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

Statement: stringlengths 7 24.3k
lemma mono_Retr: "mono Sretr" "mono ZOretr" "mono ZOretrT" "mono Wretr" "mono WretrT" "mono RetrT"
theorem fk_asymptotic_space_complexity: "fk_space_usage \<in> O[at_top \<times>\<^sub>F at_top \<times>\<^sub>F at_top \<times>\<^sub>F at_right (0::rat) \<times>\<^sub>F at_right (0::rat)](\<lambda> (k, n, m, \<epsilon>, \<delta>). real k * real n powr (1-1/ real k) / (of_rat \<delta>)\<^sup>2 * (ln (1 / of_rat \<epsilon>)) * (ln (real n) + ln (real m)))" (is "_ \<in> O[?F](?rhs)")
lemma length_Residx1: shows "length (T \<^sup>*\\\<^sup>1 u) \<le> length T"
lemma Abs_formula_inverse [simp]: assumes "hereditarily_fs t\<^sub>\<alpha>" shows "Rep_formula (Abs_formula t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
lemma get_elment_ptr_simp1 [simp]: "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr (put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr element h) = Some element"
theorem Crypt_Crypt_eq [iff]: "(Crypt K X = Crypt K X') = (X=X')"
lemma finite_Prefixes: shows "finite (Prefixes s)"
lemma mst_subgraph_inv: assumes "e \<le> v * -v\<^sup>T \<sqinter> g" and "t \<le> g" and "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>" shows "e \<le> (r\<^sup>T * g\<^sup>\<star>)\<^sup>T * (r\<^sup>T * g\<^sup>\<star>) \<sqinter> g"
lemma (in orset) rem_rem_commute: shows "\<langle>Rem i1 e1\<rangle> \<rhd> \<langle>Rem i2 e2\<rangle> = \<langle>Rem i2 e2\<rangle> \<rhd> \<langle>Rem i1 e1\<rangle>"
lemma vec_0[simp]: "vec 0 = 0"
lemma monad_writer_stateT' [locale_witness]: "monad_writer return (bind :: ('a \<times> 's, 'm) bind) (tell :: ('w, 'm) tell) \<Longrightarrow> monad_writer return (bind :: ('a, ('s, 'm) stateT) bind) (tell :: ('w, ('s, 'm) stateT) tell)"
lemma C1_differentiable_imp_piecewise: "f C1_differentiable_on S \<Longrightarrow> f piecewise_C1_differentiable_on S"
lemma invoke_split: "P (invoke ((Pred, f) # xs) ptr args) = ((\<not>(Pred ptr) \<longrightarrow> P (invoke xs ptr args)) \<and> (Pred ptr \<longrightarrow> P (do {check_in_heap ptr; f ptr args})))"
lemma restrictionVsIntersection: "{(x, f x)\| x. x \<in> X2} \|\| X1 = {(x, f x)\| x. x \<in> X2 \<inter> X1}"
lemma linkageD: "\<lbrakk> linkage \<Gamma> f; x \<in> A \<Gamma> \<rbrakk> \<Longrightarrow> d_OUT f x = weight \<Gamma> x"
lemma li_minus: assumes "locally_irrefl R A" shows "locally_irrefl R (A - B)"
lemma measurable_if_split[measurable (raw)]: "(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow> Measurable.pred M (if c then f else g)"
lemma eq_tpoly_refl[simp]: "p =\<^sub>p p"
lemma fac_acc_body2_body1_eq: "fac_acc_body2 = fac_acc_body1"
lemma (in VectorSpaceEnd) VectorSpaceHom: "VectorSpaceHom smult V smult T"
lemma order_refl_sum : fixes x :: "'a + 'b" shows "x \<le> x"
lemma instInp_fmla[simp,intro]: assumes "\<phi> \<in> fmla" and "t \<in> trm" shows "instInp \<phi> t \<in> fmla"
lemma vec_to_lin_poly_coeff_access: assumes "x < dim_vec y" shows "y $ x = coeff (vec_to_lpoly y) x"
lemma solves_ode_on_subset_domain: assumes sol: "(x solves_ode f) S Y" and iv: "x t0 = x0" and ivl: "t0 \<in> S" "is_interval S" "S \<subseteq> T" shows "(x solves_ode f) S X"
lemma sub_inc_sysI[intro]: "incidence_system \<U> \<A> \<Longrightarrow> \<U> \<subseteq> \<V> \<Longrightarrow> \<A> \<subseteq># \<B> \<Longrightarrow> sub_incidence_system \<U> \<A> \<V> \<B>"
lemma Ex_omega_sum: "\<gamma> \<in> elts (\<omega>\<up>n) \<Longrightarrow> \<exists>ns. \<gamma> = omega_sum ns \<and> length ns = n"
lemma cexpr_typing_cong: assumes "\<And>x. x \<in> free_vars e \<Longrightarrow> \<Gamma> x = \<Gamma>' x" shows "\<Gamma> \<turnstile>\<^sub>c e : t \<longleftrightarrow> \<Gamma>' \<turnstile>\<^sub>c e : t"
lemma openin_interior_of [simp]: "openin X (X interior_of S)"
lemma and_wf_dfa: assumes "wf_dfa M n" and "wf_dfa N n" shows "wf_dfa (and_dfa M N) n"
lemma subst_typ'_simulates_tsubst_gen': "distinct pairs \<Longrightarrow> tvs t \<subseteq> set pairs \<Longrightarrow> tsubst t \<rho> = subst_typ' (map (\<lambda>(x,y).((x,y), \<rho> x y)) pairs) t"
lemma if_redT_known_addrs_new: assumes redT: "mthr.if.redT s (t, ta) s'" shows "if.known_addrs_state s' \<subseteq> if.known_addrs_state s \<union> new_obs_addrs_if \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>"
lemma (in is_cf_adjunction) is_cf_adjunction_axioms'[adj_cs_intros]: assumes "\<alpha>' = \<alpha>" and "\<CC>' = \<CC>" and "\<DD>' = \<DD>" and "\<FF>' = \<FF>" and "\<GG>' = \<GG>" shows "\<Phi> : \<FF>' \<rightleftharpoons>\<^sub>C\<^sub>F \<GG>' : \<CC>' \<rightleftharpoons>\<rightleftharpoons>\<^sub>C\<^bsub>\<alpha>'\<^esub> \<DD>'"
lemma "(sep_empty imp ((A \<longrightarrow>* (B imp C)) \<longrightarrow>* ((A \<longrightarrow>* B) imp (A \<longrightarrow>* C)))) (h::'a::heap_sep_algebra)"
lemma ContraProve: "H \<turnstile> B \<Longrightarrow> insert (Neg B) H \<turnstile> A"
lemma ball_biholomorphism_exists: assumes "a \<in> ball 0 1" obtains f g where "f a = 0" "f holomorphic_on ball 0 1" "f ` ball 0 1 \<subseteq> ball 0 1" "g holomorphic_on ball 0 1" "g ` ball 0 1 \<subseteq> ball 0 1" "\<And>z. z \<in> ball 0 1 \<Longrightarrow> f (g z) = z" "\<And>z. z \<in> ball 0 1 \<Longrightarrow> g (f z) = z"
lemma (in comm_group) finite_sub_comp_iff_card_eq_mult: assumes "subgroup H G" "subgroup J G" "finite H" "finite J" shows "card (H <#> J) = card J * card H \<longleftrightarrow> complementary H J"
theorem soundness: assumes \<open>([], 0 \<turnstile> \<Psi> \<triangleright> []) \<hookrightarrow>\<^bsup>k\<^esup> \<S>\<close> shows \<open>\<lbrakk>\<lbrakk> \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<supseteq> \<lbrakk> \<S> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
theorem bernoulli_altdef: "bernoulli n = (\<Sum>m\<le>n. \<Sum>k\<le>m. (-1)^k * real (m choose k) * real k^n / real (Suc m))"
lemma singleton_union_fset_right: shows "S \|\<union>\| {\|a\|} = insert_fset a S"
lemma t_fusion_leftneutral [simp]: "t_fusion (first x, []) x = x"
lemma to_prime_props : "L (to_prime M) = L M" "observable (to_prime M)" "minimal (to_prime M)" "reachable_states (to_prime M) = states (to_prime M)" "inputs (to_prime M) = inputs M" "outputs (to_prime M) = outputs M"
lemma deriv_j_ladder_shift_j: "index < length L \<Longrightarrow> deriv_j (ladder_shift_j d L ! index) = deriv_j (L ! index) - d"
lemma setsum_cut_off_less: fixes f::"nat \<Rightarrow> nat" assumes h1: "m \<le> n" and h2: "\<forall>i \<in> {m..<n}. f i = 0" shows "(\<Sum>i < n. f i) = (\<Sum>i < m. f i)"
lemma accessible_unsat_core: shows "\<U>\<^sub>c ` {s'. s \<succ>\<^sup>* s'} = {\<U>\<^sub>c s}"
lemma balance_rbt_less[simp]: "(balance a k x b \|\<guillemotleft> v) = (a \|\<guillemotleft> v \<and> b \|\<guillemotleft> v \<and> k < v)"
lemma adv_not_knows1: assumes "out P \<subseteq> ins A" and "\<not> knows A [kE m]" shows "\<not> eout P (kE m)"
lemma var_update_out [simp]: "lens_put (out_var x) (A, A') v = (A, lens_put x A' v)"
lemma delta_split: "delta (R \<union> S) bs = delta R bs \<union> delta S bs"
lemma sorted_list_drop: "sorted_list xs \<Longrightarrow> sorted_list (drop n xs)"
lemma seq_one_simp: "\<one>\<^bsub>R\<^bsup>\<omega>\<^esup>\<^esub> k = \<one>"
lemma closure_invariant_termination: assumes WF: "wf n r" "wf n s" and result: "closure_invariant ([([], init r, init s)], [], {(post (init r), post (init s))}) = None" (is "closure_invariant ([([], ?r, ?s)], _) = None" is "?cl = None") shows "False"
lemma before_vs: "distinct vs \<Longrightarrow> before vs ram1 ram2 \<Longrightarrow> vs = fst (splitAt ram1 vs) @ ram1 # fst (splitAt ram2 (snd (splitAt ram1 vs))) @ ram2 # snd (splitAt ram2 vs)"
lemma ubx_invertible: "ubx opinl xs = Some opubx \<Longrightarrow> deubx opubx = opinl"
lemma assumes "prime n" shows prime_card_primitive_roots: "card {x\<in>totatives n. ord n x = n - 1} = totient (n - 1)" "card {x\<in>{..<n}. ord n x = n - 1} = totient (n - 1)" and prime_primitive_root_exists: "\<exists>x. residue_primroot n x"
theorem secure_equals_c_secure: "refl I \<Longrightarrow> secure (c_process step out s\<^sub>0) I (c_dom D) = c_secure step out s\<^sub>0 I D"
lemma powser_ms_aux'_MSSCons: "powser_ms_aux' (MSSCons c cs) xs = MSLCons (const_expansion c, 0) (times_ms_aux xs (powser_ms_aux' cs xs))"
lemma quad_next4_id: "\<lbrakk> \|vertices f\| = 4; distinct(vertices f); v \<in> \<V> f \<rbrakk> \<Longrightarrow> f \<bullet> (f \<bullet> (f \<bullet> (f \<bullet> v))) = v"
lemma cond_wf_Elem: assumes hyps:"\<forall>x. (\<forall>y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y) \<longrightarrow> Elem x U \<longrightarrow> P x" "Elem a U" shows "P a"
lemma closed_seqs_memE: assumes "s \<in> closed_seqs R" shows "s k \<in> carrier R"
lemma substitute_either: "substitute f T (t \<oplus>\<oplus> t') = substitute f T t \<oplus>\<oplus> substitute f T t'"
lemma iwlsFSbSwTR_imp_iwlsFSb: "iwlsFSbSwTR MOD \<Longrightarrow> iwlsFSb MOD"
lemma binA_neq_cases_swap: assumes neq: "a \<noteq> (b :: binA)" obtains "a = c" and "b = 1 - c" \| "a = 1 - c" and "b = c"
lemma ordered_list_distinct_rev : fixes xs :: "('a::preorder) list" assumes "\<And> i . Suc i < length xs \<Longrightarrow> (xs ! i) > (xs ! (Suc i))" shows "distinct xs"
lemma last_subset: assumes "A \<subseteq> {a,b}" and "a\<noteq> b" and "A \<noteq> {a, b}" and "A\<noteq> {}" and "A \<noteq> {a}" shows "A = {b}"
lemma(in UP_cring) one_over_poly_monom_add: assumes "a \<in> carrier R" assumes "a \<noteq> \<zero>" assumes "p \<in> carrier P" assumes "degree p < n" shows "one_over_poly (p \<oplus>\<^bsub>P\<^esub> monom P a n) = monom P a 0 \<oplus>\<^bsub>P\<^esub> monom P \<one> (n - degree p) \<otimes>\<^bsub>P\<^esub> one_over_poly p"
lemma form_of_lit_is_lit[simp,intro!]: "is_lit_plus (form_of_lit x)"
lemma \<psi>D_im : "T \<in> GRepHomSet (\<star>) W \<Longrightarrow> map (map (\<psi> T)) indVfbasis = aezfun_scalar_mult.negGorbit_list (\<star>) H_rcoset_reps T Vfbasis"
lemma var_dist_subst_Var: "(var_dist vs s)[Var i/j] = var_dist (map (\<lambda>v. vsubst v i j) vs) (s[Var i/j])"
lemma gen_lossless_gpvI [intro?]: "\<lbrakk> lossless_spmf (the_gpv gpv); \<And>out c input. \<lbrakk> IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out \<rbrakk> \<Longrightarrow> gen_lossless_gpv (c input) \<rbrakk> \<Longrightarrow> gen_lossless_gpv gpv"
lemma sets_pair_measure: "sets (A \<Otimes>\<^sub>M B) = sigma_sets (space A \<times> space B) {a \<times> b \| a b. a \<in> sets A \<and> b \<in> sets B}"
lemma remdups_insort[simp]: "a \<in> set xs \<Longrightarrow> remdups (insort a xs) = remdups xs"
lemma stk_convert: "Listn.le (subtype G) a b = G \<turnstile> map OK a <=l map OK b"
lemma "\<not> all_security_requirements_fulfilled security_invariants policy"
lemma (in simplification) rb_INV_finite[simp]: "rb_INV x Q \<Q> \<Longrightarrow> finite \<Q>"
lemma m_s2_rep: assumes "finite(X)" and "S1 = S2 on -X" and "\<forall>x. S1 x \<le> S2 x" and "S1 \<noteq> S2" shows "(\<Sum>x\<in>X. m (S2 x)) < (\<Sum>x\<in>X. m (S1 x))"
lemma bind_case_type_cong [fundef_cong]: assumes "x = x'" and "\<And>t. x = Value t \<Longrightarrow> f t s = f' t s" and "\<And>t. x = Calldata t \<Longrightarrow> g t s = g' t s" and "\<And>t. x = Memory t \<Longrightarrow> h t s = h' t s" and "\<And>t. x = Storage t \<Longrightarrow> i t s = i' t s" shows "(case x of Value t \<Rightarrow> f t \| Calldata t \<Rightarrow> g t \| Memory t \<Rightarrow> h t \| Storage t \<Rightarrow> i t ) s = (case x' of Value t \<Rightarrow> f' t \| Calldata t \<Rightarrow> g' t \| Memory t \<Rightarrow> h' t \| Storage t \<Rightarrow> i' t) s"
lemma indicator_sum_eq: fixes m::real and f :: "'a \<Rightarrow> real" assumes "\<bar>m\<bar> \<le> 2 ^ (2n)" "m/2^n \<le> f x" "f x < (m+1)/2^n" "m \<in> \<int>" shows "(\<Sum>k::real \| k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2n). k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x) = m/2^n" (is "sum ?f ?S = _")
lemma pmf_of_alist_aux: assumes "(s, \<mu>) \<in> set K" shows "pmf (pmf_of_alist \<mu>) t = (case map_of \<mu> t of None \<Rightarrow> 0 \| Some p \<Rightarrow> p)"
lemma ucast_ucast_eq: "\<lbrakk> ucast x = (ucast (ucast y::'a word)::'c::len word); LENGTH('a) \<le> LENGTH('b); LENGTH('b) \<le> LENGTH('c) \<rbrakk> \<Longrightarrow> x = ucast y" for x :: "'a::len word" and y :: "'b::len word"
lemma emeasure_SUP_chain: assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N" assumes ch: "Complete_Partial_Order.chain (\<le>) (M ` A)" and "A \<noteq> {}" shows "emeasure (SUP i\<in>A. M i) X = (SUP i\<in>A. emeasure (M i) X)"
lemma dom_locals_eval_mono: assumes eval: "G\<turnstile> s0 \<midarrow>t\<succ>\<rightarrow> (v,s1)" shows "dom (locals (store s0)) \<subseteq> dom (locals (store s1)) \<and> (\<forall> vv. v=In2 vv \<and> normal s1 \<longrightarrow> (\<forall> s val. dom (locals (store s)) \<subseteq> dom (locals (store ((snd vv) val s)))))"
lemma size_pf_formula14: "sum_list (map size_pf (pf_formula14 n m)) = m + 3 * n + m * (n * 16 - 21) + n * (m * 16 - 21)"
lemma nDAcharn[rule_format]: "noDenyAll p = (\<forall> r \<in> set p. \<not> member DenyAll r)"
lemma mk_deriv_intro_n[simp]: "deriv_n (mk_deriv_intro i n j) = n"
lemma vrestriction_vlrestriction[simp]: "(r \<restriction>\<^sub>\<circ> A) \<restriction>\<^sup>l\<^sub>\<circ> A = r \<restriction>\<^sub>\<circ> A"
lemma vcomp_assoc [simp]: assumes "natural_transformation A B F G \<rho>" and "natural_transformation A B G H \<sigma>" and "natural_transformation A B H K \<tau>" shows "vertical_composite.map A B (vertical_composite.map A B \<rho> \<sigma>) \<tau> = vertical_composite.map A B \<rho> (vertical_composite.map A B \<sigma> \<tau>)"
lemma triangle_F': assumes "D.ide b" shows "C.inverse_arrows (F (\<eta> b)) (\<epsilon> (F b))"
lemma af\<^sub>G_keeps_F_and_S: assumes "ys \<noteq> []" assumes "S \<Turnstile>\<^sub>P af\<^sub>G \<phi> ys" shows "S \<Turnstile>\<^sub>P af\<^sub>G (F \<phi>) (xs @ ys)"
lemma sum_assn_id[simp]: "sum_assn id_assn id_assn = id_assn"
lemma tna_ant:" tna (ant z) = z"
lemma fmdom'_add[simp]: "fmdom' (m ++\<^sub>f n) = fmdom' m \<union> fmdom' n"
lemma apply_guards_condense: "\<exists>g. apply_guards G s = (gval g s = true)"
lemma unitarily_equiv_conjugate: assumes "A\<in> carrier_mat n n" and "V\<in> carrier_mat n n" and "U \<in> carrier_mat n n" and "B\<in> carrier_mat n n" and "unitarily_equiv (mat_conj (Complex_Matrix.adjoint V) A) B U" and "unitary V" shows "unitarily_equiv A B (V * U)"
lemma norm_bound_elements_mat: "norm_bound A b = (\<forall> x \<in> elements_mat A. norm x \<le> b)"
lemma eq_0_height[simp]: "0 = height kdt \<longleftrightarrow> (\<exists>p. kdt = Leaf p)"
theorem HEndPhase2_HInv4b: assumes act: "HEndPhase2 s s' p" and inv: "HInv4b s q" shows "HInv4b s' q"
lemma append_rows_le: assumes A: "A \<in> carrier_mat nr1 nc" and B: "B \<in> carrier_mat nr2 nc" and a: "a \<in> carrier_vec nr1" and v: "v \<in> carrier_vec nc" shows "(A @\<^sub>r B) \<^sub>v v \<le> (a @\<^sub>v b) \<longleftrightarrow> A \<^sub>v v \<le> a \<and> B *\<^sub>v v \<le> b"
lemma MP: "\<forall>a b. a \<^bold>\<and> (a \<^bold>\<Rightarrow> b) \<^bold>\<preceq> b"
lemma exec_instr_xcpt: "(fst (exec_instr i G hp stk vars Cl sig pc frs) = Some xcp) = (\<exists>stk'. exec_instr i G hp stk vars Cl sig pc frs = (Some xcp, hp, (stk', vars, Cl, sig, pc)#frs))"
lemma (in list_to_set) to_set_autoref[autoref_rules]: "PREFER_id Rk \<Longrightarrow> (to_set,set)\<in>\<langle>Rk\<rangle>list_rel \<rightarrow> \<langle>Rk\<rangle>rel"

lemma mono_Retr: "mono Sretr" "mono ZOretr" "mono ZOretrT" "mono Wretr" "mono WretrT" "mono RetrT"

theorem fk_asymptotic_space_complexity: "fk_space_usage \<in> O[at_top \<times>\<^sub>F at_top \<times>\<^sub>F at_top \<times>\<^sub>F at_right (0::rat) \<times>\<^sub>F at_right (0::rat)](\<lambda> (k, n, m, \<epsilon>, \<delta>). real k * real n powr (1-1/ real k) / (of_rat \<delta>)\<^sup>2 * (ln (1 / of_rat \<epsilon>)) * (ln (real n) + ln (real m)))" (is "_ \<in> O[?F](?rhs)")

lemma length_Residx1: shows "length (T \<^sup>*\\\<^sup>1 u) \<le> length T"

lemma Abs_formula_inverse [simp]: assumes "hereditarily_fs t\<^sub>\<alpha>" shows "Rep_formula (Abs_formula t\<^sub>\<alpha>) = t\<^sub>\<alpha>"

lemma get_elment_ptr_simp1 [simp]: "get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr (put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr element h) = Some element"

theorem Crypt_Crypt_eq [iff]: "(Crypt K X = Crypt K X') = (X=X')"

lemma finite_Prefixes: shows "finite (Prefixes s)"

lemma mst_subgraph_inv: assumes "e \<le> v * -v\<^sup>T \<sqinter> g" and "t \<le> g" and "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>" shows "e \<le> (r\<^sup>T * g\<^sup>\<star>)\<^sup>T * (r\<^sup>T * g\<^sup>\<star>) \<sqinter> g"

lemma (in orset) rem_rem_commute: shows "\<langle>Rem i1 e1\<rangle> \<rhd> \<langle>Rem i2 e2\<rangle> = \<langle>Rem i2 e2\<rangle> \<rhd> \<langle>Rem i1 e1\<rangle>"

lemma vec_0[simp]: "vec 0 = 0"

lemma monad_writer_stateT' [locale_witness]: "monad_writer return (bind :: ('a \<times> 's, 'm) bind) (tell :: ('w, 'm) tell) \<Longrightarrow> monad_writer return (bind :: ('a, ('s, 'm) stateT) bind) (tell :: ('w, ('s, 'm) stateT) tell)"

lemma C1_differentiable_imp_piecewise: "f C1_differentiable_on S \<Longrightarrow> f piecewise_C1_differentiable_on S"

lemma invoke_split: "P (invoke ((Pred, f) # xs) ptr args) = ((\<not>(Pred ptr) \<longrightarrow> P (invoke xs ptr args)) \<and> (Pred ptr \<longrightarrow> P (do {check_in_heap ptr; f ptr args})))"

lemma restrictionVsIntersection: "{(x, f x)| x. x \<in> X2} || X1 = {(x, f x)| x. x \<in> X2 \<inter> X1}"

lemma linkageD: "\<lbrakk> linkage \<Gamma> f; x \<in> A \<Gamma> \<rbrakk> \<Longrightarrow> d_OUT f x = weight \<Gamma> x"

lemma li_minus: assumes "locally_irrefl R A" shows "locally_irrefl R (A - B)"

lemma measurable_if_split[measurable (raw)]: "(c \<Longrightarrow> Measurable.pred M f) \<Longrightarrow> (\<not> c \<Longrightarrow> Measurable.pred M g) \<Longrightarrow> Measurable.pred M (if c then f else g)"

lemma eq_tpoly_refl[simp]: "p =\<^sub>p p"

lemma fac_acc_body2_body1_eq: "fac_acc_body2 = fac_acc_body1"

lemma (in VectorSpaceEnd) VectorSpaceHom: "VectorSpaceHom smult V smult T"

lemma order_refl_sum : fixes x :: "'a + 'b" shows "x \<le> x"

lemma instInp_fmla[simp,intro]: assumes "\<phi> \<in> fmla" and "t \<in> trm" shows "instInp \<phi> t \<in> fmla"

lemma vec_to_lin_poly_coeff_access: assumes "x < dim_vec y" shows "y $ x = coeff (vec_to_lpoly y) x"

lemma solves_ode_on_subset_domain: assumes sol: "(x solves_ode f) S Y" and iv: "x t0 = x0" and ivl: "t0 \<in> S" "is_interval S" "S \<subseteq> T" shows "(x solves_ode f) S X"

lemma sub_inc_sysI[intro]: "incidence_system \ \<A> \<Longrightarrow> \ \<subseteq> \<V> \<Longrightarrow> \<A> \<subseteq># \ \<Longrightarrow> sub_incidence_system \ \<A> \<V> \"

lemma Ex_omega_sum: "\<gamma> \<in> elts (\<omega>\<up>n) \<Longrightarrow> \<exists>ns. \<gamma> = omega_sum ns \<and> length ns = n"

lemma cexpr_typing_cong: assumes "\<And>x. x \<in> free_vars e \<Longrightarrow> \<Gamma> x = \<Gamma>' x" shows "\<Gamma> \<turnstile>\<^sub>c e : t \<longleftrightarrow> \<Gamma>' \<turnstile>\<^sub>c e : t"

lemma openin_interior_of [simp]: "openin X (X interior_of S)"

lemma and_wf_dfa: assumes "wf_dfa M n" and "wf_dfa N n" shows "wf_dfa (and_dfa M N) n"

lemma subst_typ'_simulates_tsubst_gen': "distinct pairs \<Longrightarrow> tvs t \<subseteq> set pairs \<Longrightarrow> tsubst t \<rho> = subst_typ' (map (\<lambda>(x,y).((x,y), \<rho> x y)) pairs) t"

lemma if_redT_known_addrs_new: assumes redT: "mthr.if.redT s (t, ta) s'" shows "if.known_addrs_state s' \<subseteq> if.known_addrs_state s \<union> new_obs_addrs_if \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>"

lemma (in is_cf_adjunction) is_cf_adjunction_axioms'[adj_cs_intros]: assumes "\<alpha>' = \<alpha>" and "\<CC>' = \<CC>" and "\<DD>' = \<DD>" and "\<FF>' = \<FF>" and "\<GG>' = \<GG>" shows "\<Phi> : \<FF>' \<rightleftharpoons>\<^sub>C\<^sub>F \<GG>' : \<CC>' \<rightleftharpoons>\<rightleftharpoons>\<^sub>C\<^bsub>\<alpha>'\<^esub> \<DD>'"

lemma "(sep_empty imp ((A \<longrightarrow>* (B imp C)) \<longrightarrow>* ((A \<longrightarrow>* B) imp (A \<longrightarrow>* C)))) (h::'a::heap_sep_algebra)"

lemma ContraProve: "H \<turnstile> B \<Longrightarrow> insert (Neg B) H \<turnstile> A"

lemma ball_biholomorphism_exists: assumes "a \<in> ball 0 1" obtains f g where "f a = 0" "f holomorphic_on ball 0 1" "f ` ball 0 1 \<subseteq> ball 0 1" "g holomorphic_on ball 0 1" "g ` ball 0 1 \<subseteq> ball 0 1" "\<And>z. z \<in> ball 0 1 \<Longrightarrow> f (g z) = z" "\<And>z. z \<in> ball 0 1 \<Longrightarrow> g (f z) = z"

lemma (in comm_group) finite_sub_comp_iff_card_eq_mult: assumes "subgroup H G" "subgroup J G" "finite H" "finite J" shows "card (H <#> J) = card J * card H \<longleftrightarrow> complementary H J"

theorem soundness: assumes \<open>([], 0 \<turnstile> \<Psi> \<triangleright> []) \<hookrightarrow>\<^bsup>k\<^esup> \<S>\<close> shows \<open>\<lbrakk>\<lbrakk> \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<supseteq> \<lbrakk> \<S> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>

theorem bernoulli_altdef: "bernoulli n = (\<Sum>m\<le>n. \<Sum>k\<le>m. (-1)^k * real (m choose k) * real k^n / real (Suc m))"

lemma singleton_union_fset_right: shows "S |\<union>| {|a|} = insert_fset a S"

lemma t_fusion_leftneutral [simp]: "t_fusion (first x, []) x = x"

lemma to_prime_props : "L (to_prime M) = L M" "observable (to_prime M)" "minimal (to_prime M)" "reachable_states (to_prime M) = states (to_prime M)" "inputs (to_prime M) = inputs M" "outputs (to_prime M) = outputs M"

lemma deriv_j_ladder_shift_j: "index < length L \<Longrightarrow> deriv_j (ladder_shift_j d L ! index) = deriv_j (L ! index) - d"

lemma setsum_cut_off_less: fixes f::"nat \<Rightarrow> nat" assumes h1: "m \<le> n" and h2: "\<forall>i \<in> {m..<n}. f i = 0" shows "(\<Sum>i < n. f i) = (\<Sum>i < m. f i)"

lemma accessible_unsat_core: shows "\\<^sub>c ` {s'. s \<succ>\<^sup>* s'} = {\\<^sub>c s}"

lemma balance_rbt_less[simp]: "(balance a k x b |\<guillemotleft> v) = (a |\<guillemotleft> v \<and> b |\<guillemotleft> v \<and> k < v)"

lemma adv_not_knows1: assumes "out P \<subseteq> ins A" and "\<not> knows A [kE m]" shows "\<not> eout P (kE m)"

lemma var_update_out [simp]: "lens_put (out_var x) (A, A') v = (A, lens_put x A' v)"

lemma delta_split: "delta (R \<union> S) bs = delta R bs \<union> delta S bs"

lemma sorted_list_drop: "sorted_list xs \<Longrightarrow> sorted_list (drop n xs)"

lemma seq_one_simp: "\<one>\<^bsub>R\<^bsup>\<omega>\<^esup>\<^esub> k = \<one>"

lemma closure_invariant_termination: assumes WF: "wf n r" "wf n s" and result: "closure_invariant ([([], init r, init s)], [], {(post (init r), post (init s))}) = None" (is "closure_invariant ([([], ?r, ?s)], _) = None" is "?cl = None") shows "False"

lemma before_vs: "distinct vs \<Longrightarrow> before vs ram1 ram2 \<Longrightarrow> vs = fst (splitAt ram1 vs) @ ram1 # fst (splitAt ram2 (snd (splitAt ram1 vs))) @ ram2 # snd (splitAt ram2 vs)"

lemma ubx_invertible: "ubx opinl xs = Some opubx \<Longrightarrow> deubx opubx = opinl"

lemma assumes "prime n" shows prime_card_primitive_roots: "card {x\<in>totatives n. ord n x = n - 1} = totient (n - 1)" "card {x\<in>{..<n}. ord n x = n - 1} = totient (n - 1)" and prime_primitive_root_exists: "\<exists>x. residue_primroot n x"

theorem secure_equals_c_secure: "refl I \<Longrightarrow> secure (c_process step out s\<^sub>0) I (c_dom D) = c_secure step out s\<^sub>0 I D"

lemma powser_ms_aux'_MSSCons: "powser_ms_aux' (MSSCons c cs) xs = MSLCons (const_expansion c, 0) (times_ms_aux xs (powser_ms_aux' cs xs))"

lemma quad_next4_id: "\<lbrakk> |vertices f| = 4; distinct(vertices f); v \<in> \<V> f \<rbrakk> \<Longrightarrow> f \<bullet> (f \<bullet> (f \<bullet> (f \<bullet> v))) = v"

lemma cond_wf_Elem: assumes hyps:"\<forall>x. (\<forall>y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y) \<longrightarrow> Elem x U \<longrightarrow> P x" "Elem a U" shows "P a"

lemma closed_seqs_memE: assumes "s \<in> closed_seqs R" shows "s k \<in> carrier R"

lemma substitute_either: "substitute f T (t \<oplus>\<oplus> t') = substitute f T t \<oplus>\<oplus> substitute f T t'"

lemma iwlsFSbSwTR_imp_iwlsFSb: "iwlsFSbSwTR MOD \<Longrightarrow> iwlsFSb MOD"

lemma binA_neq_cases_swap: assumes neq: "a \<noteq> (b :: binA)" obtains "a = c" and "b = 1 - c" | "a = 1 - c" and "b = c"

lemma ordered_list_distinct_rev : fixes xs :: "('a::preorder) list" assumes "\<And> i . Suc i < length xs \<Longrightarrow> (xs ! i) > (xs ! (Suc i))" shows "distinct xs"

lemma last_subset: assumes "A \<subseteq> {a,b}" and "a\<noteq> b" and "A \<noteq> {a, b}" and "A\<noteq> {}" and "A \<noteq> {a}" shows "A = {b}"

lemma(in UP_cring) one_over_poly_monom_add: assumes "a \<in> carrier R" assumes "a \<noteq> \<zero>" assumes "p \<in> carrier P" assumes "degree p < n" shows "one_over_poly (p \<oplus>\<^bsub>P\<^esub> monom P a n) = monom P a 0 \<oplus>\<^bsub>P\<^esub> monom P \<one> (n - degree p) \<otimes>\<^bsub>P\<^esub> one_over_poly p"

lemma form_of_lit_is_lit[simp,intro!]: "is_lit_plus (form_of_lit x)"

lemma \<psi>D_im : "T \<in> GRepHomSet (\<star>) W \<Longrightarrow> map (map (\<psi> T)) indVfbasis = aezfun_scalar_mult.negGorbit_list (\<star>) H_rcoset_reps T Vfbasis"

lemma var_dist_subst_Var: "(var_dist vs s)[Var i/j] = var_dist (map (\<lambda>v. vsubst v i j) vs) (s[Var i/j])"

lemma gen_lossless_gpvI [intro?]: "\<lbrakk> lossless_spmf (the_gpv gpv); \<And>out c input. \<lbrakk> IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\ \ out \<rbrakk> \<Longrightarrow> gen_lossless_gpv (c input) \<rbrakk> \<Longrightarrow> gen_lossless_gpv gpv"

lemma sets_pair_measure: "sets (A \<Otimes>\<^sub>M B) = sigma_sets (space A \<times> space B) {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}"

lemma remdups_insort[simp]: "a \<in> set xs \<Longrightarrow> remdups (insort a xs) = remdups xs"

lemma stk_convert: "Listn.le (subtype G) a b = G \<turnstile> map OK a <=l map OK b"

lemma "\<not> all_security_requirements_fulfilled security_invariants policy"

lemma (in simplification) rb_INV_finite[simp]: "rb_INV x Q \<Q> \<Longrightarrow> finite \<Q>"

lemma m_s2_rep: assumes "finite(X)" and "S1 = S2 on -X" and "\<forall>x. S1 x \<le> S2 x" and "S1 \<noteq> S2" shows "(\<Sum>x\<in>X. m (S2 x)) < (\<Sum>x\<in>X. m (S1 x))"

lemma bind_case_type_cong [fundef_cong]: assumes "x = x'" and "\<And>t. x = Value t \<Longrightarrow> f t s = f' t s" and "\<And>t. x = Calldata t \<Longrightarrow> g t s = g' t s" and "\<And>t. x = Memory t \<Longrightarrow> h t s = h' t s" and "\<And>t. x = Storage t \<Longrightarrow> i t s = i' t s" shows "(case x of Value t \<Rightarrow> f t | Calldata t \<Rightarrow> g t | Memory t \<Rightarrow> h t | Storage t \<Rightarrow> i t ) s = (case x' of Value t \<Rightarrow> f' t | Calldata t \<Rightarrow> g' t | Memory t \<Rightarrow> h' t | Storage t \<Rightarrow> i' t) s"

lemma indicator_sum_eq: fixes m::real and f :: "'a \<Rightarrow> real" assumes "\<bar>m\<bar> \<le> 2 ^ (2*n)" "m/2^n \<le> f x" "f x < (m+1)/2^n" "m \<in> \<int>" shows "(\<Sum>k::real | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x) = m/2^n" (is "sum ?f ?S = _")

lemma pmf_of_alist_aux: assumes "(s, \<mu>) \<in> set K" shows "pmf (pmf_of_alist \<mu>) t = (case map_of \<mu> t of None \<Rightarrow> 0 | Some p \<Rightarrow> p)"

lemma ucast_ucast_eq: "\<lbrakk> ucast x = (ucast (ucast y::'a word)::'c::len word); LENGTH('a) \<le> LENGTH('b); LENGTH('b) \<le> LENGTH('c) \<rbrakk> \<Longrightarrow> x = ucast y" for x :: "'a::len word" and y :: "'b::len word"

lemma emeasure_SUP_chain: assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N" assumes ch: "Complete_Partial_Order.chain (\<le>) (M ` A)" and "A \<noteq> {}" shows "emeasure (SUP i\<in>A. M i) X = (SUP i\<in>A. emeasure (M i) X)"

lemma dom_locals_eval_mono: assumes eval: "G\<turnstile> s0 \<midarrow>t\<succ>\<rightarrow> (v,s1)" shows "dom (locals (store s0)) \<subseteq> dom (locals (store s1)) \<and> (\<forall> vv. v=In2 vv \<and> normal s1 \<longrightarrow> (\<forall> s val. dom (locals (store s)) \<subseteq> dom (locals (store ((snd vv) val s)))))"

lemma size_pf_formula14: "sum_list (map size_pf (pf_formula14 n m)) = m + 3 * n + m * (n * 16 - 21) + n * (m * 16 - 21)"

lemma nDAcharn[rule_format]: "noDenyAll p = (\<forall> r \<in> set p. \<not> member DenyAll r)"

lemma mk_deriv_intro_n[simp]: "deriv_n (mk_deriv_intro i n j) = n"

lemma vrestriction_vlrestriction[simp]: "(r \<restriction>\<^sub>\<circ> A) \<restriction>\<^sup>l\<^sub>\<circ> A = r \<restriction>\<^sub>\<circ> A"

lemma vcomp_assoc [simp]: assumes "natural_transformation A B F G \<rho>" and "natural_transformation A B G H \<sigma>" and "natural_transformation A B H K \<tau>" shows "vertical_composite.map A B (vertical_composite.map A B \<rho> \<sigma>) \<tau> = vertical_composite.map A B \<rho> (vertical_composite.map A B \<sigma> \<tau>)"

lemma triangle_F': assumes "D.ide b" shows "C.inverse_arrows (F (\<eta> b)) (\<epsilon> (F b))"

lemma af\<^sub>G_keeps_F_and_S: assumes "ys \<noteq> []" assumes "S \<Turnstile>\<^sub>P af\<^sub>G \<phi> ys" shows "S \<Turnstile>\<^sub>P af\<^sub>G (F \<phi>) (xs @ ys)"

lemma sum_assn_id[simp]: "sum_assn id_assn id_assn = id_assn"

lemma tna_ant:" tna (ant z) = z"

lemma fmdom'_add[simp]: "fmdom' (m ++\<^sub>f n) = fmdom' m \<union> fmdom' n"

lemma apply_guards_condense: "\<exists>g. apply_guards G s = (gval g s = true)"

lemma unitarily_equiv_conjugate: assumes "A\<in> carrier_mat n n" and "V\<in> carrier_mat n n" and "U \<in> carrier_mat n n" and "B\<in> carrier_mat n n" and "unitarily_equiv (mat_conj (Complex_Matrix.adjoint V) A) B U" and "unitary V" shows "unitarily_equiv A B (V * U)"

lemma norm_bound_elements_mat: "norm_bound A b = (\<forall> x \<in> elements_mat A. norm x \<le> b)"

lemma eq_0_height[simp]: "0 = height kdt \<longleftrightarrow> (\<exists>p. kdt = Leaf p)"

theorem HEndPhase2_HInv4b: assumes act: "HEndPhase2 s s' p" and inv: "HInv4b s q" shows "HInv4b s' q"

lemma append_rows_le: assumes A: "A \<in> carrier_mat nr1 nc" and B: "B \<in> carrier_mat nr2 nc" and a: "a \<in> carrier_vec nr1" and v: "v \<in> carrier_vec nc" shows "(A @\<^sub>r B) *\<^sub>v v \<le> (a @\<^sub>v b) \<longleftrightarrow> A *\<^sub>v v \<le> a \<and> B *\<^sub>v v \<le> b"

lemma MP: "\<forall>a b. a \<^bold>\<and> (a \<^bold>\<Rightarrow> b) \<^bold>\<preceq> b"

lemma exec_instr_xcpt: "(fst (exec_instr i G hp stk vars Cl sig pc frs) = Some xcp) = (\<exists>stk'. exec_instr i G hp stk vars Cl sig pc frs = (Some xcp, hp, (stk', vars, Cl, sig, pc)#frs))"

lemma (in list_to_set) to_set_autoref[autoref_rules]: "PREFER_id Rk \<Longrightarrow> (to_set,set)\<in>\<langle>Rk\<rangle>list_rel \<rightarrow> \<langle>Rk\<rangle>rel"