src/HOL/HOL.ML

isar: no_pos flag;

(*  Title:      HOL/HOL.ML
    ID:         $Id$
    Author:     Tobias Nipkow
    Copyright   1991  University of Cambridge

Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68 
*)


(** Equality **)
section "=";

qed_goal "sym" HOL.thy "s=t ==> t=s"
 (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);

(*calling "standard" reduces maxidx to 0*)
bind_thm ("ssubst", (sym RS subst));

qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t"
 (fn prems =>
        [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);

val prems = goal thy "(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*)
qed_goal "fun_cong" HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)"
  (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);

(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)"
 (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);

qed_goal "cong" HOL.thy
   "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
 (fn [prem1,prem2] =>
   [rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]);


(** 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";

qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P"
 (fn prems =>
        [rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);

qed_goal "rev_iffD2" HOL.thy "!!P. [| Q; P=Q |] ==> P"
 (fn _ => [etac iffD2 1, assume_tac 1]);

bind_thm ("iffD1", sym RS iffD2);
bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2));

qed_goal "iffE" HOL.thy
    "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
 (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);


(** True **)
section "True";

qed_goalw "TrueI" HOL.thy [True_def] "True"
  (fn _ => [(rtac refl 1)]);

qed_goal "eqTrueI" HOL.thy "P ==> P=True" 
 (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);

qed_goal "eqTrueE" HOL.thy "P=True ==> P" 
 (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);


(** Universal quantifier **)
section "!";

qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
 (fn prems => [(resolve_tac (prems RL [eqTrueI RS ext]) 1)]);

qed_goalw "spec" HOL.thy [All_def] "! x::'a. P(x) ==> P(x)"
 (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);

val major::prems= goal HOL.thy "[| !x. P(x);  P(x) ==> R |] ==> R";
by (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ;
qed "allE";

val prems = goal HOL.thy 
    "[| ! x. P(x);  [| P(x); ! 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";

qed_goalw "FalseE" HOL.thy [False_def] "False ==> P"
 (fn [major] => [rtac (major RS spec) 1]);

qed_goal "False_neq_True" HOL.thy "False=True ==> P"
 (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);


(** Negation **)
section "~";

qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P"
 (fn prems=> [rtac impI 1, eresolve_tac prems 1]);

qed_goal "False_not_True" HOL.thy "False ~= True"
  (fn _ => [rtac notI 1, etac False_neq_True 1]);

qed_goal "True_not_False" HOL.thy "True ~= False"
  (fn _ => [rtac notI 1, dtac sym 1, etac False_neq_True 1]);

qed_goalw "notE" HOL.thy [not_def] "[| ~P;  P |] ==> R"
 (fn prems => [rtac (prems MRS mp RS FalseE) 1]);

bind_thm ("classical2", notE RS notI);

qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R"
 (fn _ => [REPEAT (ares_tac [notE] 1)]);


(** 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 "?";

qed_goalw "exI" HOL.thy [Ex_def] "P x ==> ? x::'a. P x"
 (fn prems => [rtac selectI 1, resolve_tac prems 1]);

qed_goalw "exE" HOL.thy [Ex_def]
  "[| ? x::'a. P(x); !!x. P(x) ==> Q |] ==> Q"
  (fn prems => [REPEAT(resolve_tac prems 1)]);


(** Conjunction **)
section "&";

qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
 (fn prems =>
  [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);

qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
 (fn prems =>
   [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);

qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
 (fn prems =>
   [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);

qed_goal "conjE" HOL.thy "[| P&Q;  [| P; Q |] ==> R |] ==> R"
 (fn prems =>
         [cut_facts_tac prems 1, resolve_tac prems 1,
          etac conjunct1 1, etac conjunct2 1]);

qed_goal "context_conjI" HOL.thy  "[| P; P ==> Q |] ==> P & Q"
 (fn prems => [REPEAT(resolve_tac (conjI::prems) 1)]);


(** Disjunction *)
section "|";

qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
 (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);

qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
 (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);

qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
 (fn [a1,a2,a3] =>
        [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
         rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);


(** CCONTR -- classical logic **)
section "classical logic";

qed_goalw "classical" HOL.thy [not_def]  "(~P ==> P) ==> P"
 (fn [prem] =>
   [rtac (True_or_False RS (disjE RS eqTrueE)) 1,  assume_tac 1,
    rtac (impI RS prem RS eqTrueI) 1,
    etac subst 1,  assume_tac 1]);

val ccontr = FalseE RS classical;

(*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 "?!";

qed_goalw "ex1I" HOL.thy [Ex1_def]
            "[| P(a);  !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
 (fn prems =>
  [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);

(*Sometimes easier to use: the premises have no shared variables.  Safe!*)
val [ex,eq] = Goal
    "[| ? x. P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> ?! x. P(x)";
by (rtac (ex RS exE) 1);
by (REPEAT (ares_tac [ex1I,eq] 1)) ;
qed "ex_ex1I";

qed_goalw "ex1E" HOL.thy [Ex1_def]
    "[| ?! x. P(x);  !!x. [| P(x);  ! y. P(y) --> y=x |] ==> R |] ==> R"
 (fn major::prems =>
  [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);

Goal "?! x. P x ==> ? 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 selectI: 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 selectI 1);
by (resolve_tac prems 1) ;
qed "selectI2";

(*Easier to apply than selectI2 if witness ?a comes from an EX-formula*)
qed_goal "selectI2EX" HOL.thy
  "[| ? a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)"
(fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]);

val prems = Goal
    "[| P a;  !!x. P x ==> x=a |] ==> (@x. P x) = a";
by (rtac selectI2 1);
by (REPEAT (ares_tac prems 1)) ;
qed "select_equality";

Goalw [Ex1_def] "[| ?!x. P x; P a |] ==> (@x. P x) = a";
by (rtac select_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 "select1_equality";

Goal "P (@ x. P x) =  (? x. P x)";
by (rtac iffI 1);
by (etac exI 1);
by (etac exE 1);
by (etac selectI 1);
qed "select_eq_Ex";

Goal "(@y. y=x) = x";
by (rtac select_equality 1);
by (rtac refl 1);
by (atac 1);
qed "Eps_eq";

Goal "(Eps (op = x)) = x";
by (rtac select_equality 1);
by (rtac refl 1);
by (etac sym 1);
qed "Eps_sym_eq";

(** 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 "(! x. ~P(x) ==> P(a)) ==> ? x. P(x)";
by (rtac ccontr 1);
by (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ;
qed "exCI";


(* case distinction *)

qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
  (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
                  etac p2 1, etac p1 1]);

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


(** Standard abbreviations **)

(*Apply an equality or definition ONCE.
  Fails unless the substitution has an effect*)
fun stac th = 
  let val th' = th RS def_imp_eq handle THM _ => th
  in  CHANGED_GOAL (rtac (th' RS ssubst))
  end;

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


(** strip ! and --> from proved goal while preserving !-bound var names **)

local

(* Use XXX to avoid forall_intr failing because of duplicate variable name *)
val myspec = read_instantiate [("P","?XXX")] spec;
val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec;
val cvx = cterm_of (#sign(rep_thm myspec)) vx;
val aspec = forall_intr cvx myspec;

in

fun RSspec th =
  (case concl_of th of
     _ $ (Const("All",_) $ Abs(a,_,_)) =>
         let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
         in th RS forall_elim ca aspec end
  | _ => raise THM("RSspec",0,[th]));

fun RSmp th =
  (case concl_of th of
     _ $ (Const("op -->",_)$_$_) => th RS mp
  | _ => raise THM("RSmp",0,[th]));

fun normalize_thm funs =
  let fun trans [] th = th
	| trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
  in zero_var_indexes o trans funs end;

fun qed_spec_mp name =
  let val thm = normalize_thm [RSspec,RSmp] (result())
  in ThmDatabase.ml_store_thm(name, thm) end;

fun qed_goal_spec_mp name thy s p = 
	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));

fun qed_goalw_spec_mp name thy defs s p = 
	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));

end;


(* attributes *)

local

fun gen_rulify x = Attrib.no_args (Drule.rule_attribute (fn _ => (normalize_thm [RSspec, RSmp]))) x;

in

val hol_setup =
 [Attrib.add_attributes
  [("rulify", (gen_rulify, gen_rulify), "put theorem into standard rule form")]];

end;