`Tuple`s already exist in Lean; nest inside Enderton section instead.

finite-set-exercises
Joshua Potter 2023-04-26 15:39:53 -06:00
parent 40e850951c
commit e91628a828
7 changed files with 189 additions and 184 deletions

View File

@ -1,3 +1,2 @@
import Bookshelf.List
import Bookshelf.Real
import Bookshelf.Tuple

View File

@ -1,7 +1,5 @@
import Mathlib.Data.Fintype.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.LibrarySearch
namespace List

View File

@ -1 +1,2 @@
import MathematicalIntroductionLogic.Chapter0
import MathematicalIntroductionLogic.Chapter0
import MathematicalIntroductionLogic.Tuple

View File

@ -4,174 +4,7 @@ Chapter 0
Useful Facts About Sets
-/
import Bookshelf.Tuple
/--
The following describes a so-called "generic" tuple. Like in `Bookshelf.Tuple`,
an `n`-tuple is defined recursively like so:
`⟨x₁, ..., xₙ⟩ = ⟨⟨x₁, ..., xₙ₋₁⟩, xₙ⟩`
Unlike `Bookshelf.Tuple`, a "generic" tuple bends the syntax above further. For
example, both tuples above are equivalent to:
`⟨⟨x₁, ..., xₘ⟩, xₘ₊₁, ..., xₙ⟩`
for some `1 ≤ m ≤ n`. This distinction is purely syntactic, but necessary to
prove certain theorems found in [1] (e.g. `lemma_0a`).
In general, prefer `Bookshelf.Tuple`.
-/
inductive XTuple : (α : Type u) → (size : Nat × Nat) → Type u where
| nil : XTuple α (0, 0)
| snoc : XTuple α (p, q) → Tuple α r → XTuple α (p + q, r)
syntax (priority := high) "x[" term,* "]" : term
macro_rules
| `(x[]) => `(XTuple.nil)
| `(x[$x]) => `(XTuple.snoc x[] t[$x])
| `(x[x[$xs:term,*], $ys:term,*]) => `(XTuple.snoc x[$xs,*] t[$ys,*])
| `(x[$x, $xs:term,*]) => `(XTuple.snoc x[] t[$x, $xs,*])
namespace XTuple
open scoped Tuple
-- ========================================
-- Normalization
-- ========================================
/--
Converts an `XTuple` into "normal form".
-/
def norm : XTuple α (m, n) → Tuple α (m + n)
| x[] => t[]
| snoc is ts => Tuple.concat is.norm ts
/--
Normalization of an empty `XTuple` yields an empty `Tuple`.
-/
theorem norm_nil_eq_nil : @norm α 0 0 nil = Tuple.nil :=
rfl
/--
Normalization of a pseudo-empty `XTuple` yields an empty `Tuple`.
-/
theorem norm_snoc_nil_nil_eq_nil : @norm α 0 0 (snoc x[] t[]) = t[] := by
unfold norm norm
rfl
/--
Normalization elimates `snoc` when the `snd` component is `nil`.
-/
theorem norm_snoc_nil_elim {t : XTuple α (p, q)}
: norm (snoc t t[]) = norm t := by
cases t with
| nil => simp; unfold norm norm; rfl
| snoc tf tl =>
simp
conv => lhs; unfold norm
/--
Normalization eliminates `snoc` when the `fst` component is `nil`.
-/
theorem norm_nil_snoc_elim {ts : Tuple α n}
: norm (snoc x[] ts) = cast (by simp) ts := by
unfold norm norm
rw [Tuple.nil_concat_self_eq_self]
/--
Normalization distributes across `Tuple.snoc` calls.
-/
theorem norm_snoc_snoc_norm
: norm (snoc as (Tuple.snoc bs b)) = Tuple.snoc (norm (snoc as bs)) b := by
unfold norm
rw [←Tuple.concat_snoc_snoc_concat]
/--
Normalizing an `XTuple` is equivalent to concatenating the normalized `fst`
component with the `snd`.
-/
theorem norm_snoc_eq_concat {t₁ : XTuple α (p, q)} {t₂ : Tuple α n}
: norm (snoc t₁ t₂) = Tuple.concat t₁.norm t₂ := by
conv => lhs; unfold norm
-- ========================================
-- Equality
-- ========================================
/--
Implements Boolean equality for `XTuple α n` provided `α` has decidable
equality.
-/
instance BEq [DecidableEq α] : BEq (XTuple α n) where
beq t₁ t₂ := t₁.norm == t₂.norm
-- ========================================
-- Basic API
-- ========================================
/--
Returns the number of entries in the `XTuple`.
-/
def size (_ : XTuple α n) := n
/--
Returns the number of entries in the "shallowest" portion of the `XTuple`. For
example, the length of `x[x[1, 2], 3, 4]` is `3`, despite its size being `4`.
-/
def length : XTuple α n → Nat
| x[] => 0
| snoc x[] ts => ts.size
| snoc _ ts => 1 + ts.size
/--
Returns the first component of our `XTuple`. For example, the first component of
tuple `x[x[1, 2], 3, 4]` is `t[1, 2]`.
-/
def fst : XTuple α (m, n) → Tuple α m
| x[] => t[]
| snoc ts _ => ts.norm
/--
Given `XTuple α (m, n)`, the `fst` component is equal to an initial segment of
size `k` of the tuple in normal form.
-/
theorem self_fst_eq_norm_take (t : XTuple α (m, n)) : t.fst = t.norm.take m :=
match t with
| x[] => by
unfold fst
rw [Tuple.self_take_zero_eq_nil]
simp
| snoc tf tl => by
unfold fst
conv => rhs; unfold norm
rw [Tuple.eq_take_concat]
simp
/--
If the normal form of an `XTuple` is equal to a `Tuple`, the `fst` component
must be a prefix of the `Tuple`.
-/
theorem norm_eq_fst_eq_take {t₁ : XTuple α (m, n)} {t₂ : Tuple α (m + n)}
: (t₁.norm = t₂) → (t₁.fst = t₂.take m) := by
intro h
rw [self_fst_eq_norm_take, h]
/--
Returns the first component of our `XTuple`. For example, the first component of
tuple `x[x[1, 2], 3, 4]` is `t[3, 4]`.
-/
def snd : XTuple α (m, n) → Tuple α n
| x[] => t[]
| snoc _ ts => ts
-- ========================================
-- Lemma 0A
-- ========================================
section
import MathematicalIntroductionLogic.Tuple.Generic
variable {k m n : Nat}
variable (p : 1 ≤ m)
@ -217,10 +50,10 @@ private lemma n_pred_m_eq_m_k : n + (m - 1) = m + k := by
rw [←Nat.add_sub_assoc p, Nat.add_comm, Nat.add_sub_assoc (n_geq_one p q)]
conv => lhs; rw [n_pred_eq_k p q]
private def cast_norm : XTuple α (n, m - 1) → Tuple α (m + k)
private def cast_norm : GTuple α (n, m - 1) → Tuple α (m + k)
| xs => cast (by rw [q]) xs.norm
private def cast_fst : XTuple α (n, m - 1) → Tuple α (k + 1)
private def cast_fst : GTuple α (n, m - 1) → Tuple α (k + 1)
| xs => cast (by rw [n_eq_succ_k p q]) xs.fst
private def cast_take (ys : Tuple α (m + k)) :=
@ -229,14 +62,14 @@ private def cast_take (ys : Tuple α (m + k)) :=
/--
Lemma 0A
Assume that `⟨x₁, ..., xₘ⟩ = ⟨y₁, ..., yₘ, ..., yₘ₊ₖ⟩`.
Then `x₁ = ⟨y₁, ..., yₖ₊₁⟩`.
Assume that `⟨x₁, ..., xₘ⟩ = ⟨y₁, ..., yₘ, ..., yₘ₊ₖ⟩`. Then
`x₁ = ⟨y₁, ..., yₖ₊₁⟩`.
-/
theorem lemma_0a (xs : XTuple α (n, m - 1)) (ys : Tuple α (m + k))
theorem lemma_0a (xs : GTuple α (n, m - 1)) (ys : Tuple α (m + k))
: (cast_norm q xs = ys) → (cast_fst p q xs = cast_take p ys) := by
intro h
suffices HEq
(cast (_ : Tuple α n = Tuple α (k + 1)) (fst xs))
(cast (_ : Tuple α n = Tuple α (k + 1)) xs.fst)
(cast (_ : Tuple α (min (m + k) (k + 1)) = Tuple α (k + 1)) (Tuple.take ys (k + 1)))
from eq_of_heq this
congr
@ -255,7 +88,7 @@ theorem lemma_0a (xs : XTuple α (n, m - 1)) (ys : Tuple α (m + k))
· exact n_pred_eq_k p q
· exact Eq.symm (n_eq_min_comm_succ p q)
· exact n_pred_eq_k p q
· rw [self_fst_eq_norm_take]
· rw [GTuple.self_fst_eq_norm_take]
unfold cast_norm at h
simp at h
rw [←h, ←n_eq_succ_k p q]
@ -271,7 +104,3 @@ theorem lemma_0a (xs : XTuple α (n, m - 1)) (ys : Tuple α (m + k))
(Tuple.take (cast (_ : Tuple α (n + (m - 1)) = Tuple α (m + k)) xs.norm) n))
(show Tuple α (min (n + (m - 1)) n) = Tuple α n by simp)
h₂
end
end XTuple

View File

@ -0,0 +1,2 @@
import MathematicalIntroductionLogic.Tuple.Basic
import MathematicalIntroductionLogic.Tuple.Generic

View File

@ -1,12 +1,24 @@
import Mathlib.Tactic.Ring
/--
`n`-tuples are defined recursively as such:
A representation of a tuple. In particular, `n`-tuples are defined recursively
as follows:
`⟨x₁, ..., xₙ⟩ = ⟨⟨x₁, ..., xₙ₋₁⟩, xₙ⟩`
We allow empty tuples. For a `Tuple`-like type with opposite "endian", refer to
`Mathlib.Data.Vector`.
Keep in mind a tuple in Lean already exists but it differs in two ways:
1. It is right associative. That is, `(x₁, x₂, x₃)` evaluates to
`(x₁, (x₂, x₃))` instead of `((x₁, x₂), x₃)`.
2. Internally a tuple is syntactic sugar for nested `Prod` instances. Inputs
types of `Prod` are not required to be the same meaning non-homogeneous
collections are allowed.
In general, prefer using `Prod` over this `Tuple` definition. This exists solely
for proving theorems outlined in Enderton's book.
-/
inductive Tuple : (α : Type u) → (size : Nat) → Type u where
| nil : Tuple α 0

View File

@ -0,0 +1,164 @@
import MathematicalIntroductionLogic.Tuple.Basic
/--
The following describes a so-called "generic" tuple. Like a `Tuple`, an
`n`-tuple is defined recursively like so:
`⟨x₁, ..., xₙ⟩ = ⟨⟨x₁, ..., xₙ₋₁⟩, xₙ⟩`
Unlike `Tuple`, a "generic" tuple bends the syntax above further. For example,
both tuples above are equivalent to:
`⟨⟨x₁, ..., xₘ⟩, xₘ₊₁, ..., xₙ⟩`
for some `1 ≤ m ≤ n`. This distinction is purely syntactic, but necessary to
prove certain theorems (e.g. `Chapter0.lemma_0a`). In other words, `Tuple` is an
always-normalized variant of an `GTuple`. In general, prefer it over this when
working within Enderton's book.
-/
inductive GTuple : (α : Type u) → (size : Nat × Nat) → Type u where
| nil : GTuple α (0, 0)
| snoc : GTuple α (p, q) → Tuple α r → GTuple α (p + q, r)
syntax (priority := high) "g[" term,* "]" : term
macro_rules
| `(g[]) => `(GTuple.nil)
| `(g[$x]) => `(GTuple.snoc g[] t[$x])
| `(g[g[$xs:term,*], $ys:term,*]) => `(GTuple.snoc g[$xs,*] t[$ys,*])
| `(g[$x, $xs:term,*]) => `(GTuple.snoc g[] t[$x, $xs,*])
namespace GTuple
open scoped Tuple
-- ========================================
-- Normalization
-- ========================================
/--
Converts an `GTuple` into "normal form".
-/
def norm : GTuple α (m, n) → Tuple α (m + n)
| g[] => t[]
| snoc is ts => Tuple.concat is.norm ts
/--
Normalization of an empty `GTuple` yields an empty `Tuple`.
-/
theorem norm_nil_eq_nil : @norm α 0 0 nil = Tuple.nil :=
rfl
/--
Normalization of a pseudo-empty `GTuple` yields an empty `Tuple`.
-/
theorem norm_snoc_nil_nil_eq_nil : @norm α 0 0 (snoc g[] t[]) = t[] := by
unfold norm norm
rfl
/--
Normalization elimates `snoc` when the `snd` component is `nil`.
-/
theorem norm_snoc_nil_elim {t : GTuple α (p, q)}
: norm (snoc t t[]) = norm t := by
cases t with
| nil => simp; unfold norm norm; rfl
| snoc tf tl =>
simp
conv => lhs; unfold norm
/--
Normalization eliminates `snoc` when the `fst` component is `nil`.
-/
theorem norm_nil_snoc_elim {ts : Tuple α n}
: norm (snoc g[] ts) = cast (by simp) ts := by
unfold norm norm
rw [Tuple.nil_concat_self_eq_self]
/--
Normalization distributes across `Tuple.snoc` calls.
-/
theorem norm_snoc_snoc_norm
: norm (snoc as (Tuple.snoc bs b)) = Tuple.snoc (norm (snoc as bs)) b := by
unfold norm
rw [←Tuple.concat_snoc_snoc_concat]
/--
Normalizing an `GTuple` is equivalent to concatenating the normalized `fst`
component with the `snd`.
-/
theorem norm_snoc_eq_concat {t₁ : GTuple α (p, q)} {t₂ : Tuple α n}
: norm (snoc t₁ t₂) = Tuple.concat t₁.norm t₂ := by
conv => lhs; unfold norm
-- ========================================
-- Equality
-- ========================================
/--
Implements Boolean equality for `GTuple α n` provided `α` has decidable
equality.
-/
instance BEq [DecidableEq α] : BEq (GTuple α n) where
beq t₁ t₂ := t₁.norm == t₂.norm
-- ========================================
-- Basic API
-- ========================================
/--
Returns the number of entries in the `GTuple`.
-/
def size (_ : GTuple α n) := n
/--
Returns the number of entries in the "shallowest" portion of the `GTuple`. For
example, the length of `x[x[1, 2], 3, 4]` is `3`, despite its size being `4`.
-/
def length : GTuple α n → Nat
| g[] => 0
| snoc g[] ts => ts.size
| snoc _ ts => 1 + ts.size
/--
Returns the first component of our `GTuple`. For example, the first component of
tuple `x[x[1, 2], 3, 4]` is `t[1, 2]`.
-/
def fst : GTuple α (m, n) → Tuple α m
| g[] => t[]
| snoc ts _ => ts.norm
/--
Given `GTuple α (m, n)`, the `fst` component is equal to an initial segment of
size `k` of the tuple in normal form.
-/
theorem self_fst_eq_norm_take (t : GTuple α (m, n)) : t.fst = t.norm.take m :=
match t with
| g[] => by
unfold fst
rw [Tuple.self_take_zero_eq_nil]
simp
| snoc tf tl => by
unfold fst
conv => rhs; unfold norm
rw [Tuple.eq_take_concat]
simp
/--
If the normal form of an `GTuple` is equal to a `Tuple`, the `fst` component
must be a prefix of the `Tuple`.
-/
theorem norm_eq_fst_eq_take {t₁ : GTuple α (m, n)} {t₂ : Tuple α (m + n)}
: (t₁.norm = t₂) → (t₁.fst = t₂.take m) := by
intro h
rw [self_fst_eq_norm_take, h]
/--
Returns the first component of our `GTuple`. For example, the first component of
tuple `x[x[1, 2], 3, 4]` is `t[3, 4]`.
-/
def snd : GTuple α (m, n) → Tuple α n
| g[] => t[]
| snoc _ ts => ts
end GTuple