--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/nat.agda	Mon Aug 24 23:06:10 2020 +0900
@@ -0,0 +1,308 @@
+{-# OPTIONS --allow-unsolved-metas #-}
+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  logic
+
+
+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 
+
+refl-≤s : {x : ℕ } → x ≤ suc x
+refl-≤s {zero} = z≤n
+refl-≤s {suc x} = s≤s (refl-≤s {x})
+
+=→¬< : {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)
+
+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 )
+
+-- _*_ : ℕ → ℕ → ℕ
+-- _*_ zero _ = zero
+-- _*_ (suc n) m = m + ( n * m )
+
+exp : ℕ → ℕ → ℕ
+exp _ zero = 1
+exp n (suc m) = n * ( exp n m )
+
+minus : (a b : ℕ ) →  ℕ
+minus a zero = a
+minus zero (suc b) = zero
+minus (suc a) (suc b) = minus a b
+
+_-_ = minus
+
+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
+
+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
+
+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
+
+0<s : {x : ℕ } → zero < suc x
+0<s {_} = s≤s z≤n 
+
+<-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})
+
+<-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→≤ : {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)
+
+open import Data.Product
+
+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
+
+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 
+
+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