hcju/isabelleps huggingface dataset:
openin_topology\<^sub>N_I: fixes U :: "'a set"   and N :: "'a quasinorm" assumes "\<And>x. x \<in> U \<Longrightarrow> \<exists>e>0. \<forall>y. eNorm N (y - x) < e \<longrightarrow> y \<in> U" shows "openin (topology\<^sub>N N) U"
field_less_half_sum: fixes x :: "'d"   and y :: "'d" assumes "x < y" shows "x < (x + y) / (2::'d)"
ent_false_iff: fixes P :: "assn" shows "(P \<Longrightarrow>\<^sub>A false) = (\<forall>h. \<not> h \<Turnstile> P)"
sum_list_aezdeltafun_smult_distrib: fixes v :: "'v"   and fgs :: "('f \<times> 'g) list" assumes "v \<in> V"   and "set (map snd fgs) \<subseteq> G" shows "(\<Sum>(f, g)\<leftarrow>fgs. f \<delta>\<delta> g) \<cdot> v = (\<Sum>(f, g)\<leftarrow>fgs. f \<sharp>\<cdot> g *\<cdot> v)"
fmeasurable_compact: fixes S :: "'a set" assumes "compact S" shows "S \<in> fmeasurable lborel"
Units_closed: fixes x :: "'a" assumes "x \<in> Units G" shows "x \<in> carrier G"
tm_comp_step: fixes c :: "nat \<times> cell list \<times> cell list"   and A :: "(action \<times> nat) list"   and B :: "(action \<times> nat) list" assumes "\<not> is_final (step0 c A)" shows "step0 c (A |+| B) = step0 c A"
act2: shows "opnet onp p\<^sub>2 \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U (net_tree_ips p\<^sub>2) \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). castmsg (I \<sigma>) a \<and> (a = \<tau> \<or> (\<exists>i d. a = i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p\<^sub>2. U (\<sigma> i) (\<sigma>' i)) \<and> (\<forall>i. i \<notin> net_tree_ips p\<^sub>2 \<longrightarrow> \<sigma>' i = \<sigma> i)))"
fusion_character: fixes F :: "i \<Rightarrow> bool" assumes "GEM"   and "\<exists>x. F x" shows "\<forall>y. O y (\<sigma>v. F v) = (\<exists>x. F x \<and> O y x)"
opnet_bigger_than_next: fixes n :: "net_tree" shows "opnet (\<lambda>i. optoy i \<langle>\<langle>\<^bsub>i\<^esub> qmsg) n \<Turnstile> (otherwith nos_inc (net_tree_ips n) (oarrivemsg msg_ok), other nos_inc (net_tree_ips n) \<rightarrow>) global (\<lambda>\<sigma>. \<forall>i\<in>net_tree_ips n. no (\<sigma> i) \<le> no (\<sigma> (nhid (\<sigma> i))))"
shiftr_integer_code: fixes n :: "nat"   and x :: "integer" shows "drop_bit n x = integer_shiftr x (integer_of_nat n)"
fin: shows "finite {x. a * x\<^sup>2 + b * x + c = 0}"
s_learn_U0: fixes s :: "nat \<Rightarrow> nat option" assumes "r1_bc s" shows "learn_bc \<phi> U\<^sub>0 s"
parts_insert_Crypt: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" shows "parts (insert (Crypt K X) H) = insert (Crypt K X) (parts (insert X H))"
mono_mult2_alt: fixes ns :: "('a \<times> 'a) set"   and ns' :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and s' :: "('a \<times> 'a) set"   and b :: "bool" assumes "ns \<subseteq> ns'"   and "s \<subseteq> s'" shows "mult2_alt b ns s \<subseteq> mult2_alt b ns' s'"
subset_edges: shows "set (TopoS_withOffendingFlows.minimalize_offending_overapprox (\<lambda>G. sinvar_spec G nP) (edgesL G) [] (list_graph_to_graph G)) \<subseteq> edges (list_graph_to_graph G)"
igAbsRenSTR_def: fixes MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'p, 'q, 'r) model_scheme" shows "igAbsRenSTR MOD \<equiv> \<forall>xs y x X. igFresh MOD xs y X \<longrightarrow> igAbs MOD xs y (igSubst MOD xs (igVar MOD xs y) x X) = igAbs MOD xs x X"
rwof_step: fixes m0 :: "'a nres"   and cond :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a nres"   and s :: "'a" assumes "nofail (m0 \<bind> WHILE cond step)"   and "rwof m0 cond step s"   and "cond s" shows "step s \<le> SPEC (rwof m0 cond step)"
mult_submonoid_of_rng_axioms: shows "mult_submonoid_of_rng R S"
comp_fun_commute: fixes y :: "'a \<times> 'b"   and x :: "'a \<times> 'b" shows "insert_mapping y \<circ> insert_mapping x = insert_mapping x \<circ> insert_mapping y"
linear_map_def: fixes K :: "('e, 'f) ring_scheme"   and V :: "('e, 'g, 'h) module_scheme"   and W :: "('e, 'i, 'j) module_scheme"   and T :: "'g \<Rightarrow> 'i" shows "linear_map K V W T \<equiv> vectorspace K V \<and> vectorspace K W \<and> mod_hom K V W T"
\<delta>_def: shows "\<delta> \<equiv> var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t (set A)))"
monom_mult_unfold_2: fixes f :: "'b poly"   and n :: "nat" shows "f * monom (1::'b) n = monom_mult n f"
elims_3: fixes x :: "('a list \<times> 's \<times> 's) list \<times> ('s \<times> 's) list \<times> ('s \<times> 's) set" assumes "\<not> test_invariant x" obtains ws :: "('a list \<times> 's \<times> 's) list" and uu_ :: "('s \<times> 's) list" and uv_ :: "('s \<times> 's) set" where "x = (ws, uu_, uv_)" and "\<not> (case ws of [] \<Rightarrow> False | (w, p, q) # x \<Rightarrow> final p = final q)"
elims_3: fixes x :: "'edge list"   and xa :: "'edge list" assumes "\<not> valid_path_aux x xa" obtains cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "\<not> (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> valid_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> valid_path_aux [] as | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> valid_path_aux cs' as | _ \<Rightarrow> valid_path_aux cs as)"
h': shows "h3 \<turnstile> set_disconnected_nodes document_ptr (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr # disc_nodes) \<rightarrow>\<^sub>h h'"
unord_\<Gamma>_def: shows "unord_\<Gamma> = {Infer {#C#} D E |C D E. unord_resolve C D E}"
sminus_def: fixes v :: "'a"   and t :: "'a" shows "local.punit.sminus v t = fst (local.punit.pp_of_term v - t, local.punit.component_of_term v)"
assms_2: fixes n :: "nat" shows "dist (\<phi> (Suc n)) (\<phi> n) \<le> a n"
gcd_0_left: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd zero a = normalize a"
assms_1: shows "(A, B) \<in> s_mul_ext ns s"
Liminf_llist_subset_Sup_llist: fixes Xs :: "'b set llist" shows "Liminf_llist Xs \<subseteq> Sup_llist Xs"
exec_to_execn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<exists>n. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
ide_dom: fixes f :: "'a" assumes "arr f" shows "ide (local.dom f)"
order_iff: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (a = a \<squnion> b)"
vfsequence_vcons: fixes xs :: "V"   and x :: "V" assumes "vfsequence xs" shows "vfsequence (xs #\<^sub>\<circ> x)"
fmap_singleton_comm: fixes m :: "('d, 'c) fmap"   and k :: "'d"   and v :: "'c" assumes "m $$ k = None" shows "m ++\<^sub>f {k $$:= v} = {k $$:= v} ++\<^sub>f m"
A: fixes e :: "'a" shows "(case Inl e of Inl e \<Rightarrow> e) = e"
length_butlast: fixes xs :: "'b list" shows "\<^bold>|butlast xs\<^bold>| = \<^bold>|xs\<^bold>| - 1"
diff: fixes x :: "'b"   and y :: "'b" shows "g' (x - y) = g' x - g' y"
mult_of_def: fixes R :: "('c, 'd) ring_scheme" shows "Multiplicative_Group.mult_of R \<equiv> \<lparr>carrier = carrier R - {\<zero>\<^bsub>R\<^esub>}, monoid.mult = (\<otimes>\<^bsub>R\<^esub>), one = \<one>\<^bsub>R\<^esub>\<rparr>"
nth_Cons: fixes x :: "'g"   and xs :: "'g list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
translation_eq_singleton_plus: fixes a :: "'b"   and T :: "'b set" shows "(+) a ` T = {a} + T"
L_neg_def: shows "L_neg = drop (length A_pos) L"
N_pos_simps_3: shows "\<not> N \<le> Suc 0"
LID\<^sub>G_def: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option" shows "LID\<^sub>G s \<equiv> \<Union> (LID\<^sub>L ` ran s)"
degree_prod_le_max_perm_degree: fixes j :: "'b \<Rightarrow> 'b"   and A :: "('c poly, 'b) sq_matrix" assumes "j permutes UNIV" shows "degree (\<Prod>i\<in>UNIV. to_fun A i (j i)) \<le> max_perm_degree A"
diff_union_swap: fixes a :: "'c"   and b :: "'c"   and M :: "'c multiset" assumes "a \<noteq> b" shows "add_mset b (M - {#a#}) = add_mset b M - {#a#}"
cofinite_interior_finite: fixes X :: "'a cofinite set" assumes "finite X"   and "infinite UNIV" shows "interior X = {}"
fg1: fixes x :: "'a" assumes "x \<in> T" shows "f x = p (g1 x)"
length_dims0: shows "length b.dims0 = card ({0..<length dims} \<inter> b.vars0)"
Suc_length_conv: fixes n :: "nat"   and xs :: "'a list" shows "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
dvd_factorial: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "m \<le> n" shows "m dvd fact n"
gre1I: fixes n :: "nat" assumes "n = 0 \<Longrightarrow> False" shows "1 \<le> n"
Union_insert: fixes a :: "'b set"   and B :: "'b set set" shows "\<Union> ((a::'b set) \<triangleright> (B::'b set set)) = a \<union> \<Union> B"
bottom_edge_smooth: shows "(\<lambda>x. twoC (0, x)) piecewise_C1_differentiable_on {0..1}"
congruent_triangleD_21: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "angle a1 c1 b1 = angle b2 c2 a2"
Zr_add: fixes n :: "int"   and m :: "int" shows "n \<plusminus>\<^bsub>Zr\<^esub> m = n + m"
det3_same: fixes a :: "real \<times> real"   and x :: "real \<times> real" shows "coll a x x"
liftn_var_dist_lt: fixes vs :: "nat list"   and k :: "nat"   and n :: "nat"   and t :: "dB" assumes "\<forall>i\<in>set vs. i < k" shows "liftn n (var_dist vs t) k = var_dist vs (liftn n t k)"
norm_min: fixes M1 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "normalized M1"   and "[M]\<^bsub>v,n\<^esub> \<subseteq> [M1]\<^bsub>v,n\<^esub>"   and "canonical M n"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}"   and "[M]\<^bsub>v,n\<^esub> \<subseteq> V" shows "[DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub> \<subseteq> [M1]\<^bsub>v,n\<^esub>"
le_lt_convs_1: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y \<noteq> Lt) = le_of_comp comp y x"
simps_2: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
fps_compose_nth: fixes a :: "'a fps"   and b :: "'a fps"   and n :: "nat" shows "fps_nth (a oo b) n = (\<Sum>i = 0..n. fps_nth a i * fps_nth (b ^ i) n)"
compactin_empty: fixes X :: "'a topology" shows "compactin X {}"
simps_132: fixes x61 :: "poscheckvariant"   and x62 :: "('d, 'c) Term.term"   and x63 :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>x61: x62 \<in> x63\<rangle> = fv x62 \<union> fv x63"
assms_2: shows "AE x in M. 0 \<le> u x"
ded_left: fixes a :: "'b G"   and b :: "'b G"   and c :: "'b G" shows "a * b l\<rightarrow> c = a l\<rightarrow> b l\<rightarrow> c"
real_minus_le_minus: fixes b :: "'a"   and a :: "'a" shows "(- b \<le> - a) = (a \<le> b)"
TensorDiag_preserves_Ide: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Ide t"   and "Ide u"   and "Diag t"   and "Diag u" shows "Ide (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"
select_min_weight_clause_min_weight: fixes Ci :: "'a literal list \<times> nat"   and P0 :: "'a literal list \<times> nat"   and P :: "('a literal list \<times> nat) list" assumes "Ci = select_min_weight_clause P0 P" shows "weight (apfst mset Ci) = Min ((weight \<circ> apfst mset) ` set (P0 # P))"
\<psi>_eq: shows "\<psi> = complex_of_real (cos (\<theta> / 2)) \<cdot>\<^sub>v \<alpha> + complex_of_real (sin (\<theta> / 2)) \<cdot>\<^sub>v \<beta>"
measurable_linorder_from_keys_restrict: fixes A :: "'b set" assumes "finite A" shows "linorder_from_keys A \<in> Pi\<^sub>M A (\<lambda>_. borel) \<rightarrow>\<^sub>M count_space (Pow (A \<times> A))"
lub_chain_maxelem: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Y :: "'b \<Rightarrow> 'a"   and i :: "'b"   and c :: "'a" assumes "class.po below"   and "Y i = c"   and "\<forall>i. below (Y i) c" shows "po.lub below (range Y) = c"
simps_6: fixes f :: "'c \<Rightarrow> 'b"   and x1 :: "'c"   and x2 :: "'c stream" shows "smap f (x1 ## x2) = f x1 ## smap f x2"
simps_6: fixes V :: "'a set"   and t :: "('b, 'a) Term.term"   and s :: "('b, 'a) Term.term"   and S :: "('b, 'a) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (insert\<langle>t,s\<rangle> # S) = (fv t \<subseteq> V \<and> fv s \<subseteq> V \<and> wf'\<^sub>s\<^sub>s\<^sub>t V S)"
arr_iff_in_hom: fixes f :: "'c" shows "Cop.arr f = Cop.in_hom f (Cop.dom f) (Cop.cod f)"
imbal0_if_wbalanced: fixes t :: "'b tree" assumes "wbalanced t" shows "imbal t = 0"
valid_sops_prog_step: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and p' :: "'a"   and is' :: "instr list" assumes "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')"   and "valid_prog p" shows "\<forall>sop\<in>store_sops is'. valid_sop sop"
distinct_165: fixes x6 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Inj2 x6 \<noteq> Unroll x14"
zero_right_mult_decreasing: fixes x :: "'a" shows "x \<sqinter> bot \<le> x"
concrete_edge_rel_list_set_rel: fixes a :: "'cedge list"   and b :: "'edge set" assumes "(a, b) \<in> \<langle>concrete_edge_rel\<rangle>list_set_rel" shows "\<alpha> ` set a = b"
1: shows "\<eta>1 $$ X maps\<^bsub>NTCatCod \<eta>1\<^esub> NTDom \<eta>1 @@ X to NTCod \<eta>1 @@ X"
succ_of_rel_def: fixes s :: "('g \<times> 'h) set" shows "succ_of_rel s \<equiv> \<lambda>a. {b. (a, b) \<in> s}"
assms: shows "\<forall>a\<in>A. finite a \<and> (\<forall>X. X \<in> Range a \<longrightarrow> b (seller, X) = (0::'b))"
simps: fixes xs :: "(int \<times> int) list" shows "closest_pair_rec_tm xs =1 length_tm xs \<bind> (\<lambda>n. if n \<le> 3 then mergesort_tm snd xs \<bind> (\<lambda>ys. closest_pair_bf_tm xs \<bind> (\<lambda>p. return (ys, p))) else split_at_tm (n div 2) xs \<bind> (\<lambda>(xs\<^sub>L, xs\<^sub>R). closest_pair_rec_tm xs\<^sub>L \<bind> (\<lambda>(ys\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L). closest_pair_rec_tm xs\<^sub>R \<bind> (\<lambda>(ys\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R). merge_tm snd ys\<^sub>L ys\<^sub>R \<bind> (\<lambda>ys. combine_tm (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) (fst (hd xs\<^sub>R)) ys \<bind> (\<lambda>(p\<^sub>0, p\<^sub>1). return (ys, p\<^sub>0, p\<^sub>1)))))))"
fun_upd_apply: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and y :: "'b"   and z :: "'c" shows "(f(x := y)) z = (if z = x then y else f z)"
assms_1: shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> A"
ex_card: fixes n :: "nat"   and A :: "'a set" assumes "n \<le> card A" shows "\<exists>S\<subseteq>A. card S = n"
red_image_monic: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "red (monic ` B) = red B"
arg_cong2: fixes a :: "'e"   and b :: "'e"   and c :: "'f"   and d :: "'f"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "a = b"   and "c = d" shows "f a c = f b d"
LSC_NNF: fixes \<Gamma> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow>\<^sub>n" shows "image_mset nnf \<Gamma> \<Rightarrow>\<^sub>n"
simps_3: fixes Cs :: "char list list" shows "ics_classes (Called Cs) = Cs"
a_subid': fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.antidomain_pre_dioid antidomain_op plus times one zero less_eq less" shows "less_eq (antidomain_op x) one"
collect_basis_in_finite: shows "finite (Collect (basis_in \<E>))"
quotient_imp_continuous_map: fixes X :: "'d topology"   and X' :: "'e topology"   and f :: "'d \<Rightarrow> 'e" assumes "quotient_map X X' f" shows "continuous_map X X' f"
hdomain_ins: fixes x :: "hf"   and y :: "hf"   and r :: "hf" shows "hdomain (r \<triangleleft> \<langle>x, y\<rangle>) = hdomain r \<triangleleft> x"
fml_sem_or: fixes I :: "interp"   and \<phi> :: "fml"   and \<psi> :: "fml" shows "fml_sem I (\<phi> || \<psi>) = fml_sem I \<phi> \<union> fml_sem I \<psi>"
all_cong: fixes Q :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and P' :: "'b \<Rightarrow> bool" assumes "\<And>x. Q x \<Longrightarrow> P x = P' x" shows "(\<forall>x. Q x \<longrightarrow> P x) = (\<forall>x. Q x \<longrightarrow> P' x)"
admissible_targets_clocks_2: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "(l, g, \<mu>) \<in> PTA.trans_of A"   and "(X, l') \<in> set_pmf \<mu>" shows "set (SOME r. set r = X) \<subseteq> \<X>"
simps_11: fixes thread' :: "(thread_id_t \<Rightarrow> thread_t) \<Rightarrow> thread_id_t \<Rightarrow> thread_t"   and sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "thread_update thread' \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread' thread, \<dots> = more\<rparr>"
assms_1: shows "\<forall>rs\<in>ran \<Gamma>. no_call_to c rs"
nGt_le_conv: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "(key_compare ko x y \<noteq> Gt) = ko.le ko x y"
triangle_growth: fixes k :: "nat"   and A :: "'a mat" assumes "local.triangular_to k A"   and "local.triangular_column k A" shows "local.triangular_to (Suc k) A"
disc_eq_case_1: fixes option :: "'b option" shows "(option = None) = (case option of None \<Rightarrow> True | Some uu_ \<Rightarrow> False)"
inverts_components: fixes a :: "'d \<times> 'c" assumes "DopxC.ide a" shows "S.inverse_arrows (\<Phi> a) (\<Psi> a)"
s_ce_inj2: fixes e :: "nat"   and x1 :: "nat"   and x2 :: "nat" assumes "s_ce e x1 = s_ce e x2" shows "x1 = x2"
left_tree_initial: fixes s :: "'a"   and t :: "'a" assumes "s \<in> V"   and "t \<in> V" shows "s \<in> left_tree s t"
finiteI_independent: fixes B :: "'b set" assumes "independent B" shows "finite B"
gauss_int_eq_iff: fixes x :: "gauss_int"   and y :: "gauss_int" shows "(x = y) = (ReZ x = ReZ y \<and> ImZ x = ImZ y)"
elims: fixes x :: "'a bro"   and y :: "'a bro" assumes "tree x = y" obtains t :: "'a bro" where "x = N1 t" and "y = t"   | "x = N0" and "y = N0"   | v :: "'a bro" and va :: "'a" and vb :: "'a bro" where "x = N2 v__ va__ vb__" and "y = N2 v__ va__ vb__"   | v :: "'a" where "x = L2 vc__" and "y = L2 vc__"   | v :: "'a bro" and va :: "'a" and vb :: "'a bro" and vc :: "'a" and vd :: "'a bro" where "x = N3 v va vb vc vd" and "y = N3 v va vb vc vd"
1: fixes i :: "int"   and f :: "Integer \<rightarrow> 'a"   and g :: "Integer \<rightarrow> 'a"   and Pa :: "'a \<rightarrow> tr"   and xsa :: "['a]" assumes "\<forall>j\<ge>i. f\<cdot>(MkI\<cdot>j) = g\<cdot>(MkI\<cdot>j)" shows "map\<cdot>f\<cdot>(map\<cdot>snd\<cdot>(filter\<cdot>(\<Lambda> \<langle>x, i\<rangle>. Pa\<cdot>x)\<cdot>(zip\<cdot>xsa\<cdot>[MkI\<cdot>i..]))) = map\<cdot>g\<cdot>(map\<cdot>snd\<cdot>(filter\<cdot>(\<Lambda> \<langle>x, i\<rangle>. Pa\<cdot>x)\<cdot>(zip\<cdot>xsa\<cdot>[MkI\<cdot>i..])))"
Predict_item_end: fixes x :: "item"   and k :: "nat"   and Y :: "item set" assumes "x \<in> Predict k Y" shows "item_end x = k \<or> x \<in> Y"
card_subset_less: fixes i :: "nat_int"   and i' :: "nat_int" shows "i < i' \<longrightarrow> |i| < |i'|"
I_r_closed: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme"   and a :: "'e"   and x :: "'e" assumes "ideal I R"   and "a \<in> I"   and "x \<in> carrier R" shows "a \<otimes>\<^bsub>R\<^esub> x \<in> I"
PhiWhilePOp_Monotone: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "Monotone (PhiWhilePOp A b \<Phi>)"
mem_CD_on_Cd: fixes x :: "'a"   and ds :: "'b set"   and X :: "'a set" shows "(x \<in> Pds'.CD_on ds X) = (x \<in> Pds'.Cd (Xd x) X \<and> Xd x \<in> ds)"
AssociationClassCallTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and a :: "String.literal option"   and "from" :: "'a"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E AssociationClassCall src k a from : \<tau>" obtains \<C> :: "'a" and \<D> :: "'a" where "k = DotCall" and "\<tau> = class_assoc_type from" and "\<Gamma> \<turnstile>\<^sub>E src : \<langle>\<C>\<rangle>\<^sub>\<T>[1]" and "referred_by_association_class \<C> a from \<D>"
n_comm: fixes x :: "'a"   and y :: "'a" shows "n x \<cdot> n y = n y \<cdot> n x"
callee_invariant_on_plus_oracle: fixes left :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'a) spmf"   and right :: "'a \<Rightarrow> 'c \<Rightarrow> ('e \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I>l :: "('b, 'd) \<I>"   and \<I>r :: "('c, 'e) \<I>" shows "callee_invariant_on (left \<oplus>\<^sub>O right) I (\<I>l \<oplus>\<^sub>\<I> \<I>r) = (callee_invariant_on left I \<I>l \<and> callee_invariant_on right I \<I>r)"
bind_in_space: fixes A :: "'c \<Rightarrow> 'd measure"   and p :: "'c spmf"   and N :: "'d measure" assumes "A \<in> measure_spmf p \<rightarrow>\<^sub>M subprob_algebra N" shows "measure_spmf p \<bind> A \<in> space (subprob_algebra N)"
deg: shows "\<forall>d\<in>outcomes. return_pmf d \<succeq>[\<R>] b"
simps_1: fixes a :: "'a \<times> 'b" shows "sd_power a 0 = sd_unit"
is_prefix_take: fixes n :: "nat"   and a :: "'a list" shows "is_prefix (take n a) a"
mset_append: fixes xs :: "'b list"   and ys :: "'b list" shows "mset (xs @ ys) = mset xs + mset ys"
inf_le2: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<inter># y \<subseteq># y"
split_indicator: fixes P :: "'h \<Rightarrow> bool"   and S :: "'i set"   and x :: "'i" shows "P (indicator S x) = ((x \<in> S \<longrightarrow> P (1::'h)) \<and> (x \<notin> S \<longrightarrow> P (0::'h)))"
conjunctionD2: fixes A :: "prop"   and B :: "prop" assumes "PROP A &&& PROP B" shows "PROP B"
map_FGcontra_def: fixes l1 :: "'aj \<Rightarrow> 'aw"   and co1 :: "'ak \<Rightarrow> 'ax"   and co2 :: "'as \<Rightarrow> 'bf"   and co3 :: "'al \<Rightarrow> 'ay"   and co4 :: "'am \<Rightarrow> 'az"   and co5 :: "'an \<Rightarrow> 'ba"   and contra1 :: "'bb \<Rightarrow> 'ao"   and contra2 :: "'bc \<Rightarrow> 'ap"   and contra3 :: "'bd \<Rightarrow> 'aq"   and contra4 :: "'be \<Rightarrow> 'ar"   and contra5 :: "'bg \<Rightarrow> 'au" shows "map_FGcontra l1 co1 co2 co3 co4 co5 contra1 contra2 contra3 contra4 contra5 = map_F l1 co1 co3 co1 co4 co5 (map_G contra1 contra2 contra3 contra4 co1 co2) contra1 contra5"
axioms_1: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and U :: "'d set" assumes "ind_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str U" shows "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"