hcju/isabelleps huggingface dataset:
l_set_disconnected_nodes_get_disconnected_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs \<equiv> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs \<and> l_set_disconnected_nodes_get_disconnected_nodes_axioms get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
Derives1_length: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "length b = length a + length (snd r) - 1"
hyps: fixes t :: "('a, 'b) BinomialTree" assumes "invar bq" shows "invar (remove1 t bq)"
simp_1: fixes subsumes :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'a list" shows "mk_rtrancl_list_main subsumes r todo finl = (case todo of [] \<Rightarrow> finl | a # tod \<Rightarrow> if \<exists>b\<in>set finl. subsumes b a then mk_rtrancl_list_main subsumes r tod finl else mk_rtrancl_list_main subsumes r (r a @ tod) (a # finl))"
sep_conj_assoc: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" shows "P ** (Q ** R) = P ** Q ** R"
nn_integral_restrict_space: fixes \<Omega> :: "'d set"   and M :: "'d measure"   and f :: "'d \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>N (restrict_space M \<Omega>) f = set_nn_integral M \<Omega> f"
simps_3: fixes c :: "int"   and e :: "Ferrack.num" shows "isrlfm (Eq (CN 0 c e)) = (0 < c \<and> numbound0 e)"
set_mset_Union_mset: fixes MM :: "'a multiset multiset" shows "set_mset (\<Sum>\<^sub># MM) = \<Union> (set_mset ` set_mset MM)"
simps_2: fixes b :: "'d" shows "rsuml (Inl (Inr b)) = Inr (Inl b)"
axioms: fixes hom :: "'b \<Rightarrow> 'c" assumes "comm_semiring_hom hom" shows "semiring_hom hom"
no_relabel_excess: fixes v :: "nat" assumes "lo u = l u"   and "u \<noteq> v"   and "excess fo v \<noteq> excess f v" shows "(u, v) \<in> adm_edges fo lo"
distinct_map_filter: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "distinct (map f xs)" shows "distinct (map f (filter P xs))"
dvd_triv_left: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times a b)"
markedElementsAreElementsBeforeLastDecisionAndLastDecision: fixes M :: "('b \<times> bool) list" assumes "markedElements M \<noteq> []" shows "markedElements M = markedElements (prefixBeforeLastMarked M) @ [lastMarked M]"
mcont_contD: fixes luba :: "'a set \<Rightarrow> 'a"   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lubb :: "'b set \<Rightarrow> 'b"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and Y :: "'a set" assumes "mcont luba orda lubb ordb f"   and "Complete_Partial_Order.chain orda Y"   and "Y \<noteq> {}" shows "f (luba Y) = lubb (f ` Y)"
gen_dfsT_eq_assert: fixes gds :: "('b, 'a) gen_dfs_struct"   and V0 :: "'b set" shows "gen_dfs_defs.gen_dfsT gds V0 = gen_dfs_defs.gen_dfsT_assert gds V0"
cone_ordLeq_iff_Field: fixes r :: "('c \<times> 'c) set" assumes "cone \<le>o r" shows "Field r \<noteq> {}"
map_dual_def: fixes f :: "'b \<Rightarrow> 'c" shows "\<partial>\<^sub>F f = \<partial> \<circ> f \<circ> \<partial>"
prems_2: shows "rbt_sorted (Branch c t1 k v t2)"
poly_mult_zero_1: fixes p :: "'a list list" assumes "set p \<subseteq> carrier (poly_ring R)" shows "UP.poly_mult [] p = []"
servK_authentic_bis: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV_gets" shows "\<exists>A. Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"
F2set3_bd: fixes x :: "('e, 'f, 'd) F2" shows "|F2set3 x| \<le>o IFbd"
fset_of_list_append: fixes xs :: "'a list"   and ys :: "'a list" shows "fset_of_list (xs @ ys) = fset_of_list xs |\<union>| fset_of_list ys"
match_ifaceAny_eq: fixes m :: "'b simple_match"   and p :: "('b, 'c) simple_packet_scheme"   and any :: "char list" assumes "oiface m = ifaceAny" shows "simple_matches m p = simple_matches m (p\<lparr>p_oiface := any\<rparr>)"
smc_Dom_vempty_if_Arr_vempty: assumes "\<BB>\<lparr>Arr\<rparr> = []\<^sub>\<circ>" shows "\<BB>\<lparr>Dom\<rparr> = []\<^sub>\<circ>"
is_trail_intro: fixes G :: "('a, 'b) graph"   and v' :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.is_trail G v' ps v" shows "valid_graph.is_path G v' ps v"
conj_eqvt: fixes pi :: "('a \<times> 'a) list"   and A :: "bool"   and B :: "bool" shows "pi \<bullet> (A \<and> B) = (pi \<bullet> A \<and> pi \<bullet> B)"
gcd_primes64: fixes p :: "int" assumes "p \<in> set gcd_primes64" shows "prime p \<and> p \<le> 4294967295"
small_bounded_rank: fixes a :: "V" shows "small {x. rank x \<in> elts a}"
inject: fixes x1 :: "name"   and x2 :: "trm"   and y1 :: "name"   and y2 :: "trm" shows "(x1:x2 = y1:y2) = ([x1].x2 = [y1].y2)"
odd_card: fixes A :: "'a set" assumes "finite A"   and "odd (card A)" shows "\<exists>x. x \<in> A"
isStaticLoc_simps_4: fixes T :: "Arraytype"   and a :: "nat"   and i :: "nat" shows "isStaticLoc (arrLoc T a i) = False"
Su: shows "\<forall>x\<in>S. x \<le> u"
biholomorphic_to_disc_aux: fixes S :: "complex set" assumes "open S"   and "connected S"   and "0 \<in> S"   and "S \<subseteq> ball 0 1"   and "\<And>f. \<lbrakk>f holomorphic_on S; \<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0; inj_on f S\<rbrakk> \<Longrightarrow> \<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2)" shows "\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and> (\<forall>z\<in>S. f z \<in> ball 0 1 \<and> g (f z) = z) \<and> (\<forall>z\<in>ball 0 1. g z \<in> S \<and> f (g z) = z)"
singleton_set_union: fixes X :: "'a set" shows "(\<Union>Y\<in>singleton_set X. Y) = X"
bounded_simple_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "simple_path g" shows "bounded (path_image g)"
fold_fun_left_comm: fixes A :: "'d set"   and x :: "'d"   and z :: "'e" assumes "finite A" shows "g x (Finite_Set.fold g z A) = Finite_Set.fold g (g x z) A"
oracle_decrypt2_def: shows "oracle_decrypt2 = (\<lambda>key D cipher. return_spmf (None, D))"
R: assumes "A \<subseteq> carrier"   and "A \<noteq> {}" shows "R"
l_update_rightmost_eq_l: fixes f :: "'b \<Rightarrow> 'b"   and t :: "'b tree" shows "l (update_rightmost f t) = l t"
sel_transfer: fixes R1 :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "(rel_optionT' R1 R2 ===> R2) run_option run_option"
Mul_Mutator_def: fixes j :: "nat"   and n :: "nat" shows "Mul_Mutator j n \<equiv> \<lbrace>\<acute>Mul_mut_init n \<and> Z (\<acute>Muts ! j)\<rbrace>  WHILE True INV \<lbrace>\<acute>Mul_mut_init n \<and> Z (\<acute>Muts ! j)\<rbrace>  DO Mul_Redirect_Edge j n;; Mul_Color_Target j n OD"
gallery_chamber_system: fixes X :: "'b set set"   and Cs :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimpchain X Cs" shows "order.greater_eq (ChamberComplex.chamber_system X) (set Cs)"
mod_by_1: fixes a :: "'a" shows "a mod (1::'a) = (0::'a)"
that: fixes ns' :: "'node list"   and m' :: "'node" assumes "g \<turnstile> n-ns'\<rightarrow>m'"   and "P m'"   and "\<forall>x\<in>set (butlast ns'). \<not> P x"   and "prefix ns' ns" shows "thesis"
response_def: fixes \<G> :: "'a cyclic_group"   and r :: "nat"   and w :: "nat"   and c :: "nat" assumes "schnorr_base \<G>" shows "schnorr_base.response \<G> r w c = return_spmf ((w * c + r) mod order \<G>)"
distinct_8: fixes x3 :: "'f"   and x2 :: "('e, 'f, 'g) Tree" shows "tPred x3 \<noteq> tNot x2"
rfork: fixes e :: "('a, 'b, 'c) expr" shows "redex (Rfork e)"
dist_snd_le: fixes x :: "'c \<times> 'b"   and y :: "'c \<times> 'b" shows "dist (snd x) (snd y) \<le> dist x y"
updates_conv: fixes ks :: "'d list"   and vs :: "'c list"   and al :: "('d \<times> 'c) list"   and k :: "'d" shows "map_of (updates ks vs al) k = (map_of al(ks [\<mapsto>] vs)) k"
leqs: shows "P,C \<turnstile> Cs \<sqsubseteq> Cs'"
prepend_LCons: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b llist" shows "prepend xs (LCons y ys) = prepend (xs @ [y]) ys"
A: fixes i :: "'c" assumes "i \<in> J" shows "A i \<in> sets (M (f i))"
f_shrink_assoc: fixes xs :: "'b message_af list"   and a :: "nat"   and b :: "nat" shows "xs \<div> a \<div> b = xs \<div> (a * b)"
types_preserved_label_value: fixes n :: "nat"   and es0 :: "e list"   and vs :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>[Label n es0 vs]\<rparr> \<leadsto> \<lparr>vs\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [Label n es0 vs] : ts _> ts'"   and "const_list vs" shows "\<S>\<bullet>\<C> \<turnstile> vs : ts _> ts'"
update_matchP_def: fixes n :: "nat"   and I :: "\<I>"   and mr :: "mregex"   and mrs :: "mregex list"   and rels :: "event_data option list set list"   and nt :: "nat"   and aux :: "(nat \<times> (mregex, event_data option list set) mapping) list" shows "update_matchP n I mr mrs rels nt aux = (let aux = case concat (map (\<lambda>(t, rel). if enat (nt - t) \<le> right I then [(t, mrtabulate mrs (\<lambda>mr. r\<delta> id rel rels mr \<union> (if t = nt then r\<epsilon>_strict n rels mr else {})))] else []) aux) of [] \<Rightarrow> [(nt, mrtabulate mrs (r\<epsilon>_strict n rels))] | x # aux' \<Rightarrow> if fst x = nt then x # aux' else (nt, mrtabulate mrs (r\<epsilon>_strict n rels)) # x # aux' in (foldr (\<union>) (concat (map (\<lambda>(t, rel). if left I \<le> nt - t then [lookup rel mr] else []) aux)) {}, aux))"
simps: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" shows "local.union t1 t2 = (if t1 = \<langle>\<rangle> then t2 else if t2 = \<langle>\<rangle> then t1 else case t1 of \<langle>l1, (a, x), r1\<rangle> \<Rightarrow> let (l2, uu_, r2) = split t2 a; l' = local.union l1 l2 in Let (local.union r1 r2) (join l' a))"
greatest_ge_nonzero_row': fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c" assumes "reduced_row_echelon_form_upt_k A k"   and "i \<le> (GREATEST m. \<not> is_zero_row_upt_k m k A)"   and "\<not> (\<forall>a. is_zero_row_upt_k a k A)" shows "\<not> is_zero_row_upt_k i k A"
simps_9: fixes v :: "Cooper.num" shows "not (NEq v) = NOT (NEq v)"
scene_indep_compat: fixes X :: "'a scene"   and Y :: "'a scene" assumes "X \<bowtie>\<^sub>S Y" shows "X ##\<^sub>S Y"
notifyWatchesStartQIreleveant: fixes stateA :: "State"   and Wl :: "nat list"   and stateB :: "State"   and literal :: "Literal"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF stateA) (getWatch1 stateA) (getWatch2 stateA)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF stateA)"   and "getM stateA = getM stateB"   and "getF stateA = getF stateB"   and "getWatch1 stateA = getWatch1 stateB"   and "getWatch2 stateA = getWatch2 stateB"   and "getConflictFlag stateA = getConflictFlag stateB"   and "getSATFlag stateA = getSATFlag stateB" shows "let state' = notifyWatches_loop literal Wl newWl stateA; state'' = notifyWatches_loop literal Wl newWl stateB in getM state' = getM state'' \<and> getF state' = getF state'' \<and> getSATFlag state' = getSATFlag state'' \<and> getConflictFlag state' = getConflictFlag state''"
projection_is_projection_on': fixes M :: "'a set" assumes "closed_csubspace M" shows "is_projection_on (projection M) M"
freshAbs_simp: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y :: "'var"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "freshAbs ys y (Abs xs x X) = (ys = xs \<and> y = x \<or> fresh ys y X)"
filter_in_nths: fixes xs :: "'a list"   and s :: "nat set" assumes "distinct xs" shows "filter (\<lambda>x. x \<in> set (nths xs s)) xs = nths xs s"
fixp_unfold: fixes f :: "'b \<Rightarrow> 'b" assumes "monotone leq2 leq2 f" shows "c2.fixp f = f (c2.fixp f)"
hyps: shows "<\<nu>x>Q \<longmapsto> \<alpha> \<prec> Q'"
elims: fixes x :: "'b arctic_delta"   and y :: "char list \<Rightarrow> char list" assumes "Show_Arctic.shows_arctic_delta x = y" obtains i :: "'b" where "x = Num_arc_delta i" and "y = shows i"   | "x = MinInfty_delta" and "y = shows ''-inf''"
no_greater_postrec_violation: shows "\<forall>i>Max {i. postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}}. \<not> (postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j})"
nsets_doubleton_2_eq: fixes x :: "'b"   and y :: "'b" shows "[{x, y}]\<^bsup>2\<^esup> = (if x = y then {} else {{x, y}})"
Y_def: shows "Y = {x @\<^sub>v vec_of_scal (1::'a) |x. x \<in> Q}"
invertibleE: fixes u :: "'a"   and thesis :: "bool" assumes "sub.invertible u"   and "\<And>v. \<lbrakk>u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>; v \<in> N\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> N" shows "thesis"
delta_gt_zero: fixes l :: "(QDelta \<times> QDelta) list" shows "0 < \<delta>_min l"
simps: fixes f :: "'c poly \<times> 'c \<Rightarrow> 'b"   and Gi_1 :: "'c poly"   and Gi :: "'c poly"   and ni_1 :: "nat"   and d1_1 :: "nat"   and hi_2 :: "'c" shows "subresultant_prs_main_impl f Gi_1 Gi ni_1 d1_1 hi_2 = (let gi_1 = lead_coeff Gi_1; ni = degree Gi; hi_1 = if d1_1 = 1 then gi_1 else dichotomous_Lazard gi_1 hi_2 d1_1; d1 = ni_1 - ni; pmod = pseudo_mod Gi_1 Gi in if pmod = 0 then f (Gi, if d1 = 1 then lead_coeff Gi else dichotomous_Lazard (lead_coeff Gi) hi_1 d1) else let gi = lead_coeff Gi; divisor = (- (1::'c)) ^ (d1 + 1) * gi_1 * hi_1 ^ d1; Gi_p1 = sdiv_poly pmod divisor in subresultant_prs_main_impl f Gi Gi_p1 ni d1 hi_1)"
trg_vcomp: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.seq \<nu> \<mu>" shows "trg\<^sub>D (\<nu> \<cdot>\<^sub>D \<mu>) = trg\<^sub>D \<nu>"
induces_horizontal_composition: shows "horizontal_composition (\<cdot>) (\<star>) src trg"
lift_mult: fixes a :: "'a"   and b :: "'a" shows "lift (a * b) = lift a * lift b"
init_fin_red_read_typeable: fixes t :: "'thread_id"   and x :: "status \<times> 'x"   and m :: "'heap"   and ta :: "('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action"   and x' :: "status \<times> 'x"   and m' :: "'heap"   and ad :: "'addr"   and al :: "addr_loc"   and v :: "'addr val" assumes "h.mthr.if.r_syntax t x m ta x' m'"   and "init_fin_lift wfx t x m"   and "NormalAction (ReadMem ad al v) \<in> set \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>" shows "\<exists>T. h.addr_loc_type P m ad al T"
semiring_norm_159: shows "(0::'c) \<noteq> (1::'c)"
m2eq2: fixes i :: "nat" assumes "i < state_sig.d b.dims0" shows "digit_encode c.dims0 (digit_decode (nths d.dims0 (- d.vars1')) (nths (digit_encode d.dims0 i) (- d.vars1'))) = nths (digit_encode d.dims0 i) (- d.vars1')"
flow_initial_time_if: fixes x0 :: "'a" shows "rev.flow0 x0 0 = (if 0 \<in> UNIV \<and> x0 \<in> X then x0 else (0::'a))"
restrict_map_le: fixes t :: "'a \<Rightarrow> 'b option"   and S :: "'a set" shows "t |` S \<subseteq>\<^sub>m t"
od_fm_fun_indPr2: 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" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "Chain (od_fm_fun R M N f H) C"   and "C \<noteq> {}" shows "ub\<^bsub>od_fm_fun R M N f H\<^esub> C (\<Union> {a. \<exists>b. (a, b) \<in> C}, \<lambda>x\<in>fgs R M (\<Union> {a. \<exists>b. (a, b) \<in> C}). THE y. y \<in> carrier N \<and> (\<exists>a b. (a, b) \<in> C \<and> x \<in> fgs R M a \<and> y = b x))"
of_nat_class: fixes n :: "nat" shows "\<guillemotleft>n\<guillemotright>\<^sub>\<nat>\<^bsub>cring_class_ops\<^esub> = of_nat n"
num_ni_upd: fixes t :: "nat"   and ts :: "nat \<Rightarrow> thread"   and s :: "thread" assumes "t < N" shows "num_ni (ts(t := s)) = num_ni ts - ni_weight (ts t) + ni_weight s"
is_interval_path_connected: fixes S :: "'b set" assumes "is_interval S" shows "path_connected S"
card_subgrp_dvd: fixes H :: "'a set" assumes "subgroup H G" shows "card H dvd order G"
constructive_security: fixes real_resource :: "nat \<Rightarrow> ('h + 'i, 'j + 'k) resource"   and ideal_resource :: "nat \<Rightarrow> ('l + 'i, 'm + 'k) resource"   and sim :: "nat \<Rightarrow> ('h, 'j, 'l, 'm) converter"   and \<I>_real :: "nat \<Rightarrow> ('h, 'j) \<I>"   and \<I>_ideal :: "nat \<Rightarrow> ('l, 'm) \<I>"   and \<I>_common :: "nat \<Rightarrow> ('i, 'k) \<I>"   and bound :: "nat \<Rightarrow> enat"   and lossless :: "bool"   and w :: "nat \<Rightarrow> ('l \<Rightarrow> 'h) \<times> ('j \<Rightarrow> 'm)" assumes "constructive_security2 real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common bound lossless w" shows "constructive_security real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common bound lossless w"
ord_0_right_nat: fixes n :: "nat" shows "ord n 0 = (if n = 1 then 1 else 0)"
subgroup_imp_group: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G" shows "Group.group (G\<lparr>carrier := H\<rparr>)"
dom_restr: fixes A :: "'a set"   and al :: "('a \<times> 'b) list" shows "fst ` set (restrictA A al) = fst ` set al \<inter> A"
ord_eq_Eequiv_1: fixes v :: "i"   and x :: "\<kappa>" shows "[\<lparr>O!,x\<rparr> \<^bold>\<rightarrow> x \<^bold>=\<^sub>E x in v]"
neville_aitken_interpolation_poly: fixes xs_ys :: "('a \<times> 'a) list"   and p :: "'a poly"   and x :: "'a"   and y :: "'a" assumes "distinct (map fst xs_ys)"   and "p = neville_aitken_interpolation_poly xs_ys"   and "(x, y) \<in> set xs_ys" shows "poly p x = y"
non_reachable_sink: shows "\<forall>s\<in>\<RR> Tr Up s\<^sub>0. Tr s \<noteq> {}"
Inf_superset_mono: 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 B :: "'c set"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "B \<subseteq> A" shows "less_eq (Inf A) (Inf B)"
simps_4: fixes v :: "vname"   and va :: "value list" shows "enumerate_gexp_inputs (In v va) = enumerate_aexp_inputs (V v)"
ordIso_reflexive: fixes r :: "('b \<times> 'b) set" assumes "Well_order r" shows "r =o r"
projection_orthogonal: fixes M :: "'a set"   and m :: "'a"   and h :: "'a" assumes "closed_csubspace M"   and "m \<in> M" shows "is_orthogonal (h - projection M h) m"
wens_single_eq: fixes init :: "'b set"   and act :: "('b \<times> 'b) set"   and allowed :: "('b \<times> 'b) set set"   and B :: "'b set" shows "wens (mk_program (init, {act}, allowed)) act B = B \<union> wp act B"
infinite_Un: fixes S :: "'b set"   and T :: "'b set" shows "infinite (S \<union> T) = (infinite S \<or> infinite T)"
weighted_RP_complete: fixes Sts :: "(('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat) llist" assumes "full_chain (\<leadsto>\<^sub>w) Sts"   and "wrp.P_of_wstate (lhd Sts) = {}"   and "wrp.Q_of_wstate (lhd Sts) = {}"   and "\<not> satisfiable (grounding_of_clss (wrp.N_of_wstate (lhd Sts) \<union> wrp.P_of_wstate (lhd Sts) \<union> wrp.Q_of_wstate (lhd Sts)))" shows "{#} \<in> Q_of_state (wrp.Liminf_wstate Sts)"
augFlow_val: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and p :: "(nat \<times> nat) list" assumes "NPreflow c s t f"   and "NPreflow.isAugmentingPath c s t f p" shows "Flow.val (residualGraph c f) s (NPreflow.augmentingFlow c f p) = NPreflow.resCap c f p"
bs_nat: fixes n :: "nat" shows "bs_val (BNat n) (ENat n)"
d_IN_plus_flow: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and g :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "wf_residual_network"   and "flow \<Delta> f"   and "flow (residual_network f) g" shows "d_IN (f \<oplus> g) x \<le> d_IN f x + d_IN g x"
eventually_eventually: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "(\<forall>\<^sub>F y in nhds x. eventually P (nhds y)) = eventually P (nhds x)"
semiring_norm_44: shows "\<bar>0::'a\<bar> = (0::'a)"
reduced_labelling_2: fixes n :: "nat"   and x :: "nat \<Rightarrow> nat" shows "\<forall>i<reduced n x. x i = 0"
unwrapH_def: shows "unwrapH \<equiv> \<Lambda> f xs. list2H\<cdot>(f\<cdot>xs)"
prime_factorization_eqI_strong: fixes P :: "'a multiset"   and n :: "'a" assumes "\<And>p. p \<in># P \<Longrightarrow> prime p"   and "\<Prod>\<^sub># P = n" shows "prime_factorization n = P"
O_assoc: fixes R :: "('b \<times> 'e) set"   and S :: "('e \<times> 'd) set"   and T :: "('d \<times> 'c) set" shows "(R O S) O T = R O S O T"
foo: shows "edom \<rho> \<union> domA \<Delta> \<union> domA \<Gamma> - (edom \<rho> \<union> domA \<Delta> \<union> domA \<Gamma>) \<inter> - domA \<Gamma> = domA \<Gamma>"
is_num_normalize_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "plus (plus a b) c = plus a (plus b c)"
fresh_3: fixes a :: "atom"   and fm1 :: "fm"   and fm2 :: "fm" shows "a \<sharp> fm1 OR fm2 = (a \<sharp> fm1 \<and> a \<sharp> fm2)"
map_pmf_def: fixes f :: "'d \<Rightarrow> 'b"   and M :: "'d pmf" shows "map_pmf f M = M \<bind> (\<lambda>x. return_pmf (f x))"
Int_atLeastAtMostR1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linorder less_eq less" shows "ord.atMost less_eq b \<inter> ord.atLeastAtMost less_eq c d = ord.atLeastAtMost less_eq c (ord.min less_eq b d)"
assms_3: shows "FSwSbImorph h' hA' MOD' MOD''"
ev_blocks_partI: fixes m :: "nat"   and A :: "'b mat" assumes "\<And>i j k. \<lbrakk>i < j; j < k; k < m; A $$ (k, k) = A $$ (i, i)\<rbrakk> \<Longrightarrow> A $$ (j, j) = A $$ (i, i)" shows "ev_blocks_part m A"
UINF_atLeast_Suc: fixes m :: "nat"   and P :: "nat \<Rightarrow> ('b, 'c) uexpr" shows "(\<Sqinter> i \<in> {Suc m..} \<bullet> P i) = (\<Sqinter> i \<in> {m..} \<bullet> P (Suc i))"
iter_state_iterate_state: fixes f :: "'a \<Rightarrow> ('f, 'g) state"   and x :: "'a" shows "iter_state f x = iterate_state f (iterator_to_list x)"
nsmulextp_code: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "('a, nat) alist"   and ys :: "('a, nat) alist" shows "nsmulextp f (Bag xs) (Bag ys) = snd (mul_ext f (ass_list_to_single_list (alist.impl_of xs)) (ass_list_to_single_list (alist.impl_of ys)))"
is_LTLFinal_def: fixes ltl :: "'c ltl" shows "is_LTLFinal ltl = (\<exists>x9. ltl = F x9)"
assms_2: shows "list.set ms \<subseteq> {0<..}"
mset_factorsI: fixes F :: "'a multiset"   and p :: "'a" assumes "\<And>f. f \<in># F \<Longrightarrow> irreducible f"   and "F \<noteq> {#}"   and "\<Prod>\<^sub># F = p" shows "mset_factors F p"
par_beta_subset_beta: shows "(=>) \<le> (\<rightarrow>\<^sub>\<beta>\<^sup>*)"