src/HOL/HOL.thy
author wenzelm
Wed Dec 06 01:12:42 2006 +0100 (2006-12-06)
changeset 21671 f7d652ffef09
parent 21547 9c9fdf4c2949
child 22129 bb2203c93316
permissions -rw-r--r--
removed legacy ML bindings;
simplified ML setup;
tuned declarations;
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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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  by (erule iffD2)
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lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
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  by (drule sym) (rule iffD2)
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lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
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  by (drule sym) (rule rev_iffD2)
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lemma iffE:
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  assumes major: "P=Q"
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    and minor: "[| P --> Q; Q --> P |] ==> R"
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  shows R
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  by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
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subsubsection {*True*}
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lemma TrueI: "True"
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  unfolding True_def by (rule refl)
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lemma eqTrueI: "P ==> P = True"
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  by (iprover intro: iffI TrueI)
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lemma eqTrueE: "P = True ==> P"
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  by (erule iffD2) (rule TrueI)
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subsubsection {*Universal quantifier*}
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lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
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  unfolding All_def by (iprover intro: ext eqTrueI assms)
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lemma spec: "ALL x::'a. P(x) ==> P(x)"
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apply (unfold All_def)
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apply (rule eqTrueE)
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apply (erule fun_cong)
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done
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lemma allE:
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  assumes major: "ALL x. P(x)"
wenzelm@21504
   357
    and minor: "P(x) ==> R"
wenzelm@21504
   358
  shows R
wenzelm@21504
   359
  by (iprover intro: minor major [THEN spec])
paulson@15411
   360
paulson@15411
   361
lemma all_dupE:
paulson@15411
   362
  assumes major: "ALL x. P(x)"
wenzelm@21504
   363
    and minor: "[| P(x); ALL x. P(x) |] ==> R"
wenzelm@21504
   364
  shows R
wenzelm@21504
   365
  by (iprover intro: minor major major [THEN spec])
paulson@15411
   366
paulson@15411
   367
wenzelm@21504
   368
subsubsection {* False *}
wenzelm@21504
   369
wenzelm@21504
   370
text {*
wenzelm@21504
   371
  Depends upon @{text spec}; it is impossible to do propositional
wenzelm@21504
   372
  logic before quantifiers!
wenzelm@21504
   373
*}
paulson@15411
   374
paulson@15411
   375
lemma FalseE: "False ==> P"
wenzelm@21504
   376
  apply (unfold False_def)
wenzelm@21504
   377
  apply (erule spec)
wenzelm@21504
   378
  done
paulson@15411
   379
wenzelm@21504
   380
lemma False_neq_True: "False = True ==> P"
wenzelm@21504
   381
  by (erule eqTrueE [THEN FalseE])
paulson@15411
   382
paulson@15411
   383
wenzelm@21504
   384
subsubsection {* Negation *}
paulson@15411
   385
paulson@15411
   386
lemma notI:
wenzelm@21504
   387
  assumes "P ==> False"
paulson@15411
   388
  shows "~P"
wenzelm@21504
   389
  apply (unfold not_def)
wenzelm@21504
   390
  apply (iprover intro: impI assms)
wenzelm@21504
   391
  done
paulson@15411
   392
paulson@15411
   393
lemma False_not_True: "False ~= True"
wenzelm@21504
   394
  apply (rule notI)
wenzelm@21504
   395
  apply (erule False_neq_True)
wenzelm@21504
   396
  done
paulson@15411
   397
paulson@15411
   398
lemma True_not_False: "True ~= False"
wenzelm@21504
   399
  apply (rule notI)
wenzelm@21504
   400
  apply (drule sym)
wenzelm@21504
   401
  apply (erule False_neq_True)
wenzelm@21504
   402
  done
paulson@15411
   403
paulson@15411
   404
lemma notE: "[| ~P;  P |] ==> R"
wenzelm@21504
   405
  apply (unfold not_def)
wenzelm@21504
   406
  apply (erule mp [THEN FalseE])
wenzelm@21504
   407
  apply assumption
wenzelm@21504
   408
  done
paulson@15411
   409
wenzelm@21504
   410
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
wenzelm@21504
   411
  by (erule notE [THEN notI]) (erule meta_mp)
paulson@15411
   412
paulson@15411
   413
haftmann@20944
   414
subsubsection {*Implication*}
paulson@15411
   415
paulson@15411
   416
lemma impE:
paulson@15411
   417
  assumes "P-->Q" "P" "Q ==> R"
paulson@15411
   418
  shows "R"
nipkow@17589
   419
by (iprover intro: prems mp)
paulson@15411
   420
paulson@15411
   421
(* Reduces Q to P-->Q, allowing substitution in P. *)
paulson@15411
   422
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
nipkow@17589
   423
by (iprover intro: mp)
paulson@15411
   424
paulson@15411
   425
lemma contrapos_nn:
paulson@15411
   426
  assumes major: "~Q"
paulson@15411
   427
      and minor: "P==>Q"
paulson@15411
   428
  shows "~P"
nipkow@17589
   429
by (iprover intro: notI minor major [THEN notE])
paulson@15411
   430
paulson@15411
   431
(*not used at all, but we already have the other 3 combinations *)
paulson@15411
   432
lemma contrapos_pn:
paulson@15411
   433
  assumes major: "Q"
paulson@15411
   434
      and minor: "P ==> ~Q"
paulson@15411
   435
  shows "~P"
nipkow@17589
   436
by (iprover intro: notI minor major notE)
paulson@15411
   437
paulson@15411
   438
lemma not_sym: "t ~= s ==> s ~= t"
haftmann@21250
   439
  by (erule contrapos_nn) (erule sym)
haftmann@21250
   440
haftmann@21250
   441
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
haftmann@21250
   442
  by (erule subst, erule ssubst, assumption)
paulson@15411
   443
paulson@15411
   444
(*still used in HOLCF*)
paulson@15411
   445
lemma rev_contrapos:
paulson@15411
   446
  assumes pq: "P ==> Q"
paulson@15411
   447
      and nq: "~Q"
paulson@15411
   448
  shows "~P"
paulson@15411
   449
apply (rule nq [THEN contrapos_nn])
paulson@15411
   450
apply (erule pq)
paulson@15411
   451
done
paulson@15411
   452
haftmann@20944
   453
subsubsection {*Existential quantifier*}
paulson@15411
   454
paulson@15411
   455
lemma exI: "P x ==> EX x::'a. P x"
paulson@15411
   456
apply (unfold Ex_def)
nipkow@17589
   457
apply (iprover intro: allI allE impI mp)
paulson@15411
   458
done
paulson@15411
   459
paulson@15411
   460
lemma exE:
paulson@15411
   461
  assumes major: "EX x::'a. P(x)"
paulson@15411
   462
      and minor: "!!x. P(x) ==> Q"
paulson@15411
   463
  shows "Q"
paulson@15411
   464
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
nipkow@17589
   465
apply (iprover intro: impI [THEN allI] minor)
paulson@15411
   466
done
paulson@15411
   467
paulson@15411
   468
haftmann@20944
   469
subsubsection {*Conjunction*}
paulson@15411
   470
paulson@15411
   471
lemma conjI: "[| P; Q |] ==> P&Q"
paulson@15411
   472
apply (unfold and_def)
nipkow@17589
   473
apply (iprover intro: impI [THEN allI] mp)
paulson@15411
   474
done
paulson@15411
   475
paulson@15411
   476
lemma conjunct1: "[| P & Q |] ==> P"
paulson@15411
   477
apply (unfold and_def)
nipkow@17589
   478
apply (iprover intro: impI dest: spec mp)
paulson@15411
   479
done
paulson@15411
   480
paulson@15411
   481
lemma conjunct2: "[| P & Q |] ==> Q"
paulson@15411
   482
apply (unfold and_def)
nipkow@17589
   483
apply (iprover intro: impI dest: spec mp)
paulson@15411
   484
done
paulson@15411
   485
paulson@15411
   486
lemma conjE:
paulson@15411
   487
  assumes major: "P&Q"
paulson@15411
   488
      and minor: "[| P; Q |] ==> R"
paulson@15411
   489
  shows "R"
paulson@15411
   490
apply (rule minor)
paulson@15411
   491
apply (rule major [THEN conjunct1])
paulson@15411
   492
apply (rule major [THEN conjunct2])
paulson@15411
   493
done
paulson@15411
   494
paulson@15411
   495
lemma context_conjI:
paulson@15411
   496
  assumes prems: "P" "P ==> Q" shows "P & Q"
nipkow@17589
   497
by (iprover intro: conjI prems)
paulson@15411
   498
paulson@15411
   499
haftmann@20944
   500
subsubsection {*Disjunction*}
paulson@15411
   501
paulson@15411
   502
lemma disjI1: "P ==> P|Q"
paulson@15411
   503
apply (unfold or_def)
nipkow@17589
   504
apply (iprover intro: allI impI mp)
paulson@15411
   505
done
paulson@15411
   506
paulson@15411
   507
lemma disjI2: "Q ==> P|Q"
paulson@15411
   508
apply (unfold or_def)
nipkow@17589
   509
apply (iprover intro: allI impI mp)
paulson@15411
   510
done
paulson@15411
   511
paulson@15411
   512
lemma disjE:
paulson@15411
   513
  assumes major: "P|Q"
paulson@15411
   514
      and minorP: "P ==> R"
paulson@15411
   515
      and minorQ: "Q ==> R"
paulson@15411
   516
  shows "R"
nipkow@17589
   517
by (iprover intro: minorP minorQ impI
paulson@15411
   518
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
paulson@15411
   519
paulson@15411
   520
haftmann@20944
   521
subsubsection {*Classical logic*}
paulson@15411
   522
paulson@15411
   523
lemma classical:
paulson@15411
   524
  assumes prem: "~P ==> P"
paulson@15411
   525
  shows "P"
paulson@15411
   526
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
paulson@15411
   527
apply assumption
paulson@15411
   528
apply (rule notI [THEN prem, THEN eqTrueI])
paulson@15411
   529
apply (erule subst)
paulson@15411
   530
apply assumption
paulson@15411
   531
done
paulson@15411
   532
paulson@15411
   533
lemmas ccontr = FalseE [THEN classical, standard]
paulson@15411
   534
paulson@15411
   535
(*notE with premises exchanged; it discharges ~R so that it can be used to
paulson@15411
   536
  make elimination rules*)
paulson@15411
   537
lemma rev_notE:
paulson@15411
   538
  assumes premp: "P"
paulson@15411
   539
      and premnot: "~R ==> ~P"
paulson@15411
   540
  shows "R"
paulson@15411
   541
apply (rule ccontr)
paulson@15411
   542
apply (erule notE [OF premnot premp])
paulson@15411
   543
done
paulson@15411
   544
paulson@15411
   545
(*Double negation law*)
paulson@15411
   546
lemma notnotD: "~~P ==> P"
paulson@15411
   547
apply (rule classical)
paulson@15411
   548
apply (erule notE)
paulson@15411
   549
apply assumption
paulson@15411
   550
done
paulson@15411
   551
paulson@15411
   552
lemma contrapos_pp:
paulson@15411
   553
  assumes p1: "Q"
paulson@15411
   554
      and p2: "~P ==> ~Q"
paulson@15411
   555
  shows "P"
nipkow@17589
   556
by (iprover intro: classical p1 p2 notE)
paulson@15411
   557
paulson@15411
   558
haftmann@20944
   559
subsubsection {*Unique existence*}
paulson@15411
   560
paulson@15411
   561
lemma ex1I:
paulson@15411
   562
  assumes prems: "P a" "!!x. P(x) ==> x=a"
paulson@15411
   563
  shows "EX! x. P(x)"
nipkow@17589
   564
by (unfold Ex1_def, iprover intro: prems exI conjI allI impI)
paulson@15411
   565
paulson@15411
   566
text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
paulson@15411
   567
lemma ex_ex1I:
paulson@15411
   568
  assumes ex_prem: "EX x. P(x)"
paulson@15411
   569
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
paulson@15411
   570
  shows "EX! x. P(x)"
nipkow@17589
   571
by (iprover intro: ex_prem [THEN exE] ex1I eq)
paulson@15411
   572
paulson@15411
   573
lemma ex1E:
paulson@15411
   574
  assumes major: "EX! x. P(x)"
paulson@15411
   575
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
paulson@15411
   576
  shows "R"
paulson@15411
   577
apply (rule major [unfolded Ex1_def, THEN exE])
paulson@15411
   578
apply (erule conjE)
nipkow@17589
   579
apply (iprover intro: minor)
paulson@15411
   580
done
paulson@15411
   581
paulson@15411
   582
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
paulson@15411
   583
apply (erule ex1E)
paulson@15411
   584
apply (rule exI)
paulson@15411
   585
apply assumption
paulson@15411
   586
done
paulson@15411
   587
paulson@15411
   588
haftmann@20944
   589
subsubsection {*THE: definite description operator*}
paulson@15411
   590
paulson@15411
   591
lemma the_equality:
paulson@15411
   592
  assumes prema: "P a"
paulson@15411
   593
      and premx: "!!x. P x ==> x=a"
paulson@15411
   594
  shows "(THE x. P x) = a"
paulson@15411
   595
apply (rule trans [OF _ the_eq_trivial])
paulson@15411
   596
apply (rule_tac f = "The" in arg_cong)
paulson@15411
   597
apply (rule ext)
paulson@15411
   598
apply (rule iffI)
paulson@15411
   599
 apply (erule premx)
paulson@15411
   600
apply (erule ssubst, rule prema)
paulson@15411
   601
done
paulson@15411
   602
paulson@15411
   603
lemma theI:
paulson@15411
   604
  assumes "P a" and "!!x. P x ==> x=a"
paulson@15411
   605
  shows "P (THE x. P x)"
nipkow@17589
   606
by (iprover intro: prems the_equality [THEN ssubst])
paulson@15411
   607
paulson@15411
   608
lemma theI': "EX! x. P x ==> P (THE x. P x)"
paulson@15411
   609
apply (erule ex1E)
paulson@15411
   610
apply (erule theI)
paulson@15411
   611
apply (erule allE)
paulson@15411
   612
apply (erule mp)
paulson@15411
   613
apply assumption
paulson@15411
   614
done
paulson@15411
   615
paulson@15411
   616
(*Easier to apply than theI: only one occurrence of P*)
paulson@15411
   617
lemma theI2:
paulson@15411
   618
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
paulson@15411
   619
  shows "Q (THE x. P x)"
nipkow@17589
   620
by (iprover intro: prems theI)
paulson@15411
   621
wenzelm@18697
   622
lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
paulson@15411
   623
apply (rule the_equality)
paulson@15411
   624
apply  assumption
paulson@15411
   625
apply (erule ex1E)
paulson@15411
   626
apply (erule all_dupE)
paulson@15411
   627
apply (drule mp)
paulson@15411
   628
apply  assumption
paulson@15411
   629
apply (erule ssubst)
paulson@15411
   630
apply (erule allE)
paulson@15411
   631
apply (erule mp)
paulson@15411
   632
apply assumption
paulson@15411
   633
done
paulson@15411
   634
paulson@15411
   635
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
paulson@15411
   636
apply (rule the_equality)
paulson@15411
   637
apply (rule refl)
paulson@15411
   638
apply (erule sym)
paulson@15411
   639
done
paulson@15411
   640
paulson@15411
   641
haftmann@20944
   642
subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
paulson@15411
   643
paulson@15411
   644
lemma disjCI:
paulson@15411
   645
  assumes "~Q ==> P" shows "P|Q"
paulson@15411
   646
apply (rule classical)
nipkow@17589
   647
apply (iprover intro: prems disjI1 disjI2 notI elim: notE)
paulson@15411
   648
done
paulson@15411
   649
paulson@15411
   650
lemma excluded_middle: "~P | P"
nipkow@17589
   651
by (iprover intro: disjCI)
paulson@15411
   652
haftmann@20944
   653
text {*
haftmann@20944
   654
  case distinction as a natural deduction rule.
haftmann@20944
   655
  Note that @{term "~P"} is the second case, not the first
haftmann@20944
   656
*}
paulson@15411
   657
lemma case_split_thm:
paulson@15411
   658
  assumes prem1: "P ==> Q"
paulson@15411
   659
      and prem2: "~P ==> Q"
paulson@15411
   660
  shows "Q"
paulson@15411
   661
apply (rule excluded_middle [THEN disjE])
paulson@15411
   662
apply (erule prem2)
paulson@15411
   663
apply (erule prem1)
paulson@15411
   664
done
haftmann@20944
   665
lemmas case_split = case_split_thm [case_names True False]
paulson@15411
   666
paulson@15411
   667
(*Classical implies (-->) elimination. *)
paulson@15411
   668
lemma impCE:
paulson@15411
   669
  assumes major: "P-->Q"
paulson@15411
   670
      and minor: "~P ==> R" "Q ==> R"
paulson@15411
   671
  shows "R"
paulson@15411
   672
apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589
   673
apply (iprover intro: minor major [THEN mp])+
paulson@15411
   674
done
paulson@15411
   675
paulson@15411
   676
(*This version of --> elimination works on Q before P.  It works best for
paulson@15411
   677
  those cases in which P holds "almost everywhere".  Can't install as
paulson@15411
   678
  default: would break old proofs.*)
paulson@15411
   679
lemma impCE':
paulson@15411
   680
  assumes major: "P-->Q"
paulson@15411
   681
      and minor: "Q ==> R" "~P ==> R"
paulson@15411
   682
  shows "R"
paulson@15411
   683
apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589
   684
apply (iprover intro: minor major [THEN mp])+
paulson@15411
   685
done
paulson@15411
   686
paulson@15411
   687
(*Classical <-> elimination. *)
paulson@15411
   688
lemma iffCE:
paulson@15411
   689
  assumes major: "P=Q"
paulson@15411
   690
      and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
paulson@15411
   691
  shows "R"
paulson@15411
   692
apply (rule major [THEN iffE])
nipkow@17589
   693
apply (iprover intro: minor elim: impCE notE)
paulson@15411
   694
done
paulson@15411
   695
paulson@15411
   696
lemma exCI:
paulson@15411
   697
  assumes "ALL x. ~P(x) ==> P(a)"
paulson@15411
   698
  shows "EX x. P(x)"
paulson@15411
   699
apply (rule ccontr)
nipkow@17589
   700
apply (iprover intro: prems exI allI notI notE [of "\<exists>x. P x"])
paulson@15411
   701
done
paulson@15411
   702
paulson@15411
   703
wenzelm@12386
   704
subsubsection {* Intuitionistic Reasoning *}
wenzelm@12386
   705
wenzelm@12386
   706
lemma impE':
wenzelm@12937
   707
  assumes 1: "P --> Q"
wenzelm@12937
   708
    and 2: "Q ==> R"
wenzelm@12937
   709
    and 3: "P --> Q ==> P"
wenzelm@12937
   710
  shows R
wenzelm@12386
   711
proof -
wenzelm@12386
   712
  from 3 and 1 have P .
wenzelm@12386
   713
  with 1 have Q by (rule impE)
wenzelm@12386
   714
  with 2 show R .
wenzelm@12386
   715
qed
wenzelm@12386
   716
wenzelm@12386
   717
lemma allE':
wenzelm@12937
   718
  assumes 1: "ALL x. P x"
wenzelm@12937
   719
    and 2: "P x ==> ALL x. P x ==> Q"
wenzelm@12937
   720
  shows Q
wenzelm@12386
   721
proof -
wenzelm@12386
   722
  from 1 have "P x" by (rule spec)
wenzelm@12386
   723
  from this and 1 show Q by (rule 2)
wenzelm@12386
   724
qed
wenzelm@12386
   725
wenzelm@12937
   726
lemma notE':
wenzelm@12937
   727
  assumes 1: "~ P"
wenzelm@12937
   728
    and 2: "~ P ==> P"
wenzelm@12937
   729
  shows R
wenzelm@12386
   730
proof -
wenzelm@12386
   731
  from 2 and 1 have P .
wenzelm@12386
   732
  with 1 show R by (rule notE)
wenzelm@12386
   733
qed
wenzelm@12386
   734
wenzelm@15801
   735
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
wenzelm@15801
   736
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@15801
   737
  and [Pure.elim 2] = allE notE' impE'
wenzelm@15801
   738
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12386
   739
wenzelm@12386
   740
lemmas [trans] = trans
wenzelm@12386
   741
  and [sym] = sym not_sym
wenzelm@15801
   742
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@11750
   743
wenzelm@11438
   744
wenzelm@11750
   745
subsubsection {* Atomizing meta-level connectives *}
wenzelm@11750
   746
wenzelm@11750
   747
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
wenzelm@12003
   748
proof
wenzelm@9488
   749
  assume "!!x. P x"
wenzelm@10383
   750
  show "ALL x. P x" by (rule allI)
wenzelm@9488
   751
next
wenzelm@9488
   752
  assume "ALL x. P x"
wenzelm@10383
   753
  thus "!!x. P x" by (rule allE)
wenzelm@9488
   754
qed
wenzelm@9488
   755
wenzelm@11750
   756
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@12003
   757
proof
wenzelm@9488
   758
  assume r: "A ==> B"
wenzelm@10383
   759
  show "A --> B" by (rule impI) (rule r)
wenzelm@9488
   760
next
wenzelm@9488
   761
  assume "A --> B" and A
wenzelm@10383
   762
  thus B by (rule mp)
wenzelm@9488
   763
qed
wenzelm@9488
   764
paulson@14749
   765
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
paulson@14749
   766
proof
paulson@14749
   767
  assume r: "A ==> False"
paulson@14749
   768
  show "~A" by (rule notI) (rule r)
paulson@14749
   769
next
paulson@14749
   770
  assume "~A" and A
paulson@14749
   771
  thus False by (rule notE)
paulson@14749
   772
qed
paulson@14749
   773
wenzelm@11750
   774
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@12003
   775
proof
wenzelm@10432
   776
  assume "x == y"
wenzelm@10432
   777
  show "x = y" by (unfold prems) (rule refl)
wenzelm@10432
   778
next
wenzelm@10432
   779
  assume "x = y"
wenzelm@10432
   780
  thus "x == y" by (rule eq_reflection)
wenzelm@10432
   781
qed
wenzelm@10432
   782
wenzelm@12023
   783
lemma atomize_conj [atomize]:
wenzelm@19121
   784
  includes meta_conjunction_syntax
wenzelm@19121
   785
  shows "(A && B) == Trueprop (A & B)"
wenzelm@12003
   786
proof
wenzelm@19121
   787
  assume conj: "A && B"
wenzelm@19121
   788
  show "A & B"
wenzelm@19121
   789
  proof (rule conjI)
wenzelm@19121
   790
    from conj show A by (rule conjunctionD1)
wenzelm@19121
   791
    from conj show B by (rule conjunctionD2)
wenzelm@19121
   792
  qed
wenzelm@11953
   793
next
wenzelm@19121
   794
  assume conj: "A & B"
wenzelm@19121
   795
  show "A && B"
wenzelm@19121
   796
  proof -
wenzelm@19121
   797
    from conj show A ..
wenzelm@19121
   798
    from conj show B ..
wenzelm@11953
   799
  qed
wenzelm@11953
   800
qed
wenzelm@11953
   801
wenzelm@12386
   802
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18832
   803
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq
wenzelm@12386
   804
wenzelm@11750
   805
haftmann@20944
   806
subsection {* Package setup *}
haftmann@20944
   807
wenzelm@11750
   808
subsubsection {* Classical Reasoner setup *}
wenzelm@9529
   809
haftmann@20944
   810
lemma thin_refl:
haftmann@20944
   811
  "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
haftmann@20944
   812
haftmann@21151
   813
ML {*
haftmann@21151
   814
structure Hypsubst = HypsubstFun(
haftmann@21151
   815
struct
haftmann@21151
   816
  structure Simplifier = Simplifier
wenzelm@21218
   817
  val dest_eq = HOLogic.dest_eq
haftmann@21151
   818
  val dest_Trueprop = HOLogic.dest_Trueprop
haftmann@21151
   819
  val dest_imp = HOLogic.dest_imp
haftmann@21547
   820
  val eq_reflection = thm "HOL.eq_reflection"
haftmann@21547
   821
  val rev_eq_reflection = thm "HOL.def_imp_eq"
haftmann@21547
   822
  val imp_intr = thm "HOL.impI"
haftmann@21547
   823
  val rev_mp = thm "HOL.rev_mp"
haftmann@21547
   824
  val subst = thm "HOL.subst"
haftmann@21547
   825
  val sym = thm "HOL.sym"
haftmann@21151
   826
  val thin_refl = thm "thin_refl";
haftmann@21151
   827
end);
wenzelm@21671
   828
open Hypsubst;
haftmann@21151
   829
haftmann@21151
   830
structure Classical = ClassicalFun(
haftmann@21151
   831
struct
haftmann@21547
   832
  val mp = thm "HOL.mp"
haftmann@21547
   833
  val not_elim = thm "HOL.notE"
haftmann@21547
   834
  val classical = thm "HOL.classical"
haftmann@21151
   835
  val sizef = Drule.size_of_thm
haftmann@21151
   836
  val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
haftmann@21151
   837
end);
haftmann@21151
   838
haftmann@21151
   839
structure BasicClassical: BASIC_CLASSICAL = Classical; 
wenzelm@21671
   840
open BasicClassical;
haftmann@21151
   841
*}
haftmann@21151
   842
haftmann@21009
   843
setup {*
haftmann@21009
   844
let
haftmann@21009
   845
  (*prevent substitution on bool*)
haftmann@21009
   846
  fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
haftmann@21009
   847
    Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
haftmann@21009
   848
      (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
haftmann@21009
   849
in
haftmann@21151
   850
  Hypsubst.hypsubst_setup
haftmann@21151
   851
  #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
haftmann@21151
   852
  #> Classical.setup
haftmann@21151
   853
  #> ResAtpset.setup
haftmann@21009
   854
end
haftmann@21009
   855
*}
haftmann@21009
   856
haftmann@21009
   857
declare iffI [intro!]
haftmann@21009
   858
  and notI [intro!]
haftmann@21009
   859
  and impI [intro!]
haftmann@21009
   860
  and disjCI [intro!]
haftmann@21009
   861
  and conjI [intro!]
haftmann@21009
   862
  and TrueI [intro!]
haftmann@21009
   863
  and refl [intro!]
haftmann@21009
   864
haftmann@21009
   865
declare iffCE [elim!]
haftmann@21009
   866
  and FalseE [elim!]
haftmann@21009
   867
  and impCE [elim!]
haftmann@21009
   868
  and disjE [elim!]
haftmann@21009
   869
  and conjE [elim!]
haftmann@21009
   870
  and conjE [elim!]
haftmann@21009
   871
haftmann@21009
   872
declare ex_ex1I [intro!]
haftmann@21009
   873
  and allI [intro!]
haftmann@21009
   874
  and the_equality [intro]
haftmann@21009
   875
  and exI [intro]
haftmann@21009
   876
haftmann@21009
   877
declare exE [elim!]
haftmann@21009
   878
  allE [elim]
haftmann@21009
   879
haftmann@21009
   880
ML {*
haftmann@21547
   881
val HOL_cs = Classical.claset_of (the_context ());
haftmann@21009
   882
*}
mengj@19162
   883
wenzelm@20223
   884
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
wenzelm@20223
   885
  apply (erule swap)
wenzelm@20223
   886
  apply (erule (1) meta_mp)
wenzelm@20223
   887
  done
wenzelm@10383
   888
wenzelm@18689
   889
declare ex_ex1I [rule del, intro! 2]
wenzelm@18689
   890
  and ex1I [intro]
wenzelm@18689
   891
wenzelm@12386
   892
lemmas [intro?] = ext
wenzelm@12386
   893
  and [elim?] = ex1_implies_ex
wenzelm@11977
   894
haftmann@20944
   895
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
haftmann@20973
   896
lemma alt_ex1E [elim!]:
haftmann@20944
   897
  assumes major: "\<exists>!x. P x"
haftmann@20944
   898
      and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
haftmann@20944
   899
  shows R
haftmann@20944
   900
apply (rule ex1E [OF major])
haftmann@20944
   901
apply (rule prem)
haftmann@21547
   902
apply (tactic {* ares_tac [thm "allI"] 1 *})+
haftmann@21547
   903
apply (tactic {* etac (Classical.dup_elim (thm "allE")) 1 *})
haftmann@20944
   904
by iprover
haftmann@20944
   905
haftmann@21151
   906
ML {*
haftmann@21151
   907
structure Blast = BlastFun(
haftmann@21151
   908
struct
haftmann@21151
   909
  type claset = Classical.claset
haftmann@21151
   910
  val equality_name = "op ="
haftmann@21151
   911
  val not_name = "Not"
haftmann@21547
   912
  val notE = thm "HOL.notE"
haftmann@21547
   913
  val ccontr = thm "HOL.ccontr"
haftmann@21151
   914
  val contr_tac = Classical.contr_tac
haftmann@21151
   915
  val dup_intr = Classical.dup_intr
haftmann@21151
   916
  val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
wenzelm@21671
   917
  val claset = Classical.claset
haftmann@21151
   918
  val rep_cs = Classical.rep_cs
haftmann@21151
   919
  val cla_modifiers = Classical.cla_modifiers
haftmann@21151
   920
  val cla_meth' = Classical.cla_meth'
haftmann@21151
   921
end);
wenzelm@21671
   922
val Blast_tac = Blast.Blast_tac;
wenzelm@21671
   923
val blast_tac = Blast.blast_tac;
haftmann@20944
   924
*}
haftmann@20944
   925
haftmann@21151
   926
setup Blast.setup
haftmann@21151
   927
haftmann@20944
   928
haftmann@20944
   929
subsubsection {* Simplifier *}
wenzelm@12281
   930
wenzelm@12281
   931
lemma eta_contract_eq: "(%s. f s) = f" ..
wenzelm@12281
   932
wenzelm@12281
   933
lemma simp_thms:
wenzelm@12937
   934
  shows not_not: "(~ ~ P) = P"
nipkow@15354
   935
  and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
wenzelm@12937
   936
  and
berghofe@12436
   937
    "(P ~= Q) = (P = (~Q))"
berghofe@12436
   938
    "(P | ~P) = True"    "(~P | P) = True"
wenzelm@12281
   939
    "(x = x) = True"
haftmann@20944
   940
  and not_True_eq_False: "(\<not> True) = False"
haftmann@20944
   941
  and not_False_eq_True: "(\<not> False) = True"
haftmann@20944
   942
  and
berghofe@12436
   943
    "(~P) ~= P"  "P ~= (~P)"
haftmann@20944
   944
    "(True=P) = P"
haftmann@20944
   945
  and eq_True: "(P = True) = P"
haftmann@20944
   946
  and "(False=P) = (~P)"
haftmann@20944
   947
  and eq_False: "(P = False) = (\<not> P)"
haftmann@20944
   948
  and
wenzelm@12281
   949
    "(True --> P) = P"  "(False --> P) = True"
wenzelm@12281
   950
    "(P --> True) = True"  "(P --> P) = True"
wenzelm@12281
   951
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
wenzelm@12281
   952
    "(P & True) = P"  "(True & P) = P"
wenzelm@12281
   953
    "(P & False) = False"  "(False & P) = False"
wenzelm@12281
   954
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
wenzelm@12281
   955
    "(P & ~P) = False"    "(~P & P) = False"
wenzelm@12281
   956
    "(P | True) = True"  "(True | P) = True"
wenzelm@12281
   957
    "(P | False) = P"  "(False | P) = P"
berghofe@12436
   958
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
wenzelm@12281
   959
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
wenzelm@12281
   960
    -- {* needed for the one-point-rule quantifier simplification procs *}
wenzelm@12281
   961
    -- {* essential for termination!! *} and
wenzelm@12281
   962
    "!!P. (EX x. x=t & P(x)) = P(t)"
wenzelm@12281
   963
    "!!P. (EX x. t=x & P(x)) = P(t)"
wenzelm@12281
   964
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
wenzelm@12937
   965
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
nipkow@17589
   966
  by (blast, blast, blast, blast, blast, iprover+)
wenzelm@13421
   967
paulson@14201
   968
lemma disj_absorb: "(A | A) = A"
paulson@14201
   969
  by blast
paulson@14201
   970
paulson@14201
   971
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
paulson@14201
   972
  by blast
paulson@14201
   973
paulson@14201
   974
lemma conj_absorb: "(A & A) = A"
paulson@14201
   975
  by blast
paulson@14201
   976
paulson@14201
   977
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
paulson@14201
   978
  by blast
paulson@14201
   979
wenzelm@12281
   980
lemma eq_ac:
wenzelm@12937
   981
  shows eq_commute: "(a=b) = (b=a)"
wenzelm@12937
   982
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
nipkow@17589
   983
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
nipkow@17589
   984
lemma neq_commute: "(a~=b) = (b~=a)" by iprover
wenzelm@12281
   985
wenzelm@12281
   986
lemma conj_comms:
wenzelm@12937
   987
  shows conj_commute: "(P&Q) = (Q&P)"
nipkow@17589
   988
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
nipkow@17589
   989
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
wenzelm@12281
   990
paulson@19174
   991
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
paulson@19174
   992
wenzelm@12281
   993
lemma disj_comms:
wenzelm@12937
   994
  shows disj_commute: "(P|Q) = (Q|P)"
nipkow@17589
   995
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
nipkow@17589
   996
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
wenzelm@12281
   997
paulson@19174
   998
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
paulson@19174
   999
nipkow@17589
  1000
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
nipkow@17589
  1001
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
wenzelm@12281
  1002
nipkow@17589
  1003
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
nipkow@17589
  1004
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
wenzelm@12281
  1005
nipkow@17589
  1006
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
nipkow@17589
  1007
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover
nipkow@17589
  1008
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
wenzelm@12281
  1009
wenzelm@12281
  1010
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
wenzelm@12281
  1011
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
wenzelm@12281
  1012
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
wenzelm@12281
  1013
wenzelm@12281
  1014
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
wenzelm@12281
  1015
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
wenzelm@12281
  1016
haftmann@21151
  1017
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
haftmann@21151
  1018
  by iprover
haftmann@21151
  1019
nipkow@17589
  1020
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
wenzelm@12281
  1021
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
wenzelm@12281
  1022
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
wenzelm@12281
  1023
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
wenzelm@12281
  1024
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
wenzelm@12281
  1025
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
wenzelm@12281
  1026
  by blast
wenzelm@12281
  1027
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
wenzelm@12281
  1028
nipkow@17589
  1029
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
wenzelm@12281
  1030
wenzelm@12281
  1031
wenzelm@12281
  1032
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
wenzelm@12281
  1033
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
wenzelm@12281
  1034
  -- {* cases boil down to the same thing. *}
wenzelm@12281
  1035
  by blast
wenzelm@12281
  1036
wenzelm@12281
  1037
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
wenzelm@12281
  1038
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
nipkow@17589
  1039
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
nipkow@17589
  1040
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
wenzelm@12281
  1041
nipkow@17589
  1042
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
nipkow@17589
  1043
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
wenzelm@12281
  1044
wenzelm@12281
  1045
text {*
wenzelm@12281
  1046
  \medskip The @{text "&"} congruence rule: not included by default!
wenzelm@12281
  1047
  May slow rewrite proofs down by as much as 50\% *}
wenzelm@12281
  1048
wenzelm@12281
  1049
lemma conj_cong:
wenzelm@12281
  1050
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
nipkow@17589
  1051
  by iprover
wenzelm@12281
  1052
wenzelm@12281
  1053
lemma rev_conj_cong:
wenzelm@12281
  1054
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
nipkow@17589
  1055
  by iprover
wenzelm@12281
  1056
wenzelm@12281
  1057
text {* The @{text "|"} congruence rule: not included by default! *}
wenzelm@12281
  1058
wenzelm@12281
  1059
lemma disj_cong:
wenzelm@12281
  1060
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
wenzelm@12281
  1061
  by blast
wenzelm@12281
  1062
wenzelm@12281
  1063
wenzelm@12281
  1064
text {* \medskip if-then-else rules *}
wenzelm@12281
  1065
wenzelm@12281
  1066
lemma if_True: "(if True then x else y) = x"
wenzelm@12281
  1067
  by (unfold if_def) blast
wenzelm@12281
  1068
wenzelm@12281
  1069
lemma if_False: "(if False then x else y) = y"
wenzelm@12281
  1070
  by (unfold if_def) blast
wenzelm@12281
  1071
wenzelm@12281
  1072
lemma if_P: "P ==> (if P then x else y) = x"
wenzelm@12281
  1073
  by (unfold if_def) blast
wenzelm@12281
  1074
wenzelm@12281
  1075
lemma if_not_P: "~P ==> (if P then x else y) = y"
wenzelm@12281
  1076
  by (unfold if_def) blast
wenzelm@12281
  1077
wenzelm@12281
  1078
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
wenzelm@12281
  1079
  apply (rule case_split [of Q])
paulson@15481
  1080
   apply (simplesubst if_P)
paulson@15481
  1081
    prefer 3 apply (simplesubst if_not_P, blast+)
wenzelm@12281
  1082
  done
wenzelm@12281
  1083
wenzelm@12281
  1084
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
paulson@15481
  1085
by (simplesubst split_if, blast)
wenzelm@12281
  1086
wenzelm@12281
  1087
lemmas if_splits = split_if split_if_asm
wenzelm@12281
  1088
wenzelm@12281
  1089
lemma if_cancel: "(if c then x else x) = x"
paulson@15481
  1090
by (simplesubst split_if, blast)
wenzelm@12281
  1091
wenzelm@12281
  1092
lemma if_eq_cancel: "(if x = y then y else x) = x"
paulson@15481
  1093
by (simplesubst split_if, blast)
wenzelm@12281
  1094
wenzelm@12281
  1095
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
wenzelm@19796
  1096
  -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
wenzelm@12281
  1097
  by (rule split_if)
wenzelm@12281
  1098
wenzelm@12281
  1099
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
wenzelm@19796
  1100
  -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
paulson@15481
  1101
  apply (simplesubst split_if, blast)
wenzelm@12281
  1102
  done
wenzelm@12281
  1103
nipkow@17589
  1104
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
nipkow@17589
  1105
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
wenzelm@12281
  1106
schirmer@15423
  1107
text {* \medskip let rules for simproc *}
schirmer@15423
  1108
schirmer@15423
  1109
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
schirmer@15423
  1110
  by (unfold Let_def)
schirmer@15423
  1111
schirmer@15423
  1112
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
schirmer@15423
  1113
  by (unfold Let_def)
schirmer@15423
  1114
berghofe@16633
  1115
text {*
ballarin@16999
  1116
  The following copy of the implication operator is useful for
ballarin@16999
  1117
  fine-tuning congruence rules.  It instructs the simplifier to simplify
ballarin@16999
  1118
  its premise.
berghofe@16633
  1119
*}
berghofe@16633
  1120
wenzelm@17197
  1121
constdefs
wenzelm@17197
  1122
  simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1)
wenzelm@17197
  1123
  "simp_implies \<equiv> op ==>"
berghofe@16633
  1124
wenzelm@18457
  1125
lemma simp_impliesI:
berghofe@16633
  1126
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
berghofe@16633
  1127
  shows "PROP P =simp=> PROP Q"
berghofe@16633
  1128
  apply (unfold simp_implies_def)
berghofe@16633
  1129
  apply (rule PQ)
berghofe@16633
  1130
  apply assumption
berghofe@16633
  1131
  done
berghofe@16633
  1132
berghofe@16633
  1133
lemma simp_impliesE:
berghofe@16633
  1134
  assumes PQ:"PROP P =simp=> PROP Q"
berghofe@16633
  1135
  and P: "PROP P"
berghofe@16633
  1136
  and QR: "PROP Q \<Longrightarrow> PROP R"
berghofe@16633
  1137
  shows "PROP R"
berghofe@16633
  1138
  apply (rule QR)
berghofe@16633
  1139
  apply (rule PQ [unfolded simp_implies_def])
berghofe@16633
  1140
  apply (rule P)
berghofe@16633
  1141
  done
berghofe@16633
  1142
berghofe@16633
  1143
lemma simp_implies_cong:
berghofe@16633
  1144
  assumes PP' :"PROP P == PROP P'"
berghofe@16633
  1145
  and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
berghofe@16633
  1146
  shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
berghofe@16633
  1147
proof (unfold simp_implies_def, rule equal_intr_rule)
berghofe@16633
  1148
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
berghofe@16633
  1149
  and P': "PROP P'"
berghofe@16633
  1150
  from PP' [symmetric] and P' have "PROP P"
berghofe@16633
  1151
    by (rule equal_elim_rule1)
berghofe@16633
  1152
  hence "PROP Q" by (rule PQ)
berghofe@16633
  1153
  with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
berghofe@16633
  1154
next
berghofe@16633
  1155
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
berghofe@16633
  1156
  and P: "PROP P"
berghofe@16633
  1157
  from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
berghofe@16633
  1158
  hence "PROP Q'" by (rule P'Q')
berghofe@16633
  1159
  with P'QQ' [OF P', symmetric] show "PROP Q"
berghofe@16633
  1160
    by (rule equal_elim_rule1)
berghofe@16633
  1161
qed
berghofe@16633
  1162
haftmann@20944
  1163
lemma uncurry:
haftmann@20944
  1164
  assumes "P \<longrightarrow> Q \<longrightarrow> R"
haftmann@20944
  1165
  shows "P \<and> Q \<longrightarrow> R"
haftmann@20944
  1166
  using prems by blast
haftmann@20944
  1167
haftmann@20944
  1168
lemma iff_allI:
haftmann@20944
  1169
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1170
  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
haftmann@20944
  1171
  using prems by blast
haftmann@20944
  1172
haftmann@20944
  1173
lemma iff_exI:
haftmann@20944
  1174
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1175
  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
haftmann@20944
  1176
  using prems by blast
haftmann@20944
  1177
haftmann@20944
  1178
lemma all_comm:
haftmann@20944
  1179
  "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
haftmann@20944
  1180
  by blast
haftmann@20944
  1181
haftmann@20944
  1182
lemma ex_comm:
haftmann@20944
  1183
  "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
haftmann@20944
  1184
  by blast
haftmann@20944
  1185
wenzelm@9869
  1186
use "simpdata.ML"
wenzelm@21671
  1187
ML {* open Simpdata *}
wenzelm@21671
  1188
haftmann@21151
  1189
setup {*
haftmann@21151
  1190
  Simplifier.method_setup Splitter.split_modifiers
haftmann@21547
  1191
  #> (fn thy => (change_simpset_of thy (fn _ => Simpdata.simpset_simprocs); thy))
haftmann@21151
  1192
  #> Splitter.setup
haftmann@21151
  1193
  #> Clasimp.setup
haftmann@21151
  1194
  #> EqSubst.setup
haftmann@21151
  1195
*}
haftmann@21151
  1196
haftmann@21151
  1197
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
haftmann@21151
  1198
proof
haftmann@21151
  1199
  assume prem: "True \<Longrightarrow> PROP P"
haftmann@21151
  1200
  from prem [OF TrueI] show "PROP P" . 
haftmann@21151
  1201
next
haftmann@21151
  1202
  assume "PROP P"
haftmann@21151
  1203
  show "PROP P" .
haftmann@21151
  1204
qed
haftmann@21151
  1205
haftmann@21151
  1206
lemma ex_simps:
haftmann@21151
  1207
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
haftmann@21151
  1208
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
haftmann@21151
  1209
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
haftmann@21151
  1210
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
haftmann@21151
  1211
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
haftmann@21151
  1212
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
haftmann@21151
  1213
  -- {* Miniscoping: pushing in existential quantifiers. *}
haftmann@21151
  1214
  by (iprover | blast)+
haftmann@21151
  1215
haftmann@21151
  1216
lemma all_simps:
haftmann@21151
  1217
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
haftmann@21151
  1218
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
haftmann@21151
  1219
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
haftmann@21151
  1220
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
haftmann@21151
  1221
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
haftmann@21151
  1222
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
haftmann@21151
  1223
  -- {* Miniscoping: pushing in universal quantifiers. *}
haftmann@21151
  1224
  by (iprover | blast)+
paulson@15481
  1225
wenzelm@21671
  1226
lemmas [simp] =
wenzelm@21671
  1227
  triv_forall_equality (*prunes params*)
wenzelm@21671
  1228
  True_implies_equals  (*prune asms `True'*)
wenzelm@21671
  1229
  if_True
wenzelm@21671
  1230
  if_False
wenzelm@21671
  1231
  if_cancel
wenzelm@21671
  1232
  if_eq_cancel
wenzelm@21671
  1233
  imp_disjL
haftmann@20973
  1234
  (*In general it seems wrong to add distributive laws by default: they
haftmann@20973
  1235
    might cause exponential blow-up.  But imp_disjL has been in for a while
haftmann@20973
  1236
    and cannot be removed without affecting existing proofs.  Moreover,
haftmann@20973
  1237
    rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
haftmann@20973
  1238
    grounds that it allows simplification of R in the two cases.*)
wenzelm@21671
  1239
  conj_assoc
wenzelm@21671
  1240
  disj_assoc
wenzelm@21671
  1241
  de_Morgan_conj
wenzelm@21671
  1242
  de_Morgan_disj
wenzelm@21671
  1243
  imp_disj1
wenzelm@21671
  1244
  imp_disj2
wenzelm@21671
  1245
  not_imp
wenzelm@21671
  1246
  disj_not1
wenzelm@21671
  1247
  not_all
wenzelm@21671
  1248
  not_ex
wenzelm@21671
  1249
  cases_simp
wenzelm@21671
  1250
  the_eq_trivial
wenzelm@21671
  1251
  the_sym_eq_trivial
wenzelm@21671
  1252
  ex_simps
wenzelm@21671
  1253
  all_simps
wenzelm@21671
  1254
  simp_thms
wenzelm@21671
  1255
wenzelm@21671
  1256
lemmas [cong] = imp_cong simp_implies_cong
wenzelm@21671
  1257
lemmas [split] = split_if
haftmann@20973
  1258
haftmann@20973
  1259
ML {*
haftmann@21547
  1260
val HOL_ss = Simplifier.simpset_of (the_context ());
haftmann@20973
  1261
*}
haftmann@20973
  1262
haftmann@20944
  1263
text {* Simplifies x assuming c and y assuming ~c *}
haftmann@20944
  1264
lemma if_cong:
haftmann@20944
  1265
  assumes "b = c"
haftmann@20944
  1266
      and "c \<Longrightarrow> x = u"
haftmann@20944
  1267
      and "\<not> c \<Longrightarrow> y = v"
haftmann@20944
  1268
  shows "(if b then x else y) = (if c then u else v)"
haftmann@20944
  1269
  unfolding if_def using prems by simp
haftmann@20944
  1270
haftmann@20944
  1271
text {* Prevents simplification of x and y:
haftmann@20944
  1272
  faster and allows the execution of functional programs. *}
haftmann@20944
  1273
lemma if_weak_cong [cong]:
haftmann@20944
  1274
  assumes "b = c"
haftmann@20944
  1275
  shows "(if b then x else y) = (if c then x else y)"
haftmann@20944
  1276
  using prems by (rule arg_cong)
haftmann@20944
  1277
haftmann@20944
  1278
text {* Prevents simplification of t: much faster *}
haftmann@20944
  1279
lemma let_weak_cong:
haftmann@20944
  1280
  assumes "a = b"
haftmann@20944
  1281
  shows "(let x = a in t x) = (let x = b in t x)"
haftmann@20944
  1282
  using prems by (rule arg_cong)
haftmann@20944
  1283
haftmann@20944
  1284
text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}
haftmann@20944
  1285
lemma eq_cong2:
haftmann@20944
  1286
  assumes "u = u'"
haftmann@20944
  1287
  shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
haftmann@20944
  1288
  using prems by simp
haftmann@20944
  1289
haftmann@20944
  1290
lemma if_distrib:
haftmann@20944
  1291
  "f (if c then x else y) = (if c then f x else f y)"
haftmann@20944
  1292
  by simp
haftmann@20944
  1293
haftmann@20944
  1294
text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
wenzelm@21502
  1295
  side of an equality.  Used in @{text "{Integ,Real}/simproc.ML"} *}
haftmann@20944
  1296
lemma restrict_to_left:
haftmann@20944
  1297
  assumes "x = y"
haftmann@20944
  1298
  shows "(x = z) = (y = z)"
haftmann@20944
  1299
  using prems by simp
haftmann@20944
  1300
wenzelm@17459
  1301
haftmann@20944
  1302
subsubsection {* Generic cases and induction *}
wenzelm@17459
  1303
haftmann@20944
  1304
text {* Rule projections: *}
berghofe@18887
  1305
haftmann@20944
  1306
ML {*
haftmann@20944
  1307
structure ProjectRule = ProjectRuleFun
haftmann@20944
  1308
(struct
haftmann@20944
  1309
  val conjunct1 = thm "conjunct1";
haftmann@20944
  1310
  val conjunct2 = thm "conjunct2";
haftmann@20944
  1311
  val mp = thm "mp";
haftmann@20944
  1312
end)
wenzelm@17459
  1313
*}
wenzelm@17459
  1314
wenzelm@11824
  1315
constdefs
wenzelm@18457
  1316
  induct_forall where "induct_forall P == \<forall>x. P x"
wenzelm@18457
  1317
  induct_implies where "induct_implies A B == A \<longrightarrow> B"
wenzelm@18457
  1318
  induct_equal where "induct_equal x y == x = y"
wenzelm@18457
  1319
  induct_conj where "induct_conj A B == A \<and> B"
wenzelm@11824
  1320
wenzelm@11989
  1321
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
wenzelm@18457
  1322
  by (unfold atomize_all induct_forall_def)
wenzelm@11824
  1323
wenzelm@11989
  1324
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
wenzelm@18457
  1325
  by (unfold atomize_imp induct_implies_def)
wenzelm@11824
  1326
wenzelm@11989
  1327
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
wenzelm@18457
  1328
  by (unfold atomize_eq induct_equal_def)
wenzelm@18457
  1329
wenzelm@18457
  1330
lemma induct_conj_eq:
wenzelm@18457
  1331
  includes meta_conjunction_syntax
wenzelm@18457
  1332
  shows "(A && B) == Trueprop (induct_conj A B)"
wenzelm@18457
  1333
  by (unfold atomize_conj induct_conj_def)
wenzelm@18457
  1334
wenzelm@18457
  1335
lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
wenzelm@18457
  1336
lemmas induct_rulify [symmetric, standard] = induct_atomize
wenzelm@18457
  1337
lemmas induct_rulify_fallback =
wenzelm@18457
  1338
  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
wenzelm@18457
  1339
wenzelm@11824
  1340
wenzelm@11989
  1341
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
wenzelm@11989
  1342
    induct_conj (induct_forall A) (induct_forall B)"
nipkow@17589
  1343
  by (unfold induct_forall_def induct_conj_def) iprover
wenzelm@11824
  1344
wenzelm@11989
  1345
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
wenzelm@11989
  1346
    induct_conj (induct_implies C A) (induct_implies C B)"
nipkow@17589
  1347
  by (unfold induct_implies_def induct_conj_def) iprover
wenzelm@11989
  1348
berghofe@13598
  1349
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
berghofe@13598
  1350
proof
berghofe@13598
  1351
  assume r: "induct_conj A B ==> PROP C" and A B
wenzelm@18457
  1352
  show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
berghofe@13598
  1353
next
berghofe@13598
  1354
  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
wenzelm@18457
  1355
  show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
berghofe@13598
  1356
qed
wenzelm@11824
  1357
wenzelm@11989
  1358
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
wenzelm@11824
  1359
wenzelm@11989
  1360
hide const induct_forall induct_implies induct_equal induct_conj
wenzelm@11824
  1361
wenzelm@11824
  1362
text {* Method setup. *}
wenzelm@11824
  1363
wenzelm@11824
  1364
ML {*
wenzelm@11824
  1365
  structure InductMethod = InductMethodFun
wenzelm@11824
  1366
  (struct
paulson@15411
  1367
    val cases_default = thm "case_split"
paulson@15411
  1368
    val atomize = thms "induct_atomize"
wenzelm@18457
  1369
    val rulify = thms "induct_rulify"
wenzelm@18457
  1370
    val rulify_fallback = thms "induct_rulify_fallback"
wenzelm@11824
  1371
  end);
wenzelm@11824
  1372
*}
wenzelm@11824
  1373
wenzelm@11824
  1374
setup InductMethod.setup
wenzelm@11824
  1375
wenzelm@18457
  1376
haftmann@20944
  1377
haftmann@20944
  1378
subsection {* Other simple lemmas and lemma duplicates *}
haftmann@20944
  1379
haftmann@20944
  1380
lemmas eq_sym_conv = eq_commute
haftmann@20944
  1381
lemmas if_def2 = if_bool_eq_conj
haftmann@20944
  1382
haftmann@20944
  1383
lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
haftmann@20944
  1384
  by blast+
haftmann@20944
  1385
haftmann@20944
  1386
lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
haftmann@20944
  1387
  apply (rule iffI)
haftmann@20944
  1388
  apply (rule_tac a = "%x. THE y. P x y" in ex1I)
haftmann@20944
  1389
  apply (fast dest!: theI')
haftmann@20944
  1390
  apply (fast intro: ext the1_equality [symmetric])
haftmann@20944
  1391
  apply (erule ex1E)
haftmann@20944
  1392
  apply (rule allI)
haftmann@20944
  1393
  apply (rule ex1I)
haftmann@20944
  1394
  apply (erule spec)
haftmann@20944
  1395
  apply (erule_tac x = "%z. if z = x then y else f z" in allE)
haftmann@20944
  1396
  apply (erule impE)
haftmann@20944
  1397
  apply (rule allI)
haftmann@20944
  1398
  apply (rule_tac P = "xa = x" in case_split_thm)
haftmann@20944
  1399
  apply (drule_tac [3] x = x in fun_cong, simp_all)
haftmann@20944
  1400
  done
haftmann@20944
  1401
haftmann@20944
  1402
lemma mk_left_commute:
haftmann@21547
  1403
  fixes f (infix "\<otimes>" 60)
haftmann@21547
  1404
  assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and
haftmann@21547
  1405
          c: "\<And>x y. x \<otimes> y = y \<otimes> x"
haftmann@21547
  1406
  shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
haftmann@20944
  1407
  by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
haftmann@20944
  1408
wenzelm@21671
  1409
wenzelm@21671
  1410
subsection {* Basic ML bindings *}
wenzelm@21671
  1411
wenzelm@21671
  1412
ML {*
wenzelm@21671
  1413
val FalseE = thm "FalseE"
wenzelm@21671
  1414
val Let_def = thm "Let_def"
wenzelm@21671
  1415
val TrueI = thm "TrueI";
wenzelm@21671
  1416
val allE = thm "allE";
wenzelm@21671
  1417
val allI = thm "allI";
wenzelm@21671
  1418
val all_dupE = thm "all_dupE"
wenzelm@21671
  1419
val arg_cong = thm "arg_cong";
wenzelm@21671
  1420
val box_equals = thm "box_equals"
wenzelm@21671
  1421
val ccontr = thm "ccontr";
wenzelm@21671
  1422
val classical = thm "classical";
wenzelm@21671
  1423
val conjE = thm "conjE";
wenzelm@21671
  1424
val conjI = thm "conjI";
wenzelm@21671
  1425
val conjunct1 = thm "conjunct1";
wenzelm@21671
  1426
val conjunct2 = thm "conjunct2";
wenzelm@21671
  1427
val disjCI = thm "disjCI";
wenzelm@21671
  1428
val disjE = thm "disjE";
wenzelm@21671
  1429
val disjI1 = thm "disjI1"
wenzelm@21671
  1430
val disjI2 = thm "disjI2"
wenzelm@21671
  1431
val eq_reflection = thm "eq_reflection";
wenzelm@21671
  1432
val ex1E = thm "ex1E"
wenzelm@21671
  1433
val ex1I = thm "ex1I"
wenzelm@21671
  1434
val ex1_implies_ex = thm "ex1_implies_ex"
wenzelm@21671
  1435
val exE = thm "exE";
wenzelm@21671
  1436
val exI = thm "exI";
wenzelm@21671
  1437
val excluded_middle = thm "excluded_middle"
wenzelm@21671
  1438
val ext = thm "ext"
wenzelm@21671
  1439
val fun_cong = thm "fun_cong"
wenzelm@21671
  1440
val iffD1 = thm "iffD1";
wenzelm@21671
  1441
val iffD2 = thm "iffD2";
wenzelm@21671
  1442
val iffI = thm "iffI";
wenzelm@21671
  1443
val impE = thm "impE"
wenzelm@21671
  1444
val impI = thm "impI";
wenzelm@21671
  1445
val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq";
wenzelm@21671
  1446
val mp = thm "mp";
wenzelm@21671
  1447
val notE = thm "notE";
wenzelm@21671
  1448
val notI = thm "notI";
wenzelm@21671
  1449
val not_all = thm "not_all";
wenzelm@21671
  1450
val not_ex = thm "not_ex";
wenzelm@21671
  1451
val not_iff = thm "not_iff";
wenzelm@21671
  1452
val not_not = thm "not_not";
wenzelm@21671
  1453
val not_sym = thm "not_sym"
wenzelm@21671
  1454
val refl = thm "refl";
wenzelm@21671
  1455
val rev_mp = thm "rev_mp"
wenzelm@21671
  1456
val spec = thm "spec";
wenzelm@21671
  1457
val ssubst = thm "ssubst"
wenzelm@21671
  1458
val subst = thm "subst";
wenzelm@21671
  1459
val sym = thm "sym";
wenzelm@21671
  1460
val trans = thm "trans";
wenzelm@21671
  1461
*}
wenzelm@21671
  1462
wenzelm@21671
  1463
wenzelm@21671
  1464
subsection {* Legacy tactics *}
wenzelm@21671
  1465
wenzelm@21671
  1466
ML {*
wenzelm@21671
  1467
fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
wenzelm@21671
  1468
wenzelm@21671
  1469
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
wenzelm@21671
  1470
local
wenzelm@21671
  1471
  fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
wenzelm@21671
  1472
    | wrong_prem (Bound _) = true
wenzelm@21671
  1473
    | wrong_prem _ = false;
wenzelm@21671
  1474
  val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
wenzelm@21671
  1475
  val spec = thm "spec"
wenzelm@21671
  1476
  val mp = thm "mp"
wenzelm@21671
  1477
in
wenzelm@21671
  1478
  fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
wenzelm@21671
  1479
  fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
wenzelm@21671
  1480
end;
wenzelm@21671
  1481
*}
wenzelm@21671
  1482
kleing@14357
  1483
end