header {* Corecursion and coinduction up to *}
theory Stream_Corec_Upto1
imports Stream_Lift_to_Free1
begin
subsection{* The algebra associated to dd1 *}
definition "eval1 ≡ dtor_unfold_J (dd1 o ΣΣ1_map <id, dtor_J>)"
lemma eval1: "F_map eval1 o dd1 o ΣΣ1_map <id, dtor_J> = dtor_J o eval1"
unfolding eval1_def dtor_unfold_J_pointfree unfolding o_assoc ..
lemma eval1_ctor_J: "ctor_J o F_map eval1 o dd1 o ΣΣ1_map <id, dtor_J> = eval1"
unfolding o_def spec[OF eval1[unfolded o_def fun_eq_iff]] J.ctor_dtor ..
lemma eval1_leaf1: "eval1 o leaf1 = id"
proof (rule trans)
show "eval1 o leaf1 = dtor_unfold_J dtor_J"
apply(rule J.dtor_unfold_unique)
unfolding o_assoc eval1[symmetric] unfolding o_assoc[symmetric] leaf1_natural
apply(rule sym)
unfolding F_map_comp o_assoc apply (subst o_assoc[symmetric])
unfolding dd1_leaf1 unfolding o_assoc[symmetric] by simp
qed(metis F_map_id J.dtor_unfold_unique fun.map_id o_id)
lemma eval1_flat1: "eval1 o flat1 = eval1 o ΣΣ1_map eval1"
proof (rule trans)
let ?K1 = "dtor_unfold_J (dd1 o ΣΣ1_map <ΣΣ1_map fst, dd1> o ΣΣ1_map (ΣΣ1_map <id, dtor_J>))"
show "eval1 o flat1 = ?K1"
apply(rule J.dtor_unfold_unique)
unfolding F_map_comp o_assoc
apply(subst o_assoc[symmetric], subst o_assoc[symmetric])
unfolding dd1_flat1
unfolding o_assoc[symmetric] ΣΣ1.map_comp0[symmetric] snd_convol
unfolding flat1_natural
unfolding o_assoc eval1 ..
have A: "<eval1, dtor_J o eval1> = <id,dtor_J> o eval1" by simp
show "?K1 = eval1 o ΣΣ1_map eval1"
apply(rule J.dtor_unfold_unique[symmetric])
unfolding o_assoc[symmetric] ΣΣ1.map_comp0[symmetric] map_prod_o_convol id_o
unfolding F_map_comp o_assoc
apply(subst o_assoc[symmetric]) unfolding dd1_natural[symmetric]
unfolding o_assoc[symmetric] ΣΣ1.map_comp0[symmetric]
unfolding o_assoc unfolding map_prod_o_convol unfolding convol_o
unfolding o_assoc[symmetric] ΣΣ1.map_comp0[symmetric] fst_convol ΣΣ1.map_id0 o_id
unfolding o_assoc eval1 unfolding A unfolding convol_o id_o
apply(rule sym) apply(subst eval1[symmetric])
unfolding o_assoc[symmetric] ΣΣ1.map_comp0[symmetric] convol_o id_o ..
qed
subsection{* The correspondence between coalgebras up to and coalgebras *}
definition cutΣΣ1Oc :: "('a => 'a ΣΣ1 F) => ('a ΣΣ1 => 'a ΣΣ1 F)"
where "cutΣΣ1Oc s ≡ F_map flat1 o dd1 o ΣΣ1_map <leaf1, s>"
definition cΣΣ1Ocut :: "('a ΣΣ1 => 'a ΣΣ1 F) => ('a => 'a ΣΣ1 F)"
where "cΣΣ1Ocut s' ≡ s' o leaf1"
lemma cΣΣ1Ocut_cutΣΣ1Oc: "cΣΣ1Ocut (cutΣΣ1Oc s) = s"
unfolding cΣΣ1Ocut_def cutΣΣ1Oc_def
unfolding o_assoc[symmetric] unfolding leaf1_natural
unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding dd1_leaf1 unfolding o_assoc F_map_comp[symmetric] flat1_leaf1
unfolding F_map_id id_o by simp
lemma cutΣΣ1Oc_inj: "cutΣΣ1Oc s1 = cutΣΣ1Oc s2 <-> s1 = s2"
by (metis cΣΣ1Ocut_cutΣΣ1Oc)
lemma cΣΣ1Ocut_surj: "∃ s'. cΣΣ1Ocut s' = s"
using cΣΣ1Ocut_cutΣΣ1Oc by(rule exI[of _ "cutΣΣ1Oc s"])
definition extdd1 :: "('a => J) => ('a ΣΣ1 => J)"
where "extdd1 f ≡ eval1 o ΣΣ1_map f"
term eval1
definition restr :: "('a ΣΣ1 => J) => ('a => J)"
where "restr f' ≡ f' o leaf1"
lemma extdd1_mor:
assumes f: "F_map (extdd1 f) o s = dtor_J o f"
shows "F_map (extdd1 f) o cutΣΣ1Oc s = dtor_J o (extdd1 f)"
proof-
have AA: "eval1 ** F_map eval1 o (ΣΣ1_map f ** F_map (ΣΣ1_map f) o <leaf1 , s>) =
<f , F_map eval1 o (F_map (ΣΣ1_map f) o s)>"
unfolding map_prod_o_convol unfolding leaf1_natural o_assoc eval1_leaf1 id_o ..
show ?thesis
unfolding extdd1_def
unfolding o_assoc eval1[symmetric]
unfolding o_assoc[symmetric] ΣΣ1.map_comp0[symmetric]
unfolding convol_comp[symmetric] id_o
unfolding f[symmetric, unfolded extdd1_def]
unfolding o_assoc
apply(subst o_assoc[symmetric])
unfolding F_map_comp o_assoc
unfolding cutΣΣ1Oc_def
unfolding o_assoc
unfolding F_map_comp[symmetric] unfolding o_assoc[symmetric]
unfolding flat1_natural[symmetric]
unfolding o_assoc F_map_comp
apply(subst o_assoc[symmetric], subst o_assoc[symmetric], subst o_assoc[symmetric])
unfolding dd1_natural[symmetric]
unfolding o_assoc unfolding F_map_comp[symmetric] eval1_flat1
unfolding F_map_comp apply(subst o_assoc[symmetric])
unfolding dd1_natural[symmetric] unfolding o_assoc[symmetric] ΣΣ1.map_comp0[symmetric]
unfolding o_assoc[symmetric] AA[unfolded o_assoc[symmetric]] ..
qed
lemma mor_cutΣΣ1Oc_flat1:
assumes f': "F_map f' o cutΣΣ1Oc s = dtor_J o f'"
shows "eval1 o ΣΣ1_map f' = f' o flat1"
proof(rule trans)
def h ≡ "dd1 o ΣΣ1_map <id,cutΣΣ1Oc s>"
have f'_id: "f' = f' o id" by simp
show "eval1 o ΣΣ1_map f' = dtor_unfold_J h"
apply(rule J.dtor_unfold_unique, rule sym)
unfolding o_assoc eval1[symmetric]
unfolding o_assoc[symmetric] ΣΣ1.map_comp0[symmetric]
unfolding convol_comp_id1[symmetric] unfolding f'[symmetric]
apply(subst f'_id)
unfolding o_assoc ΣΣ1.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 dd1_natural[symmetric] unfolding o_assoc[symmetric]
unfolding ΣΣ1.map_comp0[symmetric] map_prod_o_convol ..
have AA: "<id , cutΣΣ1Oc s> = (flat1 ** F_map flat1) o (id ** dd1) o <leaf1, ΣΣ1_map <leaf1 , s>>"
unfolding map_prod_o_convol o_assoc map_prod.comp cutΣΣ1Oc_def o_id flat1_leaf1 ..
have BB: "flat1 ** F_map flat1 o id ** dd1 o <leaf1 , ΣΣ1_map <leaf1 , s>> = flat1 ** F_map flat1 o id ** dd1 o <ΣΣ1_map leaf1 , ΣΣ1_map <leaf1 , s>>"
unfolding map_prod.comp unfolding map_prod_o_convol unfolding o_id unfolding flat1_leaf1 leaf1_flat1 ..
show "dtor_unfold_J h = f' o flat1"
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 ΣΣ1.map_comp0 apply(rule sym)
unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding dd1_natural
unfolding o_assoc F_map_comp[symmetric]
unfolding flat1_assoc unfolding F_map_comp
unfolding cutΣΣ1Oc_def o_assoc[symmetric] apply(rule arg_cong[of _ _ "op o (F_map flat1)"])
unfolding o_assoc
unfolding o_assoc[symmetric] unfolding ΣΣ1.map_comp0[symmetric] unfolding map_prod_o_convol id_o
unfolding flat1_natural[symmetric] unfolding o_assoc
unfolding dd1_flat1[symmetric] unfolding o_assoc[symmetric] unfolding ΣΣ1.map_comp0[symmetric]
unfolding convol_o unfolding ΣΣ1.map_comp0[symmetric] unfolding fst_convol ..
qed
lemma restr_mor:
assumes f': "F_map f' o cutΣΣ1Oc s = dtor_J o f'"
shows "F_map (extdd1 (restr f')) o s = dtor_J o restr f'"
unfolding extdd1_def restr_def ΣΣ1.map_comp0
unfolding o_assoc mor_cutΣΣ1Oc_flat1[OF f']
unfolding o_assoc[symmetric] leaf1_flat1 o_id
unfolding o_assoc f'[symmetric]
unfolding o_assoc[symmetric] cΣΣ1Ocut_cutΣΣ1Oc[unfolded cΣΣ1Ocut_def] ..
lemma extdd1_restr:
assumes f': "F_map f' o cutΣΣ1Oc s = dtor_J o f'"
shows "extdd1 (restr f') = f'"
proof-
have "f' = eval1 o ΣΣ1_map f' o leaf1"
unfolding o_assoc[symmetric] leaf1_natural
unfolding o_assoc eval1_leaf1 by simp
also have "... = eval1 o ΣΣ1_map (f' o leaf1)"
unfolding ΣΣ1.map_comp0 o_assoc
unfolding mor_cutΣΣ1Oc_flat1[OF f'] unfolding o_assoc[symmetric] flat1_leaf1 leaf1_flat1 ..
finally have A: "f' = eval1 o ΣΣ1_map (f' o leaf1)" .
show ?thesis unfolding extdd1_def restr_def A[symmetric] ..
qed
lemma restr_inj:
assumes f1': "F_map f1' o cutΣΣ1Oc s = dtor_J o f1'"
and f2': "F_map f2' o cutΣΣ1Oc s = dtor_J o f2'"
shows "restr f1' = restr f2' <-> f1' = f2'"
using extdd1_restr[OF f1'] extdd1_restr[OF f2'] by metis
lemma extdd1_surj:
assumes f': "F_map f' o cutΣΣ1Oc s = dtor_J o f'"
shows "∃ f. extdd1 f = f'"
using extdd1_restr[OF f'] by(rule exI[of _ "restr f'"])
lemma restr_extdd1:
assumes f: "F_map (extdd1 f) o s = dtor_J o f"
shows "restr (extdd1 f) = f"
proof-
have "dtor_J o f = F_map (extdd1 f) o s" using assms unfolding extdd1_def by (rule sym)
also have "... = dtor_J o restr (extdd1 f)"
unfolding restr_def unfolding o_assoc extdd1_mor[OF f, symmetric]
unfolding o_assoc[symmetric] cΣΣ1Ocut_cutΣΣ1Oc[unfolded cΣΣ1Ocut_def] ..
finally have "dtor_J o f = dtor_J o restr (extdd1 f)" .
thus ?thesis unfolding dtor_J_o_inj by (rule sym)
qed
lemma extdd1_inj:
assumes f1: "F_map (extdd1 f1) o s = dtor_J o f1"
and f2: "F_map (extdd1 f2) o s = dtor_J o f2"
shows "extdd1 f1 = extdd1 f2 <-> f1 = f2"
using restr_extdd1[OF f1] restr_extdd1[OF f2] by metis
lemma restr_surj:
assumes f: "F_map (extdd1 f) o s = dtor_J o f"
shows "∃ f'. restr f' = f"
using restr_extdd1[OF f] by(rule exI[of _ "extdd1 f"])
subsection{* Coiteration up-to *}
definition "unfoldU1 s ≡ restr (dtor_unfold_J (cutΣΣ1Oc s))"
theorem unfoldU1_pointfree:
"F_map (extdd1 (unfoldU1 s)) o s = dtor_J o unfoldU1 s"
unfolding unfoldU1_def apply(rule restr_mor)
unfolding dtor_unfold_J_pointfree ..
theorem unfoldU1: "F_map (extdd1 (unfoldU1 s)) (s a) = dtor_J (unfoldU1 s a)"
using unfoldU1_pointfree unfolding o_def fun_eq_iff by(rule allE)
theorem unfoldU1_ctor_J:
"ctor_J (F_map (extdd1 (unfoldU1 s)) (s a)) = unfoldU1 s a"
using unfoldU1 by (metis J.ctor_dtor)
theorem unfoldU1_unique:
assumes "F_map (extdd1 f) o s = dtor_J o f"
shows "f = unfoldU1 s"
proof-
note f = extdd1_mor[OF assms] note co = extdd1_mor[OF unfoldU1_pointfree]
have A: "extdd1 f = extdd1 (unfoldU1 s)"
proof(rule trans)
show "extdd1 f = dtor_unfold_J (cutΣΣ1Oc s)" apply(rule J.dtor_unfold_unique) using f .
show "dtor_unfold_J (cutΣΣ1Oc s) = extdd1 (unfoldU1 s)"
apply(rule J.dtor_unfold_unique[symmetric]) using co .
qed
show ?thesis using A unfolding extdd1_inj[OF assms unfoldU1_pointfree] .
qed
lemma unfoldU1_ctor_J_pointfree:
"ctor_J o F_map (extdd1 (unfoldU1 s)) o s = unfoldU1 s"
unfolding o_def fun_eq_iff by (subst unfoldU1_ctor_J[symmetric]) (rule allI, rule refl)
definition corecU1 :: "('a => (J + 'a) ΣΣ1 F) => 'a => J" where
"corecU1 s = unfoldU1 (case_sum (dd1 o leaf1 o <Inl, F_map Inl o dtor_J>) s) o Inr"
definition extddRec1 where
"extddRec1 f ≡ eval1 o ΣΣ1_map (case_sum id f)"
lemma unfoldU1_Inl:
"unfoldU1 (case_sum (dd1 o leaf1 o <Inl , F_map Inl o dtor_J>) s) o Inl = id"
(is "?L = ?R")
proof-
have "?L = unfoldU1 (dd1 o leaf1 o <id, dtor_J>)"
apply(rule unfoldU1_unique)
unfolding o_assoc unfoldU1_pointfree[symmetric]
unfolding o_assoc[symmetric] case_sum_o_inj extdd1_def F_map_comp ΣΣ1.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 dd1_natural[symmetric]
apply(subst o_assoc[symmetric]) unfolding leaf1_natural
unfolding o_assoc[symmetric] map_prod_o_convol o_id ..
also have "... = ?R"
apply(rule sym, rule unfoldU1_unique)
unfolding extdd1_def ΣΣ1.map_id0 o_id dd1_leaf1
unfolding o_assoc[symmetric] snd_convol
unfolding o_assoc F_map_comp[symmetric] eval1_leaf1 F_map_id id_o ..
finally show ?thesis .
qed
theorem corecU1_pointfree:
"F_map (extddRec1 (corecU1 s)) o s = dtor_J o corecU1 s" (is "?L = ?R")
unfolding corecU1_def
unfolding o_assoc unfoldU1_pointfree[symmetric] extddRec1_def
unfolding o_assoc[symmetric] case_sum_o_inj
apply(subst unfoldU1_Inl[symmetric, of s])
unfolding o_assoc case_sum_Inl_Inr_L extdd1_def ..
theorem corecU1:
"F_map (extddRec1 (corecU1 s)) (s a) = dtor_J (corecU1 s a)"
using corecU1_pointfree unfolding o_def fun_eq_iff by(rule allE)
subsection{* Coinduction up-to *}
definition "cptdd1 R ≡ (ΣΣ1_rel R ===> R) eval1 eval1"
definition "cngdd1 R ≡ equivp R ∧ cptdd1 R"
lemma cngdd1_Retr: "cngdd1 R ==> cngdd1 (R \<sqinter> Retr R)"
unfolding cngdd1_def cptdd1_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 ΣΣ1_rel_inf])
apply (erule inf2E)
apply (rule inf2I)
apply (erule rel_funE)
apply assumption
apply assumption
apply (subst Retr_def)
apply (subst eval1_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 eval1_def[symmetric]
apply (rule predicate2D[OF F_rel_mono])
apply (rule predicate2I)
apply (erule rel_funD)
apply assumption
apply (rule F_rel_ΣΣ1_rel)
unfolding ΣΣ1_rel_ΣΣ1_map_ΣΣ1_map vimage2p_rel_prod vimage2p_id
unfolding vimage2p_def Retr_def[symmetric]
apply assumption
done
definition "genCngdd1 R j1 j2 ≡ ∀ R'. R ≤ R' ∧ cngdd1 R' --> R' j1 j2"
lemma cngdd1_genCngdd1: "cngdd1 (genCngdd1 R)"
unfolding cngdd1_def proof safe
show "cptdd1 (genCngdd1 R)"
unfolding cptdd1_def rel_fun_def proof safe
fix x y assume A: "ΣΣ1_rel (genCngdd1 R) x y"
show "genCngdd1 R (eval1 x) (eval1 y)"
unfolding genCngdd1_def[abs_def] proof safe
fix R' assume "R ≤ R'" and 2: "cngdd1 R'"
hence "ΣΣ1_rel R' x y" by (metis A ΣΣ1.rel_mono_strong genCngdd1_def)
thus "R' (eval1 x) (eval1 y)" using 2 unfolding cngdd1_def cptdd1_def rel_fun_def by auto
qed
qed
qed(rule equivpI, unfold reflp_def symp_def transp_def genCngdd1_def cngdd1_def equivp_def, auto)
lemma
genCngdd1_refl[intro,simp]: "genCngdd1 R j j"
and genCngdd1_sym[intro]: "genCngdd1 R j1 j2 ==> genCngdd1 R j2 j1"
and genCngdd1_trans[intro]: "[|genCngdd1 R j1 j2; genCngdd1 R j2 j3|] ==> genCngdd1 R j1 j3"
using cngdd1_genCngdd1 unfolding cngdd1_def equivp_def by auto
lemma genCngdd1_eval1_rel_fun: "(ΣΣ1_rel (genCngdd1 R) ===> genCngdd1 R) eval1 eval1"
using cngdd1_genCngdd1 unfolding cngdd1_def cptdd1_def by auto
lemma genCngdd1_eval1: "ΣΣ1_rel (genCngdd1 R) x y ==> genCngdd1 R (eval1 x) (eval1 y)"
using genCngdd1_eval1_rel_fun unfolding rel_fun_def by auto
lemma leq_genCngdd1: "R ≤ genCngdd1 R"
and imp_genCngdd1[intro]: "R j1 j2 ==> genCngdd1 R j1 j2"
unfolding genCngdd1_def[abs_def] by auto
lemma genCngdd1_minimal: "[|R ≤ R'; cngdd1 R'|] ==> genCngdd1 R ≤ R'"
unfolding genCngdd1_def[abs_def] by (metis (lifting, no_types) predicate2I)
theorem coinductionU_genCngdd1:
assumes "∀ a b. R a b --> F_rel (genCngdd1 R) (dtor_J a) (dtor_J b)"
shows "R a b --> a = b"
proof-
let ?R' = "genCngdd1 R"
have "R ≤ Retr ?R'" using assms unfolding Retr_def[abs_def] by auto
hence "R ≤ ?R' \<sqinter> Retr ?R'" using leq_genCngdd1 by auto
moreover have "cngdd1 (?R' \<sqinter> Retr ?R')" using cngdd1_Retr[OF cngdd1_genCngdd1[of R]] .
ultimately have "?R' ≤ ?R' \<sqinter> Retr ?R'" using genCngdd1_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_genCngdd1 by auto
qed
subsection{* Flattening from term algebra to "one-step" algebra *}
definition algΛ1 :: "J Σ1 => J" where "algΛ1 = eval1 o \<oo>\<pp>1 o Σ1_map leaf1"
theorem eval1_comp_\<oo>\<pp>1: "eval1 o \<oo>\<pp>1 = algΛ1 o Σ1_map eval1"
unfolding algΛ1_def
unfolding o_assoc[symmetric] Σ1.map_comp0[symmetric]
unfolding leaf1_natural[symmetric] unfolding Σ1.map_comp0
apply(rule sym) unfolding o_assoc apply(subst o_assoc[symmetric])
unfolding \<oo>\<pp>1_natural
unfolding o_assoc eval1_flat1[symmetric]
apply(subst o_assoc[symmetric]) unfolding flat1_commute[symmetric]
unfolding o_assoc[symmetric] Σ1.map_comp0[symmetric] flat1_leaf1 Σ1.map_id0 o_id ..
theorem eval1_\<oo>\<pp>1: "eval1 (\<oo>\<pp>1 t) = algΛ1 (Σ1_map eval1 t)"
using eval1_comp_\<oo>\<pp>1 unfolding o_def fun_eq_iff by (rule allE)
theorem eval1_leaf1': "eval1 (leaf1 j) = j"
using eval1_leaf1 unfolding o_def fun_eq_iff id_def by (rule allE)
theorem dtor_J_algΛ1: "dtor_J o algΛ1 = F_map eval1 o Λ1 o Σ1_map <id, dtor_J>"
unfolding algΛ1_def eval1_def o_assoc dtor_unfold_J_pointfree Λ1_dd1
unfolding o_assoc[symmetric] \<oo>\<pp>1_natural[symmetric] Σ1.map_comp0[symmetric] leaf1_natural
..
theorem dtor_J_algΛ1': "dtor_J (algΛ1 t) = F_map eval1 (Λ1 (Σ1_map (<id, dtor_J>) t))"
by (rule trans[OF o_eq_dest[OF dtor_J_algΛ1] o_apply])
theorem ctor_J_algΛ1: "algΛ1 t = ctor_J (F_map eval1 (Λ1 (Σ1_map (<id, dtor_J>) t)))"
by (rule iffD1[OF J.dtor_inject trans[OF dtor_J_algΛ1' J.dtor_ctor[symmetric]]])
definition "cptΛ1 R ≡ (Σ1_rel R ===> R) algΛ1 algΛ1"
definition "cngΛ1 R ≡ equivp R ∧ cptΛ1 R"
lemma cptdd1_iff_cptΛ1: "cptdd1 R <-> cptΛ1 R"
apply (rule iffI)
apply (unfold cptΛ1_def cptdd1_def algΛ1_def o_assoc[symmetric]) [1]
apply (erule rel_funD[OF rel_funD[OF comp_transfer]])
apply transfer_prover
apply (unfold cptΛ1_def cptdd1_def) [1]
unfolding rel_fun_def
apply (rule allI)+
apply (rule ΣΣ1_rel_induct)
apply (simp only: eval1_leaf1')
unfolding eval1_\<oo>\<pp>1
apply (drule spec2)
apply (erule mp)
apply (rule iffD2[OF Σ1_rel_Σ1_map_Σ1_map])
apply (subst vimage2p_def)
apply assumption
done
theorem genCngdd1_def2: "genCngdd1 R j1 j2 <-> (∀ R'. R ≤ R' ∧ cngΛ1 R' --> R' j1 j2)"
unfolding genCngdd1_def cngdd1_def cngΛ1_def cptdd1_iff_cptΛ1 ..
subsection {* Incremental coinduction *}
interpretation I1: Incremental where Retr = Retr and Cl = genCngdd1
proof
show "mono Retr" by (rule mono_retr)
next
show "mono genCngdd1" unfolding mono_def
unfolding genCngdd1_def[abs_def] by (smt order.trans predicate2I)
next
fix r show "genCngdd1 (genCngdd1 r) = genCngdd1 r"
by (metis cngdd1_genCngdd1 genCngdd1_minimal leq_genCngdd1 order_class.order.antisym)
next
fix r show "r ≤ genCngdd1 r" by (metis leq_genCngdd1)
next
fix r assume "genCngdd1 r = r" thus "genCngdd1 (r \<sqinter> Retr r) = r \<sqinter> Retr r"
by (metis antisym cngdd1_Retr cngdd1_genCngdd1 genCngdd1_minimal
inf.orderI inf_idem leq_genCngdd1)
qed
definition ded1 where "ded1 r s ≡ I1.ded r s"
theorems Ax = I1.Ax'[unfolded ded1_def[symmetric]]
theorems Split = I1.Split[unfolded ded1_def[symmetric]]
theorems Coind = I1.Coind[unfolded ded1_def[symmetric]]
theorem soundness_ded:
assumes "ded1 (op =) s" shows "s ≤ (op =)"
unfolding gfp_Retr_eq[symmetric] apply(rule I1.soundness_ded[unfolded ded1_def[symmetric]] )
using assms unfolding gfp_Retr_eq[symmetric] ded1_def .
unused_thms
end