hcju/isabelleps huggingface dataset:
simple_distributedI: fixes X :: "'key \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> real" assumes "simple_function (point_measure keys (\<lambda>x. ennreal (K x))) X"   and "\<And>x. 0 \<le> P x"   and "\<And>x. x \<in> X ` space (point_measure keys (\<lambda>x. ennreal (K x))) \<Longrightarrow> P x = K.prob (X -` {x} \<inter> space (point_measure keys (\<lambda>x. ennreal (K x))))" shows "simple_distributed (point_measure keys (\<lambda>x. ennreal (K x))) X P"
hm_2: shows "homeomorphic_map X' (subtopology X'' (g ` topspace X')) g"
assms: shows "comp_tfr\<^sub>s\<^sub>t arity Ana \<Gamma> M S"
ground\<^sub>l\<^sub>s_subls: fixes L :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "ground\<^sub>l\<^sub>s L" shows "L \<cdot>\<^sub>l\<^sub>s \<sigma> = L"
ccw_contr_on_line_left: fixes a :: "real"   and b :: "real"   and x :: "real"   and c :: "real"   and d :: "real" assumes "0 \<le> det3 (a, b) (x, c) (x, d)"   and "x < a" shows "d \<le> c"
MN: fixes J :: "'a set" assumes "J \<in> I" shows "A \<in> Pi\<^sub>F {J} M \<rightarrow>\<^sub>M N"
mantissa_exponent: fixes f :: "Float.float" shows "real_of_float f = real_of_int (mantissa f) * 2 powr real_of_int (Float.exponent f)"
list_rel_sv: fixes R :: "('c \<times> 'd) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>list_rel)"
simps_1: fixes e :: "'a"   and a :: "'b"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.tree_invar (SkewBinomialHeapStruc.Node e a 0 ts) = (ts = [])"
le_supI1: fixes a :: "'b"   and x :: "'b"   and b :: "'b" assumes "a \<le> x" shows "a \<sqinter> b \<le> x"
padic_one_simp_2: fixes n :: "nat"   and p :: "int" assumes "0 < n" shows "padic_one p n = 1"
productive_in_N: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset" assumes "production N C \<noteq> {}" shows "C \<in> N"
in_basis_1: fixes v :: "'b" assumes "f v \<noteq> (0::'a)" shows "v \<in> basis"
cases: fixes x :: "('a \<times> 'b) list \<times> 'a nat_term_order" obtains xs :: "('a \<times> 'b) list" and ko :: "'a nat_term_order" where "x = (xs, ko)"
degree_q: shows "q = 0 \<or> degree (q' + smult k q'') < n - d"
whiteI: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and r :: "'g" assumes "obj_at (\<lambda>obj. obj_mark obj = (\<not> sys_fM s)) r s" shows "white r s"
WCFG_edge_Exit_Skip: fixes prog :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind" assumes "prog \<turnstile> n -et\<rightarrow> (_Exit_)"   and "n \<noteq> (_Entry_)" shows "\<exists>l. n = (_ l _) \<and> labels prog l Skip \<and> et = \<Up>id"
assms_3: shows "sing_reg = (\<preceq>\<^sub>t) \<or> sing_reg = (\<prec>\<^sub>t)"
rel_\<I>_conversep: fixes C :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_\<I> C\<inverse>\<inverse> R\<inverse>\<inverse> = (rel_\<I> C R)\<inverse>\<inverse>"
arrI: fixes f :: "'s"   and a :: "'s"   and b :: "'s" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "S.arr f"
size_greater_than_median_of_medians: fixes d :: "nat"   and med :: "'b list \<Rightarrow> 'b"   and x :: "'b"   and xs :: "'b list" assumes "\<And>xs. \<lbrakk>xs \<noteq> []; length xs \<le> Suc (2 * d)\<rbrakk> \<Longrightarrow> is_median (med xs) xs"   and "is_median x (map med (chop (Suc (2 * d)) xs))" shows "size (filter_mset ((<) x) (mset xs)) \<le> nat \<lceil>(3 * real d + 1) / real (2 * (2 * d + 1)) * real (length xs)\<rceil> + 5 * d div 2 + 1"
tendsto_rabs: fixes f :: "'b \<Rightarrow> real"   and l :: "real"   and F :: "'b filter" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
m_def: shows "m = Min {(1 / 2) ^ to_nat i * e i |i. i \<in> I}"
steps_appendD2: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph_Defs.steps E (xs @ ys)"   and "ys \<noteq> []" shows "Graph_Defs.steps E ys"
h1: assumes "\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)" shows "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or> a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> ((\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<or> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0))) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
hyps: fixes f :: "nat \<Rightarrow> 'a" shows "poly_of_vec (vec n f) = Poly (rev (map f [0..<n]))"
locally_path_connected_space_product_topology: fixes X :: "'a \<Rightarrow> 'b topology"   and I :: "'a set" shows "locally_path_connected_space (product_topology X I) = (topspace (product_topology X I) = {} \<or> finite {i \<in> I. \<not> path_connected_space (X i)} \<and> (\<forall>i\<in>I. locally_path_connected_space (X i)))"
Zp_char_0': fixes n :: "nat" assumes "0 < n" shows "[n] \<cdot> \<one> \<noteq> \<zero>"
Nd: fixes n :: "nat" assumes "Nd \<le> n" shows "dist (P n) (P Nd) < 1"
g_def: shows "g = (\<lambda>x. if \<exists>n. (T ^^ n) x \<in> good_set then f ((T ^^ good_time x) x) else y0)"
simps_6: fixes v :: "Cooper.num" shows "not (Gt v) = NOT (Gt v)"
auto_local_lipschitz: shows "local_lipschitz UNIV X (\<lambda>_. - f)"
bi_unique_alt_def2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "bi_unique R = rel_fun R (rel_fun R (=)) (=) (=)"
sync_tLength_eq_trc: fixes t :: "'s Trace"   and t' :: "'s Trace"   and T :: "'s Trace set"   and as :: "'a set" assumes "(t, t') \<in> (\<Union> (relations (Clock.mkM T) ` as))\<^sup>*" shows "tLength t = tLength t'"
surjective: fixes r :: "('c, 'd, 'e) sas_plus_problem_scheme" shows "r = \<lparr>sas_plus_problem.variables_of = r\<^sub>\<V>\<^sub>+, operators_of = r\<^sub>\<O>\<^sub>+, initial_of = r\<^sub>I\<^sub>+, goal_of = r\<^sub>G\<^sub>+, range_of = range_of r, \<dots> = sas_plus_problem.more r\<rparr>"
test_d_fixpoint: fixes x :: "'a" assumes "test x" shows "d x = x"
assms: shows "[a, c]\<^sub>\<circ> \<in>\<^sub>\<circ> fconst_on A c"
FAccNull\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>1\<rangle>"
exec_step_ind_imp_exec_step: fixes si :: "step_input"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and xp' :: "nat option"   and h' :: "heap"   and frs' :: "frame list"   and sh' :: "sheap" assumes "exec_step_ind si P h stk loc C M pc ics frs sh (xp', h', frs', sh')"   and "exec_step_input P C M pc ics = si" shows "exec_step P h stk loc C M pc ics frs sh = (xp', h', frs', sh')"
Rep_phi_inject: fixes x :: "('c, 'd) phi"   and y :: "('c, 'd) phi" shows "(Rep_phi x = Rep_phi y) = (x = y)"
simps_2: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat" shows "C M2 M1 \<Omega> V m (Suc 0) = V"
Specker_aux: fixes \<alpha> :: "V" assumes "\<alpha> \<in> elts \<omega>" shows "Partitions.partn_lst pair_less UU [\<omega> \<up> ord_of_nat 2, \<alpha>] 2"
ring_distribs_1: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a * (b + c) = a * b + a * c"
matrix_to_iarray_row_add: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'d"   and q :: "'b" shows "matrix_to_iarray (row_add A i j q) = row_add_iarray (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j) q"
neg_of_aezfun_is_aezfun: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> aezfun_set" shows "- f \<in> aezfun_set"
distinct_1: fixes x2 :: "'f" shows "None \<noteq> Some x2"
has_derivative_divide: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within S)"   and "(g has_derivative g') (at x within S)"   and "g x \<noteq> (0::'b)" shows "((\<lambda>x. f x / g x) has_derivative (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within S)"
cmp_simps_3: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "trg\<^sub>D (\<Phi>\<^sub>G (f, g)) = trg\<^sub>D (G f)"
indicator_eq_0_iff: fixes A :: "'b set"   and x :: "'b" shows "(indicator A x = (0::'a)) = (x \<notin> A)"
runit'_in_hom_2: fixes f :: "'b" assumes "D.ide f" shows "\<guillemotleft>\<r>\<^sub>D\<^sup>-\<^sup>1[f] : f \<Rightarrow>\<^sub>D f \<star>\<^sub>D src\<^sub>D f\<guillemotright>"
FAddSA_HierAuto: fixes SA :: "('d, 'e, 'f) seqauto"   and F :: "('d, 'e, 'f) seqauto set"   and S :: "'d"   and D :: "'f data"   and E :: "'e set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" assumes "States SA \<inter> \<Union> (States ` F) = {}"   and "S \<in> \<Union> (States ` F)"   and "HierAuto D F E G" shows "HierAuto D (insert SA F) (E \<union> SAEvents SA) (G [f+] (S, SA))"
upd_rows_insert_rev: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and I :: "'d set"   and f :: "'d \<Rightarrow> ('c, 'd) vec" shows "upd_rows M (insert i I) f = upd_rows (upd_row M i (f i)) I f"
NTCompDefinedI: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans \<eta>1"   and "NatTrans \<eta>2"   and "NTCatDom \<eta>2 = NTCatDom \<eta>1"   and "NTCatCod \<eta>2 = NTCatCod \<eta>1"   and "NTCod \<eta>1 = NTDom \<eta>2" shows "\<eta>1 \<approx>>\<bullet> \<eta>2"
symmetric_connected_imp_strongly_connected: assumes "symmetric G"   and "Digraph_Component.connected G" shows "strongly_connected G"
UN_singleton: fixes A :: "'c set" shows "(\<Union>x\<in>A. {x}) = A"
fs_tyvrs_inst: shows "fs TYPE('a) TYPE(tyvrs)"
assms: shows "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>"
qp_close_under_unitary_operator: fixes U :: "complex mat"   and P :: "complex mat" assumes "U \<in> carrier_mat ps_P.d ps_P.d"   and "unitary U"   and "ps_P.is_quantum_predicate P" shows "ps_P.is_quantum_predicate (adjoint U * P * U)"
node_deliver_messages_Broadcast: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m :: "'c" assumes "network history msg_id" shows "trivial_network.node_deliver_messages [Broadcast m] = []"
zeta_partial_sum_le_neg: assumes "0 < s" shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto (\<lambda>n. real n powr s) x - x powr (1 + s) / (1 + s)\<bar> \<le> c * x powr s"
fsps_chain_boundTr1: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set"   and fa :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "C \<subseteq> fsps R M N f H"   and "\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a"   and "\<forall>a b. (a, b) \<in> C \<longrightarrow> (a, b) \<in> fsps R M N f H"   and "\<exists>x. (\<exists>b. (x, b) \<in> C) \<and> x \<noteq> {}" shows "fa \<in> {j. j \<le> n} \<rightarrow> \<Union> {a. \<exists>b. (a, b) \<in> C} \<longrightarrow> (\<exists>(c, d)\<in>C. fa ` {j. j \<le> n} \<subseteq> c)"
trie_lookup_update: fixes k :: "bool list"   and v :: "'b"   and t :: "'b trie_raw" shows "trie_lookup (trie_update k v t) = trie_lookup t(k \<mapsto> v)"
connected_arc_complement: fixes \<gamma> :: "real \<Rightarrow> 'b" assumes "arc \<gamma>"   and "2 \<le> DIM('b)" shows "connected (- path_image \<gamma>)"
new_shadow_root_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'bb" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
b_assn_simp: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" shows "b_assn A P x y = A x y * \<up> (P x)"
tendsto_complex_of_real_iff: fixes f :: "'b \<Rightarrow> real"   and c :: "real"   and F :: "'b filter" shows "((\<lambda>x. complex_of_real (f x)) \<longlongrightarrow> complex_of_real c) F = (f \<longlongrightarrow> c) F"
add_positive_le_reduce1: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "positive B"   and "A + B \<le>\<^sub>L C" shows "A \<le>\<^sub>L C"
gHom_comp_gsurjec: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and H :: "('i, 'j) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and g :: "'g \<Rightarrow> 'i" assumes "Group F"   and "Group G"   and "Group H"   and "gsurj\<^bsub>F,G\<^esub> f"   and "gsurj\<^bsub>G,H\<^esub> g" shows "gsurj\<^bsub>F,H\<^esub> (g \<circ>\<^bsub>F\<^esub> f)"
pprime_iff_pirreducible: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "pprime\<^bsub>K [X]\<^esub> K p = pirreducible\<^bsub>K [X]\<^esub> K p"
univ_order_rel_def: shows "univ_order_rel \<equiv> SOME r. well_order r"
f2r_Inf_pres: shows "Inf_pres \<R>"
is_sentence_concat: fixes x :: "symbol list"   and y :: "symbol list" shows "is_sentence (x @ y) = (is_sentence x \<and> is_sentence y)"
simps_1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "nat \<Rightarrow> 'b" shows "foldseq f s 0 = s 0"
ih1: assumes "\<Gamma> \<turnstile> q \<leftrightarrow> r : T\<^sub>1\<rightarrow>T\<^sub>2" shows "\<Gamma> \<turnstile> p \<leftrightarrow> r : T\<^sub>1\<rightarrow>T\<^sub>2"
eval_Mn_diverg: fixes n :: "nat"   and f :: "recf"   and xs :: "nat list" assumes "recfn (length xs) (Mn n f)" shows "(\<nexists>z. eval f (z # xs) \<down>= 0 \<and> (\<forall>y<z. eval f (y # xs) \<down>)) = (eval (Mn n f) xs \<up>)"
leg1_is_map: shows "is_left_adjoint (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1)"
rFrameAlpha: fixes A\<^sub>P :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d"   and \<Psi>\<^sub>P :: "'b" assumes "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* (p \<bullet> A\<^sub>P)"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* C"   and "set p \<subseteq> set A\<^sub>P \<times> set (p \<bullet> A\<^sub>P)"   and "distinctPerm p"   and "A\<^sub>P \<sharp>* subject \<alpha>"   and "Prop C \<Psi> P \<alpha> P' A\<^sub>P \<Psi>\<^sub>P" shows "Prop C \<Psi> P \<alpha> P' (p \<bullet> A\<^sub>P) (p \<bullet> \<Psi>\<^sub>P)"
goal_not_eval: shows "\<not> sinvar \<lparr>nodes = V, edges = insert e E\<rparr> nP"
less_eq_action: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "P \<le> Q \<equiv> P \<sqsubseteq> Q"
sel_1: fixes x21 :: "'c"   and x22 :: "'c list" shows "hd (x21 # x22) = x21"
get_shadow_root_safe_def: shows "get_shadow_root_safe = (\<lambda>element_ptr. Heap_Error_Monad.bind (get_shadow_root element_ptr) (case_option (return None) (\<lambda>shadow_root_ptr. Heap_Error_Monad.bind (get_mode shadow_root_ptr) (\<lambda>mode. if mode = Open then return (Some shadow_root_ptr) else return None))))"
cring_simprules_5: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes> y \<in> carrier R"
mult1E: fixes N :: "'c multiset"   and M :: "'c multiset"   and r :: "('c \<times> 'c) set" assumes "(N, M) \<in> mult1 r" obtains a :: "'c" and M0 :: "'c multiset" and K :: "'c multiset" where "M = add_mset a M0" and "N = M0 + K" and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r"
cases: fixes a1 :: "('a, 'b, 'c) psi"   and a2 :: "('a, 'b, 'c) psi" assumes "bangPred a1 a2" obtains P :: "('a, 'b, 'c) psi" where "a1 = Pa__" and "a2 = !Pa__"   | P :: "('a, 'b, 'c) psi" where "a1 = P" and "a2 = P \<parallel> !P"
fMax_ffilter_less: fixes x :: "'a"   and P :: "'a fset"   and n :: "'a" assumes "x |\<in>| P"   and "x < n" shows "fMax (ffilter (\<lambda>i. i < n) P) < n"
stable_def: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool" shows "stable P = ((\<exists>\<sigma>. P \<sigma>) \<and> (\<forall>\<sigma> i. P \<sigma> \<longrightarrow> P (\<sigma> |\<^sub>s i)))"
lspasl_starl_inv: fixes Gamma :: "bool"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h0 :: "'a"   and Delta :: "bool" assumes "Gamma \<and> (A \<and>* B) h0 \<longrightarrow> Delta" shows "(\<exists>h1 h2. Gamma \<and> (h1,h2\<triangleright>h0) \<and> A h1 \<and> B h2) \<longrightarrow> Delta"
intros_1: fixes F :: "'effect fs_set"   and P :: "'state"   and Q :: "'state" assumes "P \<sim>\<cdot>[F] Q" shows "L_bisimilar (EF (F, P)) (EF (F, Q))"
distinct_8: fixes x31 :: "'b"   and x32 :: "('a, 'b) form list"   and x21 :: "'a"   and x22 :: "('a, 'b) form list" shows "Modal x31 x32 \<noteq> Compound x21 x22"
distinct_Ex1: fixes xs :: "'d list"   and x :: "'d" assumes "distinct xs"   and "x \<in> set xs" shows "\<exists>!i. i < length xs \<and> xs ! i = x"
perp_down_incident: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident b l" shows "proj2_incident (perp_down a l) (drop_perp a l)"
par_beta_subst: fixes s :: "dB"   and s' :: "dB"   and t :: "dB"   and t' :: "dB"   and n :: "nat" assumes "s => s'"   and "t => t'" shows "t[s/n] => t'[s'/n]"
app: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and x :: "'v"   and e :: "('r, 'l, 'v) expr"   and v :: "('r, 'l, 'v) val" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Apply (VE (Lambda x e)) (VE v)])" shows "revision_step r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [subst (VE v) x e])))"
incident_def: fixes p :: "'a"   and b :: "'a set" shows "incident p b \<equiv> (p, b) \<in> \<I>"
u_def: fixes n :: "nat" shows "u n = snd (bisect n)"
known_ptrs_preserved: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "object_ptr_kinds h = object_ptr_kinds h'" shows "ShadowRootClass.known_ptrs h = ShadowRootClass.known_ptrs h'"
Bocardo: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set" assumes "A Z C"   and "B Q C" shows "A Z B"
uint_le_0_iff: fixes x :: "'a Word.word" shows "(uint x \<le> 0) = (uint x = 0)"
add_neg_nonpos: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less a zero"   and "less_eq b zero" shows "less (plus a b) zero"
invertible_lf_axioms: shows "invertible_lf (*a) (*b) f"
simps_2: fixes uu :: "'a"   and v :: "'b \<times> 'a"   and va :: "('b \<times> 'a) list" shows "Dioid_Models.last (uu, v # va) = snd (List.last (v # va))"
local_order_prefix_closed: fixes x :: "('a \<times> ('a, 'b) operation) event"   and i :: "nat"   and y :: "('a \<times> ('a, 'b) operation) event"   and xs :: "('a \<times> ('a, 'b) operation) event list" assumes "x \<sqsubset>\<^sup>i y"   and "xs prefix of i"   and "y \<in> set xs" shows "x \<in> set xs"
partial_magma_axioms: shows "partial_magma (\<cdot>\<^sub>A)"
\<U>: fixes S :: "'a set" assumes "S \<in> \<U>" shows "connectedin X S"
collapse: fixes optionT :: "('d, 'e) optionT" shows "OptionT (run_option optionT) = optionT"
isdus: fixes d :: "'b word \<Rightarrow> 'c word" assumes "is_down d" shows "is_up SCAST('c \<rightarrow> 'b)"
GPV_bind: fixes f :: "('h, 'f, ('h, 'f, 'g) rpv) generat spmf"   and g :: "'h \<Rightarrow> ('e, 'f, 'g) gpv" shows "GPV f \<bind> g = GPV (f \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> the_gpv (g x) | IO out c \<Rightarrow> return_spmf (IO out (\<lambda>input. c input \<bind> g))))"
leg1_simps_4: shows "trg s\<^sub>1 = trg s"
rev_vsubsetD: fixes c :: "V"   and a :: "V"   and b :: "V" assumes "c \<in> elts a"   and "a \<le> b" shows "c \<in> elts b"
Equiv_Exec_stable_set_UNIV: fixes equiv_states :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and localState1 :: "'g \<Rightarrow> 'e"   and localState2 :: "'h \<Rightarrow> 'f"   and input_fun1 :: "'d \<Rightarrow> 'i message_af"   and input_fun2 :: "'d \<Rightarrow> 'j message_af"   and output_fun1 :: "'g \<Rightarrow> 'k message_af"   and output_fun2 :: "'h \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and trans_fun2 :: "'j message_af \<Rightarrow> 'h \<Rightarrow> 'h"   and k1 :: "nat"   and k2 :: "nat"   and c1 :: "'g"   and c2 :: "'h" shows "Equiv_Exec_stable_set UNIV equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2 = Equiv_Exec_stable equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2"
norm_Pair_le: fixes x :: "'a"   and y :: "'b" shows "norm (x, y) \<le> norm x + norm y"
ComplI: fixes c :: "'g"   and A :: "'g set" assumes "c \<in> A \<Longrightarrow> False" shows "c \<in> - A"
intro: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "\<And>i p is \<O> \<R> \<D> \<theta> sb. \<lbrakk>i < length ts; ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)\<rbrakk> \<Longrightarrow> causal_program_history is sb"   and "\<And>i p is \<O> \<R> \<D> \<theta> sb. \<lbrakk>i < length ts; ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)\<rbrakk> \<Longrightarrow> last_prog p sb = p" shows "valid_program_history ts"
numeral_uint32: fixes n :: "num" shows "numeral n = Uint32 (numeral n)"