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feat(hott/types): add some theorems about operations of 2-paths

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Floris van Doorn 2015-06-23 13:21:04 -04:00
parent ea0f57aef5
commit b94b66243e
3 changed files with 227 additions and 77 deletions
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19
hott/types/cubical/square.hlean
View file

@ -10,7 +10,7 @@ open eq equiv is_equiv
namespace eq
variables {A B : Type} {a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ : A}
variables {A B : Type} {a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ a₁ a₂ a₃ a₄ : A}
/-a₀₀-/ {p₁₀ : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
{p₀₁ : a₀₀ = a₀₂} /-s₁₁-/ {p₂₁ : a₂₀ = a₂₂} /-s₃₁-/ {p₄₁ : a₄₀ = a₄₂}
/-a₀₂-/ {p₁₂ : a₀₂ = a₂₂} /-a₂₂-/ {p₃₂ : a₂₂ = a₄₂} /-a₄₂-/
@ -157,6 +157,9 @@ namespace eq
definition eq_of_vdeg_square [reducible] {p q : a = a'} (s : square p q idp idp) : p = q :=
to_fun !vdeg_square_equiv' s
definition top_deg_square (l : a₁ = a₂) (b : a₂ = a₃) (r : a₄ = a₃) : square (l ⬝ b ⬝ r⁻¹) b l r :=
by induction r;induction b;induction l;constructor
/-
the following two equivalences have as underlying inverse function the functions
hdeg_square and vdeg_square, respectively.
@ -342,4 +345,18 @@ namespace eq
--we can also do the other recursors (tl, tr, bl, br, tbl, tbr, tlr, blr), but let's postpone this until they are needed
/- squares commute with some operations on 2-paths -/
definition square_inv2 {p₁ p₂ p₃ p₄ : a = a'}
{t : p₁ = p₂} {b : p₃ = p₄} {l : p₁ = p₃} {r : p₂ = p₄} (s : square t b l r)
: square (inverse2 t) (inverse2 b) (inverse2 l) (inverse2 r) :=
by induction s;constructor
definition square_con2 {p₁ p₂ p₃ p₄ : a₁ = a₂} {q₁ q₂ q₃ q₄ : a₂ = a₃}
{t₁ : p₁ = p₂} {b₁ : p₃ = p₄} {l₁ : p₁ = p₃} {r₁ : p₂ = p₄}
{t₂ : q₁ = q₂} {b₂ : q₃ = q₄} {l₂ : q₁ = q₃} {r₂ : q₂ = q₄}
(s₁ : square t₁ b₁ l₁ r₁) (s₂ : square t₂ b₂ l₂ r₂)
: square (t₁ ◾ t₂) (b₁ ◾ b₂) (l₁ ◾ l₂) (r₁ ◾ r₂) :=
by induction s₂;induction s₁;constructor
end eq

177
hott/types/eq.hlean
View file

@ -14,49 +14,49 @@ open eq sigma sigma.ops equiv is_equiv equiv.ops
namespace eq
/- Path spaces -/
variables {A B : Type} {a a1 a2 a3 a4 : A} {b b1 b2 : B} {f g : A → B} {h : B → A}
{p p' p'' : a1 = a2}
variables {A B : Type} {a a₁ a₂ a₃ a₄ a' : A} {b b1 b2 : B} {f g : A → B} {h : B → A}
{p p' p'' : a₁ = a₂}
/- The path spaces of a path space are not, of course, determined; they are just the
higher-dimensional structure of the original space. -/
/- some lemmas about whiskering or other higher paths -/
theorem whisker_left_con_right (p : a1 = a2) {q q' q'' : a2 = a3} (r : q = q') (s : q' = q'')
theorem whisker_left_con_right (p : a₁ = a₂) {q q' q'' : a₂ = a₃} (r : q = q') (s : q' = q'')
: whisker_left p (r ⬝ s) = whisker_left p r ⬝ whisker_left p s :=
begin
induction p, induction r, induction s, reflexivity
end
theorem whisker_right_con_right (q : a2 = a3) (r : p = p') (s : p' = p'')
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 :=
begin
induction q, induction r, induction s, reflexivity
end
theorem whisker_left_con_left (p : a1 = a2) (p' : a2 = a3) {q q' : a3 = a4} (r : q = q')
theorem whisker_left_con_left (p : a₁ = a₂) (p' : a₂ = a₃) {q q' : a₃ = a₄} (r : q = q')
: whisker_left (p ⬝ p') r = !con.assoc ⬝ whisker_left p (whisker_left p' r) ⬝ !con.assoc' :=
begin
induction p', induction p, induction r, induction q, reflexivity
end
theorem whisker_right_con_left {p p' : a1 = a2} (q : a2 = a3) (q' : a3 = a4) (r : p = p')
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 :=
begin
induction q', induction q, induction r, induction p, reflexivity
end
theorem whisker_left_inv_left (p : a2 = a1) {q q' : a2 = a3} (r : q = q')
theorem whisker_left_inv_left (p : a₂ = a₁) {q q' : a₂ = a₃} (r : q = q')
: !con_inv_cancel_left⁻¹ ⬝ whisker_left p (whisker_left p⁻¹ r) ⬝ !con_inv_cancel_left = r :=
begin
induction p, induction r, induction q, reflexivity
end
theorem whisker_left_inv (p : a1 = a2) {q q' : a2 = a3} (r : q = q')
theorem whisker_left_inv (p : a₁ = a₂) {q q' : a₂ = a₃} (r : q = q')
: whisker_left p r⁻¹ = (whisker_left p r)⁻¹ :=
by induction r; reflexivity
theorem whisker_right_inv {p p' : a1 = a2} (q : a2 = a3) (r : p = p')
theorem whisker_right_inv {p p' : a₁ = a₂} (q : a₂ = a₃) (r : p = p')
: whisker_right r⁻¹ q = (whisker_right r q)⁻¹ :=
by induction r; reflexivity
@ -69,27 +69,52 @@ namespace eq
theorem inverse2_left_inv (r : p = p') : inverse2 r ◾ r ⬝ con.left_inv p' = con.left_inv p :=
by induction r;induction p;reflexivity
theorem ap_con_right_inv (f : A → B) (p : a1 = a2)
theorem ap_con_right_inv (f : A → B) (p : a₁ = a₂)
: ap_con f p p⁻¹ ⬝ whisker_left _ (ap_inv f p) ⬝ con.right_inv (ap f p)
= ap (ap f) (con.right_inv p) :=
by induction p;reflexivity
theorem ap_con_left_inv (f : A → B) (p : a1 = a2)
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 (ap f) (con.left_inv p) :=
by induction p;reflexivity
theorem idp_con_whisker_left {q q' : a2 = a3} (r : q = q') :
theorem idp_con_whisker_left {q q' : a₂ = a₃} (r : q = q') :
!idp_con⁻¹ ⬝ whisker_left idp r = r ⬝ !idp_con⁻¹ :=
by induction r;induction q;reflexivity
theorem whisker_left_idp_con {q q' : a2 = a3} (r : q = q') :
theorem whisker_left_idp_con {q q' : a₂ = a₃} (r : q = q') :
whisker_left idp r ⬝ !idp_con = !idp_con ⬝ r :=
by induction r;induction q;reflexivity
theorem idp_con_idp {p : a = a} (q : p = idp) : idp_con p ⬝ q = ap (λp, idp ⬝ p) q :=
by cases q;reflexivity
definition ap_weakly_constant [unfold-c 8] {A B : Type} {f : A → B} {b : B} (p : Πx, f x = b)
{x y : A} (q : x = y) : ap f q = p x ⬝ (p y)⁻¹ :=
by induction q;exact !con.right_inv⁻¹
definition inv2_inv {p q : a = a'} (r : p = q) : inverse2 r⁻¹ = (inverse2 r)⁻¹ :=
by induction r;reflexivity
definition inv2_con {p p' p'' : a = a'} (r : p = p') (r' : p' = p'')
: inverse2 (r ⬝ r') = inverse2 r ⬝ inverse2 r' :=
by induction r';induction r;reflexivity
definition con2_inv {p₁ q₁ : a₁ = a₂} {p₂ q₂ : a₂ = a₃} (r₁ : p₁ = q₁) (r₂ : p₂ = q₂)
: (r₁ ◾ r₂)⁻¹ = r₁⁻¹ ◾ r₂⁻¹ :=
by induction r₁;induction r₂;reflexivity
theorem eq_con_inv_of_con_eq_whisker_left {A : Type} {a a₂ a₃ : A}
{p : a = a₂} {q q' : a₂ = a₃} {r : a = a₃} (s' : q = q') (s : p ⬝ q' = r) :
eq_con_inv_of_con_eq (whisker_left p s' ⬝ s)
= eq_con_inv_of_con_eq s ⬝ whisker_left r (inverse2 s')⁻¹ :=
by induction s';induction q;induction s;reflexivity
theorem right_inv_eq_idp {A : Type} {a : A} {p : a = a} (r : p = idpath a) :
con.right_inv p = r ◾ inverse2 r :=
by cases r;reflexivity
/- Transporting in path spaces.
There are potentially a lot of these lemmas, so we adopt a uniform naming scheme:
@ -99,40 +124,40 @@ namespace eq
- `F` means application of a function to that (varying) endpoint.
-/
definition transport_eq_l (p : a1 = a2) (q : a1 = a3)
: transport (λx, x = a3) p q = p⁻¹ ⬝ q :=
definition transport_eq_l (p : a₁ = a₂) (q : a₁ = a₃)
: transport (λx, x = a₃) p q = p⁻¹ ⬝ q :=
by induction p; induction q; reflexivity
definition transport_eq_r (p : a2 = a3) (q : a1 = a2)
: transport (λx, a1 = x) p q = q ⬝ p :=
definition transport_eq_r (p : a₂ = a₃) (q : a₁ = a₂)
: transport (λx, a₁ = x) p q = q ⬝ p :=
by induction p; induction q; reflexivity
definition transport_eq_lr (p : a1 = a2) (q : a1 = a1)
definition transport_eq_lr (p : a₁ = a₂) (q : a₁ = a₁)
: transport (λx, x = x) p q = p⁻¹ ⬝ q ⬝ p :=
by induction p; rewrite [▸*,idp_con]
definition transport_eq_Fl (p : a1 = a2) (q : f a1 = b)
definition transport_eq_Fl (p : a₁ = a₂) (q : f a₁ = b)
: transport (λx, f x = b) p q = (ap f p)⁻¹ ⬝ q :=
by induction p; induction q; reflexivity
definition transport_eq_Fr (p : a1 = a2) (q : b = f a1)
definition transport_eq_Fr (p : a₁ = a₂) (q : b = f a₁)
: transport (λx, b = f x) p q = q ⬝ (ap f p) :=
by induction p; reflexivity
definition transport_eq_FlFr (p : a1 = a2) (q : f a1 = g a1)
definition transport_eq_FlFr (p : a₁ = a₂) (q : f a₁ = g a₁)
: transport (λx, f x = g x) p q = (ap f p)⁻¹ ⬝ q ⬝ (ap g p) :=
by induction p; rewrite [▸*,idp_con]
definition transport_eq_FlFr_D {B : A → Type} {f g : Πa, B a}
(p : a1 = a2) (q : f a1 = g a1)
(p : a₁ = a₂) (q : f a₁ = g a₁)
: transport (λx, f x = g x) p q = (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) :=
by induction p; rewrite [▸*,idp_con,ap_id]
definition transport_eq_FFlr (p : a1 = a2) (q : h (f a1) = a1)
definition transport_eq_FFlr (p : a₁ = a₂) (q : h (f a₁) = a₁)
: transport (λx, h (f x) = x) p q = (ap h (ap f p))⁻¹ ⬝ q ⬝ p :=
by induction p; rewrite [▸*,idp_con]
definition transport_eq_lFFr (p : a1 = a2) (q : a1 = h (f a1))
definition transport_eq_lFFr (p : a₁ = a₂) (q : a₁ = h (f a₁))
: transport (λx, x = h (f x)) p q = p⁻¹ ⬝ q ⬝ (ap h (ap f p)) :=
by induction p; rewrite [▸*,idp_con]
@ -143,44 +168,44 @@ namespace eq
-- we should probably try to do everything just with pathover_eq (defined in cubical.square),
-- the following definitions may be removed in future.
definition pathover_eq_l (p : a1 = a2) (q : a1 = a3) : q =[p] p⁻¹ ⬝ q := /-(λx, x = a3)-/
definition pathover_eq_l (p : a₁ = a₂) (q : a₁ = a₃) : q =[p] p⁻¹ ⬝ q := /-(λx, x = a₃)-/
by induction p; induction q; exact idpo
definition pathover_eq_r (p : a2 = a3) (q : a1 = a2) : q =[p] q ⬝ p := /-(λx, a1 = x)-/
definition pathover_eq_r (p : a₂ = a₃) (q : a₁ = a₂) : q =[p] q ⬝ p := /-(λx, a₁ = x)-/
by induction p; induction q; exact idpo
definition pathover_eq_lr (p : a1 = a2) (q : a1 = a1) : q =[p] p⁻¹ ⬝ q ⬝ p := /-(λx, x = x)-/
definition pathover_eq_lr (p : a₁ = a₂) (q : a₁ = a₁) : q =[p] p⁻¹ ⬝ q ⬝ p := /-(λx, x = x)-/
by induction p; rewrite [▸*,idp_con]; exact idpo
definition pathover_eq_Fl (p : a1 = a2) (q : f a1 = b) : q =[p] (ap f p)⁻¹ ⬝ q := /-(λx, f x = b)-/
definition pathover_eq_Fl (p : a₁ = a₂) (q : f a₁ = b) : q =[p] (ap f p)⁻¹ ⬝ q := /-(λx, f x = b)-/
by induction p; induction q; exact idpo
definition pathover_eq_Fr (p : a1 = a2) (q : b = f a1) : q =[p] q ⬝ (ap f p) := /-(λx, b = f x)-/
definition pathover_eq_Fr (p : a₁ = a₂) (q : b = f a₁) : q =[p] q ⬝ (ap f p) := /-(λx, b = f x)-/
by induction p; exact idpo
definition pathover_eq_FlFr (p : a1 = a2) (q : f a1 = g a1) : q =[p] (ap f p)⁻¹ ⬝ q ⬝ (ap g p) :=
definition pathover_eq_FlFr (p : a₁ = a₂) (q : f a₁ = g a₁) : q =[p] (ap f p)⁻¹ ⬝ q ⬝ (ap g p) :=
/-(λx, f x = g x)-/
by induction p; rewrite [▸*,idp_con]; exact idpo
definition pathover_eq_FlFr_D {B : A → Type} {f g : Πa, B a} (p : a1 = a2) (q : f a1 = g a1)
definition pathover_eq_FlFr_D {B : A → Type} {f g : Πa, B a} (p : a₁ = a₂) (q : f a₁ = g a₁)
: q =[p] (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) := /-(λx, f x = g x)-/
by induction p; rewrite [▸*,idp_con,ap_id];exact idpo
definition pathover_eq_FFlr (p : a1 = a2) (q : h (f a1) = a1) : q =[p] (ap h (ap f p))⁻¹ ⬝ q ⬝ p :=
definition pathover_eq_FFlr (p : a₁ = a₂) (q : h (f a₁) = a₁) : q =[p] (ap h (ap f p))⁻¹ ⬝ q ⬝ p :=
/-(λx, h (f x) = x)-/
by induction p; rewrite [▸*,idp_con];exact idpo
definition pathover_eq_lFFr (p : a1 = a2) (q : a1 = h (f a1)) : q =[p] p⁻¹ ⬝ q ⬝ (ap h (ap f p)) :=
definition pathover_eq_lFFr (p : a₁ = a₂) (q : a₁ = h (f a₁)) : q =[p] p⁻¹ ⬝ q ⬝ (ap h (ap f p)) :=
/-(λx, x = h (f x))-/
by induction p; rewrite [▸*,idp_con];exact idpo
definition pathover_eq_r_idp (p : a1 = a2) : idp =[p] p := /-(λx, a1 = x)-/
definition pathover_eq_r_idp (p : a₁ = a₂) : idp =[p] p := /-(λx, a₁ = x)-/
by induction p; exact idpo
definition pathover_eq_l_idp (p : a1 = a2) : idp =[p] p⁻¹ := /-(λx, x = a1)-/
definition pathover_eq_l_idp (p : a₁ = a₂) : idp =[p] p⁻¹ := /-(λx, x = a₁)-/
by induction p; exact idpo
definition pathover_eq_l_idp' (p : a1 = a2) : idp =[p⁻¹] p := /-(λx, x = a2)-/
definition pathover_eq_l_idp' (p : a₁ = a₂) : idp =[p⁻¹] p := /-(λx, x = a₂)-/
by induction p; exact idpo
-- The Functorial action of paths is [ap].
@ -189,46 +214,46 @@ namespace eq
/- [ap_closed] is in init.equiv -/
definition equiv_ap (f : A → B) [H : is_equiv f] (a1 a2 : A)
: (a1 = a2) ≃ (f a1 = f a2) :=
definition equiv_ap (f : A → B) [H : is_equiv f] (a₁ a₂ : A)
: (a₁ = a₂) ≃ (f a₁ = f a₂) :=
equiv.mk (ap f) _
/- Path operations are equivalences -/
definition is_equiv_eq_inverse (a1 a2 : A) : is_equiv (@inverse A a1 a2) :=
definition is_equiv_eq_inverse (a₁ a₂ : A) : is_equiv (@inverse A a₁ a₂) :=
is_equiv.mk inverse inverse inv_inv inv_inv (λp, eq.rec_on p idp)
local attribute is_equiv_eq_inverse [instance]
definition eq_equiv_eq_symm (a1 a2 : A) : (a1 = a2) ≃ (a2 = a1) :=
definition eq_equiv_eq_symm (a₁ a₂ : A) : (a₁ = a₂) ≃ (a₂ = a₁) :=
equiv.mk inverse _
definition is_equiv_concat_left [constructor] [instance] (p : a1 = a2) (a3 : A)
: is_equiv (concat p : a2 = a3 → a1 = a3) :=
definition is_equiv_concat_left [constructor] [instance] (p : a₁ = a₂) (a₃ : A)
: is_equiv (concat p : a₂ = a₃ → a₁ = a₃) :=
is_equiv.mk (concat p) (concat p⁻¹)
(con_inv_cancel_left p)
(inv_con_cancel_left p)
(λq, by induction p;induction q;reflexivity)
local attribute is_equiv_concat_left [instance]
definition equiv_eq_closed_left [constructor] (a3 : A) (p : a1 = a2) : (a1 = a3) ≃ (a2 = a3) :=
definition equiv_eq_closed_left [constructor] (a₃ : A) (p : a₁ = a₂) : (a₁ = a₃) ≃ (a₂ = a₃) :=
equiv.mk (concat p⁻¹) _
definition is_equiv_concat_right [constructor] [instance] (p : a2 = a3) (a1 : A)
: is_equiv (λq : a1 = a2, q ⬝ p) :=
definition is_equiv_concat_right [constructor] [instance] (p : a₂ = a₃) (a₁ : A)
: is_equiv (λq : a₁ = a₂, q ⬝ p) :=
is_equiv.mk (λq, q ⬝ p) (λq, q ⬝ p⁻¹)
(λq, inv_con_cancel_right q p)
(λq, con_inv_cancel_right q p)
(λq, by induction p;induction q;reflexivity)
local attribute is_equiv_concat_right [instance]
definition equiv_eq_closed_right [constructor] (a1 : A) (p : a2 = a3) : (a1 = a2) ≃ (a1 = a3) :=
definition equiv_eq_closed_right [constructor] (a₁ : A) (p : a₂ = a₃) : (a₁ = a₂) ≃ (a₁ = a₃) :=
equiv.mk (λq, q ⬝ p) _
definition eq_equiv_eq_closed [constructor] (p : a1 = a2) (q : a3 = a4) : (a1 = a3) ≃ (a2 = a4) :=
equiv.trans (equiv_eq_closed_left a3 p) (equiv_eq_closed_right a2 q)
definition eq_equiv_eq_closed [constructor] (p : a₁ = a₂) (q : a₃ = a₄) : (a₁ = a₃) ≃ (a₂ = a₄) :=
equiv.trans (equiv_eq_closed_left a₃ p) (equiv_eq_closed_right a₂ q)
definition is_equiv_whisker_left (p : a1 = a2) (q r : a2 = a3)
: is_equiv (@whisker_left A a1 a2 a3 p q r) :=
definition is_equiv_whisker_left (p : a₁ = a₂) (q r : a₂ = a₃)
: is_equiv (@whisker_left A a₁ a₂ a₃ p q r) :=
begin
fapply adjointify,
{intro s, apply (!cancel_left s)},
@ -244,11 +269,11 @@ namespace eq
{intro s, induction s, induction q, induction p, reflexivity}
end
definition eq_equiv_con_eq_con_left (p : a1 = a2) (q r : a2 = a3) : (q = r) ≃ (p ⬝ q = p ⬝ r) :=
definition eq_equiv_con_eq_con_left (p : a₁ = a₂) (q r : a₂ = a₃) : (q = r) ≃ (p ⬝ q = p ⬝ r) :=
equiv.mk _ !is_equiv_whisker_left
definition is_equiv_whisker_right {p q : a1 = a2} (r : a2 = a3)
: is_equiv (λs, @whisker_right A a1 a2 a3 p q s r) :=
definition is_equiv_whisker_right {p q : a₁ = a₂} (r : a₂ = a₃)
: is_equiv (λs, @whisker_right A a₁ a₂ a₃ p q s r) :=
begin
fapply adjointify,
{intro s, apply (!cancel_right s)},
@ -256,7 +281,7 @@ namespace eq
{intro s, induction s, induction r, induction p, reflexivity}
end
definition eq_equiv_con_eq_con_right (p q : a1 = a2) (r : a2 = a3) : (p = q) ≃ (p ⬝ r = q ⬝ r) :=
definition eq_equiv_con_eq_con_right (p q : a₁ = a₂) (r : a₂ = a₃) : (p = q) ≃ (p ⬝ r = q ⬝ r) :=
equiv.mk _ !is_equiv_whisker_right
/-
@ -264,7 +289,7 @@ namespace eq
However, these proofs have the advantage that the inverse is definitionally equal to
what we would expect
-/
definition is_equiv_con_eq_of_eq_inv_con (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
definition is_equiv_con_eq_of_eq_inv_con (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
: is_equiv (con_eq_of_eq_inv_con : p = r⁻¹ ⬝ q → r ⬝ p = q) :=
begin
fapply adjointify,
@ -275,11 +300,11 @@ namespace eq
con.assoc,con.assoc,con.right_inv,▸*,-con.assoc,con.left_inv,▸* at *,idp_con s] },
end
definition eq_inv_con_equiv_con_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
definition eq_inv_con_equiv_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
: (p = r⁻¹ ⬝ q) ≃ (r ⬝ p = q) :=
equiv.mk _ !is_equiv_con_eq_of_eq_inv_con
definition is_equiv_con_eq_of_eq_con_inv (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
definition is_equiv_con_eq_of_eq_con_inv (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
: is_equiv (con_eq_of_eq_con_inv : r = q ⬝ p⁻¹ → r ⬝ p = q) :=
begin
fapply adjointify,
@ -288,11 +313,11 @@ namespace eq
{ intro s, induction p, rewrite [↑[con_eq_of_eq_con_inv,eq_con_inv_of_con_eq]] },
end
definition eq_con_inv_equiv_con_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
definition eq_con_inv_equiv_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
: (r = q ⬝ p⁻¹) ≃ (r ⬝ p = q) :=
equiv.mk _ !is_equiv_con_eq_of_eq_con_inv
definition is_equiv_inv_con_eq_of_eq_con (p : a1 = a3) (q : a2 = a3) (r : a1 = a2)
definition is_equiv_inv_con_eq_of_eq_con (p : a₁ = a₃) (q : a₂ = a₃) (r : a₁ = a₂)
: is_equiv (inv_con_eq_of_eq_con : p = r ⬝ q → r⁻¹ ⬝ p = q) :=
begin
fapply adjointify,
@ -303,11 +328,11 @@ namespace eq
con.assoc,con.assoc,con.right_inv,▸*,-con.assoc,con.left_inv,▸* at *,idp_con s] },
end
definition eq_con_equiv_inv_con_eq (p : a1 = a3) (q : a2 = a3) (r : a1 = a2)
definition eq_con_equiv_inv_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₁ = a₂)
: (p = r ⬝ q) ≃ (r⁻¹ ⬝ p = q) :=
equiv.mk _ !is_equiv_inv_con_eq_of_eq_con
definition is_equiv_con_inv_eq_of_eq_con (p : a3 = a1) (q : a2 = a3) (r : a2 = a1)
definition is_equiv_con_inv_eq_of_eq_con (p : a₃ = a₁) (q : a₂ = a₃) (r : a₂ = a₁)
: is_equiv (con_inv_eq_of_eq_con : r = q ⬝ p → r ⬝ p⁻¹ = q) :=
begin
fapply adjointify,
@ -316,7 +341,7 @@ namespace eq
{ intro s, induction p, rewrite [↑[con_inv_eq_of_eq_con,eq_con_of_con_inv_eq]] },
end
definition eq_con_equiv_con_inv_eq (p : a3 = a1) (q : a2 = a3) (r : a2 = a1)
definition eq_con_equiv_con_inv_eq (p : a₃ = a₁) (q : a₂ = a₃) (r : a₂ = a₁)
: (r = q ⬝ p) ≃ (r ⬝ p⁻¹ = q) :=
equiv.mk _ !is_equiv_con_inv_eq_of_eq_con
@ -325,54 +350,54 @@ namespace eq
is_equiv_con_eq_of_eq_con_inv
is_equiv_con_eq_of_eq_inv_con [instance]
definition is_equiv_eq_con_of_inv_con_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
definition is_equiv_eq_con_of_inv_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
: is_equiv (eq_con_of_inv_con_eq : r⁻¹ ⬝ q = p → q = r ⬝ p) :=
is_equiv_inv inv_con_eq_of_eq_con
definition is_equiv_eq_con_of_con_inv_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
definition is_equiv_eq_con_of_con_inv_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
: is_equiv (eq_con_of_con_inv_eq : q ⬝ p⁻¹ = r → q = r ⬝ p) :=
is_equiv_inv con_inv_eq_of_eq_con
definition is_equiv_eq_con_inv_of_con_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
definition is_equiv_eq_con_inv_of_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
: is_equiv (eq_con_inv_of_con_eq : r ⬝ p = q → r = q ⬝ p⁻¹) :=
is_equiv_inv con_eq_of_eq_con_inv
definition is_equiv_eq_inv_con_of_con_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
definition is_equiv_eq_inv_con_of_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
: is_equiv (eq_inv_con_of_con_eq : r ⬝ p = q → p = r⁻¹ ⬝ q) :=
is_equiv_inv con_eq_of_eq_inv_con
/- Pathover Equivalences -/
definition pathover_eq_equiv_l (p : a1 = a2) (q : a1 = a3) (r : a2 = a3) : q =[p] r ≃ q = p ⬝ r :=
/-(λx, x = a3)-/
definition pathover_eq_equiv_l (p : a₁ = a₂) (q : a₁ = a₃) (r : a₂ = a₃) : q =[p] r ≃ q = p ⬝ r :=
/-(λx, x = a₃)-/
by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
definition pathover_eq_equiv_r (p : a2 = a3) (q : a1 = a2) (r : a1 = a3) : q =[p] r ≃ q ⬝ p = r :=
/-(λx, a1 = x)-/
definition pathover_eq_equiv_r (p : a₂ = a₃) (q : a₁ = a₂) (r : a₁ = a₃) : q =[p] r ≃ q ⬝ p = r :=
/-(λx, a₁ = x)-/
by induction p; apply pathover_idp
definition pathover_eq_equiv_lr (p : a1 = a2) (q : a1 = a1) (r : a2 = a2)
definition pathover_eq_equiv_lr (p : a₁ = a₂) (q : a₁ = a₁) (r : a₂ = a₂)
: q =[p] r ≃ q ⬝ p = p ⬝ r := /-(λx, x = x)-/
by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
definition pathover_eq_equiv_Fl (p : a1 = a2) (q : f a1 = b) (r : f a2 = b)
definition pathover_eq_equiv_Fl (p : a₁ = a₂) (q : f a₁ = b) (r : f a₂ = b)
: q =[p] r ≃ q = ap f p ⬝ r := /-(λx, f x = b)-/
by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
definition pathover_eq_equiv_Fr (p : a1 = a2) (q : b = f a1) (r : b = f a2)
definition pathover_eq_equiv_Fr (p : a₁ = a₂) (q : b = f a₁) (r : b = f a₂)
: q =[p] r ≃ q ⬝ ap f p = r := /-(λx, b = f x)-/
by induction p; apply pathover_idp
definition pathover_eq_equiv_FlFr (p : a1 = a2) (q : f a1 = g a1) (r : f a2 = g a2)
definition pathover_eq_equiv_FlFr (p : a₁ = a₂) (q : f a₁ = g a₁) (r : f a₂ = g a₂)
: q =[p] r ≃ q ⬝ ap g p = ap f p ⬝ r := /-(λx, f x = g x)-/
by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
definition pathover_eq_equiv_FFlr (p : a1 = a2) (q : h (f a1) = a1) (r : h (f a2) = a2)
definition pathover_eq_equiv_FFlr (p : a₁ = a₂) (q : h (f a₁) = a₁) (r : h (f a₂) = a₂)
: q =[p] r ≃ q ⬝ p = ap h (ap f p) ⬝ r :=
/-(λx, h (f x) = x)-/
by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
definition pathover_eq_equiv_lFFr (p : a1 = a2) (q : a1 = h (f a1)) (r : a2 = h (f a2))
definition pathover_eq_equiv_lFFr (p : a₁ = a₂) (q : a₁ = h (f a₁)) (r : a₂ = h (f a₂))
: q =[p] r ≃ q ⬝ ap h (ap f p) = p ⬝ r :=
/-(λx, x = h (f x))-/
by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹

108
hott/types/eq2.hlean Normal file
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@ -0,0 +1,108 @@
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
Theorems about 2-dimensional paths
-/
import .cubical.square
open function
namespace eq
variables {A B C : Type} {f : A → B} {a a' a₁ a₂ a₃ a₄ : A} {b b' : B}
theorem ap_weakly_constant_eq (p : Πx, f x = b) (q : a = a') :
ap_weakly_constant p q =
eq_con_inv_of_con_eq ((eq_of_square (square_of_pathover (apdo 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
end
definition ap_inv2 {p q : a = a'} (r : p = q)
: square (ap (ap f) (inverse2 r))
(inverse2 (ap (ap f) r))
(ap_inv f p)
(ap_inv f q) :=
by induction r;exact hrfl
definition ap_con2 {p₁ q₁ : a₁ = a₂} {p₂ q₂ : a₂ = a₃} (r₁ : p₁ = q₁) (r₂ : p₂ = q₂)
: square (ap (ap f) (r₁ ◾ r₂))
(ap (ap f) r₁ ◾ ap (ap f) r₂)
(ap_con f p₁ p₂)
(ap_con f q₁ q₂) :=
by induction r₂;induction r₁;exact hrfl
theorem ap_con_right_inv_sq {A B : Type} {a1 a2 : A} (f : A → B) (p : a1 = a2) :
square (ap (ap f) (con.right_inv p))
(con.right_inv (ap f p))
(ap_con f p p⁻¹ ⬝ whisker_left _ (ap_inv f p))
idp :=
by cases p;apply hrefl
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) _)
idp :=
by cases p;apply vrefl
theorem ap_ap_weakly_constant {A B C : Type} (g : B → C) {f : A → B} {b : B}
(p : Πx, f x = b) {x y : A} (q : x = y) :
square (ap (ap g) (ap_weakly_constant p q))
(ap_weakly_constant (λa, ap g (p a)) q)
(ap_compose g f q)⁻¹
(!ap_con ⬝ whisker_left _ !ap_inv) :=
begin
induction q, esimp, generalize (p x), intro p, cases p, apply ids
-- induction q, rewrite [↑ap_compose,ap_inv], apply hinverse, apply ap_con_right_inv_sq,
end
theorem ap_ap_compose {A B C D : Type} (h : C → D) (g : B → C) (f : A → B)
{x y : A} (p : x = y) :
square (ap_compose (h ∘ g) f p)
(ap (ap h) (ap_compose g f p))
(ap_compose h (g ∘ f) p)
(ap_compose h g (ap f p)) :=
by induction p;exact ids
theorem ap_compose_inv {A B C : Type} (g : B → C) (f : A → B)
{x y : A} (p : x = y) :
square (ap_compose g f p⁻¹)
(inverse2 (ap_compose g f p) ⬝ (ap_inv g (ap f p))⁻¹)
(ap_inv (g ∘ f) p)
(ap (ap g) (ap_inv f p)) :=
by induction p;exact ids
theorem ap_compose_con (g : B → C) (f : A → B) (p : a₁ = a₂) (q : a₂ = a₃) :
square (ap_compose g f (p ⬝ q))
(ap_compose g f p ◾ ap_compose g f q ⬝ (ap_con g (ap f p) (ap f q))⁻¹)
(ap_con (g ∘ f) p q)
(ap (ap g) (ap_con f p q)) :=
by induction q;induction p;exact ids
theorem ap_compose_natural {A B C : Type} (g : B → C) (f : A → B)
{x y : A} {p q : x = y} (r : p = q) :
square (ap (ap (g ∘ f)) r)
(ap (ap g ∘ ap f) r)
(ap_compose g f p)
(ap_compose g f q) :=
natural_square (ap_compose g f) r
-- definition naturality_apdo {A : Type} {B : A → Type} {a a₂ : A} {f g : Πa, B a}
-- (H : f ~ g) (p : a = a₂)
-- : squareover B vrfl (apdo f p) (apdo g p)
-- (pathover_idp_of_eq (H a)) (pathover_idp_of_eq (H a₂)) :=
-- by induction p;esimp;exact hrflo
definition naturality_apdo_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 :=
begin
induction p, esimp,
generalizes [H a, g a], intro ga Ha, induction Ha,
reflexivity
end
end eq