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refactor(hott): rename apdo to apd

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This commit is contained in:
Floris van Doorn 2016-03-19 11:25:08 -04:00 committed by Leonardo de Moura
parent 80a2e285cb
commit dc37ec954d
23 changed files with 87 additions and 87 deletions
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8
hott/algebra/e_closure.hlean
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@ -171,15 +171,15 @@ section
definition ap_e_closure_elimo_h [unfold 12] {g' : Πb, Q (g b)}
(p : Π⦃a a' : A⦄, R a a' → f a = f a')
(po : Π⦃a a' : A⦄ (s : R a a'), g' (f a) =[p s] g' (f a'))
(q : Π⦃a a' : A⦄ (s : R a a'), apdo g' (p s) = po s)
(t : T a a') : apdo g' (e_closure.elim p t) = e_closure.elimo p po t :=
(q : Π⦃a a' : A⦄ (s : R a a'), apd g' (p s) = po s)
(t : T a a') : apd g' (e_closure.elim p t) = e_closure.elimo p po t :=
begin
induction t with a a' r a a' pp a a' r IH a a' a'' r r' IH₁ IH₂,
apply q,
induction pp, reflexivity,
esimp [e_closure.elim],
exact apdo_inv g' (e_closure.elim p r) ⬝ IH⁻²ᵒ,
exact apdo_con g' (e_closure.elim p r) (e_closure.elim p r') ⬝ (IH₁ ◾o IH₂)
exact apd_inv g' (e_closure.elim p r) ⬝ IH⁻²ᵒ,
exact apd_con g' (e_closure.elim p r) (e_closure.elim p r') ⬝ (IH₁ ◾o IH₂)
end
theorem e_closure_elimo_ap {g' : Π(a : A), Q (g (f a))}

24
hott/cubical/pathover2.hlean
View file

@ -37,8 +37,8 @@ namespace eq
: pathover_of_tr_eq r = pathover_tr p b ⬝o pathover_idp_of_eq r :=
by induction r; induction p; reflexivity
definition apo011_eq_apo11_apdo (f : Πa, B a → A') (p : a = a₂) (q : b =[p] b₂)
: apo011 f p q = eq_of_pathover (apo11 (apdo f p) q) :=
definition apo011_eq_apo11_apd (f : Πa, B a → A') (p : a = a₂) (q : b =[p] b₂)
: apo011 f p q = eq_of_pathover (apo11 (apd f p) q) :=
by induction q; reflexivity
definition change_path_con (q : p = p') (q' : p' = p'') (r : b =[p] b₂) :
@ -92,23 +92,23 @@ namespace eq
{ intro r, induction q, reflexivity},
end
definition apdo_ap {B : A' → Type} (g : Πb, B b) (f : A → A') (p : a = a₂)
: apdo g (ap f p) = pathover_ap B f (apdo (λx, g (f x)) p) :=
definition apd_ap {B : A' → Type} (g : Πb, B b) (f : A → A') (p : a = a₂)
: apd g (ap f p) = pathover_ap B f (apd (λx, g (f x)) p) :=
by induction p; reflexivity
definition apdo_eq_apdo_ap {B : A' → Type} (g : Πb, B b) (f : A → A') (p : a = a₂)
: apdo (λx, g (f x)) p = pathover_of_pathover_ap B f (apdo g (ap f p)) :=
definition apd_eq_apd_ap {B : A' → Type} (g : Πb, B b) (f : A → A') (p : a = a₂)
: apd (λx, g (f x)) p = pathover_of_pathover_ap B f (apd g (ap f p)) :=
by induction p; reflexivity
definition ap_compose_ap02_constant {A B C : Type} {a a' : A} (p : a = a') (b : B) (c : C) :
ap_compose (λc, b) (λa, c) p ⬝ ap02 (λc, b) (ap_constant p c) = ap_constant p b :=
by induction p; reflexivity
theorem apdo_constant (b : B'' a') (p : a = a) :
pathover_ap B'' (λa, a') (apdo (λa, b) p) = change_path (ap_constant p a')⁻¹ idpo :=
theorem apd_constant (b : B'' a') (p : a = a) :
pathover_ap B'' (λa, a') (apd (λa, b) p) = change_path (ap_constant p a')⁻¹ idpo :=
begin
rewrite [apdo_eq_apdo_ap _ _ p],
let y := !change_path_of_pathover (apdo (apdo id) (ap_constant p b))⁻¹ᵒ,
rewrite [apd_eq_apd_ap _ _ p],
let y := !change_path_of_pathover (apd (apd id) (ap_constant p b))⁻¹ᵒ,
rewrite -y, esimp,
refine !pathover_ap_pathover_of_pathover_ap ⬝ _,
rewrite pathover_ap_change_path,
@ -117,8 +117,8 @@ namespace eq
apply ap_compose_ap02_constant
end
definition apdo_change_path {B : A → Type} {a a₂ : A} (f : Πa, B a) {p p' : a = a₂} (s : p = p')
: apdo f p' = change_path s (apdo f p) :=
definition apd_change_path {B : A → Type} {a a₂ : A} (f : Πa, B a) {p p' : a = a₂} (s : p = p')
: apd f p' = change_path s (apd f p) :=
by induction s; reflexivity
definition cono_invo_eq_idpo {q q' : b =[p] b₂} (r : q = q')

6
hott/cubical/squareover.hlean
View file

@ -49,7 +49,7 @@ namespace eq
definition idso [reducible] [constructor] := @squareover.idsquareo A B a₀₀ b₀₀
definition apds (f : Πa, B a) (s : square p₁₀ p₁₂ p₀₁ p₂₁)
: squareover B s (apdo f p₁₀) (apdo f p₁₂) (apdo f p₀₁) (apdo f p₂₁) :=
: squareover B s (apd f p₁₀) (apd f p₁₂) (apd f p₀₁) (apd f p₂₁) :=
square.rec_on s idso
definition vrflo : squareover B vrfl q₁₀ q₁₀ idpo idpo :=
@ -244,7 +244,7 @@ namespace eq
/- A version of eq_pathover where the type of the equality also varies -/
definition eq_pathover_dep {f g : Πa, B a} {p : a = a'} {q : f a = g a}
{r : f a' = g a'} (s : squareover B hrfl (pathover_idp_of_eq q) (pathover_idp_of_eq r)
(apdo f p) (apdo g p)) : q =[p] r :=
(apd f p) (apd g p)) : q =[p] r :=
begin
induction p, apply pathover_idp_of_eq, apply eq_of_vdeg_square, exact square_of_squareover_ids s
end
@ -256,7 +256,7 @@ namespace eq
(r : pathover B (b a') (q a') (b₂ a'))
(r₂ : pathover B (b a₂') (q a₂') (b₂ a₂'))
(s : squareover B (natural_square_tr q p) r r₂
(pathover_ap B f (apdo b p)) (pathover_ap B g (apdo b₂ p)))
(pathover_ap B f (apd b p)) (pathover_ap B g (apd b₂ p)))
: pathover (λa, pathover B (b a) (q a) (b₂ a)) r p r₂ :=
begin
induction p, esimp at s, apply pathover_idp_of_eq, apply eq_of_vdeg_squareover, exact s

6
hott/eq2.hlean
View file

@ -14,7 +14,7 @@ namespace eq
theorem ap_is_constant_eq (p : Πx, f x = b) (q : a = a') :
ap_is_constant p q =
eq_con_inv_of_con_eq ((eq_of_square (square_of_pathover (apdo p q)))⁻¹ ⬝
eq_con_inv_of_con_eq ((eq_of_square (square_of_pathover (apd p q)))⁻¹ ⬝
whisker_left (p a) (ap_constant q b)) :=
begin
induction q, esimp, generalize (p a), intro p, cases p, apply idpath idp
@ -105,9 +105,9 @@ namespace eq
: eq_of_con_inv_eq_idp (r ◾ inverse2 s ⬝ t) = r ⬝ eq_of_con_inv_eq_idp t ⬝ s⁻¹ :=
by induction s;induction r;induction q;reflexivity
definition naturality_apdo_eq {A : Type} {B : A → Type} {a a₂ : A} {f g : Πa, B a}
definition naturality_apd_eq {A : Type} {B : A → Type} {a a₂ : A} {f g : Πa, B a}
(H : f ~ g) (p : a = a₂)
: apdo f p = concato_eq (eq_concato (H a) (apdo g p)) (H a₂)⁻¹ :=
: apd f p = concato_eq (eq_concato (H a) (apd g p)) (H a₂)⁻¹ :=
begin
induction p, esimp,
generalizes [H a, g a], intro ga Ha, induction Ha,

4
hott/hit/coeq.hlean
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@ -44,7 +44,7 @@ parameters {A B : Type.{u}} (f g : A → B)
theorem rec_cp {P : coeq → Type} (P_i : Π(x : B), P (coeq_i x))
(Pcp : Π(x : A), P_i (f x) =[cp x] P_i (g x))
(x : A) : apdo (rec P_i Pcp) (cp x) = Pcp x :=
(x : A) : apd (rec P_i Pcp) (cp x) = Pcp x :=
!rec_eq_of_rel
protected definition elim {P : Type} (P_i : B → P)
@ -59,7 +59,7 @@ parameters {A B : Type.{u}} (f g : A → B)
(x : A) : ap (elim P_i Pcp) (cp x) = Pcp x :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (cp x)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_cp],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑elim,rec_cp],
end
protected definition elim_type (P_i : B → Type)

8
hott/hit/colimit.hlean
View file

@ -50,7 +50,7 @@ section
theorem rec_cglue {P : colimit → Type}
(Pincl : Π⦃i : I⦄ (x : A i), P (ι x))
(Pglue : Π(j : J) (x : A (dom j)), Pincl (f j x) =[cglue j x] Pincl x)
{j : J} (x : A (dom j)) : apdo (rec Pincl Pglue) (cglue j x) = Pglue j x :=
{j : J} (x : A (dom j)) : apd (rec Pincl Pglue) (cglue j x) = Pglue j x :=
!rec_eq_of_rel
protected definition elim {P : Type} (Pincl : Π⦃i : I⦄ (x : A i), P)
@ -68,7 +68,7 @@ section
{j : J} (x : A (dom j)) : ap (elim Pincl Pglue) (cglue j x) = Pglue j x :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (cglue j x)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_cglue],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑elim,rec_cglue],
end
protected definition elim_type (Pincl : Π⦃i : I⦄ (x : A i), Type)
@ -141,7 +141,7 @@ section
theorem rec_glue {P : seq_colim → Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P (sι a))
(Pglue : Π⦃n : ℕ⦄ (a : A n), Pincl (f a) =[glue a] Pincl a) {n : ℕ} (a : A n)
: apdo (rec Pincl Pglue) (glue a) = Pglue a :=
: apd (rec Pincl Pglue) (glue a) = Pglue a :=
!rec_eq_of_rel
protected definition elim {P : Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P)
@ -158,7 +158,7 @@ section
: ap (elim Pincl Pglue) (glue a) = Pglue a :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (glue a)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_glue],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑elim,rec_glue],
end
protected definition elim_type (Pincl : Π⦃n : ℕ⦄ (a : A n), Type)

4
hott/hit/pushout.hlean
View file

@ -51,7 +51,7 @@ parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
theorem rec_glue {P : pushout → Type} (Pinl : Π(x : BL), P (inl x))
(Pinr : Π(x : TR), P (inr x)) (Pglue : Π(x : TL), Pinl (f x) =[glue x] Pinr (g x))
(x : TL) : apdo (rec Pinl Pinr Pglue) (glue x) = Pglue x :=
(x : TL) : apd (rec Pinl Pinr Pglue) (glue x) = Pglue x :=
!rec_eq_of_rel
protected definition elim {P : Type} (Pinl : BL → P) (Pinr : TR → P)
@ -67,7 +67,7 @@ parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
: ap (elim Pinl Pinr Pglue) (glue x) = Pglue x :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (glue x)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑pushout.elim,rec_glue],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑pushout.elim,rec_glue],
end
protected definition elim_type (Pinl : BL → Type) (Pinr : TR → Type)

6
hott/hit/quotient.hlean
View file

@ -35,7 +35,7 @@ namespace quotient
: ap (quotient.elim Pc Pp) (eq_of_rel R H) = Pp H :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (eq_of_rel R H)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑quotient.elim,rec_eq_of_rel],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑quotient.elim,rec_eq_of_rel],
end
protected definition rec_prop {A : Type} {R : A → A → Type} {P : quotient R → Type}
@ -89,7 +89,7 @@ namespace quotient
(Σ(f : Π(a : A), C (class_of R a)), Π⦃a a' : A⦄ (H : R a a'), f a =[eq_of_rel R H] f a') :=
begin
fapply equiv.MK,
{ intro f, exact ⟨λa, f (class_of R a), λa a' H, apdo f (eq_of_rel R H)⟩},
{ intro f, exact ⟨λa, f (class_of R a), λa a' H, apd f (eq_of_rel R H)⟩},
{ intro v x, induction v with i p, induction x,
exact (i a),
exact (p H)},
@ -166,7 +166,7 @@ namespace quotient
(c : C a) : ap (elim @Qpt Qeq) (Peq r c) = Qeq r c :=
begin
refine !ap_dpair_eq_dpair ⬝ _,
rewrite [apo011_eq_apo11_apdo, rec_eq_of_rel, ▸*, apo011_arrow_pathover_constant_right,
rewrite [apo011_eq_apo11_apd, rec_eq_of_rel, ▸*, apo011_arrow_pathover_constant_right,
↑elim_type_eq_of_rel', to_right_inv !pathover_equiv_tr_eq, ap_inv],
apply inv_con_cancel_right
end

2
hott/hit/refl_quotient.hlean
View file

@ -47,7 +47,7 @@ section
definition rec_req_of_rel {P : Type} {P : refl_quotient → Type} (Pc : Π(a : A), P (rclass_of a))
(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[req_of_rel H] Pc a')
(Pr : Π(a : A), change_path (pρ a) (Pp (ρ a)) = idpo) ⦃a a' : A⦄ (r : R a a')
: apdo (rec Pc Pp Pr) (req_of_rel r) = Pp r :=
: apd (rec Pc Pp Pr) (req_of_rel r) = Pp r :=
!rec_incl1
protected definition elim {P : Type} (Pc : Π(a : A), P)

4
hott/hit/set_quotient.hlean
View file

@ -44,7 +44,7 @@ section
theorem rec_eq_of_rel {P : set_quotient → Type} [Pt : Πaa, is_set (P aa)]
(Pc : Π(a : A), P (class_of a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[eq_of_rel H] Pc a')
{a a' : A} (H : R a a') : apdo (rec Pc Pp) (eq_of_rel H) = Pp H :=
{a a' : A} (H : R a a') : apd (rec Pc Pp) (eq_of_rel H) = Pp H :=
!is_set.elimo
protected definition elim {P : Type} [Pt : is_set P] (Pc : A → P)
@ -60,7 +60,7 @@ section
: ap (elim Pc Pp) (eq_of_rel H) = Pp H :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (eq_of_rel H)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_eq_of_rel],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑elim,rec_eq_of_rel],
end
protected definition rec_prop {P : set_quotient → Type} [Pt : Πaa, is_prop (P aa)]

24
hott/hit/two_quotient.hlean
View file

@ -76,7 +76,7 @@ namespace simple_two_quotient
protected theorem rec_e {P : C → Type}
(Pj : Πa, P (j a)) (Pa : Π⦃a : A⦄ ⦃r : T a a⦄ (q : Q r), P (pre_aux q))
(Pe : Π⦃a a' : A⦄ (s : R a a'), Pj a =[e s] Pj a') ⦃a a' : A⦄ (s : R a a')
: apdo (pre_rec Pj Pa Pe) (e s) = Pe s :=
: apd (pre_rec Pj Pa Pe) (e s) = Pe s :=
!rec_eq_of_rel
protected theorem elim_e {P : Type} (Pj : A → P) (Pa : Π⦃a : A⦄ ⦃r : T a a⦄, Q r → P)
@ -84,7 +84,7 @@ namespace simple_two_quotient
: ap (pre_elim Pj Pa Pe) (e s) = Pe s :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (e s)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑pre_elim,rec_e],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑pre_elim,rec_e],
end
protected definition elim_et {P : Type} (Pj : A → P) (Pa : Π⦃a : A⦄ ⦃r : T a a⦄, Q r → P)
@ -95,7 +95,7 @@ namespace simple_two_quotient
protected definition rec_et {P : C → Type}
(Pj : Πa, P (j a)) (Pa : Π⦃a : A⦄ ⦃r : T a a⦄ (q : Q r), P (pre_aux q))
(Pe : Π⦃a a' : A⦄ (s : R a a'), Pj a =[e s] Pj a') ⦃a a' : A⦄ (t : T a a')
: apdo (pre_rec Pj Pa Pe) (et t) = e_closure.elimo e Pe t :=
: apd (pre_rec Pj Pa Pe) (et t) = e_closure.elimo e Pe t :=
ap_e_closure_elimo_h e Pe (rec_e Pj Pa Pe) t
inductive simple_two_quotient_rel : C → C → Type :=
@ -151,9 +151,9 @@ namespace simple_two_quotient
refine eq_hconcato _ _, apply _,
{ transitivity _,
{ apply ap (pathover_ap _ _),
transitivity _, apply apdo_compose2 (pre_rec P0 _ _) (f q) loop,
transitivity _, apply apd_compose2 (pre_rec P0 _ _) (f q) loop,
apply ap (pathover_of_pathover_ap _ _),
transitivity _, apply apdo_change_path, exact !elim_loop⁻¹,
transitivity _, apply apd_change_path, exact !elim_loop⁻¹,
transitivity _,
apply ap (change_path _),
transitivity _, apply rec_et,
@ -171,7 +171,7 @@ namespace simple_two_quotient
rewrite [↑incl2, con_inv], transitivity _, exact !con.assoc⁻¹,
rewrite [inv_con_cancel_right, ↑incl2w, ↑ap02, +con_inv, +ap_inv, +inv_inv, -+con.assoc,
+con_inv_cancel_right], reflexivity},
rewrite [change_path_con, apdo_constant],
rewrite [change_path_con, apd_constant],
apply squareover_change_path_left, apply squareover_change_path_right',
apply squareover_change_path_left,
refine change_square _ vrflo,
@ -189,9 +189,9 @@ namespace simple_two_quotient
(P1 : Π⦃a a' : A⦄ (s : R a a'), P0 a =[incl1 s] P0 a')
(P2 : Π⦃a : A⦄ ⦃r : T a a⦄ (q : Q r),
change_path (incl2 q) (e_closure.elimo incl1 P1 r) = idpo) ⦃a a' : A⦄ (s : R a a')
: apdo (rec P0 P1 P2) (incl1 s) = P1 s :=
: apd (rec P0 P1 P2) (incl1 s) = P1 s :=
begin
unfold [rec, incl1], refine !apdo_ap ⬝ _, esimp, rewrite rec_e,
unfold [rec, incl1], refine !apd_ap ⬝ _, esimp, rewrite rec_e,
apply to_right_inv !pathover_compose
end
@ -199,7 +199,7 @@ namespace simple_two_quotient
(P1 : Π⦃a a' : A⦄ (s : R a a'), P0 a =[incl1 s] P0 a')
(P2 : Π⦃a : A⦄ ⦃r : T a a⦄ (q : Q r),
change_path (incl2 q) (e_closure.elimo incl1 P1 r) = idpo) ⦃a a' : A⦄ (t : T a a')
: apdo (rec P0 P1 P2) (inclt t) = e_closure.elimo incl1 P1 t :=
: apd (rec P0 P1 P2) (inclt t) = e_closure.elimo incl1 P1 t :=
ap_e_closure_elimo_h incl1 P1 (rec_incl1 P0 P1 P2) t
protected definition elim {P : Type} (P0 : A → P)
@ -288,7 +288,7 @@ namespace simple_two_quotient
↑[elim_incltw]],
apply whisker_tl,
rewrite [ap_is_constant_eq],
xrewrite [naturality_apdo_eq (λx, !elim_eq_of_rel) loop],
xrewrite [naturality_apd_eq (λx, !elim_eq_of_rel) loop],
rewrite [↑elim_2,rec_loop,square_of_pathover_concato_eq,square_of_pathover_eq_concato,
eq_of_square_vconcat_eq,eq_of_square_eq_vconcat],
apply eq_vconcat,
@ -390,14 +390,14 @@ namespace two_quotient
(P1 : Π⦃a a' : A⦄ (s : R a a'), P0 a =[incl1 s] P0 a')
(P2 : Π⦃a a' : A⦄ ⦃t t' : T a a'⦄ (q : Q t t'),
change_path (incl2 q) (e_closure.elimo incl1 P1 t) = e_closure.elimo incl1 P1 t')
⦃a a' : A⦄ (s : R a a') : apdo (rec P0 P1 P2) (incl1 s) = P1 s :=
⦃a a' : A⦄ (s : R a a') : apd (rec P0 P1 P2) (incl1 s) = P1 s :=
rec_incl1 _ _ _ s
theorem rec_inclt {P : two_quotient → Type} (P0 : Π(a : A), P (incl0 a))
(P1 : Π⦃a a' : A⦄ (s : R a a'), P0 a =[incl1 s] P0 a')
(P2 : Π⦃a a' : A⦄ ⦃t t' : T a a'⦄ (q : Q t t'),
change_path (incl2 q) (e_closure.elimo incl1 P1 t) = e_closure.elimo incl1 P1 t')
⦃a a' : A⦄ (t : T a a') : apdo (rec P0 P1 P2) (inclt t) = e_closure.elimo incl1 P1 t :=
⦃a a' : A⦄ (t : T a a') : apd (rec P0 P1 P2) (inclt t) = e_closure.elimo incl1 P1 t :=
rec_inclt _ _ _ t
protected definition elim {P : Type} (P0 : A → P)

20
hott/homotopy/circle.hlean
View file

@ -42,12 +42,12 @@ namespace circle
theorem rec2_seg1 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : Pb1 =[seg1] Pb2) (Ps2 : Pb1 =[seg2] Pb2)
: apdo (rec2 Pb1 Pb2 Ps1 Ps2) seg1 = Ps1 :=
: apd (rec2 Pb1 Pb2 Ps1 Ps2) seg1 = Ps1 :=
!rec_merid
theorem rec2_seg2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : Pb1 =[seg1] Pb2) (Ps2 : Pb1 =[seg2] Pb2)
: apdo (rec2 Pb1 Pb2 Ps1 Ps2) seg2 = Ps2 :=
: apd (rec2 Pb1 Pb2 Ps1 Ps2) seg2 = Ps2 :=
!rec_merid
definition elim2 {P : Type} (Pb1 Pb2 : P) (Ps1 Ps2 : Pb1 = Pb2) (x : circle) : P :=
@ -61,14 +61,14 @@ namespace circle
: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg1 = Ps1 :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant seg1),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim2,rec2_seg1],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑elim2,rec2_seg1],
end
theorem elim2_seg2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb1 = Pb2)
: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg2 = Ps2 :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant seg2),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim2,rec2_seg2],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑elim2,rec2_seg2],
end
definition elim2_type (Pb1 Pb2 : Type) (Ps1 Ps2 : Pb1 ≃ Pb2) (x : circle) : Type :=
@ -109,9 +109,9 @@ namespace circle
eq.rec_on p idp
theorem rec_loop {P : circle → Type} (Pbase : P base) (Ploop : Pbase =[loop] Pbase) :
apdo (circle.rec Pbase Ploop) loop = Ploop :=
apd (circle.rec Pbase Ploop) loop = Ploop :=
begin
rewrite [↑loop,apdo_con,↑circle.rec,↑circle.rec2_on,↑base,rec2_seg2,apdo_inv,rec2_seg1],
rewrite [↑loop,apd_con,↑circle.rec,↑circle.rec2_on,↑base,rec2_seg2,apd_inv,rec2_seg1],
apply rec_loop_helper
end
@ -127,14 +127,14 @@ namespace circle
ap (circle.elim Pbase Ploop) loop = Ploop :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant loop),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑circle.elim,rec_loop],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑circle.elim,rec_loop],
end
theorem elim_seg1 {P : Type} (Pbase : P) (Ploop : Pbase = Pbase)
: ap (circle.elim Pbase Ploop) seg1 = (tr_constant seg1 Pbase)⁻¹ :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant seg1),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑circle.elim,↑circle.rec],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑circle.elim,↑circle.rec],
rewrite [↑circle.rec2_on,rec2_seg1], apply inverse,
apply pathover_of_eq_tr_constant_inv
end
@ -143,7 +143,7 @@ namespace circle
: ap (circle.elim Pbase Ploop) seg2 = Ploop ⬝ (tr_constant seg1 Pbase)⁻¹ :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant seg2),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑circle.elim,↑circle.rec],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑circle.elim,↑circle.rec],
rewrite [↑circle.rec2_on,rec2_seg2],
assert l : Π(A B : Type)(a a₂ a₂' : A)(b b' : B)(p : a = a₂)(p' : a₂' = a₂)
(q : b = b'),
@ -188,7 +188,7 @@ namespace circle
: (Π(x : S¹), P x) ≃ Σ(p : P base), p =[loop] p :=
begin
fapply equiv.MK,
{ intro f, exact ⟨f base, apdo f loop⟩},
{ intro f, exact ⟨f base, apd f loop⟩},
{ intro v x, induction v with p q, induction x,
{ exact p},
{ exact q}},

4
hott/homotopy/cylinder.hlean
View file

@ -52,7 +52,7 @@ parameters {A B : Type.{u}} (f : A → B)
theorem rec_seg {P : cylinder → Type}
(Pbase : Π(b : B), P (base b)) (Ptop : Π(a : A), P (top a))
(Pseg : Π(a : A), Pbase (f a) =[seg a] Ptop a)
(a : A) : apdo (rec Pbase Ptop Pseg) (seg a) = Pseg a :=
(a : A) : apd (rec Pbase Ptop Pseg) (seg a) = Pseg a :=
!rec_eq_of_rel
protected definition elim {P : Type} (Pbase : B → P) (Ptop : A → P)
@ -68,7 +68,7 @@ parameters {A B : Type.{u}} (f : A → B)
(a : A) : ap (elim Pbase Ptop Pseg) (seg a) = Pseg a :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (seg a)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_seg],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑elim,rec_seg],
end
protected definition elim_type (Pbase : B → Type) (Ptop : A → Type)

4
hott/homotopy/interval.hlean
View file

@ -31,7 +31,7 @@ namespace interval
interval.rec P0 P1 Ps x
theorem rec_seg {P : interval → Type} (P0 : P zero) (P1 : P one) (Ps : P0 =[seg] P1)
: apdo (interval.rec P0 P1 Ps) seg = Ps :=
: apd (interval.rec P0 P1 Ps) seg = Ps :=
!rec_merid
protected definition elim {P : Type} (P0 P1 : P) (Ps : P0 = P1) (x : interval) : P :=
@ -45,7 +45,7 @@ namespace interval
: ap (interval.elim P0 P1 Ps) seg = Ps :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant seg),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑interval.elim,rec_seg],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑interval.elim,rec_seg],
end
protected definition elim_type (P0 P1 : Type) (Ps : P0 ≃ P1) (x : interval) : Type :=

8
hott/homotopy/join.hlean
View file

@ -34,7 +34,7 @@ namespace join
(Pinr : Π(y : B), P (inr y))
(Pglue : Π(x : A)(y : B), Pinl x =[glue x y] Pinr y)
(x : A) (y : B)
: apdo (join.rec Pinl Pinr Pglue) (glue x y) = Pglue x y :=
: apd (join.rec Pinl Pinr Pglue) (glue x y) = Pglue x y :=
!quotient.rec_eq_of_rel
protected definition elim {P : Type} (Pinl : A → P) (Pinr : B → P)
@ -46,7 +46,7 @@ namespace join
: ap (join.elim Pinl Pinr Pglue) (glue x y) = Pglue x y :=
begin
apply equiv.eq_of_fn_eq_fn_inv !(pathover_constant (glue x y)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑join.elim],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑join.elim],
apply join.rec_glue
end
@ -404,7 +404,7 @@ section join_switch
private definition switch_inv_cube_aux2 {A B : Type} {b : B} {f : A → B}
(g : Π a, f a = b) {x y : A} (p : x = y) {sq : square (g x) (g y) (ap f p) idp}
(q : apdo g p = eq_pathover (sq ⬝hp !ap_constant⁻¹)) : square_Flr_ap_idp _ _ = sq :=
(q : apd g p = eq_pathover (sq ⬝hp !ap_constant⁻¹)) : square_Flr_ap_idp _ _ = sq :=
begin
cases p, esimp at *, apply concat, apply inverse, apply vdeg_square_idp,
apply concat, apply ap vdeg_square, exact ap eq_of_pathover_idp q,
@ -448,7 +448,7 @@ section join_switch
private definition natural_square_tr_beta {A B : Type} {f₁ f₂ : A → B}
(p : Π a, f₁ a = f₂ a) {x y : A} (q : x = y) {sq : square (p x) (p y) (ap f₁ q) (ap f₂ q)}
(e : apdo p q = eq_pathover sq) :
(e : apd p q = eq_pathover sq) :
natural_square_tr p q = sq :=
begin
cases q, esimp at *, apply concat, apply inverse, apply vdeg_square_idp,

2
hott/homotopy/red_susp.hlean
View file

@ -47,7 +47,7 @@ section
-- definition rec_merid {P : red_susp → Type} (Pb : P base) (Pm : Π(a : A), Pb =[merid a] Pb)
-- (Pe : Pm pt =[merid_pt] idpo) (a : A)
-- : apdo (rec Pb Pm Pe) (merid a) = Pm a :=
-- : apd (rec Pb Pm Pe) (merid a) = Pm a :=
-- !rec_incl1
-- theorem elim_merid_pt {P : red_susp → Type} (Pb : P base) (Pm : Π(a : A), Pb =[merid a] Pb)

6
hott/homotopy/susp.hlean
View file

@ -39,7 +39,7 @@ namespace susp
theorem rec_merid {P : susp A → Type} (PN : P north) (PS : P south)
(Pm : Π(a : A), PN =[merid a] PS) (a : A)
: apdo (susp.rec PN PS Pm) (merid a) = Pm a :=
: apd (susp.rec PN PS Pm) (merid a) = Pm a :=
!rec_glue
protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS)
@ -54,7 +54,7 @@ namespace susp
: ap (susp.elim PN PS Pm) (merid a) = Pm a :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (merid a)),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑susp.elim,rec_merid],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑susp.elim,rec_merid],
end
protected definition elim_type (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
@ -119,7 +119,7 @@ end susp
/- Flattening lemma -/
namespace susp
open prod prod.ops
section
universe variable u

4
hott/homotopy/torus.hlean
View file

@ -51,12 +51,12 @@ namespace torus
theorem rec_loop1 {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
(Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2)
: apdo (torus.rec Pb Pl1 Pl2 Ps) loop1 = Pl1 :=
: apd (torus.rec Pb Pl1 Pl2 Ps) loop1 = Pl1 :=
!rec_incl1
theorem rec_loop2 {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
(Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2)
: apdo (torus.rec Pb Pl1 Pl2 Ps) loop2 = Pl2 :=
: apd (torus.rec Pb Pl1 Pl2 Ps) loop2 = Pl2 :=
!rec_incl1
protected definition elim {P : Type} (Pb : P) (Pl1 : Pb = Pb) (Pl2 : Pb = Pb)

2
hott/init/hit.hlean
View file

@ -81,7 +81,7 @@ namespace quotient
constant rec_eq_of_rel {A : Type} {R : A → A → Type} {P : quotient R → Type}
(Pc : Π(a : A), P (class_of R a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[eq_of_rel R H] Pc a')
{a a' : A} (H : R a a') : apdo (quotient.rec Pc Pp) (eq_of_rel R H) = Pp H
{a a' : A} (H : R a a') : apd (quotient.rec Pc Pp) (eq_of_rel R H) = Pp H
end quotient
attribute quotient.class_of trunc.tr [constructor]

20
hott/init/pathover.hlean
View file

@ -70,7 +70,7 @@ namespace eq
definition inverseo [unfold 8] (r : b =[p] b₂) : b₂ =[p⁻¹] b :=
pathover.rec_on r idpo
definition apdo [unfold 6] (f : Πa, B a) (p : a = a₂) : f a =[p] f a₂ :=
definition apd [unfold 6] (f : Πa, B a) (p : a = a₂) : f a =[p] f a₂ :=
eq.rec_on p idpo
definition concato_eq [unfold 10] (r : b =[p] b₂) (q : b₂ = b₂') : b =[p] b₂' :=
@ -198,15 +198,15 @@ namespace eq
{ intro q, cases q, reflexivity},
end
definition apdo_con (f : Πa, B a) (p : a = a₂) (q : a₂ = a₃)
: apdo f (p ⬝ q) = apdo f p ⬝o apdo f q :=
definition apd_con (f : Πa, B a) (p : a = a₂) (q : a₂ = a₃)
: apd f (p ⬝ q) = apd f p ⬝o apd f q :=
by cases p; cases q; reflexivity
definition apdo_inv (f : Πa, B a) (p : a = a₂) : apdo f p⁻¹ = (apdo f p)⁻¹ᵒ :=
definition apd_inv (f : Πa, B a) (p : a = a₂) : apd f p⁻¹ = (apd f p)⁻¹ᵒ :=
by cases p; reflexivity
definition apdo_eq_pathover_of_eq_ap (f : A → A') (p : a = a₂) :
apdo f p = pathover_of_eq (ap f p) :=
definition apd_eq_pathover_of_eq_ap (f : A → A') (p : a = a₂) :
apd f p = pathover_of_eq (ap f p) :=
eq.rec_on p idp
definition pathover_of_pathover_tr (q : b =[p ⬝ p₂] p₂ ▸ b₂) : b =[p] b₂ :=
@ -273,12 +273,12 @@ namespace eq
(q : b =[p] b₂) : f b =[p] g b₂ :=
by induction q; exact apo10 r b
definition apdo_compose1 (g : Πa, B a → B' a) (f : Πa, B a) (p : a = a₂)
: apdo (g ∘' f) p = apo g (apdo f p) :=
definition apd_compose1 (g : Πa, B a → B' a) (f : Πa, B a) (p : a = a₂)
: apd (g ∘' f) p = apo g (apd f p) :=
by induction p; reflexivity
definition apdo_compose2 (g : Πa', B'' a') (f : A → A') (p : a = a₂)
: apdo (λa, g (f a)) p = pathover_of_pathover_ap B'' f (apdo g (ap f p)) :=
definition apd_compose2 (g : Πa', B'' a') (f : A → A') (p : a = a₂)
: apd (λa, g (f a)) p = pathover_of_pathover_ap B'' f (apd g (ap f p)) :=
by induction p; reflexivity
definition cono.right_inv_eq (q : b = b')

2
hott/types/sigma.hlean
View file

@ -63,7 +63,7 @@ namespace sigma
ap eq_pr1 r
definition eq2_pr2 {p q : u = v} (r : p = q) : p..2 =[eq2_pr1 r] q..2 :=
!pathover_ap (apdo eq_pr2 r)
!pathover_ap (apd eq_pr2 r)
definition tr_pr1_sigma_eq {B' : A → Type} (p : u.1 = v.1) (q : u.2 =[p] v.2)
: transport (λx, B' x.1) (sigma_eq p q) = transport B' p :=

2
hott/types/univ.hlean
View file

@ -51,7 +51,7 @@ namespace univ
begin
intro f,
have u : ¬¬bool, by exact (λg, g tt),
let H1 := apdo f eq_bnot,
let H1 := apd f eq_bnot,
note H2 := apo10 H1 u,
have p : eq_bnot ▸ u = u, from !is_prop.elim,
rewrite p at H2,

4
tests/lean/extra/755.hlean
View file

@ -31,7 +31,7 @@ section
theorem rec_tr_eq {P : one_step_tr → Type} (Pt : Π(a : A), P (tr a))
(Pe : Π(a a' : A), Pt a =[tr_eq a a'] Pt a') (a a' : A)
: apdo (rec Pt Pe) (tr_eq a a') = Pe a a' :=
: apd (rec Pt Pe) (tr_eq a a') = Pe a a' :=
!rec_eq_of_rel
theorem elim_tr_eq {P : Type} (Pt : A → P)
@ -39,7 +39,7 @@ section
: ap (elim Pt Pe) (tr_eq a a') = Pe a a' :=
begin
apply eq_of_fn_eq_fn_inv !(pathover_constant (tr_eq a a')),
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_tr_eq],
rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑elim,rec_tr_eq],
end
end