src/HOL/HOL.thy
author nipkow
Tue, 31 Mar 2015 17:29:44 +0200
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parent 59779 b6bda9140e39
child 59929 a090551e5ec8
permissions -rw-r--r--
added lemmas
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(*  Title:      HOL/HOL.thy
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    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
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*)
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section {* The basis of Higher-Order Logic *}
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theory HOL
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imports Pure "~~/src/Tools/Code_Generator"
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keywords
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  "try" "solve_direct" "quickcheck" "print_coercions" "print_claset"
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    "print_induct_rules" :: diag and
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  "quickcheck_params" :: thy_decl
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begin
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ML_file "~~/src/Tools/misc_legacy.ML"
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ML_file "~~/src/Tools/try.ML"
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ML_file "~~/src/Tools/quickcheck.ML"
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ML_file "~~/src/Tools/solve_direct.ML"
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ML_file "~~/src/Tools/IsaPlanner/zipper.ML"
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ML_file "~~/src/Tools/IsaPlanner/isand.ML"
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ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML"
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ML_file "~~/src/Provers/hypsubst.ML"
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ML_file "~~/src/Provers/splitter.ML"
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ML_file "~~/src/Provers/classical.ML"
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ML_file "~~/src/Provers/blast.ML"
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ML_file "~~/src/Provers/clasimp.ML"
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ML_file "~~/src/Tools/eqsubst.ML"
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ML_file "~~/src/Provers/quantifier1.ML"
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ML_file "~~/src/Tools/atomize_elim.ML"
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ML_file "~~/src/Tools/cong_tac.ML"
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ML_file "~~/src/Tools/intuitionistic.ML" setup \<open>Intuitionistic.method_setup @{binding iprover}\<close>
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ML_file "~~/src/Tools/project_rule.ML"
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ML_file "~~/src/Tools/subtyping.ML"
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ML_file "~~/src/Tools/case_product.ML"
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ML \<open>Plugin_Name.declare_setup @{binding extraction}\<close>
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ML \<open>
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  Plugin_Name.declare_setup @{binding quickcheck_random};
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  Plugin_Name.declare_setup @{binding quickcheck_exhaustive};
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  Plugin_Name.declare_setup @{binding quickcheck_bounded_forall};
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  Plugin_Name.declare_setup @{binding quickcheck_full_exhaustive};
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  Plugin_Name.declare_setup @{binding quickcheck_narrowing};
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\<close>
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ML \<open>
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  Plugin_Name.define_setup @{binding quickcheck}
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   [@{plugin quickcheck_exhaustive},
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    @{plugin quickcheck_random},
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    @{plugin quickcheck_bounded_forall},
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    @{plugin quickcheck_full_exhaustive},
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    @{plugin quickcheck_narrowing}]
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\<close>
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subsection {* Primitive logic *}
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subsubsection {* Core syntax *}
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setup {* Axclass.class_axiomatization (@{binding type}, []) *}
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default_sort type
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setup {* Object_Logic.add_base_sort @{sort type} *}
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axiomatization where fun_arity: "OFCLASS('a \<Rightarrow> 'b, type_class)"
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instance "fun" :: (type, type) type by (rule fun_arity)
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axiomatization where itself_arity: "OFCLASS('a itself, type_class)"
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instance itself :: (type) type by (rule itself_arity)
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typedecl bool
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judgment
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  Trueprop      :: "bool => prop"                   ("(_)" 5)
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axiomatization
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  implies       :: "[bool, bool] => bool"           (infixr "-->" 25)  and
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  eq            :: "['a, 'a] => bool"               (infixl "=" 50)  and
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  The           :: "('a => bool) => 'a"
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consts
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  True          :: bool
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  False         :: bool
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  Not           :: "bool => bool"                   ("~ _" [40] 40)
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  conj          :: "[bool, bool] => bool"           (infixr "&" 35)
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  disj          :: "[bool, bool] => bool"           (infixr "|" 30)
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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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subsubsection {* Additional concrete syntax *}
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notation (output)
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  eq  (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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  conj  (infixr "\<and>" 35) and
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  disj  (infixr "\<or>" 30) and
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  implies  (infixr "\<longrightarrow>" 25) and
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  not_equal  (infixl "\<noteq>" 50)
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notation (xsymbols output)
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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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  conj  (infixr "\<and>" 35) and
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  disj  (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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syntax "_The" :: "[pttrn, bool] => 'a"  ("(3THE _./ _)" [0, 10] 10)
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translations "THE x. P" == "CONST The (%x. P)"
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print_translation {*
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  [(@{const_syntax The}, fn _ => fn [Abs abs] =>
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      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
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      in Syntax.const @{syntax_const "_The"} $ x $ t end)]
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*}  -- {* To avoid eta-contraction of body *}
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nonterminal letbinds and letbind
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syntax
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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 (_))" [0, 10] 10)
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nonterminal case_syn and cases_syn
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syntax
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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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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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axiomatization where
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  refl: "t = (t::'a)" and
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  subst: "s = t \<Longrightarrow> P s \<Longrightarrow> P t" and
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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*} and
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  the_eq_trivial: "(THE x. x = a) = (a::'a)"
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axiomatization where
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  impI: "(P ==> Q) ==> P-->Q" and
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  mp: "[| P-->Q;  P |] ==> Q" and
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  iff: "(P-->Q) --> (Q-->P) --> (P=Q)" and
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  True_or_False: "(P=True) | (P=False)"
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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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definition If :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" [0, 0, 10] 10)
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  where "If P x y \<equiv> (THE z::'a. (P=True --> z=x) & (P=False --> z=y))"
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definition Let :: "'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"
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  where "Let s f \<equiv> f s"
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324219de6ee3 qualified constants Let and If
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translations
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  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
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  "let x = a in e"        == "CONST Let a (%x. e)"
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axiomatization undefined :: 'a
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class default = fixes default :: 'a
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subsection {* Fundamental rules *}
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subsubsection {* Equality *}
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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 trans_sym [Pure.elim?]: "r = s ==> t = s ==> r = t"
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  by (rule trans [OF _ sym])
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lemma meta_eq_to_obj_eq:
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  assumes meq: "A == B"
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  shows "A = B"
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  by (unfold meq) (rule refl)
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text {* Useful with @{text erule} for proving equalities from known equalities. *}
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     (* a = b
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        |   |
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        c = d   *)
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lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
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apply (rule trans)
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apply (rule trans)
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apply (rule sym)
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apply assumption+
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done
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text {* For calculational reasoning: *}
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lemma forw_subst: "a = b ==> P b ==> P a"
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  by (rule ssubst)
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lemma back_subst: "P a ==> a = b ==> P b"
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  by (rule subst)
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subsubsection {* Congruence rules for application *}
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text {* Similar to @{text AP_THM} in Gordon's HOL. *}
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lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
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apply (erule subst)
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apply (rule refl)
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done
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text {* Similar to @{text AP_TERM} in Gordon's HOL and FOL's @{text subst_context}. *}
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lemma arg_cong: "x=y ==> f(x)=f(y)"
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apply (erule subst)
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apply (rule refl)
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done
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lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
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apply (erule ssubst)+
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apply (rule refl)
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done
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lemma cong: "[| f = g; (x::'a) = y |] ==> f x = g y"
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apply (erule subst)+
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apply (rule refl)
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done
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ML {* fun cong_tac ctxt = Cong_Tac.cong_tac ctxt @{thm cong} *}
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subsubsection {* Equality of booleans -- iff *}
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lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
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  by (iprover intro: iff [THEN mp, THEN mp] impI assms)
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lemma iffD2: "[| P=Q; Q |] ==> P"
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  by (erule ssubst)
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lemma rev_iffD2: "[| Q; P=Q |] ==> P"
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  by (erule iffD2)
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lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
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  by (drule sym) (rule iffD2)
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lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
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  by (drule sym) (rule rev_iffD2)
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lemma iffE:
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  assumes major: "P=Q"
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    and minor: "[| P --> Q; Q --> P |] ==> R"
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  shows R
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  by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
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subsubsection {*True*}
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lemma TrueI: "True"
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  unfolding True_def by (rule refl)
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lemma eqTrueI: "P ==> P = True"
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  by (iprover intro: iffI TrueI)
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lemma eqTrueE: "P = True ==> P"
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  by (erule iffD2) (rule TrueI)
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   314
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subsubsection {*Universal quantifier*}
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lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
9c97af4a1567 tuned proofs;
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  unfolding All_def by (iprover intro: ext eqTrueI assms)
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   320
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lemma spec: "ALL x::'a. P(x) ==> P(x)"
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apply (unfold All_def)
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   323
apply (rule eqTrueE)
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   324
apply (erule fun_cong)
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   325
done
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   326
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lemma allE:
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  assumes major: "ALL x. P(x)"
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    and minor: "P(x) ==> R"
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  shows R
9c97af4a1567 tuned proofs;
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   331
  by (iprover intro: minor major [THEN spec])
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   332
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   333
lemma all_dupE:
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  assumes major: "ALL x. P(x)"
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   335
    and minor: "[| P(x); ALL x. P(x) |] ==> R"
9c97af4a1567 tuned proofs;
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   336
  shows R
9c97af4a1567 tuned proofs;
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   337
  by (iprover intro: minor major major [THEN spec])
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   338
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   339
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subsubsection {* False *}
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9c97af4a1567 tuned proofs;
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text {*
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  Depends upon @{text spec}; it is impossible to do propositional
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  logic before quantifiers!
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   345
*}
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   347
lemma FalseE: "False ==> P"
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  apply (unfold False_def)
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   349
  apply (erule spec)
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   350
  done
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   352
lemma False_neq_True: "False = True ==> P"
9c97af4a1567 tuned proofs;
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   353
  by (erule eqTrueE [THEN FalseE])
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   354
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   355
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   356
subsubsection {* Negation *}
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   357
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   358
lemma notI:
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  assumes "P ==> False"
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  shows "~P"
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  apply (unfold not_def)
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   362
  apply (iprover intro: impI assms)
9c97af4a1567 tuned proofs;
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   363
  done
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   364
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   365
lemma False_not_True: "False ~= True"
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  apply (rule notI)
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   367
  apply (erule False_neq_True)
9c97af4a1567 tuned proofs;
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parents: 21502
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   368
  done
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   369
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   370
lemma True_not_False: "True ~= False"
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   371
  apply (rule notI)
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   372
  apply (drule sym)
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   373
  apply (erule False_neq_True)
9c97af4a1567 tuned proofs;
wenzelm
parents: 21502
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   374
  done
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   375
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   376
lemma notE: "[| ~P;  P |] ==> R"
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   377
  apply (unfold not_def)
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   378
  apply (erule mp [THEN FalseE])
9c97af4a1567 tuned proofs;
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   379
  apply assumption
9c97af4a1567 tuned proofs;
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   380
  done
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   381
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   382
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
9c97af4a1567 tuned proofs;
wenzelm
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   383
  by (erule notE [THEN notI]) (erule meta_mp)
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   384
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   385
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   386
subsubsection {*Implication*}
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   387
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   388
lemma impE:
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  assumes "P-->Q" "P" "Q ==> R"
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   390
  shows "R"
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af8ae54238f5 use hologic.ML in basic HOL context;
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   391
by (iprover intro: assms mp)
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   392
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   393
(* Reduces Q to P-->Q, allowing substitution in P. *)
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   394
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
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   395
by (iprover intro: mp)
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   396
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   397
lemma contrapos_nn:
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   398
  assumes major: "~Q"
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diff changeset
   399
      and minor: "P==>Q"
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   400
  shows "~P"
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nipkow
parents: 17459
diff changeset
   401
by (iprover intro: notI minor major [THEN notE])
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diff changeset
   402
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diff changeset
   403
(*not used at all, but we already have the other 3 combinations *)
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diff changeset
   404
lemma contrapos_pn:
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   405
  assumes major: "Q"
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diff changeset
   406
      and minor: "P ==> ~Q"
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diff changeset
   407
  shows "~P"
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nipkow
parents: 17459
diff changeset
   408
by (iprover intro: notI minor major notE)
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diff changeset
   409
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diff changeset
   410
lemma not_sym: "t ~= s ==> s ~= t"
21250
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   411
  by (erule contrapos_nn) (erule sym)
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   412
a268f6288fb6 moved lemma eq_neq_eq_imp_neq to HOL
haftmann
parents: 21218
diff changeset
   413
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
   414
  by (erule subst, erule ssubst, assumption)
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diff changeset
   415
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diff changeset
   416
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   417
subsubsection {*Existential quantifier*}
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   418
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   419
lemma exI: "P x ==> EX x::'a. P x"
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   420
apply (unfold Ex_def)
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nipkow
parents: 17459
diff changeset
   421
apply (iprover intro: allI allE impI mp)
15411
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diff changeset
   422
done
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diff changeset
   423
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diff changeset
   424
lemma exE:
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   425
  assumes major: "EX x::'a. P(x)"
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diff changeset
   426
      and minor: "!!x. P(x) ==> Q"
1d195de59497 removal of HOL_Lemmas
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parents: 15380
diff changeset
   427
  shows "Q"
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paulson
parents: 15380
diff changeset
   428
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   429
apply (iprover intro: impI [THEN allI] minor)
15411
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paulson
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diff changeset
   430
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   431
1d195de59497 removal of HOL_Lemmas
paulson
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diff changeset
   432
20944
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haftmann
parents: 20833
diff changeset
   433
subsubsection {*Conjunction*}
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diff changeset
   434
1d195de59497 removal of HOL_Lemmas
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diff changeset
   435
lemma conjI: "[| P; Q |] ==> P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   436
apply (unfold and_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   437
apply (iprover intro: impI [THEN allI] mp)
15411
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paulson
parents: 15380
diff changeset
   438
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   439
1d195de59497 removal of HOL_Lemmas
paulson
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diff changeset
   440
lemma conjunct1: "[| P & Q |] ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   441
apply (unfold and_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   442
apply (iprover intro: impI dest: spec mp)
15411
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paulson
parents: 15380
diff changeset
   443
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   444
1d195de59497 removal of HOL_Lemmas
paulson
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diff changeset
   445
lemma conjunct2: "[| P & Q |] ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   446
apply (unfold and_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   447
apply (iprover intro: impI dest: spec mp)
15411
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paulson
parents: 15380
diff changeset
   448
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   449
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   450
lemma conjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   451
  assumes major: "P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   452
      and minor: "[| P; Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   453
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   454
apply (rule minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   455
apply (rule major [THEN conjunct1])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   456
apply (rule major [THEN conjunct2])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   457
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   458
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   459
lemma context_conjI:
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   460
  assumes "P" "P ==> Q" shows "P & Q"
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   461
by (iprover intro: conjI assms)
15411
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paulson
parents: 15380
diff changeset
   462
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   463
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   464
subsubsection {*Disjunction*}
15411
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paulson
parents: 15380
diff changeset
   465
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   466
lemma disjI1: "P ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   467
apply (unfold or_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   468
apply (iprover intro: allI impI mp)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   469
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   470
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   471
lemma disjI2: "Q ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   472
apply (unfold or_def)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   473
apply (iprover intro: allI impI mp)
15411
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paulson
parents: 15380
diff changeset
   474
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   475
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   476
lemma disjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   477
  assumes major: "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   478
      and minorP: "P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   479
      and minorQ: "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   480
  shows "R"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   481
by (iprover intro: minorP minorQ impI
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   482
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   483
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   484
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   485
subsubsection {*Classical logic*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   486
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   487
lemma classical:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   488
  assumes prem: "~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   489
  shows "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   490
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   491
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   492
apply (rule notI [THEN prem, THEN eqTrueI])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   493
apply (erule subst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   494
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   495
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   496
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45294
diff changeset
   497
lemmas ccontr = FalseE [THEN classical]
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   498
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   499
(*notE with premises exchanged; it discharges ~R so that it can be used to
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   500
  make elimination rules*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   501
lemma rev_notE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   502
  assumes premp: "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   503
      and premnot: "~R ==> ~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   504
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   505
apply (rule ccontr)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   506
apply (erule notE [OF premnot premp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   507
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   508
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   509
(*Double negation law*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   510
lemma notnotD: "~~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   511
apply (rule classical)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   512
apply (erule notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   513
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   514
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   515
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   516
lemma contrapos_pp:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   517
  assumes p1: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   518
      and p2: "~P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   519
  shows "P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   520
by (iprover intro: classical p1 p2 notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   521
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   522
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   523
subsubsection {*Unique existence*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   524
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   525
lemma ex1I:
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   526
  assumes "P a" "!!x. P(x) ==> x=a"
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   527
  shows "EX! x. P(x)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   528
by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   529
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   530
text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   531
lemma ex_ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   532
  assumes ex_prem: "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   533
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   534
  shows "EX! x. P(x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   535
by (iprover intro: ex_prem [THEN exE] ex1I eq)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   536
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   537
lemma ex1E:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   538
  assumes major: "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   539
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   540
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   541
apply (rule major [unfolded Ex1_def, THEN exE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   542
apply (erule conjE)
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   543
apply (iprover intro: minor)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   544
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   545
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   546
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   547
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   548
apply (rule exI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   549
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   550
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   551
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   552
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   553
subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   554
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   555
lemma disjCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   556
  assumes "~Q ==> P" shows "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   557
apply (rule classical)
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   558
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   559
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   560
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   561
lemma excluded_middle: "~P | P"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   562
by (iprover intro: disjCI)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   563
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   564
text {*
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   565
  case distinction as a natural deduction rule.
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   566
  Note that @{term "~P"} is the second case, not the first
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   567
*}
27126
3ede9103de8e eliminated obsolete case_split_thm -- use case_split;
wenzelm
parents: 27107
diff changeset
   568
lemma case_split [case_names True False]:
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   569
  assumes prem1: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   570
      and prem2: "~P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   571
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   572
apply (rule excluded_middle [THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   573
apply (erule prem2)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   574
apply (erule prem1)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   575
done
27126
3ede9103de8e eliminated obsolete case_split_thm -- use case_split;
wenzelm
parents: 27107
diff changeset
   576
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   577
(*Classical implies (-->) elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   578
lemma impCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   579
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   580
      and minor: "~P ==> R" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   581
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   582
apply (rule excluded_middle [of P, THEN disjE])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   583
apply (iprover intro: minor major [THEN mp])+
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   584
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   585
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   586
(*This version of --> elimination works on Q before P.  It works best for
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   587
  those cases in which P holds "almost everywhere".  Can't install as
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   588
  default: would break old proofs.*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   589
lemma impCE':
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   590
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   591
      and minor: "Q ==> R" "~P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   592
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   593
apply (rule excluded_middle [of P, THEN disjE])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   594
apply (iprover intro: minor major [THEN mp])+
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   595
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   596
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   597
(*Classical <-> elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   598
lemma iffCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   599
  assumes major: "P=Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   600
      and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   601
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   602
apply (rule major [THEN iffE])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   603
apply (iprover intro: minor elim: impCE notE)
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   604
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   605
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   606
lemma exCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   607
  assumes "ALL x. ~P(x) ==> P(a)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   608
  shows "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   609
apply (rule ccontr)
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   610
apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   611
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   612
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   613
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   614
subsubsection {* Intuitionistic Reasoning *}
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   615
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   616
lemma impE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   617
  assumes 1: "P --> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   618
    and 2: "Q ==> R"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   619
    and 3: "P --> Q ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   620
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   621
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   622
  from 3 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   623
  with 1 have Q by (rule impE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   624
  with 2 show R .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   625
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   626
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   627
lemma allE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   628
  assumes 1: "ALL x. P x"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   629
    and 2: "P x ==> ALL x. P x ==> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   630
  shows Q
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   631
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   632
  from 1 have "P x" by (rule spec)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   633
  from this and 1 show Q by (rule 2)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   634
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   635
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   636
lemma notE':
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   637
  assumes 1: "~ P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   638
    and 2: "~ P ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   639
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   640
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   641
  from 2 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   642
  with 1 show R by (rule notE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   643
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   644
22444
fb80fedd192d added safe intro rules for removing "True" subgoals as well as "~ False" ones.
dixon
parents: 22377
diff changeset
   645
lemma TrueE: "True ==> P ==> P" .
fb80fedd192d added safe intro rules for removing "True" subgoals as well as "~ False" ones.
dixon
parents: 22377
diff changeset
   646
lemma notFalseE: "~ False ==> P ==> P" .
fb80fedd192d added safe intro rules for removing "True" subgoals as well as "~ False" ones.
dixon
parents: 22377
diff changeset
   647
22467
c9357ef01168 TrueElim and notTrueElim tested and added as safe elim rules.
dixon
parents: 22445
diff changeset
   648
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
15801
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   649
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   650
  and [Pure.elim 2] = allE notE' impE'
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   651
  and [Pure.intro] = exI disjI2 disjI1
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   652
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   653
lemmas [trans] = trans
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   654
  and [sym] = sym not_sym
15801
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   655
  and [Pure.elim?] = iffD1 iffD2 impE
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   656
11438
3d9222b80989 declare trans [trans] (*overridden in theory Calculation*);
wenzelm
parents: 11432
diff changeset
   657
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   658
subsubsection {* Atomizing meta-level connectives *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   659
28513
b0b30fd6c264 re-introduces axiom subst
haftmann
parents: 28400
diff changeset
   660
axiomatization where
b0b30fd6c264 re-introduces axiom subst
haftmann
parents: 28400
diff changeset
   661
  eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)
b0b30fd6c264 re-introduces axiom subst
haftmann
parents: 28400
diff changeset
   662
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   663
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
   664
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   665
  assume "!!x. P x"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23263
diff changeset
   666
  then show "ALL x. P x" ..
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   667
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   668
  assume "ALL x. P x"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   669
  then show "!!x. P x" by (rule allE)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   670
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   671
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   672
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   673
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   674
  assume r: "A ==> B"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   675
  show "A --> B" by (rule impI) (rule r)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   676
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   677
  assume "A --> B" and A
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   678
  then show B by (rule mp)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   679
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   680
14749
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   681
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   682
proof
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   683
  assume r: "A ==> False"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   684
  show "~A" by (rule notI) (rule r)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   685
next
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   686
  assume "~A" and A
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   687
  then show False by (rule notE)
14749
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   688
qed
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   689
39566
87a5704673f0 Pure equality is a regular cpde operation
haftmann
parents: 39471
diff changeset
   690
lemma atomize_eq [atomize, code]: "(x == y) == Trueprop (x = y)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   691
proof
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   692
  assume "x == y"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   693
  show "x = y" by (unfold `x == y`) (rule refl)
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   694
next
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   695
  assume "x = y"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
   696
  then show "x == y" by (rule eq_reflection)
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   697
qed
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   698
28856
5e009a80fe6d Pure syntax: more coherent treatment of aprop, permanent TERM and &&&;
wenzelm
parents: 28741
diff changeset
   699
lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   700
proof
28856
5e009a80fe6d Pure syntax: more coherent treatment of aprop, permanent TERM and &&&;
wenzelm
parents: 28741
diff changeset
   701
  assume conj: "A &&& B"
19121
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   702
  show "A & B"
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   703
  proof (rule conjI)
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   704
    from conj show A by (rule conjunctionD1)
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   705
    from conj show B by (rule conjunctionD2)
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   706
  qed
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   707
next
19121
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   708
  assume conj: "A & B"
28856
5e009a80fe6d Pure syntax: more coherent treatment of aprop, permanent TERM and &&&;
wenzelm
parents: 28741
diff changeset
   709
  show "A &&& B"
19121
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   710
  proof -
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   711
    from conj show A ..
d7fd5415a781 simplified Pure conjunction;
wenzelm
parents: 19039
diff changeset
   712
    from conj show B ..
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   713
  qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   714
qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   715
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   716
lemmas [symmetric, rulify] = atomize_all atomize_imp
18832
6ab4de872a70 declare 'defn' rules;
wenzelm
parents: 18757
diff changeset
   717
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   718
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   719
26580
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   720
subsubsection {* Atomizing elimination rules *}
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   721
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   722
lemma atomize_exL[atomize_elim]: "(!!x. P x ==> Q) == ((EX x. P x) ==> Q)"
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   723
  by rule iprover+
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   724
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   725
lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   726
  by rule iprover+
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   727
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   728
lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   729
  by rule iprover+
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   730
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   731
lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop A" ..
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   732
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26555
diff changeset
   733
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   734
subsection {* Package setup *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   735
51314
eac4bb5adbf9 just one HOLogic.Trueprop_conv, with regular exception CTERM;
wenzelm
parents: 51304
diff changeset
   736
ML_file "Tools/hologic.ML"
eac4bb5adbf9 just one HOLogic.Trueprop_conv, with regular exception CTERM;
wenzelm
parents: 51304
diff changeset
   737
eac4bb5adbf9 just one HOLogic.Trueprop_conv, with regular exception CTERM;
wenzelm
parents: 51304
diff changeset
   738
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   739
subsubsection {* Sledgehammer setup *}
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   740
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   741
text {*
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   742
Theorems blacklisted to Sledgehammer. These theorems typically produce clauses
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   743
that are prolific (match too many equality or membership literals) and relate to
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   744
seldom-used facts. Some duplicate other rules.
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   745
*}
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   746
57963
cb67fac9bd89 updated to named_theorems;
wenzelm
parents: 57962
diff changeset
   747
named_theorems no_atp "theorems that should be filtered out by Sledgehammer"
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   748
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
   749
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   750
subsubsection {* Classical Reasoner setup *}
9529
d9434a9277a4 lemmas atomize = all_eq imp_eq;
wenzelm
parents: 9488
diff changeset
   751
26411
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   752
lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   753
  by (rule classical) iprover
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   754
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   755
lemma swap: "~ P ==> (~ R ==> P) ==> R"
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   756
  by (rule classical) iprover
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   757
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   758
lemma thin_refl:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   759
  "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   760
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   761
ML {*
42799
4e33894aec6d modernized functor names;
wenzelm
parents: 42795
diff changeset
   762
structure Hypsubst = Hypsubst
4e33894aec6d modernized functor names;
wenzelm
parents: 42795
diff changeset
   763
(
21218
38013c3a77a2 tuned hypsubst setup;
wenzelm
parents: 21210
diff changeset
   764
  val dest_eq = HOLogic.dest_eq
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   765
  val dest_Trueprop = HOLogic.dest_Trueprop
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   766
  val dest_imp = HOLogic.dest_imp
26411
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   767
  val eq_reflection = @{thm eq_reflection}
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   768
  val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   769
  val imp_intr = @{thm impI}
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   770
  val rev_mp = @{thm rev_mp}
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   771
  val subst = @{thm subst}
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   772
  val sym = @{thm sym}
22129
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   773
  val thin_refl = @{thm thin_refl};
42799
4e33894aec6d modernized functor names;
wenzelm
parents: 42795
diff changeset
   774
);
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
   775
open Hypsubst;
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   776
42799
4e33894aec6d modernized functor names;
wenzelm
parents: 42795
diff changeset
   777
structure Classical = Classical
4e33894aec6d modernized functor names;
wenzelm
parents: 42795
diff changeset
   778
(
26411
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   779
  val imp_elim = @{thm imp_elim}
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   780
  val not_elim = @{thm notE}
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   781
  val swap = @{thm swap}
cd74690f3bfb pass imp_elim, swap to classical prover;
wenzelm
parents: 25966
diff changeset
   782
  val classical = @{thm classical}
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   783
  val sizef = Drule.size_of_thm
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   784
  val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
42799
4e33894aec6d modernized functor names;
wenzelm
parents: 42795
diff changeset
   785
);
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   786
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   787
structure Basic_Classical: BASIC_CLASSICAL = Classical;
33308
cf62d1690d04 separate ResBlacklist, based on scalable persistent data -- avoids inefficient hashing later on;
wenzelm
parents: 33185
diff changeset
   788
open Basic_Classical;
43560
d1650e3720fd ML antiquotations are managed as theory data, with proper name space and entity markup;
wenzelm
parents: 42802
diff changeset
   789
*}
22129
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   790
21009
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   791
setup {*
35389
2be5440f7271 tuned hyp_subst_tac';
wenzelm
parents: 35364
diff changeset
   792
  (*prevent substitution on bool*)
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   793
  let
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   794
    fun non_bool_eq (@{const_name HOL.eq}, Type (_, [T, _])) = T <> @{typ bool}
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   795
      | non_bool_eq _ = false;
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   796
    fun hyp_subst_tac' ctxt =
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   797
      SUBGOAL (fn (goal, i) =>
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   798
        if Term.exists_Const non_bool_eq goal
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   799
        then Hypsubst.hyp_subst_tac ctxt i
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   800
        else no_tac);
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   801
  in
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   802
    Context_Rules.addSWrapper (fn ctxt => fn tac => hyp_subst_tac' ctxt ORELSE' tac)
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
   803
  end
21009
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   804
*}
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   805
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   806
declare iffI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   807
  and notI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   808
  and impI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   809
  and disjCI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   810
  and conjI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   811
  and TrueI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   812
  and refl [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   813
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   814
declare iffCE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   815
  and FalseE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   816
  and impCE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   817
  and disjE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   818
  and conjE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   819
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   820
declare ex_ex1I [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   821
  and allI [intro!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   822
  and exI [intro]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   823
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   824
declare exE [elim!]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   825
  allE [elim]
0eae3fb48936 lifted claset setup from ML to Isar level
haftmann
parents: 20973
diff changeset
   826
51687
3d8720271ebf discontinued obsolete ML antiquotation @{claset};
wenzelm
parents: 51314
diff changeset
   827
ML {* val HOL_cs = claset_of @{context} *}
19162
67436e2a16df Added setup for "atpset" (a rule set for ATPs).
mengj
parents: 19138
diff changeset
   828
20223
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   829
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   830
  apply (erule swap)
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   831
  apply (erule (1) meta_mp)
89d2758ecddf tuned proofs;
wenzelm
parents: 20172
diff changeset
   832
  done
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   833
18689
a50587cd8414 prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents: 18595
diff changeset
   834
declare ex_ex1I [rule del, intro! 2]
a50587cd8414 prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents: 18595
diff changeset
   835
  and ex1I [intro]
a50587cd8414 prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents: 18595
diff changeset
   836
41865
4e8483cc2cc5 declare ext [intro]: Extensionality now available by default
paulson
parents: 41827
diff changeset
   837
declare ext [intro]
4e8483cc2cc5 declare ext [intro]: Extensionality now available by default
paulson
parents: 41827
diff changeset
   838
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   839
lemmas [intro?] = ext
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   840
  and [elim?] = ex1_implies_ex
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   841
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   842
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
20973
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
   843
lemma alt_ex1E [elim!]:
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   844
  assumes major: "\<exists>!x. P x"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   845
      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
   846
  shows R
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   847
apply (rule ex1E [OF major])
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   848
apply (rule prem)
59499
14095f771781 misc tuning;
wenzelm
parents: 59498
diff changeset
   849
apply assumption
14095f771781 misc tuning;
wenzelm
parents: 59498
diff changeset
   850
apply (rule allI)+
59498
50b60f501b05 proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents: 59028
diff changeset
   851
apply (tactic {* eresolve_tac @{context} [Classical.dup_elim NONE @{thm allE}] 1 *})
22129
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   852
apply iprover
bb2203c93316 tuned ML setup;
wenzelm
parents: 21671
diff changeset
   853
done
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   854
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   855
ML {*
42477
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   856
  structure Blast = Blast
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   857
  (
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   858
    structure Classical = Classical
42802
51d7e74f6899 simplified BLAST_DATA;
wenzelm
parents: 42799
diff changeset
   859
    val Trueprop_const = dest_Const @{const Trueprop}
42477
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   860
    val equality_name = @{const_name HOL.eq}
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   861
    val not_name = @{const_name Not}
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   862
    val notE = @{thm notE}
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   863
    val ccontr = @{thm ccontr}
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   864
    val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   865
  );
52fa26b6c524 simplified Blast setup;
wenzelm
parents: 42459
diff changeset
   866
  val blast_tac = Blast.blast_tac;
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   867
*}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   868
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   869
59504
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   870
subsubsection {*THE: definite description operator*}
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   871
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   872
lemma the_equality [intro]:
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   873
  assumes "P a"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   874
      and "!!x. P x ==> x=a"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   875
  shows "(THE x. P x) = a"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   876
  by (blast intro: assms trans [OF arg_cong [where f=The] the_eq_trivial])
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   877
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   878
lemma theI:
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   879
  assumes "P a" and "!!x. P x ==> x=a"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   880
  shows "P (THE x. P x)"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   881
by (iprover intro: assms the_equality [THEN ssubst])
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   882
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   883
lemma theI': "EX! x. P x ==> P (THE x. P x)"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   884
  by (blast intro: theI)
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   885
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   886
(*Easier to apply than theI: only one occurrence of P*)
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   887
lemma theI2:
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   888
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   889
  shows "Q (THE x. P x)"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   890
by (iprover intro: assms theI)
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   891
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   892
lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   893
by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   894
           elim:allE impE)
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   895
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   896
lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   897
  by blast
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   898
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   899
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   900
  by blast
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   901
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
   902
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   903
subsubsection {* Simplifier *}
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   904
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   905
lemma eta_contract_eq: "(%s. f s) = f" ..
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   906
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   907
lemma simp_thms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   908
  shows not_not: "(~ ~ P) = P"
15354
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
   909
  and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   910
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   911
    "(P ~= Q) = (P = (~Q))"
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   912
    "(P | ~P) = True"    "(~P | P) = True"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   913
    "(x = x) = True"
32068
98acc234d683 tuned code annotations
haftmann
parents: 31998
diff changeset
   914
  and not_True_eq_False [code]: "(\<not> True) = False"
98acc234d683 tuned code annotations
haftmann
parents: 31998
diff changeset
   915
  and not_False_eq_True [code]: "(\<not> False) = True"
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   916
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   917
    "(~P) ~= P"  "P ~= (~P)"
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   918
    "(True=P) = P"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   919
  and eq_True: "(P = True) = P"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   920
  and "(False=P) = (~P)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   921
  and eq_False: "(P = False) = (\<not> P)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
   922
  and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   923
    "(True --> P) = P"  "(False --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   924
    "(P --> True) = True"  "(P --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   925
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   926
    "(P & True) = P"  "(True & P) = P"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   927
    "(P & False) = False"  "(False & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   928
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   929
    "(P & ~P) = False"    "(~P & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   930
    "(P | True) = True"  "(True | P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   931
    "(P | False) = P"  "(False | P) = P"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   932
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   933
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
31166
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 31156
diff changeset
   934
  and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   935
    "!!P. (EX x. x=t & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   936
    "!!P. (EX x. t=x & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   937
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   938
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   939
  by (blast, blast, blast, blast, blast, iprover+)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13412
diff changeset
   940
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   941
lemma disj_absorb: "(A | A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   942
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   943
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   944
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   945
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   946
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   947
lemma conj_absorb: "(A & A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   948
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   949
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   950
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   951
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   952
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   953
lemma eq_ac:
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56941
diff changeset
   954
  shows eq_commute: "a = b \<longleftrightarrow> b = a"
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56941
diff changeset
   955
    and iff_left_commute: "(P \<longleftrightarrow> (Q \<longleftrightarrow> R)) \<longleftrightarrow> (Q \<longleftrightarrow> (P \<longleftrightarrow> R))"
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56941
diff changeset
   956
    and iff_assoc: "((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longleftrightarrow> (P \<longleftrightarrow> (Q \<longleftrightarrow> R))" by (iprover, blast+)
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56941
diff changeset
   957
lemma neq_commute: "a \<noteq> b \<longleftrightarrow> b \<noteq> a" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   958
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   959
lemma conj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   960
  shows conj_commute: "(P&Q) = (Q&P)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   961
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   962
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   963
19174
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
   964
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
   965
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   966
lemma disj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   967
  shows disj_commute: "(P|Q) = (Q|P)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   968
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   969
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   970
19174
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
   971
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
df9de25e87b3 moved the "use" directive
paulson
parents: 19162
diff changeset
   972
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   973
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   974
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   975
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   976
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   977
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   978
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   979
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   980
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   981
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   982
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   983
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
   984
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   985
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   986
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   987
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   988
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   989
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   990
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   991
  by iprover
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
   992
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
   993
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   994
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   995
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   996
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   997
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   998
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   999
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1000
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1001
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1002
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1003
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1004
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1005
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1006
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1007
  -- {* cases boil down to the same thing. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1008
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1009
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1010
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1011
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1012
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1013
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
  1014
lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1015
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
  1016
declare All_def [no_atp]
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
  1017
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1018
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
  1019
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
  1020
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1021
text {*
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1022
  \medskip The @{text "&"} congruence rule: not included by default!
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1023
  May slow rewrite proofs down by as much as 50\% *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1024
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1025
lemma conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1026
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1027
  by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1028
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1029
lemma rev_conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1030
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1031
  by iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1032
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1033
text {* The @{text "|"} congruence rule: not included by default! *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1034
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1035
lemma disj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1036
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1037
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1038
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1039
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1040
text {* \medskip if-then-else rules *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1041
32068
98acc234d683 tuned code annotations
haftmann
parents: 31998
diff changeset
  1042
lemma if_True [code]: "(if True then x else y) = x"
38525
324219de6ee3 qualified constants Let and If
haftmann
parents: 37877
diff changeset
  1043
  by (unfold If_def) blast
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1044
32068
98acc234d683 tuned code annotations
haftmann
parents: 31998
diff changeset
  1045
lemma if_False [code]: "(if False then x else y) = y"
38525
324219de6ee3 qualified constants Let and If
haftmann
parents: 37877
diff changeset
  1046
  by (unfold If_def) blast
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1047
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1048
lemma if_P: "P ==> (if P then x else y) = x"
38525
324219de6ee3 qualified constants Let and If
haftmann
parents: 37877
diff changeset
  1049
  by (unfold If_def) blast
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1050
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1051
lemma if_not_P: "~P ==> (if P then x else y) = y"
38525
324219de6ee3 qualified constants Let and If
haftmann
parents: 37877
diff changeset
  1052
  by (unfold If_def) blast
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1053
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1054
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
  1055
  apply (rule case_split [of Q])
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1056
   apply (simplesubst if_P)
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1057
    prefer 3 apply (simplesubst if_not_P, blast+)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1058
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1059
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1060
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
  1061
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1062
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35808
diff changeset
  1063
lemmas if_splits [no_atp] = split_if split_if_asm
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1064
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1065
lemma if_cancel: "(if c then x else x) = x"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1066
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1067
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1068
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
  1069
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1070
41792
ff3cb0c418b7 renamed "nitpick\_def" to "nitpick_unfold" to reflect its new semantics
blanchet
parents: 41636
diff changeset
  1071
lemma if_bool_eq_conj:
ff3cb0c418b7 renamed "nitpick\_def" to "nitpick_unfold" to reflect its new semantics
blanchet
parents: 41636
diff changeset
  1072
"(if P then Q else R) = ((P-->Q) & (~P-->R))"
19796
d86e7b1fc472 quoted "if";
wenzelm
parents: 19656
diff changeset
  1073
  -- {* 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
  1074
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1075
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1076
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
19796
d86e7b1fc472 quoted "if";
wenzelm
parents: 19656
diff changeset
  1077
  -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
59504
8c6747dba731 New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents: 59028
diff changeset
  1078
  by (simplesubst split_if) blast
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1079
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1080
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17459
diff changeset
  1081
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1082
15423
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1083
text {* \medskip let rules for simproc *}
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1084
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1085
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
  1086
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1087
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1088
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1089
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1090
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1091
text {*
16999
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1092
  The following copy of the implication operator is useful for
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1093
  fine-tuning congruence rules.  It instructs the simplifier to simplify
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1094
  its premise.
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1095
*}
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1096
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
  1097
definition simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1) where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37442
diff changeset
  1098
  "simp_implies \<equiv> op ==>"
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1099
18457
356a9f711899 structure ProjectRule;
wenzelm
parents: 17992
diff changeset
  1100
lemma simp_impliesI:
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1101
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1102
  shows "PROP P =simp=> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1103
  apply (unfold simp_implies_def)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1104
  apply (rule PQ)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1105
  apply assumption
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1106
  done
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1107
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1108
lemma simp_impliesE:
25388
5cd130251825 tuned specifications of 'notation';
wenzelm
parents: 25297
diff changeset
  1109
  assumes PQ: "PROP P =simp=> PROP Q"
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1110
  and P: "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1111
  and QR: "PROP Q \<Longrightarrow> PROP R"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1112
  shows "PROP R"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1113
  apply (rule QR)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1114
  apply (rule PQ [unfolded simp_implies_def])
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1115
  apply (rule P)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1116
  done
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1117
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1118
lemma simp_implies_cong:
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1119
  assumes PP' :"PROP P == PROP P'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1120
  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
  1121
  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
  1122
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
  1123
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1124
  and P': "PROP P'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1125
  from PP' [symmetric] and P' have "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1126
    by (rule equal_elim_rule1)
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1127
  then have "PROP Q" by (rule PQ)
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1128
  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
  1129
next
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1130
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1131
  and P: "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1132
  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
  1133
  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
  1134
  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
  1135
    by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1136
qed
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1137
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1138
lemma uncurry:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1139
  assumes "P \<longrightarrow> Q \<longrightarrow> R"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1140
  shows "P \<and> Q \<longrightarrow> R"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1141
  using assms by blast
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1142
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1143
lemma iff_allI:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1144
  assumes "\<And>x. P x = Q x"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1145
  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1146
  using assms by blast
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1147
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1148
lemma iff_exI:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1149
  assumes "\<And>x. P x = Q x"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1150
  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1151
  using assms by blast
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1152
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1153
lemma all_comm:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1154
  "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1155
  by blast
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1156
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1157
lemma ex_comm:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1158
  "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1159
  by blast
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1160
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 48776
diff changeset
  1161
ML_file "Tools/simpdata.ML"
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1162
ML {* open Simpdata *}
42455
6702c984bf5a modernized Quantifier1 simproc setup;
wenzelm
parents: 42453
diff changeset
  1163
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
  1164
setup {*
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
  1165
  map_theory_simpset (put_simpset HOL_basic_ss) #>
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
  1166
  Simplifier.method_setup Splitter.split_modifiers
2ed2eaabe3df modernized setup;
wenzelm
parents: 58775
diff changeset
  1167
*}
42455
6702c984bf5a modernized Quantifier1 simproc setup;
wenzelm
parents: 42453
diff changeset
  1168
42459
38b9f023cc34 misc tuning and simplification;
wenzelm
parents: 42456
diff changeset
  1169
simproc_setup defined_Ex ("EX x. P x") = {* fn _ => Quantifier1.rearrange_ex *}
38b9f023cc34 misc tuning and simplification;
wenzelm
parents: 42456
diff changeset
  1170
simproc_setup defined_All ("ALL x. P x") = {* fn _ => Quantifier1.rearrange_all *}
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1171
24035
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1172
text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1173
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1174
simproc_setup neq ("x = y") = {* fn _ =>
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1175
let
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1176
  val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1177
  fun is_neq eq lhs rhs thm =
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1178
    (case Thm.prop_of thm of
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1179
      _ $ (Not $ (eq' $ l' $ r')) =>
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1180
        Not = HOLogic.Not andalso eq' = eq andalso
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1181
        r' aconv lhs andalso l' aconv rhs
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1182
    | _ => false);
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1183
  fun proc ss ct =
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1184
    (case Thm.term_of ct of
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1185
      eq $ lhs $ rhs =>
43597
b4a093e755db tuned signature;
wenzelm
parents: 43560
diff changeset
  1186
        (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of ss) of
24035
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1187
          SOME thm => SOME (thm RS neq_to_EQ_False)
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1188
        | NONE => NONE)
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1189
     | _ => NONE);
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1190
in proc end;
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1191
*}
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1192
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1193
simproc_setup let_simp ("Let x f") = {*
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1194
let
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1195
  val (f_Let_unfold, x_Let_unfold) =
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59507
diff changeset
  1196
    let val [(_ $ (f $ x) $ _)] = Thm.prems_of @{thm Let_unfold}
59628
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1197
    in apply2 (Thm.cterm_of @{context}) (f, x) end
24035
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1198
  val (f_Let_folded, x_Let_folded) =
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59507
diff changeset
  1199
    let val [(_ $ (f $ x) $ _)] = Thm.prems_of @{thm Let_folded}
59628
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1200
    in apply2 (Thm.cterm_of @{context}) (f, x) end;
24035
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1201
  val g_Let_folded =
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59507
diff changeset
  1202
    let val [(_ $ _ $ (g $ _))] = Thm.prems_of @{thm Let_folded}
59628
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1203
    in Thm.cterm_of @{context} g end;
28741
1b257449f804 simproc for let
haftmann
parents: 28699
diff changeset
  1204
  fun count_loose (Bound i) k = if i >= k then 1 else 0
1b257449f804 simproc for let
haftmann
parents: 28699
diff changeset
  1205
    | count_loose (s $ t) k = count_loose s k + count_loose t k
1b257449f804 simproc for let
haftmann
parents: 28699
diff changeset
  1206
    | count_loose (Abs (_, _, t)) k = count_loose  t (k + 1)
1b257449f804 simproc for let
haftmann
parents: 28699
diff changeset
  1207
    | count_loose _ _ = 0;
1b257449f804 simproc for let
haftmann
parents: 28699
diff changeset
  1208
  fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =
59628
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1209
    (case t of
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1210
      Abs (_, _, t') => count_loose t' 0 <= 1
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1211
    | _ => true);
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1212
in
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1213
  fn _ => fn ctxt => fn ct =>
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1214
    if is_trivial_let (Thm.term_of ct)
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1215
    then SOME @{thm Let_def} (*no or one ocurrence of bound variable*)
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1216
    else
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1217
      let (*Norbert Schirmer's case*)
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1218
        val t = Thm.term_of ct;
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1219
        val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1220
      in
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1221
        Option.map (hd o Variable.export ctxt' ctxt o single)
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1222
          (case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1223
            if is_Free x orelse is_Bound x orelse is_Const x
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1224
            then SOME @{thm Let_def}
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1225
            else
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1226
              let
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1227
                val n = case f of (Abs (x, _, _)) => x | _ => "x";
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1228
                val cx = Thm.cterm_of ctxt x;
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1229
                val xT = Thm.typ_of_cterm cx;
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1230
                val cf = Thm.cterm_of ctxt f;
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1231
                val fx_g = Simplifier.rewrite ctxt (Thm.apply cf cx);
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1232
                val (_ $ _ $ g) = Thm.prop_of fx_g;
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1233
                val g' = abstract_over (x, g);
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1234
                val abs_g'= Abs (n, xT, g');
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1235
              in
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1236
                if g aconv g' then
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1237
                  let
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1238
                    val rl =
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1239
                      cterm_instantiate [(f_Let_unfold, cf), (x_Let_unfold, cx)] @{thm Let_unfold};
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1240
                  in SOME (rl OF [fx_g]) end
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1241
                else if (Envir.beta_eta_contract f) aconv (Envir.beta_eta_contract abs_g')
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1242
                then NONE (*avoid identity conversion*)
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1243
                else
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1244
                  let
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1245
                    val g'x = abs_g' $ x;
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1246
                    val g_g'x = Thm.symmetric (Thm.beta_conversion false (Thm.cterm_of ctxt g'x));
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1247
                    val rl =
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1248
                      @{thm Let_folded} |> cterm_instantiate
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1249
                        [(f_Let_folded, Thm.cterm_of ctxt f),
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1250
                         (x_Let_folded, cx),
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1251
                         (g_Let_folded, Thm.cterm_of ctxt abs_g')];
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1252
                  in SOME (rl OF [Thm.transitive fx_g g_g'x]) end
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1253
              end
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1254
          | _ => NONE)
2b15625b85fc clarified context;
wenzelm
parents: 59621
diff changeset
  1255
      end
28741
1b257449f804 simproc for let
haftmann
parents: 28699
diff changeset
  1256
end *}
24035
74c032aea9ed simplified ResAtpset via NamedThmsFun;
wenzelm
parents: 23948
diff changeset
  1257
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1258
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1259
proof
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23263
diff changeset
  1260
  assume "True \<Longrightarrow> PROP P"
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23263
diff changeset
  1261
  from this [OF TrueI] show "PROP P" .
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1262
next
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1263
  assume "PROP P"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23263
diff changeset
  1264
  then show "PROP P" .
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1265
qed
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1266
59864
c777743294e1 added lemmas
nipkow
parents: 59779
diff changeset
  1267
lemma implies_True_equals: "(PROP P \<Longrightarrow> True) \<equiv> Trueprop True"
c777743294e1 added lemmas
nipkow
parents: 59779
diff changeset
  1268
by default (intro TrueI)
c777743294e1 added lemmas
nipkow
parents: 59779
diff changeset
  1269
c777743294e1 added lemmas
nipkow
parents: 59779
diff changeset
  1270
lemma False_implies_equals: "(False \<Longrightarrow> P) \<equiv> Trueprop True"
c777743294e1 added lemmas
nipkow
parents: 59779
diff changeset
  1271
by default simp_all
c777743294e1 added lemmas
nipkow
parents: 59779
diff changeset
  1272
21151
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1273
lemma ex_simps:
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1274
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1275
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1276
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1277
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1278
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1279
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1280
  -- {* Miniscoping: pushing in existential quantifiers. *}
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1281
  by (iprover | blast)+
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1282
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1283
lemma all_simps:
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1284
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1285
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1286
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1287
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1288
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1289
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1290
  -- {* Miniscoping: pushing in universal quantifiers. *}
25bd46916c12 simplified reasoning tools setup
haftmann
parents: 21112
diff changeset
  1291
  by (iprover | blast)+
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1292
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1293
lemmas [simp] =
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1294
  triv_forall_equality (*prunes params*)
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1295
  True_implies_equals  (*prune asms `True'*)
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1296
  if_True
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1297
  if_False
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1298
  if_cancel
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1299
  if_eq_cancel
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1300
  imp_disjL
20973
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1301
  (*In general it seems wrong to add distributive laws by default: they
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1302
    might cause exponential blow-up.  But imp_disjL has been in for a while
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1303
    and cannot be removed without affecting existing proofs.  Moreover,
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1304
    rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1305
    grounds that it allows simplification of R in the two cases.*)
21671
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1306
  conj_assoc
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1307
  disj_assoc
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1308
  de_Morgan_conj
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1309
  de_Morgan_disj
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1310
  imp_disj1
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1311
  imp_disj2
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1312
  not_imp
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1313
  disj_not1
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1314
  not_all
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1315
  not_ex
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1316
  cases_simp
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1317
  the_eq_trivial
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1318
  the_sym_eq_trivial
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1319
  ex_simps
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1320
  all_simps
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1321
  simp_thms
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1322
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1323
lemmas [cong] = imp_cong simp_implies_cong
f7d652ffef09 removed legacy ML bindings;
wenzelm
parents: 21547
diff changeset
  1324
lemmas [split] = split_if
20973
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1325
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51692
diff changeset
  1326
ML {* val HOL_ss = simpset_of @{context} *}
20973
0b8e436ed071 cleaned up HOL bootstrap
haftmann
parents: 20944
diff changeset
  1327
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1328
text {* Simplifies x assuming c and y assuming ~c *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1329
lemma if_cong:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1330
  assumes "b = c"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1331
      and "c \<Longrightarrow> x = u"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1332
      and "\<not> c \<Longrightarrow> y = v"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1333
  shows "(if b then x else y) = (if c then u else v)"
38525
324219de6ee3 qualified constants Let and If
haftmann
parents: 37877
diff changeset
  1334
  using assms by simp
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1335
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1336
text {* Prevents simplification of x and y:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1337
  faster and allows the execution of functional programs. *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1338
lemma if_weak_cong [cong]:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1339
  assumes "b = c"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1340
  shows "(if b then x else y) = (if c then x else y)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1341
  using assms by (rule arg_cong)
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1342
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1343
text {* Prevents simplification of t: much faster *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1344
lemma let_weak_cong:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1345
  assumes "a = b"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1346
  shows "(let x = a in t x) = (let x = b in t x)"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1347
  using assms by (rule arg_cong)
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1348
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1349
text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1350
lemma eq_cong2:
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1351
  assumes "u = u'"
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff changeset
  1352
  shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
23553
af8ae54238f5 use hologic.ML in basic HOL context;
wenzelm
parents: 23530
diff changeset
  1353
  using assms by simp
20944
34b2c1bb7178 cleanup basic HOL bootstrap
haftmann
parents: 20833
diff