hcju/isabelleps huggingface dataset:
beforexM: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and D :: "char list" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = body in D" shows "compP\<^sub>2 P,D,M \<rhd> compxE\<^sub>2 body 0 0 / {..<length (compE\<^sub>2 body)},0"
pprod_def: fixes k :: "nat"   and i :: "nat"   and j :: "nat" shows "local.pprod k i j = prod_list (map (local.xd k) [i..<j])"
exhaust: fixes y :: "'r tree" obtains "y = \<bottom>"   | "y = Null"   | a :: "'r" and tree1 :: "'r tree" and tree2 :: "'r tree" where "y = Node\<cdot>a\<cdot>tree1\<cdot>tree2" and "a \<noteq> \<bottom>"
assms_1: shows "z \<in> circline_set (chordal_circle a r)"
suffix_range: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and APs :: "'b set"   and k :: "nat" assumes "\<Union> (range \<xi>) \<subseteq> APs" shows "\<Union> (range (suffix k \<xi>)) \<subseteq> APs"
rel_spmfE: fixes P :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and p :: "'h spmf"   and q :: "'i spmf" assumes "rel_spmf P p q" obtains pq :: "('h \<times> 'i) spmf" where "\<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> P x y" and "p = map_spmf fst pq" and "q = map_spmf snd pq"
iffs: fixes nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p"   and nothinga' :: "unit"   and doctype' :: "char list"   and document_element' :: "(_) element_ptr option"   and disconnected_nodes' :: "(_) node_ptr list"   and more' :: "'p" shows "(\<lparr>RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = \<lparr>RDocument.nothing = nothinga', doctype = doctype', document_element = document_element', disconnected_nodes = disconnected_nodes', \<dots> = more'\<rparr>) = (nothinga = nothinga' \<and> doctype = doctype' \<and> document_element = document_element' \<and> disconnected_nodes = disconnected_nodes' \<and> more = more')"
pos_distrib_left: fixes c :: "real"   and q :: "'a pmf" assumes "0 < c" shows "(\<Sum>z\<in>outcomes. pmf q z * (c * u z)) = c * (\<Sum>z\<in>outcomes. pmf q z * u z)"
mult_right_sub_dist_sup_right: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y * z \<le> (x \<squnion> y) * z"
mono2: fixes a :: "'a" shows "monotone ordb ordc (\<lambda>b. f (a, b))"
vector_nth_eq: fixes A :: "'a list"   and i :: "'b" shows "vector A $ i = foldr (\<lambda>x f n. (f (n + (1::'b)))(n := x)) A (\<lambda>n x. 0::'a) (1::'b) i"
succ_tran_def: fixes G :: "'b graph"   and v :: "'b" shows "FiniteGraph.succ_tran G v = {e2. (v, e2) \<in> (edges G)\<^sup>+}"
tiebidsAlg_def: fixes a :: "('c \<times> 'd set) set"   and N :: "'c set"   and \<Omega> :: "'d set" shows "tiebidsAlg a N \<Omega> \<equiv> summedBidVectorAlg (maxbidAlg a N \<Omega>) N \<Omega>"
map_add_empty: fixes m :: "'a \<Rightarrow> 'b option" shows "m ++ Map.empty = m"
add_num_frac: 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 y :: "'a"   and z :: "'a"   and x :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero" shows "plus z (divide x y) = divide (plus x (times z y)) y"
emeasure_return: fixes A :: "'b set"   and M :: "'b measure"   and x :: "'b" assumes "A \<in> sets M" shows "emeasure (return M x) A = indicator A x"
Delta2_subsetof_E2: shows "\<Delta>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub>"
simps_1: fixes T :: "POPLmarkRecord.type" shows "type_ofB (VarB T) = T"
pre: shows "pre_splitFace g v w f [countVertices g..<countVertices g + n]"
path_snoc: fixes edges :: "(('a \<times> 'b) \<times> 'a \<times> 'c) set"   and v :: "'a"   and v' :: "'a"   and pth1 :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list"   and e :: "('a \<times> 'b) \<times> 'a \<times> 'c" shows "Pre_Port_Graph.path edges v v' (pth1 @ [e]) = (e \<in> edges \<and> Pre_Port_Graph.path edges v (Pre_Port_Graph.edge_begin e) pth1 \<and> Pre_Port_Graph.edge_end e = v')"
cases: shows "zcount (c_work c loc1) t' < 0 \<or> t' \<in>#\<^sub>z zmset_of (mset_set (set_antichain (frontier (c_pts c loc1)))) + (\<Sum>loc'\<in>UNIV. after_summary (zmset_of (mset_set (set_antichain (frontier (c_imp c loc'))))) (summary loc' loc1))"
right_uniqueI: fixes A :: "'b \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>A x y; A x z\<rbrakk> \<Longrightarrow> y = z" shows "right_unique A"
g: shows "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
non_empty_rbt_keys: fixes t :: "('b, 'c) RBT_Impl.rbt" assumes "t \<noteq> rbt.Empty" shows "RBT_Impl.keys t \<noteq> []"
sel_1: fixes x1 :: "'b" shows "un_Prim \<^bold>\<langle>x1\<^bold>\<rangle> = x1"
next_execs_consistent: shows "\<forall>s t execs. vpeq (current s) s t \<and> (\<forall>d\<in>int.involved (int.next_action s execs). vpeq d s t) \<and> current s = current t \<longrightarrow> fst (snd (CISK_control s (current s) (execs (current s)))) = fst (snd (CISK_control t (current s) (execs (current s))))"
If_transT_simp: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'a"   and c1 :: "('a, 'c) com"   and c2 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b" shows "\<not> PL.transT tval aval (if b then c1 else c2, s) s'"
union_diff_inter_eq_sup: fixes A :: "'a multiset"   and B :: "'a multiset" shows "A + B - A \<inter># B = A \<union># B"
cases: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "mm_equivp a1 a2" obtains \<R> :: "((('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" and lc\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" and lc\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" where "a1 = lc\<^sub>1" and "a2 = lc\<^sub>2" and "strong_low_bisim_mm \<R>" and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>"
simps: fixes A :: "('c, 'd) nba"   and a1 :: "('c \<times> 'd) list"   and a2 :: "'d" shows "path A a1 a2 = ((\<exists>p. a1 = [] \<and> a2 = p) \<or> (\<exists>a p r. a1 = a # r \<and> a2 = p \<and> (fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p) \<and> path A r (snd a)))"
Polygamma_real_odd_pos: fixes x :: "real"   and n :: "nat" assumes "x \<notin> \<int>\<^sub>\<le>\<^sub>0"   and "odd n" shows "0 < Polygamma n x"
emeasure_Int_set_pmf: fixes p :: "'a pmf"   and A :: "'a set" shows "emeasure (measure_pmf p) (A \<inter> set_pmf p) = emeasure (measure_pmf p) A"
simps_2: fixes a :: "'a"   and b :: "'a" shows "bothNet (DenyAllFromTo a b) = {a, b}"
init_h: shows "initial\<^sub>s\<^sub>b ts\<^sub>h \<S>"
simps: fixes p :: "real mpoly" shows "check_nonzero_const p = (case get_if_const p of None \<Rightarrow> False | Some x \<Rightarrow> x \<noteq> 0)"
equiv_class_eq_iff: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c" assumes "equiv A r" shows "((x, y) \<in> r) = (r `` {x} = r `` {y} \<and> x \<in> A \<and> y \<in> A)"
pell_valuation_uminus_uminus: fixes x :: "int"   and y :: "int" shows "pell_valuation (- x, - y) = - pell_valuation (x, y)"
that: fixes N :: "'a set" assumes "{x \<in> space M. \<not> P x} \<subseteq> N"   and "emeasure M N = 0"   and "N \<in> sets M" shows "thesis"
left_leq_right: fixes r :: "real_int" shows "left r \<le> right r"
prems_1: shows "ndiff (Suc n) f {0<..}"
assms_5: shows "\<forall>i'. n \<le> i' \<and> enat i' < n' \<and> enat i' < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t i'\<^esub> \<longrightarrow> i' = i"
assms_6: shows "a \<otimes> fst h [^] e' = \<^bold>g [^] z'"
norm_ge_zero: fixes x :: "'d" shows "0 \<le> \<parallel>x\<parallel>"
l9_4_1_aux_R21: fixes R :: "'p"   and S :: "'p"   and C :: "'p"   and A :: "'p"   and P :: "'p"   and Q :: "'p"   and M :: "'p" assumes "R \<noteq> S"   and "S C Le R A"   and "P Q TS A C"   and "Col R P Q"   and "P Q Perp A R"   and "Col S P Q"   and "P Q Perp C S"   and "M Midpoint R S" shows "\<forall>U C'. M Midpoint U C' \<longrightarrow> (R Out U A) = (S Out C C')"
iterator_to_list_def: shows "iterator_to_list \<equiv> wfrec {(nxt x, x) |x. cnt x} (\<lambda>rec x. if cnt x then x # rec (nxt x) else [])"
geometric_process: shows "geom_proc = geom_rand_walk u d init"
diff_le: fixes y :: "real" assumes "real_of_float l \<le> y"   and "y \<le> real_of_float u" shows "f' y \<le> real_of_float a"
hom: shows "\<forall>x\<in>carrier g1. \<forall>y\<in>carrier g1. h (x \<otimes>\<^bsub>g1\<^esub> y) = h x \<otimes>\<^bsub>g2\<^esub> h y"
simps_2: fixes t :: "'a tree"   and ts :: "'a tree list" shows "consistent\<^sub>F (t # ts) = (alphabet t \<inter> alphabet\<^sub>F ts = {} \<and> consistent t \<and> consistent\<^sub>F ts)"
mapi_rev_nth: fixes xs :: "'b list"   and v :: "nat"   and x :: "'b"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" assumes "xs !. v = Some x" shows "mapi f xs !. v = Some (f v x)"
qfN_pos: fixes N :: "int"   and A :: "int" assumes "1 \<le> N"   and "is_qfN A N" shows "0 \<le> A"
correctCompositionKS_def: fixes x :: "specID" shows "correctCompositionKS x \<equiv> subcomponents x \<noteq> {} \<longrightarrow> specKeysSecrets x = \<Union> (specKeysSecrets ` subcomponents x)"
minim_isMinim: fixes B :: "'b set" assumes "B \<subseteq> Field s"   and "B \<noteq> {}" shows "s.isMinim B (s.minim B)"
rHom_npow: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and x :: "'c"   and f :: "'c \<Rightarrow> 'e"   and n :: "nat" assumes "Ring A"   and "Ring R"   and "x \<in> carrier A"   and "f \<in> rHom A R" shows "f (x^\<^bsup>A n\<^esup>) = f x^\<^bsup>R n\<^esup>"
positions_in_row_unique: fixes w :: "('c \<times> bool list) list"   and i :: "nat" assumes "\<exists>!p. p < length w \<and> snd (w ! p) ! i" shows "the_elem (positions_in_row w i) = (THE p. p < length w \<and> snd (w ! p) ! i)"
fold_once_default: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'b list" assumes "\<not> rsp_fold f" shows "fold_once f xs = id"
min_union_comp: fixes a :: "'b fset"   and A :: "'b fset set"   and B :: "'b fset set" assumes "a \<in> A" shows "\<exists>c. c |\<subseteq>| a \<and> c \<in> A \<union>\<^sub>m B"
is_Elem_of_def: shows "is_Elem_of \<equiv> {(a, b) |a b. Elem a b}"
enres_lift_def: fixes m :: "'f nres" shows "enres_lift m \<equiv> m \<bind> (\<lambda>x. RETURN (Inr x))"
induct: fixes P :: "'b hp list \<Rightarrow> bool"   and a0 :: "'b hp list" assumes "P []"   and "\<And>h. P [h]"   and "\<And>h1 h2 hs. P hs \<Longrightarrow> P (h1 # h2 # hs)" shows "P a0"
UP_ring_subring_car_subset': fixes S :: "'a set"   and h :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "h \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))" shows "h \<in> carrier (UP R)"
UNIV_Times_UNIV: shows "UNIV \<times> UNIV = UNIV"
lnth_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lfinite xs"   and "\<not> lnull ys" shows "lnth (xs @\<^sub>l ys) (the_enat (llength xs)) = lhd ys"
E_wfD_1: fixes G :: "'c graph"   and v :: "'c"   and v' :: "'c" assumes "wf_graph G"   and "(v, v') \<in> edges G" shows "v \<in> nodes G"
type_I_chain_vert_bound_valid: assumes "\<forall>two_cube\<in>two_chain. typeI_twoCube two_cube" shows "\<forall>(k, \<gamma>)\<in>two_chain_vertical_boundary two_chain. valid_path \<gamma>"
simps: fixes a :: "'c set" shows "finite a = (a = {} \<or> (\<exists>A a. a = insert a A \<and> finite A))"
witness_det: shows "det (submatrix Aw' rows_with_1 rows_with_1) \<noteq> 0"
neg_divide_le_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq (divide b c) a = less_eq (times a c) b"
real_less_rsqrt: fixes x :: "real"   and y :: "real" assumes "x\<^sup>2 < y" shows "x < sqrt y"
wf_list_graph_def: fixes G :: "'c list_graph" shows "wf_list_graph G = (distinct (nodesL G) \<and> distinct (edgesL G) \<and> wf_list_graph_axioms G)"
insert_def: fixes x :: "'a"   and xs :: "'a list" shows "List.insert x xs = (if x \<in> set xs then xs else x # xs)"
remove_nth_length: fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "length (remove_nth n xs) = length xs - 1"
Pd_Xd: fixes x :: "'a"   and y :: "'a"   and d :: "'b" assumes "(x, y) \<in> Pd' d" shows "Xd x = d \<and> Xd y = d"
alphabet: fixes a :: "'c set"   and i :: "'d set"   and t :: "'c \<Rightarrow> 'd \<Rightarrow> 'd set"   and c :: "'d \<Rightarrow> bool" shows "alphabet (nba a i t c) = a"
strict_mono_Suc_iff: fixes f :: "nat \<Rightarrow> 'b" shows "strict_mono f = (\<forall>n. f n < f (Suc n))"
null_setsI: fixes M :: "'a measure"   and A :: "'a set" assumes "emeasure M A = 0"   and "A \<in> sets M" shows "A \<in> null_sets M"
nonzero_divide_divide_eq_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "b \<noteq> \<zero>"   and "c \<noteq> \<zero>" shows "a \<oslash> b \<oslash> c = a \<oslash> (b \<otimes> c)"
ground_grounding_\<rho>: fixes s :: "('s, 'v) tm" shows "ground (subst grounding_\<rho> s)"
ExcessNotAtRec_conv_Max: fixes g :: "graph"   and ps :: "(nat \<times> nat) list" assumes "minGraphProps g"   and "set (map fst ps) \<subseteq> \<V> g"   and "distinct (map fst ps)" shows "ExcessNotAtRec ps g = Max {sum snd P |P. P \<subseteq> set ps \<and> separated g (fst ` P)}"
pivotandupdate_satisfies_tableau: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s \<longrightarrow> \<langle>\<V> (pivot_and_update x\<^sub>i x\<^sub>j c s)\<rangle> \<Turnstile>\<^sub>t \<T> s"
d_sub_c: shows "fst Dl \<subset># fst Cl"
F2set3_image_Lev1: fixes xe :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xc :: "'d"   and xd :: "bd_type_F"   and x :: "'e" assumes "xe \<in> Lev1 s1 s2 n xc"   and "xe @ [Inr xd] \<in> Lev1 s1 s2 (Suc n) xc"   and "rv1 s1 s2 xe xc = Inr x" shows "xd \<in> tobd_F23 s2 x ` F2set3 (s2 x)"
max_x_impl_def: fixes a :: "nat list"   and b :: "nat list"   and y :: "nat list"   and i :: "nat" shows "max_x_impl a b y i = (if i < length a \<and> big_d a b y i \<noteq> [] then Min (set (big_d a b y i)) else Max (set b))"
uminus_lifted_pair_def: shows "uminus \<equiv> map_fun Rep_lifted_pair Abs_lifted_pair (\<lambda>xf f. triple.pairs_uminus (Rep_phi f) (xf f))"
mask_eq_iff: fixes w :: "'a word"   and n :: "nat" shows "(w && mask n = w) = (uint w < 2 ^ n)"
hb_deliver: fixes m1 :: "'a \<times> ('a, 'b) operation"   and i :: "nat"   and m2 :: "'a \<times> ('a, 'b) operation" assumes "Deliver m1 \<sqsubset>\<^sup>i Broadcast m2" shows "hb m1 m2"
prv_cnj_assoc_imp1: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (cnj \<phi> (cnj \<chi> \<psi>)) (cnj (cnj \<phi> \<chi>) \<psi>))"
is_natural_2: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<tau> :: "'c \<Rightarrow> 'd"   and f :: "'c" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (\<tau> (partial_magma.cod A f)) (F f) = \<tau> f"
del_UnEdge_odd_odd: fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and v' :: "'a" assumes "valid_unMultigraph G"   and "finite (edges G)"   and "finite (nodes G)"   and "(v, w, v') \<in> edges G"   and "odd (degree v G)"   and "odd (degree v' G)" shows "num_of_odd_nodes G = num_of_odd_nodes (del_unEdge v w v' G) + 2"
h2: shows "(\<Sum>i = 0..<n + 1. f i / B ^ i * B ^ n) = (\<Sum>i = 0..<n + 1. f i * (B ^ n / B ^ i))"
induct_step: fixes f :: "('b, 'c) urel"   and st :: "'a" assumes "(Pre \<and> (\<lceil>e\<rceil>\<^sub><, \<guillemotleft>st\<guillemotright>)\<^sub>u \<in>\<^sub>u \<guillemotleft>R\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> f"   and "\<nu> B = f" shows "(Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> B f"
vec_tensor0: fixes d :: "nat list" shows "vec (tensor0 d) = vec0 (prod_list d)"
Lt_le_LtI: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "Lt a \<le> Lt b"
field_differentiable_minus: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and F :: "'a filter" assumes "(f has_field_derivative f') F" shows "((\<lambda>z. - f z) has_field_derivative - f') F"
dvd_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" shows "dvd.dvd times b a = (\<exists>k. a = times b k)"
degree_prod_list_le: fixes ps :: "'b poly list" shows "degree (prod_list ps) \<le> sum_list (map degree ps)"
vec_def: shows "Matrix.vec \<equiv> map_fun id (map_fun id Abs_vec) (\<lambda>n f. (n, mk_vec n f))"
f': fixes y :: "nat"   and i :: "nat"   and c :: "bool"   and x :: "'a" shows "g y (snd (f i c x)) \<le> g y x"
heads_False: fixes xs :: "bool list" shows "heads (False # xs) = heads xs"
expand: fixes state :: "'b state"   and state' :: "'b state" assumes "clock state = clock state' \<and> refs state = refs state' \<and> ffi state = ffi state' \<and> defined_types state = defined_types state' \<and> defined_mods state = defined_mods state'" shows "state = state'"
assms_1: shows "poly q x \<noteq> (0::'a)"
closed_closedin: fixes S :: "'a set" shows "closed S = closedin euclidean S"
set_cases_2: fixes e :: "'a"   and a :: "('b, 'a, 'c) sum3" assumes "e \<in> set2_sum3 a" obtains "a = In2 e"
takeWhile_eq_take: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "takeWhile P xs = take (length (takeWhile P xs)) xs"
vars_term_disjoint_imp_unifier: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "fv s \<inter> fv t = {}"   and "s \<cdot> \<sigma> = t \<cdot> \<tau>" shows "\<exists>\<mu>. s \<cdot> \<mu> = t \<cdot> \<mu>"
lsimulation_this: shows "lsimulation (R\<inverse>) B A"
c: fixes s :: "complex" assumes "s \<in> complex_cone' \<alpha> - {0}" shows "cmod (stirling_integral (Suc n) s) \<le> c / cmod s ^ n"
vrange_in_VsetI: fixes r :: "V"   and \<alpha> :: "V" assumes "r \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<R>\<^sub>\<circ> r \<in>\<^sub>\<circ> Vset \<alpha>"
pairwise_def: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e set" shows "pairwise R S = (\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> R x y)"
finite_code: fixes A :: "'g set" assumes "class.finite TYPE('g)" shows "finite A = True"
list_upd_lemma: fixes n :: "'a"   and G :: "('a, 'b) form list"   and e :: "nat \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and x :: "'c list \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c list \<Rightarrow> bool" assumes "news n G" shows "list_all (eval e (f(n := x)) g) G = list_all (eval e f g) G"
tendsto_ennreal_iff: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and x :: "real" assumes "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "0 \<le> x" shows "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F = (f \<longlongrightarrow> x) F"
lm065: fixes f :: "('c \<times> 'd) set"   and X :: "'c set" assumes "runiq (f || X)" shows "sum (toFunction (f || X)) (X \<inter> Domain f) = sum snd (f || X)"
le_trans_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "le_trans t u \<equiv> \<forall>P. sound P \<longrightarrow> t P \<tturnstile> u P"
rep_eq: fixes x :: "'a dense_filter_type"   and xa :: "'a dense_filter_type" shows "(x \<le> xa) = (Rep_dense_filter_type x \<le> Rep_dense_filter_type xa)"
WTrt_elim_cases_5: 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 V :: "char list"   and e :: "expr"   and T :: "ty" assumes "P,E,h \<turnstile> V:=e : T" obtains T' :: "ty" where "E V = \<lfloor>T\<rfloor>" and "P,E,h \<turnstile> e : T'" and "P \<turnstile> T' \<le> T"
computation_toppos: shows "rat_poly_inv (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) A) = rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) B"
simps_2: fixes f1 :: "'j \<Rightarrow> 'h"   and f2 :: "'k \<Rightarrow> 'i"   and a :: "'k" shows "map_sum f1 f2 (Inr a) = Inr (f2 a)"
delay_unbounded: fixes f :: "nat \<Rightarrow> nat option"   and i :: "nat"   and n :: "nat" assumes "f \<in> \<R>"   and "\<forall>n. \<phi> i (f \<triangleright> n) \<down>" shows "\<exists>m. n < the (delay i (f \<triangleright> m))"
sterms_pinduct: fixes a0 :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and P :: "('a \<Rightarrow> ('b, 'c, 'a, 'd) seqp) \<Rightarrow> ('b, 'c, 'a, 'd) seqp \<Rightarrow> bool"   and a1 :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed a0"   and "\<And>\<Gamma> p1 p2. \<lbrakk>sterms_dom (\<Gamma>, p1 \<oplus> p2); P \<Gamma> p1; P \<Gamma> p2\<rbrakk> \<Longrightarrow> P \<Gamma> (p1 \<oplus> p2)"   and "\<And>\<Gamma> pn. \<lbrakk>sterms_dom (\<Gamma>, call(pn)); P \<Gamma> (\<Gamma> pn)\<rbrakk> \<Longrightarrow> P \<Gamma> (call(pn))"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}\<langle>va\<rangle> vb) \<Longrightarrow> P \<Gamma> ({v}\<langle>va\<rangle> vb)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}\<lbrakk>va\<rbrakk> vb) \<Longrightarrow> P \<Gamma> ({v}\<lbrakk>va\<rbrakk> vb)"   and "\<And>\<Gamma> v va vb vc vd. sterms_dom (\<Gamma>, {v}unicast(va, vb) . vc \<triangleright> vd) \<Longrightarrow> P \<Gamma> ({v}unicast(va, vb) . vc \<triangleright> vd)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}broadcast(va) . vb) \<Longrightarrow> P \<Gamma> ({v}broadcast(va) . vb)"   and "\<And>\<Gamma> v va vb vc. sterms_dom (\<Gamma>, {v}groupcast(va, vb) . vc) \<Longrightarrow> P \<Gamma> ({v}groupcast(va, vb) . vc)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}send(va) . vb) \<Longrightarrow> P \<Gamma> ({v}send(va) . vb)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}deliver(va) . vb) \<Longrightarrow> P \<Gamma> ({v}deliver(va) . vb)"   and "\<And>\<Gamma> v va vb. sterms_dom (\<Gamma>, {v}receive(va) . vb) \<Longrightarrow> P \<Gamma> ({v}receive(va) . vb)" shows "P a0 a1"
\<alpha>_ide_simp: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "\<F>C.ide a"   and "\<F>C.ide b"   and "\<F>C.ide c" shows "\<F>C.assoc a b c = \<a>[a, b, c]"