src/ZF/Cardinal.ML

 
 
 (** Equipollence is an equivalence relation **)
 
 goalw Cardinal.thy [eqpoll_def] "X eqpoll X";
-br exI 1;
-br id_bij 1;
+by (rtac exI 1);
+by (rtac id_bij 1);
 val eqpoll_refl = result();
 
 goalw Cardinal.thy [eqpoll_def] "!!X Y. X eqpoll Y ==> Y eqpoll X";
 by (fast_tac (ZF_cs addIs [bij_converse_bij]) 1);
 val eqpoll_sym = result();

 val eqpoll_trans = result();
 
 (** Le-pollence is a partial ordering **)
 
 goalw Cardinal.thy [lepoll_def] "!!X Y. X<=Y ==> X lepoll Y";
-br exI 1;
-be id_subset_inj 1;
+by (rtac exI 1);
+by (etac id_subset_inj 1);
 val subset_imp_lepoll = result();
 
 val lepoll_refl = subset_refl RS subset_imp_lepoll;
 
 goalw Cardinal.thy [eqpoll_def, bij_def, lepoll_def]

 by (REPEAT (assume_tac 1));
 val eqpollI = result();
 
 val [major,minor] = goal Cardinal.thy
     "[| X eqpoll Y; [| X lepoll Y; Y lepoll X |] ==> P |] ==> P";
-br minor 1;
+by (rtac minor 1);
 by (REPEAT (resolve_tac [major, eqpoll_imp_lepoll, eqpoll_sym] 1));
 val eqpollE = result();
 
 goal Cardinal.thy "X eqpoll Y <-> X lepoll Y & Y lepoll X";
 by (fast_tac (ZF_cs addIs [eqpollI] addSEs [eqpollE]) 1);

 val [premP,premOrd,premNot] = goalw Cardinal.thy [Least_def]
     "[| P(i);  Ord(i);  !!x. x<i ==> ~P(x) |] ==> (LEAST x.P(x)) = i";
 by (rtac the_equality 1);
 by (fast_tac (ZF_cs addSIs [premP,premOrd,premNot]) 1);
 by (REPEAT (etac conjE 1));
-be (premOrd RS Ord_linear_lt) 1;
+by (etac (premOrd RS Ord_linear_lt) 1);
 by (ALLGOALS (fast_tac (ZF_cs addSIs [premP] addSDs [premNot])));
 val Least_equality = result();
 
 goal Cardinal.thy "!!i. [| P(i);  Ord(i) |] ==> P(LEAST x.P(x))";
 by (etac rev_mp 1);

 by (fast_tac (ZF_cs addEs [ltE, lt_trans1] addIs [leI, ltI]) 1);
 val Least_le = result();
 
 (*LEAST really is the smallest*)
 goal Cardinal.thy "!!i. [| P(i);  i < (LEAST x.P(x)) |] ==> Q";
-br (Least_le RSN (2,lt_trans2) RS lt_anti_refl) 1;
+by (rtac (Least_le RSN (2,lt_trans2) RS lt_irrefl) 1);
 by (REPEAT (eresolve_tac [asm_rl, ltE] 1));
 val less_LeastE = result();
 
+(*If there is no such P then LEAST is vacuously 0*)
+goalw Cardinal.thy [Least_def]
+    "!!P. [| ~ (EX i. Ord(i) & P(i)) |] ==> (LEAST x.P(x)) = 0";
+by (rtac the_0 1);
+by (fast_tac ZF_cs 1);
+val Least_0 = result();
+
 goal Cardinal.thy "Ord(LEAST x.P(x))";
-by (res_inst_tac [("Q","EX i. Ord(i) & P(i)")] (excluded_middle RS disjE) 1);
+by (excluded_middle_tac "EX i. Ord(i) & P(i)" 1);
 by (safe_tac ZF_cs);
-br (Least_le RS ltE) 2;
+by (rtac (Least_le RS ltE) 2);
 by (REPEAT_SOME assume_tac);
-bw Least_def;
-by (rtac (the_0 RS ssubst) 1 THEN rtac Ord_0 2);
-by (fast_tac FOL_cs 1);
+by (etac (Least_0 RS ssubst) 1);
+by (rtac Ord_0 1);
 val Ord_Least = result();
 
 
 (** Basic properties of cardinals **)
 

 by (simp_tac (FOL_ss addsimps prems) 1);
 val Least_cong = result();
 
 (*Need AC to prove   X lepoll Y ==> |X| le |Y| ; see well_ord_lepoll_imp_le  *)
 goalw Cardinal.thy [eqpoll_def,cardinal_def] "!!X Y. X eqpoll Y ==> |X| = |Y|";
-br Least_cong 1;
+by (rtac Least_cong 1);
 by (fast_tac (ZF_cs addEs [comp_bij,bij_converse_bij]) 1);
 val cardinal_cong = result();
 
 (*Under AC, the premise becomes trivial; one consequence is ||A|| = |A|*)
 goalw Cardinal.thy [eqpoll_def, cardinal_def]
     "!!A. well_ord(A,r) ==> |A| eqpoll A";
-br LeastI 1;
-be Ord_ordertype 2;
-br exI 1;
-be (ordertype_bij RS bij_converse_bij) 1;
+by (rtac LeastI 1);
+by (etac Ord_ordertype 2);
+by (rtac exI 1);
+by (etac (ordertype_bij RS bij_converse_bij) 1);
 val well_ord_cardinal_eqpoll = result();
 
 val Ord_cardinal_eqpoll = well_ord_Memrel RS well_ord_cardinal_eqpoll 
                           |> standard;
 
 goal Cardinal.thy
     "!!X Y. [| well_ord(X,r);  well_ord(Y,s);  |X| = |Y| |] ==> X eqpoll Y";
-br (eqpoll_sym RS eqpoll_trans) 1;
-be well_ord_cardinal_eqpoll 1;
+by (rtac (eqpoll_sym RS eqpoll_trans) 1);
+by (etac well_ord_cardinal_eqpoll 1);
 by (asm_simp_tac (ZF_ss addsimps [well_ord_cardinal_eqpoll]) 1);
 val well_ord_cardinal_eqE = result();
 
 
 (** Observations from Kunen, page 28 **)
 
 goalw Cardinal.thy [cardinal_def] "!!i. Ord(i) ==> |i| le i";
-be (eqpoll_refl RS Least_le) 1;
+by (etac (eqpoll_refl RS Least_le) 1);
 val Ord_cardinal_le = result();
 
 goalw Cardinal.thy [Card_def] "!!i. Card(i) ==> |i| = i";
-be sym 1;
+by (etac sym 1);
 val Card_cardinal_eq = result();
 
 val prems = goalw Cardinal.thy [Card_def,cardinal_def]
     "[| Ord(i);  !!j. j<i ==> ~(j eqpoll i) |] ==> Card(i)";
-br (Least_equality RS ssubst) 1;
+by (rtac (Least_equality RS ssubst) 1);
 by (REPEAT (ares_tac ([refl,eqpoll_refl]@prems) 1));
 val CardI = result();
 
 goalw Cardinal.thy [Card_def, cardinal_def] "!!i. Card(i) ==> Ord(i)";
-be ssubst 1;
-br Ord_Least 1;
+by (etac ssubst 1);
+by (rtac Ord_Least 1);
 val Card_is_Ord = result();
 
-goalw Cardinal.thy [cardinal_def] "Ord( |i| )";
-br Ord_Least 1;
+goalw Cardinal.thy [cardinal_def] "Ord( |A| )";
+by (rtac Ord_Least 1);
 val Ord_cardinal = result();
+
+goal Cardinal.thy "Card(0)";
+by (rtac (Ord_0 RS CardI) 1);
+by (fast_tac (ZF_cs addSEs [ltE]) 1);
+val Card_0 = result();
+
+goalw Cardinal.thy [cardinal_def] "Card( |A| )";
+by (excluded_middle_tac "EX i. Ord(i) & i eqpoll A" 1);
+by (etac (Least_0 RS ssubst) 1 THEN rtac Card_0 1);
+by (rtac (Ord_Least RS CardI) 1);
+by (safe_tac ZF_cs);
+by (rtac less_LeastE 1);
+by (assume_tac 2);
+by (etac eqpoll_trans 1);
+by (REPEAT (ares_tac [LeastI] 1));
+val Card_cardinal = result();
 
 (*Kunen's Lemma 10.5*)
 goal Cardinal.thy "!!i j. [| |i| le j;  j le i |] ==> |j| = |i|";
-br (eqpollI RS cardinal_cong) 1;
-be (le_imp_subset RS subset_imp_lepoll) 1;
-br lepoll_trans 1;
-be (le_imp_subset RS subset_imp_lepoll) 2;
-br (eqpoll_sym RS eqpoll_imp_lepoll) 1;
-br Ord_cardinal_eqpoll 1;
+by (rtac (eqpollI RS cardinal_cong) 1);
+by (etac (le_imp_subset RS subset_imp_lepoll) 1);
+by (rtac lepoll_trans 1);
+by (etac (le_imp_subset RS subset_imp_lepoll) 2);
+by (rtac (eqpoll_sym RS eqpoll_imp_lepoll) 1);
+by (rtac Ord_cardinal_eqpoll 1);
 by (REPEAT (eresolve_tac [ltE, Ord_succD] 1));
 val cardinal_eq_lemma = result();
 
 goal Cardinal.thy "!!i j. i le j ==> |i| le |j|";
 by (res_inst_tac [("i","|i|"),("j","|j|")] Ord_linear_le 1);
 by (REPEAT_FIRST (ares_tac [Ord_cardinal, le_eqI]));
-br cardinal_eq_lemma 1;
-ba 2;
-be le_trans 1;
-be ltE 1;
-be Ord_cardinal_le 1;
+by (rtac cardinal_eq_lemma 1);
+by (assume_tac 2);
+by (etac le_trans 1);
+by (etac ltE 1);
+by (etac Ord_cardinal_le 1);
 val cardinal_mono = result();
 
 (*Since we have |succ(nat)| le |nat|, the converse of cardinal_mono fails!*)
 goal Cardinal.thy "!!i j. [| |i| < |j|;  Ord(i);  Ord(j) |] ==> i < j";
-br Ord_linear2 1;
+by (rtac Ord_linear2 1);
 by (REPEAT_SOME assume_tac);
-be (lt_trans2 RS lt_anti_refl) 1;
-be cardinal_mono 1;
+by (etac (lt_trans2 RS lt_irrefl) 1);
+by (etac cardinal_mono 1);
 val cardinal_lt_imp_lt = result();
 
 goal Cardinal.thy "!!i j. [| |i| < k;  Ord(i);  Card(k) |] ==> i < k";
 by (asm_simp_tac (ZF_ss addsimps 
 		  [cardinal_lt_imp_lt, Card_is_Ord, Card_cardinal_eq]) 1);

     "!!A. [| x:A;  y:A;  z:A |] ==> swap(A,x,y)`(swap(A,x,y)`z) = z";
 by (asm_simp_tac (ZF_ss setloop split_tac [expand_if]) 1);
 val swap_swap_identity = result();
 
 goal Cardinal.thy "!!A. [| x:A;  y:A |] ==> swap(A,x,y) : bij(A,A)";
-br nilpotent_imp_bijective 1;
+by (rtac nilpotent_imp_bijective 1);
 by (REPEAT (ares_tac [swap_type, comp_eq_id_iff RS iffD2,
 		      ballI, swap_swap_identity] 1));
 val swap_bij = result();
 
 (*** The finite cardinals ***)
 
 (*Lemma suggested by Mike Fourman*)
 val [prem] = goalw Cardinal.thy [inj_def]
  "f: inj(succ(m), succ(n)) ==> (lam x:m. if(f`x=n, f`m, f`x)) : inj(m,n)";
-br CollectI 1;
+by (rtac CollectI 1);
 (*Proving it's in the function space m->n*)
 by (cut_facts_tac [prem] 1);
-br (if_type RS lam_type) 1;
-by (fast_tac (ZF_cs addSEs [mem_anti_refl] addEs [apply_funtype RS succE]) 1);
-by (fast_tac (ZF_cs addSEs [mem_anti_refl] addEs [apply_funtype RS succE]) 1);
+by (rtac (if_type RS lam_type) 1);
+by (fast_tac (ZF_cs addSEs [mem_irrefl] addEs [apply_funtype RS succE]) 1);
+by (fast_tac (ZF_cs addSEs [mem_irrefl] addEs [apply_funtype RS succE]) 1);
 (*Proving it's injective*)
 by (asm_simp_tac (ZF_ss setloop split_tac [expand_if]) 1);
 (*Adding  prem  earlier would cause the simplifier to loop*)
 by (cut_facts_tac [prem] 1);
-by (fast_tac (ZF_cs addSEs [mem_anti_refl]) 1);
+by (fast_tac (ZF_cs addSEs [mem_irrefl]) 1);
 val inj_succ_succD = result();
 
 val [prem] = goalw Cardinal.thy [lepoll_def]
     "m:nat ==> ALL n: nat. m lepoll n --> m le n";
 by (nat_ind_tac "m" [prem] 1);
 by (fast_tac (ZF_cs addSIs [nat_0_le]) 1);
-br ballI 1;
+by (rtac ballI 1);
 by (eres_inst_tac [("n","n")] natE 1);
 by (asm_simp_tac (ZF_ss addsimps [inj_def, succI1 RS Pi_empty2]) 1);
 by (fast_tac (ZF_cs addSIs [succ_leI] addSDs [inj_succ_succD]) 1);
 val nat_lepoll_imp_le_lemma = result();
 val nat_lepoll_imp_le = nat_lepoll_imp_le_lemma RS bspec RS mp |> standard;
 
 goal Cardinal.thy
     "!!m n. [| m:nat; n: nat |] ==> m eqpoll n <-> m = n";
-br iffI 1;
+by (rtac iffI 1);
 by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2);
-by (fast_tac (ZF_cs addIs [nat_lepoll_imp_le, le_asym] addSEs [eqpollE]) 1);
+by (fast_tac (ZF_cs addIs [nat_lepoll_imp_le, le_anti_sym] 
+                    addSEs [eqpollE]) 1);
 val nat_eqpoll_iff = result();
 
 goalw Cardinal.thy [Card_def,cardinal_def]
     "!!n. n: nat ==> Card(n)";
-br (Least_equality RS ssubst) 1;
+by (rtac (Least_equality RS ssubst) 1);
 by (REPEAT_FIRST (ares_tac [eqpoll_refl, nat_into_Ord, refl]));
 by (asm_simp_tac (ZF_ss addsimps [lt_nat_in_nat RS nat_eqpoll_iff]) 1);
-by (fast_tac (ZF_cs addSEs [lt_anti_refl]) 1);
+by (fast_tac (ZF_cs addSEs [lt_irrefl]) 1);
 val nat_into_Card = result();
-
-val Card_0 = nat_0I RS nat_into_Card;
 
 (*Part of Kunen's Lemma 10.6*)
 goal Cardinal.thy "!!n. [| succ(n) lepoll n;  n:nat |] ==> P";
-br (nat_lepoll_imp_le RS lt_anti_refl) 1;
+by (rtac (nat_lepoll_imp_le RS lt_irrefl) 1);
 by (REPEAT (ares_tac [nat_succI] 1));
 val succ_lepoll_natE = result();
 
 
 (*** The first infinite cardinal: Omega, or nat ***)
 
 (*This implies Kunen's Lemma 10.6*)
 goal Cardinal.thy "!!n. [| n<i;  n:nat |] ==> ~ i lepoll n";
-br notI 1;
+by (rtac notI 1);
 by (rtac succ_lepoll_natE 1 THEN assume_tac 2);
 by (rtac lepoll_trans 1 THEN assume_tac 2);
-be ltE 1;
+by (etac ltE 1);
 by (REPEAT (ares_tac [Ord_succ_subsetI RS subset_imp_lepoll] 1));
 val lt_not_lepoll = result();
 
-goalw Cardinal.thy [Card_def,cardinal_def] "Card(nat)";
-br (Least_equality RS ssubst) 1;
-by (REPEAT_FIRST (ares_tac [eqpoll_refl, Ord_nat, refl]));
-be ltE 1;
-by (asm_simp_tac (ZF_ss addsimps [eqpoll_iff, lt_not_lepoll, ltI]) 1);
-val Card_nat = result();
-
 goal Cardinal.thy "!!i n. [| Ord(i);  n:nat |] ==> i eqpoll n <-> i=n";
-br iffI 1;
+by (rtac iffI 1);
 by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2);
 by (rtac Ord_linear_lt 1);
 by (REPEAT_SOME (eresolve_tac [asm_rl, nat_into_Ord]));
 by (etac (lt_nat_in_nat RS nat_eqpoll_iff RS iffD1) 1 THEN
     REPEAT (assume_tac 1));
 by (rtac (lt_not_lepoll RS notE) 1 THEN (REPEAT (assume_tac 1)));
-be eqpoll_imp_lepoll 1;
+by (etac eqpoll_imp_lepoll 1);
 val Ord_nat_eqpoll_iff = result();
 
-
+goalw Cardinal.thy [Card_def,cardinal_def] "Card(nat)";
+by (rtac (Least_equality RS ssubst) 1);
+by (REPEAT_FIRST (ares_tac [eqpoll_refl, Ord_nat, refl]));
+by (etac ltE 1);
+by (asm_simp_tac (ZF_ss addsimps [eqpoll_iff, lt_not_lepoll, ltI]) 1);
+val Card_nat = result();
+
+(*Allows showing that |i| is a limit cardinal*)
+goal Cardinal.thy  "!!i. nat le i ==> nat le |i|";
+by (rtac (Card_nat RS Card_cardinal_eq RS subst) 1);
+by (etac cardinal_mono 1);
+val nat_le_cardinal = result();
+