src/HOL/HOL_lemmas.ML

Goal: tuned pris;

(*  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.
*)

(* ML bindings *)

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


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

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";

(** Congruence rules for meta-application **)
section "Congruence";

(*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";

(** 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";

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";


(** True **)
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";


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

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

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

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

val prems = goal (the_context ()) 
    "[| ! 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" (the_context ()) [False_def] "False ==> P"
 (fn [major] => [rtac (major RS spec) 1]);

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


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

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

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

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

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

bind_thm ("classical2", notE RS notI);

qed_goal "rev_notE" (the_context ()) "!!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" (the_context ()) [Ex_def] "P x ==> ? x::'a. P x"
 (fn prems => [rtac selectI 1, resolve_tac prems 1]);

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


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

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

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

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

qed_goal "conjE" (the_context ()) "[| 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" (the_context ())  "[| P; P ==> Q |] ==> P & Q"
 (fn prems => [REPEAT(resolve_tac (conjI::prems) 1)]);


(** Disjunction *)
section "|";

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

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

qed_goalw "disjE" (the_context ()) [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" (the_context ()) [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]);

bind_thm ("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" (the_context ()) [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" (the_context ()) [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" (the_context ())
  "[| ? 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" (the_context ()) "[| P ==> Q; ~P ==> Q |] ==> Q"
  (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
                  etac p2 1, etac p1 1]);

bind_thm ("case_split", case_split_thm);

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;

(* 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); 

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

(** THIS CODE IS A MESS!!! **)

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

end;