(*  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"

  ("~~/src/HOL/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!*)