From 593faab277de53cbe2cb0c2feca5de307d9334ac Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Sat, 9 Jun 2018 00:11:39 +0200
Subject: Reorganize code

---
 Sum.thy | 78 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 78 insertions(+)
 create mode 100644 Sum.thy

(limited to 'Sum.thy')

diff --git a/Sum.thy b/Sum.thy
new file mode 100644
index 0000000..e34749a
--- /dev/null
+++ b/Sum.thy
@@ -0,0 +1,78 @@
+(*  Title:  HoTT/Sum.thy
+    Author: Josh Chen
+
+Dependent sum type.
+*)
+
+theory Sum
+  imports HoTT_Base Prod
+
+begin
+
+axiomatization
+  Sum :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" and
+  pair :: "[Term, Term] \<Rightarrow> Term"  ("(1'(_,/ _'))") and
+  indSum :: "(Term \<Rightarrow> Term) \<Rightarrow> Term"
+
+syntax
+  "_SUM" :: "[idt, Term, Term] \<Rightarrow> Term"        ("(3\<Sum>_:_./ _)" 20)
+  "_SUM_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term"  ("(3SUM _:_./ _)" 20)
+
+translations
+  "\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)"
+  "SUM x:A. B" \<rightharpoonup> "CONST Sum A (\<lambda>x. B)"
+
+axiomatization where
+  Sum_form [intro]: "\<And>A B. \<lbrakk>A : U; B: A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) : U"
+and
+  Sum_intro [intro]: "\<And>A B a b. \<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> (a, b) : \<Sum>x:A. B(x)"
+and
+  Sum_elim [elim]: "\<And>A B C f p.
+    \<lbrakk> C: \<Sum>x:A. B(x) \<rightarrow> U;
+      f : \<Prod>x:A. \<Prod>y:B(x). C((x,y));
+      p : \<Sum>x:A. B(x) \<rbrakk> \<Longrightarrow> indSum(C)`f`p : C(p)"
+and
+  Sum_comp [simp]: "\<And>(C::Term \<Rightarrow> Term) (f::Term) (a::Term) (b::Term). indSum(C)`f`(a,b) \<equiv> f`a`b"
+
+text "We choose to formulate the elimination rule by using the object-level function type and function application as much as possible.
+Hence only the type family \<open>C\<close> is left as a meta-level argument to the inductor indSum."
+
+\<comment> \<open>Nondependent pair\<close>
+abbreviation Pair :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<times>" 50)
+  where "A\<times>B \<equiv> \<Sum>_:A. B"
+
+subsubsection \<open>Projections\<close>
+
+consts
+  fst :: "[Term, 'a] \<Rightarrow> Term"  ("(1fst[/_,/ _])")
+  snd :: "[Term, 'a] \<Rightarrow> Term"  ("(1snd[/_,/ _])")
+
+overloading
+  fst_dep \<equiv> fst
+  snd_dep \<equiv> snd
+  fst_nondep \<equiv> fst
+  snd_nondep \<equiv> snd
+begin
+  definition fst_dep :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" where
+    "fst_dep A B \<equiv> indSum(\<lambda>_. A)`(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). x)"
+  
+  definition snd_dep :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" where
+    "snd_dep A B \<equiv> indSum(\<lambda>_. A)`(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). y)"
+  
+  definition fst_nondep :: "[Term, Term] \<Rightarrow> Term" where
+    "fst_nondep A B \<equiv> indSum(\<lambda>_. A)`(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)"
+  
+  definition snd_nondep :: "[Term, Term] \<Rightarrow> Term" where
+    "snd_nondep A B \<equiv> indSum(\<lambda>_. A)`(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. y)"
+end
+
+text "Simplification rules for the projections:"
+
+lemma fst_dep_comp: "\<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> fst[A,B]`(a,b) \<equiv> a" unfolding fst_dep_def by simp
+lemma snd_dep_comp: "\<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> snd[A,B]`(a,b) \<equiv> b" unfolding snd_dep_def by simp
+
+lemma fst_nondep_comp: "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> fst[A,B]`(a,b) \<equiv> a" unfolding fst_nondep_def by simp
+lemma snd_nondep_comp: "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> snd[A,B]`(a,b) \<equiv> b" unfolding snd_nondep_def by simp
+
+lemmas fst_snd_simps [simp] = fst_dep_comp snd_dep_comp fst_nondep_comp snd_nondep_comp
+end
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