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

Statement: stringlengths 7 24.3k
lemma no_\<tau>move2_\<tau>s_to_no_\<tau>move1: "\<lbrakk> s1 \<approx> s2; \<And>s2'. \<not> s2 -\<tau>2\<rightarrow> s2' \<rbrakk> \<Longrightarrow> \<exists>s1'. s1 -\<tau>1\<rightarrow>* s1' \<and> (\<forall>s1''. \<not> s1' -\<tau>1\<rightarrow> s1'') \<and> s1' \<approx> s2"
lemma prod_result_cases: obtains (val) s v where "r = (s, Rval v)" \| (err) s err where "r = (s, Rerr err)"
lemma Inputs_imp_knows_Spy_secureM_srb: "\<lbrakk> Inputs Spy C X \<in> set evs; evs \<in> srb \<rbrakk> \<Longrightarrow> X \<in> knows Spy evs"
lemma subset_UnderS_AboveS: "B \<le> Field r \<Longrightarrow> B \<le> UnderS r (AboveS r B)"
lemma kSup_distl: "f \<circ>\<^sub>K (\<Squnion>G) = (\<Squnion>g \<in> G. f \<circ>\<^sub>K g)"
lemma select_index_3_subsets [simp]: shows "{j::nat. select_index 3 0 j} = {4,5,6,7} \<and> {j::nat. j < 8 \<and> \<not> select_index 3 0 j} = {0,1,2,3} \<and> {j::nat. select_index 3 1 j} = {2,3,6,7} \<and> {j::nat. j < 8 \<and> \<not> select_index 3 1 j} = {0,1,4,5}"
lemma aux10[rule_format]: "a \<in> set (net_list p) \<longrightarrow> a \<in> set (net_list_aux p)"
lemma fermat_little_theorem: assumes "prime (P :: nat)" shows "[x^P = x] (mod P)"
lemma act_emp_var: "\<alpha> x \<bottom> = \<bottom>"
lemma set1_FGco_bound: fixes x :: "(_, 'co1, 'co2, 'co3, 'co4, 'co5, 'co6, 'contra1, 'contra2, 'contra3, 'contra4, 'f1, 'f2) FGco" shows "card_of (set1_FGco x) <o (bd_FGco :: ('co1, 'co2, 'co3, 'co4, 'co5, 'co6, 'contra1, 'contra2, 'contra3, 'contra4, 'f1, 'f2) FGcobd rel)"
lemma memberid_fdecl_simp1: "memberid (fdecl f) = fid (fst f)"
lemma disjoint_from_second_factor: "P x y \<and> \<not> O x (y \<otimes> z) \<Longrightarrow> \<not> O x z"
lemma length_corresp:"(\<exists>\<^sub>A tree_is. tree_array \<mapsto>\<^sub>a tree_is) = true \<Longrightarrow> return (length tree_is ) = Array_Time.len tree_array"
lemma outstanding_refs_non_volatile_Read\<^sub>s\<^sub>b_all_acquired_dropWhile: assumes consis: "reads_consistent pending_write \<O> m sb" assumes nvo: "non_volatile_owned_or_read_only pending_write \<S> \<O> sb" assumes out: "a \<in> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)" shows "a \<in> \<O> \<or> a \<in> all_acquired sb \<or> a \<in> read_only_reads \<O> sb"
lemma is_path_split'[simp]: "is_path v (p1@(u,w,u')#p2) v' \<longleftrightarrow> is_path v p1 u \<and> (u,w,u')\<in>E \<and> is_path u' p2 v'"
lemma acyclic_code[code] : "acyclic M = (\<not>(\<exists> p \<in> (acyclic_paths_up_to_length M (initial M) (size M - 1)) . \<exists> t \<in> transitions M . t_source t = target (initial M) p \<and> t_target t \<in> set (visited_states (initial M) p)))"
lemma mono_curry_left[simp]: "mono (curry \<circ> h) \<longleftrightarrow> mono h"
lemma conforms_upd_local: "[\|(x,(h, l))::\<preceq>(G, lT); G,h\<turnstile>v::\<preceq>T; lT va = Some T\|] ==> (x,(h, l(va\<mapsto>v)))::\<preceq>(G, lT)"
lemma supportE_single2: "supportE (\<lambda>l . P) = {}"
lemma skip_hoare_impl_r [hoare_safe]: "`p \<Rightarrow> q` \<Longrightarrow> \<lbrace>p\<rbrace>II\<lbrace>q\<rbrace>\<^sub>u"
lemma K4_imp_K2: "\<lbrakk> Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs; evs \<in> kerbV\<rbrakk> \<Longrightarrow> \<exists>Ta. Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<rbrace> \<in> set evs"
lemma restrict_relation_empty [simp]: "restrict_relation {} R = (\<lambda>_ _. False)"
lemma wcode_lemma: "args \<noteq> [] \<Longrightarrow> \<exists> stp ln rn. steps0 (Suc 0, [], <m # args>) (wcode_tm) stp = (0, [Bk], <[m ,bl_bin (<args>)]> @ Bk\<up>(rn))"
lemma record_call_below_arg: "record_call x \<cdot> f \<sqsubseteq> f"
lemma mult_eq': "(\<otimes>\<^bsub>G\<^esub>) = (\<lambda>x y. (x * y) mod n)"
lemma (in Module) has_free_generator_nonzeroring:" \<lbrakk>free_generator R M H; \<exists>p \<in> linear_span R M (carrier R) H. p \<noteq> \<zero> \<rbrakk> \<Longrightarrow> \<not> zeroring R"
lemma at_neg_neg:"\<Turnstile> (\<^bold>@c \<phi>) \<^bold>\<leftrightarrow> \<^bold>\<not>(\<^bold>@c \<^bold>\<not> \<phi>)"
lemma (in wf_digraph) subgraph_awalk_imp_awalk: assumes walk: "pre_digraph.awalk H u p v" assumes sub: "subgraph H G" shows "awalk u p v"
lemma f_image_eq_set_sublist_list: " list_all (\<lambda>i. i < length xs) ys \<Longrightarrow> xs `\<^sup>f (set ys) = set (sublist_list xs ys)"
lemma telescope_summable': fixes c :: "'a::real_normed_vector" assumes "f \<longlonglongrightarrow> c" shows "summable (\<lambda>n. f n - f (Suc n))"
lemma split0_stupid[simp]: "\<exists>x y. (x, y) = split0 p"
lemma \<phi>_\<psi>: assumes "t' \<in> S'.Univ" shows "\<phi> (\<psi> t') = t'"
lemma map_le_fun_upd2: "\<lbrakk> f \<subseteq>\<^sub>m g; x \<notin> dom f \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>m g(x := y)"
lemma PX5d_range: shows "Field (PX5d d) \<subseteq> {x. X5d x = d}"
lemma Block_\<tau>red1r_Some: "\<lbrakk> \<tau>red1gr uf P t h (e, xs[V := v]) (e', xs'); V < length xs \<rbrakk> \<Longrightarrow> \<tau>red1gr uf P t h ({V:Ty=\<lfloor>v\<rfloor>; e}, xs) ({V:Ty=None; e'}, xs')"
lemma local_sum_MAXSUM': \<open>local_sum arr = MAXSUM\<close> if \<open>k < length arr\<close> \<open>MAXSUM \<le> local_sum (take k arr) + arr ! k\<close> \<open>local_sum (take k arr) \<le> MAXSUM\<close> \<open>arr ! k \<le> MAXSUM\<close>
lemma d_dist_Sum: "ascending_chain f \<Longrightarrow> d(Sum f) = Sum (\<lambda>n . d(f n))"
lemma iFROM_inext_nth : "[n\<dots>] \<rightarrow> a = n + a"
lemma prv_subst_scnj: assumes "F \<subseteq> fmla" "finite F" "t \<in> trm" "x \<in> var" shows "prv (eqv (subst (scnj F) t x) (scnj ((\<lambda>\<phi>. subst \<phi> t x) ` F)))"
lemma scalingD[dest]: "\<lbrakk> scaling t; sound P; 0 \<le> c \<rbrakk> \<Longrightarrow> c * t P x = t (\<lambda>x. c * P x) x"
lemma filter_min_aux_relE: assumes "transp rel" and "x \<in> set xs" and "x \<notin> set (filter_min_aux xs ys)" obtains y where "y \<in> set (filter_min_aux xs ys)" and "rel y x"
lemma encode_kind_Cn: assumes "encode_kind (encode f) = 3" shows "\<exists>n f' gs. f = Cn n f' gs"
lemma iTILL_add_neg1: "k \<le> n \<Longrightarrow> [\<dots>n] \<oplus>- k = [\<dots>n-k]"
lemma mset_subset_eq_multiset_union_diff_commute: fixes A B C D :: "'a multiset" shows "B \<subseteq># A \<Longrightarrow> A - B + C = A + C - B"
lemma set_encode_vimage_Suc: "set_encode (Suc -` A) = set_encode A div 2"
lemma "\<forall>x::'a+'b. P x"
lemma (in is_tiny_ntsmcf) tiny_ntsmcf_in_Vset: "\<NN> \<in>\<^sub>\<circ> Vset \<alpha>"
lemma push_refine: assumes A: "(s,(p,D,pE))\<in>GS_rel" "(v,v')\<in>Id" assumes B: "v\<notin>\<Union>(set p)" "v\<notin>D" shows "(push_impl v s, push v' (p,D,pE))\<in>GS_rel"
lemma l12_18: assumes "Cong A B C D" and "Cong B C D A" and "\<not> Col A B C" and "B \<noteq> D" and "Col A P C" and "Col B P D" shows "A B Par C D \<and> B C Par D A \<and> B D TS A C \<and> A C TS B D"
theorem rev_is_rev [hoare_triple]: "xs \<noteq> [] \<Longrightarrow> <p \<mapsto>\<^sub>a xs> rev p 0 (length xs - 1) <\<lambda>_. p \<mapsto>\<^sub>a List.rev xs>"
lemma SLC_commute: "\<lbrakk> s\<^sub>3 = s\<^sub>3'; revision_step r' s\<^sub>2 s\<^sub>3; revision_step r s\<^sub>2' s\<^sub>3' \<rbrakk> \<Longrightarrow> s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
lemma lemma_iod: assumes "S \<subseteq> T" "S \<noteq> {}" and Tsub: "T \<subseteq> topspace(Euclidean_space n)" and S: "\<And>a b u. \<lbrakk>a \<in> S; b \<in> T; 0 < u; u < 1\<rbrakk> \<Longrightarrow> (\<lambda>i. (1 - u) * a i + u * b i) \<in> S" shows "path_connectedin (Euclidean_space n) T"
lemma two_div_sqrt_two [simp]: shows "2 * complex_of_real (sqrt (1/2)) = complex_of_real (sqrt 2)"
lemma iter_induct_isolate: "c\<^sup>\<star>;d \<sqinter> c\<^sup>\<infinity> = lfp (\<lambda> x. d \<sqinter> c;x)"
lemma dualA_iff [simp]: "pset (dual cl) = pset cl"
lemma compute_idx_range[simp,intro]: assumes "tup \<in> gamma_set" assumes "ys \<in> func_set" shows "compute_idx_set tup ys \<in> UNIV"
lemma meta_all2_eq_conv: "(\<And>a b. a = c \<Longrightarrow> PROP P a b) \<equiv> (\<And>b. PROP P c b)"
lemma rbb_rfd_dual: "\<partial> \<circ> bb\<^sub>\<R> R = fd\<^sub>\<R> (R\<inverse>) \<circ> \<partial>"
lemma Restr_tracl_comp_simps: "\<R> \<subseteq> X \<times> X \<Longrightarrow> \<L> \<subseteq> X \<times> X \<Longrightarrow> \<L>\<^sup>+ O \<R> \<subseteq> X \<times> X" "\<R> \<subseteq> X \<times> X \<Longrightarrow> \<L> \<subseteq> X \<times> X \<Longrightarrow> \<L> O \<R>\<^sup>+ \<subseteq> X \<times> X" "\<R> \<subseteq> X \<times> X \<Longrightarrow> \<L> \<subseteq> X \<times> X \<Longrightarrow> \<L>\<^sup>+ O \<R> O \<L>\<^sup>+ \<subseteq> X \<times> X"
lemma inv_P'PAQQ': assumes A: "A \<in> carrier_mat n n" and P: "P \<in> carrier_mat n n" and inv_P: "inverts_mat P' P" and inv_Q: "inverts_mat Q Q'" and Q: "Q \<in> carrier_mat n n" and P': "P' \<in> carrier_mat n n" and Q': "Q' \<in> carrier_mat n n" shows "(P'(PAQ)Q') = A"
lemma flatten_set_group_hom: assumes group:"group G" assumes inj:"inj_on rep (carrier G)" shows "rep \<in> hom G (flatten G rep)"
lemma SComplex_SReal_dense: "\<lbrakk>x \<in> SComplex; y \<in> SComplex; hcmod x < hcmod y \<rbrakk> \<Longrightarrow> \<exists>r \<in> Reals. hcmod x< r \<and> r < hcmod y"
lemma (in mpt) mpt_factor_is_qmpt_factor: assumes "mpt_factor proj M2 T2" shows "qmpt_factor proj M2 T2"
lemma Graph_T2: "Graph T2"
lemma waiting_code [code]: "waiting None = False" "\<And>w. waiting \<lfloor>PostWS w\<rfloor> = False" "\<And>w. waiting \<lfloor>InWS w\<rfloor> = True"
lemma Ia''m_calc: "$(Ia'')^{m}_n = (\<Sum>j<n. (j+1)/m * (\<Sum>k=jm..<(j+1)m. $v.^(k/m)))" if "m \<noteq> 0" for n m :: nat
lemma divide_closed: assumes "x \<in> carrier Zp" assumes "y \<in> carrier Zp" assumes "y \<noteq> \<zero>" shows "divide x y \<in> carrier Zp"
lemma BI_BC_rel: "(\<B>\<^sub>I \<phi>) = \<B>\<^sub>C(\<phi>\<^sup>d)"
lemma assumes "q' \<notin> \<int>" shows holomorphic_perzeta': "perzeta q' holomorphic_on A" and perzeta_altdef2: "Re s > 0 \<Longrightarrow> perzeta q' s = eval_fds (fds_perzeta q') s"
lemma Cons_Suc_inj2: "inj2 (\<lambda>x ys. Suc x # ys)"
lemma echelon_form_echelon_form_of: fixes A::"'a::{bezout_domain}^'cols::{mod_type}^'rows::{mod_type}" assumes ib: "is_bezout_ext bezout" shows "echelon_form (echelon_form_of A bezout)"
lemma to_metric_completion_dense': "closure (range to_metric_completion) = UNIV"
lemma wp_fun_mono [simp]: "mono wp"
lemma tree\<^sub>i_leaves: "leaves_up\<^sub>i u = leaves (tree\<^sub>i u)"
lemma T_size_tree_Node: "T_size_tree \<langle>l, x, r\<rangle> = T_size_tree l + T_size_tree r + 1"
lemma con_imp_coinitial_members_are_con: assumes "con \<T> \<U>" and "t \<in> \<T>" and "u \<in> \<U>" and "R.sources t = R.sources u" shows "t \<frown> u"
lemma alist_and_negation_type_to_match_expr_f_matches: "matches \<gamma> (alist_and (NegPos_map C spts)) a p \<longleftrightarrow> (\<forall>m\<in>set spts. matches \<gamma> (negation_type_to_match_expr_f C m) a p)"
lemma ctxt_of_gctxt_apply [simp]: "gterm_of_term (ctxt_of_gctxt C)\<langle>term_of_gterm t\<rangle> = C\<langle>t\<rangle>\<^sub>G"
lemma H_while_inv: "d p \<le> d i \<Longrightarrow> d i \<cdot> ad r \<le> d q \<Longrightarrow> H (d i \<cdot> d r) x i \<Longrightarrow> H p (while r inv i do x od) q"
lemma divisor_count_upper_bound_aux: fixes n :: nat shows "divisor_count n \<le> 2 * card {d. d dvd n \<and> d \<le> sqrt n}"
lemma epath_mono: "E \<subseteq> E' \<Longrightarrow> epath E u p v \<Longrightarrow> epath E' u p v"
lemma lemZEG: shows "z - e = g - e + (z - g)"
lemma seq_assoc [simp]: "\<turnstile> P c1;(c2;c3) Q \<longleftrightarrow> \<turnstile> P (c1;c2);c3 Q"
lemma conv_invol: "x \<rhd> 1 \<rhd> 1 = x"
lemma Bernstein_coeffs_sum: assumes "c \<noteq> d" and hP: "degree P \<le> p" shows "P = (\<Sum>j = 0..p. smult (nth_default 0 (Bernstein_coeffs p c d P) j) (Bernstein_Poly j p c d))"
lemma adjacent: "w\<in>W \<Longrightarrow> s\<in>S \<Longrightarrow> smap {w+s} \<sim> smap {w}"
lemma bin_snd_mismatch_all: assumes "xs \<in> lists {u\<^sub>0,u\<^sub>1}" shows "\<alpha> \<cdot> [c\<^sub>1] \<le>p u\<^sub>1 \<cdot> concat xs \<cdot> \<alpha>"
lemma assoc_simps [simp]: assumes "ide f" and "ide g" and "ide h" and "src f = trg g" and "src g = trg h" shows "arr (assoc\<^sub>S\<^sub>B f g h)" and "dom (assoc\<^sub>S\<^sub>B f g h) = (f \<star> g) \<star> h" and "cod (assoc\<^sub>S\<^sub>B f g h) = f \<star> g \<star> h" and "src (assoc\<^sub>S\<^sub>B f g h) = src h" and "trg (assoc\<^sub>S\<^sub>B f g h) = trg f"
lemma Times_mset_Int_distrib1: "(A \<inter># B) \<times># C = A \<times># C \<inter># B \<times># C"
lemma functor_G: shows "functor V\<^sub>D V\<^sub>C G"
lemma list_to_fun_Nil [simp]: "list_to_fun [] cs = 0"
lemma cond_hiding_empty_entails[simp]: "t hiding Map.empty under C entails R"
lemma structCongWeakCong: fixes \<Psi> :: 'b and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" assumes "P \<equiv>\<^sub>s Q" shows "P \<doteq>\<^sub>c Q"
lemma pth_eq_iff_index_eq: "pointermap_sane m \<Longrightarrow> pointermap_p_valid p1 m \<Longrightarrow> pointermap_p_valid p2 m \<Longrightarrow> (pm_pth m p1 = pm_pth m p2) \<longleftrightarrow> (p1 = p2)"
lemma (in Congruence_Rule) Ang_to_Tri : assumes "Def (Ang (An A B C))" shows "Def (Tri (Tr A B C))"
lemma beukers_key_inequality: fixes a :: int and b :: nat assumes "b > 0" and ab: "Re (zeta 3) = a / b" shows "2 * b * Re (zeta 3) * Lcm {1..n} ^ 3 / 27 ^ n \<ge> 1"
lemma Src_cod [simp]: assumes "arr \<mu>" shows "Src (cod \<mu>) = Src \<mu>"
lemma oexp_1 [simp]: "Ord \<alpha> \<Longrightarrow> 1\<up>\<alpha> = 1"
lemma filter_mset_eq_conv: "filter_mset P M = N \<longleftrightarrow> N \<subseteq># M \<and> (\<forall>b\<in>#N. P b) \<and> (\<forall>a\<in>#M - N. \<not> P a)" (is "?P \<longleftrightarrow> ?Q")
lemma zip_truncate_right : "zip xs ys = zip xs (take (length xs) ys)"

lemma no_\<tau>move2_\<tau>s_to_no_\<tau>move1: "\<lbrakk> s1 \<approx> s2; \<And>s2'. \<not> s2 -\<tau>2\<rightarrow> s2' \<rbrakk> \<Longrightarrow> \<exists>s1'. s1 -\<tau>1\<rightarrow>* s1' \<and> (\<forall>s1''. \<not> s1' -\<tau>1\<rightarrow> s1'') \<and> s1' \<approx> s2"

lemma prod_result_cases: obtains (val) s v where "r = (s, Rval v)" | (err) s err where "r = (s, Rerr err)"

lemma Inputs_imp_knows_Spy_secureM_srb: "\<lbrakk> Inputs Spy C X \<in> set evs; evs \<in> srb \<rbrakk> \<Longrightarrow> X \<in> knows Spy evs"

lemma subset_UnderS_AboveS: "B \<le> Field r \<Longrightarrow> B \<le> UnderS r (AboveS r B)"

lemma kSup_distl: "f \<circ>\<^sub>K (\<Squnion>G) = (\<Squnion>g \<in> G. f \<circ>\<^sub>K g)"

lemma select_index_3_subsets [simp]: shows "{j::nat. select_index 3 0 j} = {4,5,6,7} \<and> {j::nat. j < 8 \<and> \<not> select_index 3 0 j} = {0,1,2,3} \<and> {j::nat. select_index 3 1 j} = {2,3,6,7} \<and> {j::nat. j < 8 \<and> \<not> select_index 3 1 j} = {0,1,4,5}"

lemma aux10[rule_format]: "a \<in> set (net_list p) \<longrightarrow> a \<in> set (net_list_aux p)"

lemma fermat_little_theorem: assumes "prime (P :: nat)" shows "[x^P = x] (mod P)"

lemma act_emp_var: "\<alpha> x \<bottom> = \<bottom>"

lemma set1_FGco_bound: fixes x :: "(_, 'co1, 'co2, 'co3, 'co4, 'co5, 'co6, 'contra1, 'contra2, 'contra3, 'contra4, 'f1, 'f2) FGco" shows "card_of (set1_FGco x) <o (bd_FGco :: ('co1, 'co2, 'co3, 'co4, 'co5, 'co6, 'contra1, 'contra2, 'contra3, 'contra4, 'f1, 'f2) FGcobd rel)"

lemma memberid_fdecl_simp1: "memberid (fdecl f) = fid (fst f)"

lemma disjoint_from_second_factor: "P x y \<and> \<not> O x (y \<otimes> z) \<Longrightarrow> \<not> O x z"

lemma length_corresp:"(\<exists>\<^sub>A tree_is. tree_array \<mapsto>\<^sub>a tree_is) = true \<Longrightarrow> return (length tree_is ) = Array_Time.len tree_array"

lemma outstanding_refs_non_volatile_Read\<^sub>s\<^sub>b_all_acquired_dropWhile: assumes consis: "reads_consistent pending_write \<O> m sb" assumes nvo: "non_volatile_owned_or_read_only pending_write \<S> \<O> sb" assumes out: "a \<in> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)" shows "a \<in> \<O> \<or> a \<in> all_acquired sb \<or> a \<in> read_only_reads \<O> sb"

lemma is_path_split'[simp]: "is_path v (p1@(u,w,u')#p2) v' \<longleftrightarrow> is_path v p1 u \<and> (u,w,u')\<in>E \<and> is_path u' p2 v'"

lemma acyclic_code[code] : "acyclic M = (\<not>(\<exists> p \<in> (acyclic_paths_up_to_length M (initial M) (size M - 1)) . \<exists> t \<in> transitions M . t_source t = target (initial M) p \<and> t_target t \<in> set (visited_states (initial M) p)))"

lemma mono_curry_left[simp]: "mono (curry \<circ> h) \<longleftrightarrow> mono h"

lemma conforms_upd_local: "[|(x,(h, l))::\<preceq>(G, lT); G,h\<turnstile>v::\<preceq>T; lT va = Some T|] ==> (x,(h, l(va\<mapsto>v)))::\<preceq>(G, lT)"

lemma supportE_single2: "supportE (\<lambda>l . P) = {}"

lemma skip_hoare_impl_r [hoare_safe]: "`p \<Rightarrow> q` \<Longrightarrow> \<lbrace>p\<rbrace>II\<lbrace>q\<rbrace>\<^sub>u"

lemma K4_imp_K2: "\<lbrakk> Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs; evs \<in> kerbV\<rbrakk> \<Longrightarrow> \<exists>Ta. Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<rbrace> \<in> set evs"

lemma restrict_relation_empty [simp]: "restrict_relation {} R = (\<lambda>_ _. False)"

lemma wcode_lemma: "args \<noteq> [] \<Longrightarrow> \<exists> stp ln rn. steps0 (Suc 0, [], <m # args>) (wcode_tm) stp = (0, [Bk], <[m ,bl_bin (<args>)]> @ Bk\<up>(rn))"

lemma record_call_below_arg: "record_call x \<cdot> f \<sqsubseteq> f"

lemma mult_eq': "(\<otimes>\<^bsub>G\<^esub>) = (\<lambda>x y. (x * y) mod n)"

lemma (in Module) has_free_generator_nonzeroring:" \<lbrakk>free_generator R M H; \<exists>p \<in> linear_span R M (carrier R) H. p \<noteq> \<zero> \<rbrakk> \<Longrightarrow> \<not> zeroring R"

lemma at_neg_neg:"\<Turnstile> (\<^bold>@c \<phi>) \<^bold>\<leftrightarrow> \<^bold>\<not>(\<^bold>@c \<^bold>\<not> \<phi>)"

lemma (in wf_digraph) subgraph_awalk_imp_awalk: assumes walk: "pre_digraph.awalk H u p v" assumes sub: "subgraph H G" shows "awalk u p v"

lemma f_image_eq_set_sublist_list: " list_all (\<lambda>i. i < length xs) ys \<Longrightarrow> xs `\<^sup>f (set ys) = set (sublist_list xs ys)"

lemma telescope_summable': fixes c :: "'a::real_normed_vector" assumes "f \<longlonglongrightarrow> c" shows "summable (\<lambda>n. f n - f (Suc n))"

lemma split0_stupid[simp]: "\<exists>x y. (x, y) = split0 p"

lemma \<phi>_\<psi>: assumes "t' \<in> S'.Univ" shows "\<phi> (\<psi> t') = t'"

lemma map_le_fun_upd2: "\<lbrakk> f \<subseteq>\<^sub>m g; x \<notin> dom f \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>m g(x := y)"

lemma PX5d_range: shows "Field (PX5d d) \<subseteq> {x. X5d x = d}"

lemma Block_\<tau>red1r_Some: "\<lbrakk> \<tau>red1gr uf P t h (e, xs[V := v]) (e', xs'); V < length xs \<rbrakk> \<Longrightarrow> \<tau>red1gr uf P t h ({V:Ty=\<lfloor>v\<rfloor>; e}, xs) ({V:Ty=None; e'}, xs')"

lemma local_sum_MAXSUM': \<open>local_sum arr = MAXSUM\<close> if \<open>k < length arr\<close> \<open>MAXSUM \<le> local_sum (take k arr) + arr ! k\<close> \<open>local_sum (take k arr) \<le> MAXSUM\<close> \<open>arr ! k \<le> MAXSUM\<close>

lemma d_dist_Sum: "ascending_chain f \<Longrightarrow> d(Sum f) = Sum (\<lambda>n . d(f n))"

lemma iFROM_inext_nth : "[n\<dots>] \<rightarrow> a = n + a"

lemma prv_subst_scnj: assumes "F \<subseteq> fmla" "finite F" "t \<in> trm" "x \<in> var" shows "prv (eqv (subst (scnj F) t x) (scnj ((\<lambda>\<phi>. subst \<phi> t x) ` F)))"

lemma scalingD[dest]: "\<lbrakk> scaling t; sound P; 0 \<le> c \<rbrakk> \<Longrightarrow> c * t P x = t (\<lambda>x. c * P x) x"

lemma filter_min_aux_relE: assumes "transp rel" and "x \<in> set xs" and "x \<notin> set (filter_min_aux xs ys)" obtains y where "y \<in> set (filter_min_aux xs ys)" and "rel y x"

lemma encode_kind_Cn: assumes "encode_kind (encode f) = 3" shows "\<exists>n f' gs. f = Cn n f' gs"

lemma iTILL_add_neg1: "k \<le> n \<Longrightarrow> [\<dots>n] \<oplus>- k = [\<dots>n-k]"

lemma mset_subset_eq_multiset_union_diff_commute: fixes A B C D :: "'a multiset" shows "B \<subseteq># A \<Longrightarrow> A - B + C = A + C - B"

lemma set_encode_vimage_Suc: "set_encode (Suc -` A) = set_encode A div 2"

lemma "\<forall>x::'a+'b. P x"

lemma (in is_tiny_ntsmcf) tiny_ntsmcf_in_Vset: "\<NN> \<in>\<^sub>\<circ> Vset \<alpha>"

lemma push_refine: assumes A: "(s,(p,D,pE))\<in>GS_rel" "(v,v')\<in>Id" assumes B: "v\<notin>\<Union>(set p)" "v\<notin>D" shows "(push_impl v s, push v' (p,D,pE))\<in>GS_rel"

lemma l12_18: assumes "Cong A B C D" and "Cong B C D A" and "\<not> Col A B C" and "B \<noteq> D" and "Col A P C" and "Col B P D" shows "A B Par C D \<and> B C Par D A \<and> B D TS A C \<and> A C TS B D"

theorem rev_is_rev [hoare_triple]: "xs \<noteq> [] \<Longrightarrow> \<^sub>a xs> rev p 0 (length xs - 1) <\<lambda>_. p \<mapsto>\<^sub>a List.rev xs>"

lemma SLC_commute: "\<lbrakk> s\<^sub>3 = s\<^sub>3'; revision_step r' s\<^sub>2 s\<^sub>3; revision_step r s\<^sub>2' s\<^sub>3' \<rbrakk> \<Longrightarrow> s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"

lemma lemma_iod: assumes "S \<subseteq> T" "S \<noteq> {}" and Tsub: "T \<subseteq> topspace(Euclidean_space n)" and S: "\<And>a b u. \<lbrakk>a \<in> S; b \<in> T; 0 < u; u < 1\<rbrakk> \<Longrightarrow> (\<lambda>i. (1 - u) * a i + u * b i) \<in> S" shows "path_connectedin (Euclidean_space n) T"

lemma two_div_sqrt_two [simp]: shows "2 * complex_of_real (sqrt (1/2)) = complex_of_real (sqrt 2)"

lemma iter_induct_isolate: "c\<^sup>\<star>;d \<sqinter> c\<^sup>\<infinity> = lfp (\<lambda> x. d \<sqinter> c;x)"

lemma dualA_iff [simp]: "pset (dual cl) = pset cl"

lemma compute_idx_range[simp,intro]: assumes "tup \<in> gamma_set" assumes "ys \<in> func_set" shows "compute_idx_set tup ys \<in> UNIV"

lemma meta_all2_eq_conv: "(\<And>a b. a = c \<Longrightarrow> PROP P a b) \<equiv> (\<And>b. PROP P c b)"

lemma rbb_rfd_dual: "\<partial> \<circ> bb\<^sub>\<R> R = fd\<^sub>\<R> (R\<inverse>) \<circ> \<partial>"

lemma Restr_tracl_comp_simps: "\<R> \<subseteq> X \<times> X \<Longrightarrow> \<L> \<subseteq> X \<times> X \<Longrightarrow> \<L>\<^sup>+ O \<R> \<subseteq> X \<times> X" "\<R> \<subseteq> X \<times> X \<Longrightarrow> \<L> \<subseteq> X \<times> X \<Longrightarrow> \<L> O \<R>\<^sup>+ \<subseteq> X \<times> X" "\<R> \<subseteq> X \<times> X \<Longrightarrow> \<L> \<subseteq> X \<times> X \<Longrightarrow> \<L>\<^sup>+ O \<R> O \<L>\<^sup>+ \<subseteq> X \<times> X"

lemma inv_P'PAQQ': assumes A: "A \<in> carrier_mat n n" and P: "P \<in> carrier_mat n n" and inv_P: "inverts_mat P' P" and inv_Q: "inverts_mat Q Q'" and Q: "Q \<in> carrier_mat n n" and P': "P' \<in> carrier_mat n n" and Q': "Q' \<in> carrier_mat n n" shows "(P'*(P*A*Q)*Q') = A"

lemma flatten_set_group_hom: assumes group:"group G" assumes inj:"inj_on rep (carrier G)" shows "rep \<in> hom G (flatten G rep)"

lemma SComplex_SReal_dense: "\<lbrakk>x \<in> SComplex; y \<in> SComplex; hcmod x < hcmod y \<rbrakk> \<Longrightarrow> \<exists>r \<in> Reals. hcmod x< r \<and> r < hcmod y"

lemma (in mpt) mpt_factor_is_qmpt_factor: assumes "mpt_factor proj M2 T2" shows "qmpt_factor proj M2 T2"

lemma Graph_T2: "Graph T2"

lemma waiting_code [code]: "waiting None = False" "\<And>w. waiting \<lfloor>PostWS w\<rfloor> = False" "\<And>w. waiting \<lfloor>InWS w\<rfloor> = True"

lemma Ia''m_calc: "$(Ia'')^{m}_n = (\<Sum>j<n. (j+1)/m * (\<Sum>k=j*m..<(j+1)*m. $v.^(k/m)))" if "m \<noteq> 0" for n m :: nat

lemma divide_closed: assumes "x \<in> carrier Zp" assumes "y \<in> carrier Zp" assumes "y \<noteq> \<zero>" shows "divide x y \<in> carrier Zp"

lemma BI_BC_rel: "(\\<^sub>I \<phi>) = \\<^sub>C(\<phi>\<^sup>d)"

lemma assumes "q' \<notin> \<int>" shows holomorphic_perzeta': "perzeta q' holomorphic_on A" and perzeta_altdef2: "Re s > 0 \<Longrightarrow> perzeta q' s = eval_fds (fds_perzeta q') s"

lemma Cons_Suc_inj2: "inj2 (\<lambda>x ys. Suc x # ys)"

lemma echelon_form_echelon_form_of: fixes A::"'a::{bezout_domain}^'cols::{mod_type}^'rows::{mod_type}" assumes ib: "is_bezout_ext bezout" shows "echelon_form (echelon_form_of A bezout)"

lemma to_metric_completion_dense': "closure (range to_metric_completion) = UNIV"

lemma wp_fun_mono [simp]: "mono wp"

lemma tree\<^sub>i_leaves: "leaves_up\<^sub>i u = leaves (tree\<^sub>i u)"

lemma T_size_tree_Node: "T_size_tree \<langle>l, x, r\<rangle> = T_size_tree l + T_size_tree r + 1"

lemma con_imp_coinitial_members_are_con: assumes "con \<T> \" and "t \<in> \<T>" and "u \<in> \" and "R.sources t = R.sources u" shows "t \<frown> u"

lemma alist_and_negation_type_to_match_expr_f_matches: "matches \<gamma> (alist_and (NegPos_map C spts)) a p \<longleftrightarrow> (\<forall>m\<in>set spts. matches \<gamma> (negation_type_to_match_expr_f C m) a p)"

lemma ctxt_of_gctxt_apply [simp]: "gterm_of_term (ctxt_of_gctxt C)\<langle>term_of_gterm t\<rangle> = C\<langle>t\<rangle>\<^sub>G"

lemma H_while_inv: "d p \<le> d i \<Longrightarrow> d i \<cdot> ad r \<le> d q \<Longrightarrow> H (d i \<cdot> d r) x i \<Longrightarrow> H p (while r inv i do x od) q"

lemma divisor_count_upper_bound_aux: fixes n :: nat shows "divisor_count n \<le> 2 * card {d. d dvd n \<and> d \<le> sqrt n}"

lemma epath_mono: "E \<subseteq> E' \<Longrightarrow> epath E u p v \<Longrightarrow> epath E' u p v"

lemma lemZEG: shows "z - e = g - e + (z - g)"

lemma seq_assoc [simp]: "\<turnstile> P c1;(c2;c3) Q \<longleftrightarrow> \<turnstile> P (c1;c2);c3 Q"

lemma conv_invol: "x \<rhd> 1 \<rhd> 1 = x"

lemma Bernstein_coeffs_sum: assumes "c \<noteq> d" and hP: "degree P \<le> p" shows "P = (\<Sum>j = 0..p. smult (nth_default 0 (Bernstein_coeffs p c d P) j) (Bernstein_Poly j p c d))"

lemma adjacent: "w\<in>W \<Longrightarrow> s\<in>S \<Longrightarrow> smap {w+s} \<sim> smap {w}"

lemma bin_snd_mismatch_all: assumes "xs \<in> lists {u\<^sub>0,u\<^sub>1}" shows "\<alpha> \<cdot> [c\<^sub>1] \<le>p u\<^sub>1 \<cdot> concat xs \<cdot> \<alpha>"

lemma assoc_simps [simp]: assumes "ide f" and "ide g" and "ide h" and "src f = trg g" and "src g = trg h" shows "arr (assoc\<^sub>S\<^sub>B f g h)" and "dom (assoc\<^sub>S\<^sub>B f g h) = (f \<star> g) \<star> h" and "cod (assoc\<^sub>S\<^sub>B f g h) = f \<star> g \<star> h" and "src (assoc\<^sub>S\<^sub>B f g h) = src h" and "trg (assoc\<^sub>S\<^sub>B f g h) = trg f"

lemma Times_mset_Int_distrib1: "(A \<inter># B) \<times># C = A \<times># C \<inter># B \<times># C"

lemma functor_G: shows "functor V\<^sub>D V\<^sub>C G"

lemma list_to_fun_Nil [simp]: "list_to_fun [] cs = 0"

lemma cond_hiding_empty_entails[simp]: "t hiding Map.empty under C entails R"

lemma structCongWeakCong: fixes \<Psi> :: 'b and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" assumes "P \<equiv>\<^sub>s Q" shows "P \<doteq>\<^sub>c Q"

lemma pth_eq_iff_index_eq: "pointermap_sane m \<Longrightarrow> pointermap_p_valid p1 m \<Longrightarrow> pointermap_p_valid p2 m \<Longrightarrow> (pm_pth m p1 = pm_pth m p2) \<longleftrightarrow> (p1 = p2)"

lemma (in Congruence_Rule) Ang_to_Tri : assumes "Def (Ang (An A B C))" shows "Def (Tri (Tr A B C))"

lemma beukers_key_inequality: fixes a :: int and b :: nat assumes "b > 0" and ab: "Re (zeta 3) = a / b" shows "2 * b * Re (zeta 3) * Lcm {1..n} ^ 3 / 27 ^ n \<ge> 1"

lemma Src_cod [simp]: assumes "arr \<mu>" shows "Src (cod \<mu>) = Src \<mu>"

lemma oexp_1 [simp]: "Ord \<alpha> \<Longrightarrow> 1\<up>\<alpha> = 1"

lemma filter_mset_eq_conv: "filter_mset P M = N \<longleftrightarrow> N \<subseteq># M \<and> (\<forall>b\<in>#N. P b) \<and> (\<forall>a\<in>#M - N. \<not> P a)" (is "?P \<longleftrightarrow> ?Q")

lemma zip_truncate_right : "zip xs ys = zip xs (take (length xs) ys)"