hcju/isabelleps huggingface dataset:
higher_differentiable_on_sqrt: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real"   and n :: "nat" assumes "higher_differentiable_on S f n"   and "0 \<notin> f ` S"   and "open S" shows "higher_differentiable_on S (\<lambda>x. sqrt (f x)) n"
impl_of_inject: fixes x :: "('a, 'b) alist"   and y :: "('a, 'b) alist" shows "(impl_of x = impl_of y) = (x = y)"
the1I2: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<exists>!x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
min_maxsimpchain_rev: fixes xs :: "'a set list" assumes "min_gallery xs" shows "min_gallery (rev xs)"
assms_2: shows "h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>"
frechet_correctness: fixes I :: "('d, 'e, 'f) interp"   and \<theta> :: "('d, 'f) trm"   and \<nu> :: "(real, 'f) vec" assumes "is_interp I"   and "dfree \<theta>" shows "(sterm_sem I \<theta> has_derivative frechet I \<theta> \<nu>) (at \<nu>)"
open_Un: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "open S"   and "open T" shows "open (S \<union> T)"
nonneg: fixes t :: "'a" shows "0 \<le> zcount M t"
add_nonneg_nonneg: fixes a :: "'a"   and b :: "'a" assumes "bot \<le> a"   and "bot \<le> b" shows "bot \<le> a \<squnion> b"
hull_unique: fixes s :: "'a set"   and t :: "'a set"   and S :: "'a set \<Rightarrow> bool" assumes "s \<subseteq> t"   and "S t"   and "\<And>t'. \<lbrakk>s \<subseteq> t'; S t'\<rbrakk> \<Longrightarrow> t \<subseteq> t'" shows "S hull s = t"
OUTfromChCorrect_data2: shows "OUTfromChCorrect data2"
prems: shows "abc_notfinal (abc_steps_l (0, lm) A (Suc n)) A"
infinite_int_iff_infinite_nat_abs: fixes S :: "int set" shows "infinite S = infinite ((nat \<circ> abs) ` S)"
induct: fixes P :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'a list" assumes "\<And>eq. P eq [] []"   and "\<And>eq a l a' l'. (eq a a' \<Longrightarrow> P eq l l') \<Longrightarrow> P eq (a # l) (a' # l')"   and "\<And>uu_ v va. P uu_ (v # va) []"   and "\<And>uu_ v va. P uu_ [] (v # va)" shows "P a0 a1 a2"
set_reorder: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "set (xs @ y # ys) = set (y # xs @ ys)"
vint_assoc_law_addition: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "z \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "x +\<^sub>\<int> y +\<^sub>\<int> z = x +\<^sub>\<int> (y +\<^sub>\<int> z)"
finite_vimage_IntI: fixes F :: "'b set"   and h :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "finite F"   and "inj_on h A" shows "finite (h -` F \<inter> A)"
cInf_eq_minimum: fixes z :: "'a multiset"   and X :: "'a multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "Inf X = z"
y: shows "\<forall>x\<in>S. dist z y \<le> dist z x"
strip_while_rev: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "strip_while P (rev xs) = rev (dropWhile P xs)"
map_add_subsumed2: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "f \<subseteq>\<^sub>m g" shows "g ++ f = g"
pell_mul_commutes: fixes z1 :: "'b \<times> 'b"   and z2 :: "'b \<times> 'b" shows "pell_mul z1 z2 = pell_mul z2 z1"
cupcake_single_complete0: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env"   and s :: "'a state"   and e :: "exp0"   and res :: "'a state \<times> (v, v) result" assumes "cakeml_static_env static_cenv"   and "Big_Step_Unclocked_Single.evaluate env s e res"   and "cakeml_static_env.is_cupcake_all_env static_cenv env"   and "is_cupcake_exp e" shows "cupcake_evaluate_single env e (snd res)"
intro: fixes R :: "'c set"   and I :: "'c set"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>r a. \<lbrakk>r \<in> R; a \<in> I\<rbrakk> \<Longrightarrow> multiplication r a \<in> I" shows "lideal_axioms I R multiplication"
delete_dom_image: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "i \<in> {0..<Suc n}"   and "\<forall>i'\<in>{0..<Suc n}. f i' = f i \<longrightarrow> i' = i" shows "delete_dom f i ` {0..<n} = f ` {0..<Suc n} - {f i}"
terminates_res: shows "\<Gamma>|\<^bsub>M\<^esub>\<turnstile>c \<down> s"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (flush_instr instr s1)) \<and> t2 = snd (fst (flush_instr instr s2))"
can_def_alt: fixes f :: "'c local.term set" shows "can f = (arr f \<and> (\<forall>t. t \<in> f \<longrightarrow> Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>))"
read_only_shared_all_until_volatile_write_subset': fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and \<S> :: "nat \<Rightarrow> bool option" shows "read_only (share_all_until_volatile_write ts \<S>) \<subseteq> read_only \<S> \<union> (\<Union>(uu_, uu_, uu_, sb, uu_, uu_, uu_)\<in>set ts. all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb))"
to_fun_sub_monom: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "is_UP_monom f"   and "g \<in> carrier P"   and "a \<in> carrier R" shows "to_fun (Cring_Poly.compose R f g) a = to_fun f (to_fun g a)"
index_mat_multrow_3: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat"   and k :: "nat"   and mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "i < dim_row A"   and "j < dim_col A"   and "k \<noteq> i" shows "mat_multrow_gen mul k a A $$ (i, j) = A $$ (i, j)"
assms_3: fixes x :: "'a" assumes "x \<noteq> a"   and "dist x a < R" shows "f x = g x"
cod_char: fixes f :: "'d" shows "Dop.cod f = D.dom f"
cong_cong_half_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and A :: "'a"   and B :: "'a"   and M' :: "'a"   and A' :: "'a"   and B' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M A B"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M' A' B'"   and "Cong A B A' B'" shows "Cong A M A' M'"
Pa: shows "P l \<chi> a = (\<lambda>f. f \<in> B.hom (l a) (l a) \<and> \<chi>a.D.cones_map f (\<chi> a) = \<chi> a)"
abs_le_D1: 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"   and "less_eq (abs a) b" shows "less_eq a b"
erf_holomorphic_on: fixes A :: "complex set" shows "erf holomorphic_on A"
restrict_eq: fixes A :: "'a set" shows "restrictA A = filter (\<lambda>(k, v). k \<in> A)"
symmetric_mpoly_smult: fixes A :: "nat set"   and p :: "'a mpoly"   and c :: "'a" assumes "symmetric_mpoly A p" shows "symmetric_mpoly A (MPoly_Type.smult c p)"
take_lemma: fixes x :: "'r tree"   and y :: "'r tree" assumes "\<And>n. tree_take n\<cdot>x = tree_take n\<cdot>y" shows "x = y"
prems_2: shows "\<langle>l, y, r\<rangle> \<noteq> \<langle>\<rangle>"
norm_bcontfun_def: fixes f :: "'c \<Rightarrow>\<^sub>C 'd" shows "norm f = dist f 0"
l_get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_lemmas type_wf \<equiv> l_type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a type_wf"
simps_10: fixes var :: "nat"   and i :: "nat"   and \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "convert_fm var (ExN i \<phi>) \<Gamma> = None"
sup_lub: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L" shows "is_lub L (\<Squnion>\<^bsub>L\<^esub>A) A"
length_Cons: fixes x :: "'b"   and xs :: "'b list" shows "\<^bold>|x # xs\<^bold>| = Suc \<^bold>|xs\<^bold>|"
neighbourhood_base_of_mono: fixes P :: "'c set \<Rightarrow> bool"   and X :: "'c topology"   and Q :: "'c set \<Rightarrow> bool" assumes "neighbourhood_base_of P X"   and "\<And>S. P S \<Longrightarrow> Q S" shows "neighbourhood_base_of Q X"
mcont_def: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" shows "mcont luba orda lubb ordb f = (monotone orda ordb f \<and> cont luba orda lubb ordb f)"
to_poly_inverse: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "deg R f = 0" shows "f = to_polynomial R (f 0)"
QUPU: fixes \<sigma> :: "'a"   and \<sigma>' :: "'a" assumes "QU \<sigma> \<sigma>'" shows "PU \<sigma> \<sigma>'"
defs_in_allDefs: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> defs g n" shows "v \<in> allDefs g n"
wf': shows "wf (((subcls1 P)\<^sup>+)\<inverse>)"
index_transpose_mat_2: fixes A :: "'b mat" shows "dim_row A\<^sup>T = dim_col A"
f_Exec_Stream_Acc_Output_Nil: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun [] c = []"
Span_contains_spanset_append_left: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list"   and ns :: "'b list" assumes "Module smult M"   and "set ms \<subseteq> M"   and "set ns \<subseteq> M" shows "set ms \<subseteq> R_scalar_mult.RSpan UNIV smult (ms @ ns)"
assms_2: shows "\<Union> ((edges \<circ> snd) ` Rules) \<subseteq> L \<times> UNIV"
consensus: fixes nid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'ND"   and bc' :: "'nid list" assumes "honest nid" shows "eval nid t t' 0 (\<box>\<^sub>b[\<lambda>nd. bc' = (if \<exists>b\<in>pin nd. length (bc nd) < length b then MAX (pin nd) else bc nd)]\<^sub>b \<longrightarrow>\<^sup>b \<circle>\<^sub>b[\<lambda>nd. \<not> mining nd \<and> bc nd = bc' \<or> mining nd \<and> (\<exists>b. bc nd = bc' @ [b])]\<^sub>b)"
integral_setdiff: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'c set" assumes "f integrable_on S"   and "f integrable_on T"   and "negligible (T - S)" shows "integral (S - T) f = integral S f - integral T f"
refl_O_iff: fixes A :: "('a \<times> 'a) set" shows "refl A = (Id \<subseteq> A)"
assms_4: shows "\<forall>n'<n\<^sub>s. \<forall>nid. \<parallel>nid\<parallel>\<^bsub>t n'\<^esub> \<longrightarrow> length (bc (\<sigma>\<^bsub>nid\<^esub>t n')) < length sbc \<or> prefix sbc (bc (\<sigma>\<^bsub>nid\<^esub>t n'))"
f: fixes i :: "'a" assumes "i \<in> I" shows "f i \<in> M \<rightarrow>\<^sub>M N"
cases: fixes a1 :: "(name \<times> ty) list"   and a2 :: "lam"   and a3 :: "ty" assumes "a1 \<turnstile> a2 : a3" obtains \<Gamma> :: "(name \<times> ty) list" and x :: "name" and T :: "ty" where "a1 = \<Gamma>__" and "a2 = Var x__" and "a3 = T" and "valid \<Gamma>__" and "(x__, T) \<in> set \<Gamma>__"   | \<Gamma> :: "(name \<times> ty) list" and t\<^sub>1 :: "lam" and T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" and t\<^sub>2 :: "lam" where "a1 = \<Gamma>'__" and "a2 = App t\<^sub>1 t\<^sub>2" and "a3 = T\<^sub>2__" and "\<Gamma>'__ \<turnstile> t\<^sub>1 : T\<^sub>1__ \<rightarrow> T\<^sub>2__" and "\<Gamma>'__ \<turnstile> t\<^sub>2 : T\<^sub>1__"   | x :: "name" and \<Gamma> :: "(name \<times> ty) list" and T\<^sub>1 :: "ty" and t :: "lam" and T\<^sub>2 :: "ty" where "a1 = \<Gamma>" and "a2 = Lam [x].t" and "a3 = T\<^sub>1 \<rightarrow> T\<^sub>2" and "x \<sharp> \<Gamma>" and "(x, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2"
Diff_idemp: fixes A :: "'b set"   and B :: "'b set" shows "A\<setminus>B\<setminus>B = A\<setminus>B"
inner_scaleR_right: fixes x :: "'a"   and r :: "real"   and y :: "'a" shows "inner x (r *\<^sub>R y) = r * inner x y"
infty_eq_def: fixes eq :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and v1 :: "'a infty"   and v2 :: "'b infty" shows "infty_eq eq v1 v2 \<equiv> case (v1, v2) of (infty.Infty, infty.Infty) \<Rightarrow> True | (infty.Infty, Num a) \<Rightarrow> False | (Num a1, infty.Infty) \<Rightarrow> False | (Num a1, Num a2) \<Rightarrow> eq a1 a2"
plossless_converter_of_resource: fixes \<I> :: "('i, 'j) \<I>"   and res :: "('i, 'j) resource"   and \<I>' :: "('k, 'l) \<I>" assumes "lossless_resource \<I> res" shows "plossless_converter \<I> \<I>' (converter_of_resource res)"
right_angle_to_right_angle: fixes p :: "proj2"   and a :: "proj2"   and q :: "proj2"   and r :: "proj2"   and b :: "proj2"   and s :: "proj2" assumes "right_angle p a q"   and "right_angle r b s" shows "\<exists>J. is_K2_isometry J \<and> apply_cltn2 p J = r \<and> apply_cltn2 a J = b \<and> apply_cltn2 q J = s"
Suc_length_conv: fixes n :: "nat"   and xs :: "'b list" shows "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
tjk_eq: shows "\<forall>i l. A $ i $ Fun.swap j k id l = A $ i $ l"
bell_11_index_4: shows "|\<beta>\<^sub>1\<^sub>1\<rangle> $$ (3, 0) = 0"
map_def: fixes f :: "'a" shows "\<Omega>.map f = (if arr f then C.unity else null)"
assms_2: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "A \<in> set AL" shows "goodAbs A"
seq_tail1_2: fixes s :: "'a"   and x :: "'b \<times> 'a"   and xs :: "('b \<times> 'a) list"   and lars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, x # xs) \<in> seq lars" shows "(s, fst x, snd x) \<in> lars"
ord_le_eq_trans: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a \<le>s b"   and "b = c" shows "a \<le>s c"
base_input_length: fixes "is" :: "nat list"   and m :: "real mat convnet" assumes "is \<lhd> input_sizes m" shows "input_sizes m = map dim_vec (base_input m is)"
dom_closed: fixes f :: "'a" assumes "right a f" shows "right a (local.dom f)"
preprocess'_atoms_to_constraints': fixes cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat"   and S :: "'a istate" assumes "preprocess' cs start = S" shows "set (Atoms S) \<subseteq> {(i, qdelta_constraint_to_atom c v) |i c v. (i, c) \<in> set cs \<and> (\<not> is_monom (poly c) \<longrightarrow> Poly_Mapping S (poly c) = Some v)}"
gt: shows "A $$ (i, j) \<succ> B $$ (i, j)"
nonneg_dirichlet_series_lnD: fixes l :: "'a"   and f :: "'a fds" assumes "nonneg_dirichlet_series (fds_ln l f)"   and "exp l = fds_nth f (Suc 0)" shows "nonneg_dirichlet_series f"
card_UNIV_bit0_def: shows "card_UNIV_class.card_UNIV = Phantom('a bit0) (2 * of_phantom card_UNIV_class.card_UNIV)"
set_quicksort_by_rel: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sl :: "'a list"   and xs :: "'a list" shows "set (quicksort_by_rel R sl xs) = set (xs @ sl)"
elims_1: fixes x :: "'at Literal"   and y :: "bool" assumes "positive x = y" obtains A :: "'at" where "x = Pos Aa__" and "y"   | A :: "'at" where "x = Neg A" and "\<not> y"
assms: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "lt r = u"   and "r \<in> dgrad_sig_set' (length fs) d" shows "sig_red_zero (\<preceq>\<^sub>t) G r"
small_nstep_csmall_nstep_equiv: fixes P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat" shows "small_nstep P \<sigma> n = {\<sigma>'. \<exists>coll. (\<sigma>', coll) \<in> csmall_nstep P \<sigma> n}"
noAbruptn_startD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "'e"   and s' :: "'e" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Normal t" shows "s \<noteq> Abrupt s'"
simps_1: fixes x :: "'b ivl" shows "search \<langle>\<rangle> x = False"
final_def: fixes e :: "'a exp" shows "final e \<equiv> (\<exists>v. e = Val v) \<or> (\<exists>a. e = Throw a)"
of_nat_le_0_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) zero = (m = 0)"
affine_imp_convex: fixes s :: "'b set" assumes "affine s" shows "convex s"
Assoc_List_Mapping_def: fixes al :: "('a, 'b) alist" shows "Assoc_List_Mapping al = Mapping (DAList.lookup al)"
simps_2: fixes x :: "'a" shows "Val x + Bot = Val x"
compress_normalize_primitive_not_introduces_C_negated: fixes disc :: "'c \<Rightarrow> bool"   and m :: "'c match_expr"   and sel :: "'c \<Rightarrow> 'd"   and C :: "'d \<Rightarrow> 'c"   and f :: "'d negation_type list \<Rightarrow> ('d list \<times> 'd list) option"   and m' :: "'c match_expr" assumes "\<not> has_disc_negated disc False m"   and "wf_disc_sel (disc, sel) C"   and "normalized_nnf_match m"   and "compress_normalize_primitive (disc, sel) C f m = Some m'"   and "\<And>as as_pos as_neg. \<lbrakk>f as = Some (as_pos, as_neg); getNeg as = []\<rbrakk> \<Longrightarrow> as_neg = []" shows "\<not> has_disc_negated disc False m'"
less_imp_rlex: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs <\<^sub>v ys" shows "xs <\<^sub>r\<^sub>l\<^sub>e\<^sub>x ys"
is_dghm_axioms: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
sublens_implies_sublens': fixes Y :: "'a \<Longrightarrow> 'b"   and X :: "'c \<Longrightarrow> 'b" assumes "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "X \<subseteq>\<^sub>L' Y"
exc1: shows "abc_steps_l (s, lm) A na = (length A, lma)"
unit: fixes p :: "fm"   and q :: "fm"   and B :: "Cooper.num list"   and d :: "int"   and bbs :: "bool list"   and bs :: "int list"   and i :: "int" assumes "qfree p"   and "unit p = (q, B, d)" shows "(\<exists>x. Ifm bbs (x # bs) p) = (\<exists>x. Ifm bbs (x # bs) q) \<and> Inum (i # bs) ` set B = Inum (i # bs) ` set (\<beta> q) \<and> d_\<beta> q 1 \<and> d_\<delta> q d \<and> 0 < d \<and> iszlfm q \<and> (\<forall>b\<in>set B. numbound0 b)"
C_eq: shows "C = exp (- B - meissel_mertens)"
simps_3: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "black_depth (Node l B k v r) = black_depth l + 1"
waiting_consistent: shows "\<forall>s t u a. rvpeq (rcurrent s) s t \<and> (\<forall>d\<in>rkinvolved a. rvpeq d s t) \<and> rvpeq u s t \<longrightarrow> rwaiting s u a = rwaiting t u a"
solution: fixes s :: "'s"   and D :: "'s pmf" assumes "s \<in> S1"   and "D \<in> K s" shows "x s \<le> (\<Sum>t\<in>S. pmf D t * x t)"
proves_eq_mp_rule: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term"   and B :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "term_ok \<Theta> B"   and "typ_of B = Some propT"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq A B"   and "\<Theta>,\<Gamma> \<turnstile> A"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> B"
tendsto_eventually: fixes f :: "'c \<Rightarrow> 'a"   and l :: "'a"   and net :: "'c filter" assumes "\<forall>\<^sub>F x in net. f x = l" shows "(f \<longlongrightarrow> l) net"
length_MGF: fixes l :: "nat"   and x :: "bit list" assumes "0 < l"   and "l \<le> 2 ^ 32 * length (sha1 x)" shows "length (MGF x l) = l"
preserves_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "C.VV.seq g f" shows "fst (C.VV.comp g f) \<star>\<^sub>C snd (C.VV.comp g f) = (fst g \<star>\<^sub>C snd g) \<cdot>\<^sub>C (fst f \<star>\<^sub>C snd f)"
FGModuleHom_restrict0_GSubspace: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c"   and U :: "'c set" assumes "FGModuleEnd G smult V T"   and "RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V" shows "FGModuleHom G smult U smult (T \<down> U)"
cobounded_unique_unbounded_component: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "bounded (- S)"   and "2 \<le> DIM('a)"   and "\<not> bounded (connected_component_set S x)"   and "\<not> bounded (connected_component_set S y)" shows "connected_component_set S x = connected_component_set S y"
simps_9: fixes var :: "nat"   and i :: "nat"   and \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "convert_fm var (AllN i \<phi>) \<Gamma> = None"
backward_terminating_path_root_iff: fixes x :: "'a" shows "(\<exists>r. backward_terminating_path_root r x) = backward_terminating_path x"
joins_connected: fixes a :: "'a"   and b :: "'a"   and e :: "'edge"   and T :: "'edge set" assumes "joins a b e"   and "T \<subseteq> E"   and "e \<in> T" shows "(a, b) \<in> connected T"
hyps: shows "c \<noteq> (0::'a) \<or> p \<noteq> 0"
smult_minus_1: fixes v :: "'e" assumes "v \<in> carrier N" shows "\<ominus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>N\<^esub> v = \<ominus>\<^bsub>N\<^esub> v"
simps_127: fixes x101 :: "stmt"   and x102 :: "stmt"   and x8 :: "expr" shows "x101 Finally x102 \<noteq> Throw x8"
ctr_transfer_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_basic_type R OclVoid OclVoid"
mult_left_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "(0::'a) \<le> c" shows "c * a \<le> c * b"
R_1: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" assumes "ProofStepInduct_y_eq_new_last G v0 v1 paths P_new sep_size P_k_pre y P_k_post" shows "DisjointPathsPlusOne.new_last P_new \<leadsto>ProofStepInduct_y_eq_new_last.R G v0 v1 P_new\<leadsto>\<^bsub>ProofStepInduct_NonTrivial_P_k_pre.H G v0\<^esub> v1"
vcpower_def: fixes A :: "V"   and n :: "V" shows "A ^\<^sub>\<times> n = (\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>n. A)"
ts_inv_ok_def: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option" shows "ts_inv_ok ts I \<equiv> \<forall>t. (ts t = None) = (I t = None)"
wf_prog_stat_overridesD: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides\<^sub>S old"   and "wf_prog G" shows "G\<turnstile>resTy new\<preceq>resTy old \<and> accmodi old \<le> accmodi new \<and> \<not> is_static old"
emeasure_suntil_HLD: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s"   and t :: "'s" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "emeasure (T s) {x \<in> space (T s). ((\<lambda>xs. \<not> HLD {t} xs) suntil (\<lambda>xs. HLD {t} xs \<and> nxt P xs)) x} = emeasure (T s) {x \<in> space (T s). ev (HLD {t}) x} * emeasure (T t) {x \<in> space (T t). P x}"
simps_1: fixes g :: "'g \<Rightarrow> ('e, 'g) step"   and h :: "'h \<Rightarrow> ('f, 'h) step"   and sg :: "'g"   and sh :: "'h" shows "zip_raw g h (sg, sh, None) = (case g sg of Done \<Rightarrow> Done | Skip sg' \<Rightarrow> Skip (sg', sh, None) | Yield a sg' \<Rightarrow> Skip (sg', sh, Some a))"
simps_1: fixes xs1 :: "'b list"   and xs2 :: "'b list" shows "mergesort_by_rel_split (xs1, xs2) [] = (xs1, xs2)"
h_inject1: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and x :: "'d"   and y :: "'e"   and x' :: "'d"   and y' :: "'e" assumes "Extend h"   and "h (x, y) = h (x', y')" shows "x = x'"