diff options
Diffstat (limited to 'Prod.thy')
| -rw-r--r-- | Prod.thy | 22 |
1 files changed, 12 insertions, 10 deletions
@@ -41,20 +41,21 @@ abbreviation Function :: "[Term, Term] \<Rightarrow> Term" (infixr "\<rightarro section \<open>Type rules\<close> axiomatization where - Prod_form: "\<And>i A B. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x): U(i)" + Prod_form: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x): U(i)" and - Prod_intro: "\<And>i A B b. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i); \<And>x. x: A \<Longrightarrow> b(x): B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. b(x): \<Prod>x:A. B(x)" + Prod_intro: "\<lbrakk>A: U(i); \<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: "\<And>A B f a. \<lbrakk>f: \<Prod>x:A. B(x); a: A\<rbrakk> \<Longrightarrow> f`a: B(a)" + Prod_elim: "\<lbrakk>f: \<Prod>x:A. B(x); a: A\<rbrakk> \<Longrightarrow> f`a: B(a)" and - Prod_comp: "\<And>i A B b a. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i); \<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: "\<lbrakk>a: A; \<And>x. x: A \<Longrightarrow> b(x): B(x)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. b(x))`a \<equiv> b(a)" and - Prod_uniq: "\<And>A B f. f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x:A. (f`x) \<equiv> f" + Prod_uniq: "f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x:A. (f`x) \<equiv> f" 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). " +(* text " In addition to the usual type rules, it is a meta-theorem (*PROVE THIS!*) 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>. @@ -62,14 +63,15 @@ text " " axiomatization where - Prod_form_cond1: "\<And>i A B. (\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)" + Prod_form_cond1: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)" and - Prod_form_cond2: "\<And>i A B. (\<Prod>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)" + Prod_form_cond2: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)" +*) text "Set up the standard reasoner to use the type rules:" lemmas Prod_rules = Prod_form Prod_intro Prod_elim Prod_comp Prod_uniq -lemmas Prod_form_conds [intro (*elim, wellform*)] = Prod_form_cond1 Prod_form_cond2 +(*lemmas Prod_form_conds [intro (*elim, wellform*)] = Prod_form_cond1 Prod_form_cond2*) lemmas Prod_comps [comp] = Prod_comp Prod_uniq @@ -84,9 +86,9 @@ where and Unit_intro: "\<star>: \<one>" and - Unit_elim: "\<And>i C c a. \<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>); a: \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(a) : C(a)" + Unit_elim: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>); a: \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(a) : C(a)" and - Unit_comp: "\<And>i C c. \<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>)\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(\<star>) \<equiv> c" + Unit_comp: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>)\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(\<star>) \<equiv> c" lemmas Unit_rules [intro] = Unit_form Unit_intro Unit_elim Unit_comp lemmas Unit_comps [comp] = Unit_comp |