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

Statement: stringlengths 7 24.3k
lemma ins_type: shows "t \<in> B h \<Longrightarrow> ins x t \<in> Bp h" and "t \<in> U h \<Longrightarrow> ins x t \<in> T h"
theorem regular_emptystr: "regular {[]}"
theorem SymSystem_comp_a: "u' \<in> init' \<Longrightarrow> SymSystem init p r o SymSystem init' p' r' = {.x . \<forall> u v . u \<in> init \<and> v \<in> init' \<longrightarrow> (prec_st p r u x \<and> (\<forall>y. (\<box> lft_rel_st r) u x y \<longrightarrow> prec_st p' r' v y)) .} o [: x \<leadsto> y . \<exists> u v . u \<in> init \<and> v \<in> init' \<and> (\<box> lft_rel_st (\<lambda>(u, v) (u', v'). r u u' OO r' v v')) (u \|\| v) x y :]" (is "?p \<Longrightarrow> ?S = ?T")
lemma "VARS H x y z w {(P Q) H} SKIP {(Q P) H}"
lemma quad_part_1_less : assumes lLength : "length L > var" assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg::nat)" assumes nonzero : "D \<noteq> 0" assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) a = (A::real)" assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) b = (B::real)" assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) d = (D::real)" shows "aEval (Less p) (list_update L var ((A+B*C)/D)) = aEval (Less(quadratic_part_1 var a b d (Less p))) (list_update L var C)"
lemma lists_def2: "lists A = {l. set l \<le> A}"
lemma insert_inv[simp]: "(insert e E)\<inverse> = insert (prod.swap e) (E\<inverse>)"
lemma H_llist_hom_id: "list2H\<cdot>lnil = ID"
lemma bezout_iterate_preserves_below_n: assumes e: "echelon_form_upt_k A k" and ib: "is_bezout_ext bezout" and Aik_0: "A $ i $ from_nat k \<noteq> 0" and n: "n<nrows A" and n_less_a: "n < to_nat a" and k: "k<ncols A" and i_le_n: "to_nat i \<le> n" and zero_upt_k_i: "is_zero_row_upt_k i k A" shows "bezout_iterate A n i (from_nat k) bezout $ a $ b = A $ a $ b"
lemma rtrancl3p_converse_induct [consumes 1, case_names refl step]: assumes ih: "rtrancl3p r a bs a''" and refl: "\<And>a. P a [] a" and step: "\<And>a b a' bs a''. \<lbrakk> rtrancl3p r a' bs a''; r a b a'; P a' bs a'' \<rbrakk> \<Longrightarrow> P a (b # bs) a''" shows "P a bs a''"
lemma cf_rcomp_components: shows "cf_rcomp \<BB> \<SS> \<GG>\<lparr>HomDom\<rparr> = \<BB> \<times>\<^sub>C \<GG>\<lparr>HomDom\<rparr>" and "cf_rcomp \<BB> \<SS> \<GG>\<lparr>HomCod\<rparr> = \<SS>\<lparr>HomCod\<rparr>"
lemma rdomp_ctxt[simp]: "rdomp (hn_ctxt R) = rdomp R"
lemma a_idem [simp]: "ad x \<cdot> ad x = ad x"
lemma ENF_offending_imp_not_P: assumes "sinvar_all_edges_normal_form P" "F \<in> set_offending_flows G nP" "(e1, e2) \<in> F" shows "\<not> P (nP e1) (nP e2)"
lemma open_convex_hull[intro]: fixes S :: "'a::real_normed_vector set" assumes "open S" shows "open (convex hull S)"
lemma gen_model_kripke: shows "kripke (gen_model M w)"
lemma lvl_insert: obtains (Same) "lvl (insert x t) = lvl t" \| (Incr) "lvl (insert x t) = lvl t + 1" "sngl (insert x t)"
lemma vangle_0_left [simp]: "vangle 0 v = pi / 2" and vangle_0_right [simp]: "vangle u 0 = pi / 2"
lemma inorder_insert: "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
lemma (in Order) maxl_fun_maxl:"\<lbrakk>x \<in> carrier D; maxl_fun D x = x \<rbrakk> \<Longrightarrow> maximal x"
lemma countable_type_countable[dest]: "countable_type t \<Longrightarrow> countable (space (stock_measure t))"
lemma T_split: "T = {tmin .. t0} \<union> {t0 .. tmax}"
lemma interpolation_poly_int_def: "distinct (map fst xs_ys) \<Longrightarrow> interpolation_poly_int alg xs_ys = (let rxs_ys = map (\<lambda> (x,y). (of_int x, of_int y)) xs_ys; rp = interpolation_poly alg rxs_ys in if (\<forall> x \<in> set (coeffs rp). is_int_rat x) then Some (map_poly int_of_rat rp) else None)"
theorem set_category_is_categorical: assumes "set_category S setp" and "set_category S' setp'" and "bij_betw \<phi> (set_category_data.Univ S) (set_category_data.Univ S')" and "\<And>A. A \<subseteq> set_category_data.Univ S \<Longrightarrow> setp' (\<phi> ` A) \<longleftrightarrow> setp A" shows "\<exists>\<Phi>. invertible_functor S S' \<Phi> \<and> (\<forall>m. set_category.incl S setp m \<longrightarrow> set_category.incl S' setp' (\<Phi> m))"
lemma nsp_is_regular [iff]: "regular nsp"
lemma shift_ms_aux_MSLNil [simp]: "shift_ms_aux e MSLNil = MSLNil"
lemma map_ide [simp]: assumes "C.ide b" and "C.ide a" shows "map (b, a) = S.mkIde (set (b, a))"
lemma vydra_sound_aux: assumes "msize_fmla \<phi> \<le> n" "wf_vydra \<phi> i n v" "ru n v = Some (v', t, b)" "bounded_future_fmla \<phi>" "wf_fmla \<phi>" shows "wf_vydra \<phi> (Suc i) n v' \<and> (\<exists>es e. reaches_on run_hd init_hd es e \<and> length es = Suc i) \<and> t = \<tau> \<sigma> i \<and> b = sat \<phi> i"
lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (top_of_set S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
lemma eeqExcPID2_trans: assumes "eeqExcPID2 s s1" and "eeqExcPID2 s1 s2" shows "eeqExcPID2 s s2"
lemma eventually_deriv_compose': assumes "f twice_field_differentiable_at x" and "g twice_field_differentiable_at (f x)" shows "\<forall>\<^sub>F x in nhds x. deriv (\<lambda>x. g (f x)) x = deriv g (f x) * deriv f x"
lemma path_pd_pd0: assumes path: \<open>is_path \<pi>\<close> and lpdn: \<open>\<pi> l pd\<rightarrow> n\<close> and npd0: \<open>n pd\<rightarrow> \<pi> 0\<close> obtains k where \<open>k \<le> l\<close> \<open>\<pi> k = n\<close>
lemma TC_induct_down_lemma: assumes ab: "a \<sqsubset> b" and base: "b \<le> d" and step: "\<And>y z. \<lbrakk>y \<sqsubset> b; y \<in> elts d; z \<in> elts y\<rbrakk> \<Longrightarrow> z \<in> elts d" shows "a \<in> elts d"
lemma fst_map_elem: "(y_k, ty_k, var_k, var'_k, v_k) \<in> set y_ty_var_var'_v_list \<Longrightarrow> x_var var'_k \<in> fst ` (\<lambda>(y, ty, var, var', v). (x_var var', ty)) ` set y_ty_var_var'_v_list"
lemma g_growth2': assumes "x \<ge> x\<^sub>1" "i < k" "u \<in> {bs!ix+(hs!i) x..x}" shows "c2 g x \<ge> g u"
lemma (in finite_measure) integrable_const[intro!, simp]: "integrable M (\<lambda>x. a)"
lemma fwi_step': "fwi m n k i' j' i j = min (m i j) (m i k + m k j)" if "m k k \<ge> 0" "i' \<le> n" "j' \<le> n" "k \<le> n" "i \<le> i'" "j \<le> j'"
lemma TESL_sem_decreases_head: \<open>\<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<supseteq> \<lbrakk>\<lbrakk> \<phi> # \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<close>
lemma fic_c_reduce: assumes a: "fic M x" and b: "M \<longrightarrow>\<^sub>c M'" shows "fic M' x"
lemma set_as_mapping_image_code[code] : fixes t :: "('a1 ::ccompare \<times> 'a2 :: ccompare) set_rbt" and f1 :: "('a1 \<times> 'a2) \<Rightarrow> ('b1 :: ccompare \<times> 'b2 ::ccompare)" and xs :: "('c1 :: ceq \<times> 'c2 :: ceq) set_dlist" and f2 :: "('c1 \<times> 'c2) \<Rightarrow> ('d1 \<times> 'd2)" shows "set_as_mapping_image (RBT_set t) f1 = (case ID CCOMPARE(('a1 \<times> 'a2)) of Some _ \<Rightarrow> (RBT_Set2.fold (\<lambda> kv m1 . ( case f1 kv of (x,z) \<Rightarrow> (case Mapping.lookup m1 (x) of None \<Rightarrow> Mapping.update (x) {z} m1 \| Some zs \<Rightarrow> Mapping.update (x) (Set.insert z zs) m1))) t Mapping.empty) \| None \<Rightarrow> Code.abort (STR ''set_as_map_image RBT_set: ccompare = None'') (\<lambda>_. set_as_mapping_image (RBT_set t) f1))" (is "set_as_mapping_image (RBT_set t) f1 = ?C1 (RBT_set t)") and "set_as_mapping_image (DList_set xs) f2 = (case ID CEQ(('c1 \<times> 'c2)) of Some _ \<Rightarrow> (DList_Set.fold (\<lambda> kv m1 . ( case f2 kv of (x,z) \<Rightarrow> (case Mapping.lookup m1 (x) of None \<Rightarrow> Mapping.update (x) {z} m1 \| Some zs \<Rightarrow> Mapping.update (x) (Set.insert z zs) m1))) xs Mapping.empty) \| None \<Rightarrow> Code.abort (STR ''set_as_map_image DList_set: ccompare = None'') (\<lambda>_. set_as_mapping_image (DList_set xs) f2))" (is "set_as_mapping_image (DList_set xs) f2 = ?C2 (DList_set xs)")
lemma min_p_min: assumes "is_binqueue l xs" and "xs \<noteq> []" and "normalized xs" and "distinct (vals xs)" and "distinct (prios xs)" shows "min xs = PQ.priority (pqueue xs) (PQ.min (pqueue xs))"
lemma grey_reachable_eq_imp: "eq_imp (\<lambda>r'. (\<lambda>s. \<Union>p. WL p s) \<^bold>\<otimes> (\<lambda>s. Set.bind (Option.set_option (sys_heap s r')) (ran \<circ> obj_fields))) (grey_reachable r)"
lemma equivalence_functor_F: shows "equivalence_functor hom.comp hom'.comp F"
lemma par_distincts: assumes "A B Par C D" shows "A B Par C D \<and> A \<noteq> B \<and> C \<noteq> D"
lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
lemma L_10_5_\<upsilon>_arrow_ArrVal_app': assumes "\<KK> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" and "\<TT> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<AA>" and "f : c \<mapsto>\<^bsub>\<CC>\<^esub> \<KK>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>" shows "L_10_5_\<upsilon>_arrow \<TT> \<KK> c \<tau> a b\<lparr>ArrVal\<rparr>\<lparr>f\<rparr> = \<tau>\<lparr>NTMap\<rparr>\<lparr>0, b, f\<rparr>\<^sub>\<bullet>"
lemma take_cols_carrier_mat_strict: assumes "A \<in> carrier_mat nr nc" assumes "\<And>i. i \<in> set inds \<Longrightarrow> i < nc" shows "take_cols A inds \<in> carrier_mat nr (length inds)"
lemma (in Corps) mprod_Suc:"\<lbrakk> \<forall>j\<le>(Suc n). e j \<in> Zset; \<forall>j \<le> (Suc n). f j \<in> (carrier K - {\<zero>})\<rbrakk> \<Longrightarrow> mprod_exp K e f (Suc n) = (mprod_exp K e f n) \<cdot>\<^sub>r ((f (Suc n))\<^bsub>K\<^esub>\<^bsup>(e (Suc n))\<^esup>)"
lemma partition_on_partition_on_unique: assumes "partition_on A P" assumes "x \<in> A" shows "\<exists>!X. x \<in> X \<and> X \<in> P"
lemma inv_part_0_Suc: "m_j < n \<Longrightarrow> inv_part p A 0 m_j = inv_part p A n (Suc m_j)"
lemma \<epsilon>_hcomp': assumes "C.ide g" and "C.ide f" and "src\<^sub>C g = trg\<^sub>C f" shows "\<epsilon> (g \<star>\<^sub>C f) \<cdot>\<^sub>C GF.cmp (g, f) = \<epsilon> g \<star>\<^sub>C \<epsilon> f"
lemma simple_match_valid_alt[code_unfold]: "simple_match_valid = (\<lambda> m. (let c = (\<lambda>(s,e). (s \<noteq> 0 \<or> e \<noteq> - 1)) in ( if c (sports m) \<or> c (dports m) then proto m = Proto TCP \<or> proto m = Proto UDP \<or> proto m = Proto L4_Protocol.SCTP else True)) \<and> valid_prefix_fw (src m) \<and> valid_prefix_fw (dst m))"
lemma "case_prod (curry f) = f"
lemma gcd_ff_list_exists: "\<exists> g. gcd_ff_list (X :: 'a::ufd fract list) g"
lemma poly_mapping_eqI_proj_syz: assumes "proj_poly_syz n p = proj_poly_syz n q" and "\<And>k. k < n \<Longrightarrow> proj_poly k p = proj_poly k q" shows "p = q"
lemma real_power_down_fl: "real_of_float (power_down_fl p x n) = power_down p x n"
lemma SubobjsR_nonempty: "Subobjs\<^sub>R P C Cs \<Longrightarrow> Cs \<noteq> []"
lemma subsumes_update_equality: "subsumes t1 c t2 \<Longrightarrow> (\<forall>i. can_take_transition t2 i c \<longrightarrow> (\<forall>r'. ((evaluate_updates t1 i c $ r') = (evaluate_updates t2 i c $ r')) \<or> evaluate_updates t2 i c $ r' = None))"
lemma divideC_field_splits_simps_2 [field_split_simps]: (* In Real_Vector_Spaces, these lemmas are unnamed ) "0 < c \<Longrightarrow> a \<le> b /\<^sub>C c \<longleftrightarrow> (if c > 0 then c \<^sub>C a \<le> b else if c < 0 then b \<le> c \<^sub>C a else a \<le> 0)" "0 < c \<Longrightarrow> a < b /\<^sub>C c \<longleftrightarrow> (if c > 0 then c \<^sub>C a < b else if c < 0 then b < c \<^sub>C a else a < 0)" "0 < c \<Longrightarrow> b /\<^sub>C c \<le> a \<longleftrightarrow> (if c > 0 then b \<le> c \<^sub>C a else if c < 0 then c \<^sub>C a \<le> b else a \<ge> 0)" "0 < c \<Longrightarrow> b /\<^sub>C c < a \<longleftrightarrow> (if c > 0 then b < c \<^sub>C a else if c < 0 then c \<^sub>C a < b else a > 0)" "0 < c \<Longrightarrow> a \<le> - (b /\<^sub>C c) \<longleftrightarrow> (if c > 0 then c \<^sub>C a \<le> - b else if c < 0 then - b \<le> c \<^sub>C a else a \<le> 0)" "0 < c \<Longrightarrow> a < - (b /\<^sub>C c) \<longleftrightarrow> (if c > 0 then c \<^sub>C a < - b else if c < 0 then - b < c \<^sub>C a else a < 0)" "0 < c \<Longrightarrow> - (b /\<^sub>C c) \<le> a \<longleftrightarrow> (if c > 0 then - b \<le> c \<^sub>C a else if c < 0 then c \<^sub>C a \<le> - b else a \<ge> 0)" "0 < c \<Longrightarrow> - (b /\<^sub>C c) < a \<longleftrightarrow> (if c > 0 then - b < c \<^sub>C a else if c < 0 then c *\<^sub>C a < - b else a > 0)" for a b :: "'a :: ordered_complex_vector"
lemma of_nat_ge_1_iff: "(of_nat x :: 'a :: linordered_semidom) \<ge> 1 \<longleftrightarrow> x \<ge> 1"
lemma NSLIM_mult_zero: "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0 \<Longrightarrow> g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0 \<Longrightarrow> (\<lambda>x. f x * g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0" for f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> 1 \<le> root n y \<longleftrightarrow> 1 \<le> y"
lemma one_real_code [code]: "1 = real_of_float 1"
lemma is_bezout_ext_euclid_ext2: "is_bezout_ext (euclid_ext2)"
lemma sconcat_sfoldl: "sconcat = sfoldl\<cdot>sappend\<cdot>[::]"
lemma rc_invk_arg': "CT \<turnstile> ei \<rightarrow> ei' \<Longrightarrow> append el (ei # er) e' \<Longrightarrow> append el (ei' # er) e'' \<Longrightarrow> CT \<turnstile> MethodInvk e m e' \<rightarrow> MethodInvk e m e''"
lemma measurable_suntil[measurable]: assumes [measurable]: "Measurable.pred (stream_space M) Q" "Measurable.pred (stream_space M) P" shows "Measurable.pred (stream_space M) (Q suntil P)"
lemma switch: fixes A :: "form" and As :: "form_list" shows "y \<sharp> A \<Longrightarrow> [y,x]A = [(y,x)]\<bullet>A" and "y \<sharp> As \<Longrightarrow> [y,x]As = [(y,x)]\<bullet>As"
lemma similar_admI: "cont f \<Longrightarrow> cont g \<Longrightarrow> adm (\<lambda>x. f x \<triangleleft>\<triangleright> g x)"
lemma DBM_val_bounded_len'1: fixes v assumes "DBM_val_bounded v u m n" "0 \<notin> set vs" "v c \<le> n" "\<forall> k \<in> set vs. k > 0 \<longrightarrow> k \<le> n \<and> (\<exists> c. v c = k)" shows "dbm_entry_val u (Some c) None (len m (v c) 0 vs)"
lemma quot_set_image_times: "inj (() x) \<Longrightarrow> (() x ` A) \<div> x = A"
lemma all_strict [simp]: "all\<cdot>P\<cdot>\<bottom> = \<bottom>"
lemma supp_int: fixes i::"int" shows "supp (i) = {}"
lemma regular_bot_top_2: "regular x \<longleftrightarrow> x = bot \<or> x = top"
lemma 8: "{} \<turnstile> (P AND Neg Q\<^sub>1) IMP Q\<^sub>2"
lemma permChainSimps[simp]: fixes xvec :: "name list" and yvec :: "name list" and perm :: "name prm" and p :: "'a::pt_name" shows "((composePerm xvec yvec) @ perm) \<bullet> p = [xvec yvec] \<bullet>\<^sub>v (perm \<bullet> p)"
lemma to_cnf_0 [simp]: "to_cnf 0 = []"
lemma lemma_4_3_i_a: "a \<squnion>1 b = (a l\<rightarrow> b) r\<rightarrow> b"
lemma tagrel_sub_inv: assumes \<open>sub \<lless> r\<close> and \<open>r \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<close> shows \<open>sub \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<close>
lemma hn_if[sepref_comb_rules]: assumes P: "\<Gamma> \<Longrightarrow>\<^sub>t \<Gamma>1 * hn_val bool_rel a a'" assumes RT: "a \<Longrightarrow> hn_refine (\<Gamma>1 * hn_val bool_rel a a') b' \<Gamma>2b R b" assumes RE: "\<not>a \<Longrightarrow> hn_refine (\<Gamma>1 * hn_val bool_rel a a') c' \<Gamma>2c R c" assumes IMP: "TERM If \<Longrightarrow> \<Gamma>2b \<or>\<^sub>A \<Gamma>2c \<Longrightarrow>\<^sub>t \<Gamma>'" shows "hn_refine \<Gamma> (if a' then b' else c') \<Gamma>' R (If$a$b$c)"
lemma sources_respects_Cong: assumes "T \<^sup>\<approx>\<^sup> T'" shows "P.sources T = P.sources T'"
lemma completely_multiplicative_function_inverse: fixes f :: "nat \<Rightarrow> 'a :: field" assumes "completely_multiplicative_function f" shows "completely_multiplicative_function (\<lambda>n. inverse (f n))"
lemma mk_deriv_intro_ix[simp]: "deriv_ix (mk_deriv_intro ix n j) = ix"
lemma MPair_used [rule_format]: "MPair X Y \<in> used evs \<longrightarrow> X \<in> used evs & Y \<in> used evs"
lemma aff_dim_insert: fixes a :: "'a::euclidean_space" shows "aff_dim (insert a S) = (if a \<in> affine hull S then aff_dim S else aff_dim S + 1)"
lemma simple_function_subalgebra: assumes "simple_function N f" and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M" shows "simple_function M f"
lemma glength_greater_0_conv[simp]: "(0 < glength a) = (a \<noteq> FL [])"
lemma params_subterms_Union[simp]: "subterms\<^sub>s\<^sub>e\<^sub>t (set X) \<subseteq> subterms (Fun f X)"
lemma H\<^sub>R_preserved_along_iso: assumes "weak_unit a" and "a \<cong> a'" shows "endofunctor (Right a) (H\<^sub>R a')"
lemma distinct_sorted_intersection[simp]: "\<lbrakk> distinct xs; distinct ys; sorted xs; sorted ys \<rbrakk> \<Longrightarrow> distinct (intersection xs ys) \<and> sorted (intersection xs ys)"
lemma W_eq_L_iter: assumes "\<nu>_improving v d" shows "W d m v = (L d^^m) v"
lemma cat_sspan_composable_\<bb>\<bb>[cat_ss_cs_intros]: assumes "g = \<bb>\<^sub>S\<^sub>S" and "f = \<bb>\<^sub>S\<^sub>S" shows "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> cat_sspan_composable"
lemma (in Group) ZassenhausTr1_1:"\<lbrakk>G \<guillemotright> H; G \<guillemotright> H1; G \<guillemotright> K; G \<guillemotright> K1; Gp G H \<triangleright> H1; Gp G K \<triangleright> K1\<rbrakk> \<Longrightarrow> G \<guillemotright> (H1 \<diamondop>\<^bsub>G\<^esub> (H \<inter> K1))"
lemma sset_shift[simp]: "sset (xs @- s) = set xs \<union> sset s"
lemma iSince_iEx_conv: "(True. t' \<S> t I. P t) = (\<diamond> t I. P t)"
lemma pref_prod_longer: "u \<le>p z \<cdot> w \<Longrightarrow> v \<le>p w \<Longrightarrow> \<^bold>\|z \<cdot> v\<^bold>\| \<le> \<^bold>\|u\<^bold>\| \<Longrightarrow> z \<cdot> v \<le>p u"
theorem A_gets_good_key: "\<lbrakk>Crypt (shrK A) \<lbrace>Agent B, Key K, na\<rbrace> \<in> parts (knows Spy evs); \<forall>nb. Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs; A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> \<Longrightarrow> Key K \<notin> analz (knows Spy evs)"
lemma register_unitary: assumes "register F" assumes "unitary a" shows "unitary (F a)"
lemma all_self_eq_1[PLM]: "[\<^bold>\<box>(\<^bold>\<forall> \<alpha> :: 'a::id_eq . \<alpha> \<^bold>= \<alpha>) in v]"

lemma ins_type: shows "t \<in> B h \<Longrightarrow> ins x t \<in> Bp h" and "t \<in> U h \<Longrightarrow> ins x t \<in> T h"

theorem regular_emptystr: "regular {[]}"

theorem SymSystem_comp_a: "u' \<in> init' \<Longrightarrow> SymSystem init p r o SymSystem init' p' r' = {.x . \<forall> u v . u \<in> init \<and> v \<in> init' \<longrightarrow> (prec_st p r u x \<and> (\<forall>y. (\<box> lft_rel_st r) u x y \<longrightarrow> prec_st p' r' v y)) .} o [: x \<leadsto> y . \<exists> u v . u \<in> init \<and> v \<in> init' \<and> (\<box> lft_rel_st (\<lambda>(u, v) (u', v'). r u u' OO r' v v')) (u || v) x y :]" (is "?p \<Longrightarrow> ?S = ?T")

lemma "VARS H x y z w {(P ** Q) H} SKIP {(Q ** P) H}"

lemma quad_part_1_less : assumes lLength : "length L > var" assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg::nat)" assumes nonzero : "D \<noteq> 0" assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) a = (A::real)" assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) b = (B::real)" assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) d = (D::real)" shows "aEval (Less p) (list_update L var ((A+B*C)/D)) = aEval (Less(quadratic_part_1 var a b d (Less p))) (list_update L var C)"

lemma lists_def2: "lists A = {l. set l \<le> A}"

lemma insert_inv[simp]: "(insert e E)\<inverse> = insert (prod.swap e) (E\<inverse>)"

lemma H_llist_hom_id: "list2H\<cdot>lnil = ID"

lemma bezout_iterate_preserves_below_n: assumes e: "echelon_form_upt_k A k" and ib: "is_bezout_ext bezout" and Aik_0: "A $ i $ from_nat k \<noteq> 0" and n: "n<nrows A" and n_less_a: "n < to_nat a" and k: "k<ncols A" and i_le_n: "to_nat i \<le> n" and zero_upt_k_i: "is_zero_row_upt_k i k A" shows "bezout_iterate A n i (from_nat k) bezout $ a $ b = A $ a $ b"

lemma rtrancl3p_converse_induct [consumes 1, case_names refl step]: assumes ih: "rtrancl3p r a bs a''" and refl: "\<And>a. P a [] a" and step: "\<And>a b a' bs a''. \<lbrakk> rtrancl3p r a' bs a''; r a b a'; P a' bs a'' \<rbrakk> \<Longrightarrow> P a (b # bs) a''" shows "P a bs a''"

lemma cf_rcomp_components: shows "cf_rcomp \<BB> \<SS> \<GG>\<lparr>HomDom\<rparr> = \<BB> \<times>\<^sub>C \<GG>\<lparr>HomDom\<rparr>" and "cf_rcomp \<BB> \<SS> \<GG>\<lparr>HomCod\<rparr> = \<SS>\<lparr>HomCod\<rparr>"

lemma rdomp_ctxt[simp]: "rdomp (hn_ctxt R) = rdomp R"

lemma a_idem [simp]: "ad x \<cdot> ad x = ad x"

lemma ENF_offending_imp_not_P: assumes "sinvar_all_edges_normal_form P" "F \<in> set_offending_flows G nP" "(e1, e2) \<in> F" shows "\<not> P (nP e1) (nP e2)"

lemma open_convex_hull[intro]: fixes S :: "'a::real_normed_vector set" assumes "open S" shows "open (convex hull S)"

lemma gen_model_kripke: shows "kripke (gen_model M w)"

lemma lvl_insert: obtains (Same) "lvl (insert x t) = lvl t" | (Incr) "lvl (insert x t) = lvl t + 1" "sngl (insert x t)"

lemma vangle_0_left [simp]: "vangle 0 v = pi / 2" and vangle_0_right [simp]: "vangle u 0 = pi / 2"

lemma inorder_insert: "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"

lemma (in Order) maxl_fun_maxl:"\<lbrakk>x \<in> carrier D; maxl_fun D x = x \<rbrakk> \<Longrightarrow> maximal x"

lemma countable_type_countable[dest]: "countable_type t \<Longrightarrow> countable (space (stock_measure t))"

lemma T_split: "T = {tmin .. t0} \<union> {t0 .. tmax}"

lemma interpolation_poly_int_def: "distinct (map fst xs_ys) \<Longrightarrow> interpolation_poly_int alg xs_ys = (let rxs_ys = map (\<lambda> (x,y). (of_int x, of_int y)) xs_ys; rp = interpolation_poly alg rxs_ys in if (\<forall> x \<in> set (coeffs rp). is_int_rat x) then Some (map_poly int_of_rat rp) else None)"

theorem set_category_is_categorical: assumes "set_category S setp" and "set_category S' setp'" and "bij_betw \<phi> (set_category_data.Univ S) (set_category_data.Univ S')" and "\<And>A. A \<subseteq> set_category_data.Univ S \<Longrightarrow> setp' (\<phi> ` A) \<longleftrightarrow> setp A" shows "\<exists>\<Phi>. invertible_functor S S' \<Phi> \<and> (\<forall>m. set_category.incl S setp m \<longrightarrow> set_category.incl S' setp' (\<Phi> m))"

lemma nsp_is_regular [iff]: "regular nsp"

lemma shift_ms_aux_MSLNil [simp]: "shift_ms_aux e MSLNil = MSLNil"

lemma map_ide [simp]: assumes "C.ide b" and "C.ide a" shows "map (b, a) = S.mkIde (set (b, a))"

lemma vydra_sound_aux: assumes "msize_fmla \<phi> \<le> n" "wf_vydra \<phi> i n v" "ru n v = Some (v', t, b)" "bounded_future_fmla \<phi>" "wf_fmla \<phi>" shows "wf_vydra \<phi> (Suc i) n v' \<and> (\<exists>es e. reaches_on run_hd init_hd es e \<and> length es = Suc i) \<and> t = \<tau> \<sigma> i \<and> b = sat \<phi> i"

lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (top_of_set S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"

lemma eeqExcPID2_trans: assumes "eeqExcPID2 s s1" and "eeqExcPID2 s1 s2" shows "eeqExcPID2 s s2"

lemma eventually_deriv_compose': assumes "f twice_field_differentiable_at x" and "g twice_field_differentiable_at (f x)" shows "\<forall>\<^sub>F x in nhds x. deriv (\<lambda>x. g (f x)) x = deriv g (f x) * deriv f x"

lemma path_pd_pd0: assumes path: \<open>is_path \<pi>\<close> and lpdn: \<open>\<pi> l pd\<rightarrow> n\<close> and npd0: \<open>n pd\<rightarrow> \<pi> 0\<close> obtains k where \<open>k \<le> l\<close> \<open>\<pi> k = n\<close>

lemma TC_induct_down_lemma: assumes ab: "a \<sqsubset> b" and base: "b \<le> d" and step: "\<And>y z. \<lbrakk>y \<sqsubset> b; y \<in> elts d; z \<in> elts y\<rbrakk> \<Longrightarrow> z \<in> elts d" shows "a \<in> elts d"

lemma fst_map_elem: "(y_k, ty_k, var_k, var'_k, v_k) \<in> set y_ty_var_var'_v_list \<Longrightarrow> x_var var'_k \<in> fst ` (\<lambda>(y, ty, var, var', v). (x_var var', ty)) ` set y_ty_var_var'_v_list"

lemma g_growth2': assumes "x \<ge> x\<^sub>1" "i < k" "u \<in> {bs!i*x+(hs!i) x..x}" shows "c2 * g x \<ge> g u"

lemma (in finite_measure) integrable_const[intro!, simp]: "integrable M (\<lambda>x. a)"

lemma fwi_step': "fwi m n k i' j' i j = min (m i j) (m i k + m k j)" if "m k k \<ge> 0" "i' \<le> n" "j' \<le> n" "k \<le> n" "i \<le> i'" "j \<le> j'"

lemma TESL_sem_decreases_head: \<open>\<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<supseteq> \<lbrakk>\<lbrakk> \<phi> # \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<close>

lemma fic_c_reduce: assumes a: "fic M x" and b: "M \<longrightarrow>\<^sub>c M'" shows "fic M' x"

lemma set_as_mapping_image_code[code] : fixes t :: "('a1 ::ccompare \<times> 'a2 :: ccompare) set_rbt" and f1 :: "('a1 \<times> 'a2) \<Rightarrow> ('b1 :: ccompare \<times> 'b2 ::ccompare)" and xs :: "('c1 :: ceq \<times> 'c2 :: ceq) set_dlist" and f2 :: "('c1 \<times> 'c2) \<Rightarrow> ('d1 \<times> 'd2)" shows "set_as_mapping_image (RBT_set t) f1 = (case ID CCOMPARE(('a1 \<times> 'a2)) of Some _ \<Rightarrow> (RBT_Set2.fold (\<lambda> kv m1 . ( case f1 kv of (x,z) \<Rightarrow> (case Mapping.lookup m1 (x) of None \<Rightarrow> Mapping.update (x) {z} m1 | Some zs \<Rightarrow> Mapping.update (x) (Set.insert z zs) m1))) t Mapping.empty) | None \<Rightarrow> Code.abort (STR ''set_as_map_image RBT_set: ccompare = None'') (\<lambda>_. set_as_mapping_image (RBT_set t) f1))" (is "set_as_mapping_image (RBT_set t) f1 = ?C1 (RBT_set t)") and "set_as_mapping_image (DList_set xs) f2 = (case ID CEQ(('c1 \<times> 'c2)) of Some _ \<Rightarrow> (DList_Set.fold (\<lambda> kv m1 . ( case f2 kv of (x,z) \<Rightarrow> (case Mapping.lookup m1 (x) of None \<Rightarrow> Mapping.update (x) {z} m1 | Some zs \<Rightarrow> Mapping.update (x) (Set.insert z zs) m1))) xs Mapping.empty) | None \<Rightarrow> Code.abort (STR ''set_as_map_image DList_set: ccompare = None'') (\<lambda>_. set_as_mapping_image (DList_set xs) f2))" (is "set_as_mapping_image (DList_set xs) f2 = ?C2 (DList_set xs)")

lemma min_p_min: assumes "is_binqueue l xs" and "xs \<noteq> []" and "normalized xs" and "distinct (vals xs)" and "distinct (prios xs)" shows "min xs = PQ.priority (pqueue xs) (PQ.min (pqueue xs))"

lemma grey_reachable_eq_imp: "eq_imp (\<lambda>r'. (\<lambda>s. \<Union>p. WL p s) \<^bold>\<otimes> (\<lambda>s. Set.bind (Option.set_option (sys_heap s r')) (ran \<circ> obj_fields))) (grey_reachable r)"

lemma equivalence_functor_F: shows "equivalence_functor hom.comp hom'.comp F"

lemma par_distincts: assumes "A B Par C D" shows "A B Par C D \<and> A \<noteq> B \<and> C \<noteq> D"

lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"

lemma L_10_5_\<upsilon>_arrow_ArrVal_app': assumes "\<KK> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>" and "\<TT> : \<BB> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<AA>" and "f : c \<mapsto>\<^bsub>\<CC>\<^esub> \<KK>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>" shows "L_10_5_\<upsilon>_arrow \<TT> \<KK> c \<tau> a b\<lparr>ArrVal\<rparr>\<lparr>f\<rparr> = \<tau>\<lparr>NTMap\<rparr>\<lparr>0, b, f\<rparr>\<^sub>\<bullet>"

lemma take_cols_carrier_mat_strict: assumes "A \<in> carrier_mat nr nc" assumes "\<And>i. i \<in> set inds \<Longrightarrow> i < nc" shows "take_cols A inds \<in> carrier_mat nr (length inds)"

lemma (in Corps) mprod_Suc:"\<lbrakk> \<forall>j\<le>(Suc n). e j \<in> Zset; \<forall>j \<le> (Suc n). f j \<in> (carrier K - {\<zero>})\<rbrakk> \<Longrightarrow> mprod_exp K e f (Suc n) = (mprod_exp K e f n) \<cdot>\<^sub>r ((f (Suc n))\<^bsub>K\<^esub>\<^bsup>(e (Suc n))\<^esup>)"

lemma partition_on_partition_on_unique: assumes "partition_on A P" assumes "x \<in> A" shows "\<exists>!X. x \<in> X \<and> X \<in> P"

lemma inv_part_0_Suc: "m_j < n \<Longrightarrow> inv_part p A 0 m_j = inv_part p A n (Suc m_j)"

lemma \<epsilon>_hcomp': assumes "C.ide g" and "C.ide f" and "src\<^sub>C g = trg\<^sub>C f" shows "\<epsilon> (g \<star>\<^sub>C f) \<cdot>\<^sub>C GF.cmp (g, f) = \<epsilon> g \<star>\<^sub>C \<epsilon> f"

lemma simple_match_valid_alt[code_unfold]: "simple_match_valid = (\<lambda> m. (let c = (\<lambda>(s,e). (s \<noteq> 0 \<or> e \<noteq> - 1)) in ( if c (sports m) \<or> c (dports m) then proto m = Proto TCP \<or> proto m = Proto UDP \<or> proto m = Proto L4_Protocol.SCTP else True)) \<and> valid_prefix_fw (src m) \<and> valid_prefix_fw (dst m))"

lemma "case_prod (curry f) = f"

lemma gcd_ff_list_exists: "\<exists> g. gcd_ff_list (X :: 'a::ufd fract list) g"

lemma poly_mapping_eqI_proj_syz: assumes "proj_poly_syz n p = proj_poly_syz n q" and "\<And>k. k < n \<Longrightarrow> proj_poly k p = proj_poly k q" shows "p = q"

lemma real_power_down_fl: "real_of_float (power_down_fl p x n) = power_down p x n"

lemma SubobjsR_nonempty: "Subobjs\<^sub>R P C Cs \<Longrightarrow> Cs \<noteq> []"

lemma subsumes_update_equality: "subsumes t1 c t2 \<Longrightarrow> (\<forall>i. can_take_transition t2 i c \<longrightarrow> (\<forall>r'. ((evaluate_updates t1 i c $ r') = (evaluate_updates t2 i c $ r')) \<or> evaluate_updates t2 i c $ r' = None))"

lemma divideC_field_splits_simps_2 [field_split_simps]: (* In Real_Vector_Spaces, these lemmas are unnamed *) "0 < c \<Longrightarrow> a \<le> b /\<^sub>C c \<longleftrightarrow> (if c > 0 then c *\<^sub>C a \<le> b else if c < 0 then b \<le> c *\<^sub>C a else a \<le> 0)" "0 < c \<Longrightarrow> a < b /\<^sub>C c \<longleftrightarrow> (if c > 0 then c *\<^sub>C a < b else if c < 0 then b < c *\<^sub>C a else a < 0)" "0 < c \<Longrightarrow> b /\<^sub>C c \<le> a \<longleftrightarrow> (if c > 0 then b \<le> c *\<^sub>C a else if c < 0 then c *\<^sub>C a \<le> b else a \<ge> 0)" "0 < c \<Longrightarrow> b /\<^sub>C c < a \<longleftrightarrow> (if c > 0 then b < c *\<^sub>C a else if c < 0 then c *\<^sub>C a < b else a > 0)" "0 < c \<Longrightarrow> a \<le> - (b /\<^sub>C c) \<longleftrightarrow> (if c > 0 then c *\<^sub>C a \<le> - b else if c < 0 then - b \<le> c *\<^sub>C a else a \<le> 0)" "0 < c \<Longrightarrow> a < - (b /\<^sub>C c) \<longleftrightarrow> (if c > 0 then c *\<^sub>C a < - b else if c < 0 then - b < c *\<^sub>C a else a < 0)" "0 < c \<Longrightarrow> - (b /\<^sub>C c) \<le> a \<longleftrightarrow> (if c > 0 then - b \<le> c *\<^sub>C a else if c < 0 then c *\<^sub>C a \<le> - b else a \<ge> 0)" "0 < c \<Longrightarrow> - (b /\<^sub>C c) < a \<longleftrightarrow> (if c > 0 then - b < c *\<^sub>C a else if c < 0 then c *\<^sub>C a < - b else a > 0)" for a b :: "'a :: ordered_complex_vector"

lemma of_nat_ge_1_iff: "(of_nat x :: 'a :: linordered_semidom) \<ge> 1 \<longleftrightarrow> x \<ge> 1"

lemma NSLIM_mult_zero: "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0 \<Longrightarrow> g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0 \<Longrightarrow> (\<lambda>x. f x * g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0" for f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"

lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"

lemma real_root_ge_1_iff [simp]: "0 < n \<Longrightarrow> 1 \<le> root n y \<longleftrightarrow> 1 \<le> y"

lemma one_real_code [code]: "1 = real_of_float 1"

lemma is_bezout_ext_euclid_ext2: "is_bezout_ext (euclid_ext2)"

lemma sconcat_sfoldl: "sconcat = sfoldl\<cdot>sappend\<cdot>[::]"

lemma rc_invk_arg': "CT \<turnstile> ei \<rightarrow> ei' \<Longrightarrow> append el (ei # er) e' \<Longrightarrow> append el (ei' # er) e'' \<Longrightarrow> CT \<turnstile> MethodInvk e m e' \<rightarrow> MethodInvk e m e''"

lemma measurable_suntil[measurable]: assumes [measurable]: "Measurable.pred (stream_space M) Q" "Measurable.pred (stream_space M) P" shows "Measurable.pred (stream_space M) (Q suntil P)"

lemma switch: fixes A :: "form" and As :: "form_list" shows "y \<sharp> A \<Longrightarrow> [y,x]A = [(y,x)]\<bullet>A" and "y \<sharp> As \<Longrightarrow> [y,x]As = [(y,x)]\<bullet>As"

lemma similar_admI: "cont f \<Longrightarrow> cont g \<Longrightarrow> adm (\<lambda>x. f x \<triangleleft>\<triangleright> g x)"

lemma DBM_val_bounded_len'1: fixes v assumes "DBM_val_bounded v u m n" "0 \<notin> set vs" "v c \<le> n" "\<forall> k \<in> set vs. k > 0 \<longrightarrow> k \<le> n \<and> (\<exists> c. v c = k)" shows "dbm_entry_val u (Some c) None (len m (v c) 0 vs)"

lemma quot_set_image_times: "inj ((*) x) \<Longrightarrow> ((*) x ` A) \<div> x = A"

lemma all_strict [simp]: "all\<cdot>P\<cdot>\<bottom> = \<bottom>"

lemma supp_int: fixes i::"int" shows "supp (i) = {}"

lemma regular_bot_top_2: "regular x \<longleftrightarrow> x = bot \<or> x = top"

lemma 8: "{} \<turnstile> (P AND Neg Q\<^sub>1) IMP Q\<^sub>2"

lemma permChainSimps[simp]: fixes xvec :: "name list" and yvec :: "name list" and perm :: "name prm" and p :: "'a::pt_name" shows "((composePerm xvec yvec) @ perm) \<bullet> p = [xvec yvec] \<bullet>\<^sub>v (perm \<bullet> p)"

lemma to_cnf_0 [simp]: "to_cnf 0 = []"

lemma lemma_4_3_i_a: "a \<squnion>1 b = (a l\<rightarrow> b) r\<rightarrow> b"

lemma tagrel_sub_inv: assumes \<open>sub \<lless> r\<close> and \<open>r \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<close> shows \<open>sub \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<close>

lemma hn_if[sepref_comb_rules]: assumes P: "\<Gamma> \<Longrightarrow>\<^sub>t \<Gamma>1 * hn_val bool_rel a a'" assumes RT: "a \<Longrightarrow> hn_refine (\<Gamma>1 * hn_val bool_rel a a') b' \<Gamma>2b R b" assumes RE: "\<not>a \<Longrightarrow> hn_refine (\<Gamma>1 * hn_val bool_rel a a') c' \<Gamma>2c R c" assumes IMP: "TERM If \<Longrightarrow> \<Gamma>2b \<or>\<^sub>A \<Gamma>2c \<Longrightarrow>\<^sub>t \<Gamma>'" shows "hn_refine \<Gamma> (if a' then b' else c') \<Gamma>' R (If$a$b$c)"

lemma sources_respects_Cong: assumes "T \<^sup>*\<approx>\<^sup>* T'" shows "P.sources T = P.sources T'"

lemma completely_multiplicative_function_inverse: fixes f :: "nat \<Rightarrow> 'a :: field" assumes "completely_multiplicative_function f" shows "completely_multiplicative_function (\<lambda>n. inverse (f n))"

lemma mk_deriv_intro_ix[simp]: "deriv_ix (mk_deriv_intro ix n j) = ix"

lemma MPair_used [rule_format]: "MPair X Y \<in> used evs \<longrightarrow> X \<in> used evs & Y \<in> used evs"

lemma aff_dim_insert: fixes a :: "'a::euclidean_space" shows "aff_dim (insert a S) = (if a \<in> affine hull S then aff_dim S else aff_dim S + 1)"

lemma simple_function_subalgebra: assumes "simple_function N f" and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M" shows "simple_function M f"

lemma glength_greater_0_conv[simp]: "(0 < glength a) = (a \<noteq> FL [])"

lemma params_subterms_Union[simp]: "subterms\<^sub>s\<^sub>e\<^sub>t (set X) \<subseteq> subterms (Fun f X)"

lemma H\<^sub>R_preserved_along_iso: assumes "weak_unit a" and "a \<cong> a'" shows "endofunctor (Right a) (H\<^sub>R a')"

lemma distinct_sorted_intersection[simp]: "\<lbrakk> distinct xs; distinct ys; sorted xs; sorted ys \<rbrakk> \<Longrightarrow> distinct (intersection xs ys) \<and> sorted (intersection xs ys)"

lemma W_eq_L_iter: assumes "\<nu>_improving v d" shows "W d m v = (L d^^m) v"

lemma cat_sspan_composable_\<bb>\<bb>[cat_ss_cs_intros]: assumes "g = \<bb>\<^sub>S\<^sub>S" and "f = \<bb>\<^sub>S\<^sub>S" shows "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> cat_sspan_composable"

lemma (in Group) ZassenhausTr1_1:"\<lbrakk>G \<guillemotright> H; G \<guillemotright> H1; G \<guillemotright> K; G \<guillemotright> K1; Gp G H \<triangleright> H1; Gp G K \<triangleright> K1\<rbrakk> \<Longrightarrow> G \<guillemotright> (H1 \<diamondop>\<^bsub>G\<^esub> (H \<inter> K1))"

lemma sset_shift[simp]: "sset (xs @- s) = set xs \<union> sset s"

lemma iSince_iEx_conv: "(True. t' \<S> t I. P t) = (\<diamond> t I. P t)"

lemma pref_prod_longer: "u \<le>p z \<cdot> w \<Longrightarrow> v \<le>p w \<Longrightarrow> \<^bold>|z \<cdot> v\<^bold>| \<le> \<^bold>|u\<^bold>| \<Longrightarrow> z \<cdot> v \<le>p u"

theorem A_gets_good_key: "\<lbrakk>Crypt (shrK A) \<lbrace>Agent B, Key K, na\<rbrace> \<in> parts (knows Spy evs); \<forall>nb. Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs; A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> \<Longrightarrow> Key K \<notin> analz (knows Spy evs)"

lemma register_unitary: assumes "register F" assumes "unitary a" shows "unitary (F a)"

lemma all_self_eq_1[PLM]: "[\<^bold>\<box>(\<^bold>\<forall> \<alpha> :: 'a::id_eq . \<alpha> \<^bold>= \<alpha>) in v]"