hcju/isabelleps huggingface dataset:
smooth_on_compose2: fixes k :: "enat"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and S :: "'e set"   and g :: "'e \<Rightarrow> 'c"   and U :: "'e set" assumes "k-smooth_on T f"   and "k-smooth_on S g"   and "open U"   and "open T"   and "g ` U \<subseteq> T"   and "U \<subseteq> S" shows "k-smooth_on U (f \<circ> g)"
iteration_unfoldr: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<infinity> \<cdot> x = x\<^sup>\<infinity>"
cartesian_product_infiniteR_imp_infinite: fixes B :: "'a set"   and A :: "'b set" assumes "infinite B"   and "A \<noteq> {}" shows "infinite (A \<times> B)"
marriage_HV: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>J\<subseteq>I. card J \<le> card (\<Union> (A ` J))" shows "\<exists>R. (\<forall>i\<in>I. R i \<in> A i) \<and> inj_on R I"
quicksort_correct_case2: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'c"   and xs0 :: "'d list"   and lo :: "nat"   and hi :: "nat"   and xs :: "'d list"   and xs' :: "'d list"   and p :: "nat" assumes "quicksort_pre R h xs0 lo hi xs"   and "partition_spec R h xs lo hi xs' p"   and "\<not> hi \<le> p + 1" shows "quicksort_pre R h xs0 (Suc p) hi xs'"
\<omega>_limit_crossings: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and p :: "'a" assumes "transversal_segment a b"   and "{0..} \<subseteq> existence_ivl0 x"   and "\<omega>_limit_point x p"   and "p \<in> {a<--<b}" obtains s :: "nat \<Rightarrow> real" where "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>" and "(flow0 x \<circ> s) \<longlonglongrightarrow> p" and "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b} \<and> s n \<in> existence_ivl0 x"
sum_list_rmv1: fixes a :: "'a"   and xs :: "'a list" assumes "a \<in> set xs" shows "sum_list (remove1 a xs) = sum_list xs - a"
rel_advantage_def: fixes \<A> :: "'a \<Rightarrow> nat spmf" shows "Schnorr_\<Sigma>_commit.rel_advantage \<A> \<equiv> spmf (Schnorr_\<Sigma>_commit.rel_game \<A>) True"
vfinite_0_right: fixes a :: "V" shows "a \<times>\<^sub>\<bullet> []\<^sub>\<circ> = []\<^sub>\<circ>"
att_and_empty2: fixes s :: "'a set" shows "\<turnstile>[] \<oplus>\<^sub>\<and>\<^bsup>(s, s)\<^esup>"
Bolzano_Weierstrass_complex_disc: fixes s :: "nat \<Rightarrow> complex"   and r :: "real" assumes "\<forall>n. cmod (s n) \<le> r" shows "\<exists>f z. strict_mono f \<and> (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (s (f n) - z) < e)"
prems_1: shows "obtains 1 c1 drinks2 1 r (a # t)"
var_dist_Cons: fixes v :: "nat"   and vs :: "nat list"   and t :: "dB" shows "var_dist (v # vs) t = var_dist vs (t \<degree> Var v)"
comparator_option_def: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order" shows "comparator_option comp\<^sub>'\<^sub>a \<equiv> partial_comparator_option \<circ> map_option comp\<^sub>'\<^sub>a"
msort_sort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less"   and "distinct xs" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = linorder.msort less xs"
is_empty_set: fixes xs :: "'a list" shows "Set.is_empty (set xs) = List.null xs"
simps: fixes q :: "('b, 'c) queue_scheme" shows "first q = hd (front q)"
hyps: shows "echelon_form (foldl (Hermite_of_row_i ass res) A (map mod_type_class.from_nat [0..<k]))"
Ax: fixes k :: "'b"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "Atom k \<in># \<Gamma>"   and "Atom k \<in># \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
of_int_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "0 < z" shows "less zero (ring_1.of_int one plus zero minus z)"
demonic_def: fixes Q :: "'c \<Rightarrow> 'd"   and p :: "'d" shows "[: Q :] p = {s. Q s \<sqsubseteq> p}"
intro: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" assumes "social_decision_scheme agents alts sds"   and "ex_post_efficient_sds_axioms agents alts sds" shows "ex_post_efficient_sds agents alts sds"
Units_not_left_zero_divisor: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Units R"   and "b \<in> carrier R"   and "b \<otimes> a = \<zero>" shows "b = \<zero>"
is_nth_power_mult_coprime_nat_iff: fixes a :: "nat"   and b :: "nat"   and n :: "nat" assumes "coprime a b" shows "is_nth_power n (a * b) = (is_nth_power n a \<and> is_nth_power n b)"
less_bot_Bot_Value_code: fixes x :: "'a" shows "(Bot < Value x) = True"
content_gcd_content: fixes f :: "'b poly"   and g :: "'b poly" shows "content (gcd f g) = gcd (content f) (content g)"
sfilter_sfilter': fixes p :: "'a \<rightarrow> tr"   and q :: "'a \<rightarrow> tr"   and xs :: "[:'a:]" shows "sfilter\<cdot>p\<cdot>(sfilter\<cdot>q\<cdot>xs) = sfilter\<cdot>(\<Lambda> x. q\<cdot>x andalso p\<cdot>x)\<cdot>xs"
take_append: fixes n :: "nat"   and xs :: "'c list"   and ys :: "'c list" shows "take n (xs @ ys) = take n xs @ take (n - length xs) ys"
induct: fixes \<Gamma> :: "('a, 'b) web_scheme"   and \<epsilon> :: "ennreal" assumes "hindered_by \<Gamma> \<epsilon>" obtains f :: "'a \<times> 'a \<Rightarrow> ennreal" where "hindrance_by \<Gamma> f \<epsilon>" and "current \<Gamma> f" and "wave \<Gamma> f"
echelon_form_of_column_k_det_def: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A' :: "'b \<times> (('b, 'c) vec, 'd) vec \<times> nat"   and k :: "nat" shows "echelon_form_of_column_k_det bezout A' k = (let (det_P, A, i) = A'; from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if i \<noteq> nrows A \<and> A $ from_nat_i $ from_nat_k = (0::'b) \<and> (\<exists>m>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k \<noteq> (0::'b)) then (- (1::'b) * det_P, echelon_form_of_column_k bezout (A, i) k) else (det_P, echelon_form_of_column_k bezout (A, i) k))"
empty_Q0: shows "Q_of_state (lhd Sts) = {}"
disjoint_insert_1: fixes B :: "'d set"   and a :: "'d"   and A :: "'d set" shows "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
strict_iff_order: fixes a :: "'a"   and b :: "'a" shows "(b \<succ> a) = (b \<succeq> a \<and> a \<noteq> b)"
RG_aezdelta0fun_closed: fixes r :: "'a" shows "r \<delta>\<delta> (0::'g) \<in> ActingGroup.group_ring"
is_functor: shows "functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) Cones.map"
adjoint_transpose_right_3: fixes u :: "'a"   and v :: "'a"   and \<mu> :: "'a" assumes "ide u"   and "ide v"   and "src v = trg (tab\<^sub>0 r)\<^sup>*"   and "src u = trg (tab\<^sub>0 r)"   and "\<guillemotleft>\<mu> : v \<star> (tab\<^sub>0 r)\<^sup>* \<Rightarrow> u\<guillemotright>" shows "T0.trnr\<^sub>\<epsilon> u (T0.trnr\<^sub>\<eta> v \<mu>) = \<mu>"
bisim: shows "(strong_reduction_simulation Rel (Calculus CWB) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>Calculus CWB Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Calculus CWB P' \<and> (P', Q') \<in> Rel))) \<and> rel_respects_binary_pred Rel (\<lambda>a. hasBarb a CWB)"
assms: shows "a\<^sub>e < a\<^sub>o"
proper_map_flip_signed: fixes xs :: "('b \<times> bool) list" assumes "proper_signed_list xs" shows "proper_signed_list (map flip_signed xs)"
order_refl: fixes less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'h" assumes "class.preorder less_eq less" shows "less_eq x x"
ratio_test_divergence: fixes f :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. 0 < f n"   and "limsup (\<lambda>n. ereal (f n / f (Suc n))) < 1" shows "\<not> summable f"
mult_monoms_spec: shows "(mult_monoms, \<lambda>a b. remdups_mset (a + b)) \<in> term_poly_list_rel \<rightarrow> term_poly_list_rel \<rightarrow> term_poly_list_rel"
coeff_0_power: fixes p :: "'b poly"   and n :: "nat" shows "coeff (p ^ n) 0 = coeff p 0 ^ n"
disjoint_mono: fixes a :: "'e set"   and a' :: "'e set"   and b :: "'e set"   and b' :: "'e set" assumes "a \<subseteq> a'"   and "b \<subseteq> b'"   and "a' \<inter> b' = {}" shows "a \<inter> b = {}"
l_get\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_lemmas_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_get\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_lemmas type_wf \<equiv> l_type_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t type_wf"
Derives1_count_terminals: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "count_terminals b = count_terminals a + count_terminals (snd r)"
atMost_iff: fixes i :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and k :: "'g" shows "(i \<in> ord.atMost less_eq k) = less_eq i k"
\<omega>_gt0: shows "0 \<subset>\<^sub>\<circ> \<omega>"
fcard_flist: fixes xs :: "'a fset" shows "fcard xs = length (flist xs)"
Oops_range_spies2: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "servK \<notin> range shrK \<and> servK \<in> symKeys"
process_ret_add_match_dist: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_ret (add_match m rs), s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m (process_ret rs), s\<rangle> \<Rightarrow> t"
extremum: fixes a :: "'a" shows "bot_class.bot \<le> a"
assms_3: shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"
emeasure_pair_measure_Times: fixes A :: "'c set"   and N :: "'c measure"   and B :: "'b set" assumes "A \<in> sets N"   and "B \<in> sets M2" shows "emeasure (N \<Otimes>\<^sub>M M2) (A \<times> B) = emeasure N A * emeasure M2 B"
vert_f12: shows "vertices f12 = rev ws @ v # vs1 @ [a]"
are_arrows_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu>" shows "partial_magma.arr (span_vertical_category.vcomp C) \<nu>"
simps_1: fixes lnk :: "'index \<Rightarrow> 'node \<Rightarrow> 'node"   and lbl :: "'node \<Rightarrow> 'index"   and x :: "'node" shows "stack lnk lbl x [] = (x = nil)"
tm_induct_apps: fixes P :: "('a, 'b) tm \<Rightarrow> bool"   and s :: "('a, 'b) tm" assumes "\<And>\<zeta> ss. (\<And>s. s \<in> set ss \<Longrightarrow> P s) \<Longrightarrow> P (apps (Hd \<zeta>) ss)" shows "P s"
pvalid_left: fixes p :: "(symbol \<times> character list) list"   and x :: "item" shows "pvalid p x = pvalid_left p x"
analz_NI_E_K_analz_NI_E: fixes NI :: "msg set"   and K :: "msg set"   and I :: "msg set"   and Bad :: "agent set" assumes "NI \<subseteq> payload"   and "K \<subseteq> range LtK"   and "I \<subseteq> valid" shows "analz (extr Bad NI (local.abs I) \<union> K \<union> Tags) \<subseteq> analz (extr Bad NI (local.abs I)) \<union> K \<union> Tags"
r_nf_t_def: shows "r_nf_t \<equiv> Cn 3 r_and [Cn 3 r_eq [Cn 3 r_pdec2 [recf.Id 3 0], Cn 3 r_prod_encode [recf.Id 3 1, recf.Id 3 2]], Cn 3 r_not [Cn 3 r_result [Cn 3 r_pdec1 [recf.Id 3 0], Cn 3 r_pdec12 [recf.Id 3 0], Cn 3 r_pdec22 [recf.Id 3 0]]]]"
iface_is_wildcard_ifaceAny: shows "iface_is_wildcard ifaceAny"
append_exec_moves: fixes vs :: "'addr val list"   and es' :: "'addr expr1 list"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "length vs = length es'"   and "exec_moves ci P t es h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_moves ci P t (es' @ es) h (stk @ vs, loc, length (compEs2 es') + pc, xcp) ta h' (stk' @ vs, loc', length (compEs2 es') + pc', xcp')"
zero_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "zero_hom hom \<equiv> hom (0::'a) = (0::'b)"
a_get_tag_name_locs_def: fixes element_ptr :: "(_) element_ptr" shows "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name_locs element_ptr \<equiv> {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name)}"
call_edges_only: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "\<forall>a'. valid_edge a' \<and> targetnode a' = targetnode a \<longrightarrow> (\<exists>Qx rx fsx. kind a' = Qx:rx\<hookrightarrow>\<^bsub>p\<^esub>fsx)"
mult_minus2_left: fixes x :: "'b" shows "- (2::'b) * x = - ((2::'b) * x)"
Lim_MInfty: fixes f :: "nat \<Rightarrow> ereal" shows "f \<longlonglongrightarrow> - \<infinity> = (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<le> ereal B)"
IH: fixes K :: "'a ptrm"   and Y :: "'a ptrm"   and Z :: "'a ptrm" assumes "size K < size X"   and "K \<approx> Y"   and "Y \<approx> Z" shows "K \<approx> Z"
comm_monoid_add_hom_0_axioms: shows "comm_monoid_add_hom_0 of_int"
unity_root_eq_1_iff: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "(unity_root k (int n) = 1) = (k dvd n)"
out_VWrite\<^sub>s\<^sub>b: shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}"
simps_6: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp" shows "isVal (scrut ? e1 : e2) = False"
homf_preserves_objects: fixes A :: "'o" shows "Hom(A,_) \<^bsub>\<o>\<^esub> \<in> Ob \<rightarrow> Ob\<^bsub>Set\<^esub>"
compactE_image: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and C :: "'d set"   and f :: "'d \<Rightarrow> 'c set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "\<And>T. T \<in> C \<Longrightarrow> open (f T)"   and "S \<subseteq> \<Union> (f ` C)" obtains C' :: "'d set" where "C' \<subseteq> C" and "finite C'" and "S \<subseteq> \<Union> (f ` C')"
tr: shows "(s, a, \<sigma>', p') \<in> automaton.trans A"
intruder_synth_code: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" shows "(M \<turnstile>\<^sub>c t) = intruder_synth' public arity M t"
dom_dom: fixes f :: "'a" assumes "L'.C.arr f" shows "L'.C.dom (L'.C.dom f) = L'.C.dom f"
weakenBisimE_3: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<approx>\<^sub>w Q" shows "\<Psi> \<otimes> \<Psi>' \<rhd> P \<approx>\<^sub>w Q"
all_degree_2_convert: fixes var :: "nat"   and L :: "atom list"   and xs :: "real list" assumes "all_degree_2 var L" shows "\<exists>L'. convert_atom_list var L xs = Some L'"
mult_local_ring_at_def: shows "mult_local_ring_at \<equiv> cq.mult_rel"
rep_eq: fixes x :: "('d, 'c) sq_matrix"   and xa :: "('d, 'c) sq_matrix" shows "to_fun (x * xa) = (\<lambda>i j. \<Sum>k\<in>UNIV. to_fun x i k * to_fun xa k j)"
nres_order_simps_4: fixes M :: "'c nres" shows "(FAIL \<le> M) = (M = FAIL)"
double_add_less_zero_iff_single_add_less_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less (plus a a) zero = less a zero"
stream_all2_SCons1: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b stream"   and ys :: "'c stream" shows "stream_all2 P (x ## xs) ys = (\<exists>z zs. ys = z ## zs \<and> P x z \<and> stream_all2 P xs zs)"
linord_of_list_refl_on: fixes xs :: "'a list" shows "refl_on (set xs) (linord_of_list xs)"
on_curvep_iff_on_curve: fixes a :: "int"   and b :: "int"   and p :: "int \<times> int \<times> int" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "in_carrierp p" shows "on_curvep a b p = on_curve a b (make_affine p)"
tuple_prj: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a"   and h :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g"   and "partial_magma.arr C (C (prj1 f g) h)" shows "elementary_category_with_pullbacks.tuple C prj0 prj1 (C (prj1 f g) h) f g (C (prj0 f g) h) = h"
simps_5: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'e list \<Rightarrow> 'd"   and x21 :: "'e"   and x22 :: "'e list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
projective_family_axioms: shows "projective_family I (\<lambda>J. Pi\<^sub>M J (\<lambda>_. borel)) (\<lambda>_. borel)"
prefixI: fixes vs :: "nat list"   and f :: "nat \<Rightarrow> nat option" assumes "0 < length vs"   and "\<And>x. x < length vs \<Longrightarrow> f x \<down>= vs ! x" shows "prefix f (length vs - 1) = vs"
sum_union: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "\<Sum> (A \<union> B) = \<Sum> A + \<Sum> B"
blinding_respects_hashes_axioms: shows "blinding_respects_hashes h bo"
elts_succ: fixes x :: "V" shows "elts (ZFC_in_HOL.succ x) = insert x (elts x)"
set_minus_def: fixes X :: "'a set"   and Y :: "'a set" shows "set_minus X Y = X - Y"
sup_ty_opt_def: fixes P :: "'b prog" shows "sup_ty_opt P \<equiv> Err.le (subtype P)"
inverse_im_exists: fixes T :: "'v \<Rightarrow> 'w" assumes "T \<in> GRepHomSet (\<star>) W" shows "\<exists>!S. \<psi>_condition T S"
assms_8: shows "odd (card {f \<in> F. compo' f \<and> bnd f})"
that: fixes S :: "'a spmf \<Rightarrow> 'd spmf \<Rightarrow> bool" assumes "S p q"   and "\<And>p q a. \<lbrakk>S p q; a \<in> A\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s. map_spmf fst (callee1 s a)) = q \<bind> (\<lambda>s. map_spmf fst (callee2 s a))"   and "\<And>p q a s b s'. \<lbrakk>S p q; a \<in> A; s \<in> set_spmf q; (b, s') \<in> set_spmf (callee2 s a)\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (q \<bind> (\<lambda>s. callee2 s a)) b)" shows "thesis"
RA4_parts_spies: fixes C' :: "agent"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and X' :: "msg"   and c :: "msg"   and evs :: "event list" assumes "Says C' B \<lbrace>Crypt K X, X', c\<rbrace> \<in> set evs" shows "c \<in> parts (knows Spy evs)"
intros_2: fixes s :: "real"   and a :: "real"   and b :: "real"   and P :: "real \<Rightarrow> bool" assumes "s \<in> {a..<b}"   and "s = a \<or> P s"   and "\<forall>t\<in>{s<..<b}. P t"   and "finite_Psegments P a s" shows "finite_Psegments P a b"
countable: shows "countable (support_flow (\<Squnion> Y))"
proper_part_irreflexivity: fixes x :: "i" shows "\<not> PP x x"
bfs2_refine: fixes succ :: "nat \<Rightarrow> nat list nres"   and src :: "nat"   and dst :: "nat" assumes "\<And>ui u. \<lbrakk>(ui, u) \<in> nat_rel; u \<in> V\<rbrakk> \<Longrightarrow> succ ui \<le> (spec l. (l, E `` {u}) \<in> \<langle>nat_rel\<rangle>list_set_rel)" shows "bfs2 succ src dst \<le> \<Down> Id (bfs src dst)"
assms_1: shows "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"
expectation_geometric_pmf: fixes p :: "real" assumes "p \<in> {0<..1}" shows "measure_pmf.expectation (geometric_pmf p) real = (1 - p) / p"
\<T>covers: fixes C :: "'a set" assumes "C \<in> \<N>" shows "\<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union> F"
map_simp_ide: fixes a :: "'a \<times> 'a \<times> 'a" assumes "CCC.ide a" shows "\<alpha>.map a = \<alpha> a"
in_set_member: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> set xs) = List.member xs x"
dickson_grading_lcs: fixes d :: "'a \<Rightarrow> nat"   and s :: "'a"   and t :: "'a" assumes "dickson_grading d" shows "d (lcs s t) \<le> max (d s) (d t)"
testN_def: fixes k :: "nat" shows "testN k = mat N N (\<lambda>(i, j). if i = k \<and> j = k then 1 else 0)"
distinct_23: fixes x6 :: "Arraytype" shows "IntgT \<noteq> ArrT x6"
assms_1: shows "max_Var_floatariths xs \<le> m"
fls_nth_inject: fixes x :: "'a fls"   and y :: "'a fls" shows "(fls_nth x = fls_nth y) = (x = y)"
pow_eq: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and x :: "nat" assumes "\<exists>tid\<in>actHn (t n). length (bc (\<sigma>\<^bsub>tid\<^esub>t n)) = x"   and "pow_cond t n x" shows "PoW t n = x"
const_tm_def: fixes n :: "name" shows "const n = Hd (Sym (of_name n))"
monom_mult_eq_zero_iff: fixes c :: "'c"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "(MPoly_Type_Class.punit.monom_mult c t p = 0) = (c = (0::'c) \<or> p = 0)"
integrableI_sequence: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and s :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "\<And>i. simple_bochner_integrable M (s i)"   and "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0" shows "integrable M f"
class_ordered_semiring_def: fixes a :: "'c itself"   and gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'d" shows "class_ordered_semiring a gt b \<equiv> class_semiring a \<lparr>ordered_semiring.geq = \<lambda>x y. y \<le> x, gt = gt, max = max, \<dots> = b\<rparr>"
neg_less_divide_eq: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c < (0::'a)" shows "(a < b / c) = (b < a * c)"
part_equiv_trans_sym_1: fixes R :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "part_equiv R"   and "(a, b) \<in> R"   and "(c, b) \<in> R" shows "(a, c) \<in> R"
that: fixes xs :: "'a list" assumes "local.steps (s # xs @ [s'])"   and "length xs + 1 = n" shows "thesis"
zcf_def: fixes f :: "'d \<Rightarrow> 'c" shows "zcf f = f (0::'d)"
Torder: fixes D :: "('c, 'd) Order_scheme" assumes "Worder D" shows "Torder D"
smult_one: fixes x :: "rat vec" assumes "x \<in> Rn" shows "1 \<cdot>\<^sub>v x = x"
gState_progress_rel_gState_invI2: fixes g :: "'a gState_scheme"   and g' :: "'a gState_scheme"   and prog :: "program" assumes "(g, g') \<in> gState_progress_rel prog" shows "gState_inv prog g'"
even_and_iff: fixes a :: "'a"   and b :: "'a" shows "even (a AND b) = (even a \<or> even b)"
finite_representation: fixes basis :: "'b set"   and v :: "'b" shows "finite {b. representation basis v b \<noteq> (0::'a)}"
unwinding_implies_isecure_CISK: shows "Inst.isecure"
subst_simp_snd: fixes P :: "'a trm"   and x :: "'a"   and M :: "'a trm" shows "Snd P[x ::= M] = Snd P[x ::= M]"
build_succ_def: fixes nds :: "('a, 'b) node_scheme set" shows "build_succ nds = FOREACH nds (\<lambda>q'. FOREACH (incoming q') (\<lambda>qn s. if qn = expand_init then RETURN s else RETURN (s(qn \<mapsto> insert (name q') (the_default {} (s qn)))))) Map.empty"
similar_mat_jordan_block_smult: fixes A :: "'b mat"   and n :: "nat"   and a :: "'b"   and k :: "'b" assumes "similar_mat A (jordan_block n a)"   and "k \<noteq> (0::'b)" shows "similar_mat (k \<cdot>\<^sub>m A) (jordan_block n (k * a))"