Theory HOL.BNF_Def

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

Definition of bounded natural functors.
*)

section ‹Definition of Bounded Natural Functors›

theory BNF_Def
imports BNF_Cardinal_Arithmetic Fun_Def_Base
keywords
  "print_bnfs" :: diag and
  "bnf" :: thy_goal_defn
begin

lemma Collect_case_prodD: "x  Collect (case_prod A)  A (fst x) (snd x)"
  by auto

inductive
   rel_sum :: "('a  'c  bool)  ('b  'd  bool)  'a + 'b  'c + 'd  bool" for R1 R2
where
  "R1 a c  rel_sum R1 R2 (Inl a) (Inl c)"
| "R2 b d  rel_sum R1 R2 (Inr b) (Inr d)"

definition
  rel_fun :: "('a  'c  bool)  ('b  'd  bool)  ('a  'b)  ('c  'd)  bool"
where
  "rel_fun A B = (λf g. x y. A x y  B (f x) (g y))"

lemma rel_funI [intro]:
  assumes "x y. A x y  B (f x) (g y)"
  shows "rel_fun A B f g"
  using assms by (simp add: rel_fun_def)

lemma rel_funD:
  assumes "rel_fun A B f g" and "A x y"
  shows "B (f x) (g y)"
  using assms by (simp add: rel_fun_def)

lemma rel_fun_mono:
  " rel_fun X A f g; x y. Y x y  X x y; x y. A x y  B x y   rel_fun Y B f g"
by(simp add: rel_fun_def)

lemma rel_fun_mono' [mono]:
  " x y. Y x y  X x y; x y. A x y  B x y   rel_fun X A f g  rel_fun Y B f g"
by(simp add: rel_fun_def)

definition rel_set :: "('a  'b  bool)  'a set  'b set  bool"
  where "rel_set R = (λA B. (xA. yB. R x y)  (yB. xA. R x y))"

lemma rel_setI:
  assumes "x. x  A  yB. R x y"
  assumes "y. y  B  xA. R x y"
  shows "rel_set R A B"
  using assms unfolding rel_set_def by simp

lemma predicate2_transferD:
   "rel_fun R1 (rel_fun R2 (=)) P Q; a  A; b  B; A  {(x, y). R1 x y}; B  {(x, y). R2 x y} 
   P (fst a) (fst b)  Q (snd a) (snd b)"
  unfolding rel_fun_def by (blast dest!: Collect_case_prodD)

definition collect where
  "collect F x = (f  F. f x)"

lemma fstI: "x = (y, z)  fst x = y"
  by simp

lemma sndI: "x = (y, z)  snd x = z"
  by simp

lemma bijI': "x y. (f x = f y) = (x = y); y. x. y = f x  bij f"
  unfolding bij_def inj_on_def by auto blast

(* Operator: *)
definition "Gr A f = {(a, f a) | a. a  A}"

definition "Grp A f = (λa b. b = f a  a  A)"

definition vimage2p where
  "vimage2p f g R = (λx y. R (f x) (g y))"

lemma collect_comp: "collect F  g = collect ((λf. f  g) ` F)"
  by (rule ext) (simp add: collect_def)

definition convol ("(_,/ _)") where
  "f, g  λa. (f a, g a)"

lemma fst_convol: "fst  f, g = f"
  apply(rule ext)
  unfolding convol_def by simp

lemma snd_convol: "snd  f, g = g"
  apply(rule ext)
  unfolding convol_def by simp

lemma convol_mem_GrpI:
  "x  A  id, g x  (Collect (case_prod (Grp A g)))"
  unfolding convol_def Grp_def by auto

definition csquare where
  "csquare A f1 f2 p1 p2  ( a  A. f1 (p1 a) = f2 (p2 a))"

lemma eq_alt: "(=) = Grp UNIV id"
  unfolding Grp_def by auto

lemma leq_conversepI: "R = (=)  R  R¯¯"
  by auto

lemma leq_OOI: "R = (=)  R  R OO R"
  by auto

lemma OO_Grp_alt: "(Grp A f)¯¯ OO Grp A g = (λx y. z. z  A  f z = x  g z = y)"
  unfolding Grp_def by auto

lemma Grp_UNIV_id: "f = id  (Grp UNIV f)¯¯ OO Grp UNIV f = Grp UNIV f"
  unfolding Grp_def by auto

lemma Grp_UNIV_idI: "x = y  Grp UNIV id x y"
  unfolding Grp_def by auto

lemma Grp_mono: "A  B  Grp A f  Grp B f"
  unfolding Grp_def by auto

lemma GrpI: "f x = y; x  A  Grp A f x y"
  unfolding Grp_def by auto

lemma GrpE: "Grp A f x y  (f x = y; x  A  R)  R"
  unfolding Grp_def by auto

lemma Collect_case_prod_Grp_eqD: "z  Collect (case_prod (Grp A f))  (f  fst) z = snd z"
  unfolding Grp_def comp_def by auto

lemma Collect_case_prod_Grp_in: "z  Collect (case_prod (Grp A f))  fst z  A"
  unfolding Grp_def comp_def by auto

definition "pick_middlep P Q a c = (SOME b. P a b  Q b c)"

lemma pick_middlep:
  "(P OO Q) a c  P a (pick_middlep P Q a c)  Q (pick_middlep P Q a c) c"
  unfolding pick_middlep_def by (rule someI_ex) auto

definition fstOp where
  "fstOp P Q ac = (fst ac, pick_middlep P Q (fst ac) (snd ac))"

definition sndOp where
  "sndOp P Q ac = (pick_middlep P Q (fst ac) (snd ac), (snd ac))"

lemma fstOp_in: "ac  Collect (case_prod (P OO Q))  fstOp P Q ac  Collect (case_prod P)"
  unfolding fstOp_def mem_Collect_eq
  by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct1])

lemma fst_fstOp: "fst bc = (fst  fstOp P Q) bc"
  unfolding comp_def fstOp_def by simp

lemma snd_sndOp: "snd bc = (snd  sndOp P Q) bc"
  unfolding comp_def sndOp_def by simp

lemma sndOp_in: "ac  Collect (case_prod (P OO Q))  sndOp P Q ac  Collect (case_prod Q)"
  unfolding sndOp_def mem_Collect_eq
  by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct2])

lemma csquare_fstOp_sndOp:
  "csquare (Collect (f (P OO Q))) snd fst (fstOp P Q) (sndOp P Q)"
  unfolding csquare_def fstOp_def sndOp_def using pick_middlep by simp

lemma snd_fst_flip: "snd xy = (fst  (%(x, y). (y, x))) xy"
  by (simp split: prod.split)

lemma fst_snd_flip: "fst xy = (snd  (%(x, y). (y, x))) xy"
  by (simp split: prod.split)

lemma flip_pred: "A  Collect (case_prod (R ¯¯))  (%(x, y). (y, x)) ` A  Collect (case_prod R)"
  by auto

lemma predicate2_eqD: "A = B  A a b  B a b"
  by simp

lemma case_sum_o_inj: "case_sum f g  Inl = f" "case_sum f g  Inr = g"
  by auto

lemma map_sum_o_inj: "map_sum f g  Inl = Inl  f" "map_sum f g  Inr = Inr  g"
  by auto

lemma card_order_csum_cone_cexp_def:
  "card_order r  ( |A1| +c cone) ^c r = |Func UNIV (Inl ` A1  {Inr ()})|"
  unfolding cexp_def cone_def Field_csum Field_card_of by (auto dest: Field_card_order)

lemma If_the_inv_into_in_Func:
  "inj_on g C; C  B  {x} 
   (λi. if i  g ` C then the_inv_into C g i else x)  Func UNIV (B  {x})"
  unfolding Func_def by (auto dest: the_inv_into_into)

lemma If_the_inv_into_f_f:
  "i  C; inj_on g C  ((λi. if i  g ` C then the_inv_into C g i else x)  g) i = id i"
  unfolding Func_def by (auto elim: the_inv_into_f_f)

lemma the_inv_f_o_f_id: "inj f  (the_inv f  f) z = id z"
  by (simp add: the_inv_f_f)

lemma vimage2pI: "R (f x) (g y)  vimage2p f g R x y"
  unfolding vimage2p_def .

lemma rel_fun_iff_leq_vimage2p: "(rel_fun R S) f g = (R  vimage2p f g S)"
  unfolding rel_fun_def vimage2p_def by auto

lemma convol_image_vimage2p: "f  fst, g  snd ` Collect (case_prod (vimage2p f g R))  Collect (case_prod R)"
  unfolding vimage2p_def convol_def by auto

lemma vimage2p_Grp: "vimage2p f g P = Grp UNIV f OO P OO (Grp UNIV g)¯¯"
  unfolding vimage2p_def Grp_def by auto

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

lemma comp_apply_eq: "f (g x) = h (k x)  (f  g) x = (h  k) x"
  unfolding comp_apply by assumption

lemma refl_ge_eq: "(x. R x x)  (=)  R"
  by auto

lemma ge_eq_refl: "(=)  R  R x x"
  by auto

lemma reflp_eq: "reflp R = ((=)  R)"
  by (auto simp: reflp_def fun_eq_iff)

lemma transp_relcompp: "transp r  r OO r  r"
  by (auto simp: transp_def)

lemma symp_conversep: "symp R = (R¯¯  R)"
  by (auto simp: symp_def fun_eq_iff)

lemma diag_imp_eq_le: "(x. x  A  R x x)  x y. x  A  y  A  x = y  R x y"
  by blast

definition eq_onp :: "('a  bool)  'a  'a  bool"
  where "eq_onp R = (λx y. R x  x = y)"

lemma eq_onp_Grp: "eq_onp P = BNF_Def.Grp (Collect P) id"
  unfolding eq_onp_def Grp_def by auto

lemma eq_onp_to_eq: "eq_onp P x y  x = y"
  by (simp add: eq_onp_def)

lemma eq_onp_top_eq_eq: "eq_onp top = (=)"
  by (simp add: eq_onp_def)

lemma eq_onp_same_args: "eq_onp P x x = P x"
  by (auto simp add: eq_onp_def)

lemma eq_onp_eqD: "eq_onp P = Q  P x = Q x x"
  unfolding eq_onp_def by blast

lemma Ball_Collect: "Ball A P = (A  (Collect P))"
  by auto

lemma eq_onp_mono0: "xA. P x  Q x  xA. yA. eq_onp P x y  eq_onp Q x y"
  unfolding eq_onp_def by auto

lemma eq_onp_True: "eq_onp (λ_. True) = (=)"
  unfolding eq_onp_def by simp

lemma Ball_image_comp: "Ball (f ` A) g = Ball A (g  f)"
  by auto

lemma rel_fun_Collect_case_prodD:
  "rel_fun A B f g  X  Collect (case_prod A)  x  X  B ((f  fst) x) ((g  snd) x)"
  unfolding rel_fun_def by auto

lemma eq_onp_mono_iff: "eq_onp P  eq_onp Q  P  Q"
  unfolding eq_onp_def by auto

ML_file ‹Tools/BNF/bnf_util.ML›
ML_file ‹Tools/BNF/bnf_tactics.ML›
ML_file ‹Tools/BNF/bnf_def_tactics.ML›
ML_file ‹Tools/BNF/bnf_def.ML›

end