hcju/isabelleps huggingface dataset:
basis_left_null_space: fixes A :: "(('a, 'c) vec, 'b) vec" shows "vec.independent (basis_left_null_space A) \<and> left_null_space A = vec.span (basis_left_null_space A)"
safe_def: fixes s :: "event list"   and r :: "room" shows "safe s r = (\<exists>s\<^sub>1 s\<^sub>2 s\<^sub>3 g c. s = s\<^sub>3 @ [Enter g r c] @ s\<^sub>2 @ [Check_in g r c] @ s\<^sub>1 \<and> no_Check_in (s\<^sub>3 @ s\<^sub>2) r \<and> isin (s\<^sub>2 @ [Check_in g r c] @ s\<^sub>1) r = {})"
Act_tau_eq_iff: fixes x1 :: "('b, 'c, 'act) formula"   and \<alpha> :: "'act"   and x2 :: "('b, 'c, 'act) formula" shows "(Act \<tau> x1 = Act \<alpha> x2) = (\<alpha> = \<tau> \<and> x2 = x1)"
simps_2: fixes s :: "'a stream"   and n :: "nat" shows "s !! Suc n = stl s !! n"
r_zero: fixes x :: "int" assumes "x \<in> carrier R" shows "x \<oplus> \<zero> = x"
primitive_imp_unit_iff: fixes p :: "'a poly" assumes "primitive p" shows "(p dvd 1) = (degree p = 0)"
cross3_simps_22: fixes a :: "'e"   and c :: "'e"   and b :: "'e" shows "(a \<le> c - b) = (a + b \<le> c)"
c': shows "c = (m' + a') mod q \<and> y1 = (a' * x1 + b') mod q"
sum_bounded_above_strict: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and K :: "'b" assumes "\<And>i. i \<in> A \<Longrightarrow> f i < K"   and "0 < card A" shows "sum f A < of_nat (card A) * K"
inf_hcoords_def: shows "\<infinity>\<^sub>h\<^sub>c \<equiv> Abs_complex_homo_coords \<infinity>\<^sub>v"
a1: shows "(\<lambda>x. norm (f x)) abs_summable_on F"
tab_LEAST: fixes h :: "nat" assumes "h < length tab" shows "tab ! h = (LEAST n. p n h)"
ih_7: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state')"
FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1_dir1: fixes N :: "nat"   and fs :: "nat list" assumes "DiningPhilosophers N"   and "0 < length fs" shows "DiningPhilosophers.FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m N 1 fs \<sqsubseteq> DiningPhilosophers.FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m N 0 (fs ! 0)"
less_eq_char_def: fixes c1 :: "char"   and c2 :: "char" shows "(c1 \<le> c2) = (of_char c1 \<le> of_char c2)"
fps_X_fps_compose_startby0: fixes a :: "'a fps" assumes "fps_nth a 0 = (0::'a)" shows "fps_X oo a = a"
acts_on_subsets: fixes p :: "nat" assumes "finite (carrier G)" shows "group_action G (conjugation_action p) (subgroups_of_size p)"
defP: shows "\<tau> \<Turnstile> \<delta> P"
of_int_add: 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 w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w + z) = plus (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
ap_subset_edom_AEstack: fixes S :: "stack_elem list"   and as :: "Arity list" shows "ap S \<subseteq> edom (AEstack as S)"
cases_scheme: fixes r :: "('b, 'c) state_scheme" obtains m :: "nat" and l :: "nat" and carrots :: "nat" and hare :: "'b" and tortoise :: "'b" and more :: "'c" where "r = \<lparr>m = m, l = l, carrots = carrots, hare = hare, tortoise = tortoise, \<dots> = more\<rparr>"
binary_functor_def: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'a \<times> 'b \<Rightarrow> 'c" shows "binary_functor A1 A2 B F \<equiv> (category A1 \<and> category A2) \<and> category B \<and> product_category A1 A2 \<and> functor (product_category.comp A1 A2) B F"
case_1: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
valuep_value_eq: shows "valuep = (\<lambda>x. x \<in> value)"
Domain_int_univ: fixes A :: "'a set"   and f :: "('a \<times> 'b) set" shows "Domain (A \<times> UNIV \<inter> f) = A \<inter> Domain f"
sum_list_strict_mono: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "xs \<noteq> []"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x < g x" shows "sum_list (map f xs) < sum_list (map g xs)"
filt_equiv_space: fixes F :: "'d \<Rightarrow> 'e measure"   and M :: "'e measure"   and N :: "'e measure" assumes "filt_equiv F M N" shows "space M = space N"
getParts_nonempty: fixes rs :: "'a simple_rule list" shows "getParts rs \<noteq> []"
lottery_prob_alts_superset: fixes p :: "'alt pmf"   and A :: "'alt set" assumes "p \<in> lotteries"   and "alts \<subseteq> A" shows "lottery_prob p A = 1"
less_odlist_def: fixes xs :: "'a odlist"   and ys :: "'a odlist" shows "xs < ys \<equiv> local.less_list (toList xs) (toList ys)"
ideal_axioms: shows "ideal image.additive.induced.Ker kernel.additive.Partition ([+]) ([\<cdot>]) (kernel.additive.Class \<zero>) (kernel.additive.Class \<one>)"
ars_r_def: fixes x :: "'a" shows "r x = ar ar x"
absolutely_continuous_AE: fixes M' :: "'d measure"   and M :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "sets M' = sets M"   and "absolutely_continuous M M'"   and "almost_everywhere M P" shows "almost_everywhere M' P"
assms_5: shows "set ps = ps\<^sub>L \<union> ps\<^sub>R"
empty_Union_conv: fixes A :: "'b set set" shows "({} = \<Union> A) = (\<forall>x\<in>A. x = {})"
assms_4: shows "\<not> \<gamma> (Trans s1 a1 ou1 s1')"
m_o1: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and X :: "char list set"   and o1 :: "'a st option"   and o2 :: "'a st option" assumes "Measure m h"   and "finite X"   and "top_on\<^sub>o o1 (- X)"   and "top_on\<^sub>o o2 (- X)"   and "o1 \<le> o2" shows "Measure1.m_o m h o2 X \<le> Measure1.m_o m h o1 X"
run_step_break_imp_not_trap_const_list_2: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and es :: "e list"   and s' :: "s"   and vs' :: "v list"   and n :: "nat"   and res :: "v list" assumes "run_step d i (s, vs, es) = (s', vs', RSBreak n res)" shows "\<not> const_list es"
c2dc_dc2c: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "dcharacter n \<chi>" shows "residues_nat.c2dc n (residues_nat.dc2c n \<chi>) = \<chi>"
inf_llist_inj: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "(inf_llist f = inf_llist g) = (f = g)"
natLeq_on_well_order_on: fixes n :: "nat" shows "well_order_on {x. x < n} (natLeq_on n)"
simps_1: fixes G :: "prog"   and pt :: "prim_ty" shows "is_type G (PrimT pt) = True"
FIN_wrt_impl_CP: fixes U :: "(nat \<Rightarrow> nat option) set"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "U \<in> FIN_wrt \<psi>" shows "U \<in> CP_wrt \<psi>"
bdoubleton_eq_iff: fixes x :: "'d"   and y :: "'d"   and z :: "'d"   and w :: "'d" shows "(bdoubleton x y = bdoubleton z w) = (x = z \<and> y = w \<or> x = w \<and> y = z)"
prodag_sameTr3: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and B :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)"   and "\<forall>k\<in>I. A k = B k" shows "prod_mOp I A = prod_mOp I B"
rbt_contains: fixes a :: "'a"   and V :: "'a list" shows "rbt_contains a (rbt_fromlist V) = (a \<in> set V)"
ba3_commutative: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
invh_upd_1: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and x :: "'c"   and y :: "'d" assumes "invh t" shows "invh (upd x y t)"
set_antichain_inverse: fixes x :: "'b antichain" shows "antichain (set_antichain x) = x"
successive_arcs_extend_last: fixes P :: "nat \<times> nat \<Rightarrow> bool"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "successive P (arcs i j xs)"   and "\<not> P (i, hd xs) \<or> \<not> P (last xs, j)"   and "xs \<noteq> []" shows "successive P (arcs i j xs @ [(i, hd xs)])"
mset_update: fixes i :: "nat"   and ls :: "'b list"   and v :: "'b" assumes "i < length ls" shows "mset (ls[i := v]) = add_mset v (mset ls - {#ls ! i#})"
hyp1: assumes "distinct (map fst (deleteAround g a ps))" shows "distinct (ExcessNotAtRecList (deleteAround g a ps) g)"
subst_sends_strand_fv_to_img: fixes S :: "('d, 'c) strand_step list"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<subseteq> fv\<^sub>s\<^sub>t S \<union> range_vars \<delta>"
atLeastLessThan_shift_bounds: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "prod g {m + k..<n + k} = prod (g \<circ> (+) k) {m..<n}"
plus: shows "(R ===> R ===> R) (arith_ops_record.plus ops) (+)"
lt_max: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "lookup p v \<noteq> (0::'c)" shows "v \<preceq>\<^sub>t lt p"
i_Exec_Stream_append_last_Cons: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "i_Exec_Comp_Stream trans_fun (xs \<frown> input) c = f_Exec_Comp_Stream trans_fun xs c \<frown> i_Exec_Comp_Stream trans_fun input (last (c # f_Exec_Comp_Stream trans_fun xs c))"
bindE_cong: fixes M :: "'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool"   and M' :: "'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool"   and N :: "'e \<Rightarrow> 'c \<Rightarrow> (('d + 'f) \<times> 'c) set \<times> bool"   and N' :: "'e \<Rightarrow> 'c \<Rightarrow> (('d + 'f) \<times> 'c) set \<times> bool" assumes "M = M'"   and "\<And>v s s'. (Inr v, s') \<in> fst (M' s) \<Longrightarrow> N v s' = N' v s'" shows "M >>=E N = M' >>=E N'"
aux0_Nil: assumes "aux0 = []" shows "ne = 0 \<or> ne \<noteq> 0 \<and> (\<forall>t. t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> ne - t) \<le> right I \<longrightarrow> (\<nexists>i. \<tau> \<sigma> i = t))"
wf_tuple_change_base: fixes n :: "nat"   and X :: "nat set"   and as :: "'b option list"   and Y :: "nat set" assumes "wf_tuple n X as"   and "join_mask n X = join_mask n Y" shows "wf_tuple n Y as"
step_preserves_le: fixes S :: "(char list \<Rightarrow> int) set"   and S' :: "'av Abs_State.st option"   and cs :: "(char list \<Rightarrow> int) set acom"   and ca :: "'av Abs_State.st option acom" assumes "S \<subseteq> \<gamma>\<^sub>o S'"   and "cs \<le> \<gamma>\<^sub>c ca" shows "step S cs \<le> \<gamma>\<^sub>c (step' S' ca)"
strict_trans: fixes b :: "'a word"   and a :: "'a word"   and c :: "'a word" assumes "b <s a"   and "c <s b" shows "c <s a"
Q'_def: shows "Q' \<equiv> \<lambda>E d d'. M E d - V \<cdot>\<^sub>s\<^sub>e\<^sub>t d' \<subseteq> R d - V \<cdot>\<^sub>s\<^sub>e\<^sub>t d'"
f_bounded_below: fixes c' :: "real" assumes "0 < c'" obtains c :: "real" where "\<And>x. \<lbrakk>x\<^sub>0 \<le> x; x \<le> x\<^sub>1\<rbrakk> \<Longrightarrow> 2 * (c * f_approx x) \<le> f x" and "c \<le> c'" and "0 < c"
prime_smallest_prime_beyond: fixes n :: "nat" shows "prime (smallest_prime_beyond n)"
fset_map_snd_id: fixes cs :: "('a \<times> 'b) fset"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>a b e. (a, b) \<in> fset cs \<Longrightarrow> f a b = b" shows "(\<lambda>(a, b). (a, f a b)) |`| cs = cs"
quasi_inverses_isomorphic_right: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "quasi_inverses f g" shows "quasi_inverses f g' = (g \<cong> g')"
defval_conf: fixes G :: "prog"   and T :: "ty"   and s :: "st" assumes "is_type G T" shows "G,s\<turnstile>default_val T\<Colon>\<preceq>T"
vbrelation_vinE: fixes x :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
abs_eq: fixes x :: "real" shows "ennreal x = e2ennreal (0 \<squnion> ereal x)"
h_minus_aux_mono: fixes h' :: "'v \<times> 'v \<Rightarrow> ennreal"   and j :: "nat"   and e :: "'v \<times> 'v" assumes "\<And>e. h' e \<noteq> \<top>" shows "h_minus_aux h' j e \<le> h_minus_aux h' (Suc j) e"
valid_intv_distinct: fixes c :: "nat"   and I :: "intv"   and I' :: "intv"   and x :: "'d"   and u :: "'d \<Rightarrow> real" assumes "valid_intv c I"   and "valid_intv c I'"   and "intv_elem x u I"   and "intv_elem x u I'" shows "I = I'"
expand_singleton: fixes t :: "('b, 'sz) trm" shows "singleton t = (\<lambda>i. if i = vid1 then t else Const 0)"
from_nat_into_inject: fixes A :: "'b set"   and B :: "'b set" assumes "A \<noteq> {}"   and "countable A"   and "B \<noteq> {}"   and "countable B" shows "(from_nat_into A = from_nat_into B) = (A = B)"
set_map: fixes a :: "'d"   and b :: "'d" assumes "D.ide a"   and "D.ide b" shows "S.set (HomD.map (b, a)) = HomD.set (b, a)"
step: fixes a' :: "'a"   and bs :: "'b list"   and a :: "'a"   and b :: "'b" assumes "rtrancl3p r a' bs a''"   and "r a b a'"   and "P a' bs" shows "P a (b # bs)"
minimum_elem_mem: fixes D :: "('c, 'd) Order_scheme"   and X :: "'c set"   and a :: "'c" assumes "Order D"   and "X \<subseteq> carrier D"   and "minimum_elem D X a" shows "a \<in> X"
Lp_Lq_mult_1: fixes p :: "ennreal"   and q :: "ennreal"   and r :: "ennreal"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and g :: "'b \<Rightarrow> real" assumes "1 / p + 1 / q = 1 / r"   and "f \<in> space\<^sub>N (\<LL> p M)"   and "g \<in> space\<^sub>N (\<LL> q M)" shows "(\<lambda>x. f x * g x) \<in> space\<^sub>N (\<LL> r M)"
t0_2: fixes t :: "real" assumes "0 < t"   and "t < t0" shows "flow0 x t \<notin> P"
Y: shows "Y \<in> sets (Pi\<^sub>M J M)"
Abs_unat_inject: fixes x :: "nat option"   and y :: "nat option" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(Abs_unat x = Abs_unat y) = (x = y)"
has_derivative_transform: fixes x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b" assumes "x \<in> s"   and "\<And>x. x \<in> s \<Longrightarrow> g x = f x"   and "(f has_derivative f') (at x within s)" shows "(g has_derivative f') (at x within s)"
Gr_card: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "card (BNF_Def.Gr A f) = card A"
te: fixes e :: "real" assumes "0 < e" shows "0 < t * e"
append_exec_def: fixes e :: "'e \<times> 'f list"   and e' :: "'g \<times> 'f list" shows "append_exec e e' \<equiv> (fst e, append (snd e') (snd e))"
assms: shows "\<forall>i. P (S i) (S (Suc i))"
mapping_of_inject: fixes x :: "'a mpoly"   and y :: "'a mpoly" shows "(mapping_of x = mapping_of y) = (x = y)"
distinct_2: fixes x2 :: "nat" shows "Suc x2 \<noteq> 0"
power_Suc0_right: fixes a :: "'a" shows "a ^ Suc 0 = a"
maddux_12: fixes y :: "'a"   and x :: "'a" shows "- (y ; x) ; x\<^sup>\<smile> \<le> - y"
the_trivial_def: fixes v :: "'b"   and vs :: "'b list" shows "the_trivial v vs = (case foldl (\<lambda>(good, v') w. if w = v then (good, v') else case v' of None \<Rightarrow> (good, Some w) | Some v' \<Rightarrow> (good \<and> w = v', Some v')) (True, None) vs of (True, x) \<Rightarrow> x | (False, x) \<Rightarrow> None)"
insertions_def: fixes ops :: "('c \<times> ('c, 'd) list_op) list" shows "insertions ops \<equiv> List.map_filter (\<lambda>oper. case oper of (oid, Insert ref val) \<Rightarrow> Some (oid, ref) | (oid, Delete x) \<Rightarrow> Map.empty x) ops"
setcompr_eq_image: fixes f :: "'d \<Rightarrow> 'c"   and P :: "'d \<Rightarrow> bool" shows "{f x |x. P x} = \<P> f {x. P x}"
simps_10: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Floor a) vs = approx_un p (\<lambda>ivl. Some (floor_float_interval ivl)) (approx_floatarith p a vs)"
col_map_mat: fixes j :: "nat"   and A :: "'b mat"   and f :: "'b \<Rightarrow> 'c" assumes "j < dim_col A" shows "col (map_mat f A) j = map_vec f (col A j)"
not_icard_Diff1_less: shows "\<exists>A x. x \<in> A \<and> \<not> icard (A - {x}) < icard A"
fin_dg_Obj_vfinite: shows "vfinite (\<CC>\<lparr>Obj\<rparr>)"
Y: shows "Y = affine_unop p (real_of_float a) b (real_of_float (upper divl)) X"
ext_inject: fixes verts :: "'c set"   and arcs :: "('c \<times> 'c) set"   and more :: "'d"   and verts' :: "'c set"   and arcs' :: "('c \<times> 'c) set"   and more' :: "'d" shows "(\<lparr>Graph.verts = verts, arcs = arcs, \<dots> = more\<rparr> = \<lparr>Graph.verts = verts', arcs = arcs', \<dots> = more'\<rparr>) = (verts = verts' \<and> arcs = arcs' \<and> more = more')"
proofTree_def: fixes A :: "(nat \<times> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list) set" shows "proofTree A = (bounded A \<and> founded subs (SATAxiom \<circ> sequent) A)"
rel_optionT_simps: fixes M :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'d"   and m' :: "'e" shows "rel_optionT M (OptionT m) (OptionT m') = M m m'"
b: shows "b \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
double_diff: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "B - (C - A) = A"
rep_iso_class: fixes f :: "'a \<times> 'a" assumes "VoV.ide f" shows "VoV.isomorphic (VoV.iso_class_rep (VoV.iso_class f)) f"
base_enabled: fixes vs :: "state \<Rightarrow> 'a"   and A :: "state \<times> state \<Rightarrow> bool"   and s :: "state" assumes "basevars vs"   and "\<exists>c. \<forall>u. vs u = c \<longrightarrow> A (s, u)" shows "s \<Turnstile> Enabled A"
antisymmetric_subset: fixes B :: "'a set" assumes "B \<subseteq> UNIV" shows "antisymmetric B ((\<sqsubseteq>) \<restriction> A)"
def_homo: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "\<delta> x->excluding\<^sub>S\<^sub>e\<^sub>t(y) = (\<delta> x and \<upsilon> y)"
module_cpx_vec: fixes n :: "nat" shows "Module.module cpx_rng (module_cpx_vec n)"
assms_2: shows "\<I>1, \<I>2 \<turnstile>\<^sub>C CNV callee s \<surd>"
p_not_eq_Suc_0: fixes p :: "nat"   and a :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)" shows "p \<noteq> Suc 0"
top_injective_inf_complement: fixes x :: "'a"   and y :: "'a" assumes "injective x" shows "top * (x \<sqinter> y) \<sqinter> top * (x \<sqinter> - y) = bot"
closed_limpt: fixes S :: "'b set" shows "closed S = (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
poly_induct2: fixes P :: "'b poly \<Rightarrow> 'c poly \<Rightarrow> bool"   and p :: "'b poly"   and q :: "'c poly" assumes "P 0 0"   and "\<And>a p b q. P p q \<Longrightarrow> P (pCons a p) (pCons b q)" shows "P p q"
assms_3: shows "\<forall>x\<in>A \<inter> B. g x = \<^bold>1"
prime_dvd_mersenneD: fixes p :: "nat"   and q :: "nat" assumes "prime p"   and "p \<noteq> 2"   and "prime q"   and "q dvd 2 ^ p - 1" shows "[q = 1] (mod 2 * p)"
bex_imageD: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>f ` A. P x" shows "\<exists>x\<in>A. P (f x)"
assms: shows "\<forall>r. \<not> evaluate_top False env s tp r"
tls1: shows "tls1 = lmap (fst \<circ> snd \<circ> snd) sstls1"
hyps: assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> g k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> h k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> f k = g k + h k" shows "matrix_sum d f n = matrix_sum d g n + matrix_sum d h n"
nonempty: shows "\<forall>(uu_, x)\<in>set x. x \<noteq> []"
cases: fixes a1 :: "(att \<times> char list, att, nat) env"   and a2 :: "operation list"   and a3 :: "(att \<times> char list, att, nat) env" assumes "a1 \<Midarrow>a2\<Rightarrow> a3" obtains root :: "(att \<times> char list, att, nat) env" where "a1 = roota__" and "a2 = []" and "a3 = roota__"   | root :: "(att \<times> char list, att, nat) env" and x :: "operation" and root' :: "(att \<times> char list, att, nat) env" and xs :: "operation list" and root'' :: "(att \<times> char list, att, nat) env" where "a1 = root" and "a2 = x # xs" and "a3 = root''" and "root \<midarrow>x\<rightarrow> root'" and "root' \<Midarrow>xs\<Rightarrow> root''"
elem_def: fixes S :: "('d, 'e) eq_object_scheme"   and x :: "'d"   and A :: "'d set" shows "(x .\<in>\<^bsub>S\<^esub> A) = (\<exists>y\<in>A. x .=\<^bsub>S\<^esub> y)"
px: shows "Ifm vs (x # bs) (NEq (CNP 0 c s))"
base: shows "\<exists>b. ((S\<^sub>1, \<theta>\<^sub>1), b) \<in> {((S, \<theta>), S', \<theta>'). (S, \<theta>) \<leadsto> (S', \<theta>') \<and> \<I> \<Turnstile>\<^sub>c \<langle>S,\<theta>\<rangle> \<and> \<I> \<Turnstile>\<^sub>c \<langle>S',\<theta>'\<rangle>}"
card_subset_eq: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "order.greater_eq B A"   and "card A = card B" shows "A = B"
h4: shows "isolate_variable_sparse p x (i + 1) \<noteq> 0 \<longrightarrow> MPoly_Type.degree (f i) x = i"
cfg_onD_action: fixes K :: "'a \<Rightarrow> 'a pmf set"   and cfg :: "'a cfg"   and s :: "'a" assumes "Markov_Decision_Process K"   and "cfg \<in> Markov_Decision_Process.cfg_on K s" shows "action cfg \<in> K s"
sinks_idem: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'c list" shows "sinks I D u (ipurge_tr I D u xs) = {}"
igFreshIGAbs2_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igFreshIGAbs2 MOD \<equiv> \<forall>ys y xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD ys y X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD xs x X)"
C: shows "(Owner n, Crypt (SesK SK) (Pwd n)) \<in> s"
minimum_spanning_forest_impl_tree: fixes F :: "('c, 'd) graph"   and G :: "('c, 'd) graph" assumes "minimum_spanning_forest F G"   and "valid_graph G"   and "connected_graph F" shows "minimum_spanning_tree F G"
countable_nat_UN: fixes \<Omega> :: "'a set"   and M :: "'a set set"   and A :: "nat \<Rightarrow> 'a set" assumes "sigma_algebra \<Omega> M"   and "range A \<subseteq> M" shows "\<Union> (range A) \<in> M"
1: shows "limit (run\<^sub>t \<delta> q\<^sub>0 w) = range (suffix k (run\<^sub>t \<delta> q\<^sub>0 w))"
divide_left_mono: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq b a"   and "less_eq zero c"   and "less zero (times a b)" shows "less_eq (divide c a) (divide c b)"
conga_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C CongA D E F" shows "C B A CongA F E D"
lattice_ow_axioms: shows "lattice_ow U (\<sqinter>\<^sub>o\<^sub>w) (\<le>\<^sub>o\<^sub>w) (<\<^sub>o\<^sub>w) (\<squnion>\<^sub>o\<^sub>w)"
satisfiableFilterTautologies: fixes F :: "Literal list list" shows "satisfiable F = satisfiable (filter (\<lambda>c. \<not> clauseTautology c) F)"
onorm_scaleC_left_lemma: fixes r :: "'d \<Rightarrow> complex"   and f :: "'e" assumes "bounded_clinear r" shows "onorm (\<lambda>x. r x *\<^sub>C f) \<le> onorm r * norm f"
hyps_8: shows "y\<^sub>4 = - y\<^sub>1 - l * (x\<^sub>4 - x\<^sub>1)"
deriv_add: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "g \<in> carrier P"   and "a \<in> carrier R" shows "deriv (f \<oplus>\<^bsub>P\<^esub> g) a = deriv f a \<oplus> deriv g a"
that_2: fixes b :: "bool" assumes "z = Bool b" shows "thesis"
typing_implies_valid: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty" assumes " \<Gamma> \<turnstile> t : T " shows "valid \<Gamma>"
\<K>_def: shows "\<K> \<equiv> \<lambda>j0. nsets {j0<..}"
cut1: fixes A :: "fm"   and B :: "fm"   and H :: "fm set" assumes "{A} \<turnstile> B"   and "H \<turnstile> A" shows "H \<turnstile> B"
m_s_widen: fixes X :: "char list set"   and S1 :: "'av st"   and S2 :: "'av st" assumes "finite X"   and "fun S1 = fun S2 on - X"   and "\<not> S2 \<le> S1" shows "m\<^sub>s (S1 \<nabla> S2) X < m\<^sub>s S1 X"