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-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
-\ No newline at end of file