hcju/isabelleps huggingface dataset:
derivation_times_eq_zeroI: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and k :: "enat"   and p :: "'c"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "X \<in> manifold_eucl.tangent_space k p"   and "f \<in> manifold_eucl.diff_fun_space k"   and "g \<in> manifold_eucl.diff_fun_space k"   and "f p = 0"   and "g p = 0" shows "X (f * g) = 0"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_goon_right_moving_1 ires rs (l, r) = (\<exists>ml mr ln rn. l = Oc \<up> ml @ Bk # Bk # Bk \<up> ln @ Oc # ires \<and> r = Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc rs)"
congruent_triangle_sss_4: fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'b"   and b2 :: "'b"   and c1 :: "'a"   and c2 :: "'b" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "dist b1 a1 = dist a2 b2"
vec_lambda_eta: fixes g :: "('b, 'c) Finite_Cartesian_Product.vec" shows "(\<chi>i. g $h i) = g"
strengthen_SPEC: fixes m :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "m \<le> SPEC \<Phi>" shows "m \<le> SPEC (\<lambda>s. inres m s \<and> \<Phi> s)"
assms_2: shows "Case_Labeling.CTXT IC CT OC1 (ovalidNF P f R)"
gim: shows "g ` T \<subseteq> sphere 0 1"
word_log_esimps_9: fixes x :: "'a word" shows "0 OR x = x"
bij_\<phi>: shows "bij_betw \<phi> (HRepHomSet (\<star>) W) (GRepHomSet (\<star>) W)"
RedAAssStore: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and w :: "'addr val"   and U :: "ty"   and P :: "'addr J_prog"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id" assumes "typeof_addr (hp s) a = \<lfloor>Array_type T n\<rfloor>"   and "0 \<le>s i"   and "sint i < int n"   and "typeof\<^bsub>hp s\<^esub> w = \<lfloor>U\<rfloor>"   and "\<not> P \<turnstile> U \<le> T" shows "extTA,P,t \<turnstile> \<langle>addr a\<lfloor>Val (Intg i)\<rceil> := Val w,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt ArrayStore),s\<rangle>"
ex_br_conj_iff: fixes y :: "'d"   and a :: "'d \<Rightarrow> 'c"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" shows "(\<exists>x. (y, x) \<in> br a I \<and> P x) = (I y \<and> P (a y))"
antisym: fixes b :: "'a list"   and a :: "'a list" assumes "suffix b a"   and "suffix a b" shows "a = b"
skl1_implements_a0i_r: shows "implements med0sk1iar a0i skl1"
i_3: shows "(flow0 (flow0 x0 h) i, Dflow (flow0 x0 h) i o\<^sub>L Dflow x0 h o\<^sub>L d0) \<in> E"
assoc_props_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and h :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f"   and "partial_magma.ide (span_vertical_category.vcomp C) g"   and "partial_magma.ide (span_vertical_category.vcomp C) h"   and "span_vertical_category.src C f = span_vertical_category.trg C g"   and "span_vertical_category.src C g = span_vertical_category.trg C h" shows "partial_magma.in_hom (span_vertical_category.vcomp C) \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc C prj0 prj1 f g h, Dom = Dom (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h), Cod = Cod (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h))\<rparr> (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h) (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h))"
lookup_split: fixes l :: "'a list"   and f :: "'a \<Rightarrow> bool" shows "lookup l f = None \<or> (\<exists>h. lookup l f = Some h)"
mm_equiv_strong_low_bisim: shows "strong_low_bisim_mm mm_equiv"
uint_word_arith_bintrs_8: shows "uint 1 = take_bit LENGTH('b) 1"
no_change_eval_lt: fixes x :: "real"   and y :: "real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "x < y"   and "\<not> (\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0)" shows "aEvalUni (LessUni (a, b, c)) x = aEvalUni (LessUni (a, b, c)) y"
no_ifaces: shows "simple_firewall_without_interfaces (to_simple_firewall_without_interfaces ipassmt rtblo rs)"
numeral_tree_simps_2: fixes n :: "num" shows "left (numeral n) = numeral n"
ennexp_ge1: fixes x :: "ereal" shows "(1 \<le> ennexp x) = (0 \<le> x)"
step: shows "\<forall>i. X (Suc i) = make_step selector (X i)"
dom_map_inv: fixes f :: "'d \<Rightarrow> 'c option" shows "dom (map_inv f) = ran f"
two_equivalences_in_bicategory_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and e\<^sub>0 :: "'b"   and d\<^sub>0 :: "'b"   and \<eta>\<^sub>0 :: "'b"   and \<epsilon>\<^sub>0 :: "'b"   and e\<^sub>1 :: "'b"   and d\<^sub>1 :: "'b"   and \<eta>\<^sub>1 :: "'b"   and \<epsilon>\<^sub>1 :: "'b" shows "two_equivalences_in_bicategory V H \<a> \<i> src trg e\<^sub>0 d\<^sub>0 \<eta>\<^sub>0 \<epsilon>\<^sub>0 e\<^sub>1 d\<^sub>1 \<eta>\<^sub>1 \<epsilon>\<^sub>1 \<equiv> bicategory V H \<a> \<i> src trg \<and> equivalence_in_bicategory V H \<a> \<i> src trg e\<^sub>0 d\<^sub>0 \<eta>\<^sub>0 \<epsilon>\<^sub>0 \<and> equivalence_in_bicategory V H \<a> \<i> src trg e\<^sub>1 d\<^sub>1 \<eta>\<^sub>1 \<epsilon>\<^sub>1"
cong: fixes a :: "nat"   and b :: "nat" assumes "[a = b] (mod n)" shows "\<chi> a = \<chi> b"
subset_1: shows "F - H' \<subseteq> F \<inter> E"
norm2_slp_def: fixes n :: "nat" shows "norm2_slp n = slp_of_fas [floatarith.Inverse (norm2\<^sub>e n)]"
not_0_cCons_eq: fixes p :: "'a poly"   and a :: "'a" assumes "p \<noteq> 0" shows "cCons a (coeffs p) = a # coeffs p"
analytic_on_mult: 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"
RECT_gfp_def: fixes B :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'd"   and x :: "'e" shows "REC\<^sub>T B x = (if trimono B then gfp B x else top)"
Vandermonde_pochhammer_lemma: fixes n :: "nat"   and b :: "'a"   and a :: "'a" assumes "\<And>j. j < n \<Longrightarrow> b \<noteq> of_nat j" shows "(\<Sum>k = 0..n. pochhammer (- a) k * pochhammer (- of_nat n) k / (of_nat (fact k) * pochhammer (b - of_nat n + (1::'a)) k)) = pochhammer (- (a + b)) n / pochhammer (- b) n"
1: shows "wf\<^sub>s\<^sub>t\<^sub>s {dual\<^sub>s\<^sub>t \<A>}"
homeomorphic_map_openness_eq: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and U :: "'a set" assumes "homeomorphic_map X Y f" shows "openin X U = (U \<subseteq> topspace X \<and> openin Y (f ` U))"
pos_pi: fixes i :: "'b" assumes "i \<in> L" shows "0 \<le> fi i"
list_nonempty_induct: fixes xs :: "'c list"   and P :: "'c list \<Rightarrow> bool" assumes "xs \<noteq> []"   and "\<And>x. P [x]"   and "\<And>x xs. \<lbrakk>xs \<noteq> []; P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P xs"
inf_idempotent: fixes x :: "'a" shows "- x \<sqinter> - x = - x"
expandNonTauFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'f"   and C' :: "'g" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>P \<sharp>* C'"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* C'"   and "\<alpha> \<noteq> \<tau>" obtains p :: "(name \<times> name) list" and \<Psi>' :: "'b" and A\<^sub>P' :: "name list" and \<Psi>\<^sub>P'' :: "'b" where "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))" and "(p \<bullet> \<Psi>\<^sub>P) \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P''" and "distinctPerm p" and "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P''\<rangle>" and "A\<^sub>P' \<sharp>* P'" and "A\<^sub>P' \<sharp>* \<alpha>" and "A\<^sub>P' \<sharp>* (p \<bullet> \<alpha>)" and "A\<^sub>P' \<sharp>* C" and "bn (p \<bullet> \<alpha>) \<sharp>* C'" and "bn (p \<bullet> \<alpha>) \<sharp>* \<alpha>" and "bn (p \<bullet> \<alpha>) \<sharp>* P'" and "distinct A\<^sub>P'"
cast_def: shows "cast \<equiv> map_fun Word.rep Word.Word (take_bit LENGTH('k))"
surjective: fixes r :: "(_) Object" shows "r = \<lparr>RObject.nothing = RObject.nothing r, \<dots> = RObject.more r\<rparr>"
mop_closed: shows "mop R \<in> carrier R \<rightarrow> carrier R"
simps_3: fixes n :: "nat" shows "string_tv (Indet n) = string_of_nat n"
fmlookup_filter: fixes P :: "'d \<Rightarrow> bool"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmfilter P m $$ x = (if P x then m $$ x else None)"
tendsto_Lim: fixes net :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "net \<noteq> bot"   and "(f \<longlongrightarrow> l) net" shows "Lim net f = l"
zsplit0_I: fixes t :: "Cooper.num"   and n :: "int"   and a :: "Cooper.num"   and x :: "int"   and bs :: "int list" assumes "zsplit0 t = (n, a)" shows "Inum (x # bs) (CN 0 n a) = Inum (x # bs) t \<and> numbound0 a"
strict_is_neg_transitive: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "rational_preference carrier relation"   and "x \<in> carrier \<and> y \<in> carrier \<and> z \<in> carrier"   and "x \<succeq>[relation] y \<and> (y, x) \<notin> relation" shows "x \<succeq>[relation] z \<and> (z, x) \<notin> relation \<or> z \<succeq>[relation] y \<and> (y, z) \<notin> relation"
zhmset_of_enat_numeral: fixes n :: "num" shows "zhmset_of_enat (numeral n) = numeral n"
homeomorphic_maps_imp_map: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" assumes "homeomorphic_maps X Y f g" shows "homeomorphic_map X Y f"
simps: fixes \<Gamma> :: "form multiset"   and \<Delta> :: "form multiset"   and A :: "form" shows "( \<Gamma> \<Rightarrow>* \<Delta>) - A = ( \<Gamma> \<ominus> A \<Rightarrow>* \<Delta> \<ominus> A)"
sylvester_carrier_mat: fixes p :: "'a poly"   and q :: "'a poly" shows "sylvester_mat p q \<in> carrier_mat (degree p + degree q) (degree p + degree q)"
suci: shows "(\<pi> @\<^bsup>k\<^esup> \<pi>l) (Suc i) \<noteq> \<pi>' (Suc i')"
loop_stepC: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"   and "B\<^sub>1 \<noteq> {}"   and "u \<in> S"   and "W B\<^sub>1 + w u \<le> real c" shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 (B\<^sub>1 \<union> {u}) B\<^sub>2 (S - {u}) L"
USdBeqlen: fixes V :: "('exp, 'id) MWLfCom list"   and V' :: "('exp, 'id) MWLfCom list"   and d :: "'d" assumes "V \<approx>\<^bsub>d\<^esub> V'" shows "length V = length V'"
extract_bind: fixes x :: "'a Monomorphic_Monad.id"   and f :: "'a \<Rightarrow> 'a Monomorphic_Monad.id" shows "extract (bind_id x f) = extract (f (extract x))"
measurable_abs_UNIV: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'b \<Rightarrow> 'd measure" assumes "\<And>n. f n \<in> M \<rightarrow>\<^sub>M N n" shows "(\<lambda>\<omega> n. f n \<omega>) \<in> M \<rightarrow>\<^sub>M Pi\<^sub>M UNIV N"
comp: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'c" shows "(-`) f \<circ> (-`) g = (-`) (g \<circ> f)"
integrable_const_bound: fixes f :: "'d \<Rightarrow> 'e"   and B :: "real"   and M :: "'d pmf" assumes "AE x in measure_pmf M. norm (f x) \<le> B"   and "measure_pmf.random_variable M borel f" shows "integrable (measure_pmf M) f"
degree_pCons_le: fixes a :: "'a"   and p :: "'a poly" shows "degree (pCons a p) \<le> Suc (degree p)"
has_derivative_cos: fixes g :: "'b \<Rightarrow> real"   and g' :: "'b \<Rightarrow> real"   and x :: "'b"   and s :: "'b set" assumes "(g has_derivative g') (at x within s)" shows "((\<lambda>x. cos (g x)) has_derivative (\<lambda>x. g' x * - sin (g x))) (at x within s)"
fresh_star_def_2: fixes xs :: "'d set"   and c :: "'e" shows "xs \<sharp>* c \<equiv> \<forall>x\<in>xs. x \<sharp> c"
Compl_disjoint2: fixes A :: "'c set" shows "- A \<inter> A = {}"
fdomain_flrestriction: fixes r :: "V"   and A :: "V" shows "\<D>\<^sub>\<bullet> (r \<restriction>\<^sup>l\<^sub>\<bullet> A) = \<D>\<^sub>\<bullet> r \<inter>\<^sub>\<circ> A"
A_eq: shows "A = the_inv_into A f ` f ` A"
sel_1: fixes x1 :: "'c"   and x2 :: "'c stream" shows "shd (x1 ## x2) = x1"
word_div_eq_1_iff: fixes n :: "'a word"   and m :: "'a word" shows "(n div m = 1) = (m \<le> n \<and> unat n < 2 * unat m)"
NotD: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" assumes "\<Gamma> \<turnstile> \<^bold>\<not> F" shows "F \<triangleright> \<Gamma> \<turnstile> \<bottom>"
braun_list_Nil: fixes t :: "'b tree" shows "braun_list t [] = (t = \<langle>\<rangle>)"
ring_1_ow_def: fixes U :: "'a set"   and 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" shows "ring_1_ow U one times plus zero minus uminus \<equiv> ring_ow U plus zero minus uminus times \<and> zero_neq_one_ow U one zero \<and> monoid_mult_ow U one times"
union_commute: fixes M :: "'f multiset"   and N :: "'f multiset" shows "M + N = N + M"
ortho_span_span: fixes W :: "'a vec set"   and X :: "'a vec set"   and w :: "'a vec"   and x :: "'a vec" assumes "W \<subseteq> carrier_vec n"   and "X \<subseteq> carrier_vec n"   and "\<And>w x. \<lbrakk>w \<in> W; x \<in> X\<rbrakk> \<Longrightarrow> w \<bullet> x = (0::'a)"   and "w \<in> local.span W"   and "x \<in> local.span X" shows "w \<bullet> x = (0::'a)"
steps_z_alpha_closure_involutive': fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set"   and l'' :: "'d"   and Z'' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l', Z'\<rangle>"   and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto> \<langle>l'', Z''\<rangle>"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "Z \<subseteq> V" shows "\<exists>Z'''. A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l'', Z'''\<rangle> \<and> Closure\<^sub>\<alpha> Z'' \<subseteq> Closure\<^sub>\<alpha> Z''' \<and> Z''' \<subseteq> Z''"
Language_deltas: fixes s :: "'s"   and w :: "'a list" assumes "wellformed s"   and "w \<in> lists (set alphabet)" shows "Language (fold delta w s) = \<dd>s w (Language s)"
parPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "(P, Q) \<in> Rel"   and "\<And>S T U. (S, T) \<in> Rel \<Longrightarrow> (S \<parallel> U, T \<parallel> U) \<in> Rel'"   and "\<And>S T x. (S, T) \<in> Rel' \<Longrightarrow> (<\<nu>x>S, <\<nu>x>T) \<in> Rel'" shows "P \<parallel> R \<leadsto>\<guillemotleft>Rel'\<guillemotright> Q \<parallel> R"
gen_model_jAction_eq: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and M :: "('b, 'c, 'e) KripkeStructure"   and w :: "'e"   and M' :: "('b, 'c, 'e) KripkeStructure"   and w' :: "'e" assumes "JKBP jkbp"   and "gen_model M w = gen_model M' w"   and "w' \<in> worlds (gen_model M' w)"   and "kripke M"   and "kripke M'" shows "JKBP.jAction jkbp M w' = JKBP.jAction jkbp M' w'"
simps_13: fixes p :: "fm"   and q :: "fm" shows "decr (Iff p q) = Iff (decr p) (decr q)"
a1: shows "\<forall>x'\<ge>the_enat \<langle>c #\<^bsub>enat n\<^esub>inf_llist t\<rangle>. x' < x \<longrightarrow> \<gamma>' (lnth (\<pi>\<^bsub>c\<^esub>inf_llist t @\<^sub>l inf_llist t')) x'"
g_in_hhom: shows "\<guillemotleft>\<eta>'.map\<^sub>0 b' \<star>\<^sub>D G f \<star>\<^sub>D \<eta>\<^sub>0 b : b \<rightarrow>\<^sub>D b'\<guillemotright>"
longestImpWitnessPathEx: fixes t :: "'t"   and c :: "('loc, 't, 'a) configuration_scheme"   and loc2 :: "'loc" assumes "t \<in>\<^sub>A frontier (c_imp c loc2)" shows "\<exists>loc1 xs. longestImpWitnessPath c loc1 loc2 xs t"
distr_def: fixes M :: "'d measure"   and N :: "'c measure"   and f :: "'d \<Rightarrow> 'c" shows "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
inf_bot_right: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and x :: "'b::type" assumes "class.bounded_lattice_bot (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (bot::'b::type) = bot"
assms_2: fixes x :: "real" assumes "x \<in> A" shows "(f has_real_derivative f' x) (at x)"
red_subset: fixes G :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and F :: "('a \<Rightarrow>\<^sub>0 'd) set" assumes "punit.red G p q"   and "G \<subseteq> F" shows "punit.red F p q"
open_Times: fixes S :: "'c set"   and T :: "'d set" assumes "open S"   and "open T" shows "open (S \<times> T)"
axioms_1: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "SN_order_pair S NS" shows "order_pair S NS"
SUP_upper: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and i :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
sp_generalize: fixes F :: "'a \<Rightarrow> 'b fundef option"   and xs :: "('dyn, 'c, 'a, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list"   and \<Sigma> :: "Unboxed.type option list"   and \<Sigma>' :: "Unboxed.type option list" assumes "Subx.sp F xs \<Sigma> = Ok \<Sigma>'" shows "Subx.sp F (map Subx.generalize_instr xs) (map Map.empty \<Sigma>) = Ok (map Map.empty \<Sigma>')"
proj_def: fixes r :: "('d \<times> 'c) set"   and x :: "'d" shows "proj r x = r `` {x}"
mono2_step: fixes C1 :: "(char list \<Rightarrow> int) set acom"   and C2 :: "(char list \<Rightarrow> int) set acom"   and S1 :: "(char list \<Rightarrow> int) set"   and S2 :: "(char list \<Rightarrow> int) set" assumes "C1 \<le> C2"   and "S1 \<subseteq> S2" shows "step S1 C1 \<le> step S2 C2"
domains_comp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "A1xA2.seq g f" shows "A1xA2.domains (g \<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 f) = A1xA2.domains f"
cases: fixes a1 :: "'b Heap"   and a2 :: "heap option"   and a3 :: "heap option"   and a4 :: "'b" assumes "run a1 a2 a3 a4" obtains c :: "'b Heap" and r :: "'b" where "a1 = c__" and "a2 = None" and "a3 = None" and "a4 = r__"   | c :: "'b Heap" and h :: "heap" and r :: "'b" where "a1 = ca__" and "a2 = Some h__" and "a3 = None" and "a4 = ra__" and "execute ca__ h__ = None"   | c :: "'b Heap" and h :: "heap" and r :: "'b" and h' :: "heap" where "a1 = c" and "a2 = Some h" and "a3 = Some h'" and "a4 = r" and "execute c h = Some (r, h')"
threshold: shows "\<bar>valof a - valof b\<bar> < threshold TYPE(('e, 'f) IEEE.float)"
eq2: shows "card X - (card X - Suc 0) = 1"
assms: shows "(x, p) \<in> set (Tableau (preprocess' cs start))"
induct: fixes P :: "'b list list \<Rightarrow> bool"   and a0 :: "'b list list" assumes "P []"   and "\<And>xs. P [xs]"   and "\<And>v vb vc. P (merge_adj (v # vb # vc)) \<Longrightarrow> P (v # vb # vc)" shows "P a0"
fst_combined_restrict: fixes S :: "Vars.var set"   and p :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "fst (combined_restrict S p) = fst p f|` S"
2: shows "\<mu>.leg0 \<cdot> Prj\<^sub>1\<^sub>1 = \<nu>.leg1 \<cdot> Prj\<^sub>0\<^sub>1"
double_complement: fixes A :: "'g set" shows "- (- A) = A"
ctr_2: fixes p :: "'b \<Rightarrow> bool"   and a :: "'b"   and g21 :: "'b \<Rightarrow> 'c"   and g22 :: "'b \<Rightarrow> 'b" assumes "\<not> p a" shows "unfold_llist p g21 g22 a = LCons (g21 a) (unfold_llist p g21 g22 (g22 a))"
simps_165: fixes x5 :: "('b, 'a, 'c) expr" shows "LID\<^sub>E (Read x5) = LID\<^sub>E x5"
lm': fixes f1 :: "nat \<Rightarrow> nat"   and g1 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and h1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" shows "(f1 \<in> PrimRec1 \<longrightarrow> f1 \<in> PrimRec1') \<and> (g1 \<in> PrimRec2 \<longrightarrow> pr_conv_2_to_1 g1 \<in> PrimRec1') \<and> (h1 \<in> PrimRec3 \<longrightarrow> pr_conv_3_to_1 h1 \<in> PrimRec1')"
min_elems_def: fixes X :: "'a set" shows "min_elems X = {s \<in> X. \<forall>t. t \<in> X \<longrightarrow> \<not> t < s}"
rtrancl_O_shift: fixes S :: "('a \<times> 'b) set"   and R :: "('b \<times> 'a) set" shows "(S O R)\<^sup>* O S = S O (R O S)\<^sup>*"
fmlookup_inject: fixes x :: "('c, 'd) fmap"   and y :: "('c, 'd) fmap" shows "(($$) x = ($$) y) = (x = y)"
singular_simplex_simplex_map: fixes p :: "nat"   and X :: "'c topology"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c"   and X' :: "'d topology"   and g :: "'c \<Rightarrow> 'd" assumes "singular_simplex p X f"   and "continuous_map X X' g" shows "singular_simplex p X' (simplex_map p g f)"
domain_eq: shows "Domainp (pcr_slist (=) (=)) = (\<lambda>xs. distinct (map fst xs) \<and> rsorted (map fst xs))"
Span_eq_combine_set: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "Span K Us = {combine Ks Us |Ks. set Ks \<subseteq> K}"
is_unit_poly_iff: fixes p :: "'a poly" shows "(p dvd 1) = (\<exists>c. p = [:c:] \<and> c dvd (1::'a))"
select_convs_3: fixes NonceS :: "rat option"   and IntMapK :: "rat option"   and ExtMapK :: "rat option"   and IntAgrK :: "rat option"   and ExtAgrK :: "rat option"   and more :: "'a" shows "ExtMapK \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr> = ExtMapK"
SuccD: fixes k :: "'b"   and Kl :: "'b list set"   and kl :: "'b list" assumes "k \<in> BNF_Greatest_Fixpoint.Succ Kl kl" shows "kl @ [k] \<in> Kl"
l2_inv3_def: shows "l2_inv3 \<equiv> {s. \<forall>Ra A B K. Auth B A (Aenc K (epubKF (Ra $ kE))) \<in> chan s \<longrightarrow> A \<notin> bad s \<and> B \<notin> bad s \<longrightarrow> (\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xpkE, xsk} \<and> guessed_frame Rb xpkE = Some (epubKF (Ra $ kE)) \<and> K = NonceF (Rb $ sk))}"
omega_le_Limit: fixes \<mu> :: "V" assumes "Limit \<mu>" shows "\<omega> \<le> \<mu>"
assms_2: shows "(_High_) \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
bdd_below_def: fixes A :: "'b set" shows "bdd_below A = (\<exists>m. \<forall>x\<in>A. m \<le> x)"
length_Cons: fixes a :: "'a"   and b :: "'a list" shows "length (a # b) = length b + 1"
assms_17: shows "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"