diff options
| -rw-r--r-- | HoTT_Base.thy | 8 | ||||
| -rw-r--r-- | ex/HoTT book/Ch1.thy | 12 |
2 files changed, 18 insertions, 2 deletions
diff --git a/HoTT_Base.thy b/HoTT_Base.thy index 269a3d9..be4899c 100644 --- a/HoTT_Base.thy +++ b/HoTT_Base.thy @@ -42,7 +42,7 @@ where and Ord_lt_monotone: "\<And>m n. m < n \<Longrightarrow> S(m) < S(n)" and - Ord_leq_min: "\<And>n. n \<le> n" + Ord_leq_min: "\<And>n. O \<le> n" and Ord_leq_monotone: "\<And>m n. m \<le> n \<Longrightarrow> S(m) \<le> S(n)" @@ -57,7 +57,11 @@ where 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)" - \<comment> \<open>WARNING: \<open>rule Universe_cumulative\<close> can result in an infinite rewrite loop!\<close> + +text " + The rule \<open>U_cumulative\<close> is very unsafe: if used as-is it will usually lead to an infinite rewrite loop! + It should be instantiated with the right ordinals \<open>i\<close> and \<open>j\<close> before being applied. +" section \<open>Type families\<close> diff --git a/ex/HoTT book/Ch1.thy b/ex/HoTT book/Ch1.thy index d5f05dd..a577fca 100644 --- a/ex/HoTT book/Ch1.thy +++ b/ex/HoTT book/Ch1.thy @@ -40,4 +40,16 @@ proof (rule Sum_elim[where ?p=p]) qed (derive lems: assms) +section \<open>Exercises\<close> + +text "Exercise 1.13" + +abbreviation "not" ("\<not>'(_')") where "\<not>(A) \<equiv> A \<rightarrow> \<zero>" + +text "This proof requires the use of universe cumulativity." + +proposition assumes "A: U(i)" shows "\<^bold>\<lambda>f. f`(inr(\<^bold>\<lambda>a. f`inl(a))): \<not>(\<not>(A + \<not>(A)))" +by (derive lems: assms U_cumulative[where ?A=\<zero> and ?i=O and ?j=i]) + + end |