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theory Iterate_GPV imports
"HOL-Library.BNF_Axiomatization"
"HOL-Library.BNF_Corec"
begin
declare [[typedef_overloaded]]
datatype 'a spmf = return_spmf 'a
primrec (transfer) bind_spmf where
"bind_spmf (return_spmf a) f = f a"
datatype (generat_pures: 'a, generat_outs: 'b, generat_conts: 'c) generat
= Pure (result: 'a)
| IO ("output": 'b) (continuation: "'c")
codatatype (results'_gpv: 'a, outs'_gpv: 'out, 'in) gpv
= GPV (the_gpv: "('a, 'out, 'in \ ('a, 'out, 'in) gpv) generat spmf")
declare gpv.rel_eq [relator_eq]
primcorec bind_gpv :: "('a, 'out, 'in) gpv \ ('a \ ('b, 'out, 'in) gpv) \ ('b, 'out, 'in) gpv"
where
"bind_gpv r f = GPV (map_spmf (map_generat id id ((\) (case_sum id (\r. bind_gpv r f))))
(bind_spmf (the_gpv r)
(case_generat
(\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x)))
(\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))))"
context includes lifting_syntax begin
lemma bind_gpv_parametric [transfer_rule]:
"(rel_gpv A C ===> (A ===> rel_gpv B C) ===> rel_gpv B C) bind_gpv bind_gpv"
unfolding bind_gpv_def by transfer_prover
end
lemma [friend_of_corec_simps]:
"map_spmf f (bind_spmf x h) = bind_spmf x (map_spmf f o h)"
by (cases x) auto
lemma [friend_of_corec_simps]:
"bind_spmf (map_spmf f x) h = bind_spmf x (h o f)"
by (cases x) auto
friend_of_corec bind_gpv :: "('a, 'out, 'in) gpv \ ('a \ ('a, 'out, 'in) gpv) \ ('a, 'out, 'in) gpv"
where "bind_gpv r f = GPV (map_spmf (map_generat id id ((\) (case_sum id (\r. bind_gpv r f))))
(bind_spmf (the_gpv r)
(case_generat
(\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x)))
(\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))))"
apply(rule bind_gpv.ctr)
apply transfer_prover
apply transfer_prover
done
end
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