hcju/isabelleps huggingface dataset:
sum_list_map_filter: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. \<lbrakk>x \<in> set xs; \<not> P x\<rbrakk> \<Longrightarrow> f x = (0::'d)" shows "sum_list (map f (filter P xs)) = sum_list (map f xs)"
compact_fip_Heine_Borel: fixes \<F> :: "'a set set" assumes "\<And>T. T \<in> \<F> \<Longrightarrow> compact T"   and "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> \<Inter> \<F>' \<noteq> {}" shows "\<Inter> \<F> \<noteq> {}"
update_not_empty: fixes ks :: "'c list"   and v :: "'d"   and t :: "('c, 'd) trie" shows "\<not> is_empty_trie (update_trie ks v t)"
lookup1: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list"   and b :: "coname"   and a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list" assumes "x \<sharp> idn \<Gamma> b" shows "lookup x a (idn \<Gamma> b) \<theta>_c = lookupa x a \<theta>_c"
PROB_DOM_PROJ_DIFF: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set" shows "prob_dom (prob_proj PROB (prob_dom PROB - vs)) = prob_dom PROB - vs"
ptrm_size_alpha_equiv: fixes X :: "'b ptrm"   and Y :: "'b ptrm" assumes "X \<approx> Y" shows "size X = size Y"
index_minus_mat_2: fixes A :: "'c mat"   and B :: "'c mat" shows "dim_row (A - B) = dim_row B"
image_affinity_atLeastAtMost: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and m :: "'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(\<lambda>x. plus (times m x) c) ` ord.atLeastAtMost less_eq a b = (if ord.atLeastAtMost less_eq a b = {} then {} else if less_eq zero m then ord.atLeastAtMost less_eq (plus (times m a) c) (plus (times m b) c) else ord.atLeastAtMost less_eq (plus (times m b) c) (plus (times m a) c))"
simps_2: fixes i :: "nat"   and R :: "nat set" shows "net_tree_ips \<langle>i; R\<rangle> = {i}"
last_map: fixes xs :: "'e list"   and f :: "'e \<Rightarrow> 'f" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
u: shows "until \<phi> \<psi> (map stl \<pi>l)"
bindE: fixes R :: "'b pred"   and Q :: "'b \<Rightarrow> 'a pred"   and y :: "'a" assumes "eval (bind R Q) y" obtains x :: "'b" where "eval R x" and "eval (Q x) y"
homh: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "h \<in> hom G H"
lspasl_starl_eq: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h0 :: "'a" shows "(A \<and>* B) h0 = (\<exists>h1 h2. (h1,h2\<triangleright>h0) \<and> A h1 \<and> B h2)"
trg_cod: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "trg (cod \<mu>) = trg \<mu>"
strict_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<sqsubset> b"   and "b \<sqsubset> c" shows "a \<sqsubset> c"
at_neg2: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. \<not> (\<forall>v'. v = c > v' \<longrightarrow> \<phi> ts v') \<longrightarrow> (\<forall>v'. v = c > v' \<longrightarrow> \<not> \<phi> ts v')"
tree_map_of_invar: fixes T :: "'t"   and pres :: "('a \<times> 'b) list" assumes "invar T" shows "invar (tree_map_of' T pres)"
polymap_apply_linear: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and v :: "'b"   and c :: "'a" assumes "VectorSpaceEnd smult V T"   and "v \<in> V" shows "VectorSpaceEnd.polymap smult V T [:- c, 1::'a:] v = T v - smult c v"
Rep_Arity_inverse: fixes x :: "Arity" shows "to_Arity (Rep_Arity x) = x"
dropWhile_eq_drop: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "dropWhile P xs = drop (length (takeWhile P xs)) xs"
find_mu: shows "\<lbrace>\<lambda>s. nu s \<in> {0<..lambda + mu} \<and> (seq \<circ> (\<lambda>s. nu s + nu s)) s = (seq \<circ> nu) s \<and> hare s = (seq \<circ> nu) s\<rbrace> find_mu \<lbrace>\<lambda>s. nu s \<in> {0<..lambda + mu} \<and> tortoise s = seq mu \<and> m s = mu\<rbrace>"
indicator_UNIV: shows "indicator UNIV = (\<lambda>x. 1::'d)"
ineq: shows "s.max_fun_diff f g = s.max_fun_diff g h \<longrightarrow> f (s.max_fun_diff f g) \<noteq> h (s.max_fun_diff g h)"
startsby_zero_power_iff: fixes a :: "'a fps"   and n :: "nat" shows "(a ^ n $ 0 = (0::'a)) = (n \<noteq> 0 \<and> a $ 0 = (0::'a))"
select_convs_1: fixes iso_verts :: "'c \<Rightarrow> 'd"   and iso_arcs :: "'e \<Rightarrow> 'f"   and iso_head :: "'f \<Rightarrow> 'd"   and iso_tail :: "'f \<Rightarrow> 'd"   and more :: "'g" shows "iso_verts \<lparr>iso_verts = iso_verts, iso_arcs = iso_arcs, iso_head = iso_head, iso_tail = iso_tail, \<dots> = more\<rparr> = iso_verts"
extend_inverse: fixes F :: "'a program" shows "project h UNIV (extend h F) = F"
hfst_conv: fixes a :: "hf"   and b :: "hf" shows "hfst \<langle>a, b\<rangle> = a"
assms_3: shows "finite X \<longrightarrow> m < card X"
cubic_depressed: fixes e :: "'a"   and z :: "'a"   and y :: "'a"   and u :: "'a"   and v :: "'a"   and f :: "'a" assumes "e \<noteq> (0::'a) \<Longrightarrow> z\<^sup>2 - y * z - e / (3::'a) = (0::'a)"   and "e \<noteq> (0::'a) \<Longrightarrow> u = z ^ 3"   and "v = - (e ^ 3 / (27::'a))" shows "(y ^ 3 + e * y + f = (0::'a)) = (if e = (0::'a) then y ^ 3 = - f else u\<^sup>2 + f * u + v = (0::'a))"
norm_Pinj_0_False: fixes P :: "pol" shows "isnorm (Pinj 0 P) = False"
expect_1_I: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "P x" shows "1 \<le> \<guillemotleft> P \<guillemotright> x"
r_zero: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "x \<in> carrier (UP Zp)" shows "x \<oplus>\<^bsub>UP Zp\<^esub> \<zero>\<^bsub>UP Zp\<^esub> = x"
map\<^sub>0_in_hhom: fixes a :: "'b" assumes "D.obj a" shows "\<guillemotleft>F.counit\<^sub>0 a : src\<^sub>D (E.FG.map a) \<rightarrow>\<^sub>D src\<^sub>D (E.I\<^sub>C.map a)\<guillemotright>"
set_map: shows "set (map ((\<otimes>) b) q @ replicate (length p) \<zero>) \<subseteq> carrier R"
shiftr1_bl_of: fixes bl :: "bool list" assumes "length bl \<le> LENGTH('b)" shows "shiftr1 (of_bl bl) = of_bl (butlast bl)"
cEnum_finite_3_def: shows "CENUM(Enum.finite_3) = Some (enum_class.enum, enum_class.enum_all, enum_class.enum_ex)"
liset_suffix: fixes i :: "nat"   and A :: "'b set"   and u :: "'b llist"   and v :: "'b llist" assumes "i \<in> liset A u"   and "u \<le> v" shows "i \<in> liset A v"
add': shows "set (map of_int_hom.vec_hom fs) \<subseteq> Rn"
ty_expr_noerr: fixes \<Gamma> :: "(String.literal \<Rightarrow> unit option) \<times> (String.literal \<Rightarrow> unit option)"   and fs :: "local_state \<times> global_state"   and e :: "exp" assumes "ty_fs \<Gamma> fs" shows "(eval_exp e fs \<noteq> None) = (ty_expr \<Gamma> e \<noteq> None)"
inf_assoc: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) z = inf x (inf y z)"
interpret_floatarith_inverse: fixes f :: "floatarith"   and xs :: "real list" shows "interpret_floatarith (inverse f) xs = inverse (interpret_floatarith f xs)"
K2_isometry_cosh_dist: fixes a :: "proj2"   and b :: "proj2"   and J :: "cltn2" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "is_K2_isometry J" shows "cosh_dist (apply_cltn2 a J) (apply_cltn2 b J) = cosh_dist a b"
connected_component_refl: fixes x :: "'c"   and S :: "'c set" assumes "x \<in> S" shows "connected_component S x x"
lconf_upd: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option"   and v :: "val"   and T :: "ty"   and V :: "char list" assumes "P,h \<turnstile> l (:\<le>) E"   and "P,h \<turnstile> v :\<le> T"   and "E V = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> l(V \<mapsto> v) (:\<le>) E"
unit_SE_def: fixes e :: "'d" shows "unit\<^sub>S\<^sub>E e = (\<lambda>\<sigma>. Some (e, \<sigma>))"
D: shows "\<langle>n, Hash (ID n (Sec_PubKey m))\<rangle> \<in> spied s"
inverse_transformation_axioms: shows "inverse_transformation (\<cdot>) (\<cdot>) (MC.tensor MC.unity) local.map MC.\<ll>"
compactin_empty: fixes X :: "'c topology" shows "compactin X {}"
last: fixes z :: "'a" assumes "z \<in> set zs" shows "z \<notin> S"
assms_3: shows "Counterclockwise_2D_Arbitrary.lex s r"
select_convs_2: fixes lens_get :: "'c \<Rightarrow> 'd"   and lens_put :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and more :: "'e" shows "put\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_put"
natural_isomorphism_\<rr>: shows "natural_isomorphism (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.R B.map B.\<rr>"
get_owner_document_is_strongly_scdom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r owner_document" shows "(cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r owner_document \<in> set sc) = (ptr' \<in> set sc)"
fold_insert_idem: fixes A :: "'a set"   and z :: "'a"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold (\<^bold>*) z (insert x A) = x \<^bold>* Finite_Set.fold (\<^bold>*) z A"
iso_dghm_is_ff_dghm: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>"
normal: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "c \<Rightarrow> c'" shows "c \<Rightarrow>\<^sub>G c'"
assms: fixes x :: "'a" shows "(x \<in> T) = (x \<in> T') \<or> f x = TTree.empty"
one_final_antimono: fixes g :: "graph"   and E :: "(nat \<times> nat) set" assumes "one_final g" shows "one_final_but g E"
nLt_le_conv: fixes x :: "'b"   and y :: "'b" shows "(c1 x y \<noteq> Lt) = c1.le y x"
prems_2: shows "higher_differentiable_on S g (Suc n)"
coboundedI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> c" shows "inf a b \<le> c"
f_def: shows "f = (\<lambda>n. 1 / real (Suc n))"
dg_is_arrD_2: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
single_less_eq_eval: fixes x :: "'a"   and P :: "'a pred" shows "(single x \<le> P) = eval P x"
eq: shows "subtopology X (topspace X \<inter> S) = subtopology X S"
compress_bottom_def: fixes x :: "wall"   and y :: "wall" shows "compress_bottom x y \<equiv> \<exists>B. x = B \<circ> basic (make_vert_block (nat (codomain_wall B))) \<and> y = basic [] \<circ> B \<and> domain_wall B = 0 \<and> is_tangle_diagram B"
hyps: fixes m :: "'a"   and l' :: "'a list"   and m' :: "'a"   and t' :: "'a list" assumes "(m, l') = foldl f (m', t') xs" shows "mset (m # l') = mset (m' # t' @ xs)"
lconf_def: fixes P :: "'a prog"   and h :: "heap"   and l :: "sfields"   and E :: "char list \<Rightarrow> ty option" shows "P,h \<turnstile> l (:\<le>) E \<equiv> \<forall>V v. l V = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>T. E V = \<lfloor>T\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
ipurge_tr_aux_set: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list" shows "set (ipurge_tr_aux I D U xs) \<subseteq> set xs"
omega_continuous_def: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" shows "omega_continuous A less_eq f \<equiv> f ` A \<subseteq> A \<and> (\<forall>c. \<forall>b\<in>A. range c \<subseteq> A \<longrightarrow> monotone (\<le>) less_eq c \<longrightarrow> extreme_bound A less_eq (range c) b \<longrightarrow> extreme_bound A less_eq (f ` range c) (f b))"
abs_eq: fixes xa :: "complex \<times> complex"   and x :: "complex \<times> complex" assumes "eq_onp (\<lambda>v. v \<noteq> vec_zero) xa xa"   and "eq_onp (\<lambda>v. v \<noteq> vec_zero) x x" shows "(Abs_complex_homo_coords xa \<approx> Abs_complex_homo_coords x) = (xa \<approx>\<^sub>v x)"
spec\<^sub>0_def: fixes p :: "prog" shows "spec\<^sub>0 p \<equiv> wfp p \<and> io_vars p \<and> low_proc_non0 p \<and> low_proc_0 p \<and> high_proc p"
hn_admissible': fixes Ry :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and A :: "('c \<Rightarrow> 'a nres) set"   and x :: "'c"   and P :: "assn"   and c :: "'b Heap"   and F :: "assn" assumes "precise Ry"   and "\<forall>f\<in>A. nofail (f x) \<longrightarrow> <P> c <\<lambda>r. hn_rel Ry (f x) r * F>\<^sub>t"   and "nofail (INF f\<in>A. f x)" shows "<P> c <\<lambda>r. hn_rel Ry (INF f\<in>A. f x) r * F>\<^sub>t"
ereal_divide_ereal: fixes r :: "real" shows "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else - \<infinity>)"
aform_vals_pdevs_vals: fixes e :: "nat \<Rightarrow> real"   and X :: "('a \<times> 'a pdevs) list" shows "aform_vals e X = map2 (+) (map fst X) (pdevs_vals e (map snd X))"
f: fixes lsta :: "'b list" shows "sum_list (map f (filter ((\<noteq>) (0::'b)) lsta)) = sum_list (map f lsta)"
compactI: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.ccpo Sup less_eq less"   and "ccpo.admissible Sup less_eq (\<lambda>y. \<not> less_eq x y)" shows "ccpo.compact Sup less_eq x"
hensel_fpxs_aux_0: shows "hensel_fpxs_aux 0 = (g, h)"
Rep_star_star_n_iff: fixes X :: "nat \<Rightarrow> 'c"   and Y :: "nat \<Rightarrow> 'c" shows "(X \<in> Rep_star (star_n Y)) = (\<forall>\<^sub>F n in \<U>. Y n = X n)"
setdist_subset_right: fixes T :: "'a set"   and u :: "'a set"   and S :: "'a set" assumes "T \<noteq> {}"   and "T \<subseteq> u" shows "setdist S u \<le> setdist S T"
vonStaudt_Clausen: fixes n :: "nat" assumes "0 < n" shows "bernoulli (2 * n) + (\<Sum>p | prime p \<and> p - 1 dvd 2 * n. 1 / real p) \<in> \<int>"
hyps: fixes n :: "nat" shows "map_idx (\<lambda>_. f) xs n = map f [n..<n + length xs]"
map_drop_def: fixes a :: "'a" shows "map_drop a = map_filter (\<lambda>a'. a' \<noteq> a)"
real_code_2: fixes w :: "'b list" shows "(enc w = []) = (w = [])"
assms: shows "x \<in>\<^sub>\<circ> A ^\<^sub>\<times> 2\<^sub>\<nat>"
small_elts_of_set: fixes x :: "V set" assumes "small x" shows "elts (set x) = x"
great_Picard: fixes M :: "complex set"   and z :: "complex"   and a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "open M"   and "z \<in> M"   and "a \<noteq> b"   and "f holomorphic_on M - {z}"   and "\<And>w. w \<in> M - {z} \<Longrightarrow> f w \<noteq> a \<and> f w \<noteq> b" obtains l :: "complex" where "f \<midarrow>z\<rightarrow> l \<or> (inverse \<circ> f) \<midarrow>z\<rightarrow> l"
left_inverse: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "group f z inverse" shows "f (inverse a) a = z"
transpos_noteqTr: fixes n :: "nat"   and k :: "nat"   and j :: "nat" assumes "0 < n"   and "k \<le> n - Suc 0"   and "j \<le> n"   and "j \<noteq> n" shows "j \<noteq> (\<tau>\<^bsub>j n\<^esub>) k"
fun_upd_same: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'f"   and y :: "'e" shows "(f(x := y)) x = y"
cmp_simps_4: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.dom (\<Phi>\<^sub>G (f, g)) = G f \<star>\<^sub>D G g"
st_proj1_inv_continuous: shows "continuous_on UNIV st_proj1_inv"
assms_2: shows "create_optimal_solutions A b c = Inr X"
cInf_insert: fixes X :: "'a multiset set"   and a :: "'a multiset" assumes "X \<noteq> {}"   and "subset_mset.bdd_below X" shows "Inf (insert a X) = a \<inter># Inf X"
word_prev_unfold: fixes w :: "'b word" shows "word_prev w = (if w = 0 then 0 else w - 1)"
sortedP_right_of_append1: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zs :: "'a list"   and z :: "'a"   and y :: "'a" assumes "linorder_list0.sortedP le (zs @ [z])"   and "y \<in> set zs" shows "le y z"
mem_correct_empty_axioms: shows "mem_correct_empty lookup update P empty"
eq_commute: fixes a :: "'d"   and b :: "'d" shows "(a = b) = (b = a)"
pdom_plamda: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" shows "pdom (plambda P f) = {x. P x}"
poly_deg_max_keys: fixes t :: "'b \<Rightarrow>\<^sub>0 'c"   and p :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'd" assumes "t \<in> keys p" shows "deg_pm t \<le> poly_deg p"
finite_range_map_of_map_add: fixes f :: "'b \<Rightarrow> 'a option"   and l :: "('b \<times> 'a) list" assumes "finite (range f)" shows "finite (range (f ++ table_of l))"
null_def: shows "C.null = (THE n. \<forall>f. n \<cdot> f = n \<and> f \<cdot> n = n)"
subst_lsst_unlabel_cons: fixes l :: "'g strand_label"   and b :: "('e, 'f) stateful_strand_step"   and A :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "unlabel ((l, b) # A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>) = (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) # unlabel (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)"
polyadd_eq_const_degree: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat"   and c :: "'b" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "p +\<^sub>p q = (c)\<^sub>p" shows "Polynomial_Expression.degree p = Polynomial_Expression.degree q"
R: shows "\<forall>sl sl'. wffp sl \<and> wffp sl' \<and> hd sl = s0 \<and> hd sl' = s0 \<and> length sl = length sl' \<longrightarrow> (\<forall>i<length sl. f (sl ! i) (sl' ! i)) \<longrightarrow> g (last sl) (last sl')"
that: fixes point :: "'a" assumes "fold (\<lambda>(x, y) option. Option.bind option (f x y)) xs start = Some point"   and "fold (\<lambda>(x, y) option. Option.bind option (f x y)) ys (Some point) = Some res" shows "thesis"
finite_dom_methd: fixes G :: "prog"   and C :: "qtname" assumes "ws_prog G"   and "is_class G C" shows "finite (dom (methd G C))"
ParCR_MtransC: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c2 :: "('a, 'c) com"   and s :: "'b"   and c2' :: "('a, 'c) com"   and s' :: "'b"   and c1 :: "('a, 'c) com" assumes "PL.MtransC tval aval (c2, s) (c2', s')" shows "PL.MtransC tval aval (c1 | c2, s) (c1 | c2', s')"
aforms_err_to_aformsE: fixes x :: "real list"   and e :: "nat \<Rightarrow> real"   and X :: "((real \<times> real pdevs) \<times> real) list"   and n :: "nat" assumes "x \<in> aforms_err e X"   and "degree_aforms_err X \<le> n"   and "e \<in> UNIV \<rightarrow> {- 1..1}" obtains e' :: "nat \<Rightarrow> real" where "x = aform_vals e' (aforms_err_to_aforms n X)" and "e' \<in> UNIV \<rightarrow> {- 1..1}" and "\<And>i. i < n \<Longrightarrow> e' i = e i"
AllowedActs_eq: fixes init :: "'b set"   and acts :: "('b \<times> 'b) set set"   and allowed :: "('b \<times> 'b) set set" shows "AllowedActs (mk_program (init, acts, allowed)) = insert Id allowed"
mset_subset_eq_single: fixes a :: "'b"   and B :: "'b multiset" assumes "a \<in># B" shows "{#a#} \<subseteq># B"
p\<^sub>1_simps: shows "trg f\<^sub>0gh\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)"
sincreasing_def: fixes s :: "'a stream" shows "sincreasing s = (\<forall>x. \<exists>i. x < s !! i)"
s_mono: fixes u :: "real"   and t :: "real" assumes "u \<le> t"   and "0 \<le> u" shows "s u \<le> s t"
find_non0_2: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and l :: "nat"   and A :: "'a mat"   and m :: "nat" assumes "det_selection_fun sel_fun"   and "find_non0 sel_fun l A = Some m" shows "l < m"
set_pmf_of_set: shows "set_pmf local.pmf_of_set = S"
contains_carrier: fixes F :: "'b set set"   and T :: "'b set set" assumes "Topology.filter F T"   and "F \<noteq> {}" shows "carrier\<^bsub>T\<^esub> \<in> F"
cdleswap': shows "\<forall>j'<k'. k' + n' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> j' \<longrightarrow> (\<exists>j<k. k + n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j')"
powr_le_iff: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "1 < b"   and "0 < x" shows "(b powr y \<le> x) = (y \<le> Transcendental.log b x)"
true_cls_mset_union: fixes I :: "'a set"   and CC :: "'a literal multiset multiset"   and DD :: "'a literal multiset multiset" shows "(I \<Turnstile>m CC + DD) = (I \<Turnstile>m CC \<and> I \<Turnstile>m DD)"
finprod_singleton_set: fixes f :: "'f \<Rightarrow> 'c"   and a :: "'f" assumes "f a \<in> carrier M" shows "finsum M f {a} = f a"
simE_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and \<alpha> :: "Late_Semantics.freeRes"   and Q' :: "pi" assumes "P \<leadsto>\<^sub>l[Rel] Q"   and "Q \<longmapsto>\<^sub>l \<alpha> \<prec> Q'" shows "\<exists>P'. P \<longmapsto>\<^sub>l \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b" shows "DenyAll \<noteq> DenyAllFromTo x21 x22"
seminorm_def: fixes V :: "'c set"   and norm :: "'c \<Rightarrow> real" shows "seminorm V norm \<equiv> (\<forall>x. x \<in> V \<longrightarrow> 0 \<le> norm x) \<and> (\<forall>x a. x \<in> V \<longrightarrow> norm (a \<cdot> x) = \<bar>a\<bar> * norm x) \<and> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> norm (x + y) \<le> norm x + norm y)"
get_return_edge_call: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "get_return_edges a \<noteq> {}"
hyps: fixes pw :: "'a stream" assumes "pw \<in> streams A" shows "set (stake n pw) \<subseteq> A"
that: fixes p :: "'a list"   and y :: "'a" assumes "path \<Gamma> x p y"   and "y \<in> B \<Gamma>"   and "\<And>z. \<lbrakk>x \<noteq> y; z \<in> set p\<rbrakk> \<Longrightarrow> z = x \<or> z \<notin> S" shows "thesis"
over_tagged_division_lemma: fixes p :: "('c \<times> 'c set) set"   and i :: "'c set"   and d :: "'c set \<Rightarrow> 'a" assumes "p tagged_division_of i"   and "\<And>u v. box u v = {} \<Longrightarrow> d (cbox u v) = \<^bold>1" shows "F (\<lambda>(uu_, k). d k) p = F d (snd ` p)"
rStatEq: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "(\<Psi>', R, S) \<in> X" shows "insertAssertion (extractFrame R) \<Psi>' \<simeq>\<^sub>F insertAssertion (extractFrame S) \<Psi>'"
starP_def: fixes P :: "'b \<Rightarrow> bool" shows "*p* P = (\<lambda>x. unstar (star_of P \<star> x))"
C: shows "random_variable (subprob_algebra N) (\<lambda>(x, y). C x y)"
compute_float_uminus: fixes m1 :: "int"   and e1 :: "int" shows "- Float m1 e1 = Float (- m1) e1"
anycast_simps_1: fixes P :: "'a \<Rightarrow> bool"   and m :: "'a" shows "anycast P (broadcast m) = P m"