diff options
author | Josh Chen | 2018-07-11 19:37:07 +0200 |
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committer | Josh Chen | 2018-07-11 19:37:07 +0200 |
commit | 9723fc3ffc55b22a2a8ec09cbba80f14c40d7991 (patch) | |
tree | 57735319777d3a6423a31bd49161bf810f5b9d94 /HoTT_Base.thy | |
parent | a85feff048010fa945c0e498e45aa5626f54f352 (diff) |
Universes implemented. Type rules modified accordingly. No more automatic derivation of "A:U" from "a:A".
Diffstat (limited to '')
-rw-r--r-- | HoTT_Base.thy | 35 |
1 files changed, 12 insertions, 23 deletions
diff --git a/HoTT_Base.thy b/HoTT_Base.thy index 610e5d8..703f1aa 100644 --- a/HoTT_Base.thy +++ b/HoTT_Base.thy @@ -30,21 +30,12 @@ named_theorems comp section \<open>Judgments\<close> -text "Formalize the basic judgments. +text "Formalize the context and typing judgments \<open>a : A\<close>. -For judgmental equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it. - -The judgment \<open>is_a_type A\<close> expresses the statement ``A is an inhabitant of some universe type'', i.e. ``A is a type''." +For judgmental equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it." consts - is_of_type :: "[Term, Term] \<Rightarrow> prop" ("(1_ : _)" [0, 1000] 1000) - is_a_type :: "Term \<Rightarrow> prop" ("(1_ : U)" [0] 1000) - - -text "The following fact is used to make explicit the assumption of well-formed contexts." - -axiomatization where - inhabited_implies_type [intro, elim, wellform]: "\<And>a A. a : A \<Longrightarrow> A : U" + is_of_type :: "[Term, Term] \<Rightarrow> prop" ("(1_ : _)" [0, 0] 1000) section \<open>Universes\<close> @@ -62,31 +53,29 @@ where and Numeral_lt_trans: "\<And>m n. m <- n \<Longrightarrow> S m <- S n" -\<comment> \<open>This lets \<open>standard\<close> prove ordering statements on Numerals.\<close> -lemmas Numeral_lt_rules [intro] = Numeral_lt_min Numeral_lt_trans +lemmas Numeral_rules [intro] = Numeral_lt_min Numeral_lt_trans + \<comment> \<open>Lets \<open>standard\<close> automatically solve inequalities.\<close> text "Universe types:" axiomatization - U :: "Numeral \<Rightarrow> Term" ("U_") + U :: "Numeral \<Rightarrow> Term" ("U _") where Universe_hierarchy: "\<And>i j. i <- j \<Longrightarrow> U(i) : U(j)" and - Universe_cumulative: "\<And>a i j. \<lbrakk>a : U(i); i <- j\<rbrakk> \<Longrightarrow> a : U(j)" - -lemmas Universe_rules [intro] = Universe_hierarchy Universe_cumulative + Universe_cumulative: "\<And>A i j. \<lbrakk>A : U(i); i <- j\<rbrakk> \<Longrightarrow> A : U(j)" section \<open>Type families\<close> -text "A (one-variable) type family is a meta lambda term \<open>P :: Term \<Rightarrow> Term\<close> that further satisfies the following property." +text "We define the following abbreviation constraining the output type of a meta lambda expression when given input of certain type." -type_synonym Typefam = "Term \<Rightarrow> Term" +abbreviation (input) constrained :: "[Term \<Rightarrow> Term, Term, Term] \<Rightarrow> prop" ("_: _ \<longrightarrow> _") + where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f x : B)" -abbreviation (input) is_type_family :: "[Typefam, Term] \<Rightarrow> prop" ("(3_:/ _ \<rightarrow> U)") - where "P: A \<rightarrow> U \<equiv> (\<And>x. x : A \<Longrightarrow> P x : U)" +text "The above is used to define type families, which are just constrained meta-lambdas \<open>P: A \<longrightarrow> B\<close> where \<open>A\<close> and \<open>B\<close> are elements of some universe type." -\<comment> \<open>There is an obvious generalization to multivariate type families, but implementing such an abbreviation would probably involve writing ML code, and is for the moment not really crucial.\<close> +type_synonym Typefam = "Term \<Rightarrow> Term" end
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