hcju/isabelleps huggingface dataset:
simps_12: fixes e :: "('f, 'g, 'h) exp"   and F :: "String.literal"   and D :: "String.literal"   and A :: "'f set option" shows "\<D> (e\<bullet>F{D}) A = \<D> e A"
space_lebesgue_on: fixes S :: "'b set" shows "space (lebesgue_on S) = S"
rep_eq: shows "Rep_float \<infinity> = (0, - 1, 0)"
assms: shows "u \<le>p z \<cdot> u \<cdot> w"
eq_len_iff_less: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v" shows "(\<^bold>|x\<^bold>| < \<^bold>|u\<^bold>|) = (\<^bold>|v\<^bold>| < \<^bold>|y\<^bold>|)"
simps_1: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" shows "fwi m n k 0 0 = fw_upd m k 0 0"
INF_cong_simp: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
eq_divide_imp: fixes c :: "'a fps"   and a :: "'a fps"   and b :: "'a fps" assumes "c \<noteq> 0"   and "a * c = b" shows "a = b / c"
update_by_fun_pair_eq_update_by_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.update_by_fun_pair comp k f xs = comparator.update_by_pair comp (k, f (comparator.lookup_pair comp xs k)) xs"
dominates_antitrans: fixes g :: "'g"   and n\<^sub>1 :: "'node"   and m :: "'node"   and n\<^sub>2 :: "'node" assumes "invar g"   and "dominates g n\<^sub>1 m"   and "dominates g n\<^sub>2 m" obtains "dominates g n\<^sub>1 n\<^sub>2"   | "dominates g n\<^sub>2 n\<^sub>1"
rt_fresher_def2: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2" shows "(rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2) = (nsqn rt1 dip < nsqn rt2 dip \<or> nsqn rt1 dip = nsqn rt2 dip \<and> the (dhops rt2 dip) \<le> the (dhops rt1 dip))"
lm052: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "A \<noteq> {}" shows "Max (f ` A) \<in> f ` A"
orientable_irreflexive_inf: fixes x :: "'a" assumes "orientable x" shows "irreflexive_inf x"
axioms_2: fixes AA :: "('b, 'c, 'd) category_scheme"   and BB :: "('e, 'f, 'g) category_scheme"   and F :: "('b, 'c, 'e, 'f) functor"   and G :: "('b, 'c, 'e, 'f) functor"   and u :: "'b \<Rightarrow> 'f" assumes "u : F \<Rightarrow> G in Func (AA , BB )" shows "natural_transformation_axioms AA BB F G u"
insert_commute_assms: fixes i :: "'a"   and e :: "'a \<times> 'b \<times> bool"   and n :: "'a option"   and i' :: "'a"   and e' :: "'a \<times> 'b \<times> bool"   and n' :: "'a option"   and j :: "nat" assumes "{Deliver (i, Insert e n), Deliver (i', Insert e' n')} \<subseteq> set (history j)"   and "hb.concurrent (i, Insert e n) (i', Insert e' n')" shows "n = None \<or> n \<noteq> Some (fst e')"
bij_lens_via_comp_id_right: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'e \<Longrightarrow> 'd" assumes "wb_lens X"   and "wb_lens Y"   and "X ;\<^sub>L Y = 1\<^sub>L" shows "bij_lens Y"
bij_betw_imp_funcset: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "f \<in> A \<rightarrow> B"
Sign_Enc: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Sign X Y \<noteq> Enc X' Y'"
app_tl_empty_hd: fixes l :: "'a list"   and a :: "'a" assumes "tl (l @ [a]) = []" shows "hd (l @ [a]) = a"
simps_10: fixes f1 :: "'h \<Rightarrow> 'e"   and f2 :: "'i \<Rightarrow> 'f"   and x2 :: "('h \<times> 'i) set['g]" shows "map_var0 f1 f2 (BPS x2) = BPS (map_bset (map_prod f1 f2) x2)"
assms_4: shows "Ind (A \<inter> {a. en a s}) (executable - A)"
singleton_singleton: fixes a :: "'a" shows "fis_singleton {|a|}"
match_rtranclp_le: fixes test :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b regex"   and i :: "nat"   and j :: "nat" assumes "(match test r)\<^sup>*\<^sup>* i j" shows "i \<le> j"
add_zeror: fixes x :: "'b \<times> 'c" shows "sup_prod x bot_prod = x"
simps_5: fixes v :: "'a"   and l :: "value list"   and s :: "'a \<Rightarrow> value option" shows "gval (In v l) s = (case s v of None \<Rightarrow> invalid | Some vv \<Rightarrow> if vv \<in> set l then true else false)"
joinable_def: fixes p :: "nat list"   and q :: "nat list" shows "joinable p q = (last p = hd q)"
rbl_sizes_3: fixes bl :: "bool list"   and cl :: "bool list" shows "length (rbl_add bl cl) = length bl"
traceWhile: fixes c\<^sub>1 :: "'a"   and T :: "'a list"   and c\<^sub>4 :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "trace c\<^sub>1 T c\<^sub>4"   and "P c\<^sub>1"   and "\<not> P c\<^sub>4" obtains T\<^sub>1 :: "'a list" and c\<^sub>2 :: "'a" and c\<^sub>3 :: "'a" and T\<^sub>2 :: "'a list" where "T = T\<^sub>1 @ c\<^sub>3 # T\<^sub>2" and "trace c\<^sub>1 T\<^sub>1 c\<^sub>2" and "\<forall>x\<in>set T\<^sub>1. P x" and "P c\<^sub>2" and "c\<^sub>2 \<Rightarrow> c\<^sub>3" and "\<not> P c\<^sub>3" and "trace c\<^sub>3 T\<^sub>2 c\<^sub>4"
typeof_conf: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and v :: "'b val"   and T :: "ty"   and P :: "'c prog" assumes "heap_base.typeof_h typeof_addr h v = \<lfloor>T\<rfloor>" shows "heap_base.conf typeof_addr P h v T"
pred_inject_1: fixes P :: "'h \<Rightarrow> bool" shows "pred_option P None"
euclidean_representation_sum': fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and b :: "'c"   and f :: "'c \<Rightarrow> real" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(b = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (f i) i) Basis) = (\<forall>i\<in>Basis. f i = inner b i)"
restrict_rtrancl: fixes r :: "('b \<times> 'b) set"   and S :: "'b set" shows "(restrict r S)\<^sup>* \<subseteq> r\<^sup>*"
PPsI: fixes t :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set" assumes "keys t \<subseteq> X" shows "t \<in> .[X]"
spaceN_diff: fixes x :: "'c"   and N :: "'c quasinorm"   and y :: "'c" assumes "x \<in> space\<^sub>N N"   and "y \<in> space\<^sub>N N" shows "x - y \<in> space\<^sub>N N"
lm067: fixes R :: "('b \<times> 'a) set"   and X :: "'b set" shows "(R || X) `` X = R `` X"
b: fixes ys' :: "'a list"   and x' :: "'a" assumes "\<not> List.member ys' x'"   and "a # xs \<approx> x' # ys'" shows "thesis"
a_closure': fixes x :: "'a" shows "d (ad x) = ad x"
map_drop_fun_upd: fixes x :: "'a"   and F :: "'a \<Rightarrow> 'b option" shows "map_drop x F = F(x := None)"
sym_naturality: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g" shows "\
[cod f, cod g] \<cdot> (f \<otimes> g) = (g \<otimes> f) \<cdot> \
[local.dom f, local.dom g]"
ordIso_ordLeq_trans: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('c \<times> 'c) set" assumes "r =o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
bind_map_gpv: fixes f :: "'h \<Rightarrow> 'g"   and gpv :: "('h, 'e, 'f) gpv"   and g :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "map_gpv f id gpv \<bind> g = gpv \<bind> g \<circ> f"
minVertex_eq_if_vertices_eq: fixes f :: "face"   and f' :: "face" assumes "\<V> f = \<V> f'" shows "minVertex f = minVertex f'"
delta': fixes S :: "'b set"   and a :: "'b"   and b :: "'b \<Rightarrow> 'c" assumes "finite S" shows "(\<Sum>k\<in>S. if a = k then b k else (0::'c)) = (if a \<in> S then b a else (0::'c))"
before_irrefl: fixes a :: "'b set" shows "\<not> a \<lless> a"
termi_g: shows "\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"
R'_def: shows "R' \<equiv> Id \<inter> {(uu_, n, ns). \<forall>n'\<in>ns. name n' < name n}"
list_rel_append1: fixes as :: "'c list"   and bs :: "'c list"   and l :: "'d list"   and R :: "('c \<times> 'd) set" shows "((as @ bs, l) \<in> \<langle>R\<rangle>list_rel) = (\<exists>cs ds. l = cs @ ds \<and> (as, cs) \<in> \<langle>R\<rangle>list_rel \<and> (bs, ds) \<in> \<langle>R\<rangle>list_rel)"
i_expand_nth_interval_eq_nth_append_replicate_NoMsg: fixes k :: "nat"   and t :: "nat"   and t1 :: "nat"   and f :: "nat \<Rightarrow> 'd message_af" assumes "0 < k"   and "t * k \<le> t1"   and "t1 \<le> t * k + k - Suc 0" shows "f \<odot> k \<Down> Suc t1 \<up> (t * k) = f t # NoMsg\<^bsup>t1 - t * k\<^esup>"
d_OUT_alt_def: fixes x :: "'a"   and G :: "('a, 'b) graph_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>y. (x, y) \<notin> \<^bold>E\<^bsub>G\<^esub> \<Longrightarrow> g (x, y) = 0" shows "d_OUT g x = (\<Sum>\<^sup>+ y\<in>\<^bold>O\<^bold>U\<^bold>T\<^bsub>G\<^esub> x. g (x, y))"
convex_hull_set_plus: fixes S :: "'a set"   and T :: "'a set" shows "convex hull (S + T) = convex hull S + convex hull T"
simps: fixes s :: "bool \<times> nat" shows "local.while s = (if fst s then (case s of (b, x) \<Rightarrow> map_spmf (\<lambda>b'. (\<not> b', x + (if b' then 0 else 1))) (bernoulli p)) \<bind> local.while else return_spmf s)"
Iff_E2: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and C :: "fmla" assumes "H \<turnstile> A"   and "insert B H \<turnstile> C" shows "insert (B IFF A) H \<turnstile> C"
ncg_def: fixes x :: "'a"   and y :: "'a" shows "ncg x y = (\<nu> x \<le> \<nu> y \<and> \<nu> y \<le> \<nu> x)"
fresh_subst_if: fixes j :: "atom"   and i :: "name"   and x :: "tm"   and t :: "tm" shows "j \<sharp> subst i x t = (atom i \<sharp> t \<and> j \<sharp> t \<or> j \<sharp> x \<and> (j \<sharp> t \<or> j = atom i))"
convex_closed_interval: fixes a :: "'c"   and b :: "'c" shows "convex {a..b}"
units_group: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.monoid G" shows "Group.group (units_of G)"
hcomp_simps_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<nu> :: "'c"   and \<mu> :: "'c" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V (H \<nu> \<mu>)" shows "src (H \<nu> \<mu>) = src \<mu>"
pareto_losers_def: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool" shows "pareto_losers R = {x. \<exists>y. x \<prec>[Pareto R] y}"
subst_atm_mset_id_subst: fixes AA :: "'a multiset" shows "AA \<cdot>am id_subst = AA"
normalize_bounds_1_main_def: fixes eps :: "rat"   and rai :: "int poly \<times> rat \<times> rat" shows "normalize_bounds_1_main eps rai = (case rai of (p, l, r) \<Rightarrow> let (l', r', sr') = tighten_poly_bounds_epsilon p eps l r (sgn (ipoly p r)); fr = rat_of_int \<lfloor>r'\<rfloor>; (l'', r'', uu_) = tighten_poly_bounds_for_x p fr l' r' sr' in (p, l'', r''))"
row_to_poly_def: fixes ts :: "'c list"   and r :: "'d vec" shows "row_to_poly ts r = list_to_poly ts (list_of_vec r)"
univ_poly_units: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set" assumes "subfield K (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>)" shows "Units (K [X]\<^bsub>L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>\<^esub>) = {[k] |k. k \<in> K - {\<zero>\<^bsub>L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>\<^esub>}}"
monad_fail_stateT: fixes return :: "'b \<times> 'c \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'd) \<Rightarrow> 'd"   and fail :: "'d" assumes "monad_fail return bind fail" shows "monad_fail (return_state return) (bind_state bind) (fail_state fail)"
simps_3: fixes x2 :: "'a" shows "Some x2 \<noteq> None"
dsumMhom_mem: fixes I :: "'h set"   and M :: "'h \<Rightarrow> ('i, 'a, 'j) Module_scheme"   and N :: "'h \<Rightarrow> ('k, 'a, 'l) Module_scheme"   and S :: "'h \<Rightarrow> 'i \<Rightarrow> 'k"   and x :: "'h \<Rightarrow> 'i" assumes "\<forall>i\<in>I. R module M i"   and "\<forall>i\<in>I. R module N i"   and "\<forall>i\<in>I. S i \<in> mHom R (M i) (N i)"   and "x \<in> carr_dsumag I M" shows "dsumMhom I M N S x \<in> carr_dsumag I N"
AX10p_AX10_eq: shows "AX10p = (\<lambda>x::'b formula. x \<in> AX10)"
INF_top_conv_1: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and B :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Inf (B ` A) = top) = (\<forall>x\<in>A. B x = top)"
select_convs_3: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "rt \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = rt"
subid_aux2: fixes R :: "('b \<times> 'b set) set"   and a :: "'b"   and A :: "'b set" assumes "R \<subseteq> 1\<^sub>\<sigma>"   and "(a, A) \<in> R" shows "A = {a}"
simps: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr" shows "(\<sigma>, \<tau>, e)\<^sub>\<sigma> = \<sigma>"
set_Us: shows "set (map (([^]) x) (rev [0..<Suc n])) \<subseteq> F"
cball_empty: fixes e :: "real"   and x :: "'c" assumes "e < 0" shows "cball x e = {}"
pathStops: fixes subs :: "'b \<Rightarrow> 'b set"   and gamma :: "'b"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and n :: "nat" assumes "terminal subs (path subs gamma P n)" shows "path subs gamma P (Suc n) = path subs gamma P n"
getter_functions: fixes p :: "real poly"   and qs :: "real poly list" shows "calculate_data p qs = (get_matrix (calculate_data p qs), get_subsets (calculate_data p qs), get_signs (calculate_data p qs))"
Basic: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "'e" shows "\<Gamma>\<turnstile> (Basic f, Normal s) \<rightarrow> (Skip, Normal (f s))"
amor_del_Some: fixes x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "del x t = Some t'" shows "real (T_del x t) + \<Phi> t' - \<Phi> t \<le> (6 * e + 1) * real (height t) + 1"
altc_put: fixes C :: "'c cset"   and s :: "'s"   and f :: "'c \<Rightarrow> 'm" shows "altc C (\<lambda>c. put s (f c)) = put s (altc C f)"
Sup_empty: shows "\<Squnion>\<^sub>o\<^sub>w {} = \<bottom>\<^sub>o\<^sub>w"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (add_instr_sub1 instr_name result rs1_val operand2 s1)) \<and> t2 = snd (fst (add_instr_sub1 instr_name result rs1_val operand2 s2))"
substT_pls: fixes t1 :: "'trm"   and t2 :: "'trm"   and t :: "'trm"   and x :: "'var" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "t \<in> trm"   and "x \<in> var" shows "substT (pls t1 t2) t x = pls (substT t1 t x) (substT t2 t x)"
closure_eq_between: fixes S1 :: "'b set"   and S2 :: "'b set" assumes "convex S1"   and "convex S2" shows "(closure S1 = closure S2) = (rel_interior S1 \<subseteq> S2 \<and> S2 \<subseteq> closure S1)"
fresh_ty: fixes x :: "name"   and \<tau> :: "ty" shows "x \<sharp> \<tau>"
assms_4: shows "real_of_float (lb prec l) \<le> f (real_of_float l)"
map_of_zip_is_Some: fixes xs :: "'d list"   and ys :: "'e list"   and x :: "'d" assumes "length xs = length ys" shows "(x \<in> set xs) = (\<exists>y. map_of (zip xs ys) x = Some y)"
simps: fixes xs :: "abc_inst list"   and n :: "nat" shows "abc_shift xs n = map (\<lambda>x. abc_inst_shift x n) xs"
SUP_eq_top_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Sup (f ` A) = top) = (\<forall>x. less x top \<longrightarrow> (\<exists>i\<in>A. less x (f i)))"
2: fixes x :: "'s" assumes "\<guillemotleft>x : unity \<rightarrow> a\<guillemotright>" shows "mkArr' a b F \<cdot> x = F x"
Max_ge': fixes A :: "'d set"   and b :: "'d"   and a :: "'d" assumes "finite A"   and "A \<noteq> {}"   and "b \<in> A"   and "a \<le> b" shows "a \<le> Max A"
inconsist_converg_less_1: fixes i :: "nat"   and e :: "nat" assumes "inconsist i e \<down>"   and "the (inconsist i e) < e_length e" shows "\<exists>x<e_length e. \<phi> i x \<down>\<noteq> e_nth e x"
unwinding_theorem_BSD: fixes SES :: "('a, 'b) SES_rec"   and \<V> :: "'b V_rec"   and ur :: "('a \<times> 'a) set" assumes "Unwinding SES \<V>"   and "Unwinding.lrf SES \<V> ur"   and "Unwinding.osc SES \<V> ur" shows "BSD \<V> Tr\<^bsub>(induceES SES)\<^esub>"
rel_aext_cond: fixes b :: "'c upred"   and P :: "'c hrel"   and Q :: "'c hrel"   and a :: "'c \<Longrightarrow> 'b" shows "(P \<triangleleft> b \<triangleright>\<^sub>r Q) \<oplus>\<^sub>r a = P \<oplus>\<^sub>r a \<triangleleft> b \<oplus>\<^sub>p a \<triangleright>\<^sub>r Q \<oplus>\<^sub>r a"
carrier_mult_of: fixes R :: "('e, 'f) ring_scheme" shows "carrier (mult_of R) = carrier R - {\<zero>\<^bsub>R\<^esub>}"
assms_2: shows "\<forall>x. \<forall>y\<in>X. y \<leftarrow> x \<in> X"
abs_eq: fixes xa :: "nat"   and x :: "int" shows "drop_bit xa (Word.Word x) = Word.Word (drop_bit xa (take_bit LENGTH('b) x))"
split_paired_All: fixes P :: "'a \<times> 'b \<Rightarrow> bool" shows "(\<forall>x. P x) = (\<forall>a b. P (a, b))"
nths_project': fixes i :: "nat"   and xs :: "'b list"   and j :: "nat" assumes "i < length xs"   and "j < length xs"   and "i < j" shows "nths xs {i, j} = [xs ! i, xs ! j]"
rel_ad_inter_seq: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" shows "rel_ad X \<inter> rel_ad Y = rel_ad X ; rel_ad Y"
disj: shows "AE x in M. \<forall>i\<in>I. \<forall>j\<in>I. P i x \<longrightarrow> P j x \<longrightarrow> i = j"
f3: shows "rk {Q, B', R, B} \<le> 3"
fds_integral_def: fixes c :: "'a"   and f :: "'a fds" shows "fds_integral c f = fds (\<lambda>n. if n = 1 then c else - fds_nth f n /\<^sub>R ln (real n))"
member_def: shows "RBT_Set2.member \<equiv> map_fun mapping_rbt.impl_of id (\<lambda>t x. x \<in> dom (rbt_comp_lookup ccomp t))"
S_def: shows "S = (\<lambda>m. (\<lambda>n. complex_of_real (2 * pi * real_of_int n) * \<i>) ` {- int m..int m})"
Lambert_W_pos_iff: fixes x :: "real" shows "(0 < Lambert_W x) = (0 < x)"
integrable_alt: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set" shows "(f integrable_on s) = ((\<forall>a b. (\<lambda>x. if x \<in> s then f x else (0::'b)) integrable_on cbox a b) \<and> (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball (0::'a) B \<subseteq> cbox a b \<and> ball (0::'a) B \<subseteq> cbox c d \<longrightarrow> norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else (0::'b)) - integral (cbox c d) (\<lambda>x. if x \<in> s then f x else (0::'b))) < e))"
precedesMemberHeadMemberTail: fixes a :: "'a"   and l1 :: "'a list"   and b :: "'a"   and l2 :: "'a list" assumes "a \<in> set l1"   and "b \<notin> set l1"   and "b \<in> set l2" shows "precedes a b (l1 @ l2)"
a_conditions_def: fixes M :: "('a, 'b) insert_message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended" shows "a_conditions M a = (a \<turnstile> < a \<stileturn> \<and> (\<forall>m. m \<in> M \<longrightarrow> a (P m) < a (S m) \<and> a \<lbrakk>I m\<rbrakk> = \<lbrakk>\<Psi> (a (P m), a (S m)) (I m)\<rbrakk>))"
var_update_in: fixes x :: "'c \<Longrightarrow> 'a"   and A :: "'a"   and A' :: "'b"   and v :: "'c" shows "put\<^bsub>$x\<^esub> (A, A') v = (put\<^bsub>x\<^esub> A v, A')"
ennreal_inverse_mult: fixes a :: "ennreal"   and b :: "ennreal" assumes "a < \<top>"   and "b < \<top>" shows "inverse (a * b) = inverse a * inverse b"
path_suc: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c"   and k :: "nat" assumes "IFC prob" shows "IFC_def.path prob \<sigma> (Suc k) = fst (IFC_def.step prob (IFC_def.path prob \<sigma> k, IFC_def.kth_state prob \<sigma> k))"
ennsqrt_square: fixes x :: "ennreal" shows "ennsqrt x * ennsqrt x = x"
dim_gen_eigenspace_similar: fixes A :: "'a mat"   and B :: "'a mat" assumes "similar_mat A B" shows "local.dim_gen_eigenspace A = local.dim_gen_eigenspace B"
icard_not_empty_iff: fixes A :: "'a set" shows "(0 < icard A) = (A \<noteq> {})"
bij_betw_from_nat_into: fixes S :: "'a set" assumes "countable S"   and "infinite S" shows "bij_betw (from_nat_into S) UNIV S"
iwlsFSwSTR_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.iwlsFSwSTR Delta MOD \<equiv> FixSyn.igWlsAllDisj Delta MOD \<and> FixSyn.igWlsAbsIsInBar Delta MOD \<and> FixSyn.igConsIPresIGWlsSTR Delta MOD \<and> igSwapAllIPresIGWlsAllSTR MOD \<and> igFreshClsSTR MOD \<and> igSwapClsSTR MOD \<and> igAbsCongSSTR MOD"
smult_zero: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and r :: "'a" assumes "RModule R smult M"   and "r \<in> R" shows "smult r (0::'b) = (0::'b)"
src_port_int_def: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> 'a" shows "src_port x \<equiv> (snd \<circ> fst \<circ> snd) x"
PO_m1_inv4_rfresh_init: shows "init m1 \<subseteq> m1_inv4_rfresh"
m_up_def: shows "m_up = 1 + div_ceiling m \<bar>coeff p n\<bar>"
epiE: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "C.epi f"   and "C.seq g f"   and "C.seq g' f"   and "g \<cdot> f = g' \<cdot> f" shows "g = g'"
comp_fun_commute_filter_fold: fixes P :: "'b \<Rightarrow> bool" shows "comp_fun_commute (\<lambda>x A'. if P x then insert x A' else A')"
intgbl: fixes i :: "nat" shows "set_integrable M (A i) f"
2: shows "X \<in> sets (distr (uniform_measure (count_space A) A') (count_space B) f)"
disjoint_witness_correct_1: fixes s1 :: "'s1"   and s2 :: "'s2" assumes "invar1 s1"   and "invar2 s2"   and "disjoint_witness s1 s2 = None" shows "\<alpha>1 s1 \<inter> \<alpha>2 s2 = {}"
cases: fixes x :: "('d, 'e, 'varSort, 'var, 'f) qAbs" obtains xs :: "'varSort" and x :: "'var" and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" where "xa__ = qAbs xs x X"
reduction_language_indistinguishable: assumes "R.language \<subseteq> F.language" shows "S.language \<subseteq> F.language"
conductor_leI: fixes d :: "nat" assumes "\<Phi>.induced_modulus d" shows "\<Phi>.conductor \<le> d"
max_c_def: shows "max_c = Max (c ` {..N})"
arith_sum1: fixes t :: "nat"   and i :: "nat"   and k :: "nat"   and d :: "nat" shows "t + (i + k + d) = t + i + k + d"
reverse_phase_permutation: fixes xs :: "int list" shows "mset (concat (map (\<lambda>ys. if decr ys then rev ys else ys) (cuts xs))) = mset xs"