UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
lemma not_extf_gt_nil_singleton_if_\<delta>\<^sub>h_eq_\<epsilon>\<^sub>h: assumes wary_s: "wary s" and \<delta>_eq_\<epsilon>: "\<delta>\<^sub>h = \<epsilon>\<^sub>h" shows "\<not> extf f (>\<^sub>t) [] [s]"
lemma transpose_one[simp]: "transpose_mat (1\<^sub>m n) = (1\<^sub>m n)"
lemma eSuc_Sup: "A \<noteq> {} \<Longrightarrow> eSuc (Sup A) = Sup (eSuc ` A)"
lemma (in Ring) msub_submodule:"\<lbrakk>R module M; R module M1; msubmodule R M M1\<rbrakk> \<Longrightarrow> submodule R M (carrier M1)"
lemma big_sum_in_bigo: assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> O[F](g)" shows "(\<lambda>x. sum (\<lambda>y. f y x) A) \<in> O[F](g)"
lemma decomp_eigenvector: fixes A::"complex Matrix.mat" assumes "A \<in> carrier_mat n n" and "0 < n" and "hermitian A" and "unitary_diag A B U" and "j < n" shows "Complex_Matrix.trace (A * (rank_1_proj (Matrix.col U j))) = B $$(j,j)"
lemma priK_parts_Friend_imp_bad [rule_format,dest]: "[\| evs \<in> p2; Friend B \<noteq> A \|] ==> (Key (priK A) \<in> parts (knows (Friend B) evs)) \<longrightarrow> (A \<in> bad)"
lemma pmdl_closed_spoly: assumes "p \<in> pmdl F" and "q \<in> pmdl F" shows "spoly p q \<in> pmdl F"
lemma enum_UNIV: "set enum = UNIV"
lemma lift_option_eq_Some: "lift_option f A B = Some y \<longleftrightarrow> (\<exists>a b. A = Some a \<and> B = Some b \<and> y = f a b)"
lemma arev_arev[simp]: "arev (arev a) = a"
lemma obs_the_element: "m \<in> obs n (backward_slice S) \<Longrightarrow> (THE m. m \<in> obs n (backward_slice S)) = m"
lemma wellformed_uverts_0 : assumes "uwellformed G" and "uverts G = {}" shows "card (uedges G) = 0"
lemma star_denest_var_4: "(x\<^sup>\<star> \<cdot> y\<^sup>\<star>)\<^sup>\<star> = (y\<^sup>\<star> \<cdot> x\<^sup>\<star>)\<^sup>\<star>"
lemma Healthy_intro: "H(P) = P \<Longrightarrow> P is H"
lemma cos_zero_iff: "cos x = 0 \<longleftrightarrow> ((\<exists>n. odd n \<and> x = real n * (pi/2)) \<or> (\<exists>n. odd n \<and> x = - (real n * (pi/2))))" (is "?lhs = ?rhs")
lemma subst_subst_res [simp]: "P\<leftarrow>w\<leftarrow>v = P\<leftarrow>w"
lemma unite_sub_env: fixes e v w defines "e' \<equiv> unite v w e" assumes pre: "pre_dfss v e" and w: "w \<in> successors v" "w \<notin> vsuccs e v" "w \<in> visited e" "w \<notin> explored e" shows "sub_env e e'"
lemma (in set_delete) delete_autoref[autoref_rules]: "PREFER_id Rk \<Longrightarrow> (delete,op_set_delete)\<in>Rk\<rightarrow>\<langle>Rk\<rangle>rel\<rightarrow>\<langle>Rk\<rangle>rel"
lemma eq_diff_eq: assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V" shows "(x = z - y) = (x + y = z)"
lemma lepoll_refl [iff]: "A \<lesssim> A"
lemma sterms_not_choice [simp]: assumes "wellformed \<Gamma>" and "q \<in> sterms \<Gamma> p" shows "not_choice q"
lemma evalC_sum_list: "evalC b (C ns) = sum_list (map (\<lambda>n. b^evalC b n) ns)"
lemma fv_fo_term_setD: "n \<in> fv_fo_term_set t \<Longrightarrow> t = Var n"
lemma wf_darcs_merge1: "wf_darcs (merge1 t)"
lemma trimmed_ms_aux_imp_nz: assumes "basis_wf basis" "is_expansion_aux xs f basis" "trimmed_ms_aux xs" shows "eventually (\<lambda>x. f x \<noteq> 0) at_top"
lemma directed_ub: "directed X \<Longrightarrow> (\<forall>x \<in> X. \<forall>y \<in> X. \<exists>z \<in> X. x \<le> z \<and> y \<le> z)"
lemma uprod_split: "P (case_uprod f x) \<longleftrightarrow> (\<forall>a b. x = Upair a b \<longrightarrow> P (apply_commute f a b))"
lemma cf_map_eqI: assumes "\<FF> \<in>\<^sub>\<circ> cf_maps \<alpha> \<AA> \<BB>" and "\<GG> \<in>\<^sub>\<circ> cf_maps \<alpha> \<AA> \<BB>" and "\<FF>\<lparr>ObjMap\<rparr> = \<GG>\<lparr>ObjMap\<rparr>" and "\<FF>\<lparr>ArrMap\<rparr> = \<GG>\<lparr>ArrMap\<rparr>" shows "\<FF> = \<GG>"
lemma is_arg_max_congI: assumes "is_arg_max f P x" "\<And>x. P x \<Longrightarrow> f x = g x" shows "is_arg_max g P x"
lemma fun_type_inv: assumes "\<Gamma> t = TComp f T" shows "arity f > 0"
lemma e_length_update [simp]: "e_length (e_update b k v) = e_length b"
lemma isLbI: "x <=* S \<Longrightarrow> x \<in> R \<Longrightarrow> isLb R S x"
lemma S_mono_one: assumes S: "S s t" shows "S (Fun f (ss1 @ s # ss2)) (Fun f (ss1 @ t # ss2))"
lemma set_attribute_preserves_wellformedness: assumes "heap_is_wellformed h" and "h \<turnstile> set_attribute element_ptr k v \<rightarrow>\<^sub>h h'" shows "heap_is_wellformed h'"
lemma P2_cong: fixes tms :: "trm list" assumes sub: "\<And>i t u x. atom i \<sharp> tms \<Longrightarrow> (P t u)(i::=x) = P (subst i x t) (subst i x u)" and eq: "H \<turnstile> x EQ x'" "H \<turnstile> y EQ y'" shows "H \<turnstile> P x y IFF P x' y'"
lemma differenceset_commute [simp]: shows "minusset (differenceset B A) = differenceset A B "
lemma complex_Basis_i [iff]: "\<i> \<in> Basis"
lemma bound_main_lemma_charles_3: fixes PROB :: "'a problem" assumes "finite PROB" shows "MPLS_charles PROB \<noteq> {}"
lemma disj_conds: "(P1 \<triangleleft> b \<triangleright> Q1 \<or> P2 \<triangleleft> b \<triangleright> Q2) = (P1 \<or> P2) \<triangleleft> b \<triangleright> (Q1 \<or> Q2)"
lemma bddh_exists: "\<exists>n. bddh n B"
lemma insert_before_only1_new: assumes "\<forall>(x,e1) \<in> fset xs. \<forall>(y,e2) \<in> fset xs. (dverts x \<inter> dverts y = {} \<or> (x,e1)=(y,e2))" and "(t1,e1) \<noteq> (t2,e2)" and "(t1,e1) \<in> fset (insert_before v e y xs)" and "(t2,e2) \<in> fset (insert_before v e y xs)" shows "(t1,e1) \<in> fset xs \<or> (t2,e2) \<in> fset xs"
lemma fds_nth_0 [simp]: "fds_nth f 0 = 0"
lemma augment_augment: "augment\<cdot>g\<cdot>(augment\<cdot>h\<cdot>xs) = augment\<cdot>(g \<circ>lf h)\<cdot>xs"
lemma f_eq_zero_iff: "f x = 0 \<longleftrightarrow> x \<le> 0"
lemma lim_const [simp]: "lim (\<lambda>m. a) = a"
lemma residualFreshSimp[simp]: fixes x :: name and M :: "'a::fs_name" and N :: 'a and P :: "('a, 'b::fs_name, 'c::fs_name) psi" shows "x \<sharp> (M\<lparr>N\<rparr> \<prec> P) = (x \<sharp> M \<and> x \<sharp> N \<and> x \<sharp> P)" and "x \<sharp> (M\<lparr>\<nu>xvec\<rparr>\<langle>N\<rangle> \<prec> P) = (x \<sharp> M \<and> x \<sharp> (\<lparr>\<nu>xvec\<rparr>(N \<prec>' P)))" and "x \<sharp> (\<tau> \<prec> P) = (x \<sharp> P)"
lemma replicate_pred_Cons_length: "0 < n \<Longrightarrow> length (x # a\<^bsup>n - Suc 0\<^esup>) = n"
lemma eigenvector_pow: assumes A: "A \<in> carrier_mat n n" and ev: "eigenvector A v (k :: 'a :: comm_ring_1)" shows "A ^\<^sub>m i *\<^sub>v v = k^i \<cdot>\<^sub>v v"
lemma walk_edges_append_ss2: "set (walk_edges (xs)) \<subseteq> set (walk_edges (xs@ys))"
lemma ND_F_dir2': "\<lbrakk>s \<notin> D P; s \<in> T P; P \<sqsubseteq> S; Q \<sqsubseteq> S\<rbrakk> \<Longrightarrow> s \<in> T Q"
lemma is_iso_class_Comp: assumes "Comp G F \<noteq> {}" shows "B.is_iso_class (Comp G F)"
lemma pre_dfss_pre_dfs: assumes "pre_dfss v e" and "w \<notin> visited e" and "w \<in> successors v" shows "pre_dfs w e"
lemma setops_list\<^sub>s\<^sub>s\<^sub>t_is_setops\<^sub>s\<^sub>s\<^sub>t: "setops\<^sub>s\<^sub>s\<^sub>t S = set (setops_list\<^sub>s\<^sub>s\<^sub>t S)"
lemma semigroup_add_transfer[transfer_rule]: assumes [transfer_rule]: "bi_unique A" "right_total A" shows "((A ===> A ===> A) ===> (=)) (semigroup_add_ow (Collect (Domainp A))) class.semigroup_add"
lemma (in monoid_add) sum_list_distinct_conv_sum_set: "distinct xs \<Longrightarrow> sum_list (map f xs) = sum f (set xs)"
lemma "filterlim (\<lambda>x::real. x + sin x) at_top at_top" "filterlim (\<lambda>x::real. sin x + x) at_top at_top" "filterlim (\<lambda>x::real. x + (sin x + sin x)) at_top at_top"
lemma hom_0_iff[iff]: "hom x = 0 \<longleftrightarrow> x = 0"
lemma[code]: "wf_list_graph_impl V E = wf_list_graph_impl_rs (rbt_fromlist V) E"
lemma step_3_similar: "A \<in> carrier_mat n n \<Longrightarrow> similar_mat (step_3 A) A"
lemma maximal_repetition_sets_from_separators_cover : assumes "q \<in> states M" shows "\<exists> d \<in> (maximal_repetition_sets_from_separators M) . q \<in> fst d"
lemma iprev_empty: "iprev n {} = n"
lemma read_bijective: assumes "bijective y" and "mapping x" shows "bijective (x[[y]])"
lemma zero_carrier_vec[simp]: "0\<^sub>v n \<in> carrier_vec n"
lemma create_closure: "create v \<in> \<S>"
lemma ennreal_archimedean: assumes "x \<noteq> (\<infinity>::ennreal)" shows "\<exists>n::nat. x \<le> n"
theorem sup_state_refl [simp]: "G \<turnstile> s <=s s"
lemma ereal_open_closed_aux: fixes S :: "ereal set" assumes "open S" and "closed S" and S: "(-\<infinity>) \<notin> S" shows "S = {}"
lemma valid_len: assumes " \<turnstile>\<^sub>w\<^sub>f \<Theta>" shows "\<Theta> ; \<B> ; (x, B_int, [[x]\<^sup>v]\<^sup>c\<^sup>e == [[ L_num (int (length v)) ]\<^sup>v]\<^sup>c\<^sup>e) #\<^sub>\<Gamma> GNil \<Turnstile> [[x]\<^sup>v]\<^sup>c\<^sup>e == CE_len [[ L_bitvec v ]\<^sup>v]\<^sup>c\<^sup>e" (is "\<Theta> ; \<B> ; ?G \<Turnstile> ?c" )
lemma propT_ok[simp]: "wf_theory \<Theta> \<Longrightarrow> typ_ok \<Theta> propT"
lemma excess_pp_init_f[simp]: "excess pp_init_f = pp_init_x"
lemma substitutes_appE: assumes "substitutes (App A B) x M X" shows "\<exists>A' B'. substitutes A x M A' \<and> substitutes B x M B' \<and> X = App A' B'"
lemma finite_accent: "finite (UNIV :: accent set)"
lemma cont_D' : assumes chain:"chain Y" shows "((\<Squnion> i. Y i) \<box> S) = (\<Squnion> i. (Y i \<box> S))"
lemma interval_integral_FTC_finite: fixes f F :: "real \<Rightarrow> 'a::euclidean_space" and a b :: real assumes f: "continuous_on {min a b..max a b} f" assumes F: "\<And>x. min a b \<le> x \<Longrightarrow> x \<le> max a b \<Longrightarrow> (F has_vector_derivative (f x)) (at x within {min a b..max a b})" shows "(LBINT x=a..b. f x) = F b - F a"
lemma sparse_row_matrix_nprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"
lemma TMC_yields_num_res_unfolded_into_Hoare_halt: "TMC_yields_num_res tm ns n \<equiv> \<lbrace>(\<lambda>tap. tap = ([], <ns>))\<rbrace> tm \<lbrace>(\<lambda>tap. \<exists>k l. tap = (Bk \<up> k, <n::nat> @ Bk\<up> l))\<rbrace>"
lemma transaction_check_compI[intro]: assumes T: "transaction_check_pre (FP, OCC, TI) T \<delta>" and T_adm: "admissible_transaction T" and x1: "\<forall>x. (x \<in> fv_transaction T - set (transaction_fresh T) \<and> fst x = TAtom Value) \<longrightarrow> \<delta> x \<in> set OCC \<and> msgcs x (\<delta> x)" and x2: "\<forall>x. (x \<notin> fv_transaction T - set (transaction_fresh T) \<or> fst x \<noteq> TAtom Value) \<longrightarrow> \<delta> x = {}" shows "\<delta> \<in> abs_substs_fun ` set (transaction_check_comp msgcs (FP, OCC, TI) T)"
lemma [code]: "all_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"
lemma partial_isometryI: assumes \<open>\<And>h. h \<in> space_as_set (- kernel A) \<Longrightarrow> norm (A h) = norm h\<close> shows \<open>partial_isometry A\<close>
lemma restrictions_inwards [simp]: "x \<noteq> x' \<Longrightarrow> f(x := Some y, x' := None) = (f(x':= None, x := Some y))"
lemma tendsto_add_ereal_nonneg: fixes x y :: "ereal" assumes "x \<noteq> -\<infinity>" "y \<noteq> -\<infinity>" "(f \<longlongrightarrow> x) F" "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
lemma (in wf_digraph) comb_planar_iso: assumes "digraph_isomorphism hom" shows "comb_planar (app_iso hom G) \<longleftrightarrow> comb_planar G"
lemma replacefacesAt_nth2: "k < \|F\| \<Longrightarrow> (replacefacesAt [k] oldf newfs F) ! k = replace oldf newfs (F!k)"
lemma depth_le_height: "depth t a \<le> height t"
lemma unique_timestamp_msg2: "\<lbrakk> Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key AK, Agent Tgs, T\<rbrace>, AT\<rbrace> \<in> set evs; Says Kas A' \<lbrace>Crypt (shrK A') \<lbrace>Key AK', Agent Tgs', T\<rbrace>, AT'\<rbrace> \<in> set evs; evs \<in> kerbV \<rbrakk> \<Longrightarrow> A=A' \<and> AK=AK' \<and> Tgs=Tgs' \<and> AT=AT'"
lemma ConcAssoc2: "Cp (X \<oplus> Y \<oplus> ((A \<oplus> B) \<oplus> D)) = Cp (X \<oplus> Y \<oplus> A \<oplus> B \<oplus> D)"
lemma ipresCons_imp_ipresSubstAll: assumes : "ipresCons h hA MOD" and *: "igSubstCls MOD" and "igConsIPresIGWls MOD" and "igFreshCls MOD" shows "ipresSubstAll h hA MOD"
lemma freeword_funlift_Abs_freeword_Cons: assumes "proper_signed_list (x#xs)" shows "freeword_funlift f (Abs_freeword (x#xs)) = apply_sign f x + freeword_funlift f (Abs_freeword xs)"
lemma length_remove1_less[termination_simp]: "x \<in> set xs \<Longrightarrow> length (remove1 x xs) < length xs"
lemma (in wf_digraph) verts_app_inv_iso_subgraph: assumes hom: "digraph_isomorphism hom" and "V \<subseteq> verts G" shows "iso_verts (inv_iso hom) ` iso_verts hom ` V = V"
lemma ipassmt_disjoint_ignore_wildcard_nonempty_inj: assumes ipassmt_disjoint: "ipassmt_sanity_disjoint (ipassmt_ignore_wildcard ipassmt)" and ifce: "ipassmt ifce = Some i_ips" and a: "ipcidr_union_set (set i_ips) \<noteq> {}" and k: "(ipassmt_ignore_wildcard ipassmt) k = Some i_ips" shows "k = ifce"
lemma "w < x \<and> y \<le> z ==> w + y < x + (z::real)"
lemma servK_authentic: "\<lbrakk> Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (spies evs); Key authK \<notin> analz (spies evs); authK \<notin> range shrK; evs \<in> kerbIV \<rbrakk> \<Longrightarrow> \<exists>A. Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"
lemma integral_sin_Z [simp]: assumes "n \<in> \<int>" shows "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin(x * n)) = 0"
lemma stored_tval_is_cap: assumes "val = Cap_v v" shows "is_cap (content (store_tval obj off val)) off"
lemma insert'_code: "insert' x (set xs) = set (x # xs)"
lemma [elim_format, elim!]: "B\<cdot>b \<triangleleft>\<triangleright> y \<Longrightarrow> y = CB\<cdot>b"
lemma ex_prenex [metric_prenex]: "\<And>P Q. (\<exists>x. P x) \<and> Q \<equiv> \<exists>x. (P x \<and> Q)" "\<And>P Q. P \<and> (\<exists>x. Q x) \<equiv> \<exists>x. (P \<and> Q x)" "\<And>P Q. (\<exists>x. P x) \<or> Q \<equiv> \<exists>x. (P x \<or> Q)" "\<And>P Q. P \<or> (\<exists>x. Q x) \<equiv> \<exists>x. (P \<or> Q x)" "\<And>P. \<not>(\<exists>x. P x) \<equiv> \<forall>x. \<not>P x"
lemma sinits_sscanl: \<comment>\<open> @{cite [cite_macro=citet] \<open>Lemma~5\<close> "Bird:1987"}, @{cite [cite_macro=citet] \<open>p118 ``the scan lemma''\<close> "Bird:PearlsofFAD:2010"} \<close> shows "smap\<cdot>(sfoldl\<cdot>f\<cdot>z) oo sinits = sscanl\<cdot>f\<cdot>z"

lemma not_extf_gt_nil_singleton_if_\<delta>\<^sub>h_eq_\<epsilon>\<^sub>h: assumes wary_s: "wary s" and \<delta>_eq_\<epsilon>: "\<delta>\<^sub>h = \<epsilon>\<^sub>h" shows "\<not> extf f (>\<^sub>t) [] [s]"

lemma transpose_one[simp]: "transpose_mat (1\<^sub>m n) = (1\<^sub>m n)"

lemma eSuc_Sup: "A \<noteq> {} \<Longrightarrow> eSuc (Sup A) = Sup (eSuc ` A)"

lemma (in Ring) msub_submodule:"\<lbrakk>R module M; R module M1; msubmodule R M M1\<rbrakk> \<Longrightarrow> submodule R M (carrier M1)"

lemma big_sum_in_bigo: assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> O[F](g)" shows "(\<lambda>x. sum (\<lambda>y. f y x) A) \<in> O[F](g)"

lemma decomp_eigenvector: fixes A::"complex Matrix.mat" assumes "A \<in> carrier_mat n n" and "0 < n" and "hermitian A" and "unitary_diag A B U" and "j < n" shows "Complex_Matrix.trace (A * (rank_1_proj (Matrix.col U j))) = B $$(j,j)"

lemma priK_parts_Friend_imp_bad [rule_format,dest]: "[| evs \<in> p2; Friend B \<noteq> A |] ==> (Key (priK A) \<in> parts (knows (Friend B) evs)) \<longrightarrow> (A \<in> bad)"

lemma pmdl_closed_spoly: assumes "p \<in> pmdl F" and "q \<in> pmdl F" shows "spoly p q \<in> pmdl F"

lemma enum_UNIV: "set enum = UNIV"

lemma lift_option_eq_Some: "lift_option f A B = Some y \<longleftrightarrow> (\<exists>a b. A = Some a \<and> B = Some b \<and> y = f a b)"

lemma arev_arev[simp]: "arev (arev a) = a"

lemma obs_the_element: "m \<in> obs n (backward_slice S) \<Longrightarrow> (THE m. m \<in> obs n (backward_slice S)) = m"

lemma wellformed_uverts_0 : assumes "uwellformed G" and "uverts G = {}" shows "card (uedges G) = 0"

lemma star_denest_var_4: "(x\<^sup>\<star> \<cdot> y\<^sup>\<star>)\<^sup>\<star> = (y\<^sup>\<star> \<cdot> x\<^sup>\<star>)\<^sup>\<star>"

lemma Healthy_intro: "H(P) = P \<Longrightarrow> P is H"

lemma cos_zero_iff: "cos x = 0 \<longleftrightarrow> ((\<exists>n. odd n \<and> x = real n * (pi/2)) \<or> (\<exists>n. odd n \<and> x = - (real n * (pi/2))))" (is "?lhs = ?rhs")

lemma subst_subst_res [simp]: "P\<leftarrow>w\<leftarrow>v = P\<leftarrow>w"

lemma unite_sub_env: fixes e v w defines "e' \<equiv> unite v w e" assumes pre: "pre_dfss v e" and w: "w \<in> successors v" "w \<notin> vsuccs e v" "w \<in> visited e" "w \<notin> explored e" shows "sub_env e e'"

lemma (in set_delete) delete_autoref[autoref_rules]: "PREFER_id Rk \<Longrightarrow> (delete,op_set_delete)\<in>Rk\<rightarrow>\<langle>Rk\<rangle>rel\<rightarrow>\<langle>Rk\<rangle>rel"

lemma eq_diff_eq: assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V" shows "(x = z - y) = (x + y = z)"

lemma lepoll_refl [iff]: "A \<lesssim> A"

lemma sterms_not_choice [simp]: assumes "wellformed \<Gamma>" and "q \<in> sterms \<Gamma> p" shows "not_choice q"

lemma evalC_sum_list: "evalC b (C ns) = sum_list (map (\<lambda>n. b^evalC b n) ns)"

lemma fv_fo_term_setD: "n \<in> fv_fo_term_set t \<Longrightarrow> t = Var n"

lemma wf_darcs_merge1: "wf_darcs (merge1 t)"

lemma trimmed_ms_aux_imp_nz: assumes "basis_wf basis" "is_expansion_aux xs f basis" "trimmed_ms_aux xs" shows "eventually (\<lambda>x. f x \<noteq> 0) at_top"

lemma directed_ub: "directed X \<Longrightarrow> (\<forall>x \<in> X. \<forall>y \<in> X. \<exists>z \<in> X. x \<le> z \<and> y \<le> z)"

lemma uprod_split: "P (case_uprod f x) \<longleftrightarrow> (\<forall>a b. x = Upair a b \<longrightarrow> P (apply_commute f a b))"

lemma cf_map_eqI: assumes "\<FF> \<in>\<^sub>\<circ> cf_maps \<alpha> \<AA> \<BB>" and "\<GG> \<in>\<^sub>\<circ> cf_maps \<alpha> \<AA> \<BB>" and "\<FF>\<lparr>ObjMap\<rparr> = \<GG>\<lparr>ObjMap\<rparr>" and "\<FF>\<lparr>ArrMap\<rparr> = \<GG>\<lparr>ArrMap\<rparr>" shows "\<FF> = \<GG>"

lemma is_arg_max_congI: assumes "is_arg_max f P x" "\<And>x. P x \<Longrightarrow> f x = g x" shows "is_arg_max g P x"

lemma fun_type_inv: assumes "\<Gamma> t = TComp f T" shows "arity f > 0"

lemma e_length_update [simp]: "e_length (e_update b k v) = e_length b"

lemma isLbI: "x <=* S \<Longrightarrow> x \<in> R \<Longrightarrow> isLb R S x"

lemma S_mono_one: assumes S: "S s t" shows "S (Fun f (ss1 @ s # ss2)) (Fun f (ss1 @ t # ss2))"

lemma set_attribute_preserves_wellformedness: assumes "heap_is_wellformed h" and "h \<turnstile> set_attribute element_ptr k v \<rightarrow>\<^sub>h h'" shows "heap_is_wellformed h'"

lemma P2_cong: fixes tms :: "trm list" assumes sub: "\<And>i t u x. atom i \<sharp> tms \<Longrightarrow> (P t u)(i::=x) = P (subst i x t) (subst i x u)" and eq: "H \<turnstile> x EQ x'" "H \<turnstile> y EQ y'" shows "H \<turnstile> P x y IFF P x' y'"

lemma differenceset_commute [simp]: shows "minusset (differenceset B A) = differenceset A B "

lemma complex_Basis_i [iff]: "\<i> \<in> Basis"

lemma bound_main_lemma_charles_3: fixes PROB :: "'a problem" assumes "finite PROB" shows "MPLS_charles PROB \<noteq> {}"

lemma disj_conds: "(P1 \<triangleleft> b \<triangleright> Q1 \<or> P2 \<triangleleft> b \<triangleright> Q2) = (P1 \<or> P2) \<triangleleft> b \<triangleright> (Q1 \<or> Q2)"

lemma bddh_exists: "\<exists>n. bddh n B"

lemma insert_before_only1_new: assumes "\<forall>(x,e1) \<in> fset xs. \<forall>(y,e2) \<in> fset xs. (dverts x \<inter> dverts y = {} \<or> (x,e1)=(y,e2))" and "(t1,e1) \<noteq> (t2,e2)" and "(t1,e1) \<in> fset (insert_before v e y xs)" and "(t2,e2) \<in> fset (insert_before v e y xs)" shows "(t1,e1) \<in> fset xs \<or> (t2,e2) \<in> fset xs"

lemma fds_nth_0 [simp]: "fds_nth f 0 = 0"

lemma augment_augment: "augment\<cdot>g\<cdot>(augment\<cdot>h\<cdot>xs) = augment\<cdot>(g \<circ>lf h)\<cdot>xs"

lemma f_eq_zero_iff: "f x = 0 \<longleftrightarrow> x \<le> 0"

lemma lim_const [simp]: "lim (\<lambda>m. a) = a"

lemma residualFreshSimp[simp]: fixes x :: name and M :: "'a::fs_name" and N :: 'a and P :: "('a, 'b::fs_name, 'c::fs_name) psi" shows "x \<sharp> (M\<lparr>N\<rparr> \<prec> P) = (x \<sharp> M \<and> x \<sharp> N \<and> x \<sharp> P)" and "x \<sharp> (M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P) = (x \<sharp> M \<and> x \<sharp> (\<lparr>\<nu>*xvec\<rparr>(N \<prec>' P)))" and "x \<sharp> (\<tau> \<prec> P) = (x \<sharp> P)"

lemma replicate_pred_Cons_length: "0 < n \<Longrightarrow> length (x # a\<^bsup>n - Suc 0\<^esup>) = n"

lemma eigenvector_pow: assumes A: "A \<in> carrier_mat n n" and ev: "eigenvector A v (k :: 'a :: comm_ring_1)" shows "A ^\<^sub>m i *\<^sub>v v = k^i \<cdot>\<^sub>v v"

lemma walk_edges_append_ss2: "set (walk_edges (xs)) \<subseteq> set (walk_edges (xs@ys))"

lemma ND_F_dir2': "\<lbrakk>s \<notin> D P; s \<in> T P; P \<sqsubseteq> S; Q \<sqsubseteq> S\<rbrakk> \<Longrightarrow> s \<in> T Q"

lemma is_iso_class_Comp: assumes "Comp G F \<noteq> {}" shows "B.is_iso_class (Comp G F)"

lemma pre_dfss_pre_dfs: assumes "pre_dfss v e" and "w \<notin> visited e" and "w \<in> successors v" shows "pre_dfs w e"

lemma setops_list\<^sub>s\<^sub>s\<^sub>t_is_setops\<^sub>s\<^sub>s\<^sub>t: "setops\<^sub>s\<^sub>s\<^sub>t S = set (setops_list\<^sub>s\<^sub>s\<^sub>t S)"

lemma semigroup_add_transfer[transfer_rule]: assumes [transfer_rule]: "bi_unique A" "right_total A" shows "((A ===> A ===> A) ===> (=)) (semigroup_add_ow (Collect (Domainp A))) class.semigroup_add"

lemma (in monoid_add) sum_list_distinct_conv_sum_set: "distinct xs \<Longrightarrow> sum_list (map f xs) = sum f (set xs)"

lemma "filterlim (\<lambda>x::real. x + sin x) at_top at_top" "filterlim (\<lambda>x::real. sin x + x) at_top at_top" "filterlim (\<lambda>x::real. x + (sin x + sin x)) at_top at_top"

lemma hom_0_iff[iff]: "hom x = 0 \<longleftrightarrow> x = 0"

lemma[code]: "wf_list_graph_impl V E = wf_list_graph_impl_rs (rbt_fromlist V) E"

lemma step_3_similar: "A \<in> carrier_mat n n \<Longrightarrow> similar_mat (step_3 A) A"

lemma maximal_repetition_sets_from_separators_cover : assumes "q \<in> states M" shows "\<exists> d \<in> (maximal_repetition_sets_from_separators M) . q \<in> fst d"

lemma iprev_empty: "iprev n {} = n"

lemma read_bijective: assumes "bijective y" and "mapping x" shows "bijective (x[[y]])"

lemma zero_carrier_vec[simp]: "0\<^sub>v n \<in> carrier_vec n"

lemma create_closure: "create v \<in> \<S>"

lemma ennreal_archimedean: assumes "x \<noteq> (\<infinity>::ennreal)" shows "\<exists>n::nat. x \<le> n"

theorem sup_state_refl [simp]: "G \<turnstile> s <=s s"

lemma ereal_open_closed_aux: fixes S :: "ereal set" assumes "open S" and "closed S" and S: "(-\<infinity>) \<notin> S" shows "S = {}"

lemma valid_len: assumes " \<turnstile>\<^sub>w\<^sub>f \<Theta>" shows "\<Theta> ; \<B> ; (x, B_int, [[x]\<^sup>v]\<^sup>c\<^sup>e == [[ L_num (int (length v)) ]\<^sup>v]\<^sup>c\<^sup>e) #\<^sub>\<Gamma> GNil \<Turnstile> [[x]\<^sup>v]\<^sup>c\<^sup>e == CE_len [[ L_bitvec v ]\<^sup>v]\<^sup>c\<^sup>e" (is "\<Theta> ; \<B> ; ?G \<Turnstile> ?c" )

lemma propT_ok[simp]: "wf_theory \<Theta> \<Longrightarrow> typ_ok \<Theta> propT"

lemma excess_pp_init_f[simp]: "excess pp_init_f = pp_init_x"

lemma substitutes_appE: assumes "substitutes (App A B) x M X" shows "\<exists>A' B'. substitutes A x M A' \<and> substitutes B x M B' \<and> X = App A' B'"

lemma finite_accent: "finite (UNIV :: accent set)"

lemma cont_D' : assumes chain:"chain Y" shows "((\<Squnion> i. Y i) \<box> S) = (\<Squnion> i. (Y i \<box> S))"

lemma interval_integral_FTC_finite: fixes f F :: "real \<Rightarrow> 'a::euclidean_space" and a b :: real assumes f: "continuous_on {min a b..max a b} f" assumes F: "\<And>x. min a b \<le> x \<Longrightarrow> x \<le> max a b \<Longrightarrow> (F has_vector_derivative (f x)) (at x within {min a b..max a b})" shows "(LBINT x=a..b. f x) = F b - F a"

lemma sparse_row_matrix_nprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"

lemma TMC_yields_num_res_unfolded_into_Hoare_halt: "TMC_yields_num_res tm ns n \<equiv> \<lbrace>(\<lambda>tap. tap = ([], <ns>))\<rbrace> tm \<lbrace>(\<lambda>tap. \<exists>k l. tap = (Bk \<up> k, <n::nat> @ Bk\<up> l))\<rbrace>"

lemma transaction_check_compI[intro]: assumes T: "transaction_check_pre (FP, OCC, TI) T \<delta>" and T_adm: "admissible_transaction T" and x1: "\<forall>x. (x \<in> fv_transaction T - set (transaction_fresh T) \<and> fst x = TAtom Value) \<longrightarrow> \<delta> x \<in> set OCC \<and> msgcs x (\<delta> x)" and x2: "\<forall>x. (x \<notin> fv_transaction T - set (transaction_fresh T) \<or> fst x \<noteq> TAtom Value) \<longrightarrow> \<delta> x = {}" shows "\<delta> \<in> abs_substs_fun ` set (transaction_check_comp msgcs (FP, OCC, TI) T)"

lemma [code]: "all_n_lists P n \<longleftrightarrow> (if n = 0 then P [] else enum_all (%x. all_n_lists (%xs. P (x # xs)) (n - 1)))"

lemma partial_isometryI: assumes \<open>\<And>h. h \<in> space_as_set (- kernel A) \<Longrightarrow> norm (A h) = norm h\<close> shows \<open>partial_isometry A\<close>

lemma restrictions_inwards [simp]: "x \<noteq> x' \<Longrightarrow> f(x := Some y, x' := None) = (f(x':= None, x := Some y))"

lemma tendsto_add_ereal_nonneg: fixes x y :: "ereal" assumes "x \<noteq> -\<infinity>" "y \<noteq> -\<infinity>" "(f \<longlongrightarrow> x) F" "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"

lemma (in wf_digraph) comb_planar_iso: assumes "digraph_isomorphism hom" shows "comb_planar (app_iso hom G) \<longleftrightarrow> comb_planar G"

lemma replacefacesAt_nth2: "k < |F| \<Longrightarrow> (replacefacesAt [k] oldf newfs F) ! k = replace oldf newfs (F!k)"

lemma depth_le_height: "depth t a \<le> height t"

lemma unique_timestamp_msg2: "\<lbrakk> Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key AK, Agent Tgs, T\<rbrace>, AT\<rbrace> \<in> set evs; Says Kas A' \<lbrace>Crypt (shrK A') \<lbrace>Key AK', Agent Tgs', T\<rbrace>, AT'\<rbrace> \<in> set evs; evs \<in> kerbV \<rbrakk> \<Longrightarrow> A=A' \<and> AK=AK' \<and> Tgs=Tgs' \<and> AT=AT'"

lemma ConcAssoc2: "Cp (X \<oplus> Y \<oplus> ((A \<oplus> B) \<oplus> D)) = Cp (X \<oplus> Y \<oplus> A \<oplus> B \<oplus> D)"

lemma ipresCons_imp_ipresSubstAll: assumes *: "ipresCons h hA MOD" and **: "igSubstCls MOD" and "igConsIPresIGWls MOD" and "igFreshCls MOD" shows "ipresSubstAll h hA MOD"

lemma freeword_funlift_Abs_freeword_Cons: assumes "proper_signed_list (x#xs)" shows "freeword_funlift f (Abs_freeword (x#xs)) = apply_sign f x + freeword_funlift f (Abs_freeword xs)"

lemma length_remove1_less[termination_simp]: "x \<in> set xs \<Longrightarrow> length (remove1 x xs) < length xs"

lemma (in wf_digraph) verts_app_inv_iso_subgraph: assumes hom: "digraph_isomorphism hom" and "V \<subseteq> verts G" shows "iso_verts (inv_iso hom) ` iso_verts hom ` V = V"

lemma ipassmt_disjoint_ignore_wildcard_nonempty_inj: assumes ipassmt_disjoint: "ipassmt_sanity_disjoint (ipassmt_ignore_wildcard ipassmt)" and ifce: "ipassmt ifce = Some i_ips" and a: "ipcidr_union_set (set i_ips) \<noteq> {}" and k: "(ipassmt_ignore_wildcard ipassmt) k = Some i_ips" shows "k = ifce"

lemma "w < x \<and> y \<le> z ==> w + y < x + (z::real)"

lemma servK_authentic: "\<lbrakk> Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (spies evs); Key authK \<notin> analz (spies evs); authK \<notin> range shrK; evs \<in> kerbIV \<rbrakk> \<Longrightarrow> \<exists>A. Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"

lemma integral_sin_Z [simp]: assumes "n \<in> \<int>" shows "integral\<^sup>L (lebesgue_on {-pi..pi}) (\<lambda>x. sin(x * n)) = 0"

lemma stored_tval_is_cap: assumes "val = Cap_v v" shows "is_cap (content (store_tval obj off val)) off"

lemma insert'_code: "insert' x (set xs) = set (x # xs)"

lemma [elim_format, elim!]: "B\<cdot>b \<triangleleft>\<triangleright> y \<Longrightarrow> y = CB\<cdot>b"

lemma ex_prenex [metric_prenex]: "\<And>P Q. (\<exists>x. P x) \<and> Q \<equiv> \<exists>x. (P x \<and> Q)" "\<And>P Q. P \<and> (\<exists>x. Q x) \<equiv> \<exists>x. (P \<and> Q x)" "\<And>P Q. (\<exists>x. P x) \<or> Q \<equiv> \<exists>x. (P x \<or> Q)" "\<And>P Q. P \<or> (\<exists>x. Q x) \<equiv> \<exists>x. (P \<or> Q x)" "\<And>P. \<not>(\<exists>x. P x) \<equiv> \<forall>x. \<not>P x"

lemma sinits_sscanl: \<comment>\<open> @{cite [cite_macro=citet] \<open>Lemma~5\<close> "Bird:1987"}, @{cite [cite_macro=citet] \<open>p118 ``the scan lemma''\<close> "Bird:PearlsofFAD:2010"} \<close> shows "smap\<cdot>(sfoldl\<cdot>f\<cdot>z) oo sinits = sscanl\<cdot>f\<cdot>z"