feat: Going to const-dest fields commute timeorder
This commit is contained in:
jstoobysmith
parent
7ee877af55
commit
b98d89fb0d
4 changed files with 288 additions and 234 deletions
|
|
@ -6,65 +6,20 @@ Authors: Joseph Tooby-Smith
|
|||
import Mathlib.LinearAlgebra.PiTensorProduct
|
||||
import Mathlib.Tactic.Polyrith
|
||||
import Mathlib.Tactic.Linarith
|
||||
import HepLean.Mathematics.Fin
|
||||
/-!
|
||||
# List lemmas
|
||||
|
||||
|
||||
|
||||
-/
|
||||
namespace HepLean.List
|
||||
|
||||
open Fin
|
||||
open HepLean
|
||||
variable {n : Nat}
|
||||
|
||||
def Fin.equivCons {n m : ℕ} (e : Fin n ≃ Fin m) : Fin n.succ ≃ Fin m.succ where
|
||||
toFun := Fin.cons 0 (Fin.succ ∘ e.toFun)
|
||||
invFun := Fin.cons 0 (Fin.succ ∘ e.invFun)
|
||||
left_inv i := by
|
||||
rcases Fin.eq_zero_or_eq_succ i with hi | hi
|
||||
· subst hi
|
||||
simp
|
||||
· obtain ⟨j, hj⟩ := hi
|
||||
subst hj
|
||||
simp
|
||||
right_inv i := by
|
||||
rcases Fin.eq_zero_or_eq_succ i with hi | hi
|
||||
· subst hi
|
||||
simp
|
||||
· obtain ⟨j, hj⟩ := hi
|
||||
subst hj
|
||||
simp
|
||||
|
||||
@[simp]
|
||||
lemma Fin.equivCons_trans {n m k : ℕ} (e : Fin n ≃ Fin m) (f : Fin m ≃ Fin k) :
|
||||
Fin.equivCons (e.trans f) = (Fin.equivCons e).trans (Fin.equivCons f) := by
|
||||
refine Equiv.ext_iff.mpr ?_
|
||||
intro x
|
||||
simp [Fin.equivCons]
|
||||
match x with
|
||||
| ⟨0, h⟩ => rfl
|
||||
| ⟨i + 1, h⟩ => rfl
|
||||
|
||||
@[simp]
|
||||
lemma Fin.equivCons_castOrderIso {n m : ℕ} (h : n = m) :
|
||||
(Fin.equivCons (Fin.castOrderIso h).toEquiv) = (Fin.castOrderIso (by simp [h])).toEquiv := by
|
||||
refine Equiv.ext_iff.mpr ?_
|
||||
intro x
|
||||
simp [Fin.equivCons]
|
||||
match x with
|
||||
| ⟨0, h⟩ => rfl
|
||||
| ⟨i + 1, h⟩ => rfl
|
||||
|
||||
@[simp]
|
||||
lemma Fin.equivCons_symm_succ {n m : ℕ} (e : Fin n ≃ Fin m) (i : ℕ) (hi : i + 1 < m.succ) :
|
||||
(Fin.equivCons e).symm ⟨i + 1, hi⟩ = (e.symm ⟨i , Nat.succ_lt_succ_iff.mp hi⟩).succ := by
|
||||
simp [Fin.equivCons]
|
||||
have hi : ⟨i + 1, hi⟩ = Fin.succ ⟨i, Nat.succ_lt_succ_iff.mp hi⟩ := by rfl
|
||||
rw [hi]
|
||||
rw [Fin.cons_succ]
|
||||
simp
|
||||
|
||||
def insertEquiv {α : Type} (r : α → α → Prop) [DecidableRel r] (a : α) : (l : List α) →
|
||||
/-- The equivalence between `Fin (a :: l).length` and `Fin (List.orderedInsert r a l).length`
|
||||
mapping `0` in the former to the location of `a` in the latter. -/
|
||||
def insertEquiv {α : Type} (r : α → α → Prop) [DecidableRel r] (a : α) : (l : List α) →
|
||||
Fin (a :: l).length ≃ Fin (List.orderedInsert r a l).length
|
||||
| [] => Equiv.refl _
|
||||
| b :: l => by
|
||||
|
|
@ -74,13 +29,13 @@ def insertEquiv {α : Type} (r : α → α → Prop) [DecidableRel r] (a : α) :
|
|||
let e := insertEquiv (r := r) a l
|
||||
let e2 : Fin (a :: b :: l).length ≃ Fin (b :: a :: l).length :=
|
||||
Equiv.swap ⟨0, Nat.zero_lt_succ (b :: l).length⟩ ⟨1, Nat.one_lt_succ_succ l.length⟩
|
||||
let e3 : Fin (b :: a :: l).length ≃ Fin (b :: List.orderedInsert r a l).length :=
|
||||
let e3 : Fin (b :: a :: l).length ≃ Fin (b :: List.orderedInsert r a l).length :=
|
||||
Fin.equivCons e
|
||||
let e4 : Fin (b :: List.orderedInsert r a l).length ≃ Fin (List.orderedInsert r a (b :: l)).length :=
|
||||
let e4 : Fin (b :: List.orderedInsert r a l).length ≃
|
||||
Fin (List.orderedInsert r a (b :: l)).length :=
|
||||
(Fin.castOrderIso (by
|
||||
rw [List.orderedInsert_length]
|
||||
simpa using List.orderedInsert_length r l a
|
||||
)).toEquiv
|
||||
simpa using List.orderedInsert_length r l a)).toEquiv
|
||||
exact e2.trans (e3.trans e4)
|
||||
|
||||
lemma insertEquiv_congr {α : Type} {r : α → α → Prop} [DecidableRel r] (a : α) (l l' : List α)
|
||||
|
|
@ -99,30 +54,30 @@ lemma insertEquiv_cons_neg {α : Type} {r : α → α → Prop} [DecidableRel r]
|
|||
let e := insertEquiv r a l
|
||||
let e2 : Fin (a :: b :: l).length ≃ Fin (b :: a :: l).length :=
|
||||
Equiv.swap ⟨0, Nat.zero_lt_succ (b :: l).length⟩ ⟨1, Nat.one_lt_succ_succ l.length⟩
|
||||
let e3 : Fin (b :: a :: l).length ≃ Fin (b :: List.orderedInsert r a l).length :=
|
||||
let e3 : Fin (b :: a :: l).length ≃ Fin (b :: List.orderedInsert r a l).length :=
|
||||
Fin.equivCons e
|
||||
let e4 : Fin (b :: List.orderedInsert r a l).length ≃ Fin (List.orderedInsert r a (b :: l)).length :=
|
||||
let e4 : Fin (b :: List.orderedInsert r a l).length ≃
|
||||
Fin (List.orderedInsert r a (b :: l)).length :=
|
||||
(Fin.castOrderIso (by
|
||||
rw [List.orderedInsert_length]
|
||||
simpa using List.orderedInsert_length r l a
|
||||
)).toEquiv
|
||||
simpa using List.orderedInsert_length r l a)).toEquiv
|
||||
e2.trans (e3.trans e4) := by
|
||||
simp [insertEquiv, hab]
|
||||
|
||||
lemma insertEquiv_get {α : Type} {r : α → α → Prop} [DecidableRel r] (a : α) : (l : List α) →
|
||||
lemma insertEquiv_get {α : Type} {r : α → α → Prop} [DecidableRel r] (a : α) : (l : List α) →
|
||||
(a :: l).get ∘ (insertEquiv r a l).symm = (List.orderedInsert r a l).get
|
||||
| [] => by
|
||||
simp [insertEquiv]
|
||||
| b :: l => by
|
||||
by_cases hr : r a b
|
||||
· rw [insertEquiv_cons_pos a b hr l]
|
||||
simp_all only [List.orderedInsert.eq_2, List.length_cons, OrderIso.toEquiv_symm, symm_castOrderIso,
|
||||
RelIso.coe_fn_toEquiv]
|
||||
simp_all only [List.orderedInsert.eq_2, List.length_cons, OrderIso.toEquiv_symm,
|
||||
Fin.symm_castOrderIso, RelIso.coe_fn_toEquiv]
|
||||
ext x : 1
|
||||
simp_all only [Function.comp_apply, castOrderIso_apply, List.get_eq_getElem, List.length_cons, coe_cast,
|
||||
↓reduceIte]
|
||||
simp_all only [Function.comp_apply, Fin.castOrderIso_apply, List.get_eq_getElem,
|
||||
List.length_cons, Fin.coe_cast, ↓reduceIte]
|
||||
· rw [insertEquiv_cons_neg a b hr l]
|
||||
trans (b :: List.orderedInsert r a l).get ∘ Fin.cast (by
|
||||
trans (b :: List.orderedInsert r a l).get ∘ Fin.cast (by
|
||||
rw [List.orderedInsert_length]
|
||||
simp [List.orderedInsert_length])
|
||||
· simp
|
||||
|
|
@ -137,32 +92,37 @@ lemma insertEquiv_get {α : Type} {r : α → α → Prop} [DecidableRel r] (a :
|
|||
| ⟨0, h⟩ => rfl
|
||||
| ⟨1, h⟩ => rfl
|
||||
| ⟨Nat.succ (Nat.succ x), h⟩ => rfl
|
||||
trans (a :: b :: l).get (Equiv.swap ⟨0, by simp⟩ ⟨1, by simp⟩ ((insertEquiv r a l).symm ⟨x, by simpa [List.orderedInsert_length, hr] using h⟩).succ)
|
||||
trans (a :: b :: l).get (Equiv.swap ⟨0, by simp⟩ ⟨1, by simp⟩
|
||||
((insertEquiv r a l).symm ⟨x, by simpa [List.orderedInsert_length, hr] using h⟩).succ)
|
||||
· simp
|
||||
· rw [hswap]
|
||||
simp
|
||||
simp only [List.length_cons, List.get_eq_getElem, Fin.val_succ, List.getElem_cons_succ]
|
||||
change _ = (List.orderedInsert r a l).get _
|
||||
rw [← insertEquiv_get (r := r) a l]
|
||||
simp
|
||||
· simp_all only [List.orderedInsert.eq_2, List.length_cons]
|
||||
ext x : 1
|
||||
simp_all only [Function.comp_apply, List.get_eq_getElem, List.length_cons, coe_cast, ↓reduceIte]
|
||||
simp_all only [Function.comp_apply, List.get_eq_getElem, List.length_cons, Fin.coe_cast,
|
||||
↓reduceIte]
|
||||
|
||||
def insertionSortEquiv {α : Type} (r : α → α → Prop) [DecidableRel r] : (l : List α) →
|
||||
/-- The equivalence between `Fin l.length ≃ Fin (List.insertionSort r l).length` induced by the
|
||||
sorting algorithm. -/
|
||||
def insertionSortEquiv {α : Type} (r : α → α → Prop) [DecidableRel r] : (l : List α) →
|
||||
Fin l.length ≃ Fin (List.insertionSort r l).length
|
||||
| [] => Equiv.refl _
|
||||
| a :: l =>
|
||||
(Fin.equivCons (insertionSortEquiv r l)).trans (insertEquiv r a (List.insertionSort r l))
|
||||
|
||||
|
||||
lemma insertionSortEquiv_get {α : Type} {r : α → α → Prop} [DecidableRel r] : (l : List α) →
|
||||
lemma insertionSortEquiv_get {α : Type} {r : α → α → Prop} [DecidableRel r] : (l : List α) →
|
||||
l.get ∘ (insertionSortEquiv r l).symm = (List.insertionSort r l).get
|
||||
| [] => by
|
||||
simp [insertionSortEquiv]
|
||||
| a :: l => by
|
||||
rw [insertionSortEquiv]
|
||||
change ((a :: l).get ∘ ((Fin.equivCons (insertionSortEquiv r l))).symm) ∘ (insertEquiv r a (List.insertionSort r l)).symm = _
|
||||
have hl : (a :: l).get ∘ ((Fin.equivCons (insertionSortEquiv r l))).symm = (a :: List.insertionSort r l).get := by
|
||||
change ((a :: l).get ∘ ((Fin.equivCons (insertionSortEquiv r l))).symm) ∘
|
||||
(insertEquiv r a (List.insertionSort r l)).symm = _
|
||||
have hl : (a :: l).get ∘ ((Fin.equivCons (insertionSortEquiv r l))).symm =
|
||||
(a :: List.insertionSort r l).get := by
|
||||
ext x
|
||||
match x with
|
||||
| ⟨0, h⟩ => rfl
|
||||
|
|
@ -179,5 +139,4 @@ lemma insertionSort_eq_ofFn {α : Type} {r : α → α → Prop} [DecidableRel r
|
|||
rw [insertionSortEquiv_get (r := r)]
|
||||
exact Eq.symm (List.ofFn_get (List.insertionSort r l))
|
||||
|
||||
|
||||
end HepLean.List
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue