src/HOL/HOL.thy

added HOL-Base image

(*  Title:      HOL/HOL.thy
    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)

header {* The basis of Higher-Order Logic *}

theory HOL
imports Pure
uses
  ("Tools/hologic.ML")
  "~~/src/Tools/IsaPlanner/zipper.ML"
  "~~/src/Tools/IsaPlanner/isand.ML"
  "~~/src/Tools/IsaPlanner/rw_tools.ML"
  "~~/src/Tools/IsaPlanner/rw_inst.ML"
  "~~/src/Provers/project_rule.ML"
  "~~/src/Provers/hypsubst.ML"
  "~~/src/Provers/splitter.ML"
  "~~/src/Provers/classical.ML"
  "~~/src/Provers/blast.ML"
  "~~/src/Provers/clasimp.ML"
  "~~/src/Provers/coherent.ML"
  "~~/src/Provers/eqsubst.ML"
  "~~/src/Provers/quantifier1.ML"
  ("Tools/simpdata.ML")
  "~~/src/Tools/random_word.ML"
  "~~/src/Tools/atomize_elim.ML"
  "~~/src/Tools/induct.ML"
  ("~~/src/Tools/induct_tacs.ML")
  "~~/src/Tools/value.ML"
  "~~/src/Tools/code/code_name.ML"
  "~~/src/Tools/code/code_funcgr.ML"
  "~~/src/Tools/code/code_thingol.ML"
  "~~/src/Tools/code/code_printer.ML"
  "~~/src/Tools/code/code_target.ML"
  "~~/src/Tools/code/code_ml.ML"
  "~~/src/Tools/code/code_haskell.ML"
  "~~/src/Tools/nbe.ML"
  ("Tools/recfun_codegen.ML")
begin

subsection {* Primitive logic *}

subsubsection {* Core syntax *}

classes type
defaultsort type
setup {* ObjectLogic.add_base_sort @{sort type} *}

arities
  "fun" :: (type, type) type
  itself :: (type) type

global

typedecl bool

judgment
  Trueprop      :: "bool => prop"                   ("(_)" 5)

consts
  Not           :: "bool => bool"                   ("~ _" [40] 40)
  True          :: bool
  False         :: bool

  The           :: "('a => bool) => 'a"
  All           :: "('a => bool) => bool"           (binder "ALL " 10)
  Ex            :: "('a => bool) => bool"           (binder "EX " 10)
  Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
  Let           :: "['a, 'a => 'b] => 'b"

  "op ="        :: "['a, 'a] => bool"               (infixl "=" 50)
  "op &"        :: "[bool, bool] => bool"           (infixr "&" 35)
  "op |"        :: "[bool, bool] => bool"           (infixr "|" 30)
  "op -->"      :: "[bool, bool] => bool"           (infixr "-->" 25)

local

consts
  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)


subsubsection {* Additional concrete syntax *}

notation (output)
  "op ="  (infix "=" 50)

abbreviation
  not_equal :: "['a, 'a] => bool"  (infixl "~=" 50) where
  "x ~= y == ~ (x = y)"

notation (output)
  not_equal  (infix "~=" 50)

notation (xsymbols)
  Not  ("\<not> _" [40] 40) and
  "op &"  (infixr "\<and>" 35) and
  "op |"  (infixr "\<or>" 30) and
  "op -->"  (infixr "\<longrightarrow>" 25) and
  not_equal  (infix "\<noteq>" 50)

notation (HTML output)
  Not  ("\<not> _" [40] 40) and
  "op &"  (infixr "\<and>" 35) and
  "op |"  (infixr "\<or>" 30) and
  not_equal  (infix "\<noteq>" 50)

abbreviation (iff)
  iff :: "[bool, bool] => bool"  (infixr "<->" 25) where
  "A <-> B == A = B"

notation (xsymbols)
  iff  (infixr "\<longleftrightarrow>" 25)


nonterminals
  letbinds  letbind
  case_syn  cases_syn

syntax
  "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)

  "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
  ""            :: "letbind => letbinds"                 ("_")
  "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)

  "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
  ""            :: "case_syn => cases_syn"               ("_")
  "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")

translations
  "THE x. P"              == "The (%x. P)"
  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
  "let x = a in e"        == "Let a (%x. e)"

print_translation {*
(* To avoid eta-contraction of body: *)
[("The", fn [Abs abs] =>
     let val (x,t) = atomic_abs_tr' abs
     in Syntax.const "_The" $ x $ t end)]
*}

syntax (xsymbols)
  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)

notation (xsymbols)
  All  (binder "\<forall>" 10) and
  Ex  (binder "\<exists>" 10) and
  Ex1  (binder "\<exists>!" 10)

notation (HTML output)
  All  (binder "\<forall>" 10) and
  Ex  (binder "\<exists>" 10) and
  Ex1  (binder "\<exists>!" 10)

notation (HOL)
  All  (binder "! " 10) and
  Ex  (binder "? " 10) and
  Ex1  (binder "?! " 10)


subsubsection {* Axioms and basic definitions *}

axioms
  refl:           "t = (t::'a)"
  subst:          "s = t \<Longrightarrow> P s \<Longrightarrow> P t"
  ext:            "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
    -- {*Extensionality is built into the meta-logic, and this rule expresses
         a related property.  It is an eta-expanded version of the traditional
         rule, and similar to the ABS rule of HOL*}

  the_eq_trivial: "(THE x. x = a) = (a::'a)"

  impI:           "(P ==> Q) ==> P-->Q"
  mp:             "[| P-->Q;  P |] ==> Q"


defs
  True_def:     "True      == ((%x::bool. x) = (%x. x))"
  All_def:      "All(P)    == (P = (%x. True))"
  Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
  False_def:    "False     == (!P. P)"
  not_def:      "~ P       == P-->False"
  and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
  or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
  Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"

axioms
  iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
  True_or_False:  "(P=True) | (P=False)"

defs
  Let_def:      "Let s f == f(s)"
  if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"

finalconsts
  "op ="
  "op -->"
  The

axiomatization
  undefined :: 'a

abbreviation (input)
  "arbitrary \<equiv> undefined"


subsubsection {* Generic classes and algebraic operations *}

class default = type +
  fixes default :: 'a

class zero = type + 
  fixes zero :: 'a  ("0")

class one = type +
  fixes one  :: 'a  ("1")

hide (open) const zero one

class plus = type +
  fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "+" 65)

class minus = type +
  fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "-" 65)

class uminus = type +
  fixes uminus :: "'a \<Rightarrow> 'a"  ("- _" [81] 80)

class times = type +
  fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "*" 70)

class inverse = type +
  fixes inverse :: "'a \<Rightarrow> 'a"
    and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)

class abs = type +
  fixes abs :: "'a \<Rightarrow> 'a"
begin

notation (xsymbols)
  abs  ("\<bar>_\<bar>")

notation (HTML output)
  abs  ("\<bar>_\<bar>")

end

class sgn = type +
  fixes sgn :: "'a \<Rightarrow> 'a"

class ord = type +
  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
    and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
begin

notation
  less_eq  ("op <=") and
  less_eq  ("(_/ <= _)" [51, 51] 50) and
  less  ("op <") and
  less  ("(_/ < _)"  [51, 51] 50)
  
notation (xsymbols)
  less_eq  ("op \<le>") and
  less_eq  ("(_/ \<le> _)"  [51, 51] 50)

notation (HTML output)
  less_eq  ("op \<le>") and
  less_eq  ("(_/ \<le> _)"  [51, 51] 50)

abbreviation (input)
  greater_eq  (infix ">=" 50) where
  "x >= y \<equiv> y <= x"

notation (input)
  greater_eq  (infix "\<ge>" 50)

abbreviation (input)
  greater  (infix ">" 50) where
  "x > y \<equiv> y < x"

end

syntax
  "_index1"  :: index    ("\<^sub>1")
translations
  (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"

typed_print_translation {*
let
  fun tr' c = (c, fn show_sorts => fn T => fn ts =>
    if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
    else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
in map tr' [@{const_syntax HOL.one}, @{const_syntax HOL.zero}] end;
*} -- {* show types that are presumably too general *}


subsection {* Fundamental rules *}

subsubsection {* Equality *}

lemma sym: "s = t ==> t = s"
  by (erule subst) (rule refl)

lemma ssubst: "t = s ==> P s ==> P t"
  by (drule sym) (erule subst)

lemma trans: "[| r=s; s=t |] ==> r=t"
  by (erule subst)

lemma meta_eq_to_obj_eq: 
  assumes meq: "A == B"
  shows "A = B"
  by (unfold meq) (rule refl)

text {* Useful with @{text erule} for proving equalities from known equalities. *}
     (* a = b
        |   |
        c = d   *)
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
apply (rule trans)
apply (rule trans)
apply (rule sym)
apply assumption+
done

text {* For calculational reasoning: *}

lemma forw_subst: "a = b ==> P b ==> P a"
  by (rule ssubst)

lemma back_subst: "P a ==> a = b ==> P b"
  by (rule subst)


subsubsection {*Congruence rules for application*}

(*similar to AP_THM in Gordon's HOL*)
lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
apply (erule subst)
apply (rule refl)
done

(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
lemma arg_cong: "x=y ==> f(x)=f(y)"
apply (erule subst)
apply (rule refl)
done

lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
apply (erule ssubst)+
apply (rule refl)
done

lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
apply (erule subst)+
apply (rule refl)
done


subsubsection {*Equality of booleans -- iff*}

lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
  by (iprover intro: iff [THEN mp, THEN mp] impI assms)

lemma iffD2: "[| P=Q; Q |] ==> P"
  by (erule ssubst)

lemma rev_iffD2: "[| Q; P=Q |] ==> P"
  by (erule iffD2)

lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
  by (drule sym) (rule iffD2)

lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
  by (drule sym) (rule rev_iffD2)

lemma iffE:
  assumes major: "P=Q"
    and minor: "[| P --> Q; Q --> P |] ==> R"
  shows R
  by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])


subsubsection {*True*}

lemma TrueI: "True"
  unfolding True_def by (rule refl)

lemma eqTrueI: "P ==> P = True"
  by (iprover intro: iffI TrueI)

lemma eqTrueE: "P = True ==> P"
  by (erule iffD2) (rule TrueI)


subsubsection {*Universal quantifier*}

lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
  unfolding All_def by (iprover intro: ext eqTrueI assms)

lemma spec: "ALL x::'a. P(x) ==> P(x)"
apply (unfold All_def)
apply (rule eqTrueE)
apply (erule fun_cong)
done

lemma allE:
  assumes major: "ALL x. P(x)"
    and minor: "P(x) ==> R"
  shows R
  by (iprover intro: minor major [THEN spec])

lemma all_dupE:
  assumes major: "ALL x. P(x)"
    and minor: "[| P(x); ALL x. P(x) |] ==> R"
  shows R
  by (iprover intro: minor major major [THEN spec])


subsubsection {* False *}

text {*
  Depends upon @{text spec}; it is impossible to do propositional
  logic before quantifiers!
*}

lemma FalseE: "False ==> P"
  apply (unfold False_def)
  apply (erule spec)
  done

lemma False_neq_True: "False = True ==> P"
  by (erule eqTrueE [THEN FalseE])


subsubsection {* Negation *}

lemma notI:
  assumes "P ==> False"
  shows "~P"
  apply (unfold not_def)
  apply (iprover intro: impI assms)
  done

lemma False_not_True: "False ~= True"
  apply (rule notI)
  apply (erule False_neq_True)
  done

lemma True_not_False: "True ~= False"
  apply (rule notI)
  apply (drule sym)
  apply (erule False_neq_True)
  done

lemma notE: "[| ~P;  P |] ==> R"
  apply (unfold not_def)
  apply (erule mp [THEN FalseE])
  apply assumption
  done

lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
  by (erule notE [THEN notI]) (erule meta_mp)


subsubsection {*Implication*}

lemma impE:
  assumes "P-->Q" "P" "Q ==> R"
  shows "R"
by (iprover intro: assms mp)

(* Reduces Q to P-->Q, allowing substitution in P. *)
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
by (iprover intro: mp)

lemma contrapos_nn:
  assumes major: "~Q"
      and minor: "P==>Q"
  shows "~P"
by (iprover intro: notI minor major [THEN notE])

(*not used at all, but we already have the other 3 combinations *)
lemma contrapos_pn:
  assumes major: "Q"
      and minor: "P ==> ~Q"
  shows "~P"
by (iprover intro: notI minor major notE)

lemma not_sym: "t ~= s ==> s ~= t"
  by (erule contrapos_nn) (erule sym)

lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
  by (erule subst, erule ssubst, assumption)

(*still used in HOLCF*)
lemma rev_contrapos:
  assumes pq: "P ==> Q"
      and nq: "~Q"
  shows "~P"
apply (rule nq [THEN contrapos_nn])
apply (erule pq)
done

subsubsection {*Existential quantifier*}

lemma exI: "P x ==> EX x::'a. P x"
apply (unfold Ex_def)
apply (iprover intro: allI allE impI mp)
done

lemma exE:
  assumes major: "EX x::'a. P(x)"
      and minor: "!!x. P(x) ==> Q"
  shows "Q"
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
apply (iprover intro: impI [THEN allI] minor)
done


subsubsection {*Conjunction*}

lemma conjI: "[| P; Q |] ==> P&Q"
apply (unfold and_def)
apply (iprover intro: impI [THEN allI] mp)
done

lemma conjunct1: "[| P & Q |] ==> P"
apply (unfold and_def)
apply (iprover intro: impI dest: spec mp)
done

lemma conjunct2: "[| P & Q |] ==> Q"
apply (unfold and_def)
apply (iprover intro: impI dest: spec mp)
done

lemma conjE:
  assumes major: "P&Q"
      and minor: "[| P; Q |] ==> R"
  shows "R"
apply (rule minor)
apply (rule major [THEN conjunct1])
apply (rule major [THEN conjunct2])
done

lemma context_conjI:
  assumes "P" "P ==> Q" shows "P & Q"
by (iprover intro: conjI assms)


subsubsection {*Disjunction*}

lemma disjI1: "P ==> P|Q"
apply (unfold or_def)
apply (iprover intro: allI impI mp)
done

lemma disjI2: "Q ==> P|Q"
apply (unfold or_def)
apply (iprover intro: allI impI mp)
done

lemma disjE:
  assumes major: "P|Q"
      and minorP: "P ==> R"
      and minorQ: "Q ==> R"
  shows "R"
by (iprover intro: minorP minorQ impI
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])


subsubsection {*Classical logic*}

lemma classical:
  assumes prem: "~P ==> P"
  shows "P"
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
apply assumption
apply (rule notI [THEN prem, THEN eqTrueI])
apply (erule subst)
apply assumption
done

lemmas ccontr = FalseE [THEN classical, standard]

(*notE with premises exchanged; it discharges ~R so that it can be used to
  make elimination rules*)
lemma rev_notE:
  assumes premp: "P"
      and premnot: "~R ==> ~P"
  shows "R"
apply (rule ccontr)
apply (erule notE [OF premnot premp])
done

(*Double negation law*)
lemma notnotD: "~~P ==> P"
apply (rule classical)
apply (erule notE)
apply assumption
done

lemma contrapos_pp:
  assumes p1: "Q"
      and p2: "~P ==> ~Q"
  shows "P"
by (iprover intro: classical p1 p2 notE)


subsubsection {*Unique existence*}

lemma ex1I:
  assumes "P a" "!!x. P(x) ==> x=a"
  shows "EX! x. P(x)"
by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)

text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
lemma ex_ex1I:
  assumes ex_prem: "EX x. P(x)"
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
  shows "EX! x. P(x)"
by (iprover intro: ex_prem [THEN exE] ex1I eq)

lemma ex1E:
  assumes major: "EX! x. P(x)"
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
  shows "R"
apply (rule major [unfolded Ex1_def, THEN exE])
apply (erule conjE)
apply (iprover intro: minor)
done

lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
apply (erule ex1E)
apply (rule exI)
apply assumption
done


subsubsection {*THE: definite description operator*}

lemma the_equality:
  assumes prema: "P a"
      and premx: "!!x. P x ==> x=a"
  shows "(THE x. P x) = a"
apply (rule trans [OF _ the_eq_trivial])
apply (rule_tac f = "The" in arg_cong)
apply (rule ext)
apply (rule iffI)
 apply (erule premx)
apply (erule ssubst, rule prema)
done

lemma theI:
  assumes "P a" and "!!x. P x ==> x=a"
  shows "P (THE x. P x)"
by (iprover intro: assms the_equality [THEN ssubst])

lemma theI': "EX! x. P x ==> P (THE x. P x)"
apply (erule ex1E)
apply (erule theI)
apply (erule allE)
apply (erule mp)
apply assumption
done

(*Easier to apply than theI: only one occurrence of P*)
lemma theI2:
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
  shows "Q (THE x. P x)"
by (iprover intro: assms theI)

lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
           elim:allE impE)

lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
apply (rule the_equality)
apply  assumption
apply (erule ex1E)
apply (erule all_dupE)
apply (drule mp)
apply  assumption
apply (erule ssubst)
apply (erule allE)
apply (erule mp)
apply assumption
done

lemma the_sym_eq_trivial: "(THE y. x=y) = x"
apply (rule the_equality)
apply (rule refl)
apply (erule sym)
done


subsubsection {*Classical intro rules for disjunction and existential quantifiers*}

lemma disjCI:
  assumes "~Q ==> P" shows "P|Q"
apply (rule classical)
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
done

lemma excluded_middle: "~P | P"
by (iprover intro: disjCI)

text {*
  case distinction as a natural deduction rule.
  Note that @{term "~P"} is the second case, not the first
*}
lemma case_split [case_names True False]:
  assumes prem1: "P ==> Q"
      and prem2: "~P ==> Q"
  shows "Q"
apply (rule excluded_middle [THEN disjE])
apply (erule prem2)
apply (erule prem1)
done

(*Classical implies (-->) elimination. *)
lemma impCE:
  assumes major: "P-->Q"
      and minor: "~P ==> R" "Q ==> R"
  shows "R"
apply (rule excluded_middle [of P, THEN disjE])
apply (iprover intro: minor major [THEN mp])+
done

(*This version of --> elimination works on Q before P.  It works best for
  those cases in which P holds "almost everywhere".  Can't install as
  default: would break old proofs.*)
lemma impCE':
  assumes major: "P-->Q"
      and minor: "Q ==> R" "~P ==> R"
  shows "R"
apply (rule excluded_middle [of P, THEN disjE])
apply (iprover intro: minor major [THEN mp])+
done

(*Classical <-> elimination. *)
lemma iffCE:
  assumes major: "P=Q"
      and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
  shows "R"
apply (rule major [THEN iffE])
apply (iprover intro: minor elim: impCE notE)
done

lemma exCI:
  assumes "ALL x. ~P(x) ==> P(a)"
  shows "EX x. P(x)"
apply (rule ccontr)
apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
done


subsubsection {* Intuitionistic Reasoning *}

lemma impE':
  assumes 1: "P --> Q"
    and 2: "Q ==> R"
    and 3: "P --> Q ==> P"
  shows R
proof -
  from 3 and 1 have P .
  with 1 have Q by (rule impE)
  with 2 show R .
qed

lemma allE':
  assumes 1: "ALL x. P x"
    and 2: "P x ==> ALL x. P x ==> Q"
  shows Q
proof -
  from 1 have "P x" by (rule spec)
  from this and 1 show Q by (rule 2)
qed

lemma notE':
  assumes 1: "~ P"
    and 2: "~ P ==> P"
  shows R
proof -
  from 2 and 1 have P .
  with 1 show R by (rule notE)
qed

lemma TrueE: "True ==> P ==> P" .
lemma notFalseE: "~ False ==> P ==> P" .

lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
  and [Pure.elim 2] = allE notE' impE'
  and [Pure.intro] = exI disjI2 disjI1

lemmas [trans] = trans
  and [sym] = sym not_sym
  and [Pure.elim?] = iffD1 iffD2 impE

use "Tools/hologic.ML"


subsubsection {* Atomizing meta-level connectives *}

axiomatization where
  eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)

lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
proof
  assume "!!x. P x"
  then show "ALL x. P x" ..
next
  assume "ALL x. P x"
  then show "!!x. P x" by (rule allE)
qed

lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
proof
  assume r: "A ==> B"
  show "A --> B" by (rule impI) (rule r)
next
  assume "A --> B" and A
  then show B by (rule mp)
qed

lemma atomize_not: "(A ==> False) == Trueprop (~A)"
proof
  assume r: "A ==> False"
  show "~A" by (rule notI) (rule r)
next
  assume "~A" and A
  then show False by (rule notE)
qed

lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
proof
  assume "x == y"
  show "x = y" by (unfold `x == y`) (rule refl)
next
  assume "x = y"
  then show "x == y" by (rule eq_reflection)
qed

lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
proof
  assume conj: "A &&& B"
  show "A & B"
  proof (rule conjI)
    from conj show A by (rule conjunctionD1)
    from conj show B by (rule conjunctionD2)
  qed
next
  assume conj: "A & B"
  show "A &&& B"
  proof -
    from conj show A ..
    from conj show B ..
  qed
qed

lemmas [symmetric, rulify] = atomize_all atomize_imp
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq


subsubsection {* Atomizing elimination rules *}

setup AtomizeElim.setup

lemma atomize_exL[atomize_elim]: "(!!x. P x ==> Q) == ((EX x. P x) ==> Q)"
  by rule iprover+

lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
  by rule iprover+

lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
  by rule iprover+

lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop A" ..


subsection {* Package setup *}

subsubsection {* Classical Reasoner setup *}

lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
  by (rule classical) iprover

lemma swap: "~ P ==> (~ R ==> P) ==> R"
  by (rule classical) iprover

lemma thin_refl:
  "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .

ML {*
structure Hypsubst = HypsubstFun(
struct
  structure Simplifier = Simplifier
  val dest_eq = HOLogic.dest_eq
  val dest_Trueprop = HOLogic.dest_Trueprop
  val dest_imp = HOLogic.dest_imp
  val eq_reflection = @{thm eq_reflection}
  val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
  val imp_intr = @{thm impI}
  val rev_mp = @{thm rev_mp}
  val subst = @{thm subst}
  val sym = @{thm sym}
  val thin_refl = @{thm thin_refl};
  val prop_subst = @{lemma "PROP P t ==> PROP prop (x = t ==> PROP P x)"
                     by (unfold prop_def) (drule eq_reflection, unfold)}
end);
open Hypsubst;

structure Classical = ClassicalFun(
struct
  val imp_elim = @{thm imp_elim}
  val not_elim = @{thm notE}
  val swap = @{thm swap}
  val classical = @{thm classical}
  val sizef = Drule.size_of_thm
  val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
end);

structure BasicClassical: BASIC_CLASSICAL = Classical; 
open BasicClassical;

ML_Antiquote.value "claset"
  (Scan.succeed "Classical.local_claset_of (ML_Context.the_local_context ())");

structure ResAtpset = NamedThmsFun(val name = "atp" val description = "ATP rules");

structure ResBlacklist = NamedThmsFun(val name = "noatp" val description = "Theorems blacklisted for ATP");
*}

text {*ResBlacklist holds theorems blacklisted to sledgehammer. 
  These theorems typically produce clauses that are prolific (match too many equality or
  membership literals) and relate to seldom-used facts. Some duplicate other rules.*}

setup {*
let
  (*prevent substitution on bool*)
  fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
    Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
      (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
in
  Hypsubst.hypsubst_setup
  #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
  #> Classical.setup
  #> ResAtpset.setup
  #> ResBlacklist.setup
end
*}

declare iffI [intro!]
  and notI [intro!]
  and impI [intro!]
  and disjCI [intro!]
  and conjI [intro!]
  and TrueI [intro!]
  and refl [intro!]

declare iffCE [elim!]
  and FalseE [elim!]
  and impCE [elim!]
  and disjE [elim!]
  and conjE [elim!]
  and conjE [elim!]

declare ex_ex1I [intro!]
  and allI [intro!]
  and the_equality [intro]
  and exI [intro]

declare exE [elim!]
  allE [elim]

ML {* val HOL_cs = @{claset} *}

lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
  apply (erule swap)
  apply (erule (1) meta_mp)
  done

declare ex_ex1I [rule del, intro! 2]
  and ex1I [intro]

lemmas [intro?] = ext
  and [elim?] = ex1_implies_ex

(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
lemma alt_ex1E [elim!]:
  assumes major: "\<exists>!x. P x"
      and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
  shows R
apply (rule ex1E [OF major])
apply (rule prem)
apply (tactic {* ares_tac @{thms allI} 1 *})+
apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
apply iprover
done

ML {*
structure Blast = BlastFun
(
  type claset = Classical.claset
  val equality_name = @{const_name "op ="}
  val not_name = @{const_name Not}
  val notE = @{thm notE}
  val ccontr = @{thm ccontr}
  val contr_tac = Classical.contr_tac
  val dup_intr = Classical.dup_intr
  val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
  val claset = Classical.claset
  val rep_cs = Classical.rep_cs
  val cla_modifiers = Classical.cla_modifiers
  val cla_meth' = Classical.cla_meth'
);
val Blast_tac = Blast.Blast_tac;
val blast_tac = Blast.blast_tac;