diff options
| -rw-r--r-- | Prod.thy | 102 | ||||
| -rw-r--r-- | ProdProps.thy | 57 |
2 files changed, 51 insertions, 108 deletions
@@ -1,89 +1,89 @@ -(* Title: HoTT/Prod.thy - Author: Josh Chen +(* +Title: Prod.thy +Author: Joshua Chen +Date: 2018 -Dependent product (function) type +Dependent product type *) theory Prod - imports HoTT_Base +imports HoTT_Base HoTT_Methods + begin -section \<open>Constants and syntax\<close> +section \<open>Basic definitions\<close> axiomatization - Prod :: "[t, tf] \<Rightarrow> t" and + 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> + appl :: "[t, t] \<Rightarrow> t" ("(1_`/_)" [60, 61] 60) \<comment> \<open>Application binds tighter than abstraction.\<close> syntax - "_PROD" :: "[idt, t, t] \<Rightarrow> t" ("(3\<Prod>_:_./ _)" 30) - "_PROD_ASCII" :: "[idt, t, t] \<Rightarrow> t" ("(3PROD _:_./ _)" 30) + "_prod" :: "[idt, t, t] \<Rightarrow> t" ("(3\<Prod>_:_./ _)" 30) + "_prod_ascii" :: "[idt, t, t] \<Rightarrow> t" ("(3II _:_./ _)" 30) -text "The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>." +text \<open>The translations below bind the variable @{term x} in the expressions @{term B} and @{term b}.\<close> translations "\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod A (\<lambda>x. B)" - "PROD x:A. B" \<rightharpoonup> "CONST Prod A (\<lambda>x. B)" + "II x:A. B" \<rightharpoonup> "CONST Prod A (\<lambda>x. B)" -text "Nondependent functions are a special case." +text \<open>Non-dependent functions are a special case.\<close> -abbreviation Function :: "[t, t] \<Rightarrow> t" (infixr "\<rightarrow>" 40) +abbreviation Fun :: "[t, t] \<Rightarrow> t" (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B" +axiomatization where +\<comment> \<open>Type rules\<close> -section \<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 -axiomatization where - 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" -and - 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" -and - 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> 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. - - Note that the syntax \<open>\<^bold>\<lambda>\<close> (bold lambda) used for dependent functions clashes with the proof term syntax (cf. \<section>2.5.2 of the Isabelle/Isar Implementation). -" - -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>. - - That is to say, the following inference rules are admissible, and it simplifies proofs greatly to axiomatize them directly. -" + 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 -axiomatization where - 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_elim: "\<lbrakk>f: \<Prod>x:A. B x; a: A\<rbrakk> \<Longrightarrow> f`a: B a" and + + Prod_comp: "\<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" and +\<comment> \<open>Congruence rules\<close> -text "Rule attribute declarations---set up various methods to use the type rules." + 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_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" + +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}. +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_comp [comp] = Prod_appl Prod_uniq -lemmas Prod_wellform [wellform] = Prod_wellform1 Prod_wellform2 lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim +lemmas Prod_comps [comp] = Prod_comp Prod_uniq -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) +syntax "_compose" :: "[t, t] \<Rightarrow> t" (infixr "\<circ>" 110) translations "g \<circ> f" \<rightleftharpoons> "g o f" +declare compose_def [comp] + +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) -section \<open>Polymorphic identity function\<close> +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" +abbreviation id :: t where "id \<equiv> \<^bold>\<lambda>x. x" \<comment> \<open>Polymorphic identity function\<close> end diff --git a/ProdProps.thy b/ProdProps.thy deleted file mode 100644 index a68f79b..0000000 --- a/ProdProps.thy +++ /dev/null @@ -1,57 +0,0 @@ -(* Title: HoTT/ProdProps.thy - Author: Josh Chen - -Properties of the dependent product -*) - -theory ProdProps - imports - HoTT_Methods - Prod -begin - - -section \<open>Composition\<close> - -text " - The proof of associativity needs some guidance; it involves telling Isabelle to use the correct rule for \<Prod>-type definitional equality, and the correct substitutions in the subgoals thereafter. -" - -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)" -proof (subst (0 1 2 3) compose_def) - show "\<^bold>\<lambda>x. (\<^bold>\<lambda>y. h`(g`y))`(f`x) \<equiv> \<^bold>\<lambda>x. h`((\<^bold>\<lambda>y. g`(f`y))`x)" - proof (subst Prod_eq) - \<comment> \<open>Todo: set the Simplifier (or other simplification methods) up to use \<open>Prod_eq\<close>!\<close> - - show "\<And>x. x: A \<Longrightarrow> (\<^bold>\<lambda>y. h`(g`y))`(f`x) \<equiv> h`((\<^bold>\<lambda>y. g`(f`y))`x)" - proof compute - show "\<And>x. x: A \<Longrightarrow> h`(g`(f`x)) \<equiv> h`((\<^bold>\<lambda>y. g`(f`y))`x)" - proof compute - show "\<And>x. x: A \<Longrightarrow> g`(f`x): C" by (routine lems: assms) - qed - show "\<And>x. x: B \<Longrightarrow> h`(g`x): D (g`x)" by (routine lems: assms) - qed (routine lems: assms) - qed fact -qed - - -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)" -proof (subst compose_def, subst Prod_eq) - show "\<And>a. a: A \<Longrightarrow> (\<^bold>\<lambda>x. c x)`((\<^bold>\<lambda>x. b x)`a) \<equiv> (\<^bold>\<lambda>x. c (b x))`a" - proof compute - show "\<And>a. a: A \<Longrightarrow> c ((\<^bold>\<lambda>x. b x)`a) \<equiv> (\<^bold>\<lambda>x. c (b x))`a" - by (derive lems: assms) - qed (routine lems: assms) -qed (derive lems: assms) - - -text "Set up the \<open>compute\<close> method to automatically simplify function compositions." - -lemmas compose_comp [comp] - - -end |