src/HOL/HOL.thy
author paulson
Wed Aug 15 12:52:56 2007 +0200 (2007-08-15)
changeset 24286 7619080e49f0
parent 24280 c9867bdf2424
child 24293 7e67b9706211
permissions -rw-r--r--
ATP blacklisting is now in theory data, attribute noatp
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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/Pure/codegen.ML"
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  "~~/src/Tools/code/code_name.ML"
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  "~~/src/Tools/code/code_funcgr.ML"
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  "~~/src/Tools/code/code_thingol.ML"
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  "~~/src/Tools/code/code_target.ML"
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  "~~/src/Tools/code/code_package.ML"
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  "~~/src/Tools/nbe.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:      "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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text {* Thanks to Stephan Merz *}
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lemma subst:
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  assumes eq: "s = t" and p: "P s"
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  shows "P t"
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proof -
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  from eq have meta: "s \<equiv> t"
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    by (rule eq_reflection)
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  from p show ?thesis
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    by (unfold meta)
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qed
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lemma sym: "s = t ==> t = s"
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  by (erule subst) (rule refl)
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lemma ssubst: "t = s ==> P s ==> P t"
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  by (drule sym) (erule subst)
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lemma trans: "[| r=s; s=t |] ==> r=t"
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  by (erule subst)
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lemma meta_eq_to_obj_eq: 
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  assumes meq: "A == B"
haftmann@20944
   347
  shows "A = B"
haftmann@20944
   348
  by (unfold meq) (rule refl)
paulson@15411
   349
wenzelm@21502
   350
text {* Useful with @{text erule} for proving equalities from known equalities. *}
haftmann@20944
   351
     (* a = b
paulson@15411
   352
        |   |
paulson@15411
   353
        c = d   *)
paulson@15411
   354
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
paulson@15411
   355
apply (rule trans)
paulson@15411
   356
apply (rule trans)
paulson@15411
   357
apply (rule sym)
paulson@15411
   358
apply assumption+
paulson@15411
   359
done
paulson@15411
   360
nipkow@15524
   361
text {* For calculational reasoning: *}
nipkow@15524
   362
nipkow@15524
   363
lemma forw_subst: "a = b ==> P b ==> P a"
nipkow@15524
   364
  by (rule ssubst)
nipkow@15524
   365
nipkow@15524
   366
lemma back_subst: "P a ==> a = b ==> P b"
nipkow@15524
   367
  by (rule subst)
nipkow@15524
   368
paulson@15411
   369
haftmann@20944
   370
subsubsection {*Congruence rules for application*}
paulson@15411
   371
paulson@15411
   372
(*similar to AP_THM in Gordon's HOL*)
paulson@15411
   373
lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
paulson@15411
   374
apply (erule subst)
paulson@15411
   375
apply (rule refl)
paulson@15411
   376
done
paulson@15411
   377
paulson@15411
   378
(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
paulson@15411
   379
lemma arg_cong: "x=y ==> f(x)=f(y)"
paulson@15411
   380
apply (erule subst)
paulson@15411
   381
apply (rule refl)
paulson@15411
   382
done
paulson@15411
   383
paulson@15655
   384
lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
paulson@15655
   385
apply (erule ssubst)+
paulson@15655
   386
apply (rule refl)
paulson@15655
   387
done
paulson@15655
   388
paulson@15411
   389
lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
paulson@15411
   390
apply (erule subst)+
paulson@15411
   391
apply (rule refl)
paulson@15411
   392
done
paulson@15411
   393
paulson@15411
   394
haftmann@20944
   395
subsubsection {*Equality of booleans -- iff*}
paulson@15411
   396
wenzelm@21504
   397
lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
wenzelm@21504
   398
  by (iprover intro: iff [THEN mp, THEN mp] impI assms)
paulson@15411
   399
paulson@15411
   400
lemma iffD2: "[| P=Q; Q |] ==> P"
wenzelm@18457
   401
  by (erule ssubst)
paulson@15411
   402
paulson@15411
   403
lemma rev_iffD2: "[| Q; P=Q |] ==> P"
wenzelm@18457
   404
  by (erule iffD2)
paulson@15411
   405
wenzelm@21504
   406
lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
wenzelm@21504
   407
  by (drule sym) (rule iffD2)
wenzelm@21504
   408
wenzelm@21504
   409
lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
wenzelm@21504
   410
  by (drule sym) (rule rev_iffD2)
paulson@15411
   411
paulson@15411
   412
lemma iffE:
paulson@15411
   413
  assumes major: "P=Q"
wenzelm@21504
   414
    and minor: "[| P --> Q; Q --> P |] ==> R"
wenzelm@18457
   415
  shows R
wenzelm@18457
   416
  by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
paulson@15411
   417
paulson@15411
   418
haftmann@20944
   419
subsubsection {*True*}
paulson@15411
   420
paulson@15411
   421
lemma TrueI: "True"
wenzelm@21504
   422
  unfolding True_def by (rule refl)
paulson@15411
   423
wenzelm@21504
   424
lemma eqTrueI: "P ==> P = True"
wenzelm@18457
   425
  by (iprover intro: iffI TrueI)
paulson@15411
   426
wenzelm@21504
   427
lemma eqTrueE: "P = True ==> P"
wenzelm@21504
   428
  by (erule iffD2) (rule TrueI)
paulson@15411
   429
paulson@15411
   430
haftmann@20944
   431
subsubsection {*Universal quantifier*}
paulson@15411
   432
wenzelm@21504
   433
lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
wenzelm@21504
   434
  unfolding All_def by (iprover intro: ext eqTrueI assms)
paulson@15411
   435
paulson@15411
   436
lemma spec: "ALL x::'a. P(x) ==> P(x)"
paulson@15411
   437
apply (unfold All_def)
paulson@15411
   438
apply (rule eqTrueE)
paulson@15411
   439
apply (erule fun_cong)
paulson@15411
   440
done
paulson@15411
   441
paulson@15411
   442
lemma allE:
paulson@15411
   443
  assumes major: "ALL x. P(x)"
wenzelm@21504
   444
    and minor: "P(x) ==> R"
wenzelm@21504
   445
  shows R
wenzelm@21504
   446
  by (iprover intro: minor major [THEN spec])
paulson@15411
   447
paulson@15411
   448
lemma all_dupE:
paulson@15411
   449
  assumes major: "ALL x. P(x)"
wenzelm@21504
   450
    and minor: "[| P(x); ALL x. P(x) |] ==> R"
wenzelm@21504
   451
  shows R
wenzelm@21504
   452
  by (iprover intro: minor major major [THEN spec])
paulson@15411
   453
paulson@15411
   454
wenzelm@21504
   455
subsubsection {* False *}
wenzelm@21504
   456
wenzelm@21504
   457
text {*
wenzelm@21504
   458
  Depends upon @{text spec}; it is impossible to do propositional
wenzelm@21504
   459
  logic before quantifiers!
wenzelm@21504
   460
*}
paulson@15411
   461
paulson@15411
   462
lemma FalseE: "False ==> P"
wenzelm@21504
   463
  apply (unfold False_def)
wenzelm@21504
   464
  apply (erule spec)
wenzelm@21504
   465
  done
paulson@15411
   466
wenzelm@21504
   467
lemma False_neq_True: "False = True ==> P"
wenzelm@21504
   468
  by (erule eqTrueE [THEN FalseE])
paulson@15411
   469
paulson@15411
   470
wenzelm@21504
   471
subsubsection {* Negation *}
paulson@15411
   472
paulson@15411
   473
lemma notI:
wenzelm@21504
   474
  assumes "P ==> False"
paulson@15411
   475
  shows "~P"
wenzelm@21504
   476
  apply (unfold not_def)
wenzelm@21504
   477
  apply (iprover intro: impI assms)
wenzelm@21504
   478
  done
paulson@15411
   479
paulson@15411
   480
lemma False_not_True: "False ~= True"
wenzelm@21504
   481
  apply (rule notI)
wenzelm@21504
   482
  apply (erule False_neq_True)
wenzelm@21504
   483
  done
paulson@15411
   484
paulson@15411
   485
lemma True_not_False: "True ~= False"
wenzelm@21504
   486
  apply (rule notI)
wenzelm@21504
   487
  apply (drule sym)
wenzelm@21504
   488
  apply (erule False_neq_True)
wenzelm@21504
   489
  done
paulson@15411
   490
paulson@15411
   491
lemma notE: "[| ~P;  P |] ==> R"
wenzelm@21504
   492
  apply (unfold not_def)
wenzelm@21504
   493
  apply (erule mp [THEN FalseE])
wenzelm@21504
   494
  apply assumption
wenzelm@21504
   495
  done
paulson@15411
   496
wenzelm@21504
   497
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
wenzelm@21504
   498
  by (erule notE [THEN notI]) (erule meta_mp)
paulson@15411
   499
paulson@15411
   500
haftmann@20944
   501
subsubsection {*Implication*}
paulson@15411
   502
paulson@15411
   503
lemma impE:
paulson@15411
   504
  assumes "P-->Q" "P" "Q ==> R"
paulson@15411
   505
  shows "R"
wenzelm@23553
   506
by (iprover intro: assms mp)
paulson@15411
   507
paulson@15411
   508
(* Reduces Q to P-->Q, allowing substitution in P. *)
paulson@15411
   509
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
nipkow@17589
   510
by (iprover intro: mp)
paulson@15411
   511
paulson@15411
   512
lemma contrapos_nn:
paulson@15411
   513
  assumes major: "~Q"
paulson@15411
   514
      and minor: "P==>Q"
paulson@15411
   515
  shows "~P"
nipkow@17589
   516
by (iprover intro: notI minor major [THEN notE])
paulson@15411
   517
paulson@15411
   518
(*not used at all, but we already have the other 3 combinations *)
paulson@15411
   519
lemma contrapos_pn:
paulson@15411
   520
  assumes major: "Q"
paulson@15411
   521
      and minor: "P ==> ~Q"
paulson@15411
   522
  shows "~P"
nipkow@17589
   523
by (iprover intro: notI minor major notE)
paulson@15411
   524
paulson@15411
   525
lemma not_sym: "t ~= s ==> s ~= t"
haftmann@21250
   526
  by (erule contrapos_nn) (erule sym)
haftmann@21250
   527
haftmann@21250
   528
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
haftmann@21250
   529
  by (erule subst, erule ssubst, assumption)
paulson@15411
   530
paulson@15411
   531
(*still used in HOLCF*)
paulson@15411
   532
lemma rev_contrapos:
paulson@15411
   533
  assumes pq: "P ==> Q"
paulson@15411
   534
      and nq: "~Q"
paulson@15411
   535
  shows "~P"
paulson@15411
   536
apply (rule nq [THEN contrapos_nn])
paulson@15411
   537
apply (erule pq)
paulson@15411
   538
done
paulson@15411
   539
haftmann@20944
   540
subsubsection {*Existential quantifier*}
paulson@15411
   541
paulson@15411
   542
lemma exI: "P x ==> EX x::'a. P x"
paulson@15411
   543
apply (unfold Ex_def)
nipkow@17589
   544
apply (iprover intro: allI allE impI mp)
paulson@15411
   545
done
paulson@15411
   546
paulson@15411
   547
lemma exE:
paulson@15411
   548
  assumes major: "EX x::'a. P(x)"
paulson@15411
   549
      and minor: "!!x. P(x) ==> Q"
paulson@15411
   550
  shows "Q"
paulson@15411
   551
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
nipkow@17589
   552
apply (iprover intro: impI [THEN allI] minor)
paulson@15411
   553
done
paulson@15411
   554
paulson@15411
   555
haftmann@20944
   556
subsubsection {*Conjunction*}
paulson@15411
   557
paulson@15411
   558
lemma conjI: "[| P; Q |] ==> P&Q"
paulson@15411
   559
apply (unfold and_def)
nipkow@17589
   560
apply (iprover intro: impI [THEN allI] mp)
paulson@15411
   561
done
paulson@15411
   562
paulson@15411
   563
lemma conjunct1: "[| P & Q |] ==> P"
paulson@15411
   564
apply (unfold and_def)
nipkow@17589
   565
apply (iprover intro: impI dest: spec mp)
paulson@15411
   566
done
paulson@15411
   567
paulson@15411
   568
lemma conjunct2: "[| P & Q |] ==> Q"
paulson@15411
   569
apply (unfold and_def)
nipkow@17589
   570
apply (iprover intro: impI dest: spec mp)
paulson@15411
   571
done
paulson@15411
   572
paulson@15411
   573
lemma conjE:
paulson@15411
   574
  assumes major: "P&Q"
paulson@15411
   575
      and minor: "[| P; Q |] ==> R"
paulson@15411
   576
  shows "R"
paulson@15411
   577
apply (rule minor)
paulson@15411
   578
apply (rule major [THEN conjunct1])
paulson@15411
   579
apply (rule major [THEN conjunct2])
paulson@15411
   580
done
paulson@15411
   581
paulson@15411
   582
lemma context_conjI:
wenzelm@23553
   583
  assumes "P" "P ==> Q" shows "P & Q"
wenzelm@23553
   584
by (iprover intro: conjI assms)
paulson@15411
   585
paulson@15411
   586
haftmann@20944
   587
subsubsection {*Disjunction*}
paulson@15411
   588
paulson@15411
   589
lemma disjI1: "P ==> P|Q"
paulson@15411
   590
apply (unfold or_def)
nipkow@17589
   591
apply (iprover intro: allI impI mp)
paulson@15411
   592
done
paulson@15411
   593
paulson@15411
   594
lemma disjI2: "Q ==> P|Q"
paulson@15411
   595
apply (unfold or_def)
nipkow@17589
   596
apply (iprover intro: allI impI mp)
paulson@15411
   597
done
paulson@15411
   598
paulson@15411
   599
lemma disjE:
paulson@15411
   600
  assumes major: "P|Q"
paulson@15411
   601
      and minorP: "P ==> R"
paulson@15411
   602
      and minorQ: "Q ==> R"
paulson@15411
   603
  shows "R"
nipkow@17589
   604
by (iprover intro: minorP minorQ impI
paulson@15411
   605
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
paulson@15411
   606
paulson@15411
   607
haftmann@20944
   608
subsubsection {*Classical logic*}
paulson@15411
   609
paulson@15411
   610
lemma classical:
paulson@15411
   611
  assumes prem: "~P ==> P"
paulson@15411
   612
  shows "P"
paulson@15411
   613
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
paulson@15411
   614
apply assumption
paulson@15411
   615
apply (rule notI [THEN prem, THEN eqTrueI])
paulson@15411
   616
apply (erule subst)
paulson@15411
   617
apply assumption
paulson@15411
   618
done
paulson@15411
   619
paulson@15411
   620
lemmas ccontr = FalseE [THEN classical, standard]
paulson@15411
   621
paulson@15411
   622
(*notE with premises exchanged; it discharges ~R so that it can be used to
paulson@15411
   623
  make elimination rules*)
paulson@15411
   624
lemma rev_notE:
paulson@15411
   625
  assumes premp: "P"
paulson@15411
   626
      and premnot: "~R ==> ~P"
paulson@15411
   627
  shows "R"
paulson@15411
   628
apply (rule ccontr)
paulson@15411
   629
apply (erule notE [OF premnot premp])
paulson@15411
   630
done
paulson@15411
   631
paulson@15411
   632
(*Double negation law*)
paulson@15411
   633
lemma notnotD: "~~P ==> P"
paulson@15411
   634
apply (rule classical)
paulson@15411
   635
apply (erule notE)
paulson@15411
   636
apply assumption
paulson@15411
   637
done
paulson@15411
   638
paulson@15411
   639
lemma contrapos_pp:
paulson@15411
   640
  assumes p1: "Q"
paulson@15411
   641
      and p2: "~P ==> ~Q"
paulson@15411
   642
  shows "P"
nipkow@17589
   643
by (iprover intro: classical p1 p2 notE)
paulson@15411
   644
paulson@15411
   645
haftmann@20944
   646
subsubsection {*Unique existence*}
paulson@15411
   647
paulson@15411
   648
lemma ex1I:
wenzelm@23553
   649
  assumes "P a" "!!x. P(x) ==> x=a"
paulson@15411
   650
  shows "EX! x. P(x)"
wenzelm@23553
   651
by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
paulson@15411
   652
paulson@15411
   653
text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
paulson@15411
   654
lemma ex_ex1I:
paulson@15411
   655
  assumes ex_prem: "EX x. P(x)"
paulson@15411
   656
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
paulson@15411
   657
  shows "EX! x. P(x)"
nipkow@17589
   658
by (iprover intro: ex_prem [THEN exE] ex1I eq)
paulson@15411
   659
paulson@15411
   660
lemma ex1E:
paulson@15411
   661
  assumes major: "EX! x. P(x)"
paulson@15411
   662
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
paulson@15411
   663
  shows "R"
paulson@15411
   664
apply (rule major [unfolded Ex1_def, THEN exE])
paulson@15411
   665
apply (erule conjE)
nipkow@17589
   666
apply (iprover intro: minor)
paulson@15411
   667
done
paulson@15411
   668
paulson@15411
   669
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
paulson@15411
   670
apply (erule ex1E)
paulson@15411
   671
apply (rule exI)
paulson@15411
   672
apply assumption
paulson@15411
   673
done
paulson@15411
   674
paulson@15411
   675
haftmann@20944
   676
subsubsection {*THE: definite description operator*}
paulson@15411
   677
paulson@15411
   678
lemma the_equality:
paulson@15411
   679
  assumes prema: "P a"
paulson@15411
   680
      and premx: "!!x. P x ==> x=a"
paulson@15411
   681
  shows "(THE x. P x) = a"
paulson@15411
   682
apply (rule trans [OF _ the_eq_trivial])
paulson@15411
   683
apply (rule_tac f = "The" in arg_cong)
paulson@15411
   684
apply (rule ext)
paulson@15411
   685
apply (rule iffI)
paulson@15411
   686
 apply (erule premx)
paulson@15411
   687
apply (erule ssubst, rule prema)
paulson@15411
   688
done
paulson@15411
   689
paulson@15411
   690
lemma theI:
paulson@15411
   691
  assumes "P a" and "!!x. P x ==> x=a"
paulson@15411
   692
  shows "P (THE x. P x)"
wenzelm@23553
   693
by (iprover intro: assms the_equality [THEN ssubst])
paulson@15411
   694
paulson@15411
   695
lemma theI': "EX! x. P x ==> P (THE x. P x)"
paulson@15411
   696
apply (erule ex1E)
paulson@15411
   697
apply (erule theI)
paulson@15411
   698
apply (erule allE)
paulson@15411
   699
apply (erule mp)
paulson@15411
   700
apply assumption
paulson@15411
   701
done
paulson@15411
   702
paulson@15411
   703
(*Easier to apply than theI: only one occurrence of P*)
paulson@15411
   704
lemma theI2:
paulson@15411
   705
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
paulson@15411
   706
  shows "Q (THE x. P x)"
wenzelm@23553
   707
by (iprover intro: assms theI)
paulson@15411
   708
wenzelm@18697
   709
lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
paulson@15411
   710
apply (rule the_equality)
paulson@15411
   711
apply  assumption
paulson@15411
   712
apply (erule ex1E)
paulson@15411
   713
apply (erule all_dupE)
paulson@15411
   714
apply (drule mp)
paulson@15411
   715
apply  assumption
paulson@15411
   716
apply (erule ssubst)
paulson@15411
   717
apply (erule allE)
paulson@15411
   718
apply (erule mp)
paulson@15411
   719
apply assumption
paulson@15411
   720
done
paulson@15411
   721
paulson@15411
   722
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
paulson@15411
   723
apply (rule the_equality)
paulson@15411
   724
apply (rule refl)
paulson@15411
   725
apply (erule sym)
paulson@15411
   726
done
paulson@15411
   727
paulson@15411
   728
haftmann@20944
   729
subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
paulson@15411
   730
paulson@15411
   731
lemma disjCI:
paulson@15411
   732
  assumes "~Q ==> P" shows "P|Q"
paulson@15411
   733
apply (rule classical)
wenzelm@23553
   734
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
paulson@15411
   735
done
paulson@15411
   736
paulson@15411
   737
lemma excluded_middle: "~P | P"
nipkow@17589
   738
by (iprover intro: disjCI)
paulson@15411
   739
haftmann@20944
   740
text {*
haftmann@20944
   741
  case distinction as a natural deduction rule.
haftmann@20944
   742
  Note that @{term "~P"} is the second case, not the first
haftmann@20944
   743
*}
paulson@15411
   744
lemma case_split_thm:
paulson@15411
   745
  assumes prem1: "P ==> Q"
paulson@15411
   746
      and prem2: "~P ==> Q"
paulson@15411
   747
  shows "Q"
paulson@15411
   748
apply (rule excluded_middle [THEN disjE])
paulson@15411
   749
apply (erule prem2)
paulson@15411
   750
apply (erule prem1)
paulson@15411
   751
done
haftmann@20944
   752
lemmas case_split = case_split_thm [case_names True False]
paulson@15411
   753
paulson@15411
   754
(*Classical implies (-->) elimination. *)
paulson@15411
   755
lemma impCE:
paulson@15411
   756
  assumes major: "P-->Q"
paulson@15411
   757
      and minor: "~P ==> R" "Q ==> R"
paulson@15411
   758
  shows "R"
paulson@15411
   759
apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589
   760
apply (iprover intro: minor major [THEN mp])+
paulson@15411
   761
done
paulson@15411
   762
paulson@15411
   763
(*This version of --> elimination works on Q before P.  It works best for
paulson@15411
   764
  those cases in which P holds "almost everywhere".  Can't install as
paulson@15411
   765
  default: would break old proofs.*)
paulson@15411
   766
lemma impCE':
paulson@15411
   767
  assumes major: "P-->Q"
paulson@15411
   768
      and minor: "Q ==> R" "~P ==> R"
paulson@15411
   769
  shows "R"
paulson@15411
   770
apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589
   771
apply (iprover intro: minor major [THEN mp])+
paulson@15411
   772
done
paulson@15411
   773
paulson@15411
   774
(*Classical <-> elimination. *)
paulson@15411
   775
lemma iffCE:
paulson@15411
   776
  assumes major: "P=Q"
paulson@15411
   777
      and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
paulson@15411
   778
  shows "R"
paulson@15411
   779
apply (rule major [THEN iffE])
nipkow@17589
   780
apply (iprover intro: minor elim: impCE notE)
paulson@15411
   781
done
paulson@15411
   782
paulson@15411
   783
lemma exCI:
paulson@15411
   784
  assumes "ALL x. ~P(x) ==> P(a)"
paulson@15411
   785
  shows "EX x. P(x)"
paulson@15411
   786
apply (rule ccontr)
wenzelm@23553
   787
apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
paulson@15411
   788
done
paulson@15411
   789
paulson@15411
   790
wenzelm@12386
   791
subsubsection {* Intuitionistic Reasoning *}
wenzelm@12386
   792
wenzelm@12386
   793
lemma impE':
wenzelm@12937
   794
  assumes 1: "P --> Q"
wenzelm@12937
   795
    and 2: "Q ==> R"
wenzelm@12937
   796
    and 3: "P --> Q ==> P"
wenzelm@12937
   797
  shows R
wenzelm@12386
   798
proof -
wenzelm@12386
   799
  from 3 and 1 have P .
wenzelm@12386
   800
  with 1 have Q by (rule impE)
wenzelm@12386
   801
  with 2 show R .
wenzelm@12386
   802
qed
wenzelm@12386
   803
wenzelm@12386
   804
lemma allE':
wenzelm@12937
   805
  assumes 1: "ALL x. P x"
wenzelm@12937
   806
    and 2: "P x ==> ALL x. P x ==> Q"
wenzelm@12937
   807
  shows Q
wenzelm@12386
   808
proof -
wenzelm@12386
   809
  from 1 have "P x" by (rule spec)
wenzelm@12386
   810
  from this and 1 show Q by (rule 2)
wenzelm@12386
   811
qed
wenzelm@12386
   812
wenzelm@12937
   813
lemma notE':
wenzelm@12937
   814
  assumes 1: "~ P"
wenzelm@12937
   815
    and 2: "~ P ==> P"
wenzelm@12937
   816
  shows R
wenzelm@12386
   817
proof -
wenzelm@12386
   818
  from 2 and 1 have P .
wenzelm@12386
   819
  with 1 show R by (rule notE)
wenzelm@12386
   820
qed
wenzelm@12386
   821
dixon@22444
   822
lemma TrueE: "True ==> P ==> P" .
dixon@22444
   823
lemma notFalseE: "~ False ==> P ==> P" .
dixon@22444
   824
dixon@22467
   825
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
wenzelm@15801
   826
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@15801
   827
  and [Pure.elim 2] = allE notE' impE'
wenzelm@15801
   828
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12386
   829
wenzelm@12386
   830
lemmas [trans] = trans
wenzelm@12386
   831
  and [sym] = sym not_sym
wenzelm@15801
   832
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@11750
   833
wenzelm@23553
   834
use "hologic.ML"
wenzelm@23553
   835
wenzelm@11438
   836
wenzelm@11750
   837
subsubsection {* Atomizing meta-level connectives *}
wenzelm@11750
   838
wenzelm@11750
   839
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
wenzelm@12003
   840
proof
wenzelm@9488
   841
  assume "!!x. P x"
wenzelm@23389
   842
  then show "ALL x. P x" ..
wenzelm@9488
   843
next
wenzelm@9488
   844
  assume "ALL x. P x"
wenzelm@23553
   845
  then show "!!x. P x" by (rule allE)
wenzelm@9488
   846
qed
wenzelm@9488
   847
wenzelm@11750
   848
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@12003
   849
proof
wenzelm@9488
   850
  assume r: "A ==> B"
wenzelm@10383
   851
  show "A --> B" by (rule impI) (rule r)
wenzelm@9488
   852
next
wenzelm@9488
   853
  assume "A --> B" and A
wenzelm@23553
   854
  then show B by (rule mp)
wenzelm@9488
   855
qed
wenzelm@9488
   856
paulson@14749
   857
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
paulson@14749
   858
proof
paulson@14749
   859
  assume r: "A ==> False"
paulson@14749
   860
  show "~A" by (rule notI) (rule r)
paulson@14749
   861
next
paulson@14749
   862
  assume "~A" and A
wenzelm@23553
   863
  then show False by (rule notE)
paulson@14749
   864
qed
paulson@14749
   865
wenzelm@11750
   866
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@12003
   867
proof
wenzelm@10432
   868
  assume "x == y"
wenzelm@23553
   869
  show "x = y" by (unfold `x == y`) (rule refl)
wenzelm@10432
   870
next
wenzelm@10432
   871
  assume "x = y"
wenzelm@23553
   872
  then show "x == y" by (rule eq_reflection)
wenzelm@10432
   873
qed
wenzelm@10432
   874
wenzelm@12023
   875
lemma atomize_conj [atomize]:
wenzelm@19121
   876
  includes meta_conjunction_syntax
wenzelm@19121
   877
  shows "(A && B) == Trueprop (A & B)"
wenzelm@12003
   878
proof
wenzelm@19121
   879
  assume conj: "A && B"
wenzelm@19121
   880
  show "A & B"
wenzelm@19121
   881
  proof (rule conjI)
wenzelm@19121
   882
    from conj show A by (rule conjunctionD1)
wenzelm@19121
   883
    from conj show B by (rule conjunctionD2)
wenzelm@19121
   884
  qed
wenzelm@11953
   885
next
wenzelm@19121
   886
  assume conj: "A & B"
wenzelm@19121
   887
  show "A && B"
wenzelm@19121
   888
  proof -
wenzelm@19121
   889
    from conj show A ..
wenzelm@19121
   890
    from conj show B ..
wenzelm@11953
   891
  qed
wenzelm@11953
   892
qed
wenzelm@11953
   893
wenzelm@12386
   894
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18832
   895
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq
wenzelm@12386
   896
wenzelm@11750
   897
haftmann@20944
   898
subsection {* Package setup *}
haftmann@20944
   899
wenzelm@11750
   900
subsubsection {* Classical Reasoner setup *}
wenzelm@9529
   901
haftmann@20944
   902
lemma thin_refl:
haftmann@20944
   903
  "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
haftmann@20944
   904
haftmann@21151
   905
ML {*
haftmann@21151
   906
structure Hypsubst = HypsubstFun(
haftmann@21151
   907
struct
haftmann@21151
   908
  structure Simplifier = Simplifier
wenzelm@21218
   909
  val dest_eq = HOLogic.dest_eq
haftmann@21151
   910
  val dest_Trueprop = HOLogic.dest_Trueprop
haftmann@21151
   911
  val dest_imp = HOLogic.dest_imp
wenzelm@22129
   912
  val eq_reflection = @{thm HOL.eq_reflection}
haftmann@22218
   913
  val rev_eq_reflection = @{thm HOL.meta_eq_to_obj_eq}
wenzelm@22129
   914
  val imp_intr = @{thm HOL.impI}
wenzelm@22129
   915
  val rev_mp = @{thm HOL.rev_mp}
wenzelm@22129
   916
  val subst = @{thm HOL.subst}
wenzelm@22129
   917
  val sym = @{thm HOL.sym}
wenzelm@22129
   918
  val thin_refl = @{thm thin_refl};
haftmann@21151
   919
end);
wenzelm@21671
   920
open Hypsubst;
haftmann@21151
   921
haftmann@21151
   922
structure Classical = ClassicalFun(
haftmann@21151
   923
struct
wenzelm@22129
   924
  val mp = @{thm HOL.mp}
wenzelm@22129
   925
  val not_elim = @{thm HOL.notE}
wenzelm@22129
   926
  val classical = @{thm HOL.classical}
haftmann@21151
   927
  val sizef = Drule.size_of_thm
haftmann@21151
   928
  val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
haftmann@21151
   929
end);
haftmann@21151
   930
haftmann@21151
   931
structure BasicClassical: BASIC_CLASSICAL = Classical; 
wenzelm@21671
   932
open BasicClassical;
wenzelm@22129
   933
wenzelm@22129
   934
ML_Context.value_antiq "claset"
wenzelm@22129
   935
  (Scan.succeed ("claset", "Classical.local_claset_of (ML_Context.the_local_context ())"));
wenzelm@24035
   936
wenzelm@24035
   937
structure ResAtpset = NamedThmsFun(val name = "atp" val description = "ATP rules");
paulson@24286
   938
paulson@24286
   939
structure ResBlacklist = NamedThmsFun(val name = "noatp" val description = "Theorems blacklisted for ATP");
haftmann@21151
   940
*}
haftmann@21151
   941
paulson@24286
   942
(*ResBlacklist holds theorems blacklisted to sledgehammer. 
paulson@24286
   943
  These theorems typically produce clauses that are prolific (match too many equality or
paulson@24286
   944
  membership literals) and relate to seldom-used facts. Some duplicate other rules.*)
paulson@24286
   945
haftmann@21009
   946
setup {*
haftmann@21009
   947
let
haftmann@21009
   948
  (*prevent substitution on bool*)
haftmann@21009
   949
  fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
haftmann@21009
   950
    Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
haftmann@21009
   951
      (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
haftmann@21009
   952
in
haftmann@21151
   953
  Hypsubst.hypsubst_setup
haftmann@21151
   954
  #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
haftmann@21151
   955
  #> Classical.setup
haftmann@21151
   956
  #> ResAtpset.setup
paulson@24286
   957
  #> ResBlacklist.setup
haftmann@21009
   958
end
haftmann@21009
   959
*}
haftmann@21009
   960
haftmann@21009
   961
declare iffI [intro!]
haftmann@21009
   962
  and notI [intro!]
haftmann@21009
   963
  and impI [intro!]
haftmann@21009
   964
  and disjCI [intro!]
haftmann@21009
   965
  and conjI [intro!]
haftmann@21009
   966
  and TrueI [intro!]
haftmann@21009
   967
  and refl [intro!]
haftmann@21009
   968
haftmann@21009
   969
declare iffCE [elim!]
haftmann@21009
   970
  and FalseE [elim!]
haftmann@21009
   971
  and impCE [elim!]
haftmann@21009
   972
  and disjE [elim!]
haftmann@21009
   973
  and conjE [elim!]
haftmann@21009
   974
  and conjE [elim!]
haftmann@21009
   975
haftmann@21009
   976
declare ex_ex1I [intro!]
haftmann@21009
   977
  and allI [intro!]
haftmann@21009
   978
  and the_equality [intro]
haftmann@21009
   979
  and exI [intro]
haftmann@21009
   980
haftmann@21009
   981
declare exE [elim!]
haftmann@21009
   982
  allE [elim]
haftmann@21009
   983
wenzelm@22377
   984
ML {* val HOL_cs = @{claset} *}
mengj@19162
   985
wenzelm@20223
   986
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
wenzelm@20223
   987
  apply (erule swap)
wenzelm@20223
   988
  apply (erule (1) meta_mp)
wenzelm@20223
   989
  done
wenzelm@10383
   990
wenzelm@18689
   991
declare ex_ex1I [rule del, intro! 2]
wenzelm@18689
   992
  and ex1I [intro]
wenzelm@18689
   993
wenzelm@12386
   994
lemmas [intro?] = ext
wenzelm@12386
   995
  and [elim?] = ex1_implies_ex
wenzelm@11977
   996
haftmann@20944
   997
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
haftmann@20973
   998
lemma alt_ex1E [elim!]:
haftmann@20944
   999
  assumes major: "\<exists>!x. P x"
haftmann@20944
  1000
      and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
haftmann@20944
  1001
  shows R
haftmann@20944
  1002
apply (rule ex1E [OF major])
haftmann@20944
  1003
apply (rule prem)
wenzelm@22129
  1004
apply (tactic {* ares_tac @{thms allI} 1 *})+
wenzelm@22129
  1005
apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
wenzelm@22129
  1006
apply iprover
wenzelm@22129
  1007
done
haftmann@20944
  1008
haftmann@21151
  1009
ML {*
haftmann@21151
  1010
structure Blast = BlastFun(
haftmann@21151
  1011
struct
haftmann@21151
  1012
  type claset = Classical.claset
haftmann@22744
  1013
  val equality_name = @{const_name "op ="}
haftmann@22993
  1014
  val not_name = @{const_name Not}
wenzelm@22129
  1015
  val notE = @{thm HOL.notE}
wenzelm@22129
  1016
  val ccontr = @{thm HOL.ccontr}
haftmann@21151
  1017
  val contr_tac = Classical.contr_tac
haftmann@21151
  1018
  val dup_intr = Classical.dup_intr
haftmann@21151
  1019
  val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
wenzelm@21671
  1020
  val claset = Classical.claset
haftmann@21151
  1021
  val rep_cs = Classical.rep_cs
haftmann@21151
  1022
  val cla_modifiers = Classical.cla_modifiers
haftmann@21151
  1023
  val cla_meth' = Classical.cla_meth'
haftmann@21151
  1024
end);
wenzelm@21671
  1025
val Blast_tac = Blast.Blast_tac;
wenzelm@21671
  1026
val blast_tac = Blast.blast_tac;
haftmann@20944
  1027
*}
haftmann@20944
  1028
haftmann@21151
  1029
setup Blast.setup
haftmann@21151
  1030
haftmann@20944
  1031
haftmann@20944
  1032
subsubsection {* Simplifier *}
wenzelm@12281
  1033
wenzelm@12281
  1034
lemma eta_contract_eq: "(%s. f s) = f" ..
wenzelm@12281
  1035
wenzelm@12281
  1036
lemma simp_thms:
wenzelm@12937
  1037
  shows not_not: "(~ ~ P) = P"
nipkow@15354
  1038
  and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
wenzelm@12937
  1039
  and
berghofe@12436
  1040
    "(P ~= Q) = (P = (~Q))"
berghofe@12436
  1041
    "(P | ~P) = True"    "(~P | P) = True"
wenzelm@12281
  1042
    "(x = x) = True"
haftmann@20944
  1043
  and not_True_eq_False: "(\<not> True) = False"
haftmann@20944
  1044
  and not_False_eq_True: "(\<not> False) = True"
haftmann@20944
  1045
  and
berghofe@12436
  1046
    "(~P) ~= P"  "P ~= (~P)"
haftmann@20944
  1047
    "(True=P) = P"
haftmann@20944
  1048
  and eq_True: "(P = True) = P"
haftmann@20944
  1049
  and "(False=P) = (~P)"
haftmann@20944
  1050
  and eq_False: "(P = False) = (\<not> P)"
haftmann@20944
  1051
  and
wenzelm@12281
  1052
    "(True --> P) = P"  "(False --> P) = True"
wenzelm@12281
  1053
    "(P --> True) = True"  "(P --> P) = True"
wenzelm@12281
  1054
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
wenzelm@12281
  1055
    "(P & True) = P"  "(True & P) = P"
wenzelm@12281
  1056
    "(P & False) = False"  "(False & P) = False"
wenzelm@12281
  1057
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
wenzelm@12281
  1058
    "(P & ~P) = False"    "(~P & P) = False"
wenzelm@12281
  1059
    "(P | True) = True"  "(True | P) = True"
wenzelm@12281
  1060
    "(P | False) = P"  "(False | P) = P"
berghofe@12436
  1061
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
wenzelm@12281
  1062
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
wenzelm@12281
  1063
    -- {* needed for the one-point-rule quantifier simplification procs *}
wenzelm@12281
  1064
    -- {* essential for termination!! *} and
wenzelm@12281
  1065
    "!!P. (EX x. x=t & P(x)) = P(t)"
wenzelm@12281
  1066
    "!!P. (EX x. t=x & P(x)) = P(t)"
wenzelm@12281
  1067
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
wenzelm@12937
  1068
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
nipkow@17589
  1069
  by (blast, blast, blast, blast, blast, iprover+)
wenzelm@13421
  1070
paulson@14201
  1071
lemma disj_absorb: "(A | A) = A"
paulson@14201
  1072
  by blast
paulson@14201
  1073
paulson@14201
  1074
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
paulson@14201
  1075
  by blast
paulson@14201
  1076
paulson@14201
  1077
lemma conj_absorb: "(A & A) = A"
paulson@14201
  1078
  by blast
paulson@14201
  1079
paulson@14201
  1080
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
paulson@14201
  1081
  by blast
paulson@14201
  1082
wenzelm@12281
  1083
lemma eq_ac:
wenzelm@12937
  1084
  shows eq_commute: "(a=b) = (b=a)"
wenzelm@12937
  1085
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
nipkow@17589
  1086
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
nipkow@17589
  1087
lemma neq_commute: "(a~=b) = (b~=a)" by iprover
wenzelm@12281
  1088
wenzelm@12281
  1089
lemma conj_comms:
wenzelm@12937
  1090
  shows conj_commute: "(P&Q) = (Q&P)"
nipkow@17589
  1091
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
nipkow@17589
  1092
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
wenzelm@12281
  1093
paulson@19174
  1094
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
paulson@19174
  1095
wenzelm@12281
  1096
lemma disj_comms:
wenzelm@12937
  1097
  shows disj_commute: "(P|Q) = (Q|P)"
nipkow@17589
  1098
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
nipkow@17589
  1099
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
wenzelm@12281
  1100
paulson@19174
  1101
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
paulson@19174
  1102
nipkow@17589
  1103
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
nipkow@17589
  1104
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
wenzelm@12281
  1105
nipkow@17589
  1106
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
nipkow@17589
  1107
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
wenzelm@12281
  1108
nipkow@17589
  1109
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
nipkow@17589
  1110
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover
nipkow@17589
  1111
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
wenzelm@12281
  1112
wenzelm@12281
  1113
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
wenzelm@12281
  1114
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
wenzelm@12281
  1115
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
wenzelm@12281
  1116
wenzelm@12281
  1117
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
wenzelm@12281
  1118
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
wenzelm@12281
  1119
haftmann@21151
  1120
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
haftmann@21151
  1121
  by iprover
haftmann@21151
  1122
nipkow@17589
  1123
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
wenzelm@12281
  1124
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
wenzelm@12281
  1125
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
wenzelm@12281
  1126
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
wenzelm@12281
  1127
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
wenzelm@12281
  1128
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
wenzelm@12281
  1129
  by blast
wenzelm@12281
  1130
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
wenzelm@12281
  1131
nipkow@17589
  1132
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
wenzelm@12281
  1133
wenzelm@12281
  1134
wenzelm@12281
  1135
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
wenzelm@12281
  1136
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
wenzelm@12281
  1137
  -- {* cases boil down to the same thing. *}
wenzelm@12281
  1138
  by blast
wenzelm@12281
  1139
wenzelm@12281
  1140
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
wenzelm@12281
  1141
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
nipkow@17589
  1142
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
nipkow@17589
  1143
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
chaieb@23403
  1144
lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
wenzelm@12281
  1145
paulson@24286
  1146
declare All_def [noatp]
paulson@24286
  1147
nipkow@17589
  1148
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
nipkow@17589
  1149
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
wenzelm@12281
  1150
wenzelm@12281
  1151
text {*
wenzelm@12281
  1152
  \medskip The @{text "&"} congruence rule: not included by default!
wenzelm@12281
  1153
  May slow rewrite proofs down by as much as 50\% *}
wenzelm@12281
  1154
wenzelm@12281
  1155
lemma conj_cong:
wenzelm@12281
  1156
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
nipkow@17589
  1157
  by iprover
wenzelm@12281
  1158
wenzelm@12281
  1159
lemma rev_conj_cong:
wenzelm@12281
  1160
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
nipkow@17589
  1161
  by iprover
wenzelm@12281
  1162
wenzelm@12281
  1163
text {* The @{text "|"} congruence rule: not included by default! *}
wenzelm@12281
  1164
wenzelm@12281
  1165
lemma disj_cong:
wenzelm@12281
  1166
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
wenzelm@12281
  1167
  by blast
wenzelm@12281
  1168
wenzelm@12281
  1169
wenzelm@12281
  1170
text {* \medskip if-then-else rules *}
wenzelm@12281
  1171
wenzelm@12281
  1172
lemma if_True: "(if True then x else y) = x"
wenzelm@12281
  1173
  by (unfold if_def) blast
wenzelm@12281
  1174
wenzelm@12281
  1175
lemma if_False: "(if False then x else y) = y"
wenzelm@12281
  1176
  by (unfold if_def) blast
wenzelm@12281
  1177
wenzelm@12281
  1178
lemma if_P: "P ==> (if P then x else y) = x"
wenzelm@12281
  1179
  by (unfold if_def) blast
wenzelm@12281
  1180
wenzelm@12281
  1181
lemma if_not_P: "~P ==> (if P then x else y) = y"
wenzelm@12281
  1182
  by (unfold if_def) blast
wenzelm@12281
  1183
wenzelm@12281
  1184
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
wenzelm@12281
  1185
  apply (rule case_split [of Q])
paulson@15481
  1186
   apply (simplesubst if_P)
paulson@15481
  1187
    prefer 3 apply (simplesubst if_not_P, blast+)
wenzelm@12281
  1188
  done
wenzelm@12281
  1189
wenzelm@12281
  1190
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
paulson@15481
  1191
by (simplesubst split_if, blast)
wenzelm@12281
  1192
paulson@24286
  1193
lemmas if_splits [noatp] = split_if split_if_asm
wenzelm@12281
  1194
wenzelm@12281
  1195
lemma if_cancel: "(if c then x else x) = x"
paulson@15481
  1196
by (simplesubst split_if, blast)
wenzelm@12281
  1197
wenzelm@12281
  1198
lemma if_eq_cancel: "(if x = y then y else x) = x"
paulson@15481
  1199
by (simplesubst split_if, blast)
wenzelm@12281
  1200
wenzelm@12281
  1201
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
wenzelm@19796
  1202
  -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
wenzelm@12281
  1203
  by (rule split_if)
wenzelm@12281
  1204
wenzelm@12281
  1205
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
wenzelm@19796
  1206
  -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
paulson@15481
  1207
  apply (simplesubst split_if, blast)
wenzelm@12281
  1208
  done
wenzelm@12281
  1209
nipkow@17589
  1210
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
nipkow@17589
  1211
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
wenzelm@12281
  1212
schirmer@15423
  1213
text {* \medskip let rules for simproc *}
schirmer@15423
  1214
schirmer@15423
  1215
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
schirmer@15423
  1216
  by (unfold Let_def)
schirmer@15423
  1217
schirmer@15423
  1218
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
schirmer@15423
  1219
  by (unfold Let_def)
schirmer@15423
  1220
berghofe@16633
  1221
text {*
ballarin@16999
  1222
  The following copy of the implication operator is useful for
ballarin@16999
  1223
  fine-tuning congruence rules.  It instructs the simplifier to simplify
ballarin@16999
  1224
  its premise.
berghofe@16633
  1225
*}
berghofe@16633
  1226
wenzelm@17197
  1227
constdefs
wenzelm@17197
  1228
  simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1)
wenzelm@17197
  1229
  "simp_implies \<equiv> op ==>"
berghofe@16633
  1230
wenzelm@18457
  1231
lemma simp_impliesI:
berghofe@16633
  1232
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
berghofe@16633
  1233
  shows "PROP P =simp=> PROP Q"
berghofe@16633
  1234
  apply (unfold simp_implies_def)
berghofe@16633
  1235
  apply (rule PQ)
berghofe@16633
  1236
  apply assumption
berghofe@16633
  1237
  done
berghofe@16633
  1238
berghofe@16633
  1239
lemma simp_impliesE:
berghofe@16633
  1240
  assumes PQ:"PROP P =simp=> PROP Q"
berghofe@16633
  1241
  and P: "PROP P"
berghofe@16633
  1242
  and QR: "PROP Q \<Longrightarrow> PROP R"
berghofe@16633
  1243
  shows "PROP R"
berghofe@16633
  1244
  apply (rule QR)
berghofe@16633
  1245
  apply (rule PQ [unfolded simp_implies_def])
berghofe@16633
  1246
  apply (rule P)
berghofe@16633
  1247
  done
berghofe@16633
  1248
berghofe@16633
  1249
lemma simp_implies_cong:
berghofe@16633
  1250
  assumes PP' :"PROP P == PROP P'"
berghofe@16633
  1251
  and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
berghofe@16633
  1252
  shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
berghofe@16633
  1253
proof (unfold simp_implies_def, rule equal_intr_rule)
berghofe@16633
  1254
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
berghofe@16633
  1255
  and P': "PROP P'"
berghofe@16633
  1256
  from PP' [symmetric] and P' have "PROP P"
berghofe@16633
  1257
    by (rule equal_elim_rule1)
wenzelm@23553
  1258
  then have "PROP Q" by (rule PQ)
berghofe@16633
  1259
  with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
berghofe@16633
  1260
next
berghofe@16633
  1261
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
berghofe@16633
  1262
  and P: "PROP P"
berghofe@16633
  1263
  from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
wenzelm@23553
  1264
  then have "PROP Q'" by (rule P'Q')
berghofe@16633
  1265
  with P'QQ' [OF P', symmetric] show "PROP Q"
berghofe@16633
  1266
    by (rule equal_elim_rule1)
berghofe@16633
  1267
qed
berghofe@16633
  1268
haftmann@20944
  1269
lemma uncurry:
haftmann@20944
  1270
  assumes "P \<longrightarrow> Q \<longrightarrow> R"
haftmann@20944
  1271
  shows "P \<and> Q \<longrightarrow> R"
wenzelm@23553
  1272
  using assms by blast
haftmann@20944
  1273
haftmann@20944
  1274
lemma iff_allI:
haftmann@20944
  1275
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1276
  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
wenzelm@23553
  1277
  using assms by blast
haftmann@20944
  1278
haftmann@20944
  1279
lemma iff_exI:
haftmann@20944
  1280
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1281
  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
wenzelm@23553
  1282
  using assms by blast
haftmann@20944
  1283
haftmann@20944
  1284
lemma all_comm:
haftmann@20944
  1285
  "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
haftmann@20944
  1286
  by blast
haftmann@20944
  1287
haftmann@20944
  1288
lemma ex_comm:
haftmann@20944
  1289
  "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
haftmann@20944
  1290
  by blast
haftmann@20944
  1291
wenzelm@9869
  1292
use "simpdata.ML"
wenzelm@21671
  1293
ML {* open Simpdata *}
wenzelm@21671
  1294
haftmann@21151
  1295
setup {*
haftmann@21151
  1296
  Simplifier.method_setup Splitter.split_modifiers
haftmann@21547
  1297
  #> (fn thy => (change_simpset_of thy (fn _ => Simpdata.simpset_simprocs); thy))
haftmann@21151
  1298
  #> Splitter.setup
haftmann@21151
  1299
  #> Clasimp.setup
haftmann@21151
  1300
  #> EqSubst.setup
haftmann@21151
  1301
*}
haftmann@21151
  1302
wenzelm@24035
  1303
text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
wenzelm@24035
  1304
wenzelm@24035
  1305
simproc_setup neq ("x = y") = {* fn _ =>
wenzelm@24035
  1306
let
wenzelm@24035
  1307
  val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
wenzelm@24035
  1308
  fun is_neq eq lhs rhs thm =
wenzelm@24035
  1309
    (case Thm.prop_of thm of
wenzelm@24035
  1310
      _ $ (Not $ (eq' $ l' $ r')) =>
wenzelm@24035
  1311
        Not = HOLogic.Not andalso eq' = eq andalso
wenzelm@24035
  1312
        r' aconv lhs andalso l' aconv rhs
wenzelm@24035
  1313
    | _ => false);
wenzelm@24035
  1314
  fun proc ss ct =
wenzelm@24035
  1315
    (case Thm.term_of ct of
wenzelm@24035
  1316
      eq $ lhs $ rhs =>
wenzelm@24035
  1317
        (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of_ss ss) of
wenzelm@24035
  1318
          SOME thm => SOME (thm RS neq_to_EQ_False)
wenzelm@24035
  1319
        | NONE => NONE)
wenzelm@24035
  1320
     | _ => NONE);
wenzelm@24035
  1321
in proc end;
wenzelm@24035
  1322
*}
wenzelm@24035
  1323
wenzelm@24035
  1324
simproc_setup let_simp ("Let x f") = {*
wenzelm@24035
  1325
let
wenzelm@24035
  1326
  val (f_Let_unfold, x_Let_unfold) =
wenzelm@24035
  1327
    let val [(_$(f$x)$_)] = prems_of @{thm Let_unfold}
wenzelm@24035
  1328
    in (cterm_of @{theory} f, cterm_of @{theory} x) end
wenzelm@24035
  1329
  val (f_Let_folded, x_Let_folded) =
wenzelm@24035
  1330
    let val [(_$(f$x)$_)] = prems_of @{thm Let_folded}
wenzelm@24035
  1331
    in (cterm_of @{theory} f, cterm_of @{theory} x) end;
wenzelm@24035
  1332
  val g_Let_folded =
wenzelm@24035
  1333
    let val [(_$_$(g$_))] = prems_of @{thm Let_folded} in cterm_of @{theory} g end;
wenzelm@24035
  1334
wenzelm@24035
  1335
  fun proc _ ss ct =
wenzelm@24035
  1336
    let
wenzelm@24035
  1337
      val ctxt = Simplifier.the_context ss;
wenzelm@24035
  1338
      val thy = ProofContext.theory_of ctxt;
wenzelm@24035
  1339
      val t = Thm.term_of ct;
wenzelm@24035
  1340
      val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
wenzelm@24035
  1341
    in Option.map (hd o Variable.export ctxt' ctxt o single)
wenzelm@24035
  1342
      (case t' of Const ("Let",_) $ x $ f => (* x and f are already in normal form *)
wenzelm@24035
  1343
        if is_Free x orelse is_Bound x orelse is_Const x
wenzelm@24035
  1344
        then SOME @{thm Let_def}
wenzelm@24035
  1345
        else
wenzelm@24035
  1346
          let
wenzelm@24035
  1347
            val n = case f of (Abs (x,_,_)) => x | _ => "x";
wenzelm@24035
  1348
            val cx = cterm_of thy x;
wenzelm@24035
  1349
            val {T=xT,...} = rep_cterm cx;
wenzelm@24035
  1350
            val cf = cterm_of thy f;
wenzelm@24035
  1351
            val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
wenzelm@24035
  1352
            val (_$_$g) = prop_of fx_g;
wenzelm@24035
  1353
            val g' = abstract_over (x,g);
wenzelm@24035
  1354
          in (if (g aconv g')
wenzelm@24035
  1355
               then
wenzelm@24035
  1356
                  let
wenzelm@24035
  1357
                    val rl =
wenzelm@24035
  1358
                      cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] @{thm Let_unfold};
wenzelm@24035
  1359
                  in SOME (rl OF [fx_g]) end
wenzelm@24035
  1360
               else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
wenzelm@24035
  1361
               else let
wenzelm@24035
  1362
                     val abs_g'= Abs (n,xT,g');
wenzelm@24035
  1363
                     val g'x = abs_g'$x;
wenzelm@24035
  1364
                     val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));
wenzelm@24035
  1365
                     val rl = cterm_instantiate
wenzelm@24035
  1366
                               [(f_Let_folded,cterm_of thy f),(x_Let_folded,cx),
wenzelm@24035
  1367
                                (g_Let_folded,cterm_of thy abs_g')]
wenzelm@24035
  1368
                               @{thm Let_folded};
wenzelm@24035
  1369
                   in SOME (rl OF [transitive fx_g g_g'x])
wenzelm@24035
  1370
                   end)
wenzelm@24035
  1371
          end
wenzelm@24035
  1372
      | _ => NONE)
wenzelm@24035
  1373
    end
wenzelm@24035
  1374
in proc end *}
wenzelm@24035
  1375
wenzelm@24035
  1376
haftmann@21151
  1377
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
haftmann@21151
  1378
proof
wenzelm@23389
  1379
  assume "True \<Longrightarrow> PROP P"
wenzelm@23389
  1380
  from this [OF TrueI] show "PROP P" .
haftmann@21151
  1381
next
haftmann@21151
  1382
  assume "PROP P"
wenzelm@23389
  1383
  then show "PROP P" .
haftmann@21151
  1384
qed
haftmann@21151
  1385
haftmann@21151
  1386
lemma ex_simps:
haftmann@21151
  1387
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
haftmann@21151
  1388
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
haftmann@21151
  1389
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
haftmann@21151
  1390
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
haftmann@21151
  1391
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
haftmann@21151
  1392
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
haftmann@21151
  1393
  -- {* Miniscoping: pushing in existential quantifiers. *}
haftmann@21151
  1394
  by (iprover | blast)+
haftmann@21151
  1395
haftmann@21151
  1396
lemma all_simps:
haftmann@21151
  1397
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
haftmann@21151
  1398
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
haftmann@21151
  1399
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
haftmann@21151
  1400
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
haftmann@21151
  1401
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
haftmann@21151
  1402
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
haftmann@21151
  1403
  -- {* Miniscoping: pushing in universal quantifiers. *}
haftmann@21151
  1404
  by (iprover | blast)+
paulson@15481
  1405
wenzelm@21671
  1406
lemmas [simp] =
wenzelm@21671
  1407
  triv_forall_equality (*prunes params*)
wenzelm@21671
  1408
  True_implies_equals  (*prune asms `True'*)
wenzelm@21671
  1409
  if_True
wenzelm@21671
  1410
  if_False
wenzelm@21671
  1411
  if_cancel
wenzelm@21671
  1412
  if_eq_cancel
wenzelm@21671
  1413
  imp_disjL
haftmann@20973
  1414
  (*In general it seems wrong to add distributive laws by default: they
haftmann@20973
  1415
    might cause exponential blow-up.  But imp_disjL has been in for a while
haftmann@20973
  1416
    and cannot be removed without affecting existing proofs.  Moreover,
haftmann@20973
  1417
    rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
haftmann@20973
  1418
    grounds that it allows simplification of R in the two cases.*)
wenzelm@21671
  1419
  conj_assoc
wenzelm@21671
  1420
  disj_assoc
wenzelm@21671
  1421
  de_Morgan_conj
wenzelm@21671
  1422
  de_Morgan_disj
wenzelm@21671
  1423
  imp_disj1
wenzelm@21671
  1424
  imp_disj2
wenzelm@21671
  1425
  not_imp
wenzelm@21671
  1426
  disj_not1
wenzelm@21671
  1427
  not_all
wenzelm@21671
  1428
  not_ex
wenzelm@21671
  1429
  cases_simp
wenzelm@21671
  1430
  the_eq_trivial
wenzelm@21671
  1431
  the_sym_eq_trivial
wenzelm@21671
  1432
  ex_simps
wenzelm@21671
  1433
  all_simps
wenzelm@21671
  1434
  simp_thms
wenzelm@21671
  1435
wenzelm@21671
  1436
lemmas [cong] = imp_cong simp_implies_cong
wenzelm@21671
  1437
lemmas [split] = split_if
haftmann@20973
  1438
wenzelm@22377
  1439
ML {* val HOL_ss = @{simpset} *}
haftmann@20973
  1440
haftmann@20944
  1441
text {* Simplifies x assuming c and y assuming ~c *}
haftmann@20944
  1442
lemma if_cong:
haftmann@20944
  1443
  assumes "b = c"
haftmann@20944
  1444
      and "c \<Longrightarrow> x = u"
haftmann@20944
  1445
      and "\<not> c \<Longrightarrow> y = v"
haftmann@20944
  1446
  shows "(if b then x else y) = (if c then u else v)"
wenzelm@23553
  1447
  unfolding if_def using assms by simp
haftmann@20944
  1448
haftmann@20944
  1449
text {* Prevents simplification of x and y:
haftmann@20944
  1450
  faster and allows the execution of functional programs. *}
haftmann@20944
  1451
lemma if_weak_cong [cong]:
haftmann@20944
  1452
  assumes "b = c"
haftmann@20944
  1453
  shows "(if b then x else y) = (if c then x else y)"
wenzelm@23553
  1454
  using assms by (rule arg_cong)
haftmann@20944
  1455
haftmann@20944
  1456
text {* Prevents simplification of t: much faster *}
haftmann@20944
  1457
lemma let_weak_cong:
haftmann@20944
  1458
  assumes "a = b"
haftmann@20944
  1459
  shows "(let x = a in t x) = (let x = b in t x)"
wenzelm@23553
  1460
  using assms by (rule arg_cong)
haftmann@20944
  1461
haftmann@20944
  1462
text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}
haftmann@20944
  1463
lemma eq_cong2:
haftmann@20944
  1464
  assumes "u = u'"
haftmann@20944
  1465
  shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
wenzelm@23553
  1466
  using assms by simp
haftmann@20944
  1467
haftmann@20944
  1468
lemma if_distrib:
haftmann@20944
  1469
  "f (if c then x else y) = (if c then f x else f y)"
haftmann@20944
  1470
  by simp
haftmann@20944
  1471
haftmann@20944
  1472
text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
wenzelm@21502
  1473
  side of an equality.  Used in @{text "{Integ,Real}/simproc.ML"} *}
haftmann@20944
  1474
lemma restrict_to_left:
haftmann@20944
  1475
  assumes "x = y"
haftmann@20944
  1476
  shows "(x = z) = (y = z)"
wenzelm@23553
  1477
  using assms by simp
haftmann@20944
  1478
wenzelm@17459
  1479
haftmann@20944
  1480
subsubsection {* Generic cases and induction *}
wenzelm@17459
  1481
haftmann@20944
  1482
text {* Rule projections: *}
berghofe@18887
  1483
haftmann@20944
  1484
ML {*
haftmann@20944
  1485
structure ProjectRule = ProjectRuleFun
haftmann@20944
  1486
(struct
wenzelm@22129
  1487
  val conjunct1 = @{thm conjunct1};
wenzelm@22129
  1488
  val conjunct2 = @{thm conjunct2};
wenzelm@22129
  1489
  val mp = @{thm mp};
haftmann@20944
  1490
end)
wenzelm@17459
  1491
*}
wenzelm@17459
  1492
wenzelm@11824
  1493
constdefs
wenzelm@18457
  1494
  induct_forall where "induct_forall P == \<forall>x. P x"
wenzelm@18457
  1495
  induct_implies where "induct_implies A B == A \<longrightarrow> B"
wenzelm@18457
  1496
  induct_equal where "induct_equal x y == x = y"
wenzelm@18457
  1497
  induct_conj where "induct_conj A B == A \<and> B"
wenzelm@11824
  1498
wenzelm@11989
  1499
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
wenzelm@18457
  1500
  by (unfold atomize_all induct_forall_def)
wenzelm@11824
  1501
wenzelm@11989
  1502
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
wenzelm@18457
  1503
  by (unfold atomize_imp induct_implies_def)
wenzelm@11824
  1504
wenzelm@11989
  1505
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
wenzelm@18457
  1506
  by (unfold atomize_eq induct_equal_def)
wenzelm@18457
  1507
wenzelm@18457
  1508
lemma induct_conj_eq:
wenzelm@18457
  1509
  includes meta_conjunction_syntax
wenzelm@18457
  1510
  shows "(A && B) == Trueprop (induct_conj A B)"
wenzelm@18457
  1511
  by (unfold atomize_conj induct_conj_def)
wenzelm@18457
  1512
wenzelm@18457
  1513
lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
wenzelm@18457
  1514
lemmas induct_rulify [symmetric, standard] = induct_atomize
wenzelm@18457
  1515
lemmas induct_rulify_fallback =
wenzelm@18457
  1516
  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
wenzelm@18457
  1517
wenzelm@11824
  1518
wenzelm@11989
  1519
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
wenzelm@11989
  1520
    induct_conj (induct_forall A) (induct_forall B)"
nipkow@17589
  1521
  by (unfold induct_forall_def induct_conj_def) iprover
wenzelm@11824
  1522
wenzelm@11989
  1523
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
wenzelm@11989
  1524
    induct_conj (induct_implies C A) (induct_implies C B)"
nipkow@17589
  1525
  by (unfold induct_implies_def induct_conj_def) iprover
wenzelm@11989
  1526
berghofe@13598
  1527
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
berghofe@13598
  1528
proof
berghofe@13598
  1529
  assume r: "induct_conj A B ==> PROP C" and A B
wenzelm@18457
  1530
  show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
berghofe@13598
  1531
next
berghofe@13598
  1532
  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
wenzelm@18457
  1533
  show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
berghofe@13598
  1534
qed
wenzelm@11824
  1535
wenzelm@11989
  1536
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
wenzelm@11824
  1537
wenzelm@11989
  1538
hide const induct_forall induct_implies induct_equal induct_conj
wenzelm@11824
  1539
wenzelm@11824
  1540
text {* Method setup. *}
wenzelm@11824
  1541
wenzelm@11824
  1542
ML {*
wenzelm@11824
  1543
  structure InductMethod = InductMethodFun
wenzelm@11824
  1544
  (struct
wenzelm@22129
  1545
    val cases_default = @{thm case_split}
wenzelm@22129
  1546
    val atomize = @{thms induct_atomize}
wenzelm@22129
  1547
    val rulify = @{thms induct_rulify}
wenzelm@22129
  1548
    val rulify_fallback = @{thms induct_rulify_fallback}
wenzelm@11824
  1549
  end);
wenzelm@11824
  1550
*}
wenzelm@11824
  1551
wenzelm@11824
  1552
setup InductMethod.setup
wenzelm@11824
  1553
wenzelm@18457
  1554
haftmann@20944
  1555
haftmann@20944
  1556
subsection {* Other simple lemmas and lemma duplicates *}
haftmann@20944
  1557
haftmann@24166
  1558
lemma Let_0 [simp]: "Let 0 f = f 0"
haftmann@24166
  1559
  unfolding Let_def ..
haftmann@24166
  1560
haftmann@24166
  1561
lemma Let_1 [simp]: "Let 1 f = f 1"
haftmann@24166
  1562
  unfolding Let_def ..
haftmann@24166
  1563
haftmann@20944
  1564
lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
haftmann@20944
  1565
  by blast+
haftmann@20944
  1566
haftmann@20944
  1567
lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
haftmann@20944
  1568
  apply (rule iffI)
haftmann@20944
  1569
  apply (rule_tac a = "%x. THE y. P x y" in ex1I)
haftmann@20944
  1570
  apply (fast dest!: theI')
haftmann@20944
  1571
  apply (fast intro: ext the1_equality [symmetric])
haftmann@20944
  1572
  apply (erule ex1E)
haftmann@20944
  1573
  apply (rule allI)
haftmann@20944
  1574
  apply (rule ex1I)
haftmann@20944
  1575
  apply (erule spec)
haftmann@20944
  1576
  apply (erule_tac x = "%z. if z = x then y else f z" in allE)
haftmann@20944
  1577
  apply (erule impE)
haftmann@20944
  1578
  apply (rule allI)
haftmann@20944
  1579
  apply (rule_tac P = "xa = x" in case_split_thm)
haftmann@20944
  1580
  apply (drule_tac [3] x = x in fun_cong, simp_all)
haftmann@20944
  1581
  done
haftmann@20944
  1582
haftmann@20944
  1583
lemma mk_left_commute:
haftmann@21547
  1584
  fixes f (infix "\<otimes>" 60)
haftmann@21547
  1585
  assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and
haftmann@21547
  1586
          c: "\<And>x y. x \<otimes> y = y \<otimes> x"
haftmann@21547
  1587
  shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
haftmann@20944
  1588
  by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
haftmann@20944
  1589
haftmann@22218
  1590
lemmas eq_sym_conv = eq_commute
haftmann@22218
  1591
chaieb@23037
  1592
lemma nnf_simps:
chaieb@23037
  1593
  "(\<not>(P \<and> Q)) = (\<not> P \<or> \<not> Q)" "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)" 
chaieb@23037
  1594
  "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))" 
chaieb@23037
  1595
  "(\<not> \<not>(P)) = P"
chaieb@23037
  1596
by blast+
chaieb@23037
  1597
wenzelm@21671
  1598
wenzelm@21671
  1599
subsection {* Basic ML bindings *}
wenzelm@21671
  1600
wenzelm@21671
  1601
ML {*
wenzelm@22129
  1602
val FalseE = @{thm FalseE}
wenzelm@22129
  1603
val Let_def = @{thm Let_def}
wenzelm@22129
  1604
val TrueI = @{thm TrueI}
wenzelm@22129
  1605
val allE = @{thm allE}
wenzelm@22129
  1606
val allI = @{thm allI}
wenzelm@22129
  1607
val all_dupE = @{thm all_dupE}
wenzelm@22129
  1608
val arg_cong = @{thm arg_cong}
wenzelm@22129
  1609
val box_equals = @{thm box_equals}
wenzelm@22129
  1610
val ccontr = @{thm ccontr}
wenzelm@22129
  1611
val classical = @{thm classical}
wenzelm@22129
  1612
val conjE = @{thm conjE}
wenzelm@22129
  1613
val conjI = @{thm conjI}
wenzelm@22129
  1614
val conjunct1 = @{thm conjunct1}
wenzelm@22129
  1615
val conjunct2 = @{thm conjunct2}
wenzelm@22129
  1616
val disjCI = @{thm disjCI}
wenzelm@22129
  1617
val disjE = @{thm disjE}
wenzelm@22129
  1618
val disjI1 = @{thm disjI1}
wenzelm@22129
  1619
val disjI2 = @{thm disjI2}
wenzelm@22129
  1620
val eq_reflection = @{thm eq_reflection}
wenzelm@22129
  1621
val ex1E = @{thm ex1E}
wenzelm@22129
  1622
val ex1I = @{thm ex1I}
wenzelm@22129
  1623
val ex1_implies_ex = @{thm ex1_implies_ex}
wenzelm@22129
  1624
val exE = @{thm exE}
wenzelm@22129
  1625
val exI = @{thm exI}
wenzelm@22129
  1626
val excluded_middle = @{thm excluded_middle}
wenzelm@22129
  1627
val ext = @{thm ext}
wenzelm@22129
  1628
val fun_cong = @{thm fun_cong}
wenzelm@22129
  1629
val iffD1 = @{thm iffD1}
wenzelm@22129
  1630
val iffD2 = @{thm iffD2}
wenzelm@22129
  1631
val iffI = @{thm iffI}
wenzelm@22129
  1632
val impE = @{thm impE}
wenzelm@22129
  1633
val impI = @{thm impI}
wenzelm@22129
  1634
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
wenzelm@22129
  1635
val mp = @{thm mp}
wenzelm@22129
  1636
val notE = @{thm notE}
wenzelm@22129
  1637
val notI = @{thm notI}
wenzelm@22129
  1638
val not_all = @{thm not_all}
wenzelm@22129
  1639
val not_ex = @{thm not_ex}
wenzelm@22129
  1640
val not_iff = @{thm not_iff}
wenzelm@22129
  1641
val not_not = @{thm not_not}
wenzelm@22129
  1642
val not_sym = @{thm not_sym}
wenzelm@22129
  1643
val refl = @{thm refl}
wenzelm@22129
  1644
val rev_mp = @{thm rev_mp}
wenzelm@22129
  1645
val spec = @{thm spec}
wenzelm@22129
  1646
val ssubst = @{thm ssubst}
wenzelm@22129
  1647
val subst = @{thm subst}
wenzelm@22129
  1648
val sym = @{thm sym}
wenzelm@22129
  1649
val trans = @{thm trans}
wenzelm@21671
  1650
*}
wenzelm@21671
  1651
wenzelm@21671
  1652
haftmann@24280
  1653
subsection {* Code generator basic setup -- see further @{text Code_Setup.thy} *}
haftmann@23247
  1654
haftmann@24280
  1655
setup "CodeName.setup #> CodeTarget.setup #> Codegen.setup"
haftmann@23247
  1656
haftmann@23247
  1657
class eq (attach "op =") = type
haftmann@23247
  1658
haftmann@23247
  1659
code_datatype True False
haftmann@23247
  1660
haftmann@23247
  1661
lemma [code func]:
haftmann@24280
  1662
  shows "False \<and> x \<longleftrightarrow> False"
haftmann@24280
  1663
    and "True \<and> x \<longleftrightarrow> x"
haftmann@24280
  1664
    and "x \<and> False \<longleftrightarrow> False"
haftmann@24280
  1665
    and "x \<and> True \<longleftrightarrow> x" by simp_all
haftmann@23247
  1666
haftmann@23247
  1667
lemma [code func]:
haftmann@24280
  1668
  shows "False \<or> x \<longleftrightarrow> x"
haftmann@24280
  1669
    and "True \<or> x \<longleftrightarrow> True"
haftmann@24280
  1670
    and "x \<or> False \<longleftrightarrow> x"
haftmann@24280
  1671
    and "x \<or> True \<longleftrightarrow> True" by simp_all
haftmann@23247
  1672
haftmann@23247
  1673
lemma [code func]:
haftmann@24280
  1674
  shows "\<not> True \<longleftrightarrow> False"
haftmann@24280
  1675
    and "\<not> False \<longleftrightarrow> True" by (rule HOL.simp_thms)+
haftmann@23247
  1676
haftmann@23247
  1677
instance bool :: eq ..
haftmann@23247
  1678
haftmann@23247
  1679
lemma [code func]:
haftmann@24280
  1680
  shows "False = P \<longleftrightarrow> \<not> P"
haftmann@24280
  1681
    and "True = P \<longleftrightarrow> P" 
haftmann@24280
  1682
    and "P = False \<longleftrightarrow> \<not> P" 
haftmann@24280
  1683
    and "P = True \<longleftrightarrow> P" by simp_all
haftmann@23247
  1684
haftmann@23247
  1685
code_datatype Trueprop "prop"
haftmann@23247
  1686
haftmann@23247
  1687
code_datatype "TYPE('a)"
haftmann@23247
  1688
haftmann@23247
  1689
haftmann@22839
  1690
subsection {* Legacy tactics and ML bindings *}
wenzelm@21671
  1691
wenzelm@21671
  1692
ML {*
wenzelm@21671
  1693
fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
wenzelm@21671
  1694
wenzelm@21671
  1695
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
wenzelm@21671
  1696
local
wenzelm@21671
  1697
  fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
wenzelm@21671
  1698
    | wrong_prem (Bound _) = true
wenzelm@21671
  1699
    | wrong_prem _ = false;
wenzelm@21671
  1700
  val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
wenzelm@21671
  1701
in
wenzelm@21671
  1702
  fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
wenzelm@21671
  1703
  fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
wenzelm@21671
  1704
end;
haftmann@22839
  1705
haftmann@22839
  1706
val all_conj_distrib = thm "all_conj_distrib";
haftmann@22839
  1707
val all_simps = thms "all_simps";
haftmann@22839
  1708
val atomize_not = thm "atomize_not";
haftmann@22839
  1709
val case_split = thm "case_split_thm";
haftmann@22839
  1710
val case_split_thm = thm "case_split_thm"
haftmann@22839
  1711
val cases_simp = thm "cases_simp";
haftmann@22839
  1712
val choice_eq = thm "choice_eq"
haftmann@22839
  1713
val cong = thm "cong"
haftmann@22839
  1714
val conj_comms = thms "conj_comms";
haftmann@22839
  1715
val conj_cong = thm "conj_cong";
haftmann@22839
  1716
val de_Morgan_conj = thm "de_Morgan_conj";
haftmann@22839
  1717
val de_Morgan_disj = thm "de_Morgan_disj";
haftmann@22839
  1718
val disj_assoc = thm "disj_assoc";
haftmann@22839
  1719
val disj_comms = thms "disj_comms";
haftmann@22839
  1720
val disj_cong = thm "disj_cong";
haftmann@22839
  1721
val eq_ac = thms "eq_ac";
haftmann@22839
  1722
val eq_cong2 = thm "eq_cong2"
haftmann@22839
  1723
val Eq_FalseI = thm "Eq_FalseI";
haftmann@22839
  1724
val Eq_TrueI = thm "Eq_TrueI";
haftmann@22839
  1725
val Ex1_def = thm "Ex1_def"
haftmann@22839
  1726
val ex_disj_distrib = thm "ex_disj_distrib";
haftmann@22839
  1727
val ex_simps = thms "ex_simps";
haftmann@22839
  1728
val if_cancel = thm "if_cancel";
haftmann@22839
  1729
val if_eq_cancel = thm "if_eq_cancel";
haftmann@22839
  1730
val if_False = thm "if_False";
haftmann@22839
  1731
val iff_conv_conj_imp = thm "iff_conv_conj_imp";
haftmann@22839
  1732
val iff = thm "iff"
haftmann@22839
  1733
val if_splits = thms "if_splits";
haftmann@22839
  1734
val if_True = thm "if_True";
haftmann@22839
  1735
val if_weak_cong = thm "if_weak_cong"
haftmann@22839
  1736
val imp_all = thm "imp_all";
haftmann@22839
  1737
val imp_cong = thm "imp_cong";
haftmann@22839
  1738
val imp_conjL = thm "imp_conjL";
haftmann@22839
  1739
val imp_conjR = thm "imp_conjR";
haftmann@22839
  1740
val imp_conv_disj = thm "imp_conv_disj";
haftmann@22839
  1741
val simp_implies_def = thm "simp_implies_def";
haftmann@22839
  1742
val simp_thms = thms "simp_thms";
haftmann@22839
  1743
val split_if = thm "split_if";
haftmann@22839
  1744
val the1_equality = thm "the1_equality"
haftmann@22839
  1745
val theI = thm "theI"
haftmann@22839
  1746
val theI' = thm "theI'"
haftmann@22839
  1747
val True_implies_equals = thm "True_implies_equals";
chaieb@23037
  1748
val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms @ @{thms "nnf_simps"})
chaieb@23037
  1749
wenzelm@21671
  1750
*}
wenzelm@21671
  1751
kleing@14357
  1752
end