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

Statement: stringlengths 7 24.3k
lemma conv_no_start_points_path: "no_start_points_path x \<longleftrightarrow> no_end_points_path (x\<^sup>T)"
lemma (in infinite_coin_toss_space) nat_filtration_Suc_sets: shows "sets (nat_filtration n) \<subseteq> sets (nat_filtration (Suc n))"
lemma seq_mono_left: "c\<^sub>0 \<sqsubseteq> c\<^sub>1 \<Longrightarrow> c\<^sub>0 ; d \<sqsubseteq> c\<^sub>1 ; d"
lemma substitute_below_singlesI: assumes "t \<sqsubseteq> singles S" assumes "\<And> x. carrier (f x) \<inter> S = {}" shows "substitute f T t \<sqsubseteq> singles S"
lemma table_classes_Main [simp]: "table_of Classes Main = Some MainCl"
lemma "x - (x - y) = (y::real)"
lemma not_comp_lcp: assumes "\<not> r \<bowtie> s" shows "f (r \<and>\<^sub>p s) \<cdot> \<alpha> = f r \<cdot> f (r \<cdot> s) \<and>\<^sub>p f s \<cdot> f (r \<cdot> s)"
lemma marl_append_hnr_aux: "(uncurry marl_append,uncurry (RETURN oo op_list_append)) \<in> [\<lambda>(l,_). length l<N]\<^sub>a ((is_ms_array_list N)\<^sup>d *\<^sub>a id_assn\<^sup>k) \<rightarrow> is_ms_array_list N"
lemma (in category) cat_finite_pcategory_cat_singleton: assumes "j \<in>\<^sub>\<circ> Vset \<alpha>" shows "finite_pcategory \<alpha> (set {j}) (\<lambda>i. \<CC>)"
lemma map_in_list_rel_conv: shows "(l, l') \<in> \<langle>br \<alpha> I\<rangle>list_rel \<longleftrightarrow> ((\<forall>x\<in>set l. I x) \<and> l'=map \<alpha> l)"
lemma isCont_tan' [simp,continuous_intros]: fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a" shows "isCont f a \<Longrightarrow> cos (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
lemma InSync_\<tau>red0t_xt: "\<tau>red0t extTA P t h (e, xs) (e', xs') \<Longrightarrow> \<tau>red0t extTA P t h (insync\<^bsub>V\<^esub> (a) e, xs) (insync\<^bsub>V\<^esub> (a) e', xs')"
lemma lifting_wf_thread_conf: "lifting_wf JVM_final (mexecd P) (\<lambda>t x m. P,m \<turnstile> t \<surd>t)"
lemma comm_monoid_rng_of_frac: shows "comm_monoid (rec_monoid_rng_of_frac)"
lemma conv_concat_helper: assumes "(s,ls) \<in> conv ars" and "ss2 \<in> conv ars" and "lst_conv (s,ls) = fst ss2" shows "(s,ls@snd ss2) \<in> conv ars \<and> (lst_conv (s,ls@snd ss2) = lst_conv ss2)"
lemma under_dim:"rat_poly.row_length (brickmat under) = 4" and "length (brickmat under) = 4"
lemma cond_spmf_fst_map_prod_inj: "cond_spmf_fst (map_spmf (\<lambda>(x, y). (f x, g x y)) p) (f x) = map_spmf (g x) (cond_spmf_fst p x)" if "inj f"
lemma keys_nth [simp]: "keys (nth xs) = fst ` {(n, v) \<in> set (enumerate 0 xs). v \<noteq> 0}"
lemma transitionI: fixes P :: pi and Rs :: residual and P' :: pi shows "P \<Longrightarrow>\<^sub>l Rs \<Longrightarrow> P \<Longrightarrow>\<^sub>l\<^sup>^Rs" and "P \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P"
lemma [code]: "is_target step \<tau>s pc' = list_ex (\<lambda>pc. pc' \<noteq> pc+1 \<and> List.member (map fst (step pc (\<tau>s!pc))) pc') [0..<size \<tau>s]"
lemma Field_restrict_rel_subset: "Field (restrict_rel A R) \<subseteq> A \<inter> Field R"
lemma transpose_row_code [code abstract]: "vec_nth (transpose_row A i) = (%j. A $ j $ i)"
lemma measurable_expr_sem[measurable]: assumes "\<Gamma> \<turnstile> e : t" and "free_vars e \<subseteq> V" shows "(\<lambda>\<sigma>. expr_sem \<sigma> e) \<in> measurable (state_measure V \<Gamma>) (subprob_algebra (stock_measure t))"
lemma abc_list_crsp_steps: "\<lbrakk>abc_steps_l (0, lm @ 0\<up>m) aprog stp = (a, lm'); aprog \<noteq> []\<rbrakk> \<Longrightarrow> \<exists> lma. abc_steps_l (0, lm) aprog stp = (a, lma) \<and> abc_list_crsp lm' lma"
lemma filter_filter_mset_cond_simp: assumes "\<And> a . P a \<Longrightarrow> Q a" shows "filter_mset P A = filter_mset P (filter_mset Q A)"
lemma rel_sum_map1: "rel_sum R1 R2 (map_sum f1 f2 x) y \<longleftrightarrow> rel_sum (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
lemma val_stuck: fixes e::exp assumes val_e: "isval e" shows "stuck e"
lemma mulextp_cong[fundef_cong]: assumes "xs1 = ys1" and "xs2 = ys2" and "\<And> x x'. x \<in># ys1 \<Longrightarrow> x' \<in># ys2 \<Longrightarrow> f x x' = g x x'" shows "mulextp f xs1 xs2 = mulextp g ys1 ys2"
lemma attracting_fixed_point_neq_repelling_fixed_point: assumes "loxodromic_isometry f" shows "attracting_fixed_point f \<noteq> repelling_fixed_point f"
lemma preserve_mat_rep_num: assumes "inj_on_01_hom f" assumes "i < dim_row M" shows "mat_rep_num M i = mat_rep_num (map_mat f M) i"
lemma sq_mtx_inv_mult: assumes "mtx_invertible A" and "mtx_invertible B" shows "(A * B)\<^sup>-\<^sup>1 = B\<^sup>-\<^sup>1 * A\<^sup>-\<^sup>1"
lemma (in ring) indexed_pmult_zero [simp]: shows "indexed_pmult (indexed_const \<zero>) i = indexed_const \<zero>"
lemma correctness: "protocol (m0,m1) c = funct_OT_12 (m0,m1) c"
lemma convergent_powser'_eventually: assumes "convergent_powser' cs f" shows "eventually (\<lambda>x. powser cs x = f x) (nhds 0)"
lemma vars_llist_alt_def: \<open>vars_llist xs \<subseteq> \<V> \<longleftrightarrow> vars_of_poly_in xs \<V>\<close>
lemma term_to_sig_ctxt_apply [simp]: "ctxt_well_def_hole_path \<F> C \<Longrightarrow> term_to_sig \<F> v C\<langle>s\<rangle> = (inv_const_ctxt \<F> v C)\<langle>term_to_sig \<F> v s\<rangle>"
lemma fun_rel_mono[relator_props]: "\<lbrakk> R1\<subseteq>R2; R3\<subseteq>R4 \<rbrakk> \<Longrightarrow> R2\<rightarrow>R3 \<subseteq> R1\<rightarrow>R4"
lemma try_bind_spmf_lossless2: "lossless_spmf q \<Longrightarrow> TRY (bind_spmf p f) ELSE q = TRY (p \<bind> (\<lambda>x. TRY (f x) ELSE q)) ELSE q"
lemma setsetmapim_comp: "(f\<circ>g)\<turnstile>A = f\<turnstile>(g\<turnstile>A)"
lemma has_integral_dominated_convergence_at_top: fixes f :: "real \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space" assumes "\<And>k. (f k has_integral y k) s" "h integrable_on s" "\<And>k x. x\<in>s \<Longrightarrow> norm (f k x) \<le> h x" "\<forall>x\<in>s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) at_top" and x: "(y \<longlongrightarrow> x) at_top" shows "(g has_integral x) s"
lemma nonneg_cexprI_shift: assumes "\<And>x \<sigma>. x \<in> type_universe t \<Longrightarrow> \<sigma> \<in> space (state_measure V \<Gamma>) \<Longrightarrow> 0 \<le> extract_real (cexpr_sem (case_nat x \<sigma>) e)" shows "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) e"
lemma parCompose: fixes P :: pi and Q :: pi and R :: pi and T :: pi and Rel :: "(pi \<times> pi) set" and Rel' :: "(pi \<times> pi) set" and Rel'' :: "(pi \<times> pi) set" assumes PSimQ: "P \<leadsto>[Rel] Q" and RSimT: "R \<leadsto>[Rel'] S" and PRelQ: "(P, Q) \<in> Rel" and RRel'T: "(R, S) \<in> Rel'" and Par: "\<And>P' Q' R' S'. \<lbrakk>(P', Q') \<in> Rel; (R', S') \<in> Rel'\<rbrakk> \<Longrightarrow> (P' \<parallel> R', Q' \<parallel> S') \<in> Rel''" and Res: "\<And>S T x. (S, T) \<in> Rel'' \<Longrightarrow> (<\<nu>x>S, <\<nu>x>T) \<in> Rel''" shows "P \<parallel> R \<leadsto>[Rel''] Q \<parallel> S"
lemma list_map_autoref_delete2[param]: shows "(list_map_delete (=), op_map_delete) \<in> Id \<rightarrow> br map_of list_map_invar \<rightarrow> br map_of list_map_invar"
lemma LtK_Nonce: "LtK X \<noteq> Nonce X'"
lemma thr_conv [simp]: "thr (ls, (ts, m), ws) = ts"
lemma discr_raw_coind: assumes : "phi c" and : "\<And> c s c' s'. \<lbrakk>phi c; (c,s) \<rightarrow>c (c',s')\<rbrakk> \<Longrightarrow> s \<approx> s' \<and> phi c'" and **: "\<And> c s s'. \<lbrakk>phi c; (c,s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> s \<approx> s'" shows "discr c"
lemma ac_Diff_iff: "c \<in>\<^sub>A A - B \<longleftrightarrow> c \<in>\<^sub>A A \<and> c \<notin>\<^sub>A B"
lemma object_ptr_kinds_M_reads: "reads (\<Union>object_ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing)}) object_ptr_kinds_M h h'"
lemma tree_chop_ap_tree [simp]: "tree_chop (f \<diamondop> x) = tree_chop f \<diamondop> tree_chop x"
lemma fls_of_int: "(of_int i :: 'a::ring_1 fls) = fls_const (of_int i)"
lemma FvarsT_rawpsubstT: assumes "r \<in> atrm" and "snd ` (set txs) \<subseteq> var" and "fst ` (set txs) \<subseteq> atrm" and "distinct (map snd txs)" and "\<forall> x \<in> snd ` (set txs). \<forall> t \<in> fst ` (set txs). x \<notin> FvarsT t" shows "FvarsT (rawpsubstT r txs) = (FvarsT r - snd ` (set txs)) \<union> (\<Union> {if x \<in> FvarsT r then FvarsT t else {} \| t x . (t,x) \<in> set txs})"
lemma allDefs_ign[simp]: "CFG_SSA_base.allDefs (ign gen_ssa_defs g) (ign Mapping.lookup (gen_phis g)) ga n = CFG_SSA_base.allDefs gen_ssa_defs (\<lambda>g. Mapping.lookup (gen_phis g)) g n"
lemma new_shadow_root_get_disconnected_nodes_is_l_new_shadow_root_get_disconnected_nodes [instances]: "l_new_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs"
lemma ex_closed [simp]: \<open>\<exists>m. closed m p\<close>
lemma bres_anti: "x \<le> y \<Longrightarrow> y \<rightarrow> z \<le> x \<rightarrow> z"
lemma get_elements_by_tag_name_is_strongly_scdom_component_safe: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "h \<turnstile> get_elements_by_tag_name ptr idd \<rightarrow>\<^sub>r results" assumes "h \<turnstile> get_elements_by_tag_name ptr idd \<rightarrow>\<^sub>h h'" shows "is_strongly_scdom_component_safe {ptr} (cast ` set results) h h'"
lemma (in semicategory) op_smc_is_idem_arr[smc_op_simps]: "f : \<mapsto>\<^sub>i\<^sub>d\<^sub>e\<^bsub>op_smc \<CC>\<^esub> b \<longleftrightarrow> f : \<mapsto>\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<CC>\<^esub> b"
lemma fmap_freshness_lemma: fixes h :: "('a::at,'b::pt) fmap" assumes a: "\<exists>a. atom a \<sharp> (h, h $$ a)" shows "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h $$ a = x"
lemma last_append [simp]: "last (p, xs @ ys) = last (last (p, xs), ys)"
lemma fv\<^sub>s\<^sub>t_is_subterm_trms\<^sub>s\<^sub>t: "x \<in> fv\<^sub>s\<^sub>t A \<Longrightarrow> Var x \<in> subterms\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t A)"
lemma omap_eq_append_conv[simp]: "omap f xs = Some (ys\<^sub>1@ys\<^sub>2) \<longleftrightarrow> (\<exists>xs\<^sub>1 xs\<^sub>2. xs=xs\<^sub>1@xs\<^sub>2 \<and> omap f xs\<^sub>1 = Some ys\<^sub>1 \<and> omap f xs\<^sub>2 = Some ys\<^sub>2)"
lemma minus_Real: "cauchy X \<Longrightarrow> - Real X = Real (\<lambda>n. - X n)"
lemma assumes wf:"wf_prog wf_md P" and typeof:" P \<turnstile> typeof\<^bsub>h\<^esub> v = Some T'" and type:"is_type P T" shows sub_casts:"P \<turnstile> T' \<le> T \<Longrightarrow> \<exists>v'. P \<turnstile> T casts v to v'"
lemma udom_uupdate_pfun [simp]: fixes m :: "(('k, 'v) pfun, '\<alpha>) uexpr" shows "dom\<^sub>u(m(k \<mapsto> v)\<^sub>u) = {k}\<^sub>u \<union>\<^sub>u dom\<^sub>u(m)"
lemma test_sup_distributive: "test x \<Longrightarrow> sup_distributive x"
lemma accs_exclusive_aux: "\<lbrakk> accs \<delta>n n q; \<delta>n=\<delta>\<union>\<delta>'; \<delta>_states \<delta> \<inter> \<delta>_states \<delta>' = {}; q\<in>\<delta>_states \<delta> \<rbrakk> \<Longrightarrow> accs \<delta> n q"
lemma n_eq_1_iff: "s \<in> S \<Longrightarrow> n s = 1 \<longleftrightarrow> (\<forall>cfg\<in>cfg_on s. v cfg = 1)"
lemma bind_spmf_spmf_of_set: "\<And>A. \<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> bind_spmf (spmf_of_set A) = bind_pmf (pmf_of_set A)"
lemma sum_upd_C: "sum I (s(C := x)) = sum I s"
lemma Unassigned_botdefault: "\<forall> s r. (nP r) \<noteq> DontCare \<longrightarrow> \<not> allowed_flow (nP s) s (nP r) r \<longrightarrow> \<not> allowed_flow DontCare s (nP r) r"
lemma finite_zero_cardD [dest!]: "\<lbrakk> card A = 0; finite A \<rbrakk> \<Longrightarrow> A = {}"
lemma lconf_hext [elim]: "[\| G,h\<turnstile>l[::\<preceq>]L; h\<le>\|h' \|] ==> G,h'\<turnstile>l[::\<preceq>]L"
lemma scalar_prod_right_unit[simp]: assumes i: "i < n" shows "(v :: 'a :: semiring_1 vec) \<bullet> unit_vec n i = v $ i"
lemma "\<sim>TND \<^bold>\<not>\<^sup>I \<and> Fr_1 \<F> \<and> Fr_3 \<F> \<and> Fr_4 \<F> \<and> DM3 \<^bold>\<not>\<^sup>I"
lemma load_cond_cap_frag: assumes "load h c t = Success ret" and "\<exists> x n. ret = Cap_v_frag x n" shows "\<exists> m mc tagval tg nth_frag. Mapping.lookup (heap_map h) (block_id c) = Some (Map m) \<and> offset c \<ge> fst (bounds m) \<and> offset c + \|t\|\<^sub>\<tau> \<le> snd (bounds m) \<and> (is_contiguous_bytes (content m) (nat (offset c)) \|t\|\<^sub>\<tau> \<longrightarrow> is_contiguous_zeros (content m) (nat (offset c)) \|t\|\<^sub>\<tau> \<and> ret = Cap_v NULL) \<and> (\<not> is_contiguous_bytes (content m) (nat (offset c)) \|t\|\<^sub>\<tau> \<longrightarrow> is_cap (content m) (nat (offset c)) \<and> mc = get_cap (content m) (nat (offset c)) \<and> (t = Uint8 \<or> t = Sint8) \<and> tagval = the (Mapping.lookup (tags m) (nat (offset c))) \<and> tg = (case perm_cap_load c of False \<Rightarrow> False \| True \<Rightarrow> tagval) \<and> nth_frag = of_nth (the (Mapping.lookup (content m) (nat (offset c)))))"
lemma state_update_after_is_action: "(relation_of Ac ;; R (true \<turnstile> (\<lambda>(A, A'). sc (more A, more A') & \<not>wait A' & tr A = tr A'))) \<in> {p. is_CSP_process p}"
lemma (in group) r_inv_ex [simp]: "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"
theorem Spy_not_see_NA: "\<lbrakk>Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs; A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk> \<Longrightarrow> Nonce NA \<notin> analz (knows Spy evs)"
lemma lset_ltake: "(\<And>m. m < n \<Longrightarrow> lnth xs m \<in> A) \<Longrightarrow> lset (ltake (enat n) xs) \<subseteq> A"
lemma sig_trd_aux_irred: assumes "fst args0 \<in> keys (rep_list (snd args0))" and "\<And>b s. b \<in> set bs \<Longrightarrow> rep_list b \<noteq> 0 \<Longrightarrow> fst args0 \<prec> s + punit.lt (rep_list b) \<Longrightarrow> s \<oplus> lt b \<prec>\<^sub>t lt (snd (args0)) \<Longrightarrow> lookup (rep_list (snd args0)) (s + punit.lt (rep_list b)) = 0" shows "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) (set bs) (sig_trd_aux args0)"
lemma plus_adds_pp_0: assumes "(s + t) adds\<^sub>p v" shows "s adds\<^sub>p (v \<ominus> t)"
lemma SINK_plus_current: "SINK (plus_current f g) = SINK f \<inter> SINK g"
lemma vector_derivative_diff_chain_within: assumes Df: "(f has_vector_derivative f') (at x within S)" and Dg: "(g has_derivative g') (at (f x) within f`S)" shows "((g \<circ> f) has_vector_derivative (g' f')) (at x within S)"
lemma top_on_opt_widen: "top_on_opt o1 X \<Longrightarrow> top_on_opt o2 X \<Longrightarrow> top_on_opt (o1 \<nabla> o2 :: _ st option) X"
lemma "\<sim>TND \<^bold>\<not>\<^sup>I \<and> Fr_1 \<F> \<and> Fr_3 \<F> \<and> Fr_4 \<F> \<and> CoP2 \<^bold>\<not>\<^sup>I"
lemma swapnorm_isnpoly [simp]: assumes "SORT_CONSTRAINT('a::field_char_0)" shows "isnpoly (swapnorm n m p)"
lemma create_document_is_weakly_dom_component_safe_step: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "h \<turnstile> create_document \<rightarrow>\<^sub>h h'" assumes "ptr \<noteq> cast \|h \<turnstile> create_document\|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
lemma constant_function_comp_is_constant_seq: assumes "a \<in> carrier R" assumes "s \<in> closed_seqs R" shows "is_constant_seq R ((const a) \<circ> s)"
lemma set_gC[simp]: "set (gC c) = gL ` (set c) \<union> glitOfC c"
lemma (in CRR_market) delta_hedging_trading_strat: assumes "N = bernoulli_stream q" and "0 < q" and "q < 1" and "der \<in> borel_measurable (G matur)" shows "trading_strategy (delta_hedging N der matur)"
lemma "ECQm \<^bold>\<not>"
lemma ide_prod [intro, simp]: assumes "ide a" and "ide b" shows "ide (a \<otimes> b)"
lemma (in Module) indmhom_someTr:"\<lbrakk>R module N; f \<in> mHom R M N; X \<in> set_mr_cos M (ker\<^bsub>M,N\<^esub> f)\<rbrakk> \<Longrightarrow> f (SOME xa. xa \<in> X) \<in> f `(carrier M)"
lemma length_filter_conv_size_filter_mset: "length (filter P xs) = size (filter_mset P (mset xs))"
lemma mu_isotone: "has_least_prefixpoint f \<Longrightarrow> has_least_prefixpoint g \<Longrightarrow> isotone f \<Longrightarrow> isotone g \<Longrightarrow> f \<le>\<le> g \<Longrightarrow> \<mu> f \<le> \<mu> g"
lemma jordan_matrix_concat_diag_block_mat: "jordan_matrix (concat jbs) = diag_block_mat (map jordan_matrix jbs)"
lemma comp_by_index_inj: "comp_by_index x1 y1 = comp_by_index x2 y2 \<Longrightarrow> x1=x2 \<and> y1=y2"
lemma ntcf_ntsmcf_ntcf_0: "ntcf_ntsmcf (ntcf_0 \<AA>) = ntsmcf_0 (cat_smc \<AA>)"
lemma bigtheta_const_ln_pow' [landau_simp]: "0 < a \<Longrightarrow> (\<lambda>x::real. ln (x * a) ^ p) \<in> \<Theta>(\<lambda>x. ln x ^ p)"
lemma sep_conj_exists1: "((EXS x. P x) Q) = (EXS x. (P x Q))"

lemma conv_no_start_points_path: "no_start_points_path x \<longleftrightarrow> no_end_points_path (x\<^sup>T)"

lemma (in infinite_coin_toss_space) nat_filtration_Suc_sets: shows "sets (nat_filtration n) \<subseteq> sets (nat_filtration (Suc n))"

lemma seq_mono_left: "c\<^sub>0 \<sqsubseteq> c\<^sub>1 \<Longrightarrow> c\<^sub>0 ; d \<sqsubseteq> c\<^sub>1 ; d"

lemma substitute_below_singlesI: assumes "t \<sqsubseteq> singles S" assumes "\<And> x. carrier (f x) \<inter> S = {}" shows "substitute f T t \<sqsubseteq> singles S"

lemma table_classes_Main [simp]: "table_of Classes Main = Some MainCl"

lemma "x - (x - y) = (y::real)"

lemma not_comp_lcp: assumes "\<not> r \<bowtie> s" shows "f (r \<and>\<^sub>p s) \<cdot> \<alpha> = f r \<cdot> f (r \<cdot> s) \<and>\<^sub>p f s \<cdot> f (r \<cdot> s)"

lemma marl_append_hnr_aux: "(uncurry marl_append,uncurry (RETURN oo op_list_append)) \<in> [\<lambda>(l,_). length l<N]\<^sub>a ((is_ms_array_list N)\<^sup>d *\<^sub>a id_assn\<^sup>k) \<rightarrow> is_ms_array_list N"

lemma (in category) cat_finite_pcategory_cat_singleton: assumes "j \<in>\<^sub>\<circ> Vset \<alpha>" shows "finite_pcategory \<alpha> (set {j}) (\<lambda>i. \<CC>)"

lemma map_in_list_rel_conv: shows "(l, l') \<in> \<langle>br \<alpha> I\<rangle>list_rel \<longleftrightarrow> ((\<forall>x\<in>set l. I x) \<and> l'=map \<alpha> l)"

lemma isCont_tan' [simp,continuous_intros]: fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a" shows "isCont f a \<Longrightarrow> cos (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"

lemma InSync_\<tau>red0t_xt: "\<tau>red0t extTA P t h (e, xs) (e', xs') \<Longrightarrow> \<tau>red0t extTA P t h (insync\<^bsub>V\<^esub> (a) e, xs) (insync\<^bsub>V\<^esub> (a) e', xs')"

lemma lifting_wf_thread_conf: "lifting_wf JVM_final (mexecd P) (\<lambda>t x m. P,m \<turnstile> t \<surd>t)"

lemma comm_monoid_rng_of_frac: shows "comm_monoid (rec_monoid_rng_of_frac)"

lemma conv_concat_helper: assumes "(s,ls) \<in> conv ars" and "ss2 \<in> conv ars" and "lst_conv (s,ls) = fst ss2" shows "(s,ls@snd ss2) \<in> conv ars \<and> (lst_conv (s,ls@snd ss2) = lst_conv ss2)"

lemma under_dim:"rat_poly.row_length (brickmat under) = 4" and "length (brickmat under) = 4"

lemma cond_spmf_fst_map_prod_inj: "cond_spmf_fst (map_spmf (\<lambda>(x, y). (f x, g x y)) p) (f x) = map_spmf (g x) (cond_spmf_fst p x)" if "inj f"

lemma keys_nth [simp]: "keys (nth xs) = fst ` {(n, v) \<in> set (enumerate 0 xs). v \<noteq> 0}"

lemma transitionI: fixes P :: pi and Rs :: residual and P' :: pi shows "P \<Longrightarrow>\<^sub>l Rs \<Longrightarrow> P \<Longrightarrow>\<^sub>l\<^sup>^Rs" and "P \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P"

lemma [code]: "is_target step \<tau>s pc' = list_ex (\<lambda>pc. pc' \<noteq> pc+1 \<and> List.member (map fst (step pc (\<tau>s!pc))) pc') [0..<size \<tau>s]"

lemma Field_restrict_rel_subset: "Field (restrict_rel A R) \<subseteq> A \<inter> Field R"

lemma transpose_row_code [code abstract]: "vec_nth (transpose_row A i) = (%j. A $ j $ i)"

lemma measurable_expr_sem[measurable]: assumes "\<Gamma> \<turnstile> e : t" and "free_vars e \<subseteq> V" shows "(\<lambda>\<sigma>. expr_sem \<sigma> e) \<in> measurable (state_measure V \<Gamma>) (subprob_algebra (stock_measure t))"

lemma abc_list_crsp_steps: "\<lbrakk>abc_steps_l (0, lm @ 0\<up>m) aprog stp = (a, lm'); aprog \<noteq> []\<rbrakk> \<Longrightarrow> \<exists> lma. abc_steps_l (0, lm) aprog stp = (a, lma) \<and> abc_list_crsp lm' lma"

lemma filter_filter_mset_cond_simp: assumes "\<And> a . P a \<Longrightarrow> Q a" shows "filter_mset P A = filter_mset P (filter_mset Q A)"

lemma rel_sum_map1: "rel_sum R1 R2 (map_sum f1 f2 x) y \<longleftrightarrow> rel_sum (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"

lemma val_stuck: fixes e::exp assumes val_e: "isval e" shows "stuck e"

lemma mulextp_cong[fundef_cong]: assumes "xs1 = ys1" and "xs2 = ys2" and "\<And> x x'. x \<in># ys1 \<Longrightarrow> x' \<in># ys2 \<Longrightarrow> f x x' = g x x'" shows "mulextp f xs1 xs2 = mulextp g ys1 ys2"

lemma attracting_fixed_point_neq_repelling_fixed_point: assumes "loxodromic_isometry f" shows "attracting_fixed_point f \<noteq> repelling_fixed_point f"

lemma preserve_mat_rep_num: assumes "inj_on_01_hom f" assumes "i < dim_row M" shows "mat_rep_num M i = mat_rep_num (map_mat f M) i"

lemma sq_mtx_inv_mult: assumes "mtx_invertible A" and "mtx_invertible B" shows "(A * B)\<^sup>-\<^sup>1 = B\<^sup>-\<^sup>1 * A\<^sup>-\<^sup>1"

lemma (in ring) indexed_pmult_zero [simp]: shows "indexed_pmult (indexed_const \<zero>) i = indexed_const \<zero>"

lemma correctness: "protocol (m0,m1) c = funct_OT_12 (m0,m1) c"

lemma convergent_powser'_eventually: assumes "convergent_powser' cs f" shows "eventually (\<lambda>x. powser cs x = f x) (nhds 0)"

lemma vars_llist_alt_def: \<open>vars_llist xs \<subseteq> \<V> \<longleftrightarrow> vars_of_poly_in xs \<V>\<close>

lemma term_to_sig_ctxt_apply [simp]: "ctxt_well_def_hole_path \<F> C \<Longrightarrow> term_to_sig \<F> v C\<langle>s\<rangle> = (inv_const_ctxt \<F> v C)\<langle>term_to_sig \<F> v s\<rangle>"

lemma fun_rel_mono[relator_props]: "\<lbrakk> R1\<subseteq>R2; R3\<subseteq>R4 \<rbrakk> \<Longrightarrow> R2\<rightarrow>R3 \<subseteq> R1\<rightarrow>R4"

lemma try_bind_spmf_lossless2: "lossless_spmf q \<Longrightarrow> TRY (bind_spmf p f) ELSE q = TRY (p \<bind> (\<lambda>x. TRY (f x) ELSE q)) ELSE q"

lemma setsetmapim_comp: "(f\<circ>g)\<turnstile>A = f\<turnstile>(g\<turnstile>A)"

lemma has_integral_dominated_convergence_at_top: fixes f :: "real \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space" assumes "\<And>k. (f k has_integral y k) s" "h integrable_on s" "\<And>k x. x\<in>s \<Longrightarrow> norm (f k x) \<le> h x" "\<forall>x\<in>s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) at_top" and x: "(y \<longlongrightarrow> x) at_top" shows "(g has_integral x) s"

lemma nonneg_cexprI_shift: assumes "\<And>x \<sigma>. x \<in> type_universe t \<Longrightarrow> \<sigma> \<in> space (state_measure V \<Gamma>) \<Longrightarrow> 0 \<le> extract_real (cexpr_sem (case_nat x \<sigma>) e)" shows "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) e"

lemma parCompose: fixes P :: pi and Q :: pi and R :: pi and T :: pi and Rel :: "(pi \<times> pi) set" and Rel' :: "(pi \<times> pi) set" and Rel'' :: "(pi \<times> pi) set" assumes PSimQ: "P \<leadsto>[Rel] Q" and RSimT: "R \<leadsto>[Rel'] S" and PRelQ: "(P, Q) \<in> Rel" and RRel'T: "(R, S) \<in> Rel'" and Par: "\<And>P' Q' R' S'. \<lbrakk>(P', Q') \<in> Rel; (R', S') \<in> Rel'\<rbrakk> \<Longrightarrow> (P' \<parallel> R', Q' \<parallel> S') \<in> Rel''" and Res: "\<And>S T x. (S, T) \<in> Rel'' \<Longrightarrow> (<\<nu>x>S, <\<nu>x>T) \<in> Rel''" shows "P \<parallel> R \<leadsto>[Rel''] Q \<parallel> S"

lemma list_map_autoref_delete2[param]: shows "(list_map_delete (=), op_map_delete) \<in> Id \<rightarrow> br map_of list_map_invar \<rightarrow> br map_of list_map_invar"

lemma LtK_Nonce: "LtK X \<noteq> Nonce X'"

lemma thr_conv [simp]: "thr (ls, (ts, m), ws) = ts"

lemma discr_raw_coind: assumes *: "phi c" and **: "\<And> c s c' s'. \<lbrakk>phi c; (c,s) \<rightarrow>c (c',s')\<rbrakk> \<Longrightarrow> s \<approx> s' \<and> phi c'" and ***: "\<And> c s s'. \<lbrakk>phi c; (c,s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> s \<approx> s'" shows "discr c"

lemma ac_Diff_iff: "c \<in>\<^sub>A A - B \<longleftrightarrow> c \<in>\<^sub>A A \<and> c \<notin>\<^sub>A B"

lemma object_ptr_kinds_M_reads: "reads (\<Union>object_ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing)}) object_ptr_kinds_M h h'"

lemma tree_chop_ap_tree [simp]: "tree_chop (f \<diamondop> x) = tree_chop f \<diamondop> tree_chop x"

lemma fls_of_int: "(of_int i :: 'a::ring_1 fls) = fls_const (of_int i)"

lemma FvarsT_rawpsubstT: assumes "r \<in> atrm" and "snd ` (set txs) \<subseteq> var" and "fst ` (set txs) \<subseteq> atrm" and "distinct (map snd txs)" and "\<forall> x \<in> snd ` (set txs). \<forall> t \<in> fst ` (set txs). x \<notin> FvarsT t" shows "FvarsT (rawpsubstT r txs) = (FvarsT r - snd ` (set txs)) \<union> (\<Union> {if x \<in> FvarsT r then FvarsT t else {} | t x . (t,x) \<in> set txs})"

lemma allDefs_ign[simp]: "CFG_SSA_base.allDefs (ign gen_ssa_defs g) (ign Mapping.lookup (gen_phis g)) ga n = CFG_SSA_base.allDefs gen_ssa_defs (\<lambda>g. Mapping.lookup (gen_phis g)) g n"

lemma new_shadow_root_get_disconnected_nodes_is_l_new_shadow_root_get_disconnected_nodes [instances]: "l_new_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs"

lemma ex_closed [simp]: \<open>\<exists>m. closed m p\<close>

lemma bres_anti: "x \<le> y \<Longrightarrow> y \<rightarrow> z \<le> x \<rightarrow> z"

lemma get_elements_by_tag_name_is_strongly_scdom_component_safe: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "h \<turnstile> get_elements_by_tag_name ptr idd \<rightarrow>\<^sub>r results" assumes "h \<turnstile> get_elements_by_tag_name ptr idd \<rightarrow>\<^sub>h h'" shows "is_strongly_scdom_component_safe {ptr} (cast ` set results) h h'"

lemma (in semicategory) op_smc_is_idem_arr[smc_op_simps]: "f : \<mapsto>\<^sub>i\<^sub>d\<^sub>e\<^bsub>op_smc \<CC>\<^esub> b \<longleftrightarrow> f : \<mapsto>\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<CC>\<^esub> b"

lemma fmap_freshness_lemma: fixes h :: "('a::at,'b::pt) fmap" assumes a: "\<exists>a. atom a \<sharp> (h, h $$ a)" shows "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h $$ a = x"

lemma last_append [simp]: "last (p, xs @ ys) = last (last (p, xs), ys)"

lemma fv\<^sub>s\<^sub>t_is_subterm_trms\<^sub>s\<^sub>t: "x \<in> fv\<^sub>s\<^sub>t A \<Longrightarrow> Var x \<in> subterms\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t A)"

lemma omap_eq_append_conv[simp]: "omap f xs = Some (ys\<^sub>1@ys\<^sub>2) \<longleftrightarrow> (\<exists>xs\<^sub>1 xs\<^sub>2. xs=xs\<^sub>1@xs\<^sub>2 \<and> omap f xs\<^sub>1 = Some ys\<^sub>1 \<and> omap f xs\<^sub>2 = Some ys\<^sub>2)"

lemma minus_Real: "cauchy X \<Longrightarrow> - Real X = Real (\<lambda>n. - X n)"

lemma assumes wf:"wf_prog wf_md P" and typeof:" P \<turnstile> typeof\<^bsub>h\<^esub> v = Some T'" and type:"is_type P T" shows sub_casts:"P \<turnstile> T' \<le> T \<Longrightarrow> \<exists>v'. P \<turnstile> T casts v to v'"

lemma udom_uupdate_pfun [simp]: fixes m :: "(('k, 'v) pfun, '\<alpha>) uexpr" shows "dom\<^sub>u(m(k \<mapsto> v)\<^sub>u) = {k}\<^sub>u \<union>\<^sub>u dom\<^sub>u(m)"

lemma test_sup_distributive: "test x \<Longrightarrow> sup_distributive x"

lemma accs_exclusive_aux: "\<lbrakk> accs \<delta>n n q; \<delta>n=\<delta>\<union>\<delta>'; \<delta>_states \<delta> \<inter> \<delta>_states \<delta>' = {}; q\<in>\<delta>_states \<delta> \<rbrakk> \<Longrightarrow> accs \<delta> n q"

lemma n_eq_1_iff: "s \<in> S \<Longrightarrow> n s = 1 \<longleftrightarrow> (\<forall>cfg\<in>cfg_on s. v cfg = 1)"

lemma bind_spmf_spmf_of_set: "\<And>A. \<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> bind_spmf (spmf_of_set A) = bind_pmf (pmf_of_set A)"

lemma sum_upd_C: "sum I (s(C := x)) = sum I s"

lemma Unassigned_botdefault: "\<forall> s r. (nP r) \<noteq> DontCare \<longrightarrow> \<not> allowed_flow (nP s) s (nP r) r \<longrightarrow> \<not> allowed_flow DontCare s (nP r) r"

lemma finite_zero_cardD [dest!]: "\<lbrakk> card A = 0; finite A \<rbrakk> \<Longrightarrow> A = {}"

lemma lconf_hext [elim]: "[| G,h\<turnstile>l[::\<preceq>]L; h\<le>|h' |] ==> G,h'\<turnstile>l[::\<preceq>]L"

lemma scalar_prod_right_unit[simp]: assumes i: "i < n" shows "(v :: 'a :: semiring_1 vec) \<bullet> unit_vec n i = v $ i"

lemma "\<sim>TND \<^bold>\<not>\<^sup>I \<and> Fr_1 \<F> \<and> Fr_3 \<F> \<and> Fr_4 \<F> \<and> DM3 \<^bold>\<not>\<^sup>I"

lemma load_cond_cap_frag: assumes "load h c t = Success ret" and "\<exists> x n. ret = Cap_v_frag x n" shows "\<exists> m mc tagval tg nth_frag. Mapping.lookup (heap_map h) (block_id c) = Some (Map m) \<and> offset c \<ge> fst (bounds m) \<and> offset c + |t|\<^sub>\<tau> \<le> snd (bounds m) \<and> (is_contiguous_bytes (content m) (nat (offset c)) |t|\<^sub>\<tau> \<longrightarrow> is_contiguous_zeros (content m) (nat (offset c)) |t|\<^sub>\<tau> \<and> ret = Cap_v NULL) \<and> (\<not> is_contiguous_bytes (content m) (nat (offset c)) |t|\<^sub>\<tau> \<longrightarrow> is_cap (content m) (nat (offset c)) \<and> mc = get_cap (content m) (nat (offset c)) \<and> (t = Uint8 \<or> t = Sint8) \<and> tagval = the (Mapping.lookup (tags m) (nat (offset c))) \<and> tg = (case perm_cap_load c of False \<Rightarrow> False | True \<Rightarrow> tagval) \<and> nth_frag = of_nth (the (Mapping.lookup (content m) (nat (offset c)))))"

lemma state_update_after_is_action: "(relation_of Ac ;; R (true \<turnstile> (\<lambda>(A, A'). sc (more A, more A') & \<not>wait A' & tr A = tr A'))) \<in> {p. is_CSP_process p}"

lemma (in group) r_inv_ex [simp]: "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"

theorem Spy_not_see_NA: "\<lbrakk>Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs; A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk> \<Longrightarrow> Nonce NA \<notin> analz (knows Spy evs)"

lemma lset_ltake: "(\<And>m. m < n \<Longrightarrow> lnth xs m \<in> A) \<Longrightarrow> lset (ltake (enat n) xs) \<subseteq> A"

lemma sig_trd_aux_irred: assumes "fst args0 \<in> keys (rep_list (snd args0))" and "\<And>b s. b \<in> set bs \<Longrightarrow> rep_list b \<noteq> 0 \<Longrightarrow> fst args0 \<prec> s + punit.lt (rep_list b) \<Longrightarrow> s \<oplus> lt b \<prec>\<^sub>t lt (snd (args0)) \<Longrightarrow> lookup (rep_list (snd args0)) (s + punit.lt (rep_list b)) = 0" shows "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) (set bs) (sig_trd_aux args0)"

lemma plus_adds_pp_0: assumes "(s + t) adds\<^sub>p v" shows "s adds\<^sub>p (v \<ominus> t)"

lemma SINK_plus_current: "SINK (plus_current f g) = SINK f \<inter> SINK g"

lemma vector_derivative_diff_chain_within: assumes Df: "(f has_vector_derivative f') (at x within S)" and Dg: "(g has_derivative g') (at (f x) within f`S)" shows "((g \<circ> f) has_vector_derivative (g' f')) (at x within S)"

lemma top_on_opt_widen: "top_on_opt o1 X \<Longrightarrow> top_on_opt o2 X \<Longrightarrow> top_on_opt (o1 \<nabla> o2 :: _ st option) X"

lemma "\<sim>TND \<^bold>\<not>\<^sup>I \<and> Fr_1 \<F> \<and> Fr_3 \<F> \<and> Fr_4 \<F> \<and> CoP2 \<^bold>\<not>\<^sup>I"

lemma swapnorm_isnpoly [simp]: assumes "SORT_CONSTRAINT('a::field_char_0)" shows "isnpoly (swapnorm n m p)"

lemma create_document_is_weakly_dom_component_safe_step: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "h \<turnstile> create_document \<rightarrow>\<^sub>h h'" assumes "ptr \<noteq> cast |h \<turnstile> create_document|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"

lemma constant_function_comp_is_constant_seq: assumes "a \<in> carrier R" assumes "s \<in> closed_seqs R" shows "is_constant_seq R ((const a) \<circ> s)"

lemma set_gC[simp]: "set (gC c) = gL ` (set c) \<union> glitOfC c"

lemma (in CRR_market) delta_hedging_trading_strat: assumes "N = bernoulli_stream q" and "0 < q" and "q < 1" and "der \<in> borel_measurable (G matur)" shows "trading_strategy (delta_hedging N der matur)"

lemma "ECQm \<^bold>\<not>"

lemma ide_prod [intro, simp]: assumes "ide a" and "ide b" shows "ide (a \<otimes> b)"

lemma (in Module) indmhom_someTr:"\<lbrakk>R module N; f \<in> mHom R M N; X \<in> set_mr_cos M (ker\<^bsub>M,N\<^esub> f)\<rbrakk> \<Longrightarrow> f (SOME xa. xa \<in> X) \<in> f `(carrier M)"

lemma length_filter_conv_size_filter_mset: "length (filter P xs) = size (filter_mset P (mset xs))"

lemma mu_isotone: "has_least_prefixpoint f \<Longrightarrow> has_least_prefixpoint g \<Longrightarrow> isotone f \<Longrightarrow> isotone g \<Longrightarrow> f \<le>\<le> g \<Longrightarrow> \<mu> f \<le> \<mu> g"

lemma jordan_matrix_concat_diag_block_mat: "jordan_matrix (concat jbs) = diag_block_mat (map jordan_matrix jbs)"

lemma comp_by_index_inj: "comp_by_index x1 y1 = comp_by_index x2 y2 \<Longrightarrow> x1=x2 \<and> y1=y2"

lemma ntcf_ntsmcf_ntcf_0: "ntcf_ntsmcf (ntcf_0 \<AA>) = ntsmcf_0 (cat_smc \<AA>)"

lemma bigtheta_const_ln_pow' [landau_simp]: "0 < a \<Longrightarrow> (\<lambda>x::real. ln (x * a) ^ p) \<in> \<Theta>(\<lambda>x. ln x ^ p)"

lemma sep_conj_exists1: "((EXS x. P x) ** Q) = (EXS x. (P x ** Q))"