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(* Author: Joshua Schneider, ETH Zurich *)
section \<open>Common applicative functors\<close>
subsection \<open>Environment functor\<close>
theory Applicative_Environment imports
Applicative
"HOL-Library.Adhoc_Overloading"
begin
definition "const x = (\<lambda>_. x)"
definition "apf x y = (\<lambda>z. x z (y z))"
adhoc_overloading Applicative.pure const
adhoc_overloading Applicative.ap apf
text \<open>The declaration below demonstrates that applicative functors which lift the reductions
for combinators K and W also lift C. However, the interchange law must be supplied in this case.\<close>
applicative env (K, W)
for
pure: const
ap: apf
rel: "rel_fun (=)"
set: range
by(simp_all add: const_def apf_def rel_fun_def)
lemma
includes applicative_syntax
shows "const (\<lambda>f x y. f y x) \<diamondop> f \<diamondop> x \<diamondop> y = f \<diamondop> y \<diamondop> x"
by applicative_lifting simp
end
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