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src/HOL/HOL.thy

author | paulson |

Thu, 02 Mar 2006 18:49:13 +0100 | |

changeset 19174 | df9de25e87b3 |

parent 19162 | 67436e2a16df |

child 19233 | 77ca20b0ed77 |

permissions | -rw-r--r-- |

moved the "use" directive

(* Title: HOL/HOL.thy ID: $Id$ Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson *) header {* The basis of Higher-Order Logic *} theory HOL imports CPure uses ("cladata.ML") ("blastdata.ML") ("simpdata.ML") "Tools/res_atpset.ML" begin subsection {* Primitive logic *} subsubsection {* Core syntax *} classes type defaultsort type global typedecl bool arities bool :: type fun :: (type, type) type judgment Trueprop :: "bool => prop" ("(_)" 5) consts Not :: "bool => bool" ("~ _" [40] 40) True :: bool False :: bool arbitrary :: 'a The :: "('a => bool) => 'a" All :: "('a => bool) => bool" (binder "ALL " 10) Ex :: "('a => bool) => bool" (binder "EX " 10) Ex1 :: "('a => bool) => bool" (binder "EX! " 10) Let :: "['a, 'a => 'b] => 'b" "=" :: "['a, 'a] => bool" (infixl 50) & :: "[bool, bool] => bool" (infixr 35) "|" :: "[bool, bool] => bool" (infixr 30) --> :: "[bool, bool] => bool" (infixr 25) local consts If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10) subsubsection {* Additional concrete syntax *} nonterminals letbinds letbind case_syn cases_syn syntax "_not_equal" :: "['a, 'a] => bool" (infixl "~=" 50) "_The" :: "[pttrn, bool] => 'a" ("(3THE _./ _)" [0, 10] 10) "_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10) "" :: "letbind => letbinds" ("_") "_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _") "_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10) "_case_syntax":: "['a, cases_syn] => 'b" ("(case _ of/ _)" 10) "_case1" :: "['a, 'b] => case_syn" ("(2_ =>/ _)" 10) "" :: "case_syn => cases_syn" ("_") "_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ | _") translations "x ~= y" == "~ (x = y)" "THE x. P" == "The (%x. P)" "_Let (_binds b bs) e" == "_Let b (_Let bs e)" "let x = a in e" == "Let a (%x. e)" print_translation {* (* To avoid eta-contraction of body: *) [("The", fn [Abs abs] => let val (x,t) = atomic_abs_tr' abs in Syntax.const "_The" $ x $ t end)] *} syntax (output) "=" :: "['a, 'a] => bool" (infix 50) "_not_equal" :: "['a, 'a] => bool" (infix "~=" 50) syntax (xsymbols) Not :: "bool => bool" ("\<not> _" [40] 40) "op &" :: "[bool, bool] => bool" (infixr "\<and>" 35) "op |" :: "[bool, bool] => bool" (infixr "\<or>" 30) "op -->" :: "[bool, bool] => bool" (infixr "\<longrightarrow>" 25) "_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50) "ALL " :: "[idts, bool] => bool" ("(3\<forall>_./ _)" [0, 10] 10) "EX " :: "[idts, bool] => bool" ("(3\<exists>_./ _)" [0, 10] 10) "EX! " :: "[idts, bool] => bool" ("(3\<exists>!_./ _)" [0, 10] 10) "_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10) (*"_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ \<orelse> _")*) syntax (xsymbols output) "_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50) syntax (HTML output) "_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50) Not :: "bool => bool" ("\<not> _" [40] 40) "op &" :: "[bool, bool] => bool" (infixr "\<and>" 35) "op |" :: "[bool, bool] => bool" (infixr "\<or>" 30) "_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50) "ALL " :: "[idts, bool] => bool" ("(3\<forall>_./ _)" [0, 10] 10) "EX " :: "[idts, bool] => bool" ("(3\<exists>_./ _)" [0, 10] 10) "EX! " :: "[idts, bool] => bool" ("(3\<exists>!_./ _)" [0, 10] 10) syntax (HOL) "ALL " :: "[idts, bool] => bool" ("(3! _./ _)" [0, 10] 10) "EX " :: "[idts, bool] => bool" ("(3? _./ _)" [0, 10] 10) "EX! " :: "[idts, bool] => bool" ("(3?! _./ _)" [0, 10] 10) syntax "_iff" :: "bool => bool => bool" (infixr "<->" 25) syntax (xsymbols) "_iff" :: "bool => bool => bool" (infixr "\<longleftrightarrow>" 25) translations "op <->" => "op = :: bool => bool => bool" typed_print_translation {* let fun iff_tr' _ (Type ("fun", (Type ("bool", _) :: _))) ts = if Output.has_mode "iff" then Term.list_comb (Syntax.const "_iff", ts) else raise Match | iff_tr' _ _ _ = raise Match; in [("op =", iff_tr')] end *} subsubsection {* Axioms and basic definitions *} axioms eq_reflection: "(x=y) ==> (x==y)" refl: "t = (t::'a)" ext: "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)" -- {*Extensionality is built into the meta-logic, and this rule expresses a related property. It is an eta-expanded version of the traditional rule, and similar to the ABS rule of HOL*} the_eq_trivial: "(THE x. x = a) = (a::'a)" impI: "(P ==> Q) ==> P-->Q" mp: "[| P-->Q; P |] ==> Q" text{*Thanks to Stephan Merz*} theorem subst: assumes eq: "s = t" and p: "P(s)" shows "P(t::'a)" proof - from eq have meta: "s \<equiv> t" by (rule eq_reflection) from p show ?thesis by (unfold meta) qed defs True_def: "True == ((%x::bool. x) = (%x. x))" All_def: "All(P) == (P = (%x. True))" Ex_def: "Ex(P) == !Q. (!x. P x --> Q) --> Q" False_def: "False == (!P. P)" not_def: "~ P == P-->False" and_def: "P & Q == !R. (P-->Q-->R) --> R" or_def: "P | Q == !R. (P-->R) --> (Q-->R) --> R" Ex1_def: "Ex1(P) == ? x. P(x) & (! y. P(y) --> y=x)" axioms iff: "(P-->Q) --> (Q-->P) --> (P=Q)" True_or_False: "(P=True) | (P=False)" defs Let_def: "Let s f == f(s)" if_def: "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)" finalconsts "op =" "op -->" The arbitrary subsubsection {* Generic algebraic operations *} axclass zero < type axclass one < type axclass plus < type axclass minus < type axclass times < type axclass inverse < type global consts "0" :: "'a::zero" ("0") "1" :: "'a::one" ("1") "+" :: "['a::plus, 'a] => 'a" (infixl 65) - :: "['a::minus, 'a] => 'a" (infixl 65) uminus :: "['a::minus] => 'a" ("- _" [81] 80) * :: "['a::times, 'a] => 'a" (infixl 70) syntax "_index1" :: index ("\<^sub>1") translations (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>" local typed_print_translation {* let fun tr' c = (c, fn show_sorts => fn T => fn ts => if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T); in [tr' "0", tr' "1"] end; *} -- {* show types that are presumably too general *} consts abs :: "'a::minus => 'a" inverse :: "'a::inverse => 'a" divide :: "['a::inverse, 'a] => 'a" (infixl "'/" 70) syntax (xsymbols) abs :: "'a::minus => 'a" ("\<bar>_\<bar>") syntax (HTML output) abs :: "'a::minus => 'a" ("\<bar>_\<bar>") subsection {*Equality*} lemma sym: "s = t ==> t = s" by (erule subst) (rule refl) lemma ssubst: "t = s ==> P s ==> P t" by (drule sym) (erule subst) lemma trans: "[| r=s; s=t |] ==> r=t" by (erule subst) lemma def_imp_eq: assumes meq: "A == B" shows "A = B" by (unfold meq) (rule refl) (*Useful with eresolve_tac for proving equalties from known equalities. a = b | | c = d *) lemma box_equals: "[| a=b; a=c; b=d |] ==> c=d" apply (rule trans) apply (rule trans) apply (rule sym) apply assumption+ done text {* For calculational reasoning: *} lemma forw_subst: "a = b ==> P b ==> P a" by (rule ssubst) lemma back_subst: "P a ==> a = b ==> P b" by (rule subst) subsection {*Congruence rules for application*} (*similar to AP_THM in Gordon's HOL*) lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)" apply (erule subst) apply (rule refl) done (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*) lemma arg_cong: "x=y ==> f(x)=f(y)" apply (erule subst) apply (rule refl) done lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d" apply (erule ssubst)+ apply (rule refl) done lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)" apply (erule subst)+ apply (rule refl) done subsection {*Equality of booleans -- iff*} lemma iffI: assumes prems: "P ==> Q" "Q ==> P" shows "P=Q" by (iprover intro: iff [THEN mp, THEN mp] impI prems) lemma iffD2: "[| P=Q; Q |] ==> P" by (erule ssubst) lemma rev_iffD2: "[| Q; P=Q |] ==> P" by (erule iffD2) lemmas iffD1 = sym [THEN iffD2, standard] lemmas rev_iffD1 = sym [THEN [2] rev_iffD2, standard] lemma iffE: assumes major: "P=Q" and minor: "[| P --> Q; Q --> P |] ==> R" shows R by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1]) subsection {*True*} lemma TrueI: "True" by (unfold True_def) (rule refl) lemma eqTrueI: "P ==> P=True" by (iprover intro: iffI TrueI) lemma eqTrueE: "P=True ==> P" apply (erule iffD2) apply (rule TrueI) done subsection {*Universal quantifier*} lemma allI: assumes p: "!!x::'a. P(x)" shows "ALL x. P(x)" apply (unfold All_def) apply (iprover intro: ext eqTrueI p) done lemma spec: "ALL x::'a. P(x) ==> P(x)" apply (unfold All_def) apply (rule eqTrueE) apply (erule fun_cong) done lemma allE: assumes major: "ALL x. P(x)" and minor: "P(x) ==> R" shows "R" by (iprover intro: minor major [THEN spec]) lemma all_dupE: assumes major: "ALL x. P(x)" and minor: "[| P(x); ALL x. P(x) |] ==> R" shows "R" by (iprover intro: minor major major [THEN spec]) subsection {*False*} (*Depends upon spec; it is impossible to do propositional logic before quantifiers!*) lemma FalseE: "False ==> P" apply (unfold False_def) apply (erule spec) done lemma False_neq_True: "False=True ==> P" by (erule eqTrueE [THEN FalseE]) subsection {*Negation*} lemma notI: assumes p: "P ==> False" shows "~P" apply (unfold not_def) apply (iprover intro: impI p) done lemma False_not_True: "False ~= True" apply (rule notI) apply (erule False_neq_True) done lemma True_not_False: "True ~= False" apply (rule notI) apply (drule sym) apply (erule False_neq_True) done lemma notE: "[| ~P; P |] ==> R" apply (unfold not_def) apply (erule mp [THEN FalseE]) apply assumption done (* Alternative ~ introduction rule: [| P ==> ~ Pa; P ==> Pa |] ==> ~ P *) lemmas notI2 = notE [THEN notI, standard] subsection {*Implication*} lemma impE: assumes "P-->Q" "P" "Q ==> R" shows "R" by (iprover intro: prems mp) (* Reduces Q to P-->Q, allowing substitution in P. *) lemma rev_mp: "[| P; P --> Q |] ==> Q" by (iprover intro: mp) lemma contrapos_nn: assumes major: "~Q" and minor: "P==>Q" shows "~P" by (iprover intro: notI minor major [THEN notE]) (*not used at all, but we already have the other 3 combinations *) lemma contrapos_pn: assumes major: "Q" and minor: "P ==> ~Q" shows "~P" by (iprover intro: notI minor major notE) lemma not_sym: "t ~= s ==> s ~= t" apply (erule contrapos_nn) apply (erule sym) done (*still used in HOLCF*) lemma rev_contrapos: assumes pq: "P ==> Q" and nq: "~Q" shows "~P" apply (rule nq [THEN contrapos_nn]) apply (erule pq) done subsection {*Existential quantifier*} lemma exI: "P x ==> EX x::'a. P x" apply (unfold Ex_def) apply (iprover intro: allI allE impI mp) done lemma exE: assumes major: "EX x::'a. P(x)" and minor: "!!x. P(x) ==> Q" shows "Q" apply (rule major [unfolded Ex_def, THEN spec, THEN mp]) apply (iprover intro: impI [THEN allI] minor) done subsection {*Conjunction*} lemma conjI: "[| P; Q |] ==> P&Q" apply (unfold and_def) apply (iprover intro: impI [THEN allI] mp) done lemma conjunct1: "[| P & Q |] ==> P" apply (unfold and_def) apply (iprover intro: impI dest: spec mp) done lemma conjunct2: "[| P & Q |] ==> Q" apply (unfold and_def) apply (iprover intro: impI dest: spec mp) done lemma conjE: assumes major: "P&Q" and minor: "[| P; Q |] ==> R" shows "R" apply (rule minor) apply (rule major [THEN conjunct1]) apply (rule major [THEN conjunct2]) done lemma context_conjI: assumes prems: "P" "P ==> Q" shows "P & Q" by (iprover intro: conjI prems) subsection {*Disjunction*} lemma disjI1: "P ==> P|Q" apply (unfold or_def) apply (iprover intro: allI impI mp) done lemma disjI2: "Q ==> P|Q" apply (unfold or_def) apply (iprover intro: allI impI mp) done lemma disjE: assumes major: "P|Q" and minorP: "P ==> R" and minorQ: "Q ==> R" shows "R" by (iprover intro: minorP minorQ impI major [unfolded or_def, THEN spec, THEN mp, THEN mp]) subsection {*Classical logic*} lemma classical: assumes prem: "~P ==> P" shows "P" apply (rule True_or_False [THEN disjE, THEN eqTrueE]) apply assumption apply (rule notI [THEN prem, THEN eqTrueI]) apply (erule subst) apply assumption done lemmas ccontr = FalseE [THEN classical, standard] (*notE with premises exchanged; it discharges ~R so that it can be used to make elimination rules*) lemma rev_notE: assumes premp: "P" and premnot: "~R ==> ~P" shows "R" apply (rule ccontr) apply (erule notE [OF premnot premp]) done (*Double negation law*) lemma notnotD: "~~P ==> P" apply (rule classical) apply (erule notE) apply assumption done lemma contrapos_pp: assumes p1: "Q" and p2: "~P ==> ~Q" shows "P" by (iprover intro: classical p1 p2 notE) subsection {*Unique existence*} lemma ex1I: assumes prems: "P a" "!!x. P(x) ==> x=a" shows "EX! x. P(x)" by (unfold Ex1_def, iprover intro: prems exI conjI allI impI) text{*Sometimes easier to use: the premises have no shared variables. Safe!*} lemma ex_ex1I: assumes ex_prem: "EX x. P(x)" and eq: "!!x y. [| P(x); P(y) |] ==> x=y" shows "EX! x. P(x)" by (iprover intro: ex_prem [THEN exE] ex1I eq) lemma ex1E: assumes major: "EX! x. P(x)" and minor: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R" shows "R" apply (rule major [unfolded Ex1_def, THEN exE]) apply (erule conjE) apply (iprover intro: minor) done lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x" apply (erule ex1E) apply (rule exI) apply assumption done subsection {*THE: definite description operator*} lemma the_equality: assumes prema: "P a" and premx: "!!x. P x ==> x=a" shows "(THE x. P x) = a" apply (rule trans [OF _ the_eq_trivial]) apply (rule_tac f = "The" in arg_cong) apply (rule ext) apply (rule iffI) apply (erule premx) apply (erule ssubst, rule prema) done lemma theI: assumes "P a" and "!!x. P x ==> x=a" shows "P (THE x. P x)" by (iprover intro: prems the_equality [THEN ssubst]) lemma theI': "EX! x. P x ==> P (THE x. P x)" apply (erule ex1E) apply (erule theI) apply (erule allE) apply (erule mp) apply assumption done (*Easier to apply than theI: only one occurrence of P*) lemma theI2: assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x" shows "Q (THE x. P x)" by (iprover intro: prems theI) lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a" apply (rule the_equality) apply assumption apply (erule ex1E) apply (erule all_dupE) apply (drule mp) apply assumption apply (erule ssubst) apply (erule allE) apply (erule mp) apply assumption done lemma the_sym_eq_trivial: "(THE y. x=y) = x" apply (rule the_equality) apply (rule refl) apply (erule sym) done subsection {*Classical intro rules for disjunction and existential quantifiers*} lemma disjCI: assumes "~Q ==> P" shows "P|Q" apply (rule classical) apply (iprover intro: prems disjI1 disjI2 notI elim: notE) done lemma excluded_middle: "~P | P" by (iprover intro: disjCI) text{*case distinction as a natural deduction rule. Note that @{term "~P"} is the second case, not the first.*} lemma case_split_thm: assumes prem1: "P ==> Q" and prem2: "~P ==> Q" shows "Q" apply (rule excluded_middle [THEN disjE]) apply (erule prem2) apply (erule prem1) done (*Classical implies (-->) elimination. *) lemma impCE: assumes major: "P-->Q" and minor: "~P ==> R" "Q ==> R" shows "R" apply (rule excluded_middle [of P, THEN disjE]) apply (iprover intro: minor major [THEN mp])+ done (*This version of --> elimination works on Q before P. It works best for those cases in which P holds "almost everywhere". Can't install as default: would break old proofs.*) lemma impCE': assumes major: "P-->Q" and minor: "Q ==> R" "~P ==> R" shows "R" apply (rule excluded_middle [of P, THEN disjE]) apply (iprover intro: minor major [THEN mp])+ done (*Classical <-> elimination. *) lemma iffCE: assumes major: "P=Q" and minor: "[| P; Q |] ==> R" "[| ~P; ~Q |] ==> R" shows "R" apply (rule major [THEN iffE]) apply (iprover intro: minor elim: impCE notE) done lemma exCI: assumes "ALL x. ~P(x) ==> P(a)" shows "EX x. P(x)" apply (rule ccontr) apply (iprover intro: prems exI allI notI notE [of "\<exists>x. P x"]) done subsection {* Theory and package setup *} ML {* val eq_reflection = thm "eq_reflection" val refl = thm "refl" val subst = thm "subst" val ext = thm "ext" val impI = thm "impI" val mp = thm "mp" val True_def = thm "True_def" val All_def = thm "All_def" val Ex_def = thm "Ex_def" val False_def = thm "False_def" val not_def = thm "not_def" val and_def = thm "and_def" val or_def = thm "or_def" val Ex1_def = thm "Ex1_def" val iff = thm "iff" val True_or_False = thm "True_or_False" val Let_def = thm "Let_def" val if_def = thm "if_def" val sym = thm "sym" val ssubst = thm "ssubst" val trans = thm "trans" val def_imp_eq = thm "def_imp_eq" val box_equals = thm "box_equals" val fun_cong = thm "fun_cong" val arg_cong = thm "arg_cong" val cong = thm "cong" val iffI = thm "iffI" val iffD2 = thm "iffD2" val rev_iffD2 = thm "rev_iffD2" val iffD1 = thm "iffD1" val rev_iffD1 = thm "rev_iffD1" val iffE = thm "iffE" val TrueI = thm "TrueI" val eqTrueI = thm "eqTrueI" val eqTrueE = thm "eqTrueE" val allI = thm "allI" val spec = thm "spec" val allE = thm "allE" val all_dupE = thm "all_dupE" val FalseE = thm "FalseE" val False_neq_True = thm "False_neq_True" val notI = thm "notI" val False_not_True = thm "False_not_True" val True_not_False = thm "True_not_False" val notE = thm "notE" val notI2 = thm "notI2" val impE = thm "impE" val rev_mp = thm "rev_mp" val contrapos_nn = thm "contrapos_nn" val contrapos_pn = thm "contrapos_pn" val not_sym = thm "not_sym" val rev_contrapos = thm "rev_contrapos" val exI = thm "exI" val exE = thm "exE" val conjI = thm "conjI" val conjunct1 = thm "conjunct1" val conjunct2 = thm "conjunct2" val conjE = thm "conjE" val context_conjI = thm "context_conjI" val disjI1 = thm "disjI1" val disjI2 = thm "disjI2" val disjE = thm "disjE" val classical = thm "classical" val ccontr = thm "ccontr" val rev_notE = thm "rev_notE" val notnotD = thm "notnotD" val contrapos_pp = thm "contrapos_pp" val ex1I = thm "ex1I" val ex_ex1I = thm "ex_ex1I" val ex1E = thm "ex1E" val ex1_implies_ex = thm "ex1_implies_ex" val the_equality = thm "the_equality" val theI = thm "theI" val theI' = thm "theI'" val theI2 = thm "theI2" val the1_equality = thm "the1_equality" val the_sym_eq_trivial = thm "the_sym_eq_trivial" val disjCI = thm "disjCI" val excluded_middle = thm "excluded_middle" val case_split_thm = thm "case_split_thm" val impCE = thm "impCE" val impCE = thm "impCE" val iffCE = thm "iffCE" val exCI = thm "exCI" (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *) local fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t | wrong_prem (Bound _) = true | wrong_prem _ = false val filter_right = List.filter (fn t => not (wrong_prem (HOLogic.dest_Trueprop (hd (Thm.prems_of t))))) in fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]) fun smp_tac j = EVERY'[dresolve_tac (smp j), atac] end fun strip_tac i = REPEAT(resolve_tac [impI,allI] i) (*Obsolete form of disjunctive case analysis*) fun excluded_middle_tac sP = res_inst_tac [("Q",sP)] (excluded_middle RS disjE) fun case_tac a = res_inst_tac [("P",a)] case_split_thm *} theorems case_split = case_split_thm [case_names True False] ML {* structure ProjectRule = ProjectRuleFun (struct val conjunct1 = thm "conjunct1"; val conjunct2 = thm "conjunct2"; val mp = thm "mp"; end) *} subsubsection {* Intuitionistic Reasoning *} lemma impE': assumes 1: "P --> Q" and 2: "Q ==> R" and 3: "P --> Q ==> P" shows R proof - from 3 and 1 have P . with 1 have Q by (rule impE) with 2 show R . qed lemma allE': assumes 1: "ALL x. P x" and 2: "P x ==> ALL x. P x ==> Q" shows Q proof - from 1 have "P x" by (rule spec) from this and 1 show Q by (rule 2) qed lemma notE': assumes 1: "~ P" and 2: "~ P ==> P" shows R proof - from 2 and 1 have P . with 1 show R by (rule notE) qed lemmas [Pure.elim!] = disjE iffE FalseE conjE exE and [Pure.intro!] = iffI conjI impI TrueI notI allI refl and [Pure.elim 2] = allE notE' impE' and [Pure.intro] = exI disjI2 disjI1 lemmas [trans] = trans and [sym] = sym not_sym and [Pure.elim?] = iffD1 iffD2 impE subsubsection {* Atomizing meta-level connectives *} lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)" proof assume "!!x. P x" show "ALL x. P x" by (rule allI) next assume "ALL x. P x" thus "!!x. P x" by (rule allE) qed lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)" proof assume r: "A ==> B" show "A --> B" by (rule impI) (rule r) next assume "A --> B" and A thus B by (rule mp) qed lemma atomize_not: "(A ==> False) == Trueprop (~A)" proof assume r: "A ==> False" show "~A" by (rule notI) (rule r) next assume "~A" and A thus False by (rule notE) qed lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)" proof assume "x == y" show "x = y" by (unfold prems) (rule refl) next assume "x = y" thus "x == y" by (rule eq_reflection) qed lemma atomize_conj [atomize]: includes meta_conjunction_syntax shows "(A && B) == Trueprop (A & B)" proof assume conj: "A && B" show "A & B" proof (rule conjI) from conj show A by (rule conjunctionD1) from conj show B by (rule conjunctionD2) qed next assume conj: "A & B" show "A && B" proof - from conj show A .. from conj show B .. qed qed lemmas [symmetric, rulify] = atomize_all atomize_imp and [symmetric, defn] = atomize_all atomize_imp atomize_eq subsubsection {* Classical Reasoner setup *} use "cladata.ML" setup hypsubst_setup setup {* ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac) *} setup Classical.setup setup ResAtpSet.setup setup clasetup declare ex_ex1I [rule del, intro! 2] and ex1I [intro] lemmas [intro?] = ext and [elim?] = ex1_implies_ex use "blastdata.ML" setup Blast.setup subsubsection {* Simplifier setup *} lemma meta_eq_to_obj_eq: "x == y ==> x = y" proof - assume r: "x == y" show "x = y" by (unfold r) (rule refl) qed lemma eta_contract_eq: "(%s. f s) = f" .. lemma simp_thms: shows not_not: "(~ ~ P) = P" and Not_eq_iff: "((~P) = (~Q)) = (P = Q)" and "(P ~= Q) = (P = (~Q))" "(P | ~P) = True" "(~P | P) = True" "(x = x) = True" "(~True) = False" "(~False) = True" "(~P) ~= P" "P ~= (~P)" "(True=P) = P" "(P=True) = P" "(False=P) = (~P)" "(P=False) = (~P)" "(True --> P) = P" "(False --> P) = True" "(P --> True) = True" "(P --> P) = True" "(P --> False) = (~P)" "(P --> ~P) = (~P)" "(P & True) = P" "(True & P) = P" "(P & False) = False" "(False & P) = False" "(P & P) = P" "(P & (P & Q)) = (P & Q)" "(P & ~P) = False" "(~P & P) = False" "(P | True) = True" "(True | P) = True" "(P | False) = P" "(False | P) = P" "(P | P) = P" "(P | (P | Q)) = (P | Q)" and "(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x" -- {* needed for the one-point-rule quantifier simplification procs *} -- {* essential for termination!! *} and "!!P. (EX x. x=t & P(x)) = P(t)" "!!P. (EX x. t=x & P(x)) = P(t)" "!!P. (ALL x. x=t --> P(x)) = P(t)" "!!P. (ALL x. t=x --> P(x)) = P(t)" by (blast, blast, blast, blast, blast, iprover+) lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))" by iprover lemma ex_simps: "!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)" "!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))" "!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)" "!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))" "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)" "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))" -- {* Miniscoping: pushing in existential quantifiers. *} by (iprover | blast)+ lemma all_simps: "!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)" "!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))" "!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)" "!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))" "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)" "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))" -- {* Miniscoping: pushing in universal quantifiers. *} by (iprover | blast)+ lemma disj_absorb: "(A | A) = A" by blast lemma disj_left_absorb: "(A | (A | B)) = (A | B)" by blast lemma conj_absorb: "(A & A) = A" by blast lemma conj_left_absorb: "(A & (A & B)) = (A & B)" by blast lemma eq_ac: shows eq_commute: "(a=b) = (b=a)" and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))" and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+) lemma neq_commute: "(a~=b) = (b~=a)" by iprover lemma conj_comms: shows conj_commute: "(P&Q) = (Q&P)" and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+ lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover lemmas conj_ac = conj_commute conj_left_commute conj_assoc lemma disj_comms: shows disj_commute: "(P|Q) = (Q|P)" and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+ lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover lemmas disj_ac = disj_commute disj_left_commute disj_assoc lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by iprover lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *} lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast lemma not_iff: "(P~=Q) = (P = (~Q))" by blast lemma disj_not1: "(~P | Q) = (P --> Q)" by blast lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *} by blast lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q" -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *} -- {* cases boil down to the same thing. *} by blast lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover text {* \medskip The @{text "&"} congruence rule: not included by default! May slow rewrite proofs down by as much as 50\% *} lemma conj_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))" by iprover lemma rev_conj_cong: "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))" by iprover text {* The @{text "|"} congruence rule: not included by default! *} lemma disj_cong: "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))" by blast lemma eq_sym_conv: "(x = y) = (y = x)" by iprover text {* \medskip if-then-else rules *} lemma if_True: "(if True then x else y) = x" by (unfold if_def) blast lemma if_False: "(if False then x else y) = y" by (unfold if_def) blast lemma if_P: "P ==> (if P then x else y) = x" by (unfold if_def) blast lemma if_not_P: "~P ==> (if P then x else y) = y" by (unfold if_def) blast lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))" apply (rule case_split [of Q]) apply (simplesubst if_P) prefer 3 apply (simplesubst if_not_P, blast+) done lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))" by (simplesubst split_if, blast) lemmas if_splits = split_if split_if_asm lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))" by (rule split_if) lemma if_cancel: "(if c then x else x) = x" by (simplesubst split_if, blast) lemma if_eq_cancel: "(if x = y then y else x) = x" by (simplesubst split_if, blast) lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))" -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *} by (rule split_if) lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))" -- {* And this form is useful for expanding @{text if}s on the LEFT. *} apply (simplesubst split_if, blast) done lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover text {* \medskip let rules for simproc *} lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g" by (unfold Let_def) lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g" by (unfold Let_def) text {* The following copy of the implication operator is useful for fine-tuning congruence rules. It instructs the simplifier to simplify its premise. *} constdefs simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1) "simp_implies \<equiv> op ==>" lemma simp_impliesI: assumes PQ: "(PROP P \<Longrightarrow> PROP Q)" shows "PROP P =simp=> PROP Q" apply (unfold simp_implies_def) apply (rule PQ) apply assumption done lemma simp_impliesE: assumes PQ:"PROP P =simp=> PROP Q" and P: "PROP P" and QR: "PROP Q \<Longrightarrow> PROP R" shows "PROP R" apply (rule QR) apply (rule PQ [unfolded simp_implies_def]) apply (rule P) done lemma simp_implies_cong: assumes PP' :"PROP P == PROP P'" and P'QQ': "PROP P' ==> (PROP Q == PROP Q')" shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')" proof (unfold simp_implies_def, rule equal_intr_rule) assume PQ: "PROP P \<Longrightarrow> PROP Q" and P': "PROP P'" from PP' [symmetric] and P' have "PROP P" by (rule equal_elim_rule1) hence "PROP Q" by (rule PQ) with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1) next assume P'Q': "PROP P' \<Longrightarrow> PROP Q'" and P: "PROP P" from PP' and P have P': "PROP P'" by (rule equal_elim_rule1) hence "PROP Q'" by (rule P'Q') with P'QQ' [OF P', symmetric] show "PROP Q" by (rule equal_elim_rule1) qed text {* \medskip Actual Installation of the Simplifier. *} use "simpdata.ML" setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup setup Splitter.setup setup Clasimp.setup setup EqSubst.setup subsubsection {* Code generator setup *} types_code "bool" ("bool") attach (term_of) {* fun term_of_bool b = if b then HOLogic.true_const else HOLogic.false_const; *} attach (test) {* fun gen_bool i = one_of [false, true]; *} "prop" ("bool") attach (term_of) {* fun term_of_prop b = HOLogic.mk_Trueprop (if b then HOLogic.true_const else HOLogic.false_const); *} consts_code "Trueprop" ("(_)") "True" ("true") "False" ("false") "Not" ("not") "op |" ("(_ orelse/ _)") "op &" ("(_ andalso/ _)") "HOL.If" ("(if _/ then _/ else _)") ML {* local fun eq_codegen thy defs gr dep thyname b t = (case strip_comb t of (Const ("op =", Type (_, [Type ("fun", _), _])), _) => NONE | (Const ("op =", _), [t, u]) => let val (gr', pt) = Codegen.invoke_codegen thy defs dep thyname false (gr, t); val (gr'', pu) = Codegen.invoke_codegen thy defs dep thyname false (gr', u); val (gr''', _) = Codegen.invoke_tycodegen thy defs dep thyname false (gr'', HOLogic.boolT) in SOME (gr''', Codegen.parens (Pretty.block [pt, Pretty.str " =", Pretty.brk 1, pu])) end | (t as Const ("op =", _), ts) => SOME (Codegen.invoke_codegen thy defs dep thyname b (gr, Codegen.eta_expand t ts 2)) | _ => NONE); exception Evaluation of term; fun evaluation_oracle (thy, Evaluation t) = Logic.mk_equals (t, Codegen.eval_term thy t); fun evaluation_conv ct = let val {sign, t, ...} = rep_cterm ct in Thm.invoke_oracle_i sign "HOL.Evaluation" (sign, Evaluation t) end; fun evaluation_tac i = Tactical.PRIMITIVE (Drule.fconv_rule (Drule.goals_conv (equal i) evaluation_conv)); val evaluation_meth = Method.no_args (Method.METHOD (fn _ => evaluation_tac 1 THEN rtac TrueI 1)); in val eq_codegen_setup = Codegen.add_codegen "eq_codegen" eq_codegen; val evaluation_oracle_setup = Theory.add_oracle ("Evaluation", evaluation_oracle) #> Method.add_method ("evaluation", evaluation_meth, "solve goal by evaluation"); end; *} setup eq_codegen_setup setup evaluation_oracle_setup subsection {* Other simple lemmas *} declare disj_absorb [simp] conj_absorb [simp] lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x" by blast+ theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))" apply (rule iffI) apply (rule_tac a = "%x. THE y. P x y" in ex1I) apply (fast dest!: theI') apply (fast intro: ext the1_equality [symmetric]) apply (erule ex1E) apply (rule allI) apply (rule ex1I) apply (erule spec) apply (erule_tac x = "%z. if z = x then y else f z" in allE) apply (erule impE) apply (rule allI) apply (rule_tac P = "xa = x" in case_split_thm) apply (drule_tac [3] x = x in fun_cong, simp_all) done text{*Needs only HOL-lemmas:*} lemma mk_left_commute: assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and c: "\<And>x y. f x y = f y x" shows "f x (f y z) = f y (f x z)" by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]]) subsection {* Generic cases and induction *} constdefs induct_forall where "induct_forall P == \<forall>x. P x" induct_implies where "induct_implies A B == A \<longrightarrow> B" induct_equal where "induct_equal x y == x = y" induct_conj where "induct_conj A B == A \<and> B" lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))" by (unfold atomize_all induct_forall_def) lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)" by (unfold atomize_imp induct_implies_def) lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)" by (unfold atomize_eq induct_equal_def) lemma induct_conj_eq: includes meta_conjunction_syntax shows "(A && B) == Trueprop (induct_conj A B)" by (unfold atomize_conj induct_conj_def) lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq lemmas induct_rulify [symmetric, standard] = induct_atomize lemmas induct_rulify_fallback = induct_forall_def induct_implies_def induct_equal_def induct_conj_def lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) = induct_conj (induct_forall A) (induct_forall B)" by (unfold induct_forall_def induct_conj_def) iprover lemma induct_implies_conj: "induct_implies C (induct_conj A B) = induct_conj (induct_implies C A) (induct_implies C B)" by (unfold induct_implies_def induct_conj_def) iprover lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)" proof assume r: "induct_conj A B ==> PROP C" and A B show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`) next assume r: "A ==> B ==> PROP C" and "induct_conj A B" show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def]) qed lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry hide const induct_forall induct_implies induct_equal induct_conj text {* Method setup. *} ML {* structure InductMethod = InductMethodFun (struct val cases_default = thm "case_split" val atomize = thms "induct_atomize" val rulify = thms "induct_rulify" val rulify_fallback = thms "induct_rulify_fallback" end); *} setup InductMethod.setup subsubsection {*Tags, for the ATP Linkup *} constdefs tag :: "bool => bool" "tag P == P" text{*These label the distinguished literals of introduction and elimination rules.*} lemma tagI: "P ==> tag P" by (simp add: tag_def) lemma tagD: "tag P ==> P" by (simp add: tag_def) text{*Applications of "tag" to True and False must go!*} lemma tag_True: "tag True = True" by (simp add: tag_def) lemma tag_False: "tag False = False" by (simp add: tag_def) subsection {* Code generator setup *} code_alias bool "HOL.bool" True "HOL.True" False "HOL.False" "op =" "HOL.op_eq" "op -->" "HOL.op_implies" "op &" "HOL.op_and" "op |" "HOL.op_or" "op +" "HOL.op_add" "op -" "HOL.op_minus" "op *" "HOL.op_times" Not "HOL.not" uminus "HOL.uminus" arbitrary "HOL.arbitrary" code_syntax_const "op =" (* an intermediate solution for polymorphic equality *) ml (infixl 6 "=") haskell (infixl 4 "==") arbitrary ml ("raise/ (Fail/ \"non-defined-result\")") haskell ("error/ \"non-defined result\"") end