diff options
| -rw-r--r-- | Univalence.thy | 161 |
1 files changed, 115 insertions, 46 deletions
diff --git a/Univalence.thy b/Univalence.thy index c6733c6..e073e50 100644 --- a/Univalence.thy +++ b/Univalence.thy @@ -1,72 +1,141 @@ -(* -Title: Univalence.thy -Author: Joshua Chen -Date: 2018 +(******** +Isabelle/HoTT: Univalence +Feb 2019 -Definitions of homotopy, equivalence and the univalence axiom. -*) +********) theory Univalence -imports HoTT_Methods Equality Prod Sum +imports HoTT_Methods Prod Sum Eq begin -section \<open>Homotopy and equivalence\<close> - -definition homotopic :: "[t, tf, t, t] \<Rightarrow> t" where "homotopic A B f g \<equiv> \<Prod>x:A. (f`x) =[B x] (g`x)" +section \<open>Homotopy\<close> -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" +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] -definition isequiv :: "[t, t, t] \<Rightarrow> t" ("(3isequiv[_, _]/ _)") where - "isequiv[A, B] f \<equiv> (\<Sum>g: B \<rightarrow> A. g \<circ> f ~[x:A. A] id) \<times> (\<Sum>g: B \<rightarrow> A. f \<circ> g ~[x:B. B] id)" +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]: + "A: U i" "B: A \<leadsto> U i" + "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x" + "p: f =[\<Prod>x: A. B x] g" + 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" ("(2happly[_, _] _ _ _)" [0, 0, 1000, 1000, 1000]) +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" + +corollary happly_type: + assumes [intro]: + "A: U i" "B: A \<leadsto> U i" + "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x" + "p: f =[\<Prod>x: A. B x] g" + shows "happly[A, B] f g p: f ~[x: A. B x] g" +unfolding happly_def by (derive lems: fun_eq_imp_homotopic) + + +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 "isequiv[A, B] f"} expresses that the non-dependent function @{term f} of type @{term "A \<rightarrow> B"} is an equivalence. +\<^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> -definition equivalence :: "[t, t] \<Rightarrow> t" (infix "\<simeq>" 100) - where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. isequiv[A, B] f" +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 -lemma id_isequiv: - assumes "A: U i" "id: A \<rightarrow> A" - shows "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv[A, A] id" -unfolding isequiv_def proof (routine add: assms) - show "\<And>g. g: A \<rightarrow> A \<Longrightarrow> id \<circ> g ~[x:A. A] id: U i" by (derive lems: assms) - show "<id, \<^bold>\<lambda>x. refl x>: \<Sum>g:A \<rightarrow> A. (g \<circ> id) ~[x:A. A] id" by (derive lems: assms) - show "<id, \<^bold>\<lambda>x. refl x>: \<Sum>g:A \<rightarrow> A. (id \<circ> g) ~[x:A. A] id" by (derive lems: assms) -qed +declare biinv_type [intro] -lemma equivalence_symm: - assumes "A: U i" and "id: A \<rightarrow> A" - shows "<id, <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>>: A \<simeq> A" -unfolding equivalence_def proof - show "\<And>f. f: A \<rightarrow> A \<Longrightarrow> isequiv[A, A] f: U i" by (derive lems: assms isequiv_def) - show "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv[A, A] id" using assms by (rule id_isequiv) -qed fact +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>idtoeqv\<close> -definition idtoeqv :: t where "idtoeqv \<equiv> \<^bold>\<lambda>p. <transport p, ind\<^sub>= (\<lambda>_. <<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>) p>" +section \<open>Univalence\<close> -text \<open>We prove that equal types are equivalent. The proof involves universe types.\<close> +(* +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)" +unfolding biinv_def +apply (rule Sum_routine) defer +apply (rule Sum_intro) + have + "transport[id(U i), A, B] p: (id (U i))`A \<rightarrow> (id (U i))`B" + by (derive lems: transport_type[where ?A="U i"] + moreover have + "(id (U i))`A \<rightarrow> (id (U i))`B \<equiv> A \<rightarrow> B" by deriv + ultimately have [intro]: + "transport[id(U i), A, B] p: A \<rightarrow> B" by simp + + show "\<Sum>g: B \<rightarrow> A. (transport[id(U i), A, B] p o[B] g) ~[x: B. B] (id B) : U i" + by deriv + -theorem - assumes "A: U i" and "B: U i" - shows "idtoeqv: (A =[U i] B) \<rightarrow> A \<simeq> B" -unfolding idtoeqv_def equivalence_def proof (routine add: assms) +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) + 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> + 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) @@ -75,14 +144,14 @@ unfolding idtoeqv_def equivalence_def proof (routine add: assms) \<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) + unfolding isequiv_def by (derive lems: assms transport_type qed fact+ -qed (derive lems: assms) +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 |