From 9a8dc821469836bce507e276059502ee921949e7 Mon Sep 17 00:00:00 2001
From: Chloe Brown <chloe.brown.00@outlook.com>
Date: Thu, 23 Sep 2021 13:46:05 +0100
Subject: Define Motzkin numbers.

---
 src/Data/List/Relation/Binary/Prefix.agda | 118 ++++++++++++++++++++++++++++++
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 create mode 100644 src/Data/List/Relation/Binary/Prefix.agda

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+module Data.List.Relation.Binary.Prefix where
+
+open import Data.Empty using (⊥-elim)
+open import Data.List.Base using (List; []; _∷_; _++_; length; inits; map)
+open import Data.List.Properties using (length-++)
+open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_; setoid)
+open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
+open import Data.Nat using (_≤_; z≤n; s≤s; _+_) renaming (_<_ to _<ⁿ_)
+open import Data.Nat.Properties using (<⇒≱; m<m+n; module ≤-Reasoning)
+open import Data.Product using (_,_; uncurry)
+open import Function.Equivalence using (_⇔_; equivalence)
+open import Level using (Level; _⊔_)
+open import Relation.Binary
+open import Relation.Nullary using (Dec; yes; no; ¬_)
+import Relation.Nullary.Decidable as Dec
+open import Relation.Nullary.Product using (_×-dec_)
+open import Relation.Unary using (Pred)
+
+private
+  variable
+    a ℓ : Level
+
+module _
+  {A : Set a} (P : Set) (_≈_ : Rel A ℓ)
+  where
+
+  data Prefix : Rel (List A) ℓ where
+    base : P                                    → Prefix [] []
+    halt : ∀ {y ys}                             → Prefix [] (y ∷ ys)
+    next : ∀ {x xs y ys}
+           (x≈y : x ≈ y) (xs<ys : Prefix xs ys) → Prefix (x ∷ xs) (y ∷ ys)
+
+module _
+  {A : Set a} {P : Set} {_≈_ : Rel A ℓ}
+  where
+
+  private
+    _≋_ = Pointwise _≈_
+    _<_ = Prefix P _≈_
+
+  initsPrefixes : P → Reflexive _≈_ → ∀ xs → All (_< xs) (inits xs)
+  initsPrefixes p refl []       = base p ∷ []
+  initsPrefixes p refl (x ∷ xs) = halt ∷ map′ (x ∷_ ) (next refl) (initsPrefixes p refl xs)
+    where
+    map′ : ∀ {a b p q} {A : Set a} {B : Set b} {P : Pred A p} {Q : Pred B q} →
+           (f : A → B) → (∀ {x} → P x → Q (f x)) → ∀ {xs} → All P xs → All Q (map f xs)
+    map′ f pres []         = []
+    map′ f pres (px ∷ pxs) = pres px ∷ map′ f pres pxs
+
+  module _
+    (equivalence : IsEquivalence _≈_)
+    where
+
+    open IsEquivalence equivalence
+    open import Data.List.Membership.Setoid (setoid (record { isEquivalence = equivalence })) using (_∈_)
+    open import Data.List.Relation.Unary.Any using (Any; here; there)
+
+    prefix∈inits : ∀ {xs ys} → xs < ys → xs ∈ inits ys
+    prefix∈inits (base x)         = here []
+    prefix∈inits halt             = here []
+    prefix∈inits (next x≈y xs<ys) = there (map′ (_∷_ _) (x≈y ∷_) (prefix∈inits xs<ys))
+      where
+      map′ : ∀ {a b p q} {A : Set a} {B : Set b} {P : Pred A p} {Q : Pred B q} →
+           (f : A → B) → (∀ {x} → P x → Q (f x)) → ∀ {xs} → Any P xs → Any Q (map f xs)
+      map′ f pres (here px) = here (pres px)
+      map′ f pres (there pxs) = there (map′ f pres pxs)
+
+  reflexive : P → Reflexive _≈_ → Reflexive _<_
+  reflexive p refl {[]}      = base p
+  reflexive p refl {x ∷ xs} = next refl (reflexive p refl)
+
+  irreflexive : ¬ P → Irreflexive _≋_ _<_
+  irreflexive ¬p []          (base p)       = ¬p p
+  irreflexive ¬p (_ ∷ xs≋ys) (next _ xs<ys) = irreflexive ¬p xs≋ys xs<ys
+
+  transitive : Transitive _≈_ → Transitive _<_
+  transitive trans (base _)         ys<zs            = ys<zs
+  transitive trans halt             (next _ _)       = halt
+  transitive trans (next x≈y xs<ys) (next y≈z ys<zs) =
+    next (trans x≈y y≈z) (transitive trans xs<ys ys<zs)
+
+  antisymmetric : Symmetric _≈_ → Antisymmetric _≋_ _<_
+  antisymmetric sym (base _)         (base _)         = []
+  antisymmetric sym (next x≈y xs<ys) (next y≈x ys<xs) =
+    x≈y ∷ antisymmetric sym xs<ys ys<xs
+
+  decidable : Dec P → Decidable _≈_ → Decidable _<_
+  decidable P? _≈?_ []       []       = Dec.map′ base (λ { (base p) → p }) P?
+  decidable P? _≈?_ []       (x ∷ ys) = yes halt
+  decidable P? _≈?_ (x ∷ xs) []       = no λ ()
+  decidable P? _≈?_ (x ∷ xs) (y ∷ ys) =
+    Dec.map′ (uncurry next)
+             (λ { (next x≈y xs<ys) → x≈y , xs<ys })
+             ((x ≈? y) ×-dec (decidable P? _≈?_ xs ys))
+
+  length≤ : ∀ {xs} {ys} → xs < ys → length xs ≤ length ys
+  length≤ (base x)         = z≤n
+  length≤ halt             = z≤n
+  length≤ (next x≈y xs<ys) = s≤s (length≤ xs<ys)
+
+  length< : ∀ {xs} {ys} → ¬ P → xs < ys → length xs <ⁿ length ys
+  length< ¬p (base p)         = ⊥-elim (¬p p)
+  length< ¬p halt             = s≤s z≤n
+  length< ¬p (next x≈y xs<ys) = s≤s (length< ¬p xs<ys)
+
+  minimum : P → Minimum _<_ []
+  minimum p []       = base p
+  minimum p (x ∷ xs) = halt
+
+  ¬maximum : A → ∀ xs → ¬ Maximum _<_ xs
+  ¬maximum a xs max =
+    <⇒≱
+      (begin-strict
+        length xs                   <⟨  m<m+n (length xs) (s≤s z≤n) ⟩
+        length xs + length (a ∷ []) ≡˘⟨ length-++ xs ⟩
+        length (xs ++ a ∷ [])       ∎)
+      (length≤ (max (xs ++ a ∷ [])))
+    where open ≤-Reasoning
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