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

Statement: stringlengths 7 24.3k
lemma (in semiring_1) last_linear_mul: assumes p: "p \<noteq> []" shows "last ([a, 1] *** p) = last p"
lemma llist_of_tllist_tappend: "llist_of_tllist (tappend xs f) = lappend (llist_of_tllist xs) (llist_of_tllist (f (terminal xs)))"
lemma complete_digraph_pair_def: "K\<^bsub>n\<^esub> (with_proj G) \<longleftrightarrow> finite (pverts G) \<and> card (pverts G) = n \<and> parcs G = {(u,v). (u,v) \<in> (pverts G \<times> pverts G) \<and> u \<noteq> v}" (is "_ = ?R")
lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
lemma add_leaf_is_leaf: assumes "T' = \<lparr>verts = V, arcs = A, tail = t, head = h\<rparr>" and "T = \<lparr>verts = V \<union> {v}, arcs = A \<union> {a}, tail = t(a := u), head = h(a := v)\<rparr>" and "u \<in> V" and "v \<notin> V" and "a \<notin> A" and "directed_tree T' root'" shows "leaf v"
lemma upright_obtain_support: assumes "upright a" and "zcount a t > 0" obtains s where "s < t" "zcount a s < 0" "nonpos_upto a s"
lemma SA: "Fr_1b \<F> \<Longrightarrow> Fr_2 \<F> \<Longrightarrow> Fr_3 \<F> \<Longrightarrow> \<forall>a b c. (a \<^bold>\<Rightarrow> c) \<^bold>\<Rightarrow> ((a \<^bold>\<and> b) \<^bold>\<Rightarrow> c) \<^bold>\<approx> \<^bold>\<top>"
lemma m2pi_less_pi: "- (2*pi) < pi"
lemma interesting_subst1: assumes a: "x\<noteq>y" "x\<sharp>P" "y\<sharp>P" shows "N{y:=<c>.P}{x:=<c>.P} = N{x:=<c>.Ax y c}{y:=<c>.P}"
lemma E_thm13: "Lift \<in> (moving \<inter> Req n \<inter> {s. metric n s = N} \<inter> {s. floor s \<in> req s}) LeadsTo (stopped \<inter> Req n \<inter> {s. metric n s = N} \<inter> {s. floor s \<in> req s})"
lemma sdp_SetPC: fixes p::"'a \<Rightarrow> 's prog" assumes sdp: "\<And>s a. a \<in> supp (P s) \<Longrightarrow> sub_distrib_pconj (p a)" and fin: "\<And>s. finite (supp (P s))" and nnp: "\<And>s a. 0 \<le> P s a" and sub: "\<And>s. sum (P s) (supp (P s)) \<le> 1" shows "sub_distrib_pconj (SetPC p P)"
lemma K_in_space[simp]: "K x \<in> space (prob_algebra S)"
lemma sumResRight: fixes x :: name and P :: pi and Q :: pi assumes Id: "Id \<subseteq> Rel" and Eqvt: "eqvt Rel" shows "<\<nu>x>(P \<oplus> Q) \<leadsto>[Rel] (<\<nu>x>P) \<oplus> (<\<nu>x>Q)"
lemma cat_comma_Hom_def': "cat_comma_Hom \<GG> \<HH> A B \<equiv> set { [A, B, [g, h]\<^sub>\<circ>]\<^sub>\<circ> \| g h. A \<in>\<^sub>\<circ> cat_comma_Obj \<GG> \<HH> \<and> B \<in>\<^sub>\<circ> cat_comma_Obj \<GG> \<HH> \<and> g : A\<lparr>0\<rparr> \<mapsto>\<^bsub>\<AA>\<^esub> B\<lparr>0\<rparr> \<and> h : A\<lparr>1\<^sub>\<nat>\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> B\<lparr>1\<^sub>\<nat>\<rparr> \<and> B\<lparr>2\<^sub>\<nat>\<rparr> \<circ>\<^sub>A\<^bsub>\<CC>\<^esub> \<GG>\<lparr>ArrMap\<rparr>\<lparr>g\<rparr> = \<HH>\<lparr>ArrMap\<rparr>\<lparr>h\<rparr> \<circ>\<^sub>A\<^bsub>\<CC>\<^esub> A\<lparr>2\<^sub>\<nat>\<rparr> }"
lemma ni_thin_locs_keep_atE: "\<lbrakk>at proc l s \<longrightarrow> P; AT s' = (AT s)(p := lfn, q := lfn'); at proc l s'; proc \<noteq> p; proc \<noteq> q; P \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" "\<lbrakk>at proc l s \<longrightarrow> P; AT s' = (AT s)(p := lfn); at proc l s'; proc \<noteq> p; P \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
lemma lzip_ltakeWhile_snd: "lzip xs (ltakeWhile P ys) = ltakeWhile (P \<circ> snd) (lzip xs ys)"
lemma stable_pair_on_allocation: assumes "stable_pair_on ds XD_XH" shows "allocation (match XD_XH)"
lemma mult_left_isotone: "x \<le> y \<Longrightarrow> x * z \<le> y * z"
lemma \<phi>_zero: "x \<notin> {real_of_int (i - 1) / 2^(l + 1) <..< real_of_int (i + 1) / 2^(l + 1)} \<Longrightarrow> \<phi> (l,i) x = 0"
lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {\|\|}"
lemma product_run: assumes "length rs = length AA" "length ps = length AA" assumes "b.run (product AA) (w \|\|\| stranspose rs) ps" shows "k < length AA \<Longrightarrow> a.run (AA ! k) (w \|\|\| rs ! k) (ps ! k)"
lemma lemma_2_9_i1: "b \<in> supremum A \<Longrightarrow> b * a \<in> supremum ((\<lambda> x . x * a) ` A)"
lemma eval_v_restrict: fixes c::v and B::"bv fset" assumes "i = i' \|` d" and "supp c \<subseteq> atom ` d \<union> supp B " and "i' \<lbrakk> c \<rbrakk> ~ s" shows "i \<lbrakk> c \<rbrakk> ~ s"
lemma eval_monom_mult[simp]: "eval_monom \<alpha> (m * n) = eval_monom \<alpha> m * eval_monom \<alpha> n"
lemma build_append[simp]: "(w @ a # u) \<frown> v = w \<frown> a ## u \<frown> v"
lemma ANR_imp_closed_neighbourhood_retract: fixes S :: "'a::euclidean_space set" assumes "ANR S" "closedin (top_of_set U) S" obtains V W where "openin (top_of_set U) V" "closedin (top_of_set U) W" "S \<subseteq> V" "V \<subseteq> W" "S retract_of W"
lemma t_buildup_cnt: "T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p n \<le> cnt (vebt_buildup n) * 13"
lemma of_nat_monomial: "of_nat p = monomial p 0"
lemma bconfs_map_throw[iff]: "P,sh \<turnstile>\<^sub>b (map Val vs @ throw e # es',b) \<surd> \<longleftrightarrow> P,sh \<turnstile>\<^sub>b (e,b) \<surd>"
lemma unique_awalk_All: "\<exists>p. awalk u p v \<Longrightarrow> \<exists>!p. awalk u p v"
lemma infinite_part1: "infinite V \<Longrightarrow> infinite (part1 V)"
lemma algebraic_power_iff [simp]: assumes "n > 0" shows "algebraic (x ^ n) \<longleftrightarrow> algebraic x"
lemma mult_col_div_row_similar: assumes A: "A \<in> carrier_mat n n" and ak: "k < n" "a \<noteq> 0" shows "similar_mat (mult_col_div_row a k A) A"
lemma col_elems_ss01: assumes "j < dim_col M" shows "vec_set (col M j) \<subseteq> {0, 1}"
lemma in_interval_interval_bij: fixes a b u v x :: "'a::euclidean_space" assumes "x \<in> cbox a b" and "cbox u v \<noteq> {}" shows "interval_bij (a, b) (u, v) x \<in> cbox u v"
lemma memberid_pair_simp1: "memberid p = memberid (snd p)"
lemma cong: "(\<And>s. P s = P' s) \<Longrightarrow> \<lceil>P\<rceil> = \<lceil>P'\<rceil>" "(\<And>\<sigma>. P \<sigma> = P' \<sigma>) \<Longrightarrow> (\<box>P) = (\<box>P')" "(\<And>\<sigma>. P \<sigma> = P' \<sigma>) \<Longrightarrow> (\<diamond>P) = (\<diamond>P')" "(\<And>\<sigma>. P \<sigma> = P' \<sigma>) \<Longrightarrow> (\<circle>P) = (\<circle>P')" "\<lbrakk>\<And>\<sigma>. P \<sigma> = P' \<sigma>; \<And>\<sigma>. Q \<sigma> = Q' \<sigma>\<rbrakk> \<Longrightarrow> (P \<U> Q) = (P' \<U> Q')" "\<lbrakk>\<And>\<sigma>. P \<sigma> = P' \<sigma>; \<And>\<sigma>. Q \<sigma> = Q' \<sigma>\<rbrakk> \<Longrightarrow> (P \<W> Q) = (P' \<W> Q')"
theorem "DA.accepts (na2da(rexp2na r)) w = (w : lang r)"
lemma cblinfun_power_Suc: \<open>cblinfun_power A (Suc n) = cblinfun_power A n o\<^sub>C\<^sub>L A\<close>
lemma [quot_preserve]: assumes a: "Quotient3 R abs1 rep1" shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
lemma hd_lq_conv_nth: assumes "u <p v" shows "hd(u\<inverse>\<^sup>>v) = v!\<^bold>\|u\<^bold>\|"
lemma par_qmsg_oreachable_statelessassm: assumes "(\<sigma>, \<zeta>) \<in> oreachable (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg) (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>) (other (\<lambda>_ _. True) {i})" and ustutter: "\<And>\<xi>. U \<xi> \<xi>" shows "(\<sigma>, fst \<zeta>) \<in> oreachable A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>) (other (\<lambda>_ _. True) {i}) \<and> snd \<zeta> \<in> reachable qmsg (recvmsg R) \<and> (\<forall>m\<in>set (fst (snd \<zeta>)). R m)"
lemma (in pseudo_hoop_lattice_a) supremum_pair [simp]: "supremum {a, b} = {a \<squnion> b}"
lemma foundation20 : "\<tau> \<Turnstile> (\<delta> X) \<Longrightarrow> \<tau> \<Turnstile> \<upsilon> X"
lemma ta_det'_ground_id: "t \|\<in>\| ta_der' \<A> s \<Longrightarrow> ground t \<Longrightarrow> t = s"
lemma ces_append_one : "ces v1 (es @ [e]) v2 subs = (ces v1 es (src e) subs \<and> ces (src e) [e] v2 subs)"
lemma less_eq_bot_Bot_is_Bot: "x \<le> Bot \<Longrightarrow> x = Bot"
lemma step_augmenter_eval : assumes steph : "\<And>xs var L F \<Gamma>. length xs = var \<Longrightarrow> (\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (step var L F) (xs @ x # \<Gamma>))" assumes heuristic: "\<And>n var L F. heuristic n L F = var \<Longrightarrow> var \<le> n" shows "\<And>var amount L F \<Gamma>. amount \<le> var + 1 \<Longrightarrow> (\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. (length xs = (var + 1)) \<and> eval (step_augment step heuristic amount var L F) (xs @ \<Gamma>))"
lemma compatible_refl: assumes coms_secure: "list_all com_sifum_secure cmds" assumes low_eq: "mem\<^sub>1 =\<^sup>l mem\<^sub>2" shows "makes_compatible (cmds, mem\<^sub>1) (cmds, mem\<^sub>2) (replicate (length cmds) (mem\<^sub>1, mem\<^sub>2))"
lemma of_class_linorder: \<open>OFCLASS('a, linorder_class)\<close>
lemma heap_clone_NewHeapElemD: assumes "heap_clone P h a h' \<lfloor>(obs, a')\<rfloor>" and "ad \<in> allocated h'" and "ad \<notin> allocated h" shows "\<exists>CTn. NewHeapElem ad CTn \<in> set obs"
lemma iMODb_div_self: " 0 < m \<Longrightarrow> [r, mod m, c] \<oslash> m = [r div m\<dots>,c]"
lemma unitarily_equiv_commute: assumes "unitarily_equiv A B U" and "AC = CA" shows "B * (Complex_Matrix.adjoint U * C * U) = Complex_Matrix.adjoint U * C * U * B"
lemma flatten_fcall [simp]: "flatten (fcall init p return result c) = [fcall init p return result c]"
lemma Abort_refines[intro]: "well_def a \<Longrightarrow> Abort \<sqsubseteq> a"
lemma \<R>\<^sub>SI [intro]: assumes bij_\<beta>: "bij \<beta>" and none_case: "\<And>l. \<sigma> l = None \<Longrightarrow> \<sigma>' (\<beta> l) = None" and some_case: "\<And>l v. \<sigma> l = Some v \<Longrightarrow> \<sigma>' (\<beta> l) = Some (\<R>\<^sub>V \<alpha> \<beta> v)" shows "\<R>\<^sub>S \<alpha> \<beta> \<sigma> = \<sigma>'"
lemma proves_eq_abstract_rule_derived_rule: assumes thy: "wf_theory \<Theta>" assumes x: "(x, \<tau>) \<notin> FV \<Gamma>" "typ_ok \<Theta> \<tau>" assumes ctxt: "finite \<Gamma>" "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A" "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" assumes eq: "\<Theta>, \<Gamma> \<turnstile> mk_eq s t" shows "\<Theta>, \<Gamma> \<turnstile> mk_eq (Abs \<tau> (bind_fv (x, \<tau>) s)) (Abs \<tau> (bind_fv (x, \<tau>) t))"
lemma STAR_subset_closed: "x \<in> s A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> x \<in> s B"
lemma cat_parallel_composable_\<aa>\<aa>[cat_parallel_cs_intros]: assumes "g = \<aa>" and "f = \<aa>" shows "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> cat_parallel_composable \<aa> \<bb> F"
lemma steps_SCons_iff: "steps (x # y # xs) \<longleftrightarrow> E x y \<and> steps (y # xs)"
lemma phi_equiv_mappingE: assumes "g \<turnstile> m\<^sub>1 \<approx>\<^sub>\<phi> ssa.phiNodes_of g" and "b \<in> Mapping.keys (phis g)" and "Mapping.lookup (phis g) x = Some vs" and "snd b \<in> set vs" obtains ns where "Mapping.lookup m\<^sub>1 (snd b) = Some {n \<in> Mapping.keys (phis g). snd b \<in> set (the (Mapping.lookup (phis g) n))}"
lemma adm_subst2: "cont f \<Longrightarrow> cont g \<Longrightarrow> adm (\<lambda>x. f (fst x) = g (snd x))"
lemma kparts_insert_Key [iff]: "kparts (insert (Key K) H) = insert (Key K) (kparts H)"
lemma int_conseq_seq: " {(m::nat)..n} \<inter> {n+1..l} = {}"
lemma mult_le_cancel_iff2: "0 < z \<Longrightarrow> z * x \<le> z * y \<longleftrightarrow> x \<le> y" for x y z :: "'a::linordered_idom"
lemma powrat_add_neg_pos: assumes pos_x: "0 < x" and neg_r: "r < 0" and pos_s: "0 < s" shows "x pow\<^sub>\<rat> (r + s) = (x pow\<^sub>\<rat> r) * (x pow\<^sub>\<rat> s)"
lemma \<open>- (1705 :: int) AND 1 = 1\<close>
lemma object_clean_fields_twice [simp]: "(object_clean_fields (object_clean_fields obj cmp') cmp) = object_clean_fields obj (cmp \<inter> cmp')"
lemma normFace_neq: "a \<in> \<V> f \<Longrightarrow> a \<notin> \<V> f' \<Longrightarrow> vertices f' \<noteq> [] \<Longrightarrow> normFace f \<noteq> normFace f'"
lemma (in lbvc) wti_mono: assumes less: "s2 <=_r s1" assumes pc: "pc < length \<phi>" assumes s1: "s1 \<in> A" assumes s2: "s2 \<in> A" shows "wti c pc s2 <=_r wti c pc s1" (is "?s2' <=_r ?s1'")
lemma col_of_dagger [simp]: assumes "j < dim_row M" shows "col (M\<^sup>\<dagger>) j = vec (dim_col M) (\<lambda>i. cnj (M $$ (j,i)))"
lemma assign_eval\<^sub>w_load\<^sub>A: shows "(\<langle>x \<leftarrow> Load y, mds, mem\<rangle>\<^sub>A, \<langle>Stop, mds, mem (x := mem y)\<rangle>\<^sub>A) \<in> A.eval\<^sub>w"
lemma ran_ran_restrict [simp]: "ran(f\<upharpoonleft>\<^bsub>B\<^esub>) = ran(f) \<inter> B"
lemma replacefacesAt2_in: "i \<in> set is \<Longrightarrow> distinct is \<Longrightarrow> i < \|Fss\| \<Longrightarrow> (replacefacesAt2 is olfF newFs Fss)!i = replace olfF newFs (Fss !i)"
lemma card_equiv_class_restricted_same_size: assumes "equiv A R" assumes "P respects R" assumes "\<And>F. F \<in> {x \<in> A. P x} // R \<Longrightarrow> card F = k" shows "card {x \<in> A. P x} = k * card ({x \<in> A. P x} // R)"
theorem impl_model_check_correct_no_ce: assumes "(sysi,sys)\<in>sa_rel" assumes SA: "sa sys" "finite ((g_E sys)\<^sup>* `` g_V0 sys)" shows "impl_model_check cfg sysi \<phi> = None \<longleftrightarrow> sa.lang sys \<subseteq> language_ltlc \<phi>"
lemma ArrayIndexOutOfBounds_not_Object[simp]: "ArrayIndexOutOfBounds \<noteq> Object"
lemma OclNot_defargs: "\<tau> \<Turnstile> (not P) \<Longrightarrow> \<tau> \<Turnstile> \<delta> P"
lemma sints32:"sints 32 = {i. - (2 ^ 31) \<le> i \<and> i < 2 ^ 31}"
lemma T_eq_T': "T s = T' (K s)"
lemma "\<turnstile> \<lbrace>i = \<acute>I \<and> \<acute>time = 0\<rbrace> (timeit (WHILE \<acute>I \<noteq> 0 INV \<lbrace>2 \<acute> time + \<acute>I \<acute>I + 5 * \<acute>I = i * i + 5 * i\<rbrace> DO \<acute>J := \<acute>I; WHILE \<acute>J \<noteq> 0 INV \<lbrace>0 < \<acute>I \<and> 2 * \<acute>time + \<acute>I * \<acute>I + 3 * \<acute>I + 2 * \<acute>J - 2 = i * i + 5 * i\<rbrace> DO \<acute>J := \<acute>J - 1 OD; \<acute>I := \<acute>I - 1 OD)) \<lbrace>2 * \<acute>time = i * i + 5 * i\<rbrace>"
lemma signed_mod_arith: "sint ((a::('a::len) word) smod b) = signed_take_bit (LENGTH('a) - 1) (sint a smod sint b)"
lemma [code_unfold]: "((x::Person) \<doteq> y) = StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y"
lemma weak_ranking_unique: assumes "is_weak_ranking As" "of_weak_ranking As = le" shows "As = weak_ranking le"
lemma renaming_distr_global [simp]: "bij \<alpha> \<Longrightarrow> \<R>\<^sub>G \<alpha> \<beta> (s(r \<mapsto> ls)) = \<R>\<^sub>G \<alpha> \<beta> s(\<alpha> r \<mapsto> \<R>\<^sub>L \<alpha> \<beta> ls)" "bij \<alpha> \<Longrightarrow> \<R>\<^sub>G \<alpha> \<beta> (s(r := None)) = (\<R>\<^sub>G \<alpha> \<beta> s)(\<alpha> r := None)"
lemma image_image_id_if[simp]: "(\<And>x. f(f x) = x) \<Longrightarrow> f ` f ` M = M"
lemma rewrite_MultiportPorts_removes_MultiportsPorts: assumes n: "normalized_nnf_match m" shows "m' \<in> set (rewrite_MultiportPorts m) \<Longrightarrow> \<not> has_disc is_MultiportPorts m'"
lemma wf_sig_iff_exe_wf_sig': "exe_sig_conds \<Sigma> \<Longrightarrow> wf_sig (translate_signature \<Sigma>) \<longleftrightarrow> exe_wf_sig \<Sigma>"
lemma not_authKeys_not_AKcryptSK: "\<lbrakk> K \<notin> authKeys evs; K \<notin> range shrK; evs \<in> kerbIV \<rbrakk> \<Longrightarrow> \<forall>SK. \<not> AKcryptSK K SK evs"
lemma Says_Tgs_AKcryptSK: "\<lbrakk> Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, X \<rbrace>) \<in> set evs; authK' \<noteq> authK; evs \<in> kerbIV_gets \<rbrakk> \<Longrightarrow> \<not> AKcryptSK authK' servK evs"
lemma job_lower_bound_makespan: assumes "lb T A j" "x \<in> {1..j}" shows "t x \<le> makespan T"
lemma conv_radius_eqI_smallomega_smallo: fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra, banach}" assumes "\<And>\<epsilon>. \<epsilon> > l \<Longrightarrow> \<epsilon> < inverse C \<Longrightarrow> (\<lambda>n. norm (f n)) \<in> \<omega>(\<lambda>n. \<epsilon> ^ n)" assumes "\<And>\<epsilon>. \<epsilon> < u \<Longrightarrow> \<epsilon> > inverse C \<Longrightarrow> (\<lambda>n. norm (f n)) \<in> o(\<lambda>n. \<epsilon> ^ n)" assumes C: "C > 0" and lu: "l > 0" "l < inverse C" "u > inverse C" shows "conv_radius f = ereal C"
lemma add_replicate: "foldl (+) k (replicate m n) = k + m * n"
lemma Infinitesimal_diff: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x - y \<in> Infinitesimal"
theorem VSLuckiestRepeat : "\<forall>xs. eval (VSLuckiestRepeat \<phi>) xs = eval \<phi> xs"
lemma inverse_less_iff_less_neg [simp]: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
lemma arc_expanded_1: "arc e \<Longrightarrow> e * x * e\<^sup>T \<le> 1"
lemma abs_conv_abscissa_mono: assumes "\<And>s. fds_abs_converges g s \<Longrightarrow> fds_abs_converges f s" shows "abs_conv_abscissa f \<le> abs_conv_abscissa g"
lemma cop_not_par_other_side: assumes "C \<noteq> D" and "Col A B I" and "Col C D I" and "\<not> Col A B C" and "\<not> Col A B P" and "Coplanar A B C P" shows "\<exists> Q. Col C D Q \<and> A B TS P Q"
lemma iT_Plus_inext_nth: "I \<noteq> {} \<Longrightarrow> (I \<oplus> k) \<rightarrow> n = (I \<rightarrow> n) + k"

lemma (in semiring_1) last_linear_mul: assumes p: "p \<noteq> []" shows "last ([a, 1] *** p) = last p"

lemma llist_of_tllist_tappend: "llist_of_tllist (tappend xs f) = lappend (llist_of_tllist xs) (llist_of_tllist (f (terminal xs)))"

lemma complete_digraph_pair_def: "K\<^bsub>n\<^esub> (with_proj G) \<longleftrightarrow> finite (pverts G) \<and> card (pverts G) = n \<and> parcs G = {(u,v). (u,v) \<in> (pverts G \<times> pverts G) \<and> u \<noteq> v}" (is "_ = ?R")

lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"

lemma add_leaf_is_leaf: assumes "T' = \<lparr>verts = V, arcs = A, tail = t, head = h\<rparr>" and "T = \<lparr>verts = V \<union> {v}, arcs = A \<union> {a}, tail = t(a := u), head = h(a := v)\<rparr>" and "u \<in> V" and "v \<notin> V" and "a \<notin> A" and "directed_tree T' root'" shows "leaf v"

lemma upright_obtain_support: assumes "upright a" and "zcount a t > 0" obtains s where "s < t" "zcount a s < 0" "nonpos_upto a s"

lemma SA: "Fr_1b \<F> \<Longrightarrow> Fr_2 \<F> \<Longrightarrow> Fr_3 \<F> \<Longrightarrow> \<forall>a b c. (a \<^bold>\<Rightarrow> c) \<^bold>\<Rightarrow> ((a \<^bold>\<and> b) \<^bold>\<Rightarrow> c) \<^bold>\<approx> \<^bold>\<top>"

lemma m2pi_less_pi: "- (2*pi) < pi"

lemma interesting_subst1: assumes a: "x\<noteq>y" "x\<sharp>P" "y\<sharp>P" shows "N{y:=<c>.P}{x:=<c>.P} = N{x:=<c>.Ax y c}{y:=<c>.P}"

lemma E_thm13: "Lift \<in> (moving \<inter> Req n \<inter> {s. metric n s = N} \<inter> {s. floor s \<in> req s}) LeadsTo (stopped \<inter> Req n \<inter> {s. metric n s = N} \<inter> {s. floor s \<in> req s})"

lemma sdp_SetPC: fixes p::"'a \<Rightarrow> 's prog" assumes sdp: "\<And>s a. a \<in> supp (P s) \<Longrightarrow> sub_distrib_pconj (p a)" and fin: "\<And>s. finite (supp (P s))" and nnp: "\<And>s a. 0 \<le> P s a" and sub: "\<And>s. sum (P s) (supp (P s)) \<le> 1" shows "sub_distrib_pconj (SetPC p P)"

lemma K_in_space[simp]: "K x \<in> space (prob_algebra S)"

lemma sumResRight: fixes x :: name and P :: pi and Q :: pi assumes Id: "Id \<subseteq> Rel" and Eqvt: "eqvt Rel" shows "<\<nu>x>(P \<oplus> Q) \<leadsto>[Rel] (<\<nu>x>P) \<oplus> (<\<nu>x>Q)"

lemma cat_comma_Hom_def': "cat_comma_Hom \<GG> \<HH> A B \<equiv> set { [A, B, [g, h]\<^sub>\<circ>]\<^sub>\<circ> | g h. A \<in>\<^sub>\<circ> cat_comma_Obj \<GG> \<HH> \<and> B \<in>\<^sub>\<circ> cat_comma_Obj \<GG> \<HH> \<and> g : A\<lparr>0\<rparr> \<mapsto>\<^bsub>\<AA>\<^esub> B\<lparr>0\<rparr> \<and> h : A\<lparr>1\<^sub>\<nat>\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> B\<lparr>1\<^sub>\<nat>\<rparr> \<and> B\<lparr>2\<^sub>\<nat>\<rparr> \<circ>\<^sub>A\<^bsub>\<CC>\<^esub> \<GG>\<lparr>ArrMap\<rparr>\<lparr>g\<rparr> = \<HH>\<lparr>ArrMap\<rparr>\<lparr>h\<rparr> \<circ>\<^sub>A\<^bsub>\<CC>\<^esub> A\<lparr>2\<^sub>\<nat>\<rparr> }"

lemma ni_thin_locs_keep_atE: "\<lbrakk>at proc l s \<longrightarrow> P; AT s' = (AT s)(p := lfn, q := lfn'); at proc l s'; proc \<noteq> p; proc \<noteq> q; P \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" "\<lbrakk>at proc l s \<longrightarrow> P; AT s' = (AT s)(p := lfn); at proc l s'; proc \<noteq> p; P \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"

lemma lzip_ltakeWhile_snd: "lzip xs (ltakeWhile P ys) = ltakeWhile (P \<circ> snd) (lzip xs ys)"

lemma stable_pair_on_allocation: assumes "stable_pair_on ds XD_XH" shows "allocation (match XD_XH)"

lemma mult_left_isotone: "x \<le> y \<Longrightarrow> x * z \<le> y * z"

lemma \<phi>_zero: "x \<notin> {real_of_int (i - 1) / 2^(l + 1) <..< real_of_int (i + 1) / 2^(l + 1)} \<Longrightarrow> \<phi> (l,i) x = 0"

lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"

lemma product_run: assumes "length rs = length AA" "length ps = length AA" assumes "b.run (product AA) (w ||| stranspose rs) ps" shows "k < length AA \<Longrightarrow> a.run (AA ! k) (w ||| rs ! k) (ps ! k)"

lemma lemma_2_9_i1: "b \<in> supremum A \<Longrightarrow> b * a \<in> supremum ((\<lambda> x . x * a) ` A)"

lemma eval_v_restrict: fixes c::v and B::"bv fset" assumes "i = i' |` d" and "supp c \<subseteq> atom ` d \<union> supp B " and "i' \<lbrakk> c \<rbrakk> ~ s" shows "i \<lbrakk> c \<rbrakk> ~ s"

lemma eval_monom_mult[simp]: "eval_monom \<alpha> (m * n) = eval_monom \<alpha> m * eval_monom \<alpha> n"

lemma build_append[simp]: "(w @ a # u) \<frown> v = w \<frown> a ## u \<frown> v"

lemma ANR_imp_closed_neighbourhood_retract: fixes S :: "'a::euclidean_space set" assumes "ANR S" "closedin (top_of_set U) S" obtains V W where "openin (top_of_set U) V" "closedin (top_of_set U) W" "S \<subseteq> V" "V \<subseteq> W" "S retract_of W"

lemma t_buildup_cnt: "T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p n \<le> cnt (vebt_buildup n) * 13"

lemma of_nat_monomial: "of_nat p = monomial p 0"

lemma bconfs_map_throw[iff]: "P,sh \<turnstile>\<^sub>b (map Val vs @ throw e # es',b) \<surd> \<longleftrightarrow> P,sh \<turnstile>\<^sub>b (e,b) \<surd>"

lemma unique_awalk_All: "\<exists>p. awalk u p v \<Longrightarrow> \<exists>!p. awalk u p v"

lemma infinite_part1: "infinite V \<Longrightarrow> infinite (part1 V)"

lemma algebraic_power_iff [simp]: assumes "n > 0" shows "algebraic (x ^ n) \<longleftrightarrow> algebraic x"

lemma mult_col_div_row_similar: assumes A: "A \<in> carrier_mat n n" and ak: "k < n" "a \<noteq> 0" shows "similar_mat (mult_col_div_row a k A) A"

lemma col_elems_ss01: assumes "j < dim_col M" shows "vec_set (col M j) \<subseteq> {0, 1}"

lemma in_interval_interval_bij: fixes a b u v x :: "'a::euclidean_space" assumes "x \<in> cbox a b" and "cbox u v \<noteq> {}" shows "interval_bij (a, b) (u, v) x \<in> cbox u v"

lemma memberid_pair_simp1: "memberid p = memberid (snd p)"

lemma cong: "(\<And>s. P s = P' s) \<Longrightarrow> \<lceil>P\<rceil> = \<lceil>P'\<rceil>" "(\<And>\<sigma>. P \<sigma> = P' \<sigma>) \<Longrightarrow> (\<box>P) = (\<box>P')" "(\<And>\<sigma>. P \<sigma> = P' \<sigma>) \<Longrightarrow> (\<diamond>P) = (\<diamond>P')" "(\<And>\<sigma>. P \<sigma> = P' \<sigma>) \<Longrightarrow> (\<circle>P) = (\<circle>P')" "\<lbrakk>\<And>\<sigma>. P \<sigma> = P' \<sigma>; \<And>\<sigma>. Q \<sigma> = Q' \<sigma>\<rbrakk> \<Longrightarrow> (P \<U> Q) = (P' \<U> Q')" "\<lbrakk>\<And>\<sigma>. P \<sigma> = P' \<sigma>; \<And>\<sigma>. Q \<sigma> = Q' \<sigma>\<rbrakk> \<Longrightarrow> (P \<W> Q) = (P' \<W> Q')"

theorem "DA.accepts (na2da(rexp2na r)) w = (w : lang r)"

lemma cblinfun_power_Suc: \<open>cblinfun_power A (Suc n) = cblinfun_power A n o\<^sub>C\<^sub>L A\<close>

lemma [quot_preserve]: assumes a: "Quotient3 R abs1 rep1" shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"

lemma hd_lq_conv_nth: assumes "u <p v" shows "hd(u\<inverse>\<^sup>>v) = v!\<^bold>|u\<^bold>|"

lemma par_qmsg_oreachable_statelessassm: assumes "(\<sigma>, \<zeta>) \<in> oreachable (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg) (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>) (other (\<lambda>_ _. True) {i})" and ustutter: "\<And>\<xi>. U \<xi> \<xi>" shows "(\<sigma>, fst \<zeta>) \<in> oreachable A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>) (other (\<lambda>_ _. True) {i}) \<and> snd \<zeta> \<in> reachable qmsg (recvmsg R) \<and> (\<forall>m\<in>set (fst (snd \<zeta>)). R m)"

lemma (in pseudo_hoop_lattice_a) supremum_pair [simp]: "supremum {a, b} = {a \<squnion> b}"

lemma foundation20 : "\<tau> \<Turnstile> (\<delta> X) \<Longrightarrow> \<tau> \<Turnstile> \<upsilon> X"

lemma ta_det'_ground_id: "t |\<in>| ta_der' \<A> s \<Longrightarrow> ground t \<Longrightarrow> t = s"

lemma ces_append_one : "ces v1 (es @ [e]) v2 subs = (ces v1 es (src e) subs \<and> ces (src e) [e] v2 subs)"

lemma less_eq_bot_Bot_is_Bot: "x \<le> Bot \<Longrightarrow> x = Bot"

lemma step_augmenter_eval : assumes steph : "\<And>xs var L F \<Gamma>. length xs = var \<Longrightarrow> (\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (step var L F) (xs @ x # \<Gamma>))" assumes heuristic: "\<And>n var L F. heuristic n L F = var \<Longrightarrow> var \<le> n" shows "\<And>var amount L F \<Gamma>. amount \<le> var + 1 \<Longrightarrow> (\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. (length xs = (var + 1)) \<and> eval (step_augment step heuristic amount var L F) (xs @ \<Gamma>))"

lemma compatible_refl: assumes coms_secure: "list_all com_sifum_secure cmds" assumes low_eq: "mem\<^sub>1 =\<^sup>l mem\<^sub>2" shows "makes_compatible (cmds, mem\<^sub>1) (cmds, mem\<^sub>2) (replicate (length cmds) (mem\<^sub>1, mem\<^sub>2))"

lemma of_class_linorder: \<open>OFCLASS('a, linorder_class)\<close>

lemma heap_clone_NewHeapElemD: assumes "heap_clone P h a h' \<lfloor>(obs, a')\<rfloor>" and "ad \<in> allocated h'" and "ad \<notin> allocated h" shows "\<exists>CTn. NewHeapElem ad CTn \<in> set obs"

lemma iMODb_div_self: " 0 < m \<Longrightarrow> [r, mod m, c] \<oslash> m = [r div m\<dots>,c]"

lemma unitarily_equiv_commute: assumes "unitarily_equiv A B U" and "A*C = C*A" shows "B * (Complex_Matrix.adjoint U * C * U) = Complex_Matrix.adjoint U * C * U * B"

lemma flatten_fcall [simp]: "flatten (fcall init p return result c) = [fcall init p return result c]"

lemma Abort_refines[intro]: "well_def a \<Longrightarrow> Abort \<sqsubseteq> a"

lemma \<R>\<^sub>SI [intro]: assumes bij_\<beta>: "bij \<beta>" and none_case: "\<And>l. \<sigma> l = None \<Longrightarrow> \<sigma>' (\<beta> l) = None" and some_case: "\<And>l v. \<sigma> l = Some v \<Longrightarrow> \<sigma>' (\<beta> l) = Some (\<R>\<^sub>V \<alpha> \<beta> v)" shows "\<R>\<^sub>S \<alpha> \<beta> \<sigma> = \<sigma>'"

lemma proves_eq_abstract_rule_derived_rule: assumes thy: "wf_theory \<Theta>" assumes x: "(x, \<tau>) \<notin> FV \<Gamma>" "typ_ok \<Theta> \<tau>" assumes ctxt: "finite \<Gamma>" "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A" "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" assumes eq: "\<Theta>, \<Gamma> \<turnstile> mk_eq s t" shows "\<Theta>, \<Gamma> \<turnstile> mk_eq (Abs \<tau> (bind_fv (x, \<tau>) s)) (Abs \<tau> (bind_fv (x, \<tau>) t))"

lemma STAR_subset_closed: "x \<in> *s* A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> x \<in> *s* B"

lemma cat_parallel_composable_\<aa>\<aa>[cat_parallel_cs_intros]: assumes "g = \<aa>" and "f = \<aa>" shows "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> cat_parallel_composable \<aa> \<bb> F"

lemma steps_SCons_iff: "steps (x # y # xs) \<longleftrightarrow> E x y \<and> steps (y # xs)"

lemma phi_equiv_mappingE: assumes "g \<turnstile> m\<^sub>1 \<approx>\<^sub>\<phi> ssa.phiNodes_of g" and "b \<in> Mapping.keys (phis g)" and "Mapping.lookup (phis g) x = Some vs" and "snd b \<in> set vs" obtains ns where "Mapping.lookup m\<^sub>1 (snd b) = Some {n \<in> Mapping.keys (phis g). snd b \<in> set (the (Mapping.lookup (phis g) n))}"

lemma adm_subst2: "cont f \<Longrightarrow> cont g \<Longrightarrow> adm (\<lambda>x. f (fst x) = g (snd x))"

lemma kparts_insert_Key [iff]: "kparts (insert (Key K) H) = insert (Key K) (kparts H)"

lemma int_conseq_seq: " {(m::nat)..n} \<inter> {n+1..l} = {}"

lemma mult_le_cancel_iff2: "0 < z \<Longrightarrow> z * x \<le> z * y \<longleftrightarrow> x \<le> y" for x y z :: "'a::linordered_idom"

lemma powrat_add_neg_pos: assumes pos_x: "0 < x" and neg_r: "r < 0" and pos_s: "0 < s" shows "x pow\<^sub>\<rat> (r + s) = (x pow\<^sub>\<rat> r) * (x pow\<^sub>\<rat> s)"

lemma \<open>- (1705 :: int) AND 1 = 1\<close>

lemma object_clean_fields_twice [simp]: "(object_clean_fields (object_clean_fields obj cmp') cmp) = object_clean_fields obj (cmp \<inter> cmp')"

lemma normFace_neq: "a \<in> \<V> f \<Longrightarrow> a \<notin> \<V> f' \<Longrightarrow> vertices f' \<noteq> [] \<Longrightarrow> normFace f \<noteq> normFace f'"

lemma (in lbvc) wti_mono: assumes less: "s2 <=_r s1" assumes pc: "pc < length \<phi>" assumes s1: "s1 \<in> A" assumes s2: "s2 \<in> A" shows "wti c pc s2 <=_r wti c pc s1" (is "?s2' <=_r ?s1'")

lemma col_of_dagger [simp]: assumes "j < dim_row M" shows "col (M\<^sup>\<dagger>) j = vec (dim_col M) (\<lambda>i. cnj (M $$ (j,i)))"

lemma assign_eval\<^sub>w_load\<^sub>A: shows "(\<langle>x \<leftarrow> Load y, mds, mem\<rangle>\<^sub>A, \<langle>Stop, mds, mem (x := mem y)\<rangle>\<^sub>A) \<in> A.eval\<^sub>w"

lemma ran_ran_restrict [simp]: "ran(f\<upharpoonleft>\<^bsub>B\<^esub>) = ran(f) \<inter> B"

lemma replacefacesAt2_in: "i \<in> set is \<Longrightarrow> distinct is \<Longrightarrow> i < |Fss| \<Longrightarrow> (replacefacesAt2 is olfF newFs Fss)!i = replace olfF newFs (Fss !i)"

lemma card_equiv_class_restricted_same_size: assumes "equiv A R" assumes "P respects R" assumes "\<And>F. F \<in> {x \<in> A. P x} // R \<Longrightarrow> card F = k" shows "card {x \<in> A. P x} = k * card ({x \<in> A. P x} // R)"

theorem impl_model_check_correct_no_ce: assumes "(sysi,sys)\<in>sa_rel" assumes SA: "sa sys" "finite ((g_E sys)\<^sup>* `` g_V0 sys)" shows "impl_model_check cfg sysi \<phi> = None \<longleftrightarrow> sa.lang sys \<subseteq> language_ltlc \<phi>"

lemma ArrayIndexOutOfBounds_not_Object[simp]: "ArrayIndexOutOfBounds \<noteq> Object"

lemma OclNot_defargs: "\<tau> \<Turnstile> (not P) \<Longrightarrow> \<tau> \<Turnstile> \<delta> P"

lemma sints32:"sints 32 = {i. - (2 ^ 31) \<le> i \<and> i < 2 ^ 31}"

lemma T_eq_T': "T s = T' (K s)"

lemma "\<turnstile> \<lbrace>i = \<acute>I \<and> \<acute>time = 0\<rbrace> (timeit (WHILE \<acute>I \<noteq> 0 INV \<lbrace>2 *\<acute> time + \<acute>I * \<acute>I + 5 * \<acute>I = i * i + 5 * i\<rbrace> DO \<acute>J := \<acute>I; WHILE \<acute>J \<noteq> 0 INV \<lbrace>0 < \<acute>I \<and> 2 * \<acute>time + \<acute>I * \<acute>I + 3 * \<acute>I + 2 * \<acute>J - 2 = i * i + 5 * i\<rbrace> DO \<acute>J := \<acute>J - 1 OD; \<acute>I := \<acute>I - 1 OD)) \<lbrace>2 * \<acute>time = i * i + 5 * i\<rbrace>"

lemma signed_mod_arith: "sint ((a::('a::len) word) smod b) = signed_take_bit (LENGTH('a) - 1) (sint a smod sint b)"

lemma [code_unfold]: "((x::Person) \<doteq> y) = StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y"

lemma weak_ranking_unique: assumes "is_weak_ranking As" "of_weak_ranking As = le" shows "As = weak_ranking le"

lemma renaming_distr_global [simp]: "bij \<alpha> \<Longrightarrow> \<R>\<^sub>G \<alpha> \<beta> (s(r \<mapsto> ls)) = \<R>\<^sub>G \<alpha> \<beta> s(\<alpha> r \<mapsto> \<R>\<^sub>L \<alpha> \<beta> ls)" "bij \<alpha> \<Longrightarrow> \<R>\<^sub>G \<alpha> \<beta> (s(r := None)) = (\<R>\<^sub>G \<alpha> \<beta> s)(\<alpha> r := None)"

lemma image_image_id_if[simp]: "(\<And>x. f(f x) = x) \<Longrightarrow> f ` f ` M = M"

lemma rewrite_MultiportPorts_removes_MultiportsPorts: assumes n: "normalized_nnf_match m" shows "m' \<in> set (rewrite_MultiportPorts m) \<Longrightarrow> \<not> has_disc is_MultiportPorts m'"

lemma wf_sig_iff_exe_wf_sig': "exe_sig_conds \<Sigma> \<Longrightarrow> wf_sig (translate_signature \<Sigma>) \<longleftrightarrow> exe_wf_sig \<Sigma>"

lemma not_authKeys_not_AKcryptSK: "\<lbrakk> K \<notin> authKeys evs; K \<notin> range shrK; evs \<in> kerbIV \<rbrakk> \<Longrightarrow> \<forall>SK. \<not> AKcryptSK K SK evs"

lemma Says_Tgs_AKcryptSK: "\<lbrakk> Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, X \<rbrace>) \<in> set evs; authK' \<noteq> authK; evs \<in> kerbIV_gets \<rbrakk> \<Longrightarrow> \<not> AKcryptSK authK' servK evs"

lemma job_lower_bound_makespan: assumes "lb T A j" "x \<in> {1..j}" shows "t x \<le> makespan T"

lemma conv_radius_eqI_smallomega_smallo: fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra, banach}" assumes "\<And>\<epsilon>. \<epsilon> > l \<Longrightarrow> \<epsilon> < inverse C \<Longrightarrow> (\<lambda>n. norm (f n)) \<in> \<omega>(\<lambda>n. \<epsilon> ^ n)" assumes "\<And>\<epsilon>. \<epsilon> < u \<Longrightarrow> \<epsilon> > inverse C \<Longrightarrow> (\<lambda>n. norm (f n)) \<in> o(\<lambda>n. \<epsilon> ^ n)" assumes C: "C > 0" and lu: "l > 0" "l < inverse C" "u > inverse C" shows "conv_radius f = ereal C"

lemma add_replicate: "foldl (+) k (replicate m n) = k + m * n"

lemma Infinitesimal_diff: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x - y \<in> Infinitesimal"

theorem VSLuckiestRepeat : "\<forall>xs. eval (VSLuckiestRepeat \<phi>) xs = eval \<phi> xs"

lemma inverse_less_iff_less_neg [simp]: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"

lemma arc_expanded_1: "arc e \<Longrightarrow> e * x * e\<^sup>T \<le> 1"

lemma abs_conv_abscissa_mono: assumes "\<And>s. fds_abs_converges g s \<Longrightarrow> fds_abs_converges f s" shows "abs_conv_abscissa f \<le> abs_conv_abscissa g"

lemma cop_not_par_other_side: assumes "C \<noteq> D" and "Col A B I" and "Col C D I" and "\<not> Col A B C" and "\<not> Col A B P" and "Coplanar A B C P" shows "\<exists> Q. Col C D Q \<and> A B TS P Q"

lemma iT_Plus_inext_nth: "I \<noteq> {} \<Longrightarrow> (I \<oplus> k) \<rightarrow> n = (I \<rightarrow> n) + k"