(*  Title:      HOL/HOL.thy

    ID:         $Id$

    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson

*)

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

theory HOL

imports CPure

uses ("cladata.ML") ("blastdata.ML") ("simpdata.ML")

    "Tools/res_atpset.ML"

begin

subsection {* Primitive logic *}

subsubsection {* Core syntax *}

classes type

defaultsort type

global

typedecl bool

arities

  bool :: type

  fun :: (type, type) type

judgment

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

consts

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

  True          :: bool

  False         :: bool

  arbitrary     :: 'a

  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"

  "="           :: "['a, 'a] => bool"               (infixl 50)

  &             :: "[bool, bool] => bool"           (infixr 35)

  "|"           :: "[bool, bool] => bool"           (infixr 30)

  -->           :: "[bool, bool] => bool"           (infixr 25)

local

consts

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

subsubsection {* Additional concrete syntax *}

nonterminals

  letbinds  letbind

  case_syn  cases_syn

syntax

  "_not_equal"  :: "['a, 'a] => bool"                    (infixl "~=" 50)

  "_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

  "x ~= y"                == "~ (x = y)"

  "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 (output)

  "="           :: "['a, 'a] => bool"                    (infix 50)

  "_not_equal"  :: "['a, 'a] => bool"                    (infix "~=" 50)

syntax (xsymbols)

  Not           :: "bool => bool"                        ("\<not> _" [40] 40)

  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)

  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)

  "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)

  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)

  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)

  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)

  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)

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

(*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \<orelse> _")*)

syntax (xsymbols output)

  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)

syntax (HTML output)

  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)

  Not           :: "bool => bool"                        ("\<not> _" [40] 40)

  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)

  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)

  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)

  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)

  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)

  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)

syntax (HOL)

  "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)

  "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)

  "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)

syntax

  "_iff" :: "bool => bool => bool"                       (infixr "<->" 25)

syntax (xsymbols)

  "_iff" :: "bool => bool => bool"                       (infixr "\<longleftrightarrow>" 25)

translations

  "op <->" => "op = :: bool => bool => bool"

typed_print_translation {*

  let

    fun iff_tr' _ (Type ("fun", (Type ("bool", _) :: _))) ts =

          if Output.has_mode "iff" then Term.list_comb (Syntax.const "_iff", ts)

          else raise Match

      | iff_tr' _ _ _ = raise Match;

  in [("op =", iff_tr')] end

*}

subsubsection {* Axioms and basic definitions *}

axioms

  eq_reflection:  "(x=y) ==> (x==y)"

  refl:           "t = (t::'a)"

  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"

text{*Thanks to Stephan Merz*}

theorem subst:

  assumes eq: "s = t" and p: "P(s)"

  shows "P(t::'a)"

proof -

  from eq have meta: "s \<equiv> t"

    by (rule eq_reflection)

  from p show ?thesis

    by (unfold meta)

qed

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

  arbitrary

subsubsection {* Generic algebraic operations *}

axclass zero < type

axclass one < type

axclass plus < type

axclass minus < type

axclass times < type

axclass inverse < type

consts

  plus             :: "['a::plus, 'a]  => 'a"       (infixl "+" 65)

  uminus           :: "'a::minus => 'a"             ("- _" [81] 80)

  minus            :: "['a::minus, 'a] => 'a"       (infixl "-" 65)

  abs              :: "'a::minus => 'a"

  times            :: "['a::times, 'a] => 'a"       (infixl "*" 70)

  inverse          :: "'a::inverse => 'a"

  divide           :: "['a::inverse, 'a] => 'a"     (infixl "'/" 70)

global

consts

  "0"           :: "'a::zero"                       ("0")

  "1"           :: "'a::one"                        ("1")

syntax

  "_index1"  :: index    ("\<^sub>1")

translations

  (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"

local

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 [tr' "0", tr' "1"] end;

*} -- {* show types that are presumably too general *}

syntax (xsymbols)

  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")

syntax (HTML output)

  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")

subsection {*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 def_imp_eq: assumes meq: "A == B" shows "A = B"

  by (unfold meq) (rule refl)

(*Useful with eresolve_tac for proving equalties 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)

subsection {*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

subsection {*Equality of booleans -- iff*}

lemma iffI: assumes prems: "P ==> Q" "Q ==> P" shows "P=Q"

  by (iprover intro: iff [THEN mp, THEN mp] impI prems)

lemma iffD2: "[| P=Q; Q |] ==> P"

  by (erule ssubst)

lemma rev_iffD2: "[| Q; P=Q |] ==> P"

  by (erule iffD2)

lemmas iffD1 = sym [THEN iffD2, standard]

lemmas rev_iffD1 = sym [THEN [2] rev_iffD2, standard]

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])

subsection {*True*}

lemma TrueI: "True"

  by (unfold True_def) (rule refl)

lemma eqTrueI: "P ==> P=True"

  by (iprover intro: iffI TrueI)

lemma eqTrueE: "P=True ==> P"

apply (erule iffD2)

apply (rule TrueI)

done

subsection {*Universal quantifier*}

lemma allI: assumes p: "!!x::'a. P(x)" shows "ALL x. P(x)"

apply (unfold All_def)

apply (iprover intro: ext eqTrueI p)

done

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])

subsection {*False*}

(*Depends upon 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])

subsection {*Negation*}

lemma notI:

  assumes p: "P ==> False"

  shows "~P"

apply (unfold not_def)

apply (iprover intro: impI p)

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

(* Alternative ~ introduction rule: [| P ==> ~ Pa; P ==> Pa |] ==> ~ P *)

lemmas notI2 = notE [THEN notI, standard]

subsection {*Implication*}

lemma impE:

  assumes "P-->Q" "P" "Q ==> R"

  shows "R"

by (iprover intro: prems 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"

apply (erule contrapos_nn)

apply (erule sym)

done

(*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

subsection {*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

subsection {*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 prems: "P" "P ==> Q" shows "P & Q"

by (iprover intro: conjI prems)

subsection {*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])

subsection {*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)

subsection {*Unique existence*}

lemma ex1I:

  assumes prems: "P a" "!!x. P(x) ==> x=a"

  shows "EX! x. P(x)"

by (unfold Ex1_def, iprover intro: prems 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

subsection {*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: prems 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: prems theI)

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

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

lemma disjCI:

  assumes "~Q ==> P" shows "P|Q"

apply (rule classical)

apply (iprover intro: prems 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_thm:

  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: prems exI allI notI notE [of "\<exists>x. P x"])

done

subsection {* Theory and package setup *}

ML

{*

val eq_reflection = thm "eq_reflection"

val refl = thm "refl"

val subst = thm "subst"

val ext = thm "ext"

val impI = thm "impI"

val mp = thm "mp"

val True_def = thm "True_def"

val All_def = thm "All_def"

val Ex_def = thm "Ex_def"

val False_def = thm "False_def"

val not_def = thm "not_def"

val and_def = thm "and_def"

val or_def = thm "or_def"

val Ex1_def = thm "Ex1_def"

val iff = thm "iff"

val True_or_False = thm "True_or_False"

val Let_def = thm "Let_def"

val if_def = thm "if_def"

val sym = thm "sym"

val ssubst = thm "ssubst"

val trans = thm "trans"

val def_imp_eq = thm "def_imp_eq"

val box_equals = thm "box_equals"

val fun_cong = thm "fun_cong"

val arg_cong = thm "arg_cong"

val cong = thm "cong"

val iffI = thm "iffI"

val iffD2 = thm "iffD2"

val rev_iffD2 = thm "rev_iffD2"

val iffD1 = thm "iffD1"

val rev_iffD1 = thm "rev_iffD1"

val iffE = thm "iffE"

val TrueI = thm "TrueI"

val eqTrueI = thm "eqTrueI"

val eqTrueE = thm "eqTrueE"

val allI = thm "allI"

val spec = thm "spec"

val allE = thm "allE"

val all_dupE = thm "all_dupE"

val FalseE = thm "FalseE"

val False_neq_True = thm "False_neq_True"

val notI = thm "notI"

val False_not_True = thm "False_not_True"

val True_not_False = thm "True_not_False"

val notE = thm "notE"

val notI2 = thm "notI2"

val impE = thm "impE"

val rev_mp = thm "rev_mp"

val contrapos_nn = thm "contrapos_nn"

val contrapos_pn = thm "contrapos_pn"

val not_sym = thm "not_sym"

val rev_contrapos = thm "rev_contrapos"

val exI = thm "exI"

val exE = thm "exE"

val conjI = thm "conjI"

val conjunct1 = thm "conjunct1"

val conjunct2 = thm "conjunct2"

val conjE = thm "conjE"

val context_conjI = thm "context_conjI"

val disjI1 = thm "disjI1"

val disjI2 = thm "disjI2"

val disjE = thm "disjE"

val classical = thm "classical"

val ccontr = thm "ccontr"

val rev_notE = thm "rev_notE"

val notnotD = thm "notnotD"

val contrapos_pp = thm "contrapos_pp"

val ex1I = thm "ex1I"

val ex_ex1I = thm "ex_ex1I"

val ex1E = thm "ex1E"

val ex1_implies_ex = thm "ex1_implies_ex"

val the_equality = thm "the_equality"

val theI = thm "theI"

val theI' = thm "theI'"

val theI2 = thm "theI2"

val the1_equality = thm "the1_equality"

val the_sym_eq_trivial = thm "the_sym_eq_trivial"

val disjCI = thm "disjCI"

val excluded_middle = thm "excluded_middle"

val case_split_thm = thm "case_split_thm"

val impCE = thm "impCE"

val impCE = thm "impCE"

val iffCE = thm "iffCE"

val exCI = thm "exCI"

(* 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 = List.filter (fn t => not (wrong_prem (HOLogic.dest_Trueprop (hd (Thm.prems_of t)))))

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

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

(*Obsolete form of disjunctive case analysis*)

fun excluded_middle_tac sP =

    res_inst_tac [("Q",sP)] (excluded_middle RS disjE)

fun case_tac a = res_inst_tac [("P",a)] case_split_thm

*}

theorems case_split = case_split_thm [case_names True False]

ML {*

structure ProjectRule = ProjectRuleFun

(struct

  val conjunct1 = thm "conjunct1";

  val conjunct2 = thm "conjunct2";

  val mp = thm "mp";

end)

*}

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

lemmas [Pure.elim!] = disjE iffE FalseE conjE exE

  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

subsubsection {* Atomizing meta-level connectives *}

lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"

proof

  assume "!!x. P x"

  show "ALL x. P x" by (rule allI)

next

  assume "ALL x. P x"

  thus "!!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

  thus 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

  thus False by (rule notE)

qed

lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"

proof

  assume "x == y"

  show "x = y" by (unfold prems) (rule refl)

next

  assume "x = y"

  thus "x == y" by (rule eq_reflection)

qed

lemma atomize_conj [atomize]:

  includes meta_conjunction_syntax

  shows "(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 {* Classical Reasoner setup *}

use "cladata.ML"

setup hypsubst_setup

setup {* ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac) *}

setup Classical.setup

setup ResAtpSet.setup

setup clasetup

declare ex_ex1I [rule del, intro! 2]

  and ex1I [intro]

lemmas [intro?] = ext

  and [elim?] = ex1_implies_ex

use "blastdata.ML"

setup Blast.setup

subsubsection {* Simplifier setup *}

lemma meta_eq_to_obj_eq: "x == y ==> x = y"

proof -

  assume r: "x == y"

  show "x = y" by (unfold r) (rule refl)

qed

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"

    "(~True) = False"  "(~False) = True"

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

    "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"

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

  by iprover

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)"