mirror of https://github.com/nmvdw/HITs-Examples
114 lines
2.9 KiB
Plaintext
114 lines
2.9 KiB
Plaintext
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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open import Mod2
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module Thms where
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paths_set : (A B : Set) (m : is-set B) (f g : A -> B) (a : A) -> is-set (f a == g a)
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paths_set A B m f g a = \c₁ c₂ q₁ q₂ ->
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prop-has-level-S
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(contr-is-prop (m (f a) (g a) c₁ c₂))
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q₁
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q₂
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trunc_paths : (A : Set) (Y : A -> Set) {x y : A} (p : x == y) (t : is-prop (Y x)) (c₁ : Y x) (c₂ : Y y) -> PathOver Y p c₁ c₂
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trunc_paths A Y p t c₁ c₂ = from-transp! Y p ((prop-has-all-paths t) c₁ (transport! Y p c₂))
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plus : M -> M -> M
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plus n = M-rec
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n
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Succ
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mod
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trunc
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plus_0n : (n : M) -> plus z n == n
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plus_0n = M-ind
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(\n -> plus z n == n)
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idp
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(\x -> \p -> ap Succ p)
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(\x -> \c ->
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trunc_paths M (\ n → plus z n == n) (mod x) (trunc (plus z x) x) c (ap Succ (ap Succ c))
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)
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(\m ->
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paths_set M M trunc (\x -> plus z x) (\x -> x) m
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)
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plus_n0 : (n : M) -> plus n z == n
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plus_n0 = M-ind
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(\n -> plus n z == n)
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idp
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(\x p -> idp)
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(\x c ->
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trunc_paths M (\x -> plus x z == x) (mod x) (trunc x x) c idp
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)
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(\m -> paths_set M M trunc (\x -> plus x z) (\x -> x) m )
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plus_Sn : (n m : M) -> plus (Succ n) m == Succ (plus n m)
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plus_Sn n = M-ind
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(\m -> plus (Succ n) m == Succ (plus n m))
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idp
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(\x p -> ap Succ p)
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(\x c ->
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trunc_paths M (\x -> plus (Succ n) x == Succ (plus n x)) (mod x) (trunc (plus (Succ n) x) (Succ (plus n x))) c (ap Succ (ap Succ c))
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)
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(\m -> paths_set M M trunc (\x -> plus (Succ x) m) (\x -> Succ(plus x m)) n)
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plus_nS : (n m : M) -> plus n (Succ m) == Succ (plus n m)
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plus_nS n m = idp
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not : Bool -> Bool
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not true = false
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not false = true
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not-not : (x : Bool) -> x == not (not x)
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not-not true = idp
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not-not false = idp
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toBool : M -> Bool
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toBool = M-rec
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true
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not
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((\x -> not-not x))
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Bool-is-set
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toBoolS : (n : M) -> toBool (Succ n) == not (toBool n)
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toBoolS = M-ind
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(\n -> toBool (Succ n) == not (toBool n))
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idp
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(\x p -> idp)
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(\n c ->
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trunc_paths M (\x -> toBool (Succ x) == not (toBool x)) (mod n) (Bool-is-set (not (toBool n)) (not (toBool n))) c idp)
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(\m -> paths_set M Bool Bool-is-set (\n -> toBool(Succ n)) (\n -> not(toBool n)) m)
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fromBool : Bool -> M
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fromBool true = z
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fromBool false = Succ z
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fromBoolNot : (b : Bool) -> fromBool (not b) == Succ (fromBool b)
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fromBoolNot true = idp
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fromBoolNot false = mod z
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iso₁ : (b : Bool) -> toBool (fromBool b) == b
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iso₁ true = idp
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iso₁ false = idp
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iso₂ : (n : M) -> fromBool (toBool n) == n
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iso₂ = M-ind
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(\n -> fromBool (toBool n) == n)
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idp
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(\x p ->
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ap fromBool (toBoolS x)
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∙ fromBoolNot (toBool x)
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∙ ap Succ p)
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(\n p -> trunc_paths M
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(λ z₁ → fromBool (toBool z₁) == z₁)
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(mod n)
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(trunc (fromBool (toBool n)) n)
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p
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(ap fromBool (toBoolS (Succ n))
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∙ fromBoolNot (toBool (Succ n))
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∙ ap Succ (ap fromBool (toBoolS n) ∙ fromBoolNot (toBool n) ∙ ap Succ p))
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)
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(\m -> paths_set M M trunc (\n -> fromBool (toBool n)) (\n -> n) m)
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