module Record.Prelude where

open import Data.Nat public
open import Data.Nat.Properties public
open import Data.String using (String) public
open import Relation.Nullary using (yes; no; Dec; ¬_) public
open import Relation.Nullary.Decidable using (True; toWitness; fromWitness) public
open import Function.Base using (case_of_; case_return_of_) public
open import Relation.Binary.PropositionalEquality using (_≡_; _≢_; refl; cong; sym; ≢-sym) public
open import Data.Empty public
open import Data.Product using (_×_; proj₁; proj₂; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩) public
open import Data.List using (List; []; _∷_; _++_; reverse; map; foldr; downFrom) renaming (length to len) public
open import Data.List.Properties using (map-++) public
open import Agda.Builtin.Float renaming (Float to 𝔽; primFloatPlus to _++f_) public
open import Data.Sum using (_⊎_; inj₁; inj₂) renaming ([_,_] to case-⊎) public
open import Data.Nat.Tactic.RingSolver public

m+1≤n→m≤n : ∀ {m n}
  → suc m ≤ n
  → m ≤ n
m+1≤n→m≤n (s≤s m+1≤n) = m≤n⇒m≤1+n m+1≤n

n-1+1≡n+1-1 : ∀ {n}
  →  < n
  → suc (pred n) ≡ pred (suc n)
n-1+1≡n+1-1 (s≤s 0<n) = refl

m+1≰n+1⇒m≰n : ∀ {m n}
  → suc m ≰ suc n
  → m ≰ n
m+1≰n+1⇒m≰n m+1≰n+1 = λ m≤n → m+1≰n+1 (s≤s m≤n)  
  
m≰n⇒n<m : ∀ {m n}
  → m ≰ n
  → n < m
m≰n⇒n<m {zero} {zero} m≰n = ⊥-elim (m≰n z≤n)
m≰n⇒n<m {zero} {suc n} m≰n = ⊥-elim (m≰n z≤n)
m≰n⇒n<m {suc m} {zero} m≰n = s≤s z≤n
m≰n⇒n<m {suc m} {suc n} m≰n = s≤s (m≰n⇒n<m {m} {n} (m+1≰n+1⇒m≰n m≰n))

n<m⇒m≰n : ∀ {m n}
  → n < m
  → m ≰ n
n<m⇒m≰n {suc m} {zero} n<m = λ ()
n<m⇒m≰n {suc m} {suc n} (s≤s n<m) (s≤s m≤n) = n<m⇒m≰n {m} {n} n<m m≤n

m+0≡m : ∀ m
  → m +  ≡ m
m+0≡m m rewrite +-comm m  = refl

m≤m : ∀ {m}
  → m ≤ m
m≤m {zero} = z≤n
m≤m {suc m} = s≤s m≤m

pattern ⟦_⟧ z = z ∷ []
pattern ⟦_,_⟧ a b = a ∷ b ∷ []
pattern ⟦_,_,_⟧ a b c = a ∷ b ∷ c ∷ []
pattern ⟦_,_,_,_⟧ a b c d = a ∷ b ∷ c ∷ d ∷ []

data Singleton {a} {A : Set a} (x : A) : Set a where
  _with≡_ : (y : A) → x ≡ y → Singleton x

inspect : ∀ {a} {A : Set a} (x : A) → Singleton x
inspect x = x with≡ refl

m+n<o⇒m<o : ∀ {m n o}
  → m + n < o
  → m < o
m+n<o⇒m<o {m} {n} {o} m+n<o = ≤-trans (s≤s (m≤m+n m n)) m+n<o

m+n<o⇒n<o : ∀ {m n o}
  → m + n < o
  → n < o
m+n<o⇒n<o {m} {n} {o} m+n<o = ≤-trans (s≤s (m≤n+m n m)) m+n<o