ZF/Cardinal: some results moved here from CardinalArith
ZF/CardinalArith/nat_succ_lepoll: removed obsolete proof
ZF/CardinalArith/nat_cons_eqpoll: new
(* Title: ZF/Cardinal.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Cardinals in Zermelo-Fraenkel Set Theory
This theory does NOT assume the Axiom of Choice
*)
open Cardinal;
(*** The Schroeder-Bernstein Theorem -- see Davey & Priestly, page 106 ***)
(** Lemma: Banach's Decomposition Theorem **)
goal Cardinal.thy "bnd_mono(X, %W. X - g``(Y - f``W))";
by (rtac bnd_monoI 1);
by (REPEAT (ares_tac [Diff_subset, subset_refl, Diff_mono, image_mono] 1));
val decomp_bnd_mono = result();
val [gfun] = goal Cardinal.thy
"g: Y->X ==> \
\ g``(Y - f`` lfp(X, %W. X - g``(Y - f``W))) = \
\ X - lfp(X, %W. X - g``(Y - f``W)) ";
by (res_inst_tac [("P", "%u. ?v = X-u")]
(decomp_bnd_mono RS lfp_Tarski RS ssubst) 1);
by (simp_tac (ZF_ss addsimps [subset_refl, double_complement,
gfun RS fun_is_rel RS image_subset]) 1);
val Banach_last_equation = result();
val prems = goal Cardinal.thy
"[| f: X->Y; g: Y->X |] ==> \
\ EX XA XB YA YB. (XA Int XB = 0) & (XA Un XB = X) & \
\ (YA Int YB = 0) & (YA Un YB = Y) & \
\ f``XA=YA & g``YB=XB";
by (REPEAT
(FIRSTGOAL
(resolve_tac [refl, exI, conjI, Diff_disjoint, Diff_partition])));
by (rtac Banach_last_equation 3);
by (REPEAT (resolve_tac (prems@[fun_is_rel, image_subset, lfp_subset]) 1));
val decomposition = result();
val prems = goal Cardinal.thy
"[| f: inj(X,Y); g: inj(Y,X) |] ==> EX h. h: bij(X,Y)";
by (cut_facts_tac prems 1);
by (cut_facts_tac [(prems RL [inj_is_fun]) MRS decomposition] 1);
by (fast_tac (ZF_cs addSIs [restrict_bij,bij_disjoint_Un]
addIs [bij_converse_bij]) 1);
(* The instantiation of exI to "restrict(f,XA) Un converse(restrict(g,YB))"
is forced by the context!! *)
val schroeder_bernstein = result();
(** Equipollence is an equivalence relation **)
goalw Cardinal.thy [eqpoll_def] "X eqpoll X";
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();
goalw Cardinal.thy [eqpoll_def]
"!!X Y. [| X eqpoll Y; Y eqpoll Z |] ==> X eqpoll Z";
by (fast_tac (ZF_cs addIs [comp_bij]) 1);
val eqpoll_trans = result();
(** Le-pollence is a partial ordering **)
goalw Cardinal.thy [lepoll_def] "!!X Y. X<=Y ==> X lepoll Y";
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]
"!!X Y. X eqpoll Y ==> X lepoll Y";
by (fast_tac ZF_cs 1);
val eqpoll_imp_lepoll = result();
goalw Cardinal.thy [lepoll_def]
"!!X Y. [| X lepoll Y; Y lepoll Z |] ==> X lepoll Z";
by (fast_tac (ZF_cs addIs [comp_inj]) 1);
val lepoll_trans = result();
(*Asymmetry law*)
goalw Cardinal.thy [lepoll_def,eqpoll_def]
"!!X Y. [| X lepoll Y; Y lepoll X |] ==> X eqpoll Y";
by (REPEAT (etac exE 1));
by (rtac schroeder_bernstein 1);
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";
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 eqpoll_iff = result();
(** LEAST -- the least number operator [from HOL/Univ.ML] **)
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));
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 (trans_ind_tac "i" [] 1);
by (rtac impI 1);
by (rtac classical 1);
by (EVERY1 [rtac (Least_equality RS ssubst), assume_tac, assume_tac]);
by (assume_tac 2);
by (fast_tac (ZF_cs addSEs [ltE]) 1);
val LeastI = result();
(*Proof is almost identical to the one above!*)
goal Cardinal.thy "!!i. [| P(i); Ord(i) |] ==> (LEAST x.P(x)) le i";
by (etac rev_mp 1);
by (trans_ind_tac "i" [] 1);
by (rtac impI 1);
by (rtac classical 1);
by (EVERY1 [rtac (Least_equality RS ssubst), assume_tac, assume_tac]);
by (etac le_refl 2);
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";
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 (excluded_middle_tac "EX i. Ord(i) & P(i)" 1);
by (safe_tac ZF_cs);
by (rtac (Least_le RS ltE) 2);
by (REPEAT_SOME assume_tac);
by (etac (Least_0 RS ssubst) 1);
by (rtac Ord_0 1);
val Ord_Least = result();
(** Basic properties of cardinals **)
(*Not needed for simplification, but helpful below*)
val prems = goal Cardinal.thy
"[| !!y. P(y) <-> Q(y) |] ==> (LEAST x.P(x)) = (LEAST x.Q(x))";
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|";
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";
by (rtac LeastI 1);
by (etac Ord_ordertype 2);
by (rtac exI 1);
by (etac (ordermap_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";
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";
by (etac (eqpoll_refl RS Least_le) 1);
val Ord_cardinal_le = result();
goalw Cardinal.thy [Card_def] "!!K. Card(K) ==> |K| = K";
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)";
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)";
by (etac ssubst 1);
by (rtac Ord_Least 1);
val Card_is_Ord = result();
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();
val [premK,premL] = goal Cardinal.thy
"[| Card(K); Card(L) |] ==> Card(K Un L)";
by (rtac ([premK RS Card_is_Ord, premL RS Card_is_Ord] MRS Ord_linear_le) 1);
by (asm_simp_tac
(ZF_ss addsimps [premL, le_imp_subset, subset_Un_iff RS iffD1]) 1);
by (asm_simp_tac
(ZF_ss addsimps [premK, le_imp_subset, subset_Un_iff2 RS iffD1]) 1);
val Card_Un = result();
(*Infinite unions of cardinals? See Devlin, Lemma 6.7, page 98*)
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|";
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]));
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";
by (rtac Ord_linear2 1);
by (REPEAT_SOME assume_tac);
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);
val Card_lt_imp_lt = result();
goal Cardinal.thy "!!i j. [| Ord(i); Card(K) |] ==> (|i| < K) <-> (i < K)";
by (fast_tac (ZF_cs addEs [Card_lt_imp_lt, Ord_cardinal_le RS lt_trans1]) 1);
val Card_lt_iff = result();
goal Cardinal.thy "!!i j. [| Ord(i); Card(K) |] ==> (K le |i|) <-> (K le i)";
by (asm_simp_tac (ZF_ss addsimps
[Card_lt_iff, Card_is_Ord, Ord_cardinal,
not_lt_iff_le RS iff_sym]) 1);
val Card_le_iff = result();
(** The swap operator [NOT USED] **)
goalw Cardinal.thy [swap_def]
"!!A. [| x:A; y:A |] ==> swap(A,x,y) : A->A";
by (REPEAT (ares_tac [lam_type,if_type] 1));
val swap_type = result();
goalw Cardinal.thy [swap_def]
"!!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)";
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)";
by (rtac CollectI 1);
(*Proving it's in the function space m->n*)
by (cut_facts_tac [prem] 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_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);
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";
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_anti_sym]
addSEs [eqpollE]) 1);
val nat_eqpoll_iff = result();
goalw Cardinal.thy [Card_def,cardinal_def]
"!!n. n: nat ==> Card(n)";
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_irrefl]) 1);
val nat_into_Card = result();
(*Part of Kunen's Lemma 10.6*)
goal Cardinal.thy "!!n. [| succ(n) lepoll n; n:nat |] ==> P";
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";
by (rtac notI 1);
by (rtac succ_lepoll_natE 1 THEN assume_tac 2);
by (rtac lepoll_trans 1 THEN assume_tac 2);
by (etac ltE 1);
by (REPEAT (ares_tac [Ord_succ_subsetI RS subset_imp_lepoll] 1));
val lt_not_lepoll = result();
goal Cardinal.thy "!!i n. [| Ord(i); n:nat |] ==> i eqpoll n <-> i=n";
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)));
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();
(*** Towards Cardinal Arithmetic ***)
(** Congruence laws for successor, cardinal addition and multiplication **)
val case_ss = ZF_ss addsimps [Inl_iff, Inl_Inr_iff, Inr_iff, Inr_Inl_iff,
case_Inl, case_Inr, InlI, InrI];
val bij_inverse_ss =
case_ss addsimps [bij_is_fun RS apply_type,
bij_converse_bij RS bij_is_fun RS apply_type,
left_inverse_bij, right_inverse_bij];
(*Congruence law for cons under equipollence*)
goalw Cardinal.thy [lepoll_def]
"!!A B. [| A lepoll B; b ~: B |] ==> cons(a,A) lepoll cons(b,B)";
by (safe_tac ZF_cs);
by (res_inst_tac [("x", "lam y: cons(a,A).if(y=a, b, f`y)")] exI 1);
by (res_inst_tac [("d","%z.if(z:B, converse(f)`z, a)")]
lam_injective 1);
by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_type, cons_iff]
setloop etac consE') 1);
by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_type, left_inverse]
setloop etac consE') 1);
val cons_lepoll_cong = result();
goal Cardinal.thy
"!!A B. [| A eqpoll B; a ~: A; b ~: B |] ==> cons(a,A) eqpoll cons(b,B)";
by (asm_full_simp_tac (ZF_ss addsimps [eqpoll_iff, cons_lepoll_cong]) 1);
val cons_eqpoll_cong = result();
(*Congruence law for succ under equipollence*)
goalw Cardinal.thy [succ_def]
"!!A B. A eqpoll B ==> succ(A) eqpoll succ(B)";
by (REPEAT (ares_tac [cons_eqpoll_cong, mem_not_refl] 1));
val succ_eqpoll_cong = result();
(*Congruence law for + under equipollence*)
goalw Cardinal.thy [eqpoll_def]
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A+B eqpoll C+D";
by (safe_tac ZF_cs);
by (rtac exI 1);
by (res_inst_tac [("c", "case(%x. Inl(f`x), %y. Inr(fa`y))"),
("d", "case(%x. Inl(converse(f)`x), %y. Inr(converse(fa)`y))")]
lam_bijective 1);
by (safe_tac (ZF_cs addSEs [sumE]));
by (ALLGOALS (asm_simp_tac bij_inverse_ss));
val sum_eqpoll_cong = result();
(*Congruence law for * under equipollence*)
goalw Cardinal.thy [eqpoll_def]
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A*B eqpoll C*D";
by (safe_tac ZF_cs);
by (rtac exI 1);
by (res_inst_tac [("c", "split(%x y. <f`x, fa`y>)"),
("d", "split(%x y. <converse(f)`x, converse(fa)`y>)")]
lam_bijective 1);
by (safe_tac ZF_cs);
by (ALLGOALS (asm_simp_tac bij_inverse_ss));
val prod_eqpoll_cong = result();
goalw Cardinal.thy [eqpoll_def]
"!!f. [| f: inj(A,B); A Int B = 0 |] ==> A Un (B - range(f)) eqpoll B";
by (rtac exI 1);
by (res_inst_tac [("c", "%x. if(x:A, f`x, x)"),
("d", "%y. if(y: range(f), converse(f)`y, y)")]
lam_bijective 1);
by (fast_tac (ZF_cs addSIs [if_type, apply_type] addIs [inj_is_fun]) 1);
by (asm_simp_tac
(ZF_ss addsimps [inj_converse_fun RS apply_funtype]
setloop split_tac [expand_if]) 1);
by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_rangeI, left_inverse]
setloop etac UnE') 1);
by (asm_simp_tac
(ZF_ss addsimps [inj_converse_fun RS apply_funtype, right_inverse]
setloop split_tac [expand_if]) 1);
by (fast_tac (ZF_cs addEs [equals0D]) 1);
val inj_disjoint_eqpoll = result();