From 76deb7ae15fa00b5498ab43db020a0364499251e Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Fri, 6 Jul 2018 10:48:41 +0200
Subject: Added attributes to type elimination rules, not sure if these will
 actually be needed or useful later. Added [comp] attribute to be used by
 simplification met methods.

---
 scratch.thy | 55 ++++++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 54 insertions(+), 1 deletion(-)

(limited to 'scratch.thy')

diff --git a/scratch.thy b/scratch.thy
index b90fd59..1c921bd 100644
--- a/scratch.thy
+++ b/scratch.thy
@@ -3,6 +3,59 @@ theory scratch
 
 begin
 
-schematic_goal "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> ?x : A \<times> B" by simple
+(* Typechecking *)
+schematic_goal "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> (a,b) : ?A" by simple
+
+
+(* Simplification *)
+
+notepad begin
+
+assume asm:
+  "f`a \<equiv> g"
+  "g`b \<equiv> \<^bold>\<lambda>x:A. d"
+  "c : A"
+  "d : B"
+
+have "f`a`b`c \<equiv> d" by (simp add: asm | rule comp | derive lems: asm)+
+
+end
+
+lemma "a : A \<Longrightarrow> indEqual[A] (\<lambda>x y _. y =\<^sub>A x) (\<lambda>x. refl(x)) a a refl(a) \<equiv> refl(a)"
+  by verify_simp
+
+lemma "\<lbrakk>a : A; \<And>x. x : A \<Longrightarrow> b x : X\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. b x)`a \<equiv> b a"
+  by verify_simp
+
+lemma
+  assumes "a : A" and "\<And>x. x : A \<Longrightarrow> b x : B x"
+  shows "(\<^bold>\<lambda>x:A. b x)`a \<equiv> b a"
+  by (verify_simp lems: assms)
+
+lemma "\<lbrakk>a : A; b : B a; B: A \<rightarrow> U\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B x. c x y)`a`b \<equiv> c a b"
+oops
+
+lemma
+  assumes
+    "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B x. c x y)`a \<equiv> \<^bold>\<lambda>y:B a. c a y"
+    "(\<^bold>\<lambda>y:B a. c a y)`b \<equiv> c a b"
+  shows "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B x. c x y)`a`b \<equiv> c a b"
+apply (simp add: assms)
+done
+
+lemmas lems =
+  Prod_comp[where ?A = "B a" and ?b = "\<lambda>b. c a b" and ?a = b]
+  Prod_comp[where ?A = A and ?b = "\<lambda>x. \<^bold>\<lambda>y:B x. c x y" and ?a = a]
+
+lemma
+  assumes
+    "a : A"
+    "b : B a"
+    "B: A \<rightarrow> U"
+    "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> c x y : D x y"
+  shows "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B x. c x y)`a`b \<equiv> c a b"
+apply (verify_simp lems: assms)
+
+
 
 end
\ No newline at end of file
-- 
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