26 lam <x,y>:K*K. <x \<union> y, x, y>, |
26 lam <x,y>:K*K. <x \<union> y, x, y>, |
27 rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))" |
27 rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))" |
28 |
28 |
29 definition |
29 definition |
30 jump_cardinal :: "i=>i" where |
30 jump_cardinal :: "i=>i" where |
31 --\<open>This def is more complex than Kunen's but it more easily proved to |
31 \<comment>\<open>This def is more complex than Kunen's but it more easily proved to |
32 be a cardinal\<close> |
32 be a cardinal\<close> |
33 "jump_cardinal(K) == |
33 "jump_cardinal(K) == |
34 \<Union>X\<in>Pow(K). {z. r \<in> Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}" |
34 \<Union>X\<in>Pow(K). {z. r \<in> Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}" |
35 |
35 |
36 definition |
36 definition |
37 csucc :: "i=>i" where |
37 csucc :: "i=>i" where |
38 --\<open>needed because @{term "jump_cardinal(K)"} might not be the successor |
38 \<comment>\<open>needed because @{term "jump_cardinal(K)"} might not be the successor |
39 of @{term K}\<close> |
39 of @{term K}\<close> |
40 "csucc(K) == \<mu> L. Card(L) & K<L" |
40 "csucc(K) == \<mu> L. Card(L) & K<L" |
41 |
41 |
42 |
42 |
43 lemma Card_Union [simp,intro,TC]: |
43 lemma Card_Union [simp,intro,TC]: |
572 also have "... \<approx> |succ(z)| \<times> |succ(z)|" using oz |
572 also have "... \<approx> |succ(z)| \<times> |succ(z)|" using oz |
573 by (blast intro: prod_eqpoll_cong Ord_succ Ord_cardinal_eqpoll eqpoll_sym) |
573 by (blast intro: prod_eqpoll_cong Ord_succ Ord_cardinal_eqpoll eqpoll_sym) |
574 finally show "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> |succ(z)| \<times> |succ(z)|" . |
574 finally show "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> |succ(z)| \<times> |succ(z)|" . |
575 qed |
575 qed |
576 |
576 |
577 text\<open>Kunen: "... so the order type is @{text"\<le>"} K"\<close> |
577 text\<open>Kunen: "... so the order type is \<open>\<le>\<close> K"\<close> |
578 lemma ordertype_csquare_le: |
578 lemma ordertype_csquare_le: |
579 assumes IK: "InfCard(K)" and eq: "\<And>y. y\<in>K \<Longrightarrow> InfCard(y) \<Longrightarrow> y \<otimes> y = y" |
579 assumes IK: "InfCard(K)" and eq: "\<And>y. y\<in>K \<Longrightarrow> InfCard(y) \<Longrightarrow> y \<otimes> y = y" |
580 shows "ordertype(K*K, csquare_rel(K)) \<le> K" |
580 shows "ordertype(K*K, csquare_rel(K)) \<le> K" |
581 proof - |
581 proof - |
582 have CK: "Card(K)" using IK by (rule InfCard_is_Card) |
582 have CK: "Card(K)" using IK by (rule InfCard_is_Card) |