hcju/isabelleps huggingface dataset:
gauss_cnj_power: fixes a :: "gauss_int"   and n :: "nat" shows "gauss_cnj (a ^ n) = gauss_cnj a ^ n"
eq: fixes i :: "nat"   and x :: "nat \<Rightarrow> real" shows "(if i = k then x k * x k else x i * x i) = x i * x i"
parallel_converter2_id_id: fixes \<I>1 :: "('i, 'k) \<I>"   and \<I>2 :: "('j, 'l) \<I>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2,\<I>1 \<oplus>\<^sub>\<I> \<I>2 \<turnstile>\<^sub>C 1\<^sub>C |\<^sub>= 1\<^sub>C \<sim> 1\<^sub>C"
closure_of_product_topology: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and S :: "'c \<Rightarrow> 'd set" shows "product_topology X I closure_of Pi\<^sub>E I S = (\<Pi>\<^sub>E i\<in>I. X i closure_of S i)"
shared_eq: shows "\<forall>a\<in>\<O> \<union> all_acquired sb. \<S>' a = \<S> a"
assms: shows "P,h \<turnstile> v :\<le> T"
simps_3: fixes \<sigma> :: "nat \<Rightarrow> exp option"   and e :: "exp"   and m :: "nat"   and es :: "exp list" shows "substs \<sigma> (MethodInvk e m es) = MethodInvk (substs \<sigma> e) m (subst_list1 \<sigma> es)"
AE_cong: fixes M :: "'d measure"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x. x \<in> space M \<Longrightarrow> P x = Q x" shows "almost_everywhere M P = almost_everywhere M Q"
carrier_subgroup_generated: fixes G :: "('f, 'g) monoid_scheme"   and S :: "'f set" shows "carrier (subgroup_generated G S) = generate G (carrier G \<inter> S)"
monoid_def: fixes G :: "('c, 'd) monoid_scheme" shows "Group.monoid G \<equiv> ((\<forall>x y. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G) \<and> (\<forall>x y z. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> z \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z))) \<and> \<one>\<^bsub>G\<^esub> \<in> carrier G \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x) \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x)"
elims_2: fixes x :: "('b, 'c) Combinators list" assumes "wellformed_policy1 x" obtains "x = []"   | xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "DenyAll \<notin> set xs"
transp_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "transp less_eq"
valid_path_ltl': fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and Ps :: "'c llist" assumes "Digraph G"   and "Digraph.valid_path G (LCons v Ps)" shows "Digraph.valid_path G Ps"
R: shows "rel_set R (set_pmf p) (set_pmf q)"
bigo: shows "(\<lambda>n. sum f {1..n} - g (real n)) \<in> O(\<lambda>n. h (real n))"
Inf_closed': shows "\<forall>x\<subseteq>U. \<Sqinter>\<^sub>o\<^sub>w x \<in> U"
uniformly_convergent_mult: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and c :: "'b" assumes "uniformly_convergent_on A f" shows "uniformly_convergent_on A (\<lambda>k x. c * f k x)"
spmf_rel_mono_strong: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a spmf"   and g :: "'b spmf"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. \<lbrakk>x \<in> set_spmf f; y \<in> set_spmf g; A x y\<rbrakk> \<Longrightarrow> B x y" shows "rel_spmf B f g"
crit_pair_cbelow_imp_confluent_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "\<And>p q. \<lbrakk>p \<in> F; q \<in> F; p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m F p q" shows "relation.is_confluent_on (red F) (dgrad_p_set d m)"
eq0: shows "a * x\<^sup>2 + b * x + c = 0"
pprod_component_of_term: shows "pprod.component_of_term = snd"
ASSUME_def: shows "ASSUME \<equiv> iASSUME RETURN"
linear: fixes x :: "'a"   and y :: "'a" shows "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
cInf_atLeast: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Inf (ord.atLeast less_eq x) = x"
assms_2: shows "k_\<Lambda>_PBD V B \<Lambda> (int k)"
l2_inv5I: fixes x :: "l2_state" assumes "\<And>Rb A B gnx Ni. \<lbrakk>hmac \<langle>Number 1, Exp Gen (NonceF (Rb $ ny)), gnx, Messages.Agent A, Messages.Agent B\<rangle> (hmac Ni (NonceF (Rb $ nr))) \<in> parts (extr (bad x) (ik x) (chan x)); guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; A \<notin> bad x \<and> B \<notin> bad x\<rbrakk> \<Longrightarrow> \<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress x Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Ra xgnx = Some gnx \<and> sklvl1.guessed_frame Ra xni = Some Ni \<and> sklvl1.guessed_frame Ra xnr = Some (NonceF (Rb $ nr)) \<and> sklvl1.guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny)))" shows "x \<in> l2_inv5"
last_index_Cons: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" shows "last_index (x # xs) y = (if x = y then if x \<in> set xs then last_index xs y + 1 else 0 else last_index xs y + 1)"
select_convs_1: fixes lens_get :: "'d \<Rightarrow> 'e"   and lens_put :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and more :: "'f" shows "get\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_get"
finite_dom_iff_fsubst: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "finite (subst_domain \<theta>) = (\<theta> \<in> fsubst)"
Ord_vpair_in_Vset_succI: fixes \<alpha> :: "V"   and a :: "V"   and b :: "V" assumes "Ord \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "b \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ (ZFC_in_HOL.succ \<alpha>))"
m1_inv2r_servD: fixes x :: "'a m1r_state_scheme"   and Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and nlb :: "atom list" assumes "runs x Rb = Some (Resp, [A, B], aKey Kab # aNum Ts # nlb)"   and "x \<in> m1_inv2r_serv"   and "B \<notin> bad" shows "\<exists>Rs Na. Kab = sesK (Rs $ sk) \<and> runs x Rs = Some (Serv, [A, B], [aNon Na, aNum Ts])"
simpleDef: assumes "v \<in> defs g (defNode g v)" shows "thesis"
pref_lcp_pref': fixes w :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "w \<le>p u \<and>\<^sub>p v" shows "w \<le>p v"
statEqvt': fixes p :: "(name \<times> name) list"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "p \<bullet> \<Psi> \<otimes> \<Psi>' = (p \<bullet> \<Psi>) \<otimes> p \<bullet> \<Psi>'"
simps_1: shows "sys 0 = CH (VAR (Suc 0)) (ACT ''b'' (VAR 0))"
bset_of_option_def: shows "bset_of_option = case_option bempty bsingleton"
deepR_inv: fixes n :: "nat"   and m :: "('b, 'c) FingerTreeStruc"   and pr :: "('b, 'c) Digit"   and sf :: "('b, 'c) Node list" assumes "is_leveln_ftree (Suc n) m \<and> is_measured_ftree m"   and "is_leveln_digit n pr \<and> is_measured_digit pr"   and "\<forall>x\<in>set sf. (is_measured_node x \<and> is_leveln_node n x) \<and> length sf \<le> 4" shows "is_leveln_ftree n (deepR pr m sf) \<and> is_measured_ftree (deepR pr m sf)"
less_cons: fixes s :: "'a list"   and t :: "'a list"   and a :: "'a" assumes "s < t" shows "a # s < a # t"
module_hom_with_overloaded: shows "module_hom = (\<lambda>scale\<^sub>1. module_hom_with (+) (0::'e) (-) uminus scale\<^sub>1 (+) (0::'f) (-) uminus)"
Sup_finite_2_def: fixes A :: "finite_2 set" shows "\<Squnion> A = (if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
d_increasing_sub_identity: fixes x :: "'a" assumes "x \<le> top" shows "x \<le> - - x"
uncurry0_apply: fixes c :: "'e"   and x :: "unit" shows "uncurry0 c x = c"
fields_f_def: fixes P :: "cld list"   and ty :: "ty" shows "fields_f P ty = (case find_path_ty_f P ty of None \<Rightarrow> None | Some ctxclds \<Rightarrow> Some (fields_in_path_f ctxclds))"
assms_2: shows "KD_Tree.size_kdt kdt \<le> KD_Tree.size_kdt kdt'"
extensions_are_safe: fixes x' :: "nat"   and p :: "int mpoly"   and \<V> :: "nat set"   and A :: "int mpoly multiset"   and q :: "int mpoly" assumes "x' \<in> vars p"   and "x' \<notin> \<V>"   and "\<Union> (vars ` set_mset A) \<subseteq> \<V>"   and "MPoly_Type.coeff p (monomial (Suc 0) x') = 1"   and "vars q \<subseteq> \<V>"   and "q \<in> More_Modules.ideal (insert p (set_mset A \<union> polynomial_bool))"   and "x' \<notin> vars (p - Var x')"   and "(Var x' - p)\<^sup>2 - (Var x' - p) \<in> More_Modules.ideal polynomial_bool" shows "q \<in> More_Modules.ideal (set_mset A \<union> polynomial_bool)"
ppos: fixes s :: "'b list" assumes "True" shows "0 \<le> real (length s)"
simps_1: fixes uu :: "'a infty" shows "(Infty < uu) = False"
simps: fixes uu :: "'d list"   and XL :: "('a, 'b, 'c, 'd, 'e) qTerm set list"   and uv :: "('a, 'b, 'c, 'd, 'e) qAbs set list"   and uw :: "('c \<Rightarrow> 'd \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set option) list" shows "termsOf (Par uu XL uv uw) = set XL"
class_wf: fixes P :: "'a prog"   and C :: "char list"   and c :: "'a class"   and wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool" assumes "class P C = \<lfloor>c\<rfloor>"   and "wf_prog wf_md P" shows "wf_cdecl wf_md P (C, c)"
find_base_vectors_2: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "row_echelon_form A"   and "A \<in> carrier_mat nr nc" shows "0\<^sub>v nc \<notin> set (find_base_vectors A)"
map_constr_eval: fixes S :: "nat \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "map_constr S f n = Map (\<lambda>i. if i < n then if S i then Some (f i) else None else None)"
identity_rules_graph_rule: fixes x :: "('b Standard_Constant, nat) labeled_graph \<times> ('b Standard_Constant, nat) labeled_graph"   and L :: "'b Standard_Constant set" assumes "x \<in> identity_rules L" shows "graph_rule x"
contk: shows "continuous_on ({0..1} \<times> U) k"
i_Exec_Stream_nth: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d"   and n :: "nat" shows "i_Exec_Comp_Stream trans_fun input c n = f_Exec_Comp trans_fun (input \<Down> Suc n) c"
le_inf_reflp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a stream" assumes "traces ind" shows "traces.le_inf ind w w"
ddu: shows "du * cnj du - nu * cnj nu = ddu * kk"
setdist_empty1: fixes t :: "'b set" shows "setdist {} t = 0"
no_state_change_if_nonregular_event: fixes ev :: "('a, 'b, 'c) event"   and c :: "('a, 'b, 'c) configuration"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a" assumes "\<not> regular_event ev"   and "c \<turnstile> ev \<mapsto> c'" shows "states c p = states c' p"
PiF_gen_subset: fixes I :: "'c set set"   and M :: "'c \<Rightarrow> 'd measure" shows "{Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} \<subseteq> Pow (\<Union>J\<in>I. \<Pi>' j\<in>J. space (M j))"
lnull_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lnull (lappend xs ys) = (lnull xs \<and> lnull ys)"
minus_mult_distrib_mat: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and C :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat nr n"   and "C \<in> carrier_mat n nc" shows "(A - B) * C = A * C - B * C"
add_left_cancel: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.cancel_semigroup_add plus" shows "(plus a b = plus a c) = (b = c)"
ih\<^sub>2: fixes \<Delta> :: "binding list" assumes "\<turnstile> \<Delta> ok"   and "\<Delta> extends (TVarB X T\<^sub>1 # \<Gamma>)" shows "\<Delta>\<turnstile>S\<^sub>2<:T\<^sub>2"
distinct_47: fixes x3 :: "nop_type"   and x6 :: "arith_type" shows "nop_type x3 \<noteq> arith_type x6"
abs_eq: fixes x :: "('b, 'c) oalist_ntm" shows "MP_oalist x = Abs_poly_mapping (OAlist_lookup_ntm x)"
comp_arr_inv: fixes f :: "'b"   and g :: "'b" assumes "D.inverse_arrows f g" shows "f \<cdot>\<^sub>D g = D.dom g"
minus_ms_aux: fixes basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and G :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "is_expansion_aux F f basis"   and "is_expansion_aux G g basis" shows "is_expansion_aux (minus_ms_aux F G) (\<lambda>x. f x - g x) basis"
common_matcher_SrcDst_defined_2: fixes m :: "'a ipt_iprange"   and p :: "('a, 'b) tagged_packet_scheme" shows "common_matcher (Dst m) p \<noteq> TernaryUnknown"
v\<^sub>m_def_1: shows "v\<^sub>m \<in> set fs"
extended_distance_basepoint: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open" shows "metric_space.extended_distance dist (basepoint x) (basepoint y) = dist x y"
locally_full: fixes f :: "'b"   and f' :: "'b"   and \<nu> :: "'a" assumes "D.ide f"   and "D.ide f'"   and "src\<^sub>D f = src\<^sub>D f'"   and "trg\<^sub>D f = trg\<^sub>D f'"   and "\<guillemotleft>\<nu> : F.right_map f \<Rightarrow>\<^sub>C F.right_map f'\<guillemotright>" shows "\<exists>\<mu>. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D f'\<guillemotright> \<and> F.right_map \<mu> = \<nu>"
F_def: fixes s :: "'s"   and t :: "'s" shows "F s t = T.prob s {\<omega> \<in> space (T s). ev (HLD {t}) (s ## \<omega>)}"
PQ_def: shows "PQ = Pos ''Q'' [Fun ''d'' []]"
s_s'_3: shows "\<R>\<^sub>G \<alpha> \<beta> s = s'"
ord_le_eq_subst: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "a \<le> b"   and "f b = c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
inf_def: fixes a :: "'b G"   and b :: "'b G" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
twom: fixes e :: "real" assumes "0 < e" shows "\<forall>\<^sub>F n in at w. \<forall>x\<in>path_image \<gamma>. cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - inverse (x - w) ^ Suc k) < e"
total_llt: fixes A :: "nat list set" shows "total_on A (lenlex less_than)"
list: shows "[0..<i] = [0..<i - 1] @ [i - 1]"
subtype_def: fixes t :: "'BExp set"   and P :: "'BExp set"   and t' :: "'BExp set" shows "t \<le>:\<^sub>P t' \<equiv> (P \<union> t') \<turnstile> t"
Main: fixes n :: "nat" assumes "N \<le> n"   and "2 * Cf / a \<le> B (Suc n)" shows "B (Suc n) \<le> 2 * C0 / a * B n"
set_alist_product: fixes f :: "'c \<Rightarrow> 'f \<Rightarrow> 'd \<Rightarrow> 'g \<Rightarrow> 'e"   and xs :: "('c \<times> 'f) list"   and ys :: "('d \<times> 'g) list" shows "set (alist_product f xs ys) = (\<lambda>((a, b), c, d). ((a, c), f a b c d)) ` (set xs \<times> set ys)"
bisim': shows "t \<turnstile> (x1', m1') \<approx> (x2', m2')"
weakBisim_def: shows "weakBisim \<equiv> {(xb, xa, x). weakBisimp xb xa x}"
supp_def: fixes x :: "'e" shows "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
min_plus_max: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.add_linorder plus minus zero less_eq less" shows "plus (ord.min less_eq a b) (ord.max less_eq a b) = plus a b"
module_hom_ident: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "module scale" shows "module_hom scale scale (\<lambda>x. x)"
cod_simp: fixes f :: "'c local.term set \<times> 'c local.term set" assumes "\<F>C.CC.arr f" shows "\<F>C.CC.cod f = (\<F>C.cod (fst f), \<F>C.cod (snd f))"
hyps: fixes i :: "nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "share_all_until_volatile_write ts \<S> = share_all_until_volatile_write (ts[i := (p', is', \<theta>', sb', \<D>', \<O>', \<R>')]) \<S>"
rw3: shows "fst (snd (snd (foldl echelon_form_of_column_k_det_iarrays (1::'a, matrix_to_iarray A, 0, bezout) [0..<Suc k]))) = snd (snd (foldl (echelon_form_of_column_k_det bezout) (1::'a, A, 0) [0..<Suc k]))"
Nil_is_append_conv: fixes xs :: "'k list"   and ys :: "'k list" shows "([] = xs @ ys) = (xs = [] \<and> ys = [])"
triv_subgroup: shows "subgroup {\<one>\<^bsub>H\<^esub>} H"
additive_def: fixes f :: "'a \<Rightarrow> 'a" shows "additive f \<equiv> \<forall>x y. f (x \<squnion> y) = f x \<squnion> f y"
frp_algo_def: fixes T1 :: "('a, 'c) tree_automaton_rec"   and T2 :: "('b, 'c) tree_automaton_rec" shows "frp_algo T1 T2 \<equiv> \<lparr>wa_cond = frp_cond, wa_step = frp_step (ta_rules T1) (ta_rules T2), wa_initial = frp_initial (ta_initial T1) (ta_initial T2), wa_invar = frp_invar T1 T2\<rparr>"
order_0I: fixes p :: "'a poly"   and a :: "'a" assumes "poly p a \<noteq> (0::'a)" shows "order a p = 0"
matrix_to_iarray_ncols: fixes A :: "(('b, 'c) vec, 'd) vec" shows "ncols A = ncols_iarray (matrix_to_iarray A)"
undefined_def: shows "Set_Monad.undefined = Collect HOL.undefined"
framed_uncross_def: fixes x :: "wall"   and y :: "wall" shows "framed_uncross x y \<equiv> uncross_positive_flip x y \<or> uncross_negative_flip x y"
assms: shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"
ta_seq_consist_lappend: fixes obs :: "('a, 'b) obs_event action llist"   and P :: "'c prog"   and vs :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option"   and obs' :: "('a, 'b) obs_event action llist" assumes "lfinite obs" shows "ta_seq_consist P vs (lappend obs obs') = (ta_seq_consist P vs obs \<and> ta_seq_consist P (mrw_values P vs (list_of obs)) obs')"
parts_image_Key: fixes N :: "key set" shows "parts (Key ` N) = Key ` N"
finsum_empty: fixes f :: "'b \<Rightarrow> 'a" shows "sum f {} = (0::'a)"
simps: fixes a :: "enat" shows "(a \<in> enat_set) = (a = 0 \<or> (\<exists>n. a = eSuc n \<and> n \<in> enat_set))"
hyp2_S_cart2_inj: fixes p :: "proj2"   and q :: "proj2" assumes "p \<in> hyp2 \<union> S"   and "q \<in> hyp2 \<union> S"   and "cart2_pt p = cart2_pt q" shows "p = q"
simps: fixes a :: "'a"   and K :: "'a set"   and x :: "'a" shows "(a \<in> simple_extension K x) = (a = \<zero> \<or> (\<exists>k1 k2. a = k1 \<otimes> x \<oplus> k2 \<and> k1 \<in> simple_extension K x \<and> k2 \<in> K))"
elims: fixes x :: "nat"   and y :: "(char list, nat) process_term" assumes "sys x = y" obtains "x = 0" and "y = CH (VAR (Suc 0)) (ACT ''b'' (VAR 0))"   | "x = Suc 0" and "y = ACT ''a'' (VAR 0)"   | va :: "nat" where "x = Suc (Suc va)" and "y = ACT ''a'' (VAR 0)"
nat_filtration_borel_measurable_integrable: fixes p :: "real"   and M :: "bool stream measure"   and f :: "bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "infinite_coin_toss_space p M"   and "f \<in> borel_measurable (infinite_coin_toss_space.nat_filtration M n)" shows "integrable M f"
take_below: fixes n :: "nat"   and x :: "C" shows "C_take n\<cdot>x \<sqsubseteq> x"
step_step': fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and test_num' :: "int \<Rightarrow> 'b \<Rightarrow> bool"   and inv_plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b"   and inv_less' :: "bool \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and S :: "'b st option"   and C :: "'b st option acom" assumes "Abs_Int_inv num' plus' test_num' inv_plus' inv_less' \<gamma>" shows "step (\<gamma>_option (\<gamma>_st \<gamma>) S) (ACom.map_acom (\<gamma>_option (\<gamma>_st \<gamma>)) C) \<le> ACom.map_acom (\<gamma>_option (\<gamma>_st \<gamma>)) (Abs_Int_inv.step' num' plus' test_num' inv_plus' inv_less' S C)"
try_bind_assert_gpv: fixes b :: "bool"   and f :: "unit \<Rightarrow> ('d, 'e, 'f) gpv"   and gpv :: "('d, 'e, 'f) gpv" shows "TRY assert_gpv b \<bind> f ELSE gpv = (if b then TRY f () ELSE gpv else gpv)"
TR2_sub_carr: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR2 R M N P \<subseteq> carrier P"
filter_set: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "Set.filter P (set xs) = set (filter P xs)"
rot_scaleR: fixes u :: "real"   and v :: "'a" shows "rot (u *\<^sub>R v) = u *\<^sub>R rot v"
similar_matI: fixes A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat"   and n :: "nat" assumes "{A, B, P, Q} \<subseteq> carrier_mat n n"   and "P * Q = 1\<^sub>m n"   and "Q * P = 1\<^sub>m n"   and "A = P * B * Q" shows "similar_mat A B"
measurable_PairVal: fixes t1 :: "pdf_type"   and t2 :: "pdf_type" shows "(\<lambda>(x, y). <|x, y|>) \<in> stock_measure t1 \<Otimes>\<^sub>M stock_measure t2 \<rightarrow>\<^sub>M stock_measure (PRODUCT t1 t2)"
initial_abs': fixes s :: "'a" assumes "s \<in> wa_initial WC" shows "\<alpha>1 s \<in> wa_initial WB"
assms_1: shows "squarefree_part m = squarefree_part n"
my_U_is_linear_function: fixes p :: "'a pmf"   and q :: "'a pmf"   and \<alpha> :: "real" assumes "p \<in> local.\<P>"   and "q \<in> local.\<P>"   and "\<alpha> \<in> {0..1}"   and "local.some_best \<succeq>[\<R>] local.some_worst \<and> (local.some_worst, local.some_best) \<notin> \<R>" shows "local.my_U (mix_pmf \<alpha> p q) = \<alpha> * local.my_U p + (1 - \<alpha>) * local.my_U q"
landau_product_preprocess_38: shows "neg_numeral_class.dbl_inc (1::'a) = (3::'a)"
iD: shows "isFDERIV DIM('a) [0..<DIM('a)] fas (list_of_eucl x @ ramsch)"
methd_declclass: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "class G C = Some c"   and "wf_prog G"   and "methd G C sig = Some m" shows "methd G (declclass m) sig = Some m"
selectlike_antimon: fixes V :: "variable set"   and W :: "variable set"   and X :: "(variable \<Rightarrow> real) set"   and \<nu> :: "variable \<Rightarrow> real" assumes "V \<subseteq> W" shows "selectlike X \<nu> W \<subseteq> selectlike X \<nu> V"
not_alw: fixes \<phi> :: "'b stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> alw \<phi> xs) = ev (\<lambda>xs. \<not> \<phi> xs)"
wftbind: fixes \<rho> :: "val list"   and \<eta> :: "val set list"   and \<Gamma> :: "(ty \<times> nat) list \<times> nat"   and V :: "val set" assumes "\<turnstile> \<rho>,\<eta> : \<Gamma>" shows "\<turnstile> \<rho>,V # \<eta> : push_tyvar \<Gamma>"
extremum: fixes a :: "'b multiset" shows "{#} \<subseteq># a"
radicals_empty_rational: fixes e :: "expr" assumes "radicals e = {}" shows "\<lbrace>e\<rbrace> \<in> \<rat>"
llist_all2_lappendI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and xs' :: "'c llist"   and ys' :: "'d llist" assumes "llist_all2 P xs ys"   and "\<lbrakk>lfinite xs; lfinite ys\<rbrakk> \<Longrightarrow> llist_all2 P xs' ys'" shows "llist_all2 P (lappend xs xs') (lappend ys ys')"
intro: fixes step1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and step2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and final1 :: "'a \<Rightarrow> bool"   and final2 :: "'b \<Rightarrow> bool"   and order :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and match :: "'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "forward_simulation step1 step2 final1 final2 order match"   and "backward_simulation step1 step2 final1 final2 order match" shows "bisimulation step1 step2 final1 final2 order match"
prems_1: shows "\<forall>i\<in>insert x I. subgroup (S i) G"
Inl_iff: fixes a :: "V"   and b :: "V" shows "(ZFC_Cardinals.Inl a = ZFC_Cardinals.Inl b) = (a = b)"
Y2_A: fixes X :: "'v set"   and n :: "nat" assumes "X \<subseteq> A_n n" shows "Y2 n X \<subseteq> \<^bold>E `` X"
assms_2: shows "openin (top_of_set (affine hull S)) S"