From f0fab6e197510ce0e6d23a669f69de966701d495 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Wed, 8 Jul 2020 15:54:04 +0200
Subject: minor

---
 hott/Identity.thy | 4 ++--
 hott/Nat.thy      | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

(limited to 'hott')

diff --git a/hott/Identity.thy b/hott/Identity.thy
index 8675134..308e664 100644
--- a/hott/Identity.thy
+++ b/hott/Identity.thy
@@ -473,7 +473,7 @@ Lemma (derive) right_whisker:
       have "t \<bullet> refl x = t" by (rule pathcomp_refl)
       also have ".. = s" by fact
       also have ".. = s \<bullet> refl x" by (rule pathcomp_refl[symmetric])
-      finally show "t \<bullet> refl x = s \<bullet> refl x" by this
+      finally show "{}" by this
     qed
   done
 
@@ -486,7 +486,7 @@ Lemma (derive) left_whisker:
       have "refl x \<bullet> t = t" by (rule refl_pathcomp)
       also have ".. = s" by fact
       also have ".. = refl x \<bullet> s" by (rule refl_pathcomp[symmetric])
-      finally show "refl x \<bullet> t = refl x \<bullet> s" by this
+      finally show "{}" by this
     qed
   done
 
diff --git a/hott/Nat.thy b/hott/Nat.thy
index 22c1860..50b7996 100644
--- a/hott/Nat.thy
+++ b/hott/Nat.thy
@@ -151,7 +151,7 @@ Lemma (derive) zero_mul:
     proof reduce
       have "0 + 0 * n = 0 + 0 " by (eq ih) refl
       also have ".. = 0" by reduce refl
-      finally show "0 + 0 * n = 0" by this
+      finally show "{}" by this
     qed
   done
 
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