Theory Basis

(*  Title:      HOL/Bali/Basis.thy
    Author:     David von Oheimb
subsection ‹Definitions extending HOL as logical basis of Bali›

theory Basis
imports Main

subsubsection "misc"

ML fun strip_tac ctxt i = REPEAT (resolve_tac ctxt [impI, allI] i)

declare if_split_asm  [split] option.split [split] option.split_asm [split]
setup map_theory_simpset (fn ctxt => ctxt addloop ("split_all_tac", split_all_tac))
declare if_weak_cong [cong del] option.case_cong_weak [cong del]
declare length_Suc_conv [iff]

lemma Collect_split_eq: "{p. P (case_prod f p)} = {(a,b). P (f a b)}"
  by auto

lemma subset_insertD: "A  insert x B  A  B  x  A  (B'. A = insert x B'  B'  B)"
  apply (case_tac "x  A")
   apply (rule disjI2)
   apply (rule_tac x = "A - {x}" in exI)
   apply fast+

abbreviation nat3 :: nat  ("3") where "3  Suc 2"
abbreviation nat4 :: nat  ("4") where "4  Suc 3"

(* irrefl_tranclI in Transitive_Closure.thy is more general *)
lemma irrefl_tranclI': "r¯  r+ = {}  x. (x, x)  r+"
  by (blast elim: tranclE dest: trancl_into_rtrancl)

lemma trancl_rtrancl_trancl: "(x, y)  r+; (y, z)  r*  (x, z)  r+"
  by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)

lemma rtrancl_into_trancl3: "(a, b)  r*; a  b  (a, b)  r+"
  apply (drule rtranclD)
  apply auto

lemma rtrancl_into_rtrancl2: "(a, b)   r; (b, c)  r*  (a, c)  r*"
  by (auto intro: rtrancl_trans)

lemma triangle_lemma:
  assumes unique: "a b c. (a,b)r; (a,c)r  b = c"
    and ax: "(a,x)r*" and ay: "(a,y)r*"
  shows "(x,y)r*  (y,x)r*"
  using ax ay
proof (induct rule: converse_rtrancl_induct)
  assume "(x,y)r*"
  then show ?thesis by blast
  fix a v
  assume a_v_r: "(a, v)  r"
    and v_x_rt: "(v, x)  r*"
    and a_y_rt: "(a, y)  r*"
    and hyp: "(v, y)  r*  (x, y)  r*  (y, x)  r*"
  from a_y_rt show "(x, y)  r*  (y, x)  r*"
  proof (cases rule: converse_rtranclE)
    assume "a = y"
    with a_v_r v_x_rt have "(y,x)  r*"
      by (auto intro: rtrancl_trans)
    then show ?thesis by blast
    fix w
    assume a_w_r: "(a, w)  r"
      and w_y_rt: "(w, y)  r*"
    from a_v_r a_w_r unique have "v=w" by auto
    with w_y_rt hyp show ?thesis by blast

lemma rtrancl_cases:
  assumes "(a,b)r*"
  obtains (Refl) "a = b"
    | (Trancl) "(a,b)r+"
  apply (rule rtranclE [OF assms])
   apply (auto dest: rtrancl_into_trancl1)

lemma Ball_weaken: "Ball s P;  x. P xQ xBall s Q"
  by auto

lemma finite_SetCompr2:
  "finite {f y x |x y. P y}" if "finite (Collect P)"
    "y. P y  finite (range (f y))"
proof -
  have "{f y x |x y. P y} = (yCollect P. range (f y))"
    by auto
  with that show ?thesis by simp

lemma list_all2_trans: "a b c. P1 a b  P2 b c  P3 a c 
    xs2 xs3. list_all2 P1 xs1 xs2  list_all2 P2 xs2 xs3  list_all2 P3 xs1 xs3"
  apply (induct_tac xs1)
   apply simp
  apply (rule allI)
  apply (induct_tac xs2)
   apply simp
  apply (rule allI)
  apply (induct_tac xs3)
   apply auto

subsubsection "pairs"

lemma surjective_pairing5:
  "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
    snd (snd (snd (snd p))))"
  by auto

lemma fst_splitE [elim!]:
  assumes "fst s' = x'"
  obtains x s where "s' = (x,s)" and "x = x'"
  using assms by (cases s') auto

lemma fst_in_set_lemma: "(x, y)  set l  x  fst ` set l"
  by (induct l) auto

subsubsection "quantifiers"

lemma All_Ex_refl_eq2 [simp]: "(x. (b. x = f b  Q b)  P x) = (b. Q b  P (f b))"
  by auto

lemma ex_ex_miniscope1 [simp]: "(w v. P w v  Q v) = (v. (w. P w v)  Q v)"
  by auto

lemma ex_miniscope2 [simp]: "(v. P v  Q  R v) = (Q  (v. P v  R v))"
  by auto

lemma ex_reorder31: "(z x y. P x y z) = (x y z. P x y z)"
  by auto

lemma All_Ex_refl_eq1 [simp]: "(x. (b. x = f b)  P x) = (b. P (f b))"
  by auto

subsubsection "sums"

notation case_sum  (infixr "'(+')" 80)

primrec the_Inl :: "'a + 'b  'a"
  where "the_Inl (Inl a) = a"

primrec the_Inr :: "'a + 'b  'b"
  where "the_Inr (Inr b) = b"

datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c

primrec the_In1 :: "('a, 'b, 'c) sum3  'a"
  where "the_In1 (In1 a) = a"

primrec the_In2 :: "('a, 'b, 'c) sum3  'b"
  where "the_In2 (In2 b) = b"

primrec the_In3 :: "('a, 'b, 'c) sum3  'c"
  where "the_In3 (In3 c) = c"

abbreviation In1l :: "'al  ('al + 'ar, 'b, 'c) sum3"
  where "In1l e  In1 (Inl e)"

abbreviation In1r :: "'ar  ('al + 'ar, 'b, 'c) sum3"
  where "In1r c  In1 (Inr c)"

abbreviation the_In1l :: "('al + 'ar, 'b, 'c) sum3  'al"
  where "the_In1l  the_Inl  the_In1"

abbreviation the_In1r :: "('al + 'ar, 'b, 'c) sum3  'ar"
  where "the_In1r  the_Inr  the_In1"

ML fun sum3_instantiate ctxt thm =
  map (fn s =>
    simplify (ctxt delsimps @{thms not_None_eq})
      (Rule_Insts.read_instantiate ctxt [((("t", 0), Position.none), "In" ^ s ^ " x")] ["x"] thm))
(* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)

subsubsection "quantifiers for option type"

  "_Oall" :: "[pttrn, 'a option, bool]  bool"   ("(3! _:_:/ _)" [0,0,10] 10)
  "_Oex"  :: "[pttrn, 'a option, bool]  bool"   ("(3? _:_:/ _)" [0,0,10] 10)

syntax (symbols)
  "_Oall" :: "[pttrn, 'a option, bool]  bool"   ("(3__:/ _)"  [0,0,10] 10)
  "_Oex"  :: "[pttrn, 'a option, bool]  bool"   ("(3__:/ _)"  [0,0,10] 10)

  "xA: P"  "xCONST set_option A. P"
  "xA: P"  "xCONST set_option A. P"

subsubsection "Special map update"

text‹Deemed too special for theory Map.›

definition chg_map :: "('b  'b)  'a  ('a  'b)  ('a  'b)"
  where "chg_map f a m = (case m a of None  m | Some b  m(af b))"

lemma chg_map_new[simp]: "m a = None  chg_map f a m = m"
  unfolding chg_map_def by auto

lemma chg_map_upd[simp]: "m a = Some b  chg_map f a m = m(af b)"
  unfolding chg_map_def by auto

lemma chg_map_other [simp]: "a  b  chg_map f a m b = m b"
  by (auto simp: chg_map_def)

subsubsection "unique association lists"

definition unique :: "('a × 'b) list  bool"
  where "unique = distinct  map fst"

lemma uniqueD: "unique l  (x, y)  set l  (x', y')  set l  x = x'  y = y'"
  unfolding unique_def o_def
  by (induct l) (auto dest: fst_in_set_lemma)

lemma unique_Nil [simp]: "unique []"
  by (simp add: unique_def)

lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l  (y. (x,y)  set l))"
  by (auto simp: unique_def dest: fst_in_set_lemma)

lemma unique_ConsD: "unique (x#xs)  unique xs"
  by (simp add: unique_def)

lemma unique_single [simp]: "p. unique [p]"
  by simp

lemma unique_append [rule_format (no_asm)]: "unique l'  unique l 
    ((x,y)set l. (x',y')set l'. x'  x)  unique (l @ l')"
  by (induct l) (auto dest: fst_in_set_lemma)

lemma unique_map_inj: "unique l  inj f  unique (map (λ(k,x). (f k, g k x)) l)"
  by (induct l) (auto dest: fst_in_set_lemma simp add: inj_eq)

lemma map_of_SomeI: "unique l  (k, x)  set l  map_of l k = Some x"
  by (induct l) auto

subsubsection "list patterns"

definition lsplit :: "[['a, 'a list]  'b, 'a list]  'b"
  where "lsplit = (λf l. f (hd l) (tl l))"

text ‹list patterns -- extends pre-defined type "pttrn" used in abstractions›
  "_lpttrn" :: "[pttrn, pttrn]  pttrn"    ("_#/_" [901,900] 900)
  "λy # x # xs. b"  "CONST lsplit (λy x # xs. b)"
  "λx # xs. b"  "CONST lsplit (λx xs. b)"

lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
  by (simp add: lsplit_def)

lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
  by (simp add: lsplit_def)