diff options
Diffstat (limited to 'Equal.thy')
-rw-r--r-- | Equal.thy | 32 |
1 files changed, 16 insertions, 16 deletions
@@ -12,13 +12,13 @@ begin section \<open>Constants and syntax\<close> axiomatization - Equal :: "[Term, Term, Term] \<Rightarrow> Term" and - refl :: "Term \<Rightarrow> Term" and - indEqual :: "[Term \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>=)") + Equal :: "[t, t, t] \<Rightarrow> t" and + refl :: "t \<Rightarrow> t" and + indEqual :: "[t \<Rightarrow> t, t] \<Rightarrow> t" ("(1ind\<^sub>=)") syntax - "_EQUAL" :: "[Term, Term, Term] \<Rightarrow> Term" ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100) - "_EQUAL_ASCII" :: "[Term, Term, Term] \<Rightarrow> Term" ("(3_ =[_]/ _)" [101, 0, 101] 100) + "_EQUAL" :: "[t, t, t] \<Rightarrow> t" ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100) + "_EQUAL_ASCII" :: "[t, t, t] \<Rightarrow> t" ("(3_ =[_]/ _)" [101, 0, 101] 100) translations "a =[A] b" \<rightleftharpoons> "CONST Equal A a b" "a =\<^sub>A b" \<rightharpoonup> "CONST Equal A a b" @@ -27,33 +27,33 @@ translations section \<open>Type rules\<close> axiomatization where - Equal_form: "\<lbrakk>A: U(i); a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U(i)" + Equal_form: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U i" and - Equal_intro: "a : A \<Longrightarrow> refl(a): a =\<^sub>A a" + Equal_intro: "a : A \<Longrightarrow> (refl a): a =\<^sub>A a" and Equal_elim: "\<lbrakk> x: A; y: A; p: x =\<^sub>A y; - \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x); - \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i) - \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(p) : C(x)(y)(p)" + \<And>x. x: A \<Longrightarrow> f x: C x x (refl x); + \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i + \<rbrakk> \<Longrightarrow> ind\<^sub>= f p : C x y p" and Equal_comp: "\<lbrakk> a: A; - \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x); - \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i) - \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(refl(a)) \<equiv> f(a)" + \<And>x. x: A \<Longrightarrow> f x: C x x (refl x); + \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i + \<rbrakk> \<Longrightarrow> ind\<^sub>= f (refl a) \<equiv> f a" text "Admissible inference rules for equality type formation:" axiomatization where - Equal_wellform1: "a =\<^sub>A b: U(i) \<Longrightarrow> A: U(i)" + Equal_wellform1: "a =\<^sub>A b: U i \<Longrightarrow> A: U i" and - Equal_wellform2: "a =\<^sub>A b: U(i) \<Longrightarrow> a: A" + Equal_wellform2: "a =\<^sub>A b: U i \<Longrightarrow> a: A" and - Equal_wellform3: "a =\<^sub>A b: U(i) \<Longrightarrow> b: A" + Equal_wellform3: "a =\<^sub>A b: U i \<Longrightarrow> b: A" text "Rule attribute declarations:" |