--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HOL_lemmas.ML Wed Aug 25 20:49:02 1999 +0200
@@ -0,0 +1,484 @@
+(* Title: HOL/HOL_lemmas.ML
+ ID: $Id$
+ Author: Tobias Nipkow
+ Copyright 1991 University of Cambridge
+
+Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68.
+*)
+
+(* ML bindings *)
+
+val plusI = thm "plusI";
+val minusI = thm "minusI";
+val timesI = thm "timesI";
+val powerI = thm "powerI";
+val eq_reflection = thm "eq_reflection";
+val refl = thm "refl";
+val subst = thm "subst";
+val ext = thm "ext";
+val selectI = thm "selectI";
+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 arbitrary_def = thm "arbitrary_def";
+
+
+(** Equality **)
+section "=";
+
+qed_goal "sym" (the_context ()) "s=t ==> t=s"
+ (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);
+
+(*calling "standard" reduces maxidx to 0*)
+bind_thm ("ssubst", (sym RS subst));
+
+qed_goal "trans" (the_context ()) "[| r=s; s=t |] ==> r=t"
+ (fn prems =>
+ [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);
+
+val prems = goal (the_context ()) "(A == B) ==> A = B";
+by (rewrite_goals_tac prems);
+by (rtac refl 1);
+qed "def_imp_eq";
+
+(*Useful with eresolve_tac for proving equalties from known equalities.
+ a = b
+ | |
+ c = d *)
+Goal "[| a=b; a=c; b=d |] ==> c=d";
+by (rtac trans 1);
+by (rtac trans 1);
+by (rtac sym 1);
+by (REPEAT (assume_tac 1)) ;
+qed "box_equals";
+
+
+(** Congruence rules for meta-application **)
+section "Congruence";
+
+(*similar to AP_THM in Gordon's HOL*)
+qed_goal "fun_cong" (the_context ()) "(f::'a=>'b) = g ==> f(x)=g(x)"
+ (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
+
+(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
+qed_goal "arg_cong" (the_context ()) "x=y ==> f(x)=f(y)"
+ (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
+
+qed_goal "cong" (the_context ())
+ "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
+ (fn [prem1,prem2] =>
+ [rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]);
+
+
+(** Equality of booleans -- iff **)
+section "iff";
+
+val prems = Goal
+ "[| P ==> Q; Q ==> P |] ==> P=Q";
+by (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1));
+qed "iffI";
+
+qed_goal "iffD2" (the_context ()) "[| P=Q; Q |] ==> P"
+ (fn prems =>
+ [rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);
+
+qed_goal "rev_iffD2" (the_context ()) "!!P. [| Q; P=Q |] ==> P"
+ (fn _ => [etac iffD2 1, assume_tac 1]);
+
+bind_thm ("iffD1", sym RS iffD2);
+bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2));
+
+qed_goal "iffE" (the_context ())
+ "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
+ (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);
+
+
+(** True **)
+section "True";
+
+qed_goalw "TrueI" (the_context ()) [True_def] "True"
+ (fn _ => [(rtac refl 1)]);
+
+qed_goal "eqTrueI" (the_context ()) "P ==> P=True"
+ (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
+
+qed_goal "eqTrueE" (the_context ()) "P=True ==> P"
+ (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
+
+
+(** Universal quantifier **)
+section "!";
+
+qed_goalw "allI" (the_context ()) [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
+ (fn prems => [(resolve_tac (prems RL [eqTrueI RS ext]) 1)]);
+
+qed_goalw "spec" (the_context ()) [All_def] "! x::'a. P(x) ==> P(x)"
+ (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
+
+val major::prems= goal (the_context ()) "[| !x. P(x); P(x) ==> R |] ==> R";
+by (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ;
+qed "allE";
+
+val prems = goal (the_context ())
+ "[| ! x. P(x); [| P(x); ! x. P(x) |] ==> R |] ==> R";
+by (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ;
+qed "all_dupE";
+
+
+(** False ** Depends upon spec; it is impossible to do propositional logic
+ before quantifiers! **)
+section "False";
+
+qed_goalw "FalseE" (the_context ()) [False_def] "False ==> P"
+ (fn [major] => [rtac (major RS spec) 1]);
+
+qed_goal "False_neq_True" (the_context ()) "False=True ==> P"
+ (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
+
+
+(** Negation **)
+section "~";
+
+qed_goalw "notI" (the_context ()) [not_def] "(P ==> False) ==> ~P"
+ (fn prems=> [rtac impI 1, eresolve_tac prems 1]);
+
+qed_goal "False_not_True" (the_context ()) "False ~= True"
+ (fn _ => [rtac notI 1, etac False_neq_True 1]);
+
+qed_goal "True_not_False" (the_context ()) "True ~= False"
+ (fn _ => [rtac notI 1, dtac sym 1, etac False_neq_True 1]);
+
+qed_goalw "notE" (the_context ()) [not_def] "[| ~P; P |] ==> R"
+ (fn prems => [rtac (prems MRS mp RS FalseE) 1]);
+
+bind_thm ("classical2", notE RS notI);
+
+qed_goal "rev_notE" (the_context ()) "!!P R. [| P; ~P |] ==> R"
+ (fn _ => [REPEAT (ares_tac [notE] 1)]);
+
+
+(** Implication **)
+section "-->";
+
+val prems = Goal "[| P-->Q; P; Q ==> R |] ==> R";
+by (REPEAT (resolve_tac (prems@[mp]) 1));
+qed "impE";
+
+(* Reduces Q to P-->Q, allowing substitution in P. *)
+Goal "[| P; P --> Q |] ==> Q";
+by (REPEAT (ares_tac [mp] 1)) ;
+qed "rev_mp";
+
+val [major,minor] = Goal "[| ~Q; P==>Q |] ==> ~P";
+by (rtac (major RS notE RS notI) 1);
+by (etac minor 1) ;
+qed "contrapos";
+
+val [major,minor] = Goal "[| P==>Q; ~Q |] ==> ~P";
+by (rtac (minor RS contrapos) 1);
+by (etac major 1) ;
+qed "rev_contrapos";
+
+(* ~(?t = ?s) ==> ~(?s = ?t) *)
+bind_thm("not_sym", sym COMP rev_contrapos);
+
+
+(** Existential quantifier **)
+section "?";
+
+qed_goalw "exI" (the_context ()) [Ex_def] "P x ==> ? x::'a. P x"
+ (fn prems => [rtac selectI 1, resolve_tac prems 1]);
+
+qed_goalw "exE" (the_context ()) [Ex_def]
+ "[| ? x::'a. P(x); !!x. P(x) ==> Q |] ==> Q"
+ (fn prems => [REPEAT(resolve_tac prems 1)]);
+
+
+(** Conjunction **)
+section "&";
+
+qed_goalw "conjI" (the_context ()) [and_def] "[| P; Q |] ==> P&Q"
+ (fn prems =>
+ [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
+
+qed_goalw "conjunct1" (the_context ()) [and_def] "[| P & Q |] ==> P"
+ (fn prems =>
+ [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
+
+qed_goalw "conjunct2" (the_context ()) [and_def] "[| P & Q |] ==> Q"
+ (fn prems =>
+ [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
+
+qed_goal "conjE" (the_context ()) "[| P&Q; [| P; Q |] ==> R |] ==> R"
+ (fn prems =>
+ [cut_facts_tac prems 1, resolve_tac prems 1,
+ etac conjunct1 1, etac conjunct2 1]);
+
+qed_goal "context_conjI" (the_context ()) "[| P; P ==> Q |] ==> P & Q"
+ (fn prems => [REPEAT(resolve_tac (conjI::prems) 1)]);
+
+
+(** Disjunction *)
+section "|";
+
+qed_goalw "disjI1" (the_context ()) [or_def] "P ==> P|Q"
+ (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
+
+qed_goalw "disjI2" (the_context ()) [or_def] "Q ==> P|Q"
+ (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
+
+qed_goalw "disjE" (the_context ()) [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
+ (fn [a1,a2,a3] =>
+ [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
+ rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
+
+
+(** CCONTR -- classical logic **)
+section "classical logic";
+
+qed_goalw "classical" (the_context ()) [not_def] "(~P ==> P) ==> P"
+ (fn [prem] =>
+ [rtac (True_or_False RS (disjE RS eqTrueE)) 1, assume_tac 1,
+ rtac (impI RS prem RS eqTrueI) 1,
+ etac subst 1, assume_tac 1]);
+
+val ccontr = FalseE RS classical;
+
+(*Double negation law*)
+Goal "~~P ==> P";
+by (rtac classical 1);
+by (etac notE 1);
+by (assume_tac 1);
+qed "notnotD";
+
+val [p1,p2] = Goal "[| Q; ~ P ==> ~ Q |] ==> P";
+by (rtac classical 1);
+by (dtac p2 1);
+by (etac notE 1);
+by (rtac p1 1);
+qed "contrapos2";
+
+val [p1,p2] = Goal "[| P; Q ==> ~ P |] ==> ~ Q";
+by (rtac notI 1);
+by (dtac p2 1);
+by (etac notE 1);
+by (rtac p1 1);
+qed "swap2";
+
+(** Unique existence **)
+section "?!";
+
+qed_goalw "ex1I" (the_context ()) [Ex1_def]
+ "[| P(a); !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
+ (fn prems =>
+ [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
+
+(*Sometimes easier to use: the premises have no shared variables. Safe!*)
+val [ex,eq] = Goal
+ "[| ? x. P(x); !!x y. [| P(x); P(y) |] ==> x=y |] ==> ?! x. P(x)";
+by (rtac (ex RS exE) 1);
+by (REPEAT (ares_tac [ex1I,eq] 1)) ;
+qed "ex_ex1I";
+
+qed_goalw "ex1E" (the_context ()) [Ex1_def]
+ "[| ?! x. P(x); !!x. [| P(x); ! y. P(y) --> y=x |] ==> R |] ==> R"
+ (fn major::prems =>
+ [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
+
+Goal "?! x. P x ==> ? x. P x";
+by (etac ex1E 1);
+by (rtac exI 1);
+by (assume_tac 1);
+qed "ex1_implies_ex";
+
+
+(** Select: Hilbert's Epsilon-operator **)
+section "@";
+
+(*Easier to apply than selectI: conclusion has only one occurrence of P*)
+val prems = Goal
+ "[| P a; !!x. P x ==> Q x |] ==> Q (@x. P x)";
+by (resolve_tac prems 1);
+by (rtac selectI 1);
+by (resolve_tac prems 1) ;
+qed "selectI2";
+
+(*Easier to apply than selectI2 if witness ?a comes from an EX-formula*)
+qed_goal "selectI2EX" (the_context ())
+ "[| ? a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)"
+(fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]);
+
+val prems = Goal
+ "[| P a; !!x. P x ==> x=a |] ==> (@x. P x) = a";
+by (rtac selectI2 1);
+by (REPEAT (ares_tac prems 1)) ;
+qed "select_equality";
+
+Goalw [Ex1_def] "[| ?!x. P x; P a |] ==> (@x. P x) = a";
+by (rtac select_equality 1);
+by (atac 1);
+by (etac exE 1);
+by (etac conjE 1);
+by (rtac allE 1);
+by (atac 1);
+by (etac impE 1);
+by (atac 1);
+by (etac ssubst 1);
+by (etac allE 1);
+by (etac mp 1);
+by (atac 1);
+qed "select1_equality";
+
+Goal "P (@ x. P x) = (? x. P x)";
+by (rtac iffI 1);
+by (etac exI 1);
+by (etac exE 1);
+by (etac selectI 1);
+qed "select_eq_Ex";
+
+Goal "(@y. y=x) = x";
+by (rtac select_equality 1);
+by (rtac refl 1);
+by (atac 1);
+qed "Eps_eq";
+
+Goal "(Eps (op = x)) = x";
+by (rtac select_equality 1);
+by (rtac refl 1);
+by (etac sym 1);
+qed "Eps_sym_eq";
+
+(** Classical intro rules for disjunction and existential quantifiers *)
+section "classical intro rules";
+
+val prems= Goal "(~Q ==> P) ==> P|Q";
+by (rtac classical 1);
+by (REPEAT (ares_tac (prems@[disjI1,notI]) 1));
+by (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ;
+qed "disjCI";
+
+Goal "~P | P";
+by (REPEAT (ares_tac [disjCI] 1)) ;
+qed "excluded_middle";
+
+(*For disjunctive case analysis*)
+fun excluded_middle_tac sP =
+ res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
+
+(*Classical implies (-->) elimination. *)
+val major::prems = Goal "[| P-->Q; ~P ==> R; Q ==> R |] ==> R";
+by (rtac (excluded_middle RS disjE) 1);
+by (REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1)));
+qed "impCE";
+
+(*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.*)
+val major::prems = Goal
+ "[| P-->Q; Q ==> R; ~P ==> R |] ==> R";
+by (resolve_tac [excluded_middle RS disjE] 1);
+by (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ;
+qed "impCE'";
+
+(*Classical <-> elimination. *)
+val major::prems = Goal
+ "[| P=Q; [| P; Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R";
+by (rtac (major RS iffE) 1);
+by (REPEAT (DEPTH_SOLVE_1
+ (eresolve_tac ([asm_rl,impCE,notE]@prems) 1)));
+qed "iffCE";
+
+val prems = Goal "(! x. ~P(x) ==> P(a)) ==> ? x. P(x)";
+by (rtac ccontr 1);
+by (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1)) ;
+qed "exCI";
+
+
+(* case distinction *)
+
+qed_goal "case_split_thm" (the_context ()) "[| P ==> Q; ~P ==> Q |] ==> Q"
+ (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
+ etac p2 1, etac p1 1]);
+
+fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
+
+
+(** Standard abbreviations **)
+
+(*Apply an equality or definition ONCE.
+ Fails unless the substitution has an effect*)
+fun stac th =
+ let val th' = th RS def_imp_eq handle THM _ => th
+ in CHANGED_GOAL (rtac (th' RS ssubst))
+ end;
+
+fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);
+
+
+(** strip ! and --> from proved goal while preserving !-bound var names **)
+
+local
+
+(* Use XXX to avoid forall_intr failing because of duplicate variable name *)
+val myspec = read_instantiate [("P","?XXX")] spec;
+val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec;
+val cvx = cterm_of (#sign(rep_thm myspec)) vx;
+val aspec = forall_intr cvx myspec;
+
+in
+
+fun RSspec th =
+ (case concl_of th of
+ _ $ (Const("All",_) $ Abs(a,_,_)) =>
+ let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
+ in th RS forall_elim ca aspec end
+ | _ => raise THM("RSspec",0,[th]));
+
+fun RSmp th =
+ (case concl_of th of
+ _ $ (Const("op -->",_)$_$_) => th RS mp
+ | _ => raise THM("RSmp",0,[th]));
+
+fun normalize_thm funs =
+ let fun trans [] th = th
+ | trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
+ in zero_var_indexes o trans funs end;
+
+fun qed_spec_mp name =
+ let val thm = normalize_thm [RSspec,RSmp] (result())
+ in ThmDatabase.ml_store_thm(name, thm) end;
+
+fun qed_goal_spec_mp name thy s p =
+ bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));
+
+fun qed_goalw_spec_mp name thy defs s p =
+ bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));
+
+end;
+
+
+(* attributes *)
+
+local
+
+fun gen_rulify x =
+ Attrib.no_args (Drule.rule_attribute (fn _ => (normalize_thm [RSspec, RSmp]))) x;
+
+in
+
+val attrib_setup =
+ [Attrib.add_attributes
+ [("rulify", (gen_rulify, gen_rulify), "put theorem into standard rule form")]];
+
+end;