From 76b57317d7568f4dcd673b1b8085601c6c723355 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Mon, 11 Feb 2019 22:44:21 +0100
Subject: Organize this commit as a backup of the work on type inference done
 so far; learnt that I probably need to take a different approach. In
 particular, should first make the constants completely monomorphic, and then
 work on full proper type inference, rather than the heuristic approach taken
 here.

---
 Sum.thy | 51 ++++++++++++++++++++++++---------------------------
 1 file changed, 24 insertions(+), 27 deletions(-)

(limited to 'Sum.thy')

diff --git a/Sum.thy b/Sum.thy
index 2646c97..9549a5e 100644
--- a/Sum.thy
+++ b/Sum.thy
@@ -1,59 +1,56 @@
-(*
-Title:  Sum.thy
-Author: Joshua Chen
-Date:   2018
+(********
+Isabelle/HoTT: Dependent sum (dependent pair)
+Feb 2019
 
-Dependent sum type
-*)
+********)
 
 theory Sum
 imports HoTT_Base
 
 begin
 
-
 axiomatization
-  Sum    :: "[t, tf] \<Rightarrow> t" and
+  Sum    :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and
   pair   :: "[t, t] \<Rightarrow> t"  ("(1<_,/ _>)") and
-  indSum :: "[[t, t] \<Rightarrow> t, t] \<Rightarrow> t"  ("(1ind\<^sub>\<Sum>)")
+  indSum :: "[t \<Rightarrow> t, [t, t] \<Rightarrow> t, t] \<Rightarrow> t"
 
 syntax
-  "_sum" :: "[idt, t, t] \<Rightarrow> t"        ("(3\<Sum>_:_./ _)" 20)
-  "_sum_ascii" :: "[idt, t, t] \<Rightarrow> t"  ("(3SUM _:_./ _)" 20)
-
+  "_Sum"  :: "[idt, t, t] \<Rightarrow> t"  ("(3\<Sum>'(_: _')./ _)" 20)
+  "_Sum'" :: "[idt, t, t] \<Rightarrow> t"  ("(3\<Sum>_: _./ _)" 20)
 translations
-  "\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)"
-  "SUM x:A. B" \<rightharpoonup> "CONST Sum A (\<lambda>x. B)"
+  "\<Sum>(x: A). B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)"
+  "\<Sum>x: A. B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)"
 
 abbreviation Pair :: "[t, t] \<Rightarrow> t"  (infixr "\<times>" 50)
-  where "A \<times> B \<equiv> \<Sum>_:A. B"
+  where "A \<times> B \<equiv> \<Sum>_: A. B"
 
 axiomatization where
 \<comment> \<open>Type rules\<close>
 
-  Sum_form: "\<lbrakk>A: U i; B: A \<longrightarrow> U i\<rbrakk> \<Longrightarrow> \<Sum>x:A. B x: U i" and
+  Sum_form: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and
 
-  Sum_intro: "\<lbrakk>B: A \<longrightarrow> U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a,b>: \<Sum>x:A. B x" and
+  Sum_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a, b>: \<Sum>x: A. B x" and
 
   Sum_elim: "\<lbrakk>
-    p: \<Sum>x:A. B x;
-    C: \<Sum>x:A. B x \<longrightarrow> U i;
-    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x,y> \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum> f p: C p" and
+    p: \<Sum>x: A. B x;
+    C: \<Sum>x: A. B x \<leadsto> U i;
+    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x,y> \<rbrakk> \<Longrightarrow> indSum C f p: C p" and
 
-  Sum_comp: "\<lbrakk>
+  Sum_cmp: "\<lbrakk>
     a: A;
     b: B a;
-    B: A \<longrightarrow> U i;
-    C: \<Sum>x:A. B x \<longrightarrow> U i;
-    \<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f x y: C <x,y> \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum> f <a,b> \<equiv> f a b" and
+    B: A \<leadsto> U i;
+    C: \<Sum>x: A. B x \<leadsto> U i;
+    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x,y> \<rbrakk> \<Longrightarrow> indSum C f <a, b> \<equiv> f a b" and
 
 \<comment> \<open>Congruence rules\<close>
 
-  Sum_form_eq: "\<lbrakk>A: U i; B: A \<longrightarrow> U i; C: A \<longrightarrow> U i; \<And>x. x: A \<Longrightarrow> B x \<equiv> C x\<rbrakk> \<Longrightarrow> \<Sum>x:A. B x \<equiv> \<Sum>x:A. C x"
+  Sum_form_eq: "\<lbrakk>A: U i; B: A \<leadsto> U i; C: A \<leadsto> U i; \<And>x. x: A \<Longrightarrow> B x \<equiv> C x\<rbrakk>
+    \<Longrightarrow> \<Sum>x: A. B x \<equiv> \<Sum>x: A. C x"
 
 lemmas Sum_form [form]
 lemmas Sum_routine [intro] = Sum_form Sum_intro Sum_elim
-lemmas Sum_comp [comp]
-
+lemmas Sum_comp [comp] = Sum_cmp
+lemmas Sum_cong [cong] = Sum_form_eq
 
 end
-- 
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