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| (* Author: Andreas Lochbihler, ETH Zurich *) | |
| subsection \<open>Probability mass functions\<close> | |
| theory Applicative_PMF imports | |
| Applicative | |
| "HOL-Probability.Probability" | |
| "HOL-Library.Adhoc_Overloading" | |
| begin | |
| abbreviation (input) pure_pmf :: "'a \<Rightarrow> 'a pmf" | |
| where "pure_pmf \<equiv> return_pmf" | |
| definition ap_pmf :: "('a \<Rightarrow> 'b) pmf \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" | |
| where "ap_pmf f x = map_pmf (\<lambda>(f, x). f x) (pair_pmf f x)" | |
| adhoc_overloading Applicative.ap ap_pmf | |
| context includes applicative_syntax | |
| begin | |
| lemma ap_pmf_id: "pure_pmf (\<lambda>x. x) \<diamondop> x = x" | |
| by(simp add: ap_pmf_def pair_return_pmf1 pmf.map_comp o_def) | |
| lemma ap_pmf_comp: "pure_pmf (\<circ>) \<diamondop> u \<diamondop> v \<diamondop> w = u \<diamondop> (v \<diamondop> w)" | |
| by(simp add: ap_pmf_def pair_return_pmf1 pair_map_pmf1 pair_map_pmf2 pmf.map_comp o_def split_def pair_pair_pmf) | |
| lemma ap_pmf_homo: "pure_pmf f \<diamondop> pure_pmf x = pure_pmf (f x)" | |
| by(simp add: ap_pmf_def pair_return_pmf1) | |
| lemma ap_pmf_interchange: "u \<diamondop> pure_pmf x = pure_pmf (\<lambda>f. f x) \<diamondop> u" | |
| by(simp add: ap_pmf_def pair_return_pmf1 pair_return_pmf2 pmf.map_comp o_def) | |
| lemma ap_pmf_K: "return_pmf (\<lambda>x _. x) \<diamondop> x \<diamondop> y = x" | |
| by(simp add: ap_pmf_def pair_map_pmf1 pmf.map_comp pair_return_pmf1 o_def split_def map_fst_pair_pmf) | |
| lemma ap_pmf_C: "return_pmf (\<lambda>f x y. f y x) \<diamondop> f \<diamondop> x \<diamondop> y = f \<diamondop> y \<diamondop> x" | |
| apply(simp add: ap_pmf_def pair_map_pmf1 pmf.map_comp pair_return_pmf1 pair_pair_pmf o_def split_def) | |
| apply(subst (2) pair_commute_pmf) | |
| apply(simp add: pair_map_pmf2 pmf.map_comp o_def split_def) | |
| done | |
| lemma ap_pmf_transfer[transfer_rule]: | |
| "rel_fun (rel_pmf (rel_fun A B)) (rel_fun (rel_pmf A) (rel_pmf B)) ap_pmf ap_pmf" | |
| unfolding ap_pmf_def[abs_def] pair_pmf_def | |
| by transfer_prover | |
| applicative pmf (C, K) | |
| for | |
| pure: pure_pmf | |
| ap: ap_pmf | |
| rel: rel_pmf | |
| set: set_pmf | |
| proof - | |
| fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" | |
| show "rel_fun R (rel_pmf R) pure_pmf pure_pmf" by transfer_prover | |
| next | |
| fix R and f :: "('a \<Rightarrow> 'b) pmf" and g :: "('a \<Rightarrow> 'c) pmf" and x | |
| assume [transfer_rule]: "rel_pmf (rel_fun (eq_on (set_pmf x)) R) f g" | |
| have [transfer_rule]: "rel_pmf (eq_on (set_pmf x)) x x" by (simp add: pmf.rel_refl_strong) | |
| show "rel_pmf R (ap_pmf f x) (ap_pmf g x)" by transfer_prover | |
| qed(rule ap_pmf_comp[unfolded o_def[abs_def]] ap_pmf_homo ap_pmf_C ap_pmf_K)+ | |
| end | |
| end | |