Statement:
stringlengths 7
24.3k
|
---|
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" |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.