--- a/Set.ML Fri Nov 11 10:35:03 1994 +0100
+++ b/Set.ML Mon Nov 21 17:50:34 1994 +0100
@@ -11,21 +11,21 @@
val [prem] = goal Set.thy "[| P(a) |] ==> a : {x.P(x)}";
by (rtac (mem_Collect_eq RS ssubst) 1);
by (rtac prem 1);
-val CollectI = result();
+qed "CollectI";
val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)";
by (resolve_tac (prems RL [mem_Collect_eq RS subst]) 1);
-val CollectD = result();
+qed "CollectD";
val [prem] = goal Set.thy "[| !!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);
-val set_ext = result();
+qed "set_ext";
val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}";
by (rtac (prem RS ext RS arg_cong) 1);
-val Collect_cong = result();
+qed "Collect_cong";
val CollectE = make_elim CollectD;
@@ -34,18 +34,18 @@
val prems = goalw Set.thy [Ball_def]
"[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)";
by (REPEAT (ares_tac (prems @ [allI,impI]) 1));
-val ballI = result();
+qed "ballI";
val [major,minor] = goalw Set.thy [Ball_def]
"[| ! x:A. P(x); x:A |] ==> P(x)";
by (rtac (minor RS (major RS spec RS mp)) 1);
-val bspec = result();
+qed "bspec";
val major::prems = goalw Set.thy [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));
-val ballE = result();
+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);
@@ -53,7 +53,7 @@
val prems = goalw Set.thy [Bex_def]
"[| P(x); x:A |] ==> ? x:A. P(x)";
by (REPEAT (ares_tac (prems @ [exI,conjI]) 1));
-val bexI = result();
+qed "bexI";
val bexCI = prove_goal Set.thy
"[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A.P(x)"
@@ -65,13 +65,13 @@
"[| ? 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));
-val bexE = result();
+qed "bexE";
(*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*)
val prems = goal Set.thy
"(! x:A. True) = True";
by (REPEAT (ares_tac [TrueI,ballI,iffI] 1));
-val ball_rew = result();
+qed "ball_rew";
(** Congruence rules **)
@@ -81,7 +81,7 @@
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));
-val ball_cong = result();
+qed "ball_cong";
val prems = goal Set.thy
"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \
@@ -89,19 +89,19 @@
by (resolve_tac (prems RL [ssubst]) 1);
by (REPEAT (etac bexE 1
ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1));
-val bex_cong = result();
+qed "bex_cong";
(*** Subsets ***)
val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B";
by (REPEAT (ares_tac (prems @ [ballI]) 1));
-val subsetI = result();
+qed "subsetI";
(*Rule in Modus Ponens style*)
val major::prems = goalw Set.thy [subset_def] "[| A <= B; c:A |] ==> c:B";
by (rtac (major RS bspec) 1);
by (resolve_tac prems 1);
-val subsetD = result();
+qed "subsetD";
(*The same, with reversed premises for use with etac -- cf rev_mp*)
val rev_subsetD = prove_goal Set.thy "[| c:A; A <= B |] ==> c:B"
@@ -112,7 +112,7 @@
"[| A <= B; c~:A ==> P; c:B ==> P |] ==> P";
by (rtac (major RS ballE) 1);
by (REPEAT (eresolve_tac prems 1));
-val subsetCE = result();
+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;
@@ -123,7 +123,7 @@
val prems = goal Set.thy "[| A<=B; B<=C |] ==> A<=(C::'a set)";
by (cut_facts_tac prems 1);
by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1));
-val subset_trans = result();
+qed "subset_trans";
(*** Equality ***)
@@ -132,31 +132,31 @@
val prems = goal Set.thy "[| A <= B; B <= A |] ==> A = (B::'a set)";
by (rtac (iffI RS set_ext) 1);
by (REPEAT (ares_tac (prems RL [subsetD]) 1));
-val subset_antisym = result();
+qed "subset_antisym";
val equalityI = subset_antisym;
(* Equality rules from ZF set theory -- are they appropriate here? *)
val prems = goal Set.thy "A = B ==> A<=(B::'a set)";
by (resolve_tac (prems RL [subst]) 1);
by (rtac subset_refl 1);
-val equalityD1 = result();
+qed "equalityD1";
val prems = goal Set.thy "A = B ==> B<=(A::'a set)";
by (resolve_tac (prems RL [subst]) 1);
by (rtac subset_refl 1);
-val equalityD2 = result();
+qed "equalityD2";
val prems = goal Set.thy
"[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P";
by (resolve_tac prems 1);
by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1));
-val equalityE = result();
+qed "equalityE";
val major::prems = goal Set.thy
"[| 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));
-val equalityCE = result();
+qed "equalityCE";
(*Lemma for creating induction formulae -- for "pattern matching" on p
To make the induction hypotheses usable, apply "spec" or "bspec" to
@@ -165,7 +165,7 @@
"[| p:A; !!z. z:A ==> p=z --> R |] ==> R";
by (rtac mp 1);
by (REPEAT (resolve_tac (refl::prems) 1));
-val setup_induction = result();
+qed "setup_induction";
(*** Set complement -- Compl ***)
@@ -173,7 +173,7 @@
val prems = goalw Set.thy [Compl_def]
"[| c:A ==> False |] ==> c : Compl(A)";
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1));
-val ComplI = result();
+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
@@ -181,7 +181,7 @@
val major::prems = goalw Set.thy [Compl_def]
"[| c : Compl(A) |] ==> c~:A";
by (rtac (major RS CollectD) 1);
-val ComplD = result();
+qed "ComplD";
val ComplE = make_elim ComplD;
@@ -190,11 +190,11 @@
val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B";
by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1));
-val UnI1 = result();
+qed "UnI1";
val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B";
by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1));
-val UnI2 = result();
+qed "UnI2";
(*Classical introduction rule: no commitment to A vs B*)
val UnCI = prove_goal Set.thy "(c~:B ==> c:A) ==> c : A Un B"
@@ -207,7 +207,7 @@
"[| 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));
-val UnE = result();
+qed "UnE";
(*** Binary intersection -- Int ***)
@@ -215,22 +215,22 @@
val prems = goalw Set.thy [Int_def]
"[| c:A; c:B |] ==> c : A Int B";
by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1));
-val IntI = result();
+qed "IntI";
val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A";
by (rtac (major RS CollectD RS conjunct1) 1);
-val IntD1 = result();
+qed "IntD1";
val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B";
by (rtac (major RS CollectD RS conjunct2) 1);
-val IntD2 = result();
+qed "IntD2";
val [major,minor] = goal Set.thy
"[| 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);
-val IntE = result();
+qed "IntE";
(*** Set difference ***)
@@ -311,14 +311,14 @@
goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a";
by(fast_tac (HOL_cs addSEs [emptyE,CollectE,UnE]) 1);
-val singletonD = result();
+qed "singletonD";
val singletonE = make_elim singletonD;
val [major] = goal Set.thy "{a}={b} ==> a=b";
by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1);
by (rtac singletonI 1);
-val singleton_inject = result();
+qed "singleton_inject";
(*** Unions of families -- UNION x:A. B(x) is Union(B``A) ***)
@@ -326,13 +326,13 @@
val prems = goalw Set.thy [UNION_def]
"[| a:A; b: B(a) |] ==> b: (UN x:A. B(x))";
by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1));
-val UN_I = result();
+qed "UN_I";
val major::prems = goalw Set.thy [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));
-val UN_E = result();
+qed "UN_E";
val prems = goal Set.thy
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \
@@ -340,7 +340,7 @@
by (REPEAT (etac UN_E 1
ORELSE ares_tac ([UN_I,equalityI,subsetI] @
(prems RL [equalityD1,equalityD2] RL [subsetD])) 1));
-val UN_cong = result();
+qed "UN_cong";
(*** Intersections of families -- INTER x:A. B(x) is Inter(B``A) *)
@@ -348,20 +348,20 @@
val prems = goalw Set.thy [INTER_def]
"(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))";
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1));
-val INT_I = result();
+qed "INT_I";
val major::prems = goalw Set.thy [INTER_def]
"[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)";
by (rtac (major RS CollectD RS bspec) 1);
by (resolve_tac prems 1);
-val INT_D = result();
+qed "INT_D";
(*"Classical" elimination -- by the Excluded Middle on a:A *)
val major::prems = goalw Set.thy [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));
-val INT_E = result();
+qed "INT_E";
val prems = goal Set.thy
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \
@@ -369,7 +369,7 @@
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));
-val INT_cong = result();
+qed "INT_cong";
(*** Unions over a type; UNION1(B) = Union(range(B)) ***)
@@ -378,13 +378,13 @@
val prems = goalw Set.thy [UNION1_def]
"b: B(x) ==> b: (UN x. B(x))";
by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1));
-val UN1_I = result();
+qed "UN1_I";
val major::prems = goalw Set.thy [UNION1_def]
"[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R";
by (rtac (major RS UN_E) 1);
by (REPEAT (ares_tac prems 1));
-val UN1_E = result();
+qed "UN1_E";
(*** Intersections over a type; INTER1(B) = Inter(range(B)) *)
@@ -392,12 +392,12 @@
val prems = goalw Set.thy [INTER1_def]
"(!!x. b: B(x)) ==> b : (INT x. B(x))";
by (REPEAT (ares_tac (INT_I::prems) 1));
-val INT1_I = result();
+qed "INT1_I";
val [major] = goalw Set.thy [INTER1_def]
"b : (INT x. B(x)) ==> b: B(a)";
by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1);
-val INT1_D = result();
+qed "INT1_D";
(*** Unions ***)
@@ -405,20 +405,20 @@
val prems = goalw Set.thy [Union_def]
"[| X:C; A:X |] ==> A : Union(C)";
by (REPEAT (resolve_tac (prems @ [UN_I]) 1));
-val UnionI = result();
+qed "UnionI";
val major::prems = goalw Set.thy [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));
-val UnionE = result();
+qed "UnionE";
(*** Inter ***)
val prems = goalw Set.thy [Inter_def]
"[| !!X. X:C ==> A:X |] ==> A : Inter(C)";
by (REPEAT (ares_tac ([INT_I] @ prems) 1));
-val InterI = result();
+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". *)
@@ -426,14 +426,14 @@
"[| A : Inter(C); X:C |] ==> A:X";
by (rtac (major RS INT_D) 1);
by (resolve_tac prems 1);
-val InterD = result();
+qed "InterD";
(*"Classical" elimination rule -- does not require proving X:C *)
val major::prems = goalw Set.thy [Inter_def]
"[| A : Inter(C); A:X ==> R; X~:C ==> R |] ==> R";
by (rtac (major RS INT_E) 1);
by (REPEAT (eresolve_tac prems 1));
-val InterE = result();
+qed "InterE";
(*** Powerset ***)