Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
| theory More_Missing_Multiset | |
| imports | |
| "HOL-Combinatorics.Permutations" | |
| Polynomial_Factorization.Missing_Multiset | |
| begin | |
| lemma rel_mset_free: | |
| assumes rel: "rel_mset rel X Y" and xs: "mset xs = X" | |
| shows "\<exists>ys. mset ys = Y \<and> list_all2 rel xs ys" | |
| proof- | |
| from rel[unfolded rel_mset_def] obtain xs' ys' | |
| where xs': "mset xs' = X" and ys': "mset ys' = Y" and xsys': "list_all2 rel xs' ys'" by auto | |
| from xs' xs have "mset xs = mset xs'" by auto | |
| from mset_eq_permutation[OF this] | |
| obtain f where perm: "f permutes {..<length xs'}" and xs': "permute_list f xs' = xs". | |
| then have [simp]: "length xs' = length xs" by auto | |
| from permute_list_nth[OF perm, unfolded xs'] have *: "\<And>i. i < length xs \<Longrightarrow> xs ! i = xs' ! f i" by auto | |
| note [simp] = list_all2_lengthD[OF xsys',symmetric] | |
| note [simp] = atLeast0LessThan[symmetric] | |
| note bij = permutes_bij[OF perm] | |
| define ys where "ys \<equiv> map (nth ys' \<circ> f) [0..<length ys']" | |
| then have [simp]: "length ys = length ys'" by auto | |
| have "mset ys = mset (map (nth ys') (map f [0..<length ys']))" | |
| unfolding ys_def by auto | |
| also have "... = image_mset (nth ys') (image_mset f (mset [0..<length ys']))" | |
| by (simp add: multiset.map_comp) | |
| also have "(mset [0..<length ys']) = mset_set {0..<length ys'}" | |
| by (metis mset_sorted_list_of_multiset sorted_list_of_mset_set sorted_list_of_set_range) | |
| also have "image_mset f (...) = mset_set (f ` {..<length ys'})" | |
| using subset_inj_on[OF bij_is_inj[OF bij]] by (subst image_mset_mset_set, auto) | |
| also have "... = mset [0..<length ys']" using perm by (simp add: permutes_image) | |
| also have "image_mset (nth ys') ... = mset ys'" by(fold mset_map, unfold map_nth, auto) | |
| finally have "mset ys = Y" using ys' by auto | |
| moreover have "list_all2 rel xs ys" | |
| proof(rule list_all2_all_nthI) | |
| fix i assume i: "i < length xs" | |
| with * have "xs ! i = xs' ! f i" by auto | |
| also from i permutes_in_image[OF perm] | |
| have "rel (xs' ! f i) (ys' ! f i)" by (intro list_all2_nthD[OF xsys'], auto) | |
| finally show "rel (xs ! i) (ys ! i)" unfolding ys_def using i by simp | |
| qed simp | |
| ultimately show ?thesis by auto | |
| qed | |
| lemma rel_mset_split: | |
| assumes rel: "rel_mset rel (X1+X2) Y" | |
| shows "\<exists>Y1 Y2. Y = Y1 + Y2 \<and> rel_mset rel X1 Y1 \<and> rel_mset rel X2 Y2" | |
| proof- | |
| obtain xs1 where xs1: "mset xs1 = X1" using ex_mset by auto | |
| obtain xs2 where xs2: "mset xs2 = X2" using ex_mset by auto | |
| from xs1 xs2 have "mset (xs1 @ xs2) = X1 + X2" by auto | |
| from rel_mset_free[OF rel this] obtain ys | |
| where ys: "mset ys = Y" "list_all2 rel (xs1 @ xs2) ys" by auto | |
| then obtain ys1 ys2 | |
| where ys12: "ys = ys1 @ ys2" | |
| and xs1ys1: "list_all2 rel xs1 ys1" | |
| and xs2ys2: "list_all2 rel xs2 ys2" | |
| using list_all2_append1 by blast | |
| from ys12 ys have "Y = mset ys1 + mset ys2" by auto | |
| moreover from xs1 xs1ys1 have "rel_mset rel X1 (mset ys1)" unfolding rel_mset_def by auto | |
| moreover from xs2 xs2ys2 have "rel_mset rel X2 (mset ys2)" unfolding rel_mset_def by auto | |
| ultimately show ?thesis by (subst exI[of _ "mset ys1"], subst exI[of _ "mset ys2"],auto) | |
| qed | |
| lemma rel_mset_OO: | |
| assumes AB: "rel_mset R A B" and BC: "rel_mset S B C" | |
| shows "rel_mset (R OO S) A C" | |
| proof- | |
| from AB obtain as bs where A_as: "A = mset as" and B_bs: "B = mset bs" and as_bs: "list_all2 R as bs" | |
| by (auto simp: rel_mset_def) | |
| from rel_mset_free[OF BC] B_bs obtain cs where C_cs: "C = mset cs" and bs_cs: "list_all2 S bs cs" | |
| by auto | |
| from list_all2_trans[OF _ as_bs bs_cs, of "R OO S"] A_as C_cs | |
| show ?thesis by (auto simp: rel_mset_def) | |
| qed | |
| (* a variant for "right" *) | |
| lemma ex_mset_zip_right: | |
| assumes "length xs = length ys" "mset ys' = mset ys" | |
| shows "\<exists>xs'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)" | |
| using assms | |
| proof (induct xs ys arbitrary: ys' rule: list_induct2) | |
| case Nil | |
| thus ?case | |
| by auto | |
| next | |
| case (Cons x xs y ys ys') | |
| obtain j where j_len: "j < length ys'" and nth_j: "ys' ! j = y" | |
| by (metis Cons.prems in_set_conv_nth list.set_intros(1) mset_eq_setD) | |
| define ysa where "ysa = take j ys' @ drop (Suc j) ys'" | |
| have "mset ys' = {#y#} + mset ysa" | |
| unfolding ysa_def using j_len nth_j | |
| by (metis Cons_nth_drop_Suc union_mset_add_mset_right add_mset_remove_trivial add_diff_cancel_left' | |
| append_take_drop_id mset.simps(2) mset_append) | |
| hence ms_y: "mset ysa = mset ys" | |
| by (simp add: Cons.prems) | |
| then obtain xsa where | |
| len_a: "length ysa = length xsa" and ms_a: "mset (zip xsa ysa) = mset (zip xs ys)" | |
| using Cons.hyps(2) by blast | |
| define xs' where "xs' = take j xsa @ x # drop j xsa" | |
| have ys': "ys' = take j ysa @ y # drop j ysa" | |
| using ms_y j_len nth_j Cons.prems ysa_def | |
| by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons | |
| length_drop size_mset) | |
| have j_len': "j \<le> length ysa" | |
| using j_len ys' ysa_def | |
| by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less) | |
| have "length ys' = length xs'" | |
| unfolding xs'_def using Cons.prems len_a ms_y | |
| by (metis add_Suc_right append_take_drop_id length_Cons length_append mset_eq_length) | |
| moreover have "mset (zip xs' ys') = mset (zip (x # xs) (y # ys))" | |
| unfolding ys' xs'_def | |
| apply (rule HOL.trans[OF mset_zip_take_Cons_drop_twice]) | |
| using j_len' by (auto simp: len_a ms_a) | |
| ultimately show ?case | |
| by blast | |
| qed | |
| lemma list_all2_reorder_right_invariance: | |
| assumes rel: "list_all2 R xs ys" and ms_y: "mset ys' = mset ys" | |
| shows "\<exists>xs'. list_all2 R xs' ys' \<and> mset xs' = mset xs" | |
| proof - | |
| have len: "length xs = length ys" | |
| using rel list_all2_conv_all_nth by auto | |
| obtain xs' where | |
| len': "length xs' = length ys'" and ms_xy: "mset (zip xs' ys') = mset (zip xs ys)" | |
| using len ms_y by (metis ex_mset_zip_right) | |
| have "list_all2 R xs' ys'" | |
| using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: mset_eq_setD) | |
| moreover have "mset xs' = mset xs" | |
| using len len' ms_xy map_fst_zip mset_map by metis | |
| ultimately show ?thesis | |
| by blast | |
| qed | |
| lemma rel_mset_via_perm: "rel_mset rel (mset xs) (mset ys) \<longleftrightarrow> (\<exists>zs. mset xs = mset zs \<and> list_all2 rel zs ys)" | |
| proof (unfold rel_mset_def, intro iffI, goal_cases) | |
| case 1 | |
| then obtain zs ws where zs: "mset zs = mset xs" and ws: "mset ws = mset ys" and zsws: "list_all2 rel zs ws" by auto | |
| note list_all2_reorder_right_invariance[OF zsws ws[symmetric], unfolded zs] | |
| then show ?case by (auto dest: sym) | |
| next | |
| case 2 | |
| from this show ?case by force | |
| qed | |
| end | |