Theory Basis

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theory Basis = Main: 
(*  Title:      isabelle/Bali/Basis.thy
    ID:         $Id: Basis.thy,v 1.30 2001/05/11 14:41:56 oheimb Exp $
    Author:     David von Oheimb
    Copyright   1997 Technische Universitaet Muenchen

Definitions extending HOL as logical basis of Bali
*)

theory Basis = Main:

ML_setup {*
Unify.search_bound := 40;
Unify.trace_bound  := 40;

quick_and_dirty:=true;

Pretty.setmargin 198;
goals_limit:=2;
*}
(*print_depth 100;*)
(*Syntax.ambiguity_level := 1;*)

section "misc"

declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)

(* ###TO HOL/???.ML?? *)
ML {*
fun make_simproc name pat pred thm = Simplifier.mk_simproc name
   [Thm.read_cterm (Thm.sign_of_thm thm) (pat, HOLogic.termT)] 
   (K (K (fn s => if pred s then None else Some (standard (mk_meta_eq thm)))))
*}

declare split_if_asm  [split] option.split [split] option.split_asm [split]
ML {*
simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)
*}
declare if_weak_cong [cong del] option.weak_case_cong [cong del]
declare length_Suc_conv [iff];

(* just an optimization *)
ML {*
Delsimprocs [record_simproc]; 
*}

(*###to be phased out *)
ML {*
bind_thm ("make_imp", rearrange_prems [1,0] mp)
*}

lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
apply auto
done

lemma subset_insertD: 
  "A <= insert x B ==> A <= B & x ~: A | (EX 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+
done

syntax
  "3" :: nat   ("3")
  "4" :: nat   ("4")
translations
 "3" == "Suc 2"
 "4" == "Suc 3"

(*unused*)
lemma range_bool_domain: "range f = {f True, f False}"
apply auto
apply (case_tac "xa")
apply auto
done

(* context (theory "Transitive_Closure") *)
lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
apply (rule allI)
apply (erule irrefl_tranclI)
done

(* context (theory "Finite") *)
lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>  
  finite {f y x |x y. P y}"
apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
prefer 2 apply  fast
apply (erule ssubst)
apply (erule finite_UN_I)
apply fast
done

(* ### TO theory "List" *)
lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
 \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> 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
done


section "pairs"

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

lemma fst_splitE [elim!]: 
"[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
apply (cut_tac p = "s'" in surjective_pairing)
apply auto
done

lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
apply (induct_tac "l")
apply  auto
done


section "quantifiers"

(*###to be phased out *)
ML {* 
fun noAll_simpset () = simpset() setmksimps 
        mksimps (filter (fn (x,_) => x<>"All") mksimps_pairs)
*}

lemma All_Ex_refl_eq2 [simp]: 
 "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
apply auto
done

lemma ex_ex_miniscope1 [simp]:
  "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
apply auto
done

lemma ex_miniscope2 [simp]:
  "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))" 
apply auto
done

lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
apply auto
done

lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
apply auto
done


section "sums"

hide const In0 In1

syntax
  fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
translations
 "fun_sum" == "sum_case"

consts    the_Inl  :: "'a + 'b \<Rightarrow> 'a"
          the_Inr  :: "'a + 'b \<Rightarrow> 'b"
primrec  "the_Inl (Inl a) = a"
primrec  "the_Inr (Inr b) = b"

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

consts    the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
          the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
          the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
primrec  "the_In1 (In1 a) = a"
primrec  "the_In2 (In2 b) = b"
primrec  "the_In3 (In3 c) = c"

syntax
         In1l   :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
         In1r   :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
translations
        "In1l e" == "In1 (Inl e)"
        "In1r c" == "In1 (Inr c)"

ML {*
fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[not_None_eq])
 (read_instantiate [("t","In"^s^" ?x")] thm)) ["1l","2","3","1r"]
*}

translations
  "option"<= (type) "Option.option"
  "list"  <= (type) "List.list"
  "sum3"  <= (type) "Basis.sum3"


section "quantifiers for option type"

syntax
  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\<forall>_\<in>_:/ _)"  [0,0,10] 10)
  Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)

translations
  "! x:A: P"    == "! x:o2s A. P"
  "? x:A: P"    == "? x:o2s A. P"


section "unique association lists"

constdefs
  unique   :: "('a × 'b) list \<Rightarrow> bool"
 "unique \<equiv> nodups \<circ> map fst"

lemma uniqueD [rule_format (no_asm)]: 
"unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
apply (unfold unique_def o_def)
apply (induct_tac "l")
apply  (auto dest: fst_in_set_lemma)
done

lemma unique_Nil [simp]: "unique []"
apply (unfold unique_def)
apply (simp (no_asm))
done

lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
apply (unfold unique_def)
apply  (auto dest: fst_in_set_lemma)
done

lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]

lemma unique_single [simp]: "!!p. unique [p]"
apply auto
done

lemma unique_ConsD: "unique (x#xs) ==> unique xs"
apply (simp add: unique_def)
done

lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->  
  (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
apply (induct_tac "l")
apply  (auto dest: fst_in_set_lemma)
done

lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
apply (induct_tac "l")
apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
done

lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
apply (induct_tac "l")
apply auto
done


section "list patterns"

consts
  lsplit         :: "[['a, 'a list] => 'b, 'a list] => 'b"
defs
  lsplit_def:    "lsplit == %f l. f (hd l) (tl l)"
(*  list patterns -- extends pre-defined type "pttrn" used in abstractions *)
syntax
  "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
translations
  "%y#x#xs. b"  == "lsplit (%y x#xs. b)"
  "%x#xs  . b"  == "lsplit (%x xs  . b)"

lemma lsplit [iff]: "lsplit c (x#xs) = c x xs"
apply (unfold lsplit_def)
apply (simp (no_asm))
done

lemma lsplit2: "lsplit P (x#xs) y z = P x xs y z"
apply (unfold lsplit_def)
apply simp
done
declare lsplit2 [iff]; 


section "dummy pattern for quantifiers, let, etc."

syntax
  "@dummy_pat"   :: pttrn    ("'_")

parse_translation {*
let fun dummy_pat_tr [] = Free ("_",dummyT)
  | dummy_pat_tr ts = raise TERM ("dummy_pat_tr", ts);
in [("@dummy_pat", dummy_pat_tr)] 
end
*}

end

misc

theorem make_imp: 

  [| P; P --> Q |] ==> Q

lemma Collect_split_eq: 

  {p. P (split f p)} = {(a, b). P (f a b)}

lemma subset_insertD: 

  A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)

lemma range_bool_domain: 

  range f = {f True, f False}

lemma irrefl_tranclI': 

  r^-1 Int r^+ = {} ==> ALL x. (x, x) ~: r^+

lemma finite_SetCompr2: 

  [| finite (Collect P); ALL y. P y --> finite (range (f y)) |]
  ==> finite {f y x |x y. P y}

lemma list_all2_trans: 

  ALL a b c. P1 a b --> P2 b c --> P3 a c
  ==> ALL xs2 xs3.
         list_all2 P1 xs1 xs2 --> list_all2 P2 xs2 xs3 --> list_all2 P3 xs1 xs3

pairs

lemma surjective_pairing5: 

  p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
       snd (snd (snd (snd p))))

lemma fst_splitE: 

  [| fst s' = x'; !!x s. [| s' = (x, s); x = x' |] ==> Q |] ==> Q

lemma fst_in_set_lemma: 

  (x, y) : set l ==> x : fst ` set l

quantifiers

lemma All_Ex_refl_eq2: 

  (ALL x. (EX b. x = f b & Q b) --> P x) = (ALL b. Q b --> P (f b))

lemma ex_ex_miniscope1: 

  (EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)

lemma ex_miniscope2: 

  (EX v. P v & Q & R v) = (Q & (EX v. P v & R v))

lemma ex_reorder31: 

  (EX z x y. P x y z) = (EX x y z. P x y z)

lemma All_Ex_refl_eq1: 

  (ALL x. (EX b. x = f b) --> P x) = (ALL b. P (f b))

sums

quantifiers for option type

unique association lists

lemma uniqueD: 

  [| unique l; (x, y) : set l; (x', y') : set l; x = x' |] ==> y = y'

lemma unique_Nil: 

  unique []

lemma unique_Cons: 

  unique ((x, y) # l) = (unique l & (ALL y. (x, y) ~: set l))

lemmas unique_ConsI: 

  [| unique l; ALL y. (x, y) ~: set l |] ==> unique ((x, y) # l)

lemma unique_single: 

  unique [p]

lemma unique_ConsD: 

  unique (x # xs) ==> unique xs

lemma unique_append: 

  [| unique l'; unique l; ALL (x, y):set l. ALL (x', y'):set l'. x' ~= x |]
  ==> unique (l @ l')

lemma unique_map_inj: 

  [| unique l; inj f |] ==> unique (map (%(k, x). (f k, g k x)) l)

lemma map_of_SomeI: 

  [| unique l; (k, x) : set l |] ==> map_of l k = Some x

list patterns

lemma lsplit: 

  lsplit c (x # xs) = c x xs

lemma lsplit2: 

  lsplit P (x # xs) y z = P x xs y z

dummy pattern for quantifiers, let, etc.