header {* Lifting of the distributive law to the free algebra *}
theory Stream_Lift_to_Free3
imports Stream_Distributive_Law3
begin
subsection{* The lifting *}
definition ddd3 :: "('a × 'a F) ΣΣ3 => 'a ΣΣ3 × 'a ΣΣ3 F" where
"ddd3 = ext3 <\<oo>\<pp>3 o Σ3_map fst, F_map flat3 o Λ3> (leaf3 ** F_map leaf3)"
definition dd3 :: "('a × 'a F) ΣΣ3 => 'a ΣΣ3 F" where
"dd3 = snd o ddd3"
lemma ddd3_transfer[transfer_rule]:
"(ΣΣ3_rel (rel_prod R (F_rel R)) ===> rel_prod (ΣΣ3_rel R) (F_rel (ΣΣ3_rel R))) ddd3 ddd3"
unfolding ddd3_def ext3_alt by transfer_prover
lemma dd3_transfer[transfer_rule]:
"(ΣΣ3_rel (rel_prod R (F_rel R)) ===> F_rel (ΣΣ3_rel R)) dd3 dd3"
unfolding dd3_def by transfer_prover
lemma F_rel_ΣΣ3_rel: "ΣΣ3_rel (rel_prod R (F_rel R)) x y ==> F_rel (ΣΣ3_rel R) (dd3 x) (dd3 y)"
by (erule rel_funD[OF dd3_transfer])
theorem dd3_leaf3: "dd3 o leaf3 = F_map leaf3 o snd"
unfolding dd3_def ddd3_def o_assoc[symmetric] ext3_comp_leaf3 snd_comp_map_prod ..
lemma ddd3_natural: "ddd3 o ΣΣ3_map (f ** F_map f) = (ΣΣ3_map f ** F_map (ΣΣ3_map f)) o ddd3"
using ddd3_transfer[of "BNF_Def.Grp UNIV f"]
unfolding F_rel_Grp prod.rel_Grp ΣΣ3.rel_Grp
unfolding Grp_def rel_fun_def by auto
theorem dd3_natural: "dd3 o ΣΣ3_map (f ** F_map f) = F_map (ΣΣ3_map f) o dd3"
using dd3_transfer[of "BNF_Def.Grp UNIV f"]
unfolding F_rel_Grp prod.rel_Grp ΣΣ3.rel_Grp
unfolding Grp_def rel_fun_def by auto
lemma Λ3_dd3: "Λ3 = dd3 o \<oo>\<pp>3 o Σ3_map leaf3"
unfolding dd3_def ddd3_def o_assoc[symmetric] Σ3.map_comp0[symmetric] ext3_commute
unfolding o_assoc snd_convol ext3_comp_leaf3
unfolding o_assoc[symmetric] Λ3_natural
unfolding o_assoc F_map_comp[symmetric] leaf3_flat3 F_map_id id_o
..
lemma fst_ddd3: "fst o ddd3 = ΣΣ3_map fst"
proof-
have "fst o ddd3 = ext3 \<oo>\<pp>3 (leaf3 o fst)"
apply(rule ext3_unique) unfolding ddd3_def o_assoc[symmetric] ext3_comp_leaf3 ext3_commute
unfolding o_assoc fst_comp_map_prod fst_convol
unfolding o_assoc[symmetric] Σ3.map_comp0 by(rule refl, rule refl)
also have "... = ΣΣ3_map fst"
apply(rule sym, rule ext3_unique)
unfolding leaf3_natural \<oo>\<pp>3_natural by(rule refl, rule refl)
finally show ?thesis .
qed
lemma ddd3_flat3: "(flat3 ** F_map flat3) o ddd3 o ΣΣ3_map ddd3 = ddd3 o flat3" (is "?L = ?R")
proof-
have "?L = ext3 <\<oo>\<pp>3 o Σ3_map fst, F_map flat3 o Λ3> ddd3"
proof(rule ext3_unique)
show "(flat3 ** F_map flat3) o ddd3 o ΣΣ3_map ddd3 o leaf3 = ddd3"
unfolding ddd3_def unfolding o_assoc[symmetric] leaf3_natural
unfolding o_assoc
apply(subst o_assoc[symmetric]) unfolding ext3_comp_leaf3
unfolding map_prod.comp F_map_comp[symmetric] flat3_leaf3 F_map_id map_prod.id id_o ..
next
have A: "<flat3 o (\<oo>\<pp>3 o Σ3_map fst) , F_map flat3 o (F_map flat3 o Λ3)> =
<\<oo>\<pp>3 o Σ3_map fst , F_map flat3 o Λ3> o Σ3_map (flat3 ** F_map flat3)"
unfolding o_assoc unfolding flat3_commute[symmetric]
apply(rule fst_snd_cong) unfolding o_assoc fst_convol snd_convol
unfolding o_assoc[symmetric] Σ3.map_comp0[symmetric] fst_comp_map_prod snd_comp_map_prod
unfolding Λ3_natural unfolding o_assoc F_map_comp[symmetric] flat3_assoc by(rule refl, rule refl)
show "(flat3 ** F_map flat3) o ddd3 o ΣΣ3_map ddd3 o \<oo>\<pp>3 =
<\<oo>\<pp>3 o Σ3_map fst , F_map flat3 o Λ3> o Σ3_map (flat3 ** F_map flat3 o ddd3 o ΣΣ3_map ddd3)"
unfolding ddd3_def unfolding o_assoc[symmetric] unfolding \<oo>\<pp>3_natural[symmetric]
unfolding o_assoc
apply(subst o_assoc[symmetric]) unfolding ext3_commute
unfolding o_assoc[symmetric] Σ3.map_comp0[symmetric]
unfolding Σ3.map_comp0
unfolding o_assoc unfolding map_prod_o_convol
unfolding ext3_ΣΣ3_map[symmetric] A ..
qed
also have "... = ?R"
proof(rule sym, rule ext3_unique)
show "ddd3 o flat3 o leaf3 = ddd3" unfolding o_assoc[symmetric] flat3_leaf3 o_id ..
next
show "ddd3 o flat3 o \<oo>\<pp>3 = <\<oo>\<pp>3 o Σ3_map fst , F_map flat3 o Λ3> o Σ3_map (ddd3 o flat3)"
unfolding ddd3_def unfolding o_assoc[symmetric] unfolding flat3_commute[symmetric]
unfolding o_assoc unfolding ext3_commute Σ3.map_comp0 unfolding o_assoc ..
qed
finally show ?thesis .
qed
theorem dd3_flat3: "F_map flat3 o dd3 o ΣΣ3_map <ΣΣ3_map fst, dd3> = dd3 o flat3"
proof-
have A: "snd o ((flat3 ** F_map flat3) o ddd3 o ΣΣ3_map ddd3) = snd o (ddd3 o flat3)"
unfolding ddd3_flat3 ..
have B: "ddd3 = <ΣΣ3_map fst , snd o ddd3>" apply(rule fst_snd_cong)
unfolding fst_ddd3 by auto
show ?thesis unfolding dd3_def
unfolding A[symmetric, unfolded o_assoc snd_comp_map_prod] o_assoc B[symmetric] ..
qed
lemma dd3_leaf32: "<ΣΣ3_map fst, dd3> o leaf3 = leaf3 ** F_map leaf3"
apply (rule fst_snd_cong) unfolding o_assoc by (simp_all add: leaf3_natural dd3_leaf3)
lemma ddd3_leaf3: "ddd3 o leaf3 = leaf3 ** F_map leaf3"
unfolding ddd3_def ext3_comp_leaf3 ..
lemma ddd3_\<oo>\<pp>3: "ddd3 o \<oo>\<pp>3 = <\<oo>\<pp>3 o Σ3_map fst , F_map flat3 o Λ3> o Σ3_map ddd3"
unfolding ddd3_def ext3_commute ..
lemma ΣΣ3_rel_induct_pointfree:
assumes leaf3: "!! x1 x2. R x1 x2 ==> phi (leaf3 x1) (leaf3 x2)"
and \<oo>\<pp>3: "!! y1 y2. [|Σ3_rel (ΣΣ3_rel R) y1 y2; Σ3_rel phi y1 y2|] ==> phi (\<oo>\<pp>3 y1) (\<oo>\<pp>3 y2)"
shows "ΣΣ3_rel R ≤ phi"
proof-
have "ΣΣ3_rel R ≤ phi \<sqinter> ΣΣ3_rel R"
apply(induct rule: ΣΣ3.ctor_rel_induct)
using assms ΣΣ3.rel_inject[of R] unfolding rel_pre_ΣΣ3_def ΣΣ3.leaf3_def ΣΣ3.\<oo>\<pp>3_def
using inf_greatest[OF Σ3.rel_mono[OF inf_le1] Σ3.rel_mono[OF inf_le2]]
unfolding rel_sum_def BNF_Comp.id_bnf_comp_def vimage2p_def by (auto split: sum.splits) blast+
thus ?thesis by simp
qed
lemma ΣΣ3_rel_induct[case_names leaf3 \<oo>\<pp>3]:
assumes leaf3: "!! x1 x2. R x1 x2 ==> phi (leaf3 x1) (leaf3 x2)"
and \<oo>\<pp>3: "!! y1 y2. [|Σ3_rel (ΣΣ3_rel R) y1 y2; Σ3_rel phi y1 y2|] ==> phi (\<oo>\<pp>3 y1) (\<oo>\<pp>3 y2)"
shows "ΣΣ3_rel R t1 t2 --> phi t1 t2"
using ΣΣ3_rel_induct_pointfree[of R, OF assms] by auto
end