lean2/hott/init/types.hlean

/-
Copyright (c) 2014-2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Module: init.types
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn, Jakob von Raumer
-/

prelude
import .logic .num .wf

-- Empty type
-- ----------

namespace empty

  protected theorem elim (A : Type) (H : empty) : A :=
  empty.rec (λe, A) H

end empty

protected definition empty.has_decidable_eq [instance] : decidable_eq empty :=
take (a b : empty), decidable.inl (!empty.elim a)

-- Unit type
-- ---------

namespace unit

  notation `⋆` := star

end unit

-- Sigma type
-- ----------

notation `Σ` binders `,` r:(scoped P, sigma P) := r

namespace sigma
  notation `⟨`:max t:(foldr `,` (e r, mk e r)) `⟩`:0 := t --input ⟨ ⟩ as \< \>

  namespace ops
  postfix `.1`:(max+1) := pr1
  postfix `.2`:(max+1) := pr2
  abbreviation pr₁ := @pr1
  abbreviation pr₂ := @pr2
  end ops
end sigma

-- Sum type
-- --------

namespace sum
  infixr ⊎ := sum
  infixr + := sum
  namespace low_precedence_plus
    reserve infixr `+`:25  -- conflicts with notation for addition
    infixr `+` := sum
  end low_precedence_plus
end sum

-- Product type
-- ------------

definition pair := @prod.mk

namespace prod

  -- notation for n-ary tuples
  notation `(` h `,` t:(foldl `,` (e r, prod.mk r e) h) `)` := t
  infixr × := prod

  namespace ops
  infixr * := prod
  postfix `.1`:(max+1) := pr1
  postfix `.2`:(max+1) := pr2
  abbreviation pr₁ := @pr1
  abbreviation pr₂ := @pr2
  end ops

  namespace low_precedence_times

    reserve infixr `*`:30  -- conflicts with notation for multiplication
    infixr `*` := prod

  end low_precedence_times

  open prod.ops

  definition flip {A B : Type} (a : A × B) : B × A := pair (pr2 a) (pr1 a)

  open well_founded

  section
  variables {A B : Type}
  variable  (Ra  : A → A → Type)
  variable  (Rb  : B → B → Type)

  -- Lexicographical order based on Ra and Rb
  inductive lex : A × B → A × B → Type :=
  | left  : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
  | right : ∀a {b₁ b₂},     Rb b₁ b₂ → lex (a, b₁)  (a, b₂)

  -- Relational product based on Ra and Rb
  inductive rprod : A × B → A × B → Type :=
  intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
  end

  section
  parameters {A B : Type}
  parameters {Ra  : A → A → Type} {Rb  : B → B → Type}
  local infix `≺`:50 := lex Ra Rb

  definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
  acc.rec_on aca
    (λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
      λb, acc.rec_on (acb b)
        (λxb acb
          (iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
          acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
            have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
              @prod.lex.rec_on A B Ra Rb (λp₁ p₂ h, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
                               p (xa, xb) lt
                (λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
                  show acc (lex Ra Rb) (a₁, b₁), from
                  have Ra₁  : Ra a₁ xa, from eq.rec_on eq₂ H,
                  iHa a₁ Ra₁ b₁)
                (λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
                  show acc (lex Ra Rb) (a, b₁), from
                  have Rb₁  : Rb b₁ xb, from eq.rec_on eq₃ H,
                  have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
                  eq.rec_on eq₂' (iHb b₁ Rb₁)),
            aux rfl rfl)))

  -- The lexicographical order of well founded relations is well-founded
  definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
  well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))

  -- Relational product is a subrelation of the lex
  definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
  λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)

  -- The relational product of well founded relations is well-founded
  definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
  subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)

  end

end prod