src/HOL/HOL_lemmas.ML

lemma Least_mono moved from Typedef.thy to Set.thy;

(*  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.
*)

(* legacy ML bindings *)

val plusI = thm "plusI";
val minusI = thm "minusI";
val timesI = thm "timesI";
val eq_reflection = thm "eq_reflection";
val refl = thm "refl";
val subst = thm "subst";
val ext = thm "ext";
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";


section "Equality";

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";


section "Congruence rules for application";

(*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";


section "Equality of booleans -- 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";


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";


section "Universal quantifier";

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";


section "False";
(*Depends upon spec; it is impossible to do propositional logic before quantifiers!*)

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";


section "Negation";

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);


section "Implication";

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_nn";

(*not used at all, but we already have the other 3 combinations *)
val [major,minor] = Goal "[| Q;  P ==> ~Q |] ==> ~P";
by (rtac (minor RS notE RS notI) 1);
by (assume_tac 1);
by (rtac major 1) ;
qed "contrapos_pn";

Goal "t ~= s ==> s ~= t";
by (etac contrapos_nn 1); 
by (etac sym 1); 
qed "not_sym";

(*still used in HOLCF*)
val [major,minor] = Goal "[| P==>Q; ~Q |] ==> ~P";
by (rtac (minor RS contrapos_nn) 1);
by (etac major 1) ;
qed "rev_contrapos";

section "Existential quantifier";

Goalw [Ex_def] "P x ==> EX x::'a. P x";
by (rtac allI 1); 
by (rtac impI 1); 
by (etac allE 1); 
by (etac mp 1) ;
by (assume_tac 1); 
qed "exI";

val [major,minor] =
Goalw [Ex_def] "[| EX x::'a. P(x); !!x. P(x) ==> Q |] ==> Q";
by (rtac (major RS spec RS mp) 1); 
by (rtac (impI RS allI) 1); 
by (etac minor 1); 
qed "exE";


section "Conjunction";

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";


section "Disjunction";

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";


section "Classical logic";
(*CCONTR -- 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 "contrapos_pp";


section "Unique existence";

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";


section "THE: definite description operator";

val [prema,premx] = Goal "[| P a;  !!x. P x ==> x=a |] ==> (THE x. P x) = a";
by (rtac trans 1); 
 by (rtac (thm "the_eq_trivial") 2);
by (res_inst_tac [("f","The")] arg_cong 1); 
by (rtac ext 1); 
 by (rtac iffI 1); 
by (etac premx 1); 
by (etac ssubst 1 THEN rtac prema 1);
qed "the_equality";

val [prema,premx] = Goal "[| P a;  !!x. P x ==> x=a |] ==> P (THE x. P x)";
by (rtac (the_equality RS ssubst) 1);
by (REPEAT (ares_tac [prema,premx] 1));
qed "theI";

Goal "EX! x. P x ==> P (THE x. P x)";
by (etac ex1E 1);
by (etac theI 1);
by (etac allE 1);
by (etac mp 1);
by (atac 1);
qed "theI'";

(*Easier to apply than theI: only one occurrence of P*)
val [prema,premx,premq] = Goal
     "[| P a;  !!x. P x ==> x=a;  !!x. P x ==> Q x |] \
\     ==> Q (THE x. P x)";
by (rtac premq 1); 
by (rtac theI 1); 
by (REPEAT (ares_tac [prema,premx] 1));
qed "theI2";

Goal "[| EX!x. P x; P a |] ==> (THE x. P x) = a";
by (rtac the_equality 1);
by  (atac 1);
by (etac ex1E 1);
by (etac all_dupE 1);
by (dtac mp 1);
by  (atac 1);
by (etac ssubst 1);
by (etac allE 1);
by (etac mp 1);
by (atac 1);
qed "the1_equality";

Goal "(THE y. x=y) = x";
by (rtac the_equality 1);
by (rtac refl 1);
by (etac sym 1);
qed "the_sym_eq_trivial";


section "Classical intro rules for disjunction and existential quantifiers";

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);