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1 
(* Title: HOL/HOL_lemmas.ML


2 
ID: $Id$


3 
Author: Tobias Nipkow


4 
Copyright 1991 University of Cambridge


5 


6 
Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68.


7 
*)


8 


9 
(* ML bindings *)


10 


11 
val plusI = thm "plusI";


12 
val minusI = thm "minusI";


13 
val timesI = thm "timesI";


14 
val powerI = thm "powerI";


15 
val eq_reflection = thm "eq_reflection";


16 
val refl = thm "refl";


17 
val subst = thm "subst";


18 
val ext = thm "ext";

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val someI = thm "someI";

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val impI = thm "impI";


21 
val mp = thm "mp";


22 
val True_def = thm "True_def";


23 
val All_def = thm "All_def";


24 
val Ex_def = thm "Ex_def";


25 
val False_def = thm "False_def";


26 
val not_def = thm "not_def";


27 
val and_def = thm "and_def";


28 
val or_def = thm "or_def";


29 
val Ex1_def = thm "Ex1_def";


30 
val iff = thm "iff";


31 
val True_or_False = thm "True_or_False";


32 
val Let_def = thm "Let_def";


33 
val if_def = thm "if_def";


34 
val arbitrary_def = thm "arbitrary_def";


35 


36 


37 
(** Equality **)


38 
section "=";


39 

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Goal "s=t ==> t=s";


41 
by (etac subst 1);


42 
by (rtac refl 1);


43 
qed "sym";

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44 


45 
(*calling "standard" reduces maxidx to 0*)

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46 
bind_thm ("ssubst", sym RS subst);

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47 

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Goal "[ r=s; s=t ] ==> r=t";


49 
by (etac subst 1 THEN assume_tac 1);


50 
qed "trans";

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51 

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val prems = goal (the_context()) "(A == B) ==> A = B";

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by (rewrite_goals_tac prems);


54 
by (rtac refl 1);


55 
qed "def_imp_eq";


56 


57 
(*Useful with eresolve_tac for proving equalties from known equalities.


58 
a = b


59 
 


60 
c = d *)


61 
Goal "[ a=b; a=c; b=d ] ==> c=d";


62 
by (rtac trans 1);


63 
by (rtac trans 1);


64 
by (rtac sym 1);


65 
by (REPEAT (assume_tac 1)) ;


66 
qed "box_equals";


67 


68 
(** Congruence rules for metaapplication **)


69 
section "Congruence";


70 


71 
(*similar to AP_THM in Gordon's HOL*)

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Goal "(f::'a=>'b) = g ==> f(x)=g(x)";


73 
by (etac subst 1);


74 
by (rtac refl 1);


75 
qed "fun_cong";

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77 
(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)

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Goal "x=y ==> f(x)=f(y)";


79 
by (etac subst 1);


80 
by (rtac refl 1);


81 
qed "arg_cong";

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82 

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Goal "[ f = g; (x::'a) = y ] ==> f(x) = g(y)";


84 
by (etac subst 1);


85 
by (etac subst 1);


86 
by (rtac refl 1);


87 
qed "cong";

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88 


89 
(** Equality of booleans  iff **)


90 
section "iff";


91 

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val prems = Goal "[ P ==> Q; Q ==> P ] ==> P=Q";

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by (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1));


94 
qed "iffI";


95 

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Goal "[ P=Q; Q ] ==> P";


97 
by (etac ssubst 1);


98 
by (assume_tac 1);


99 
qed "iffD2";

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Goal "[ Q; P=Q ] ==> P";


102 
by (etac iffD2 1);


103 
by (assume_tac 1);


104 
qed "rev_iffD2";

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bind_thm ("iffD1", sym RS iffD2);


107 
bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2));


108 

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val [p1,p2] = Goal "[ P=Q; [ P > Q; Q > P ] ==> R ] ==> R";


110 
by (REPEAT (ares_tac [p1 RS iffD2, p1 RS iffD1, p2, impI] 1));


111 
qed "iffE";

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112 


113 


114 
(** True **)


115 
section "True";


116 

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Goalw [True_def] "True";


118 
by (rtac refl 1);


119 
qed "TrueI";

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120 

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Goal "P ==> P=True";


122 
by (REPEAT (ares_tac [iffI,TrueI] 1));


123 
qed "eqTrueI";

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Goal "P=True ==> P";


126 
by (etac iffD2 1);


127 
by (rtac TrueI 1);


128 
qed "eqTrueE";

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129 


130 


131 
(** Universal quantifier **)


132 
section "!";


133 

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val prems = Goalw [All_def] "(!!x::'a. P(x)) ==> ALL x. P(x)";

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by (resolve_tac (prems RL [eqTrueI RS ext]) 1);


136 
qed "allI";

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137 

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Goalw [All_def] "ALL x::'a. P(x) ==> P(x)";

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by (rtac eqTrueE 1);


140 
by (etac fun_cong 1);


141 
qed "spec";

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val major::prems = Goal "[ ALL x. P(x); P(x) ==> R ] ==> R";

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by (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ;


145 
qed "allE";


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val prems = Goal

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"[ ALL x. P(x); [ P(x); ALL x. P(x) ] ==> R ] ==> R";

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by (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ;


150 
qed "all_dupE";


151 


152 


153 
(** False ** Depends upon spec; it is impossible to do propositional logic


154 
before quantifiers! **)


155 
section "False";


156 

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Goalw [False_def] "False ==> P";


158 
by (etac spec 1);


159 
qed "FalseE";

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Goal "False=True ==> P";


162 
by (etac (eqTrueE RS FalseE) 1);


163 
qed "False_neq_True";

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164 


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166 
(** Negation **)


167 
section "~";


168 

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val prems = Goalw [not_def] "(P ==> False) ==> ~P";


170 
by (rtac impI 1);


171 
by (eresolve_tac prems 1);


172 
qed "notI";

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Goal "False ~= True";


175 
by (rtac notI 1);


176 
by (etac False_neq_True 1);


177 
qed "False_not_True";

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Goal "True ~= False";


180 
by (rtac notI 1);


181 
by (dtac sym 1);


182 
by (etac False_neq_True 1);


183 
qed "True_not_False";

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Goalw [not_def] "[ ~P; P ] ==> R";


186 
by (etac (mp RS FalseE) 1);


187 
by (assume_tac 1);


188 
qed "notE";

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(* Alternative ~ introduction rule: [ P ==> ~ Pa; P ==> Pa ] ==> ~ P *)


191 
bind_thm ("notI2", notE RS notI);

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192 


193 


194 
(** Implication **)


195 
section ">";


196 


197 
val prems = Goal "[ P>Q; P; Q ==> R ] ==> R";


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by (REPEAT (resolve_tac (prems@[mp]) 1));


199 
qed "impE";


200 


201 
(* Reduces Q to P>Q, allowing substitution in P. *)


202 
Goal "[ P; P > Q ] ==> Q";


203 
by (REPEAT (ares_tac [mp] 1)) ;


204 
qed "rev_mp";


205 


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val [major,minor] = Goal "[ ~Q; P==>Q ] ==> ~P";


207 
by (rtac (major RS notE RS notI) 1);


208 
by (etac minor 1) ;


209 
qed "contrapos";


210 


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val [major,minor] = Goal "[ P==>Q; ~Q ] ==> ~P";


212 
by (rtac (minor RS contrapos) 1);


213 
by (etac major 1) ;


214 
qed "rev_contrapos";


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(* t ~= s ==> s ~= t *)

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bind_thm("not_sym", sym COMP rev_contrapos);


218 


219 


220 
(** Existential quantifier **)

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section "EX ";

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Goalw [Ex_def] "P x ==> EX x::'a. P x";

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by (etac someI 1) ;

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qed "exI";

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val [major,minor] =

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Goalw [Ex_def] "[ EX x::'a. P(x); !!x. P(x) ==> Q ] ==> Q";

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by (rtac (major RS minor) 1);


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qed "exE";

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232 


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(** Conjunction **)


234 
section "&";


235 

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Goalw [and_def] "[ P; Q ] ==> P&Q";


237 
by (rtac (impI RS allI) 1);


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by (etac (mp RS mp) 1);


239 
by (REPEAT (assume_tac 1));


240 
qed "conjI";

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Goalw [and_def] "[ P & Q ] ==> P";


243 
by (dtac spec 1) ;


244 
by (etac mp 1);


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by (REPEAT (ares_tac [impI] 1));


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qed "conjunct1";

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Goalw [and_def] "[ P & Q ] ==> Q";


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by (dtac spec 1) ;


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by (etac mp 1);


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by (REPEAT (ares_tac [impI] 1));


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qed "conjunct2";

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val [major,minor] =


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Goal "[ P&Q; [ P; Q ] ==> R ] ==> R";


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by (rtac minor 1);


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by (rtac (major RS conjunct1) 1);


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by (rtac (major RS conjunct2) 1);


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qed "conjE";

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val prems =


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Goal "[ P; P ==> Q ] ==> P & Q";


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by (REPEAT (resolve_tac (conjI::prems) 1));


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qed "context_conjI";

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266 


267 
(** Disjunction *)


268 
section "";


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Goalw [or_def] "P ==> PQ";


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by (REPEAT (resolve_tac [allI,impI] 1));


272 
by (etac mp 1 THEN assume_tac 1);


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qed "disjI1";

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Goalw [or_def] "Q ==> PQ";


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by (REPEAT (resolve_tac [allI,impI] 1));


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by (etac mp 1 THEN assume_tac 1);


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qed "disjI2";

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val [major,minorP,minorQ] =


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Goalw [or_def] "[ P  Q; P ==> R; Q ==> R ] ==> R";


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by (rtac (major RS spec RS mp RS mp) 1);


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by (DEPTH_SOLVE (ares_tac [impI,minorP,minorQ] 1));


284 
qed "disjE";

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286 


287 
(** CCONTR  classical logic **)


288 
section "classical logic";


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val [prem] = Goal "(~P ==> P) ==> P";


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by (rtac (True_or_False RS disjE RS eqTrueE) 1);


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by (assume_tac 1);


293 
by (rtac (notI RS prem RS eqTrueI) 1);


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by (etac subst 1);


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by (assume_tac 1);


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qed "classical";

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bind_thm ("ccontr", FalseE RS classical);

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(*notE with premises exchanged; it discharges ~R so that it can be used to


301 
make elimination rules*)


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val [premp,premnot] = Goal "[ P; ~R ==> ~P ] ==> R";


303 
by (rtac ccontr 1);


304 
by (etac ([premnot,premp] MRS notE) 1);


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qed "rev_notE";


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(*Double negation law*)


308 
Goal "~~P ==> P";


309 
by (rtac classical 1);


310 
by (etac notE 1);


311 
by (assume_tac 1);


312 
qed "notnotD";


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val [p1,p2] = Goal "[ Q; ~ P ==> ~ Q ] ==> P";


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by (rtac classical 1);


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by (dtac p2 1);


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by (etac notE 1);


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by (rtac p1 1);


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qed "contrapos2";


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val [p1,p2] = Goal "[ P; Q ==> ~ P ] ==> ~ Q";


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by (rtac notI 1);


323 
by (dtac p2 1);


324 
by (etac notE 1);


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by (rtac p1 1);


326 
qed "swap2";


327 


328 
(** Unique existence **)

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section "EX!";

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val prems = Goalw [Ex1_def] "[ P(a); !!x. P(x) ==> x=a ] ==> EX! x. P(x)";

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by (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1));


333 
qed "ex1I";

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(*Sometimes easier to use: the premises have no shared variables. Safe!*)

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val [ex_prem,eq] = Goal

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"[ EX x. P(x); !!x y. [ P(x); P(y) ] ==> x=y ] ==> EX! x. P(x)";

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by (rtac (ex_prem RS exE) 1);

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by (REPEAT (ares_tac [ex1I,eq] 1)) ;


340 
qed "ex_ex1I";


341 

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val major::prems = Goalw [Ex1_def]

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"[ EX! x. P(x); !!x. [ P(x); ALL y. P(y) > y=x ] ==> R ] ==> R";

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by (rtac (major RS exE) 1);


345 
by (REPEAT (etac conjE 1 ORELSE ares_tac prems 1));


346 
qed "ex1E";

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Goal "EX! x. P x ==> EX x. P x";

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by (etac ex1E 1);


350 
by (rtac exI 1);


351 
by (assume_tac 1);


352 
qed "ex1_implies_ex";


353 


354 


355 
(** Select: Hilbert's Epsilonoperator **)


356 
section "@";


357 

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(*Easier to apply than someI if witness ?a comes from an EXformula*)

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Goal "EX x. P x ==> P (SOME x. P x)";

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by (etac exE 1);

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by (etac someI 1);

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qed "ex_someI";


363 

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(*Easier to apply than someI: conclusion has only one occurrence of P*)

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val prems = Goal


366 
"[ P a; !!x. P x ==> Q x ] ==> Q (@x. P x)";


367 
by (resolve_tac prems 1);

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by (rtac someI 1);

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by (resolve_tac prems 1) ;

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qed "someI2";

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371 

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(*Easier to apply than someI2 if witness ?a comes from an EXformula*)

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val [major,minor] = Goal "[ EX a. P a; !!x. P x ==> Q x ] ==> Q (Eps P)";

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by (rtac (major RS exE) 1);

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by (etac someI2 1 THEN etac minor 1);

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qed "someI2_ex";

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378 
val prems = Goal


379 
"[ P a; !!x. P x ==> x=a ] ==> (@x. P x) = a";

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by (rtac someI2 1);

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by (REPEAT (ares_tac prems 1)) ;

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qed "some_equality";

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Goalw [Ex1_def] "[ EX!x. P x; P a ] ==> (@x. P x) = a";

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by (rtac some_equality 1);

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by (atac 1);


387 
by (etac exE 1);


388 
by (etac conjE 1);


389 
by (rtac allE 1);


390 
by (atac 1);


391 
by (etac impE 1);


392 
by (atac 1);


393 
by (etac ssubst 1);


394 
by (etac allE 1);


395 
by (etac mp 1);


396 
by (atac 1);

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qed "some1_equality";

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398 

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Goal "P (@ x. P x) = (EX x. P x)";

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400 
by (rtac iffI 1);


401 
by (etac exI 1);


402 
by (etac exE 1);

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403 
by (etac someI 1);

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404 
qed "some_eq_ex";

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405 


406 
Goal "(@y. y=x) = x";

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407 
by (rtac some_equality 1);

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408 
by (rtac refl 1);


409 
by (atac 1);

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410 
qed "some_eq_trivial";

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411 

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Goal "(@y. x=y) = x";


413 
by (rtac some_equality 1);

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by (rtac refl 1);


415 
by (etac sym 1);

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416 
qed "some_sym_eq_trivial";

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417 


418 
(** Classical intro rules for disjunction and existential quantifiers *)


419 
section "classical intro rules";


420 

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421 
val prems = Goal "(~Q ==> P) ==> PQ";

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422 
by (rtac classical 1);


423 
by (REPEAT (ares_tac (prems@[disjI1,notI]) 1));


424 
by (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ;


425 
qed "disjCI";


426 


427 
Goal "~P  P";


428 
by (REPEAT (ares_tac [disjCI] 1)) ;


429 
qed "excluded_middle";


430 


431 
(*For disjunctive case analysis*)


432 
fun excluded_middle_tac sP =


433 
res_inst_tac [("Q",sP)] (excluded_middle RS disjE);


434 


435 
(*Classical implies (>) elimination. *)


436 
val major::prems = Goal "[ P>Q; ~P ==> R; Q ==> R ] ==> R";


437 
by (rtac (excluded_middle RS disjE) 1);


438 
by (REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1)));


439 
qed "impCE";


440 


441 
(*This version of > elimination works on Q before P. It works best for


442 
those cases in which P holds "almost everywhere". Can't install as


443 
default: would break old proofs.*)


444 
val major::prems = Goal


445 
"[ P>Q; Q ==> R; ~P ==> R ] ==> R";


446 
by (resolve_tac [excluded_middle RS disjE] 1);


447 
by (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ;


448 
qed "impCE'";


449 


450 
(*Classical <> elimination. *)


451 
val major::prems = Goal


452 
"[ P=Q; [ P; Q ] ==> R; [ ~P; ~Q ] ==> R ] ==> R";


453 
by (rtac (major RS iffE) 1);

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454 
by (REPEAT (DEPTH_SOLVE_1


455 
(eresolve_tac ([asm_rl,impCE,notE]@prems) 1)));

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456 
qed "iffCE";


457 

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458 
val prems = Goal "(ALL x. ~P(x) ==> P(a)) ==> EX x. P(x)";

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459 
by (rtac ccontr 1);


460 
by (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1)) ;


461 
qed "exCI";


462 

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463 
Goal "x + (y+z) = y + ((x+z)::'a::plus_ac0)";


464 
by (rtac (thm"plus_ac0.commute" RS trans) 1);


465 
by (rtac (thm"plus_ac0.assoc" RS trans) 1);


466 
by (rtac (thm"plus_ac0.commute" RS arg_cong) 1);


467 
qed "plus_ac0_left_commute";


468 


469 
Goal "x + 0 = (x ::'a::plus_ac0)";


470 
by (rtac (thm"plus_ac0.commute" RS trans) 1);


471 
by (rtac (thm"plus_ac0.zero") 1);


472 
qed "plus_ac0_zero_right";


473 

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474 
bind_thms ("plus_ac0", [thm"plus_ac0.assoc", thm"plus_ac0.commute",


475 
plus_ac0_left_commute,


476 
thm"plus_ac0.zero", plus_ac0_zero_right]);

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477 


478 
(* case distinction *)


479 

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480 
val [prem1,prem2] = Goal "[ P ==> Q; ~P ==> Q ] ==> Q";


481 
by (rtac (excluded_middle RS disjE) 1);


482 
by (etac prem2 1);


483 
by (etac prem1 1);


484 
qed "case_split_thm";

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485 


486 
fun case_tac a = res_inst_tac [("P",a)] case_split_thm;


487 


488 


489 
(** Standard abbreviations **)


490 

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491 
(* combination of (spec RS spec RS ...(j times) ... spec RS mp *)

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492 
local


493 
fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t


494 
 wrong_prem (Bound _) = true


495 
 wrong_prem _ = false;

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496 
val filter_right = filter (fn t => not (wrong_prem (HOLogic.dest_Trueprop (hd (Thm.prems_of t)))));

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497 
in


498 
fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);


499 
fun smp_tac j = EVERY'[dresolve_tac (smp j), atac]


500 
end;


501 


502 

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503 
fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);
