278 lines
7.9 KiB
Plaintext
278 lines
7.9 KiB
Plaintext
import Bookshelf.LTuple.Basic
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/-! # Exercises.Enderton.Chapter0
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Useful Facts About Sets
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-/
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namespace Exercises.Enderton.Chapter0
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/--
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The following describes a so-called "generic" tuple. Like an `LTuple`, a generic
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`n`-tuple is defined recursively like so:
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`⟨x₁, ..., xₙ⟩ = ⟨⟨x₁, ..., xₙ₋₁⟩, xₙ⟩`
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Unlike `LTuple`, this tuple bends the syntax above further. For example,
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both tuples above are equivalent to:
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`⟨⟨x₁, ..., xₘ⟩, xₘ₊₁, ..., xₙ⟩`
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for some `1 ≤ m ≤ n`. This distinction is purely syntactic, and introduced
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solely to prove `lemma_0a`. In other words, `LTuple` is an always-normalized
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variant of an `GTuple`. In general, prefer it over this when working within
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Enderton's book.
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-/
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inductive GTuple : (α : Type u) → (size : Nat × Nat) → Type u where
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| nil : GTuple α (0, 0)
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| snoc : GTuple α (p, q) → LTuple α r → GTuple α (p + q, r)
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syntax (priority := high) "t[" term,* "]" : term
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macro_rules
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| `(t[]) => `(LTuple.nil)
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| `(t[$x]) => `(LTuple.snoc t[] $x)
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| `(t[$xs:term,*, $x]) => `(LTuple.snoc t[$xs,*] $x)
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syntax (priority := high) "g[" term,* "]" : term
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macro_rules
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| `(g[]) => `(GTuple.nil)
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| `(g[$x]) => `(GTuple.snoc g[] t[$x])
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| `(g[g[$xs:term,*], $ys:term,*]) => `(GTuple.snoc g[$xs,*] t[$ys,*])
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| `(g[$x, $xs:term,*]) => `(GTuple.snoc g[] t[$x, $xs,*])
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namespace GTuple
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open scoped LTuple
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/-! ## Normalization -/
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/--
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Converts an `GTuple` into "normal form".
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-/
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def norm : GTuple α (m, n) → LTuple α (m + n)
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| g[] => t[]
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| snoc is ts => LTuple.concat is.norm ts
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/--
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Normalization of an empty `GTuple` yields an empty `Tuple`.
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-/
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theorem norm_nil_eq_nil : @norm α 0 0 nil = LTuple.nil :=
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rfl
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/--
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Normalization of a pseudo-empty `GTuple` yields an empty `Tuple`.
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-/
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theorem norm_snoc_nil_nil_eq_nil : @norm α 0 0 (snoc g[] t[]) = t[] := by
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unfold norm norm
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rfl
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/--
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Normalization elimates `snoc` when the `snd` component is `nil`.
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-/
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theorem norm_snoc_nil_elim {t : GTuple α (p, q)}
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: norm (snoc t t[]) = norm t := by
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cases t with
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| nil => simp; unfold norm norm; rfl
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| snoc tf tl =>
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simp
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conv => lhs; unfold norm
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/--
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Normalization eliminates `snoc` when the `fst` component is `nil`.
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-/
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theorem norm_nil_snoc_elim {ts : LTuple α n}
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: norm (snoc g[] ts) = cast (by simp) ts := by
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unfold norm norm
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rw [LTuple.nil_concat_self_eq_self]
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/--
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Normalization distributes across `Tuple.snoc` calls.
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-/
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theorem norm_snoc_snoc_norm
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: norm (snoc as (LTuple.snoc bs b)) = LTuple.snoc (norm (snoc as bs)) b := by
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unfold norm
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rw [← LTuple.concat_snoc_snoc_concat]
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/--
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Normalizing an `GTuple` is equivalent to concatenating the normalized `fst`
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component with the `snd`.
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-/
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theorem norm_snoc_eq_concat {t₁ : GTuple α (p, q)} {t₂ : LTuple α n}
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: norm (snoc t₁ t₂) = LTuple.concat t₁.norm t₂ := by
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conv => lhs; unfold norm
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/-! ## Equality -/
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/--
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Implements Boolean equality for `GTuple α n` provided `α` has decidable
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equality.
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-/
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instance BEq [DecidableEq α] : BEq (GTuple α n) where
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beq t₁ t₂ := t₁.norm == t₂.norm
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/-! ## Basic API -/
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/--
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Returns the number of entries in the `GTuple`.
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-/
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def size (_ : GTuple α n) := n
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/--
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Returns the number of entries in the "shallowest" portion of the `GTuple`. For
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example, the length of `x[x[1, 2], 3, 4]` is `3`, despite its size being `4`.
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-/
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def length : GTuple α n → Nat
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| g[] => 0
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| snoc g[] ts => ts.size
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| snoc _ ts => 1 + ts.size
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/--
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Returns the first component of our `GTuple`. For example, the first component of
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tuple `x[x[1, 2], 3, 4]` is `t[1, 2]`.
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-/
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def fst : GTuple α (m, n) → LTuple α m
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| g[] => t[]
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| snoc ts _ => ts.norm
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/--
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Given `GTuple α (m, n)`, the `fst` component is equal to an initial segment of
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size `k` of the tuple in normal form.
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-/
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theorem self_fst_eq_norm_take (t : GTuple α (m, n)) : t.fst = t.norm.take m :=
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match t with
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| g[] => by
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unfold fst
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rw [LTuple.self_take_zero_eq_nil]
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simp
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| snoc tf tl => by
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unfold fst
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conv => rhs; unfold norm
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rw [LTuple.eq_take_concat]
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simp
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/--
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If the normal form of an `GTuple` is equal to a `Tuple`, the `fst` component
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must be a prefix of the `Tuple`.
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-/
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theorem norm_eq_fst_eq_take {t₁ : GTuple α (m, n)} {t₂ : LTuple α (m + n)}
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: (t₁.norm = t₂) → (t₁.fst = t₂.take m) := by
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intro h
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rw [self_fst_eq_norm_take, h]
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/--
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Returns the first component of our `GTuple`. For example, the first component of
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tuple `x[x[1, 2], 3, 4]` is `t[3, 4]`.
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-/
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def snd : GTuple α (m, n) → LTuple α n
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| g[] => t[]
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| snoc _ ts => ts
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end GTuple
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/-! ## Lemma 0A -/
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section
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variable {k m n : Nat}
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variable (p : 1 ≤ m)
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variable (q : n + (m - 1) = m + k)
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private lemma n_eq_succ_k : n = k + 1 := by
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let ⟨m', h⟩ := Nat.exists_eq_succ_of_ne_zero $ show m ≠ 0 by
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intro h
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have ff : 1 ≤ 0 := h ▸ p
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ring_nf at ff
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exact ff.elim
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calc
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n = n + (m - 1) - (m - 1) := by rw [Nat.add_sub_cancel]
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_ = m' + 1 + k - (m' + 1 - 1) := by rw [q, h]
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_ = m' + 1 + k - m' := by simp
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_ = 1 + k + m' - m' := by rw [Nat.add_assoc, Nat.add_comm]
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_ = 1 + k := by simp
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_ = k + 1 := by rw [Nat.add_comm]
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private lemma n_pred_eq_k : n - 1 = k := by
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have h : k + 1 - 1 = k + 1 - 1 := rfl
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conv at h => lhs; rw [←n_eq_succ_k p q]
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simp at h
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exact h
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private lemma n_geq_one : 1 ≤ n := by
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rw [n_eq_succ_k p q]
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simp
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private lemma min_comm_succ_eq : min (m + k) (k + 1) = k + 1 :=
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Nat.recOn k
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(by simp; exact p)
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(fun k' ih => calc min (m + (k' + 1)) (k' + 1 + 1)
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_ = min (m + k' + 1) (k' + 1 + 1) := by conv => rw [Nat.add_assoc]
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_ = min (m + k') (k' + 1) + 1 := Nat.min_succ_succ (m + k') (k' + 1)
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_ = k' + 1 + 1 := by rw [ih])
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private lemma n_eq_min_comm_succ : n = min (m + k) (k + 1) := by
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rw [min_comm_succ_eq p]
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exact n_eq_succ_k p q
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private lemma n_pred_m_eq_m_k : n + (m - 1) = m + k := by
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rw [←Nat.add_sub_assoc p, Nat.add_comm, Nat.add_sub_assoc (n_geq_one p q)]
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conv => lhs; rw [n_pred_eq_k p q]
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private def cast_norm : GTuple α (n, m - 1) → LTuple α (m + k)
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| xs => cast (by rw [q]) xs.norm
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private def cast_fst : GTuple α (n, m - 1) → LTuple α (k + 1)
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| xs => cast (by rw [n_eq_succ_k p q]) xs.fst
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private def cast_take (ys : LTuple α (m + k)) :=
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cast (by rw [min_comm_succ_eq p]) (ys.take (k + 1))
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/-- #### Lemma 0A
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Assume that `⟨x₁, ..., xₘ⟩ = ⟨y₁, ..., yₘ, ..., yₘ₊ₖ⟩`. Then
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`x₁ = ⟨y₁, ..., yₖ₊₁⟩`.
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-/
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theorem lemma_0a (xs : GTuple α (n, m - 1)) (ys : LTuple α (m + k))
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: (cast_norm q xs = ys) → (cast_fst p q xs = cast_take p ys) := by
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intro h
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suffices HEq
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(cast (_ : LTuple α n = LTuple α (k + 1)) xs.fst)
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(cast (_ : LTuple α (min (m + k) (k + 1)) = LTuple α (k + 1)) (LTuple.take ys (k + 1)))
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from eq_of_heq this
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congr
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· exact n_eq_min_comm_succ p q
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· rfl
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· exact n_eq_min_comm_succ p q
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· exact HEq.rfl
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· exact Eq.recOn
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(motive := fun _ h => HEq
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(_ : n + (n - 1) = n + k)
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(cast h (show n + (n - 1) = n + k by rw [n_pred_eq_k p q])))
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(show (n + (n - 1) = n + k) = (min (m + k) (k + 1) + (n - 1) = n + k) by
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rw [n_eq_min_comm_succ p q])
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HEq.rfl
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· exact n_geq_one p q
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· exact n_pred_eq_k p q
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· exact Eq.symm (n_eq_min_comm_succ p q)
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· exact n_pred_eq_k p q
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· rw [GTuple.self_fst_eq_norm_take]
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unfold cast_norm at h
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simp at h
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rw [←h, ←n_eq_succ_k p q]
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have h₂ := Eq.recOn
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(motive := fun x h => HEq
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(LTuple.take xs.norm n)
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(LTuple.take (cast (show LTuple α (n + (m - 1)) = LTuple α x by rw [h]) xs.norm) n))
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(show n + (m - 1) = m + k by rw [n_pred_m_eq_m_k p q])
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HEq.rfl
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exact Eq.recOn
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(motive := fun x h => HEq
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(cast h (LTuple.take xs.norm n))
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(LTuple.take (cast (_ : LTuple α (n + (m - 1)) = LTuple α (m + k)) xs.norm) n))
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(show LTuple α (min (n + (m - 1)) n) = LTuple α n by simp)
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h₂
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end
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end Exercises.Enderton.Chapter0 |