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lemma "eval (ExQ (Or (Atom A) (Atom B))) xs = eval (Or (ExQ(Atom A)) (ExQ(Atom B))) xs"
lemma param_array_of_list[param]: "(array_of_list, array_of_list) \<in> \<langle>R\<rangle> list_rel \<rightarrow> \<langle>R\<rangle> array_rel"
lemma unreachable_bounded_path_only: assumes d'_def: "d'\<notin> unreach-on ab from e" "d'\<in>ab" "d'\<noteq>e" and e_event: "e \<in> \<E>" and path_ab: "ab \<in> \<P>" and e_notin_S: "e \<notin> ab" shows "\<exists>d'e. path d'e d' e"
lemma \<Delta>_Atr_infer0: "infer0 = fset Q"
lemma ccspan_Times_sing1: \<open>ccspan ({0::'a::complex_normed_vector} \<times> B) = ccsubspace_Times 0 (ccspan B)\<close>
lemma lspasl_orr: "Gamma \<longrightarrow> (A h) \<or> (B h) \<or> Delta \<Longrightarrow> Gamma \<longrightarrow> ((A or B) h) \<or> Delta"
lemma monomial_is_monomial: assumes "c \<noteq> 0" shows "is_monomial (monomial c t)"
lemma cat_RK23_is_functor'[cat_Kan_cs_intros]: assumes "\<FF> : cat_ordinal (2\<^sub>\<nat>) \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" and "\<AA>' = cat_ordinal (3\<^sub>\<nat>)" shows "RK23 \<FF> : \<AA>' \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>"
lemma gtrancl_rel_subseteq_trancl_gctxtcl_funas: assumes "\<R> \<subseteq> \<T>\<^sub>G \<F> \<times> \<T>\<^sub>G \<F>" shows "gtrancl_rel \<F> \<R> \<subseteq> (gctxtcl_funas \<F> \<R>)\<^sup>+"
lemma paths_with_new_start_in_v0: "xs \<in> paths_with_new \<Longrightarrow> hd xs = v0"
lemma NSDERIV_chain: "NSDERIV f (g x) :> Da \<Longrightarrow> NSDERIV g x :> Db \<Longrightarrow> NSDERIV (f \<circ> g) x :> Da * Db"
lemma RECT_eq_REC': "nofail (RECT B x) \<Longrightarrow> RECT B x = REC B x"
lemma conj_var_subst: assumes "vwb_lens x" shows "(P \<and> var x =\<^sub>u v) = (P\<lbrakk>v/x\<rbrakk> \<and> var x =\<^sub>u v)"
lemma r2f_rtrancl_hom_var: "\<F> \<circ> rtrancl = kstar \<circ> \<F>"
lemma Gleason9_34: assumes "cut A" "0 < u" shows "\<exists>r \<in> A. r + u \<notin> A"
lemma dom_lc3 [simp]: "d x \<cdot> d (x \<cdot> y) = d (x \<cdot> y)"
lemma sorted_spmat_minus_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (minus_spmat A)"
lemma ImplE[PLM_elim, PLM_dest]: "[\<phi> \<^bold>\<rightarrow> \<psi> in v] \<Longrightarrow> ([\<phi> in v] \<Longrightarrow> [\<psi> in v])"
lemma not_know_s_not_eout: assumes "m \<notin> specSecrets A" and "\<not> know A (sKS m)" and "eoutKnowCorrect A (sKS m)" shows "\<not> eout A (sE m)"
lemma get_M_Element_preserved6 [simp]: "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h' \<Longrightarrow> (\<And>x. getter (cast (setter (\<lambda>_. v) x)) = getter (cast x)) \<Longrightarrow> preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr getter) h h'"
lemma frange_vinsert[simp]: "\<R>\<^sub>\<bullet> (vinsert [a, b]\<^sub>\<circ> r) = vinsert b (\<R>\<^sub>\<bullet> r)"
lemma cauchy_in\<^sub>N_I: assumes "\<And>e. e > 0 \<Longrightarrow> (\<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. Norm N (u n - u m) < e)" shows "cauchy_in\<^sub>N N u"
lemma all_eqa: assumes "A B C Ang a1" and "A B C Ang a2" shows "a1 = a2"
lemma lm089: "Range(f outside X) \<supseteq> (Range f)-(f``X)"
lemma map_index_no_index[simp]: "map_index (\<lambda>n x. f x) xs = map f xs"
lemma qbs_integrable_iff_bounded: assumes "qbs_prob_space_qbs s = X" shows "qbs_integrable s f \<longleftrightarrow> f \<in> X \<rightarrow>\<^sub>Q \<real>\<^sub>Q \<and> qbs_prob_ennintegral s (\<lambda>x. ennreal \<bar>f x \<bar>) < \<infinity>" (is "?lhs = ?rhs")
lemma mset_link[simp]: "mset_tree (link t\<^sub>1 t\<^sub>2) = mset_tree t\<^sub>1 + mset_tree t\<^sub>2"
lemma ID1: "Der_1 \<D> \<Longrightarrow> Int_1 (\<I>\<^sub>D \<D>)"
lemma fv_incr_boundvars [simp]: "fv (incr_boundvars inc t) = fv t"
lemma bit_word_rotr_iff [bit_simps]: \<open>bit (word_rotr m w) n \<longleftrightarrow> n < LENGTH('a) \<and> bit w ((n + m) mod LENGTH('a))\<close> for w :: \<open>'a::len word\<close>
lemma atomicity_refinement: assumes "s = s * q" and "x = q * x" and "q * b = bot" and "r * b \<le> b * r" and "r * l \<le> l * r" and "x * l \<le> l * x" and "b * l \<le> l * b" and "q * l \<le> l * q" and "r\<^sup>\<circ> * q \<le> q * r\<^sup>\<circ>" and "q \<le> 1" shows "s * (x \<squnion> b \<squnion> r \<squnion> l)\<^sup>\<circ> * q \<le> s * (x * b\<^sup>\<circ> * q \<squnion> r \<squnion> l)\<^sup>\<circ>"
lemma swap_o_Snd: "swap o Snd = Fst"
lemma subst_with_ax2: shows "M{b:=(x).Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]"
lemma observable_preamble_paths : assumes "is_preamble P M q'" and "observable M" and "path M q p" and "p_io p \<in> LS P q" and "q \<in> reachable_states P" shows "path P q p"
lemma "\<FF> \<F> \<Longrightarrow> \<forall>a. (a \<^bold>\<Rightarrow> \<^bold>\<not>a) \<^bold>\<Rightarrow> \<^bold>\<not>a \<^bold>\<approx> \<^bold>\<top>"
lemma stopping_time_0: assumes T: "stopping_time (stream_filtration M) T" and x: "x \<in> space M" and \<omega>: "\<omega> \<in> streams (space M)" "T (x ## \<omega>) > 0" and \<omega>': "\<omega>' \<in> streams (space M)" shows "T (x ## \<omega>') > 0"
lemma mpoly_induct [case_names monom sum]:\<comment> \<open>TODO: overwrites @{thm mpoly_induct}\<close> assumes monom:"\<And>m a. P (MPoly_Type.monom m a)" and sum:"(\<And>p1 p2 m a. P p1 \<Longrightarrow> P p2 \<Longrightarrow> p2 = (MPoly_Type.monom m a) \<Longrightarrow> m \<notin> monomials p1 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> P (p1+p2))" shows "P p"
lemma comparator_lex_comp: "comparator lex_comp"
lemma \<open>- (1705 :: int) OR 42 = - 1665\<close>
lemma extended_hyperb_ineq [mono_intros]: "extended_Gromov_product_at (e::'a::Gromov_hyperbolic_space) x z \<ge> min (extended_Gromov_product_at e x y) (extended_Gromov_product_at e y z) - deltaG(TYPE('a))"
lemma inv_detectI: assumes "\<And>m x . soup m s \<Longrightarrow> prefix (history m) (past m)" shows "inv_detect s"
lemma m_inv_chain_group [simp]: "Poly_Mapping.keys a \<subseteq> singular_simplex_set p X \<Longrightarrow> inv\<^bsub>chain_group p X\<^esub> a = -a"
lemma mulex1_union: "mulex1 P M N \<Longrightarrow> mulex1 P (K + M) (K + N)"
lemma integrable_on_affinity: assumes "m \<noteq> 0" "f integrable_on (cbox a b)" shows "(\<lambda>x. f (m *\<^sub>R x + c)) integrable_on ((\<lambda>x. (1 / m) *\<^sub>R x - ((1 / m) *\<^sub>R c)) ` cbox a b)"
lemma f_nxt_cong: "f x' = f' x' \<Longrightarrow> f_nxt f T x x' = f_nxt f' T x x'"
lemma fps_nth_deriv_linear[simp]: "fps_nth_deriv n (fps_const a * f + fps_const b * g) = fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g"
lemma gencode_permute: assumes "set ps = {0..<length ts}" shows "gencode (map ((!) ts) ps) = map_gterm (\<lambda>xs. map ((!) xs) ps) (gencode ts)"
lemma domID[rule_format]: "p \<noteq> [] \<and> x \<in> dom(Cp(list2FWpolicy p)) \<longrightarrow> x \<in> dom (Cp(list2FWpolicy(insertDenies p)))"
lemma eigenvector_hom: assumes A: "A \<in> carrier_mat n n" and ev: "eigenvector A v ev" shows "eigenvector (mat\<^sub>h A) (vec\<^sub>h v) (hom ev)"
lemma B_authenticates_and_keydist_to_A_r: "\<lbrakk> Crypt servK \<lbrace>Agent A, Number T3\<rbrace> \<in> parts (spies evs); Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (spies evs); Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (spies evs); Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace> \<in> parts (spies evs); \<not> expiredSK Ts evs; \<not> expiredAK Ta evs; B \<noteq> Tgs; A \<notin> bad; B \<notin> bad; evs \<in> kerbIV \<rbrakk> \<Longrightarrow> A Issues B with (Crypt servK \<lbrace>Agent A, Number T3\<rbrace>) on evs"
lemma (in orset) added_ids_Broadcast_collapse [simp]: shows "added_ids ([Broadcast e]) e' = []"
lemma m\<tau>move_False: "\<tau>multithreaded.m\<tau>move (\<lambda>s ta s'. False) = (\<lambda>s ta s'. False)"
lemma mapping_of_mpoly_of_poly [simp]: "mapping_of (mpoly_of_poly i p) = mpoly_of_poly_aux i p"
lemma twice_field_differentiable_at_sqrt_fun [intro]: assumes "f twice_field_differentiable_at x" and "f x > 0" shows "(\<lambda>x. sqrt (f x)) twice_field_differentiable_at x"
lemma has_derivative_inverse_dieudonne: fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" assumes "open S" and "open (f ` S)" and "continuous_on S f" and "continuous_on (f ` S) g" and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x" and "x \<in> S" and "(f has_derivative f') (at x)" and "bounded_linear g'" and "g' \<circ> f' = id" shows "(g has_derivative g') (at (f x))"
lemma carrier_one_not_zero: "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
lemma Ker_closed [intro, simp]: "a \<in> Ker \<Longrightarrow> a \<in> G"
theorem Nullstellensatz: assumes "finite X" and "F \<subseteq> P[X]" and "(f::(_::{countable,linorder} \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 _::alg_closed_field) \<in> \<I> (\<V> F)" shows "f \<in> \<surd>ideal F"
lemma density_context_empty[simp]: "density_context {} (V\<union>V') \<Gamma> (\<lambda>_. 1)"
lemma \<Sigma>_pos_id [summation]: "0 \<le> k \<Longrightarrow> 0 \<le> l \<Longrightarrow> \<Sigma> (\<lambda>r. f (pos_id r)) k l = \<Sigma> f k l"
lemma analz_pparts_kparts: "X \<in> analz H \<Longrightarrow> X \<in> pparts H \<or> X \<in> analz (kparts H)"
lemma cnv_I: "cnv x x"
lemma (in Order) Chain_sub_Chain:"\<lbrakk>Chain D X; Y \<subseteq> X \<rbrakk> \<Longrightarrow> Chain D Y"
lemma DOM_in_dom [intro]: assumes "arr f" shows "DOM f \<in> dom f"
lemma nths'_all[simp]: "nths' 0 (length xs) xs = xs"
lemma zero_vector_b: "zero_vector x \<longleftrightarrow> -x * bot \<le> -x"
lemma new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_new_ptr: assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast new_element_ptr|}"
lemma enum_not_empty[simp]: "Enum.enum \<noteq> []" (is "?enum \<noteq> []")
lemma partial_term_of_code: "partial_term_of (ty :: 'a itself) (Quickcheck_Narrowing.Narrowing_variable p t) \<equiv> Code_Evaluation.Free (STR ''_'') tr" "partial_term_of (ty :: 'a itself) (Quickcheck_Narrowing.Narrowing_constructor i []) \<equiv> Code_Evaluation.term_of (partial_term_of_sample i)"
lemma (in infinite_coin_toss_space) nat_filtration_AE_zero: fixes f::"bool stream \<Rightarrow> real" assumes "AE w in M. f w = 0" and "f\<in> borel_measurable (nat_filtration n)" and "0 < p" and "p < 1" shows "\<forall>w. f w = 0"
lemma cat_equalizer_is_cat_equalizer_2: assumes "\<epsilon> : E <\<^sub>C\<^sub>F\<^sub>.\<^sub>e\<^sub>q (\<aa>,\<bb>,set {\<gg>\<^sub>P\<^sub>L, \<ff>\<^sub>P\<^sub>L},(\<lambda>f\<in>\<^sub>\<circ>set {\<gg>\<^sub>P\<^sub>L, \<ff>\<^sub>P\<^sub>L}. (f = \<ff>\<^sub>P\<^sub>L ? \<ff> : \<gg>))) : \<Up>\<^sub>C \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" shows "\<epsilon> : E <\<^sub>C\<^sub>F\<^sub>.\<^sub>e\<^sub>q (\<aa>,\<bb>,\<gg>,\<ff>) : \<up>\<up>\<^sub>C \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>"
lemma reaches1_steps_iff: "x \<rightarrow>\<^sup>+ y \<longleftrightarrow> (\<exists> xs. steps (x # xs @ [y]))"
lemma ik_dyn_mono: "\<lbrakk>x \<in> ik_dyn s; \<And>m . soup2 m s \<Longrightarrow> soup2 m s'\<rbrakk> \<Longrightarrow> x \<in> ik_dyn s'"
lemma sqrt_inverse_power2 [simp]: "sqrt (n\<^sup>2) = n"
lemma id_image_two[simp]: "(\<lambda>(x,y). (x,y)) ` set list = set list"
lemma (in flowgraph) return_return_same_proc[simp]: "return fg p = return fg p' \<Longrightarrow> p=p'"
lemma cat_Funct_is_iso_arrD: assumes "tiny_category \<alpha> \<AA>" and "category \<alpha> \<BB>" and "\<NN> : \<FF> \<mapsto>\<^sub>i\<^sub>s\<^sub>o\<^bsub>cat_Funct \<alpha> \<AA> \<BB>\<^esub> \<GG>" (is \<open>\<NN> : \<FF> \<mapsto>\<^sub>i\<^sub>s\<^sub>o\<^bsub>?Funct\<^esub> \<GG>\<close>) shows "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN> : cf_of_cf_map \<AA> \<BB> \<FF> \<mapsto>\<^sub>C\<^sub>F\<^sub>.\<^sub>t\<^sub>m\<^sub>.\<^sub>i\<^sub>s\<^sub>o cf_of_cf_map \<AA> \<BB> \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" and "\<NN> = ntcf_arrow (ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>)" and "\<FF> = cf_map (cf_of_cf_map \<AA> \<BB> \<FF>)" and "\<GG> = cf_map (cf_of_cf_map \<AA> \<BB> \<GG>)"
lemma (in bounded_linear) Zfun: assumes g: "Zfun g F" shows "Zfun (\<lambda>x. f (g x)) F"
lemma unstream_append_trans [stream_fusion]: "unstream (append_trans g h sh) (Inl sg) = append (unstream g sg) (unstream h sh)"
lemma notin_closed: "(\<not> ((c::eint) \<le> x \<and> x \<le> d)) = (x < c \<or> d < x)"
lemma "eq (Const (1::nat)) (Var (1::nat)) \<longleftrightarrow> False"
lemma fmapU_cast_eq: "fmapU\<cdot>(cast\<cdot>A) = PRJ(udom\<cdot>'f) oo cast\<cdot>(TC('f::functor)\<cdot>A) oo emb"
lemma Z_inv [simp]: "Z * Z = 1\<^sub>m 2"
lemma either_or: fixes r :: "real" assumes a: "(\<exists>y'>r. \<forall>x\<in>{r<..y'}. (aEvalUni (EqUni (a, b, c)) x) \<or> (aEvalUni (LessUni (a, b, c)) x))" shows "(\<exists>y'>r. \<forall>x\<in>{r<..y'}. (aEvalUni (EqUni (a, b, c)) x)) \<or> (\<exists>y'>r. \<forall>x\<in>{r<..y'}. (aEvalUni (LessUni (a, b, c)) x))"
lemma mcont2mcont_lset[THEN mcont2mcont, cont_intro, simp]: shows mcont_lset: "mcont lSup (\<sqsubseteq>) Union (\<subseteq>) lset"
lemma inv_tm_skip_first_arg_len_eq_1_step: assumes "inv_tm_skip_first_arg_len_eq_1 n cf" shows "inv_tm_skip_first_arg_len_eq_1 n (step0 cf tm_skip_first_arg)"
lemma analytically_valid_y: assumes "analytically_valid s F i" shows "(\<lambda>x. integral UNIV (\<lambda>y. (partial_vector_derivative F i) (y, x) * (indicator s (y, x)))) \<in> borel_measurable lborel"
lemma grid_transitive: "\<lbrakk> a \<in> grid b ds ; b \<in> grid c ds' ; ds' \<subseteq> ds'' ; ds \<subseteq> ds'' \<rbrakk> \<Longrightarrow> a \<in> grid c ds''"
lemma psi_ubound_log: "psi n \<le> 551 / 256 * ln 2 * n"
lemma square_dvd_squarefree_part_iff: "x^2 dvd \<Prod>(squarefree_part n) \<longleftrightarrow> x = 1"
lemma i_State_Change_Init_exists_set: " \<lbrakk> n1 \<le> n2; n1 \<in> I; n2 \<in> I; \<not> P (i_Exec_Comp_Stream_Init trans_fun input c n1); P (i_Exec_Comp_Stream_Init trans_fun input c n2) \<rbrakk> \<Longrightarrow> \<exists>n\<in>I. n1 \<le> n \<and> n < n2 \<and> \<not> P (i_Exec_Comp_Stream_Init trans_fun input c n) \<and> P (i_Exec_Comp_Stream_Init trans_fun input c (inext n I))"
lemma zero_lens_scene: "\<lbrakk>0\<^sub>L\<rbrakk>\<^sub>\<sim> = \<bottom>\<^sub>S"
lemma s_has_field_derivative[derivative_intros]: assumes "t \<ge> 0" "v / a \<le> 0" "a \<noteq> 0" shows "(s has_field_derivative s' t) (at t within {0..})"
lemma tt_mult_scalar: assumes "p \<noteq> 0" and "q \<noteq> (0::'t \<Rightarrow>\<^sub>0 'b::semiring_no_zero_divisors)" shows "tt (p \<odot> q) = punit.tt p \<oplus> tt q"
lemma tm_erase_right_then_dblBk_left_erp_partial_correctness_CL_ew_Oc: assumes "\<exists>stp. is_final (steps0 (1, [Bk,Oc] @ CL, CR) tm_erase_right_then_dblBk_left stp)" and "noDblBk CL" and "noDblBk CR" and "CL \<noteq> []" and "last CL = Oc" shows "\<lbrace> \<lambda>tap. tap = ([Bk,Oc] @ CL, CR) \<rbrace> tm_erase_right_then_dblBk_left \<lbrace> \<lambda>tap. \<exists>rex. tap = ([], [Bk, Bk] @ (rev CL) @ [Oc, Bk] @ Bk \<up> rex ) \<rbrace>"
lemma splitFace_add_vertices_direct[simp]: "vertices (snd (snd (splitFace g ram1 ram2 oldF [countVertices g ..< countVertices g + n]))) = vertices g @ [countVertices g ..< countVertices g + n]"
lemma clinear_continuous_within: assumes \<open>bounded_clinear f\<close> shows \<open>continuous (at x within s) f\<close>
lemma quotI [intro]: "{x. a \<sim> x} \<in> quot"
lemma height_balL: "\<lbrakk> hbt l; hbt r; height l = height r + m + 1 \<rbrakk> \<Longrightarrow> height (balL l a r) \<in> {height r + m + 1, height r + m + 2}"
lemma listset_closed_map2: assumes "ys1 \<in> listset xs" and "ys2 \<in> listset xs" and "\<And>x y1 y2. x \<in> set xs \<Longrightarrow> y1 \<in> x \<Longrightarrow> y2 \<in> x \<Longrightarrow> f y1 y2 \<in> x" shows "map2 f ys1 ys2 \<in> listset xs"