src/HOL/HOL.thy
author paulson
Wed, 15 Dec 2004 10:19:01 +0100
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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") ("antisym_setup.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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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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lemma allI: assumes p: "!!x::'a. P(x)" shows "ALL x. P(x)"
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apply (unfold All_def)
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apply (rules intro: ext eqTrueI p)
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done
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lemma spec: "ALL x::'a. P(x) ==> P(x)"
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apply (unfold All_def)
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apply (rule eqTrueE)
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apply (erule fun_cong)
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done
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1d195de59497 removal of HOL_Lemmas
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diff changeset
   325
lemma allE:
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   326
  assumes major: "ALL x. P(x)"
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   327
      and minor: "P(x) ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   328
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   329
by (rules intro: minor major [THEN spec])
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   330
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   331
lemma all_dupE:
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parents: 15380
diff changeset
   332
  assumes major: "ALL x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   333
      and minor: "[| P(x); ALL x. P(x) |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   334
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   335
by (rules intro: minor major major [THEN spec])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   336
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   337
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   338
subsection {*False*}
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parents: 15380
diff changeset
   339
(*Depends upon spec; it is impossible to do propositional logic before quantifiers!*)
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parents: 15380
diff changeset
   340
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   341
lemma FalseE: "False ==> P"
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parents: 15380
diff changeset
   342
apply (unfold False_def)
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   343
apply (erule spec)
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   344
done
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   345
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   346
lemma False_neq_True: "False=True ==> P"
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parents: 15380
diff changeset
   347
by (erule eqTrueE [THEN FalseE])
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   348
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   349
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   350
subsection {*Negation*}
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diff changeset
   351
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   352
lemma notI:
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parents: 15380
diff changeset
   353
  assumes p: "P ==> False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   354
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   355
apply (unfold not_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   356
apply (rules intro: impI p)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   357
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   358
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   359
lemma False_not_True: "False ~= True"
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parents: 15380
diff changeset
   360
apply (rule notI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   361
apply (erule False_neq_True)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   362
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   363
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   364
lemma True_not_False: "True ~= False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   365
apply (rule notI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   366
apply (drule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   367
apply (erule False_neq_True)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   368
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   369
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   370
lemma notE: "[| ~P;  P |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   371
apply (unfold not_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   372
apply (erule mp [THEN FalseE])
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   373
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   374
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   375
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   376
(* Alternative ~ introduction rule: [| P ==> ~ Pa; P ==> Pa |] ==> ~ P *)
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parents: 15380
diff changeset
   377
lemmas notI2 = notE [THEN notI, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   378
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   379
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   380
subsection {*Implication*}
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   381
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   382
lemma impE:
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parents: 15380
diff changeset
   383
  assumes "P-->Q" "P" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   384
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   385
by (rules intro: prems mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   386
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   387
(* Reduces Q to P-->Q, allowing substitution in P. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   388
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   389
by (rules intro: mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   390
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   391
lemma contrapos_nn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   392
  assumes major: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   393
      and minor: "P==>Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   394
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   395
by (rules intro: notI minor major [THEN notE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   396
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   397
(*not used at all, but we already have the other 3 combinations *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   398
lemma contrapos_pn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   399
  assumes major: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   400
      and minor: "P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   401
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   402
by (rules intro: notI minor major notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   403
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   404
lemma not_sym: "t ~= s ==> s ~= t"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   405
apply (erule contrapos_nn)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   406
apply (erule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   407
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   408
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   409
(*still used in HOLCF*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   410
lemma rev_contrapos:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   411
  assumes pq: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   412
      and nq: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   413
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   414
apply (rule nq [THEN contrapos_nn])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   415
apply (erule pq)
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
subsection {*Existential quantifier*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   419
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   420
lemma exI: "P x ==> EX x::'a. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   421
apply (unfold Ex_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   422
apply (rules intro: allI allE impI mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   423
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   424
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   425
lemma exE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   426
  assumes major: "EX x::'a. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   427
      and minor: "!!x. P(x) ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   428
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   429
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   430
apply (rules intro: impI [THEN allI] minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   431
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   432
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   433
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   434
subsection {*Conjunction*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   435
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   436
lemma conjI: "[| P; Q |] ==> P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   437
apply (unfold and_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   438
apply (rules intro: impI [THEN allI] mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   439
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   440
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   441
lemma conjunct1: "[| P & Q |] ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   442
apply (unfold and_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   443
apply (rules intro: impI dest: spec mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   444
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   445
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   446
lemma conjunct2: "[| P & Q |] ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   447
apply (unfold and_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   448
apply (rules intro: impI dest: spec mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   449
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   450
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   451
lemma conjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   452
  assumes major: "P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   453
      and minor: "[| P; Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   454
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   455
apply (rule minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   456
apply (rule major [THEN conjunct1])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   457
apply (rule major [THEN conjunct2])
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 context_conjI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   461
  assumes prems: "P" "P ==> Q" shows "P & Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   462
by (rules intro: conjI prems)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   463
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   464
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   465
subsection {*Disjunction*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   466
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   467
lemma disjI1: "P ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   468
apply (unfold or_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   469
apply (rules intro: allI impI mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   470
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   471
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   472
lemma disjI2: "Q ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   473
apply (unfold or_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   474
apply (rules intro: allI impI mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   475
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   476
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   477
lemma disjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   478
  assumes major: "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   479
      and minorP: "P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   480
      and minorQ: "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   481
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   482
by (rules intro: minorP minorQ impI
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   483
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   484
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   485
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   486
subsection {*Classical logic*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   487
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   488
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   489
lemma classical:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   490
  assumes prem: "~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   491
  shows "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   492
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   493
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   494
apply (rule notI [THEN prem, THEN eqTrueI])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   495
apply (erule subst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   496
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   497
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   498
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   499
lemmas ccontr = FalseE [THEN classical, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   500
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   501
(*notE with premises exchanged; it discharges ~R so that it can be used to
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   502
  make elimination rules*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   503
lemma rev_notE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   504
  assumes premp: "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   505
      and premnot: "~R ==> ~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   506
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   507
apply (rule ccontr)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   508
apply (erule notE [OF premnot premp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   509
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   510
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   511
(*Double negation law*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   512
lemma notnotD: "~~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   513
apply (rule classical)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   514
apply (erule notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   515
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   516
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   517
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   518
lemma contrapos_pp:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   519
  assumes p1: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   520
      and p2: "~P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   521
  shows "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   522
by (rules intro: classical p1 p2 notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   523
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   524
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   525
subsection {*Unique existence*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   526
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   527
lemma ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   528
  assumes prems: "P a" "!!x. P(x) ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   529
  shows "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   530
by (unfold Ex1_def, rules intro: prems exI conjI allI impI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   531
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   532
text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   533
lemma ex_ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   534
  assumes ex_prem: "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   535
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   536
  shows "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   537
by (rules intro: ex_prem [THEN exE] ex1I eq)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   538
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   539
lemma ex1E:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   540
  assumes major: "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   541
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   542
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   543
apply (rule major [unfolded Ex1_def, THEN exE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   544
apply (erule conjE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   545
apply (rules intro: minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   546
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   547
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   548
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   549
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   550
apply (rule exI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   551
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   552
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   553
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   554
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   555
subsection {*THE: definite description operator*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   556
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   557
lemma the_equality:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   558
  assumes prema: "P a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   559
      and premx: "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   560
  shows "(THE x. P x) = a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   561
apply (rule trans [OF _ the_eq_trivial])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   562
apply (rule_tac f = "The" in arg_cong)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   563
apply (rule ext)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   564
apply (rule iffI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   565
 apply (erule premx)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   566
apply (erule ssubst, rule prema)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   567
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   568
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   569
lemma theI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   570
  assumes "P a" and "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   571
  shows "P (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   572
by (rules intro: prems the_equality [THEN ssubst])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   573
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   574
lemma theI': "EX! x. P x ==> P (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   575
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   576
apply (erule theI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   577
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   578
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   579
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   580
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   581
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   582
(*Easier to apply than theI: only one occurrence of P*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   583
lemma theI2:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   584
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   585
  shows "Q (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   586
by (rules intro: prems theI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   587
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   588
lemma the1_equality: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   589
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   590
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   591
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   592
apply (erule all_dupE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   593
apply (drule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   594
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   595
apply (erule ssubst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   596
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   597
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   598
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   599
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   600
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   601
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   602
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   603
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   604
apply (erule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   605
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   606
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   607
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   608
subsection {*Classical intro rules for disjunction and existential quantifiers*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   609
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   610
lemma disjCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   611
  assumes "~Q ==> P" shows "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   612
apply (rule classical)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   613
apply (rules intro: prems disjI1 disjI2 notI elim: notE)
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
lemma excluded_middle: "~P | P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   617
by (rules intro: disjCI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   618
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   619
text{*case distinction as a natural deduction rule. Note that @{term "~P"}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   620
   is the second case, not the first.*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   621
lemma case_split_thm:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   622
  assumes prem1: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   623
      and prem2: "~P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   624
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   625
apply (rule excluded_middle [THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   626
apply (erule prem2)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   627
apply (erule prem1)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   628
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   629
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   630
(*Classical implies (-->) elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   631
lemma impCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   632
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   633
      and minor: "~P ==> R" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   634
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   635
apply (rule excluded_middle [of P, THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   636
apply (rules intro: minor major [THEN mp])+
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
(*This version of --> elimination works on Q before P.  It works best for
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   640
  those cases in which P holds "almost everywhere".  Can't install as
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   641
  default: would break old proofs.*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   642
lemma impCE':
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   643
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   644
      and minor: "Q ==> R" "~P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   645
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   646
apply (rule excluded_middle [of P, THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   647
apply (rules intro: minor major [THEN mp])+
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   648
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   649
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   650
(*Classical <-> elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   651
lemma iffCE:
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: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> 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 major [THEN iffE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   656
apply (rules intro: minor elim: impCE notE)
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
lemma exCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   660
  assumes "ALL x. ~P(x) ==> P(a)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   661
  shows "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   662
apply (rule ccontr)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   663
apply (rules intro: prems exI allI notI notE [of "\<exists>x. P x"])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   664
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   665
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   666
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   667
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   668
subsection {* Theory and package setup *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   669
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   670
ML
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   671
{*
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   672
val plusI = thm "plusI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   673
val minusI = thm "minusI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   674
val timesI = thm "timesI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   675
val eq_reflection = thm "eq_reflection"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   676
val refl = thm "refl"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   677
val subst = thm "subst"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   678
val ext = thm "ext"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   679
val impI = thm "impI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   680
val mp = thm "mp"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   681
val True_def = thm "True_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   682
val All_def = thm "All_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   683
val Ex_def = thm "Ex_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   684
val False_def = thm "False_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   685
val not_def = thm "not_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   686
val and_def = thm "and_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   687
val or_def = thm "or_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   688
val Ex1_def = thm "Ex1_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   689
val iff = thm "iff"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   690
val True_or_False = thm "True_or_False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   691
val Let_def = thm "Let_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   692
val if_def = thm "if_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   693
val sym = thm "sym"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   694
val ssubst = thm "ssubst"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   695
val trans = thm "trans"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   696
val def_imp_eq = thm "def_imp_eq"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   697
val box_equals = thm "box_equals"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   698
val fun_cong = thm "fun_cong"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   699
val arg_cong = thm "arg_cong"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   700
val cong = thm "cong"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   701
val iffI = thm "iffI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   702
val iffD2 = thm "iffD2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   703
val rev_iffD2 = thm "rev_iffD2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   704
val iffD1 = thm "iffD1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   705
val rev_iffD1 = thm "rev_iffD1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   706
val iffE = thm "iffE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   707
val TrueI = thm "TrueI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   708
val eqTrueI = thm "eqTrueI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   709
val eqTrueE = thm "eqTrueE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   710
val allI = thm "allI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   711
val spec = thm "spec"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   712
val allE = thm "allE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   713
val all_dupE = thm "all_dupE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   714
val FalseE = thm "FalseE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   715
val False_neq_True = thm "False_neq_True"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   716
val notI = thm "notI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   717
val False_not_True = thm "False_not_True"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   718
val True_not_False = thm "True_not_False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   719
val notE = thm "notE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   720
val notI2 = thm "notI2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   721
val impE = thm "impE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   722
val rev_mp = thm "rev_mp"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   723
val contrapos_nn = thm "contrapos_nn"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   724
val contrapos_pn = thm "contrapos_pn"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   725
val not_sym = thm "not_sym"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   726
val rev_contrapos = thm "rev_contrapos"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   727
val exI = thm "exI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   728
val exE = thm "exE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   729
val conjI = thm "conjI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   730
val conjunct1 = thm "conjunct1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   731
val conjunct2 = thm "conjunct2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   732
val conjE = thm "conjE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   733
val context_conjI = thm "context_conjI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   734
val disjI1 = thm "disjI1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   735
val disjI2 = thm "disjI2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   736
val disjE = thm "disjE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   737
val classical = thm "classical"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   738
val ccontr = thm "ccontr"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   739
val rev_notE = thm "rev_notE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   740
val notnotD = thm "notnotD"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   741
val contrapos_pp = thm "contrapos_pp"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   742
val ex1I = thm "ex1I"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   743
val ex_ex1I = thm "ex_ex1I"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   744
val ex1E = thm "ex1E"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   745
val ex1_implies_ex = thm "ex1_implies_ex"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   746
val the_equality = thm "the_equality"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   747
val theI = thm "theI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   748
val theI' = thm "theI'"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   749
val theI2 = thm "theI2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   750
val the1_equality = thm "the1_equality"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   751
val the_sym_eq_trivial = thm "the_sym_eq_trivial"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   752
val disjCI = thm "disjCI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   753
val excluded_middle = thm "excluded_middle"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   754
val case_split_thm = thm "case_split_thm"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   755
val impCE = thm "impCE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   756
val impCE = thm "impCE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   757
val iffCE = thm "iffCE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   758
val exCI = thm "exCI"
4868
843a9f5b3c3d nonterminals;
wenzelm
parents: 4793
diff changeset
   759
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   760
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   761
local
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   762
  fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   763
  |   wrong_prem (Bound _) = true
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   764
  |   wrong_prem _ = false
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   765
  val filter_right = filter (fn t => not (wrong_prem (HOLogic.dest_Trueprop (hd (Thm.prems_of t)))))
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   766
in
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   767
  fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   768
  fun smp_tac j = EVERY'[dresolve_tac (smp j), atac]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   769
end
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   770
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   771
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   772
fun strip_tac i = REPEAT(resolve_tac [impI,allI] i)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   773
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   774
(*Obsolete form of disjunctive case analysis*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   775
fun excluded_middle_tac sP =
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   776
    res_inst_tac [("Q",sP)] (excluded_middle RS disjE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   777
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   778
fun case_tac a = res_inst_tac [("P",a)] case_split_thm
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   779
*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   780
11687
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
   781
theorems case_split = case_split_thm [case_names True False]
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   782
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   783
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   784
subsubsection {* Intuitionistic Reasoning *}
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   785
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   786
lemma impE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   787
  assumes 1: "P --> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   788
    and 2: "Q ==> R"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   789
    and 3: "P --> Q ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   790
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   791
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   792
  from 3 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   793
  with 1 have Q by (rule impE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   794
  with 2 show R .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   795
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   796
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   797
lemma allE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   798
  assumes 1: "ALL x. P x"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   799
    and 2: "P x ==> ALL x. P x ==> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   800
  shows Q
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   801
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   802
  from 1 have "P x" by (rule spec)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   803
  from this and 1 show Q by (rule 2)
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
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   806
lemma notE':
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   807
  assumes 1: "~ P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   808
    and 2: "~ P ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   809
  shows R
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 2 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   812
  with 1 show R by (rule notE)
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
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   815
lemmas [CPure.elim!] = disjE iffE FalseE conjE exE
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   816
  and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   817
  and [CPure.elim 2] = allE notE' impE'
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   818
  and [CPure.intro] = exI disjI2 disjI1
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   819
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   820
lemmas [trans] = trans
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   821
  and [sym] = sym not_sym
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   822
  and [CPure.elim?] = iffD1 iffD2 impE
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   823
11438
3d9222b80989 declare trans [trans] (*overridden in theory Calculation*);
wenzelm
parents: 11432
diff changeset
   824
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   825
subsubsection {* Atomizing meta-level connectives *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   826
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   827
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
   828
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   829
  assume "!!x. P x"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   830
  show "ALL x. P x" by (rule allI)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   831
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   832
  assume "ALL x. P x"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   833
  thus "!!x. P x" by (rule allE)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   834
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   835
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   836
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
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 r: "A ==> B"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   839
  show "A --> B" by (rule impI) (rule r)
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 "A --> B" and A
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   842
  thus B by (rule mp)
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
14749
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   845
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   846
proof
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   847
  assume r: "A ==> False"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   848
  show "~A" by (rule notI) (rule r)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   849
next
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   850
  assume "~A" and A
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   851
  thus False by (rule notE)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   852
qed
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   853
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   854
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   855
proof
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   856
  assume "x == y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   857
  show "x = y" by (unfold prems) (rule refl)
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   858
next
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   859
  assume "x = y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   860
  thus "x == y" by (rule eq_reflection)
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   861
qed
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   862
12023
wenzelm
parents: 12003
diff changeset
   863
lemma atomize_conj [atomize]:
wenzelm
parents: 12003
diff changeset
   864
  "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   865
proof
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   866
  assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   867
  show "A & B" by (rule conjI)
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   868
next
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   869
  fix C
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   870
  assume "A & B"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   871
  assume "A ==> B ==> PROP C"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   872
  thus "PROP C"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   873
  proof this
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   874
    show A by (rule conjunct1)
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   875
    show B by (rule conjunct2)
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   876
  qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   877
qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   878
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   879
lemmas [symmetric, rulify] = atomize_all atomize_imp
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   880
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   881
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   882
subsubsection {* Classical Reasoner setup *}
9529
d9434a9277a4 lemmas atomize = all_eq imp_eq;
wenzelm
parents: 9488
diff changeset
   883
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   884
use "cladata.ML"
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   885
setup hypsubst_setup
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   886
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   887
ML_setup {*
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   888
  Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   889
*}
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   890
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   891
setup Classical.setup
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   892
setup clasetup
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   893
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   894
lemmas [intro?] = ext
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   895
  and [elim?] = ex1_implies_ex
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   896
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   897
use "blastdata.ML"
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   898
setup Blast.setup
4868
843a9f5b3c3d nonterminals;
wenzelm
parents: 4793
diff changeset
   899
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   900
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   901
subsubsection {* Simplifier setup *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   902
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   903
lemma meta_eq_to_obj_eq: "x == y ==> x = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   904
proof -
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   905
  assume r: "x == y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   906
  show "x = y" by (unfold r) (rule refl)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   907
qed
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   908
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   909
lemma eta_contract_eq: "(%s. f s) = f" ..
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   910
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   911
lemma simp_thms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   912
  shows not_not: "(~ ~ P) = P"
15354
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
   913
  and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   914
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   915
    "(P ~= Q) = (P = (~Q))"
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   916
    "(P | ~P) = True"    "(~P | P) = True"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   917
    "(x = x) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   918
    "(~True) = False"  "(~False) = True"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   919
    "(~P) ~= P"  "P ~= (~P)"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   920
    "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   921
    "(True --> P) = P"  "(False --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   922
    "(P --> True) = True"  "(P --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   923
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   924
    "(P & True) = P"  "(True & P) = P"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   925
    "(P & False) = False"  "(False & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   926
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   927
    "(P & ~P) = False"    "(~P & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   928
    "(P | True) = True"  "(True | P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   929
    "(P | False) = P"  "(False | P) = P"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   930
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   931
    "(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
   932
    -- {* needed for the one-point-rule quantifier simplification procs *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   933
    -- {* essential for termination!! *} and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   934
    "!!P. (EX x. x=t & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   935
    "!!P. (EX x. t=x & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   936
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   937
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   938
  by (blast, blast, blast, blast, blast, rules+)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13412
diff changeset
   939
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   940
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   941
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   942
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   943
lemma ex_simps:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   944
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   945
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   946
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   947
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   948
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   949
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   950
  -- {* Miniscoping: pushing in existential quantifiers. *}
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   951
  by (rules | blast)+
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   952
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   953
lemma all_simps:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   954
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   955
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   956
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   957
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   958
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   959
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   960
  -- {* Miniscoping: pushing in universal quantifiers. *}
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   961
  by (rules | blast)+
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   962
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   963
lemma disj_absorb: "(A | A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   964
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   965
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   966
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   967
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   968
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   969
lemma conj_absorb: "(A & A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   970
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   971
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   972
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   973
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   974
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   975
lemma eq_ac:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   976
  shows eq_commute: "(a=b) = (b=a)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   977
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   978
    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
   979
lemma neq_commute: "(a~=b) = (b~=a)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   980
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   981
lemma conj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   982
  shows conj_commute: "(P&Q) = (Q&P)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   983
    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
   984
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   985
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   986
lemma disj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   987
  shows disj_commute: "(P|Q) = (Q|P)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   988
    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
   989
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   990
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   991
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
   992
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   993
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   994
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
   995
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   996
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   997
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
   998
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   999
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1000
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1001
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
  1002
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1003
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1004
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1005
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1006
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1007
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1008
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1009
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1010
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1011
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1012
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1013
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1014
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1015
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" 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 iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1018
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1019
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1020
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1021
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1022
  -- {* cases boil down to the same thing. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1023
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1024
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1025
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1026
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
  1027
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1028
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1029
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1030
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
  1031
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
  1032
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1033
text {*
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1034
  \medskip The @{text "&"} congruence rule: not included by default!
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1035
  May slow rewrite proofs down by as much as 50\% *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1036
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1037
lemma conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1038
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1039
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1040
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1041
lemma rev_conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1042
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1043
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1044
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1045
text {* The @{text "|"} congruence rule: not included by default! *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1046
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1047
lemma disj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1048
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1049
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1050
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1051
lemma eq_sym_conv: "(x = y) = (y = x)"
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
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1055
text {* \medskip if-then-else rules *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1056
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1057
lemma if_True: "(if True then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1058
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1059
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1060
lemma if_False: "(if False then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1061
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1062
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1063
lemma if_P: "P ==> (if P then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1064
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1065
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1066
lemma if_not_P: "~P ==> (if P then x else y) = y"
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 split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1070
  apply (rule case_split [of Q])
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1071
   apply (subst if_P)
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1072
    prefer 3 apply (subst if_not_P, blast+)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1073
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1074
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1075
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1076
by (subst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1077
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1078
lemmas if_splits = split_if split_if_asm
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1079
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1080
lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1081
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1082
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1083
lemma if_cancel: "(if c then x else x) = x"
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1084
by (subst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1085
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1086
lemma if_eq_cancel: "(if x = y then y else x) = x"
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1087
by (subst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1088
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1089
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1090
  -- {* 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
  1091
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1092
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1093
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1094
  -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1095
  apply (subst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1096
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1097
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1098
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1099
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1100
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1101
subsubsection {* Actual Installation of the Simplifier *}
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1102
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1103
use "simpdata.ML"
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1104
setup Simplifier.setup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1105
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1106
setup Splitter.setup setup Clasimp.setup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1107
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1108
declare disj_absorb [simp] conj_absorb [simp]
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1109
13723
c7d104550205 *** empty log message ***
nipkow
parents: 13638
diff changeset
  1110
lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
c7d104550205 *** empty log message ***
nipkow
parents: 13638
diff changeset
  1111
by blast+
c7d104550205 *** empty log message ***
nipkow
parents: 13638
diff changeset
  1112
13638
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1113
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
  1114
  apply (rule iffI)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1115
  apply (rule_tac a = "%x. THE y. P x y" in ex1I)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1116
  apply (fast dest!: theI')
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1117
  apply (fast intro: ext the1_equality [symmetric])
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1118
  apply (erule ex1E)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1119
  apply (rule allI)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1120
  apply (rule ex1I)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1121
  apply (erule spec)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1122
  apply (erule_tac x = "%z. if z = x then y else f z" in allE)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1123
  apply (erule impE)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1124
  apply (rule allI)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1125
  apply (rule_tac P = "xa = x" in case_split_thm)
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1126
  apply (drule_tac [3] x = x in fun_cong, simp_all)
13638
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1127
  done
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1128
13438
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1129
text{*Needs only HOL-lemmas:*}
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1130
lemma mk_left_commute:
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1131
  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
  1132
          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
  1133
  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
  1134
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
  1135
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1136
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1137
subsubsection {* Generic cases and induction *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1138
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1139
constdefs
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1140
  induct_forall :: "('a => bool) => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1141
  "induct_forall P == \<forall>x. P x"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1142
  induct_implies :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1143
  "induct_implies A B == A --> B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1144
  induct_equal :: "'a => 'a => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1145
  "induct_equal x y == x = y"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1146
  induct_conj :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1147
  "induct_conj A B == A & B"
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1148
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1149
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
  1150
  by (simp only: atomize_all induct_forall_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1151
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1152
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1153
  by (simp only: atomize_imp induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1154
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1155
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1156
  by (simp only: atomize_eq induct_equal_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1157
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1158
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
  1159
    induct_conj (induct_forall A) (induct_forall B)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1160
  by (unfold induct_forall_def induct_conj_def) rules
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1161
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1162
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1163
    induct_conj (induct_implies C A) (induct_implies C B)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1164
  by (unfold induct_implies_def induct_conj_def) rules
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1165
13598
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1166
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
  1167
proof
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1168
  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
  1169
  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
  1170
next
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1171
  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
  1172
  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
  1173
qed
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1174
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1175
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1176
  by (simp add: induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1177
12161
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
  1178
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
  1179
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
  1180
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
  1181
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
  1182
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1183
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
  1184
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1185
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1186
text {* Method setup. *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1187
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1188
ML {*
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1189
  structure InductMethod = InductMethodFun
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1190
  (struct
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1191
    val dest_concls = HOLogic.dest_concls
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1192
    val cases_default = thm "case_split"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1193
    val local_impI = thm "induct_impliesI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1194
    val conjI = thm "conjI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1195
    val atomize = thms "induct_atomize"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1196
    val rulify1 = thms "induct_rulify1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1197
    val rulify2 = thms "induct_rulify2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1198
    val localize = [Thm.symmetric (thm "induct_implies_def")]
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1199
  end);
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1200
*}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1201
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1202
setup InductMethod.setup
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1203
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1204
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1205
subsection {* Order signatures and orders *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1206
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1207
axclass
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
  1208
  ord < type
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1209
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1210
syntax
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1211
  "op <"        :: "['a::ord, 'a] => bool"             ("op <")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1212
  "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1213
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1214
global
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1215
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1216
consts
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1217
  "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1218
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1219
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1220
local
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1221
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
  1222
syntax (xsymbols)
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1223
  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1224
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1225
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
  1226
syntax (HTML output)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
  1227
  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
  1228
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
  1229
15354
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1230
text{* Syntactic sugar: *}
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1231
15354
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1232
consts
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1233
  "_gt" :: "'a::ord => 'a => bool"             (infixl ">" 50)
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1234
  "_ge" :: "'a::ord => 'a => bool"             (infixl ">=" 50)
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1235
translations
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1236
  "x > y"  => "y < x"
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1237
  "x >= y" => "y <= x"
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1238
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1239
syntax (xsymbols)
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1240
  "_ge"       :: "'a::ord => 'a => bool"             (infixl "\<ge>" 50)
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1241
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1242
syntax (HTML output)
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1243
  "_ge"       :: "['a::ord, 'a] => bool"             (infixl "\<ge>" 50)
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1244
14295
7f115e5c5de4 more general lemmas for Ring_and_Field
paulson
parents: 14248
diff changeset
  1245
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1246
subsubsection {* Monotonicity *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1247
13412
666137b488a4 predicate defs via locales;
wenzelm
parents: 12937
diff changeset
  1248
locale mono =
666137b488a4 predicate defs via locales;
wenzelm
parents: 12937
diff changeset
  1249
  fixes f
666137b488a4 predicate defs via locales;
wenzelm
parents: 12937
diff changeset
  1250
  assumes mono: "A <= B ==> f A <= f B"
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1251
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13412
diff changeset
  1252
lemmas monoI [intro?] = mono.intro
13412
666137b488a4 predicate defs via locales;
wenzelm
parents: 12937
diff changeset
  1253
  and monoD [dest?] = mono.mono
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1254
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1255
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1256
  min :: "['a::ord, 'a] => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1257
  "min a b == (if a <= b then a else b)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1258
  max :: "['a::ord, 'a] => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1259
  "max a b == (if a <= b then b else a)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1260
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1261
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1262
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1263
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1264
lemma min_of_mono:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1265
    "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1266
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1267
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1268
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1269
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1270
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1271
lemma max_of_mono:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1272
    "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1273
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1274
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1275
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1276
subsubsection "Orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1277
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1278
axclass order < ord
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1279
  order_refl [iff]: "x <= x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1280
  order_trans: "x <= y ==> y <= z ==> x <= z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1281
  order_antisym: "x <= y ==> y <= x ==> x = y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1282
  order_less_le: "(x < y) = (x <= y & x ~= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1283
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1284
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1285
text {* Reflexivity. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1286
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1287
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1288
    -- {* This form is useful with the classical reasoner. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1289
  apply (erule ssubst)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1290
  apply (rule order_refl)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1291
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1292
13553
855f6bae851e order_less_irrefl: [simp] -> [iff]
nipkow
parents: 13550
diff changeset
  1293
lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1294
  by (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1295
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1296
lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1297
    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1298
  apply (simp add: order_less_le, blast)
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1299
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1300
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1301
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1302
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1303
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1304
  by (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1305
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1306
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1307
text {* Asymmetry. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1308
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1309
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1310
  by (simp add: order_less_le order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1311
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1312
lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1313
  apply (drule order_less_not_sym)
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1314
  apply (erule contrapos_np, simp)
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1315
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1316
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1317
lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"
14295
7f115e5c5de4 more general lemmas for Ring_and_Field
paulson
parents: 14248
diff changeset
  1318
by (blast intro: order_antisym)
7f115e5c5de4 more general lemmas for Ring_and_Field
paulson
parents: 14248
diff changeset
  1319
15197
19e735596e51 Added antisymmetry simproc
nipkow
parents: 15140
diff changeset
  1320
lemma order_antisym_conv: "(y::'a::order) <= x ==> (x <= y) = (x = y)"
19e735596e51 Added antisymmetry simproc
nipkow
parents: 15140
diff changeset
  1321
by(blast intro:order_antisym)
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1322
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1323
text {* Transitivity. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1324
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1325
lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1326
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1327
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1328
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1329
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1330
lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1331
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1332
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1333
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1334
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1335
lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1336
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1337
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1338
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1339
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1340
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1341
text {* Useful for simplification, but too risky to include by default. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1342
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1343
lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1344
  by (blast elim: order_less_asym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1345
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1346
lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1347
  by (blast elim: order_less_asym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1348
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1349
lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1350
  by auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1351
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1352
lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1353
  by auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1354
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1355
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1356
text {* Other operators. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1357
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1358
lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1359
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1360
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1361
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1362
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1363
lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1364
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1365
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1366
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1367
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1368
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1369
subsubsection {* Least value operator *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1370
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1371
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1372
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1373
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1374
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1375
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1376
lemma LeastI2:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1377
  "[| P (x::'a::order);
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1378
      !!y. P y ==> x <= y;
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1379
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1380
   ==> Q (Least P)"
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1381
  apply (unfold Least_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1382
  apply (rule theI2)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1383
    apply (blast intro: order_antisym)+
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1384
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1385
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1386
lemma Least_equality:
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1387
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1388
  apply (simp add: Least_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1389
  apply (rule the_equality)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1390
  apply (auto intro!: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: