From ba0205cddb0205dcc5d9f7bba5702b11a791e6c7 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Wed, 27 Jun 2018 16:11:20 +0200
Subject: Move projection function definitions out of Sum.thy

---
 Proj.thy | 196 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 196 insertions(+)
 create mode 100644 Proj.thy

(limited to 'Proj.thy')

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
+
-- 
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