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src/HOL/HOL_lemmas.ML

author | wenzelm |

Fri, 15 Sep 2000 20:22:00 +0200 | |

changeset 9998 | 09bf8fcd1c6e |

parent 9970 | dfe4747c8318 |

child 10063 | 947ee8608b90 |

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

fixed someI2_ex;

(* 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 someI = thm "someI"; 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 "="; Goal "s=t ==> t=s"; by (etac subst 1); by (rtac refl 1); qed "sym"; (*calling "standard" reduces maxidx to 0*) bind_thm ("ssubst", sym RS subst); Goal "[| r=s; s=t |] ==> r=t"; by (etac subst 1 THEN assume_tac 1); qed "trans"; 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*) Goal "(f::'a=>'b) = g ==> f(x)=g(x)"; by (etac subst 1); by (rtac refl 1); qed "fun_cong"; (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*) Goal "x=y ==> f(x)=f(y)"; by (etac subst 1); by (rtac refl 1); qed "arg_cong"; Goal "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"; by (etac subst 1); by (etac subst 1); by (rtac refl 1); qed "cong"; (** 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"; Goal "[| P=Q; Q |] ==> P"; by (etac ssubst 1); by (assume_tac 1); qed "iffD2"; Goal "[| Q; P=Q |] ==> P"; by (etac iffD2 1); by (assume_tac 1); qed "rev_iffD2"; bind_thm ("iffD1", sym RS iffD2); bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2)); val [p1,p2] = Goal "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"; by (REPEAT (ares_tac [p1 RS iffD2, p1 RS iffD1, p2, impI] 1)); qed "iffE"; (** True **) section "True"; Goalw [True_def] "True"; by (rtac refl 1); qed "TrueI"; Goal "P ==> P=True"; by (REPEAT (ares_tac [iffI,TrueI] 1)); qed "eqTrueI"; Goal "P=True ==> P"; by (etac iffD2 1); by (rtac TrueI 1); qed "eqTrueE"; (** Universal quantifier **) section "!"; val prems = Goalw [All_def] "(!!x::'a. P(x)) ==> ALL x. P(x)"; by (resolve_tac (prems RL [eqTrueI RS ext]) 1); qed "allI"; Goalw [All_def] "ALL x::'a. P(x) ==> P(x)"; by (rtac eqTrueE 1); by (etac fun_cong 1); qed "spec"; val major::prems = Goal "[| ALL x. P(x); P(x) ==> R |] ==> R"; by (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ; qed "allE"; val prems = Goal "[| ALL x. P(x); [| P(x); ALL 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"; Goalw [False_def] "False ==> P"; by (etac spec 1); qed "FalseE"; Goal "False=True ==> P"; by (etac (eqTrueE RS FalseE) 1); qed "False_neq_True"; (** Negation **) section "~"; val prems = Goalw [not_def] "(P ==> False) ==> ~P"; by (rtac impI 1); by (eresolve_tac prems 1); qed "notI"; Goal "False ~= True"; by (rtac notI 1); by (etac False_neq_True 1); qed "False_not_True"; Goal "True ~= False"; by (rtac notI 1); by (dtac sym 1); by (etac False_neq_True 1); qed "True_not_False"; Goalw [not_def] "[| ~P; P |] ==> R"; by (etac (mp RS FalseE) 1); by (assume_tac 1); qed "notE"; (* Alternative ~ introduction rule: [| P ==> ~ Pa; P ==> Pa |] ==> ~ P *) bind_thm ("notI2", notE RS notI); (** 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 "EX "; Goalw [Ex_def] "P x ==> EX x::'a. P x"; by (etac someI 1) ; qed "exI"; val [major,minor] = Goalw [Ex_def] "[| EX x::'a. P(x); !!x. P(x) ==> Q |] ==> Q"; by (rtac (major RS minor) 1); qed "exE"; (** Conjunction **) section "&"; Goalw [and_def] "[| P; Q |] ==> P&Q"; by (rtac (impI RS allI) 1); by (etac (mp RS mp) 1); by (REPEAT (assume_tac 1)); qed "conjI"; Goalw [and_def] "[| P & Q |] ==> P"; by (dtac spec 1) ; by (etac mp 1); by (REPEAT (ares_tac [impI] 1)); qed "conjunct1"; Goalw [and_def] "[| P & Q |] ==> Q"; by (dtac spec 1) ; by (etac mp 1); by (REPEAT (ares_tac [impI] 1)); qed "conjunct2"; val [major,minor] = Goal "[| P&Q; [| P; Q |] ==> R |] ==> R"; by (rtac minor 1); by (rtac (major RS conjunct1) 1); by (rtac (major RS conjunct2) 1); qed "conjE"; val prems = Goal "[| P; P ==> Q |] ==> P & Q"; by (REPEAT (resolve_tac (conjI::prems) 1)); qed "context_conjI"; (** Disjunction *) section "|"; Goalw [or_def] "P ==> P|Q"; by (REPEAT (resolve_tac [allI,impI] 1)); by (etac mp 1 THEN assume_tac 1); qed "disjI1"; Goalw [or_def] "Q ==> P|Q"; by (REPEAT (resolve_tac [allI,impI] 1)); by (etac mp 1 THEN assume_tac 1); qed "disjI2"; val [major,minorP,minorQ] = Goalw [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"; by (rtac (major RS spec RS mp RS mp) 1); by (DEPTH_SOLVE (ares_tac [impI,minorP,minorQ] 1)); qed "disjE"; (** CCONTR -- classical logic **) section "classical logic"; val [prem] = Goal "(~P ==> P) ==> P"; by (rtac (True_or_False RS disjE RS eqTrueE) 1); by (assume_tac 1); by (rtac (notI RS prem RS eqTrueI) 1); by (etac subst 1); by (assume_tac 1); qed "classical"; bind_thm ("ccontr", FalseE RS classical); (*notE with premises exchanged; it discharges ~R so that it can be used to make elimination rules*) val [premp,premnot] = Goal "[| P; ~R ==> ~P |] ==> R"; by (rtac ccontr 1); by (etac ([premnot,premp] MRS notE) 1); qed "rev_notE"; (*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 "EX!"; val prems = Goalw [Ex1_def] "[| P(a); !!x. P(x) ==> x=a |] ==> EX! x. P(x)"; by (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)); qed "ex1I"; (*Sometimes easier to use: the premises have no shared variables. Safe!*) val [ex_prem,eq] = Goal "[| EX x. P(x); !!x y. [| P(x); P(y) |] ==> x=y |] ==> EX! x. P(x)"; by (rtac (ex_prem RS exE) 1); by (REPEAT (ares_tac [ex1I,eq] 1)) ; qed "ex_ex1I"; val major::prems = Goalw [Ex1_def] "[| EX! x. P(x); !!x. [| P(x); ALL y. P(y) --> y=x |] ==> R |] ==> R"; by (rtac (major RS exE) 1); by (REPEAT (etac conjE 1 ORELSE ares_tac prems 1)); qed "ex1E"; Goal "EX! x. P x ==> EX 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 someI if witness ?a comes from an EX-formula*) Goal "EX x. P x ==> P (SOME x. P x)"; by (etac exE 1); by (etac someI 1); qed "ex_someI"; (*Easier to apply than someI: 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 someI 1); by (resolve_tac prems 1) ; qed "someI2"; (*Easier to apply than someI2 if witness ?a comes from an EX-formula*) val [major,minor] = Goal "[| EX a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)"; by (rtac (major RS exE) 1); by (etac someI2 1 THEN etac minor 1); qed "someI2_ex"; val prems = Goal "[| P a; !!x. P x ==> x=a |] ==> (@x. P x) = a"; by (rtac someI2 1); by (REPEAT (ares_tac prems 1)) ; qed "some_equality"; Goalw [Ex1_def] "[| EX!x. P x; P a |] ==> (@x. P x) = a"; by (rtac some_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 "some1_equality"; Goal "P (@ x. P x) = (EX x. P x)"; by (rtac iffI 1); by (etac exI 1); by (etac exE 1); by (etac someI 1); qed "some_eq_ex"; Goal "(@y. y=x) = x"; by (rtac some_equality 1); by (rtac refl 1); by (atac 1); qed "some_eq_trivial"; Goal "(@y. x=y) = x"; by (rtac some_equality 1); by (rtac refl 1); by (etac sym 1); qed "some_sym_eq_trivial"; (** 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 "(ALL x. ~P(x) ==> P(a)) ==> EX x. P(x)"; by (rtac ccontr 1); by (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1)) ; qed "exCI"; Goal "x + (y+z) = y + ((x+z)::'a::plus_ac0)"; by (rtac (thm"plus_ac0.commute" RS trans) 1); by (rtac (thm"plus_ac0.assoc" RS trans) 1); by (rtac (thm"plus_ac0.commute" RS arg_cong) 1); qed "plus_ac0_left_commute"; Goal "x + 0 = (x ::'a::plus_ac0)"; by (rtac (thm"plus_ac0.commute" RS trans) 1); by (rtac (thm"plus_ac0.zero") 1); qed "plus_ac0_zero_right"; bind_thms ("plus_ac0", [thm"plus_ac0.assoc", thm"plus_ac0.commute", plus_ac0_left_commute, thm"plus_ac0.zero", plus_ac0_zero_right]); (* case distinction *) val [prem1,prem2] = Goal "[| P ==> Q; ~P ==> Q |] ==> Q"; by (rtac (excluded_middle RS disjE) 1); by (etac prem2 1); by (etac prem1 1); qed "case_split_thm"; fun case_tac a = res_inst_tac [("P",a)] case_split_thm; (** Standard abbreviations **) (* 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 = 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);