From 97f3c05e0511a1ed9a95babc800a8e3f3b6a2ea8 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Fri, 3 Apr 2020 01:13:29 +0200
Subject: 1. Base theory. 2. Fix Nat axioms, addition.

---
 hott/Base.thy        | 99 ++++++++++++++++++++++++++++++++++++++++++++++++++++
 hott/Basic_Types.thy | 90 -----------------------------------------------
 hott/Nat.thy         | 19 +++++-----
 3 files changed, 109 insertions(+), 99 deletions(-)
 create mode 100644 hott/Base.thy
 delete mode 100644 hott/Basic_Types.thy

diff --git a/hott/Base.thy b/hott/Base.thy
new file mode 100644
index 0000000..2a4ff9c
--- /dev/null
+++ b/hott/Base.thy
@@ -0,0 +1,99 @@
+theory Base
+imports Spartan.Equivalence
+
+begin
+
+
+section \<open>Notation\<close>
+
+syntax "_dollar" :: \<open>logic \<Rightarrow> logic \<Rightarrow> logic\<close> (infixr "$" 3)
+translations "a $ b" \<rightharpoonup> "a (b)"
+
+abbreviation (input) K where "K x \<equiv> \<lambda>_. x"
+
+
+section \<open>Sum type\<close>
+
+axiomatization
+  Sum    :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> and
+  inl    :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
+  inr    :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
+  SumInd :: \<open>o \<Rightarrow> o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
+
+notation Sum (infixl "\<or>" 50)
+
+axiomatization where
+  SumF: "\<lbrakk>A: U i; B: U i\<rbrakk> \<Longrightarrow> A \<or> B: U i" and
+
+  Sum_inl: "\<lbrakk>a: A; B: U i\<rbrakk> \<Longrightarrow> inl A B a: A \<or> B" and
+
+  Sum_inr: "\<lbrakk>b: B; A: U i\<rbrakk> \<Longrightarrow> inr A B b: A \<or> B" and
+
+  SumE: "\<lbrakk>
+    s: A \<or> B;
+    \<And>s. s: A \<or> B \<Longrightarrow> C s: U i;
+    \<And>a. a: A \<Longrightarrow> c a: C (inl A B a);
+    \<And>b. b: B \<Longrightarrow> d b: C (inr A B b)
+    \<rbrakk> \<Longrightarrow> SumInd A B (\<lambda>s. C s) (\<lambda>a. c a) (\<lambda>b. d b) s: C s" and
+
+  Sum_comp_inl: "\<lbrakk>
+    a: A;
+    \<And>s. s: A \<or> B \<Longrightarrow> C s: U i;
+    \<And>a. a: A \<Longrightarrow> c a: C (inl A B a);
+    \<And>b. b: B \<Longrightarrow> d b: C (inr A B b)
+    \<rbrakk> \<Longrightarrow> SumInd A B (\<lambda>s. C s) (\<lambda>a. c a) (\<lambda>b. d b) (inl A B a) \<equiv> c a" and
+
+  Sum_comp_inr: "\<lbrakk>
+    b: B;
+    \<And>s. s: A \<or> B \<Longrightarrow> C s: U i;
+    \<And>a. a: A \<Longrightarrow> c a: C (inl A B a);
+    \<And>b. b: B \<Longrightarrow> d b: C (inr A B b)
+    \<rbrakk> \<Longrightarrow> SumInd A B (\<lambda>s. C s) (\<lambda>a. c a) (\<lambda>b. d b) (inr A B b) \<equiv> d b"
+
+lemmas
+  [intros] = SumF Sum_inl Sum_inr and
+  [elims] = SumE and
+  [comps] = Sum_comp_inl Sum_comp_inr
+
+
+section \<open>Empty and unit types\<close>
+
+axiomatization
+  Top    :: \<open>o\<close> and
+  tt     :: \<open>o\<close> and
+  TopInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
+and
+  Bot    :: \<open>o\<close> and
+  BotInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
+
+notation Top ("\<top>") and Bot ("\<bottom>")
+
+axiomatization where
+  TopF: "\<top>: U i" and
+
+  TopI: "tt: \<top>" and
+
+  TopE: "\<lbrakk>a: \<top>; \<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (\<lambda>x. C x) c a: C a" and
+
+  Top_comp: "\<lbrakk>\<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (\<lambda>x. C x) c tt \<equiv> c"
+and
+  BotF: "\<bottom>: U i" and
+
+  BotE: "\<lbrakk>x: \<bottom>; \<And>x. x: \<bottom> \<Longrightarrow> C x: U i\<rbrakk> \<Longrightarrow> BotInd (\<lambda>x. C x) x: C x"
+
+lemmas
+  [intros] = TopF TopI BotF and
+  [elims] = TopE BotE and
+  [comps] = Top_comp
+
+
+section \<open>Booleans\<close>
+
+definition "Bool \<equiv> \<top> \<or> \<top>"
+definition "true \<equiv> inl \<top> \<top> tt"
+definition "false \<equiv> inr \<top> \<top> tt"
+
+\<comment> \<open>Can define if-then-else etc.\<close>
+
+
+end
diff --git a/hott/Basic_Types.thy b/hott/Basic_Types.thy
deleted file mode 100644
index 85f22a8..0000000
--- a/hott/Basic_Types.thy
+++ /dev/null
@@ -1,90 +0,0 @@
-theory Basic_Types
-imports Spartan
-
-begin
-
-section \<open>Sum type\<close>
-
-axiomatization
-  Sum    :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> and
-  inl    :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
-  inr    :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
-  SumInd :: \<open>o \<Rightarrow> o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
-
-notation Sum (infixl "\<or>" 50)
-
-axiomatization where
-  SumF: "\<lbrakk>A: U i; B: U i\<rbrakk> \<Longrightarrow> A \<or> B: U i" and
-
-  Sum_inl: "\<lbrakk>a: A; B: U i\<rbrakk> \<Longrightarrow> inl A B a: A \<or> B" and
-
-  Sum_inr: "\<lbrakk>b: B; A: U i\<rbrakk> \<Longrightarrow> inr A B b: A \<or> B" and
-
-  SumE: "\<lbrakk>
-    s: A \<or> B;
-    \<And>s. s: A \<or> B \<Longrightarrow> C s: U i;
-    \<And>a. a: A \<Longrightarrow> c a: C (inl A B a);
-    \<And>b. b: B \<Longrightarrow> d b: C (inr A B b)
-    \<rbrakk> \<Longrightarrow> SumInd A B (\<lambda>s. C s) (\<lambda>a. c a) (\<lambda>b. d b) s: C s" and
-
-  Sum_comp_inl: "\<lbrakk>
-    a: A;
-    \<And>s. s: A \<or> B \<Longrightarrow> C s: U i;
-    \<And>a. a: A \<Longrightarrow> c a: C (inl A B a);
-    \<And>b. b: B \<Longrightarrow> d b: C (inr A B b)
-    \<rbrakk> \<Longrightarrow> SumInd A B (\<lambda>s. C s) (\<lambda>a. c a) (\<lambda>b. d b) (inl A B a) \<equiv> c a" and
-
-  Sum_comp_inr: "\<lbrakk>
-    b: B;
-    \<And>s. s: A \<or> B \<Longrightarrow> C s: U i;
-    \<And>a. a: A \<Longrightarrow> c a: C (inl A B a);
-    \<And>b. b: B \<Longrightarrow> d b: C (inr A B b)
-    \<rbrakk> \<Longrightarrow> SumInd A B (\<lambda>s. C s) (\<lambda>a. c a) (\<lambda>b. d b) (inr A B b) \<equiv> d b"
-
-lemmas
-  [intros] = SumF Sum_inl Sum_inr and
-  [elims] = SumE and
-  [comps] = Sum_comp_inl Sum_comp_inr
-
-
-section \<open>Empty and unit types\<close>
-
-axiomatization
-  Top    :: \<open>o\<close> and
-  tt     :: \<open>o\<close> and
-  TopInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
-and
-  Bot    :: \<open>o\<close> and
-  BotInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
-
-notation Top ("\<top>") and Bot ("\<bottom>")
-
-axiomatization where
-  TopF: "\<top>: U i" and
-
-  TopI: "tt: \<top>" and
-
-  TopE: "\<lbrakk>a: \<top>; \<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (\<lambda>x. C x) c a: C a" and
-
-  Top_comp: "\<lbrakk>\<And>x. x: \<top> \<Longrightarrow> C x: U i; c: C tt\<rbrakk> \<Longrightarrow> TopInd (\<lambda>x. C x) c tt \<equiv> c"
-and
-  BotF: "\<bottom>: U i" and
-
-  BotE: "\<lbrakk>x: \<bottom>; \<And>x. x: \<bottom> \<Longrightarrow> C x: U i\<rbrakk> \<Longrightarrow> BotInd (\<lambda>x. C x) x: C x"
-
-lemmas
-  [intros] = TopF TopI BotF and
-  [elims] = TopE BotE and
-  [comps] = Top_comp
-
-
-section \<open>Booleans\<close>
-
-definition "Bool \<equiv> \<top> \<or> \<top>"
-definition "true \<equiv> inl \<top> \<top> tt"
-definition "false \<equiv> inr \<top> \<top> tt"
-
-\<comment> \<open>Can define if-then-else etc.\<close>
-
-
-end
diff --git a/hott/Nat.thy b/hott/Nat.thy
index f846dbf..269d6bb 100644
--- a/hott/Nat.thy
+++ b/hott/Nat.thy
@@ -1,5 +1,5 @@
 theory Nat
-imports Spartan
+imports Base
 
 begin
 
@@ -17,22 +17,22 @@ where
 
   NatE: "\<lbrakk>
     n: Nat;
-    n\<^sub>0: Nat;
+    n\<^sub>0: C 0;
     \<And>n. n: Nat \<Longrightarrow> C n: U i;
     \<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (suc n)
     \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>n c. f n c) n: C n" and
 
   Nat_comp_zero: "\<lbrakk>
-    n\<^sub>0: Nat;
-    \<And>n. n: Nat \<Longrightarrow> C n: U i;
-    \<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (suc n)
+    n\<^sub>0: C 0;
+    \<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (suc n);
+    \<And>n. n: Nat \<Longrightarrow> C n: U i
     \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>n c. f n c) 0 \<equiv> n\<^sub>0" and
 
   Nat_comp_suc: "\<lbrakk>
     n: Nat;
-    n\<^sub>0: Nat;
-    \<And>n. n: Nat \<Longrightarrow> C n: U i;
-    \<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (suc n)
+    n\<^sub>0: C 0;
+    \<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (suc n);
+    \<And>n. n: Nat \<Longrightarrow> C n: U i
     \<rbrakk> \<Longrightarrow>
       NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>n c. f n c) (suc n) \<equiv>
         f n (NatInd (\<lambda>n. C n) n\<^sub>0 (\<lambda>n c. f n c) n)"
@@ -45,7 +45,8 @@ lemmas
 
 section \<open>Basic arithmetic\<close>
 
-\<comment> \<open>TODO\<close>
+definition add (infixl "+" 50) where
+  [comps]: "m + n \<equiv> NatInd (K Nat) n (K $ \<lambda>n. suc n) m"
 
 
 end
-- 
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