diff options
| -rw-r--r-- | HoTT_Base.thy | 65 | ||||
| -rw-r--r-- | Prod.thy | 101 | ||||
| -rw-r--r-- | ROOT | 3 |
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> @@ -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 @@ -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. |