diff options
Diffstat (limited to 'Prod.thy')
| -rw-r--r-- | Prod.thy | 31 |
1 files changed, 19 insertions, 12 deletions
@@ -16,40 +16,47 @@ axiomatization \<comment> \<open>Application binds tighter than abstraction.\<close> appl :: "[Term, Term] \<Rightarrow> Term" (infixl "`" 60) + +section \<open>Syntax\<close> + syntax "_PROD" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Prod>_:_./ _)" 30) "_LAMBDA" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<^bold>\<lambda>_:_./ _)" 30) "_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3PROD _:_./ _)" 30) "_LAMBDA_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3%%_:_./ _)" 30) -\<comment> \<open>The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>.\<close> +text "The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>." + translations "\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod A (\<lambda>x. B)" "\<^bold>\<lambda>x:A. b" \<rightleftharpoons> "CONST lambda A (\<lambda>x. b)" "PROD x:A. B" \<rightharpoonup> "CONST Prod A (\<lambda>x. B)" "%%x:A. b" \<rightharpoonup> "CONST lambda A (\<lambda>x. b)" -\<comment> \<open>Type rules\<close> + +section \<open>Type rules\<close> + axiomatization where - Prod_form [intro]: "\<And>A B. \<lbrakk>A : U; B : A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Prod>x:A. B x : U" + Prod_form: "\<And>A B. \<lbrakk>A : U; B : A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Prod>x:A. B x : U" and - Prod_intro [intro]: "\<And>A B b. \<lbrakk>A : U; \<And>x. x : A \<Longrightarrow> b x : B x\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. b x : \<Prod>x:A. B x" + Prod_intro: "\<And>A B b. \<lbrakk>A : U; \<And>x. x : A \<Longrightarrow> b x : B x\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. b x : \<Prod>x:A. B x" and - Prod_elim [elim]: "\<And>A B f a. \<lbrakk>f : \<Prod>x:A. B x; a : A\<rbrakk> \<Longrightarrow> f`a : B a" + Prod_elim: "\<And>A B f a. \<lbrakk>f : \<Prod>x:A. B x; a : A\<rbrakk> \<Longrightarrow> f`a : B a" and - Prod_comp [simp]: "\<And>A B b a. \<lbrakk>\<And>x. x : A \<Longrightarrow> b x : B x; a : A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. b x)`a \<equiv> b a" + Prod_comp: "\<And>A B b a. \<lbrakk>\<And>x. x : A \<Longrightarrow> b x : B x; a : A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. b x)`a \<equiv> b a" and - Prod_uniq [simp]: "\<And>A B f. f : \<Prod>x:A. B x \<Longrightarrow> \<^bold>\<lambda>x:A. (f`x) \<equiv> f" + Prod_uniq: "\<And>A B f. f : \<Prod>x:A. B x \<Longrightarrow> \<^bold>\<lambda>x:A. (f`x) \<equiv> f" + +text "The type rules should be able to be used as introduction rules by the standard reasoner:" -\<comment> \<open>The funny thing about the first premises of the computation and uniqueness rules is that they introduce a variable B that doesn't actually explicitly appear in the statement of the conclusion. -In a sense, they say something like "if this condition holds for some type family B... (then we can apply the rule)". -This forces the theorem prover to search for a suitable B. Is this additional overhead necessary? -It *is* a safety check for well-formedness...\<close> +lemmas Prod_rules [intro] = Prod_form Prod_intro Prod_elim Prod_comp Prod_uniq text "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)." -\<comment> \<open>Nondependent functions are a special case.\<close> +text "Nondependent functions are a special case." + abbreviation Function :: "[Term, Term] \<Rightarrow> Term" (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_:A. B" + end
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