src/HOL/HOL.thy
author skalberg
Thu, 03 Mar 2005 12:43:01 +0100
changeset 15570 8d8c70b41bab
parent 15524 2ef571f80a55
child 15655 157f3988f775
permissions -rw-r--r--
Move towards standard functions.
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(*  Title:      HOL/HOL.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
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*)
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header {* The basis of Higher-Order Logic *}
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theory HOL
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imports CPure
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files ("cladata.ML") ("blastdata.ML") ("simpdata.ML") ("eqrule_HOL_data.ML")
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      ("~~/src/Provers/eqsubst.ML")
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begin
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subsection {* Primitive logic *}
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subsubsection {* Core syntax *}
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classes type
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defaultsort type
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global
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typedecl bool
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arities
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  bool :: type
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  fun :: (type, type) type
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judgment
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  Trueprop      :: "bool => prop"                   ("(_)" 5)
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consts
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  Not           :: "bool => bool"                   ("~ _" [40] 40)
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  True          :: bool
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  False         :: bool
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  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
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  arbitrary     :: 'a
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  The           :: "('a => bool) => 'a"
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  All           :: "('a => bool) => bool"           (binder "ALL " 10)
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  Ex            :: "('a => bool) => bool"           (binder "EX " 10)
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  Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
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  Let           :: "['a, 'a => 'b] => 'b"
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  "="           :: "['a, 'a] => bool"               (infixl 50)
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  &             :: "[bool, bool] => bool"           (infixr 35)
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  "|"           :: "[bool, bool] => bool"           (infixr 30)
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  -->           :: "[bool, bool] => bool"           (infixr 25)
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local
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subsubsection {* Additional concrete syntax *}
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nonterminals
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  letbinds  letbind
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  case_syn  cases_syn
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syntax
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  "_not_equal"  :: "['a, 'a] => bool"                    (infixl "~=" 50)
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  "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
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  "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
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  ""            :: "letbind => letbinds"                 ("_")
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  "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
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  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
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  "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
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  ""            :: "case_syn => cases_syn"               ("_")
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  "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
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translations
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  "x ~= y"                == "~ (x = y)"
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  "THE x. P"              == "The (%x. P)"
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  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
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  "let x = a in e"        == "Let a (%x. e)"
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print_translation {*
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(* To avoid eta-contraction of body: *)
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[("The", fn [Abs abs] =>
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     let val (x,t) = atomic_abs_tr' abs
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     in Syntax.const "_The" $ x $ t end)]
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*}
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syntax (output)
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  "="           :: "['a, 'a] => bool"                    (infix 50)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "~=" 50)
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syntax (xsymbols)
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  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
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  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
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  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
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  "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
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(*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \<orelse> _")*)
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syntax (xsymbols output)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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syntax (HTML output)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
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  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
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  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
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syntax (HOL)
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  "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
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subsubsection {* Axioms and basic definitions *}
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axioms
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  eq_reflection:  "(x=y) ==> (x==y)"
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  refl:           "t = (t::'a)"
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  ext:            "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
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    -- {*Extensionality is built into the meta-logic, and this rule expresses
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         a related property.  It is an eta-expanded version of the traditional
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         rule, and similar to the ABS rule of HOL*}
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  the_eq_trivial: "(THE x. x = a) = (a::'a)"
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  impI:           "(P ==> Q) ==> P-->Q"
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  mp:             "[| P-->Q;  P |] ==> Q"
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text{*Thanks to Stephan Merz*}
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theorem subst:
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  assumes eq: "s = t" and p: "P(s)"
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  shows "P(t::'a)"
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proof -
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  from eq have meta: "s \<equiv> t"
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    by (rule eq_reflection)
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  from p show ?thesis
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    by (unfold meta)
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qed
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defs
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  True_def:     "True      == ((%x::bool. x) = (%x. x))"
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  All_def:      "All(P)    == (P = (%x. True))"
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  Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
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  False_def:    "False     == (!P. P)"
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  not_def:      "~ P       == P-->False"
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  and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
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  or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
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  Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
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axioms
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  iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
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  True_or_False:  "(P=True) | (P=False)"
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defs
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  Let_def:      "Let s f == f(s)"
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  if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
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finalconsts
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  "op ="
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  "op -->"
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  The
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  arbitrary
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subsubsection {* Generic algebraic operations *}
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axclass zero < type
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axclass one < type
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axclass plus < type
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axclass minus < type
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axclass times < type
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axclass inverse < type
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global
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consts
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  "0"           :: "'a::zero"                       ("0")
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  "1"           :: "'a::one"                        ("1")
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  "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
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  -             :: "['a::minus, 'a] => 'a"          (infixl 65)
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  uminus        :: "['a::minus] => 'a"              ("- _" [81] 80)
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  *             :: "['a::times, 'a] => 'a"          (infixl 70)
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syntax
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  "_index1"  :: index    ("\<^sub>1")
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translations
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  (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
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local
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typed_print_translation {*
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  let
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    fun tr' c = (c, fn show_sorts => fn T => fn ts =>
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      if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
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      else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
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  in [tr' "0", tr' "1"] end;
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*} -- {* show types that are presumably too general *}
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consts
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  abs           :: "'a::minus => 'a"
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  inverse       :: "'a::inverse => 'a"
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  divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
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syntax (xsymbols)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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syntax (HTML output)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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subsection {*Equality*}
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lemma sym: "s=t ==> t=s"
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apply (erule subst)
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apply (rule refl)
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done
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(*calling "standard" reduces maxidx to 0*)
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lemmas ssubst = sym [THEN subst, standard]
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lemma trans: "[| r=s; s=t |] ==> r=t"
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apply (erule subst , assumption)
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done
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lemma def_imp_eq:  assumes meq: "A == B" shows "A = B"
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apply (unfold meq)
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apply (rule refl)
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done
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(*Useful with eresolve_tac for proving equalties from known equalities.
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        a = b
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        |   |
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        c = d   *)
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lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
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apply (rule trans)
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apply (rule trans)
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apply (rule sym)
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apply assumption+
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done
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text {* For calculational reasoning: *}
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lemma forw_subst: "a = b ==> P b ==> P a"
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  by (rule ssubst)
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lemma back_subst: "P a ==> a = b ==> P b"
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  by (rule subst)
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subsection {*Congruence rules for application*}
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(*similar to AP_THM in Gordon's HOL*)
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lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
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apply (erule subst)
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apply (rule refl)
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done
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(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
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lemma arg_cong: "x=y ==> f(x)=f(y)"
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apply (erule subst)
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apply (rule refl)
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done
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lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
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apply (erule subst)+
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apply (rule refl)
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done
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subsection {*Equality of booleans -- iff*}
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lemma iffI: assumes prems: "P ==> Q" "Q ==> P" shows "P=Q"
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apply (rules intro: iff [THEN mp, THEN mp] impI prems)
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done
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lemma iffD2: "[| P=Q; Q |] ==> P"
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apply (erule ssubst)
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apply assumption
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done
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lemma rev_iffD2: "[| Q; P=Q |] ==> P"
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apply (erule iffD2)
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apply assumption
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done
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lemmas iffD1 = sym [THEN iffD2, standard]
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lemmas rev_iffD1 = sym [THEN [2] rev_iffD2, standard]
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lemma iffE:
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  assumes major: "P=Q"
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      and minor: "[| P --> Q; Q --> P |] ==> R"
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  shows "R"
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by (rules intro: minor impI major [THEN iffD2] major [THEN iffD1])
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subsection {*True*}
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lemma TrueI: "True"
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apply (unfold True_def)
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apply (rule refl)
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done
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lemma eqTrueI: "P ==> P=True"
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by (rules intro: iffI TrueI)
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lemma eqTrueE: "P=True ==> P"
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apply (erule iffD2)
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apply (rule TrueI)
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done
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subsection {*Universal quantifier*}
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diff changeset
   323
lemma allI: assumes p: "!!x::'a. P(x)" shows "ALL x. P(x)"
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parents: 15380
diff changeset
   324
apply (unfold All_def)
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parents: 15380
diff changeset
   325
apply (rules intro: ext eqTrueI p)
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parents: 15380
diff changeset
   326
done
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   327
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   328
lemma spec: "ALL x::'a. P(x) ==> P(x)"
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parents: 15380
diff changeset
   329
apply (unfold All_def)
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   330
apply (rule eqTrueE)
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   331
apply (erule fun_cong)
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   332
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   333
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   334
lemma allE:
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parents: 15380
diff changeset
   335
  assumes major: "ALL x. P(x)"
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   336
      and minor: "P(x) ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   337
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   338
by (rules intro: minor major [THEN spec])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   339
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   340
lemma all_dupE:
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   341
  assumes major: "ALL x. P(x)"
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   342
      and minor: "[| P(x); ALL x. P(x) |] ==> R"
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   343
  shows "R"
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   344
by (rules intro: minor major major [THEN spec])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   345
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   346
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   347
subsection {*False*}
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parents: 15380
diff changeset
   348
(*Depends upon spec; it is impossible to do propositional logic before quantifiers!*)
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parents: 15380
diff changeset
   349
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   350
lemma FalseE: "False ==> P"
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parents: 15380
diff changeset
   351
apply (unfold False_def)
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   352
apply (erule spec)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   353
done
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   354
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   355
lemma False_neq_True: "False=True ==> P"
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parents: 15380
diff changeset
   356
by (erule eqTrueE [THEN FalseE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   357
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   358
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   359
subsection {*Negation*}
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   360
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   361
lemma notI:
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   362
  assumes p: "P ==> False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   363
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   364
apply (unfold not_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   365
apply (rules intro: impI p)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   366
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   367
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   368
lemma False_not_True: "False ~= True"
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parents: 15380
diff changeset
   369
apply (rule notI)
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   370
apply (erule False_neq_True)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   371
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   372
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   373
lemma True_not_False: "True ~= False"
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   374
apply (rule notI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   375
apply (drule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   376
apply (erule False_neq_True)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   377
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   378
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   379
lemma notE: "[| ~P;  P |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   380
apply (unfold not_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   381
apply (erule mp [THEN FalseE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   382
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   383
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   384
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   385
(* Alternative ~ introduction rule: [| P ==> ~ Pa; P ==> Pa |] ==> ~ P *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   386
lemmas notI2 = notE [THEN notI, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   387
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   388
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   389
subsection {*Implication*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   390
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   391
lemma impE:
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   392
  assumes "P-->Q" "P" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   393
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   394
by (rules intro: prems mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   395
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   396
(* Reduces Q to P-->Q, allowing substitution in P. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   397
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   398
by (rules intro: mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   399
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   400
lemma contrapos_nn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   401
  assumes major: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   402
      and minor: "P==>Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   403
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   404
by (rules intro: notI minor major [THEN notE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   405
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   406
(*not used at all, but we already have the other 3 combinations *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   407
lemma contrapos_pn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   408
  assumes major: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   409
      and minor: "P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   410
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   411
by (rules intro: notI minor major notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   412
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   413
lemma not_sym: "t ~= s ==> s ~= t"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   414
apply (erule contrapos_nn)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   415
apply (erule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   416
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   417
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   418
(*still used in HOLCF*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   419
lemma rev_contrapos:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   420
  assumes pq: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   421
      and nq: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   422
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   423
apply (rule nq [THEN contrapos_nn])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   424
apply (erule pq)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   425
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   426
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   427
subsection {*Existential quantifier*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   428
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   429
lemma exI: "P x ==> EX x::'a. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   430
apply (unfold Ex_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   431
apply (rules intro: allI allE impI mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   432
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   433
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   434
lemma exE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   435
  assumes major: "EX x::'a. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   436
      and minor: "!!x. P(x) ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   437
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   438
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   439
apply (rules intro: impI [THEN allI] minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   440
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   441
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   442
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   443
subsection {*Conjunction*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   444
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   445
lemma conjI: "[| P; Q |] ==> P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   446
apply (unfold and_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   447
apply (rules intro: impI [THEN allI] mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   448
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   449
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   450
lemma conjunct1: "[| P & Q |] ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   451
apply (unfold and_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   452
apply (rules intro: impI dest: spec mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   453
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   454
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   455
lemma conjunct2: "[| P & Q |] ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   456
apply (unfold and_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   457
apply (rules intro: impI dest: spec mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   458
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   459
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   460
lemma conjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   461
  assumes major: "P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   462
      and minor: "[| P; Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   463
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   464
apply (rule minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   465
apply (rule major [THEN conjunct1])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   466
apply (rule major [THEN conjunct2])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   467
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   468
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   469
lemma context_conjI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   470
  assumes prems: "P" "P ==> Q" shows "P & Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   471
by (rules intro: conjI prems)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   472
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   473
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   474
subsection {*Disjunction*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   475
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   476
lemma disjI1: "P ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   477
apply (unfold or_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   478
apply (rules intro: allI impI mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   479
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   480
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   481
lemma disjI2: "Q ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   482
apply (unfold or_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   483
apply (rules intro: allI impI mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   484
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   485
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   486
lemma disjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   487
  assumes major: "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   488
      and minorP: "P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   489
      and minorQ: "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   490
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   491
by (rules intro: minorP minorQ impI
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   492
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   493
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   494
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   495
subsection {*Classical logic*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   496
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   497
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   498
lemma classical:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   499
  assumes prem: "~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   500
  shows "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   501
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   502
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   503
apply (rule notI [THEN prem, THEN eqTrueI])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   504
apply (erule subst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   505
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   506
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   507
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   508
lemmas ccontr = FalseE [THEN classical, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   509
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   510
(*notE with premises exchanged; it discharges ~R so that it can be used to
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   511
  make elimination rules*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   512
lemma rev_notE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   513
  assumes premp: "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   514
      and premnot: "~R ==> ~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   515
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   516
apply (rule ccontr)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   517
apply (erule notE [OF premnot premp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   518
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   519
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   520
(*Double negation law*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   521
lemma notnotD: "~~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   522
apply (rule classical)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   523
apply (erule notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   524
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   525
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   526
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   527
lemma contrapos_pp:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   528
  assumes p1: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   529
      and p2: "~P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   530
  shows "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   531
by (rules intro: classical p1 p2 notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   532
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   533
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   534
subsection {*Unique existence*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   535
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   536
lemma ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   537
  assumes prems: "P a" "!!x. P(x) ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   538
  shows "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   539
by (unfold Ex1_def, rules intro: prems exI conjI allI impI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   540
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   541
text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   542
lemma ex_ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   543
  assumes ex_prem: "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   544
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   545
  shows "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   546
by (rules intro: ex_prem [THEN exE] ex1I eq)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   547
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   548
lemma ex1E:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   549
  assumes major: "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   550
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   551
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   552
apply (rule major [unfolded Ex1_def, THEN exE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   553
apply (erule conjE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   554
apply (rules intro: minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   555
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   556
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   557
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   558
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   559
apply (rule exI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   560
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   561
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   562
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   563
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   564
subsection {*THE: definite description operator*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   565
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   566
lemma the_equality:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   567
  assumes prema: "P a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   568
      and premx: "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   569
  shows "(THE x. P x) = a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   570
apply (rule trans [OF _ the_eq_trivial])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   571
apply (rule_tac f = "The" in arg_cong)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   572
apply (rule ext)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   573
apply (rule iffI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   574
 apply (erule premx)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   575
apply (erule ssubst, rule prema)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   576
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   577
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   578
lemma theI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   579
  assumes "P a" and "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   580
  shows "P (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   581
by (rules intro: prems the_equality [THEN ssubst])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   582
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   583
lemma theI': "EX! x. P x ==> P (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   584
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   585
apply (erule theI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   586
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   587
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   588
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   589
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   590
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   591
(*Easier to apply than theI: only one occurrence of P*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   592
lemma theI2:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   593
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   594
  shows "Q (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   595
by (rules intro: prems theI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   596
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   597
lemma the1_equality: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   598
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   599
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   600
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   601
apply (erule all_dupE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   602
apply (drule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   603
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   604
apply (erule ssubst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   605
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   606
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   607
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   608
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   609
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   610
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   611
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   612
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   613
apply (erule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   614
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   615
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   616
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   617
subsection {*Classical intro rules for disjunction and existential quantifiers*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   618
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   619
lemma disjCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   620
  assumes "~Q ==> P" shows "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   621
apply (rule classical)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   622
apply (rules intro: prems disjI1 disjI2 notI elim: notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   623
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   624
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   625
lemma excluded_middle: "~P | P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   626
by (rules intro: disjCI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   627
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   628
text{*case distinction as a natural deduction rule. Note that @{term "~P"}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   629
   is the second case, not the first.*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   630
lemma case_split_thm:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   631
  assumes prem1: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   632
      and prem2: "~P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   633
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   634
apply (rule excluded_middle [THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   635
apply (erule prem2)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   636
apply (erule prem1)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   637
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   638
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   639
(*Classical implies (-->) elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   640
lemma impCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   641
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   642
      and minor: "~P ==> R" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   643
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   644
apply (rule excluded_middle [of P, THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   645
apply (rules intro: minor major [THEN mp])+
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   646
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   647
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   648
(*This version of --> elimination works on Q before P.  It works best for
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   649
  those cases in which P holds "almost everywhere".  Can't install as
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   650
  default: would break old proofs.*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   651
lemma impCE':
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   652
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   653
      and minor: "Q ==> R" "~P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   654
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   655
apply (rule excluded_middle [of P, THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   656
apply (rules intro: minor major [THEN mp])+
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   657
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   658
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   659
(*Classical <-> elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   660
lemma iffCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   661
  assumes major: "P=Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   662
      and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   663
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   664
apply (rule major [THEN iffE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   665
apply (rules intro: minor elim: impCE notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   666
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   667
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   668
lemma exCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   669
  assumes "ALL x. ~P(x) ==> P(a)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   670
  shows "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   671
apply (rule ccontr)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   672
apply (rules intro: prems exI allI notI notE [of "\<exists>x. P x"])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   673
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   674
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   675
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   676
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   677
subsection {* Theory and package setup *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   678
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   679
ML
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   680
{*
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   681
val plusI = thm "plusI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   682
val minusI = thm "minusI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   683
val timesI = thm "timesI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   684
val eq_reflection = thm "eq_reflection"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   685
val refl = thm "refl"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   686
val subst = thm "subst"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   687
val ext = thm "ext"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   688
val impI = thm "impI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   689
val mp = thm "mp"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   690
val True_def = thm "True_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   691
val All_def = thm "All_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   692
val Ex_def = thm "Ex_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   693
val False_def = thm "False_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   694
val not_def = thm "not_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   695
val and_def = thm "and_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   696
val or_def = thm "or_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   697
val Ex1_def = thm "Ex1_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   698
val iff = thm "iff"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   699
val True_or_False = thm "True_or_False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   700
val Let_def = thm "Let_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   701
val if_def = thm "if_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   702
val sym = thm "sym"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   703
val ssubst = thm "ssubst"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   704
val trans = thm "trans"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   705
val def_imp_eq = thm "def_imp_eq"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   706
val box_equals = thm "box_equals"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   707
val fun_cong = thm "fun_cong"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   708
val arg_cong = thm "arg_cong"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   709
val cong = thm "cong"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   710
val iffI = thm "iffI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   711
val iffD2 = thm "iffD2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   712
val rev_iffD2 = thm "rev_iffD2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   713
val iffD1 = thm "iffD1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   714
val rev_iffD1 = thm "rev_iffD1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   715
val iffE = thm "iffE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   716
val TrueI = thm "TrueI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   717
val eqTrueI = thm "eqTrueI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   718
val eqTrueE = thm "eqTrueE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   719
val allI = thm "allI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   720
val spec = thm "spec"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   721
val allE = thm "allE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   722
val all_dupE = thm "all_dupE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   723
val FalseE = thm "FalseE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   724
val False_neq_True = thm "False_neq_True"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   725
val notI = thm "notI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   726
val False_not_True = thm "False_not_True"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   727
val True_not_False = thm "True_not_False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   728
val notE = thm "notE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   729
val notI2 = thm "notI2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   730
val impE = thm "impE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   731
val rev_mp = thm "rev_mp"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   732
val contrapos_nn = thm "contrapos_nn"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   733
val contrapos_pn = thm "contrapos_pn"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   734
val not_sym = thm "not_sym"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   735
val rev_contrapos = thm "rev_contrapos"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   736
val exI = thm "exI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   737
val exE = thm "exE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   738
val conjI = thm "conjI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   739
val conjunct1 = thm "conjunct1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   740
val conjunct2 = thm "conjunct2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   741
val conjE = thm "conjE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   742
val context_conjI = thm "context_conjI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   743
val disjI1 = thm "disjI1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   744
val disjI2 = thm "disjI2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   745
val disjE = thm "disjE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   746
val classical = thm "classical"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   747
val ccontr = thm "ccontr"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   748
val rev_notE = thm "rev_notE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   749
val notnotD = thm "notnotD"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   750
val contrapos_pp = thm "contrapos_pp"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   751
val ex1I = thm "ex1I"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   752
val ex_ex1I = thm "ex_ex1I"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   753
val ex1E = thm "ex1E"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   754
val ex1_implies_ex = thm "ex1_implies_ex"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   755
val the_equality = thm "the_equality"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   756
val theI = thm "theI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   757
val theI' = thm "theI'"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   758
val theI2 = thm "theI2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   759
val the1_equality = thm "the1_equality"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   760
val the_sym_eq_trivial = thm "the_sym_eq_trivial"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   761
val disjCI = thm "disjCI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   762
val excluded_middle = thm "excluded_middle"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   763
val case_split_thm = thm "case_split_thm"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   764
val impCE = thm "impCE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   765
val impCE = thm "impCE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   766
val iffCE = thm "iffCE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   767
val exCI = thm "exCI"
4868
843a9f5b3c3d nonterminals;
wenzelm
parents: 4793
diff changeset
   768
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   769
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   770
local
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   771
  fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   772
  |   wrong_prem (Bound _) = true
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   773
  |   wrong_prem _ = false
15570
8d8c70b41bab Move towards standard functions.
skalberg
parents: 15524
diff changeset
   774
  val filter_right = List.filter (fn t => not (wrong_prem (HOLogic.dest_Trueprop (hd (Thm.prems_of t)))))
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   775
in
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   776
  fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   777
  fun smp_tac j = EVERY'[dresolve_tac (smp j), atac]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   778
end
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   779
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   780
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   781
fun strip_tac i = REPEAT(resolve_tac [impI,allI] i)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   782
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   783
(*Obsolete form of disjunctive case analysis*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   784
fun excluded_middle_tac sP =
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   785
    res_inst_tac [("Q",sP)] (excluded_middle RS disjE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   786
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   787
fun case_tac a = res_inst_tac [("P",a)] case_split_thm
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   788
*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   789
11687
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
   790
theorems case_split = case_split_thm [case_names True False]
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   791
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   792
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   793
subsubsection {* Intuitionistic Reasoning *}
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   794
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   795
lemma impE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   796
  assumes 1: "P --> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   797
    and 2: "Q ==> R"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   798
    and 3: "P --> Q ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   799
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   800
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   801
  from 3 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   802
  with 1 have Q by (rule impE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   803
  with 2 show R .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   804
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   805
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   806
lemma allE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   807
  assumes 1: "ALL x. P x"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   808
    and 2: "P x ==> ALL x. P x ==> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   809
  shows Q
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   810
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   811
  from 1 have "P x" by (rule spec)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   812
  from this and 1 show Q by (rule 2)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   813
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   814
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   815
lemma notE':
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   816
  assumes 1: "~ P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   817
    and 2: "~ P ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   818
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   819
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   820
  from 2 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   821
  with 1 show R by (rule notE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   822
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   823
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   824
lemmas [CPure.elim!] = disjE iffE FalseE conjE exE
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   825
  and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   826
  and [CPure.elim 2] = allE notE' impE'
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   827
  and [CPure.intro] = exI disjI2 disjI1
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   828
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   829
lemmas [trans] = trans
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   830
  and [sym] = sym not_sym
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   831
  and [CPure.elim?] = iffD1 iffD2 impE
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   832
11438
3d9222b80989 declare trans [trans] (*overridden in theory Calculation*);
wenzelm
parents: 11432
diff changeset
   833
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   834
subsubsection {* Atomizing meta-level connectives *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   835
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   836
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   837
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   838
  assume "!!x. P x"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   839
  show "ALL x. P x" by (rule allI)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   840
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   841
  assume "ALL x. P x"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   842
  thus "!!x. P x" by (rule allE)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   843
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   844
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   845
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   846
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   847
  assume r: "A ==> B"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   848
  show "A --> B" by (rule impI) (rule r)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   849
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   850
  assume "A --> B" and A
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   851
  thus B by (rule mp)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   852
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   853
14749
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   854
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   855
proof
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   856
  assume r: "A ==> False"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   857
  show "~A" by (rule notI) (rule r)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   858
next
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   859
  assume "~A" and A
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   860
  thus False by (rule notE)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   861
qed
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   862
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   863
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   864
proof
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   865
  assume "x == y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   866
  show "x = y" by (unfold prems) (rule refl)
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   867
next
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   868
  assume "x = y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   869
  thus "x == y" by (rule eq_reflection)
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   870
qed
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   871
12023
wenzelm
parents: 12003
diff changeset
   872
lemma atomize_conj [atomize]:
wenzelm
parents: 12003
diff changeset
   873
  "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   874
proof
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   875
  assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   876
  show "A & B" by (rule conjI)
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   877
next
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   878
  fix C
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   879
  assume "A & B"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   880
  assume "A ==> B ==> PROP C"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   881
  thus "PROP C"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   882
  proof this
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   883
    show A by (rule conjunct1)
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   884
    show B by (rule conjunct2)
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   885
  qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   886
qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   887
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   888
lemmas [symmetric, rulify] = atomize_all atomize_imp
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   889
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   890
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   891
subsubsection {* Classical Reasoner setup *}
9529
d9434a9277a4 lemmas atomize = all_eq imp_eq;
wenzelm
parents: 9488
diff changeset
   892
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   893
use "cladata.ML"
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   894
setup hypsubst_setup
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   895
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   896
ML_setup {*
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   897
  Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   898
*}
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   899
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   900
setup Classical.setup
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   901
setup clasetup
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   902
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   903
lemmas [intro?] = ext
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   904
  and [elim?] = ex1_implies_ex
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   905
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   906
use "blastdata.ML"
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   907
setup Blast.setup
4868
843a9f5b3c3d nonterminals;
wenzelm
parents: 4793
diff changeset
   908
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   909
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
   910
subsection {* Simplifier setup *}
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   911
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   912
lemma meta_eq_to_obj_eq: "x == y ==> x = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   913
proof -
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   914
  assume r: "x == y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   915
  show "x = y" by (unfold r) (rule refl)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   916
qed
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   917
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   918
lemma eta_contract_eq: "(%s. f s) = f" ..
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   919
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   920
lemma simp_thms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   921
  shows not_not: "(~ ~ P) = P"
15354
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
   922
  and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   923
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   924
    "(P ~= Q) = (P = (~Q))"
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   925
    "(P | ~P) = True"    "(~P | P) = True"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   926
    "(x = x) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   927
    "(~True) = False"  "(~False) = True"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   928
    "(~P) ~= P"  "P ~= (~P)"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   929
    "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   930
    "(True --> P) = P"  "(False --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   931
    "(P --> True) = True"  "(P --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   932
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   933
    "(P & True) = P"  "(True & P) = P"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   934
    "(P & False) = False"  "(False & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   935
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   936
    "(P & ~P) = False"    "(~P & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   937
    "(P | True) = True"  "(True | P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   938
    "(P | False) = P"  "(False | P) = P"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   939
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   940
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   941
    -- {* needed for the one-point-rule quantifier simplification procs *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   942
    -- {* essential for termination!! *} and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   943
    "!!P. (EX x. x=t & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   944
    "!!P. (EX x. t=x & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   945
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   946
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   947
  by (blast, blast, blast, blast, blast, rules+)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13412
diff changeset
   948
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   949
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   950
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   951
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   952
lemma ex_simps:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   953
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   954
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   955
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   956
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   957
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   958
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   959
  -- {* Miniscoping: pushing in existential quantifiers. *}
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   960
  by (rules | blast)+
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   961
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   962
lemma all_simps:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   963
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   964
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   965
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   966
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   967
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   968
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   969
  -- {* Miniscoping: pushing in universal quantifiers. *}
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   970
  by (rules | blast)+
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   971
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   972
lemma disj_absorb: "(A | A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   973
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   974
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   975
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   976
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   977
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   978
lemma conj_absorb: "(A & A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   979
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   980
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   981
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   982
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   983
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   984
lemma eq_ac:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   985
  shows eq_commute: "(a=b) = (b=a)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   986
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   987
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   988
lemma neq_commute: "(a~=b) = (b~=a)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   989
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   990
lemma conj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   991
  shows conj_commute: "(P&Q) = (Q&P)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   992
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   993
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   994
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   995
lemma disj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   996
  shows disj_commute: "(P|Q) = (Q|P)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   997
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   998
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   999
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1000
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1001
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1002
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1003
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1004
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1005
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1006
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1007
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1008
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1009
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1010
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1011
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1012
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1013
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1014
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1015
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1016
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1017
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1018
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1019
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1020
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1021
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1022
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1023
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1024
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1025
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1026
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1027
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1028
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1029
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1030
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1031
  -- {* cases boil down to the same thing. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1032
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1033
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1034
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1035
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1036
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1037
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1038
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1039
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1040
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1041
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1042
text {*
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1043
  \medskip The @{text "&"} congruence rule: not included by default!
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1044
  May slow rewrite proofs down by as much as 50\% *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1045
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1046
lemma conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1047
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1048
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1049
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1050
lemma rev_conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1051
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1052
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1053
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1054
text {* The @{text "|"} congruence rule: not included by default! *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1055
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1056
lemma disj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1057
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1058
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1059
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1060
lemma eq_sym_conv: "(x = y) = (y = x)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1061
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1062
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1063
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1064
text {* \medskip if-then-else rules *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1065
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1066
lemma if_True: "(if True then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1067
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1068
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1069
lemma if_False: "(if False then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1070
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1071
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1072
lemma if_P: "P ==> (if P then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1073
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1074
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1075
lemma if_not_P: "~P ==> (if P then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1076
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1077
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1078
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1079
  apply (rule case_split [of Q])
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1080
   apply (simplesubst if_P)
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1081
    prefer 3 apply (simplesubst if_not_P, blast+)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1082
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1083
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1084
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1085
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1086
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1087
lemmas if_splits = split_if split_if_asm
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1088
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1089
lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1090
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1091
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1092
lemma if_cancel: "(if c then x else x) = x"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1093
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1094
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1095
lemma if_eq_cancel: "(if x = y then y else x) = x"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1096
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1097
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1098
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1099
  -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1100
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1101
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1102
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1103
  -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1104
  apply (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1105
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1106
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1107
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1108
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1109
15423
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1110
text {* \medskip let rules for simproc *}
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1111
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1112
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1113
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1114
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1115
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1116
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1117
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1118
subsubsection {* Actual Installation of the Simplifier *}
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1119
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1120
use "simpdata.ML"
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1121
setup Simplifier.setup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1122
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1123
setup Splitter.setup setup Clasimp.setup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1124
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1125
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1126
subsubsection {* Lucas Dixon's eqstep tactic *}
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1127
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1128
use "~~/src/Provers/eqsubst.ML";
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1129
use "eqrule_HOL_data.ML";
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1130
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1131
setup EQSubstTac.setup
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1132
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1133
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1134
subsection {* Other simple lemmas *}
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1135
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1136
declare disj_absorb [simp] conj_absorb [simp]
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1137
13723
c7d104550205 *** empty log message ***
nipkow
parents: 13638
diff changeset
  1138
lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
c7d104550205 *** empty log message ***
nipkow
parents: 13638
diff changeset
  1139
by blast+
c7d104550205 *** empty log message ***
nipkow
parents: 13638
diff changeset
  1140
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1141
13638
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1142
theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1143
  apply (rule iffI)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1144
  apply (rule_tac a = "%x. THE y. P x y" in ex1I)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1145
  apply (fast dest!: theI')
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1146
  apply (fast intro: ext the1_equality [symmetric])
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1147
  apply (erule ex1E)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1148
  apply (rule allI)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1149
  apply (rule ex1I)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1150
  apply (erule spec)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1151
  apply (erule_tac x = "%z. if z = x then y else f z" in allE)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1152
  apply (erule impE)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1153
  apply (rule allI)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1154
  apply (rule_tac P = "xa = x" in case_split_thm)
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1155
  apply (drule_tac [3] x = x in fun_cong, simp_all)
13638
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1156
  done
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1157
13438
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1158
text{*Needs only HOL-lemmas:*}
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1159
lemma mk_left_commute:
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1160
  assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1161
          c: "\<And>x y. f x y = f y x"
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1162
  shows "f x (f y z) = f y (f x z)"
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1163
by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1164
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1165
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1166
subsection {* Generic cases and induction *}
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1167
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1168
constdefs
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1169
  induct_forall :: "('a => bool) => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1170
  "induct_forall P == \<forall>x. P x"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1171
  induct_implies :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1172
  "induct_implies A B == A --> B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1173
  induct_equal :: "'a => 'a => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1174
  "induct_equal x y == x = y"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1175
  induct_conj :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1176
  "induct_conj A B == A & B"
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1177
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1178
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1179
  by (simp only: atomize_all induct_forall_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1180
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1181
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1182
  by (simp only: atomize_imp induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1183
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1184
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1185
  by (simp only: atomize_eq induct_equal_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1186
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1187
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1188
    induct_conj (induct_forall A) (induct_forall B)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1189
  by (unfold induct_forall_def induct_conj_def) rules
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1190
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1191
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1192
    induct_conj (induct_implies C A) (induct_implies C B)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1193
  by (unfold induct_implies_def induct_conj_def) rules
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1194
13598
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1195
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1196
proof
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1197
  assume r: "induct_conj A B ==> PROP C" and A B
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1198
  show "PROP C" by (rule r) (simp! add: induct_conj_def)
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1199
next
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1200
  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1201
  show "PROP C" by (rule r) (simp! add: induct_conj_def)+
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1202
qed
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1203
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1204
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1205
  by (simp add: induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1206
12161
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
  1207
lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
  1208
lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
  1209
lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1210
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1211
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1212
hide const induct_forall induct_implies induct_equal induct_conj
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1213
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1214
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1215
text {* Method setup. *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1216
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1217
ML {*
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1218
  structure InductMethod = InductMethodFun
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1219
  (struct
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1220
    val dest_concls = HOLogic.dest_concls
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1221
    val cases_default = thm "case_split"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1222
    val local_impI = thm "induct_impliesI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1223
    val conjI = thm "conjI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1224
    val atomize = thms "induct_atomize"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1225
    val rulify1 = thms "induct_rulify1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1226
    val rulify2 = thms "induct_rulify2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1227
    val localize = [Thm.symmetric (thm "induct_implies_def")]
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1228
  end);
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1229
*}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1230
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1231
setup InductMethod.setup
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1232
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1233
14357
e49d5d5ae66a print translation for ALL x <= n. P x
kleing
parents: 14295
diff changeset
  1234
end
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1235