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 (symbols)
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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 "\<midarrow>\<rightarrow>" 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 (symbols output)
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  "op ~="       :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
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syntax (xsymbols)
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  "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
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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"
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axclass minus < "term"
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axclass times < "term"
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axclass inverse < "term"
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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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local
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typed_print_translation {*
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  let
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    fun tr' c = (c, fn show_sorts => fn T => fn ts =>
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      if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
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      else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
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  in [tr' "0", tr' "1"] end;
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*} -- {* show types that are presumably too general *}
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consts
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  abs           :: "'a::minus => 'a"
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  inverse       :: "'a::inverse => 'a"
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  divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
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syntax (xsymbols)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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syntax (HTML output)
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  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
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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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subsection {* Theory and package setup *}
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subsubsection {* Basic lemmas *}
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use "HOL_lemmas.ML"
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theorems case_split = case_split_thm [case_names True False]
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declare trans [trans]
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declare impE [CPure.elim]  iffD1 [CPure.elim]  iffD2 [CPure.elim]
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subsubsection {* Atomizing meta-level connectives *}
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lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
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proof (rule equal_intr_rule)
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  assume "!!x. P x"
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  show "ALL x. P x" by (rule allI)
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next
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  assume "ALL x. P x"
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  thus "!!x. P x" by (rule allE)
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qed
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lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
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proof (rule equal_intr_rule)
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  assume r: "A ==> B"
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  show "A --> B" by (rule impI) (rule r)
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next
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  assume "A --> B" and A
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  thus B by (rule mp)
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qed
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lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
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proof (rule equal_intr_rule)
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  assume "x == y"
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  show "x = y" by (unfold prems) (rule refl)
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next
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  assume "x = y"
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  thus "x == y" by (rule eq_reflection)
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qed
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lemma atomize_conj [atomize]: "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
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proof (rule equal_intr_rule)
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  assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
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  show "A & B" by (rule conjI)
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next
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  fix C
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  assume "A & B"
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  assume "A ==> B ==> PROP C"
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  thus "PROP C"
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  proof this
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    show A by (rule conjunct1)
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    show B by (rule conjunct2)
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  qed
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qed
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subsubsection {* Classical Reasoner setup *}
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use "cladata.ML"
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setup hypsubst_setup
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declare atomize_all [symmetric, rulify]  atomize_imp [symmetric, rulify]
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setup Classical.setup
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setup clasetup
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use "blastdata.ML"
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setup Blast.setup
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subsubsection {* Simplifier setup *}
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use "simpdata.ML"
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setup Simplifier.setup
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setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
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setup Splitter.setup setup Clasimp.setup
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subsubsection {* Generic cases and induction *}
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constdefs
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  inductive_forall :: "('a => bool) => bool"
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  "inductive_forall P == \<forall>x. P x"
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  inductive_implies :: "bool => bool => bool"
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  "inductive_implies A B == A --> B"
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  inductive_equal :: "'a => 'a => bool"
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  "inductive_equal x y == x = y"
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  inductive_conj :: "bool => bool => bool"
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  "inductive_conj A B == A & B"
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lemma inductive_forall_eq: "(!!x. P x) == Trueprop (inductive_forall (\<lambda>x. P x))"
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  by (simp only: atomize_all inductive_forall_def)
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lemma inductive_implies_eq: "(A ==> B) == Trueprop (inductive_implies A B)"
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  by (simp only: atomize_imp inductive_implies_def)
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lemma inductive_equal_eq: "(x == y) == Trueprop (inductive_equal x y)"
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  by (simp only: atomize_eq inductive_equal_def)
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lemma inductive_forall_conj: "inductive_forall (\<lambda>x. inductive_conj (A x) (B x)) =
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    inductive_conj (inductive_forall A) (inductive_forall B)"
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  by (unfold inductive_forall_def inductive_conj_def) blast
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lemma inductive_implies_conj: "inductive_implies C (inductive_conj A B) =
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    inductive_conj (inductive_implies C A) (inductive_implies C B)"
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  by (unfold inductive_implies_def inductive_conj_def) blast
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lemma inductive_conj_curry: "(inductive_conj A B ==> C) == (A ==> B ==> C)"
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  by (simp only: atomize_imp atomize_eq inductive_conj_def) (rule equal_intr_rule, blast+)
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lemmas inductive_atomize = inductive_forall_eq inductive_implies_eq inductive_equal_eq
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lemmas inductive_rulify1 = inductive_atomize [symmetric, standard]
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lemmas inductive_rulify2 =
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  inductive_forall_def inductive_implies_def inductive_equal_def inductive_conj_def
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lemmas inductive_conj = inductive_forall_conj inductive_implies_conj inductive_conj_curry
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hide const inductive_forall inductive_implies inductive_equal inductive_conj
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diff changeset
   310
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   311
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   312
text {* Method setup. *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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   313
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   314
ML {*
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   315
  structure InductMethod = InductMethodFun
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   316
  (struct
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   317
    val dest_concls = HOLogic.dest_concls;
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   318
    val cases_default = thm "case_split";
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   319
    val conjI = thm "conjI";
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   320
    val atomize = thms "inductive_atomize";
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   321
    val rulify1 = thms "inductive_rulify1";
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   322
    val rulify2 = thms "inductive_rulify2";
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   323
  end);
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   324
*}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   325
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   326
setup InductMethod.setup
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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   327
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
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diff changeset
   328
11750
3e400964893e judgment Trueprop;
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diff changeset
   329
subsection {* Order signatures and orders *}
3e400964893e judgment Trueprop;
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diff changeset
   330
3e400964893e judgment Trueprop;
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diff changeset
   331
axclass
3e400964893e judgment Trueprop;
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diff changeset
   332
  ord < "term"
3e400964893e judgment Trueprop;
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diff changeset
   333
3e400964893e judgment Trueprop;
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diff changeset
   334
syntax
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   335
  "op <"        :: "['a::ord, 'a] => bool"             ("op <")
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   336
  "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   337
3e400964893e judgment Trueprop;
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diff changeset
   338
global
3e400964893e judgment Trueprop;
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diff changeset
   339
3e400964893e judgment Trueprop;
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diff changeset
   340
consts
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   341
  "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   342
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   343
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   344
local
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   345
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   346
syntax (symbols)
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   347
  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   348
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   349
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   350
(*Tell blast about overloading of < and <= to reduce the risk of
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   351
  its applying a rule for the wrong type*)
3e400964893e judgment Trueprop;
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diff changeset
   352
ML {*
3e400964893e judgment Trueprop;
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diff changeset
   353
Blast.overloaded ("op <" , domain_type);
3e400964893e judgment Trueprop;
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diff changeset
   354
Blast.overloaded ("op <=", domain_type);
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   355
*}
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   356
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   357
3e400964893e judgment Trueprop;
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diff changeset
   358
subsubsection {* Monotonicity *}
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   359
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   360
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   361
  mono :: "['a::ord => 'b::ord] => bool"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   362
  "mono f == ALL A B. A <= B --> f A <= f B"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   363
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   364
lemma monoI [intro?]: "(!!A B. A <= B ==> f A <= f B) ==> mono f"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   365
  by (unfold mono_def) blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   366
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   367
lemma monoD [dest?]: "mono f ==> A <= B ==> f A <= f B"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   368
  by (unfold mono_def) blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   369
3e400964893e judgment Trueprop;
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parents: 11724
diff changeset
   370
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   371
  min :: "['a::ord, 'a] => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   372
  "min a b == (if a <= b then a else b)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   373
  max :: "['a::ord, 'a] => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   374
  "max a b == (if a <= b then b else a)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   375
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   376
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   377
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   378
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   379
lemma min_of_mono:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   380
    "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
   381
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   382
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   383
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   384
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   385
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   386
lemma max_of_mono:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   387
    "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
   388
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   389
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   390
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   391
subsubsection "Orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   392
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   393
axclass order < ord
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   394
  order_refl [iff]: "x <= x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   395
  order_trans: "x <= y ==> y <= z ==> x <= z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   396
  order_antisym: "x <= y ==> y <= x ==> x = y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   397
  order_less_le: "(x < y) = (x <= y & x ~= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   398
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   399
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   400
text {* Reflexivity. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   401
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   402
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   403
    -- {* This form is useful with the classical reasoner. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   404
  apply (erule ssubst)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   405
  apply (rule order_refl)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   406
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   407
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   408
lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   409
  by (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   410
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   411
lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   412
    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   413
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   414
  apply (blast intro!: order_refl)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   415
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   416
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   417
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   418
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   419
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   420
  by (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   421
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   422
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   423
text {* Asymmetry. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   424
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   425
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   426
  by (simp add: order_less_le order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   427
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   428
lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   429
  apply (drule order_less_not_sym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   430
  apply (erule contrapos_np)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   431
  apply simp
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   432
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   433
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   434
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   435
text {* Transitivity. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   436
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   437
lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   438
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   439
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   440
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   441
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   442
lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   443
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   444
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   445
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   446
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   447
lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   448
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   449
  apply (blast intro: order_trans order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   450
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   451
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   452
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   453
text {* Useful for simplification, but too risky to include by default. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   454
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   455
lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   456
  by (blast elim: order_less_asym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   457
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   458
lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   459
  by (blast elim: order_less_asym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   460
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   461
lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   462
  by auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   463
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   464
lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   465
  by auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   466
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   467
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   468
text {* Other operators. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   469
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   470
lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   471
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   472
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   473
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   474
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   475
lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   476
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   477
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   478
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   479
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   480
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   481
subsubsection {* Least value operator *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   482
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   483
constdefs
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   484
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   485
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   486
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   487
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   488
lemma LeastI2:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   489
  "[| P (x::'a::order);
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   490
      !!y. P y ==> x <= y;
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   491
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   492
   ==> Q (Least P)";
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   493
  apply (unfold Least_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   494
  apply (rule theI2)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   495
    apply (blast intro: order_antisym)+
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   496
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   497
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   498
lemma Least_equality:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   499
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k";
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   500
  apply (simp add: Least_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   501
  apply (rule the_equality)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   502
  apply (auto intro!: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   503
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   504
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   505
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   506
subsubsection "Linear / total orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   507
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   508
axclass linorder < order
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   509
  linorder_linear: "x <= y | y <= x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   510
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   511
lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   512
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   513
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   514
  apply blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   515
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   516
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   517
lemma linorder_cases [case_names less equal greater]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   518
    "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   519
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   520
  apply blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   521
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   522
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   523
lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   524
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   525
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   526
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   527
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   528
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   529
lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   530
  apply (simp add: order_less_le)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   531
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   532
  apply (blast intro: order_antisym)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   533
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   534
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   535
lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   536
  apply (cut_tac x = x and y = y in linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   537
  apply auto
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   538
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   539
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   540
lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   541
  apply (simp add: linorder_neq_iff)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   542
  apply blast
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   543
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   544
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   545
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   546
subsubsection "Min and max on (linear) orders"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   547
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   548
lemma min_same [simp]: "min (x::'a::order) x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   549
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   550
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   551
lemma max_same [simp]: "max (x::'a::order) x = x"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   552
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   553
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   554
lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   555
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   556
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   557
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   558
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   559
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   560
lemma le_maxI1: "(x::'a::linorder) <= max x y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   561
  by (simp add: le_max_iff_disj)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   562
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   563
lemma le_maxI2: "(y::'a::linorder) <= max x y"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   564
    -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   565
  by (simp add: le_max_iff_disj)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   566
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   567
lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   568
  apply (simp add: max_def order_le_less)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   569
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   570
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   571
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   572
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   573
lemma max_le_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   574
    "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   575
  apply (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   576
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   577
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   578
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   579
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   580
lemma max_less_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   581
    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   582
  apply (simp add: order_le_less max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   583
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   584
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   585
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   586
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   587
lemma le_min_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   588
    "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   589
    -- {* @{text "[iff]"} screws up a Q{text blast} in MiniML *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   590
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   591
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   592
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   593
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   594
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   595
lemma min_less_iff_conj [simp]:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   596
    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   597
  apply (simp add: order_le_less min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   598
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   599
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   600
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   601
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   602
lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   603
  apply (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   604
  apply (insert linorder_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   605
  apply (blast intro: order_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   606
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   607
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   608
lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   609
  apply (simp add: min_def order_le_less)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   610
  apply (insert linorder_less_linear)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   611
  apply (blast intro: order_less_trans)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   612
  done
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   613
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   614
lemma split_min:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   615
    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   616
  by (simp add: min_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   617
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   618
lemma split_max:
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   619
    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   620
  by (simp add: max_def)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   621
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   622
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   623
subsubsection "Bounded quantifiers"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   624
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   625
syntax
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   626
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   627
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   628
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   629
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   630
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   631
syntax (symbols)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   632
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   633
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   634
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   635
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   636
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   637
syntax (HOL)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   638
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   639
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   640
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   641
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   642
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   643
translations
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   644
 "ALL x<y. P"   =>  "ALL x. x < y --> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   645
 "EX x<y. P"    =>  "EX x. x < y  & P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   646
 "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   647
 "EX x<=y. P"   =>  "EX x. x <= y & P"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   648
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   649
end