(*  Title:      HOL/HOL.thy

     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson

 *)

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

 theory HOL

 imports Pure "~~/src/Tools/Code_Generator"

 uses

   ("Tools/hologic.ML")

   "~~/src/Tools/auto_solve.ML"

   "~~/src/Tools/IsaPlanner/zipper.ML"

   "~~/src/Tools/IsaPlanner/isand.ML"

   "~~/src/Tools/IsaPlanner/rw_tools.ML"

   "~~/src/Tools/IsaPlanner/rw_inst.ML"

   "~~/src/Tools/intuitionistic.ML"

   "~~/src/Tools/project_rule.ML"

   "~~/src/Provers/hypsubst.ML"

   "~~/src/Provers/splitter.ML"

   "~~/src/Provers/classical.ML"

   "~~/src/Provers/blast.ML"

   "~~/src/Provers/clasimp.ML"

   "~~/src/Tools/coherent.ML"

   "~~/src/Tools/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")

   ("Tools/recfun_codegen.ML")

 begin

 setup {* Intuitionistic.method_setup "iprover" *}

 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 =

   fixes default :: 'a

 class zero = 

   fixes zero :: 'a  ("0")

 class one =

   fixes one  :: 'a  ("1")

 hide (open) const zero one

 class plus =

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

 class minus =

   fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "-" 65)

 class uminus =

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

 class times =

   fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "*" 70)

 class inverse =

   fixes inverse :: "'a \<Rightarrow> 'a"

     and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)

 class abs =

   fixes abs :: "'a \<Rightarrow> 'a"

 begin

 notation (xsymbols)

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

 notation (HTML output)

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

 end

 class sgn =

   fixes sgn :: "'a \<Rightarrow> 'a"

 class ord =

   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 (not o null) ts orelse 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 rep_cs = Classical.rep_cs

   val cla_modifiers = Classical.cla_modifiers

   val cla_meth' = Classical.cla_meth'

 );

 val blast_tac = Blast.blast_tac;

 *}

 setup Blast.setup

 subsubsection {* Simplifier *}

 lemma eta_contract_eq: "(%s. f s) = f" ..

 lemma simp_thms:

   shows not_not: "(~ ~ P) = P"

   and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"

   and

     "(P ~= Q) = (P = (~Q))"

     "(P | ~P) = True"    "(~P | P) = True"

     "(x = x) = True"

   and not_True_eq_False: "(\<not> True) = False"

   and not_False_eq_True: "(\<not> False) = True"

   and

     "(~P) ~= P"  "P ~= (~P)"

     "(True=P) = P"

   and eq_True: "(P = True) = P"

   and "(False=P) = (~P)"

   and eq_False: "(P = False) = (\<not> P)"

   and

     "(True --> P) = P"  "(False --> P) = True"

     "(P --> True) = True"  "(P --> P) = True"

     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"

     "(P & True) = P"  "(True & P) = P"

     "(P & False) = False"  "(False & P) = False"

     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"

     "(P & ~P) = False"    "(~P & P) = False"

     "(P | True) = True"  "(True | P) = True"

     "(P | False) = P"  "(False | P) = P"

     "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and

     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"

     -- {* needed for the one-point-rule quantifier simplification procs *}

     -- {* essential for termination!! *} and

     "!!P. (EX x. x=t & P(x)) = P(t)"

     "!!P. (EX x. t=x & P(x)) = P(t)"

     "!!P. (ALL x. x=t --> P(x)) = P(t)"

     "!!P. (ALL x. t=x --> P(x)) = P(t)"

   by (blast, blast, blast, blast, blast, iprover+)

 lemma disj_absorb: "(A | A) = A"

   by blast

 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"

   by blast

 lemma conj_absorb: "(A & A) = A"

   by blast

 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"

   by blast

 lemma eq_ac:

   shows eq_commute: "(a=b) = (b=a)"

     and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"

     and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)

 lemma neq_commute: "(a~=b) = (b~=a)" by iprover

 lemma conj_comms:

   shows conj_commute: "(P&Q) = (Q&P)"

     and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+

 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover

 lemmas conj_ac = conj_commute conj_left_commute conj_assoc

 lemma disj_comms:

   shows disj_commute: "(P|Q) = (Q|P)"

     and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+

 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover

 lemmas disj_ac = disj_commute disj_left_commute disj_assoc

 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover

 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover

 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover

 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover

 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover

 lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover

 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover

 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}

 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast

 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast

 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast

 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast

 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"

   by iprover

 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover

 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast

 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast

 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast

 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast

 lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}

   by blast

 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast

 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover

 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"

   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}

   -- {* cases boil down to the same thing. *}

   by blast

 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast

 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast

 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover

 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover

 lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast

 declare All_def [noatp]

 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover

 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover

 text {*

   \medskip The @{text "&"} congruence rule: not included by default!

   May slow rewrite proofs down by as much as 50\% *}

 lemma conj_cong:

     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"

   by iprover

 lemma rev_conj_cong:

     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"

   by iprover

 text {* The @{text "|"} congruence rule: not included by default! *}

 lemma disj_cong:

     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"

   by blast

 text {* \medskip if-then-else rules *}

 lemma if_True: "(if True then x else y) = x"

   by (unfold if_def) blast

 lemma if_False: "(if False then x else y) = y"

   by (unfold if_def) blast

 lemma if_P: "P ==> (if P then x else y) = x"

   by (unfold if_def) blast

 lemma if_not_P: "~P ==> (if P then x else y) = y"

   by (unfold if_def) blast

 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"

   apply (rule case_split [of Q])

    apply (simplesubst if_P)

     prefer 3 apply (simplesubst if_not_P, blast+)

   done

 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"

 by (simplesubst split_if, blast)

 lemmas if_splits [noatp] = split_if split_if_asm

 lemma if_cancel: "(if c then x else x) = x"

 by (simplesubst split_if, blast)

 lemma if_eq_cancel: "(if x = y then y else x) = x"

 by (simplesubst split_if, blast)

 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"

   -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}

   by (rule split_if)

 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"

   -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}

   apply (simplesubst split_if, blast)

   done

 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover

 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover

 text {* \medskip let rules for simproc *}

 lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"

   by (unfold Let_def)

 lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"

   by (unfold Let_def)

 text {*

   The following copy of the implication operator is useful for

   fine-tuning congruence rules.  It instructs the simplifier to simplify

   its premise.

 *}

 constdefs

   simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1)

   [code del]: "simp_implies \<equiv> op ==>"

 lemma simp_impliesI:

   assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"

   shows "PROP P =simp=> PROP Q"

   apply (unfold simp_implies_def)

   apply (rule PQ)

   apply assumption

   done

 lemma simp_impliesE:

   assumes PQ: "PROP P =simp=> PROP Q"

   and P: "PROP P"

   and QR: "PROP Q \<Longrightarrow> PROP R"

   shows "PROP R"

   apply (rule QR)

   apply (rule PQ [unfolded simp_implies_def])

   apply (rule P)

   done

 lemma simp_implies_cong:

   assumes PP' :"PROP P == PROP P'"

   and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"

   shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"

 proof (unfold simp_implies_def, rule equal_intr_rule)

   assume PQ: "PROP P \<Longrightarrow> PROP Q"

   and P': "PROP P'"

   from PP' [symmetric] and P' have "PROP P"

     by (rule equal_elim_rule1)

   then have "PROP Q" by (rule PQ)

   with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)

 next

   assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"

   and P: "PROP P"

   from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)

   then have "PROP Q'" by (rule P'Q')

   with P'QQ' [OF P', symmetric] show "PROP Q"

     by (rule equal_elim_rule1)

 qed

 lemma uncurry:

   assumes "P \<longrightarrow> Q \<longrightarrow> R"

   shows "P \<and> Q \<longrightarrow> R"

   using assms by blast

 lemma iff_allI:

   assumes "\<And>x. P x = Q x"

   shows "(\<forall>x. P x) = (\<forall>x. Q x)"

   using assms by blast

 lemma iff_exI:

   assumes "\<And>x. P x = Q x"

   shows "(\<exists>x. P x) = (\<exists>x. Q x)"

   using assms by blast

 lemma all_comm:

   "(\<forall>x y. P x y) = (\<forall>y x. P x y)"

   by blast

 lemma ex_comm:

   "(\<exists>x y. P x y) = (\<exists>y x. P x y)"

   by blast

 use "Tools/simpdata.ML"

 ML {* open Simpdata *}

 setup {*

   Simplifier.method_setup Splitter.split_modifiers

   #> Simplifier.map_simpset (K Simpdata.simpset_simprocs)

   #> Splitter.setup

   #> clasimp_setup

   #> EqSubst.setup

 *}

 text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}

 simproc_setup neq ("x = y") = {* fn _ =>

 let

   val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};

   fun is_neq eq lhs rhs thm =

     (case Thm.prop_of thm of

       _ $ (Not $ (eq' $ l' $ r')) =>

         Not = HOLogic.Not andalso eq' = eq andalso

         r' aconv lhs andalso l' aconv rhs

     | _ => false);

   fun proc ss ct =

     (case Thm.term_of ct of

       eq $ lhs $ rhs =>

         (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of_ss ss) of

           SOME thm => SOME (thm RS neq_to_EQ_False)

         | NONE => NONE)

      | _ => NONE);

 in proc end;

 *}

 simproc_setup let_simp ("Let x f") = {*

 let

   val (f_Let_unfold, x_Let_unfold) =

     let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_unfold}

     in (cterm_of @{theory} f, cterm_of @{theory} x) end

   val (f_Let_folded, x_Let_folded) =

     let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_folded}

     in (cterm_of @{theory} f, cterm_of @{theory} x) end;

   val g_Let_folded =

     let val [(_ $ _ $ (g $ _))] = prems_of @{thm Let_folded}

     in cterm_of @{theory} g end;

   fun count_loose (Bound i) k = if i >= k then 1 else 0

     | count_loose (s $ t) k = count_loose s k + count_loose t k

     | count_loose (Abs (_, _, t)) k = count_loose  t (k + 1)

     | count_loose _ _ = 0;

   fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =

    case t

     of Abs (_, _, t') => count_loose t' 0 <= 1

      | _ => true;

 in fn _ => fn ss => fn ct => if is_trivial_let (Thm.term_of ct)

   then SOME @{thm Let_def} (*no or one ocurrenc of bound variable*)

   else let (*Norbert Schirmer's case*)

     val ctxt = Simplifier.the_context ss;

     val thy = ProofContext.theory_of ctxt;

     val t = Thm.term_of ct;

     val ([t'], ctxt') = Variable.import_terms false [t] ctxt;

   in Option.map (hd o Variable.export ctxt' ctxt o single)

     (case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)

       if is_Free x orelse is_Bound x orelse is_Const x

       then SOME @{thm Let_def}

       else

         let

           val n = case f of (Abs (x, _, _)) => x | _ => "x";

           val cx = cterm_of thy x;

           val {T = xT, ...} = rep_cterm cx;

           val cf = cterm_of thy f;

           val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);

           val (_ $ _ $ g) = prop_of fx_g;

           val g' = abstract_over (x,g);

         in (if (g aconv g')

              then

                 let

                   val rl =

                     cterm_instantiate [(f_Let_unfold, cf), (x_Let_unfold, cx)] @{thm Let_unfold};

                 in SOME (rl OF [fx_g]) end

              else if Term.betapply (f, x) aconv g then NONE (*avoid identity conversion*)

              else let

                    val abs_g'= Abs (n,xT,g');

                    val g'x = abs_g'$x;

                    val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));

                    val rl = cterm_instantiate

                              [(f_Let_folded, cterm_of thy f), (x_Let_folded, cx),

                               (g_Let_folded, cterm_of thy abs_g')]

                              @{thm Let_folded};

                  in SOME (rl OF [transitive fx_g g_g'x])

                  end)

         end

     | _ => NONE)

   end

 end *}

 lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"

 proof

   assume "True \<Longrightarrow> PROP P"

   from this [OF TrueI] show "PROP P" .

 next

   assume "PROP P"

   then show "PROP P" .

 qed

 lemma ex_simps:

   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"

   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"

   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"

   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"

   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"

   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"

   -- {* Miniscoping: pushing in existential quantifiers. *}

   by (iprover | blast)+

 lemma all_simps:

   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"

   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"

   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"

   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"

   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"

   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"

   -- {* Miniscoping: pushing in universal quantifiers. *}

   by (iprover | blast)+

 lemmas [simp] =

   triv_forall_equality (*prunes params*)

   True_implies_equals  (*prune asms `True'*)

   if_True

   if_False

   if_cancel

   if_eq_cancel

   imp_disjL

   (*In general it seems wrong to add distributive laws by default: they

     might cause exponential blow-up.  But imp_disjL has been in for a while

     and cannot be removed without affecting existing proofs.  Moreover,

     rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the

     grounds that it allows simplification of R in the two cases.*)

   conj_assoc

   disj_assoc

   de_Morgan_conj

   de_Morgan_disj

   imp_disj1

   imp_disj2

   not_imp

   disj_not1

   not_all

   not_ex

   cases_simp

   the_eq_trivial

   the_sym_eq_trivial

   ex_simps

   all_simps

   simp_thms

 lemmas [cong] = imp_cong simp_implies_cong

 lemmas [split] = split_if

 ML {* val HOL_ss = @{simpset} *}

 text {* Simplifies x assuming c and y assuming ~c *}

 lemma if_cong:

   assumes "b = c"

       and "c \<Longrightarrow> x = u"

       and "\<not> c \<Longrightarrow> y = v"

   shows "(if b then x else y) = (if c then u else v)"

   unfolding if_def using assms by simp

 text {* Prevents simplification of x and y:

   faster and allows the execution of functional programs. *}

 lemma if_weak_cong [cong]:

   assumes "b = c"

   shows "(if b then x else y) = (if c then x else y)"

   using assms by (rule arg_cong)

 text {* Prevents simplification of t: much faster *}

 lemma let_weak_cong:

   assumes "a = b"

   shows "(let x = a in t x) = (let x = b in t x)"

   using assms by (rule arg_cong)

 text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}

 lemma eq_cong2:

   assumes "u = u'"

   shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"

   using assms by simp

 lemma if_distrib:

   "f (if c then x else y) = (if c then f x else f y)"

   by simp

 text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand

   side of an equality.  Used in @{text "{Integ,Real}/simproc.ML"} *}

 lemma restrict_to_left:

   assumes "x = y"

   shows "(x = z) = (y = z)"

   using assms by simp

 subsubsection {* Generic cases and induction *}

 text {* Rule projections: *}

 ML {*

 structure ProjectRule = ProjectRuleFun

 (

   val conjunct1 = @{thm conjunct1}

   val conjunct2 = @{thm conjunct2}

   val mp = @{thm mp}

 )

 *}

 constdefs

   induct_forall where "induct_forall P == \<forall>x. P x"

   induct_implies where "induct_implies A B == A \<longrightarrow> B"

   induct_equal where "induct_equal x y == x = y"

   induct_conj where "induct_conj A B == A \<and> B"

 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"

   by (unfold atomize_all induct_forall_def)

 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"

   by (unfold atomize_imp induct_implies_def)

 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"

   by (unfold atomize_eq induct_equal_def)

 lemma induct_conj_eq: "(A &&& B) == Trueprop (induct_conj A B)"

   by (unfold atomize_conj induct_conj_def)

 lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq

 lemmas induct_rulify [symmetric, standard] = induct_atomize

 lemmas induct_rulify_fallback =

   induct_forall_def induct_implies_def induct_equal_def induct_conj_def

 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =

     induct_conj (induct_forall A) (induct_forall B)"

   by (unfold induct_forall_def induct_conj_def) iprover

 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =

     induct_conj (induct_implies C A) (induct_implies C B)"

   by (unfold induct_implies_def induct_conj_def) iprover

 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"

 proof

   assume r: "induct_conj A B ==> PROP C" and A B

   show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)

 next

   assume r: "A ==> B ==> PROP C" and "induct_conj A B"

   show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])

 qed

 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry

 hide const induct_forall induct_implies induct_equal induct_conj

 text {* Method setup. *}

 ML {*

 structure Induct = InductFun

 (

   val cases_default = @{thm case_split}

   val atomize = @{thms induct_atomize}

   val rulify = @{thms induct_rulify}

   val rulify_fallback = @{thms induct_rulify_fallback}

 )

 *}

 setup Induct.setup

 use "~~/src/Tools/induct_tacs.ML"

 setup InductTacs.setup

 subsubsection {* Coherent logic *}

 ML {*

 structure Coherent = CoherentFun

 (

   val atomize_elimL = @{thm atomize_elimL}

   val atomize_exL = @{thm atomize_exL}

   val atomize_conjL = @{thm atomize_conjL}

   val atomize_disjL = @{thm atomize_disjL}

   val operator_names =

     [@{const_name "op |"}, @{const_name "op &"}, @{const_name "Ex"}]

 );

 *}

 setup Coherent.setup

 subsubsection {* Reorienting equalities *}

 ML {*

 signature REORIENT_PROC =

 sig

   val init : theory -> theory

   val add : (term -> bool) -> theory -> theory

   val proc : morphism -> simpset -> cterm -> thm option

 end;

 structure ReorientProc : REORIENT_PROC =

 struct

   structure Data = TheoryDataFun

   (

     type T = term -> bool;

     val empty = (fn _ => false);

     val copy = I;

     val extend = I;

     fun merge _ (m1, m2) = (fn t => m1 t orelse m2 t);

   )

   val init = Data.init;

   fun add m = Data.map (fn matches => fn t => matches t orelse m t);

   val meta_reorient = @{thm eq_commute [THEN eq_reflection]};

   fun proc phi ss ct =

     let

       val ctxt = Simplifier.the_context ss;

       val thy = ProofContext.theory_of ctxt;

       val matches = Data.get thy;

     in

       case Thm.term_of ct of

         (_ $ t $ u) => if matches u then NONE else SOME meta_reorient

       | _ => NONE

     end;

 end;

 *}

 setup ReorientProc.init

 setup {*

   ReorientProc.add

     (fn Const(@{const_name HOL.zero}, _) => true

       | Const(@{const_name HOL.one}, _) => true

       | _ => false)

 *}

 simproc_setup reorient_zero ("0 = x") = ReorientProc.proc

 simproc_setup reorient_one ("1 = x") = ReorientProc.proc

 subsection {* Other simple lemmas and lemma duplicates *}

 lemma Let_0 [simp]: "Let 0 f = f 0"

   unfolding Let_def ..

 lemma Let_1 [simp]: "Let 1 f = f 1"

   unfolding Let_def ..

 lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"

   by blast+

 lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"

   apply (rule iffI)

   apply (rule_tac a = "%x. THE y. P x y" in ex1I)

   apply (fast dest!: theI')

   apply (fast intro: ext the1_equality [symmetric])

   apply (erule ex1E)

   apply (rule allI)

   apply (rule ex1I)

   apply (erule spec)

   apply (erule_tac x = "%z. if z = x then y else f z" in allE)

   apply (erule impE)

   apply (rule allI)

   apply (case_tac "xa = x")

   apply (drule_tac [3] x = x in fun_cong, simp_all)

   done

 lemma mk_left_commute:

   fixes f (infix "\<otimes>" 60)

   assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and

           c: "\<And>x y. x \<otimes> y = y \<otimes> x"

   shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"

   by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])

 lemmas eq_sym_conv = eq_commute

 lemma nnf_simps:

   "(\<not>(P \<and> Q)) = (\<not> P \<or> \<not> Q)" "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)" 

   "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))" 

   "(\<not> \<not>(P)) = P"

 by blast+

 subsection {* Basic ML bindings *}

 ML {*

 val FalseE = @{thm FalseE}

 val Let_def = @{thm Let_def}

 val TrueI = @{thm TrueI}

 val allE = @{thm allE}

 val allI = @{thm allI}

 val all_dupE = @{thm all_dupE}

 val arg_cong = @{thm arg_cong}

 val box_equals = @{thm box_equals}

 val ccontr = @{thm ccontr}

 val classical = @{thm classical}

 val conjE = @{thm conjE}

 val conjI = @{thm conjI}

 val conjunct1 = @{thm conjunct1}

 val conjunct2 = @{thm conjunct2}

 val disjCI = @{thm disjCI}

 val disjE = @{thm disjE}

 val disjI1 = @{thm disjI1}

 val disjI2 = @{thm disjI2}

 val eq_reflection = @{thm eq_reflection}

 val ex1E = @{thm ex1E}

 val ex1I = @{thm ex1I}

 val ex1_implies_ex = @{thm ex1_implies_ex}

 val exE = @{thm exE}

 val exI = @{thm exI}

 val excluded_middle = @{thm excluded_middle}

 val ext = @{thm ext}

 val fun_cong = @{thm fun_cong}

 val iffD1 = @{thm iffD1}

 val iffD2 = @{thm iffD2}

 val iffI = @{thm iffI}

 val impE = @{thm impE}

 val impI = @{thm impI}

 val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}

 val mp = @{thm mp}

 val notE = @{thm notE}

 val notI = @{thm notI}

 val not_all = @{thm not_all}

 val not_ex = @{thm not_ex}

 val not_iff = @{thm not_iff}

 val not_not = @{thm not_not}

 val not_sym = @{thm not_sym}

 val refl = @{thm refl}

 val rev_mp = @{thm rev_mp}

 val spec = @{thm spec}

 val ssubst = @{thm ssubst}

 val subst = @{thm subst}

 val sym = @{thm sym}

 val trans = @{thm trans}

 *}

 subsection {* Code generator setup *}

 subsubsection {* SML code generator setup *}

 use "Tools/recfun_codegen.ML"

 setup {*

   Codegen.setup

   #> RecfunCodegen.setup

 *}

 types_code

   "bool"  ("bool")

 attach (term_of) {*

 fun term_of_bool b = if b then HOLogic.true_const else HOLogic.false_const;

 *}

 attach (test) {*

 fun gen_bool i =

   let val b = one_of [false, true]

   in (b, fn () => term_of_bool b) end;

 *}

   "prop"  ("bool")

 attach (term_of) {*

 fun term_of_prop b =

   HOLogic.mk_Trueprop (if b then HOLogic.true_const else HOLogic.false_const);

 *}

 consts_code

   "Trueprop" ("(_)")

   "True"    ("true")

   "False"   ("false")

   "Not"     ("Bool.not")

   "op |"    ("(_ orelse/ _)")

   "op &"    ("(_ andalso/ _)")

   "If"      ("(if _/ then _/ else _)")

 setup {*

 let

 fun eq_codegen thy defs dep thyname b t gr =

     (case strip_comb t of

        (Const ("op =", Type (_, [Type ("fun", _), _])), _) => NONE

      | (Const ("op =", _), [t, u]) =>

           let

             val (pt, gr') = Codegen.invoke_codegen thy defs dep thyname false t gr;

             val (pu, gr'') = Codegen.invoke_codegen thy defs dep thyname false u gr';

             val (_, gr''') = Codegen.invoke_tycodegen thy defs dep thyname false HOLogic.boolT gr'';

           in

             SOME (Codegen.parens

               (Pretty.block [pt, Codegen.str " =", Pretty.brk 1, pu]), gr''')

           end

      | (t as Const ("op =", _), ts) => SOME (Codegen.invoke_codegen

          thy defs dep thyname b (Codegen.eta_expand t ts 2) gr)

      | _ => NONE);

 in

   Codegen.add_codegen "eq_codegen" eq_codegen

 end

 *}

 subsubsection {* Equality *}

 class eq =

   fixes eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

   assumes eq_equals: "eq x y \<longleftrightarrow> x = y"

 begin

 lemma eq [code unfold, code inline del]: "eq = (op =)"

   by (rule ext eq_equals)+

 lemma eq_refl: "eq x x \<longleftrightarrow> True"

   unfolding eq by rule+

 lemma equals_eq [code inline]: "(op =) \<equiv> eq"

   by (rule eq_reflection) (rule ext, rule ext, rule sym, rule eq_equals)

 declare equals_eq [symmetric, code post]

 end

 declare equals_eq [code]

 subsubsection {* Generic code generator foundation *}

 text {* Datatypes *}

 code_datatype True False

 code_datatype "TYPE('a\<Colon>{})"

 code_datatype Trueprop "prop"

 text {* Code equations *}

 lemma [code]:

   shows "(True \<Longrightarrow> PROP P) \<equiv> PROP P" 

     and "(False \<Longrightarrow> Q) \<equiv> Trueprop True" 

     and "(PROP P \<Longrightarrow> True) \<equiv> Trueprop True" 

     and "(Q \<Longrightarrow> False) \<equiv> Trueprop (\<not> Q)" by (auto intro!: equal_intr_rule)

 lemma [code]:

   shows "False \<and> x \<longleftrightarrow> False"

     and "True \<and> x \<longleftrightarrow> x"

     and "x \<and> False \<longleftrightarrow> False"

     and "x \<and> True \<longleftrightarrow> x" by simp_all

 lemma [code]:

   shows "False \<or> x \<longleftrightarrow> x"

     and "True \<or> x \<longleftrightarrow> True"

     and "x \<or> False \<longleftrightarrow> x"

     and "x \<or> True \<longleftrightarrow> True" by simp_all

 lemma [code]:

   shows "\<not> True \<longleftrightarrow> False"

     and "\<not> False \<longleftrightarrow> True" by (rule HOL.simp_thms)+

 lemmas [code] = Let_def if_True if_False

 lemmas [code, code unfold, symmetric, code post] = imp_conv_disj

 instantiation itself :: (type) eq

 begin

 definition eq_itself :: "'a itself \<Rightarrow> 'a itself \<Rightarrow> bool" where

   "eq_itself x y \<longleftrightarrow> x = y"

 instance proof

 qed (fact eq_itself_def)

 end

 lemma eq_itself_code [code]:

   "eq_class.eq TYPE('a) TYPE('a) \<longleftrightarrow> True"

   by (simp add: eq)

 text {* Equality *}

 declare simp_thms(6) [code nbe]

 hide (open) const eq

 hide const eq

 setup {*

   Code_Unit.add_const_alias @{thm equals_eq}

 *}

 text {* Cases *}

 lemma Let_case_cert:

   assumes "CASE \<equiv> (\<lambda>x. Let x f)"

   shows "CASE x \<equiv> f x"

   using assms by simp_all

 lemma If_case_cert:

   assumes "CASE \<equiv> (\<lambda>b. If b f g)"

   shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"

   using assms by simp_all

 setup {*

   Code.add_case @{thm Let_case_cert}

   #> Code.add_case @{thm If_case_cert}

   #> Code.add_undefined @{const_name undefined}

 *}

 code_abort undefined

 subsubsection {* Generic code generator preprocessor *}

 setup {*

   Code_Preproc.map_pre (K HOL_basic_ss)

   #> Code_Preproc.map_post (K HOL_basic_ss)

 *}

 subsubsection {* Generic code generator target languages *}

 text {* type bool *}

 code_type bool

   (SML "bool")

   (OCaml "bool")

   (Haskell "Bool")

 code_const True and False and Not and "op &" and "op |" and If

   (SML "true" and "false" and "not"

     and infixl 1 "andalso" and infixl 0 "orelse"

     and "!(if (_)/ then (_)/ else (_))")

   (OCaml "true" and "false" and "not"

     and infixl 4 "&&" and infixl 2 "||"

     and "!(if (_)/ then (_)/ else (_))")

   (Haskell "True" and "False" and "not"

     and infixl 3 "&&" and infixl 2 "||"

     and "!(if (_)/ then (_)/ else (_))")

 code_reserved SML

   bool true false not

 code_reserved OCaml

   bool not

 text {* using built-in Haskell equality *}

 code_class eq

   (Haskell "Eq")

 code_const "eq_class.eq"

   (Haskell infixl 4 "==")

 code_const "op ="

   (Haskell infixl 4 "==")

 text {* undefined *}

 code_const undefined

   (SML "!(raise/ Fail/ \"undefined\")")

   (OCaml "failwith/ \"undefined\"")

   (Haskell "error/ \"undefined\"")

 subsubsection {* Evaluation and normalization by evaluation *}

 setup {*

   Value.add_evaluator ("SML", Codegen.eval_term o ProofContext.theory_of)

 *}

 ML {*

 structure Eval_Method =

 struct

 val eval_ref : (unit -> bool) option ref = ref NONE;

 end;

 *}

 oracle eval_oracle = {* fn ct =>

   let

     val thy = Thm.theory_of_cterm ct;

     val t = Thm.term_of ct;

     val dummy = @{cprop True};

   in case try HOLogic.dest_Trueprop t

    of SOME t' => if Code_ML.eval NONE

          ("Eval_Method.eval_ref", Eval_Method.eval_ref) (K I) thy t' [] 

        then Thm.capply (Thm.capply @{cterm "op \<equiv> \<Colon> prop \<Rightarrow> prop \<Rightarrow> prop"} ct) dummy

        else dummy

     | NONE => dummy

   end

 *}

 ML {*

 fun gen_eval_method conv ctxt = SIMPLE_METHOD'

   (CONVERSION (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt)

     THEN' rtac TrueI)

 *}

 method_setup eval = {* Scan.succeed (gen_eval_method eval_oracle) *}

   "solve goal by evaluation"

 method_setup evaluation = {* Scan.succeed (gen_eval_method Codegen.evaluation_conv) *}

   "solve goal by evaluation"

 method_setup normalization = {*

   Scan.succeed (K (SIMPLE_METHOD' (CONVERSION Nbe.norm_conv THEN' (fn k => TRY (rtac TrueI k)))))

 *} "solve goal by normalization"

 subsubsection {* Quickcheck *}

 setup {*

   Quickcheck.add_generator ("SML", Codegen.test_term)

 *}

 quickcheck_params [size = 5, iterations = 50]

 subsection {* Nitpick setup *}

 text {* This will be relocated once Nitpick is moved to HOL. *}

 ML {*

 structure Nitpick_Const_Def_Thms = NamedThmsFun

 (

   val name = "nitpick_const_def"

   val description = "alternative definitions of constants as needed by Nitpick"

 )

 structure Nitpick_Const_Simp_Thms = NamedThmsFun

 (

   val name = "nitpick_const_simp"

   val description = "equational specification of constants as needed by Nitpick"

 )

 structure Nitpick_Const_Psimp_Thms = NamedThmsFun

 (

   val name = "nitpick_const_psimp"

   val description = "partial equational specification of constants as needed by Nitpick"

 )

 structure Nitpick_Ind_Intro_Thms = NamedThmsFun

 (

   val name = "nitpick_ind_intro"

   val description = "introduction rules for (co)inductive predicates as needed by Nitpick"

 )

 *}

 setup {*

   Nitpick_Const_Def_Thms.setup

   #> Nitpick_Const_Simp_Thms.setup

   #> Nitpick_Const_Psimp_Thms.setup

   #> Nitpick_Ind_Intro_Thms.setup

 *}

 subsection {* Legacy tactics and ML bindings *}

 ML {*

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

 (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)

 local

   fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t

     | wrong_prem (Bound _) = true

     | wrong_prem _ = false;

   val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);

 in

   fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);

   fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];

 end;

 val all_conj_distrib = thm "all_conj_distrib";

 val all_simps = thms "all_simps";

 val atomize_not = thm "atomize_not";

 val case_split = thm "case_split";

 val cases_simp = thm "cases_simp";

 val choice_eq = thm "choice_eq"

 val cong = thm "cong"

 val conj_comms = thms "conj_comms";

 val conj_cong = thm "conj_cong";

 val de_Morgan_conj = thm "de_Morgan_conj";

 val de_Morgan_disj = thm "de_Morgan_disj";

 val disj_assoc = thm "disj_assoc";

 val disj_comms = thms "disj_comms";

 val disj_cong = thm "disj_cong";

 val eq_ac = thms "eq_ac";

 val eq_cong2 = thm "eq_cong2"

 val Eq_FalseI = thm "Eq_FalseI";

 val Eq_TrueI = thm "Eq_TrueI";

 val Ex1_def = thm "Ex1_def"

 val ex_disj_distrib = thm "ex_disj_distrib";

 val ex_simps = thms "ex_simps";

 val if_cancel = thm "if_cancel";

 val if_eq_cancel = thm "if_eq_cancel";

 val if_False = thm "if_False";

 val iff_conv_conj_imp = thm "iff_conv_conj_imp";

 val iff = thm "iff"

 val if_splits = thms "if_splits";

 val if_True = thm "if_True";

 val if_weak_cong = thm "if_weak_cong"

 val imp_all = thm "imp_all";

 val imp_cong = thm "imp_cong";

 val imp_conjL = thm "imp_conjL";

 val imp_conjR = thm "imp_conjR";

 val imp_conv_disj = thm "imp_conv_disj";

 val simp_implies_def = thm "simp_implies_def";

 val simp_thms = thms "simp_thms";

 val split_if = thm "split_if";

 val the1_equality = thm "the1_equality"

 val theI = thm "theI"

 val theI' = thm "theI'"

 val True_implies_equals = thm "True_implies_equals";

 val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms @ @{thms "nnf_simps"})

 *}

 end