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{-# OPTIONS --without-K --safe #-}
module Cfe.Fin.Properties where
open import Cfe.Fin.Base
open import Data.Empty using (⊥-elim)
open import Data.Fin using (zero; suc; toℕ; punchIn; punchOut; inject₁)
open import Data.Nat using (suc; pred; _≤_; _<_; _≥_; z≤n; s≤s)
open import Data.Nat.Properties using (suc-injective; pred-mono; module ≤-Reasoning)
open import Function using (_∘_)
open import Relation.Binary.PropositionalEquality
------------------------------------------------------------------------
-- Properties missing from Data.Fin.Properties
------------------------------------------------------------------------
inject₁-mono : ∀ {n i j} → toℕ {n} i ≤ toℕ {n} j → toℕ (inject₁ i) ≤ toℕ (inject₁ j)
inject₁-mono {i = zero} i≤j = z≤n
inject₁-mono {i = suc i} {suc j} (s≤s i≤j) = s≤s (inject₁-mono i≤j)
------------------------------------------------------------------------
-- Properties of toℕ<
------------------------------------------------------------------------
toℕ<<i : ∀ {n i} j → toℕ< {n} {i} j < toℕ i
toℕ<<i zero = s≤s z≤n
toℕ<<i (suc j) = s≤s (toℕ<<i j)
------------------------------------------------------------------------
-- Properties of toℕ>
toℕ>≥i : ∀ {n i} j → toℕ> {n} {i} j ≥ toℕ i
toℕ>≥i zero = z≤n
toℕ>≥i (suc j) = z≤n
toℕ>≥i (inj j) = s≤s (toℕ>≥i j)
------------------------------------------------------------------------
-- Properties of inject!<
------------------------------------------------------------------------
toℕ-inject!< : ∀ {n i} j → toℕ (inject!< {n} {i} j) ≡ toℕ< j
toℕ-inject!< {suc _} zero = refl
toℕ-inject!< {suc _} (suc j) = cong suc (toℕ-inject!< j)
inject!<-mono :
∀ {m n i j k l} → toℕ< k ≤ toℕ< l → toℕ (inject!< {m} {i} k) ≤ toℕ (inject!< {n} {j} l)
inject!<-mono {k = k} {l} k≤l = begin
toℕ (inject!< k) ≡⟨ toℕ-inject!< k ⟩
toℕ< k ≤⟨ k≤l ⟩
toℕ< l ≡˘⟨ toℕ-inject!< l ⟩
toℕ (inject!< l) ∎
where open ≤-Reasoning
inject!<-cong :
∀ {m n i j k l} → toℕ< k ≡ toℕ< l → toℕ (inject!< {m} {i} k) ≡ toℕ (inject!< {n} {j} l)
inject!<-cong {k = k} {l} k≡l = begin
toℕ (inject!< k) ≡⟨ toℕ-inject!< k ⟩
toℕ< k ≡⟨ k≡l ⟩
toℕ< l ≡˘⟨ toℕ-inject!< l ⟩
toℕ (inject!< l) ∎
where open ≡-Reasoning
------------------------------------------------------------------------
-- Properties of inject*<*
------------------------------------------------------------------------
inject-square : ∀ {n i j} k → inject< (inject!<< {n} {i} {j} k) ≡ inject!< (inject<< k)
inject-square {suc n} {suc i} zero = refl
inject-square {suc n} {suc i} (suc k) = cong suc (inject-square k)
------------------------------------------------------------------------
-- Properties of strengthen<
------------------------------------------------------------------------
toℕ-strengthen< : ∀ {n} i → toℕ< (strengthen< {n} i) ≡ toℕ i
toℕ-strengthen< zero = refl
toℕ-strengthen< (suc i) = cong suc (toℕ-strengthen< i)
strengthen<-inject!< : ∀ {n i} j → toℕ< (strengthen< (inject!< {n} {i} j)) ≡ toℕ< j
strengthen<-inject!< {suc _} zero = refl
strengthen<-inject!< {suc _} (suc j) = cong suc (strengthen<-inject!< j)
------------------------------------------------------------------------
-- Properties of cast<
------------------------------------------------------------------------
toℕ-cast< : ∀ {m n i j} i≡j k → toℕ< (cast< {m} {n} {i} {j} i≡j k) ≡ toℕ< k
toℕ-cast< {i = suc _} {suc _} i≡j zero = refl
toℕ-cast< {i = suc _} {suc _} i≡j (suc k) = cong suc (toℕ-cast< (cong pred i≡j) k)
------------------------------------------------------------------------
-- Properties of cast>
------------------------------------------------------------------------
toℕ-cast> : ∀ {n i j} j≤i k → toℕ> (cast> {n} {i} {j} j≤i k) ≡ toℕ> k
toℕ-cast> {_} {zero} {zero} j≤i zero = refl
toℕ-cast> {_} {zero} {zero} j≤i (suc k) = cong suc (toℕ-cast> j≤i k)
toℕ-cast> {suc (suc n)} {suc i} {zero} j≤i (inj k) = cong suc (toℕ-cast> z≤n k)
toℕ-cast> {suc (suc n)} {suc i} {suc j} j≤i (inj k) = cong suc (toℕ-cast> (pred-mono j≤i) k)
------------------------------------------------------------------------
-- Properties of raise!>
------------------------------------------------------------------------
toℕ-raise!> : ∀ {n i} j → toℕ (raise!> {n} {i} j) ≡ toℕ> j
toℕ-raise!> zero = refl
toℕ-raise!> (suc j) = cong suc (toℕ-raise!> j)
toℕ-raise!> {suc n} (inj j) = cong suc (toℕ-raise!> j)
raise!>-cong : ∀ {m n i j k l} → toℕ> k ≡ toℕ> l → toℕ (raise!> {m} {i} k) ≡ toℕ (raise!> {n} {j} l)
raise!>-cong {k = k} {l} k≡l = begin
toℕ (raise!> k) ≡⟨ toℕ-raise!> k ⟩
toℕ> k ≡⟨ k≡l ⟩
toℕ> l ≡˘⟨ toℕ-raise!> l ⟩
toℕ (raise!> l) ∎
where open ≡-Reasoning
------------------------------------------------------------------------
-- Properties of suc>
------------------------------------------------------------------------
toℕ-suc> : ∀ {n i} j → toℕ> (suc> {n} {i} j) ≡ suc (toℕ> j)
toℕ-suc> zero = refl
toℕ-suc> (suc j) = cong suc (toℕ-suc> j)
toℕ-suc> (inj j) = cong suc (toℕ-suc> j)
------------------------------------------------------------------------
-- Properties of predⁱ<
------------------------------------------------------------------------
toℕ-predⁱ< : ∀ {n i} j → suc (toℕ (predⁱ< {n} {i} j)) ≡ toℕ i
toℕ-predⁱ< {i = suc _} _ = refl
predⁱ<-mono :
∀ {n i j} k l → toℕ i ≤ toℕ j → toℕ (predⁱ< {n} {i} k) ≤ toℕ (predⁱ< {n} {j} l)
predⁱ<-mono {i = i} {j} k l i≤j = pred-mono (begin
suc (toℕ (predⁱ< k)) ≡⟨ toℕ-predⁱ< k ⟩
toℕ i ≤⟨ i≤j ⟩
toℕ j ≡˘⟨ toℕ-predⁱ< l ⟩
suc (toℕ (predⁱ< l)) ∎)
where open ≤-Reasoning
predⁱ<-cong :
∀ {n i j} k l → toℕ i ≡ toℕ j → toℕ (predⁱ< {n} {i} k) ≡ toℕ (predⁱ< {n} {j} l)
predⁱ<-cong {i = i} {j} k l i≡j = suc-injective (begin
suc (toℕ (predⁱ< k)) ≡⟨ toℕ-predⁱ< k ⟩
toℕ i ≡⟨ i≡j ⟩
toℕ j ≡˘⟨ toℕ-predⁱ< l ⟩
suc (toℕ (predⁱ< l)) ∎)
where open ≡-Reasoning
------------------------------------------------------------------------
-- Properties of inject₁ⁱ>
------------------------------------------------------------------------
toℕ-inject₁ⁱ> : ∀ {n i} j → toℕ (inject₁ⁱ> {n} {i} j) ≡ toℕ i
toℕ-inject₁ⁱ> {suc _} zero = refl
toℕ-inject₁ⁱ> {suc _} (suc k) = refl
toℕ-inject₁ⁱ> {suc _} (inj k) = cong suc (toℕ-inject₁ⁱ> k)
inject₁ⁱ>-mono :
∀ {n i j} k l → toℕ i ≤ toℕ j → toℕ (inject₁ⁱ> {n} {i} k) ≤ toℕ (inject₁ⁱ> {n} {j} l)
inject₁ⁱ>-mono {i = i} {j} k l i≤j = begin
toℕ (inject₁ⁱ> k) ≡⟨ toℕ-inject₁ⁱ> k ⟩
toℕ i ≤⟨ i≤j ⟩
toℕ j ≡˘⟨ toℕ-inject₁ⁱ> l ⟩
toℕ (inject₁ⁱ> l) ∎
where open ≤-Reasoning
inject₁ⁱ>-cong :
∀ {n i j} k l → toℕ i ≡ toℕ j → toℕ (inject₁ⁱ> {n} {i} k) ≡ toℕ (inject₁ⁱ> {n} {j} l)
inject₁ⁱ>-cong {i = i} {j} k l i≡j = begin
toℕ (inject₁ⁱ> k) ≡⟨ toℕ-inject₁ⁱ> k ⟩
toℕ i ≡⟨ i≡j ⟩
toℕ j ≡˘⟨ toℕ-inject₁ⁱ> l ⟩
toℕ (inject₁ⁱ> l) ∎
where open ≡-Reasoning
------------------------------------------------------------------------
-- Properties of punchIn>
------------------------------------------------------------------------
toℕ-punchIn> : ∀ {n i} j k → toℕ> (punchIn> {suc n} {i} j k) ≡ toℕ (punchIn (raise!> j) (raise!> k))
toℕ-punchIn> {_} {zero} zero k = sym (cong suc (toℕ-raise!> k))
toℕ-punchIn> {_} {zero} (suc j) zero = refl
toℕ-punchIn> {_} {zero} (suc j) (suc k) = cong suc (toℕ-punchIn> j k)
toℕ-punchIn> {suc _} {suc i} (inj j) (inj k) = cong suc (toℕ-punchIn> j k)
------------------------------------------------------------------------
-- Properties of punchOut>
------------------------------------------------------------------------
toℕ-punchOut> : ∀ {n i j k} j≢k → toℕ> (punchOut> {suc n} {i} {j} {k} j≢k) ≡ toℕ (punchOut j≢k)
toℕ-punchOut> {_} {_} {zero} {zero} j≢k = ⊥-elim (j≢k refl)
toℕ-punchOut> {_} {_} {zero} {suc k} j≢k = sym (toℕ-raise!> k)
toℕ-punchOut> {suc _} {_} {suc j} {zero} j≢k = refl
toℕ-punchOut> {suc _} {_} {suc zero} {suc k} j≢k =
cong suc (toℕ-punchOut> (j≢k ∘ cong suc))
toℕ-punchOut> {suc _} {_} {suc (suc j)} {suc k} j≢k =
cong suc (toℕ-punchOut> {j = suc j} (j≢k ∘ cong suc))
toℕ-punchOut> {suc _} {suc zero} {inj j} {inj k} j≢k =
cong suc (toℕ-punchOut> (j≢k ∘ cong suc))
toℕ-punchOut> {suc _} {suc (suc _)} {inj j} {inj k} j≢k =
cong suc (toℕ-punchOut> (j≢k ∘ cong suc))
------------------------------------------------------------------------
-- Properties of reflect!
------------------------------------------------------------------------
toℕ-reflect! : ∀ {n i} j k → toℕ> (reflect! {n} {i} j k) ≡ toℕ< j
toℕ-reflect! {suc _} zero zero = refl
toℕ-reflect! {suc (suc _)} {suc _} (suc j) zero = cong suc (toℕ-reflect! j zero)
toℕ-reflect! {suc (suc _)} {suc _} (suc j) (suc k) = cong suc (toℕ-reflect! j k)
------------------------------------------------------------------------
-- Properties of reflect
------------------------------------------------------------------------
toℕ-reflect : ∀ {n i} j k → toℕ> (reflect {n} {i} j k) ≡ toℕ< j
toℕ-reflect {suc (suc _)} zero zero = refl
toℕ-reflect {_} {suc (suc _)} (suc j) zero = cong suc (toℕ-reflect j zero)
toℕ-reflect {_} {suc (suc _)} (suc j) (suc k) = cong suc (toℕ-reflect j k)
------------------------------------------------------------------------
-- Other properties
------------------------------------------------------------------------
inj-punchOut :
∀ {n i j k} → (j≢k : inject!< {suc n} {suc i} j ≢ raise!> (inj {suc n} {i} k)) →
toℕ (punchOut j≢k) ≡ toℕ> k
inj-punchOut {j = zero} {k} j≢k = toℕ-raise!> k
inj-punchOut {suc n} {j = suc j} {inj k} j≢k = cong suc (inj-punchOut (j≢k ∘ cong suc))
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