hcju/isabelleps huggingface dataset:
nfv_regex_def: fixes fv :: "'a \<Rightarrow> nat set"   and r :: "'a Regex.regex" shows "Regex.nfv_regex fv r = Max (insert 0 (Suc ` Regex.fv_regex fv r))"
has_codomain_iff_arr: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "(codomains f \<noteq> {}) = arr f"
boundedE: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "semilattice_order f less_eq less"   and "less_eq a (f b c)" obtains "less_eq a b" and "less_eq a c"
LIM_fun_gt_zero: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and c :: "real" assumes "f \<midarrow>c\<rightarrow> l"   and "0 < l" shows "\<exists>r>0. \<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x"
mask_out_sub_mask: fixes x :: "'b word"   and n :: "nat" shows "x && ~~ mask n = x - (x && mask n)"
fp_cop_F_code: fixes ds :: "'b set" shows "fp_cop_F ds = while (\<lambda>A. cop_F ds A \<noteq> A) (cop_F ds) {}"
lmap_lappend_distrib: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist"   and ys :: "'b llist" shows "lmap f (xs @\<^sub>l ys) = lmap f xs @\<^sub>l lmap f ys"
g_assoc: shows "associative (\<lambda>x y. if x = One \<or> y = One then One else Zero)"
that: fixes T :: "('a, 'b, 'c) psi" assumes "P = T \<parallel> R"   and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' T = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q" shows "thesis"
EquivE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<equiv> \<psi> in v]" shows "([\<phi> in v] \<longrightarrow> [\<psi> in v]) \<and> ([\<psi> in v] \<longrightarrow> [\<phi> in v])"
inject: fixes x2 :: "'a"   and y2 :: "'a" shows "(Inner x2 = Inner y2) = (x2 = y2)"
upper: fixes y :: "'a" assumes "y \<in> M" shows "\<exists>z\<in>M. (y, z) \<in> leq \<and> z \<in> iterates_above a"
euclidean_representation_sum: fixes f :: "'a \<Rightarrow> real"   and b :: "'a" shows "((\<Sum>i\<in>Basis. f i *\<^sub>R i) = b) = (\<forall>i\<in>Basis. f i = b \<bullet> i)"
nat_pow_one: fixes n :: "nat" shows "\<one>\<^bsub>units_of R\<^esub> [^]\<^bsub>units_of R\<^esub> n = \<one>\<^bsub>units_of R\<^esub>"
gt_ex: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.no_top less_eq less" shows "\<exists>y. less x y"
summable_comparison_test': fixes g :: "nat \<Rightarrow> real"   and N :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "summable g"   and "\<And>n. N \<le> n \<Longrightarrow> norm (f n) \<le> g n" shows "summable f"
part_equivp_refl_symp_transp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "part_equivp R = ((\<exists>x. R x x) \<and> symp R \<and> transp R)"
inject: fixes x1 :: "bool"   and x2 :: "nat"   and y1 :: "bool"   and y2 :: "nat" shows "(Lit x1 x2 = Lit y1 y2) = (x1 = y1 \<and> x2 = y2)"
divide_nonneg_nonneg: fixes x :: "'b"   and y :: "'b" assumes "(0::'b) \<le> x"   and "(0::'b) \<le> y" shows "(0::'b) \<le> x / y"
that: fixes a :: "nat \<Rightarrow> complex"   and n :: "nat" assumes "\<And>z. f z = (\<Sum>i\<le>n. a i * z ^ i)" shows "thesis"
trms\<^sub>s\<^sub>s\<^sub>t_unlabel_dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_eq: fixes A :: "('f strand_label \<times> ('d, 'e) stateful_strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
induct: fixes x1 :: "'a OCL_Types.type"   and x2 :: "'a OCL_Types.type"   and P :: "'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "to_ordered_collection x1 x2"   and "\<And>\<tau>. P (Collection \<tau>) (Sequence \<tau>)"   and "\<And>\<tau>. P (Set \<tau>) (OrderedSet \<tau>)"   and "\<And>\<tau>. P (OrderedSet \<tau>) (OrderedSet \<tau>)"   and "\<And>\<tau>. P (Bag \<tau>) (Sequence \<tau>)"   and "\<And>\<tau>. P (Sequence \<tau>) (Sequence \<tau>)" shows "P x1 x2"
Rep_inject: fixes x :: "'b word"   and y :: "'b word" shows "(to_bl x = to_bl y) = (x = y)"
pCons_eq_0_iff: fixes a :: "'a"   and p :: "'a poly" shows "(pCons a p = 0) = (a = (0::'a) \<and> p = 0)"
polyadd_normh: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "isnpolyh (p +\<^sub>p q) (min n0 n1)"
reach_nil_iff: fixes tr :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a" shows "reach tr p [] q = (p = q)"
pos_oriented_of_circline: fixes H :: "circline" shows "pos_oriented (of_circline H)"
unit_char_2: fixes a :: "'c" assumes "C.obj a" shows "D.iso (unit a)"
less_0_prodinf: fixes f :: "nat \<Rightarrow> real" assumes "convergent_prod f"   and "\<And>i. 0 < f i" shows "0 < prodinf f"
sees_field_def: fixes P :: "'c prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" shows "P \<turnstile> C sees F:T (fm) in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of (map (\<lambda>((F, D), Tm). (F, D, Tm)) FDTs) F = \<lfloor>(D, T, fm)\<rfloor>"
cases: fixes x :: "'c list" obtains "xa__ = []"   | x :: "'c" where "xa__ = [xb__]"   | x :: "'c" and y :: "'c" and xs :: "'c list" where "xa__ = x # y # xs"
SUP_const: fixes A :: "'b set"   and f :: "'a" assumes "A \<noteq> {}" shows "(\<Squnion>i\<in>A. f) = f"
int_pow_neg: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and i :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> - i = inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> i)"
bound_aux: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real"   and x :: "real"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>f l - f m\<bar> \<le> x"   and "p \<in> C"   and "q \<in> C" shows "\<bar>fiX f p r - f q\<bar> \<le> x + \<Delta>"
fls_X_power_base_factor: fixes n :: "nat" shows "fls_base_factor (fls_X ^ n) = 1"
openI: fixes m :: "'a set" assumes "m \<in> S" shows "m open"
cod_runit: fixes a :: "'c local.term set" assumes "ide a" shows "cod \<r>[a] = a"
K2_abs: fixes v :: "(real, 3) vec" assumes "v \<noteq> 0" shows "(proj2_abs v \<in> K2) = (v \<bullet> (M *v v) < 0)"
path_split_Cons: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog n as n'"   and "as \<noteq> []" obtains a' :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node" and as' :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list" where "as = a' # as'" and "n = sourcenode a'" and "valid_edge prog a'" and "While_CFG.path prog (targetnode a') as' n'"
pair_acc_modi_simp: fixes x :: "'a"   and a :: "'b" shows "accmodi (x, a) = accmodi a"
Rep_map_def: fixes f :: "'c \<Rightarrow> ('b, unit) Old_Datatype.node set"   and xs :: "'c SList.list" shows "Rep_map f xs = list_rec xs NIL (\<lambda>x l. CONS (f x))"
trans_less_add1: fixes i :: "nat"   and j :: "nat"   and m :: "nat" assumes "i < j" shows "i < j + m"
UP_ring_def: fixes R :: "('a, 'b) ring_scheme" shows "UP_ring R \<equiv> ring R"
old: assumes "x \<in> cipher_queries ose"   and "x \<notin> set ciphers" shows "thesis"
inner_scaleR_left: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and r :: "real"   and x :: "'b"   and y :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (scaleR r x) y = r * inner x y"
card2_either_elt1_or_elt2: fixes X :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "card X = 2"   and "x \<in> X"   and "y \<in> X"   and "x \<noteq> y"   and "z \<in> X"   and "z \<noteq> x" shows "z = y"
keys_proj_poly: fixes k :: "'k"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "keys (proj_poly k p) = pp_of_term ` {x \<in> keys p. component_of_term x = k}"
mC_C_part_def: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_C_part c d P F \<equiv> {} \<notin> P \<and> {} \<notin> F ` P \<and> part {..<brn c} P \<and> part {..<brn d} (F ` P)"
reflp_rel_option: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflp R" shows "reflp (rel_option R)"
ideE: fixes F :: "('a E.term, 'a) arr" assumes "ide F" obtains "arr F" and "Dom F = Cod F" and "B.ide (Map F)" and "Map F = EVAL (Dom F)" and "Map F = EVAL (Cod F)"
simps_3: fixes x :: "floatarith"   and xs :: "floatarith list"   and y :: "floatarith"   and ys :: "floatarith list" shows "inner_floatariths (x # xs) (y # ys) = Add (Mult x y) (inner_floatariths xs ys)"
sel_4: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real"   and x5 :: "real"   and x6 :: "real"   and x7 :: "real"   and x8 :: "real" shows "Im3 (Octo x1 x2 x3 x4 x5 x6 x7 x8) = x4"
N_fs: fixes i :: "nat" assumes "i < length fs" shows "\<parallel>fs ! i\<parallel>\<^sup>2 \<le> N"
assms_4: shows "revision_step r' s\<^sub>1 s\<^sub>2'"
fds_euler_product_LIMSEQ': fixes f :: "'a fds"   and s :: "'a" assumes "completely_multiplicative_function (fds_nth f)"   and "fds_abs_converges f s" shows "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse ((1::'a) - fds_nth f p / nat_power p s) else (1::'a)) \<longlonglongrightarrow> eval_fds f s"
prems_1: shows "ins3 n d x \<langle>l, y, r\<rangle> = Bal2 t'"
simps_2: fixes P :: "'a list \<Rightarrow> bool"   and res :: "'a list \<times> 'a list"   and ps :: "'a list"   and q :: "'a"   and qs :: "'a list" shows "maxsplit P res ps (q # qs) = maxsplit P (if P ps then (ps, q # qs) else res) (ps @ [q]) qs"
abc_only_cba_3: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c" shows "\<not> betw b c a"
dim_subset_UNIV: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and Basis :: "'b set"   and S :: "'b set" assumes "finite_dimensional_vector_space scale Basis" shows "vector_space.dim scale S \<le> finite_dimensional_vector_space.dimension Basis"
minus_add_distrib: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ab_group_add plus zero minus uminus" shows "uminus (plus a b) = plus (uminus a) (uminus b)"
nhop_of_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" assumes "rt \<noteq> update rt dip (dsn, dsk, flag, hops, nhip)" shows "the (nhop (update rt dip (dsn, dsk, flag, hops, nhip)) dip) = nhip"
min_algs_mono2: fixes s1 :: "('f, 'e, 'd) F1 \<Rightarrow> 'e"   and s2 :: "('f, 'e, 'd) F2 \<Rightarrow> 'd" shows "relChain SucFbd (\<lambda>i. snd (min_algs s1 s2 i))"
rep_preserves_seq: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "seq g f" shows "Seq (rep g) (rep f)"
disjointpD: fixes xs :: "('c \<Rightarrow> bool) list"   and n :: "nat"   and x :: "'c"   and m :: "nat" assumes "disjointp xs"   and "(xs ! n) x"   and "(xs ! m) x"   and "n < length xs"   and "m < length xs" shows "n = m"
mset_set_upto_eq_mset_upto: fixes n :: "nat" shows "mset_set {..<n} = mset [0..<n]"
comp_null_2: fixes f :: "'b" shows "f \<cdot>\<^sub>D D.null = D.null"
L_lower_bound_card: fixes P :: "'a set set" assumes "bp P" shows "card {u \<in> U. real c / 2 < w u} \<le> card P"
Ri_Sup_subset_Ri_Liminf: fixes Ns :: "'a literal multiset set llist" assumes "chain (\<triangleright>) Ns" shows "Ri (Sup_llist Ns) \<subseteq> Ri (Liminf_llist Ns)"
convergent_one_step_def: fixes t0 :: "real"   and t1 :: "real"   and x :: "real \<Rightarrow> 'b"   and incr :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "nat"   and B :: "real"   and r :: "real"   and L :: "real"   and t :: "nat \<Rightarrow> real" shows "convergent_one_step t0 t1 x incr p B r L t \<equiv> consistent_one_step t0 t1 x incr p B r L \<and> max_step t t1 p L B r \<and> convergent_one_step_axioms t0 t"
oVeblen_oSuc: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal" assumes "normal F" shows "oVeblen F (oSuc x) = oDeriv (oVeblen F x)"
stab_typed_some_imp_member: fixes s :: "s"   and i :: "nat"   and c :: "nat"   and cl :: "cl"   and \<S> :: "s_context" assumes "stab s i c = Some cl"   and "store_typing s \<S>"   and "i < length (inst s)" shows "Some cl \<in> set (concat (s.tab s))"
pmf_set_integral_code_alt_finite: fixes A :: "'b set"   and p :: "'b pmf"   and f :: "'b \<Rightarrow> real" assumes "finite A" shows "pmf_set_integral p f A = (\<Sum>x\<in>A. pmf p x * f x)"
iNext_iff_singleton: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNext t0 I P = (\<box> t {inext t0 I}. P t)"
product_open_generates_sets_PiF_single: fixes I :: "'d set" assumes "I \<noteq> {}"   and "finite I" shows "sets (Pi\<^sub>F {I} (\<lambda>_. borel)) = sigma_sets (space (Pi\<^sub>F {I} (\<lambda>_. borel))) {Pi' I F |F. \<forall>i\<in>I. F i \<in> Collect open}"
ran_m_fmdrop: fixes C :: "'a"   and N :: "('a, 'b) fmap" assumes "C \<in># dom_m N" shows "ran_m (fmdrop C N) = remove1_mset (the (fmlookup N C)) (ran_m N)"
Gauss_Jordan_in_ij_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'e" shows "Gauss_Jordan_in_ij A i j = (let n = LEAST n. A $ n $ j \<noteq> (0::'d) \<and> i \<le> n; interchange_A = interchange_rows A i n; A' = mult_row interchange_A i ((1::'d) / interchange_A $ i $ j) in \<chi>s. if s = i then A' $ s else row_add A' s i (- interchange_A $ s $ j) $ s)"
kAD: shows "k \<in> mat_kernel (four_block_mat A (0\<^sub>m n m) (0\<^sub>m m n) D)"
cgenideal_self: fixes i :: "('c multiset \<Rightarrow> 'a) list" assumes "i \<in> carrier (poly_ring L)" shows "i \<in> PIdl\<^bsub>poly_ring L\<^esub> i"
apex_def: shows "\<mu>\<nu>_\<pi>.apex \<equiv> C.dom \<mu>\<nu>_\<pi>.leg0"
assms: shows "\<theta> \<circ>\<^sub>s \<delta> supports \<I>"
hyps_2: fixes U :: "POPLmark.type"   and U' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> S' <: (\<forall><:U. U')" obtains S' :: "POPLmark.type" where "\<Gamma> \<turnstile> U <: S" and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'" and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'"
assms_1: shows "q $ 0 \<noteq> (0::'a)"
h0: fixes x :: "'c" shows "h (0, x) = f x"
pr_comp1_1: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1"   and "g \<in> PrimRec1" shows "(\<lambda>x. f (g x)) \<in> PrimRec1"
Entry_no_call_source: fixes a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "knd a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "src a = NewEntry" shows "False"
A: shows "M - 4 * delta + dist x' y' \<le> dist px y'"
ord_option_eq_simps_2: fixes x :: "'a"   and y :: "'a option" shows "le_option (Some x) y = (y = Some x)"
accD: fixes x :: "'s"   and y :: "'s" assumes "(x, y) \<in> acc"   and "x \<in> X" shows "y \<in> X"
pdevs_inner_def: fixes x :: "'b pdevs"   and b :: "'b" shows "pdevs_inner x b = unop_pdevs (\<lambda>x. x \<bullet> b) x"
conj_qf: fixes p :: "fm"   and q :: "fm" assumes "qfree p"   and "qfree q" shows "qfree (MIR.conj p q)"
trans_opar_comp: fixes s :: "((nat \<Rightarrow> 'a) \<times> 'b, 'd seq_action) automaton"   and i :: "nat"   and t :: "('c, 'd seq_action) automaton" shows "automaton.trans (s \<langle>\<langle>\<^bsub>i\<^esub> t) = oparp_sos i (automaton.trans s) (automaton.trans t)"
cr_option_le_spmf_def: fixes x :: "'c option"   and p :: "'c spmf" shows "cr_option_le_spmf x p = ord_spmf (=) (return_pmf x) p"
dependent_wellorder_choice: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<And>r f g x. (\<And>y. less y x \<Longrightarrow> f y = g y) \<Longrightarrow> P f x r = P g x r"   and "\<And>x f. (\<And>y. less y x \<Longrightarrow> P f y (f y)) \<Longrightarrow> \<exists>r. P f x r" shows "\<exists>f. \<forall>x. P f x (f x)"
downset_set_prop: shows "\<Down> = \<Union> \<circ> (`) \<down>"
tuple_simps_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.commutative_square f g h k" shows "C.cod \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = f \<down>\<down> g"
that: fixes a :: "ereal"   and b :: "ereal" assumes "a < x"   and "x < b"   and "{a<..<b} \<subseteq> S" shows "thesis"
normalize_gcd: fixes a :: "'c"   and b :: "'c" shows "normalize (normalization_euclidean_semiring_class.gcd a b) = normalization_euclidean_semiring_class.gcd a b"
vector_space_ow_axioms: fixes U\<^sub>V\<^sub>S :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" assumes "vector_space_on U\<^sub>V\<^sub>S scale" shows "vector_space_ow U\<^sub>V\<^sub>S (+) (0::'d) (-) uminus scale"
F_weight_OUT: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "f \<in> Field leq" shows "d_OUT f x \<le> weight \<Gamma> x"
IHP_3: shows "l2 \<in> w12 \<otimes> w22"
natural_transformation_axioms: shows "natural_transformation A B H G \<tau>'"
exp_similiar_sq_mtx_diag_eq: fixes P :: "'b sq_mtx"   and A :: "'b sq_mtx"   and f :: "'b \<Rightarrow> real" assumes "mtx_invertible P"   and "A = P\<^sup>-\<^sup>1 * (\<d>\<i>\<a>\<g> i. f i) * P" shows "exp A = P\<^sup>-\<^sup>1 * exp (\<d>\<i>\<a>\<g> i. f i) * P"
set_subseqs_eq: fixes ys :: "'a list" shows "set (subseqs ys) = {xs. subseq xs ys}"
simps_6: fixes \<phi> :: "formula" shows "future_bounded (Neg \<phi>) = future_bounded \<phi>"
concat_eq_Nil_conv: fixes xss :: "'d list list" shows "(concat xss = []) = (\<forall>xs\<in>set xss. xs = [])"
of_int_0_eq_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(zero = ring_1.of_int one plus zero minus z) = (z = 0)"
induct: fixes P :: "llam \<Rightarrow> bool"   and llam :: "llam" assumes "\<And>name. P (lPar name)"   and "\<And>nat. P (lVar nat)"   and "\<And>llam1 llam2. \<lbrakk>P llam1; P llam2\<rbrakk> \<Longrightarrow> P (lApp llam1 llam2)"   and "\<And>llam. P llam \<Longrightarrow> P (lLam llam)" shows "P llam"
smult_distrib_left: fixes r :: "'f"   and m :: "'v"   and n :: "'v" assumes "r \<in> UNIV"   and "m \<in> V"   and "n \<in> V" shows "(r \<cdot>\<cdot> (+) m) n = r \<cdot> m + r \<cdot> n"
strict_sorted_equal: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less"   and "linorder.strict_sorted less xs"   and "linorder.strict_sorted less ys"   and "set ys = set xs" shows "ys = xs"
piecewise_C1_differentiable_add: fixes f :: "real \<Rightarrow> 'a"   and i :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "f piecewise_C1_differentiable_on i"   and "g piecewise_C1_differentiable_on i" shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
push_bit_0_id: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit 0 = id"
op_dghm_components_3: fixes \<FF> :: "V" shows "op_dghm \<FF>\<lparr>HomDom\<rparr> = op_dg (\<FF>\<lparr>HomDom\<rparr>)"
uminus_carrier_vec: fixes v :: "'b vec"   and n :: "nat" shows "(- v \<in> carrier_vec n) = (v \<in> carrier_vec n)"
locally_sound_mode_use_def: fixes lc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" shows "abs.locally_sound_mode_use lc = (\<forall>c' mds' mem'. \<langle>c', mds', mem'\<rangle>\<^sub>A \<in> abs.loc_reach lc \<longrightarrow> (\<forall>x. (x \<in> mds' GuarNoReadOrWrite \<longrightarrow> abs.doesnt_read_or_modify c' x) \<and> (x \<in> mds' GuarNoWrite \<longrightarrow> abs.doesnt_modify c' x)))"
intro: fixes M :: "real measure" assumes "sets M = sets borel" shows "finite_borel_measure_axioms M"
Union_subsetI: fixes A :: "'a set set"   and B :: "'a set set" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> B \<and> x \<subseteq> y" shows "\<Union> A \<subseteq> \<Union> B"
le_imp_diff_le: fixes j :: "nat"   and k :: "nat"   and n :: "nat" assumes "j \<le> k" shows "j - n \<le> k"
no_empty_conclusions: shows "\<forall>xs\<in>i.R rules. consequent xs \<noteq> []"
abc_Hoare_halt_def: fixes P :: "nat list \<Rightarrow> bool"   and p :: "abc_inst list"   and Q :: "nat list \<Rightarrow> bool" shows "{P} p {Q} \<equiv> \<forall>lm. P lm \<longrightarrow> (\<exists>n. abc_final (abc_steps_l (0, lm) p n) p \<and> Q abc_holds_for abc_steps_l (0, lm) p n)"
reachable_def: shows "G'.reachable = G'.reaches s\<^sub>0"
inv': fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" assumes "inverse_functors A B F G" shows "F \<circ> G = identity_functor.map A"
i_def: shows "i = (LEAST i. \<forall>j\<ge>i. P j)"
substitutionts: fixes E :: "char list \<Rightarrow> 'b"   and F :: "char list \<Rightarrow> 'b list \<Rightarrow> 'b"   and \<sigma> :: "char list \<Rightarrow> fterm"   and ts :: "fterm list" shows "eval\<^sub>t\<^sub>s E F (ts \<cdot>\<^sub>t\<^sub>s \<sigma>) = eval\<^sub>t\<^sub>s (evalsub E F \<sigma>) F ts"
continuous_map_frontier_frontier_preimage_subset: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_map X Y f" shows "X frontier_of {x \<in> topspace X. f x \<in> T} \<subseteq> {x \<in> topspace X. f x \<in> Y frontier_of T}"
injI: fixes f :: "'e \<Rightarrow> 'f" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f"
reachable_modes_subset: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(cms, mem) \<rightarrow> (cms', mem')" shows "reachable_mode_states (cms', mem') \<subseteq> reachable_mode_states (cms, mem)"
comp_isotone: fixes x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "x \<le> y"   and "w \<le> z" shows "x \<sqinter> w \<le> y \<sqinter> z"
M_dim_3: shows "M (length fs) \<in> carrier_mat (length fs) (length fs)"
updownI_def: fixes a :: "rat array" shows "updownI a = updownI' dm a"
mremove_def: fixes i :: "int"   and t :: "(int \<times> 'b) Tree" shows "mremove i t \<equiv> remove fst (i, undefined) t"