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| (* Author: Andreas Lochbihler, ETH Zurich *) | |
| section \<open>Examples of applicative lifting\<close> | |
| subsection \<open>Algebraic operations for the environment functor\<close> | |
| theory Applicative_Environment_Algebra imports | |
| Applicative_Environment | |
| "HOL-Library.Function_Division" | |
| begin | |
| text \<open> Link between applicative instance of the environment functor with the pointwise operations | |
| for the algebraic type classes \<close> | |
| context includes applicative_syntax | |
| begin | |
| lemma plus_fun_af [applicative_unfold]: "f + g = pure (+) \<diamondop> f \<diamondop> g" | |
| unfolding plus_fun_def const_def apf_def .. | |
| lemma zero_fun_af [applicative_unfold]: "0 = pure 0" | |
| unfolding zero_fun_def const_def .. | |
| lemma times_fun_af [applicative_unfold]: "f * g = pure (*) \<diamondop> f \<diamondop> g" | |
| unfolding times_fun_def const_def apf_def .. | |
| lemma one_fun_af [applicative_unfold]: "1 = pure 1" | |
| unfolding one_fun_def const_def .. | |
| lemma of_nat_fun_af [applicative_unfold]: "of_nat n = pure (of_nat n)" | |
| unfolding of_nat_fun const_def .. | |
| lemma inverse_fun_af [applicative_unfold]: "inverse f = pure inverse \<diamondop> f" | |
| unfolding inverse_fun_def o_def const_def apf_def .. | |
| lemma divide_fun_af [applicative_unfold]: "divide f g = pure divide \<diamondop> f \<diamondop> g" | |
| unfolding divide_fun_def const_def apf_def .. | |
| end | |
| end | |