110 lines
2.8 KiB
Plaintext
110 lines
2.8 KiB
Plaintext
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/-
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# References
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1. Enderton, Herbert B. A Mathematical Introduction to Logic. 2nd ed. San Diego:
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Harcourt/Academic Press, 2001.
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-/
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import Bookshelf.Tuple
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/--
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The following describes a so-called "generic" tuple. Like in `Bookshelf.Tuple`,
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an `n`-tuple is defined recursively like so:
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`⟨x₁, ..., xₙ⟩ = ⟨⟨x₁, ..., xₙ₋₁⟩, xₙ⟩`
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Unlike `Bookshelf.Tuple`, a "generic" tuple bends the syntax above further. For
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example, 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, but necessary to
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prove certain theorems found in [1] (e.g. `lemma_0a`).
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In general, prefer `Bookshelf.Tuple`.
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-/
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inductive XTuple : (α : Type u) → (size : Nat × Nat) → Type u where
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| nil : XTuple α (0, 0)
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| snoc : XTuple α (p, q) → Tuple α r → XTuple α (p + q, r)
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syntax (priority := high) "x[" term,* "]" : term
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macro_rules
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| `(x[]) => `(XTuple.nil)
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| `(x[$x]) => `(XTuple.snoc x[] t[$x])
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| `(x[x[$xs:term,*], $ys:term,*]) => `(XTuple.snoc x[$xs,*] t[$ys,*])
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| `(x[$x, $xs:term,*]) => `(XTuple.snoc x[] t[$x, $xs,*])
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namespace XTuple
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/--
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Converts an `XTuple` into "normal form".
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-/
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def norm : XTuple α (m, n) → Tuple α (m + n)
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| x[] => t[]
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| snoc x[] ts => cast (by simp) ts
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| snoc is ts => is.norm.concat ts
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/--
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Casts a tuple indexed by `m` to one indexed by `n`.
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-/
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theorem cast_eq_size : (m = n) → (Tuple α m = Tuple α n) :=
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fun h => by rw [h]
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/--
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Implements Boolean equality for `XTuple α n` provided `α` has decidable
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equality.
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-/
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instance BEq [DecidableEq α] : BEq (XTuple α n) where
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beq t₁ t₂ := t₁.norm == t₂.norm
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/--
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Returns the number of entries in the `XTuple`.
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-/
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def size (_ : XTuple α n) := n
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/--
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Returns the number of entries in the "shallowest" portion of the `XTuple`. 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 : XTuple α n → Nat
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| x[] => 0
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| snoc x[] 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 `XTuple`. 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 first : XTuple α (m, n) → 1 ≤ m → Tuple α m
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| snoc ts _, _ => ts.norm
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section
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variable {k m n : Nat}
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variable (p : n + (m - 1) = m + k)
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variable (qₙ : 1 ≤ n)
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variable (qₘ : 1 ≤ m)
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namespace Lemma_0a
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lemma aux1 : n = k + 1 := sorry
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lemma aux2 : 1 ≤ m → 1 ≤ k + 1 ∧ k + 1 ≤ m + k := sorry
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end Lemma_0a
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open Lemma_0a
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/--[1]
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Assume that ⟨x₁, ..., xₘ⟩ = ⟨y₁, ..., yₘ, ..., yₘ₊ₖ⟩. Then x₁ = ⟨y₁, ..., yₖ₊₁⟩.
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-/
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theorem lemma_0a (xs : XTuple α (n, m - 1)) (ys : Tuple α (m + k))
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: (cast (cast_eq_size p) xs.norm = ys)
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→ (cast (cast_eq_size aux1) (xs.first qₙ) = ys.take (k + 1) (aux2 qₘ))
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:= sorry
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end
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end XTuple
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