--- a/src/ZF/CardinalArith.ML Fri Aug 12 12:51:34 1994 +0200
+++ b/src/ZF/CardinalArith.ML Fri Aug 12 18:45:33 1994 +0200
@@ -619,6 +619,11 @@
val lt_csucc = csucc_basic RS conjunct2 |> standard;
+goal CardinalArith.thy "!!K. Ord(K) ==> 0 < csucc(K)";
+by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1);
+by (REPEAT (assume_tac 1));
+val Ord_0_lt_csucc = result();
+
goalw CardinalArith.thy [csucc_def]
"!!K L. [| Card(L); K<L |] ==> csucc(K) le L";
by (rtac Least_le 1);
@@ -650,3 +655,5 @@
by (asm_simp_tac (ZF_ss addsimps [Card_csucc, Card_is_Ord,
lt_csucc RS leI RSN (2,le_trans)]) 1);
val InfCard_csucc = result();
+
+val Limit_csucc = InfCard_csucc RS InfCard_is_Limit |> standard;
--- a/src/ZF/Cardinal_AC.ML Fri Aug 12 12:51:34 1994 +0200
+++ b/src/ZF/Cardinal_AC.ML Fri Aug 12 18:45:33 1994 +0200
@@ -4,6 +4,8 @@
Copyright 1994 University of Cambridge
Cardinal arithmetic WITH the Axiom of Choice
+
+These results help justify infinite-branching datatypes
*)
open Cardinal_AC;
@@ -135,15 +137,34 @@
by (eresolve_tac [InfCard_is_Card RS Card_is_Ord RS Card_csucc] 1);
val cardinal_UN_Ord_lt_csucc = result();
+
+(*Saves checking Ord(j) below*)
+goal Ordinal.thy "!!i j. [| i <= j; j<k; Ord(i) |] ==> i<k";
+by (resolve_tac [subset_imp_le RS lt_trans1] 1);
+by (REPEAT (eresolve_tac [asm_rl, ltE] 1));
+val lt_subset_trans = result();
+
goal Cardinal_AC.thy
- "!!K. [| InfCard(K); |I| le K; ALL i:I. j(i) < csucc(K) |] ==> \
-\ (UN i:I. j(i)) < csucc(K)";
+ "!!K. [| InfCard(K); |W| le K; ALL w:W. j(w) < csucc(K) |] ==> \
+\ (UN w:W. j(w)) < csucc(K)";
+by (excluded_middle_tac "W=0" 1);
+by (asm_simp_tac
+ (ZF_ss addsimps [UN_0, InfCard_is_Card, Card_is_Ord RS Card_csucc,
+ Card_is_Ord, Ord_0_lt_csucc]) 2);
by (asm_full_simp_tac
(ZF_ss addsimps [InfCard_is_Card, le_Card_iff, lepoll_def]) 1);
-by (eresolve_tac [exE] 1);
-by (resolve_tac [lt_trans1] 1);
-by (resolve_tac [cardinal_UN_Ord_lt_csucc] 2);
+by (safe_tac eq_cs);
+by (eresolve_tac [notE] 1);
+by (res_inst_tac [("j1", "%i. j(if(i: range(f), converse(f)`i, x))")]
+ (cardinal_UN_Ord_lt_csucc RSN (2,lt_subset_trans)) 1);
by (assume_tac 2);
+by (resolve_tac [UN_least] 1);
+by (res_inst_tac [("x1", "f`xa")] (UN_upper RSN (2,subset_trans)) 1);
+by (eresolve_tac [inj_is_fun RS apply_type] 2 THEN assume_tac 2);
+by (asm_simp_tac
+ (ZF_ss addsimps [inj_is_fun RS apply_rangeI, left_inverse]) 1);
+by (fast_tac (ZF_cs addSIs [Ord_UN] addEs [ltE]) 2);
+by (asm_simp_tac (ZF_ss addsimps [inj_converse_fun RS apply_type]
+ setloop split_tac [expand_if]) 1);
+val le_UN_Ord_lt_csucc = result();
-val ?cardinal_UN_Ord_lt_csucc = result();
-
--- a/src/ZF/InfDatatype.ML Fri Aug 12 12:51:34 1994 +0200
+++ b/src/ZF/InfDatatype.ML Fri Aug 12 18:45:33 1994 +0200
@@ -70,72 +70,89 @@
|> standard;
goal InfDatatype.thy
- "!!K. [| f: I -> Vfrom(A,csucc(K)); |I| le K; InfCard(K) \
-\ |] ==> EX j. f: I -> Vfrom(A,j) & j < csucc(K)";
-by (res_inst_tac [("x", "UN x:I. LEAST i. f`x : Vfrom(A,i)")] exI 1);
+ "!!K. [| f: W -> Vfrom(A,csucc(K)); |W| le K; InfCard(K) \
+\ |] ==> EX j. f: W -> Vfrom(A,j) & j < csucc(K)";
+by (res_inst_tac [("x", "UN w:W. LEAST i. f`w : Vfrom(A,i)")] exI 1);
by (resolve_tac [conjI] 1);
-by (resolve_tac [ballI RSN (2,cardinal_UN_Ord_lt_csucc)] 2);
-by (eresolve_tac [fun_Limit_VfromE] 3 THEN REPEAT_SOME assume_tac);
+by (resolve_tac [le_UN_Ord_lt_csucc] 2);
+by (rtac ballI 4 THEN
+ eresolve_tac [fun_Limit_VfromE] 4 THEN REPEAT_SOME assume_tac);
by (fast_tac (ZF_cs addEs [Least_le RS lt_trans1, ltE]) 2);
by (resolve_tac [Pi_type] 1);
-by (rename_tac "k" 2);
+by (rename_tac "w" 2);
by (eresolve_tac [fun_Limit_VfromE] 2 THEN REPEAT_SOME assume_tac);
-by (subgoal_tac "f`k : Vfrom(A, LEAST i. f`k : Vfrom(A,i))" 1);
+by (subgoal_tac "f`w : Vfrom(A, LEAST i. f`w : Vfrom(A,i))" 1);
by (fast_tac (ZF_cs addEs [LeastI, ltE]) 2);
by (eresolve_tac [[subset_refl, UN_upper] MRS Vfrom_mono RS subsetD] 1);
by (assume_tac 1);
val fun_Vcsucc_lemma = result();
goal InfDatatype.thy
- "!!K. [| f: K -> Vfrom(A,csucc(K)); InfCard(K) \
-\ |] ==> EX j. f: K -> Vfrom(A,j) & j < csucc(K)";
-by (res_inst_tac [("x", "UN k:K. LEAST i. f`k : Vfrom(A,i)")] exI 1);
-by (resolve_tac [conjI] 1);
-by (resolve_tac [ballI RSN (2,cardinal_UN_Ord_lt_csucc)] 2);
-by (eresolve_tac [fun_Limit_VfromE] 3 THEN REPEAT_SOME assume_tac);
-by (fast_tac (ZF_cs addEs [Least_le RS lt_trans1, ltE]) 2);
-by (resolve_tac [Pi_type] 1);
-by (rename_tac "k" 2);
-by (eresolve_tac [fun_Limit_VfromE] 2 THEN REPEAT_SOME assume_tac);
-by (subgoal_tac "f`k : Vfrom(A, LEAST i. f`k : Vfrom(A,i))" 1);
-by (fast_tac (ZF_cs addEs [LeastI, ltE]) 2);
-by (eresolve_tac [[subset_refl, UN_upper] MRS Vfrom_mono RS subsetD] 1);
-by (assume_tac 1);
-val fun_Vcsucc_lemma = result();
+ "!!K. [| W <= Vfrom(A,csucc(K)); |W| le K; InfCard(K) \
+\ |] ==> EX j. W <= Vfrom(A,j) & j < csucc(K)";
+by (asm_full_simp_tac (ZF_ss addsimps [subset_iff_id, fun_Vcsucc_lemma]) 1);
+val subset_Vcsucc = result();
+(*Version for arbitrary index sets*)
goal InfDatatype.thy
- "!!K. InfCard(K) ==> K -> Vfrom(A,csucc(K)) <= Vfrom(A,csucc(K))";
-by (safe_tac (ZF_cs addSDs [fun_Vcsucc_lemma]));
+ "!!K. [| |W| le K; W <= Vfrom(A,csucc(K)); InfCard(K) |] ==> \
+\ W -> Vfrom(A,csucc(K)) <= Vfrom(A,csucc(K))";
+by (safe_tac (ZF_cs addSDs [fun_Vcsucc_lemma, subset_Vcsucc]));
by (resolve_tac [Vfrom RS ssubst] 1);
by (eresolve_tac [PiE] 1);
(*This level includes the function, and is below csucc(K)*)
-by (res_inst_tac [("a1", "succ(succ(K Un j))")] (UN_I RS UnI2) 1);
+by (res_inst_tac [("a1", "succ(succ(j Un ja))")] (UN_I RS UnI2) 1);
by (eresolve_tac [subset_trans RS PowI] 2);
-by (safe_tac (ZF_cs addSIs [Pair_in_Vfrom]));
-by (fast_tac (ZF_cs addIs [i_subset_Vfrom RS subsetD]) 2);
-by (eresolve_tac [[subset_refl, Un_upper2] MRS Vfrom_mono RS subsetD] 2);
+by (fast_tac (ZF_cs addIs [Pair_in_Vfrom, Vfrom_UnI1, Vfrom_UnI2]) 2);
+
by (REPEAT (ares_tac [ltD, InfCard_csucc, InfCard_is_Limit,
Limit_has_succ, Un_least_lt] 1));
-by (eresolve_tac [InfCard_is_Card RS Card_is_Ord RS lt_csucc] 1);
-by (assume_tac 1);
val fun_Vcsucc = result();
goal InfDatatype.thy
+ "!!K. [| f: W -> Vfrom(A, csucc(K)); |W| le K; InfCard(K); \
+\ W <= Vfrom(A,csucc(K)) \
+\ |] ==> f: Vfrom(A,csucc(K))";
+by (REPEAT (ares_tac [fun_Vcsucc RS subsetD] 1));
+val fun_in_Vcsucc = result();
+
+goal InfDatatype.thy
+ "!!K. [| W <= Vfrom(A,csucc(K)); B <= Vfrom(A,csucc(K)); \
+\ |W| le K; InfCard(K) \
+\ |] ==> W -> B <= Vfrom(A, csucc(K))";
+by (REPEAT (ares_tac [[Pi_mono, fun_Vcsucc] MRS subset_trans] 1));
+val fun_subset_Vcsucc = result();
+
+goal InfDatatype.thy
+ "!!f. [| f: W -> B; W <= Vfrom(A,csucc(K)); B <= Vfrom(A,csucc(K)); \
+\ |W| le K; InfCard(K) \
+\ |] ==> f: Vfrom(A,csucc(K))";
+by (DEPTH_SOLVE (ares_tac [fun_subset_Vcsucc RS subsetD] 1));
+val fun_into_Vcsucc = result();
+
+(*Version where K itself is the index set*)
+goal InfDatatype.thy
+ "!!K. InfCard(K) ==> K -> Vfrom(A,csucc(K)) <= Vfrom(A,csucc(K))";
+by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
+by (REPEAT (ares_tac [fun_Vcsucc, Ord_cardinal_le,
+ i_subset_Vfrom,
+ lt_csucc RS leI RS le_imp_subset RS subset_trans] 1));
+val Card_fun_Vcsucc = result();
+
+goal InfDatatype.thy
"!!K. [| f: K -> Vfrom(A, csucc(K)); InfCard(K) \
\ |] ==> f: Vfrom(A,csucc(K))";
-by (REPEAT (ares_tac [fun_Vcsucc RS subsetD] 1));
-val fun_in_Vcsucc = result();
+by (REPEAT (ares_tac [Card_fun_Vcsucc RS subsetD] 1));
+val Card_fun_in_Vcsucc = result();
-val fun_subset_Vcsucc =
- [Pi_mono, fun_Vcsucc] MRS subset_trans |> standard;
+val Card_fun_subset_Vcsucc =
+ [Pi_mono, Card_fun_Vcsucc] MRS subset_trans |> standard;
goal InfDatatype.thy
"!!f. [| f: K -> B; B <= Vfrom(A,csucc(K)); InfCard(K) \
\ |] ==> f: Vfrom(A,csucc(K))";
-by (REPEAT (ares_tac [fun_subset_Vcsucc RS subsetD] 1));
-val fun_into_Vcsucc = result();
-
-val Limit_csucc = InfCard_csucc RS InfCard_is_Limit |> standard;
+by (REPEAT (ares_tac [Card_fun_subset_Vcsucc RS subsetD] 1));
+val Card_fun_into_Vcsucc = result();
val Pair_in_Vcsucc = Limit_csucc RSN (3, Pair_in_VLimit) |> standard;
val Inl_in_Vcsucc = Limit_csucc RSN (2, Inl_in_VLimit) |> standard;
@@ -145,7 +162,7 @@
(*For most K-branching datatypes with domain Vfrom(A, csucc(K)) *)
val inf_datatype_intrs =
- [fun_in_Vcsucc, InfCard_nat, Pair_in_Vcsucc,
+ [Card_fun_in_Vcsucc, fun_in_Vcsucc, InfCard_nat, Pair_in_Vcsucc,
Inl_in_Vcsucc, Inr_in_Vcsucc,
zero_in_Vcsucc, A_into_Vfrom, nat_into_Vcsucc] @ datatype_intrs;
--- a/src/ZF/Perm.ML Fri Aug 12 12:51:34 1994 +0200
+++ b/src/ZF/Perm.ML Fri Aug 12 18:45:33 1994 +0200
@@ -135,6 +135,14 @@
by (fast_tac (ZF_cs addIs [id_inj,id_surj]) 1);
val id_bij = result();
+goalw Perm.thy [id_def] "A <= B <-> id(A) : A->B";
+by (safe_tac ZF_cs);
+by (fast_tac (ZF_cs addSIs [lam_type]) 1);
+by (dtac apply_type 1);
+by (assume_tac 1);
+by (asm_full_simp_tac ZF_ss 1);
+val subset_iff_id = result();
+
(*** Converse of a function ***)
--- a/src/ZF/equalities.ML Fri Aug 12 12:51:34 1994 +0200
+++ b/src/ZF/equalities.ML Fri Aug 12 18:45:33 1994 +0200
@@ -200,6 +200,10 @@
by (fast_tac eq_cs 1);
val Inter_eq_INT = result();
+goal ZF.thy "(UN i:0. A(i)) = 0";
+by (fast_tac eq_cs 1);
+val UN_0 = result();
+
(*Halmos, Naive Set Theory, page 35.*)
goal ZF.thy "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))";
by (fast_tac eq_cs 1);
--- a/src/ZF/func.ML Fri Aug 12 12:51:34 1994 +0200
+++ b/src/ZF/func.ML Fri Aug 12 18:45:33 1994 +0200
@@ -71,6 +71,13 @@
by (REPEAT (assume_tac 1));
val apply_equality = result();
+(*Applying a function outside its domain yields 0*)
+goalw ZF.thy [apply_def]
+ "!!a b f. [| a ~: domain(f); f: Pi(A,B) |] ==> f`a = 0";
+by (rtac the_0 1);
+by (fast_tac ZF_cs 1);
+val apply_0 = result();
+
val prems = goal ZF.thy
"[| f: Pi(A,B); c: f; !!x. [| x:A; c = <x,f`x> |] ==> P \
\ |] ==> P";
@@ -338,9 +345,14 @@
by (fast_tac ZF_cs 1);
val domain_of_fun = result();
+goal ZF.thy "!!f. [| f : Pi(A,B); a: A |] ==> f`a : range(f)";
+by (etac (apply_Pair RS rangeI) 1);
+by (assume_tac 1);
+val apply_rangeI = result();
+
val [major] = goal ZF.thy "f : Pi(A,B) ==> f : A->range(f)";
by (rtac (major RS Pi_type) 1);
-by (etac (major RS apply_Pair RS rangeI) 1);
+by (etac (major RS apply_rangeI) 1);
val range_of_fun = result();
(*** Extensions of functions ***)