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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    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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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           :: "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 < 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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3e400964893e judgment Trueprop;
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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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3e400964893e judgment Trueprop;
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local
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   167
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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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3e400964893e judgment Trueprop;
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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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   181
3e400964893e judgment Trueprop;
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syntax (xsymbols)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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   184
syntax (HTML output)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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3e400964893e judgment Trueprop;
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axclass plus_ac0 < plus, zero
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  commute: "x + y = y + x"
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  assoc:   "(x + y) + z = x + (y + z)"
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  zero:    "0 + x = x"
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   191
3e400964893e judgment Trueprop;
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subsection {* Theory and package setup *}
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subsubsection {* Basic lemmas *}
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   196
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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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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   213
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   214
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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   217
  shows Q
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   218
proof -
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   219
  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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   222
12937
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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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   226
  shows R
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   227
proof -
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  from 2 and 1 have P .
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   229
  with 1 show R by (rule notE)
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   230
qed
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   231
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   232
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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   236
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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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   240
11438
3d9222b80989 declare trans [trans] (*overridden in theory Calculation*);
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   241
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   242
subsubsection {* Atomizing meta-level connectives *}
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   243
3e400964893e judgment Trueprop;
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   244
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
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   245
proof
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   246
  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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   248
next
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   249
  assume "ALL x. P x"
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  thus "!!x. P x" by (rule allE)
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   251
qed
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   252
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   253
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
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   254
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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   257
next
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   258
  assume "A --> B" and A
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   259
  thus B by (rule mp)
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   260
qed
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   261
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   262
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
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   263
proof
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   264
  assume "x == y"
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   265
  show "x = y" by (unfold prems) (rule refl)
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   266
next
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
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   267
  assume "x = y"
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   268
  thus "x == y" by (rule eq_reflection)
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   269
qed
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
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   270
12023
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   271
lemma atomize_conj [atomize]:
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  "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
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   273
proof
11953
f98623fdf6ef atomize_conj;
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   274
  assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
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   275
  show "A & B" by (rule conjI)
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   276
next
f98623fdf6ef atomize_conj;
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   277
  fix C
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   278
  assume "A & B"
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   279
  assume "A ==> B ==> PROP C"
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   280
  thus "PROP C"
f98623fdf6ef atomize_conj;
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   281
  proof this
f98623fdf6ef atomize_conj;
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   282
    show A by (rule conjunct1)
f98623fdf6ef atomize_conj;
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   283
    show B by (rule conjunct2)
f98623fdf6ef atomize_conj;
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   284
  qed
f98623fdf6ef atomize_conj;
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   285
qed
f98623fdf6ef atomize_conj;
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   286
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lemmas [symmetric, rulify] = atomize_all atomize_imp
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   288
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   289
3e400964893e judgment Trueprop;
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   290
subsubsection {* Classical Reasoner setup *}
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   291
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   292
use "cladata.ML"
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   293
setup hypsubst_setup
11977
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   294
12386
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   295
ML_setup {*
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   296
  Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
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   297
*}
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   298
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   299
setup Classical.setup
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   300
setup clasetup
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   301
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lemmas [intro?] = ext
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   303
  and [elim?] = ex1_implies_ex
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   304
9869
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   305
use "blastdata.ML"
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   306
setup Blast.setup
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   307
11750
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   308
3e400964893e judgment Trueprop;
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   309
subsubsection {* Simplifier setup *}
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   310
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   311
lemma meta_eq_to_obj_eq: "x == y ==> x = y"
3bd113b8f7a6 converted simp lemmas;
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   312
proof -
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   313
  assume r: "x == y"
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   314
  show "x = y" by (unfold r) (rule refl)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   315
qed
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   316
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   317
lemma eta_contract_eq: "(%s. f s) = f" ..
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   318
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   319
lemma simp_thms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   320
  shows not_not: "(~ ~ P) = P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   321
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   322
    "(P ~= Q) = (P = (~Q))"
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   323
    "(P | ~P) = True"    "(~P | P) = True"
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   324
    "((~P) = (~Q)) = (P=Q)"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   325
    "(x = x) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   326
    "(~True) = False"  "(~False) = True"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   327
    "(~P) ~= P"  "P ~= (~P)"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   328
    "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   329
    "(True --> P) = P"  "(False --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   330
    "(P --> True) = True"  "(P --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   331
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   332
    "(P & True) = P"  "(True & P) = P"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   333
    "(P & False) = False"  "(False & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   334
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   335
    "(P & ~P) = False"    "(~P & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   336
    "(P | True) = True"  "(True | P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   337
    "(P | False) = P"  "(False | P) = P"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   338
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   339
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   340
    -- {* needed for the one-point-rule quantifier simplification procs *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   341
    -- {* essential for termination!! *} and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   342
    "!!P. (EX x. x=t & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   343
    "!!P. (EX x. t=x & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   344
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   345
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   346
  by (blast, blast, blast, blast, blast, rules+)
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   347
 
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   348
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   349
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   350
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   351
lemma ex_simps:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   352
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   353
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   354
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   355
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   356
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   357
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   358
  -- {* Miniscoping: pushing in existential quantifiers. *}
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   359
  by (rules | blast)+
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   360
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   361
lemma all_simps:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   362
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   363
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   364
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   365
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   366
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   367
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   368
  -- {* Miniscoping: pushing in universal quantifiers. *}
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   369
  by (rules | blast)+
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   370
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   371
lemma eq_ac:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   372
  shows eq_commute: "(a=b) = (b=a)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   373
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   374
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   375
lemma neq_commute: "(a~=b) = (b~=a)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   376
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   377
lemma conj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   378
  shows conj_commute: "(P&Q) = (Q&P)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   379
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   380
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   381
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   382
lemma disj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   383
  shows disj_commute: "(P|Q) = (Q|P)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   384
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   385
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   386
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   387
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   388
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   389
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   390
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   391
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   392
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   393
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   394
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   395
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   396
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   397
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
   398
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   399
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   400
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   401
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   402
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   403
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   404
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   405
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   406
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   407
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   408
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   409
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   410
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   411
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   412
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   413
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   414
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   415
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   416
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   417
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   418
  -- {* cases boil down to the same thing. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   419
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   420
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   421
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   422
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   423
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   424
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   425
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   426
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   427
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   428
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   429
text {*
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   430
  \medskip The @{text "&"} congruence rule: not included by default!
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   431
  May slow rewrite proofs down by as much as 50\% *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   432
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   433
lemma conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   434
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   435
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   436
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   437
lemma rev_conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   438
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   439
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   440
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   441
text {* The @{text "|"} congruence rule: not included by default! *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   442
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   443
lemma disj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   444
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   445
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   446
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   447
lemma eq_sym_conv: "(x = y) = (y = x)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   448
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   449
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   450
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   451
text {* \medskip if-then-else rules *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   452
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   453
lemma if_True: "(if True then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   454
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   455
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   456
lemma if_False: "(if False then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   457
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   458
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   459
lemma if_P: "P ==> (if P then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   460
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   461
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   462
lemma if_not_P: "~P ==> (if P then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   463
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   464
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   465
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
   466
  apply (rule case_split [of Q])
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   467
   apply (subst if_P)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   468
    prefer 3 apply (subst if_not_P)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   469
     apply blast+
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   470
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   471
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   472
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
   473
  apply (subst split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   474
  apply blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   475
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   476
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   477
lemmas if_splits = split_if split_if_asm
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   478
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   479
lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   480
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   481
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   482
lemma if_cancel: "(if c then x else x) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   483
  apply (subst split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   484
  apply blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   485
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   486
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   487
lemma if_eq_cancel: "(if x = y then y else x) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   488
  apply (subst split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   489
  apply blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   490
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   491
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   492
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   493
  -- {* 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
   494
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   495
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   496
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   497
  -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   498
  apply (subst split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   499
  apply blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   500
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   501
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   502
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   503
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   504
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   505
use "simpdata.ML"
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   506
setup Simplifier.setup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   507
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   508
setup Splitter.setup setup Clasimp.setup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   509
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   510
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   511
subsubsection {* Generic cases and induction *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   512
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   513
constdefs
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   514
  induct_forall :: "('a => bool) => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   515
  "induct_forall P == \<forall>x. P x"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   516
  induct_implies :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   517
  "induct_implies A B == A --> B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   518
  induct_equal :: "'a => 'a => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   519
  "induct_equal x y == x = y"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   520
  induct_conj :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   521
  "induct_conj A B == A & B"
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   522
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   523
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
   524
  by (simp only: atomize_all induct_forall_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   525
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   526
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   527
  by (simp only: atomize_imp induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   528
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   529
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   530
  by (simp only: atomize_eq induct_equal_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   531
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   532
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
   533
    induct_conj (induct_forall A) (induct_forall B)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   534
  by (unfold induct_forall_def induct_conj_def) rules
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   535
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   536
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   537
    induct_conj (induct_implies C A) (induct_implies C B)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   538
  by (unfold induct_implies_def induct_conj_def) rules
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   539
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   540
lemma induct_conj_curry: "(induct_conj A B ==> C) == (A ==> B ==> C)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   541
  by (simp only: atomize_imp atomize_eq induct_conj_def) (rules intro: equal_intr_rule)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   542
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   543
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   544
  by (simp add: induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   545
12161
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
   546
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
   547
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
   548
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
   549
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
   550
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   551
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
   552
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   553
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   554
text {* Method setup. *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   555
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   556
ML {*
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   557
  structure InductMethod = InductMethodFun
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   558
  (struct
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   559
    val dest_concls = HOLogic.dest_concls;
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   560
    val cases_default = thm "case_split";
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   561
    val local_impI = thm "induct_impliesI";
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   562
    val conjI = thm "conjI";
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   563
    val atomize = thms "induct_atomize";
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   564
    val rulify1 = thms "induct_rulify1";
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
   565
    val rulify2 = thms "induct_rulify2";
12240
0760eda193c4 induct method: localize rews for rule;
wenzelm
parents: 12161
diff changeset
   566
    val localize = [Thm.symmetric (thm "induct_implies_def")];
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   567
  end);
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   568
*}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   569
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   570
setup InductMethod.setup
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   571
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
   572
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   573
subsection {* Order signatures and orders *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   574
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   575
axclass
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
   576
  ord < type
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   577
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   578
syntax
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   579
  "op <"        :: "['a::ord, 'a] => bool"             ("op <")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   580
  "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   581
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   582
global
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   583
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   584
consts
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   585
  "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   586
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   587
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   588
local
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   589
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
   590
syntax (xsymbols)
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   591
  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   592
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   593
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   594
(*Tell blast about overloading of < and <= to reduce the risk of
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   595
  its applying a rule for the wrong type*)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   596
ML {*
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   597
Blast.overloaded ("op <" , domain_type);
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   598
Blast.overloaded ("op <=", domain_type);
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   599
*}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   600
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   601
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   602
subsubsection {* Monotonicity *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   603
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   604
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   605
  mono :: "['a::ord => 'b::ord] => bool"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   606
  "mono f == ALL A B. A <= B --> f A <= f B"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   607
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   608
lemma monoI [intro?]: "(!!A B. A <= B ==> f A <= f B) ==> mono f"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   609
  by (unfold mono_def) rules
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   610
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   611
lemma monoD [dest?]: "mono f ==> A <= B ==> f A <= f B"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   612
  by (unfold mono_def) rules
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   613
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   614
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   615
  min :: "['a::ord, 'a] => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   616
  "min a b == (if a <= b then a else b)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   617
  max :: "['a::ord, 'a] => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   618
  "max a b == (if a <= b then b else a)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   619
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   620
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   621
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   622
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   623
lemma min_of_mono:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   624
    "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
   625
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   626
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   627
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   628
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   629
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   630
lemma max_of_mono:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   631
    "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
   632
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   633
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   634
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   635
subsubsection "Orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   636
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   637
axclass order < ord
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   638
  order_refl [iff]: "x <= x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   639
  order_trans: "x <= y ==> y <= z ==> x <= z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   640
  order_antisym: "x <= y ==> y <= x ==> x = y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   641
  order_less_le: "(x < y) = (x <= y & x ~= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   642
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   643
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   644
text {* Reflexivity. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   645
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   646
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   647
    -- {* This form is useful with the classical reasoner. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   648
  apply (erule ssubst)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   649
  apply (rule order_refl)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   650
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   651
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   652
lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   653
  by (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   654
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   655
lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   656
    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   657
  apply (simp add: order_less_le)
12256
wenzelm
parents: 12240
diff changeset
   658
  apply blast
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   659
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   660
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   661
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   662
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   663
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   664
  by (simp add: order_less_le)
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 {* Asymmetry. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   668
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   669
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   670
  by (simp add: order_less_le order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   671
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   672
lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   673
  apply (drule order_less_not_sym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   674
  apply (erule contrapos_np)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   675
  apply simp
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   676
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   677
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   678
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   679
text {* Transitivity. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   680
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   681
lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   682
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   683
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   684
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   685
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   686
lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   687
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   688
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   689
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   690
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   691
lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   692
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   693
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   694
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   695
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   696
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   697
text {* Useful for simplification, but too risky to include by default. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   698
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   699
lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   700
  by (blast elim: order_less_asym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   701
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   702
lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   703
  by (blast elim: order_less_asym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   704
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   705
lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   706
  by auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   707
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   708
lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   709
  by auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   710
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   711
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   712
text {* Other operators. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   713
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   714
lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   715
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   716
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   717
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   718
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   719
lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   720
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   721
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   722
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   723
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   724
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   725
subsubsection {* Least value operator *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   726
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   727
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   728
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   729
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   730
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   731
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   732
lemma LeastI2:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   733
  "[| P (x::'a::order);
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   734
      !!y. P y ==> x <= y;
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   735
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   736
   ==> Q (Least P)"
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   737
  apply (unfold Least_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   738
  apply (rule theI2)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   739
    apply (blast intro: order_antisym)+
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   740
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   741
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   742
lemma Least_equality:
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   743
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   744
  apply (simp add: Least_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   745
  apply (rule the_equality)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   746
  apply (auto intro!: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   747
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   748
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   749
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   750
subsubsection "Linear / total orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   751
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   752
axclass linorder < order
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   753
  linorder_linear: "x <= y | y <= x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   754
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   755
lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   756
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   757
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   758
  apply blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   759
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   760
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   761
lemma linorder_cases [case_names less equal greater]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   762
    "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   763
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   764
  apply blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   765
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   766
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   767
lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   768
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   769
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   770
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   771
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   772
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   773
lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   774
  apply (simp add: order_less_le)
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_antisym)
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 linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   780
  apply (cut_tac x = x and y = y in linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   781
  apply auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   782
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   783
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   784
lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   785
  apply (simp add: linorder_neq_iff)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   786
  apply blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   787
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   788
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   789
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   790
subsubsection "Min and max on (linear) orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   791
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   792
lemma min_same [simp]: "min (x::'a::order) x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   793
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   794
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   795
lemma max_same [simp]: "max (x::'a::order) x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   796
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   797
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   798
lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   799
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   800
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   801
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   802
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   803
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   804
lemma le_maxI1: "(x::'a::linorder) <= max x y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   805
  by (simp add: le_max_iff_disj)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   806
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   807
lemma le_maxI2: "(y::'a::linorder) <= max x y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   808
    -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   809
  by (simp add: le_max_iff_disj)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   810
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   811
lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   812
  apply (simp add: max_def order_le_less)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   813
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   814
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   815
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   816
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   817
lemma max_le_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   818
    "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   819
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   820
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   821
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   822
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   823
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   824
lemma max_less_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   825
    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   826
  apply (simp add: order_le_less max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   827
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   828
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   829
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   830
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   831
lemma le_min_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   832
    "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
12892
wenzelm
parents: 12650
diff changeset
   833
    -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   834
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   835
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   836
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   837
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   838
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   839
lemma min_less_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   840
    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   841
  apply (simp add: order_le_less min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   842
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   843
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   844
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   845
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   846
lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   847
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   848
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   849
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   850
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   851
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   852
lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   853
  apply (simp add: min_def order_le_less)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   854
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   855
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   856
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   857
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   858
lemma split_min:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   859
    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   860
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   861
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   862
lemma split_max:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   863
    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   864
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   865
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   866
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   867
subsubsection "Bounded quantifiers"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   868
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   869
syntax
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   870
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   871
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   872
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   873
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   874
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
   875
syntax (xsymbols)
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   876
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   877
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   878
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   879
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   880
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   881
syntax (HOL)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   882
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   883
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   884
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   885
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   886
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   887
translations
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   888
 "ALL x<y. P"   =>  "ALL x. x < y --> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   889
 "EX x<y. P"    =>  "EX x. x < y  & P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   890
 "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   891
 "EX x<=y. P"   =>  "EX x. x <= y & P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   892
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   893
end