(*  Title:      HOL/set

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1991  University of Cambridge

Set theory for higher-order logic.  A set is simply a predicate.

*)

open Set;

section "Relating predicates and sets";

Addsimps [Collect_mem_eq];

AddIffs  [mem_Collect_eq];

Goal "P(a) ==> a : {x. P(x)}";

by (Asm_simp_tac 1);

qed "CollectI";

Goal "a : {x. P(x)} ==> P(a)";

by (Asm_full_simp_tac 1);

qed "CollectD";

val [prem] = Goal "[| !!x. (x:A) = (x:B) |] ==> A = B";

by (rtac (prem RS ext RS arg_cong RS box_equals) 1);

by (rtac Collect_mem_eq 1);

by (rtac Collect_mem_eq 1);

qed "set_ext";

val [prem] = Goal "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";

by (rtac (prem RS ext RS arg_cong) 1);

qed "Collect_cong";

val CollectE = make_elim CollectD;

AddSIs [CollectI];

AddSEs [CollectE];

section "Bounded quantifiers";

val prems = Goalw [Ball_def]

    "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";

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

qed "ballI";

Goalw [Ball_def] "[| ! x:A. P(x);  x:A |] ==> P(x)";

by (Blast_tac 1);

qed "bspec";

val major::prems = Goalw [Ball_def]

    "[| ! x:A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q";

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

by (REPEAT (eresolve_tac prems 1));

qed "ballE";

(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*)

fun ball_tac i = etac ballE i THEN contr_tac (i+1);

AddSIs [ballI];

AddEs  [ballE];

Goalw [Bex_def] "[| P(x);  x:A |] ==> ? x:A. P(x)";

by (Blast_tac 1);

qed "bexI";

qed_goal "bexCI" Set.thy 

   "[| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A. P(x)"

 (fn prems=>

  [ (rtac classical 1),

    (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1))  ]);

val major::prems = Goalw [Bex_def]

    "[| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q";

by (rtac (major RS exE) 1);

by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1));

qed "bexE";

AddIs  [bexI];

AddSEs [bexE];

(*Trival rewrite rule*)

Goal "(! x:A. P) = ((? x. x:A) --> P)";

by (simp_tac (simpset() addsimps [Ball_def]) 1);

qed "ball_triv";

(*Dual form for existentials*)

Goal "(? x:A. P) = ((? x. x:A) & P)";

by (simp_tac (simpset() addsimps [Bex_def]) 1);

qed "bex_triv";

Addsimps [ball_triv, bex_triv];

(** Congruence rules **)

val prems = Goal

    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \

\    (! x:A. P(x)) = (! x:B. Q(x))";

by (resolve_tac (prems RL [ssubst]) 1);

by (REPEAT (ares_tac [ballI,iffI] 1

     ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1));

qed "ball_cong";

val prems = Goal

    "[| A=B;  !!x. x:B ==> P(x) = Q(x) |] ==> \

\    (? x:A. P(x)) = (? x:B. Q(x))";

by (resolve_tac (prems RL [ssubst]) 1);

by (REPEAT (etac bexE 1

     ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));

qed "bex_cong";

section "Subsets";

val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B";

by (REPEAT (ares_tac (prems @ [ballI]) 1));

qed "subsetI";

Blast.overloaded ("op <=", domain_type); (*The <= relation is overloaded*)

(*While (:) is not, its type must be kept

  for overloading of = to work.*)

Blast.overloaded ("op :", domain_type);

seq (fn a => Blast.overloaded (a, HOLogic.dest_setT o domain_type))

    ["Ball", "Bex"];

(*need UNION, INTER also?*)

(*Image: retain the type of the set being expressed*)

Blast.overloaded ("op ``", domain_type o domain_type);

(*Rule in Modus Ponens style*)

Goalw [subset_def] "[| A <= B;  c:A |] ==> c:B";

by (Blast_tac 1);

qed "subsetD";

(*The same, with reversed premises for use with etac -- cf rev_mp*)

qed_goal "rev_subsetD" Set.thy "[| c:A;  A <= B |] ==> c:B"

 (fn prems=>  [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]);

(*Converts A<=B to x:A ==> x:B*)

fun impOfSubs th = th RSN (2, rev_subsetD);

qed_goal "contra_subsetD" Set.thy "!!c. [| A <= B; c ~: B |] ==> c ~: A"

 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);

qed_goal "rev_contra_subsetD" Set.thy "!!c. [| c ~: B;  A <= B |] ==> c ~: A"

 (fn prems=>  [ (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ]);

(*Classical elimination rule*)

val major::prems = Goalw [subset_def] 

    "[| A <= B;  c~:A ==> P;  c:B ==> P |] ==> P";

by (rtac (major RS ballE) 1);

by (REPEAT (eresolve_tac prems 1));

qed "subsetCE";

(*Takes assumptions A<=B; c:A and creates the assumption c:B *)

fun set_mp_tac i = etac subsetCE i  THEN  mp_tac i;

AddSIs [subsetI];

AddEs  [subsetD, subsetCE];

qed_goal "subset_refl" Set.thy "A <= (A::'a set)"

 (fn _=> [Fast_tac 1]);		(*Blast_tac would try order_refl and fail*)

Goal "[| A<=B;  B<=C |] ==> A<=(C::'a set)";

by (Blast_tac 1);

qed "subset_trans";

section "Equality";

(*Anti-symmetry of the subset relation*)

Goal "[| A <= B;  B <= A |] ==> A = (B::'a set)";

by (rtac set_ext 1);

by (blast_tac (claset() addIs [subsetD]) 1);

qed "subset_antisym";

val equalityI = subset_antisym;

AddSIs [equalityI];

(* Equality rules from ZF set theory -- are they appropriate here? *)

Goal "A = B ==> A<=(B::'a set)";

by (etac ssubst 1);

by (rtac subset_refl 1);

qed "equalityD1";

Goal "A = B ==> B<=(A::'a set)";

by (etac ssubst 1);

by (rtac subset_refl 1);

qed "equalityD2";

val prems = Goal

    "[| A = B;  [| A<=B; B<=(A::'a set) |] ==> P |]  ==>  P";

by (resolve_tac prems 1);

by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));

qed "equalityE";

val major::prems = Goal

    "[| A = B;  [| c:A; c:B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P";

by (rtac (major RS equalityE) 1);

by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1));

qed "equalityCE";

(*Lemma for creating induction formulae -- for "pattern matching" on p

  To make the induction hypotheses usable, apply "spec" or "bspec" to

  put universal quantifiers over the free variables in p. *)

val prems = Goal 

    "[| p:A;  !!z. z:A ==> p=z --> R |] ==> R";

by (rtac mp 1);

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

qed "setup_induction";

section "The universal set -- UNIV";

qed_goalw "UNIV_I" Set.thy [UNIV_def] "x : UNIV"

  (fn _ => [rtac CollectI 1, rtac TrueI 1]);

Addsimps [UNIV_I];

AddIs    [UNIV_I];  (*unsafe makes it less likely to cause problems*)

qed_goal "subset_UNIV" Set.thy "A <= UNIV"

  (fn _ => [rtac subsetI 1, rtac UNIV_I 1]);

(** Eta-contracting these two rules (to remove P) causes them to be ignored

    because of their interaction with congruence rules. **)

Goalw [Ball_def] "Ball UNIV P = All P";

by (Simp_tac 1);

qed "ball_UNIV";

Goalw [Bex_def] "Bex UNIV P = Ex P";

by (Simp_tac 1);

qed "bex_UNIV";

Addsimps [ball_UNIV, bex_UNIV];

section "The empty set -- {}";

qed_goalw "empty_iff" Set.thy [empty_def] "(c : {}) = False"

 (fn _ => [ (Blast_tac 1) ]);

Addsimps [empty_iff];

qed_goal "emptyE" Set.thy "!!a. a:{} ==> P"

 (fn _ => [Full_simp_tac 1]);

AddSEs [emptyE];

qed_goal "empty_subsetI" Set.thy "{} <= A"

 (fn _ => [ (Blast_tac 1) ]);

(*One effect is to delete the ASSUMPTION {} <= A*)

AddIffs [empty_subsetI];

qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}"

 (fn [prem]=>

  [ (blast_tac (claset() addIs [prem RS FalseE]) 1) ]);

(*Use for reasoning about disjointness: A Int B = {} *)

qed_goal "equals0E" Set.thy "!!a. [| A={};  a:A |] ==> P"

 (fn _ => [ (Blast_tac 1) ]);

AddEs [equals0E, sym RS equals0E];

Goalw [Ball_def] "Ball {} P = True";

by (Simp_tac 1);

qed "ball_empty";

Goalw [Bex_def] "Bex {} P = False";

by (Simp_tac 1);

qed "bex_empty";

Addsimps [ball_empty, bex_empty];

Goal "UNIV ~= {}";

by (blast_tac (claset() addEs [equalityE]) 1);

qed "UNIV_not_empty";

AddIffs [UNIV_not_empty];

section "The Powerset operator -- Pow";

qed_goalw "Pow_iff" Set.thy [Pow_def] "(A : Pow(B)) = (A <= B)"

 (fn _ => [ (Asm_simp_tac 1) ]);

AddIffs [Pow_iff]; 

qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)"

 (fn _ => [ (etac CollectI 1) ]);

qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B)  ==>  A<=B"

 (fn _=> [ (etac CollectD 1) ]);

val Pow_bottom = empty_subsetI RS PowI;        (* {}: Pow(B) *)

val Pow_top = subset_refl RS PowI;             (* A : Pow(A) *)

section "Set complement -- Compl";

qed_goalw "Compl_iff" Set.thy [Compl_def] "(c : Compl(A)) = (c~:A)"

 (fn _ => [ (Blast_tac 1) ]);

Addsimps [Compl_iff];

val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : Compl(A)";

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

qed "ComplI";

(*This form, with negated conclusion, works well with the Classical prover.

  Negated assumptions behave like formulae on the right side of the notional

  turnstile...*)

Goalw [Compl_def] "c : Compl(A) ==> c~:A";

by (etac CollectD 1);

qed "ComplD";

val ComplE = make_elim ComplD;

AddSIs [ComplI];

AddSEs [ComplE];

section "Binary union -- Un";

qed_goalw "Un_iff" Set.thy [Un_def] "(c : A Un B) = (c:A | c:B)"

 (fn _ => [ Blast_tac 1 ]);

Addsimps [Un_iff];

Goal "c:A ==> c : A Un B";

by (Asm_simp_tac 1);

qed "UnI1";

Goal "c:B ==> c : A Un B";

by (Asm_simp_tac 1);

qed "UnI2";

(*Classical introduction rule: no commitment to A vs B*)

qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B"

 (fn prems=>

  [ (Simp_tac 1),

    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);

val major::prems = Goalw [Un_def]

    "[| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P";

by (rtac (major RS CollectD RS disjE) 1);

by (REPEAT (eresolve_tac prems 1));

qed "UnE";

AddSIs [UnCI];

AddSEs [UnE];

section "Binary intersection -- Int";

qed_goalw "Int_iff" Set.thy [Int_def] "(c : A Int B) = (c:A & c:B)"

 (fn _ => [ (Blast_tac 1) ]);

Addsimps [Int_iff];

Goal "[| c:A;  c:B |] ==> c : A Int B";

by (Asm_simp_tac 1);

qed "IntI";

Goal "c : A Int B ==> c:A";

by (Asm_full_simp_tac 1);

qed "IntD1";

Goal "c : A Int B ==> c:B";

by (Asm_full_simp_tac 1);

qed "IntD2";

val [major,minor] = Goal

    "[| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P";

by (rtac minor 1);

by (rtac (major RS IntD1) 1);

by (rtac (major RS IntD2) 1);

qed "IntE";

AddSIs [IntI];

AddSEs [IntE];

section "Set difference";

qed_goalw "Diff_iff" Set.thy [set_diff_def] "(c : A-B) = (c:A & c~:B)"

 (fn _ => [ (Blast_tac 1) ]);

Addsimps [Diff_iff];

qed_goal "DiffI" Set.thy "!!c. [| c : A;  c ~: B |] ==> c : A - B"

 (fn _=> [ Asm_simp_tac 1 ]);

qed_goal "DiffD1" Set.thy "!!c. c : A - B ==> c : A"

 (fn _=> [ (Asm_full_simp_tac 1) ]);

qed_goal "DiffD2" Set.thy "!!c. [| c : A - B;  c : B |] ==> P"

 (fn _=> [ (Asm_full_simp_tac 1) ]);

qed_goal "DiffE" Set.thy "[| c : A - B;  [| c:A; c~:B |] ==> P |] ==> P"

 (fn prems=>

  [ (resolve_tac prems 1),

    (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]);

AddSIs [DiffI];

AddSEs [DiffE];

section "Augmenting a set -- insert";

qed_goalw "insert_iff" Set.thy [insert_def] "a : insert b A = (a=b | a:A)"

 (fn _ => [Blast_tac 1]);

Addsimps [insert_iff];

qed_goal "insertI1" Set.thy "a : insert a B"

 (fn _ => [Simp_tac 1]);

qed_goal "insertI2" Set.thy "!!a. a : B ==> a : insert b B"

 (fn _=> [Asm_simp_tac 1]);

qed_goalw "insertE" Set.thy [insert_def]

    "[| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P"

 (fn major::prems=>

  [ (rtac (major RS UnE) 1),

    (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]);

(*Classical introduction rule*)

qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert b B"

 (fn prems=>

  [ (Simp_tac 1),

    (REPEAT (ares_tac (prems@[disjCI]) 1)) ]);

AddSIs [insertCI]; 

AddSEs [insertE];

section "Singletons, using insert";

qed_goal "singletonI" Set.thy "a : {a}"

 (fn _=> [ (rtac insertI1 1) ]);

Goal "b : {a} ==> b=a";

by (Blast_tac 1);

qed "singletonD";

bind_thm ("singletonE", make_elim singletonD);

qed_goal "singleton_iff" thy "(b : {a}) = (b=a)" 

(fn _ => [Blast_tac 1]);

Goal "{a}={b} ==> a=b";

by (blast_tac (claset() addEs [equalityE]) 1);

qed "singleton_inject";

(*Redundant? But unlike insertCI, it proves the subgoal immediately!*)

AddSIs [singletonI];   

AddSDs [singleton_inject];

AddSEs [singletonE];

Goal "{x. x=a} = {a}";

by (Blast_tac 1);

qed "singleton_conv";

Addsimps [singleton_conv];

section "Unions of families -- UNION x:A. B(x) is Union(B``A)";

Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))";

by (Blast_tac 1);

qed "UN_iff";

Addsimps [UN_iff];

(*The order of the premises presupposes that A is rigid; b may be flexible*)

Goal "[| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))";

by Auto_tac;

qed "UN_I";

val major::prems = Goalw [UNION_def]

    "[| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R";

by (rtac (major RS CollectD RS bexE) 1);

by (REPEAT (ares_tac prems 1));

qed "UN_E";

AddIs  [UN_I];

AddSEs [UN_E];

val prems = Goal

    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \

\    (UN x:A. C(x)) = (UN x:B. D(x))";

by (REPEAT (etac UN_E 1

     ORELSE ares_tac ([UN_I,equalityI,subsetI] @ 

                      (prems RL [equalityD1,equalityD2] RL [subsetD])) 1));

qed "UN_cong";

section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)";

Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))";

by Auto_tac;

qed "INT_iff";

Addsimps [INT_iff];

val prems = Goalw [INTER_def]

    "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";

by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));

qed "INT_I";

Goal "[| b : (INT x:A. B(x));  a:A |] ==> b: B(a)";

by Auto_tac;

qed "INT_D";

(*"Classical" elimination -- by the Excluded Middle on a:A *)

val major::prems = Goalw [INTER_def]

    "[| b : (INT x:A. B(x));  b: B(a) ==> R;  a~:A ==> R |] ==> R";

by (rtac (major RS CollectD RS ballE) 1);

by (REPEAT (eresolve_tac prems 1));

qed "INT_E";

AddSIs [INT_I];

AddEs  [INT_D, INT_E];

val prems = Goal

    "[| A=B;  !!x. x:B ==> C(x) = D(x) |] ==> \

\    (INT x:A. C(x)) = (INT x:B. D(x))";

by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI]));

by (REPEAT (dtac INT_D 1

     ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1));

qed "INT_cong";

section "Union";

Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)";

by (Blast_tac 1);

qed "Union_iff";

Addsimps [Union_iff];

(*The order of the premises presupposes that C is rigid; A may be flexible*)

Goal "[| X:C;  A:X |] ==> A : Union(C)";

by Auto_tac;

qed "UnionI";

val major::prems = Goalw [Union_def]

    "[| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R";

by (rtac (major RS UN_E) 1);

by (REPEAT (ares_tac prems 1));

qed "UnionE";

AddIs  [UnionI];

AddSEs [UnionE];

section "Inter";

Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)";

by (Blast_tac 1);

qed "Inter_iff";

Addsimps [Inter_iff];

val prems = Goalw [Inter_def]

    "[| !!X. X:C ==> A:X |] ==> A : Inter(C)";

by (REPEAT (ares_tac ([INT_I] @ prems) 1));

qed "InterI";

(*A "destruct" rule -- every X in C contains A as an element, but

  A:X can hold when X:C does not!  This rule is analogous to "spec". *)

Goal "[| A : Inter(C);  X:C |] ==> A:X";

by Auto_tac;

qed "InterD";

(*"Classical" elimination rule -- does not require proving X:C *)

val major::prems = Goalw [Inter_def]

    "[| A : Inter(C);  X~:C ==> R;  A:X ==> R |] ==> R";

by (rtac (major RS INT_E) 1);

by (REPEAT (eresolve_tac prems 1));

qed "InterE";

AddSIs [InterI];

AddEs  [InterD, InterE];

(*** Image of a set under a function ***)

(*Frequently b does not have the syntactic form of f(x).*)

Goalw [image_def] "[| b=f(x);  x:A |] ==> b : f``A";

by (Blast_tac 1);

qed "image_eqI";

Addsimps [image_eqI];

bind_thm ("imageI", refl RS image_eqI);

(*The eta-expansion gives variable-name preservation.*)

val major::prems = Goalw [image_def]

    "[| b : (%x. f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 

by (rtac (major RS CollectD RS bexE) 1);

by (REPEAT (ares_tac prems 1));

qed "imageE";

AddIs  [image_eqI];

AddSEs [imageE]; 

Goal "f``(A Un B) = f``A Un f``B";

by (Blast_tac 1);

qed "image_Un";

Goal "(z : f``A) = (EX x:A. z = f x)";

by (Blast_tac 1);

qed "image_iff";

(*This rewrite rule would confuse users if made default.*)

Goal "(f``A <= B) = (ALL x:A. f(x): B)";

by (Blast_tac 1);

qed "image_subset_iff";

(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too

  many existing proofs.*)

val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B";

by (blast_tac (claset() addIs prems) 1);

qed "image_subsetI";

(*** Range of a function -- just a translation for image! ***)

Goal "b=f(x) ==> b : range(f)";

by (EVERY1 [etac image_eqI, rtac UNIV_I]);

bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));

bind_thm ("rangeI", UNIV_I RS imageI);

val [major,minor] = Goal 

    "[| b : range(%x. f(x));  !!x. b=f(x) ==> P |] ==> P"; 

by (rtac (major RS imageE) 1);

by (etac minor 1);

qed "rangeE";

(*** Set reasoning tools ***)

(** Rewrite rules for boolean case-splitting: faster than 

	addsplits[split_if]

**)

bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if);

bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if);

(*Split ifs on either side of the membership relation.

	Not for Addsimps -- can cause goals to blow up!*)

bind_thm ("split_if_mem1", 

    read_instantiate_sg (sign_of Set.thy) [("P", "%x. x : ?b")] split_if);

bind_thm ("split_if_mem2", 

    read_instantiate_sg (sign_of Set.thy) [("P", "%x. ?a : x")] split_if);

val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2,

		  split_if_mem1, split_if_mem2];

(*Each of these has ALREADY been added to simpset() above.*)

val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, 

                 mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff];

val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs;

simpset_ref() := simpset() addcongs [ball_cong,bex_cong]

                    setmksimps (mksimps mksimps_pairs);

Addsimps[subset_UNIV, subset_refl];

(*** < ***)

Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B";

by (Blast_tac 1);

qed "psubsetI";

Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B";

by Auto_tac;

qed "psubset_insertD";

bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq);