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alexaricealexarice
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Add some conversions
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‎src/Function/Bijection/Strict.agda

+33-2
Original file line numberDiff line numberDiff line change
@@ -15,8 +15,16 @@ open import Relation.Binary.PropositionalEquality as P
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open import Function.Equality as F
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using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)
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open import Function.Injection as Inj hiding (id; _∘_; injection)
18-
open import Function.Surjection.Strict as Surj hiding (id; _∘_; surjection)
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open import Function.LeftInverse.Strict as Left hiding (id; _∘_; leftInverse)
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open import Function.Surjection.Strict as Surj hiding (id; _∘_;
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surjection;
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non-strict-to-strict;
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strict-to-non-strict)
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open import Function.LeftInverse.Strict as Left hiding (id; _∘_;
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leftInverse;
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non-strict-to-strict;
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strict-to-non-strict)
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import Function.Bijection as NonStrict
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------------------------------------------------------------------------
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-- Bijective functions.
@@ -34,6 +42,29 @@ record Bijective {f₁ f₂ t₁ t₂}
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left-inverse-of : from LeftInverseOf to
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left-inverse-of x y y≈tox = injective (right-inverse-of (to ⟨$⟩ x) (from ⟨$⟩ y) (F.cong from y≈tox))
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------------------------------------------------------------------------
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-- Relation to non-strict versions
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non-strict-to-strict :
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∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
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(f : To ⟶ From) →
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NonStrict.Bijective f →
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Bijective f
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non-strict-to-strict f ns = record
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{ injective = injective
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; surjective = Surj.non-strict-to-strict f surjective
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} where open NonStrict.Bijective ns
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strict-to-non-strict :
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∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
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(f : From ⟶ To) →
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Bijective f →
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NonStrict.Bijective f
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strict-to-non-strict f ns = record
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{ injective = injective
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; surjective = Surj.strict-to-non-strict f surjective
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} where open Bijective ns
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------------------------------------------------------------------------
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-- The set of all bijections between two setoids.
3970

‎src/Function/Inverse/Strict.agda

+33-2
Original file line numberDiff line numberDiff line change
@@ -10,14 +10,20 @@ module Function.Inverse.Strict where
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open import Level
1212
open import Function.Base using (flip)
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open import Function.Bijection.Strict hiding (id; _∘_; bijection)
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open import Function.Bijection.Strict hiding (id; _∘_; bijection;
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non-strict-to-strict;
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strict-to-non-strict)
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open import Function.Equality as F
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using (_⟶_) renaming (_∘_ to _⟪∘⟫_)
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open import Function.LeftInverse.Strict as Left hiding (id; _∘_)
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open import Function.LeftInverse.Strict as Left hiding (id; _∘_;
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non-strict-to-strict;
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strict-to-non-strict)
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open import Relation.Binary
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open import Relation.Binary.PropositionalEquality as P using (_≗_; _≡_)
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open import Relation.Unary using (Pred)
2024

25+
import Function.Inverse as NonStrict
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2127
------------------------------------------------------------------------
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-- Inverses
2329

@@ -29,6 +35,31 @@ record _InverseOf_ {f₁ f₂ t₁ t₂}
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left-inverse-of : from LeftInverseOf to
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right-inverse-of : from RightInverseOf to
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------------------------------------------------------------------------
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-- Relation to non-strict versions
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non-strict-to-strict :
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∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
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(f : To ⟶ From) →
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(g : From ⟶ To) →
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f NonStrict.InverseOf g →
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f InverseOf g
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non-strict-to-strict f g ns = record
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{ left-inverse-of = Left.non-strict-to-strict f g left-inverse-of
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; right-inverse-of = Left.non-strict-to-strict g f right-inverse-of
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} where open NonStrict._InverseOf_ ns
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strict-to-non-strict :
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∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
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(f : To ⟶ From) →
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(g : From ⟶ To) →
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f InverseOf g →
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f NonStrict.InverseOf g
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strict-to-non-strict f g ns = record
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{ left-inverse-of = Left.strict-to-non-strict f g left-inverse-of
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; right-inverse-of = Left.strict-to-non-strict g f right-inverse-of
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} where open _InverseOf_ ns
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3263
------------------------------------------------------------------------
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-- The set of all inverses between two setoids
3465

‎src/Function/LeftInverse/Strict.agda

+13-9
Original file line numberDiff line numberDiff line change
@@ -18,7 +18,7 @@ open import Function.Equivalence using (Equivalence)
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open import Function.Injection using (Injective; Injection)
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open import Relation.Binary.PropositionalEquality as P using (_≡_)
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21-
import Function.LeftInverse as LI
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import Function.LeftInverse as NonStrict
2222

2323
------------------------------------------------------------------------
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-- Left and right inverses.
@@ -41,19 +41,23 @@ f RightInverseOf g = g LeftInverseOf f
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4242
non-strict-to-strict :
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∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
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LI._LeftInverseOf_ {From = From} {To = To} ⇒
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_LeftInverseOf_ {From = From} {To = To}
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non-strict-to-strict {From = From} {x = f} {y = g} ns x y y∼gx = begin
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(f : To ⟶ From) →
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(g : From ⟶ To) →
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f NonStrict.LeftInverseOf g →
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f LeftInverseOf g
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non-strict-to-strict {From = From} f g ns x y y∼gx = begin
4749
f ⟨$⟩ y ≈⟨ Eq.cong f y∼gx ⟩
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f ⟨$⟩ (g ⟨$⟩ x) ≈⟨ ns x ⟩
4951
x ∎
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where open EqReasoning From
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5254
strict-to-non-strict :
5355
∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
54-
_LeftInverseOf_ {From = From} {To = To} ⇒
55-
LI._LeftInverseOf_ {From = From} {To = To}
56-
strict-to-non-strict {To = To} {x = f} {y = g} str x = str x (g ⟨$⟩ x) refl
56+
(f : To ⟶ From) →
57+
(g : From ⟶ To) →
58+
f LeftInverseOf g →
59+
f NonStrict.LeftInverseOf g
60+
strict-to-non-strict {To = To} f g str x = str x (g ⟨$⟩ x) refl
5761
where open Setoid To
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5963
------------------------------------------------------------------------
@@ -70,8 +74,8 @@ record LeftInverse {f₁ f₂ t₁ t₂}
7074
private
7175
open module F = Setoid From
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open module T = Setoid To
73-
non-strict : from LI.LeftInverseOf to
74-
non-strict = strict-to-non-strict {x = from} {y = to} left-inverse-of
77+
non-strict : from NonStrict.LeftInverseOf to
78+
non-strict = strict-to-non-strict from to left-inverse-of
7579
open EqReasoning From
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7781
injective : Injective to

‎src/Function/Surjection/Strict.agda

+28-1
Original file line numberDiff line numberDiff line change
@@ -13,11 +13,15 @@ open import Function.Equality as F
1313
using (_⟶_) renaming (_∘_ to _⟪∘⟫_)
1414
open import Function.Equivalence using (Equivalence)
1515
open import Function.Injection hiding (id; _∘_; injection)
16-
open import Function.LeftInverse.Strict as Left hiding (id; _∘_)
16+
open import Function.LeftInverse.Strict as Left hiding (id; _∘_;
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non-strict-to-strict;
18+
strict-to-non-strict)
1719
open import Data.Product
1820
open import Relation.Binary
1921
open import Relation.Binary.PropositionalEquality as P using (_≡_)
2022

23+
import Function.Surjection as NonStrict
24+
2125
------------------------------------------------------------------------
2226
-- Surjective functions.
2327

@@ -29,6 +33,29 @@ record Surjective {f₁ f₂ t₁ t₂}
2933
from : To ⟶ From
3034
right-inverse-of : from RightInverseOf to
3135

36+
------------------------------------------------------------------------
37+
-- Relation to non-strict versions
38+
39+
non-strict-to-strict :
40+
∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
41+
(f : To ⟶ From) →
42+
NonStrict.Surjective f →
43+
Surjective f
44+
non-strict-to-strict f ns = record
45+
{ from = from
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; right-inverse-of = Left.non-strict-to-strict f from right-inverse-of
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} where open NonStrict.Surjective ns
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49+
strict-to-non-strict :
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∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
51+
(f : From ⟶ To) →
52+
Surjective f →
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NonStrict.Surjective f
54+
strict-to-non-strict f ns = record
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{ from = from
56+
; right-inverse-of = Left.strict-to-non-strict f from right-inverse-of
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} where open Surjective ns
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3259
------------------------------------------------------------------------
3360
-- The set of all surjections from one setoid to another.
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