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

Statement: stringlengths 7 24.3k
lemma (in Ring) ele_n_prodTr0:"\<lbrakk>\<forall>k \<le> (Suc n). ideal R (J k); a \<in> i\<Pi>\<^bsub>R,(Suc n)\<^esub> J \<rbrakk> \<Longrightarrow> a \<in> (i\<Pi>\<^bsub>R,n\<^esub> J) \<and> a \<in> (J (Suc n))"
lemma (in \<Z>) M\<alpha>_Rel_arrow_lr_is_cat_Par_arr: assumes "A \<in>\<^sub>\<circ> cat_Par \<alpha>\<lparr>Obj\<rparr>" and "B \<in>\<^sub>\<circ> cat_Par \<alpha>\<lparr>Obj\<rparr>" and "C \<in>\<^sub>\<circ> cat_Par \<alpha>\<lparr>Obj\<rparr>" shows "M\<alpha>_Rel_arrow_lr A B C : (A \<times>\<^sub>\<circ> B) \<times>\<^sub>\<circ> C \<mapsto>\<^bsub>cat_Par \<alpha>\<^esub> A \<times>\<^sub>\<circ> (B \<times>\<^sub>\<circ> C)"
lemma le_replicateI: "\<forall>x\<in>set xs. x \<le> b \<Longrightarrow> xs \<le>\<^sub>v replicate (length xs) b"
lemma prog_not_eq_in_par_ctran [simp]: "\<not> (P,s) -pc\<rightarrow> (P,t)"
lemma crename_coname_eqvt[eqvt]: fixes pi::"coname prm" shows "pi\<bullet>(M[d\<turnstile>c>e]) = (pi\<bullet>M)[(pi\<bullet>d)\<turnstile>c>(pi\<bullet>e)]"
lemma ghole_poss_num_gholes_zero: "ghole_poss D = {} \<Longrightarrow> num_gholes D = 0"
lemma higher_differentiable_Taylor: fixes f::"'a::real_normed_vector \<Rightarrow> 'b::banach" and H::"'a" and Df::"'a \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b" assumes "n > 0" assumes hd: "higher_differentiable_on S f n" "open S" assumes cs: "closed_segment X (X + H) \<subseteq> S" defines "i \<equiv> \<lambda>x. ((1 - x) ^ (n - 1) / fact (n - 1)) \<^sub>R nth_derivative n f (X + x \<^sub>R H) H" shows "(i has_integral f (X + H) - (\<Sum>i<n. (1 / fact i) \<^sub>R nth_derivative i f X H)) {0..1}" (is ?th1) and "f (X + H) = (\<Sum>i<n. (1 / fact i) \<^sub>R nth_derivative i f X H) + integral {0..1} i" (is ?th2) and "i integrable_on {0..1}" (is ?th3)
lemma supset_glbound_in_of_lcoset_shift: fixes P :: "'a::group_add set set" assumes "supset_glbound_in_of P X Y B" shows "supset_glbound_in_of ((+o) a ` P) (a +o X) (a +o Y) (a +o B)"
lemma rbt_sorted_rbt_join2: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> \<forall>x \<in> set (RBT_Impl.keys l). \<forall>y \<in> set (RBT_Impl.keys r). x < y \<Longrightarrow> rbt_sorted (rbt_join2 l r)"
lemma p2ndf_simps[simp]: "\<lceil>P\<rceil> \<le> \<lceil>Q\<rceil> = (\<forall>s. P s \<longrightarrow> Q s)" "(\<lceil>P\<rceil> = \<lceil>Q\<rceil>) = (\<forall>s. P s = Q s)" "(\<lceil>P\<rceil> \<cdot> \<lceil>Q\<rceil>) = \<lceil>\<lambda>s. P s \<and> Q s\<rceil>" "(\<lceil>P\<rceil> + \<lceil>Q\<rceil>) = \<lceil>\<lambda>s. P s \<or> Q s\<rceil>" "\<tt>\<tt> \<lceil>P\<rceil> = \<lceil>P\<rceil>" "n \<lceil>P\<rceil> = \<lceil>\<lambda>s. \<not> P s\<rceil>"
lemma invimage_of_vempty[simp]: "r -`\<^sub>\<circ> 0 = 0"
lemma partition_on_transform: assumes P: "partition_on A P" assumes F_UN: "\<Union>(F ` P) = F (\<Union>P)" and F_disjnt: "\<And>p q. p \<in> P \<Longrightarrow> q \<in> P \<Longrightarrow> disjnt p q \<Longrightarrow> disjnt (F p) (F q)" shows "partition_on (F A) (F ` P - {{}})"
lemma complete_dual: "UNIV-complete A (\<sqsubseteq>) \<Longrightarrow> UNIV-complete A (\<sqsupseteq>)"
lemma is_constant_seqI: fixes a assumes "s \<in> closed_seqs R" assumes "\<And>k. s k = a" shows "is_constant_seq R s"
lemma mono_prover_monoI[refine_mono]: "monotone (fun_ord (\<le>)) (fun_ord (\<le>)) B \<Longrightarrow> mono B"
lemma development_map_App_1: shows "\<lbrakk>development t T; \<Lambda>.Arr u\<rbrakk> \<Longrightarrow> development (t \<^bold>\<circ> u) (map (\<lambda>x. x \<^bold>\<circ> \<Lambda>.Src u) T)"
lemma range_vars_alt_def: "range_vars s \<equiv> fv\<^sub>s\<^sub>e\<^sub>t (subst_range s)"
lemma itop_sub_ttop_base: fixes A :: "'a set" and B :: "'a llist set set" and C :: "'a llist set set" defines [simp]: "B \<equiv> \<Union>s\<in>A\<^sup>\<star>. {suff A s}" and [simp]: "C \<equiv> \<Union>s\<in>A\<^sup>\<star>. {infsuff A s}" shows "C = (\<Union> t\<in>B. {t \<inter> \<Union>C})"
lemma frameChainEqSuppEmpty[dest]: fixes xvec :: "name list" and \<Psi> :: "'a::fs_name" and yvec :: "name list" and \<Psi>' :: "'a::fs_name" assumes "\<langle>xvec, \<Psi>\<rangle> = \<langle>yvec, \<Psi>'\<rangle>" and "supp \<Psi> = ({}::name set)" shows "\<Psi> = \<Psi>'"
lemma lcs_adds_fun: assumes "s adds u" and "t adds (u::'a \<Rightarrow> 'b)" shows "(lcs s t) adds u"
lemma concrete_edge_rel_list_set_rel: "(a, b) \<in> \<langle>concrete_edge_rel\<rangle>list_set_rel \<Longrightarrow> \<alpha> ` (set a) = b"
lemma bv_sadd_type1 [simp]: "bv_sadd (norm_signed w1) w2 = bv_sadd w1 w2"
lemma fps_divide_nth_0 [simp]: "g $ 0 \<noteq> 0 \<Longrightarrow> (f div g) $ 0 = f $ 0 / (g $ 0 :: _ :: division_ring)"
lemma iteratei_rule_P: assumes "I S0 \<sigma>0" "\<And>S \<sigma> x. \<lbrakk> c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S - {x}. R x y; \<forall>y\<in>S0 - S. R y x\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)" "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" "\<And>\<sigma> S. \<lbrakk> S \<subseteq> S0; S \<noteq> {}; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S. \<forall>y\<in>S0-S. R y x \<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iti c f \<sigma>0)"
lemma (in group) int_pow_diff: "x \<in> carrier G \<Longrightarrow> x [^] (n - m :: int) = x [^] n \<otimes> inv (x [^] m)"
lemma set_pred_eq_transfer[transfer_rule]: assumes "right_total A" shows "((rel_set A ===> (=)) ===> (rel_set A ===> (=)) ===> (=)) (\<lambda>X Y. \<forall>s\<subseteq>Collect (Domainp A). X s = Y s) ((=)::['b set \<Rightarrow> bool, 'b set \<Rightarrow> bool] \<Rightarrow> bool)"
lemma fls_left_inverse_idempotent_comm_ring1: fixes f :: "'a::comm_ring_1 fls" assumes "x * f $$ fls_subdegree f = 1" shows "fls_left_inverse (fls_left_inverse f x) (f $$ fls_subdegree f) = f"
lemma n_omega_L_below_zero: "n(x\<^sup>\<omega>) * L \<le> x * x\<^sup>\<star> * bot \<squnion> x * n(x\<^sup>\<omega>) * L"
lemma fresh_star_unit_elim: shows "((a::'a set)\<sharp>() \<Longrightarrow> PROP C) \<equiv> PROP C" and "((b::'a list)\<sharp>() \<Longrightarrow> PROP C) \<equiv> PROP C"
lemma normal_upper_triangular_matrix_is_diagonal: fixes A :: "'a::conjugatable_ordered_field mat" assumes "A \<in> carrier_mat n n" and tri: "upper_triangular A" and norm: "A * adjoint A = adjoint A * A" shows "diagonal_mat A"
lemma pad_disjoint: assumes A: "A \<subseteq> carrier_vec n" and A0: "0\<^sub>v n \<notin> A" and B: "B \<subseteq> carrier_vec m" shows "padr m ` A \<inter> padl n ` B = {}" (is "?A \<inter> ?B = _")
lemma ntsmcf_hcomp_components: shows "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTMap\<rparr> = ( \<lambda>a\<in>\<^sub>\<circ>\<NN>\<lparr>NTDGDom\<rparr>\<lparr>Obj\<rparr>. ( \<MM>\<lparr>NTCod\<rparr>\<lparr>ArrMap\<rparr>\<lparr>\<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr>\<rparr> \<circ>\<^sub>A\<^bsub>\<MM>\<lparr>NTDGCod\<rparr>\<^esub> \<MM>\<lparr>NTMap\<rparr>\<lparr>\<NN>\<lparr>NTDom\<rparr>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<rparr> ) )" and [dg_shared_cs_simps, smc_cs_simps]: "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTDom\<rparr> = \<MM>\<lparr>NTDom\<rparr> \<circ>\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>\<lparr>NTDom\<rparr>" and [dg_shared_cs_simps, smc_cs_simps]: "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTCod\<rparr> = \<MM>\<lparr>NTCod\<rparr> \<circ>\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>\<lparr>NTCod\<rparr>" and [dg_shared_cs_simps, smc_cs_simps]: "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTDGDom\<rparr> = \<NN>\<lparr>NTDGDom\<rparr>" and [dg_shared_cs_simps, smc_cs_simps]: "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTDGCod\<rparr> = \<MM>\<lparr>NTDGCod\<rparr>"
lemma appMEnter[simp]: "app\<^sub>i (MEnter,P,pc,mxs,T\<^sub>r,s) = (\<exists>T ST LT. s=(T#ST,LT) \<and> is_refT T)"
lemma Vars_indep_foldr: assumes "x \<in> set Vars" "set xs \<subseteq> set Vars" shows "x \<bowtie>\<^sub>S \<Squnion>\<^sub>S (removeAll x xs)"
lemma last_index_eq_index_conv[simp]: "x \<in> set xs \<or> y \<in> set xs \<Longrightarrow> (last_index xs x = last_index xs y) = (x = y)"
lemma maxLemma: assumes "x \<in> X" "finite X" shows "Max (f`X) >= f x" (is "?L >= ?R")
lemma config_config_length: "length (fst (config A init qs)) = length init"
lemma sdp_Bind: "\<lbrakk> \<And>s. sub_distrib_pconj (p (f s)) \<rbrakk> \<Longrightarrow> sub_distrib_pconj (Bind f p)"
lemma cones_map_is_composition: assumes "\<guillemotleft>g : a' \<rightarrow> a\<guillemotright>" and "cone J C D a \<chi>" shows "J_C.MkArr (constant_functor.map J C a') D (diagram.cones_map J C D g \<chi>) = J_C.MkArr (constant_functor.map J C a) D \<chi> \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] map g"
lemma RUN_subset_DT: "A \<subseteq> B \<Longrightarrow> RUN B \<sqsubseteq>\<^sub>D\<^sub>T RUN A"
lemma rcut_eq: "rcut = tr0 \<longleftrightarrow> reg H tr0"
lemma iT_Div_not_empty: "I \<noteq> {} \<Longrightarrow> I \<oslash> k \<noteq> {}"
lemma PO_m1a_step3_refines_m1x_step3: "{R1x1a} (m1x_step3 Rs A B Kab), (m1a_step3 Rs A B Kab nls) {> R1x1a}"
lemma HFun_Sigma_subst [simp]: "(HFun_Sigma r)(i::=t) = HFun_Sigma (subst i t r)"
lemma root_in_start_points_2: assumes "backward_finite_path_root r x" and "start_points x \<noteq> 0" shows "r \<le> start_points x"
lemma fls_regpart_const [simp]: "fls_regpart (fls_const c) = fps_const c"
lemma blinfunpow_nonneg: assumes "\<And>v. 0 \<le> v \<Longrightarrow> 0 \<le> blinfun_apply (f :: ('b :: {ord, real_normed_vector} \<Rightarrow>\<^sub>L 'b)) v" shows "0 \<le> v \<Longrightarrow> 0 \<le> (f^^n) v"
lemma before_C_unique: assumes \<omega>: "before_C I1 \<omega>" "before_C I2 \<omega>" shows "I1 \<inter> I2 \<noteq> {}"
lemma permutes_empty [simp]: \<open>p permutes {} \<longleftrightarrow> p = id\<close>
lemma alwaysWFI: "\<turnstile> WF(A)_v \<longrightarrow> \<box>WF(A)_v"
lemma divide_poly_list[code]: "f div g = divide_poly_list f g"
lemma the_elemI: assumes "is_singleton A" shows "the_elem A \<in> A"
lemma deleteMin_correct_aux: assumes I: "invar q" assumes NE: "q\<noteq>[]" shows "invar (deleteMin q)" "queue_to_multiset_aux (deleteMin q) = queue_to_multiset_aux q - {# (findMin q, eprio (findMin q)) #}"
lemma assert_disch3 :" \<not> P \<sigma> \<Longrightarrow> \<not> (\<sigma> \<Turnstile> (assert\<^sub>S\<^sub>E P))"
lemma right_commute: "f (f a b) c = f (f a c) b"
lemma norm_pair_fst0[simp]: "norm (0, x) = norm x"
lemma finite_llast_V [simp]: "lfinite P \<Longrightarrow> llast P \<in> V"
lemma sup_apx_left_isotone_2: assumes "x \<sqsubseteq> y" shows "x \<squnion> z \<sqsubseteq> y \<squnion> z"
lemma unit_quasi_borel_terminal: "\<exists>! f. f \<in> X \<rightarrow>\<^sub>Q unit_quasi_borel"
lemma snd_sum_list: "snd (sum_list xs) = sum_list (map snd xs)"
lemma LIST_FRAG_DICHOTOMY: fixes l la x lb assumes "sublist l (la @ [x] @ lb)" "\<not>ListMem x l" shows "sublist l la \<or> sublist l lb"
lemma borel_measurable_power_ennreal [measurable (raw)]: fixes f :: "_ \<Rightarrow> ennreal" assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. (f x) ^ n) \<in> borel_measurable M"
lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = undefined"
lemma bdoubleton_eq_iff[simp]: "bdoubleton x y = bdoubleton z w \<longleftrightarrow> (x = z \<and> y = w \<or> x = w \<and> y = z)"
lemma clop_preorder: "clop f \<Longrightarrow> class.preorder (\<lambda>x y. f {x} \<subseteq> f {y}) (\<lambda>x y. f {x} \<subset> f {y})"
lemma (in ring) subdomain_iff: assumes "H \<subseteq> carrier R" shows "subdomain H R \<longleftrightarrow> domain (R \<lparr> carrier := H \<rparr>)"
lemma ok'_conv: "\<sigma>t OKAY' = true"
lemma kp_5x7_ul: "knights_path b5x7 kp5x7ul"
lemma VarP_sf [iff]: "Sigma_fm (VarP x)"
lemma normalize1_hd_root_eq[simp]: assumes "root t1 \<noteq> []" shows "hd (root (normalize1 t1)) = hd (root t1)"
lemma not_in_pow : "var\<notin>(vars(p::real mpoly)) \<Longrightarrow> var\<notin>(vars(p^i))"
lemma [code]: "enum 0 = {}" "enum (Suc n) = insert n (enum n)"
lemma path_decomp: assumes "path (xs @ ys)" shows "path xs" "path ys"
lemma exp_times_has_integral: "((\<lambda>t. exp (c * t)) has_integral (if c = 0 then t else exp (c * t) / c) - (if c = 0 then t0 else exp (c * t0) / c)) {t0 .. t}" if "t0 \<le> t" for c t::real
lemma (in bar) bar1: shows "(s\<langle>b:=True\<rangle>)\<cdot>c = s\<cdot>c"
lemma idempotent_subset: assumes "idempotent R" "S \<subseteq> R" shows "S O R \<subseteq> R" "R O S \<subseteq> R" "S O R O S \<subseteq> R"
lemma monad_state_alt_stateT' [locale_witness]: "monad_alt return (bind :: ('a \<times> 's, 'm) bind) alt \<Longrightarrow> monad_state_alt return (bind :: ('a, ('s, 'm) stateT) bind) (get :: ('s, ('s, 'm) stateT) get) put alt"
lemma Confidentiality_Kas_lemma [rule_format]: "\<lbrakk> authK \<in> symKeys; A \<notin> bad; evs \<in> kerbIV \<rbrakk> \<Longrightarrow> Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs \<longrightarrow> Key authK \<in> analz (spies evs) \<longrightarrow> expiredAK Ta evs"
lemma IF_alg_derivable_from_If: "\<lbrakk>(G,b,p):HS_B; (p,G,c1,H):HS; (p,G,c2,H):HS\<rbrakk> \<Longrightarrow> (p,G,Iff b c1 c2,H):HS"
lemma map_us_func1: "map_us id = (id::'a::complete_lattice_with_dual upset \<Rightarrow> 'a upset)"
lemma class_add_relevant_entries: "\<not> is_class P C \<Longrightarrow> set (relevant_entries P i pc xt) \<subseteq> set (relevant_entries (class_add P (C, cdec)) i pc xt)"
lemma "\<^bold>\<exists>\<pi>\<^sup>c = \<^bold>\<midarrow>(\<^bold>\<forall>\<pi>)"
theorem wf_termi_call_steps: "wf (termi_call_steps \<Gamma>)"
lemma enum_match_cong[sepref_frame_match_rules]: "\<lbrakk>\<And>x y. \<lbrakk>x\<in>set_enum e; y\<in>set_enum e'\<rbrakk> \<Longrightarrow> hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y\<rbrakk> \<Longrightarrow> hn_ctxt (enum_assn A) e e' \<Longrightarrow>\<^sub>t hn_ctxt (enum_assn A') e e'"
lemma dtree_from_list_root_r[simp]: "root (dtree_from_list r xs) = r"
lemma member_inv: assumes "vebt_member (Node (Some (mi, ma)) deg treeList summary) x " shows "deg \<ge> 2 \<and> (x = mi \<or> x = ma \<or> (x < ma \<and> x > mi \<and> high x (deg div 2) < length treeList \<and> vebt_member (treeList ! ( high x (deg div 2))) (low x (deg div 2))))"
lemma del_emp_concat: "concat us = concat (filter (\<lambda>x. x \<noteq> \<epsilon>) us)"
lemma llist_of_tllist_transfer2 [transfer_rule]: "(tllist_all2 A B ===> llist_all2 A) llist_of_tllist llist_of_tllist"
lemma diamond_n_an: "\|n(x)>an(y) = n(x) * an(y)"
lemma path_entry_append: "\<lbrakk> path_entry E l v; (v,w)\<in>E \<rbrakk> \<Longrightarrow> path_entry E (v#l) w"
lemma prodrg_mOp:"\<forall>k\<in>I. Ring (A k) \<Longrightarrow> mop (prodrg I A) = prod_mOp I A"
lemma range_None: assumes "MaxR_opt A = None" shows "A \<inter> Field r = {}"
lemma Checkcast_correct: "\<lbrakk> wf_jvm_prog\<^bsub>\<Phi>\<^esub> P; P \<turnstile> C sees M:Ts\<rightarrow>T=(mxs,mxl\<^sub>0,ins,xt) in C; ins!pc = Checkcast D; P,T,mxs,size ins,xt \<turnstile> ins!pc,pc :: \<Phi> C M; Some \<sigma>' = exec (P, None, h, (stk,loc,C,M,pc)#frs) ; P,\<Phi> \<turnstile> (None, h, (stk,loc,C,M,pc)#frs)\<surd>; fst (exec_instr (ins!pc) P h stk loc C M pc frs) = None \<rbrakk> \<Longrightarrow> P,\<Phi> \<turnstile> \<sigma>'\<surd>"
lemma mangoldt_primepow' [simp]: "prime p \<Longrightarrow> k > 0 \<Longrightarrow> mangoldt (p ^ k) = of_real (ln (real p))"
lemma NSLIMSEQ_inverse_zero: "\<forall>y::real. \<exists>N. \<forall>n \<ge> N. y < f n \<Longrightarrow> (\<lambda>n. inverse (f n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S 0"
lemma pred_pmf_of_set [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> pred_pmf P (pmf_of_set A) = Ball A P"
lemma (in Group) rcs_Unit2:"\<lbrakk>G \<guillemotright> H; h \<in> H\<rbrakk> \<Longrightarrow> H \<bullet> h = H"
lemma fds_shift_inverse [simp]: "fds_shift (a :: 'a :: {field, nat_power}) (inverse f) = inverse (fds_shift a f)"
lemma subprob_space_bind: assumes "subprob_space M" "f \<in> measurable M (subprob_algebra N)" shows "subprob_space (M \<bind> f)"
lemma mapRuleI: "[\| A = map f a; B = (map f) ` b \|] ==> (A,B) = mapRule f (a,b)"

lemma (in Ring) ele_n_prodTr0:"\<lbrakk>\<forall>k \<le> (Suc n). ideal R (J k); a \<in> i\<Pi>\<^bsub>R,(Suc n)\<^esub> J \<rbrakk> \<Longrightarrow> a \<in> (i\<Pi>\<^bsub>R,n\<^esub> J) \<and> a \<in> (J (Suc n))"

lemma (in \<Z>) M\<alpha>_Rel_arrow_lr_is_cat_Par_arr: assumes "A \<in>\<^sub>\<circ> cat_Par \<alpha>\<lparr>Obj\<rparr>" and "B \<in>\<^sub>\<circ> cat_Par \<alpha>\<lparr>Obj\<rparr>" and "C \<in>\<^sub>\<circ> cat_Par \<alpha>\<lparr>Obj\<rparr>" shows "M\<alpha>_Rel_arrow_lr A B C : (A \<times>\<^sub>\<circ> B) \<times>\<^sub>\<circ> C \<mapsto>\<^bsub>cat_Par \<alpha>\<^esub> A \<times>\<^sub>\<circ> (B \<times>\<^sub>\<circ> C)"

lemma le_replicateI: "\<forall>x\<in>set xs. x \<le> b \<Longrightarrow> xs \<le>\<^sub>v replicate (length xs) b"

lemma prog_not_eq_in_par_ctran [simp]: "\<not> (P,s) -pc\<rightarrow> (P,t)"

lemma crename_coname_eqvt[eqvt]: fixes pi::"coname prm" shows "pi\<bullet>(M[d\<turnstile>c>e]) = (pi\<bullet>M)[(pi\<bullet>d)\<turnstile>c>(pi\<bullet>e)]"

lemma ghole_poss_num_gholes_zero: "ghole_poss D = {} \<Longrightarrow> num_gholes D = 0"

lemma higher_differentiable_Taylor: fixes f::"'a::real_normed_vector \<Rightarrow> 'b::banach" and H::"'a" and Df::"'a \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b" assumes "n > 0" assumes hd: "higher_differentiable_on S f n" "open S" assumes cs: "closed_segment X (X + H) \<subseteq> S" defines "i \<equiv> \<lambda>x. ((1 - x) ^ (n - 1) / fact (n - 1)) *\<^sub>R nth_derivative n f (X + x *\<^sub>R H) H" shows "(i has_integral f (X + H) - (\<Sum>i<n. (1 / fact i) *\<^sub>R nth_derivative i f X H)) {0..1}" (is ?th1) and "f (X + H) = (\<Sum>i<n. (1 / fact i) *\<^sub>R nth_derivative i f X H) + integral {0..1} i" (is ?th2) and "i integrable_on {0..1}" (is ?th3)

lemma supset_glbound_in_of_lcoset_shift: fixes P :: "'a::group_add set set" assumes "supset_glbound_in_of P X Y B" shows "supset_glbound_in_of ((+o) a ` P) (a +o X) (a +o Y) (a +o B)"

lemma rbt_sorted_rbt_join2: "rbt_sorted l \<Longrightarrow> rbt_sorted r \<Longrightarrow> \<forall>x \<in> set (RBT_Impl.keys l). \<forall>y \<in> set (RBT_Impl.keys r). x < y \<Longrightarrow> rbt_sorted (rbt_join2 l r)"

lemma p2ndf_simps[simp]: "\<lceil>P\<rceil> \<le> \<lceil>Q\<rceil> = (\<forall>s. P s \<longrightarrow> Q s)" "(\<lceil>P\<rceil> = \<lceil>Q\<rceil>) = (\<forall>s. P s = Q s)" "(\<lceil>P\<rceil> \<cdot> \<lceil>Q\<rceil>) = \<lceil>\<lambda>s. P s \<and> Q s\<rceil>" "(\<lceil>P\<rceil> + \<lceil>Q\<rceil>) = \<lceil>\<lambda>s. P s \<or> Q s\<rceil>" "\<tt>\<tt> \<lceil>P\<rceil> = \<lceil>P\<rceil>" "n \<lceil>P\<rceil> = \<lceil>\<lambda>s. \<not> P s\<rceil>"

lemma invimage_of_vempty[simp]: "r -`\<^sub>\<circ> 0 = 0"

lemma partition_on_transform: assumes P: "partition_on A P" assumes F_UN: "\<Union>(F ` P) = F (\<Union>P)" and F_disjnt: "\<And>p q. p \<in> P \<Longrightarrow> q \<in> P \<Longrightarrow> disjnt p q \<Longrightarrow> disjnt (F p) (F q)" shows "partition_on (F A) (F ` P - {{}})"

lemma complete_dual: "UNIV-complete A (\<sqsubseteq>) \<Longrightarrow> UNIV-complete A (\<sqsupseteq>)"

lemma is_constant_seqI: fixes a assumes "s \<in> closed_seqs R" assumes "\<And>k. s k = a" shows "is_constant_seq R s"

lemma mono_prover_monoI[refine_mono]: "monotone (fun_ord (\<le>)) (fun_ord (\<le>)) B \<Longrightarrow> mono B"

lemma development_map_App_1: shows "\<lbrakk>development t T; \<Lambda>.Arr u\<rbrakk> \<Longrightarrow> development (t \<^bold>\<circ> u) (map (\<lambda>x. x \<^bold>\<circ> \<Lambda>.Src u) T)"

lemma range_vars_alt_def: "range_vars s \<equiv> fv\<^sub>s\<^sub>e\<^sub>t (subst_range s)"

lemma itop_sub_ttop_base: fixes A :: "'a set" and B :: "'a llist set set" and C :: "'a llist set set" defines [simp]: "B \<equiv> \<Union>s\<in>A\<^sup>\<star>. {suff A s}" and [simp]: "C \<equiv> \<Union>s\<in>A\<^sup>\<star>. {infsuff A s}" shows "C = (\<Union> t\<in>B. {t \<inter> \<Union>C})"

lemma frameChainEqSuppEmpty[dest]: fixes xvec :: "name list" and \<Psi> :: "'a::fs_name" and yvec :: "name list" and \<Psi>' :: "'a::fs_name" assumes "\<langle>xvec, \<Psi>\<rangle> = \<langle>yvec, \<Psi>'\<rangle>" and "supp \<Psi> = ({}::name set)" shows "\<Psi> = \<Psi>'"

lemma lcs_adds_fun: assumes "s adds u" and "t adds (u::'a \<Rightarrow> 'b)" shows "(lcs s t) adds u"

lemma concrete_edge_rel_list_set_rel: "(a, b) \<in> \<langle>concrete_edge_rel\<rangle>list_set_rel \<Longrightarrow> \<alpha> ` (set a) = b"

lemma bv_sadd_type1 [simp]: "bv_sadd (norm_signed w1) w2 = bv_sadd w1 w2"

lemma fps_divide_nth_0 [simp]: "g $ 0 \<noteq> 0 \<Longrightarrow> (f div g) $ 0 = f $ 0 / (g $ 0 :: _ :: division_ring)"

lemma iteratei_rule_P: assumes "I S0 \<sigma>0" "\<And>S \<sigma> x. \<lbrakk> c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S - {x}. R x y; \<forall>y\<in>S0 - S. R y x\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)" "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" "\<And>\<sigma> S. \<lbrakk> S \<subseteq> S0; S \<noteq> {}; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S. \<forall>y\<in>S0-S. R y x \<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iti c f \<sigma>0)"

lemma (in group) int_pow_diff: "x \<in> carrier G \<Longrightarrow> x [^] (n - m :: int) = x [^] n \<otimes> inv (x [^] m)"

lemma set_pred_eq_transfer[transfer_rule]: assumes "right_total A" shows "((rel_set A ===> (=)) ===> (rel_set A ===> (=)) ===> (=)) (\<lambda>X Y. \<forall>s\<subseteq>Collect (Domainp A). X s = Y s) ((=)::['b set \<Rightarrow> bool, 'b set \<Rightarrow> bool] \<Rightarrow> bool)"

lemma fls_left_inverse_idempotent_comm_ring1: fixes f :: "'a::comm_ring_1 fls" assumes "x * f $$ fls_subdegree f = 1" shows "fls_left_inverse (fls_left_inverse f x) (f $$ fls_subdegree f) = f"

lemma n_omega_L_below_zero: "n(x\<^sup>\<omega>) * L \<le> x * x\<^sup>\<star> * bot \<squnion> x * n(x\<^sup>\<omega>) * L"

lemma fresh_star_unit_elim: shows "((a::'a set)\<sharp>*() \<Longrightarrow> PROP C) \<equiv> PROP C" and "((b::'a list)\<sharp>*() \<Longrightarrow> PROP C) \<equiv> PROP C"

lemma normal_upper_triangular_matrix_is_diagonal: fixes A :: "'a::conjugatable_ordered_field mat" assumes "A \<in> carrier_mat n n" and tri: "upper_triangular A" and norm: "A * adjoint A = adjoint A * A" shows "diagonal_mat A"

lemma pad_disjoint: assumes A: "A \<subseteq> carrier_vec n" and A0: "0\<^sub>v n \<notin> A" and B: "B \<subseteq> carrier_vec m" shows "padr m ` A \<inter> padl n ` B = {}" (is "?A \<inter> ?B = _")

lemma ntsmcf_hcomp_components: shows "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTMap\<rparr> = ( \<lambda>a\<in>\<^sub>\<circ>\<NN>\<lparr>NTDGDom\<rparr>\<lparr>Obj\<rparr>. ( \<MM>\<lparr>NTCod\<rparr>\<lparr>ArrMap\<rparr>\<lparr>\<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr>\<rparr> \<circ>\<^sub>A\<^bsub>\<MM>\<lparr>NTDGCod\<rparr>\<^esub> \<MM>\<lparr>NTMap\<rparr>\<lparr>\<NN>\<lparr>NTDom\<rparr>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<rparr> ) )" and [dg_shared_cs_simps, smc_cs_simps]: "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTDom\<rparr> = \<MM>\<lparr>NTDom\<rparr> \<circ>\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>\<lparr>NTDom\<rparr>" and [dg_shared_cs_simps, smc_cs_simps]: "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTCod\<rparr> = \<MM>\<lparr>NTCod\<rparr> \<circ>\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>\<lparr>NTCod\<rparr>" and [dg_shared_cs_simps, smc_cs_simps]: "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTDGDom\<rparr> = \<NN>\<lparr>NTDGDom\<rparr>" and [dg_shared_cs_simps, smc_cs_simps]: "(\<MM> \<circ>\<^sub>N\<^sub>T\<^sub>S\<^sub>M\<^sub>C\<^sub>F \<NN>)\<lparr>NTDGCod\<rparr> = \<MM>\<lparr>NTDGCod\<rparr>"

lemma appMEnter[simp]: "app\<^sub>i (MEnter,P,pc,mxs,T\<^sub>r,s) = (\<exists>T ST LT. s=(T#ST,LT) \<and> is_refT T)"

lemma Vars_indep_foldr: assumes "x \<in> set Vars" "set xs \<subseteq> set Vars" shows "x \<bowtie>\<^sub>S \<Squnion>\<^sub>S (removeAll x xs)"

lemma last_index_eq_index_conv[simp]: "x \<in> set xs \<or> y \<in> set xs \<Longrightarrow> (last_index xs x = last_index xs y) = (x = y)"

lemma maxLemma: assumes "x \<in> X" "finite X" shows "Max (f`X) >= f x" (is "?L >= ?R")

lemma config_config_length: "length (fst (config A init qs)) = length init"

lemma sdp_Bind: "\<lbrakk> \<And>s. sub_distrib_pconj (p (f s)) \<rbrakk> \<Longrightarrow> sub_distrib_pconj (Bind f p)"

lemma cones_map_is_composition: assumes "\<guillemotleft>g : a' \<rightarrow> a\<guillemotright>" and "cone J C D a \<chi>" shows "J_C.MkArr (constant_functor.map J C a') D (diagram.cones_map J C D g \<chi>) = J_C.MkArr (constant_functor.map J C a) D \<chi> \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] map g"

lemma RUN_subset_DT: "A \<subseteq> B \<Longrightarrow> RUN B \<sqsubseteq>\<^sub>D\<^sub>T RUN A"

lemma rcut_eq: "rcut = tr0 \<longleftrightarrow> reg H tr0"

lemma iT_Div_not_empty: "I \<noteq> {} \<Longrightarrow> I \<oslash> k \<noteq> {}"

lemma PO_m1a_step3_refines_m1x_step3: "{R1x1a} (m1x_step3 Rs A B Kab), (m1a_step3 Rs A B Kab nls) {> R1x1a}"

lemma HFun_Sigma_subst [simp]: "(HFun_Sigma r)(i::=t) = HFun_Sigma (subst i t r)"

lemma root_in_start_points_2: assumes "backward_finite_path_root r x" and "start_points x \<noteq> 0" shows "r \<le> start_points x"

lemma fls_regpart_const [simp]: "fls_regpart (fls_const c) = fps_const c"

lemma blinfunpow_nonneg: assumes "\<And>v. 0 \<le> v \<Longrightarrow> 0 \<le> blinfun_apply (f :: ('b :: {ord, real_normed_vector} \<Rightarrow>\<^sub>L 'b)) v" shows "0 \<le> v \<Longrightarrow> 0 \<le> (f^^n) v"

lemma before_C_unique: assumes \<omega>: "before_C I1 \<omega>" "before_C I2 \<omega>" shows "I1 \<inter> I2 \<noteq> {}"

lemma permutes_empty [simp]: \<open>p permutes {} \<longleftrightarrow> p = id\<close>

lemma alwaysWFI: "\<turnstile> WF(A)_v \<longrightarrow> \<box>WF(A)_v"

lemma divide_poly_list[code]: "f div g = divide_poly_list f g"

lemma the_elemI: assumes "is_singleton A" shows "the_elem A \<in> A"

lemma deleteMin_correct_aux: assumes I: "invar q" assumes NE: "q\<noteq>[]" shows "invar (deleteMin q)" "queue_to_multiset_aux (deleteMin q) = queue_to_multiset_aux q - {# (findMin q, eprio (findMin q)) #}"

lemma assert_disch3 :" \<not> P \<sigma> \<Longrightarrow> \<not> (\<sigma> \<Turnstile> (assert\<^sub>S\<^sub>E P))"

lemma right_commute: "f (f a b) c = f (f a c) b"

lemma norm_pair_fst0[simp]: "norm (0, x) = norm x"

lemma finite_llast_V [simp]: "lfinite P \<Longrightarrow> llast P \<in> V"

lemma sup_apx_left_isotone_2: assumes "x \<sqsubseteq> y" shows "x \<squnion> z \<sqsubseteq> y \<squnion> z"

lemma unit_quasi_borel_terminal: "\<exists>! f. f \<in> X \<rightarrow>\<^sub>Q unit_quasi_borel"

lemma snd_sum_list: "snd (sum_list xs) = sum_list (map snd xs)"

lemma LIST_FRAG_DICHOTOMY: fixes l la x lb assumes "sublist l (la @ [x] @ lb)" "\<not>ListMem x l" shows "sublist l la \<or> sublist l lb"

lemma borel_measurable_power_ennreal [measurable (raw)]: fixes f :: "_ \<Rightarrow> ennreal" assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. (f x) ^ n) \<in> borel_measurable M"

lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = undefined"

lemma bdoubleton_eq_iff[simp]: "bdoubleton x y = bdoubleton z w \<longleftrightarrow> (x = z \<and> y = w \<or> x = w \<and> y = z)"

lemma clop_preorder: "clop f \<Longrightarrow> class.preorder (\<lambda>x y. f {x} \<subseteq> f {y}) (\<lambda>x y. f {x} \<subset> f {y})"

lemma (in ring) subdomain_iff: assumes "H \<subseteq> carrier R" shows "subdomain H R \<longleftrightarrow> domain (R \<lparr> carrier := H \<rparr>)"

lemma ok'_conv: "\<sigma>t OKAY' = true"

lemma kp_5x7_ul: "knights_path b5x7 kp5x7ul"

lemma VarP_sf [iff]: "Sigma_fm (VarP x)"

lemma normalize1_hd_root_eq[simp]: assumes "root t1 \<noteq> []" shows "hd (root (normalize1 t1)) = hd (root t1)"

lemma not_in_pow : "var\<notin>(vars(p::real mpoly)) \<Longrightarrow> var\<notin>(vars(p^i))"

lemma [code]: "enum 0 = {}" "enum (Suc n) = insert n (enum n)"

lemma path_decomp: assumes "path (xs @ ys)" shows "path xs" "path ys"

lemma exp_times_has_integral: "((\<lambda>t. exp (c * t)) has_integral (if c = 0 then t else exp (c * t) / c) - (if c = 0 then t0 else exp (c * t0) / c)) {t0 .. t}" if "t0 \<le> t" for c t::real

lemma (in bar) bar1: shows "(s\<langle>b:=True\<rangle>)\<cdot>c = s\<cdot>c"

lemma idempotent_subset: assumes "idempotent R" "S \<subseteq> R" shows "S O R \<subseteq> R" "R O S \<subseteq> R" "S O R O S \<subseteq> R"

lemma monad_state_alt_stateT' [locale_witness]: "monad_alt return (bind :: ('a \<times> 's, 'm) bind) alt \<Longrightarrow> monad_state_alt return (bind :: ('a, ('s, 'm) stateT) bind) (get :: ('s, ('s, 'm) stateT) get) put alt"

lemma Confidentiality_Kas_lemma [rule_format]: "\<lbrakk> authK \<in> symKeys; A \<notin> bad; evs \<in> kerbIV \<rbrakk> \<Longrightarrow> Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs \<longrightarrow> Key authK \<in> analz (spies evs) \<longrightarrow> expiredAK Ta evs"

lemma IF_alg_derivable_from_If: "\<lbrakk>(G,b,p):HS_B; (p,G,c1,H):HS; (p,G,c2,H):HS\<rbrakk> \<Longrightarrow> (p,G,Iff b c1 c2,H):HS"

lemma map_us_func1: "map_us id = (id::'a::complete_lattice_with_dual upset \<Rightarrow> 'a upset)"

lemma class_add_relevant_entries: "\<not> is_class P C \<Longrightarrow> set (relevant_entries P i pc xt) \<subseteq> set (relevant_entries (class_add P (C, cdec)) i pc xt)"

lemma "\<^bold>\<exists>\<pi>\<^sup>c = \<^bold>\<midarrow>(\<^bold>\<forall>\<pi>)"

theorem wf_termi_call_steps: "wf (termi_call_steps \<Gamma>)"

lemma enum_match_cong[sepref_frame_match_rules]: "\<lbrakk>\<And>x y. \<lbrakk>x\<in>set_enum e; y\<in>set_enum e'\<rbrakk> \<Longrightarrow> hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y\<rbrakk> \<Longrightarrow> hn_ctxt (enum_assn A) e e' \<Longrightarrow>\<^sub>t hn_ctxt (enum_assn A') e e'"

lemma dtree_from_list_root_r[simp]: "root (dtree_from_list r xs) = r"

lemma member_inv: assumes "vebt_member (Node (Some (mi, ma)) deg treeList summary) x " shows "deg \<ge> 2 \<and> (x = mi \<or> x = ma \<or> (x < ma \<and> x > mi \<and> high x (deg div 2) < length treeList \<and> vebt_member (treeList ! ( high x (deg div 2))) (low x (deg div 2))))"

lemma del_emp_concat: "concat us = concat (filter (\<lambda>x. x \<noteq> \<epsilon>) us)"

lemma llist_of_tllist_transfer2 [transfer_rule]: "(tllist_all2 A B ===> llist_all2 A) llist_of_tllist llist_of_tllist"

lemma diamond_n_an: "|n(x)>an(y) = n(x) * an(y)"

lemma path_entry_append: "\<lbrakk> path_entry E l v; (v,w)\<in>E \<rbrakk> \<Longrightarrow> path_entry E (v#l) w"

lemma prodrg_mOp:"\<forall>k\<in>I. Ring (A k) \<Longrightarrow> mop (prodrg I A) = prod_mOp I A"

lemma range_None: assumes "MaxR_opt A = None" shows "A \<inter> Field r = {}"

lemma Checkcast_correct: "\<lbrakk> wf_jvm_prog\<^bsub>\<Phi>\<^esub> P; P \<turnstile> C sees M:Ts\<rightarrow>T=(mxs,mxl\<^sub>0,ins,xt) in C; ins!pc = Checkcast D; P,T,mxs,size ins,xt \<turnstile> ins!pc,pc :: \<Phi> C M; Some \<sigma>' = exec (P, None, h, (stk,loc,C,M,pc)#frs) ; P,\<Phi> \<turnstile> (None, h, (stk,loc,C,M,pc)#frs)\<surd>; fst (exec_instr (ins!pc) P h stk loc C M pc frs) = None \<rbrakk> \<Longrightarrow> P,\<Phi> \<turnstile> \<sigma>'\<surd>"

lemma mangoldt_primepow' [simp]: "prime p \<Longrightarrow> k > 0 \<Longrightarrow> mangoldt (p ^ k) = of_real (ln (real p))"

lemma NSLIMSEQ_inverse_zero: "\<forall>y::real. \<exists>N. \<forall>n \<ge> N. y < f n \<Longrightarrow> (\<lambda>n. inverse (f n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S 0"

lemma pred_pmf_of_set [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> pred_pmf P (pmf_of_set A) = Ball A P"

lemma (in Group) rcs_Unit2:"\<lbrakk>G \<guillemotright> H; h \<in> H\<rbrakk> \<Longrightarrow> H \<bullet> h = H"

lemma fds_shift_inverse [simp]: "fds_shift (a :: 'a :: {field, nat_power}) (inverse f) = inverse (fds_shift a f)"

lemma subprob_space_bind: assumes "subprob_space M" "f \<in> measurable M (subprob_algebra N)" shows "subprob_space (M \<bind> f)"

lemma mapRuleI: "[| A = map f a; B = (map f) ` b |] ==> (A,B) = mapRule f (a,b)"