diff options
Diffstat (limited to 'Prod.thy')
-rw-r--r-- | Prod.thy | 22 |
1 files changed, 11 insertions, 11 deletions
@@ -36,17 +36,17 @@ abbreviation Function :: "[Term, Term] \<Rightarrow> Term" (infixr "\<rightarro 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)" + Prod_form: "\<lbrakk>A: U i; B: A \<longrightarrow> U i\<rbrakk> \<Longrightarrow> \<Prod>x:A. B x: U i" and - Prod_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x): B(x); A: U(i)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x): \<Prod>x:A. B(x)" + Prod_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<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)" + Prod_elim: "\<lbrakk>f: \<Prod>x:A. B x; a: A\<rbrakk> \<Longrightarrow> f`a: B a" and - Prod_appl: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x): B(x); a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b(x))`a \<equiv> b(a)" + Prod_appl: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b x)`a \<equiv> b a" and - Prod_uniq: "f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f" + Prod_uniq: "f : \<Prod>x:A. B x \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f" and - Prod_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x) \<equiv> b'(x); A: U(i)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) \<equiv> \<^bold>\<lambda>x. b'(x)" + Prod_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b x \<equiv> \<^bold>\<lambda>x. c x" text " The Pure rules for \<open>\<equiv>\<close> only let us judge strict syntactic equality of object lambda expressions; Prod_eq is the actual definitional equality rule. @@ -55,15 +55,15 @@ text " " text " - 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>. + 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. " axiomatization where - Prod_wellform1: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)" + Prod_wellform1: "(\<Prod>x:A. B x: U i) \<Longrightarrow> A: U i" and - Prod_wellform2: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)" + Prod_wellform2: "(\<Prod>x:A. B x: U i) \<Longrightarrow> B: A \<longrightarrow> U i" text "Rule attribute declarations---set up various methods to use the type rules." @@ -75,9 +75,9 @@ lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim section \<open>Function composition\<close> -definition compose :: "[Term, Term] \<Rightarrow> Term" (infixr "o" 70) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)" +definition compose :: "[Term, Term] \<Rightarrow> Term" (infixr "o" 110) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)" -syntax "_COMPOSE" :: "[Term, Term] \<Rightarrow> Term" (infixr "\<circ>" 70) +syntax "_COMPOSE" :: "[Term, Term] \<Rightarrow> Term" (infixr "\<circ>" 110) translations "g \<circ> f" \<rightleftharpoons> "g o f" |