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September 7, 2023 12:33
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{-# OPTIONS --cubical --safe #-} | |
open import Cubical.Foundations.Prelude | |
open import Cubical.Data.List | |
private variable X : Type | |
rev-ind : (P : List X → Type) | |
→ P [] | |
→ (∀ x xs → P xs → P (xs ∷ʳ x)) | |
→ (xs : List X) → P xs | |
rev-ind P bc ic [] = bc | |
-- this is filled by pressing Ctrl-C Ctrl-A, I didn't review it | |
-- maybe there is a better implementation | |
rev-ind P bc ic (x ∷ xs) = rev-ind (λ z → P (x ∷ z)) (ic x [] bc) (λ x xs → ic x (_ ∷ xs)) xs | |
-- parameters are written here so they're all fixed in the following definitions | |
module _ {A : Type} {a : A} where | |
-- random theorem that is probably not suitable for rev-ind | |
demo : (xs : List A) → xs ++ [ a ] ≡ rev (a ∷ rev xs) | |
demo = rev-ind | |
-- the motive, created by "copying" the goal | |
-- and rename xs to the parameter | |
-- no need of those implicit parameters | |
(λ z → z ++ [ a ] ≡ rev (a ∷ rev z)) | |
-- base case is just refl | |
refl | |
-- I don't see how to fill this, but I don't think you need this either | |
{!!} |
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