From e94784953a751b0720689b686e607c95ba0f0592 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Wed, 15 Aug 2018 00:00:09 +0200
Subject: Function composition, rename path composition to differentiate the
 two.

---
 Prod.thy | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)

(limited to 'Prod.thy')

diff --git a/Prod.thy b/Prod.thy
index 76b66dd..496bf3e 100644
--- a/Prod.thy
+++ b/Prod.thy
@@ -68,6 +68,22 @@ lemmas Prod_rules [intro] = Prod_form Prod_intro Prod_elim Prod_comp Prod_uniq
 lemmas Prod_wellform [wellform] = Prod_form_cond1 Prod_form_cond2
 lemmas Prod_comps [comp] = Prod_comp Prod_uniq
 
+subsection \<open>Composition\<close>
+
+axiomatization fncompose :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<circ>" 70) where
+  fncompose_type: "\<lbrakk>
+    g: \<Prod>x:B. C(x);
+    f: A \<rightarrow> B;
+    (\<Prod>x:B. C(x)): U(i);
+    A \<rightarrow> B: U(i)
+  \<rbrakk> \<Longrightarrow> g \<circ> f: \<Prod>x:A. C(f`x)"
+and
+  fncompose_comp: "\<lbrakk>
+    A: U(i);
+    \<And>x. x: A \<Longrightarrow> b(x): B;
+    \<And>x. x: A \<Longrightarrow> c(x): C(x)
+  \<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. c(x)) \<circ> (\<^bold>\<lambda>x. b(x)) \<equiv> \<^bold>\<lambda>x. c(b(x))"
+
 
 section \<open>Unit type\<close>
 
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