From c2dfffffb7586662c67e44a2d255a1a97ab0398b Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Thu, 2 Apr 2020 17:57:48 +0200
Subject: Brand-spanking new version using Spartan infrastructure

---
 hott/Basic_Types.thy | 90 ++++++++++++++++++++++++++++++++++++++++++++++++++++
 hott/Nat.thy         | 51 +++++++++++++++++++++++++++++
 2 files changed, 141 insertions(+)
 create mode 100644 hott/Basic_Types.thy
 create mode 100644 hott/Nat.thy

(limited to 'hott')

diff --git a/hott/Basic_Types.thy b/hott/Basic_Types.thy
new file mode 100644
index 0000000..85f22a8
--- /dev/null
+++ b/hott/Basic_Types.thy
@@ -0,0 +1,90 @@
+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
new file mode 100644
index 0000000..f846dbf
--- /dev/null
+++ b/hott/Nat.thy
@@ -0,0 +1,51 @@
+theory Nat
+imports Spartan
+
+begin
+
+axiomatization
+  Nat    :: \<open>o\<close> and
+  zero   :: \<open>o\<close> ("0") and
+  suc    :: \<open>o \<Rightarrow> o\<close> and
+  NatInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
+where
+  NatF: "Nat: U i" and
+
+  Nat_zero: "0: Nat" and
+
+  Nat_suc: "n: Nat \<Longrightarrow> suc n: Nat" and
+
+  NatE: "\<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)
+    \<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)
+    \<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)
+    \<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)"
+
+lemmas
+  [intros] = NatF Nat_zero Nat_suc and
+  [elims] = NatE and
+  [comps] = Nat_comp_zero Nat_comp_suc
+
+
+section \<open>Basic arithmetic\<close>
+
+\<comment> \<open>TODO\<close>
+
+
+end
-- 
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