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Progress towards UFDs (#88)

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Patrick Stevens
2019-12-08 11:18:39 +00:00
committed by GitHub
parent 99c38495ce
commit 33098e94b0
20 changed files with 462 additions and 253 deletions
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55
Maybe.agda
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@@ -4,36 +4,41 @@ open import LogicalFormulae
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Maybe where
data Maybe {a : _} (A : Set a) : Set a where
no : Maybe A
yes : A → Maybe A
joinMaybe : {a : _} → {A : Set a} → Maybe (Maybe A) → Maybe A
joinMaybe no = no
joinMaybe (yes s) = s
data Maybe {a : _} (A : Set a) : Set a where
no : Maybe A
yes : A → Maybe A
bindMaybe : {a b : _} → {A : Set a} → {B : Set b} → Maybe A → (A → Maybe B) → Maybe B
bindMaybe no f = no
bindMaybe (yes x) f = f x
joinMaybe : {a : _} → {A : Set a} → Maybe (Maybe A) → Maybe A
joinMaybe no = no
joinMaybe (yes s) = s
applyMaybe : {a b : _} → {A : Set a} → {B : Set b} → Maybe (A → B) → Maybe A → Maybe B
applyMaybe f no = no
applyMaybe no (yes x) = no
applyMaybe (yes f) (yes x) = yes (f x)
bindMaybe : {a b : _} → {A : Set a} → {B : Set b} → Maybe A → (A → Maybe B) → Maybe B
bindMaybe no f = no
bindMaybe (yes x) f = f x
yesInjective : {a : _} → {A : Set a} → {x y : A} → (yes x ≡ yes y) → x ≡ y
yesInjective {a} {A} {x} {.x} refl = refl
applyMaybe : {a b : _} → {A : Set a} → {B : Set b} → Maybe (A → B) → Maybe A → Maybe B
applyMaybe f no = no
applyMaybe no (yes x) = no
applyMaybe (yes f) (yes x) = yes (f x)
mapMaybe : {a b : _} → {A : Set a} → {B : Set b} → (f : A → B) → Maybe A → Maybe B
mapMaybe f no = no
mapMaybe f (yes x) = yes (f x)
yesInjective : {a : _} → {A : Set a} → {x y : A} → (yes x ≡ yes y) → x ≡ y
yesInjective {a} {A} {x} {.x} refl = refl
defaultValue : {a : _} → {A : Set a} → (default : A) → Maybe A → A
defaultValue default no = default
defaultValue default (yes x) = x
mapMaybe : {a b : _} → {A : Set a} → {B : Set b} → (f : A → B) → Maybe A → Maybe B
mapMaybe f no = no
mapMaybe f (yes x) = yes (f x)
noNotYes : {a : _} {A : Set a} {b : A} → (no ≡ yes b) → False
noNotYes ()
defaultValue : {a : _} → {A : Set a} → (default : A) → Maybe A → A
defaultValue default no = default
defaultValue default (yes x) = x
mapMaybePreservesNo : {a b : _} {A : Set a} {B : Set b} {f : A → B} {x : Maybe A} → mapMaybe f x ≡ no → x ≡ no
mapMaybePreservesNo {f = f} {no} pr = refl
noNotYes : {a : _} {A : Set a} {b : A} → (no ≡ yes b) → False
noNotYes ()
mapMaybePreservesNo : {a b : _} {A : Set a} {B : Set b} {f : A → B} {x : Maybe A} → mapMaybe f x ≡ no → x ≡ no
mapMaybePreservesNo {f = f} {no} pr = refl
mapMaybePreservesYes : {a b : _} {A : Set a} {B : Set b} {f : A → B} {x : Maybe A} {y : B} → mapMaybe f x ≡ yes y → Sg A (λ z → (x ≡ yes z) && (f z ≡ y))
mapMaybePreservesYes {f = f} {x} {y} map with x
mapMaybePreservesYes {f = f} {x} {y} map | yes z = z , (refl ,, yesInjective map)