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

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lemma wprepare_goto_start_pos_Bk_nonempty_fst[simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
lemma awalk_decomp: assumes "awalk u p v" assumes "w \<in> set (awalk_verts u p)" shows "\<exists>q r. p = q @ r \<and> awalk u q w \<and> awalk w r v"
lemma curry_uncurry: assumes "functor A1 A2_B.comp F" and "functor A1 A2_B.comp G" and "natural_transformation A1 A2_B.comp F G \<tau>" shows "curry (uncurry F) (uncurry G) (uncurry \<tau>) = \<tau>"
lemma connected_impl_maximally_connected: assumes "connected_graph H" assumes subgraph: "subgraph H G" shows "maximally_connected H G"
lemma torder_sync_order: "torder_on (sactions P E) (sync_order P E)"
lemma mset_ran_mem[simp, intro]: "finite r \<Longrightarrow> i\<in>r \<Longrightarrow> a i \<in># mset_ran a r"
lemma numsubst0_numbound0: assumes nb: "numbound0 t" shows "numbound0 (numsubst0 t a)"
lemma set_sel'_symm: "sel_symm f \<Longrightarrow> set_sel' f X Y = set_sel' f Y X"
lemma deduct_proj_priv_term_prefix_ex: assumes A: "ik\<^sub>s\<^sub>t (proj_unl l A) \<cdot>\<^sub>s\<^sub>e\<^sub>t I \<turnstile> t" and t: "\<not>{} \<turnstile>\<^sub>c t" shows "\<exists>B k s. (k = \<star> \<or> k = ln l) \<and> prefix (B@[(k,receive\<langle>s\<rangle>\<^sub>s\<^sub>t)]) A \<and> declassified\<^sub>l\<^sub>s\<^sub>t ((B@[(k,receive\<langle>s\<rangle>\<^sub>s\<^sub>t)])) I = declassified\<^sub>l\<^sub>s\<^sub>t A I \<and> ik\<^sub>s\<^sub>t (proj_unl l (B@[(k,receive\<langle>s\<rangle>\<^sub>s\<^sub>t)])) = ik\<^sub>s\<^sub>t (proj_unl l A)"
lemma wprepare_loop_goon_Bk_nonempty[simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"
lemma rev_bl_order_simps: "rev_bl_order F [] [] = F" "rev_bl_order F (x # xs) (y # ys) = rev_bl_order ((y \<and> \<not> x) \<or> ((y \<or> \<not> x) \<and> F)) xs ys"
lemma consistent_signs_atw: assumes "\<And>p. p \<in> set fs \<Longrightarrow> poly p x \<noteq> 0" shows "consistent_sign_vec_copr fs x = signs_at fs x"
lemma wf_rulesetD: assumes "wf_ruleset \<gamma> p (r # rs)" shows "wf_ruleset \<gamma> p [r]" and "wf_ruleset \<gamma> p rs"
lemma arr_runit [simp]: assumes "ide a" shows "arr \<r>[a]"
lemma sublist_appendI [simp, intro]: "sublist xs (ps @ xs @ ss)"
lemma orth_at2_tsp__ts: assumes "P \<noteq> Q" and "P OrthAt A B C P X" and "Q OrthAt A B C Q Y" and "A B C TSP X Y" shows "P Q TS X Y"
theorem normal_form: assumes "recfn n f" obtains i where "\<forall>x. e_length x = n \<longrightarrow> eval r_normal_form [i, x] = eval f (list_decode x)"
lemma trms\<^sub>s\<^sub>s\<^sub>t_proj_subset[simp]: "trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n A) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (unlabel A)" (is ?A) "trms\<^sub>s\<^sub>s\<^sub>t (proj_unl m (proj n A)) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n A)" (is ?B) "trms\<^sub>s\<^sub>s\<^sub>t (proj_unl m (proj n A)) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (proj_unl m A)" (is ?C)
lemma ctrm_trunc: assumes "p \<in> carrier P" assumes "degree p >0" shows "zcf(trunc p) = zcf p"
lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f :: 'a::comm_semiring_1 fps)) $ 0 = of_nat (fact k) * f $ k"
lemma tLength_g0_conv: "(tLength t > 0) \<longleftrightarrow> (\<exists>s t'. t = t' \<leadsto> s \<and> tLength t = Suc (tLength t'))"
lemma sum_emeasure': assumes [simp]: "finite A" assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> B x \<in> sets M" assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> emeasure M (B x \<inter> B y) = 0" shows "(\<Sum>x\<in>A. emeasure M (B x)) = emeasure M (\<Union>x\<in>A. B x)"
lemma ratfps_inverse_code [code abstract]: "quot_of_ratfps (inverse x) = (let (a,b) = quot_of_ratfps x in if coeff a 0 = 0 then (0, 1) else let u = unit_factor a in (b div u, a div u))"
lemma nextVertex_in_edges: "v \<in> \<V> f \<Longrightarrow> (v, f \<bullet> v) \<in> edges f"
lemma r_unit_add_rng_of_frac: assumes "(r, s) \<in> carrier rel" shows "(r \|\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> \<zero>\<^bsub>rec_rng_of_frac\<^esub> = (r \|\<^bsub>rel\<^esub> s)"
lemma assumes deriv: "\<And>y. a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {a..})" assumes deriv': "\<And>z t x. z \<in> U \<Longrightarrow> x \<ge> a \<Longrightarrow> t \<in> {a..x} \<Longrightarrow> ((\<lambda>z. f z t) has_field_derivative f' z t) (at z within U)" assumes cont: "continuous_on (U \<times> {of_int a..}) (\<lambda>(z, t). f' z t)" assumes int: "\<And>b c z e. a \<le> b \<Longrightarrow> z \<in> U \<Longrightarrow> (\<lambda>t. of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}" assumes int': "\<And>a' b y. y \<in> U \<Longrightarrow> a \<le> a' \<Longrightarrow> a' \<le> b \<Longrightarrow> (\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}" assumes conv: "convergent (\<lambda>y. G (real y))" assumes bound: "eventually (\<lambda>x. \<forall>y\<in>U. norm (f y x) \<le> g x) at_top" assumes "open U" shows analytic_EM_remainder: "(\<lambda>s::complex. EM_remainder n (f s) a) analytic_on U" and holomorphic_EM_remainder: "(\<lambda>s::complex. EM_remainder n (f s) a) holomorphic_on U"
lemma iUntil_disj_distrib: " (P t1. t1 \<U> t2 I. (Q1 t2 \<or> Q2 t2)) = ((P t1. t1 \<U> t2 I. Q1 t2) \<or> (P t1. t1 \<U> t2 I. Q2 t2))"
lemma i_drop_the_conv: " f \<Up> k = (THE g. (\<exists>xs. length xs = k \<and> xs \<frown> g = f))"
lemma is_process3: "is_process P \<Longrightarrow> \<forall> s t. (s @ t,{}) \<in> FAILURES P \<longrightarrow> (s, {}) \<in> FAILURES P"
lemma inj_map_prim: assumes "inj_on f A" and "u \<in> lists A" and "primitive u" shows "primitive (map f u)"
lemma connecting_paths_sym_length: "i \<in> connecting_paths u v \<Longrightarrow> \<exists>j\<in>connecting_paths v u. (walk_length j) = (walk_length i)"
lemma join_distr: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
lemma inner_node_Entry_edge: assumes "inner_node n" obtains a where "valid_edge a" and "inner_node (targetnode a)" and "sourcenode a = (_Entry_)"
lemma asIMOD_igAbs2: fixes SEM :: "('index,'bindex,'varSort,'sort,'opSym,'sTerm)semDom" assumes : "sWlsDisj SEM" and *: "igWls (asIMOD SEM) s X" shows "igAbs (asIMOD SEM) xs x X = (\<lambda>val. if sWlsVal SEM val then sAbs xs (\<lambda>sX. if sWls SEM (asSort xs) sX then X (val (x := sX)_xs) else sDummy SEM s) else undefined)"
lemma fpxs_of_fls_compose_power [simp]: "fpxs_of_fls (fls_compose_power f d) = fpxs_compose_power (fpxs_of_fls f) (of_nat d)"
lemma both_mono2: "paths t \<subseteq> paths t' \<Longrightarrow> paths (t'' \<otimes>\<otimes> t) \<subseteq> paths (t'' \<otimes>\<otimes> t')"
lemma F_rec: "F n = F (n-1) + 1 + (\<Sum>i=m..<n. F (n-i-1))" if \<open>n>m\<close> "m > 0"
lemma callee_of_rest_simps [simp]: "callee_of_rest rest s (Inl iadv_rest) = map_spmf (apfst Inl) (rfunc_adv rest s iadv_rest)" "callee_of_rest rest s (Inr iusr_rest) = map_spmf (apfst Inr) (rfunc_usr rest s iusr_rest)"
lemma conditionalize: assumes "finite A" shows "F.F g A = G (\<lambda>a. if a \<in> A then g a else \<^bold>1)"
lemma bind_single: "P \<bind> single = P"
lemma [code]: \<open>mask (Suc n) = push_bit n (1 :: 'a) OR mask n\<close> \<open>mask 0 = (0 :: 'a)\<close>
lemma last_ladder_\<gamma>: assumes is_ladder: "is_ladder D L" assumes ladder_last_n: "ladder_last_n L = length D" shows "ladder_\<gamma> a D L (length L - Suc 0) = Derive a D"
lemma USUP_empty [simp]: "(\<Squnion> i \<in> {} \<bullet> P(i)) = true"
lemma rbt_eq_iff: "t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2"
lemma dsteps_to_steps: "a \<in> st.dstep ^^ n \<Longrightarrow> a \<in> st.step ^^ n"
lemma PO_m1a_leak_refines_m1x_leak: "{R1x1a} (m1x_leak Rs), (m1a_leak Rs) {> R1x1a}"
lemma vec_contains_row_elements_mat: assumes "i < dim_row M" assumes "a \<in>$ row M i" shows "a \<in> elements_mat M"
lemma seq_filter_UNIV [simp]: "xs \<restriction>\<^sub>l UNIV = xs"
lemma simplePath_empty_conv[simp]: "isSimplePath s [] t \<longleftrightarrow> s=t"
lemma Some_in_opt [iff]: "(Some x \<in> opt A) = (x\<in>A)"
lemma interior_of_eq_empty: "X interior_of S = {} \<longleftrightarrow> (\<forall>T. openin X T \<and> T \<subseteq> S \<longrightarrow> T = {})"
lemma deg_not_zero_imp_not_unit: fixes f:: "'a::{idom_divide,semidom_divide_unit_factor} poly" assumes deg_f: "degree f > 0" shows "\<not> is_unit f"
lemma wf_PLUS[simp]: "wf n (PLUS xs) = (\<forall>r \<in> set xs. wf n r)"
lemma inj_axis: assumes "c \<noteq> 0" shows "inj (\<lambda>k. axis k c :: ('a :: {zero}) ^ 'n)"
lemma assumes ms: "measure_space M" and un: "(\<Union>i\<in>R. B i) = UNIV" and fin: "finite (R::nat set)" and dis: "\<forall>j1\<in>R. \<forall>j2\<in>R. j1 \<noteq> j2 \<longrightarrow> (B j1) \<inter> (B j2) = {}" and meas: "\<forall>j\<in>R. B j \<in> measurable_sets M" and Ameas: "A \<in> measurable_sets M" shows measure_split: "measure M A = (\<Sum>j\<in>R. measure M (A \<inter> B j))"
lemma itypeI: "(c::'t) ::\<^sub>i I"
lemma analz_insert_Crypt: "Key (invKey K) \<notin> analz H \<Longrightarrow> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
lemma complement_p: "x \<sqinter> y = bot \<Longrightarrow> x \<squnion> y = top \<Longrightarrow> -x = y"
lemma trms\<^sub>s\<^sub>s\<^sub>t_memI[intro?]: "send\<langle>ts\<rangle> \<in> set S \<Longrightarrow> t \<in> set ts \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "receive\<langle>ts\<rangle> \<in> set S \<Longrightarrow> t \<in> set ts \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "\<langle>ac: t \<doteq> s\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "\<langle>ac: t \<doteq> s\<rangle> \<in> set S \<Longrightarrow> s \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "insert\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "insert\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> s \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "delete\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "delete\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> s \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"
lemma it_context_true: "-p * (-p \<rhd> x) = -p * x"
lemma maddux2c: "(a \<lhd> x) \<sqinter> y \<le> a \<lhd> (x \<sqinter> (y \<rhd> a))"
lemma valid_insert_both_member_options_pres: "invar_vebt t n \<Longrightarrow> x<2^n \<Longrightarrow> y < 2^n \<Longrightarrow> both_member_options t x \<Longrightarrow> both_member_options (vebt_insert t y) x"
lemma higher_deriv_diff: fixes z::complex assumes "f holomorphic_on S" "g holomorphic_on S" "open S" "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
theorem soundness: "ir_hoare P c c' Q \<Longrightarrow> ir_valid P c c' Q"
lemma no_microstep_broadcast: "\<not> (({l}broadcast(s\<^sub>m\<^sub>s\<^sub>g).p) \<leadsto>\<^bsub>\<Gamma>\<^esub> q)"
lemma get_root_node_si_same_no_parent: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "h \<turnstile> get_root_node_si ptr \<rightarrow>\<^sub>r cast child" shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r None"
lemma hd_map: "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
lemma asprod_pos_mono:"0 < w \<Longrightarrow> ((w \<^sub>a x) \<le> (w \<^sub>a y)) = (x \<le> y)"
lemma lift_opts_exists: "\<forall>x\<in>set ty_ty'_list. (\<lambda>(ty, ty'). is_sty_one P ty ty' = Some True) x \<Longrightarrow> \<exists>bools. lift_opts (map (\<lambda>(ty, ty'). is_sty_one P ty ty') ty_ty'_list) = Some bools"
lemma before_vs_until: "(before p q) = ((\<box>\<guillemotleft>p\<guillemotright>) U \<guillemotleft>q\<guillemotright>)"
lemma (in SecurityInvariant_withOffendingFlows) fixes "default_node_properties" :: "'a" ("\<bottom>") shows "\<not> sinvar G nP \<Longrightarrow> sinvar_all_edges_normal_form P \<Longrightarrow> (\<forall> (nP::'v \<Rightarrow> 'a) e1 e2. \<not> (P (nP e1) (nP e2)) \<longrightarrow> \<not> (P \<bottom> (nP e2))) \<Longrightarrow> (\<forall> (nP::'v \<Rightarrow> 'a) e1 e2. \<not> (P (nP e1) (nP e2)) \<longrightarrow> \<not> (P (nP e1) \<bottom>)) \<Longrightarrow> (\<forall> (nP::'v \<Rightarrow> 'a) e1 e2. \<not> P \<bottom> \<bottom>) \<Longrightarrow> \<not> sinvar G (nP(i := \<bottom>))"
lemma "matches (\<beta>, \<alpha>) (remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot (Match A))) a p = matches (\<beta>, \<alpha>) (MatchNot (Match A)) a p"
lemma fields_declC: "\<lbrakk>table_of (fields G C) efn = Some f; ws_prog G; is_class G C\<rbrakk> \<Longrightarrow> (\<exists>d. class G (declclassf efn) = Some d \<and> table_of (cfields d) (fname efn)=Some f) \<and> G\<turnstile>C \<preceq>\<^sub>C (declclassf efn) \<and> table_of (fields G (declclassf efn)) efn = Some f"
lemma ET_target_source: "\<lbrakk> TS \<subseteq> ET ST; t \<in> TS; target t \<in> States A; A \<in> SAs (HA ST) \<rbrakk> \<Longrightarrow> source t \<in> States A"
lemma mcont_contD: "\<lbrakk> mcont luba orda lubb ordb f; Complete_Partial_Order.chain orda Y; Y \<noteq> {} \<rbrakk> \<Longrightarrow> f (luba Y) = lubb (f ` Y)"
lemma d_delta_lnexp_cf3_nonneg: "numer_cf3 x > 0 \<Longrightarrow> numer_cf3 (-x) > 0 \<Longrightarrow> diff_delta_lnexp_cf3 x \<ge> 0"
lemma "share_correct.sharing_correct input"
lemma NormalizeD_sound: assumes valid: "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P (normalize c) Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
lemma FD1: "Der_1 \<D> \<Longrightarrow> Fr_1(\<F>\<^sub>D \<D>)"
lemma vector_in_orthogonal_basis: fixes a :: "'a::euclidean_space" assumes "a \<noteq> 0" obtains S where "a \<in> S" "0 \<notin> S" "pairwise orthogonal S" "independent S" "finite S" "span S = UNIV" "card S = DIM('a)"
lemma wlp_is_weakest_liberal_precondition: assumes "well_com S" and "is_quantum_predicate P" shows "is_weakest_liberal_precondition (wlp S P) S P"
lemma usemantics_UNIV: \<open>usemantics UNIV e f g p \<longleftrightarrow> semantics e f g p\<close>
lemma stream_all_iff[iff]: "stream_all P s \<longleftrightarrow> Ball (sset s) P"
lemma SN_on_imp_acc: assumes "SN_on {(y, z). (z, y) \<in> r} {x}" shows "x \<in> Wellfounded.acc r"
lemma subset_single [rewrite]: "{a} \<subseteq> B \<longleftrightarrow> a \<in> B"
lemma inv_loop6_loop_Bk_Bk_drop[elim]: "\<lbrakk>0 < x; inv_loop6_loop x (b, Bk # list); b \<noteq> []; hd b = Bk\<rbrakk> \<Longrightarrow> inv_loop6_loop x (tl b, Bk # Bk # list)"
lemma solves_odeD: assumes "(y solves_ode f) T X" shows solves_ode_vderivD: "(y has_vderiv_on (\<lambda>t. f t (y t))) T" and solves_ode_domainD: "\<And>t. t \<in> T \<Longrightarrow> y t \<in> X"
lemma SAT_deduction: "SATAxiom x ==> x : deductions CutFreePC"
lemma order_refl_type [iff]: "\<tau> \<le> \<tau>" for \<tau> :: "'a type"
lemma connected_empty [simp]: "connected {}"
lemma secureEQUIV: "secure p G c H = (\<forall> q . secure1 p G c H \<and> secure2 q G c H)"
lemma CE_symmetric: "CE r s \<Longrightarrow> CE s r"
lemma const_transfer: "rel_fun (\<lambda>x y. x = c) (=) f (\<lambda>_. f c)"
lemma COND_extr_prefix_path: "\<lbrakk>hfs_valid ainfo uinfo l nxt; nxt = None\<rbrakk> \<Longrightarrow> prefix (extr_from_hd l) (AHIS l)"
lemma segm_morph: "snd (Rep_segment x::('a::lip_order \<times> 'a::lip_order)) = fst (Rep_segment y) \<Longrightarrow> segm x \<union> segm y = segm (x \<cdot> y)"
lemma lcp_pref_monotone: assumes "w \<le>p r" and "w' \<le>p s" shows "w \<and>\<^sub>p w' \<le>p (r \<and>\<^sub>p s)"
lemma list_max_is_max : "q \<in> set xs \<Longrightarrow> q \<le> list_max xs"
lemma count_sum_mset_if_1_0: \<open>count M a = (\<Sum>x\<in>#M. if x = a then 1 else 0)\<close>
lemma drop_upd_irrelevant: "m < n \<Longrightarrow> drop n (l[m:=x]) = drop n l"
lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2 * pi * n) = 0"

lemma wprepare_goto_start_pos_Bk_nonempty_fst[simp]: "\<lbrakk>lm \<noteq> []; wprepare_goto_start_pos m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"

lemma awalk_decomp: assumes "awalk u p v" assumes "w \<in> set (awalk_verts u p)" shows "\<exists>q r. p = q @ r \<and> awalk u q w \<and> awalk w r v"

lemma curry_uncurry: assumes "functor A1 A2_B.comp F" and "functor A1 A2_B.comp G" and "natural_transformation A1 A2_B.comp F G \<tau>" shows "curry (uncurry F) (uncurry G) (uncurry \<tau>) = \<tau>"

lemma connected_impl_maximally_connected: assumes "connected_graph H" assumes subgraph: "subgraph H G" shows "maximally_connected H G"

lemma torder_sync_order: "torder_on (sactions P E) (sync_order P E)"

lemma mset_ran_mem[simp, intro]: "finite r \<Longrightarrow> i\<in>r \<Longrightarrow> a i \<in># mset_ran a r"

lemma numsubst0_numbound0: assumes nb: "numbound0 t" shows "numbound0 (numsubst0 t a)"

lemma set_sel'_symm: "sel_symm f \<Longrightarrow> set_sel' f X Y = set_sel' f Y X"

lemma deduct_proj_priv_term_prefix_ex: assumes A: "ik\<^sub>s\<^sub>t (proj_unl l A) \<cdot>\<^sub>s\<^sub>e\<^sub>t I \<turnstile> t" and t: "\<not>{} \<turnstile>\<^sub>c t" shows "\<exists>B k s. (k = \<star> \<or> k = ln l) \<and> prefix (B@[(k,receive\<langle>s\<rangle>\<^sub>s\<^sub>t)]) A \<and> declassified\<^sub>l\<^sub>s\<^sub>t ((B@[(k,receive\<langle>s\<rangle>\<^sub>s\<^sub>t)])) I = declassified\<^sub>l\<^sub>s\<^sub>t A I \<and> ik\<^sub>s\<^sub>t (proj_unl l (B@[(k,receive\<langle>s\<rangle>\<^sub>s\<^sub>t)])) = ik\<^sub>s\<^sub>t (proj_unl l A)"

lemma wprepare_loop_goon_Bk_nonempty[simp]: "\<lbrakk>lm \<noteq> []; wprepare_loop_goon m lm (b, Bk # list)\<rbrakk> \<Longrightarrow> b \<noteq> []"

lemma rev_bl_order_simps: "rev_bl_order F [] [] = F" "rev_bl_order F (x # xs) (y # ys) = rev_bl_order ((y \<and> \<not> x) \<or> ((y \<or> \<not> x) \<and> F)) xs ys"

lemma consistent_signs_atw: assumes "\<And>p. p \<in> set fs \<Longrightarrow> poly p x \<noteq> 0" shows "consistent_sign_vec_copr fs x = signs_at fs x"

lemma wf_rulesetD: assumes "wf_ruleset \<gamma> p (r # rs)" shows "wf_ruleset \<gamma> p [r]" and "wf_ruleset \<gamma> p rs"

lemma arr_runit [simp]: assumes "ide a" shows "arr \<r>[a]"

lemma sublist_appendI [simp, intro]: "sublist xs (ps @ xs @ ss)"

lemma orth_at2_tsp__ts: assumes "P \<noteq> Q" and "P OrthAt A B C P X" and "Q OrthAt A B C Q Y" and "A B C TSP X Y" shows "P Q TS X Y"

theorem normal_form: assumes "recfn n f" obtains i where "\<forall>x. e_length x = n \<longrightarrow> eval r_normal_form [i, x] = eval f (list_decode x)"

lemma trms\<^sub>s\<^sub>s\<^sub>t_proj_subset[simp]: "trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n A) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (unlabel A)" (is ?A) "trms\<^sub>s\<^sub>s\<^sub>t (proj_unl m (proj n A)) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n A)" (is ?B) "trms\<^sub>s\<^sub>s\<^sub>t (proj_unl m (proj n A)) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (proj_unl m A)" (is ?C)

lemma ctrm_trunc: assumes "p \<in> carrier P" assumes "degree p >0" shows "zcf(trunc p) = zcf p"

lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f :: 'a::comm_semiring_1 fps)) $ 0 = of_nat (fact k) * f $ k"

lemma tLength_g0_conv: "(tLength t > 0) \<longleftrightarrow> (\<exists>s t'. t = t' \<leadsto> s \<and> tLength t = Suc (tLength t'))"

lemma sum_emeasure': assumes [simp]: "finite A" assumes [measurable]: "\<And>x. x \<in> A \<Longrightarrow> B x \<in> sets M" assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> emeasure M (B x \<inter> B y) = 0" shows "(\<Sum>x\<in>A. emeasure M (B x)) = emeasure M (\<Union>x\<in>A. B x)"

lemma ratfps_inverse_code [code abstract]: "quot_of_ratfps (inverse x) = (let (a,b) = quot_of_ratfps x in if coeff a 0 = 0 then (0, 1) else let u = unit_factor a in (b div u, a div u))"

lemma nextVertex_in_edges: "v \<in> \<V> f \<Longrightarrow> (v, f \<bullet> v) \<in> edges f"

lemma r_unit_add_rng_of_frac: assumes "(r, s) \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> \<zero>\<^bsub>rec_rng_of_frac\<^esub> = (r |\<^bsub>rel\<^esub> s)"

lemma assumes deriv: "\<And>y. a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {a..})" assumes deriv': "\<And>z t x. z \<in> U \<Longrightarrow> x \<ge> a \<Longrightarrow> t \<in> {a..x} \<Longrightarrow> ((\<lambda>z. f z t) has_field_derivative f' z t) (at z within U)" assumes cont: "continuous_on (U \<times> {of_int a..}) (\<lambda>(z, t). f' z t)" assumes int: "\<And>b c z e. a \<le> b \<Longrightarrow> z \<in> U \<Longrightarrow> (\<lambda>t. of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}" assumes int': "\<And>a' b y. y \<in> U \<Longrightarrow> a \<le> a' \<Longrightarrow> a' \<le> b \<Longrightarrow> (\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}" assumes conv: "convergent (\<lambda>y. G (real y))" assumes bound: "eventually (\<lambda>x. \<forall>y\<in>U. norm (f y x) \<le> g x) at_top" assumes "open U" shows analytic_EM_remainder: "(\<lambda>s::complex. EM_remainder n (f s) a) analytic_on U" and holomorphic_EM_remainder: "(\<lambda>s::complex. EM_remainder n (f s) a) holomorphic_on U"

lemma iUntil_disj_distrib: " (P t1. t1 \<U> t2 I. (Q1 t2 \<or> Q2 t2)) = ((P t1. t1 \<U> t2 I. Q1 t2) \<or> (P t1. t1 \<U> t2 I. Q2 t2))"

lemma i_drop_the_conv: " f \<Up> k = (THE g. (\<exists>xs. length xs = k \<and> xs \<frown> g = f))"

lemma is_process3: "is_process P \<Longrightarrow> \<forall> s t. (s @ t,{}) \<in> FAILURES P \<longrightarrow> (s, {}) \<in> FAILURES P"

lemma inj_map_prim: assumes "inj_on f A" and "u \<in> lists A" and "primitive u" shows "primitive (map f u)"

lemma connecting_paths_sym_length: "i \<in> connecting_paths u v \<Longrightarrow> \<exists>j\<in>connecting_paths v u. (walk_length j) = (walk_length i)"

lemma join_distr: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"

lemma inner_node_Entry_edge: assumes "inner_node n" obtains a where "valid_edge a" and "inner_node (targetnode a)" and "sourcenode a = (_Entry_)"

lemma asIMOD_igAbs2: fixes SEM :: "('index,'bindex,'varSort,'sort,'opSym,'sTerm)semDom" assumes *: "sWlsDisj SEM" and **: "igWls (asIMOD SEM) s X" shows "igAbs (asIMOD SEM) xs x X = (\<lambda>val. if sWlsVal SEM val then sAbs xs (\<lambda>sX. if sWls SEM (asSort xs) sX then X (val (x := sX)_xs) else sDummy SEM s) else undefined)"

lemma fpxs_of_fls_compose_power [simp]: "fpxs_of_fls (fls_compose_power f d) = fpxs_compose_power (fpxs_of_fls f) (of_nat d)"

lemma both_mono2: "paths t \<subseteq> paths t' \<Longrightarrow> paths (t'' \<otimes>\<otimes> t) \<subseteq> paths (t'' \<otimes>\<otimes> t')"

lemma F_rec: "F n = F (n-1) + 1 + (\<Sum>i=m..<n. F (n-i-1))" if \<open>n>m\<close> "m > 0"

lemma callee_of_rest_simps [simp]: "callee_of_rest rest s (Inl iadv_rest) = map_spmf (apfst Inl) (rfunc_adv rest s iadv_rest)" "callee_of_rest rest s (Inr iusr_rest) = map_spmf (apfst Inr) (rfunc_usr rest s iusr_rest)"

lemma conditionalize: assumes "finite A" shows "F.F g A = G (\<lambda>a. if a \<in> A then g a else \<^bold>1)"

lemma bind_single: "P \<bind> single = P"

lemma [code]: \<open>mask (Suc n) = push_bit n (1 :: 'a) OR mask n\<close> \<open>mask 0 = (0 :: 'a)\<close>

lemma last_ladder_\<gamma>: assumes is_ladder: "is_ladder D L" assumes ladder_last_n: "ladder_last_n L = length D" shows "ladder_\<gamma> a D L (length L - Suc 0) = Derive a D"

lemma USUP_empty [simp]: "(\<Squnion> i \<in> {} \<bullet> P(i)) = true"

lemma rbt_eq_iff: "t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2"

lemma dsteps_to_steps: "a \<in> st.dstep ^^ n \<Longrightarrow> a \<in> st.step ^^ n"

lemma PO_m1a_leak_refines_m1x_leak: "{R1x1a} (m1x_leak Rs), (m1a_leak Rs) {> R1x1a}"

lemma vec_contains_row_elements_mat: assumes "i < dim_row M" assumes "a \<in>$ row M i" shows "a \<in> elements_mat M"

lemma seq_filter_UNIV [simp]: "xs \<restriction>\<^sub>l UNIV = xs"

lemma simplePath_empty_conv[simp]: "isSimplePath s [] t \<longleftrightarrow> s=t"

lemma Some_in_opt [iff]: "(Some x \<in> opt A) = (x\<in>A)"

lemma interior_of_eq_empty: "X interior_of S = {} \<longleftrightarrow> (\<forall>T. openin X T \<and> T \<subseteq> S \<longrightarrow> T = {})"

lemma deg_not_zero_imp_not_unit: fixes f:: "'a::{idom_divide,semidom_divide_unit_factor} poly" assumes deg_f: "degree f > 0" shows "\<not> is_unit f"

lemma wf_PLUS[simp]: "wf n (PLUS xs) = (\<forall>r \<in> set xs. wf n r)"

lemma inj_axis: assumes "c \<noteq> 0" shows "inj (\<lambda>k. axis k c :: ('a :: {zero}) ^ 'n)"

lemma assumes ms: "measure_space M" and un: "(\<Union>i\<in>R. B i) = UNIV" and fin: "finite (R::nat set)" and dis: "\<forall>j1\<in>R. \<forall>j2\<in>R. j1 \<noteq> j2 \<longrightarrow> (B j1) \<inter> (B j2) = {}" and meas: "\<forall>j\<in>R. B j \<in> measurable_sets M" and Ameas: "A \<in> measurable_sets M" shows measure_split: "measure M A = (\<Sum>j\<in>R. measure M (A \<inter> B j))"

lemma itypeI: "(c::'t) ::\<^sub>i I"

lemma analz_insert_Crypt: "Key (invKey K) \<notin> analz H \<Longrightarrow> analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"

lemma complement_p: "x \<sqinter> y = bot \<Longrightarrow> x \<squnion> y = top \<Longrightarrow> -x = y"

lemma trms\<^sub>s\<^sub>s\<^sub>t_memI[intro?]: "send\<langle>ts\<rangle> \<in> set S \<Longrightarrow> t \<in> set ts \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "receive\<langle>ts\<rangle> \<in> set S \<Longrightarrow> t \<in> set ts \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "\<langle>ac: t \<doteq> s\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "\<langle>ac: t \<doteq> s\<rangle> \<in> set S \<Longrightarrow> s \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "insert\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "insert\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> s \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "delete\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "delete\<langle>t,s\<rangle> \<in> set S \<Longrightarrow> s \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S" "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set S \<Longrightarrow> t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G \<Longrightarrow> t \<in> trms\<^sub>s\<^sub>s\<^sub>t S"

lemma it_context_true: "-p * (-p \<rhd> x) = -p * x"

lemma maddux2c: "(a \<lhd> x) \<sqinter> y \<le> a \<lhd> (x \<sqinter> (y \<rhd> a))"

lemma valid_insert_both_member_options_pres: "invar_vebt t n \<Longrightarrow> x<2^n \<Longrightarrow> y < 2^n \<Longrightarrow> both_member_options t x \<Longrightarrow> both_member_options (vebt_insert t y) x"

lemma higher_deriv_diff: fixes z::complex assumes "f holomorphic_on S" "g holomorphic_on S" "open S" "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"

theorem soundness: "ir_hoare P c c' Q \<Longrightarrow> ir_valid P c c' Q"

lemma no_microstep_broadcast: "\<not> (({l}broadcast(s\<^sub>m\<^sub>s\<^sub>g).p) \<leadsto>\<^bsub>\<Gamma>\<^esub> q)"

lemma get_root_node_si_same_no_parent: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "h \<turnstile> get_root_node_si ptr \<rightarrow>\<^sub>r cast child" shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r None"

lemma hd_map: "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"

lemma asprod_pos_mono:"0 < w \<Longrightarrow> ((w *\<^sub>a x) \<le> (w *\<^sub>a y)) = (x \<le> y)"

lemma lift_opts_exists: "\<forall>x\<in>set ty_ty'_list. (\<lambda>(ty, ty'). is_sty_one P ty ty' = Some True) x \<Longrightarrow> \<exists>bools. lift_opts (map (\<lambda>(ty, ty'). is_sty_one P ty ty') ty_ty'_list) = Some bools"

lemma before_vs_until: "(before p q) = ((\<box>\<guillemotleft>p\<guillemotright>) U \<guillemotleft>q\<guillemotright>)"

lemma (in SecurityInvariant_withOffendingFlows) fixes "default_node_properties" :: "'a" ("\<bottom>") shows "\<not> sinvar G nP \<Longrightarrow> sinvar_all_edges_normal_form P \<Longrightarrow> (\<forall> (nP::'v \<Rightarrow> 'a) e1 e2. \<not> (P (nP e1) (nP e2)) \<longrightarrow> \<not> (P \<bottom> (nP e2))) \<Longrightarrow> (\<forall> (nP::'v \<Rightarrow> 'a) e1 e2. \<not> (P (nP e1) (nP e2)) \<longrightarrow> \<not> (P (nP e1) \<bottom>)) \<Longrightarrow> (\<forall> (nP::'v \<Rightarrow> 'a) e1 e2. \<not> P \<bottom> \<bottom>) \<Longrightarrow> \<not> sinvar G (nP(i := \<bottom>))"

lemma "matches (\<beta>, \<alpha>) (remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot (Match A))) a p = matches (\<beta>, \<alpha>) (MatchNot (Match A)) a p"

lemma fields_declC: "\<lbrakk>table_of (fields G C) efn = Some f; ws_prog G; is_class G C\<rbrakk> \<Longrightarrow> (\<exists>d. class G (declclassf efn) = Some d \<and> table_of (cfields d) (fname efn)=Some f) \<and> G\<turnstile>C \<preceq>\<^sub>C (declclassf efn) \<and> table_of (fields G (declclassf efn)) efn = Some f"

lemma ET_target_source: "\<lbrakk> TS \<subseteq> ET ST; t \<in> TS; target t \<in> States A; A \<in> SAs (HA ST) \<rbrakk> \<Longrightarrow> source t \<in> States A"

lemma mcont_contD: "\<lbrakk> mcont luba orda lubb ordb f; Complete_Partial_Order.chain orda Y; Y \<noteq> {} \<rbrakk> \<Longrightarrow> f (luba Y) = lubb (f ` Y)"

lemma d_delta_lnexp_cf3_nonneg: "numer_cf3 x > 0 \<Longrightarrow> numer_cf3 (-x) > 0 \<Longrightarrow> diff_delta_lnexp_cf3 x \<ge> 0"

lemma "share_correct.sharing_correct input"

lemma NormalizeD_sound: assumes valid: "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P (normalize c) Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"

lemma FD1: "Der_1 \<D> \<Longrightarrow> Fr_1(\<F>\<^sub>D \<D>)"

lemma vector_in_orthogonal_basis: fixes a :: "'a::euclidean_space" assumes "a \<noteq> 0" obtains S where "a \<in> S" "0 \<notin> S" "pairwise orthogonal S" "independent S" "finite S" "span S = UNIV" "card S = DIM('a)"

lemma wlp_is_weakest_liberal_precondition: assumes "well_com S" and "is_quantum_predicate P" shows "is_weakest_liberal_precondition (wlp S P) S P"

lemma usemantics_UNIV: \<open>usemantics UNIV e f g p \<longleftrightarrow> semantics e f g p\<close>

lemma stream_all_iff[iff]: "stream_all P s \<longleftrightarrow> Ball (sset s) P"

lemma SN_on_imp_acc: assumes "SN_on {(y, z). (z, y) \<in> r} {x}" shows "x \<in> Wellfounded.acc r"

lemma subset_single [rewrite]: "{a} \<subseteq> B \<longleftrightarrow> a \<in> B"

lemma inv_loop6_loop_Bk_Bk_drop[elim]: "\<lbrakk>0 < x; inv_loop6_loop x (b, Bk # list); b \<noteq> []; hd b = Bk\<rbrakk> \<Longrightarrow> inv_loop6_loop x (tl b, Bk # Bk # list)"

lemma solves_odeD: assumes "(y solves_ode f) T X" shows solves_ode_vderivD: "(y has_vderiv_on (\<lambda>t. f t (y t))) T" and solves_ode_domainD: "\<And>t. t \<in> T \<Longrightarrow> y t \<in> X"

lemma SAT_deduction: "SATAxiom x ==> x : deductions CutFreePC"

lemma order_refl_type [iff]: "\<tau> \<le> \<tau>" for \<tau> :: "'a type"

lemma connected_empty [simp]: "connected {}"

lemma secureEQUIV: "secure p G c H = (\<forall> q . secure1 p G c H \<and> secure2 q G c H)"

lemma CE_symmetric: "CE r s \<Longrightarrow> CE s r"

lemma const_transfer: "rel_fun (\<lambda>x y. x = c) (=) f (\<lambda>_. f c)"

lemma COND_extr_prefix_path: "\<lbrakk>hfs_valid ainfo uinfo l nxt; nxt = None\<rbrakk> \<Longrightarrow> prefix (extr_from_hd l) (AHIS l)"

lemma segm_morph: "snd (Rep_segment x::('a::lip_order \<times> 'a::lip_order)) = fst (Rep_segment y) \<Longrightarrow> segm x \<union> segm y = segm (x \<cdot> y)"

lemma lcp_pref_monotone: assumes "w \<le>p r" and "w' \<le>p s" shows "w \<and>\<^sub>p w' \<le>p (r \<and>\<^sub>p s)"

lemma list_max_is_max : "q \<in> set xs \<Longrightarrow> q \<le> list_max xs"

lemma count_sum_mset_if_1_0: \<open>count M a = (\<Sum>x\<in>#M. if x = a then 1 else 0)\<close>

lemma drop_upd_irrelevant: "m < n \<Longrightarrow> drop n (l[m:=x]) = drop n l"

lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2 * pi * n) = 0"