/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
-/
prelude
import init.datatypes init.num init.wf init.logic init.tactic

definition dpair := @sigma.mk
notation `Σ` binders `, ` r:(scoped P, sigma P) := r
-- notation for n-ary tuples; input ⟨ ⟩ as \< \>
notation `⟨`:max t:(foldr `, ` (e r, sigma.mk e r)) `⟩`:0 := t

lemma ex_of_sig {A : Type} {P : A → Prop} : (Σ x, P x) → ∃ x, P x :=
assume h, obtain x hx, from h, exists.intro x hx

namespace sigma
  notation `pr₁`  := pr1
  notation `pr₂`  := pr2

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

  open ops well_founded

  section
  variables {A : Type} {B : A → Type}
  variable  (Ra  : A → A → Prop)
  variable  (Rb  : ∀ a, B a → B a → Prop)

  theorem dpair_eq : ∀ {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} (H₁ : a₁ = a₂), eq.rec_on H₁ b₁ = b₂ → ⟨a₁, b₁⟩ = ⟨a₂, b₂⟩
  | a₁ a₁ b₁ b₁ (eq.refl a₁) (eq.refl b₁) := rfl

  protected theorem eq {p₁ p₂ : Σa : A, B a} :
    ∀(H₁ : p₁.1 = p₂.1) (H₂ : eq.rec_on H₁ p₁.2 = p₂.2), p₁ = p₂ :=
  destruct p₁ (take a₁ b₁, destruct p₂ (take a₂ b₂ H₁ H₂, dpair_eq H₁ H₂))

  -- Lexicographical order based on Ra and Rb
  inductive lex : sigma B → sigma B → Prop :=
  | left  : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂   → lex ⟨a₁, b₁⟩ ⟨a₂, b₂⟩
  | right : ∀a {b₁ b₂},     Rb a b₁ b₂ → lex ⟨a, b₁⟩  ⟨a, b₂⟩
  end

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

  definition lex.accessible {a} (aca : acc Ra a) (acb : ∀a, well_founded (Rb a)) : ∀ (b : B a), acc (lex Ra Rb) ⟨a, b⟩ :=
  acc.rec_on aca
    (λxa aca (iHa : ∀y, Ra y xa → ∀b : B y, acc (lex Ra Rb) ⟨y, b⟩),
      λb : B xa, acc.rec_on (acb xa b)
        (λxb acb
          (iHb : ∀ (y : B xa), Rb xa 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
              @sigma.lex.rec_on A B Ra Rb (λp₁ p₂, p₂.1 = xa → p₂.2 == xb → acc (lex Ra Rb) p₁)
                                p ⟨xa, xb⟩ lt
                (λ (a₁ : A) (b₁ : B a₁) (a₂ : A) (b₂ : B a₂) (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ == xb),
                  begin cases eq₂, exact (iHa a₁ H b₁) end)
                (λ (a : A) (b₁ b₂ : B a) (H : Rb a b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ == xb),
                  begin cases eq₂, cases eq₃, exact (iHb b₁ H) end),
            aux rfl !heq.refl)))

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