From 64d2a5c60acce40113362c9d7eca8cd633362d23 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Tue, 5 Feb 2019 18:32:43 +0100
Subject: Add cong named theorem for congruence rules

---
 HoTT_Base.thy | 9 ++++++---
 Prod.thy      | 5 +++--
 2 files changed, 9 insertions(+), 5 deletions(-)

diff --git a/HoTT_Base.thy b/HoTT_Base.thy
index 88cf986..e74ef31 100644
--- a/HoTT_Base.thy
+++ b/HoTT_Base.thy
@@ -68,16 +68,19 @@ text \<open>We use the notation @{prop "B: A \<leadsto> U i"} to abbreviate type
 
 section \<open>Named theorems\<close>
 
-named_theorems comp
 named_theorems form
+named_theorems comp
+named_theorems cong
 
 text \<open>
 The named theorems above will be used by proof methods defined in @{file HoTT_Methods.thy}.
 
-@{attribute comp} declares computation rules, which are used by the \<open>compute\<close> method, and may also be passed to invocations of the method \<open>subst\<close> to perform equational rewriting.
-
 @{attribute form} declares type formation rules.
 These are mainly used by the \<open>cumulativity\<close> method, which lifts types into higher universes.
+
+@{attribute comp} declares computation rules, which are used by the \<open>compute\<close> method, and may also be passed to invocations of the method \<open>subst\<close> to perform equational rewriting.
+
+@{attribute cong} declares congruence rules, the definitional equality rules of the type theory.
 \<close>
 
 (* Todo: Set up the Simplifier! *)
diff --git a/Prod.thy b/Prod.thy
index a28c52a..0de7d89 100644
--- a/Prod.thy
+++ b/Prod.thy
@@ -66,6 +66,7 @@ Note that this is a separate rule from function extensionality.
 lemmas Prod_form [form]
 lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim
 lemmas Prod_comp [comp] = Prod_cmp Prod_uniq
+lemmas Prod_cong [cong] = Prod_form_eq Prod_intro_eq
 
 section \<open>Function composition\<close>
 
@@ -98,12 +99,12 @@ end
 lemma compose_assoc:
   assumes "A: U i" "f: A \<rightarrow> B" "g: B \<rightarrow> C" "h: TT x: C. D x"
   shows "compose A (compose B h g) f \<equiv> compose A h (compose A g f)"
-by (derive lems: assms Prod_intro_eq)
+by (derive lems: assms cong)
 
 lemma compose_comp:
   assumes "A: U i" and "\<And>x. x: A \<Longrightarrow> b x: B" and "\<And>x. x: B \<Longrightarrow> c x: C x"
   shows "(,\\x: B. c x) o (,\\x: A. b x) \<equiv> ,\\x: A. c (b x)"
-by (derive lems: assms Prod_intro_eq)
+by (derive lems: assms cong)
 
 abbreviation id :: "t \<Rightarrow> t" where "id A \<equiv> ,\\x: A. x"
 
-- 
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