hcju/isabelleps huggingface dataset:
hyps: assumes "distinct xs" shows "card (set (Gram_Schmidt xs)) \<le> card (set xs)"
assms_3: shows "fixespointwise (f \<circ> g) (\<Union> Y)"
Empty_WordInterval_def: shows "Empty_WordInterval \<equiv> WordInterval 1 0"
continuous_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U}))"
map_spp_of_vec_of: fixes sps :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list" assumes "list_all spp_inv sps" shows "map (spp_of \<circ> vec_of) sps = sps"
filter_remove1: fixes Q :: "'k \<Rightarrow> bool"   and x :: "'k"   and xs :: "'k list" shows "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
card_ge_dim_independent: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "vs1.independent B"   and "vs1.dim V \<le> card B" shows "V \<subseteq> vs1.span B"
lnth_lzip: fixes n :: "nat"   and xs :: "'b llist"   and ys :: "'c llist" assumes "enat n < llength xs"   and "enat n < llength ys" shows "(xs \<bar>\<bar> ys) ?! n = (xs ?! n, ys ?! n)"
1: shows "(c1, s) \<Rightarrow> p \<Down> t"
winding_number_shiftpath: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and a :: "real" assumes "path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "a \<in> {0..1}" shows "winding_number (shiftpath a \<gamma>) z = winding_number \<gamma> z"
maxVnameLen_newname: fixes c :: "cmd"   and V :: "char list" assumes "maxVnameLen c < length V" shows "newname c V"
not_Cons_self: fixes xs :: "'d list"   and x :: "'d" shows "xs \<noteq> x # xs"
read_only_unowned_axioms: shows "read_only_unowned \<S> (t # ts)"
assms_2: fixes x :: "'s" assumes "x \<in> A" shows "F x = F' x"
assms_5: shows "val_process Mkt pf n w \<le> 0"
oth_class_taut_3_g: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<equiv> \<psi> \<^bold>\<equiv> (\<psi> \<^bold>\<equiv> \<phi>) in v]"
S_def: fixes m :: "((nat \<times> nat) \<times> transition) fset" shows "S m = (\<lambda>((s, s'), t). s) |`| m |\<union>| (\<lambda>((s, s'), t). s') |`| m"
gcg_subdivide_eq: fixes e :: "'a \<times> 'a"   and w :: "'a"   and V :: "'a set" assumes "e \<in> parcs G'"   and "w \<notin> pverts G'"   and "V \<subseteq> pverts G'" shows "gen_contr_graph (with_proj (subdivide G' e w)) V = gen_contr_graph (with_proj G') V"
initial_valueI: fixes pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "trading_strategy pf"   and "support_adapt Mkt pf" shows "\<forall>w\<in>space M. val_process Mkt pf 0 w = initial_value pf"
sound_monitor: fixes i :: "nat"   and v :: "'a option list"   and \<pi> :: "(char list \<times> 'a list) prefix"   and \<sigma> :: "(char list \<times> 'a list) trace" assumes "(i, v) \<in> joiner (\<lambda>k. M (splitter \<pi> k))"   and "prefix_of \<pi> \<sigma>" shows "MFOTL.sat \<sigma> (map the v) i \<phi>"
g2dvdxy: shows "(gcd c d)\<^sup>2 dvd y * x"
mono_aval': fixes S :: "char list \<Rightarrow> 'a"   and S' :: "char list \<Rightarrow> 'a"   and e :: "aexp" assumes "S \<sqsubseteq> S'" shows "aval' e S \<sqsubseteq> aval' e S'"
card_Suc_Diff1: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "Suc (card (A - {x})) = card A"
analz_Un_implSecureSet: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "G \<subseteq> payload"   and "Enc_keys_clean H" shows "analz ({implSecure A B M |A B M. (A, B) \<in> Ag \<and> M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
notPAB: fixes n :: "nat" shows "\<not> P (cbox (A (Suc n)) (B (Suc n)))"
ereal_uminus_reorder_1: fixes a :: "ereal"   and b :: "ereal" shows "(- a = b) = (a = - b)"
not_less_iff_gr_or_eq: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" shows "(\<not> less x y) = (less y x \<or> x = y)"
is_root_def: fixes p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list"   and x :: "('a list \<times> nat) multiset \<Rightarrow> 'a" shows "Id.S.is_root p x = (x \<in> carrier L \<and> Id.S.eval p x = \<zero>\<^bsub>L\<^esub> \<and> p \<noteq> [])"
icdi_path_swap_le: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and j :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and n :: "nat" assumes "is_path \<pi>'"   and "j icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> =\<^bsub>n\<^esub> \<pi>'"   and "j \<le> n" shows "j icd\<^bsup>\<pi>'\<^esup>\<rightarrow> k"
all_formulas_entailed: fixes N2 :: "'b literal multiset set"   and N1 :: "'b literal multiset set" assumes "\<forall>C\<in>N2. N1 \<TTurnstile>e {C}" shows "N1 \<TTurnstile>e N2"
1: fixes x :: "'node"   and n :: "'node" assumes "x \<in> set un" shows "phiDefNodes_aux g v (removeAll x un) n \<subseteq> set (removeAll x un)"
disj_absorb: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "disj x x = x"
assms_2: shows "sum_list (map2 (\<odot>) (cofactor_list_syz (length bs) s) bs) = 0"
interval_lebesgue_integral_by_parts: assumes "0 \<le> 1"   and "continuous_on {0..1} f"   and "continuous_on {0..1} g"   and "\<And>x. x \<in> {0..1} \<Longrightarrow> (F has_real_derivative f x) (at x within {0..1})"   and "\<And>x. x \<in> {0..1} \<Longrightarrow> (G has_real_derivative g x) (at x within {0..1})" shows "LBINT x=ereal 0..ereal 1. F x * g x = F 1 * G 1 - F 0 * G 0 - (LBINT x=ereal 0..ereal 1. f x * G x)"
intro: fixes R :: "('e, 'f) ring_scheme" assumes "ring R" shows "UP_ring R"
preserves_reflects_arr: fixes f :: "(bool, bool list) concrete_category.arr" shows "C.arr (\<chi> f) = J.arr f"
infinite_Ioo: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less"   and "less a b" shows "infinite (ord.greaterThanLessThan less a b)"
progress_mono_gen: fixes j :: "nat"   and j' :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and P' :: "String.literal \<Rightarrow> nat option"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<phi> :: "Formula.formula" assumes "j \<le> j'"   and "rel_mapping (\<le>) P P'" shows "Monitor.progress \<sigma> P \<phi> j \<le> Monitor.progress \<sigma> P' \<phi> j'"
braun_heapify: fixes n :: "nat"   and xs :: "'b list" assumes "n \<le> length xs" shows "braun (fst (heapify n xs))"
compl_bot_eq: shows "\<^bold>- \<bottom> = \<top>"
ide_char: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VV.ide a = (VV.arr a \<and> VxV.ide a)"
induct: fixes x1 :: "('a \<times> 'b) list"   and x2 :: "'a"   and x3 :: "'b"   and P :: "('a \<times> 'b) list \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "srewrite_step x1 x2 x3"   and "\<And>name rhs rest. P ((name, rhs) # rest) name rhs"   and "\<And>name name' rs rhs rhs'. \<lbrakk>name \<noteq> name'; srewrite_step rs name rhs; P rs name rhs\<rbrakk> \<Longrightarrow> P ((name', rhs') # rs) name rhs" shows "P x1 x2 x3"
card_UNIV_num0_def: shows "card_UNIV_class.card_UNIV = Phantom(num0) 0"
fun_upd_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and x :: "'b"   and y :: "'d" shows "f \<circ> g(x := y) = (f \<circ> g)(x := f y)"
Compl_step1: fixes I :: "'a set"   and s :: "'a set" assumes "Kripke {s. \<exists>i\<in>I. i \<rightarrow>\<^sub>i* s} I \<turnstile> EF s" shows "\<forall>x\<in>I. \<exists>y\<in>s. x \<rightarrow>\<^sub>i* y"
id_hom_singleton: fixes G :: "('b, 'c) monoid_scheme" assumes "Group.group G" shows "id \<in> hom (singleton_group \<one>\<^bsub>G\<^esub>) G"
s3_correct: fixes ard :: "(16 word \<times> 32 simple_match \<times> char list \<times> simple_action) list"   and p :: "(32, 'b) simple_packet_ext_scheme"   and ifs :: "char list list"   and ao :: "of_action list" assumes "list_all simple_match_valid (map (fst \<circ> snd) ard)"   and "p_l2type p = 2048"   and "p_iiface p \<in> set ifs"   and "list_all (\<lambda>m. oiface (fst (snd m)) = ifaceAny) ard" shows "(OF_match_linear OF_match_fields_safe (pack_OF_entries ifs ard) p = Action ao) = (\<exists>r af. generalized_sfw (map snd ard) p = Some (r, af) \<and> (if snd af = Drop then ao = [] else ao = [Forward (fst af)]))"
simps_2: fixes C :: "('a, nat) Term.term literal multiset"   and Cs :: "('a, nat) Term.term literal multiset list" shows "renamings_apart (C # Cs) = (let \<sigma>s = renamings_apart Cs in (\<lambda>v. Var (v + Max (vars_clause_list (subst_cls_lists Cs \<sigma>s) \<union> {0}) + 1)) # \<sigma>s)"
preserves_reflects_arr: fixes f :: "'b" shows "D.arr (trg\<^sub>D f) = D.arr f"
ftree_no_ipath: fixes t :: "'a tree"   and steps :: "'a stream" assumes "tfinite t" shows "\<not> ipath t steps"
simps_1: fixes carrier :: "'d set"   and mult :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and more :: "'e" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
con_g2: shows "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` S)) (\<lambda>x. vector_derivative g2 (at x))"
lifting_intersection_axioms_def: fixes Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Q :: "'c set"   and Inf_G_q :: "'c \<Rightarrow> 'b inference set"   and entails_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b inference set"   and Red_F_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set"   and Bot_F :: "'a set"   and \<G>_F_q :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and \<G>_I_q :: "'c \<Rightarrow> 'a inference \<Rightarrow> 'b inference set option"   and Prec_F_g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "lifting_intersection_axioms Inf_F Bot_G Q Inf_G_q entails_q Red_I_q Red_F_q Bot_F \<G>_F_q \<G>_I_q Prec_F_g \<equiv> \<forall>q\<in>Q. tiebreaker_lifting Bot_F Inf_F Bot_G (entails_q q) (Inf_G_q q) (Red_I_q q) (Red_F_q q) (\<G>_F_q q) (\<G>_I_q q) Prec_F_g"
simps_1: fixes s1 :: "'b stream"   and s2 :: "'c stream" shows "shd (szip s1 s2) = (shd s1, shd s2)"
sum_mset_conv_Sum_any: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" shows "\<Sum>\<^sub># (image_mset f A) = (\<Sum>x. of_nat (count A x) * f x)"
that: fixes \<I>\<^sub>\<tau> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)"   and "\<I>\<^sub>\<tau> \<Turnstile>\<^sub>c \<langle>S,\<theta>\<rangle>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>)" shows "thesis"
cr_vint_transfer_domain_rule: shows "Domainp cr_vint = (\<lambda>x. x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>)"
hyp: fixes t :: "real" assumes "t \<in>\<^sub>r ivlx" shows "(\<Sum>j = 0..<n. inverse (fact j) * F j (real_of_float c) * (xs ! x - real_of_float c) ^ j) + inverse (fact n) * F n t * (xs ! x - real_of_float c) ^ n \<in>\<^sub>r ivl"
finite_eq_wqo_on: fixes A :: "'a set" assumes "finite A" shows "wqo_on (=) A"
p_prob_1: fixes p :: "real" assumes "edge_space p" shows "0 \<le> p"
inj_image_mem_iff: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and A :: "'c set" assumes "inj f" shows "(f a \<in> f ` A) = (a \<in> A)"
less_mult_imp_div_less: fixes m :: "nat"   and i :: "nat"   and n :: "nat" assumes "m < i * n" shows "m div n < i"
Adec_lt: fixes X :: "msg"   and Y :: "agent"   and H :: "msg set" assumes "Aenc X (pubK Y) \<in> analz H"   and "priK Y \<in> analz H" shows "X \<in> analz H"
the_\<nu>_simps_1: shows "arr \<chi>.the_\<nu>"
ZObis_pres_discr_L: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "c \<approx>01 d"   and "discr d" shows "discr c"
ts\<^sub>1\<^sub>2: shows "instance_of\<^sub>t\<^sub>s ts\<^sub>1 ts\<^sub>2"
Re_Ln_pos_lt_imp: fixes z :: "complex" assumes "0 < Re z" shows "\<bar>Im (Ln z)\<bar> < pi / 2"
contour_integral_uniform_limit_circlepath_1: fixes f :: "'a \<Rightarrow> complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real"   and F :: "'a filter"   and l :: "complex \<Rightarrow> complex" assumes "\<forall>\<^sub>F n in F. f n contour_integrable_on circlepath z r"   and "uniform_limit (sphere z r) f l F"   and "F \<noteq> bot"   and "0 < r" shows "l contour_integrable_on circlepath z r"
supp_2: fixes x2 :: "char list"   and x1 :: "form_list" shows "supp (Cpd0 x2 x1) = supp x2 \<union> supp x1"
set_eq_iff: fixes A :: "'b set"   and B :: "'b set" shows "(A = B) = (\<forall>x. (x \<in> A) = (x \<in> B))"
ind: fixes tp :: "(action \<times> nat) list" assumes "st \<le> length tp div 2"   and "0 < st" shows "modify_tprog tp ! (4 * (st - Suc 0)) = action_map (fst (tp ! (2 * (st - Suc 0))))"
eq_\<I>_converter_mono: fixes \<I>1 :: "('h, 'i) \<I>"   and \<I>2 :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter"   and conv' :: "('h, 'i, 'j, 'k) converter"   and \<I>1' :: "('h, 'i) \<I>"   and \<I>2' :: "('j, 'k) \<I>" assumes "\<I>1,\<I>2 \<turnstile>\<^sub>C conv \<sim> conv'"   and "\<I>1' \<le> \<I>1"   and "\<I>2 \<le> \<I>2'" shows "\<I>1',\<I>2' \<turnstile>\<^sub>C conv \<sim> conv'"
substT_num1: fixes t :: "'trm"   and y :: "'var"   and n :: "'trm" assumes "t \<in> trm"   and "y \<in> var"   and "n \<in> num" shows "substT n t y = n"
iso_fiter_fun2: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "mono f"   and "mono g" shows "mono (fiter_fun f g)"
induct: fixes P :: "('a, 'b) strand_step list \<Rightarrow> bool"   and a0 :: "('a, 'b) strand_step list" assumes "P []"   and "\<And>t t' S. P S \<Longrightarrow> P (\<langle>assign: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v S. P S \<Longrightarrow> P (send\<langle>v\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v S. P S \<Longrightarrow> P (receive\<langle>v\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>va vb S. P S \<Longrightarrow> P (\<langle>check: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v va S. P S \<Longrightarrow> P (\<forall>v\<langle>\<or>\<noteq>: va\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0"
surjective: fixes r :: "'b vars_scheme" shows "r = \<lparr>vars.I = vars.I r, M = M r, N = N r, S = S r, \<dots> = vars.more r\<rparr>"
lappend_assoc: fixes xs :: "'a llist"   and ys :: "'a llist"   and zs :: "'a llist" shows "lappend (lappend xs ys) zs = lappend xs (lappend ys zs)"
hom_deg_set_sum_list_listset: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and ss :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set list"   and z :: "nat" assumes "A = sum_list ` listset ss" shows "hom_deg_set z A = sum_list ` listset (map (hom_deg_set z) ss)"
preservation: shows "rel_preserves_binary_pred (indRelSTEQ SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
correctCompositionDiffLevelsA92: shows "correctCompositionDiffLevels sA92"
part_aux_induct: fixes P :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0a :: "'b"   and a1a :: "'a list"   and a2a :: "'a list"   and a2b :: "'a list"   and a2c :: "'a list" assumes "\<And>p ls eq gs. P p [] ls eq gs"   and "\<And>x p xs ls eq gs. \<lbrakk>f x < p; P p xs (x # ls) eq gs\<rbrakk> \<Longrightarrow> P p (x # xs) ls eq gs"   and "\<And>x p xs ls eq gs. \<lbrakk>p < f x; P p xs ls eq (x # gs)\<rbrakk> \<Longrightarrow> P p (x # xs) ls eq gs"   and "\<And>x p xs ls eqa gs. \<lbrakk>f x = p; P p xs ls (eqa @ [x]) gs\<rbrakk> \<Longrightarrow> P p (x # xs) ls eqa gs" shows "P a0a a1a a2a a2b a2c"
Crypt: fixes K :: "nat"   and U :: "freemsg" shows "Crypt K (Abs_Msg (msgrel `` {U})) = Abs_Msg (msgrel `` {CRYPT K U})"
rep_in_span: fixes r :: "'c \<Rightarrow>\<^sub>0 'c" shows "ideal.rep r \<in> ideal (keys r)"
sum_mono: fixes K :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i" shows "sum f K \<le> sum g K"
or_one_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "or a one = plus a (zero_neq_one.of_bool one zero (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a))"
fv_subst: fixes s :: "'subst"   and f :: "'form" shows "lconsts (subst s f) \<subseteq> lconsts f \<union> subst_lconsts s"
SN_on_Image: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "SN_on r A" shows "SN_on r (r `` A)"
min_wpl_simps_2: fixes i :: "int"   and j :: "int" assumes "i \<le> j" shows "local.min_wpl i j = Min (min_wpl3 i j ` {i..j})"
fin_1: shows "xs \<in> UNIV\<^sup>\<star>"
ndet_cond_distr: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<or> (Q \<triangleleft> b \<triangleright>  R)) = (P \<or> Q \<triangleleft> b \<triangleright>  P \<or> R)"
sSplice_pubActs: fixes a :: "'a"   and s :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState"   and s' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" shows "pubActs (sSplice a s s') = pubActs s"
pre_between_rotate1: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2" shows "pre_between (rotate1 vs) ram1 ram2"
induct: fixes P :: "tm \<Rightarrow> bool"   and tm :: "tm" assumes "P Zero"   and "\<And>x. P (Var x)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (Eats x1 x2a)" shows "P tm"
generated_ringI_Basic: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M" shows "A \<in> sets.generated_ring M"
infinity_ileE: fixes m :: "int"   and R :: "bool" assumes "\<infinity> \<le> eint m" shows "R"
mult_assoc_hyp: shows "matr_option (dim_row (M_mat_R signs subsets)) (mat_inverse (M_mat_R signs subsets)) *\<^sub>v (M_mat_R signs subsets *\<^sub>v w_vec_R p qs signs) = matr_option (dim_row (M_mat_R signs subsets)) (mat_inverse (M_mat_R signs subsets)) * M_mat_R signs subsets *\<^sub>v w_vec_R p qs signs"
strict2: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)" shows "(x \<doteq> invalid) = invalid"
interval_inner_leI_1: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and a :: "'e"   and b :: "'e"   and i :: "'e" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "x \<in> ord.atLeastAtMost less_eq a b"   and "less_eq zero i" shows "inner a i \<le> inner x i"
measure_preserving_distr: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> measure_preserving M N" shows "distr M N f = N"
Card_order_iff_ordLeq_card_of: fixes r :: "('a \<times> 'a) set" shows "Card_order r = (r \<le>o |Field r|)"
ARR_iff_has_domain: fixes f :: "'c local.term set" shows "ARR f = (domains f \<noteq> {})"
wf_dfa_def: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat" shows "wf_dfa A n = (list_all (bddh n) (fst A) \<and> list_all (bdd_all (dfa_is_node A)) (fst A) \<and> length (snd A) = length (fst A) \<and> 0 < length (fst A))"
elims: fixes x :: "complex \<times> complex"   and xa :: "complex \<times> complex \<times> complex \<times> complex"   and y :: "complex \<times> complex" assumes "x *\<^sub>v\<^sub>m xa = y" obtains xb :: "complex" and ya :: "complex" and a :: "complex" and b :: "complex" and c :: "complex" and d :: "complex" where "x = (xb, ya)" and "xa = (a, b, c, d)" and "y = (xb * a + ya * c, xb * b + ya * d)"
val_of_spec: fixes e :: "'b exp"   and v :: "val" assumes "val_of e = \<lfloor>v\<rfloor>" shows "e = Val v"
either_both_distr: fixes t' :: "'b ttree"   and t :: "'b ttree"   and t'' :: "'b ttree" shows "t' \<otimes>\<otimes> t \<oplus>\<oplus> t' \<otimes>\<otimes> t'' = t' \<otimes>\<otimes> (t \<oplus>\<oplus> t'')"
erfc_aux_def: fixes x :: "real" shows "erfc_aux x = exp (x\<^sup>2) * sqrt pi * erfc x"
inject_7: fixes x7 :: "rat"   and y7 :: "rat" shows "(Pri_AgrK x7 = Pri_AgrK y7) = (x7 = y7)"
is_extensional: fixes f :: "'d" assumes "\<not> D.arr f" shows "D.map f = D.null"
simps_2: fixes x :: "nat" shows "c_fold [x] = x"
Units_assoc: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Units G"   and "b \<in> Units G" shows "a \<sim> b"
FG_advice_b2_helper: fixes Y :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "\<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n G\<^sub>n \<psi>"   and "w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<mu>" shows "w \<Turnstile>\<^sub>n \<phi>"
preserves_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "S.trg (S.UP \<mu>) = UP.map\<^sub>0 (trg \<mu>)"
diamond_right_edges_def: shows "diamond_right_edges = (1::'a, \<lambda>y. (diamond_y (diamond_x y), diamond_x y))"
MClkFwd_enabled: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState" assumes "basevars (rtrner send!p, caller rcv!p, cst!p)" shows "\<turnstile> Calling send p \<and> \<not> Calling rcv p \<and> cst!p = #clkA \<longrightarrow> Enabled MClkFwd send rcv cst p"
sum_bounded_above: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and K :: "'c" assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> K" shows "sum f A \<le> of_nat (card A) * K"
inj_on': fixes a :: "'a"   and b :: "'a"   and m :: "'a \<Rightarrow>\<^sub>0 'b" shows "inj_on (swap a b) (keys m)"
assms_2: shows "(a, b) \<noteq> (i, x)"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<in>\<^sub>\<circ> Vset \<alpha>" shows "T\<lparr>ArrVal\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
zncarr: shows "zn \<in> carrier (relative_homology_group (int n) (usphere n) (equator n))"
E'_prop1: shows "backflows (filternew_flows_state \<T>) - E' = E"
oarrivemsg_True: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and a :: "'b node_action" shows "oarrivemsg (\<lambda>_ _. True) \<sigma> a"
derg: fixes y :: "'a" assumes "y \<in> V" shows "(g has_derivative g' y) (at y)"