module prime where

open import Data.Nat 
open import Data.Nat.Properties
open import Data.Empty
open import Data.Unit using (⊤ ; tt)
open import Relation.Nullary
open import Relation.Binary.PropositionalEquality
open import Relation.Binary.Definitions

open import gcd
open import nat

record Prime (i : ℕ ) : Set where
   field
      p>1 : i > 1
      isPrime : ( j : ℕ ) → j < i → 0 < j → gcd i j ≡ 1


record NonPrime ( n : ℕ ) : Set where
   field
      factor : ℕ
      prime : Prime factor
      dividable : Dividable factor n

PrimeP : ( n : ℕ ) → Dec ( Prime n )
PrimeP 0 = no (λ p → ⊥-elim ( nat-<> (Prime.p>1 p) (s≤s z≤n))) 
PrimeP 1 = no (λ p → ⊥-elim ( nat-≤> (Prime.p>1 p) (s≤s (≤-refl))))
PrimeP (suc (suc n)) = isPrime1 (suc (suc n)) (suc n) (s≤s (s≤s z≤n)) a<sa (λ i m<i i<n → isp0 (suc n) i (<to≤ m<i) i<n ) where  
   isp0 : (n : ℕ) (i : ℕ) ( n<i : n ≤ i) ( i<n : i < suc n ) →  gcd (suc n) i ≡ 1
   isp0  n i n<i i<n with <-cmp i n
   ... | tri< a ¬b ¬c = ⊥-elim ( nat-≤> n<i a) 
   ... | tri≈ ¬a refl ¬c = gcd203 i
   ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c i<n )
   isPrime1 : ( n m : ℕ ) → n > 1 → m < n → ( (i : ℕ) → m < i → i < n  →  gcd n i ≡ 1 )  → Dec ( Prime n )
   isPrime1 n zero n>1 m<n lt = yes record { isPrime = λ j j<i 0<j → lt j 0<j j<i ; p>1 = n>1 } 
   isPrime1 n (suc m) n>1 m<n lt with <-cmp (gcd n (suc m)) 1
   ... | tri< a ¬b ¬c = ⊥-elim ( nat-≤> ( gcd>0 n (suc m) (<-trans (s≤s z≤n) n>1) (s≤s z≤n)) a )
   ... | tri≈ ¬a b ¬c = isPrime1 n m n>1 (<-trans a<sa m<n) isp1 where
    --  lt : (i : ℕ) → suc m ≤ i → suc i ≤ n → gcd1 n n i i ≡ 1
        isp1 :  (i : ℕ) → m < i → i < n → gcd n i ≡ 1
        isp1 i m<i i<n with <-cmp (suc m) i
        ... | tri< a ¬b ¬c = lt i a i<n
        ... | tri≈ ¬a m=i  ¬c = subst (λ k → gcd n k ≡ 1) m=i b -- gcd n (suc m) ≡ 1 →  gcd n i ≡ 1
        ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> m<i c) -- suc i ≤ suc m →  i < m
   ... | tri> ¬a ¬b c = no ( λ p → nat-≡< (sym (Prime.isPrime p (suc m) m<n (s≤s z≤n) )) c )

open import logic
open _∧_

record Pair≤ : Set where
  field 
    ox oy : ℕ
    ox<oy : ox ≤ oy 

open Pair≤

nonPrime1 : { n : ℕ } → 1 < n → ¬ Prime n → NonPrime n
nonPrime1 {n} 1<n np = f-induction {_} {_} {Pair≤} {λ op → G (ox op) (oy op) → NonPrime n } (λ op → oy op - ox op ) I
         record { ox = n ; oy = n ; ox<oy = ≤-refl } {!!} where
    record G (p g : ℕ) : Set where
      field
        1<p : 1 < p 
        div : Dividable p n
        npr : g > 0 → ¬ Prime p
        is-g : (j : ℕ) → g ≤ j → j < p → gcd p j ≡ 1
    is-g= : {g p : ℕ} → g ≡ p →  (j : ℕ) → g ≤ j → j < p → gcd p j ≡ 1
    is-g= eq _ g≤j j<p = ⊥-elim ( nat-≡< eq (≤-trans (s≤s g≤j ) j<p))
    next : Pair≤ → Pair≤
    next record { ox = ox ; oy = oy ; ox<oy = ox<oy } = record { ox = ox ; oy = gcd oy ox ; ox<oy = {!!} } 
    ind : {p : Pair≤ } → (G (ox (next p)) (oy (next p)) → NonPrime n) → G (ox p) (oy p) → NonPrime n
    ind {p} prev g = {!!} where
      next1  : ( m : ℕ ) →  ℕ  → {!!} → {!!} -- G.g g ≡ m  → G 
      next1  0 _ _ = g
      next1  _ 0  _ = g
      next1  (suc m) 1 sm=g = {!!} --record g { g = m ;  npr = λ lt → G.npr g {!!} ; is-g = {!!} }
      next1  (suc m) (suc (suc f)) _ with PrimeP (suc (suc f))
      ... | yes p = {!!} -- record { p = suc (suc f) ; g = 0 ; 1<p = Prime.p>1 p ; div = {!!} ; npr = λ () ; is-g = {!!} }
      ... | no np = {!!} -- record { p = suc (suc f) ; g = suc (suc f)  ; 1<p = s≤s (s≤s z≤n) ; div = {!!} ; npr = λ lt → np ; is-g = is-g= refl }
    pzero : (p : Pair≤ ) →  (oy p - ox p) ≡ zero → G (ox p) (oy p) → NonPrime n
    pzero p eq g = record { factor = {!!} ; prime = record { p>1 = G.1<p g ; isPrime = {!!} } ;  dividable = G.div g }
    decline : (p : Pair≤ ) → 0 < (oy p - ox p) → (oy (next p) - ox (next p)) < (oy p - ox p)
    decline = {!!}
    I : Finduction Pair≤  _  (λ op → oy op - ox op )
    I = record {
              pnext = next
            ; fzero = λ {p} → pzero p
            ; decline = λ {p} → decline p
            ; ind = λ {p} → ind {p}
          }


nonPrime : { n : ℕ } → 1 < n → ¬ Prime n → NonPrime n
nonPrime {n} 1<n np = np1 n (λ j n≤j j<n → ⊥-elim (nat-≤>  n≤j j<n ) ) where
    mg1 :  (m : ℕ )→ ( (j : ℕ ) → suc m ≤ j → j < n → gcd n j ≡ 1  ) →  gcd n (suc m) ≡ 1 → (j : ℕ) → m ≤ j → j < n → gcd n j ≡ 1
    mg1 m mg gcd j m<j j<n with <-cmp j (suc m)
    ... | tri< a ¬b ¬c = ⊥-elim ( nat-≤> (≤-trans {!!} m<j) a)
    ... | tri≈ ¬a refl ¬c = gcd
    ... | tri> ¬a ¬b c = mg j {!!} j<n
    np1 : ( m : ℕ ) → ( (j : ℕ ) → m ≤ j → j < n → gcd n j ≡ 1  ) → NonPrime n
    np1 zero mg = ⊥-elim ( np record { isPrime = λ j lt _ → mg j z≤n lt ; p>1 = 1<n } ) -- zero < j , j < n
    np1 (suc m) mg with <-cmp ( gcd n (suc m) ) 1
    ... | tri< a ¬b ¬c = ⊥-elim ( nat-≤> ( gcd>0 n (suc m) (<-trans (s≤s z≤n) 1<n) (s≤s z≤n)) a )
    ... | tri≈ ¬a b ¬c = np1 m (mg1 m mg b )
    ... | tri> ¬a ¬b c with PrimeP ( gcd n (suc m) )
    ... | yes y = record { factor = gcd n (suc m) ; prime = y ;  dividable = proj1 (gcd-divable n (suc m) ) }
    ... | no ngcd = np1 m (mg1 m mg {!!} )


prime-is-infinite : (max-prime : ℕ ) → ¬ ( (j : ℕ) → max-prime < j → ¬ Prime j ) 
prime-is-infinite zero pmax = pmax 3 (s≤s z≤n) record { isPrime = λ n lt 0<j → pif3 n lt 0<j  ; p>1 = s≤s (s≤s z≤n) } where
  pif3 : (n : ℕ) →  n < 3 →  0 < n → gcd 3 n ≡ 1
  pif3 .1 (s≤s (s≤s z≤n)) _ = refl
  pif3 .2 (s≤s (s≤s (s≤s z≤n))) _ = refl
prime-is-infinite (suc m) pmax = getPrime where 
  factorial : (n : ℕ) → ℕ
  factorial zero = 1
  factorial (suc n) = (suc n) * (factorial n)
  prime<max : (n : ℕ ) → Prime n → n < suc (suc m)
  prime<max n p with <-cmp n (suc m) 
  ... | tri< a ¬b ¬c = ≤-trans a refl-≤s 
  ... | tri≈ ¬a refl ¬c = ≤-refl 
  ... | tri> ¬a ¬b c = ⊥-elim ( pmax n c p ) 
  factorial-mono : (n : ℕ) → factorial n ≤ factorial (suc n)
  factorial-mono n = begin
     factorial n  ≤⟨ x≤x+y ⟩
     factorial n + n * factorial n ≡⟨ refl ⟩
     (suc n) * factorial n  ≡⟨ refl ⟩
     factorial (suc n)  ∎  where open ≤-Reasoning
  factorial≥1 : {m : ℕ} → 1 ≤ factorial m
  factorial≥1 {zero} = ≤-refl
  factorial≥1 {suc m} = begin
     1 ≤⟨ s≤s z≤n ⟩
     (suc m) * 1 ≤⟨  *-monoʳ-≤ (suc m) (factorial≥1 {m}) ⟩
     (suc m) * factorial m ≡⟨ refl ⟩
     factorial (suc m)  ∎  where open ≤-Reasoning
  m<factorial : (m : ℕ) → m ≤ factorial m
  m<factorial zero = z≤n
  m<factorial (suc m) = begin
     suc m ≡⟨ cong suc (+-comm 0 _) ⟩
     1 * suc m ≡⟨ *-comm 1 _ ⟩
     (suc m) * 1 ≤⟨  *-monoʳ-≤ (suc m) (factorial≥1 {m}) ⟩
     (suc m) * factorial m  ≡⟨ refl ⟩
     factorial (suc m)  ∎  where open ≤-Reasoning
  -- *-monoˡ-≤ (suc m) {!!}
  f>m :  suc m < suc (factorial (suc m))
  f>m = begin
     suc (suc m)  ≡⟨ cong (λ k → 1 + suc k ) (+-comm _ m) ⟩
     suc (1 + 1 * m)  ≡⟨ cong (λ k → suc (1 + k )) (*-comm 1 m)  ⟩
     suc (1 + m * 1)  ≤⟨ s≤s (s≤s (*-monoʳ-≤ m  (factorial≥1 {m}) )) ⟩
     suc (1 + m * factorial m) ≤⟨ s≤s  (+-monoˡ-≤ _ (factorial≥1 {m})) ⟩
     suc (factorial m + m * factorial m)  ≡⟨ refl ⟩
     suc (factorial (suc m)) ∎  where open ≤-Reasoning
  fact< : (m n : ℕ) → 0 < n → n < suc (suc m) → Dividable n ( factorial (suc m) )
  fact< zero 1 0<n (s≤s (s≤s z≤n)) = record { factor = 1 ; is-factor = refl }
  fact< (suc m) (suc zero) 0<n n<m = record { factor = factorial (suc (suc m)) ; is-factor = begin
     factorial (suc (suc m)) * 1 + 0  ≡⟨ +-comm _ 0 ⟩
     factorial (suc (suc m)) * 1   ≡⟨ m*1=m  ⟩
     (suc (suc m)) * factorial (suc m)  ≡⟨ refl ⟩
     factorial (suc (suc m))  ∎  } where open ≡-Reasoning
  fact< (suc m) (suc (suc n)) 0<n n<m with <-cmp (suc (suc n)) (suc (suc m))
  ... | tri< a ¬b ¬c = record { factor = suc (suc m) * Dividable.factor fact1 ; is-factor = fact2 } where
      fact1 : Dividable (suc (suc n))  (factorial (suc m ))
      fact1 = fact< m (suc (suc n)) 0<n a 
      d =  (fact< m (suc (suc n)) 0<n a)
      fact2 : suc (suc m) * Dividable.factor d * suc (suc n) + 0 ≡ factorial (suc (suc m))
      fact2 = begin
        suc (suc m) * Dividable.factor d * suc (suc n) + 0  ≡⟨ +-comm _ 0 ⟩
        suc (suc m) * Dividable.factor d * suc (suc n)   ≡⟨ *-assoc (suc (suc m)) (Dividable.factor d) ( suc (suc n)) ⟩
        suc (suc m) * (Dividable.factor d * suc (suc n))   ≡⟨ cong (λ k →  suc (suc m) * k ) ( +-comm 0 (Dividable.factor d * suc (suc n)) ) ⟩
        suc (suc m) * (Dividable.factor d * suc (suc n) + 0)   ≡⟨ cong (λ k → suc (suc m) * k ) (Dividable.is-factor d)  ⟩
        suc (suc m) * factorial (suc m)  ≡⟨ refl ⟩
        factorial (suc (suc m))  ∎   where open ≡-Reasoning
  ... | tri≈ ¬a b ¬c = record { factor = factorial (suc m)  ; is-factor = begin
     factorial (suc m) * suc (suc n) + 0 ≡⟨ +-comm _ 0 ⟩
     factorial (suc m) * suc (suc n)  ≡⟨ *-comm (factorial (suc m)) (suc (suc n))  ⟩
     (suc (suc n)) * factorial (suc m)  ≡⟨ cong (λ k → k * factorial (suc m) ) b ⟩
     (suc (suc m)) * factorial (suc m)  ≡⟨ refl ⟩
     factorial (suc (suc m))  ∎  } where open ≡-Reasoning
  ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n<m) 
  fact : (n : ℕ) → Prime n → Dividable n ( factorial (suc m) )
  fact n p = fact< m n (<-trans (s≤s z≤n) (Prime.p>1 p)) ( prime<max n p )
  -- div+1 : { i k : ℕ } → k > 1 →  Dividable k i →  ¬ Dividable k (suc i)
  getPrime : ⊥
  getPrime with PrimeP ( suc (factorial (suc m)) )
  ... | yes p = pmax _ f>m p 
  ... | no np = div+1 (Prime.p>1 (NonPrime.prime p1)) (fact (NonPrime.factor p1) (NonPrime.prime p1) ) (NonPrime.dividable p1) where
      p1 : NonPrime  ( suc (factorial (suc m)) )
      p1 = nonPrime (begin
       2 ≤⟨ s≤s ( s≤s z≤n) ⟩
       suc (suc m) ≤⟨ s≤s (m<factorial (suc m))  ⟩
       suc (factorial (suc m)) ∎ ) np where open ≤-Reasoning