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

Statement: stringlengths 7 24.3k
lemma topo_open_imp: fixes A and S (structure) defines "S \<equiv> topo A" fixes B and T (structure) defines "T \<equiv> topo B" shows "\<lbrakk> A \<subseteq> B; x open\<^bsub>S\<^esub> \<rbrakk> \<Longrightarrow> x open\<^bsub>T\<^esub>" (is "PROP ?P")
lemma fp_cop_F_preferred: assumes "y \<in> CD_on ds (CH (fp_cop_F ds) \<union> X'')" assumes "x \<in> CH (fp_cop_F ds)" assumes "Xd x = Xd y" shows "(x, y) \<in> Pd (Xd x)"
lemma bst_iff_sorted_wrt_less: "bst t \<longleftrightarrow> sorted_wrt (<) (inorder t)"
lemma mdeg_child_if_wedge: "\<lbrakk>max_deg (Node r xs) > n; fcard xs \<le> n \<or> \<not>(\<forall>t \<in> fst ` fset xs. max_deg t \<le> n)\<rbrakk> \<Longrightarrow> \<exists>t \<in> fst ` fset xs. max_deg t > n"
lemma exposed_face_of_Int_supporting_hyperplane_ge: "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"
lemma intvs_decr_h: "{l::int..<h - 1} = {l..<h} - {h-1}"
lemma inv_baldR: "\<lbrakk> invh l; invh r; bheight l = bheight r + 1; invc l; invc2 r \<rbrakk> \<Longrightarrow> invh (baldR l a r) \<and> bheight (baldR l a r) = bheight l \<and> invc2 (baldR l a r) \<and> (color l = Black \<longrightarrow> invc (baldR l a r))"
lemma uint32_of_nat_code [code]: "uint32_of_nat = uint32_of_int \<circ> int"
lemma safety_invariant: shows "safety (\<lambda>\<sigma>. \<forall>i. P (\<sigma> i))"
lemma sum_not_less_zero[simp, dest]: "(s::'lbl) < 0 \<Longrightarrow> False"
lemma prime_factorization_0 [simp]: "prime_factorization 0 = {#}"
lemma ins_impl: "set_ins \<alpha> invar ins"
lemma W_endomorphism: "w\<in>W \<Longrightarrow> ChamberComplexEndomorphism X (permutation w)"
lemma free_incorrect_cap_offset: assumes "c \<noteq> NULL" and "tag c = True" and "perm_global c = False" and "Mapping.lookup (heap_map h) (block_id c) = Some (Map m)" and "offset c \<noteq> 0" shows "free h c = Error (LogicErr (Unhandled 0))"
lemma degen_path_sound: assumes "path (degen.E T m) u p v" shows "path E (fst u) (map fst p) (fst v)"
lemma acyclic_5a_5e: "acyclic_5a x \<longleftrightarrow> acyclic_5e x"
lemma insert_type: "t \<in> B h \<Longrightarrow> insert x t \<in> B h \<union> B (Suc h)"
lemma mcont_id' [cont_intro, simp]: "mcont lub ord lub ord (\<lambda>x. x)"
lemma bdt_Some_Node_iff [simp]: "(bdt t var = Some (Bdt_Node bdt1 v bdt2)) = (\<exists> p l r. t = Node l p r \<and> bdt l var = Some bdt1 \<and> bdt r var = Some bdt2 \<and> 1 < v \<and> var p = v )"
lemma field_cont_on_typeI_region_cont_on_edges: assumes typeI_twoC: "typeI_twoCube twoC" and field_cont: "continuous_on (cubeImage twoC) F" and member_of_boundary: "(k,\<gamma>) \<in> boundary twoC" shows "continuous_on (\<gamma> ` {0 .. 1}) F"
lemma flow_initial_time[simp]: "t0 \<in> T \<Longrightarrow> x0 \<in> X \<Longrightarrow> flow t0 x0 t0 = x0"
lemma ordinal_plus_oSuc [simp]: "x + oSuc y = oSuc (x + y)"
lemma mutator_reachable_tso: "sys_mem_store_buffers (mutator m) s = mw_Mutate r f opt_r' # ws \<Longrightarrow> mut_m.reachable m r s \<and> (\<forall>r'. opt_r' = Some r' \<longrightarrow> mut_m.reachable m r' s)" "sys_mem_store_buffers (mutator m) s = mw_Mutate_Payload r f pl # ws \<Longrightarrow> mut_m.reachable m r s"
lemma approx_diff: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> a - c \<approx> b - d"
lemma finite_natarray2: "finite {(x, y). x < (m::nat) & y < (n::nat)}"
lemma exits_simps [simp]: "exits [ADD] = {1}" "exits [LOADI v] = {1}" "exits [LOAD x] = {1}" "exits [STORE x] = {1}" "i \<noteq> -1 \<Longrightarrow> exits [JMP i] = {1 + i}" "i \<noteq> -1 \<Longrightarrow> exits [JMPGE i] = {1 + i, 1}" "i \<noteq> -1 \<Longrightarrow> exits [JMPLESS i] = {1 + i, 1}"
lemma dom_restrict_inter[simp] : "T \<triangleleft> S \<triangleleft> p = T \<inter> S \<triangleleft> p"
lemma total_on_lexord_less_than_char_linear2: \<open>xs \<noteq> ys \<Longrightarrow> (xs, ys) \<notin> lexord (less_than_char) \<longleftrightarrow> (ys, xs) \<in> lexord less_than_char\<close>
lemma assumes A: "\<forall>xs A. (\<forall>X \<in> A. (xs, X) \<in> failures P) \<longrightarrow> (xs, \<Union>X \<in> A. X) \<in> failures P" shows "\<forall>xs. xs \<in> traces P"
lemma split_invar: "invar t \<Longrightarrow> split t = t"
lemma mult_Limit: "Limit \<gamma> \<Longrightarrow> x * \<gamma> = \<Squnion> ((*) x ` elts \<gamma>)"
lemma geod_dist3: assumes "a \<le> 2^n" "b \<le> 2^p" shows "dist (geod n a) (geod p b) = abs(b/2^p - a/2^n) * dist x0 y0"
lemma L2_set_empty [simp]: "L2_set f {} = 0"
lemma assertion_fun_disj_less_one: "assertion_fun = Apply.disjunctive \<inter> {x::'a::boolean_algebra \<Rightarrow> 'a . x \<le> id}"
lemma drop_bit_word_minus_numeral [simp]: \<open>drop_bit (numeral n) (- numeral k) = (word_of_int (drop_bit (numeral n) (take_bit LENGTH('a) (- numeral k))) :: 'a::len word)\<close>
lemma stream_rel_pred_szip: "stream_all2 P u v \<longleftrightarrow> pred_stream (case_prod P) (szip u v)"
lemma reg_funas: "\<L> \<A> \<subseteq> \<T>\<^sub>G (fset (ta_sig (ta \<A>)))"
lemma at_least_one_step:"steps0 (1, [], r) tm n = (0,tap) \<Longrightarrow> 0 < n"
lemma epsclo_nextl [simp]: "epsclo (nextl Q xs) = nextl Q xs"
lemma valid_adv_start_bounds': assumes "valid_window args t0 sub rho w" "w_run_t args (w_ti w) = Some (ti', t)" "w_run_sub args (w_si w) = Some (si', bs)" shows "w_ti (adv_start args w) = ti'" "w_si (adv_start args w) = si'"
lemma "(R'::'a::complete_lattice \<Rightarrow> 'b::complete_lattice) \<in> Apply.Disjunctive \<Longrightarrow> DataRefinement S R R' S' \<Longrightarrow> R (- grd S) \<le> - grd S'"
lemma apply_swap_same [simp]: "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> \<langle>a \<leftrightarrow> b\<rangle> \<langle>$\<rangle> c = c"
lemma (in Order) BNTr4:"\<lbrakk>f \<in> carrier D \<rightarrow> carrier D; a \<in> carrier D; \<forall>x\<in>carrier D. x \<preceq> (f x); W1 \<in> WWa D f a; W2 \<in> WWa D f a; \<exists>b\<in>carrier D. ord_equiv (Iod D W1) (Iod D (segment (Iod D W2) b))\<rbrakk> \<Longrightarrow> W1 \<subseteq> W2"
lemma holomorphic_injective_imp_regular: assumes holf: "f holomorphic_on S" and "open S" and injf: "inj_on f S" and "\<xi> \<in> S" shows "deriv f \<xi> \<noteq> 0"
lemma all_bi_edges_alt2: "all_bi_edges X Y = {{x, y} \| x y. x \<in> X \<and> y \<in> Y }"
lemma vdisjnt_singleton0[simp]: "vdisjnt (set {a}) (set {b}) \<longleftrightarrow> a \<noteq> b" and vdisjnt_singleton1[simp]: "vdisjnt (set {a}) A \<longleftrightarrow> a \<notin>\<^sub>\<circ> A" and vdisjnt_singleton2[simp]: "vdisjnt A (set {a}) \<longleftrightarrow> a \<notin>\<^sub>\<circ> A"
lemma \<Delta>\<^sub>\<epsilon>_swap: "prod.swap p \|\<in>\| \<Delta>\<^sub>\<epsilon> \<A> \<B> \<longleftrightarrow> p \|\<in>\| \<Delta>\<^sub>\<epsilon> \<B> \<A>"
lemma bounded_closure[intro]: assumes "bounded S" shows "bounded (closure S)"
theorem secure_implies_c_secure: assumes S: "secure (c_process step out s\<^sub>0) I (c_dom D)" shows "c_secure step out s\<^sub>0 I D"
lemma L_alt: "L idx I = to_language {xs. \<exists>\<AA> \<in> I. \<exists>\<BB>. \<AA> = fold CONS (rev xs) \<BB> \<and> Length \<BB> = 0 \<and> #\<^sub>V \<BB> = idx \<and> (\<forall>x \<in> set xs. size x = idx)}"
lemma Map_comp [simp]: assumes "arr f" and "arr g" and "Dom g = Cod f" shows "Map (COMP g f) = Comp (Cod g) (Dom g) (Dom f) (Map g) (Map f)"
lemma subst_plus [usubst]: "\<sigma> \<dagger> (x + y) = \<sigma> \<dagger> x + \<sigma> \<dagger> y"
lemma suffix_imp_sublist [simp, intro]: "suffix xs ys \<Longrightarrow> sublist xs ys"
lemma assumes asms: "A" "B" shows "True"
lemma closed_cspan_range_ket[simp]: \<open>closure (cspan (range ket)) = UNIV\<close>
lemma contour_integral_subpath_combine: assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}" shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f = contour_integral (subpath u w g) f"
lemma rstep_imp_C_s_r: assumes "(s,t) \<in> rstep R" shows "\<exists>C \<sigma> l r. (l,r) \<in> R \<and> s = C\<langle>l\<cdot>\<sigma>\<rangle> \<and> t = C\<langle>r\<cdot>\<sigma>\<rangle>"
lemma absolutely_integrable_component_ubound: fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space" assumes f: "f integrable_on A" and g: "g absolutely_integrable_on A" and comp: "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> f x \<bullet> b \<le> g x \<bullet> b" shows "f absolutely_integrable_on A"
lemma iteci_lu_rule: " ( mi.ite_impl_lu i t e bdd = Some (p,bdd')) \<longrightarrow> <is_bdd_impl bdd bddi> iteci_lu i t e bddi <\<lambda>(pi,bddi'). is_bdd_impl bdd' bddi' * \<up>(pi=p )>\<^sub>t"
lemma FL_bisimilar_symp: "symp (FL_bisimilar F)"
lemma card_one[elim]: assumes "card A = 1" obtains a where "A = {a}"
lemma AKcryptSK_not_AKcryptSK: "\<lbrakk> AKcryptSK authK servK evs; evs \<in> kerbIV_gets \<rbrakk> \<Longrightarrow> \<not> AKcryptSK servK K evs"
lemma add_node_wf[simp]: "wf_graph g \<Longrightarrow> wf_graph (add_node v g)"
lemma neg_lt: "neg\<cdot>(lt\<cdot>x\<cdot>y) = le\<cdot>y\<cdot>x"
lemma input_unconstrained_aval_input_swap: "\<forall>i. \<not> aexp_constrains a (V (I i)) \<Longrightarrow> aval a (join_ir i r) = aval a (join_ir i' r)"
lemma l3: assumes "x \<in> S2" shows "finite x"
lemma B_trusts_NS3: "\<lbrakk>Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace> \<in> parts (spies evs); B \<notin> bad; evs \<in> ns_shared\<rbrakk> \<Longrightarrow> \<exists>NA. Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>\<rbrace>) \<in> set evs"
lemma signed_take_bit_eq_if_positive: \<open>signed_take_bit n a = take_bit n a\<close> if \<open>\<not> bit a n\<close>
lemma findnth_inv_layout_of_via_crsp[simp]: "crsp (layout_of ap) (as, lm) (s, l, r) ires \<Longrightarrow> findnth_inv (layout_of ap) n (as, lm) (Suc 0, l, r) ires"
lemma %invisible F2'_antimono: shows "antimono (\<lambda>XD. - RD_on (ds-{d'}) XD)"
lemma lookup_pair_eq_valueI: assumes "oalist_inv_raw xs" and "(k, v) \<in> set xs" shows "lookup_pair xs k = v"
lemma b_assn_invalid_merge5: "hn_ctxt (b_assn A P') x y \<or>\<^sub>A hn_invalid (b_assn A P) x y \<Longrightarrow>\<^sub>t hn_invalid (b_assn A (\<lambda>x. P x \<or> P' x)) x y"
lemma cm_all_eq_cm_all': assumes "\<forall>(loc,t)\<in>set_zmset \<Delta>. \<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t" shows "cm_all c0 \<Delta> = cm_all' c0 \<Delta>"
lemma weakCongSum1: fixes P :: ccs and \<alpha> :: act and P' :: ccs and Q :: ccs assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "P \<oplus> Q \<Longrightarrow>\<alpha> \<prec> P'"
lemma msetext_dersh_trans: assumes zs_a: "zs \<in> lists A" and ys_a: "ys \<in> lists A" and xs_a: "xs \<in> lists A" and trans: "\<forall>z \<in> A. \<forall>y \<in> A. \<forall>x \<in> A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x" and zs_gt_ys: "msetext_dersh gt zs ys" and ys_gt_xs: "msetext_dersh gt ys xs" shows "msetext_dersh gt zs xs"
lemma lasso_run_rel_def: "\<langle>R\<rangle>lasso_run_rel = br run_of_lasso (\<lambda>_. True) O (nat_rel \<rightarrow> R)"
lemma filternew_flows_state_alt2: "filternew_flows_state \<T> = {e \<in> flows_state \<T>. e \<notin> backflows (flows_fix \<T>)}"
lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
lemma Col_dep2: "real_euclid.Col a b c \<longleftrightarrow> dep2 (b - a) (c - a)"
lemma reachable_append: "reachable g (xs @ ys) = reachable g xs \<union> reachable g ys"
lemma (in \<Z>) \<KK>23_is_tiny_functor: "\<KK>23 : cat_ordinal (2\<^sub>\<nat>) \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> cat_ordinal (3\<^sub>\<nat>)"
lemma continuous_on_inverse_ennreal': "continuous_on (UNIV :: ennreal set) inverse"
lemma bisimI: fixes P :: "ccs" and Q :: "ccs" assumes "P \<leadsto>[bisim] Q" and "Q \<sim> P" shows "P \<sim> Q"
lemma SSpec_strong: "\<Turnstile> c :A \<Longrightarrow> \<forall> s t . SSpec c s t \<longrightarrow> A s t"
lemma Trivially_fulfilled_FCD_Nabla_\<Upsilon>_empty: "\<lbrakk>\<nabla>\<^bsub>\<Gamma>\<^esub>={} \<or> \<Upsilon>\<^bsub>\<Gamma>\<^esub>={}\<rbrakk>\<Longrightarrow> FCD \<Gamma> \<V> Tr\<^bsub>ES\<^esub>"
lemma bin_basis_code: "code {\<zero>,\<one>}"
lemma (in fl_subdigraph) fl_subdg_Hom_eq: assumes "A \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" and "B \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "Hom \<BB> A B = Hom \<CC> A B"
lemma (in Corps) PCauchy_lPCauchy:"\<lbrakk>valuation K v; PolynRg R (Vr K v) X; \<forall>n. F n \<in> carrier R; \<forall>n. deg R (Vr K v) X (F n) \<le> an (Suc d); P_mod R (Vr K v) X (vp K v\<^bsup>(Vr K v) (an N)\<^esup>) (F n \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> (F m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (vp K v\<^bsup>(Vr K v) (an N)\<^esup>) (((Pseql\<^bsub>R X K v d\<^esub> F) n) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> ((Pseql\<^bsub>R X K v d\<^esub> F) m))"
lemma Skip_Sim: "Skip \<preceq>S Skip"
lemma has_vector_derivative_eq_has_derivative_blinfun: "(f has_vector_derivative f') (at x within U) \<longleftrightarrow> (f has_derivative blinfun_scaleR_left f') (at x within U)"
lemma [simp]: "fields E ClassCast = []"
lemma unital_quantale_homset_iff: "f \<in> unital_quantale_homset = (comp_pres f \<and> Sup_pres f \<and> un_pres f)"
lemma siso_cont_indis[simp]: assumes : "siso c" and *: "s \<approx> t" "i < brn c" shows "eff c s i \<approx> eff c t i \<and> wt c s i = wt c t i \<and> cont c s i = cont c t i"
lemma vrangeD[dest]: assumes "\<langle>r, s\<rangle> \<in>\<^sub>\<circ> vrange A" shows "r \<in>\<^sub>\<circ> A" and "s = \<R>\<^sub>\<circ> r"
lemma bind_assoc_SBE: "(y :\<equiv> (x :\<equiv> m; k); h) = (x :\<equiv> m; (y :\<equiv> k; h))"
theorem gb_sig_z_is_min_sig_GB: assumes "p \<in> set (fst (gb_sig_z rw_rat_strict fs))" and "q \<in> set (fst (gb_sig_z rw_rat_strict fs))" and "p \<noteq> q" and "punit.lt (snd p) adds punit.lt (snd q)" shows "punit.lt (snd p) \<oplus> fst q \<prec>\<^sub>t punit.lt (snd q) \<oplus> fst p"
lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
lemma (in imap) added_ids_Deliver_Expunge_diff_collapse [simp]: shows "e \<noteq> e' \<Longrightarrow> added_ids ([Deliver (i, Expunge e mo j)]) e' = []"
lemma valuesum_rec: assumes fin: "finite (dom (Mapping.lookup m))" shows "valuesum m = (if Mapping.is_empty m then 0 else let l = (SOME l. l \<in> Mapping.keys m) in the (Mapping.lookup m l) + valuesum (Mapping.delete l m))"
lemma apply_guards_double_cons: "apply_guards (y # x # G) s = (gval (gAnd y x) s = true \<and> apply_guards G s)"

lemma topo_open_imp: fixes A and S (structure) defines "S \<equiv> topo A" fixes B and T (structure) defines "T \<equiv> topo B" shows "\<lbrakk> A \<subseteq> B; x open\<^bsub>S\<^esub> \<rbrakk> \<Longrightarrow> x open\<^bsub>T\<^esub>" (is "PROP ?P")

lemma fp_cop_F_preferred: assumes "y \<in> CD_on ds (CH (fp_cop_F ds) \<union> X'')" assumes "x \<in> CH (fp_cop_F ds)" assumes "Xd x = Xd y" shows "(x, y) \<in> Pd (Xd x)"

lemma bst_iff_sorted_wrt_less: "bst t \<longleftrightarrow> sorted_wrt (<) (inorder t)"

lemma mdeg_child_if_wedge: "\<lbrakk>max_deg (Node r xs) > n; fcard xs \<le> n \<or> \<not>(\<forall>t \<in> fst ` fset xs. max_deg t \<le> n)\<rbrakk> \<Longrightarrow> \<exists>t \<in> fst ` fset xs. max_deg t > n"

lemma exposed_face_of_Int_supporting_hyperplane_ge: "\<lbrakk>convex S; \<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b\<rbrakk> \<Longrightarrow> (S \<inter> {x. a \<bullet> x = b}) exposed_face_of S"

lemma intvs_decr_h: "{l::int..<h - 1} = {l..<h} - {h-1}"

lemma inv_baldR: "\<lbrakk> invh l; invh r; bheight l = bheight r + 1; invc l; invc2 r \<rbrakk> \<Longrightarrow> invh (baldR l a r) \<and> bheight (baldR l a r) = bheight l \<and> invc2 (baldR l a r) \<and> (color l = Black \<longrightarrow> invc (baldR l a r))"

lemma uint32_of_nat_code [code]: "uint32_of_nat = uint32_of_int \<circ> int"

lemma safety_invariant: shows "safety (\<lambda>\<sigma>. \<forall>i. P (\<sigma> i))"

lemma sum_not_less_zero[simp, dest]: "(s::'lbl) < 0 \<Longrightarrow> False"

lemma prime_factorization_0 [simp]: "prime_factorization 0 = {#}"

lemma ins_impl: "set_ins \<alpha> invar ins"

lemma W_endomorphism: "w\<in>W \<Longrightarrow> ChamberComplexEndomorphism X (permutation w)"

lemma free_incorrect_cap_offset: assumes "c \<noteq> NULL" and "tag c = True" and "perm_global c = False" and "Mapping.lookup (heap_map h) (block_id c) = Some (Map m)" and "offset c \<noteq> 0" shows "free h c = Error (LogicErr (Unhandled 0))"

lemma degen_path_sound: assumes "path (degen.E T m) u p v" shows "path E (fst u) (map fst p) (fst v)"

lemma acyclic_5a_5e: "acyclic_5a x \<longleftrightarrow> acyclic_5e x"

lemma insert_type: "t \<in> B h \<Longrightarrow> insert x t \<in> B h \<union> B (Suc h)"

lemma mcont_id' [cont_intro, simp]: "mcont lub ord lub ord (\<lambda>x. x)"

lemma bdt_Some_Node_iff [simp]: "(bdt t var = Some (Bdt_Node bdt1 v bdt2)) = (\<exists> p l r. t = Node l p r \<and> bdt l var = Some bdt1 \<and> bdt r var = Some bdt2 \<and> 1 < v \<and> var p = v )"

lemma field_cont_on_typeI_region_cont_on_edges: assumes typeI_twoC: "typeI_twoCube twoC" and field_cont: "continuous_on (cubeImage twoC) F" and member_of_boundary: "(k,\<gamma>) \<in> boundary twoC" shows "continuous_on (\<gamma> ` {0 .. 1}) F"

lemma flow_initial_time[simp]: "t0 \<in> T \<Longrightarrow> x0 \<in> X \<Longrightarrow> flow t0 x0 t0 = x0"

lemma ordinal_plus_oSuc [simp]: "x + oSuc y = oSuc (x + y)"

lemma mutator_reachable_tso: "sys_mem_store_buffers (mutator m) s = mw_Mutate r f opt_r' # ws \<Longrightarrow> mut_m.reachable m r s \<and> (\<forall>r'. opt_r' = Some r' \<longrightarrow> mut_m.reachable m r' s)" "sys_mem_store_buffers (mutator m) s = mw_Mutate_Payload r f pl # ws \<Longrightarrow> mut_m.reachable m r s"

lemma approx_diff: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> a - c \<approx> b - d"

lemma finite_natarray2: "finite {(x, y). x < (m::nat) & y < (n::nat)}"

lemma exits_simps [simp]: "exits [ADD] = {1}" "exits [LOADI v] = {1}" "exits [LOAD x] = {1}" "exits [STORE x] = {1}" "i \<noteq> -1 \<Longrightarrow> exits [JMP i] = {1 + i}" "i \<noteq> -1 \<Longrightarrow> exits [JMPGE i] = {1 + i, 1}" "i \<noteq> -1 \<Longrightarrow> exits [JMPLESS i] = {1 + i, 1}"

lemma dom_restrict_inter[simp] : "T \<triangleleft> S \<triangleleft> p = T \<inter> S \<triangleleft> p"

lemma total_on_lexord_less_than_char_linear2: \<open>xs \<noteq> ys \<Longrightarrow> (xs, ys) \<notin> lexord (less_than_char) \<longleftrightarrow> (ys, xs) \<in> lexord less_than_char\<close>

lemma assumes A: "\<forall>xs A. (\<forall>X \<in> A. (xs, X) \<in> failures P) \<longrightarrow> (xs, \<Union>X \<in> A. X) \<in> failures P" shows "\<forall>xs. xs \<in> traces P"

lemma split_invar: "invar t \<Longrightarrow> split t = t"

lemma mult_Limit: "Limit \<gamma> \<Longrightarrow> x * \<gamma> = \<Squnion> ((*) x ` elts \<gamma>)"

lemma geod_dist3: assumes "a \<le> 2^n" "b \<le> 2^p" shows "dist (geod n a) (geod p b) = abs(b/2^p - a/2^n) * dist x0 y0"

lemma L2_set_empty [simp]: "L2_set f {} = 0"

lemma assertion_fun_disj_less_one: "assertion_fun = Apply.disjunctive \<inter> {x::'a::boolean_algebra \<Rightarrow> 'a . x \<le> id}"

lemma drop_bit_word_minus_numeral [simp]: \<open>drop_bit (numeral n) (- numeral k) = (word_of_int (drop_bit (numeral n) (take_bit LENGTH('a) (- numeral k))) :: 'a::len word)\<close>

lemma stream_rel_pred_szip: "stream_all2 P u v \<longleftrightarrow> pred_stream (case_prod P) (szip u v)"

lemma reg_funas: "\<L> \<A> \<subseteq> \<T>\<^sub>G (fset (ta_sig (ta \<A>)))"

lemma at_least_one_step:"steps0 (1, [], r) tm n = (0,tap) \<Longrightarrow> 0 < n"

lemma epsclo_nextl [simp]: "epsclo (nextl Q xs) = nextl Q xs"

lemma valid_adv_start_bounds': assumes "valid_window args t0 sub rho w" "w_run_t args (w_ti w) = Some (ti', t)" "w_run_sub args (w_si w) = Some (si', bs)" shows "w_ti (adv_start args w) = ti'" "w_si (adv_start args w) = si'"

lemma "(R'::'a::complete_lattice \<Rightarrow> 'b::complete_lattice) \<in> Apply.Disjunctive \<Longrightarrow> DataRefinement S R R' S' \<Longrightarrow> R (- grd S) \<le> - grd S'"

lemma apply_swap_same [simp]: "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> \<langle>a \<leftrightarrow> b\<rangle> \<langle>$\<rangle> c = c"

lemma (in Order) BNTr4:"\<lbrakk>f \<in> carrier D \<rightarrow> carrier D; a \<in> carrier D; \<forall>x\<in>carrier D. x \<preceq> (f x); W1 \<in> WWa D f a; W2 \<in> WWa D f a; \<exists>b\<in>carrier D. ord_equiv (Iod D W1) (Iod D (segment (Iod D W2) b))\<rbrakk> \<Longrightarrow> W1 \<subseteq> W2"

lemma holomorphic_injective_imp_regular: assumes holf: "f holomorphic_on S" and "open S" and injf: "inj_on f S" and "\<xi> \<in> S" shows "deriv f \<xi> \<noteq> 0"

lemma all_bi_edges_alt2: "all_bi_edges X Y = {{x, y} | x y. x \<in> X \<and> y \<in> Y }"

lemma vdisjnt_singleton0[simp]: "vdisjnt (set {a}) (set {b}) \<longleftrightarrow> a \<noteq> b" and vdisjnt_singleton1[simp]: "vdisjnt (set {a}) A \<longleftrightarrow> a \<notin>\<^sub>\<circ> A" and vdisjnt_singleton2[simp]: "vdisjnt A (set {a}) \<longleftrightarrow> a \<notin>\<^sub>\<circ> A"

lemma \<Delta>\<^sub>\<epsilon>_swap: "prod.swap p |\<in>| \<Delta>\<^sub>\<epsilon> \<A> \<B> \<longleftrightarrow> p |\<in>| \<Delta>\<^sub>\<epsilon> \<B> \<A>"

lemma bounded_closure[intro]: assumes "bounded S" shows "bounded (closure S)"

theorem secure_implies_c_secure: assumes S: "secure (c_process step out s\<^sub>0) I (c_dom D)" shows "c_secure step out s\<^sub>0 I D"

lemma L_alt: "L idx I = to_language {xs. \<exists>\<AA> \<in> I. \<exists>\<BB>. \<AA> = fold CONS (rev xs) \<BB> \<and> Length \<BB> = 0 \<and> #\<^sub>V \<BB> = idx \<and> (\<forall>x \<in> set xs. size x = idx)}"

lemma Map_comp [simp]: assumes "arr f" and "arr g" and "Dom g = Cod f" shows "Map (COMP g f) = Comp (Cod g) (Dom g) (Dom f) (Map g) (Map f)"

lemma subst_plus [usubst]: "\<sigma> \<dagger> (x + y) = \<sigma> \<dagger> x + \<sigma> \<dagger> y"

lemma suffix_imp_sublist [simp, intro]: "suffix xs ys \<Longrightarrow> sublist xs ys"

lemma assumes asms: "A" "B" shows "True"

lemma closed_cspan_range_ket[simp]: \<open>closure (cspan (range ket)) = UNIV\<close>

lemma contour_integral_subpath_combine: assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}" shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f = contour_integral (subpath u w g) f"

lemma rstep_imp_C_s_r: assumes "(s,t) \<in> rstep R" shows "\<exists>C \<sigma> l r. (l,r) \<in> R \<and> s = C\<langle>l\<cdot>\<sigma>\<rangle> \<and> t = C\<langle>r\<cdot>\<sigma>\<rangle>"

lemma absolutely_integrable_component_ubound: fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space" assumes f: "f integrable_on A" and g: "g absolutely_integrable_on A" and comp: "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> f x \<bullet> b \<le> g x \<bullet> b" shows "f absolutely_integrable_on A"

lemma iteci_lu_rule: " ( mi.ite_impl_lu i t e bdd = Some (p,bdd')) \<longrightarrow> <is_bdd_impl bdd bddi> iteci_lu i t e bddi <\<lambda>(pi,bddi'). is_bdd_impl bdd' bddi' * \<up>(pi=p )>\<^sub>t"

lemma FL_bisimilar_symp: "symp (FL_bisimilar F)"

lemma card_one[elim]: assumes "card A = 1" obtains a where "A = {a}"

lemma AKcryptSK_not_AKcryptSK: "\<lbrakk> AKcryptSK authK servK evs; evs \<in> kerbIV_gets \<rbrakk> \<Longrightarrow> \<not> AKcryptSK servK K evs"

lemma add_node_wf[simp]: "wf_graph g \<Longrightarrow> wf_graph (add_node v g)"

lemma neg_lt: "neg\<cdot>(lt\<cdot>x\<cdot>y) = le\<cdot>y\<cdot>x"

lemma input_unconstrained_aval_input_swap: "\<forall>i. \<not> aexp_constrains a (V (I i)) \<Longrightarrow> aval a (join_ir i r) = aval a (join_ir i' r)"

lemma l3: assumes "x \<in> S2" shows "finite x"

lemma B_trusts_NS3: "\<lbrakk>Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace> \<in> parts (spies evs); B \<notin> bad; evs \<in> ns_shared\<rbrakk> \<Longrightarrow> \<exists>NA. Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>\<rbrace>) \<in> set evs"

lemma signed_take_bit_eq_if_positive: \<open>signed_take_bit n a = take_bit n a\<close> if \<open>\<not> bit a n\<close>

lemma findnth_inv_layout_of_via_crsp[simp]: "crsp (layout_of ap) (as, lm) (s, l, r) ires \<Longrightarrow> findnth_inv (layout_of ap) n (as, lm) (Suc 0, l, r) ires"

lemma %invisible F2'_antimono: shows "antimono (\<lambda>XD. - RD_on (ds-{d'}) XD)"

lemma lookup_pair_eq_valueI: assumes "oalist_inv_raw xs" and "(k, v) \<in> set xs" shows "lookup_pair xs k = v"

lemma b_assn_invalid_merge5: "hn_ctxt (b_assn A P') x y \<or>\<^sub>A hn_invalid (b_assn A P) x y \<Longrightarrow>\<^sub>t hn_invalid (b_assn A (\<lambda>x. P x \<or> P' x)) x y"

lemma cm_all_eq_cm_all': assumes "\<forall>(loc,t)\<in>set_zmset \<Delta>. \<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t" shows "cm_all c0 \<Delta> = cm_all' c0 \<Delta>"

lemma weakCongSum1: fixes P :: ccs and \<alpha> :: act and P' :: ccs and Q :: ccs assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "P \<oplus> Q \<Longrightarrow>\<alpha> \<prec> P'"

lemma msetext_dersh_trans: assumes zs_a: "zs \<in> lists A" and ys_a: "ys \<in> lists A" and xs_a: "xs \<in> lists A" and trans: "\<forall>z \<in> A. \<forall>y \<in> A. \<forall>x \<in> A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x" and zs_gt_ys: "msetext_dersh gt zs ys" and ys_gt_xs: "msetext_dersh gt ys xs" shows "msetext_dersh gt zs xs"

lemma lasso_run_rel_def: "\<langle>R\<rangle>lasso_run_rel = br run_of_lasso (\<lambda>_. True) O (nat_rel \<rightarrow> R)"

lemma filternew_flows_state_alt2: "filternew_flows_state \<T> = {e \<in> flows_state \<T>. e \<notin> backflows (flows_fix \<T>)}"

lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"

lemma Col_dep2: "real_euclid.Col a b c \<longleftrightarrow> dep2 (b - a) (c - a)"

lemma reachable_append: "reachable g (xs @ ys) = reachable g xs \<union> reachable g ys"

lemma (in \<Z>) \<KK>23_is_tiny_functor: "\<KK>23 : cat_ordinal (2\<^sub>\<nat>) \<mapsto>\<mapsto>\<^sub>C\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> cat_ordinal (3\<^sub>\<nat>)"

lemma continuous_on_inverse_ennreal': "continuous_on (UNIV :: ennreal set) inverse"

lemma bisimI: fixes P :: "ccs" and Q :: "ccs" assumes "P \<leadsto>[bisim] Q" and "Q \<sim> P" shows "P \<sim> Q"

lemma SSpec_strong: "\<Turnstile> c :A \<Longrightarrow> \<forall> s t . SSpec c s t \<longrightarrow> A s t"

lemma Trivially_fulfilled_FCD_Nabla_\<Upsilon>_empty: "\<lbrakk>\<nabla>\<^bsub>\<Gamma>\<^esub>={} \<or> \<Upsilon>\<^bsub>\<Gamma>\<^esub>={}\<rbrakk>\<Longrightarrow> FCD \<Gamma> \<V> Tr\<^bsub>ES\<^esub>"

lemma bin_basis_code: "code {\<zero>,\<one>}"

lemma (in fl_subdigraph) fl_subdg_Hom_eq: assumes "A \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" and "B \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "Hom \<BB> A B = Hom \<CC> A B"

lemma (in Corps) PCauchy_lPCauchy:"\<lbrakk>valuation K v; PolynRg R (Vr K v) X; \<forall>n. F n \<in> carrier R; \<forall>n. deg R (Vr K v) X (F n) \<le> an (Suc d); P_mod R (Vr K v) X (vp K v\<^bsup>(Vr K v) (an N)\<^esup>) (F n \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> (F m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (vp K v\<^bsup>(Vr K v) (an N)\<^esup>) (((Pseql\<^bsub>R X K v d\<^esub> F) n) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> ((Pseql\<^bsub>R X K v d\<^esub> F) m))"

lemma Skip_Sim: "Skip \<preceq>S Skip"

lemma has_vector_derivative_eq_has_derivative_blinfun: "(f has_vector_derivative f') (at x within U) \<longleftrightarrow> (f has_derivative blinfun_scaleR_left f') (at x within U)"

lemma [simp]: "fields E ClassCast = []"

lemma unital_quantale_homset_iff: "f \<in> unital_quantale_homset = (comp_pres f \<and> Sup_pres f \<and> un_pres f)"

lemma siso_cont_indis[simp]: assumes *: "siso c" and **: "s \<approx> t" "i < brn c" shows "eff c s i \<approx> eff c t i \<and> wt c s i = wt c t i \<and> cont c s i = cont c t i"

lemma vrangeD[dest]: assumes "\<langle>r, s\<rangle> \<in>\<^sub>\<circ> vrange A" shows "r \<in>\<^sub>\<circ> A" and "s = \<R>\<^sub>\<circ> r"

lemma bind_assoc_SBE: "(y :\<equiv> (x :\<equiv> m; k); h) = (x :\<equiv> m; (y :\<equiv> k; h))"

theorem gb_sig_z_is_min_sig_GB: assumes "p \<in> set (fst (gb_sig_z rw_rat_strict fs))" and "q \<in> set (fst (gb_sig_z rw_rat_strict fs))" and "p \<noteq> q" and "punit.lt (snd p) adds punit.lt (snd q)" shows "punit.lt (snd p) \<oplus> fst q \<prec>\<^sub>t punit.lt (snd q) \<oplus> fst p"

lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"

lemma (in imap) added_ids_Deliver_Expunge_diff_collapse [simp]: shows "e \<noteq> e' \<Longrightarrow> added_ids ([Deliver (i, Expunge e mo j)]) e' = []"

lemma valuesum_rec: assumes fin: "finite (dom (Mapping.lookup m))" shows "valuesum m = (if Mapping.is_empty m then 0 else let l = (SOME l. l \<in> Mapping.keys m) in the (Mapping.lookup m l) + valuesum (Mapping.delete l m))"

lemma apply_guards_double_cons: "apply_guards (y # x # G) s = (gval (gAnd y x) s = true \<and> apply_guards G s)"