Implementing univalence

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+(*  Title:  HoTT/Univalence.thy
+    Author: Josh Chen
+
+Definitions of homotopy, equivalence and the univalence axiom.
+*)
+
+theory Univalence
+  imports
+    HoTT_Methods
+    Equal
+    ProdProps
+    Sum
+begin
+
+
+section \<open>Homotopy and equivalence\<close>
+
+axiomatization homotopic :: "[Term, Term] \<Rightarrow> Term"  (infix "~" 100) where
+  homotopic_def: "\<lbrakk>
+    f: \<Prod>x:A. B x;
+    g: \<Prod>x:A. B x
+  \<rbrakk> \<Longrightarrow> f ~ g \<equiv> \<Prod>x:A. (f`x) =[B x] (g`x)"
+
+axiomatization isequiv :: "Term \<Rightarrow> Term" where
+  isequiv_def: "f: A \<rightarrow> B \<Longrightarrow> isequiv f \<equiv> (\<Sum>g: A \<rightarrow> B. g \<circ> f ~ id) \<times> (\<Sum>g: A \<rightarrow> B. f \<circ> g ~ id)"
+
+definition equivalence :: "[Term, Term] \<Rightarrow> Term"  (infix "\<simeq>" 100)
+  where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. isequiv f"
+
+
+text "The identity function is an equivalence:"
+
+lemma isequiv_id:
+  assumes "A: U i" and "id: A \<rightarrow> A"
+  shows "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv id"
+proof (derive lems: assms isequiv_def homotopic_def)
+  fix g assume asm: "g: A \<rightarrow> A"
+  show "id \<circ> g: A \<rightarrow> A"
+    unfolding compose_def by (derive lems: asm assms)
+
+  show "\<Prod>x:A. ((id \<circ> g)`x) =\<^sub>A (id`x): U i"
+    unfolding compose_def by (derive lems: asm assms)
+
+  next
+  show "<\<^bold>\<lambda>x. x, \<^bold>\<lambda>x. refl x>: \<Sum>g:A \<rightarrow> A. (g \<circ> id) ~ id"
+    unfolding compose_def by (derive lems: assms homotopic_def)
+
+  show "<\<^bold>\<lambda>x. x, lambda refl>: \<Sum>g:A \<rightarrow> A. (id \<circ> g) ~ id"
+    unfolding compose_def by (derive lems: assms homotopic_def)
+qed (rule assms)
+
+
+text "The equivalence relation \<open>\<simeq>\<close> is symmetric:"
+
+lemma
+  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 "<<id, \<^bold>\<lambda>x. refl x>, <id, \<^bold>\<lambda>x. refl x>>: isequiv id" using assms by (rule isequiv_id)
+  
+  fix f assume asm: "f: A \<rightarrow> A" show "isequiv f: U i"
+    by (derive lems: asm assms homotopic_def isequiv_def compose_def)
+qed (rule assms)
+
+
+text "Definition of \<open>idtoeqv\<close>:"
+
+
+
+
+end