From fa25173a38557238a0f1f0dd868cbcc0c7a3aa16 Mon Sep 17 00:00:00 2001 From: Egbert Rijke Date: Thu, 10 Dec 2015 16:36:10 -0500 Subject: [PATCH 1/7] kernels --- group_theory/basic.hlean | 12 +++-------- group_theory/constructions.hlean | 36 +++++++++++++++++++++++++++++++- 2 files changed, 38 insertions(+), 10 deletions(-) diff --git a/group_theory/basic.hlean b/group_theory/basic.hlean index 9da1d98..f1dc834 100644 --- a/group_theory/basic.hlean +++ b/group_theory/basic.hlean @@ -54,9 +54,9 @@ namespace group theorem respect_inv (φ : G₁ →g G₂) (g : G₁) : φ g⁻¹ = (φ g)⁻¹ := eq_inv_of_mul_eq_one (!respect_mul⁻¹ ⬝ ap φ !mul.left_inv ⬝ !respect_one) - local attribute Pointed_of_Group [coercion] - definition pmap_of_homomorphism [constructor] (φ : G₁ →g G₂) : G₁ →* G₂ := - pmap.mk φ !respect_one + --local attribute Pointed_of_Group [coercion] + --definition pmap_of_homomorphism [constructor] (φ : G₁ →g G₂) : G₁ →* G₂ := + --pmap.mk φ !respect_one definition homomorphism_eq (p : group_fun φ ~ group_fun φ') : φ = φ' := begin @@ -72,10 +72,4 @@ namespace group definition homomorphism_id [constructor] (G : Group) : G → G := homomorphism.mk id (λg h, idp) - - - -- TODO: maybe define this in more generality for pointed types? - definition kernel [constructor] (φ : G₁ →g G₂) (g : G₁) : hprop := trunctype.mk (φ g = 1) _ - - end group diff --git a/group_theory/constructions.hlean b/group_theory/constructions.hlean index 39dd861..2e1b242 100644 --- a/group_theory/constructions.hlean +++ b/group_theory/constructions.hlean @@ -8,7 +8,7 @@ Constructions of groups import .basic hit.set_quotient types.sigma types.list types.sum -open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function +open eq algebra is_trunc pi pointed set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function equiv set_option class.force_new true namespace group @@ -597,5 +597,39 @@ namespace group exact !respect_inv⁻¹}} end + section kernels + variables {G₁ G₂ : Group} + + -- TODO: maybe define this in more generality for pointed types? + definition kernel [constructor] (φ : G₁ →g G₂) (g : G₁) : hprop := trunctype.mk (φ g = 1) _ + + definition kernel_mul (φ : G₁ →g G₂) (g h : G₁) (H₁ : kernel φ g) (H₂ : kernel φ h) : kernel φ (g *[G₁] h) := + begin + esimp at *, + exact calc + φ (g * h) = (φ g) * (φ h) : respect_mul + ... = 1 * (φ h) : H₁ + ... = 1 * 1 : H₂ + ... = 1 : one_mul + end + + definition kernel_inv (φ : G₁ →g G₂) (g : G₁) (H : kernel φ g) : kernel φ (g⁻¹) := + begin + esimp at *, + exact calc + φ g⁻¹ = (φ g)⁻¹ : respect_inv + ... = 1⁻¹ : H + ... = 1 : one_inv + end + + definition kernel_subgroup (φ : G₁ →g G₂) : subgroup_rel G₁ := + ⦃ subgroup_rel, + R := kernel φ, + Rone := respect_one φ, + Rmul := kernel_mul φ, + Rinv := kernel_inv φ + ⦄ + + end kernels end group From 304bcc472a3ec532d55f756a643d40ac5b9d4828 Mon Sep 17 00:00:00 2001 From: Egbert Rijke Date: Thu, 10 Dec 2015 18:37:30 -0500 Subject: [PATCH 2/7] homework accomplished --- group_theory/constructions.hlean | 19 +++++++++++++++++++ 1 file changed, 19 insertions(+) diff --git a/group_theory/constructions.hlean b/group_theory/constructions.hlean index 2e1b242..7f017b8 100644 --- a/group_theory/constructions.hlean +++ b/group_theory/constructions.hlean @@ -631,5 +631,24 @@ namespace group Rinv := kernel_inv φ ⦄ + definition kernel_subgroup_isnormal (φ : G₁ →g G₂) (g : G₁) (h : G₁) : kernel φ g → kernel φ (h * g * h⁻¹) := + begin + esimp at *, + intro p, + exact calc + φ (h * g * h⁻¹) = (φ (h * g)) * φ (h⁻¹) : respect_mul + ... = (φ h) * (φ g) * (φ h⁻¹) : respect_mul + ... = (φ h) * 1 * (φ h⁻¹) : p + ... = (φ h) * (φ h⁻¹) : mul_one + ... = (φ h) * (φ h)⁻¹ : respect_inv + ... = 1 : mul.right_inv + end + + definition kernel_normal_subgroup (φ : G₁ →g G₂) : normal_subgroup_rel G₁ := + ⦃ normal_subgroup_rel, + kernel_subgroup φ, + is_normal := kernel_subgroup_isnormal φ + ⦄ + end kernels end group From 492b433cc612d3f78805363a8243c1784c06839f Mon Sep 17 00:00:00 2001 From: Floris van Doorn Date: Thu, 10 Dec 2015 15:45:49 -0500 Subject: [PATCH 3/7] show that groups form a precategory (category todo) --- group_theory/basic.hlean | 87 +++++++++++++++++++++++++++++++++++++--- 1 file changed, 82 insertions(+), 5 deletions(-) diff --git a/group_theory/basic.hlean b/group_theory/basic.hlean index f1dc834..07700bc 100644 --- a/group_theory/basic.hlean +++ b/group_theory/basic.hlean @@ -12,14 +12,15 @@ Basic group theory However, there is currently no group theory. -/ -import types.pointed types.pi algebra.bundled +import types.pointed types.pi algebra.bundled algebra.category.category -open eq algebra pointed function is_trunc pi +open eq algebra pointed function is_trunc pi category equiv is_equiv set_option class.force_new true namespace group definition pointed_Group [instance] (G : Group) : pointed G := pointed.mk one definition Pointed_of_Group (G : Group) : Type* := pointed.mk' G + definition hset_of_Group (G : Group) : hset := trunctype.mk G _ -- print Type* -- print Pointed @@ -42,7 +43,7 @@ namespace group abbreviation respect_mul := @homomorphism.p infix ` →g `:55 := homomorphism - variables {G₁ G₂ G₃ : Group} {g h : G₁} {ψ : G₂ →g G₃} {φ φ' : G₁ →g G₂} + variables {G G₁ G₂ G₃ : Group} {g h : G₁} {ψ : G₂ →g G₃} {φ φ' : G₁ →g G₂} theorem respect_one (φ : G₁ →g G₂) : φ 1 = 1 := mul.right_cancel @@ -54,6 +55,17 @@ namespace group theorem respect_inv (φ : G₁ →g G₂) (g : G₁) : φ g⁻¹ = (φ g)⁻¹ := eq_inv_of_mul_eq_one (!respect_mul⁻¹ ⬝ ap φ !mul.left_inv ⬝ !respect_one) + definition is_hset_homomorphism [instance] (G₁ G₂ : Group) : is_hset (homomorphism G₁ G₂) := + begin + assert H : G₁ →g G₂ ≃ Σ(f : G₁ → G₂), Π(g₁ g₂ : G₁), f (g₁ * g₂) = f g₁ * f g₂, + { fapply equiv.MK, + { intro φ, induction φ, constructor, assumption}, + { intro v, induction v, constructor, assumption}, + { intro v, induction v, reflexivity}, + { intro φ, induction φ, reflexivity}}, + apply is_trunc_equiv_closed_rev, exact H + end + --local attribute Pointed_of_Group [coercion] --definition pmap_of_homomorphism [constructor] (φ : G₁ →g G₂) : G₁ →* G₂ := --pmap.mk φ !respect_one @@ -66,10 +78,75 @@ namespace group /- categorical structure of groups + homomorphisms -/ - definition homomorphism_compose [constructor] (ψ : G₂ →g G₃) (φ : G₁ →g G₂) : G₁ → G₃ := + definition homomorphism_compose [constructor] (ψ : G₂ →g G₃) (φ : G₁ →g G₂) : G₁ →g G₃ := homomorphism.mk (ψ ∘ φ) (λg h, ap ψ !respect_mul ⬝ !respect_mul) - definition homomorphism_id [constructor] (G : Group) : G → G := + definition homomorphism_id [constructor] (G : Group) : G →g G := homomorphism.mk id (λg h, idp) + + + infixr ` ∘g `:75 := homomorphism_compose + notation 1 := homomorphism_id _ + + structure isomorphism (A B : Group) := + (to_hom : A →g B) + (is_equiv_to_hom : is_equiv to_hom) + + infix ` ≃g `:25 := isomorphism + attribute isomorphism.to_hom [coercion] + attribute isomorphism.is_equiv_to_hom [instance] + + -- definition equiv_of_isomorphism [constructor] (φ : G₁ ≃g G₂) : G₁ ≃ G₂ := + -- equiv.mk φ sorry + + definition to_ginv [constructor] (φ : G₁ ≃g G₂) : G₂ →g G₁ := + homomorphism.mk φ⁻¹ + abstract begin + intro g₁ g₂, apply eq_of_fn_eq_fn' φ, + rewrite [respect_mul, +right_inv φ] + end end + + definition isomorphism.refl [refl] [constructor] (G : Group) : G ≃g G := + isomorphism.mk 1 !is_equiv_id + + definition isomorphism.symm [symm] [constructor] (φ : G₁ ≃g G₂) : G₂ ≃g G₁ := + isomorphism.mk (to_ginv φ) !is_equiv_inv + + definition isomorphism.trans [trans] [constructor] (φ : G₁ ≃g G₂) (ψ : G₂ ≃g G₃) + : G₁ ≃g G₃ := + isomorphism.mk (ψ ∘g φ) !is_equiv_compose + + postfix `⁻¹ᵍ`:(max + 1) := isomorphism.symm + infixl ` ⬝g `:75 := isomorphism.trans + + -- definition Group_univalence (G₁ G₂ : Group) : (G₁ ≃g G₂) ≃ (G₁ = G₂) := + -- begin + -- fapply equiv.MK, + -- { intro φ, fapply Group_eq, apply equiv_of_isomorphism φ, apply respect_mul}, + -- { intro p, apply transport _ p, reflexivity}, + -- { intro p, induction p, esimp, }, + -- { } + -- end + + /- category of groups -/ + + definition precategory_group [constructor] : precategory Group := + precategory.mk homomorphism + @homomorphism_compose + @homomorphism_id + (λG₁ G₂ G₃ G₄ φ₃ φ₂ φ₁, homomorphism_eq (λg, idp)) + (λG₁ G₂ φ, homomorphism_eq (λg, idp)) + (λG₁ G₂ φ, homomorphism_eq (λg, idp)) + + -- definition category_group : category Group := + -- category.mk precategory_group + -- begin + -- intro G₁ G₂, + -- fapply adjointify, + -- { intro φ, fapply Group_eq, }, + -- { }, + -- { } + -- end + end group From 3c9ca4b51c9711df54549c6b3df4d33e3e3ddfff Mon Sep 17 00:00:00 2001 From: Floris van Doorn Date: Thu, 10 Dec 2015 15:46:09 -0500 Subject: [PATCH 4/7] working on tensor group --- group_theory/constructions.hlean | 128 ++++++++++++++++++++++++++++++- 1 file changed, 126 insertions(+), 2 deletions(-) diff --git a/group_theory/constructions.hlean b/group_theory/constructions.hlean index 7f017b8..f7a3103 100644 --- a/group_theory/constructions.hlean +++ b/group_theory/constructions.hlean @@ -8,7 +8,7 @@ Constructions of groups import .basic hit.set_quotient types.sigma types.list types.sum -open eq algebra is_trunc pi pointed set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function +open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function equiv set_option class.force_new true namespace group @@ -32,7 +32,7 @@ namespace group abbreviation is_normal_subgroup [unfold 2] := @normal_subgroup_rel.is_normal variables {G G' : Group} (H : subgroup_rel G) (N : normal_subgroup_rel G) {g g' h h' k : G} - {A : CommGroup} + {A B : CommGroup} theorem is_normal_subgroup' (h : G) (r : N g) : N (h⁻¹ * g * h) := inv_inv h ▸ is_normal_subgroup N h⁻¹ r @@ -597,6 +597,130 @@ namespace group exact !respect_inv⁻¹}} end + /- Free Commutative Group of a set -/ + namespace tensor_group + + local abbreviation ι := @free_comm_group_inclusion + + inductive tensor_rel : free_comm_group (hset_of_Group A ×t hset_of_Group B) → Type := + | mul_left : Π(a₁ a₂ : A) (b : B), tensor_rel (ι (a₁, b) * ι (a₂, b) * (ι (a₁ * a₂, b))⁻¹) + | mul_right : Π(a : A) (b₁ b₂ : B), tensor_rel (ι (a, b₁) * ι (a, b₂) * (ι (a, b₁ * b₂))⁻¹) + +exit + open tensor_rel + local abbreviation R [reducible] := tensor_rel + attribute tensor_rel.rrefl [refl] + attribute tensor_rel.rtrans [trans] + + definition tensor_carrier [reducible] : Type := set_quotient (λx y, ∥R X x y∥) + local abbreviation FG := tensor_carrier + + definition is_reflexive_R : is_reflexive (λx y, ∥R X x y∥) := + begin constructor, intro s, apply tr, unfold R end + local attribute is_reflexive_R [instance] + + variable {X} + theorem rel_respect_flip (r : R X l l') : R X (map sum.flip l) (map sum.flip l') := + begin + induction r with l x x x y l₁ l₂ l₃ l₄ r₁ r₂ IH₁ IH₂ l₁ l₂ l₃ r₁ r₂ IH₁ IH₂, + { reflexivity}, + { repeat esimp [map], exact cancel2 x}, + { repeat esimp [map], exact cancel1 x}, + { repeat esimp [map], apply rflip}, + { rewrite [+map_append], exact resp_append IH₁ IH₂}, + { exact rtrans IH₁ IH₂} + end + + theorem rel_respect_reverse (r : R X l l') : R X (reverse l) (reverse l') := + begin + induction r with l x x x y l₁ l₂ l₃ l₄ r₁ r₂ IH₁ IH₂ l₁ l₂ l₃ r₁ r₂ IH₁ IH₂, + { reflexivity}, + { repeat esimp [map], exact cancel2 x}, + { repeat esimp [map], exact cancel1 x}, + { repeat esimp [map], apply rflip}, + { rewrite [+reverse_append], exact resp_append IH₂ IH₁}, + { exact rtrans IH₁ IH₂} + end + + theorem rel_cons_concat (l s) : R X (s :: l) (concat s l) := + begin + induction l with t l IH, + { reflexivity}, + { rewrite [concat_cons], transitivity (t :: s :: l), + { exact resp_append !rflip !rrefl}, + { exact resp_append (rrefl [t]) IH}} + end + + definition tensor_one [constructor] : FG X := class_of [] + definition tensor_inv [unfold 3] : FG X → FG X := + quotient_unary_map (reverse ∘ map sum.flip) + (λl l', trunc_functor -1 (rel_respect_reverse ∘ rel_respect_flip)) + definition tensor_mul [unfold 3 4] : FG X → FG X → FG X := + quotient_binary_map append (λl l', trunc.elim (λr m m', trunc.elim (λs, tr (resp_append r s)))) + + section + local notation 1 := tensor_one + local postfix ⁻¹ := tensor_inv + local infix * := tensor_mul + + theorem tensor_mul_assoc (g₁ g₂ g₃ : FG X) : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) := + begin + refine set_quotient.rec_hprop _ g₁, + refine set_quotient.rec_hprop _ g₂, + refine set_quotient.rec_hprop _ g₃, + clear g₁ g₂ g₃, intro g₁ g₂ g₃, + exact ap class_of !append.assoc + end + + theorem tensor_one_mul (g : FG X) : 1 * g = g := + begin + refine set_quotient.rec_hprop _ g, clear g, intro g, + exact ap class_of !append_nil_left + end + + theorem tensor_mul_one (g : FG X) : g * 1 = g := + begin + refine set_quotient.rec_hprop _ g, clear g, intro g, + exact ap class_of !append_nil_right + end + + theorem tensor_mul_left_inv (g : FG X) : g⁻¹ * g = 1 := + begin + refine set_quotient.rec_hprop _ g, clear g, intro g, + apply eq_of_rel, apply tr, + induction g with s l IH, + { reflexivity}, + { rewrite [▸*, map_cons, reverse_cons, concat_append], + refine rtrans _ IH, + apply resp_append, reflexivity, + change R X ([flip s, s] ++ l) ([] ++ l), + apply resp_append, + induction s, apply cancel2, apply cancel1, + reflexivity} + end + + theorem tensor_mul_comm (g h : FG X) : g * h = h * g := + begin + refine set_quotient.rec_hprop _ g, clear g, intro g, + refine set_quotient.rec_hprop _ h, clear h, intro h, + apply eq_of_rel, apply tr, + revert h, induction g with s l IH: intro h, + { rewrite [append_nil_left, append_nil_right]}, + { rewrite [append_cons,-concat_append], + transitivity concat s (l ++ h), apply rel_cons_concat, + rewrite [-append_concat], apply IH} + end + end + end tensor_group open tensor_group + + variables (X) + definition group_tensor_group [constructor] : comm_group (tensor_carrier X) := + comm_group.mk tensor_mul _ tensor_mul_assoc tensor_one tensor_one_mul tensor_mul_one + tensor_inv tensor_mul_left_inv tensor_mul_comm + + definition tensor_group [constructor] : CommGroup := + CommGroup.mk _ (group_tensor_group X) + section kernels variables {G₁ G₂ : Group} From 01e0d1897bc7827176e9a006d3d00d4e12ae13ea Mon Sep 17 00:00:00 2001 From: Floris van Doorn Date: Thu, 10 Dec 2015 21:17:29 -0500 Subject: [PATCH 5/7] some cleanup --- group_theory/basic.hlean | 14 ++-- group_theory/constructions.hlean | 140 +++++-------------------------- 2 files changed, 28 insertions(+), 126 deletions(-) diff --git a/group_theory/basic.hlean b/group_theory/basic.hlean index 07700bc..ea49e81 100644 --- a/group_theory/basic.hlean +++ b/group_theory/basic.hlean @@ -66,9 +66,9 @@ namespace group apply is_trunc_equiv_closed_rev, exact H end - --local attribute Pointed_of_Group [coercion] - --definition pmap_of_homomorphism [constructor] (φ : G₁ →g G₂) : G₁ →* G₂ := - --pmap.mk φ !respect_one + definition pmap_of_homomorphism [constructor] (φ : G₁ →g G₂) + : Pointed_of_Group G₁ →* Pointed_of_Group G₂ := + pmap.mk φ !respect_one definition homomorphism_eq (p : group_fun φ ~ group_fun φ') : φ = φ' := begin @@ -84,8 +84,6 @@ namespace group definition homomorphism_id [constructor] (G : Group) : G →g G := homomorphism.mk id (λg h, idp) - - infixr ` ∘g `:75 := homomorphism_compose notation 1 := homomorphism_id _ @@ -97,8 +95,8 @@ namespace group attribute isomorphism.to_hom [coercion] attribute isomorphism.is_equiv_to_hom [instance] - -- definition equiv_of_isomorphism [constructor] (φ : G₁ ≃g G₂) : G₁ ≃ G₂ := - -- equiv.mk φ sorry + definition equiv_of_isomorphism [constructor] (φ : G₁ ≃g G₂) : G₁ ≃ G₂ := + equiv.mk φ _ definition to_ginv [constructor] (φ : G₁ ≃g G₂) : G₂ →g G₁ := homomorphism.mk φ⁻¹ @@ -120,6 +118,7 @@ namespace group postfix `⁻¹ᵍ`:(max + 1) := isomorphism.symm infixl ` ⬝g `:75 := isomorphism.trans + -- TODO -- definition Group_univalence (G₁ G₂ : Group) : (G₁ ≃g G₂) ≃ (G₁ = G₂) := -- begin -- fapply equiv.MK, @@ -139,6 +138,7 @@ namespace group (λG₁ G₂ φ, homomorphism_eq (λg, idp)) (λG₁ G₂ φ, homomorphism_eq (λg, idp)) + -- TODO -- definition category_group : category Group := -- category.mk precategory_group -- begin diff --git a/group_theory/constructions.hlean b/group_theory/constructions.hlean index f7a3103..b2b7f47 100644 --- a/group_theory/constructions.hlean +++ b/group_theory/constructions.hlean @@ -598,128 +598,28 @@ namespace group end /- Free Commutative Group of a set -/ - namespace tensor_group +/- namespace tensor_group + variables {A B} local abbreviation ι := @free_comm_group_inclusion - inductive tensor_rel : free_comm_group (hset_of_Group A ×t hset_of_Group B) → Type := - | mul_left : Π(a₁ a₂ : A) (b : B), tensor_rel (ι (a₁, b) * ι (a₂, b) * (ι (a₁ * a₂, b))⁻¹) - | mul_right : Π(a : A) (b₁ b₂ : B), tensor_rel (ι (a, b₁) * ι (a, b₂) * (ι (a, b₁ * b₂))⁻¹) + inductive tensor_rel_type : free_comm_group (hset_of_Group A ×t hset_of_Group B) → Type := + | mul_left : Π(a₁ a₂ : A) (b : B), tensor_rel_type (ι (a₁, b) * ι (a₂, b) * (ι (a₁ * a₂, b))⁻¹) + | mul_right : Π(a : A) (b₁ b₂ : B), tensor_rel_type (ι (a, b₁) * ι (a, b₂) * (ι (a, b₁ * b₂))⁻¹) -exit - open tensor_rel - local abbreviation R [reducible] := tensor_rel - attribute tensor_rel.rrefl [refl] - attribute tensor_rel.rtrans [trans] + open tensor_rel_type - definition tensor_carrier [reducible] : Type := set_quotient (λx y, ∥R X x y∥) - local abbreviation FG := tensor_carrier + definition tensor_rel' (x : free_comm_group (hset_of_Group A ×t hset_of_Group B)) : hprop := + ∥ tensor_rel_type x ∥ - definition is_reflexive_R : is_reflexive (λx y, ∥R X x y∥) := - begin constructor, intro s, apply tr, unfold R end - local attribute is_reflexive_R [instance] - - variable {X} - theorem rel_respect_flip (r : R X l l') : R X (map sum.flip l) (map sum.flip l') := - begin - induction r with l x x x y l₁ l₂ l₃ l₄ r₁ r₂ IH₁ IH₂ l₁ l₂ l₃ r₁ r₂ IH₁ IH₂, - { reflexivity}, - { repeat esimp [map], exact cancel2 x}, - { repeat esimp [map], exact cancel1 x}, - { repeat esimp [map], apply rflip}, - { rewrite [+map_append], exact resp_append IH₁ IH₂}, - { exact rtrans IH₁ IH₂} - end - - theorem rel_respect_reverse (r : R X l l') : R X (reverse l) (reverse l') := - begin - induction r with l x x x y l₁ l₂ l₃ l₄ r₁ r₂ IH₁ IH₂ l₁ l₂ l₃ r₁ r₂ IH₁ IH₂, - { reflexivity}, - { repeat esimp [map], exact cancel2 x}, - { repeat esimp [map], exact cancel1 x}, - { repeat esimp [map], apply rflip}, - { rewrite [+reverse_append], exact resp_append IH₂ IH₁}, - { exact rtrans IH₁ IH₂} - end - - theorem rel_cons_concat (l s) : R X (s :: l) (concat s l) := - begin - induction l with t l IH, - { reflexivity}, - { rewrite [concat_cons], transitivity (t :: s :: l), - { exact resp_append !rflip !rrefl}, - { exact resp_append (rrefl [t]) IH}} - end - - definition tensor_one [constructor] : FG X := class_of [] - definition tensor_inv [unfold 3] : FG X → FG X := - quotient_unary_map (reverse ∘ map sum.flip) - (λl l', trunc_functor -1 (rel_respect_reverse ∘ rel_respect_flip)) - definition tensor_mul [unfold 3 4] : FG X → FG X → FG X := - quotient_binary_map append (λl l', trunc.elim (λr m m', trunc.elim (λs, tr (resp_append r s)))) - - section - local notation 1 := tensor_one - local postfix ⁻¹ := tensor_inv - local infix * := tensor_mul - - theorem tensor_mul_assoc (g₁ g₂ g₃ : FG X) : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) := - begin - refine set_quotient.rec_hprop _ g₁, - refine set_quotient.rec_hprop _ g₂, - refine set_quotient.rec_hprop _ g₃, - clear g₁ g₂ g₃, intro g₁ g₂ g₃, - exact ap class_of !append.assoc - end - - theorem tensor_one_mul (g : FG X) : 1 * g = g := - begin - refine set_quotient.rec_hprop _ g, clear g, intro g, - exact ap class_of !append_nil_left - end - - theorem tensor_mul_one (g : FG X) : g * 1 = g := - begin - refine set_quotient.rec_hprop _ g, clear g, intro g, - exact ap class_of !append_nil_right - end - - theorem tensor_mul_left_inv (g : FG X) : g⁻¹ * g = 1 := - begin - refine set_quotient.rec_hprop _ g, clear g, intro g, - apply eq_of_rel, apply tr, - induction g with s l IH, - { reflexivity}, - { rewrite [▸*, map_cons, reverse_cons, concat_append], - refine rtrans _ IH, - apply resp_append, reflexivity, - change R X ([flip s, s] ++ l) ([] ++ l), - apply resp_append, - induction s, apply cancel2, apply cancel1, - reflexivity} - end - - theorem tensor_mul_comm (g h : FG X) : g * h = h * g := - begin - refine set_quotient.rec_hprop _ g, clear g, intro g, - refine set_quotient.rec_hprop _ h, clear h, intro h, - apply eq_of_rel, apply tr, - revert h, induction g with s l IH: intro h, - { rewrite [append_nil_left, append_nil_right]}, - { rewrite [append_cons,-concat_append], - transitivity concat s (l ++ h), apply rel_cons_concat, - rewrite [-append_concat], apply IH} - end - end - end tensor_group open tensor_group - - variables (X) - definition group_tensor_group [constructor] : comm_group (tensor_carrier X) := - comm_group.mk tensor_mul _ tensor_mul_assoc tensor_one tensor_one_mul tensor_mul_one - tensor_inv tensor_mul_left_inv tensor_mul_comm + definition tensor_group_rel (A B : CommGroup) + : normal_subgroup_rel (free_comm_group (hset_of_Group A ×t hset_of_Group B)) := + sorry /- relation generated by tensor_rel'-/ definition tensor_group [constructor] : CommGroup := - CommGroup.mk _ (group_tensor_group X) + quotient_comm_group (tensor_group_rel A B) + + end tensor_group-/ section kernels @@ -728,7 +628,7 @@ exit -- TODO: maybe define this in more generality for pointed types? definition kernel [constructor] (φ : G₁ →g G₂) (g : G₁) : hprop := trunctype.mk (φ g = 1) _ - definition kernel_mul (φ : G₁ →g G₂) (g h : G₁) (H₁ : kernel φ g) (H₂ : kernel φ h) : kernel φ (g *[G₁] h) := + theorem kernel_mul (φ : G₁ →g G₂) (g h : G₁) (H₁ : kernel φ g) (H₂ : kernel φ h) : kernel φ (g *[G₁] h) := begin esimp at *, exact calc @@ -738,7 +638,7 @@ exit ... = 1 : one_mul end - definition kernel_inv (φ : G₁ →g G₂) (g : G₁) (H : kernel φ g) : kernel φ (g⁻¹) := + theorem kernel_inv (φ : G₁ →g G₂) (g : G₁) (H : kernel φ g) : kernel φ (g⁻¹) := begin esimp at *, exact calc @@ -747,7 +647,7 @@ exit ... = 1 : one_inv end - definition kernel_subgroup (φ : G₁ →g G₂) : subgroup_rel G₁ := + definition kernel_subgroup [constructor] (φ : G₁ →g G₂) : subgroup_rel G₁ := ⦃ subgroup_rel, R := kernel φ, Rone := respect_one φ, @@ -755,7 +655,8 @@ exit Rinv := kernel_inv φ ⦄ - definition kernel_subgroup_isnormal (φ : G₁ →g G₂) (g : G₁) (h : G₁) : kernel φ g → kernel φ (h * g * h⁻¹) := + theorem kernel_subgroup_isnormal (φ : G₁ →g G₂) (g : G₁) (h : G₁) + : kernel φ g → kernel φ (h * g * h⁻¹) := begin esimp at *, intro p, @@ -768,11 +669,12 @@ exit ... = 1 : mul.right_inv end - definition kernel_normal_subgroup (φ : G₁ →g G₂) : normal_subgroup_rel G₁ := + definition kernel_normal_subgroup [constructor] (φ : G₁ →g G₂) : normal_subgroup_rel G₁ := ⦃ normal_subgroup_rel, kernel_subgroup φ, is_normal := kernel_subgroup_isnormal φ ⦄ end kernels + end group From 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14:25:17 -0500 Subject: [PATCH 7/7] added HoTT Book Chapter 8 to do --- README.md | 1 + 1 file changed, 1 insertion(+) diff --git a/README.md b/README.md index e839737..e1732ce 100644 --- a/README.md +++ b/README.md @@ -28,6 +28,7 @@ Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Floris van Doorn, Clive Newstead, - [convergence of spectral sequences](http://ncatlab.org/nlab/show/spectral+sequence#ConvergenceOfSpectralSequences) ### Topology To Do: +- HoTT Book chapter 8 - fiber and cofiber sequences (is this in the library already?) - [prespectra](http://ncatlab.org/nlab/show/spectrum+object) and [spectra](http://ncatlab.org/nlab/show/spectrum), suspension - [spectrification](http://ncatlab.org/nlab/show/higher+inductive+type#spectrification)

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