diff options
Diffstat (limited to '')
| -rw-r--r-- | Prod.thy | 10 | ||||
| -rw-r--r-- | Sum.thy | 18 |
2 files changed, 14 insertions, 14 deletions
@@ -78,15 +78,15 @@ section \<open>Unit type\<close> axiomatization Unit :: Term ("\<one>") and pt :: Term ("\<star>") and - indUnit :: "[Typefam, Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<one>)") + indUnit :: "[Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<one>)") where - Unit_form: "\<one> : U(O)" + Unit_form: "\<one>: U(O)" and - Unit_intro: "\<star> : \<one>" + 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, c, a) : C(a)" + 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)" and - Unit_comp: "\<And>i C c. \<lbrakk>C: \<one> \<longrightarrow> U(i); c : C(\<star>)\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(C, c, \<star>) \<equiv> c" + Unit_comp: "\<And>i C c. \<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 @@ -15,15 +15,15 @@ section \<open>Constants and syntax\<close> axiomatization Sum :: "[Term, Typefam] \<Rightarrow> Term" and pair :: "[Term, Term] \<Rightarrow> Term" ("(1'(_,/ _'))") and - indSum :: "[Term, Typefam, Typefam, [Term, Term] \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<Sum>[_,/ _])") + indSum :: "[[Term, Term] \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<Sum>)") syntax "_SUM" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Sum>_:_./ _)" 20) "_SUM_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3SUM _:_./ _)" 20) translations - "\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)" - "SUM x:A. B" \<rightharpoonup> "CONST Sum A (\<lambda>x. B)" + "\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum(A, \<lambda>x. B)" + "SUM x:A. B" \<rightharpoonup> "CONST Sum(A, \<lambda>x. B)" text "Nondependent pair." @@ -34,15 +34,15 @@ abbreviation Pair :: "[Term, Term] \<Rightarrow> Term" (infixr "\<times>" 50) section \<open>Type rules\<close> axiomatization where - Sum_form: "\<And>i A B. \<lbrakk>A : U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B x : U(i)" + Sum_form: "\<And>i A B. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x): U(i)" and - Sum_intro: "\<And>i A B a b. \<lbrakk>A : U(i); B: A \<longrightarrow> U(i); a : A; b : B a\<rbrakk> \<Longrightarrow> (a,b) : \<Sum>x:A. B x" + Sum_intro: "\<And>i A B a b. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i); a: A; b: B(a)\<rbrakk> \<Longrightarrow> (a,b) : \<Sum>x:A. B(x)" and Sum_elim: "\<And>i A B C f p. \<lbrakk> - C: \<Sum>x:A. B x \<longrightarrow> U(i); - \<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> f x y : C (x,y); - p : \<Sum>x:A. B x - \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>[A,B] C f p : C p" + C: \<Sum>x:A. B(x) \<longrightarrow> U(i); + \<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f(x, y) : C((x,y)); + p : \<Sum>x:A. B(x) + \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>(f, p) : C(p)" (* What does writing \<lambda>x y. f(x, y) change? *) and Sum_comp: "\<And>i A B C f a b. \<lbrakk> A : U(i); |