diff options
Diffstat (limited to '')
-rw-r--r-- | Prod.thy | 22 |
1 files changed, 8 insertions, 14 deletions
@@ -14,23 +14,19 @@ section \<open>Constants and syntax\<close> axiomatization Prod :: "[Term, Typefam] \<Rightarrow> Term" and - lambda :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" and - appl :: "[Term, Term] \<Rightarrow> Term" ("(1_`_)" [61, 60] 60) + lambda :: "(Term \<Rightarrow> Term) \<Rightarrow> Term" (binder "\<^bold>\<lambda>" 30) and + appl :: "[Term, Term] \<Rightarrow> Term" (infixl "`" 60) \<comment> \<open>Application binds tighter than abstraction.\<close> syntax "_PROD" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Prod>_:_./ _)" 30) - "_LAMBDA" :: "[idt, Term, Term] \<Rightarrow> Term" ("(1\<^bold>\<lambda>_:_./ _)" 30) "_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3PROD _:_./ _)" 30) - "_LAMBDA_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3%%_:_./ _)" 30) 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)" text "Nondependent functions are a special case." @@ -43,21 +39,20 @@ section \<open>Type rules\<close> 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>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)" + Prod_intro: "\<lbrakk>A: U(i); \<And>x. x: A \<Longrightarrow> b(x): B(x)\<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_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)" + 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_uniq: "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. (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>. + 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. " @@ -66,12 +61,11 @@ axiomatization where Prod_form_cond1: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)" and 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_rules [intro] = Prod_form Prod_intro Prod_elim Prod_comp Prod_uniq +lemmas Prod_wellform [wellform] = Prod_form_cond1 Prod_form_cond2 lemmas Prod_comps [comp] = Prod_comp Prod_uniq |