
clasohm@923: (*  Title:      HOL/HOL.thy
wenzelm@11750:     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
wenzelm@11750: *)
clasohm@923: 
wenzelm@58889: section {* The basis of Higher-Order Logic *}
clasohm@923: 
nipkow@15131: theory HOL
haftmann@30929: imports Pure "~~/src/Tools/Code_Generator"
wenzelm@46950: keywords
wenzelm@52432:   "try" "solve_direct" "quickcheck" "print_coercions" "print_claset"
wenzelm@52432:     "print_induct_rules" :: diag and
haftmann@47657:   "quickcheck_params" :: thy_decl
nipkow@15131: begin
wenzelm@2260: 
wenzelm@48891: ML_file "~~/src/Tools/misc_legacy.ML"
wenzelm@48891: ML_file "~~/src/Tools/try.ML"
wenzelm@48891: ML_file "~~/src/Tools/quickcheck.ML"
wenzelm@48891: ML_file "~~/src/Tools/solve_direct.ML"
wenzelm@48891: ML_file "~~/src/Tools/IsaPlanner/zipper.ML"
wenzelm@48891: ML_file "~~/src/Tools/IsaPlanner/isand.ML"
wenzelm@48891: ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML"
wenzelm@48891: ML_file "~~/src/Provers/hypsubst.ML"
wenzelm@48891: ML_file "~~/src/Provers/splitter.ML"
wenzelm@48891: ML_file "~~/src/Provers/classical.ML"
wenzelm@48891: ML_file "~~/src/Provers/blast.ML"
wenzelm@48891: ML_file "~~/src/Provers/clasimp.ML"
wenzelm@48891: ML_file "~~/src/Tools/eqsubst.ML"
wenzelm@48891: ML_file "~~/src/Provers/quantifier1.ML"
wenzelm@48891: ML_file "~~/src/Tools/atomize_elim.ML"
wenzelm@48891: ML_file "~~/src/Tools/cong_tac.ML"
wenzelm@58826: ML_file "~~/src/Tools/intuitionistic.ML" setup \<open>Intuitionistic.method_setup @{binding iprover}\<close>
wenzelm@48891: ML_file "~~/src/Tools/project_rule.ML"
wenzelm@48891: ML_file "~~/src/Tools/subtyping.ML"
wenzelm@48891: ML_file "~~/src/Tools/case_product.ML"
wenzelm@48891: 
wenzelm@30165: 
wenzelm@58659: ML \<open>Plugin_Name.declare_setup @{binding extraction}\<close>
wenzelm@58659: 
wenzelm@58659: ML \<open>
wenzelm@58659:   Plugin_Name.declare_setup @{binding quickcheck_random};
wenzelm@58659:   Plugin_Name.declare_setup @{binding quickcheck_exhaustive};
wenzelm@58659:   Plugin_Name.declare_setup @{binding quickcheck_bounded_forall};
wenzelm@58659:   Plugin_Name.declare_setup @{binding quickcheck_full_exhaustive};
wenzelm@58659:   Plugin_Name.declare_setup @{binding quickcheck_narrowing};
wenzelm@58659: \<close>
wenzelm@58659: ML \<open>
wenzelm@58659:   Plugin_Name.define_setup @{binding quickcheck}
wenzelm@58659:    [@{plugin quickcheck_exhaustive},
wenzelm@58659:     @{plugin quickcheck_random},
wenzelm@58659:     @{plugin quickcheck_bounded_forall},
wenzelm@58659:     @{plugin quickcheck_full_exhaustive},
wenzelm@58659:     @{plugin quickcheck_narrowing}]
wenzelm@58659: \<close>
wenzelm@58659: 
wenzelm@58659: 
wenzelm@11750: subsection {* Primitive logic *}
wenzelm@11750: 
wenzelm@11750: subsubsection {* Core syntax *}
wenzelm@2260: 
wenzelm@56941: setup {* Axclass.class_axiomatization (@{binding type}, []) *}
wenzelm@36452: default_sort type
wenzelm@35625: setup {* Object_Logic.add_base_sort @{sort type} *}
haftmann@25460: 
wenzelm@55383: axiomatization where fun_arity: "OFCLASS('a \<Rightarrow> 'b, type_class)"
wenzelm@55383: instance "fun" :: (type, type) type by (rule fun_arity)
wenzelm@55383: 
wenzelm@55383: axiomatization where itself_arity: "OFCLASS('a itself, type_class)"
wenzelm@55383: instance itself :: (type) type by (rule itself_arity)
haftmann@25460: 
wenzelm@7357: typedecl bool
clasohm@923: 
wenzelm@11750: judgment
wenzelm@11750:   Trueprop      :: "bool => prop"                   ("(_)" 5)
clasohm@923: 
wenzelm@46973: axiomatization
wenzelm@46973:   implies       :: "[bool, bool] => bool"           (infixr "-->" 25)  and
wenzelm@46973:   eq            :: "['a, 'a] => bool"               (infixl "=" 50)  and
wenzelm@46973:   The           :: "('a => bool) => 'a"
wenzelm@46973: 
wenzelm@11750: consts
wenzelm@7357:   True          :: bool
wenzelm@7357:   False         :: bool
haftmann@38547:   Not           :: "bool => bool"                   ("~ _" [40] 40)
haftmann@38795: 
haftmann@38795:   conj          :: "[bool, bool] => bool"           (infixr "&" 35)
haftmann@38795:   disj          :: "[bool, bool] => bool"           (infixr "|" 30)
haftmann@38555: 
wenzelm@7357:   All           :: "('a => bool) => bool"           (binder "ALL " 10)
wenzelm@7357:   Ex            :: "('a => bool) => bool"           (binder "EX " 10)
wenzelm@7357:   Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
clasohm@923: 
wenzelm@19656: 
wenzelm@11750: subsubsection {* Additional concrete syntax *}
wenzelm@2260: 
wenzelm@21210: notation (output)
haftmann@38864:   eq  (infix "=" 50)
wenzelm@19656: 
wenzelm@19656: abbreviation
wenzelm@21404:   not_equal :: "['a, 'a] => bool"  (infixl "~=" 50) where
wenzelm@19656:   "x ~= y == ~ (x = y)"
wenzelm@19656: 
wenzelm@21210: notation (output)
wenzelm@19656:   not_equal  (infix "~=" 50)
wenzelm@19656: 
wenzelm@21210: notation (xsymbols)
wenzelm@21404:   Not  ("\<not> _" [40] 40) and
haftmann@38864:   conj  (infixr "\<and>" 35) and
haftmann@38864:   disj  (infixr "\<or>" 30) and
haftmann@38864:   implies  (infixr "\<longrightarrow>" 25) and
nipkow@50360:   not_equal  (infixl "\<noteq>" 50)
nipkow@50360: 
nipkow@50360: notation (xsymbols output)
wenzelm@19656:   not_equal  (infix "\<noteq>" 50)
wenzelm@19656: 
wenzelm@21210: notation (HTML output)
wenzelm@21404:   Not  ("\<not> _" [40] 40) and
haftmann@38864:   conj  (infixr "\<and>" 35) and
haftmann@38864:   disj  (infixr "\<or>" 30) and
wenzelm@19656:   not_equal  (infix "\<noteq>" 50)
wenzelm@19656: 
wenzelm@19656: abbreviation (iff)
wenzelm@21404:   iff :: "[bool, bool] => bool"  (infixr "<->" 25) where
wenzelm@19656:   "A <-> B == A = B"
wenzelm@19656: 
wenzelm@21210: notation (xsymbols)
wenzelm@19656:   iff  (infixr "\<longleftrightarrow>" 25)
wenzelm@19656: 
wenzelm@46973: syntax "_The" :: "[pttrn, bool] => 'a"  ("(3THE _./ _)" [0, 10] 10)
wenzelm@46973: translations "THE x. P" == "CONST The (%x. P)"
wenzelm@46125: print_translation {*
wenzelm@52143:   [(@{const_syntax The}, fn _ => fn [Abs abs] =>
wenzelm@46125:       let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
wenzelm@46125:       in Syntax.const @{syntax_const "_The"} $ x $ t end)]
wenzelm@46125: *}  -- {* To avoid eta-contraction of body *}
clasohm@923: 
wenzelm@46125: nonterminal letbinds and letbind
clasohm@923: syntax
wenzelm@7357:   "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
wenzelm@7357:   ""            :: "letbind => letbinds"                 ("_")
wenzelm@7357:   "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
huffman@36363:   "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" [0, 10] 10)
clasohm@923: 
wenzelm@46125: nonterminal case_syn and cases_syn
wenzelm@46125: syntax
wenzelm@46125:   "_case_syntax" :: "['a, cases_syn] => 'b"  ("(case _ of/ _)" 10)
wenzelm@46125:   "_case1" :: "['a, 'b] => case_syn"  ("(2_ =>/ _)" 10)
wenzelm@46125:   "" :: "case_syn => cases_syn"  ("_")
wenzelm@46125:   "_case2" :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
wenzelm@42057: syntax (xsymbols)
wenzelm@46125:   "_case1" :: "['a, 'b] => case_syn"  ("(2_ \<Rightarrow>/ _)" 10)
nipkow@13763: 
wenzelm@21524: notation (xsymbols)
wenzelm@21524:   All  (binder "\<forall>" 10) and
wenzelm@21524:   Ex  (binder "\<exists>" 10) and
wenzelm@21524:   Ex1  (binder "\<exists>!" 10)
wenzelm@2372: 
wenzelm@21524: notation (HTML output)
wenzelm@21524:   All  (binder "\<forall>" 10) and
wenzelm@21524:   Ex  (binder "\<exists>" 10) and
wenzelm@21524:   Ex1  (binder "\<exists>!" 10)
wenzelm@6340: 
wenzelm@21524: notation (HOL)
wenzelm@21524:   All  (binder "! " 10) and
wenzelm@21524:   Ex  (binder "? " 10) and
wenzelm@21524:   Ex1  (binder "?! " 10)
wenzelm@7238: 
wenzelm@7238: 
wenzelm@11750: subsubsection {* Axioms and basic definitions *}
wenzelm@2260: 
wenzelm@46973: axiomatization where
wenzelm@46973:   refl: "t = (t::'a)" and
wenzelm@46973:   subst: "s = t \<Longrightarrow> P s \<Longrightarrow> P t" and
wenzelm@46973:   ext: "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
paulson@15380:     -- {*Extensionality is built into the meta-logic, and this rule expresses
paulson@15380:          a related property.  It is an eta-expanded version of the traditional
wenzelm@46973:          rule, and similar to the ABS rule of HOL*} and
paulson@6289: 
wenzelm@11432:   the_eq_trivial: "(THE x. x = a) = (a::'a)"
clasohm@923: 
wenzelm@46973: axiomatization where
wenzelm@46973:   impI: "(P ==> Q) ==> P-->Q" and
wenzelm@46973:   mp: "[| P-->Q;  P |] ==> Q" and
paulson@15380: 
wenzelm@46973:   iff: "(P-->Q) --> (Q-->P) --> (P=Q)" and
wenzelm@46973:   True_or_False: "(P=True) | (P=False)"
paulson@15380: 
clasohm@923: defs
wenzelm@7357:   True_def:     "True      == ((%x::bool. x) = (%x. x))"
wenzelm@7357:   All_def:      "All(P)    == (P = (%x. True))"
paulson@11451:   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
wenzelm@7357:   False_def:    "False     == (!P. P)"
wenzelm@7357:   not_def:      "~ P       == P-->False"
wenzelm@7357:   and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
wenzelm@7357:   or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
wenzelm@7357:   Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
clasohm@923: 
wenzelm@46973: definition If :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" [0, 0, 10] 10)
wenzelm@46973:   where "If P x y \<equiv> (THE z::'a. (P=True --> z=x) & (P=False --> z=y))"
clasohm@923: 
wenzelm@46973: definition Let :: "'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"
wenzelm@46973:   where "Let s f \<equiv> f s"
haftmann@38525: 
haftmann@38525: translations
haftmann@38525:   "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
haftmann@38525:   "let x = a in e"        == "CONST Let a (%x. e)"
haftmann@38525: 
wenzelm@46973: axiomatization undefined :: 'a
haftmann@22481: 
wenzelm@46973: class default = fixes default :: 'a
wenzelm@4868: 
wenzelm@11750: 
haftmann@20944: subsection {* Fundamental rules *}
haftmann@20944: 
haftmann@20973: subsubsection {* Equality *}
haftmann@20944: 
wenzelm@18457: lemma sym: "s = t ==> t = s"
wenzelm@18457:   by (erule subst) (rule refl)
paulson@15411: 
wenzelm@18457: lemma ssubst: "t = s ==> P s ==> P t"
wenzelm@18457:   by (drule sym) (erule subst)
paulson@15411: 
paulson@15411: lemma trans: "[| r=s; s=t |] ==> r=t"
wenzelm@18457:   by (erule subst)
paulson@15411: 
wenzelm@40715: lemma trans_sym [Pure.elim?]: "r = s ==> t = s ==> r = t"
wenzelm@40715:   by (rule trans [OF _ sym])
wenzelm@40715: 
wenzelm@58826: lemma meta_eq_to_obj_eq:
haftmann@20944:   assumes meq: "A == B"
haftmann@20944:   shows "A = B"
haftmann@20944:   by (unfold meq) (rule refl)
paulson@15411: 
wenzelm@21502: text {* Useful with @{text erule} for proving equalities from known equalities. *}
haftmann@20944:      (* a = b
paulson@15411:         |   |
paulson@15411:         c = d   *)
paulson@15411: lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
paulson@15411: apply (rule trans)
paulson@15411: apply (rule trans)
paulson@15411: apply (rule sym)
paulson@15411: apply assumption+
paulson@15411: done
paulson@15411: 
nipkow@15524: text {* For calculational reasoning: *}
nipkow@15524: 
nipkow@15524: lemma forw_subst: "a = b ==> P b ==> P a"
nipkow@15524:   by (rule ssubst)
nipkow@15524: 
nipkow@15524: lemma back_subst: "P a ==> a = b ==> P b"
nipkow@15524:   by (rule subst)
nipkow@15524: 
paulson@15411: 
wenzelm@32733: subsubsection {* Congruence rules for application *}
paulson@15411: 
wenzelm@32733: text {* Similar to @{text AP_THM} in Gordon's HOL. *}
paulson@15411: lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
paulson@15411: apply (erule subst)
paulson@15411: apply (rule refl)
paulson@15411: done
paulson@15411: 
wenzelm@32733: text {* Similar to @{text AP_TERM} in Gordon's HOL and FOL's @{text subst_context}. *}
paulson@15411: lemma arg_cong: "x=y ==> f(x)=f(y)"
paulson@15411: apply (erule subst)
paulson@15411: apply (rule refl)
paulson@15411: done
paulson@15411: 
paulson@15655: lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
paulson@15655: apply (erule ssubst)+
paulson@15655: apply (rule refl)
paulson@15655: done
paulson@15655: 
wenzelm@32733: lemma cong: "[| f = g; (x::'a) = y |] ==> f x = g y"
paulson@15411: apply (erule subst)+
paulson@15411: apply (rule refl)
paulson@15411: done
paulson@15411: 
wenzelm@58956: ML {* fun cong_tac ctxt = Cong_Tac.cong_tac ctxt @{thm cong} *}
paulson@15411: 
wenzelm@32733: 
wenzelm@32733: subsubsection {* Equality of booleans -- iff *}
paulson@15411: 
wenzelm@21504: lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
wenzelm@21504:   by (iprover intro: iff [THEN mp, THEN mp] impI assms)
paulson@15411: 
paulson@15411: lemma iffD2: "[| P=Q; Q |] ==> P"
wenzelm@18457:   by (erule ssubst)
paulson@15411: 
paulson@15411: lemma rev_iffD2: "[| Q; P=Q |] ==> P"
wenzelm@18457:   by (erule iffD2)
paulson@15411: 
wenzelm@21504: lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
wenzelm@21504:   by (drule sym) (rule iffD2)
wenzelm@21504: 
wenzelm@21504: lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
wenzelm@21504:   by (drule sym) (rule rev_iffD2)
paulson@15411: 
paulson@15411: lemma iffE:
paulson@15411:   assumes major: "P=Q"
wenzelm@21504:     and minor: "[| P --> Q; Q --> P |] ==> R"
wenzelm@18457:   shows R
wenzelm@18457:   by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
paulson@15411: 
paulson@15411: 
haftmann@20944: subsubsection {*True*}
paulson@15411: 
paulson@15411: lemma TrueI: "True"
wenzelm@21504:   unfolding True_def by (rule refl)
paulson@15411: 
wenzelm@21504: lemma eqTrueI: "P ==> P = True"
wenzelm@18457:   by (iprover intro: iffI TrueI)
paulson@15411: 
wenzelm@21504: lemma eqTrueE: "P = True ==> P"
wenzelm@21504:   by (erule iffD2) (rule TrueI)
paulson@15411: 
paulson@15411: 
haftmann@20944: subsubsection {*Universal quantifier*}
paulson@15411: 
wenzelm@21504: lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
wenzelm@21504:   unfolding All_def by (iprover intro: ext eqTrueI assms)
paulson@15411: 
paulson@15411: lemma spec: "ALL x::'a. P(x) ==> P(x)"
paulson@15411: apply (unfold All_def)
paulson@15411: apply (rule eqTrueE)
paulson@15411: apply (erule fun_cong)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma allE:
paulson@15411:   assumes major: "ALL x. P(x)"
wenzelm@21504:     and minor: "P(x) ==> R"
wenzelm@21504:   shows R
wenzelm@21504:   by (iprover intro: minor major [THEN spec])
paulson@15411: 
paulson@15411: lemma all_dupE:
paulson@15411:   assumes major: "ALL x. P(x)"
wenzelm@21504:     and minor: "[| P(x); ALL x. P(x) |] ==> R"
wenzelm@21504:   shows R
wenzelm@21504:   by (iprover intro: minor major major [THEN spec])
paulson@15411: 
paulson@15411: 
wenzelm@21504: subsubsection {* False *}
wenzelm@21504: 
wenzelm@21504: text {*
wenzelm@21504:   Depends upon @{text spec}; it is impossible to do propositional
wenzelm@21504:   logic before quantifiers!
wenzelm@21504: *}
paulson@15411: 
paulson@15411: lemma FalseE: "False ==> P"
wenzelm@21504:   apply (unfold False_def)
wenzelm@21504:   apply (erule spec)
wenzelm@21504:   done
paulson@15411: 
wenzelm@21504: lemma False_neq_True: "False = True ==> P"
wenzelm@21504:   by (erule eqTrueE [THEN FalseE])
paulson@15411: 
paulson@15411: 
wenzelm@21504: subsubsection {* Negation *}
paulson@15411: 
paulson@15411: lemma notI:
wenzelm@21504:   assumes "P ==> False"
paulson@15411:   shows "~P"
wenzelm@21504:   apply (unfold not_def)
wenzelm@21504:   apply (iprover intro: impI assms)
wenzelm@21504:   done
paulson@15411: 
paulson@15411: lemma False_not_True: "False ~= True"
wenzelm@21504:   apply (rule notI)
wenzelm@21504:   apply (erule False_neq_True)
wenzelm@21504:   done
paulson@15411: 
paulson@15411: lemma True_not_False: "True ~= False"
wenzelm@21504:   apply (rule notI)
wenzelm@21504:   apply (drule sym)
wenzelm@21504:   apply (erule False_neq_True)
wenzelm@21504:   done
paulson@15411: 
paulson@15411: lemma notE: "[| ~P;  P |] ==> R"
wenzelm@21504:   apply (unfold not_def)
wenzelm@21504:   apply (erule mp [THEN FalseE])
wenzelm@21504:   apply assumption
wenzelm@21504:   done
paulson@15411: 
wenzelm@21504: lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
wenzelm@21504:   by (erule notE [THEN notI]) (erule meta_mp)
paulson@15411: 
paulson@15411: 
haftmann@20944: subsubsection {*Implication*}
paulson@15411: 
paulson@15411: lemma impE:
paulson@15411:   assumes "P-->Q" "P" "Q ==> R"
paulson@15411:   shows "R"
wenzelm@23553: by (iprover intro: assms mp)
paulson@15411: 
paulson@15411: (* Reduces Q to P-->Q, allowing substitution in P. *)
paulson@15411: lemma rev_mp: "[| P;  P --> Q |] ==> Q"
nipkow@17589: by (iprover intro: mp)
paulson@15411: 
paulson@15411: lemma contrapos_nn:
paulson@15411:   assumes major: "~Q"
paulson@15411:       and minor: "P==>Q"
paulson@15411:   shows "~P"
nipkow@17589: by (iprover intro: notI minor major [THEN notE])
paulson@15411: 
paulson@15411: (*not used at all, but we already have the other 3 combinations *)
paulson@15411: lemma contrapos_pn:
paulson@15411:   assumes major: "Q"
paulson@15411:       and minor: "P ==> ~Q"
paulson@15411:   shows "~P"
nipkow@17589: by (iprover intro: notI minor major notE)
paulson@15411: 
paulson@15411: lemma not_sym: "t ~= s ==> s ~= t"
haftmann@21250:   by (erule contrapos_nn) (erule sym)
haftmann@21250: 
haftmann@21250: lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
haftmann@21250:   by (erule subst, erule ssubst, assumption)
paulson@15411: 
paulson@15411: 
haftmann@20944: subsubsection {*Existential quantifier*}
paulson@15411: 
paulson@15411: lemma exI: "P x ==> EX x::'a. P x"
paulson@15411: apply (unfold Ex_def)
nipkow@17589: apply (iprover intro: allI allE impI mp)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma exE:
paulson@15411:   assumes major: "EX x::'a. P(x)"
paulson@15411:       and minor: "!!x. P(x) ==> Q"
paulson@15411:   shows "Q"
paulson@15411: apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
nipkow@17589: apply (iprover intro: impI [THEN allI] minor)
paulson@15411: done
paulson@15411: 
paulson@15411: 
haftmann@20944: subsubsection {*Conjunction*}
paulson@15411: 
paulson@15411: lemma conjI: "[| P; Q |] ==> P&Q"
paulson@15411: apply (unfold and_def)
nipkow@17589: apply (iprover intro: impI [THEN allI] mp)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma conjunct1: "[| P & Q |] ==> P"
paulson@15411: apply (unfold and_def)
nipkow@17589: apply (iprover intro: impI dest: spec mp)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma conjunct2: "[| P & Q |] ==> Q"
paulson@15411: apply (unfold and_def)
nipkow@17589: apply (iprover intro: impI dest: spec mp)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma conjE:
paulson@15411:   assumes major: "P&Q"
paulson@15411:       and minor: "[| P; Q |] ==> R"
paulson@15411:   shows "R"
paulson@15411: apply (rule minor)
paulson@15411: apply (rule major [THEN conjunct1])
paulson@15411: apply (rule major [THEN conjunct2])
paulson@15411: done
paulson@15411: 
paulson@15411: lemma context_conjI:
wenzelm@23553:   assumes "P" "P ==> Q" shows "P & Q"
wenzelm@23553: by (iprover intro: conjI assms)
paulson@15411: 
paulson@15411: 
haftmann@20944: subsubsection {*Disjunction*}
paulson@15411: 
paulson@15411: lemma disjI1: "P ==> P|Q"
paulson@15411: apply (unfold or_def)
nipkow@17589: apply (iprover intro: allI impI mp)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma disjI2: "Q ==> P|Q"
paulson@15411: apply (unfold or_def)
nipkow@17589: apply (iprover intro: allI impI mp)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma disjE:
paulson@15411:   assumes major: "P|Q"
paulson@15411:       and minorP: "P ==> R"
paulson@15411:       and minorQ: "Q ==> R"
paulson@15411:   shows "R"
nipkow@17589: by (iprover intro: minorP minorQ impI
paulson@15411:                  major [unfolded or_def, THEN spec, THEN mp, THEN mp])
paulson@15411: 
paulson@15411: 
haftmann@20944: subsubsection {*Classical logic*}
paulson@15411: 
paulson@15411: lemma classical:
paulson@15411:   assumes prem: "~P ==> P"
paulson@15411:   shows "P"
paulson@15411: apply (rule True_or_False [THEN disjE, THEN eqTrueE])
paulson@15411: apply assumption
paulson@15411: apply (rule notI [THEN prem, THEN eqTrueI])
paulson@15411: apply (erule subst)
paulson@15411: apply assumption
paulson@15411: done
paulson@15411: 
wenzelm@45607: lemmas ccontr = FalseE [THEN classical]
paulson@15411: 
paulson@15411: (*notE with premises exchanged; it discharges ~R so that it can be used to
paulson@15411:   make elimination rules*)
paulson@15411: lemma rev_notE:
paulson@15411:   assumes premp: "P"
paulson@15411:       and premnot: "~R ==> ~P"
paulson@15411:   shows "R"
paulson@15411: apply (rule ccontr)
paulson@15411: apply (erule notE [OF premnot premp])
paulson@15411: done
paulson@15411: 
paulson@15411: (*Double negation law*)
paulson@15411: lemma notnotD: "~~P ==> P"
paulson@15411: apply (rule classical)
paulson@15411: apply (erule notE)
paulson@15411: apply assumption
paulson@15411: done
paulson@15411: 
paulson@15411: lemma contrapos_pp:
paulson@15411:   assumes p1: "Q"
paulson@15411:       and p2: "~P ==> ~Q"
paulson@15411:   shows "P"
nipkow@17589: by (iprover intro: classical p1 p2 notE)
paulson@15411: 
paulson@15411: 
haftmann@20944: subsubsection {*Unique existence*}
paulson@15411: 
paulson@15411: lemma ex1I:
wenzelm@23553:   assumes "P a" "!!x. P(x) ==> x=a"
paulson@15411:   shows "EX! x. P(x)"
wenzelm@23553: by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
paulson@15411: 
paulson@15411: text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
paulson@15411: lemma ex_ex1I:
paulson@15411:   assumes ex_prem: "EX x. P(x)"
paulson@15411:       and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
paulson@15411:   shows "EX! x. P(x)"
nipkow@17589: by (iprover intro: ex_prem [THEN exE] ex1I eq)
paulson@15411: 
paulson@15411: lemma ex1E:
paulson@15411:   assumes major: "EX! x. P(x)"
paulson@15411:       and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
paulson@15411:   shows "R"
paulson@15411: apply (rule major [unfolded Ex1_def, THEN exE])
paulson@15411: apply (erule conjE)
nipkow@17589: apply (iprover intro: minor)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
paulson@15411: apply (erule ex1E)
paulson@15411: apply (rule exI)
paulson@15411: apply assumption
paulson@15411: done
paulson@15411: 
paulson@15411: 
haftmann@20944: subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
paulson@15411: 
paulson@15411: lemma disjCI:
paulson@15411:   assumes "~Q ==> P" shows "P|Q"
paulson@15411: apply (rule classical)
wenzelm@23553: apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma excluded_middle: "~P | P"
nipkow@17589: by (iprover intro: disjCI)
paulson@15411: 
haftmann@20944: text {*
haftmann@20944:   case distinction as a natural deduction rule.
haftmann@20944:   Note that @{term "~P"} is the second case, not the first
haftmann@20944: *}
wenzelm@27126: lemma case_split [case_names True False]:
paulson@15411:   assumes prem1: "P ==> Q"
paulson@15411:       and prem2: "~P ==> Q"
paulson@15411:   shows "Q"
paulson@15411: apply (rule excluded_middle [THEN disjE])
paulson@15411: apply (erule prem2)
paulson@15411: apply (erule prem1)
paulson@15411: done
wenzelm@27126: 
paulson@15411: (*Classical implies (-->) elimination. *)
paulson@15411: lemma impCE:
paulson@15411:   assumes major: "P-->Q"
paulson@15411:       and minor: "~P ==> R" "Q ==> R"
paulson@15411:   shows "R"
paulson@15411: apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589: apply (iprover intro: minor major [THEN mp])+
paulson@15411: done
paulson@15411: 
paulson@15411: (*This version of --> elimination works on Q before P.  It works best for
paulson@15411:   those cases in which P holds "almost everywhere".  Can't install as
paulson@15411:   default: would break old proofs.*)
paulson@15411: lemma impCE':
paulson@15411:   assumes major: "P-->Q"
paulson@15411:       and minor: "Q ==> R" "~P ==> R"
paulson@15411:   shows "R"
paulson@15411: apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589: apply (iprover intro: minor major [THEN mp])+
paulson@15411: done
paulson@15411: 
paulson@15411: (*Classical <-> elimination. *)
paulson@15411: lemma iffCE:
paulson@15411:   assumes major: "P=Q"
paulson@15411:       and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
paulson@15411:   shows "R"
paulson@15411: apply (rule major [THEN iffE])
nipkow@17589: apply (iprover intro: minor elim: impCE notE)
paulson@15411: done
paulson@15411: 
paulson@15411: lemma exCI:
paulson@15411:   assumes "ALL x. ~P(x) ==> P(a)"
paulson@15411:   shows "EX x. P(x)"
paulson@15411: apply (rule ccontr)
wenzelm@23553: apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
paulson@15411: done
paulson@15411: 
paulson@15411: 
wenzelm@12386: subsubsection {* Intuitionistic Reasoning *}
wenzelm@12386: 
wenzelm@12386: lemma impE':
wenzelm@12937:   assumes 1: "P --> Q"
wenzelm@12937:     and 2: "Q ==> R"
wenzelm@12937:     and 3: "P --> Q ==> P"
wenzelm@12937:   shows R
wenzelm@12386: proof -
wenzelm@12386:   from 3 and 1 have P .
wenzelm@12386:   with 1 have Q by (rule impE)
wenzelm@12386:   with 2 show R .
wenzelm@12386: qed
wenzelm@12386: 
wenzelm@12386: lemma allE':
wenzelm@12937:   assumes 1: "ALL x. P x"
wenzelm@12937:     and 2: "P x ==> ALL x. P x ==> Q"
wenzelm@12937:   shows Q
wenzelm@12386: proof -
wenzelm@12386:   from 1 have "P x" by (rule spec)
wenzelm@12386:   from this and 1 show Q by (rule 2)
wenzelm@12386: qed
wenzelm@12386: 
wenzelm@12937: lemma notE':
wenzelm@12937:   assumes 1: "~ P"
wenzelm@12937:     and 2: "~ P ==> P"
wenzelm@12937:   shows R
wenzelm@12386: proof -
wenzelm@12386:   from 2 and 1 have P .
wenzelm@12386:   with 1 show R by (rule notE)
wenzelm@12386: qed
wenzelm@12386: 
dixon@22444: lemma TrueE: "True ==> P ==> P" .
dixon@22444: lemma notFalseE: "~ False ==> P ==> P" .
dixon@22444: 
dixon@22467: lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
wenzelm@15801:   and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@15801:   and [Pure.elim 2] = allE notE' impE'
wenzelm@15801:   and [Pure.intro] = exI disjI2 disjI1
wenzelm@12386: 
wenzelm@12386: lemmas [trans] = trans
wenzelm@12386:   and [sym] = sym not_sym
wenzelm@15801:   and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@11750: 
wenzelm@11438: 
wenzelm@11750: subsubsection {* Atomizing meta-level connectives *}
wenzelm@11750: 
haftmann@28513: axiomatization where
haftmann@28513:   eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)
haftmann@28513: 
wenzelm@11750: lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
wenzelm@12003: proof
wenzelm@9488:   assume "!!x. P x"
wenzelm@23389:   then show "ALL x. P x" ..
wenzelm@9488: next
wenzelm@9488:   assume "ALL x. P x"
wenzelm@23553:   then show "!!x. P x" by (rule allE)
wenzelm@9488: qed
wenzelm@9488: 
wenzelm@11750: lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@12003: proof
wenzelm@9488:   assume r: "A ==> B"
wenzelm@10383:   show "A --> B" by (rule impI) (rule r)
wenzelm@9488: next
wenzelm@9488:   assume "A --> B" and A
wenzelm@23553:   then show B by (rule mp)
wenzelm@9488: qed
wenzelm@9488: 
paulson@14749: lemma atomize_not: "(A ==> False) == Trueprop (~A)"
paulson@14749: proof
paulson@14749:   assume r: "A ==> False"
paulson@14749:   show "~A" by (rule notI) (rule r)
paulson@14749: next
paulson@14749:   assume "~A" and A
wenzelm@23553:   then show False by (rule notE)
paulson@14749: qed
paulson@14749: 
haftmann@39566: lemma atomize_eq [atomize, code]: "(x == y) == Trueprop (x = y)"
wenzelm@12003: proof
wenzelm@10432:   assume "x == y"
wenzelm@23553:   show "x = y" by (unfold `x == y`) (rule refl)
wenzelm@10432: next
wenzelm@10432:   assume "x = y"
wenzelm@23553:   then show "x == y" by (rule eq_reflection)
wenzelm@10432: qed
wenzelm@10432: 
wenzelm@28856: lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
wenzelm@12003: proof
wenzelm@28856:   assume conj: "A &&& B"
wenzelm@19121:   show "A & B"
wenzelm@19121:   proof (rule conjI)
wenzelm@19121:     from conj show A by (rule conjunctionD1)
wenzelm@19121:     from conj show B by (rule conjunctionD2)
wenzelm@19121:   qed
wenzelm@11953: next
wenzelm@19121:   assume conj: "A & B"
wenzelm@28856:   show "A &&& B"
wenzelm@19121:   proof -
wenzelm@19121:     from conj show A ..
wenzelm@19121:     from conj show B ..
wenzelm@11953:   qed
wenzelm@11953: qed
wenzelm@11953: 
wenzelm@12386: lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18832:   and [symmetric, defn] = atomize_all atomize_imp atomize_eq
wenzelm@12386: 
wenzelm@11750: 
krauss@26580: subsubsection {* Atomizing elimination rules *}
krauss@26580: 
krauss@26580: lemma atomize_exL[atomize_elim]: "(!!x. P x ==> Q) == ((EX x. P x) ==> Q)"
krauss@26580:   by rule iprover+
krauss@26580: 
krauss@26580: lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
krauss@26580:   by rule iprover+
krauss@26580: 
krauss@26580: lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
krauss@26580:   by rule iprover+
krauss@26580: 
krauss@26580: lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop A" ..
krauss@26580: 
krauss@26580: 
haftmann@20944: subsection {* Package setup *}
haftmann@20944: 
wenzelm@51314: ML_file "Tools/hologic.ML"
wenzelm@51314: 
wenzelm@51314: 
blanchet@35828: subsubsection {* Sledgehammer setup *}
blanchet@35828: 
blanchet@35828: text {*
blanchet@35828: Theorems blacklisted to Sledgehammer. These theorems typically produce clauses
blanchet@35828: that are prolific (match too many equality or membership literals) and relate to
blanchet@35828: seldom-used facts. Some duplicate other rules.
blanchet@35828: *}
blanchet@35828: 
wenzelm@57963: named_theorems no_atp "theorems that should be filtered out by Sledgehammer"
blanchet@35828: 
blanchet@35828: 
wenzelm@11750: subsubsection {* Classical Reasoner setup *}
wenzelm@9529: 
wenzelm@26411: lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
wenzelm@26411:   by (rule classical) iprover
wenzelm@26411: 
wenzelm@26411: lemma swap: "~ P ==> (~ R ==> P) ==> R"
wenzelm@26411:   by (rule classical) iprover
wenzelm@26411: 
haftmann@20944: lemma thin_refl:
haftmann@20944:   "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
haftmann@20944: 
haftmann@21151: ML {*
wenzelm@42799: structure Hypsubst = Hypsubst
wenzelm@42799: (
wenzelm@21218:   val dest_eq = HOLogic.dest_eq
haftmann@21151:   val dest_Trueprop = HOLogic.dest_Trueprop
haftmann@21151:   val dest_imp = HOLogic.dest_imp
wenzelm@26411:   val eq_reflection = @{thm eq_reflection}
wenzelm@26411:   val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
wenzelm@26411:   val imp_intr = @{thm impI}
wenzelm@26411:   val rev_mp = @{thm rev_mp}
wenzelm@26411:   val subst = @{thm subst}
wenzelm@26411:   val sym = @{thm sym}
wenzelm@22129:   val thin_refl = @{thm thin_refl};
wenzelm@42799: );
wenzelm@21671: open Hypsubst;
haftmann@21151: 
wenzelm@42799: structure Classical = Classical
wenzelm@42799: (
wenzelm@26411:   val imp_elim = @{thm imp_elim}
wenzelm@26411:   val not_elim = @{thm notE}
wenzelm@26411:   val swap = @{thm swap}
wenzelm@26411:   val classical = @{thm classical}
haftmann@21151:   val sizef = Drule.size_of_thm
haftmann@21151:   val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
wenzelm@42799: );
haftmann@21151: 
wenzelm@58826: structure Basic_Classical: BASIC_CLASSICAL = Classical;
wenzelm@33308: open Basic_Classical;
wenzelm@43560: *}
wenzelm@22129: 
haftmann@21009: setup {*
wenzelm@35389:   (*prevent substitution on bool*)
wenzelm@58826:   let
wenzelm@58826:     fun non_bool_eq (@{const_name HOL.eq}, Type (_, [T, _])) = T <> @{typ bool}
wenzelm@58826:       | non_bool_eq _ = false;
wenzelm@58826:     fun hyp_subst_tac' ctxt =
wenzelm@58826:       SUBGOAL (fn (goal, i) =>
wenzelm@58826:         if Term.exists_Const non_bool_eq goal
wenzelm@58826:         then Hypsubst.hyp_subst_tac ctxt i
wenzelm@58826:         else no_tac);
wenzelm@58826:   in
wenzelm@58826:     Context_Rules.addSWrapper (fn ctxt => fn tac => hyp_subst_tac' ctxt ORELSE' tac)
wenzelm@58826:   end
haftmann@21009: *}
haftmann@21009: 
haftmann@21009: declare iffI [intro!]
haftmann@21009:   and notI [intro!]
haftmann@21009:   and impI [intro!]
haftmann@21009:   and disjCI [intro!]
haftmann@21009:   and conjI [intro!]
haftmann@21009:   and TrueI [intro!]
haftmann@21009:   and refl [intro!]
haftmann@21009: 
haftmann@21009: declare iffCE [elim!]
haftmann@21009:   and FalseE [elim!]
haftmann@21009:   and impCE [elim!]
haftmann@21009:   and disjE [elim!]
haftmann@21009:   and conjE [elim!]
haftmann@21009: 
haftmann@21009: declare ex_ex1I [intro!]
haftmann@21009:   and allI [intro!]
haftmann@21009:   and exI [intro]
haftmann@21009: 
haftmann@21009: declare exE [elim!]
haftmann@21009:   allE [elim]
haftmann@21009: 
wenzelm@51687: ML {* val HOL_cs = claset_of @{context} *}
mengj@19162: 
wenzelm@20223: lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
wenzelm@20223:   apply (erule swap)
wenzelm@20223:   apply (erule (1) meta_mp)
wenzelm@20223:   done
wenzelm@10383: 
wenzelm@18689: declare ex_ex1I [rule del, intro! 2]
wenzelm@18689:   and ex1I [intro]
wenzelm@18689: 
paulson@41865: declare ext [intro]
paulson@41865: 
wenzelm@12386: lemmas [intro?] = ext
wenzelm@12386:   and [elim?] = ex1_implies_ex
wenzelm@11977: 
haftmann@20944: (*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
haftmann@20973: lemma alt_ex1E [elim!]:
haftmann@20944:   assumes major: "\<exists>!x. P x"
haftmann@20944:       and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
haftmann@20944:   shows R
haftmann@20944: apply (rule ex1E [OF major])
haftmann@20944: apply (rule prem)
wenzelm@59499: apply assumption
wenzelm@59499: apply (rule allI)+
wenzelm@59498: apply (tactic {* eresolve_tac @{context} [Classical.dup_elim NONE @{thm allE}] 1 *})
wenzelm@22129: apply iprover
wenzelm@22129: done
haftmann@20944: 
haftmann@21151: ML {*
wenzelm@42477:   structure Blast = Blast
wenzelm@42477:   (
wenzelm@42477:     structure Classical = Classical
wenzelm@42802:     val Trueprop_const = dest_Const @{const Trueprop}
wenzelm@42477:     val equality_name = @{const_name HOL.eq}
wenzelm@42477:     val not_name = @{const_name Not}
wenzelm@42477:     val notE = @{thm notE}
wenzelm@42477:     val ccontr = @{thm ccontr}
wenzelm@42477:     val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
wenzelm@42477:   );
wenzelm@42477:   val blast_tac = Blast.blast_tac;
haftmann@20944: *}
haftmann@20944: 
haftmann@20944: 
lp15@59504: subsubsection {*THE: definite description operator*}
lp15@59504: 
lp15@59504: lemma the_equality [intro]:
lp15@59504:   assumes "P a"
lp15@59504:       and "!!x. P x ==> x=a"
lp15@59504:   shows "(THE x. P x) = a"
lp15@59504:   by (blast intro: assms trans [OF arg_cong [where f=The] the_eq_trivial])
lp15@59504: 
lp15@59504: lemma theI:
lp15@59504:   assumes "P a" and "!!x. P x ==> x=a"
lp15@59504:   shows "P (THE x. P x)"
lp15@59504: by (iprover intro: assms the_equality [THEN ssubst])
lp15@59504: 
lp15@59504: lemma theI': "EX! x. P x ==> P (THE x. P x)"
lp15@59504:   by (blast intro: theI)
lp15@59504: 
lp15@59504: (*Easier to apply than theI: only one occurrence of P*)
lp15@59504: lemma theI2:
lp15@59504:   assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
lp15@59504:   shows "Q (THE x. P x)"
lp15@59504: by (iprover intro: assms theI)
lp15@59504: 
lp15@59504: lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
lp15@59504: by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
lp15@59504:            elim:allE impE)
lp15@59504: 
lp15@59504: lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
lp15@59504:   by blast
lp15@59504: 
lp15@59504: lemma the_sym_eq_trivial: "(THE y. x=y) = x"
lp15@59504:   by blast
lp15@59504: 
lp15@59504: 
haftmann@20944: subsubsection {* Simplifier *}
wenzelm@12281: 
wenzelm@12281: lemma eta_contract_eq: "(%s. f s) = f" ..
wenzelm@12281: 
wenzelm@12281: lemma simp_thms:
wenzelm@12937:   shows not_not: "(~ ~ P) = P"
nipkow@15354:   and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
wenzelm@12937:   and
berghofe@12436:     "(P ~= Q) = (P = (~Q))"
berghofe@12436:     "(P | ~P) = True"    "(~P | P) = True"
wenzelm@12281:     "(x = x) = True"
haftmann@32068:   and not_True_eq_False [code]: "(\<not> True) = False"
haftmann@32068:   and not_False_eq_True [code]: "(\<not> False) = True"
haftmann@20944:   and
berghofe@12436:     "(~P) ~= P"  "P ~= (~P)"
haftmann@20944:     "(True=P) = P"
haftmann@20944:   and eq_True: "(P = True) = P"
haftmann@20944:   and "(False=P) = (~P)"
haftmann@20944:   and eq_False: "(P = False) = (\<not> P)"
haftmann@20944:   and
wenzelm@12281:     "(True --> P) = P"  "(False --> P) = True"
wenzelm@12281:     "(P --> True) = True"  "(P --> P) = True"
wenzelm@12281:     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
wenzelm@12281:     "(P & True) = P"  "(True & P) = P"
wenzelm@12281:     "(P & False) = False"  "(False & P) = False"
wenzelm@12281:     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
wenzelm@12281:     "(P & ~P) = False"    "(~P & P) = False"
wenzelm@12281:     "(P | True) = True"  "(True | P) = True"
wenzelm@12281:     "(P | False) = P"  "(False | P) = P"
berghofe@12436:     "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
wenzelm@12281:     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
nipkow@31166:   and
wenzelm@12281:     "!!P. (EX x. x=t & P(x)) = P(t)"
wenzelm@12281:     "!!P. (EX x. t=x & P(x)) = P(t)"
wenzelm@12281:     "!!P. (ALL x. x=t --> P(x)) = P(t)"
wenzelm@12937:     "!!P. (ALL x. t=x --> P(x)) = P(t)"
nipkow@17589:   by (blast, blast, blast, blast, blast, iprover+)
wenzelm@13421: 
paulson@14201: lemma disj_absorb: "(A | A) = A"
paulson@14201:   by blast
paulson@14201: 
paulson@14201: lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
paulson@14201:   by blast
paulson@14201: 
paulson@14201: lemma conj_absorb: "(A & A) = A"
paulson@14201:   by blast
paulson@14201: 
paulson@14201: lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
paulson@14201:   by blast
paulson@14201: 
wenzelm@12281: lemma eq_ac:
haftmann@57512:   shows eq_commute: "a = b \<longleftrightarrow> b = a"
haftmann@57512:     and iff_left_commute: "(P \<longleftrightarrow> (Q \<longleftrightarrow> R)) \<longleftrightarrow> (Q \<longleftrightarrow> (P \<longleftrightarrow> R))"
haftmann@57512:     and iff_assoc: "((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longleftrightarrow> (P \<longleftrightarrow> (Q \<longleftrightarrow> R))" by (iprover, blast+)
haftmann@57512: lemma neq_commute: "a \<noteq> b \<longleftrightarrow> b \<noteq> a" by iprover
wenzelm@12281: 
wenzelm@12281: lemma conj_comms:
wenzelm@12937:   shows conj_commute: "(P&Q) = (Q&P)"
nipkow@17589:     and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
nipkow@17589: lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
wenzelm@12281: 
paulson@19174: lemmas conj_ac = conj_commute conj_left_commute conj_assoc
paulson@19174: 
wenzelm@12281: lemma disj_comms:
wenzelm@12937:   shows disj_commute: "(P|Q) = (Q|P)"
nipkow@17589:     and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
nipkow@17589: lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
wenzelm@12281: 
paulson@19174: lemmas disj_ac = disj_commute disj_left_commute disj_assoc
paulson@19174: 
nipkow@17589: lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
nipkow@17589: lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
wenzelm@12281: 
nipkow@17589: lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
nipkow@17589: lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
wenzelm@12281: 
nipkow@17589: lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
nipkow@17589: lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover
nipkow@17589: lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
wenzelm@12281: 
wenzelm@12281: text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
wenzelm@12281: lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
wenzelm@12281: lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
wenzelm@12281: 
wenzelm@12281: lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
wenzelm@12281: lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
wenzelm@12281: 
haftmann@21151: lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
haftmann@21151:   by iprover
haftmann@21151: 
nipkow@17589: lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
wenzelm@12281: lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
wenzelm@12281: lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
wenzelm@12281: lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
wenzelm@12281: lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
wenzelm@12281: lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
wenzelm@12281:   by blast
wenzelm@12281: lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
wenzelm@12281: 
nipkow@17589: lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
wenzelm@12281: 
wenzelm@12281: 
wenzelm@12281: lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
wenzelm@12281:   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
wenzelm@12281:   -- {* cases boil down to the same thing. *}
wenzelm@12281:   by blast
wenzelm@12281: 
wenzelm@12281: lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
wenzelm@12281: lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
nipkow@17589: lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
nipkow@17589: lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
chaieb@23403: lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
wenzelm@12281: 
blanchet@35828: declare All_def [no_atp]
paulson@24286: 
nipkow@17589: lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
nipkow@17589: lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
wenzelm@12281: 
wenzelm@12281: text {*
wenzelm@12281:   \medskip The @{text "&"} congruence rule: not included by default!
wenzelm@12281:   May slow rewrite proofs down by as much as 50\% *}
wenzelm@12281: 
wenzelm@12281: lemma conj_cong:
wenzelm@12281:     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
nipkow@17589:   by iprover
wenzelm@12281: 
wenzelm@12281: lemma rev_conj_cong:
wenzelm@12281:     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
nipkow@17589:   by iprover
wenzelm@12281: 
wenzelm@12281: text {* The @{text "|"} congruence rule: not included by default! *}
wenzelm@12281: 
wenzelm@12281: lemma disj_cong:
wenzelm@12281:     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
wenzelm@12281:   by blast
wenzelm@12281: 
wenzelm@12281: 
wenzelm@12281: text {* \medskip if-then-else rules *}
wenzelm@12281: 
haftmann@32068: lemma if_True [code]: "(if True then x else y) = x"
haftmann@38525:   by (unfold If_def) blast
wenzelm@12281: 
haftmann@32068: lemma if_False [code]: "(if False then x else y) = y"
haftmann@38525:   by (unfold If_def) blast
wenzelm@12281: 
wenzelm@12281: lemma if_P: "P ==> (if P then x else y) = x"
haftmann@38525:   by (unfold If_def) blast
wenzelm@12281: 
wenzelm@12281: lemma if_not_P: "~P ==> (if P then x else y) = y"
haftmann@38525:   by (unfold If_def) blast
wenzelm@12281: 
wenzelm@12281: lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
wenzelm@12281:   apply (rule case_split [of Q])
paulson@15481:    apply (simplesubst if_P)
paulson@15481:     prefer 3 apply (simplesubst if_not_P, blast+)
wenzelm@12281:   done
wenzelm@12281: 
wenzelm@12281: lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
paulson@15481: by (simplesubst split_if, blast)
wenzelm@12281: 
blanchet@35828: lemmas if_splits [no_atp] = split_if split_if_asm
wenzelm@12281: 
wenzelm@12281: lemma if_cancel: "(if c then x else x) = x"
paulson@15481: by (simplesubst split_if, blast)
wenzelm@12281: 
wenzelm@12281: lemma if_eq_cancel: "(if x = y then y else x) = x"
paulson@15481: by (simplesubst split_if, blast)
wenzelm@12281: 
blanchet@41792: lemma if_bool_eq_conj:
blanchet@41792: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
wenzelm@19796:   -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
wenzelm@12281:   by (rule split_if)
wenzelm@12281: 
wenzelm@12281: lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
wenzelm@19796:   -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
lp15@59504:   by (simplesubst split_if) blast
wenzelm@12281: 
nipkow@17589: lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
nipkow@17589: lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
wenzelm@12281: 
schirmer@15423: text {* \medskip let rules for simproc *}
schirmer@15423: 
schirmer@15423: lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
schirmer@15423:   by (unfold Let_def)
schirmer@15423: 
schirmer@15423: lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
schirmer@15423:   by (unfold Let_def)
schirmer@15423: 
berghofe@16633: text {*
ballarin@16999:   The following copy of the implication operator is useful for
ballarin@16999:   fine-tuning congruence rules.  It instructs the simplifier to simplify
ballarin@16999:   its premise.
berghofe@16633: *}
berghofe@16633: 
haftmann@35416: definition simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1) where
haftmann@37767:   "simp_implies \<equiv> op ==>"
berghofe@16633: 
wenzelm@18457: lemma simp_impliesI:
berghofe@16633:   assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
berghofe@16633:   shows "PROP P =simp=> PROP Q"
berghofe@16633:   apply (unfold simp_implies_def)
berghofe@16633:   apply (rule PQ)
berghofe@16633:   apply assumption
berghofe@16633:   done
berghofe@16633: 
berghofe@16633: lemma simp_impliesE:
wenzelm@25388:   assumes PQ: "PROP P =simp=> PROP Q"
berghofe@16633:   and P: "PROP P"
berghofe@16633:   and QR: "PROP Q \<Longrightarrow> PROP R"
berghofe@16633:   shows "PROP R"
berghofe@16633:   apply (rule QR)
berghofe@16633:   apply (rule PQ [unfolded simp_implies_def])
berghofe@16633:   apply (rule P)
berghofe@16633:   done
berghofe@16633: 
berghofe@16633: lemma simp_implies_cong:
berghofe@16633:   assumes PP' :"PROP P == PROP P'"
berghofe@16633:   and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
berghofe@16633:   shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
berghofe@16633: proof (unfold simp_implies_def, rule equal_intr_rule)
berghofe@16633:   assume PQ: "PROP P \<Longrightarrow> PROP Q"
berghofe@16633:   and P': "PROP P'"
berghofe@16633:   from PP' [symmetric] and P' have "PROP P"
berghofe@16633:     by (rule equal_elim_rule1)
wenzelm@23553:   then have "PROP Q" by (rule PQ)
berghofe@16633:   with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
berghofe@16633: next
berghofe@16633:   assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
berghofe@16633:   and P: "PROP P"
berghofe@16633:   from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
wenzelm@23553:   then have "PROP Q'" by (rule P'Q')
berghofe@16633:   with P'QQ' [OF P', symmetric] show "PROP Q"
berghofe@16633:     by (rule equal_elim_rule1)
berghofe@16633: qed
berghofe@16633: 
haftmann@20944: lemma uncurry:
haftmann@20944:   assumes "P \<longrightarrow> Q \<longrightarrow> R"
haftmann@20944:   shows "P \<and> Q \<longrightarrow> R"
wenzelm@23553:   using assms by blast
haftmann@20944: 
haftmann@20944: lemma iff_allI:
haftmann@20944:   assumes "\<And>x. P x = Q x"
haftmann@20944:   shows "(\<forall>x. P x) = (\<forall>x. Q x)"
wenzelm@23553:   using assms by blast
haftmann@20944: 
haftmann@20944: lemma iff_exI:
haftmann@20944:   assumes "\<And>x. P x = Q x"
haftmann@20944:   shows "(\<exists>x. P x) = (\<exists>x. Q x)"
wenzelm@23553:   using assms by blast
haftmann@20944: 
haftmann@20944: lemma all_comm:
haftmann@20944:   "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
haftmann@20944:   by blast
haftmann@20944: 
haftmann@20944: lemma ex_comm:
haftmann@20944:   "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
haftmann@20944:   by blast
haftmann@20944: 
wenzelm@48891: ML_file "Tools/simpdata.ML"
wenzelm@21671: ML {* open Simpdata *}
wenzelm@42455: 
wenzelm@58826: setup {*
wenzelm@58826:   map_theory_simpset (put_simpset HOL_basic_ss) #>
wenzelm@58826:   Simplifier.method_setup Splitter.split_modifiers
wenzelm@58826: *}
wenzelm@42455: 
wenzelm@42459: simproc_setup defined_Ex ("EX x. P x") = {* fn _ => Quantifier1.rearrange_ex *}
wenzelm@42459: simproc_setup defined_All ("ALL x. P x") = {* fn _ => Quantifier1.rearrange_all *}
wenzelm@21671: 
wenzelm@24035: text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
wenzelm@24035: 
wenzelm@24035: simproc_setup neq ("x = y") = {* fn _ =>
wenzelm@24035: let
wenzelm@24035:   val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
wenzelm@24035:   fun is_neq eq lhs rhs thm =
wenzelm@24035:     (case Thm.prop_of thm of
wenzelm@24035:       _ $ (Not $ (eq' $ l' $ r')) =>
wenzelm@24035:         Not = HOLogic.Not andalso eq' = eq andalso
wenzelm@24035:         r' aconv lhs andalso l' aconv rhs
wenzelm@24035:     | _ => false);
wenzelm@24035:   fun proc ss ct =
wenzelm@24035:     (case Thm.term_of ct of
wenzelm@24035:       eq $ lhs $ rhs =>
wenzelm@43597:         (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of ss) of
wenzelm@24035:           SOME thm => SOME (thm RS neq_to_EQ_False)
wenzelm@24035:         | NONE => NONE)
wenzelm@24035:      | _ => NONE);
wenzelm@24035: in proc end;
wenzelm@24035: *}
wenzelm@24035: 
wenzelm@24035: simproc_setup let_simp ("Let x f") = {*
wenzelm@24035: let
wenzelm@24035:   val (f_Let_unfold, x_Let_unfold) =
wenzelm@59582:     let val [(_ $ (f $ x) $ _)] = Thm.prems_of @{thm Let_unfold}
wenzelm@59628:     in apply2 (Thm.cterm_of @{context}) (f, x) end
wenzelm@24035:   val (f_Let_folded, x_Let_folded) =
wenzelm@59582:     let val [(_ $ (f $ x) $ _)] = Thm.prems_of @{thm Let_folded}
wenzelm@59628:     in apply2 (Thm.cterm_of @{context}) (f, x) end;
wenzelm@24035:   val g_Let_folded =
wenzelm@59582:     let val [(_ $ _ $ (g $ _))] = Thm.prems_of @{thm Let_folded}
wenzelm@59628:     in Thm.cterm_of @{context} g end;
haftmann@28741:   fun count_loose (Bound i) k = if i >= k then 1 else 0
haftmann@28741:     | count_loose (s $ t) k = count_loose s k + count_loose t k
haftmann@28741:     | count_loose (Abs (_, _, t)) k = count_loose  t (k + 1)
haftmann@28741:     | count_loose _ _ = 0;
haftmann@28741:   fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =
wenzelm@59628:     (case t of
wenzelm@59628:       Abs (_, _, t') => count_loose t' 0 <= 1
wenzelm@59628:     | _ => true);
wenzelm@59628: in
wenzelm@59628:   fn _ => fn ctxt => fn ct =>
wenzelm@59628:     if is_trivial_let (Thm.term_of ct)
wenzelm@59628:     then SOME @{thm Let_def} (*no or one ocurrence of bound variable*)
wenzelm@59628:     else
wenzelm@59628:       let (*Norbert Schirmer's case*)
wenzelm@59628:         val t = Thm.term_of ct;
wenzelm@59628:         val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
wenzelm@59628:       in
wenzelm@59628:         Option.map (hd o Variable.export ctxt' ctxt o single)
wenzelm@59628:           (case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)
wenzelm@59628:             if is_Free x orelse is_Bound x orelse is_Const x
wenzelm@59628:             then SOME @{thm Let_def}
wenzelm@59628:             else
wenzelm@59628:               let
wenzelm@59628:                 val n = case f of (Abs (x, _, _)) => x | _ => "x";
wenzelm@59628:                 val cx = Thm.cterm_of ctxt x;
wenzelm@59628:                 val xT = Thm.typ_of_cterm cx;
wenzelm@59628:                 val cf = Thm.cterm_of ctxt f;
wenzelm@59628:                 val fx_g = Simplifier.rewrite ctxt (Thm.apply cf cx);
wenzelm@59628:                 val (_ $ _ $ g) = Thm.prop_of fx_g;
wenzelm@59628:                 val g' = abstract_over (x, g);
wenzelm@59628:                 val abs_g'= Abs (n, xT, g');
wenzelm@59628:               in
wenzelm@59628:                 if g aconv g' then
wenzelm@59628:                   let
wenzelm@59628:                     val rl =
wenzelm@59628:                       cterm_instantiate [(f_Let_unfold, cf), (x_Let_unfold, cx)] @{thm Let_unfold};
wenzelm@59628:                   in SOME (rl OF [fx_g]) end
wenzelm@59628:                 else if (Envir.beta_eta_contract f) aconv (Envir.beta_eta_contract abs_g')
wenzelm@59628:                 then NONE (*avoid identity conversion*)
wenzelm@59628:                 else
wenzelm@59628:                   let
wenzelm@59628:                     val g'x = abs_g' $ x;
wenzelm@59628:                     val g_g'x = Thm.symmetric (Thm.beta_conversion false (Thm.cterm_of ctxt g'x));
wenzelm@59628:                     val rl =
wenzelm@59628:                       @{thm Let_folded} |> cterm_instantiate
wenzelm@59628:                         [(f_Let_folded, Thm.cterm_of ctxt f),
wenzelm@59628:                          (x_Let_folded, cx),
wenzelm@59628:                          (g_Let_folded, Thm.cterm_of ctxt abs_g')];
wenzelm@59628:                   in SOME (rl OF [Thm.transitive fx_g g_g'x]) end
wenzelm@59628:               end
wenzelm@59628:           | _ => NONE)
wenzelm@59628:       end
haftmann@28741: end *}
wenzelm@24035: 
haftmann@21151: lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
haftmann@21151: proof
wenzelm@23389:   assume "True \<Longrightarrow> PROP P"
wenzelm@23389:   from this [OF TrueI] show "PROP P" .
haftmann@21151: next
haftmann@21151:   assume "PROP P"
wenzelm@23389:   then show "PROP P" .
haftmann@21151: qed
haftmann@21151: 
nipkow@59864: lemma implies_True_equals: "(PROP P \<Longrightarrow> True) \<equiv> Trueprop True"
nipkow@59864: by default (intro TrueI)
nipkow@59864: 
nipkow@59864: lemma False_implies_equals: "(False \<Longrightarrow> P) \<equiv> Trueprop True"
nipkow@59864: by default simp_all
nipkow@59864: 
haftmann@21151: lemma ex_simps:
haftmann@21151:   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
haftmann@21151:   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
haftmann@21151:   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
haftmann@21151:   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
haftmann@21151:   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
haftmann@21151:   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
haftmann@21151:   -- {* Miniscoping: pushing in existential quantifiers. *}
haftmann@21151:   by (iprover | blast)+
haftmann@21151: 
haftmann@21151: lemma all_simps:
haftmann@21151:   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
haftmann@21151:   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
haftmann@21151:   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
haftmann@21151:   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
haftmann@21151:   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
haftmann@21151:   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
haftmann@21151:   -- {* Miniscoping: pushing in universal quantifiers. *}
haftmann@21151:   by (iprover | blast)+
paulson@15481: 
wenzelm@21671: lemmas [simp] =
wenzelm@21671:   triv_forall_equality (*prunes params*)
wenzelm@21671:   True_implies_equals  (*prune asms `True'*)
wenzelm@21671:   if_True
wenzelm@21671:   if_False
wenzelm@21671:   if_cancel
wenzelm@21671:   if_eq_cancel
wenzelm@21671:   imp_disjL
haftmann@20973:   (*In general it seems wrong to add distributive laws by default: they
haftmann@20973:     might cause exponential blow-up.  But imp_disjL has been in for a while
haftmann@20973:     and cannot be removed without affecting existing proofs.  Moreover,
haftmann@20973:     rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
haftmann@20973:     grounds that it allows simplification of R in the two cases.*)
wenzelm@21671:   conj_assoc
wenzelm@21671:   disj_assoc
wenzelm@21671:   de_Morgan_conj
wenzelm@21671:   de_Morgan_disj
wenzelm@21671:   imp_disj1
wenzelm@21671:   imp_disj2
wenzelm@21671:   not_imp
wenzelm@21671:   disj_not1
wenzelm@21671:   not_all
wenzelm@21671:   not_ex
wenzelm@21671:   cases_simp
wenzelm@21671:   the_eq_trivial
wenzelm@21671:   the_sym_eq_trivial
wenzelm@21671:   ex_simps
wenzelm@21671:   all_simps
wenzelm@21671:   simp_thms
wenzelm@21671: 
wenzelm@21671: lemmas [cong] = imp_cong simp_implies_cong
wenzelm@21671: lemmas [split] = split_if
haftmann@20973: 
wenzelm@51717: ML {* val HOL_ss = simpset_of @{context} *}
haftmann@20973: 
haftmann@20944: text {* Simplifies x assuming c and y assuming ~c *}
haftmann@20944: lemma if_cong:
haftmann@20944:   assumes "b = c"
haftmann@20944:       and "c \<Longrightarrow> x = u"
haftmann@20944:       and "\<not> c \<Longrightarrow> y = v"
haftmann@20944:   shows "(if b then x else y) = (if c then u else v)"
haftmann@38525:   using assms by simp
haftmann@20944: 
haftmann@20944: text {* Prevents simplification of x and y:
haftmann@20944:   faster and allows the execution of functional programs. *}
haftmann@20944: lemma if_weak_cong [cong]:
haftmann@20944:   assumes "b = c"
haftmann@20944:   shows "(if b then x else y) = (if c then x else y)"
wenzelm@23553:   using assms by (rule arg_cong)
haftmann@20944: 
haftmann@20944: text {* Prevents simplification of t: much faster *}
haftmann@20944: lemma let_weak_cong:
haftmann@20944:   assumes "a = b"
haftmann@20944:   shows "(let x = a in t x) = (let x = b in t x)"
wenzelm@23553:   using assms by (rule arg_cong)
haftmann@20944: 
haftmann@20944: text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}
haftmann@20944: lemma eq_cong2:
haftmann@20944:   assumes "u = u'"
haftmann@20944:   shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
wenzelm@23553:   using assms by simp
haftmann@20944: 
haftmann@20944: lemma if_distrib:
haftmann@20944:   "f (if c then x else y) = (if c then f x else f y)"
haftmann@20944:   by simp
haftmann@20944: 
haftmann@44277: text{*As a simplification rule, it replaces all function equalities by
haftmann@44277:   first-order equalities.*}
haftmann@44277: lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
haftmann@44277:   by auto
haftmann@44277: 
wenzelm@17459: 
haftmann@20944: subsubsection {* Generic cases and induction *}
wenzelm@17459: 
haftmann@20944: text {* Rule projections: *}
haftmann@20944: ML {*
wenzelm@32172: structure Project_Rule = Project_Rule
wenzelm@25388: (
wenzelm@27126:   val conjunct1 = @{thm conjunct1}
wenzelm@27126:   val conjunct2 = @{thm conjunct2}
wenzelm@27126:   val mp = @{thm mp}
wenzelm@59929: );
wenzelm@17459: *}
wenzelm@17459: 
wenzelm@59940: context
wenzelm@59940: begin
wenzelm@59940: 
wenzelm@59940: restricted definition "induct_forall P \<equiv> \<forall>x. P x"
wenzelm@59940: restricted definition "induct_implies A B \<equiv> A \<longrightarrow> B"
wenzelm@59940: restricted definition "induct_equal x y \<equiv> x = y"
wenzelm@59940: restricted definition "induct_conj A B \<equiv> A \<and> B"
wenzelm@59940: restricted definition "induct_true \<equiv> True"
wenzelm@59940: restricted definition "induct_false \<equiv> False"
haftmann@35416: 
wenzelm@59929: lemma induct_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (induct_forall (\<lambda>x. P x))"
wenzelm@18457:   by (unfold atomize_all induct_forall_def)
wenzelm@11824: 
wenzelm@59929: lemma induct_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (induct_implies A B)"
wenzelm@18457:   by (unfold atomize_imp induct_implies_def)
wenzelm@11824: 
wenzelm@59929: lemma induct_equal_eq: "(x \<equiv> y) \<equiv> Trueprop (induct_equal x y)"
wenzelm@18457:   by (unfold atomize_eq induct_equal_def)
wenzelm@18457: 
wenzelm@59929: lemma induct_conj_eq: "(A &&& B) \<equiv> Trueprop (induct_conj A B)"
wenzelm@18457:   by (unfold atomize_conj induct_conj_def)
wenzelm@18457: 
berghofe@34908: lemmas induct_atomize' = induct_forall_eq induct_implies_eq induct_conj_eq
berghofe@34908: lemmas induct_atomize = induct_atomize' induct_equal_eq
wenzelm@45607: lemmas induct_rulify' [symmetric] = induct_atomize'
wenzelm@45607: lemmas induct_rulify [symmetric] = induct_atomize
wenzelm@18457: lemmas induct_rulify_fallback =
wenzelm@18457:   induct_forall_def induct_implies_def induct_equal_def induct_conj_def
berghofe@34908:   induct_true_def induct_false_def
wenzelm@18457: 
wenzelm@11989: lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
wenzelm@11989:     induct_conj (induct_forall A) (induct_forall B)"
nipkow@17589:   by (unfold induct_forall_def induct_conj_def) iprover
wenzelm@11824: 
wenzelm@11989: lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
wenzelm@11989:     induct_conj (induct_implies C A) (induct_implies C B)"
nipkow@17589:   by (unfold induct_implies_def induct_conj_def) iprover
wenzelm@11989: 
wenzelm@59929: lemma induct_conj_curry: "(induct_conj A B \<Longrightarrow> PROP C) \<equiv> (A \<Longrightarrow> B \<Longrightarrow> PROP C)"
berghofe@13598: proof
wenzelm@59929:   assume r: "induct_conj A B \<Longrightarrow> PROP C"
wenzelm@59929:   assume ab: A B
wenzelm@59929:   show "PROP C" by (rule r) (simp add: induct_conj_def ab)
berghofe@13598: next
wenzelm@59929:   assume r: "A \<Longrightarrow> B \<Longrightarrow> PROP C"
wenzelm@59929:   assume ab: "induct_conj A B"
wenzelm@59929:   show "PROP C" by (rule r) (simp_all add: ab [unfolded induct_conj_def])
berghofe@13598: qed
wenzelm@11824: 
wenzelm@11989: lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
wenzelm@11824: 
berghofe@34908: lemma induct_trueI: "induct_true"
berghofe@34908:   by (simp add: induct_true_def)
wenzelm@11824: 
wenzelm@11824: text {* Method setup. *}
wenzelm@11824: 
wenzelm@58826: ML_file "~~/src/Tools/induct.ML"
wenzelm@11824: ML {*
wenzelm@32171: structure Induct = Induct
wenzelm@27126: (
wenzelm@27126:   val cases_default = @{thm case_split}
wenzelm@27126:   val atomize = @{thms induct_atomize}
berghofe@34908:   val rulify = @{thms induct_rulify'}
wenzelm@27126:   val rulify_fallback = @{thms induct_rulify_fallback}
berghofe@34988:   val equal_def = @{thm induct_equal_def}
berghofe@34908:   fun dest_def (Const (@{const_name induct_equal}, _) $ t $ u) = SOME (t, u)
berghofe@34908:     | dest_def _ = NONE
wenzelm@58957:   fun trivial_tac ctxt = match_tac ctxt @{thms induct_trueI}
wenzelm@27126: )
wenzelm@11824: *}
wenzelm@11824: 
wenzelm@48891: ML_file "~~/src/Tools/induction.ML"
nipkow@45014: 
wenzelm@59940: declaration {*
wenzelm@59940:   fn _ => Induct.map_simpset (fn ss => ss
berghofe@34908:     addsimprocs
wenzelm@38715:       [Simplifier.simproc_global @{theory} "swap_induct_false"
berghofe@34908:          ["induct_false ==> PROP P ==> PROP Q"]
wenzelm@51717:          (fn _ =>
berghofe@34908:             (fn _ $ (P as _ $ @{const induct_false}) $ (_ $ Q $ _) =>
berghofe@34908:                   if P <> Q then SOME Drule.swap_prems_eq else NONE
berghofe@34908:               | _ => NONE)),
wenzelm@38715:        Simplifier.simproc_global @{theory} "induct_equal_conj_curry"
berghofe@34908:          ["induct_conj P Q ==> PROP R"]
wenzelm@51717:          (fn _ =>
berghofe@34908:             (fn _ $ (_ $ P) $ _ =>
berghofe@34908:                 let
berghofe@34908:                   fun is_conj (@{const induct_conj} $ P $ Q) =
berghofe@34908:                         is_conj P andalso is_conj Q
berghofe@34908:                     | is_conj (Const (@{const_name induct_equal}, _) $ _ $ _) = true
berghofe@34908:                     | is_conj @{const induct_true} = true
berghofe@34908:                     | is_conj @{const induct_false} = true
berghofe@34908:                     | is_conj _ = false
berghofe@34908:                 in if is_conj P then SOME @{thm induct_conj_curry} else NONE end
wenzelm@45625:               | _ => NONE))]
wenzelm@54742:     |> Simplifier.set_mksimps (fn ctxt =>
wenzelm@54742:         Simpdata.mksimps Simpdata.mksimps_pairs ctxt #>
wenzelm@59940:         map (rewrite_rule ctxt (map Thm.symmetric @{thms induct_rulify_fallback}))))
berghofe@34908: *}
berghofe@34908: 
berghofe@34908: text {* Pre-simplification of induction and cases rules *}
berghofe@34908: 
wenzelm@59929: lemma [induct_simp]: "(\<And>x. induct_equal x t \<Longrightarrow> PROP P x) \<equiv> PROP P t"
berghofe@34908:   unfolding induct_equal_def
berghofe@34908: proof
wenzelm@59929:   assume r: "\<And>x. x = t \<Longrightarrow> PROP P x"
wenzelm@59929:   show "PROP P t" by (rule r [OF refl])
berghofe@34908: next
wenzelm@59929:   fix x
wenzelm@59929:   assume "PROP P t" "x = t"
berghofe@34908:   then show "PROP P x" by simp
berghofe@34908: qed
berghofe@34908: 
wenzelm@59929: lemma [induct_simp]: "(\<And>x. induct_equal t x \<Longrightarrow> PROP P x) \<equiv> PROP P t"
berghofe@34908:   unfolding induct_equal_def
berghofe@34908: proof
wenzelm@59929:   assume r: "\<And>x. t = x \<Longrightarrow> PROP P x"
wenzelm@59929:   show "PROP P t" by (rule r [OF refl])
berghofe@34908: next
wenzelm@59929:   fix x
wenzelm@59929:   assume "PROP P t" "t = x"
berghofe@34908:   then show "PROP P x" by simp
berghofe@34908: qed
berghofe@34908: 
wenzelm@59929: lemma [induct_simp]: "(induct_false \<Longrightarrow> P) \<equiv> Trueprop induct_true"
berghofe@34908:   unfolding induct_false_def induct_true_def
berghofe@34908:   by (iprover intro: equal_intr_rule)
berghofe@34908: 
wenzelm@59929: lemma [induct_simp]: "(induct_true \<Longrightarrow> PROP P) \<equiv> PROP P"
berghofe@34908:   unfolding induct_true_def
berghofe@34908: proof
wenzelm@59929:   assume "True \<Longrightarrow> PROP P"
wenzelm@59929:   then show "PROP P" using TrueI .
berghofe@34908: next
berghofe@34908:   assume "PROP P"
berghofe@34908:   then show "PROP P" .
berghofe@34908: qed
berghofe@34908: 
wenzelm@59929: lemma [induct_simp]: "(PROP P \<Longrightarrow> induct_true) \<equiv> Trueprop induct_true"
berghofe@34908:   unfolding induct_true_def
berghofe@34908:   by (iprover intro: equal_intr_rule)
berghofe@34908: 
wenzelm@59929: lemma [induct_simp]: "(\<And>x. induct_true) \<equiv> Trueprop induct_true"
berghofe@34908:   unfolding induct_true_def
berghofe@34908:   by (iprover intro: equal_intr_rule)
berghofe@34908: 
wenzelm@59929: lemma [induct_simp]: "induct_implies induct_true P \<equiv> P"
berghofe@34908:   by (simp add: induct_implies_def induct_true_def)
berghofe@34908: 
wenzelm@59929: lemma [induct_simp]: "x = x \<longleftrightarrow> True"
berghofe@34908:   by (rule simp_thms)
berghofe@34908: 
wenzelm@59940: end
wenzelm@18457: 
wenzelm@48891: ML_file "~~/src/Tools/induct_tacs.ML"
wenzelm@27126: 
haftmann@20944: 
berghofe@28325: subsubsection {* Coherent logic *}
berghofe@28325: 
wenzelm@55632: ML_file "~~/src/Tools/coherent.ML"
berghofe@28325: ML {*
wenzelm@32734: structure Coherent = Coherent
berghofe@28325: (
wenzelm@55632:   val atomize_elimL = @{thm atomize_elimL};
wenzelm@55632:   val atomize_exL = @{thm atomize_exL};
wenzelm@55632:   val atomize_conjL = @{thm atomize_conjL};
wenzelm@55632:   val atomize_disjL = @{thm atomize_disjL};
wenzelm@55632:   val operator_names = [@{const_name HOL.disj}, @{const_name HOL.conj}, @{const_name Ex}];
berghofe@28325: );
berghofe@28325: *}
berghofe@28325: 
berghofe@28325: 
huffman@31024: subsubsection {* Reorienting equalities *}
huffman@31024: 
huffman@31024: ML {*
huffman@31024: signature REORIENT_PROC =
huffman@31024: sig
huffman@31024:   val add : (term -> bool) -> theory -> theory
wenzelm@51717:   val proc : morphism -> Proof.context -> cterm -> thm option
huffman@31024: end;
huffman@31024: 
wenzelm@33523: structure Reorient_Proc : REORIENT_PROC =
huffman@31024: struct
wenzelm@33523:   structure Data = Theory_Data
huffman@31024:   (
wenzelm@33523:     type T = ((term -> bool) * stamp) list;
wenzelm@33523:     val empty = [];
huffman@31024:     val extend = I;
wenzelm@33523:     fun merge data : T = Library.merge (eq_snd op =) data;
wenzelm@33523:   );
wenzelm@33523:   fun add m = Data.map (cons (m, stamp ()));
wenzelm@33523:   fun matches thy t = exists (fn (m, _) => m t) (Data.get thy);
huffman@31024: 
huffman@31024:   val meta_reorient = @{thm eq_commute [THEN eq_reflection]};
wenzelm@51717:   fun proc phi ctxt ct =
huffman@31024:     let
wenzelm@42361:       val thy = Proof_Context.theory_of ctxt;
huffman@31024:     in
huffman@31024:       case Thm.term_of ct of
wenzelm@33523:         (_ $ t $ u) => if matches thy u then NONE else SOME meta_reorient
huffman@31024:       | _ => NONE
huffman@31024:     end;
huffman@31024: end;
huffman@31024: *}
huffman@31024: 
huffman@31024: 
haftmann@20944: subsection {* Other simple lemmas and lemma duplicates *}
haftmann@20944: 
haftmann@20944: lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
haftmann@20944:   by blast+
haftmann@20944: 
haftmann@20944: lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
haftmann@20944:   apply (rule iffI)
haftmann@20944:   apply (rule_tac a = "%x. THE y. P x y" in ex1I)
haftmann@20944:   apply (fast dest!: theI')
huffman@44921:   apply (fast intro: the1_equality [symmetric])
haftmann@20944:   apply (erule ex1E)
haftmann@20944:   apply (rule allI)
haftmann@20944:   apply (rule ex1I)
haftmann@20944:   apply (erule spec)
haftmann@20944:   apply (erule_tac x = "%z. if z = x then y else f z" in allE)
haftmann@20944:   apply (erule impE)
haftmann@20944:   apply (rule allI)
wenzelm@27126:   apply (case_tac "xa = x")
haftmann@20944:   apply (drule_tac [3] x = x in fun_cong, simp_all)
haftmann@20944:   done
haftmann@20944: 
haftmann@22218: lemmas eq_sym_conv = eq_commute
haftmann@22218: 
chaieb@23037: lemma nnf_simps:
wenzelm@58826:   "(\<not>(P \<and> Q)) = (\<not> P \<or> \<not> Q)" "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
wenzelm@58826:   "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))"
chaieb@23037:   "(\<not> \<not>(P)) = P"
chaieb@23037: by blast+
chaieb@23037: 
wenzelm@21671: subsection {* Basic ML bindings *}
wenzelm@21671: 
wenzelm@21671: ML {*
wenzelm@22129: val FalseE = @{thm FalseE}
wenzelm@22129: val Let_def = @{thm Let_def}
wenzelm@22129: val TrueI = @{thm TrueI}
wenzelm@22129: val allE = @{thm allE}
wenzelm@22129: val allI = @{thm allI}
wenzelm@22129: val all_dupE = @{thm all_dupE}
wenzelm@22129: val arg_cong = @{thm arg_cong}
wenzelm@22129: val box_equals = @{thm box_equals}
wenzelm@22129: val ccontr = @{thm ccontr}
wenzelm@22129: val classical = @{thm classical}
wenzelm@22129: val conjE = @{thm conjE}
wenzelm@22129: val conjI = @{thm conjI}
wenzelm@22129: val conjunct1 = @{thm conjunct1}
wenzelm@22129: val conjunct2 = @{thm conjunct2}
wenzelm@22129: val disjCI = @{thm disjCI}
wenzelm@22129: val disjE = @{thm disjE}
wenzelm@22129: val disjI1 = @{thm disjI1}
wenzelm@22129: val disjI2 = @{thm disjI2}
wenzelm@22129: val eq_reflection = @{thm eq_reflection}
wenzelm@22129: val ex1E = @{thm ex1E}
wenzelm@22129: val ex1I = @{thm ex1I}
wenzelm@22129: val ex1_implies_ex = @{thm ex1_implies_ex}
wenzelm@22129: val exE = @{thm exE}
wenzelm@22129: val exI = @{thm exI}
wenzelm@22129: val excluded_middle = @{thm excluded_middle}
wenzelm@22129: val ext = @{thm ext}
wenzelm@22129: val fun_cong = @{thm fun_cong}
wenzelm@22129: val iffD1 = @{thm iffD1}
wenzelm@22129: val iffD2 = @{thm iffD2}
wenzelm@22129: val iffI = @{thm iffI}
wenzelm@22129: val impE = @{thm impE}
wenzelm@22129: val impI = @{thm impI}
wenzelm@22129: val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
wenzelm@22129: val mp = @{thm mp}
wenzelm@22129: val notE = @{thm notE}
wenzelm@22129: val notI = @{thm notI}
wenzelm@22129: val not_all = @{thm not_all}
wenzelm@22129: val not_ex = @{thm not_ex}
wenzelm@22129: val not_iff = @{thm not_iff}
wenzelm@22129: val not_not = @{thm not_not}
wenzelm@22129: val not_sym = @{thm not_sym}
wenzelm@22129: val refl = @{thm refl}
wenzelm@22129: val rev_mp = @{thm rev_mp}
wenzelm@22129: val spec = @{thm spec}
wenzelm@22129: val ssubst = @{thm ssubst}
wenzelm@22129: val subst = @{thm subst}
wenzelm@22129: val sym = @{thm sym}
wenzelm@22129: val trans = @{thm trans}
wenzelm@21671: *}
wenzelm@21671: 
wenzelm@55239: ML_file "Tools/cnf.ML"
wenzelm@55239: 
wenzelm@21671: 
hoelzl@58775: section {* @{text NO_MATCH} simproc *}
hoelzl@58775: 
hoelzl@58775: text {*
hoelzl@58775:  The simplification procedure can be used to avoid simplification of terms of a certain form
hoelzl@58775: *}
hoelzl@58775: 
hoelzl@59779: definition NO_MATCH :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where "NO_MATCH pat val \<equiv> True"
hoelzl@58830: 
hoelzl@59779: lemma NO_MATCH_cong[cong]: "NO_MATCH pat val = NO_MATCH pat val" by (rule refl)
hoelzl@58775: 
hoelzl@58830: declare [[coercion_args NO_MATCH - -]]
hoelzl@58830: 
hoelzl@59779: simproc_setup NO_MATCH ("NO_MATCH pat val") = {* fn _ => fn ctxt => fn ct =>
hoelzl@58775:   let
hoelzl@58775:     val thy = Proof_Context.theory_of ctxt
hoelzl@58775:     val dest_binop = Term.dest_comb #> apfst (Term.dest_comb #> snd)
hoelzl@58775:     val m = Pattern.matches thy (dest_binop (Thm.term_of ct))
hoelzl@58775:   in if m then NONE else SOME @{thm NO_MATCH_def} end
hoelzl@58775: *}
hoelzl@58775: 
hoelzl@58775: text {*
hoelzl@59779:   This setup ensures that a rewrite rule of the form @{term "NO_MATCH pat val \<Longrightarrow> t"}
hoelzl@58775:   is only applied, if the pattern @{term pat} does not match the value @{term val}.
hoelzl@58775: *}
hoelzl@58775: 
hoelzl@58775: 
haftmann@30929: subsection {* Code generator setup *}
haftmann@30929: 
haftmann@31151: subsubsection {* Generic code generator preprocessor setup *}
haftmann@31151: 
haftmann@53146: lemma conj_left_cong:
haftmann@53146:   "P \<longleftrightarrow> Q \<Longrightarrow> P \<and> R \<longleftrightarrow> Q \<and> R"
haftmann@53146:   by (fact arg_cong)
haftmann@53146: 
haftmann@53146: lemma disj_left_cong:
haftmann@53146:   "P \<longleftrightarrow> Q \<Longrightarrow> P \<or> R \<longleftrightarrow> Q \<or> R"
haftmann@53146:   by (fact arg_cong)
haftmann@53146: 
haftmann@31151: setup {*
wenzelm@58826:   Code_Preproc.map_pre (put_simpset HOL_basic_ss) #>
wenzelm@58826:   Code_Preproc.map_post (put_simpset HOL_basic_ss) #>
wenzelm@58826:   Code_Simp.map_ss (put_simpset HOL_basic_ss #>
wenzelm@58826:   Simplifier.add_cong @{thm conj_left_cong} #>
wenzelm@58826:   Simplifier.add_cong @{thm disj_left_cong})
haftmann@31151: *}
haftmann@31151: 
haftmann@53146: 
haftmann@30929: subsubsection {* Equality *}
haftmann@24844: 
haftmann@38857: class equal =
haftmann@38857:   fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@38857:   assumes equal_eq: "equal x y \<longleftrightarrow> x = y"
haftmann@26513: begin
haftmann@26513: 
bulwahn@45231: lemma equal: "equal = (op =)"
haftmann@38857:   by (rule ext equal_eq)+
haftmann@28346: 
haftmann@38857: lemma equal_refl: "equal x x \<longleftrightarrow> True"
haftmann@38857:   unfolding equal by rule+
haftmann@28346: 
haftmann@38857: lemma eq_equal: "(op =) \<equiv> equal"
haftmann@38857:   by (rule eq_reflection) (rule ext, rule ext, rule sym, rule equal_eq)
haftmann@30929: 
haftmann@26513: end
haftmann@26513: 
haftmann@38857: declare eq_equal [symmetric, code_post]
haftmann@38857: declare eq_equal [code]
haftmann@30966: 
haftmann@31151: setup {*
wenzelm@51717:   Code_Preproc.map_pre (fn ctxt =>
wenzelm@51717:     ctxt addsimprocs [Simplifier.simproc_global_i @{theory} "equal" [@{term HOL.eq}]
wenzelm@51717:       (fn _ => fn Const (_, Type ("fun", [Type _, _])) => SOME @{thm eq_equal} | _ => NONE)])
haftmann@31151: *}
haftmann@31151: 
haftmann@30966: 
haftmann@30929: subsubsection {* Generic code generator foundation *}
haftmann@30929: 
haftmann@39421: text {* Datatype @{typ bool} *}
haftmann@30929: 
haftmann@30929: code_datatype True False
haftmann@30929: 
haftmann@30929: lemma [code]:
haftmann@33185:   shows "False \<and> P \<longleftrightarrow> False"
haftmann@33185:     and "True \<and> P \<longleftrightarrow> P"
haftmann@33185:     and "P \<and> False \<longleftrightarrow> False"
haftmann@33185:     and "P \<and> True \<longleftrightarrow> P" by simp_all
haftmann@30929: 
haftmann@30929: lemma [code]:
haftmann@33185:   shows "False \<or> P \<longleftrightarrow> P"
haftmann@33185:     and "True \<or> P \<longleftrightarrow> True"
haftmann@33185:     and "P \<or> False \<longleftrightarrow> P"
haftmann@33185:     and "P \<or> True \<longleftrightarrow> True" by simp_all
haftmann@30929: 
haftmann@33185: lemma [code]:
haftmann@33185:   shows "(False \<longrightarrow> P) \<longleftrightarrow> True"
haftmann@33185:     and "(True \<longrightarrow> P) \<longleftrightarrow> P"
haftmann@33185:     and "(P \<longrightarrow> False) \<longleftrightarrow> \<not> P"
haftmann@33185:     and "(P \<longrightarrow> True) \<longleftrightarrow> True" by simp_all
haftmann@30929: 
haftmann@39421: text {* More about @{typ prop} *}
haftmann@39421: 
haftmann@39421: lemma [code nbe]:
wenzelm@58826:   shows "(True \<Longrightarrow> PROP Q) \<equiv> PROP Q"
haftmann@39421:     and "(PROP Q \<Longrightarrow> True) \<equiv> Trueprop True"
haftmann@39421:     and "(P \<Longrightarrow> R) \<equiv> Trueprop (P \<longrightarrow> R)" by (auto intro!: equal_intr_rule)
haftmann@39421: 
haftmann@39421: lemma Trueprop_code [code]:
haftmann@39421:   "Trueprop True \<equiv> Code_Generator.holds"
haftmann@39421:   by (auto intro!: equal_intr_rule holds)
haftmann@39421: 
haftmann@39421: declare Trueprop_code [symmetric, code_post]
haftmann@39421: 
haftmann@39421: text {* Equality *}
haftmann@39421: 
haftmann@39421: declare simp_thms(6) [code nbe]
haftmann@39421: 
haftmann@38857: instantiation itself :: (type) equal
haftmann@31132: begin
haftmann@31132: 
haftmann@38857: definition equal_itself :: "'a itself \<Rightarrow> 'a itself \<Rightarrow> bool" where
haftmann@38857:   "equal_itself x y \<longleftrightarrow> x = y"
haftmann@31132: 
haftmann@31132: instance proof
haftmann@38857: qed (fact equal_itself_def)
haftmann@31132: 
haftmann@31132: end
haftmann@31132: 
haftmann@38857: lemma equal_itself_code [code]:
haftmann@38857:   "equal TYPE('a) TYPE('a) \<longleftrightarrow> True"
haftmann@38857:   by (simp add: equal)
haftmann@31132: 
wenzelm@58826: setup {* Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a\<Colon>type \<Rightarrow> 'a \<Rightarrow> bool"}) *}
haftmann@31956: 
haftmann@38857: lemma equal_alias_cert: "OFCLASS('a, equal_class) \<equiv> ((op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool) \<equiv> equal)" (is "?ofclass \<equiv> ?equal")
haftmann@31956: proof
haftmann@31956:   assume "PROP ?ofclass"
haftmann@38857:   show "PROP ?equal"
wenzelm@59498:     by (tactic {* ALLGOALS (resolve_tac @{context} [Thm.unconstrainT @{thm eq_equal}]) *})
haftmann@31956:       (fact `PROP ?ofclass`)
haftmann@31956: next
haftmann@38857:   assume "PROP ?equal"
haftmann@31956:   show "PROP ?ofclass" proof
haftmann@38857:   qed (simp add: `PROP ?equal`)
haftmann@31956: qed
haftmann@31956: 
wenzelm@58826: setup {* Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a\<Colon>equal \<Rightarrow> 'a \<Rightarrow> bool"}) *}
wenzelm@58826: 
wenzelm@58826: setup {* Nbe.add_const_alias @{thm equal_alias_cert} *}
haftmann@30929: 
haftmann@30929: text {* Cases *}
haftmann@30929: 
haftmann@30929: lemma Let_case_cert:
haftmann@30929:   assumes "CASE \<equiv> (\<lambda>x. Let x f)"
haftmann@30929:   shows "CASE x \<equiv> f x"
haftmann@30929:   using assms by simp_all
haftmann@30929: 
haftmann@30929: setup {*
wenzelm@58826:   Code.add_case @{thm Let_case_cert} #>
wenzelm@58826:   Code.add_undefined @{const_name undefined}
haftmann@30929: *}
haftmann@30929: 
haftmann@54890: declare [[code abort: undefined]]
haftmann@30929: 
haftmann@38972: 
haftmann@30929: subsubsection {* Generic code generator target languages *}
haftmann@30929: 
haftmann@38972: text {* type @{typ bool} *}
haftmann@30929: 
haftmann@52435: code_printing
haftmann@52435:   type_constructor bool \<rightharpoonup>
haftmann@52435:     (SML) "bool" and (OCaml) "bool" and (Haskell) "Bool" and (Scala) "Boolean"
haftmann@52435: | constant True \<rightharpoonup>
haftmann@52435:     (SML) "true" and (OCaml) "true" and (Haskell) "True" and (Scala) "true"
haftmann@52435: | constant False \<rightharpoonup>
wenzelm@58826:     (SML) "false" and (OCaml) "false" and (Haskell) "False" and (Scala) "false"
haftmann@34294: 
haftmann@30929: code_reserved SML
haftmann@52435:   bool true false
haftmann@30929: 
haftmann@30929: code_reserved OCaml
haftmann@52435:   bool
haftmann@30929: 
haftmann@34294: code_reserved Scala
haftmann@34294:   Boolean
haftmann@34294: 
haftmann@52435: code_printing
haftmann@52435:   constant Not \<rightharpoonup>
haftmann@52435:     (SML) "not" and (OCaml) "not" and (Haskell) "not" and (Scala) "'! _"
haftmann@52435: | constant HOL.conj \<rightharpoonup>
haftmann@52435:     (SML) infixl 1 "andalso" and (OCaml) infixl 3 "&&" and (Haskell) infixr 3 "&&" and (Scala) infixl 3 "&&"
haftmann@52435: | constant HOL.disj \<rightharpoonup>
haftmann@52435:     (SML) infixl 0 "orelse" and (OCaml) infixl 2 "||" and (Haskell) infixl 2 "||" and (Scala) infixl 1 "||"
haftmann@52435: | constant HOL.implies \<rightharpoonup>
haftmann@52435:     (SML) "!(if (_)/ then (_)/ else true)"
haftmann@52435:     and (OCaml) "!(if (_)/ then (_)/ else true)"
haftmann@52435:     and (Haskell) "!(if (_)/ then (_)/ else True)"
haftmann@52435:     and (Scala) "!(if ((_))/ (_)/ else true)"
haftmann@52435: | constant If \<rightharpoonup>
haftmann@52435:     (SML) "!(if (_)/ then (_)/ else (_))"
haftmann@52435:     and (OCaml) "!(if (_)/ then (_)/ else (_))"
haftmann@52435:     and (Haskell) "!(if (_)/ then (_)/ else (_))"
haftmann@52435:     and (Scala) "!(if ((_))/ (_)/ else (_))"
haftmann@52435: 
haftmann@52435: code_reserved SML
haftmann@52435:   not
haftmann@52435: 
haftmann@52435: code_reserved OCaml
haftmann@52435:   not
haftmann@52435: 
haftmann@52435: code_identifier
haftmann@52435:   code_module Pure \<rightharpoonup>
haftmann@52435:     (SML) HOL and (OCaml) HOL and (Haskell) HOL and (Scala) HOL
haftmann@39026: 
haftmann@30929: text {* using built-in Haskell equality *}
haftmann@30929: 
haftmann@52435: code_printing
haftmann@52435:   type_class equal \<rightharpoonup> (Haskell) "Eq"
haftmann@52435: | constant HOL.equal \<rightharpoonup> (Haskell) infix 4 "=="
haftmann@52435: | constant HOL.eq \<rightharpoonup> (Haskell) infix 4 "=="
haftmann@30929: 
haftmann@30929: text {* undefined *}
haftmann@30929: 
haftmann@52435: code_printing
haftmann@52435:   constant undefined \<rightharpoonup>
haftmann@52435:     (SML) "!(raise/ Fail/ \"undefined\")"
haftmann@52435:     and (OCaml) "failwith/ \"undefined\""
haftmann@52435:     and (Haskell) "error/ \"undefined\""
haftmann@52435:     and (Scala) "!sys.error(\"undefined\")"
haftmann@52435: 
haftmann@30929: 
haftmann@30929: subsubsection {* Evaluation and normalization by evaluation *}
haftmann@30929: 
haftmann@55757: method_setup eval = {*
wenzelm@58826:   let
wenzelm@58826:     fun eval_tac ctxt =
wenzelm@58826:       let val conv = Code_Runtime.dynamic_holds_conv ctxt
wenzelm@58839:       in
wenzelm@58839:         CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1 conv)) ctxt) THEN'
wenzelm@59498:         resolve_tac ctxt [TrueI]
wenzelm@58839:       end
wenzelm@58826:   in
wenzelm@58826:     Scan.succeed (SIMPLE_METHOD' o eval_tac)
wenzelm@58826:   end
haftmann@55757: *} "solve goal by evaluation"
haftmann@30929: 
haftmann@30929: method_setup normalization = {*
wenzelm@46190:   Scan.succeed (fn ctxt =>
wenzelm@46190:     SIMPLE_METHOD'
wenzelm@46190:       (CHANGED_PROP o
haftmann@55757:         (CONVERSION (Nbe.dynamic_conv ctxt)
wenzelm@59498:           THEN_ALL_NEW (TRY o resolve_tac ctxt [TrueI]))))
haftmann@30929: *} "solve goal by normalization"
haftmann@30929: 
wenzelm@31902: 
haftmann@33084: subsection {* Counterexample Search Units *}
haftmann@33084: 
haftmann@30929: subsubsection {* Quickcheck *}
haftmann@30929: 
haftmann@33084: quickcheck_params [size = 5, iterations = 50]
haftmann@33084: 
haftmann@30929: 
haftmann@33084: subsubsection {* Nitpick setup *}
blanchet@30309: 
wenzelm@59028: named_theorems nitpick_unfold "alternative definitions of constants as needed by Nitpick"
wenzelm@59028:   and nitpick_simp "equational specification of constants as needed by Nitpick"
wenzelm@59028:   and nitpick_psimp "partial equational specification of constants as needed by Nitpick"
wenzelm@59028:   and nitpick_choice_spec "choice specification of constants as needed by Nitpick"
wenzelm@30980: 
blanchet@41792: declare if_bool_eq_conj [nitpick_unfold, no_atp]
blanchet@41792:         if_bool_eq_disj [no_atp]
blanchet@41792: 
blanchet@29863: 
haftmann@33084: subsection {* Preprocessing for the predicate compiler *}
haftmann@33084: 
wenzelm@59028: named_theorems code_pred_def "alternative definitions of constants for the Predicate Compiler"
wenzelm@59028:   and code_pred_inline "inlining definitions for the Predicate Compiler"
wenzelm@59028:   and code_pred_simp "simplification rules for the optimisations in the Predicate Compiler"
haftmann@33084: 
haftmann@33084: 
haftmann@22839: subsection {* Legacy tactics and ML bindings *}
wenzelm@21671: 
wenzelm@21671: ML {*
wenzelm@58826:   (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
wenzelm@58826:   local
wenzelm@58826:     fun wrong_prem (Const (@{const_name All}, _) $ Abs (_, _, t)) = wrong_prem t
wenzelm@58826:       | wrong_prem (Bound _) = true
wenzelm@58826:       | wrong_prem _ = false;
wenzelm@58826:     val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
wenzelm@58826:   in
wenzelm@58826:     fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
wenzelm@59498:     fun smp_tac ctxt j = EVERY' [dresolve_tac ctxt (smp j), assume_tac ctxt];
wenzelm@58826:   end;
haftmann@22839: 
wenzelm@58826:   local
wenzelm@58826:     val nnf_ss =
wenzelm@58826:       simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms simp_thms nnf_simps});
wenzelm@58826:   in
wenzelm@58826:     fun nnf_conv ctxt = Simplifier.rewrite (put_simpset nnf_ss ctxt);
wenzelm@58826:   end
wenzelm@21671: *}
wenzelm@21671: 
haftmann@38866: hide_const (open) eq equal
haftmann@38866: 
kleing@14357: end




