src/HOL/HOL.thy
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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 = 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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global
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classes "term" < logic
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defaultsort "term"
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typedecl bool
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arities
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  bool :: "term"
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  fun :: ("term", "term") "term"
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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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  ~=            :: "['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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syntax ("" output)
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  "="           :: "['a, 'a] => bool"                    (infix 50)
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  "~="          :: "['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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  "op ~="       :: "['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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  "op ~="       :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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syntax (HTML output)
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  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
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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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  arbitrary_def:  "False ==> arbitrary == (THE x. False)"
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    -- {* @{term arbitrary} is completely unspecified, but is made to appear as a
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    definition syntactically *}
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subsubsection {* Generic algebraic operations *}
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axclass zero < "term"
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axclass one < "term"
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axclass plus < "term"
3e400964893e judgment Trueprop;
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   151
axclass minus < "term"
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axclass times < "term"
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axclass inverse < "term"
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   154
3e400964893e judgment Trueprop;
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global
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3e400964893e judgment Trueprop;
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consts
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  "0"           :: "'a::zero"                       ("0")
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   159
  "1"           :: "'a::one"                        ("1")
3e400964893e judgment Trueprop;
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   160
  "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
3e400964893e judgment Trueprop;
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   161
  -             :: "['a::minus, 'a] => 'a"          (infixl 65)
3e400964893e judgment Trueprop;
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   162
  uminus        :: "['a::minus] => 'a"              ("- _" [81] 80)
3e400964893e judgment Trueprop;
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  *             :: "['a::times, 'a] => 'a"          (infixl 70)
3e400964893e judgment Trueprop;
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   164
3e400964893e judgment Trueprop;
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   165
local
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   166
3e400964893e judgment Trueprop;
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   167
typed_print_translation {*
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   168
  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
3e400964893e judgment Trueprop;
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   171
      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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   173
*} -- {* show types that are presumably too general *}
3e400964893e judgment Trueprop;
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   174
3e400964893e judgment Trueprop;
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   175
3e400964893e judgment Trueprop;
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consts
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   177
  abs           :: "'a::minus => 'a"
3e400964893e judgment Trueprop;
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  inverse       :: "'a::inverse => 'a"
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   179
  divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
3e400964893e judgment Trueprop;
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   180
3e400964893e judgment Trueprop;
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   181
syntax (xsymbols)
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   182
  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
3e400964893e judgment Trueprop;
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   183
syntax (HTML output)
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   184
  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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   185
3e400964893e judgment Trueprop;
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axclass plus_ac0 < plus, zero
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  commute: "x + y = y + x"
3e400964893e judgment Trueprop;
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  assoc:   "(x + y) + z = x + (y + z)"
3e400964893e judgment Trueprop;
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  zero:    "0 + x = x"
3e400964893e judgment Trueprop;
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   190
3e400964893e judgment Trueprop;
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   191
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subsection {* Theory and package setup *}
3e400964893e judgment Trueprop;
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3e400964893e judgment Trueprop;
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subsubsection {* Basic lemmas *}
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9736
332fab43628f Fixed rulify.
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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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   198
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declare trans [trans]
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declare impE [CPure.elim]  iffD1 [CPure.elim]  iffD2 [CPure.elim]
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11438
3d9222b80989 declare trans [trans] (*overridden in theory Calculation*);
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11750
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   203
subsubsection {* Atomizing meta-level connectives *}
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   204
3e400964893e judgment Trueprop;
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   205
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
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   206
proof
9488
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   207
  assume "!!x. P x"
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  show "ALL x. P x" by (rule allI)
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f11bece4e2db added all_eq, imp_eq (for blast);
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   209
next
f11bece4e2db added all_eq, imp_eq (for blast);
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   210
  assume "ALL x. P x"
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   211
  thus "!!x. P x" by (rule allE)
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   212
qed
f11bece4e2db added all_eq, imp_eq (for blast);
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   213
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   214
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
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   215
proof
9488
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   216
  assume r: "A ==> B"
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   217
  show "A --> B" by (rule impI) (rule r)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
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   218
next
f11bece4e2db added all_eq, imp_eq (for blast);
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   219
  assume "A --> B" and A
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   220
  thus B by (rule mp)
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f11bece4e2db added all_eq, imp_eq (for blast);
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   221
qed
f11bece4e2db added all_eq, imp_eq (for blast);
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   222
11750
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   223
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
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   224
proof
10432
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   225
  assume "x == y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
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  show "x = y" by (unfold prems) (rule refl)
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   227
next
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
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   228
  assume "x = y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
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   229
  thus "x == y" by (rule eq_reflection)
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
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   230
qed
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
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   231
12023
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   232
lemma atomize_conj [atomize]:
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   233
  "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
12003
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   234
proof
11953
f98623fdf6ef atomize_conj;
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   235
  assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
f98623fdf6ef atomize_conj;
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   236
  show "A & B" by (rule conjI)
f98623fdf6ef atomize_conj;
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   237
next
f98623fdf6ef atomize_conj;
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   238
  fix C
f98623fdf6ef atomize_conj;
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   239
  assume "A & B"
f98623fdf6ef atomize_conj;
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   240
  assume "A ==> B ==> PROP C"
f98623fdf6ef atomize_conj;
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   241
  thus "PROP C"
f98623fdf6ef atomize_conj;
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   242
  proof this
f98623fdf6ef atomize_conj;
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   243
    show A by (rule conjunct1)
f98623fdf6ef atomize_conj;
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   244
    show B by (rule conjunct2)
f98623fdf6ef atomize_conj;
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   245
  qed
f98623fdf6ef atomize_conj;
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   246
qed
f98623fdf6ef atomize_conj;
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   247
11750
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   248
3e400964893e judgment Trueprop;
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   249
subsubsection {* Classical Reasoner setup *}
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d9434a9277a4 lemmas atomize = all_eq imp_eq;
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   250
10383
a092ae7bb2a6 "atomize" for classical tactics;
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   251
use "cladata.ML"
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   252
setup hypsubst_setup
11977
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   253
11770
b6bb7a853dd2 moved rulify to ObjectLogic;
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   254
declare atomize_all [symmetric, rulify]  atomize_imp [symmetric, rulify]
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   255
10383
a092ae7bb2a6 "atomize" for classical tactics;
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   256
setup Classical.setup
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   257
setup clasetup
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   258
11977
2e7c54b86763 tuned declaration of rules;
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   259
declare ext [intro?]
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   260
declare disjI1 [elim?]  disjI2 [elim?]  ex1_implies_ex [elim?]  sym [elim?]
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   261
9869
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   262
use "blastdata.ML"
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   263
setup Blast.setup
4868
843a9f5b3c3d nonterminals;
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   264
11750
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   265
3e400964893e judgment Trueprop;
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   266
subsubsection {* Simplifier setup *}
3e400964893e judgment Trueprop;
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   267
12281
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   268
lemma meta_eq_to_obj_eq: "x == y ==> x = y"
3bd113b8f7a6 converted simp lemmas;
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   269
proof -
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   270
  assume r: "x == y"
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   271
  show "x = y" by (unfold r) (rule refl)
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   272
qed
3bd113b8f7a6 converted simp lemmas;
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   273
3bd113b8f7a6 converted simp lemmas;
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   274
lemma eta_contract_eq: "(%s. f s) = f" ..
3bd113b8f7a6 converted simp lemmas;
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   275
3bd113b8f7a6 converted simp lemmas;
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   276
lemma simp_thms:
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   277
  (not_not: "(~ ~ P) = P" and
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   278
    "(x = x) = True"
3bd113b8f7a6 converted simp lemmas;
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   279
    "(~True) = False"  "(~False) = True"
3bd113b8f7a6 converted simp lemmas;
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   280
    "(~P) ~= P"  "P ~= (~P)"  "(P ~= Q) = (P = (~Q))"
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   281
    "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
3bd113b8f7a6 converted simp lemmas;
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   282
    "(True --> P) = P"  "(False --> P) = True"
3bd113b8f7a6 converted simp lemmas;
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   283
    "(P --> True) = True"  "(P --> P) = True"
3bd113b8f7a6 converted simp lemmas;
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   284
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
3bd113b8f7a6 converted simp lemmas;
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   285
    "(P & True) = P"  "(True & P) = P"
3bd113b8f7a6 converted simp lemmas;
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   286
    "(P & False) = False"  "(False & P) = False"
3bd113b8f7a6 converted simp lemmas;
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   287
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
3bd113b8f7a6 converted simp lemmas;
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   288
    "(P & ~P) = False"    "(~P & P) = False"
3bd113b8f7a6 converted simp lemmas;
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   289
    "(P | True) = True"  "(True | P) = True"
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   290
    "(P | False) = P"  "(False | P) = P"
3bd113b8f7a6 converted simp lemmas;
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   291
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)"
3bd113b8f7a6 converted simp lemmas;
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   292
    "(P | ~P) = True"    "(~P | P) = True"
3bd113b8f7a6 converted simp lemmas;
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   293
    "((~P) = (~Q)) = (P=Q)" and
3bd113b8f7a6 converted simp lemmas;
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   294
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
3bd113b8f7a6 converted simp lemmas;
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   295
    -- {* needed for the one-point-rule quantifier simplification procs *}
3bd113b8f7a6 converted simp lemmas;
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   296
    -- {* essential for termination!! *} and
3bd113b8f7a6 converted simp lemmas;
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   297
    "!!P. (EX x. x=t & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   298
    "!!P. (EX x. t=x & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   299
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   300
    "!!P. (ALL x. t=x --> P(x)) = P(t)")
3bd113b8f7a6 converted simp lemmas;
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   301
  by blast+
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   302
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   303
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   304
  by blast
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   305
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   306
lemma ex_simps:
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   307
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   308
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   309
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   310
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
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parents: 12256
diff changeset
   311
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
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diff changeset
   312
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   313
  -- {* Miniscoping: pushing in existential quantifiers. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   314
  by blast+
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   315
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   316
lemma all_simps:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   317
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   318
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   319
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   320
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   321
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   322
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   323
  -- {* Miniscoping: pushing in universal quantifiers. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   324
  by blast+
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   325
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   326
lemma eq_ac:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   327
 (eq_commute: "(a=b) = (b=a)" and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   328
  eq_left_commute: "(P=(Q=R)) = (Q=(P=R))" and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   329
  eq_assoc: "((P=Q)=R) = (P=(Q=R))") by blast+
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   330
lemma neq_commute: "(a~=b) = (b~=a)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   331
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   332
lemma conj_comms:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   333
 (conj_commute: "(P&Q) = (Q&P)" and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   334
  conj_left_commute: "(P&(Q&R)) = (Q&(P&R))") by blast+
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   335
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   336
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   337
lemma disj_comms:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   338
 (disj_commute: "(P|Q) = (Q|P)" and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   339
  disj_left_commute: "(P|(Q|R)) = (Q|(P|R))") by blast+
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   340
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   341
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   342
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   343
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   344
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   345
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   346
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   347
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   348
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   349
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   350
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   351
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   352
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   353
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   354
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   355
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   356
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   357
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   358
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   359
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   360
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   361
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   362
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   363
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   364
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   365
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   366
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   367
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   368
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   369
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   370
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   371
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   372
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   373
  -- {* cases boil down to the same thing. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   374
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   375
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   376
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   377
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   378
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   379
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   380
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   381
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   382
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   383
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   384
text {*
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   385
  \medskip The @{text "&"} congruence rule: not included by default!
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   386
  May slow rewrite proofs down by as much as 50\% *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   387
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   388
lemma conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   389
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   390
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   391
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   392
lemma rev_conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   393
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   394
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   395
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   396
text {* The @{text "|"} congruence rule: not included by default! *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   397
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   398
lemma disj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   399
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   400
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   401
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   402
lemma eq_sym_conv: "(x = y) = (y = x)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   403
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   404
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   405
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   406
text {* \medskip if-then-else rules *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   407
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   408
lemma if_True: "(if True then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   409
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   410
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   411
lemma if_False: "(if False then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   412
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   413
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   414
lemma if_P: "P ==> (if P then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   415
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   416
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   417
lemma if_not_P: "~P ==> (if P then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   418
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   419
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   420
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   421
  apply (rule case_split [of Q])
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   422
   apply (subst if_P)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   423
    prefer 3 apply (subst if_not_P)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   424
     apply blast+
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   425
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   426
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   427
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   428
  apply (subst split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   429
  apply blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   430
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   431
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   432
lemmas if_splits = split_if split_if_asm
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   433
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   434
lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   435
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   436
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   437
lemma if_cancel: "(if c then x else x) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   438
  apply (subst split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   439
  apply blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   440
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   441
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   442
lemma if_eq_cancel: "(if x = y then y else x) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   443
  apply (subst split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   444
  apply blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   445
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   446
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   447
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   448
  -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   449
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   450
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   451
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   452
  -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   453
  apply (subst split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   454
  apply blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   455
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   456
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   457
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   458
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   459
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   460
use "simpdata.ML"
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   461
setup Simplifier.setup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   462
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   463
setup Splitter.setup setup Clasimp.setup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   464
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   465
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   466
subsubsection {* Generic cases and induction *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   467
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   468
constdefs
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   469
  induct_forall :: "('a => bool) => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   470
  "induct_forall P == \<forall>x. P x"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   471
  induct_implies :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   472
  "induct_implies A B == A --> B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   473
  induct_equal :: "'a => 'a => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   474
  "induct_equal x y == x = y"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   475
  induct_conj :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   476
  "induct_conj A B == A & B"
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   477
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   478
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   479
  by (simp only: atomize_all induct_forall_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   480
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   481
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   482
  by (simp only: atomize_imp induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   483
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   484
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   485
  by (simp only: atomize_eq induct_equal_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   486
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   487
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   488
    induct_conj (induct_forall A) (induct_forall B)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   489
  by (unfold induct_forall_def induct_conj_def) blast
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   490
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   491
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   492
    induct_conj (induct_implies C A) (induct_implies C B)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   493
  by (unfold induct_implies_def induct_conj_def) blast
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   494
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   495
lemma induct_conj_curry: "(induct_conj A B ==> C) == (A ==> B ==> C)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   496
  by (simp only: atomize_imp atomize_eq induct_conj_def) (rule equal_intr_rule, blast+)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   497
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   498
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   499
  by (simp add: induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   500
12161
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
   501
lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
   502
lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
   503
lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   504
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   505
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   506
hide const induct_forall induct_implies induct_equal induct_conj
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   507
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   508
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   509
text {* Method setup. *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   510
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   511
ML {*
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   512
  structure InductMethod = InductMethodFun
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   513
  (struct
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   514
    val dest_concls = HOLogic.dest_concls;
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   515
    val cases_default = thm "case_split";
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   516
    val local_impI = thm "induct_impliesI";
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   517
    val conjI = thm "conjI";
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   518
    val atomize = thms "induct_atomize";
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   519
    val rulify1 = thms "induct_rulify1";
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   520
    val rulify2 = thms "induct_rulify2";
12240
0760eda193c4 induct method: localize rews for rule;
wenzelm
parents: 12161
diff changeset
   521
    val localize = [Thm.symmetric (thm "induct_implies_def")];
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   522
  end);
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   523
*}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   524
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   525
setup InductMethod.setup
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   526
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   527
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   528
subsection {* Order signatures and orders *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   529
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   530
axclass
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   531
  ord < "term"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   532
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   533
syntax
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   534
  "op <"        :: "['a::ord, 'a] => bool"             ("op <")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   535
  "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   536
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   537
global
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   538
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   539
consts
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   540
  "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   541
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   542
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   543
local
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   544
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
   545
syntax (xsymbols)
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   546
  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   547
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   548
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   549
(*Tell blast about overloading of < and <= to reduce the risk of
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   550
  its applying a rule for the wrong type*)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   551
ML {*
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   552
Blast.overloaded ("op <" , domain_type);
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   553
Blast.overloaded ("op <=", domain_type);
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   554
*}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   555
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   556
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   557
subsubsection {* Monotonicity *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   558
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   559
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   560
  mono :: "['a::ord => 'b::ord] => bool"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   561
  "mono f == ALL A B. A <= B --> f A <= f B"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   562
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   563
lemma monoI [intro?]: "(!!A B. A <= B ==> f A <= f B) ==> mono f"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   564
  by (unfold mono_def) blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   565
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   566
lemma monoD [dest?]: "mono f ==> A <= B ==> f A <= f B"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   567
  by (unfold mono_def) blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   568
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   569
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   570
  min :: "['a::ord, 'a] => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   571
  "min a b == (if a <= b then a else b)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   572
  max :: "['a::ord, 'a] => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   573
  "max a b == (if a <= b then b else a)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   574
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   575
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   576
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   577
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   578
lemma min_of_mono:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   579
    "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   580
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   581
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   582
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   583
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   584
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   585
lemma max_of_mono:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   586
    "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   587
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   588
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   589
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   590
subsubsection "Orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   591
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   592
axclass order < ord
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   593
  order_refl [iff]: "x <= x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   594
  order_trans: "x <= y ==> y <= z ==> x <= z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   595
  order_antisym: "x <= y ==> y <= x ==> x = y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   596
  order_less_le: "(x < y) = (x <= y & x ~= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   597
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   598
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   599
text {* Reflexivity. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   600
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   601
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   602
    -- {* This form is useful with the classical reasoner. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   603
  apply (erule ssubst)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   604
  apply (rule order_refl)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   605
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   606
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   607
lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   608
  by (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   609
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   610
lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   611
    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   612
  apply (simp add: order_less_le)
12256
wenzelm
parents: 12240
diff changeset
   613
  apply blast
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   614
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   615
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   616
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   617
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   618
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   619
  by (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   620
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   621
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   622
text {* Asymmetry. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   623
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   624
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   625
  by (simp add: order_less_le order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   626
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   627
lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   628
  apply (drule order_less_not_sym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   629
  apply (erule contrapos_np)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   630
  apply simp
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   631
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   632
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   633
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   634
text {* Transitivity. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   635
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   636
lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   637
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   638
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   639
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   640
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   641
lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   642
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   643
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   644
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   645
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   646
lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   647
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   648
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   649
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   650
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   651
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   652
text {* Useful for simplification, but too risky to include by default. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   653
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   654
lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   655
  by (blast elim: order_less_asym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   656
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   657
lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   658
  by (blast elim: order_less_asym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   659
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   660
lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   661
  by auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   662
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   663
lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   664
  by auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   665
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   666
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   667
text {* Other operators. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   668
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   669
lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   670
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   671
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   672
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   673
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   674
lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   675
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   676
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   677
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   678
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   679
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   680
subsubsection {* Least value operator *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   681
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   682
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   683
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   684
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   685
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   686
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   687
lemma LeastI2:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   688
  "[| P (x::'a::order);
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   689
      !!y. P y ==> x <= y;
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   690
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   691
   ==> Q (Least P)"
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   692
  apply (unfold Least_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   693
  apply (rule theI2)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   694
    apply (blast intro: order_antisym)+
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   695
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   696
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   697
lemma Least_equality:
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   698
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   699
  apply (simp add: Least_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   700
  apply (rule the_equality)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   701
  apply (auto intro!: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   702
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   703
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   704
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   705
subsubsection "Linear / total orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   706
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   707
axclass linorder < order
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   708
  linorder_linear: "x <= y | y <= x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   709
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   710
lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   711
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   712
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   713
  apply blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   714
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   715
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   716
lemma linorder_cases [case_names less equal greater]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   717
    "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   718
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   719
  apply blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   720
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   721
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   722
lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   723
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   724
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   725
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   726
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   727
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   728
lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   729
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   730
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   731
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   732
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   733
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   734
lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   735
  apply (cut_tac x = x and y = y in linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   736
  apply auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   737
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   738
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   739
lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   740
  apply (simp add: linorder_neq_iff)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   741
  apply blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   742
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   743
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   744
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   745
subsubsection "Min and max on (linear) orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   746
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   747
lemma min_same [simp]: "min (x::'a::order) x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   748
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   749
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   750
lemma max_same [simp]: "max (x::'a::order) x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   751
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   752
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   753
lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   754
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   755
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   756
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   757
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   758
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   759
lemma le_maxI1: "(x::'a::linorder) <= max x y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   760
  by (simp add: le_max_iff_disj)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   761
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   762
lemma le_maxI2: "(y::'a::linorder) <= max x y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   763
    -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   764
  by (simp add: le_max_iff_disj)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   765
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   766
lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   767
  apply (simp add: max_def order_le_less)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   768
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   769
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   770
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   771
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   772
lemma max_le_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   773
    "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   774
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   775
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   776
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   777
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   778
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   779
lemma max_less_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   780
    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   781
  apply (simp add: order_le_less max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   782
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   783
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   784
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   785
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   786
lemma le_min_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   787
    "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   788
    -- {* @{text "[iff]"} screws up a Q{text blast} in MiniML *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   789
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   790
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   791
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   792
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   793
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   794
lemma min_less_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   795
    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   796
  apply (simp add: order_le_less min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   797
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   798
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   799
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   800
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   801
lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   802
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   803
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   804
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   805
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   806
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   807
lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   808
  apply (simp add: min_def order_le_less)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   809
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   810
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   811
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   812
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   813
lemma split_min:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   814
    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   815
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   816
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   817
lemma split_max:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   818
    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   819
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   820
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   821
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   822
subsubsection "Bounded quantifiers"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   823
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   824
syntax
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   825
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   826
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   827
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   828
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   829
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
   830
syntax (xsymbols)
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   831
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   832
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   833
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   834
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   835
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   836
syntax (HOL)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   837
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   838
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   839
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   840
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   841
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   842
translations
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   843
 "ALL x<y. P"   =>  "ALL x. x < y --> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   844
 "EX x<y. P"    =>  "EX x. x < y  & P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   845
 "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   846
 "EX x<=y. P"   =>  "EX x. x <= y & P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   847
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   848
end