diff -r f04b33ce250f -r a4dc62a46ee4 HOL.ML --- a/HOL.ML Tue Oct 24 14:59:17 1995 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,266 +0,0 @@ -(* Title: HOL/hol.ML - ID: $Id$ - Author: Tobias Nipkow - Copyright 1991 University of Cambridge - -For hol.thy -Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68 -*) - -open HOL; - - -(** Equality **) - -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*) -bind_thm ("ssubst", (sym RS subst)); - -qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t" - (fn prems => - [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]); - -(*Useful with eresolve_tac for proving equalties from known equalities. - a = b - | | - c = d *) -qed_goal "box_equals" HOL.thy - "[| a=b; a=c; b=d |] ==> c=d" - (fn prems=> - [ (rtac trans 1), - (rtac trans 1), - (rtac sym 1), - (REPEAT (resolve_tac prems 1)) ]); - -(** Congruence rules for meta-application **) - -(*similar to AP_THM in Gordon's HOL*) -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*) -qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)" - (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]); - -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 **) - -qed_goal "iffI" HOL.thy - "[| P ==> Q; Q ==> P |] ==> P=Q" - (fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]); - -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; - -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 **) - -qed_goalw "TrueI" HOL.thy [True_def] "True" - (fn _ => [rtac refl 1]); - -qed_goal "eqTrueI " HOL.thy "P ==> P=True" - (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]); - -qed_goal "eqTrueE" HOL.thy "P=True ==> P" - (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]); - -(** Universal quantifier **) - -qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)" - (fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]); - -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]); - -qed_goal "allE" HOL.thy "[| !x.P(x); P(x) ==> R |] ==> R" - (fn major::prems=> - [ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]); - -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)) ]); - - -(** False ** Depends upon spec; it is impossible to do propositional logic - before quantifiers! **) - -qed_goalw "FalseE" HOL.thy [False_def] "False ==> P" - (fn [major] => [rtac (major RS spec) 1]); - -qed_goal "False_neq_True" HOL.thy "False=True ==> P" - (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]); - - -(** Negation **) - -qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P" - (fn prems=> [rtac impI 1, eresolve_tac prems 1]); - -qed_goalw "notE" HOL.thy [not_def] "[| ~P; P |] ==> R" - (fn prems => [rtac (prems MRS mp RS FalseE) 1]); - -(** Implication **) - -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. *) -qed_goal "rev_mp" HOL.thy "[| P; P --> Q |] ==> Q" - (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]); - -qed_goal "contrapos" HOL.thy "[| ~Q; P==>Q |] ==> ~P" - (fn [major,minor]=> - [ (rtac (major RS notE RS notI) 1), - (etac minor 1) ]); - -(* ~(?t = ?s) ==> ~(?s = ?t) *) -val [not_sym] = compose(sym,2,contrapos); - - -(** Existential quantifier **) - -qed_goalw "exI" HOL.thy [Ex_def] "P(x) ==> ? x::'a.P(x)" - (fn prems => [rtac selectI 1, resolve_tac prems 1]); - -qed_goalw "exE" HOL.thy [Ex_def] - "[| ? x::'a.P(x); !!x. P(x) ==> Q |] ==> Q" - (fn prems => [REPEAT(resolve_tac prems 1)]); - - -(** Conjunction **) - -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)]); - -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)]); - -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)]); - -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 *) - -qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q" - (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]); - -qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q" - (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]); - -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 **) - -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, - etac subst 1, assume_tac 1]); - -val ccontr = FalseE RS classical; - -(*Double negation law*) -qed_goal "notnotD" HOL.thy "~~P ==> P" - (fn [major]=> - [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]); - - -(** Unique existence **) - -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)]); - -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)]); - - -(** Select: Hilbert's Epsilon-operator **) - -(*Easier to apply than selectI: conclusion has only one occurrence of P*) -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 ]); - -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) ]); - - -(** Classical intro rules for disjunction and existential quantifiers *) - -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)) ]); - -qed_goal "excluded_middle" HOL.thy "~P | P" - (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]); - -(*For disjunctive case analysis*) -fun excluded_middle_tac sP = - res_inst_tac [("Q",sP)] (excluded_middle RS disjE); - -(*Classical implies (-->) elimination. *) -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. *) -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))) ]); - -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)) ]); - - -(* case distinction *) - -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]); - -fun case_tac a = res_inst_tac [("P",a)] case_split_thm; - -(** Standard abbreviations **) - -fun stac th = rtac(th RS ssubst); -fun sstac ths = EVERY' (map stac ths); -fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);