HOL.ML

HOL/Sum.thy: generalized the type of Part

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

For hol.thy
Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68 
*)

open HOL;

signature HOL_LEMMAS =
  sig
  val allE: thm
  val all_dupE: thm
  val allI: thm
  val arg_cong: thm
  val fun_cong: thm
  val box_equals: thm
  val case_tac: string -> int -> tactic
  val ccontr: thm
  val classical: thm
  val cong: thm
  val conjunct1: thm
  val conjunct2: thm
  val conjE: thm
  val conjI: thm
  val contrapos: thm
  val disjCI: thm
  val disjE: thm
  val disjI1: thm
  val disjI2: thm
  val eqTrueI: thm
  val eqTrueE: thm
  val ex1E: thm
  val ex1I: thm
  val exCI: thm
  val exI: thm
  val exE: thm
  val excluded_middle: thm
  val FalseE: thm
  val False_neq_True: thm
  val iffCE : thm
  val iffD1: thm
  val iffD2: thm
  val iffE: thm
  val iffI: thm
  val impCE: thm
  val impE: thm
  val not_sym: thm
  val notE: thm
  val notI: thm
  val notnotD : thm
  val rev_mp: thm
  val select_equality: thm
  val spec: thm
  val sstac: thm list -> int -> tactic
  val ssubst: thm
  val stac: thm -> int -> tactic
  val strip_tac: int -> tactic
  val swap: thm
  val sym: thm
  val trans: thm
  val TrueI: thm
  end;

structure HOL_Lemmas : HOL_LEMMAS =

struct

(** Equality **)

val sym = prove_goal 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*)
val ssubst = standard (sym RS subst);

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

(*Useful with eresolve_tac for proving equalties from known equalities.
	a = b
	|   |
	c = d	*)
val box_equals = prove_goal HOL.thy
    "[| a=b;  a=c;  b=d |] ==> c=d"  
 (fn prems=>
  [ (rtac trans 1),
    (rtac trans 1),
    (rtac sym 1),
    (REPEAT (resolve_tac prems 1)) ]);

(** Congruence rules for meta-application **)

(*similar to AP_THM in Gordon's HOL*)
val fun_cong = prove_goal 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*)
val arg_cong = prove_goal HOL.thy "x=y ==> f(x)=f(y)"
 (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);

val cong = prove_goal 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 **)

val iffI = prove_goal HOL.thy
   "[| P ==> Q;  Q ==> P |] ==> P=Q"
 (fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]);

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

val iffD1 = sym RS iffD2;

val iffE = prove_goal 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 **)

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

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

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

(** Universal quantifier **)

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

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

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

val all_dupE = prove_goal HOL.thy 
    "[| ! x.P(x);  [| P(x); ! x.P(x) |] ==> R |] ==> R"
 (fn prems =>
  [ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]);


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

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

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


(** Negation **)

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

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

(** Implication **)

val impE = prove_goal HOL.thy "[| P-->Q;  P;  Q ==> R |] ==> R"
 (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);

(* Reduces Q to P-->Q, allowing substitution in P. *)
val rev_mp = prove_goal HOL.thy "[| P;  P --> Q |] ==> Q"
 (fn prems=>  [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);

val contrapos = prove_goal HOL.thy "[| ~Q;  P==>Q |] ==> ~P"
 (fn [major,minor]=> 
  [ (rtac (major RS notE RS notI) 1), 
    (etac minor 1) ]);

(* ~(?t = ?s) ==> ~(?s = ?t) *)
val [not_sym] = compose(sym,2,contrapos);


(** Existential quantifier **)

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

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


(** Conjunction **)

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

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

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

val conjE = prove_goal 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]);

(** Disjunction *)

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

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

val disjE = prove_goalw 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, atac 1, rtac (a3 RS impI) 1, atac 1]);

(** CCONTR -- classical logic **)

val ccontr = prove_goal HOL.thy "(~P ==> False) ==> P"
 (fn prems =>
   [rtac (True_or_False RS (disjE RS eqTrueE)) 1, atac 1,
    rtac spec 1, fold_tac [False_def], resolve_tac prems 1,
    rtac ssubst 1, atac 1, rewtac not_def,
    REPEAT (ares_tac [impI] 1) ]);

val ccontr = prove_goalw HOL.thy [not_def] "(~P ==> False) ==> P"
 (fn prems =>
   [rtac (True_or_False RS (disjE RS eqTrueE)) 1, atac 1,
    rtac spec 1, fold_tac [False_def], resolve_tac prems 1,
    rtac ssubst 1, atac 1, REPEAT (ares_tac [impI] 1) ]);

val classical = prove_goal HOL.thy "(~P ==> P) ==> P"
 (fn prems => [rtac ccontr 1, REPEAT (ares_tac (prems@[notE]) 1)]);


(*Double negation law*)
val notnotD = prove_goal HOL.thy "~~P ==> P"
 (fn [major]=>
  [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);


(** Unique existence **)

val ex1I = prove_goalw 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)]);

val ex1E = prove_goalw 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)]);


(** Select: Hilbert's Epsilon-operator **)

val select_equality = prove_goal HOL.thy
    "[| P(a);  !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a"
 (fn prems => [ resolve_tac prems 1, 
	        rtac selectI 1, 
		resolve_tac prems 1 ]);

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

val disjCI = prove_goal HOL.thy "(~Q ==> P) ==> P|Q"
 (fn prems=>
  [ (rtac classical 1),
    (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
    (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);

val excluded_middle = prove_goal HOL.thy "~P | P"
 (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);

(*Classical implies (-->) elimination. *)
val impCE = prove_goal HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R" 
 (fn major::prems=>
  [ rtac (excluded_middle RS disjE) 1,
    REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);

(*Classical <-> elimination. *)
val iffCE = prove_goal HOL.thy
    "[| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"
 (fn major::prems =>
  [ (rtac (major RS iffE) 1),
    (REPEAT (DEPTH_SOLVE_1 
	(eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);

val exCI = prove_goal HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x.P(x)"
 (fn prems=>
  [ (rtac ccontr 1),
    (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ]);

(*Required by the "classical" module*)
val swap = prove_goal HOL.thy "~P ==> (~Q ==> P) ==> Q"
 (fn major::prems=>
  [ rtac ccontr 1, rtac (major RS notE) 1, REPEAT (ares_tac prems 1)]);

(* case distinction *)

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

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

(** Standard abbreviations **)

fun stac th = rtac(th RS ssubst);
fun sstac ths = EVERY' (map stac ths);
fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); 


end;

open HOL_Lemmas;