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string | meta
dict |
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data Unit : Set where
unit : Unit
P : Unit β Set
P unit = Unit
postulate
Q : (u : Unit) β P u β Set
variable
u : Unit
p : P u
postulate
q : P u β Q u p
q' : (u : Unit) (p : P u) β P u β Q u p
q' u p = q {u} {p}
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data β : Set where
zero : β
suc : β β β
{-# BUILTIN NATURAL β #-}
infixl 6 _+_
infix 6 _βΈ_
_+_ : β β β β β
zero + n = n
suc m + n = suc (m + n)
_βΈ_ : β β β β β
m βΈ zero = m
zero βΈ suc n = zero
suc m βΈ suc n = m βΈ n
should-be-rejected : β
should-be-rejected = 1 + 0 βΈ 1
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-- {-# OPTIONS -v term:20 #-}
-- Andreas, 2011-04-19 (Agda list post by Leonard Rodriguez)
module TerminationSubExpression where
infixr 3 _β¨_
data Type : Set where
int : Type
_β¨_ : Type β Type β Type
test : Type β Type
test int = int
test (Ο β¨ int) = test Ο
test (Ο β¨ (Οβ² β¨ Οβ³)) = test (Οβ² β¨ Οβ³)
-- this should terminate since rec. call on subterm
test' : Type β Type
test' int = int
test' (Ο β¨ int) = test' Ο
test' (Ο β¨ Οβ²) = test' Οβ²
ok : Type β Type
ok int = int
ok (Ο β¨ Οβ²) with Οβ²
... | int = ok Ο
... | (Οβ³ β¨ Οβ΄) = ok (Οβ³ β¨ Οβ΄)
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Sets.EquivalenceRelations
open import Setoids.Setoids
open import Functions.Definition
open import Groups.Definition
open import Groups.Homomorphisms.Definition
open import Groups.Subgroups.Definition
open import Groups.Subgroups.Normal.Definition
module Groups.Cosets {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A β A β A} (G : Group S _+_) {c : _} {pred : A β Set c} (subgrp : Subgroup G pred) where
open Equivalence (Setoid.eq S)
open import Groups.Lemmas G
open Group G
open Subgroup subgrp
cosetSetoid : Setoid A
Setoid._βΌ_ cosetSetoid g h = pred ((Group.inverse G h) + g)
Equivalence.reflexive (Setoid.eq cosetSetoid) = isSubset (symmetric (Group.invLeft G)) containsIdentity
Equivalence.symmetric (Setoid.eq cosetSetoid) yx = isSubset (transitive invContravariant (+WellDefined reflexive invInv)) (closedUnderInverse yx)
Equivalence.transitive (Setoid.eq cosetSetoid) yx zy = isSubset (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) reflexive)) (closedUnderPlus zy yx)
cosetGroup : normalSubgroup G subgrp β Group cosetSetoid _+_
Group.+WellDefined (cosetGroup norm) {m} {n} {x} {y} m=x n=y = ans
where
t : pred (inverse y + n)
t = n=y
u : pred (inverse x + m)
u = m=x
v : pred (m + inverse x)
v = isSubset (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) (norm u)
ans' : pred ((inverse y) + ((inverse x + m) + inverse (inverse y)))
ans' = norm u
ans'' : pred ((inverse y) + ((inverse x + m) + y))
ans'' = isSubset (+WellDefined reflexive (+WellDefined reflexive (invTwice y))) ans'
ans : pred (inverse (x + y) + (m + n))
ans = isSubset (transitive (transitive +Associative (transitive (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) +Associative)) reflexive) (symmetric +Associative))) (symmetric (+WellDefined invContravariant reflexive))) (closedUnderPlus ans'' t)
Group.0G (cosetGroup norm) = 0G
Group.inverse (cosetGroup norm) = inverse
Group.+Associative (cosetGroup norm) {a} {b} {c} = isSubset (symmetric (transitive (+WellDefined (inverseWellDefined (symmetric +Associative)) reflexive) (invLeft {a + (b + c)}))) containsIdentity
Group.identRight (cosetGroup norm) = isSubset (symmetric (transitive +Associative (transitive (+WellDefined invLeft reflexive) identRight))) containsIdentity
Group.identLeft (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive identLeft) invLeft)) containsIdentity
Group.invLeft (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive invLeft) invLeft)) containsIdentity
Group.invRight (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive invRight) invLeft)) containsIdentity
cosetGroupHom : (norm : normalSubgroup G subgrp) β GroupHom G (cosetGroup norm) id
GroupHom.groupHom (cosetGroupHom norm) = isSubset (symmetric (transitive (+WellDefined invContravariant reflexive) (transitive +Associative (transitive (+WellDefined (transitive (symmetric +Associative) (+WellDefined reflexive invLeft)) reflexive) (transitive (+WellDefined identRight reflexive) invLeft))))) (Subgroup.containsIdentity subgrp)
GroupHom.wellDefined (cosetGroupHom norm) {x} {y} x=y = isSubset (symmetric (transitive (+WellDefined reflexive x=y) invLeft)) (Subgroup.containsIdentity subgrp)
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{-# BUILTIN NATURAL β #-}
module the-naturals where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_β‘_; refl)
open Eq.β‘-Reasoning using (begin_; _β‘β¨β©_; _β)
infixl 6 _+_ _βΈ_
infixl 7 _*_
-- the naturals
data β : Set where
zero : β
suc : β β β
-- addition
_+_ : β β β β β
zero + n = n
(suc m) + n = suc (m + n)
-- multiplication
_*_ : β β β β β
zero * n = zero
(suc m) * n = n + (m * n)
-- monus ( subtraction for the naturals )
_βΈ_ : β β β β β
m βΈ zero = m
zero βΈ suc n = zero
suc m βΈ suc n = m βΈ n
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module Category.Functor.Arr where
open import Agda.Primitive using (_β_)
open import Category.Functor using (RawFunctor ; module RawFunctor )
open import Category.Applicative using (RawApplicative; module RawApplicative)
open import Function using (_β_)
open import Category.Functor.Lawful
open import Relation.Binary.PropositionalEquality using (refl)
Arr : β {lβ lβ} β Set lβ β Set lβ β Set (lβ β lβ)
Arr A B = A β B
arrFunctor : β {lβ lβ} {B : Set lβ} β RawFunctor (Arr {lβ} {lβ} B)
arrFunctor = record { _<$>_ = Ξ» z zβ x β z (zβ x) } -- auto-found
arrLawfulFunctor : β {lβ lβ} {B : Set lβ} β LawfulFunctorImp (arrFunctor {lβ} {lβ} {B})
arrLawfulFunctor = record
{ <$>-identity = refl
; <$>-compose = refl
}
arrApplicative : β {lβ} {B : Set lβ} β RawApplicative (Arr {lβ} {lβ} B)
arrApplicative = record { pure = Ξ» z x β z ; _β_ = Ξ» z zβ x β z x (zβ x) } -- auto-found
arrLawfulApplicative : β {lβ} {B : Set lβ} β LawfulApplicativeImp (arrApplicative {lβ} {B})
arrLawfulApplicative = record
{ β-identity = refl
; β-homomorphism = refl
; β-interchange = refl
; β-composition = refl
}
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Universe levels
------------------------------------------------------------------------
module Level where
-- Levels.
open import Agda.Primitive public
using (Level; _β_)
renaming (lzero to zero; lsuc to suc)
-- Lifting.
record Lift {a β} (A : Set a) : Set (a β β) where
constructor lift
field lower : A
open Lift public
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{-# OPTIONS --without-K #-}
open import lib.Basics
open import lib.NConnected
open import lib.types.Nat
open import lib.types.TLevel
open import lib.types.Empty
open import lib.types.Group
open import lib.types.Pi
open import lib.types.Pointed
open import lib.types.Paths
open import lib.types.Sigma
open import lib.types.Truncation
open import lib.cubical.Square
module lib.types.LoopSpace where
module _ {i} where
βΞ© : Ptd i β Ptd i
βΞ© (A , a) = β[ (a == a) , idp ]
Ξ© : Ptd i β Type i
Ξ© = fst β βΞ©
βΞ©^ : (n : β) β Ptd i β Ptd i
βΞ©^ O X = X
βΞ©^ (S n) X = βΞ© (βΞ©^ n X)
Ξ©^ : (n : β) β Ptd i β Type i
Ξ©^ n X = fst (βΞ©^ n X)
idp^ : β {i} (n : β) {X : Ptd i} β Ξ©^ n X
idp^ n {X} = snd (βΞ©^ n X)
{- for n β₯ 1, we have a group structure on the loop space -}
module _ {i} where
!^ : (n : β) (t : n β O) {X : Ptd i} β Ξ©^ n X β Ξ©^ n X
!^ O t = β₯-rec (t idp)
!^ (S n) _ = !
conc^ : (n : β) (t : n β O) {X : Ptd i} β Ξ©^ n X β Ξ©^ n X β Ξ©^ n X
conc^ O t = β₯-rec (t idp)
conc^ (S n) _ = _β_
{- ap and ap2 for pointed functions -}
private
pt-lemma : β {i} {A : Type i} {x y : A} (p : x == y)
β ! p β (idp β' p) == idp
pt-lemma idp = idp
βap : β {i j} {X : Ptd i} {Y : Ptd j}
β fst (X ββ Y) β fst (βΞ© X ββ βΞ© Y)
βap (f , fpt) = ((Ξ» p β ! fpt β ap f p β' fpt) , pt-lemma fpt)
βap2 : β {i j k} {X : Ptd i} {Y : Ptd j} {Z : Ptd k}
β fst (X βΓ Y ββ Z) β fst (βΞ© X βΓ βΞ© Y ββ βΞ© Z)
βap2 (f , fpt) = ((Ξ» {(p , q) β ! fpt β ap2 (curry f) p q β' fpt}) ,
pt-lemma fpt)
βap-β : β {i j k} {X : Ptd i} {Y : Ptd j} {Z : Ptd k}
(g : fst (Y ββ Z)) (f : fst (X ββ Y))
β βap (g ββ f) == βap g ββ βap f
βap-β (g , idp) (f , idp) = βΞ»= (Ξ» p β ap-β g f p) idp
βap-idf : β {i} {X : Ptd i} β βap (βidf X) == βidf _
βap-idf = βΞ»= ap-idf idp
βap2-fst : β {i j} {X : Ptd i} {Y : Ptd j}
β βap2 {X = X} {Y = Y} βfst == βfst
βap2-fst = βΞ»= (uncurry ap2-fst) idp
βap2-snd : β {i j} {X : Ptd i} {Y : Ptd j}
β βap2 {X = X} {Y = Y} βsnd == βsnd
βap2-snd = βΞ»= (uncurry ap2-snd) idp
βap-ap2 : β {i j k l} {X : Ptd i} {Y : Ptd j} {Z : Ptd k} {W : Ptd l}
(G : fst (Z ββ W)) (F : fst (X βΓ Y ββ Z))
β βap G ββ βap2 F == βap2 (G ββ F)
βap-ap2 (g , idp) (f , idp) =
βΞ»= (uncurry (ap-ap2 g (curry f))) idp
βap2-ap : β {i j k l m}
{X : Ptd i} {Y : Ptd j} {U : Ptd k} {V : Ptd l} {Z : Ptd m}
(G : fst ((U βΓ V) ββ Z)) (Fβ : fst (X ββ U)) (Fβ : fst (Y ββ V))
β βap2 G ββ pairββ (βap Fβ) (βap Fβ) == βap2 (G ββ pairββ Fβ Fβ)
βap2-ap (g , idp) (fβ , idp) (fβ , idp) =
βΞ»= (Ξ» {(p , q) β ap2-ap-l (curry g) fβ p (ap fβ q)
β ap2-ap-r (Ξ» x v β g (fβ x , v)) fβ p q})
idp
βap2-diag : β {i j} {X : Ptd i} {Y : Ptd j} (F : fst (X βΓ X ββ Y))
β βap2 F ββ βdiag == βap (F ββ βdiag)
βap2-diag (f , idp) = βΞ»= (ap2-diag (curry f)) idp
{- ap and ap2 for higher loop spaces -}
ap^ : β {i j} (n : β) {X : Ptd i} {Y : Ptd j}
β fst (X ββ Y) β fst (βΞ©^ n X ββ βΞ©^ n Y)
ap^ O F = F
ap^ (S n) F = βap (ap^ n F)
ap2^ : β {i j k} (n : β) {X : Ptd i} {Y : Ptd j} {Z : Ptd k}
β fst ((X βΓ Y) ββ Z)
β fst ((βΞ©^ n X βΓ βΞ©^ n Y) ββ βΞ©^ n Z)
ap2^ O F = F
ap2^ (S n) F = βap2 (ap2^ n F)
ap^-idf : β {i} (n : β) {X : Ptd i} β ap^ n (βidf X) == βidf _
ap^-idf O = idp
ap^-idf (S n) = ap βap (ap^-idf n) β βap-idf
ap^-ap2^ : β {i j k l} (n : β) {X : Ptd i} {Y : Ptd j} {Z : Ptd k} {W : Ptd l}
(G : fst (Z ββ W)) (F : fst ((X βΓ Y) ββ Z))
β ap^ n G ββ ap2^ n F == ap2^ n (G ββ F)
ap^-ap2^ O G F = idp
ap^-ap2^ (S n) G F = βap-ap2 (ap^ n G) (ap2^ n F) β ap βap2 (ap^-ap2^ n G F)
ap2^-fst : β {i j} (n : β) {X : Ptd i} {Y : Ptd j}
β ap2^ n {X} {Y} βfst == βfst
ap2^-fst O = idp
ap2^-fst (S n) = ap βap2 (ap2^-fst n) β βap2-fst
ap2^-snd : β {i j} (n : β) {X : Ptd i} {Y : Ptd j}
β ap2^ n {X} {Y} βsnd == βsnd
ap2^-snd O = idp
ap2^-snd (S n) = ap βap2 (ap2^-snd n) β βap2-snd
ap2^-ap^ : β {i j k l m} (n : β)
{X : Ptd i} {Y : Ptd j} {U : Ptd k} {V : Ptd l} {Z : Ptd m}
(G : fst ((U βΓ V) ββ Z)) (Fβ : fst (X ββ U)) (Fβ : fst (Y ββ V))
β ap2^ n G ββ pairββ (ap^ n Fβ) (ap^ n Fβ) == ap2^ n (G ββ pairββ Fβ Fβ)
ap2^-ap^ O G Fβ Fβ = idp
ap2^-ap^ (S n) G Fβ Fβ =
βap2-ap (ap2^ n G) (ap^ n Fβ) (ap^ n Fβ) β ap βap2 (ap2^-ap^ n G Fβ Fβ)
ap2^-diag : β {i j} (n : β) {X : Ptd i} {Y : Ptd j} (F : fst (X βΓ X ββ Y))
β ap2^ n F ββ βdiag == ap^ n (F ββ βdiag)
ap2^-diag O F = idp
ap2^-diag (S n) F = βap2-diag (ap2^ n F) β ap βap (ap2^-diag n F)
module _ {i} {X : Ptd i} where
{- Prove these as lemmas now
- so we don't have to deal with the n = O case later -}
conc^-unit-l : (n : β) (t : n β O) (q : Ξ©^ n X)
β (conc^ n t (idp^ n) q) == q
conc^-unit-l O t _ = β₯-rec (t idp)
conc^-unit-l (S n) _ _ = idp
conc^-unit-r : (n : β) (t : n β O) (q : Ξ©^ n X)
β (conc^ n t q (idp^ n)) == q
conc^-unit-r O t = β₯-rec (t idp)
conc^-unit-r (S n) _ = β-unit-r
conc^-assoc : (n : β) (t : n β O) (p q r : Ξ©^ n X)
β conc^ n t (conc^ n t p q) r == conc^ n t p (conc^ n t q r)
conc^-assoc O t = β₯-rec (t idp)
conc^-assoc (S n) _ = β-assoc
!^-inv-l : (n : β) (t : n β O) (p : Ξ©^ n X)
β conc^ n t (!^ n t p) p == idp^ n
!^-inv-l O t = β₯-rec (t idp)
!^-inv-l (S n) _ = !-inv-l
!^-inv-r : (n : β) (t : n β O) (p : Ξ©^ n X)
β conc^ n t p (!^ n t p) == idp^ n
!^-inv-r O t = β₯-rec (t idp)
!^-inv-r (S n) _ = !-inv-r
abstract
ap^-conc^ : β {i j} (n : β) (t : n β O)
{X : Ptd i} {Y : Ptd j} (F : fst (X ββ Y)) (p q : Ξ©^ n X)
β fst (ap^ n F) (conc^ n t p q)
== conc^ n t (fst (ap^ n F) p) (fst (ap^ n F) q)
ap^-conc^ O t _ _ _ = β₯-rec (t idp)
ap^-conc^ (S n) _ {X = X} {Y = Y} F p q =
! gpt β ap g (p β q) β' gpt
=β¨ ap-β g p q |in-ctx (Ξ» w β ! gpt β w β' gpt) β©
! gpt β (ap g p β ap g q) β' gpt
=β¨ lemma (ap g p) (ap g q) gpt β©
(! gpt β ap g p β' gpt) β (! gpt β ap g q β' gpt) β
where
g : Ξ©^ n X β Ξ©^ n Y
g = fst (ap^ n F)
gpt : g (idp^ n) == idp^ n
gpt = snd (ap^ n F)
lemma : β {i} {A : Type i} {x y : A}
β (p q : x == x) (r : x == y)
β ! r β (p β q) β' r == (! r β p β' r) β (! r β q β' r)
lemma p q idp = idp
{- ap^ preserves (pointed) equivalences -}
module _ {i j} {X : Ptd i} {Y : Ptd j} where
is-equiv-ap^ : (n : β) (F : fst (X ββ Y)) (e : is-equiv (fst F))
β is-equiv (fst (ap^ n F))
is-equiv-ap^ O F e = e
is-equiv-ap^ (S n) F e =
preβ-is-equiv (! (snd (ap^ n F)))
βise postβ'-is-equiv (snd (ap^ n F))
βise snd (equiv-ap (_ , is-equiv-ap^ n F e) _ _)
equiv-ap^ : (n : β) (F : fst (X ββ Y)) (e : is-equiv (fst F))
β Ξ©^ n X β Ξ©^ n Y
equiv-ap^ n F e = (fst (ap^ n F) , is-equiv-ap^ n F e)
Ξ©^-level-in : β {i} (m : βββ) (n : β) (X : Ptd i)
β (has-level ((n -2) +2+ m) (fst X) β has-level m (Ξ©^ n X))
Ξ©^-level-in m O X pX = pX
Ξ©^-level-in m (S n) X pX =
Ξ©^-level-in (S m) n X
(transport (Ξ» k β has-level k (fst X)) (! (+2+-Ξ²r (n -2) m)) pX)
(idp^ n) (idp^ n)
Ξ©^-conn-in : β {i} (m : βββ) (n : β) (X : Ptd i)
β (is-connected ((n -2) +2+ m) (fst X)) β is-connected m (Ξ©^ n X)
Ξ©^-conn-in m O X pX = pX
Ξ©^-conn-in m (S n) X pX =
path-conn $ Ξ©^-conn-in (S m) n X $
transport (Ξ» k β is-connected k (fst X)) (! (+2+-Ξ²r (n -2) m)) pX
{- Eckmann-Hilton argument -}
module _ {i} {X : Ptd i} where
conc^2-comm : (Ξ± Ξ² : Ξ©^ 2 X) β conc^ 2 (β-Sβ O _) Ξ± Ξ² == conc^ 2 (β-Sβ O _) Ξ² Ξ±
conc^2-comm Ξ± Ξ² = ! (β2=conc^ Ξ± Ξ²) β β2=β'2 Ξ± Ξ² β β'2=conc^ Ξ± Ξ²
where
β2=conc^ : (Ξ± Ξ² : Ξ©^ 2 X) β Ξ± β2 Ξ² == conc^ 2 (β-Sβ O _) Ξ± Ξ²
β2=conc^ Ξ± Ξ² = ap (Ξ» Ο β Ο β Ξ²) (β-unit-r Ξ±)
β'2=conc^ : (Ξ± Ξ² : Ξ©^ 2 X) β Ξ± β'2 Ξ² == conc^ 2 (β-Sβ O _) Ξ² Ξ±
β'2=conc^ Ξ± Ξ² = ap (Ξ» Ο β Ξ² β Ο) (β-unit-r Ξ±)
{- Pushing truncation through loop space -}
module _ {i} where
Trunc-Ξ©^ : (m : βββ) (n : β) (X : Ptd i)
β βTrunc m (βΞ©^ n X) == βΞ©^ n (βTrunc ((n -2) +2+ m) X)
Trunc-Ξ©^ m O X = idp
Trunc-Ξ©^ m (S n) X =
βTrunc m (βΞ©^ (S n) X)
=β¨ ! (pair= (Trunc=-path [ _ ] [ _ ]) (β-idf-ua-in _ idp)) β©
βΞ© (βTrunc (S m) (βΞ©^ n X))
=β¨ ap βΞ© (Trunc-Ξ©^ (S m) n X) β©
βΞ©^ (S n) (βTrunc ((n -2) +2+ S m) X)
=β¨ +2+-Ξ²r (n -2) m |in-ctx (Ξ» k β βΞ©^ (S n) (βTrunc k X)) β©
βΞ©^ (S n) (βTrunc (S (n -2) +2+ m) X) β
Ξ©-Trunc-equiv : (m : βββ) (X : Ptd i)
β Ξ© (βTrunc (S m) X) β Trunc m (Ξ© X)
Ξ©-Trunc-equiv m X = Trunc=-equiv [ snd X ] [ snd X ]
{- A loop space is a pregroup, and a group if it has the right level -}
module _ {i} (n : β) (t : n β O) (X : Ptd i) where
Ξ©^-group-structure : GroupStructure (Ξ©^ n X)
Ξ©^-group-structure = record {
ident = idp^ n;
inv = !^ n t;
comp = conc^ n t;
unitl = conc^-unit-l n t;
unitr = conc^-unit-r n t;
assoc = conc^-assoc n t;
invr = !^-inv-r n t;
invl = !^-inv-l n t
}
Ξ©^-Group : has-level β¨ n β© (fst X) β Group i
Ξ©^-Group pX = group
(Ξ©^ n X)
(Ξ©^-level-in β¨0β© n X $
transport (Ξ» t β has-level t (fst X)) (+2+-comm β¨0β© (n -2)) pX)
Ξ©^-group-structure
{- Our definition of Ξ©^ builds up loops on the outside,
- but this is equivalent to building up on the inside -}
module _ {i} where
βΞ©^-inner-path : (n : β) (X : Ptd i)
β βΞ©^ (S n) X == βΞ©^ n (βΞ© X)
βΞ©^-inner-path O X = idp
βΞ©^-inner-path (S n) X = ap βΞ© (βΞ©^-inner-path n X)
βΞ©^-inner-out : (n : β) (X : Ptd i)
β fst (βΞ©^ (S n) X ββ βΞ©^ n (βΞ© X))
βΞ©^-inner-out O _ = (idf _ , idp)
βΞ©^-inner-out (S n) X = ap^ 1 (βΞ©^-inner-out n X)
Ξ©^-inner-out : (n : β) (X : Ptd i)
β (Ξ©^ (S n) X β Ξ©^ n (βΞ© X))
Ξ©^-inner-out n X = fst (βΞ©^-inner-out n X)
Ξ©^-inner-out-conc^ : (n : β) (t : n β O)
(X : Ptd i) (p q : Ξ©^ (S n) X)
β Ξ©^-inner-out n X (conc^ (S n) (β-Sβ O _) p q)
== conc^ n t (Ξ©^-inner-out n X p) (Ξ©^-inner-out n X q)
Ξ©^-inner-out-conc^ O t X _ _ = β₯-rec (t idp)
Ξ©^-inner-out-conc^ (S n) t X p q =
ap^-conc^ 1 (β-Sβ O _) (βΞ©^-inner-out n X) p q
Ξ©^-inner-is-equiv : (n : β) (X : Ptd i)
β is-equiv (fst (βΞ©^-inner-out n X))
Ξ©^-inner-is-equiv O X = is-eq (idf _) (idf _) (Ξ» _ β idp) (Ξ» _ β idp)
Ξ©^-inner-is-equiv (S n) X =
is-equiv-ap^ 1 (βΞ©^-inner-out n X) (Ξ©^-inner-is-equiv n X)
Ξ©^-inner-equiv : (n : β) (X : Ptd i) β Ξ©^ (S n) X β Ξ©^ n (βΞ© X)
Ξ©^-inner-equiv n X = _ , Ξ©^-inner-is-equiv n X
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open import Prelude
module Implicits.Syntax.Term where
open import Implicits.Syntax.Type
infixl 9 _[_] _Β·_
data Term (Ξ½ n : β) : Set where
var : (x : Fin n) β Term Ξ½ n
Ξ : Term (suc Ξ½) n β Term Ξ½ n
Ξ»' : Type Ξ½ β Term Ξ½ (suc n) β Term Ξ½ n
_[_] : Term Ξ½ n β Type Ξ½ β Term Ξ½ n
_Β·_ : Term Ξ½ n β Term Ξ½ n β Term Ξ½ n
-- rule abstraction and application
Ο : Type Ξ½ β Term Ξ½ (suc n) β Term Ξ½ n
_with'_ : Term Ξ½ n β Term Ξ½ n β Term Ξ½ n
-- implicit rule application
_β¨β© : Term Ξ½ n β Term Ξ½ n
ClosedTerm : Set
ClosedTerm = Term 0 0
-----------------------------------------------------------------------------
-- syntactic sugar
let'_βΆ_in'_ : β {Ξ½ n} β Term Ξ½ n β Type Ξ½ β Term Ξ½ (suc n) β Term Ξ½ n
let' eβ βΆ r in' eβ = (Ξ»' r eβ) Β· eβ
implicit_βΆ_in'_ : β {Ξ½ n} β Term Ξ½ n β Type Ξ½ β Term Ξ½ (suc n) β Term Ξ½ n
implicit eβ βΆ r in' eβ = (Ο r eβ) with' eβ
ΒΏ_ : β {Ξ½ n} β Type Ξ½ β Term Ξ½ n
ΒΏ r = (Ο r (var zero)) β¨β©
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{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-}
module Light.Library where
module Literals where open import Light.Literals public
module Data where
module Empty where open import Light.Library.Data.Empty public
module Either where open import Light.Library.Data.Either public
module Natural where open import Light.Library.Data.Natural public
module Unit where open import Light.Library.Data.Unit public
module Integer where open import Light.Library.Data.Integer public
module Boolean where open import Light.Library.Data.Boolean public
module Both where open import Light.Library.Data.Both public
module Product where open import Light.Library.Data.Product public
module These where open import Light.Library.Data.These public
module Relation where
module Binary where
open import Light.Library.Relation.Binary public
module Equality where
open import Light.Library.Relation.Binary.Equality public
module Decidable where open import Light.Library.Relation.Binary.Equality.Decidable public
module Decidable where open import Light.Library.Relation.Binary.Decidable public
open import Light.Library.Relation public
module Decidable where open import Light.Library.Relation.Decidable public
module Action where open import Light.Library.Action public
module Arithmetic where open import Light.Library.Arithmetic public
module Level where open import Light.Level public
module Subtyping where open import Light.Subtyping public
module Variable where
module Levels where open import Light.Variable.Levels public
module Sets where open import Light.Variable.Sets public
module Other {β} (π : Set β) where open import Light.Variable.Other π public
module Package where open import Light.Package public
open Package using (Package) hiding (module Package) public
-- module Indexed where open import Light.Indexed
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{-# OPTIONS --safe #-}
module Cubical.Algebra.CommRing.Instances.Polynomials where
open import Cubical.Foundations.Prelude
open import Cubical.Algebra.CommRing
open import Cubical.Algebra.Polynomials
private
variable
β : Level
Poly : (CommRing β) β CommRing β
Poly R = (PolyMod.Poly R) , str
where
open CommRingStr --(snd R)
str : CommRingStr (PolyMod.Poly R)
0r str = PolyMod.0P R
1r str = PolyMod.1P R
_+_ str = PolyMod._Poly+_ R
_Β·_ str = PolyMod._Poly*_ R
- str = PolyMod.Poly- R
isCommRing str = makeIsCommRing (PolyMod.isSetPoly R)
(PolyMod.Poly+Assoc R)
(PolyMod.Poly+Rid R)
(PolyMod.Poly+Inverses R)
(PolyMod.Poly+Comm R)
(PolyMod.Poly*Associative R)
(PolyMod.Poly*Rid R)
(PolyMod.Poly*LDistrPoly+ R)
(PolyMod.Poly*Commutative R)
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module empty where
open import level
----------------------------------------------------------------------
-- datatypes
----------------------------------------------------------------------
data β₯ {β : Level} : Set β where
----------------------------------------------------------------------
-- syntax
----------------------------------------------------------------------
----------------------------------------------------------------------
-- theorems
----------------------------------------------------------------------
β₯-elim : β{β} β β₯ {β} β β{β'}{P : Set β'} β P
β₯-elim ()
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-- Andreas, 2015-08-11, issue reported by G.Allais
-- The `a` record field of `Pack` is identified as a function
-- (coloured blue, put in a \AgdaFunction in the LaTeX backend)
-- when it should be coloured pink.
-- The problem does not show up when dropping the second record
-- type or removing the module declaration.
record Pack (A : Set) : Set where
field
a : A
record Packed {A : Set} (p : Pack A) : Set where
module PP = Pack p
module Synchronised {A : Set} {p : Pack A} (rel : Packed p) where
module M = Packed rel
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Data.Nat.GCD where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Isomorphism
open import Cubical.Induction.WellFounded
open import Cubical.Data.Fin
open import Cubical.Data.Sigma as Ξ£
open import Cubical.Data.NatPlusOne
open import Cubical.HITs.PropositionalTruncation as PropTrunc
open import Cubical.Data.Nat.Base
open import Cubical.Data.Nat.Properties
open import Cubical.Data.Nat.Order
open import Cubical.Data.Nat.Divisibility
private
variable
m n d : β
-- common divisors
isCD : β β β β β β Typeβ
isCD m n d = (d β£ m) Γ (d β£ n)
isPropIsCD : isProp (isCD m n d)
isPropIsCD = isPropΓ isPropβ£ isPropβ£
symCD : isCD m n d β isCD n m d
symCD (dβ£m , dβ£n) = (dβ£n , dβ£m)
-- greatest common divisors
isGCD : β β β β β β Typeβ
isGCD m n d = (isCD m n d) Γ (β d' β isCD m n d' β d' β£ d)
GCD : β β β β Typeβ
GCD m n = Ξ£ β (isGCD m n)
isPropIsGCD : isProp (isGCD m n d)
isPropIsGCD = isPropΓ isPropIsCD (isPropΞ 2 (Ξ» _ _ β isPropβ£))
isPropGCD : isProp (GCD m n)
isPropGCD (d , dCD , gr) (d' , d'CD , gr') =
Ξ£β‘Prop (Ξ» _ β isPropIsGCD) (antisymβ£ (gr' d dCD) (gr d' d'CD))
symGCD : isGCD m n d β isGCD n m d
symGCD (dCD , gr) = symCD dCD , Ξ» { d' d'CD β gr d' (symCD d'CD) }
divsGCD : m β£ n β isGCD m n m
divsGCD p = (β£-refl refl , p) , Ξ» { d (dβ£m , _) β dβ£m }
oneGCD : β m β isGCD m 1 1
oneGCD m = symGCD (divsGCD (β£-oneΛ‘ m))
-- The base case of the Euclidean algorithm
zeroGCD : β m β isGCD m 0 m
zeroGCD m = divsGCD (β£-zeroΚ³ m)
private
lemβ : prediv d (suc n) β prediv d (m % suc n) β prediv d m
lemβ {d} {n} {m} (cβ , pβ) (cβ , pβ) = (q Β· cβ + cβ) , p
where r = m % suc n; q = n%kβ‘n[modk] m (suc n) .fst
p = (q Β· cβ + cβ) Β· d β‘β¨ sym (Β·-distribΚ³ (q Β· cβ) cβ d) β©
(q Β· cβ) Β· d + cβ Β· d β‘β¨ cong (_+ cβ Β· d) (sym (Β·-assoc q cβ d)) β©
q Β· (cβ Β· d) + cβ Β· d β‘[ i ]β¨ q Β· (pβ i) + (pβ i) β©
q Β· (suc n) + r β‘β¨ n%kβ‘n[modk] m (suc n) .snd β©
m β
lemβ : prediv d (suc n) β prediv d m β prediv d (m % suc n)
lemβ {d} {n} {m} (cβ , pβ) (cβ , pβ) = cβ βΈ q Β· cβ , p
where r = m % suc n; q = n%kβ‘n[modk] m (suc n) .fst
p = (cβ βΈ q Β· cβ) Β· d β‘β¨ βΈ-distribΚ³ cβ (q Β· cβ) d β©
cβ Β· d βΈ (q Β· cβ) Β· d β‘β¨ cong (cβ Β· d βΈ_) (sym (Β·-assoc q cβ d)) β©
cβ Β· d βΈ q Β· (cβ Β· d) β‘[ i ]β¨ pβ i βΈ q Β· (pβ i) β©
m βΈ q Β· (suc n) β‘β¨ cong (_βΈ q Β· (suc n)) (sym (n%kβ‘n[modk] m (suc n) .snd)) β©
(q Β· (suc n) + r) βΈ q Β· (suc n) β‘β¨ cong (_βΈ q Β· (suc n)) (+-comm (q Β· (suc n)) r) β©
(r + q Β· (suc n)) βΈ q Β· (suc n) β‘β¨ βΈ-cancelΚ³ r zero (q Β· (suc n)) β©
r β
-- The inductive step of the Euclidean algorithm
stepGCD : isGCD (suc n) (m % suc n) d
β isGCD m (suc n) d
fst (stepGCD ((dβ£n , dβ£m%n) , gr)) = PropTrunc.map2 lemβ dβ£n dβ£m%n , dβ£n
snd (stepGCD ((dβ£n , dβ£m%n) , gr)) d' (d'β£m , d'β£n) = gr d' (d'β£n , PropTrunc.map2 lemβ d'β£n d'β£m)
-- putting it all together using well-founded induction
euclid< : β m n β n < m β GCD m n
euclid< = WFI.induction <-wellfounded Ξ» {
m rec zero p β m , zeroGCD m ;
m rec (suc n) p β let d , dGCD = rec (suc n) p (m % suc n) (n%sk<sk m n)
in d , stepGCD dGCD }
euclid : β m n β GCD m n
euclid m n with n β m
... | lt p = euclid< m n p
... | gt p = Ξ£.map-snd symGCD (euclid< n m p)
... | eq p = m , divsGCD (β£-refl (sym p))
isContrGCD : β m n β isContr (GCD m n)
isContrGCD m n = euclid m n , isPropGCD _
-- the gcd operator on β
gcd : β β β β β
gcd m n = euclid m n .fst
gcdIsGCD : β m n β isGCD m n (gcd m n)
gcdIsGCD m n = euclid m n .snd
isGCDβgcdβ‘ : isGCD m n d β gcd m n β‘ d
isGCDβgcdβ‘ dGCD = cong fst (isContrGCD _ _ .snd (_ , dGCD))
gcdβ‘βisGCD : gcd m n β‘ d β isGCD m n d
gcdβ‘βisGCD p = subst (isGCD _ _) p (gcdIsGCD _ _)
-- multiplicative properties of the gcd
isCD-cancelΚ³ : β k β isCD (m Β· suc k) (n Β· suc k) (d Β· suc k)
β isCD m n d
isCD-cancelΚ³ k (dkβ£mk , dkβ£nk) = (β£-cancelΚ³ k dkβ£mk , β£-cancelΚ³ k dkβ£nk)
isCD-multΚ³ : β k β isCD m n d
β isCD (m Β· k) (n Β· k) (d Β· k)
isCD-multΚ³ k (dβ£m , dβ£n) = (β£-multΚ³ k dβ£m , β£-multΚ³ k dβ£n)
isGCD-cancelΚ³ : β k β isGCD (m Β· suc k) (n Β· suc k) (d Β· suc k)
β isGCD m n d
isGCD-cancelΚ³ {m} {n} {d} k (dCD , gr) =
isCD-cancelΚ³ k dCD , Ξ» d' d'CD β β£-cancelΚ³ k (gr (d' Β· suc k) (isCD-multΚ³ (suc k) d'CD))
gcd-factorΚ³ : β m n k β gcd (m Β· k) (n Β· k) β‘ gcd m n Β· k
gcd-factorΚ³ m n zero = (Ξ» i β gcd (0β‘mΒ·0 m (~ i)) (0β‘mΒ·0 n (~ i))) β 0β‘mΒ·0 (gcd m n)
gcd-factorΚ³ m n (suc k) = sym p β cong (_Β· suc k) (sym q)
where kβ£gcd : suc k β£ gcd (m Β· suc k) (n Β· suc k)
kβ£gcd = gcdIsGCD (m Β· suc k) (n Β· suc k) .snd (suc k) (β£-right m , β£-right n)
d' = β£-untrunc kβ£gcd .fst
p : d' Β· suc k β‘ gcd (m Β· suc k) (n Β· suc k)
p = β£-untrunc kβ£gcd .snd
d'GCD : isGCD m n d'
d'GCD = isGCD-cancelΚ³ _ (subst (isGCD _ _) (sym p) (gcdIsGCD (m Β· suc k) (n Β· suc k)))
q : gcd m n β‘ d'
q = isGCDβgcdβ‘ d'GCD
-- Q: Can this be proved directly? (i.e. without a transport)
isGCD-multΚ³ : β k β isGCD m n d
β isGCD (m Β· k) (n Β· k) (d Β· k)
isGCD-multΚ³ {m} {n} {d} k dGCD = gcdβ‘βisGCD (gcd-factorΚ³ m n k β cong (_Β· k) r)
where r : gcd m n β‘ d
r = isGCDβgcdβ‘ dGCD
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{-# OPTIONS --prop --without-K --rewriting #-}
module Calf.Types.Bool where
open import Calf.Prelude
open import Calf.Metalanguage
open import Data.Bool public using (Bool; true; false; if_then_else_)
bool : tp pos
bool = U (meta Bool)
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{-# OPTIONS --safe #-}
module MissingDefinition where
T : Set -> Set
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{- Cubical Agda with K
This file demonstrates the incompatibility of the --cubical
and --with-K flags, relying on the well-known incosistency of K with
univalence.
The --safe flag can be used to prevent accidentally mixing such
incompatible flags.
-}
{-# OPTIONS --with-K #-}
module Cubical.WithK where
open import Cubical.Data.Equality
open import Cubical.Data.Bool
open import Cubical.Data.Empty
private
variable
β : Level
A : Type β
x y : A
uip : (prf : x β‘ x) β Path _ prf refl
uip refl i = refl
transport-uip : (prf : A β‘ A) β Path _ (transportPath (eqToPath prf) x) x
transport-uip {x = x} prf =
compPath (congPath (Ξ» p β transportPath (eqToPath p) x) (uip prf)) (transportRefl x)
transport-not : Path _ (transportPath (eqToPath (pathToEq notEq)) true) false
transport-not = congPath (Ξ» a β transportPath a true) (eqToPath-pathToEq notEq)
false-true : Path _ false true
false-true = compPath (symPath transport-not) (transport-uip (pathToEq notEq))
absurd : (X : Type) β X
absurd X = transportPath (congPath sel false-true) true
where
sel : Bool β Type
sel false = Bool
sel true = X
inconsistency : β₯
inconsistency = absurd β₯
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module WrongNamedArgument2 where
postulate
f : {A : Setβ} β A
test : Set
test = f {B = Set}
-- Unsolved meta.
-- It is not an error since A could be instantiated to a function type
-- accepting hidden argument with name B.
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Argument information used in the reflection machinery
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Reflection.Argument.Information where
open import Data.Product
open import Relation.Nullary
import Relation.Nullary.Decidable as Dec
open import Relation.Nullary.Product using (_Γ-dec_)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality
open import Reflection.Argument.Relevance as Relevance using (Relevance)
open import Reflection.Argument.Visibility as Visibility using (Visibility)
------------------------------------------------------------------------
-- Re-exporting the builtins publically
open import Agda.Builtin.Reflection public using (ArgInfo)
open ArgInfo public
------------------------------------------------------------------------
-- Operations
visibility : ArgInfo β Visibility
visibility (arg-info v _) = v
relevance : ArgInfo β Relevance
relevance (arg-info _ r) = r
------------------------------------------------------------------------
-- Decidable equality
arg-info-injectiveβ : β {v r vβ² rβ²} β arg-info v r β‘ arg-info vβ² rβ² β v β‘ vβ²
arg-info-injectiveβ refl = refl
arg-info-injectiveβ : β {v r vβ² rβ²} β arg-info v r β‘ arg-info vβ² rβ² β r β‘ rβ²
arg-info-injectiveβ refl = refl
arg-info-injective : β {v r vβ² rβ²} β arg-info v r β‘ arg-info vβ² rβ² β v β‘ vβ² Γ r β‘ rβ²
arg-info-injective = < arg-info-injectiveβ , arg-info-injectiveβ >
_β_ : DecidableEquality ArgInfo
arg-info v r β arg-info vβ² rβ² =
Dec.mapβ² (uncurry (congβ arg-info))
arg-info-injective
(v Visibility.β vβ² Γ-dec r Relevance.β rβ²)
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{-# OPTIONS --rewriting #-}
module FFI.Data.Vector where
open import Agda.Builtin.Equality using (_β‘_)
open import Agda.Builtin.Equality.Rewrite using ()
open import Agda.Builtin.Int using (Int; pos; negsuc)
open import Agda.Builtin.Nat using (Nat)
open import Agda.Builtin.Bool using (Bool; false; true)
open import FFI.Data.HaskellInt using (HaskellInt; haskellIntToInt; intToHaskellInt)
open import FFI.Data.Maybe using (Maybe; just; nothing)
open import Properties.Equality using (_β’_)
{-# FOREIGN GHC import qualified Data.Vector #-}
postulate Vector : Set β Set
{-# POLARITY Vector ++ #-}
{-# COMPILE GHC Vector = type Data.Vector.Vector #-}
postulate
empty : β {A} β (Vector A)
null : β {A} β (Vector A) β Bool
unsafeHead : β {A} β (Vector A) β A
unsafeTail : β {A} β (Vector A) β (Vector A)
length : β {A} β (Vector A) β Nat
lookup : β {A} β (Vector A) β Nat β (Maybe A)
snoc : β {A} β (Vector A) β A β (Vector A)
{-# COMPILE GHC empty = \_ -> Data.Vector.empty #-}
{-# COMPILE GHC null = \_ -> Data.Vector.null #-}
{-# COMPILE GHC unsafeHead = \_ -> Data.Vector.unsafeHead #-}
{-# COMPILE GHC unsafeTail = \_ -> Data.Vector.unsafeTail #-}
{-# COMPILE GHC length = \_ -> (fromIntegral . Data.Vector.length) #-}
{-# COMPILE GHC lookup = \_ v -> ((v Data.Vector.!?) . fromIntegral) #-}
{-# COMPILE GHC snoc = \_ -> Data.Vector.snoc #-}
postulate length-empty : β {A} β (length (empty {A}) β‘ 0)
postulate lookup-empty : β {A} n β (lookup (empty {A}) n β‘ nothing)
postulate lookup-snoc : β {A} (x : A) (v : Vector A) β (lookup (snoc v x) (length v) β‘ just x)
postulate lookup-length : β {A} (v : Vector A) β (lookup v (length v) β‘ nothing)
postulate lookup-snoc-empty : β {A} (x : A) β (lookup (snoc empty x) 0 β‘ just x)
postulate lookup-snoc-not : β {A n} (x : A) (v : Vector A) β (n β’ length v) β (lookup v n β‘ lookup (snoc v x) n)
{-# REWRITE length-empty lookup-snoc lookup-length lookup-snoc-empty lookup-empty #-}
head : β {A} β (Vector A) β (Maybe A)
head vec with null vec
head vec | false = just (unsafeHead vec)
head vec | true = nothing
tail : β {A} β (Vector A) β Vector A
tail vec with null vec
tail vec | false = unsafeTail vec
tail vec | true = empty
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open import Function using (_β_)
open import Category.Functor
open import Category.Monad
open import Data.Empty using (β₯; β₯-elim)
open import Data.Fin as Fin using (Fin; zero; suc)
open import Data.Fin.Props as FinProps using ()
open import Data.Maybe as Maybe using (Maybe; maybe; just; nothing)
open import Data.Nat using (β; zero; suc)
open import Data.Product using (Ξ£; β; _,_; projβ; projβ) renaming (_Γ_ to _β§_)
open import Data.Sum using (_β_; injβ; injβ; [_,_])
open import Data.Vec as Vec using (Vec; []; _β·_; head; tail)
open import Data.Vec.Equality as VecEq
open import Relation.Nullary using (Dec; yes; no; Β¬_)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as PropEq using (_β‘_; _β’_; refl; sym; trans; cong; congβ; inspect; Reveal_is_; [_])
module Unification.Correctness (Symbol : β -> Set) (decEqSym : β {k} (f g : Symbol k) β Dec (f β‘ g)) where
open import Unification Symbol decEqSym
open RawFunctor {{...}}
open DecSetoid {{...}} using (_β_)
private maybeFunctor = Maybe.functor
private finDecSetoid : β {n} β DecSetoid _ _
finDecSetoid {n} = FinProps.decSetoid n
-- * proving correctness of replacement function
mutual
-- | proof that var is the identity of replace
replace-thmβ : β {n} (t : Term n) β replace var t β‘ t
replace-thmβ (var x) = refl
replace-thmβ (con s ts) = cong (con s) (replaceChildren-thmβ ts)
-- | proof that var is the identity of replaceChildren
replaceChildren-thmβ : β {n k} (ts : Vec (Term n) k) β replaceChildren var ts β‘ ts
replaceChildren-thmβ [] = refl
replaceChildren-thmβ (t β· ts) rewrite replace-thmβ t = cong (_β·_ _) (replaceChildren-thmβ ts)
-- * proving correctness of substitution/replacement composition
-- | proof that `var β _` is the identity of β
compose-thmβ
: β {m n l} (f : Fin m β Term n) (r : Fin l β Fin m) (t : Term l)
β f β (var β r) β‘ f β r
compose-thmβ f r t = refl
mutual
-- | proof that _β_ behaves as composition of replacements
compose-thmβ
: β {m n l} (f : Fin m β Term n) (g : Fin l β Term m) (t : Term l)
β replace (f β g) t β‘ replace f (replace g t)
compose-thmβ f g (var x) = refl
compose-thmβ f g (con s ts) = cong (con s) (composeChildren-thmβ f g ts)
-- | proof that _β_ behaves as composition of replacements
composeChildren-thmβ
: β {m n l k} (f : Fin m β Term n) (g : Fin l β Term m) (ts : Vec (Term l) k)
β replaceChildren (f β g) ts β‘ replaceChildren f (replaceChildren g ts)
composeChildren-thmβ f g [] = refl
composeChildren-thmβ f g (t β· ts) rewrite compose-thmβ f g t = cong (_β·_ _) (composeChildren-thmβ f g ts)
-- * proving correctness of thick and thin
-- | predecessor function over finite numbers
pred : β {n} β Fin (suc (suc n)) β Fin (suc n)
pred zero = zero
pred (suc x) = x
-- | proof of injectivity of thin
thin-injective
: β {n} (x : Fin (suc n)) (y z : Fin n)
β thin x y β‘ thin x z β y β‘ z
thin-injective {zero} zero () _ _
thin-injective {zero} (suc _) () _ _
thin-injective {suc _} zero zero zero refl = refl
thin-injective {suc _} zero zero (suc _) ()
thin-injective {suc _} zero (suc _) zero ()
thin-injective {suc _} zero (suc y) (suc .y) refl = refl
thin-injective {suc _} (suc _) zero zero refl = refl
thin-injective {suc _} (suc _) zero (suc _) ()
thin-injective {suc _} (suc _) (suc _) zero ()
thin-injective {suc n} (suc x) (suc y) (suc z) p
= cong suc (thin-injective x y z (cong pred p))
-- | proof that thin x will never map anything to x
thinxyβ’x
: β {n} (x : Fin (suc n)) (y : Fin n)
β thin x y β’ x
thinxyβ’x zero zero ()
thinxyβ’x zero (suc _) ()
thinxyβ’x (suc _) zero ()
thinxyβ’x (suc x) (suc y) p
= thinxyβ’x x y (cong pred p)
-- | proof that `thin x` reaches all y where x β’ y
xβ’yβthinxzβ‘y
: β {n} (x y : Fin (suc n))
β x β’ y β β (Ξ» z β thin x z β‘ y)
xβ’yβthinxzβ‘y zero zero 0β’0 with 0β’0 refl
xβ’yβthinxzβ‘y zero zero 0β’0 | ()
xβ’yβthinxzβ‘y {zero} (suc ()) _ _
xβ’yβthinxzβ‘y {zero} zero (suc ()) _
xβ’yβthinxzβ‘y {suc _} zero (suc y) _ = y , refl
xβ’yβthinxzβ‘y {suc _} (suc x) zero _ = zero , refl
xβ’yβthinxzβ‘y {suc _} (suc x) (suc y) sxβ’sy
= (suc (projβ prf)) , (lem x y (projβ prf) (projβ prf))
where
xβ’y = sxβ’sy β cong suc
prf = xβ’yβthinxzβ‘y x y xβ’y
lem : β {n} (x y : Fin (suc n)) (z : Fin n)
β thin x z β‘ y β thin (suc x) (suc z) β‘ suc y
lem zero zero _ ()
lem zero (suc .z) z refl = refl
lem (suc _) zero zero refl = refl
lem (suc _) zero (suc _) ()
lem (suc _) (suc _) zero ()
lem (suc x) (suc .(thin x z)) (suc z) refl = refl
-- | proof that thick x composed with thin x is the identity
thickxβthinxβ‘yes
: β {n} (x : Fin (suc n)) (y : Fin n)
β thick x (thin x y) β‘ just y
thickxβthinxβ‘yes zero zero = refl
thickxβthinxβ‘yes zero (suc _) = refl
thickxβthinxβ‘yes (suc _) zero = refl
thickxβthinxβ‘yes (suc x) (suc y) = cong (_<$>_ suc) (thickxβthinxβ‘yes x y)
-- | proof that `thin` is a partial inverse of `thick`
thinβ‘thickβ»ΒΉ
: β {n} (x : Fin (suc n)) (y : Fin n) (z : Fin (suc n))
β thin x y β‘ z
β thick x z β‘ just y
thinβ‘thickβ»ΒΉ x y z p with p
thinβ‘thickβ»ΒΉ x y .(thin x y) _ | refl = thickxβthinxβ‘yes x y
-- | proof that `thick x x` returns nothing
thickxxβ‘no
: β {n} (x : Fin (suc n))
β thick x x β‘ nothing
thickxxβ‘no zero = refl
thickxxβ‘no {zero} (suc ())
thickxxβ‘no {suc n} (suc x)
= cong (maybe (Ξ» x β just (suc x)) nothing) (thickxxβ‘no x)
-- | proof that `thick x y` returns something when x β’ y
xβ’yβthickxyβ‘yes
: β {n} (x y : Fin (suc n))
β x β’ y β β (Ξ» z β thick x y β‘ just z)
xβ’yβthickxyβ‘yes zero zero 0β’0 with 0β’0 refl
xβ’yβthickxyβ‘yes zero zero 0β’0 | ()
xβ’yβthickxyβ‘yes zero (suc y) p = y , refl
xβ’yβthickxyβ‘yes {zero} (suc ()) _ _
xβ’yβthickxyβ‘yes {suc n} (suc x) zero _ = zero , refl
xβ’yβthickxyβ‘yes {suc n} (suc x) (suc y) sxβ’sy
= (suc (projβ prf)) , (cong (_<$>_ suc) (projβ prf))
where
xβ’y = sxβ’sy β cong suc
prf = xβ’yβthickxyβ‘yes {n} x y xβ’y
-- | proof that `thick` is the partial inverse of `thin`
thickβ‘thinβ»ΒΉ : β {n} (x y : Fin (suc n)) (r : Maybe (Fin n))
β thick x y β‘ r
β x β‘ y β§ r β‘ nothing
β β (Ξ» z β thin x z β‘ y β§ r β‘ just z)
thickβ‘thinβ»ΒΉ x y _ thickxyβ‘r with x β y | thickxyβ‘r
thickβ‘thinβ»ΒΉ x .x .(thick x x) _ | yes refl | refl
= injβ (refl , thickxxβ‘no x)
thickβ‘thinβ»ΒΉ x y .(thick x y) _ | no xβ’y | refl
= injβ (projβ prfβ , (projβ prfβ) , prfβ)
where
prfβ = xβ’yβthinxzβ‘y x y xβ’y
prfβ = thinβ‘thickβ»ΒΉ x (projβ prfβ) y (projβ prfβ)
-- | proof that if check returns nothing, checkChildren will too
checkβ‘noβcheckChildrenβ‘no
: β {n} (x : Fin (suc n)) (s : Symbol (suc n)) (ts : Vec (Term (suc n)) (suc n))
β check x (con s ts) β‘ nothing β checkChildren x ts β‘ nothing
checkβ‘noβcheckChildrenβ‘no x s ts p with checkChildren x ts
checkβ‘noβcheckChildrenβ‘no x s ts p | nothing = refl
checkβ‘noβcheckChildrenβ‘no x s ts () | just _
-- | proof that if check returns something, checkChildren will too
checkβ‘yesβcheckChildrenβ‘yes
: β {n} (x : Fin (suc n)) (s : Symbol (suc n)) (ts : Vec (Term (suc n)) (suc n)) (ts' : Vec (Term n) (suc n))
β check x (con s ts) β‘ just (con s ts') β checkChildren x ts β‘ just ts'
checkβ‘yesβcheckChildrenβ‘yes x s ts ts' p with checkChildren x ts
checkβ‘yesβcheckChildrenβ‘yes x s ts ts' refl | just .ts' = refl
checkβ‘yesβcheckChildrenβ‘yes x s ts ts' () | nothing
-- | occurs predicate that is only inhabited if x occurs in t
mutual
data Occurs {n : β} (x : Fin n) : Term n β Set where
Here : Occurs x (var x)
Further : β {k ts} {s : Symbol k} β OccursChildren x {k} ts β Occurs x (con s ts)
data OccursChildren {n : β} (x : Fin n) : {k : β} β Vec (Term n) k β Set where
Here : β {k t ts} β Occurs x t β OccursChildren x {suc k} (t β· ts)
Further : β {k t ts} β OccursChildren x {k} ts β OccursChildren x {suc k} (t β· ts)
-- | proof of decidability for the occurs predicate
mutual
occurs? : β {n} (x : Fin n) (t : Term n) β Dec (Occurs x t)
occurs? xβ (var xβ) with xβ β xβ
occurs? .xβ (var xβ) | yes refl = yes Here
occurs? xβ (var xβ) | no xββ’xβ = no (xββ’xβ β lem xβ xβ)
where
lem : β {n} (x y : Fin n) β Occurs x (var y) β x β‘ y
lem zero zero _ = refl
lem zero (suc _) ()
lem (suc x) zero ()
lem (suc x) (suc .x) Here = refl
occurs? xβ (con s ts) with occursChildren? xβ ts
occurs? xβ (con s ts) | yes xββts = yes (Further xββts)
occurs? xβ (con s ts) | no xββts = no (xββts β lem xβ)
where
lem : β {n s ts} (x : Fin n) β Occurs x (con s ts) β OccursChildren x ts
lem x (Further xβ) = xβ
occursChildren? : β {n k} (x : Fin n) (ts : Vec (Term n) k) β Dec (OccursChildren x ts)
occursChildren? xβ [] = no (Ξ» ())
occursChildren? xβ (t β· ts) with occurs? xβ t
occursChildren? xβ (t β· ts) | yes h = yes (Here h)
occursChildren? xβ (t β· ts) | no Β¬h with occursChildren? xβ ts
occursChildren? xβ (t β· ts) | no Β¬h | yes f = yes (Further f)
occursChildren? xβ (t β· ts) | no Β¬h | no Β¬f = no lem
where
lem : OccursChildren xβ (t β· ts) β β₯
lem (Here p) = Β¬h p
lem (Further p) = Β¬f p
-- * proving correctness of check
mutual
-- | proving that if x occurs in t, check returns nothing
occursβcheckβ‘no
: β {n} (x : Fin (suc n)) (t : Term (suc n))
β Occurs x t β check x t β‘ nothing
occursβcheckβ‘no x .(Unification.var x) Here
rewrite thickxxβ‘no x = refl
occursβcheckβ‘no x .(Unification.con s ts) (Further {k} {ts} {s} p)
rewrite occursChildrenβcheckChildrenβ‘no x ts p = refl
-- | proving that if x occurs in ts, checkChildren returns nothing
occursChildrenβcheckChildrenβ‘no
: β {n k} (x : Fin (suc n)) (ts : Vec (Term (suc n)) k)
β OccursChildren x ts β checkChildren x ts β‘ nothing
occursChildrenβcheckChildrenβ‘no x .(t β· ts) (Here {k} {t} {ts} p)
rewrite occursβcheckβ‘no x t p = refl
occursChildrenβcheckChildrenβ‘no x .(t β· ts) (Further {k} {t} {ts} p)
with check x t
... | just _ rewrite occursChildrenβcheckChildrenβ‘no x ts p = refl
... | nothing rewrite occursChildrenβcheckChildrenβ‘no x ts p = refl
mutual
-- | proof that if check x t returns nothing, x occurs in t
checkβ‘noβoccurs
: β {n} (x : Fin (suc n)) (t : Term (suc n))
β check x t β‘ nothing β Occurs x t
checkβ‘noβoccurs xβ (var xβ) p with xβ β xβ
checkβ‘noβoccurs .xβ (var xβ) p | yes refl = Here
checkβ‘noβoccurs xβ (var xβ) p | no xββ’xβ = β₯-elim (lemβ p)
where
lemβ : β (Ξ» z β thick xβ xβ β‘ just z)
lemβ = xβ’yβthickxyβ‘yes xβ xβ xββ’xβ
lemβ : var <$> thick xβ xβ β‘ nothing β β₯
lemβ rewrite projβ lemβ = Ξ» ()
checkβ‘noβoccurs {n} xβ (con s ts) p
= Further (checkChildrenβ‘noβoccursChildren xβ ts (lem p))
where
lem : con s <$> checkChildren xβ ts β‘ nothing β checkChildren xβ ts β‘ nothing
lem p with checkChildren xβ ts | inspect (checkChildren xβ) ts
lem () | just _ | [ eq ]
lem p | nothing | [ eq ] = refl
-- | proof that if checkChildren x ts returns nothing, x occurs in ts
checkChildrenβ‘noβoccursChildren
: β {n k} (x : Fin (suc n)) (ts : Vec (Term (suc n)) k)
β checkChildren x ts β‘ nothing β OccursChildren x ts
checkChildrenβ‘noβoccursChildren x [] ()
checkChildrenβ‘noβoccursChildren x (t β· ts) p with check x t | inspect (check x) t
... | nothing | [ eβ ] = Here (checkβ‘noβoccurs x t eβ)
... | just _ | [ eβ ] with checkChildren x ts | inspect (checkChildren x) ts
... | nothing | [ eβ ] = Further (checkChildrenβ‘noβoccursChildren x ts eβ)
checkChildrenβ‘noβoccursChildren x (t β· ts) () | just _ | [ eβ ] | just _ | [ eβ ]
-- | proof that if check returns just, x does not occur in t
checkβ‘yesβΒ¬occurs
: β {n} (x : Fin (suc n)) (t : Term (suc n)) (t' : Term n)
β check x t β‘ just t' β Β¬ (Occurs x t)
checkβ‘yesβΒ¬occurs x t t' pβ xβt with occursβcheckβ‘no x t xβt
checkβ‘yesβΒ¬occurs x t t' pβ _ | pβ with check x t
checkβ‘yesβΒ¬occurs x t t' pβ _ | () | just _
checkβ‘yesβΒ¬occurs x t t' () _ | pβ | nothing
-- | proof that x does not occur in t, check returns just
Β¬occursβcheckβ‘yes
: β {n} (x : Fin (suc n)) (t : Term (suc n))
β Β¬ (Occurs x t) β β (Ξ» t' β check x t β‘ just t')
Β¬occursβcheckβ‘yes x t xβt with check x t | inspect (check x) t
Β¬occursβcheckβ‘yes x t xβt | nothing | [ eq ] with xβt (checkβ‘noβoccurs x t eq)
Β¬occursβcheckβ‘yes x t xβt | nothing | [ eq ] | ()
Β¬occursβcheckβ‘yes x t xβt | just t' | [ eq ] = t' , refl
-- * proving correctness of _for_
-- | proof that if there is nothing to unify, _for_ is the identity
for-thmβ
: β {n} (t : Term n) (x : Fin (suc n)) (y : Fin n)
β (t for x) (thin x y) β‘ var y
for-thmβ t x y rewrite thickxβthinxβ‘yes x y = refl
mutual
-- | proof that if there is something to unify, _for_ unifies
for-thmβ
: β {n} (x : Fin (suc n)) (t : Term (suc n)) (t' : Term n)
β check x t β‘ just t' β replace (t' for x) t β‘ (t' for x) x
for-thmβ x (var y) _ _ with x β y
for-thmβ .y (var y) _ _ | yes refl = refl
for-thmβ x (var y) _ _ | no xβ’y
with thick x y | xβ’yβthickxyβ‘yes x y xβ’y
| thick x x | thickxxβ‘no x
for-thmβ x (var y) .(var z) refl | no _
| .(just z) | z , refl
| .nothing | refl = refl
for-thmβ x (con s ts) _ _ with checkChildren x ts | inspect (checkChildren x) ts
for-thmβ x (con s ts) _ () | nothing | _
for-thmβ x (con s ts) .(con s ts') refl | just ts' | [ checkChildrenβ‘yes ]
rewrite thickxxβ‘no x = cong (con s) (forChildren-thmβ x s ts ts' checkChildrenβ‘yes)
forChildren-thmβ : β {n k} -> (x : Fin (suc n)) (s : Symbol k)
(ts : Vec (Term (suc n)) k) (ts' : Vec (Term n) k) ->
checkChildren x ts β‘ just ts' ->
replaceChildren (con s ts' for x) ts β‘ ts'
forChildren-thmβ x s [] [] eq rewrite thickxxβ‘no x = refl
forChildren-thmβ x s (t1 β· ts) (t2 β· ts') eq
with check x t1 | inspect (check x) t1 | checkChildren x ts | inspect (checkChildren x) ts
forChildren-thmβ x s (t1 β· ts) (t2 β· ts') refl | just .t2 | [ eq1 ] | just .ts' | [ eq2 ]
= congβ _β·_ {!!} {!!}
where
lemmaβ = for-thmβ x t1 t2 eq1
forChildren-thmβ x s (t1 β· ts) (t2 β· ts') () | just xβ | _ | nothing | _
forChildren-thmβ x s (t1 β· ts) (t2 β· ts') () | nothing | _ | cs | _
-- * proving correctness of apply, concat and compose
++-thmβ : β {m n} (s : Subst m n) β nil ++ s β‘ s
++-thmβ nil = refl
++-thmβ (snoc s t x) = cong (Ξ» s β snoc s t x) (++-thmβ s)
mutual
replace-var-id : β {m} (t : Term m) -> replace var t β‘ t
replace-var-id (Unification.var x) = refl
replace-var-id (Unification.con s ts) = cong (con s) (replaceChildren-var-id ts)
replaceChildren-var-id : β {m n} -> (ts : Vec (Term m) n) -> replaceChildren var ts β‘ ts
replaceChildren-var-id [] = refl
replaceChildren-var-id (x β· ts) = congβ _β·_ (replace-var-id x) (replaceChildren-var-id ts)
mutual
replace-var-id' : β {n m} (f : Fin n -> Term m) (t : Term n) ->
replace (\x -> replace var (f x)) t β‘ replace f t
replace-var-id' f (Unification.var x) = replace-var-id (f x)
replace-var-id' f (Unification.con s ts) = cong (con s) (replaceChildren-var-id' f ts)
replaceChildren-var-id' : β {m n k} -> (f : Fin m -> Term k) (ts : Vec (Term m) n) ->
replaceChildren (\x -> replace var (f x)) ts ββ‘ replaceChildren f ts
replaceChildren-var-id' f [] = refl
replaceChildren-var-id' f (x β· ts) = congβ _β·_ (replace-var-id' f x) (replaceChildren-var-id' f ts)
++-lemβ : β {m n} (s : Subst m n) (t : Term (suc m)) (t' : Term m) (x : Fin (suc m)) ->
replace (apply s) (replace (t' for x) t) β‘ replace (\x' -> replace (apply s) (_for_ t' x x')) t
++-lemβ Unification.nil t t' x
rewrite replace-var-id (replace (t' for x) t)
| replace-var-id' (t' for x) t = refl
++-lemβ (Unification.snoc s t x) tβ t' xβ = {!!}
++-lemβ
: β {l m n} (sβ : Subst m n) (sβ : Subst l m) (t : Term l)
β replace (apply (sβ ++ sβ)) t β‘ replace (apply sβ) (replace (apply sβ) t)
++-lemβ sβ nil t rewrite replace-thmβ t = refl
++-lemβ {.(suc k)} {m} {n} sβ (snoc {k} sβ tβ x) t = {!!}
where
lem = ++-lemβ sβ sβ (replace (tβ for x) t)
++-thmβ
: β {l m n} (sβ : Subst m n) (sβ : Subst l m) (x : Fin l)
β apply (sβ ++ sβ) x β‘ (apply sβ β apply sβ) x
++-thmβ sβ nil x = refl
++-thmβ sβ (snoc sβ t y) x with thick y x
++-thmβ sβ (snoc sβ t y) x | just t' = ++-thmβ sβ sβ t'
++-thmβ sβ (snoc sβ t y) x | nothing = ++-lemβ sβ sβ t
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{-# OPTIONS --safe --experimental-lossy-unification #-}
module Cubical.ZCohomology.RingStructure.GradedCommutativity where
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Function
open import Cubical.Foundations.Transport
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Pointed.Homogeneous
open import Cubical.Foundations.GroupoidLaws hiding (assoc)
open import Cubical.Foundations.Path
open import Cubical.Data.Empty as β₯
open import Cubical.Data.Nat
open import Cubical.Data.Int
renaming (_+_ to _β€+_ ; _Β·_ to _β€β_ ; +Comm to +β€-comm ; Β·Comm to β-comm ; +Assoc to β€+-assoc ; -_ to -β€_)
hiding (_+'_ ; +'β‘+)
open import Cubical.Data.Sigma
open import Cubical.Data.Sum
open import Cubical.HITs.SetTruncation as ST
open import Cubical.HITs.Truncation as T
open import Cubical.HITs.S1 hiding (_Β·_)
open import Cubical.HITs.Sn
open import Cubical.HITs.Susp
open import Cubical.Homotopy.Loopspace
open import Cubical.ZCohomology.Base
open import Cubical.ZCohomology.GroupStructure
open import Cubical.ZCohomology.RingStructure.CupProduct
open import Cubical.ZCohomology.RingStructure.RingLaws
open import Cubical.ZCohomology.Properties
private
variable
β : Level
open PlusBis
natTranspLem : β {β} {A B : β β Type β} {n m : β} (a : A n)
(f : (n : β) β (a : A n) β B n) (p : n β‘ m)
β f m (subst A p a) β‘ subst B p (f n a)
natTranspLem {A = A} {B = B} a f p = sym (substCommSlice A B f p a)
transp0β : (n : β) β subst coHomK (+'-comm 1 (suc n)) (0β _) β‘ 0β _
transp0β zero = refl
transp0β (suc n) = refl
transp0β : (n m : β) β subst coHomK (+'-comm (suc (suc n)) (suc m)) (0β _) β‘ 0β _
transp0β n zero = refl
transp0β n (suc m) = refl
-- Recurring expressions
private
Ξ©Kn+1βΩ²Kn+2 : {k : β} β typ (Ξ© (coHomK-ptd k)) β typ ((Ξ©^ 2) (coHomK-ptd (suc k)))
Ξ©Kn+1βΩ²Kn+2 x = sym (KnβΞ©Kn+10β _) ββ cong (KnβΞ©Kn+1 _) x ββ KnβΞ©Kn+10β _
Ξ©Kn+1βΩ²Kn+2' : {k : β} β KnβΞ©Kn+1 k (0β k) β‘ KnβΞ©Kn+1 k (0β k) β typ ((Ξ©^ 2) (coHomK-ptd (suc k)))
Ξ©Kn+1βΩ²Kn+2' p = sym (KnβΞ©Kn+10β _) ββ p ββ KnβΞ©Kn+10β _
KnβΩ²Kn+2 : {k : β} β coHomK k β typ ((Ξ©^ 2) (coHomK-ptd (2 + k)))
KnβΩ²Kn+2 x = Ξ©Kn+1βΩ²Kn+2 (KnβΞ©Kn+1 _ x)
-- Definition of of -β'βΏΜ*α΅
-- This definition is introduced to facilite the proofs
-β'-helper : {k : β} (n m : β)
β isEvenT n β isOddT n β isEvenT m β isOddT m
β coHomKType k β coHomKType k
-β'-helper {k = zero} n m (inl xβ) q x = x
-β'-helper {k = zero} n m (inr xβ) (inl xβ) x = x
-β'-helper {k = zero} n m (inr xβ) (inr xβ) x = 0 - x
-β'-helper {k = suc zero} n m p q base = base
-β'-helper {k = suc zero} n m (inl x) q (loop i) = loop i
-β'-helper {k = suc zero} n m (inr x) (inl xβ) (loop i) = loop i
-β'-helper {k = suc zero} n m (inr x) (inr xβ) (loop i) = loop (~ i)
-β'-helper {k = suc (suc k)} n m p q north = north
-β'-helper {k = suc (suc k)} n m p q south = north
-β'-helper {k = suc (suc k)} n m (inl x) q (merid a i) =
(merid a β sym (merid (ptSn (suc k)))) i
-β'-helper {k = suc (suc k)} n m (inr x) (inl xβ) (merid a i) =
(merid a β sym (merid (ptSn (suc k)))) i
-β'-helper {k = suc (suc k)} n m (inr x) (inr xβ) (merid a i) =
(merid a β sym (merid (ptSn (suc k)))) (~ i)
-β'-gen : {k : β} (n m : β)
(p : isEvenT n β isOddT n)
(q : isEvenT m β isOddT m)
β coHomK k β coHomK k
-β'-gen {k = zero} = -β'-helper {k = zero}
-β'-gen {k = suc k} n m p q = T.map (-β'-helper {k = suc k} n m p q)
-- -β'βΏΜ*α΅
-β'^_Β·_ : {k : β} (n m : β) β coHomK k β coHomK k
-β'^_Β·_ {k = k} n m = -β'-gen n m (evenOrOdd n) (evenOrOdd m)
-- cohomology version
-β'^_Β·_ : {k : β} {A : Type β} (n m : β) β coHom k A β coHom k A
-β'^_Β·_ n m = ST.map Ξ» f x β (-β'^ n Β· m) (f x)
-- -β'βΏΜ*α΅ = -β' for n m odd
-β'-gen-inrβ‘-β' : {k : β} (n m : β) (p : _) (q : _) (x : coHomK k)
β -β'-gen n m (inr p) (inr q) x β‘ (-β x)
-β'-gen-inrβ‘-β' {k = zero} n m p q _ = refl
-β'-gen-inrβ‘-β' {k = suc zero} n m p q =
T.elim ((Ξ» _ β isOfHLevelTruncPath))
Ξ» { base β refl
; (loop i) β refl}
-β'-gen-inrβ‘-β' {k = suc (suc k)} n m p q =
T.elim ((Ξ» _ β isOfHLevelTruncPath))
Ξ» { north β refl
; south β refl
; (merid a i) k β β£ symDistr (merid (ptSn _)) (sym (merid a)) (~ k) (~ i) β£β}
-- -β'βΏΜ*α΅ x = x for n even
-β'-gen-inl-left : {k : β} (n m : β) (p : _) (q : _) (x : coHomK k)
β -β'-gen n m (inl p) q x β‘ x
-β'-gen-inl-left {k = zero} n m p q x = refl
-β'-gen-inl-left {k = suc zero} n m p q =
T.elim (Ξ» _ β isOfHLevelTruncPath)
Ξ» { base β refl ; (loop i) β refl}
-β'-gen-inl-left {k = suc (suc k)} n m p q =
T.elim (Ξ» _ β isOfHLevelPath (4 + k) (isOfHLevelTrunc (4 + k)) _ _)
Ξ» { north β refl
; south β cong β£_β£β (merid (ptSn _))
; (merid a i) k β β£ compPath-filler (merid a) (sym (merid (ptSn _))) (~ k) i β£β}
-- -β'βΏΜ*α΅ x = x for m even
-β'-gen-inl-right : {k : β} (n m : β) (p : _) (q : _) (x : coHomK k)
β -β'-gen n m p (inl q) x β‘ x
-β'-gen-inl-right {k = zero} n m (inl xβ) q x = refl
-β'-gen-inl-right {k = zero} n m (inr xβ) q x = refl
-β'-gen-inl-right {k = suc zero} n m (inl xβ) q =
T.elim (Ξ» _ β isOfHLevelTruncPath)
Ξ» { base β refl ; (loop i) β refl}
-β'-gen-inl-right {k = suc zero} n m (inr xβ) q =
T.elim (Ξ» _ β isOfHLevelTruncPath)
Ξ» { base β refl ; (loop i) β refl}
-β'-gen-inl-right {k = suc (suc k)} n m (inl xβ) q =
T.elim (Ξ» _ β isOfHLevelTruncPath)
Ξ» { north β refl
; south β cong β£_β£β (merid (ptSn _))
; (merid a i) k β β£ compPath-filler (merid a) (sym (merid (ptSn _))) (~ k) i β£β}
-β'-gen-inl-right {k = suc (suc k)} n m (inr xβ) q =
T.elim (Ξ» _ β isOfHLevelTruncPath)
Ξ» { north β refl
; south β cong β£_β£β (merid (ptSn _))
; (merid a i) k β β£ compPath-filler (merid a) (sym (merid (ptSn _))) (~ k) i β£β}
-β'-genΒ² : {k : β} (n m : β)
(p : isEvenT n β isOddT n)
(q : isEvenT m β isOddT m)
β (x : coHomK k) β -β'-gen n m p q (-β'-gen n m p q x) β‘ x
-β'-genΒ² {k = zero} n m (inl xβ) q x = refl
-β'-genΒ² {k = zero} n m (inr xβ) (inl xβ) x = refl
-β'-genΒ² {k = zero} n m (inr xβ) (inr xβ) x =
cong (pos 0 -_) (-AntiComm (pos 0) x)
ββ -AntiComm (pos 0) (-β€ (x - pos 0))
ββ h x
where
h : (x : _) β -β€ (-β€ (x - pos 0) - pos 0) β‘ x
h (pos zero) = refl
h (pos (suc n)) = refl
h (negsuc n) = refl
-β'-genΒ² {k = suc k} n m (inl xβ) q x i =
-β'-gen-inl-left n m xβ q (-β'-gen-inl-left n m xβ q x i) i
-β'-genΒ² {k = suc k} n m (inr xβ) (inl xβ) x i =
-β'-gen-inl-right n m (inr xβ) xβ (-β'-gen-inl-right n m (inr xβ) xβ x i) i
-β'-genΒ² {k = suc k} n m (inr xβ) (inr xβ) x =
(Ξ» i β -β'-gen-inrβ‘-β' n m xβ xβ (-β'-gen-inrβ‘-β' n m xβ xβ x i) i) β -β^2 x
-β'-genIso : {k : β} (n m : β)
(p : isEvenT n β isOddT n)
(q : isEvenT m β isOddT m)
β Iso (coHomK k) (coHomK k)
Iso.fun (-β'-genIso {k = k} n m p q) = -β'-gen n m p q
Iso.inv (-β'-genIso {k = k} n m p q) = -β'-gen n m p q
Iso.rightInv (-β'-genIso {k = k} n m p q) = -β'-genΒ² n m p q
Iso.leftInv (-β'-genIso {k = k} n m p q) = -β'-genΒ² n m p q
-- action of cong on -β'βΏΜ*α΅
cong-β'-gen-inr : {k : β} (n m : β) (p : _) (q : _) (P : Path (coHomK (2 + k)) (0β _) (0β _))
β cong (-β'-gen n m (inr p) (inr q)) P β‘ sym P
cong-β'-gen-inr {k = k} n m p q P = codeβ‘sym (0β _) P
where
code : (x : coHomK (2 + k)) β 0β _ β‘ x β x β‘ 0β _
code = T.elim (Ξ» _ β isOfHLevelΞ (4 + k) Ξ» _ β isOfHLevelTruncPath)
Ξ» { north β cong (-β'-gen n m (inr p) (inr q))
; south P β cong β£_β£β (sym (merid (ptSn _))) β (cong (-β'-gen n m (inr p) (inr q)) P)
; (merid a i) β t a i}
where
t : (a : Sβ (suc k)) β PathP (Ξ» i β 0β (2 + k) β‘ β£ merid a i β£β β β£ merid a i β£β β‘ 0β (2 + k))
(cong (-β'-gen n m (inr p) (inr q)))
(Ξ» P β cong β£_β£β (sym (merid (ptSn _))) β (cong (-β'-gen n m (inr p) (inr q)) P))
t a = toPathP (funExt Ξ» P β cong (transport (Ξ» i β β£ merid a i β£ β‘ 0β (suc (suc k))))
(cong (cong (-β'-gen n m (inr p) (inr q))) (Ξ» i β (transp (Ξ» j β 0β (suc (suc k)) β‘ β£ merid a (~ j β§ ~ i) β£) i
(compPath-filler P (Ξ» j β β£ merid a (~ j) β£β) i))))
ββ cong (transport (Ξ» i β β£ merid a i β£ β‘ 0β (suc (suc k)))) (congFunct (-β'-gen n m (inr p) (inr q)) P (sym (cong β£_β£β (merid a))))
ββ (Ξ» j β transp (Ξ» i β β£ merid a (i β¨ j) β£ β‘ 0β (suc (suc k))) j
(compPath-filler' (cong β£_β£β (sym (merid a)))
(cong (-β'-gen n m (inr p) (inr q)) P
β cong (-β'-gen n m (inr p) (inr q)) (sym (cong β£_β£β (merid a)))) j))
ββ (Ξ» i β sym (cong β£_β£β (merid a))
β isCommΞ©K (2 + k) (cong (-β'-gen n m (inr p) (inr q)) P)
(cong (-β'-gen n m (inr p) (inr q)) (sym (cong β£_β£β (merid a)))) i)
ββ (Ξ» j β (Ξ» i β β£ merid a (~ i β¨ j) β£)
β (cong β£_β£β (compPath-filler' (merid a) (sym (merid (ptSn _))) (~ j)) β (Ξ» i β -β'-gen n m (inr p) (inr q) (P i))))
β sym (lUnit _))
codeβ‘sym : (x : coHomK (2 + k)) β (p : 0β _ β‘ x) β code x p β‘ sym p
codeβ‘sym x = J (Ξ» x p β code x p β‘ sym p) refl
cong-cong-β'-gen-inr : {k : β} (n m : β) (p : _) (q : _)
(P : Square (refl {x = 0β (suc (suc k))}) refl refl refl)
β cong (cong (-β'-gen n m (inr p) (inr q))) P β‘ sym P
cong-cong-β'-gen-inr n m p q P =
rUnit _
ββ (Ξ» k β (Ξ» i β cong-β'-gen-inr n m p q refl (i β§ k))
ββ (Ξ» i β cong-β'-gen-inr n m p q (P i) k)
ββ Ξ» i β cong-β'-gen-inr n m p q refl (~ i β§ k))
ββ (Ξ» k β transportRefl refl k
ββ cong sym P
ββ transportRefl refl k)
ββ sym (rUnit (cong sym P))
ββ sym (symβ‘cong-sym P)
KnβΞ©Kn+1-β'' : {k : β} (n m : β) (p : _) (q : _) (x : coHomK k)
β KnβΞ©Kn+1 k (-β'-gen n m (inr p) (inr q) x) β‘ sym (KnβΞ©Kn+1 k x)
KnβΞ©Kn+1-β'' n m p q x = cong (KnβΞ©Kn+1 _) (-β'-gen-inrβ‘-β' n m p q x) β KnβΞ©Kn+1-β _ x
transpΩ² : {n m : β} (p q : n β‘ m) β (P : _)
β transport (Ξ» i β refl {x = 0β (p i)} β‘ refl {x = 0β (p i)}) P
β‘ transport (Ξ» i β refl {x = 0β (q i)} β‘ refl {x = 0β (q i)}) P
transpΩ² p q P k = subst (Ξ» n β refl {x = 0β n} β‘ refl {x = 0β n}) (isSetβ _ _ p q k) P
-- Some technical lemmas about KnβΩ²Kn+2 and its interaction with -β'βΏΜ*α΅ and transports
-- TODO : Check if this can be cleaned up more by having more general lemmas
private
lemβ : (n : β) (a : _)
β (cong (cong (subst coHomK (+'-comm (suc zero) (suc (suc n)))))
(KnβΩ²Kn+2 β£ a β£β))
β‘ Ξ©Kn+1βΩ²Kn+2
(sym (transp0β n) ββ cong (subst coHomK (+'-comm (suc zero) (suc n))) (KnβΞ©Kn+1 (suc n) β£ a β£β) ββ transp0β n)
lemβ zero a =
(Ξ» k i j β transportRefl (KnβΩ²Kn+2 β£ a β£β i j) k)
β cong Ξ©Kn+1βΩ²Kn+2 Ξ» k β rUnit (Ξ» i β transportRefl (KnβΞ©Kn+1 1 β£ a β£ i) (~ k)) k
lemβ (suc n) a =
(Ξ» k β transp (Ξ» i β refl {x = 0β (+'-comm 1 (suc (suc (suc n))) (i β¨ ~ k))}
β‘ refl {x = 0β (+'-comm 1 (suc (suc (suc n))) (i β¨ ~ k))}) (~ k)
(Ξ» i j β transp (Ξ» i β coHomK (+'-comm 1 (suc (suc (suc n))) (i β§ ~ k))) k
(KnβΩ²Kn+2 β£ a β£β i j)))
ββ transpΩ² (+'-comm 1 (suc (suc (suc n))))
(cong suc (+'-comm (suc zero) (suc (suc n))))
(KnβΩ²Kn+2 β£ a β£β)
ββ sym (natTranspLem {A = Ξ» n β 0β n β‘ 0β n}
(KnβΞ©Kn+1 (suc (suc n)) β£ a β£)
(Ξ» _ β Ξ©Kn+1βΩ²Kn+2)
(+'-comm 1 (suc (suc n))))
ββ cong Ξ©Kn+1βΩ²Kn+2
(Ξ» k β transp (Ξ» i β 0β (+'-comm (suc zero) (suc (suc n)) (i β¨ k))
β‘ 0β (+'-comm (suc zero) (suc (suc n)) (i β¨ k))) k
(Ξ» i β transp (Ξ» i β coHomK (+'-comm (suc zero) (suc (suc n)) (i β§ k))) (~ k)
(KnβΞ©Kn+1 _ β£ a β£β i)))
ββ cong Ξ©Kn+1βΩ²Kn+2 (rUnit (cong (subst coHomK (+'-comm (suc zero) (suc (suc n))))
(KnβΞ©Kn+1 (suc (suc n)) β£ a β£β)))
lemβ : (n : β) (a : _) (p : _) (q : _)
β (cong (cong (-β'-gen (suc (suc n)) (suc zero) p q
β (subst coHomK (+'-comm 1 (suc (suc n))))))
(KnβΩ²Kn+2 (β£ a β£β)))
β‘ Ξ©Kn+1βΩ²Kn+2
(sym (transp0β n)
ββ cong (subst coHomK (+'-comm (suc zero) (suc n)))
(cong (-β'-gen (suc (suc n)) (suc zero) p q)
(KnβΞ©Kn+1 (suc n) β£ a β£β))
ββ transp0β n)
lemβ n a (inl x) (inr y) =
(Ξ» k i j β (-β'-gen-inl-left (suc (suc n)) 1 x (inr y) (
subst coHomK (+'-comm 1 (suc (suc n)))
(KnβΩ²Kn+2 β£ a β£β i j))) k)
ββ lemβ n a
ββ cong Ξ©Kn+1βΩ²Kn+2 (cong (sym (transp0β n) ββ_ββ transp0β n)
Ξ» k i β subst coHomK (+'-comm 1 (suc n))
(-β'-gen-inl-left (suc (suc n)) 1 x (inr y) (KnβΞ©Kn+1 (suc n) β£ a β£ i) (~ k)))
lemβ n a (inr x) (inr y) =
cong-cong-β'-gen-inr (suc (suc n)) 1 x y
(cong
(cong
(subst coHomK (+'-comm 1 (suc (suc n)))))
(KnβΩ²Kn+2 β£ a β£β))
ββ cong sym (lemβ n a)
ββ Ξ» k β Ξ©Kn+1βΩ²Kn+2
(sym (transp0β n) ββ
cong (subst coHomK (+'-comm 1 (suc n)))
(cong-β'-gen-inr (suc (suc n)) 1 x y
(KnβΞ©Kn+1 (suc n) β£ a β£) (~ k))
ββ transp0β n)
lemβ : (n m : β) (q : _) (p : isEvenT (suc (suc n)) β isOddT (suc (suc n))) (x : _)
β (((sym (cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n))
ββ (Ξ» j β -β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (+'-comm (suc (suc m)) (suc n)) (KnβΞ©Kn+1 _ x j)))
ββ cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n))))
β‘ (KnβΞ©Kn+1 _ (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n)) x)))
lemβ n m q p x =
sym (cong-ββ (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (sym (transp0β m n))
(Ξ» j β subst coHomK (+'-comm (suc (suc m)) (suc n)) (KnβΞ©Kn+1 _ x j))
(transp0β m n))
β h n m p q x
where
help : (n m : β) (x : _)
β ((sym (transp0β m n))
ββ (Ξ» j β subst coHomK (+'-comm (suc (suc m)) (suc n))
(KnβΞ©Kn+1 (suc (suc (m + n))) x j))
ββ transp0β m n)
β‘ KnβΞ©Kn+1 (suc (n + suc m))
(subst coHomK (cong suc (+-comm (suc m) n)) x)
help zero m x =
sym (rUnit _)
ββ (Ξ» k i β transp (Ξ» i β coHomK (+'-comm (suc (suc m)) 1 (i β¨ k))) k
(KnβΞ©Kn+1 _
(transp (Ξ» i β coHomK (predβ (+'-comm (suc (suc m)) 1 (i β§ k)))) (~ k) x) i))
ββ cong (KnβΞ©Kn+1 _)
Ξ» k β subst coHomK (isSetβ _ _ (cong predβ (+'-comm (suc (suc m)) 1))
(cong suc (+-comm (suc m) zero)) k) x
help (suc n) m x =
sym (rUnit _)
ββ ((Ξ» k i β transp (Ξ» i β coHomK (+'-comm (suc (suc m)) (suc (suc n)) (i β¨ k))) k
(KnβΞ©Kn+1 _
(transp (Ξ» i β coHomK (predβ (+'-comm (suc (suc m)) (suc (suc n)) (i β§ k)))) (~ k) x) i)))
ββ cong (KnβΞ©Kn+1 _)
(Ξ» k β subst coHomK (isSetβ _ _ (cong predβ (+'-comm (suc (suc m)) (suc (suc n))))
(cong suc (+-comm (suc m) (suc n))) k) x)
h : (n m : β) (p : isEvenT (suc (suc n)) β isOddT (suc (suc n))) (q : _) (x : _)
β cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q)
(sym (transp0β m n)
ββ (Ξ» j β subst coHomK (+'-comm (suc (suc m)) (suc n))
(KnβΞ©Kn+1 (suc (suc (m + n))) x j))
ββ transp0β m n)
β‘ KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n)) x))
h n m (inl p) (inl q) x =
(Ξ» k β cong (-β'-gen (suc n) (suc (suc m))
(isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inr p) k) (inl q))
(help n m x k))
ββ ((Ξ» k i β -β'-gen-inl-right (suc n) (suc (suc m)) (inr p) q (help n m x i1 i) k))
ββ Ξ» i β KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen-inl-right (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (inr p) (evenOrOdd (suc n)) i) q
(subst coHomK (cong suc (+-comm (suc m) n)) x) (~ i))
h n m (inl p) (inr q) x =
(Ξ» k β cong (-β'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inr p) k) (inr q))
(help n m x k))
ββ cong-β'-gen-inr (suc n) (suc (suc m)) p q (help n m x i1)
ββ sym (KnβΞ©Kn+1-β'' (suc n) (suc (suc m)) p q
(subst coHomK (Ξ» i β suc (+-comm (suc m) n i)) x))
β Ξ» k β KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (inr p) (evenOrOdd (suc n)) k) (inr q)
(subst coHomK (cong suc (+-comm (suc m) n)) x))
h n m (inr p) (inl q) x =
(Ξ» k β cong (-β'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inl p) k) (inl q))
(help n m x k))
ββ (Ξ» k i β -β'-gen-inl-left (suc n) (suc (suc m)) p (inl q) (help n m x i1 i) k)
ββ Ξ» k β KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen-inl-right (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n)) x) (~ k))
h n m (inr p) (inr q) x =
(Ξ» k β cong (-β'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inl p) k) (inr q))
(help n m x k))
ββ (Ξ» k i β -β'-gen-inl-left (suc n) (suc (suc m)) p (inr q) (help n m x i1 i) k)
ββ cong (KnβΞ©Kn+1 (suc (n + suc m)))
(sym (-β'-gen-inl-left (suc n) (suc (suc m)) p (inr q)
(subst coHomK (Ξ» i β suc (+-comm (suc m) n i)) x)))
β Ξ» k β KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen (suc n) (suc (suc m)) (isPropEvenOrOdd (suc n) (inl p) (evenOrOdd (suc n)) k) (inr q)
(subst coHomK (cong suc (+-comm (suc m) n)) x))
lemβ : (n m : β) (q : _) (p : isEvenT (suc (suc n)) β isOddT (suc (suc n))) (a : _) (b : _)
β cong (KnβΞ©Kn+1 _) (((sym (cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n))
ββ (Ξ» j β -β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (+'-comm (suc (suc m)) (suc n))
(_β£β_ {n = suc (suc m)} {m = (suc n)} β£ merid b j β£β β£ a β£)))
ββ cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n))))
β‘ cong (KnβΞ©Kn+1 _) (KnβΞ©Kn+1 _ (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n))
(_β£β_ {n = suc m} {m = (suc n)} β£ b β£β β£ a β£))))
lemβ n m q p a b = cong (cong (KnβΞ©Kn+1 _)) (lemβ n m q p (_β£β_ {n = suc m} {m = (suc n)} β£ b β£β β£ a β£))
lemβ
: (n m : β) (p : _) (q : _) (a : _) (b : _)
β cong (cong (-β'-gen (suc (suc n)) (suc (suc m)) p q
β (subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))))
(Ξ©Kn+1βΩ²Kn+2 (sym (cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0β n m))
ββ (Ξ» i β -β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (+'-comm (suc (suc n)) (suc m))
(_β£β_ {n = suc (suc n)} {m = suc m} β£ merid a i β£β β£ b β£β)))
ββ cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0β n m)))
β‘ KnβΩ²Kn+2 (-β'-gen (suc (suc n)) (suc (suc m)) p q
(-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m))) (_β£β_ {n = suc n} {m = suc m} β£ a β£β β£ b β£β))))
lemβ
n m p q a b =
cong (cong (cong (-β'-gen (suc (suc n)) (suc (suc m)) p q
β (subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))))))
(cong (sym (KnβΞ©Kn+10β _) ββ_ββ KnβΞ©Kn+10β _)
(lemβ m n p q b a))
β help p q (_β£β_ {n = suc n} {m = suc m} β£ a β£ β£ b β£)
where
annoying : (x : _)
β cong (cong (subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))))
(KnβΩ²Kn+2 (subst coHomK (cong suc (+-comm (suc n) m)) x))
β‘ KnβΩ²Kn+2 (subst coHomK (cong suc (sym (+-suc n m))) x)
annoying x =
((Ξ» k β transp (Ξ» i β refl {x = 0β ((+'-comm (suc (suc m)) (suc (suc n))) (i β¨ ~ k))}
β‘ refl {x = 0β ((+'-comm (suc (suc m)) (suc (suc n))) (i β¨ ~ k))}) (~ k)
Ξ» i j β transp (Ξ» i β coHomK (+'-comm (suc (suc m)) (suc (suc n)) (i β§ ~ k))) k
(KnβΩ²Kn+2 (subst coHomK (cong suc (+-comm (suc n) m)) x) i j)))
ββ cong (transport (Ξ» i β refl {x = 0β ((+'-comm (suc (suc m)) (suc (suc n))) i)}
β‘ refl {x = 0β ((+'-comm (suc (suc m)) (suc (suc n))) i)}))
(natTranspLem {A = coHomK} x (Ξ» _ β KnβΩ²Kn+2) (cong suc (+-comm (suc n) m)))
ββ sym (substComposite (Ξ» n β refl {x = 0β n} β‘ refl {x = 0β n})
(cong (suc β suc β suc) (+-comm (suc n) m)) (+'-comm (suc (suc m)) (suc (suc n)))
(KnβΩ²Kn+2 x))
ββ (Ξ» k β subst (Ξ» n β refl {x = 0β n} β‘ refl {x = 0β n})
(isSetβ _ _
(cong (suc β suc β suc) (+-comm (suc n) m) β (+'-comm (suc (suc m)) (suc (suc n))))
(cong (suc β suc β suc) (sym (+-suc n m))) k)
(KnβΩ²Kn+2 x))
ββ sym (natTranspLem {A = coHomK} x (Ξ» _ β KnβΩ²Kn+2) (cong suc (sym (+-suc n m))))
help : (p : _) (q : _) (x : _) β
cong (cong (-β'-gen (suc (suc n)) (suc (suc m)) p q
β subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))))
(KnβΩ²Kn+2 (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (+-comm (suc n) m)) x)))
β‘ KnβΩ²Kn+2
(-β'-gen (suc (suc n)) (suc (suc m)) p q
(-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m))) x)))
help (inl x) (inl y) z =
(Ξ» k i j β
-β'-gen-inl-right (suc (suc n)) (suc (suc m)) (inl x) y
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
((Ξ©Kn+1βΩ²Kn+2
(KnβΞ©Kn+1 _ (-β'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(subst coHomK (cong suc (+-comm (suc n) m)) z) k))) i j)) k)
ββ annoying z
ββ cong KnβΩ²Kn+2
Ξ» k β (-β'-gen-inl-right (suc (suc n)) (suc (suc m)) (inl x) y
(-β'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(subst coHomK (cong suc (sym (+-suc n m))) z) (~ k)) (~ k))
help (inl x) (inr y) z =
(Ξ» k i j β
-β'-gen-inl-left (suc (suc n)) (suc (suc m)) x (inr y)
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(KnβΩ²Kn+2 (-β'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(subst coHomK (cong suc (+-comm (suc n) m)) z) k) i j)) k)
ββ annoying z
ββ cong KnβΩ²Kn+2
(Ξ» k β (-β'-gen-inl-left (suc (suc n)) (suc (suc m)) x (inr y)
(-β'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(subst coHomK (cong suc (sym (+-suc n m))) z) (~ k)) (~ k)))
help (inr x) (inl y) z =
(Ξ» k i j β -β'-gen-inl-right (suc (suc n)) (suc (suc m)) (inr x) y
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(KnβΩ²Kn+2
(-β'-gen (suc m) (suc (suc n)) (isPropEvenOrOdd (suc m) (evenOrOdd (suc m)) (inr y) k) (inr x)
(subst coHomK (cong suc (+-comm (suc n) m)) z)) i j)) k)
ββ cong (cong (cong (subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))) β Ξ©Kn+1βΩ²Kn+2)
(KnβΞ©Kn+1-β'' (suc m) (suc (suc n)) y x
(subst coHomK (cong suc (+-comm (suc n) m)) z))
ββ cong sym (annoying z)
ββ cong Ξ©Kn+1βΩ²Kn+2 (sym (KnβΞ©Kn+1-β'' (suc m) (suc (suc n)) y x
(subst coHomK (cong suc (sym (+-suc n m))) z)))
ββ cong KnβΩ²Kn+2 Ξ» k β (-β'-gen-inl-right (suc (suc n)) (suc (suc m)) (inr x) y
(-β'-gen (suc m) (suc (suc n)) (isPropEvenOrOdd (suc m) (evenOrOdd (suc m)) (inr y) (~ k)) (inr x)
(subst coHomK (cong suc (sym (+-suc n m))) z)) (~ k))
help (inr x) (inr y) z =
(Ξ» k β cong-cong-β'-gen-inr (suc (suc n)) (suc (suc m)) x y
(Ξ» i j β subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(KnβΩ²Kn+2
(-β'-gen (suc m) (suc (suc n))
(isPropEvenOrOdd (suc m) (evenOrOdd (suc m)) (inl y) k) (inr x)
(subst coHomK (cong suc (+-comm (suc n) m)) z)) i j)) k)
ββ cong (sym β cong (cong (subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))) β KnβΩ²Kn+2)
(-β'-gen-inl-left (suc m) (suc (suc n)) y (inr x)
(subst coHomK (cong suc (+-comm (suc n) m)) z))
ββ cong sym (annoying z)
ββ cong (sym β KnβΩ²Kn+2)
(sym (-β'-gen-inl-left (suc m) (suc (suc n)) y (inr x)
(subst coHomK (cong suc (sym (+-suc n m))) z)))
ββ cong Ξ©Kn+1βΩ²Kn+2
Ξ» k β (KnβΞ©Kn+1-β'' (suc (suc n)) (suc (suc m)) x y
(-β'-gen (suc m) (suc (suc n)) (
isPropEvenOrOdd (suc m) (evenOrOdd (suc m)) (inl y) (~ k)) (inr x)
(subst coHomK (cong suc (sym (+-suc n m))) z))) (~ k)
lemβ : (n m : β) (p : _) (q : _) (a : _) (b : _)
β flipSquare
(KnβΩ²Kn+2 (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n))
(_β£β_ {n = suc m} {m = (suc n)} β£ b β£β β£ a β£))))
β‘ KnβΩ²Kn+2
(-β'-gen (suc (suc n)) (suc (suc m)) p q
(-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m)))
(-β'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m))
(subst coHomK (+'-comm (suc m) (suc n)) (β£ b β£β β£β β£ a β£β))))))
lemβ n m p q a b =
sym (symβ‘flipSquare (KnβΩ²Kn+2 (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (cong suc (+-comm (suc m) n))
(_β£β_ {n = suc m} {m = (suc n)} β£ b β£β β£ a β£)))))
β cong Ξ©Kn+1βΩ²Kn+2
(helpβ
(subst coHomK (cong suc (+-comm (suc m) n))
(_β£β_ {n = suc m} {m = (suc n)} β£ b β£ β£ a β£)) p q
β cong (KnβΞ©Kn+1 _) (sym (helpβ (β£ b β£ β£β β£ a β£))))
where
helpβ : (x : _) (p : _) (q : _)
β sym (KnβΞ©Kn+1 (suc (n + suc m)) (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q x))
β‘ KnβΞ©Kn+1 (suc (n + suc m)) ((-β'-gen (suc (suc n)) (suc (suc m)) p q
(-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(-β'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m)) x))))
helpβ z (inl x) (inl y) =
cong (Ξ» x β sym (KnβΞ©Kn+1 (suc (n + suc m)) x))
(-β'-gen-inl-right (suc n) (suc (suc m)) (evenOrOdd (suc n)) y z)
ββ sym (KnβΞ©Kn+1-β'' (suc n) (suc m) x y z)
ββ Ξ» k β KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen-inl-right (suc (suc n)) (suc (suc m)) (inl x) y
(-β'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(-β'-gen (suc n) (suc m)
(isPropEvenOrOdd (suc n) (inr x) (evenOrOdd (suc n)) k)
(isPropEvenOrOdd (suc m) (inr y) (evenOrOdd (suc m)) k) z) (~ k)) (~ k))
helpβ z (inl x) (inr y) =
(Ξ» k β sym (KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen (suc n) (suc (suc m))
(isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inr x) k) (inr y) z)))
ββ cong sym (KnβΞ©Kn+1-β'' (suc n) (suc (suc m)) x y z)
ββ cong (KnβΞ©Kn+1 (suc (n + suc m))) (sym (-β'-gen-inl-right (suc n) (suc m) (inr x) y z))
β Ξ» k β KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen-inl-left (suc (suc n)) (suc (suc m)) x (inr y)
(-β'-gen-inl-right (suc m) (suc (suc n)) (evenOrOdd (suc m)) x
(-β'-gen (suc n) (suc m)
(isPropEvenOrOdd (suc n) (inr x) (evenOrOdd (suc n)) k)
(isPropEvenOrOdd (suc m) (inl y) (evenOrOdd (suc m)) k) z) (~ k)) (~ k))
helpβ z (inr x) (inl y) =
cong (Ξ» x β sym (KnβΞ©Kn+1 (suc (n + suc m)) x))
(-β'-gen-inl-right (suc n) (suc (suc m)) (evenOrOdd (suc n)) y z)
ββ (Ξ» k β KnβΞ©Kn+1-β'' (suc m) (suc (suc n)) y x
(-β'-gen-inl-left (suc n) (suc m) x (inr y) z (~ k)) (~ k))
ββ Ξ» k β KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen-inl-right (suc (suc n)) (suc (suc m)) (inr x) y
(-β'-gen (suc m) (suc (suc n))
(isPropEvenOrOdd (suc m) (inr y) (evenOrOdd (suc m)) k) (inr x)
(-β'-gen (suc n) (suc m)
(isPropEvenOrOdd (suc n) (inl x) (evenOrOdd (suc n)) k)
(isPropEvenOrOdd (suc m) (inr y) (evenOrOdd (suc m)) k) z)) (~ k))
helpβ z (inr x) (inr y) =
((Ξ» k β sym (KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen (suc n) (suc (suc m))
(isPropEvenOrOdd (suc n) (evenOrOdd (suc n)) (inl x) k) (inr y) z))))
ββ cong sym (cong (KnβΞ©Kn+1 (suc (n + suc m))) (-β'-gen-inl-left (suc n) (suc (suc m)) x (inr y) z))
ββ (Ξ» k β sym (KnβΞ©Kn+1 (suc (n + suc m))
(-β'-gen-inl-left (suc m) (suc (suc n)) y (inr x)
(-β'-gen-inl-right (suc n) (suc m) (inl x) y z (~ k)) (~ k))))
β Ξ» k β KnβΞ©Kn+1-β'' (suc (suc n)) (suc (suc m)) x y
(-β'-gen (suc m) (suc (suc n)) (isPropEvenOrOdd (suc m) (inl y) (evenOrOdd (suc m)) k) (inr x)
(-β'-gen (suc n) (suc m)
(isPropEvenOrOdd (suc n) (inl x) (evenOrOdd (suc n)) k)
(isPropEvenOrOdd (suc m) (inl y) (evenOrOdd (suc m)) k) z)) (~ k)
helpβ : (x : _) β
(-β'-gen (suc (suc n)) (suc (suc m)) p q
(-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m)))
(-β'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m))
(subst coHomK (+'-comm (suc m) (suc n)) x)))))
β‘ -β'-gen (suc (suc n)) (suc (suc m)) p q
(-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(-β'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m))
(subst coHomK (cong suc (+-comm (suc m) n)) x)))
helpβ x =
(Ξ» k β -β'-gen (suc (suc n)) (suc (suc m)) p q
(-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(transp (Ξ» i β coHomK ((cong suc (sym (+-suc n m))) (i β¨ k))) k
(-β'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m))
(transp (Ξ» i β coHomK ((cong suc (sym (+-suc n m))) (i β§ k))) (~ k)
(subst coHomK (+'-comm (suc m) (suc n)) x))))))
β cong (-β'-gen (suc (suc n)) (suc (suc m)) p q
β -β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
β -β'-gen (suc n) (suc m) (evenOrOdd (suc n)) (evenOrOdd (suc m)))
(sym (substComposite coHomK (+'-comm (suc m) (suc n)) ((cong suc (sym (+-suc n m)))) x)
β Ξ» k β subst coHomK (isSetβ _ _ (+'-comm (suc m) (suc n) β cong suc (sym (+-suc n m)))
((cong suc (+-comm (suc m) n))) k) x)
lemβ : (n : β) (a : _) (p : _) (q : _)
β ((Ξ» i β KnβΞ©Kn+1 _ (-β'-gen (suc n) (suc zero) (evenOrOdd (suc n)) (inr tt)
(transp0β n (~ i))))
ββ (Ξ» i j β KnβΞ©Kn+1 _ (-β'-gen (suc n) (suc zero) (evenOrOdd (suc n)) (inr tt)
(subst coHomK (+'-comm (suc zero) (suc n)) (KnβΞ©Kn+1 (suc n) β£ a β£β i))) j)
ββ (Ξ» i β KnβΞ©Kn+1 _ (-β'-gen (suc n) (suc zero) (evenOrOdd (suc n)) (inr tt)
(transp0β n i))))
β‘ (cong (KnβΞ©Kn+1 (suc (suc (n + zero))))
(sym (transp0β n)
ββ sym (cong (subst coHomK (+'-comm (suc zero) (suc n)))
(cong (-β'-gen (suc (suc n)) (suc zero) p q) (KnβΞ©Kn+1 (suc n) β£ a β£β)))
ββ transp0β n))
lemβ zero a (inl x) (inr tt) =
(Ξ» k β rUnit ((cong (KnβΞ©Kn+1 _) (cong-β'-gen-inr (suc zero) (suc zero) tt tt
(Ξ» i β (subst coHomK (+'-comm (suc zero) (suc zero))
(KnβΞ©Kn+1 (suc zero) β£ a β£β i))) k))) (~ k))
β Ξ» k β ((cong (KnβΞ©Kn+1 (suc (suc zero)))
(rUnit (Ξ» i β subst coHomK (+'-comm (suc zero) (suc zero))
(-β'-gen-inl-left (suc (suc zero)) (suc zero) tt (inr tt)
(KnβΞ©Kn+1 (suc zero) β£ a β£β (~ i)) (~ k))) k)))
lemβ (suc n) a (inl x) (inr tt) =
((Ξ» k β rUnit (cong (KnβΞ©Kn+1 _)
(Ξ» i β -β'-gen (suc (suc n)) (suc zero)
(isPropEvenOrOdd n (evenOrOdd (suc (suc n))) (inr x) k) (inr tt)
(subst coHomK (+'-comm (suc zero) (suc (suc n)))
(KnβΞ©Kn+1 (suc (suc n)) β£ a β£β i)))) (~ k)))
ββ (((Ξ» k β ((cong (KnβΞ©Kn+1 _) (cong-β'-gen-inr (suc (suc n)) (suc zero) x tt
(Ξ» i β (subst coHomK (+'-comm (suc zero) (suc (suc n)))
(KnβΞ©Kn+1 (suc (suc n)) β£ a β£β i))) k))))))
ββ Ξ» k β ((cong (KnβΞ©Kn+1 (suc (suc (suc n + zero))))
(rUnit (Ξ» i β subst coHomK (+'-comm (suc zero) (suc (suc n)))
(-β'-gen-inl-left (suc (suc (suc n))) (suc zero) x (inr tt)
(KnβΞ©Kn+1 (suc (suc n)) β£ a β£β (~ i)) (~ k))) k)))
lemβ (suc n) a (inr x) (inr tt) =
(Ξ» k β rUnit (Ξ» i j β KnβΞ©Kn+1 _
(-β'-gen (suc (suc n)) (suc zero)
(isPropEvenOrOdd (suc (suc n)) (evenOrOdd (suc (suc n))) (inl x) k) (inr tt)
(subst coHomK (+'-comm (suc zero) (suc (suc n)))
(KnβΞ©Kn+1 (suc (suc n)) β£ a β£β i))) j) (~ k))
ββ (Ξ» k i j β KnβΞ©Kn+1 _ (-β'-gen-inl-left (suc (suc n)) (suc zero) x (inr tt)
(subst coHomK (+'-comm (suc zero) (suc (suc n)))
(KnβΞ©Kn+1 (suc (suc n)) β£ a β£β i)) k) j)
ββ Ξ» k β cong (KnβΞ©Kn+1 _)
(rUnit (sym (cong (subst coHomK (+'-comm (suc zero) (suc (suc n))))
(cong-β'-gen-inr (suc (suc (suc n))) (suc zero) x tt (KnβΞ©Kn+1 (suc (suc n)) β£ a β£β) (~ k)))) k)
-- β£ a β£ β£β β£ b β£ β‘ -β'βΏ*α΅ (β£ b β£ β£β β£ a β£) for n β₯ 1, m = 1
gradedComm'-elimCase-left : (n : β) (p : _) (q : _) (a : Sβ (suc n)) (b : SΒΉ) β
(_β£β_ {n = suc n} {m = (suc zero)} β£ a β£β β£ b β£β)
β‘ (-β'-gen (suc n) (suc zero) p q)
(subst coHomK (+'-comm (suc zero) (suc n))
(_β£β_ {n = suc zero} {m = suc n} β£ b β£β β£ a β£β))
gradedComm'-elimCase-left zero (inr tt) (inr tt) a b =
proof a b
β cong (-β'-gen 1 1 (inr tt) (inr tt))
(sym (transportRefl ((_β£β_ {n = suc zero} {m = suc zero} β£ b β£ β£ a β£))))
where
help : flipSquare (Ξ©Kn+1βΩ²Kn+2' (Ξ» j i β _β£β_ {n = suc zero} {m = suc zero} β£ loop i β£β β£ loop j β£β)) β‘
cong (cong (-β'-gen 1 1 (inr tt) (inr tt)))
(Ξ©Kn+1βΩ²Kn+2' (Ξ» i j β _β£β_ {n = suc zero} {m = suc zero} β£ loop j β£β β£ loop i β£β))
help = sym (symβ‘flipSquare _)
β sym (cong-cong-β'-gen-inr 1 1 tt tt
(Ξ©Kn+1βΩ²Kn+2' (Ξ» i j β _β£β_ {n = suc zero} {m = suc zero} β£ loop j β£ β£ loop i β£)))
proof : (a b : SΒΉ) β _β£β_ {n = suc zero} {m = suc zero} β£ a β£β β£ b β£β β‘
-β'-gen 1 1 (inr tt) (inr tt) (_β£β_ {n = suc zero} {m = suc zero} β£ b β£ β£ a β£)
proof base base = refl
proof base (loop i) k = -β'-gen 1 1 (inr tt) (inr tt) (KnβΞ©Kn+10β _ (~ k) i)
proof (loop i) base k = KnβΞ©Kn+10β _ k i
proof (loop i) (loop j) k =
hcomp (Ξ» r β Ξ» { (i = i0) β -β'-gen 1 1 (inr tt) (inr tt) (KnβΞ©Kn+10β _ (~ k β¨ ~ r) j)
; (i = i1) β -β'-gen 1 1 (inr tt) (inr tt) (KnβΞ©Kn+10β _ (~ k β¨ ~ r) j)
; (j = i0) β KnβΞ©Kn+10β _ (k β¨ ~ r) i
; (j = i1) β KnβΞ©Kn+10β _ (k β¨ ~ r) i
; (k = i0) β doubleCompPath-filler
(sym (KnβΞ©Kn+10β _))
(Ξ» j i β _β£β_ {n = suc zero} {m = suc zero} β£ loop i β£β β£ loop j β£β)
(KnβΞ©Kn+10β _) (~ r) j i
; (k = i1) β (-β'-gen 1 1 (inr tt) (inr tt)
(doubleCompPath-filler
(sym (KnβΞ©Kn+10β _))
(Ξ» i j β _β£β_ {n = suc zero} {m = suc zero} β£ loop j β£β β£ loop i β£β)
(KnβΞ©Kn+10β _) (~ r) i j))})
(help k i j)
gradedComm'-elimCase-left (suc n) p q north b =
cong (-β'-gen (suc (suc n)) 1 p q β
(subst coHomK (+'-comm 1 (suc (suc n)))))
(sym (β£β-0β _ (suc (suc n)) β£ b β£β))
gradedComm'-elimCase-left (suc n) p q south b =
cong (-β'-gen (suc (suc n)) 1 p q β
(subst coHomK (+'-comm 1 (suc (suc n)))))
((sym (β£β-0β _ (suc (suc n)) β£ b β£β)) β Ξ» i β β£ b β£ β£β β£ merid (ptSn (suc n)) i β£β)
gradedComm'-elimCase-left (suc n) p q (merid a i) base k =
hcomp (Ξ» j β Ξ» {(i = i0) β (-β'-gen (suc (suc n)) 1 p q β
(subst coHomK (+'-comm 1 (suc (suc n)))))
(0β _)
; (i = i1) β (-β'-gen (suc (suc n)) 1 p q β
(subst coHomK (+'-comm 1 (suc (suc n)))))
(compPath-filler (sym (β£β-0β _ (suc (suc n)) β£ base β£β))
(Ξ» i β β£ base β£ β£β β£ merid a i β£β) j k)
; (k = i0) β _β£β_ {n = suc (suc n)} {m = suc zero} β£ merid a i β£β β£ base β£β
; (k = i1) β -β'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(β£ base β£β β£β β£ merid a i β£β))})
(hcomp (Ξ» j β Ξ» {(i = i0) β β£ north β£
; (i = i1) β β£ north β£
; (k = i0) β (sym (KnβΞ©Kn+10β _)
β (Ξ» j β KnβΞ©Kn+1 _
(sym (gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a base
β cong (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)) (transp0β n)) j))) j i
; (k = i1) β β£ north β£})
β£ north β£)
gradedComm'-elimCase-left (suc n) p q (merid a i) (loop j) k =
hcomp (Ξ» r β
Ξ» { (i = i0) β (-β'-gen (suc (suc n)) 1 p q β
(subst coHomK (+'-comm 1 (suc (suc n)))))
(sym (β£β-0β _ (suc (suc n)) β£ (loop j) β£β) k)
; (i = i1) β (-β'-gen (suc (suc n)) 1 p q β
(subst coHomK (+'-comm 1 (suc (suc n)))))
(compPath-filler (sym (β£β-0β _ (suc (suc n)) β£ (loop j) β£β))
(Ξ» i β β£ loop j β£ β£β β£ merid (ptSn (suc n)) i β£β) r k)
; (k = i0) β _β£β_ {n = suc (suc n)} {m = suc zero} β£ merid a i β£β β£ loop j β£β
; (k = i1) β -β'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(_β£β_ {n = suc zero} {m = suc (suc n)} β£ loop j β£β
β£ compPath-filler (merid a) (sym (merid (ptSn (suc n)))) (~ r) i β£β))})
(hcomp (Ξ» r β
Ξ» { (i = i0) β (-β'-gen (suc (suc n)) 1 p q β
(subst coHomK (+'-comm 1 (suc (suc n)))))
β£ rCancel (merid (ptSn (suc (suc n)))) (~ k β§ r) j β£
; (i = i1) β (-β'-gen (suc (suc n)) 1 p q β
(subst coHomK (+'-comm 1 (suc (suc n)))))
β£ rCancel (merid (ptSn (suc (suc n)))) (~ k β§ r) j β£
; (k = i0) β helpβ r i j
; (k = i1) β -β'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(_β£β_ {n = suc zero} {m = suc (suc n)} β£ loop j β£β
(KnβΞ©Kn+1 _ β£ a β£β i)))})
(-β'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(_β£β_ {n = suc zero} {m = suc (suc n)} β£ loop j β£β
(KnβΞ©Kn+1 _ β£ a β£β i)))))
where
P : Path _ (KnβΞ©Kn+1 (suc (suc (n + 0))) (0β _))
(KnβΞ©Kn+1 (suc (suc (n + 0))) (_β£β_ {n = (suc n)} {m = suc zero} β£ a β£ β£ base β£))
P i = KnβΞ©Kn+1 (suc (suc (n + 0)))
((sym (gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a base
β cong (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)) (transp0β n)) i))
helpβ : (P ββ ((Ξ» i j β _β£β_ {n = suc (suc n)} {m = suc zero} β£ merid a j β£β β£ loop i β£β)) ββ sym P)
β‘ ((Ξ» i β KnβΞ©Kn+1 _ (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(transp0β n (~ i))))
ββ (Ξ» i j β KnβΞ©Kn+1 _ (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(subst coHomK (+'-comm 1 (suc n)) (β£ loop i β£β β£β β£ a β£β))) j)
ββ (Ξ» i β KnβΞ©Kn+1 _ (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(transp0β n i))))
helpβ k i j =
((Ξ» i β (KnβΞ©Kn+1 (suc (suc (n + 0))))
(compPath-filler'
((gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a base))
(cong (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt))
(transp0β n)) (~ k) (~ i)))
ββ (Ξ» i j β (KnβΞ©Kn+1 _
(gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a (loop i) k) j))
ββ Ξ» i β (KnβΞ©Kn+1 (suc (suc (n + 0))))
(compPath-filler'
((gradedComm'-elimCase-left n (evenOrOdd (suc n)) (inr tt) a base))
(cong (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt))
(transp0β n)) (~ k) i)) i j
helpβ : I β I β I β coHomK _
helpβ r i j =
hcomp (Ξ» k β
Ξ» { (i = i0) β (-β'-gen (suc (suc n)) 1 p q β
subst coHomK (+'-comm 1 (suc (suc n))))
β£ rCancel (merid (ptSn (suc (suc n)))) (~ k β¨ r) j β£
; (i = i1) β (-β'-gen (suc (suc n)) 1 p q β
subst coHomK (+'-comm 1 (suc (suc n))))
β£ rCancel (merid (ptSn (suc (suc n)))) (~ k β¨ r) j β£
; (j = i0) β compPath-filler (sym (KnβΞ©Kn+10β (suc (suc (n + 0)))))
P k r i
; (j = i1) β compPath-filler (sym (KnβΞ©Kn+10β (suc (suc (n + 0)))))
P k r i
; (r = i0) β -β'-gen (suc (suc n)) 1 p q
(subst coHomK (+'-comm 1 (suc (suc n)))
(doubleCompPath-filler (sym (KnβΞ©Kn+10β _))
(Ξ» i j β _β£β_ {n = suc zero} {m = suc (suc n)} β£ loop j β£β
(KnβΞ©Kn+1 (suc n) β£ a β£β i)) (KnβΞ©Kn+10β _) (~ k) i j))
; (r = i1) β doubleCompPath-filler P
(Ξ» i j β _β£β_ {n = suc (suc n)} {m = suc zero} β£ merid a j β£β β£ loop i β£β)
(sym P) (~ k) j i})
(hcomp (Ξ» k β
Ξ» { (i = i0) β β£ north β£
; (i = i1) β β£ north β£
; (j = i0) β (KnβΞ©Kn+10β (suc (suc (n + 0)))) (~ r) i
; (j = i1) β (KnβΞ©Kn+10β (suc (suc (n + 0)))) (~ r) i
; (r = i0) β lemβ n a p q (~ k) i j
; (r = i1) β helpβ (~ k) j i})
(hcomp (Ξ» k β
Ξ» { (i = i0) β β£ north β£
; (i = i1) β β£ north β£
; (j = i0) β KnβΞ©Kn+10β (suc (suc (n + 0))) (~ r) i
; (j = i1) β KnβΞ©Kn+10β (suc (suc (n + 0))) (~ r) i
; (r = i0) β flipSquareβ‘cong-sym (flipSquare (Ξ©Kn+1βΩ²Kn+2
(sym (transp0β n)
ββ cong (subst coHomK (+'-comm 1 (suc n)))
(cong (-β'-gen (suc (suc n)) 1 p q)
(KnβΞ©Kn+1 (suc n) β£ a β£β))
ββ transp0β n))) (~ k) i j
; (r = i1) β ((Ξ» i β KnβΞ©Kn+1 _ (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(transp0β n (~ i))))
ββ (Ξ» i j β KnβΞ©Kn+1 _ (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(subst coHomK (+'-comm 1 (suc n)) (KnβΞ©Kn+1 (suc n) β£ a β£β i))) j)
ββ (Ξ» i β KnβΞ©Kn+1 _ (-β'-gen (suc n) 1 (evenOrOdd (suc n)) (inr tt)
(transp0β n i)))) j i})
(hcomp (Ξ» k β
Ξ» { (i = i0) β β£ north β£
; (i = i1) β β£ north β£
; (j = i0) β KnβΞ©Kn+10β (suc (suc (n + 0))) (~ r β§ k) i
; (j = i1) β KnβΞ©Kn+10β (suc (suc (n + 0))) (~ r β§ k) i
; (r = i0) β doubleCompPath-filler
(sym (KnβΞ©Kn+10β _))
(cong (KnβΞ©Kn+1 (suc (suc (n + 0))))
(sym (transp0β n)
ββ sym (cong (subst coHomK (+'-comm 1 (suc n)))
(cong (-β'-gen (suc (suc n)) 1 p q)
(KnβΞ©Kn+1 (suc n) β£ a β£β)))
ββ transp0β n))
(KnβΞ©Kn+10β _) k j i
; (r = i1) β lemβ n a p q (~ k) j i})
(lemβ n a p q i1 j i))))
-- β£ a β£ β£β β£ b β£ β‘ -β'βΏ*α΅ (β£ b β£ β£β β£ a β£) for all n, m β₯ 1
gradedComm'-elimCase : (k n m : β) (term : n + m β‘ k) (p : _) (q : _) (a : _) (b : _) β
(_β£β_ {n = suc n} {m = (suc m)} β£ a β£β β£ b β£β)
β‘ (-β'-gen (suc n) (suc m) p q)
(subst coHomK (+'-comm (suc m) (suc n))
(_β£β_ {n = suc m} {m = suc n} β£ b β£β β£ a β£β))
gradedComm'-elimCase k zero zero term p q a b = gradedComm'-elimCase-left zero p q a b
gradedComm'-elimCase k zero (suc m) term (inr tt) q a b =
help q
β sym (cong (-β'-gen 1 (suc (suc m)) (inr tt) q
β (subst coHomK (+'-comm (suc (suc m)) 1)))
(gradedComm'-elimCase-left (suc m) q (inr tt) b a))
where
help : (q : _) β β£ a β£β β£β β£ b β£β β‘
-β'-gen 1 (suc (suc m)) (inr tt) q
(subst coHomK (+'-comm (suc (suc m)) 1)
(-β'-gen (suc (suc m)) 1 q (inr tt)
(subst coHomK (+'-comm 1 (suc (suc m))) (β£ a β£β β£β β£ b β£β))))
help (inl x) =
(sym (transportRefl _)
β (Ξ» i β subst coHomK (isSetβ _ _ refl (+'-comm 1 (suc (suc m)) β +'-comm (suc (suc m)) 1) i)
(β£ a β£β β£β β£ b β£β)))
ββ substComposite coHomK
(+'-comm 1 (suc (suc m)))
(+'-comm (suc (suc m)) 1)
((β£ a β£β β£β β£ b β£β))
ββ Ξ» i β -β'-gen-inl-right (suc zero) (suc (suc m)) (inr tt) x
((subst coHomK (+'-comm (suc (suc m)) 1)
(-β'-gen-inl-left (suc (suc m)) 1 x (inr tt)
(subst coHomK (+'-comm 1 (suc (suc m))) (β£ a β£β β£β β£ b β£β)) (~ i)))) (~ i)
help (inr x) =
(sym (transportRefl _)
ββ (Ξ» k β subst coHomK (isSetβ _ _ refl (+'-comm 1 (suc (suc m)) β +'-comm (suc (suc m)) 1) k) (β£ a β£β β£β β£ b β£β))
ββ sym (-β^2 (subst coHomK (+'-comm 1 (suc (suc m)) β +'-comm (suc (suc m)) 1) (β£ a β£β β£β β£ b β£β))))
ββ (Ξ» i β -β'-gen-inrβ‘-β' 1 (suc (suc m)) tt x
(-β'-gen-inrβ‘-β' (suc (suc m)) 1 x tt
(substComposite coHomK (+'-comm 1 (suc (suc m))) (+'-comm (suc (suc m)) 1) (β£ a β£β β£β β£ b β£β) i)
(~ i)) (~ i))
ββ Ξ» i β (-β'-gen 1 (suc (suc m)) (inr tt) (inr x)
(transp (Ξ» j β coHomK ((+'-comm (suc (suc m)) 1) (j β¨ ~ i))) (~ i)
(-β'-gen (suc (suc m)) 1 (inr x) (inr tt)
(transp (Ξ» j β coHomK ((+'-comm (suc (suc m)) 1) (j β§ ~ i))) i
((subst coHomK (+'-comm 1 (suc (suc m))) (β£ a β£β β£β β£ b β£β)))))))
gradedComm'-elimCase k (suc n) zero term p q a b =
gradedComm'-elimCase-left (suc n) p q a b
gradedComm'-elimCase zero (suc n) (suc m) term p q a b =
β₯.rec (snotz (sym (+-suc n m) β cong predβ term))
gradedComm'-elimCase (suc zero) (suc n) (suc m) term p q a b =
β₯.rec (snotz (sym (+-suc n m) β cong predβ term))
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q north north = refl
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q north south = refl
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q north (merid a i) r =
-β'-gen (suc (suc n)) (suc (suc m)) p q (
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))
((sym (KnβΞ©Kn+10β _)
β cong (KnβΞ©Kn+1 _)
(cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (sym (transp0β n m))
β sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n β +-comm (suc m) n β cong predβ term)
(evenOrOdd (suc m)) p a north))) r i))
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q south north = refl
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q south south = refl
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q south (merid a i) r =
-β'-gen (suc (suc n)) (suc (suc m)) p q (
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))
((sym (KnβΞ©Kn+10β _)
β cong (KnβΞ©Kn+1 _)
(cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (sym (transp0β n m))
β sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n β +-comm (suc m) n β cong predβ term)
(evenOrOdd (suc m)) p a south))) r i))
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q (merid a i) north r =
(cong (KnβΞ©Kn+1 (suc (suc (n + suc m))))
(gradedComm'-elimCase (suc k) n (suc m) (cong predβ term) (evenOrOdd (suc n)) q a north
β cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n))
β' KnβΞ©Kn+10β _) r i
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q (merid a i) south r =
(cong (KnβΞ©Kn+1 (suc (suc (n + suc m))))
(gradedComm'-elimCase (suc k) n (suc m) (cong predβ term) (evenOrOdd (suc n)) q a south
β cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n))
β' KnβΞ©Kn+10β _) r i
gradedComm'-elimCase (suc (suc k)) (suc n) (suc m) term p q (merid a i) (merid b j) r =
hcomp (Ξ» l β
Ξ» { (i = i0) β -β'-gen (suc (suc n)) (suc (suc m)) p q (
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))
((compPath-filler (sym (KnβΞ©Kn+10β _))
(cong (KnβΞ©Kn+1 _)
(cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (sym (transp0β n m))
β sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n β +-comm (suc m) n β cong predβ term)
(evenOrOdd (suc m)) p b north))) l r j)))
; (i = i1) β -β'-gen (suc (suc n)) (suc (suc m)) p q (
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n))))
((compPath-filler (sym (KnβΞ©Kn+10β _))
(cong (KnβΞ©Kn+1 _)
(cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (sym (transp0β n m))
β sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n β +-comm (suc m) n β cong predβ term)
(evenOrOdd (suc m)) p b south))) l r j)))
; (r = i0) β helpβ l i j
; (r = i1) β -β'-gen (suc (suc n)) (suc (suc m)) p q
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(helpβ l i j))})
(hcomp (Ξ» l β
Ξ» { (i = i0) β -β'-gen (suc (suc n)) (suc (suc m)) p q
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(KnβΞ©Kn+10β _ (~ r β¨ ~ l) j))
; (i = i1) β -β'-gen (suc (suc n)) (suc (suc m)) p q
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(KnβΞ©Kn+10β _ (~ r β¨ ~ l) j))
; (j = i0) β KnβΞ©Kn+10β _ r i
; (j = i1) β KnβΞ©Kn+10β _ r i
; (r = i0) β lemβ n m q p a b (~ l) j i
; (r = i1) β -β'-gen (suc (suc n)) (suc (suc m)) p q
(subst coHomK (+'-comm (suc (suc m)) (suc (suc n)))
(doubleCompPath-filler
(sym (KnβΞ©Kn+10β _))
(Ξ» i j β KnβΞ©Kn+1 _ ((sym (cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0β n m))
ββ (Ξ» i β -β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (+'-comm (suc (suc n)) (suc m))
(_β£β_ {n = suc (suc n)} {m = suc m} β£ merid a i β£β β£ b β£β)))
ββ cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0β n m)) i) j)
(KnβΞ©Kn+10β _) (~ l) i j))})
(hcomp (Ξ» l β
Ξ» { (i = i0) β β£ north β£
; (i = i1) β β£ north β£
; (j = i0) β KnβΞ©Kn+10β _ r i
; (j = i1) β KnβΞ©Kn+10β _ r i
; (r = i0) β lemβ n m q p a b i1 j i
; (r = i1) β lemβ
n m p q a b (~ l) i j})
(hcomp (Ξ» l β
Ξ» { (i = i0) β β£ north β£
; (i = i1) β β£ north β£
; (j = i0) β KnβΞ©Kn+10β _ (r β¨ ~ l) i
; (j = i1) β KnβΞ©Kn+10β _ (r β¨ ~ l) i
; (r = i0) β doubleCompPath-filler
(sym (KnβΞ©Kn+10β _))
(lemβ n m q p a b i1)
(KnβΞ©Kn+10β _) (~ l) j i
; (r = i1) β KnβΩ²Kn+2 (-β'-gen (suc (suc n)) (suc (suc m)) p q
(-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (cong suc (sym (+-suc n m)))
(gradedComm'-elimCase k n m
(+-comm n m ββ cong predβ (+-comm (suc m) n) ββ cong (predβ β predβ) term)
(evenOrOdd (suc n)) (evenOrOdd (suc m)) a b (~ l))))) i j})
(lemβ n m p q a b r i j))))
where
helpβ : I β I β I β coHomK _
helpβ l i j =
KnβΞ©Kn+1 _
(hcomp (Ξ» r
β Ξ» { (i = i0) β compPath-filler' (cong ((-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p)) (sym (transp0β n m)))
(sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n β +-comm (suc m) n β cong predβ term)
(evenOrOdd (suc m)) p b north)) r l
; (i = i1) β compPath-filler' (cong ((-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p)) (sym (transp0β n m)))
(sym (gradedComm'-elimCase (suc k) m (suc n) (+-suc m n β +-comm (suc m) n β cong predβ term)
(evenOrOdd (suc m)) p b south)) r l
; (l = i0) β doubleCompPath-filler (sym (cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0β n m)))
(Ξ» i β -β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p
(subst coHomK (+'-comm (suc (suc n)) (suc m))
(_β£β_ {n = suc (suc n)} {m = suc m} β£ merid a i β£β β£ b β£β)))
(cong (-β'-gen (suc m) (suc (suc n)) (evenOrOdd (suc m)) p) (transp0β n m)) r i
; (l = i1) β _β£β_ {n = suc m} {m = suc (suc n)} β£ b β£β β£ merid a i β£β})
(gradedComm'-elimCase (suc k) m (suc n) (+-suc m n β +-comm (suc m) n β cong predβ term)
(evenOrOdd (suc m)) p b (merid a i) (~ l))) j
helpβ : I β I β I β coHomK _
helpβ l i j =
hcomp (Ξ» r β
Ξ» { (i = i0) β β£ north β£
; (i = i1) β β£ north β£
; (j = i0) β
KnβΞ©Kn+1 (suc (suc (n + suc m)))
(compPath-filler (gradedComm'-elimCase (suc k) n (suc m)
(cong predβ term) (evenOrOdd (suc n)) q a north)
(cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n)) r (~ l)) i
; (j = i1) β
KnβΞ©Kn+1 (suc (suc (n + suc m)))
(compPath-filler (gradedComm'-elimCase (suc k) n (suc m)
(cong predβ term) (evenOrOdd (suc n)) q a south)
(cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n)) r (~ l)) i
; (l = i0) β
KnβΞ©Kn+1 _
(doubleCompPath-filler (sym (cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n)))
(Ξ» j β -β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q
(subst coHomK (+'-comm (suc (suc m)) (suc n))
(_β£β_ {n = suc (suc m)} {m = (suc n)} β£ merid b j β£β β£ a β£)))
(cong (-β'-gen (suc n) (suc (suc m)) (evenOrOdd (suc n)) q) (transp0β m n)) r j) i
; (l = i1) β KnβΞ©Kn+1 _ (_β£β_ {n = (suc n)} {m = suc (suc m)} β£ a β£ β£ merid b j β£β) i})
(hcomp (Ξ» r β
Ξ» { (i = i0) β β£ north β£
; (i = i1) β β£ north β£
; (j = i0) β KnβΞ©Kn+1 (suc (suc (n + suc m)))
(gradedComm'-elimCase (suc k) n (suc m)
(cong predβ term) (evenOrOdd (suc n)) q a north (~ l β¨ ~ r)) i
; (j = i1) β KnβΞ©Kn+1 (suc (suc (n + suc m)))
(gradedComm'-elimCase (suc k) n (suc m)
(cong predβ term) (evenOrOdd (suc n)) q a south (~ l β¨ ~ r)) i
; (l = i0) β KnβΞ©Kn+1 (suc (suc (n + suc m)))
(gradedComm'-elimCase (suc k) n (suc m)
(cong predβ term) (evenOrOdd (suc n)) q a (merid b j) i1) i
; (l = i1) β KnβΞ©Kn+1 _ (gradedComm'-elimCase (suc k) n (suc m) (cong predβ term)
(evenOrOdd (suc n)) q a (merid b j) (~ r)) i})
(KnβΞ©Kn+1 (suc (suc (n + suc m)))
(gradedComm'-elimCase (suc k) n (suc m) (cong predβ term)
(evenOrOdd (suc n)) q a (merid b j) i1) i))
private
coherence-transp : (n m : β) (p : _) (q : _)
β -β'-gen (suc n) (suc m) p q
(subst coHomK (+'-comm (suc m) (suc n)) (0β (suc m +' suc n))) β‘ 0β _
coherence-transp zero zero p q = refl
coherence-transp zero (suc m) p q = refl
coherence-transp (suc n) zero p q = refl
coherence-transp (suc n) (suc m) p q = refl
gradedComm'-β£ββ : (n m : β) (p : _) (q : _) (a : _)
β β£ββ (suc n) (suc m) a
β‘ ((Ξ» b β -β'-gen (suc n) (suc m) p q (subst coHomK (+'-comm (suc m) (suc n)) (b β£β a)))
, (cong (-β'-gen (suc n) (suc m) p q)
(cong (subst coHomK (+'-comm (suc m) (suc n)))
(0β-β£β (suc m) (suc n) a))
β coherence-transp n m p q))
gradedComm'-β£ββ n m p q =
T.elim (Ξ» _ β isOfHLevelPath (3 + n) ((isOfHLevelββ (suc n) m)) _ _)
Ξ» a β ββHomogeneousβ‘ (isHomogeneousKn _) (funExt Ξ» b β funExtβ» (cong fst (fββ‘fβ b)) a)
where
fβ : coHomK (suc m) β Sββ (suc n) ββ coHomK-ptd (suc n +' suc m)
fst (fβ b) a = _β£β_ {n = suc n} {m = suc m} β£ a β£β b
snd (fβ b) = 0β-β£β (suc n) (suc m) b
fβ : coHomK (suc m) β Sββ (suc n) ββ coHomK-ptd (suc n +' suc m)
fst (fβ b) a =
-β'-gen (suc n) (suc m) p q (subst coHomK (+'-comm (suc m) (suc n))
(_β£β_ {n = suc m} {m = suc n} b β£ a β£β))
snd (fβ b) =
(cong (-β'-gen (suc n) (suc m) p q)
(cong (subst coHomK (+'-comm (suc m) (suc n)))
(β£β-0β (suc m) (suc n) b))
β coherence-transp n m p q)
fββ‘fβ : (b : _) β fβ b β‘ fβ b
fββ‘fβ =
T.elim (Ξ» _ β isOfHLevelPath (3 + m)
(subst (isOfHLevel (3 + m))
(Ξ» i β Sββ (suc n) ββ coHomK-ptd (+'-comm (suc n) (suc m) (~ i)))
(isOfHLevelββ' (suc m) n)) _ _)
Ξ» b β ββHomogeneousβ‘ (isHomogeneousKn _)
(funExt Ξ» a β gradedComm'-elimCase (n + m) n m refl p q a b)
-- Finally, graded commutativity:
gradedComm'-β£β : (n m : β) (a : coHomK n) (b : coHomK m)
β a β£β b β‘ (-β'^ n Β· m) (subst coHomK (+'-comm m n) (b β£β a))
gradedComm'-β£β zero zero a b = sym (transportRefl _) β cong (transport refl) (comm-Β·β a b)
gradedComm'-β£β zero (suc m) a b =
sym (transportRefl _)
ββ (Ξ» k β subst coHomK (isSetβ _ _ refl (+'-comm (suc m) zero) k) (b β£β a))
ββ sym (-β'-gen-inl-left zero (suc m) tt (evenOrOdd (suc m))
(subst coHomK (+'-comm (suc m) zero) (b β£β a)))
gradedComm'-β£β (suc n) zero a b =
sym (transportRefl _)
ββ ((Ξ» k β subst coHomK (isSetβ _ _ refl (+'-comm zero (suc n)) k) (b β£β a)))
ββ sym (-β'-gen-inl-right (suc n) zero (evenOrOdd (suc n)) tt
(subst coHomK (+'-comm zero (suc n)) (b β£β a)))
gradedComm'-β£β (suc n) (suc m) a b =
funExtβ» (cong fst (gradedComm'-β£ββ n m (evenOrOdd (suc n)) (evenOrOdd (suc m)) a)) b
gradedComm'-β£ : {A : Type β} (n m : β) (a : coHom n A) (b : coHom m A)
β a β£ b β‘ (-β'^ n Β· m) (subst (Ξ» n β coHom n A) (+'-comm m n) (b β£ a))
gradedComm'-β£ n m =
ST.elim2 (Ξ» _ _ β isOfHLevelPath 2 squashβ _ _)
Ξ» f g β
cong β£_β£β (funExt (Ξ» x β
gradedComm'-β£β n m (f x) (g x)
β cong ((-β'^ n Β· m) β (subst coHomK (+'-comm m n)))
Ξ» i β g (transportRefl x (~ i)) β£β f (transportRefl x (~ i))))
-----------------------------------------------------------------------------
-- The previous code introduces another - to facilitate proof
-- This a reformulation with the usual -β' definition (the one of the ring) of the results
-β^-gen : {k : β} β {A : Type β} β (n m : β)
β (p : isEvenT n β isOddT n)
β (q : isEvenT m β isOddT m)
β (a : coHom k A) β coHom k A
-β^-gen n m (inl p) q a = a
-β^-gen n m (inr p) (inl q) a = a
-β^-gen n m (inr p) (inr q) a = -β a
-β^_Β·_ : {k : β} β {A : Type β} β (n m : β) β (a : coHom k A) β coHom k A
-β^_Β·_ n m a = -β^-gen n m (evenOrOdd n) (evenOrOdd m) a
-β^-gen-eq : β {β} {k : β} {A : Type β} (n m : β)
β (p : isEvenT n β isOddT n) (q : isEvenT m β isOddT m)
β (x : coHom k A)
β -β^-gen n m p q x β‘ (ST.map Ξ» f x β (-β'-gen n m p q) (f x)) x
-β^-gen-eq {k = k} n m (inl p) q = ST.elim (Ξ» _ β isSetPathImplicit) Ξ» f β cong β£_β£β (funExt Ξ» x β sym (-β'-gen-inl-left n m p q (f x)))
-β^-gen-eq {k = k} n m (inr p) (inl q) = ST.elim (Ξ» _ β isSetPathImplicit) Ξ» f β cong β£_β£β (funExt Ξ» z β sym (-β'-gen-inl-right n m (inr p) q (f z)))
-β^-gen-eq {k = k} n m (inr p) (inr q) = ST.elim (Ξ» _ β isSetPathImplicit) Ξ» f β cong β£_β£β (funExt Ξ» z β sym (-β'-gen-inrβ‘-β' n m p q (f z)))
-β^-eq : β {β} {k : β} {A : Type β} (n m : β) β (a : coHom k A)
β (-β^ n Β· m) a β‘ (-β'^ n Β· m) a
-β^-eq n m a = -β^-gen-eq n m (evenOrOdd n) (evenOrOdd m) a
gradedComm-β£ : β {β} {A : Type β} (n m : β) (a : coHom n A) (b : coHom m A)
β a β£ b β‘ (-β^ n Β· m) (subst (Ξ» n β coHom n A) (+'-comm m n) (b β£ a))
gradedComm-β£ n m a b = (gradedComm'-β£ n m a b) β (sym (-β^-eq n m (subst (Ξ» nβ β coHom nβ _) (+'-comm m n) (b β£ a))))
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Divisibility
------------------------------------------------------------------------
module Data.Nat.Divisibility where
open import Data.Nat as Nat
open import Data.Nat.DivMod
import Data.Nat.Properties as NatProp
open import Data.Fin as Fin using (Fin; zero; suc)
import Data.Fin.Properties as FP
open NatProp.SemiringSolver
open import Algebra
private
module CS = CommutativeSemiring NatProp.commutativeSemiring
open import Data.Product
open import Relation.Nullary
open import Relation.Binary
import Relation.Binary.PartialOrderReasoning as PartialOrderReasoning
open import Relation.Binary.PropositionalEquality as PropEq
using (_β‘_; _β’_; refl; sym; cong; subst)
open import Function
-- m β£ n is inhabited iff m divides n. Some sources, like Hardy and
-- Wright's "An Introduction to the Theory of Numbers", require m to
-- be non-zero. However, some things become a bit nicer if m is
-- allowed to be zero. For instance, _β£_ becomes a partial order, and
-- the gcd of 0 and 0 becomes defined.
infix 4 _β£_
data _β£_ : β β β β Set where
divides : {m n : β} (q : β) (eq : n β‘ q * m) β m β£ n
-- Extracts the quotient.
quotient : β {m n} β m β£ n β β
quotient (divides q _) = q
-- If m divides n, and n is positive, then m β€ n.
β£ββ€ : β {m n} β m β£ suc n β m β€ suc n
β£ββ€ (divides zero ())
β£ββ€ {m} {n} (divides (suc q) eq) = begin
m β€β¨ NatProp.mβ€m+n m (q * m) β©
suc q * m β‘β¨ sym eq β©
suc n β
where open β€-Reasoning
-- _β£_ is a partial order.
poset : Poset _ _ _
poset = record
{ Carrier = β
; _β_ = _β‘_
; _β€_ = _β£_
; isPartialOrder = record
{ isPreorder = record
{ isEquivalence = PropEq.isEquivalence
; reflexive = reflexive
; trans = trans
}
; antisym = antisym
}
}
where
module DTO = DecTotalOrder Nat.decTotalOrder
open PropEq.β‘-Reasoning
reflexive : _β‘_ β _β£_
reflexive {n} refl = divides 1 (sym $ projβ CS.*-identity n)
antisym : Antisymmetric _β‘_ _β£_
antisym (divides {n = zero} qβ eqβ) (divides {n = nβ} qβ eqβ) = begin
nβ β‘β¨ eqβ β©
qβ * 0 β‘β¨ CS.*-comm qβ 0 β©
0 β
antisym (divides {n = nβ} qβ eqβ) (divides {n = zero} qβ eqβ) = begin
0 β‘β¨ CS.*-comm 0 qβ β©
qβ * 0 β‘β¨ sym eqβ β©
nβ β
antisym (divides {n = suc nβ} qβ eqβ) (divides {n = suc nβ} qβ eqβ) =
DTO.antisym (β£ββ€ (divides qβ eqβ)) (β£ββ€ (divides qβ eqβ))
trans : Transitive _β£_
trans (divides qβ refl) (divides qβ refl) =
divides (qβ * qβ) (sym (CS.*-assoc qβ qβ _))
module β£-Reasoning = PartialOrderReasoning poset
renaming (_β€β¨_β©_ to _β£β¨_β©_; _ββ¨_β©_ to _β‘β¨_β©_)
private module P = Poset poset
-- 1 divides everything.
1β£_ : β n β 1 β£ n
1β£ n = divides n (sym $ projβ CS.*-identity n)
-- Everything divides 0.
_β£0 : β n β n β£ 0
n β£0 = divides 0 refl
-- 0 only divides 0.
0β£ββ‘0 : β {n} β 0 β£ n β n β‘ 0
0β£ββ‘0 {n} 0β£n = P.antisym (n β£0) 0β£n
-- Only 1 divides 1.
β£1ββ‘1 : β {n} β n β£ 1 β n β‘ 1
β£1ββ‘1 {n} nβ£1 = P.antisym nβ£1 (1β£ n)
-- If i divides m and n, then i divides their sum.
β£-+ : β {i m n} β i β£ m β i β£ n β i β£ m + n
β£-+ (divides {m = i} q refl) (divides q' refl) =
divides (q + q') (sym $ projβ CS.distrib i q q')
-- If i divides m and n, then i divides their difference.
β£-βΈ : β {i m n} β i β£ m + n β i β£ m β i β£ n
β£-βΈ (divides {m = i} q' eq) (divides q refl) =
divides (q' βΈ q)
(sym $ NatProp.imβ‘jm+nβ[iβΈj]mβ‘n q' q i _ $ sym eq)
-- A simple lemma: n divides kn.
β£-* : β k {n} β n β£ k * n
β£-* k = divides k refl
-- If i divides j, then ki divides kj.
*-cong : β {i j} k β i β£ j β k * i β£ k * j
*-cong {i} {j} k (divides q eq) = divides q lemma
where
open PropEq.β‘-Reasoning
lemma = begin
k * j β‘β¨ cong (_*_ k) eq β©
k * (q * i) β‘β¨ solve 3 (Ξ» k q i β k :* (q :* i)
:= q :* (k :* i))
refl k q i β©
q * (k * i) β
-- If ki divides kj, and k is positive, then i divides j.
/-cong : β {i j} k β suc k * i β£ suc k * j β i β£ j
/-cong {i} {j} k (divides q eq) = divides q lemma
where
open PropEq.β‘-Reasoning
kβ² = suc k
lemma = NatProp.cancel-*-right j (q * i) (begin
j * kβ² β‘β¨ CS.*-comm j kβ² β©
kβ² * j β‘β¨ eq β©
q * (kβ² * i) β‘β¨ solve 3 (Ξ» q k i β q :* (k :* i)
:= q :* i :* k)
refl q kβ² i β©
q * i * kβ² β)
-- If the remainder after division is non-zero, then the divisor does
-- not divide the dividend.
nonZeroDivisor-lemma
: β m q (r : Fin (1 + m)) β Fin.toβ r β’ 0 β
Β¬ (1 + m) β£ (Fin.toβ r + q * (1 + m))
nonZeroDivisor-lemma m zero r rβ’zero (divides zero eq) = rβ’zero $ begin
Fin.toβ r
β‘β¨ sym $ projβ CS.*-identity (Fin.toβ r) β©
1 * Fin.toβ r
β‘β¨ eq β©
0
β
where open PropEq.β‘-Reasoning
nonZeroDivisor-lemma m zero r rβ’zero (divides (suc q) eq) =
NatProp.Β¬i+1+jβ€i m $ begin
m + suc (q * suc m)
β‘β¨ solve 2 (Ξ» m q β m :+ (con 1 :+ q) := con 1 :+ m :+ q)
refl m (q * suc m) β©
suc (m + q * suc m)
β‘β¨ sym eq β©
1 * Fin.toβ r
β‘β¨ projβ CS.*-identity (Fin.toβ r) β©
Fin.toβ r
β€β¨ β€-pred $ FP.bounded r β©
m
β
where open β€-Reasoning
nonZeroDivisor-lemma m (suc q) r rβ’zero d =
nonZeroDivisor-lemma m q r rβ’zero (β£-βΈ d' P.refl)
where
lem = solve 3 (Ξ» m r q β r :+ (m :+ q) := m :+ (r :+ q))
refl (suc m) (Fin.toβ r) (q * suc m)
d' = subst (Ξ» x β (1 + m) β£ x) lem d
-- Divisibility is decidable.
_β£?_ : Decidable _β£_
zero β£? zero = yes (0 β£0)
zero β£? suc n = no ((Ξ» ()) ββ² 0β£ββ‘0)
suc m β£? n with n divMod suc m
suc m β£? .(q * suc m) | result q zero refl =
yes $ divides q refl
suc m β£? .(1 + Fin.toβ r + q * suc m) | result q (suc r) refl =
no $ nonZeroDivisor-lemma m q (suc r) (Ξ»())
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------------------------------------------------------------------------
-- An alternative (non-standard) classical definition of weak
-- bisimilarity
------------------------------------------------------------------------
-- This definition is based on the function "wb" in Section 6.5.1 of
-- Pous and Sangiorgi's "Enhancements of the bisimulation proof
-- method".
{-# OPTIONS --sized-types #-}
open import Labelled-transition-system
module Bisimilarity.Weak.Alternative.Classical {β} (lts : LTS β) where
open import Prelude
import Bisimilarity.Classical
open LTS lts
-- We get weak bisimilarity by instantiating strong bisimilarity with
-- a different LTS.
private
module WB = Bisimilarity.Classical (weak lts)
open WB public
using (βͺ_,_β«)
renaming ( Bisimulation to Weak-bisimulation
; Bisimilarityβ² to Weak-bisimilarityβ²
; Bisimilarity to Weak-bisimilarity
; _βΌ_ to _β_
)
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------------------------------------------------------------------------------
-- Properties related with the group commutator
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module GroupTheory.Commutator.PropertiesATP where
open import GroupTheory.Base
open import GroupTheory.Commutator
------------------------------------------------------------------------------
-- From: A. G. Kurosh. The Theory of Groups, vol. 1. Chelsea Publising
-- Company, 2nd edition, 1960. p. 99.
postulate commutatorInverse : β a b β [ a , b ] Β· [ b , a ] β‘ Ξ΅
{-# ATP prove commutatorInverse #-}
-- If the commutator is associative, then commutator of any two
-- elements lies in the center of the group, i.e. a [b,c] = [b,c] a.
-- From: TPTP 6.4.0 problem GRP/GRP024-5.p.
postulate commutatorAssocCenter : (β a b c β commutatorAssoc a b c) β
(β a b c β a Β· [ b , c ] β‘ [ b , c ] Β· a)
{-# ATP prove commutatorAssocCenter #-}
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-- 2014-01-01 Andreas, test case constructed by Christian Sattler
{-# OPTIONS --allow-unsolved-metas #-}
-- unguarded recursive record
record R : Set where
constructor cons
field
r : R
postulate F : (R β Set) β Set
q : (β P β F P) β (β P β F P)
q h P = h (Ξ» {(cons x) β {!!}})
-- ISSUE WAS: Bug in implementation of eta-expansion of projected var,
-- leading to loop in Agda.
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Examples of format strings and printf
------------------------------------------------------------------------
{-# OPTIONS --safe --without-K #-}
module README.Text.Printf where
open import Data.Nat.Base
open import Data.Char.Base
open import Data.List.Base
open import Data.String.Base
open import Data.Sum.Base
open import Relation.Binary.PropositionalEquality
------------------------------------------------------------------------
-- Format strings
open import Text.Format
-- We can specify a format by writing a string which will get interpreted
-- by a lexer into a list of formatting directives.
-- The specification types are always started with a '%' character:
-- Integers (%d or %i)
-- Naturals (%u)
-- Floats (%f)
-- Chars (%c)
-- Strings (%s)
-- Anything which is not a type specification is a raw string to be spliced
-- in the output of printf.
-- For instance the following format alternates types and raw strings
_ : lexer "%s: %u + %u β‘ %u"
β‘ injβ (`String β· Raw ": " β· `β β· Raw " + " β· `β β· Raw " β‘ " β· `β β· [])
_ = refl
-- Lexing can fail. There are two possible errors:
-- If we start a specification type with a '%' but the string ends then
-- we get an UnexpectedEndOfString error
_ : lexer "%s: %u + %u β‘ %"
β‘ injβ (UnexpectedEndOfString "%s: %u + %u β‘ %")
_ = refl
-- If we start a specification type with a '%' and the following character
-- does not correspond to an existing type, we get an InvalidType error
-- together with a focus highlighting the position of the problematic type.
_ : lexer "%s: %u + %a β‘ %u"
β‘ injβ (InvalidType "%s: %u + %" 'a' " β‘ %u")
_ = refl
------------------------------------------------------------------------
-- Printf
open import Text.Printf
-- printf is a function which takes a format string as an argument and
-- returns a function expecting a value for each type specification present
-- in the format and returns a string splicing in these values into the
-- format string.
-- For instance `printf "%s: %u + %u β‘ %u"` is a
-- `String β β β β β β β String` function.
_ : String β β β β β β β String
_ = printf "%s: %u + %u β‘ %u"
_ : printf "%s: %u + %u β‘ %u" "example" 3 2 5
β‘ "example: 3 + 2 β‘ 5"
_ = refl
-- If the format string str is invalid then `printf str` will have type
-- `Error e` where `e` is the lexing error.
_ : Text.Printf.Error (UnexpectedEndOfString "%s: %u + %u β‘ %")
_ = printf "%s: %u + %u β‘ %"
_ : Text.Printf.Error (InvalidType "%s: %u + %" 'a' " β‘ %u")
_ = printf "%s: %u + %a β‘ %u"
-- Trying to pass arguments to such an ΜError` type will lead to a
-- unification error which hopefully makes the problem clear e.g.
-- `printf "%s: %u + %a β‘ %u" "example" 3 2 5` fails with the error:
-- Text.Printf.Error (InvalidType "%s: %u + %" 'a' " β‘ %u") should be
-- a function type, but it isn't
-- when checking that "example" 3 2 5 are valid arguments to a
-- function of type Text.Printf.Printf (lexer "%s: %u + %a β‘ %u")
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module Inference-of-implicit-function-space where
postulate
_β_ : Set β Set β Set
equivalence : {A B : Set} β (A β B) β (B β A) β A β B
A : Set
P : Set
P = {x : A} β A β A
works : P β P
works = equivalence (Ξ» r {x} β r {x = x}) (Ξ» r {x} β r {x = x})
worksβ : P β P
worksβ = equivalence {A = P} (Ξ» r {x} β r {x = x}) (Ξ» r {y} β r {y})
fails : P β P
fails = equivalence (Ξ» r {x} β r {x = x}) (Ξ» r {y} β r {y})
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{-# OPTIONS --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution.Introductions.Transp {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped as U hiding (wk)
open import Definition.Untyped.Properties
open import Definition.Typed
open import Definition.Typed.Properties
open import Definition.Typed.Weakening as T hiding (wk; wkTerm; wkEqTerm)
open import Definition.Typed.RedSteps
open import Definition.LogicalRelation
open import Definition.LogicalRelation.ShapeView
open import Definition.LogicalRelation.Irrelevance as I
open import Definition.LogicalRelation.Weakening
open import Definition.LogicalRelation.Properties
open import Definition.LogicalRelation.Application
open import Definition.LogicalRelation.Substitution
open import Definition.LogicalRelation.Substitution.Properties
open import Definition.LogicalRelation.Substitution.Irrelevance as S
open import Definition.LogicalRelation.Substitution.Reflexivity
open import Definition.LogicalRelation.Substitution.Introductions.Sigma
open import Definition.LogicalRelation.Substitution.Introductions.Fst
open import Definition.LogicalRelation.Substitution.Introductions.Pi
open import Definition.LogicalRelation.Substitution.Introductions.Lambda
open import Definition.LogicalRelation.Substitution.Introductions.Application
open import Definition.LogicalRelation.Substitution.Introductions.Cast
open import Definition.LogicalRelation.Substitution.Introductions.Id
open import Definition.LogicalRelation.Substitution.Introductions.SingleSubst
open import Definition.LogicalRelation.Substitution.MaybeEmbed
open import Definition.LogicalRelation.Substitution.Escape
open import Definition.LogicalRelation.Substitution.Introductions.Universe
open import Definition.LogicalRelation.Substitution.Reduction
open import Definition.LogicalRelation.Substitution.Weakening
open import Definition.LogicalRelation.Substitution.ProofIrrelevance
open import Tools.Product
import Tools.PropositionalEquality as PE
IdSymα΅α΅ : β {A l t u e Ξ}
([Ξ] : β©α΅ Ξ)
([U] : Ξ β©α΅β¨ β β© U l ^ [ ! , next l ] / [Ξ])
([AU] : Ξ β©α΅β¨ β β© A β· U l ^ [ ! , next l ] / [Ξ] / [U])
([A] : Ξ β©α΅β¨ β β© A ^ [ ! , ΞΉ l ] / [Ξ])
([t] : Ξ β©α΅β¨ β β© t β· A ^ [ ! , ΞΉ l ] / [Ξ] / [A])
([u] : Ξ β©α΅β¨ β β© u β· A ^ [ ! , ΞΉ l ] / [Ξ] / [A])
([Id] : Ξ β©α΅β¨ β β© Id A t u ^ [ % , ΞΉ l ] / [Ξ]) β
([Idinv] : Ξ β©α΅β¨ β β© Id A u t ^ [ % , ΞΉ l ] / [Ξ]) β
([e] : Ξ β©α΅β¨ β β© e β· Id A t u ^ [ % , ΞΉ l ] / [Ξ] / [Id] ) β
Ξ β©α΅β¨ β β© Idsym A t u e β· Id A u t ^ [ % , ΞΉ l ] / [Ξ] / [Idinv]
IdSymα΅α΅ {A} {l} {t} {u} {e} {Ξ} [Ξ] [U] [AU] [A] [t] [u] [Id] [Idinv] [e] = validityIrr {A = Id A u t} {t = Idsym A t u e} [Ξ] [Idinv] Ξ» {Ξ} {Ο} β’Ξ [Ο] β
PE.subst (Ξ» X β Ξ β’ X β· subst Ο (Id A u t) ^ [ % , ΞΉ l ] ) (PE.sym (subst-Idsym Ο A t u e))
(Idsymβ±Ό {A = subst Ο A} {x = subst Ο t} {y = subst Ο u} (escapeTerm (projβ ([U] {Ξ} {Ο} β’Ξ [Ο])) (projβ ([AU] β’Ξ [Ο])))
(escapeTerm (projβ ([A] {Ξ} {Ο} β’Ξ [Ο])) (projβ ([t] β’Ξ [Ο])))
(escapeTerm (projβ ([A] {Ξ} {Ο} β’Ξ [Ο])) (projβ ([u] β’Ξ [Ο])))
(escapeTerm (projβ ([Id] {Ξ} {Ο} β’Ξ [Ο])) (projβ ([e] β’Ξ [Ο]))))
abstract
transpα΅α΅ : β {A P l t s u e Ξ}
([Ξ] : β©α΅ Ξ)
([A] : Ξ β©α΅β¨ β β© A ^ [ ! , l ] / [Ξ])
([P] : Ξ β A ^ [ ! , l ] β©α΅β¨ β β© P ^ [ % , l ] / (_β_ {A = A} [Ξ] [A]))
([t] : Ξ β©α΅β¨ β β© t β· A ^ [ ! , l ] / [Ξ] / [A])
([s] : Ξ β©α΅β¨ β β© s β· P [ t ] ^ [ % , l ] / [Ξ] / substS {A} {P} {t} [Ξ] [A] [P] [t])
([u] : Ξ β©α΅β¨ β β© u β· A ^ [ ! , l ] / [Ξ] / [A])
([Id] : Ξ β©α΅β¨ β β© Id A t u ^ [ % , l ] / [Ξ]) β
([e] : Ξ β©α΅β¨ β β© e β· Id A t u ^ [ % , l ] / [Ξ] / [Id] ) β
Ξ β©α΅β¨ β β© transp A P t s u e β· P [ u ] ^ [ % , l ] / [Ξ] / substS {A} {P} {u} [Ξ] [A] [P] [u]
transpα΅α΅ {A} {P} {l} {t} {s} {u} {e} {Ξ} [Ξ] [A] [P] [t] [s] [u] [Id] [e] =
validityIrr {A = P [ u ]} {t = transp A P t s u e } [Ξ] (substS {A} {P} {u} [Ξ] [A] [P] [u]) Ξ» {Ξ} {Ο} β’Ξ [Ο] β
let [liftΟ] = liftSubstS {F = A} [Ξ] β’Ξ [A] [Ο]
[A]Ο = projβ ([A] {Ξ} {Ο} β’Ξ [Ο])
[P[t]]Ο = I.irrelevanceβ² (singleSubstLift P t) (projβ (substS {A} {P} {t} [Ξ] [A] [P] [t] {Ξ} {Ο} β’Ξ [Ο]))
X = transpβ±Ό (escape [A]Ο) (escape (projβ ([P] {Ξ β subst Ο A ^ [ ! , l ]} {liftSubst Ο} (β’Ξ β (escape [A]Ο)) [liftΟ])))
(escapeTerm [A]Ο (projβ ([t] β’Ξ [Ο]))) (escapeTerm [P[t]]Ο (I.irrelevanceTermβ² (singleSubstLift P t) PE.refl PE.refl (projβ (substS {A} {P} {t} [Ξ] [A] [P] [t] {Ξ} {Ο} β’Ξ [Ο])) [P[t]]Ο (projβ ([s] β’Ξ [Ο]))))
(escapeTerm [A]Ο (projβ ([u] β’Ξ [Ο]))) (escapeTerm (projβ ([Id] {Ξ} {Ο} β’Ξ [Ο])) (projβ ([e] β’Ξ [Ο])))
in PE.subst (Ξ» X β Ξ β’ transp (subst Ο A) ( subst (liftSubst Ο) P) (subst Ο t) (subst Ο s) (subst Ο u) (subst Ο e) β· X ^ [ % , l ] ) (PE.sym (singleSubstLift P u)) X
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--------------------------------------------------------------------------------
-- This is part of Agda Inference Systems
{-# OPTIONS --sized-types --guardedness #-}
open import Data.Product
open import Data.Vec
open import Codata.Colist as Colist
open import Agda.Builtin.Equality
open import Size
open import Codata.Thunk
open import Data.Fin
open import Data.Nat
open import Data.Maybe
open import Examples.Colists.Auxiliary.Colist_member
open import is-lib.InfSys
module Examples.Colists.member {A : Set} where
U = A Γ Colist A β
data memberRN : Set where
mem-h mem-t : memberRN
mem-h-r : FinMetaRule U
mem-h-r .Ctx = A Γ Thunk (Colist A) β
mem-h-r .comp (x , xs) =
[] ,
----------------
(x , x β· xs)
mem-t-r : FinMetaRule U
mem-t-r .Ctx = A Γ A Γ Thunk (Colist A) β
mem-t-r .comp (x , y , xs) =
((x , xs .force) β· []) ,
----------------
(x , y β· xs)
memberIS : IS U
memberIS .Names = memberRN
memberIS .rules mem-h = from mem-h-r
memberIS .rules mem-t = from mem-t-r
_member_ : A β Colist A β β Set
x member xs = Ind⦠memberIS ⧠(x , xs)
memSpec : U β Set
memSpec (x , xs) = Ξ£[ i β β ] (Colist.lookup i xs β‘ just x)
memSpecClosed : ISClosed memberIS memSpec
memSpecClosed mem-h _ _ = zero , refl
memSpecClosed mem-t _ pr =
let (i , proof) = pr Fin.zero in
(suc i) , proof
memberSound : β{x xs} β x member xs β memSpec (x , xs)
memberSound = ind[ memberIS ] memSpec memSpecClosed
-- Completeness using memSpec does not terminate
-- Product implemented as record. Record projections do not decrease
memSpec' : U β β β Set
memSpec' (x , xs) i = Colist.lookup i xs β‘ just x
memberCompl : β{x xs i} β memSpec' (x , xs) i β x member xs
memberCompl {.x} {x β· _} {zero} refl = apply-ind mem-h _ Ξ» ()
memberCompl {x} {y β· xs} {suc i} eq = apply-ind mem-t _ Ξ»{zero β memberCompl eq}
memberComplete : β{x xs} β memSpec (x , xs) β x member xs
memberComplete (i , eq) = memberCompl eq
{- Correctness wrt to Agda DataType -}
β-sound : β{x xs} β x β xs β x member xs
β-sound here = apply-ind mem-h _ Ξ» ()
β-sound (there mem) = apply-ind mem-t _ Ξ»{zero β β-sound mem}
β-complete : β{x xs} β x member xs β x β xs
β-complete (fold (mem-h , _ , refl , _)) = here
β-complete (fold (mem-t , _ , refl , prem)) = there (β-complete (prem zero)) | {
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{-# OPTIONS --cubical-compatible #-}
open import Common.Prelude
open import Common.Equality
open import Common.Product
data _β
_ {A : Set} (a : A) : {B : Set} (b : B) β Setβ where
refl : a β
a
data D : Bool β Set where
x : D true
y : D false
P : Set -> Setβ
P S = Ξ£ S (\s β s β
x)
pbool : P (D true)
pbool = _ , refl
Β¬pfin : P (D false) β β₯
Β¬pfin (y , ())
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End of preview.
NOTE: Please see EleutherAI/proof-pile-2
This is a cherry-picked repackaging of the algebraic-stack segment from the proof-pile-2 dataset as parquet files
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