Theory Function_Division

(*  Title:      HOL/Library/Function_Division.thy
    Author:     Florian Haftmann, TUM

section ‹Pointwise instantiation of functions to division›

theory Function_Division
imports Function_Algebras

subsection ‹Syntactic with division›

instantiation "fun" :: (type, inverse) inverse

definition "inverse f = inverse  f"

definition "f div g = (λx. f x / g x)"

instance ..


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 ‹
  Unfortunately, we cannot lift this operations to algebraic type
  classes for division: being different from the constant
  zero function termf  0 is too weak as precondition.
  So we must introduce our own set of lemmas.

abbreviation zero_free :: "('b  'a::field)  bool" where
  "zero_free f  ¬ (x. f x = 0)"

lemma fun_left_inverse:
  fixes f :: "'b  'a::field"
  shows "zero_free f  inverse f * f = 1"
  by (simp add: fun_eq_iff)

lemma fun_right_inverse:
  fixes f :: "'b  'a::field"
  shows "zero_free f  f * inverse f = 1"
  by (simp add: fun_eq_iff)

lemma fun_divide_inverse:
  fixes f g :: "'b  'a::field"
  shows "f / g = f * inverse g"
  by (simp add: fun_eq_iff divide_inverse)

text ‹Feel free to extend this.›

text ‹
  Another possibility would be a reformulation of the division type
  classes to user a termzero_free predicate rather than
  a direct terma  0 condition.