header {* Corecursion and coinduction up to *}
theory Stream_Corec_Upto3
imports Stream_Lift_to_Free3
begin
subsection{* The algebra associated to dd3 *}
definition "eval3 ≡ dtor_unfold_J (dd3 o ΣΣ3_map <id, dtor_J>)"
lemma eval3: "F_map eval3 o dd3 o ΣΣ3_map <id, dtor_J> = dtor_J o eval3"
unfolding eval3_def dtor_unfold_J_pointfree unfolding o_assoc ..
lemma eval3_ctor_J: "ctor_J o F_map eval3 o dd3 o ΣΣ3_map <id, dtor_J> = eval3"
unfolding o_def spec[OF eval3[unfolded o_def fun_eq_iff]] J.ctor_dtor ..
lemma eval3_leaf3: "eval3 o leaf3 = id"
proof (rule trans)
show "eval3 o leaf3 = dtor_unfold_J dtor_J"
apply(rule J.dtor_unfold_unique)
unfolding o_assoc eval3[symmetric] unfolding o_assoc[symmetric] leaf3_natural
apply(rule sym)
unfolding F_map_comp o_assoc apply (subst o_assoc[symmetric])
unfolding dd3_leaf3 unfolding o_assoc[symmetric] by simp
qed(metis F_map_id J.dtor_unfold_unique fun.map_id o_id)
lemma eval3_flat3: "eval3 o flat3 = eval3 o ΣΣ3_map eval3"
proof (rule trans)
let ?K3 = "dtor_unfold_J (dd3 o ΣΣ3_map <ΣΣ3_map fst, dd3> o ΣΣ3_map (ΣΣ3_map <id, dtor_J>))"
show "eval3 o flat3 = ?K3"
apply(rule J.dtor_unfold_unique)
unfolding F_map_comp o_assoc
apply(subst o_assoc[symmetric], subst o_assoc[symmetric])
unfolding dd3_flat3
unfolding o_assoc[symmetric] ΣΣ3.map_comp0[symmetric] snd_convol
unfolding flat3_natural
unfolding o_assoc eval3 ..
have A: "<eval3, dtor_J o eval3> = <id,dtor_J> o eval3" by simp
show "?K3 = eval3 o ΣΣ3_map eval3"
apply(rule J.dtor_unfold_unique[symmetric])
unfolding o_assoc[symmetric] ΣΣ3.map_comp0[symmetric] map_prod_o_convol id_o
unfolding F_map_comp o_assoc
apply(subst o_assoc[symmetric]) unfolding dd3_natural[symmetric]
unfolding o_assoc[symmetric] ΣΣ3.map_comp0[symmetric]
unfolding o_assoc unfolding map_prod_o_convol unfolding convol_o
unfolding o_assoc[symmetric] ΣΣ3.map_comp0[symmetric] fst_convol ΣΣ3.map_id0 o_id
unfolding o_assoc eval3 unfolding A unfolding convol_o id_o
apply(rule sym) apply(subst eval3[symmetric])
unfolding o_assoc[symmetric] ΣΣ3.map_comp0[symmetric] convol_o id_o ..
qed
subsection{* The correspondence between coalgebras up to and coalgebras *}
definition cutΣΣ3Oc :: "('a => 'a ΣΣ3 F) => ('a ΣΣ3 => 'a ΣΣ3 F)"
where "cutΣΣ3Oc s ≡ F_map flat3 o dd3 o ΣΣ3_map <leaf3, s>"
definition cΣΣ3Ocut :: "('a ΣΣ3 => 'a ΣΣ3 F) => ('a => 'a ΣΣ3 F)"
where "cΣΣ3Ocut s' ≡ s' o leaf3"
lemma cΣΣ3Ocut_cutΣΣ3Oc: "cΣΣ3Ocut (cutΣΣ3Oc s) = s"
unfolding cΣΣ3Ocut_def cutΣΣ3Oc_def
unfolding o_assoc[symmetric] unfolding leaf3_natural
unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding dd3_leaf3 unfolding o_assoc F_map_comp[symmetric] flat3_leaf3
unfolding F_map_id id_o by simp
lemma cutΣΣ3Oc_inj: "cutΣΣ3Oc s3 = cutΣΣ3Oc s2 <-> s3 = s2"
by (metis cΣΣ3Ocut_cutΣΣ3Oc)
lemma cΣΣ3Ocut_surj: "∃ s'. cΣΣ3Ocut s' = s"
using cΣΣ3Ocut_cutΣΣ3Oc by(rule exI[of _ "cutΣΣ3Oc s"])
definition extdd3 :: "('a => J) => ('a ΣΣ3 => J)"
where "extdd3 f ≡ eval3 o ΣΣ3_map f"
term eval3
definition restr :: "('a ΣΣ3 => J) => ('a => J)"
where "restr f' ≡ f' o leaf3"
lemma extdd3_mor:
assumes f: "F_map (extdd3 f) o s = dtor_J o f"
shows "F_map (extdd3 f) o cutΣΣ3Oc s = dtor_J o (extdd3 f)"
proof-
have AA: "eval3 ** F_map eval3 o (ΣΣ3_map f ** F_map (ΣΣ3_map f) o <leaf3 , s>) =
<f , F_map eval3 o (F_map (ΣΣ3_map f) o s)>"
unfolding map_prod_o_convol unfolding leaf3_natural o_assoc eval3_leaf3 id_o ..
show ?thesis
unfolding extdd3_def
unfolding o_assoc eval3[symmetric]
unfolding o_assoc[symmetric] ΣΣ3.map_comp0[symmetric]
unfolding convol_comp[symmetric] id_o
unfolding f[symmetric, unfolded extdd3_def]
unfolding o_assoc
apply(subst o_assoc[symmetric])
unfolding F_map_comp o_assoc
unfolding cutΣΣ3Oc_def
unfolding o_assoc
unfolding F_map_comp[symmetric] unfolding o_assoc[symmetric]
unfolding flat3_natural[symmetric]
unfolding o_assoc F_map_comp
apply(subst o_assoc[symmetric], subst o_assoc[symmetric], subst o_assoc[symmetric])
unfolding dd3_natural[symmetric]
unfolding o_assoc unfolding F_map_comp[symmetric] eval3_flat3
unfolding F_map_comp apply(subst o_assoc[symmetric])
unfolding dd3_natural[symmetric] unfolding o_assoc[symmetric] ΣΣ3.map_comp0[symmetric]
unfolding o_assoc[symmetric] AA[unfolded o_assoc[symmetric]] ..
qed
lemma mor_cutΣΣ3Oc_flat3:
assumes f': "F_map f' o cutΣΣ3Oc s = dtor_J o f'"
shows "eval3 o ΣΣ3_map f' = f' o flat3"
proof(rule trans)
def h ≡ "dd3 o ΣΣ3_map <id,cutΣΣ3Oc s>"
have f'_id: "f' = f' o id" by simp
show "eval3 o ΣΣ3_map f' = dtor_unfold_J h"
apply(rule J.dtor_unfold_unique, rule sym)
unfolding o_assoc eval3[symmetric]
unfolding o_assoc[symmetric] ΣΣ3.map_comp0[symmetric]
unfolding convol_comp_id1[symmetric] unfolding f'[symmetric]
apply(subst f'_id)
unfolding o_assoc ΣΣ3.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 dd3_natural[symmetric] unfolding o_assoc[symmetric]
unfolding ΣΣ3.map_comp0[symmetric] map_prod_o_convol ..
have AA: "<id , cutΣΣ3Oc s> = (flat3 ** F_map flat3) o (id ** dd3) o <leaf3, ΣΣ3_map <leaf3 , s>>"
unfolding map_prod_o_convol o_assoc map_prod.comp cutΣΣ3Oc_def o_id flat3_leaf3 ..
have BB: "flat3 ** F_map flat3 o id ** dd3 o <leaf3 , ΣΣ3_map <leaf3 , s>> = flat3 ** F_map flat3 o id ** dd3 o <ΣΣ3_map leaf3 , ΣΣ3_map <leaf3 , s>>"
unfolding map_prod.comp unfolding map_prod_o_convol unfolding o_id unfolding flat3_leaf3 leaf3_flat3 ..
show "dtor_unfold_J h = f' o flat3"
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 ΣΣ3.map_comp0 apply(rule sym)
unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding dd3_natural
unfolding o_assoc F_map_comp[symmetric]
unfolding flat3_assoc unfolding F_map_comp
unfolding cutΣΣ3Oc_def o_assoc[symmetric] apply(rule arg_cong[of _ _ "op o (F_map flat3)"])
unfolding o_assoc
unfolding o_assoc[symmetric] unfolding ΣΣ3.map_comp0[symmetric] unfolding map_prod_o_convol id_o
unfolding flat3_natural[symmetric] unfolding o_assoc
unfolding dd3_flat3[symmetric] unfolding o_assoc[symmetric] unfolding ΣΣ3.map_comp0[symmetric]
unfolding convol_o unfolding ΣΣ3.map_comp0[symmetric] unfolding fst_convol ..
qed
lemma restr_mor:
assumes f': "F_map f' o cutΣΣ3Oc s = dtor_J o f'"
shows "F_map (extdd3 (restr f')) o s = dtor_J o restr f'"
unfolding extdd3_def restr_def ΣΣ3.map_comp0
unfolding o_assoc mor_cutΣΣ3Oc_flat3[OF f']
unfolding o_assoc[symmetric] leaf3_flat3 o_id
unfolding o_assoc f'[symmetric]
unfolding o_assoc[symmetric] cΣΣ3Ocut_cutΣΣ3Oc[unfolded cΣΣ3Ocut_def] ..
lemma extdd3_restr:
assumes f': "F_map f' o cutΣΣ3Oc s = dtor_J o f'"
shows "extdd3 (restr f') = f'"
proof-
have "f' = eval3 o ΣΣ3_map f' o leaf3"
unfolding o_assoc[symmetric] leaf3_natural
unfolding o_assoc eval3_leaf3 by simp
also have "... = eval3 o ΣΣ3_map (f' o leaf3)"
unfolding ΣΣ3.map_comp0 o_assoc
unfolding mor_cutΣΣ3Oc_flat3[OF f'] unfolding o_assoc[symmetric] flat3_leaf3 leaf3_flat3 ..
finally have A: "f' = eval3 o ΣΣ3_map (f' o leaf3)" .
show ?thesis unfolding extdd3_def restr_def A[symmetric] ..
qed
lemma restr_inj:
assumes f3': "F_map f3' o cutΣΣ3Oc s = dtor_J o f3'"
and f2': "F_map f2' o cutΣΣ3Oc s = dtor_J o f2'"
shows "restr f3' = restr f2' <-> f3' = f2'"
using extdd3_restr[OF f3'] extdd3_restr[OF f2'] by metis
lemma extdd3_surj:
assumes f': "F_map f' o cutΣΣ3Oc s = dtor_J o f'"
shows "∃ f. extdd3 f = f'"
using extdd3_restr[OF f'] by(rule exI[of _ "restr f'"])
lemma restr_extdd3:
assumes f: "F_map (extdd3 f) o s = dtor_J o f"
shows "restr (extdd3 f) = f"
proof-
have "dtor_J o f = F_map (extdd3 f) o s" using assms unfolding extdd3_def by (rule sym)
also have "... = dtor_J o restr (extdd3 f)"
unfolding restr_def unfolding o_assoc extdd3_mor[OF f, symmetric]
unfolding o_assoc[symmetric] cΣΣ3Ocut_cutΣΣ3Oc[unfolded cΣΣ3Ocut_def] ..
finally have "dtor_J o f = dtor_J o restr (extdd3 f)" .
thus ?thesis unfolding dtor_J_o_inj by (rule sym)
qed
lemma extdd3_inj:
assumes f1: "F_map (extdd3 f1) o s = dtor_J o f1"
and f2: "F_map (extdd3 f2) o s = dtor_J o f2"
shows "extdd3 f1 = extdd3 f2 <-> f1 = f2"
using restr_extdd3[OF f1] restr_extdd3[OF f2] by metis
lemma restr_surj:
assumes f: "F_map (extdd3 f) o s = dtor_J o f"
shows "∃ f'. restr f' = f"
using restr_extdd3[OF f] by(rule exI[of _ "extdd3 f"])
subsection{* Coiteration up-to *}
definition "unfoldU3 s ≡ restr (dtor_unfold_J (cutΣΣ3Oc s))"
theorem unfoldU3_pointfree:
"F_map (extdd3 (unfoldU3 s)) o s = dtor_J o unfoldU3 s"
unfolding unfoldU3_def apply(rule restr_mor)
unfolding dtor_unfold_J_pointfree ..
theorem unfoldU3: "F_map (extdd3 (unfoldU3 s)) (s a) = dtor_J (unfoldU3 s a)"
using unfoldU3_pointfree unfolding o_def fun_eq_iff by(rule allE)
theorem unfoldU3_ctor_J:
"ctor_J (F_map (extdd3 (unfoldU3 s)) (s a)) = unfoldU3 s a"
using unfoldU3 by (metis J.ctor_dtor)
theorem unfoldU3_unique:
assumes "F_map (extdd3 f) o s = dtor_J o f"
shows "f = unfoldU3 s"
proof-
note f = extdd3_mor[OF assms] note co = extdd3_mor[OF unfoldU3_pointfree]
have A: "extdd3 f = extdd3 (unfoldU3 s)"
proof(rule trans)
show "extdd3 f = dtor_unfold_J (cutΣΣ3Oc s)" apply(rule J.dtor_unfold_unique) using f .
show "dtor_unfold_J (cutΣΣ3Oc s) = extdd3 (unfoldU3 s)"
apply(rule J.dtor_unfold_unique[symmetric]) using co .
qed
show ?thesis using A unfolding extdd3_inj[OF assms unfoldU3_pointfree] .
qed
lemma unfoldU3_ctor_J_pointfree:
"ctor_J o F_map (extdd3 (unfoldU3 s)) o s = unfoldU3 s"
unfolding o_def fun_eq_iff by (subst unfoldU3_ctor_J[symmetric]) (rule allI, rule refl)
definition corecU3 :: "('a => (J + 'a) ΣΣ3 F) => 'a => J" where
"corecU3 s = unfoldU3 (case_sum (dd3 o leaf3 o <Inl, F_map Inl o dtor_J>) s) o Inr"
definition extddRec3 where
"extddRec3 f ≡ eval3 o ΣΣ3_map (case_sum id f)"
lemma unfoldU3_Inl:
"unfoldU3 (case_sum (dd3 o leaf3 o <Inl , F_map Inl o dtor_J>) s) o Inl = id"
(is "?L = ?R")
proof-
have "?L = unfoldU3 (dd3 o leaf3 o <id, dtor_J>)"
apply(rule unfoldU3_unique)
unfolding o_assoc unfoldU3_pointfree[symmetric]
unfolding o_assoc[symmetric] case_sum_o_inj extdd3_def F_map_comp ΣΣ3.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 dd3_natural[symmetric]
apply(subst o_assoc[symmetric]) unfolding leaf3_natural
unfolding o_assoc[symmetric] map_prod_o_convol o_id ..
also have "... = ?R"
apply(rule sym, rule unfoldU3_unique)
unfolding extdd3_def ΣΣ3.map_id0 o_id dd3_leaf3
unfolding o_assoc[symmetric] snd_convol
unfolding o_assoc F_map_comp[symmetric] eval3_leaf3 F_map_id id_o ..
finally show ?thesis .
qed
theorem corecU3_pointfree:
"F_map (extddRec3 (corecU3 s)) o s = dtor_J o corecU3 s" (is "?L = ?R")
unfolding corecU3_def
unfolding o_assoc unfoldU3_pointfree[symmetric] extddRec3_def
unfolding o_assoc[symmetric] case_sum_o_inj
apply(subst unfoldU3_Inl[symmetric, of s])
unfolding o_assoc case_sum_Inl_Inr_L extdd3_def ..
theorem corecU3:
"F_map (extddRec3 (corecU3 s)) (s a) = dtor_J (corecU3 s a)"
using corecU3_pointfree unfolding o_def fun_eq_iff by(rule allE)
subsection{* Coinduction up-to *}
definition "cptdd3 R ≡ (ΣΣ3_rel R ===> R) eval3 eval3"
definition "cngdd3 R ≡ equivp R ∧ cptdd3 R"
lemma cngdd3_Retr: "cngdd3 R ==> cngdd3 (R \<sqinter> Retr R)"
unfolding cngdd3_def cptdd3_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 ΣΣ3_rel_inf])
apply (erule inf2E)
apply (rule inf2I)
apply (erule rel_funE)
apply assumption
apply assumption
apply (subst Retr_def)
apply (subst eval3_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 eval3_def[symmetric]
apply (rule predicate2D[OF F_rel_mono])
apply (rule predicate2I)
apply (erule rel_funD)
apply assumption
apply (rule F_rel_ΣΣ3_rel)
unfolding ΣΣ3_rel_ΣΣ3_map_ΣΣ3_map vimage2p_rel_prod vimage2p_id
unfolding vimage2p_def Retr_def[symmetric]
apply assumption
done
definition "genCngdd3 R j1 j2 ≡ ∀ R'. R ≤ R' ∧ cngdd3 R' --> R' j1 j2"
lemma cngdd3_genCngdd3: "cngdd3 (genCngdd3 R)"
unfolding cngdd3_def proof safe
show "cptdd3 (genCngdd3 R)"
unfolding cptdd3_def rel_fun_def proof safe
fix x y assume A: "ΣΣ3_rel (genCngdd3 R) x y"
show "genCngdd3 R (eval3 x) (eval3 y)"
unfolding genCngdd3_def[abs_def] proof safe
fix R' assume "R ≤ R'" and 2: "cngdd3 R'"
hence "ΣΣ3_rel R' x y" by (metis A ΣΣ3.rel_mono_strong genCngdd3_def)
thus "R' (eval3 x) (eval3 y)" using 2 unfolding cngdd3_def cptdd3_def rel_fun_def by auto
qed
qed
qed(rule equivpI, unfold reflp_def symp_def transp_def genCngdd3_def cngdd3_def equivp_def, auto)
lemma
genCngdd3_refl[intro,simp]: "genCngdd3 R j j"
and genCngdd3_sym[intro]: "genCngdd3 R j1 j2 ==> genCngdd3 R j2 j1"
and genCngdd3_trans[intro]: "[|genCngdd3 R j1 j2; genCngdd3 R j2 j3|] ==> genCngdd3 R j1 j3"
using cngdd3_genCngdd3 unfolding cngdd3_def equivp_def by auto
lemma genCngdd3_eval3_rel_fun: "(ΣΣ3_rel (genCngdd3 R) ===> genCngdd3 R) eval3 eval3"
using cngdd3_genCngdd3 unfolding cngdd3_def cptdd3_def by auto
lemma genCngdd3_eval3: "ΣΣ3_rel (genCngdd3 R) x y ==> genCngdd3 R (eval3 x) (eval3 y)"
using genCngdd3_eval3_rel_fun unfolding rel_fun_def by auto
lemma leq_genCngdd3: "R ≤ genCngdd3 R"
and imp_genCngdd3[intro]: "R j1 j2 ==> genCngdd3 R j1 j2"
unfolding genCngdd3_def[abs_def] by auto
lemma genCngdd3_minimal: "[|R ≤ R'; cngdd3 R'|] ==> genCngdd3 R ≤ R'"
unfolding genCngdd3_def[abs_def] by (metis (lifting, no_types) predicate2I)
theorem coinductionU_genCngdd3:
assumes "∀ a b. R a b --> F_rel (genCngdd3 R) (dtor_J a) (dtor_J b)"
shows "R a b --> a = b"
proof-
let ?R' = "genCngdd3 R"
have "R ≤ Retr ?R'" using assms unfolding Retr_def[abs_def] by auto
hence "R ≤ ?R' \<sqinter> Retr ?R'" using leq_genCngdd3 by auto
moreover have "cngdd3 (?R' \<sqinter> Retr ?R')" using cngdd3_Retr[OF cngdd3_genCngdd3[of R]] .
ultimately have "?R' ≤ ?R' \<sqinter> Retr ?R'" using genCngdd3_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_genCngdd3 by auto
qed
subsection{* Flattening from term algebra to "one-step" algebra *}
definition algΛ3 :: "J Σ3 => J" where "algΛ3 = eval3 o \<oo>\<pp>3 o Σ3_map leaf3"
theorem eval3_comp_\<oo>\<pp>3: "eval3 o \<oo>\<pp>3 = algΛ3 o Σ3_map eval3"
unfolding algΛ3_def
unfolding o_assoc[symmetric] Σ3.map_comp0[symmetric]
unfolding leaf3_natural[symmetric] unfolding Σ3.map_comp0
apply(rule sym) unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding \<oo>\<pp>3_natural
unfolding o_assoc eval3_flat3[symmetric]
apply(subst o_assoc[symmetric]) unfolding flat3_commute[symmetric]
unfolding o_assoc[symmetric] Σ3.map_comp0[symmetric] flat3_leaf3 Σ3.map_id0 o_id ..
theorem eval3_\<oo>\<pp>3: "eval3 (\<oo>\<pp>3 t) = algΛ3 (Σ3_map eval3 t)"
using eval3_comp_\<oo>\<pp>3 unfolding o_def fun_eq_iff by (rule allE)
theorem eval3_leaf3': "eval3 (leaf3 j) = j"
using eval3_leaf3 unfolding o_def fun_eq_iff id_def by (rule allE)
theorem dtor_J_algΛ3: "dtor_J o algΛ3 = F_map eval3 o Λ3 o Σ3_map <id, dtor_J>"
unfolding algΛ3_def eval3_def o_assoc dtor_unfold_J_pointfree Λ3_dd3
unfolding o_assoc[symmetric] \<oo>\<pp>3_natural[symmetric] Σ3.map_comp0[symmetric] leaf3_natural
..
theorem dtor_J_algΛ3': "dtor_J (algΛ3 t) = F_map eval3 (Λ3 (Σ3_map (<id, dtor_J>) t))"
by (rule trans[OF o_eq_dest[OF dtor_J_algΛ3] o_apply])
theorem ctor_J_algΛ3: "algΛ3 t = ctor_J (F_map eval3 (Λ3 (Σ3_map (<id, dtor_J>) t)))"
by (rule iffD1[OF J.dtor_inject trans[OF dtor_J_algΛ3' J.dtor_ctor[symmetric]]])
definition "cptΛ3 R ≡ (Σ3_rel R ===> R) algΛ3 algΛ3"
definition "cngΛ3 R ≡ equivp R ∧ cptΛ3 R"
lemma cptdd3_iff_cptΛ3: "cptdd3 R <-> cptΛ3 R"
apply (rule iffI)
apply (unfold cptΛ3_def cptdd3_def algΛ3_def o_assoc[symmetric]) [1]
apply (erule rel_funD[OF rel_funD[OF comp_transfer]])
apply transfer_prover
apply (unfold cptΛ3_def cptdd3_def) [1]
unfolding rel_fun_def
apply (rule allI)+
apply (rule ΣΣ3_rel_induct)
apply (simp only: eval3_leaf3')
unfolding eval3_\<oo>\<pp>3
apply (drule spec2)
apply (erule mp)
apply (rule iffD2[OF Σ3_rel_Σ3_map_Σ3_map])
apply (subst vimage2p_def)
apply assumption
done
theorem genCngdd3_def2: "genCngdd3 R j1 j2 <-> (∀ R'. R ≤ R' ∧ cngΛ3 R' --> R' j1 j2)"
unfolding genCngdd3_def cngdd3_def cngΛ3_def cptdd3_iff_cptΛ3 ..
subsection {* Incremental coinduction *}
interpretation I3: Incremental where Retr = Retr and Cl = genCngdd3
proof
show "mono Retr" by (rule mono_retr)
next
show "mono genCngdd3" unfolding mono_def
unfolding genCngdd3_def[abs_def] by (smt order.trans predicate2I)
next
fix r show "genCngdd3 (genCngdd3 r) = genCngdd3 r"
by (metis cngdd3_genCngdd3 genCngdd3_minimal leq_genCngdd3 order_class.order.antisym)
next
fix r show "r ≤ genCngdd3 r" by (metis leq_genCngdd3)
next
fix r assume "genCngdd3 r = r" thus "genCngdd3 (r \<sqinter> Retr r) = r \<sqinter> Retr r"
by (metis antisym cngdd3_Retr cngdd3_genCngdd3 genCngdd3_minimal
inf.orderI inf_idem leq_genCngdd3)
qed
definition ded3 where "ded3 r s ≡ I3.ded r s"
theorems Ax = I3.Ax'[unfolded ded3_def[symmetric]]
theorems Split = I3.Split[unfolded ded3_def[symmetric]]
theorems Coind = I3.Coind[unfolded ded3_def[symmetric]]
theorem soundness_ded:
assumes "ded3 (op =) s" shows "s ≤ (op =)"
unfolding gfp_Retr_eq[symmetric] apply(rule I3.soundness_ded[unfolded ded3_def[symmetric]] )
using assms unfolding gfp_Retr_eq[symmetric] ded3_def .
unused_thms
end