hcju/isabelleps huggingface dataset:
wf_Un: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "wf r"   and "wf s"   and "Domain r \<inter> Range s = {}" shows "wf (r \<union> s)"
b: shows "discounted_value r (prices Mkt asset) (Suc n) \<in> borel_measurable N"
InterE: fixes A :: "'c"   and C :: "'c set set"   and X :: "'c set" assumes "s2p (\<Inter> C) A" obtains "X \<notin> C"   | "s2p X A"
Ord_less_TC_mem: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "\<beta> \<sqsubset> \<alpha>" shows "\<beta> \<in> elts \<alpha>"
match_list_matches: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and ms :: "'b match_expr list"   and a :: "action"   and p :: "'c" shows "match_list \<gamma> ms a p = (\<exists>m\<in>set ms. matches \<gamma> m a p)"
bisim1_ThrowD: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option" assumes "P,e,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "pc < length (compE2 e) \<and> (xcp = \<lfloor>a\<rfloor> \<or> xcp = None) \<and> xs = loc"
CFG_edge_transfer_uses_only_Use: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and a :: "'f"   and s :: "'h"   and s' :: "'h" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val"   and "valid_edge a"   and "\<forall>V\<in>Use (sourcenode a). state_val s V = state_val s' V"   and "pred (kind a) s"   and "pred (kind a) s'" shows "\<forall>V\<in>Def (sourcenode a). state_val (transfer (kind a) s) V = state_val (transfer (kind a) s') V"
intersects_y_axis_positive_iff: fixes H :: "circline" assumes "is_poincare_line H"   and "H \<noteq> y_axis" shows "(\<exists>y\<in>unit_disc. y \<in> circline_set H \<inter> positive_y_axis) = intersects_y_axis_positive H"
mpadd_def: fixes m :: "int"   and a :: "int"   and p\<^sub>1 :: "int \<times> int \<times> int"   and p\<^sub>2 :: "int \<times> int \<times> int" shows "mpadd m a p\<^sub>1 p\<^sub>2 = (let (x\<^sub>1, y\<^sub>1, z\<^sub>1) = p\<^sub>1; (x\<^sub>2, y\<^sub>2, z\<^sub>2) = p\<^sub>2 in if z\<^sub>1 = 0 then p\<^sub>2 else if z\<^sub>2 = 0 then p\<^sub>1 else let d\<^sub>1 = x\<^sub>2 **\<^bsub>m\<^esub> z\<^sub>1; d\<^sub>2 = x\<^sub>1 **\<^bsub>m\<^esub> z\<^sub>2; l = d\<^sub>1 --\<^bsub>m\<^esub> d\<^sub>2; n = y\<^sub>2 **\<^bsub>m\<^esub> z\<^sub>1 --\<^bsub>m\<^esub> y\<^sub>1 **\<^bsub>m\<^esub> z\<^sub>2 in if l = 0 then if n = 0 then mpdouble m a p\<^sub>1 else (0, 0, 0) else let h = n ^^^\<^bsub>m\<^esub> 2 **\<^bsub>m\<^esub> z\<^sub>1 **\<^bsub>m\<^esub> z\<^sub>2 --\<^bsub>m\<^esub> (d\<^sub>1 ++\<^bsub>m\<^esub> d\<^sub>2) **\<^bsub>m\<^esub> l ^^^\<^bsub>m\<^esub> 2 in (l **\<^bsub>m\<^esub> h, (d\<^sub>2 **\<^bsub>m\<^esub> l ^^^\<^bsub>m\<^esub> 2 --\<^bsub>m\<^esub> h) **\<^bsub>m\<^esub> n --\<^bsub>m\<^esub> l ^^^\<^bsub>m\<^esub> 3 **\<^bsub>m\<^esub> y\<^sub>1 **\<^bsub>m\<^esub> z\<^sub>2, l ^^^\<^bsub>m\<^esub> 3 **\<^bsub>m\<^esub> z\<^sub>1 **\<^bsub>m\<^esub> z\<^sub>2))"
rel_pmf_joinI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf pmf"   and q :: "'d pmf pmf" assumes "rel_pmf (rel_pmf P) p q" shows "rel_pmf P (join_pmf p) (join_pmf q)"
bal_i_ins_Unbal: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins n d x t = Unbal t'" shows "bal_i (size t') (height t')"
None_notin_image_Some: fixes A :: "'a set" shows "None \<notin> Some ` A"
canon_term_pp_cancel: fixes a :: "'t \<times> 'a" shows "canon_term_pp_pair (cancel_term_pp_pair a)"
finite3: shows "set_nn_integral (count_space UNIV) (X - {source \<Delta>}) (d_OUT (h_diff i)) \<noteq> \<top>"
disj_matrices_add_zero_le: fixes A :: "'a matrix"   and B :: "'a matrix" assumes "disj_matrices A B" shows "(0 \<le> A + B) = (0 \<le> A \<and> 0 \<le> B)"
unique_letter_wordE: fixes w :: "'b list"   and a :: "'b" assumes "\<forall>c. c \<in> set w \<longrightarrow> c = a" obtains k :: "nat" where "w = [a] \<^sup>@ k"
intact_def: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and I :: "'b set" shows "intact slices W I \<equiv> stellar slices W \<and> stellar (project slices I) W \<and> intact_axioms slices W I"
eq: fixes U :: "'a set" assumes "openin X U" shows "{x \<in> topspace Y. g x \<in> U} = f ` U"
simps_1: fixes "next" :: "bool list \<Rightarrow> bool list" shows "buildchain next 0 = []"
S_def: fixes h :: "'a"   and e :: "nat" shows "local.\<Sigma>_protocols_base.S h e = map_spmf (\<lambda>(a, z). (a, e, z)) (S2 h e)"
lens_comp_right_id: fixes X :: "'b \<Longrightarrow> 'c" shows "X ;\<^sub>L 1\<^sub>L = X"
res: shows "local.tighten_poly_bounds l r sr = (l', r', sr')"
lambdas1: fixes l :: "nat"   and vs :: "(nat \<times> char list) list"   and c :: "call"   and x :: "lambda" assumes "Lambda l vs c \<in> lambdas x" shows "c \<in> calls x"
UN_ball_bex_simps_2: fixes B :: "'f \<Rightarrow> 'e set"   and A :: "'f set"   and P :: "'e \<Rightarrow> bool" shows "(\<forall>x\<in>\<Union> (B ` A). P x) = (\<forall>a\<in>A. \<forall>x\<in>B a. P x)"
ln_Gamma_bounds_aux_1: fixes x :: "real" assumes "1 \<le> x" shows "c + (x - 1 / 2) * ln x - x \<le> ln_Gamma x"
nonneg_bounded: shows "\<exists>K\<ge>0. \<forall>a b. norm (blinfun_apply (f a) b) \<le> norm a * norm b * K"
snapshot_state: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and p :: "'d"   and u :: "'e" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "states (distributed_system.s channel trans send recv init t i) p = u"   and "\<not> ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "ps (distributed_system.s channel trans send recv init t (i + 1)) p \<noteq> None" shows "ps (distributed_system.s channel trans send recv init t (i + 1)) p = Some u"
residue_primroot_def: fixes n :: "nat"   and x :: "nat" shows "residue_primroot n x = (0 < n \<and> coprime n x \<and> ord n x = totient n)"
less_imp_le: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "x \<le> y"
mat_ordered_semiring_def: fixes n :: "nat"   and sd :: "nat"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'c" shows "mat_ordered_semiring n sd gt b \<equiv> ring_mat TYPE('b) n \<lparr>ordered_semiring.geq = (\<ge>\<^sub>m), gt = mat_gt gt sd, max = max\<^sub>m, \<dots> = b\<rparr>"
back_edge_def: fixes u :: "'v"   and v :: "'v"   and s :: "('v, 'es) state_scheme" shows "back_edge u v s \<equiv> let b = insert (u, v) (back_edges s) in s\<lparr>back_edges := b\<rparr>"
cases_scheme: fixes r :: "('o, 'm, 'n, 'p) RDocument_scheme" obtains nothing :: "unit" and nothinga :: "unit" and doctype :: "char list" and document_element :: "(_) element_ptr option" and disconnected_nodes :: "(_) node_ptr list" and more :: "'p" where "r = \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr>"
INF_eq_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and I :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and c :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> less_eq (f i) c" shows "(Inf (f ` I) = c) = (\<forall>i\<in>I. f i = c)"
lt_def: fixes compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering" shows "Ord.lt compare = (\<Lambda> x y. case compare\<cdot>x\<cdot>y of LT \<Rightarrow> TT | EQ \<Rightarrow> FF | GT \<Rightarrow> FF)"
norm_Basis: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and u :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "norm u = 1"
word_and_1_shiftls'_1: fixes x :: "'b word" shows "x AND (1 << 0) = (if x !! 0 then 1 << 0 else 0)"
simps_1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" shows "power.power one times a 0 = one"
simps: fixes n :: "nat"   and m :: "nat"   and c :: "int"   and bs :: "int list" shows "bin_rsplit_aux n m c bs = (if m = 0 \<or> n = 0 then bs else let (a, b) = bin_split n c in bin_rsplit_aux n (m - n) a (b # bs))"
zip_Cons_Cons: fixes x :: "'a"   and xs :: "'a list"   and y :: "'b"   and ys :: "'b list" shows "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
wpl_ab_def: fixes a :: "int \<Rightarrow> nat"   and b :: "int \<Rightarrow> nat" shows "wpl_ab a b \<equiv> wpl.wpl (w_ab a b)"
lt_le: fixes x :: "'a"   and y :: "'a" shows "lt\<cdot>x\<cdot>y = (le\<cdot>x\<cdot>y andalso neq\<cdot>x\<cdot>y)"
Let_def: fixes s :: "'d"   and f :: "'d \<Rightarrow> 'b" shows "Let s f \<equiv> f s"
is_category_with_binary_products: shows "category_with_binary_products C"
s: shows "s \<approx> s1 \<and> s \<approx> s2"
Iff_reduce: fixes p :: "'a pfm" shows "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! reduce p)"
order_tensor_deep_model: fixes m :: "real mat convnet" assumes "remove_weights m = deep_model_l rs" shows "order (tensors_from_net m $ y) = 2 * N_half"
YMapInv'_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and F :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) Functor"   and x :: "ZF" shows "YMapInv' C X F x \<equiv> \<lparr>NTDom = YFtor C @@ X, NTCod = F, NatTransMap = \<lambda>B. ZFfun (Hom\<^bsub>C\<^esub> B X) (F @@ B) (\<lambda>f. (F ## z2m\<^bsub>C\<^esub>f) |@| x)\<rparr>"
arcs_symmetric: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> parcs (subdivide G e w)" shows "(b, a) \<in> parcs (subdivide G e w)"
sorted_iff_nth_Suc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq xs = (\<forall>i. Suc i < length xs \<longrightarrow> less_eq (xs ! i) (xs ! Suc i))"
map_def: shows "Mapping.map \<equiv> map_fun id (map_fun id (map_fun mapping.rep Mapping)) (\<lambda>f g m. map_option g \<circ> m \<circ> f)"
simps_1: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "matpow A 0 = mat (1::'a)"
simps: fixes x :: "real"   and y :: "real" shows "(Rmin x \<le> Rmin y) = (x \<le> y)"
to_mregex_empty_progress: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and mr :: "mregex"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat" assumes "safe_regex m g r"   and "to_mregex r = (mr, [])" shows "progress_regex \<sigma> P r j = j"
monom_finsum_decomp: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "p = (\<Oplus>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub>t\<in>set (UP.dense_repr p). UP.monom (fst t) (snd t))"
G_perm_apply_identity: fixes g :: "'a" assumes "g \<in> G" shows "\<pp> g \<rightarrow> e = g"
is_map_from_is_homomorphism: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and V :: "'c set" assumes "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "is_open U"   and "is_open V"   and "V \<subseteq> U" shows "Set_Theory.map (\<rho> U V) (\<FF> U) (\<FF> V)"
semiring_simprules_9: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<one> \<otimes> x = x"
SUP_le_iff: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and u :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (f ` A)) u = (\<forall>i\<in>A. less_eq (f i) u)"
restrict_compose_in_diff_fun_space: fixes x :: "'b \<Rightarrow> real" assumes "x \<in> dest.diff_fun_space" shows "restrict0 src.carrier (x \<circ> f) \<in> src.diff_fun_space"
simps_3: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "sym.implAuth A B M = \<langle>M, hmac \<langle>AuthTag, M\<rangle> (shrK A B)\<rangle>"
fun_rep_map_of: fixes ps :: "(char list \<times> 'a) list" shows "fun_rep ps = (\<lambda>x. case map_of ps x of None \<Rightarrow> \<top> | Some a \<Rightarrow> a)"
abs_def: fixes S :: "msg set" shows "local.abs S \<equiv> {Chan x A B M |x A B M. M \<in> payload \<and> implem (Chan x A B M) \<in> S}"
Gromov_extension_preserves_extended_Gromov_product: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and xi :: "'b Gromov_completion"   and eta :: "'b Gromov_completion" assumes "isometry f" shows "extended_Gromov_product_at (f x) (Gromov_extension f xi) (Gromov_extension f eta) = extended_Gromov_product_at x xi eta"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat list"   and a5 :: "nat" assumes "mtype(a1,a2,a3) = a4 \<rightarrow> a5" obtains CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and CDef :: "classDef" and m :: "nat" and mDef :: "methodDef" and Bs :: "nat list" and B :: "nat" where "a1 = CTa__" and "a2 = m__" and "a3 = Ca__" and "a4 = Bs__" and "a5 = B__" and "CTa__ Ca__ = Some CDef__" and "lookup (cMethods CDef__) (\<lambda>md. mName md = m__) = Some mDef" and "varDefs_types (mParams mDef) = Bs__" and "mReturn mDef = B__"   | CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and CDef :: "classDef" and m :: "nat" and D :: "nat" and Bs :: "nat list" and B :: "nat" where "a1 = CT" and "a2 = m" and "a3 = C" and "a4 = Bs" and "a5 = B" and "CT C = Some CDef" and "lookup (cMethods CDef) (\<lambda>md. mName md = m) = None" and "cSuper CDef = D" and "mtype(CT,m,D) = Bs \<rightarrow> B"
neq_Nil_revE: fixes l :: "'a list" assumes "l \<noteq> []" obtains ll :: "'a list" and e :: "'a" where "l = ll @ [e]"
R: fixes p :: "'a spmf"   and q :: "'b spmf" assumes "(p, q) \<in> Y" shows "rel_spmf R p q"
semiring_norm_77: fixes n :: "Num.num" shows "(num.One < num.Bit1 n) = True"
expand_set: fixes g :: "'c \<Rightarrow> 'b" shows "Sum_any g = sum g {a. g a \<noteq> (0::'b)}"
same_cs_implies_same_resulting_cs: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and c :: "('a, 'c, 'b) configuration"   and i :: "nat"   and d :: "('a, 'c, 'b) configuration"   and trans :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and ev :: "('a, 'c, 'b) event"   and c' :: "('a, 'c, 'b) configuration"   and d' :: "('a, 'c, 'b) configuration" assumes "distributed_system channel"   and "cs c i = cs d i"   and "distributed_system.next channel trans send recv c ev c'"   and "distributed_system.next channel trans send recv d ev d'"   and "regular_event ev" shows "cs c' i = cs d' i"
red_monom_mult: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'c"   and s :: "'a" assumes "red F p q"   and "c \<noteq> (0::'c)" shows "red F (monom_mult c s p) (monom_mult c s q)"
Domainp_ZN: shows "Domainp ZN = (\<le>) 0"
real_sqrt_sum_squares_eq_cancel: fixes x :: "real"   and y :: "real" assumes "sqrt (x\<^sup>2 + y\<^sup>2) = x" shows "y = 0"
h': fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "(g has_derivative g' x) (at x within S) \<and> linear (h' x) \<and> g' x \<circ> h' x = id \<and> h' x \<circ> g' x = id"
hdomain_def: fixes r :: "hf" shows "hdomain r = \<lbrace>x . w \<^bold>\<in> r, \<exists>y. w = \<langle>x, y\<rangle>\<rbrace>"
Bignorabimus_empty: fixes \<alpha> :: "game" shows "{} \<in> Bignorabimus \<alpha>"
fps_square_nth: fixes f :: "'a fps"   and n :: "nat" shows "fps_nth (f\<^sup>2) n = (\<Sum>k\<le>n. fps_nth f k * fps_nth f (n - k))"
arr_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.mkarr C t)"
lookup_minus_pp: fixes s :: "('b, 'c) pp"   and t :: "('b, 'c) pp" shows "lookup_pp (s - t) = lookup_pp s - lookup_pp t"
simps_6: fixes f :: "'c \<Rightarrow> 'a"   and p :: "('c, 'b) form"   and q :: "('c, 'b) form" shows "psubst f (Impl p q) = Impl (psubst f p) (psubst f q)"
inv_unique': fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<oplus> y = \<zero>"   and "y \<oplus> x = \<zero>" shows "y = inv\<^bsub>add_monoid R\<^esub> x"
sum_fun_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite A" shows "\<Sum> ((\<lambda>x. f x + g x) ` A) = \<Sum> (f ` A) + \<Sum> (g ` A)"
ENC_FNot: fixes n :: "nat"   and \<phi> :: "'b formula" shows "ENC n (FOV (FNot \<phi>)) = ENC n (FOV \<phi>)"
zero_less_power: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero a" shows "less zero (power.power one times a n)"
sign_simps_29: fixes am :: "'c"   and bm :: "'c" shows "am + bm = bm + am"
ordinal_le_plusR: fixes x :: "ordinal"   and y :: "ordinal" shows "x \<le> x + y"
rQuots_rev_lQuots: fixes x :: "'c list"   and A :: "'c list set" shows "rQuots x A = rev ` lQuots x (rev ` A)"
assms_7: shows "V = (\<Oplus>U\<leftarrow>Us. U)"
strand_subst_hom_1: fixes S :: "('a, 'b) strand_step list"   and S' :: "('a, 'b) strand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "S @ S' \<cdot>\<^sub>s\<^sub>t \<theta> = (S \<cdot>\<^sub>s\<^sub>t \<theta>) @ (S' \<cdot>\<^sub>s\<^sub>t \<theta>)"
br_run_conv: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j) graph_rec_scheme"   and B :: "('i, 'k) graph_rec_scheme"   and \<alpha> :: "'h \<Rightarrow> 'i"   and I :: "'h \<Rightarrow> bool"   and rb :: "nat \<Rightarrow> 'i" assumes "bisimulation R A B"   and "R = br \<alpha> I" shows "graph_defs.is_run B rb = (\<exists>ra. rb = \<alpha> \<circ> ra \<and> graph_defs.is_run A ra)"
a5: shows "p \<cdot> y = y \<cdot> p"
inorder_n2: fixes l :: "'b bro"   and a :: "'b"   and r :: "'b bro" shows "inorder (n2 l a r) = inorder l @ a # inorder r"
TS_pairwise: shows "List_Factoring.pairwise (Partial_Cost_Model.embed (rTS []))"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "nat"   and a3 :: "nat" assumes "\<And>m k acc n. (\<And>x. \<lbrakk>x = (k + 1) ^ m; \<not> (n \<le> k \<or> n < x)\<rbrakk> \<Longrightarrow> P m (k + 1) x n) \<Longrightarrow> P m k acc n" shows "P a0 a1 a2 a3"
Rep_Program: fixes x :: "'b program" shows "Rep_Program x \<in> Program"
IdD: fixes a :: "'c"   and b :: "'c" assumes "(a, b) \<in> Id" shows "a = b"
cod_simps_3: shows "\<mu>\<nu>.cod.leg1 = \<mu>\<nu>.leg1"
bound: fixes n :: "nat"   and x :: "'a" shows "F n x \<le> ennreal (real n)"
kinds_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and xs :: "'a list" assumes "CFG sourcenode targetnode valid_edge Entry" shows "CFG.kinds kind xs \<equiv> map kind xs"
inv: shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
vlrestriction_vrrestriction: fixes r :: "V"   and A :: "V" shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A) \<restriction>\<^sup>r\<^sub>\<circ> A = r \<restriction>\<^sub>\<circ> A"
amax_ge_l: fixes x :: "ant"   and y :: "ant" shows "x \<le> amax x y"
g: shows "\<guillemotleft>g : b \<rightarrow> c\<guillemotright> \<and> equivalence_map g"
reachable_adjI: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs (subdivide G e w)" shows "u \<rightarrow>\<^sup>*\<^bsub>with_proj (subdivide G e w)\<^esub> v"
ordered_fmap_list_all: fixes S :: "('a \<times> 'b) fset"   and P :: "'a \<times> 'b \<Rightarrow> bool" assumes "is_fmap S" shows "list_all P (ordered_fmap S) = fBall S P"
alist_product_simps_1: fixes f :: "'c \<Rightarrow> 'f \<Rightarrow> 'd \<Rightarrow> 'g \<Rightarrow> 'e"   and ys :: "('d \<times> 'g) list" shows "alist_product f [] ys = []"
maxsimpD_simplex: fixes x :: "'b set" assumes "codomain.chamber x" shows "x \<in> Y"
Int_lower2: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> B \<subseteq> B"
intro: fixes G :: "('c, 'd) pre_digraph" assumes "fin_digraph G"   and "loopfree_digraph G"   and "nomulti_digraph G" shows "digraph G"
pos: fixes b :: "real" assumes "\<bar>b\<bar> < 1" shows "0 < 2 powr real_of_int k + (real_of_int r + b)"
mat_zero_l: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_zero *\<^sub>m\<^sub>m A = mat_zero"
source_target_relation_with_trans_symcl_impl_full_abstraction: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "trans (symcl Rel)" shows "\<forall>S1 S2. ((S1, S2) \<in> _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> symcl Rel)) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> symcl Rel))"
insertCI: fixes a :: "'b"   and B :: "'b set"   and b :: "'b" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> b \<triangleright> B"
map_add_def: fixes m1 :: "'b \<Rightarrow> 'c option"   and m2 :: "'b \<Rightarrow> 'c option" shows "m2 \<Oplus> m1 = (\<lambda>x. case m2 x of \<bottom> \<Rightarrow> m1 x | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>y\<rfloor>)"