p : α × α, min p.1 p.2) := continuous_fst.min continuous_snd
lemma continuous_max : continuous (λ p : α × α, max p.1 p.2) := continuous_fst.max continuous_snd
lemma tendsto.max {b : filter β} {a₁ a₂ : α} (hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) :
tendsto (λb, max (f b) (g b)) b (𝓝 (max a₁ a₂)) :=
(continuous_max.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)
lemma tendsto.min {b : filter β} {a₁ a₂ : α} (hf : tendsto f b (𝓝 a₁)) (hg : tendsto g b (𝓝 a₂)) :
tendsto (λb, min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=
(continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)
end linear_order
end order_closed_topology
/-- The order topology on an ordered type is the topology generated by open intervals. We register
it on a preorder, but it is mostly interesting in linear orders, where it is also order-closed.
We define it as a mixin. If you want to introduce the order topology on a preorder, use
`preorder.topology`. -/
class order_topology (α : Type*) [t : topological_space α] [preorder α] : Prop :=
(topology_eq_generate_intervals : t = generate_from {s | ∃a, s = Ioi a ∨ s = Iio a})
/-- (Order) topology on a partial order `α` generated by the subbase of open intervals
`(a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an
instance as many ordered sets are already endowed with the same topology, most often in a non-defeq
way though. Register as a local instance when necessary. -/
def preorder.topology (α : Type*) [preorder α] : topological_space α :=
generate_from {s : set α | ∃ (a : α), s = {b : α | a < b} ∨ s = {b : α | b < a}}
section order_topology
instance {α : Type*} [topological_space α] [partial_order α] [order_topology α] :
order_topology (order_dual α) :=
⟨by convert @order_topology.topology_eq_generate_intervals α _ _ _;
conv in (_ ∨ _) { rw or.comm }; refl⟩
section partial_order
variables [topological_space α] [partial_order α] [t : order_topology α]
include t
lemma is_open_iff_generate_intervals {s : set α} :
is_open s ↔ generate_open {s | ∃a, s = Ioi a ∨ s = Iio a} s :=
by rw [t.topology_eq_generate_intervals]; refl
lemma is_open_lt' (a : α) : is_open {b:α | a < b} :=
by rw [@is_open_iff_generate_intervals α _ _ t]; exact generate_open.basic _ ⟨a, or.inl rfl⟩
lemma is_open_gt' (a : α) : is_open {b:α | b < a} :=
by rw [@is_open_iff_generate_intervals α _ _ t]; exact generate_open.basic _ ⟨a, or.inr rfl⟩
lemma lt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x :=
mem_nhds_sets (is_open_lt' _) h
lemma le_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x :=
(𝓝 b).sets_of_superset (lt_mem_nhds h) $ assume b hb, le_of_lt hb
lemma gt_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b :=
mem_nhds_sets (is_open_gt' _) h
lemma ge_mem_nhds {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b :=
(𝓝 a).sets_of_superset (gt_mem_nhds h) $ assume b hb, le_of_lt hb
lemma nhds_eq_order (a : α) :
𝓝 a = (⨅b ∈ Iio a, 𝓟 (Ioi b)) ⊓ (⨅b ∈ Ioi a, 𝓟 (Iio b)) :=
by rw [t.topology_eq_generate_intervals, nhds_generate_from];
from le_antisymm
(le_inf
(le_binfi $ assume b hb, infi_le_of_le {c : α | b < c} $ infi_le _ ⟨hb, b, or.inl rfl⟩)
(le_binfi $ assume b hb, infi_le_of_le {c : α | c < b} $ infi_le _ ⟨hb, b, or.inr rfl⟩))
(le_infi $ assume s, le_infi $ assume ⟨ha, b, hs⟩,
match s, ha, hs with
| _, h, (or.inl rfl) := inf_le_left_of_le $ infi_le_of_le b $ infi_le _ h
| _, h, (or.inr rfl) := inf_le_right_of_le $ infi_le_of_le b $ infi_le _ h
end)
lemma tendsto_order {f : β → α} {a : α} {x : filter β} :
tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ (∀ a' > a, ∀ᶠ b in x, f b < a') :=
by simp [nhds_eq_order a, tendsto_inf, tendsto_infi, tendsto_principal]
instance tendsto_Icc_class_nhds (a : α) : tendsto_Ixx_class Icc (𝓝 a) (𝓝 a) :=
begin
simp only [nhds_eq_order, infi_subtype'],
refine ((has_basis_infi_principal_finite _).inf
(has_basis_infi_principal_finite _)).tendsto_Ixx_class (λ s hs, _),
refine (ord_connected_bInter _).inter (ord_connected_bInter _); intros _ _,
exacts [ord_connected_Ioi, ord_connected_Iio]
end
instance tendsto_Ico_class_nhds (a : α) : tendsto_Ixx_class Ico (𝓝 a) (𝓝 a) :=
tendsto_Ixx_class_of_subset (λ _ _, Ico_subset_Icc_self)
instance tendsto_Ioc_class_nhds (a : α) : tendsto_Ixx_class Ioc (𝓝 a) (𝓝 a) :=
tendsto_Ixx_class_of_subset (λ _ _, Ioc_subset_Icc_self)
instance tendsto_Ioo_class_nhds (a : α) : tendsto_Ixx_class Ioo (𝓝 a) (𝓝 a) :=
tendsto_Ixx_class_of_subset (λ _ _, Ioo_subset_Icc_self)
/-- Also known as squeeze or sandwich theorem. This version assumes that inequalities hold
eventually for the filter. -/
lemma tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : β → α} {b : filter β} {a : α}
(hg : tendsto g b (𝓝 a)) (hh : tendsto h b (𝓝 a))
(hgf : ∀ᶠ b in b, g b ≤ f b) (hfh : ∀ᶠ b in b, f b ≤ h b) :
tendsto f b (𝓝 a) :=
tendsto_order.2
⟨assume a' h',
have ∀ᶠ b in b, a' < g b, from (tendsto_order.1 hg).left a' h',
by filter_upwards [this, hgf] assume a, lt_of_lt_of_le,
assume a' h',
have ∀ᶠ b in b, h b < a', from (tendsto_order.1 hh).right a' h',
by filter_upwards [this, hfh] assume a h₁ h₂, lt_of_le_of_lt h₂ h₁⟩
/-- Also known as squeeze or sandwich theorem. This version assumes that inequalities hold
everywhere. -/
lemma tendsto_of_tendsto_of_tendsto_of_le_of_le {f g h : β → α} {b : filter β} {a : α}
(hg : tendsto g b (𝓝 a)) (hh : tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) :
tendsto f b (𝓝 a) :=
tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh
(eventually_of_forall hgf) (eventually_of_forall hfh)
lemma nhds_order_unbounded {a : α} (hu : ∃u, a < u) (hl : ∃l, l < a) :
𝓝 a = (⨅l (h₂ : l < a) u (h₂ : a < u), 𝓟 (Ioo l u)) :=
have ∃ u, u ∈ Ioi a, from hu, have ∃ l, l ∈ Iio a, from hl,
by { simp only [nhds_eq_order, inf_binfi, binfi_inf, *, inf_principal, Ioi_inter_Iio], refl }
lemma tendsto_order_unbounded {f : β → α} {a : α} {x : filter β}
(hu : ∃u, a < u) (hl : ∃l, l < a) (h : ∀l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) :
tendsto f x (𝓝 a) :=
by rw [nhds_order_unbounded hu hl];
from (tendsto_infi.2 $ assume l, tendsto_infi.2 $ assume hl,
tendsto_infi.2 $ assume u, tendsto_infi.2 $ assume hu, tendsto_principal.2 $ h l u hl hu)
end partial_order
instance tendsto_Ixx_nhds_within {α : Type*} [preorder α] [topological_space α]
(a : α) {s t : set α} {Ixx}
[tendsto_Ixx_class Ixx (𝓝 a) (𝓝 a)] [tendsto_Ixx_class Ixx (𝓟 s) (𝓟 t)]:
tendsto_Ixx_class Ixx (𝓝[s] a) (𝓝[t] a) :=
filter.tendsto_Ixx_class_inf
instance tendsto_Icc_class_nhds_pi {ι : Type*} {α : ι → Type*} [nonempty ι]
[Π i, partial_order (α i)] [Π i, topological_space (α i)] [∀ i, order_topology (α i)]
(f : Π i, α i) :
tendsto_Ixx_class Icc (𝓝 f) (𝓝 f) :=
begin
constructor,
conv in ((𝓝 f).lift' powerset) { rw [nhds_pi] },
simp only [lift'_infi_powerset, comap_lift'_eq2 monotone_powerset, tendsto_infi, tendsto_lift',
mem_powerset_iff, subset_def, mem_preimage],
intros i s hs,
have : tendsto (λ g : Π i, α i, g i) (𝓝 f) (𝓝 (f i)) := ((continuous_apply i).tendsto f),
refine (tendsto_lift'.1 ((this.comp tendsto_fst).Icc (this.comp tendsto_snd)) s hs).mono _,
exact λ p hp g hg, hp ⟨hg.1 _, hg.2 _⟩
end
theorem induced_order_topology' {α : Type u} {β : Type v}
[partial_order α] [ta : topological_space β] [partial_order β] [order_topology β]
(f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
(H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b)
(H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) :
@order_topology _ (induced f ta) _ :=
begin
letI := induced f ta,
refine ⟨eq_of_nhds_eq_nhds (λ a, _)⟩,
rw [nhds_induced, nhds_generate_from, nhds_eq_order (f a)],
apply le_antisymm,
{ refine le_infi (λ s, le_infi $ λ hs, le_principal_iff.2 _),
rcases hs with ⟨ab, b, rfl|rfl⟩,
{ exact mem_comap_sets.2 ⟨{x | f b < x},
mem_inf_sets_of_left $ mem_infi_sets _ $ mem_infi_sets (hf.2 ab) $ mem_principal_self _,
λ x, hf.1⟩ },
{ exact mem_comap_sets.2 ⟨{x | x < f b},
mem_inf_sets_of_right $ mem_infi_sets _ $ mem_infi_sets (hf.2 ab) $ mem_principal_self _,
λ x, hf.1⟩ } },
{ rw [← map_le_iff_le_comap],
refine le_inf _ _; refine le_infi (λ x, le_infi $ λ h, le_principal_iff.2 _); simp,
{ rcases H₁ h with ⟨b, ab, xb⟩,
refine mem_infi_sets _ (mem_infi_sets ⟨ab, b, or.inl rfl⟩ (mem_principal_sets.2 _)),
exact λ c hc, lt_of_le_of_lt xb (hf.2 hc) },
{ rcases H₂ h with ⟨b, ab, xb⟩,
refine mem_infi_sets _ (mem_infi_sets ⟨ab, b, or.inr rfl⟩ (mem_principal_sets.2 _)),
exact λ c hc, lt_of_lt_of_le (hf.2 hc) xb } },
end
theorem induced_order_topology {α : Type u} {β : Type v}
[partial_order α] [ta : topological_space β] [partial_order β] [order_topology β]
(f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y)
(H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) :
@order_topology _ (induced f ta) _ :=
induced_order_topology' f @hf
(λ a x xa, let ⟨b, xb, ba⟩ := H xa in ⟨b, hf.1 ba, le_of_lt xb⟩)
(λ a x ax, let ⟨b, ab, bx⟩ := H ax in ⟨b, hf.1 ab, le_of_lt bx⟩)
/-- On an `ord_connected` subset of a linear order, the order topology for the restriction of the
order is the same as the restriction to the subset of the order topology. -/
instance order_topology_of_ord_connected {α : Type u}
[ta : topological_space α] [linear_order α] [order_topology α]
{t : set α} [ht : ord_connected t] :
order_topology t :=
begin
letI := induced (coe : t → α) ta,
refine ⟨eq_of_nhds_eq_nhds (λ a, _)⟩,
rw [nhds_induced, nhds_generate_from, nhds_eq_order (a : α)],
apply le_antisymm,
{ refine le_infi (λ s, le_infi $ λ hs, le_principal_iff.2 _),
rcases hs with ⟨ab, b, rfl|rfl⟩,
{ refine ⟨Ioi b, _, λ _, id⟩,
refine mem_inf_sets_of_left (mem_infi_sets b _),
exact mem_infi_sets ab (mem_principal_self (Ioi ↑b)) },
{ refine ⟨Iio b, _, λ _, id⟩,
refine mem_inf_sets_of_right (mem_infi_sets b _),
exact mem_infi_sets ab (mem_principal_self (Iio b)) } },
{ rw [← map_le_iff_le_comap],
refine le_inf _ _,
{ refine le_infi (λ x, le_infi $ λ h, le_principal_iff.2 _),
by_cases hx : x ∈ t,
{ refine mem_infi_sets (Ioi ⟨x, hx⟩) (mem_infi_sets ⟨h, ⟨⟨x, hx⟩, or.inl rfl⟩⟩ _),
exact λ _, id },
simp only [set_coe.exists, mem_set_of_eq, mem_map],
convert univ_sets _,
suffices hx' : ∀ (y : t), ↑y ∈ Ioi x,
{ simp [hx'] },
intros y,
revert hx,
contrapose!,
-- here we use the `ord_connected` hypothesis
exact λ hx, ht y.2 a.2 ⟨le_of_not_gt hx, le_of_lt h⟩ },
{ refine le_infi (λ x, le_infi $ λ h, le_principal_iff.2 _),
by_cases hx : x ∈ t,
{ refine mem_infi_sets (Iio ⟨x, hx⟩) (mem_infi_sets ⟨h, ⟨⟨x, hx⟩, or.inr rfl⟩⟩ _),
exact λ _, id },
simp only [set_coe.exists, mem_set_of_eq, mem_map],
convert univ_sets _,
suffices hx' : ∀ (y : t), ↑y ∈ Iio x,
{ simp [hx'] },
intros y,
revert hx,
contrapose!,
-- here we use the `ord_connected` hypothesis
exact λ hx, ht a.2 y.2 ⟨le_of_lt h, le_of_not_gt hx⟩ } }
end
lemma nhds_top_order [topological_space α] [order_top α] [order_topology α] :
𝓝 (⊤:α) = (⨅l (h₂ : l < ⊤), 𝓟 (Ioi l)) :=
by simp [nhds_eq_order (⊤:α)]
lemma nhds_bot_order [topological_space α] [order_bot α] [order_topology α] :
𝓝 (⊥:α) = (⨅l (h₂ : ⊥ < l), 𝓟 (Iio l)) :=
by simp [nhds_eq_order (⊥:α)]
lemma tendsto_nhds_top_mono [topological_space β] [order_top β] [order_topology β] {l : filter α}
{f g : α → β} (hf : tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) :
tendsto g l (𝓝 ⊤) :=
begin
simp only [nhds_top_order, tendsto_infi, tendsto_principal] at hf ⊢,
intros x hx,
filter_upwards [hf x hx, hg],
exact λ x, lt_of_lt_of_le
end
lemma tendsto_nhds_bot_mono [topological_space β] [order_bot β] [order_topology β] {l : filter α}
{f g : α → β} (hf : tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) :
tendsto g l (𝓝 ⊥) :=
@tendsto_nhds_top_mono α (order_dual β) _ _ _ _ _ _ hf hg
lemma tendsto_nhds_top_mono' [topological_space β] [order_top β] [order_topology β] {l : filter α}
{f g : α → β} (hf : tendsto f l (𝓝 ⊤)) (hg : f ≤ g) :
tendsto g l (𝓝 ⊤) :=
tendsto_nhds_top_mono hf (eventually_of_forall hg)
lemma tendsto_nhds_bot_mono' [topological_space β] [order_bot β] [order_topology β] {l : filter α}
{f g : α → β} (hf : tendsto f l (𝓝 ⊥)) (hg : g ≤ f) :
tendsto g l (𝓝 ⊥) :=
tendsto_nhds_bot_mono hf (eventually_of_forall hg)
section linear_order
variables [topological_space α] [linear_order α] [order_topology α]
lemma exists_Ioc_subset_of_mem_nhds' {a : α} {s : set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) :
∃ l' ∈ Ico l a, Ioc l' a ⊆ s :=
begin
rw [nhds_eq_order a] at hs,
rcases hs with ⟨t₁, ht₁, t₂, ht₂, hts⟩,
-- First we show that `t₂` includes `(-∞, a]`, so it suffices to show `(l', ∞) ⊆ t₁`
suffices : ∃ l' ∈ Ico l a, Ioi l' ⊆ t₁,
{ have A : 𝓟 (Iic a) ≤ ⨅ b ∈ Ioi a, 𝓟 (Iio b),
from (le_infi $ λ b, le_infi $ λ hb, principal_mono.2 $ Iic_subset_Iio.2 hb),
have B : t₁ ∩ Iic a ⊆ s,
from subset.trans (inter_subset_inter_right _ (A ht₂)) hts,
from this.imp (λ l', Exists.imp $ λ hl' hl x hx, B ⟨hl hx.1, hx.2⟩) },
clear hts ht₂ t₂,
-- Now we find `l` such that `(l', ∞) ⊆ t₁`
rw [mem_binfi] at ht₁,
{ rcases ht₁ with ⟨b, hb, hb'⟩,
exact ⟨max b l, ⟨le_max_right _ _, max_lt hb hl⟩,
λ x hx, hb' $ Ioi_subset_Ioi (le_max_left _ _) hx⟩ },
{ intros b hb b' hb', simp only [mem_Iio] at hb hb',
use [max b b', max_lt hb hb'],
simp [le_refl] },
exact ⟨l, hl⟩
end
lemma exists_Ico_subset_of_mem_nhds' {a : α} {s : set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) :
∃ u' ∈ Ioc a u, Ico a u' ⊆ s :=
begin
convert @exists_Ioc_subset_of_mem_nhds' (order_dual α) _ _ _ _ _ hs _ hu,
ext, rw [dual_Ico, dual_Ioc]
end
lemma exists_Ioc_subset_of_mem_nhds {a : α} {s : set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) :
∃ l < a, Ioc l a ⊆ s :=
let ⟨l', hl'⟩ := h in let ⟨l, hl⟩ := exists_Ioc_subset_of_mem_nhds' hs hl' in ⟨l, hl.fst.2, hl.snd⟩
lemma exists_Ico_subset_of_mem_nhds {a : α} {s : set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) :
∃ u (_ : a < u), Ico a u ⊆ s :=
let ⟨l', hl'⟩ := h in let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl' in ⟨l, hl.fst.1, hl.snd⟩
lemma order_separated {a₁ a₂ : α} (h : a₁ < a₂) :
∃u v : set α, is_open u ∧ is_open v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ (∀b₁∈u, ∀b₂∈v, b₁ < b₂) :=
match dense_or_discrete a₁ a₂ with
| or.inl ⟨a, ha₁, ha₂⟩ := ⟨{a' | a' < a}, {a' | a < a'}, is_open_gt' a, is_open_lt' a, ha₁, ha₂,
assume b₁ h₁ b₂ h₂, lt_trans h₁ h₂⟩
| or.inr ⟨h₁, h₂⟩ := ⟨{a | a < a₂}, {a | a₁ < a}, is_open_gt' a₂, is_open_lt' a₁, h, h,
assume b₁ hb₁ b₂ hb₂,
calc b₁ ≤ a₁ : h₂ _ hb₁
... < a₂ : h
... ≤ b₂ : h₁ _ hb₂⟩
end
@[priority 100] -- see Note [lower instance priority]
instance order_topology.to_order_closed_topology : order_closed_topology α :=
{ is_closed_le' :=
is_open_prod_iff.mpr $ assume a₁ a₂ (h : ¬ a₁ ≤ a₂),
have h : a₂ < a₁, from lt_of_not_ge h,
let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h in
⟨v, u, hv, hu, ha₂, ha₁, assume ⟨b₁, b₂⟩ ⟨h₁, h₂⟩, not_le_of_gt $ h b₂ h₂ b₁ h₁⟩ }
lemma order_topology.t2_space : t2_space α := by apply_instance
@[priority 100] -- see Note [lower instance priority]
instance order_topology.regular_space : regular_space α :=
{ regular := assume s a hs ha,
have hs' : sᶜ ∈ 𝓝 a, from mem_nhds_sets hs ha,
have ∃t:set α, is_open t ∧ (∀l∈ s, l < a → l ∈ t) ∧ 𝓝[t] a = ⊥,
from by_cases
(assume h : ∃l, l < a,
let ⟨l, hl, h⟩ := exists_Ioc_subset_of_mem_nhds hs' h in
match dense_or_discrete l a with
| or.inl ⟨b, hb₁, hb₂⟩ := ⟨{a | a < b}, is_open_gt' _,
assume c hcs hca, show c < b,
from lt_of_not_ge $ assume hbc, h ⟨lt_of_lt_of_le hb₁ hbc, le_of_lt hca⟩ hcs,
inf_principal_eq_bot $ (𝓝 a).sets_of_superset (mem_nhds_sets (is_open_lt' _) hb₂) $
assume x (hx : b < x), show ¬ x < b, from not_lt.2 $ le_of_lt hx⟩
| or.inr ⟨h₁, h₂⟩ := ⟨{a' | a' < a}, is_open_gt' _, assume b hbs hba, hba,
inf_principal_eq_bot $ (𝓝 a).sets_of_superset (mem_nhds_sets (is_open_lt' _) hl) $
assume x (hx : l < x), show ¬ x < a, from not_lt.2 $ h₁ _ hx⟩
end)
(assume : ¬ ∃l, l < a, ⟨∅, is_open_empty, assume l _ hl, (this ⟨l, hl⟩).elim,
nhds_within_empty _⟩),
let ⟨t₁, ht₁o, ht₁s, ht₁a⟩ := this in
have ∃t:set α, is_open t ∧ (∀u∈ s, u>a → u ∈ t) ∧ 𝓝[t] a = ⊥,
from by_cases
(assume h : ∃u, u > a,
let ⟨u, hu, h⟩ := exists_Ico_subset_of_mem_nhds hs' h in
match dense_or_discrete a u with
| or.inl ⟨b, hb₁, hb₂⟩ := ⟨{a | b < a}, is_open_lt' _,
assume c hcs hca, show c > b,
from lt_of_not_ge $ assume hbc, h ⟨le_of_lt hca, lt_of_le_of_lt hbc hb₂⟩ hcs,
inf_principal_eq_bot $ (𝓝 a).sets_of_superset (mem_nhds_sets (is_open_gt' _) hb₁) $
assume x (hx : b > x), show ¬ x > b, from not_lt.2 $ le_of_lt hx⟩
| or.inr ⟨h₁, h₂⟩ := ⟨{a' | a' > a}, is_open_lt' _, assume b hbs hba, hba,
inf_principal_eq_bot $ (𝓝 a).sets_of_superset (mem_nhds_sets (is_open_gt' _) hu) $
assume x (hx : u > x), show ¬ x > a, from not_lt.2 $ h₂ _ hx⟩
end)
(assume : ¬ ∃u, u > a, ⟨∅, is_open_empty, assume l _ hl, (this ⟨l, hl⟩).elim,
nhds_within_empty _⟩),
let ⟨t₂, ht₂o, ht₂s, ht₂a⟩ := this in
⟨t₁ ∪ t₂, is_open_union ht₁o ht₂o,
assume x hx,
have x ≠ a, from assume eq, ha $ eq ▸ hx,
(ne_iff_lt_or_gt.mp this).imp (ht₁s _ hx) (ht₂s _ hx),
by rw [nhds_within_union, ht₁a, ht₂a, bot_sup_eq]⟩,
..order_topology.t2_space }
/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`,
provided `a` is neither a bottom element nor a top element. -/
lemma mem_nhds_iff_exists_Ioo_subset' {a : α} {s : set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) :
s ∈ 𝓝 a ↔ ∃l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=
begin
split,
{ assume h,
rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩,
rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩,
refine ⟨l, u, ⟨la, au⟩, λx hx, _⟩,
cases le_total a x with hax hax,
{ exact hu ⟨hax, hx.2⟩ },
{ exact hl ⟨hx.1, hax⟩ } },
{ rintros ⟨l, u, ha, h⟩,
apply mem_sets_of_superset (mem_nhds_sets is_open_Ioo ha) h }
end
/-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.
-/
lemma mem_nhds_iff_exists_Ioo_subset [no_top_order α] [no_bot_order α] {a : α} {s : set α} :
s ∈ 𝓝 a ↔ ∃l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=
mem_nhds_iff_exists_Ioo_subset' (no_bot a) (no_top a)
lemma nhds_basis_Ioo' {a : α} (hl : ∃l, l < a) (hu : ∃u, a < u) :
(𝓝 a).has_basis (λ b : α × α, b.1 < a ∧ a < b.2) (λ b, Ioo b.1 b.2) :=
⟨λ s, (mem_nhds_iff_exists_Ioo_subset' hl hu).trans $ by simp⟩
lemma nhds_basis_Ioo [no_top_order α] [no_bot_order α] {a : α} :
(𝓝 a).has_basis (λ b : α × α, b.1 < a ∧ a < b.2) (λ b, Ioo b.1 b.2) :=
nhds_basis_Ioo' (no_bot a) (no_top a)
lemma filter.eventually.exists_Ioo_subset [no_top_order α] [no_bot_order α] {a : α} {p : α → Prop}
(hp : ∀ᶠ x in 𝓝 a, p x) :
∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ {x | p x} :=
mem_nhds_iff_exists_Ioo_subset.1 hp
lemma Iio_mem_nhds {a b : α} (h : a < b) : Iio b ∈ 𝓝 a :=
mem_nhds_sets is_open_Iio h
lemma Ioi_mem_nhds {a b : α} (h : a < b) : Ioi a ∈ 𝓝 b :=
mem_nhds_sets is_open_Ioi h
lemma Iic_mem_nhds {a b : α} (h : a < b) : Iic b ∈ 𝓝 a :=
mem_sets_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self
lemma Ici_mem_nhds {a b : α} (h : a < b) : Ici a ∈ 𝓝 b :=
mem_sets_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self
lemma Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x :=
mem_nhds_sets is_open_Ioo ⟨ha, hb⟩
lemma Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x :=
mem_sets_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self
lemma Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x :=
mem_sets_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self
lemma Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x :=
mem_sets_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self
section pi
/-!
### Intervals in `Π i, π i` belong to `𝓝 x`
For each leamma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because
sometimes Lean fails to unify different instances while trying to apply the dependent version to,
e.g., `ι → ℝ`.
-/
variables {ι : Type*} {π : ι → Type*} [fintype ι] [Π i, linear_order (π i)]
[Π i, topological_space (π i)] [∀ i, order_topology (π i)] {a b x : Π i, π i} {a' b' x' : ι → α}
lemma pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x :=
pi_univ_Iic a ▸ set_pi_mem_nhds (finite.of_fintype _) (λ i _, Iic_mem_nhds (ha _))
lemma pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' :=
pi_Iic_mem_nhds ha
lemma pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x :=
pi_univ_Ici a ▸ set_pi_mem_nhds (finite.of_fintype _) (λ i _, Ici_mem_nhds (ha _))
lemma pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' :=
pi_Ici_mem_nhds ha
lemma pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x :=
pi_univ_Icc a b ▸ set_pi_mem_nhds (finite.of_fintype _) (λ i _, Icc_mem_nhds (ha _) (hb _))
lemma pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' :=
pi_Icc_mem_nhds ha hb
variables [nonempty ι]
lemma pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x :=
begin
refine mem_sets_of_superset (set_pi_mem_nhds (finite.of_fintype _) (λ i _, _))
(pi_univ_Iio_subset a),
exact Iio_mem_nhds (ha i)
end
lemma pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' :=
pi_Iio_mem_nhds ha
lemma pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x :=
@pi_Iio_mem_nhds ι (λ i, order_dual (π i)) _ _ _ _ _ _ _ ha
lemma pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' :=
pi_Ioi_mem_nhds ha
lemma pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x :=
begin
refine mem_sets_of_superset (set_pi_mem_nhds (finite.of_fintype _) (λ i _, _))
(pi_univ_Ioc_subset a b),
exact Ioc_mem_nhds (ha i) (hb i)
end
lemma pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' :=
pi_Ioc_mem_nhds ha hb
lemma pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x :=
begin
refine mem_sets_of_superset (set_pi_mem_nhds (finite.of_fintype _) (λ i _, _))
(pi_univ_Ico_subset a b),
exact Ico_mem_nhds (ha i) (hb i)
end
lemma pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' :=
pi_Ico_mem_nhds ha hb
lemma pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x :=
begin
refine mem_sets_of_superset (set_pi_mem_nhds (finite.of_fintype _) (λ i _, _))
(pi_univ_Ioo_subset a b),
exact Ioo_mem_nhds (ha i) (hb i)
end
lemma pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' :=
pi_Ioo_mem_nhds ha hb
end pi
lemma disjoint_nhds_at_top [no_top_order α] (x : α) :
disjoint (𝓝 x) at_top :=
begin
rw filter.disjoint_iff,
cases no_top x with a ha,
use [Iio a, Iio_mem_nhds ha, Ici a, mem_at_top a],
rw [inter_comm, Ici_inter_Iio, Ico_self]
end
@[simp] lemma inf_nhds_at_top [no_top_order α] (x : α) :
𝓝 x ⊓ at_top = ⊥ :=
disjoint_iff.1 (disjoint_nhds_at_top x)
lemma disjoint_nhds_at_bot [no_bot_order α] (x : α) :
disjoint (𝓝 x) at_bot :=
@disjoint_nhds_at_top (order_dual α) _ _ _ _ x
@[simp] lemma inf_nhds_at_bot [no_bot_order α] (x : α) :
𝓝 x ⊓ at_bot = ⊥ :=
@inf_nhds_at_top (order_dual α) _ _ _ _ x
lemma not_tendsto_nhds_of_tendsto_at_top [no_top_order α]
{F : filter β} [ne_bot F] {f : β → α} (hf : tendsto f F at_top) (x : α) :
¬ tendsto f F (𝓝 x) :=
hf.not_tendsto (disjoint_nhds_at_top x).symm
lemma not_tendsto_at_top_of_tendsto_nhds [no_top_order α]
{F : filter β} [ne_bot F] {f : β → α} {x : α} (hf : tendsto f F (𝓝 x)) :
¬ tendsto f F at_top :=
hf.not_tendsto (disjoint_nhds_at_top x)
lemma not_tendsto_nhds_of_tendsto_at_bot [no_bot_order α]
{F : filter β} [ne_bot F] {f : β → α} (hf : tendsto f F at_bot) (x : α) :
¬ tendsto f F (𝓝 x) :=
hf.not_tendsto (disjoint_nhds_at_bot x).symm
lemma not_tendsto_at_bot_of_tendsto_nhds [no_bot_order α]
{F : filter β} [ne_bot F] {f : β → α} {x : α} (hf : tendsto f F (𝓝 x)) :
¬ tendsto f F at_bot :=
hf.not_tendsto (disjoint_nhds_at_bot x)
/-!
### Neighborhoods to the left and to the right on an `order_topology`
We've seen some properties of left and right neighborhood of a point in an `order_closed_topology`.
In an `order_topology`, such neighborhoods can be characterized as the sets containing suitable
intervals to the right or to the left of `a`. We give now these characterizations. -/
-- NB: If you extend the list, append to the end please to avoid breaking the API
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `(a, +∞)`
1. `s` is a neighborhood of `a` within `(a, b]`
2. `s` is a neighborhood of `a` within `(a, b)`
3. `s` includes `(a, u)` for some `u ∈ (a, b]`
4. `s` includes `(a, u)` for some `u > a` -/
lemma tfae_mem_nhds_within_Ioi {a b : α} (hab : a < b) (s : set α) :
tfae [s ∈ 𝓝[Ioi a] a, -- 0 : `s` is a neighborhood of `a` within `(a, +∞)`
s ∈ 𝓝[Ioc a b] a, -- 1 : `s` is a neighborhood of `a` within `(a, b]`
s ∈ 𝓝[Ioo a b] a, -- 2 : `s` is a neighborhood of `a` within `(a, b)`
∃ u ∈ Ioc a b, Ioo a u ⊆ s, -- 3 : `s` includes `(a, u)` for some `u ∈ (a, b]`
∃ u ∈ Ioi a, Ioo a u ⊆ s] := -- 4 : `s` includes `(a, u)` for some `u > a`
begin
tfae_have : 1 ↔ 2, by rw [nhds_within_Ioc_eq_nhds_within_Ioi hab],
tfae_have : 1 ↔ 3, by rw [nhds_within_Ioo_eq_nhds_within_Ioi hab],
tfae_have : 4 → 5, from λ ⟨u, umem, hu⟩, ⟨u, umem.1, hu⟩,
tfae_have : 5 → 1,
{ rintros ⟨u, hau, hu⟩,
exact mem_sets_of_superset (Ioo_mem_nhds_within_Ioi ⟨le_refl a, hau⟩) hu },
tfae_have : 1 → 4,
{ assume h,
rcases mem_nhds_within_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩,
rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩,
refine ⟨u, au, λx hx, _⟩,
refine hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, _⟩,
exact hx.1 },
tfae_finish
end
lemma mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[Ioi a] a ↔ ∃u ∈ Ioc a u', Ioo a u ⊆ s :=
(tfae_mem_nhds_within_Ioi hu' s).out 0 3
/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u < u'`, provided `a` is not a top element. -/
lemma mem_nhds_within_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[Ioi a] a ↔ ∃u ∈ Ioi a, Ioo a u ⊆ s :=
(tfae_mem_nhds_within_Ioi hu' s).out 0 4
/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u`. -/
lemma mem_nhds_within_Ioi_iff_exists_Ioo_subset [no_top_order α] {a : α} {s : set α} :
s ∈ 𝓝[Ioi a] a ↔ ∃u ∈ Ioi a, Ioo a u ⊆ s :=
let ⟨u', hu'⟩ := no_top a in mem_nhds_within_Ioi_iff_exists_Ioo_subset' hu'
/-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
lemma mem_nhds_within_Ioi_iff_exists_Ioc_subset [no_top_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Ioi a] a ↔ ∃u ∈ Ioi a, Ioc a u ⊆ s :=
begin
rw mem_nhds_within_Ioi_iff_exists_Ioo_subset,
split,
{ rintros ⟨u, au, as⟩,
rcases exists_between au with ⟨v, hv⟩,
exact ⟨v, hv.1, λx hx, as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ },
{ rintros ⟨u, au, as⟩,
exact ⟨u, au, subset.trans Ioo_subset_Ioc_self as⟩ }
end
/-- The following statements are equivalent:
0. `s` is a neighborhood of `b` within `(-∞, b)`
1. `s` is a neighborhood of `b` within `[a, b)`
2. `s` is a neighborhood of `b` within `(a, b)`
3. `s` includes `(l, b)` for some `l ∈ [a, b)`
4. `s` includes `(l, b)` for some `l < b` -/
lemma tfae_mem_nhds_within_Iio {a b : α} (h : a < b) (s : set α) :
tfae [s ∈ 𝓝[Iio b] b, -- 0 : `s` is a neighborhood of `b` within `(-∞, b)`
s ∈ 𝓝[Ico a b] b, -- 1 : `s` is a neighborhood of `b` within `[a, b)`
s ∈ 𝓝[Ioo a b] b, -- 2 : `s` is a neighborhood of `b` within `(a, b)`
∃ l ∈ Ico a b, Ioo l b ⊆ s, -- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`
∃ l ∈ Iio b, Ioo l b ⊆ s] := -- 4 : `s` includes `(l, b)` for some `l < b`
begin
have := @tfae_mem_nhds_within_Ioi (order_dual α) _ _ _ _ _ h s,
-- If we call `convert` here, it generates wrong equations, so we need to simplify first
simp only [exists_prop] at this ⊢,
rw [dual_Ioi, dual_Ioc, dual_Ioo] at this,
convert this; ext l; rw [dual_Ioo]
end
lemma mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[Iio a] a ↔ ∃l ∈ Ico l' a, Ioo l a ⊆ s :=
(tfae_mem_nhds_within_Iio hl' s).out 0 3
/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`, provided `a` is not a bottom element. -/
lemma mem_nhds_within_Iio_iff_exists_Ioo_subset' {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[Iio a] a ↔ ∃l ∈ Iio a, Ioo l a ⊆ s :=
(tfae_mem_nhds_within_Iio hl' s).out 0 4
/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`. -/
lemma mem_nhds_within_Iio_iff_exists_Ioo_subset [no_bot_order α] {a : α} {s : set α} :
s ∈ 𝓝[Iio a] a ↔ ∃l ∈ Iio a, Ioo l a ⊆ s :=
let ⟨l', hl'⟩ := no_bot a in mem_nhds_within_Iio_iff_exists_Ioo_subset' hl'
/-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`
with `l < a`. -/
lemma mem_nhds_within_Iio_iff_exists_Ico_subset [no_bot_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Iio a] a ↔ ∃l ∈ Iio a, Ico l a ⊆ s :=
begin
convert @mem_nhds_within_Ioi_iff_exists_Ioc_subset (order_dual α) _ _ _ _ _ _ _,
simp only [dual_Ioc], refl
end
/-- The following statements are equivalent:
0. `s` is a neighborhood of `a` within `[a, +∞)`
1. `s` is a neighborhood of `a` within `[a, b]`
2. `s` is a neighborhood of `a` within `[a, b)`
3. `s` includes `[a, u)` for some `u ∈ (a, b]`
4. `s` includes `[a, u)` for some `u > a` -/
lemma tfae_mem_nhds_within_Ici {a b : α} (hab : a < b) (s : set α) :
tfae [s ∈ 𝓝[Ici a] a, -- 0 : `s` is a neighborhood of `a` within `[a, +∞)`
s ∈ 𝓝[Icc a b] a, -- 1 : `s` is a neighborhood of `a` within `[a, b]`
s ∈ 𝓝[Ico a b] a, -- 2 : `s` is a neighborhood of `a` within `[a, b)`
∃ u ∈ Ioc a b, Ico a u ⊆ s, -- 3 : `s` includes `[a, u)` for some `u ∈ (a, b]`
∃ u ∈ Ioi a, Ico a u ⊆ s] := -- 4 : `s` includes `[a, u)` for some `u > a`
begin
tfae_have : 1 ↔ 2, by rw [nhds_within_Icc_eq_nhds_within_Ici hab],
tfae_have : 1 ↔ 3, by rw [nhds_within_Ico_eq_nhds_within_Ici hab],
tfae_have : 4 → 5, from λ ⟨u, umem, hu⟩, ⟨u, umem.1, hu⟩,
tfae_have : 5 → 1,
{ rintros ⟨u, hau, hu⟩,
exact mem_sets_of_superset (Ico_mem_nhds_within_Ici ⟨le_refl a, hau⟩) hu },
tfae_have : 1 → 4,
{ assume h,
rcases mem_nhds_within_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩,
rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩,
refine ⟨u, au, λx hx, _⟩,
refine hv ⟨hu ⟨hx.1, hx.2⟩, _⟩,
exact hx.1 },
tfae_finish
end
lemma mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[Ici a] a ↔ ∃u ∈ Ioc a u', Ico a u ⊆ s :=
(tfae_mem_nhds_within_Ici hu' s).out 0 3 (by norm_num) (by norm_num)
/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u < u'`, provided `a` is not a top element. -/
lemma mem_nhds_within_Ici_iff_exists_Ico_subset' {a u' : α} {s : set α} (hu' : a < u') :
s ∈ 𝓝[Ici a] a ↔ ∃u ∈ Ioi a, Ico a u ⊆ s :=
(tfae_mem_nhds_within_Ici hu' s).out 0 4 (by norm_num) (by norm_num)
/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u`. -/
lemma mem_nhds_within_Ici_iff_exists_Ico_subset [no_top_order α] {a : α} {s : set α} :
s ∈ 𝓝[Ici a] a ↔ ∃u ∈ Ioi a, Ico a u ⊆ s :=
let ⟨u', hu'⟩ := no_top a in mem_nhds_within_Ici_iff_exists_Ico_subset' hu'
/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
with `a < u`. -/
lemma mem_nhds_within_Ici_iff_exists_Icc_subset' [no_top_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Ici a] a ↔ ∃u ∈ Ioi a, Icc a u ⊆ s :=
begin
rw mem_nhds_within_Ici_iff_exists_Ico_subset,
split,
{ rintros ⟨u, au, as⟩,
rcases exists_between au with ⟨v, hv⟩,
exact ⟨v, hv.1, λx hx, as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ },
{ rintros ⟨u, au, as⟩,
exact ⟨u, au, subset.trans Ico_subset_Icc_self as⟩ }
end
/-- The following statements are equivalent:
0. `s` is a neighborhood of `b` within `(-∞, b]`
1. `s` is a neighborhood of `b` within `[a, b]`
2. `s` is a neighborhood of `b` within `(a, b]`
3. `s` includes `(l, b]` for some `l ∈ [a, b)`
4. `s` includes `(l, b]` for some `l < b` -/
lemma tfae_mem_nhds_within_Iic {a b : α} (h : a < b) (s : set α) :
tfae [s ∈ 𝓝[Iic b] b, -- 0 : `s` is a neighborhood of `b` within `(-∞, b]`
s ∈ 𝓝[Icc a b] b, -- 1 : `s` is a neighborhood of `b` within `[a, b]`
s ∈ 𝓝[Ioc a b] b, -- 2 : `s` is a neighborhood of `b` within `(a, b]`
∃ l ∈ Ico a b, Ioc l b ⊆ s, -- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`
∃ l ∈ Iio b, Ioc l b ⊆ s] := -- 4 : `s` includes `(l, b]` for some `l < b`
begin
have := @tfae_mem_nhds_within_Ici (order_dual α) _ _ _ _ _ h s,
-- If we call `convert` here, it generates wrong equations, so we need to simplify first
simp only [exists_prop] at this ⊢,
rw [dual_Icc, dual_Ioc, dual_Ioi] at this,
convert this; ext l; rw [dual_Ico]
end
lemma mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[Iic a] a ↔ ∃l ∈ Ico l' a, Ioc l a ⊆ s :=
(tfae_mem_nhds_within_Iic hl' s).out 0 3 (by norm_num) (by norm_num)
/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`, provided `a` is not a bottom element. -/
lemma mem_nhds_within_Iic_iff_exists_Ioc_subset' {a l' : α} {s : set α} (hl' : l' < a) :
s ∈ 𝓝[Iic a] a ↔ ∃l ∈ Iio a, Ioc l a ⊆ s :=
(tfae_mem_nhds_within_Iic hl' s).out 0 4 (by norm_num) (by norm_num)
/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`. -/
lemma mem_nhds_within_Iic_iff_exists_Ioc_subset [no_bot_order α] {a : α} {s : set α} :
s ∈ 𝓝[Iic a] a ↔ ∃l ∈ Iio a, Ioc l a ⊆ s :=
let ⟨l', hl'⟩ := no_bot a in mem_nhds_within_Iic_iff_exists_Ioc_subset' hl'
/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
with `l < a`. -/
lemma mem_nhds_within_Iic_iff_exists_Icc_subset' [no_bot_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Iic a] a ↔ ∃l ∈ Iio a, Icc l a ⊆ s :=
begin
convert @mem_nhds_within_Ici_iff_exists_Icc_subset' (order_dual α) _ _ _ _ _ _ _,
simp_rw (show ∀ u : order_dual α, @Icc (order_dual α) _ a u = @Icc α _ u a, from λ u, dual_Icc),
refl,
end
/-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
with `a < u`. -/
lemma mem_nhds_within_Ici_iff_exists_Icc_subset [no_top_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Ici a] a ↔ ∃u, a < u ∧ Icc a u ⊆ s :=
begin
rw mem_nhds_within_Ici_iff_exists_Ico_subset,
split,
{ rintros ⟨u, au, as⟩,
rcases exists_between au with ⟨v, hv⟩,
exact ⟨v, hv.1, λx hx, as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ },
{ rintros ⟨u, au, as⟩,
exact ⟨u, au, subset.trans Ico_subset_Icc_self as⟩ }
end
/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
with `l < a`. -/
lemma mem_nhds_within_Iic_iff_exists_Icc_subset [no_bot_order α] [densely_ordered α]
{a : α} {s : set α} : s ∈ 𝓝[Iic a] a ↔ ∃l, l < a ∧ Icc l a ⊆ s :=
begin
rw mem_nhds_within_Iic_iff_exists_Ioc_subset,
split,
{ rintros ⟨l, la, as⟩,
rcases exists_between la with ⟨v, hv⟩,
refine ⟨v, hv.2, λx hx, as ⟨lt_of_lt_of_le hv.1 hx.1, hx.2⟩⟩, },
{ rintros ⟨l, la, as⟩,
exact ⟨l, la, subset.trans Ioc_subset_Icc_self as⟩ }
end
end linear_order
section linear_ordered_add_comm_group
variables [topological_space α] [linear_ordered_add_comm_group α] [order_topology α]
variables {l : filter β} {f g : β → α}
local notation `|` x `|` := abs x
lemma nhds_eq_infi_abs_sub (a : α) : 𝓝 a = (⨅r>0, 𝓟 {b | |a - b| < r}) :=
begin
simp only [le_antisymm_iff, nhds_eq_order, le_inf_iff, le_infi_iff, le_principal_iff, mem_Ioi,
mem_Iio, abs_sub_lt_iff, @sub_lt_iff_lt_add _ _ _ _ a, @sub_lt _ _ a, set_of_and],
refine ⟨_, _, _⟩,
{ intros ε ε0,
exact inter_mem_inf_sets
(mem_infi_sets (a - ε) $ mem_infi_sets (sub_lt_self a ε0) (mem_principal_self _))
(mem_infi_sets (ε + a) $ mem_infi_sets (by simpa) (mem_principal_self _)) },
{ intros b hb,
exact mem_infi_sets (a - b) (mem_infi_sets (sub_pos.2 hb) (by simp [Ioi])) },
{ intros b hb,
exact mem_infi_sets (b - a) (mem_infi_sets (sub_pos.2 hb) (by simp [Iio])) }
end
lemma order_topology_of_nhds_abs {α : Type*} [topological_space α] [linear_ordered_add_comm_group α]
(h_nhds : ∀a:α, 𝓝 a = (⨅r>0, 𝓟 {b | |a - b| < r})) : order_topology α :=
begin
refine ⟨eq_of_nhds_eq_nhds $ λ a, _⟩,
rw [h_nhds],
letI := preorder.topology α, letI : order_topology α := ⟨rfl⟩,
exact (nhds_eq_infi_abs_sub a).symm
end
lemma linear_ordered_add_comm_group.tendsto_nhds {x : filter β} {a : α} :
tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε :=
by simp [nhds_eq_infi_abs_sub, abs_sub a]
lemma eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε :=
(nhds_eq_infi_abs_sub a).symm ▸ mem_infi_sets ε
(mem_infi_sets hε $ by simp only [abs_sub, mem_principal_self])
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_add_comm_group.topological_add_group : topological_add_group α :=
{ continuous_add :=
begin
refine continuous_iff_continuous_at.2 _,
rintro ⟨a, b⟩,
refine linear_ordered_add_comm_group.tendsto_nhds.2 (λ ε ε0, _),
rcases dense_or_discrete 0 ε with (⟨δ, δ0, δε⟩|⟨h₁, h₂⟩),
{ -- If there exists `δ ∈ (0, ε)`, then we choose `δ`-nhd of `a` and `(ε-δ)`-nhd of `b`
filter_upwards [prod_mem_nhds_sets (eventually_abs_sub_lt a δ0)
(eventually_abs_sub_lt b (sub_pos.2 δε))],
rintros ⟨x, y⟩ ⟨hx : |x - a| < δ, hy : |y - b| < ε - δ⟩,
rw [add_sub_comm],
calc |x - a + (y - b)| ≤ |x - a| + |y - b| : abs_add _ _
... < δ + (ε - δ) : add_lt_add hx hy
... = ε : add_sub_cancel'_right _ _ },
{ -- Otherewise `ε`-nhd of each point `a` is `{a}`
have hε : ∀ {x y}, abs (x - y) < ε → x = y,
{ intros x y h,
simpa [sub_eq_zero] using h₂ _ h },
filter_upwards [prod_mem_nhds_sets (eventually_abs_sub_lt a ε0)
(eventually_abs_sub_lt b ε0)],
rintros ⟨x, y⟩ ⟨hx : |x - a| < ε, hy : |y - b| < ε⟩,
simpa [hε hx, hε hy] }
end,
continuous_neg := continuous_iff_continuous_at.2 $ λ a,
linear_ordered_add_comm_group.tendsto_nhds.2 $ λ ε ε0,
(eventually_abs_sub_lt a ε0).mono $ λ x hx, by rwa [neg_sub_neg, abs_sub] }
@[continuity]
lemma continuous_abs : continuous (abs : α → α) := continuous_id.max continuous_neg
lemma filter.tendsto.abs {f : β → α} {a : α} {l : filter β} (h : tendsto f l (𝓝 a)) :
tendsto (λ x, |f x|) l (𝓝 (|a|)) :=
(continuous_abs.tendsto _).comp h
section
variables [topological_space β] {b : β} {a : α} {s : set β}
lemma continuous.abs (h : continuous f) : continuous (λ x, |f x|) := continuous_abs.comp h
lemma continuous_at.abs (h : continuous_at f b) : continuous_at (λ x, |f x|) b := h.abs
lemma continuous_within_at.abs (h : continuous_within_at f s b) :
continuous_within_at (λ x, |f x|) s b := h.abs
lemma continuous_on.abs (h : continuous_on f s) : continuous_on (λ x, |f x|) s :=
λ x hx, (h x hx).abs
lemma tendsto_abs_nhds_within_zero : tendsto (abs : α → α) (𝓝[{0}ᶜ] 0) (𝓝[Ioi 0] 0) :=
(continuous_abs.tendsto' (0 : α) 0 abs_zero).inf $ tendsto_principal_principal.2 $ λ x, abs_pos.2
end
/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`
and `g` tends to `at_top` then `f + g` tends to `at_top`. -/
lemma filter.tendsto.add_at_top {C : α} (hf : tendsto f l (𝓝 C)) (hg : tendsto g l at_top) :
tendsto (λ x, f x + g x) l at_top :=
begin
nontriviality α,
obtain ⟨C', hC'⟩ : ∃ C', C' < C := no_bot C,
refine tendsto_at_top_add_left_of_le' _ C' _ hg,
exact (hf.eventually (lt_mem_nhds hC')).mono (λ x, le_of_lt)
end
/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`
and `g` tends to `at_bot` then `f + g` tends to `at_bot`. -/
lemma filter.tendsto.add_at_bot {C : α} (hf : tendsto f l (𝓝 C)) (hg : tendsto g l at_bot) :
tendsto (λ x, f x + g x) l at_bot :=
@filter.tendsto.add_at_top (order_dual α) _ _ _ _ _ _ _ _ hf hg
/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to
`at_top` and `g` tends to `C` then `f + g` tends to `at_top`. -/
lemma filter.tendsto.at_top_add {C : α} (hf : tendsto f l at_top) (hg : tendsto g l (𝓝 C)) :
tendsto (λ x, f x + g x) l at_top :=
by { conv in (_ + _) { rw add_comm }, exact hg.add_at_top hf }
/-- In a linearly ordered additive commutative group with the order topology, if `f` tends to
`at_bot` and `g` tends to `C` then `f + g` tends to `at_bot`. -/
lemma filter.tendsto.at_bot_add {C : α} (hf : tendsto f l at_bot) (hg : tendsto g l (𝓝 C)) :
tendsto (λ x, f x + g x) l at_bot :=
by { conv in (_ + _) { rw add_comm }, exact hg.add_at_bot hf }
end linear_ordered_add_comm_group
section linear_ordered_field
variables [linear_ordered_field α] [topological_space α] [order_topology α]
variables {l : filter β} {f g : β → α}
/-- In a linearly ordered field with the order topology, if `f` tends to `at_top` and `g` tends to
a positive constant `C` then `f * g` tends to `at_top`. -/
lemma filter.tendsto.at_top_mul {C : α} (hC : 0 < C) (hf : tendsto f l at_top)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_top :=
begin
refine tendsto_at_top_mono' _ _ (hf.at_top_mul_const (half_pos hC)),
filter_upwards [hg.eventually (lt_mem_nhds (half_lt_self hC)),
hf.eventually (eventually_ge_at_top 0)],
exact λ x hg hf, mul_le_mul_of_nonneg_left hg.le hf
end
/-- In a linearly ordered field with the order topology, if `f` tends to a positive constant `C` and
`g` tends to `at_top` then `f * g` tends to `at_top`. -/
lemma filter.tendsto.mul_at_top {C : α} (hC : 0 < C) (hf : tendsto f l (𝓝 C))
(hg : tendsto g l at_top) :
tendsto (λ x, (f x * g x)) l at_top :=
by simpa only [mul_comm] using hg.at_top_mul hC hf
/-- In a linearly ordered field with the order topology, if `f` tends to `at_top` and `g` tends to
a negative constant `C` then `f * g` tends to `at_bot`. -/
lemma filter.tendsto.at_top_mul_neg {C : α} (hC : C < 0) (hf : tendsto f l at_top)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_bot :=
by simpa only [(∘), neg_mul_eq_mul_neg, neg_neg]
using tendsto_neg_at_top_at_bot.comp (hf.at_top_mul (neg_pos.2 hC) hg.neg)
/-- In a linearly ordered field with the order topology, if `f` tends to a negative constant `C` and
`g` tends to `at_top` then `f * g` tends to `at_bot`. -/
lemma filter.tendsto.neg_mul_at_top {C : α} (hC : C < 0) (hf : tendsto f l (𝓝 C))
(hg : tendsto g l at_top) :
tendsto (λ x, (f x * g x)) l at_bot :=
by simpa only [mul_comm] using hg.at_top_mul_neg hC hf
/-- In a linearly ordered field with the order topology, if `f` tends to `at_bot` and `g` tends to
a positive constant `C` then `f * g` tends to `at_bot`. -/
lemma filter.tendsto.at_bot_mul {C : α} (hC : 0 < C) (hf : tendsto f l at_bot)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_bot :=
by simpa [(∘)]
using tendsto_neg_at_top_at_bot.comp ((tendsto_neg_at_bot_at_top.comp hf).at_top_mul hC hg)
/-- In a linearly ordered field with the order topology, if `f` tends to `at_bot` and `g` tends to
a negative constant `C` then `f * g` tends to `at_top`. -/
lemma filter.tendsto.at_bot_mul_neg {C : α} (hC : C < 0) (hf : tendsto f l at_bot)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_top :=
by simpa [(∘)]
using tendsto_neg_at_bot_at_top.comp ((tendsto_neg_at_bot_at_top.comp hf).at_top_mul_neg hC hg)
/-- In a linearly ordered field with the order topology, if `f` tends to a positive constant `C` and
`g` tends to `at_bot` then `f * g` tends to `at_bot`. -/
lemma filter.tendsto.mul_at_bot {C : α} (hC : 0 < C) (hf : tendsto f l (𝓝 C))
(hg : tendsto g l at_bot) :
tendsto (λ x, (f x * g x)) l at_bot :=
by simpa only [mul_comm] using hg.at_bot_mul hC hf
/-- In a linearly ordered field with the order topology, if `f` tends to a negative constant `C` and
`g` tends to `at_bot` then `f * g` tends to `at_top`. -/
lemma filter.tendsto.neg_mul_at_bot {C : α} (hC : C < 0) (hf : tendsto f l (𝓝 C))
(hg : tendsto g l at_bot) :
tendsto (λ x, (f x * g x)) l at_top :=
by simpa only [mul_comm] using hg.at_bot_mul_neg hC hf
/-- The function `x ↦ x⁻¹` tends to `+∞` on the right of `0`. -/
lemma tendsto_inv_zero_at_top : tendsto (λx:α, x⁻¹) (𝓝[set.Ioi (0:α)] 0) at_top :=
begin
refine (at_top_basis' 1).tendsto_right_iff.2 (λ b hb, _),
have hb' : 0 < b := zero_lt_one.trans_le hb,
filter_upwards [Ioc_mem_nhds_within_Ioi ⟨le_rfl, inv_pos.2 hb'⟩],
exact λ x hx, (le_inv hx.1 hb').1 hx.2
end
/-- The function `r ↦ r⁻¹` tends to `0` on the right as `r → +∞`. -/
lemma tendsto_inv_at_top_zero' : tendsto (λr:α, r⁻¹) at_top (𝓝[set.Ioi (0:α)] 0) :=
begin
refine (has_basis.tendsto_iff at_top_basis ⟨λ s, mem_nhds_within_Ioi_iff_exists_Ioc_subset⟩).2 _,
refine λ b hb, ⟨b⁻¹, trivial, λ x hx, _⟩,
have : 0 < x := lt_of_lt_of_le (inv_pos.2 hb) hx,
exact ⟨inv_pos.2 this, (inv_le this hb).2 hx⟩
end
lemma tendsto_inv_at_top_zero : tendsto (λr:α, r⁻¹) at_top (𝓝 0) :=
tendsto_inv_at_top_zero'.mono_right inf_le_left
lemma filter.tendsto.div_at_top [has_continuous_mul α] {f g : β → α} {l : filter β} {a : α}
(h : tendsto f l (𝓝 a)) (hg : tendsto g l at_top) : tendsto (λ x, f x / g x) l (𝓝 0) :=
by { simp only [div_eq_mul_inv], exact mul_zero a ▸ h.mul (tendsto_inv_at_top_zero.comp hg) }
lemma tendsto.inv_tendsto_at_top (h : tendsto f l at_top) : tendsto (f⁻¹) l (𝓝 0) :=
tendsto_inv_at_top_zero.comp h
lemma tendsto.inv_tendsto_zero (h : tendsto f l (𝓝[set.Ioi 0] 0)) : tendsto (f⁻¹) l at_top :=
tendsto_inv_zero_at_top.comp h
/-- The function `x^(-n)` tends to `0` at `+∞` for any positive natural `n`.
A version for positive real powers exists as `tendsto_rpow_neg_at_top`. -/
lemma tendsto_pow_neg_at_top {n : ℕ} (hn : 1 ≤ n) : tendsto (λ x : α, x ^ (-(n:ℤ))) at_top (𝓝 0) :=
tendsto.congr (λ x, (fpow_neg x n).symm) (tendsto.inv_tendsto_at_top (tendsto_pow_at_top hn))
end linear_ordered_field
lemma preimage_neg [add_group α] : preimage (has_neg.neg : α → α) = image (has_neg.neg : α → α) :=
(image_eq_preimage_of_inverse neg_neg neg_neg).symm
lemma filter.map_neg [add_group α] : map (has_neg.neg : α → α) = comap (has_neg.neg : α → α) :=
funext $ assume f, map_eq_comap_of_inverse (funext neg_neg) (funext neg_neg)
section order_topology
variables [topological_space α] [topological_space β]
[linear_order α] [linear_order β] [order_topology α] [order_topology β]
lemma is_lub.nhds_within_ne_bot {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :
ne_bot (𝓝[s] a) :=
let ⟨a', ha'⟩ := hs in
forall_sets_nonempty_iff_ne_bot.mp $ assume t ht,
let ⟨t₁, ht₁, t₂, ht₂, ht⟩ := mem_inf_sets.mp ht in
by_cases
(assume h : a = a',
have a ∈ t₁, from mem_of_nhds ht₁,
have a ∈ t₂, from ht₂ $ by rwa [h],
⟨a, ht ⟨‹a ∈ t₁›, ‹a ∈ t₂›⟩⟩)
(assume : a ≠ a',
have a' < a, from lt_of_le_of_ne (ha.left ‹a' ∈ s›) this.symm,
let ⟨l, hl, hlt₁⟩ := exists_Ioc_subset_of_mem_nhds ht₁ ⟨a', this⟩ in
have ∃a'∈s, l < a',
from classical.by_contradiction $ assume : ¬ ∃a'∈s, l < a',
have ∀a'∈s, a' ≤ l, from assume a ha, not_lt.1 $ assume ha', this ⟨a, ha, ha'⟩,
have ¬ l < a, from not_lt.2 $ ha.right this,
this ‹l < a›,
let ⟨a', ha', ha'l⟩ := this in
have a' ∈ t₁, from hlt₁ ⟨‹l < a'›, ha.left ha'⟩,
⟨a', ht ⟨‹a' ∈ t₁›, ht₂ ‹a' ∈ s›⟩⟩)
lemma is_glb.nhds_within_ne_bot : ∀ {a : α} {s : set α}, is_glb s a → s.nonempty →
ne_bot (𝓝[s] a) :=
@is_lub.nhds_within_ne_bot (order_dual α) _ _ _
lemma is_lub_of_mem_nhds {s : set α} {a : α} {f : filter α}
(hsa : a ∈ upper_bounds s) (hsf : s ∈ f) [ne_bot (f ⊓ 𝓝 a)] : is_lub s a :=
⟨hsa, assume b hb,
not_lt.1 $ assume hba,
have s ∩ {a | b < a} ∈ f ⊓ 𝓝 a,
from inter_mem_inf_sets hsf (mem_nhds_sets (is_open_lt' _) hba),
let ⟨x, ⟨hxs, hxb⟩⟩ := nonempty_of_mem_sets this in
have b < b, from lt_of_lt_of_le hxb $ hb hxs,
lt_irrefl b this⟩
lemma is_glb_of_mem_nhds : ∀ {s : set α} {a : α} {f : filter α},
a ∈ lower_bounds s → s ∈ f → ne_bot (f ⊓ 𝓝 a) → is_glb s a :=
@is_lub_of_mem_nhds (order_dual α) _ _ _
lemma is_lub_of_is_lub_of_tendsto {f : α → β} {s : set α} {a : α} {b : β}
(hf : ∀x∈s, ∀y∈s, x ≤ y → f x ≤ f y) (ha : is_lub s a) (hs : s.nonempty)
(hb : tendsto f (𝓝[s] a) (𝓝 b)) : is_lub (f '' s) b :=
have hnbot : ne_bot (𝓝[s] a), from ha.nhds_within_ne_bot hs,
have ∀a'∈s, ¬ b < f a',
from assume a' ha' h,
have ∀ᶠ x in 𝓝 b, x < f a', from mem_nhds_sets (is_open_gt' _) h,
let ⟨t₁, ht₁, t₂, ht₂, hs⟩ := mem_inf_sets.mp (hb this) in
by_cases
(assume h : a = a',
have a ∈ t₁ ∩ t₂, from ⟨mem_of_nhds ht₁, ht₂ $ by rwa [h]⟩,
have f a < f a', from hs this,
lt_irrefl (f a') $ by rwa [h] at this)
(assume h : a ≠ a',
have a' < a, from lt_of_le_of_ne (ha.left ha') h.symm,
have {x | a' < x} ∈ 𝓝 a, from mem_nhds_sets (is_open_lt' _) this,
have {x | a' < x} ∩ t₁ ∈ 𝓝 a, from inter_mem_sets this ht₁,
have ({x | a' < x} ∩ t₁) ∩ s ∈ 𝓝[s] a,
from inter_mem_inf_sets this (subset.refl s),
let ⟨x, ⟨hx₁, hx₂⟩, hx₃⟩ := hnbot.nonempty_of_mem this in
have hxa' : f x < f a', from hs ⟨hx₂, ht₂ hx₃⟩,
have ha'x : f a' ≤ f x, from hf _ ha' _ hx₃ $ le_of_lt hx₁,
lt_irrefl _ (lt_of_le_of_lt ha'x hxa')),
and.intro
(assume b' ⟨a', ha', h_eq⟩, h_eq ▸ not_lt.1 $ this _ ha')
(assume b' hb', by exactI (le_of_tendsto hb $
mem_inf_sets_of_right $ assume x hx, hb' $ mem_image_of_mem _ hx))
lemma is_glb_of_is_glb_of_tendsto {f : α → β} {s : set α} {a : α} {b : β}
(hf : ∀x∈s, ∀y∈s, x ≤ y → f x ≤ f y) : is_glb s a → s.nonempty →
tendsto f (𝓝[s] a) (𝓝 b) → is_glb (f '' s) b :=
@is_lub_of_is_lub_of_tendsto (order_dual α) (order_dual β) _ _ _ _ _ _ f s a b
(λ x hx y hy, hf y hy x hx)
lemma is_glb_of_is_lub_of_tendsto : ∀ {f : α → β} {s : set α} {a : α} {b : β},
(∀x∈s, ∀y∈s, x ≤ y → f y ≤ f x) → is_lub s a → s.nonempty →
tendsto f (𝓝[s] a) (𝓝 b) → is_glb (f '' s) b :=
@is_lub_of_is_lub_of_tendsto α (order_dual β) _ _ _ _ _ _
lemma is_lub_of_is_glb_of_tendsto : ∀ {f : α → β} {s : set α} {a : α} {b : β},
(∀x∈s, ∀y∈s, x ≤ y → f y ≤ f x) → is_glb s a → s.nonempty →
tendsto f (𝓝[s] a) (𝓝 b) → is_lub (f '' s) b :=
@is_glb_of_is_glb_of_tendsto α (order_dual β) _ _ _ _ _ _
lemma mem_closure_of_is_lub {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :
a ∈ closure s :=
by rw closure_eq_cluster_pts; exact ha.nhds_within_ne_bot hs
lemma mem_of_is_lub_of_is_closed {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty)
(sc : is_closed s) : a ∈ s :=
by rw ←sc.closure_eq; exact mem_closure_of_is_lub ha hs
lemma mem_closure_of_is_glb {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) :
a ∈ closure s :=
by rw closure_eq_cluster_pts; exact ha.nhds_within_ne_bot hs
lemma mem_of_is_glb_of_is_closed {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty)
(sc : is_closed s) : a ∈ s :=
by rw ←sc.closure_eq; exact mem_closure_of_is_glb ha hs
/-- A compact set is bounded below -/
lemma is_compact.bdd_below {α : Type u} [topological_space α] [linear_order α]
[order_closed_topology α] [nonempty α] {s : set α} (hs : is_compact s) : bdd_below s :=
begin
by_contra H,
rcases hs.elim_finite_subcover_image (λ x (_ : x ∈ s), @is_open_Ioi _ _ _ _ x) _
with ⟨t, st, ft, ht⟩,
{ refine H (ft.bdd_below.imp $ λ C hC y hy, _),
rcases mem_bUnion_iff.1 (ht hy) with ⟨x, hx, xy⟩,
exact le_trans (hC hx) (le_of_lt xy) },
{ refine λ x hx, mem_bUnion_iff.2 (not_imp_comm.1 _ H),
exact λ h, ⟨x, λ y hy, le_of_not_lt (h.imp $ λ ys, ⟨_, hy, ys⟩)⟩ }
end
/-- A compact set is bounded above -/
lemma is_compact.bdd_above {α : Type u} [topological_space α] [linear_order α]
[order_topology α] : Π [nonempty α] {s : set α}, is_compact s → bdd_above s :=
@is_compact.bdd_below (order_dual α) _ _ _
end order_topology
section linear_order
variables [topological_space α] [linear_order α] [order_topology α] [densely_ordered α]
/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top
element. -/
lemma closure_Ioi' {a b : α} (hab : a < b) :
closure (Ioi a) = Ici a :=
begin
apply subset.antisymm,
{ exact closure_minimal Ioi_subset_Ici_self is_closed_Ici },
{ assume x hx,
by_cases h : x = a,
{ rw h, exact mem_closure_of_is_glb is_glb_Ioi ⟨_, hab⟩ },
{ exact subset_closure (lt_of_le_of_ne hx (ne.symm h)) } }
end
/-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/
@[simp] lemma closure_Ioi (a : α) [no_top_order α] :
closure (Ioi a) = Ici a :=
let ⟨b, hb⟩ := no_top a in closure_Ioi' hb
/-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom
element. -/
lemma closure_Iio' {a b : α} (hab : b < a) :
closure (Iio a) = Iic a :=
begin
apply subset.antisymm,
{ exact closure_minimal Iio_subset_Iic_self is_closed_Iic },
{ assume x hx,
by_cases h : x = a,
{ rw h, exact mem_closure_of_is_lub is_lub_Iio ⟨_, hab⟩ },
{ apply subset_closure, by simpa [h] using lt_or_eq_of_le hx } }
end
/-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/
@[simp] lemma closure_Iio (a : α) [no_bot_order α] :
closure (Iio a) = Iic a :=
let ⟨b, hb⟩ := no_bot a in closure_Iio' hb
/-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/
@[simp] lemma closure_Ioo {a b : α} (hab : a < b) :
closure (Ioo a b) = Icc a b :=
begin
apply subset.antisymm,
{ exact closure_minimal Ioo_subset_Icc_self is_closed_Icc },
{ have hab' : (Ioo a b).nonempty, from nonempty_Ioo.2 hab,
assume x hx,
by_cases h : x = a,
{ rw h, exact mem_closure_of_is_glb (is_glb_Ioo hab) hab' },
by_cases h' : x = b,
{ rw h', refine mem_closure_of_is_lub (is_lub_Ioo hab) hab' },
exact subset_closure ⟨lt_of_le_of_ne hx.1 (ne.symm h),
by simpa [h'] using lt_or_eq_of_le hx.2⟩ }
end
/-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/
@[simp] lemma closure_Ioc {a b : α} (hab : a < b) :
closure (Ioc a b) = Icc a b :=
begin
apply subset.antisymm,
{ exact closure_minimal Ioc_subset_Icc_self is_closed_Icc },
{ apply subset.trans _ (closure_mono Ioo_subset_Ioc_self),
rw closure_Ioo hab }
end
/-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/
@[simp] lemma closure_Ico {a b : α} (hab : a < b) :
closure (Ico a b) = Icc a b :=
begin
apply subset.antisymm,
{ exact closure_minimal Ico_subset_Icc_self is_closed_Icc },
{ apply subset.trans _ (closure_mono Ioo_subset_Ico_self),
rw closure_Ioo hab }
end
@[simp] lemma interior_Ici [no_bot_order α] {a : α} : interior (Ici a) = Ioi a :=
by rw [← compl_Iio, interior_compl, closure_Iio, compl_Iic]
@[simp] lemma interior_Iic [no_top_order α] {a : α} : interior (Iic a) = Iio a :=
by rw [← compl_Ioi, interior_compl, closure_Ioi, compl_Ici]
@[simp] lemma interior_Icc [no_bot_order α] [no_top_order α] {a b : α}:
interior (Icc a b) = Ioo a b :=
by rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]
@[simp] lemma interior_Ico [no_bot_order α] {a b : α} : interior (Ico a b) = Ioo a b :=
by rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]
@[simp] lemma interior_Ioc [no_top_order α] {a b : α} : interior (Ioc a b) = Ioo a b :=
by rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]
@[simp] lemma frontier_Ici [no_bot_order α] {a : α} : frontier (Ici a) = {a} :=
by simp [frontier]
@[simp] lemma frontier_Iic [no_top_order α] {a : α} : frontier (Iic a) = {a} :=
by simp [frontier]
@[simp] lemma frontier_Ioi [no_top_order α] {a : α} : frontier (Ioi a) = {a} :=
by simp [frontier]
@[simp] lemma frontier_Iio [no_bot_order α] {a : α} : frontier (Iio a) = {a} :=
by simp [frontier]
@[simp] lemma frontier_Icc [no_bot_order α] [no_top_order α] {a b : α} (h : a < b) :
frontier (Icc a b) = {a, b} :=
by simp [frontier, le_of_lt h, Icc_diff_Ioo_same]
@[simp] lemma frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} :=
by simp [frontier, h, le_of_lt h, Icc_diff_Ioo_same]
@[simp] lemma frontier_Ico [no_bot_order α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} :=
by simp [frontier, h, le_of_lt h, Icc_diff_Ioo_same]
@[simp] lemma frontier_Ioc [no_top_order α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} :=
by simp [frontier, h, le_of_lt h, Icc_diff_Ioo_same]
lemma nhds_within_Ioi_ne_bot' {a b c : α} (H₁ : a < c) (H₂ : a ≤ b) :
ne_bot (𝓝[Ioi a] b) :=
mem_closure_iff_nhds_within_ne_bot.1 $ by { rw [closure_Ioi' H₁], exact H₂ }
lemma nhds_within_Ioi_ne_bot [no_top_order α] {a b : α} (H : a ≤ b) :
ne_bot (𝓝[Ioi a] b) :=
let ⟨c, hc⟩ := no_top a in nhds_within_Ioi_ne_bot' hc H
lemma nhds_within_Ioi_self_ne_bot' {a b : α} (H : a < b) :
ne_bot (𝓝[Ioi a] a) :=
nhds_within_Ioi_ne_bot' H (le_refl a)
@[instance]
lemma nhds_within_Ioi_self_ne_bot [no_top_order α] (a : α) :
ne_bot (𝓝[Ioi a] a) :=
nhds_within_Ioi_ne_bot (le_refl a)
lemma nhds_within_Iio_ne_bot' {a b c : α} (H₁ : a < c) (H₂ : b ≤ c) :
ne_bot (𝓝[Iio c] b) :=
mem_closure_iff_nhds_within_ne_bot.1 $ by { rw [closure_Iio' H₁], exact H₂ }
lemma nhds_within_Iio_ne_bot [no_bot_order α] {a b : α} (H : a ≤ b) :
ne_bot (𝓝[Iio b] a) :=
let ⟨c, hc⟩ := no_bot b in nhds_within_Iio_ne_bot' hc H
lemma nhds_within_Iio_self_ne_bot' {a b : α} (H : a < b) :
ne_bot (𝓝[Iio b] b) :=
nhds_within_Iio_ne_bot' H (le_refl b)
@[instance]
lemma nhds_within_Iio_self_ne_bot [no_bot_order α] (a : α) :
ne_bot (𝓝[Iio a] a) :=
nhds_within_Iio_ne_bot (le_refl a)
end linear_order
section linear_order
variables [topological_space α] [linear_order α] [order_topology α] [densely_ordered α]
{a b : α} {s : set α}
lemma comap_coe_nhds_within_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.nonempty → ∃ a < b, Ioo a b ⊆ s) :
comap (coe : s → α) (𝓝[Iio b] b) = at_top :=
begin
nontriviality,
haveI : nonempty s := nontrivial_iff_nonempty.1 ‹_›,
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩,
ext u, split,
{ rintros ⟨t, ht, hts⟩,
obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=
(mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht,
obtain ⟨y, hxy, hyb⟩ := exists_between hxb,
refine mem_sets_of_superset (mem_at_top ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) _,
rintros ⟨z, hzs⟩ (hyz : y ≤ z),
refine hts (hxt ⟨hxy.trans_le _, hb _⟩); assumption },
{ intros hu,
obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_at_top_sets.1 hu,
exact ⟨Ioo x b, Ioo_mem_nhds_within_Iio (right_mem_Ioc.2 $ hb x.2), λ z hz, hx _ hz.1.le⟩ }
end
lemma comap_coe_nhds_within_Ioi_of_Ioo_subset (ha : s ⊆ Ioi a)
(hs : s.nonempty → ∃ b > a, Ioo a b ⊆ s) :
comap (coe : s → α) (𝓝[Ioi a] a) = at_bot :=
begin
refine @comap_coe_nhds_within_Iio_of_Ioo_subset (order_dual α) _ _ _ _ _ _ ha (λ h, _),
rcases hs h with ⟨b, hab, h⟩,
use [b, hab],
rwa dual_Ioo
end
lemma map_coe_at_top_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map (coe : s → α) at_top = 𝓝[Iio b] b :=
begin
rcases eq_empty_or_nonempty (Iio b) with (hb'|⟨a, ha⟩),
{ rw [filter_eq_bot_of_not_nonempty at_top, map_bot, hb', nhds_within_empty],
exact λ ⟨⟨x, hx⟩⟩, not_nonempty_iff_eq_empty.2 hb' ⟨x, hb hx⟩ },
{ rw [← comap_coe_nhds_within_Iio_of_Ioo_subset hb (λ _, hs a ha), map_comap],
rw subtype.range_coe,
exact (mem_nhds_within_Iio_iff_exists_Ioo_subset' ha).2 (hs a ha) },
end
lemma map_coe_at_bot_of_Ioo_subset (ha : s ⊆ Ioi a)
(hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) :
map (coe : s → α) at_bot = (𝓝[Ioi a] a) :=
begin
refine @map_coe_at_top_of_Ioo_subset (order_dual α) _ _ _ _ a s ha (λ b' hb', _),
rcases hs b' hb' with ⟨b, hab, hbs⟩,
use [b, hab],
rwa dual_Ioo
end
/-- The `at_top` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at
the right endpoint in the ambient order. -/
lemma comap_coe_Ioo_nhds_within_Iio (a b : α) :
comap (coe : Ioo a b → α) (𝓝[Iio b] b) = at_top :=
comap_coe_nhds_within_Iio_of_Ioo_subset Ioo_subset_Iio_self $
λ h, ⟨a, nonempty_Ioo.1 h, subset.refl _⟩
/-- The `at_bot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at
the left endpoint in the ambient order. -/
lemma comap_coe_Ioo_nhds_within_Ioi (a b : α) :
comap (coe : Ioo a b → α) (𝓝[Ioi a] a) = at_bot :=
comap_coe_nhds_within_Ioi_of_Ioo_subset Ioo_subset_Ioi_self $
λ h, ⟨b, nonempty_Ioo.1 h, subset.refl _⟩
lemma comap_coe_Ioi_nhds_within_Ioi (a : α) : comap (coe : Ioi a → α) (𝓝[Ioi a] a) = at_bot :=
comap_coe_nhds_within_Ioi_of_Ioo_subset (subset.refl _) $
λ ⟨x, hx⟩, ⟨x, hx, Ioo_subset_Ioi_self⟩
lemma comap_coe_Iio_nhds_within_Iio (a : α) :
comap (coe : Iio a → α) (𝓝[Iio a] a) = at_top :=
@comap_coe_Ioi_nhds_within_Ioi (order_dual α) _ _ _ _ a
@[simp] lemma map_coe_Ioo_at_top {a b : α} (h : a < b) :
map (coe : Ioo a b → α) at_top = 𝓝[Iio b] b :=
map_coe_at_top_of_Ioo_subset Ioo_subset_Iio_self $ λ _ _, ⟨_, h, subset.refl _⟩
@[simp] lemma map_coe_Ioo_at_bot {a b : α} (h : a < b) :
map (coe : Ioo a b → α) at_bot = 𝓝[Ioi a] a :=
map_coe_at_bot_of_Ioo_subset Ioo_subset_Ioi_self $ λ _ _, ⟨_, h, subset.refl _⟩
@[simp] lemma map_coe_Ioi_at_bot (a : α) :
map (coe : Ioi a → α) at_bot = 𝓝[Ioi a] a :=
map_coe_at_bot_of_Ioo_subset (subset.refl _) $ λ b hb, ⟨b, hb, Ioo_subset_Ioi_self⟩
@[simp] lemma map_coe_Iio_at_top (a : α) :
map (coe : Iio a → α) at_top = 𝓝[Iio a] a :=
@map_coe_Ioi_at_bot (order_dual α) _ _ _ _ _
variables {l : filter β} {f : α → β}
@[simp] lemma tendsto_comp_coe_Ioo_at_top (h : a < b) :
tendsto (λ x : Ioo a b, f x) at_top l ↔ tendsto f (𝓝[Iio b] b) l :=
by rw [← map_coe_Ioo_at_top h, tendsto_map'_iff]
@[simp] lemma tendsto_comp_coe_Ioo_at_bot (h : a < b) :
tendsto (λ x : Ioo a b, f x) at_bot l ↔ tendsto f (𝓝[Ioi a] a) l :=
by rw [← map_coe_Ioo_at_bot h, tendsto_map'_iff]
@[simp] lemma tendsto_comp_coe_Ioi_at_bot :
tendsto (λ x : Ioi a, f x) at_bot l ↔ tendsto f (𝓝[Ioi a] a) l :=
by rw [← map_coe_Ioi_at_bot, tendsto_map'_iff]
@[simp] lemma tendsto_comp_coe_Iio_at_top :
tendsto (λ x : Iio a, f x) at_top l ↔ tendsto f (𝓝[Iio a] a) l :=
by rw [← map_coe_Iio_at_top, tendsto_map'_iff]
@[simp] lemma tendsto_Ioo_at_top {f : β → Ioo a b} :
tendsto f l at_top ↔ tendsto (λ x, (f x : α)) l (𝓝[Iio b] b) :=
by rw [← comap_coe_Ioo_nhds_within_Iio, tendsto_comap_iff]
@[simp] lemma tendsto_Ioo_at_bot {f : β → Ioo a b} :
tendsto f l at_bot ↔ tendsto (λ x, (f x : α)) l (𝓝[Ioi a] a) :=
by rw [← comap_coe_Ioo_nhds_within_Ioi, tendsto_comap_iff]
@[simp] lemma tendsto_Ioi_at_bot {f : β → Ioi a} :
tendsto f l at_bot ↔ tendsto (λ x, (f x : α)) l (𝓝[Ioi a] a) :=
by rw [← comap_coe_Ioi_nhds_within_Ioi, tendsto_comap_iff]
@[simp] lemma tendsto_Iio_at_top {f : β → Iio a} :
tendsto f l at_top ↔ tendsto (λ x, (f x : α)) l (𝓝[Iio a] a) :=
by rw [← comap_coe_Iio_nhds_within_Iio, tendsto_comap_iff]
end linear_order
section complete_linear_order
variables [complete_linear_order α] [topological_space α] [order_topology α]
[complete_linear_order β] [topological_space β] [order_topology β] [nonempty γ]
lemma Sup_mem_closure {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) :
Sup s ∈ closure s :=
mem_closure_of_is_lub (is_lub_Sup _) hs
lemma Inf_mem_closure {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) :
Inf s ∈ closure s :=
mem_closure_of_is_glb (is_glb_Inf _) hs
lemma is_closed.Sup_mem {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) (hc : is_closed s) :
Sup s ∈ s :=
mem_of_is_lub_of_is_closed (is_lub_Sup _) hs hc
lemma is_closed.Inf_mem {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) (hc : is_closed s) :
Inf s ∈ s :=
mem_of_is_glb_of_is_closed (is_glb_Inf _) hs hc
/-- A monotone function continuous at the supremum of a nonempty set sends this supremum to
the supremum of the image of this set. -/
lemma map_Sup_of_continuous_at_of_monotone' {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Mf : monotone f) (hs : s.nonempty) :
f (Sup s) = Sup (f '' s) :=
--This is a particular case of the more general is_lub_of_is_lub_of_tendsto
(is_lub_of_is_lub_of_tendsto (λ x hx y hy xy, Mf xy) (is_lub_Sup _) hs $
Cf.mono_left inf_le_left).Sup_eq.symm
/-- A monotone function `s` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. -/
lemma map_Sup_of_continuous_at_of_monotone {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Mf : monotone f) (fbot : f ⊥ = ⊥) :
f (Sup s) = Sup (f '' s) :=
begin
cases s.eq_empty_or_nonempty with h h,
{ simp [h, fbot] },
{ exact map_Sup_of_continuous_at_of_monotone' Cf Mf h }
end
/-- A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. -/
lemma map_supr_of_continuous_at_of_monotone' {ι : Sort*} [nonempty ι] {f : α → β} {g : ι → α}
(Cf : continuous_at f (supr g)) (Mf : monotone f) :
f (⨆ i, g i) = ⨆ i, f (g i) :=
by rw [supr, map_Sup_of_continuous_at_of_monotone' Cf Mf (range_nonempty g), ← range_comp, supr]
/-- If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/
lemma map_supr_of_continuous_at_of_monotone {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : continuous_at f (supr g)) (Mf : monotone f) (fbot : f ⊥ = ⊥) :
f (⨆ i, g i) = ⨆ i, f (g i) :=
by rw [supr, map_Sup_of_continuous_at_of_monotone Cf Mf fbot, ← range_comp, supr]
/-- A monotone function continuous at the infimum of a nonempty set sends this infimum to
the infimum of the image of this set. -/
lemma map_Inf_of_continuous_at_of_monotone' {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Mf : monotone f) (hs : s.nonempty) :
f (Inf s) = Inf (f '' s) :=
@map_Sup_of_continuous_at_of_monotone' (order_dual α) (order_dual β) _ _ _ _ _ _ f s Cf
Mf.order_dual hs
/-- A monotone function `s` sending `top` to `top` and continuous at the infimum of a set sends
this infimum to the infimum of the image of this set. -/
lemma map_Inf_of_continuous_at_of_monotone {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Mf : monotone f) (ftop : f ⊤ = ⊤) :
f (Inf s) = Inf (f '' s) :=
@map_Sup_of_continuous_at_of_monotone (order_dual α) (order_dual β) _ _ _ _ _ _ f s Cf
Mf.order_dual ftop
/-- A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. -/
lemma map_infi_of_continuous_at_of_monotone' {ι : Sort*} [nonempty ι] {f : α → β} {g : ι → α}
(Cf : continuous_at f (infi g)) (Mf : monotone f) :
f (⨅ i, g i) = ⨅ i, f (g i) :=
@map_supr_of_continuous_at_of_monotone' (order_dual α) (order_dual β) _ _ _ _ _ _ ι _ f g Cf
Mf.order_dual
/-- If a monotone function sending `top` to `top` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed infimum of the composition. -/
lemma map_infi_of_continuous_at_of_monotone {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : continuous_at f (infi g)) (Mf : monotone f) (ftop : f ⊤ = ⊤) :
f (infi g) = infi (f ∘ g) :=
@map_supr_of_continuous_at_of_monotone (order_dual α) (order_dual β) _ _ _ _ _ _ ι f g Cf
Mf.order_dual ftop
end complete_linear_order
section conditionally_complete_linear_order
variables [conditionally_complete_linear_order α] [topological_space α] [order_topology α]
[conditionally_complete_linear_order β] [topological_space β] [order_topology β] [nonempty γ]
lemma cSup_mem_closure {s : set α} (hs : s.nonempty) (B : bdd_above s) : Sup s ∈ closure s :=
mem_closure_of_is_lub (is_lub_cSup hs B) hs
lemma cInf_mem_closure {s : set α} (hs : s.nonempty) (B : bdd_below s) : Inf s ∈ closure s :=
mem_closure_of_is_glb (is_glb_cInf hs B) hs
lemma is_closed.cSup_mem {s : set α} (hc : is_closed s) (hs : s.nonempty) (B : bdd_above s) :
Sup s ∈ s :=
mem_of_is_lub_of_is_closed (is_lub_cSup hs B) hs hc
lemma is_closed.cInf_mem {s : set α} (hc : is_closed s) (hs : s.nonempty) (B : bdd_below s) :
Inf s ∈ s :=
mem_of_is_glb_of_is_closed (is_glb_cInf hs B) hs hc
/-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. -/
lemma map_cSup_of_continuous_at_of_monotone {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Mf : monotone f) (ne : s.nonempty) (H : bdd_above s) :
f (Sup s) = Sup (f '' s) :=
begin
refine ((is_lub_cSup (ne.image f) (Mf.map_bdd_above H)).unique _).symm,
refine is_lub_of_is_lub_of_tendsto (λx hx y hy xy, Mf xy) (is_lub_cSup ne H) ne _,
exact Cf.mono_left inf_le_left
end
/-- If a monotone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the supremum of the composition. -/
lemma map_csupr_of_continuous_at_of_monotone {f : α → β} {g : γ → α}
(Cf : continuous_at f (⨆ i, g i)) (Mf : monotone f) (H : bdd_above (range g)) :
f (⨆ i, g i) = ⨆ i, f (g i) :=
by rw [supr, map_cSup_of_continuous_at_of_monotone Cf Mf (range_nonempty _) H, ← range_comp, supr]
/-- If a monotone function is continuous at the infimum of a nonempty bounded below set `s`,
then it sends this infimum to the infimum of the image of `s`. -/
lemma map_cInf_of_continuous_at_of_monotone {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Mf : monotone f) (ne : s.nonempty) (H : bdd_below s) :
f (Inf s) = Inf (f '' s) :=
@map_cSup_of_continuous_at_of_monotone (order_dual α) (order_dual β) _ _ _ _ _ _ f s Cf
Mf.order_dual ne H
/-- A continuous monotone function sends indexed infimum to indexed infimum in conditionally
complete linear order, under a boundedness assumption. -/
lemma map_cinfi_of_continuous_at_of_monotone {f : α → β} {g : γ → α}
(Cf : continuous_at f (⨅ i, g i)) (Mf : monotone f) (H : bdd_below (range g)) :
f (⨅ i, g i) = ⨅ i, f (g i) :=
@map_csupr_of_continuous_at_of_monotone (order_dual α) (order_dual β) _ _ _ _ _ _ _ _ _ _
Cf Mf.order_dual H
/-- A bounded connected subset of a conditionally complete linear order includes the open interval
`(Inf s, Sup s)`. -/
lemma is_connected.Ioo_cInf_cSup_subset {s : set α} (hs : is_connected s) (hb : bdd_below s)
(ha : bdd_above s) :
Ioo (Inf s) (Sup s) ⊆ s :=
λ x hx, let ⟨y, ys, hy⟩ := (is_glb_lt_iff (is_glb_cInf hs.nonempty hb)).1 hx.1 in
let ⟨z, zs, hz⟩ := (lt_is_lub_iff (is_lub_cSup hs.nonempty ha)).1 hx.2 in
hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩
lemma eq_Icc_cInf_cSup_of_connected_bdd_closed {s : set α} (hc : is_connected s) (hb : bdd_below s)
(ha : bdd_above s) (hcl : is_closed s) :
s = Icc (Inf s) (Sup s) :=
subset.antisymm (subset_Icc_cInf_cSup hb ha) $
hc.Icc_subset (hcl.cInf_mem hc.nonempty hb) (hcl.cSup_mem hc.nonempty ha)
lemma is_preconnected.Ioi_cInf_subset {s : set α} (hs : is_preconnected s) (hb : bdd_below s)
(ha : ¬bdd_above s) :
Ioi (Inf s) ⊆ s :=
begin
have sne : s.nonempty := @nonempty_of_not_bdd_above α _ s ⟨Inf ∅⟩ ha,
intros x hx,
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := (is_glb_lt_iff (is_glb_cInf sne hb)).1 hx,
obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bdd_above_iff.1 ha x,
exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩
end
lemma is_preconnected.Iio_cSup_subset {s : set α} (hs : is_preconnected s) (hb : ¬bdd_below s)
(ha : bdd_above s) :
Iio (Sup s) ⊆ s :=
@is_preconnected.Ioi_cInf_subset (order_dual α) _ _ _ s hs ha hb
/-- A preconnected set in a conditionally complete linear order is either one of the intervals
`[Inf s, Sup s]`, `[Inf s, Sup s)`, `(Inf s, Sup s]`, `(Inf s, Sup s)`, `[Inf s, +∞)`,
`(Inf s, +∞)`, `(-∞, Sup s]`, `(-∞, Sup s)`, `(-∞, +∞)`, or `∅`. The converse statement requires
`α` to be densely ordererd. -/
lemma is_preconnected.mem_intervals {s : set α} (hs : is_preconnected s) :
s ∈ ({Icc (Inf s) (Sup s), Ico (Inf s) (Sup s), Ioc (Inf s) (Sup s), Ioo (Inf s) (Sup s),
Ici (Inf s), Ioi (Inf s), Iic (Sup s), Iio (Sup s), univ, ∅} : set (set α)) :=
begin
rcases s.eq_empty_or_nonempty with rfl|hne,
{ apply_rules [or.inr, mem_singleton] },
have hs' : is_connected s := ⟨hne, hs⟩,
by_cases hb : bdd_below s; by_cases ha : bdd_above s,
{ rcases mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset (hs'.Ioo_cInf_cSup_subset hb ha)
(subset_Icc_cInf_cSup hb ha) with hs|hs|hs|hs,
{ exact (or.inl hs) },
{ exact (or.inr $ or.inl hs) },
{ exact (or.inr $ or.inr $ or.inl hs) },
{ exact (or.inr $ or.inr $ or.inr $ or.inl hs) } },
{ refine (or.inr $ or.inr $ or.inr $ or.inr _),
cases mem_Ici_Ioi_of_subset_of_subset (hs.Ioi_cInf_subset hb ha) (λ x hx, cInf_le hb hx)
with hs hs,
{ exact or.inl hs },
{ exact or.inr (or.inl hs) } },
{ iterate 6 { apply or.inr },
cases mem_Iic_Iio_of_subset_of_subset (hs.Iio_cSup_subset hb ha) (λ x hx, le_cSup ha hx)
with hs hs,
{ exact or.inl hs },
{ exact or.inr (or.inl hs) } },
{ iterate 8 { apply or.inr },
exact or.inl (hs.eq_univ_of_unbounded hb ha) }
end
/-- A preconnected set is either one of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`,
`Iic`, `Iio`, or `univ`, or `∅`. The converse statement requires `α` to be densely ordererd. Though
one can represent `∅` as `(Inf s, Inf s)`, we include it into the list of possible cases to improve
readability. -/
lemma set_of_is_preconnected_subset_of_ordered :
{s : set α | is_preconnected s} ⊆
-- bounded intervals
(range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪
-- unbounded intervals and `univ`
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) :=
begin
intros s hs,
rcases hs.mem_intervals with hs|hs|hs|hs|hs|hs|hs|hs|hs|hs,
{ exact (or.inl $ or.inl $ or.inl $ or.inl ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inl $ or.inl $ or.inl $ or.inr ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inl $ or.inl $ or.inr ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inl $ or.inr ⟨(Inf s, Sup s), hs.symm⟩) },
{ exact (or.inr $ or.inl $ or.inl $ or.inl $ or.inl ⟨Inf s, hs.symm⟩) },
{ exact (or.inr $ or.inl $ or.inl $ or.inl $ or.inr ⟨Inf s, hs.symm⟩) },
{ exact (or.inr $ or.inl $ or.inl $ or.inr ⟨Sup s, hs.symm⟩) },
{ exact (or.inr $ or.inl $ or.inr ⟨Sup s, hs.symm⟩) },
{ exact (or.inr $ or.inr $ or.inl hs) },
{ exact (or.inr $ or.inr $ or.inr hs) }
end
/-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and the set `s ∩ [a, b)` has no maximal point, then `b ∈ s`. -/
lemma is_closed.mem_of_ge_of_forall_exists_gt {a b : α} {s : set α} (hs : is_closed (s ∩ Icc a b))
(ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).nonempty) :
b ∈ s :=
begin
let S := s ∩ Icc a b,
replace ha : a ∈ S, from ⟨ha, left_mem_Icc.2 hab⟩,
have Sbd : bdd_above S, from ⟨b, λ z hz, hz.2.2⟩,
let c := Sup (s ∩ Icc a b),
have c_mem : c ∈ S, from hs.cSup_mem ⟨_, ha⟩ Sbd,
have c_le : c ≤ b, from cSup_le ⟨_, ha⟩ (λ x hx, hx.2.2),
cases eq_or_lt_of_le c_le with hc hc, from hc ▸ c_mem.1,
exfalso,
rcases hgt c ⟨c_mem.1, c_mem.2.1, hc⟩ with ⟨x, xs, cx, xb⟩,
exact not_lt_of_le (le_cSup Sbd ⟨xs, le_trans (le_cSup Sbd ha) (le_of_lt cx), xb⟩) cx
end
/-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `a ≤ x < y ≤ b`, `x ∈ s`, the set `s ∩ (x, y]`
is not empty, then `[a, b] ⊆ s`. -/
lemma is_closed.Icc_subset_of_forall_exists_gt {a b : α} {s : set α} (hs : is_closed (s ∩ Icc a b))
(ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).nonempty) :
Icc a b ⊆ s :=
begin
assume y hy,
have : is_closed (s ∩ Icc a y),
{ suffices : s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y,
{ rw this, exact is_closed_inter hs is_closed_Icc },
rw [inter_assoc],
congr,
exact (inter_eq_self_of_subset_right $ Icc_subset_Icc_right hy.2).symm },
exact is_closed.mem_of_ge_of_forall_exists_gt this ha hy.1
(λ x hx, hgt x ⟨hx.1, Ico_subset_Ico_right hy.2 hx.2⟩ y hx.2.2)
end
section densely_ordered
variables [densely_ordered α] {a b : α}
/-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]`
on a closed subset, contains `a`, and for any `x ∈ s ∩ [a, b)` the set `s` includes some open
neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. -/
lemma is_closed.Icc_subset_of_forall_mem_nhds_within {a b : α} {s : set α}
(hs : is_closed (s ∩ Icc a b)) (ha : a ∈ s)
(hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[Ioi x] x) :
Icc a b ⊆ s :=
begin
apply hs.Icc_subset_of_forall_exists_gt ha,
rintros x ⟨hxs, hxab⟩ y hyxb,
have : s ∩ Ioc x y ∈ 𝓝[Ioi x] x,
from inter_mem_sets (hgt x ⟨hxs, hxab⟩) (Ioc_mem_nhds_within_Ioi ⟨le_refl _, hyxb⟩),
exact (nhds_within_Ioi_self_ne_bot' hxab.2).nonempty_of_mem this
end
/-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/
lemma is_preconnected_Icc : is_preconnected (Icc a b) :=
is_preconnected_closed_iff.2
begin
rintros s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩,
wlog hxy : x ≤ y := le_total x y using [x y s t, y x t s],
have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2,
by_contradiction hst,
suffices : Icc x y ⊆ s,
from hst ⟨y, xyab $ right_mem_Icc.2 hxy, this $ right_mem_Icc.2 hxy, hy.2⟩,
apply (is_closed_inter hs is_closed_Icc).Icc_subset_of_forall_mem_nhds_within hx.2,
rintros z ⟨zs, hz⟩,
have zt : z ∈ tᶜ, from λ zt, hst ⟨z, xyab $ Ico_subset_Icc_self hz, zs, zt⟩,
have : tᶜ ∩ Ioc z y ∈ 𝓝[Ioi z] z,
{ rw [← nhds_within_Ioc_eq_nhds_within_Ioi hz.2],
exact mem_nhds_within.2 ⟨tᶜ, ht, zt, subset.refl _⟩},
apply mem_sets_of_superset this,
have : Ioc z y ⊆ s ∪ t, from λ w hw, hab (xyab ⟨le_trans hz.1 (le_of_lt hw.1), hw.2⟩),
exact λ w ⟨wt, wzy⟩, (this wzy).elim id (λ h, (wt h).elim)
end
lemma is_preconnected_interval : is_preconnected (interval a b) := is_preconnected_Icc
lemma is_preconnected_iff_ord_connected {s : set α} :
is_preconnected s ↔ ord_connected s :=
⟨λ h x hx y hy, h.Icc_subset hx hy, λ h, is_preconnected_of_forall_pair $ λ x y hx hy,
⟨interval x y, h.interval_subset hx hy, left_mem_interval, right_mem_interval,
is_preconnected_interval⟩⟩
alias is_preconnected_iff_ord_connected ↔
is_preconnected.ord_connected set.ord_connected.is_preconnected
lemma is_preconnected_Ici : is_preconnected (Ici a) := ord_connected_Ici.is_preconnected
lemma is_preconnected_Iic : is_preconnected (Iic a) := ord_connected_Iic.is_preconnected
lemma is_preconnected_Iio : is_preconnected (Iio a) := ord_connected_Iio.is_preconnected
lemma is_preconnected_Ioi : is_preconnected (Ioi a) := ord_connected_Ioi.is_preconnected
lemma is_preconnected_Ioo : is_preconnected (Ioo a b) := ord_connected_Ioo.is_preconnected
lemma is_preconnected_Ioc : is_preconnected (Ioc a b) := ord_connected_Ioc.is_preconnected
lemma is_preconnected_Ico : is_preconnected (Ico a b) := ord_connected_Ico.is_preconnected
@[priority 100]
instance ordered_connected_space : preconnected_space α :=
⟨ord_connected_univ.is_preconnected⟩
/-- In a dense conditionally complete linear order, the set of preconnected sets is exactly
the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-∞, +∞)`,
or `∅`. Though one can represent `∅` as `(Inf s, Inf s)`, we include it into the list of
possible cases to improve readability. -/
lemma set_of_is_preconnected_eq_of_ordered :
{s : set α | is_preconnected s} =
-- bounded intervals
(range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪
-- unbounded intervals and `univ`
(range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) :=
begin
refine subset.antisymm set_of_is_preconnected_subset_of_ordered _,
simp only [subset_def, -mem_range, forall_range_iff, uncurry, or_imp_distrib, forall_and_distrib,
mem_union, mem_set_of_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true,
is_preconnected_Icc, is_preconnected_Ico, is_preconnected_Ioc,
is_preconnected_Ioo, is_preconnected_Ioi, is_preconnected_Iio, is_preconnected_Ici,
is_preconnected_Iic, is_preconnected_univ, is_preconnected_empty],
end
variables {δ : Type*} [linear_order δ] [topological_space δ] [order_closed_topology δ]
/--Intermediate Value Theorem for continuous functions on closed intervals, case `f a ≤ t ≤ f b`.-/
lemma intermediate_value_Icc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Icc (f a) (f b) ⊆ f '' (Icc a b) :=
is_preconnected_Icc.intermediate_value (left_mem_Icc.2 hab) (right_mem_Icc.2 hab) hf
/--Intermediate Value Theorem for continuous functions on closed intervals, case `f a ≥ t ≥ f b`.-/
lemma intermediate_value_Icc' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : continuous_on f (Icc a b)) :
Icc (f b) (f a) ⊆ f '' (Icc a b) :=
is_preconnected_Icc.intermediate_value (right_mem_Icc.2 hab) (left_mem_Icc.2 hab) hf
/-- A continuous function which tendsto `at_top` `at_top` and to `at_bot` `at_bot` is surjective. -/
lemma continuous.surjective {f : α → δ} (hf : continuous f) (h_top : tendsto f at_top at_top)
(h_bot : tendsto f at_bot at_bot) :
function.surjective f :=
λ p, mem_range_of_exists_le_of_exists_ge hf
(h_bot.eventually (eventually_le_at_bot p)).exists
(h_top.eventually (eventually_ge_at_top p)).exists
/-- A continuous function which tendsto `at_bot` `at_top` and to `at_top` `at_bot` is surjective. -/
lemma continuous.surjective' {f : α → δ} (hf : continuous f) (h_top : tendsto f at_bot at_top)
(h_bot : tendsto f at_top at_bot) :
function.surjective f :=
@continuous.surjective (order_dual α) _ _ _ _ _ _ _ _ _ hf h_top h_bot
/-- If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s`
tends to `at_bot : filter β` along `at_bot : filter ↥s` and tends to `at_top : filter β` along
`at_top : filter ↥s`, then the restriction of `f` to `s` is surjective. We formulate the
conclusion as `surj_on f s univ`. -/
lemma continuous_on.surj_on_of_tendsto {f : α → β} {s : set α} [ord_connected s]
(hs : s.nonempty) (hf : continuous_on f s) (hbot : tendsto (λ x : s, f x) at_bot at_bot)
(htop : tendsto (λ x : s, f x) at_top at_top) :
surj_on f s univ :=
by haveI := inhabited_of_nonempty hs.to_subtype;
exact (surj_on_iff_surjective.2 $
(continuous_on_iff_continuous_restrict.1 hf).surjective htop hbot)
/-- If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s`
tends to `at_top : filter β` along `at_bot : filter ↥s` and tends to `at_bot : filter β` along
`at_top : filter ↥s`, then the restriction of `f` to `s` is surjective. We formulate the
conclusion as `surj_on f s univ`. -/
lemma continuous_on.surj_on_of_tendsto' {f : α → β} {s : set α} [ord_connected s]
(hs : s.nonempty) (hf : continuous_on f s) (hbot : tendsto (λ x : s, f x) at_bot at_top)
(htop : tendsto (λ x : s, f x) at_top at_bot) :
surj_on f s univ :=
@continuous_on.surj_on_of_tendsto α (order_dual β) _ _ _ _ _ _ _ _ _ _ hs hf hbot htop
end densely_ordered
lemma is_compact.Inf_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
Inf s ∈ s :=
hs.is_closed.cInf_mem ne_s hs.bdd_below
lemma is_compact.Sup_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
Sup s ∈ s :=
@is_compact.Inf_mem (order_dual α) _ _ _ _ hs ne_s
lemma is_compact.is_glb_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_glb s (Inf s) :=
is_glb_cInf ne_s hs.bdd_below
lemma is_compact.is_lub_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_lub s (Sup s) :=
@is_compact.is_glb_Inf (order_dual α) _ _ _ _ hs ne_s
lemma is_compact.is_least_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_least s (Inf s) :=
⟨hs.Inf_mem ne_s, (hs.is_glb_Inf ne_s).1⟩
lemma is_compact.is_greatest_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
is_greatest s (Sup s) :=
@is_compact.is_least_Inf (order_dual α) _ _ _ _ hs ne_s
lemma is_compact.exists_is_least {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x, is_least s x :=
⟨_, hs.is_least_Inf ne_s⟩
lemma is_compact.exists_is_greatest {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x, is_greatest s x :=
⟨_, hs.is_greatest_Sup ne_s⟩
lemma is_compact.exists_is_glb {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x ∈ s, is_glb s x :=
⟨_, hs.Inf_mem ne_s, hs.is_glb_Inf ne_s⟩
lemma is_compact.exists_is_lub {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
∃ x ∈ s, is_lub s x :=
⟨_, hs.Sup_mem ne_s, hs.is_lub_Sup ne_s⟩
lemma is_compact.exists_Inf_image_eq {α : Type u} [topological_space α]
{s : set α} (hs : is_compact s) (ne_s : s.nonempty) {f : α → β} (hf : continuous_on f s) :
∃ x ∈ s, Inf (f '' s) = f x :=
let ⟨x, hxs, hx⟩ := (hs.image_of_continuous_on hf).Inf_mem (ne_s.image f)
in ⟨x, hxs, hx.symm⟩
lemma is_compact.exists_Sup_image_eq {α : Type u} [topological_space α]:
∀ {s : set α}, is_compact s → s.nonempty → ∀ {f : α → β}, continuous_on f s →
∃ x ∈ s, Sup (f '' s) = f x :=
@is_compact.exists_Inf_image_eq (order_dual β) _ _ _ _ _
lemma eq_Icc_of_connected_compact {s : set α} (h₁ : is_connected s) (h₂ : is_compact s) :
s = Icc (Inf s) (Sup s) :=
eq_Icc_cInf_cSup_of_connected_bdd_closed h₁ h₂.bdd_below h₂.bdd_above h₂.is_closed
/-- The extreme value theorem: a continuous function realizes its minimum on a compact set -/
lemma is_compact.exists_forall_le {α : Type u} [topological_space α]
{s : set α} (hs : is_compact s) (ne_s : s.nonempty) {f : α → β} (hf : continuous_on f s) :
∃x∈s, ∀y∈s, f x ≤ f y :=
begin
rcases hs.exists_Inf_image_eq ne_s hf with ⟨x, hxs, hx⟩,
refine ⟨x, hxs, λ y hy, _⟩,
rw ← hx,
exact ((hs.image_of_continuous_on hf).is_glb_Inf (ne_s.image f)).1 (mem_image_of_mem _ hy)
end
/-- The extreme value theorem: a continuous function realizes its maximum on a compact set -/
lemma is_compact.exists_forall_ge {α : Type u} [topological_space α]:
∀ {s : set α}, is_compact s → s.nonempty → ∀ {f : α → β}, continuous_on f s →
∃x∈s, ∀y∈s, f y ≤ f x :=
@is_compact.exists_forall_le (order_dual β) _ _ _ _ _
/-- The extreme value theorem: if a continuous function `f` tends to infinity away from compact
sets, then it has a global minimum. -/
lemma continuous.exists_forall_le {α : Type*} [topological_space α] [nonempty α] {f : α → β}
(hf : continuous f) (hlim : tendsto f (cocompact α) at_top) :
∃ x, ∀ y, f x ≤ f y :=
begin
inhabit α,
obtain ⟨s : set α, hsc : is_compact s, hsf : ∀ x ∉ s, f (default α) ≤ f x⟩ :=
(has_basis_cocompact.tendsto_iff at_top_basis).1 hlim (f $ default α) trivial,
obtain ⟨x, -, hx⟩ :=
(hsc.insert (default α)).exists_forall_le (nonempty_insert _ _) hf.continuous_on,
refine ⟨x, λ y, _⟩,
by_cases hy : y ∈ s,
exacts [hx y (or.inr hy), (hx _ (or.inl rfl)).trans (hsf y hy)]
end
/-- The extreme value theorem: if a continuous function `f` tends to negative infinity away from
compactx sets, then it has a global maximum. -/
lemma continuous.exists_forall_ge {α : Type*} [topological_space α] [nonempty α] {f : α → β}
(hf : continuous f) (hlim : tendsto f (cocompact α) at_bot) :
∃ x, ∀ y, f y ≤ f x :=
@continuous.exists_forall_le (order_dual β) _ _ _ _ _ _ _ hf hlim
end conditionally_complete_linear_order
section liminf_limsup
section order_closed_topology
variables [semilattice_sup α] [topological_space α] [order_topology α]
lemma is_bounded_le_nhds (a : α) : (𝓝 a).is_bounded (≤) :=
match forall_le_or_exists_lt_sup a with
| or.inl h := ⟨a, eventually_of_forall h⟩
| or.inr ⟨b, hb⟩ := ⟨b, ge_mem_nhds hb⟩
end
lemma filter.tendsto.is_bounded_under_le {f : filter β} {u : β → α} {a : α}
(h : tendsto u f (𝓝 a)) : f.is_bounded_under (≤) u :=
(is_bounded_le_nhds a).mono h
lemma is_cobounded_ge_nhds (a : α) : (𝓝 a).is_cobounded (≥) :=
(is_bounded_le_nhds a).is_cobounded_flip
lemma filter.tendsto.is_cobounded_under_ge {f : filter β} {u : β → α} {a : α}
[ne_bot f] (h : tendsto u f (𝓝 a)) : f.is_cobounded_under (≥) u :=
h.is_bounded_under_le.is_cobounded_flip
end order_closed_topology
section order_closed_topology
variables [semilattice_inf α] [topological_space α] [order_topology α]
lemma is_bounded_ge_nhds (a : α) : (𝓝 a).is_bounded (≥) :=
@is_bounded_le_nhds (order_dual α) _ _ _ a
lemma filter.tendsto.is_bounded_under_ge {f : filter β} {u : β → α} {a : α}
(h : tendsto u f (𝓝 a)) : f.is_bounded_under (≥) u :=
(is_bounded_ge_nhds a).mono h
lemma is_cobounded_le_nhds (a : α) : (𝓝 a).is_cobounded (≤) :=
(is_bounded_ge_nhds a).is_cobounded_flip
lemma filter.tendsto.is_cobounded_under_le {f : filter β} {u : β → α} {a : α}
[ne_bot f] (h : tendsto u f (𝓝 a)) : f.is_cobounded_under (≤) u :=
h.is_bounded_under_ge.is_cobounded_flip
end order_closed_topology
section conditionally_complete_linear_order
variables [conditionally_complete_linear_order α]
theorem lt_mem_sets_of_Limsup_lt {f : filter α} {b} (h : f.is_bounded (≤)) (l : f.Limsup < b) :
∀ᶠ a in f, a < b :=
let ⟨c, (h : ∀ᶠ a in f, a ≤ c), hcb⟩ := exists_lt_of_cInf_lt h l in
mem_sets_of_superset h $ assume a hac, lt_of_le_of_lt hac hcb
theorem gt_mem_sets_of_Liminf_gt : ∀ {f : filter α} {b}, f.is_bounded (≥) → b < f.Liminf →
∀ᶠ a in f, b < a :=
@lt_mem_sets_of_Limsup_lt (order_dual α) _
variables [topological_space α] [order_topology α]
/-- If the liminf and the limsup of a filter coincide, then this filter converges to
their common value, at least if the filter is eventually bounded above and below. -/
theorem le_nhds_of_Limsup_eq_Liminf {f : filter α} {a : α}
(hl : f.is_bounded (≤)) (hg : f.is_bounded (≥)) (hs : f.Limsup = a) (hi : f.Liminf = a) :
f ≤ 𝓝 a :=
tendsto_order.2 $ and.intro
(assume b hb, gt_mem_sets_of_Liminf_gt hg $ hi.symm ▸ hb)
(assume b hb, lt_mem_sets_of_Limsup_lt hl $ hs.symm ▸ hb)
theorem Limsup_nhds (a : α) : Limsup (𝓝 a) = a :=
cInf_intro (is_bounded_le_nhds a)
(assume a' (h : {n : α | n ≤ a'} ∈ 𝓝 a), show a ≤ a', from @mem_of_nhds α _ a _ h)
(assume b (hba : a < b), show ∃c (h : {n : α | n ≤ c} ∈ 𝓝 a), c < b, from
match dense_or_discrete a b with
| or.inl ⟨c, hac, hcb⟩ := ⟨c, ge_mem_nhds hac, hcb⟩
| or.inr ⟨_, h⟩ := ⟨a, (𝓝 a).sets_of_superset (gt_mem_nhds hba) h, hba⟩
end)
theorem Liminf_nhds : ∀ (a : α), Liminf (𝓝 a) = a :=
@Limsup_nhds (order_dual α) _ _ _
/-- If a filter is converging, its limsup coincides with its limit. -/
theorem Liminf_eq_of_le_nhds {f : filter α} {a : α} [ne_bot f] (h : f ≤ 𝓝 a) : f.Liminf = a :=
have hb_ge : is_bounded (≥) f, from (is_bounded_ge_nhds a).mono h,
have hb_le : is_bounded (≤) f, from (is_bounded_le_nhds a).mono h,
le_antisymm
(calc f.Liminf ≤ f.Limsup : Liminf_le_Limsup hb_le hb_ge
... ≤ (𝓝 a).Limsup :
Limsup_le_Limsup_of_le h hb_ge.is_cobounded_flip (is_bounded_le_nhds a)
... = a : Limsup_nhds a)
(calc a = (𝓝 a).Liminf : (Liminf_nhds a).symm
... ≤ f.Liminf :
Liminf_le_Liminf_of_le h (is_bounded_ge_nhds a) hb_le.is_cobounded_flip)
/-- If a filter is converging, its liminf coincides with its limit. -/
theorem Limsup_eq_of_le_nhds : ∀ {f : filter α} {a : α} [ne_bot f], f ≤ 𝓝 a → f.Limsup = a :=
@Liminf_eq_of_le_nhds (order_dual α) _ _ _
/-- If a function has a limit, then its limsup coincides with its limit. -/
theorem filter.tendsto.limsup_eq {f : filter β} {u : β → α} {a : α} [ne_bot f]
(h : tendsto u f (𝓝 a)) : limsup f u = a :=
Limsup_eq_of_le_nhds h
/-- If a function has a limit, then its liminf coincides with its limit. -/
theorem filter.tendsto.liminf_eq {f : filter β} {u : β → α} {a : α} [ne_bot f]
(h : tendsto u f (𝓝 a)) : liminf f u = a :=
Liminf_eq_of_le_nhds h
end conditionally_complete_linear_order
section complete_linear_order
variables [complete_linear_order α] [topological_space α] [order_topology α]
-- In complete_linear_order, the above theorems take a simpler form
/-- If the liminf and the limsup of a function coincide, then the limit of the function
exists and has the same value -/
theorem tendsto_of_liminf_eq_limsup {f : filter β} {u : β → α} {a : α}
(hinf : liminf f u = a) (hsup : limsup f u = a) : tendsto u f (𝓝 a) :=
le_nhds_of_Limsup_eq_Liminf is_bounded_le_of_top is_bounded_ge_of_bot hsup hinf
/-- If a number `a` is less than or equal to the `liminf` of a function `f` at some filter
and is greater than or equal to the `limsup` of `f`, then `f` tends to `a` along this filter. -/
theorem tendsto_of_le_liminf_of_limsup_le {f : filter β} {u : β → α} {a : α}
(hinf : a ≤ liminf f u) (hsup : limsup f u ≤ a) :
tendsto u f (𝓝 a) :=
if hf : f = ⊥ then hf.symm ▸ tendsto_bot
else by haveI : ne_bot f := hf; exact tendsto_of_liminf_eq_limsup
(le_antisymm (le_trans liminf_le_limsup hsup) hinf)
(le_antisymm hsup (le_trans hinf liminf_le_limsup))
end complete_linear_order
end liminf_limsup
end order_topology
/-!
Here is a counter-example to a version of the following with `conditionally_complete_lattice α`.
Take `α = [0, 1) → ℝ` with the natural lattice structure, `ι = ℕ`. Put `f n x = -x^n`. Then
`⨆ n, f n = 0` while none of `f n` is strictly greater than the constant function `-0.5`.
-/
lemma tendsto_at_top_csupr {ι α : Type*} [preorder ι] [topological_space α]
[conditionally_complete_linear_order α] [order_topology α]
{f : ι → α} (h_mono : monotone f) (hbdd : bdd_above $ range f) :
tendsto f at_top (𝓝 (⨆i, f i)) :=
begin
by_cases hi : nonempty ι,
{ resetI,
rw tendsto_order,
split,
{ intros a h,
cases exists_lt_of_lt_csupr h with N hN,
apply eventually.mono (mem_at_top N),
exact λ i hi, lt_of_lt_of_le hN (h_mono hi) },
{ exact λ a h, eventually_of_forall (λ n, lt_of_le_of_lt (le_csupr hbdd n) h) } },
{ exact tendsto_of_not_nonempty hi }
end
lemma tendsto_at_top_cinfi {ι α : Type*} [preorder ι] [topological_space α]
[conditionally_complete_linear_order α] [order_topology α]
{f : ι → α} (h_mono : ∀ ⦃i j⦄, i ≤ j → f j ≤ f i) (hbdd : bdd_below $ range f) :
tendsto f at_top (𝓝 (⨅i, f i)) :=
@tendsto_at_top_csupr _ (order_dual α) _ _ _ _ _ @h_mono hbdd
lemma tendsto_at_top_supr {ι α : Type*} [preorder ι] [topological_space α]
[complete_linear_order α] [order_topology α] {f : ι → α} (h_mono : monotone f) :
tendsto f at_top (𝓝 (⨆i, f i)) :=
tendsto_at_top_csupr h_mono (order_top.bdd_above _)
lemma tendsto_at_top_infi {ι α : Type*} [preorder ι] [topological_space α]
[complete_linear_order α] [order_topology α] {f : ι → α} (h_mono : ∀ ⦃i j⦄, i ≤ j → f j ≤ f i) :
tendsto f at_top (𝓝 (⨅i, f i)) :=
tendsto_at_top_cinfi @h_mono (order_bot.bdd_below _)
lemma tendsto_of_monotone {ι α : Type*} [preorder ι] [topological_space α]
[conditionally_complete_linear_order α] [order_topology α] {f : ι → α} (h_mono : monotone f) :
tendsto f at_top at_top ∨ (∃ l, tendsto f at_top (𝓝 l)) :=
if H : bdd_above (range f) then or.inr ⟨_, tendsto_at_top_csupr h_mono H⟩
else or.inl $ tendsto_at_top_at_top_of_monotone' h_mono H
lemma supr_eq_of_tendsto {α β} [topological_space α] [complete_linear_order α] [order_topology α]
[nonempty β] [semilattice_sup β] {f : β → α} {a : α} (hf : monotone f) :
tendsto f at_top (𝓝 a) → supr f = a :=
tendsto_nhds_unique (tendsto_at_top_supr hf)
lemma infi_eq_of_tendsto {α} [topological_space α] [complete_linear_order α] [order_topology α]
[nonempty β] [semilattice_sup β] {f : β → α} {a : α} (hf : ∀n m, n ≤ m → f m ≤ f n) :
tendsto f at_top (𝓝 a) → infi f = a :=
tendsto_nhds_unique (tendsto_at_top_infi hf)
@[to_additive] lemma tendsto_inv_nhds_within_Ioi [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Ioi a] a) (𝓝[Iio (a⁻¹)] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Iio [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Iio a] a) (𝓝[Ioi (a⁻¹)] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Ioi_inv [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Ioi (a⁻¹)] (a⁻¹)) (𝓝[Iio a] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Ioi _ _ _ _ (a⁻¹)
@[to_additive] lemma tendsto_inv_nhds_within_Iio_inv [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Iio (a⁻¹)] (a⁻¹)) (𝓝[Ioi a] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Iio _ _ _ _ (a⁻¹)
@[to_additive] lemma tendsto_inv_nhds_within_Ici [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Ici a] a) (𝓝[Iic (a⁻¹)] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Iic [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Iic a] a) (𝓝[Ici (a⁻¹)] (a⁻¹)) :=
(continuous_inv.tendsto a).inf $ by simp [tendsto_principal_principal]
@[to_additive] lemma tendsto_inv_nhds_within_Ici_inv [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Ici (a⁻¹)] (a⁻¹)) (𝓝[Iic a] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Ici _ _ _ _ (a⁻¹)
@[to_additive] lemma tendsto_inv_nhds_within_Iic_inv [ordered_comm_group α]
[topological_space α] [topological_group α] {a : α} :
tendsto has_inv.inv (𝓝[Iic (a⁻¹)] (a⁻¹)) (𝓝[Ici a] a) :=
by simpa only [inv_inv] using @tendsto_inv_nhds_within_Iic _ _ _ _ (a⁻¹)
lemma nhds_left_sup_nhds_right (a : α) [topological_space α] [linear_order α] :
𝓝[Iic a] a ⊔ 𝓝[Ici a] a = 𝓝 a :=
by rw [← nhds_within_union, Iic_union_Ici, nhds_within_univ]
lemma nhds_left'_sup_nhds_right (a : α) [topological_space α] [linear_order α] :
𝓝[Iio a] a ⊔ 𝓝[Ici a] a = 𝓝 a :=
by rw [← nhds_within_union, Iio_union_Ici, nhds_within_univ]
lemma nhds_left_sup_nhds_right' (a : α) [topological_space α] [linear_order α] :
𝓝[Iic a] a ⊔ 𝓝[Ioi a] a = 𝓝 a :=
by rw [← nhds_within_union, Iic_union_Ioi, nhds_within_univ]
lemma continuous_at_iff_continuous_left_right [topological_space α] [linear_order α]
[topological_space β] {a : α} {f : α → β} :
continuous_at f a ↔ continuous_within_at f (Iic a) a ∧ continuous_within_at f (Ici a) a :=
by simp only [continuous_within_at, continuous_at, ← tendsto_sup, nhds_left_sup_nhds_right]
lemma continuous_on_Icc_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α]
[order_topology α] [topological_space β] [regular_space β] {f : α → β} {a b : α}
{la lb : β} (hab : a < b) (hf : continuous_on f (Ioo a b))
(ha : tendsto f (𝓝[Ioi a] a) (𝓝 la)) (hb : tendsto f (𝓝[Iio b] b) (𝓝 lb)) :
continuous_on (extend_from (Ioo a b) f) (Icc a b) :=
begin
apply continuous_on_extend_from,
{ rw closure_Ioo hab, },
{ intros x x_in,
rcases mem_Ioo_or_eq_endpoints_of_mem_Icc x_in with rfl | rfl | h,
{ use la,
simpa [hab] },
{ use lb,
simpa [hab] },
{ use [f x, hf x h] } }
end
lemma eq_lim_at_left_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α]
[order_topology α] [topological_space β] [t2_space β] {f : α → β} {a b : α}
{la : β} (hab : a < b) (ha : tendsto f (𝓝[Ioi a] a) (𝓝 la)) :
extend_from (Ioo a b) f a = la :=
begin
apply extend_from_eq,
{ rw closure_Ioo hab,
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc] },
{ simpa [hab] }
end
lemma eq_lim_at_right_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α]
[order_topology α] [topological_space β] [t2_space β] {f : α → β} {a b : α}
{lb : β} (hab : a < b) (hb : tendsto f (𝓝[Iio b] b) (𝓝 lb)) :
extend_from (Ioo a b) f b = lb :=
begin
apply extend_from_eq,
{ rw closure_Ioo hab,
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc] },
{ simpa [hab] }
end
lemma continuous_on_Ico_extend_from_Ioo [topological_space α]
[linear_order α] [densely_ordered α] [order_topology α] [topological_space β]
[regular_space β] {f : α → β} {a b : α} {la : β} (hab : a < b) (hf : continuous_on f (Ioo a b))
(ha : tendsto f (𝓝[Ioi a] a) (𝓝 la)) :
continuous_on (extend_from (Ioo a b) f) (Ico a b) :=
begin
apply continuous_on_extend_from,
{ rw [closure_Ioo hab], exact Ico_subset_Icc_self, },
{ intros x x_in,
rcases mem_Ioo_or_eq_left_of_mem_Ico x_in with rfl | h,
{ use la,
simpa [hab] },
{ use [f x, hf x h] } }
end
lemma continuous_on_Ioc_extend_from_Ioo [topological_space α]
[linear_order α] [densely_ordered α] [order_topology α] [topological_space β]
[regular_space β] {f : α → β} {a b : α} {lb : β} (hab : a < b) (hf : continuous_on f (Ioo a b))
(hb : tendsto f (𝓝[Iio b] b) (𝓝 lb)) :
continuous_on (extend_from (Ioo a b) f) (Ioc a b) :=
begin
have := @continuous_on_Ico_extend_from_Ioo (order_dual α) _ _ _ _ _ _ _ f _ _ _ hab,
erw [dual_Ico, dual_Ioi, dual_Ioo] at this,
exact this hf hb
end
lemma continuous_within_at_Ioi_iff_Ici {α β : Type*} [topological_space α] [partial_order α]
[topological_space β] {a : α} {f : α → β} :
continuous_within_at f (Ioi a) a ↔ continuous_within_at f (Ici a) a :=
by simp only [← Ici_diff_left, continuous_within_at_diff_self]
lemma continuous_within_at_Iio_iff_Iic
{α β : Type*} [topological_space α] [linear_order α] [topological_space β] {a : α} {f : α → β} :
continuous_within_at f (Iio a) a ↔ continuous_within_at f (Iic a) a :=
begin
have := @continuous_within_at_Ioi_iff_Ici (order_dual α) _ _ _ _ _ f,
erw [dual_Ici, dual_Ioi] at this,
exact this,
end
lemma continuous_at_iff_continuous_left'_right' [topological_space α] [linear_order α]
[topological_space β] {a : α} {f : α → β} :
continuous_at f a ↔ continuous_within_at f (Iio a) a ∧ continuous_within_at f (Ioi a) a :=
by rw [continuous_within_at_Ioi_iff_Ici, continuous_within_at_Iio_iff_Iic,
continuous_at_iff_continuous_left_right]
/-!
### Continuity of monotone functions
In this section we prove the following fact: if `f` is a monotone function on a neighborhood of `a`
and the image of this neighborhood is a neighborhood of `f a`, then `f` is continuous at `a`, see
`continuous_at_of_mono_incr_on_of_image_mem_nhds`, as well as several similar facts.
-/
section linear_order
variables [linear_order α] [topological_space α] [order_topology α]
variables [linear_order β] [topological_space β] [order_topology β]
/-- If `f` is a function strictly monotonically increasing on a right neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `(f a, b]`, `b > f a`, then `f` is
continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` is required because otherwise the
function `f : ℝ → ℝ` given by `f x = if x ≤ 0 then x else x + 1` would be a counter-example at
`a = 0`. -/
lemma strict_mono_incr_on.continuous_at_right_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Ici a] a)
(hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
continuous_within_at f (Ici a) a :=
begin
have ha : a ∈ Ici a := left_mem_Ici,
have has : a ∈ s := mem_of_mem_nhds_within ha hs,
refine tendsto_order.2 ⟨λ b hb, _, λ b hb, _⟩,
{ filter_upwards [hs, self_mem_nhds_within],
intros x hxs hxa,
exact hb.trans_le ((h_mono.le_iff_le has hxs).2 hxa) },
{ rcases hfs b hb with ⟨c, hcs, hac, hcb⟩,
rw [h_mono.lt_iff_lt has hcs] at hac,
filter_upwards [hs, Ico_mem_nhds_within_Ici (left_mem_Ico.2 hac)],
rintros x hx ⟨hax, hxc⟩,
exact ((h_mono.lt_iff_lt hx hcs).2 hxc).trans_le hcb }
end
/-- If `f` is a function monotonically increasing function on a right neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `(f a, b)`, `b > f a`, then `f` is
continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b` cannot be replaced by the weaker
assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` we use for strictly monotone functions
because otherwise the function `ceil : ℝ → ℤ` would be a counter-example at `a = 0`. -/
lemma continuous_at_right_of_mono_incr_on_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝[Ici a] a)
(hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
continuous_within_at f (Ici a) a :=
begin
have ha : a ∈ Ici a := left_mem_Ici,
have has : a ∈ s := mem_of_mem_nhds_within ha hs,
refine tendsto_order.2 ⟨λ b hb, _, λ b hb, _⟩,
{ filter_upwards [hs, self_mem_nhds_within],
intros x hxs hxa,
exact hb.trans_le (h_mono _ has _ hxs hxa) },
{ rcases hfs b hb with ⟨c, hcs, hac, hcb⟩,
have : a < c, from not_le.1 (λ h, hac.not_le $ h_mono _ hcs _ has h),
filter_upwards [hs, Ico_mem_nhds_within_Ici (left_mem_Ico.2 this)],
rintros x hx ⟨hax, hxc⟩,
exact (h_mono _ hx _ hcs hxc.le).trans_lt hcb }
end
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a right
neighborhood of `a` and the closure of the image of this neighborhood under `f` is a right
neighborhood of `f a`, then `f` is continuous at `a` from the right. -/
lemma continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[Ici a] a) (hfs : closure (f '' s) ∈ 𝓝[Ici (f a)] (f a)) :
continuous_within_at f (Ici a) a :=
begin
refine continuous_at_right_of_mono_incr_on_of_exists_between h_mono hs (λ b hb, _),
rcases (mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset hb).1 hfs with ⟨b', ⟨hab', hbb'⟩, hb'⟩,
rcases exists_between hab' with ⟨c', hc'⟩,
rcases mem_closure_iff.1 (hb' ⟨hc'.1.le, hc'.2⟩) (Ioo (f a) b') is_open_Ioo hc'
with ⟨_, hc, ⟨c, hcs, rfl⟩⟩,
exact ⟨c, hcs, hc.1, hc.2.trans_le hbb'⟩
end
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a right
neighborhood of `a` and the image of this neighborhood under `f` is a right neighborhood of `f a`,
then `f` is continuous at `a` from the right. -/
lemma continuous_at_right_of_mono_incr_on_of_image_mem_nhds_within [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝[Ici a] a)
(hfs : f '' s ∈ 𝓝[Ici (f a)] (f a)) :
continuous_within_at f (Ici a) a :=
continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within h_mono hs $
mem_sets_of_superset hfs subset_closure
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
right neighborhood of `a` and the closure of the image of this neighborhood under `f` is a right
neighborhood of `f a`, then `f` is continuous at `a` from the right. -/
lemma strict_mono_incr_on.continuous_at_right_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Ici a] a)
(hfs : closure (f '' s) ∈ 𝓝[Ici (f a)] (f a)) :
continuous_within_at f (Ici a) a :=
continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within
(λ x hx y hy, (h_mono.le_iff_le hx hy).2) hs hfs
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
right neighborhood of `a` and the image of this neighborhood under `f` is a right neighborhood of
`f a`, then `f` is continuous at `a` from the right. -/
lemma strict_mono_incr_on.continuous_at_right_of_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Ici a] a)
(hfs : f '' s ∈ 𝓝[Ici (f a)] (f a)) :
continuous_within_at f (Ici a) a :=
h_mono.continuous_at_right_of_closure_image_mem_nhds_within hs
(mem_sets_of_superset hfs subset_closure)
/-- If a function `f` is strictly monotonically increasing on a right neighborhood of `a` and the
image of this neighborhood under `f` includes `Ioi (f a)`, then `f` is continuous at `a` from the
right. -/
lemma strict_mono_incr_on.continuous_at_right_of_surj_on {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Ici a] a) (hfs : surj_on f s (Ioi (f a))) :
continuous_within_at f (Ici a) a :=
h_mono.continuous_at_right_of_exists_between hs $ λ b hb, let ⟨c, hcs, hcb⟩ := hfs hb in
⟨c, hcs, hcb.symm ▸ hb, hcb.le⟩
/-- If `f` is a function strictly monotonically increasing on a left neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, then `f` is
continuous at `a` from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` is required because otherwise the
function `f : ℝ → ℝ` given by `f x = if x < 0 then x else x + 1` would be a counter-example at
`a = 0`. -/
lemma strict_mono_incr_on.continuous_at_left_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Iic a] a)
(hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) :
continuous_within_at f (Iic a) a :=
h_mono.dual.continuous_at_right_of_exists_between hs $
λ b hb, let ⟨c, hcs, hcb, hca⟩ := hfs b hb in ⟨c, hcs, hca, hcb⟩
/-- If `f` is a function monotonically increasing function on a left neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `(b, f a)`, `b < f a`, then `f` is
continuous at `a` from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)` cannot be replaced by the weaker
assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` we use for strictly monotone functions
because otherwise the function `floor : ℝ → ℤ` would be a counter-example at `a = 0`. -/
lemma continuous_at_left_of_mono_incr_on_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝[Iic a] a)
(hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) :
continuous_within_at f (Iic a) a :=
@continuous_at_right_of_mono_incr_on_of_exists_between (order_dual α) (order_dual β) _ _ _ _ _ _
f s a (λ x hx y hy, h_mono y hy x hx) hs $
λ b hb, let ⟨c, hcs, hcb, hca⟩ := hfs b hb in ⟨c, hcs, hca, hcb⟩
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a left
neighborhood of `a` and the closure of the image of this neighborhood under `f` is a left
neighborhood of `f a`, then `f` is continuous at `a` from the left -/
lemma continuous_at_left_of_mono_incr_on_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[Iic a] a) (hfs : closure (f '' s) ∈ 𝓝[Iic (f a)] (f a)) :
continuous_within_at f (Iic a) a :=
@continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within (order_dual α) (order_dual β)
_ _ _ _ _ _ _ f s a (λ x hx y hy, h_mono y hy x hx) hs hfs
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a left
neighborhood of `a` and the image of this neighborhood under `f` is a left neighborhood of `f a`,
then `f` is continuous at `a` from the left. -/
lemma continuous_at_left_of_mono_incr_on_of_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y)
(hs : s ∈ 𝓝[Iic a] a) (hfs : f '' s ∈ 𝓝[Iic (f a)] (f a)) :
continuous_within_at f (Iic a) a :=
continuous_at_left_of_mono_incr_on_of_closure_image_mem_nhds_within h_mono hs
(mem_sets_of_superset hfs subset_closure)
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
left neighborhood of `a` and the closure of the image of this neighborhood under `f` is a left
neighborhood of `f a`, then `f` is continuous at `a` from the left. -/
lemma strict_mono_incr_on.continuous_at_left_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Iic a] a)
(hfs : closure (f '' s) ∈ 𝓝[Iic (f a)] (f a)) :
continuous_within_at f (Iic a) a :=
h_mono.dual.continuous_at_right_of_closure_image_mem_nhds_within hs hfs
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
left neighborhood of `a` and the image of this neighborhood under `f` is a left neighborhood of
`f a`, then `f` is continuous at `a` from the left. -/
lemma strict_mono_incr_on.continuous_at_left_of_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Iic a] a)
(hfs : f '' s ∈ 𝓝[Iic (f a)] (f a)) :
continuous_within_at f (Iic a) a :=
h_mono.dual.continuous_at_right_of_image_mem_nhds_within hs hfs
/-- If a function `f` is strictly monotonically increasing on a left neighborhood of `a` and the
image of this neighborhood under `f` includes `Iio (f a)`, then `f` is continuous at `a` from the
left. -/
lemma strict_mono_incr_on.continuous_at_left_of_surj_on {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝[Iic a] a) (hfs : surj_on f s (Iio (f a))) :
continuous_within_at f (Iic a) a :=
h_mono.dual.continuous_at_right_of_surj_on hs hfs
/-- If a function `f` is strictly monotonically increasing on a neighborhood of `a` and the image of
this neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, and every interval
`(f a, b]`, `b > f a`, then `f` is continuous at `a`. -/
lemma strict_mono_incr_on.continuous_at_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝 a)
(hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
continuous_at f a :=
continuous_at_iff_continuous_left_right.2
⟨h_mono.continuous_at_left_of_exists_between (mem_nhds_within_of_mem_nhds hs) hfs_l,
h_mono.continuous_at_right_of_exists_between (mem_nhds_within_of_mem_nhds hs) hfs_r⟩
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
neighborhood of `a` and the closure of the image of this neighborhood under `f` is a neighborhood of
`f a`, then `f` is continuous at `a`. -/
lemma strict_mono_incr_on.continuous_at_of_closure_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) :
continuous_at f a :=
continuous_at_iff_continuous_left_right.2
⟨h_mono.continuous_at_left_of_closure_image_mem_nhds_within (mem_nhds_within_of_mem_nhds hs)
(mem_nhds_within_of_mem_nhds hfs),
h_mono.continuous_at_right_of_closure_image_mem_nhds_within (mem_nhds_within_of_mem_nhds hs)
(mem_nhds_within_of_mem_nhds hfs)⟩
/-- If a function `f` with a densely ordered codomain is strictly monotonically increasing on a
neighborhood of `a` and the image of this set under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. -/
lemma strict_mono_incr_on.continuous_at_of_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : strict_mono_incr_on f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) :
continuous_at f a :=
h_mono.continuous_at_of_closure_image_mem_nhds hs (mem_sets_of_superset hfs subset_closure)
/-- If `f` is a function monotonically increasing function on a neighborhood of `a` and the image of
this neighborhood under `f` meets every interval `(b, f a)`, `b < f a`, and every interval `(f a,
b)`, `b > f a`, then `f` is continuous at `a`. -/
lemma continuous_at_of_mono_incr_on_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝 a)
(hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
continuous_at f a :=
continuous_at_iff_continuous_left_right.2
⟨continuous_at_left_of_mono_incr_on_of_exists_between h_mono
(mem_nhds_within_of_mem_nhds hs) hfs_l,
continuous_at_right_of_mono_incr_on_of_exists_between h_mono
(mem_nhds_within_of_mem_nhds hs) hfs_r⟩
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a neighborhood
of `a` and the closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then
`f` is continuous at `a`. -/
lemma continuous_at_of_mono_incr_on_of_closure_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) :
continuous_at f a :=
continuous_at_iff_continuous_left_right.2
⟨continuous_at_left_of_mono_incr_on_of_closure_image_mem_nhds_within h_mono
(mem_nhds_within_of_mem_nhds hs) (mem_nhds_within_of_mem_nhds hfs),
continuous_at_right_of_mono_incr_on_of_closure_image_mem_nhds_within h_mono
(mem_nhds_within_of_mem_nhds hs) (mem_nhds_within_of_mem_nhds hfs)⟩
/-- If a function `f` with a densely ordered codomain is monotonically increasing on a neighborhood
of `a` and the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. -/
lemma continuous_at_of_mono_incr_on_of_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : ∀ (x ∈ s) (y ∈ s), x ≤ y → f x ≤ f y) (hs : s ∈ 𝓝 a)
(hfs : f '' s ∈ 𝓝 (f a)) :
continuous_at f a :=
continuous_at_of_mono_incr_on_of_closure_image_mem_nhds h_mono hs
(mem_sets_of_superset hfs subset_closure)
/-- A monotone function with densely ordered codomain and a dense range is continuous. -/
lemma monotone.continuous_of_dense_range [densely_ordered β] {f : α → β}
(h_mono : monotone f) (h_dense : dense_range f) :
continuous f :=
continuous_iff_continuous_at.mpr $ λ a,
continuous_at_of_mono_incr_on_of_closure_image_mem_nhds (λ x hx y hy hxy, h_mono hxy)
univ_mem_sets $ by simp only [image_univ, h_dense.closure_eq, univ_mem_sets]
/-- A monotone surjective function with a densely ordered codomain is surjective. -/
lemma monotone.continuous_of_surjective [densely_ordered β] {f : α → β} (h_mono : monotone f)
(h_surj : function.surjective f) :
continuous f :=
h_mono.continuous_of_dense_range h_surj.dense_range
end linear_order
/-!
### Continuity of order isomorphisms
In this section we prove that an `order_iso` is continuous, hence it is a `homeomorph`. We prove
this for an `order_iso` between to partial orders with order topology.
-/
namespace order_iso
variables [partial_order α] [partial_order β] [topological_space α] [topological_space β]
[order_topology α] [order_topology β]
protected lemma continuous (e : α ≃o β) : continuous e :=
begin
rw [‹order_topology β›.topology_eq_generate_intervals],
refine continuous_generated_from (λ s hs, _),
rcases hs with ⟨a, rfl|rfl⟩,
{ rw e.preimage_Ioi, apply is_open_lt' },
{ rw e.preimage_Iio, apply is_open_gt' }
end
/-- An order isomorphism between two linear order `order_topology` spaces is a homeomorphism. -/
def to_homeomorph (e : α ≃o β) : α ≃ₜ β :=
{ continuous_to_fun := e.continuous,
continuous_inv_fun := e.symm.continuous,
.. e }
@[simp] lemma coe_to_homeomorph (e : α ≃o β) : ⇑e.to_homeomorph = e := rfl
@[simp] lemma coe_to_homeomorph_symm (e : α ≃o β) : ⇑e.to_homeomorph.symm = e.symm := rfl
end order_iso
|
153fe90a135f18d4099158a62ec70abdd76278ef | a0e23cfdd129a671bf3154ee1a8a3a72bf4c7940 | /src/Lean/Server/FileSource.lean | 362f7e50d589e69eea96baa0d80b0423ea2ae207 | [
"Apache-2.0"
] | permissive | WojciechKarpiel/lean4 | 7f89706b8e3c1f942b83a2c91a3a00b05da0e65b | f6e1314fa08293dea66a329e05b6c196a0189163 | refs/heads/master | 1,686,633,402,214 | 1,625,821,189,000 | 1,625,821,258,000 | 384,640,886 | 0 | 0 | Apache-2.0 | 1,625,903,617,000 | 1,625,903,026,000 | null | UTF-8 | Lean | false | false | 2,906 | lean | /-
Copyright (c) 2020 Marc Huisinga. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Marc Huisinga
-/
import Lean.Data.Lsp
namespace Lean.Lsp
class FileSource (α : Type) where
fileSource : α → DocumentUri
export FileSource (fileSource)
instance Location.hasFileSource : FileSource Location :=
⟨fun l => l.uri⟩
instance TextDocumentIdentifier.hasFileSource : FileSource TextDocumentIdentifier :=
⟨fun i => i.uri⟩
instance VersionedTextDocumentIdentifier.hasFileSource : FileSource VersionedTextDocumentIdentifier :=
⟨fun i => i.uri⟩
instance TextDocumentEdit.hasFileSource : FileSource TextDocumentEdit :=
⟨fun e => fileSource e.textDocument⟩
instance TextDocumentItem.hasFileSource : FileSource TextDocumentItem :=
⟨fun i => i.uri⟩
instance TextDocumentPositionParams.hasFileSource : FileSource TextDocumentPositionParams :=
⟨fun p => fileSource p.textDocument⟩
instance DidOpenTextDocumentParams.hasFileSource : FileSource DidOpenTextDocumentParams :=
⟨fun p => fileSource p.textDocument⟩
instance DidChangeTextDocumentParams.hasFileSource : FileSource DidChangeTextDocumentParams :=
⟨fun p => fileSource p.textDocument⟩
instance DidCloseTextDocumentParams.hasFileSource : FileSource DidCloseTextDocumentParams :=
⟨fun p => fileSource p.textDocument⟩
instance CompletionParams.hasFileSource : FileSource CompletionParams :=
⟨fun h => fileSource h.toTextDocumentPositionParams⟩
instance HoverParams.hasFileSource : FileSource HoverParams :=
⟨fun h => fileSource h.toTextDocumentPositionParams⟩
instance DeclarationParams.hasFileSource : FileSource DeclarationParams :=
⟨fun h => fileSource h.toTextDocumentPositionParams⟩
instance DefinitionParams.hasFileSource : FileSource DefinitionParams :=
⟨fun h => fileSource h.toTextDocumentPositionParams⟩
instance TypeDefinitionParams.hasFileSource : FileSource TypeDefinitionParams :=
⟨fun h => fileSource h.toTextDocumentPositionParams⟩
instance WaitForDiagnosticsParams.hasFileSource : FileSource WaitForDiagnosticsParams :=
⟨fun p => p.uri⟩
instance DocumentHighlightParams.hasFileSource : FileSource DocumentHighlightParams :=
⟨fun h => fileSource h.toTextDocumentPositionParams⟩
instance DocumentSymbolParams.hasFileSource : FileSource DocumentSymbolParams :=
⟨fun p => fileSource p.textDocument⟩
instance SemanticTokensParams.hasFileSource : FileSource SemanticTokensParams :=
⟨fun p => fileSource p.textDocument⟩
instance SemanticTokensRangeParams.hasFileSource : FileSource SemanticTokensRangeParams :=
⟨fun p => fileSource p.textDocument⟩
instance PlainGoalParams.hasFileSource : FileSource PlainGoalParams :=
⟨fun p => fileSource p.textDocument⟩
instance : FileSource PlainTermGoalParams where
fileSource p := fileSource p.textDocument
end Lean.Lsp
|
7623751e5207dd7aa622e7aea1d46b9feb3aa849 | 947fa6c38e48771ae886239b4edce6db6e18d0fb | /src/number_theory/legendre_symbol/gauss_sum.lean | b60b5163673807360446b1c51ccaad9c6fe7a709 | [
"Apache-2.0"
] | permissive | ramonfmir/mathlib | c5dc8b33155473fab97c38bd3aa6723dc289beaa | 14c52e990c17f5a00c0cc9e09847af16fabbed25 | refs/heads/master | 1,661,979,343,526 | 1,660,830,384,000 | 1,660,830,384,000 | 182,072,989 | 0 | 0 | null | 1,555,585,876,000 | 1,555,585,876,000 | null | UTF-8 | Lean | false | false | 13,518 | lean | /-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import number_theory.legendre_symbol.add_character
import number_theory.legendre_symbol.zmod_char
import algebra.char_p.char_and_card
/-!
# Gauss sums
We define the Gauss sum associated to a multiplicative and an additive
character of a finite field and prove some results about them.
## Main definition
Let `R` be a finite commutative ring and let `R'` be another commutative ring.
If `χ` is a multiplicative character `R → R'` (type `mul_char R R'`) and `ψ`
is an additive character `R → R'` (type `add_char R R'`, which abbreviates
`(multiplicative R) →* R'`), then the *Gauss sum* of `χ` and `ψ` is `∑ a, χ a * ψ a`.
## Main results
Some important results are as follows.
* `gauss_sum_mul_gauss_sum_eq_card`: The product of the Gauss
sums of `χ` and `ψ` and that of `χ⁻¹` and `ψ⁻¹` is the cardinality
of the source ring `R` (if `χ` is nontrivial, `ψ` is primitive and `R` is a field).
* `gauss_sum_sq`: The square of the Gauss sum is `χ(-1)` times
the cardinality of `R` if in addition `χ` is a quadratic character.
* `quad_gauss_sum_frob`: For a quadratic character `χ`, raising
the Gauss sum to the `p`th power (where `p` is the characteristic of
the target ring `R'`) multiplies it by `χ p`.
* `char.card_pow_card`: When `F` and `F'` are finite fields and `χ : F → F'`
is a nontrivial quadratic character, then `(χ (-1) * #F)^(#F'/2) = χ (#F')`.
* `finite_field.two_pow_card`: For every finite field `F` of odd characteristic,
we have `2^(#F/2) = χ₈(#F)` in `F`.
This machinery can be used to derive (a generalization of) the Law of
Quadratic Reciprocity.
## Tags
additive character, multiplicative character, Gauss sum
-/
universes u v
open_locale big_operators
open add_char mul_char
section gauss_sum_def
-- `R` is the domain of the characters
variables {R : Type u} [comm_ring R] [fintype R]
-- `R'` is the target of the characters
variables {R' : Type v} [comm_ring R']
/-!
### Definition and first properties
-/
/-- Definition of the Gauss sum associated to a multiplicative and an additive character. -/
def gauss_sum (χ : mul_char R R') (ψ : add_char R R') : R' := ∑ a, χ a * ψ a
/-- Replacing `ψ` by `mul_shift ψ a` and multiplying the Gauss sum by `χ a` does not change it. -/
lemma gauss_sum_mul_shift (χ : mul_char R R') (ψ : add_char R R') (a : Rˣ) :
χ a * gauss_sum χ (mul_shift ψ a) = gauss_sum χ ψ :=
begin
simp only [gauss_sum, mul_shift_apply, finset.mul_sum],
simp_rw [← mul_assoc, ← map_mul],
exact fintype.sum_bijective _ a.mul_left_bijective _ _ (λ x, rfl),
end
end gauss_sum_def
/-!
### The product of two Gauss sums
-/
section gauss_sum_prod
-- In the following, we need `R` to be a finite field and `R'` to be a domain.
variables {R : Type u} [field R] [fintype R] {R' : Type v} [comm_ring R'] [is_domain R']
-- A helper lemma for `gauss_sum_mul_gauss_sum_eq_card` below
-- Is this useful enough in other contexts to be public?
private
lemma gauss_sum_mul_aux {χ : mul_char R R'} (hχ : is_nontrivial χ) (ψ : add_char R R') (b : R) :
∑ a, χ (a * b⁻¹) * ψ (a - b) = ∑ c, χ c * ψ (b * (c - 1)) :=
begin
cases eq_or_ne b 0 with hb hb,
{ -- case `b = 0`
simp only [hb, inv_zero, mul_zero, mul_char.map_zero, zero_mul, finset.sum_const_zero,
map_zero_one, mul_one],
exact hχ.sum_eq_zero.symm, },
{ -- case `b ≠ 0`
refine (fintype.sum_bijective _ (mul_left_bijective₀ b hb) _ _ $ λ x, _).symm,
rw [mul_assoc, mul_comm x, ← mul_assoc, mul_inv_cancel hb, one_mul, mul_sub, mul_one] },
end
/-- We have `gauss_sum χ ψ * gauss_sum χ⁻¹ ψ⁻¹ = fintype.card R`
when `χ` is nontrivial and `ψ` is primitive (and `R` is a field). -/
lemma gauss_sum_mul_gauss_sum_eq_card {χ : mul_char R R'} (hχ : is_nontrivial χ)
{ψ : add_char R R'} (hψ : is_primitive ψ) :
gauss_sum χ ψ * gauss_sum χ⁻¹ ψ⁻¹ = fintype.card R :=
begin
simp only [gauss_sum, add_char.inv_apply, finset.sum_mul, finset.mul_sum, mul_char.inv_apply'],
conv in (_ * _ * (_ * _))
{ rw [mul_mul_mul_comm, ← map_mul, ← map_add_mul, ← sub_eq_add_neg], },
simp_rw gauss_sum_mul_aux hχ ψ,
rw [finset.sum_comm],
classical, -- to get `[decidable_eq R]` for `sum_mul_shift`
simp_rw [← finset.mul_sum, sum_mul_shift _ hψ, sub_eq_zero, mul_ite, mul_zero],
rw [finset.sum_ite_eq' finset.univ (1 : R)],
simp only [finset.mem_univ, map_one, one_mul, if_true],
end
/-- When `χ` is a nontrivial quadratic character, then the square of `gauss_sum χ ψ`
is `χ(-1)` times the cardinality of `R`. -/
lemma gauss_sum_sq {χ : mul_char R R'} (hχ₁ : is_nontrivial χ) (hχ₂ : is_quadratic χ)
{ψ : add_char R R'} (hψ : is_primitive ψ) :
(gauss_sum χ ψ) ^ 2 = χ (-1) * fintype.card R :=
begin
rw [pow_two, ← gauss_sum_mul_gauss_sum_eq_card hχ₁ hψ, hχ₂.inv, mul_rotate'],
congr,
rw [mul_comm, ← gauss_sum_mul_shift _ _ (-1 : Rˣ), inv_mul_shift],
refl,
end
end gauss_sum_prod
/-!
### Gauss sums and Frobenius
-/
section gauss_sum_frob
variables {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R']
-- We assume that the target ring `R'` has prime characteristic `p`.
variables (p : ℕ) [fp : fact p.prime] [hch : char_p R' p]
include fp hch
/-- When `R'` has prime characteristic `p`, then the `p`th power of the Gauss sum
of `χ` and `ψ` is the Gauss sum of `χ^p` and `ψ^p`. -/
lemma gauss_sum_frob (χ : mul_char R R') (ψ : add_char R R') :
gauss_sum χ ψ ^ p = gauss_sum (χ ^ p) (ψ ^ p) :=
begin
rw [← frobenius_def, gauss_sum, gauss_sum, map_sum],
simp_rw [pow_apply' χ fp.1.pos, map_mul, frobenius_def],
refl,
end
/-- For a quadratic character `χ` and when the characteristic `p` of the target ring
is a unit in the source ring, the `p`th power of the Gauss sum of`χ` and `ψ` is
`χ p` times the original Gauss sum. -/
lemma mul_char.is_quadratic.gauss_sum_frob (hp : is_unit (p : R)) {χ : mul_char R R'}
(hχ : is_quadratic χ) (ψ : add_char R R') :
gauss_sum χ ψ ^ p = χ p * gauss_sum χ ψ :=
by rw [gauss_sum_frob, pow_mul_shift, hχ.pow_char p, ← gauss_sum_mul_shift χ ψ hp.unit,
← mul_assoc, hp.unit_spec, ← pow_two, ← pow_apply' _ (by norm_num : 0 < 2),
hχ.sq_eq_one, ← hp.unit_spec, one_apply_coe, one_mul]
/-- For a quadratic character `χ` and when the characteristic `p` of the target ring
is a unit in the source ring and `n` is a natural number, the `p^n`th power of the Gauss
sum of`χ` and `ψ` is `χ (p^n)` times the original Gauss sum. -/
lemma mul_char.is_quadratic.gauss_sum_frob_iter (n : ℕ) (hp : is_unit (p : R))
{χ : mul_char R R'} (hχ : is_quadratic χ) (ψ : add_char R R') :
gauss_sum χ ψ ^ (p ^ n) = χ (p ^ n) * gauss_sum χ ψ :=
begin
induction n with n ih,
{ rw [pow_zero, pow_one, pow_zero, mul_char.map_one, one_mul], },
{ rw [pow_succ, mul_comm p, pow_mul, ih, mul_pow, hχ.gauss_sum_frob _ hp,
← mul_assoc, pow_succ, mul_comm (p : R), map_mul,
← pow_apply' χ fp.1.pos (p ^ n), hχ.pow_char p], },
end
end gauss_sum_frob
/-!
### Values of quadratic characters
-/
section gauss_sum_values
variables {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R'] [is_domain R']
/-- If the square of the Gauss sum of a quadratic character is `χ(-1) * #R`,
then we get, for all `n : ℕ`, the relation `(χ(-1) * #R) ^ (p^n/2) = χ(p^n)`,
where `p` is the (odd) characteristic of the target ring `R'`.
This version can be used when `R` is not a field, e.g., `ℤ/8ℤ`. -/
lemma char.card_pow_char_pow {χ : mul_char R R'} (hχ : is_quadratic χ) (ψ : add_char R R') (p n : ℕ)
[fp : fact p.prime] [hch : char_p R' p] (hp : is_unit (p : R)) (hp' : p ≠ 2)
(hg : (gauss_sum χ ψ) ^ 2 = χ (-1) * fintype.card R) :
(χ (-1) * fintype.card R) ^ (p ^ n / 2) = χ (p ^ n) :=
begin
have : gauss_sum χ ψ ≠ 0,
{ intro hf, rw [hf, zero_pow (by norm_num : 0 < 2), eq_comm, mul_eq_zero] at hg,
exact not_is_unit_prime_of_dvd_card p
((char_p.cast_eq_zero_iff R' p _).mp $ hg.resolve_left (is_unit_one.neg.map χ).ne_zero) hp },
rw ← hg, apply mul_right_cancel₀ this,
rw [← hχ.gauss_sum_frob_iter p n hp ψ, ← pow_mul, mul_comm, ← pow_succ,
nat.two_mul_div_two_add_one_of_odd ((fp.1.eq_two_or_odd').resolve_left hp').pow],
end
/-- When `F` and `F'` are finite fields and `χ : F → F'` is a nontrivial quadratic character,
then `(χ(-1) * #F)^(#F'/2) = χ(#F')`. -/
lemma char.card_pow_card {F : Type} [field F] [fintype F] {F' : Type} [field F'] [fintype F']
{χ : mul_char F F'} (hχ₁ : is_nontrivial χ) (hχ₂ : is_quadratic χ)
(hch₁ : ring_char F' ≠ ring_char F) (hch₂ : ring_char F' ≠ 2) :
(χ (-1) * fintype.card F) ^ (fintype.card F' / 2) = χ (fintype.card F') :=
begin
obtain ⟨n, hp, hc⟩ := finite_field.card F (ring_char F),
obtain ⟨n', hp', hc'⟩ := finite_field.card F' (ring_char F'),
let ψ := primitive_char_finite_field F F' hch₁,
let FF' := cyclotomic_field ψ.n F',
have hchar := algebra.ring_char_eq F' FF',
apply (algebra_map F' FF').injective,
rw [map_pow, map_mul, map_nat_cast, hc', hchar, nat.cast_pow],
simp only [← mul_char.ring_hom_comp_apply],
haveI := fact.mk hp',
haveI := fact.mk (hchar.subst hp'),
rw [ne, ← nat.prime_dvd_prime_iff_eq hp' hp, ← is_unit_iff_not_dvd_char, hchar] at hch₁,
exact char.card_pow_char_pow (hχ₂.comp _) ψ.char (ring_char FF') n' hch₁ (hchar ▸ hch₂)
(gauss_sum_sq (hχ₁.comp $ ring_hom.injective _) (hχ₂.comp _) ψ.prim),
end
end gauss_sum_values
section gauss_sum_two
/-!
### The quadratic character of 2
This section proves the following result.
For every finite field `F` of odd characteristic, we have `2^(#F/2) = χ₈(#F)` in `F`.
This can be used to show that the quadratic character of `F` takes the value
`χ₈(#F)` at `2`.
The proof uses the Gauss sum of `χ₈` and a primitive additive character on `ℤ/8ℤ`;
in this way, the result is reduced to `card_pow_char_pow`.
-/
open zmod
/-- For every finite field `F` of odd characteristic, we have `2^(#F/2) = χ₈(#F)` in `F`. -/
lemma finite_field.two_pow_card {F : Type*} [fintype F] [field F] (hF : ring_char F ≠ 2) :
(2 : F) ^ (fintype.card F / 2) = χ₈ (fintype.card F) :=
begin
have hp2 : ∀ (n : ℕ), (2 ^ n : F) ≠ 0 := λ n, pow_ne_zero n (ring.two_ne_zero hF),
obtain ⟨n, hp, hc⟩ := finite_field.card F (ring_char F),
-- we work in `FF`, the eighth cyclotomic field extension of `F`
let FF := (polynomial.cyclotomic 8 F).splitting_field,
haveI : finite_dimensional F FF :=
polynomial.is_splitting_field.finite_dimensional FF (polynomial.cyclotomic 8 F),
haveI : fintype FF := finite_dimensional.fintype_of_fintype F FF,
have hchar := algebra.ring_char_eq F FF,
have FFp := hchar.subst hp,
haveI := fact.mk FFp,
have hFF := ne_of_eq_of_ne hchar.symm hF, -- `ring_char FF ≠ 2`
have hu : is_unit (ring_char FF : zmod 8),
{ rw [is_unit_iff_not_dvd_char, ring_char_zmod_n],
rw [ne, ← nat.prime_dvd_prime_iff_eq FFp nat.prime_two] at hFF,
change ¬ _ ∣ 2 ^ 3,
exact mt FFp.dvd_of_dvd_pow hFF },
-- there is a primitive additive character `ℤ/8ℤ → FF`, sending `a + 8ℤ ↦ τ^a`
-- with a primitive eighth root of unity `τ`
let ψ₈ := primitive_zmod_char 8 F (by convert hp2 3; norm_num),
let τ : FF := ψ₈.char 1,
have τ_spec : τ ^ 4 = -1,
{ refine (sq_eq_one_iff.1 _).resolve_left _;
{ simp only [τ, ← map_nsmul_pow],
erw add_char.is_primitive.zmod_char_eq_one_iff 8 ψ₈.prim,
dec_trivial } },
-- we consider `χ₈` as a multiplicative character `ℤ/8ℤ → FF`
let χ := χ₈.ring_hom_comp (int.cast_ring_hom FF),
have hχ : χ (-1) = 1 := norm_num.int_cast_one,
have hq : is_quadratic χ := is_quadratic_χ₈.comp _,
-- we now show that the Gauss sum of `χ` and `ψ₈` has the relevant property
have hg : gauss_sum χ ψ₈.char ^ 2 = χ (-1) * fintype.card (zmod 8),
{ rw [hχ, one_mul, card, gauss_sum],
convert ← congr_arg (^ 2) (fin.sum_univ_eight $ λ x, (χ₈ x : FF) * τ ^ x.val),
{ ext, congr, apply pow_one },
convert_to (0 + 1 * τ ^ 1 + 0 + (-1) * τ ^ 3 + 0 + (-1) * τ ^ 5 + 0 + 1 * τ ^ 7) ^ 2 = _,
{ simp only [χ₈_apply, matrix.cons_val_zero, matrix.cons_val_one, matrix.head_cons,
matrix.cons_vec_bit0_eq_alt0, matrix.cons_vec_bit1_eq_alt1, matrix.cons_append,
matrix.cons_vec_alt0, matrix.cons_vec_alt1, int.cast_zero, int.cast_one, int.cast_neg,
zero_mul], refl },
convert_to 8 + (τ ^ 4 + 1) * (τ ^ 10 - 2 * τ ^ 8 - 2 * τ ^ 6 + 6 * τ ^ 4 + τ ^ 2 - 8) = _,
{ ring }, { rw τ_spec, norm_num } },
-- this allows us to apply `card_pow_char_pow` to our situation
have h := char.card_pow_char_pow hq ψ₈.char (ring_char FF) n hu hFF hg,
rw [card, ← hchar, hχ, one_mul, ← hc, ← nat.cast_pow (ring_char F), ← hc] at h,
-- finally, we change `2` to `8` on the left hand side
convert_to (8 : F) ^ (fintype.card F / 2) = _,
{ rw [(by norm_num : (8 : F) = 2 ^ 2 * 2), mul_pow,
(finite_field.is_square_iff hF $ hp2 2).mp ⟨2, pow_two 2⟩, one_mul] },
apply (algebra_map F FF).injective,
simp only [map_pow, map_bit0, map_one, ring_hom.map_int_cast],
convert h, norm_num,
end
end gauss_sum_two
|
aa265fdaa40cbdace112dd4d31ef6305a1a05981 | 968e2f50b755d3048175f176376eff7139e9df70 | /examples/prop_logic_lean_summary/unnamed_583.lean | 7dc15759fbf1ea39a25eadedcd8251bba58246bd | [] | no_license | gihanmarasingha/mth1001_sphinx | 190a003269ba5e54717b448302a27ca26e31d491 | 05126586cbf5786e521be1ea2ef5b4ba3c44e74a | refs/heads/master | 1,672,913,933,677 | 1,604,516,583,000 | 1,604,516,583,000 | 309,245,750 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 201 | lean | variables p : Prop
-- BEGIN
example (k : false) : ¬p :=
begin
intro h, -- This is equivalent to 'assume h : p' in mathematics.
exact k, -- We close the goal using our proof of `false`.
end
-- END |
191cb0f1dcdee44cc69161f58c797172c28da0ed | 022215fec0be87ac6243b0f4fa3cc2939361d7d0 | /src/category_theory/instances/TopCommRing/basic.lean | 40e3206dd76476ccc9d879d0cbe9df772c9ff2f4 | [
"Apache-2.0"
] | permissive | PaulGustafson/mathlib | 4aa7bc81ca971fdd7b6e50bf3a245fade2978391 | c49ac06ff9fa1371e9b6050a121df618cfd3fb80 | refs/heads/master | 1,590,798,947,521 | 1,559,220,227,000 | 1,559,220,227,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,053 | lean | import category_theory.instances.CommRing.basic
import category_theory.instances.Top.basic
import topology.instances.complex
universes u
open category_theory
namespace category_theory.instances
structure TopCommRing :=
(α : Type u)
[is_comm_ring : comm_ring α]
[is_topological_space : topological_space α]
[is_topological_ring : topological_ring α]
instance : has_coe_to_sort TopCommRing :=
{ S := Type u, coe := TopCommRing.α }
instance TopCommRing_comm_ring (R : TopCommRing) : comm_ring R := R.is_comm_ring
instance TopCommRing_topological_space (R : TopCommRing) : topological_space R := R.is_topological_space
instance TopCommRing_topological_ring (R : TopCommRing) : topological_ring R := R.is_topological_ring
instance TopCommRing_category : category TopCommRing :=
{ hom := λ R S, {f : R → S // is_ring_hom f ∧ continuous f },
id := λ R, ⟨id, by obviously⟩,
comp := λ R S T f g, ⟨g.val ∘ f.val,
begin -- TODO automate
cases f, cases g, cases f_property, cases g_property, split,
dsimp, resetI, apply_instance,
dsimp, apply continuous.comp ; assumption
end⟩ }.
namespace TopCommRing
def of (X : Type u) [comm_ring X] [topological_space X] [topological_ring X] : TopCommRing :=
{ α := X }
noncomputable example : TopCommRing := TopCommRing.of ℚ
noncomputable example : TopCommRing := TopCommRing.of ℝ
noncomputable example : TopCommRing := TopCommRing.of ℂ
/-- The forgetful functor to CommRing. -/
def forget_to_CommRing : TopCommRing ⥤ CommRing :=
{ obj := λ R, { α := R, str := instances.TopCommRing_comm_ring R },
map := λ R S f, ⟨ f.1, f.2.left ⟩ }
instance forget_to_CommRing_faithful : faithful (forget_to_CommRing) := by tidy
instance forget_to_CommRing_topological_space (R : TopCommRing) : topological_space (forget_to_CommRing.obj R) :=
R.is_topological_space
/-- The forgetful functor to Top. -/
def forget_to_Top : TopCommRing ⥤ Top :=
{ obj := λ R, { α := R, str := instances.TopCommRing_topological_space R },
map := λ R S f, ⟨ f.1, f.2.right ⟩ }
instance forget_to_Top_faithful : faithful (forget_to_Top) := by tidy
instance forget_to_Top_comm_ring (R : TopCommRing) : comm_ring (forget_to_Top.obj R) :=
R.is_comm_ring
instance forget_to_Top_topological_ring (R : TopCommRing) : topological_ring (forget_to_Top.obj R) :=
R.is_topological_ring
def forget : TopCommRing ⥤ (Type u) :=
{ obj := λ R, R,
map := λ R S f, f.1 }
instance forget_faithful : faithful (forget) := by tidy
instance forget_topological_space (R : TopCommRing) : topological_space (forget.obj R) :=
R.is_topological_space
instance forget_comm_ring (R : TopCommRing) : comm_ring (forget.obj R) :=
R.is_comm_ring
instance forget_topological_ring (R : TopCommRing) : topological_ring (forget.obj R) :=
R.is_topological_ring
def forget_to_Type_via_Top : forget_to_Top ⋙ Top.forget ≅ forget := iso.refl _
def forget_to_Type_via_CommRing : forget_to_CommRing ⋙ CommRing.forget ≅ forget := iso.refl _
end TopCommRing
end category_theory.instances
|
e472066923541d12f252965cb50d6a3cc21c222c | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/algebra/module/pointwise_pi.lean | 3a6335a55f771c71620ff5f0b1aa88de1e0adbe9 | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 1,729 | lean | /-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best
-/
import data.set.pointwise.smul
import group_theory.group_action.pi
/-!
# Pointwise actions on sets in Pi types
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file contains lemmas about pointwise actions on sets in Pi types.
## Tags
set multiplication, set addition, pointwise addition, pointwise multiplication, pi
-/
open_locale pointwise
open set
variables {K ι : Type*} {R : ι → Type*}
@[to_additive]
lemma smul_pi_subset [∀ i, has_smul K (R i)] (r : K) (s : set ι) (t : Π i, set (R i)) :
r • pi s t ⊆ pi s (r • t) :=
begin
rintros x ⟨y, h, rfl⟩ i hi,
exact smul_mem_smul_set (h i hi),
end
@[to_additive]
lemma smul_univ_pi [∀ i, has_smul K (R i)] (r : K) (t : Π i, set (R i)) :
r • pi (univ : set ι) t = pi (univ : set ι) (r • t) :=
subset.antisymm (smul_pi_subset _ _ _) $ λ x h, begin
refine ⟨λ i, classical.some (h i $ set.mem_univ _), λ i hi, _, funext $ λ i, _⟩,
{ exact (classical.some_spec (h i _)).left, },
{ exact (classical.some_spec (h i _)).right, },
end
@[to_additive]
lemma smul_pi [group K] [∀ i, mul_action K (R i)] (r : K) (S : set ι) (t : Π i, set (R i)) :
r • S.pi t = S.pi (r • t) :=
subset.antisymm (smul_pi_subset _ _ _) $ λ x h,
⟨r⁻¹ • x, λ i hiS, mem_smul_set_iff_inv_smul_mem.mp (h i hiS), smul_inv_smul _ _⟩
lemma smul_pi₀ [group_with_zero K] [∀ i, mul_action K (R i)] {r : K} (S : set ι)
(t : Π i, set (R i)) (hr : r ≠ 0) : r • S.pi t = S.pi (r • t) :=
smul_pi (units.mk0 r hr) S t
|
3ecf30c2c8963bd5597d6dba685da89c1e71c39a | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/data/mv_polynomial/equiv.lean | a28475c1f03281131114bcc45fdd96b2d7358f58 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 18,793 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import data.mv_polynomial.rename
import data.polynomial.algebra_map
import data.mv_polynomial.variables
import data.finsupp.fin
import logic.equiv.fin
import algebra.big_operators.fin
/-!
# Equivalences between polynomial rings
This file establishes a number of equivalences between polynomial rings,
based on equivalences between the underlying types.
## Notation
As in other polynomial files, we typically use the notation:
+ `σ : Type*` (indexing the variables)
+ `R : Type*` `[comm_semiring R]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `mv_polynomial σ R` which mathematicians might call `X^s`
+ `a : R`
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : mv_polynomial σ R`
## Tags
equivalence, isomorphism, morphism, ring hom, hom
-/
noncomputable theory
open_locale classical big_operators polynomial
open set function finsupp add_monoid_algebra
universes u v w x
variables {R : Type u} {S₁ : Type v} {S₂ : Type w} {S₃ : Type x}
namespace mv_polynomial
variables {σ : Type*} {a a' a₁ a₂ : R} {e : ℕ} {s : σ →₀ ℕ}
section equiv
variables (R) [comm_semiring R]
/--
The ring isomorphism between multivariable polynomials in a single variable and
polynomials over the ground ring.
-/
@[simps]
def punit_alg_equiv : mv_polynomial punit R ≃ₐ[R] R[X] :=
{ to_fun := eval₂ polynomial.C (λu:punit, polynomial.X),
inv_fun := polynomial.eval₂ mv_polynomial.C (X punit.star),
left_inv :=
begin
let f : R[X] →+* mv_polynomial punit R :=
(polynomial.eval₂_ring_hom mv_polynomial.C (X punit.star)),
let g : mv_polynomial punit R →+* R[X] :=
(eval₂_hom polynomial.C (λu:punit, polynomial.X)),
show ∀ p, f.comp g p = p,
apply is_id,
{ ext a, dsimp, rw [eval₂_C, polynomial.eval₂_C] },
{ rintros ⟨⟩, dsimp, rw [eval₂_X, polynomial.eval₂_X] }
end,
right_inv := assume p, polynomial.induction_on p
(assume a, by rw [polynomial.eval₂_C, mv_polynomial.eval₂_C])
(assume p q hp hq, by rw [polynomial.eval₂_add, mv_polynomial.eval₂_add, hp, hq])
(assume p n hp,
by rw [polynomial.eval₂_mul, polynomial.eval₂_pow, polynomial.eval₂_X, polynomial.eval₂_C,
eval₂_mul, eval₂_C, eval₂_pow, eval₂_X]),
map_mul' := λ _ _, eval₂_mul _ _,
map_add' := λ _ _, eval₂_add _ _,
commutes' := λ _, eval₂_C _ _ _}
section map
variables {R} (σ)
/-- If `e : A ≃+* B` is an isomorphism of rings, then so is `map e`. -/
@[simps apply]
def map_equiv [comm_semiring S₁] [comm_semiring S₂] (e : S₁ ≃+* S₂) :
mv_polynomial σ S₁ ≃+* mv_polynomial σ S₂ :=
{ to_fun := map (e : S₁ →+* S₂),
inv_fun := map (e.symm : S₂ →+* S₁),
left_inv := map_left_inverse e.left_inv,
right_inv := map_right_inverse e.right_inv,
..map (e : S₁ →+* S₂) }
@[simp] lemma map_equiv_refl :
map_equiv σ (ring_equiv.refl R) = ring_equiv.refl _ :=
ring_equiv.ext map_id
@[simp] lemma map_equiv_symm [comm_semiring S₁] [comm_semiring S₂] (e : S₁ ≃+* S₂) :
(map_equiv σ e).symm = map_equiv σ e.symm := rfl
@[simp] lemma map_equiv_trans [comm_semiring S₁] [comm_semiring S₂] [comm_semiring S₃]
(e : S₁ ≃+* S₂) (f : S₂ ≃+* S₃) :
(map_equiv σ e).trans (map_equiv σ f) = map_equiv σ (e.trans f) :=
ring_equiv.ext (map_map e f)
variables {A₁ A₂ A₃ : Type*} [comm_semiring A₁] [comm_semiring A₂] [comm_semiring A₃]
variables [algebra R A₁] [algebra R A₂] [algebra R A₃]
/-- If `e : A ≃ₐ[R] B` is an isomorphism of `R`-algebras, then so is `map e`. -/
@[simps apply]
def map_alg_equiv (e : A₁ ≃ₐ[R] A₂) :
mv_polynomial σ A₁ ≃ₐ[R] mv_polynomial σ A₂ :=
{ to_fun := map (e : A₁ →+* A₂),
..map_alg_hom (e : A₁ →ₐ[R] A₂),
..map_equiv σ (e : A₁ ≃+* A₂) }
@[simp] lemma map_alg_equiv_refl :
map_alg_equiv σ (alg_equiv.refl : A₁ ≃ₐ[R] A₁) = alg_equiv.refl :=
alg_equiv.ext map_id
@[simp] lemma map_alg_equiv_symm (e : A₁ ≃ₐ[R] A₂) :
(map_alg_equiv σ e).symm = map_alg_equiv σ e.symm := rfl
@[simp] lemma map_alg_equiv_trans (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
(map_alg_equiv σ e).trans (map_alg_equiv σ f) = map_alg_equiv σ (e.trans f) :=
alg_equiv.ext (map_map e f)
end map
section
variables (S₁ S₂ S₃)
/--
The function from multivariable polynomials in a sum of two types,
to multivariable polynomials in one of the types,
with coefficents in multivariable polynomials in the other type.
See `sum_ring_equiv` for the ring isomorphism.
-/
def sum_to_iter : mv_polynomial (S₁ ⊕ S₂) R →+* mv_polynomial S₁ (mv_polynomial S₂ R) :=
eval₂_hom (C.comp C) (λbc, sum.rec_on bc X (C ∘ X))
@[simp]
lemma sum_to_iter_C (a : R) : sum_to_iter R S₁ S₂ (C a) = C (C a) :=
eval₂_C _ _ a
@[simp]
lemma sum_to_iter_Xl (b : S₁) : sum_to_iter R S₁ S₂ (X (sum.inl b)) = X b :=
eval₂_X _ _ (sum.inl b)
@[simp]
lemma sum_to_iter_Xr (c : S₂) : sum_to_iter R S₁ S₂ (X (sum.inr c)) = C (X c) :=
eval₂_X _ _ (sum.inr c)
/--
The function from multivariable polynomials in one type,
with coefficents in multivariable polynomials in another type,
to multivariable polynomials in the sum of the two types.
See `sum_ring_equiv` for the ring isomorphism.
-/
def iter_to_sum : mv_polynomial S₁ (mv_polynomial S₂ R) →+* mv_polynomial (S₁ ⊕ S₂) R :=
eval₂_hom (eval₂_hom C (X ∘ sum.inr)) (X ∘ sum.inl)
lemma iter_to_sum_C_C (a : R) : iter_to_sum R S₁ S₂ (C (C a)) = C a :=
eq.trans (eval₂_C _ _ (C a)) (eval₂_C _ _ _)
lemma iter_to_sum_X (b : S₁) : iter_to_sum R S₁ S₂ (X b) = X (sum.inl b) :=
eval₂_X _ _ _
lemma iter_to_sum_C_X (c : S₂) : iter_to_sum R S₁ S₂ (C (X c)) = X (sum.inr c) :=
eq.trans (eval₂_C _ _ (X c)) (eval₂_X _ _ _)
variable (σ)
/-- The algebra isomorphism between multivariable polynomials in no variables
and the ground ring. -/
@[simps] def is_empty_alg_equiv [he : is_empty σ] : mv_polynomial σ R ≃ₐ[R] R :=
alg_equiv.of_alg_hom
(aeval (is_empty.elim he))
(algebra.of_id _ _)
(by { ext, simp [algebra.of_id_apply, algebra_map_eq] })
(by { ext i m, exact is_empty.elim' he i })
/-- The ring isomorphism between multivariable polynomials in no variables
and the ground ring. -/
@[simps] def is_empty_ring_equiv [he : is_empty σ] : mv_polynomial σ R ≃+* R :=
(is_empty_alg_equiv R σ).to_ring_equiv
variable {σ}
/-- A helper function for `sum_ring_equiv`. -/
@[simps]
def mv_polynomial_equiv_mv_polynomial [comm_semiring S₃]
(f : mv_polynomial S₁ R →+* mv_polynomial S₂ S₃)
(g : mv_polynomial S₂ S₃ →+* mv_polynomial S₁ R)
(hfgC : (f.comp g).comp C = C)
(hfgX : ∀n, f (g (X n)) = X n)
(hgfC : (g.comp f).comp C = C)
(hgfX : ∀n, g (f (X n)) = X n) :
mv_polynomial S₁ R ≃+* mv_polynomial S₂ S₃ :=
{ to_fun := f, inv_fun := g,
left_inv := is_id (ring_hom.comp _ _) hgfC hgfX,
right_inv := is_id (ring_hom.comp _ _) hfgC hfgX,
map_mul' := f.map_mul,
map_add' := f.map_add }
/--
The ring isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficents in multivariable polynomials in the other type.
-/
def sum_ring_equiv : mv_polynomial (S₁ ⊕ S₂) R ≃+* mv_polynomial S₁ (mv_polynomial S₂ R) :=
begin
apply @mv_polynomial_equiv_mv_polynomial R (S₁ ⊕ S₂) _ _ _ _
(sum_to_iter R S₁ S₂) (iter_to_sum R S₁ S₂),
{ refine ring_hom.ext (λ p, _),
rw [ring_hom.comp_apply],
convert hom_eq_hom ((sum_to_iter R S₁ S₂).comp ((iter_to_sum R S₁ S₂).comp C)) C _ _ p,
{ ext1 a, dsimp, rw [iter_to_sum_C_C R S₁ S₂, sum_to_iter_C R S₁ S₂] },
{ assume c, dsimp, rw [iter_to_sum_C_X R S₁ S₂, sum_to_iter_Xr R S₁ S₂] } },
{ assume b, rw [iter_to_sum_X R S₁ S₂, sum_to_iter_Xl R S₁ S₂] },
{ ext1 a, rw [ring_hom.comp_apply, ring_hom.comp_apply,
sum_to_iter_C R S₁ S₂, iter_to_sum_C_C R S₁ S₂] },
{ assume n, cases n with b c,
{ rw [sum_to_iter_Xl, iter_to_sum_X] },
{ rw [sum_to_iter_Xr, iter_to_sum_C_X] } },
end
/--
The algebra isomorphism between multivariable polynomials in a sum of two types,
and multivariable polynomials in one of the types,
with coefficents in multivariable polynomials in the other type.
-/
def sum_alg_equiv : mv_polynomial (S₁ ⊕ S₂) R ≃ₐ[R]
mv_polynomial S₁ (mv_polynomial S₂ R) :=
{ commutes' := begin
intro r,
have A : algebra_map R (mv_polynomial S₁ (mv_polynomial S₂ R)) r = (C (C r) : _), by refl,
have B : algebra_map R (mv_polynomial (S₁ ⊕ S₂) R) r = C r, by refl,
simp only [sum_ring_equiv, sum_to_iter_C, mv_polynomial_equiv_mv_polynomial_apply,
ring_equiv.to_fun_eq_coe, A, B],
end,
..sum_ring_equiv R S₁ S₂ }
section
-- this speeds up typeclass search in the lemma below
local attribute [instance, priority 2000] is_scalar_tower.right
/--
The algebra isomorphism between multivariable polynomials in `option S₁` and
polynomials with coefficients in `mv_polynomial S₁ R`.
-/
@[simps] def option_equiv_left :
mv_polynomial (option S₁) R ≃ₐ[R] polynomial (mv_polynomial S₁ R) :=
alg_equiv.of_alg_hom
(mv_polynomial.aeval (λ o, o.elim polynomial.X (λ s, polynomial.C (X s))))
(polynomial.aeval_tower (mv_polynomial.rename some) (X none))
(by ext : 2; simp [← polynomial.C_eq_algebra_map])
(by ext i : 2; cases i; simp)
end
/--
The algebra isomorphism between multivariable polynomials in `option S₁` and
multivariable polynomials with coefficients in polynomials.
-/
def option_equiv_right : mv_polynomial (option S₁) R ≃ₐ[R] mv_polynomial S₁ R[X] :=
alg_equiv.of_alg_hom
(mv_polynomial.aeval (λ o, o.elim (C polynomial.X) X))
(mv_polynomial.aeval_tower (polynomial.aeval (X none)) (λ i, X (option.some i)))
begin
ext : 2;
simp only [mv_polynomial.algebra_map_eq, option.elim, alg_hom.coe_comp, alg_hom.id_comp,
is_scalar_tower.coe_to_alg_hom', comp_app, aeval_tower_C, polynomial.aeval_X, aeval_X,
option.elim, aeval_tower_X, alg_hom.coe_id, id.def, eq_self_iff_true, implies_true_iff],
end
begin
ext ⟨i⟩ : 2;
simp only [option.elim, alg_hom.coe_comp, comp_app, aeval_X, aeval_tower_C,
polynomial.aeval_X, alg_hom.coe_id, id.def, aeval_tower_X],
end
variables (n : ℕ)
/--
The algebra isomorphism between multivariable polynomials in `fin (n + 1)` and
polynomials over multivariable polynomials in `fin n`.
-/
def fin_succ_equiv :
mv_polynomial (fin (n + 1)) R ≃ₐ[R] polynomial (mv_polynomial (fin n) R) :=
(rename_equiv R (fin_succ_equiv n)).trans
(option_equiv_left R (fin n))
lemma fin_succ_equiv_eq :
(fin_succ_equiv R n : mv_polynomial (fin (n + 1)) R →+* polynomial (mv_polynomial (fin n) R)) =
eval₂_hom (polynomial.C.comp (C : R →+* mv_polynomial (fin n) R))
(λ i : fin (n+1), fin.cases polynomial.X (λ k, polynomial.C (X k)) i) :=
begin
ext : 2,
{ simp only [fin_succ_equiv, option_equiv_left_apply, aeval_C, alg_equiv.coe_trans,
ring_hom.coe_coe, coe_eval₂_hom, comp_app, rename_equiv_apply, eval₂_C, ring_hom.coe_comp,
rename_C],
refl },
{ intro i,
refine fin.cases _ _ i;
simp [fin_succ_equiv] }
end
@[simp] lemma fin_succ_equiv_apply (p : mv_polynomial (fin (n + 1)) R) :
fin_succ_equiv R n p =
eval₂_hom (polynomial.C.comp (C : R →+* mv_polynomial (fin n) R))
(λ i : fin (n+1), fin.cases polynomial.X (λ k, polynomial.C (X k)) i) p :=
by { rw ← fin_succ_equiv_eq, refl }
lemma fin_succ_equiv_comp_C_eq_C {R : Type u} [comm_semiring R] (n : ℕ) :
(↑(mv_polynomial.fin_succ_equiv R n).symm : polynomial (mv_polynomial (fin n) R) →+* _).comp
((polynomial.C).comp (mv_polynomial.C))
= (mv_polynomial.C : R →+* mv_polynomial (fin n.succ) R) :=
begin
refine ring_hom.ext (λ x, _),
rw ring_hom.comp_apply,
refine (mv_polynomial.fin_succ_equiv R n).injective
(trans ((mv_polynomial.fin_succ_equiv R n).apply_symm_apply _) _),
simp only [mv_polynomial.fin_succ_equiv_apply, mv_polynomial.eval₂_hom_C],
end
variables {n} {R}
lemma fin_succ_equiv_X_zero :
fin_succ_equiv R n (X 0) = polynomial.X := by simp
lemma fin_succ_equiv_X_succ {j : fin n} :
fin_succ_equiv R n (X j.succ) = polynomial.C (X j) := by simp
/-- The coefficient of `m` in the `i`-th coefficient of `fin_succ_equiv R n f` equals the
coefficient of `finsupp.cons i m` in `f`. -/
lemma fin_succ_equiv_coeff_coeff (m : fin n →₀ ℕ)
(f : mv_polynomial (fin (n + 1)) R) (i : ℕ) :
coeff m (polynomial.coeff (fin_succ_equiv R n f) i) = coeff (m.cons i) f :=
begin
induction f using mv_polynomial.induction_on' with j r p q hp hq generalizing i m,
swap,
{ simp only [(fin_succ_equiv R n).map_add, polynomial.coeff_add, coeff_add, hp, hq] },
simp only [fin_succ_equiv_apply, coe_eval₂_hom, eval₂_monomial, ring_hom.coe_comp, prod_pow,
polynomial.coeff_C_mul, coeff_C_mul, coeff_monomial,
fin.prod_univ_succ, fin.cases_zero, fin.cases_succ, ← ring_hom.map_prod, ← ring_hom.map_pow],
rw [← mul_boole, mul_comm (polynomial.X ^ j 0), polynomial.coeff_C_mul_X_pow], congr' 1,
obtain rfl | hjmi := eq_or_ne j (m.cons i),
{ simpa only [cons_zero, cons_succ, if_pos rfl, monomial_eq, C_1, one_mul, prod_pow]
using coeff_monomial m m (1:R), },
{ simp only [hjmi, if_false],
obtain hij | rfl := ne_or_eq i (j 0),
{ simp only [hij, if_false, coeff_zero] },
simp only [eq_self_iff_true, if_true],
have hmj : m ≠ j.tail, { rintro rfl, rw [cons_tail] at hjmi, contradiction },
simpa only [monomial_eq, C_1, one_mul, prod_pow, finsupp.tail_apply, if_neg hmj.symm]
using coeff_monomial m j.tail (1:R), }
end
lemma eval_eq_eval_mv_eval' (s : fin n → R) (y : R) (f : mv_polynomial (fin (n + 1)) R) :
eval (fin.cons y s : fin (n + 1) → R) f =
polynomial.eval y (polynomial.map (eval s) (fin_succ_equiv R n f)) :=
begin
-- turn this into a def `polynomial.map_alg_hom`
let φ : (mv_polynomial (fin n) R)[X] →ₐ[R] R[X] :=
{ commutes' := λ r, by { convert polynomial.map_C _, exact (eval_C _).symm },
.. polynomial.map_ring_hom (eval s) },
show aeval (fin.cons y s : fin (n + 1) → R) f =
(polynomial.aeval y).comp (φ.comp (fin_succ_equiv R n).to_alg_hom) f,
congr' 2,
apply mv_polynomial.alg_hom_ext,
rw fin.forall_fin_succ,
simp only [aeval_X, fin.cons_zero, alg_equiv.to_alg_hom_eq_coe, alg_hom.coe_comp,
polynomial.coe_aeval_eq_eval, polynomial.map_C, alg_hom.coe_mk, ring_hom.to_fun_eq_coe,
polynomial.coe_map_ring_hom, alg_equiv.coe_alg_hom, comp_app, fin_succ_equiv_apply,
eval₂_hom_X', fin.cases_zero, polynomial.map_X, polynomial.eval_X, eq_self_iff_true,
fin.cons_succ, fin.cases_succ, eval_X, polynomial.eval_C, implies_true_iff, and_self],
end
lemma coeff_eval_eq_eval_coeff (s' : fin n → R) (f : polynomial (mv_polynomial (fin n) R))
(i : ℕ) : polynomial.coeff (polynomial.map (eval s') f) i = eval s' (polynomial.coeff f i) :=
by simp only [polynomial.coeff_map]
lemma support_coeff_fin_succ_equiv {f : mv_polynomial (fin (n + 1)) R} {i : ℕ}
{m : fin n →₀ ℕ } : m ∈ (polynomial.coeff ((fin_succ_equiv R n) f) i).support
↔ (finsupp.cons i m) ∈ f.support :=
begin
apply iff.intro,
{ intro h,
simpa [←fin_succ_equiv_coeff_coeff] using h },
{ intro h,
simpa [mem_support_iff, ←fin_succ_equiv_coeff_coeff m f i] using h },
end
lemma fin_succ_equiv_support (f : mv_polynomial (fin (n + 1)) R) :
(fin_succ_equiv R n f).support = finset.image (λ m : fin (n + 1)→₀ ℕ, m 0) f.support :=
begin
ext i,
rw [polynomial.mem_support_iff, finset.mem_image, nonzero_iff_exists],
split,
{ rintro ⟨m, hm⟩,
refine ⟨cons i m, _, cons_zero _ _⟩,
rw ← support_coeff_fin_succ_equiv,
simpa using hm, },
{ rintro ⟨m, h, rfl⟩,
refine ⟨tail m, _⟩,
rwa [← coeff, ← mem_support_iff, support_coeff_fin_succ_equiv, cons_tail] },
end
lemma fin_succ_equiv_support' {f : mv_polynomial (fin (n + 1)) R} {i : ℕ} :
finset.image (finsupp.cons i) (polynomial.coeff ((fin_succ_equiv R n) f) i).support
= f.support.filter(λ m, m 0 = i) :=
begin
ext m,
rw [finset.mem_filter, finset.mem_image, mem_support_iff],
conv_lhs
{ congr,
funext,
rw [mem_support_iff, fin_succ_equiv_coeff_coeff, ne.def] },
split,
{ rintros ⟨m',⟨h, hm'⟩⟩,
simp only [←hm'],
exact ⟨h, by rw cons_zero⟩ },
{ intro h,
use tail m,
rw [← h.2, cons_tail],
simp [h.1] }
end
lemma support_fin_succ_equiv_nonempty {f : mv_polynomial (fin (n + 1)) R} (h : f ≠ 0) :
(fin_succ_equiv R n f).support.nonempty :=
begin
by_contradiction c,
simp only [finset.not_nonempty_iff_eq_empty, polynomial.support_eq_empty] at c,
have t'' : (fin_succ_equiv R n f) ≠ 0,
{ let ii := (fin_succ_equiv R n).symm,
have h' : f = 0 :=
calc f = ii (fin_succ_equiv R n f) : by simpa only [ii, ←alg_equiv.inv_fun_eq_symm]
using ((fin_succ_equiv R n).left_inv f).symm
... = ii 0 : by rw c
... = 0 : by simp,
simpa [h'] using h },
simpa [c] using h,
end
lemma degree_fin_succ_equiv {f : mv_polynomial (fin (n + 1)) R} (h : f ≠ 0) :
(fin_succ_equiv R n f).degree = degree_of 0 f :=
begin
have h' : (fin_succ_equiv R n f).support.sup (λ x , x) = degree_of 0 f,
{ rw [degree_of_eq_sup, fin_succ_equiv_support f, finset.sup_image] },
rw [polynomial.degree, ← h', finset.coe_sup_of_nonempty (support_fin_succ_equiv_nonempty h),
finset.max_eq_sup_coe],
end
lemma nat_degree_fin_succ_equiv (f : mv_polynomial (fin (n + 1)) R) :
(fin_succ_equiv R n f).nat_degree = degree_of 0 f :=
begin
by_cases c : f = 0,
{ rw [c, (fin_succ_equiv R n).map_zero, polynomial.nat_degree_zero, degree_of_zero] },
{ rw [polynomial.nat_degree, degree_fin_succ_equiv (by simpa only [ne.def]) ],
simp },
end
lemma degree_of_coeff_fin_succ_equiv (p : mv_polynomial (fin (n + 1)) R) (j : fin n)
(i : ℕ) : degree_of j (polynomial.coeff (fin_succ_equiv R n p) i) ≤ degree_of j.succ p :=
begin
rw [degree_of_eq_sup, degree_of_eq_sup, finset.sup_le_iff],
intros m hm,
rw ← finsupp.cons_succ j i m,
convert finset.le_sup (support_coeff_fin_succ_equiv.1 hm),
refl,
end
end
end equiv
end mv_polynomial
|
3173664b4f92c94b1154ddfa4266cc6adf9e0c20 | dd0f5513e11c52db157d2fcc8456d9401a6cd9da | /12_Axioms.org.7.lean | c93f08f9189f27510712d40a62d46e8fc2712f4f | [] | no_license | cjmazey/lean-tutorial | ba559a49f82aa6c5848b9bf17b7389bf7f4ba645 | 381f61c9fcac56d01d959ae0fa6e376f2c4e3b34 | refs/heads/master | 1,610,286,098,832 | 1,447,124,923,000 | 1,447,124,923,000 | 43,082,433 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 581 | lean | import standard
namespace hide
-- BEGIN
structure setoid [class] (A : Type) :=
(r : A → A → Prop) (iseqv : equivalence r)
namespace setoid
infix `≈` := setoid.r
variable {A : Type}
variable [s : setoid A]
include s
theorem refl (a : A) : a ≈ a :=
and.elim_left (@setoid.iseqv A s) a
theorem symm {a b : A} : a ≈ b → b ≈ a :=
λ H, and.elim_left (and.elim_right (@setoid.iseqv A s)) a b H
theorem trans {a b c : A} : a ≈ b → b ≈ c → a ≈ c :=
λ H₁ H₂, and.elim_right (and.elim_right (@setoid.iseqv A s)) a b c H₁ H₂
end setoid
-- END
end hide
|
e6f85bcfe70f90c42a17c59b8214b069e2904f63 | e39f04f6ff425fe3b3f5e26a8998b817d1dba80f | /algebra/ordered_group.lean | 05cff0dec04fac316a39bc85fd32ba6d8f300961 | [
"Apache-2.0"
] | permissive | kristychoi/mathlib | c504b5e8f84e272ea1d8966693c42de7523bf0ec | 257fd84fe98927ff4a5ffe044f68c4e9d235cc75 | refs/heads/master | 1,586,520,722,896 | 1,544,030,145,000 | 1,544,031,933,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 24,045 | lean | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
Ordered monoids and groups.
-/
import algebra.group order.bounded_lattice tactic.basic
universe u
variable {α : Type u}
section old_structure_cmd
set_option old_structure_cmd true
/-- An ordered (additive) commutative monoid is a commutative monoid
with a partial order such that addition is an order embedding, i.e.
`a + b ≤ a + c ↔ b ≤ c`. These monoids are automatically cancellative. -/
class ordered_comm_monoid (α : Type*) extends add_comm_monoid α, partial_order α :=
(add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b)
(lt_of_add_lt_add_left : ∀ a b c : α, a + b < a + c → b < c)
/-- A canonically ordered monoid is an ordered commutative monoid
in which the ordering coincides with the divisibility relation,
which is to say, `a ≤ b` iff there exists `c` with `b = a + c`.
This is satisfied by the natural numbers, for example, but not
the integers or other ordered groups. -/
class canonically_ordered_monoid (α : Type*) extends ordered_comm_monoid α :=
(le_iff_exists_add : ∀a b:α, a ≤ b ↔ ∃c, b = a + c)
end old_structure_cmd
section ordered_comm_monoid
variables [ordered_comm_monoid α] {a b c d : α}
lemma add_le_add_left' (h : a ≤ b) : c + a ≤ c + b :=
ordered_comm_monoid.add_le_add_left a b h c
lemma add_le_add_right' (h : a ≤ b) : a + c ≤ b + c :=
add_comm c a ▸ add_comm c b ▸ add_le_add_left' h
lemma lt_of_add_lt_add_left' : a + b < a + c → b < c :=
ordered_comm_monoid.lt_of_add_lt_add_left a b c
lemma add_le_add' (h₁ : a ≤ b) (h₂ : c ≤ d) : a + c ≤ b + d :=
le_trans (add_le_add_right' h₁) (add_le_add_left' h₂)
lemma le_add_of_nonneg_right' (h : b ≥ 0) : a ≤ a + b :=
have a + b ≥ a + 0, from add_le_add_left' h,
by rwa add_zero at this
lemma le_add_of_nonneg_left' (h : b ≥ 0) : a ≤ b + a :=
have 0 + a ≤ b + a, from add_le_add_right' h,
by rwa zero_add at this
lemma lt_of_add_lt_add_right' (h : a + b < c + b) : a < c :=
lt_of_add_lt_add_left'
(show b + a < b + c, begin rw [add_comm b a, add_comm b c], assumption end)
-- here we start using properties of zero.
lemma le_add_of_nonneg_of_le' (ha : 0 ≤ a) (hbc : b ≤ c) : b ≤ a + c :=
zero_add b ▸ add_le_add' ha hbc
lemma le_add_of_le_of_nonneg' (hbc : b ≤ c) (ha : 0 ≤ a) : b ≤ c + a :=
add_zero b ▸ add_le_add' hbc ha
lemma add_nonneg' (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a + b :=
le_add_of_nonneg_of_le' ha hb
lemma add_pos_of_pos_of_nonneg' (ha : 0 < a) (hb : 0 ≤ b) : 0 < a + b :=
lt_of_lt_of_le ha $ le_add_of_nonneg_right' hb
lemma add_pos' (ha : 0 < a) (hb : 0 < b) : 0 < a + b :=
add_pos_of_pos_of_nonneg' ha $ le_of_lt hb
lemma add_pos_of_nonneg_of_pos' (ha : 0 ≤ a) (hb : 0 < b) : 0 < a + b :=
lt_of_lt_of_le hb $ le_add_of_nonneg_left' ha
lemma add_nonpos' (ha : a ≤ 0) (hb : b ≤ 0) : a + b ≤ 0 :=
zero_add (0:α) ▸ (add_le_add' ha hb)
lemma add_le_of_nonpos_of_le' (ha : a ≤ 0) (hbc : b ≤ c) : a + b ≤ c :=
zero_add c ▸ add_le_add' ha hbc
lemma add_le_of_le_of_nonpos' (hbc : b ≤ c) (ha : a ≤ 0) : b + a ≤ c :=
add_zero c ▸ add_le_add' hbc ha
lemma add_neg_of_neg_of_nonpos' (ha : a < 0) (hb : b ≤ 0) : a + b < 0 :=
lt_of_le_of_lt (add_le_of_le_of_nonpos' (le_refl _) hb) ha
lemma add_neg_of_nonpos_of_neg' (ha : a ≤ 0) (hb : b < 0) : a + b < 0 :=
lt_of_le_of_lt (add_le_of_nonpos_of_le' ha (le_refl _)) hb
lemma add_neg' (ha : a < 0) (hb : b < 0) : a + b < 0 :=
add_neg_of_nonpos_of_neg' (le_of_lt ha) hb
lemma lt_add_of_nonneg_of_lt' (ha : 0 ≤ a) (hbc : b < c) : b < a + c :=
lt_of_lt_of_le hbc $ le_add_of_nonneg_left' ha
lemma lt_add_of_lt_of_nonneg' (hbc : b < c) (ha : 0 ≤ a) : b < c + a :=
lt_of_lt_of_le hbc $ le_add_of_nonneg_right' ha
lemma lt_add_of_pos_of_lt' (ha : 0 < a) (hbc : b < c) : b < a + c :=
lt_add_of_nonneg_of_lt' (le_of_lt ha) hbc
lemma lt_add_of_lt_of_pos' (hbc : b < c) (ha : 0 < a) : b < c + a :=
lt_add_of_lt_of_nonneg' hbc (le_of_lt ha)
lemma add_lt_of_nonpos_of_lt' (ha : a ≤ 0) (hbc : b < c) : a + b < c :=
lt_of_le_of_lt (add_le_of_nonpos_of_le' ha (le_refl _)) hbc
lemma add_lt_of_lt_of_nonpos' (hbc : b < c) (ha : a ≤ 0) : b + a < c :=
lt_of_le_of_lt (add_le_of_le_of_nonpos' (le_refl _) ha) hbc
lemma add_lt_of_neg_of_lt' (ha : a < 0) (hbc : b < c) : a + b < c :=
add_lt_of_nonpos_of_lt' (le_of_lt ha) hbc
lemma add_lt_of_lt_of_neg' (hbc : b < c) (ha : a < 0) : b + a < c :=
add_lt_of_lt_of_nonpos' hbc (le_of_lt ha)
lemma add_eq_zero_iff' (ha : 0 ≤ a) (hb : 0 ≤ b) : a + b = 0 ↔ a = 0 ∧ b = 0 :=
iff.intro
(assume hab : a + b = 0,
have a ≤ 0, from hab ▸ le_add_of_le_of_nonneg' (le_refl _) hb,
have a = 0, from le_antisymm this ha,
have b ≤ 0, from hab ▸ le_add_of_nonneg_of_le' ha (le_refl _),
have b = 0, from le_antisymm this hb,
and.intro ‹a = 0› ‹b = 0›)
(assume ⟨ha', hb'⟩, by rw [ha', hb', add_zero])
lemma bit0_pos {a : α} (h : 0 < a) : 0 < bit0 a :=
add_pos' h h
end ordered_comm_monoid
namespace units
instance [monoid α] [preorder α] : preorder (units α) :=
{ le := λ a b, (a:α) ≤ b,
lt := λ a b, (a:α) < b,
le_refl := λ a, @le_refl α _ _,
le_trans := λ a b c, @le_trans α _ _ _ _,
lt_iff_le_not_le := λ a b, @lt_iff_le_not_le α _ _ _ }
@[simp] theorem coe_le_coe [monoid α] [preorder α] {a b : units α} :
(a : α) ≤ b ↔ a ≤ b := iff.rfl
@[simp] theorem coe_lt_coe [monoid α] [preorder α] {a b : units α} :
(a : α) < b ↔ a < b := iff.rfl
instance [monoid α] [partial_order α] : partial_order (units α) :=
{ le_antisymm := λ a b h₁ h₂, ext $ le_antisymm h₁ h₂, ..units.preorder }
instance [monoid α] [linear_order α] : linear_order (units α) :=
{ le_total := λ a b, @le_total α _ _ _, ..units.partial_order }
instance [monoid α] [decidable_linear_order α] : decidable_linear_order (units α) :=
{ decidable_le := by apply_instance,
decidable_lt := by apply_instance,
decidable_eq := by apply_instance,
..units.linear_order }
theorem max_coe [monoid α] [decidable_linear_order α] {a b : units α} :
(↑(max a b) : α) = max a b :=
by by_cases a ≤ b; simp [max, h]
theorem min_coe [monoid α] [decidable_linear_order α] {a b : units α} :
(↑(min a b) : α) = min a b :=
by by_cases a ≤ b; simp [min, h]
end units
namespace with_zero
open lattice
instance [partial_order α] : partial_order (with_zero α) := with_bot.partial_order
instance [partial_order α] : order_bot (with_zero α) := with_bot.order_bot
instance [lattice α] : lattice (with_zero α) := with_bot.lattice
instance [linear_order α] : linear_order (with_zero α) := with_bot.linear_order
instance [decidable_linear_order α] :
decidable_linear_order (with_zero α) := with_bot.decidable_linear_order
def ordered_comm_monoid [ordered_comm_monoid α]
(zero_le : ∀ a : α, 0 ≤ a) : ordered_comm_monoid (with_zero α) :=
begin
suffices, refine {
add_le_add_left := this,
..with_zero.partial_order,
..with_zero.add_comm_monoid, .. },
{ intros a b c h,
have h' := lt_iff_le_not_le.1 h,
rw lt_iff_le_not_le at ⊢,
refine ⟨λ b h₂, _, λ h₂, h'.2 $ this _ _ h₂ _⟩,
cases h₂, cases c with c,
{ cases h'.2 (this _ _ bot_le a) },
{ refine ⟨_, rfl, _⟩,
cases a with a,
{ exact with_bot.some_le_some.1 h'.1 },
{ exact le_of_lt (lt_of_add_lt_add_left' $
with_bot.some_lt_some.1 h), } } },
{ intros a b h c ca h₂,
cases b with b,
{ rw le_antisymm h bot_le at h₂,
exact ⟨_, h₂, le_refl _⟩ },
cases a with a,
{ change c + 0 = some ca at h₂,
simp at h₂, simp [h₂],
exact ⟨_, rfl, by simpa using add_le_add_left' (zero_le b)⟩ },
{ simp at h,
cases c with c; change some _ = _ at h₂;
simp [-add_comm] at h₂; subst ca; refine ⟨_, rfl, _⟩,
{ exact h },
{ exact add_le_add_left' h } } }
end
end with_zero
namespace with_top
open lattice
instance [add_semigroup α] : add_semigroup (with_top α) :=
{ add := λ o₁ o₂, o₁.bind (λ a, o₂.map (λ b, a + b)),
..@additive.add_semigroup _ $ @with_zero.semigroup (multiplicative α) _ }
lemma coe_add [add_semigroup α] {a b : α} : ((a + b : α) : with_top α) = a + b := rfl
instance [add_comm_semigroup α] : add_comm_semigroup (with_top α) :=
{ ..@additive.add_comm_semigroup _ $
@with_zero.comm_semigroup (multiplicative α) _ }
instance [add_monoid α] : add_monoid (with_top α) :=
{ zero := some 0,
add := (+),
..@additive.add_monoid _ $ @with_zero.monoid (multiplicative α) _ }
instance [add_comm_monoid α] : add_comm_monoid (with_top α) :=
{ zero := 0,
add := (+),
..@additive.add_comm_monoid _ $
@with_zero.comm_monoid (multiplicative α) _ }
instance [ordered_comm_monoid α] : ordered_comm_monoid (with_top α) :=
begin
suffices, refine {
add_le_add_left := this,
..with_top.partial_order,
..with_top.add_comm_monoid, ..},
{ intros a b c h,
refine ⟨λ c h₂, _, λ h₂, h.2 $ this _ _ h₂ _⟩,
cases h₂, cases a with a,
{ exact (not_le_of_lt h).elim le_top },
cases b with b,
{ exact (not_le_of_lt h).elim le_top },
{ exact ⟨_, rfl, le_of_lt (lt_of_add_lt_add_left' $
with_top.some_lt_some.1 h)⟩ } },
{ intros a b h c cb h₂,
cases c with c, {cases h₂},
cases b with b; cases h₂,
cases a with a, {cases le_antisymm h le_top},
simp at h,
exact ⟨_, rfl, add_le_add_left' h⟩, }
end
@[simp] lemma zero_lt_top [ordered_comm_monoid α] : (0 : with_top α) < ⊤ :=
coe_lt_top 0
@[simp] lemma zero_lt_coe [ordered_comm_monoid α] (a : α) : (0 : with_top α) < a ↔ 0 < a :=
coe_lt_coe
@[simp] lemma add_top [ordered_comm_monoid α] : ∀{a : with_top α}, a + ⊤ = ⊤
| none := rfl
| (some a) := rfl
@[simp] lemma top_add [ordered_comm_monoid α] {a : with_top α} : ⊤ + a = ⊤ := rfl
lemma add_eq_top [ordered_comm_monoid α] (a b : with_top α) : a + b = ⊤ ↔ a = ⊤ ∨ b = ⊤ :=
by cases a; cases b; simp [none_eq_top, some_eq_coe, coe_add.symm]
instance [canonically_ordered_monoid α] : canonically_ordered_monoid (with_top α) :=
{ le_iff_exists_add := assume a b,
match a, b with
| a, none := show a ≤ ⊤ ↔ ∃c, ⊤ = a + c, by simp; refine ⟨⊤, _⟩; cases a; refl
| (some a), (some b) := show (a:with_top α) ≤ ↑b ↔ ∃c:with_top α, ↑b = ↑a + c,
begin
simp [canonically_ordered_monoid.le_iff_exists_add, -add_comm],
split,
{ rintro ⟨c, rfl⟩, refine ⟨c, _⟩, simp [coe_add] },
{ exact assume h, match b, h with _, ⟨some c, rfl⟩ := ⟨_, rfl⟩ end }
end
| none, some b := show (⊤ : with_top α) ≤ b ↔ ∃c:with_top α, ↑b = ⊤ + c, by simp
end,
.. with_top.ordered_comm_monoid }
end with_top
namespace with_bot
open lattice
instance [add_semigroup α] : add_semigroup (with_bot α) := with_top.add_semigroup
instance [add_comm_semigroup α] : add_comm_semigroup (with_bot α) := with_top.add_comm_semigroup
instance [add_monoid α] : add_monoid (with_bot α) := with_top.add_monoid
instance [add_comm_monoid α] : add_comm_monoid (with_bot α) := with_top.add_comm_monoid
instance [ordered_comm_monoid α] : ordered_comm_monoid (with_bot α) :=
begin
suffices, refine {
add_le_add_left := this,
..with_bot.partial_order,
..with_bot.add_comm_monoid, ..},
{ intros a b c h,
have h' := h,
rw lt_iff_le_not_le at h' ⊢,
refine ⟨λ b h₂, _, λ h₂, h'.2 $ this _ _ h₂ _⟩,
cases h₂, cases a with a,
{ exact (not_le_of_lt h).elim bot_le },
cases c with c,
{ exact (not_le_of_lt h).elim bot_le },
{ exact ⟨_, rfl, le_of_lt (lt_of_add_lt_add_left' $
with_bot.some_lt_some.1 h)⟩ } },
{ intros a b h c ca h₂,
cases c with c, {cases h₂},
cases a with a; cases h₂,
cases b with b, {cases le_antisymm h bot_le},
simp at h,
exact ⟨_, rfl, add_le_add_left' h⟩, }
end
@[simp] lemma coe_zero [add_monoid α] : ((0 : α) : with_bot α) = 0 := rfl
@[simp] lemma coe_add [add_semigroup α] (a b : α) : ((a + b : α) : with_bot α) = a + b := rfl
@[simp] lemma bot_add [ordered_comm_monoid α] (a : with_bot α) : ⊥ + a = ⊥ := rfl
@[simp] lemma add_bot [ordered_comm_monoid α] (a : with_bot α) : a + ⊥ = ⊥ := by cases a; refl
instance has_one [has_one α] : has_one (with_bot α) := ⟨(1 : α)⟩
end with_bot
section canonically_ordered_monoid
variables [canonically_ordered_monoid α] {a b c d : α}
lemma le_iff_exists_add : a ≤ b ↔ ∃c, b = a + c :=
canonically_ordered_monoid.le_iff_exists_add a b
@[simp] lemma zero_le (a : α) : 0 ≤ a := le_iff_exists_add.mpr ⟨a, by simp⟩
@[simp] lemma add_eq_zero_iff : a + b = 0 ↔ a = 0 ∧ b = 0 :=
add_eq_zero_iff' (zero_le _) (zero_le _)
@[simp] lemma le_zero_iff_eq : a ≤ 0 ↔ a = 0 :=
iff.intro
(assume h, le_antisymm h (zero_le a))
(assume h, h ▸ le_refl a)
protected lemma zero_lt_iff_ne_zero : 0 < a ↔ a ≠ 0 :=
iff.intro ne_of_gt $ assume hne, lt_of_le_of_ne (zero_le _) hne.symm
lemma le_add_left (h : a ≤ c) : a ≤ b + c :=
calc a = 0 + a : by simp
... ≤ b + c : add_le_add' (zero_le _) h
lemma le_add_right (h : a ≤ b) : a ≤ b + c :=
calc a = a + 0 : by simp
... ≤ b + c : add_le_add' h (zero_le _)
instance with_zero.canonically_ordered_monoid :
canonically_ordered_monoid (with_zero α) :=
{ le_iff_exists_add := λ a b, begin
cases a with a,
{ exact iff_of_true lattice.bot_le ⟨b, (zero_add b).symm⟩ },
cases b with b,
{ exact iff_of_false
(mt (le_antisymm lattice.bot_le) (by simp))
(λ ⟨c, h⟩, by cases c; cases h) },
{ simp [le_iff_exists_add, -add_comm],
split; intro h; rcases h with ⟨c, h⟩,
{ exact ⟨some c, congr_arg some h⟩ },
{ cases c; cases h,
{ exact ⟨_, (add_zero _).symm⟩ },
{ exact ⟨_, rfl⟩ } } }
end,
..with_zero.ordered_comm_monoid zero_le }
end canonically_ordered_monoid
instance ordered_cancel_comm_monoid.to_ordered_comm_monoid
[H : ordered_cancel_comm_monoid α] : ordered_comm_monoid α :=
{ lt_of_add_lt_add_left := @lt_of_add_lt_add_left _ _, ..H }
section ordered_cancel_comm_monoid
variables [ordered_cancel_comm_monoid α] {a b c : α}
@[simp] lemma add_le_add_iff_left (a : α) {b c : α} : a + b ≤ a + c ↔ b ≤ c :=
⟨le_of_add_le_add_left, λ h, add_le_add_left h _⟩
@[simp] lemma add_le_add_iff_right (c : α) : a + c ≤ b + c ↔ a ≤ b :=
add_comm c a ▸ add_comm c b ▸ add_le_add_iff_left c
@[simp] lemma add_lt_add_iff_left (a : α) {b c : α} : a + b < a + c ↔ b < c :=
⟨lt_of_add_lt_add_left, λ h, add_lt_add_left h _⟩
@[simp] lemma add_lt_add_iff_right (c : α) : a + c < b + c ↔ a < b :=
add_comm c a ▸ add_comm c b ▸ add_lt_add_iff_left c
@[simp] lemma le_add_iff_nonneg_right (a : α) {b : α} : a ≤ a + b ↔ 0 ≤ b :=
have a + 0 ≤ a + b ↔ 0 ≤ b, from add_le_add_iff_left a,
by rwa add_zero at this
@[simp] lemma le_add_iff_nonneg_left (a : α) {b : α} : a ≤ b + a ↔ 0 ≤ b :=
by rw [add_comm, le_add_iff_nonneg_right]
@[simp] lemma lt_add_iff_pos_right (a : α) {b : α} : a < a + b ↔ 0 < b :=
have a + 0 < a + b ↔ 0 < b, from add_lt_add_iff_left a,
by rwa add_zero at this
@[simp] lemma lt_add_iff_pos_left (a : α) {b : α} : a < b + a ↔ 0 < b :=
by rw [add_comm, lt_add_iff_pos_right]
lemma add_eq_zero_iff_eq_zero_of_nonneg
(ha : 0 ≤ a) (hb : 0 ≤ b) : a + b = 0 ↔ a = 0 ∧ b = 0 :=
⟨λ hab : a + b = 0,
by split; apply le_antisymm; try {assumption};
rw ← hab; simp [ha, hb],
λ ⟨ha', hb'⟩, by rw [ha', hb', add_zero]⟩
lemma with_top.add_lt_add_iff_left :
∀{a b c : with_top α}, a < ⊤ → (a + c < a + b ↔ c < b)
| none := assume b c h, (lt_irrefl ⊤ h).elim
| (some a) :=
begin
assume b c h,
cases b; cases c;
simp [with_top.none_eq_top, with_top.some_eq_coe, with_top.coe_lt_top, with_top.coe_lt_coe],
{ rw [← with_top.coe_add], exact with_top.coe_lt_top _ },
{ rw [← with_top.coe_add, ← with_top.coe_add, with_top.coe_lt_coe],
exact add_lt_add_iff_left _ }
end
end ordered_cancel_comm_monoid
section ordered_comm_group
variables [ordered_comm_group α] {a b c : α}
@[simp] lemma neg_le_neg_iff : -a ≤ -b ↔ b ≤ a :=
have a + b + -a ≤ a + b + -b ↔ -a ≤ -b, from add_le_add_iff_left _,
by simp at this; simp [this]
lemma neg_le : -a ≤ b ↔ -b ≤ a :=
have -a ≤ -(-b) ↔ -b ≤ a, from neg_le_neg_iff,
by rwa neg_neg at this
lemma le_neg : a ≤ -b ↔ b ≤ -a :=
have -(-a) ≤ -b ↔ b ≤ -a, from neg_le_neg_iff,
by rwa neg_neg at this
@[simp] lemma neg_nonpos : -a ≤ 0 ↔ 0 ≤ a :=
have -a ≤ -0 ↔ 0 ≤ a, from neg_le_neg_iff,
by rwa neg_zero at this
@[simp] lemma neg_nonneg : 0 ≤ -a ↔ a ≤ 0 :=
have -0 ≤ -a ↔ a ≤ 0, from neg_le_neg_iff,
by rwa neg_zero at this
@[simp] lemma neg_lt_neg_iff : -a < -b ↔ b < a :=
have a + b + -a < a + b + -b ↔ -a < -b, from add_lt_add_iff_left _,
by simp at this; simp [this]
lemma neg_lt_zero : -a < 0 ↔ 0 < a :=
have -a < -0 ↔ 0 < a, from neg_lt_neg_iff,
by rwa neg_zero at this
lemma neg_pos : 0 < -a ↔ a < 0 :=
have -0 < -a ↔ a < 0, from neg_lt_neg_iff,
by rwa neg_zero at this
lemma neg_lt : -a < b ↔ -b < a :=
have -a < -(-b) ↔ -b < a, from neg_lt_neg_iff,
by rwa neg_neg at this
lemma lt_neg : a < -b ↔ b < -a :=
have -(-a) < -b ↔ b < -a, from neg_lt_neg_iff,
by rwa neg_neg at this
lemma sub_le_sub_iff_left (a : α) {b c : α} : a - b ≤ a - c ↔ c ≤ b :=
(add_le_add_iff_left _).trans neg_le_neg_iff
lemma sub_le_sub_iff_right (c : α) : a - c ≤ b - c ↔ a ≤ b :=
add_le_add_iff_right _
lemma sub_lt_sub_iff_left (a : α) {b c : α} : a - b < a - c ↔ c < b :=
(add_lt_add_iff_left _).trans neg_lt_neg_iff
lemma sub_lt_sub_iff_right (c : α) : a - c < b - c ↔ a < b :=
add_lt_add_iff_right _
@[simp] lemma sub_nonneg : 0 ≤ a - b ↔ b ≤ a :=
have a - a ≤ a - b ↔ b ≤ a, from sub_le_sub_iff_left a,
by rwa sub_self at this
@[simp] lemma sub_nonpos : a - b ≤ 0 ↔ a ≤ b :=
have a - b ≤ b - b ↔ a ≤ b, from sub_le_sub_iff_right b,
by rwa sub_self at this
@[simp] lemma sub_pos : 0 < a - b ↔ b < a :=
have a - a < a - b ↔ b < a, from sub_lt_sub_iff_left a,
by rwa sub_self at this
@[simp] lemma sub_lt_zero : a - b < 0 ↔ a < b :=
have a - b < b - b ↔ a < b, from sub_lt_sub_iff_right b,
by rwa sub_self at this
lemma le_neg_add_iff_add_le : b ≤ -a + c ↔ a + b ≤ c :=
have -a + (a + b) ≤ -a + c ↔ a + b ≤ c, from add_le_add_iff_left _,
by rwa neg_add_cancel_left at this
lemma le_sub_iff_add_le' : b ≤ c - a ↔ a + b ≤ c :=
by rw [sub_eq_add_neg, add_comm, le_neg_add_iff_add_le]
lemma le_sub_iff_add_le : a ≤ c - b ↔ a + b ≤ c :=
by rw [le_sub_iff_add_le', add_comm]
@[simp] lemma neg_add_le_iff_le_add : -b + a ≤ c ↔ a ≤ b + c :=
have -b + a ≤ -b + (b + c) ↔ a ≤ b + c, from add_le_add_iff_left _,
by rwa neg_add_cancel_left at this
lemma sub_le_iff_le_add' : a - b ≤ c ↔ a ≤ b + c :=
by rw [sub_eq_add_neg, add_comm, neg_add_le_iff_le_add]
lemma sub_le_iff_le_add : a - c ≤ b ↔ a ≤ b + c :=
by rw [sub_le_iff_le_add', add_comm]
@[simp] lemma add_neg_le_iff_le_add : a + -c ≤ b ↔ a ≤ b + c :=
sub_le_iff_le_add
@[simp] lemma add_neg_le_iff_le_add' : a + -b ≤ c ↔ a ≤ b + c :=
sub_le_iff_le_add'
lemma neg_add_le_iff_le_add' : -c + a ≤ b ↔ a ≤ b + c :=
by rw [neg_add_le_iff_le_add, add_comm]
@[simp] lemma neg_le_sub_iff_le_add : -b ≤ a - c ↔ c ≤ a + b :=
le_sub_iff_add_le.trans neg_add_le_iff_le_add'
lemma neg_le_sub_iff_le_add' : -a ≤ b - c ↔ c ≤ a + b :=
by rw [neg_le_sub_iff_le_add, add_comm]
lemma sub_le : a - b ≤ c ↔ a - c ≤ b :=
sub_le_iff_le_add'.trans sub_le_iff_le_add.symm
theorem le_sub : a ≤ b - c ↔ c ≤ b - a :=
le_sub_iff_add_le'.trans le_sub_iff_add_le.symm
@[simp] lemma lt_neg_add_iff_add_lt : b < -a + c ↔ a + b < c :=
have -a + (a + b) < -a + c ↔ a + b < c, from add_lt_add_iff_left _,
by rwa neg_add_cancel_left at this
lemma lt_sub_iff_add_lt' : b < c - a ↔ a + b < c :=
by rw [sub_eq_add_neg, add_comm, lt_neg_add_iff_add_lt]
lemma lt_sub_iff_add_lt : a < c - b ↔ a + b < c :=
by rw [lt_sub_iff_add_lt', add_comm]
@[simp] lemma neg_add_lt_iff_lt_add : -b + a < c ↔ a < b + c :=
have -b + a < -b + (b + c) ↔ a < b + c, from add_lt_add_iff_left _,
by rwa neg_add_cancel_left at this
lemma sub_lt_iff_lt_add' : a - b < c ↔ a < b + c :=
by rw [sub_eq_add_neg, add_comm, neg_add_lt_iff_lt_add]
lemma sub_lt_iff_lt_add : a - c < b ↔ a < b + c :=
by rw [sub_lt_iff_lt_add', add_comm]
lemma neg_add_lt_iff_lt_add_right : -c + a < b ↔ a < b + c :=
by rw [neg_add_lt_iff_lt_add, add_comm]
@[simp] lemma neg_lt_sub_iff_lt_add : -b < a - c ↔ c < a + b :=
lt_sub_iff_add_lt.trans neg_add_lt_iff_lt_add_right
lemma neg_lt_sub_iff_lt_add' : -a < b - c ↔ c < a + b :=
by rw [neg_lt_sub_iff_lt_add, add_comm]
lemma sub_lt : a - b < c ↔ a - c < b :=
sub_lt_iff_lt_add'.trans sub_lt_iff_lt_add.symm
theorem lt_sub : a < b - c ↔ c < b - a :=
lt_sub_iff_add_lt'.trans lt_sub_iff_add_lt.symm
lemma sub_le_self_iff (a : α) {b : α} : a - b ≤ a ↔ 0 ≤ b :=
sub_le_iff_le_add'.trans (le_add_iff_nonneg_left _)
lemma sub_lt_self_iff (a : α) {b : α} : a - b < a ↔ 0 < b :=
sub_lt_iff_lt_add'.trans (lt_add_iff_pos_left _)
end ordered_comm_group
set_option old_structure_cmd true
/-- This is not so much a new structure as a construction mechanism
for ordered groups, by specifying only the "positive cone" of the group. -/
class nonneg_comm_group (α : Type*) extends add_comm_group α :=
(nonneg : α → Prop)
(pos : α → Prop := λ a, nonneg a ∧ ¬ nonneg (neg a))
(pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬ nonneg (-a) . order_laws_tac)
(zero_nonneg : nonneg 0)
(add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b))
(nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0)
namespace nonneg_comm_group
variable [s : nonneg_comm_group α]
include s
@[reducible] instance to_ordered_comm_group : ordered_comm_group α :=
{ le := λ a b, nonneg (b - a),
lt := λ a b, pos (b - a),
lt_iff_le_not_le := λ a b, by simp; rw [pos_iff]; simp,
le_refl := λ a, by simp [zero_nonneg],
le_trans := λ a b c nab nbc, by simp [-sub_eq_add_neg];
rw ← sub_add_sub_cancel; exact add_nonneg nbc nab,
le_antisymm := λ a b nab nba, eq_of_sub_eq_zero $
nonneg_antisymm nba (by rw neg_sub; exact nab),
add_le_add_left := λ a b nab c, by simpa [(≤), preorder.le] using nab,
add_lt_add_left := λ a b nab c, by simpa [(<), preorder.lt] using nab, ..s }
theorem nonneg_def {a : α} : nonneg a ↔ 0 ≤ a :=
show _ ↔ nonneg _, by simp
theorem pos_def {a : α} : pos a ↔ 0 < a :=
show _ ↔ pos _, by simp
theorem not_zero_pos : ¬ pos (0 : α) :=
mt pos_def.1 (lt_irrefl _)
theorem zero_lt_iff_nonneg_nonneg {a : α} :
0 < a ↔ nonneg a ∧ ¬ nonneg (-a) :=
pos_def.symm.trans (pos_iff α _)
theorem nonneg_total_iff :
(∀ a : α, nonneg a ∨ nonneg (-a)) ↔
(∀ a b : α, a ≤ b ∨ b ≤ a) :=
⟨λ h a b, by have := h (b - a); rwa [neg_sub] at this,
λ h a, by rw [nonneg_def, nonneg_def, neg_nonneg]; apply h⟩
def to_decidable_linear_ordered_comm_group
[decidable_pred (@nonneg α _)]
(nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a))
: decidable_linear_ordered_comm_group α :=
{ le := (≤),
lt := (<),
lt_iff_le_not_le := @lt_iff_le_not_le _ _,
le_refl := @le_refl _ _,
le_trans := @le_trans _ _,
le_antisymm := @le_antisymm _ _,
le_total := nonneg_total_iff.1 nonneg_total,
decidable_le := by apply_instance,
decidable_eq := by apply_instance,
decidable_lt := by apply_instance,
..@nonneg_comm_group.to_ordered_comm_group _ s }
end nonneg_comm_group
|
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