hcju/isabelleps huggingface dataset:
\<G>_\<F>\<G>_subset: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<G> \<phi> w \<subseteq> \<F>\<G> \<phi> w"
LIMSEQ_inv_powr: fixes c :: "real"   and d :: "real" assumes "0 < c"   and "0 < d" shows "(\<lambda>n. (c / real n) powr d) \<longlonglongrightarrow> 0"
no_head_sub: fixes C :: "'b list set"   and b :: "'b list" shows "no_head C b \<subseteq> \<langle>C\<rangle>"
simw: shows "similar_mat_wit A (jordan_matrix n_as) P Q"
hd_or_tl: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and xs :: "'a list"   and y :: "'a"   and x :: "'a" assumes "\<forall>z y x. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "\<forall>y x. gt y x \<or> gt x y \<or> y = x"   and "length ys = length xs"   and "ext gt (y # ys) (x # xs)" shows "gt y x \<or> ext gt ys xs"
lower_sup: fixes A :: "'b interval"   and B :: "'b interval" shows "lower (sup A B) = inf (lower A) (lower B)"
permutes_invI: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and p' :: "'b \<Rightarrow> 'b" assumes "p permutes S"   and "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x"   and "\<And>x. x \<notin> S \<Longrightarrow> p' x = x" shows "inv p = p'"
d2_2: fixes K :: "'a set" assumes "K \<in> p2" shows "K \<subseteq> s2"
idem_normalize: fixes a :: "'a" shows "lcm a a = normalize a"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "monoid_add_hom hom" shows "comm_monoid_add_hom hom"
sub2_wit_norms_mus': fixes n_gs' :: "('a vec \<times> 'a) list"   and gs' :: "'a vec list"   and fs :: "'a vec list"   and mus :: "'a list list"   and gs :: "'a vec list"   and mus_acc :: "'a list list" assumes "n_gs' = map (\<lambda>v. (v, \<parallel>v\<parallel>\<^sup>2)) gs'"   and "sub2_wit gs' fs = (mus, gs)"   and "set fs \<subseteq> carrier_vec n"   and "set gs' \<subseteq> carrier_vec n" shows "norms_mus' fs n_gs' mus_acc = (map sq_norm (rev gs @ gs'), rev mus @ mus_acc)"
set_char: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "local.set a = UP ` Dom a"
tendsto_at_within_iff_tendsto_nhds: fixes g :: "'b \<Rightarrow> 'c"   and l :: "'b"   and S :: "'b set" shows "(g \<longlongrightarrow> g l) (at l within S) = (g \<longlongrightarrow> g l) (inf (nhds l) (principal S))"
Spec: fixes s :: "'a"   and t :: "'a"   and r :: "('a \<times> 'a) set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option" assumes "(s, t) \<in> r" shows "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> \<Rightarrow> Normal t"
lcf_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and p :: "'e" shows "lcf R S X p = snd (s_cf R S X p) (fst (s_cf R S X p))"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b * (a * c) = a * (b * c)"
signed_take_bit_eq_if_positive: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "\<not> bit a n" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = take_bit n a"
nat_pow_one: fixes n :: "nat" shows "\<one>\<^bsub>H\<^esub> [^]\<^bsub>H\<^esub> n = \<one>\<^bsub>H\<^esub>"
word_add_le_iff: fixes i :: "'b word"   and k :: "'b word"   and j :: "'b word" assumes "unat i + unat k < 2 ^ LENGTH('b)"   and "unat j + unat k < 2 ^ LENGTH('b)" shows "(i + k \<le> j + k) = (i \<le> j)"
map_prod_def: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
range_permutation_def: fixes A :: "'c set"   and B :: "'d set" shows "range_permutation A B = {(f, f'). (f, f') \<in> (A \<rightarrow>\<^sub>E B) \<times> (A \<rightarrow>\<^sub>E B) \<and> (\<exists>p. p permutes B \<and> (\<forall>x\<in>A. f x = p (f' x)))}"
meta_adjunction_axioms: shows "meta_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<phi> \<psi>"
uniformly_convergent_eq_cauchy: fixes P :: "'c \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" shows "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e) = (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e)"
if_distrib: fixes x :: "bool"   and y :: "'a"   and z :: "'a" shows "(if x then y else (0::'a)) * z = (if x then y * z else (0::'a))"
borel_measurable_if_I: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue" shows "(\<lambda>x. if x \<in> S then f x else (0::'c)) \<in> borel_measurable lebesgue"
DBM_val_bounded_def: fixes v :: "'b \<Rightarrow> nat"   and u :: "'b \<Rightarrow> 'd"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat" shows "u \<turnstile>\<^bsub>v,n\<^esub> m \<equiv> Le (0::'d) \<preceq> m 0 0 \<and> (\<forall>c. v c \<le> n \<longrightarrow> dbm_entry_val u None (Some c) (m 0 (v c)) \<and> dbm_entry_val u (Some c) None (m (v c) 0)) \<and> (\<forall>c1 c2. v c1 \<le> n \<and> v c2 \<le> n \<longrightarrow> dbm_entry_val u (Some c1) (Some c2) (m (v c1) (v c2)))"
RelRel': fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> Rel" shows "(P, Q) \<in> Rel'"
distinct_45: fixes x51 :: "expr"   and x52 :: "stmt"   and x53 :: "stmt"   and x61 :: "expr"   and x62 :: "stmt" shows "Cond x51 x52 x53 \<noteq> While x61 x62"
triangle_contour_integrals_convex_primitive: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and a :: "complex"   and x :: "complex" assumes "continuous_on S f"   and "a \<in> S"   and "convex S"   and "x \<in> S"   and "\<And>b c. \<lbrakk>b \<in> S; c \<in> S\<rbrakk> \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0" shows "((\<lambda>x. contour_integral (linepath a x) f) has_field_derivative f x) (at x within S)"
gf: shows "(\<lambda>x. g x - f 0 x) integrable_on S \<and> (\<lambda>k. integral S (\<lambda>x. f (Suc k) x - f 0 x)) \<longlonglongrightarrow> integral S (\<lambda>x. g x - f 0 x)"
to_nat_i_le_m: shows "mod_type_class.to_nat i \<le> mod_type_class.to_nat m"
trans: fixes A :: "'e set"   and h :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "blinding_of_on A h bo"   and "bo x y"   and "bo y z"   and "x \<in> A" shows "bo x z"
Rep_formula_cases: fixes y :: "'a set set" assumes "y \<in> fba" obtains x :: "'a formula" where "y = Rep_formula x"
converse_rtrancl_induct3: fixes r :: "'d \<times> 'e \<times> 'f \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> bool"   and cs :: "'d"   and css :: "'e"   and s :: "'f"   and cs' :: "'d"   and css' :: "'e"   and t :: "'f"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (cs, css, s) (cs', css', t)"   and "P cs' css' t"   and "\<And>a aa b ab ac ba. \<lbrakk>r (a, aa, b) (ab, ac, ba); r\<^sup>*\<^sup>* (ab, ac, ba) (cs', css', t); P ab ac ba\<rbrakk> \<Longrightarrow> P a aa b" shows "P cs css s"
iso_is_section: fixes f :: "'b" assumes "\<phi>.B.iso f" shows "\<phi>.B.section f"
map_replicate_const: fixes k :: "'a"   and lst :: "'b list" shows "map (\<lambda>x. k) lst = k \<up> length lst"
SD_pref_profileD_2: fixes i :: "'agent"   and q :: "'alt pmf"   and p :: "'alt pmf" assumes "i \<in> agents'"   and "SD (R i) q p" shows "q \<in> lowered.lotteries"
dual\<^sub>s\<^sub>t_tfr\<^sub>s\<^sub>t\<^sub>p: fixes S :: "('fun, 'var) strand_step list" assumes "list_all tfr\<^sub>s\<^sub>t\<^sub>p S" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (dual\<^sub>s\<^sub>t S)"
t_fourtimes_len_def: shows "t_fourtimes_len = length t_fourtimes div 2"
apath_awalk_to_apath: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.awalk u p v" shows "G.apath u (G.awalk_to_apath p) v"
ss_67: fixes s :: "form list" shows "ns_of_s s = map (Pair 0) s"
retractionI: fixes g :: "'s"   and f :: "'s" assumes "ide (g \<cdot> f)" shows "retraction g"
emeasure_lborel_cbox_finite: fixes a :: "'c"   and b :: "'c" shows "emeasure lborel (cbox a b) < \<infinity>"
congruent_triangle_sas_4: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "angle a1 b1 c1 = angle a2 b2 c2" shows "dist b1 a1 = dist a2 b2"
bind_writer_def: fixes return :: "'e \<times> 'd list \<Rightarrow> 'f"   and bind :: "'f \<Rightarrow> ('e \<times> 'd list \<Rightarrow> 'f) \<Rightarrow> 'f"   and m :: "('d, 'e, 'f) writerT"   and f :: "'e \<Rightarrow> ('d, 'e, 'f) writerT" shows "bind_writer return bind m f = WriterT (bind (run_writer m) (\<lambda>(a, ws). bind (run_writer (f a)) (\<lambda>(b, ws'). return (b, ws @ ws'))))"
map_in_list_rel_conv: fixes l :: "'c list"   and \<alpha> :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool" shows "((l, map \<alpha> l) \<in> \<langle>br \<alpha> I\<rangle>list_rel) = (\<forall>x\<in>set l. I x)"
simps_6: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'e \<Rightarrow> 'd"   and x2 :: "'e" shows "(case deny x2 of allow x \<Rightarrow> f1 x | deny x \<Rightarrow> f2 x) = f2 x2"
sys_block_sizes_def: shows "source.sys_block_sizes \<equiv> {int (card bl) |bl. bl \<in># \<B>}"
unaffected_domains_subset: fixes U :: "'a set"   and D :: "'b \<Rightarrow> 'a"   and I :: "('a \<times> 'a) set"   and xs :: "'b list" assumes "U \<subseteq> range D"   and "U \<noteq> {}" shows "unaffected_domains I D U xs \<subseteq> range D \<inter> (- I) `` range D"
normalize_mult_unit_left: fixes a :: "'b poly"   and b :: "'b poly" assumes "is_unit a" shows "smult (inverse (lead_coeff (a * b))) (a * b) = smult (inverse (lead_coeff b)) b"
intros_42: fixes p :: "pat"   and ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and v1 :: "v"   and e :: "exp0"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" assumes "\<not> allDistinct (pat_bindings p [])" shows "evaluate_match ck env s v1 ((p, e) # pes) err_v (s, Rerr (Rabort Rtype_error))"
Nat_ss_UNIV: shows "\<guillemotleft>\<nat>\<guillemotright> \<subseteq> U"
invrules_4: shows "2 * card P\<^sub>2 \<le> card (\<Union> P\<^sub>2)"
ordinal_less_exp_oLog: fixes b :: "ordinal"   and x :: "ordinal" assumes "oSuc 0 < b" shows "x < b ** oLog b x * b"
rep_eq: fixes x :: "'e"   and xa :: "'d"   and xb :: "('b, 'c, 'd, 'e) formula" shows "Rep_formula (Act x xa xb) = Act\<^sub>\<alpha> x xa (Rep_formula xb)"
matrix_to_iarray_Gauss_Jordan: fixes A :: "(('b, 'c) vec, 'd) vec" shows "matrix_to_iarray (Gauss_Jordan A) = Gauss_Jordan_iarrays (matrix_to_iarray A)"
nsqn\<^sub>r_val: fixes dsn :: "nat"   and dsk :: "k"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "nsqn\<^sub>r (dsn, dsk, val, hops, nhip, pre) = dsn"
exists: fixes p :: "assertion"   and c :: "command"   and q :: "assertion"   and x :: "char list" assumes "prov_triple (p, c, q)" shows "prov_triple (Exists x p, c, Exists x q)"
simple_bochner_integral_eq_nn_integral: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "simple_bochner_integrable M f"   and "\<And>x. 0 \<le> f x" shows "ennreal (simple_bochner_integral M f) = \<integral>\<^sup>+ x. ennreal (f x) \<partial>M"
Max_less_iff: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(Max A < x) = (\<forall>a\<in>A. a < x)"
intPF_def: shows "intPF \<equiv> InfModel.intPF GE_parOf intTI intPI"
fps_div_by_zero': fixes g :: "'a fps" assumes "inverse (0::'a) = (0::'a)" shows "g / 0 = 0"
src_cod: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.src (S.cod \<mu>) = S.src \<mu>"
af_equals_af\<^sub>G_base_cases_3: fixes a :: "'a"   and w :: "'a set list" shows "af p(a) w = af\<^sub>G p(a) w"
dsj_EH_2: fixes A :: "fmla"   and A1 :: "fmla"   and H :: "fmla set"   and B :: "fmla"   and Ba :: "fmla" assumes "insert A (insert A1 H) \<turnstile> B"   and "insert Ba (insert A1 H) \<turnstile> B" shows "insert A1 (insert (A OR Ba) H) \<turnstile> B"
clauseFalseAppendValuation: fixes clause :: "Literal list"   and valuation :: "Literal list"   and valuation' :: "Literal list" assumes "clauseFalse clause valuation" shows "clauseFalse clause (valuation @ valuation')"
Rep_zet: fixes x :: "'b zet" shows "Rep_zet x \<in> zet"
Skip: fixes A :: "triple set"   and P :: "assn" shows "A \<turnstile> {P} Skip {P}"
dims_A: fixes rs :: "nat list"   and y :: "nat"   and shared_weights :: "bool"   and ws :: "nat \<Rightarrow> real" assumes "deep_model_correct_params_y rs y" shows "dims (deep_model_correct_params_y.A shared_weights rs y ws) = replicate (2 * deep_model_correct_params.N_half rs) (last rs)"
succg_rel_def: fixes R :: "('b \<times> 'c) set" shows "\<langle>R\<rangle>succg_rel \<equiv> (R \<rightarrow> \<langle>R\<rangle>list_set_rel) O br E_of_succ (\<lambda>_. True)"
SETfunDomAppCod: fixes f :: "ZF"   and X :: "ZF"   and Y :: "ZF"   and x :: "ZF" assumes "f maps\<^bsub>SET\<^esub> X to Y"   and "x |\<in>| X" shows "f |@| x |\<in>| Y"
is_natural_transformation: shows "natural_transformation A B F F \<tau>\<tau>'.map"
tss_list'_def_2: shows "tss_list' = dropWhile (\<lambda>t. enat t + right I < enat nt) (linearize tss)"
3: shows "|\<Union>xs. \<Union> (range (AS xs))| <o |UNIV|"
vsv_is_vconst_onI: fixes A :: "V"   and a :: "V" assumes "\<D>\<^sub>\<circ> r = A"   and "\<R>\<^sub>\<circ> r = set {a}" shows "r = vconst_on A a"
rel_\<I>I: fixes C :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and \<I>1 :: "('b, 'e) \<I>"   and \<I>2 :: "('d, 'f) \<I>"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "rel_set C (outs_\<I> \<I>1) (outs_\<I> \<I>2)"   and "\<And>x y. C x y \<Longrightarrow> rel_set R (responses_\<I> \<I>1 x) (responses_\<I> \<I>2 y)" shows "rel_\<I> C R \<I>1 \<I>2"
simps_8: fixes f :: "'b \<Rightarrow> 'a" shows "map_tree f Node = Node"
\<S>_abss_S: fixes s :: "'s \<times> ('c \<Rightarrow> real)" assumes "abss s \<in> \<S>" shows "s \<in> S"
left_right_m: fixes m :: "real"   and a :: "real"   and b :: "real" assumes "m \<in> {a..b}"   and "{a..b} \<subseteq> {0..1}" shows "leftcut a b m \<le> m \<and> m \<le> rightcut a b m"
simps_2: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int option" shows "pbval' (bexp.Not b) s = optnot (pbval' b s)"
s_AuthP_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Auth e\<ggreater> P\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Auth e'a" and "\<lless>\<pi>, e\<ggreater> P\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"   | "\<pi>' = \<pi>" and "e' = Hashed (hash \<lparr>e\<rparr>) e" and "closed \<lparr>e\<rparr>" and "value e"
assms_1: shows "v \<leadsto>(xs @ x # xs')\<leadsto> w"
bitlen_pow2: fixes b :: "int"   and c :: "nat" assumes "0 < b" shows "bitlen (b * 2 ^ c) = bitlen b + int c"
matrix_to_iarray_Gram_Schmidt_column_k: fixes k :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "k < ncols A" shows "matrix_to_iarray (Gram_Schmidt_column_k A k) = Gram_Schmidt_column_k_iarrays (matrix_to_iarray A) k"
disc_nodes: shows "h \<turnstile> get_disconnected_nodes document \<rightarrow>\<^sub>r disc_nodes"
igSwapBinp_def: fixes MOD :: "('d, 'e, 'f, 'g, 'h, 'i, 'j, 'c, 'k) model_scheme"   and zs :: "'f"   and z1 :: "'i"   and z2 :: "'i"   and binp :: "'b \<Rightarrow> 'c option" shows "igSwapBinp MOD zs z1 z2 binp \<equiv> lift (igSwapAbs MOD zs z1 z2) binp"
2: shows "monom_mult (- c / lc f) (0::'a) (p \<odot> tail f) = - local.punit.monom_mult (c / lc f) (0::'a) p \<odot> tail f"
outstanding_refs_append: fixes vol :: "'b memref \<Rightarrow> bool"   and xs :: "'b memref list"   and ys :: "'b memref list" shows "outstanding_refs vol (xs @ ys) = outstanding_refs vol xs \<union> outstanding_refs vol ys"
MVT: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a < b"   and "continuous_on {a..b} f"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> f differentiable at x" shows "\<exists>l z. a < z \<and> z < b \<and> (f has_real_derivative l) (at z) \<and> f b - f a = (b - a) * l"
mod_less_eq_dividend: fixes a :: "'c"   and b :: "'c" assumes "(0::'c) \<le> a" shows "a mod b \<le> a"
that: fixes N :: "'a"   and Q :: "('a, 'b, 'c) psi" assumes "inputChain xvec M P = inputChain yvec N Q" shows "thesis"
vars_com_def: fixes c :: "com" shows "vars c = lvars c \<union> rvars c"
simps_2: fixes ts :: "('c btree \<times> 'c) list"   and t :: "'c btree" shows "nonempty_lasttreebal (Node ts t) = ((\<exists>ls tsub tsep. ts = ls @ [(tsub, tsep)] \<and> height tsub = height t) \<and> nonempty_lasttreebal t)"
semiring_0_cancel_ow_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "semiring_0_cancel_ow U plus minus zero times \<equiv> semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero"
simps: fixes a :: "'a llist" shows "lsorted a = (a = LNil \<or> (\<exists>x. a = LCons x LNil) \<or> (\<exists>x y xs. a = LCons x (LCons y xs) \<and> x \<le> y \<and> lsorted (LCons y xs)))"
valid_success'': fixes ioprog :: "'d \<Rightarrow> 'c \<Rightarrow> ('b \<times> 'c) option"   and a :: "'d"   and \<sigma> :: "'c"   and b :: "'b"   and \<sigma>' :: "'c"   and S :: "'d list"   and P :: "'b list \<Rightarrow> bool" assumes "ioprog a \<sigma> = Some (b, \<sigma>')" shows "(\<sigma> \<Turnstile> ( s \<leftarrow> mbind (a # S) ioprog; unit\<^sub>S\<^sub>E (P s))) = (\<sigma>' \<Turnstile> ( s \<leftarrow> mbind S ioprog; unit\<^sub>S\<^sub>E (P (b # s))))"
vec_zero_r: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "vec_zero *\<^sub>v\<^sub>m A = vec_zero"
subprob_density_distr_real_exp: fixes M :: "real measure"   and f :: "real \<Rightarrow> ennreal" assumes "has_subprob_density M lborel f" shows "has_subprob_density (distr M borel exp) lborel (\<lambda>x. if 0 < x then f (ln x) * ennreal (inverse x) else 0)"
prob0_fst_qubits_leq_1: shows "(cmod (\<Sum>k<2 ^ n. (- 1) ^ f k))\<^sup>2 \<le> real (2 ^ (2 * n))"
leq: shows "(C, D) \<in> (subcls1 P)\<^sup>+"
assms: shows "low_equal s1 s2 \<and> t1 = store_barrier_pending_mod False s1 \<and> t2 = store_barrier_pending_mod False s2"
divisors: fixes x :: "nat"   and n :: "nat"   and d :: "nat" assumes "x \<in> {1..n}"   and "d dvd x" shows "\<exists>k\<in>{1..n div d}. x = d * k"
simps_7: fixes x31 :: "'a"   and x32 :: "'a ifex"   and x33 :: "'a ifex" shows "IF x31 x32 x33 \<noteq> Falseif"
degree_isnpolyh_Suc': fixes n :: "nat"   and p :: "poly" assumes "0 < n"   and "isnpolyh p n" shows "Reflected_Multivariate_Polynomial.degree p = 0"
limit_upperbound: fixes G :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and I :: "'a set" assumes "\<And>I. I \<subseteq> G \<Longrightarrow> f I \<subseteq> G"   and "I \<subseteq> G" shows "limit f I \<subseteq> G"
seqseq_holds: fixes n :: "nat" shows "\<exists>l. (\<lambda>i. (((\<lambda>k. fm (Suc k) (y (Suc k))) \<circ> seqseq (Suc n)) i)\<^sub>F n) \<longlonglongrightarrow> l"
c_ratio_tendsto: shows "(\<lambda>n. c (n + 1) / c n) \<longlonglongrightarrow> 0"
preorder_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "class.preorder less_eq less \<equiv> (\<forall>x y. less x y = Restricted_Predicates.strict less_eq x y) \<and> (\<forall>x. less_eq x x) \<and> (\<forall>x y z. less_eq x y \<longrightarrow> less_eq y z \<longrightarrow> less_eq x z)"
Sbis_finished_discr_L: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "c \<approx>s d"   and "proper d"   and "finished d" shows "discr c"
snd_lens_plus: fixes x :: "'c \<Longrightarrow> 'd"   and y :: "'e \<Longrightarrow> 'd" assumes "wb_lens x"   and "x \<bowtie> y" shows "snd\<^sub>L ;\<^sub>L (x +\<^sub>L y) = y"
elims: fixes x :: "sterm"   and y :: "(sterm \<times> sterm) option" assumes "unapp x = y" obtains t :: "sterm" and u :: "sterm" where "x = t $\<^sub>s u" and "y = Some (t, u)"   | v :: "name" where "x = Sconst v__" and "y = None"   | v :: "name" where "x = Svar va__" and "y = None"   | v :: "(Term_Class.term \<times> sterm) list" where "x = Sabs v" and "y = None"
numgcdh_pos: fixes g :: "int"   and t :: "MIR.num" assumes "0 \<le> g" shows "0 \<le> numgcdh t g"
cases: fixes a1 :: "'a \<rightharpoonup>\<^sub>f 'b \<rightharpoonup>\<^sub>f 'c"   and a2 :: "'a"   and a3 :: "'b"   and a4 :: "'a"   and a5 :: "'c" assumes "closest_attribute a1 a2 a3 a4 a5" obtains attributes :: "'a \<rightharpoonup>\<^sub>f 'b \<rightharpoonup>\<^sub>f 'c" and \<D> :: "'a" and attr :: "'b" and \<tau> :: "'c" and \<C> :: "'a" where "a1 = attributes" and "a2 = \<C>" and "a3 = attr" and "a4 = \<D>" and "a5 = \<tau>" and "owned_attribute' attributes \<D> attr \<tau>" and "\<C> \<le> \<D>" and "\<not> attribute_not_closest attributes \<C> attr \<D> \<tau>"
mini_maxi_nths: fixes xs :: "'c list"   and key :: "'c \<Rightarrow> 'd" shows "length (nths xs (- {mini xs key, maxi xs key})) = (case length xs of 0 \<Rightarrow> 0 | Suc 0 \<Rightarrow> 0 | Suc (Suc n) \<Rightarrow> n)"
is_pseudonatural_transformation: shows "pseudonatural_transformation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G G \<Phi>\<^sub>G G.map\<^sub>0 (\<lambda>f. \<l>\<^sub>D\<^sup>-\<^sup>1[G f] \<cdot>\<^sub>D \<r>\<^sub>D[G f])"
match: shows "mC_C Sbis c1 d1 s t P F"
put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def: fixes node_ptr :: "(_) node_ptr"   and node :: "'o RNode_scheme" shows "put node_ptr node = put (cast node_ptr) (cast node)"
simps_2: fixes bs :: "'a list"   and n :: "nat" shows "\<lparr>Bound n\<rparr>\<^sub>p\<^bsup>bs\<^esup> = bs ! n"
ars_r_def: fixes x :: "('a \<times> 'a) set" shows "rel_antirange_kleene_algebra.ars_r x = rel_ar (rel_ar x)"
root_rat_ceiling: fixes p :: "nat"   and x :: "rat" shows "root_rat_ceiling p x = \<lceil>root p (real_of_rat x)\<rceil>"
cod_char: fixes f :: "'a" shows "S.cod f = (if S.arr f then C.cod f else C.null)"
DERIV_const: fixes k :: "'c"   and F :: "'c filter" shows "((\<lambda>x. k) has_field_derivative (0::'c)) F"
r_state_def: shows "r_state \<equiv> Pr 1 (Cn 1 r_prod_encode [Cn 1 r_snoc [Cn 1 r_singleton_encode [recf.Id 1 0], Z], r_const 1]) (Cn 3 r_next [recf.Id 3 1])"
axioms_2: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.group G" shows "Group.group_axioms G"
msfg: shows "matrix_seq d (\<lambda>n. f n + g n)"
satisfies0_eqI: fixes \<BB> :: "'i"   and a :: "'a"   and \<AA> :: "'i" assumes "wf0 (#\<^sub>V \<BB>) a"   and "#\<^sub>V \<AA> = #\<^sub>V \<BB>"   and "lformula0 a"   and "\<And>m k. LESS k m (#\<^sub>V \<BB>) \<Longrightarrow> m\<^bsup>\<AA>\<^esup>k = m\<^bsup>\<BB>\<^esup>k" shows "(\<AA> \<Turnstile>\<^sub>0 a) = (\<BB> \<Turnstile>\<^sub>0 a)"
simps_4: fixes V :: "'d set"   and s :: "('e, 'd) Term.term"   and t :: "('e, 'd) Term.term"   and S :: "('e, 'd) strand_step list" shows "wf\<^sub>s\<^sub>t V (\<langle>assign: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # S) = (fv t \<subseteq> V \<and> wf\<^sub>s\<^sub>t (V \<union> fv s) S)"
rank_ge_card_indpt: fixes A :: "'a mat"   and nc :: "nat"   and U :: "'a vec set" assumes "A \<in> carrier_mat n nc"   and "U \<subseteq> set (cols A)"   and "lin_indpt U" shows "card U \<le> rank A"