Reorganized HoTT_Base, updated theories

1 files changed, 19 insertions, 23 deletions

diff --git a/Prod.thy b/Prod.thy
index 3b74531..a3cc347 100644
--- a/Prod.thy
+++ b/Prod.thy
@@ -12,14 +12,14 @@ begin
 section \<open>Constants and syntax\<close>
 
 axiomatization
-  Prod :: "[Term, Typefam] \<Rightarrow> Term" and
-  lambda :: "(Term \<Rightarrow> Term) \<Rightarrow> Term"  (binder "\<^bold>\<lambda>" 30) and
-  appl :: "[Term, Term] \<Rightarrow> Term"  (infixl "`" 60)
+  Prod :: "[t, tf] \<Rightarrow> t" and
+  lambda :: "(t \<Rightarrow> t) \<Rightarrow> t"  (binder "\<^bold>\<lambda>" 30) and
+  appl :: "[t, t] \<Rightarrow> t"  (infixl "`" 60)
     \<comment> \<open>Application binds tighter than abstraction.\<close>
 
 syntax
-  "_PROD" :: "[idt, Term, Term] \<Rightarrow> Term"          ("(3\<Prod>_:_./ _)" 30)
-  "_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term"    ("(3PROD _:_./ _)" 30)
+  "_PROD" :: "[idt, t, t] \<Rightarrow> t"          ("(3\<Prod>_:_./ _)" 30)
+  "_PROD_ASCII" :: "[idt, t, t] \<Rightarrow> t"    ("(3PROD _:_./ _)" 30)
 
 text "The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>."
 
@@ -29,24 +29,24 @@ translations
 
 text "Nondependent functions are a special case."
 
-abbreviation Function :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<rightarrow>" 40)
+abbreviation Function :: "[t, t] \<Rightarrow> t"  (infixr "\<rightarrow>" 40)
   where "A \<rightarrow> B \<equiv> \<Prod>_: A. B"
 
 
 section \<open>Type rules\<close>
 
 axiomatization where
-  Prod_form: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x): U(i)"
+  Prod_form: "\<lbrakk>A: U i; B: A \<longrightarrow> U i\<rbrakk> \<Longrightarrow> \<Prod>x:A. B x: U i"
 and
-  Prod_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x): B(x); A: U(i)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x): \<Prod>x:A. B(x)"
+  Prod_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b x: \<Prod>x:A. B x"
 and
-  Prod_elim: "\<lbrakk>f: \<Prod>x:A. B(x); a: A\<rbrakk> \<Longrightarrow> f`a: B(a)"
+  Prod_elim: "\<lbrakk>f: \<Prod>x:A. B x; a: A\<rbrakk> \<Longrightarrow> f`a: B a"
 and
-  Prod_appl: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x): B(x); a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b(x))`a \<equiv> b(a)"
+  Prod_appl: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b x)`a \<equiv> b a"
 and
-  Prod_uniq: "f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f"
+  Prod_uniq: "f : \<Prod>x:A. B x \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f"
 and
-  Prod_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x) \<equiv> b'(x); A: U(i)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) \<equiv> \<^bold>\<lambda>x. b'(x)"
+  Prod_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b x \<equiv> \<^bold>\<lambda>x. c x"
 
 text "
   The Pure rules for \<open>\<equiv>\<close> only let us judge strict syntactic equality of object lambda expressions; Prod_eq is the actual definitional equality rule.
@@ -55,15 +55,15 @@ text "
 "
 
 text "
-  In addition to the usual type rules, it is a meta-theorem that whenever \<open>\<Prod>x:A. B x: U(i)\<close> is derivable from some set of premises \<Gamma>, then so are \<open>A: U(i)\<close> and \<open>B: A \<longrightarrow> U(i)\<close>.
+  In addition to the usual type rules, it is a meta-theorem that whenever \<open>\<Prod>x:A. B x: U i\<close> is derivable from some set of premises \<Gamma>, then so are \<open>A: U i\<close> and \<open>B: A \<longrightarrow> U i\<close>.
   
   That is to say, the following inference rules are admissible, and it simplifies proofs greatly to axiomatize them directly.
 "
 
 axiomatization where
-  Prod_wellform1: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)"
+  Prod_wellform1: "(\<Prod>x:A. B x: U i) \<Longrightarrow> A: U i"
 and
-  Prod_wellform2: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
+  Prod_wellform2: "(\<Prod>x:A. B x: U i) \<Longrightarrow> B: A \<longrightarrow> U i"
 
 
 text "Rule attribute declarations---set up various methods to use the type rules."
@@ -75,19 +75,15 @@ lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim
 
 section \<open>Function composition\<close>
 
-definition compose :: "[Term, Term] \<Rightarrow> Term"  (infixr "o" 70) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)"
+definition compose :: "[t, t] \<Rightarrow> t"  (infixr "o" 110) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)"
 
-syntax "_COMPOSE" :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<circ>" 70)
+syntax "_COMPOSE" :: "[t, t] \<Rightarrow> t"  (infixr "\<circ>" 110)
 translations "g \<circ> f" \<rightleftharpoons> "g o f"
 
 
-section \<open>Atomization\<close>
+section \<open>Polymorphic identity function\<close>
 
-text "
-  Universal statements can be internalized within the theory; the following rule is admissible.
-" (* PROOF NEEDED *)
-
-axiomatization where Prod_atomize: "(\<^bold>\<lambda>x. b(x): \<Prod>x:A. B(x)) \<Longrightarrow> (\<And>x. x: A \<Longrightarrow> b(x): B(x))"
+abbreviation id :: t where "id \<equiv> \<^bold>\<lambda>x. x"
 
 
 end