diff options
author | Josh Chen | 2018-09-16 11:03:48 +0200 |
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committer | Josh Chen | 2018-09-16 11:03:48 +0200 |
commit | d4900ced2e071927d81a21a9127034941f258ec3 (patch) | |
tree | c0289b3fd8337a05baa7740ca3f5e84c57f539ca /Prod.thy | |
parent | 515872533295e8464799467303fff923b52a2c01 (diff) | |
parent | f0999d07a0f41284ba84fae725a0186e0ec9ff5f (diff) |
Reorganized HoTT_Base, updated theories
Diffstat (limited to '')
-rw-r--r-- | Prod.thy | 42 |
1 files changed, 19 insertions, 23 deletions
@@ -12,14 +12,14 @@ begin section \<open>Constants and syntax\<close> axiomatization - Prod :: "[Term, Typefam] \<Rightarrow> Term" and - lambda :: "(Term \<Rightarrow> Term) \<Rightarrow> Term" (binder "\<^bold>\<lambda>" 30) and - appl :: "[Term, Term] \<Rightarrow> Term" (infixl "`" 60) + Prod :: "[t, tf] \<Rightarrow> t" and + lambda :: "(t \<Rightarrow> t) \<Rightarrow> t" (binder "\<^bold>\<lambda>" 30) and + appl :: "[t, t] \<Rightarrow> t" (infixl "`" 60) \<comment> \<open>Application binds tighter than abstraction.\<close> syntax - "_PROD" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Prod>_:_./ _)" 30) - "_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3PROD _:_./ _)" 30) + "_PROD" :: "[idt, t, t] \<Rightarrow> t" ("(3\<Prod>_:_./ _)" 30) + "_PROD_ASCII" :: "[idt, t, t] \<Rightarrow> t" ("(3PROD _:_./ _)" 30) text "The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>." @@ -29,24 +29,24 @@ translations text "Nondependent functions are a special case." -abbreviation Function :: "[Term, Term] \<Rightarrow> Term" (infixr "\<rightarrow>" 40) +abbreviation Function :: "[t, t] \<Rightarrow> t" (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B" 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." @@ -75,19 +75,15 @@ 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 :: "[t, t] \<Rightarrow> t" (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" :: "[t, t] \<Rightarrow> t" (infixr "\<circ>" 110) translations "g \<circ> f" \<rightleftharpoons> "g o f" -section \<open>Atomization\<close> +section \<open>Polymorphic identity function\<close> -text " - Universal statements can be internalized within the theory; the following rule is admissible. -" (* PROOF NEEDED *) - -axiomatization where Prod_atomize: "(\<^bold>\<lambda>x. b(x): \<Prod>x:A. B(x)) \<Longrightarrow> (\<And>x. x: A \<Longrightarrow> b(x): B(x))" +abbreviation id :: t where "id \<equiv> \<^bold>\<lambda>x. x" end |