diff options
Diffstat (limited to '')
| -rw-r--r-- | HoTT_Base.thy | 71 | ||||
| -rw-r--r-- | HoTT_Methods.thy | 75 | ||||
| -rw-r--r-- | Prod.thy | 108 | ||||
| -rw-r--r-- | ROOT | 3 | ||||
| -rw-r--r-- | typing.ML | 91 |
5 files changed, 226 insertions, 122 deletions
diff --git a/HoTT_Base.thy b/HoTT_Base.thy index 9e8fee7..e74ef31 100644 --- a/HoTT_Base.thy +++ b/HoTT_Base.thy @@ -1,23 +1,34 @@ -(* -Title: HoTT_Base.thy -Author: Joshua Chen -Date: 2018 +(******** +Isabelle/HoTT: Foundational definitions and notation +Jan 2019 -Basic setup of a homotopy type theory object logic with a cumulative Russell-style universe hierarchy. -*) +This file is the starting point of the Isabelle/HoTT object logic, and contains the basic setup and definitions. +Among other things, it: + +* Sets up object-level type inference functionality (via typing.ML). +* Defines the universe hierarchy and its governing rules. +* Defines named theorems for later use by proof methods. + +********) theory HoTT_Base imports Pure begin - -section \<open>Basic setup\<close> +section \<open>Terms and typing\<close> declare[[eta_contract=false]] -typedecl t \<comment> \<open>Type of object types and terms\<close> -typedecl ord \<comment> \<open>Type of meta-level numerals\<close> +typedecl t \<comment> \<open>Type of object-logic terms (which includes the types)\<close> + +judgment has_type :: "[t, t] \<Rightarrow> prop" ("(3_:/ _)") + +ML_file "typing.ML" + +section \<open>Universes\<close> + +typedecl ord \<comment> \<open>Type of meta-numerals\<close> axiomatization O :: ord and @@ -34,56 +45,44 @@ declare leq_min [intro!] lt_trans [intro] - -section \<open>Judgment\<close> - -judgment hastype :: "[t, t] \<Rightarrow> prop" ("(3_:/ _)") - - -section \<open>Universes\<close> - axiomatization U :: "ord \<Rightarrow> t" where - U_hierarchy: "i < j \<Longrightarrow> U i: U j" and + U_hierarchy: "i < j \<Longrightarrow> U i: U j" and U_cumulative: "\<lbrakk>A: U i; i < j\<rbrakk> \<Longrightarrow> A: U j" and U_cumulative': "\<lbrakk>A: U i; i \<le> j\<rbrakk> \<Longrightarrow> A: U j" text \<open> -Using method @{method rule} with @{thm U_cumulative} is unsafe, if applied blindly it will typically lead to non-termination. -One should instead use @{method elim}, or instantiate @{thm U_cumulative} suitably. +Using method @{method rule} with @{thm U_cumulative} and @{thm U_cumulative'} is unsafe: if applied blindly it will very easily lead to non-termination. +Instead use @{method elim}, or instantiate the rules suitably. -@{thm U_cumulative'} is an alternative rule used by the method \<open>cumulativity\<close> in @{file HoTT_Methods.thy}. +@{thm U_cumulative'} is an alternative rule used by the method @{theory_text cumulativity} in @{file HoTT_Methods.thy}. \<close> - section \<open>Type families\<close> -abbreviation (input) constrained :: "[t \<Rightarrow> t, t, t] \<Rightarrow> prop" ("(1_:/ _ \<longrightarrow> _)") - where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f x: B)" - -text \<open> -The abbreviation @{abbrev constrained} is used to define type families, which are constrained expressions @{term "P: A \<longrightarrow> B"}, where @{term "A::t"} and @{term "B::t"} are small types. -\<close> - -type_synonym tf = "t \<Rightarrow> t" \<comment> \<open>Type of type families\<close> +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> -named_theorems comp named_theorems form +named_theorems comp +named_theorems cong text \<open> -Declare named theorems to be used by proof methods defined in @{file HoTT_Methods.thy}. - -@{attribute comp} declares computation rules, which 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. +The named theorems above will be used by proof methods defined in @{file HoTT_Methods.thy}. @{attribute form} declares type formation rules. These are mainly used by the \<open>cumulativity\<close> method, which lifts types into higher universes. + +@{attribute comp} declares computation rules, which 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. + +@{attribute cong} declares congruence rules, the definitional equality rules of the type theory. \<close> (* Todo: Set up the Simplifier! *) - end diff --git a/HoTT_Methods.thy b/HoTT_Methods.thy index f0cee6c..99f3446 100644 --- a/HoTT_Methods.thy +++ b/HoTT_Methods.thy @@ -1,80 +1,73 @@ -(* -Title: HoTT_Methods.thy -Author: Joshua Chen -Date: 2018 +(******** +Isabelle/HoTT: Proof methods +Jan 2019 -Method setup for the HoTT logic. -*) +********) theory HoTT_Methods imports HoTT_Base "HOL-Eisbach.Eisbach" "HOL-Eisbach.Eisbach_Tools" begin +section \<open>Substitution and simplification\<close> -section \<open>Handling universes\<close> +ML_file "~~/src/Tools/misc_legacy.ML" +ML_file "~~/src/Tools/IsaPlanner/isand.ML" +ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML" +ML_file "~~/src/Tools/IsaPlanner/zipper.ML" +ML_file "~~/src/Tools/eqsubst.ML" +\<comment> \<open>Import the @{method subst} method, used for substituting definitional equalities.\<close> -method provelt = (rule lt_Suc | (rule lt_trans, (rule lt_Suc)+)+) +text \<open> +Method @{theory_text compute} performs single-step simplifications, using any rules declared @{attribute comp} in the context. +Premises of the rule(s) applied are added as new subgoals. +\<close> -method hierarchy = (rule U_hierarchy, provelt) +method compute declares comp = (subst comp) -method cumulativity declares form = ( - ((elim U_cumulative' | (rule U_cumulative', rule form)), rule leq_min) | - ((elim U_cumulative | (rule U_cumulative, rule form)), provelt) -) +section \<open>Handling universes\<close> text \<open> -Methods @{method provelt}, @{method hierarchy}, and @{method cumulativity} prove statements of the form +Methods @{theory_text provelt}, @{theory_text hierarchy}, and @{theory_text cumulativity} prove propositions of the form + \<^item> \<open>n < (Suc (... (Suc n) ...))\<close>, \<^item> \<open>U i: U (Suc (... (Suc i) ...))\<close>, and -\<^item> @{prop "A: U i \<Longrightarrow> A: U j"}, where @{prop "i \<le> j"} -respectively. -\<close> - - -section \<open>Deriving typing judgments\<close> +\<^item> @{prop "A: U i \<Longrightarrow> A: U j"}, where @{prop "i \<le> j"}, -method routine uses add = (assumption | rule add | rule)+ - -text \<open> -The method @{method routine} proves type judgments @{prop "a : A"} using the rules declared @{attribute intro} in the respective theory files, as well as additional provided lemmas passed using the modifier \<open>add\<close>. +respectively. \<close> +method provelt = (rule lt_Suc | (rule lt_trans, (rule lt_Suc)+)+) -section \<open>Substitution and simplification\<close> +method hierarchy = (rule U_hierarchy, provelt) -ML_file "~~/src/Tools/misc_legacy.ML" -ML_file "~~/src/Tools/IsaPlanner/isand.ML" -ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML" -ML_file "~~/src/Tools/IsaPlanner/zipper.ML" -ML_file "~~/src/Tools/eqsubst.ML" +method cumulativity declares form = ( + ((elim U_cumulative' | (rule U_cumulative', rule form)), rule leq_min) | + ((elim U_cumulative | (rule U_cumulative, rule form)), provelt) +) -\<comment> \<open>Import the @{method subst} method, used for substituting definitional equalities.\<close> +section \<open>Deriving typing judgments\<close> -method compute declares comp = (subst comp) +method routine uses add = (assumption | rule add | rule)+ text \<open> -Method @{method compute} performs single-step simplifications, using any rules declared @{attribute comp} in the context. -Premises of the rule(s) applied are added as new subgoals. +The method @{method routine} proves typing judgments @{prop "a: A"} using the rules declared @{attribute intro} in the respective theory files, as well as any additional lemmas provided with the modifier @{theory_text add}. \<close> - section \<open>Derivation search\<close> text \<open> -Combine @{method routine} and @{method compute} to search for derivations of judgments. -Also handle universes using @{method hierarchy} and @{method cumulativity}. +The method @{theory_text derive} combines @{method routine} and @{method compute} to search for derivations of judgments. +It also handles universes using @{method hierarchy} and @{method cumulativity}. \<close> method derive uses lems = (routine add: lems | compute comp: lems | cumulativity form: lems | hierarchy)+ - section \<open>Induction\<close> text \<open> -Placeholder section for the automation of induction, i.e. using the elimination rules. -At the moment one must directly apply them with \<open>rule X_elim\<close>. +Placeholder section for the automation of induction, i.e. using the type elimination rules. +At the moment one must directly apply them with @{method rule} or @{method elim}. \<close> - end @@ -1,91 +1,111 @@ -(* -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>Non-dependent functions are a special case.\<close> +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> 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 +lemmas Prod_cong [cong] = Prod_form_eq Prod_intro_eq +section \<open>Function composition\<close> -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 g :: f :: _ = tms |> map (Typing.tm_of_ptm ctxt) + val dom = + case f of + Const ("Prod.lam", _) $ T $ _ => T + | _ => (case Typing.get_typing f (Typing.typing_assms ctxt) of + SOME (Const ("Prod.Prod", _) $ T $ _) => T + | SOME _ => Exn.error "Can't compose with a non-function" + | NONE => Exn.error "Cannot infer domain of composition: please state this explicitly") + in + @{const compose} $ dom $ g $ f + end +in + [("_compose", compose_tr)] +end +\<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 "compose A (compose B h g) f \<equiv> compose A h (compose A g f)" +by (derive lems: assms cong) 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> + shows "(,\\x: B. c x) o (,\\x: A. b x) \<equiv> ,\\x: A. c (b x)" +by (derive lems: assms cong) +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. diff --git a/typing.ML b/typing.ML new file mode 100644 index 0000000..bfe8409 --- /dev/null +++ b/typing.ML @@ -0,0 +1,91 @@ +(******** +Isabelle/HoTT: typing.ML +Feb 2019 + +Functionality for object-level type inference. + +********) + +signature TYPING = +sig + type jmt = term + val term_of_jmt: jmt -> term + val type_of_jmt: jmt -> term + val typing_assms: Proof.context -> jmt list + val get_typing: term -> jmt list -> term option + val get_global_typing: theory -> Proof.context -> term -> term option + val tm_of_ptm: Proof.context -> term -> term +end + +structure Typing: TYPING = +struct + +type jmt = term + +(* Functions to extract a and A from propositions "a: A" *) +fun term_of_jmt (@{const has_type} $ t $ _) = t + | term_of_jmt _ = Exn.error "Not a typing judgment" + +fun type_of_jmt (@{const has_type} $ _ $ T) = T + | type_of_jmt _ = Exn.error "Not a typing judgment" + +(* Get the typing assumptions in the current context *) +fun typing_assms ctxt = + let + fun is_typing_jmt (@{const has_type} $ _ $ _) = true + | is_typing_jmt _ = false + in + Assumption.all_prems_of ctxt |> map Thm.prop_of |> filter is_typing_jmt + end + +(* Search for a term typing in a list of judgments*) +fun get_typing tm jmts = + let val find_type = + fn jmt => if Term.aconv_untyped (tm, term_of_jmt jmt) then SOME (type_of_jmt jmt) else NONE + in get_first find_type jmts end + +(* Private storage space for global typing data *) +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 +) + +(* Accessor for the above data *) +fun get_global_typing thy ctxt tm = Symtab.lookup (Typing_Data.get thy) (Syntax.string_of_term ctxt tm) + +(* Convert a Pure pre-term to an unchecked Pure term *) +fun tm_of_ptm ctxt ptm = + let val ptm = Term_Position.strip_positions ptm + in + (case ptm of + Const ("_constrain", _) $ Free (t, _) $ (Const ("\<^type>fun", _) $ Const(n1, _) $ Const (n2, _)) => + let + val typ1 = unprefix "\<^type>" n1 + val typ2 = unprefix "\<^type>" n2 + val typ = typ1 ^ "\<Rightarrow>" ^ typ2 |> Syntax.read_typ ctxt + in Free (t, typ) end + | Const ("_constrain", _) $ Free (t, _) $ Const (T, _) => + Free (t, Syntax.read_typ ctxt (unprefix "\<^type>" T)) + | ptm1 $ ptm2 => + (tm_of_ptm ctxt ptm1) $ (tm_of_ptm ctxt ptm2) + | Const (t, T) => + let val t' = + unprefix "\<^const>" t handle Fail "unprefix" => + unprefix "\<^type>" t handle Fail "unprefix" => + t + in Const (t', T) end + | Abs (t, T, ptm') => + Abs (t, T, tm_of_ptm ctxt ptm') + | t => t) + end + +end |