author | clasohm |
Tue, 04 Oct 1994 13:00:20 +0100 | |
changeset 149 | 7cfa79d92a83 |
parent 128 | 89669c58e506 |
child 153 | c0ff8f1ebc16 |
permissions | -rw-r--r-- |
0 | 1 |
(* 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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signature HOL_LEMMAS = |
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sig |
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val allE : thm |
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val all_dupE : thm |
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val allI : thm |
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val arg_cong : thm |
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val fun_cong : thm |
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val box_equals: thm |
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val case_tac : string -> int -> tactic |
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val ccontr : thm |
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val classical : thm |
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val cong : thm |
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val conjunct1 : thm |
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val conjunct2 : thm |
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val conjE : thm |
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val conjI : thm |
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val contrapos : thm |
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val disjCI : thm |
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val disjE : thm |
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val disjI1 : thm |
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val disjI2 : thm |
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val eqTrueI : thm |
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val eqTrueE : thm |
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val ex1E : thm |
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val ex1I : thm |
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val exCI : thm |
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val exI : thm |
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val exE : thm |
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val excluded_middle : thm |
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val excluded_middle_tac : string -> int -> tactic |
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val False_neq_True : thm |
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val FalseE : thm |
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val iffCE : thm |
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val iffD1 : thm |
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val iffD2 : thm |
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val iffE : thm |
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val iffI : thm |
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val impCE : thm |
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val impE : thm |
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val not_sym : thm |
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val notE : thm |
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val notI : thm |
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val notnotD : thm |
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val rev_mp : thm |
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val select_equality : thm |
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val spec : thm |
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val sstac : thm list -> int -> tactic |
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val ssubst : thm |
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val stac : thm -> int -> tactic |
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val strip_tac : int -> tactic |
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val swap : thm |
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val sym : thm |
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val trans : thm |
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val TrueI : thm |
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end; |
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structure HOL_Lemmas : HOL_LEMMAS = |
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struct |
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(** Equality **) |
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val sym = prove_goal 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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val ssubst = standard (sym RS subst); |
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val trans = prove_goal 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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c = d *) |
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val box_equals = prove_goal 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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(** Congruence rules for meta-application **) |
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(*similar to AP_THM in Gordon's HOL*) |
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val fun_cong = prove_goal 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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val arg_cong = prove_goal 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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val cong = prove_goal 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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val iffI = prove_goal 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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val iffD2 = prove_goal 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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val iffD1 = sym RS iffD2; |
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val iffE = prove_goal 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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val TrueI = prove_goalw HOL.thy [True_def] "True" |
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(fn _ => [rtac refl 1]); |
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val eqTrueI = prove_goal HOL.thy "P ==> P=True" |
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(fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]); |
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val eqTrueE = prove_goal HOL.thy "P=True ==> P" |
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(fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]); |
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(** Universal quantifier **) |
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val allI = prove_goalw 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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val spec = prove_goalw 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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val allE = prove_goal 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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val all_dupE = prove_goal 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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val FalseE = prove_goalw HOL.thy [False_def] "False ==> P" |
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(fn [major] => [rtac (major RS spec) 1]); |
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val False_neq_True = prove_goal 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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val notI = prove_goalw HOL.thy [not_def] "(P ==> False) ==> ~P" |
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(fn prems=> [rtac impI 1, eresolve_tac prems 1]); |
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val notE = prove_goalw HOL.thy [not_def] "[| ~P; P |] ==> R" |
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(fn prems => [rtac (prems MRS mp RS FalseE) 1]); |
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(** Implication **) |
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val impE = prove_goal 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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val rev_mp = prove_goal HOL.thy "[| P; P --> Q |] ==> Q" |
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(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]); |
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val contrapos = prove_goal 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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(* ~(?t = ?s) ==> ~(?s = ?t) *) |
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val [not_sym] = compose(sym,2,contrapos); |
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(** Existential quantifier **) |
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val exI = prove_goalw 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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val exE = prove_goalw 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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val conjI = prove_goalw 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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val conjunct1 = prove_goalw 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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val conjunct2 = prove_goalw 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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val conjE = prove_goal 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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val disjI1 = prove_goalw 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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val disjI2 = prove_goalw 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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val disjE = prove_goalw 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, atac 1, rtac (a3 RS impI) 1, atac 1]); |
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(** CCONTR -- classical logic **) |
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val ccontr = prove_goal HOL.thy "(~P ==> False) ==> P" |
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(fn prems => |
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[rtac (True_or_False RS (disjE RS eqTrueE)) 1, atac 1, |
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rtac spec 1, fold_tac [False_def], resolve_tac prems 1, |
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rtac ssubst 1, atac 1, rewtac not_def, |
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REPEAT (ares_tac [impI] 1) ]); |
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val ccontr = prove_goalw HOL.thy [not_def] "(~P ==> False) ==> P" |
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(fn prems => |
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[rtac (True_or_False RS (disjE RS eqTrueE)) 1, atac 1, |
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rtac spec 1, fold_tac [False_def], resolve_tac prems 1, |
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rtac ssubst 1, atac 1, REPEAT (ares_tac [impI] 1) ]); |
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val classical = prove_goal HOL.thy "(~P ==> P) ==> P" |
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(fn prems => [rtac ccontr 1, REPEAT (ares_tac (prems@[notE]) 1)]); |
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(*Double negation law*) |
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val notnotD = prove_goal 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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(** Unique existence **) |
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val ex1I = prove_goalw 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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val ex1E = prove_goalw 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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val select_equality = prove_goal HOL.thy |
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"[| P(a); !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a" |
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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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(** Classical intro rules for disjunction and existential quantifiers *) |
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val disjCI = prove_goal 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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val excluded_middle = prove_goal 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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(*Classical implies (-->) elimination. *) |
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val impCE = prove_goal 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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(*Classical <-> elimination. *) |
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val iffCE = prove_goal 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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val exCI = prove_goal 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)) ]); |
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(*Required by the "classical" module*) |
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val swap = prove_goal HOL.thy "~P ==> (~Q ==> P) ==> Q" |
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(fn major::prems=> |
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[ rtac ccontr 1, rtac (major RS notE) 1, REPEAT (ares_tac prems 1)]); |
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(* case distinction *) |
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val case_split_thm = prove_goal 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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(** Standard abbreviations **) |
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fun stac th = rtac(th RS ssubst); |
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fun sstac ths = EVERY' (map stac ths); |
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fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); |
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changed "." to "$" and Cons to infix "#" to eliminate ambiguity
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0 | 334 |
end; |
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open HOL_Lemmas; |