--- a/src/ZF/CardinalArith.ML Sat Jun 15 22:57:33 2002 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,909 +0,0 @@
-(* Title: ZF/CardinalArith.ML
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1994 University of Cambridge
-
-Cardinal arithmetic -- WITHOUT the Axiom of Choice
-
-Note: Could omit proving the algebraic laws for cardinal addition and
-multiplication. On finite cardinals these operations coincide with
-addition and multiplication of natural numbers; on infinite cardinals they
-coincide with union (maximum). Either way we get most laws for free.
-*)
-
-val InfCard_def = thm "InfCard_def";
-val cmult_def = thm "cmult_def";
-val cadd_def = thm "cadd_def";
-val csquare_rel_def = thm "csquare_rel_def";
-val jump_cardinal_def = thm "jump_cardinal_def";
-val csucc_def = thm "csucc_def";
-
-
-(*** Cardinal addition ***)
-
-(** Cardinal addition is commutative **)
-
-Goalw [eqpoll_def] "A+B eqpoll B+A";
-by (rtac exI 1);
-by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")]
- lam_bijective 1);
-by Auto_tac;
-qed "sum_commute_eqpoll";
-
-Goalw [cadd_def] "i |+| j = j |+| i";
-by (rtac (sum_commute_eqpoll RS cardinal_cong) 1);
-qed "cadd_commute";
-
-(** Cardinal addition is associative **)
-
-Goalw [eqpoll_def] "(A+B)+C eqpoll A+(B+C)";
-by (rtac exI 1);
-by (rtac sum_assoc_bij 1);
-qed "sum_assoc_eqpoll";
-
-(*Unconditional version requires AC*)
-Goalw [cadd_def]
- "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \
-\ (i |+| j) |+| k = i |+| (j |+| k)";
-by (rtac cardinal_cong 1);
-by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS
- eqpoll_trans) 1);
-by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2);
-by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS
- eqpoll_sym) 2);
-by (REPEAT (ares_tac [well_ord_radd] 1));
-qed "well_ord_cadd_assoc";
-
-(** 0 is the identity for addition **)
-
-Goalw [eqpoll_def] "0+A eqpoll A";
-by (rtac exI 1);
-by (rtac bij_0_sum 1);
-qed "sum_0_eqpoll";
-
-Goalw [cadd_def] "Card(K) ==> 0 |+| K = K";
-by (asm_simp_tac (simpset() addsimps [sum_0_eqpoll RS cardinal_cong,
- Card_cardinal_eq]) 1);
-qed "cadd_0";
-Addsimps [cadd_0];
-
-(** Addition by another cardinal **)
-
-Goalw [lepoll_def, inj_def] "A lepoll A+B";
-by (res_inst_tac [("x", "lam x:A. Inl(x)")] exI 1);
-by (Asm_simp_tac 1);
-qed "sum_lepoll_self";
-
-(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
-Goalw [cadd_def]
- "[| Card(K); Ord(L) |] ==> K le (K |+| L)";
-by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1);
-by (rtac sum_lepoll_self 3);
-by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Card_is_Ord] 1));
-qed "cadd_le_self";
-
-(** Monotonicity of addition **)
-
-Goalw [lepoll_def]
- "[| A lepoll C; B lepoll D |] ==> A + B lepoll C + D";
-by (REPEAT (etac exE 1));
-by (res_inst_tac [("x", "lam z:A+B. case(%w. Inl(f`w), %y. Inr(fa`y), z)")]
- exI 1);
-by (res_inst_tac
- [("d", "case(%w. Inl(converse(f)`w), %y. Inr(converse(fa)`y))")]
- lam_injective 1);
-by (typecheck_tac (tcset() addTCs [inj_is_fun]));
-by Auto_tac;
-qed "sum_lepoll_mono";
-
-Goalw [cadd_def]
- "[| K' le K; L' le L |] ==> (K' |+| L') le (K |+| L)";
-by (safe_tac (claset() addSDs [le_subset_iff RS iffD1]));
-by (rtac well_ord_lepoll_imp_Card_le 1);
-by (REPEAT (ares_tac [sum_lepoll_mono, subset_imp_lepoll] 2));
-by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
-qed "cadd_le_mono";
-
-(** Addition of finite cardinals is "ordinary" addition **)
-
-Goalw [eqpoll_def] "succ(A)+B eqpoll succ(A+B)";
-by (rtac exI 1);
-by (res_inst_tac [("c", "%z. if z=Inl(A) then A+B else z"),
- ("d", "%z. if z=A+B then Inl(A) else z")]
- lam_bijective 1);
-by (ALLGOALS
- (asm_simp_tac (simpset() addsimps [succI2, mem_imp_not_eq]
- setloop eresolve_tac [sumE,succE])));
-qed "sum_succ_eqpoll";
-
-(*Pulling the succ(...) outside the |...| requires m, n: nat *)
-(*Unconditional version requires AC*)
-Goalw [cadd_def]
- "[| Ord(m); Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|";
-by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1);
-by (rtac (succ_eqpoll_cong RS cardinal_cong) 1);
-by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1);
-by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
-qed "cadd_succ_lemma";
-
-Goal "[| m: nat; n: nat |] ==> m |+| n = m#+n";
-by (induct_tac "m" 1);
-by (asm_simp_tac (simpset() addsimps [nat_into_Card RS cadd_0]) 1);
-by (asm_simp_tac (simpset() addsimps [cadd_succ_lemma,
- nat_into_Card RS Card_cardinal_eq]) 1);
-qed "nat_cadd_eq_add";
-
-
-(*** Cardinal multiplication ***)
-
-(** Cardinal multiplication is commutative **)
-
-(*Easier to prove the two directions separately*)
-Goalw [eqpoll_def] "A*B eqpoll B*A";
-by (rtac exI 1);
-by (res_inst_tac [("c", "%<x,y>.<y,x>"), ("d", "%<x,y>.<y,x>")]
- lam_bijective 1);
-by Safe_tac;
-by (ALLGOALS (Asm_simp_tac));
-qed "prod_commute_eqpoll";
-
-Goalw [cmult_def] "i |*| j = j |*| i";
-by (rtac (prod_commute_eqpoll RS cardinal_cong) 1);
-qed "cmult_commute";
-
-(** Cardinal multiplication is associative **)
-
-Goalw [eqpoll_def] "(A*B)*C eqpoll A*(B*C)";
-by (rtac exI 1);
-by (rtac prod_assoc_bij 1);
-qed "prod_assoc_eqpoll";
-
-(*Unconditional version requires AC*)
-Goalw [cmult_def]
- "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \
-\ (i |*| j) |*| k = i |*| (j |*| k)";
-by (rtac cardinal_cong 1);
-by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
- eqpoll_trans) 1);
-by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2);
-by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS
- eqpoll_sym) 2);
-by (REPEAT (ares_tac [well_ord_rmult] 1));
-qed "well_ord_cmult_assoc";
-
-(** Cardinal multiplication distributes over addition **)
-
-Goalw [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)";
-by (rtac exI 1);
-by (rtac sum_prod_distrib_bij 1);
-qed "sum_prod_distrib_eqpoll";
-
-Goalw [cadd_def, cmult_def]
- "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> \
-\ (i |+| j) |*| k = (i |*| k) |+| (j |*| k)";
-by (rtac cardinal_cong 1);
-by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
- eqpoll_trans) 1);
-by (rtac (sum_prod_distrib_eqpoll RS eqpoll_trans) 2);
-by (rtac ([well_ord_cardinal_eqpoll, well_ord_cardinal_eqpoll] MRS
- sum_eqpoll_cong RS eqpoll_sym) 2);
-by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd] 1));
-qed "well_ord_cadd_cmult_distrib";
-
-(** Multiplication by 0 yields 0 **)
-
-Goalw [eqpoll_def] "0*A eqpoll 0";
-by (rtac exI 1);
-by (rtac lam_bijective 1);
-by Safe_tac;
-qed "prod_0_eqpoll";
-
-Goalw [cmult_def] "0 |*| i = 0";
-by (asm_simp_tac (simpset() addsimps [prod_0_eqpoll RS cardinal_cong,
- Card_0 RS Card_cardinal_eq]) 1);
-qed "cmult_0";
-Addsimps [cmult_0];
-
-(** 1 is the identity for multiplication **)
-
-Goalw [eqpoll_def] "{x}*A eqpoll A";
-by (rtac exI 1);
-by (resolve_tac [singleton_prod_bij RS bij_converse_bij] 1);
-qed "prod_singleton_eqpoll";
-
-Goalw [cmult_def, succ_def] "Card(K) ==> 1 |*| K = K";
-by (asm_simp_tac (simpset() addsimps [prod_singleton_eqpoll RS cardinal_cong,
- Card_cardinal_eq]) 1);
-qed "cmult_1";
-Addsimps [cmult_1];
-
-(*** Some inequalities for multiplication ***)
-
-Goalw [lepoll_def, inj_def] "A lepoll A*A";
-by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1);
-by (Simp_tac 1);
-qed "prod_square_lepoll";
-
-(*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
-Goalw [cmult_def] "Card(K) ==> K le K |*| K";
-by (rtac le_trans 1);
-by (rtac well_ord_lepoll_imp_Card_le 2);
-by (rtac prod_square_lepoll 3);
-by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2));
-by (asm_simp_tac (simpset()
- addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
-qed "cmult_square_le";
-
-(** Multiplication by a non-zero cardinal **)
-
-Goalw [lepoll_def, inj_def] "b: B ==> A lepoll A*B";
-by (res_inst_tac [("x", "lam x:A. <x,b>")] exI 1);
-by (Asm_simp_tac 1);
-qed "prod_lepoll_self";
-
-(*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
-Goalw [cmult_def]
- "[| Card(K); Ord(L); 0<L |] ==> K le (K |*| L)";
-by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1);
-by (rtac prod_lepoll_self 3);
-by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord, ltD] 1));
-qed "cmult_le_self";
-
-(** Monotonicity of multiplication **)
-
-Goalw [lepoll_def]
- "[| A lepoll C; B lepoll D |] ==> A * B lepoll C * D";
-by (REPEAT (etac exE 1));
-by (res_inst_tac [("x", "lam <w,y>:A*B. <f`w, fa`y>")] exI 1);
-by (res_inst_tac [("d", "%<w,y>.<converse(f)`w, converse(fa)`y>")]
- lam_injective 1);
-by (typecheck_tac (tcset() addTCs [inj_is_fun]));
-by Auto_tac;
-qed "prod_lepoll_mono";
-
-Goalw [cmult_def]
- "[| K' le K; L' le L |] ==> (K' |*| L') le (K |*| L)";
-by (safe_tac (claset() addSDs [le_subset_iff RS iffD1]));
-by (rtac well_ord_lepoll_imp_Card_le 1);
-by (REPEAT (ares_tac [prod_lepoll_mono, subset_imp_lepoll] 2));
-by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
-qed "cmult_le_mono";
-
-(*** Multiplication of finite cardinals is "ordinary" multiplication ***)
-
-Goalw [eqpoll_def] "succ(A)*B eqpoll B + A*B";
-by (rtac exI 1);
-by (res_inst_tac [("c", "%<x,y>. if x=A then Inl(y) else Inr(<x,y>)"),
- ("d", "case(%y. <A,y>, %z. z)")]
- lam_bijective 1);
-by Safe_tac;
-by (ALLGOALS
- (asm_simp_tac (simpset() addsimps [succI2, if_type, mem_imp_not_eq])));
-qed "prod_succ_eqpoll";
-
-(*Unconditional version requires AC*)
-Goalw [cmult_def, cadd_def]
- "[| Ord(m); Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)";
-by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1);
-by (rtac (cardinal_cong RS sym) 1);
-by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1);
-by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
-qed "cmult_succ_lemma";
-
-Goal "[| m: nat; n: nat |] ==> m |*| n = m#*n";
-by (induct_tac "m" 1);
-by (Asm_simp_tac 1);
-by (asm_simp_tac (simpset() addsimps [cmult_succ_lemma, nat_cadd_eq_add]) 1);
-qed "nat_cmult_eq_mult";
-
-Goal "Card(n) ==> 2 |*| n = n |+| n";
-by (asm_simp_tac
- (simpset() addsimps [cmult_succ_lemma, Card_is_Ord,
- inst "j" "0" cadd_commute]) 1);
-qed "cmult_2";
-
-
-bind_thm ("sum_lepoll_prod", [sum_eq_2_times RS equalityD1 RS subset_imp_lepoll,
- asm_rl, lepoll_refl] MRS (prod_lepoll_mono RSN (2, lepoll_trans))
- |> standard);
-
-Goal "[| A lepoll B; 2 lepoll A |] ==> A+B lepoll A*B";
-by (REPEAT (ares_tac [[sum_lepoll_mono, sum_lepoll_prod]
- MRS lepoll_trans, lepoll_refl] 1));
-qed "lepoll_imp_sum_lepoll_prod";
-
-
-(*** Infinite Cardinals are Limit Ordinals ***)
-
-(*This proof is modelled upon one assuming nat<=A, with injection
- lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z
- and inverse %y. if y:nat then nat_case(u, %z. z, y) else y. \
- If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
-Goalw [lepoll_def] "nat lepoll A ==> cons(u,A) lepoll A";
-by (etac exE 1);
-by (res_inst_tac [("x",
- "lam z:cons(u,A). if z=u then f`0 \
-\ else if z: range(f) then f`succ(converse(f)`z) \
-\ else z")] exI 1);
-by (res_inst_tac [("d", "%y. if y: range(f) \
-\ then nat_case(u, %z. f`z, converse(f)`y) \
-\ else y")]
- lam_injective 1);
-by (fast_tac (claset() addSIs [if_type, apply_type]
- addIs [inj_is_fun, inj_converse_fun]) 1);
-by (asm_simp_tac
- (simpset() addsimps [inj_is_fun RS apply_rangeI,
- inj_converse_fun RS apply_rangeI,
- inj_converse_fun RS apply_funtype]) 1);
-qed "nat_cons_lepoll";
-
-Goal "nat lepoll A ==> cons(u,A) eqpoll A";
-by (etac (nat_cons_lepoll RS eqpollI) 1);
-by (rtac (subset_consI RS subset_imp_lepoll) 1);
-qed "nat_cons_eqpoll";
-
-(*Specialized version required below*)
-Goalw [succ_def] "nat <= A ==> succ(A) eqpoll A";
-by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1);
-qed "nat_succ_eqpoll";
-
-Goalw [InfCard_def] "InfCard(nat)";
-by (blast_tac (claset() addIs [Card_nat, le_refl, Card_is_Ord]) 1);
-qed "InfCard_nat";
-
-Goalw [InfCard_def] "InfCard(K) ==> Card(K)";
-by (etac conjunct1 1);
-qed "InfCard_is_Card";
-
-Goalw [InfCard_def]
- "[| InfCard(K); Card(L) |] ==> InfCard(K Un L)";
-by (asm_simp_tac (simpset() addsimps [Card_Un, Un_upper1_le RSN (2,le_trans),
- Card_is_Ord]) 1);
-qed "InfCard_Un";
-
-(*Kunen's Lemma 10.11*)
-Goalw [InfCard_def] "InfCard(K) ==> Limit(K)";
-by (etac conjE 1);
-by (ftac Card_is_Ord 1);
-by (rtac (ltI RS non_succ_LimitI) 1);
-by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1);
-by (safe_tac (claset() addSDs [Limit_nat RS Limit_le_succD]));
-by (rewtac Card_def);
-by (dtac trans 1);
-by (etac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1);
-by (etac (Ord_cardinal_le RS lt_trans2 RS lt_irrefl) 1);
-by (REPEAT (ares_tac [le_eqI, Ord_cardinal] 1));
-qed "InfCard_is_Limit";
-
-
-(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
-
-(*A general fact about ordermap*)
-Goalw [eqpoll_def]
- "[| well_ord(A,r); x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)";
-by (rtac exI 1);
-by (asm_simp_tac (simpset() addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
-by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1);
-by (rtac pred_subset 1);
-qed "ordermap_eqpoll_pred";
-
-(** Establishing the well-ordering **)
-
-Goalw [inj_def] "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)";
-by (force_tac (claset() addIs [lam_type, Un_least_lt RS ltD, ltI],
- simpset()) 1);
-qed "csquare_lam_inj";
-
-Goalw [csquare_rel_def] "Ord(K) ==> well_ord(K*K, csquare_rel(K))";
-by (rtac (csquare_lam_inj RS well_ord_rvimage) 1);
-by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
-qed "well_ord_csquare";
-
-(** Characterising initial segments of the well-ordering **)
-
-Goalw [csquare_rel_def]
- "[| <<x,y>, <z,z>> : csquare_rel(K); x<K; y<K; z<K |] ==> x le z & y le z";
-by (etac rev_mp 1);
-by (REPEAT (etac ltE 1));
-by (asm_simp_tac (simpset() addsimps [rvimage_iff,
- Un_absorb, Un_least_mem_iff, ltD]) 1);
-by (safe_tac (claset() addSEs [mem_irrefl]
- addSIs [Un_upper1_le, Un_upper2_le]));
-by (ALLGOALS (asm_simp_tac (simpset() addsimps [lt_def, succI2])));
-qed "csquareD";
-
-Goalw [thm "Order.pred_def"]
- "z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)";
-by (safe_tac (claset() delrules [SigmaI, succCI]));
-by (etac (csquareD RS conjE) 1);
-by (rewtac lt_def);
-by (ALLGOALS Blast_tac);
-qed "pred_csquare_subset";
-
-Goalw [csquare_rel_def]
- "[| x<z; y<z; z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K)";
-by (subgoals_tac ["x<K", "y<K"] 1);
-by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2));
-by (REPEAT (etac ltE 1));
-by (asm_simp_tac (simpset() addsimps [rvimage_iff,
- Un_absorb, Un_least_mem_iff, ltD]) 1);
-qed "csquare_ltI";
-
-(*Part of the traditional proof. UNUSED since it's harder to prove & apply *)
-Goalw [csquare_rel_def]
- "[| x le z; y le z; z<K |] ==> \
-\ <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z";
-by (subgoals_tac ["x<K", "y<K"] 1);
-by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2));
-by (REPEAT (etac ltE 1));
-by (asm_simp_tac (simpset() addsimps [rvimage_iff,
- Un_absorb, Un_least_mem_iff, ltD]) 1);
-by (REPEAT_FIRST (etac succE));
-by (ALLGOALS
- (asm_simp_tac (simpset() addsimps [subset_Un_iff RS iff_sym,
- subset_Un_iff2 RS iff_sym, OrdmemD])));
-qed "csquare_or_eqI";
-
-(** The cardinality of initial segments **)
-
-Goal "[| Limit(K); x<K; y<K; z=succ(x Un y) |] ==> \
-\ ordermap(K*K, csquare_rel(K)) ` <x,y> < \
-\ ordermap(K*K, csquare_rel(K)) ` <z,z>";
-by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1);
-by (etac (Limit_is_Ord RS well_ord_csquare) 2);
-by (blast_tac (claset() addSIs [Un_least_lt, Limit_has_succ]) 2);
-by (rtac (csquare_ltI RS ordermap_mono RS ltI) 1);
-by (etac well_ord_is_wf 4);
-by (ALLGOALS
- (blast_tac (claset() addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap]
- addSEs [ltE])));
-qed "ordermap_z_lt";
-
-(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
-Goalw [cmult_def]
- "[| Limit(K); x<K; y<K; z=succ(x Un y) |] ==> \
-\ | ordermap(K*K, csquare_rel(K)) ` <x,y> | le |succ(z)| |*| |succ(z)|";
-by (rtac (well_ord_rmult RS well_ord_lepoll_imp_Card_le) 1);
-by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1));
-by (subgoals_tac ["z<K"] 1);
-by (blast_tac (claset() addSIs [Un_least_lt, Limit_has_succ]) 2);
-by (rtac (ordermap_z_lt RS leI RS le_imp_lepoll RS lepoll_trans) 1);
-by (REPEAT_SOME assume_tac);
-by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1);
-by (etac (Limit_is_Ord RS well_ord_csquare) 1);
-by (blast_tac (claset() addIs [ltD]) 1);
-by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN
- assume_tac 1);
-by (REPEAT_FIRST (etac ltE));
-by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
-by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll)));
-qed "ordermap_csquare_le";
-
-(*Kunen: "... so the order type <= K" *)
-Goal "[| InfCard(K); ALL y:K. InfCard(y) --> y |*| y = y |] ==> \
-\ ordertype(K*K, csquare_rel(K)) le K";
-by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
-by (rtac all_lt_imp_le 1);
-by (assume_tac 1);
-by (etac (well_ord_csquare RS Ord_ordertype) 1);
-by (rtac Card_lt_imp_lt 1);
-by (etac InfCard_is_Card 3);
-by (etac ltE 2 THEN assume_tac 2);
-by (asm_full_simp_tac (simpset() addsimps [ordertype_unfold]) 1);
-by (safe_tac (claset() addSEs [ltE]));
-by (subgoals_tac ["Ord(xa)", "Ord(ya)"] 1);
-by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2));
-by (rtac (InfCard_is_Limit RS ordermap_csquare_le RS lt_trans1) 1 THEN
- REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1));
-by (res_inst_tac [("i","xa Un ya"), ("j","nat")] Ord_linear2 1 THEN
- REPEAT (ares_tac [Ord_Un, Ord_nat] 1));
-(*the finite case: xa Un ya < nat *)
-by (res_inst_tac [("j", "nat")] lt_trans2 1);
-by (asm_full_simp_tac (simpset() addsimps [InfCard_def]) 2);
-by (asm_full_simp_tac
- (simpset() addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
- nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1);
-(*case nat le (xa Un ya) *)
-by (asm_full_simp_tac
- (simpset() addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
- le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt,
- Ord_Un, ltI, nat_le_cardinal,
- Ord_cardinal_le RS lt_trans1 RS ltD]) 1);
-qed "ordertype_csquare_le";
-
-(*Main result: Kunen's Theorem 10.12*)
-Goal "InfCard(K) ==> K |*| K = K";
-by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
-by (etac rev_mp 1);
-by (trans_ind_tac "K" [] 1);
-by (rtac impI 1);
-by (rtac le_anti_sym 1);
-by (etac (InfCard_is_Card RS cmult_square_le) 2);
-by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1);
-by (assume_tac 2);
-by (assume_tac 2);
-by (asm_simp_tac
- (simpset() addsimps [cmult_def, Ord_cardinal_le,
- well_ord_csquare RS ordermap_bij RS
- bij_imp_eqpoll RS cardinal_cong,
- well_ord_csquare RS Ord_ordertype]) 1);
-qed "InfCard_csquare_eq";
-
-(*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
-Goal "[| well_ord(A,r); InfCard(|A|) |] ==> A*A eqpoll A";
-by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1);
-by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1));
-by (rtac well_ord_cardinal_eqE 1);
-by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1));
-by (asm_simp_tac (simpset()
- addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1);
-qed "well_ord_InfCard_square_eq";
-
-(** Toward's Kunen's Corollary 10.13 (1) **)
-
-Goal "[| InfCard(K); L le K; 0<L |] ==> K |*| L = K";
-by (rtac le_anti_sym 1);
-by (etac ltE 2 THEN
- REPEAT (ares_tac [cmult_le_self, InfCard_is_Card] 2));
-by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
-by (resolve_tac [cmult_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
-by (asm_full_simp_tac (simpset() addsimps [InfCard_csquare_eq]) 1);
-qed "InfCard_le_cmult_eq";
-
-(*Corollary 10.13 (1), for cardinal multiplication*)
-Goal "[| InfCard(K); InfCard(L) |] ==> K |*| L = K Un L";
-by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
-by (typecheck_tac (tcset() addTCs [InfCard_is_Card, Card_is_Ord]));
-by (resolve_tac [cmult_commute RS ssubst] 1);
-by (resolve_tac [Un_commute RS ssubst] 1);
-by (ALLGOALS
- (asm_simp_tac
- (simpset() addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq,
- subset_Un_iff2 RS iffD1, le_imp_subset])));
-qed "InfCard_cmult_eq";
-
-Goal "InfCard(K) ==> K |+| K = K";
-by (asm_simp_tac
- (simpset() addsimps [cmult_2 RS sym, InfCard_is_Card, cmult_commute]) 1);
-by (asm_simp_tac
- (simpset() addsimps [InfCard_le_cmult_eq, InfCard_is_Limit,
- Limit_has_0, Limit_has_succ]) 1);
-qed "InfCard_cdouble_eq";
-
-(*Corollary 10.13 (1), for cardinal addition*)
-Goal "[| InfCard(K); L le K |] ==> K |+| L = K";
-by (rtac le_anti_sym 1);
-by (etac ltE 2 THEN
- REPEAT (ares_tac [cadd_le_self, InfCard_is_Card] 2));
-by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
-by (resolve_tac [cadd_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
-by (asm_full_simp_tac (simpset() addsimps [InfCard_cdouble_eq]) 1);
-qed "InfCard_le_cadd_eq";
-
-Goal "[| InfCard(K); InfCard(L) |] ==> K |+| L = K Un L";
-by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
-by (typecheck_tac (tcset() addTCs [InfCard_is_Card, Card_is_Ord]));
-by (resolve_tac [cadd_commute RS ssubst] 1);
-by (resolve_tac [Un_commute RS ssubst] 1);
-by (ALLGOALS
- (asm_simp_tac
- (simpset() addsimps [InfCard_le_cadd_eq,
- subset_Un_iff2 RS iffD1, le_imp_subset])));
-qed "InfCard_cadd_eq";
-
-(*The other part, Corollary 10.13 (2), refers to the cardinality of the set
- of all n-tuples of elements of K. A better version for the Isabelle theory
- might be InfCard(K) ==> |list(K)| = K.
-*)
-
-(*** For every cardinal number there exists a greater one
- [Kunen's Theorem 10.16, which would be trivial using AC] ***)
-
-Goalw [jump_cardinal_def] "Ord(jump_cardinal(K))";
-by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
-by (blast_tac (claset() addSIs [Ord_ordertype]) 2);
-by (rewtac Transset_def);
-by (safe_tac subset_cs);
-by (asm_full_simp_tac (simpset() addsimps [ordertype_pred_unfold]) 1);
-by Safe_tac;
-by (rtac UN_I 1);
-by (rtac ReplaceI 2);
-by (ALLGOALS (blast_tac (claset() addIs [well_ord_subset] addSEs [predE])));
-qed "Ord_jump_cardinal";
-
-(*Allows selective unfolding. Less work than deriving intro/elim rules*)
-Goalw [jump_cardinal_def]
- "i : jump_cardinal(K) <-> \
-\ (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))";
-by (fast_tac subset_cs 1); (*It's vital to avoid reasoning about <=*)
-qed "jump_cardinal_iff";
-
-(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
-Goal "Ord(K) ==> K < jump_cardinal(K)";
-by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1);
-by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
-by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel]));
-by (rtac subset_refl 2);
-by (asm_simp_tac (simpset() addsimps [Memrel_def, subset_iff]) 1);
-by (asm_simp_tac (simpset() addsimps [ordertype_Memrel]) 1);
-qed "K_lt_jump_cardinal";
-
-(*The proof by contradiction: the bijection f yields a wellordering of X
- whose ordertype is jump_cardinal(K). *)
-Goal "[| well_ord(X,r); r <= K * K; X <= K; \
-\ f : bij(ordertype(X,r), jump_cardinal(K)) \
-\ |] ==> jump_cardinal(K) : jump_cardinal(K)";
-by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1);
-by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2));
-by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
-by (REPEAT_FIRST (resolve_tac [exI, conjI]));
-by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1);
-by (REPEAT (assume_tac 1));
-by (etac (bij_is_inj RS well_ord_rvimage) 1);
-by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1);
-by (asm_simp_tac
- (simpset() addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage),
- ordertype_Memrel, Ord_jump_cardinal]) 1);
-qed "Card_jump_cardinal_lemma";
-
-(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
-Goal "Card(jump_cardinal(K))";
-by (rtac (Ord_jump_cardinal RS CardI) 1);
-by (rewtac eqpoll_def);
-by (safe_tac (claset() addSDs [ltD, jump_cardinal_iff RS iffD1]));
-by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1));
-qed "Card_jump_cardinal";
-
-(*** Basic properties of successor cardinals ***)
-
-Goalw [csucc_def]
- "Ord(K) ==> Card(csucc(K)) & K < csucc(K)";
-by (rtac LeastI 1);
-by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal,
- Ord_jump_cardinal] 1));
-qed "csucc_basic";
-
-bind_thm ("Card_csucc", csucc_basic RS conjunct1);
-
-bind_thm ("lt_csucc", csucc_basic RS conjunct2);
-
-Goal "Ord(K) ==> 0 < csucc(K)";
-by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1);
-by (REPEAT (assume_tac 1));
-qed "Ord_0_lt_csucc";
-
-Goalw [csucc_def]
- "[| Card(L); K<L |] ==> csucc(K) le L";
-by (rtac Least_le 1);
-by (REPEAT (ares_tac [conjI, Card_is_Ord] 1));
-qed "csucc_le";
-
-Goal "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K";
-by (rtac iffI 1);
-by (rtac Card_lt_imp_lt 2);
-by (etac lt_trans1 2);
-by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2));
-by (resolve_tac [notI RS not_lt_imp_le] 1);
-by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1);
-by (assume_tac 1);
-by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1);
-by (REPEAT (ares_tac [Ord_cardinal] 1
- ORELSE eresolve_tac [ltE, Card_is_Ord] 1));
-qed "lt_csucc_iff";
-
-Goal "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K";
-by (asm_simp_tac
- (simpset() addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1);
-qed "Card_lt_csucc_iff";
-
-Goalw [InfCard_def]
- "InfCard(K) ==> InfCard(csucc(K))";
-by (asm_simp_tac (simpset() addsimps [Card_csucc, Card_is_Ord,
- lt_csucc RS leI RSN (2,le_trans)]) 1);
-qed "InfCard_csucc";
-
-
-(*** Finite sets ***)
-
-Goal "n: nat ==> ALL A. A eqpoll n --> A : Fin(A)";
-by (induct_tac "n" 1);
-by (simp_tac (simpset() addsimps [eqpoll_0_iff]) 1);
-by (Clarify_tac 1);
-by (subgoal_tac "EX u. u:A" 1);
-by (etac exE 1);
-by (resolve_tac [Diff_sing_eqpoll RS revcut_rl] 1);
-by (assume_tac 2);
-by (assume_tac 1);
-by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
-by (assume_tac 1);
-by (resolve_tac [thm "Fin.consI"] 1);
-by (Blast_tac 1);
-by (blast_tac (claset() addIs [subset_consI RS Fin_mono RS subsetD]) 1);
-(*Now for the lemma assumed above*)
-by (rewtac eqpoll_def);
-by (blast_tac (claset() addIs [bij_converse_bij RS bij_is_fun RS apply_type]) 1);
-val lemma = result();
-
-Goalw [Finite_def] "Finite(A) ==> A : Fin(A)";
-by (blast_tac (claset() addIs [lemma RS spec RS mp]) 1);
-qed "Finite_into_Fin";
-
-Goal "A : Fin(U) ==> Finite(A)";
-by (fast_tac (claset() addSIs [Finite_0, Finite_cons] addEs [Fin_induct]) 1);
-qed "Fin_into_Finite";
-
-Goal "Finite(A) <-> A : Fin(A)";
-by (blast_tac (claset() addIs [Finite_into_Fin, Fin_into_Finite]) 1);
-qed "Finite_Fin_iff";
-
-Goal "[| Finite(A); Finite(B) |] ==> Finite(A Un B)";
-by (blast_tac (claset() addSIs [Fin_into_Finite, Fin_UnI]
- addSDs [Finite_into_Fin]
- addIs [Un_upper1 RS Fin_mono RS subsetD,
- Un_upper2 RS Fin_mono RS subsetD]) 1);
-qed "Finite_Un";
-
-Goal "[| ALL y:X. Finite(y); Finite(X) |] ==> Finite(Union(X))";
-by (asm_full_simp_tac (simpset() addsimps [Finite_Fin_iff]) 1);
-by (rtac Fin_UnionI 1);
-by (etac Fin_induct 1);
-by (Simp_tac 1);
-by (blast_tac (claset() addIs [thm "Fin.consI", impOfSubs Fin_mono]) 1);
-qed "Finite_Union";
-
-(* Induction principle for Finite(A), by Sidi Ehmety *)
-val major::prems = Goal
-"[| Finite(A); P(0); \
-\ !! x B. [| Finite(B); x ~: B; P(B) |] \
-\ ==> P(cons(x, B)) |] \
-\ ==> P(A)";
-by (resolve_tac [major RS Finite_into_Fin RS Fin_induct] 1);
-by (ALLGOALS(resolve_tac prems));
-by (ALLGOALS(asm_simp_tac (simpset() addsimps [Fin_into_Finite])));
-qed "Finite_induct";
-
-
-(** Removing elements from a finite set decreases its cardinality **)
-
-Goal "A: Fin(U) ==> x~:A --> ~ cons(x,A) lepoll A";
-by (etac Fin_induct 1);
-by (simp_tac (simpset() addsimps [lepoll_0_iff]) 1);
-by (subgoal_tac "cons(x,cons(xa,y)) = cons(xa,cons(x,y))" 1);
-by (Asm_simp_tac 1);
-by (blast_tac (claset() addSDs [cons_lepoll_consD]) 1);
-by (Blast_tac 1);
-qed "Fin_imp_not_cons_lepoll";
-
-Goal "[| Finite(A); a~:A |] ==> |cons(a,A)| = succ(|A|)";
-by (rewtac cardinal_def);
-by (rtac Least_equality 1);
-by (fold_tac [cardinal_def]);
-by (simp_tac (simpset() addsimps [succ_def]) 1);
-by (blast_tac (claset() addIs [cons_eqpoll_cong, well_ord_cardinal_eqpoll]
- addSEs [mem_irrefl]
- addSDs [Finite_imp_well_ord]) 1);
-by (blast_tac (claset() addIs [Card_cardinal, Card_is_Ord]) 1);
-by (rtac notI 1);
-by (resolve_tac [Finite_into_Fin RS Fin_imp_not_cons_lepoll RS mp RS notE] 1);
-by (assume_tac 1);
-by (assume_tac 1);
-by (eresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_trans] 1);
-by (eresolve_tac [le_imp_lepoll RS lepoll_trans] 1);
-by (blast_tac (claset() addIs [well_ord_cardinal_eqpoll RS eqpoll_imp_lepoll]
- addSDs [Finite_imp_well_ord]) 1);
-qed "Finite_imp_cardinal_cons";
-
-
-Goal "[| Finite(A); a:A |] ==> succ(|A-{a}|) = |A|";
-by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
-by (assume_tac 1);
-by (asm_simp_tac (simpset() addsimps [Finite_imp_cardinal_cons,
- Diff_subset RS subset_Finite]) 1);
-by (asm_simp_tac (simpset() addsimps [cons_Diff]) 1);
-qed "Finite_imp_succ_cardinal_Diff";
-
-Goal "[| Finite(A); a:A |] ==> |A-{a}| < |A|";
-by (rtac succ_leE 1);
-by (asm_simp_tac (simpset() addsimps [Finite_imp_succ_cardinal_Diff]) 1);
-qed "Finite_imp_cardinal_Diff";
-
-
-(** Theorems by Krzysztof Grabczewski, proofs by lcp **)
-
-bind_thm ("nat_implies_well_ord",
- (transfer (the_context ()) nat_into_Ord) RS well_ord_Memrel);
-
-Goal "[| m:nat; n:nat |] ==> m + n eqpoll m #+ n";
-by (rtac eqpoll_trans 1);
-by (resolve_tac [well_ord_radd RS well_ord_cardinal_eqpoll RS eqpoll_sym] 1);
-by (REPEAT (etac nat_implies_well_ord 1));
-by (asm_simp_tac (simpset()
- addsimps [nat_cadd_eq_add RS sym, cadd_def, eqpoll_refl]) 1);
-qed "nat_sum_eqpoll_sum";
-
-
-(*** Theorems by Sidi Ehmety ***)
-
-(*The contrapositive says ~Finite(A) ==> ~Finite(A-{a}) *)
-Goalw [Finite_def] "Finite(A - {a}) ==> Finite(A)";
-by (case_tac "a:A" 1);
-by (subgoal_tac "A-{a}=A" 2);
-by Auto_tac;
-by (res_inst_tac [("x", "succ(n)")] bexI 1);
-by (subgoal_tac "cons(a, A - {a}) = A & cons(n, n) = succ(n)" 1);
-by (dres_inst_tac [("a", "a"), ("b", "n")] cons_eqpoll_cong 1);
-by (auto_tac (claset() addDs [mem_irrefl], simpset()));
-qed "Diff_sing_Finite";
-
-(*And the contrapositive of this says
- [| ~Finite(A); Finite(B) |] ==> ~Finite(A-B) *)
-Goal "Finite(B) ==> Finite(A-B) --> Finite(A)";
-by (etac Finite_induct 1);
-by Auto_tac;
-by (case_tac "x:A" 1);
- by (subgoal_tac "A-cons(x, B) = A - B" 2);
-by (subgoal_tac "A - cons(x, B) = (A - B) - {x}" 1);
-by (rotate_tac ~1 1);
-by (Asm_full_simp_tac 1);
-by (dtac Diff_sing_Finite 1);
-by Auto_tac;
-qed_spec_mp "Diff_Finite";
-
-Goal "Ord(i) ==> i <= nat --> i : nat | i=nat";
-by (etac trans_induct3 1);
-by Auto_tac;
-by (blast_tac (claset() addSDs [nat_le_Limit RS le_imp_subset]) 1);
-qed_spec_mp "Ord_subset_natD";
-
-Goal "[| Ord(i); i <= nat |] ==> Card(i)";
-by (blast_tac (claset() addDs [Ord_subset_natD]
- addIs [Card_nat, nat_into_Card]) 1);
-qed "Ord_nat_subset_into_Card";
-
-Goal "Finite(A) ==> |A| : nat";
-by (etac Finite_induct 1);
-by (auto_tac (claset(),
- simpset() addsimps
- [cardinal_0, Finite_imp_cardinal_cons]));
-qed "Finite_cardinal_in_nat";
-Addsimps [Finite_cardinal_in_nat];
-
-Goal "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1";
-by (rtac succ_inject 1);
-by (res_inst_tac [("b", "|A|")] trans 1);
-by (asm_simp_tac (simpset() addsimps
- [Finite_imp_succ_cardinal_Diff]) 1);
-by (subgoal_tac "1 lepoll A" 1);
-by (blast_tac (claset() addIs [not_0_is_lepoll_1]) 2);
-by (forward_tac [Finite_imp_well_ord] 1);
-by (Clarify_tac 1);
-by (rotate_tac ~1 1);
-by (dtac well_ord_lepoll_imp_Card_le 1);
-by (auto_tac (claset(), simpset() addsimps [cardinal_1]));
-by (rtac trans 1);
-by (rtac diff_succ 2);
-by (auto_tac (claset(), simpset() addsimps [Finite_cardinal_in_nat]));
-qed "Finite_Diff_sing_eq_diff_1";
-
-Goal "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0";
-by (etac Finite_induct 1);
-by (ALLGOALS(Clarify_tac));
-by (ALLGOALS(asm_full_simp_tac (simpset() addsimps
- [cardinal_0,Finite_imp_cardinal_cons])));
-by (case_tac "Finite(A)" 1);
-by (subgoal_tac "Finite(cons(x, B))" 2);
-by (dres_inst_tac [("B", "cons(x, B)")] Diff_Finite 2);
-by (auto_tac (claset(), simpset() addsimps [Finite_0, Finite_cons]));
-by (subgoal_tac "|B|<|A|" 1);
-by (blast_tac (claset() addIs [lt_trans, Ord_cardinal]) 2);
-by (case_tac "x:A" 1);
-by (subgoal_tac "A - cons(x, B)= A - B" 2);
-by Auto_tac;
-by (subgoal_tac "|A| le |cons(x, B)|" 1);
-by (blast_tac (claset() addDs [Finite_cons RS Finite_imp_well_ord]
- addIs [well_ord_lepoll_imp_Card_le, subset_imp_lepoll]) 2);
-by (auto_tac (claset(), simpset() addsimps [Finite_imp_cardinal_cons]));
-by (auto_tac (claset() addSDs [Finite_cardinal_in_nat],
- simpset() addsimps [le_iff]));
-by (blast_tac (claset() addIs [lt_trans]) 1);
-qed_spec_mp "cardinal_lt_imp_Diff_not_0";