src/HOL/HOL.thy
author kleing
Wed Apr 14 14:13:05 2004 +0200 (2004-04-14)
changeset 14565 c6dc17aab88a
parent 14444 24724afce166
child 14590 276ef51cedbf
permissions -rw-r--r--
use more symbols in HTML output
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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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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* The basis of Higher-Order Logic *}
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theory HOL = CPure
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files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML"):
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subsection {* Primitive logic *}
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subsubsection {* Core syntax *}
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classes type < logic
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defaultsort type
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global
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typedecl bool
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arities
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  bool :: type
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  fun :: (type, type) type
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judgment
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  Trueprop      :: "bool => prop"                   ("(_)" 5)
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consts
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  Not           :: "bool => bool"                   ("~ _" [40] 40)
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  True          :: bool
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  False         :: bool
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  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
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  arbitrary     :: 'a
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  The           :: "('a => bool) => 'a"
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  All           :: "('a => bool) => bool"           (binder "ALL " 10)
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  Ex            :: "('a => bool) => bool"           (binder "EX " 10)
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  Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
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  Let           :: "['a, 'a => 'b] => 'b"
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  "="           :: "['a, 'a] => bool"               (infixl 50)
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  &             :: "[bool, bool] => bool"           (infixr 35)
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  "|"           :: "[bool, bool] => bool"           (infixr 30)
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  -->           :: "[bool, bool] => bool"           (infixr 25)
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local
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subsubsection {* Additional concrete syntax *}
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nonterminals
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  letbinds  letbind
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  case_syn  cases_syn
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syntax
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  "_not_equal"  :: "['a, 'a] => bool"                    (infixl "~=" 50)
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  "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
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  "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
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  ""            :: "letbind => letbinds"                 ("_")
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  "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
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  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
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  "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
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  ""            :: "case_syn => cases_syn"               ("_")
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  "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
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translations
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  "x ~= y"                == "~ (x = y)"
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  "THE x. P"              == "The (%x. P)"
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  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
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  "let x = a in e"        == "Let a (%x. e)"
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print_translation {*
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(* To avoid eta-contraction of body: *)
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[("The", fn [Abs abs] =>
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     let val (x,t) = atomic_abs_tr' abs
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     in Syntax.const "_The" $ x $ t end)]
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*}
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syntax (output)
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  "="           :: "['a, 'a] => bool"                    (infix 50)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "~=" 50)
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syntax (xsymbols)
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  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
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  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
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  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
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  "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
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(*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \<orelse> _")*)
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syntax (xsymbols output)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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syntax (HTML output)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
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  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
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  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
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  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
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syntax (HOL)
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  "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
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subsubsection {* Axioms and basic definitions *}
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axioms
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  eq_reflection: "(x=y) ==> (x==y)"
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  refl:         "t = (t::'a)"
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  subst:        "[| s = t; P(s) |] ==> P(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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subsubsection {* Generic algebraic operations *}
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axclass zero < type
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axclass one < type
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axclass plus < type
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axclass minus < type
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axclass times < type
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axclass inverse < type
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global
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consts
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  "0"           :: "'a::zero"                       ("0")
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  "1"           :: "'a::one"                        ("1")
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  "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
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  -             :: "['a::minus, 'a] => 'a"          (infixl 65)
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  uminus        :: "['a::minus] => 'a"              ("- _" [81] 80)
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  *             :: "['a::times, 'a] => 'a"          (infixl 70)
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syntax
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  "_index1"  :: index    ("\<^sub>1")
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translations
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  (index) "\<^sub>1" == "_index 1"
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local
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typed_print_translation {*
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  let
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    fun tr' c = (c, fn show_sorts => fn T => fn ts =>
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      if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
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      else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
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  in [tr' "0", tr' "1"] end;
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*} -- {* show types that are presumably too general *}
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consts
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  abs           :: "'a::minus => 'a"
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  inverse       :: "'a::inverse => 'a"
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  divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
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syntax (xsymbols)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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syntax (HTML output)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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subsection {* Theory and package setup *}
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subsubsection {* Basic lemmas *}
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use "HOL_lemmas.ML"
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theorems case_split = case_split_thm [case_names True False]
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subsubsection {* Intuitionistic Reasoning *}
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lemma impE':
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  assumes 1: "P --> Q"
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    and 2: "Q ==> R"
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    and 3: "P --> Q ==> P"
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  shows R
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proof -
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  from 3 and 1 have P .
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  with 1 have Q by (rule impE)
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  with 2 show R .
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qed
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lemma allE':
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  assumes 1: "ALL x. P x"
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    and 2: "P x ==> ALL x. P x ==> Q"
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  shows Q
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proof -
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  from 1 have "P x" by (rule spec)
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  from this and 1 show Q by (rule 2)
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qed
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lemma notE':
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  assumes 1: "~ P"
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    and 2: "~ P ==> P"
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  shows R
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proof -
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  from 2 and 1 have P .
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  with 1 show R by (rule notE)
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qed
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lemmas [CPure.elim!] = disjE iffE FalseE conjE exE
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  and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
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  and [CPure.elim 2] = allE notE' impE'
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  and [CPure.intro] = exI disjI2 disjI1
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lemmas [trans] = trans
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  and [sym] = sym not_sym
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  and [CPure.elim?] = iffD1 iffD2 impE
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subsubsection {* Atomizing meta-level connectives *}
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lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
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proof
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  assume "!!x. P x"
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  show "ALL x. P x" by (rule allI)
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next
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  assume "ALL x. P x"
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  thus "!!x. P x" by (rule allE)
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qed
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lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
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proof
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  assume r: "A ==> B"
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  show "A --> B" by (rule impI) (rule r)
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next
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  assume "A --> B" and A
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  thus B by (rule mp)
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qed
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lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
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proof
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  assume "x == y"
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  show "x = y" by (unfold prems) (rule refl)
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next
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  assume "x = y"
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  thus "x == y" by (rule eq_reflection)
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qed
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lemma atomize_conj [atomize]:
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  "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
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proof
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  assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
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  show "A & B" by (rule conjI)
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next
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  fix C
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  assume "A & B"
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  assume "A ==> B ==> PROP C"
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  thus "PROP C"
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  proof this
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    show A by (rule conjunct1)
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    show B by (rule conjunct2)
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  qed
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qed
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lemmas [symmetric, rulify] = atomize_all atomize_imp
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subsubsection {* Classical Reasoner setup *}
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use "cladata.ML"
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setup hypsubst_setup
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ML_setup {*
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  Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
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*}
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setup Classical.setup
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setup clasetup
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lemmas [intro?] = ext
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  and [elim?] = ex1_implies_ex
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use "blastdata.ML"
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setup Blast.setup
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subsubsection {* Simplifier setup *}
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lemma meta_eq_to_obj_eq: "x == y ==> x = y"
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proof -
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  assume r: "x == y"
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  show "x = y" by (unfold r) (rule refl)
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qed
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lemma eta_contract_eq: "(%s. f s) = f" ..
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lemma simp_thms:
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  shows not_not: "(~ ~ P) = P"
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  and
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    "(P ~= Q) = (P = (~Q))"
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   338
    "(P | ~P) = True"    "(~P | P) = True"
berghofe@12436
   339
    "((~P) = (~Q)) = (P=Q)"
wenzelm@12281
   340
    "(x = x) = True"
wenzelm@12281
   341
    "(~True) = False"  "(~False) = True"
berghofe@12436
   342
    "(~P) ~= P"  "P ~= (~P)"
wenzelm@12281
   343
    "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
wenzelm@12281
   344
    "(True --> P) = P"  "(False --> P) = True"
wenzelm@12281
   345
    "(P --> True) = True"  "(P --> P) = True"
wenzelm@12281
   346
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
wenzelm@12281
   347
    "(P & True) = P"  "(True & P) = P"
wenzelm@12281
   348
    "(P & False) = False"  "(False & P) = False"
wenzelm@12281
   349
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
wenzelm@12281
   350
    "(P & ~P) = False"    "(~P & P) = False"
wenzelm@12281
   351
    "(P | True) = True"  "(True | P) = True"
wenzelm@12281
   352
    "(P | False) = P"  "(False | P) = P"
berghofe@12436
   353
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
wenzelm@12281
   354
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
wenzelm@12281
   355
    -- {* needed for the one-point-rule quantifier simplification procs *}
wenzelm@12281
   356
    -- {* essential for termination!! *} and
wenzelm@12281
   357
    "!!P. (EX x. x=t & P(x)) = P(t)"
wenzelm@12281
   358
    "!!P. (EX x. t=x & P(x)) = P(t)"
wenzelm@12281
   359
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
wenzelm@12937
   360
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
berghofe@12436
   361
  by (blast, blast, blast, blast, blast, rules+)
wenzelm@13421
   362
wenzelm@12281
   363
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
wenzelm@12354
   364
  by rules
wenzelm@12281
   365
wenzelm@12281
   366
lemma ex_simps:
wenzelm@12281
   367
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
wenzelm@12281
   368
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
wenzelm@12281
   369
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
wenzelm@12281
   370
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
wenzelm@12281
   371
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
wenzelm@12281
   372
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
wenzelm@12281
   373
  -- {* Miniscoping: pushing in existential quantifiers. *}
berghofe@12436
   374
  by (rules | blast)+
wenzelm@12281
   375
wenzelm@12281
   376
lemma all_simps:
wenzelm@12281
   377
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
wenzelm@12281
   378
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
wenzelm@12281
   379
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
wenzelm@12281
   380
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
wenzelm@12281
   381
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
wenzelm@12281
   382
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
wenzelm@12281
   383
  -- {* Miniscoping: pushing in universal quantifiers. *}
berghofe@12436
   384
  by (rules | blast)+
wenzelm@12281
   385
paulson@14201
   386
lemma disj_absorb: "(A | A) = A"
paulson@14201
   387
  by blast
paulson@14201
   388
paulson@14201
   389
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
paulson@14201
   390
  by blast
paulson@14201
   391
paulson@14201
   392
lemma conj_absorb: "(A & A) = A"
paulson@14201
   393
  by blast
paulson@14201
   394
paulson@14201
   395
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
paulson@14201
   396
  by blast
paulson@14201
   397
wenzelm@12281
   398
lemma eq_ac:
wenzelm@12937
   399
  shows eq_commute: "(a=b) = (b=a)"
wenzelm@12937
   400
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
wenzelm@12937
   401
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
berghofe@12436
   402
lemma neq_commute: "(a~=b) = (b~=a)" by rules
wenzelm@12281
   403
wenzelm@12281
   404
lemma conj_comms:
wenzelm@12937
   405
  shows conj_commute: "(P&Q) = (Q&P)"
wenzelm@12937
   406
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
berghofe@12436
   407
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
wenzelm@12281
   408
wenzelm@12281
   409
lemma disj_comms:
wenzelm@12937
   410
  shows disj_commute: "(P|Q) = (Q|P)"
wenzelm@12937
   411
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
berghofe@12436
   412
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
wenzelm@12281
   413
berghofe@12436
   414
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
berghofe@12436
   415
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
wenzelm@12281
   416
berghofe@12436
   417
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
berghofe@12436
   418
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
wenzelm@12281
   419
berghofe@12436
   420
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
berghofe@12436
   421
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
berghofe@12436
   422
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
wenzelm@12281
   423
wenzelm@12281
   424
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
wenzelm@12281
   425
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
wenzelm@12281
   426
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
wenzelm@12281
   427
wenzelm@12281
   428
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
wenzelm@12281
   429
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
wenzelm@12281
   430
berghofe@12436
   431
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
wenzelm@12281
   432
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
wenzelm@12281
   433
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
wenzelm@12281
   434
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
wenzelm@12281
   435
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
wenzelm@12281
   436
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
wenzelm@12281
   437
  by blast
wenzelm@12281
   438
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
wenzelm@12281
   439
berghofe@12436
   440
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
wenzelm@12281
   441
wenzelm@12281
   442
wenzelm@12281
   443
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
wenzelm@12281
   444
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
wenzelm@12281
   445
  -- {* cases boil down to the same thing. *}
wenzelm@12281
   446
  by blast
wenzelm@12281
   447
wenzelm@12281
   448
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
wenzelm@12281
   449
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
berghofe@12436
   450
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
berghofe@12436
   451
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
wenzelm@12281
   452
berghofe@12436
   453
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
berghofe@12436
   454
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
wenzelm@12281
   455
wenzelm@12281
   456
text {*
wenzelm@12281
   457
  \medskip The @{text "&"} congruence rule: not included by default!
wenzelm@12281
   458
  May slow rewrite proofs down by as much as 50\% *}
wenzelm@12281
   459
wenzelm@12281
   460
lemma conj_cong:
wenzelm@12281
   461
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
wenzelm@12354
   462
  by rules
wenzelm@12281
   463
wenzelm@12281
   464
lemma rev_conj_cong:
wenzelm@12281
   465
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
wenzelm@12354
   466
  by rules
wenzelm@12281
   467
wenzelm@12281
   468
text {* The @{text "|"} congruence rule: not included by default! *}
wenzelm@12281
   469
wenzelm@12281
   470
lemma disj_cong:
wenzelm@12281
   471
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
wenzelm@12281
   472
  by blast
wenzelm@12281
   473
wenzelm@12281
   474
lemma eq_sym_conv: "(x = y) = (y = x)"
wenzelm@12354
   475
  by rules
wenzelm@12281
   476
wenzelm@12281
   477
wenzelm@12281
   478
text {* \medskip if-then-else rules *}
wenzelm@12281
   479
wenzelm@12281
   480
lemma if_True: "(if True then x else y) = x"
wenzelm@12281
   481
  by (unfold if_def) blast
wenzelm@12281
   482
wenzelm@12281
   483
lemma if_False: "(if False then x else y) = y"
wenzelm@12281
   484
  by (unfold if_def) blast
wenzelm@12281
   485
wenzelm@12281
   486
lemma if_P: "P ==> (if P then x else y) = x"
wenzelm@12281
   487
  by (unfold if_def) blast
wenzelm@12281
   488
wenzelm@12281
   489
lemma if_not_P: "~P ==> (if P then x else y) = y"
wenzelm@12281
   490
  by (unfold if_def) blast
wenzelm@12281
   491
wenzelm@12281
   492
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
wenzelm@12281
   493
  apply (rule case_split [of Q])
wenzelm@12281
   494
   apply (subst if_P)
paulson@14208
   495
    prefer 3 apply (subst if_not_P, blast+)
wenzelm@12281
   496
  done
wenzelm@12281
   497
wenzelm@12281
   498
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
paulson@14208
   499
by (subst split_if, blast)
wenzelm@12281
   500
wenzelm@12281
   501
lemmas if_splits = split_if split_if_asm
wenzelm@12281
   502
wenzelm@12281
   503
lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
wenzelm@12281
   504
  by (rule split_if)
wenzelm@12281
   505
wenzelm@12281
   506
lemma if_cancel: "(if c then x else x) = x"
paulson@14208
   507
by (subst split_if, blast)
wenzelm@12281
   508
wenzelm@12281
   509
lemma if_eq_cancel: "(if x = y then y else x) = x"
paulson@14208
   510
by (subst split_if, blast)
wenzelm@12281
   511
wenzelm@12281
   512
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
wenzelm@12281
   513
  -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
wenzelm@12281
   514
  by (rule split_if)
wenzelm@12281
   515
wenzelm@12281
   516
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
wenzelm@12281
   517
  -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
paulson@14208
   518
  apply (subst split_if, blast)
wenzelm@12281
   519
  done
wenzelm@12281
   520
berghofe@12436
   521
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
berghofe@12436
   522
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
wenzelm@12281
   523
paulson@14201
   524
subsubsection {* Actual Installation of the Simplifier *}
paulson@14201
   525
wenzelm@9869
   526
use "simpdata.ML"
wenzelm@9869
   527
setup Simplifier.setup
wenzelm@9869
   528
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
wenzelm@9869
   529
setup Splitter.setup setup Clasimp.setup
wenzelm@9869
   530
paulson@14201
   531
declare disj_absorb [simp] conj_absorb [simp] 
paulson@14201
   532
nipkow@13723
   533
lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
nipkow@13723
   534
by blast+
nipkow@13723
   535
berghofe@13638
   536
theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
berghofe@13638
   537
  apply (rule iffI)
berghofe@13638
   538
  apply (rule_tac a = "%x. THE y. P x y" in ex1I)
berghofe@13638
   539
  apply (fast dest!: theI')
berghofe@13638
   540
  apply (fast intro: ext the1_equality [symmetric])
berghofe@13638
   541
  apply (erule ex1E)
berghofe@13638
   542
  apply (rule allI)
berghofe@13638
   543
  apply (rule ex1I)
berghofe@13638
   544
  apply (erule spec)
berghofe@13638
   545
  apply (erule_tac x = "%z. if z = x then y else f z" in allE)
berghofe@13638
   546
  apply (erule impE)
berghofe@13638
   547
  apply (rule allI)
berghofe@13638
   548
  apply (rule_tac P = "xa = x" in case_split_thm)
paulson@14208
   549
  apply (drule_tac [3] x = x in fun_cong, simp_all)
berghofe@13638
   550
  done
berghofe@13638
   551
nipkow@13438
   552
text{*Needs only HOL-lemmas:*}
nipkow@13438
   553
lemma mk_left_commute:
nipkow@13438
   554
  assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
nipkow@13438
   555
          c: "\<And>x y. f x y = f y x"
nipkow@13438
   556
  shows "f x (f y z) = f y (f x z)"
nipkow@13438
   557
by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
nipkow@13438
   558
wenzelm@11750
   559
wenzelm@11824
   560
subsubsection {* Generic cases and induction *}
wenzelm@11824
   561
wenzelm@11824
   562
constdefs
wenzelm@11989
   563
  induct_forall :: "('a => bool) => bool"
wenzelm@11989
   564
  "induct_forall P == \<forall>x. P x"
wenzelm@11989
   565
  induct_implies :: "bool => bool => bool"
wenzelm@11989
   566
  "induct_implies A B == A --> B"
wenzelm@11989
   567
  induct_equal :: "'a => 'a => bool"
wenzelm@11989
   568
  "induct_equal x y == x = y"
wenzelm@11989
   569
  induct_conj :: "bool => bool => bool"
wenzelm@11989
   570
  "induct_conj A B == A & B"
wenzelm@11824
   571
wenzelm@11989
   572
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
wenzelm@11989
   573
  by (simp only: atomize_all induct_forall_def)
wenzelm@11824
   574
wenzelm@11989
   575
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
wenzelm@11989
   576
  by (simp only: atomize_imp induct_implies_def)
wenzelm@11824
   577
wenzelm@11989
   578
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
wenzelm@11989
   579
  by (simp only: atomize_eq induct_equal_def)
wenzelm@11824
   580
wenzelm@11989
   581
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
wenzelm@11989
   582
    induct_conj (induct_forall A) (induct_forall B)"
wenzelm@12354
   583
  by (unfold induct_forall_def induct_conj_def) rules
wenzelm@11824
   584
wenzelm@11989
   585
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
wenzelm@11989
   586
    induct_conj (induct_implies C A) (induct_implies C B)"
wenzelm@12354
   587
  by (unfold induct_implies_def induct_conj_def) rules
wenzelm@11989
   588
berghofe@13598
   589
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
berghofe@13598
   590
proof
berghofe@13598
   591
  assume r: "induct_conj A B ==> PROP C" and A B
berghofe@13598
   592
  show "PROP C" by (rule r) (simp! add: induct_conj_def)
berghofe@13598
   593
next
berghofe@13598
   594
  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
berghofe@13598
   595
  show "PROP C" by (rule r) (simp! add: induct_conj_def)+
berghofe@13598
   596
qed
wenzelm@11824
   597
wenzelm@11989
   598
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
wenzelm@11989
   599
  by (simp add: induct_implies_def)
wenzelm@11824
   600
wenzelm@12161
   601
lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
wenzelm@12161
   602
lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
wenzelm@12161
   603
lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
wenzelm@11989
   604
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
wenzelm@11824
   605
wenzelm@11989
   606
hide const induct_forall induct_implies induct_equal induct_conj
wenzelm@11824
   607
wenzelm@11824
   608
wenzelm@11824
   609
text {* Method setup. *}
wenzelm@11824
   610
wenzelm@11824
   611
ML {*
wenzelm@11824
   612
  structure InductMethod = InductMethodFun
wenzelm@11824
   613
  (struct
wenzelm@11824
   614
    val dest_concls = HOLogic.dest_concls;
wenzelm@11824
   615
    val cases_default = thm "case_split";
wenzelm@11989
   616
    val local_impI = thm "induct_impliesI";
wenzelm@11824
   617
    val conjI = thm "conjI";
wenzelm@11989
   618
    val atomize = thms "induct_atomize";
wenzelm@11989
   619
    val rulify1 = thms "induct_rulify1";
wenzelm@11989
   620
    val rulify2 = thms "induct_rulify2";
wenzelm@12240
   621
    val localize = [Thm.symmetric (thm "induct_implies_def")];
wenzelm@11824
   622
  end);
wenzelm@11824
   623
*}
wenzelm@11824
   624
wenzelm@11824
   625
setup InductMethod.setup
wenzelm@11824
   626
wenzelm@11824
   627
wenzelm@11750
   628
subsection {* Order signatures and orders *}
wenzelm@11750
   629
wenzelm@11750
   630
axclass
wenzelm@12338
   631
  ord < type
wenzelm@11750
   632
wenzelm@11750
   633
syntax
wenzelm@11750
   634
  "op <"        :: "['a::ord, 'a] => bool"             ("op <")
wenzelm@11750
   635
  "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
wenzelm@11750
   636
wenzelm@11750
   637
global
wenzelm@11750
   638
wenzelm@11750
   639
consts
wenzelm@11750
   640
  "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
wenzelm@11750
   641
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
wenzelm@11750
   642
wenzelm@11750
   643
local
wenzelm@11750
   644
wenzelm@12114
   645
syntax (xsymbols)
wenzelm@11750
   646
  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
wenzelm@11750
   647
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
wenzelm@11750
   648
kleing@14565
   649
syntax (HTML output)
kleing@14565
   650
  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
kleing@14565
   651
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
kleing@14565
   652
wenzelm@11750
   653
paulson@14295
   654
lemma Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
paulson@14295
   655
by blast
paulson@14295
   656
wenzelm@11750
   657
subsubsection {* Monotonicity *}
wenzelm@11750
   658
wenzelm@13412
   659
locale mono =
wenzelm@13412
   660
  fixes f
wenzelm@13412
   661
  assumes mono: "A <= B ==> f A <= f B"
wenzelm@11750
   662
wenzelm@13421
   663
lemmas monoI [intro?] = mono.intro
wenzelm@13412
   664
  and monoD [dest?] = mono.mono
wenzelm@11750
   665
wenzelm@11750
   666
constdefs
wenzelm@11750
   667
  min :: "['a::ord, 'a] => 'a"
wenzelm@11750
   668
  "min a b == (if a <= b then a else b)"
wenzelm@11750
   669
  max :: "['a::ord, 'a] => 'a"
wenzelm@11750
   670
  "max a b == (if a <= b then b else a)"
wenzelm@11750
   671
wenzelm@11750
   672
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
wenzelm@11750
   673
  by (simp add: min_def)
wenzelm@11750
   674
wenzelm@11750
   675
lemma min_of_mono:
wenzelm@11750
   676
    "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
wenzelm@11750
   677
  by (simp add: min_def)
wenzelm@11750
   678
wenzelm@11750
   679
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
wenzelm@11750
   680
  by (simp add: max_def)
wenzelm@11750
   681
wenzelm@11750
   682
lemma max_of_mono:
wenzelm@11750
   683
    "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
wenzelm@11750
   684
  by (simp add: max_def)
wenzelm@11750
   685
wenzelm@11750
   686
wenzelm@11750
   687
subsubsection "Orders"
wenzelm@11750
   688
wenzelm@11750
   689
axclass order < ord
wenzelm@11750
   690
  order_refl [iff]: "x <= x"
wenzelm@11750
   691
  order_trans: "x <= y ==> y <= z ==> x <= z"
wenzelm@11750
   692
  order_antisym: "x <= y ==> y <= x ==> x = y"
wenzelm@11750
   693
  order_less_le: "(x < y) = (x <= y & x ~= y)"
wenzelm@11750
   694
wenzelm@11750
   695
wenzelm@11750
   696
text {* Reflexivity. *}
wenzelm@11750
   697
wenzelm@11750
   698
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
wenzelm@11750
   699
    -- {* This form is useful with the classical reasoner. *}
wenzelm@11750
   700
  apply (erule ssubst)
wenzelm@11750
   701
  apply (rule order_refl)
wenzelm@11750
   702
  done
wenzelm@11750
   703
nipkow@13553
   704
lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
wenzelm@11750
   705
  by (simp add: order_less_le)
wenzelm@11750
   706
wenzelm@11750
   707
lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
wenzelm@11750
   708
    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
paulson@14208
   709
  apply (simp add: order_less_le, blast)
wenzelm@11750
   710
  done
wenzelm@11750
   711
wenzelm@11750
   712
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
wenzelm@11750
   713
wenzelm@11750
   714
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
wenzelm@11750
   715
  by (simp add: order_less_le)
wenzelm@11750
   716
wenzelm@11750
   717
wenzelm@11750
   718
text {* Asymmetry. *}
wenzelm@11750
   719
wenzelm@11750
   720
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
wenzelm@11750
   721
  by (simp add: order_less_le order_antisym)
wenzelm@11750
   722
wenzelm@11750
   723
lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
wenzelm@11750
   724
  apply (drule order_less_not_sym)
paulson@14208
   725
  apply (erule contrapos_np, simp)
wenzelm@11750
   726
  done
wenzelm@11750
   727
paulson@14295
   728
lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"  
paulson@14295
   729
by (blast intro: order_antisym)
paulson@14295
   730
wenzelm@11750
   731
wenzelm@11750
   732
text {* Transitivity. *}
wenzelm@11750
   733
wenzelm@11750
   734
lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
wenzelm@11750
   735
  apply (simp add: order_less_le)
wenzelm@11750
   736
  apply (blast intro: order_trans order_antisym)
wenzelm@11750
   737
  done
wenzelm@11750
   738
wenzelm@11750
   739
lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
wenzelm@11750
   740
  apply (simp add: order_less_le)
wenzelm@11750
   741
  apply (blast intro: order_trans order_antisym)
wenzelm@11750
   742
  done
wenzelm@11750
   743
wenzelm@11750
   744
lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
wenzelm@11750
   745
  apply (simp add: order_less_le)
wenzelm@11750
   746
  apply (blast intro: order_trans order_antisym)
wenzelm@11750
   747
  done
wenzelm@11750
   748
wenzelm@11750
   749
wenzelm@11750
   750
text {* Useful for simplification, but too risky to include by default. *}
wenzelm@11750
   751
wenzelm@11750
   752
lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
wenzelm@11750
   753
  by (blast elim: order_less_asym)
wenzelm@11750
   754
wenzelm@11750
   755
lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
wenzelm@11750
   756
  by (blast elim: order_less_asym)
wenzelm@11750
   757
wenzelm@11750
   758
lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
wenzelm@11750
   759
  by auto
wenzelm@11750
   760
wenzelm@11750
   761
lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
wenzelm@11750
   762
  by auto
wenzelm@11750
   763
wenzelm@11750
   764
wenzelm@11750
   765
text {* Other operators. *}
wenzelm@11750
   766
wenzelm@11750
   767
lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
wenzelm@11750
   768
  apply (simp add: min_def)
wenzelm@11750
   769
  apply (blast intro: order_antisym)
wenzelm@11750
   770
  done
wenzelm@11750
   771
wenzelm@11750
   772
lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
wenzelm@11750
   773
  apply (simp add: max_def)
wenzelm@11750
   774
  apply (blast intro: order_antisym)
wenzelm@11750
   775
  done
wenzelm@11750
   776
wenzelm@11750
   777
wenzelm@11750
   778
subsubsection {* Least value operator *}
wenzelm@11750
   779
wenzelm@11750
   780
constdefs
wenzelm@11750
   781
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
wenzelm@11750
   782
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
wenzelm@11750
   783
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
wenzelm@11750
   784
wenzelm@11750
   785
lemma LeastI2:
wenzelm@11750
   786
  "[| P (x::'a::order);
wenzelm@11750
   787
      !!y. P y ==> x <= y;
wenzelm@11750
   788
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
wenzelm@12281
   789
   ==> Q (Least P)"
wenzelm@11750
   790
  apply (unfold Least_def)
wenzelm@11750
   791
  apply (rule theI2)
wenzelm@11750
   792
    apply (blast intro: order_antisym)+
wenzelm@11750
   793
  done
wenzelm@11750
   794
wenzelm@11750
   795
lemma Least_equality:
wenzelm@12281
   796
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
wenzelm@11750
   797
  apply (simp add: Least_def)
wenzelm@11750
   798
  apply (rule the_equality)
wenzelm@11750
   799
  apply (auto intro!: order_antisym)
wenzelm@11750
   800
  done
wenzelm@11750
   801
wenzelm@11750
   802
wenzelm@11750
   803
subsubsection "Linear / total orders"
wenzelm@11750
   804
wenzelm@11750
   805
axclass linorder < order
wenzelm@11750
   806
  linorder_linear: "x <= y | y <= x"
wenzelm@11750
   807
wenzelm@11750
   808
lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
wenzelm@11750
   809
  apply (simp add: order_less_le)
paulson@14208
   810
  apply (insert linorder_linear, blast)
wenzelm@11750
   811
  done
wenzelm@11750
   812
paulson@14365
   813
lemma linorder_le_cases [case_names le ge]:
paulson@14365
   814
    "((x::'a::linorder) \<le> y ==> P) ==> (y \<le> x ==> P) ==> P"
paulson@14365
   815
  by (insert linorder_linear, blast)
paulson@14365
   816
wenzelm@11750
   817
lemma linorder_cases [case_names less equal greater]:
wenzelm@11750
   818
    "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
paulson@14365
   819
  by (insert linorder_less_linear, blast)
wenzelm@11750
   820
wenzelm@11750
   821
lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
wenzelm@11750
   822
  apply (simp add: order_less_le)
wenzelm@11750
   823
  apply (insert linorder_linear)
wenzelm@11750
   824
  apply (blast intro: order_antisym)
wenzelm@11750
   825
  done
wenzelm@11750
   826
wenzelm@11750
   827
lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
wenzelm@11750
   828
  apply (simp add: order_less_le)
wenzelm@11750
   829
  apply (insert linorder_linear)
wenzelm@11750
   830
  apply (blast intro: order_antisym)
wenzelm@11750
   831
  done
wenzelm@11750
   832
wenzelm@11750
   833
lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
paulson@14208
   834
by (cut_tac x = x and y = y in linorder_less_linear, auto)
wenzelm@11750
   835
wenzelm@11750
   836
lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
paulson@14208
   837
by (simp add: linorder_neq_iff, blast)
wenzelm@11750
   838
wenzelm@11750
   839
wenzelm@11750
   840
subsubsection "Min and max on (linear) orders"
wenzelm@11750
   841
wenzelm@11750
   842
lemma min_same [simp]: "min (x::'a::order) x = x"
wenzelm@11750
   843
  by (simp add: min_def)
wenzelm@11750
   844
wenzelm@11750
   845
lemma max_same [simp]: "max (x::'a::order) x = x"
wenzelm@11750
   846
  by (simp add: max_def)
wenzelm@11750
   847
wenzelm@11750
   848
lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
wenzelm@11750
   849
  apply (simp add: max_def)
wenzelm@11750
   850
  apply (insert linorder_linear)
wenzelm@11750
   851
  apply (blast intro: order_trans)
wenzelm@11750
   852
  done
wenzelm@11750
   853
wenzelm@11750
   854
lemma le_maxI1: "(x::'a::linorder) <= max x y"
wenzelm@11750
   855
  by (simp add: le_max_iff_disj)
wenzelm@11750
   856
wenzelm@11750
   857
lemma le_maxI2: "(y::'a::linorder) <= max x y"
wenzelm@11750
   858
    -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
wenzelm@11750
   859
  by (simp add: le_max_iff_disj)
wenzelm@11750
   860
wenzelm@11750
   861
lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
wenzelm@11750
   862
  apply (simp add: max_def order_le_less)
wenzelm@11750
   863
  apply (insert linorder_less_linear)
wenzelm@11750
   864
  apply (blast intro: order_less_trans)
wenzelm@11750
   865
  done
wenzelm@11750
   866
wenzelm@11750
   867
lemma max_le_iff_conj [simp]:
wenzelm@11750
   868
    "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
wenzelm@11750
   869
  apply (simp add: max_def)
wenzelm@11750
   870
  apply (insert linorder_linear)
wenzelm@11750
   871
  apply (blast intro: order_trans)
wenzelm@11750
   872
  done
wenzelm@11750
   873
wenzelm@11750
   874
lemma max_less_iff_conj [simp]:
wenzelm@11750
   875
    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
wenzelm@11750
   876
  apply (simp add: order_le_less max_def)
wenzelm@11750
   877
  apply (insert linorder_less_linear)
wenzelm@11750
   878
  apply (blast intro: order_less_trans)
wenzelm@11750
   879
  done
wenzelm@11750
   880
wenzelm@11750
   881
lemma le_min_iff_conj [simp]:
wenzelm@11750
   882
    "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
wenzelm@12892
   883
    -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
wenzelm@11750
   884
  apply (simp add: min_def)
wenzelm@11750
   885
  apply (insert linorder_linear)
wenzelm@11750
   886
  apply (blast intro: order_trans)
wenzelm@11750
   887
  done
wenzelm@11750
   888
wenzelm@11750
   889
lemma min_less_iff_conj [simp]:
wenzelm@11750
   890
    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
wenzelm@11750
   891
  apply (simp add: order_le_less min_def)
wenzelm@11750
   892
  apply (insert linorder_less_linear)
wenzelm@11750
   893
  apply (blast intro: order_less_trans)
wenzelm@11750
   894
  done
wenzelm@11750
   895
wenzelm@11750
   896
lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
wenzelm@11750
   897
  apply (simp add: min_def)
wenzelm@11750
   898
  apply (insert linorder_linear)
wenzelm@11750
   899
  apply (blast intro: order_trans)
wenzelm@11750
   900
  done
wenzelm@11750
   901
wenzelm@11750
   902
lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
wenzelm@11750
   903
  apply (simp add: min_def order_le_less)
wenzelm@11750
   904
  apply (insert linorder_less_linear)
wenzelm@11750
   905
  apply (blast intro: order_less_trans)
wenzelm@11750
   906
  done
wenzelm@11750
   907
nipkow@13438
   908
lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
nipkow@13438
   909
apply(simp add:max_def)
nipkow@13438
   910
apply(rule conjI)
nipkow@13438
   911
apply(blast intro:order_trans)
nipkow@13438
   912
apply(simp add:linorder_not_le)
nipkow@13438
   913
apply(blast dest: order_less_trans order_le_less_trans)
nipkow@13438
   914
done
nipkow@13438
   915
nipkow@13438
   916
lemma max_commute: "!!x::'a::linorder. max x y = max y x"
nipkow@13438
   917
apply(simp add:max_def)
nipkow@13438
   918
apply(rule conjI)
nipkow@13438
   919
apply(blast intro:order_antisym)
nipkow@13438
   920
apply(simp add:linorder_not_le)
nipkow@13438
   921
apply(blast dest: order_less_trans)
nipkow@13438
   922
done
nipkow@13438
   923
nipkow@13438
   924
lemmas max_ac = max_assoc max_commute
nipkow@13438
   925
                mk_left_commute[of max,OF max_assoc max_commute]
nipkow@13438
   926
nipkow@13438
   927
lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
nipkow@13438
   928
apply(simp add:min_def)
nipkow@13438
   929
apply(rule conjI)
nipkow@13438
   930
apply(blast intro:order_trans)
nipkow@13438
   931
apply(simp add:linorder_not_le)
nipkow@13438
   932
apply(blast dest: order_less_trans order_le_less_trans)
nipkow@13438
   933
done
nipkow@13438
   934
nipkow@13438
   935
lemma min_commute: "!!x::'a::linorder. min x y = min y x"
nipkow@13438
   936
apply(simp add:min_def)
nipkow@13438
   937
apply(rule conjI)
nipkow@13438
   938
apply(blast intro:order_antisym)
nipkow@13438
   939
apply(simp add:linorder_not_le)
nipkow@13438
   940
apply(blast dest: order_less_trans)
nipkow@13438
   941
done
nipkow@13438
   942
nipkow@13438
   943
lemmas min_ac = min_assoc min_commute
nipkow@13438
   944
                mk_left_commute[of min,OF min_assoc min_commute]
nipkow@13438
   945
wenzelm@11750
   946
lemma split_min:
wenzelm@11750
   947
    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
wenzelm@11750
   948
  by (simp add: min_def)
wenzelm@11750
   949
wenzelm@11750
   950
lemma split_max:
wenzelm@11750
   951
    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
wenzelm@11750
   952
  by (simp add: max_def)
wenzelm@11750
   953
wenzelm@11750
   954
ballarin@14398
   955
subsubsection {* Transitivity rules for calculational reasoning *}
ballarin@14398
   956
ballarin@14398
   957
ballarin@14398
   958
lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
ballarin@14398
   959
  by (simp add: order_less_le)
ballarin@14398
   960
ballarin@14398
   961
lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
ballarin@14398
   962
  by (simp add: order_less_le)
ballarin@14398
   963
ballarin@14398
   964
lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
ballarin@14398
   965
  by (rule order_less_asym)
ballarin@14398
   966
ballarin@14398
   967
ballarin@14444
   968
subsubsection {* Setup of transitivity reasoner as Solver *}
ballarin@14398
   969
ballarin@14398
   970
lemma less_imp_neq: "[| (x::'a::order) < y |] ==> x ~= y"
ballarin@14398
   971
  by (erule contrapos_pn, erule subst, rule order_less_irrefl)
ballarin@14398
   972
ballarin@14398
   973
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
ballarin@14398
   974
  by (erule subst, erule ssubst, assumption)
ballarin@14398
   975
ballarin@14398
   976
ML_setup {*
ballarin@14398
   977
ballarin@14398
   978
structure Trans_Tac = Trans_Tac_Fun (
ballarin@14398
   979
  struct
ballarin@14398
   980
    val less_reflE = thm "order_less_irrefl" RS thm "notE";
ballarin@14398
   981
    val le_refl = thm "order_refl";
ballarin@14398
   982
    val less_imp_le = thm "order_less_imp_le";
ballarin@14398
   983
    val not_lessI = thm "linorder_not_less" RS thm "iffD2";
ballarin@14398
   984
    val not_leI = thm "linorder_not_le" RS thm "iffD2";
ballarin@14398
   985
    val not_lessD = thm "linorder_not_less" RS thm "iffD1";
ballarin@14398
   986
    val not_leD = thm "linorder_not_le" RS thm "iffD1";
ballarin@14398
   987
    val eqI = thm "order_antisym";
ballarin@14398
   988
    val eqD1 = thm "order_eq_refl";
ballarin@14398
   989
    val eqD2 = thm "sym" RS thm "order_eq_refl";
ballarin@14398
   990
    val less_trans = thm "order_less_trans";
ballarin@14398
   991
    val less_le_trans = thm "order_less_le_trans";
ballarin@14398
   992
    val le_less_trans = thm "order_le_less_trans";
ballarin@14398
   993
    val le_trans = thm "order_trans";
ballarin@14398
   994
    val le_neq_trans = thm "order_le_neq_trans";
ballarin@14398
   995
    val neq_le_trans = thm "order_neq_le_trans";
ballarin@14398
   996
    val less_imp_neq = thm "less_imp_neq";
ballarin@14398
   997
    val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
ballarin@14398
   998
ballarin@14398
   999
    fun decomp_gen sort sign (Trueprop $ t) =
ballarin@14398
  1000
      let fun of_sort t = Sign.of_sort sign (type_of t, sort)
ballarin@14398
  1001
      fun dec (Const ("Not", _) $ t) = (
ballarin@14398
  1002
              case dec t of
ballarin@14398
  1003
                None => None
ballarin@14398
  1004
              | Some (t1, rel, t2) => Some (t1, "~" ^ rel, t2))
ballarin@14398
  1005
            | dec (Const ("op =",  _) $ t1 $ t2) = 
ballarin@14398
  1006
                if of_sort t1
ballarin@14398
  1007
                then Some (t1, "=", t2)
ballarin@14398
  1008
                else None
ballarin@14398
  1009
            | dec (Const ("op <=",  _) $ t1 $ t2) = 
ballarin@14398
  1010
                if of_sort t1
ballarin@14398
  1011
                then Some (t1, "<=", t2)
ballarin@14398
  1012
                else None
ballarin@14398
  1013
            | dec (Const ("op <",  _) $ t1 $ t2) = 
ballarin@14398
  1014
                if of_sort t1
ballarin@14398
  1015
                then Some (t1, "<", t2)
ballarin@14398
  1016
                else None
ballarin@14398
  1017
            | dec _ = None
ballarin@14398
  1018
      in dec t end;
ballarin@14398
  1019
ballarin@14398
  1020
    val decomp_part = decomp_gen ["HOL.order"];
ballarin@14398
  1021
    val decomp_lin = decomp_gen ["HOL.linorder"];
ballarin@14398
  1022
ballarin@14398
  1023
  end);  (* struct *)
ballarin@14398
  1024
ballarin@14398
  1025
Context.>> (fn thy => (simpset_ref_of thy :=
ballarin@14398
  1026
  simpset_of thy
ballarin@14398
  1027
    addSolver (mk_solver "Trans_linear" (fn _ => Trans_Tac.linear_tac))
ballarin@14398
  1028
    addSolver (mk_solver "Trans_partial" (fn _ => Trans_Tac.partial_tac));
ballarin@14444
  1029
  (* Adding the transitivity reasoners also as safe solvers showed a slight
ballarin@14444
  1030
     speed up, but the reasoning strength appears to be not higher (at least
ballarin@14444
  1031
     no breaking of additional proofs in the entire HOL distribution, as
ballarin@14444
  1032
     of 5 March 2004, was observed). *)
ballarin@14398
  1033
  thy))
ballarin@14398
  1034
*}
ballarin@14398
  1035
ballarin@14398
  1036
(* Optional methods
ballarin@14398
  1037
ballarin@14398
  1038
method_setup trans_partial =
ballarin@14398
  1039
  {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trans_tac_partial)) *}
ballarin@14398
  1040
  {* simple transitivity reasoner *}	    
ballarin@14398
  1041
method_setup trans_linear =
ballarin@14398
  1042
  {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trans_tac_linear)) *}
ballarin@14398
  1043
  {* simple transitivity reasoner *}
ballarin@14398
  1044
*)
ballarin@14398
  1045
ballarin@14444
  1046
(*
ballarin@14444
  1047
declare order.order_refl [simp del] order_less_irrefl [simp del]
ballarin@14444
  1048
*)
ballarin@14444
  1049
wenzelm@11750
  1050
subsubsection "Bounded quantifiers"
wenzelm@11750
  1051
wenzelm@11750
  1052
syntax
wenzelm@11750
  1053
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
  1054
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
  1055
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@11750
  1056
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
wenzelm@11750
  1057
wenzelm@12114
  1058
syntax (xsymbols)
wenzelm@11750
  1059
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
  1060
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
  1061
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@11750
  1062
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@11750
  1063
wenzelm@11750
  1064
syntax (HOL)
wenzelm@11750
  1065
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
  1066
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
  1067
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@11750
  1068
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
wenzelm@11750
  1069
kleing@14565
  1070
syntax (HTML output)
kleing@14565
  1071
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
kleing@14565
  1072
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
kleing@14565
  1073
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
kleing@14565
  1074
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
kleing@14565
  1075
wenzelm@11750
  1076
translations
wenzelm@11750
  1077
 "ALL x<y. P"   =>  "ALL x. x < y --> P"
wenzelm@11750
  1078
 "EX x<y. P"    =>  "EX x. x < y  & P"
wenzelm@11750
  1079
 "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
wenzelm@11750
  1080
 "EX x<=y. P"   =>  "EX x. x <= y & P"
wenzelm@11750
  1081
kleing@14357
  1082
print_translation {*
kleing@14357
  1083
let
kleing@14357
  1084
  fun all_tr' [Const ("_bound",_) $ Free (v,_), 
kleing@14357
  1085
               Const("op -->",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
kleing@14357
  1086
  (if v=v' then Syntax.const "_lessAll" $ Syntax.mark_bound v' $ n $ P else raise Match)
kleing@14357
  1087
kleing@14357
  1088
  | all_tr' [Const ("_bound",_) $ Free (v,_), 
kleing@14357
  1089
               Const("op -->",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
kleing@14357
  1090
  (if v=v' then Syntax.const "_leAll" $ Syntax.mark_bound v' $ n $ P else raise Match);
kleing@14357
  1091
kleing@14357
  1092
  fun ex_tr' [Const ("_bound",_) $ Free (v,_), 
kleing@14357
  1093
               Const("op &",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
kleing@14357
  1094
  (if v=v' then Syntax.const "_lessEx" $ Syntax.mark_bound v' $ n $ P else raise Match)
kleing@14357
  1095
kleing@14357
  1096
  | ex_tr' [Const ("_bound",_) $ Free (v,_), 
kleing@14357
  1097
               Const("op &",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
kleing@14357
  1098
  (if v=v' then Syntax.const "_leEx" $ Syntax.mark_bound v' $ n $ P else raise Match)
kleing@14357
  1099
in
kleing@14357
  1100
[("ALL ", all_tr'), ("EX ", ex_tr')]
clasohm@923
  1101
end
kleing@14357
  1102
*}
kleing@14357
  1103
kleing@14357
  1104
end