(* Title: Miscellaneous Definitions and Lemmas Author: Peter Lammich <peter.lammich@uni-muenster.de> Maintainer: Peter Lammich <peter.lammich@uni-muenster.de> Thomas Tuerk <tuerk@in.tum.de> *) (* CHANGELOG: 2010-05-09: Removed AC, AI locales, they are superseeded by concepts from OrderedGroups 2010-09-22: Merges with ext/Aux 2017-06-12: Added a bunch of lemmas from various other projects *) section ‹Miscellaneous Definitions and Lemmas› theory Misc imports Main "HOL-Library.Multiset" "HOL-ex.Quicksort" "HOL-Library.Option_ord" "HOL-Library.Infinite_Set" "HOL-Eisbach.Eisbach" begin text_raw ‹\label{thy:Misc}› subsection ‹Isabelle Distribution Move Proposals› subsubsection ‹Pure› lemma TERMI: "TERM x" unfolding Pure.term_def . subsubsection ‹HOL› (* Stronger disjunction elimination rules. *) lemma disjE1: "⟦ P ∨ Q; P ⟹ R; ⟦¬P;Q⟧ ⟹ R ⟧ ⟹ R" by metis lemma disjE2: "⟦ P ∨ Q; ⟦P; ¬Q⟧ ⟹ R; Q ⟹ R ⟧ ⟹ R" by blast lemma imp_mp_iff[simp]: "a ∧ (a ⟶ b) ⟷ a ∧ b" "(a ⟶ b) ∧ a ⟷ a ∧ b" (* is Inductive.imp_conj_iff, but not in simpset by default *) by blast+ lemma atomize_Trueprop_eq[atomize]: "(Trueprop x ≡ Trueprop y) ≡ Trueprop (x=y)" apply rule apply (rule) apply (erule equal_elim_rule1) apply assumption apply (erule equal_elim_rule2) apply assumption apply simp done subsubsection ‹Set› lemma inter_compl_diff_conv[simp]: "A ∩ -B = A - B" by auto lemma pair_set_inverse[simp]: "{(a,b). P a b}¯ = {(b,a). P a b}" by auto lemma card_doubleton_eq_2_iff[simp]: "card {a,b} = 2 ⟷ a≠b" by auto subsubsection ‹List› (* TODO: Move, analogous to List.length_greater_0_conv *) thm List.length_greater_0_conv lemma length_ge_1_conv[iff]: "Suc 0 ≤ length l ⟷ l≠[]" by (cases l) auto ― ‹Obtains a list from the pointwise characterization of its elements› lemma obtain_list_from_elements: assumes A: "∀i<n. (∃li. P li i)" obtains l where "length l = n" "∀i<n. P (l!i) i" proof - from A have "∃l. length l=n ∧ (∀i<n. P (l!i) i)" proof (induct n) case 0 thus ?case by simp next case (Suc n) then obtain l where IH: "length l = n" "(∀i<n. P(l!i) i)" by auto moreover from Suc.prems obtain ln where "P ln n" by auto ultimately have "length (l@[ln]) = Suc n" "(∀i<Suc n. P((l@[ln])!i) i)" by (auto simp add: nth_append dest: less_antisym) thus ?case by blast qed thus ?thesis using that by (blast) qed lemma distinct_sorted_mono: assumes S: "sorted l" assumes D: "distinct l" assumes B: "i<j" "j<length l" shows "l!i < l!j" proof - from S B have "l!i ≤ l!j" by (simp add: sorted_iff_nth_mono) also from nth_eq_iff_index_eq[OF D] B have "l!i ≠ l!j" by auto finally show ?thesis . qed lemma distinct_sorted_strict_mono_iff: assumes "distinct l" "sorted l" assumes "i<length l" "j<length l" shows "l!i<l!j ⟷ i<j" using assms by (metis distinct_sorted_mono leI less_le_not_le order.strict_iff_order) lemma distinct_sorted_mono_iff: assumes "distinct l" "sorted l" assumes "i<length l" "j<length l" shows "l!i≤l!j ⟷ i≤j" by (metis assms distinct_sorted_strict_mono_iff leD le_less_linear) (* List.thy has: declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!] We could, analogously, declare rules for "map _ _ = _@_" as dest!, or use elim!, or declare the _conv-rule as simp *) lemma map_eq_appendE: assumes "map f ls = fl@fl'" obtains l l' where "ls=l@l'" and "map f l=fl" and "map f l' = fl'" using assms proof (induction fl arbitrary: ls thesis) case (Cons x xs) then obtain l ls' where [simp]: "ls = l#ls'" "f l = x" by force with Cons.prems(2) have "map f ls' = xs @ fl'" by simp from Cons.IH[OF _ this] obtain ll ll' where "ls' = ll @ ll'" "map f ll = xs" "map f ll' = fl'" . with Cons.prems(1)[of "l#ll" ll'] show thesis by simp qed simp lemma map_eq_append_conv: "map f ls = fl@fl' ⟷ (∃l l'. ls = l@l' ∧ map f l = fl ∧ map f l' = fl')" by (auto elim!: map_eq_appendE) lemmas append_eq_mapE = map_eq_appendE[OF sym] lemma append_eq_map_conv: "fl@fl' = map f ls ⟷ (∃l l'. ls = l@l' ∧ map f l = fl ∧ map f l' = fl')" by (auto elim!: append_eq_mapE) lemma distinct_mapI: "distinct (map f l) ⟹ distinct l" by (induct l) auto lemma map_distinct_upd_conv: "⟦i<length l; distinct l⟧ ⟹ (map f l)[i := x] = map (f(l!i := x)) l" ― ‹Updating a mapped distinct list is equal to updating the mapping function› by (auto simp: nth_eq_iff_index_eq intro: nth_equalityI) lemma zip_inj: "⟦length a = length b; length a' = length b'; zip a b = zip a' b'⟧ ⟹ a=a' ∧ b=b'" proof (induct a b arbitrary: a' b' rule: list_induct2) case Nil then show ?case by (cases a'; cases b'; auto) next case (Cons x xs y ys) then show ?case by (cases a'; cases b'; auto) qed lemma zip_eq_zip_same_len[simp]: "⟦ length a = length b; length a' = length b' ⟧ ⟹ zip a b = zip a' b' ⟷ a=a' ∧ b=b'" by (auto dest: zip_inj) lemma upt_merge[simp]: "i≤j ∧ j≤k ⟹ [i..<j]@[j..<k] = [i..<k]" by (metis le_Suc_ex upt_add_eq_append) (* Maybe this should go into List.thy, next to snoc_eq_iff_butlast *) lemma snoc_eq_iff_butlast': "(ys = xs @ [x]) ⟷ (ys ≠ [] ∧ butlast ys = xs ∧ last ys = x)" by auto (* Case distinction how two elements of a list can be related to each other *) lemma list_match_lel_lel: assumes "c1 @ qs # c2 = c1' @ qs' # c2'" obtains (left) c21' where "c1 = c1' @ qs' # c21'" "c2' = c21' @ qs # c2" | (center) "c1' = c1" "qs' = qs" "c2' = c2" | (right) c21 where "c1' = c1 @ qs # c21" "c2 = c21 @ qs' # c2'" using assms apply (clarsimp simp: append_eq_append_conv2) subgoal for us by (cases us) auto done lemma xy_in_set_cases[consumes 2, case_names EQ XY YX]: assumes A: "x∈set l" "y∈set l" and C: "!!l1 l2. ⟦ x=y; l=l1@y#l2 ⟧ ⟹ P" "!!l1 l2 l3. ⟦ x≠y; l=l1@x#l2@y#l3 ⟧ ⟹ P" "!!l1 l2 l3. ⟦ x≠y; l=l1@y#l2@x#l3 ⟧ ⟹ P" shows P proof (cases "x=y") case True with A(1) obtain l1 l2 where "l=l1@y#l2" by (blast dest: split_list) with C(1) True show ?thesis by blast next case False from A(1) obtain l1 l2 where S1: "l=l1@x#l2" by (blast dest: split_list) from A(2) obtain l1' l2' where S2: "l=l1'@y#l2'" by (blast dest: split_list) from S1 S2 have M: "l1@x#l2 = l1'@y#l2'" by simp thus P proof (cases rule: list_match_lel_lel[consumes 1, case_names 1 2 3]) case (1 c) with S1 have "l=l1'@y#c@x#l2" by simp with C(3) False show ?thesis by blast next case 2 with False have False by blast thus ?thesis .. next case (3 c) with S1 have "l=l1@x#c@y#l2'" by simp with C(2) False show ?thesis by blast qed qed lemma list_e_eq_lel[simp]: "[e] = l1@e'#l2 ⟷ l1=[] ∧ e'=e ∧ l2=[]" "l1@e'#l2 = [e] ⟷ l1=[] ∧ e'=e ∧ l2=[]" apply (cases l1, auto) [] apply (cases l1, auto) [] done lemma list_ee_eq_leel[simp]: "([e1,e2] = l1@e1'#e2'#l2) ⟷ (l1=[] ∧ e1=e1' ∧ e2=e2' ∧ l2=[])" "(l1@e1'#e2'#l2 = [e1,e2]) ⟷ (l1=[] ∧ e1=e1' ∧ e2=e2' ∧ l2=[])" apply (cases l1, auto) [] apply (cases l1, auto) [] done subsubsection ‹Transitive Closure› text ‹A point-free induction rule for elements reachable from an initial set› lemma rtrancl_reachable_induct[consumes 0, case_names base step]: assumes I0: "I ⊆ INV" assumes IS: "E``INV ⊆ INV" shows "E⇧^{*}``I ⊆ INV" by (metis I0 IS Image_closed_trancl Image_mono subset_refl) lemma acyclic_empty[simp, intro!]: "acyclic {}" by (unfold acyclic_def) auto lemma acyclic_union: "acyclic (A∪B) ⟹ acyclic A" "acyclic (A∪B) ⟹ acyclic B" by (metis Un_upper1 Un_upper2 acyclic_subset)+ text ‹Here we provide a collection of miscellaneous definitions and helper lemmas› subsection "Miscellaneous (1)" text ‹This stuff is used in this theory itself, and thus occurs in first place or is simply not sorted into any other section of this theory.› lemma IdD: "(a,b)∈Id ⟹ a=b" by simp text ‹Conversion Tag› definition [simp]: "CNV x y ≡ x=y" lemma CNV_I: "CNV x x" by simp lemma CNV_eqD: "CNV x y ⟹ x=y" by simp lemma CNV_meqD: "CNV x y ⟹ x≡y" by simp (* TODO: Move. Shouldn't this be detected by simproc? *) lemma ex_b_b_and_simp[simp]: "(∃b. b ∧ Q b) ⟷ Q True" by auto lemma ex_b_not_b_and_simp[simp]: "(∃b. ¬b ∧ Q b) ⟷ Q False" by auto method repeat_all_new methods m = m;(repeat_all_new ‹m›)? subsubsection "AC-operators" text ‹Locale to declare AC-laws as simplification rules› locale Assoc = fixes f assumes assoc[simp]: "f (f x y) z = f x (f y z)" locale AC = Assoc + assumes commute[simp]: "f x y = f y x" lemma (in AC) left_commute[simp]: "f x (f y z) = f y (f x z)" by (simp only: assoc[symmetric]) simp lemmas (in AC) AC_simps = commute assoc left_commute text ‹Locale to define functions from surjective, unique relations› locale su_rel_fun = fixes F and f assumes unique: "⟦(A,B)∈F; (A,B')∈F⟧ ⟹ B=B'" assumes surjective: "⟦!!B. (A,B)∈F ⟹ P⟧ ⟹ P" assumes f_def: "f A == THE B. (A,B)∈F" lemma (in su_rel_fun) repr1: "(A,f A)∈F" proof (unfold f_def) obtain B where "(A,B)∈F" by (rule surjective) with theI[where P="λB. (A,B)∈F", OF this] show "(A, THE x. (A, x) ∈ F) ∈ F" by (blast intro: unique) qed lemma (in su_rel_fun) repr2: "(A,B)∈F ⟹ B=f A" using repr1 by (blast intro: unique) lemma (in su_rel_fun) repr: "(f A = B) = ((A,B)∈F)" using repr1 repr2 by (blast) ― ‹Contract quantification over two variables to pair› lemma Ex_prod_contract: "(∃a b. P a b) ⟷ (∃z. P (fst z) (snd z))" by auto lemma All_prod_contract: "(∀a b. P a b) ⟷ (∀z. P (fst z) (snd z))" by auto lemma nat_geq_1_eq_neqz: "x≥1 ⟷ x≠(0::nat)" by auto lemma nat_in_between_eq: "(a<b ∧ b≤Suc a) ⟷ b = Suc a" "(a≤b ∧ b<Suc a) ⟷ b = a" by auto lemma Suc_n_minus_m_eq: "⟦ n≥m; m>1 ⟧ ⟹ Suc (n - m) = n - (m - 1)" by simp lemma Suc_to_right: "Suc n = m ⟹ n = m - Suc 0" by simp lemma Suc_diff[simp]: "⋀n m. n≥m ⟹ m≥1 ⟹ Suc (n - m) = n - (m - 1)" by simp lemma if_not_swap[simp]: "(if ¬c then a else b) = (if c then b else a)" by auto lemma all_to_meta: "Trueprop (∀a. P a) ≡ (⋀a. P a)" apply rule by auto lemma imp_to_meta: "Trueprop (P⟶Q) ≡ (P⟹Q)" apply rule by auto (* for some reason, there is no such rule in HOL *) lemma iffI2: "⟦P ⟹ Q; ¬ P ⟹ ¬ Q⟧ ⟹ P ⟷ Q" by metis lemma iffExI: "⟦ ⋀x. P x ⟹ Q x; ⋀x. Q x ⟹ P x ⟧ ⟹ (∃x. P x) ⟷ (∃x. Q x)" by metis lemma bex2I[intro?]: "⟦ (a,b)∈S; (a,b)∈S ⟹ P a b ⟧ ⟹ ∃a b. (a,b)∈S ∧ P a b" by blast (* TODO: Move lemma to HOL! *) lemma cnv_conj_to_meta: "(P ∧ Q ⟹ PROP X) ≡ (⟦P;Q⟧ ⟹ PROP X)" by (rule BNF_Fixpoint_Base.conj_imp_eq_imp_imp) subsection ‹Sets› lemma remove_subset: "x∈S ⟹ S-{x} ⊂ S" by auto lemma subset_minus_empty: "A⊆B ⟹ A-B = {}" by auto lemma insert_minus_eq: "x≠y ⟹ insert x A - {y} = insert x (A - {y})" by auto lemma set_notEmptyE: "⟦S≠{}; !!x. x∈S ⟹ P⟧ ⟹ P" by (metis equals0I) lemma subset_Collect_conv: "S ⊆ Collect P ⟷ (∀x∈S. P x)" by auto lemma memb_imp_not_empty: "x∈S ⟹ S≠{}" by auto lemma disjoint_mono: "⟦ a⊆a'; b⊆b'; a'∩b'={} ⟧ ⟹ a∩b={}" by auto lemma disjoint_alt_simp1: "A-B = A ⟷ A∩B = {}" by auto lemma disjoint_alt_simp2: "A-B ≠ A ⟷ A∩B ≠ {}" by auto lemma disjoint_alt_simp3: "A-B ⊂ A ⟷ A∩B ≠ {}" by auto lemma disjointI[intro?]: "⟦ ⋀x. ⟦x∈a; x∈b⟧ ⟹ False ⟧ ⟹ a∩b={}" by auto lemmas set_simps = subset_minus_empty disjoint_alt_simp1 disjoint_alt_simp2 disjoint_alt_simp3 Un_absorb1 Un_absorb2 lemma set_minus_singleton_eq: "x∉X ⟹ X-{x} = X" by auto lemma set_diff_diff_left: "A-B-C = A-(B∪C)" by auto lemma image_update[simp]: "x∉A ⟹ f(x:=n)`A = f`A" by auto lemma eq_or_mem_image_simp[simp]: "{f l |l. l = a ∨ l∈B} = insert (f a) {f l|l. l∈B}" by blast lemma set_union_code [code_unfold]: "set xs ∪ set ys = set (xs @ ys)" by auto lemma in_fst_imageE: assumes "x ∈ fst`S" obtains y where "(x,y)∈S" using assms by auto lemma in_snd_imageE: assumes "y ∈ snd`S" obtains x where "(x,y)∈S" using assms by auto lemma fst_image_mp: "⟦fst`A ⊆ B; (x,y)∈A ⟧ ⟹ x∈B" by (metis Domain.DomainI fst_eq_Domain in_mono) lemma snd_image_mp: "⟦snd`A ⊆ B; (x,y)∈A ⟧ ⟹ y∈B" by (metis Range.intros rev_subsetD snd_eq_Range) lemma inter_eq_subsetI: "⟦ S⊆S'; A∩S' = B∩S' ⟧ ⟹ A∩S = B∩S" by auto text ‹ Decompose general union over sum types. › lemma Union_plus: "(⋃ x ∈ A <+> B. f x) = (⋃ a ∈ A. f (Inl a)) ∪ (⋃b ∈ B. f (Inr b))" by auto lemma Union_sum: "(⋃x. f (x::'a+'b)) = (⋃l. f (Inl l)) ∪ (⋃r. f (Inr r))" (is "?lhs = ?rhs") proof - have "?lhs = (⋃x ∈ UNIV <+> UNIV. f x)" by simp thus ?thesis by (simp only: Union_plus) qed subsubsection ‹Finite Sets› lemma card_1_singletonI: "⟦finite S; card S = 1; x∈S⟧ ⟹ S={x}" proof (safe, rule ccontr, goal_cases) case prems: (1 x') hence "finite (S-{x})" "S-{x} ≠ {}" by auto hence "card (S-{x}) ≠ 0" by auto moreover from prems(1-3) have "card (S-{x}) = 0" by auto ultimately have False by simp thus ?case .. qed lemma card_insert_disjoint': "⟦finite A; x ∉ A⟧ ⟹ card (insert x A) - Suc 0 = card A" by (drule (1) card_insert_disjoint) auto lemma card_eq_UNIV[simp]: "card (S::'a::finite set) = card (UNIV::'a set) ⟷ S=UNIV" proof (auto) fix x assume A: "card S = card (UNIV::'a set)" show "x∈S" proof (rule ccontr) assume "x∉S" hence "S⊂UNIV" by auto with psubset_card_mono[of UNIV S] have "card S < card (UNIV::'a set)" by auto with A show False by simp qed qed lemma card_eq_UNIV2[simp]: "card (UNIV::'a set) = card (S::'a::finite set) ⟷ S=UNIV" using card_eq_UNIV[of S] by metis lemma card_ge_UNIV[simp]: "card (UNIV::'a::finite set) ≤ card (S::'a set) ⟷ S=UNIV" using card_mono[of "UNIV::'a::finite set" S, simplified] by auto lemmas length_remdups_card = length_remdups_concat[of "[l]", simplified] for l lemma fs_contract: "fst ` { p | p. f (fst p) (snd p) ∈ S } = { a . ∃b. f a b ∈ S }" by (simp add: image_Collect) ― ‹Finite sets have an injective mapping to an initial segments of the natural numbers› (* This lemma is also in the standard library (from Isabelle2009-1 on) as @{thm [source] Finite_Set.finite_imp_inj_to_nat_seg}. However, it is formulated with HOL's ∃ there rather then with the meta-logic obtain *) lemma finite_imp_inj_to_nat_seg': fixes A :: "'a set" assumes A: "finite A" obtains f::"'a ⇒ nat" and n::"nat" where "f`A = {i. i<n}" "inj_on f A" by (metis A finite_imp_inj_to_nat_seg) lemma lists_of_len_fin1: "finite P ⟹ finite (lists P ∩ { l. length l = n })" proof (induct n) case 0 thus ?case by auto next case (Suc n) have "lists P ∩ { l. length l = Suc n } = (λ(a,l). a#l) ` (P × (lists P ∩ {l. length l = n}))" apply auto apply (case_tac x) apply auto done moreover from Suc have "finite …" by auto ultimately show ?case by simp qed lemma lists_of_len_fin2: "finite P ⟹ finite (lists P ∩ { l. n = length l })" proof - assume A: "finite P" have S: "{ l. n = length l } = { l. length l = n }" by auto have "finite (lists P ∩ { l. n = length l }) ⟷ finite (lists P ∩ { l. length l = n })" by (subst S) simp thus ?thesis using lists_of_len_fin1[OF A] by auto qed lemmas lists_of_len_fin = lists_of_len_fin1 lists_of_len_fin2 (* Try (simp only: cset_fin_simps, fastforce intro: cset_fin_intros) when reasoning about finiteness of collected sets *) lemmas cset_fin_simps = Ex_prod_contract fs_contract[symmetric] image_Collect[symmetric] lemmas cset_fin_intros = finite_imageI finite_inverse_image inj_onI lemma Un_interval: fixes b1 :: "'a::linorder" assumes "b1≤b2" and "b2≤b3" shows "{ f i | i. b1≤i ∧ i<b2 } ∪ { f i | i. b2≤i ∧ i<b3 } = {f i | i. b1≤i ∧ i<b3}" using assms apply - apply rule apply safe [] apply (rule_tac x=i in exI, auto) [] apply (rule_tac x=i in exI, auto) [] apply rule apply simp apply (elim exE, simp) apply (case_tac "i<b2") apply (rule disjI1) apply (rule_tac x=i in exI, auto) [] apply (rule disjI2) apply (rule_tac x=i in exI, auto) [] done text ‹ The standard library proves that a generalized union is finite if the index set is finite and if for every index the component set is itself finite. Conversely, we show that every component set must be finite when the union is finite. › lemma finite_UNION_then_finite: "finite (⋃(B ` A)) ⟹ a ∈ A ⟹ finite (B a)" by (metis Set.set_insert UN_insert Un_infinite) lemma finite_if_eq_beyond_finite: "finite S ⟹ finite {s. s - S = s' - S}" proof (rule finite_subset[where B="(λs. s ∪ (s' - S)) ` Pow S"], clarsimp) fix s have "s = (s ∩ S) ∪ (s - S)" by auto also assume "s - S = s' - S" finally show "s ∈ (λs. s ∪ (s' - S)) ` Pow S" by blast qed blast lemma distinct_finite_set: shows "finite {ys. set ys = x ∧ distinct ys}" (is "finite ?S") proof (cases "finite x") case False hence "{ys. set ys = x} = {}" by auto thus ?thesis by simp next case True show ?thesis proof (rule finite_subset) show "?S ⊆ {ys. set ys ⊆ x ∧ length ys ≤ card x}" using distinct_card by force from True show "finite ..." by (rule finite_lists_length_le) qed qed lemma finite_set_image: assumes f: "finite (set ` A)" and dist: "⋀xs. xs ∈ A ⟹ distinct xs" shows "finite A" proof (rule finite_subset) from f show "finite (set -` (set ` A) ∩ {xs. distinct xs})" proof (induct rule: finite_induct) case (insert x F) have "finite (set -` {x} ∩ {xs. distinct xs})" apply (simp add: vimage_def) by (metis Collect_conj_eq distinct_finite_set) with insert show ?case apply (subst vimage_insert) apply (subst Int_Un_distrib2) apply (rule finite_UnI) apply simp_all done qed simp moreover from dist show "A ⊆ ..." by (auto simp add: vimage_image_eq) qed subsubsection ‹Infinite Set› lemma INFM_nat_inductI: assumes P0: "P (0::nat)" assumes PS: "⋀i. P i ⟹ ∃j>i. P j ∧ Q j" shows "∃⇩_{∞}i. Q i" proof - have "∀i. ∃j>i. P j ∧ Q j" proof fix i show "∃j>i. P j ∧ Q j" apply (induction i) using PS[OF P0] apply auto [] by (metis PS Suc_lessI) qed thus ?thesis unfolding INFM_nat by blast qed subsection ‹Functions› lemma fun_neq_ext_iff: "m≠m' ⟷ (∃x. m x ≠ m' x)" by auto definition "inv_on f A x == SOME y. y∈A ∧ f y = x" lemma inv_on_f_f[simp]: "⟦inj_on f A; x∈A⟧ ⟹ inv_on f A (f x) = x" by (auto simp add: inv_on_def inj_on_def) lemma f_inv_on_f: "⟦ y∈f`A ⟧ ⟹ f (inv_on f A y) = y" by (auto simp add: inv_on_def intro: someI2) lemma inv_on_f_range: "⟦ y ∈ f`A ⟧ ⟹ inv_on f A y ∈ A" by (auto simp add: inv_on_def intro: someI2) lemma inj_on_map_inv_f [simp]: "⟦set l ⊆ A; inj_on f A⟧ ⟹ map (inv_on f A) (map f l) = l" apply (simp) apply (induct l) apply auto done lemma comp_cong_right: "x = y ⟹ f o x = f o y" by (simp) lemma comp_cong_left: "x = y ⟹ x o f = y o f" by (simp) lemma fun_comp_eq_conv: "f o g = fg ⟷ (∀x. f (g x) = fg x)" by auto abbreviation comp2 (infixl "oo" 55) where "f oo g ≡ λx. f o (g x)" abbreviation comp3 (infixl "ooo" 55) where "f ooo g ≡ λx. f oo (g x)" notation comp2 (infixl "∘∘" 55) and comp3 (infixl "∘∘∘" 55) definition [code_unfold, simp]: "swap_args2 f x y ≡ f y x" subsection ‹Multisets› (* The following is a syntax extension for multisets. Unfortunately, it depends on a change in the Library/Multiset.thy, so it is commented out here, until it will be incorporated into Library/Multiset.thy by its maintainers. The required change in Library/Multiset.thy is removing the syntax for single: - single :: "'a => 'a multiset" ("{#_#}") + single :: "'a => 'a multiset" And adding the following translations instead: + syntax + "_multiset" :: "args ⇒ 'a multiset" ("{#(_)#}") + translations + "{#x, xs#}" == "{#x#} + {#xs#}" + "{# x #}" == "single x" This translates "{# … #}" into a sum of singletons, that is parenthesized to the right. ?? Can we also achieve left-parenthesizing ?? *) (* Let's try what happens if declaring AC-rules for multiset union as simp-rules *) (*declare union_ac[simp] -- don't do it !*) lemma count_mset_set_finite_iff: "finite S ⟹ count (mset_set S) a = (if a ∈ S then 1 else 0)" by simp lemma ex_Melem_conv: "(∃x. x ∈# A) = (A ≠ {#})" by (simp add: ex_in_conv) subsubsection ‹Count› lemma count_ne_remove: "⟦ x ~= t⟧ ⟹ count S x = count (S-{#t#}) x" by (auto) lemma mset_empty_count[simp]: "(∀p. count M p = 0) = (M={#})" by (auto simp add: multiset_eq_iff) subsubsection ‹Union, difference and intersection› lemma size_diff_se: "t ∈# S ⟹ size S = size (S - {#t#}) + 1" proof (unfold size_multiset_overloaded_eq) let ?SIZE = "sum (count S) (set_mset S)" assume A: "t ∈# S" from A have SPLITPRE: "finite (set_mset S) & {t}⊆(set_mset S)" by auto hence "?SIZE = sum (count S) (set_mset S - {t}) + sum (count S) {t}" by (blast dest: sum.subset_diff) hence "?SIZE = sum (count S) (set_mset S - {t}) + count (S) t" by auto moreover with A have "count S t = count (S-{#t#}) t + 1" by auto ultimately have D: "?SIZE = sum (count S) (set_mset S - {t}) + count (S-{#t#}) t + 1" by (arith) moreover have "sum (count S) (set_mset S - {t}) = sum (count (S-{#t#})) (set_mset S - {t})" proof - have "∀x∈(set_mset S - {t}). count S x = count (S-{#t#}) x" by (auto iff add: count_ne_remove) thus ?thesis by simp qed ultimately have D: "?SIZE = sum (count (S-{#t#})) (set_mset S - {t}) + count (S-{#t#}) t + 1" by (simp) moreover { assume CASE: "t ∉# S - {#t#}" from CASE have "set_mset S - {t} = set_mset (S - {#t#})" by (auto simp add: in_diff_count split: if_splits) with CASE D have "?SIZE = sum (count (S-{#t#})) (set_mset (S - {#t#})) + 1" by (simp add: not_in_iff) } moreover { assume CASE: "t ∈# S - {#t#}" from CASE have "t ∈# S" by (rule in_diffD) with CASE have 1: "set_mset S = set_mset (S-{#t#})" by (auto simp add: in_diff_count split: if_splits) moreover from D have "?SIZE = sum (count (S-{#t#})) (set_mset S - {t}) + sum (count (S-{#t#})) {t} + 1" by simp moreover from SPLITPRE sum.subset_diff have "sum (count (S-{#t#})) (set_mset S) = sum (count (S-{#t#})) (set_mset S - {t}) + sum (count (S-{#t#})) {t}" by (blast) ultimately have "?SIZE = sum (count (S-{#t#})) (set_mset (S-{#t#})) + 1" by simp } ultimately show "?SIZE = sum (count (S-{#t#})) (set_mset (S - {#t#})) + 1" by blast qed (* TODO: Check whether this proof can be done simpler *) lemma mset_union_diff_comm: "t ∈# S ⟹ T + (S - {#t#}) = (T + S) - {#t#}" proof - assume "t ∈# S" then obtain S' where S: "S = add_mset t S'" by (metis insert_DiffM) then show ?thesis by auto qed (* lemma mset_diff_diff_left: "A-B-C = A-((B::'a multiset)+C)" proof - have "∀e . count (A-B-C) e = count (A-(B+C)) e" by auto thus ?thesis by (simp add: multiset_eq_conv_count_eq) qed lemma mset_diff_commute: "A-B-C = A-C-(B::'a multiset)" proof - have "A-B-C = A-(B+C)" by (simp add: mset_diff_diff_left) also have "… = A-(C+B)" by (simp add: union_commute) thus ?thesis by (simp add: mset_diff_diff_left) qed lemma mset_diff_same_empty[simp]: "(S::'a multiset) - S = {#}" proof - have "∀e . count (S-S) e = 0" by auto hence "∀e . ~ (e : set_mset (S-S))" by auto hence "set_mset (S-S) = {}" by blast thus ?thesis by (auto) qed *) lemma mset_right_cancel_union: "⟦a ∈# A+B; ~(a ∈# B)⟧ ⟹ a∈#A" by (simp) lemma mset_left_cancel_union: "⟦a ∈# A+B; ~(a ∈# A)⟧ ⟹ a∈#B" by (simp) lemmas mset_cancel_union = mset_right_cancel_union mset_left_cancel_union lemma mset_right_cancel_elem: "⟦a ∈# A+{#b#}; a~=b⟧ ⟹ a∈#A" by simp lemma mset_left_cancel_elem: "⟦a ∈# {#b#}+A; a~=b⟧ ⟹ a∈#A" by simp lemmas mset_cancel_elem = mset_right_cancel_elem mset_left_cancel_elem lemma mset_diff_cancel1elem[simp]: "~(a ∈# B) ⟹ {#a#}-B = {#a#}" by (auto simp add: not_in_iff intro!: multiset_eqI) (* lemma diff_union_inverse[simp]: "A + B - B = (A::'a multiset)" by (auto iff add: multiset_eq_conv_count_eq) lemma diff_union_inverse2[simp]: "B + A - B = (A::'a multiset)" by (auto iff add: multiset_eq_conv_count_eq) *) (*lemma union_diff_assoc_se2: "t ∈# A ⟹ (A+B)-{#t#} = (A-{#t#}) + B" by (auto iff add: multiset_eq_conv_count_eq) lemmas union_diff_assoc_se = union_diff_assoc_se1 union_diff_assoc_se2*) lemma union_diff_assoc: "C-B={#} ⟹ (A+B)-C = A + (B-C)" by (simp add: multiset_eq_iff) lemmas mset_neutral_cancel1 = union_left_cancel[where N="{#}", simplified] union_right_cancel[where N="{#}", simplified] declare mset_neutral_cancel1[simp] lemma mset_union_2_elem: "{#a, b#} = add_mset c M ⟹ {#a#}=M & b=c | a=c & {#b#}=M" by (auto simp: add_eq_conv_diff) lemma mset_un_cases[cases set, case_names left right]: "⟦a ∈# A + B; a ∈# A ⟹ P; a ∈# B ⟹ P⟧ ⟹ P" by (auto) lemma mset_unplusm_dist_cases[cases set, case_names left right]: assumes A: "{#s#}+A = B+C" assumes L: "⟦B={#s#}+(B-{#s#}); A=(B-{#s#})+C⟧ ⟹ P" assumes R: "⟦C={#s#}+(C-{#s#}); A=B+(C-{#s#})⟧ ⟹ P" shows P proof - from A[symmetric] have "s ∈# B+C" by simp thus ?thesis proof (cases rule: mset_un_cases) case left hence 1: "B={#s#}+(B-{#s#})" by simp with A have "{#s#}+A = {#s#}+((B-{#s#})+C)" by (simp add: union_ac) hence 2: "A = (B-{#s#})+C" by (simp) from L[OF 1 2] show ?thesis . next case right hence 1: "C={#s#}+(C-{#s#})" by simp with A have "{#s#}+A = {#s#}+(B+(C-{#s#}))" by (simp add: union_ac) hence 2: "A = B+(C-{#s#})" by (simp) from R[OF 1 2] show ?thesis . qed qed lemma mset_unplusm_dist_cases2[cases set, case_names left right]: assumes A: "B+C = {#s#}+A" assumes L: "⟦B={#s#}+(B-{#s#}); A=(B-{#s#})+C⟧ ⟹ P" assumes R: "⟦C={#s#}+(C-{#s#}); A=B+(C-{#s#})⟧ ⟹ P" shows P using mset_unplusm_dist_cases[OF A[symmetric]] L R by blast lemma mset_single_cases[cases set, case_names loc env]: assumes A: "add_mset s c = add_mset r' c'" assumes CASES: "⟦s=r'; c=c'⟧ ⟹ P" "⟦c'={#s#}+(c'-{#s#}); c={#r'#}+(c-{#r'#}); c-{#r'#} = c'-{#s#} ⟧ ⟹ P" shows "P" proof - { assume CASE: "s=r'" with A have "c=c'" by simp with CASE CASES have ?thesis by auto } moreover { assume CASE: "s≠r'" have "s ∈# {#s#}+c" by simp with A have "s ∈# {#r'#}+c'" by simp with CASE have "s ∈# c'" by simp hence 1: "c' = {#s#} + (c' - {#s#})" by simp with A have "{#s#}+c = {#s#}+({#r'#}+(c' - {#s#}))" by (auto simp add: union_ac) hence 2: "c={#r'#}+(c' - {#s#})" by (auto) hence 3: "c-{#r'#} = (c' - {#s#})" by auto from 1 2 3 CASES have ?thesis by auto } ultimately show ?thesis by blast qed lemma mset_single_cases'[cases set, case_names loc env]: assumes A: "add_mset s c = add_mset r' c'" assumes CASES: "⟦s=r'; c=c'⟧ ⟹ P" "!!cc. ⟦c'={#s#}+cc; c={#r'#}+cc; c'-{#s#}=cc; c-{#r'#}=cc⟧ ⟹ P" shows "P" using A CASES by (auto elim!: mset_single_cases) lemma mset_single_cases2[cases set, case_names loc env]: assumes A: "add_mset s c = add_mset r' c'" assumes CASES: "⟦s=r'; c=c'⟧ ⟹ P" "⟦c'=(c'-{#s#})+{#s#}; c=(c-{#r'#})+{#r'#}; c-{#r'#} = c'-{#s#} ⟧ ⟹ P" shows "P" proof - from A have "add_mset s c = add_mset r' c'" by (simp add: union_ac) thus ?thesis using CASES by (cases rule: mset_single_cases) simp_all qed lemma mset_single_cases2'[cases set, case_names loc env]: assumes A: "add_mset s c = add_mset r' c'" assumes CASES: "⟦s=r'; c=c'⟧ ⟹ P" "!!cc. ⟦c'=cc+{#s#}; c=cc+{#r'#}; c'-{#s#}=cc; c-{#r'#}=cc⟧ ⟹ P" shows "P" using A CASES by (auto elim!: mset_single_cases2) lemma mset_un_single_un_cases [consumes 1, case_names left right]: assumes A: "add_mset a A = B + C" and CASES: "a ∈# B ⟹ A = (B - {#a#}) + C ⟹ P" "a ∈# C ⟹ A = B + (C - {#a#}) ⟹ P" shows "P" proof - have "a ∈# A+{#a#}" by simp with A have "a ∈# B+C" by auto thus ?thesis proof (cases rule: mset_un_cases) case left hence "B=B-{#a#}+{#a#}" by auto with A have "A+{#a#} = (B-{#a#})+C+{#a#}" by (auto simp add: union_ac) hence "A=(B-{#a#})+C" by simp with CASES(1)[OF left] show ?thesis by blast next case right hence "C=C-{#a#}+{#a#}" by auto with A have "A+{#a#} = B+(C-{#a#})+{#a#}" by (auto simp add: union_ac) hence "A=B+(C-{#a#})" by simp with CASES(2)[OF right] show ?thesis by blast qed qed (* TODO: Can this proof be done more automatically ? *) lemma mset_distrib[consumes 1, case_names dist]: assumes A: "(A::'a multiset)+B = M+N" "!!Am An Bm Bn. ⟦A=Am+An; B=Bm+Bn; M=Am+Bm; N=An+Bn⟧ ⟹ P" shows "P" proof - have BN_MA: "B - N = M - A" by (metis (no_types) add_diff_cancel_right assms(1) union_commute) have H: "A = A∩# C + (A - C) ∩# D" if "A + B = C + D" for A B C D :: "'a multiset" by (metis add.commute diff_intersect_left_idem mset_subset_eq_add_left subset_eq_diff_conv subset_mset.add_diff_inverse subset_mset.inf_absorb1 subset_mset.inf_le1 that) have A': "A = A∩# M + (A - M) ∩# N" using A(1) H by blast moreover have B': "B = (B - N) ∩# M + B∩# N" using A(1) H[of B A N M] by (auto simp: ac_simps) moreover have "M = A ∩# M + (B - N) ∩# M" using H[of M N A B] BN_MA[symmetric] A(1) by (metis (no_types) diff_intersect_left_idem diff_union_cancelR multiset_inter_commute subset_mset.diff_add subset_mset.inf.cobounded1 union_commute) moreover have "N = (A - M) ∩# N + B ∩# N" by (metis A' assms(1) diff_union_cancelL inter_union_distrib_left inter_union_distrib_right mset_subset_eq_multiset_union_diff_commute subset_mset.inf.cobounded1 subset_mset.inf.commute) ultimately show P using A(2) by blast qed subsubsection ‹Singleton multisets› lemma mset_size_le1_cases[case_names empty singleton,consumes 1]: "⟦ size M ≤ Suc 0; M={#} ⟹ P; !!m. M={#m#} ⟹ P ⟧ ⟹ P" by (cases M) auto lemma diff_union_single_conv2: "a ∈# J ⟹ J + I - {#a#} = (J - {#a#}) + I" by simp lemmas diff_union_single_convs = diff_union_single_conv diff_union_single_conv2 lemma mset_contains_eq: "(m ∈# M) = ({#m#}+(M-{#m#})=M)" using diff_single_trivial by fastforce subsubsection ‹Pointwise ordering› (*declare mset_le_trans[trans] Seems to be in there now. Why is this not done in Multiset.thy or order-class ? *) lemma mset_le_incr_right1: "a⊆#(b::'a multiset) ⟹ a⊆#b+c" using mset_subset_eq_mono_add[of a b "{#}" c, simplified] . lemma mset_le_incr_right2: "a⊆#(b::'a multiset) ⟹ a⊆#c+b" using mset_le_incr_right1 by (auto simp add: union_commute) lemmas mset_le_incr_right = mset_le_incr_right1 mset_le_incr_right2 lemma mset_le_decr_left1: "a+c⊆#(b::'a multiset) ⟹ a⊆#b" using mset_le_incr_right1 mset_subset_eq_mono_add_right_cancel by blast lemma mset_le_decr_left2: "c+a⊆#(b::'a multiset) ⟹ a⊆#b" using mset_le_decr_left1 by (auto simp add: union_ac) lemma mset_le_add_mset_decr_left1: "add_mset c a⊆#(b::'a multiset) ⟹ a⊆#b" by (simp add: mset_subset_eq_insertD subset_mset.dual_order.strict_implies_order) lemma mset_le_add_mset_decr_left2: "add_mset c a⊆#(b::'a multiset) ⟹ {#c#}⊆#b" by (simp add: mset_subset_eq_insertD subset_mset.dual_order.strict_implies_order) lemmas mset_le_decr_left = mset_le_decr_left1 mset_le_decr_left2 mset_le_add_mset_decr_left1 mset_le_add_mset_decr_left2 lemma mset_le_subtract: "A⊆#B ⟹ A-C ⊆# B-(C::'a multiset)" apply (unfold subseteq_mset_def count_diff) apply clarify apply (subgoal_tac "count A a ≤ count B a") apply arith apply simp done lemma mset_union_subset: "A+B ⊆# C ⟹ A⊆#C ∧ B⊆#(C::'a multiset)" by (auto dest: mset_le_decr_left) lemma mset_le_add_mset: "add_mset x B ⊆# C ⟹ {#x#}⊆#C ∧ B⊆#(C::'a multiset)" by (auto dest: mset_le_decr_left) lemma mset_le_subtract_left: "A+B ⊆# (X::'a multiset) ⟹ B ⊆# X-A ∧ A⊆#X" by (auto dest: mset_le_subtract[of "A+B" "X" "A"] mset_union_subset) lemma mset_le_subtract_right: "A+B ⊆# (X::'a multiset) ⟹ A ⊆# X-B ∧ B⊆#X" by (auto dest: mset_le_subtract[of "A+B" "X" "B"] mset_union_subset) lemma mset_le_subtract_add_mset_left: "add_mset x B ⊆# (X::'a multiset) ⟹ B ⊆# X-{#x#} ∧ {#x#}⊆#X" by (auto dest: mset_le_subtract[of "add_mset x B" "X" "{#x#}"] mset_le_add_mset) lemma mset_le_subtract_add_mset_right: "add_mset x B ⊆# (X::'a multiset) ⟹ {#x#} ⊆# X-B ∧ B⊆#X" by (auto dest: mset_le_subtract[of "add_mset x B" "X" "B"] mset_le_add_mset) lemma mset_le_addE: "⟦ xs ⊆# (ys::'a multiset); !!zs. ys=xs+zs ⟹ P ⟧ ⟹ P" using mset_subset_eq_exists_conv by blast declare subset_mset.diff_add[simp, intro] lemma mset_2dist2_cases: assumes A: "{#a#}+{#b#} ⊆# A+B" assumes CASES: "{#a#}+{#b#} ⊆# A ⟹ P" "{#a#}+{#b#} ⊆# B ⟹ P" "⟦a ∈# A; b ∈# B⟧ ⟹ P" "⟦a ∈# B; b ∈# A⟧ ⟹ P" shows "P" proof - { assume C: "a ∈# A" "b ∈# A-{#a#}" with mset_subset_eq_mono_add[of "{#a#}" "{#a#}" "{#b#}" "A-{#a#}"] have "{#a#}+{#b#} ⊆# A" by auto } moreover { assume C: "a ∈# A" "¬ (b ∈# A-{#a#})" with A have "b ∈# B" by (metis diff_union_single_conv2 mset_le_subtract_left mset_subset_eq_insertD mset_un_cases) } moreover { assume C: "¬ (a ∈# A)" "b ∈# B-{#a#}" with A have "a ∈# B" by (auto dest: mset_subset_eqD) with C mset_subset_eq_mono_add[of "{#a#}" "{#a#}" "{#b#}" "B-{#a#}"] have "{#a#}+{#b#} ⊆# B" by auto } moreover { assume C: "¬ (a ∈# A)" "¬ (b ∈# B-{#a#})" with A have "a ∈# B ∧ b ∈# A" apply (intro conjI) apply (auto dest!: mset_subset_eq_insertD simp: insert_union_subset_iff; fail)[] by (metis mset_diff_cancel1elem mset_le_subtract_left multiset_diff_union_assoc single_subset_iff subset_eq_diff_conv) } ultimately show P using CASES by blast qed lemma mset_union_subset_s: "{#a#}+B ⊆# C ⟹ a ∈# C ∧ B ⊆# C" by (drule mset_union_subset) simp (* TODO: Check which of these lemmas are already introduced by order-classes ! *) lemma mset_le_single_cases[consumes 1, case_names empty singleton]: "⟦M⊆#{#a#}; M={#} ⟹ P; M={#a#} ⟹ P⟧ ⟹ P" by (induct M) auto lemma mset_le_distrib[consumes 1, case_names dist]: "⟦(X::'a multiset)⊆#A+B; !!Xa Xb. ⟦X=Xa+Xb; Xa⊆#A; Xb⊆#B⟧ ⟹ P ⟧ ⟹ P" by (auto elim!: mset_le_addE mset_distrib) lemma mset_le_mono_add_single: "⟦a ∈# ys; b ∈# ws⟧ ⟹ {#a#} + {#b#} ⊆# ys + ws" by (meson mset_subset_eq_mono_add single_subset_iff) lemma mset_size1elem: "⟦size P ≤ 1; q ∈# P⟧ ⟹ P={#q#}" by (auto elim: mset_size_le1_cases) lemma mset_size2elem: "⟦size P ≤ 2; {#q#}+{#q'#} ⊆# P⟧ ⟹ P={#q#}+{#q'#}" by (auto elim: mset_le_addE) subsubsection ‹Image under function› notation image_mset (infixr "`#" 90) lemma mset_map_split_orig: "!!M1 M2. ⟦f `# P = M1+M2; !!P1 P2. ⟦P=P1+P2; f `# P1 = M1; f `# P2 = M2⟧ ⟹ Q ⟧ ⟹ Q" by (induct P) (force elim!: mset_un_single_un_cases)+ lemma mset_map_id: "⟦!!x. f (g x) = x⟧ ⟹ f `# g `# X = X" by (induct X) auto text ‹The following is a very specialized lemma. Intuitively, it splits the original multiset by a splitting of some pointwise supermultiset of its image. Application: This lemma came in handy when proving the correctness of a constraint system that collects at most k sized submultisets of the sets of spawned threads. › lemma mset_map_split_orig_le: assumes A: "f `# P ⊆# M1+M2" and EX: "!!P1 P2. ⟦P=P1+P2; f `# P1 ⊆# M1; f `# P2 ⊆# M2⟧ ⟹ Q" shows "Q" using A EX by (auto elim: mset_le_distrib mset_map_split_orig) subsection ‹Lists› lemma len_greater_imp_nonempty[simp]: "length l > x ⟹ l≠[]" by auto lemma list_take_induct_tl2: "⟦length xs = length ys; ∀n<length xs. P (ys ! n) (xs ! n)⟧ ⟹ ∀n < length (tl xs). P ((tl ys) ! n) ((tl xs) ! n)" by (induct xs ys rule: list_induct2) auto lemma not_distinct_split_distinct: assumes "¬ distinct xs" obtains y ys zs where "distinct ys" "y ∈ set ys" "xs = ys@[y]@zs" using assms proof (induct xs rule: rev_induct) case Nil thus ?case by simp next case (snoc x xs) thus ?case by (cases "distinct xs") auto qed lemma distinct_length_le: assumes d: "distinct ys" and eq: "set ys = set xs" shows "length ys ≤ length xs" proof - from d have "length ys = card (set ys)" by (simp add: distinct_card) also from eq List.card_set have "card (set ys) = length (remdups xs)" by simp also have "... ≤ length xs" by simp finally show ?thesis . qed lemma find_SomeD: "List.find P xs = Some x ⟹ P x" "List.find P xs = Some x ⟹ x∈set xs" by (auto simp add: find_Some_iff) lemma in_hd_or_tl_conv[simp]: "l≠[] ⟹ x=hd l ∨ x∈set (tl l) ⟷ x∈set l" by (cases l) auto lemma length_dropWhile_takeWhile: assumes "x < length (dropWhile P xs)" shows "x + length (takeWhile P xs) < length xs" using assms by (induct xs) auto text ‹Elim-version of @{thm neq_Nil_conv}.› lemma neq_NilE: assumes "l≠[]" obtains x xs where "l=x#xs" using assms by (metis list.exhaust) lemma length_Suc_rev_conv: "length xs = Suc n ⟷ (∃ys y. xs=ys@[y] ∧ length ys = n)" by (cases xs rule: rev_cases) auto subsubsection ‹List Destructors› lemma not_hd_in_tl: "x ≠ hd xs ⟹ x ∈ set xs ⟹ x ∈ set (tl xs)" by (induct xs) simp_all lemma distinct_hd_tl: "distinct xs ⟹ x = hd xs ⟹ x ∉ set (tl (xs))" by (induct xs) simp_all lemma in_set_tlD: "x ∈ set (tl xs) ⟹ x ∈ set xs" by (induct xs) simp_all lemma nth_tl: "xs ≠ [] ⟹ tl xs ! n = xs ! Suc n" by (induct xs) simp_all lemma tl_subset: "xs ≠ [] ⟹ set xs ⊆ A ⟹ set (tl xs) ⊆ A" by (metis in_set_tlD rev_subsetD subsetI) lemma tl_last: "tl xs ≠ [] ⟹ last xs = last (tl xs)" by (induct xs) simp_all lemma tl_obtain_elem: assumes "xs ≠ []" "tl xs = []" obtains e where "xs = [e]" using assms by (induct xs rule: list_nonempty_induct) simp_all lemma butlast_subset: "xs ≠ [] ⟹ set xs ⊆ A ⟹ set (butlast xs) ⊆ A" by (metis in_set_butlastD rev_subsetD subsetI) lemma butlast_rev_tl: "xs ≠ [] ⟹ butlast (rev xs) = rev (tl xs)" by (induct xs rule: rev_induct) simp_all lemma hd_butlast: "length xs > 1 ⟹ hd (butlast xs) = hd xs" by (induct xs) simp_all lemma butlast_upd_last_eq[simp]: "length l ≥ 2 ⟹ (butlast l) [ length l - 2 := x ] = take (length l - 2) l @ [x]" apply (case_tac l rule: rev_cases) apply simp apply simp apply (case_tac ys rule: rev_cases) apply simp apply simp done lemma distinct_butlast_swap[simp]: "distinct pq ⟹ distinct (butlast (pq[i := last pq]))" apply (cases pq rule: rev_cases) apply (auto simp: list_update_append distinct_list_update split: nat.split) done subsubsection ‹Splitting list according to structure of other list› context begin private definition "SPLIT_ACCORDING m l ≡ length l = length m" private lemma SPLIT_ACCORDINGE: assumes "length m = length l" obtains "SPLIT_ACCORDING m l" unfolding SPLIT_ACCORDING_def using assms by auto private lemma SPLIT_ACCORDING_simp: "SPLIT_ACCORDING m (l1@l2) ⟷ (∃m1 m2. m=m1@m2 ∧ SPLIT_ACCORDING m1 l1 ∧ SPLIT_ACCORDING m2 l2)" "SPLIT_ACCORDING m (x#l') ⟷ (∃y m'. m=y#m' ∧ SPLIT_ACCORDING m' l')" apply (fastforce simp: SPLIT_ACCORDING_def intro: exI[where x = "take (length l1) m"] exI[where x = "drop (length l1) m"]) apply (cases m;auto simp: SPLIT_ACCORDING_def) done text ‹Split structure of list @{term m} according to structure of list @{term l}.› method split_list_according for m :: "'a list" and l :: "'b list" = (rule SPLIT_ACCORDINGE[of m l], (simp; fail), ( simp only: SPLIT_ACCORDING_simp, elim exE conjE, simp only: SPLIT_ACCORDING_def ) ) end subsubsection ‹‹list_all2›› lemma list_all2_induct[consumes 1, case_names Nil Cons]: assumes "list_all2 P l l'" assumes "Q [] []" assumes "⋀x x' ls ls'. ⟦ P x x'; list_all2 P ls ls'; Q ls ls' ⟧ ⟹ Q (x#ls) (x'#ls')" shows "Q l l'" using list_all2_lengthD[OF assms(1)] assms apply (induct rule: list_induct2) apply auto done subsubsection ‹Indexing› lemma ran_nth_set_encoding_conv[simp]: "ran (λi. if i<length l then Some (l!i) else None) = set l" apply safe apply (auto simp: ran_def split: if_split_asm) [] apply (auto simp: in_set_conv_nth intro: ranI) [] done lemma nth_image_indices[simp]: "(!) l ` {0..<length l} = set l" by (auto simp: in_set_conv_nth) lemma nth_update_invalid[simp]:"¬i<length l ⟹ l[j:=x]!i = l!i" apply (induction l arbitrary: i j) apply (auto split: nat.splits) done lemma nth_list_update': "l[i:=x]!j = (if i=j ∧ i<length l then x else l!j)" by auto lemma last_take_nth_conv: "n ≤ length l ⟹ n≠0 ⟹ last (take n l) = l!(n - 1)" apply (induction l arbitrary: n) apply (auto simp: take_Cons split: nat.split) done lemma nth_append_first[simp]: "i<length l ⟹ (l@l')!i = l!i" by (simp add: nth_append) lemma in_set_image_conv_nth: "f x ∈ f`set l ⟷ (∃i<length l. f (l!i) = f x)" by (auto simp: in_set_conv_nth) (metis image_eqI nth_mem) lemma set_image_eq_pointwiseI: assumes "length l = length l'" assumes "⋀i. i<length l ⟹ f (l!i) = f (l'!i)" shows "f`set l = f`set l'" using assms by (fastforce simp: in_set_conv_nth in_set_image_conv_nth) lemma insert_swap_set_eq: "i<length l ⟹ insert (l!i) (set (l[i:=x])) = insert x (set l)" by (auto simp: in_set_conv_nth nth_list_update split: if_split_asm) subsubsection ‹Reverse lists› lemma neq_Nil_revE: assumes "l≠[]" obtains ll e where "l = ll@[e]" using assms by (cases l rule: rev_cases) auto lemma neq_Nil_rev_conv: "l≠[] ⟷ (∃xs x. l = xs@[x])" by (cases l rule: rev_cases) auto text ‹Caution: Same order of case variables in snoc-case as @{thm [source] rev_exhaust}, the other way round than @{thm [source] rev_induct} !› lemma length_compl_rev_induct[case_names Nil snoc]: "⟦P []; !! l e . ⟦!! ll . length ll <= length l ⟹ P ll⟧ ⟹ P (l@[e])⟧ ⟹ P l" apply(induct_tac l rule: length_induct) apply(case_tac "xs" rule: rev_cases) apply(auto) done lemma list_append_eq_Cons_cases[consumes 1]: "⟦ys@zs = x#xs; ⟦ys=[]; zs=x#xs⟧ ⟹ P; !!ys'. ⟦ ys=x#ys'; ys'@zs=xs ⟧ ⟹ P ⟧ ⟹ P" by (auto iff add: append_eq_Cons_conv) lemma list_Cons_eq_append_cases[consumes 1]: "⟦x#xs = ys@zs; ⟦ys=[]; zs=x#xs⟧ ⟹ P; !!ys'. ⟦ ys=x#ys'; ys'@zs=xs ⟧ ⟹ P ⟧ ⟹ P" by (auto iff add: Cons_eq_append_conv) lemma map_of_rev_distinct[simp]: "distinct (map fst m) ⟹ map_of (rev m) = map_of m" apply (induct m) apply simp apply simp apply (subst map_add_comm) apply force apply simp done ― ‹Tail-recursive, generalized @{const rev}. May also be used for tail-recursively getting a list with all elements of the two operands, if the order does not matter, e.g. when implementing sets by lists.› fun revg where "revg [] b = b" | "revg (a#as) b = revg as (a#b)" lemma revg_fun[simp]: "revg a b = rev a @ b" by (induct a arbitrary: b) auto lemma rev_split_conv[simp]: "l ≠ [] ⟹ rev (tl l) @ [hd l] = rev l" by (induct l) simp_all lemma rev_butlast_is_tl_rev: "rev (butlast l) = tl (rev l)" by (induct l) auto lemma hd_last_singletonI: "⟦xs ≠ []; hd xs = last xs; distinct xs⟧ ⟹ xs = [hd xs]" by (induct xs rule: list_nonempty_induct) auto lemma last_filter: "⟦xs ≠ []; P (last xs)⟧ ⟹ last (filter P xs) = last xs" by (induct xs rule: rev_nonempty_induct) simp_all (* As the following two rules are similar in nature to list_induct2', they are named accordingly. *) lemma rev_induct2' [case_names empty snocl snocr snoclr]: assumes empty: "P [] []" and snocl: "⋀x xs. P (xs@[x]) []" and snocr: "⋀y ys. P [] (ys@[y])" and snoclr: "⋀x xs y ys. P xs ys ⟹ P (xs@[x]) (ys@[y])" shows "P xs ys" proof (induct xs arbitrary: ys rule: rev_induct) case Nil thus ?case using empty snocr by (cases ys rule: rev_exhaust) simp_all next case snoc thus ?case using snocl snoclr by (cases ys rule: rev_exhaust) simp_all qed lemma rev_nonempty_induct2' [case_names single snocl snocr snoclr, consumes 2]: assumes "xs ≠ []" "ys ≠ []" assumes single': "⋀x y. P [x] [y]" and snocl: "⋀x xs y. xs ≠ [] ⟹ P (xs@[x]) [y]" and snocr: "⋀x y ys. ys ≠ [] ⟹ P [x] (ys@[y])" and snoclr: "⋀x xs y ys. ⟦P xs ys; xs ≠ []; ys≠[]⟧ ⟹ P (xs@[x]) (ys@[y])" shows "P xs ys" using assms(1,2) proof (induct xs arbitrary: ys rule: rev_nonempty_induct) case single then obtain z zs where "ys = zs@[z]" by (metis rev_exhaust) thus ?case using single' snocr by (cases "zs = []") simp_all next case (snoc x xs) then obtain z zs where zs: "ys = zs@[z]" by (metis rev_exhaust) thus ?case using snocl snoclr snoc by (cases "zs = []") simp_all qed subsubsection "Folding" text "Ugly lemma about foldl over associative operator with left and right neutral element" lemma foldl_A1_eq: "!!i. ⟦ !! e. f n e = e; !! e. f e n = e; !!a b c. f a (f b c) = f (f a b) c ⟧ ⟹ foldl f i ww = f i (foldl f n ww)" proof (induct ww) case Nil thus ?case by simp next case (Cons a ww i) note IHP[simplified]=this have "foldl f i (a # ww) = foldl f (f i a) ww" by simp also from IHP have "… = f (f i a) (foldl f n ww)" by blast also from IHP(4) have "… = f i (f a (foldl f n ww))" by simp also from IHP(1)[OF IHP(2,3,4), where i=a] have "… = f i (foldl f a ww)" by simp also from IHP(2)[of a] have "… = f i (foldl f (f n a) ww)" by simp also have "… = f i (foldl f n (a#ww))" by simp finally show ?case . qed lemmas foldl_conc_empty_eq = foldl_A1_eq[of "(@)" "[]", simplified] lemmas foldl_un_empty_eq = foldl_A1_eq[of "(∪)" "{}", simplified, OF Un_assoc[symmetric]] lemma foldl_set: "foldl (∪) {} l = ⋃{x. x∈set l}" apply (induct l) apply simp_all apply (subst foldl_un_empty_eq) apply auto done lemma (in monoid_mult) foldl_absorb1: "x*foldl (*) 1 zs = foldl (*) x zs" apply (rule sym) apply (rule foldl_A1_eq) apply (auto simp add: mult.assoc) done text ‹Towards an invariant rule for foldl› lemma foldl_rule_aux: fixes I :: "'σ ⇒ 'a list ⇒ bool" assumes initial: "I σ0 l0" assumes step: "!!l1 l2 x σ. ⟦ l0=l1@x#l2; I σ (x#l2) ⟧ ⟹ I (f σ x) l2" shows "I (foldl f σ0 l0) []" using initial step apply (induct l0 arbitrary: σ0) apply auto done lemma foldl_rule_aux_P: fixes I :: "'σ ⇒ 'a list ⇒ bool" assumes initial: "I σ0 l0" assumes step: "!!l1 l2 x σ. ⟦ l0=l1@x#l2; I σ (x#l2) ⟧ ⟹ I (f σ x) l2" assumes final: "!!σ. I σ [] ⟹ P σ" shows "P (foldl f σ0 l0)" using foldl_rule_aux[of I σ0 l0, OF initial, OF step] final by simp lemma foldl_rule: fixes I :: "'σ ⇒ 'a list ⇒ 'a list ⇒ bool" assumes initial: "I σ0 [] l0" assumes step: "!!l1 l2 x σ. ⟦ l0=l1@x#l2; I σ l1 (x#l2) ⟧ ⟹ I (f σ x) (l1@[x]) l2" shows "I (foldl f σ0 l0) l0 []" using initial step apply (rule_tac I="λσ lr. ∃ll. l0=ll@lr ∧ I σ ll lr" in foldl_rule_aux_P) apply auto done text ‹ Invariant rule for foldl. The invariant is parameterized with the state, the list of items that have already been processed and the list of items that still have to be processed. › lemma foldl_rule_P: fixes I :: "'σ ⇒ 'a list ⇒ 'a list ⇒ bool" ― ‹The invariant holds for the initial state, no items processed yet and all items to be processed:› assumes initial: "I σ0 [] l0" ― ‹The invariant remains valid if one item from the list is processed› assumes step: "!!l1 l2 x σ. ⟦ l0=l1@x#l2; I σ l1 (x#l2) ⟧ ⟹ I (f σ x) (l1@[x]) l2" ― ‹The proposition follows from the invariant in the final state, i.e. all items processed and nothing to be processed› assumes final: "!!σ. I σ l0 [] ⟹ P σ" shows "P (foldl f σ0 l0)" using foldl_rule[of I, OF initial step] by (simp add: final) text ‹Invariant reasoning over @{const foldl} for distinct lists. Invariant rule makes no assumptions about ordering.› lemma distinct_foldl_invar: "⟦ distinct S; I (set S) σ0; ⋀x it σ. ⟦x ∈ it; it ⊆ set S; I it σ⟧ ⟹ I (it - {x}) (f σ x) ⟧ ⟹ I {} (foldl f σ0 S)" proof (induct S arbitrary: σ0) case Nil thus ?case by auto next case (Cons x S) note [simp] = Cons.prems(1)[simplified] show ?case apply simp apply (rule Cons.hyps) proof - from Cons.prems(1) show "distinct S" by simp from Cons.prems(3)[of x "set (x#S)", simplified, OF Cons.prems(2)[simplified]] show "I (set S) (f σ0 x)" . fix xx it σ assume A: "xx∈it" "it ⊆ set S" "I it σ" show "I (it - {xx}) (f σ xx)" using A(2) apply (rule_tac Cons.prems(3)) apply (simp_all add: A(1,3)) apply blast done qed qed lemma foldl_length_aux: "foldl (λi x. Suc i) a l = a + length l" by (induct l arbitrary: a) auto lemmas foldl_length[simp] = foldl_length_aux[where a=0, simplified] lemma foldr_length_aux: "foldr (λx i. Suc i) l a = a + length l" by (induct l arbitrary: a rule: rev_induct) auto lemmas foldr_length[simp] = foldr_length_aux[where a=0, simplified] context comp_fun_commute begin lemma foldl_f_commute: "f a (foldl (λa b. f b a) b xs) = foldl (λa b. f b a) (f a b) xs" by(induct xs arbitrary: b)(simp_all add: fun_left_comm) lemma foldr_conv_foldl: "foldr f xs a = foldl (λa b. f b a) a xs" by(induct xs arbitrary: a)(simp_all add: foldl_f_commute) end lemma filter_conv_foldr: "filter P xs = foldr (λx xs. if P x then x # xs else xs) xs []" by(induct xs) simp_all lemma foldr_Cons: "foldr Cons xs [] = xs" by(induct xs) simp_all lemma foldr_snd_zip: "length xs ≥ length ys ⟹ foldr (λ(x, y). f y) (zip xs ys) b = foldr f ys b" proof(induct ys arbitrary: xs) case (Cons y ys) thus ?case by(cases xs) simp_all qed simp lemma foldl_snd_zip: "length xs ≥ length ys ⟹ foldl (λb (x, y). f b y) b (zip xs ys) = foldl f b ys" proof(induct ys arbitrary: xs b) case (Cons y ys) thus ?case by(cases xs) simp_all qed simp lemma fst_foldl: "fst (foldl (λ(a, b) x. (f a x, g a b x)) (a, b) xs) = foldl f a xs" by(induct xs arbitrary: a b) simp_all lemma foldl_foldl_conv_concat: "foldl (foldl f) a xs = foldl f a (concat xs)" by(induct xs arbitrary: a) simp_all lemma foldl_list_update: "n < length xs ⟹ foldl f a (xs[n := x]) = foldl f (f (foldl f a (take n xs)) x) (drop (Suc n) xs)" by(simp add: upd_conv_take_nth_drop) lemma map_by_foldl: fixes l :: "'a list" and f :: "'a ⇒ 'b" shows "foldl (λl x. l@[f x]) [] l = map f l" proof - { fix l' have "foldl (λl x. l@[f x]) l' l = l'@map f l" by (induct l arbitrary: l') auto } thus ?thesis by simp qed subsubsection ‹Sorting› lemma sorted_in_between: assumes A: "0≤i" "i<j" "j<length l" assumes S: "sorted l" assumes E: "l!i ≤ x" "x<l!j" obtains k where "i≤k" and "k<j" and "l!k≤x" and "x<l!(k+1)" proof - from A E have "∃k. i≤k ∧ k<j ∧ l!k≤x ∧ x<l!(k+1)" proof (induct "j-i" arbitrary: i j) case (Suc d) show ?case proof (cases "l!(i+1) ≤ x") case True from True Suc.hyps have "d = j - (i + 1)" by simp moreover from True have "i+1 < j" by (metis Suc.prems Suc_eq_plus1 Suc_lessI not_less) moreover from True have "0≤i+1" by simp ultimately obtain k where "i+1≤k" "k<j" "l!k ≤ x" "x<l!(k+1)" using Suc.hyps(1)[of j "i+1"] Suc.prems True by auto thus ?thesis by (auto dest: Suc_leD) next case False show ?thesis proof (cases "x<(l!(j - 1))") case True from True Suc.hyps have "d = j - (i + 1)" by simp moreover from True Suc.prems have "i < j - 1" by (metis Suc_eq_plus1 Suc_lessI diff_Suc_1 less_diff_conv not_le) moreover from True Suc.prems have "j - 1 < length l" by simp ultimately obtain k where "i≤k" "k<j - 1" "l!k ≤ x" "x<l!(k+1)" using Suc.hyps(1)[of "j - 1" i] Suc.prems True by auto thus ?thesis by (auto dest: Suc_leD) next case False thus ?thesis using Suc apply clarsimp by (metis Suc_leI add_0_iff add_diff_inverse diff_Suc_1 le_add2 lessI not0_implies_Suc not_less) qed qed qed simp thus ?thesis by (blast intro: that) qed lemma sorted_hd_last: "⟦sorted l; l≠[]⟧ ⟹ hd l ≤ last l" by (metis eq_iff hd_Cons_tl last_in_set not_hd_in_tl sorted_wrt.simps(2)) lemma (in linorder) sorted_hd_min: "⟦xs ≠ []; sorted xs⟧ ⟹ ∀x ∈ set xs. hd xs ≤ x" by (induct xs, auto) lemma sorted_append_bigger: "⟦sorted xs; ∀x ∈ set xs. x ≤ y⟧ ⟹ sorted (xs @ [y])" proof (induct xs) case Nil then show ?case by simp next case (Cons x xs) then have s: "sorted xs" by (cases xs) simp_all from Cons have a: "∀x∈set xs. x ≤ y" by simp from Cons(1)[OF s a] Cons(2-) show ?case by (cases xs) simp_all qed lemma sorted_filter': "sorted l ⟹ sorted (filter P l)" using sorted_filter[where f=id, simplified] . subsubsection ‹Map› (* List.thy has: declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!] *) lemma map_eq_consE: "⟦map f ls = fa#fl; !!a l. ⟦ ls=a#l; f a=fa; map f l = fl ⟧ ⟹ P⟧ ⟹ P" by auto lemma map_fst_mk_snd[simp]: "map fst (map (λx. (x,k)) l) = l" by (induct l) auto lemma map_snd_mk_fst[simp]: "map snd (map (λx. (k,x)) l) = l" by (induct l) auto lemma map_fst_mk_fst[simp]: "map fst (map (λx. (k,x)) l) = replicate (length l) k" by (induct l) auto lemma map_snd_mk_snd[