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-rw-r--r--Equal.thy59
-rw-r--r--HoTT.thy7
-rw-r--r--HoTT_Base.thy2
3 files changed, 28 insertions, 40 deletions
diff --git a/Equal.thy b/Equal.thy
index 7b6acb5..f9bc223 100644
--- a/Equal.thy
+++ b/Equal.thy
@@ -1,66 +1,51 @@
-(* Title: HoTT/Equal.thy
- Author: Josh Chen
+(*
+Title: Equal.thy
+Author: Joshua Chen
+Date: 2018
Equality type
*)
theory Equal
- imports HoTT_Base
+imports HoTT_Base HoTT_Methods
+
begin
-section \<open>Constants and syntax\<close>
+section \<open>Basic definitions\<close>
axiomatization
- Equal :: "[t, t, t] \<Rightarrow> t" and
- refl :: "t \<Rightarrow> t" and
+ Equal :: "[t, t, t] \<Rightarrow> t" and
+ refl :: "t \<Rightarrow> t" and
indEqual :: "[t \<Rightarrow> t, t] \<Rightarrow> t" ("(1ind\<^sub>=)")
syntax
- "_EQUAL" :: "[t, t, t] \<Rightarrow> t" ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100)
- "_EQUAL_ASCII" :: "[t, t, t] \<Rightarrow> t" ("(3_ =[_]/ _)" [101, 0, 101] 100)
+ "_equal" :: "[t, t, t] \<Rightarrow> t" ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100)
+ "_equal_ascii" :: "[t, t, t] \<Rightarrow> t" ("(3_ =[_]/ _)" [101, 0, 101] 100)
+
translations
"a =[A] b" \<rightleftharpoons> "CONST Equal A a b"
"a =\<^sub>A b" \<rightharpoonup> "CONST Equal A a b"
+axiomatization where
+ Equal_form: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U i" and
-section \<open>Type rules\<close>
+ Equal_intro: "a : A \<Longrightarrow> (refl a): a =\<^sub>A a" and
-axiomatization where
- Equal_form: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U i"
-and
- Equal_intro: "a : A \<Longrightarrow> (refl a): a =\<^sub>A a"
-and
Equal_elim: "\<lbrakk>
+ p: x =\<^sub>A y;
x: A;
y: A;
- p: x =\<^sub>A y;
- \<And>x. x: A \<Longrightarrow> f x: C x x (refl x);
- \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i
- \<rbrakk> \<Longrightarrow> ind\<^sub>= f p : C x y p"
-and
+ \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i;
+ \<And>x. x: A \<Longrightarrow> f x: C x x (refl x) \<rbrakk> \<Longrightarrow> ind\<^sub>= (\<lambda>x. f x) p : C x y p" and
+
Equal_comp: "\<lbrakk>
a: A;
- \<And>x. x: A \<Longrightarrow> f x: C x x (refl x);
- \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i
- \<rbrakk> \<Longrightarrow> ind\<^sub>= f (refl a) \<equiv> f a"
-
+ \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i;
+ \<And>x. x: A \<Longrightarrow> f x: C x x (refl x) \<rbrakk> \<Longrightarrow> ind\<^sub>= (\<lambda>x. f x) (refl a) \<equiv> f a"
-text "Admissible inference rules for equality type formation:"
-
-axiomatization where
- Equal_wellform1: "a =\<^sub>A b: U i \<Longrightarrow> A: U i"
-and
- Equal_wellform2: "a =\<^sub>A b: U i \<Longrightarrow> a: A"
-and
- Equal_wellform3: "a =\<^sub>A b: U i \<Longrightarrow> b: A"
-
-
-text "Rule attribute declarations:"
-
-lemmas Equal_comp [comp]
-lemmas Equal_wellform [wellform] = Equal_wellform1 Equal_wellform2 Equal_wellform3
lemmas Equal_routine [intro] = Equal_form Equal_intro Equal_elim
+lemmas Equal_comp [comp]
end
diff --git a/HoTT.thy b/HoTT.thy
index 3f2d475..e2a7e35 100644
--- a/HoTT.thy
+++ b/HoTT.thy
@@ -6,7 +6,6 @@ Homotopy type theory
theory HoTT
imports
-
(* Basic theories *)
HoTT_Base
HoTT_Methods
@@ -22,18 +21,22 @@ Unit
(* Derived definitions and properties *)
EqualProps
-ProdProps
Proj
begin
+
lemmas forms =
Nat_form Prod_form Sum_form Coprod_form Equal_form Unit_form Empty_form
+
lemmas intros =
Nat_intro_0 Nat_intro_succ Prod_intro Sum_intro Equal_intro Coprod_intro_inl Coprod_intro_inr Unit_intro
+
lemmas elims =
Nat_elim Prod_elim Sum_elim Equal_elim Coprod_elim Unit_elim Empty_elim
+
lemmas routines =
Nat_routine Prod_routine Sum_routine Equal_routine Coprod_routine Unit_routine Empty_routine
+
end
diff --git a/HoTT_Base.thy b/HoTT_Base.thy
index efc6182..17e3142 100644
--- a/HoTT_Base.thy
+++ b/HoTT_Base.thy
@@ -77,7 +77,7 @@ Declare named theorems to be used by proof methods defined in @{file HoTT_Method
These are used by the \<open>compute\<close> method, and may also be passed to invocations of the method \<open>subst\<close> to perform equational rewriting.
\<close>
-(* Todo: Set up the simplifier! *)
+(* Todo: Set up the Simplifier! *)
end