hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "('p, 's, 'm) configuration"   and x2 :: "('p, 's, 'm) event list"   and x3 :: "('p, 's, 'm) configuration"   and P :: "('p, 's, 'm) configuration \<Rightarrow> ('p, 's, 'm) event list \<Rightarrow> ('p, 's, 'm) configuration \<Rightarrow> bool" assumes "trace x1 x2 x3"   and "\<And>c. P c [] c"   and "\<And>c ev c' t c''. \<lbrakk>c \<turnstile> ev \<mapsto> c'; trace c' t c''; P c' t c''\<rbrakk> \<Longrightarrow> P c (ev # t) c''" shows "P x1 x2 x3"
fold_Nil: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" shows "fold f [] = id"
poly_mult_rat_main_def: fixes n :: "'a"   and d :: "'a"   and f :: "'a poly" shows "poly_mult_rat_main n d f = (let fs = coeffs f; k = length fs in poly_of_list (map2 (\<lambda>fi i. fi * d ^ i * n ^ (k - Suc i)) fs [0..<k]))"
SETmor: fixes f :: "ZF" shows "isZFfun f = (f \<in> mor\<^bsub>SET\<^esub>)"
that: fixes y :: "'x" assumes "y \<in> keys t"   and "y \<le> x" shows "thesis"
normal_moment_abs_odd: shows "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * \<bar>x - \<mu>\<bar> ^ (2 * k + 1)) (2 ^ k * \<sigma> ^ (2 * k + 1) * fact k * sqrt (2 / pi))"
triangle_reindex_eq: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat" shows "(\<Sum>(i, j)\<in>{(i, j). i + j \<le> n}. g i j) = (\<Sum>k\<le>n. \<Sum>i\<le>k. g i (k - i))"
simps_2: fixes s :: "'state_t"   and d :: "'dom_t" shows "CISK_control s d [[]] = (None, [], s)"
bind_spmf_const: fixes p :: "'b spmf"   and q :: "'a spmf" shows "p \<bind> (\<lambda>x. q) = scale_spmf (weight_spmf p) q"
fsubsetI: fixes A :: "'b fset"   and B :: "'b fset" assumes "\<And>x. x |\<in>| A \<Longrightarrow> x |\<in>| B" shows "A |\<subseteq>| B"
less_not_sym: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.preorder less_eq less"   and "less x y" shows "\<not> less y x"
assigns_Extend: fixes k :: "'k"   and i :: "nat"   and \<AA> :: "'i"   and m :: "nat"   and P :: "'v"   and k' :: "'k" assumes "LEQ k i (#\<^sub>V \<AA>)" shows "m\<^bsup>Extend k i \<AA> P\<^esup>k' = (if k = k' then if m = i then P else dec i m\<^bsup>\<AA>\<^esup>k else m\<^bsup>\<AA>\<^esup>k')"
degree_mult_eq: fixes p :: "'a poly"   and q :: "'a poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "Polynomial.degree (p * q) = Polynomial.degree p + Polynomial.degree q"
left_null_space_eq_null_space_transpose: fixes A :: "(('a, 'c) vec, 'b) vec" shows "left_null_space A = null_space (Finite_Cartesian_Product.transpose A)"
map: fixes f :: "'c \<Rightarrow> 'b"   and x1 :: "'c"   and x2 :: "'c stream" shows "smap f (x1 ## x2) = f x1 ## smap f x2"
ipoly_roots_finite: fixes p :: "int poly" assumes "p \<noteq> 0" shows "finite {x. ipoly p x = (0::'a)}"
Spy_not_see_NB: fixes B :: "agent"   and A :: "agent"   and NA :: "nat"   and NB :: "nat"   and evs :: "event list" assumes "Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> \<in> set evs"   and "\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Nonce NB \<notin> analz (knows Spy evs)"
not_arg_cong_Inr: fixes x :: "'b"   and y :: "'b" assumes "x \<noteq> y" shows "Inr x \<noteq> Inr y"
space_lim_stream: fixes x :: "real \<times> 'a" shows "space (K.lim_stream x) = streams (space S)"
assms_2: shows "g piecewise_C1_differentiable_on s"
prime_dvd_multD: 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 p :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "dvd.dvd times p (times a b)" shows "dvd.dvd times p a \<or> dvd.dvd times p b"
prod_orA_def: fixes p1 :: "'e \<Rightarrow> 'g decision option"   and p2 :: "'f \<Rightarrow> 'h decision option" shows "p1 \<Otimes>\<^sub>\<or>\<^sub>A p2 = (\<lambda>(x, y). case p1 x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor>)"
sub_designII: fixes \<U> :: "'a set"   and \<A> :: "'a set multiset" assumes "design \<U> \<A>"   and "sub_incidence_system \<V> \<B> \<U> \<A>" shows "sub_design \<V> \<B> \<U> \<A>"
assms: fixes t :: "'a"   and x :: "'b" assumes "t \<in> T"   and "x \<in> X" shows "\<exists>u>0. \<exists>L. \<forall>t\<in>cball t u \<inter> T. L-lipschitz_on (cball x u \<inter> X) (f t)"
F0: shows "pderiv f \<bullet> a \<noteq> \<zero>"
flrestrictionD_2: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<bullet> A" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
rel_inject_3: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x3 :: "'a \<times> nat"   and y3 :: "'b \<times> nat" shows "rel_SDG_node R (Formal_out x3) (Formal_out y3) = rel_prod R (=) x3 y3"
distinct_take_cols_id: fixes inds :: "nat list" assumes "distinct inds" shows "distinct (cols (take_cols (1\<^sub>m n) inds))"
frac_le_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and 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 y :: "'a"   and z :: "'a"   and x :: "'a"   and w :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "y \<noteq> zero"   and "z \<noteq> zero" shows "less_eq (divide x y) (divide w z) = less_eq (divide (minus (times x z) (times w y)) (times y z)) zero"
gamma4_smooth: shows "gamma4 piecewise_C1_differentiable_on {0..1}"
cases: fixes a1 :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and a2 :: "'f set"   and a3 :: "'f \<Rightarrow> 'g" assumes "valid_abstraction a1 a2 a3" obtains A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)" and k :: "'f \<Rightarrow> 'g" and X :: "'f set" where "a1 = A" and "a2 = X" and "a3 = k" and "\<forall>(x, m)\<in>clkp_set A. m \<le> k x \<and> x \<in> X \<and> m \<in> \<nat>" and "collect_clkvt (trans_of A) \<subseteq> X" and "finite X"
set_butlast: shows "(c, d) \<in> set (butlast (half_segments_of_aform X))"
JF_cind1: fixes R1 :: "('d JF1 \<times> 'd JF1) set"   and R2 :: "('d JF2 \<times> 'd JF2) set" assumes "sbis UNIV UNIV dtor1 dtor2 R1 R2" shows "R1 \<subseteq> Id"
SBHReadUnbuffered: fixes sb :: "'b memref list"   and a :: "nat"   and volatile :: "bool"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" assumes "buffered_val sb a = None" shows "(Read volatile a t # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>(t \<mapsto> m a), sb @ [Read\<^sub>s\<^sub>b volatile a t (m a)], m, \<D>, \<O>, \<R>, \<S>)"
filter_prop: fixes x :: "'c"   and P :: "'c \<Rightarrow> bool"   and ys :: "'c list" assumes "x \<in> set (filter P ys)" shows "P x"
real_interval_abs_interval: fixes x :: "float interval" shows "real_interval (abs_interval x) = abs_interval (real_interval x)"
Osum_def: fixes r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set" shows "r \<union>o r' = r \<union> r' \<union> {(a, a'). a \<in> Field r \<and> a' \<in> Field r'}"
signof_id_2: shows "signof (\<lambda>x. x) = (1::'b)"
zeros_One_pdevs_raw: shows "One_pdevs_raw -` {0::'b} = {length Basis_list..}"
prems_1: shows "spec_ops (xs @ ys @ [y]) deps"
nlistListLCons: fixes xs :: "('c, 'd) Node list"   and t :: "('c, 'd) FingerTreeStruc" shows "toList (lconsNlist xs t) = nlistToList xs @ toList t"
fun_ran_wb_lens: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "('c \<Rightarrow> 'b) \<Longrightarrow> 'd" assumes "wb_lens X"   and "wb_lens Y" shows "wb_lens (fun_ran_lens X Y)"
less_1: fixes p :: "'a list" assumes "length p < length pa__"   and "polynomial K p" shows "p = poly_of_dense (dense_repr p)"
Iter_comm: fixes f :: "'a \<Rightarrow> 'a"   and n :: "nat"   and x :: "'a" shows "Iter f n (f x) = f (Iter f n x)"
ide_dom_lam: shows "ide (local.dom (\<Lambda> a b c g))"
wfp_on_accessible_on_iff: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "wfp_on P A = (\<forall>x\<in>A. accessible_on P A x)"
advantage1_def: fixes \<A> :: "(bool list \<times> 'hash, bool list, 'hash) gpv" shows "UPF.advantage1 \<A> = spmf (UPF.game1 \<A>) True"
path_component_path_image_pathstart: fixes p :: "real \<Rightarrow> 'a"   and x :: "'a" assumes "path p"   and "x \<in> path_image p" shows "path_component (path_image p) (pathstart p) x"
runProc_def: fixes name :: "String.literal"   and args :: "expr list"   and prog :: "program"   and g :: "'b gState_scheme"   and p :: "pState" shows "runProc name args prog g p = (if max_procs \<le> length (procs g) then abort STR ''Too many processes'' (\<lambda>_. (g, p)) else let pid = length (procs g) + 1 in case lm.lookup name (proc_data prog) of None \<Rightarrow> abortv STR ''No such process: '' name (\<lambda>_. (g, p)) | Some proc_idx \<Rightarrow> let (g', proc) = mkProc g p name args (processes prog !! proc_idx) pid in (g'\<lparr>procs := procs g @ [proc]\<rparr>, p))"
sums_emeasure': fixes B :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "\<And>x. B x \<in> sets M"   and "\<And>x y. x \<noteq> y \<Longrightarrow> emeasure M (B x \<inter> B y) = 0" shows "(\<lambda>x. emeasure M (B x)) sums emeasure M (\<Union> (range B))"
updates: shows "fs'.gs.\<mu> i' j' = (if i' = i \<and> j' \<le> j then fs.gs.\<mu> i j' - rat_of_int c * fs.gs.\<mu> j j' else fs.gs.\<mu> i' j')"
paths_with_new_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list" assumes "DisjointPathsPlusOne G v0 v1 paths P_new" shows "DisjointPathsPlusOne.paths_with_new paths P_new \<equiv> insert P_new paths"
set_discriminating_def: fixes n :: "nat"   and f :: "bool stream \<Rightarrow> 'a"   and N :: "'a measure" shows "set_discriminating n f N \<equiv> \<forall>w. f w \<noteq> f (pseudo_proj_True n w) \<longrightarrow> (\<exists>A\<in>sets N. (f w \<in> A) = (f (pseudo_proj_True n w) \<notin> A))"
V_2: fixes a :: "'a set" assumes "a \<in> M" shows "\<mu> a = \<mu>_r a"
deg_h': shows "poly_deg (s * h') = poly_deg s + poly_deg h'"
lincomb_empty: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and a :: "'d \<Rightarrow> 'b" assumes "module R M" shows "module.lincomb M a {} = \<zero>\<^bsub>M\<^esub>"
indis_def: shows "indis \<equiv> {(s, t). \<forall>x. sec x = Lo \<longrightarrow> s x = t x}"
fls_integral_of_int': fixes i :: "int" assumes "inverse (1::'a) = (1::'a)" shows "fls_integral (of_int i) = of_int i * fls_X"
step_4: shows "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"
affine_scheme_axioms: shows "affine_scheme R (+) (\<cdot>) \<zero> \<one> X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str f \<phi>\<^sub>f"
rel_return_pmf: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'b" shows "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
size_nat_def: fixes n :: "nat" shows "??.Nat.size_class.size n = n"
rtrancl_on_trans: fixes a :: "'e"   and b :: "'e"   and F :: "'e set"   and r :: "('e \<times> 'e) set"   and c :: "'e" assumes "(a, b) \<in> rtrancl_on F r"   and "(b, c) \<in> rtrancl_on F r" shows "(a, c) \<in> rtrancl_on F r"
oexp_1: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "1 \<up> \<alpha> = 1"
simps: fixes M :: "('d, 'e, 'f) FSM"   and f :: "'f \<Rightarrow> 'd list" shows "is_det_state_cover_ass M f = (f (initial M) = [] \<and> (\<forall>s\<in>d_reachable M (initial M). d_reaches M (initial M) (f s) s))"
plane_vec_eq_def: shows "plane_vec_eq \<equiv> map_fun Rep_plane_vec (map_fun Rep_plane_vec id) plane_vec_eq_r4"
assms_1: shows "directed_complete A (\<sqsubseteq>)"
minkowski: fixes B :: "(real, 'a) vec set" assumes "convex B"   and "uminus ` B \<subseteq> B"   and "B \<in> sets lebesgue"   and "2 ^ CARD('a) < emeasure lebesgue B" obtains x :: "(real, 'a) vec" where "x \<in> B" and "x \<noteq> 0" and "\<And>i. x $ i \<in> \<int>"
insert_index_2: fixes i :: "nat"   and i' :: "nat" assumes "i \<le> i'" shows "insert_index i i' = Suc i'"
sheaf_spec_def: fixes U :: "('a \<times> 'a) set set set" shows "local.local.sheaf_spec U \<equiv> {s \<in> \<Pi>\<^sub>E \<pp>\<in>U. local.carrier_quotient_ring \<^bsub>\<pp> local.add_rel local.mult_rel local.zero_rel\<^esub>. local.local.is_regular s U}"
extend_set_Un_distrib: fixes A :: "'a set"   and B :: "'a set" shows "extend_set h (A \<union> B) = extend_set h A \<union> extend_set h B"
assms: shows "ltl_convert \<phi> = (APs, \<phi>\<^sub>i)"
TensorDiag_preserves_Can: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Can t"   and "Can u"   and "Diag t"   and "Diag u" shows "Can (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"
appr_rell_internal: shows "appr_rell \<equiv> br set_of_appr top"
right_closed: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). x \<sim> y}" shows "b \<in> Sigma I \<FF>"
matrix_vector_mult_0_right: fixes A :: "(('a, 'c) vec, 'b) vec" shows "A *v 0 = 0"
StaticCastRed: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow> \<langle>\<lparr>C\<rparr>e',s'\<rangle>"
intros_1: fixes M :: "lam" shows " M \<longrightarrow>\<^sub>1\<^sup>* M"
map_cong0: fixes x :: "'c pmf"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>z. z \<in> set_pmf x \<Longrightarrow> f z = g z" shows "map_pmf f x = map_pmf g x"
simps_6: fixes x8 :: "'b ltl"   and y8 :: "'b ltl" shows "(G x8 = G y8) = (x8 = y8)"
Bex_fold: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "finite A" shows "Bex A P = Finite_Set.fold (\<lambda>k s. s \<or> P k) False A"
measurable_cong_sets: fixes M :: "'b measure"   and M' :: "'b measure"   and N :: "'c measure"   and N' :: "'c measure" assumes "sets M = sets M'"   and "sets N = sets N'" shows "M \<rightarrow>\<^sub>M N = M' \<rightarrow>\<^sub>M N'"
List_unique1: fixes h :: "'c \<Rightarrow> 'c ref"   and p :: "'c ref"   and as :: "'c list" assumes "List h p as" shows "\<exists>!as. List h p as"
M1_def: shows "M1 \<equiv> x$ = Suc<$x> \<and> y$ = $y"
o_id: fixes f :: "'i \<Rightarrow> 'j" shows "f \<circ> id = f"
Cc: shows "extreme_bound A (\<sqsubseteq>) C c"
multrow_mat_inv: fixes k :: "nat"   and n :: "nat"   and a :: "'c" assumes "k < n"   and "a \<noteq> (0::'c)" shows "multrow_mat n k a * multrow_mat n k (inverse a) = 1\<^sub>m n"
ch2ch_monofun: fixes f :: "'b \<Rightarrow> 'c"   and Y :: "nat \<Rightarrow> 'b" assumes "monofun f"   and "chain Y" shows "chain (\<lambda>i. f (Y i))"
is_fpxs_def: fixes f :: "rat \<Rightarrow> 'b" shows "is_fpxs f = (bdd_below (Formal_Puiseux_Series.supp f) \<and> (LCM r\<in>Formal_Puiseux_Series.supp f. snd (quotient_of r)) \<noteq> 0)"
tt_alt: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "tt p = ord_term_lin.Min (keys p)"
dvd_power_minus_1_conv1: fixes x :: "nat"   and a :: "nat"   and b :: "nat" assumes "1 < x"   and "0 < a"   and "x ^ a - 1 dvd x ^ b - 1"   and "0 < b" shows "a dvd b"
prime_power_iff: 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 n :: "nat" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalization_semidom.prime zero times one normalize (power.power one times p n) = (normalization_semidom.prime zero times one normalize p \<and> n = 1)"
essential_class_def: fixes K :: "'a \<Rightarrow> 'a pmf"   and C :: "'a set" shows "MC_syntax.essential_class K C = (C \<in> UNIV // MC_syntax.communicating K \<and> (SIGMA x:UNIV. set_pmf (K x))\<^sup>* `` C \<subseteq> C)"
parse_ipv6_address_compressed_identity2: fixes ipv6_syntax :: "ipv6addr_syntax_compressed"   and ls :: "16 word option list" shows "(ipv6addr_syntax_compressed_to_list ipv6_syntax = ls) = (parse_ipv6_address_compressed ls = Some ipv6_syntax)"
bit_horner_sum_bit_word_iff: fixes bs :: "bool list"   and n :: "nat" shows "bit (horner_sum of_bool 2 bs) n = (n < min LENGTH('a) (length bs) \<and> bs ! n)"
gfp_exp_lemma3: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "\<And>P Q. \<lbrakk>unitary P; unitary Q; P \<tturnstile> Q\<rbrakk> \<Longrightarrow> t P \<tturnstile> t Q" shows "t (gfp_exp t) \<tturnstile> gfp_exp t"
simps_3: fixes \<Sigma> :: "'b list"   and n :: "nat"   and m :: "nat"   and M :: "nat" assumes "formula \<Sigma>" shows "formula.rexp_of \<Sigma> n (FIn m M) = rexp.Inter (TIMES [rexp.Not Zero, Atom (Arbitrary_Except2 m M), rexp.Not Zero]) (formula.ENC n (FOV (FIn m M)))"
component_of_term_def: fixes v :: "'b \<times> 'a" shows "pprod.component_of_term v = snd v"
homotopic_paths: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" shows "homotopic_paths s p q = (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and> h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and> (\<forall>x\<in>{0..1}. h (0, x) = p x) \<and> (\<forall>x\<in>{0..1}. h (1, x) = q x) \<and> (\<forall>t\<in>{0..1}. pathstart (h \<circ> Pair t) = pathstart p \<and> pathfinish (h \<circ> Pair t) = pathfinish p))"
rank_uniqify: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_skew_invar q" shows "SkewBinomialHeapStruc.rank_invar (SkewBinomialHeapStruc.uniqify q)"
timpl_closure'_timpls_trancl_eq: fixes c :: "('g set \<times> 'g set) set" shows "timpl_closure' (c\<^sup>+) = timpl_closure' c"
nested_include_restrict: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and I :: "nat set"   and z :: "'c option list"   and t :: "'c option list"   and A :: "nat set" assumes "getIJ getIJ"   and "restrict I z = t"   and "A \<subseteq> I" shows "restrict A z = restrict A t"
responses_\<I>_parametric: fixes C :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_fun (rel_\<I> C R) (rel_fun C (rel_set R)) responses_\<I> responses_\<I>"
is_elementary_category_with_pullbacks: shows "elementary_category_with_pullbacks (\<odot>) PRJ\<^sub>0 PRJ\<^sub>1"
filter_mset_add_mset: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and A :: "'b multiset" shows "filter_mset P (add_mset x A) = (if P x then add_mset x (filter_mset P A) else filter_mset P A)"
v: shows "v (s (Suc n)) \<noteq> v (s n)"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> 'ag" assumes "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
uminus_1: fixes x :: "int poly \<times> rat \<times> rat" assumes "invariant_1 x" shows "invariant_1 (uminus_1 x) \<and> real_of_1 (uminus_1 x) = - real_of_1 x"
list_length_hint3: fixes l :: "'d list" assumes "length l = Suc 0" shows "l \<noteq> []"
sum_relE: fixes x :: "'a + 'b"   and x' :: "'c + 'd"   and Rl :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set" assumes "(x, x') \<in> \<langle>Rl, Rr\<rangle>sum_rel" obtains l :: "'a" and l' :: "'c" where "x = Inl l" and "x' = Inl l'" and "(l, l') \<in> Rl"   | r :: "'b" and r' :: "'d" where "x = Inr r" and "x' = Inr r'" and "(r, r') \<in> Rr"
Lim_transform2: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "(f \<longlongrightarrow> a) F"   and "((\<lambda>x. f x - g x) \<longlongrightarrow> (0::'b)) F" shows "(g \<longlongrightarrow> a) F"
span_li_not_depend_1: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and S :: "'d set"   and N :: "'d set" assumes "module R M"   and "S \<subseteq> N"   and "LinearCombinations.submodule R N M" shows "LinearCombinations.module.span R (M\<lparr>carrier := N\<rparr>) S = LinearCombinations.module.span R M S"
simps: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "prognosis ae as a (\<Gamma>, e, S) = (\<lambda>_. many) f|` (edom (ABinds \<Gamma>\<cdot>ae) \<union> edom (\<A>\<^bsub>a\<^esub> e) \<union> edom (AEstack as S))"
same_one_same_zero_2: fixes R :: "'a ring" assumes "R \<in> C" shows "\<zero>\<^bsub>union_ring C\<^esub> = \<zero>\<^bsub>R\<^esub>"
Knaster_Tarski_idem_extremes_2: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L"   and "Mono\<^bsub>L\<^esub> f"   and "Idem\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "\<bottom>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> f \<bottom>\<^bsub>L\<^esub>"
mix_pmf_preserves_finite_support: fixes a :: "real"   and p :: "'b pmf"   and q :: "'b pmf" assumes "a \<in> {0..1}"   and "finite (set_pmf p)"   and "finite (set_pmf q)" shows "finite (set_pmf (mix_pmf a p q))"
good_ruleset_append: fixes rs\<^sub>1 :: "'b rule list"   and rs\<^sub>2 :: "'b rule list" shows "good_ruleset (rs\<^sub>1 @ rs\<^sub>2) = (good_ruleset rs\<^sub>1 \<and> good_ruleset rs\<^sub>2)"
eq_fin_end: fixes u\<^sub>1 :: "'action list"   and u\<^sub>2 :: "'action list"   and v\<^sub>1 :: "'action list"   and v\<^sub>2 :: "'action list" assumes "u\<^sub>1 =\<^sub>F u\<^sub>2"   and "u\<^sub>1 @ v\<^sub>1 =\<^sub>F u\<^sub>2 @ v\<^sub>2" shows "v\<^sub>1 =\<^sub>F v\<^sub>2"
lmap_strict: fixes f :: "'c \<rightarrow> 'b" shows "lmap\<cdot>f\<cdot>\<bottom> = \<bottom>"
IH_3: shows "map f xs = (y # ys) @ ys2"