diff options
Diffstat (limited to '')
-rw-r--r-- | HoTT_Base.thy | 14 |
1 files changed, 6 insertions, 8 deletions
diff --git a/HoTT_Base.thy b/HoTT_Base.thy index 4e1a9be..a5b88fd 100644 --- a/HoTT_Base.thy +++ b/HoTT_Base.thy @@ -16,8 +16,6 @@ text "Meta syntactic type for object-logic types and terms." typedecl Term -section \<open>Judgments\<close> - text " Formalize the typing judgment \<open>a: A\<close>. For judgmental/definitional equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it. @@ -38,13 +36,13 @@ axiomatization lt :: "[Ord, Ord] \<Rightarrow> prop" (infix "<" 999) and leq :: "[Ord, Ord] \<Rightarrow> prop" (infix "\<le>" 999) where - Ord_lt_min: "\<And>n. O < S(n)" + Ord_lt_min: "\<And>n. O < S n" and - Ord_lt_monotone: "\<And>m n. m < n \<Longrightarrow> S(m) < S(n)" + Ord_lt_monotone: "\<And>m n. m < n \<Longrightarrow> S m < S n" and Ord_leq_min: "\<And>n. O \<le> n" and - Ord_leq_monotone: "\<And>m n. m \<le> n \<Longrightarrow> S(m) \<le> S(n)" + Ord_leq_monotone: "\<And>m n. m \<le> n \<Longrightarrow> S m \<le> S n" lemmas Ord_rules [intro] = Ord_lt_min Ord_lt_monotone Ord_leq_min Ord_leq_monotone \<comment> \<open>Enables \<open>standard\<close> to automatically solve inequalities.\<close> @@ -54,9 +52,9 @@ text "Define the universe types." axiomatization U :: "Ord \<Rightarrow> Term" where - U_hierarchy: "\<And>i j. i < j \<Longrightarrow> U(i): U(j)" + U_hierarchy: "\<And>i j. i < j \<Longrightarrow> U i: U j" and - U_cumulative: "\<And>A i j. \<lbrakk>A: U(i); i \<le> j\<rbrakk> \<Longrightarrow> A: U(j)" + U_cumulative: "\<And>A i j. \<lbrakk>A: U i; i \<le> j\<rbrakk> \<Longrightarrow> A: U j" text " The rule \<open>U_cumulative\<close> is very unsafe: if used as-is it will usually lead to an infinite rewrite loop! @@ -71,7 +69,7 @@ text " " abbreviation (input) constrained :: "[Term \<Rightarrow> Term, Term, Term] \<Rightarrow> prop" ("(1_: _ \<longrightarrow> _)") - where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f(x): B)" + where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f x: B)" text " The above is used to define type families, which are constrained meta-lambdas \<open>P: A \<longrightarrow> B\<close> where \<open>A\<close> and \<open>B\<close> are small types. |