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-rw-r--r-- | Proj.thy | 196 |
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diff --git a/Proj.thy b/Proj.thy new file mode 100644 index 0000000..9387ffb --- /dev/null +++ b/Proj.thy @@ -0,0 +1,196 @@ +(* Title: HoTT/Proj.thy + Author: Josh Chen + Date: Jun 2018 + +Projection functions \<open>fst\<close> and \<open>snd\<close> for the dependent sum type. +*) + +theory Proj + imports + HoTT_Methods + Prod + Sum +begin + +consts + fst :: "[Term, 'a] \<Rightarrow> Term" ("(1fst[/_,/ _])") + snd :: "[Term, 'a] \<Rightarrow> Term" ("(1snd[/_,/ _])") + + +section \<open>Overloaded syntax for dependent and nondependent pairs\<close> + +overloading + fst_dep \<equiv> fst + fst_nondep \<equiv> fst +begin + definition fst_dep :: "[Term, Typefam] \<Rightarrow> Term" where + "fst_dep A B \<equiv> \<^bold>\<lambda>p: (\<Sum>x:A. B x). indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) p" + + definition fst_nondep :: "[Term, Term] \<Rightarrow> Term" where + "fst_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. indSum[A, \<lambda>_. B] (\<lambda>_. A) (\<lambda>x y. x) p" +end + +overloading + snd_dep \<equiv> snd + snd_nondep \<equiv> snd +begin + definition snd_dep :: "[Term, Typefam] \<Rightarrow> Term" where + "snd_dep A B \<equiv> \<^bold>\<lambda>p: (\<Sum>x:A. B x). indSum[A,B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) p" + + definition snd_nondep :: "[Term, Term] \<Rightarrow> Term" where + "snd_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) p" +end + + +section \<open>Properties\<close> + +text "Typing judgments and computation rules for the dependent and non-dependent projection functions." + +lemma fst_dep_type: + assumes "p : \<Sum>x:A. B x" + shows "fst[A,B]`p : A" + +proof \<comment> \<open>The standard reasoner knows to backchain with the product elimination rule here...\<close> + \<comment> \<open>Also write about this proof: compare the effect of letting the standard reasoner do simplifications, as opposed to using the minus switch and writing everything out explicitly from first principles.\<close> + + have *: "\<Sum>x:A. B x : U" using assms .. + + show "fst[A,B]: (\<Sum>x:A. B x) \<rightarrow> A" + + proof (unfold fst_dep_def, standard) \<comment> \<open>...and with the product introduction rule here...\<close> + show "\<And>p. p : \<Sum>x:A. B x \<Longrightarrow> indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) p : A" + + proof \<comment> \<open>...and with sum elimination here.\<close> + show "A : U" using * .. + qed + + qed (rule *) + +qed (rule assms) + + +lemma fst_dep_comp: (* Potential for automation *) + assumes "B: A \<rightarrow> U" and "a : A" and "b : B a" + shows "fst[A,B]`(a,b) \<equiv> a" + +proof - + \<comment> "Write about this proof: unfolding, how we set up the introduction rules (explain \<open>..\<close>), do a trace of the proof, explain the meaning of keywords, etc." + + have *: "A : U" using assms(2) .. (* I keep thinking this should not have to be done explicitly, but rather automated. *) + + have "fst[A,B]`(a,b) \<equiv> indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) (a,b)" + proof (unfold fst_dep_def, standard) + show "\<And>p. p : \<Sum>x:A. B x \<Longrightarrow> indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) p : A" using * .. + show "(a,b) : \<Sum>x:A. B x" using assms .. + qed + + also have "indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) (a,b) \<equiv> a" + by (rule Sum_comp) (rule *, assumption, (rule assms)+) + + finally show "fst[A,B]`(a,b) \<equiv> a" . +qed + + +text "In proving results about the second dependent projection function we often use the following two lemmas." + +lemma snd_dep_welltyped: + assumes "p : \<Sum>x:A. B x" + shows "B (fst[A,B]`p) : U" + +proof - + have "\<Sum>x:A. B x : U" using assms .. + then have *: "B: A \<rightarrow> U" .. + + have "fst[A,B]`p : A" using assms by (rule fst_dep_type) + then show "B (fst[A,B]`p) : U" by (rule *) +qed + + +lemma snd_dep_const_type: + assumes "B: A \<rightarrow> U" and "x : A" and "y : B x" + shows "y : B (fst[A,B]`(x,y))" + +proof - + have "fst[A,B]`(x,y) \<equiv> x" using assms by (rule fst_dep_comp) + then show "y : B (fst[A,B]`(x,y))" using assms by simp +qed + + +lemma snd_dep_type: + assumes "p : \<Sum>x:A. B x" + shows "snd[A,B]`p : B (fst[A,B]`p)" + +proof + have *: "\<Sum>x:A. B x : U" using assms .. + + show "snd[A, B] : \<Prod>p:(\<Sum>x:A. B x). B (fst[A,B]`p)" + + proof (unfold snd_dep_def, standard) + show "\<And>p. p : \<Sum>x:A. B x \<Longrightarrow> indSum[A,B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) p : B (fst[A,B]`p)" + + proof (standard, elim snd_dep_welltyped) + fix x y assume 1: "x : A" and 2: "y : B x" + show "y : B (fst[A,B]`(x,y))" + proof - + have "B: A \<rightarrow> U" using * .. + then show "y : B (fst[A,B]`(x,y))" using 1 2 by (rule snd_dep_const_type) + qed + qed + + qed (rule *) + +qed (rule assms) + + +lemma snd_dep_comp: + assumes "B: A \<rightarrow> U" and "a : A" and "b : B a" + shows "snd[A,B]`(a,b) \<equiv> b" + +proof - + have "snd[A,B]`(a,b) \<equiv> indSum[A, B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) (a,b)" + proof (unfold snd_dep_def, standard) + show "(a,b) : \<Sum>x:A. B x" using assms .. + + fix p assume *: "p : \<Sum>x:A. B x" + show "indSum[A,B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) p : B (fst[A,B]`p)" + proof (standard, elim snd_dep_welltyped) + show "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> y : B (fst[A,B]`(x,y))" using assms + by (elim snd_dep_const_type) + qed (rule *) + qed + + also have "indSum[A, B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) (a,b) \<equiv> b" + proof (standard, elim snd_dep_welltyped) + show "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> y : B (fst[A,B]`(x,y))" using assms + by (elim snd_dep_const_type) + qed (rule assms)+ + + finally show "snd[A,B]`(a,b) \<equiv> b" . +qed + + +text "For non-dependent projection functions:" + +print_statement fst_dep_type +print_statement fst_dep_type[where ?p = p and ?A = A and ?B = "\<lambda>_. B"] + + +lemma fst_nondep_type: "p : A \<times> B \<Longrightarrow> fst[A,B]`p : A" +by (rule fst_dep_type[where ?B = "\<lambda>_. B"]) + + +lemma fst_nondep_comp: + assumes "a : A" and "b : B" + shows "fst[A,B]`(a,b) \<equiv> a" +proof (unfold fst_nondep_def) + have "A : U" using assms(1) .. + then show "fst[A,B]`(a,b) \<equiv> a" unfolding fst_nondep_def +qed + +lemma snd_nondep_comp: "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> snd[A,B]`(a,b) \<equiv> b" +proof - + assume "a : A" and "b : B" + then have "(a, b) : A \<times> B" .. + then show "snd[A,B]`(a,b) \<equiv> b" unfolding snd_nondep_def by simp +qed + |