header {* Copy of lift_to_free, but for the initial distributive law *}
theory Stream_Lift_to_Free0
imports Stream_Distributive_Law0
begin
subsection{* The lifting *}
definition ddd0 :: "('a × 'a F) ΣΣ0 => 'a ΣΣ0 × 'a ΣΣ0 F" where
"ddd0 = ext0 <\<oo>\<pp>0 o Σ0_map fst, F_map flat0 o Λ0> (leaf0 ** F_map leaf0)"
definition dd0 :: "('a × 'a F) ΣΣ0 => 'a ΣΣ0 F" where
"dd0 = snd o ddd0"
lemma ddd0_transfer[transfer_rule]:
"(ΣΣ0_rel (rel_prod R (F_rel R)) ===> rel_prod (ΣΣ0_rel R) (F_rel (ΣΣ0_rel R))) ddd0 ddd0"
unfolding ddd0_def ext0_alt by transfer_prover
lemma dd0_transfer[transfer_rule]:
"(ΣΣ0_rel (rel_prod R (F_rel R)) ===> F_rel (ΣΣ0_rel R)) dd0 dd0"
unfolding dd0_def by transfer_prover
lemma F_rel_ΣΣ0_rel: "ΣΣ0_rel (rel_prod R (F_rel R)) x y ==> F_rel (ΣΣ0_rel R) (dd0 x) (dd0 y)"
by (erule rel_funD[OF dd0_transfer])
theorem dd0_leaf0: "dd0 o leaf0 = F_map leaf0 o snd"
unfolding dd0_def ddd0_def by auto
lemma ddd0_natural: "ddd0 o ΣΣ0_map (f ** F_map f) = (ΣΣ0_map f ** F_map (ΣΣ0_map f)) o ddd0"
using ddd0_transfer[of "BNF_Def.Grp UNIV f"]
unfolding F_rel_Grp prod.rel_Grp ΣΣ0.rel_Grp
unfolding Grp_def rel_fun_def by auto
theorem dd0_natural: "dd0 o ΣΣ0_map (f ** F_map f) = F_map (ΣΣ0_map f) o dd0"
using dd0_transfer[of "BNF_Def.Grp UNIV f"]
unfolding F_rel_Grp prod.rel_Grp ΣΣ0.rel_Grp
unfolding Grp_def rel_fun_def by auto
lemma Λ0_dd0: "Λ0 = dd0 o \<oo>\<pp>0 o Σ0_map leaf0"
unfolding dd0_def ddd0_def o_assoc[symmetric] Σ0.map_comp0[symmetric] ext0_commute
unfolding o_assoc snd_convol ext0_comp_leaf0
unfolding o_assoc[symmetric] Λ0_natural
unfolding o_assoc F_map_comp[symmetric] leaf0_flat0 F_map_id id_o
..
lemma fst_ddd0: "fst o ddd0 = ΣΣ0_map fst"
proof-
have "fst o ddd0 = ext0 \<oo>\<pp>0 (leaf0 o fst)"
apply(rule ext0_unique) unfolding ddd0_def o_assoc[symmetric] ext0_comp_leaf0 ext0_commute
unfolding o_assoc fst_comp_map_prod fst_convol
unfolding o_assoc[symmetric] Σ0.map_comp0 by(rule refl, rule refl)
also have "... = ΣΣ0_map fst"
apply(rule sym, rule ext0_unique)
unfolding leaf0_natural \<oo>\<pp>0_natural by(rule refl, rule refl)
finally show ?thesis .
qed
lemma ddd0_flat0: "(flat0 ** F_map flat0) o ddd0 o ΣΣ0_map ddd0 = ddd0 o flat0" (is "?L = ?R")
proof-
have "?L = ext0 <\<oo>\<pp>0 o Σ0_map fst, F_map flat0 o Λ0> ddd0"
proof(rule ext0_unique)
show "(flat0 ** F_map flat0) o ddd0 o ΣΣ0_map ddd0 o leaf0 = ddd0"
unfolding ddd0_def unfolding o_assoc[symmetric] leaf0_natural
unfolding o_assoc
apply(subst o_assoc[symmetric]) unfolding ext0_comp_leaf0
unfolding map_prod.comp F_map_comp[symmetric] flat0_leaf0 F_map_id map_prod.id id_o ..
next
have 1: "<flat0 o (\<oo>\<pp>0 o Σ0_map fst) , F_map flat0 o (F_map flat0 o Λ0)> =
<\<oo>\<pp>0 o Σ0_map fst , F_map flat0 o Λ0> o Σ0_map (flat0 ** F_map flat0)"
unfolding o_assoc unfolding flat0_commute[symmetric]
apply(rule fst_snd_cong) unfolding o_assoc fst_convol snd_convol
unfolding o_assoc[symmetric] Σ0.map_comp0[symmetric] fst_comp_map_prod snd_comp_map_prod
unfolding Λ0_natural unfolding o_assoc F_map_comp[symmetric] flat0_assoc
by(rule refl, rule refl)
show "(flat0 ** F_map flat0) o ddd0 o ΣΣ0_map ddd0 o \<oo>\<pp>0 =
<\<oo>\<pp>0 o Σ0_map fst , F_map flat0 o Λ0> o Σ0_map (flat0 ** F_map flat0 o ddd0 o ΣΣ0_map ddd0)"
unfolding ddd0_def unfolding o_assoc[symmetric] unfolding \<oo>\<pp>0_natural[symmetric]
unfolding o_assoc
apply(subst o_assoc[symmetric]) unfolding ext0_commute
unfolding o_assoc[symmetric] Σ0.map_comp0[symmetric]
unfolding Σ0.map_comp0
unfolding o_assoc unfolding map_prod_o_convol
unfolding ext0_ΣΣ0_map[symmetric] 1 ..
qed
also have "... = ?R"
proof(rule sym, rule ext0_unique)
show "ddd0 o flat0 o leaf0 = ddd0" unfolding o_assoc[symmetric] flat0_leaf0 o_id ..
next
show "ddd0 o flat0 o \<oo>\<pp>0 = <\<oo>\<pp>0 o Σ0_map fst , F_map flat0 o Λ0> o Σ0_map (ddd0 o flat0)"
unfolding ddd0_def unfolding o_assoc[symmetric] unfolding flat0_commute[symmetric]
unfolding o_assoc unfolding ext0_commute Σ0.map_comp0 unfolding o_assoc ..
qed
finally show ?thesis .
qed
theorem dd0_flat0: "F_map flat0 o dd0 o ΣΣ0_map <ΣΣ0_map fst, dd0> = dd0 o flat0"
proof-
have 1: "snd o ((flat0 ** F_map flat0) o ddd0 o ΣΣ0_map ddd0) = snd o (ddd0 o flat0)"
unfolding ddd0_flat0 ..
have 2: "ddd0 = <ΣΣ0_map fst , snd o ddd0>" apply(rule fst_snd_cong)
unfolding fst_ddd0 by auto
show ?thesis unfolding dd0_def
unfolding 1[symmetric, unfolded o_assoc snd_comp_map_prod] o_assoc 2[symmetric] ..
qed
lemma dd0_leaf02: "<ΣΣ0_map fst, dd0> o leaf0 = leaf0 ** F_map leaf0"
apply (rule fst_snd_cong) unfolding o_assoc by (simp_all add: leaf0_natural dd0_leaf0)
lemma ddd0_leaf0: "ddd0 o leaf0 = leaf0 ** F_map leaf0"
unfolding ddd0_def ext0_comp_leaf0 ..
lemma ddd0_\<oo>\<pp>0:
"ddd0 o \<oo>\<pp>0 = <\<oo>\<pp>0 o Σ0_map fst , F_map flat0 o Λ0> o Σ0_map ddd0"
unfolding ddd0_def ext0_commute ..
lemma ΣΣ0_rel_induct_pointfree:
assumes leaf: "!! x1 x2. R x1 x2 ==> phi (leaf0 x1) (leaf0 x2)"
and \<oo>\<pp>: "!! y1 y2. [|Σ0_rel (ΣΣ0_rel R) y1 y2; Σ0_rel phi y1 y2|] ==> phi (\<oo>\<pp>0 y1) (\<oo>\<pp>0 y2)"
shows "ΣΣ0_rel R ≤ phi"
proof-
have "ΣΣ0_rel R ≤ phi \<sqinter> ΣΣ0_rel R"
apply(induct rule: ΣΣ0.ctor_rel_induct)
using assms ΣΣ0.rel_inject[of R] unfolding rel_pre_ΣΣ0_def ΣΣ0.leaf0_def ΣΣ0.\<oo>\<pp>0_def
using inf_greatest[OF Σ0.rel_mono[OF inf_le1] Σ0.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 ΣΣ0_rel_induct[case_names leaf \<oo>\<pp>]:
assumes leaf: "!! x1 x2. R x1 x2 ==> phi (leaf0 x1) (leaf0 x2)"
and \<oo>\<pp>: "!! y1 y2. [|Σ0_rel (ΣΣ0_rel R) y1 y2; Σ0_rel phi y1 y2|] ==> phi (\<oo>\<pp>0 y1) (\<oo>\<pp>0 y2)"
shows "ΣΣ0_rel R t1 t2 --> phi t1 t2"
using ΣΣ0_rel_induct_pointfree[of R, OF assms] by auto
end