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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
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