215 [well_ord_is_wf RS ordermap_pred_unfold]) 1); |
215 [well_ord_is_wf RS ordermap_pred_unfold]) 1); |
216 by (asm_simp_tac (ZF_ss addsimps [ordertype_unfold, |
216 by (asm_simp_tac (ZF_ss addsimps [ordertype_unfold, |
217 ordermap_pred_eq_ordermap]) 1); |
217 ordermap_pred_eq_ordermap]) 1); |
218 qed "ordertype_subset"; |
218 qed "ordertype_subset"; |
219 |
219 |
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220 |
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221 (*Kunen's Theorem 7.3 (i), page 16; see also Ordinal/Ord_in_Ord |
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222 The smaller ordinal is an initial segment of the larger *) |
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223 goalw OrderType.thy [pred_def, lt_def] |
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224 "!!i j. j<i ==> j = pred(i, j, Memrel(i))"; |
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225 by (asm_simp_tac (ZF_ss addsimps [Memrel_iff]) 1); |
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226 by (fast_tac (eq_cs addEs [Ord_trans]) 1); |
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227 val lt_eq_pred = result(); |
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228 |
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229 goal OrderType.thy |
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230 "!!i. [| j<i; f: ord_iso(i,Memrel(i),j,Memrel(j)) \ |
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231 \ |] ==> R"; |
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232 by (forward_tac [lt_eq_pred] 1); |
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233 be ltE 1; |
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234 by (rtac (well_ord_Memrel RS not_well_ord_iso_pred RS notE) 1 THEN |
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235 assume_tac 1 THEN assume_tac 1); |
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236 be subst 1; |
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237 by (asm_full_simp_tac (ZF_ss addsimps [ord_iso_def]) 1); |
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238 (*Combining the two simplifications causes looping*) |
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239 by (asm_simp_tac (ZF_ss addsimps [Memrel_iff]) 1); |
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240 by (fast_tac (ZF_cs addSEs [bij_is_fun RS apply_type] |
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241 addEs [Ord_trans]) 1); |
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242 val Ord_iso_implies_eq_lemma = result(); |
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243 |
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244 (*Kunen's Theorem 7.3 (ii), page 16. Isomorphic ordinals are equal*) |
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245 goal OrderType.thy |
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246 "!!i. [| Ord(i); Ord(j); f: ord_iso(i,Memrel(i),j,Memrel(j)) \ |
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247 \ |] ==> i=j"; |
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248 by (res_inst_tac [("i","i"),("j","j")] Ord_linear_lt 1); |
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249 by (REPEAT (eresolve_tac [asm_rl, ord_iso_sym, Ord_iso_implies_eq_lemma] 1)); |
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250 val Ord_iso_implies_eq = result(); |