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(* Title: HoTT/ex/Methods.thy
Author: Josh Chen
Date: Jul 2018
HoTT method usage examples
*)
theory Methods
imports "../HoTT"
begin
lemma
assumes "A : U" "B: A \<rightarrow> U" "\<And>x. x : A \<Longrightarrow> C x: B x \<rightarrow> U"
shows "\<Sum>x:A. \<Prod>y:B x. \<Sum>z:C x y. \<Prod>w:A. x =\<^sub>A w : U"
by (simple lems: assms)
lemma
assumes "f : \<Sum>x:A. \<Prod>y: B x. \<Sum>z: C x y. D x y z"
shows
"A : U" and
"B: A \<rightarrow> U" and
"\<And>x. x : A \<Longrightarrow> C x: B x \<rightarrow> U" and
"\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> D x y: C x y \<rightarrow> U"
proof -
show "A : U" by (wellformed jdgmt: assms)
show "B: A \<rightarrow> U" by (wellformed jdgmt: assms)
show "\<And>x. x : A \<Longrightarrow> C x: B x \<rightarrow> U" by (wellformed jdgmt: assms)
show "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> D x y: C x y \<rightarrow> U" by (wellformed jdgmt: assms)
qed
text "Typechecking:"
\<comment> \<open>Correctly determines the type of the pair\<close>
schematic_goal "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> (a, b) : ?A" by simple
\<comment> \<open>Finds element\<close>
schematic_goal "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> ?x : A \<times> B" by simple
end
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