hcju/isabelleps huggingface dataset:
asymp_equiv_imp_diff_smallo: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<sim>[F] g" shows "(\<lambda>x. f x - g x) \<in> o[F](g)"
A_dest_1: fixes A :: "'a set"   and y :: "'a" assumes "A \<in> {{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}"   and "y \<in> A" shows "(SOME y. y \<in> A, y) \<in> r"
A'block: shows "A' = four_block_mat A1 A2 A0 A3"
subset_imp_msubset_mset_set: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "finite B" shows "mset_set A \<subseteq># mset_set B"
amult_one_r: fixes x :: "ant" shows "x * 1 = x"
frac_add: fixes x :: "'b"   and y :: "'b" shows "frac (x + y) = (if frac x + frac y < (1::'b) then frac x + frac y else frac x + frac y - (1::'b))"
wls_Abs_simp: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s :: "'sort"   and x :: "'var" assumes "good X" shows "wlsAbs (xs, s) (Abs xs x X) = (isInBar (xs, s) \<and> wls s X)"
absolutely_integrable_on_iff_nonneg: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x" shows "(f absolutely_integrable_on S) = (f integrable_on S)"
real_non_neg_mat_def: fixes A :: "((complex, 'a) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "real_non_neg_mat A \<equiv> \<forall>a\<in>elements_mat_h A. a \<in> \<real> \<and> 0 \<le> Re a"
rev_finite_subset: fixes B :: "'e set"   and A :: "'e set" assumes "finite B"   and "A \<subseteq> B" shows "finite A"
Upper_def: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" shows "Upper L A = {u. \<forall>x. x \<in> A \<inter> carrier L \<longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> u} \<inter> carrier L"
\<psi>_3: fixes p :: "'a"   and j :: "nat" assumes "p \<in> V j" shows "domain (\<psi> p) \<subseteq> V j"
strict_mono_def: fixes f :: "'d \<Rightarrow> 'e" shows "strict_mono f = (\<forall>x y. x < y \<longrightarrow> f x < f y)"
dfree_is_dsafe: fixes \<theta> :: "('a, 'b) trm" assumes "dfree \<theta>" shows "dsafe \<theta>"
has_domain_iff_arr: fixes f :: "'a1" shows "(C1.domains f \<noteq> {}) = C1.arr f"
ordinal_plus_left_cancel: fixes w :: "ordinal"   and x :: "ordinal"   and y :: "ordinal" shows "(w + x = w + y) = (x = y)"
sorted_rev_nth_mono: fixes xs :: "'t list"   and i :: "nat"   and j :: "nat" assumes "ord_term_lin.sorted (rev xs)"   and "i \<le> j"   and "j < length xs" shows "xs ! j \<preceq>\<^sub>t xs ! i"
monom_add: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and b :: "'b" shows "monom m (a + b) = monom m a + monom m b"
induct: fixes P :: "('Var \<Rightarrow> Sec option) \<Rightarrow> 'Var ModeUpd \<Rightarrow> bool"   and a0 :: "'Var \<Rightarrow> Sec option"   and a1 :: "'Var ModeUpd" assumes "\<And>\<Gamma> v. P \<Gamma> (v +=\<^sub>m AsmNoRead)"   and "\<And>\<Gamma> v. P \<Gamma> (v +=\<^sub>m AsmNoWrite)"   and "\<And>\<Gamma> v. P \<Gamma> (v +=\<^sub>m GuarNoRead)"   and "\<And>\<Gamma> v. P \<Gamma> (v +=\<^sub>m GuarNoWrite)"   and "\<And>\<Gamma> v. P \<Gamma> (v -=\<^sub>m AsmNoRead)"   and "\<And>\<Gamma> v. P \<Gamma> (v -=\<^sub>m AsmNoWrite)"   and "\<And>\<Gamma> v. P \<Gamma> (v -=\<^sub>m GuarNoRead)"   and "\<And>\<Gamma> v. P \<Gamma> (v -=\<^sub>m GuarNoWrite)" shows "P a0 a1"
insert_code_2: fixes x :: "'b"   and xs :: "'b list" shows "insert x (List.coset xs) = List.coset (removeAll x xs)"
connectedD: fixes "open" :: "'b set \<Rightarrow> bool"   and A :: "'b set"   and U :: "'b set"   and V :: "'b set" assumes "class.topological_space open"   and "topological_space.connected open A"   and "open U"   and "open V"   and "U \<inter> V \<inter> A = {}"   and "A \<subseteq> U \<union> V" shows "U \<inter> A = {} \<or> V \<inter> A = {}"
enum_all_D2_def: fixes P :: "D2 \<Rightarrow> bool" shows "enum_class.enum_all P = (P D1 \<and> P D2)"
Word_compareTr1: fixes E :: "('e, 'f) Order_scheme" assumes "Worder E"   and "\<exists>a\<in>carrier D. ord_equiv (Iod D (segment D a)) E"   and "ord_equiv D E" shows "False"
subst_id: fixes env :: "(name, 'a) fmap"   and t :: "'a" assumes "id_env env" shows "subst t env = t"
comp_in_homE: fixes g :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr"   and a :: "('a, 'a set) Maps.arr"   and c :: "('a, 'a set) Maps.arr" assumes "Maps.in_hom (g \<odot> f) a c" obtains b :: "('a, 'a set) Maps.arr" where "Maps.in_hom f a b" and "Maps.in_hom g b c"
R: fixes u :: "'a set"   and v :: "'a set" assumes "u \<in> nhds x"   and "v \<in> nhds y"   and "u \<inter> v = {}" shows "R"
bags_exist: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set"   and v :: "'c" assumes "TreeDecomposition G T bag"   and "v \<in> V\<^bsub>G\<^esub>" shows "\<exists>t\<in>V\<^bsub>T\<^esub>. v \<in> bag t"
character_in_order_1: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" assumes "character G \<chi>"   and "order G = 1" shows "\<chi> = principal_char G"
del_arc_def: fixes a :: "'b" shows "del_arc a = \<lparr>verts = verts G, arcs = arcs G - {a}, tail = tail G, head = head G\<rparr>"
eq_refl: fixes x :: "'c"   and y :: "'c" assumes "x = y" shows "y \<le> x"
eq: shows "(\<lambda>ys. \<not> LCons x xs \<sqsubseteq> ys) = (\<lambda>ys. \<not> xs \<sqsubseteq> ltl ys \<or> ys = LNil \<or> \<not> lnull ys \<and> lhd ys \<noteq> x)"
c_diff_def: shows "c_diff = (\<lambda>u v. set_to_nat (nat_to_set u - nat_to_set v))"
that: fixes p :: "'a set set" assumes "p division_of S \<union> T" shows "thesis"
extremum_strict: fixes a :: "'d list" shows "\<not> strict_prefix a []"
p_ge_0: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "0 \<le> p n x"
xor_def2: fixes x :: "'a"   and y :: "'a" shows "x XOR y = (x OR y) AND (NOT x OR NOT y)"
fixp_lfp_parametric_eq: fixes F :: "('d \<Rightarrow> 'e) \<Rightarrow> 'd \<Rightarrow> 'e"   and G :: "('f \<Rightarrow> 'e) \<Rightarrow> 'f \<Rightarrow> 'e"   and A :: "'d \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<And>x. lfp.mono_body (\<lambda>f. F f x)"   and "\<And>x. lfp.mono_body (\<lambda>f. G f x)"   and "rel_fun (rel_fun A (=)) (rel_fun A (=)) F G" shows "rel_fun A (=) (lfp.fixp_fun F) (lfp.fixp_fun G)"
suc_rw: shows "[0..<Suc (Suc k)] = [0..<Suc k] @ [Suc k]"
Mf: shows "(\<lambda>(x, y). f x y) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure t)"
Ll_length: shows "length Ll = length (prems_of \<iota>F)"
nonzero_inverse_scaleR_distrib: fixes a :: "real"   and x :: "'c" assumes "a \<noteq> 0"   and "x \<noteq> (0::'c)" shows "inverse (a *\<^sub>R x) = inverse x /\<^sub>R a"
piecewise_continuous_on_def: fixes a :: "'a"   and b :: "'a"   and I :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "piecewise_continuous_on a b I f = (continuous_on ({a..b} - I) f \<and> finite I \<and> (\<forall>i\<in>I. (i \<in> {a<..b} \<longrightarrow> (\<exists>l. (f \<longlongrightarrow> l) (at_left i))) \<and> (i \<in> {a..<b} \<longrightarrow> (\<exists>u. (f \<longlongrightarrow> u) (at_right i)))))"
leq_pathRep: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and Ds :: "char list list" assumes "Subobjs P C Cs"   and "Subobjs P C Ds"   and "Cs = butlast Ds" shows "P,C \<turnstile> Cs \<sqsubset>\<^sup>1 Ds"
that: fixes c :: "real" assumes "X - Y = c *\<^sub>R (S - Y)" shows "thesis"
assms_2: fixes k :: "natural" shows "random_aux (Code_Numeral.Suc k) = rhs (Code_Numeral.Suc k)"
bd_cts_scale: fixes a :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and c :: "real" assumes "bd_cts a"   and "healthy a"   and "0 \<le> c" shows "bd_cts (\<lambda>P s. c * a P s)"
has_real_derivative_pos_inc_right: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real"   and S :: "real set" assumes "(f has_real_derivative l) (at x within S)"   and "0 < l" shows "\<exists>d>0. \<forall>h>0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x + h)"
cases: fixes a1 :: "nat \<Rightarrow> nat"   and a2 :: "((nat \<times> nat) \<times> transition) fset"   and a3 :: "nat"   and a4 :: "nat \<Rightarrow>f value option"   and a5 :: "((nat \<times> nat) \<times> transition) fset"   and a6 :: "nat"   and a7 :: "nat \<Rightarrow>f value option"   and a8 :: "(String.literal \<times> value list \<times> value list) list" assumes "trace_simulation a1 a2 a3 a4 a5 a6 a7 a8" obtains s2 :: "nat" and f :: "nat \<Rightarrow> nat" and s1 :: "nat" and e1 :: "((nat \<times> nat) \<times> transition) fset" and r1 :: "nat \<Rightarrow>f value option" and e2 :: "((nat \<times> nat) \<times> transition) fset" and r2 :: "nat \<Rightarrow>f value option" where "a1 = f__" and "a2 = e1__" and "a3 = s1__" and "a4 = r1__" and "a5 = e2__" and "a6 = s2__" and "a7 = r2__" and "a8 = []" and "s2__ = f__ s1__"   | s2 :: "nat" and f :: "nat \<Rightarrow> nat" and s1 :: "nat" and i :: "value list" and r1 :: "nat \<Rightarrow>f value option" and o :: "value list" and e1 :: "((nat \<times> nat) \<times> transition) fset" and l :: "String.literal" and e2 :: "((nat \<times> nat) \<times> transition) fset" and r2 :: "nat \<Rightarrow>f value option" and es :: "(String.literal \<times> value list \<times> value list) list" where "a1 = f" and "a2 = e1" and "a3 = s1" and "a4 = r1" and "a5 = e2" and "a6 = s2" and "a7 = r2" and "a8 = (l, i, o) # es" and "s2 = f s1" and "\<forall>(s1', t1)|\<in>|ffilter (\<lambda>(s1', t1). evaluate_outputs t1 i r1 = map Some o) (possible_steps e1 s1 r1 l i). \<exists>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. evaluate_outputs t2 i r2 = map Some o \<and> trace_simulation f e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es"
monoid_vec_def: fixes ty :: "'b itself"   and n :: "nat" shows "monoid_vec ty n \<equiv> \<lparr>carrier = carrier_vec n, mult = (+), one = 0\<^sub>v n\<rparr>"
normalize_primitive_extract: fixes m :: "'b match_expr"   and disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and C :: "'c \<Rightarrow> 'b"   and \<gamma> :: "('b \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and f :: "'c negation_type list \<Rightarrow> 'c list"   and a :: "action"   and p :: "'d" assumes "normalized_nnf_match m"   and "wf_disc_sel disc_sel C"   and "\<forall>ml. match_list \<gamma> (map (Match \<circ> C) (f ml)) a p = matches \<gamma> (alist_and (NegPos_map C ml)) a p" shows "match_list \<gamma> (normalize_primitive_extract disc_sel C f m) a p = matches \<gamma> m a p"
rep_eq: shows "poly_mapping.lookup 0 = (\<lambda>k. 0::'c)"
pp_prod_def: fixes X :: "'a ppath set"   and Y :: "'a ppath set" shows "pp_prod X Y = {pp_fusion u v |u v. u \<in> X \<and> v \<in> Y \<and> pp_last u = pp_first v}"
cl6: fixes x :: "('a \<times> 'a set) set"   and z :: "('a \<times> 'a set) set" shows "x \<cdot> {} \<cdot> z = x \<cdot> {}"
split_rule: fixes xs :: "('a btnode ref option \<times> 'a) list"   and c :: "nat"   and a :: "('a btnode ref option \<times> 'a) array"   and n :: "nat"   and p :: "'a" assumes "sorted_less (separators xs)" shows "<is_pfa c xs (a, n)> split_fun (a, n) p <\<lambda>r. is_pfa c xs (a, n) * \<up> (r \<le> n \<and> (\<forall>j<r. snd (xs ! j) < p) \<and> (r < n \<longrightarrow> p \<le> snd (xs ! r)))>\<^sub>t"
has_density_equal_on_space: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal"   and g :: "'c \<Rightarrow> ennreal" assumes "has_density M N f"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "has_density M N g"
UNlift_shift: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and n :: "nat"   and I :: "nat set"   and dl :: "('test, 'atom, 'choice) cmd list"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "n < length cl"   and "I \<in> P n" shows "UNlift cl dl P F (local.shift cl n I) = local.shift dl n (F n I)"
abs_triangle_ineq2_sym: fixes abs :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (minus (abs a) (abs b)) (abs (minus b a))"
induct: fixes P :: "(('a, 'b) vec, 'c) vec \<Rightarrow> nat list \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<Rightarrow> bool"   and a0 :: "(('a, 'b) vec, 'c) vec"   and a1 :: "nat list"   and a2 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "\<And>A bezout. P A [] bezout"   and "\<And>A i xs bezout. P (Diagonal_to_Smith_row_i A i bezout) xs bezout \<Longrightarrow> P A (i # xs) bezout" shows "P a0 a1 a2"
interp_ins_maybe_grow: fixes xs :: "('b \<times> 'b option) list"   and oid :: "'b"   and ref :: "'b option" assumes "insert_ops (xs @ [(oid, ref)])" shows "set (interp_ins (xs @ [(oid, ref)])) = set (interp_ins xs) \<or> set (interp_ins (xs @ [(oid, ref)])) = set (interp_ins xs) \<union> {oid}"
assms: shows "A \<in> sets (M1 \<Otimes>\<^sub>M M2)"
round_Re_minimises_norm: fixes z :: "complex"   and m :: "int" shows "cmod (z - of_int (round (Re z))) \<le> cmod (z - of_int m)"
mem_carrier: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "subgroup H G"   and "x \<in> H" shows "x \<in> carrier G"
existence_ivl_eq_rev0: fixes y :: "'a" shows "existence_ivl0 y = uminus ` rev.existence_ivl0 y"
stutterd_axiom_def: shows "stutterd_axiom \<equiv> \<langle> xid1 := Var xid1 \<rangle> pusall \<leftrightarrow> pusall"
simps_1: fixes p :: "perm"   and tset :: "('a, 'b, 'c, 'd) Tree set['a]" shows "p \<bullet> tConj tset = tConj (map_bset (permute p) tset)"
b_e_type_comp: assumes "\<C> \<turnstile> [e1] @ [e2] : t1s _> t2s" shows "\<exists>ts'. \<C> \<turnstile> [e1] : t1s _> ts' \<and> \<C> \<turnstile> [e2] : ts' _> t2s"
multi_member_split: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "\<exists>A. M = x, A"
assms_2: shows "\<FF> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
rep_eq: fixes x :: "nat" shows "generator (upt_prod x) = upt_raw x"
extend_volume: fixes \<mu> :: "'a set \<Rightarrow> ennreal" assumes "volume generated_ring \<mu>" shows "\<exists>\<mu>'. volume G.generated_ring \<mu>' \<and> (\<forall>a\<in>generated_ring. \<mu>' a = \<mu> a)"
tmin_1: fixes t :: "real" assumes "t \<in> cball t__ et" shows "tmin \<le> t"
pair_loopfree_digraph_axioms: shows "pair_loopfree_digraph G"
subsuming_states_append: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list"   and s :: "'a \<Rightarrow> 'b option" assumes "distinct (map fst (xs @ ys))" shows "(map_of (xs @ ys) \<subseteq>\<^sub>m s) = (map_of ys \<subseteq>\<^sub>m s \<and> map_of xs \<subseteq>\<^sub>m s)"
auxhyp_in_R1: shows "auxhyp \<in> \<R>"
degree_less_sum: fixes p :: "real mpoly"   and var :: "nat"   and n :: "nat"   and q :: "real mpoly"   and m :: "nat" assumes "MPoly_Type.degree p var = n"   and "MPoly_Type.degree q var = m"   and "m < n" shows "MPoly_Type.degree (p + q) var = n"
finite_imp_inj_to_nat_seg: fixes A :: "'a set" assumes "finite A" shows "\<exists>f n. f ` A = {i. i < n} \<and> inj_on f A"
rstar_fixpoint: fixes y :: "'a" assumes "has_least_prefixpoint (Rf y)" shows "rstar y = \<mu> Rf y"
SourcesLoop: fixes i :: "AbstrLevelsID"   and C :: "CSet"   and XS :: "CSet set"   and S :: "CSet"   and ZS :: "CSet set" shows "Sources i C = XS \<union> Sources i S \<and> Sources i S = ZS \<union> Sources i C \<longrightarrow> Sources i C = XS \<union> ZS \<union> {C, S}"
swap_def: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" shows "swap xs i j = xs[i := xs ! j, j := xs ! i]"
indspace_def: fixes G :: "'g set"   and FH :: "('r, 'g) aezfun set"   and V :: "'v set" shows "indspace G FH V = addfunset FH V \<inter> smultfunset G FH"
bhv2cnf_lActive: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "\<exists>i. active c (t i)"   and "lfinite (dynamic_component.proj tCMP active c (inf_llist t))" shows "dynamic_component.bhv2cnf tCMP active c t (the_enat (llength (dynamic_component.proj tCMP active c (inf_llist t)))) = Suc (dynamic_component.lActive active c t)"
fwi_canonical_id: fixes n :: "nat"   and k :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "canonical_subs n {k} M"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "fwi M n k i j = M"
tsp__nosp: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B CTSP P Q" shows "\<not> A B C OSP P Q"
assms_1: shows "A \<in> decomps\<^sub>e\<^sub>s\<^sub>t S N \<I>"
bit_concat_bit_iff: fixes m :: "nat"   and k :: "int"   and l :: "int"   and n :: "nat" shows "bit (concat_bit m k l) n = (n < m \<and> bit k n \<or> m \<le> n \<and> bit l (n - m))"
fresh_11: fixes a :: "name"   and x2 :: "name"   and x3 :: "trm"   and x1 :: "name" shows "a \<sharp> AndL1 x2.x3 x1 = (a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
w1Lower: shows "- (2 ^ 31) \<le> sint (SCAST(32 \<rightarrow> 64) w1)"
ncloses_subset: fixes \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<Gamma> :: "(name \<times> ty) list"   and \<Gamma>' :: "(name \<times> ty) list" assumes "\<theta>_n ncloses \<Gamma>"   and "set \<Gamma>' \<subseteq> set \<Gamma>" shows "\<theta>_n ncloses \<Gamma>'"
simps: fixes P :: "'b" shows "Pc P = InContext P (Geq (Const 0) (Const 0))"
inner_prod_with_itself_nonneg_reals_non0: fixes u :: "complex Matrix.vec" assumes "u \<noteq> 0\<^sub>v (dim_vec u)" shows "\<langle>u|u\<rangle> \<noteq> 0"
idempotency_L: fixes Cn :: "'b set \<Rightarrow> 'b set"   and A :: "'b set" assumes "Tarskian_logic Cn" shows "Cn (Cn A) = Cn A"
fimageE: fixes b :: "'m"   and f :: "'n \<Rightarrow> 'm"   and A :: "'n fset" assumes "b |\<in>| f |`| A" obtains x :: "'n" where "b = f x" and "x |\<in>| A"
mem_case_prodI2: fixes p :: "'a \<times> 'b"   and z :: "'c"   and c :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set" assumes "\<And>a b. p = (a, b) \<Longrightarrow> z \<in> c a b" shows "z \<in> (case p of (a, b) \<Rightarrow> c a b)"
bounded_measure: fixes A :: "real set" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M (space M)"
a: shows "generate G (\<Union>S\<in>A. generate G (f S)) \<subseteq> generate G (\<Union> (f ` A))"
Lp_space_numeral: fixes n :: "num"   and M :: "'a measure" assumes "0 < numeral n" shows "lspace M (numeral n) = {f \<in> borel_measurable M. integrable M (\<lambda>x. \<bar>f x\<bar> ^ numeral n)}"
p: fixes n :: "nat" shows "positive (c \<cdot>\<^sub>m (f n - f 0))"
lfp_def: fixes f :: "'d \<Rightarrow> 'd" shows "lfp f = \<Sqinter> {u. f u \<le> u}"
span_diff: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> vs1.span S"   and "y \<in> vs1.span S" shows "x - y \<in> vs1.span S"
extendRule_def: fixes forms :: "'b sequent"   and R :: "'b sequent list \<times> 'b sequent" shows "extendRule forms R \<equiv> (map (extend forms) (fst R), extend forms (snd R))"
trans: fixes a :: "'b Word.word"   and b :: "'b Word.word"   and c :: "'b Word.word" assumes "a \<le>s b"   and "b \<le>s c" shows "a \<le>s c"
rep_inverse: fixes x :: "'a" shows "abs_nat (rep_nat x) = x"
hyps_1: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and u :: "trm" assumes "\<Gamma> \<turnstile> s is t : T\<^sub>1"   and "\<Gamma> \<turnstile> t is u : T\<^sub>1" shows "\<Gamma> \<turnstile> s is u : T\<^sub>1"
det_transpose: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "Determinant.det A\<^sup>T = Determinant.det A"
t1_space_derived_set_of_singleton: fixes X :: "'c topology" shows "t1_space X = (\<forall>x\<in>topspace X. X derived_set_of {x} = {})"
inner_simps_5: fixes r :: "real"   and x :: "'c"   and y :: "'c" shows "r *\<^sub>R x \<bullet> y = r * (x \<bullet> y)"
forests_bot_1: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and e :: "'b"   and f :: "'b" assumes "class.pd_kleene_allegory star one times sup less_eq less bot inf top conv uminus"   and "(less_eq one e \<and> less_eq (times e e) e) \<and> conv e = e"   and "less_eq (times f (conv f)) one \<and> less_eq (times f (star f)) (uminus one)" shows "times (inf (uminus e) f) (conv (inf e f)) = bot"
A: fixes x :: "'b list"   and y :: "'b list" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
supr_under: fixes a :: "'a" assumes "a \<in> Field r" shows "a = local.supr (local.under a)"
distl: shows "\<forall>s. dist (l s) x = \<bar>s\<bar>"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "size_tree_tm \<langle>l, x, r\<rangle> =1 size_tree_tm l \<bind> (\<lambda>m. size_tree_tm r \<bind> (\<lambda>n. return (m + n + 1)))"
pair_pseudo_graph_def: fixes G :: "'b pair_pre_digraph" shows "pair_pseudo_graph G \<equiv> pair_fin_digraph G \<and> pair_sym_digraph G"
Inf_superset_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and B :: "'b set"   and A :: "'b 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)"
well_formedE_cases: fixes B :: "binding"   and \<Gamma> :: "binding list" assumes "B \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f" obtains "\<Gamma> \<turnstile>\<^sub>w\<^sub>f\<^sub>B B" and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"
simps_1: fixes l :: "'b tree" shows "merge l \<langle>\<rangle> = l"
simps: fixes \<Sigma> :: "'a set set"   and \<phi> :: "'a ltl" shows "rabin_pairs \<Sigma> \<phi> = {(M_fin \<pi> \<union> \<Union> {Acc_fin \<Sigma> \<pi> \<chi> |\<chi>. \<chi> \<in> dom \<pi>}, {Acc_inf \<pi> \<chi> |\<chi>. \<chi> \<in> dom \<pi>}) |\<pi>. dom \<pi> \<subseteq> \<^bold>G \<phi> \<and> (\<forall>\<chi>\<in>dom \<pi>. the (\<pi> \<chi>) < max_rank_of \<Sigma> \<chi>)}"
leD: fixes y :: "nat list"   and x :: "nat list" assumes "y \<le>\<^sub>v x" shows "\<not> x <\<^sub>v y"
Polygamma_approx_1_complex: fixes x :: "complex"   and m :: "nat" assumes "0 < Re x" shows "Polygamma_approx (Suc 0) m x = Ln x - 1 / (2 * x) + stirling_sum (Suc 0) m x"
Sim: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> Rel" shows "P \<leadsto>[Rel] Q"
language_state_split: fixes w1 :: "('d \<times> 'e) list"   and w2 :: "('d \<times> 'e) list"   and M :: "('d, 'e, 'f) FSM"   and q :: "'f" assumes "w1 @ w2 \<in> LS M q" obtains tr1 :: "'f list" and tr2 :: "'f list" where "path M (w1 || tr1) q" and "length w1 = length tr1" and "path M (w2 || tr2) (target (w1 || tr1) q)" and "length w2 = length tr2"
strip_while_append_rec: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "P x" shows "strip_while P (xs @ [x]) = strip_while P xs"