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

Statement: stringlengths 7 24.3k
lemma eqExcPID_N_step_\<phi>_imp: assumes ss1: "eqExcPID_N s s1" and step: "step s a = (ou,s')" and step1: "step s1 a = (ou1,s1')" and \<phi>: "\<phi> (Trans s a ou s')" shows "\<phi> (Trans s1 a ou1 s1')"
lemma blocks_index_simp_unique: "i1 < length \<B>s \<Longrightarrow> i2 < length \<B>s \<Longrightarrow> i1 \<noteq> i2 \<Longrightarrow> \<B>s ! i1 \<noteq> \<B>s ! i2"
lemma opar_unicastTE [elim]: "\<lbrakk>((\<sigma>, (s, t)), unicast i m, (\<sigma>', (s', t'))) \<in> oparp_sos i S T; \<lbrakk>((\<sigma>, s), unicast i m, (\<sigma>', s')) \<in> S; t' = t\<rbrakk> \<Longrightarrow> P; \<lbrakk>(t, unicast i m, t') \<in> T; s' = s; \<sigma>' i = \<sigma> i\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
lemma start_prog_Start_sees_start_method: "P \<turnstile> Object sees_methods Mm \<Longrightarrow> start_prog P C M \<turnstile> Start sees start_m, Static : []\<rightarrow>Void = (1, 0, [Invokestatic C M 0,Return], []) in Start"
lemma integer_less_eq_iff: "k \<le> l \<longleftrightarrow> int_of_integer k \<le> int_of_integer l"
lemma extend_invariant: "(extend h F \<in> invariant (extend_set h A)) = (F \<in> invariant A)"
lemma LIM_imp_LIM: fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector" assumes f: "f \<midarrow>a\<rightarrow> l" and le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)" shows "g \<midarrow>a\<rightarrow> m"
lemma ntcf_of_ntcf_arrow_components: shows "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTMap\<rparr> = \<NN>\<lparr>NTMap\<rparr>" and "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTDom\<rparr> = cf_of_cf_map \<AA> \<BB> (\<NN>\<lparr>NTDom\<rparr>)" and "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTCod\<rparr> = cf_of_cf_map \<AA> \<BB> (\<NN>\<lparr>NTCod\<rparr>)" and "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTDGDom\<rparr> = \<AA>" and "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTDGCod\<rparr> = \<BB>"
lemma sum_list_rmv1: "a \<in> set xs \<Longrightarrow> \<Sum>:(remove1 a xs) = \<Sum>:xs - (a :: 'a :: ab_group_add)"
lemma trms\<^sub>s\<^sub>s\<^sub>t_unlabel_suffix_subset: "trms\<^sub>s\<^sub>s\<^sub>t (unlabel B) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (unlabel (A@B))" "trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n B) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n (A@B))"
lemma lm04_15_register_masking: fixes c :: nat and l :: register and ic :: configuration and p :: program and q :: nat defines "b == B c" defines "d == D q c b" assumes cells_bounded: "cells_bounded ic p c" assumes l: "l < length (snd ic)" defines "r == RLe ic p b q" shows "r l \<preceq> d"
lemma (in Functor) CodFunctor: "f \<in> mor\<^bsub>CatDom F\<^esub> \<Longrightarrow> cod\<^bsub>CatCod F\<^esub> (F ## f) = F @@ (cod\<^bsub>CatDom F\<^esub> f)"
lemma mset_le_trans: "K < M \<Longrightarrow> M < N \<Longrightarrow> K < (N::'a::preorder multiset)"
lemma mono_id[simp, intro!]: "mono id" "mono (\<lambda>x. x)"
lemma ereal_inverse_antimono: fixes x y :: ereal shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> inverse y \<le> inverse x"
lemma minus_nonzero: "x : carrier G \<Longrightarrow> x \<noteq> \<zero> \<Longrightarrow> \<ominus> x \<noteq> \<zero>"
lemma listsp_mono [mono]: "A \<le> B \<Longrightarrow> listsp A \<le> listsp B"
theorem "(c,s) \<Rightarrow> t \<Longrightarrow> (c,s) \<Rightarrow> t' \<Longrightarrow> t' = t"
lemma int_nat_div: "(int a) div (int b) = int ((a::nat) div b)"
lemma range_to_fract_dvd_iff: assumes b: "b \<noteq> 0" shows "Fract a b \<in> range to_fract \<longleftrightarrow> b dvd a"
lemma cf_array_ArrMap_vsv: "vsv (cf_array \<BB> \<CC> \<DD> \<FF> \<GG>\<lparr>ArrMap\<rparr>)"
lemma [simp]: "p \<in> set ps"
lemma fun_rep_map2_rep[simp]: "f \<top> \<top> = \<top> \<Longrightarrow> fun_rep (map2_st_rep f ps1 ps2) = (\<lambda>x. f (fun_rep ps1 x) (fun_rep ps2 x))"
lemma f_bounded_below: assumes c': "c' > 0" obtains c where "\<And>x. x \<ge> x\<^sub>0 \<Longrightarrow> x \<le> x\<^sub>1 \<Longrightarrow> 2 * (c * f_approx x) \<le> f x" "c \<le> c'" "c > 0"
lemma conj_qf[simp]: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (conj p q)"
lemma Infinitesimal_FreeUltrafilterNat_iff: "(star_n X \<in> Infinitesimal) = (\<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U>)" (is "?lhs = ?rhs")
lemma set_tag_name_get_disconnected_nodes_is_l_set_tag_name_get_disconnected_nodes [instances]: "l_set_tag_name_get_disconnected_nodes type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs"
lemma proper_interval_integer_simps [code]: includes integer.lifting fixes x y :: integer and xo yo :: "integer option" shows "proper_interval (Some x) (Some y) = (1 < y - x)" "proper_interval None yo = True" "proper_interval xo None = True"
lemma suffix_zero: "\<sigma> \|\<^sub>s 0 = \<sigma>"
lemma Busemann_function_le_Gromov_product: "- Busemann_function_at xi y x/2 \<le> extended_Gromov_product_at x xi (to_Gromov_completion y)"
lemma update_assoc_list_with_default_key_not_found : assumes "distinct (map fst xys)" and "k \<notin> set (map fst xys)" shows "update_assoc_list_with_default k f d xys = xys @ [(k,f d)]"
lemma real_polynomial_function_imp_sum: assumes "real_polynomial_function f" shows "\<exists>a n::nat. f = (\<lambda>x. \<Sum>i\<le>n. a i * x^i)"
lemma fmdom'_fmupd[simp]: "fmdom' (fmupd a b m) = insert a (fmdom' m)"
lemma repv_selectlike_other: "x\<noteq>y \<Longrightarrow> (repv \<omega> x d \<in> selectlike X \<omega> {y}) = (repv \<omega> x d \<in> X)"
lemma Outpts_imp_knows_agents_secureM_sr: "\<lbrakk> Outpts (Card A) A X \<in> set evs; evs \<in> sr \<rbrakk> \<Longrightarrow> X \<in> knows A evs"
lemma fmpred_of_list[intro]: assumes "\<And>k v. (k, v) \<in> set xs \<Longrightarrow> P k v" shows "fmpred P (fmap_of_list xs)"
lemma PAR_least: assumes y: "y \<in> I x" shows "PAR x {p\<in>space (PAR x). t \<le> p y \<and> select_first x p y} = emeasure (exponential (escape_rate x)) {t ..} * ennreal (pmf (J x) y)"
lemma zero_lens_compat [simp]: "0\<^sub>L ##\<^sub>L X"
lemma continuous_on_sgn[continuous_intros]: fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0" shows "continuous_on s (\<lambda>x. sgn (f x))"
lemma content_division_of: assumes "K \<in> \<D>" "\<D> division_of S" shows "content K = (\<Prod>i \<in> Basis. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)"
lemma compare_lists_Cons [simp]: "a < b \<Longrightarrow> compare_lists (a # as) (b # bs) = LT" "a > b \<Longrightarrow> compare_lists (a # as) (b # bs) = GT" "a = b \<Longrightarrow> compare_lists (a # as) (b # bs) = compare_lists as bs"
lemma ordertype_nat_\<omega>: assumes "infinite N" shows "ordertype N less_than = \<omega>"
lemma Limsup_compose_continuous_mono: fixes f :: "'a::{complete_linorder, linorder_topology} \<Rightarrow> 'b::{complete_linorder, linorder_topology}" assumes c: "continuous_on UNIV f" and am: "mono f" and F: "F \<noteq> bot" shows "Limsup F (\<lambda>n. f (g n)) = f (Limsup F g)"
lemma lift_Pareto_SD_strict_iff: assumes "p \<in> lotteries_on alts" "q \<in> lotteries_on alts" shows "p \<prec>[Pareto(SD \<circ> R')] q \<longleftrightarrow> p \<prec>[Pareto(SD \<circ> R)] q"
lemma interior_of_Int: "X interior_of (S \<inter> T) = X interior_of S \<inter> X interior_of T" (is "?lhs = ?rhs")
lemma "\<exists>n. even n \<and> even (Suc n)"
lemma restrict_map_disj': "S \<inter> T = {} \<Longrightarrow> h \|` S \<bottom> h' \|` T"
lemma Ord_cases [cases type: hf, case_names 0 succ]: assumes Ok: "Ord(k)" obtains "k = 0" \| l where "Ord l" "succ l = k"
lemma idx_of_uniq: assumes p_disjoint_sym: "\<forall>i j v. i<length p \<and> j<length p \<and> v\<in>p!i \<and> v\<in>p!j \<longrightarrow> i=j" assumes A: "i<length p" "v\<in>p!i" shows "idx_of p v = i"
lemma null_strict[simp,code_unfold]: " f null = invalid"
lemma fun_lub_simps[simp]: "fun_lub lub {} = (\<lambda>x. lub {})" "fun_lub lub {f} = (\<lambda>x. lub {f x})"
lemma snd_readVariableRecursive: assumes "v \<in> vars g" "n \<in> set (\<alpha>n g)" "finite (Mapping.keys phis)" "\<And>n. (n,v) \<in> Mapping.keys phis \<Longrightarrow> length (predecessors g n) \<noteq> 1" "Mapping.lookup phis (Entry g,v) \<in> {None, Some []}" shows "phis'_aux g v {n} phis = snd (readVariableRecursive g v n phis)" "set ms \<subseteq> set (\<alpha>n g) \<Longrightarrow> (phis'_aux g v (set ms) phis, map (\<lambda>m. lookupDef g m v) ms) = readArgs g v n phis ms"
theorem safe: "s \<in> reach \<Longrightarrow> safe s r \<Longrightarrow> g \<in> isin s r \<Longrightarrow> owns s r = Some g"
lemma lg_safe: "lg 0 = 0" "lg (Suc 0) = 0" "lg (Suc (Suc 0)) = 1" "0 < x \<Longrightarrow> lg (x + x) = 1 + lg x"
lemma le_minus_power_downI: "0 \<le> x \<Longrightarrow> x ^ n \<le> - a \<Longrightarrow> a \<le> - power_down prec x n"
lemma select_fold_exec\<^sub>S\<^sub>e\<^sub>q: assumes "list_all (\<lambda>f. (\<tau> \<Turnstile> \<upsilon> f)) l" shows "\<lceil>\<lceil>Rep_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e (foldl UML_Sequence.OclIncluding Sequence{} l \<tau>)\<rceil>\<rceil> = List.map (\<lambda>f. f \<tau>) l"
lemma higher_id_iff: "higher p v = p \<longleftrightarrow> (p = 0 \<or> v \<prec>\<^sub>t tt p)" (is "?L \<longleftrightarrow> ?R")
lemma (in vec_space) rank_card_indpt: assumes "A \<in> carrier_mat n nc" assumes "maximal S (\<lambda>T. T \<subseteq> set (cols A) \<and> lin_indpt T)" shows "rank A = card S"
lemma poly_roots_eq_imp_eq: fixes p q :: "complex poly" assumes "Polynomial.lead_coeff p = Polynomial.lead_coeff q" assumes "poly_roots p = poly_roots q" shows "p = q"
lemma small_cfs[simp]: "small {\<FF>. \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>}"
lemma map_int_t[simp]: assumes Tl: "list_all Ik.Ik.wt Tl" and \<xi>: "Ik.wtE \<xi>" shows "map2 ntsem (map Ik.Ik.tpOf Tl) (map (Ik.int \<xi>) Tl) = map (TE.int (eenv \<xi>) \<circ> tT) Tl"
lemma ereal_between: fixes x e :: ereal assumes "\<bar>x\<bar> \<noteq> \<infinity>" and "0 < e" shows "x - e < x" and "x < x + e"
lemma frestriction_fconverse: "r\<inverse>\<^sub>\<bullet> \<restriction>\<^sub>\<bullet> A = (r \<restriction>\<^sub>\<bullet> A)\<inverse>\<^sub>\<bullet>"
lemma downset_iso_iff: "(\<down>x \<subseteq> \<down>y) = (x \<le> y)"
lemma mat_of_rows_index: "i < length rs \<Longrightarrow> j < n \<Longrightarrow> mat_of_rows n rs $$ (i,j) = rs ! i $ j"
lemma i_Exec_Stream_input_map: " i_Exec_Comp_Stream trans_fun (f \<circ> input) c = i_Exec_Comp_Stream (trans_fun \<circ> f) input c"
lemma lesssub_list_impl_same_size: "xs <_(Listn.le r) ys \<Longrightarrow> size ys = size xs"
lemma vars_of_subterm : assumes "x \<in> vars_of s" shows "\<And> t. subterm t p s \<Longrightarrow> x \<in> vars_of t"
lemma n_distrib_alt: "n x \<cdot> n z = n y \<cdot> n z \<Longrightarrow> n x \<oplus> n z = n y \<oplus> n z \<Longrightarrow> n x = n y"
lemma sorted_list_of_set_UN_lessThan: fixes k::nat assumes sm: "strict_mono_sets {..<k} A" and "\<And>i. i < k \<Longrightarrow> finite (A i)" shows "sorted_list_of_set (\<Union>i<k. A i) = concat (map (sorted_list_of_set \<circ> A) (sorted_list_of_set {..<k}))"
lemma chop_leq_max:"N_Chop(i,j,k) \<and> consec j k \<longrightarrow> (\<forall>n . n \<in> Rep_nat_int i \<and> n \<le> maximum j \<longrightarrow> n \<in> Rep_nat_int j)"
lemma dagger_of_id_is_id [simp]: "(1\<^sub>m n)\<^sup>\<dagger> = 1\<^sub>m n"
lemma list_set_insert[param]: assumes "y \<notin> Y" assumes "(x, y) \<in> A" "(xs, Y) \<in> \<langle>A\<rangle> list_set_rel" shows "(x # xs, insert y Y) \<in> \<langle>A\<rangle> list_set_rel"
lemma dbl_dec_neg_numeral: "Num.dbl_dec (- Num.numeral k) = - Num.numeral (Num.Bit1 k)"
lemma qp_Neg[dest]: "qp (Neg Q) \<Longrightarrow> False"
lemma eq_acom_iff_strip_anno: "C1=C2 \<longleftrightarrow> strip C1 = strip C2 \<and> (\<forall>p<size(annos C1). anno C1 p = anno C2 p)"
lemma extend_filter: "frequently P F \<Longrightarrow> inf F (principal {x. P x}) \<noteq> bot"
lemma maybe_arith_int_Not_Num: "(\<forall>n. maybe_arith_int f a1 a2 \<noteq> Some (Num n)) = (maybe_arith_int f a1 a2 = None)"
lemma WT_implies_WTrt: "P,E \<turnstile> e :: T \<Longrightarrow> P,E,h \<turnstile> e : T" and WTs_implies_WTrts: "P,E \<turnstile> es [::] Ts \<Longrightarrow> P,E,h \<turnstile> es [:] Ts"
lemma singleton_appends_dotP [simp]: "dim_vec x = dim_vec y \<Longrightarrow> (x @\<^sub>v [v]\<^sub>v) \<bullet> (y @\<^sub>v [w]\<^sub>v) = x \<bullet> y + v * w"
lemma real_root_commute: "root m (root n x) = root n (root m x)"
lemma val_size_mem_r: "(a, b) \<in> set t \<Longrightarrow> val_size b < fun_size t"
lemma terminates_strip_to_terminates: assumes termi_strip: "strip F \<Gamma>\<turnstile>c\<down>s" shows "\<Gamma>\<turnstile>c\<down>s"
lemma True_eqvt [eqvt]: shows "p \<bullet> True = True"
lemma weakSimE: fixes P :: ccs and Rel :: "(ccs \<times> ccs) set" and Q :: ccs and \<alpha> :: act and Q' :: ccs assumes "P \<leadsto><Rel> Q" and "Q \<longmapsto>\<alpha> \<prec> Q'" obtains P' where "P \<Longrightarrow>\<alpha> \<prec> P'" and "(P', Q') \<in> Rel"
lemma Sigma_triv: "(a, b) \<in> Sigma A B \<Longrightarrow> a \<in> A & b \<in> B a"
lemma [simp]: "a \<in> G \<Longrightarrow> Rep_G (Abs_G a) = a"
lemma circ_one_L: "1\<^sup>\<circ> = L \<squnion> 1"
lemma split_set_path_length:"{Cs. Subobjs P C Cs \<and> length Cs \<le> Suc(n)} = {Cs. Subobjs P C Cs \<and> length Cs \<le> n} \<union> {Cs. Subobjs P C Cs \<and> length Cs = Suc(n)}"
lemma lookup_monom_of_set: "Poly_Mapping.lookup (monom_of_set X) i = (if finite X \<and> i \<in> X then 1 else 0)"
lemma perp_in_col: assumes "X PerpAt A B C D" shows "Col A B X \<and> Col C D X"
lemma pred_prod_conj [simp]: shows pred_prod_conj1: "\<And>P Q R. pred_prod (\<lambda>x. P x \<and> Q x) R = (\<lambda>x. pred_prod P R x \<and> pred_prod Q R x)" and pred_prod_conj2: "\<And>P Q R. pred_prod P (\<lambda>x. Q x \<and> R x) = (\<lambda>x. pred_prod P Q x \<and> pred_prod P R x)"
lemma mono_D2: assumes ordered: "P \<sqsubseteq> Q" shows "(\<forall> s. s \<notin> D (P \<box> S) \<longrightarrow> Ra (P \<box> S) s = Ra (Q \<box> S) s)"
lemma LID\<^sub>LI [intro]: assumes "s = (\<sigma>, \<tau>, e)" shows "r \<in> LID\<^sub>S \<sigma> \<Longrightarrow> r \<in> LID\<^sub>L s" "r \<in> LID\<^sub>S \<tau> \<Longrightarrow> r \<in> LID\<^sub>L s" "r \<in> LID\<^sub>E e \<Longrightarrow> r \<in> LID\<^sub>L s"
lemma distincts_inj_on_set: assumes "distincts xss" shows "inj_on set (set xss)"
lemma lang_ENC_split: assumes "finite X" "X = Y1 \<union> Y2" "n = 0 \<or> (\<forall>p \<in> X. p < n)" shows "lang n (ENC n X) = lang n (ENC n Y1) \<inter> lang n (ENC n Y2)"
lemma ordertype_eqI: assumes "wf r" "total_on A r" "small A" "wf s" "bij_betw f A B" "(\<forall>x \<in> A. \<forall>y \<in> A. (f x, f y) \<in> s \<longleftrightarrow> (x,y) \<in> r)" shows "ordertype A r = ordertype B s"
lemma partial_fraction_decomposition: fixes ys :: "('a \<times> nat) list" defines "ys' \<equiv> map (\<lambda>(x,n). x ^ Suc n) ys :: 'a list" assumes unit: "\<And>y. y \<in> fst ` set ys \<Longrightarrow> is_unit (lift y)" assumes coprime: "pairwise coprime (set ys')" assumes distinct: "distinct ys'" assumes "partial_fraction_decomposition x ys = (a, zs)" shows "lift a + (\<Sum>i<length ys. \<Sum>j\<le>snd (ys!i). from_decomp (zs!i!j) (fst (ys!i)) (snd (ys!i)+1 - j)) = lift x div lift (prod_list ys')"
lemma Bossy_stable: shows "Bossy.stable_on ds X \<longleftrightarrow> X = {d1h1, d3h3}" (<) (is "?lhs = ?rhs")
lemma suc_least: "\<lbrakk>B \<le> Field r; a \<in> Field r; (\<And> b. b \<in> B \<Longrightarrow> a \<noteq> b \<and> (b,a) \<in> r)\<rbrakk> \<Longrightarrow> (suc B, a) \<in> r"

lemma eqExcPID_N_step_\<phi>_imp: assumes ss1: "eqExcPID_N s s1" and step: "step s a = (ou,s')" and step1: "step s1 a = (ou1,s1')" and \<phi>: "\<phi> (Trans s a ou s')" shows "\<phi> (Trans s1 a ou1 s1')"

lemma blocks_index_simp_unique: "i1 < length \s \<Longrightarrow> i2 < length \s \<Longrightarrow> i1 \<noteq> i2 \<Longrightarrow> \s ! i1 \<noteq> \s ! i2"

lemma opar_unicastTE [elim]: "\<lbrakk>((\<sigma>, (s, t)), unicast i m, (\<sigma>', (s', t'))) \<in> oparp_sos i S T; \<lbrakk>((\<sigma>, s), unicast i m, (\<sigma>', s')) \<in> S; t' = t\<rbrakk> \<Longrightarrow> P; \<lbrakk>(t, unicast i m, t') \<in> T; s' = s; \<sigma>' i = \<sigma> i\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"

lemma start_prog_Start_sees_start_method: "P \<turnstile> Object sees_methods Mm \<Longrightarrow> start_prog P C M \<turnstile> Start sees start_m, Static : []\<rightarrow>Void = (1, 0, [Invokestatic C M 0,Return], []) in Start"

lemma integer_less_eq_iff: "k \<le> l \<longleftrightarrow> int_of_integer k \<le> int_of_integer l"

lemma extend_invariant: "(extend h F \<in> invariant (extend_set h A)) = (F \<in> invariant A)"

lemma LIM_imp_LIM: fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector" assumes f: "f \<midarrow>a\<rightarrow> l" and le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)" shows "g \<midarrow>a\<rightarrow> m"

lemma ntcf_of_ntcf_arrow_components: shows "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTMap\<rparr> = \<NN>\<lparr>NTMap\<rparr>" and "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTDom\<rparr> = cf_of_cf_map \<AA> \<BB> (\<NN>\<lparr>NTDom\<rparr>)" and "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTCod\<rparr> = cf_of_cf_map \<AA> \<BB> (\<NN>\<lparr>NTCod\<rparr>)" and "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTDGDom\<rparr> = \<AA>" and "ntcf_of_ntcf_arrow \<AA> \<BB> \<NN>\<lparr>NTDGCod\<rparr> = \<BB>"

lemma sum_list_rmv1: "a \<in> set xs \<Longrightarrow> \<Sum>:(remove1 a xs) = \<Sum>:xs - (a :: 'a :: ab_group_add)"

lemma trms\<^sub>s\<^sub>s\<^sub>t_unlabel_suffix_subset: "trms\<^sub>s\<^sub>s\<^sub>t (unlabel B) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (unlabel (A@B))" "trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n B) \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t (proj_unl n (A@B))"

lemma lm04_15_register_masking: fixes c :: nat and l :: register and ic :: configuration and p :: program and q :: nat defines "b == B c" defines "d == D q c b" assumes cells_bounded: "cells_bounded ic p c" assumes l: "l < length (snd ic)" defines "r == RLe ic p b q" shows "r l \<preceq> d"

lemma (in Functor) CodFunctor: "f \<in> mor\<^bsub>CatDom F\<^esub> \<Longrightarrow> cod\<^bsub>CatCod F\<^esub> (F ## f) = F @@ (cod\<^bsub>CatDom F\<^esub> f)"

lemma mset_le_trans: "K < M \<Longrightarrow> M < N \<Longrightarrow> K < (N::'a::preorder multiset)"

lemma mono_id[simp, intro!]: "mono id" "mono (\<lambda>x. x)"

lemma ereal_inverse_antimono: fixes x y :: ereal shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> inverse y \<le> inverse x"

lemma minus_nonzero: "x : carrier G \<Longrightarrow> x \<noteq> \<zero> \<Longrightarrow> \<ominus> x \<noteq> \<zero>"

lemma listsp_mono [mono]: "A \<le> B \<Longrightarrow> listsp A \<le> listsp B"

theorem "(c,s) \<Rightarrow> t \<Longrightarrow> (c,s) \<Rightarrow> t' \<Longrightarrow> t' = t"

lemma int_nat_div: "(int a) div (int b) = int ((a::nat) div b)"

lemma range_to_fract_dvd_iff: assumes b: "b \<noteq> 0" shows "Fract a b \<in> range to_fract \<longleftrightarrow> b dvd a"

lemma cf_array_ArrMap_vsv: "vsv (cf_array \<BB> \<CC> \<DD> \<FF> \<GG>\<lparr>ArrMap\<rparr>)"

lemma [simp]: "p \<in> set ps"

lemma fun_rep_map2_rep[simp]: "f \<top> \<top> = \<top> \<Longrightarrow> fun_rep (map2_st_rep f ps1 ps2) = (\<lambda>x. f (fun_rep ps1 x) (fun_rep ps2 x))"

lemma f_bounded_below: assumes c': "c' > 0" obtains c where "\<And>x. x \<ge> x\<^sub>0 \<Longrightarrow> x \<le> x\<^sub>1 \<Longrightarrow> 2 * (c * f_approx x) \<le> f x" "c \<le> c'" "c > 0"

lemma conj_qf[simp]: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (conj p q)"

lemma Infinitesimal_FreeUltrafilterNat_iff: "(star_n X \<in> Infinitesimal) = (\<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \)" (is "?lhs = ?rhs")

lemma set_tag_name_get_disconnected_nodes_is_l_set_tag_name_get_disconnected_nodes [instances]: "l_set_tag_name_get_disconnected_nodes type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs"

lemma proper_interval_integer_simps [code]: includes integer.lifting fixes x y :: integer and xo yo :: "integer option" shows "proper_interval (Some x) (Some y) = (1 < y - x)" "proper_interval None yo = True" "proper_interval xo None = True"

lemma suffix_zero: "\<sigma> |\<^sub>s 0 = \<sigma>"

lemma Busemann_function_le_Gromov_product: "- Busemann_function_at xi y x/2 \<le> extended_Gromov_product_at x xi (to_Gromov_completion y)"

lemma update_assoc_list_with_default_key_not_found : assumes "distinct (map fst xys)" and "k \<notin> set (map fst xys)" shows "update_assoc_list_with_default k f d xys = xys @ [(k,f d)]"

lemma real_polynomial_function_imp_sum: assumes "real_polynomial_function f" shows "\<exists>a n::nat. f = (\<lambda>x. \<Sum>i\<le>n. a i * x^i)"

lemma fmdom'_fmupd[simp]: "fmdom' (fmupd a b m) = insert a (fmdom' m)"

lemma repv_selectlike_other: "x\<noteq>y \<Longrightarrow> (repv \<omega> x d \<in> selectlike X \<omega> {y}) = (repv \<omega> x d \<in> X)"

lemma Outpts_imp_knows_agents_secureM_sr: "\<lbrakk> Outpts (Card A) A X \<in> set evs; evs \<in> sr \<rbrakk> \<Longrightarrow> X \<in> knows A evs"

lemma fmpred_of_list[intro]: assumes "\<And>k v. (k, v) \<in> set xs \<Longrightarrow> P k v" shows "fmpred P (fmap_of_list xs)"

lemma PAR_least: assumes y: "y \<in> I x" shows "PAR x {p\<in>space (PAR x). t \<le> p y \<and> select_first x p y} = emeasure (exponential (escape_rate x)) {t ..} * ennreal (pmf (J x) y)"

lemma zero_lens_compat [simp]: "0\<^sub>L ##\<^sub>L X"

lemma continuous_on_sgn[continuous_intros]: fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0" shows "continuous_on s (\<lambda>x. sgn (f x))"

lemma content_division_of: assumes "K \<in> \<D>" "\<D> division_of S" shows "content K = (\<Prod>i \<in> Basis. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)"

lemma compare_lists_Cons [simp]: "a compare_lists (a # as) (b # bs) = LT" "a > b \<Longrightarrow> compare_lists (a # as) (b # bs) = GT" "a = b \<Longrightarrow> compare_lists (a # as) (b # bs) = compare_lists as bs"

lemma ordertype_nat_\<omega>: assumes "infinite N" shows "ordertype N less_than = \<omega>"

lemma Limsup_compose_continuous_mono: fixes f :: "'a::{complete_linorder, linorder_topology} \<Rightarrow> 'b::{complete_linorder, linorder_topology}" assumes c: "continuous_on UNIV f" and am: "mono f" and F: "F \<noteq> bot" shows "Limsup F (\<lambda>n. f (g n)) = f (Limsup F g)"

lemma lift_Pareto_SD_strict_iff: assumes "p \<in> lotteries_on alts" "q \<in> lotteries_on alts" shows "p \<prec>[Pareto(SD \<circ> R')] q \<longleftrightarrow> p \<prec>[Pareto(SD \<circ> R)] q"

lemma interior_of_Int: "X interior_of (S \<inter> T) = X interior_of S \<inter> X interior_of T" (is "?lhs = ?rhs")

lemma "\<exists>n. even n \<and> even (Suc n)"

lemma restrict_map_disj': "S \<inter> T = {} \<Longrightarrow> h |` S \<bottom> h' |` T"

lemma Ord_cases [cases type: hf, case_names 0 succ]: assumes Ok: "Ord(k)" obtains "k = 0" | l where "Ord l" "succ l = k"

lemma idx_of_uniq: assumes p_disjoint_sym: "\<forall>i j v. i<length p \<and> j<length p \<and> v\<in>p!i \<and> v\<in>p!j \<longrightarrow> i=j" assumes A: "i<length p" "v\<in>p!i" shows "idx_of p v = i"

lemma null_strict[simp,code_unfold]: " f null = invalid"

lemma fun_lub_simps[simp]: "fun_lub lub {} = (\<lambda>x. lub {})" "fun_lub lub {f} = (\<lambda>x. lub {f x})"

lemma snd_readVariableRecursive: assumes "v \<in> vars g" "n \<in> set (\<alpha>n g)" "finite (Mapping.keys phis)" "\<And>n. (n,v) \<in> Mapping.keys phis \<Longrightarrow> length (predecessors g n) \<noteq> 1" "Mapping.lookup phis (Entry g,v) \<in> {None, Some []}" shows "phis'_aux g v {n} phis = snd (readVariableRecursive g v n phis)" "set ms \<subseteq> set (\<alpha>n g) \<Longrightarrow> (phis'_aux g v (set ms) phis, map (\<lambda>m. lookupDef g m v) ms) = readArgs g v n phis ms"

theorem safe: "s \<in> reach \<Longrightarrow> safe s r \<Longrightarrow> g \<in> isin s r \<Longrightarrow> owns s r = Some g"

lemma lg_safe: "lg 0 = 0" "lg (Suc 0) = 0" "lg (Suc (Suc 0)) = 1" "0 < x \<Longrightarrow> lg (x + x) = 1 + lg x"

lemma le_minus_power_downI: "0 \<le> x \<Longrightarrow> x ^ n \<le> - a \<Longrightarrow> a \<le> - power_down prec x n"

lemma select_fold_exec\<^sub>S\<^sub>e\<^sub>q: assumes "list_all (\<lambda>f. (\<tau> \<Turnstile> \<upsilon> f)) l" shows "\<lceil>\<lceil>Rep_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e (foldl UML_Sequence.OclIncluding Sequence{} l \<tau>)\<rceil>\<rceil> = List.map (\<lambda>f. f \<tau>) l"

lemma higher_id_iff: "higher p v = p \<longleftrightarrow> (p = 0 \<or> v \<prec>\<^sub>t tt p)" (is "?L \<longleftrightarrow> ?R")

lemma (in vec_space) rank_card_indpt: assumes "A \<in> carrier_mat n nc" assumes "maximal S (\<lambda>T. T \<subseteq> set (cols A) \<and> lin_indpt T)" shows "rank A = card S"

lemma poly_roots_eq_imp_eq: fixes p q :: "complex poly" assumes "Polynomial.lead_coeff p = Polynomial.lead_coeff q" assumes "poly_roots p = poly_roots q" shows "p = q"

lemma small_cfs[simp]: "small {\<FF>. \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>}"

lemma map_int_t[simp]: assumes Tl: "list_all Ik.Ik.wt Tl" and \<xi>: "Ik.wtE \<xi>" shows "map2 ntsem (map Ik.Ik.tpOf Tl) (map (Ik.int \<xi>) Tl) = map (TE.int (eenv \<xi>) \<circ> tT) Tl"

lemma ereal_between: fixes x e :: ereal assumes "\<bar>x\<bar> \<noteq> \<infinity>" and "0 < e" shows "x - e < x" and "x < x + e"

lemma frestriction_fconverse: "r\<inverse>\<^sub>\<bullet> \<restriction>\<^sub>\<bullet> A = (r \<restriction>\<^sub>\<bullet> A)\<inverse>\<^sub>\<bullet>"

lemma downset_iso_iff: "(\<down>x \<subseteq> \<down>y) = (x \<le> y)"

lemma mat_of_rows_index: "i < length rs \<Longrightarrow> j < n \<Longrightarrow> mat_of_rows n rs $$ (i,j) = rs ! i $ j"

lemma i_Exec_Stream_input_map: " i_Exec_Comp_Stream trans_fun (f \<circ> input) c = i_Exec_Comp_Stream (trans_fun \<circ> f) input c"

lemma lesssub_list_impl_same_size: "xs <_(Listn.le r) ys \<Longrightarrow> size ys = size xs"

lemma vars_of_subterm : assumes "x \<in> vars_of s" shows "\<And> t. subterm t p s \<Longrightarrow> x \<in> vars_of t"

lemma n_distrib_alt: "n x \<cdot> n z = n y \<cdot> n z \<Longrightarrow> n x \<oplus> n z = n y \<oplus> n z \<Longrightarrow> n x = n y"

lemma sorted_list_of_set_UN_lessThan: fixes k::nat assumes sm: "strict_mono_sets {..<k} A" and "\<And>i. i < k \<Longrightarrow> finite (A i)" shows "sorted_list_of_set (\<Union>i<k. A i) = concat (map (sorted_list_of_set \<circ> A) (sorted_list_of_set {..<k}))"

lemma chop_leq_max:"N_Chop(i,j,k) \<and> consec j k \<longrightarrow> (\<forall>n . n \<in> Rep_nat_int i \<and> n \<le> maximum j \<longrightarrow> n \<in> Rep_nat_int j)"

lemma dagger_of_id_is_id [simp]: "(1\<^sub>m n)\<^sup>\<dagger> = 1\<^sub>m n"

lemma list_set_insert[param]: assumes "y \<notin> Y" assumes "(x, y) \<in> A" "(xs, Y) \<in> \<langle>A\<rangle> list_set_rel" shows "(x # xs, insert y Y) \<in> \<langle>A\<rangle> list_set_rel"

lemma dbl_dec_neg_numeral: "Num.dbl_dec (- Num.numeral k) = - Num.numeral (Num.Bit1 k)"

lemma qp_Neg[dest]: "qp (Neg Q) \<Longrightarrow> False"

lemma eq_acom_iff_strip_anno: "C1=C2 \<longleftrightarrow> strip C1 = strip C2 \<and> (\<forall>p<size(annos C1). anno C1 p = anno C2 p)"

lemma extend_filter: "frequently P F \<Longrightarrow> inf F (principal {x. P x}) \<noteq> bot"

lemma maybe_arith_int_Not_Num: "(\<forall>n. maybe_arith_int f a1 a2 \<noteq> Some (Num n)) = (maybe_arith_int f a1 a2 = None)"

lemma WT_implies_WTrt: "P,E \<turnstile> e :: T \<Longrightarrow> P,E,h \<turnstile> e : T" and WTs_implies_WTrts: "P,E \<turnstile> es [::] Ts \<Longrightarrow> P,E,h \<turnstile> es [:] Ts"

lemma singleton_appends_dotP [simp]: "dim_vec x = dim_vec y \<Longrightarrow> (x @\<^sub>v [v]\<^sub>v) \<bullet> (y @\<^sub>v [w]\<^sub>v) = x \<bullet> y + v * w"

lemma real_root_commute: "root m (root n x) = root n (root m x)"

lemma val_size_mem_r: "(a, b) \<in> set t \<Longrightarrow> val_size b < fun_size t"

lemma terminates_strip_to_terminates: assumes termi_strip: "strip F \<Gamma>\<turnstile>c\<down>s" shows "\<Gamma>\<turnstile>c\<down>s"

lemma True_eqvt [eqvt]: shows "p \<bullet> True = True"

lemma weakSimE: fixes P :: ccs and Rel :: "(ccs \<times> ccs) set" and Q :: ccs and \<alpha> :: act and Q' :: ccs assumes "P \<leadsto><Rel> Q" and "Q \<longmapsto>\<alpha> \<prec> Q'" obtains P' where "P \<Longrightarrow>\<alpha> \<prec> P'" and "(P', Q') \<in> Rel"

lemma Sigma_triv: "(a, b) \<in> Sigma A B \<Longrightarrow> a \<in> A & b \<in> B a"

lemma [simp]: "a \<in> G \<Longrightarrow> Rep_G (Abs_G a) = a"

lemma circ_one_L: "1\<^sup>\<circ> = L \<squnion> 1"

lemma split_set_path_length:"{Cs. Subobjs P C Cs \<and> length Cs \<le> Suc(n)} = {Cs. Subobjs P C Cs \<and> length Cs \<le> n} \<union> {Cs. Subobjs P C Cs \<and> length Cs = Suc(n)}"

lemma lookup_monom_of_set: "Poly_Mapping.lookup (monom_of_set X) i = (if finite X \<and> i \<in> X then 1 else 0)"

lemma perp_in_col: assumes "X PerpAt A B C D" shows "Col A B X \<and> Col C D X"

lemma pred_prod_conj [simp]: shows pred_prod_conj1: "\<And>P Q R. pred_prod (\<lambda>x. P x \<and> Q x) R = (\<lambda>x. pred_prod P R x \<and> pred_prod Q R x)" and pred_prod_conj2: "\<And>P Q R. pred_prod P (\<lambda>x. Q x \<and> R x) = (\<lambda>x. pred_prod P Q x \<and> pred_prod P R x)"

lemma mono_D2: assumes ordered: "P \<sqsubseteq> Q" shows "(\<forall> s. s \<notin> D (P \<box> S) \<longrightarrow> Ra (P \<box> S) s = Ra (Q \<box> S) s)"

lemma LID\<^sub>LI [intro]: assumes "s = (\<sigma>, \<tau>, e)" shows "r \<in> LID\<^sub>S \<sigma> \<Longrightarrow> r \<in> LID\<^sub>L s" "r \<in> LID\<^sub>S \<tau> \<Longrightarrow> r \<in> LID\<^sub>L s" "r \<in> LID\<^sub>E e \<Longrightarrow> r \<in> LID\<^sub>L s"

lemma distincts_inj_on_set: assumes "distincts xss" shows "inj_on set (set xss)"

lemma lang_ENC_split: assumes "finite X" "X = Y1 \<union> Y2" "n = 0 \<or> (\<forall>p \<in> X. p < n)" shows "lang n (ENC n X) = lang n (ENC n Y1) \<inter> lang n (ENC n Y2)"

lemma ordertype_eqI: assumes "wf r" "total_on A r" "small A" "wf s" "bij_betw f A B" "(\<forall>x \<in> A. \<forall>y \<in> A. (f x, f y) \<in> s \<longleftrightarrow> (x,y) \<in> r)" shows "ordertype A r = ordertype B s"

lemma partial_fraction_decomposition: fixes ys :: "('a \<times> nat) list" defines "ys' \<equiv> map (\<lambda>(x,n). x ^ Suc n) ys :: 'a list" assumes unit: "\<And>y. y \<in> fst ` set ys \<Longrightarrow> is_unit (lift y)" assumes coprime: "pairwise coprime (set ys')" assumes distinct: "distinct ys'" assumes "partial_fraction_decomposition x ys = (a, zs)" shows "lift a + (\<Sum>i<length ys. \<Sum>j\<le>snd (ys!i). from_decomp (zs!i!j) (fst (ys!i)) (snd (ys!i)+1 - j)) = lift x div lift (prod_list ys')"

lemma Bossy_stable: shows "Bossy.stable_on ds X \<longleftrightarrow> X = {d1h1, d3h3}" (*<*) (is "?lhs = ?rhs")

lemma suc_least: "\<lbrakk>B \<le> Field r; a \<in> Field r; (\<And> b. b \<in> B \<Longrightarrow> a \<noteq> b \<and> (b,a) \<in> r)\<rbrakk> \<Longrightarrow> (suc B, a) \<in> r"