diff -r ad45e477926c -r c480add17d52 HOL.ML --- a/HOL.ML Thu Dec 08 12:50:38 1994 +0100 +++ b/HOL.ML Fri Dec 09 13:39:05 1994 +0100 @@ -9,75 +9,16 @@ open HOL; -signature HOL_LEMMAS = - sig - val allE : thm - val all_dupE : thm - val allI : thm - val arg_cong : thm - val fun_cong : thm - val box_equals: thm - val case_tac : string -> int -> tactic - val ccontr : thm - val classical : thm - val cong : thm - val conjunct1 : thm - val conjunct2 : thm - val conjE : thm - val conjI : thm - val contrapos : thm - val disjCI : thm - val disjE : thm - val disjI1 : thm - val disjI2 : thm - val eqTrueI : thm - val eqTrueE : thm - val ex1E : thm - val ex1I : thm - val exCI : thm - val exI : thm - val exE : thm - val excluded_middle : thm - val excluded_middle_tac : string -> int -> tactic - val False_neq_True : thm - val FalseE : thm - val iffCE : thm - val iffD1 : thm - val iffD2 : thm - val iffE : thm - val iffI : thm - val impCE : thm - val impE : thm - val not_sym : thm - val notE : thm - val notI : thm - val notnotD : thm - val rev_mp : thm - val select_equality : thm - val selectI2 : thm - val spec : thm - val sstac : thm list -> int -> tactic - val ssubst : thm - val stac : thm -> int -> tactic - val strip_tac : int -> tactic - val sym : thm - val trans : thm - val TrueI : thm - end; - -structure HOL_Lemmas : HOL_LEMMAS = - -struct (** Equality **) -val sym = prove_goal HOL.thy "s=t ==> t=s" +qed_goal "sym" HOL.thy "s=t ==> t=s" (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]); (*calling "standard" reduces maxidx to 0*) val ssubst = standard (sym RS subst); -val trans = prove_goal HOL.thy "[| r=s; s=t |] ==> r=t" +qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t" (fn prems => [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]); @@ -85,7 +26,7 @@ a = b | | c = d *) -val box_equals = prove_goal HOL.thy +qed_goal "box_equals" HOL.thy "[| a=b; a=c; b=d |] ==> c=d" (fn prems=> [ (rtac trans 1), @@ -96,58 +37,58 @@ (** Congruence rules for meta-application **) (*similar to AP_THM in Gordon's HOL*) -val fun_cong = prove_goal HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)" +qed_goal "fun_cong" HOL.thy "(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*) -val arg_cong = prove_goal HOL.thy "x=y ==> f(x)=f(y)" +qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)" (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]); -val cong = prove_goal HOL.thy +qed_goal "cong" HOL.thy "[| 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 **) -val iffI = prove_goal HOL.thy +qed_goal "iffI" HOL.thy "[| P ==> Q; Q ==> P |] ==> P=Q" (fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]); -val iffD2 = prove_goal HOL.thy "[| P=Q; Q |] ==> P" +qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P" (fn prems => [rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]); val iffD1 = sym RS iffD2; -val iffE = prove_goal HOL.thy +qed_goal "iffE" HOL.thy "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R" (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]); (** True **) -val TrueI = prove_goalw HOL.thy [True_def] "True" +qed_goalw "TrueI" HOL.thy [True_def] "True" (fn _ => [rtac refl 1]); -val eqTrueI = prove_goal HOL.thy "P ==> P=True" +qed_goal "eqTrueI " HOL.thy "P ==> P=True" (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]); -val eqTrueE = prove_goal HOL.thy "P=True ==> P" +qed_goal "eqTrueE" HOL.thy "P=True ==> P" (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]); (** Universal quantifier **) -val allI = prove_goalw HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)" +qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)" (fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]); -val spec = prove_goalw HOL.thy [All_def] "! x::'a.P(x) ==> P(x)" +qed_goalw "spec" HOL.thy [All_def] "! x::'a.P(x) ==> P(x)" (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]); -val allE = prove_goal HOL.thy "[| !x.P(x); P(x) ==> R |] ==> R" +qed_goal "allE" HOL.thy "[| !x.P(x); P(x) ==> R |] ==> R" (fn major::prems=> [ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]); -val all_dupE = prove_goal HOL.thy +qed_goal "all_dupE" HOL.thy "[| ! x.P(x); [| P(x); ! x.P(x) |] ==> R |] ==> R" (fn prems => [ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]); @@ -156,31 +97,31 @@ (** False ** Depends upon spec; it is impossible to do propositional logic before quantifiers! **) -val FalseE = prove_goalw HOL.thy [False_def] "False ==> P" +qed_goalw "FalseE" HOL.thy [False_def] "False ==> P" (fn [major] => [rtac (major RS spec) 1]); -val False_neq_True = prove_goal HOL.thy "False=True ==> P" +qed_goal "False_neq_True" HOL.thy "False=True ==> P" (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]); (** Negation **) -val notI = prove_goalw HOL.thy [not_def] "(P ==> False) ==> ~P" +qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P" (fn prems=> [rtac impI 1, eresolve_tac prems 1]); -val notE = prove_goalw HOL.thy [not_def] "[| ~P; P |] ==> R" +qed_goalw "notE" HOL.thy [not_def] "[| ~P; P |] ==> R" (fn prems => [rtac (prems MRS mp RS FalseE) 1]); (** Implication **) -val impE = prove_goal HOL.thy "[| P-->Q; P; Q ==> R |] ==> R" +qed_goal "impE" HOL.thy "[| P-->Q; P; Q ==> R |] ==> R" (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]); (* Reduces Q to P-->Q, allowing substitution in P. *) -val rev_mp = prove_goal HOL.thy "[| P; P --> Q |] ==> Q" +qed_goal "rev_mp" HOL.thy "[| P; P --> Q |] ==> Q" (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]); -val contrapos = prove_goal HOL.thy "[| ~Q; P==>Q |] ==> ~P" +qed_goal "contrapos" HOL.thy "[| ~Q; P==>Q |] ==> ~P" (fn [major,minor]=> [ (rtac (major RS notE RS notI) 1), (etac minor 1) ]); @@ -191,49 +132,49 @@ (** Existential quantifier **) -val exI = prove_goalw HOL.thy [Ex_def] "P(x) ==> ? x::'a.P(x)" +qed_goalw "exI" HOL.thy [Ex_def] "P(x) ==> ? x::'a.P(x)" (fn prems => [rtac selectI 1, resolve_tac prems 1]); -val exE = prove_goalw HOL.thy [Ex_def] +qed_goalw "exE" HOL.thy [Ex_def] "[| ? x::'a.P(x); !!x. P(x) ==> Q |] ==> Q" (fn prems => [REPEAT(resolve_tac prems 1)]); (** Conjunction **) -val conjI = prove_goalw HOL.thy [and_def] "[| P; Q |] ==> P&Q" +qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q" (fn prems => [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]); -val conjunct1 = prove_goalw HOL.thy [and_def] "[| P & Q |] ==> P" +qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P" (fn prems => [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]); -val conjunct2 = prove_goalw HOL.thy [and_def] "[| P & Q |] ==> Q" +qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q" (fn prems => [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]); -val conjE = prove_goal HOL.thy "[| P&Q; [| P; Q |] ==> R |] ==> R" +qed_goal "conjE" HOL.thy "[| P&Q; [| P; Q |] ==> R |] ==> R" (fn prems => [cut_facts_tac prems 1, resolve_tac prems 1, etac conjunct1 1, etac conjunct2 1]); (** Disjunction *) -val disjI1 = prove_goalw HOL.thy [or_def] "P ==> P|Q" +qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q" (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]); -val disjI2 = prove_goalw HOL.thy [or_def] "Q ==> P|Q" +qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q" (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]); -val disjE = prove_goalw HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R" +qed_goalw "disjE" HOL.thy [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 **) -val classical = prove_goalw HOL.thy [not_def] "(~P ==> P) ==> P" +qed_goalw "classical" HOL.thy [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, @@ -242,19 +183,19 @@ val ccontr = FalseE RS classical; (*Double negation law*) -val notnotD = prove_goal HOL.thy "~~P ==> P" +qed_goal "notnotD" HOL.thy "~~P ==> P" (fn [major]=> [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]); (** Unique existence **) -val ex1I = prove_goalw HOL.thy [Ex1_def] +qed_goalw "ex1I" HOL.thy [Ex1_def] "[| P(a); !!x. P(x) ==> x=a |] ==> ?! x. P(x)" (fn prems => [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]); -val ex1E = prove_goalw HOL.thy [Ex1_def] +qed_goalw "ex1E" HOL.thy [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)]); @@ -263,13 +204,13 @@ (** Select: Hilbert's Epsilon-operator **) (*Easier to apply than selectI: conclusion has only one occurrence of P*) -val selectI2 = prove_goal HOL.thy +qed_goal "selectI2" HOL.thy "[| P(a); !!x. P(x) ==> Q(x) |] ==> Q(@x.P(x))" (fn prems => [ resolve_tac prems 1, rtac selectI 1, resolve_tac prems 1 ]); -val select_equality = prove_goal HOL.thy +qed_goal "select_equality" HOL.thy "[| P(a); !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a" (fn prems => [ rtac selectI2 1, REPEAT (ares_tac prems 1) ]); @@ -277,13 +218,13 @@ (** Classical intro rules for disjunction and existential quantifiers *) -val disjCI = prove_goal HOL.thy "(~Q ==> P) ==> P|Q" +qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q" (fn prems=> [ (rtac classical 1), (REPEAT (ares_tac (prems@[disjI1,notI]) 1)), (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]); -val excluded_middle = prove_goal HOL.thy "~P | P" +qed_goal "excluded_middle" HOL.thy "~P | P" (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]); (*For disjunctive case analysis*) @@ -291,20 +232,20 @@ res_inst_tac [("Q",sP)] (excluded_middle RS disjE); (*Classical implies (-->) elimination. *) -val impCE = prove_goal HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R" +qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R" (fn major::prems=> [ rtac (excluded_middle RS disjE) 1, REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]); (*Classical <-> elimination. *) -val iffCE = prove_goal HOL.thy +qed_goal "iffCE" HOL.thy "[| P=Q; [| P; Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R" (fn major::prems => [ (rtac (major RS iffE) 1), (REPEAT (DEPTH_SOLVE_1 (eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]); -val exCI = prove_goal HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x.P(x)" +qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x.P(x)" (fn prems=> [ (rtac ccontr 1), (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1)) ]); @@ -312,7 +253,7 @@ (* case distinction *) -val case_split_thm = prove_goal HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q" +qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q" (fn [p1,p2] => [cut_facts_tac [excluded_middle] 1, etac disjE 1, etac p2 1, etac p1 1]); @@ -323,8 +264,3 @@ fun stac th = rtac(th RS ssubst); fun sstac ths = EVERY' (map stac ths); fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); - - -end; - -open HOL_Lemmas;