header {* Corecursion and coinduction up to *}
theory Stream_Corec_Upto4
imports Stream_Lift_to_Free4
begin
subsection{* The algebra associated to dd4 *}
definition "eval4 ≡ dtor_unfold_J (dd4 o ΣΣ4_map <id, dtor_J>)"
lemma eval4: "F_map eval4 o dd4 o ΣΣ4_map <id, dtor_J> = dtor_J o eval4"
unfolding eval4_def dtor_unfold_J_pointfree unfolding o_assoc ..
lemma eval4_ctor_J: "ctor_J o F_map eval4 o dd4 o ΣΣ4_map <id, dtor_J> = eval4"
unfolding o_def spec[OF eval4[unfolded o_def fun_eq_iff]] J.ctor_dtor ..
lemma eval4_leaf4: "eval4 o leaf4 = id"
proof (rule trans)
show "eval4 o leaf4 = dtor_unfold_J dtor_J"
apply(rule J.dtor_unfold_unique)
unfolding o_assoc eval4[symmetric] unfolding o_assoc[symmetric] leaf4_natural
apply(rule sym)
unfolding F_map_comp o_assoc apply (subst o_assoc[symmetric])
unfolding dd4_leaf4 unfolding o_assoc[symmetric] by simp
qed(metis F_map_id J.dtor_unfold_unique fun.map_id o_id)
lemma eval4_flat4: "eval4 o flat4 = eval4 o ΣΣ4_map eval4"
proof (rule trans)
let ?K4 = "dtor_unfold_J (dd4 o ΣΣ4_map <ΣΣ4_map fst, dd4> o ΣΣ4_map (ΣΣ4_map <id, dtor_J>))"
show "eval4 o flat4 = ?K4"
apply(rule J.dtor_unfold_unique)
unfolding F_map_comp o_assoc
apply(subst o_assoc[symmetric], subst o_assoc[symmetric])
unfolding dd4_flat4
unfolding o_assoc[symmetric] ΣΣ4.map_comp0[symmetric] snd_convol
unfolding flat4_natural
unfolding o_assoc eval4 ..
have A: "<eval4, dtor_J o eval4> = <id,dtor_J> o eval4" by simp
show "?K4 = eval4 o ΣΣ4_map eval4"
apply(rule J.dtor_unfold_unique[symmetric])
unfolding o_assoc[symmetric] ΣΣ4.map_comp0[symmetric] map_prod_o_convol id_o
unfolding F_map_comp o_assoc
apply(subst o_assoc[symmetric]) unfolding dd4_natural[symmetric]
unfolding o_assoc[symmetric] ΣΣ4.map_comp0[symmetric]
unfolding o_assoc unfolding map_prod_o_convol unfolding convol_o
unfolding o_assoc[symmetric] ΣΣ4.map_comp0[symmetric] fst_convol ΣΣ4.map_id0 o_id
unfolding o_assoc eval4 unfolding A unfolding convol_o id_o
apply(rule sym) apply(subst eval4[symmetric])
unfolding o_assoc[symmetric] ΣΣ4.map_comp0[symmetric] convol_o id_o ..
qed
subsection{* The correspondence between coalgebras up to and coalgebras *}
definition cutΣΣ4Oc :: "('a => 'a ΣΣ4 F) => ('a ΣΣ4 => 'a ΣΣ4 F)"
where "cutΣΣ4Oc s ≡ F_map flat4 o dd4 o ΣΣ4_map <leaf4, s>"
definition cΣΣ4Ocut :: "('a ΣΣ4 => 'a ΣΣ4 F) => ('a => 'a ΣΣ4 F)"
where "cΣΣ4Ocut s' ≡ s' o leaf4"
lemma cΣΣ4Ocut_cutΣΣ4Oc: "cΣΣ4Ocut (cutΣΣ4Oc s) = s"
unfolding cΣΣ4Ocut_def cutΣΣ4Oc_def
unfolding o_assoc[symmetric] unfolding leaf4_natural
unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding dd4_leaf4 unfolding o_assoc F_map_comp[symmetric] flat4_leaf4
unfolding F_map_id id_o by simp
lemma cutΣΣ4Oc_inj: "cutΣΣ4Oc s4 = cutΣΣ4Oc s2 <-> s4 = s2"
by (metis cΣΣ4Ocut_cutΣΣ4Oc)
lemma cΣΣ4Ocut_surj: "∃ s'. cΣΣ4Ocut s' = s"
using cΣΣ4Ocut_cutΣΣ4Oc by(rule exI[of _ "cutΣΣ4Oc s"])
definition extdd4 :: "('a => J) => ('a ΣΣ4 => J)"
where "extdd4 f ≡ eval4 o ΣΣ4_map f"
term eval4
definition restr :: "('a ΣΣ4 => J) => ('a => J)"
where "restr f' ≡ f' o leaf4"
lemma extdd4_mor:
assumes f: "F_map (extdd4 f) o s = dtor_J o f"
shows "F_map (extdd4 f) o cutΣΣ4Oc s = dtor_J o (extdd4 f)"
proof-
have AA: "eval4 ** F_map eval4 o (ΣΣ4_map f ** F_map (ΣΣ4_map f) o <leaf4 , s>) =
<f , F_map eval4 o (F_map (ΣΣ4_map f) o s)>"
unfolding map_prod_o_convol unfolding leaf4_natural o_assoc eval4_leaf4 id_o ..
show ?thesis
unfolding extdd4_def
unfolding o_assoc eval4[symmetric]
unfolding o_assoc[symmetric] ΣΣ4.map_comp0[symmetric]
unfolding convol_comp[symmetric] id_o
unfolding f[symmetric, unfolded extdd4_def]
unfolding o_assoc
apply(subst o_assoc[symmetric])
unfolding F_map_comp o_assoc
unfolding cutΣΣ4Oc_def
unfolding o_assoc
unfolding F_map_comp[symmetric] unfolding o_assoc[symmetric]
unfolding flat4_natural[symmetric]
unfolding o_assoc F_map_comp
apply(subst o_assoc[symmetric], subst o_assoc[symmetric], subst o_assoc[symmetric])
unfolding dd4_natural[symmetric]
unfolding o_assoc unfolding F_map_comp[symmetric] eval4_flat4
unfolding F_map_comp apply(subst o_assoc[symmetric])
unfolding dd4_natural[symmetric] unfolding o_assoc[symmetric] ΣΣ4.map_comp0[symmetric]
unfolding o_assoc[symmetric] AA[unfolded o_assoc[symmetric]] ..
qed
lemma mor_cutΣΣ4Oc_flat4:
assumes f': "F_map f' o cutΣΣ4Oc s = dtor_J o f'"
shows "eval4 o ΣΣ4_map f' = f' o flat4"
proof(rule trans)
def h ≡ "dd4 o ΣΣ4_map <id,cutΣΣ4Oc s>"
have f'_id: "f' = f' o id" by simp
show "eval4 o ΣΣ4_map f' = dtor_unfold_J h"
apply(rule J.dtor_unfold_unique, rule sym)
unfolding o_assoc eval4[symmetric]
unfolding o_assoc[symmetric] ΣΣ4.map_comp0[symmetric]
unfolding convol_comp_id1[symmetric] unfolding f'[symmetric]
apply(subst f'_id)
unfolding o_assoc ΣΣ4.map_comp0
apply(subst o_assoc[symmetric])
unfolding o_assoc[symmetric] F_map_comp
unfolding h_def apply(rule sym)
unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding dd4_natural[symmetric] unfolding o_assoc[symmetric]
unfolding ΣΣ4.map_comp0[symmetric] map_prod_o_convol ..
have AA: "<id , cutΣΣ4Oc s> = (flat4 ** F_map flat4) o (id ** dd4) o <leaf4, ΣΣ4_map <leaf4 , s>>"
unfolding map_prod_o_convol o_assoc map_prod.comp cutΣΣ4Oc_def o_id flat4_leaf4 ..
have BB: "flat4 ** F_map flat4 o id ** dd4 o <leaf4 , ΣΣ4_map <leaf4 , s>> = flat4 ** F_map flat4 o id ** dd4 o <ΣΣ4_map leaf4 , ΣΣ4_map <leaf4 , s>>"
unfolding map_prod.comp unfolding map_prod_o_convol unfolding o_id unfolding flat4_leaf4 leaf4_flat4 ..
show "dtor_unfold_J h = f' o flat4"
apply(rule J.dtor_unfold_unique[symmetric], rule sym)
unfolding o_assoc f'[symmetric]
unfolding F_map_comp o_assoc[symmetric]
apply(rule arg_cong[of _ _ "op o (F_map f')"])
unfolding h_def
unfolding AA BB
unfolding ΣΣ4.map_comp0 apply(rule sym)
unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding dd4_natural
unfolding o_assoc F_map_comp[symmetric]
unfolding flat4_assoc unfolding F_map_comp
unfolding cutΣΣ4Oc_def o_assoc[symmetric] apply(rule arg_cong[of _ _ "op o (F_map flat4)"])
unfolding o_assoc
unfolding o_assoc[symmetric] unfolding ΣΣ4.map_comp0[symmetric] unfolding map_prod_o_convol id_o
unfolding flat4_natural[symmetric] unfolding o_assoc
unfolding dd4_flat4[symmetric] unfolding o_assoc[symmetric] unfolding ΣΣ4.map_comp0[symmetric]
unfolding convol_o unfolding ΣΣ4.map_comp0[symmetric] unfolding fst_convol ..
qed
lemma restr_mor:
assumes f': "F_map f' o cutΣΣ4Oc s = dtor_J o f'"
shows "F_map (extdd4 (restr f')) o s = dtor_J o restr f'"
unfolding extdd4_def restr_def ΣΣ4.map_comp0
unfolding o_assoc mor_cutΣΣ4Oc_flat4[OF f']
unfolding o_assoc[symmetric] leaf4_flat4 o_id
unfolding o_assoc f'[symmetric]
unfolding o_assoc[symmetric] cΣΣ4Ocut_cutΣΣ4Oc[unfolded cΣΣ4Ocut_def] ..
lemma extdd4_restr:
assumes f': "F_map f' o cutΣΣ4Oc s = dtor_J o f'"
shows "extdd4 (restr f') = f'"
proof-
have "f' = eval4 o ΣΣ4_map f' o leaf4"
unfolding o_assoc[symmetric] leaf4_natural
unfolding o_assoc eval4_leaf4 by simp
also have "... = eval4 o ΣΣ4_map (f' o leaf4)"
unfolding ΣΣ4.map_comp0 o_assoc
unfolding mor_cutΣΣ4Oc_flat4[OF f'] unfolding o_assoc[symmetric] flat4_leaf4 leaf4_flat4 ..
finally have A: "f' = eval4 o ΣΣ4_map (f' o leaf4)" .
show ?thesis unfolding extdd4_def restr_def A[symmetric] ..
qed
lemma restr_inj:
assumes f4': "F_map f4' o cutΣΣ4Oc s = dtor_J o f4'"
and f2': "F_map f2' o cutΣΣ4Oc s = dtor_J o f2'"
shows "restr f4' = restr f2' <-> f4' = f2'"
using extdd4_restr[OF f4'] extdd4_restr[OF f2'] by metis
lemma extdd4_surj:
assumes f': "F_map f' o cutΣΣ4Oc s = dtor_J o f'"
shows "∃ f. extdd4 f = f'"
using extdd4_restr[OF f'] by(rule exI[of _ "restr f'"])
lemma restr_extdd4:
assumes f: "F_map (extdd4 f) o s = dtor_J o f"
shows "restr (extdd4 f) = f"
proof-
have "dtor_J o f = F_map (extdd4 f) o s" using assms unfolding extdd4_def by (rule sym)
also have "... = dtor_J o restr (extdd4 f)"
unfolding restr_def unfolding o_assoc extdd4_mor[OF f, symmetric]
unfolding o_assoc[symmetric] cΣΣ4Ocut_cutΣΣ4Oc[unfolded cΣΣ4Ocut_def] ..
finally have "dtor_J o f = dtor_J o restr (extdd4 f)" .
thus ?thesis unfolding dtor_J_o_inj by (rule sym)
qed
lemma extdd4_inj:
assumes f1: "F_map (extdd4 f1) o s = dtor_J o f1"
and f2: "F_map (extdd4 f2) o s = dtor_J o f2"
shows "extdd4 f1 = extdd4 f2 <-> f1 = f2"
using restr_extdd4[OF f1] restr_extdd4[OF f2] by metis
lemma restr_surj:
assumes f: "F_map (extdd4 f) o s = dtor_J o f"
shows "∃ f'. restr f' = f"
using restr_extdd4[OF f] by(rule exI[of _ "extdd4 f"])
subsection{* Coiteration up-to *}
definition "unfoldU4 s ≡ restr (dtor_unfold_J (cutΣΣ4Oc s))"
theorem unfoldU4_pointfree:
"F_map (extdd4 (unfoldU4 s)) o s = dtor_J o unfoldU4 s"
unfolding unfoldU4_def apply(rule restr_mor)
unfolding dtor_unfold_J_pointfree ..
theorem unfoldU4: "F_map (extdd4 (unfoldU4 s)) (s a) = dtor_J (unfoldU4 s a)"
using unfoldU4_pointfree unfolding o_def fun_eq_iff by(rule allE)
theorem unfoldU4_ctor_J:
"ctor_J (F_map (extdd4 (unfoldU4 s)) (s a)) = unfoldU4 s a"
using unfoldU4 by (metis J.ctor_dtor)
theorem unfoldU4_unique:
assumes "F_map (extdd4 f) o s = dtor_J o f"
shows "f = unfoldU4 s"
proof-
note f = extdd4_mor[OF assms] note co = extdd4_mor[OF unfoldU4_pointfree]
have A: "extdd4 f = extdd4 (unfoldU4 s)"
proof(rule trans)
show "extdd4 f = dtor_unfold_J (cutΣΣ4Oc s)" apply(rule J.dtor_unfold_unique) using f .
show "dtor_unfold_J (cutΣΣ4Oc s) = extdd4 (unfoldU4 s)"
apply(rule J.dtor_unfold_unique[symmetric]) using co .
qed
show ?thesis using A unfolding extdd4_inj[OF assms unfoldU4_pointfree] .
qed
lemma unfoldU4_ctor_J_pointfree:
"ctor_J o F_map (extdd4 (unfoldU4 s)) o s = unfoldU4 s"
unfolding o_def fun_eq_iff by (subst unfoldU4_ctor_J[symmetric]) (rule allI, rule refl)
definition corecU4 :: "('a => (J + 'a) ΣΣ4 F) => 'a => J" where
"corecU4 s = unfoldU4 (case_sum (dd4 o leaf4 o <Inl, F_map Inl o dtor_J>) s) o Inr"
definition extddRec4 where
"extddRec4 f ≡ eval4 o ΣΣ4_map (case_sum id f)"
lemma unfoldU4_Inl:
"unfoldU4 (case_sum (dd4 o leaf4 o <Inl , F_map Inl o dtor_J>) s) o Inl = id"
(is "?L = ?R")
proof-
have "?L = unfoldU4 (dd4 o leaf4 o <id, dtor_J>)"
apply(rule unfoldU4_unique)
unfolding o_assoc unfoldU4_pointfree[symmetric]
unfolding o_assoc[symmetric] case_sum_o_inj extdd4_def F_map_comp ΣΣ4.map_comp0
unfolding o_assoc apply(subst o_assoc[symmetric], subst o_assoc[symmetric], subst o_assoc[symmetric],
subst o_assoc[symmetric], subst o_assoc[symmetric], subst o_assoc[symmetric])
unfolding dd4_natural[symmetric]
apply(subst o_assoc[symmetric]) unfolding leaf4_natural
unfolding o_assoc[symmetric] map_prod_o_convol o_id ..
also have "... = ?R"
apply(rule sym, rule unfoldU4_unique)
unfolding extdd4_def ΣΣ4.map_id0 o_id dd4_leaf4
unfolding o_assoc[symmetric] snd_convol
unfolding o_assoc F_map_comp[symmetric] eval4_leaf4 F_map_id id_o ..
finally show ?thesis .
qed
theorem corecU4_pointfree:
"F_map (extddRec4 (corecU4 s)) o s = dtor_J o corecU4 s" (is "?L = ?R")
unfolding corecU4_def
unfolding o_assoc unfoldU4_pointfree[symmetric] extddRec4_def
unfolding o_assoc[symmetric] case_sum_o_inj
apply(subst unfoldU4_Inl[symmetric, of s])
unfolding o_assoc case_sum_Inl_Inr_L extdd4_def ..
theorem corecU4:
"F_map (extddRec4 (corecU4 s)) (s a) = dtor_J (corecU4 s a)"
using corecU4_pointfree unfolding o_def fun_eq_iff by(rule allE)
subsection{* Coinduction up-to *}
definition "cptdd4 R ≡ (ΣΣ4_rel R ===> R) eval4 eval4"
definition "cngdd4 R ≡ equivp R ∧ cptdd4 R"
lemma cngdd4_Retr: "cngdd4 R ==> cngdd4 (R \<sqinter> Retr R)"
unfolding cngdd4_def cptdd4_def
apply (erule conjE)
apply (rule conjI[OF equivp_inf[OF _ equivp_retr]])
apply assumption
apply assumption
apply (rule rel_funI)
apply (frule predicate2D[OF ΣΣ4_rel_inf])
apply (erule inf2E)
apply (rule inf2I)
apply (erule rel_funE)
apply assumption
apply assumption
apply (subst Retr_def)
apply (subst eval4_def)+
apply (subst J.dtor_unfold)+
unfolding F_rel_F_map_F_map Grp_def relcompp.simps[abs_def] conversep.simps[abs_def]
apply auto
unfolding eval4_def[symmetric]
apply (rule predicate2D[OF F_rel_mono])
apply (rule predicate2I)
apply (erule rel_funD)
apply assumption
apply (rule F_rel_ΣΣ4_rel)
unfolding ΣΣ4_rel_ΣΣ4_map_ΣΣ4_map vimage2p_rel_prod vimage2p_id
unfolding vimage2p_def Retr_def[symmetric]
apply assumption
done
definition "genCngdd4 R j1 j2 ≡ ∀ R'. R ≤ R' ∧ cngdd4 R' --> R' j1 j2"
lemma cngdd4_genCngdd4: "cngdd4 (genCngdd4 R)"
unfolding cngdd4_def proof safe
show "cptdd4 (genCngdd4 R)"
unfolding cptdd4_def rel_fun_def proof safe
fix x y assume A: "ΣΣ4_rel (genCngdd4 R) x y"
show "genCngdd4 R (eval4 x) (eval4 y)"
unfolding genCngdd4_def[abs_def] proof safe
fix R' assume "R ≤ R'" and 2: "cngdd4 R'"
hence "ΣΣ4_rel R' x y" by (metis A ΣΣ4.rel_mono_strong genCngdd4_def)
thus "R' (eval4 x) (eval4 y)" using 2 unfolding cngdd4_def cptdd4_def rel_fun_def by auto
qed
qed
qed(rule equivpI, unfold reflp_def symp_def transp_def genCngdd4_def cngdd4_def equivp_def, auto)
lemma
genCngdd4_refl[intro,simp]: "genCngdd4 R j j"
and genCngdd4_sym[intro]: "genCngdd4 R j1 j2 ==> genCngdd4 R j2 j1"
and genCngdd4_trans[intro]: "[|genCngdd4 R j1 j2; genCngdd4 R j2 j3|] ==> genCngdd4 R j1 j3"
using cngdd4_genCngdd4 unfolding cngdd4_def equivp_def by auto
lemma genCngdd4_eval4_rel_fun: "(ΣΣ4_rel (genCngdd4 R) ===> genCngdd4 R) eval4 eval4"
using cngdd4_genCngdd4 unfolding cngdd4_def cptdd4_def by auto
lemma genCngdd4_eval4: "ΣΣ4_rel (genCngdd4 R) x y ==> genCngdd4 R (eval4 x) (eval4 y)"
using genCngdd4_eval4_rel_fun unfolding rel_fun_def by auto
lemma leq_genCngdd4: "R ≤ genCngdd4 R"
and imp_genCngdd4[intro]: "R j1 j2 ==> genCngdd4 R j1 j2"
unfolding genCngdd4_def[abs_def] by auto
lemma genCngdd4_minimal: "[|R ≤ R'; cngdd4 R'|] ==> genCngdd4 R ≤ R'"
unfolding genCngdd4_def[abs_def] by (metis (lifting, no_types) predicate2I)
theorem coinductionU_genCngdd4:
assumes "∀ a b. R a b --> F_rel (genCngdd4 R) (dtor_J a) (dtor_J b)"
shows "R a b --> a = b"
proof-
let ?R' = "genCngdd4 R"
have "R ≤ Retr ?R'" using assms unfolding Retr_def[abs_def] by auto
hence "R ≤ ?R' \<sqinter> Retr ?R'" using leq_genCngdd4 by auto
moreover have "cngdd4 (?R' \<sqinter> Retr ?R')" using cngdd4_Retr[OF cngdd4_genCngdd4[of R]] .
ultimately have "?R' ≤ ?R' \<sqinter> Retr ?R'" using genCngdd4_minimal by metis
hence "?R' ≤ Retr ?R'" by auto
hence "?R' a b --> a = b" unfolding Retr_def[abs_def] by (intro J.dtor_coinduct) auto
thus ?thesis using leq_genCngdd4 by auto
qed
subsection{* Flattening from term algebra to "one-step" algebra *}
definition algΛ4 :: "J Σ4 => J" where "algΛ4 = eval4 o \<oo>\<pp>4 o Σ4_map leaf4"
theorem eval4_comp_\<oo>\<pp>4: "eval4 o \<oo>\<pp>4 = algΛ4 o Σ4_map eval4"
unfolding algΛ4_def
unfolding o_assoc[symmetric] Σ4.map_comp0[symmetric]
unfolding leaf4_natural[symmetric] unfolding Σ4.map_comp0
apply(rule sym) unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding \<oo>\<pp>4_natural
unfolding o_assoc eval4_flat4[symmetric]
apply(subst o_assoc[symmetric]) unfolding flat4_commute[symmetric]
unfolding o_assoc[symmetric] Σ4.map_comp0[symmetric] flat4_leaf4 Σ4.map_id0 o_id ..
theorem eval4_\<oo>\<pp>4: "eval4 (\<oo>\<pp>4 t) = algΛ4 (Σ4_map eval4 t)"
using eval4_comp_\<oo>\<pp>4 unfolding o_def fun_eq_iff by (rule allE)
theorem eval4_leaf4': "eval4 (leaf4 j) = j"
using eval4_leaf4 unfolding o_def fun_eq_iff id_def by (rule allE)
theorem dtor_J_algΛ4: "dtor_J o algΛ4 = F_map eval4 o Λ4 o Σ4_map <id, dtor_J>"
unfolding algΛ4_def eval4_def o_assoc dtor_unfold_J_pointfree Λ4_dd4
unfolding o_assoc[symmetric] \<oo>\<pp>4_natural[symmetric] Σ4.map_comp0[symmetric] leaf4_natural
..
theorem dtor_J_algΛ4': "dtor_J (algΛ4 t) = F_map eval4 (Λ4 (Σ4_map (<id, dtor_J>) t))"
by (rule trans[OF o_eq_dest[OF dtor_J_algΛ4] o_apply])
theorem ctor_J_algΛ4: "algΛ4 t = ctor_J (F_map eval4 (Λ4 (Σ4_map (<id, dtor_J>) t)))"
by (rule iffD1[OF J.dtor_inject trans[OF dtor_J_algΛ4' J.dtor_ctor[symmetric]]])
definition "cptΛ4 R ≡ (Σ4_rel R ===> R) algΛ4 algΛ4"
definition "cngΛ4 R ≡ equivp R ∧ cptΛ4 R"
lemma cptdd4_iff_cptΛ4: "cptdd4 R <-> cptΛ4 R"
apply (rule iffI)
apply (unfold cptΛ4_def cptdd4_def algΛ4_def o_assoc[symmetric]) [1]
apply (erule rel_funD[OF rel_funD[OF comp_transfer]])
apply transfer_prover
apply (unfold cptΛ4_def cptdd4_def) [1]
unfolding rel_fun_def
apply (rule allI)+
apply (rule ΣΣ4_rel_induct)
apply (simp only: eval4_leaf4')
unfolding eval4_\<oo>\<pp>4
apply (drule spec2)
apply (erule mp)
apply (rule iffD2[OF Σ4_rel_Σ4_map_Σ4_map])
apply (subst vimage2p_def)
apply assumption
done
theorem genCngdd4_def2: "genCngdd4 R j1 j2 <-> (∀ R'. R ≤ R' ∧ cngΛ4 R' --> R' j1 j2)"
unfolding genCngdd4_def cngdd4_def cngΛ4_def cptdd4_iff_cptΛ4 ..
subsection {* Incremental coinduction *}
interpretation I4: Incremental where Retr = Retr and Cl = genCngdd4
proof
show "mono Retr" by (rule mono_retr)
next
show "mono genCngdd4" unfolding mono_def
unfolding genCngdd4_def[abs_def] by (smt order.trans predicate2I)
next
fix r show "genCngdd4 (genCngdd4 r) = genCngdd4 r"
by (metis cngdd4_genCngdd4 genCngdd4_minimal leq_genCngdd4 order_class.order.antisym)
next
fix r show "r ≤ genCngdd4 r" by (metis leq_genCngdd4)
next
fix r assume "genCngdd4 r = r" thus "genCngdd4 (r \<sqinter> Retr r) = r \<sqinter> Retr r"
by (metis antisym cngdd4_Retr cngdd4_genCngdd4 genCngdd4_minimal
inf.orderI inf_idem leq_genCngdd4)
qed
definition ded4 where "ded4 r s ≡ I4.ded r s"
theorems Ax = I4.Ax'[unfolded ded4_def[symmetric]]
theorems Split = I4.Split[unfolded ded4_def[symmetric]]
theorems Coind = I4.Coind[unfolded ded4_def[symmetric]]
theorem soundness_ded:
assumes "ded4 (op =) s" shows "s ≤ (op =)"
unfolding gfp_Retr_eq[symmetric] apply(rule I4.soundness_ded[unfolded ded4_def[symmetric]] )
using assms unfolding gfp_Retr_eq[symmetric] ded4_def .
unused_thms
end