------------------------------------------------------------------------------
-- The gcd is divisible by any common divisor
------------------------------------------------------------------------------

{-# OPTIONS --exact-split              #-}
{-# OPTIONS --no-sized-types           #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K                #-}

module LTC-PCF.Program.GCD.Partial.Divisible where

open import LTC-PCF.Base
open import LTC-PCF.Base.Properties
open import LTC-PCF.Data.Nat
open import LTC-PCF.Data.Nat.Divisibility.NotBy0
open import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties
open import LTC-PCF.Data.Nat.Induction.NonAcc.Lexicographic
open import LTC-PCF.Data.Nat.Inequalities
open import LTC-PCF.Data.Nat.Inequalities.EliminationProperties
open import LTC-PCF.Data.Nat.Inequalities.Properties
open import LTC-PCF.Data.Nat.Properties
open import LTC-PCF.Program.GCD.Partial.ConversionRules
open import LTC-PCF.Program.GCD.Partial.Definitions
open import LTC-PCF.Program.GCD.Partial.GCD

------------------------------------------------------------------------------
-- The gcd 0 (succ n) is Divisible.
gcd-0S-Divisible : ∀ {n} → N n → Divisible zero (succ₁ n) (gcd zero (succ₁ n))
gcd-0S-Divisible {n} _ c _ (c∣0 , c∣Sn) = subst (_∣_ c) (sym (gcd-0S n)) c∣Sn

------------------------------------------------------------------------------
-- The gcd (succ₁ n) 0 is Divisible.
gcd-S0-Divisible : ∀ {n} → N n → Divisible (succ₁ n) zero (gcd (succ₁ n) zero)
gcd-S0-Divisible {n} _ c _ (c∣Sn , c∣0) = subst (_∣_ c) (sym (gcd-S0 n)) c∣Sn

------------------------------------------------------------------------------
-- The gcd (succ₁ m) (succ₁ n) when succ₁ m > succ₁ n is Divisible.
gcd-S>S-Divisible :
  ∀ {m n} → N m → N n →
  (Divisible (succ₁ m ∸ succ₁ n) (succ₁ n) (gcd (succ₁ m ∸ succ₁ n) (succ₁ n))) →
  succ₁ m > succ₁ n →
  Divisible (succ₁ m) (succ₁ n) (gcd (succ₁ m) (succ₁ n))
gcd-S>S-Divisible {m} {n} Nm Nn acc Sm>Sn c Nc (c∣Sm , c∣Sn) =
{-
Proof:
   ----------------- (Hip.)
     c | m    c | n
   ---------------------- (Thm.)   -------- (Hip.)
       c | (m ∸ n)                   c | n
     ------------------------------------------ (IH)
              c | gcd m (n ∸ m)                          m > n
             --------------------------------------------------- (gcd def.)
                             c | gcd m n
-}
 subst (_∣_ c) (sym (gcd-S>S m n Sm>Sn)) (acc c Nc (c|Sm-Sn , c∣Sn))
 where
 c|Sm-Sn : c ∣ succ₁ m ∸ succ₁ n
 c|Sm-Sn = x∣y→x∣z→x∣y∸z Nc (nsucc Nm) (nsucc Nn) c∣Sm c∣Sn

------------------------------------------------------------------------------
-- The gcd (succ₁ m) (succ₁ n) when succ₁ m ≯ succ₁ n is Divisible.
gcd-S≯S-Divisible :
  ∀ {m n} → N m → N n →
  (Divisible (succ₁ m) (succ₁ n ∸ succ₁ m) (gcd (succ₁ m) (succ₁ n ∸ succ₁ m))) →
  succ₁ m ≯ succ₁ n →
  Divisible (succ₁ m) (succ₁ n) (gcd (succ₁ m) (succ₁ n))
gcd-S≯S-Divisible {m} {n} Nm Nn acc Sm≯Sn c Nc (c∣Sm , c∣Sn) =
{-
Proof
                            ----------------- (Hip.)
                                c | m    c | n
        -------- (Hip.)       ---------------------- (Thm.)
         c | m                      c | n ∸ m
     ------------------------------------------ (IH)
              c | gcd m (n ∸ m)                          m ≯ n
             --------------------------------------------------- (gcd def.)
                             c | gcd m n
-}

  subst (_∣_ c) (sym (gcd-S≯S m n Sm≯Sn)) (acc c Nc (c∣Sm , c|Sn-Sm))
  where
  c|Sn-Sm : c ∣ succ₁ n ∸ succ₁ m
  c|Sn-Sm = x∣y→x∣z→x∣y∸z Nc (nsucc Nn) (nsucc Nm) c∣Sn c∣Sm

------------------------------------------------------------------------------
-- The gcd m n when m > n is Divisible.
gcd-x>y-Divisible :
  ∀ {m n} → N m → N n →
  (∀ {o p} → N o → N p → Lexi o p m n → x≢0≢y o p →
             Divisible o p (gcd o p)) →
  m > n →
  x≢0≢y m n →
  Divisible m n (gcd m n)
gcd-x>y-Divisible nzero Nn _ 0>n _ _ _ = ⊥-elim (0>x→⊥ Nn 0>n)
gcd-x>y-Divisible (nsucc Nm) nzero _ _ _ c Nc = gcd-S0-Divisible Nm c Nc
gcd-x>y-Divisible (nsucc {m} Nm) (nsucc {n} Nn) ah Sm>Sn _ c Nc =
  gcd-S>S-Divisible Nm Nn ih Sm>Sn c Nc
  where
  -- Inductive hypothesis.
  ih : Divisible (succ₁ m ∸ succ₁ n) (succ₁ n) (gcd (succ₁ m ∸ succ₁ n) (succ₁ n))
  ih = ah {succ₁ m ∸ succ₁ n}
          {succ₁ n}
          (∸-N (nsucc Nm) (nsucc Nn))
          (nsucc Nn)
          ([Sx∸Sy,Sy]<[Sx,Sy] Nm Nn)
          (λ p → ⊥-elim (S≢0 (∧-proj₂ p)))

------------------------------------------------------------------------------
-- The gcd m n when m ≯ n is Divisible.
gcd-x≯y-Divisible :
  ∀ {m n} → N m → N n →
  (∀ {o p} → N o → N p → Lexi o p m n → x≢0≢y o p →
             Divisible o p (gcd o p)) →
  m ≯ n →
  x≢0≢y m n →
  Divisible m n (gcd m n)
gcd-x≯y-Divisible nzero nzero _ _ h _ _ = ⊥-elim (h (refl , refl))
gcd-x≯y-Divisible nzero (nsucc Nn) _ _ _ c Nc = gcd-0S-Divisible Nn c Nc
gcd-x≯y-Divisible (nsucc _) nzero _ Sm≯0 _ _ _ = ⊥-elim (S≯0→⊥ Sm≯0)
gcd-x≯y-Divisible (nsucc {m} Nm) (nsucc {n} Nn) ah Sm≯Sn _ c Nc =
  gcd-S≯S-Divisible Nm Nn ih Sm≯Sn c Nc
  where
  -- Inductive hypothesis.
  ih : Divisible (succ₁ m) (succ₁ n ∸ succ₁ m) (gcd (succ₁ m) (succ₁ n ∸ succ₁ m))
  ih = ah {succ₁ m}
          {succ₁ n ∸ succ₁ m}
          (nsucc Nm)
          (∸-N (nsucc Nn) (nsucc Nm))
          ([Sx,Sy∸Sx]<[Sx,Sy] Nm Nn)
          (λ p → ⊥-elim (S≢0 (∧-proj₁ p)))

------------------------------------------------------------------------------
-- The gcd is Divisible.
gcdDivisible : ∀ {m n} → N m → N n → x≢0≢y m n → Divisible m n (gcd m n)
gcdDivisible = Lexi-wfind A h
  where
  A : D → D → Set
  A i j = x≢0≢y i j → Divisible i j (gcd i j)

  h : ∀ {i j} → N i → N j → (∀ {k l} → N k → N l → Lexi k l i j → A k l) →
      A i j
  h Ni Nj ah = case (gcd-x>y-Divisible Ni Nj ah)
                    (gcd-x≯y-Divisible Ni Nj ah)
                    (x>y∨x≯y Ni Nj)