From 607c3971e08d1ded22bd9f1cabdd309653af1248 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Tue, 29 May 2018 12:28:13 +0200
Subject: More rigorous rules for Product type. Propositions and proofs all
 working, but have to think about maybe relaxing the computation rule, or else
 automating the currying of dependent functions.

---
 HoTT_Theorems.thy | 67 +++++++++++++++++++++++++++++++++++++++++++++----------
 1 file changed, 55 insertions(+), 12 deletions(-)

(limited to 'HoTT_Theorems.thy')

diff --git a/HoTT_Theorems.thy b/HoTT_Theorems.thy
index 33b0957..d83a08c 100644
--- a/HoTT_Theorems.thy
+++ b/HoTT_Theorems.thy
@@ -6,23 +6,25 @@ text "A bunch of theorems and other statements for sanity-checking, as well as t
 
 Things that *should* be automated:
   \<bullet> Checking that \<open>A\<close> is a well-formed type, when writing things like \<open>x : A\<close> and \<open>A : U\<close>.
-"
+  \<bullet> Checking that the argument to a (dependent/non-dependent) function matches the type? Also the arguments to a pair?"
 
 \<comment> \<open>Turn on trace for unification and the simplifier, for debugging.\<close>
-declare[[unify_trace_simp, unify_trace_types, simp_trace]]
+declare[[unify_trace_simp, unify_trace_types, simp_trace, simp_trace_depth_limit=2]]
 
 section \<open>Functions\<close>
 
+subsection \<open>Typing functions\<close>
+
 text "Declaring \<open>Prod_intro\<close> with the \<open>intro\<close> attribute (in HoTT.thy) enables \<open>standard\<close> to prove the following."
 
-lemma id_function: "A : U \<Longrightarrow> \<^bold>\<lambda>x. x : A\<rightarrow>A" ..
+lemma id_function: "A : U \<Longrightarrow> \<^bold>\<lambda>x:A. x : A\<rightarrow>A" ..
 
 text "Here is the same result, stated and proved differently.
 The standard method invoked after the keyword \<open>proof\<close> is applied to the goal \<open>\<^bold>\<lambda>x. x: A\<rightarrow>A\<close>, and so we need to show the prover how to continue, as opposed to the previous lemma."
 
 lemma
   assumes "A : U"
-  shows "\<^bold>\<lambda>x. x : A\<rightarrow>A"
+  shows "\<^bold>\<lambda>x:A. x : A\<rightarrow>A"
 proof
   show "A : U" using assms .
   show "\<lambda>x. A : A \<rightarrow> U" using assms ..
@@ -31,29 +33,29 @@ qed
 text "Note that there is no provision for declaring the type of bound variables outside of the scope of a lambda expression.
 More generally, we cannot write an assumption stating 'Let \<open>x\<close> be a variable of type \<open>A\<close>'."
 
-proposition "\<lbrakk>A : U; A \<equiv> B\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. x : B\<rightarrow>A"
+proposition "\<lbrakk>A : U; A \<equiv> B\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. x : B\<rightarrow>A"
 proof -
   assume
     1: "A : U" and
     2: "A \<equiv> B"
-  from id_function[OF 1] have 3: "\<^bold>\<lambda>x. x : A\<rightarrow>A" .
+  from id_function[OF 1] have 3: "\<^bold>\<lambda>x:A. x : A\<rightarrow>A" .
   from 2 have "A\<rightarrow>A \<equiv> B\<rightarrow>A" by simp
-  with 3 show "\<^bold>\<lambda>x. x : B\<rightarrow>A" ..
+  with 3 show "\<^bold>\<lambda>x:A. x : B\<rightarrow>A" ..
 qed
 
 text "It is instructive to try to prove \<open>\<lbrakk>A : U; B : U\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A\<close>.
 First we prove an intermediate step."
 
-lemma constant_function: "\<lbrakk>A : U; B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y. x : B\<rightarrow>A" ..
+lemma constant_function: "\<lbrakk>A : U; B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y:B. x : B\<rightarrow>A" ..
 
 text "And now the actual result:"
 
 proposition
   assumes 1: "A : U" and 2: "B : U"
-  shows "\<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A"
+  shows "\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x : A\<rightarrow>B\<rightarrow>A"
 proof
   show "A : U" using assms(1) .
-  show "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y. x : B \<rightarrow> A" using assms by (rule constant_function)
+  show "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y:B. x : B \<rightarrow> A" using assms by (rule constant_function)
 
   from assms have "B \<rightarrow> A : U" by (rule Prod_formation)
   then show "\<lambda>x. B \<rightarrow> A: A \<rightarrow> U" using assms(1) by (rule constant_type_family)
@@ -61,13 +63,54 @@ qed
 
 text "Maybe a nicer way to write it:"
 
-proposition "\<lbrakk>A : U; B: U\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A"
+proposition alternating_function: "\<lbrakk>A : U; B: U\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x : A\<rightarrow>B\<rightarrow>A"
 proof
   fix x
-  show "\<lbrakk>A : U;  B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y. x : B \<rightarrow> A" by (rule constant_function)
+  show "\<lbrakk>A : U;  B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y:B. x : B \<rightarrow> A" by (rule constant_function)
   show "\<lbrakk>A : U; B : U\<rbrakk> \<Longrightarrow> B\<rightarrow>A : U" by (rule Prod_formation)
 qed
 
+subsection \<open>Function application\<close>
+
+lemma "\<lbrakk>A : U; a : A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. x)`a \<equiv> a" by simp
+
+lemma
+  assumes
+    "A:U" and
+    "B:U" and
+    "a:A" and
+    "b:B"
+  shows "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)`a`b \<equiv> a"
+proof -
+  have "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)`a \<equiv> \<^bold>\<lambda>y:B. a" 
+    proof (rule Prod_comp[of A "\<lambda>_. B\<rightarrow>A"])
+      have "B \<rightarrow> A : U" using constant_type_family[OF assms(1) assms(2)] assms(2) by (rule Prod_formation)
+      then show "\<lambda>x. B \<rightarrow> A: A \<rightarrow> U" using assms(1) by (rule constant_type_family[of "B\<rightarrow>A"])
+      
+      show "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y:B. x : B \<rightarrow> A" using assms(2) assms(1) ..
+      show "A:U" using assms(1) .
+      show "a:A" using assms(3) .
+    qed  (* Why do I need to do the above for the last two goals? Can't Isabelle do it automatically? *)
+
+  then have "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)`a`b \<equiv> (\<^bold>\<lambda>y:B. a)`b" by simp
+
+  also have "(\<^bold>\<lambda>y:B. a)`b \<equiv> a"
+    proof (rule Prod_comp[of B "\<lambda>_. A"])
+      show "\<lambda>y. A: B \<rightarrow> U" using assms(1) assms(2) by (rule constant_type_family)
+      show "\<And>y. y : B \<Longrightarrow> a : A" using assms(3) .
+      show "B:U" using assms(2) .
+      show "b:B" using assms(4) .
+    qed
+
+  finally show "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)`a`b \<equiv> a" .
+qed
+
+text "Polymorphic identity function."
+
+consts Ui::Term
+definition Id where "Id \<equiv> \<^bold>\<lambda>A:Ui. \<^bold>\<lambda>x:A. x"
+(* Have to think about universes... *)
+
 section \<open>Nats\<close>
 
 text "Here's a dumb proof that 2 is a natural number."
-- 
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