Theory BNF_Greatest_Fixpoint

(*  Title:      HOL/BNF_Greatest_Fixpoint.thy
    Author:     Dmitriy Traytel, TU Muenchen
    Author:     Lorenz Panny, TU Muenchen
    Author:     Jasmin Blanchette, TU Muenchen
    Copyright   2012, 2013, 2014

Greatest fixpoint (codatatype) operation on bounded natural functors.

section ‹Greatest Fixpoint (Codatatype) Operation on Bounded Natural Functors›

theory BNF_Greatest_Fixpoint
imports BNF_Fixpoint_Base String
  "codatatype" :: thy_defn and
  "primcorecursive" :: thy_goal_defn and
  "primcorec" :: thy_defn

alias proj = Equiv_Relations.proj

lemma one_pointE: "x. s = x  P  P"
  by simp

lemma obj_sumE: "x. s = Inl x  P; x. s = Inr x  P  P"
  by (cases s) auto

lemma not_TrueE: "¬ True  P"
  by (erule notE, rule TrueI)

lemma neq_eq_eq_contradict: "t  u; s = t; s = u  P"
  by fast

lemma converse_Times: "(A × B)¯ = B × A"
  by fast

lemma equiv_proj:
  assumes e: "equiv A R" and m: "z  R"
  shows "(proj R  fst) z = (proj R  snd) z"
proof -
  from m have z: "(fst z, snd z)  R" by auto
  with e have "x. (fst z, x)  R  (snd z, x)  R" "x. (snd z, x)  R  (fst z, x)  R"
    unfolding equiv_def sym_def trans_def by blast+
  then show ?thesis unfolding proj_def[abs_def] by auto

(* Operators: *)
definition image2 where "image2 A f g = {(f a, g a) | a. a  A}"

lemma Id_on_Gr: "Id_on A = Gr A id"
  unfolding Id_on_def Gr_def by auto

lemma image2_eqI: "b = f x; c = g x; x  A  (b, c)  image2 A f g"
  unfolding image2_def by auto

lemma IdD: "(a, b)  Id  a = b"
  by auto

lemma image2_Gr: "image2 A f g = (Gr A f)¯ O (Gr A g)"
  unfolding image2_def Gr_def by auto

lemma GrD1: "(x, fx)  Gr A f  x  A"
  unfolding Gr_def by simp

lemma GrD2: "(x, fx)  Gr A f  f x = fx"
  unfolding Gr_def by simp

lemma Gr_incl: "Gr A f  A × B  f ` A  B"
  unfolding Gr_def by auto

lemma subset_Collect_iff: "B  A  (B  {x  A. P x}) = (x  B. P x)"
  by blast

lemma subset_CollectI: "B  A  (x. x  B  Q x  P x)  ({x  B. Q x}  {x  A. P x})"
  by blast

lemma in_rel_Collect_case_prod_eq: "in_rel (Collect (case_prod X)) = X"
  unfolding fun_eq_iff by auto

lemma Collect_case_prod_in_rel_leI: "X  Y  X  Collect (case_prod (in_rel Y))"
  by auto

lemma Collect_case_prod_in_rel_leE: "X  Collect (case_prod (in_rel Y))  (X  Y  R)  R"
  by force

lemma conversep_in_rel: "(in_rel R)¯¯ = in_rel (R¯)"
  unfolding fun_eq_iff by auto

lemma relcompp_in_rel: "in_rel R OO in_rel S = in_rel (R O S)"
  unfolding fun_eq_iff by auto

lemma in_rel_Gr: "in_rel (Gr A f) = Grp A f"
  unfolding Gr_def Grp_def fun_eq_iff by auto

definition relImage where
  "relImage R f  {(f a1, f a2) | a1 a2. (a1,a2)  R}"

definition relInvImage where
  "relInvImage A R f  {(a1, a2) | a1 a2. a1  A  a2  A  (f a1, f a2)  R}"

lemma relImage_Gr:
  "R  A × A  relImage R f = (Gr A f)¯ O R O Gr A f"
  unfolding relImage_def Gr_def relcomp_def by auto

lemma relInvImage_Gr: "R  B × B  relInvImage A R f = Gr A f O R O (Gr A f)¯"
  unfolding Gr_def relcomp_def image_def relInvImage_def by auto

lemma relImage_mono:
  "R1  R2  relImage R1 f  relImage R2 f"
  unfolding relImage_def by auto

lemma relInvImage_mono:
  "R1  R2  relInvImage A R1 f  relInvImage A R2 f"
  unfolding relInvImage_def by auto

lemma relInvImage_Id_on:
  "(a1 a2. f a1 = f a2  a1 = a2)  relInvImage A (Id_on B) f  Id"
  unfolding relInvImage_def Id_on_def by auto

lemma relInvImage_UNIV_relImage:
  "R  relInvImage UNIV (relImage R f) f"
  unfolding relInvImage_def relImage_def by auto

lemma relImage_proj:
  assumes "equiv A R"
  shows "relImage R (proj R)  Id_on (A//R)"
  unfolding relImage_def Id_on_def
  using proj_iff[OF assms] equiv_class_eq_iff[OF assms]
  by (auto simp: proj_preserves)

lemma relImage_relInvImage:
  assumes "R  f ` A × f ` A"
  shows "relImage (relInvImage A R f) f = R"
  using assms unfolding relImage_def relInvImage_def by fast

lemma subst_Pair: "P x y  a = (x, y)  P (fst a) (snd a)"
  by simp

lemma fst_diag_id: "(fst  (λx. (x, x))) z = id z" by simp
lemma snd_diag_id: "(snd  (λx. (x, x))) z = id z" by simp

lemma fst_diag_fst: "fst  ((λx. (x, x))  fst) = fst" by auto
lemma snd_diag_fst: "snd  ((λx. (x, x))  fst) = fst" by auto
lemma fst_diag_snd: "fst  ((λx. (x, x))  snd) = snd" by auto
lemma snd_diag_snd: "snd  ((λx. (x, x))  snd) = snd" by auto

definition Succ where "Succ Kl kl = {k . kl @ [k]  Kl}"
definition Shift where "Shift Kl k = {kl. k # kl  Kl}"
definition shift where "shift lab k = (λkl. lab (k # kl))"

lemma empty_Shift: "[]  Kl; k  Succ Kl []  []  Shift Kl k"
  unfolding Shift_def Succ_def by simp

lemma SuccD: "k  Succ Kl kl  kl @ [k]  Kl"
  unfolding Succ_def by simp

lemmas SuccE = SuccD[elim_format]

lemma SuccI: "kl @ [k]  Kl  k  Succ Kl kl"
  unfolding Succ_def by simp

lemma ShiftD: "kl  Shift Kl k  k # kl  Kl"
  unfolding Shift_def by simp

lemma Succ_Shift: "Succ (Shift Kl k) kl = Succ Kl (k # kl)"
  unfolding Succ_def Shift_def by auto

lemma length_Cons: "length (x # xs) = Suc (length xs)"
  by simp

lemma length_append_singleton: "length (xs @ [x]) = Suc (length xs)"
  by simp

(*injection into the field of a cardinal*)
definition "toCard_pred A r f  inj_on f A  f ` A  Field r  Card_order r"
definition "toCard A r  SOME f. toCard_pred A r f"

lemma ex_toCard_pred:
  "|A| ≤o r; Card_order r   f. toCard_pred A r f"
  unfolding toCard_pred_def
  using card_of_ordLeq[of A "Field r"]
    ordLeq_ordIso_trans[OF _ card_of_unique[of "Field r" r], of "|A|"]
  by blast

lemma toCard_pred_toCard:
  "|A| ≤o r; Card_order r  toCard_pred A r (toCard A r)"
  unfolding toCard_def using someI_ex[OF ex_toCard_pred] .

lemma toCard_inj: "|A| ≤o r; Card_order r; x  A; y  A  toCard A r x = toCard A r y  x = y"
  using toCard_pred_toCard unfolding inj_on_def toCard_pred_def by blast

definition "fromCard A r k  SOME b. b  A  toCard A r b = k"

lemma fromCard_toCard:
  "|A| ≤o r; Card_order r; b  A  fromCard A r (toCard A r b) = b"
  unfolding fromCard_def by (rule some_equality) (auto simp add: toCard_inj)

lemma Inl_Field_csum: "a  Field r  Inl a  Field (r +c s)"
  unfolding Field_card_of csum_def by auto

lemma Inr_Field_csum: "a  Field s  Inr a  Field (r +c s)"
  unfolding Field_card_of csum_def by auto

lemma rec_nat_0_imp: "f = rec_nat f1 (λn rec. f2 n rec)  f 0 = f1"
  by auto

lemma rec_nat_Suc_imp: "f = rec_nat f1 (λn rec. f2 n rec)  f (Suc n) = f2 n (f n)"
  by auto

lemma rec_list_Nil_imp: "f = rec_list f1 (λx xs rec. f2 x xs rec)  f [] = f1"
  by auto

lemma rec_list_Cons_imp: "f = rec_list f1 (λx xs rec. f2 x xs rec)  f (x # xs) = f2 x xs (f xs)"
  by auto

lemma not_arg_cong_Inr: "x  y  Inr x  Inr y"
  by simp

definition image2p where
  "image2p f g R = (λx y. x' y'. R x' y'  f x' = x  g y' = y)"

lemma image2pI: "R x y  image2p f g R (f x) (g y)"
  unfolding image2p_def by blast

lemma image2pE: "image2p f g R fx gy; (x y. fx = f x  gy = g y  R x y  P)  P"
  unfolding image2p_def by blast

lemma rel_fun_iff_geq_image2p: "rel_fun R S f g = (image2p f g R  S)"
  unfolding rel_fun_def image2p_def by auto

lemma rel_fun_image2p: "rel_fun R (image2p f g R) f g"
  unfolding rel_fun_def image2p_def by auto

subsection ‹Equivalence relations, quotients, and Hilbert's choice›

lemma equiv_Eps_in:
"equiv A r; X  A//r  Eps (λx. x  X)  X"
  apply (rule someI2_ex)
  using in_quotient_imp_non_empty by blast

lemma equiv_Eps_preserves:
  assumes ECH: "equiv A r" and X: "X  A//r"
  shows "Eps (λx. x  X)  A"
  apply (rule in_mono[rule_format])
   using assms apply (rule in_quotient_imp_subset)
  by (rule equiv_Eps_in) (rule assms)+

lemma proj_Eps:
  assumes "equiv A r" and "X  A//r"
  shows "proj r (Eps (λx. x  X)) = X"
unfolding proj_def
proof auto
  fix x assume x: "x  X"
  thus "(Eps (λx. x  X), x)  r" using assms equiv_Eps_in in_quotient_imp_in_rel by fast
  fix x assume "(Eps (λx. x  X),x)  r"
  thus "x  X" using in_quotient_imp_closed[OF assms equiv_Eps_in[OF assms]] by fast

definition univ where "univ f X == f (Eps (λx. x  X))"

lemma univ_commute:
assumes ECH: "equiv A r" and RES: "f respects r" and x: "x  A"
shows "(univ f) (proj r x) = f x"
proof (unfold univ_def)
  have prj: "proj r x  A//r" using x proj_preserves by fast
  hence "Eps (λy. y  proj r x)  A" using ECH equiv_Eps_preserves by fast
  moreover have "proj r (Eps (λy. y  proj r x)) = proj r x" using ECH prj proj_Eps by fast
  ultimately have "(x, Eps (λy. y  proj r x))  r" using x ECH proj_iff by fast
  thus "f (Eps (λy. y  proj r x)) = f x" using RES unfolding congruent_def by fastforce

lemma univ_preserves:
  assumes ECH: "equiv A r" and RES: "f respects r" and PRES: "x  A. f x  B"
  shows "X  A//r. univ f X  B"
  fix X assume "X  A//r"
  then obtain x where x: "x  A" and X: "X = proj r x" using ECH proj_image[of r A] by blast
  hence "univ f X = f x" using ECH RES univ_commute by fastforce
  thus "univ f X  B" using x PRES by simp

lemma card_suc_ordLess_imp_ordLeq:
  assumes ORD: "Card_order r" "Card_order r'" "card_order r'"
  and LESS: "r <o card_suc r'"
shows "r ≤o r'"
proof -
  have "Card_order (card_suc r')" by (rule Card_order_card_suc[OF ORD(3)])
  then have "cardSuc r ≤o card_suc r'" using cardSuc_least ORD LESS by blast
  then have "cardSuc r ≤o cardSuc r'" using cardSuc_ordIso_card_suc ordIso_symmetric
      ordLeq_ordIso_trans ORD(3) by blast
  then show ?thesis using cardSuc_mono_ordLeq ORD by blast

lemma natLeq_ordLess_cinfinite: "Cinfinite r; card_order r  natLeq <o card_suc r"
  using natLeq_ordLeq_cinfinite card_suc_greater ordLeq_ordLess_trans by blast

corollary natLeq_ordLess_cinfinite': "Cinfinite r'; card_order r'; r  card_suc r'  natLeq <o r"
  using natLeq_ordLess_cinfinite by blast

ML_file ‹Tools/BNF/bnf_gfp_util.ML›
ML_file ‹Tools/BNF/bnf_gfp_tactics.ML›
ML_file ‹Tools/BNF/bnf_gfp.ML›
ML_file ‹Tools/BNF/bnf_gfp_rec_sugar_tactics.ML›
ML_file ‹Tools/BNF/bnf_gfp_rec_sugar.ML›