fix(hott/init/path): reorder arguments of whisker_right
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
a9fc853985
commit
e87a27cb4b
31 changed files with 196 additions and 195 deletions
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@ -63,29 +63,29 @@ Group.struct G
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attribute AddGroup.struct Group.struct [instance] [priority 2000]
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structure CommGroup :=
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(carrier : Type) (struct : comm_group carrier)
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structure AbGroup :=
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(carrier : Type) (struct : ab_group carrier)
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attribute CommGroup.carrier [coercion]
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attribute AbGroup.carrier [coercion]
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definition AddCommGroup : Type := CommGroup
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definition AddAbGroup : Type := AbGroup
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definition AddCommGroup.mk [constructor] [reducible] (G : Type) (H : add_comm_group G) :
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AddCommGroup :=
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CommGroup.mk G H
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definition AddAbGroup.mk [constructor] [reducible] (G : Type) (H : add_ab_group G) :
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AddAbGroup :=
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AbGroup.mk G H
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definition AddCommGroup.struct [reducible] (G : AddCommGroup) : add_comm_group G :=
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CommGroup.struct G
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definition AddAbGroup.struct [reducible] (G : AddAbGroup) : add_ab_group G :=
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AbGroup.struct G
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attribute AddCommGroup.struct CommGroup.struct [instance] [priority 2000]
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attribute AddAbGroup.struct AbGroup.struct [instance] [priority 2000]
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definition Group_of_CommGroup [coercion] [constructor] (G : CommGroup) : Group :=
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definition Group_of_AbGroup [coercion] [constructor] (G : AbGroup) : Group :=
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Group.mk G _
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attribute algebra._trans_of_Group_of_CommGroup_1
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algebra._trans_of_Group_of_CommGroup
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algebra._trans_of_Group_of_CommGroup_3 [constructor]
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attribute algebra._trans_of_Group_of_CommGroup_2 [unfold 1]
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attribute algebra._trans_of_Group_of_AbGroup_1
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algebra._trans_of_Group_of_AbGroup
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algebra._trans_of_Group_of_AbGroup_3 [constructor]
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attribute algebra._trans_of_Group_of_AbGroup_2 [unfold 1]
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-- structure AddSemigroup :=
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@ -118,9 +118,9 @@ attribute algebra._trans_of_Group_of_CommGroup_2 [unfold 1]
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-- attribute AddGroup.carrier [coercion]
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-- attribute AddGroup.struct [instance]
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-- structure AddCommGroup :=
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-- (carrier : Type) (struct : add_comm_group carrier)
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-- structure AddAbGroup :=
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-- (carrier : Type) (struct : add_ab_group carrier)
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-- attribute AddCommGroup.carrier [coercion]
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-- attribute AddCommGroup.struct [instance]
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-- attribute AddAbGroup.carrier [coercion]
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-- attribute AddAbGroup.struct [instance]
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end algebra
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@ -158,7 +158,7 @@ section
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ap_e_closure_elim_h e p t =
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ap_e_closure_elim_h (λa a' s, ap g (e s)) (λa a' s, (ap_compose h g (e s))⁻¹ ⬝ p s) t :=
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begin
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refine whisker_right (eq_of_square (ap_ap_e_closure_elim g h e t)⁻¹ʰ)⁻¹ _ ⬝ _,
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refine whisker_right _ (eq_of_square (ap_ap_e_closure_elim g h e t)⁻¹ʰ)⁻¹ ⬝ _,
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refine !con.assoc ⬝ _, apply inv_con_eq_of_eq_con, apply eq_of_square,
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apply transpose,
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-- the rest of the proof is almost the same as the proof of ap_ap_e_closure_elim[_h].
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@ -346,7 +346,7 @@ section group
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end group
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structure comm_group [class] (A : Type) extends group A, comm_monoid A
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structure ab_group [class] (A : Type) extends group A, comm_monoid A
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/- additive group -/
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@ -551,21 +551,21 @@ section add_group
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end add_group
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definition add_comm_group [class] : Type → Type := comm_group
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definition add_ab_group [class] : Type → Type := ab_group
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definition add_group_of_add_comm_group [reducible] [trans_instance] (A : Type)
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[H : add_comm_group A] : add_group A :=
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@comm_group.to_group A H
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definition add_group_of_add_ab_group [reducible] [trans_instance] (A : Type)
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[H : add_ab_group A] : add_group A :=
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@ab_group.to_group A H
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definition add_comm_monoid_of_add_comm_group [reducible] [trans_instance] (A : Type)
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[H : add_comm_group A] : add_comm_monoid A :=
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@comm_group.to_comm_monoid A H
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definition add_comm_monoid_of_add_ab_group [reducible] [trans_instance] (A : Type)
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[H : add_ab_group A] : add_comm_monoid A :=
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@ab_group.to_comm_monoid A H
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definition add_comm_group.to_comm_group {A : Type} [s : add_comm_group A] : comm_group A := s
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definition comm_group.to_add_comm_group {A : Type} [s : comm_group A] : add_comm_group A := s
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definition add_ab_group.to_ab_group {A : Type} [s : add_ab_group A] : ab_group A := s
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definition ab_group.to_add_ab_group {A : Type} [s : ab_group A] : add_ab_group A := s
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section add_comm_group
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variable [s : add_comm_group A]
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section add_ab_group
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variable [s : add_ab_group A]
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include s
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theorem sub_add_eq_sub_sub (a b c : A) : a - (b + c) = a - b - c :=
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@ -606,7 +606,7 @@ section add_comm_group
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theorem neg_neg_sub_neg (a b : A) : - (-a - -b) = a - b :=
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by rewrite [neg_sub, sub_neg_eq_add, neg_add_eq_sub]
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end add_comm_group
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end add_ab_group
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definition group_of_add_group (A : Type) [G : add_group A] : group A :=
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⦃group,
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@ -21,12 +21,12 @@ namespace group
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definition Group.struct' [instance] [reducible] (G : Group) : group G :=
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Group.struct G
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definition comm_group_Group_of_CommGroup [instance] [constructor] [priority 900]
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(G : CommGroup) : comm_group (Group_of_CommGroup G) :=
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definition ab_group_Group_of_AbGroup [instance] [constructor] [priority 900]
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(G : AbGroup) : ab_group (Group_of_AbGroup G) :=
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begin esimp, exact _ end
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definition comm_group_pSet_of_Group [instance] (G : CommGroup) : comm_group (pSet_of_Group G) :=
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CommGroup.struct G
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definition ab_group_pSet_of_Group [instance] (G : AbGroup) : ab_group (pSet_of_Group G) :=
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AbGroup.struct G
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definition group_pSet_of_Group [instance] [priority 900] (G : Group) :
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group (pSet_of_Group G) :=
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@ -463,14 +463,14 @@ namespace group
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definition eq_of_isomorphism {G₁ G₂ : Group} (φ : G₁ ≃g G₂) : G₁ = G₂ :=
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Group_eq (equiv_of_isomorphism φ) (respect_mul φ)
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definition comm_group.to_has_mul {A : Type} (H : comm_group A) : has_mul A := _
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local attribute comm_group.to_has_mul [coercion]
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definition ab_group.to_has_mul {A : Type} (H : ab_group A) : has_mul A := _
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local attribute ab_group.to_has_mul [coercion]
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definition comm_group_eq {A : Type} {G H : comm_group A}
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definition ab_group_eq {A : Type} {G H : ab_group A}
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(same_mul : Π(g h : A), @mul A G g h = @mul A H g h)
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: G = H :=
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begin
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have g_eq : @comm_group.to_group A G = @comm_group.to_group A H, from group_eq same_mul,
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have g_eq : @ab_group.to_group A G = @ab_group.to_group A H, from group_eq same_mul,
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cases G with Gm Gs Gh1 G1 Gh2 Gh3 Gi Gh4 Gh5,
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cases H with Hm Hs Hh1 H1 Hh2 Hh3 Hi Hh4 Hh5,
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have pm : Gm = Hm, from ap (@mul _ ∘ group.to_has_mul) g_eq,
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@ -487,18 +487,18 @@ namespace group
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reflexivity
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end
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definition comm_group_pathover {A B : Type} {G : comm_group A} {H : comm_group B} {p : A = B}
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definition ab_group_pathover {A B : Type} {G : ab_group A} {H : ab_group B} {p : A = B}
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(resp_mul : Π(g h : A), cast p (g * h) = cast p g * cast p h) : G =[p] H :=
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begin
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induction p,
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apply pathover_idp_of_eq, exact comm_group_eq (resp_mul)
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apply pathover_idp_of_eq, exact ab_group_eq (resp_mul)
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end
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definition CommGroup_eq_of_isomorphism {G₁ G₂ : CommGroup} (φ : G₁ ≃g G₂) : G₁ = G₂ :=
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definition AbGroup_eq_of_isomorphism {G₁ G₂ : AbGroup} (φ : G₁ ≃g G₂) : G₁ = G₂ :=
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begin
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induction G₁, induction G₂,
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apply apd011 CommGroup.mk (ua (equiv_of_isomorphism φ)),
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apply comm_group_pathover,
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apply apd011 AbGroup.mk (ua (equiv_of_isomorphism φ)),
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apply ab_group_pathover,
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intro g h, exact !cast_ua ⬝ respect_mul φ g h ⬝ ap011 mul !cast_ua⁻¹ !cast_ua⁻¹
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end
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@ -26,18 +26,18 @@ namespace eq
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group (carrier (ptrunctype.to_pType (π[k + 1] A))) :=
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group_homotopy_group k A
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definition comm_group_homotopy_group [constructor] [reducible] (n : ℕ) (A : Type*)
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: comm_group (π[succ (succ n)] A) :=
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trunc_comm_group concat inverse idp con.assoc idp_con con_idp con.left_inv eckmann_hilton
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definition ab_group_homotopy_group [constructor] [reducible] (n : ℕ) (A : Type*)
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: ab_group (π[succ (succ n)] A) :=
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trunc_ab_group concat inverse idp con.assoc idp_con con_idp con.left_inv eckmann_hilton
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local attribute comm_group_homotopy_group [instance]
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local attribute ab_group_homotopy_group [instance]
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definition ghomotopy_group [constructor] : Π(n : ℕ) [is_succ n] (A : Type*), Group
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| (succ n) x A := Group.mk (π[succ n] A) _
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definition cghomotopy_group [constructor] :
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Π(n : ℕ) [is_at_least_two n] (A : Type*), CommGroup
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| (succ (succ n)) x A := CommGroup.mk (π[succ (succ n)] A) _
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Π(n : ℕ) [is_at_least_two n] (A : Type*), AbGroup
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| (succ (succ n)) x A := AbGroup.mk (π[succ (succ n)] A) _
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definition fundamental_group [constructor] (A : Type*) : Group :=
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ghomotopy_group 1 A
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@ -213,40 +213,40 @@ end
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/- partially ordered groups -/
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structure ordered_mul_comm_group [class] (A : Type) extends comm_group A, order_pair A :=
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structure ordered_mul_ab_group [class] (A : Type) extends ab_group A, order_pair A :=
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(mul_le_mul_left : Πa b, le a b → Πc, le (mul c a) (mul c b))
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(mul_lt_mul_left : Πa b, lt a b → Π c, lt (mul c a) (mul c b))
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definition ordered_comm_group [class] : Type → Type := ordered_mul_comm_group
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definition ordered_ab_group [class] : Type → Type := ordered_mul_ab_group
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definition add_comm_group_of_ordered_comm_group [reducible] [trans_instance] (A : Type)
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[H : ordered_comm_group A] : add_comm_group A :=
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@ordered_mul_comm_group.to_comm_group A H
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definition add_ab_group_of_ordered_ab_group [reducible] [trans_instance] (A : Type)
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[H : ordered_ab_group A] : add_ab_group A :=
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@ordered_mul_ab_group.to_ab_group A H
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theorem ordered_mul_comm_group.le_of_mul_le_mul_left [s : ordered_mul_comm_group A] {a b c : A}
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theorem ordered_mul_ab_group.le_of_mul_le_mul_left [s : ordered_mul_ab_group A] {a b c : A}
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(H : a * b ≤ a * c) : b ≤ c :=
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have H' : a⁻¹ * (a * b) ≤ a⁻¹ * (a * c), from ordered_mul_comm_group.mul_le_mul_left _ _ H _,
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have H' : a⁻¹ * (a * b) ≤ a⁻¹ * (a * c), from ordered_mul_ab_group.mul_le_mul_left _ _ H _,
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by rewrite *inv_mul_cancel_left at H'; exact H'
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theorem ordered_mul_comm_group.lt_of_mul_lt_mul_left [s : ordered_mul_comm_group A] {a b c : A}
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theorem ordered_mul_ab_group.lt_of_mul_lt_mul_left [s : ordered_mul_ab_group A] {a b c : A}
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(H : a * b < a * c) : b < c :=
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have H' : a⁻¹ * (a * b) < a⁻¹ * (a * c), from ordered_mul_comm_group.mul_lt_mul_left _ _ H _,
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have H' : a⁻¹ * (a * b) < a⁻¹ * (a * c), from ordered_mul_ab_group.mul_lt_mul_left _ _ H _,
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by rewrite *inv_mul_cancel_left at H'; exact H'
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definition ordered_mul_comm_group.to_ordered_mul_cancel_comm_monoid [reducible] [trans_instance]
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[s : ordered_mul_comm_group A] : ordered_mul_cancel_comm_monoid A :=
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definition ordered_mul_ab_group.to_ordered_mul_cancel_comm_monoid [reducible] [trans_instance]
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[s : ordered_mul_ab_group A] : ordered_mul_cancel_comm_monoid A :=
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⦃ ordered_mul_cancel_comm_monoid, s,
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mul_left_cancel := @mul.left_cancel A _,
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mul_right_cancel := @mul.right_cancel A _,
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le_of_mul_le_mul_left := @ordered_mul_comm_group.le_of_mul_le_mul_left A _,
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lt_of_mul_lt_mul_left := @ordered_mul_comm_group.lt_of_mul_lt_mul_left A _⦄
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le_of_mul_le_mul_left := @ordered_mul_ab_group.le_of_mul_le_mul_left A _,
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lt_of_mul_lt_mul_left := @ordered_mul_ab_group.lt_of_mul_lt_mul_left A _⦄
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definition ordered_comm_group.to_ordered_cancel_comm_monoid [reducible] [trans_instance]
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[s : ordered_comm_group A] : ordered_cancel_comm_monoid A :=
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@ordered_mul_comm_group.to_ordered_mul_cancel_comm_monoid A s
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definition ordered_ab_group.to_ordered_cancel_comm_monoid [reducible] [trans_instance]
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[s : ordered_ab_group A] : ordered_cancel_comm_monoid A :=
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@ordered_mul_ab_group.to_ordered_mul_cancel_comm_monoid A s
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section
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variables [s : ordered_comm_group A] (a b c d e : A)
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variables [s : ordered_ab_group A] (a b c d e : A)
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include s
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theorem neg_le_neg {a b : A} (H : a ≤ b) : -b ≤ -a :=
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@ -603,37 +603,37 @@ end
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/- linear ordered group with decidable order -/
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structure decidable_linear_ordered_mul_comm_group [class] (A : Type)
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extends comm_group A, decidable_linear_order A :=
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structure decidable_linear_ordered_mul_ab_group [class] (A : Type)
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extends ab_group A, decidable_linear_order A :=
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(mul_le_mul_left : Π a b, le a b → Π c, le (mul c a) (mul c b))
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(mul_lt_mul_left : Π a b, lt a b → Π c, lt (mul c a) (mul c b))
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definition decidable_linear_ordered_comm_group [class] : Type → Type :=
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decidable_linear_ordered_mul_comm_group
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definition decidable_linear_ordered_ab_group [class] : Type → Type :=
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decidable_linear_ordered_mul_ab_group
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definition add_comm_group_of_decidable_linear_ordered_comm_group [reducible] [trans_instance] (A : Type)
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[H : decidable_linear_ordered_comm_group A] : add_comm_group A :=
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@decidable_linear_ordered_mul_comm_group.to_comm_group A H
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definition add_ab_group_of_decidable_linear_ordered_ab_group [reducible] [trans_instance] (A : Type)
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[H : decidable_linear_ordered_ab_group A] : add_ab_group A :=
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@decidable_linear_ordered_mul_ab_group.to_ab_group A H
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definition decidable_linear_order_of_decidable_linear_ordered_comm_group [reducible]
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[trans_instance] (A : Type) [H : decidable_linear_ordered_comm_group A] :
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definition decidable_linear_order_of_decidable_linear_ordered_ab_group [reducible]
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[trans_instance] (A : Type) [H : decidable_linear_ordered_ab_group A] :
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decidable_linear_order A :=
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@decidable_linear_ordered_mul_comm_group.to_decidable_linear_order A H
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@decidable_linear_ordered_mul_ab_group.to_decidable_linear_order A H
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definition decidable_linear_ordered_mul_comm_group.to_ordered_mul_comm_group [reducible]
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[trans_instance] (A : Type) [s : decidable_linear_ordered_mul_comm_group A] :
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ordered_mul_comm_group A :=
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⦃ ordered_mul_comm_group, s,
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definition decidable_linear_ordered_mul_ab_group.to_ordered_mul_ab_group [reducible]
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[trans_instance] (A : Type) [s : decidable_linear_ordered_mul_ab_group A] :
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ordered_mul_ab_group A :=
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⦃ ordered_mul_ab_group, s,
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le_of_lt := @le_of_lt A _,
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lt_of_le_of_lt := @lt_of_le_of_lt A _,
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lt_of_lt_of_le := @lt_of_lt_of_le A _ ⦄
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definition decidable_linear_ordered_comm_group.to_ordered_comm_group [reducible] [trans_instance]
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(A : Type) [s : decidable_linear_ordered_comm_group A] : ordered_comm_group A :=
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@decidable_linear_ordered_mul_comm_group.to_ordered_mul_comm_group A s
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definition decidable_linear_ordered_ab_group.to_ordered_ab_group [reducible] [trans_instance]
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(A : Type) [s : decidable_linear_ordered_ab_group A] : ordered_ab_group A :=
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@decidable_linear_ordered_mul_ab_group.to_ordered_mul_ab_group A s
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section
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variables [s : decidable_linear_ordered_comm_group A]
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variables [s : decidable_linear_ordered_ab_group A]
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variables {a b c d e : A}
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include s
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@ -34,7 +34,7 @@ structure ordered_semiring (A : Type)
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-- ordered_semiring.to_ordered_mul_cancel_comm_monoid to be an instance -/
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attribute ordered_semiring [class]
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definition add_comm_group_of_ordered_semiring [trans_instance] [reducible] (A : Type)
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definition add_ab_group_of_ordered_semiring [trans_instance] [reducible] (A : Type)
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[H : ordered_semiring A] : semiring A :=
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@ordered_semiring.to_semiring A H
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@ -209,7 +209,7 @@ structure decidable_linear_ordered_semiring [class] (A : Type)
|
|||
|
||||
/- ring structures -/
|
||||
|
||||
structure ordered_ring (A : Type) extends ring A, ordered_mul_comm_group A renaming
|
||||
structure ordered_ring (A : Type) extends ring A, ordered_mul_ab_group A renaming
|
||||
mul→add mul_assoc→add_assoc one→zero one_mul→zero_add mul_one→add_zero inv→neg
|
||||
mul_left_inv→add_left_inv mul_comm→add_comm mul_le_mul_left→add_le_add_left
|
||||
mul_lt_mul_left→add_lt_add_left,
|
||||
|
|
@ -218,16 +218,16 @@ structure ordered_ring (A : Type) extends ring A, ordered_mul_comm_group A renam
|
|||
(mul_pos : Πa b, lt zero a → lt zero b → lt zero (mul a b))
|
||||
|
||||
-- /- we make it a class now (and not as part of the structure) to avoid
|
||||
-- ordered_ring.to_ordered_mul_comm_group to be an instance -/
|
||||
-- ordered_ring.to_ordered_mul_ab_group to be an instance -/
|
||||
attribute ordered_ring [class]
|
||||
|
||||
definition add_comm_group_of_ordered_ring [reducible] [trans_instance] (A : Type)
|
||||
definition add_ab_group_of_ordered_ring [reducible] [trans_instance] (A : Type)
|
||||
[H : ordered_ring A] : ring A :=
|
||||
@ordered_ring.to_ring A H
|
||||
|
||||
definition monoid_of_ordered_ring [reducible] [trans_instance] (A : Type)
|
||||
[H : ordered_ring A] : ordered_comm_group A :=
|
||||
@ordered_ring.to_ordered_mul_comm_group A H
|
||||
[H : ordered_ring A] : ordered_ab_group A :=
|
||||
@ordered_ring.to_ordered_mul_ab_group A H
|
||||
|
||||
definition zero_ne_one_class_of_ordered_ring [reducible] [trans_instance] (A : Type)
|
||||
[H : ordered_ring A] : zero_ne_one_class A :=
|
||||
|
|
@ -486,7 +486,7 @@ end
|
|||
Search on mult_le_cancel_right1 in Rings.thy. -/
|
||||
|
||||
structure decidable_linear_ordered_comm_ring [class] (A : Type) extends linear_ordered_comm_ring A,
|
||||
decidable_linear_ordered_mul_comm_group A renaming
|
||||
decidable_linear_ordered_mul_ab_group A renaming
|
||||
mul→add mul_assoc→add_assoc one→zero one_mul→zero_add mul_one→add_zero inv→neg
|
||||
mul_left_inv→add_left_inv mul_comm→add_comm mul_le_mul_left→add_le_add_left
|
||||
mul_lt_mul_left→add_lt_add_left
|
||||
|
|
@ -500,10 +500,10 @@ definition linear_ordered_comm_ring_of_decidable_linear_ordered_comm_ring [reduc
|
|||
linear_ordered_comm_ring A :=
|
||||
@decidable_linear_ordered_comm_ring.to_linear_ordered_comm_ring A H
|
||||
|
||||
definition decidable_linear_ordered_comm_group_of_decidable_linear_ordered_comm_ring [reducible]
|
||||
definition decidable_linear_ordered_ab_group_of_decidable_linear_ordered_comm_ring [reducible]
|
||||
[trans_instance] (A : Type) [H : decidable_linear_ordered_comm_ring A] :
|
||||
decidable_linear_ordered_comm_group A :=
|
||||
@decidable_linear_ordered_comm_ring.to_decidable_linear_ordered_mul_comm_group A H
|
||||
decidable_linear_ordered_ab_group A :=
|
||||
@decidable_linear_ordered_comm_ring.to_decidable_linear_ordered_mul_ab_group A H
|
||||
|
||||
section
|
||||
variable [s : decidable_linear_ordered_comm_ring A]
|
||||
|
|
|
|||
|
|
@ -186,17 +186,17 @@ end comm_semiring
|
|||
|
||||
/- ring -/
|
||||
|
||||
structure ring (A : Type) extends comm_group A renaming mul→add mul_assoc→add_assoc
|
||||
structure ring (A : Type) extends ab_group A renaming mul→add mul_assoc→add_assoc
|
||||
one→zero one_mul→zero_add mul_one→add_zero inv→neg mul_left_inv→add_left_inv mul_comm→add_comm,
|
||||
monoid A, distrib A
|
||||
|
||||
/- we make it a class now (and not as part of the structure) to avoid
|
||||
ring.to_comm_group to be an instance -/
|
||||
ring.to_ab_group to be an instance -/
|
||||
attribute ring [class]
|
||||
|
||||
definition add_comm_group_of_ring [reducible] [trans_instance] (A : Type)
|
||||
[H : ring A] : add_comm_group A :=
|
||||
@ring.to_comm_group A H
|
||||
definition add_ab_group_of_ring [reducible] [trans_instance] (A : Type)
|
||||
[H : ring A] : add_ab_group A :=
|
||||
@ring.to_ab_group A H
|
||||
|
||||
definition monoid_of_ring [reducible] [trans_instance] (A : Type)
|
||||
[H : ring A] : monoid A :=
|
||||
|
|
@ -485,7 +485,7 @@ theorem mk_cong (op : A → A) (a b : A) (H : a = b) : op a = op b :=
|
|||
|
||||
theorem mk_eq (a : A) : a = a := rfl
|
||||
|
||||
theorem neg_add_neg_eq_of_add_add_eq_zero [s : add_comm_group A] (a b c : A) (H : c + a + b = 0) :
|
||||
theorem neg_add_neg_eq_of_add_add_eq_zero [s : add_ab_group A] (a b c : A) (H : c + a + b = 0) :
|
||||
-a + -b = c :=
|
||||
begin
|
||||
apply add_neg_eq_of_eq_add,
|
||||
|
|
@ -493,26 +493,26 @@ theorem neg_add_neg_eq_of_add_add_eq_zero [s : add_comm_group A] (a b c : A) (H
|
|||
rewrite [add.comm, add.assoc, add.comm b, -add.assoc, H]
|
||||
end
|
||||
|
||||
theorem neg_add_neg_helper [s : add_comm_group A] (a b c : A) (H : a + b = c) : -a + -b = -c :=
|
||||
theorem neg_add_neg_helper [s : add_ab_group A] (a b c : A) (H : a + b = c) : -a + -b = -c :=
|
||||
begin apply iff.mp !neg_eq_neg_iff_eq, rewrite [neg_add, *neg_neg, H] end
|
||||
|
||||
theorem neg_add_pos_eq_of_eq_add [s : add_comm_group A] (a b c : A) (H : b = c + a) : -a + b = c :=
|
||||
theorem neg_add_pos_eq_of_eq_add [s : add_ab_group A] (a b c : A) (H : b = c + a) : -a + b = c :=
|
||||
begin apply neg_add_eq_of_eq_add, rewrite add.comm, exact H end
|
||||
|
||||
theorem neg_add_pos_helper1 [s : add_comm_group A] (a b c : A) (H : b + c = a) : -a + b = -c :=
|
||||
theorem neg_add_pos_helper1 [s : add_ab_group A] (a b c : A) (H : b + c = a) : -a + b = -c :=
|
||||
begin apply neg_add_eq_of_eq_add, apply eq_add_neg_of_add_eq H end
|
||||
|
||||
theorem neg_add_pos_helper2 [s : add_comm_group A] (a b c : A) (H : a + c = b) : -a + b = c :=
|
||||
theorem neg_add_pos_helper2 [s : add_ab_group A] (a b c : A) (H : a + c = b) : -a + b = c :=
|
||||
begin apply neg_add_eq_of_eq_add, rewrite H end
|
||||
|
||||
theorem pos_add_neg_helper [s : add_comm_group A] (a b c : A) (H : b + a = c) : a + b = c :=
|
||||
theorem pos_add_neg_helper [s : add_ab_group A] (a b c : A) (H : b + a = c) : a + b = c :=
|
||||
by rewrite [add.comm, H]
|
||||
|
||||
theorem sub_eq_add_neg_helper [s : add_comm_group A] (t₁ t₂ e w₁ w₂: A) (H₁ : t₁ = w₁)
|
||||
theorem sub_eq_add_neg_helper [s : add_ab_group A] (t₁ t₂ e w₁ w₂: A) (H₁ : t₁ = w₁)
|
||||
(H₂ : t₂ = w₂) (H : w₁ + -w₂ = e) : t₁ - t₂ = e :=
|
||||
by rewrite [sub_eq_add_neg, H₁, H₂, H]
|
||||
|
||||
theorem pos_add_pos_helper [s : add_comm_group A] (a b c h₁ h₂ : A) (H₁ : a = h₁) (H₂ : b = h₂)
|
||||
theorem pos_add_pos_helper [s : add_ab_group A] (a b c h₁ h₂ : A) (H₁ : a = h₁) (H₂ : b = h₂)
|
||||
(H : h₁ + h₂ = c) : a + b = c :=
|
||||
by rewrite [H₁, H₂, H]
|
||||
|
||||
|
|
|
|||
|
|
@ -96,9 +96,9 @@ namespace algebra
|
|||
mul_left_inv := algebra.trunc_mul_left_inv 0 mul_left_inv,
|
||||
is_set_carrier := _⦄
|
||||
|
||||
definition trunc_comm_group [constructor] (mul_comm : ∀a b, mul a b = mul b a)
|
||||
: comm_group (trunc 0 A) :=
|
||||
⦃comm_group, trunc_group, mul_comm := algebra.trunc_mul_comm 0 mul_comm⦄
|
||||
definition trunc_ab_group [constructor] (mul_comm : ∀a b, mul a b = mul b a)
|
||||
: ab_group (trunc 0 A) :=
|
||||
⦃ab_group, trunc_group, mul_comm := algebra.trunc_mul_comm 0 mul_comm⦄
|
||||
|
||||
end
|
||||
end algebra
|
||||
|
|
|
|||
|
|
@ -378,11 +378,11 @@ namespace eq
|
|||
by induction r;induction p;reflexivity
|
||||
|
||||
definition eq_of_square_eq_vconcat {p : a₀₀ = a₂₀} (r : p = p₁₀) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
|
||||
: eq_of_square (r ⬝pv s₁₁) = whisker_right r p₂₁ ⬝ eq_of_square s₁₁ :=
|
||||
: eq_of_square (r ⬝pv s₁₁) = whisker_right p₂₁ r ⬝ eq_of_square s₁₁ :=
|
||||
by induction s₁₁;cases r;reflexivity
|
||||
|
||||
definition eq_of_square_eq_hconcat {p : a₀₀ = a₀₂} (r : p = p₀₁) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
|
||||
: eq_of_square (r ⬝ph s₁₁) = eq_of_square s₁₁ ⬝ (whisker_right r p₁₂)⁻¹ :=
|
||||
: eq_of_square (r ⬝ph s₁₁) = eq_of_square s₁₁ ⬝ (whisker_right p₁₂ r)⁻¹ :=
|
||||
by induction r;reflexivity
|
||||
|
||||
definition eq_of_square_vconcat_eq {p : a₀₂ = a₂₂} (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₁₂ = p)
|
||||
|
|
|
|||
|
|
@ -44,7 +44,7 @@ namespace eq
|
|||
theorem ap_con_left_inv_sq {A B : Type} {a1 a2 : A} (f : A → B) (p : a1 = a2) :
|
||||
square (ap (ap f) (con.left_inv p))
|
||||
(con.left_inv (ap f p))
|
||||
(ap_con f p⁻¹ p ⬝ whisker_right (ap_inv f p) _)
|
||||
(ap_con f p⁻¹ p ⬝ whisker_right _ (ap_inv f p))
|
||||
idp :=
|
||||
by cases p;apply vrefl
|
||||
|
||||
|
|
@ -91,7 +91,7 @@ namespace eq
|
|||
natural_square_tr (ap_compose g f) r
|
||||
|
||||
theorem whisker_right_eq_of_con_inv_eq_idp {p q : a₁ = a₂} (r : p ⬝ q⁻¹ = idp) :
|
||||
whisker_right (eq_of_con_inv_eq_idp r) q⁻¹ ⬝ con.right_inv q = r :=
|
||||
whisker_right q⁻¹ (eq_of_con_inv_eq_idp r) ⬝ con.right_inv q = r :=
|
||||
by induction q; esimp at r; cases r; reflexivity
|
||||
|
||||
theorem ap_eq_of_con_inv_eq_idp (f : A → B) {p q : a₁ = a₂} (r : p ⬝ q⁻¹ = idp)
|
||||
|
|
|
|||
|
|
@ -279,6 +279,7 @@ namespace simple_two_quotient
|
|||
⦃a : A⦄ ⦃r : T a a⦄ (q : Q r) : ap (elim P0 P1 P2) (incl2' q base) = idpath (P0 a) :=
|
||||
!elim_eq_of_rel
|
||||
|
||||
local attribute whisker_right [reducible]
|
||||
protected theorem elim_incl2w {P : Type} (P0 : A → P)
|
||||
(P1 : Π⦃a a' : A⦄ (s : R a a'), P0 a = P0 a')
|
||||
(P2 : Π⦃a : A⦄ ⦃r : T a a⦄ (q : Q r), e_closure.elim P1 r = idp)
|
||||
|
|
|
|||
|
|
@ -192,7 +192,7 @@ end EM
|
|||
open hopf susp
|
||||
namespace EM
|
||||
/- EM1 G is an h-space if G is an abelian group. This allows us to construct K(G,n) for n ≥ 2 -/
|
||||
variables {G : CommGroup} (n : ℕ)
|
||||
variables {G : AbGroup} (n : ℕ)
|
||||
|
||||
definition EM1_mul [unfold 2 3] (x x' : EM1' G) : EM1' G :=
|
||||
begin
|
||||
|
|
@ -260,7 +260,7 @@ namespace EM
|
|||
refine freudenthal_pequiv _ this }
|
||||
end
|
||||
|
||||
definition loopn_EMadd1_pequiv_EM1 (G : CommGroup) (n : ℕ) : EM1 G ≃* Ω[n] (EMadd1 G n) :=
|
||||
definition loopn_EMadd1_pequiv_EM1 (G : AbGroup) (n : ℕ) : EM1 G ≃* Ω[n] (EMadd1 G n) :=
|
||||
begin
|
||||
induction n with n e,
|
||||
{ reflexivity },
|
||||
|
|
@ -270,7 +270,7 @@ namespace EM
|
|||
end
|
||||
|
||||
-- use loopn_EMadd1_pequiv_EM1 in this definition?
|
||||
definition loopn_EMadd1 (G : CommGroup) (n : ℕ) : G ≃* Ω[succ n] (EMadd1 G n) :=
|
||||
definition loopn_EMadd1 (G : AbGroup) (n : ℕ) : G ≃* Ω[succ n] (EMadd1 G n) :=
|
||||
begin
|
||||
induction n with n e,
|
||||
{ apply loop_EM1 },
|
||||
|
|
@ -279,15 +279,15 @@ namespace EM
|
|||
exact e }
|
||||
end
|
||||
|
||||
definition loopn_EMadd1_succ [unfold_full] (G : CommGroup) (n : ℕ) : loopn_EMadd1 G (succ n) ~*
|
||||
definition loopn_EMadd1_succ [unfold_full] (G : AbGroup) (n : ℕ) : loopn_EMadd1 G (succ n) ~*
|
||||
!loopn_succ_in⁻¹ᵉ* ∘* apn (succ n) !loop_EMadd1 ∘* loopn_EMadd1 G n :=
|
||||
by reflexivity
|
||||
|
||||
definition EM_up {G : CommGroup} {X : Type*} {n : ℕ} (e : Ω[succ (succ n)] X ≃* G)
|
||||
definition EM_up {G : AbGroup} {X : Type*} {n : ℕ} (e : Ω[succ (succ n)] X ≃* G)
|
||||
: Ω[succ n] (Ω X) ≃* G :=
|
||||
!loopn_succ_in⁻¹ᵉ* ⬝e* e
|
||||
|
||||
definition is_homomorphism_EM_up {G : CommGroup} {X : Type*} {n : ℕ}
|
||||
definition is_homomorphism_EM_up {G : AbGroup} {X : Type*} {n : ℕ}
|
||||
(e : Ω[succ (succ n)] X ≃* G)
|
||||
(r : Π(p q : Ω[succ (succ n)] X), e (p ⬝ q) = e p * e q)
|
||||
(p q : Ω[succ n] (Ω X)) : EM_up e (p ⬝ q) = EM_up e p * EM_up e q :=
|
||||
|
|
@ -295,7 +295,7 @@ namespace EM
|
|||
refine _ ⬝ !r, apply ap e, esimp, apply apn_con
|
||||
end
|
||||
|
||||
definition EMadd1_pmap [unfold 8] {G : CommGroup} {X : Type*} (n : ℕ)
|
||||
definition EMadd1_pmap [unfold 8] {G : AbGroup} {X : Type*} (n : ℕ)
|
||||
(e : Ω[succ n] X ≃* G)
|
||||
(r : Πp q, e (p ⬝ q) = e p * e q)
|
||||
[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EMadd1 G n →* X :=
|
||||
|
|
@ -307,12 +307,12 @@ namespace EM
|
|||
(psusp.elim (f _ (EM_up e) (is_homomorphism_EM_up e r) _ _)),
|
||||
end
|
||||
|
||||
definition EMadd1_pmap_succ {G : CommGroup} {X : Type*} (n : ℕ) (e : Ω[succ (succ n)] X ≃* G)
|
||||
definition EMadd1_pmap_succ {G : AbGroup} {X : Type*} (n : ℕ) (e : Ω[succ (succ n)] X ≃* G)
|
||||
r [H1 : is_conn (succ n) X] [H2 : is_trunc ((succ n).+1) X] : EMadd1_pmap (succ n) e r =
|
||||
ptrunc.elim ((succ n).+1) (psusp.elim (EMadd1_pmap n (EM_up e) (is_homomorphism_EM_up e r))) :=
|
||||
by reflexivity
|
||||
|
||||
definition loop_EMadd1_pmap {G : CommGroup} {X : Type*} {n : ℕ} (e : Ω[succ (succ n)] X ≃* G)
|
||||
definition loop_EMadd1_pmap {G : AbGroup} {X : Type*} {n : ℕ} (e : Ω[succ (succ n)] X ≃* G)
|
||||
(r : Πp q, e (p ⬝ q) = e p * e q)
|
||||
[H1 : is_conn (succ n) X] [H2 : is_trunc ((succ n).+1) X] :
|
||||
Ω→(EMadd1_pmap (succ n) e r) ∘* loop_EMadd1 G n ~*
|
||||
|
|
@ -329,7 +329,7 @@ namespace EM
|
|||
reflexivity }
|
||||
end
|
||||
|
||||
definition loopn_EMadd1_pmap' {G : CommGroup} {X : Type*} {n : ℕ} (e : Ω[succ n] X ≃* G)
|
||||
definition loopn_EMadd1_pmap' {G : AbGroup} {X : Type*} {n : ℕ} (e : Ω[succ n] X ≃* G)
|
||||
(r : Πp q, e (p ⬝ q) = e p * e q)
|
||||
[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] :
|
||||
Ω→[succ n](EMadd1_pmap n e r) ∘* loopn_EMadd1 G n ~* e⁻¹ᵉ* :=
|
||||
|
|
@ -346,7 +346,7 @@ namespace EM
|
|||
apply pinv_pcompose_cancel_left
|
||||
end
|
||||
|
||||
definition EMadd1_pequiv' {G : CommGroup} {X : Type*} (n : ℕ) (e : Ω[succ n] X ≃* G)
|
||||
definition EMadd1_pequiv' {G : AbGroup} {X : Type*} (n : ℕ) (e : Ω[succ n] X ≃* G)
|
||||
(r : Π(p q : Ω[succ n] X), e (p ⬝ q) = e p * e q)
|
||||
[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EMadd1 G n ≃* X :=
|
||||
begin
|
||||
|
|
@ -365,7 +365,7 @@ namespace EM
|
|||
do 2 exact trivial_homotopy_group_of_is_trunc _ H}
|
||||
end
|
||||
|
||||
definition EMadd1_pequiv {G : CommGroup} {X : Type*} (n : ℕ) (e : πg[n+1] X ≃g G)
|
||||
definition EMadd1_pequiv {G : AbGroup} {X : Type*} (n : ℕ) (e : πg[n+1] X ≃g G)
|
||||
[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EMadd1 G n ≃* X :=
|
||||
begin
|
||||
have is_set (Ω[succ n] X), from !is_set_loopn,
|
||||
|
|
@ -373,7 +373,7 @@ namespace EM
|
|||
intro p q, esimp, exact to_respect_mul e (tr p) (tr q)
|
||||
end
|
||||
|
||||
definition EMadd1_pequiv_succ {G : CommGroup} {X : Type*} (n : ℕ) (e : πag[n+2] X ≃g G)
|
||||
definition EMadd1_pequiv_succ {G : AbGroup} {X : Type*} (n : ℕ) (e : πag[n+2] X ≃g G)
|
||||
[H1 : is_conn (n.+1) X] [H2 : is_trunc (n.+2) X] : EMadd1 G (succ n) ≃* X :=
|
||||
EMadd1_pequiv (succ n) e
|
||||
|
||||
|
|
@ -389,7 +389,7 @@ namespace EM
|
|||
EMadd1_pequiv_succ n !isomorphism.refl
|
||||
|
||||
/- K(G, n) -/
|
||||
definition EM (G : CommGroup) : ℕ → Type*
|
||||
definition EM (G : AbGroup) : ℕ → Type*
|
||||
| 0 := G
|
||||
| (k+1) := EMadd1 G k
|
||||
|
||||
|
|
@ -449,7 +449,7 @@ namespace EM
|
|||
{ reflexivity }
|
||||
end
|
||||
|
||||
definition EMadd1_functor [constructor] {G H : CommGroup} (φ : G →g H) (n : ℕ) :
|
||||
definition EMadd1_functor [constructor] {G H : AbGroup} (φ : G →g H) (n : ℕ) :
|
||||
EMadd1 G n →* EMadd1 H n :=
|
||||
begin
|
||||
induction n with n ψ,
|
||||
|
|
@ -457,7 +457,7 @@ namespace EM
|
|||
{ apply ptrunc_functor, apply psusp_functor, exact ψ }
|
||||
end
|
||||
|
||||
definition EM_functor [unfold 4] {G H : CommGroup} (φ : G →g H) (n : ℕ) :
|
||||
definition EM_functor [unfold 4] {G H : AbGroup} (φ : G →g H) (n : ℕ) :
|
||||
K G n →* K H n :=
|
||||
begin
|
||||
cases n with n,
|
||||
|
|
@ -481,7 +481,7 @@ namespace EM
|
|||
begin intro X, apply ptruncconntype_eq, esimp, exact EM1_pequiv_type X end
|
||||
begin intro G, apply eq_of_isomorphism, apply fundamental_group_EM1 end
|
||||
|
||||
/- Equivalence of CommGroups and pointed n-connected (n+1)-truncated types (n ≥ 1) -/
|
||||
/- Equivalence of AbGroups and pointed n-connected (n+1)-truncated types (n ≥ 1) -/
|
||||
|
||||
open trunc_index
|
||||
definition ptruncconntype_pequiv : Π(n : ℕ) (X Y : (n.+1)-Type*[n])
|
||||
|
|
@ -497,16 +497,16 @@ namespace EM
|
|||
EMadd1 (πag[n+2] X) (succ n) ≃* X :=
|
||||
EMadd1_pequiv_type X n
|
||||
|
||||
definition CommGroup_equiv_ptruncconntype' [constructor] (n : ℕ) :
|
||||
CommGroup ≃ (n + 1 + 1)-Type*[n+1] :=
|
||||
definition AbGroup_equiv_ptruncconntype' [constructor] (n : ℕ) :
|
||||
AbGroup ≃ (n + 1 + 1)-Type*[n+1] :=
|
||||
equiv.MK
|
||||
(λG, ptruncconntype.mk (EMadd1 G (n+1)) _ pt _)
|
||||
(λX, πag[n+2] X)
|
||||
begin intro X, apply ptruncconntype_eq, apply EMadd1_pequiv_type end
|
||||
begin intro G, apply CommGroup_eq_of_isomorphism, exact ghomotopy_group_EMadd1 G (n+1) end
|
||||
begin intro G, apply AbGroup_eq_of_isomorphism, exact ghomotopy_group_EMadd1 G (n+1) end
|
||||
|
||||
definition CommGroup_equiv_ptruncconntype [constructor] (n : ℕ) :
|
||||
CommGroup ≃ (n.+2)-Type*[n.+1] :=
|
||||
CommGroup_equiv_ptruncconntype' n
|
||||
definition AbGroup_equiv_ptruncconntype [constructor] (n : ℕ) :
|
||||
AbGroup ≃ (n.+2)-Type*[n.+1] :=
|
||||
AbGroup_equiv_ptruncconntype' n
|
||||
|
||||
end EM
|
||||
|
|
|
|||
|
|
@ -716,20 +716,20 @@ namespace chain_complex
|
|||
| (fin.mk 2 H) := begin rexact group_homotopy_group n (pfiber f) end
|
||||
| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||||
|
||||
definition comm_group_LES_of_homotopy_groups (n : ℕ) : Π(x : fin (succ 2)),
|
||||
comm_group (LES_of_homotopy_groups (n + 2, x))
|
||||
| (fin.mk 0 H) := proof comm_group_homotopy_group n Y qed
|
||||
| (fin.mk 1 H) := proof comm_group_homotopy_group n X qed
|
||||
| (fin.mk 2 H) := proof comm_group_homotopy_group n (pfiber f) qed
|
||||
definition ab_group_LES_of_homotopy_groups (n : ℕ) : Π(x : fin (succ 2)),
|
||||
ab_group (LES_of_homotopy_groups (n + 2, x))
|
||||
| (fin.mk 0 H) := proof ab_group_homotopy_group n Y qed
|
||||
| (fin.mk 1 H) := proof ab_group_homotopy_group n X qed
|
||||
| (fin.mk 2 H) := proof ab_group_homotopy_group n (pfiber f) qed
|
||||
| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||||
|
||||
definition Group_LES_of_homotopy_groups (x : +3ℕ) : Group.{u} :=
|
||||
Group.mk (LES_of_homotopy_groups (nat.succ (pr1 x), pr2 x))
|
||||
(group_LES_of_homotopy_groups (pr1 x) (pr2 x))
|
||||
|
||||
definition CommGroup_LES_of_homotopy_groups (n : +3ℕ) : CommGroup.{u} :=
|
||||
CommGroup.mk (LES_of_homotopy_groups (pr1 n + 2, pr2 n))
|
||||
(comm_group_LES_of_homotopy_groups (pr1 n) (pr2 n))
|
||||
definition AbGroup_LES_of_homotopy_groups (n : +3ℕ) : AbGroup.{u} :=
|
||||
AbGroup.mk (LES_of_homotopy_groups (pr1 n + 2, pr2 n))
|
||||
(ab_group_LES_of_homotopy_groups (pr1 n) (pr2 n))
|
||||
|
||||
definition homomorphism_LES_of_homotopy_groups_fun : Π(k : +3ℕ),
|
||||
Group_LES_of_homotopy_groups (S k) →g Group_LES_of_homotopy_groups k
|
||||
|
|
|
|||
|
|
@ -110,7 +110,7 @@ namespace freudenthal section
|
|||
apply trunc_pathover,
|
||||
apply eq_pathover_constant_left_id_right,
|
||||
apply square_of_eq,
|
||||
exact whisker_right !con.right_inv (merid a)
|
||||
exact whisker_right (merid a) !con.right_inv
|
||||
end
|
||||
|
||||
definition decode_coh_g (a' : A) : tr (up a') =[merid pt] decode_south (code_merid pt (tr a')) :=
|
||||
|
|
@ -122,7 +122,7 @@ namespace freudenthal section
|
|||
end
|
||||
|
||||
definition decode_coh_lem {A : Type} {a a' : A} (p : a = a')
|
||||
: whisker_right (con.right_inv p) p = inv_con_cancel_right p p ⬝ (idp_con p)⁻¹ :=
|
||||
: whisker_right p (con.right_inv p) = inv_con_cancel_right p p ⬝ (idp_con p)⁻¹ :=
|
||||
by induction p; reflexivity
|
||||
|
||||
theorem decode_coh (a : A) : decode_north =[merid a] decode_south :=
|
||||
|
|
|
|||
|
|
@ -63,7 +63,7 @@ namespace is_trunc
|
|||
end
|
||||
|
||||
/- Corollaries of the LES of homotopy groups -/
|
||||
local attribute comm_group.to_group [coercion]
|
||||
local attribute ab_group.to_group [coercion]
|
||||
local attribute is_equiv_tinverse [instance]
|
||||
open prod chain_complex group fin equiv function is_equiv lift
|
||||
|
||||
|
|
|
|||
|
|
@ -50,7 +50,7 @@ namespace smash
|
|||
gluel' a pt ⬝ gluer' pt b
|
||||
|
||||
definition glue_pt_left (b : B) : glue (Point A) b = gluer' pt b :=
|
||||
whisker_right !con.right_inv _ ⬝ !idp_con
|
||||
whisker_right _ !con.right_inv ⬝ !idp_con
|
||||
|
||||
definition glue_pt_right (a : A) : glue a (Point B) = gluel' a pt :=
|
||||
proof whisker_left _ !con.right_inv qed
|
||||
|
|
@ -82,11 +82,11 @@ namespace smash
|
|||
{ exact gluel pt ▸ Pmk pt pt },
|
||||
{ exact gluer pt ▸ Pmk pt pt },
|
||||
{ refine change_path _ (Pg a pt ⬝o !pathover_tr),
|
||||
refine whisker_right !glue_pt_right _ ⬝ _, esimp, refine !con.assoc ⬝ _,
|
||||
refine whisker_right _ !glue_pt_right ⬝ _, esimp, refine !con.assoc ⬝ _,
|
||||
apply whisker_left, apply con.left_inv },
|
||||
{ refine change_path _ ((Pg pt b)⁻¹ᵒ ⬝o !pathover_tr),
|
||||
refine whisker_right !glue_pt_left⁻² _ ⬝ _,
|
||||
refine whisker_right !inv_con_inv_right _ ⬝ _, refine !con.assoc ⬝ _,
|
||||
refine whisker_right _ !glue_pt_left⁻² ⬝ _,
|
||||
refine whisker_right _ !inv_con_inv_right ⬝ _, refine !con.assoc ⬝ _,
|
||||
apply whisker_left, apply con.left_inv }
|
||||
end
|
||||
|
||||
|
|
|
|||
|
|
@ -174,12 +174,12 @@ namespace is_equiv
|
|||
(left_inv f x)⁻¹ ⬝ ap f⁻¹ q ⬝ left_inv f y
|
||||
|
||||
definition ap_eq_of_fn_eq_fn' {x y : A} (q : f x = f y) : ap f (eq_of_fn_eq_fn' f q) = q :=
|
||||
!ap_con ⬝ whisker_right !ap_con _
|
||||
!ap_con ⬝ whisker_right _ !ap_con
|
||||
⬝ ((!ap_inv ⬝ inverse2 (adj f _)⁻¹)
|
||||
◾ (inverse (ap_compose f f⁻¹ _))
|
||||
◾ (adj f _)⁻¹)
|
||||
⬝ con_ap_con_eq_con_con (right_inv f) _ _
|
||||
⬝ whisker_right !con.left_inv _
|
||||
⬝ whisker_right _ !con.left_inv
|
||||
⬝ !idp_con
|
||||
|
||||
definition eq_of_fn_eq_fn'_ap {x y : A} (q : x = y) : eq_of_fn_eq_fn' f (ap f q) = q :=
|
||||
|
|
|
|||
|
|
@ -631,7 +631,7 @@ namespace eq
|
|||
definition whisker_left [unfold 8] (p : x = y) {q r : y = z} (h : q = r) : p ⬝ q = p ⬝ r :=
|
||||
idp ◾ h
|
||||
|
||||
definition whisker_right [unfold 7] {p q : x = y} (h : p = q) (r : y = z) : p ⬝ r = q ⬝ r :=
|
||||
definition whisker_right [unfold 7] {p q : x = y} (r : y = z) (h : p = q) : p ⬝ r = q ⬝ r :=
|
||||
h ◾ idp
|
||||
|
||||
-- Unwhiskering, a.k.a. cancelling
|
||||
|
|
@ -640,16 +640,16 @@ namespace eq
|
|||
λs, !inv_con_cancel_left⁻¹ ⬝ whisker_left p⁻¹ s ⬝ !inv_con_cancel_left
|
||||
|
||||
definition cancel_right {x y z : A} {p q : x = y} (r : y = z) : (p ⬝ r = q ⬝ r) → (p = q) :=
|
||||
λs, !con_inv_cancel_right⁻¹ ⬝ whisker_right s r⁻¹ ⬝ !con_inv_cancel_right
|
||||
λs, !con_inv_cancel_right⁻¹ ⬝ whisker_right r⁻¹ s ⬝ !con_inv_cancel_right
|
||||
|
||||
-- Whiskering and identity paths.
|
||||
|
||||
definition whisker_right_idp {p q : x = y} (h : p = q) :
|
||||
whisker_right h idp = h :=
|
||||
whisker_right idp h = h :=
|
||||
by induction h; induction p; reflexivity
|
||||
|
||||
definition whisker_right_idp_left [unfold_full] (p : x = y) (q : y = z) :
|
||||
whisker_right idp q = idp :> (p ⬝ q = p ⬝ q) :=
|
||||
whisker_right q idp = idp :> (p ⬝ q = p ⬝ q) :=
|
||||
idp
|
||||
|
||||
definition whisker_left_idp_right [unfold_full] (p : x = y) (q : y = z) :
|
||||
|
|
@ -668,7 +668,7 @@ namespace eq
|
|||
end
|
||||
|
||||
definition con2_idp [unfold_full] {p q : x = y} (h : p = q) :
|
||||
h ◾ idp = whisker_right h idp :> (p ⬝ idp = q ⬝ idp) :=
|
||||
h ◾ idp = whisker_right idp h :> (p ⬝ idp = q ⬝ idp) :=
|
||||
idp
|
||||
|
||||
definition idp_con2 [unfold_full] {p q : x = y} (h : p = q) :
|
||||
|
|
@ -686,16 +686,16 @@ namespace eq
|
|||
by induction d; induction c; induction b;induction a; reflexivity
|
||||
|
||||
definition con2_eq_rl {A : Type} {x y z : A} {p p' : x = y} {q q' : y = z}
|
||||
(a : p = p') (b : q = q') : a ◾ b = whisker_right a q ⬝ whisker_left p' b :=
|
||||
(a : p = p') (b : q = q') : a ◾ b = whisker_right q a ⬝ whisker_left p' b :=
|
||||
by induction b; induction a; reflexivity
|
||||
|
||||
definition con2_eq_lf {A : Type} {x y z : A} {p p' : x = y} {q q' : y = z}
|
||||
(a : p = p') (b : q = q') : a ◾ b = whisker_left p b ⬝ whisker_right a q' :=
|
||||
(a : p = p') (b : q = q') : a ◾ b = whisker_left p b ⬝ whisker_right q' a :=
|
||||
by induction b; induction a; reflexivity
|
||||
|
||||
definition whisker_right_con_whisker_left {x y z : A} {p p' : x = y} {q q' : y = z}
|
||||
(a : p = p') (b : q = q') :
|
||||
(whisker_right a q) ⬝ (whisker_left p' b) = (whisker_left p b) ⬝ (whisker_right a q') :=
|
||||
(whisker_right q a) ⬝ (whisker_left p' b) = (whisker_left p b) ⬝ (whisker_right q' a) :=
|
||||
by induction b; induction a; reflexivity
|
||||
|
||||
-- Structure corresponding to the coherence equations of a bicategory.
|
||||
|
|
@ -704,13 +704,13 @@ namespace eq
|
|||
definition pentagon {v w x y z : A} (p : v = w) (q : w = x) (r : x = y) (s : y = z) :
|
||||
whisker_left p (con.assoc' q r s)
|
||||
⬝ con.assoc' p (q ⬝ r) s
|
||||
⬝ whisker_right (con.assoc' p q r) s
|
||||
⬝ whisker_right s (con.assoc' p q r)
|
||||
= con.assoc' p q (r ⬝ s) ⬝ con.assoc' (p ⬝ q) r s :=
|
||||
by induction s;induction r;induction q;induction p;reflexivity
|
||||
|
||||
-- The 3-cell witnessing the left unit triangle.
|
||||
definition triangulator (p : x = y) (q : y = z) :
|
||||
con.assoc' p idp q ⬝ whisker_right (con_idp p) q = whisker_left p (idp_con q) :=
|
||||
con.assoc' p idp q ⬝ whisker_right q (con_idp p) = whisker_left p (idp_con q) :=
|
||||
by induction q; induction p; reflexivity
|
||||
|
||||
definition eckmann_hilton (p q : idp = idp :> a = a) : p ⬝ q = q ⬝ p :=
|
||||
|
|
|
|||
|
|
@ -158,7 +158,7 @@ namespace arrow
|
|||
(p (f⁻¹ b))⁻¹],
|
||||
apply eq_of_square,
|
||||
refine vconcat_eq _
|
||||
(whisker_right (ap_inv β (right_inv f b)) (p (f⁻¹ b))⁻¹)⁻¹,
|
||||
(whisker_right (p (f⁻¹ b))⁻¹ (ap_inv β (right_inv f b)))⁻¹,
|
||||
refine vconcat_eq _
|
||||
(con_inv (p (f⁻¹ b)) (ap β (right_inv f b))),
|
||||
refine vconcat_eq _
|
||||
|
|
|
|||
|
|
@ -28,7 +28,7 @@ namespace eq
|
|||
end
|
||||
|
||||
theorem whisker_right_con_right (q : a₂ = a₃) (r : p = p') (s : p' = p'')
|
||||
: whisker_right (r ⬝ s) q = whisker_right r q ⬝ whisker_right s q :=
|
||||
: whisker_right q (r ⬝ s) = whisker_right q r ⬝ whisker_right q s :=
|
||||
begin
|
||||
induction q, induction r, induction s, reflexivity
|
||||
end
|
||||
|
|
@ -40,7 +40,7 @@ namespace eq
|
|||
end
|
||||
|
||||
theorem whisker_right_con_left {p p' : a₁ = a₂} (q : a₂ = a₃) (q' : a₃ = a₄) (r : p = p')
|
||||
: whisker_right r (q ⬝ q') = !con.assoc' ⬝ whisker_right (whisker_right r q) q' ⬝ !con.assoc :=
|
||||
: whisker_right (q ⬝ q') r = !con.assoc' ⬝ whisker_right q' (whisker_right q r) ⬝ !con.assoc :=
|
||||
begin
|
||||
induction q', induction q, induction r, induction p, reflexivity
|
||||
end
|
||||
|
|
@ -56,7 +56,7 @@ namespace eq
|
|||
by induction r; reflexivity
|
||||
|
||||
theorem whisker_right_inv {p p' : a₁ = a₂} (q : a₂ = a₃) (r : p = p')
|
||||
: whisker_right r⁻¹ q = (whisker_right r q)⁻¹ :=
|
||||
: whisker_right q r⁻¹ = (whisker_right q r)⁻¹ :=
|
||||
by induction r; reflexivity
|
||||
|
||||
theorem ap_eq_apd10 {B : A → Type} {f g : Πa, B a} (p : f = g) (a : A) :
|
||||
|
|
@ -75,7 +75,7 @@ namespace eq
|
|||
by induction p;reflexivity
|
||||
|
||||
theorem ap_con_left_inv (f : A → B) (p : a₁ = a₂)
|
||||
: ap_con f p⁻¹ p ⬝ whisker_right (ap_inv f p) _ ⬝ con.left_inv (ap f p)
|
||||
: ap_con f p⁻¹ p ⬝ whisker_right _ (ap_inv f p) ⬝ con.left_inv (ap f p)
|
||||
= ap (ap f) (con.left_inv p) :=
|
||||
by induction p;reflexivity
|
||||
|
||||
|
|
@ -282,7 +282,7 @@ namespace eq
|
|||
equiv.mk _ !is_equiv_whisker_left
|
||||
|
||||
definition is_equiv_whisker_right [constructor] {p q : a₁ = a₂} (r : a₂ = a₃)
|
||||
: is_equiv (λs, whisker_right s r : p = q → p ⬝ r = q ⬝ r) :=
|
||||
: is_equiv (λs, whisker_right r s : p = q → p ⬝ r = q ⬝ r) :=
|
||||
begin
|
||||
fapply adjointify,
|
||||
{intro s, apply (!cancel_right s)},
|
||||
|
|
|
|||
|
|
@ -164,10 +164,10 @@ namespace is_equiv
|
|||
intro q, apply eq_of_homotopy, intro c,
|
||||
unfold inv_homotopy_of_homotopy_post,
|
||||
unfold homotopy_of_inv_homotopy_post,
|
||||
apply trans (whisker_right
|
||||
apply trans (whisker_right (left_inv f (α c))
|
||||
(ap_con f⁻¹ (right_inv f (β c))⁻¹ (ap f (q c))
|
||||
⬝ whisker_right (ap_inv f⁻¹ (right_inv f (β c)))
|
||||
(ap f⁻¹ (ap f (q c)))) (left_inv f (α c))),
|
||||
⬝ whisker_right (ap f⁻¹ (ap f (q c)))
|
||||
(ap_inv f⁻¹ (right_inv f (β c))))),
|
||||
apply inverse, apply eq_bot_of_square,
|
||||
apply eq_hconcat (adj_inv f (β c))⁻¹,
|
||||
apply eq_vconcat (ap_compose f⁻¹ f (q c))⁻¹,
|
||||
|
|
|
|||
|
|
@ -299,11 +299,11 @@ lemma madd_left_inv : Π i : fin (succ n), madd (minv i) i = fin.zero n
|
|||
| (mk iv ilt) := eq_of_veq (by
|
||||
rewrite [val_madd, ↑minv, mod_add_mod, nat.sub_add_cancel (le_of_lt ilt), mod_self])
|
||||
|
||||
definition madd_is_comm_group [instance] : add_comm_group (fin (succ n)) :=
|
||||
comm_group.mk madd _ madd_assoc (fin.zero n) zero_madd madd_zero minv madd_left_inv madd_comm
|
||||
definition madd_is_ab_group [instance] : add_ab_group (fin (succ n)) :=
|
||||
ab_group.mk madd _ madd_assoc (fin.zero n) zero_madd madd_zero minv madd_left_inv madd_comm
|
||||
|
||||
definition gfin (n : ℕ) [H : is_succ n] : AddCommGroup.{0} :=
|
||||
by induction H with n; exact AddCommGroup.mk (fin (succ n)) _
|
||||
definition gfin (n : ℕ) [H : is_succ n] : AddAbGroup.{0} :=
|
||||
by induction H with n; exact AddAbGroup.mk (fin (succ n)) _
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end madd
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@ -24,10 +24,10 @@ namespace int
|
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||||
notation `gℤ` := group_integers
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|
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definition CommGroup_int [reducible] [constructor] : AddCommGroup :=
|
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AddCommGroup.mk ℤ _
|
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definition AbGroup_int [reducible] [constructor] : AddAbGroup :=
|
||||
AddAbGroup.mk ℤ _
|
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|
||||
notation `agℤ` := CommGroup_int
|
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notation `agℤ` := AbGroup_int
|
||||
end
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||||
|
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definition is_equiv_succ [constructor] [instance] : is_equiv succ :=
|
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|
|
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@ -383,7 +383,7 @@ namespace pointed
|
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(respect_pt g))
|
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... ≃ Σ(p : pmap.to_fun f = pmap.to_fun g), respect_pt f = ap10 p pt ⬝ respect_pt g
|
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: sigma_equiv_sigma_right
|
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(λp, equiv_eq_closed_right _ (whisker_right (ap_eq_apd10 p _) _))
|
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(λp, equiv_eq_closed_right _ (whisker_right _ (ap_eq_apd10 p _)))
|
||||
... ≃ Σ(p : pmap.to_fun f ~ pmap.to_fun g), respect_pt f = p pt ⬝ respect_pt g
|
||||
: sigma_equiv_sigma_left' eq_equiv_homotopy
|
||||
... ≃ Σ(p : pmap.to_fun f ~ pmap.to_fun g), p pt ⬝ respect_pt g = respect_pt f
|
||||
|
|
@ -448,7 +448,7 @@ namespace pointed
|
|||
begin
|
||||
fconstructor,
|
||||
{ intro p, esimp, refine !con.assoc ⬝ _ ⬝ !con_inv⁻¹, apply whisker_left,
|
||||
refine whisker_right !ap_inv _ ⬝ _ ⬝ !con_inv⁻¹, apply whisker_left,
|
||||
refine whisker_right _ !ap_inv ⬝ _ ⬝ !con_inv⁻¹, apply whisker_left,
|
||||
exact !inv_inv⁻¹},
|
||||
{ induction B with B b, induction f with f pf, esimp at *, induction pf, reflexivity},
|
||||
end
|
||||
|
|
|
|||
|
|
@ -764,8 +764,8 @@ namespace trunc
|
|||
begin
|
||||
fapply phomotopy.mk,
|
||||
{ apply trunc_functor_compose},
|
||||
{ esimp, refine !idp_con ⬝ _, refine whisker_right !ap_compose'⁻¹ᵖ _ ⬝ _,
|
||||
esimp, refine whisker_right (ap_compose' tr g _) _ ⬝ _, exact !ap_con⁻¹},
|
||||
{ esimp, refine !idp_con ⬝ _, refine whisker_right _ !ap_compose'⁻¹ᵖ ⬝ _,
|
||||
esimp, refine whisker_right _ (ap_compose' tr g _) ⬝ _, exact !ap_con⁻¹},
|
||||
end
|
||||
|
||||
definition ptrunc_functor_pid [constructor] (X : Type*) (n : ℕ₋₂) :
|
||||
|
|
@ -851,7 +851,7 @@ namespace trunc
|
|||
begin
|
||||
fapply phomotopy.mk,
|
||||
{ intro x, induction x with a, reflexivity },
|
||||
{ esimp, exact !idp_con ⬝ whisker_right !ap_compose _ },
|
||||
{ esimp, exact !idp_con ⬝ whisker_right _ !ap_compose },
|
||||
end
|
||||
|
||||
end trunc open trunc
|
||||
|
|
|
|||
|
|
@ -8,7 +8,7 @@ protected definition my_decode {x : circle} (c : circle.code x) : base = x :=
|
|||
begin
|
||||
induction x,
|
||||
{ revert c, exact power loop },
|
||||
{ apply arrow_pathover_left, intro b, apply concato_eq, apply pathover_eq_r,
|
||||
{ apply arrow_pathover_left, intro b, apply concato_eq, apply eq_pathover_r,
|
||||
rewrite [power_con,transport_code_loop]},
|
||||
end
|
||||
|
||||
|
|
@ -16,6 +16,6 @@ protected definition my_decode' {x : circle} : circle.code x → base = x :=
|
|||
begin
|
||||
induction x,
|
||||
{ exact power loop},
|
||||
{ apply arrow_pathover_left, intro b, apply concato_eq, apply pathover_eq_r,
|
||||
{ apply arrow_pathover_left, intro b, apply concato_eq, apply eq_pathover_r,
|
||||
rewrite [power_con,transport_code_loop]},
|
||||
end
|
||||
|
|
|
|||
|
|
@ -15,6 +15,6 @@ protected definition my_decode {x : circle} : my_code x → base = x :=
|
|||
begin
|
||||
induction x,
|
||||
{ exact power loop},
|
||||
{ apply arrow_pathover_left, intro b, apply concato_eq, apply pathover_eq_r,
|
||||
{ apply arrow_pathover_left, intro b, apply concato_eq, apply eq_pathover_r,
|
||||
xrewrite [power_con, transport_code_loop]},
|
||||
end
|
||||
|
|
|
|||
|
|
@ -2,7 +2,7 @@ open eq tactic
|
|||
open eq (rec_on)
|
||||
|
||||
definition concat_whisker2 {A} {x y z : A} (p p' : x = y) (q q' : y = z) (a : p = p') (b : q = q') :
|
||||
(whisker_right a q) ⬝ (whisker_left p' b) = (whisker_left p b) ⬝ (whisker_right a q') :=
|
||||
(whisker_right q a) ⬝ (whisker_left p' b) = (whisker_left p b) ⬝ (whisker_right q' a) :=
|
||||
begin
|
||||
apply (rec_on b),
|
||||
apply (rec_on a),
|
||||
|
|
|
|||
|
|
@ -3,7 +3,7 @@ open algebra
|
|||
|
||||
constant f {A : Type} : A → A → A
|
||||
|
||||
theorem test1 {A : Type} [s : add_comm_group A] (a b c : A) : f (a + 0) (f (a + 0) (a + 0)) = f a (f (0 + a) a) :=
|
||||
theorem test1 {A : Type} [s : add_ab_group A] (a b c : A) : f (a + 0) (f (a + 0) (a + 0)) = f a (f (0 + a) a) :=
|
||||
begin
|
||||
rewrite [
|
||||
add_zero at {1, 3}, -- rewrite 1st and 3rd occurrences
|
||||
|
|
@ -12,7 +12,7 @@ end
|
|||
|
||||
axiom Ax {A : Type} [s₁ : has_mul A] [s₂ : has_one A] (a : A) : f (a * 1) (a * 1) = 1
|
||||
|
||||
theorem test2 {A : Type} [s : comm_group A] (a b c : A) : f a a = 1 :=
|
||||
theorem test2 {A : Type} [s : ab_group A] (a b c : A) : f a a = 1 :=
|
||||
begin
|
||||
rewrite [-(mul_one a), -- - means apply symmetry, rewrite 0 ==> a * 0 at 1st and 2nd occurrences
|
||||
Ax] -- use Ax as rewrite rule
|
||||
|
|
|
|||
|
|
@ -5,5 +5,5 @@ variables {A : Type} {a1 a2 a3 : A}
|
|||
definition my_transport_eq_l (p : a1 = a2) (q : a1 = a3)
|
||||
: transport (λx, x = a3) p q = p⁻¹ ⬝ q :=
|
||||
begin
|
||||
rewrite transport_eq_l,
|
||||
rewrite eq_transport_l,
|
||||
end
|
||||
|
|
|
|||
Loading…
Reference in a new issue