src/HOL/HOL.thy
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Sun, 29 Jul 2007 14:29:49 +0200
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(*  Title:      HOL/HOL.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
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*)
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header {* The basis of Higher-Order Logic *}
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theory HOL
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imports CPure
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uses
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  "~~/src/Tools/integer.ML"
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  ("hologic.ML")
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  "~~/src/Tools/IsaPlanner/zipper.ML"
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  "~~/src/Tools/IsaPlanner/isand.ML"
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  "~~/src/Tools/IsaPlanner/rw_tools.ML"
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  "~~/src/Tools/IsaPlanner/rw_inst.ML"
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  "~~/src/Provers/project_rule.ML"
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  "~~/src/Provers/induct_method.ML"
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  "~~/src/Provers/hypsubst.ML"
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  "~~/src/Provers/splitter.ML"
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  "~~/src/Provers/classical.ML"
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  "~~/src/Provers/blast.ML"
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  "~~/src/Provers/clasimp.ML"
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  "~~/src/Provers/eqsubst.ML"
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  "~~/src/Provers/quantifier1.ML"
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  ("simpdata.ML")
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  ("~~/src/HOL/Tools/recfun_codegen.ML")
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begin
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subsection {* Primitive logic *}
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subsubsection {* Core syntax *}
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classes type
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defaultsort type
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global
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typedecl bool
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arities
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  bool :: type
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  "fun" :: (type, type) type
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judgment
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  Trueprop      :: "bool => prop"                   ("(_)" 5)
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consts
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  Not           :: "bool => bool"                   ("~ _" [40] 40)
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  True          :: bool
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  False         :: bool
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  arbitrary     :: 'a
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  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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  "op ="        :: "['a, 'a] => bool"               (infixl "=" 50)
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  "op &"        :: "[bool, bool] => bool"           (infixr "&" 35)
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  "op |"        :: "[bool, bool] => bool"           (infixr "|" 30)
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  "op -->"      :: "[bool, bool] => bool"           (infixr "-->" 25)
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local
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consts
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  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
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subsubsection {* Additional concrete syntax *}
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notation (output)
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  "op ="  (infix "=" 50)
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abbreviation
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  not_equal :: "['a, 'a] => bool"  (infixl "~=" 50) where
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  "x ~= y == ~ (x = y)"
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notation (output)
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  not_equal  (infix "~=" 50)
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notation (xsymbols)
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  Not  ("\<not> _" [40] 40) and
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  "op &"  (infixr "\<and>" 35) and
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  "op |"  (infixr "\<or>" 30) and
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  "op -->"  (infixr "\<longrightarrow>" 25) and
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  not_equal  (infix "\<noteq>" 50)
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notation (HTML output)
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  Not  ("\<not> _" [40] 40) and
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  "op &"  (infixr "\<and>" 35) and
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  "op |"  (infixr "\<or>" 30) and
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  not_equal  (infix "\<noteq>" 50)
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abbreviation (iff)
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  iff :: "[bool, bool] => bool"  (infixr "<->" 25) where
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  "A <-> B == A = B"
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notation (xsymbols)
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  iff  (infixr "\<longleftrightarrow>" 25)
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nonterminals
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  letbinds  letbind
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  case_syn  cases_syn
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syntax
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  "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
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  "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
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  ""            :: "letbind => letbinds"                 ("_")
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  "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
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  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
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  "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
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  ""            :: "case_syn => cases_syn"               ("_")
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  "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
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translations
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  "THE x. P"              == "The (%x. P)"
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  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
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  "let x = a in e"        == "Let a (%x. e)"
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print_translation {*
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(* To avoid eta-contraction of body: *)
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[("The", fn [Abs abs] =>
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     let val (x,t) = atomic_abs_tr' abs
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     in Syntax.const "_The" $ x $ t end)]
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*}
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syntax (xsymbols)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
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notation (xsymbols)
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  All  (binder "\<forall>" 10) and
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  Ex  (binder "\<exists>" 10) and
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  Ex1  (binder "\<exists>!" 10)
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notation (HTML output)
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  All  (binder "\<forall>" 10) and
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  Ex  (binder "\<exists>" 10) and
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  Ex1  (binder "\<exists>!" 10)
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notation (HOL)
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  All  (binder "! " 10) and
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  Ex  (binder "? " 10) and
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  Ex1  (binder "?! " 10)
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subsubsection {* Axioms and basic definitions *}
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axioms
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  eq_reflection:  "(x=y) ==> (x==y)"
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  refl:           "t = (t::'a)"
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  ext:            "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
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    -- {*Extensionality is built into the meta-logic, and this rule expresses
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         a related property.  It is an eta-expanded version of the traditional
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         rule, and similar to the ABS rule of HOL*}
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  the_eq_trivial: "(THE x. x = a) = (a::'a)"
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  impI:           "(P ==> Q) ==> P-->Q"
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  mp:             "[| P-->Q;  P |] ==> Q"
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defs
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  True_def:     "True      == ((%x::bool. x) = (%x. x))"
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  All_def:      "All(P)    == (P = (%x. True))"
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  Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
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  False_def:    "False     == (!P. P)"
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  not_def:      "~ P       == P-->False"
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  and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
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  or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
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  Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
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axioms
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  iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
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  True_or_False:  "(P=True) | (P=False)"
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defs
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  Let_def [code func]: "Let s f == f(s)"
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  if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
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finalconsts
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  "op ="
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  "op -->"
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  The
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  arbitrary
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axiomatization
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  undefined :: 'a
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axiomatization where
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  undefined_fun: "undefined x = undefined"
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subsubsection {* Generic classes and algebraic operations *}
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class default = type +
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  fixes default :: "'a"
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class zero = type + 
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  fixes zero :: "'a"  ("\<^loc>0")
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class one = type +
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  fixes one  :: "'a"  ("\<^loc>1")
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hide (open) const zero one
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class plus = type +
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  fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>+" 65)
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class minus = type +
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  fixes uminus :: "'a \<Rightarrow> 'a" 
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    and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>-" 65)
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class times = type +
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  fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>*" 70)
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class inverse = type +
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  fixes inverse :: "'a \<Rightarrow> 'a"
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    and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>'/" 70)
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class abs = type +
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  fixes abs :: "'a \<Rightarrow> 'a"
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notation
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  uminus  ("- _" [81] 80)
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notation (xsymbols)
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  abs  ("\<bar>_\<bar>")
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notation (HTML output)
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  abs  ("\<bar>_\<bar>")
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class ord = type +
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  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubseteq>" 50)
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    and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
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begin
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notation
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  less_eq  ("op \<^loc><=") and
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  less_eq  ("(_/ \<^loc><= _)" [51, 51] 50) and
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  less  ("op \<^loc><") and
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  less  ("(_/ \<^loc>< _)"  [51, 51] 50)
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notation (xsymbols)
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  less_eq  ("op \<^loc>\<le>") and
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  less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
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notation (HTML output)
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  less_eq  ("op \<^loc>\<le>") and
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  less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
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abbreviation (input)
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  greater  (infix "\<^loc>>" 50) where
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  "x \<^loc>> y \<equiv> y \<^loc>< x"
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abbreviation (input)
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  greater_eq  (infix "\<^loc>>=" 50) where
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  "x \<^loc>>= y \<equiv> y \<^loc><= x"
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notation (input)
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  greater_eq  (infix "\<^loc>\<ge>" 50)
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definition
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  Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "\<^loc>LEAST " 10)
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where
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  "Least P == (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<^loc>\<le> y))"
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end
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notation
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  less_eq  ("op <=") and
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  less_eq  ("(_/ <= _)" [51, 51] 50) and
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  less  ("op <") and
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  less  ("(_/ < _)"  [51, 51] 50)
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notation (xsymbols)
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  less_eq  ("op \<le>") and
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  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
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notation (HTML output)
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  less_eq  ("op \<le>") and
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  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
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abbreviation (input)
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  greater  (infix ">" 50) where
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  "x > y \<equiv> y < x"
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abbreviation (input)
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  greater_eq  (infix ">=" 50) where
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  "x >= y \<equiv> y <= x"
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notation (input)
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  greater_eq  (infix "\<ge>" 50)
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syntax
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  "_index1"  :: index    ("\<^sub>1")
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translations
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  (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
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typed_print_translation {*
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let
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  fun tr' c = (c, fn show_sorts => fn T => fn ts =>
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    if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
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    else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
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in map tr' [@{const_syntax HOL.one}, @{const_syntax HOL.zero}] end;
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*} -- {* show types that are presumably too general *}
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subsection {* Fundamental rules *}
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subsubsection {* Equality *}
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   318
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text {* Thanks to Stephan Merz *}
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   320
lemma subst:
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   321
  assumes eq: "s = t" and p: "P s"
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   322
  shows "P t"
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   323
proof -
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   324
  from eq have meta: "s \<equiv> t"
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   325
    by (rule eq_reflection)
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   326
  from p show ?thesis
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   327
    by (unfold meta)
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   328
qed
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   329
18457
356a9f711899 structure ProjectRule;
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   330
lemma sym: "s = t ==> t = s"
356a9f711899 structure ProjectRule;
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   331
  by (erule subst) (rule refl)
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   332
18457
356a9f711899 structure ProjectRule;
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   333
lemma ssubst: "t = s ==> P s ==> P t"
356a9f711899 structure ProjectRule;
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parents: 17992
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   334
  by (drule sym) (erule subst)
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   335
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   336
lemma trans: "[| r=s; s=t |] ==> r=t"
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356a9f711899 structure ProjectRule;
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diff changeset
   337
  by (erule subst)
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   338
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   339
lemma meta_eq_to_obj_eq: 
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   340
  assumes meq: "A == B"
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   341
  shows "A = B"
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   342
  by (unfold meq) (rule refl)
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   343
21502
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   344
text {* Useful with @{text erule} for proving equalities from known equalities. *}
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   345
     (* a = b
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   346
        |   |
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   347
        c = d   *)
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   348
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
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   349
apply (rule trans)
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   350
apply (rule trans)
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   351
apply (rule sym)
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   352
apply assumption+
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   353
done
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   354
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
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   355
text {* For calculational reasoning: *}
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   356
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
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   357
lemma forw_subst: "a = b ==> P b ==> P a"
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
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   358
  by (rule ssubst)
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
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parents: 15481
diff changeset
   359
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
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   360
lemma back_subst: "P a ==> a = b ==> P b"
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
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   361
  by (rule subst)
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   362
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   363
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   364
subsubsection {*Congruence rules for application*}
15411
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   365
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   366
(*similar to AP_THM in Gordon's HOL*)
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   367
lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
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diff changeset
   368
apply (erule subst)
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parents: 15380
diff changeset
   369
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
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diff changeset
   370
done
1d195de59497 removal of HOL_Lemmas
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diff changeset
   371
1d195de59497 removal of HOL_Lemmas
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   372
(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
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   373
lemma arg_cong: "x=y ==> f(x)=f(y)"
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diff changeset
   374
apply (erule subst)
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diff changeset
   375
apply (rule refl)
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diff changeset
   376
done
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diff changeset
   377
15655
157f3988f775 arg_cong2 by Norbert Voelker
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   378
lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
157f3988f775 arg_cong2 by Norbert Voelker
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   379
apply (erule ssubst)+
157f3988f775 arg_cong2 by Norbert Voelker
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   380
apply (rule refl)
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paulson
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diff changeset
   381
done
157f3988f775 arg_cong2 by Norbert Voelker
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diff changeset
   382
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diff changeset
   383
lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
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diff changeset
   384
apply (erule subst)+
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   385
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   386
done
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   387
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   388
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   389
subsubsection {*Equality of booleans -- iff*}
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paulson
parents: 15380
diff changeset
   390
21504
9c97af4a1567 tuned proofs;
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parents: 21502
diff changeset
   391
lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   392
  by (iprover intro: iff [THEN mp, THEN mp] impI assms)
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diff changeset
   393
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   394
lemma iffD2: "[| P=Q; Q |] ==> P"
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
   395
  by (erule ssubst)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   396
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   397
lemma rev_iffD2: "[| Q; P=Q |] ==> P"
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
   398
  by (erule iffD2)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   399
21504
9c97af4a1567 tuned proofs;
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parents: 21502
diff changeset
   400
lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   401
  by (drule sym) (rule iffD2)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   402
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   403
lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   404
  by (drule sym) (rule rev_iffD2)
15411
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   405
1d195de59497 removal of HOL_Lemmas
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diff changeset
   406
lemma iffE:
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   407
  assumes major: "P=Q"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   408
    and minor: "[| P --> Q; Q --> P |] ==> R"
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
   409
  shows R
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
   410
  by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
15411
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parents: 15380
diff changeset
   411
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   412
20944
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   413
subsubsection {*True*}
15411
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diff changeset
   414
1d195de59497 removal of HOL_Lemmas
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diff changeset
   415
lemma TrueI: "True"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   416
  unfolding True_def by (rule refl)
15411
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diff changeset
   417
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   418
lemma eqTrueI: "P ==> P = True"
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
   419
  by (iprover intro: iffI TrueI)
15411
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   420
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   421
lemma eqTrueE: "P = True ==> P"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   422
  by (erule iffD2) (rule TrueI)
15411
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paulson
parents: 15380
diff changeset
   423
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   424
20944
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   425
subsubsection {*Universal quantifier*}
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parents: 15380
diff changeset
   426
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   427
lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   428
  unfolding All_def by (iprover intro: ext eqTrueI assms)
15411
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parents: 15380
diff changeset
   429
1d195de59497 removal of HOL_Lemmas
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diff changeset
   430
lemma spec: "ALL x::'a. P(x) ==> P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   431
apply (unfold All_def)
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paulson
parents: 15380
diff changeset
   432
apply (rule eqTrueE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   433
apply (erule fun_cong)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   434
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   435
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   436
lemma allE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   437
  assumes major: "ALL x. P(x)"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   438
    and minor: "P(x) ==> R"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   439
  shows R
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   440
  by (iprover intro: minor major [THEN spec])
15411
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paulson
parents: 15380
diff changeset
   441
1d195de59497 removal of HOL_Lemmas
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diff changeset
   442
lemma all_dupE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   443
  assumes major: "ALL x. P(x)"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   444
    and minor: "[| P(x); ALL x. P(x) |] ==> R"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   445
  shows R
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   446
  by (iprover intro: minor major major [THEN spec])
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   447
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   448
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   449
subsubsection {* False *}
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   450
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   451
text {*
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   452
  Depends upon @{text spec}; it is impossible to do propositional
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   453
  logic before quantifiers!
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   454
*}
15411
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paulson
parents: 15380
diff changeset
   455
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   456
lemma FalseE: "False ==> P"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   457
  apply (unfold False_def)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   458
  apply (erule spec)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   459
  done
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   460
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   461
lemma False_neq_True: "False = True ==> P"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   462
  by (erule eqTrueE [THEN FalseE])
15411
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paulson
parents: 15380
diff changeset
   463
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   464
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   465
subsubsection {* Negation *}
15411
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paulson
parents: 15380
diff changeset
   466
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   467
lemma notI:
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   468
  assumes "P ==> False"
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   469
  shows "~P"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   470
  apply (unfold not_def)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   471
  apply (iprover intro: impI assms)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   472
  done
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   473
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   474
lemma False_not_True: "False ~= True"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   475
  apply (rule notI)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   476
  apply (erule False_neq_True)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   477
  done
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   478
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   479
lemma True_not_False: "True ~= False"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   480
  apply (rule notI)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   481
  apply (drule sym)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   482
  apply (erule False_neq_True)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   483
  done
15411
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paulson
parents: 15380
diff changeset
   484
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   485
lemma notE: "[| ~P;  P |] ==> R"
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   486
  apply (unfold not_def)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   487
  apply (erule mp [THEN FalseE])
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   488
  apply assumption
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   489
  done
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   490
21504
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   491
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
diff changeset
   492
  by (erule notE [THEN notI]) (erule meta_mp)
15411
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paulson
parents: 15380
diff changeset
   493
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   494
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   495
subsubsection {*Implication*}
15411
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paulson
parents: 15380
diff changeset
   496
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   497
lemma impE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   498
  assumes "P-->Q" "P" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   499
  shows "R"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   500
by (iprover intro: assms mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   501
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   502
(* Reduces Q to P-->Q, allowing substitution in P. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   503
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   504
by (iprover intro: mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   505
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   506
lemma contrapos_nn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   507
  assumes major: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   508
      and minor: "P==>Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   509
  shows "~P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   510
by (iprover intro: notI minor major [THEN notE])
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   511
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   512
(*not used at all, but we already have the other 3 combinations *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   513
lemma contrapos_pn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   514
  assumes major: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   515
      and minor: "P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   516
  shows "~P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   517
by (iprover intro: notI minor major notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   518
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   519
lemma not_sym: "t ~= s ==> s ~= t"
21250
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   520
  by (erule contrapos_nn) (erule sym)
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   521
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   522
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   523
  by (erule subst, erule ssubst, assumption)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   524
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   525
(*still used in HOLCF*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   526
lemma rev_contrapos:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   527
  assumes pq: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   528
      and nq: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   529
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   530
apply (rule nq [THEN contrapos_nn])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   531
apply (erule pq)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   532
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   533
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   534
subsubsection {*Existential quantifier*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   535
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   536
lemma exI: "P x ==> EX x::'a. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   537
apply (unfold Ex_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   538
apply (iprover intro: allI allE impI mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   539
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   540
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   541
lemma exE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   542
  assumes major: "EX x::'a. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   543
      and minor: "!!x. P(x) ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   544
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   545
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   546
apply (iprover intro: impI [THEN allI] minor)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   547
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   548
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   549
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   550
subsubsection {*Conjunction*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   551
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   552
lemma conjI: "[| P; Q |] ==> P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   553
apply (unfold and_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   554
apply (iprover intro: impI [THEN allI] mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   555
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   556
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   557
lemma conjunct1: "[| P & Q |] ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   558
apply (unfold and_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   559
apply (iprover intro: impI dest: spec mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   560
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   561
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   562
lemma conjunct2: "[| P & Q |] ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   563
apply (unfold and_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   564
apply (iprover intro: impI dest: spec mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   565
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   566
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   567
lemma conjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   568
  assumes major: "P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   569
      and minor: "[| P; Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   570
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   571
apply (rule minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   572
apply (rule major [THEN conjunct1])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   573
apply (rule major [THEN conjunct2])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   574
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   575
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   576
lemma context_conjI:
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   577
  assumes "P" "P ==> Q" shows "P & Q"
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   578
by (iprover intro: conjI assms)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   579
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   580
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   581
subsubsection {*Disjunction*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   582
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   583
lemma disjI1: "P ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   584
apply (unfold or_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   585
apply (iprover intro: allI impI mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   586
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   587
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   588
lemma disjI2: "Q ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   589
apply (unfold or_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   590
apply (iprover intro: allI impI mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   591
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   592
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   593
lemma disjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   594
  assumes major: "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   595
      and minorP: "P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   596
      and minorQ: "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   597
  shows "R"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   598
by (iprover intro: minorP minorQ impI
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   599
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   600
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   601
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   602
subsubsection {*Classical logic*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   603
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   604
lemma classical:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   605
  assumes prem: "~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   606
  shows "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   607
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   608
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   609
apply (rule notI [THEN prem, THEN eqTrueI])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   610
apply (erule subst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   611
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   612
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   613
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   614
lemmas ccontr = FalseE [THEN classical, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   615
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   616
(*notE with premises exchanged; it discharges ~R so that it can be used to
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   617
  make elimination rules*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   618
lemma rev_notE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   619
  assumes premp: "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   620
      and premnot: "~R ==> ~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   621
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   622
apply (rule ccontr)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   623
apply (erule notE [OF premnot premp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   624
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   625
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   626
(*Double negation law*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   627
lemma notnotD: "~~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   628
apply (rule classical)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   629
apply (erule notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   630
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   631
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   632
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   633
lemma contrapos_pp:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   634
  assumes p1: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   635
      and p2: "~P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   636
  shows "P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   637
by (iprover intro: classical p1 p2 notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   638
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   639
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   640
subsubsection {*Unique existence*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   641
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   642
lemma ex1I:
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   643
  assumes "P a" "!!x. P(x) ==> x=a"
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   644
  shows "EX! x. P(x)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   645
by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   646
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   647
text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   648
lemma ex_ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   649
  assumes ex_prem: "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   650
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   651
  shows "EX! x. P(x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   652
by (iprover intro: ex_prem [THEN exE] ex1I eq)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   653
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   654
lemma ex1E:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   655
  assumes major: "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   656
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   657
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   658
apply (rule major [unfolded Ex1_def, THEN exE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   659
apply (erule conjE)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   660
apply (iprover intro: minor)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   661
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   662
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   663
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   664
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   665
apply (rule exI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   666
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   667
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   668
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   669
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   670
subsubsection {*THE: definite description operator*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   671
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   672
lemma the_equality:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   673
  assumes prema: "P a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   674
      and premx: "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   675
  shows "(THE x. P x) = a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   676
apply (rule trans [OF _ the_eq_trivial])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   677
apply (rule_tac f = "The" in arg_cong)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   678
apply (rule ext)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   679
apply (rule iffI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   680
 apply (erule premx)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   681
apply (erule ssubst, rule prema)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   682
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   683
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   684
lemma theI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   685
  assumes "P a" and "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   686
  shows "P (THE x. P x)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   687
by (iprover intro: assms the_equality [THEN ssubst])
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   688
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   689
lemma theI': "EX! x. P x ==> P (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   690
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   691
apply (erule theI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   692
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   693
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   694
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   695
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   696
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   697
(*Easier to apply than theI: only one occurrence of P*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   698
lemma theI2:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   699
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   700
  shows "Q (THE x. P x)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   701
by (iprover intro: assms theI)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   702
18697
86b3f73e3fd5 declare the1_equality [elim?];
wenzelm
parents: 18689
diff changeset
   703
lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   704
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   705
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   706
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   707
apply (erule all_dupE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   708
apply (drule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   709
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   710
apply (erule ssubst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   711
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   712
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   713
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   714
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   715
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   716
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   717
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   718
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   719
apply (erule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   720
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   721
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   722
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   723
subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   724
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   725
lemma disjCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   726
  assumes "~Q ==> P" shows "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   727
apply (rule classical)
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   728
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   729
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   730
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   731
lemma excluded_middle: "~P | P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   732
by (iprover intro: disjCI)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   733
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   734
text {*
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   735
  case distinction as a natural deduction rule.
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   736
  Note that @{term "~P"} is the second case, not the first
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   737
*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   738
lemma case_split_thm:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   739
  assumes prem1: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   740
      and prem2: "~P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   741
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   742
apply (rule excluded_middle [THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   743
apply (erule prem2)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   744
apply (erule prem1)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   745
done
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   746
lemmas case_split = case_split_thm [case_names True False]
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   747
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   748
(*Classical implies (-->) elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   749
lemma impCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   750
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   751
      and minor: "~P ==> R" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   752
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   753
apply (rule excluded_middle [of P, THEN disjE])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   754
apply (iprover intro: minor major [THEN mp])+
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   755
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   756
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   757
(*This version of --> elimination works on Q before P.  It works best for
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   758
  those cases in which P holds "almost everywhere".  Can't install as
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   759
  default: would break old proofs.*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   760
lemma impCE':
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   761
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   762
      and minor: "Q ==> R" "~P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   763
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   764
apply (rule excluded_middle [of P, THEN disjE])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   765
apply (iprover intro: minor major [THEN mp])+
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   766
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   767
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   768
(*Classical <-> elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   769
lemma iffCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   770
  assumes major: "P=Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   771
      and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   772
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   773
apply (rule major [THEN iffE])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   774
apply (iprover intro: minor elim: impCE notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   775
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   776
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   777
lemma exCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   778
  assumes "ALL x. ~P(x) ==> P(a)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   779
  shows "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   780
apply (rule ccontr)
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   781
apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   782
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   783
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   784
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   785
subsubsection {* Intuitionistic Reasoning *}
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   786
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   787
lemma impE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   788
  assumes 1: "P --> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   789
    and 2: "Q ==> R"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   790
    and 3: "P --> Q ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   791
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   792
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   793
  from 3 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   794
  with 1 have Q by (rule impE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   795
  with 2 show R .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   796
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   797
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   798
lemma allE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   799
  assumes 1: "ALL x. P x"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   800
    and 2: "P x ==> ALL x. P x ==> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   801
  shows Q
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   802
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   803
  from 1 have "P x" by (rule spec)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   804
  from this and 1 show Q by (rule 2)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   805
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   806
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   807
lemma notE':
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   808
  assumes 1: "~ P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   809
    and 2: "~ P ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   810
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   811
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   812
  from 2 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   813
  with 1 show R by (rule notE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   814
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   815
22444
fb80fedd192d added safe intro rules for removing "True" subgoals as well as "~ False" ones.
dixon
parents: 22377
diff changeset
   816
lemma TrueE: "True ==> P ==> P" .
fb80fedd192d added safe intro rules for removing "True" subgoals as well as "~ False" ones.
dixon
parents: 22377
diff changeset
   817
lemma notFalseE: "~ False ==> P ==> P" .
fb80fedd192d added safe intro rules for removing "True" subgoals as well as "~ False" ones.
dixon
parents: 22377
diff changeset
   818
22467
c9357ef01168 TrueElim and notTrueElim tested and added as safe elim rules.
dixon
parents: 22445
diff changeset
   819
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
15801
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   820
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   821
  and [Pure.elim 2] = allE notE' impE'
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   822
  and [Pure.intro] = exI disjI2 disjI1
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   823
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   824
lemmas [trans] = trans
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   825
  and [sym] = sym not_sym
15801
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   826
  and [Pure.elim?] = iffD1 iffD2 impE
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   827
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   828
use "hologic.ML"
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   829
11438
3d9222b80989 declare trans [trans] (*overridden in theory Calculation*);
wenzelm
parents: 11432
diff changeset
   830
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   831
subsubsection {* Atomizing meta-level connectives *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   832
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   833
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   834
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   835
  assume "!!x. P x"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23263
diff changeset
   836
  then show "ALL x. P x" ..
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   837
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   838
  assume "ALL x. P x"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   839
  then show "!!x. P x" by (rule allE)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   840
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   841
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   842
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   843
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   844
  assume r: "A ==> B"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   845
  show "A --> B" by (rule impI) (rule r)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   846
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   847
  assume "A --> B" and A
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   848
  then show B by (rule mp)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   849
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   850
14749
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   851
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   852
proof
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   853
  assume r: "A ==> False"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   854
  show "~A" by (rule notI) (rule r)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   855
next
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   856
  assume "~A" and A
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   857
  then show False by (rule notE)
14749
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   858
qed
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   859
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   860
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   861
proof
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   862
  assume "x == y"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   863
  show "x = y" by (unfold `x == y`) (rule refl)
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   864
next
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   865
  assume "x = y"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   866
  then show "x == y" by (rule eq_reflection)
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   867
qed
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   868
12023
wenzelm
parents: 12003
diff changeset
   869
lemma atomize_conj [atomize]:
19121
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   870
  includes meta_conjunction_syntax
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   871
  shows "(A && B) == Trueprop (A & B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   872
proof
19121
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   873
  assume conj: "A && B"
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   874
  show "A & B"
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   875
  proof (rule conjI)
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   876
    from conj show A by (rule conjunctionD1)
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   877
    from conj show B by (rule conjunctionD2)
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   878
  qed
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   879
next
19121
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   880
  assume conj: "A & B"
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   881
  show "A && B"
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   882
  proof -
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   883
    from conj show A ..
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   884
    from conj show B ..
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   885
  qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   886
qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   887
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   888
lemmas [symmetric, rulify] = atomize_all atomize_imp
18832
6ab4de872a70 declare 'defn' rules;
wenzelm
parents: 18757
diff changeset
   889
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   890
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   891
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   892
subsection {* Package setup *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   893
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   894
subsubsection {* Classical Reasoner setup *}
9529
d9434a9277a4 lemmas atomize = all_eq imp_eq;
wenzelm
parents: 9488
diff changeset
   895
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   896
lemma thin_refl:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   897
  "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   898
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   899
ML {*
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   900
structure Hypsubst = HypsubstFun(
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   901
struct
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   902
  structure Simplifier = Simplifier
21218
38013c3a77a2 tuned hypsubst setup;
wenzelm
parents: 21210
diff changeset
   903
  val dest_eq = HOLogic.dest_eq
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   904
  val dest_Trueprop = HOLogic.dest_Trueprop
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   905
  val dest_imp = HOLogic.dest_imp
22129
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   906
  val eq_reflection = @{thm HOL.eq_reflection}
22218
30a8890d2967 dropped lemma duplicates in HOL.thy
haftmann
parents: 22129
diff changeset
   907
  val rev_eq_reflection = @{thm HOL.meta_eq_to_obj_eq}
22129
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   908
  val imp_intr = @{thm HOL.impI}
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   909
  val rev_mp = @{thm HOL.rev_mp}
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   910
  val subst = @{thm HOL.subst}
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   911
  val sym = @{thm HOL.sym}
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   912
  val thin_refl = @{thm thin_refl};
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   913
end);
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
   914
open Hypsubst;
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   915
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   916
structure Classical = ClassicalFun(
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   917
struct
22129
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   918
  val mp = @{thm HOL.mp}
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   919
  val not_elim = @{thm HOL.notE}
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   920
  val classical = @{thm HOL.classical}
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   921
  val sizef = Drule.size_of_thm
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   922
  val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   923
end);
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   924
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   925
structure BasicClassical: BASIC_CLASSICAL = Classical; 
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
   926
open BasicClassical;
22129
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   927
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   928
ML_Context.value_antiq "claset"
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   929
  (Scan.succeed ("claset", "Classical.local_claset_of (ML_Context.the_local_context ())"));
24035
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
   930
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
   931
structure ResAtpset = NamedThmsFun(val name = "atp" val description = "ATP rules");
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   932
*}
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   933
21009
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   934
setup {*
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   935
let
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   936
  (*prevent substitution on bool*)
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   937
  fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   938
    Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   939
      (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   940
in
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   941
  Hypsubst.hypsubst_setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   942
  #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   943
  #> Classical.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   944
  #> ResAtpset.setup
21009
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   945
end
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   946
*}
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   947
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   948
declare iffI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   949
  and notI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   950
  and impI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   951
  and disjCI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   952
  and conjI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   953
  and TrueI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   954
  and refl [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   955
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   956
declare iffCE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   957
  and FalseE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   958
  and impCE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   959
  and disjE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   960
  and conjE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   961
  and conjE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   962
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   963
declare ex_ex1I [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   964
  and allI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   965
  and the_equality [intro]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   966
  and exI [intro]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   967
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   968
declare exE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   969
  allE [elim]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   970
22377
61610b1beedf tuned ML setup;
wenzelm
parents: 22218
diff changeset
   971
ML {* val HOL_cs = @{claset} *}
19162
67436e2a16df Added setup for "atpset" (a rule set for ATPs).
mengj
parents: 19138
diff changeset
   972
20223
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   973
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   974
  apply (erule swap)
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   975
  apply (erule (1) meta_mp)
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   976
  done
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   977
18689
a50587cd8414 prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents: 18595
diff changeset
   978
declare ex_ex1I [rule del, intro! 2]
a50587cd8414 prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents: 18595
diff changeset
   979
  and ex1I [intro]
a50587cd8414 prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents: 18595
diff changeset
   980
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   981
lemmas [intro?] = ext
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   982
  and [elim?] = ex1_implies_ex
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   983
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   984
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
20973
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
   985
lemma alt_ex1E [elim!]:
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   986
  assumes major: "\<exists>!x. P x"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   987
      and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   988
  shows R
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   989
apply (rule ex1E [OF major])
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   990
apply (rule prem)
22129
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   991
apply (tactic {* ares_tac @{thms allI} 1 *})+
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   992
apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   993
apply iprover
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   994
done
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   995
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   996
ML {*
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   997
structure Blast = BlastFun(
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   998
struct
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   999
  type claset = Classical.claset
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22481
diff changeset
  1000
  val equality_name = @{const_name "op ="}
22993
haftmann
parents: 22839
diff changeset
  1001
  val not_name = @{const_name Not}
22129
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
  1002
  val notE = @{thm HOL.notE}
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
  1003
  val ccontr = @{thm HOL.ccontr}
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1004
  val contr_tac = Classical.contr_tac
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1005
  val dup_intr = Classical.dup_intr
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1006
  val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1007
  val claset = Classical.claset
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1008
  val rep_cs = Classical.rep_cs
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1009
  val cla_modifiers = Classical.cla_modifiers
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1010
  val cla_meth' = Classical.cla_meth'
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1011
end);
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1012
val Blast_tac = Blast.Blast_tac;
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1013
val blast_tac = Blast.blast_tac;
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1014
*}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1015
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1016
setup Blast.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1017
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1018
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1019
subsubsection {* Simplifier *}
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1020
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1021
lemma eta_contract_eq: "(%s. f s) = f" ..
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1022
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1023
lemma simp_thms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1024
  shows not_not: "(~ ~ P) = P"
15354
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
  1025
  and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1026
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1027
    "(P ~= Q) = (P = (~Q))"
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1028
    "(P | ~P) = True"    "(~P | P) = True"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1029
    "(x = x) = True"
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1030
  and not_True_eq_False: "(\<not> True) = False"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1031
  and not_False_eq_True: "(\<not> False) = True"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1032
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1033
    "(~P) ~= P"  "P ~= (~P)"
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1034
    "(True=P) = P"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1035
  and eq_True: "(P = True) = P"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1036
  and "(False=P) = (~P)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1037
  and eq_False: "(P = False) = (\<not> P)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1038
  and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1039
    "(True --> P) = P"  "(False --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1040
    "(P --> True) = True"  "(P --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1041
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1042
    "(P & True) = P"  "(True & P) = P"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1043
    "(P & False) = False"  "(False & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1044
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1045
    "(P & ~P) = False"    "(~P & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1046
    "(P | True) = True"  "(True | P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1047
    "(P | False) = P"  "(False | P) = P"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1048
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1049
    "(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
  1050
    -- {* needed for the one-point-rule quantifier simplification procs *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1051
    -- {* essential for termination!! *} and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1052
    "!!P. (EX x. x=t & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1053
    "!!P. (EX x. t=x & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1054
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1055
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1056
  by (blast, blast, blast, blast, blast, iprover+)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13412
diff changeset
  1057
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1058
lemma disj_absorb: "(A | A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1059
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1060
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1061
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1062
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1063
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1064
lemma conj_absorb: "(A & A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1065
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1066
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1067
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1068
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1069
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1070
lemma eq_ac:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1071
  shows eq_commute: "(a=b) = (b=a)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1072
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1073
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1074
lemma neq_commute: "(a~=b) = (b~=a)" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1075
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1076
lemma conj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1077
  shows conj_commute: "(P&Q) = (Q&P)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1078
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1079
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1080
19174
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
  1081
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
  1082
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1083
lemma disj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1084
  shows disj_commute: "(P|Q) = (Q|P)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1085
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1086
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1087
19174
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
  1088
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
  1089
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1090
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1091
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1092
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1093
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1094
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1095
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1096
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1097
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1098
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1099
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1100
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
  1101
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1102
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1103
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1104
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1105
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1106
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1107
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1108
  by iprover
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1109
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1110
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1111
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1112
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1113
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1114
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1115
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1116
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1117
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1118
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1119
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1120
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1121
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1122
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1123
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1124
  -- {* cases boil down to the same thing. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1125
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1126
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1127
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1128
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1129
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1130
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
23403
9e1edc15ef52 added Theorem all_not_ex
chaieb
parents: 23389
diff changeset
  1131
lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1132
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1133
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1134
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1135
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1136
text {*
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1137
  \medskip The @{text "&"} congruence rule: not included by default!
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1138
  May slow rewrite proofs down by as much as 50\% *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1139
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1140
lemma conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1141
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1142
  by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1143
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1144
lemma rev_conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1145
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1146
  by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1147
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1148
text {* The @{text "|"} congruence rule: not included by default! *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1149
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1150
lemma disj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1151
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1152
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1153
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1154
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1155
text {* \medskip if-then-else rules *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1156
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1157
lemma if_True: "(if True then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1158
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1159
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1160
lemma if_False: "(if False then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1161
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1162
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1163
lemma if_P: "P ==> (if P then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1164
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1165
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1166
lemma if_not_P: "~P ==> (if P then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1167
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1168
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1169
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
  1170
  apply (rule case_split [of Q])
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1171
   apply (simplesubst if_P)
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1172
    prefer 3 apply (simplesubst if_not_P, blast+)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1173
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1174
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1175
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1176
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1177
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1178
lemmas if_splits = split_if split_if_asm
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1179
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1180
lemma if_cancel: "(if c then x else x) = x"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1181
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1182
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1183
lemma if_eq_cancel: "(if x = y then y else x) = x"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1184
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1185
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1186
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
19796
d86e7b1fc472 quoted "if";
wenzelm
parents: 19656
diff changeset
  1187
  -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1188
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1189
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1190
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
19796
d86e7b1fc472 quoted "if";
wenzelm
parents: 19656
diff changeset
  1191
  -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1192
  apply (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1193
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1194
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1195
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1196
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1197
15423
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1198
text {* \medskip let rules for simproc *}
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1199
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1200
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1201
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1202
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1203
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1204
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1205
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1206
text {*
16999
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1207
  The following copy of the implication operator is useful for
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1208
  fine-tuning congruence rules.  It instructs the simplifier to simplify
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1209
  its premise.
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1210
*}
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1211
17197
917c6e7ca28d simp_implies: proper named infix;
wenzelm
parents: 16999
diff changeset
  1212
constdefs
917c6e7ca28d simp_implies: proper named infix;
wenzelm
parents: 16999
diff changeset
  1213
  simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1)
917c6e7ca28d simp_implies: proper named infix;
wenzelm
parents: 16999
diff changeset
  1214
  "simp_implies \<equiv> op ==>"
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1215
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1216
lemma simp_impliesI:
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1217
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1218
  shows "PROP P =simp=> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1219
  apply (unfold simp_implies_def)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1220
  apply (rule PQ)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1221
  apply assumption
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1222
  done
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1223
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1224
lemma simp_impliesE:
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1225
  assumes PQ:"PROP P =simp=> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1226
  and P: "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1227
  and QR: "PROP Q \<Longrightarrow> PROP R"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1228
  shows "PROP R"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1229
  apply (rule QR)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1230
  apply (rule PQ [unfolded simp_implies_def])
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1231
  apply (rule P)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1232
  done
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1233
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1234
lemma simp_implies_cong:
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1235
  assumes PP' :"PROP P == PROP P'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1236
  and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1237
  shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1238
proof (unfold simp_implies_def, rule equal_intr_rule)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1239
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1240
  and P': "PROP P'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1241
  from PP' [symmetric] and P' have "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1242
    by (rule equal_elim_rule1)
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1243
  then have "PROP Q" by (rule PQ)
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1244
  with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1245
next
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1246
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1247
  and P: "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1248
  from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1249
  then have "PROP Q'" by (rule P'Q')
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1250
  with P'QQ' [OF P', symmetric] show "PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1251
    by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1252
qed
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1253
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1254
lemma uncurry:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1255
  assumes "P \<longrightarrow> Q \<longrightarrow> R"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1256
  shows "P \<and> Q \<longrightarrow> R"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1257
  using assms by blast
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1258
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1259
lemma iff_allI:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1260
  assumes "\<And>x. P x = Q x"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1261
  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1262
  using assms by blast
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1263
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1264
lemma iff_exI:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1265
  assumes "\<And>x. P x = Q x"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1266
  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1267
  using assms by blast
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1268
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1269
lemma all_comm:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1270
  "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1271
  by blast
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1272
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1273
lemma ex_comm:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1274
  "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1275
  by blast
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1276
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1277
use "simpdata.ML"
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1278
ML {* open Simpdata *}
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1279
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1280
setup {*
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1281
  Simplifier.method_setup Splitter.split_modifiers
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21524
diff changeset
  1282
  #> (fn thy => (change_simpset_of thy (fn _ => Simpdata.simpset_simprocs); thy))
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1283
  #> Splitter.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1284
  #> Clasimp.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1285
  #> EqSubst.setup
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1286
*}
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1287
24035
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1288
text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1289
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1290
simproc_setup neq ("x = y") = {* fn _ =>
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1291
let
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1292
  val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1293
  fun is_neq eq lhs rhs thm =
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1294
    (case Thm.prop_of thm of
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1295
      _ $ (Not $ (eq' $ l' $ r')) =>
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1296
        Not = HOLogic.Not andalso eq' = eq andalso
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1297
        r' aconv lhs andalso l' aconv rhs
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1298
    | _ => false);
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1299
  fun proc ss ct =
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1300
    (case Thm.term_of ct of
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1301
      eq $ lhs $ rhs =>
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1302
        (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of_ss ss) of
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1303
          SOME thm => SOME (thm RS neq_to_EQ_False)
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1304
        | NONE => NONE)
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1305
     | _ => NONE);
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1306
in proc end;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1307
*}
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1308
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1309
simproc_setup let_simp ("Let x f") = {*
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1310
let
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1311
  val (f_Let_unfold, x_Let_unfold) =
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1312
    let val [(_$(f$x)$_)] = prems_of @{thm Let_unfold}
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1313
    in (cterm_of @{theory} f, cterm_of @{theory} x) end
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1314
  val (f_Let_folded, x_Let_folded) =
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1315
    let val [(_$(f$x)$_)] = prems_of @{thm Let_folded}
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1316
    in (cterm_of @{theory} f, cterm_of @{theory} x) end;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1317
  val g_Let_folded =
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1318
    let val [(_$_$(g$_))] = prems_of @{thm Let_folded} in cterm_of @{theory} g end;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1319
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1320
  fun proc _ ss ct =
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1321
    let
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1322
      val ctxt = Simplifier.the_context ss;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1323
      val thy = ProofContext.theory_of ctxt;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1324
      val t = Thm.term_of ct;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1325
      val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1326
    in Option.map (hd o Variable.export ctxt' ctxt o single)
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1327
      (case t' of Const ("Let",_) $ x $ f => (* x and f are already in normal form *)
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1328
        if is_Free x orelse is_Bound x orelse is_Const x
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1329
        then SOME @{thm Let_def}
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1330
        else
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1331
          let
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1332
            val n = case f of (Abs (x,_,_)) => x | _ => "x";
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1333
            val cx = cterm_of thy x;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1334
            val {T=xT,...} = rep_cterm cx;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1335
            val cf = cterm_of thy f;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1336
            val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1337
            val (_$_$g) = prop_of fx_g;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1338
            val g' = abstract_over (x,g);
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1339
          in (if (g aconv g')
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1340
               then
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1341
                  let
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1342
                    val rl =
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1343
                      cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] @{thm Let_unfold};
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1344
                  in SOME (rl OF [fx_g]) end
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1345
               else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1346
               else let
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1347
                     val abs_g'= Abs (n,xT,g');
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1348
                     val g'x = abs_g'$x;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1349
                     val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1350
                     val rl = cterm_instantiate
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1351
                               [(f_Let_folded,cterm_of thy f),(x_Let_folded,cx),
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1352
                                (g_Let_folded,cterm_of thy abs_g')]
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1353
                               @{thm Let_folded};
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1354
                   in SOME (rl OF [transitive fx_g g_g'x])
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1355
                   end)
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1356
          end
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1357
      | _ => NONE)
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1358
    end
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1359
in proc end *}
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1360
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1361
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1362
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1363
proof
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23263
diff changeset
  1364
  assume "True \<Longrightarrow> PROP P"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23263
diff changeset
  1365
  from this [OF TrueI] show "PROP P" .
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1366
next
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1367
  assume "PROP P"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23263
diff changeset
  1368
  then show "PROP P" .
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112