hcju/isabelleps huggingface dataset:
mult_nonneg_nonpos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "Rings.class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq b zero" shows "less_eq (times a b) zero"
S3: shows "c_fst (c_pair 4 (c_pair n1 n2)) = 4"
sup_least_type: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<le> \<rho>"   and "\<sigma> \<le> \<rho>" shows "\<tau> \<squnion> \<sigma> \<le> \<rho>"
semiring_norm_4: fixes n :: "num" shows "num.One + num.Bit1 n = num.Bit0 (n + num.One)"
a_z_z: fixes m :: "int"   and n :: "int" shows "ant m * ant n = ant (m * n)"
comp_right_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "times x bot = bot"
prems_1: shows "x \<in> set (remdups_wrt f (a # xs))"
padic_add_assoc: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "prime p"   and "x \<in> carrier (padic_int p)"   and "y \<in> carrier (padic_int p)"   and "z \<in> carrier (padic_int p)" shows "x \<oplus>\<^bsub>padic_int p\<^esub> y \<oplus>\<^bsub>padic_int p\<^esub> z = x \<oplus>\<^bsub>padic_int p\<^esub> (y \<oplus>\<^bsub>padic_int p\<^esub> z)"
Rep_JF2_inject: fixes x :: "'d JF2"   and y :: "'d JF2" shows "(Rep_JF2 x = Rep_JF2 y) = (x = y)"
list_update_incr: fixes x :: "'a"   and xs :: "'a list"   and i :: "nat" assumes "x \<in> A"   and "set xs \<subseteq> A"   and "i < length xs" shows "xs [\<sqsubseteq>\<^bsub>r\<^esub>] xs[i := x \<squnion>\<^bsub>f\<^esub> xs ! i]"
starfun_lambda_cancel: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and x' :: "'b star" shows "(*f* (\<lambda>h. f (x + h))) x' = (*f* f) (star_of x + x')"
semiring_normalization_rules_25: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "plus a (plus c d) = plus (plus a c) d"
NextRecvCapD_2: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c0 :: "('d, 'c, 'a) Combined.configuration"   and c1 :: "('d, 'c, 'a) Combined.configuration"   and p :: "'d"   and t :: "'a \<times> 'c" assumes "dataflow_topology summary results_in"   and "dataflow_topology.NextRecvCap' c0 c1 p t" shows "prop_config c1 = prop_config c0"
hyps_1: assumes "consistent t\<^sub>1"   and "sibling t\<^sub>1 b \<noteq> b"   and "a \<noteq> b" shows "sibling (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>s (sibling t\<^sub>1 b)) a = b"
hcomp_simps_3: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.hseq \<nu> \<mu>" shows "D.dom (\<nu> \<star>\<^sub>D \<mu>) = D.dom \<nu> \<star>\<^sub>D D.dom \<mu>"
IS_BELOW_ID_def: fixes R :: "('c \<times> 'c) set" shows "IS_BELOW_ID R \<equiv> R \<subseteq> Id"
sub_inc_sysI: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and \<U> :: "'b set"   and \<A> :: "'b set multiset" assumes "incidence_system point_set block_collection"   and "incidence_system \<U> \<A>"   and "\<U> \<subseteq> point_set"   and "\<A> \<subseteq># block_collection" shows "sub_incidence_system \<U> \<A> point_set block_collection"
root_unity_def: fixes n :: "nat" shows "root_unity n = monom (1::'a) n - 1"
c_simps: fixes x :: "nat"   and y :: "nat" shows "c (x, y) = (if x = s \<and> y \<le> n then f y else if x \<le> n \<and> n < y \<and> y \<le> 2 * n + 1 \<and> (x, y - n - 1) \<in> R' then LARGE else if n < x \<and> x \<le> 2 * n + 1 \<and> y = t then g (x - n - 1) else 0)"
disconnected_nodes_h2: shows "h2 \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disconnected_nodes_h2"
fresh_transform: fixes a :: "atom"   and e :: "exp"   and n :: "'a" assumes "a \<sharp> e" shows "a \<sharp> trans n e"
add_left_mono_trans: fixes x :: "'h multiset"   and a :: "'h multiset"   and b :: "'h multiset"   and c :: "'h multiset" assumes "x \<subseteq># a + b"   and "b \<subseteq># c" shows "x \<subseteq># a + c"
upred_ref_iff: fixes Q :: "'a upred"   and P :: "'a upred" shows "(P \<sqsubseteq> Q) = (\<forall>b. \<lbrakk>Q\<rbrakk>\<^sub>e b \<longrightarrow> \<lbrakk>P\<rbrakk>\<^sub>e b)"
in_rel: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "('d, 'f, 'h) F1"   and b :: "('e, 'g, 'i) F1" shows "F1rel R1 R2 R3 a b = (\<exists>z. z \<in> F1in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F1map fst fst fst z = a \<and> F1map snd snd snd z = b)"
wfP_trancl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "wfP r" shows "wfP r\<^sup>+\<^sup>+"
assms_1: shows "A \<approx> [a \<leftrightarrow> b] \<bullet> B"
rtrancl_edges_iff_path: fixes u :: "'b"   and v :: "'b"   and g :: "'b ugraph" shows "((u, v) \<in> (edges g)\<^sup>*) = (\<exists>p. path g u p v)"
lt_of_comp_less_prod: fixes c_a :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and c_b :: "'d \<Rightarrow> 'd \<Rightarrow> order" shows "lt_of_comp (comparator_prod c_a c_b) = less_prod (le_of_comp c_a) (lt_of_comp c_a) (lt_of_comp c_b)"
codomains_def: fixes f :: "'a arrow_of_spans_data" shows "V.codomains f \<equiv> {b. V.ide b \<and> b \<bullet> f \<noteq> V.null}"
nterm_to_term_bounds_dupl: fixes i :: "nat"   and \<Gamma> :: "name list"   and j :: "nat"   and t :: "nterm" assumes "i < length \<Gamma>"   and "j < length \<Gamma>"   and "i < j"   and "\<Gamma> ! i = \<Gamma> ! j" shows "j |\<notin>| bounds (nterm_to_term \<Gamma> t)"
induct: fixes x1 :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and x2 :: "'d"   and x3 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and x4 :: "nat \<Rightarrow> nat"   and x5 :: "'e \<Rightarrow> nat"   and x6 :: "nat"   and x7 :: "'d"   and x8 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and P :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint) \<Rightarrow> 'd \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> real DBMEntry) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> ('e \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> real DBMEntry) \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<leadsto>\<^bsub>x4,x5,x6\<^esub>* \<langle>x7, x8\<rangle>"   and "\<And>A l Z k v n. P A l Z k v n l Z"   and "\<And>A l Z k v n l' Z' l'' Z''. \<lbrakk>A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^bsub>k,v,n\<^esub>* \<langle>l', Z'\<rangle>; P A l Z k v n l' Z'; A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^bsub>k,v,n\<^esub> \<langle>l'', Z''\<rangle>\<rbrakk> \<Longrightarrow> P A l Z k v n l'' Z''" shows "P x1 x2 x3 x4 x5 x6 x7 x8"
r2f_pow: fixes i :: "nat"   and R :: "('a \<times> 'a) set" shows "\<F> (R ^^ i) = kpower (\<F> R) i"
mod_mult_div_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (modulo a b) (times b (divide a b)) = a"
invertible_left_inverse2: fixes u :: "'a"   and v :: "'a" assumes "additive.invertible u"   and "u \<in> R"   and "v \<in> R" shows "- u + (u + v) = v"
fin_def: fixes A :: "'c \<times> 'd \<times> 'b" shows "fin A = snd (snd A)"
list_slice_nth_eq_sublist_list: fixes m :: "nat"   and xs :: "'b list"   and k :: "nat" assumes "m < length xs div k" shows "list_slice xs k ! m = sublist_list xs [m * k..<m * k + k]"
asym: fixes A :: "'b set"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "asymmetric A less"   and "less x y"   and "less y x"   and "x \<in> A"   and "y \<in> A" shows "False"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (add_instr instr s1)) \<and> t2 = snd (fst (add_instr instr s2))"
find_csa_at_index: fixes i :: "nat"   and fss :: "real poly list" assumes "i < length fss"   and "\<And>p q. \<lbrakk>p \<in> set fss; q \<in> set fss; p \<noteq> q\<rbrakk> \<Longrightarrow> comm_monoid_mult_class.coprime p q"   and "\<And>p. p \<in> set fss \<Longrightarrow> p \<noteq> 0"   and "distinct fss" shows "set (map (insertAt i 0) (find_consistent_signs_at_roots (fss ! i) (removeAt i fss))) = {v \<in> consistent_sign_vectors_r fss UNIV. v ! i = 0}"
Step1_5_2b: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> $ImpInv rmhist p \<and> (\<forall>l. $MemInv mm l)) \<longrightarrow> \<box>[UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)"
distinct_map_filter: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "distinct (list.map f xs)" shows "distinct (list.map f (List.filter P xs))"
inner_verts_def: fixes G :: "('b, 'c) pre_digraph"   and p :: "'c list" shows "pre_digraph.inner_verts G p \<equiv> tl (map (tail G) p)"
bitxor_cancel: fixes b :: "bit" shows "(b \<oplus>\<^sub>b b) = \<zero>"
card_Pow: fixes A :: "'c set" assumes "finite A" shows "card (Pow A) = 2 ^ card A"
locally_path_connected_space_open_subset: fixes X :: "'a topology"   and s :: "'a set" assumes "locally_path_connected_space X"   and "openin X s" shows "locally_path_connected_space (subtopology X s)"
setmonotone_implies_chain_funpower: fixes f :: "'a set \<Rightarrow> 'a set"   and I :: "'a set" assumes "setmonotone f" shows "chain (\<lambda>n. funpower f n I)"
inheritedBounded: fixes subs :: "'a \<Rightarrow> 'a set" shows "inherited subs bounded"
lucas_lehmer_to_add_pow_nat: fixes n :: "nat"   and x :: "int \<times> int" shows "lucas_lehmer_to_real ([n] \<cdot>\<^bsub>lucas_lehmer_ring\<^esub> x) = real n * lucas_lehmer_to_real x"
N: fixes n :: "nat" assumes "N \<le> n" shows "f n \<noteq> (0::'a)"
elims_2: fixes x :: "'a tree" assumes "braun x" obtains "x = \<langle>\<rangle>"   | l :: "'a tree" and xa :: "'a" and r :: "'a tree" where "x = \<langle>l, xa, r\<rangle>" and "(size l = size r \<or> size l = size r + 1) \<and> braun l \<and> braun r"
selsort_eq_empty_iff: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list" shows "(ccw.selsort x xs = []) = (xs = [])"
supp_sum_divide_distrib: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and r :: "'c" shows "supp_sum f A / r = supp_sum (\<lambda>n. f n / r) A"
fin: shows "finite (poly_deg ` fst ` set (shift_list (h, U) x qs))"
r_cancel_one': fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier M"   and "a \<in> carrier M" shows "(x = a \<oplus>\<^bsub>M\<^esub> x) = (a = \<zero>\<^bsub>M\<^esub>)"
closest_pair_rec_set_length_sorted_snd: fixes ys :: "(int \<times> int) list"   and p :: "(int \<times> int) \<times> int \<times> int"   and xs :: "(int \<times> int) list" assumes "(ys, p) = closest_pair_rec xs" shows "set ys = set xs \<and> length ys = length xs \<and> sorted_snd ys"
wt'_iff_wt: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T :: "ty" shows "(P,E,h \<turnstile> e :' T) = (P,E,h \<turnstile> e : T)"
prems_2: shows "\<forall>jj<length (xs @ [x]). ys ! (i + jj) = (xs @ [x]) ! jj"
u: shows "u = fst (snd (snd (bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k))))"
U_c: shows "U \<cdot> 1\<^sub>\<pi> = 1\<^sub>\<pi>"
cut_thin: fixes HA :: "fm set"   and A :: "fm"   and HB :: "fm set"   and B :: "fm"   and H :: "fm set" assumes "HA \<turnstile> A"   and "insert A HB \<turnstile> B"   and "HA \<union> HB \<subseteq> H" shows "H \<turnstile> B"
assms_2: fixes k :: "nat" assumes "k < n" shows "proj_poly k p = proj_poly k q"
cfuncset_def: fixes l :: "'e"   and u :: "'e"   and X :: "'f set" shows "Abstract_Rigorous_Numerics.cfuncset l u X = ({l..u} \<rightarrow> X) \<inter> Collect (continuous_on {l..u})"
mult_eSuc_right: fixes m :: "enat"   and n :: "enat" shows "m * eSuc n = m + m * n"
subset_inj_on: fixes f :: "'e \<Rightarrow> 'f"   and B :: "'e set"   and A :: "'e set" assumes "inj_on f B"   and "A \<subseteq> B" shows "inj_on f A"
div_minus_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "divide a (uminus b) = divide (uminus a) b"
map_leI: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "\<And>k v. f k = \<lfloor>v\<rfloor> \<Longrightarrow> g k = \<lfloor>v\<rfloor>" shows "f \<subseteq>\<^sub>m g"
\<Gamma>_\<delta>x: shows "\<Gamma> (\<delta> x) = Fun f T"
prefs_from_table_undef: fixes agents :: "'c set"   and alts :: "'d set"   and xss :: "('c \<times> 'd set list) list"   and i :: "'c" assumes "prefs_from_table_wf agents alts xss"   and "i \<notin> agents" shows "prefs_from_table xss i = (\<lambda>_ _. False)"
AbstrLevels_S10_A71: fixes i :: "AbstrLevelsID" assumes "sS10 \<in> AbstrLevel i" shows "sA71 \<notin> AbstrLevel i"
sublens'_def: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" shows "X \<subseteq>\<^sub>L' Y = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>3 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> (s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>3))"
long_ch_Y: shows "[f[a\<^sub>0 .. a ..  a\<^sub>n]Y]"
gval_reg_some_superset: fixes r :: "nat \<Rightarrow>f value option"   and r' :: "nat \<Rightarrow>f value option"   and x :: "trilean"   and a :: "vname gexp"   and i :: "value list" assumes "\<forall>a. r $ a \<noteq> None \<longrightarrow> r $ a = r' $ a"   and "x \<noteq> invalid"   and "gval a (join_ir i r) = x" shows "gval a (join_ir i r') = x"
subsequence_singleton: fixes w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "w [i \<rightarrow> Suc i] = [w i]"
d_indistinguishable_sym: fixes d :: "'d"   and e :: "'exp"   and e' :: "'exp" assumes "e \<equiv>\<^bsub>d\<^esub> e'" shows "e' \<equiv>\<^bsub>d\<^esub> e"
assms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (f \<phi>) \<subseteq> atoms_ltln \<phi>"
path_edge: fixes prog :: "cmd"   and a :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node" assumes "valid_edge prog a" shows "While_CFG.path prog (sourcenode a) [a] (targetnode a)"
pair_loopfree_digraph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs (subdivide G e w)"   and "w \<notin> pverts (subdivide G e w)" shows "pair_loopfree_digraph (subdivide (subdivide G e w) e w)"
pred_def: fixes P :: "bexp\<^sub>C set" shows "C.pred P \<equiv> \<lambda>mem. \<forall>p\<in>P. ev\<^sub>B\<^sub>C mem p"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (local.dom \<mu>) \<mu>.r.tab (tab\<^sub>0 (local.dom \<mu>)) (tab\<^sub>1 (local.dom \<mu>)) (local.dom \<tau>) \<tau>.r.tab (tab\<^sub>0 (local.dom \<tau>)) (tab\<^sub>1 (local.dom \<tau>)) \<mu>"
cauchy_root_bound: fixes p :: "'a poly"   and x :: "'a" assumes "poly p x = (0::'a)"   and "p \<noteq> 0" shows "norm x \<le> 1 + max_list_non_empty (map (\<lambda>i. norm (coeff p i)) [0..<degree p]) / norm (lead_coeff p)"
Compl_iff: fixes c :: "'b"   and A :: "'b set" shows "(c \<in> - A) = (c \<notin> A)"
cs_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" shows "relation.cs r a b \<equiv> \<exists>s. r\<^sup>*\<^sup>* a s \<and> r\<^sup>*\<^sup>* b s"
C: shows "Crypt (sesK (c * f)) \<lbrace>pubAK (c * (s + a * b)), Auth_Data g' b, pubAK g', Crypt (priSK CA) (Hash (pubAK g'))\<rbrace> \<in> synth (analz (A \<union> spies evsFR4))"
map_filter_map_filter: fixes f :: "'e \<Rightarrow> 'd"   and P :: "'e \<Rightarrow> bool"   and xs :: "'e list" shows "map f (filter P xs) = List.map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
ideal_pOp_closed: fixes I :: "'a set"   and x :: "'a"   and y :: "'a" assumes "ideal R I"   and "x \<in> I"   and "y \<in> I" shows "x \<plusminus> y \<in> I"
eval_subst_2: fixes \<phi> :: "'a ltln"   and m :: "'a ltln \<Rightarrow> 'a ltln option" assumes "eval \<phi> = No" shows "eval (subst \<phi> m) = No"
tendsto_inverse: fixes f :: "'a \<Rightarrow> 'c"   and a :: "'c"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F"   and "a \<noteq> (0::'c)" shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
adjoint_pair_antipar_1: fixes f :: "'b"   and g :: "'b" assumes "D.adjoint_pair f g" shows "D.ide f"
tuple_in'_def: shows "tuple_in' \<equiv> fold (\<lambda>(t, X) tuple_in. Mapping.filter (filter_cond X tuple_in t) tuple_in) discard tuple_in"
lnorm_minus_commute: fixes M :: "'a measure"   and p :: "real"   and g :: "'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> real" shows "lnorm M p (g - f) = lnorm M p (f - g)"
c_failures_ref: fixes xps :: "('e \<times> 'f) list"   and X :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" assumes "(xps, X) \<in> c_failures step out s\<^sub>0" shows "X \<subseteq> {(x, p). p \<noteq> out (foldl step s\<^sub>0 (map fst xps)) x}"
edges_join: fixes g\<^sub>1 :: "'b ugraph"   and g\<^sub>2 :: "'b ugraph" shows "edges (graph_join g\<^sub>1 g\<^sub>2) = edges g\<^sub>1 \<union> edges g\<^sub>2"
transp_rel_fun: fixes Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "is_equality Q"   and "transp R" shows "transp (rel_fun Q R)"
B_nth_prime: shows "\<forall>\<^sub>F n in sequentially. B < int (nth_prime n)"
sifum_security_axioms: shows "sifum_security stop eval some_val some_val'"
roots_of_poly_f_def: fixes qs :: "real poly list" shows "roots_of_poly_f qs = {x. poly (poly_f qs) x = 0}"
inj_idom_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "inj_idom_hom hom \<equiv> idom_hom hom \<and> zero_hom_0 hom"
consis: shows "history_consistent \<theta> (hd_prog p ((r # sb\<^sub>1) @ sb')) ((r # sb\<^sub>1) @ sb')"
Approximation1_5: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "vals_nonequiv K (Suc n) vv"   and "\<forall>j\<le>Suc n. n_val K (vv j) = vv j" shows "\<exists>x. (\<forall>j\<le>Suc n. x j \<in> carrier K) \<and> (\<forall>i\<le>Suc n. \<forall>j\<le>Suc n. vv i (x j) = \<delta>\<^bsub>i j\<^esub>)"
simps_1: fixes uu :: "bool \<times> (('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e \<Rightarrow> bool \<times> (('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e \<Rightarrow> bool"   and uv :: "bool"   and uw :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "add_pairs_single_sorted uu uv [] [] uw ps = ps"
set_intros_1: fixes x21 :: "'a"   and x22 :: "'a list" shows "x21 \<in> list.set (x21 # x22)"
w_def: shows "w = (\<lambda>n. u n + v n)"
arg_mult_of_real_left: fixes r :: "real"   and z :: "complex" assumes "0 < r" shows "arg (complex_of_real r * z) = arg z"
RPHIL_def: fixes i :: "nat" shows "RPHIL i = (\<mu> x. picks i i \<rightarrow> picks i (i - 1) \<rightarrow> putsdown i (i - 1) \<rightarrow> putsdown i i \<rightarrow> x)"
rep_eq: fixes x :: "('c, 'b) ffun"   and xa :: "('a, 'c) ffun" shows "pfun_of (x \<circ>\<^sub>f xa) = pfun_of x \<circ>\<^sub>p pfun_of xa"
case_prod_beta: fixes f :: "'b \<Rightarrow> 'd \<Rightarrow> 'a"   and p :: "'b \<times> 'd" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
orient_lit_def: fixes L :: "'a literal"   and u :: "'a trm"   and v :: "'a trm"   and s :: "sign" shows "orient_lit L u v s = ((L = Pos (Eq u v) \<or> L = Pos (Eq v u)) \<and> (u, v) \<notin> trm_ord \<and> s = pos \<or> (L = Neg (Eq u v) \<or> L = Neg (Eq v u)) \<and> (u, v) \<notin> trm_ord \<and> s = neg)"
mdl_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set" shows "mdl M H = \<lparr>carrier = H, pop = (\<plusminus>\<^bsub>M\<^esub>), mop = mop M, zero = \<zero>\<^bsub>M\<^esub>, sprod = \<lambda>a. restrict ((\<cdot>\<^sub>s\<^bsub>M\<^esub>) a) H\<rparr>"
higher_deriv_cong_ev: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "\<forall>\<^sub>F x in nhds x. f x = g x"   and "x = y" shows "(deriv ^^ n) f x = (deriv ^^ n) g y"
same_diag_ev_block: fixes A :: "'a mat"   and B :: "'a mat" assumes "local.same_diag A B"   and "local.ev_block n A" shows "local.ev_block n B"
group_isomorphisms_imp_iso: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'c" assumes "group_isomorphisms G H f g" shows "f \<in> Group.iso G H"
null_eqI: fixes n :: "'a" assumes "\<And>f. n \<cdot>\<^sub>B f = n \<and> f \<cdot>\<^sub>B n = n" shows "n = B.null"
c_tr_ipurge_tr: fixes ys :: "'e list"   and I :: "('g \<times> 'g) set"   and D :: "'e \<Rightarrow> 'g"   and u :: "'g"   and step :: "'h \<Rightarrow> 'e \<Rightarrow> 'h"   and out :: "'h \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'h"   and xs :: "'e list" assumes "\<And>n. n \<in> {..length ys} \<Longrightarrow> \<exists>Y. (ipurge_tr I (c_dom D) u (c_tr step out (foldl step s\<^sub>0 xs) (take n ys)), Y) \<in> futures (c_process step out s\<^sub>0) (c_tr step out s\<^sub>0 xs)" shows "ipurge_tr I (c_dom D) u (c_tr step out (foldl step s\<^sub>0 xs) ys) = c_tr step out (foldl step s\<^sub>0 xs) (ipurge_tr I D u ys)"
restrict_eq_lan_subs: fixes v :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "|restrict v f c| = |lan v| \<and> restrict v f c \<le> lan v \<longrightarrow> restrict v f c = lan v"
t_stop_nonneg: shows "0 \<le> ego.t_stop"
less_top_ennreal: fixes x :: "ennreal" shows "(x < \<top>) = (\<exists>r\<ge>0. x = ennreal r)"
ffUnion_alt_def: fixes x :: "'c"   and A :: "'c fset fset" shows "(x |\<in>| ffUnion A) = fBex A ((|\<in>|) x)"
assoc: fixes a :: "'d upred"   and b :: "'d upred"   and c :: "'d upred" shows "((a \<and> b) \<and> c) = (a \<and> b \<and> c)"
linear: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less_eq y x"
assms: shows "spr_jview a t = spr_jview a t'"