hcju/isabelleps huggingface dataset:
cases: fixes S :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and E :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and IA :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and IU :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and OA :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and OU :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and a1 :: "('m, 'o, 'q, 's, 'u, 'w) core"   and a2 :: "('n, 'p, 'r, 't, 'v, 'x) core" assumes "rel_core' S E IA IU OA OU a1 a2" obtains cpoke :: "'m \<Rightarrow> 'o \<Rightarrow> 'm spmf" and cfunc_adv :: "'m \<Rightarrow> 'q \<Rightarrow> ('u \<times> 'm) spmf" and cfunc_usr :: "'m \<Rightarrow> 's \<Rightarrow> ('w \<times> 'm) spmf" and cpoke' :: "'n \<Rightarrow> 'p \<Rightarrow> 'n spmf" and cfunc_adv' :: "'n \<Rightarrow> 'r \<Rightarrow> ('v \<times> 'n) spmf" and cfunc_usr' :: "'n \<Rightarrow> 't \<Rightarrow> ('x \<times> 'n) spmf" where "a1 = Core cpoke cfunc_adv cfunc_usr" and "a2 = Core cpoke' cfunc_adv' cfunc_usr'" and "(S ===> E ===> rel_spmf S) cpoke cpoke'" and "(S ===> IA ===> rel_spmf (rel_prod OA S)) cfunc_adv cfunc_adv'" and "(S ===> IU ===> rel_spmf (rel_prod OU S)) cfunc_usr cfunc_usr'"
hyps: assumes "p \<noteq> 0" shows "order (of_real t) (map_poly of_real p) = order t p"
assms: fixes v :: "nat" shows "coeff p v = coeff q v"
set_in_cone: fixes Vs :: "'a vec set" assumes "Vs \<subseteq> carrier_vec (n + 1)" shows "Vs \<subseteq> next_dim.cone Vs"
list_of_vec_vCons: fixes a :: "'a"   and v :: "'a vec" shows "list_of_vec (vCons a v) = a # list_of_vec v"
fun_Compl_def: fixes A :: "'a \<Rightarrow> 'b" shows "- A = (\<lambda>x. - A x)"
PO_hoare_def: fixes pre :: "'a set"   and R :: "('a \<times> 'a) set"   and post :: "'a set" shows "{pre} R {> post} \<equiv> R `` pre \<subseteq> post"
plug_decomposition_equivalence: fixes r :: "('a, 'b, 'c) expr"   and e :: "('a, 'b, 'c) expr"   and \<E> :: "('a, 'b, 'c) cntxt" assumes "redex r" shows "decompose e \<E> r = (\<E> [r] = e)"
type_definition_uexpr: shows "type_definition Rep_uexpr mk\<^sub>e UNIV"
tyenv_wellformed_subset: fixes mds :: "Mode \<Rightarrow> 'Var set"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and P' :: "'BExp set" assumes "tyenv_wellformed mds \<Gamma> \<S> P"   and "P' \<subseteq> P" shows "tyenv_wellformed mds \<Gamma> \<S> P'"
tensor_from_vec_simp: fixes A :: "'b tensor" shows "tensor_from_vec (dims A) (vec A) = A"
hoare_triple_D1: shows "\<turnstile>\<^sub>p {Q} Utrans_P vars1 mat_O {Q1}"
simple_ownership_distinct_nth_update: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<O>' :: "nat set"   and p' :: "'p"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "unit"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<forall>j<length ts. i \<noteq> j \<longrightarrow> (let (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j) = ts ! j in \<O>' \<inter> \<O>\<^sub>j = {})" shows "simple_ownership_distinct (ts[i := (p', is', \<theta>', sb', \<D>', \<O>', \<R>')])"
bounded_bilinear_blinop_apply: shows "bounded_bilinear ($)"
le_finI: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and v\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* (w\<^sub>1 @ v\<^sub>1) w\<^sub>2" shows "traces.le_fin ind w\<^sub>1 w\<^sub>2"
dmining_def: fixes t :: "nat \<Rightarrow> cnf" shows "dmining t \<equiv> \<lambda>n. \<exists>nid\<in>actDn (t n). mining (\<sigma>\<^bsub>nid\<^esub>t n)"
GFP_closed: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L" shows "GFP\<^bsub>L\<^esub> f \<in> carrier L"
cos_one_2pi_int: fixes x :: "real" shows "(cos x = 1) = (\<exists>x. x = real_of_int x * 2 * pi)"
wordinterval_lowest_element_correct_A: fixes r :: "'b wordinterval"   and x :: "'b word" assumes "wordinterval_lowest_element r = Some x" shows "is_lowest_element x (wordinterval_to_set r)"
path_ex_obs: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and S :: "'node set" assumes "n -as\<rightarrow>* n'"   and "n' \<in> S" obtains m :: "'node" where "m \<in> obs n S"
hyps_2: fixes S :: "'node SDG_node set" assumes "n'' \<in> set ns"   and "n\<^sub>1 \<in> S" shows "n'' \<in> HRB_slice S"
finite_pderivs_lang: fixes A :: "'b list set"   and r :: "'b rexp" shows "finite (pderivs_lang A r)"
rep_eq: fixes x :: "perm"   and xa :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "FL_Formula.formula.Rep_formula (x \<bullet> xa) = x \<bullet> FL_Formula.formula.Rep_formula xa"
32: shows "bprv (imp (NN \<langle>\<phi>\<rangle> \<langle>neg \<phi>\<rangle>) (all yy' (imp (NN \<langle>\<phi>\<rangle> (Var yy')) (eql \<langle>neg \<phi>\<rangle> (Var yy')))))"
a_2: shows "\<forall>stp. \<not> TSTD (steps0 (Suc 0, Bk \<up> l, <lm>) tp stp)"
enum_latin_type_def: shows "enum_class.enum \<equiv> [Row, Col, Num]"
rel_frontier_cball: fixes a :: "'c"   and r :: "real" shows "rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)"
polyfun_evaluate_net_plus_a: fixes inputs :: "real vec list"   and m :: "(nat \<times> nat) convnet"   and j :: "nat"   and a :: "nat"   and s :: "bool" assumes "map dim_vec inputs = input_sizes m"   and "valid_net m"   and "j < output_size m" shows "polyfun {..<a + count_weights s m} (\<lambda>f. evaluate_net (insert_weights s m (\<lambda>i. f (i + a))) inputs $ j)"
aux_Cauchy: fixes x :: "complex vec"   and n :: "nat"   and y :: "complex vec"   and a :: "complex" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "0 \<le> inner_prod x x + a * inner_prod x y + cnj a * (cnj (inner_prod x y) + a * inner_prod y y)"
case_prodI: fixes f :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and a :: "'x"   and b :: "'y" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
sel_1: fixes x1 :: "'a"   and x2 :: "'c" shows "fst (x1, x2) = x1"
tauActionChain_1: fixes P :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'" shows "P \<Longrightarrow>\<^sub>\<tau> P'"
square_free_factorizationI': fixes p :: "'c poly"   and c :: "'c"   and bs :: "('c poly \<times> nat) list" assumes "p = smult c (\<Prod>(a, i)\<leftarrow>bs. a ^ Suc i)"   and "square_free (prod_list (map fst bs))"   and "\<And>b i. (b, i) \<in> set bs \<Longrightarrow> 0 < degree b"   and "p = 0 \<Longrightarrow> c = (0::'c) \<and> bs = []" shows "square_free_factorization p (c, bs)"
map_def: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b pred" shows "Predicate.map f P = bind P (single \<circ> f)"
g_unhalt: fixes anything :: "nat list" shows "{\<lambda>nl. nl = args @ 0 \<up> (gft - gar) @ anything} gap \<up>"
distinct_65: fixes x9 :: "('a, String.literal) phantom" shows "UnlimitedNatural \<noteq> Enum x9"
left_add_mult_distrib_zmset: fixes i :: "nat"   and u :: "'b zmultiset"   and j :: "nat"   and k :: "'b zmultiset" shows "repeat_zmset i u + (repeat_zmset j u + k) = repeat_zmset (i + j) u + k"
pbinop_def: fixes A :: "V"   and f :: "V" shows "pbinop A f \<equiv> pnop A (2\<^sub>\<nat>) f"
is_unitE: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" obtains b :: "'a" where "a \<noteq> zero" and "b \<noteq> zero" and "dvd.dvd times b one" and "divide one a = b" and "divide one b = a" and "times a b = one" and "divide c a = times c b"
get_disconnected_document_locs_impl: shows "get_disconnected_document_locs = local.a_get_disconnected_document_locs"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'd"   and xa :: "'c \<Rightarrow>\<^sub>0 'd" shows "lookup (x + xa) = (\<lambda>k. lookup x k + lookup xa k)"
mat_zerol: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "f \<oplus> \<delta> = f"
point_ordinate_diff: fixes A :: "point"   and B :: "point" shows "ordinate (A - B) = ordinate A - ordinate B"
RedTryCatch: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and a :: "'addr"   and D :: "String.literal"   and P :: "'addr J_prog"   and C :: "String.literal"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id"   and V :: "String.literal"   and e2 :: "'addr expr" assumes "typeof_addr (hp s) a = \<lfloor>Class_type D\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "extTA,P,t \<turnstile> \<langle>try Throw a catch(C V) e2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>{V:Class C=\<lfloor>Addr a\<rfloor>; e2},s\<rangle>"
pure_assn_rule: fixes h :: "pheap"   and b :: "bool" shows "(h \<Turnstile> \<up> b) = (addrOf h = {} \<and> b)"
mult_imp_less_div_pos: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "(0::'a) < y"   and "z * y < x" shows "z < x / y"
arbitrage_processE: fixes Mkt :: "(bool stream, 'b) discrete_market"   and p :: "'b \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real" assumes "arbitrage_process Mkt p" shows "\<exists>m. self_financing Mkt p \<and> trading_strategy p \<and> (\<forall>w\<in>space M. cls_val_process Mkt p 0 w = 0) \<and> (AE w in M. 0 \<le> cls_val_process Mkt p m w) \<and> 0 < prob {w \<in> space M. 0 < cls_val_process Mkt p m w}"
pl_lemma: fixes w1 :: "32 Word.word"   and r1 :: "real"   and w2 :: "32 Word.word"   and r2 :: "real" assumes "w1 \<equiv>\<^sub>L r1"   and "w2 \<equiv>\<^sub>L r2" shows "pl w1 w2 \<equiv>\<^sub>L r1 + r2"
greaterThan_def: fixes l :: "'e word" shows "signed.greaterThan l \<equiv> {x. l <s x}"
simps: fixes a :: "'e"   and T :: "('e, 'f) TS_scheme" shows "(a \<in> reach T) = ((\<exists>s. a = s \<and> s \<in> init T) \<or> (\<exists>s t. a = t \<and> (s, t) \<in> TS.trans T \<and> s \<in> reach T))"
N': shows "N' \<in> null_sets M \<and> null_part M (\<Union> (range S)) \<subseteq> N'"
tllist_all2_tmap2: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and f :: "'k \<Rightarrow> 'h"   and g :: "'l \<Rightarrow> 'j"   and ys :: "('k, 'l) tllist" shows "tllist_all2 P Q xs (tmap f g ys) = tllist_all2 (\<lambda>x y. P x (f y)) (\<lambda>x y. Q x (g y)) xs ys"
simps_5: shows "cospan_shape.Cod cospan_shape.BT = cospan_shape.TT"
1: shows "continuous_on ({c..d} - g ` (s \<union> g -` s')) f"
ctr: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv" shows "Generative_Probabilistic_Value.Pause out c = GPV (return_spmf (IO out c))"
r_one: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP R)" shows "x \<oplus>\<^bsub>UP R\<^esub> \<zero>\<^bsub>UP R\<^esub> = x"
assms_2: shows "A \<approx>\<^sub>\<circ> B"
lborel_distr_plus: fixes c :: "'b" shows "distr lborel borel ((+) c) = lborel"
trailing_monomial_higher: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "p = punit.higher p (punit.tt p) + monomial (punit.tc p) (punit.tt p)"
trimmed_imp_dominant_term_nz: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "trimmed F" shows "fst (dominant_term F) \<noteq> 0"
component_in_v: fixes e :: "'a"   and v :: "'a" shows "choose_component e v \<le> - - v"
PP: fixes t :: "'trm" assumes "t \<in> trm" shows "wrepr.PP t \<in> fmla"
that: fixes B :: "real" assumes "B-lipschitz_on A f" shows "thesis"
orthogonal_matrix_intro: fixes A :: "((real, 'b) vec, 'b) vec" assumes "pairwise orthogonal (columns A)"   and "\<forall>i. norm (column i A) = 1"   and "card (columns A) = ncols A" shows "orthogonal_matrix A"
AllOther: shows "\<forall>m. msgsCJ' m = (msgs cJ1 -# <pJ, inM True>) m"
diff_delta_ln_upper_7_def: shows "diff_delta_ln_upper_7 \<equiv> \<lambda>x. (x - 1) ^ 7 / ((4 * x ^ 3 + 18 * x\<^sup>2 + 12 * x + 1)\<^sup>2 * x)"
IH: fixes p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'c" shows "simple_fw rs \<lparr>p_iiface = pc_iiface c, p_oiface = pc_oiface c, p_src = s, p_dst = d, p_proto = pc_proto c, p_sport = pc_sport c, p_dport = pc_dport c, p_tcp_flags = {TCP_SYN}, p_payload = []\<rparr> = simple_fw rs \<lparr>p_iiface = pc_iiface c, p_oiface = pc_oiface c, p_src = s, p_dst = d, p_proto = pc_proto c, p_sport = pc_sport c, p_dport = pc_dport c, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr>"
B_mod_greater_zero: fixes x :: "int" assumes "x \<in> B" shows "0 < x mod int p"
dens_ctxt_\<alpha>_def: shows "dens_ctxt_\<alpha> \<equiv> \<lambda>x. case case x of (vs, vs', \<Gamma>, \<delta>) \<Rightarrow> (set vs, set vs', \<Gamma>, \<lambda>\<sigma>. extract_real (cexpr_sem \<sigma> \<delta>)) of (x, y) \<Rightarrow> (x, case y of (x, y) \<Rightarrow> (x, case y of (x, y) \<Rightarrow> (x, \<lambda>x. ennreal (y x))))"
tl_def: fixes xs :: "'x" shows "local.tl xs = oalist_of_list (tl_raw (list_of_oalist xs))"
lc_monomial: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c"   and c :: "'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.lc term_of_pair ord_term (monomial c t) = c"
Restr_natLeq: fixes n :: "nat" shows "Restr natLeq {x. x < n} = natLeq_on n"
infinite_inj_infinite_image: fixes f :: "'a \<Rightarrow> 'b"   and Z :: "'a set" assumes "inj_on f Z" shows "infinite (f ` Z) = infinite Z"
embed2_def: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and wf_atom :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and project :: "'c \<Rightarrow> 'c"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and embed :: "'c \<Rightarrow> 'c list"   and singleton :: "'c \<Rightarrow> 'd" shows "embed2 \<Sigma> wf_atom project lookup embed singleton \<equiv> Pi_Derivatives.embed \<Sigma> project embed \<and> embed2_axioms \<Sigma> wf_atom lookup singleton"
Ag_ind_def: fixes A :: "('d, 'e) aGroup_scheme"   and f :: "'d \<Rightarrow> 'c" shows "Ag_ind A f = \<lparr>carrier = f ` carrier A, pop = \<lambda>x\<in>f ` carrier A. \<lambda>y\<in>f ` carrier A. f ((f\<inverse>\<^bsub>f ` carrier A,carrier A\<^esub>) x \<plusminus>\<^bsub>A\<^esub> (f\<inverse>\<^bsub>f ` carrier A,carrier A\<^esub>) y), mop = \<lambda>x\<in>f ` carrier A. f (-\<^sub>a\<^bsub>A\<^esub> (f\<inverse>\<^bsub>f ` carrier A,carrier A\<^esub>) x), zero = f \<zero>\<^bsub>A\<^esub>\<rparr>"
outstanding_volatile_writes_unowned_by_others_axioms: shows "outstanding_volatile_writes_unowned_by_others ts"
uncountable_def: fixes A :: "'b set" shows "uncountable A = (A \<noteq> {} \<and> (\<nexists>f. range f = A))"
\<psi>: shows "\<guillemotleft>\<psi> : src f' \<Rightarrow> w \<star> w'\<guillemotright> \<and> local.iso \<psi>"
map_update_chain_if: fixes chain :: "'c"   and rs :: "'d"   and \<Gamma> :: "'c \<Rightarrow> 'd option" shows "(\<lambda>b. if b = chain then Some rs else \<Gamma> b) = \<Gamma>(chain \<mapsto> rs)"
strict_implies_not_eq: fixes a :: "'c list"   and b :: "'c list" assumes "strict_prefix a b" shows "a \<noteq> b"
poly_of_vec_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "poly_of_vec (vec n f) = Poly (rev (map f [0..<n]))"
typeII_twoCube_smooth_edges: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "typeII_twoCube twoC"   and "(k, \<gamma>) \<in> boundary twoC" shows "\<gamma> piecewise_C1_differentiable_on {0..1}"
outs_\<I>_map_\<I>: fixes f :: "'h \<Rightarrow> 'j"   and g :: "'k \<Rightarrow> 'i"   and \<I> :: "('j, 'k) \<I>" shows "outs_\<I> (map_\<I> f g \<I>) = f -` outs_\<I> \<I>"
trees_cyl_map_treeI: fixes t :: "'d tree"   and A :: "'d set" assumes "t \<in> trees A" shows "t \<in> trees_cyl (map_tree (\<lambda>x. A) t)"
mono_D22: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (S `;` P) \<longrightarrow> Ra (S `;` P) s = Ra (S `;` Q) s"
isin_sorted_split: fixes ts :: "('a btree \<times> 'a) list"   and t :: "'a btree"   and x :: "'a"   and ls :: "('a btree \<times> 'a) list"   and rs :: "('a btree \<times> 'a) list" assumes "sorted_less (inorder (Node ts t))"   and "split ts x = (ls, rs)" shows "(x \<in> set (inorder (Node ts t))) = (x \<in> set (inorder_list rs @ inorder t))"
ratpolys_def: shows "ratpolys = {p. \<forall>i. poly.coeff p i \<in> \<rat>}"
bad_sticky2: fixes s1 :: "'s1" assumes "bad1 s1"   and "I1 s1" shows "callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"
raa_cor_3: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v]"   and "[\<^bold>\<not>\<psi> \<^bold>\<rightarrow> \<^bold>\<not>\<phi> in v]"   and "[\<phi> in v]" shows "[\<psi> in v]"
effect_while_post: fixes a :: "'c array"   and key :: "'c"   and i :: "nat ref"   and f :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "unit" assumes "effect (while (shiftr_p a key i) f) h h' r" shows "success (shiftr_p a key i) h' \<and> \<not> cond (shiftr_p a key i) h'"
assms: shows "bo \<le> vimage2p rh rh (=)"
inspan: shows "lincomb l1 X + lincomb l2 W \<in> local.span X"
fim: shows "f ` rel_frontier S \<subseteq> rel_frontier T"
edom_perm: fixes \<pi> :: "perm"   and f :: "'c \<Rightarrow> 'd" shows "edom (\<pi> \<bullet> f) = \<pi> \<bullet> edom f"
psubst_upd: fixes x :: "'a"   and P :: "('a, 'b) form"   and f :: "'a \<Rightarrow> 'c"   and y :: "'c" assumes "new x P" shows "psubst (f(x := y)) P = psubst f P"
entails_def: fixes \<phi> :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and \<psi> :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "\<phi> \<Turnstile>\<^sub>B \<psi> \<equiv> \<forall>\<sigma>. \<phi> \<sigma> \<longrightarrow> \<psi> \<sigma>"
derived_set_of_subtopology: fixes X :: "'c topology"   and U :: "'c set"   and S :: "'c set" shows "subtopology X U derived_set_of S = U \<inter> X derived_set_of (U \<inter> S)"
update_convs_1: fixes sp_impl_subj_subj' :: "(partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool) \<Rightarrow> partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "sp_impl_subj_subj_update sp_impl_subj_subj' \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = \<lparr>sp_impl_subj_subj = sp_impl_subj_subj' sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr>"
eval_hom': fixes a :: "('c multiset \<Rightarrow> 'a) list set"   and p :: "('c multiset \<Rightarrow> 'a) list set list" assumes "a \<in> carrier (Rupt\<^bsub>L\<^esub> (carrier L) q)"   and "set p \<subseteq> carrier (Rupt\<^bsub>L\<^esub> (carrier L) q)" shows "the_elem (eval_pmod q ` Rupt.eval p a) = Hom.eval (map (\<lambda>x. the_elem (eval_pmod q ` x)) p) (the_elem (eval_pmod q ` a))"
power_down: fixes x :: "real"   and p :: "nat"   and n :: "nat" assumes "0 \<le> x" shows "power_down p x n \<le> x ^ n"
pfa_append_grow_full_rule: fixes n :: "nat"   and c :: "nat"   and l :: "'b list"   and a :: "'b array"   and x :: "'b" assumes "n = c" shows "<is_pfa c l (a, n)> array_grow a (c + 1) x <\<lambda>r. is_pfa (c + 1) (l @ [x]) (r, n + 1)>\<^sub>t"
FullModel: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "FullModel wtFsym wtPsym arOf resOf parOf \<Phi> InfModel.intTF (InfModel.intFF arOf resOf intT intF) (InfModel.intPF parOf intT intP)"
degree_frame: fixes x :: "'v"   and v :: "'v"   and v' :: "'v"   and w :: "'w" assumes "finite E"   and "x \<notin> {v, v'}" shows "degree x (del_unEdge v w v' G) = degree x G"
dvd_prime_factors: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "y \<noteq> zero"   and "dvd.dvd times x y" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize x) \<subseteq> set_mset (factorial_semiring.prime_factorization zero times one normalize y)"
elims: fixes x :: "varID"   and y :: "chanID set" assumes "VARto x = y" obtains "x = stA1" and "y = {data10}"   | "x = stA2" and "y = {data4, data12}"   | "x = stA4" and "y = {data3}"   | "x = stA6" and "y = {data15, data16}"
integrable_distr_eq: fixes g :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "g \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> borel_measurable N" shows "integrable (distr M N g) f = integrable M (\<lambda>x. f (g x))"
closed_csubspace_UNIV: shows "closed_csubspace UNIV"
JVM_le_unfold: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and m :: "nat"   and n :: "nat" shows "JVM_SemiType.le P m n \<equiv> Err.le (Opt.le (Product.le (Listn.le (subtype P)) (Listn.le (Err.le (subtype P)))))"
almost0_canonical: fixes f :: "nat \<Rightarrow> nat option"   and vs :: "nat list" assumes "f = vs \<odot> 0\<^sup>\<infinity>"   and "f \<noteq> 0\<^sup>\<infinity>" obtains ws :: "nat list" where "0 < length ws" and "last ws \<noteq> 0" and "f = ws \<odot> 0\<^sup>\<infinity>"
sylow_def: fixes G :: "('c, 'd) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" shows "sylow G p a m \<equiv> Group.group G \<and> sylow_axioms G p a m"
complex_vector_affinity_eq: fixes m :: "complex"   and x :: "'b"   and c :: "'b"   and y :: "'b" assumes "m \<noteq> 0" shows "(m *\<^sub>C x + c = y) = (x = y /\<^sub>C m - c /\<^sub>C m)"