hcju/isabelleps huggingface dataset:
measure_split: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and B :: "nat \<Rightarrow> 'b set"   and R :: "nat set"   and A :: "'b set" assumes "measure_space M"   and "\<Union> (B ` R) = UNIV"   and "finite R"   and "\<forall>j1\<in>R. \<forall>j2\<in>R. j1 \<noteq> j2 \<longrightarrow> B j1 \<inter> B j2 = {}"   and "\<forall>j\<in>R. B j \<in> measurable_sets M"   and "A \<in> measurable_sets M" shows "Measure.measure M A = (\<Sum>j\<in>R. Measure.measure M (A \<inter> B j))"
unI: shows "(uni, (\<union>)) \<in> L \<rightarrow> L \<rightarrow> L"
simps_6: fixes \<alpha> :: "('b + 'e, 'c, 'd) hp"   and \<phi> :: "('b + 'e, 'c, 'd) formula"   and \<sigma> :: "'e \<Rightarrow> ('b, 'd) trm" shows "FsubstFO (\<langle> \<alpha> \<rangle> \<phi>) \<sigma> = (\<langle> PsubstFO \<alpha> \<sigma> \<rangle> FsubstFO \<phi> \<sigma>)"
set_mset_add_mset_insert: fixes a :: "'b"   and A :: "'b multiset" shows "set_mset (add_mset a A) = Set.insert a (set_mset A)"
sscan_smap: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and xs :: "'c stream"   and a :: "'a" shows "sscan f (smap g xs) a = sscan (f \<circ> g) xs a"
swap_self: fixes a :: "atom" shows "(a \<rightleftharpoons> a) = 0"
trace_divergence_refine_def: fixes P :: "'a process"   and Q :: "'a process" shows "P \<sqsubseteq>\<^sub>D\<^sub>T Q \<equiv> P \<sqsubseteq>\<^sub>T Q \<and> P \<sqsubseteq>\<^sub>D Q"
abs_majoritiesE': fixes E :: "'b set"   and S :: "'b set"   and T :: "'b set" assumes "card E div 2 < card S"   and "card E div 2 < card T"   and "S \<subseteq> E"   and "T \<subseteq> E"   and "finite E" obtains p :: "'b" where "p \<in> S" and "p \<in> T"
edges_irrefl': fixes u :: "'b"   and g :: "'b ugraph" shows "(u, u) \<notin> edges g"
K_def: shows "K = 1 - (1 - cmod z) / 2"
aux_1_def: fixes g :: "'g"   and n :: "'node" shows "aux_1 g n = (\<lambda>v (uses, phis). let (use, phis') = readVariableRecursive g v n phis in (Mapping.update n (insert use (lookup_multimap uses n)) uses, phis'))"
simps_4: fixes b :: "bexp"   and c1 :: "'a acom"   and c2 :: "'a acom"   and P :: "'a" shows "post (IF b THEN c1 ELSE c2 {P}) = P"
ipresIGSwapAll_def: fixes h :: "'w \<Rightarrow> 'x"   and hA :: "'y \<Rightarrow> 'z"   and MOD :: "('aa, 'ab, 'ac, 'ad, 'ae, 'af, 'w, 'y, 'ag) model_scheme"   and MOD' :: "('ah, 'ai, 'ac, 'aj, 'ak, 'af, 'x, 'z, 'al) model_scheme" shows "ipresIGSwapAll h hA MOD MOD' \<equiv> ipresIGSwap h MOD MOD' \<and> ipresIGSwapAbs hA MOD MOD'"
abs_of_nonneg: fixes a :: "'a" assumes "(0::'a) \<le> a" shows "\<bar>a\<bar> = a"
sgn_vec_def: fixes x :: "('d, 'e) vec" shows "sgn x = x /\<^sub>R (\<parallel>x\<parallel>)"
elims: fixes x :: "parity"   and xa :: "parity"   and y :: "parity" assumes "plus_parity x xa = y" obtains "x = Even" and "xa = Even" and "ya__ = Even"   | "x = Odd" and "xa = Odd" and "ya__ = Even"   | "x = Even" and "xa = Odd" and "ya__ = Odd"   | "x = Odd" and "xa = Even" and "ya__ = Odd"   | y :: "parity" where "x = Either" and "xa = y" and "ya__ = Either"   | "x = Even" and "xa = Either" and "ya__ = Either"   | "x = Odd" and "xa = Either" and "ya__ = Either"
card_UNIV_natural_def: shows "card_UNIV_class.card_UNIV = Phantom(natural) 0"
second_summand_in_sum: fixes y :: "i"   and x :: "i" assumes "CEM" shows "P y (x \<oplus> y)"
IH: fixes y :: "nat" assumes "y < n" shows "f y \<le> C\<^sub>3 * real y + C\<^sub>4"
distinct_83: fixes x71 :: "'d"   and x72 :: "'a \<Rightarrow> 'b"   and x73 :: "('a, 'b, 'c, 'd) seqp"   and x10 :: "'c" shows "{x71}send(x72) . x73 \<noteq> call(x10)"
i: shows "i < length (dropWhile (\<lambda>x. t < f x) xs)"
simps_1: shows "Nat_Bijection.list_decode 0 = []"
getEnv_updEnv_idEnv: fixes x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and y :: "'var" shows "(idEnv [x \<leftarrow> X]_xs) ys y = (if ys = xs \<and> y = x then Some X else None)"
iT_Minus_less_Min_empty: fixes k :: "nat"   and I :: "nat set" assumes "k < iMin I" shows "k \<ominus> I = {}"
\<BB>_Arr: shows "\<BB>\<lparr>Arr\<rparr> = ZFC_in_HOL.set {f}"
arr_lunit: fixes a :: "'d" assumes "D.ide a" shows "D.arr (D.lunit a)"
bij_dual: fixes dual :: "'b \<Rightarrow> 'b" assumes "class.dual dual" shows "bij dual"
sin_cos_eq_iff: fixes y :: "real"   and x :: "real" shows "(sin y = sin x \<and> cos y = cos x) = (\<exists>xa. y = x + 2 * pi * real_of_int xa)"
has_derivative_linear: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "(f has_derivative f') F" shows "linear f'"
elims_3: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list" assumes "class.linorder less_eq less"   and "\<not> linorder.sorted less_eq x" obtains xa :: "'c" and ys :: "'c list" where "x = xa # ys" and "\<not> (Ball (set ys) (less_eq xa) \<and> linorder.sorted less_eq ys)"
rreq_dip_in_vD_dip_eq_ip: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:14} \<longrightarrow> dip \<xi> \<in> vD (rt \<xi>)) \<and> (l \<in> {PRreq-:5, PRreq-:6} \<longrightarrow> dip \<xi> = ip \<xi>) \<and> (l \<in> {PRreq-:13..PRreq-:14} \<longrightarrow> dip \<xi> \<noteq> ip \<xi>))"
identify_block_main: fixes A :: "'b mat"   and j :: "nat"   and i :: "nat" assumes "??.Jordan_Normal_Form_Existence.identify_block A j = i" shows "i \<le> j \<and> (i = 0 \<or> A $$ (i - 1, i) \<noteq> (1::'b)) \<and> (\<forall>k\<ge>i. k < j \<longrightarrow> A $$ (k, Suc k) = (1::'b))"
leq_antisym: fixes x :: "'f llist"   and y :: "'f llist" assumes "lprefix x y"   and "lprefix y x" shows "x = y"
le_ASSERTI_pres: fixes \<Phi> :: "bool"   and S :: "'b nres"   and S' :: "'b nres" assumes "\<Phi> \<Longrightarrow> S \<le> ASSERT \<Phi> \<bind> (\<lambda>_. S')" shows "S \<le> ASSERT \<Phi> \<bind> (\<lambda>_. S')"
invertible_right_inverse2: fixes u :: "'a"   and v :: "'a" assumes "invertible u"   and "u \<in> G"   and "v \<in> G" shows "u \<cdot> (local.inverse u \<cdot> v) = v"
clock_monotone_3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and r3 :: "'b state \<times> (v, v) result" assumes "evaluate ck env s e r3" shows "\<forall>s' r'. r3 = (s', r') \<and> ck = True \<longrightarrow> clock s' \<le> clock s"
T_on_rand'_as_sum: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and s0 :: "('c list \<times> 'd) pmf"   and rs :: "'c list" shows "T\<^sub>p_on_rand' A s0 rs = sum (Partial_Cost_Model.T_on_rand'_n A s0 rs) {..<length rs}"
pow_generator_eq_iff_cong: fixes x :: "nat"   and y :: "nat" assumes "finite (carrier \<G>)" shows "(\<^bold>g [^] x = \<^bold>g [^] y) = [x = y] (mod order \<G>)"
lvl_insert_incr_iff: fixes a :: "'b"   and t :: "('b \<times> nat) tree" shows "(lvl (insert a t) = lvl t + 1) = (\<exists>l x r. insert a t = \<langle>l, (x, lvl t + 1), r\<rangle> \<and> lvl l = lvl r)"
select_convs_1: fixes access :: "acc_modi"   and more :: "'b" shows "access \<lparr>access = access, \<dots> = more\<rparr> = access"
deg: fixes factor :: "int poly" assumes "p.dvdm u factor"   and "factor dvd f" shows "j' \<le> degree factor"
mon: shows "monic (q.Mp (Polynomial.smult ilc f))"
x_val: shows "x = int (\<B>' index \<pi> ` ps)"
exists_QuoteP: fixes j :: "name"   and x :: "tm" assumes "atom j \<sharp> x" shows "{} \<turnstile> SyntaxN.Ex j (QuoteP x (Var j))"
ordered_negative_resolvent_def: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set" shows "ordered_negative_resolvent P1 P2 C = (\<exists>L. C = P1 - {L} \<union> (P2 - {complement L}) \<and> (positive_part P1 = {} \<and> strictly_maximal_literal P1 L) \<and> strictly_maximal_literal (positive_part P2) (complement L))"
vars2'_def: shows "st1.vars2' = ind_in_set st1.vars0 ` vars2"
poly_prod_list: fixes ps :: "'c poly list"   and x :: "'c" shows "poly (prod_list ps) x = (\<Prod>p\<leftarrow>ps. poly p x)"
fps_const_nonzero_eq_nonzero: fixes c :: "'b" assumes "c \<noteq> (0::'b)" shows "fps_const c \<noteq> 0"
prems_6: shows "(l, g, \<mu>) \<in> PTA.trans_of A"
Cons_2: shows "convert_atom_list var (a # L) (xs' @ x # xs) = Some L'"
Rep_bit1_inject: fixes x :: "'a bit1"   and y :: "'a bit1" shows "(Rep_bit1 x = Rep_bit1 y) = (x = y)"
limproc_is_thelub_fun: fixes S :: "nat \<Rightarrow> 'b \<Rightarrow> 'c process"   and c :: "'b" assumes "chain S" shows "Lub S c = lim_proc (range (\<lambda>x. S x c))"
lcm_list_least: fixes bs :: "'a list"   and a :: "'a" assumes "\<And>b. b \<in> set bs \<Longrightarrow> b dvd a" shows "lcm_list bs dvd a"
fold_min_eqI1: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and y :: "'b" assumes "fold (linorder_list0.min_for le) ys y \<notin> set ys" shows "fold (linorder_list0.min_for le) ys y = y"
poly_no_roots_leq: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x\<le>a. poly p x \<noteq> 0) = (p \<noteq> 0 \<and> count_roots_below p a = 0)"
spoly_repI: fixes g1 :: "'t \<Rightarrow>\<^sub>0 'c"   and g2 :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "spoly g1 g2 = (\<Sum>g\<in>G. q g \<odot> g)"   and "\<And>g. q g \<in> punit.dgrad_p_set d m"   and "\<And>g. q g \<odot> g \<noteq> 0 \<Longrightarrow> lt (q g \<odot> g) \<prec>\<^sub>t term_of_pair (lcs (lp g1) (lp g2), component_of_term (lt g2))" shows "spoly_rep d m G g1 g2"
foldl_append: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and a :: "'c"   and xs :: "'d list"   and ys :: "'d list" shows "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
M_neg_imp_z_squared_big: fixes v :: "(real, 3) vec" assumes "v \<bullet> (M *v v) < 0" shows "(v $ 1)\<^sup>2 + (v $ 2)\<^sup>2 < (v $ 3)\<^sup>2"
sgn_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (sgn a) zero = less a zero"
fvu: fixes i :: "nat" assumes "i < n" shows "v $ i = a (unit_vec n i)"
iT_inext_1: fixes t :: "nat"   and n :: "nat" assumes "t \<in> [n\<dots>]" shows "inext t [n\<dots>] = Suc t"
multpw_split1L: fixes X :: "'a multiset"   and y :: "'a"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set" assumes "(X, add_mset y Y) \<in> multpw ns" obtains z :: "'a" and Z :: "'a multiset" where "X = add_mset z Z" and "(z, y) \<in> ns" and "(Z, Y) \<in> multpw ns"
NatTrans_def: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "NatTrans \<eta> \<equiv> NatTransP \<eta> \<and> NatTransExt \<eta>"
inf_bool_def: fixes P :: "bool"   and Q :: "bool" shows "P \<sqinter> Q = (P \<and> Q)"
g: fixes j :: "nat"   and x :: "nat"   and r :: "nat"   and i :: "nat" assumes "j < e_length x" shows "eval g [j, r, i, x] \<down>= (if 0 < the (eval r_result1 [e_length x, i, e_take (Suc j) x]) then Suc j else r)"
prime_factorization_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.prime_factorization zero times one normalize zero = {#}"
trans2: fixes p :: "'a"   and t :: "'a"   and r :: "'a"   and q :: "'a" assumes "p \<parallel> t"   and "t \<parallel> r"   and "r \<parallel> q" shows "\<not> p \<parallel> q"
eq_refl: fixes x :: "'c list"   and y :: "'c list" assumes "x = y" shows "subseq x y"
length_vmuaux_def: shows "length_vmuaux = (\<lambda>_ (uu_, y). length y)"
valid_prefixes_alt_def: fixes r :: "('h, 'i) routing_rule_scheme list" shows "valid_prefixes r = (\<forall>e\<in>set r. valid_prefix (routing_match e))"
get_proc_get_return_edges: fixes a :: "'a"   and a' :: "'a" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "get_proc (targetnode a) = get_proc (sourcenode a')"
vp1: shows "val_process Mkt pf1 n w = (\<Sum>x\<in>support_set pf1 \<union> support_set pf2. prices Mkt x n w * pf1 x (Suc n) w)"
fset_eq_fsubset: fixes A :: "'a fset"   and B :: "'a fset" shows "(A = B) = (A |\<subseteq>| B \<and> B |\<subseteq>| A)"
rel_simps_73: fixes n :: "num" shows "(numeral n = (1::'a)) = (n = num.One)"
ceuclidean_eqI: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and CBasis :: "'a set"   and x :: "'a"   and y :: "'a" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis"   and "\<And>b. b \<in> CBasis \<Longrightarrow> cinner x b = cinner y b" shows "x = y"
select_convs_2: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
natural_transformation_axioms: shows "natural_transformation CC.comp (\<cdot>) local.Prod Prod' \<sigma>"
sup_left_idem: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup x y) = sup x y"
unat_numeral: fixes b :: "num" shows "unat (numeral b) = numeral b mod 2 ^ LENGTH('a)"
mono: fixes x :: "'a"   and y :: "'a" assumes "x \<prec>[le] y" shows "u x + \<epsilon> * f x < u y + \<epsilon> * f y"
set_times_def: fixes A :: "'a set"   and B :: "'a set" shows "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
ennreal_indicator: fixes A :: "'d set"   and x :: "'d" shows "ennreal (indicat_real A x) = indicator A x"
list_append_eq_Cons_cases: fixes ys :: "'b list"   and zs :: "'b list"   and x :: "'b"   and xs :: "'b list" assumes "ys @ zs = x # xs" obtains "ys = []" and "zs = x # xs"   | ys' :: "'b list" where "ys = x # ys'" and "ys' @ zs = xs"
h_def: fixes a :: "nat"   and b :: "nat"   and c :: "nat" shows "h a b c = a * b"
insert_if: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "finite A" shows "comm_monoid_add.sum (\<squnion>) bot g (insert x A) = (if x \<in> A then comm_monoid_add.sum (\<squnion>) bot g A else g x \<squnion> comm_monoid_add.sum (\<squnion>) bot g A)"
monoE: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "mono f"   and "x \<le> y" obtains "f x \<le> f y"
quot_monomial_ideal_monomial: fixes S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and x :: "'x" shows "ideal (monomial (1::'b) ` S) \<div> monomial (1::'b) (monomial 1 x) = ideal (monomial (1::'b) ` (\<lambda>s. s - monomial 1 x) ` S)"
lookup_raw_map2_val_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ys :: "('a \<times> 'd) list \<times> 'o"   and g :: "('a \<times> 'c) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o"   and h :: "('a \<times> 'd) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e"   and k0 :: "'a" assumes "oalist_inv xs"   and "oalist_inv ys"   and "map2_val_compat' g"   and "map2_val_compat' h"   and "\<And>zs. oalist_inv zs \<Longrightarrow> g zs = map_val_raw (\<lambda>k v. f k v (0::'d)) zs"   and "\<And>zs. oalist_inv zs \<Longrightarrow> h zs = map_val_raw (\<lambda>k. f k (0::'c)) zs"   and "\<And>k. f k (0::'c) (0::'d) = (0::'e)" shows "lookup_raw (map2_val_raw f g h xs ys) k0 = f k0 (lookup_raw xs k0) (lookup_raw ys k0)"
assms_1: shows "finite {x. jump f x \<noteq> 0 \<and> a < x \<and> x < c}"
nn_integral_const: fixes M :: "'d measure"   and c :: "ennreal" shows "\<integral>\<^sup>+ x. c \<partial>M = c * emeasure M (space M)"
simps_2: fixes x :: "char list"   and G :: "diagram" shows "bot_ass (Exists_dia x G) = Exists_int x (bot_dia G)"
mult1_def: fixes r :: "('k \<times> 'k) set" shows "mult1 r = {(N, M). \<exists>a M0 K. M = add_mset a M0 \<and> N = M0 + K \<and> (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
knows_Spy_Gets: fixes A :: "agent"   and X :: "msg"   and evs :: "event list" shows "knows Spy (Gets A X # evs) = knows Spy evs"
thin_set_def: fixes \<F> :: "nat set set" shows "thin_set \<F> \<equiv> \<F> \<subseteq> Collect finite \<and> (\<forall>S\<in>\<F>. \<forall>T\<in>\<F>. init_segment S T \<longrightarrow> S = T)"
TopoS_modelLibrary_axioms: shows "TopoS_modelLibrary SINVAR_LIB_BLPtrusted SINVAR_BLPtrusted.sinvar"
axioms: fixes R :: "('a, 'b) ring_scheme" assumes "UP_ring R" shows "ring R"
continuous_on_closed_vimage: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "closed s" shows "continuous_on s f = (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
Red1NewArrayFail: fixes i :: "32 word"   and h :: "'heap"   and T :: "ty"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and l :: "'addr val list" assumes "0 \<le>s i"   and "allocate h (Array_type T (nat (sint i))) = {}" shows "uf,P,t \<turnstile>1 \<langle>newA T\<lfloor>Val (Intg i)\<rceil>,(h, l)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt OutOfMemory),(h, l)\<rangle>"
infinite_finite_induct: fixes P :: "'a set \<Rightarrow> bool"   and A :: "'a set" assumes "\<And>A. infinite A \<Longrightarrow> P A"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P A"
iso_class_eq: fixes F :: "'a set"   and G :: "'a set" assumes "B.is_iso_class F"   and "B.is_iso_class G"   and "F \<inter> G \<noteq> {}" shows "F = G"
intros_2: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and seq :: "('a, 'b) sequent"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" assumes "r \<in> R"   and "case r of (x, xa) \<Rightarrow> (x, xa) \<in> upRules" shows "extendRule seq r \<in> ext R R' M N"
cvalid: shows "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
assms_3: shows "(spec, spec') \<in> fully_unsorted_poly_rel O mset_poly_rel"
id_iso: fixes r :: "('d \<times> 'd) set" shows "iso r r id"
approx_int: fixes n :: "nat" shows "\<bar>x - real_of_int (p n) / real_of_int (q n)\<bar> < 1 / real_of_int (q n) ^ n"
b'c: fixes z :: "complex" assumes "z \<in> closed_segment b' c" shows "k \<le> d \<bullet> z"
varsPB_def: fixes \<Phi> :: "('c, 'd) lit list set" shows "varsPB \<Phi> = \<Union> {varsC c |c. c \<in> \<Phi>}"
eq_dff_trans: fixes x :: "'b fract"   and y :: "'b fract"   and z :: "'b fract" assumes "x =dff y"   and "y =dff z" shows "x =dff z"
simps_1: fixes i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec" shows "diagonal_to_Smith_i_PQ [] i bezout (P, A, Q) = (P, A, Q)"
absorb_iff1: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (a \<sqinter> b = a)"
errMOD_igVarIPresIGWlsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igVarIPresIGWls MOD" shows "igVarIPresIGWls (errMOD MOD)"
star_rtc3: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "(t_one \<union> t_prod x x = x) = (x = t_star x)"
naturality: fixes f :: "'a" assumes "\<phi>.A.arr f" shows "B (\<phi> (\<phi>.A.cod f)) (F f) = B (G f) (\<phi> (\<phi>.A.dom f))"
ide_def: fixes a :: "'j" shows "J.ide a \<equiv> a \<cdot>\<^sub>J a \<noteq> J.null \<and> (\<forall>f. (f \<cdot>\<^sub>J a \<noteq> J.null \<longrightarrow> f \<cdot>\<^sub>J a = f) \<and> (a \<cdot>\<^sub>J f \<noteq> J.null \<longrightarrow> a \<cdot>\<^sub>J f = f))"
Says_imp_knows: fixes A' :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list"   and A :: "agent" assumes "Says A' B X \<in> set evs" shows "X \<in> knows A evs"
steps: shows "(m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f\<^sup>* (m', sb', \<O>', \<R>', \<S>')"
emeasure_count_space_eq_0: fixes A :: "'k set"   and X :: "'k set" shows "(emeasure (count_space A) X = 0) = (X \<subseteq> A \<longrightarrow> X = {})"
integral_iff: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "(a \<otimes>\<^bsub>K [X]\<^esub> b = \<zero>\<^bsub>K [X]\<^esub>) = (a = \<zero>\<^bsub>K [X]\<^esub> \<or> b = \<zero>\<^bsub>K [X]\<^esub>)"
Vinv3_inductive_2: shows "{Vinv3 \<inter> Vinv1} TS.trans v_TS {> Vinv3}"
in_smallo_zero_iff: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "(f \<in> o[F](\<lambda>_. 0::'c)) = (\<forall>\<^sub>F x in F. f x = (0::'c))"
normal_form_uminus: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "normal_form F (- p) = - normal_form F p"
rng_to_rng_of_frac_is_ring_hom: shows "rng_to_rng_of_frac \<in> ring_hom R rec_rng_of_frac"
intro: fixes R :: "('b, 'c) ring_scheme" assumes "semiring R"   and "Group.comm_monoid R" shows "csemiring R"
diff_left: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and a' :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b" shows "cblinfun_apply (a - a') b = cblinfun_apply a b - cblinfun_apply a' b"
True: shows "poly p st \<noteq> 0 \<and> poly p tt \<noteq> 0"
comm: fixes a :: "int poly fract"   and b :: "int poly fract" shows "rat_poly_times a b = rat_poly_times b a"
simps_3: fixes c :: "int"   and e :: "MIR.num" shows "isrlfm (Eq (CN 0 c e)) = (0 < c \<and> numbound0 e)"
rel_gpv''_map_gpv2: fixes A :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and C :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and R :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and gpv :: "('q, 's, 'u) gpv"   and f :: "'w \<Rightarrow> 'r"   and g :: "'x \<Rightarrow> 't"   and gpv' :: "('w, 'x, 'v) gpv" shows "rel_gpv'' A C R gpv (map_gpv f g gpv') = rel_gpv'' (\<lambda>a b. A a (f b)) (\<lambda>c d. C c (g d)) R gpv gpv'"
node_in_heap: shows "node |\<in>| node_ptr_kinds h"
FvarsT_suc: fixes t :: "'trm" assumes "t \<in> trm" shows "FvarsT (suc t) = FvarsT t"
qFreshAll_simps_2: fixes xs :: "'a"   and x :: "'b"   and delta :: "'e"   and inp :: "'c \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qTerm option"   and binp :: "'d \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qAbs option" shows "qFresh xs x (qOp delta inp binp) = (liftAll (qFresh xs x) inp \<and> liftAll (qFreshAbs xs x) binp)"
beukers_integral3_pos: shows "0 < local.beukers_integral3"
cases: fixes a :: "'b list"   and A :: "'b set" assumes "a \<in> lists A" obtains "a = []"   | aa :: "'b" and l :: "'b list" where "a = aa # l" and "aa \<in> A" and "l \<in> lists A"
Sink_def: shows "GM_sec_model.Sink = {Idle st |st. True}"
normalized_normalize: fixes xs :: "('c, 'd) bintree option list" shows "normalized (Binomial_Queue.normalize xs)"
terminal_tllist_of_llist_lfinite: fixes xs :: "'c llist"   and b :: "'d" assumes "lfinite xs" shows "terminal (tllist_of_llist b xs) = b"
emeasure_countably_additive: fixes M :: "'b measure" shows "countably_additive (sets M) (emeasure M)"
x1: shows "x\<^sub>1 = (- b + sqrt D) / (2 * a)"
l_p_2: shows "nat_of_fatom (get_atom l) = length B"