{-# OPTIONS --cubical-compatible --safe #-}

module nat where

open import Data.Nat 
open import Data.Nat.Properties
open import Data.Empty
open import Relation.Nullary
open import  Relation.Binary.PropositionalEquality
open import  Relation.Binary.Core
open import  Relation.Binary.Definitions
open import  logic
open import Level hiding ( zero ; suc ) 

nat-<> : { x y : ℕ } → x < y → y < x → ⊥
nat-<>  (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x

nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥
nat-≤>  (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x

nat-<≡ : { x : ℕ } → x < x → ⊥
nat-<≡  (s≤s lt) = nat-<≡ lt

nat-≡< : { x y : ℕ } → x ≡ y → x < y → ⊥
nat-≡< refl lt = nat-<≡ lt

¬a≤a : {la : ℕ} → suc la ≤ la → ⊥
¬a≤a  (s≤s lt) = ¬a≤a  lt

a<sa : {la : ℕ} → la < suc la 
a<sa {zero} = s≤s z≤n
a<sa {suc la} = s≤s a<sa 

=→¬< : {x : ℕ  } → ¬ ( x < x )
=→¬< {zero} ()
=→¬< {suc x} (s≤s lt) = =→¬< lt

>→¬< : {x y : ℕ  } → (x < y ) → ¬ ( y < x )
>→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x

<-∨ : { x y : ℕ } → x < suc y → ( (x ≡ y ) ∨ (x < y) )
<-∨ {zero} {zero} (s≤s z≤n) = case1 refl
<-∨ {zero} {suc y} (s≤s lt) = case2 (s≤s z≤n)
<-∨ {suc x} {zero} (s≤s ())
<-∨ {suc x} {suc y} (s≤s lt) with <-∨ {x} {y} lt
<-∨ {suc x} {suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → suc k ) eq)
<-∨ {suc x} {suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)

≤-∨ : { x y : ℕ } → x ≤ y → ( (x ≡ y ) ∨ (x < y) )
≤-∨ {zero} {zero} z≤n = case1 refl
≤-∨ {zero} {suc y} z≤n = case2 (s≤s z≤n)
≤-∨ {suc x} {zero} ()
≤-∨ {suc x} {suc y} (s≤s lt) with ≤-∨ {x} {y} lt
≤-∨ {suc x} {suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → suc k ) eq)
≤-∨ {suc x} {suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)

max : (x y : ℕ) → ℕ
max zero zero = zero
max zero (suc x) = (suc x)
max (suc x) zero = (suc x)
max (suc x) (suc y) = suc ( max x y )

x≤max : (x y : ℕ) → x ≤ max x y
x≤max zero zero = ≤-refl
x≤max zero (suc x) = z≤n
x≤max (suc x) zero = ≤-refl
x≤max (suc x) (suc y) = s≤s( x≤max x y )

y≤max : (x y : ℕ) → y ≤ max x y
y≤max zero zero = ≤-refl
y≤max zero (suc x) = ≤-refl
y≤max (suc x) zero = z≤n
y≤max (suc x) (suc y) = s≤s( y≤max x y )

x≤y→max=y : (x y : ℕ) → x ≤ y → max x y ≡ y
x≤y→max=y zero zero x≤y = refl
x≤y→max=y zero (suc y) x≤y = refl
x≤y→max=y (suc x) (suc y) (s≤s x≤y) = cong suc (x≤y→max=y x y x≤y )

y≤x→max=x : (x y : ℕ) → y ≤ x → max x y ≡ x
y≤x→max=x zero zero y≤x = refl
y≤x→max=x zero (suc y) ()
y≤x→max=x (suc x) zero lt = refl
y≤x→max=x (suc x) (suc y) (s≤s y≤x) = cong suc (y≤x→max=x x y y≤x )

-- _*_ : ℕ → ℕ → ℕ
-- _*_ zero _ = zero
-- _*_ (suc n) m = m + ( n * m )

-- x ^ y
exp : ℕ → ℕ → ℕ
exp _ zero = 1
exp n (suc m) = n * ( exp n m )

div2 : ℕ → (ℕ ∧ Bool )
div2 zero =  ⟪ 0 , false ⟫
div2 (suc zero) =  ⟪ 0 , true ⟫
div2 (suc (suc n)) =  ⟪ suc (proj1 (div2 n)) , proj2 (div2 n) ⟫ where
    open _∧_

div2-rev : (ℕ ∧ Bool ) → ℕ
div2-rev ⟪ x , true ⟫ = suc (x + x)
div2-rev ⟪ x , false ⟫ = x + x

div2-eq : (x : ℕ ) → div2-rev ( div2 x ) ≡ x
div2-eq zero = refl
div2-eq (suc zero) = refl
div2-eq (suc (suc x)) with div2 x in eq1 
... | ⟪ x1 , true ⟫ = begin -- eq1 : div2 x ≡ ⟪ x1 , true ⟫
     div2-rev ⟪ suc x1 , true ⟫ ≡⟨⟩
     suc (suc (x1 + suc x1)) ≡⟨ cong (λ k → suc (suc k )) (+-comm x1  _ ) ⟩
     suc (suc (suc (x1 + x1))) ≡⟨⟩    
     suc (suc (div2-rev ⟪ x1 , true ⟫)) ≡⟨ cong (λ k → suc (suc (div2-rev k ))) (sym eq1) ⟩ 
     suc (suc (div2-rev (div2 x)))      ≡⟨ cong (λ k → suc (suc k)) (div2-eq x) ⟩ 
     suc (suc x) ∎  where open ≡-Reasoning
... | ⟪ x1 , false ⟫ = begin
     div2-rev ⟪ suc x1 , false ⟫ ≡⟨⟩
     suc (x1 + suc x1) ≡⟨ cong (λ k → (suc k )) (+-comm x1  _ ) ⟩
     suc (suc (x1 + x1)) ≡⟨⟩    
     suc (suc (div2-rev ⟪ x1 , false ⟫)) ≡⟨ cong (λ k → suc (suc (div2-rev k ))) (sym eq1) ⟩ 
     suc (suc (div2-rev (div2 x)))      ≡⟨ cong (λ k → suc (suc k)) (div2-eq x) ⟩ 
     suc (suc x) ∎  where open ≡-Reasoning

sucprd : {i : ℕ } → 0 < i  → suc (pred i) ≡ i
sucprd {suc i} 0<i = refl

0<s : {x : ℕ } → zero < suc x
0<s {_} = s≤s z≤n 

px<py : {x y : ℕ } → pred x  < pred y → x < y
px<py {zero} {suc y} lt = 0<s
px<py {suc zero} {suc (suc y)} (s≤s lt) = s≤s 0<s
px<py {suc (suc x)} {suc (suc y)} (s≤s lt) = s≤s (px<py {suc x} {suc y} lt)

minus : (a b : ℕ ) →  ℕ
minus a zero = a
minus zero (suc b) = zero
minus (suc a) (suc b) = minus a b

_-_ = minus

sn-m=sn-m : {m n : ℕ } →  m ≤ n → suc n - m ≡ suc ( n - m )
sn-m=sn-m {0} {n} z≤n = refl
sn-m=sn-m {suc m} {suc n} (s≤s m<n) = sn-m=sn-m m<n

si-sn=i-n : {i n : ℕ } → n < i  → suc (i - suc n) ≡ (i - n)
si-sn=i-n {i} {n} n<i = begin
   suc (i - suc n) ≡⟨ sym (sn-m=sn-m n<i )  ⟩
   suc i - suc n ≡⟨⟩
   i - n
   ∎  where
      open ≡-Reasoning

refl-≤s : {x : ℕ } → x ≤ suc x
refl-≤s {zero} = z≤n
refl-≤s {suc x} = s≤s (refl-≤s {x})

a≤sa = refl-≤s

n-m<n : (n m : ℕ ) →  n - m ≤ n
n-m<n zero zero = z≤n
n-m<n (suc n) zero = s≤s (n-m<n n zero)
n-m<n zero (suc m) = z≤n
n-m<n (suc n) (suc m) = ≤-trans (n-m<n n m ) refl-≤s

n-n-m=m : {m n : ℕ } → m ≤ n  → m ≡ (n - (n - m))
n-n-m=m {0} {zero} z≤n = refl
n-n-m=m {0} {suc n} z≤n = n-n-m=m {0} {n} z≤n
n-n-m=m {suc m} {suc n} (s≤s m≤n) = sym ( begin
   suc n - ( n - m )    ≡⟨ sn-m=sn-m (n-m<n  n m) ⟩
   suc (n - ( n - m ))  ≡⟨ cong (λ k → suc k ) (sym (n-n-m=m m≤n)) ⟩
   suc m
   ∎  ) where
      open ≡-Reasoning

m+= : {i j  m : ℕ } → m + i ≡ m + j → i ≡ j
m+= {i} {j} {zero} refl = refl
m+= {i} {j} {suc m} eq = m+= {i} {j} {m} ( cong (λ k → pred k ) eq )

+m= : {i j  m : ℕ } → i + m ≡ j + m → i ≡ j
+m= {i} {j} {m} eq = m+= ( subst₂ (λ j k → j ≡ k ) (+-comm i _ ) (+-comm j _ ) eq )

less-1 :  { n m : ℕ } → suc n < m → n < m
less-1 {zero} {suc (suc _)} (s≤s (s≤s z≤n)) = s≤s z≤n
less-1 {suc n} {suc m} (s≤s lt) = s≤s (less-1 {n} {m} lt)

sa=b→a<b :  { n m : ℕ } → suc n ≡ m → n < m
sa=b→a<b {0} {suc zero} refl = s≤s z≤n
sa=b→a<b {suc n} {suc (suc n)} refl = s≤s (sa=b→a<b refl)

minus+n : {x y : ℕ } → suc x > y  → minus x y + y ≡ x
minus+n {x} {zero} _ = trans (sym (+-comm zero  _ )) refl
minus+n {zero} {suc y} (s≤s ())
minus+n {suc x} {suc y} (s≤s lt) = begin
           minus (suc x) (suc y) + suc y
        ≡⟨ +-comm _ (suc y)    ⟩
           suc y + minus x y 
        ≡⟨ cong ( λ k → suc k ) (
           begin
                 y + minus x y 
              ≡⟨ +-comm y  _ ⟩
                 minus x y + y
              ≡⟨ minus+n {x} {y} lt ⟩
                 x 
           ∎  
           ) ⟩
           suc x
        ∎  where open ≡-Reasoning

<-minus-0 : {x y z : ℕ } → z + x < z + y → x < y
<-minus-0 {x} {suc _} {zero} lt = lt
<-minus-0 {x} {y} {suc z} (s≤s lt) = <-minus-0 {x} {y} {z} lt

<-minus : {x y z : ℕ } → x + z < y + z → x < y
<-minus {x} {y} {z} lt = <-minus-0 ( subst₂ ( λ j k → j < k ) (+-comm x _) (+-comm y _ ) lt )

x≤x+y : {z y : ℕ } → z ≤ z + y
x≤x+y {zero} {y} = z≤n
x≤x+y {suc z} {y} = s≤s  (x≤x+y {z} {y})

x≤y+x : {z y : ℕ } → z ≤ y + z
x≤y+x {z} {y} = subst (λ k → z ≤ k ) (+-comm _ y ) x≤x+y

x≤x+sy : {x y : ℕ} → x < x + suc y
x≤x+sy {x} {y} = begin
        suc x ≤⟨ x≤x+y ⟩
        suc x + y ≡⟨ cong (λ k → k + y) (+-comm 1 x ) ⟩
        (x + 1) + y ≡⟨ (+-assoc x 1 _) ⟩
        x + suc y ∎  where open ≤-Reasoning

<-plus : {x y z : ℕ } → x < y → x + z < y + z 
<-plus {zero} {suc y} {z} (s≤s z≤n) = s≤s (subst (λ k → z ≤ k ) (+-comm z _ ) x≤x+y  )
<-plus {suc x} {suc y} {z} (s≤s lt) = s≤s (<-plus {x} {y} {z} lt)

<-plus-0 : {x y z : ℕ } → x < y → z + x < z + y 
<-plus-0 {x} {y} {z} lt = subst₂ (λ j k → j < k ) (+-comm _ z) (+-comm _ z) ( <-plus {x} {y} {z} lt )

≤-plus : {x y z : ℕ } → x ≤ y → x + z ≤ y + z
≤-plus {0} {y} {zero} z≤n = z≤n
≤-plus {0} {y} {suc z} z≤n = subst (λ k → z < k ) (+-comm _ y ) x≤x+y 
≤-plus {suc x} {suc y} {z} (s≤s lt) = s≤s ( ≤-plus {x} {y} {z} lt )

≤-plus-0 : {x y z : ℕ } → x ≤ y → z + x ≤ z + y 
≤-plus-0 {x} {y} {zero} lt = lt
≤-plus-0 {x} {y} {suc z} lt = s≤s ( ≤-plus-0 {x} {y} {z} lt )

x+y<z→x<z : {x y z : ℕ } → x + y < z → x < z 
x+y<z→x<z {zero} {y} {suc z} (s≤s lt1) = s≤s z≤n
x+y<z→x<z {suc x} {y} {suc z} (s≤s lt1) = s≤s ( x+y<z→x<z {x} {y} {z} lt1 )

*≤ : {x y z : ℕ } → x ≤ y → x * z ≤ y * z 
*≤ lt = *-mono-≤ lt ≤-refl

*< : {x y z : ℕ } → x < y → x * suc z < y * suc z 
*< {zero} {suc y} lt = s≤s z≤n
*< {suc x} {suc y} (s≤s lt) = <-plus-0 (*< lt)

<to<s : {x y  : ℕ } → x < y → x < suc y
<to<s {zero} {suc y} (s≤s lt) = s≤s z≤n
<to<s {suc x} {suc y} (s≤s lt) = s≤s (<to<s {x} {y} lt)

<tos<s : {x y  : ℕ } → x < y → suc x < suc y
<tos<s {zero} {suc y} (s≤s z≤n) = s≤s (s≤s z≤n)
<tos<s {suc x} {suc y} (s≤s lt) = s≤s (<tos<s {x} {y} lt)

<to≤ : {x y  : ℕ } → x < y → x ≤ y 
<to≤ {zero} {suc y} (s≤s z≤n) = z≤n
<to≤ {suc x} {suc y} (s≤s lt) = s≤s (<to≤ {x} {y}  lt)

<∨≤ : ( x y : ℕ ) →  (x < y ) ∨ (y ≤ x) 
<∨≤ x y with <-cmp x y
... | tri< a ¬b ¬c = case1 a
... | tri≈ ¬a refl ¬c = case2 ≤-refl
... | tri> ¬a ¬b c = case2 (<to≤ c)

refl-≤ : {x : ℕ } → x ≤ x
refl-≤ {zero} = z≤n
refl-≤ {suc x} = s≤s (refl-≤ {x})

refl-≤≡ : {x y : ℕ } → x ≡ y → x ≤ y
refl-≤≡ refl = refl-≤ 

x<y→≤ : {x y : ℕ } → x < y →  x ≤ suc y
x<y→≤ {zero} {.(suc _)} (s≤s z≤n) = z≤n
x<y→≤ {suc x} {suc y} (s≤s lt) = s≤s (x<y→≤ {x} {y} lt)

≤→= : {i j : ℕ} → i ≤ j → j ≤ i → i ≡ j
≤→= {0} {0} z≤n z≤n = refl
≤→= {suc i} {suc j} (s≤s i<j) (s≤s j<i) = cong suc ( ≤→= {i} {j} i<j j<i )

px≤x : {x  : ℕ } → pred x ≤ x 
px≤x {zero} = refl-≤
px≤x {suc x} = refl-≤s

px≤py : {x y : ℕ } → x ≤ y → pred x  ≤ pred y 
px≤py {zero} {zero} lt = refl-≤
px≤py {zero} {suc y} lt = z≤n
px≤py {suc x} {suc y} (s≤s lt) = lt 

sx≤py→x≤y : {x y : ℕ } → suc x ≤ suc y → x  ≤ y 
sx≤py→x≤y (s≤s lt) = lt

sx<py→x<y : {x y : ℕ } → suc x < suc y → x  < y 
sx<py→x<y (s≤s lt) = lt

sx≤y→x≤y : {x y : ℕ } → suc x ≤ y → x  ≤ y 
sx≤y→x≤y {zero} {suc y} (s≤s le) = z≤n
sx≤y→x≤y {suc x} {suc y} (s≤s le) = s≤s (sx≤y→x≤y {x} {y} le)

x<sy→x≤y : {x y : ℕ } → x < suc y → x  ≤ y 
x<sy→x≤y {zero} {suc y} (s≤s le) = z≤n
x<sy→x≤y {suc x} {suc y} (s≤s le) = s≤s (x<sy→x≤y {x} {y} le)
x<sy→x≤y {zero} {zero} (s≤s z≤n) = ≤-refl

x≤y→x<sy : {x y : ℕ } → x ≤ y → x < suc y 
x≤y→x<sy {.zero} {y} z≤n = ≤-trans a<sa (s≤s z≤n)
x≤y→x<sy {.(suc _)} {.(suc _)} (s≤s le) = s≤s ( x≤y→x<sy le) 

sx≤y→x<y : {x y : ℕ } → suc x ≤ y → x < y 
sx≤y→x<y {zero} {suc y} (s≤s le) = s≤s z≤n
sx≤y→x<y {suc x} {suc y} (s≤s le) = s≤s ( sx≤y→x<y {x} {y} le )

open import Data.Product

i-j=0→i=j : {i j  : ℕ } → j ≤ i  → i - j ≡ 0 → i ≡ j
i-j=0→i=j {zero} {zero} _ refl = refl
i-j=0→i=j {zero} {suc j} () refl
i-j=0→i=j {suc i} {zero} z≤n ()
i-j=0→i=j {suc i} {suc j} (s≤s lt) eq = cong suc (i-j=0→i=j {i} {j} lt eq)

m*n=0⇒m=0∨n=0 : {i j : ℕ} → i * j ≡ 0 → (i ≡ 0) ∨ ( j ≡ 0 )
m*n=0⇒m=0∨n=0 {zero} {j} refl = case1 refl
m*n=0⇒m=0∨n=0 {suc i} {zero} eq = case2 refl


minus+1 : {x y  : ℕ } → y ≤ x  → suc (minus x y)  ≡ minus (suc x) y 
minus+1 {zero} {zero} y≤x = refl
minus+1 {suc x} {zero} y≤x = refl
minus+1 {suc x} {suc y} (s≤s y≤x) = minus+1 {x} {y} y≤x 

minus+yz : {x y z : ℕ } → z ≤ y  → x + minus y z  ≡ minus (x + y) z
minus+yz {zero} {y} {z} _ = refl
minus+yz {suc x} {y} {z} z≤y = begin
         suc x + minus y z ≡⟨ cong suc ( minus+yz z≤y ) ⟩
         suc (minus (x + y) z) ≡⟨ minus+1 {x + y} {z} (≤-trans z≤y (subst (λ g → y ≤ g) (+-comm y x) x≤x+y) ) ⟩
         minus (suc x + y) z ∎  where open ≡-Reasoning

minus<=0 : {x y : ℕ } → x ≤ y → minus x y ≡ 0
minus<=0 {0} {zero} z≤n = refl
minus<=0 {0} {suc y} z≤n = refl
minus<=0 {suc x} {suc y} (s≤s le) = minus<=0 {x} {y} le

minus>0 : {x y : ℕ } → x < y → 0 < minus y x 
minus>0 {zero} {suc _} (s≤s z≤n) = s≤s z≤n
minus>0 {suc x} {suc y} (s≤s lt) = minus>0 {x} {y} lt

minus>0→x<y : {x y : ℕ } → 0 < minus y x  → x < y
minus>0→x<y {x} {y} lt with <-cmp x y
... | tri< a ¬b ¬c = a
... | tri≈ ¬a refl ¬c = ⊥-elim ( nat-≡< (sym (minus<=0 {x} ≤-refl)) lt )
... | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (sym (minus<=0 {y} (≤-trans refl-≤s c ))) lt )

minus+y-y : {x y : ℕ } → (x + y) - y  ≡ x
minus+y-y {zero} {y} = minus<=0 {zero + y} {y} ≤-refl 
minus+y-y {suc x} {y} = begin
         (suc x + y) - y ≡⟨ sym (minus+1 {_} {y} x≤y+x) ⟩
         suc ((x + y) - y) ≡⟨ cong suc (minus+y-y {x} {y}) ⟩
         suc x ∎  where open ≡-Reasoning

minus+yx-yz : {x y z : ℕ } → (y + x) - (y + z)  ≡ x - z
minus+yx-yz {x} {zero} {z} = refl
minus+yx-yz {x} {suc y} {z} = minus+yx-yz {x} {y} {z} 

minus+xy-zy : {x y z : ℕ } → (x + y) - (z + y)  ≡ x - z
minus+xy-zy {x} {y} {z} = subst₂ (λ j k → j - k ≡ x - z  ) (+-comm y x) (+-comm y z) (minus+yx-yz {x} {y} {z})

+cancel<l : (x z : ℕ ) {y : ℕ} → y + x < y + z → x < z
+cancel<l x z {zero} lt = lt
+cancel<l x z {suc y} (s≤s lt) = +cancel<l x z {y} lt

+cancel<r : (x z : ℕ ) {y : ℕ} → x + y < z + y → x < z
+cancel<r x z {y} lt = +cancel<l x z (subst₂ (λ j k → j < k ) (+-comm x _) (+-comm z _) lt ) 

y-x<y : {x y : ℕ } → 0 < x → 0 < y  → y - x  <  y
y-x<y {x} {y} 0<x 0<y with <-cmp x (suc y)
... | tri< a ¬b ¬c = +cancel<r (y - x) _ ( begin
         suc ((y - x) + x) ≡⟨ cong suc (minus+n {y} {x} a ) ⟩
         suc y  ≡⟨ +-comm 1 _ ⟩
         y + suc 0  ≤⟨ +-mono-≤ ≤-refl 0<x ⟩
         y + x ∎ )  where open ≤-Reasoning
... | tri≈ ¬a refl ¬c = subst ( λ k → k < y ) (sym (minus<=0 {y} {x} refl-≤s )) 0<y
... | tri> ¬a ¬b c = subst ( λ k → k < y ) (sym (minus<=0 {y} {x} (≤-trans (≤-trans refl-≤s refl-≤s) c))) 0<y -- suc (suc y) ≤ x → y ≤ x

open import Relation.Binary.Definitions

distr-minus-* : {x y z : ℕ } → (minus x y) * z ≡ minus (x * z) (y * z) 
distr-minus-* {x} {zero} {z} = refl
distr-minus-* {x} {suc y} {z} with <-cmp x y
distr-minus-* {x} {suc y} {z} | tri< a ¬b ¬c = begin
          minus x (suc y) * z
        ≡⟨ cong (λ k → k * z ) (minus<=0 {x} {suc y} (x<y→≤ a)) ⟩
           0 * z 
        ≡⟨ sym (minus<=0 {x * z} {z + y * z} le ) ⟩
          minus (x * z) (z + y * z) 
        ∎  where
            open ≡-Reasoning
            le : x * z ≤ z + y * z
            le  = ≤-trans lemma (subst (λ k → y * z ≤ k ) (+-comm _ z ) (x≤x+y {y * z} {z} ) ) where
               lemma : x * z ≤ y * z
               lemma = *≤ {x} {y} {z} (<to≤ a)
distr-minus-* {x} {suc y} {z} | tri≈ ¬a refl ¬c = begin
          minus x (suc y) * z
        ≡⟨ cong (λ k → k * z ) (minus<=0 {x} {suc y} refl-≤s ) ⟩
           0 * z 
        ≡⟨ sym (minus<=0 {x * z} {z + y * z} (lt {x} {z} )) ⟩
          minus (x * z) (z + y * z) 
        ∎  where
            open ≡-Reasoning
            lt : {x z : ℕ } →  x * z ≤ z + x * z
            lt {zero} {zero} = z≤n
            lt {suc x} {zero} = lt {x} {zero}
            lt {x} {suc z} = ≤-trans lemma refl-≤s where
               lemma : x * suc z ≤   z + x * suc z
               lemma = subst (λ k → x * suc z ≤ k ) (+-comm _ z) (x≤x+y {x * suc z} {z}) 
distr-minus-* {x} {suc y} {z} | tri> ¬a ¬b c = +m= {_} {_} {suc y * z} ( begin
           minus x (suc y) * z + suc y * z
        ≡⟨ sym (proj₂ *-distrib-+ z  (minus x (suc y) )  _) ⟩
           ( minus x (suc y) + suc y ) * z
        ≡⟨ cong (λ k → k * z) (minus+n {x} {suc y} (s≤s c))  ⟩
           x * z 
        ≡⟨ sym (minus+n {x * z} {suc y * z} (s≤s (lt c))) ⟩
           minus (x * z) (suc y * z) + suc y * z
        ∎ ) where
            open ≡-Reasoning
            lt : {x y z : ℕ } → suc y ≤ x → z + y * z ≤ x * z
            lt {x} {y} {z} le = *≤ le 

distr-minus-*' : {z x y : ℕ } → z * (minus x y)  ≡ minus (z * x) (z * y) 
distr-minus-*' {z} {x} {y} = begin
        z * (minus x y) ≡⟨ *-comm _ (x - y) ⟩
        (minus x y) * z ≡⟨ distr-minus-* {x} {y} {z} ⟩
        minus (x * z) (y * z) ≡⟨ cong₂ (λ j k → j - k ) (*-comm x z ) (*-comm y z) ⟩
        minus (z * x) (z * y) ∎  where open ≡-Reasoning

minus- : {x y z : ℕ } → suc x > z + y → minus (minus x y) z ≡ minus x (y + z)
minus- {x} {y} {z} gt = +m= {_} {_} {z} ( begin
           minus (minus x y) z + z
        ≡⟨ minus+n {_} {z} lemma ⟩
           minus x y
        ≡⟨ +m= {_} {_} {y} ( begin
              minus x y + y 
           ≡⟨ minus+n {_} {y} lemma1 ⟩
              x
           ≡⟨ sym ( minus+n {_} {z + y} gt ) ⟩
              minus x (z + y) + (z + y)
           ≡⟨ sym ( +-assoc (minus x (z + y)) _  _ ) ⟩
              minus x (z + y) + z + y
           ∎ ) ⟩
           minus x (z + y) + z
        ≡⟨ cong (λ k → minus x k + z ) (+-comm _ y )  ⟩
           minus x (y + z) + z
        ∎  ) where
             open ≡-Reasoning
             lemma1 : suc x > y
             lemma1 = x+y<z→x<z (subst (λ k → k < suc x ) (+-comm z _ ) gt )
             lemma : suc (minus x y) > z
             lemma = <-minus {_} {_} {y} ( subst ( λ x → z + y < suc x ) (sym (minus+n {x} {y}  lemma1 ))  gt )

minus-* : {M k n : ℕ } → n < k  → minus k (suc n) * M ≡ minus (minus k n * M ) M
minus-* {zero} {k} {n} lt = begin
           minus k (suc n) * zero
        ≡⟨ *-comm (minus k (suc n)) zero ⟩
           zero * minus k (suc n) 
        ≡⟨⟩
           0 * minus k n 
        ≡⟨ *-comm 0 (minus k n) ⟩
           minus (minus k n * 0 ) 0
        ∎  where
        open ≡-Reasoning
minus-* {suc m} {k} {n} lt with <-cmp k 1
minus-* {suc m} {.0} {zero} lt | tri< (s≤s z≤n) ¬b ¬c = refl
minus-* {suc m} {.0} {suc n} lt | tri< (s≤s z≤n) ¬b ¬c = refl
minus-* {suc zero} {.1} {zero} lt | tri≈ ¬a refl ¬c = refl
minus-* {suc (suc m)} {.1} {zero} lt | tri≈ ¬a refl ¬c = minus-* {suc m} {1} {zero} lt 
minus-* {suc m} {.1} {suc n} (s≤s ()) | tri≈ ¬a refl ¬c
minus-* {suc m} {k} {n} lt | tri> ¬a ¬b c = begin
           minus k (suc n) * M
        ≡⟨ distr-minus-* {k} {suc n} {M}  ⟩
           minus (k * M ) ((suc n) * M)
        ≡⟨⟩
           minus (k * M ) (M + n * M  )
        ≡⟨ cong (λ x → minus (k * M) x) (+-comm M _ ) ⟩
           minus (k * M ) ((n * M) + M )
        ≡⟨ sym ( minus- {k * M} {n * M} (lemma lt) ) ⟩
           minus (minus (k * M ) (n * M)) M
        ≡⟨ cong (λ x → minus x M ) ( sym ( distr-minus-* {k} {n} )) ⟩
           minus (minus k n * M ) M
        ∎  where
             M = suc m
             lemma : {n k m : ℕ } → n < k  → suc (k * suc m) > suc m + n * suc m
             lemma {zero} {suc k} {m} (s≤s lt) = s≤s (s≤s (subst (λ x → x ≤ m + k * suc m) (+-comm 0 _ ) x≤x+y ))
             lemma {suc n} {suc k} {m} lt = begin
                         suc (suc m + suc n * suc m) 
                      ≡⟨⟩
                         suc ( suc (suc n) * suc m)
                      ≤⟨ ≤-plus-0 {_} {_} {1} (*≤ lt ) ⟩
                         suc (suc k * suc m)
                      ∎   where open ≤-Reasoning
             open ≡-Reasoning

x=y+z→x-z=y : {x y z : ℕ } → x ≡ y + z → x - z ≡ y
x=y+z→x-z=y {x} {zero} {.x} refl = minus<=0 {x} {x} refl-≤ -- x ≡ suc (y + z) → (x ≡ y + z → x - z ≡ y)   → (x - z) ≡ suc y
x=y+z→x-z=y {suc x} {suc y} {zero} eq = begin -- suc x ≡ suc (y + zero) → (suc x - zero) ≡ suc y
       suc x - zero ≡⟨ refl ⟩
       suc x  ≡⟨ eq ⟩
       suc y + zero ≡⟨ +-comm _ zero ⟩
       suc y ∎  where open ≡-Reasoning
x=y+z→x-z=y {suc x} {suc y} {suc z} eq = x=y+z→x-z=y {x} {suc y} {z} ( begin
       x ≡⟨ cong pred eq ⟩
       pred (suc y + suc z) ≡⟨ +-comm _ (suc z)  ⟩
       suc z + y ≡⟨ cong suc ( +-comm _ y ) ⟩
       suc y + z ∎  ) where open ≡-Reasoning

m*1=m : {m : ℕ } → m * 1 ≡ m
m*1=m {zero} = refl
m*1=m {suc m} = cong suc m*1=m

+-cancel-1 : (x y z : ℕ ) → x + y  ≡ x + z  → y ≡ z
+-cancel-1 zero y z eq = eq
+-cancel-1 (suc x) y z eq = +-cancel-1 x y z (cong pred eq )

+-cancel-0 : (x y z : ℕ ) → y + x ≡ z + x → y ≡ z
+-cancel-0 x y z eq = +-cancel-1 x y z (trans (+-comm x y) (trans eq (sym (+-comm x z)) ))

*-cancel-left : {x y z : ℕ } → x > 0 → x * y ≡ x * z → y ≡ z
*-cancel-left {suc x} {zero} {zero} lt eq = refl
*-cancel-left {suc x} {zero} {suc z} lt eq = ⊥-elim ( nat-≡< eq (s≤s (begin
  x * zero  ≡⟨ *-comm x _ ⟩ 
  zero  ≤⟨ z≤n ⟩ 
  z + x * suc z ∎ ))) where open ≤-Reasoning
*-cancel-left {suc x} {suc y} {zero} lt eq = ⊥-elim ( nat-≡< (sym eq) (s≤s (begin
  x * zero  ≡⟨ *-comm x _ ⟩ 
  zero  ≤⟨ z≤n ⟩ 
  _ ∎ ))) where open ≤-Reasoning
*-cancel-left {suc x} {suc y} {suc z} lt eq with cong pred eq
... | eq1 =  cong suc (*-cancel-left {suc x} {y} {z} lt (+-cancel-0 x _ _ (begin
   y + x * y + x ≡⟨ +-assoc y _ _ ⟩ 
   y + (x * y + x) ≡⟨ cong (λ k → y + (k + x)) (*-comm x _)  ⟩ 
   y + (y * x + x) ≡⟨ cong (_+_ y) (+-comm _ x) ⟩ 
   y + (x + y * x ) ≡⟨ refl ⟩ 
   y + suc y * x ≡⟨ cong (_+_ y) (*-comm (suc y) _)  ⟩ 
   y + x * suc y ≡⟨ eq1 ⟩ 
   z + x * suc z ≡⟨ refl ⟩ 
   _ ≡⟨ sym ( cong (_+_ z) (*-comm (suc z) _) ) ⟩ 
   _ ≡⟨ sym ( cong (_+_ z) (+-comm _ x)) ⟩ 
   z + (z * x + x) ≡⟨ sym ( cong (λ k → z + (k + x)) (*-comm x _) ) ⟩ 
   z + (x * z + x) ≡⟨ sym ( +-assoc z _ _) ⟩ 
   z + x * z + x  ∎ ))) where open ≡-Reasoning

record Finduction {n m : Level} (P : Set n ) (Q : P → Set m ) (f : P → ℕ) : Set  (n Level.⊔ m) where
  field
    fzero   : {p : P} → f p ≡ zero → Q p
    pnext : (p : P ) → P
    decline : {p : P} → 0 < f p  → f (pnext p) < f p
    ind : {p : P} → Q (pnext p) → Q p

y<sx→y≤x : {x y : ℕ} → y < suc x → y ≤ x
y<sx→y≤x (s≤s lt) = lt 

fi0 : (x : ℕ) → x ≤ zero → x ≡ zero
fi0 .0 z≤n = refl

f-induction : {n m : Level} {P : Set n } → {Q : P → Set m }
  → (f : P → ℕ) 
  → Finduction P Q f
  → (p : P ) → Q p
f-induction {n} {m} {P} {Q} f I p with <-cmp 0 (f p)
... | tri> ¬a ¬b ()
... | tri≈ ¬a b ¬c = Finduction.fzero I (sym b) 
... | tri< lt _ _ = f-induction0 p (f p) (<to≤ (Finduction.decline I lt)) where 
   f-induction0 : (p : P) → (x : ℕ) → (f (Finduction.pnext I p)) ≤ x → Q p
   f-induction0 p zero le = Finduction.ind I (Finduction.fzero I (fi0 _ le)) 
   f-induction0 p (suc x) le with <-cmp (f (Finduction.pnext I p)) (suc x)
   ... | tri< (s≤s a) ¬b ¬c = f-induction0 p x a 
   ... | tri≈ ¬a b ¬c = Finduction.ind I (f-induction0 (Finduction.pnext I p) x (y<sx→y≤x f1)) where
       f1 : f (Finduction.pnext I (Finduction.pnext I p)) < suc x
       f1 = subst (λ k → f (Finduction.pnext I (Finduction.pnext I p)) < k ) b ( Finduction.decline I {Finduction.pnext I p}
         (subst (λ k → 0 < k ) (sym b) (s≤s z≤n ) ))
   ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> le c ) 


record Ninduction {n m : Level} (P : Set n ) (Q : P → Set m ) (f : P → ℕ) : Set  (n Level.⊔ m) where
  field
    pnext : (p : P ) → P
    fzero   : {p : P} → f (pnext p) ≡ zero → Q p
    decline : {p : P} → 0 < f p  → f (pnext p) < f p
    ind : {p : P} → Q (pnext p) → Q p

s≤s→≤ : { i j : ℕ} → suc i ≤ suc j → i ≤ j
s≤s→≤ (s≤s lt) = lt

n-induction : {n m : Level} {P : Set n } → {Q : P → Set m }
  → (f : P → ℕ) 
  → Ninduction P Q f
  → (p : P ) → Q p
n-induction {n} {m} {P} {Q} f I p  = f-induction0 p (f (Ninduction.pnext I p)) ≤-refl where 
   f-induction0 : (p : P) → (x : ℕ) → (f (Ninduction.pnext I p)) ≤ x  →  Q p
   f-induction0 p zero lt = Ninduction.fzero I {p} (fi0 _ lt) 
   f-induction0 p (suc x) le with <-cmp (f (Ninduction.pnext I p)) (suc x) 
   ... | tri< (s≤s a)  ¬b ¬c = f-induction0 p x a
   ... | tri≈ ¬a b ¬c = Ninduction.ind I (f-induction0 (Ninduction.pnext I p) x (s≤s→≤ nle) ) where
      f>0 :  0 < f (Ninduction.pnext I p)
      f>0 = subst (λ k → 0 < k ) (sym b) ( s≤s z≤n ) 
      nle : suc (f (Ninduction.pnext I (Ninduction.pnext I p))) ≤ suc x
      nle = subst (λ k → suc (f (Ninduction.pnext I (Ninduction.pnext I p))) ≤ k) b (Ninduction.decline I {Ninduction.pnext I p} f>0 ) 
   ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> le c )  


record Factor (n m : ℕ ) : Set where
   field 
      factor : ℕ
      remain : ℕ
      is-factor : factor * n + remain ≡ m

record Dividable (n m : ℕ ) : Set where
   field 
      factor : ℕ
      is-factor : factor * n + 0 ≡ m 

open Factor

DtoF : {n m : ℕ} → Dividable n m → Factor n m
DtoF {n} {m} record { factor = f ; is-factor = fa } = record { factor = f ; remain = 0 ; is-factor = fa }

FtoD : {n m : ℕ} → (fc : Factor n m) → remain fc ≡ 0 → Dividable n m 
FtoD {n} {m} record { factor = f ; remain = r ; is-factor = fa } refl = record { factor = f ; is-factor = fa }

--divdable^2 : ( n k : ℕ ) → Dividable k ( n * n ) → Dividable k n
--divdable^2 n k dn2 = {!!}

decf : { n k : ℕ } → ( x : Factor k (suc n) ) → Factor k n
decf {n} {k} record { factor = f ; remain = r ; is-factor = fa } = 
 decf1 {n} {k} f r fa where
  decf1 : { n k : ℕ } → (f r : ℕ) → (f * k + r ≡ suc n)  → Factor k n 
  decf1 {n} {k} f (suc r) fa  =  -- this case must be the first
     record { factor = f ; remain = r ; is-factor = ( begin -- fa : f * k + suc r ≡ suc n
        f * k + r ≡⟨ cong pred ( begin
          suc ( f * k + r ) ≡⟨ +-comm _ r ⟩
          r + suc (f * k)  ≡⟨ sym (+-assoc r 1 _) ⟩
          (r + 1) + f * k ≡⟨ cong (λ t → t + f * k ) (+-comm r 1) ⟩
          (suc r ) + f * k ≡⟨ +-comm (suc r) _ ⟩
          f * k + suc r  ≡⟨ fa ⟩
          suc n ∎ ) ⟩ 
        n ∎ ) }  where open ≡-Reasoning
  decf1 {n} {zero} (suc f) zero fa  = ⊥-elim ( nat-≡< fa (
        begin suc (suc f * zero + zero) ≡⟨ cong suc (+-comm _ zero)  ⟩
        suc (f * 0) ≡⟨ cong suc (*-comm f zero)  ⟩
        suc zero ≤⟨ s≤s z≤n ⟩
        suc n ∎ )) where open ≤-Reasoning
  decf1 {n} {suc k} (suc f) zero fa  = 
     record { factor = f ; remain = k ; is-factor = ( begin -- fa : suc (k + f * suc k + zero) ≡ suc n
        f * suc k + k ≡⟨ +-comm _ k ⟩
        k + f * suc k ≡⟨ +-comm zero _ ⟩
        (k + f * suc k) + zero  ≡⟨ cong pred fa ⟩
        n ∎ ) }  where open ≡-Reasoning

div0 :  {k : ℕ} → Dividable k 0
div0 {k} = record { factor = 0; is-factor = refl }

div= :  {k : ℕ} → Dividable k k
div= {k} = record { factor = 1; is-factor = ( begin
        k + 0 * k + 0  ≡⟨ trans ( +-comm _ 0) ( +-comm _ 0) ⟩
        k ∎ ) }  where open ≡-Reasoning

div1 : { k : ℕ } → k > 1 →  ¬  Dividable k 1
div1 {k} k>1 record { factor = (suc f) ; is-factor = fa } = ⊥-elim ( nat-≡< (sym fa) ( begin
    2 ≤⟨ k>1 ⟩
    k ≡⟨ +-comm 0 _ ⟩
    k + 0 ≡⟨ refl  ⟩
    1 * k ≤⟨ *-mono-≤ {1} {suc f} (s≤s z≤n ) ≤-refl ⟩
    suc f * k ≡⟨ +-comm 0 _ ⟩
    suc f * k + 0 ∎  )) where open ≤-Reasoning  

div+div : { i j k : ℕ } →  Dividable k i →  Dividable k j → Dividable k (i + j) ∧ Dividable k (j + i)
div+div {i} {j} {k} di dj = ⟪ div+div1 , subst (λ g → Dividable k g) (+-comm i j) div+div1 ⟫ where
      fki = Dividable.factor di
      fkj = Dividable.factor dj
      div+div1 : Dividable k (i + j)
      div+div1 = record { factor = fki + fkj  ; is-factor = ( begin 
          (fki + fkj) * k + 0 ≡⟨ +-comm _ 0 ⟩
          (fki + fkj) * k  ≡⟨ *-distribʳ-+ k fki _ ⟩
          fki * k + fkj * k  ≡⟨ cong₂ ( λ i j → i + j ) (+-comm 0 (fki * k)) (+-comm 0 (fkj * k)) ⟩
          (fki * k + 0) + (fkj * k + 0) ≡⟨ cong₂ ( λ i j → i + j ) (Dividable.is-factor di) (Dividable.is-factor dj) ⟩
          i + j  ∎ ) } where
             open ≡-Reasoning  

div-div : { i j k : ℕ } → k > 1 →  Dividable k i →  Dividable k j → Dividable k (i - j) ∧ Dividable k (j - i)
div-div {i} {j} {k} k>1 di dj = ⟪ div-div1 di dj , div-div1 dj di ⟫ where
      div-div1 : {i j : ℕ } → Dividable k i →  Dividable k j → Dividable k (i - j)
      div-div1 {i} {j} di dj = record { factor = fki - fkj  ; is-factor = ( begin 
          (fki - fkj) * k + 0 ≡⟨ +-comm _ 0 ⟩
          (fki - fkj) * k  ≡⟨ distr-minus-* {fki} {fkj}  ⟩
          (fki * k) - (fkj * k)  ≡⟨ cong₂ ( λ i j → i - j ) (+-comm 0 (fki * k)) (+-comm 0 (fkj * k)) ⟩
          (fki * k + 0) - (fkj * k + 0) ≡⟨ cong₂ ( λ i j → i - j ) (Dividable.is-factor di) (Dividable.is-factor dj) ⟩
          i - j  ∎ ) } where
             open ≡-Reasoning  
             fki = Dividable.factor di
             fkj = Dividable.factor dj

open _∧_

div+1 : { i k : ℕ } → k > 1 →  Dividable k i →  ¬ Dividable k (suc i)  
div+1 {i} {k} k>1 d d1 = div1 k>1 div+11 where
   div+11 : Dividable k 1
   div+11 = subst (λ g → Dividable k g) (minus+y-y {1} {i} ) ( proj2 (div-div k>1 d d1  ) )

div<k : { m k : ℕ } → k > 1 → m > 0 →  m < k →  ¬ Dividable k m
div<k {m} {k} k>1 m>0 m<k d = ⊥-elim ( nat-≤> (div<k1 (Dividable.factor d) (Dividable.is-factor d)) m<k ) where
    div<k1 : (f : ℕ ) → f * k + 0 ≡ m → k ≤ m
    div<k1 zero eq = ⊥-elim (nat-≡< eq m>0 )
    div<k1 (suc f) eq = begin
          k ≤⟨ x≤x+y ⟩
          k + (f * k + 0) ≡⟨ sym (+-assoc k _ _) ⟩
          k + f * k + 0 ≡⟨ eq ⟩
          m ∎  where open ≤-Reasoning  

0<factor : { m k : ℕ } → k > 0 → m > 0 →  (d :  Dividable k m ) → Dividable.factor d > 0
0<factor {m} {k} k>0 m>0 d with Dividable.factor d in eq1 
... | zero = ⊥-elim ( nat-≡< ff1 m>0 ) where
    ff1 : 0 ≡ m 
    ff1 = begin
          0 ≡⟨⟩
          0 * k + 0 ≡⟨ cong  (λ j → j * k + 0) (sym eq1) ⟩
          Dividable.factor d * k + 0 ≡⟨ Dividable.is-factor d  ⟩
          m ∎  where open ≡-Reasoning  
... | suc t = s≤s z≤n 

div→k≤m : { m k : ℕ } → k > 1 → m > 0 →  Dividable k m → m ≥ k
div→k≤m {m} {k} k>1 m>0 d with <-cmp m k
... | tri< a ¬b ¬c = ⊥-elim ( div<k k>1 m>0 a d )
... | tri≈ ¬a refl ¬c = ≤-refl
... | tri> ¬a ¬b c = <to≤ c

div1*k+0=k : {k : ℕ } → 1 * k + 0 ≡ k
div1*k+0=k {k} =  begin
   1 * k + 0 ≡⟨ cong (λ g → g + 0) (+-comm _ 0) ⟩
   k + 0 ≡⟨ +-comm _ 0 ⟩
   k  ∎ where open ≡-Reasoning

decD : {k m : ℕ} → k > 1 → Dec (Dividable k m )
decD {k} {m} k>1 = n-induction {_} {_} {ℕ} {λ m → Dec (Dividable k m ) } F I m where
        F : ℕ → ℕ
        F m = m
        F0 : ( m : ℕ ) → F (m - k) ≡ 0 →  Dec  (Dividable k m  )
        F0 0 eq = yes record { factor = 0 ; is-factor = refl }
        F0 (suc m) eq with <-cmp k (suc m)
        ... | tri< a ¬b ¬c = yes  record { factor = 1 ; is-factor =
            subst (λ g → 1 * k + 0 ≡ g ) (sym (i-j=0→i=j (<to≤ a) eq )) div1*k+0=k }  -- (suc m - k) ≡ 0 → k ≡ suc m, k ≤ suc m
        ... | tri≈ ¬a refl ¬c =  yes   record { factor = 1 ; is-factor = div1*k+0=k } 
        ... | tri> ¬a ¬b c = no ( λ d →  ⊥-elim (div<k k>1 (s≤s z≤n ) c d) )
        decl : {m  : ℕ } → 0 < m → m - k < m
        decl {m} 0<m = y-x<y (<-trans a<sa k>1 ) 0<m 
        ind : (p : ℕ ) → Dec (Dividable k (p - k) ) → Dec (Dividable k p )
        ind p (yes y) with <-cmp p k
        ... | tri≈ ¬a refl ¬c = yes (subst (λ g → Dividable k g) (minus+n ≤-refl ) (proj1 ( div+div y div= ))) 
        ... | tri> ¬a ¬b k<p  = yes (subst (λ g → Dividable k g) (minus+n (<-trans k<p a<sa)) (proj1 ( div+div y div= ))) 
        ... | tri< a ¬b ¬c with <-cmp p 0
        ... | tri≈ ¬a refl ¬c₁ = yes div0
        ... | tri> ¬a ¬b₁ c = no (λ d → not-div p (Dividable.factor d) a c (Dividable.is-factor d) ) where
            not-div : (p f : ℕ) → p < k  → 0 < p → f * k + 0 ≡ p → ⊥
            not-div (suc p) (suc f) p<k 0<p eq = nat-≡< (sym eq) ( begin -- ≤-trans p<k {!!}) -- suc p ≤ k
              suc (suc p) ≤⟨ p<k ⟩
              k ≤⟨ x≤x+y  ⟩
              k + (f * k + 0) ≡⟨ sym (+-assoc k _ _) ⟩
              suc f * k + 0 ∎  ) where open ≤-Reasoning  
        ind p (no n) = no (λ d → n (proj1 (div-div k>1 d div=))  ) 
        I : Ninduction ℕ  _  F
        I = record {
              pnext = λ p → p - k
            ; fzero = λ {m} eq → F0 m eq
            ; decline = λ {m} lt → decl lt 
            ; ind = λ {p} prev → ind p prev
          }