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

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lemma inheritedPropagateEq[rule_format]: assumes a: "inherited subs P" and b: "fans subs" and c: "~(terminal subs delta)" shows "P(tree subs delta) = (!sigma:subs delta. P(tree subs sigma))"
lemma mono2mono2: assumes f: "monotone (rel_prod ordb ordc) leq (\<lambda>(x, y). f x y)" and t: "monotone orda ordb (\<lambda>x. t x)" and t': "monotone orda ordc (\<lambda>x. t' x)" shows "monotone orda leq (\<lambda>x. f (t x) (t' x))"
lemma inj_on_add'[simp]: "inj_on (\<lambda>b. b \<oplus>\<^sub>a a) A"
lemma tabulate_alt: "tabulate f x n = map f [x ..< x + n]"
lemma flush_all_until_volatile_write_append_Prog\<^sub>s\<^sub>b_commute: "\<And>i m. \<lbrakk>i < length ts; ts!i=(p,is,\<theta>,sb,\<D>,\<O>,\<R>)\<rbrakk> \<Longrightarrow> flush_all_until_volatile_write (ts[i := (p\<^sub>2,is@mis, \<theta>, sb@[Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis],\<D>', \<O>,\<R>')]) m = flush_all_until_volatile_write ts m"
lemma butlast_take: "n \<le> length xs \<Longrightarrow> butlast (take n xs) = take (n - 1) xs"
lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"
lemma cons_lcop1: "[\<^bold>\<circ>b, a \<^bold>\<rightarrow> b \<^bold>\<turnstile> \<^bold>\<not>b \<^bold>\<rightarrow> \<^bold>\<not>a]"
lemma vec_times_mat_eqD[dest]: assumes "[y \<^sub>v* A]=c" shows "(\<forall>i < dim_vec c. (A\<^sup>T *\<^sub>v y)$i = c$i)" "(dim_col A\<^sup>T = dim_vec y)" "(dim_row A\<^sup>T = dim_vec c)"
lemma sup_le_union [simp]: "a \<le> b \<Longrightarrow> supremum (A \<union> {a, b}) = supremum (A \<union> {b})"
lemma base_in_grid: assumes p_grid: "p \<in> sparsegrid' dm" shows "base ds p \<in> grid (start dm) {0..<dm}"
lemma sq_mtx_zero_nth[simp]: "0 $$ i $ j = 0"
lemma word_mult_less_mono1: fixes i :: "'a :: len word" assumes ij: "i < j" and knz: "0 < k" and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)" shows "i * k < j * k"
lemma transpose_nil: "ps = [] \<longleftrightarrow> transpose ps = []"
lemma fmdom'_notI: "fmlookup m x = None \<Longrightarrow> x \<notin> fmdom' m"
lemma "Fr_6 \<F> \<Longrightarrow> \<forall>A. Cl(A) \<longrightarrow> DNE\<^sup>A \<^bold>\<not>\<^sup>C"
lemma increasingD: "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"
lemma fps_mult_fps_X_deriv_shift: "fps_X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)"
lemma TSO_inv1_invariant: "reach tso_TS \<subseteq> TSO_inv1"
lemma coeff_dvd_poly: "[:coeff a (degree a):] dvd (a::'a::{field} poly)"
lemma expectation_Y_\<Delta>: "expectation (\<lambda>x. (Y x)^2) = \<mu> + \<Delta>\<^sub>a"
lemma GuardStrip: "\<lbrakk>P \<subseteq> R; \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c Q,A; f \<in> F\<rbrakk> \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P (Guard f g c) Q,A"
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
lemma "i < j ==> nat (i - j) = 0"
lemma length_greater_Suc_0_conv: "Suc 0 < length xs \<longleftrightarrow> (\<exists>x x' xs'. xs = x # x' # xs')"
lemma unsat_farkas_coefficients: assumes "\<nexists> v. v \<Turnstile>\<^sub>c\<^sub>s cs" and fin: "finite cs" shows "\<exists> C. farkas_coefficients cs C"
lemma join_raw_Nil [simp]: "join_raw f xs [] = xs"
lemma imirror_eq_mirror_elem_image: " imirror I = (\<lambda>x. mirror_elem x I) ` I"
lemma oppositeLiteralListIdempotency [simp]: fixes literalList :: "Literal list" shows "oppositeLiteralList (oppositeLiteralList literalList) = literalList"
lemma sum_distrib: fixes SX :: "program \<Rightarrow> nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> nat" and p :: program assumes SX_simps: "\<And>h. SX p x h = (\<Sum>k = 0..length p-1. if g x k then h k else 0)" shows "SX p x h1 + SX p x h2 = SX p x (\<lambda>k. h1 k + h2 k)"
lemma reachable_constraints_var_types_in_transactions: fixes \<A>::"('fun,'atom,'sets,'lbl) prot_constr" assumes \<A>: "\<A> \<in> reachable_constraints P" and P: "\<forall>T \<in> set P. \<forall>x \<in> set (transaction_fresh T). \<Gamma>\<^sub>v x = TAtom Value \<or> (\<exists>a. \<Gamma>\<^sub>v x = TAtom (Atom a))" shows "\<Gamma>\<^sub>v ` fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<subseteq> (\<Union>T \<in> set P. \<Gamma>\<^sub>v ` fv_transaction T)" (is "?A \<A>") and "\<Gamma>\<^sub>v ` bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<subseteq> (\<Union>T \<in> set P. \<Gamma>\<^sub>v ` bvars_transaction T)" (is "?B \<A>") and "\<Gamma>\<^sub>v ` vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<subseteq> (\<Union>T \<in> set P. \<Gamma>\<^sub>v ` vars_transaction T)" (is "?C \<A>")
lemma "attack\<langle>ln 0\<rangle> \<in> set (fst ATTACK_UNSET_fixpoint)"
lemma convolution_transfer : assumes "f \<in> aezfun_set" "g \<in> aezfun_set" shows "Abs_aezfun (convolution f g) = Abs_aezfun f * Abs_aezfun g"
lemma raw_analz_image_freshK: "evs \<in> recur ==> \<forall>K KK. KK \<subseteq> - (range shrK) \<longrightarrow> (Key K \<in> analz (Key`KK \<union> (spies evs))) = (K \<in> KK \| Key K \<in> analz (spies evs))"
lemma step_eps_cong_SQ: "q \<in> nfa'.SQ \<Longrightarrow> step_eps bs q q' \<longleftrightarrow> nfa'.step_eps bs q q'"
lemma not_greater_Max: "\<lbrakk> finite A; Max A < k \<rbrakk> \<Longrightarrow> k \<notin> A"
lemma inc_of_zero: "\<iota> \<zero>\<^bsub>Z\<^sub>p\<^esub> = \<zero>"
lemma processedD2: assumes "processed (a, b) xs ps" shows "b \<in> set xs"
lemma returns_to_outsideI: assumes t: "t \<ge> 0" "t \<in> existence_ivl0 x" "flow0 x t \<in> P" assumes ev: "x \<notin> P" assumes "closed P" shows "returns_to P x"
lemma mat_pow_ring_pow: assumes mat: "mat n n (m :: ('a :: semiring_1)mat)" shows "mat_pow n m k = m [^]\<^bsub>mat_ring n b\<^esub> k" (is "_ = m [^]\<^bsub>?C\<^esub> k")
lemma restrict_spmf_UNIV [simp]: "p \<upharpoonleft> UNIV = p"
lemma set_sel_aux_1_if_notfin: "\<not>finite Y \<Longrightarrow> set_sel_aux f x Y = 1"
lemma dvd_lcm2 [iff]: "b dvd lcm a b"
lemma Nonce_req_notin_spies: "[\| evs \<in> p2; req A r n I B \<in> set evs; A \<notin> bad \|] ==> Nonce n \<notin> analz (spies evs)"
lemma hypnat_add_is_0 [iff]: "\<And>m n::hypnat. m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
lemma loc_srj_5: "\<lbrakk>nat_to_sch (sch_to_nat sch1) = sch1; nat_to_sch (sch_to_nat sch2) = sch2\<rbrakk> \<Longrightarrow> nat_to_sch (c_pair 6 (c_pair (sch_to_nat sch1) (sch_to_nat sch2))) = Rec_op sch1 sch2"
lemma \<I>_trivial_\<I>_full [simp]: "\<I>_trivial \<I>_full"
lemma purgeIdle_Cons_iff: "purgeIdle sl = s # sll \<longleftrightarrow> (\<exists> sl1 sl2. sl = sl1 @ s # sl2 \<and> (\<forall>s1\<in>set sl1. \<not> isState s1) \<and> isState s \<and> purgeIdle sl2 = sll)"
lemma "q2 n"
lemma merge_split_supset_fst: assumes "as@(r,e)#bs = (Sorting_Algorithms.merge cmp xs ys)" shows "\<exists>as' bs'. set bs' \<subseteq> set bs \<and> (as'@(r,e)#bs' = xs \<or> as'@(r,e)#bs' = ys)"
lemma log_Suc_zero [simp]: "log (Suc 0) = 0"
lemma length_listsetD: "vs \<in> listset VS \<Longrightarrow> length vs = length VS"
lemma frac_le_eq: "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
lemma varInFormulaVars: fixes variable :: Variable and formula :: Formula shows "variable \<in> vars formula = (\<exists> literal. literal el formula \<and> var literal = variable)" (is "?lhs formula = ?rhs formula")
lemma map_graph_countable [simp]: "countable (dom f) \<Longrightarrow> countable (map_graph f)"
lemma 22: \<open>\<turnstile> (q \<rightarrow> r) \<rightarrow> (p \<rightarrow> q) \<rightarrow> p \<rightarrow> r\<close>
lemma norm_notMT_manual: "DenyAll \<in> set (policy2list p) \<Longrightarrow> normalize_manual_order p l \<noteq> []"
lemma last_coeff_is_hd: "xs \<noteq> [] \<Longrightarrow> coeff (Poly xs) (length xs - 1) = hd (rev xs)"
lemma [sepref_fr_rules]: \<open>((uncurry3 (\<lambda>x y. return oo (check_extension_l_side_cond_err_impl x y))), uncurry3 (check_extension_l_side_cond_err)) \<in> string_assn\<^sup>k \<^sub>a poly_assn\<^sup>k \<^sub>a poly_assn\<^sup>k *\<^sub>a poly_assn\<^sup>k \<rightarrow>\<^sub>a raw_string_assn\<close>
lemma (in flowgraph) ntrp_mon_env_e_no_ctx: "((s,c),ENV e,(s',c'))\<in>ntrp fg \<Longrightarrow> mon_w fg e \<inter> mon_s fg s = {}"
lemma sees_methods_compP: "P \<turnstile> C sees_methods Mm \<Longrightarrow> compP f P \<turnstile> C sees_methods (map_option (\<lambda>((Ts,T,m),D). ((Ts,T,f m),D)) \<circ> Mm)" (<)(is "?P \<Longrightarrow> compP f P \<turnstile> C sees_methods (?map Mm)")
lemma basis_wf_snoc: assumes "bs \<noteq> []" assumes "basis_wf bs" "filterlim b at_top at_top" assumes "(\<lambda>x. ln (b x)) \<in> o(\<lambda>x. ln (last bs x))" shows "basis_wf (bs @ [b])"
lemma ocomplete_no_change [elim]: assumes "((\<sigma>, s), a, (\<sigma>', s')) \<in> ocnet_sos T" and "j \<notin> net_ips s" shows "\<sigma>' j = \<sigma> j"
lemma to_fun_unit_is_unit: assumes "f \<in> carrier (SA n)" shows "to_fun_unit n f \<in> Units (SA n)"
lemma mset_le_add_iff2: "i \<le> (j::nat) \<Longrightarrow> (repeat_mset i u + m \<le> repeat_mset j u + n) = (m \<le> repeat_mset (j-i) u + n)"
lemma countableI_bij: "bij_betw f (C::nat set) S \<Longrightarrow> countable S"
lemma items_crypt_1: "items (insert (Crypt K X) H) \<subseteq> insert (Crypt K X) (items (insert X H))"
lemma conjuncts_list_nFAND: "\<lbrakk>list_all (\<lambda>x. \<not> is_FAnd x) \<phi>s; \<phi>s \<noteq> []\<rbrakk> \<Longrightarrow> conjuncts_list (nFAND \<phi>s) = \<phi>s"
lemma S: assumes "H \<turnstile> A IMP (B IMP C)" "H' \<turnstile> A IMP B" shows "H \<union> H' \<turnstile> A IMP C"
lemma fun_comp_eq_conv: "f o g = fg \<longleftrightarrow> (\<forall>x. f (g x) = fg x)"
lemma iwlsFSwSTR_alphaAll_qInitAll: assumes "iwlsFSwSTR MOD" shows "(\<forall> qX'. qX #= qX' \<longrightarrow> qInit MOD qX = qInit MOD qX') \<and> (\<forall> qA'. qA $= qA' \<longrightarrow> qInitAbs MOD qA = qInitAbs MOD qA')"
lemma invertible_times_non_zero: fixes M :: "real^'n^'n" assumes "invertible M" and "v \<noteq> 0" shows "M *v v \<noteq> 0"
lemma fls_inverse_subdegree_base_nonzero: assumes "f \<noteq> 0" "inverse (f $$ fls_subdegree f) \<noteq> 0" shows "inverse f $$ (fls_subdegree (inverse f)) = inverse (f $$ fls_subdegree f)"
lemma q1: "int q > 1"
lemma assumes "(IF b THEN SKIP ELSE SKIP, s) \<Rightarrow> x \<Down> t" shows "t = s"
lemma simple_cg_insert'_invar : "convergence_graph_insert_invar M1 M2 simple_cg_lookup simple_cg_insert'"
lemma NF_stepI [intro]: "s \<notin> fst ` M \<Longrightarrow> (s, vs) \<in> NF (step M)"
lemma cis_minus_pi_half [simp]: "cis (- (pi / 2)) = -\<i>"
lemma size_lower_result': "size (lower_result C) = size C - (if x \<in># C \<or> y \<in># C then 1 else 0)"
lemma inner_loop_refine[refine]: (assumes NSS: "N \<subseteq> dom PRED") assumes [simp]: "finite succ" assumes [simplified, simp]: "(succi,succ)\<in>Id" "(ui,u)\<in>Id" "(PREDi,PRED)\<in>Id" "(Ni,N)\<in>Id" shows "inner_loop dst succi ui PREDi Ni \<le> \<Down>Id (add_succ_spec dst succ u PRED N)"
lemma ultrametric_equal_eq: assumes "x \<in> carrier Q\<^sub>p" assumes "y \<in> carrier Q\<^sub>p" assumes "val (y \<ominus> x) > val x" shows "val x = val y"
lemma rename_Call: "(rename h c = Call q) = (\<exists>p. c = Call p \<and> q=h p)"
lemma ran_empty [simp]: "ran empty = {}"
lemma power: "VARS (p::int) i { True } p := 1; i := 0; WHILE i < n INV { p = x^i \<and> i \<le> n } DO p := p * x; i := i + 1 OD { p = x^n }"
lemma distinct_ExcessTable: "distinct vs \<Longrightarrow> distinct [fst p. p \<leftarrow> ExcessTable g vs]"
lemma set_enforce_spmf [simp]: "set_spmf (enforce_spmf P p) = {a \<in> set_spmf p. P a}"
lemma (in lbv) merge_mono: assumes less: "ss2 \<le>\|r\| ss1" assumes x: "x \<in> A" assumes ss1: "snd`set ss1 \<subseteq> A" assumes ss2: "snd`set ss2 \<subseteq> A" shows "merge c pc ss2 x <=_r merge c pc ss1 x" (is "?s2 <=_r ?s1")
lemma W_fun_correct: "W_fun i j = W i j"
theorem syn_sen_mult_same: "\<sigma>, s, \<Delta> \<Turnstile> syn_mult \<pi> A \<longleftrightarrow> \<sigma>, s, \<Delta> \<Turnstile> Mult \<pi> A"
lemma Resid_by_members: assumes "N.is_Cong_class \<T>" and "N.is_Cong_class \<U>" and "t \<in> \<T>" and "u \<in> \<U>" and "t \<frown> u" shows "\<T> \<lbrace>\\\<rbrace> \<U> = \<lbrace>t \\ u\<rbrace>"
lemma fold_graph_fold: assumes f: "finite B" and BA: "B\<subseteq>A" and z: "z \<in> A" shows "fold_graph f z B (Finite_Set.fold f z B)"
lemma wf_trm_param: assumes "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)" "t \<in> set T" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
lemma insert_ops_split: assumes "insert_ops ops" and "(oid, ref) \<in> set ops" shows "\<exists>pre suf. ops = pre @ [(oid, ref)] @ suf \<and> (\<forall>i \<in> set (map fst pre). i < oid) \<and> (\<forall>i \<in> set (map fst suf). oid < i)"
lemma interval_integrable_abs_iff: fixes f :: "real \<Rightarrow> real" shows "f \<in> borel_measurable lborel \<Longrightarrow> interval_lebesgue_integrable lborel a b (\<lambda>x. \<bar>f x\<bar>) = interval_lebesgue_integrable lborel a b f"
lemma doesnt_read_or_modify_subst: assumes noread: "doesnt_read_or_modify c x" assumes step: "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" assumes subset: "X \<subseteq> {x} \<union> \<C>_vars x" shows "\<And> \<sigma>. dom \<sigma> = X \<Longrightarrow> \<langle>c, mds, mem[\<mapsto> \<sigma>]\<rangle> \<leadsto> \<langle>c', mds', mem'[\<mapsto> \<sigma>]\<rangle>"
lemma all_tuples_setD: "vs \<in> all_tuples xs n \<Longrightarrow> set vs \<subseteq> xs"
lemma append_rows_mat_mul: assumes "dim_col A = dim_col B" shows "(A @\<^sub>r B) * C = A * C @\<^sub>r B * C"
lemma LCons_mono [partial_function_mono, cont_intro]: "mono_tllist A \<Longrightarrow> mono_tllist (\<lambda>f. TCons x (A f))"
lemma compute_higherPoly\<^sub>f[code]: "higherPoly\<^sub>f n i (fmap_of_list xs) = fmap_of_list (filter (\<lambda>(mon, v). \<forall>j\<in>{n..<n+i}. lookup mon j = 0) (map (\<lambda>(mon, c). (lowerPowers n i mon, c)) xs))"
lemma typ_ok_contained_tvars_typ_ok: "typ_ok thy ty \<Longrightarrow> (idn, S) \<in> tvsT ty \<Longrightarrow> typ_ok thy (Tv idn S)"

lemma inheritedPropagateEq[rule_format]: assumes a: "inherited subs P" and b: "fans subs" and c: "~(terminal subs delta)" shows "P(tree subs delta) = (!sigma:subs delta. P(tree subs sigma))"

lemma mono2mono2: assumes f: "monotone (rel_prod ordb ordc) leq (\<lambda>(x, y). f x y)" and t: "monotone orda ordb (\<lambda>x. t x)" and t': "monotone orda ordc (\<lambda>x. t' x)" shows "monotone orda leq (\<lambda>x. f (t x) (t' x))"

lemma inj_on_add'[simp]: "inj_on (\<lambda>b. b \<oplus>\<^sub>a a) A"

lemma tabulate_alt: "tabulate f x n = map f [x ..< x + n]"

lemma flush_all_until_volatile_write_append_Prog\<^sub>s\<^sub>b_commute: "\<And>i m. \<lbrakk>i < length ts; ts!i=(p,is,\<theta>,sb,\<D>,\<O>,\<R>)\<rbrakk> \<Longrightarrow> flush_all_until_volatile_write (ts[i := (p\<^sub>2,is@mis, \<theta>, sb@[Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis],\<D>', \<O>,\<R>')]) m = flush_all_until_volatile_write ts m"

lemma butlast_take: "n \<le> length xs \<Longrightarrow> butlast (take n xs) = take (n - 1) xs"

lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"

lemma cons_lcop1: "[\<^bold>\<circ>b, a \<^bold>\<rightarrow> b \<^bold>\<turnstile> \<^bold>\<not>b \<^bold>\<rightarrow> \<^bold>\<not>a]"

lemma vec_times_mat_eqD[dest]: assumes "[y \<^sub>v* A]=c" shows "(\<forall>i < dim_vec c. (A\<^sup>T *\<^sub>v y)$i = c$i)" "(dim_col A\<^sup>T = dim_vec y)" "(dim_row A\<^sup>T = dim_vec c)"

lemma sup_le_union [simp]: "a \<le> b \<Longrightarrow> supremum (A \<union> {a, b}) = supremum (A \<union> {b})"

lemma base_in_grid: assumes p_grid: "p \<in> sparsegrid' dm" shows "base ds p \<in> grid (start dm) {0..<dm}"

lemma sq_mtx_zero_nth[simp]: "0 $$ i $ j = 0"

lemma word_mult_less_mono1: fixes i :: "'a :: len word" assumes ij: "i < j" and knz: "0 < k" and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)" shows "i * k < j * k"

lemma transpose_nil: "ps = [] \<longleftrightarrow> transpose ps = []"

lemma fmdom'_notI: "fmlookup m x = None \<Longrightarrow> x \<notin> fmdom' m"

lemma "Fr_6 \<F> \<Longrightarrow> \<forall>A. Cl(A) \<longrightarrow> DNE\<^sup>A \<^bold>\<not>\<^sup>C"

lemma increasingD: "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"

lemma fps_mult_fps_X_deriv_shift: "fps_X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)"

lemma TSO_inv1_invariant: "reach tso_TS \<subseteq> TSO_inv1"

lemma coeff_dvd_poly: "[:coeff a (degree a):] dvd (a::'a::{field} poly)"

lemma expectation_Y_\<Delta>: "expectation (\<lambda>x. (Y x)^2) = \<mu> + \<Delta>\<^sub>a"

lemma GuardStrip: "\<lbrakk>P \<subseteq> R; \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c Q,A; f \<in> F\<rbrakk> \<Longrightarrow> \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P (Guard f g c) Q,A"

lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"

lemma "i < j ==> nat (i - j) = 0"

lemma length_greater_Suc_0_conv: "Suc 0 < length xs \<longleftrightarrow> (\<exists>x x' xs'. xs = x # x' # xs')"

lemma unsat_farkas_coefficients: assumes "\<nexists> v. v \<Turnstile>\<^sub>c\<^sub>s cs" and fin: "finite cs" shows "\<exists> C. farkas_coefficients cs C"

lemma join_raw_Nil [simp]: "join_raw f xs [] = xs"

lemma imirror_eq_mirror_elem_image: " imirror I = (\<lambda>x. mirror_elem x I) ` I"

lemma oppositeLiteralListIdempotency [simp]: fixes literalList :: "Literal list" shows "oppositeLiteralList (oppositeLiteralList literalList) = literalList"

lemma sum_distrib: fixes SX :: "program \<Rightarrow> nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> nat" and p :: program assumes SX_simps: "\<And>h. SX p x h = (\<Sum>k = 0..length p-1. if g x k then h k else 0)" shows "SX p x h1 + SX p x h2 = SX p x (\<lambda>k. h1 k + h2 k)"

lemma reachable_constraints_var_types_in_transactions: fixes \<A>::"('fun,'atom,'sets,'lbl) prot_constr" assumes \<A>: "\<A> \<in> reachable_constraints P" and P: "\<forall>T \<in> set P. \<forall>x \<in> set (transaction_fresh T). \<Gamma>\<^sub>v x = TAtom Value \<or> (\<exists>a. \<Gamma>\<^sub>v x = TAtom (Atom a))" shows "\<Gamma>\<^sub>v ` fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<subseteq> (\<Union>T \<in> set P. \<Gamma>\<^sub>v ` fv_transaction T)" (is "?A \<A>") and "\<Gamma>\<^sub>v ` bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<subseteq> (\<Union>T \<in> set P. \<Gamma>\<^sub>v ` bvars_transaction T)" (is "?B \<A>") and "\<Gamma>\<^sub>v ` vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<subseteq> (\<Union>T \<in> set P. \<Gamma>\<^sub>v ` vars_transaction T)" (is "?C \<A>")

lemma "attack\<langle>ln 0\<rangle> \<in> set (fst ATTACK_UNSET_fixpoint)"

lemma convolution_transfer : assumes "f \<in> aezfun_set" "g \<in> aezfun_set" shows "Abs_aezfun (convolution f g) = Abs_aezfun f * Abs_aezfun g"

lemma raw_analz_image_freshK: "evs \<in> recur ==> \<forall>K KK. KK \<subseteq> - (range shrK) \<longrightarrow> (Key K \<in> analz (Key`KK \<union> (spies evs))) = (K \<in> KK | Key K \<in> analz (spies evs))"

lemma step_eps_cong_SQ: "q \<in> nfa'.SQ \<Longrightarrow> step_eps bs q q' \<longleftrightarrow> nfa'.step_eps bs q q'"

lemma not_greater_Max: "\<lbrakk> finite A; Max A < k \<rbrakk> \<Longrightarrow> k \<notin> A"

lemma inc_of_zero: "\<iota> \<zero>\<^bsub>Z\<^sub>p\<^esub> = \<zero>"

lemma processedD2: assumes "processed (a, b) xs ps" shows "b \<in> set xs"

lemma returns_to_outsideI: assumes t: "t \<ge> 0" "t \<in> existence_ivl0 x" "flow0 x t \<in> P" assumes ev: "x \<notin> P" assumes "closed P" shows "returns_to P x"

lemma mat_pow_ring_pow: assumes mat: "mat n n (m :: ('a :: semiring_1)mat)" shows "mat_pow n m k = m [^]\<^bsub>mat_ring n b\<^esub> k" (is "_ = m [^]\<^bsub>?C\<^esub> k")

lemma restrict_spmf_UNIV [simp]: "p \<upharpoonleft> UNIV = p"

lemma set_sel_aux_1_if_notfin: "\<not>finite Y \<Longrightarrow> set_sel_aux f x Y = 1"

lemma dvd_lcm2 [iff]: "b dvd lcm a b"

lemma Nonce_req_notin_spies: "[| evs \<in> p2; req A r n I B \<in> set evs; A \<notin> bad |] ==> Nonce n \<notin> analz (spies evs)"

lemma hypnat_add_is_0 [iff]: "\<And>m n::hypnat. m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"

lemma loc_srj_5: "\<lbrakk>nat_to_sch (sch_to_nat sch1) = sch1; nat_to_sch (sch_to_nat sch2) = sch2\<rbrakk> \<Longrightarrow> nat_to_sch (c_pair 6 (c_pair (sch_to_nat sch1) (sch_to_nat sch2))) = Rec_op sch1 sch2"

lemma \_trivial_\_full [simp]: "\_trivial \_full"

lemma purgeIdle_Cons_iff: "purgeIdle sl = s # sll \<longleftrightarrow> (\<exists> sl1 sl2. sl = sl1 @ s # sl2 \<and> (\<forall>s1\<in>set sl1. \<not> isState s1) \<and> isState s \<and> purgeIdle sl2 = sll)"

lemma "q2 n"

lemma merge_split_supset_fst: assumes "as@(r,e)#bs = (Sorting_Algorithms.merge cmp xs ys)" shows "\<exists>as' bs'. set bs' \<subseteq> set bs \<and> (as'@(r,e)#bs' = xs \<or> as'@(r,e)#bs' = ys)"

lemma log_Suc_zero [simp]: "log (Suc 0) = 0"

lemma length_listsetD: "vs \<in> listset VS \<Longrightarrow> length vs = length VS"

lemma frac_le_eq: "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"

lemma varInFormulaVars: fixes variable :: Variable and formula :: Formula shows "variable \<in> vars formula = (\<exists> literal. literal el formula \<and> var literal = variable)" (is "?lhs formula = ?rhs formula")

lemma map_graph_countable [simp]: "countable (dom f) \<Longrightarrow> countable (map_graph f)"

lemma 22: \<open>\<turnstile> (q \<rightarrow> r) \<rightarrow> (p \<rightarrow> q) \<rightarrow> p \<rightarrow> r\<close>

lemma norm_notMT_manual: "DenyAll \<in> set (policy2list p) \<Longrightarrow> normalize_manual_order p l \<noteq> []"

lemma last_coeff_is_hd: "xs \<noteq> [] \<Longrightarrow> coeff (Poly xs) (length xs - 1) = hd (rev xs)"

lemma [sepref_fr_rules]: \<open>((uncurry3 (\<lambda>x y. return oo (check_extension_l_side_cond_err_impl x y))), uncurry3 (check_extension_l_side_cond_err)) \<in> string_assn\<^sup>k *\<^sub>a poly_assn\<^sup>k *\<^sub>a poly_assn\<^sup>k *\<^sub>a poly_assn\<^sup>k \<rightarrow>\<^sub>a raw_string_assn\<close>

lemma (in flowgraph) ntrp_mon_env_e_no_ctx: "((s,c),ENV e,(s',c'))\<in>ntrp fg \<Longrightarrow> mon_w fg e \<inter> mon_s fg s = {}"

lemma sees_methods_compP: "P \<turnstile> C sees_methods Mm \<Longrightarrow> compP f P \<turnstile> C sees_methods (map_option (\<lambda>((Ts,T,m),D). ((Ts,T,f m),D)) \<circ> Mm)" (*<*)(is "?P \<Longrightarrow> compP f P \<turnstile> C sees_methods (?map Mm)")

lemma basis_wf_snoc: assumes "bs \<noteq> []" assumes "basis_wf bs" "filterlim b at_top at_top" assumes "(\<lambda>x. ln (b x)) \<in> o(\<lambda>x. ln (last bs x))" shows "basis_wf (bs @ [b])"

lemma ocomplete_no_change [elim]: assumes "((\<sigma>, s), a, (\<sigma>', s')) \<in> ocnet_sos T" and "j \<notin> net_ips s" shows "\<sigma>' j = \<sigma> j"

lemma to_fun_unit_is_unit: assumes "f \<in> carrier (SA n)" shows "to_fun_unit n f \<in> Units (SA n)"

lemma mset_le_add_iff2: "i \<le> (j::nat) \<Longrightarrow> (repeat_mset i u + m \<le> repeat_mset j u + n) = (m \<le> repeat_mset (j-i) u + n)"

lemma countableI_bij: "bij_betw f (C::nat set) S \<Longrightarrow> countable S"

lemma items_crypt_1: "items (insert (Crypt K X) H) \<subseteq> insert (Crypt K X) (items (insert X H))"

lemma conjuncts_list_nFAND: "\<lbrakk>list_all (\<lambda>x. \<not> is_FAnd x) \<phi>s; \<phi>s \<noteq> []\<rbrakk> \<Longrightarrow> conjuncts_list (nFAND \<phi>s) = \<phi>s"

lemma S: assumes "H \<turnstile> A IMP (B IMP C)" "H' \<turnstile> A IMP B" shows "H \<union> H' \<turnstile> A IMP C"

lemma fun_comp_eq_conv: "f o g = fg \<longleftrightarrow> (\<forall>x. f (g x) = fg x)"

lemma iwlsFSwSTR_alphaAll_qInitAll: assumes "iwlsFSwSTR MOD" shows "(\<forall> qX'. qX #= qX' \<longrightarrow> qInit MOD qX = qInit MOD qX') \<and> (\<forall> qA'. qA $= qA' \<longrightarrow> qInitAbs MOD qA = qInitAbs MOD qA')"

lemma invertible_times_non_zero: fixes M :: "real^'n^'n" assumes "invertible M" and "v \<noteq> 0" shows "M *v v \<noteq> 0"

lemma fls_inverse_subdegree_base_nonzero: assumes "f \<noteq> 0" "inverse (f $$ fls_subdegree f) \<noteq> 0" shows "inverse f $$ (fls_subdegree (inverse f)) = inverse (f $$ fls_subdegree f)"

lemma q1: "int q > 1"

lemma assumes "(IF b THEN SKIP ELSE SKIP, s) \<Rightarrow> x \<Down> t" shows "t = s"

lemma simple_cg_insert'_invar : "convergence_graph_insert_invar M1 M2 simple_cg_lookup simple_cg_insert'"

lemma NF_stepI [intro]: "s \<notin> fst ` M \<Longrightarrow> (s, vs) \<in> NF (step M)"

lemma cis_minus_pi_half [simp]: "cis (- (pi / 2)) = -\"

lemma size_lower_result': "size (lower_result C) = size C - (if x \<in># C \<or> y \<in># C then 1 else 0)"

lemma inner_loop_refine[refine]: (*assumes NSS: "N \<subseteq> dom PRED"*) assumes [simp]: "finite succ" assumes [simplified, simp]: "(succi,succ)\<in>Id" "(ui,u)\<in>Id" "(PREDi,PRED)\<in>Id" "(Ni,N)\<in>Id" shows "inner_loop dst succi ui PREDi Ni \<le> \<Down>Id (add_succ_spec dst succ u PRED N)"

lemma ultrametric_equal_eq: assumes "x \<in> carrier Q\<^sub>p" assumes "y \<in> carrier Q\<^sub>p" assumes "val (y \<ominus> x) > val x" shows "val x = val y"

lemma rename_Call: "(rename h c = Call q) = (\<exists>p. c = Call p \<and> q=h p)"

lemma ran_empty [simp]: "ran empty = {}"

lemma power: "VARS (p::int) i { True } p := 1; i := 0; WHILE i < n INV { p = x^i \<and> i \<le> n } DO p := p * x; i := i + 1 OD { p = x^n }"

lemma distinct_ExcessTable: "distinct vs \<Longrightarrow> distinct [fst p. p \<leftarrow> ExcessTable g vs]"

lemma set_enforce_spmf [simp]: "set_spmf (enforce_spmf P p) = {a \<in> set_spmf p. P a}"

lemma (in lbv) merge_mono: assumes less: "ss2 \<le>|r| ss1" assumes x: "x \<in> A" assumes ss1: "snd`set ss1 \<subseteq> A" assumes ss2: "snd`set ss2 \<subseteq> A" shows "merge c pc ss2 x <=_r merge c pc ss1 x" (is "?s2 <=_r ?s1")

lemma W_fun_correct: "W_fun i j = W i j"

theorem syn_sen_mult_same: "\<sigma>, s, \<Delta> \<Turnstile> syn_mult \<pi> A \<longleftrightarrow> \<sigma>, s, \<Delta> \<Turnstile> Mult \<pi> A"

lemma Resid_by_members: assumes "N.is_Cong_class \<T>" and "N.is_Cong_class \" and "t \<in> \<T>" and "u \<in> \" and "t \<frown> u" shows "\<T> \<lbrace>\\\<rbrace> \ = \<lbrace>t \\ u\<rbrace>"

lemma fold_graph_fold: assumes f: "finite B" and BA: "B\<subseteq>A" and z: "z \<in> A" shows "fold_graph f z B (Finite_Set.fold f z B)"

lemma wf_trm_param: assumes "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)" "t \<in> set T" shows "wf\<^sub>t\<^sub>r\<^sub>m t"

lemma insert_ops_split: assumes "insert_ops ops" and "(oid, ref) \<in> set ops" shows "\<exists>pre suf. ops = pre @ [(oid, ref)] @ suf \<and> (\<forall>i \<in> set (map fst pre). i < oid) \<and> (\<forall>i \<in> set (map fst suf). oid < i)"

lemma interval_integrable_abs_iff: fixes f :: "real \<Rightarrow> real" shows "f \<in> borel_measurable lborel \<Longrightarrow> interval_lebesgue_integrable lborel a b (\<lambda>x. \<bar>f x\<bar>) = interval_lebesgue_integrable lborel a b f"

lemma doesnt_read_or_modify_subst: assumes noread: "doesnt_read_or_modify c x" assumes step: "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" assumes subset: "X \<subseteq> {x} \<union> \<C>_vars x" shows "\<And> \<sigma>. dom \<sigma> = X \<Longrightarrow> \<langle>c, mds, mem[\<mapsto> \<sigma>]\<rangle> \<leadsto> \<langle>c', mds', mem'[\<mapsto> \<sigma>]\<rangle>"

lemma all_tuples_setD: "vs \<in> all_tuples xs n \<Longrightarrow> set vs \<subseteq> xs"

lemma append_rows_mat_mul: assumes "dim_col A = dim_col B" shows "(A @\<^sub>r B) * C = A * C @\<^sub>r B * C"

lemma LCons_mono [partial_function_mono, cont_intro]: "mono_tllist A \<Longrightarrow> mono_tllist (\<lambda>f. TCons x (A f))"

lemma compute_higherPoly\<^sub>f[code]: "higherPoly\<^sub>f n i (fmap_of_list xs) = fmap_of_list (filter (\<lambda>(mon, v). \<forall>j\<in>{n..<n+i}. lookup mon j = 0) (map (\<lambda>(mon, c). (lowerPowers n i mon, c)) xs))"

lemma typ_ok_contained_tvars_typ_ok: "typ_ok thy ty \<Longrightarrow> (idn, S) \<in> tvsT ty \<Longrightarrow> typ_ok thy (Tv idn S)"