hcju/isabelleps huggingface dataset:
round_stab_inv: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and q :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and n :: "nat"   and f :: "'d \<Rightarrow> 'c list"   and r :: "nat" assumes "index_less index key"   and "index_same index key"   and "bn_inv p q t"   and "add_inv n t"   and "stab_inv f key t" shows "stab_inv f key (round index key p q r t)"
corec: fixes g1 :: "'g \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a spmf"   and g2 :: "'g \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> ('e \<times> 'a) spmf"   and g3 :: "'g \<Rightarrow> 'a \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'a) spmf"   and a :: "'g" shows "corec_core g1 g2 g3 a = Core (g1 a) (g2 a) (g3 a)"
subcls_is_class2: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<preceq>\<^sub>C D"   and "is_class G D" shows "is_class G C"
while_option_stop2: fixes b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c" assumes "while_option b c s = Some t" shows "\<exists>k. t = (c ^^ k) s \<and> \<not> b t"
less_linear: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" shows "less x y \<or> x = y \<or> less y x"
Impl: fixes s :: "state"   and Cm :: "cname \<times> mname"   and n :: "nat"   and s' :: "state" assumes "s -body Cm-n\<rightarrow> s'" shows "s -Impl Cm-Suc n\<rightarrow> s'"
exitI2: fixes s :: "'state"   and K :: "'state \<Rightarrow> bool"   and v :: "'value" assumes "reachNT s"   and "K s"   and "invarNT K"   and "noVal K v" shows "exit s v"
Cauchy_next_derivative_circlepath_2: fixes z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and k :: "nat"   and g :: "complex \<Rightarrow> complex"   and w :: "complex" assumes "continuous_on (path_image (circlepath z r)) f"   and "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ k) has_contour_integral g w) (circlepath z r)"   and "k \<noteq> 0"   and "w \<in> ball z r" shows "(g has_field_derivative of_nat k * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc k)) (at w)"
sorted_wrt_dist_last: fixes p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list" assumes "sorted_wrt_dist p ps" shows "\<forall>q\<in>set ps. dist q p \<le> dist (last ps) p"
rev_induct: fixes P :: "'c list \<Rightarrow> bool"   and xs :: "'c list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (xs @ [x])" shows "P xs"
rel_generat_resultD: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Out :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "('g, 'i, 'k) generat"   and y :: "('h, 'j, 'l) generat" assumes "rel_generat A Out R x y"   and "is_Pure x \<or> is_Pure y" shows "A (generat.result x) (generat.result y)"
psimps: fixes l :: "nat list"   and i :: "nat" assumes "rep_of_dom (l, i)" shows "rep_of l i = (if l ! i = i then i else rep_of l (l ! i))"
a: shows "abc_notfinal (abc_steps_l (0, lm) A na) A \<and> abc_final (abc_steps_l (0, lm) A (Suc na)) A"
ordertype_image_ordermap: fixes A :: "'a set"   and X :: "'a set"   and r :: "('a \<times> 'a) set" assumes "small A"   and "X \<subseteq> A"   and "wf r"   and "trans r"   and "total_on X r" shows "ordertype (ordermap A r ` X) VWF = ordertype X r"
wt_u_ge_s: shows "wt u \<ge>\<^sub>p wt s"
inf_synlr_def: fixes f :: "('a, 'b) synlr"   and g :: "('a, 'b) synlr" shows "inf f g \<equiv> mksynlr (unsynlr f \<inter> unsynlr g)"
permutes_conv_has_dom: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set" shows "(f permutes S) = (bij f \<and> has_dom f S)"
sublens_def: fixes X :: "'c \<Longrightarrow> 'd"   and Y :: "'e \<Longrightarrow> 'd" shows "X \<subseteq>\<^sub>L Y = (\<exists>Z. vwb_lens Z \<and> X = Z ;\<^sub>L Y)"
\<tau>red1t_1step: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list" assumes "uf,P,t \<turnstile>1 \<langle>e,(h, xs)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>e',(h, xs')\<rangle>"   and "\<tau>move1 P h e" shows "\<tau>red1gt uf P t h (e, xs) (e', xs')"
ring_tOp_closed: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and y :: "'e" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<in> carrier R"
adjunct_ord_def: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" shows "adjunct_ord D a = D\<lparr>carrier := carrier D \<union> {a}, rel := {(x, y). (x, y) \<in> rel D \<or> x \<in> carrier D \<union> {a} \<and> y = a}\<rparr>"
b2: shows "Re (infsetsum g A) = (\<Sum>\<^sub>ax\<in>A. Re (g x))"
fds_shift_prod: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a fds"   and A :: "'b set" shows "fds_shift a (prod f A) = (\<Prod>x\<in>A. fds_shift a (f x))"
countable_Int1: fixes A :: "'c set"   and B :: "'c set" assumes "countable A" shows "countable (A \<inter> B)"
is_Var_def: fixes "term" :: "('a, 'b) Term.term" shows "is_Var term = (\<exists>x1. term = Var x1)"
inner_zero_right: fixes x :: "'a" shows "x \<bullet> (0::'a) = 0"
of_rat_le_0_iff: fixes r :: "rat" shows "(of_rat r \<le> (0::'a)) = (r \<le> 0)"
pid_system_automaton'_def: shows "pid_system_automaton' \<equiv> \<lparr>g_V = UNIV, g_E = pid_gstep, g_V0 = {pid_init}, sa_L = pid_label\<rparr>"
norm1: shows "norm1 (a *\<^sub>R v + b *\<^sub>R w) = 1"
Bseq_offset: fixes X :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "Bseq (\<lambda>n. X (n + k))" shows "Bseq X"
natural_transformation_axioms: shows "natural_transformation B D (K \<circ> F) (M \<circ> H) \<mu>o\<nu>_\<sigma>o\<tau>.map"
neq_Nil_conv: fixes xs :: "'c list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
rs: shows "rs = repeat_stabilize 1000 (optimize_matches opt_MatchAny_match_expr) (optimize_matches optimize (rw_Reject (rm_LogEmpty (repeat_stabilize 10000 (process_call (map_of \<Gamma>)) [Rule MatchAny (Call chain_name), Rule MatchAny default_action]))))"
bind_ask2: fixes m :: "'m"   and f :: "'s \<Rightarrow> 'r \<Rightarrow> 'm" shows "bind m (\<lambda>x. ask (f x)) = ask (\<lambda>r. bind m (\<lambda>x. f x r))"
mult_mono': fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero a"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
prime_divisorE: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero"   and "\<not> dvd.dvd times a one" obtains p :: "'b" where "normalization_semidom.prime zero times one normalize p" and "dvd.dvd times p a"
isUbD2a: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isUb R S x" shows "x \<in> R"
adjuster_carrier': fixes w :: "'b vec"   and n :: "nat"   and us :: "'b vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n" shows "adjuster n w us \<in> carrier_vec n"
restrict_continuous_map: fixes X :: "'b topology"   and S :: "'b set"   and X' :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "topspace X \<subseteq> S" shows "continuous_map X X' (restrict f S) = continuous_map X X' f"
prime_factors_fact: fixes n :: "nat" shows "prime_factors (fact n) = {p \<in> {2..n}. prime p}"
l_lagrange: fixes H :: "'a set" assumes "finite (carrier G)"   and "subgroup H G" shows "card (lcosets H) * card H = order G"
max_filter_ultrafilter: fixes F :: "'b filter" assumes "F \<noteq> bot"   and "\<And>G. \<lbrakk>G \<noteq> bot; G \<le> F\<rbrakk> \<Longrightarrow> F = G" shows "ultrafilter F"
restrict_to_singleton: fixes P :: "('c \<times> 'd) set"   and x :: "'c" shows "P || {x} = {x} \<times> P `` {x}"
elims_2: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "('d, 'e, 'f) FSM"   and xb :: "('d \<times> 'e) list" assumes "sequence_to_failure x xa xb" obtains M1 :: "('d, 'e, 'f) FSM" and M2 :: "('d, 'e, 'f) FSM" and xs :: "('d \<times> 'e) list" where "x = M1" and "xa = M2" and "xb = xs" and "butlast xs \<in> L M2 \<inter> L M1 \<and> xs \<in> L M1 - L M2"
nextElem_prevElem: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x \<in> set xs" shows "nextElem xs (hd xs) (nextElem (rev xs) (last xs) x) = x"
presAbs_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)" shows "presAbs (fromIMor h) (fromIMorAbs hA) MOD"
integral_has_vector_derivative: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and x :: "real" assumes "continuous_on {a..b} f"   and "x \<in> {a..b}" shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b})"
f_cd_mw: fixes w :: "complex"   and m :: "nat" assumes "w \<in> s" shows "f ^^ m field_differentiable at w"
aless_le_trans: fixes x :: "ant"   and y :: "ant"   and z :: "ant" assumes "x < y"   and "y \<le> z" shows "x < z"
sep_ne: fixes M :: "(nat \<times> 'a) set"   and g :: "graph" shows "\<exists>P\<subseteq>M. separated g (fst ` P)"
converse_Rel_def: fixes T :: "V" shows "T\<inverse>\<^sub>R\<^sub>e\<^sub>l = [(T\<lparr>ArrVal\<rparr>)\<inverse>\<^sub>\<circ>, T\<lparr>ArrCod\<rparr>, T\<lparr>ArrDom\<rparr>]\<^sub>\<circ>"
countable_UN: fixes A :: "'b \<Rightarrow> 'a set"   and X :: "'b set" assumes "A ` X \<subseteq> M" shows "\<Union> (A ` X) \<in> M"
matrix_index: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zer :: "'a"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inver :: "'a \<Rightarrow> 'a"   and m1 :: "'a list list"   and n :: "nat"   and nc :: "nat"   and m2 :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "plus_mult id f zer g inver"   and "Matrix_Legacy.mat (mult.row_length m1) n m1"   and "Matrix_Legacy.mat n nc m2"   and "i < mult.row_length m1"   and "j < nc" shows "mat_multI zer g f (mult.row_length m1) m1 m2 ! j ! i = plus_mult.scalar_product f zer g (Matrix_Legacy.row m1 i) (Matrix_Legacy.col m2 j)"
monomial_0_iff: fixes t :: "'b"   and c :: "'c" shows "(monomial c t = 0) = (c = (0::'c))"
poly_monom: fixes a :: "'c"   and n :: "nat"   and x :: "'c" shows "poly (monom a n) x = a * x ^ n"
min_int_poly_complex_of_real: fixes x :: "real" shows "min_int_poly (complex_of_real x) = min_int_poly x"
ring_irreducibleI: fixes r :: "'a" assumes "r \<in> carrier R - {\<zero>}"   and "r \<notin> Units R"   and "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; r = a \<otimes> b\<rbrakk> \<Longrightarrow> a \<in> Units R \<or> b \<in> Units R" shows "ring_irreducible r"
rvars_eq_subst_var_eq: fixes x\<^sub>j :: "nat"   and lp :: "linear_poly"   and eq :: "nat \<times> linear_poly" shows "rvars_eq (subst_var_eq x\<^sub>j lp eq) \<subseteq> rvars_eq eq - {x\<^sub>j} \<union> vars lp"
intros: fixes ra :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and adal :: "'b \<times> addr_loc"   and P :: "'c prog"   and wa :: "nat" assumes "ra \<in> read_actions E"   and "adal \<in> action_loc P E ra"   and "E \<turnstile> wa \<le>a ra"   and "wa \<in> write_actions E"   and "adal \<in> action_loc P E wa"   and "\<And>wa'. \<lbrakk>wa' \<in> write_actions E; adal \<in> action_loc P E wa'\<rbrakk> \<Longrightarrow> E \<turnstile> wa' \<le>a wa \<or> E \<turnstile> ra \<le>a wa'" shows "P,E \<turnstile> ra \<leadsto>mrw wa"
sum_eq: fixes x :: "i"   and y :: "i" assumes "CM" shows "x \<oplus> y = (THE z. \<forall>v. O v z = (O v x \<or> O v y))"
num_abs_bintr: fixes x :: "num" shows "numeral x = word_of_int (take_bit LENGTH('b) (numeral x))"
mono_def: fixes Retr :: "'b \<Rightarrow> 'c" shows "Bisim.mono Retr \<equiv> \<forall>theta theta'. theta \<le> theta' \<longrightarrow> Retr theta \<le> Retr theta'"
children_def: fixes SkewBinomialTree :: "('c, 'd) SkewBinomialTree" shows "children SkewBinomialTree = (case SkewBinomialTree of Node x1 x2 x3 x4 \<Rightarrow> x4)"
Suc_2: shows "Suc n = 1 + length r - length d"
space_stream_filtration: fixes M :: "'b measure"   and n :: "enat" shows "space (stream_filtration M n) = space (stream_space M)"
wqo_on_imp_almost_full_on: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wqo_on P A" shows "almost_full_on P A"
iT_Mult_finite_iff: fixes k :: "nat"   and I :: "nat set" assumes "0 < k" shows "finite (I \<otimes> k) = finite I"
find_Some_D: fixes f :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and y :: "'a" assumes "find f xs = Some y" shows "y \<in> set xs \<and> f y"
to_partial_def: fixes f :: "'c \<Rightarrow> 'd" shows "to_partial f = (\<lambda>x. Some (f x))"
rbt_insert_rbt_sorted: fixes t :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c" assumes "rbt_sorted t" shows "rbt_sorted (rbt_insert k v t)"
fodd_def: shows "PROP fodd \<equiv> (\<And>n kvs t k v kvs'. \<lbrakk>0 < n; n \<le> length kvs; P n kvs; rbtreeify_f n kvs = (t, (k, v) # kvs'); n \<le> length kvs'; P n kvs'\<rbrakk> \<Longrightarrow> P (Suc (2 * n)) kvs)"
X_2: shows "add_mset x' X' = add_mset x X"
a_consistent_lamvar: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and S :: "stack_elem list"   and e :: "exp"   and u :: "Arity" assumes "a_consistent (ae, a, as) (\<Gamma>, Var x, S)"   and "map_of \<Gamma> x = Some e"   and "ae x = up\<cdot>u" shows "a_consistent (ae, u, as) ((x, e) # delete x \<Gamma>, e, S)"
simps_2: fixes a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and vs :: "'a set"   and s :: "('a, 'b) fmap" shows "vars_change (a # as) vs s = (if fmrestrict_set vs (state_succ s a) \<noteq> fmrestrict_set vs s then state_succ s a # vars_change as vs (state_succ s a) else vars_change as vs (state_succ s a))"
Cauchy: fixes x :: "'a" assumes "x \<in> R" shows "Cauchy (\<lambda>n. \<F> (k n) x)"
induct: fixes P :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b tree" assumes "\<And>t. P t \<langle>\<rangle>"   and "\<And>v va vb. P \<langle>\<rangle> \<langle>v, va, vb\<rangle>"   and "\<And>t1 a t2 t3 b t4. P t2 t3 \<Longrightarrow> P \<langle>t1, a, t2\<rangle> \<langle>t3, b, t4\<rangle>" shows "P a0 a1"
Says_tgs_message_form: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and servTicket :: "msg"   and evs :: "event list" assumes "Gets A (Crypt authK \<lbrace>Key servK, Agent B, Ts, servTicket\<rbrace>) \<in> set evs"   and "authK \<in> symKeys"   and "evs \<in> kerbIV_gets" shows "servK \<notin> range shrK \<and> (\<exists>A. servTicket = Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace>) \<or> servTicket \<in> analz (knows Spy evs)"
fusion_spec_local_a: fixes a :: "'a"   and init :: "'a set"   and p :: "'a \<times> 'c \<Rightarrow> bool"   and r :: "'a \<times> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "a \<in> init" shows "([: x \<leadsto> (u, y) . u \<in> init \<and> x = y :] \<circ> {. p .} \<circ> [: r :]) \<parallel> [: r' :] = [: x \<leadsto> (u, y) . u \<in> init \<and> x = y :] \<circ> {. p .} \<circ> [: (u, x) \<leadsto> y . r (u, x) y \<and> r' x y :]"
assms: shows "y * x\<^sup>\<star> * y \<le> y"
urel_integer_def: fixes x :: "integer"   and y :: "int" shows "urel_integer x y = (y = int_of_integer x \<and> 0 \<le> y \<and> y < p)"
label_wrt_special_coset1: fixes \<phi> :: "'a \<Rightarrow> 'a"   and w0 :: "'a permutation"   and s :: "'a permutation" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0"   and "w0 \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"   and "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "{w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> (w0 \<rightarrow> fundantivertex s) = w0 \<rightarrow> fundantivertex s} = w0 +o \<langle>(\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) - {s}\<rangle>"
zero_mat_def: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<equiv> mat nr nc (\<lambda>ij. 0::'d)"
Map_in_Hom: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" shows "Map f \<in> Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom f) (Cod f))"
prime_dvd_fact_iff: fixes p :: "nat"   and n :: "nat" assumes "prime p" shows "(p dvd fact n) = (p \<le> n)"
relation_of_Assign: fixes x :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)"   and e :: "'d \<Rightarrow> 'e" shows "action.relation_of (ASSIGN x e) = R (true \<turnstile> \<lambda>(S, S'). tr S' = tr S \<and> \<not> wait S' \<and> alpha_rp.more S' = update x (\<lambda>_. e (alpha_rp.more S)) (alpha_rp.more S))"
res: fixes z :: "'a option list" shows "(restrict A_x z \<in> join x False y) = (restrict A_x z \<in> x \<and> restrict A_y z \<notin> y)"
arr_lunit: fixes a :: "'d" assumes "D.ide a" shows "D.arr \<l>\<^sub>D[a]"
simps_2: fixes x :: "bool" shows "sign Neg x = (\<not> x)"
plus_float: fixes r :: "real"   and p :: "real" assumes "r \<in> float"   and "p \<in> float" shows "r + p \<in> float"
face_cycle_set_def: fixes M :: "'b pre_map"   and a :: "'b" shows "pre_digraph_map.face_cycle_set M a = orbit (pre_digraph_map.face_cycle_succ M) a"
ax_valids2_def: fixes G :: "prog"   and A :: "'b triple set"   and ts :: "'b triple set" shows "G,A|\<Turnstile>\<Colon>ts = (\<forall>n. (\<forall>t\<in>A. G\<Turnstile>n\<Colon>t) \<longrightarrow> (\<forall>t\<in>ts. G\<Turnstile>n\<Colon>t))"
charts_sphere_def: shows "charts_sphere \<equiv> {st_proj1_chart, st_proj2_chart}"
one_xor_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "xor one a = minus (plus a (zero_neq_one.of_bool one zero (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a))) (zero_neq_one.of_bool one zero (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a))"
lookup_keys: fixes t :: "('b, 'c) RBT.rbt" shows "dom (RBT.lookup t) = set (RBT.keys t)"
not_infinity_eq: fixes x :: "enat" shows "(x \<noteq> \<infinity>) = (\<exists>i. x = enat i)"
Sim_def: fixes A :: "'c DFIG_alphabet_scheme" shows "Sim A = FIG_alphabet.make (DFIG_alphabet.v A) (DFIG_alphabet.x A) {a. a < DFIG_alphabet.max A \<and> a \<notin> retidS A}"
inorder_balance_tree: fixes t :: "'a tree" shows "inorder (balance_tree t) = inorder t"
free_generator_generator: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "free_generator R M H" shows "generator R M H"
satisfies_bounded_lderiv: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula"   and x :: "'x" assumes "local.wf (#\<^sub>V \<AA>) \<phi>"   and "#\<^sub>V \<AA> = size x"   and "lformula \<phi>" shows "(\<AA> \<Turnstile>\<^sub>b lderiv x \<phi>) = (CONS x \<AA> \<Turnstile>\<^sub>b \<phi>)"
io_10: shows "inside \<union> outside = - path_image (complex_of \<circ> c)"
less_ordinal_of_nat: fixes x :: "ordinal"   and n :: "nat" shows "(x < ordinal_of_nat n) = (\<exists>m. x = ordinal_of_nat m \<and> m < n)"
prems: shows "{$$} \<turnstile>\<^sub>W e : \<tau>"
assms: fixes n :: "nat"   and d :: "Value"   and d' :: "Value" shows "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>(d \<down>Fn d') = \<psi>\<^sup>D\<^bsub>Suc n\<^esub>\<cdot>d \<down>Fn \<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>d'"
2: shows "\<forall>f\<in>set fs. PRIMREC f \<and> (\<exists>kf. \<forall>l. f l < ack kf (sum_list l))"
gbm_invar_def: fixes m1_ops :: "('d, 'e, 'f, 'g) map_ops_scheme"   and m2_ops :: "('d, 'h, 'e, 'i) map_ops_scheme"   and s3_ops :: "('j, 'h, 'k) set_ops_scheme"   and m1 :: "'f" shows "GraphByMapDefs.gbm_invar m1_ops m2_ops s3_ops m1 \<equiv> map_op_invar m1_ops m1 \<and> (\<forall>m2\<in>ran (map_op_\<alpha> m1_ops m1). map_op_invar m2_ops m2 \<and> (\<forall>s3\<in>ran (map_op_\<alpha> m2_ops m2). set_op_invar s3_ops s3)) \<and> valid_graph (GraphByMapDefs.gbm_\<alpha> m1_ops m2_ops s3_ops m1)"
pickInp_def: fixes inp :: "'i \<Rightarrow> 'j set option" shows "pickInp inp \<equiv> lift pick inp"
i_p_2: fixes j :: "nat" assumes "i \<le> j"   and "enat j < llength Sts" shows "C \<in> N_of_wstate (lnth Sts j) \<union> P_of_wstate (lnth Sts j)"
G_def: shows "G = F \<inter> H - (F - H')"
funpow_mult: fixes n :: "nat"   and m :: "nat"   and f :: "'a \<Rightarrow> 'a" shows "(f ^^ m) ^^ n = f ^^ (m * n)"
toList_inject: fixes x :: "'b odlist"   and y :: "'b odlist" shows "(toList x = toList y) = (x = y)"
multiplicity_prime_elem_times_other: 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 p :: "'a"   and q :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "\<not> dvd.dvd times p q" shows "factorial_semiring.multiplicity times one p (times q x) = factorial_semiring.multiplicity times one p x"
not_ordLeq_iff_ordLess: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Well_order r"   and "Well_order r'" shows "((r', r) \<notin> ordLeq) = (r <o r')"
hom_mult: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a" assumes "x \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"   and "y \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)" shows "(case x \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> y of (x, y) \<Rightarrow> x \<otimes> y) = (case x of (x, y) \<Rightarrow> x \<otimes> y) \<otimes> (case y of (x, y) \<Rightarrow> x \<otimes> y)"
rec_inseparable_many_reducible: fixes f :: "nat \<Rightarrow> nat"   and A :: "nat set"   and B :: "nat set" assumes "total_recursive f"   and "rec_inseparable (f -` A) (f -` B)" shows "rec_inseparable A B"
fresh_1: fixes a :: "name"   and x1 :: "char list" shows "a \<sharp> tVAR x1 = a \<sharp> x1"