mirror of https://github.com/nmvdw/HITs-Examples
60 lines
1.5 KiB
Plaintext
60 lines
1.5 KiB
Plaintext
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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module Mod2 where
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private
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data M' : Set where
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Zero : M'
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S : M' -> M'
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M : Set
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M = M'
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z : M
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z = Zero
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Succ : M -> M
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Succ = S
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postulate
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mod : (n : M) -> n == Succ(Succ n)
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trunc : is-set M
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M-ind : (C : M -> Set)
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-> (a : C Zero)
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-> (sC : (x : M) -> C x -> C (S x))
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-> (p : (n : M) (c : C n) -> PathOver C (mod n) c (sC (Succ n) (sC n c)))
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-> (t : (m : M) -> is-set (C m))
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-> (x : M) -> C x
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M-ind C a sC _ t Zero = a
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M-ind C a sC p t (S x) = sC x (M-ind C a sC p t x)
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postulate
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M-ind-βmod : (C : M -> Set)
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-> (a : C Zero)
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-> (sC : (x : M) -> C x -> C (S x))
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-> (p : (n : M) (c : C n) -> PathOver C (mod n) c (sC (Succ n) (sC n c)))
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-> (t : (m : M) -> is-set (C m))
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-> (n : M)
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-> apd (M-ind C a sC p t) (mod n) == p n (M-ind C a sC p t n)
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M-rec : {C : Set}
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-> (a : C)
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-> (sC : C -> C)
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-> (p : (c : C) -> c == sC (sC c))
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-> (t : is-set C)
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-> M -> C
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M-rec a sC _ _ Zero = a
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M-rec a sC p t (S x) = sC (M-rec a sC p t x)
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postulate
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M-rec-βmod : {C : Set}
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-> (a : C)
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-> (sC : C -> C)
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-> (p : (c : C) -> c == sC (sC c))
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-> (t : is-set C)
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-> (n : M)
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-> ap (M-rec a sC p t) (mod n) == p (M-rec a sC p t n)
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