hcju/isabelleps huggingface dataset:
isSubobj_eq_Subobjs: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list" assumes "wf_prog wf_md P" shows "is_subobj P (C, Cs) = Subobjs P C Cs"
black_heap_valid_ref_marked_insertions: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state"   and m :: "'e" assumes "black_heap s"   and "valid_refs_inv s" shows "mut_m.marked_insertions m s"
r_universal: fixes f :: "recf"   and n :: "nat"   and xs :: "nat list" assumes "recfn n f"   and "length xs = n" shows "eval (r_universal n) (encode f # xs) = eval f xs"
simps: fixes rt :: "recf"   and rf :: "recf" shows "rec_all rt rf = (let vl = arity rf in Cn (vl - 1) rec_sg [Cn (vl - 1) (rec_accum rf) (get_fstn_args (vl - 1) (vl - 1) @ [rt])])"
chain_subset: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c set" assumes "Complete_Partial_Order.chain ord A"   and "B \<subseteq> A" shows "Complete_Partial_Order.chain ord B"
induct: fixes x :: "'c llist"   and P :: "'c llist \<Rightarrow> bool" assumes "lfinite x"   and "P LNil"   and "\<And>x xs. \<lbrakk>lfinite xs; P xs\<rbrakk> \<Longrightarrow> P (LCons x xs)" shows "P x"
globally_sound_mode_use_def: fixes gc :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" shows "conc.globally_sound_mode_use gc \<equiv> \<forall>mdss. mdss \<in> conc.reachable_mode_states gc \<longrightarrow> conc.compatible_modes mdss"
it_a: shows "set_iterator_genord it_a S_a R_a"
equiv_up_to_refl: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com" shows "P \<Turnstile> c \<sim> c"
hyps_3: shows "y \<preceq>\<^sub>p p"
valid_Pred: fixes P :: "'state"   and \<phi> :: "unit" shows "P \<Turnstile>\<^sub>S Formula.Pred \<phi> = P \<turnstile>\<^sub>S \<phi>"
N: shows "x [^] N = \<one> \<and> 0 < N"
bind_pure_returns_result_I: fixes f :: "('l, 'm, 'n) prog"   and h :: "'l"   and x :: "'n"   and g :: "'n \<Rightarrow> ('l, 'm, 'o) prog"   and y :: "'o" assumes "pure f h"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
IH: fixes i :: "nat" shows "\<exists>J F X. X \<in> sets M \<and> F \<in> J \<rightarrow> sets M \<and> countable J \<and> A i = (UNIV - J) \<times> X \<union> Sigma J F"
igSubstIGVar1_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSubstIGVar1 MOD \<equiv> \<forall>ys y Y xs x. igWls MOD (asSort ys) Y \<longrightarrow> ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igSubst MOD ys Y y (igVar MOD xs x) = igVar MOD xs x"
alt_fail2: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and fail :: "'e"   and alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and m :: "'e" assumes "monad_fail_alt return bind fail alt" shows "alt m fail = m"
Card_order_Pow: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "r <o |Pow (Field r)|"
cardofpairs: fixes S :: "'c list" assumes "S \<noteq> []"   and "sorted S"   and "distinct S" shows "real (card {(x, y). x \<in> set S \<and> y \<in> set S \<and> x < y}) = real (length S * (length S - 1)) / 2"
intros_6: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and k :: "nat"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}"   and "\<And>s. P s \<le> enat k * P' s"   and "\<And>s. enat k * Q' s \<le> Q s"   and "0 < k" shows "\<turnstile>\<^sub>2\<^sub>' {P'} c {Q'}"
ordermap_in_ordertype: fixes a :: "'c"   and A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "a \<in> A"   and "small A" shows "ordermap A r a \<in> elts (ordertype A r)"
argmax_1: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "fst (argmax f xs) \<in> set xs"
outer_transitivity_between2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Bet A B C"   and "Bet B C D"   and "B \<noteq> C" shows "Bet A C D"
cbilinear_def: shows "cbilinear = (\<lambda>f. (\<forall>y. clinear (\<lambda>x. f x y)) \<and> (\<forall>x. clinear (f x)))"
ritem_not_in_ltree: fixes n :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat" assumes "avl (MKT n l r h)"   and "is_ord (MKT n l r h)"   and "r \<noteq> ET" shows "rightmost_item r \<notin> set_of l"
amor_le_long: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "int (T_merge t1 t2) + \<Phi> (merge t1 t2) - \<Phi> t1 - \<Phi> t2 \<le> int (lrh (merge t1 t2) + rlh t1 + rlh t2 + 1)"
that: fixes q :: "'b list"   and r :: "'b list"   and s :: "'b list" assumes "p = q @ r @ s"   and "distinct (awalk_verts u q)"   and "\<exists>w. awalk u q w \<and> awalk w r w \<and> awalk w s v"   and "closed_w r" shows "thesis"
a_r_congruent_def: fixes G :: "('c, 'd) ring_scheme" shows "racong\<^bsub>G\<^esub> = r_congruent (add_monoid G)"
err1_def: shows "err1 \<equiv> \<lambda>s r. contour_integral (part_circlepath 0 r pi 0) (f s)"
sharing_consistent_share_all_shared: fixes sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option" shows "dom (share sb \<S>) \<subseteq> dom \<S> \<union> all_shared sb"
ereal_eq_1_1: fixes r :: "real" shows "(ereal r = 1) = (r = 1)"
nonsat_potential_bound: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" assumes "Height_Bounded_Labeling c s t f l" shows "Network.nonsat_potential c f l \<le> 2 * (card (Graph.V c))\<^sup>2"
WW_def: shows "WW \<equiv> {l. strict_sorted l}"
null_space_eq_solution_set: fixes A :: "(('b, 'c) vec, 'd) vec" shows "null_space A = solution_set A 0"
inf_Int_eq: fixes R :: "'a set"   and S :: "'a set" shows "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
vertical_chop_leq2: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u--w \<longrightarrow> w \<le> v"
Rep_one_matrix: fixes n :: "nat"   and j :: "nat"   and i :: "nat" shows "Rep_matrix (one_matrix n) j i = (if j = i \<and> j < n then 1::'b else (0::'b))"
g_def: fixes x :: "'a" shows "g x = lim (\<lambda>i. f (seq x i))"
Join_left_absorb: fixes F :: "'b program"   and G :: "'b program" shows "F \<squnion> (F \<squnion> G) = F \<squnion> G"
Figure_6: fixes k :: "nat"   and js :: "nat list"   and i :: "nat" assumes "\<And>i. i \<le> k \<Longrightarrow> foldl1 f1 (map (h1 k i) js) = One"   and "k < i" shows "i \<notin> set js"
assms_2: shows "termFSwSbImorph h hA MOD"
remove_heap_is_wellformed_preserved: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.remove child \<rightarrow>\<^sub>h h'" shows "Shadow_DOM.heap_is_wellformed h'"
assms: shows "mgu\<^sub>t\<^sub>s \<theta> (literal_to_term ` L)"
dvd_def: fixes times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and b :: "'k"   and a :: "'k" shows "dvd.dvd times b a = (\<exists>k. a = times b k)"
agree_on_comm: fixes X :: "('e, 'f) labeled_graph"   and f :: "('f \<times> 'g) set"   and g :: "('f \<times> 'g) set" shows "agree_on X f g = agree_on X g f"
diff: fixes x :: "complex"   and y :: "complex"   and xa :: "'b" shows "(x - y) *\<^sub>C xa = x *\<^sub>C xa - y *\<^sub>C xa"
Guard: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and f :: "'f"   and c :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "s \<in> g" shows "\<Gamma>\<turnstile> (Guard f g c # cs, css, Normal s) \<rightarrow> (c # cs, css, Normal s)"
del_invalid_point: fixes p :: "'a"   and bl :: "'a set" assumes "p \<notin> \<V> \<union> bl" shows "add_block_sys.del_point bl p = \<V> \<union> bl"
f_def: shows "f \<equiv> \<lambda>s z. exp (Ln' z * - s + complex_of_real a * z) / (1 - exp z)"
induced_image: shows "induced ` kernel.Partition = \<eta> ` G"
\<Psi>_DF: fixes l :: "nat" shows "\<Psi> (DF (Suc l)) l"
elims: fixes x :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat"   and y :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "rem_implicit_pres x = y" obtains name :: "char list" and preconds :: "(nat \<times> nat) list" and effects :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list" and cost :: "nat" where "x = (name, preconds, effects, cost)" and "y = (name, implicit_pres effects @ preconds, map rem_effect_implicit_pres effects, cost)"
set_disconnected_nodes_get_parent: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_parent_locs. r h h')"
mat_onel: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "\<epsilon> \<otimes> f = f"
prob_space_distr: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b"   and M' :: "'b measure" assumes "M'.random_variable M' f" shows "prob_space (distr M'__ M' f)"
7: shows "bijective ((d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top) * top)"
map_map_snd: fixes f :: "'c \<rightarrow> 'b"   and xs :: "[\<langle>'d, 'c\<rangle>]" assumes "f\<cdot>\<bottom> = \<bottom>" shows "map\<cdot>f\<cdot>(map\<cdot>snd\<cdot>xs) = map\<cdot>snd\<cdot>(map\<cdot>(\<Lambda> \<langle>x, y\<rangle>. \<langle>x, f\<cdot>y\<rangle>)\<cdot>xs)"
Q: shows "Q' = Polynomial.smult (inverse (of_nat c)) (of_int_poly Q)"
double_fold_C0: fixes v :: "'a" assumes "v \<in> f ` D0 - D0" shows "f (g v) = v"
simps_4: fixes r :: "real" shows "val_type (RealVal r) = REAL"
lmap_eq_LNil: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "(lmap f xs = LNil) = (xs = LNil)"
LIMSEQ_le_const2: fixes X :: "nat \<Rightarrow> 'b"   and x :: "'b"   and a :: "'b" assumes "X \<longlonglongrightarrow> x"   and "\<exists>N. \<forall>n\<ge>N. X n \<le> a" shows "x \<le> a"
subst4Chain: fixes xvec :: "name list"   and Tvec :: "'a list"   and T :: "'a" assumes "length xvec = length Tvec"   and "distinct xvec"   and "xvec \<sharp>* Tvec" shows "xvec \<sharp>* substTerm T xvec Tvec"
read_only: fixes h :: "heap" assumes "success p h" shows "snd (the (execute p h)) = h"
assms_3: shows "\<forall>x. poly (char_poly A) x = 0 \<longrightarrow> x \<le> 1"
k0: fixes x :: "'a" shows "k (0, x) = f x"
n_t_compl: fixes x :: "'a" shows "!x \<oplus> t x = (1::'a)"
compl_eq_compl_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "(uminus x = uminus y) = (x = y)"
W_conv_sum_lists: shows "\<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle> = (\<Union>ss\<in>lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}). {sum_list ss})"
r_one: fixes x :: "'a" assumes "x \<in> carrier \<G>" shows "x \<otimes> \<one> = x"
indicator_sum_eq: fixes m :: "real"   and n :: "nat"   and f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "\<bar>m\<bar> \<le> 2 ^ (2 * n)"   and "m / 2 ^ n \<le> f x"   and "f x < (m + 1) / 2 ^ n"   and "m \<in> \<int>" shows "(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2 * n). k / 2 ^ n * indicat_real {y. k / 2 ^ n \<le> f y \<and> f y < (k + 1) / 2 ^ n} x) = m / 2 ^ n"
dH_indistinguishable_def: fixes E :: "'c \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and DA :: "'d \<Rightarrow> 'f"   and d :: "'f"   and H :: "('f \<times> 'c) set"   and e1 :: "'c"   and e2 :: "'c" shows "WHATWHERE_Secure_Programs.dH_indistinguishable E DA d H e1 e2 \<equiv> \<forall>m m'. WHATWHERE.dH_equal E DA d H m m' \<longrightarrow> E e1 m = E e2 m'"
rel_interior_scaleR: fixes c :: "real"   and S :: "'a set" assumes "c \<noteq> 0" shows "(*\<^sub>R) c ` rel_interior S = rel_interior ((*\<^sub>R) c ` S)"
reduced_GB_pmdl_finite: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite F" shows "local.punit.pmdl (punit.reduced_GB F) = local.punit.pmdl F"
wffp_isState_doo_purgeIdle: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list" assumes "GM_sec_model.wffp do sl"   and "isState (hd sl)" shows "GM_sec_model.doo do (local.getGMState (hd sl)) (map getGMUserCom (tl (GM_sec_model.purgeIdle sl))) = local.getGMState (last sl)"
SC_WbisT_imp_SC_BisT: fixes c :: "('test, 'atom) com" assumes "SC_WbisT c" shows "SC_BisT c"
forward_terminating_path_end_points_1: fixes x :: "'a" assumes "forward_terminating_path x" shows "x \<le> x\<^sup>+ ; end_points x"
inclusion_preserves_inverse: fixes f :: "'a"   and g :: "'a" assumes "inverse_arrows f g" shows "C.inverse_arrows f g"
geodesic_segment_between_real: fixes x :: "real"   and y :: "real"   and G :: "real set" assumes "x \<le> y" shows "geodesic_segment_between G x y = (G = {x..y})"
retraction_openin_vimage_iff: fixes S :: "'a set"   and T :: "'a set"   and r :: "'a \<Rightarrow> 'a"   and U :: "'a set" assumes "retraction S T r"   and "U \<subseteq> T" shows "openin (top_of_set S) (S \<inter> r -` U) = openin (top_of_set T) U"
iso_class_is_nonempty: fixes F :: "'a set" assumes "is_iso_class F" shows "F \<noteq> {}"
f_join_all: fixes xs :: "'b list"   and I :: "nat set" assumes "{..<length xs} \<subseteq> I" shows "xs \<Join>\<^sub> I = xs"
indiff_set: shows "indiff_set (R i) (Max_wrt_among (R i) (rsd_winners R alts agents'))"
simps_2: fixes nodesL :: "'b list"   and edgesL :: "('b \<times> 'b) list"   and more :: "'c" shows "edgesL \<lparr>nodesL = nodesL, edgesL = edgesL, \<dots> = more\<rparr> = edgesL"
matrix_vector_mul_lid: fixes x :: "('e, 'f) vec" shows "mat (1::'e) *v x = x"
hrealpow_two: fixes r :: "real star" shows "r ^ Suc (Suc 0) = r * r"
finite_set_mset: fixes M :: "'k multiset" shows "finite (set_mset M)"
expl_cond_expect_prop_sets2: fixes Y :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and D :: "'c set"   and P :: "real \<Rightarrow> bool"   and X :: "'c \<Rightarrow> real" assumes "disc_fct Y"   and "point_measurable (fct_gen_subalgebra M N Y) (space N) Y"   and "D = {w \<in> space M. Y w \<in> space N \<and> P (expl_cond_expect M Y X w)}" shows "D \<in> sets (fct_gen_subalgebra M N Y)"
cross_alt_def: fixes p :: "real poly"   and q :: "real poly"   and a :: "real"   and b :: "real" shows "cross_alt p q a b = \<bar>sign (poly p a) - sign (poly q a)\<bar> - \<bar>sign (poly p b) - sign (poly q b)\<bar>"
extensional_funcset_def: fixes S :: "'c set"   and T :: "'d set" shows "S \<rightarrow>\<^sub>E T = (S \<rightarrow> T) \<inter> extensional S"
hm_\<alpha>_def: shows "hm_\<alpha> \<equiv> ahm_\<alpha> \<circ> hm_\<alpha>' \<circ> impl_of_RBT_HM"
vars_def: fixes e :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" shows "Bexp.vars e = {v. \<exists>\<sigma> val. e (\<sigma>(v := val)) \<noteq> e \<sigma>}"
WbisT_MtransT: fixes indis :: "('a \<times> 'a) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a"   and s' :: "'a" assumes "PL_Indis indis"   and "(c, d) \<in> PL_Indis.WbisT tval aval indis"   and "(s, t) \<in> indis"   and "PL.MtransT tval aval (c, s) s'" obtains t' :: "'a" where "PL.MtransT tval aval (d, t) t'" and "(s', t') \<in> indis"
stable_def: fixes A :: "'c set" shows "stable A \<equiv> A co A"
simps: fixes p :: "'a Point"   and q :: "'a Point"   and r :: "'a Point" shows "collinear p q r = (\<exists>\<alpha> \<beta>. \<alpha> + \<beta> = (1::'a) \<and> positionVector p = \<alpha> ** positionVector q \<oplus> \<beta> ** positionVector r)"
underS_subset_under: fixes r :: "('c \<times> 'c) set"   and a :: "'c" shows "underS r a \<subseteq> under r a"
vrrestriction_vsubset_vpairs: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>r\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> vpairs r"
prems_2: shows "INIT_class e0 = \<lfloor>C\<rfloor>"
trancl_list_impl: fixes r :: "('a \<times> 'a) list"   and as :: "'a list" shows "set (trancl_list_impl r as) = {b. \<exists>a\<in>set as. (a, b) \<in> (set r)\<^sup>+}"
cobounded1: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "signed.min a b \<le>s a"
cond_symm: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<triangleleft> b \<triangleright>  Q) = (Q \<triangleleft> \<not> b \<triangleright>  P)"
left_commute: fixes b :: "'e"   and a :: "'e"   and c :: "'e" shows "sup b (sup a c) = sup a (sup b c)"
M_def: shows "M = {g \<in> G. q g \<odot> g \<noteq> 0 \<and> lt (q g \<odot> g) = v}"
arctan_bounds: shows "(\<Sum>i = 0..<n. (- 1) ^ i * (1 / real (i * 2 + 1) * sqrt (real_of_float y) ^ (i * 2 + 1))) \<le> arctan (sqrt (real_of_float y)) \<and> arctan (sqrt (real_of_float y)) \<le> (\<Sum>i = 0..<Suc n. (- 1) ^ i * (1 / real (i * 2 + 1) * sqrt (real_of_float y) ^ (i * 2 + 1)))"
list_le_def: fixes xs :: "'b list"   and ys :: "'b list" shows "list_le xs ys = (list_less xs ys \<or> xs = ys)"
matrix_to_iarray_morph: fixes A :: "(('a, 'b) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'c) vec" shows "(A = B) = (matrix_to_iarray A = matrix_to_iarray B)"
assms_1: shows "insert_ops (ops @ [(oid, Some ref)])"
calculatePrios_def: fixes ess :: "edge list list" shows "calculatePrios ess = map (\<lambda>es. (min_prio es 0, es)) ess"
nth_prefixes: fixes n :: "nat"   and p :: "'c list" assumes "n < length p"   and "\<not> Suc n < length p" shows "tl (prefixes p) ! n = p"
dom_llp: fixes x :: "'a"   and y :: "'a" shows "(x \<le> x \<cdot> ar ar y) = (ar ar x \<le> ar ar y)"
depthInput: fixes a :: "name"   and x :: "name"   and P :: "pi" shows "depth (a<x>.P) = 1 + depth P"
product_in_factor_by_sum: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CEM"   and "O x y" shows "P (x \<otimes> y) (x \<otimes> (y \<oplus> z))"
assms_2: fixes n :: "nat" shows "ValidTC (i \<inter> b \<inter> {s. v s = n}) c (i \<inter> {s. v s < n})"
\<tau>diverge_bisim_inv: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1 \<approx> s2" shows "r1.\<tau>diverge t s1 = r2.\<tau>diverge t s2"
rep_assoc: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "\<F>C.ide a"   and "\<F>C.ide b"   and "\<F>C.ide c" shows "\<F>C.rep \<a>[a, b, c] = \<^bold>\<parallel>\<^bold>\<a>\<^bold>[\<F>C.rep a, \<F>C.rep b, \<F>C.rep c\<^bold>]\<^bold>\<parallel>"
carrier_not_empty: shows "carrier (residue_ring (int p)) \<noteq> {}"
dvd_prod_eqI: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and A :: "'b set"   and a :: "'b"   and b :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "a \<in> A"   and "b = f a" shows "dvd.dvd times b (comm_monoid_mult.prod times one f A)"
add_less_cancel_left: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus c a) (plus c b) = less a b"
satisfies_bounded_nFOr: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile>\<^sub>b nFOr \<phi> \<psi>) = (\<AA> \<Turnstile>\<^sub>b FOr \<phi> \<psi>)"
inv_closed: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<in> carrier G"
rec_inseparable_mono: fixes A :: "nat set"   and B :: "nat set"   and A' :: "nat set"   and B' :: "nat set" assumes "rec_inseparable A B"   and "A \<subseteq> A'"   and "B \<subseteq> B'" shows "rec_inseparable A' B'"
step_3_c_similar: fixes A :: "'b mat"   and n :: "nat"   and l :: "nat"   and k :: "nat"   and blocks :: "(nat \<times> nat) list"   and x :: "'b" assumes "A \<in> carrier_mat n n"   and "l < k"   and "k < n"   and "\<And>i_begin i_end. (i_begin, i_end) \<in> set blocks \<Longrightarrow> i_end \<le> k \<and> i_end - i_begin \<le> l" shows "similar_mat (step_3_c x l k blocks A) A"
induct: fixes P :: "nat \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'a list" assumes "\<And>i xs. (\<And>x. \<lbrakk>i \<noteq> 0; x \<in> set_pmf (pmf_of_set {..i})\<rbrakk> \<Longrightarrow> P (i - 1) (swap xs i x)) \<Longrightarrow> P i xs" shows "P a0 a1"
add_edge_is_path: fixes x :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and y :: "'v"   and a :: "'v"   and b :: "'w"   and c :: "'v" assumes "is_path_undir G x p y" shows "is_path_undir (add_edge a b c G) x p y"
assms_2: shows "x \<in> MPLS_s PROB s"
between_midpoint_1: fixes a :: "'a"   and b :: "'a" shows "between (a, b) (midpoint a b)"
one_dim_prod_scale1: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and canonical_basis :: "'b list"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "complex"   and b :: "complex" assumes "class.one_dim scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner canonical_basis divide inverse one times" shows "times (scaleC a one) (scaleC b one) = scaleC (a * b) one"
caratheodory_aff_dim: fixes p :: "'b set" shows "convex hull p = {x. \<exists>S. finite S \<and> S \<subseteq> p \<and> int (card S) \<le> aff_dim p + 1 \<and> x \<in> convex hull S}"
gt_trans: fixes u :: "('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary u"   and "wary t"   and "wary s"   and "u >\<^sub>t t"   and "t >\<^sub>t s" shows "u >\<^sub>t s"
A_def: shows "A = {real k / 2 ^ n * dist x0 y0 |k n. k \<le> 2 ^ n}"
right_neutral: fixes a :: "'a" shows "a \<cdot> (1::'a) = a"
Sin: shows "((\<lambda>x. sin (c x)) expands_to Sin) basis"
of_weak_ranking_Collect_ge_Cons_1: fixes y :: "'c"   and x :: "'c set"   and xs :: "'c set list" assumes "y \<in> x" shows "of_weak_ranking_Collect_ge (x # xs) y = \<Union> (set (x # xs))"