From d8699451025a3bd5e8955e07fa879ed248418949 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Thu, 16 Aug 2018 16:28:50 +0200
Subject: Some comments and reorganization

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 ex/HoTT book/Ch1.thy | 44 ++++++++++++++++++++++++++++++++++++++++++++
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+(*  Title:  HoTT/ex/HoTT book/Ch1.thy
+    Author: Josh Chen
+    Date:   Aug 2018
+
+A formalization of some content of Chapter 1 of the Homotopy Type Theory book.
+*)
+
+theory Ch1
+  imports "../../HoTT"
+begin
+
+chapter \<open>HoTT Book, Chapter 1\<close>
+
+section \<open>1.6 Dependent pair types (\<Sigma>-types)\<close>
+
+text "Prove that the only inhabitants of the \<Sigma>-type are those given by the pair constructor."
+
+schematic_goal
+  assumes "(\<Sum>x:A. B(x)): U(i)" and "p: \<Sum>x:A. B(x)"
+  shows "?a: p =[\<Sum>x:A. B(x)] <fst p, snd p>"
+
+text "Proof by induction on \<open>p: \<Sum>x:A. B(x)\<close>:"
+
+proof (rule Sum_elim[where ?p=p])
+  text "We just need to prove the base case; the rest will be taken care of automatically."
+
+  fix x y assume asm: "x: A" "y: B(x)" show
+    "refl(<x,y>): <x,y> =[\<Sum>x:A. B(x)] <fst <x,y>, snd <x,y>>"
+  proof (subst (0 1) comp)
+    text "
+      The computation rules for \<open>fst\<close> and \<open>snd\<close> require that \<open>x\<close> and \<open>y\<close> have appropriate types.
+      The automatic proof methods have trouble picking the appropriate types, so we state them explicitly,
+    "
+    show "x: A" and "y: B(x)" by (fact asm)+
+
+    text "...twice, once each for the substitutions of \<open>fst\<close> and \<open>snd\<close>."
+    show "x: A" and "y: B(x)" by (fact asm)+
+
+  qed (derive lems: assms asm)
+
+qed (derive lems: assms)
+
+
+end
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