header {* Lifting of the distributive law to the free algebra *}
theory Tree_Lift_to_Free1
imports Tree_Distributive_Law1
begin
subsection{* The lifting *}
definition ddd1 :: "('a × 'a F) ΣΣ1 => 'a ΣΣ1 × 'a ΣΣ1 F" where
"ddd1 = ext1 <\<oo>\<pp>1 o Σ1_map fst, F_map flat1 o Λ1> (leaf1 ** F_map leaf1)"
definition dd1 :: "('a × 'a F) ΣΣ1 => 'a ΣΣ1 F" where
"dd1 = snd o ddd1"
lemma ddd1_transfer[transfer_rule]:
"(ΣΣ1_rel (rel_prod R (F_rel R)) ===> rel_prod (ΣΣ1_rel R) (F_rel (ΣΣ1_rel R))) ddd1 ddd1"
unfolding ddd1_def ext1_alt by transfer_prover
lemma dd1_transfer[transfer_rule]:
"(ΣΣ1_rel (rel_prod R (F_rel R)) ===> F_rel (ΣΣ1_rel R)) dd1 dd1"
unfolding dd1_def by transfer_prover
lemma F_rel_ΣΣ1_rel: "ΣΣ1_rel (rel_prod R (F_rel R)) x y ==> F_rel (ΣΣ1_rel R) (dd1 x) (dd1 y)"
by (erule rel_funD[OF dd1_transfer])
theorem dd1_leaf1: "dd1 o leaf1 = F_map leaf1 o snd"
unfolding dd1_def ddd1_def o_assoc[symmetric] ext1_comp_leaf1 snd_comp_map_prod ..
lemma ddd1_natural: "ddd1 o ΣΣ1_map (f ** F_map f) = (ΣΣ1_map f ** F_map (ΣΣ1_map f)) o ddd1"
using ddd1_transfer[of "BNF_Def.Grp UNIV f"]
unfolding F_rel_Grp prod.rel_Grp ΣΣ1.rel_Grp
unfolding Grp_def rel_fun_def by auto
theorem dd1_natural: "dd1 o ΣΣ1_map (f ** F_map f) = F_map (ΣΣ1_map f) o dd1"
using dd1_transfer[of "BNF_Def.Grp UNIV f"]
unfolding F_rel_Grp prod.rel_Grp ΣΣ1.rel_Grp
unfolding Grp_def rel_fun_def by auto
lemma Λ1_dd1: "Λ1 = dd1 o \<oo>\<pp>1 o Σ1_map leaf1"
unfolding dd1_def ddd1_def o_assoc[symmetric] Σ1.map_comp0[symmetric] ext1_commute
unfolding o_assoc snd_convol ext1_comp_leaf1
unfolding o_assoc[symmetric] Λ1_natural
unfolding o_assoc F_map_comp[symmetric] leaf1_flat1 F_map_id id_o
..
lemma fst_ddd1: "fst o ddd1 = ΣΣ1_map fst"
proof-
have "fst o ddd1 = ext1 \<oo>\<pp>1 (leaf1 o fst)"
apply(rule ext1_unique) unfolding ddd1_def o_assoc[symmetric] ext1_comp_leaf1 ext1_commute
unfolding o_assoc fst_comp_map_prod fst_convol
unfolding o_assoc[symmetric] Σ1.map_comp0 by(rule refl, rule refl)
also have "... = ΣΣ1_map fst"
apply(rule sym, rule ext1_unique)
unfolding leaf1_natural \<oo>\<pp>1_natural by(rule refl, rule refl)
finally show ?thesis .
qed
lemma ddd1_flat1: "(flat1 ** F_map flat1) o ddd1 o ΣΣ1_map ddd1 = ddd1 o flat1" (is "?L = ?R")
proof-
have "?L = ext1 <\<oo>\<pp>1 o Σ1_map fst, F_map flat1 o Λ1> ddd1"
proof(rule ext1_unique)
show "(flat1 ** F_map flat1) o ddd1 o ΣΣ1_map ddd1 o leaf1 = ddd1"
unfolding ddd1_def unfolding o_assoc[symmetric] leaf1_natural
unfolding o_assoc
apply(subst o_assoc[symmetric]) unfolding ext1_comp_leaf1
unfolding map_prod.comp F_map_comp[symmetric] flat1_leaf1 F_map_id map_prod.id id_o ..
next
have A: "<flat1 o (\<oo>\<pp>1 o Σ1_map fst) , F_map flat1 o (F_map flat1 o Λ1)> =
<\<oo>\<pp>1 o Σ1_map fst , F_map flat1 o Λ1> o Σ1_map (flat1 ** F_map flat1)"
unfolding o_assoc unfolding flat1_commute[symmetric]
apply(rule fst_snd_cong) unfolding o_assoc fst_convol snd_convol
unfolding o_assoc[symmetric] Σ1.map_comp0[symmetric] fst_comp_map_prod snd_comp_map_prod
unfolding Λ1_natural unfolding o_assoc F_map_comp[symmetric] flat1_assoc by(rule refl, rule refl)
show "(flat1 ** F_map flat1) o ddd1 o ΣΣ1_map ddd1 o \<oo>\<pp>1 =
<\<oo>\<pp>1 o Σ1_map fst , F_map flat1 o Λ1> o Σ1_map (flat1 ** F_map flat1 o ddd1 o ΣΣ1_map ddd1)"
unfolding ddd1_def unfolding o_assoc[symmetric] unfolding \<oo>\<pp>1_natural[symmetric]
unfolding o_assoc
apply(subst o_assoc[symmetric]) unfolding ext1_commute
unfolding o_assoc[symmetric] Σ1.map_comp0[symmetric]
unfolding Σ1.map_comp0
unfolding o_assoc unfolding map_prod_o_convol
unfolding ext1_ΣΣ1_map[symmetric] A ..
qed
also have "... = ?R"
proof(rule sym, rule ext1_unique)
show "ddd1 o flat1 o leaf1 = ddd1" unfolding o_assoc[symmetric] flat1_leaf1 o_id ..
next
show "ddd1 o flat1 o \<oo>\<pp>1 = <\<oo>\<pp>1 o Σ1_map fst , F_map flat1 o Λ1> o Σ1_map (ddd1 o flat1)"
unfolding ddd1_def unfolding o_assoc[symmetric] unfolding flat1_commute[symmetric]
unfolding o_assoc unfolding ext1_commute Σ1.map_comp0 unfolding o_assoc ..
qed
finally show ?thesis .
qed
theorem dd1_flat1: "F_map flat1 o dd1 o ΣΣ1_map <ΣΣ1_map fst, dd1> = dd1 o flat1"
proof-
have A: "snd o ((flat1 ** F_map flat1) o ddd1 o ΣΣ1_map ddd1) = snd o (ddd1 o flat1)"
unfolding ddd1_flat1 ..
have B: "ddd1 = <ΣΣ1_map fst , snd o ddd1>" apply(rule fst_snd_cong)
unfolding fst_ddd1 by auto
show ?thesis unfolding dd1_def
unfolding A[symmetric, unfolded o_assoc snd_comp_map_prod] o_assoc B[symmetric] ..
qed
lemma dd1_leaf12: "<ΣΣ1_map fst, dd1> o leaf1 = leaf1 ** F_map leaf1"
apply (rule fst_snd_cong) unfolding o_assoc by (simp_all add: leaf1_natural dd1_leaf1)
lemma ddd1_leaf1: "ddd1 o leaf1 = leaf1 ** F_map leaf1"
unfolding ddd1_def ext1_comp_leaf1 ..
lemma ddd1_\<oo>\<pp>1: "ddd1 o \<oo>\<pp>1 = <\<oo>\<pp>1 o Σ1_map fst , F_map flat1 o Λ1> o Σ1_map ddd1"
unfolding ddd1_def ext1_commute ..
lemma ΣΣ1_rel_induct_pointfree:
assumes leaf1: "!! x1 x2. R x1 x2 ==> phi (leaf1 x1) (leaf1 x2)"
and \<oo>\<pp>1: "!! y1 y2. [|Σ1_rel (ΣΣ1_rel R) y1 y2; Σ1_rel phi y1 y2|] ==> phi (\<oo>\<pp>1 y1) (\<oo>\<pp>1 y2)"
shows "ΣΣ1_rel R ≤ phi"
proof-
have "ΣΣ1_rel R ≤ phi \<sqinter> ΣΣ1_rel R"
apply(induct rule: ΣΣ1.ctor_rel_induct)
using assms ΣΣ1.rel_inject[of R] unfolding rel_pre_ΣΣ1_def ΣΣ1.leaf1_def ΣΣ1.\<oo>\<pp>1_def
using inf_greatest[OF Σ1.rel_mono[OF inf_le1] Σ1.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 ΣΣ1_rel_induct[case_names leaf1 \<oo>\<pp>1]:
assumes leaf1: "!! x1 x2. R x1 x2 ==> phi (leaf1 x1) (leaf1 x2)"
and \<oo>\<pp>1: "!! y1 y2. [|Σ1_rel (ΣΣ1_rel R) y1 y2; Σ1_rel phi y1 y2|] ==> phi (\<oo>\<pp>1 y1) (\<oo>\<pp>1 y2)"
shows "ΣΣ1_rel R t1 t2 --> phi t1 t2"
using ΣΣ1_rel_induct_pointfree[of R, OF assms] by auto
end