(* Title: HOL/Library/Function_Division.thy
Author: Florian Haftmann, TUM
*)
section \<open>Pointwise instantiation of functions to division\<close>
theory Function_Division
imports Function_Algebras
begin
subsection \<open>Syntactic with division\<close>
instantiation "fun" :: (type, inverse) inverse
begin
definition "inverse f = inverse \<circ> f"
definition "f div g = (\<lambda>x. f x / g x)"
instance ..
end
lemma inverse_fun_apply [simp]:
"inverse f x = inverse (f x)"
by (simp add: inverse_fun_def)
lemma divide_fun_apply [simp]:
"(f / g) x = f x / g x"
by (simp add: divide_fun_def)
text \<open>
Unfortunately, we cannot lift this operations to algebraic type
classes for division: being different from the constant
zero function @{term "f \<noteq> 0"} is too weak as precondition.
So we must introduce our own set of lemmas.
\<close>
abbreviation zero_free :: "('b \<Rightarrow> 'a::field) \<Rightarrow> bool" where
"zero_free f \<equiv> \<not> (\<exists>x. f x = 0)"
lemma fun_left_inverse:
fixes f :: "'b \<Rightarrow> 'a::field"
shows "zero_free f \<Longrightarrow> inverse f * f = 1"
by (simp add: fun_eq_iff)
lemma fun_right_inverse:
fixes f :: "'b \<Rightarrow> 'a::field"
shows "zero_free f \<Longrightarrow> f * inverse f = 1"
by (simp add: fun_eq_iff)
lemma fun_divide_inverse:
fixes f g :: "'b \<Rightarrow> 'a::field"
shows "f / g = f * inverse g"
by (simp add: fun_eq_iff divide_inverse)
text \<open>Feel free to extend this.\<close>
text \<open>
Another possibility would be a reformulation of the division type
classes to user a @{term zero_free} predicate rather than
a direct @{term "a \<noteq> 0"} condition.
\<close>
end