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-rw-r--r--HoTT_Base.thy65
-rw-r--r--Prod.thy101
-rw-r--r--ROOT3
3 files changed, 116 insertions, 53 deletions
diff --git a/HoTT_Base.thy b/HoTT_Base.thy
index bb47a74..a5280ce 100644
--- a/HoTT_Base.thy
+++ b/HoTT_Base.thy
@@ -5,7 +5,7 @@ Jan 2019
This file is the starting point of the Isabelle/HoTT object logic, and contains the basic setup and definitions.
Among other things, it defines:
-* Meta-numerals to index the universe hierarchy, and related axioms.
+* Containers for storing type information of object-level terms.
* The universe hierarchy and its governing rules.
* Named theorems for later use by proof methods.
@@ -16,12 +16,56 @@ imports Pure
begin
-section \<open>Basic setup\<close>
+section \<open>Terms and typing\<close>
-declare[[eta_contract=false]] (* Does this option actually do anything? *)
+typedecl t \<comment> \<open>Type of object-logic terms (which includes the types)\<close>
-typedecl t \<comment> \<open>Type of object-logic terms (which includes the types)\<close>
-typedecl ord \<comment> \<open>Type of meta-numerals\<close>
+consts has_type :: "[t, t] \<Rightarrow> prop" \<comment> \<open>The judgment coercion\<close>
+
+text \<open>
+We create a dedicated container in which to store object-level typing information whenever a typing judgment that does not depend on assumptions is proved.
+\<close>
+
+ML \<open>
+signature TYPINGS =
+sig
+ val get: term -> term
+end
+
+structure Typings: TYPINGS =
+struct
+
+structure Typing_Data = Theory_Data
+(
+ type T = term Symtab.table
+ val empty = Symtab.empty
+ val extend = I
+
+ fun merge (tbl, tbl') =
+ let
+ val key_vals = map (Symtab.lookup_key tbl') (Symtab.keys tbl')
+ val updates = map (fn SOME kv => Symtab.update kv) key_vals
+ fun f u t = u t
+ in fold f updates tbl end
+)
+
+fun get tm =
+ case Symtab.lookup (Typing_Data.get @{theory}) (Syntax.string_of_term @{context} tm) of
+ SOME tm' => tm'
+ | NONE => raise Fail "No typing information available"
+
+end
+\<close>
+
+syntax "_has_type" :: "[t, t] \<Rightarrow> prop" ("3(_:/ _)")
+
+parse_translation \<open>
+
+\<close>
+
+section \<open>Universes\<close>
+
+typedecl ord \<comment> \<open>Type of meta-numerals\<close>
axiomatization
O :: ord and
@@ -38,12 +82,6 @@ declare
leq_min [intro!]
lt_trans [intro]
-section \<open>Typing judgment\<close>
-
-judgment hastype :: "[t, t] \<Rightarrow> prop" ("(3_:/ _)")
-
-section \<open>Universes\<close>
-
axiomatization
U :: "ord \<Rightarrow> t"
where
@@ -60,7 +98,10 @@ Instead use @{method elim}, or instantiate @{thm U_cumulative} suitably.
section \<open>Type families\<close>
-abbrev (input)
+abbreviation (input) constraint :: "[t \<Rightarrow> t, t, t] \<Rightarrow> prop" ("(1_:/ _ \<leadsto> _)")
+where "f: A \<leadsto> B \<equiv> (\<And>x. x: A \<Longrightarrow> f x: B)"
+
+text \<open>We use the notation @{prop "B: A \<leadsto> U i"} to abbreviate type families.\<close>
section \<open>Named theorems\<close>
diff --git a/Prod.thy b/Prod.thy
index 8d840bd..68c9405 100644
--- a/Prod.thy
+++ b/Prod.thy
@@ -1,91 +1,112 @@
-(*
-Title: Prod.thy
-Author: Joshua Chen
-Date: 2018
+(********
+Isabelle/HoTT: Dependent product (dependent function)
+Jan 2019
-Dependent product type
-*)
+********)
theory Prod
imports HoTT_Base HoTT_Methods
begin
-
-section \<open>Basic definitions\<close>
+section \<open>Basic type definitions\<close>
axiomatization
- Prod :: "[t, tf] \<Rightarrow> t" and
- lambda :: "(t \<Rightarrow> t) \<Rightarrow> t" (binder "\<^bold>\<lambda>" 30) and
- appl :: "[t, t] \<Rightarrow> t" ("(1_`_)" [120, 121] 120) \<comment> \<open>Application binds tighter than abstraction.\<close>
+ Prod :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and
+ lam :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and
+ app :: "[t, t] \<Rightarrow> t" ("(1_ ` _)" [120, 121] 120)
+ \<comment> \<open>Application should bind tighter than abstraction.\<close>
syntax
- "_prod" :: "[idt, t, t] \<Rightarrow> t" ("(3\<Prod>_: _./ _)" 30)
- "_prod_ascii" :: "[idt, t, t] \<Rightarrow> t" ("(3II _: _./ _)" 30)
-
-text \<open>The translations below bind the variable @{term x} in the expressions @{term B} and @{term b}.\<close>
-
+ "_Prod" :: "[idt, t, t] \<Rightarrow> t" ("(3TT '(_: _')./ _)" 30)
+ "_Prod'" :: "[idt, t, t] \<Rightarrow> t" ("(3TT _: _./ _)" 30)
+ "_lam" :: "[idt, t, t] \<Rightarrow> t" ("(3,\\ '(_: _')./ _)" 30)
+ "_lam'" :: "[idt, t, t] \<Rightarrow> t" ("(3,\\ _: _./ _)" 30)
translations
- "\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod A (\<lambda>x. B)"
- "II x:A. B" \<rightharpoonup> "CONST Prod A (\<lambda>x. B)"
+ "TT(x: A). B" \<rightleftharpoons> "(CONST Prod) A (\<lambda>x. B)"
+ "TT x: A. B" \<rightleftharpoons> "(CONST Prod) A (\<lambda>x. B)"
+ ",\\(x: A). b" \<rightleftharpoons> "(CONST lam) A (\<lambda>x. b)"
+ ",\\x: A. b" \<rightleftharpoons> "(CONST lam) A (\<lambda>x. b)"
+
+text \<open>
+The syntax translations above bind the variable @{term x} in the expressions @{term B} and @{term b}.
+\<close>
-text \<open>Non-dependent functions are a special case.\<close>
+text \<open>Non-dependent functions are a special case:\<close>
abbreviation Fun :: "[t, t] \<Rightarrow> t" (infixr "\<rightarrow>" 40)
- where "A \<rightarrow> B \<equiv> \<Prod>_: A. B"
+where "A \<rightarrow> B \<equiv> TT(_: A). B"
axiomatization where
\<comment> \<open>Type rules\<close>
- Prod_form: "\<lbrakk>A: U i; B: A \<longrightarrow> U i\<rbrakk> \<Longrightarrow> \<Prod>x:A. B x: U i" and
+ Prod_form: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: U i\<rbrakk> \<Longrightarrow> TT 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" and
+ Prod_intro: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> b x: B x\<rbrakk> \<Longrightarrow> ,\\x: A. b x: TT x: A. B x" and
- Prod_elim: "\<lbrakk>f: \<Prod>x:A. B x; a: A\<rbrakk> \<Longrightarrow> f`a: B a" and
+ Prod_elim: "\<lbrakk>f: TT x: A. B x; a: A\<rbrakk> \<Longrightarrow> f`a: B a" and
- Prod_comp: "\<lbrakk>a: A; \<And>x. x: A \<Longrightarrow> b x: B x\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b x)`a \<equiv> b a" and
+ Prod_cmp: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; a: A\<rbrakk> \<Longrightarrow> (,\\x: A. b x)`a \<equiv> b a" and
- Prod_uniq: "f: \<Prod>x:A. B x \<Longrightarrow> \<^bold>\<lambda>x. f`x \<equiv> f" and
+ Prod_uniq: "f: TT x: A. B x \<Longrightarrow> ,\\x: A. f`x \<equiv> f" and
\<comment> \<open>Congruence rules\<close>
- Prod_form_eq: "\<lbrakk>A: U i; B: A \<longrightarrow> U i; C: A \<longrightarrow> U i; \<And>x. x: A \<Longrightarrow> B x \<equiv> C x\<rbrakk> \<Longrightarrow> \<Prod>x:A. B x \<equiv> \<Prod>x:A. C x" and
+ Prod_form_eq: "\<lbrakk>A: U i; B: A \<leadsto> U i; C: A \<leadsto> U i; \<And>x. x: A \<Longrightarrow> B x \<equiv> C x\<rbrakk>
+ \<Longrightarrow> TT x: A. B x \<equiv> TT x: A. C x" and
- Prod_intro_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"
+ Prod_intro_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> ,\\x: A. b x \<equiv> ,\\x: A. c x"
text \<open>
The Pure rules for \<open>\<equiv>\<close> only let us judge strict syntactic equality of object lambda expressions.
-The actual definitional equality rule is @{thm Prod_intro_eq}.
+The actual definitional equality rule in the type theory is @{thm Prod_intro_eq}.
Note that this is a separate rule from function extensionality.
-
-Note that the bold lambda symbol \<open>\<^bold>\<lambda>\<close> used for dependent functions clashes with the proof term syntax (cf. \<section>2.5.2 of the Isabelle/Isar Implementation).
\<close>
lemmas Prod_form [form]
lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim
-lemmas Prod_comps [comp] = Prod_comp Prod_uniq
-
+lemmas Prod_comp [comp] = Prod_cmp Prod_uniq
section \<open>Additional definitions\<close>
-definition compose :: "[t, t] \<Rightarrow> t" (infixr "o" 110) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)"
-
-syntax "_compose" :: "[t, t] \<Rightarrow> t" (infixr "\<circ>" 110)
-translations "g \<circ> f" \<rightleftharpoons> "g o f"
+definition compose :: "[t, t, t] \<Rightarrow> t"
+where "compose A g f \<equiv> ,\\x: A. g`(f`x)"
declare compose_def [comp]
+syntax "_compose" :: "[t, t] \<Rightarrow> t" (infixr "o" 110)
+
+parse_translation \<open>
+let fun compose_tr ctxt tms =
+ let
+ val f::g::_ = tms;
+ fun domain f = @{term "A :: t"}
+ in
+ @{const compose} $ (domain f) $ f $ g
+ end
+in
+ [("_compose", compose_tr)]
+end
+\<close>
+
+ML_val \<open>
+Proof_Context.get_thms @{context} "compose_def"
+\<close>
+
lemma compose_assoc:
- assumes "A: U i" and "f: A \<rightarrow> B" "g: B \<rightarrow> C" "h: \<Prod>x:C. D x"
- shows "(h \<circ> g) \<circ> f \<equiv> h \<circ> (g \<circ> f)"
-by (derive lems: assms Prod_intro_eq)
+ assumes "A: U i" "f: A \<rightarrow> B" "g: B \<rightarrow> C" "h: TT x: C. D x"
+ shows "(h o g) o f \<equiv> h o (g o f)"
+ML_prf \<open>
+@{thms assms};
+\<close>
+(* (derive lems: assms Prod_intro_eq) *)
lemma compose_comp:
assumes "A: U i" and "\<And>x. x: A \<Longrightarrow> b x: B" and "\<And>x. x: B \<Longrightarrow> c x: C x"
shows "(\<^bold>\<lambda>x. c x) \<circ> (\<^bold>\<lambda>x. b x) \<equiv> \<^bold>\<lambda>x. c (b x)"
by (derive lems: assms Prod_intro_eq)
-abbreviation id :: t where "id \<equiv> \<^bold>\<lambda>x. x" \<comment> \<open>Polymorphic identity function\<close>
+abbreviation id :: "t \<Rightarrow> t" where "id A \<equiv> ,\\x: A. x"
end
diff --git a/ROOT b/ROOT
index cecdd32..b05d022 100644
--- a/ROOT
+++ b/ROOT
@@ -6,7 +6,8 @@ session HoTT = Pure +
hierarchy à la Russell.
Follows the development of the theory in
- The Univalent Foundations Program, Homotopy Type Theory: Univalent Foundations of Mathematics,
+ The Univalent Foundations Program,
+ Homotopy Type Theory: Univalent Foundations of Mathematics,
Institute for Advanced Study, (2013).
Available online at https://homotopytypetheory.org/book.