hcju/isabelleps huggingface dataset:
simps_2: fixes x :: "'a"   and n :: "'a"   and l :: "'a tree\<^sub>0"   and r :: "'a tree\<^sub>0" shows "insrt\<^sub>0 x (MKT\<^sub>0 n l r) = (if x = n then MKT\<^sub>0 n l r else if x < n then let l' = insrt\<^sub>0 x l in if height l' = 2 + height r then l_bal\<^sub>0 n l' r else MKT\<^sub>0 n l' r else let r' = insrt\<^sub>0 x r in if height r' = 2 + height l then r_bal\<^sub>0 n l r' else MKT\<^sub>0 n l r')"
simps_4: fixes a :: "'b"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<^sub>F\<^sub>G (np(a)) w = {}"
kcon: shows "continuous_on (sphere z 1 \<inter> U) k"
ncoplanar_perm_16: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet C D A B"
term_brackets_Cons_SomeE: fixes term_bracket :: "nat \<Rightarrow> dB \<Rightarrow> dB option"   and v :: "nat"   and vs :: "nat list"   and t :: "dB"   and t' :: "dB" assumes "bracket_abstraction term_bracket"   and "bracket_abstraction.term_brackets term_bracket (v # vs) t = Some t'" obtains s' :: "dB" where "bracket_abstraction.term_brackets term_bracket vs t = Some s'" and "term_bracket v s' = Some t'"
alw_False: fixes \<omega> :: "'a stream" shows "alw (\<lambda>x. False) \<omega> = False"
add_left_comm: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "y + (x + z) = x + (y + z)"
prems_4: shows "set ps \<inter>\<^sub>p D = {}"
pos_mat_lim_is_pos: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and m :: "nat" assumes "limit_mat X A m"   and "\<forall>n. positive (X n)" shows "positive A"
ty_eq_def: fixes A :: "ty"   and B :: "ty" shows "A \<approx> B \<equiv> A <: B \<and> B <: A"
composition_3: shows "\<C> \<turnstile> [e] : t2s _> t1s @ ts"
aforms_err_aform_valsI: fixes vs :: "real list"   and e :: "nat \<Rightarrow> real"   and XS :: "(real \<times> real pdevs) list" assumes "vs = aform_vals e XS" shows "vs \<in> aforms_err e (map (\<lambda>x. (x, 0)) XS)"
absorb_iff1: fixes b :: "'a Word.word"   and a :: "'a Word.word" shows "(b \<le>s a) = (signed.max a b = a)"
applyDecide_def: fixes state :: "State"   and Vbl :: "nat set" shows "applyDecide state Vbl = assertLiteral (selectLiteral state Vbl) True state"
9413: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> y) \<sqinter> y = - x \<sqinter> y"
g': shows "\<forall>\<^sub>F x in at_top. g' x \<noteq> 0"
axioms_1: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah" assumes "functor A B F" shows "category A"
flatten_prod: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b"   and I :: "'c set" shows "flatten (prod f I) = (\<Prod>i\<in>I. flatten (f i))"
cancel_separate_2: fixes x :: "'b"   and y :: "'b" assumes "bounded_distrib_lattice_class.comp_inf.coreflexive (x \<sqinter> y)" shows "top = bounded_distrib_lattice_class.comp_inf.tc top"
assms_4: shows "zs \<noteq> ys \<or> z \<noteq> y"
alw_mono: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream"   and \<psi> :: "'b stream \<Rightarrow> bool" assumes "alw \<phi> xs"   and "\<And>xs. \<phi> xs \<Longrightarrow> \<psi> xs" shows "alw \<psi> xs"
simps: fixes a1 :: "'a sequent list \<times> 'a sequent"   and a2 :: "'a form" shows "leftPrincipal a1 a2 = (\<exists>C F Fs Ps. a1 = (Ps, C) \<and> a2 = Compound F Fs \<and> C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* \<Empt>))"
esssup_cmult: fixes c :: "real"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ereal" assumes "0 < c" shows "esssup M (\<lambda>x. ereal c * f x) = ereal c * esssup M f"
simps: fixes E :: "'b set set"   and v :: "'b" assumes "Graph_E E" shows "Graph_E.degree_vertex E v = card (Graph_E.neighbors E v)"
to_rat_complex_def: fixes x :: "complex" shows "to_rat x = (if is_rat (Re x) \<and> Im x = 0 then to_rat (Re x) else 0)"
convex_hull_eq: fixes s :: "'b set" shows "(convex hull s = s) = convex s"
csorted_list_of_set_split: fixes P :: "'b list \<Rightarrow> bool"   and A :: "'b set" shows "P (csorted_list_of_set A) = ((\<forall>xs. ID ccompare \<noteq> None \<longrightarrow> finite A \<longrightarrow> A = set xs \<longrightarrow> distinct xs \<longrightarrow> linorder.sorted cless_eq xs \<longrightarrow> P xs) \<and> (ID ccompare = None \<or> infinite A \<longrightarrow> P undefined))"
conv_abscissa_def: fixes f :: "'c fds" shows "conv_abscissa f = (INF s\<in>{s. fds_converges f s}. ereal (s \<bullet> (1::'c)))"
signed_take_bit_eq_take_bit_shift: fixes n :: "nat"   and k :: "int" shows "signed_take_bit n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n"
Max_mono2: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "mono f"   and "A \<noteq> {}"   and "finite A" shows "Max (f ` A) = f (Max A)"
intro: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state" assumes "EqForLVar eq_idx_for_lvar"   and "PivotAndUpdate_axioms eq_idx_for_lvar pivot_and_update" shows "PivotAndUpdate eq_idx_for_lvar pivot_and_update"
sums_iff_shift': fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "'a" shows "(\<lambda>i. f (i + n)) sums (s - sum f {..<n}) = f sums s"
simps_4: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and L :: "rrule"   and j :: "nat" shows "step_result (SG, C) (i, Rrule L j) = (close (SG @ Rrule_result L j (SG ! i)) (SG ! i), C)"
algebra_generator: shows "algebra (space (Pi\<^sub>M I (\<lambda>_. borel))) generator"
zero_order_1: fixes x :: "'k multiset" shows "{#} \<subseteq># x"
equivalence_in_bicategory_axioms: shows "internal_equivalence e d \<eta> \<epsilon>"
approx_from_above_dense_linorder: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "\<exists>u. (\<forall>n. x < u n) \<and> u \<longlonglongrightarrow> x"
that: fixes B' :: "'a set" assumes "B' \<in> B"   and "x \<in> B'"   and "B' \<subseteq> O'" shows "thesis"
Gromov_product_at_diff: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" shows "\<bar>Gromov_product_at x y z - Gromov_product_at a b c\<bar> \<le> dist x a + dist y b + dist z c"
iso_group_mul_alt: shows "((\<lambda>(x, y). x \<otimes> y) \<in> Group.iso (subgroup_generated G A \<times>\<times> subgroup_generated G B) G) = (A \<inter> B = {\<one>} \<and> A <#> B = carrier G \<and> (\<forall>x\<in>A. \<forall>y\<in>B. x \<otimes> y = y \<otimes> x))"
cong_mult: 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 division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b"   and d :: "'b"   and e :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo d e a" shows "unique_euclidean_semiring.cong modulo (times b d) (times c e) a"
monoid_axioms: shows "Group.monoid (ring_mat TYPE('a) n ())"
word_neg_numeral_alt: fixes b :: "num" shows "- numeral b = word_of_int (- numeral b)"
spmf_map: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf"   and x :: "'c" shows "spmf (map_spmf f p) x = Sigma_Algebra.measure (measure_spmf p) (f -` {x})"
H2: fixes x11 :: "nat"   and x12 :: "nat"   and r :: "int mpoly" assumes "x11 \<in># dom_m A"   and "x12 \<in># dom_m A"   and "the (fmlookup A x11) + the (fmlookup A x12) - r \<in> More_Modules.ideal polynomial_bool"   and "r - spec \<in> More_Modules.ideal polynomial_bool" shows "spec \<in> pac_ideal (set_mset B)"
distinct_5: shows "AsmNoReadOrWrite \<noteq> GuarNoWrite"
fn_not_pair: fixes x :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b trm"   and C :: "'b trm"   and D :: "'b trm" shows "Fn x T A \<noteq> SimplyTyped.Pair C D"
assms_1: shows "Undirected_Graph.connected g\<^sub>1"
basis_subset: shows "basis \<subseteq> U\<^sub>V\<^sub>S"
vrs_of_subst: fixes B :: "binding"   and X :: "tyvrs"   and T :: "ty" shows "vrs_of (B[X \<mapsto> T]\<^sub>b) = vrs_of B"
set_iterator_map_rev_linord_def: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'd \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "('a \<times> 'd) set" shows "set_iterator_map_rev_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). k' \<le> k)"
j_is_y_axis: fixes i :: "real \<times> real"   and j :: "real \<times> real" assumes "R2 i j" shows "j = (0, 1)"
lem2: fixes G :: "msg set"   and H :: "msg set" shows "{a. a \<in> analz (G \<union> H) \<or> a \<in> synth G} = analz (G \<union> H) \<union> synth G"
xss_ne: fixes x :: "'a list" assumes "x \<in> set xss" shows "x \<noteq> []"
sum2: shows "(\<lambda>(p, k). - ((fds_nth f p / nat_power p s) ^ Suc k * of_real (ln (real p)))) abs_summable_on {p. prime p} \<times> UNIV"
sat_preserving_inference_system_def: fixes \<Gamma> :: "'b inference set" shows "sat_preserving_inference_system \<Gamma> \<equiv> \<forall>N. satisfiable N \<longrightarrow> satisfiable (N \<union> concls_of (inference_system.inferences_from \<Gamma> N))"
set_reset_trap_def: fixes b :: "bool" shows "set_reset_trap b \<equiv> do modify (reset_trap_mod b); return () od"
Imagep_def: fixes r :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d set" shows "r ``` A \<equiv> {b. \<exists>a\<in>A. r a b}"
threshold: shows "\<bar>valof a + valof b\<bar> < threshold TYPE(('e, 'f) IEEE.float)"
size_13: fixes x51 :: "expr"   and x52 :: "stmt"   and x53 :: "stmt" shows "size (Cond x51 x52 x53) = size x52 + size x53 + Suc 0"
unsandwich_left_inverse: fixes A :: "'o"   and a :: "'a" assumes "A \<in> Ob"   and "a \<in> F \<^bsub>\<o>\<^esub> A" shows "\<sigma>\<^sup>\<leftarrow>(A,\<sigma>(A,a)) = a"
simps: fixes n :: "int" shows "get_arctic_num (Num_arc n) = n"
product_concat_map: fixes xs :: "'b list"   and ys :: "'c list" shows "List.product xs ys = concat (map (\<lambda>x. map (Pair x) ys) xs)"
dvd_0_left: fixes times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times zero a" shows "a = zero"
vector_def: fixes l :: "'c list" shows "vector l = (\<chi>i. foldr (\<lambda>x f n. (f (n + (1::'d)))(n := x)) l (\<lambda>n x. 0::'c) (1::'d) i)"
del_list_split_right: fixes ts :: "('a btree \<times> 'a) list"   and x :: "'a"   and ls :: "('a btree \<times> 'a) list"   and sub :: "'a btree"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list"   and t :: "'a btree" assumes "split ts x = (ls, (sub, sep) # rs)"   and "sorted_less (inorder (Node ts t))"   and "sep \<noteq> x" shows "del_list x (inorder_list ((sub, sep) # rs) @ inorder t) = del_list x (inorder sub) @ sep # inorder_list rs @ inorder t"
unl_lift: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'c \<Rightarrow> 'b"   and w :: "'c" shows "(f<x>) w \<equiv> f (x w)"
bind_eq_Some_conv: fixes f :: "'b option"   and g :: "'b \<Rightarrow> 'a option"   and x :: "'a" shows "(f \<bind> g = \<lfloor>x\<rfloor>) = (\<exists>y. f = \<lfloor>y\<rfloor> \<and> g y = \<lfloor>x\<rfloor>)"
split_list: fixes x :: "'e"   and xs :: "'e list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs"
trans: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "(a, b) \<in> rel D"   and "(b, c) \<in> rel D" shows "(a, c) \<in> rel D"
that: fixes a :: "'c" assumes "a \<in> as"   and "f a = (b, c)" shows "thesis"
h_def: shows "h = (\<lambda>t. f t / g t)"
Det_commute: fixes P :: "'a process"   and Q :: "'a process" shows "(P [+] Q) = (Q [+] P)"
prems_2: shows "number_partition (card A) (add_mset x N)"
add_closed: fixes G :: "'a set"   and g :: "'a"   and h :: "'a" assumes "Group G"   and "g \<in> G"   and "h \<in> G" shows "g + h \<in> G"
flow_residual_add: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and g :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "wf_residual_network"   and "flow \<Delta> f"   and "flow (residual_network f) g" shows "flow \<Delta> (f \<oplus> g)"
assoc_bpp_restrict: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "A \<subseteq> B"   and "assoc_bpp B f" shows "assoc_bpp A f"
total: shows "\<forall>y x. gt y x \<or> gt x y \<or> y = x"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c Q,A"
termMOD_igConsIPresIGWls: shows "igConsIPresIGWls termMOD"
cl_invI: fixes p :: "pState"   and g :: "'a gState_scheme" assumes "length (pState.channels p) \<le> length (gState.channels g)" shows "cl_inv (g, p)"
disjoint_Aux: fixes c :: "('a, 'b) sequent"   and A :: "('a, 'b) form"   and S :: "('a, 'b) sequent" assumes "ModalSequents.mset c = \<LM> A  \<RM>" shows "A \<in># ModalSequents.mset (extend S c)"
function_mult_eval_closed: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "a \<in> S"   and "f \<in> carrier F"   and "g \<in> carrier F" shows "(f \<otimes>\<^bsub>F\<^esub> g) a \<in> carrier R"
Spy_see_priK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> zg" shows "(Key (priEK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
Conj_E2: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A AND B" shows "H \<turnstile> B"
tendsto_arctan_at_bot: shows "(arctan \<longlongrightarrow> - (pi / 2)) at_bot"
set_take: shows "set (take i as') \<subseteq> carrier G"
SUP_set_fold: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` set xs) = fold (sup \<circ> f) xs bot"
assms_3: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "S n \<subseteq> S m"
div_int_poly_step_def: fixes q :: "int poly" shows "div_int_poly_step q = (\<lambda>a sro. case sro of None \<Rightarrow> None | Some (s, r) \<Rightarrow> let ar = pCons a r; (b, m) = divmod_int (coeff ar (degree q)) (lead_coeff q) in if m = 0 then Some (pCons b s, ar - smult b q) else None)"
degree_reciprocal: fixes P :: "'b poly"   and p :: "nat" assumes "degree P \<le> p" shows "degree (reciprocal_poly p P) \<le> p"
ant_inf_in_Ainteg: shows "(0, 1) \<in> Ainteg"
lmap_lprefix: fixes xs :: "'c llist"   and ys :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "xs \<sqsubseteq> ys" shows "lmap f xs \<sqsubseteq> lmap f ys"
mid_def: fixes i :: "float interval" shows "mid i = (lower i + upper i) * Float 1 (- 1)"
reflexive_conv_closed: fixes x :: "'a" assumes "reflexive x" shows "reflexive (x\<^sup>T)"
prod_acc: fixes r :: "nat \<Rightarrow> 'q \<times> 's" assumes "range (snd \<circ> r) \<subseteq> sa.V" shows "prod.is_acc r = igba.is_acc (fst \<circ> r)"
connected_component_maximal: fixes x :: "'a"   and T :: "'a set"   and S :: "'a set" assumes "x \<in> T"   and "connected T"   and "T \<subseteq> S" shows "T \<subseteq> connected_component_set S x"
relcomppI: fixes r :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "'h"   and b :: "'i"   and s :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and c :: "'j" assumes "r a b"   and "s b c" shows "(r OO s) a c"
right_hom_axioms: shows "right_hom (\<cdot>) (\<star>) a"
l2_nostep_defs_2: shows "l2_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0, signalsInit2 = \<lambda>x. 0, signalsResp2 = \<lambda>x. 0, chan = {}, bad = bad_init\<rparr>}"
compatible_stream: fixes \<phi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> bool"   and xs :: "('s \<times> ('c \<Rightarrow> real)) stream"   and x :: "'s \<times> ('c \<Rightarrow> real)" assumes "\<And>x y. \<lbrakk>x \<in> S; x ~ y\<rbrakk> \<Longrightarrow> \<phi> x = \<phi> y"   and "pred_stream (\<lambda>s. s \<in> S) xs"   and "x \<in> S" shows "pred_stream (\<lambda>s. \<phi> (reps (abss s)) = \<phi> s) (x ## xs)"
section_retraction_of_iso_2: fixes f :: "'d"   and g :: "'d" assumes "D.iso f"   and "D.ide (f \<cdot>\<^sub>D g)" shows "D.inverse_arrows f g"
module_vec_def: fixes ty :: "'b itself"   and n :: "nat" shows "module_vec ty n \<equiv> \<lparr>carrier = carrier_vec n, mult = undefined, one = undefined, zero = 0\<^sub>v n, add = (+), smult = (\<cdot>\<^sub>v)\<rparr>"
list_ball_nth: fixes n :: "nat"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "n < length xs"   and "\<forall>x\<in>set xs. P x" shows "P (xs ! n)"
share_all_until_volatile_write_share_acquired: fixes \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "sharing_consis \<S> ts\<^sub>s\<^sub>b"   and "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "a \<in> dom (share_all_until_volatile_write ts\<^sub>s\<^sub>b \<S>)" shows "a \<in> dom (share sb \<S>) \<or> a \<in> acquired True sb \<O> \<or> (\<exists>j<length ts\<^sub>s\<^sub>b. j \<noteq> i \<and> (let (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ts\<^sub>s\<^sub>b ! j in a \<in> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j)))"
pfun_app_def: shows "pfun_app \<equiv> map_fun Rep_pfun id (\<lambda>f x. if x \<in> dom f then the (f x) else undefined)"
integrable_mult_right: fixes c :: "'a"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. c * f x)"
brw_pref_bre': fixes Qi :: "'c set"   and \<delta> :: "('c, 'd) ta_rule set" shows "wa_precise_refine (brw_algo Qi \<delta>) (bre'_algo Qi \<delta>) brw_\<alpha>"
not_all_zero: shows "\<not> (\<forall>m\<ge>i. A $ m $ j = (0::'a))"
that: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" assumes "{u. u \<turnstile> cc} = [M]\<^bsub>v,n\<^esub>"   and "dbm_int M n" shows "thesis"
NOT_VS_DEP_IMP_DEP_PROJ: fixes v :: "'d"   and vs :: "'d set"   and v' :: "'d"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "v \<notin> vs"   and "v' \<notin> vs"   and "dep PROB v v'" shows "dep (prob_proj PROB (prob_dom PROB - vs)) v v'"
coeff_mult_X: fixes p :: "'b poly"   and i :: "nat" shows "coeff (p * X) (Suc i) = coeff p i"
ifac_rec: fixes i :: "int" shows "ifac i = (if i = 0 then 1 else i * ifac (i - 1))"
suminf_divide: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "summable f" shows "(\<Sum>n. f n / c) = suminf f / c"
CFG_call_edge_length: 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"   and ins :: "'var list"   and outs :: "'var list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs" shows "length fs = length ins"
tl_prefixes_idx: fixes i :: "nat"   and p :: "'b list" assumes "i < length p" shows "tl (prefixes p) ! i = take (Suc i) p"
lenlex_def: fixes r :: "('g \<times> 'g) set" shows "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
VectorSpaceHom: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleIso G smult V smult' T W" shows "VectorSpaceHom (aezfun_scalar_mult.fsmult smult) V (aezfun_scalar_mult.fsmult smult') T"
plen_take_prefix: fixes i :: "nat"   and \<sigma> :: "'a trace" shows "plen (take_prefix i \<sigma>) = i"
new: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and v :: "('r, 'l, 'v) val"   and l :: "'l" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"   and "l \<notin> LID\<^sub>G s" shows "revision_step r s (s(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (Loc l)])))"
commute: fixes a :: "'m"   and b :: "'m" shows "a + b = b + a"
cong_square_alt_int: fixes p :: "'b"   and a :: "'b" assumes "prime p"   and "[a * a = 1::'b] (mod p)" shows "[a = 1::'b] (mod p) \<or> [a = p - (1::'b)] (mod p)"
cltn2_rep_invertible: fixes A :: "cltn2" shows "invertible (cltn2_rep A)"
unit_imp_no_irreducible_divisors: fixes x :: "'a"   and p :: "'a" assumes "is_unit x"   and "irreducible p" shows "\<not> p dvd x"
samp_uni_minus_one_time_pad: fixes y :: "nat"   and q :: "nat" shows "map_spmf (\<lambda>b. (y + (q - b)) mod q) (sample_uniform q) = sample_uniform q"
tendsto_mult_right_zero: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" assumes "(f \<longlongrightarrow> (0::'b)) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> (0::'b)) F"
supporting_hyperplane_rel_boundary: fixes S :: "'a set"   and x :: "'a" assumes "convex S"   and "x \<in> S"   and "x \<notin> rel_interior S" obtains a :: "'a" where "a \<noteq> (0::'a)" and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y" and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
q_def: shows "q = Polynomial.smult (rat_of_nat c) p'"
EMN: shows "E * M' * (mat_of_rows n (map gs1.gso [0..<m]) * (mat_of_rows n (map gs1.gso [0..<m]))\<^sup>T) = N' * (mat_of_rows n (map gs1.gso [0..<m]) * (mat_of_rows n (map gs1.gso [0..<m]))\<^sup>T)"
isomorphic_hcomp_REP: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and F :: "('b, 'b set) Maps.arr"   and G :: "('b, 'b set) Maps.arr" assumes "maps_category V H \<a> \<i> src trg"   and "partial_magma.arr (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H)) (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H) F G)" shows "category.isomorphic V (H (maps_category.REP V H \<a> \<i> src trg F) (maps_category.REP V H \<a> \<i> src trg G)) (maps_category.REP V H \<a> \<i> src trg (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H) F G))"
drop_all_iff: fixes n :: "nat"   and xs :: "'a list" shows "(drop n xs = []) = (|xs| \<le> n)"