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(********
Isabelle/HoTT: Univalence
Feb 2019
********)
theory Univalence
imports HoTT_Methods Prod Sum Eq
begin
section \<open>Homotopy\<close>
definition homotopic :: "[t, t \<Rightarrow> t, t, t] \<Rightarrow> t" ("(2homotopic[_, _] _ _)" [0, 0, 1000, 1000])
where "homotopic[A, B] f g \<equiv> \<Prod>x: A. f`x =[B x] g`x"
declare homotopic_def [comp]
syntax "_homotopic" :: "[t, idt, t, t, t] \<Rightarrow> t" ("(1_ ~[_: _. _]/ _)" [101, 0, 0, 0, 101] 100)
translations "f ~[x: A. B] g" \<rightleftharpoons> "(CONST homotopic) A (\<lambda>x. B) f g"
(*
syntax "_homotopic'" :: "[t, t] \<Rightarrow> t" ("(2_ ~ _)" [1000, 1000])
ML \<open>val pretty_homotopic = Attrib.setup_config_bool @{binding "pretty_homotopic"} (K true)\<close>
print_translation \<open>
let fun homotopic_tr' ctxt [A, B, f, g] =
if Config.get ctxt pretty_homotopic
then Syntax.const @{syntax_const "_homotopic'"} $ f $ g
else @{const homotopic} $ A $ B $ f $ g
in
[(@{const_syntax homotopic}, homotopic_tr')]
end
\<close>
*)
lemma homotopic_type:
assumes [intro]: "A: U i" "B: A \<leadsto> U i" "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x"
shows "f ~[x: A. B x] g: U i"
by derive
declare homotopic_type [intro]
schematic_goal fun_eq_imp_homotopic:
assumes [intro]:
"p: f =[\<Prod>x: A. B x] g"
"f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x"
"A: U i" "B: A \<leadsto> U i"
shows "?prf: f ~[x: A. B x] g"
proof (path_ind' f g p)
show "\<And>f. f : \<Prod>(x: A). B x \<Longrightarrow> \<lambda>x: A. refl(f`x): f ~[x: A. B x] f" by derive
qed routine
definition happly :: "[t, t \<Rightarrow> t, t, t, t] \<Rightarrow> t"
where "happly A B f g p \<equiv> indEq (\<lambda>f g. & f ~[x: A. B x] g) (\<lambda>f. \<lambda>(x: A). refl(f`x)) f g p"
syntax "_happly" :: "[idt, t, t, t, t, t] \<Rightarrow> t"
("(2happly[_: _. _] _ _ _)" [0, 0, 0, 1000, 1000, 1000])
translations "happly[x: A. B] f g p" \<rightleftharpoons> "(CONST happly) A (\<lambda>x. B) f g p"
corollary happly_type:
assumes [intro]:
"p: f =[\<Prod>x: A. B x] g"
"f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x"
"A: U i" "B: A \<leadsto> U i"
shows "happly[x: A. B x] f g p: f ~[x: A. B x] g"
unfolding happly_def by (derive lems: fun_eq_imp_homotopic)
section \<open>Transport and homotopy\<close>
schematic_goal transport_invl_hom:
assumes [intro]:
"A: U i" "P: A \<rightarrow> U j"
"x: A" "y: A" "p: x =[A] y"
shows "?prf:
(transport[P, y, x] (inv[A, x, y] p)) o[P`x] (transport[P, x, y] p) ~[w: P`x. P`x] id(P`x)"
proof (rule happly_type[OF transport_invl])
show "(transport[P, y, x] (inv[A, x, y] p)) o[P`x] (transport[P, x, y] p): P`x \<rightarrow> P`x"
proof show "P`y: U j" by routine qed routine
qed routine
schematic_goal transport_invr_hom:
assumes [intro]:
"A: U i" "P: A \<rightarrow> U j"
"x: A" "y: A" "p: x =[A] y"
shows "?prf:
(transport[P, x, y] p) o[P`y] (transport[P, y, x] (inv[A, x, y] p)) ~[w: P`y. P`y] id(P`y)"
proof (rule happly_type[OF transport_invr])
show "(transport[P, x, y] p) o[P`y] (transport[P, y, x] (inv[A, x, y] p)): P`y \<rightarrow> P`y"
proof show "P`x: U j" by routine qed routine
qed routine
declare
transport_invl_hom [intro]
transport_invr_hom [intro]
section \<open>Equivalence\<close>
text \<open>For now, we define equivalence in terms of bi-invertibility.\<close>
definition biinv :: "[t, t, t] \<Rightarrow> t" ("(2biinv[_, _]/ _)")
where "biinv[A, B] f \<equiv>
(\<Sum>g: B \<rightarrow> A. g o[A] f ~[x:A. A] id A) \<times> (\<Sum>g: B \<rightarrow> A. f o[B] g ~[x: B. B] id B)"
text \<open>
The meanings of the syntax defined above are:
\<^item> @{term "f ~[x: A. B x] g"} expresses that @{term f} and @{term g} are homotopic functions of type @{term "\<Prod>x:A. B x"}.
\<^item> @{term "biinv[A, B] f"} expresses that the function @{term f} of type @{term "A \<rightarrow> B"} is bi-invertible.
\<close>
lemma biinv_type:
assumes [intro]: "A: U i" "B: U i" "f: A \<rightarrow> B"
shows "biinv[A, B] f: U i"
unfolding biinv_def by derive
declare biinv_type [intro]
schematic_goal id_is_biinv:
assumes [intro]: "A: U i"
shows "?prf: biinv[A, A] (id A)"
unfolding biinv_def proof (rule Sum_routine, compute)
show "<id A, \<lambda>x: A. refl x>: \<Sum>(g: A \<rightarrow> A). (g o[A] id A) ~[x: A. A] (id A)" by derive
show "<id A, \<lambda>x: A. refl x>: \<Sum>(g: A \<rightarrow> A). (id A o[A] g) ~[x: A. A] (id A)" by derive
qed routine
definition equivalence :: "[t, t] \<Rightarrow> t" (infix "\<simeq>" 100)
where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. biinv[A, B] f"
schematic_goal equivalence_symmetric:
assumes [intro]: "A: U i"
shows "?prf: A \<simeq> A"
unfolding equivalence_def proof (rule Sum_routine)
show "\<And>f. f : A \<rightarrow> A \<Longrightarrow> biinv[A, A] f : U i" unfolding biinv_def by derive
show "id A: A \<rightarrow> A" by routine
qed (routine add: id_is_biinv)
section \<open>Univalence\<close>
declare[[goals_limit=100, pretty_transport=true]]
lemma lem: "\<And>A. A: U i \<Longrightarrow> (id (U i))`A \<equiv> A" by derive
schematic_goal
assumes [intro]: "A: U i" "B: U i" "p: A =[U i] B"
shows "?prf: biinv[A, B] (transport[id(U i), A, B] p)"
apply (path_ind' A B p)
apply (unfold biinv_def)
show "?a : biinv[A, B] (transport[id(U i), A, B] p)"
oops
schematic_goal
assumes [intro]: "A: U i" "B: U i" "p: A =[U i] B"
shows "?prf: biinv[id(U i)`A, id(U i)`B] (transport[id(U i), A, B] p)"
unfolding biinv_def
apply (rule Sum_routine)
prefer 2 apply (rule Sum_routine)
prefer 3 apply (rule transport_invl_hom)
prefer 9 apply (rule Sum_routine)
prefer 3 apply (rule transport_invr_hom)
proof -
show "U i: U (Suc i)" by hierarchy
show "U i: U (Suc i)" by hierarchy
show "\<And>g. g : id (U i)`B \<rightarrow> id (U i)`A \<Longrightarrow> (transport[id(U i), A, B] p) o[id (U i)`B] g ~[x: id (U i)`B. id (U i)`B] id (id (U i)`B) : U i"
proof
show "id (U i)`B : U i" by derive
show "id (U i)`B : U i" by derive
show "id (id (U i)`B) : id (U i)`B \<rightarrow> id (U i)`B" by derive
show
"\<And>g. g : id (U i)`B \<rightarrow> id (U i)`A \<Longrightarrow>
(transport[id(U i), A, B] p) o[id (U i)`B] g: id (U i)`B \<rightarrow> id (U i)`B"
apply rule
prefer 4 apply assumption
proof -
show "id (U i)`B : U i" by derive
show "id (U i)`A : U i" by derive
show "id (U i)`B : U i" by derive
oops
schematic_goal type_eq_imp_equiv:
assumes [intro]: "A: U i" "B: U i"
shows "?prf: (A =[U i] B) \<rightarrow> A \<simeq> B"
unfolding equivalence_def
apply (rule Prod_routine, rule Sum_routine)
prefer 2
show *: "\<And>f. f: A \<rightarrow> B \<Longrightarrow> isequiv[A, B] f: U i"
unfolding isequiv_def by (derive lems: assms
show "\<And>p. p: A =[U i] B \<Longrightarrow> transport p: A \<rightarrow> B"
by (derive lems: assms transport_type[where ?i="Suc i"] \<comment> \<open>Instantiate @{thm transport_type} with a suitable universe level here...\<close>
show "\<And>p. p: A =[U i] B \<Longrightarrow> ind\<^sub>= (\<lambda>_. <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>) p: isequiv[A, B] (transport p)"
proof (elim Equal_elim)
show "\<And>T. T: U i \<Longrightarrow> <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv[T, T] (transport (refl T))"
by (derive lems: transport_comp[where ?i="Suc i" and ?A="U i"] id_isequiv)
\<comment> \<open>...and also here.\<close>
show "\<And>A B p. \<lbrakk>A: U i; B: U i; p: A =[U i] B\<rbrakk> \<Longrightarrow> isequiv[A, B] (transport p): U i"
unfolding isequiv_def by (derive lems: assms transport_type
qed fact+
qed (derive lems: assms
(*
section \<open>The univalence axiom\<close>
axiomatization univalence :: "[t, t] \<Rightarrow> t" where UA: "univalence A B: isequiv[A, B] idtoeqv"
*)
end
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