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(* Title: HoTT/ex/Methods.thy
Author: Josh Chen
Date: Aug 2018
HoTT method usage examples
*)
theory Methods
imports "../HoTT"
begin
lemma
assumes "A : U(i)" "B: A \<longrightarrow> U(i)" "\<And>x. x : A \<Longrightarrow> C x: B x \<longrightarrow> U(i)"
shows "\<Sum>x:A. \<Prod>y:B x. \<Sum>z:C x y. \<Prod>w:A. x =\<^sub>A w : U(i)"
by (simple lem: assms)
lemma
assumes "\<Sum>x:A. \<Prod>y: B x. \<Sum>z: C x y. D x y z: U(i)"
shows
"A : U(i)" and
"B: A \<longrightarrow> U(i)" and
"\<And>x. x : A \<Longrightarrow> C x: B x \<longrightarrow> U(i)" and
"\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> D x y: C x y \<longrightarrow> U(i)"
proof -
show
"A : U(i)" and
"B: A \<longrightarrow> U(i)" and
"\<And>x. x : A \<Longrightarrow> C x: B x \<longrightarrow> U(i)" and
"\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> D x y: C x y \<longrightarrow> U(i)"
by (derive lem: assms)
qed
text "Typechecking and constructing inhabitants:"
\<comment> \<open>Correctly determines the type of the pair\<close>
schematic_goal "\<lbrakk>A: U(i); B: U(i); a : A; b : B\<rbrakk> \<Longrightarrow> <a, b> : ?A"
by simple
\<comment> \<open>Finds pair (too easy).\<close>
schematic_goal "\<lbrakk>A: U(i); B: U(i); a : A; b : B\<rbrakk> \<Longrightarrow> ?x : A \<times> B"
apply (rule Sum_intro)
apply assumption+
done
end
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