hcju/isabelleps huggingface dataset:
sup_continuous_lfp: fixes F :: "'c \<Rightarrow> 'c" assumes "sup_continuous F" shows "lfp F = (\<Squnion>i. (F ^^ i) \<bottom>)"
linear_injective_left_inverse: fixes f :: "('a \<Rightarrow> real) \<Rightarrow> real" assumes "\<forall>x. x \<in> diff_fun_space \<longrightarrow> f x \<in> UNIV"   and "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) f"   and "inj_on f diff_fun_space" shows "\<exists>g. (\<forall>x. x \<in> UNIV \<longrightarrow> g x \<in> diff_fun_space) \<and> linear_on UNIV diff_fun_space (*\<^sub>R) (*\<^sub>R) g \<and> (\<forall>x\<in>diff_fun_space. g (f x) = x)"
gen_dfs_fe_inv_def: fixes E :: "('a \<times> 'a) set"   and U :: "'b"   and S :: "'a set"   and V0 :: "'a set"   and u0 :: "'a"   and it :: "'a set"   and V :: "'a set"   and brk :: "bool" shows "gen_dfs_fe_inv E U S V0 u0 it V brk \<equiv> (\<not> brk \<longrightarrow> E `` (V - S) \<subseteq> V) \<and> E `` {u0} - it \<subseteq> V \<and> V0 \<subseteq> V \<and> V \<subseteq> V0 \<union> (E - UNIV \<times> S)\<^sup>* `` (E `` {u0} - it - S)"
pderiv_diff: fixes p :: "'c poly"   and q :: "'c poly" shows "pderiv (p - q) = pderiv p - pderiv q"
inf_lt: fixes x :: "'a DBMEntry" assumes "\<infinity> < x" shows "False"
\<tau>exec_move_conv_\<tau>exec_meth: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1" shows "\<tau>exec_move ci P t e = \<tau>exec_meth ci (compP2 P) (compE2 e) (compxE2 e 0 0) t"
gState_progress_refl: fixes prog :: "program"   and g :: "'a gState_scheme" assumes "gState_inv prog g" shows "(g, g) \<in> gState_progress_rel prog"
gcd_impl_primitive_def: fixes G1 :: "'b poly"   and G2 :: "'b poly" shows "gcd_impl_primitive G1 G2 = normalize (primitive_part (fst (subresultant_prs dichotomous_Lazard G1 G2)))"
that: fixes a :: "'edge" assumes "valid_edge a"   and "intra_kind (kind a)"   and "inner_node (targetnode a)"   and "sourcenode a = (_Entry_)" shows "thesis"
pre_subdivFace'_None: fixes g :: "graph"   and f :: "face"   and v' :: "nat"   and v :: "nat"   and n :: "nat"   and vol :: "nat option list" assumes "pre_subdivFace' g f v' v n (None # vol)" shows "pre_subdivFace' g f v' v (Suc n) vol"
has_as_limit_iff_terminal: fixes a :: "'c" shows "D.has_as_limit a = terminal a"
injectivity: fixes x :: "real"   and y :: "real" assumes "of_real x = of_real y" shows "x = y"
term_variants_pred_const: fixes b :: "'c"   and P :: "'c \<Rightarrow> 'c list"   and a :: "'c" assumes "b \<in> set (P a)" shows "term_variants_pred P (Fun a []) (Fun b [])"
linf_norm_poly_def: fixes f :: "'b poly" shows "\<parallel>f\<parallel>\<^sub>\<infinity> \<equiv> Missing_Lemmas.max_list (map abs (coeffs f) @ [0::'b])"
inf_rr_inf: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r" shows "pseudo_hoop_algebra.inf_rr times imp_r = inf"
commutes: shows "f \<circ> \<phi> = \<phi> \<circ> g"
\<tau>\<phi>: shows "\<guillemotleft>\<rho> \<cdot> \<phi> : g' \<Rightarrow> r \<star> f\<guillemotright>"
reduce_fps_poly_def: fixes F :: "'b fps poly" shows "reduce_fps_poly F = fps_poly_swap1 F $ 0"
no_call_in_path_set_conv: fixes x :: "'a"   and p :: "'a list" shows "no_call_in_path x p = (x \<notin> set p)"
summedBidVectorAlg_def: fixes bids :: "'c \<times> 'd set \<Rightarrow> 'e"   and N :: "'c set"   and \<Omega> :: "'d set" shows "summedBidVectorAlg bids N \<Omega> \<equiv> summedBidVectorRel bids N \<Omega> Elsee (0::'e)"
concat_bit_Suc: fixes n :: "nat"   and k :: "int"   and l :: "int" shows "concat_bit (Suc n) k l = k mod 2 + 2 * concat_bit n (k div 2) l"
filter_M_holds_for_result: fixes h :: "'a"   and P :: "'c \<Rightarrow> ('a, 'b, bool) prog"   and xs :: "'c list"   and ys :: "'c list"   and x :: "'c" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys"   and "x \<in> set ys"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "h \<turnstile> P x \<rightarrow>\<^sub>r True"
uminus_Pos: fixes A :: "'b" shows "- Pos A = Neg A"
amax_le: fixes x :: "ant"   and y :: "ant" shows "amax x y \<le> x \<or> amax x y \<le> y"
strips_equivalent_to_sas_plus_i: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> \<psi>" shows "(\<phi> \<Psi> )\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan ((\<phi> \<Psi> )\<^sub>I) (\<phi>\<^sub>P \<Psi> \<psi>)"
awalk_Nil_iff: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G" shows "pre_digraph.awalk G u [] v = (u = v \<and> u \<in> verts G)"
ord_eq_le_subst: fixes a :: "'b::ord"   and f :: "'c::ord \<Rightarrow> 'b::ord"   and b :: "'c::ord"   and c :: "'c::ord" assumes "(a::'b::ord) = (f::'c::ord \<Rightarrow> 'b::ord) (b::'c::ord)"   and "(b::'c::ord) \<le> (c::'c::ord)"   and "\<And>(x::'c::ord) y::'c::ord. x \<le> y \<Longrightarrow> (f::'c::ord \<Rightarrow> 'b::ord) x \<le> f y" shows "(a::'b::ord) \<le> (f::'c::ord \<Rightarrow> 'b::ord) (c::'c::ord)"
compose_def: fixes A :: "'e set"   and g :: "'g \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'g" shows "compose A g f = (\<lambda>x\<in>A. g (f x))"
abs_eq_iff: fixes x :: "'b ltln"   and y :: "'b ltln" shows "(abs_ltln\<^sub>P x = abs_ltln\<^sub>P y) = x \<sim>\<^sub>P y"
simps_3: fixes x21 :: "'k"   and x22 :: "'k list" shows "x21 # x22 \<noteq> []"
hv_diff: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "Hv (x - y) = Hv x - Hv y"
self_domain_iff_ide: fixes a :: "(bool, 'arr list) concrete_category.arr" shows "(a \<in> domains a) = ide a"
grid_level_eq: fixes ds :: "nat set"   and p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list" assumes "\<forall>d\<in>ds. lv p d = lv b d"   and "p \<in> grid b ds" shows "level p = level b"
matrix_vector_mult_linear_continuous_on: fixes S :: "('b, 'c) vec set"   and A :: "(('b, 'c) vec, 'd) vec" shows "continuous_on S ((*v) A)"
2: shows "card {z \<in> A. z \<le> y} < card {z \<in> A. z \<le> x}"
less_le: fixes x :: "'c"   and y :: "'c" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
satisfiableAppendTautology: fixes F :: "Literal list list"   and c :: "Literal list" assumes "satisfiable F"   and "clauseTautology c" shows "satisfiable (F @ [c])"
rep_eq: fixes x :: "CoCalls set"   and xa :: "CoCalls" shows "is_cc_lub x xa = (Rep_CoCall xa = \<Union> (Rep_CoCall ` x))"
hoare2_sound: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2 {P} c {Q}" shows "\<Turnstile>\<^sub>2 {P} c {Q}"
not_Some_eq: fixes x :: "'f option" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = None)"
init_locs_def: fixes C :: "cname"   and m :: "mname"   and s :: "state" shows "init_locs C m s \<equiv> s\<lparr>locals := locals s ++ init_vars (map_of (lcl (the (method C m))))\<rparr>"
delay_unbounded_monotone: fixes f :: "nat \<Rightarrow> nat option"   and i :: "nat"   and n :: "nat" assumes "f \<in> \<R>"   and "\<forall>n. \<phi> i (f \<triangleright> n) \<down>" shows "\<exists>m\<^sub>0. \<forall>m\<ge>m\<^sub>0. n < the (delay i (f \<triangleright> m))"
l1_step3_refines_a0i_running_skip_r: fixes A :: "agent"   and B :: "agent"   and gny :: "msg"   and Ra :: "fid_t" shows "{R01iar} (a0i_running A B (Exp gny (NonceF (Ra $ nx))))\<^sup>=, l1_step3 Ra A B gny {> R01iar}"
Join_def: fixes F :: "'b program"   and G :: "'b program" shows "F \<squnion> G = mk_program (Init F \<inter> Init G, Acts F \<union> Acts G, AllowedActs F \<inter> AllowedActs G)"
span_minimal: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "Modules.module scale"   and "S \<subseteq> T"   and "module.subspace scale T" shows "module.span scale S \<subseteq> T"
composition_series_axioms_def: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list" shows "composition_series_axioms G \<GG> \<equiv> \<forall>i. i + 1 < length \<GG> \<longrightarrow> simple_group (G\<lparr>carrier := \<GG> ! (i + 1)\<rparr> Mod \<GG> ! i)"
pps_to_list_sorted_wrt: fixes S :: "'t set" shows "sorted_wrt (\<succ>\<^sub>t) (pps_to_list S)"
order_iff_strict: fixes a :: "nat"   and b :: "nat" shows "(a dvd b) = (a dvd b \<and> a \<noteq> b \<or> a = b)"
inject_2: fixes x21 :: "(char list \<times> tid_or_exn) option"   and x22 :: "v list"   and y21 :: "(char list \<times> tid_or_exn) option"   and y22 :: "v list" shows "(Conv x21 x22 = Conv y21 y22) = (x21 = y21 \<and> x22 = y22)"
hom_prod_list: fixes xs :: "int list" shows "of_int (prod_list xs) = prod_list (map of_int xs)"
eval_Pr_0: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list" assumes "recfn (Suc n) (Pr n f g)"   and "n = length xs" shows "eval (Pr n f g) (0 # xs) = eval f xs"
thetaSSeqD_def: shows "thetaSSeqD \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. proper c1 \<and> proper d1 \<and> proper c2 \<and> proper d2 \<and> discr c2 \<and> discr d2 \<and> c1 \<approx>s d1}"
pred_pmf_def: shows "pred_pmf \<equiv> \<lambda>P x. Ball (set_pmf x) P"
cring_simprules_22: fixes R :: "('c, 'd) ring_scheme" assumes "cring R" shows "\<ominus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
integrable_continuous: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on (cbox a b) f" shows "f integrable_on cbox a b"
mojmir_accept_iff_rabin_accept: shows "\<MM>.accept = accept\<^sub>R (\<MM>.\<delta>\<^sub>\<R>, \<MM>.q\<^sub>\<R>, {(\<MM>.fail\<^sub>R \<union> \<MM>.merge\<^sub>R j, \<MM>.succeed\<^sub>R j) |j. j < \<MM>.max_rank}) w"
iter_hcomp_insert: fixes v :: "'b"   and ws :: "'b fset"   and p :: "'b \<Rightarrow> interface" assumes "v |\<notin>| ws" shows "iter_hcomp (finsert v ws) p = (p v \<otimes> iter_hcomp ws p)"
execute_opr'_def: shows "execute_opr' \<equiv> \<lambda>(name, uu_, effs, uu_) s. let effs = filter (eff_enabled' s) effs in fold (\<lambda>(uu_, x, uu_, v) s. s(x \<mapsto> v)) effs s"
closure_halfspace_gt: fixes a :: "'a"   and b :: "real" assumes "a \<noteq> (0::'a)" shows "closure {x. b < a \<bullet> x} = {x. b \<le> a \<bullet> x}"
domintros: fixes A :: "'a mat" assumes "\<lbrakk>dim_row A \<noteq> Suc 0; dim_row A \<noteq> 2; dim_col A \<noteq> Suc 0; dim_col A \<noteq> 2; 0 < dim_row A; 0 < dim_col A\<rbrakk> \<Longrightarrow> Smith_mxn_dom (mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]))"   and "\<And>a aa b xc ab ba ac ad bb xg ae bc af bd xk yd ag ah ai be xm aj ak bf. \<lbrakk>dim_row A \<noteq> Suc 0; dim_row A \<noteq> 2; dim_col A \<noteq> Suc 0; dim_col A \<noteq> 2; (a, aa, b) = Smith_mxn (mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A])); (xc, ab, ba) = Smith_mxn (mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A])); (ac, ad, bb) = Smith_2xn (mat_of_rows (dim_col A) [Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba) 0, Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba) (Suc 0)]); (xg, ae, bc) = Smith_2xn (mat_of_rows (dim_col A) [Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba) 0, Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba) (Suc 0)]); (af, bd) = reduce_column div_op (xg * mat_of_rows (dim_col A) [Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba) 0, Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba) (Suc 0)] * bc @\<^sub>r mat_of_rows (dim_col A) (map (Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba)) [2..<dim_row A]) * bc); (xk, yd) = reduce_column div_op (xg * mat_of_rows (dim_col A) [Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba) 0, Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba) (Suc 0)] * bc @\<^sub>r mat_of_rows (dim_col A) (map (Matrix.row (mat_of_row (Matrix.row A 0) * ba @\<^sub>r xc * mat_of_rows (dim_col A) (map (Matrix.row A) [Suc 0..<dim_row A]) * ba)) [2..<dim_row A]) * bc); (ag, ah, ai, be) = split_block yd (Suc 0) (Suc 0); (xm, aj, ak, bf) = split_block yd (Suc 0) (Suc 0); 0 < dim_row A; 0 < dim_col A\<rbrakk> \<Longrightarrow> Smith_mxn_dom bf" shows "Smith_mxn_dom A"
iT_Minus_mem_iff: fixes x :: "nat"   and k :: "nat"   and I :: "nat set" shows "(x \<in> k \<ominus> I) = (x \<le> k \<and> k - x \<in> I)"
inf_idem: fixes x :: "'a" shows "x \<sqinter> x = x"
someI_ex: fixes P :: "'n \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (SOME x. P x)"
assms_3: shows "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'a)"
nat_distrib_2: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a * (b + c) = a * b + a * c"
hcomp_in_hhomE: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and a :: "'a"   and c :: "'a" assumes "\<guillemotleft>\<nu> \<star> \<mu> : a \<rightarrow>\<^sub>B c\<guillemotright>" obtains "B.arr \<mu>" and "B.arr \<nu>" and "src \<nu> = trg \<mu>" and "src \<mu> = a" and "trg \<nu> = c"
ext_inject: fixes cx :: "int"   and cy :: "int"   and more :: "'a"   and cx' :: "int"   and cy' :: "int"   and more' :: "'a" shows "(\<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> = \<lparr>cx = cx', cy = cy', \<dots> = more'\<rparr>) = (cx = cx' \<and> cy = cy' \<and> more = more')"
ind: shows "UF.Sigma (rec_exec (rec_all (recf.id (Suc (length xs)) (length xs)) (Cn (Suc (Suc (length xs))) rec_not [Cn (Suc (Suc (length xs))) rf (get_fstn_args (Suc (Suc (length xs))) (length xs) @ [recf.id (Suc (Suc (length xs))) (Suc (length xs))])]))) (xs @ [w]) = Minr (\<lambda>args. 0 < rec_exec rf args) xs w"
0: shows "reflp (\<lambda>a\<^sub>1 a\<^sub>2. P a\<^sub>1 \<longrightarrow> P a\<^sub>2)"
calc_atm_3: fixes a :: "name"   and b :: "name"   and c :: "name" shows "swap (a, b) c = (if a = c then b else if b = c then a else c)"
simps_7: fixes p :: "('a, 'b) form"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "Neg p[s/k] = Neg (p[s/k])"
simps_2: fixes a :: "'a"   and as :: "'a list"   and b :: "'a list" shows "revg (a # as) b = revg as (a # b)"
in_p: fixes x :: "'b" shows "(p x \<in> B) = (x \<in> B)"
ABind_nonrec_eq: fixes x :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "ABind_nonrec x e\<cdot>(ae, G) = (if isVal e \<or> x--x\<notin>G then ae x else up\<cdot>0)"
that: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) list" assumes "valid_decomp X qs"   and "cone_decomp T qs"   and "standard_decomp d qs"   and "monomial_decomp ps \<Longrightarrow> monomial_decomp qs"   and "hom_decomp ps \<Longrightarrow> hom_decomp qs" shows "thesis"
is_Prim0_Trg: fixes t :: "'c local.term" shows "is_Prim\<^sub>0 (Trg t)"
llength_ltakeWhile_all: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(llength (ltakeWhile P xs) = llength xs) = (ltakeWhile P xs = xs)"
Abs_Msg_inverse: fixes y :: "freemsg set" assumes "y \<in> Msg" shows "Rep_Msg (Abs_Msg y) = y"
lp: shows "0 \<le> log 2 (real_of_int x)"
fine_Cons: fixes \<delta> :: "real \<Rightarrow> real"   and b :: "real"   and c :: "real"   and D :: "(real \<times> real \<times> real) list"   and a :: "real"   and x :: "real" assumes "fine \<delta> (b, c) D"   and "a < b"   and "a \<le> x"   and "x \<le> b"   and "b - a < \<delta> x" shows "fine \<delta> (a, c) ((a, x, b) # D)"
\<alpha>_strict: shows "nres_of dSUCCEED = SUCCEED"
path_connectedin_imp_connectedin: fixes X :: "'c topology"   and S :: "'c set" assumes "path_connectedin X S" shows "connectedin X S"
stat_is_stat_red: shows "statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F = statically_complete_calculus Bot Inf (\<Turnstile>) reduc_calc.Red_Red_I Red_F"
inf_p_eq: fixes x :: "'a" shows "x \<sqinter> - x = bot"
composite_leg1_is_map: shows "is_left_adjoint (s\<^sub>1 \<star> w)"
map\<^sub>0_in_hom_1: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<tau>\<^sub>0 a : F.map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 a\<guillemotright>"
colossless_gpv_try: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and gpv' :: "('f, 'd, 'e) gpv" shows "colossless_gpv \<I> (TRY gpv ELSE gpv') = (colossless_gpv \<I> gpv \<or> colossless_gpv \<I> gpv')"
WhileTrue_5: fixes l :: "char list \<Rightarrow> nat" assumes "Q l s\<^sub>3_" shows "\<exists>n. wpw b c n Q l s\<^sub>2_"
empty: shows "\<Sum>\<^sub># {#} = (0::'b)"
order_prod_list: fixes ps :: "'a poly list"   and x :: "'a" assumes "\<And>p. p \<in> set ps \<Longrightarrow> p \<noteq> 0" shows "order x (prod_list ps) = sum_list (map (order x) ps)"
sd_pos: fixes x :: "'a" assumes "x \<in> U"   and "x \<notin> S" shows "0 < setdist {x} S"
R39_d: shows "pmf (sds R39) d = 1 / 2 - pmf (sds R39) c"
ennexp_special_values_2: fixes x :: "ereal" shows "(ennexp x = 1) = (x = 0)"
zero_closed: shows "\<zero>\<^bsub>N\<^esub> \<in> carrier N"
subset_entailment: fixes P' :: "bexp set"   and P :: "bexp set" assumes "P' \<subseteq> P" shows "pred_entailment P P'"
open_cballE: fixes t\<^sub>0 :: "'b"   and T :: "'b set" assumes "t\<^sub>0 \<in> T"   and "open T" shows "\<exists>e>0. cball t\<^sub>0 e \<subseteq> T"
fin: fixes s :: "'a rexp" shows "finite {fold delta w (init s) |w. True}"
hext_objD: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and h' :: "'a"   and a :: "'b"   and C :: "String.literal" assumes "heap_base.hext typeof_addr h h'"   and "typeof_addr h a = \<lfloor>Class_type C\<rfloor>" shows "typeof_addr h' a = \<lfloor>Class_type C\<rfloor>"
mset_list_of_B: fixes t :: "'b tree" assumes "braun t" shows "mset (list_of_B t) = mset_tree t"
comp_in_homI': fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "VoV.arr f"   and "VoV.arr g"   and "VoV.dom f = a"   and "VoV.cod g = c"   and "VoV.dom g = VoV.cod f" shows "VoV.in_hom (VoV.comp g f) a c"
prod_mset_singleton: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one {#x#} = x"
combine_options_cases: fixes x :: "'c option"   and P :: "'c option \<Rightarrow> 'd option \<Rightarrow> bool"   and y :: "'d option" assumes "x = None \<Longrightarrow> P x y"   and "y = None \<Longrightarrow> P x y"   and "\<And>a b. \<lbrakk>x = Some a; y = Some b\<rbrakk> \<Longrightarrow> P x y" shows "P x y"
dropWhile_neq_rev: fixes xs :: "'g list"   and x :: "'g" assumes "distinct xs"   and "x \<in> set xs" shows "dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
simps_3: fixes committed :: "nat set"   and justifying_exec :: "('b \<times> ('a, 'b) obs_event action) llist"   and justifying_ws :: "nat \<Rightarrow> nat"   and more :: "'c" shows "justifying_ws \<lparr>committed = committed, justifying_exec = justifying_exec, justifying_ws = justifying_ws, \<dots> = more\<rparr> = justifying_ws"
sl1: shows "2 * e * b * d \<in> radical_sqrt"
exhaust: fixes y :: "dining_event" obtains x11 :: "nat" and x12 :: "nat" where "y = picks x11 x12"   | x21 :: "nat" and x22 :: "nat" where "y = putsdown x21 x22"
sup_pred_def: fixes P :: "'a pred"   and Q :: "'a pred" shows "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
assms_3: shows "\<forall>x. d1 \<le> x \<and> x < c \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)"
refl_r: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "Semilat A r f" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x"
abs_eq: fixes x :: "int poly" shows "roots_of_3 x = map Real_Alg_Invariant (roots_of_2 x)"
thmD10: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and X :: "symbol \<times> character list"   and T :: "(symbol \<times> character list) set"   and N :: "symbol"   and \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and r :: "nat"   and \<gamma> :: "symbol list"   and k' :: "nat"   and x :: "item"   and I :: "item set" assumes "p \<in> \<PP>"   and "charslength p = k"   and "X \<in> T"   and "T \<subseteq> \<X> k"   and "(N, \<alpha> @ \<beta>) \<in> \<RR>"   and "r \<le> length p"   and "leftderives [\<SS>] (terminals (take r p) @ [N] @ \<gamma>)"   and "leftderives \<alpha> (terminals (drop r p @ [X]))"   and "k' = k + length (chars_of_token X)"   and "x = Item (N, \<alpha> @ \<beta>) (length \<alpha>) (charslength (take r p)) k'"   and "I = items_le k' (\<pi> k' {} (Scan T k (Gen (Prefixes p))))" shows "x \<in> I"
rminusinf_bound0: fixes p :: "fm" assumes "isrlfm p" shows "bound0 (minusinf p)"
exhaust_disc: fixes expr :: "'a expr" obtains "is_Literal expr"   | "is_Let expr"   | "is_Var expr"   | "is_If expr"   | "is_MetaOperationCall expr"   | "is_StaticOperationCall expr"   | "is_Call expr"
1: shows "pfp_wn (local.step' \<top>) (bot c) = Some C"
mix_5: shows "local.general_U q = measure_pmf.expectation q local.ocU"
uminus_finfun_def: fixes A :: "'a \<Rightarrow>f 'b" shows "- A = uminus \<circ>$ A"
r7_trans: fixes t :: "trm"   and t' :: "trm"   and s :: "trm"   and x :: "name" assumes " t \<mapsto>\<^sup>* t' " shows " s to x in t \<mapsto>\<^sup>* s to x in t' "
filter_def: fixes F :: "'c set" shows "ord_class.filter F \<equiv> F \<noteq> {} \<and> (\<forall>x\<in>F. \<forall>y\<in>F. \<exists>z\<in>F. z \<le> x \<and> z \<le> y) \<and> is_up_set F"
absolutely_integrable_component_ubound: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "f integrable_on A"   and "g absolutely_integrable_on A"   and "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> f x \<bullet> b \<le> g x \<bullet> b" shows "f absolutely_integrable_on A"
stream_space_eq_distr: fixes M :: "'b measure" shows "stream_space M = distr (Pi\<^sub>M UNIV (\<lambda>i. M)) (stream_space M) to_stream"
subprob_emeasure_le_1: fixes M :: "'c measure"   and X :: "'c set" assumes "subprob_space M" shows "emeasure M X \<le> 1"
g: shows "g \<in> [WW]\<^bsup>2\<^esup> \<rightarrow> {0, 1}"
Image_runiq_eq_eval: fixes x :: "'c"   and R :: "('c \<times> 'd) set" assumes "x \<in> Domain R"   and "runiq R" shows "R `` {x} = {R ,, x}"
conth: shows "continuous_on ({0..1} \<times> U) h"
ket_plus_dim: shows "ket_plus \<in> carrier_vec 2"
upper_split_float_interval2: fixes X :: "float interval" shows "upper (snd (split_float_interval X)) = upper X"
is_offending_flows_def: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and f :: "('b \<times> 'b) set"   and G :: "'b graph"   and nP :: "'b \<Rightarrow> 'c" shows "SecurityInvariant_withOffendingFlows.is_offending_flows sinvar f G nP \<equiv> \<not> sinvar G nP \<and> sinvar (delete_edges G f) nP"
keys_diff: fixes a :: "'a \<Rightarrow>\<^sub>0 'b"   and b :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys (a - b) \<subseteq> keys a \<union> keys b"
times_monomial_monomial: fixes t :: "'a"   and c :: "'b"   and v :: "'a"   and d :: "'b" shows "monomial c t * monomial d v = monomial (c * d) (t + v)"
linorder_not_less: fixes x :: "'c"   and y :: "'c" shows "(\<not> x < y) = (y \<le> x)"
vardict_inv_def: fixes ss :: "(integer \<times> edge list) iarray"   and p :: "nat \<times> edgeIndex \<times> procArg list \<times> varDecl list"   and vs :: "(String.literal, variable) assoc_list" shows "vardict_inv ss p vs = lm.ball vs (\<lambda>(k, v). k \<in> process_names ss p \<and> variable_inv v)"