diff -r 934d4aed8497 -r 2510e6f7b11c src/CTT/ex/Equality.thy
--- a/src/CTT/ex/Equality.thy	Wed Oct 26 17:22:12 2022 +0100
+++ b/src/CTT/ex/Equality.thy	Wed Oct 26 18:08:44 2022 +0100
@@ -6,63 +6,62 @@
 section "Equality reasoning by rewriting"
 
 theory Equality
-imports "../CTT"
+  imports "../CTT"
 begin
 
 lemma split_eq: "p : Sum(A,B) \<Longrightarrow> split(p,pair) = p : Sum(A,B)"
-apply (rule EqE)
-apply (rule elim_rls, assumption)
-apply rew
-done
+  apply (rule EqE)
+  apply (rule elim_rls, assumption)
+  apply rew
+  done
 
 lemma when_eq: "\<lbrakk>A type; B type; p : A+B\<rbrakk> \<Longrightarrow> when(p,inl,inr) = p : A + B"
-apply (rule EqE)
-apply (rule elim_rls, assumption)
-apply rew
-done
+  apply (rule EqE)
+  apply (rule elim_rls, assumption)
+   apply rew
+  done
 
-(*in the "rec" formulation of addition, 0+n=n *)
+text \<open>in the "rec" formulation of addition, $0+n=n$\<close>
 lemma "p:N \<Longrightarrow> rec(p,0, \<lambda>y z. succ(y)) = p : N"
-apply (rule EqE)
-apply (rule elim_rls, assumption)
-apply rew
-done
+  apply (rule EqE)
+  apply (rule elim_rls, assumption)
+   apply rew
+  done
 
-(*the harder version, n+0=n: recursive, uses induction hypothesis*)
+text \<open>the harder version, $n+0=n$: recursive, uses induction hypothesis\<close>
 lemma "p:N \<Longrightarrow> rec(p,0, \<lambda>y z. succ(z)) = p : N"
-apply (rule EqE)
-apply (rule elim_rls, assumption)
-apply hyp_rew
-done
+  apply (rule EqE)
+  apply (rule elim_rls, assumption)
+   apply hyp_rew
+  done
 
-(*Associativity of addition*)
+text \<open>Associativity of addition\<close>
 lemma "\<lbrakk>a:N; b:N; c:N\<rbrakk>
   \<Longrightarrow> rec(rec(a, b, \<lambda>x y. succ(y)), c, \<lambda>x y. succ(y)) =
     rec(a, rec(b, c, \<lambda>x y. succ(y)), \<lambda>x y. succ(y)) : N"
-apply (NE a)
-apply hyp_rew
-done
+  apply (NE a)
+    apply hyp_rew
+  done
 
-(*Martin-Löf (1984) page 62: pairing is surjective*)
+text \<open>Martin-Löf (1984) page 62: pairing is surjective\<close>
 lemma "p : Sum(A,B) \<Longrightarrow> <split(p,\<lambda>x y. x), split(p,\<lambda>x y. y)> = p : Sum(A,B)"
-apply (rule EqE)
-apply (rule elim_rls, assumption)
-apply (tactic \<open>DEPTH_SOLVE_1 (rew_tac \<^context> [])\<close>) (*!!!!!!!*)
-done
+  apply (rule EqE)
+  apply (rule elim_rls, assumption)
+  apply (tactic \<open>DEPTH_SOLVE_1 (rew_tac \<^context> [])\<close>) (*!!!!!!!*)
+  done
 
 lemma "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>u. split(u, \<lambda>v w.<w,v>)) ` <a,b> = <b,a> : \<Sum>x:B. A"
-apply rew
-done
+  by rew
 
-(*a contrived, complicated simplication, requires sum-elimination also*)
+text \<open>a contrived, complicated simplication, requires sum-elimination also\<close>
 lemma "(\<^bold>\<lambda>f. \<^bold>\<lambda>x. f`(f`x)) ` (\<^bold>\<lambda>u. split(u, \<lambda>v w.<w,v>)) =
       \<^bold>\<lambda>x. x  :  \<Prod>x:(\<Sum>y:N. N). (\<Sum>y:N. N)"
-apply (rule reduction_rls)
-apply (rule_tac [3] intrL_rls)
-apply (rule_tac [4] EqE)
-apply (erule_tac [4] SumE)
-(*order of unifiers is essential here*)
-apply rew
-done
+  apply (rule reduction_rls)
+    apply (rule_tac [3] intrL_rls)
+     apply (rule_tac [4] EqE)
+     apply (erule_tac [4] SumE)
+    (*order of unifiers is essential here*)
+     apply rew
+  done
 
 end