diff options
Diffstat (limited to 'Prod.thy')
| -rw-r--r-- | Prod.thy | 108 |
1 files changed, 64 insertions, 44 deletions
@@ -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 |