src/HOL/HOL.thy
author wenzelm
Wed, 06 Dec 2006 01:12:42 +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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uses ("simpdata.ML") "Tools/res_atpset.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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  arbitrary     :: 'a
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  undefined     :: '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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consts
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  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
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subsubsection {* Additional concrete syntax *}
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notation (output)
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  "op ="  (infix "=" 50)
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abbreviation
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  not_equal :: "['a, 'a] => bool"  (infixl "~=" 50) where
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  "x ~= y == ~ (x = y)"
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notation (output)
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  not_equal  (infix "~=" 50)
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notation (xsymbols)
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  Not  ("\<not> _" [40] 40) and
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  "op &"  (infixr "\<and>" 35) and
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  "op |"  (infixr "\<or>" 30) and
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  "op -->"  (infixr "\<longrightarrow>" 25) and
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  not_equal  (infix "\<noteq>" 50)
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notation (HTML output)
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  Not  ("\<not> _" [40] 40) and
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  "op &"  (infixr "\<and>" 35) and
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  "op |"  (infixr "\<or>" 30) and
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  not_equal  (infix "\<noteq>" 50)
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abbreviation (iff)
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  iff :: "[bool, bool] => bool"  (infixr "<->" 25) where
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  "A <-> B == A = B"
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notation (xsymbols)
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  iff  (infixr "\<longleftrightarrow>" 25)
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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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  "_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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  "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 (xsymbols)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
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notation (xsymbols)
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  All  (binder "\<forall>" 10) and
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  Ex  (binder "\<exists>" 10) and
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  Ex1  (binder "\<exists>!" 10)
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notation (HTML output)
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  All  (binder "\<forall>" 10) and
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  Ex  (binder "\<exists>" 10) and
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  Ex1  (binder "\<exists>!" 10)
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notation (HOL)
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  All  (binder "! " 10) and
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  Ex  (binder "? " 10) and
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  Ex1  (binder "?! " 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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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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  undefined
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subsubsection {* Generic algebraic operations *}
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class zero =
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  fixes zero :: "'a"  ("\<^loc>0")
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class one =
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  fixes one  :: "'a"  ("\<^loc>1")
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hide (open) const zero one
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class plus =
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  fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>+" 65)
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class minus =
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  fixes uminus :: "'a \<Rightarrow> 'a" 
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    and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>-" 65)
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    and abs :: "'a \<Rightarrow> 'a"
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class times =
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  fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>*" 70)
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class inverse = 
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  fixes inverse :: "'a \<Rightarrow> 'a"
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    and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>'/" 70)
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notation
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  uminus  ("- _" [81] 80)
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notation (xsymbols)
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  abs  ("\<bar>_\<bar>")
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notation (HTML output)
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  abs  ("\<bar>_\<bar>")
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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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typed_print_translation {*
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let
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  val thy = the_context ();
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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 map (tr' o Sign.const_syntax_name thy) ["HOL.one", "HOL.zero"] end;
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*} -- {* show types that are presumably too general *}
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subsection {* Fundamental rules *}
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subsubsection {* Equality *}
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text {* Thanks to Stephan Merz *}
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lemma subst:
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  assumes eq: "s = t" and p: "P s"
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  shows "P t"
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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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lemma sym: "s = t ==> t = s"
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  by (erule subst) (rule refl)
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lemma ssubst: "t = s ==> P s ==> P t"
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  by (drule sym) (erule subst)
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lemma trans: "[| r=s; s=t |] ==> r=t"
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  by (erule subst)
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lemma def_imp_eq:
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  assumes meq: "A == B"
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  shows "A = B"
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  by (unfold meq) (rule refl)
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(*a mere copy*)
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lemma meta_eq_to_obj_eq: 
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  assumes meq: "A == B"
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  shows "A = B"
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  by (unfold meq) (rule refl)
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text {* Useful with @{text erule} for proving equalities from known equalities. *}
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     (* a = b
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        |   |
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        c = d   *)
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lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
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apply (rule trans)
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apply (rule trans)
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apply (rule sym)
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apply assumption+
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done
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text {* For calculational reasoning: *}
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lemma forw_subst: "a = b ==> P b ==> P a"
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  by (rule ssubst)
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lemma back_subst: "P a ==> a = b ==> P b"
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  by (rule subst)
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subsubsection {*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 arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
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apply (erule ssubst)+
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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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subsubsection {*Equality of booleans -- iff*}
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lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
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  by (iprover intro: iff [THEN mp, THEN mp] impI assms)
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lemma iffD2: "[| P=Q; Q |] ==> P"
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  by (erule ssubst)
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lemma rev_iffD2: "[| Q; P=Q |] ==> P"
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   317
  by (erule iffD2)
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   319
lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
9c97af4a1567 tuned proofs;
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   320
  by (drule sym) (rule iffD2)
9c97af4a1567 tuned proofs;
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   321
9c97af4a1567 tuned proofs;
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   322
lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
9c97af4a1567 tuned proofs;
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   323
  by (drule sym) (rule rev_iffD2)
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   324
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   325
lemma iffE:
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  assumes major: "P=Q"
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9c97af4a1567 tuned proofs;
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   327
    and minor: "[| P --> Q; Q --> P |] ==> R"
18457
356a9f711899 structure ProjectRule;
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parents: 17992
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   328
  shows R
356a9f711899 structure ProjectRule;
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parents: 17992
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   329
  by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
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   330
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   331
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   332
subsubsection {*True*}
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   333
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   334
lemma TrueI: "True"
21504
9c97af4a1567 tuned proofs;
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   335
  unfolding True_def by (rule refl)
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   336
21504
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   337
lemma eqTrueI: "P ==> P = True"
18457
356a9f711899 structure ProjectRule;
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parents: 17992
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   338
  by (iprover intro: iffI TrueI)
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   340
lemma eqTrueE: "P = True ==> P"
9c97af4a1567 tuned proofs;
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parents: 21502
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   341
  by (erule iffD2) (rule TrueI)
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parents: 15380
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   342
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   343
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   344
subsubsection {*Universal quantifier*}
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   345
21504
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   346
lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
9c97af4a1567 tuned proofs;
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   347
  unfolding All_def by (iprover intro: ext eqTrueI assms)
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   348
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   349
lemma spec: "ALL x::'a. P(x) ==> P(x)"
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   350
apply (unfold All_def)
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   351
apply (rule eqTrueE)
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   352
apply (erule fun_cong)
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   353
done
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   354
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   355
lemma allE:
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parents: 15380
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   356
  assumes major: "ALL x. P(x)"
21504
9c97af4a1567 tuned proofs;
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parents: 21502
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   357
    and minor: "P(x) ==> R"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   358
  shows R
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   359
  by (iprover intro: minor major [THEN spec])
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parents: 15380
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   360
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   361
lemma all_dupE:
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   362
  assumes major: "ALL x. P(x)"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   363
    and minor: "[| P(x); ALL x. P(x) |] ==> R"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   364
  shows R
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   365
  by (iprover intro: minor major major [THEN spec])
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parents: 15380
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   366
1d195de59497 removal of HOL_Lemmas
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   367
21504
9c97af4a1567 tuned proofs;
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   368
subsubsection {* False *}
9c97af4a1567 tuned proofs;
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parents: 21502
diff changeset
   369
9c97af4a1567 tuned proofs;
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parents: 21502
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   370
text {*
9c97af4a1567 tuned proofs;
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parents: 21502
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   371
  Depends upon @{text spec}; it is impossible to do propositional
9c97af4a1567 tuned proofs;
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   372
  logic before quantifiers!
9c97af4a1567 tuned proofs;
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   373
*}
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   374
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   375
lemma FalseE: "False ==> P"
21504
9c97af4a1567 tuned proofs;
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   376
  apply (unfold False_def)
9c97af4a1567 tuned proofs;
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parents: 21502
diff changeset
   377
  apply (erule spec)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   378
  done
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parents: 15380
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   379
21504
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   380
lemma False_neq_True: "False = True ==> P"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   381
  by (erule eqTrueE [THEN FalseE])
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   382
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   383
21504
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wenzelm
parents: 21502
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   384
subsubsection {* Negation *}
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   385
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   386
lemma notI:
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   387
  assumes "P ==> False"
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parents: 15380
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   388
  shows "~P"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   389
  apply (unfold not_def)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   390
  apply (iprover intro: impI assms)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   391
  done
15411
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diff changeset
   392
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diff changeset
   393
lemma False_not_True: "False ~= True"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   394
  apply (rule notI)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   395
  apply (erule False_neq_True)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   396
  done
15411
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paulson
parents: 15380
diff changeset
   397
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parents: 15380
diff changeset
   398
lemma True_not_False: "True ~= False"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   399
  apply (rule notI)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   400
  apply (drule sym)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   401
  apply (erule False_neq_True)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   402
  done
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   403
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   404
lemma notE: "[| ~P;  P |] ==> R"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   405
  apply (unfold not_def)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   406
  apply (erule mp [THEN FalseE])
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   407
  apply assumption
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   408
  done
15411
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paulson
parents: 15380
diff changeset
   409
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   410
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   411
  by (erule notE [THEN notI]) (erule meta_mp)
15411
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paulson
parents: 15380
diff changeset
   412
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   413
20944
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haftmann
parents: 20833
diff changeset
   414
subsubsection {*Implication*}
15411
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parents: 15380
diff changeset
   415
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parents: 15380
diff changeset
   416
lemma impE:
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   417
  assumes "P-->Q" "P" "Q ==> R"
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parents: 15380
diff changeset
   418
  shows "R"
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58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   419
by (iprover intro: prems mp)
15411
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paulson
parents: 15380
diff changeset
   420
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   421
(* Reduces Q to P-->Q, allowing substitution in P. *)
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   422
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
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58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   423
by (iprover intro: mp)
15411
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paulson
parents: 15380
diff changeset
   424
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   425
lemma contrapos_nn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   426
  assumes major: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   427
      and minor: "P==>Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   428
  shows "~P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   429
by (iprover intro: notI minor major [THEN notE])
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   430
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   431
(*not used at all, but we already have the other 3 combinations *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   432
lemma contrapos_pn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   433
  assumes major: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   434
      and minor: "P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   435
  shows "~P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   436
by (iprover intro: notI minor major notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   437
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   438
lemma not_sym: "t ~= s ==> s ~= t"
21250
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   439
  by (erule contrapos_nn) (erule sym)
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   440
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   441
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   442
  by (erule subst, erule ssubst, assumption)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   443
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   444
(*still used in HOLCF*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   445
lemma rev_contrapos:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   446
  assumes pq: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   447
      and nq: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   448
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   449
apply (rule nq [THEN contrapos_nn])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   450
apply (erule pq)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   451
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   452
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   453
subsubsection {*Existential quantifier*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   454
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   455
lemma exI: "P x ==> EX x::'a. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   456
apply (unfold Ex_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   457
apply (iprover intro: allI allE impI mp)
15411
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 exE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   461
  assumes major: "EX x::'a. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   462
      and minor: "!!x. P(x) ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   463
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   464
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   465
apply (iprover intro: impI [THEN allI] minor)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   466
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   467
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   468
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   469
subsubsection {*Conjunction*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   470
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   471
lemma conjI: "[| P; Q |] ==> P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   472
apply (unfold and_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   473
apply (iprover intro: impI [THEN allI] mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   474
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   475
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   476
lemma conjunct1: "[| P & Q |] ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   477
apply (unfold and_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   478
apply (iprover intro: impI dest: spec mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   479
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   480
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   481
lemma conjunct2: "[| P & Q |] ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   482
apply (unfold and_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   483
apply (iprover intro: impI dest: spec mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   484
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   485
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   486
lemma conjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   487
  assumes major: "P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   488
      and minor: "[| P; Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   489
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   490
apply (rule minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   491
apply (rule major [THEN conjunct1])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   492
apply (rule major [THEN conjunct2])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   493
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   494
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   495
lemma context_conjI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   496
  assumes prems: "P" "P ==> Q" shows "P & Q"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   497
by (iprover intro: conjI prems)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   498
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   499
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   500
subsubsection {*Disjunction*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   501
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   502
lemma disjI1: "P ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   503
apply (unfold or_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   504
apply (iprover intro: allI impI mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   505
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   506
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   507
lemma disjI2: "Q ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   508
apply (unfold or_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   509
apply (iprover intro: allI impI mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   510
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   511
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   512
lemma disjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   513
  assumes major: "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   514
      and minorP: "P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   515
      and minorQ: "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   516
  shows "R"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   517
by (iprover intro: minorP minorQ impI
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   518
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   519
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   520
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   521
subsubsection {*Classical logic*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   522
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   523
lemma classical:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   524
  assumes prem: "~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   525
  shows "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   526
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   527
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   528
apply (rule notI [THEN prem, THEN eqTrueI])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   529
apply (erule subst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   530
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   531
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   532
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   533
lemmas ccontr = FalseE [THEN classical, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   534
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   535
(*notE with premises exchanged; it discharges ~R so that it can be used to
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   536
  make elimination rules*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   537
lemma rev_notE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   538
  assumes premp: "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   539
      and premnot: "~R ==> ~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   540
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   541
apply (rule ccontr)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   542
apply (erule notE [OF premnot premp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   543
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   544
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   545
(*Double negation law*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   546
lemma notnotD: "~~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   547
apply (rule classical)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   548
apply (erule notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   549
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   550
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   551
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   552
lemma contrapos_pp:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   553
  assumes p1: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   554
      and p2: "~P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   555
  shows "P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   556
by (iprover intro: classical p1 p2 notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   557
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   558
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   559
subsubsection {*Unique existence*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   560
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   561
lemma ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   562
  assumes prems: "P a" "!!x. P(x) ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   563
  shows "EX! x. P(x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   564
by (unfold Ex1_def, iprover intro: prems exI conjI allI impI)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   565
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   566
text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   567
lemma ex_ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   568
  assumes ex_prem: "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   569
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   570
  shows "EX! x. P(x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   571
by (iprover intro: ex_prem [THEN exE] ex1I eq)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   572
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   573
lemma ex1E:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   574
  assumes major: "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   575
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   576
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   577
apply (rule major [unfolded Ex1_def, THEN exE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   578
apply (erule conjE)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   579
apply (iprover intro: minor)
15411
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
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   583
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   584
apply (rule exI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   585
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   586
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   587
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   588
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   589
subsubsection {*THE: definite description operator*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   590
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   591
lemma the_equality:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   592
  assumes prema: "P a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   593
      and premx: "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   594
  shows "(THE x. P x) = a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   595
apply (rule trans [OF _ the_eq_trivial])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   596
apply (rule_tac f = "The" in arg_cong)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   597
apply (rule ext)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   598
apply (rule iffI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   599
 apply (erule premx)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   600
apply (erule ssubst, rule prema)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   601
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   602
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   603
lemma theI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   604
  assumes "P a" and "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   605
  shows "P (THE x. P x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   606
by (iprover intro: prems the_equality [THEN ssubst])
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   607
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   608
lemma theI': "EX! x. P x ==> P (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   609
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   610
apply (erule theI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   611
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   612
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   613
apply assumption
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
(*Easier to apply than theI: only one occurrence of P*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   617
lemma theI2:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   618
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   619
  shows "Q (THE x. P x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   620
by (iprover intro: prems theI)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   621
18697
86b3f73e3fd5 declare the1_equality [elim?];
wenzelm
parents: 18689
diff changeset
   622
lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   623
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   624
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   625
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   626
apply (erule all_dupE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   627
apply (drule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   628
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   629
apply (erule ssubst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   630
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   631
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   632
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   633
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   634
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   635
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   636
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   637
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   638
apply (erule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   639
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   640
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   641
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   642
subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   643
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   644
lemma disjCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   645
  assumes "~Q ==> P" shows "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   646
apply (rule classical)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   647
apply (iprover intro: prems disjI1 disjI2 notI elim: notE)
15411
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
lemma excluded_middle: "~P | P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   651
by (iprover intro: disjCI)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   652
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   653
text {*
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   654
  case distinction as a natural deduction rule.
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   655
  Note that @{term "~P"} is the second case, not the first
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   656
*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   657
lemma case_split_thm:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   658
  assumes prem1: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   659
      and prem2: "~P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   660
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   661
apply (rule excluded_middle [THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   662
apply (erule prem2)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   663
apply (erule prem1)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   664
done
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   665
lemmas case_split = case_split_thm [case_names True False]
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   666
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   667
(*Classical implies (-->) elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   668
lemma impCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   669
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   670
      and minor: "~P ==> R" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   671
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   672
apply (rule excluded_middle [of P, THEN disjE])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   673
apply (iprover intro: minor major [THEN mp])+
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   674
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   675
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   676
(*This version of --> elimination works on Q before P.  It works best for
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   677
  those cases in which P holds "almost everywhere".  Can't install as
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   678
  default: would break old proofs.*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   679
lemma impCE':
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   680
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   681
      and minor: "Q ==> R" "~P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   682
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   683
apply (rule excluded_middle [of P, THEN disjE])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   684
apply (iprover intro: minor major [THEN mp])+
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   685
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   686
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   687
(*Classical <-> elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   688
lemma iffCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   689
  assumes major: "P=Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   690
      and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   691
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   692
apply (rule major [THEN iffE])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   693
apply (iprover intro: minor elim: impCE notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   694
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   695
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   696
lemma exCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   697
  assumes "ALL x. ~P(x) ==> P(a)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   698
  shows "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   699
apply (rule ccontr)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   700
apply (iprover intro: prems exI allI notI notE [of "\<exists>x. P x"])
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   701
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   702
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   703
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   704
subsubsection {* Intuitionistic Reasoning *}
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   705
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   706
lemma impE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   707
  assumes 1: "P --> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   708
    and 2: "Q ==> R"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   709
    and 3: "P --> Q ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   710
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   711
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   712
  from 3 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   713
  with 1 have Q by (rule impE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   714
  with 2 show R .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   715
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   716
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   717
lemma allE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   718
  assumes 1: "ALL x. P x"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   719
    and 2: "P x ==> ALL x. P x ==> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   720
  shows Q
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   721
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   722
  from 1 have "P x" by (rule spec)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   723
  from this and 1 show Q by (rule 2)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   724
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   725
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   726
lemma notE':
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   727
  assumes 1: "~ P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   728
    and 2: "~ P ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   729
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   730
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   731
  from 2 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   732
  with 1 show R by (rule notE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   733
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   734
15801
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   735
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   736
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   737
  and [Pure.elim 2] = allE notE' impE'
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   738
  and [Pure.intro] = exI disjI2 disjI1
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   739
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   740
lemmas [trans] = trans
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   741
  and [sym] = sym not_sym
15801
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   742
  and [Pure.elim?] = iffD1 iffD2 impE
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   743
11438
3d9222b80989 declare trans [trans] (*overridden in theory Calculation*);
wenzelm
parents: 11432
diff changeset
   744
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   745
subsubsection {* Atomizing meta-level connectives *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   746
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   747
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
   748
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   749
  assume "!!x. P x"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   750
  show "ALL x. P x" by (rule allI)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   751
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   752
  assume "ALL x. P x"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   753
  thus "!!x. P x" by (rule allE)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   754
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   755
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   756
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   757
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   758
  assume r: "A ==> B"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   759
  show "A --> B" by (rule impI) (rule r)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   760
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   761
  assume "A --> B" and A
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   762
  thus B by (rule mp)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   763
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   764
14749
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   765
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   766
proof
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   767
  assume r: "A ==> False"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   768
  show "~A" by (rule notI) (rule r)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   769
next
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   770
  assume "~A" and A
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   771
  thus False by (rule notE)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   772
qed
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   773
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   774
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   775
proof
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   776
  assume "x == y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   777
  show "x = y" by (unfold prems) (rule refl)
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   778
next
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   779
  assume "x = y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   780
  thus "x == y" by (rule eq_reflection)
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   781
qed
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   782
12023
wenzelm
parents: 12003
diff changeset
   783
lemma atomize_conj [atomize]:
19121
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   784
  includes meta_conjunction_syntax
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   785
  shows "(A && B) == Trueprop (A & B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   786
proof
19121
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   787
  assume conj: "A && B"
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   788
  show "A & B"
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   789
  proof (rule conjI)
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   790
    from conj show A by (rule conjunctionD1)
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   791
    from conj show B by (rule conjunctionD2)
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   792
  qed
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   793
next
19121
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   794
  assume conj: "A & B"
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   795
  show "A && B"
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   796
  proof -
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   797
    from conj show A ..
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   798
    from conj show B ..
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   799
  qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   800
qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   801
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   802
lemmas [symmetric, rulify] = atomize_all atomize_imp
18832
6ab4de872a70 declare 'defn' rules;
wenzelm
parents: 18757
diff changeset
   803
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   804
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   805
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   806
subsection {* Package setup *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   807
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   808
subsubsection {* Classical Reasoner setup *}
9529
d9434a9277a4 lemmas atomize = all_eq imp_eq;
wenzelm
parents: 9488
diff changeset
   809
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   810
lemma thin_refl:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   811
  "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   812
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   813
ML {*
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   814
structure Hypsubst = HypsubstFun(
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   815
struct
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   816
  structure Simplifier = Simplifier
21218
38013c3a77a2 tuned hypsubst setup;
wenzelm
parents: 21210
diff changeset
   817
  val dest_eq = HOLogic.dest_eq
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   818
  val dest_Trueprop = HOLogic.dest_Trueprop
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   819
  val dest_imp = HOLogic.dest_imp
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   820
  val eq_reflection = thm "HOL.eq_reflection"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   821
  val rev_eq_reflection = thm "HOL.def_imp_eq"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   822
  val imp_intr = thm "HOL.impI"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   823
  val rev_mp = thm "HOL.rev_mp"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   824
  val subst = thm "HOL.subst"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   825
  val sym = thm "HOL.sym"
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   826
  val thin_refl = thm "thin_refl";
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   827
end);
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
   828
open Hypsubst;
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   829
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   830
structure Classical = ClassicalFun(
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   831
struct
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   832
  val mp = thm "HOL.mp"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   833
  val not_elim = thm "HOL.notE"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   834
  val classical = thm "HOL.classical"
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   835
  val sizef = Drule.size_of_thm
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   836
  val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   837
end);
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   838
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   839
structure BasicClassical: BASIC_CLASSICAL = Classical; 
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
   840
open BasicClassical;
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   841
*}
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   842
21009
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   843
setup {*
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   844
let
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   845
  (*prevent substitution on bool*)
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   846
  fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   847
    Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   848
      (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   849
in
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   850
  Hypsubst.hypsubst_setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   851
  #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   852
  #> Classical.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   853
  #> ResAtpset.setup
21009
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   854
end
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   855
*}
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   856
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   857
declare iffI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   858
  and notI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   859
  and impI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   860
  and disjCI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   861
  and conjI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   862
  and TrueI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   863
  and refl [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   864
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   865
declare iffCE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   866
  and FalseE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   867
  and impCE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   868
  and disjE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   869
  and conjE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   870
  and conjE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   871
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   872
declare ex_ex1I [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   873
  and allI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   874
  and the_equality [intro]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   875
  and exI [intro]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   876
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   877
declare exE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   878
  allE [elim]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   879
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   880
ML {*
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   881
val HOL_cs = Classical.claset_of (the_context ());
21009
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   882
*}
19162
67436e2a16df Added setup for "atpset" (a rule set for ATPs).
mengj
parents: 19138
diff changeset
   883
20223
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   884
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   885
  apply (erule swap)
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   886
  apply (erule (1) meta_mp)
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   887
  done
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   888
18689
a50587cd8414 prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents: 18595
diff changeset
   889
declare ex_ex1I [rule del, intro! 2]
a50587cd8414 prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents: 18595
diff changeset
   890
  and ex1I [intro]
a50587cd8414 prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents: 18595
diff changeset
   891
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   892
lemmas [intro?] = ext
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   893
  and [elim?] = ex1_implies_ex
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   894
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   895
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
20973
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
   896
lemma alt_ex1E [elim!]:
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   897
  assumes major: "\<exists>!x. P x"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   898
      and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   899
  shows R
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   900
apply (rule ex1E [OF major])
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   901
apply (rule prem)
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   902
apply (tactic {* ares_tac [thm "allI"] 1 *})+
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   903
apply (tactic {* etac (Classical.dup_elim (thm "allE")) 1 *})
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   904
by iprover
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   905
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   906
ML {*
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   907
structure Blast = BlastFun(
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   908
struct
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   909
  type claset = Classical.claset
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   910
  val equality_name = "op ="
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   911
  val not_name = "Not"
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   912
  val notE = thm "HOL.notE"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
   913
  val ccontr = thm "HOL.ccontr"
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   914
  val contr_tac = Classical.contr_tac
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   915
  val dup_intr = Classical.dup_intr
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   916
  val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
   917
  val claset = Classical.claset
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   918
  val rep_cs = Classical.rep_cs
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   919
  val cla_modifiers = Classical.cla_modifiers
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   920
  val cla_meth' = Classical.cla_meth'
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   921
end);
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
   922
val Blast_tac = Blast.Blast_tac;
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
   923
val blast_tac = Blast.blast_tac;
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   924
*}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   925
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   926
setup Blast.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   927
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   928
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   929
subsubsection {* Simplifier *}
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   930
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   931
lemma eta_contract_eq: "(%s. f s) = f" ..
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   932
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   933
lemma simp_thms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   934
  shows not_not: "(~ ~ P) = P"
15354
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
   935
  and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   936
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   937
    "(P ~= Q) = (P = (~Q))"
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   938
    "(P | ~P) = True"    "(~P | P) = True"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   939
    "(x = x) = True"
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   940
  and not_True_eq_False: "(\<not> True) = False"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   941
  and not_False_eq_True: "(\<not> False) = True"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   942
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   943
    "(~P) ~= P"  "P ~= (~P)"
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   944
    "(True=P) = P"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   945
  and eq_True: "(P = True) = P"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   946
  and "(False=P) = (~P)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   947
  and eq_False: "(P = False) = (\<not> P)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   948
  and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   949
    "(True --> P) = P"  "(False --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   950
    "(P --> True) = True"  "(P --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   951
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   952
    "(P & True) = P"  "(True & P) = P"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   953
    "(P & False) = False"  "(False & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   954
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   955
    "(P & ~P) = False"    "(~P & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   956
    "(P | True) = True"  "(True | P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   957
    "(P | False) = P"  "(False | P) = P"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   958
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   959
    "(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
   960
    -- {* needed for the one-point-rule quantifier simplification procs *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   961
    -- {* essential for termination!! *} and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   962
    "!!P. (EX x. x=t & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   963
    "!!P. (EX x. t=x & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   964
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   965
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   966
  by (blast, blast, blast, blast, blast, iprover+)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13412
diff changeset
   967
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   968
lemma disj_absorb: "(A | A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   969
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   970
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   971
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   972
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   973
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   974
lemma conj_absorb: "(A & A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   975
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   976
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   977
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   978
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   979
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   980
lemma eq_ac:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   981
  shows eq_commute: "(a=b) = (b=a)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   982
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   983
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   984
lemma neq_commute: "(a~=b) = (b~=a)" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   985
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   986
lemma conj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   987
  shows conj_commute: "(P&Q) = (Q&P)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   988
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   989
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   990
19174
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
   991
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
   992
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   993
lemma disj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   994
  shows disj_commute: "(P|Q) = (Q|P)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   995
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   996
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   997
19174
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
   998
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
   999
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1000
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1001
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1002
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1003
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1004
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1005
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1006
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1007
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1008
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1009
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1010
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1011
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1012
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1013
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1014
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1015
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1016
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1017
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1018
  by iprover
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1019
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1020
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1021
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1022
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1023
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1024
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1025
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1026
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1027
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1028
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1029
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1030
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1031
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1032
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1033
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1034
  -- {* cases boil down to the same thing. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1035
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1036
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1037
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1038
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1039
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1040
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1041
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1042
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1043
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1044
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1045
text {*
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1046
  \medskip The @{text "&"} congruence rule: not included by default!
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1047
  May slow rewrite proofs down by as much as 50\% *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1048
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1049
lemma conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1050
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1051
  by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1052
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1053
lemma rev_conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1054
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1055
  by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1056
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1057
text {* The @{text "|"} congruence rule: not included by default! *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1058
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1059
lemma disj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1060
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1061
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1062
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1063
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1064
text {* \medskip if-then-else rules *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1065
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1066
lemma if_True: "(if True then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1067
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1068
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1069
lemma if_False: "(if False then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1070
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1071
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1072
lemma if_P: "P ==> (if P then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1073
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1074
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1075
lemma if_not_P: "~P ==> (if P then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1076
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1077
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1078
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1079
  apply (rule case_split [of Q])
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1080
   apply (simplesubst if_P)
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1081
    prefer 3 apply (simplesubst if_not_P, blast+)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1082
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1083
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1084
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1085
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1086
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1087
lemmas if_splits = split_if split_if_asm
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1088
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1089
lemma if_cancel: "(if c then x else x) = x"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1090
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1091
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1092
lemma if_eq_cancel: "(if x = y then y else x) = x"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1093
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1094
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1095
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
19796
d86e7b1fc472 quoted "if";
wenzelm
parents: 19656
diff changeset
  1096
  -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1097
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1098
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1099
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
19796
d86e7b1fc472 quoted "if";
wenzelm
parents: 19656
diff changeset
  1100
  -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1101
  apply (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1102
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1103
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1104
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1105
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1106
15423
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1107
text {* \medskip let rules for simproc *}
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1108
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1109
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1110
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1111
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1112
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1113
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1114
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1115
text {*
16999
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1116
  The following copy of the implication operator is useful for
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1117
  fine-tuning congruence rules.  It instructs the simplifier to simplify
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1118
  its premise.
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1119
*}
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1120
17197
917c6e7ca28d simp_implies: proper named infix;
wenzelm
parents: 16999
diff changeset
  1121
constdefs
917c6e7ca28d simp_implies: proper named infix;
wenzelm
parents: 16999
diff changeset
  1122
  simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1)
917c6e7ca28d simp_implies: proper named infix;
wenzelm
parents: 16999
diff changeset
  1123
  "simp_implies \<equiv> op ==>"
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1124
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1125
lemma simp_impliesI:
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1126
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1127
  shows "PROP P =simp=> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1128
  apply (unfold simp_implies_def)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1129
  apply (rule PQ)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1130
  apply assumption
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1131
  done
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1132
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1133
lemma simp_impliesE:
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1134
  assumes PQ:"PROP P =simp=> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1135
  and P: "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1136
  and QR: "PROP Q \<Longrightarrow> PROP R"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1137
  shows "PROP R"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1138
  apply (rule QR)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1139
  apply (rule PQ [unfolded simp_implies_def])
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1140
  apply (rule P)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1141
  done
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1142
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1143
lemma simp_implies_cong:
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1144
  assumes PP' :"PROP P == PROP P'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1145
  and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1146
  shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1147
proof (unfold simp_implies_def, rule equal_intr_rule)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1148
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1149
  and P': "PROP P'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1150
  from PP' [symmetric] and P' have "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1151
    by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1152
  hence "PROP Q" by (rule PQ)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1153
  with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1154
next
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1155
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1156
  and P: "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1157
  from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1158
  hence "PROP Q'" by (rule P'Q')
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1159
  with P'QQ' [OF P', symmetric] show "PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1160
    by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1161
qed
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1162
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1163
lemma uncurry:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1164
  assumes "P \<longrightarrow> Q \<longrightarrow> R"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1165
  shows "P \<and> Q \<longrightarrow> R"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1166
  using prems by blast
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1167
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1168
lemma iff_allI:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1169
  assumes "\<And>x. P x = Q x"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1170
  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1171
  using prems by blast
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1172
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1173
lemma iff_exI:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1174
  assumes "\<And>x. P x = Q x"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1175
  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1176
  using prems by blast
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1177
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1178
lemma all_comm:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1179
  "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1180
  by blast
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1181
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1182
lemma ex_comm:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1183
  "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1184
  by blast
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1185
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1186
use "simpdata.ML"
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1187
ML {* open Simpdata *}
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1188
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1189
setup {*
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1190
  Simplifier.method_setup Splitter.split_modifiers
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
  1191
  #> (fn thy => (change_simpset_of thy (fn _ => Simpdata.simpset_simprocs); thy))
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1192
  #> Splitter.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1193
  #> Clasimp.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1194
  #> EqSubst.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1195
*}
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1196
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1197
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1198
proof
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1199
  assume prem: "True \<Longrightarrow> PROP P"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1200
  from prem [OF TrueI] show "PROP P" . 
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1201
next
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1202
  assume "PROP P"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1203
  show "PROP P" .
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1204
qed
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1205
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1206
lemma ex_simps:
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1207
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1208
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1209
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1210
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1211
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1212
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1213
  -- {* Miniscoping: pushing in existential quantifiers. *}
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1214
  by (iprover | blast)+
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1215
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1216
lemma all_simps:
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1217
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1218
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1219
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1220
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1221
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1222
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1223
  -- {* Miniscoping: pushing in universal quantifiers. *}
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1224
  by (iprover | blast)+
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1225
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1226
lemmas [simp] =
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1227
  triv_forall_equality (*prunes params*)
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1228
  True_implies_equals  (*prune asms `True'*)
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1229
  if_True
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1230
  if_False
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1231
  if_cancel
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1232
  if_eq_cancel
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1233
  imp_disjL
20973
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1234
  (*In general it seems wrong to add distributive laws by default: they
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1235
    might cause exponential blow-up.  But imp_disjL has been in for a while
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1236
    and cannot be removed without affecting existing proofs.  Moreover,
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1237
    rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1238
    grounds that it allows simplification of R in the two cases.*)
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1239
  conj_assoc
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1240
  disj_assoc
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1241
  de_Morgan_conj
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1242
  de_Morgan_disj
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1243
  imp_disj1
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1244
  imp_disj2
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1245
  not_imp
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1246
  disj_not1
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1247
  not_all
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1248
  not_ex
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1249
  cases_simp
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1250
  the_eq_trivial
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1251
  the_sym_eq_trivial
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1252
  ex_simps
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1253
  all_simps
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1254
  simp_thms
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1255
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1256
lemmas [cong] = imp_cong simp_implies_cong
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1257
lemmas [split] = split_if
20973
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1258
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1259
ML {*
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
  1260
val HOL_ss = Simplifier.simpset_of (the_context ());
20973
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1261
*}
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1262
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1263
text {* Simplifies x assuming c and y assuming ~c *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1264
lemma if_cong:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1265
  assumes "b = c"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1266
      and "c \<Longrightarrow> x = u"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1267
      and "\<not> c \<Longrightarrow> y = v"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1268
  shows "(if b then x else y) = (if c then u else v)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1269
  unfolding if_def using prems by simp
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1270
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1271
text {* Prevents simplification of x and y:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1272
  faster and allows the execution of functional programs. *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1273
lemma if_weak_cong [cong]:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1274
  assumes "b = c"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1275
  shows "(if b then x else y) = (if c then x else y)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1276
  using prems by (rule arg_cong)
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1277
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1278
text {* Prevents simplification of t: much faster *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1279
lemma let_weak_cong:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1280
  assumes "a = b"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1281
  shows "(let x = a in t x) = (let x = b in t x)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1282
  using prems by (rule arg_cong)
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1283
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1284
text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1285
lemma eq_cong2:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1286
  assumes "u = u'"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1287
  shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1288
  using prems by simp
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1289
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1290
lemma if_distrib:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1291
  "f (if c then x else y) = (if c then f x else f y)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1292
  by simp
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1293
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1294
text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
21502
7f3ea2b3bab6 prefer antiquotations over LaTeX macros;
wenzelm
parents: 21486
diff changeset
  1295
  side of an equality.  Used in @{text "{Integ,Real}/simproc.ML"} *}
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1296
lemma restrict_to_left:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1297
  assumes "x = y"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1298
  shows "(x = z) = (y = z)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1299
  using prems by simp
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1300
17459
9a3925c07392 added code generator setup (from Main.thy);
wenzelm
parents: 17404
diff changeset
  1301
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1302
subsubsection {* Generic cases and induction *}
17459
9a3925c07392 added code generator setup (from Main.thy);
wenzelm
parents: 17404
diff changeset
  1303
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1304
text {* Rule projections: *}
18887
6ad81e3fa478 Added "evaluation" method and oracle.
berghofe
parents: 18867
diff changeset
  1305
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1306
ML {*
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1307
structure ProjectRule = ProjectRuleFun
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1308
(struct
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1309
  val conjunct1 = thm "conjunct1";
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1310
  val conjunct2 = thm "conjunct2";
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1311
  val mp = thm "mp";
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1312
end)
17459
9a3925c07392 added code generator setup (from Main.thy);
wenzelm
parents: 17404
diff changeset
  1313
*}
9a3925c07392 added code generator setup (from Main.thy);
wenzelm
parents: 17404
diff changeset
  1314
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1315
constdefs
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1316
  induct_forall where "induct_forall P == \<forall>x. P x"
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1317
  induct_implies where "induct_implies A B == A \<longrightarrow> B"
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1318
  induct_equal where "induct_equal x y == x = y"
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1319
  induct_conj where "induct_conj A B == A \<and> B"
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1320
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1321
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1322
  by (unfold atomize_all induct_forall_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1323
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1324
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1325
  by (unfold atomize_imp induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1326
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1327
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1328
  by (unfold atomize_eq induct_equal_def)
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1329
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1330
lemma induct_conj_eq:
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1331
  includes meta_conjunction_syntax
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1332
  shows "(A && B) == Trueprop (induct_conj A B)"
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1333
  by (unfold atomize_conj induct_conj_def)
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1334
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1335
lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1336
lemmas induct_rulify [symmetric, standard] = induct_atomize
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1337
lemmas induct_rulify_fallback =
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1338
  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1339
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1340
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1341
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
  1342
    induct_conj (induct_forall A) (induct_forall B)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1343
  by (unfold induct_forall_def induct_conj_def) iprover
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1344
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1345
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1346
    induct_conj (induct_implies C A) (induct_implies C B)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1347
  by (unfold induct_implies_def induct_conj_def) iprover
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1348
13598
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1349
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
  1350
proof
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1351
  assume r: "induct_conj A B ==> PROP C" and A B
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1352
  show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
13598
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1353
next
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1354
  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1355
  show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
13598
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1356
qed
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1357
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1358
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
  1359
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1360
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
  1361
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1362
text {* Method setup. *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1363
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1364
ML {*
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1365
  structure InductMethod = InductMethodFun
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1366
  (struct
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1367
    val cases_default = thm "case_split"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1368
    val atomize = thms "induct_atomize"
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1369