From 62c1c8f306bff84b74b3b1c935d0d6722e1251a2 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Wed, 27 May 2020 21:31:35 +0200
Subject: 1. Define Maybe in terms of other types. 2. Move More_Types to
 Spartan

---
 hott/More_Types.thy         | 91 --------------------------------------------
 spartan/data/Maybe.thy      | 70 ++++++++++++++++++++--------------
 spartan/data/More_Types.thy | 92 +++++++++++++++++++++++++++++++++++++++++++++
 3 files changed, 134 insertions(+), 119 deletions(-)
 delete mode 100644 hott/More_Types.thy
 create mode 100644 spartan/data/More_Types.thy

diff --git a/hott/More_Types.thy b/hott/More_Types.thy
deleted file mode 100644
index d29d794..0000000
--- a/hott/More_Types.thy
+++ /dev/null
@@ -1,91 +0,0 @@
-theory More_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/spartan/data/Maybe.thy b/spartan/data/Maybe.thy
index 532879a..dc3ad04 100644
--- a/spartan/data/Maybe.thy
+++ b/spartan/data/Maybe.thy
@@ -1,43 +1,57 @@
 theory Maybe
-imports Spartan
+imports More_Types
 
 begin
 
-axiomatization
-  Maybe    :: \<open>o \<Rightarrow> o\<close> and
-  none     :: \<open>o \<Rightarrow> o\<close> and
-  some     :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> and
-  MaybeInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
-where
-  MaybeF: "A: U i \<Longrightarrow> Maybe A: U i" and
+definition "Maybe A \<equiv> A \<or> \<top>"
+definition "none A \<equiv> inr A \<top> tt"
+definition "some A a \<equiv> inl A \<top> a"
 
+lemma
+  MaybeF: "A: U i \<Longrightarrow> Maybe A: U i" and
   Maybe_none: "A: U i \<Longrightarrow> none A: Maybe A" and
+  Maybe_some: "a: A \<Longrightarrow> some A a: Maybe A"
+  unfolding Maybe_def none_def some_def by typechk+
 
-  Maybe_some: "a: A \<Longrightarrow> some A a: Maybe A" and
-
-  MaybeE: "\<lbrakk>
-    m: Maybe A;
-    c\<^sub>0: C (none A);
-    \<And>a. a: A \<Longrightarrow> f a: C (some A a);
-    \<And>m. m: Maybe A \<Longrightarrow> C m: U i
-    \<rbrakk> \<Longrightarrow> MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) m: C m" and
+Lemma (derive) MaybeInd:
+  assumes
+    "A: U i"
+    "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
+    "c\<^sub>0: C (none A)"
+    "\<And>a. a: A \<Longrightarrow> f a: C (some A a)"
+    "m: Maybe A"
+  shows "?MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) m: C m"
+  supply assms[unfolded Maybe_def none_def some_def]
+  apply (elim m)
+  \<guillemotright> unfolding Maybe_def .
+  \<guillemotright> by (rule \<open>_ \<Longrightarrow> _: C (inl _ _ _)\<close>)
+  \<guillemotright> by elim (rule \<open>_: C (inr _ _ _)\<close>)
+  done
 
-  Maybe_comp_none: "\<lbrakk>
-    c\<^sub>0: C (none A);
-    \<And>a. a: A \<Longrightarrow> f a: C (some A a);
-    \<And>m. m: Maybe A \<Longrightarrow> C m: U i
-    \<rbrakk> \<Longrightarrow> MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) (none A) \<equiv> c\<^sub>0" and
+Lemma Maybe_comp_none:
+  assumes
+    "A: U i"
+    "c\<^sub>0: C (none A)"
+    "\<And>a. a: A \<Longrightarrow> f a: C (some A a)"
+    "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
+  shows "MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) (none A) \<equiv> c\<^sub>0"
+  supply assms[unfolded Maybe_def some_def none_def]
+  unfolding MaybeInd_def none_def by reduce
 
-  Maybe_comp_some: "\<lbrakk>
-    m: Maybe A;
-    c\<^sub>0: C (none A);
-    \<And>a. a: A \<Longrightarrow> f a: C (some A a);
-    \<And>m. m: Maybe A \<Longrightarrow> C m: U i
-    \<rbrakk> \<Longrightarrow> MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) (some A a) \<equiv> f a"
+Lemma Maybe_comp_some:
+  assumes
+    "A: U i"
+    "a: A"
+    "c\<^sub>0: C (none A)"
+    "\<And>a. a: A \<Longrightarrow> f a: C (some A a)"
+    "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
+  shows "MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) (some A a) \<equiv> f a"
+  supply assms[unfolded Maybe_def some_def none_def]
+  unfolding MaybeInd_def some_def by (reduce add: Maybe_def)
 
 lemmas
   [intros] = MaybeF Maybe_none Maybe_some and
-  [elims "?m"] = MaybeE and
+  MaybeE [elims "?m"] = MaybeInd and
   [comps] = Maybe_comp_none Maybe_comp_some
 
 abbreviation "MaybeRec A C \<equiv> MaybeInd A (K C)"
diff --git a/spartan/data/More_Types.thy b/spartan/data/More_Types.thy
new file mode 100644
index 0000000..f0cee6d
--- /dev/null
+++ b/spartan/data/More_Types.thy
@@ -0,0 +1,92 @@
+theory More_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 ?s] = 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 ?a] = TopE and
+  [elims ?x] = 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
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