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Cleanup finset and modulo (#92)

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Patrick Stevens
2020-01-01 10:14:55 +00:00
committed by GitHub
parent b6ef9b46f2
commit 019a9d9a07
66 changed files with 1154 additions and 1431 deletions
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16
Sets/FinSetWithK.agda
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@@ -2,12 +2,18 @@
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Sets.FinSet
open import Sets.FinSet.Definition
open import LogicalFormulae
open import Numbers.Naturals.WithK
open import Sets.FinSet.Lemmas
module Sets.FinSetWithK where
private
sgEq : {l m : _} {L : Set l} → {pr : L → Set m} → {a b : Sg L pr} → (underlying a ≡ underlying b) → ({c : L} → (r s : pr c) → r ≡ s) → (a ≡ b)
sgEq {l} {m} {L} {prop} {(a , b1)} {(.a , b)} refl pr2 with pr2 {a} b b1
sgEq {l} {m} {L} {prop} {(a , b1)} {(.a , .b1)} refl pr2 | refl = refl
finNotEqualsRefl : {n : ℕ} {a b : FinSet (succ n)} → (p1 p2 : FinNotEquals a b) → p1 ≡ p2
finNotEqualsRefl {.1} {.fzero} {.(fsucc fzero)} (fne2 .fzero .(fsucc fzero) (inl (refl ,, refl))) (fne2 .fzero .(fsucc fzero) (inl (refl ,, refl))) = refl
finNotEqualsRefl {.1} {.fzero} {.(fsucc fzero)} (fne2 .fzero .(fsucc fzero) (inl (refl ,, refl))) (fne2 .fzero .(fsucc fzero) (inr (() ,, snd)))
@@ -75,11 +81,3 @@ finNotEqualsRefl {.(succ (succ _))} {.(fsucc a)} {.(fsucc b)} (fneN (fsucc a) (f
r rewrite equalityCommutative t | equalityCommutative t' | equalityCommutative u | equalityCommutative u' = refl
ans : fneN (fsucc a) (fsucc b) (inr ((fst ,, snd1) , snd)) ≡ fneN (fsucc a) (fsucc b) (inr ((fst1 ,, snd2) , b1))
ans = applyEquality (λ t → fneN (fsucc a) (fsucc b) (inr t)) (sgEq r equality)
ofNatToNat : {n : ℕ} → (a : FinSet n) → ofNat (toNat a) (toNatSmaller a) ≡ a
ofNatToNat {zero} ()
ofNatToNat {succ n} fzero = refl
ofNatToNat {succ n} (fsucc a) = applyEquality fsucc indHyp
where
indHyp : ofNat (toNat a) (canRemoveSuccFrom<N (succPreservesInequality (toNatSmaller a))) ≡ a
indHyp rewrite <NRefl (canRemoveSuccFrom<N (succPreservesInequality (toNatSmaller a))) (toNatSmaller a) = ofNatToNat a