mirror of https://github.com/nmvdw/HITs-Examples
Added interface of finite stes
This commit is contained in:
Niels
parent
376efbf2e9
commit
d9cde16f5a
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@ -1,98 +0,0 @@
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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module CL where
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private
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data CL' : Set where
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K' : CL'
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S' : CL'
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app' : CL' -> CL' -> CL'
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CL : Set
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CL = CL'
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K : CL
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K = K'
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Sc : CL
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Sc = S'
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app : CL -> CL -> CL
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app = app'
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postulate
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KConv : {x y : CL} -> app (app K x) y == x
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SConv : {x y z : CL} -> app (app (app Sc x) y) z == app (app x z) (app y z)
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CLind : (Y : CL -> Set)
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(KY : Y K)
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(SY : Y Sc)
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(appY : (x y : CL) -> Y x -> Y y -> Y (app x y))
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(KConvY : (x y : CL) (a : Y x) (b : Y y) -> PathOver Y KConv (appY (app K x) y (appY K x KY a) b) a)
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(SConvY : (x y z : CL) (a : Y x) (b : Y y) (c : Y z) ->
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PathOver Y SConv
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(appY
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(app (app Sc x) y)
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z
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(appY
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(app Sc x)
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y
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(appY Sc x SY a)
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b
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)
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c
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)
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(appY (app x z) (app y z) (appY x z a c) (appY y z b c))
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)
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(x : CL) -> Y x
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CLind Y KY SY appY _ _ K' = KY
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CLind Y KY SY appY _ _ S' = SY
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CLind Y KY SY appY KConvY SConvY (app' x x₁) = appY x x₁ (CLind Y KY SY appY KConvY SConvY x) (CLind Y KY SY appY KConvY SConvY x₁)
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postulate
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CLind_βKConv : (Y : CL -> Set)
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(KY : Y K)
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(SY : Y Sc)
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(appY : (x y : CL) -> Y x -> Y y -> Y (app x y))
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(KConvY : (x y : CL) (a : Y x) (b : Y y) -> PathOver Y KConv (appY (app K x) y (appY K x KY a) b) a)
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(SConvY : (x y z : CL) (a : Y x) (b : Y y) (c : Y z) ->
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PathOver Y SConv
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(appY
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(app (app Sc x) y)
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z
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(appY
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(app Sc x)
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y
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(appY Sc x SY a)
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b
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)
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c
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)
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(appY (app x z) (app y z) (appY x z a c) (appY y z b c))
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)
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(x y : CL) ->
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apd (CLind Y KY SY appY KConvY SConvY) KConv == KConvY x y (CLind Y KY SY appY KConvY SConvY x) (CLind Y KY SY appY KConvY SConvY y)
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CLind_βSConv : (Y : CL -> Set)
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(KY : Y K)
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(SY : Y Sc)
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(appY : (x y : CL) -> Y x -> Y y -> Y (app x y))
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(KConvY : (x y : CL) (a : Y x) (b : Y y) -> PathOver Y KConv (appY (app K x) y (appY K x KY a) b) a)
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(SConvY : (x y z : CL) (a : Y x) (b : Y y) (c : Y z) ->
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PathOver Y SConv
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(appY
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(app (app Sc x) y)
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z
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(appY
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(app Sc x)
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y
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(appY Sc x SY a)
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b
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)
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c
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)
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(appY (app x z) (app y z) (appY x z a c) (appY y z b c))
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)
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(x y z : CL) ->
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apd (CLind Y KY SY appY KConvY SConvY) SConv == SConvY x y z (CLind Y KY SY appY KConvY SConvY x) (CLind Y KY SY appY KConvY SConvY y) (CLind Y KY SY appY KConvY SConvY z)
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@ -1,122 +0,0 @@
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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open import CL
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module Thms where
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trans-cst : (A : Set) {x y : A} (B : Set) (p : x == y) (z : B) -> transport (\x -> B) p z == z
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trans-cst A B idp z = idp
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I : CL
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I = app (app Sc K) K
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IConv : {x : CL} -> app I x == x
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IConv {x} = SConv ∙ KConv
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B : CL
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B = app (app Sc (app K Sc)) K
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BConv : {x y z : CL} -> app (app (app B x) y) z == app x (app y z)
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BConv {x} {y} {z} =
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ap (λ p -> app (app p y) z) SConv
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∙ ap (λ p -> app (app (app (p) (app K x)) y) z) KConv
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∙ SConv
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∙ ap (λ p -> app p (app y z)) KConv
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M : CL
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M = app (app Sc I) I
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MConv : {x : CL} -> app M x == app x x
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MConv {x} =
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SConv
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∙ ap (λ p -> app p (app I x)) IConv
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∙ ap (app x) IConv
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T : CL
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T = app (app B (app Sc I)) K
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TConv : {x y : CL} -> app (app T x) y == app y x
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TConv {x} {y} =
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ap (λ p -> app p y) BConv
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∙ SConv
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∙ ap (λ p -> app p (app (app K x) y)) IConv
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∙ ap (app y) KConv
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C : CL
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C =
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app
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(app
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B
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(app
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T
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(app
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(app
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B
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B
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)
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T
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)
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)
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)
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(app
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(app
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B
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B
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)
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T
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)
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CConv : {x y z : CL} -> app (app (app C x) y) z == app (app x z) y
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CConv {x} {y} {z} =
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ap (λ p -> app (app p y) z) BConv
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∙ ap (λ p -> app (app p y) z) TConv
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∙ ap (λ p -> app (app (app p (app (app B B) T)) y) z) BConv
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∙ ap (λ p -> app p z) BConv
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∙ ap (λ p -> app p z) TConv
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∙ ap (λ p -> app (app p x) z) BConv
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∙ BConv
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∙ TConv
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W : CL
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W = app (app C Sc) I
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WConv : {x y : CL} -> app (app W x) y == app (app x y) y
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WConv {x} {y} =
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ap (λ p -> app p y) CConv
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∙ SConv
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∙ ap (app (app x y)) IConv
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B' : CL
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B' = app C B
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B'Conv : {x y z : CL} -> app (app (app B' x) y) z == app y (app x z)
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B'Conv {x} {y} {z} =
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ap (λ p -> app p z) CConv
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∙ BConv
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V : CL
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V = app (app B C) T
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VConv : {x y z : CL} -> app (app (app V x) y) z == app (app z x) y
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VConv {x} {y} {z} =
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ap (λ p -> app (app p y) z) BConv
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∙ CConv
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∙ ap (λ p -> app p y) TConv
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Y : CL
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Y = app (app B' (app B' M)) M
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YConv : {x : CL} -> app Y x == app x (app Y x)
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YConv {x} =
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B'Conv
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∙ MConv
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∙ B'Conv
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∙ ap (app x) (! B'Conv)
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fixpoint : (x : CL) -> Σ CL (λ y -> app x y == y)
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fixpoint x = app Y x , ! YConv
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S' : CL
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S' = app C Sc
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@ -1,59 +0,0 @@
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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module Expressions where
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private
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data Exp' : Set where
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value : Nat -> Exp'
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addition : Exp' -> Exp' -> Exp'
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Exp : Set
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Exp = Exp'
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val : Nat -> Exp
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val = value
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plus : Exp -> Exp -> Exp
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plus = addition
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postulate
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add : (n m : Nat) -> plus (val n) (val m) == val (n + m)
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trunc : is-set Exp
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Exp-ind : (C : Exp -> Set)
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-> (vC : (n : Nat) -> C (val n))
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-> (pC : (e₁ e₂ : Exp) -> C e₁ -> C e₂ -> C(plus e₁ e₂))
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-> (addC : (n m : Nat) -> PathOver C (add n m) (pC (val n) (val m) (vC n) (vC m)) (vC (n + m)))
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-> (t : (e : Exp) -> is-set (C e))
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-> (x : Exp) -> C x
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Exp-ind C vC pC addC t (value n) = vC n
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Exp-ind C vC pC addC t (addition e₁ e₂) = pC e₁ e₂ (Exp-ind C vC pC addC t e₁) (Exp-ind C vC pC addC t e₂)
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postulate
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Exp-ind-βadd : (C : Exp -> Set)
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-> (vC : (n : Nat) -> C (val n))
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-> (pC : (e₁ e₂ : Exp) -> C e₁ -> C e₂ -> C(plus e₁ e₂))
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-> (addC : (n m : Nat) -> PathOver C (add n m) (pC (val n) (val m) (vC n) (vC m)) (vC (n + m)))
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-> (t : (e : Exp) -> is-set (C e))
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-> (n m : Nat)
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-> apd (Exp-ind C vC pC addC t) (add n m) == addC n m
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Exp-rec : {C : Set}
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-> (vC : Nat -> C)
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-> (pC : C -> C -> C)
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-> (addC : (n m : Nat) -> pC (vC n) (vC m) == vC (n + m))
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-> (t : is-set C)
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-> Exp -> C
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Exp-rec vC pC addC t (value n) = vC n
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Exp-rec vC pC addC t (addition e₁ e₂) = pC (Exp-rec vC pC addC t e₁) (Exp-rec vC pC addC t e₂)
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postulate
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Exp-rec-βadd : {C : Set}
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-> (vC : Nat -> C)
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-> (pC : C -> C -> C)
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-> (addC : (n m : Nat) -> pC (vC n) (vC m) == vC (n + m))
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-> (t : is-set C)
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-> (n m : Nat)
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-> ap (Exp-rec vC pC addC t) (add n m) == addC n m
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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open import Expressions
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module Thms where
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value : (e : Exp) -> Σ Nat (\n -> e == val n)
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value = Exp-ind
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(\e -> Σ Nat (\n -> e == val n))
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(\n -> n , idp)
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(\e₁ e₂ v₁ v₂ -> fst v₁ + fst v₂ ,
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(ap (\e -> plus e e₂) (snd v₁) ∙ ap (plus (val (fst v₁))) (snd v₂)) ∙ add (fst v₁) (fst v₂)
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)
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(\n m -> from-transp! (\e -> Σ Nat (\n -> e == val n)) (add n m) (pair= {!!} {!!}))
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(\e -> {!!})
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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open import Syntax
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module Language where
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data Program : Set where
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fail : Program
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exec : Syntax -> State -> Program
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postulate
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assignp : (z : State) (x n : Nat) -> exec (x := n) z == exec skip ( z [ x :== n ])
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comp₁ : (z : State) (S : Syntax) -> exec (conc skip S) z == exec S z
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comp₂ : (z z' : State) (S₁ S₂ S₁' : Syntax) -> exec S₁ z == exec S₁' z' -> exec (conc S₁ S₂) z == exec (conc S₁' S₂) z'
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while₁ : (z : State) (x n : Nat) (S : Syntax) -> defined z x -> equals z x n -> exec (while x == n do S) z == exec (conc S (while x == n do S)) z
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while₂ : (z : State) (x n : Nat) (S : Syntax) -> defined z x -> unequals z x n -> exec (while x == n do S) z == exec skip z
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while₃ : (z : State) (x n : Nat) (S : Syntax) -> undefined z x -> exec (while x == n do S) z == fail
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Program-elim :
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(Y : Set)
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-> (failY : Y)
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-> (execY : Syntax -> State -> Y)
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-> (assignY : (z : State) (x n : Nat) -> execY (x := n) z == execY skip ( z [ x :== n ]) )
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-> (compY₁ : (z : State) (S : Syntax) -> execY (conc skip S) z == execY S z )
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-> (compY₂ : (z z' : State) (S₁ S₂ S₁' : Syntax) -> execY S₁ z == execY S₁' z' -> execY (conc S₁ S₂) z == execY (conc S₁' S₂) z')
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-> (whileY₁ : (z : State) (x n : Nat) (S : Syntax) -> defined z x -> equals z x n -> execY (while x == n do S) z == execY (conc S (while x == n do S)) z)
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-> (whileY₂ : (z : State) (x n : Nat) (S : Syntax) -> defined z x -> unequals z x n -> execY (while x == n do S) z == execY skip z)
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-> (whileY₃ : (z : State) (x n : Nat) (S : Syntax) -> undefined z x -> execY (while x == n do S) z == failY)
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-> Program -> Y
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Program-elim _ failY _ _ _ _ _ _ _ fail = failY
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Program-elim _ _ execY _ _ _ _ _ _ (exec s z) = execY s z
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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module Semantics where
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data koe : Set where
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a : koe
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b : koe
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postulate
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kek : a ↦ b
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{-# REWRITE kek #-}
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Y : koe -> Set
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Y a = Nat
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Y b = Bool
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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module Syntax where
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data Maybe (A : Set) : Set where
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Just : A -> Maybe A
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Nothing : Maybe A
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eqN : Nat -> Nat -> Bool
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eqN 0 0 = true
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eqN 0 _ = false
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eqN (S _) 0 = false
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eqN (S n) (S m) = eqN n m
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-- first coordinate represents the variable x_i, second the value
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State : Set
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State = List (Nat × Nat)
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_[_:==_] : State -> Nat -> Nat -> State
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nil [ x :== n ] = (x , n) :: nil
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((y , m) :: s) [ x :== n ] =
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if eqN x y
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then (x , n) :: s
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else ((y , m) :: (s [ x :== n ]))
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equals : State -> Nat -> Nat -> Set
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equals nil _ _ = Empty
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equals ((x , n) :: s) y m =
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if eqN x y
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then
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if eqN n m
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then Unit
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else Empty
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else equals s y m
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unequals : State -> Nat -> Nat -> Set
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unequals nil _ _ = Unit
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unequals ((x , n) :: s) y m =
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if eqN x y
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then
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if eqN n m
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then Empty
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else Unit
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else unequals s y m
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defined : State -> Nat -> Set
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defined nil y = Empty
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defined ((x , n) :: s) y =
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if eqN x y
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then Unit
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else defined s y
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undefined : State -> Nat -> Set
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undefined nil y = Unit
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undefined ((x , n) :: s) y =
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if eqN x y
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then Empty
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else undefined s y
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data Syntax : Set where
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skip : Syntax
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_:=_ : Nat -> Nat -> Syntax
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conc : Syntax -> Syntax -> Syntax
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while_==_do_ : Nat -> Nat -> Syntax -> Syntax
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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module Integers where
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private
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data Integers : Set where
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z : Integers
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S : Integers -> Integers
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P : Integers -> Integers
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Ints : Set
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Ints = Integers
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nul : Ints
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nul = z
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Succ : Ints -> Ints
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Succ = S
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Pred : Ints -> Ints
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Pred = P
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postulate
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invl : (x : Integers) -> P(S x) == x
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invr : (x : Integers) -> S(P x) == x
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trunc : is-set Ints
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Zind : (Y : Integers -> Set)
|
||||
-> (zY : Y z)
|
||||
-> (SY : (x : Integers) -> Y x -> Y(S x))
|
||||
-> (PY : (x : Integers) -> Y x -> Y(P x))
|
||||
-> (invYl : (x : Integers) (y : Y x) -> PathOver Y (invl x) (PY (S x) (SY x y)) y)
|
||||
-> (invYr : (x : Integers) (y : Y x) -> PathOver Y (invr x) (SY (P x) (PY x y)) y)
|
||||
-> (t : (x : Integers) -> is-set (Y x))
|
||||
-> (x : Integers) -> Y x
|
||||
Zind Y zY SY PY invYl invYr t z = zY
|
||||
Zind Y zY SY PY invYl invYr t (S x) = SY x (Zind Y zY SY PY invYl invYr t x)
|
||||
Zind Y zY SY PY invYl invYr t (P x) = PY x (Zind Y zY SY PY invYl invYr t x)
|
||||
|
||||
postulate
|
||||
Zind-βinvl :
|
||||
(Y : Integers -> Set)
|
||||
-> (zY : Y z)
|
||||
-> (SY : (x : Integers) -> Y x -> Y(S x))
|
||||
-> (PY : (x : Integers) -> Y x -> Y(P x))
|
||||
-> (invYl : (x : Integers) (y : Y x) -> PathOver Y (invl x) (PY (S x) (SY x y)) y)
|
||||
-> (invYr : (x : Integers) (y : Y x) -> PathOver Y (invr x) (SY (P x) (PY x y)) y)
|
||||
-> (t : (x : Integers) -> is-set (Y x))
|
||||
-> (x : Integers)
|
||||
-> apd (Zind Y zY SY PY invYl invYr t) (invl x) == invYl x (Zind Y zY SY PY invYl invYr t x)
|
||||
|
||||
Zind-βinvr :
|
||||
(Y : Integers -> Set)
|
||||
-> (zY : Y z)
|
||||
-> (SY : (x : Integers) -> Y x -> Y(S x))
|
||||
-> (PY : (x : Integers) -> Y x -> Y(P x))
|
||||
-> (invYl : (x : Integers) (y : Y x) -> PathOver Y (invl x) (PY (S x) (SY x y)) y)
|
||||
-> (invYr : (x : Integers) (y : Y x) -> PathOver Y (invr x) (SY (P x) (PY x y)) y)
|
||||
-> (t : (x : Integers) -> is-set (Y x))
|
||||
-> (x : Integers)
|
||||
-> apd (Zind Y zY SY PY invYl invYr t) (invr x) == invYr x (Zind Y zY SY PY invYl invYr t x)
|
||||
|
|
@ -1,205 +0,0 @@
|
|||
{-# OPTIONS --without-K --rewriting #-}
|
||||
|
||||
open import HoTT
|
||||
open import Integers
|
||||
|
||||
module Thms where
|
||||
|
||||
paths_set : (A B : Set) (m : is-set B) (f g : A -> B) (a : A) -> is-set (f a == g a)
|
||||
paths_set A B m f g a = \c₁ c₂ q₁ q₂ ->
|
||||
prop-has-level-S
|
||||
(contr-is-prop (m (f a) (g a) c₁ c₂))
|
||||
q₁
|
||||
q₂
|
||||
|
||||
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
|
||||
trunc_paths A Y p t c₁ c₂ = from-transp! Y p ((prop-has-all-paths t) c₁ (transport! Y p c₂))
|
||||
|
||||
trans-cst : (A : Set) {x y : A} (B : Set) (p : x == y) (z : B) -> transport (\x -> B) p z == z
|
||||
trans-cst A B idp z = idp
|
||||
|
||||
plus : Ints -> Ints -> Ints
|
||||
plus n = Zind
|
||||
(\m -> Ints)
|
||||
n
|
||||
(\m -> Succ)
|
||||
(\m -> Pred)
|
||||
(\x y -> from-transp (λ _ → Ints) (invl x) (trans-cst Ints Ints (invl x) (Pred (Succ y)) ∙ invl y))
|
||||
(\x y -> from-transp (λ _ → Ints) (invr x) (trans-cst Ints Ints (invr x) (Succ (Pred y)) ∙ invr y))
|
||||
(\x -> trunc)
|
||||
|
||||
negate : Ints -> Ints
|
||||
negate = Zind
|
||||
(λ _ → Ints)
|
||||
nul
|
||||
(λ _ -> Pred)
|
||||
(λ _ -> Succ)
|
||||
(λ x y -> from-transp (λ _ -> Ints) (invl x) (trans-cst Ints Ints (invl x) (Succ (Pred y)) ∙ invr y))
|
||||
(λ x y -> from-transp (λ _ -> Ints) (invr x) (trans-cst Ints Ints (invr x) (Pred (Succ y)) ∙ invl y))
|
||||
(\x -> trunc)
|
||||
|
||||
min : Ints -> Ints -> Ints
|
||||
min x y = plus x (negate y)
|
||||
|
||||
plus_0n : (x : Ints) -> plus x nul == x
|
||||
plus_0n x = idp
|
||||
|
||||
plus_n0 : (x : Ints) -> plus nul x == x
|
||||
plus_n0 = Zind
|
||||
(\x -> plus nul x == x)
|
||||
idp
|
||||
(\x p -> ap Succ p)
|
||||
(\x p -> ap Pred p)
|
||||
(\x y ->
|
||||
trunc_paths
|
||||
Ints
|
||||
(\m -> plus nul m == m)
|
||||
(invl x)
|
||||
(trunc (plus nul (Pred (Succ x)))
|
||||
(Pred(Succ x)))
|
||||
(ap Pred (ap Succ y))
|
||||
y
|
||||
)
|
||||
(\x y ->
|
||||
trunc_paths
|
||||
Ints
|
||||
(\m -> plus nul m == m)
|
||||
(invr x)
|
||||
(trunc (plus nul (Succ (Pred x)))
|
||||
(Succ(Pred x)))
|
||||
(ap Succ (ap Pred y))
|
||||
y
|
||||
)
|
||||
(\x -> paths_set Ints Ints trunc (\m -> plus nul m) (\m -> m) x)
|
||||
|
||||
plus_assoc : (x y z : Ints) -> plus x (plus y z) == plus (plus x y) z
|
||||
plus_assoc x = Zind
|
||||
(λ y -> (z : Ints) -> plus x (plus y z) == plus (plus x y) z)
|
||||
(
|
||||
Zind
|
||||
(λ z -> plus x (plus nul z) == plus (plus x nul) z)
|
||||
idp
|
||||
(λ x p -> ap Succ p)
|
||||
(λ x p -> ap Pred p)
|
||||
{!!}
|
||||
{!!}
|
||||
{!!}
|
||||
)
|
||||
(λ y p ->
|
||||
Zind
|
||||
(λ z -> plus x (plus (Succ y) z) == plus (plus x (Succ y)) z)
|
||||
(p (Succ nul))
|
||||
(λ y' p' -> ap Succ p')
|
||||
(λ y' p' -> ap Pred p')
|
||||
{!!}
|
||||
{!!}
|
||||
{!!}
|
||||
)
|
||||
(λ y p ->
|
||||
Zind
|
||||
(λ z -> plus x (plus (Pred y) z) == plus (plus x (Pred y)) z)
|
||||
(p (Pred nul))
|
||||
(λ y' p' -> ap Succ p')
|
||||
(λ y' p' -> ap Pred p')
|
||||
{!!}
|
||||
{!!}
|
||||
{!!}
|
||||
)
|
||||
{!!}
|
||||
{!!}
|
||||
{!!}
|
||||
|
||||
plus_Succ : (x y : Ints) -> plus x (Succ y) == Succ(plus x y)
|
||||
plus_Succ x y = idp
|
||||
|
||||
Succ_plus : (x y : Ints) -> plus (Succ x) y == Succ(plus x y)
|
||||
Succ_plus x = Zind
|
||||
(λ y -> plus (Succ x) y == Succ(plus x y))
|
||||
idp
|
||||
(λ y' p -> ap Succ p)
|
||||
(λ y' p -> ap Pred p ∙ invl (plus x y') ∙ ! (invr (plus x y')))
|
||||
{!!}
|
||||
{!!}
|
||||
{!!}
|
||||
|
||||
plus_Pred : (x y : Ints) -> plus x (Pred y) == Pred(plus x y)
|
||||
plus_Pred x y = idp
|
||||
|
||||
Pred_plus : (x y : Ints) -> plus (Pred x) y == Pred(plus x y)
|
||||
Pred_plus x = Zind
|
||||
(λ y -> plus (Pred x) y == Pred(plus x y))
|
||||
idp
|
||||
(λ y' p -> ap Succ p ∙ invr (plus x y') ∙ ! (invl (plus x y')))
|
||||
(λ y' p -> ap Pred p)
|
||||
{!!}
|
||||
{!!}
|
||||
{!!}
|
||||
|
||||
plus_negr : (y : Ints) -> plus y (negate y) == nul
|
||||
plus_negr = Zind
|
||||
(λ y -> plus y (negate y) == nul)
|
||||
idp
|
||||
(λ x p ->
|
||||
Succ_plus x (negate (Succ x))
|
||||
∙ invr (plus x (negate x))
|
||||
∙ p
|
||||
)
|
||||
(λ x p ->
|
||||
Pred_plus x (negate (Pred x))
|
||||
∙ invl (plus x (negate x))
|
||||
∙ p
|
||||
)
|
||||
{!!}
|
||||
{!!}
|
||||
{!!}
|
||||
|
||||
plus_negl : (y : Ints) -> plus (negate y) y == nul
|
||||
plus_negl = Zind
|
||||
(λ y -> plus (negate y) y == nul)
|
||||
idp
|
||||
(λ y' p ->
|
||||
Pred_plus (negate y') (Succ y')
|
||||
∙ invl (plus (negate y') y')
|
||||
∙ p
|
||||
)
|
||||
(λ y' p ->
|
||||
Succ_plus (negate y') (Pred y')
|
||||
∙ invr (plus (negate y') y')
|
||||
∙ p
|
||||
)
|
||||
{!!}
|
||||
{!!}
|
||||
{!!}
|
||||
|
||||
plus_com : (x y : Ints) -> plus x y == plus y x
|
||||
plus_com x = Zind
|
||||
(λ y -> plus x y == plus y x)
|
||||
(plus_0n x ∙ ! (plus_n0 x))
|
||||
(λ y' p ->
|
||||
plus_Succ x y'
|
||||
∙ ap Succ p
|
||||
∙ ! (Succ_plus y' x))
|
||||
(λ y' p ->
|
||||
plus_Pred x y'
|
||||
∙ ap Pred p
|
||||
∙ ! (Pred_plus y' x)
|
||||
)
|
||||
{!!}
|
||||
{!!}
|
||||
{!!}
|
||||
|
||||
times : Ints -> Ints -> Ints
|
||||
times n = Zind
|
||||
(λ _ → Ints)
|
||||
nul
|
||||
(\x y -> plus y n)
|
||||
(\x y -> min y n)
|
||||
(λ x y -> from-transp (λ _ → Ints) (invl x) (trans-cst Ints Ints (invl x) (min (plus y n) n)
|
||||
∙ ! (plus_assoc y n (negate n))
|
||||
∙ ap (plus y) (plus_negr n)
|
||||
∙ plus_0n y))
|
||||
(λ x y -> from-transp (λ _ → Ints) (invr x) (trans-cst Ints Ints (invr x) (plus (min y n) n)
|
||||
∙ ! (plus_assoc y (negate n) n)
|
||||
∙ ap (λ z -> plus y z) (plus_negl n)
|
||||
∙ plus_0n y))
|
||||
(\x -> trunc)
|
||||
|
|
@ -1,49 +0,0 @@
|
|||
{-# OPTIONS --without-K --rewriting #-}
|
||||
|
||||
open import HoTT
|
||||
|
||||
module Interval where
|
||||
|
||||
postulate
|
||||
I : Set
|
||||
z : I
|
||||
o : I
|
||||
s : z == o
|
||||
I-ind : (Y : I -> Set)
|
||||
(zY : Y z)
|
||||
(oY : Y o)
|
||||
(sY : PathOver Y s zY oY)
|
||||
(x : I)
|
||||
-> Y x
|
||||
I-ind-βz : (Y : I -> Set)
|
||||
(zY : Y z)
|
||||
(oY : Y o)
|
||||
(sY : PathOver Y s zY oY)
|
||||
-> I-ind Y zY oY sY z ↦ zY
|
||||
{-# REWRITE I-ind-βz #-}
|
||||
I-ind-βo : (Y : I -> Set)
|
||||
(zY : Y z)
|
||||
(oY : Y o)
|
||||
(sY : PathOver Y s zY oY)
|
||||
-> I-ind Y zY oY sY o ↦ oY
|
||||
{-# REWRITE I-ind-βo #-}
|
||||
I-ind-βs : (Y : I -> Set)
|
||||
(zY : Y z)
|
||||
(oY : Y o)
|
||||
(sY : PathOver Y s zY oY)
|
||||
-> apd (I-ind Y zY oY sY) s == sY
|
||||
|
||||
transp-cst : (A : Set) {x y : A} (B : Set) (p : x == y) (z : B) -> transport (\x -> B) p z == z
|
||||
transp-cst A B idp z = idp
|
||||
|
||||
transp-fun : (A B : Set) (a b : A) (p : a == b) (f : A -> B) -> transport (λ _ -> A -> B) p f == transport (λ _ -> B) p (f a)
|
||||
transp-fun = ?
|
||||
|
||||
fe : {A B : Set} (f g : A -> B) -> ( (x : A) -> f x == g x) -> f == g
|
||||
fe {A} {B} f g p =
|
||||
ap
|
||||
(I-ind (λ _ → (x : A) → B) f g
|
||||
(from-transp (λ _ → (x : A) → B) s (
|
||||
{!!}
|
||||
)))
|
||||
s
|
||||
|
|
@ -1,59 +0,0 @@
|
|||
{-# OPTIONS --without-K --rewriting #-}
|
||||
|
||||
open import HoTT
|
||||
|
||||
module Mod2 where
|
||||
|
||||
private
|
||||
data M' : Set where
|
||||
Zero : M'
|
||||
S : M' -> M'
|
||||
|
||||
M : Set
|
||||
M = M'
|
||||
|
||||
z : M
|
||||
z = Zero
|
||||
|
||||
Succ : M -> M
|
||||
Succ = S
|
||||
|
||||
postulate
|
||||
mod : (n : M) -> n == Succ(Succ n)
|
||||
trunc : is-set M
|
||||
|
||||
M-ind : (C : M -> Set)
|
||||
-> (a : C Zero)
|
||||
-> (sC : (x : M) -> C x -> C (S x))
|
||||
-> (p : (n : M) (c : C n) -> PathOver C (mod n) c (sC (Succ n) (sC n c)))
|
||||
-> (t : (m : M) -> is-set (C m))
|
||||
-> (x : M) -> C x
|
||||
M-ind C a sC _ t Zero = a
|
||||
M-ind C a sC p t (S x) = sC x (M-ind C a sC p t x)
|
||||
|
||||
postulate
|
||||
M-ind-βmod : (C : M -> Set)
|
||||
-> (a : C Zero)
|
||||
-> (sC : (x : M) -> C x -> C (S x))
|
||||
-> (p : (n : M) (c : C n) -> PathOver C (mod n) c (sC (Succ n) (sC n c)))
|
||||
-> (t : (m : M) -> is-set (C m))
|
||||
-> (n : M)
|
||||
-> apd (M-ind C a sC p t) (mod n) == p n (M-ind C a sC p t n)
|
||||
|
||||
M-rec : {C : Set}
|
||||
-> (a : C)
|
||||
-> (sC : C -> C)
|
||||
-> (p : (c : C) -> c == sC (sC c))
|
||||
-> (t : is-set C)
|
||||
-> M -> C
|
||||
M-rec a sC _ _ Zero = a
|
||||
M-rec a sC p t (S x) = sC (M-rec a sC p t x)
|
||||
|
||||
postulate
|
||||
M-rec-βmod : {C : Set}
|
||||
-> (a : C)
|
||||
-> (sC : C -> C)
|
||||
-> (p : (c : C) -> c == sC (sC c))
|
||||
-> (t : is-set C)
|
||||
-> (n : M)
|
||||
-> ap (M-rec a sC p t) (mod n) == p (M-rec a sC p t n)
|
||||
|
|
@ -1,113 +0,0 @@
|
|||
{-# OPTIONS --without-K --rewriting #-}
|
||||
|
||||
open import HoTT
|
||||
open import Mod2
|
||||
|
||||
module Thms where
|
||||
|
||||
paths_set : (A B : Set) (m : is-set B) (f g : A -> B) (a : A) -> is-set (f a == g a)
|
||||
paths_set A B m f g a = \c₁ c₂ q₁ q₂ ->
|
||||
prop-has-level-S
|
||||
(contr-is-prop (m (f a) (g a) c₁ c₂))
|
||||
q₁
|
||||
q₂
|
||||
|
||||
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
|
||||
trunc_paths A Y p t c₁ c₂ = from-transp! Y p ((prop-has-all-paths t) c₁ (transport! Y p c₂))
|
||||
|
||||
plus : M -> M -> M
|
||||
plus n = M-rec
|
||||
n
|
||||
Succ
|
||||
mod
|
||||
trunc
|
||||
|
||||
plus_0n : (n : M) -> plus z n == n
|
||||
plus_0n = M-ind
|
||||
(\n -> plus z n == n)
|
||||
idp
|
||||
(\x -> \p -> ap Succ p)
|
||||
(\x -> \c ->
|
||||
trunc_paths M (\ n → plus z n == n) (mod x) (trunc (plus z x) x) c (ap Succ (ap Succ c))
|
||||
)
|
||||
(\m ->
|
||||
paths_set M M trunc (\x -> plus z x) (\x -> x) m
|
||||
)
|
||||
|
||||
plus_n0 : (n : M) -> plus n z == n
|
||||
plus_n0 = M-ind
|
||||
(\n -> plus n z == n)
|
||||
idp
|
||||
(\x p -> idp)
|
||||
(\x c ->
|
||||
trunc_paths M (\x -> plus x z == x) (mod x) (trunc x x) c idp
|
||||
)
|
||||
(\m -> paths_set M M trunc (\x -> plus x z) (\x -> x) m )
|
||||
|
||||
plus_Sn : (n m : M) -> plus (Succ n) m == Succ (plus n m)
|
||||
plus_Sn n = M-ind
|
||||
(\m -> plus (Succ n) m == Succ (plus n m))
|
||||
idp
|
||||
(\x p -> ap Succ p)
|
||||
(\x c ->
|
||||
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))
|
||||
)
|
||||
(\m -> paths_set M M trunc (\x -> plus (Succ x) m) (\x -> Succ(plus x m)) n)
|
||||
|
||||
plus_nS : (n m : M) -> plus n (Succ m) == Succ (plus n m)
|
||||
plus_nS n m = idp
|
||||
|
||||
not : Bool -> Bool
|
||||
not true = false
|
||||
not false = true
|
||||
|
||||
not-not : (x : Bool) -> x == not (not x)
|
||||
not-not true = idp
|
||||
not-not false = idp
|
||||
|
||||
toBool : M -> Bool
|
||||
toBool = M-rec
|
||||
true
|
||||
not
|
||||
((\x -> not-not x))
|
||||
Bool-is-set
|
||||
|
||||
toBoolS : (n : M) -> toBool (Succ n) == not (toBool n)
|
||||
toBoolS = M-ind
|
||||
(\n -> toBool (Succ n) == not (toBool n))
|
||||
idp
|
||||
(\x p -> idp)
|
||||
(\n c ->
|
||||
trunc_paths M (\x -> toBool (Succ x) == not (toBool x)) (mod n) (Bool-is-set (not (toBool n)) (not (toBool n))) c idp)
|
||||
(\m -> paths_set M Bool Bool-is-set (\n -> toBool(Succ n)) (\n -> not(toBool n)) m)
|
||||
|
||||
fromBool : Bool -> M
|
||||
fromBool true = z
|
||||
fromBool false = Succ z
|
||||
|
||||
fromBoolNot : (b : Bool) -> fromBool (not b) == Succ (fromBool b)
|
||||
fromBoolNot true = idp
|
||||
fromBoolNot false = mod z
|
||||
|
||||
iso₁ : (b : Bool) -> toBool (fromBool b) == b
|
||||
iso₁ true = idp
|
||||
iso₁ false = idp
|
||||
|
||||
iso₂ : (n : M) -> fromBool (toBool n) == n
|
||||
iso₂ = M-ind
|
||||
(\n -> fromBool (toBool n) == n)
|
||||
idp
|
||||
(\x p ->
|
||||
ap fromBool (toBoolS x)
|
||||
∙ fromBoolNot (toBool x)
|
||||
∙ ap Succ p)
|
||||
(\n p -> trunc_paths M
|
||||
(λ z₁ → fromBool (toBool z₁) == z₁)
|
||||
(mod n)
|
||||
(trunc (fromBool (toBool n)) n)
|
||||
p
|
||||
(ap fromBool (toBoolS (Succ n))
|
||||
∙ fromBoolNot (toBool (Succ n))
|
||||
∙ ap Succ (ap fromBool (toBoolS n) ∙ fromBoolNot (toBool n) ∙ ap Succ p))
|
||||
)
|
||||
(\m -> paths_set M M trunc (\n -> fromBool (toBool n)) (\n -> n) m)
|
||||
|
|
@ -18,6 +18,7 @@ fsets/monad.v
|
|||
FSets.v
|
||||
Sub.v
|
||||
representations/T.v
|
||||
implementations/interface.v
|
||||
implementations/lists.v
|
||||
variations/enumerated.v
|
||||
variations/k_finite.v
|
||||
|
|
|
|||
|
|
@ -0,0 +1,43 @@
|
|||
Require Import HoTT.
|
||||
Require Import FSets.
|
||||
|
||||
Section structure.
|
||||
Variable (T : Type -> Type).
|
||||
|
||||
Class hasMembership : Type :=
|
||||
member : forall A : Type, A -> T A -> hProp.
|
||||
|
||||
Class hasEmpty : Type :=
|
||||
empty : forall A, T A.
|
||||
|
||||
Class hasSingleton : Type :=
|
||||
singleton : forall A, A -> T A.
|
||||
|
||||
Class hasUnion : Type :=
|
||||
union : forall A, T A -> T A -> T A.
|
||||
|
||||
Class hasComprehension : Type :=
|
||||
filter : forall A, (A -> Bool) -> T A -> T A.
|
||||
End structure.
|
||||
|
||||
Arguments member {_} {_} {_} _ _.
|
||||
Arguments empty {_} {_} {_}.
|
||||
Arguments singleton {_} {_} {_} _.
|
||||
Arguments union {_} {_} {_} _ _.
|
||||
Arguments filter {_} {_} {_} _ _.
|
||||
|
||||
Section interface.
|
||||
Context `{Univalence}.
|
||||
Variable (T : Type -> Type)
|
||||
(f : forall A, T A -> FSet A).
|
||||
Context `{hasMembership T, hasEmpty T, hasSingleton T, hasUnion T, hasComprehension T}.
|
||||
|
||||
Class sets :=
|
||||
{
|
||||
f_empty : forall A, f A empty = E ;
|
||||
f_singleton : forall A a, f A (singleton a) = L a;
|
||||
f_union : forall A X Y, f A (union X Y) = U (f A X) (f A Y);
|
||||
f_filter : forall A ϕ X, f A (filter ϕ X) = comprehension ϕ (f A X);
|
||||
f_member : forall A a X, member a X = isIn a (f A X)
|
||||
}.
|
||||
End interface.
|
||||
|
|
@ -1,113 +1,69 @@
|
|||
(* Implementation of [FSet A] using lists *)
|
||||
Require Import HoTT HitTactics.
|
||||
Require Import representations.cons_repr FSets.
|
||||
From fsets Require Import operations_cons_repr isomorphism length.
|
||||
Require Import FSets implementations.interface.
|
||||
|
||||
Section Operations.
|
||||
Variable A : Type.
|
||||
Context {A_deceq : DecidablePaths A}.
|
||||
Context `{Univalence}.
|
||||
|
||||
Global Instance list_empty : hasEmpty list := fun A => nil.
|
||||
|
||||
Global Instance list_single : hasSingleton list := fun A a => cons a nil.
|
||||
|
||||
Fixpoint member (l : list A) (a : A) : Bool :=
|
||||
Global Instance list_union : hasUnion list.
|
||||
Proof.
|
||||
intros A l1 l2.
|
||||
induction l1.
|
||||
* apply l2.
|
||||
* apply (cons a IHl1).
|
||||
Defined.
|
||||
|
||||
Global Instance list_membership : hasMembership list.
|
||||
Proof.
|
||||
intros A.
|
||||
intros a l.
|
||||
induction l as [ | b l IHl].
|
||||
- apply False_hp.
|
||||
- apply (hor (a = b) IHl).
|
||||
Defined.
|
||||
|
||||
Global Instance list_comprehension : hasComprehension list.
|
||||
Proof.
|
||||
intros A ϕ l.
|
||||
induction l as [ | b l IHl].
|
||||
- apply nil.
|
||||
- apply (if ϕ b then cons b IHl else IHl).
|
||||
Defined.
|
||||
|
||||
Fixpoint list_to_set A (l : list A) : FSet A :=
|
||||
match l with
|
||||
| nil => false
|
||||
| cons b l => if (dec (a = b)) then true else member l a
|
||||
end.
|
||||
|
||||
Fixpoint append (l1 l2 : list A) :=
|
||||
match l1 with
|
||||
| nil => l2
|
||||
| cons a l => cons a (append l l2)
|
||||
end.
|
||||
|
||||
Definition empty : list A := nil.
|
||||
|
||||
Definition singleton (a : A) : list A := cons a nil.
|
||||
|
||||
Fixpoint filter (ϕ : A -> Bool) (l : list A) : list A :=
|
||||
match l with
|
||||
| nil => nil
|
||||
| cons a l => if ϕ a then cons a (filter ϕ l) else filter ϕ l
|
||||
end.
|
||||
|
||||
Fixpoint cardinality (l : list A) : nat :=
|
||||
match l with
|
||||
| nil => 0
|
||||
| cons a l => if member l a then cardinality l else 1 + cardinality l
|
||||
| nil => E
|
||||
| cons a l => U (L a) (list_to_set A l)
|
||||
end.
|
||||
|
||||
End Operations.
|
||||
|
||||
Arguments nil {_}.
|
||||
Arguments cons {_} _ _.
|
||||
Arguments member {_} {_} _ _.
|
||||
Arguments singleton {_} _.
|
||||
Arguments append {_} _ _.
|
||||
Arguments empty {_}.
|
||||
Arguments filter {_} _ _.
|
||||
Arguments cardinality {_} {_} _.
|
||||
|
||||
Section ListToSet.
|
||||
Variable A : Type.
|
||||
Context {A_deceq : DecidablePaths A} `{Univalence}.
|
||||
Context `{Univalence}.
|
||||
|
||||
Lemma member_isIn (a : A) (l : list A) :
|
||||
member a l = isIn a (list_to_set A l).
|
||||
Proof.
|
||||
induction l ; unfold member in * ; simpl in *.
|
||||
- reflexivity.
|
||||
- rewrite IHl.
|
||||
unfold hor, merely, lor.
|
||||
apply path_iff_hprop ; intros z ; strip_truncations ; destruct z as [z1 | z2].
|
||||
* apply (tr (inl (tr z1))).
|
||||
* apply (tr (inr z2)).
|
||||
* strip_truncations ; apply (tr (inl z1)).
|
||||
* apply (tr (inr z2)).
|
||||
Defined.
|
||||
|
||||
Fixpoint list_to_setC (l : list A) : FSetC A :=
|
||||
match l with
|
||||
| nil => Nil
|
||||
| cons a l => Cns a (list_to_setC l)
|
||||
end.
|
||||
Definition empty_empty : list_to_set A empty = E := idpath.
|
||||
|
||||
Definition list_to_set (l : list A) := FSetC_to_FSet(list_to_setC l).
|
||||
|
||||
Lemma list_to_setC_surj : forall X : FSetC A, Trunc (-1) ({l : list A & list_to_setC l = X}).
|
||||
Proof.
|
||||
hrecursion ; try (intros ; apply hprop_allpath ; apply (istrunc_truncation (-1) _)).
|
||||
- apply tr ; exists nil ; cbn. reflexivity.
|
||||
- intros a x P.
|
||||
simple refine (Trunc_rec _ P).
|
||||
intros [l Q].
|
||||
apply tr.
|
||||
exists (cons a l).
|
||||
simpl.
|
||||
apply (ap (fun y => a;;y) Q).
|
||||
Defined.
|
||||
|
||||
Lemma member_isIn (l : list A) (a : A) :
|
||||
member l a = isIn_b a (FSetC_to_FSet (list_to_setC l)).
|
||||
Proof.
|
||||
induction l ; cbn in *.
|
||||
- reflexivity.
|
||||
- destruct (dec (a = a0)) ; cbn.
|
||||
* rewrite ?p. simplify_isIn. reflexivity.
|
||||
* rewrite IHl. simplify_isIn. rewrite L_isIn_b_false ; auto.
|
||||
Defined.
|
||||
|
||||
Lemma append_FSetCappend (l1 l2 : list A) :
|
||||
list_to_setC (append l1 l2) = operations_cons_repr.append (list_to_setC l1) (list_to_setC l2).
|
||||
Proof.
|
||||
induction l1 ; simpl in *.
|
||||
- reflexivity.
|
||||
- apply (ap (fun y => a ;; y) IHl1).
|
||||
Defined.
|
||||
|
||||
Lemma append_FSetappend (l1 l2 : list A) :
|
||||
list_to_set (append l1 l2) = U (list_to_set l1) (list_to_set l2).
|
||||
Proof.
|
||||
induction l1 ; cbn in *.
|
||||
- symmetry. apply nl.
|
||||
- rewrite <- assoc.
|
||||
refine (ap (fun y => U {|a|} y) _).
|
||||
rewrite <- append_union.
|
||||
rewrite <- append_FSetCappend.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Lemma empty_empty : list_to_set empty = E.
|
||||
Proof.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Lemma filter_comprehension (l : list A) (ϕ : A -> Bool) :
|
||||
list_to_set (filter ϕ l) = comprehension ϕ (list_to_set l).
|
||||
Lemma filter_comprehension (ϕ : A -> Bool) (l : list A) :
|
||||
list_to_set A (filter ϕ l) = comprehension ϕ (list_to_set A l).
|
||||
Proof.
|
||||
induction l ; cbn in *.
|
||||
- reflexivity.
|
||||
|
|
@ -118,32 +74,33 @@ Section ListToSet.
|
|||
apply IHl.
|
||||
Defined.
|
||||
|
||||
Lemma length_sizeC (l : list A) :
|
||||
cardinality l = length (list_to_setC l).
|
||||
Definition singleton_single (a : A) : list_to_set A (singleton a) = L a :=
|
||||
nr (L a).
|
||||
|
||||
Lemma append_union (l1 l2 : list A) :
|
||||
list_to_set A (union l1 l2) = U (list_to_set A l1) (list_to_set A l2).
|
||||
Proof.
|
||||
induction l.
|
||||
- cbn.
|
||||
reflexivity.
|
||||
- simpl.
|
||||
rewrite IHl.
|
||||
rewrite member_isIn.
|
||||
reflexivity.
|
||||
induction l1 ; induction l2 ; cbn.
|
||||
- apply (union_idem _)^.
|
||||
- apply (nl _)^.
|
||||
- rewrite IHl1.
|
||||
apply assoc.
|
||||
- rewrite IHl1.
|
||||
cbn.
|
||||
apply assoc.
|
||||
Defined.
|
||||
End ListToSet.
|
||||
|
||||
Lemma length_size (l : list A) :
|
||||
cardinality l = length_FSet (list_to_set l).
|
||||
Section lists_are_sets.
|
||||
Context `{Univalence}.
|
||||
|
||||
Instance lists_sets : sets list list_to_set.
|
||||
Proof.
|
||||
unfold length_FSet.
|
||||
unfold list_to_set.
|
||||
rewrite repr_iso_id_r.
|
||||
apply length_sizeC.
|
||||
split ; intros.
|
||||
- apply empty_empty.
|
||||
- apply singleton_single.
|
||||
- apply append_union.
|
||||
- apply filter_comprehension.
|
||||
- apply member_isIn.
|
||||
Defined.
|
||||
|
||||
Lemma singleton_single (a : A) :
|
||||
list_to_set (singleton a) = L a.
|
||||
Proof.
|
||||
cbn.
|
||||
apply nr.
|
||||
Defined.
|
||||
|
||||
End ListToSet.
|
||||
End lists_are_sets.
|
||||
|
|
|
|||
Loading…
Reference in New Issue