author | oheimb |
Thu, 08 Jan 1998 17:47:22 +0100 | |
changeset 4527 | 4878fb3d0ac5 |
parent 4467 | bd05e2a28602 |
child 5139 | 013ea0f023e3 |
permissions | -rw-r--r-- |
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(* Title: HOL/HOL.ML |
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ID: $Id$ |
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Author: Tobias Nipkow |
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Copyright 1991 University of Cambridge |
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For HOL.thy |
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Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68 |
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*) |
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open HOL; |
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(** Equality **) |
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section "="; |
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qed_goal "sym" HOL.thy "s=t ==> t=s" |
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(fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]); |
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(*calling "standard" reduces maxidx to 0*) |
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bind_thm ("ssubst", (sym RS subst)); |
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qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t" |
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(fn prems => |
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[rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]); |
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(*Useful with eresolve_tac for proving equalties from known equalities. |
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a = b |
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| | |
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c = d *) |
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qed_goal "box_equals" HOL.thy |
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"[| a=b; a=c; b=d |] ==> c=d" |
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(fn prems=> |
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[ (rtac trans 1), |
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(rtac trans 1), |
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(rtac sym 1), |
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(REPEAT (resolve_tac prems 1)) ]); |
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||
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|
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(** Congruence rules for meta-application **) |
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section "Congruence"; |
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(*similar to AP_THM in Gordon's HOL*) |
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qed_goal "fun_cong" HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)" |
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(fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]); |
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(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*) |
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qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)" |
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(fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]); |
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qed_goal "cong" HOL.thy |
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"[| f = g; (x::'a) = y |] ==> f(x) = g(y)" |
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(fn [prem1,prem2] => |
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[rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]); |
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(** Equality of booleans -- iff **) |
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section "iff"; |
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qed_goal "iffI" HOL.thy |
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"[| P ==> Q; Q ==> P |] ==> P=Q" |
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(fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]); |
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qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P" |
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(fn prems => |
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[rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]); |
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qed_goal "rev_iffD2" HOL.thy "!!P. [| Q; P=Q |] ==> P" |
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(fn _ => [etac iffD2 1, assume_tac 1]); |
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bind_thm ("iffD1", sym RS iffD2); |
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bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2)); |
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qed_goal "iffE" HOL.thy |
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"[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R" |
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(fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]); |
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(** True **) |
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section "True"; |
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qed_goalw "TrueI" HOL.thy [True_def] "True" |
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(fn _ => [rtac refl 1]); |
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qed_goal "eqTrueI" HOL.thy "P ==> P=True" |
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(fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]); |
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qed_goal "eqTrueE" HOL.thy "P=True ==> P" |
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(fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]); |
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||
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(** Universal quantifier **) |
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section "!"; |
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qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)" |
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(fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]); |
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||
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qed_goalw "spec" HOL.thy [All_def] "! x::'a. P(x) ==> P(x)" |
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(fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]); |
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qed_goal "allE" HOL.thy "[| !x. P(x); P(x) ==> R |] ==> R" |
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(fn major::prems=> |
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[ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]); |
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qed_goal "all_dupE" HOL.thy |
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"[| ! x. P(x); [| P(x); ! x. P(x) |] ==> R |] ==> R" |
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(fn prems => |
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[ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]); |
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(** False ** Depends upon spec; it is impossible to do propositional logic |
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before quantifiers! **) |
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section "False"; |
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qed_goalw "FalseE" HOL.thy [False_def] "False ==> P" |
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(fn [major] => [rtac (major RS spec) 1]); |
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qed_goal "False_neq_True" HOL.thy "False=True ==> P" |
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(fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]); |
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(** Negation **) |
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section "~"; |
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qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P" |
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(fn prems=> [rtac impI 1, eresolve_tac prems 1]); |
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qed_goalw "notE" HOL.thy [not_def] "[| ~P; P |] ==> R" |
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(fn prems => [rtac (prems MRS mp RS FalseE) 1]); |
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bind_thm ("classical2", notE RS notI); |
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qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R" |
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(fn _ => [REPEAT (ares_tac [notE] 1)]); |
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||
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(** Implication **) |
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section "-->"; |
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qed_goal "impE" HOL.thy "[| P-->Q; P; Q ==> R |] ==> R" |
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(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]); |
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(* Reduces Q to P-->Q, allowing substitution in P. *) |
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qed_goal "rev_mp" HOL.thy "[| P; P --> Q |] ==> Q" |
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(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]); |
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qed_goal "contrapos" HOL.thy "[| ~Q; P==>Q |] ==> ~P" |
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(fn [major,minor]=> |
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[ (rtac (major RS notE RS notI) 1), |
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(etac minor 1) ]); |
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qed_goal "rev_contrapos" HOL.thy "[| P==>Q; ~Q |] ==> ~P" |
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(fn [major,minor]=> |
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[ (rtac (minor RS contrapos) 1), (etac major 1) ]); |
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(* ~(?t = ?s) ==> ~(?s = ?t) *) |
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bind_thm("not_sym", sym COMP rev_contrapos); |
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(** Existential quantifier **) |
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section "?"; |
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qed_goalw "exI" HOL.thy [Ex_def] "P x ==> ? x::'a. P x" |
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(fn prems => [rtac selectI 1, resolve_tac prems 1]); |
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qed_goalw "exE" HOL.thy [Ex_def] |
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"[| ? x::'a. P(x); !!x. P(x) ==> Q |] ==> Q" |
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(fn prems => [REPEAT(resolve_tac prems 1)]); |
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(** Conjunction **) |
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section "&"; |
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qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q" |
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(fn prems => |
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[REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]); |
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qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P" |
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(fn prems => |
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[resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]); |
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qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q" |
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(fn prems => |
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[resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]); |
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qed_goal "conjE" HOL.thy "[| P&Q; [| P; Q |] ==> R |] ==> R" |
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(fn prems => |
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[cut_facts_tac prems 1, resolve_tac prems 1, |
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etac conjunct1 1, etac conjunct2 1]); |
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(** Disjunction *) |
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section "|"; |
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qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q" |
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(fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]); |
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qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q" |
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(fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]); |
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qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R" |
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(fn [a1,a2,a3] => |
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[rtac (mp RS mp) 1, rtac spec 1, rtac a1 1, |
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rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]); |
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(** CCONTR -- classical logic **) |
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section "classical logic"; |
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qed_goalw "classical" HOL.thy [not_def] "(~P ==> P) ==> P" |
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(fn [prem] => |
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[rtac (True_or_False RS (disjE RS eqTrueE)) 1, assume_tac 1, |
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rtac (impI RS prem RS eqTrueI) 1, |
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etac subst 1, assume_tac 1]); |
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val ccontr = FalseE RS classical; |
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(*Double negation law*) |
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qed_goal "notnotD" HOL.thy "~~P ==> P" |
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(fn [major]=> |
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[ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]); |
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qed_goal "contrapos2" HOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [ |
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rtac classical 1, |
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dtac p2 1, |
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etac notE 1, |
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rtac p1 1]); |
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qed_goal "swap2" HOL.thy "[| P; Q ==> ~ P |] ==> ~ Q" (fn [p1,p2] => [ |
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rtac notI 1, |
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dtac p2 1, |
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etac notE 1, |
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rtac p1 1]); |
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(** Unique existence **) |
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section "?!"; |
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qed_goalw "ex1I" HOL.thy [Ex1_def] |
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"[| P(a); !!x. P(x) ==> x=a |] ==> ?! x. P(x)" |
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(fn prems => |
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[REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]); |
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||
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(*Sometimes easier to use: the premises have no shared variables. Safe!*) |
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qed_goal "ex_ex1I" HOL.thy |
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"[| ? x. P(x); !!x y. [| P(x); P(y) |] ==> x=y |] ==> ?! x. P(x)" |
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(fn [ex,eq] => [ (rtac (ex RS exE) 1), |
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(REPEAT (ares_tac [ex1I,eq] 1)) ]); |
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||
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qed_goalw "ex1E" HOL.thy [Ex1_def] |
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"[| ?! x. P(x); !!x. [| P(x); ! y. P(y) --> y=x |] ==> R |] ==> R" |
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(fn major::prems => |
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[rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]); |
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(** Select: Hilbert's Epsilon-operator **) |
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section "@"; |
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(*Easier to apply than selectI: conclusion has only one occurrence of P*) |
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qed_goal "selectI2" HOL.thy |
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"[| P a; !!x. P x ==> Q x |] ==> Q (@x. P x)" |
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(fn prems => [ resolve_tac prems 1, |
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rtac selectI 1, |
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resolve_tac prems 1 ]); |
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(*Easier to apply than selectI2 if witness ?a comes from an EX-formula*) |
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qed_goal "selectI2EX" HOL.thy |
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"[| ? a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)" |
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(fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]); |
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||
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qed_goal "select_equality" HOL.thy |
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"[| P a; !!x. P x ==> x=a |] ==> (@x. P x) = a" |
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(fn prems => [ rtac selectI2 1, |
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REPEAT (ares_tac prems 1) ]); |
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|
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qed_goalw "select1_equality" HOL.thy [Ex1_def] |
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"!!P. [| ?!x. P x; P a |] ==> (@x. P x) = a" (K [ |
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rtac select_equality 1, atac 1, |
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etac exE 1, etac conjE 1, |
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rtac allE 1, atac 1, |
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etac impE 1, atac 1, etac ssubst 1, |
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etac allE 1, etac impE 1, atac 1, etac ssubst 1, |
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rtac refl 1]); |
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qed_goal "select_eq_Ex" HOL.thy "P (@ x. P x) = (? x. P x)" (K [ |
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rtac iffI 1, |
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etac exI 1, |
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etac exE 1, |
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etac selectI 1]); |
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||
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|
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(** Classical intro rules for disjunction and existential quantifiers *) |
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section "classical intro rules"; |
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|
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qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q" |
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(fn prems=> |
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[ (rtac classical 1), |
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(REPEAT (ares_tac (prems@[disjI1,notI]) 1)), |
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(REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]); |
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||
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qed_goal "excluded_middle" HOL.thy "~P | P" |
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(fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]); |
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(*For disjunctive case analysis*) |
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fun excluded_middle_tac sP = |
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res_inst_tac [("Q",sP)] (excluded_middle RS disjE); |
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||
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(*Classical implies (-->) elimination. *) |
|
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qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R" |
|
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(fn major::prems=> |
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[ rtac (excluded_middle RS disjE) 1, |
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REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]); |
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||
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(*This version of --> elimination works on Q before P. It works best for |
313 |
those cases in which P holds "almost everywhere". Can't install as |
|
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default: would break old proofs.*) |
|
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qed_goal "impCE'" thy |
|
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"[| P-->Q; Q ==> R; ~P ==> R |] ==> R" |
|
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(fn major::prems=> |
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[ (resolve_tac [excluded_middle RS disjE] 1), |
|
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(DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]); |
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||
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(*Classical <-> elimination. *) |
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qed_goal "iffCE" HOL.thy |
|
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"[| P=Q; [| P; Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R" |
|
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(fn major::prems => |
|
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[ (rtac (major RS iffE) 1), |
|
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(REPEAT (DEPTH_SOLVE_1 |
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(eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]); |
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|
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qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x. P(x)" |
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(fn prems=> |
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[ (rtac ccontr 1), |
|
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(REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1)) ]); |
|
333 |
||
334 |
||
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(* case distinction *) |
|
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||
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qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q" |
|
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(fn [p1,p2] => [cut_facts_tac [excluded_middle] 1, etac disjE 1, |
|
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etac p2 1, etac p1 1]); |
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fun case_tac a = res_inst_tac [("P",a)] case_split_thm; |
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|
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(** Standard abbreviations **) |
345 |
||
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fun stac th = CHANGED o rtac (th RS ssubst); |
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fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); |
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|
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|
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(** strip ! and --> from proved goal while preserving !-bound var names **) |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
351 |
|
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
352 |
local |
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
353 |
|
1515 | 354 |
(* Use XXX to avoid forall_intr failing because of duplicate variable name *) |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
355 |
val myspec = read_instantiate [("P","?XXX")] spec; |
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
356 |
val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec; |
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
357 |
val cvx = cterm_of (#sign(rep_thm myspec)) vx; |
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
358 |
val aspec = forall_intr cvx myspec; |
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
359 |
|
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
360 |
in |
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
361 |
|
1515 | 362 |
fun RSspec th = |
363 |
(case concl_of th of |
|
364 |
_ $ (Const("All",_) $ Abs(a,_,_)) => |
|
365 |
let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT)) |
|
366 |
in th RS forall_elim ca aspec end |
|
367 |
| _ => raise THM("RSspec",0,[th])); |
|
368 |
||
369 |
fun RSmp th = |
|
370 |
(case concl_of th of |
|
371 |
_ $ (Const("op -->",_)$_$_) => th RS mp |
|
372 |
| _ => raise THM("RSmp",0,[th])); |
|
373 |
||
374 |
fun normalize_thm funs = |
|
375 |
let fun trans [] th = th |
|
376 |
| trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th |
|
377 |
in trans funs end; |
|
378 |
||
3615
e5322197cfea
Moved some functions which used to be part of thy_data.ML
berghofe
parents:
3436
diff
changeset
|
379 |
fun qed_spec_mp name = |
e5322197cfea
Moved some functions which used to be part of thy_data.ML
berghofe
parents:
3436
diff
changeset
|
380 |
let val thm = normalize_thm [RSspec,RSmp] (result()) |
e5322197cfea
Moved some functions which used to be part of thy_data.ML
berghofe
parents:
3436
diff
changeset
|
381 |
in bind_thm(name, thm) end; |
e5322197cfea
Moved some functions which used to be part of thy_data.ML
berghofe
parents:
3436
diff
changeset
|
382 |
|
e5322197cfea
Moved some functions which used to be part of thy_data.ML
berghofe
parents:
3436
diff
changeset
|
383 |
fun qed_goal_spec_mp name thy s p = |
e5322197cfea
Moved some functions which used to be part of thy_data.ML
berghofe
parents:
3436
diff
changeset
|
384 |
bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p)); |
e5322197cfea
Moved some functions which used to be part of thy_data.ML
berghofe
parents:
3436
diff
changeset
|
385 |
|
e5322197cfea
Moved some functions which used to be part of thy_data.ML
berghofe
parents:
3436
diff
changeset
|
386 |
fun qed_goalw_spec_mp name thy defs s p = |
e5322197cfea
Moved some functions which used to be part of thy_data.ML
berghofe
parents:
3436
diff
changeset
|
387 |
bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p)); |
1660 | 388 |
|
3621 | 389 |
end; |