From 8e4ca285430c7bcdabbd4ea34da38e0770f4a832 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Mon, 17 Sep 2018 15:33:42 +0200
Subject: Tweak proof

---
 HoTT_Methods.thy |  4 ++--
 Proj.thy         | 13 +++++--------
 2 files changed, 7 insertions(+), 10 deletions(-)

diff --git a/HoTT_Methods.thy b/HoTT_Methods.thy
index 9849195..411176e 100644
--- a/HoTT_Methods.thy
+++ b/HoTT_Methods.thy
@@ -50,10 +50,10 @@ method derive uses lems = (routine add: lems | compute comp: lems)+
 
 section \<open>Induction\<close>
 
-text "
+text \<open>
   Placeholder section for the automation of induction, i.e. using the elimination rules.
   At the moment one must directly apply them with \<open>rule X_elim\<close>.
-"
+\<close>
 
 
 end
diff --git a/Proj.thy b/Proj.thy
index f272224..e76e8d3 100644
--- a/Proj.thy
+++ b/Proj.thy
@@ -7,10 +7,7 @@ Projection functions \<open>fst\<close> and \<open>snd\<close> for the dependent
 *)
 
 theory Proj
-imports
-  HoTT_Methods
-  Prod
-  Sum
+imports HoTT_Methods Prod Sum
 
 begin
 
@@ -28,12 +25,12 @@ unfolding fst_def by compute (derive lems: assms)
 
 lemma snd_type:
   assumes "A: U i" and "B: A \<longrightarrow> U i" and "p: \<Sum>x:A. B x" shows "snd p: B (fst p)"
-unfolding snd_def proof
-  show "\<And>p. p: \<Sum>x:A. B x \<Longrightarrow> B (fst p): U i" by (derive lems: assms fst_type)
-  
+unfolding snd_def proof (derive lems: assms)
+  show "\<And>p. p: \<Sum>x:A. B x \<Longrightarrow> fst p: A" using assms(1-2) by (rule fst_type)
+
   fix x y assume asm: "x: A" "y: B x"
   show "y: B (fst <x,y>)" by (derive lems: asm assms fst_comp)
-qed fact
+qed
 
 lemma snd_comp:
   assumes "A: U i" and "B: A \<longrightarrow> U i" and "a: A" and "b: B a" shows "snd <a,b> \<equiv> b"
-- 
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