hcju/isabelleps huggingface dataset:
reachable_refl: fixes v :: "'a" assumes "v \<in> verts H" shows "v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"
until_must_release_ev: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and \<psi> :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" assumes "(\<phi> until \<psi>) \<omega>"   and "ev (\<lambda>xs. \<not> \<phi> xs) \<omega>" shows "ev \<psi> \<omega>"
y_axis_def: shows "y_axis \<equiv> abs_circline y_axis_clmat"
self_in_downset: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.order less_eq less" shows "x \<in> {y. less_eq y x}"
transversal_segment_commute: fixes x :: "'a"   and y :: "'a" shows "rev.transversal_segment x y = rev.transversal_segment y x"
counit_simps_3: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>C (\<epsilon>'.counit a) = I\<^sub>C.map\<^sub>0 a"
CondThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e' :: "nat exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>1 :: "nat exp"   and e\<^sub>2 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
removeNones_hd: fixes a :: "'a"   and ls :: "'a option list" shows "removeNones (Some a # ls) = a # removeNones ls"
integrable_bound1: shows "interval_lebesgue_integrable lborel 0 (ereal (exp (- 1))) g1"
\<omega>: fixes n :: "nat" shows "(restrict \<omega> {0..<n})(n := \<omega> n) \<in> X (Suc n)"
spr_commonKnowledge_relation_image: fixes s :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState"   and cec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and as :: "'a list" assumes "s \<in> toSet cec" shows "toSet (spr_commonKnowledge envObs cec as s) = (\<Union> (relations (spr_repMC (toSet cec)) ` set as))\<^sup>+ `` {s}"
in_language_to_language: fixes L :: "'b list set" shows "Collect (in_language (to_language L)) = L"
simps_6: fixes m :: "modality"   and g :: "safety"   and r :: "'a regex" shows "local.safe_regex m g (Star r) = ((g = Lax \<or> fv_regex fv r = {}) \<and> local.safe_regex m g r)"
abs_ast_variable_section_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.abs_ast_variable_section problem = [0..<length (ast_problem.astDom problem)]"
le_cases: fixes y :: "'a list"   and x :: "'a list" obtains "y \<le>lex x"   | "x \<le>lex y"
up_set_prop_var2: fixes x :: "'a"   and X :: "'a set"   and y :: "'a" assumes "x \<in> \<Up> X"   and "x \<le> y" shows "y \<in> \<Up> X"
evalO_exp\<omega>: fixes b :: "nat"   and n :: "Ord" shows "evalO b (exp\<omega> n) = b ^ evalO b n"
hyps: assumes "finals es" shows "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es,s\<rangle>"
divideC_field_splits_simps_1_2: fixes c :: "complex"   and b :: "'c"   and a :: "'c" shows "(b /\<^sub>C c = a) = (if c = 0 then a = (0::'c) else b = c *\<^sub>C a)"
iterate_size_abort_def: fixes it :: "(nat \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat"   and n :: "nat" shows "iterate_size_abort it n = it (\<lambda>\<sigma>. \<sigma> < n) (\<lambda>x. Suc) 0"
finite_inverse_real_of_posnat_ge_real: fixes u :: "real" assumes "0 < u" shows "finite {n. u \<le> inverse (real (Suc n))}"
frontier_halfspace_ge: fixes a :: "'c"   and b :: "real" assumes "a \<noteq> (0::'c) \<or> b \<noteq> 0" shows "frontier {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
def_coinduct: fixes A :: "'a"   and f :: "'a \<Rightarrow> 'a"   and X :: "'a" assumes "A \<equiv> gfp f"   and "mono f"   and "X \<le> f (sup X A)" shows "X \<le> A"
mult_right_le_one_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x"   and "less_eq zero y"   and "less_eq y one" shows "less_eq (times x y) x"
satC_iff_set: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('fsym, 'psym) lit list" shows "I.satC \<xi> c = (\<exists>l\<in>set c. I.satL \<xi> l)"
Rep_covector_inverse: fixes x :: "'b covector" shows "Abs_covector (Rep_covector x) = x"
cart_loc_def: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('d, 'e, 'f) set_ops_scheme"   and ops3 :: "('a \<times> 'd, 'g, 'h) set_ops_scheme" shows "cart_loc ops1 ops2 ops3 \<equiv> inj_image_filter_cp_loc ops1 ops2 ops3"
eq_refl: fixes x :: "'a Word.word"   and y :: "'a Word.word" assumes "x = y" shows "x \<le>s y"
eq_\<I>_converter_reflI: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv"
neg_0_equal_iff_equal: fixes a :: "'a" shows "((0::'a) = - a) = ((0::'a) = a)"
slpa_split_2: fixes cs :: "'edge list"   and as :: "'edge list"   and xs :: "'edge list"   and ys :: "'edge list" assumes "same_level_path_aux cs as"   and "as = xs @ ys"   and "upd_cs cs xs = []" shows "same_level_path_aux [] ys"
vquot_tm_def: fixes t :: "tm"   and V :: "name set" shows "\<lfloor>t\<rfloor>V = vquot_dbtm V (trans_tm [] t)"
lookup_lift_poly_syz: fixes n :: "nat"   and b :: "('a, 'b) pp \<times> nat \<Rightarrow>\<^sub>0 'c"   and i :: "nat" shows "lookup (lift_poly_syz_pprod n b i) = (\<lambda>x. if x = (0, i) then 1::'c else if n \<le> snd x then lookup b (map_component_pprod (\<lambda>k. k - n) x) else (0::'c))"
ipresFresh_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'h, 'm, 'n) model_scheme" shows "ipresFresh h MOD \<equiv> \<forall>ys y s X. wls s X \<longrightarrow> fresh ys y X \<longrightarrow> igFresh MOD ys y (h X)"
scale_cancel_right: fixes a :: "real"   and x :: "'c"   and b :: "real" shows "(a *\<^sub>R x = b *\<^sub>R x) = (a = b \<or> x = (0::'c))"
ok_commute: fixes F :: "'b program"   and G :: "'b program" shows "F ok G = G ok F"
from_N_to_P_or_Q: fixes Gs :: "'a literal multiset set llist"   and C :: "'a literal multiset"   and D :: "'a literal multiset"   and i :: "nat" assumes "chain (\<leadsto>) Sts"   and "fair_state_seq Sts"   and "Gs = lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts"   and "C \<in> Liminf_llist Gs - sr.Rf (Liminf_llist Gs)"   and "D \<in> N_of_state (lnth Sts i)"   and "enat i < llength Sts"   and "subsumes D C"   and "\<forall>E\<in>{E \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts). subsumes E C}. \<not> strictly_subsumes E D" shows "\<exists>l D' \<sigma>'. D' \<in> P_of_state (lnth Sts l) \<union> Q_of_state (lnth Sts l) \<and> enat l < llength Sts \<and> (\<forall>E\<in>{E \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts). subsumes E C}. \<not> strictly_subsumes E D') \<and> D' \<cdot> \<sigma>' = C \<and> is_ground_subst \<sigma>' \<and> subsumes D' C"
two_mutual_lists_induction: fixes P :: "'b list \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>xs. P xs []"   and "\<And>x xs y ys. \<lbrakk>P xs (y # ys); P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
env_restr_env_restr_subset: fixes d1 :: "'c set"   and d2 :: "'c set"   and x :: "'c \<Rightarrow> 'd" assumes "d1 \<subseteq> d2" shows "x f|` d2 f|` d1 = x f|` d1"
distrib: fixes g :: "'b \<Rightarrow> 'a"   and h :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" shows "(\<Prod>x\<in>#A. g x * h x) = \<Prod>\<^sub># (image_mset g A) * \<Prod>\<^sub># (image_mset h A)"
ivl_disj_int_one_2: fixes l :: "'b"   and u :: "'b" shows "{..<l} \<inter> {l..<u} = {}"
right_inverse: fixes a :: "'a" shows "a \<cdot> inverse a = (1::'a)"
join_pres_def: fixes X :: "('a, 'b) gorder_scheme"   and Y :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c" shows "join_pres X Y f \<equiv> lattice X \<and> lattice Y \<and> (\<forall>x\<in>carrier X. \<forall>y\<in>carrier X. f (x \<squnion>\<^bsub>X\<^esub> y) = f x \<squnion>\<^bsub>Y\<^esub> f y)"
sum_norm_le: fixes 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 S :: "'f set"   and f :: "'f \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x" shows "norm (comm_monoid_add.sum plus zero f S) \<le> sum g S"
covering_space_lift_homotopy_alt: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and U :: "'e set"   and h :: "'e \<times> real \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'c" assumes "covering_space C p S"   and "continuous_on (U \<times> {0..1}) h"   and "h ` (U \<times> {0..1}) \<subseteq> S"   and "\<And>y. y \<in> U \<Longrightarrow> h (y, 0) = p (f y)"   and "continuous_on U f"   and "f ` U \<subseteq> C" obtains k :: "'e \<times> real \<Rightarrow> 'c" where "continuous_on (U \<times> {0..1}) k" and "k ` (U \<times> {0..1}) \<subseteq> C" and "\<And>y. y \<in> U \<Longrightarrow> k (y, 0) = f y" and "\<And>z. z \<in> U \<times> {0..1} \<Longrightarrow> h z = p (k z)"
GreenThm_typeII_divisible: assumes "gen_division s (cubeImage ` two_chain)" shows "integral s (partial_vector_derivative (\<lambda>x. F x \<bullet> j) i) = one_chain_line_integral F {j} (two_chain_boundary two_chain)"
epair_def: fixes e :: "'a set" shows "epair e = (if card e = 2 then Some (SOME (u, v). e = {u, v}) else None)"
assms: fixes a :: "'a"   and b :: "'a" assumes "a \<in> set_list_R1 x"   and "b \<in> set_list_R1 y" shows "m a b = m' a b"
le_take: fixes x :: "nat list"   and y :: "nat list"   and i :: "nat" assumes "x \<le>\<^sub>v y"   and "i \<le> length x" shows "take i x \<le>\<^sub>v take i y"
Inter_eq: fixes A :: "'b set set" shows "\<Inter> A = {x. \<forall>B\<in>A. x \<in> B}"
fall2_imp_alw: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "fall2 (\<lambda>\<pi>' \<pi> \<pi>l. imp (alw (\<phi> \<pi>l)) (alw (\<psi> \<pi>l)) (\<pi>l @ [\<pi>, \<pi>'])) [] = (\<forall>\<pi> \<pi>'. wfp AP' \<pi> \<and> wfp AP' \<pi>' \<and> stateOf \<pi> = s0 \<and> stateOf \<pi>' = s0 \<longrightarrow> imp (alw (\<phi> [])) (alw (\<psi> [])) [\<pi>, \<pi>'])"
poly_roots_finite: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "finite {x. poly p x = (0::'b)}"
simps_2: fixes x :: "'a"   and t :: "'a tree \<times> nat \<times> nat" shows "t\<^sub>s (Delete x) t = real (T_delete2 x t)"
kat_3_equiv_opp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(r x \<cdot> (y \<cdot> ar z) = \<bottom>) = (r x \<cdot> y = r x \<cdot> (y \<cdot> r z))"
Aizerman_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>B C. \<lbrakk>B \<subseteq> A; C \<subseteq> B; f B \<subseteq> C\<rbrakk> \<Longrightarrow> f C \<subseteq> f B" shows "Aizerman_on A f"
distinct_1: fixes x2 :: "n" shows "Z \<noteq> S x2"
canon_ang_ex: fixes \<alpha> :: "real" shows "\<exists>\<alpha>''. canon_ang_P \<alpha> \<alpha>''"
tendsto_def: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter" shows "(f \<longlongrightarrow> l) F = (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> S))"
mreq_end2: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and x :: "'b"   and b :: "'a list"   and c :: "'a list"   and a :: "'a" assumes "applied_rule_rev C x b = applied_rule_rev C x c" shows "applied_rule_rev C x (a # b) = applied_rule_rev C x (a # c)"
zero_interval: shows "(0::'b) \<in>\<^sub>i 0"
weird: fixes D :: "real"   and a' :: "real"   and A :: "real"   and B :: "real"   and C :: "real"   and b' :: "real"   and c' :: "real" assumes "D \<noteq> 0" shows "(a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' < 0 \<or> a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' = 0 \<and> (b' + a' * (A + B * sqrt C) * 2 / D < 0 \<or> b' + a' * (A + B * sqrt C) * 2 / D = 0 \<and> 2 * a' < 0)) = (a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' < 0 \<or> a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' = 0 \<and> (2 * a' * (A + B * sqrt C) / D + b' < 0 \<or> 2 * a' * (A + B * sqrt C) / D + b' = 0 \<and> a' < 0))"
Rep_formula': fixes x :: "('b, 'c, 'd, 'e) formula" shows "hereditarily_fs (Rep_formula x)"
norm_10: shows "\<circle>(\<lambda>s. False) = (\<lambda>s. False)"
the_\<nu>_props_1: shows "\<guillemotleft>\<chi>.the_\<nu> : s\<^sub>1 \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine\<guillemotright>"
PiE_eq_iff: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'e set"   and F' :: "'d \<Rightarrow> 'e set" shows "(Pi\<^sub>E I F = Pi\<^sub>E I F') = ((\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
takeWhile_map: fixes P :: "'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
rep_eq: fixes x :: "'d" shows "blinfun_apply (blinfun_inner_right x) = (\<bullet>) x"
realizable_call: fixes n\<^sub>0 :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n\<^sub>1 :: "'node SDG_node"   and p :: "'pname"   and n\<^sub>2 :: "'node SDG_node"   and V :: "'var"   and ns' :: "'node SDG_node list"   and n\<^sub>3 :: "'node SDG_node" assumes "realizable n\<^sub>0 ns n\<^sub>1"   and "n\<^sub>1 -p\<rightarrow>\<^bsub>call\<^esub> n\<^sub>2 \<or> n\<^sub>1 -p:V\<rightarrow>\<^bsub>in\<^esub> n\<^sub>2"   and "matched n\<^sub>2 ns' n\<^sub>3" shows "realizable n\<^sub>0 (ns @ n\<^sub>1 # ns') n\<^sub>3"
empty_is_image: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "({} = f ` A) = (A = {})"
subfrmlsn_size: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "size \<psi> < size \<phi> \<or> \<psi> = \<phi>"
closed_interval_as_convex_hull: fixes a :: "'b"   and b :: "'b" obtains S :: "'b set" where "finite S" and "cbox a b = convex hull S"
coeff_map_poly: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly"   and n :: "nat" assumes "f (0::'b) = (0::'a)" shows "coeff (map_poly f p) n = f (coeff p n)"
closures_necessitation: fixes \<phi> :: "\<o>" assumes "[[\<phi>]]" shows "[[\<^bold>\<box>\<phi>]]"
hyps: assumes "abc_final (abc_steps_l (0, lm) p n) p"   and "p \<noteq> []" shows "\<exists>n'. abc_notfinal (abc_steps_l (0, lm) p n') p \<and> abc_final (abc_steps_l (0, lm) p (Suc n')) p"
simps_4: fixes v :: "'b poly"   and va :: "'b poly" shows "isnpolyh (Add v va) = (\<lambda>k. False)"
f_def: shows "\<forall>B\<in>M. f B \<in> L \<and> f B \<in> B"
StrongEq_L_subst3: fixes P :: "('d state \<times> 'd state \<Rightarrow> 'e) \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> 'e"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" assumes "cp P"   and "\<tau> \<Turnstile> x \<triangleq> y" shows "(\<tau> \<Turnstile> P x) = (\<tau> \<Turnstile> P y)"
kills_e1: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>"
list_of_code: fixes as :: "'c iarray" shows "IArray.list_of as = map ((!!) as) [0..<IArray.length as]"
ex_has_greatest_nat: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat"   and b :: "nat" assumes "P k"   and "\<forall>y. P y \<longrightarrow> y \<le> b" shows "\<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> y \<le> x)"
hyp2_abs_rep: fixes p :: "hyp2" shows "hyp2_abs (hyp2_rep p) = p"
Expr_lemma1: fixes e :: "Expr"   and X :: "Var set" assumes "Expr_noLoc e"   and "EVars e \<subseteq> X"   and "\<And>x. x \<in> X \<Longrightarrow> CONTEXT x = low" shows "Expr_low e"
approximating_bigstep_fun_induct_wf: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs :: "'b rule list"   and P :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> 'b rule list \<Rightarrow> state \<Rightarrow> bool"   and s :: "state" assumes "wf_ruleset \<gamma> p rs"   and "\<And>\<gamma> p s. P \<gamma> p [] s"   and "\<And>\<gamma> p r rs X. P \<gamma> p (r # rs) (Decision X)"   and "\<And>\<gamma> p m a rs. \<lbrakk>\<not> matches \<gamma> m a p; P \<gamma> p rs Undecided\<rbrakk> \<Longrightarrow> P \<gamma> p (Rule m a # rs) Undecided"   and "\<And>\<gamma> p m a rs. \<lbrakk>matches \<gamma> m a p; a = Accept\<rbrakk> \<Longrightarrow> P \<gamma> p (Rule m a # rs) Undecided"   and "\<And>\<gamma> p m a rs. \<lbrakk>matches \<gamma> m a p; a = Drop\<rbrakk> \<Longrightarrow> P \<gamma> p (Rule m a # rs) Undecided"   and "\<And>\<gamma> p m a rs. \<lbrakk>matches \<gamma> m a p; a = Reject\<rbrakk> \<Longrightarrow> P \<gamma> p (Rule m a # rs) Undecided"   and "\<And>\<gamma> p m a rs. \<lbrakk>matches \<gamma> m a p; a = Log; P \<gamma> p rs Undecided\<rbrakk> \<Longrightarrow> P \<gamma> p (Rule m a # rs) Undecided"   and "\<And>\<gamma> p m a rs. \<lbrakk>matches \<gamma> m a p; a = Empty; P \<gamma> p rs Undecided\<rbrakk> \<Longrightarrow> P \<gamma> p (Rule m a # rs) Undecided" shows "P \<gamma> p rs s"
p43: shows "\<forall>D\<in>MajoritySet. \<exists>d\<in>D. bal bk \<le> mbal (disk s d p)"
mult_eq_is_mul_eq: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "mult_eq r = mul_eq r"
exec_remcondless_cons: fixes s :: "('c, 'd) fmap"   and h :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and pfx :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "exec_plan s (rem_condless_act s (h # pfx) as) = exec_plan (state_succ s h) (rem_condless_act (state_succ s h) pfx as)"
red_lA_incr: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and e :: "'addr expr"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" assumes "extTA,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "\<lfloor>dom (lcl s)\<rfloor> \<squnion> \<A> e \<sqsubseteq> \<lfloor>dom (lcl s')\<rfloor> \<squnion> \<A> e'"
measurable_id: fixes M :: "'b measure" shows "(\<lambda>x. x) \<in> M \<rightarrow>\<^sub>M M"
sscan_const: fixes xs :: "'a stream"   and a :: "'a" shows "sscan (\<lambda>x _. x) xs a = xs"
comp_null_2: fixes f :: "'b" shows "J.comp f J.null = J.null"
that_1: shows "(g has_derivative g') (at x within {..x})"
enabled_iff: fixes s :: "('test, 'atom, 'choice) cmd \<times> 'state"   and \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" shows "enabled s \<omega> = (enabled (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> G s)"
upd_rev_cs_Cons_Call_Cons_Empty: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and cs :: "'edge list"   and as :: "'edge list" assumes "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "upd_rev_cs cs as = []" shows "upd_rev_cs cs (a # as) = []"
rev_H_eq_rev_G: fixes x :: "'b" assumes "x \<in> arcs G - {uv, rev_G uv}" shows "rev_H x = rev_G x"
conv_radius_eqI': fixes R :: "ereal"   and f :: "nat \<Rightarrow> 'b" assumes "0 \<le> R"   and "\<And>r. \<lbrakk>0 < r; ereal r < R\<rbrakk> \<Longrightarrow> summable (\<lambda>n. f n * of_real r ^ n)"   and "\<And>r. \<lbrakk>0 < r; R < ereal r\<rbrakk> \<Longrightarrow> \<not> summable (\<lambda>n. norm (f n * of_real r ^ n))" shows "conv_radius f = R"
diff_eq: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "minus x y = inf x (uminus y)"
unwind_decomp_secure: fixes \<Delta>s :: "('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) list"   and \<Delta>e :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" assumes "\<Delta>s \<noteq> []"   and "\<And>vl vl1. B vl vl1 \<Longrightarrow> hd \<Delta>s istate vl istate vl1"   and "\<And>\<Delta>1 \<Delta>2. (\<Delta>1, \<Delta>2) \<in> allConsec \<Delta>s \<Longrightarrow> unwind_cont \<Delta>1 {\<Delta>1, \<Delta>2, \<Delta>e}"   and "unwind_cont (last \<Delta>s) {last \<Delta>s, \<Delta>e}"   and "unwind_exit \<Delta>e" shows "secure"
divide_nonpos_nonneg: fixes x :: "'a"   and y :: "'a" assumes "x \<le> (0::'a)"   and "(0::'a) \<le> y" shows "x / y \<le> (0::'a)"
extremum: fixes a :: "'e multiset" shows "{#} \<subseteq># a"
simps: fixes a1 :: "('d, 'e) Term.term"   and a2 :: "('d, 'e) Term.term" shows "a1 \<rhd> a2 = ((\<exists>s ss f. a1 = Fun f ss \<and> a2 = s \<and> s \<in> set ss) \<or> (\<exists>s ss t f. a1 = Fun f ss \<and> a2 = t \<and> s \<in> set ss \<and> s \<rhd> t))"
cblinfun_norm_approx_witness: fixes \<epsilon> :: "real"   and A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "0 < \<epsilon>" shows "\<exists>\<psi>. norm A - \<epsilon> \<le> norm (A *\<^sub>V \<psi>) \<and> norm \<psi> = 1"
valid_sops: shows "\<forall>sop\<in>write_sops sb. valid_sop sop"
ccw_system_axioms: fixes S :: "(real \<times> real) set" shows "ccw_system ccw S"
mod_eq_imp_diff_mod_eq: fixes x :: "nat"   and m :: "nat"   and y :: "nat"   and t :: "nat" assumes "x mod m = y mod m"   and "x \<le> t"   and "y \<le> t" shows "(t - x) mod m = (t - y) mod m"
top_apply: fixes x :: "'b" shows "\<top> x = \<top>"