header {* Corecursion and coinduction up to *}
theory Tree_Corec_Upto2
imports Tree_Lift_to_Free2
begin
subsection{* The algebra associated to dd2 *}
definition "eval2 ≡ dtor_unfold_J (dd2 o ΣΣ2_map <id, dtor_J>)"
lemma eval2: "F_map eval2 o dd2 o ΣΣ2_map <id, dtor_J> = dtor_J o eval2"
unfolding eval2_def dtor_unfold_J_pointfree unfolding o_assoc ..
lemma eval2_ctor_J: "ctor_J o F_map eval2 o dd2 o ΣΣ2_map <id, dtor_J> = eval2"
unfolding o_def spec[OF eval2[unfolded o_def fun_eq_iff]] J.ctor_dtor ..
lemma eval2_leaf2: "eval2 o leaf2 = id"
proof (rule trans)
show "eval2 o leaf2 = dtor_unfold_J dtor_J"
apply(rule J.dtor_unfold_unique)
unfolding o_assoc eval2[symmetric] unfolding o_assoc[symmetric] leaf2_natural
apply(rule sym)
unfolding F_map_comp o_assoc apply (subst o_assoc[symmetric])
unfolding dd2_leaf2 unfolding o_assoc[symmetric] by simp
qed(metis F_map_id J.dtor_unfold_unique fun.map_id o_id)
lemma eval2_flat2: "eval2 o flat2 = eval2 o ΣΣ2_map eval2"
proof (rule trans)
let ?K2 = "dtor_unfold_J (dd2 o ΣΣ2_map <ΣΣ2_map fst, dd2> o ΣΣ2_map (ΣΣ2_map <id, dtor_J>))"
show "eval2 o flat2 = ?K2"
apply(rule J.dtor_unfold_unique)
unfolding F_map_comp o_assoc
apply(subst o_assoc[symmetric], subst o_assoc[symmetric])
unfolding dd2_flat2
unfolding o_assoc[symmetric] ΣΣ2.map_comp0[symmetric] snd_convol
unfolding flat2_natural
unfolding o_assoc eval2 ..
have A: "<eval2, dtor_J o eval2> = <id,dtor_J> o eval2" by simp
show "?K2 = eval2 o ΣΣ2_map eval2"
apply(rule J.dtor_unfold_unique[symmetric])
unfolding o_assoc[symmetric] ΣΣ2.map_comp0[symmetric] map_prod_o_convol id_o
unfolding F_map_comp o_assoc
apply(subst o_assoc[symmetric]) unfolding dd2_natural[symmetric]
unfolding o_assoc[symmetric] ΣΣ2.map_comp0[symmetric]
unfolding o_assoc unfolding map_prod_o_convol unfolding convol_o
unfolding o_assoc[symmetric] ΣΣ2.map_comp0[symmetric] fst_convol ΣΣ2.map_id0 o_id
unfolding o_assoc eval2 unfolding A unfolding convol_o id_o
apply(rule sym) apply(subst eval2[symmetric])
unfolding o_assoc[symmetric] ΣΣ2.map_comp0[symmetric] convol_o id_o ..
qed
subsection{* The correspondence between coalgebras up to and coalgebras *}
definition cutΣΣ2Oc :: "('a => 'a ΣΣ2 F) => ('a ΣΣ2 => 'a ΣΣ2 F)"
where "cutΣΣ2Oc s ≡ F_map flat2 o dd2 o ΣΣ2_map <leaf2, s>"
definition cΣΣ2Ocut :: "('a ΣΣ2 => 'a ΣΣ2 F) => ('a => 'a ΣΣ2 F)"
where "cΣΣ2Ocut s' ≡ s' o leaf2"
lemma cΣΣ2Ocut_cutΣΣ2Oc: "cΣΣ2Ocut (cutΣΣ2Oc s) = s"
unfolding cΣΣ2Ocut_def cutΣΣ2Oc_def
unfolding o_assoc[symmetric] unfolding leaf2_natural
unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding dd2_leaf2 unfolding o_assoc F_map_comp[symmetric] flat2_leaf2
unfolding F_map_id id_o by simp
lemma cutΣΣ2Oc_inj: "cutΣΣ2Oc s2 = cutΣΣ2Oc s2 <-> s2 = s2"
by (metis cΣΣ2Ocut_cutΣΣ2Oc)
lemma cΣΣ2Ocut_surj: "∃ s'. cΣΣ2Ocut s' = s"
using cΣΣ2Ocut_cutΣΣ2Oc by(rule exI[of _ "cutΣΣ2Oc s"])
definition extdd2 :: "('a => J) => ('a ΣΣ2 => J)"
where "extdd2 f ≡ eval2 o ΣΣ2_map f"
term eval2
definition restr :: "('a ΣΣ2 => J) => ('a => J)"
where "restr f' ≡ f' o leaf2"
lemma extdd2_mor:
assumes f: "F_map (extdd2 f) o s = dtor_J o f"
shows "F_map (extdd2 f) o cutΣΣ2Oc s = dtor_J o (extdd2 f)"
proof-
have AA: "eval2 ** F_map eval2 o (ΣΣ2_map f ** F_map (ΣΣ2_map f) o <leaf2 , s>) =
<f , F_map eval2 o (F_map (ΣΣ2_map f) o s)>"
unfolding map_prod_o_convol unfolding leaf2_natural o_assoc eval2_leaf2 id_o ..
show ?thesis
unfolding extdd2_def
unfolding o_assoc eval2[symmetric]
unfolding o_assoc[symmetric] ΣΣ2.map_comp0[symmetric]
unfolding convol_comp[symmetric] id_o
unfolding f[symmetric, unfolded extdd2_def]
unfolding o_assoc
apply(subst o_assoc[symmetric])
unfolding F_map_comp o_assoc
unfolding cutΣΣ2Oc_def
unfolding o_assoc
unfolding F_map_comp[symmetric] unfolding o_assoc[symmetric]
unfolding flat2_natural[symmetric]
unfolding o_assoc F_map_comp
apply(subst o_assoc[symmetric], subst o_assoc[symmetric], subst o_assoc[symmetric])
unfolding dd2_natural[symmetric]
unfolding o_assoc unfolding F_map_comp[symmetric] eval2_flat2
unfolding F_map_comp apply(subst o_assoc[symmetric])
unfolding dd2_natural[symmetric] unfolding o_assoc[symmetric] ΣΣ2.map_comp0[symmetric]
unfolding o_assoc[symmetric] AA[unfolded o_assoc[symmetric]] ..
qed
lemma mor_cutΣΣ2Oc_flat2:
assumes f': "F_map f' o cutΣΣ2Oc s = dtor_J o f'"
shows "eval2 o ΣΣ2_map f' = f' o flat2"
proof(rule trans)
def h ≡ "dd2 o ΣΣ2_map <id,cutΣΣ2Oc s>"
have f'_id: "f' = f' o id" by simp
show "eval2 o ΣΣ2_map f' = dtor_unfold_J h"
apply(rule J.dtor_unfold_unique, rule sym)
unfolding o_assoc eval2[symmetric]
unfolding o_assoc[symmetric] ΣΣ2.map_comp0[symmetric]
unfolding convol_comp_id1[symmetric] unfolding f'[symmetric]
apply(subst f'_id)
unfolding o_assoc ΣΣ2.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 dd2_natural[symmetric] unfolding o_assoc[symmetric]
unfolding ΣΣ2.map_comp0[symmetric] map_prod_o_convol ..
have AA: "<id , cutΣΣ2Oc s> = (flat2 ** F_map flat2) o (id ** dd2) o <leaf2, ΣΣ2_map <leaf2 , s>>"
unfolding map_prod_o_convol o_assoc map_prod.comp cutΣΣ2Oc_def o_id flat2_leaf2 ..
have BB: "flat2 ** F_map flat2 o id ** dd2 o <leaf2 , ΣΣ2_map <leaf2 , s>> = flat2 ** F_map flat2 o id ** dd2 o <ΣΣ2_map leaf2 , ΣΣ2_map <leaf2 , s>>"
unfolding map_prod.comp unfolding map_prod_o_convol unfolding o_id unfolding flat2_leaf2 leaf2_flat2 ..
show "dtor_unfold_J h = f' o flat2"
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 ΣΣ2.map_comp0 apply(rule sym)
unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding dd2_natural
unfolding o_assoc F_map_comp[symmetric]
unfolding flat2_assoc unfolding F_map_comp
unfolding cutΣΣ2Oc_def o_assoc[symmetric] apply(rule arg_cong[of _ _ "op o (F_map flat2)"])
unfolding o_assoc
unfolding o_assoc[symmetric] unfolding ΣΣ2.map_comp0[symmetric] unfolding map_prod_o_convol id_o
unfolding flat2_natural[symmetric] unfolding o_assoc
unfolding dd2_flat2[symmetric] unfolding o_assoc[symmetric] unfolding ΣΣ2.map_comp0[symmetric]
unfolding convol_o unfolding ΣΣ2.map_comp0[symmetric] unfolding fst_convol ..
qed
lemma restr_mor:
assumes f': "F_map f' o cutΣΣ2Oc s = dtor_J o f'"
shows "F_map (extdd2 (restr f')) o s = dtor_J o restr f'"
unfolding extdd2_def restr_def ΣΣ2.map_comp0
unfolding o_assoc mor_cutΣΣ2Oc_flat2[OF f']
unfolding o_assoc[symmetric] leaf2_flat2 o_id
unfolding o_assoc f'[symmetric]
unfolding o_assoc[symmetric] cΣΣ2Ocut_cutΣΣ2Oc[unfolded cΣΣ2Ocut_def] ..
lemma extdd2_restr:
assumes f': "F_map f' o cutΣΣ2Oc s = dtor_J o f'"
shows "extdd2 (restr f') = f'"
proof-
have "f' = eval2 o ΣΣ2_map f' o leaf2"
unfolding o_assoc[symmetric] leaf2_natural
unfolding o_assoc eval2_leaf2 by simp
also have "... = eval2 o ΣΣ2_map (f' o leaf2)"
unfolding ΣΣ2.map_comp0 o_assoc
unfolding mor_cutΣΣ2Oc_flat2[OF f'] unfolding o_assoc[symmetric] flat2_leaf2 leaf2_flat2 ..
finally have A: "f' = eval2 o ΣΣ2_map (f' o leaf2)" .
show ?thesis unfolding extdd2_def restr_def A[symmetric] ..
qed
lemma restr_inj:
assumes f2': "F_map f2' o cutΣΣ2Oc s = dtor_J o f2'"
and f2': "F_map f2' o cutΣΣ2Oc s = dtor_J o f2'"
shows "restr f2' = restr f2' <-> f2' = f2'"
using extdd2_restr[OF f2'] extdd2_restr[OF f2'] by metis
lemma extdd2_surj:
assumes f': "F_map f' o cutΣΣ2Oc s = dtor_J o f'"
shows "∃ f. extdd2 f = f'"
using extdd2_restr[OF f'] by(rule exI[of _ "restr f'"])
lemma restr_extdd2:
assumes f: "F_map (extdd2 f) o s = dtor_J o f"
shows "restr (extdd2 f) = f"
proof-
have "dtor_J o f = F_map (extdd2 f) o s" using assms unfolding extdd2_def by (rule sym)
also have "... = dtor_J o restr (extdd2 f)"
unfolding restr_def unfolding o_assoc extdd2_mor[OF f, symmetric]
unfolding o_assoc[symmetric] cΣΣ2Ocut_cutΣΣ2Oc[unfolded cΣΣ2Ocut_def] ..
finally have "dtor_J o f = dtor_J o restr (extdd2 f)" .
thus ?thesis unfolding dtor_J_o_inj by (rule sym)
qed
lemma extdd2_inj:
assumes f1: "F_map (extdd2 f1) o s = dtor_J o f1"
and f2: "F_map (extdd2 f2) o s = dtor_J o f2"
shows "extdd2 f1 = extdd2 f2 <-> f1 = f2"
using restr_extdd2[OF f1] restr_extdd2[OF f2] by metis
lemma restr_surj:
assumes f: "F_map (extdd2 f) o s = dtor_J o f"
shows "∃ f'. restr f' = f"
using restr_extdd2[OF f] by(rule exI[of _ "extdd2 f"])
subsection{* Coiteration up-to *}
definition "unfoldU2 s ≡ restr (dtor_unfold_J (cutΣΣ2Oc s))"
theorem unfoldU2_pointfree:
"F_map (extdd2 (unfoldU2 s)) o s = dtor_J o unfoldU2 s"
unfolding unfoldU2_def apply(rule restr_mor)
unfolding dtor_unfold_J_pointfree ..
theorem unfoldU2: "F_map (extdd2 (unfoldU2 s)) (s a) = dtor_J (unfoldU2 s a)"
using unfoldU2_pointfree unfolding o_def fun_eq_iff by(rule allE)
theorem unfoldU2_ctor_J:
"ctor_J (F_map (extdd2 (unfoldU2 s)) (s a)) = unfoldU2 s a"
using unfoldU2 by (metis J.ctor_dtor)
theorem unfoldU2_unique:
assumes "F_map (extdd2 f) o s = dtor_J o f"
shows "f = unfoldU2 s"
proof-
note f = extdd2_mor[OF assms] note co = extdd2_mor[OF unfoldU2_pointfree]
have A: "extdd2 f = extdd2 (unfoldU2 s)"
proof(rule trans)
show "extdd2 f = dtor_unfold_J (cutΣΣ2Oc s)" apply(rule J.dtor_unfold_unique) using f .
show "dtor_unfold_J (cutΣΣ2Oc s) = extdd2 (unfoldU2 s)"
apply(rule J.dtor_unfold_unique[symmetric]) using co .
qed
show ?thesis using A unfolding extdd2_inj[OF assms unfoldU2_pointfree] .
qed
lemma unfoldU2_ctor_J_pointfree:
"ctor_J o F_map (extdd2 (unfoldU2 s)) o s = unfoldU2 s"
unfolding o_def fun_eq_iff by (subst unfoldU2_ctor_J[symmetric]) (rule allI, rule refl)
definition corecU2 :: "('a => (J + 'a) ΣΣ2 F) => 'a => J" where
"corecU2 s = unfoldU2 (case_sum (dd2 o leaf2 o <Inl, F_map Inl o dtor_J>) s) o Inr"
definition extddRec2 where
"extddRec2 f ≡ eval2 o ΣΣ2_map (case_sum id f)"
lemma unfoldU2_Inl:
"unfoldU2 (case_sum (dd2 o leaf2 o <Inl , F_map Inl o dtor_J>) s) o Inl = id"
(is "?L = ?R")
proof-
have "?L = unfoldU2 (dd2 o leaf2 o <id, dtor_J>)"
apply(rule unfoldU2_unique)
unfolding o_assoc unfoldU2_pointfree[symmetric]
unfolding o_assoc[symmetric] case_sum_o_inj extdd2_def F_map_comp ΣΣ2.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 dd2_natural[symmetric]
apply(subst o_assoc[symmetric]) unfolding leaf2_natural
unfolding o_assoc[symmetric] map_prod_o_convol o_id ..
also have "... = ?R"
apply(rule sym, rule unfoldU2_unique)
unfolding extdd2_def ΣΣ2.map_id0 o_id dd2_leaf2
unfolding o_assoc[symmetric] snd_convol
unfolding o_assoc F_map_comp[symmetric] eval2_leaf2 F_map_id id_o ..
finally show ?thesis .
qed
theorem corecU2_pointfree:
"F_map (extddRec2 (corecU2 s)) o s = dtor_J o corecU2 s" (is "?L = ?R")
unfolding corecU2_def
unfolding o_assoc unfoldU2_pointfree[symmetric] extddRec2_def
unfolding o_assoc[symmetric] case_sum_o_inj
apply(subst unfoldU2_Inl[symmetric, of s])
unfolding o_assoc case_sum_Inl_Inr_L extdd2_def ..
theorem corecU2:
"F_map (extddRec2 (corecU2 s)) (s a) = dtor_J (corecU2 s a)"
using corecU2_pointfree unfolding o_def fun_eq_iff by(rule allE)
subsection{* Coinduction up-to *}
definition "cptdd2 R ≡ (ΣΣ2_rel R ===> R) eval2 eval2"
definition "cngdd2 R ≡ equivp R ∧ cptdd2 R"
lemma cngdd2_Retr: "cngdd2 R ==> cngdd2 (R \<sqinter> Retr R)"
unfolding cngdd2_def cptdd2_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 ΣΣ2_rel_inf])
apply (erule inf2E)
apply (rule inf2I)
apply (erule rel_funE)
apply assumption
apply assumption
apply (subst Retr_def)
apply (subst eval2_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 eval2_def[symmetric]
apply (rule predicate2D[OF F_rel_mono])
apply (rule predicate2I)
apply (erule rel_funD)
apply assumption
apply (rule F_rel_ΣΣ2_rel)
unfolding ΣΣ2_rel_ΣΣ2_map_ΣΣ2_map vimage2p_rel_prod vimage2p_id
unfolding vimage2p_def Retr_def[symmetric]
apply assumption
done
definition "genCngdd2 R j1 j2 ≡ ∀ R'. R ≤ R' ∧ cngdd2 R' --> R' j1 j2"
lemma cngdd2_genCngdd2: "cngdd2 (genCngdd2 R)"
unfolding cngdd2_def proof safe
show "cptdd2 (genCngdd2 R)"
unfolding cptdd2_def rel_fun_def proof safe
fix x y assume A: "ΣΣ2_rel (genCngdd2 R) x y"
show "genCngdd2 R (eval2 x) (eval2 y)"
unfolding genCngdd2_def[abs_def] proof safe
fix R' assume "R ≤ R'" and 2: "cngdd2 R'"
hence "ΣΣ2_rel R' x y" by (metis A ΣΣ2.rel_mono_strong genCngdd2_def)
thus "R' (eval2 x) (eval2 y)" using 2 unfolding cngdd2_def cptdd2_def rel_fun_def by auto
qed
qed
qed(rule equivpI, unfold reflp_def symp_def transp_def genCngdd2_def cngdd2_def equivp_def, auto)
lemma
genCngdd2_refl[intro,simp]: "genCngdd2 R j j"
and genCngdd2_sym[intro]: "genCngdd2 R j1 j2 ==> genCngdd2 R j2 j1"
and genCngdd2_trans[intro]: "[|genCngdd2 R j1 j2; genCngdd2 R j2 j3|] ==> genCngdd2 R j1 j3"
using cngdd2_genCngdd2 unfolding cngdd2_def equivp_def by auto
lemma genCngdd2_eval2_rel_fun: "(ΣΣ2_rel (genCngdd2 R) ===> genCngdd2 R) eval2 eval2"
using cngdd2_genCngdd2 unfolding cngdd2_def cptdd2_def by auto
lemma genCngdd2_eval2: "ΣΣ2_rel (genCngdd2 R) x y ==> genCngdd2 R (eval2 x) (eval2 y)"
using genCngdd2_eval2_rel_fun unfolding rel_fun_def by auto
lemma leq_genCngdd2: "R ≤ genCngdd2 R"
and imp_genCngdd2[intro]: "R j1 j2 ==> genCngdd2 R j1 j2"
unfolding genCngdd2_def[abs_def] by auto
lemma genCngdd2_minimal: "[|R ≤ R'; cngdd2 R'|] ==> genCngdd2 R ≤ R'"
unfolding genCngdd2_def[abs_def] by (metis (lifting, no_types) predicate2I)
theorem coinductionU_genCngdd2:
assumes "∀ a b. R a b --> F_rel (genCngdd2 R) (dtor_J a) (dtor_J b)"
shows "R a b --> a = b"
proof-
let ?R' = "genCngdd2 R"
have "R ≤ Retr ?R'" using assms unfolding Retr_def[abs_def] by auto
hence "R ≤ ?R' \<sqinter> Retr ?R'" using leq_genCngdd2 by auto
moreover have "cngdd2 (?R' \<sqinter> Retr ?R')" using cngdd2_Retr[OF cngdd2_genCngdd2[of R]] .
ultimately have "?R' ≤ ?R' \<sqinter> Retr ?R'" using genCngdd2_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_genCngdd2 by auto
qed
subsection{* Flattening from term algebra to "one-step" algebra *}
definition algΛ2 :: "J Σ2 => J" where "algΛ2 = eval2 o \<oo>\<pp>2 o Σ2_map leaf2"
theorem eval2_comp_\<oo>\<pp>2: "eval2 o \<oo>\<pp>2 = algΛ2 o Σ2_map eval2"
unfolding algΛ2_def
unfolding o_assoc[symmetric] Σ2.map_comp0[symmetric]
unfolding leaf2_natural[symmetric] unfolding Σ2.map_comp0
apply(rule sym) unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding \<oo>\<pp>2_natural
unfolding o_assoc eval2_flat2[symmetric]
apply(subst o_assoc[symmetric]) unfolding flat2_commute[symmetric]
unfolding o_assoc[symmetric] Σ2.map_comp0[symmetric] flat2_leaf2 Σ2.map_id0 o_id ..
theorem eval2_\<oo>\<pp>2: "eval2 (\<oo>\<pp>2 t) = algΛ2 (Σ2_map eval2 t)"
using eval2_comp_\<oo>\<pp>2 unfolding o_def fun_eq_iff by (rule allE)
theorem eval2_leaf2': "eval2 (leaf2 j) = j"
using eval2_leaf2 unfolding o_def fun_eq_iff id_def by (rule allE)
theorem dtor_J_algΛ2: "dtor_J o algΛ2 = F_map eval2 o Λ2 o Σ2_map <id, dtor_J>"
unfolding algΛ2_def eval2_def o_assoc dtor_unfold_J_pointfree Λ2_dd2
unfolding o_assoc[symmetric] \<oo>\<pp>2_natural[symmetric] Σ2.map_comp0[symmetric] leaf2_natural
..
theorem dtor_J_algΛ2': "dtor_J (algΛ2 t) = F_map eval2 (Λ2 (Σ2_map (<id, dtor_J>) t))"
by (rule trans[OF o_eq_dest[OF dtor_J_algΛ2] o_apply])
theorem ctor_J_algΛ2: "algΛ2 t = ctor_J (F_map eval2 (Λ2 (Σ2_map (<id, dtor_J>) t)))"
by (rule iffD1[OF J.dtor_inject trans[OF dtor_J_algΛ2' J.dtor_ctor[symmetric]]])
definition "cptΛ2 R ≡ (Σ2_rel R ===> R) algΛ2 algΛ2"
definition "cngΛ2 R ≡ equivp R ∧ cptΛ2 R"
lemma cptdd2_iff_cptΛ2: "cptdd2 R <-> cptΛ2 R"
apply (rule iffI)
apply (unfold cptΛ2_def cptdd2_def algΛ2_def o_assoc[symmetric]) [1]
apply (erule rel_funD[OF rel_funD[OF comp_transfer]])
apply transfer_prover
apply (unfold cptΛ2_def cptdd2_def) [1]
unfolding rel_fun_def
apply (rule allI)+
apply (rule ΣΣ2_rel_induct)
apply (simp only: eval2_leaf2')
unfolding eval2_\<oo>\<pp>2
apply (drule spec2)
apply (erule mp)
apply (rule iffD2[OF Σ2_rel_Σ2_map_Σ2_map])
apply (subst vimage2p_def)
apply assumption
done
theorem genCngdd2_def2: "genCngdd2 R j1 j2 <-> (∀ R'. R ≤ R' ∧ cngΛ2 R' --> R' j1 j2)"
unfolding genCngdd2_def cngdd2_def cngΛ2_def cptdd2_iff_cptΛ2 ..
subsection {* Incremental coinduction *}
interpretation I2: Incremental where Retr = Retr and Cl = genCngdd2
proof
show "mono Retr" by (rule mono_retr)
next
show "mono genCngdd2" unfolding mono_def
unfolding genCngdd2_def[abs_def] by (smt order.trans predicate2I)
next
fix r show "genCngdd2 (genCngdd2 r) = genCngdd2 r"
by (metis cngdd2_genCngdd2 genCngdd2_minimal leq_genCngdd2 order_class.order.antisym)
next
fix r show "r ≤ genCngdd2 r" by (metis leq_genCngdd2)
next
fix r assume "genCngdd2 r = r" thus "genCngdd2 (r \<sqinter> Retr r) = r \<sqinter> Retr r"
by (metis antisym cngdd2_Retr cngdd2_genCngdd2 genCngdd2_minimal
inf.orderI inf_idem leq_genCngdd2)
qed
definition ded2 where "ded2 r s ≡ I2.ded r s"
theorems Ax = I2.Ax'[unfolded ded2_def[symmetric]]
theorems Split = I2.Split[unfolded ded2_def[symmetric]]
theorems Coind = I2.Coind[unfolded ded2_def[symmetric]]
theorem soundness_ded:
assumes "ded2 (op =) s" shows "s ≤ (op =)"
unfolding gfp_Retr_eq[symmetric] apply(rule I2.soundness_ded[unfolded ded2_def[symmetric]] )
using assms unfolding gfp_Retr_eq[symmetric] ded2_def .
unused_thms
end