Theory BNF_Least_Fixpoint

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

Least fixpoint (datatype) operation on bounded natural functors.
*)

section Least Fixpoint (Datatype) Operation on Bounded Natural Functors

theory BNF_Least_Fixpoint
imports BNF_Fixpoint_Base
keywords
  "datatype" :: thy_defn and
  "datatype_compat" :: thy_defn
begin

lemma subset_emptyI: "(x. x  A  False)  A  {}"
  by blast

lemma image_Collect_subsetI: "(x. P x  f x  B)  f ` {x. P x}  B"
  by blast

lemma Collect_restrict: "{x. x  X  P x}  X"
  by auto

lemma prop_restrict: "x  Z; Z  {x. x  X  P x}  P x"
  by auto

lemma underS_I: "i  j; (i, j)  R  i  underS R j"
  unfolding underS_def by simp

lemma underS_E: "i  underS R j  i  j  (i, j)  R"
  unfolding underS_def by simp

lemma underS_Field: "i  underS R j  i  Field R"
  unfolding underS_def Field_def by auto

lemma ex_bij_betw: "|A| ≤o (r :: 'b rel)  f B :: 'b set. bij_betw f B A"
  by (subst (asm) internalize_card_of_ordLeq) (auto dest!: iffD2[OF card_of_ordIso ordIso_symmetric])

lemma bij_betwI':
  "x y. x  X; y  X  (f x = f y) = (x = y);
    x. x  X  f x  Y;
    y. y  Y  x  X. y = f x  bij_betw f X Y"
  unfolding bij_betw_def inj_on_def by blast

lemma surj_fun_eq:
  assumes surj_on: "f ` X = UNIV" and eq_on: "x  X. (g1  f) x = (g2  f) x"
  shows "g1 = g2"
proof (rule ext)
  fix y
  from surj_on obtain x where "x  X" and "y = f x" by blast
  thus "g1 y = g2 y" using eq_on by simp
qed

lemma Card_order_wo_rel: "Card_order r  wo_rel r"
  unfolding wo_rel_def card_order_on_def by blast

lemma Cinfinite_limit: "x  Field r; Cinfinite r  y  Field r. x  y  (x, y)  r"
  unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)

lemma Card_order_trans:
  "Card_order r; x  y; (x, y)  r; y  z; (y, z)  r  x  z  (x, z)  r"
  unfolding card_order_on_def well_order_on_def linear_order_on_def
    partial_order_on_def preorder_on_def trans_def antisym_def by blast

lemma Cinfinite_limit2:
  assumes x1: "x1  Field r" and x2: "x2  Field r" and r: "Cinfinite r"
  shows "y  Field r. (x1  y  (x1, y)  r)  (x2  y  (x2, y)  r)"
proof -
  from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
    unfolding card_order_on_def well_order_on_def linear_order_on_def
      partial_order_on_def preorder_on_def by auto
  obtain y1 where y1: "y1  Field r" "x1  y1" "(x1, y1)  r"
    using Cinfinite_limit[OF x1 r] by blast
  obtain y2 where y2: "y2  Field r" "x2  y2" "(x2, y2)  r"
    using Cinfinite_limit[OF x2 r] by blast
  show ?thesis
  proof (cases "y1 = y2")
    case True with y1 y2 show ?thesis by blast
  next
    case False
    with y1(1) y2(1) total have "(y1, y2)  r  (y2, y1)  r"
      unfolding total_on_def by auto
    thus ?thesis
    proof
      assume *: "(y1, y2)  r"
      with trans y1(3) have "(x1, y2)  r" unfolding trans_def by blast
      with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
    next
      assume *: "(y2, y1)  r"
      with trans y2(3) have "(x2, y1)  r" unfolding trans_def by blast
      with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
    qed
  qed
qed

lemma Cinfinite_limit_finite:
  "finite X; X  Field r; Cinfinite r  y  Field r. x  X. (x  y  (x, y)  r)"
proof (induct X rule: finite_induct)
  case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
next
  case (insert x X)
  then obtain y where y: "y  Field r" "x  X. (x  y  (x, y)  r)" by blast
  then obtain z where z: "z  Field r" "x  z  (x, z)  r" "y  z  (y, z)  r"
    using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
  show ?case
    apply (intro bexI ballI)
    apply (erule insertE)
    apply hypsubst
    apply (rule z(2))
    using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
    apply blast
    apply (rule z(1))
    done
qed

lemma insert_subsetI: "x  A; X  A  insert x X  A"
  by auto

lemmas well_order_induct_imp = wo_rel.well_order_induct[of r "λx. x  Field r  P x" for r P]

lemma meta_spec2:
  assumes "(x y. PROP P x y)"
  shows "PROP P x y"
  by (rule assms)

lemma nchotomy_relcomppE:
  assumes "y. x. y = f x" "(r OO s) a c" "b. r a (f b)  s (f b) c  P"
  shows P
proof (rule relcompp.cases[OF assms(2)], hypsubst)
  fix b assume "r a b" "s b c"
  moreover from assms(1) obtain b' where "b = f b'" by blast
  ultimately show P by (blast intro: assms(3))
qed

lemma predicate2D_vimage2p: "R  vimage2p f g S; R x y  S (f x) (g y)"
  unfolding vimage2p_def by auto

lemma ssubst_Pair_rhs: "(r, s)  R; s' = s  (r, s')  R"
  by (rule ssubst)

lemma all_mem_range1:
  "(y. y  range f  P y)  (x. P (f x)) "
  by (rule equal_intr_rule) fast+

lemma all_mem_range2:
  "(fa y. fa  range f  y  range fa  P y)  (x xa. P (f x xa))"
  by (rule equal_intr_rule) fast+

lemma all_mem_range3:
  "(fa fb y. fa  range f  fb  range fa  y  range fb  P y)  (x xa xb. P (f x xa xb))"
  by (rule equal_intr_rule) fast+

lemma all_mem_range4:
  "(fa fb fc y. fa  range f  fb  range fa  fc  range fb  y  range fc  P y) 
   (x xa xb xc. P (f x xa xb xc))"
  by (rule equal_intr_rule) fast+

lemma all_mem_range5:
  "(fa fb fc fd y. fa  range f  fb  range fa  fc  range fb  fd  range fc 
     y  range fd  P y) 
   (x xa xb xc xd. P (f x xa xb xc xd))"
  by (rule equal_intr_rule) fast+

lemma all_mem_range6:
  "(fa fb fc fd fe ff y. fa  range f  fb  range fa  fc  range fb  fd  range fc 
     fe  range fd  ff  range fe  y  range ff  P y) 
   (x xa xb xc xd xe xf. P (f x xa xb xc xd xe xf))"
  by (rule equal_intr_rule) (fastforce, fast)

lemma all_mem_range7:
  "(fa fb fc fd fe ff fg y. fa  range f  fb  range fa  fc  range fb  fd  range fc 
     fe  range fd  ff  range fe  fg  range ff  y  range fg  P y) 
   (x xa xb xc xd xe xf xg. P (f x xa xb xc xd xe xf xg))"
  by (rule equal_intr_rule) (fastforce, fast)

lemma all_mem_range8:
  "(fa fb fc fd fe ff fg fh y. fa  range f  fb  range fa  fc  range fb  fd  range fc 
     fe  range fd  ff  range fe  fg  range ff  fh  range fg  y  range fh  P y) 
   (x xa xb xc xd xe xf xg xh. P (f x xa xb xc xd xe xf xg xh))"
  by (rule equal_intr_rule) (fastforce, fast)

lemmas all_mem_range = all_mem_range1 all_mem_range2 all_mem_range3 all_mem_range4 all_mem_range5
  all_mem_range6 all_mem_range7 all_mem_range8

lemma pred_fun_True_id: "NO_MATCH id p  pred_fun (λx. True) p f = pred_fun (λx. True) id (p  f)"
  unfolding fun.pred_map unfolding comp_def id_def ..

ML_file Tools/BNF/bnf_lfp_util.ML
ML_file Tools/BNF/bnf_lfp_tactics.ML
ML_file Tools/BNF/bnf_lfp.ML
ML_file Tools/BNF/bnf_lfp_compat.ML
ML_file Tools/BNF/bnf_lfp_rec_sugar_more.ML
ML_file Tools/BNF/bnf_lfp_size.ML

ML_file Tools/datatype_simprocs.ML

simproc_setup datatype_no_proper_subterm
  ("(x :: 'a :: size) = y") = K Datatype_Simprocs.no_proper_subterm_simproc

end