hcju/isabelleps huggingface dataset:
memo_rbt_rtrancl_def: fixes r :: "('b \<times> 'b) list" shows "memo_rbt_rtrancl r = (let tr = rtrancl_rbt_impl r; rm = rm.to_map (map (\<lambda>a. (a, tr [a])) (((rs.to_list \<circ>\<circ> set_op_from_list) rs_ops \<circ>\<circ> map) fst r)) in (\<lambda>a. case rm.lookup a rm of None \<Rightarrow> rs.from_list [a] | Some as \<Rightarrow> as))"
exist_reduce: fixes f :: "nat \<Rightarrow> real" shows "\<forall>c\<in>reduce f PR. \<exists>i\<in>PR - (kmax f PR \<union> kmin f PR). f i = c"
succ_edge: fixes n :: "nat" shows "\<forall>v\<in>succs n. (n, v) \<in> g_E G"
IH: assumes "y \<le> x"   and "eval (Pr n f g) (y # xs) \<down>" shows "reachable ((Pr n f g, x # xs, []) # rest, None) ((Pr n f g, x # xs, trace n f g xs y) # rest, None)"
assms_8: shows "\<not> (\<exists>(a', b', c')\<in>set d. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
split_spec_ivl_def: fixes M :: "nat"   and x :: "'b set" shows "split_spec_ivl M x = ivl_rep x \<bind> (\<lambda>(i, s). let d = s - i; b = max_coord M d; m = (i \<bullet> b + s \<bullet> b) / 2 in split_ivl_at_halfspace (Sctn b m) x)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "inj_comm_semiring_hom hom" shows "map_poly_inj_comm_semiring_hom hom"
1: shows "x \<notin> S \<or> s x \<noteq> 0"
lossless_funct_OT12: fixes funct_OT12 :: "bool \<times> bool \<Rightarrow> bool \<Rightarrow> (unit \<times> bool) spmf"   and m0 :: "bool"   and m1 :: "bool"   and c :: "bool" shows "lossless_spmf (funct_OT12 (m0, m1) c)"
bfsi'_def: fixes N :: "nat"   and s :: "nat"   and t :: "nat"   and psi :: "nat list Heap.array"   and cfi :: "int Heap.array" shows "bfsi' N s t psi cfi \<equiv> bfs_impl (\<lambda>(am, cf). succ_imp N am cf) (psi, cfi) s t"
design_defs_3: shows "J \<equiv> \<lambda>(A, A'). (ok A \<longrightarrow> ok A') \<and> alpha_d.more A = alpha_d.more A'"
tt: shows "exi out \<tau> = exi zz (subst \<tau> (Var zz) out)"
external_equiv_internal_invariant_vimage: fixes P :: "'e set"   and S :: "('e, 'f) spec"   and Q :: "'f set" assumes "P = obs S -` Q" shows "(oreach S \<subseteq> Q) = (reach S \<subseteq> P)"
ind_sheaf_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and V :: "'c set" assumes "ind_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str U" shows "ind_sheaf.ind_sheaf \<FF> U V \<equiv> \<FF> (U \<inter> V)"
closed_subspace: fixes s :: "'a set" assumes "subspace s" shows "closed s"
next_commutes_3: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<circle>(F \<longrightarrow> G) \<equiv> \<circle>F \<longrightarrow> \<circle>G"
simps_2: fixes x :: "'a" shows "psums [x] = [x]"
lemma5_2: fixes r :: "lam"   and r' :: "lam" assumes "r \<rightarrow>\<^sub>l r'" shows "NF r'"
subgroup_def: fixes G :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "subgroup G M composition unit \<equiv> submonoid G M composition unit \<and> group G composition unit"
snd_map_prod: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'c"   and x :: "'e \<times> 'f" shows "snd (map_prod f g x) = g (snd x)"
rsemimodule4: fixes s :: "'b" shows "\<lambda> (s, 1::'a) = s"
contractible_space_empty: fixes X :: "'b topology" assumes "topspace X = {}" shows "contractible_space X"
head_def: fixes tm :: "('a, 'b) tm" shows "head tm = (case tm of Hd x1 \<Rightarrow> x1 | App s _dummy_ \<Rightarrow> head0 s)"
bs_env_inv1: fixes x :: "nat"   and w :: "bval"   and \<rho> :: "(nat \<times> bval) list"   and \<rho>' :: "(nat \<times> exp) list" assumes "bs_env ((x, w) # \<rho>) \<rho>'" obtains v :: "exp" and \<rho>' :: "(nat \<times> exp) list" where "\<rho>'''__ = (x, v) # \<rho>'" and "bs_val w v" and "bs_env \<rho> \<rho>'"
cset_Coll_def: shows "cset_Coll \<equiv> map_fun rcset (map_fun id acset) (\<lambda>A P. {x \<in> A. P x})"
eventually_imp_iff: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x) = (eventually P F \<longrightarrow> eventually Q F)"
sp_lift_instr: fixes instr :: "('f, 'g, 'fun, 'op, 'opinl) Inca.instr"   and \<Sigma>1 :: "Unboxed.type option list"   and instr' :: "('f, 'g, 'fun, 'op, 'opinl, 'opubx, 'h, 'i) Ubx.instr"   and \<Sigma>2 :: "Unboxed.type option list"   and F :: "'fun \<Rightarrow> 'j fundef option" assumes "lift_instr instr \<Sigma>1 = Ok (instr', \<Sigma>2)"   and "\<And>x. rel_option (\<lambda>ar fd. arity fd = ar) (get_arity x) (F x)" shows "Subx.sp_instr F instr' \<Sigma>1 = Ok \<Sigma>2"
card_rcosets_equal: fixes R :: "'a set"   and H :: "'a set" assumes "R \<in> rcosets H"   and "H \<subseteq> carrier G" shows "card H = card R"
card_dom_random_oracle: fixes \<A> :: "('b, 'a, 'c) gpv"   and q :: "nat"   and y :: "'b"   and \<sigma>' :: "'a \<Rightarrow> 'c option"   and p :: "'c spmf"   and \<sigma> :: "'a \<Rightarrow> 'c option" assumes "interaction_bounded_by' (\<lambda>_. True) \<A> q"   and "(y, \<sigma>') \<in> set_spmf (exec_gpv (random_function.random_oracle p) \<A> \<sigma>)"   and "finite (dom \<sigma>)" shows "card (dom \<sigma>') \<le> q + card (dom \<sigma>)"
par_strict_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "B A ParStrict D C"
impl_safe: fixes c :: "('loc, 't) Propagate.configuration" assumes "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "\<And>loc. c_work c loc = {#}\<^sub>z" shows "impl_safe c"
intro: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real" assumes "\<And>x y. \<lbrakk>x \<in> carrier; y \<in> carrier\<rbrakk> \<Longrightarrow> x \<succeq>[relation] y = (u y \<le> u x)"   and "\<And>x y. x \<succeq>[relation] y \<Longrightarrow> x \<in> carrier"   and "\<And>x y. x \<succeq>[relation] y \<Longrightarrow> y \<in> carrier" shows "ordinal_utility carrier relation u"
GR: shows "(Gi, G) \<in> igba_impl_rel_ext unit_rel Rq Rl"
pdt_trancl_pdt: shows "pdt\<^sup>+ = pdt"
check_certD: fixes P :: "jvm_prog"   and mxs :: "nat"   and mxl :: "nat"   and n :: "nat"   and cert :: "(ty list \<times> ty err list) option err list" assumes "check_cert P mxs mxl n cert" shows "cert_ok cert n Err (OK None) (states P mxs mxl)"
lessThan_subset_iff: fixes x :: "'a"   and y :: "'a" shows "({..<x} \<subseteq> {..<y}) = (x \<le> y)"
const_onI: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "\<And>y. y \<in> S \<Longrightarrow> f y = x" shows "const_on f S x"
vimage_image_subset: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "A \<subseteq> f -` f ` A"
a5: fixes x :: "'b set"   and y :: "'b set" shows "a (x \<union> y) = a x \<inter> a y"
bernoulli_stream_pseudo_prob: fixes w :: "bool stream"   and n :: "nat" assumes "M = bernoulli_stream p"   and "p \<le> 1"   and "0 \<le> p"   and "w \<in> range (pseudo_proj_True n)" shows "prob (pseudo_proj_True n -` {w} \<inter> space M) = prod (prob_component p w) {0..<n}"
Smith_1xn_works: fixes A :: "'a mat"   and n :: "nat"   and S :: "'a mat"   and Q :: "'a mat" assumes "A \<in> carrier_mat 1 n"   and "(S, Q) = Smith_1xn A" shows "is_SNF A (1\<^sub>m 1, S, Q)"
finite_FactGroup: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "finite_group G"   and "H \<lhd> G" shows "finite_group (G Mod H)"
surjD: fixes f :: "'c \<Rightarrow> 'b"   and y :: "'b" assumes "top_pres ((`) f)" shows "\<exists>x. y = f x"
stat_acc: shows "G \<turnstile>Methd sig statM of statC accessible_from accC"
trg_hcomp: fixes \<mu> :: "'d"   and \<nu> :: "'d" assumes "D.hseq \<mu> \<nu>" shows "trg\<^sub>D (\<mu> \<star>\<^sub>D \<nu>) = trg\<^sub>D \<mu>"
intros: fixes rh :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "('e, 'f, ('e, 'f) T\<^sub>m, 'f T\<^sub>h) F\<^sub>m"   and y :: "('e, 'f, ('e, 'f) T\<^sub>m, 'f T\<^sub>h) F\<^sub>m" assumes "blinding_of_F rh bo (root_hash_T rh) (blinding_of_T rh bo) x y" shows "blinding_of_T rh bo (T\<^sub>m x) (T\<^sub>m y)"
eq: fixes x :: "'a"   and y :: "'a" shows "(comp x y = Eq) = (x = y)"
WHILE_annotRVI_def: fixes R :: "('a \<times> 'a) set"   and V :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'a"   and I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "WHILE_annotRVI R V I \<equiv> While"
subset_antisym: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
vec_real_inner_axioms: shows "vec_real_inner inner_field inner"
imp: fixes x :: "'a" assumes "x \<in># \<Sum>\<^sub># (mset_set (mset_set ` P))" shows "x \<in># mset_set A"
alpha_beta_orth: shows "inner_prod \<alpha> \<beta> = 0"
L4_6: fixes \<phi> :: "'b ltlr"   and qs :: "'b node set"   and q :: "'b node" assumes "create_gba_from_nodes_precond \<phi> qs"   and "q \<in> g_V0 (create_gba_from_nodes \<phi> qs)" shows "\<phi> \<in> old q"
bernoulli_sigma_finite: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "\<forall>n. sigma_finite_subalgebra N (G n)"
to_ct_list_append_2: fixes ts :: "t list"   and as :: "ct list"   and bs :: "ct list" assumes "to_ct_list ts = as @ bs" shows "\<exists>bs'. to_ct_list bs' = bs"
ord_rec_Seq_def: fixes T :: "hf"   and G :: "hf \<Rightarrow> hf"   and s :: "hf"   and k :: "hf"   and y :: "hf" shows "ord_rec_Seq T G s k y = (Seq s k \<and> y = G (app s (\<Squnion>k)) \<and> app s 0 = T \<and> (\<forall>n. succ n \<^bold>\<in> k \<longrightarrow> app s (succ n) = G (app s n)))"
input2state_within_bounds: fixes i :: "value list"   and x :: "nat"   and a :: "value" assumes "input2state i $ x = Some a" shows "x < length i"
in_path_event: fixes Q :: "'a set"   and a :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q" shows "a \<in> \<E>"
analytic_on_add: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "f analytic_on S"   and "g analytic_on S" shows "(\<lambda>z. f z + g z) analytic_on S"
Un_insert_left: fixes a :: "'a"   and B :: "'a set"   and C :: "'a set" shows "(a \<triangleright> B) \<union> C = a \<triangleright> B \<union> C"
bind_runit: fixes x :: "'f option" shows "x \<bind> Some = x"
card_of_min_alg1: fixes s1 :: "('e, 'd, 'f) F1 \<Rightarrow> 'd"   and s2 :: "('e, 'd, 'f) F2 \<Rightarrow> 'f" shows "|min_alg1 s1 s2| \<le>o ASucFbd"
assms: shows "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
transrec3_succ: fixes a :: "V"   and b :: "V \<Rightarrow> V \<Rightarrow> V"   and c :: "V \<Rightarrow> (V \<Rightarrow> V) \<Rightarrow> V"   and i :: "V" shows "transrec3 a b c (ZFC_in_HOL.succ i) = b i (transrec3 a b c i)"
primerec_rec_add_2: shows "primerec rec_add (Suc (Suc 0))"
prop3: fixes xs :: "letter list list"   and a :: "letter"   and zs :: "letter list list" assumes "bar xs"   and "xs \<noteq> []"   and "R a xs zs" shows "bar zs"
sourcesA81: shows "Sources level1 sA81 = {sA71, sA91, sA6} \<union> Sources level1 sA91"
lookup_union1: fixes k :: "'a"   and ys :: "'a \<rightharpoonup>\<^sub>f 'b"   and xs :: "'a \<rightharpoonup>\<^sub>f 'b" assumes "k |\<in>| fmdom ys" shows "lookup (xs ++\<^sub>f ys) k = lookup ys k"
join_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
pole_theorem_analytic: fixes g :: "complex \<Rightarrow> complex"   and S :: "complex set"   and a :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "g analytic_on S"   and "\<And>z. z \<in> S \<Longrightarrow> \<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w" shows "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) analytic_on S"
simps_26: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "nat"   and x13 :: "nat"   and x14 :: "k"   and x15 :: "nat"   and x16 :: "nat"   and x17 :: "nat" shows "(case Rreq x11 x12 x13 x14 x15 x16 x17 of Rreq x xa xb xc xd xe xf \<Rightarrow> f1 x xa xb xc xd xe xf | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f1 x11 x12 x13 x14 x15 x16 x17"
divide_eq_eq_numeral1_2: fixes b :: "'a"   and w :: "num"   and a :: "'a" shows "(b / - numeral w = a) = (if - numeral w \<noteq> (0::'a) then b = a * - numeral w else a = (0::'a))"
part_equivp_Gromov_completion_rel: shows "part_equivp Gromov_completion_rel"
Lindelof_space_perfect_map_image: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "Lindelof_space X"   and "perfect_map X Y f" shows "Lindelof_space Y"
list_3_cases: fixes xs :: "'b list" obtains "xs = []"   | x :: "'b" where "xs = [x__]"   | x :: "'b" and y :: "'b" and ys :: "'b list" where "xs = x # y # ys"
is_ang_conga: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C Ang a"   and "A' B' C' Ang a" shows "A B C CongA A' B' C'"
image_Nsetn_card_pos: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat" shows "0 < card (f ` {i. i \<le> n})"
pr_index_enumerator_mono2: fixes m1 :: "nat"   and m2 :: "nat"   and n :: "nat" assumes "m1 < m2" shows "pr_index_enumerator n m1 < pr_index_enumerator n m2"
seq_semD': fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp"   and \<Gamma> :: "('sf, 'sc, 'sz) formula list"   and \<Delta> :: "('sf, 'sc, 'sz) formula list" assumes "\<nu> \<in> seq_sem I (\<Gamma>, \<Delta>)" obtains "\<nu> \<in> fml_sem I (foldr (&&) \<Gamma> TT) \<Longrightarrow> \<nu> \<in> fml_sem I (foldr (||) \<Delta> FF)"
funpow_dist_prop: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "y \<in> orbit f x" shows "(f ^^ funpow_dist f x y) x = y"
ceiling_le_iff: fixes x :: "'b"   and z :: "int" shows "(\<lceil>x\<rceil> \<le> z) = (x \<le> of_int z)"
norm_cblinfun_eqI: fixes n :: "real"   and f :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and x :: "'b" assumes "n \<le> norm (f *\<^sub>V x) / norm x"   and "\<And>x. norm (f *\<^sub>V x) \<le> n * norm x"   and "0 \<le> n" shows "norm f = n"
card_const_lt_component_le: fixes B :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "finite B" shows "card (const_lt_component ` (B - {0}) - {None}) \<le> card (component_of_term ` Keys B)"
genby_lcoset_subgroup_imp_eq_reps: fixes w :: "'b"   and Sa :: "'b set"   and w' :: "'b"   and S :: "'b set" assumes "order.greater_eq (w' +o \<langle>S\<rangle>) (w +o \<langle>Sa\<rangle>)" shows "w' +o \<langle>S\<rangle> = w +o \<langle>S\<rangle>"
intros_2: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
continuous_map_minus: fixes X :: "'c topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X euclidean f" shows "continuous_map X euclidean (\<lambda>x. - f x)"
wls_fresh_Op_simp: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and xs :: "'varSort"   and x :: "'var" assumes "wlsInp delta inp"   and "wlsBinp delta binp" shows "fresh xs x (Op delta inp binp) = (freshInp xs x inp \<and> freshBinp xs x binp)"
a1: shows "s' = snd (fst (load_sub1 instr rd s_val s)) \<and> get_S (cpu_reg_val PSR s) = 0"
chart_last_inverse: fixes x :: "'b proj_space" assumes "chart_last_domainP x" shows "chart_last_inv (chart_last x) = x"
eq: shows "\<Union> \<F> = \<Union> (U ` S)"
simps: fixes P :: "'b \<Rightarrow> bool" shows "Preliminaries2.index P = (if Alm_all P then Some (LEAST i. \<forall>j\<ge>i. P j) else None)"
Dom_cod: fixes f :: "('o, 'a) arr" assumes "arr f" shows "Dom (cod f) = Cod f"
\<psi>\<^sub>2_values_odd: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"   and "j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"   and "odd i" shows "Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (i, j) = complex_of_real ((- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))"
Cv_inter_Ei_subsetof_Cvi: shows "C\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ESi\<^esub> \<subseteq> C\<^bsub>\<V>i\<^esub>"
Cons_1: fixes l1 :: "'a list"   and h :: "'a"   and l1' :: "'a list"   and l2 :: "'a list" assumes "l1 = h # l1'"   and "l2 = l2a @ [h] @ l2b"   and "subseq l1 l2"   and "\<not> ListMem h l2a" shows "subseq l1 (h # l2b)"
map\<^sub>0_simp: fixes a :: "'d" assumes "D.obj a" shows "GF.map\<^sub>0 a = G.map\<^sub>0 (F.map\<^sub>0 a)"
y: shows "y \<in> aform_err e ((X, xs), d)"
is_type_void: fixes P :: "'a prog" shows "is_type P Void = True"
allI: fixes P :: "'a \<Rightarrow> bool" assumes "\<And>x. P x" shows "\<forall>x. P x"
l6_21: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C"   and "C \<noteq> D"   and "Tarski_neutral_dimensionless.Col Bet A B P"   and "Tarski_neutral_dimensionless.Col Bet A B Q"   and "Tarski_neutral_dimensionless.Col Bet C D P"   and "Tarski_neutral_dimensionless.Col Bet C D Q" shows "P = Q"
good2: shows "is_interp (local.adjoint I \<sigma> \<omega>)"
undeff_equiv: fixes \<phi> :: "fml option" shows "(\<phi> \<noteq> undeff) = (\<exists>f. \<phi> = Afml f)"
distinct_1: fixes x1 :: "int"   and x2 :: "String.literal" shows "Num x1 \<noteq> value.Str x2"
matches_oiface_rewrite_ipassmt: fixes m :: "'c common_primitive match_expr"   and ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and p :: "('c, 'd) tagged_packet_scheme"   and \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action" assumes "normalized_nnf_match m"   and "ipassmt_sanity_nowildcards ipassmt"   and "ipassmt_sanity_disjoint ipassmt"   and "\<exists>p_ips. ipassmt (Iface (p_oiface p)) = Some p_ips \<and> p_dst p \<in> ipcidr_union_set (set p_ips)" shows "matches (common_matcher, \<alpha>) (oiface_rewrite ipassmt m) a p = matches (common_matcher, \<alpha>) m a p"
not_iszero_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and w :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "\<not> ring_1.iszero zero (numeral.numeral one plus w)"
that: fixes g :: "'c \<Rightarrow> 'a" assumes "continuous_on U g"   and "g ` U \<subseteq> C"   and "\<And>y. y \<in> U \<Longrightarrow> p (g y) = f y" shows "thesis"
order_linear': fixes a :: "'b"   and b :: "'b" shows "order a [:b, 1::'b:] = (if b = - a then 1 else 0)"
choice_Basis_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and P :: "'b \<Rightarrow> real \<Rightarrow> bool" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(\<forall>i\<in>Basis. \<exists>x. P i x) = (\<exists>x. \<forall>i\<in>Basis. P i (inner x i))"
ce_rel_lm_14: fixes x :: "nat"   and y :: "nat"   and A :: "nat set" assumes "c_pair x y \<in> A" shows "(x, y) \<in> ce_set_to_rel A"
vsv_vimage_iff: fixes b :: "V"   and Aa :: "V"   and A :: "V" shows "(b \<in>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> Aa) `\<^sub>\<circ> A) = (\<exists>a. a \<in>\<^sub>\<circ> A \<and> a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> Aa) \<and> (r \<restriction>\<^sup>l\<^sub>\<circ> Aa)\<lparr>a\<rparr> = b)"
recv_marker_means_snapshotted: fixes t :: "('a, 'b, 'c) event list"   and ev :: "('a, 'b, 'c) event"   and cid :: "nat"   and p :: "'a"   and q :: "'a"   and i :: "nat" assumes "trace init t final"   and "ev = RecvMarker cid p q"   and "S t i \<turnstile> ev \<mapsto> S t (Suc i)" shows "ps (S t i) q \<noteq> None"
clm_add: shows "|restrict v (clm ts) c| = |restrict vu (clm ts) c| + |restrict vd (clm ts) c|"
simps: fixes a1 :: "'e"   and a2 :: "'e" shows "((a1, a2) \<in> hull_rel) = (\<exists>p x. a1 = p \<bullet> x \<and> a2 = x)"
ph2: fixes t :: "real" assumes "t \<in> {0..1}" shows "p (h2 t) = g2 t"
ideal_generated_UNIV_insert: fixes S :: "'a set"   and a :: "'a" assumes "ideal_generated S = UNIV" shows "ideal_generated (insert a S) = UNIV"
sscan_stake: fixes k :: "nat"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c stream"   and a :: "'b" shows "stake k (sscan f xs a) = scan f (stake k xs) a"
convex_negations: fixes S :: "'c set" assumes "convex S" shows "convex (uminus ` S)"
assms: shows "\<guillemotleft>p : unity \<rightarrow> a\<guillemotright>"
flip_commute: fixes a :: "'e"   and b :: "'e" shows "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
all_nstri_imp_lex_nstri: fixes ys :: "'a list"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "'a list"   and m :: "nat" assumes "\<forall>i<length ys. snd (f (xs ! i) (ys ! i))"   and "length ys \<le> length xs"   and "length xs = length ys \<or> length ys \<le> m" shows "snd (lex_ext f m xs ys)"
unstream_dropWhile_trans_False: fixes P :: "'e \<Rightarrow> bool"   and g :: "('e, 'f) generator"   and s :: "'f" shows "unstream (dropWhile_trans P g) (False, s) = unstream g s"
closed_sequential_limits: fixes S :: "'b set" shows "closed S = (\<forall>x l. (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> l \<longrightarrow> l \<in> S)"