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diff --git a/ex/Methods.thy b/ex/Methods.thy new file mode 100644 index 0000000..d174dde --- /dev/null +++ b/ex/Methods.thy @@ -0,0 +1,41 @@ +(* 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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