hcju/isabelleps huggingface dataset:
existence_ivl_initial_time: fixes x0 :: "'a" assumes "0 \<in> UNIV"   and "x0 \<in> X" shows "0 \<in> existence_ivl0 x0"
prv_prv_imp_trans: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> \<chi>)"   and "prv (imp \<chi> \<psi>)" shows "prv (imp \<phi> \<psi>)"
IH: shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
tr: shows "((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A"
eWls_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and s :: "'sort"   and X :: "'a" shows "eWls MOD s (OK X) = igWls MOD s X"
count_roots_below: fixes ps :: "real poly list"   and p :: "real poly"   and a :: "real" assumes "sturm_seq ps p"   and "p \<noteq> 0" shows "sign_changes_neg_inf ps - sign_changes ps a = card {x. x \<le> a \<and> poly p x = 0}"
Ag_ind_carrier: fixes f :: "'a \<Rightarrow> 'c"   and D :: "'c set" assumes "bij_to f (carrier A) D" shows "carrier (Ag_ind A f) = f ` carrier A"
assms_4: fixes n :: "nat" shows "\<integral>\<^sup>+ x. ennreal (norm (F n x)) \<partial>M \<le> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M"
open_affinity: fixes S :: "'c set"   and c :: "real"   and a :: "'c" assumes "open S"   and "c \<noteq> 0" shows "open ((\<lambda>x. a + c *\<^sub>R x) ` S)"
last_list_update: fixes xs :: "'g list"   and k :: "nat"   and x :: "'g" assumes "xs \<noteq> []" shows "last (xs[k := x]) = (if k = length xs - 1 then x else last xs)"
simps_1: fixes x1 :: "'d"   and x2 :: "'d stream"   and y1 :: "'d"   and y2 :: "'d stream" shows "(x1 ## x2 = y1 ## y2) = (x1 = y1 \<and> x2 = y2)"
prems: fixes i :: "'i" assumes "i \<in> insert i__ I" shows "A i \<in> sets (M i)"
real_vector_space_on_axioms: fixes p :: "'a" shows "real_vector_space_on (src.tangent_space p)"
ord_spmf_exec_gpv_stop: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and stop :: "'a \<Rightarrow> bool"   and callee1 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c option \<times> 'a) spmf"   and callee2 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c option \<times> 'a) spmf"   and s1 :: "'a"   and s2 :: "'a"   and gpv :: "('d, 'b, 'c) gpv" assumes "\<And>s1 s2 x. \<lbrakk>S s1 s2; \<not> stop s2\<rbrakk> \<Longrightarrow> ord_spmf (\<lambda>(r1, s1') (r2, s2'). le_option r2 r1 \<and> S s1' s2' \<and> (r2 = None \<and> r1 \<noteq> None) = stop s2') (callee1 s1 x) (callee2 s2 x)"   and "S s1 s2"   and "\<not> stop s2"   and "\<And>s1 s2 x y s1'. \<lbrakk>(y, s1') \<in> set_spmf (callee1 s1 x); S s1 s2; stop s2\<rbrakk> \<Longrightarrow> S s1' s2" shows "ord_spmf (rel_prod (ord_option \<top>)\<inverse>\<inverse> S) (exec_gpv_stop callee1 gpv s1) (exec_gpv_stop callee2 gpv s2)"
uniform_unfold_point_set: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "int (card {p \<in> \<V>. p \<in> bl}) = \<k>"
inject_1: fixes x1 :: "\<omega>"   and y1 :: "\<omega>" shows "(\<omega>\<nu> x1 = \<omega>\<nu> y1) = (x1 = y1)"
right_idem: fixes a :: "'a"   and b :: "'a" shows "max (max a b) b = max a b"
12383: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) \<sqinter> - y = - (x \<squnion> y)"
rot_circle_right_edge_def: shows "rot_circle_right_edge = (1, \<lambda>t. (d * circle_y (x_coord t), x_coord t * d))"
subst_upd_comp: fixes \<rho> :: "'d \<Rightarrow> 'c"   and x :: "'e \<Longrightarrow> 'c"   and v :: "('e, 'd) uexpr"   and \<sigma> :: "'b \<Rightarrow> 'd" shows "\<rho>(x \<mapsto>\<^sub>s v) \<circ> \<sigma> = (\<rho> \<circ> \<sigma>)(x \<mapsto>\<^sub>s \<sigma> \<dagger> v)"
qOp_pickInp_pick_Op: fixes inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs set option"   and delta :: "'f" assumes "goodInp inp"   and "goodBinp binp" shows "qOp delta (pickInp inp) (pickBinp binp) #= pick (Op delta inp binp)"
att_or: fixes As :: "'a attree list"   and s :: "'a set \<times> 'a set" shows "(\<turnstile>As \<oplus>\<^sub>\<or>\<^bsup>s\<^esup>) = (case As of [] \<Rightarrow> fst s \<subseteq> snd s | [a] \<Rightarrow> \<turnstile>a \<and> fst s \<subseteq> fst (attack a) \<and> snd (attack a) \<subseteq> snd s | a # ab # list \<Rightarrow> \<turnstile>a \<and> fst (attack a) \<subseteq> fst s \<and> snd (attack a) \<subseteq> snd s \<and> \<turnstile>ab # list \<oplus>\<^sub>\<or>\<^bsup>(fst s - fst (attack a), snd s)\<^esup>)"
lprefix_ltakeI: fixes a :: "'c llist"   and b :: "'c llist"   and n :: "enat" assumes "lprefix a b"   and "llength a \<le> n" shows "lprefix a (ltake n b)"
parent_child_rel_def: shows "Shadow_DOM.parent_child_rel \<equiv> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_parent_child_rel Shadow_DOM.get_child_nodes"
simps: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'d" shows "r\<^sup>*\<^sup>* a1 a2 = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> r\<^sup>*\<^sup>* a b \<and> r b c))"
i_cut_empty_iff_1: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>\<le> t = {}) = (\<forall>x\<in>I. t < x)"
Inv_swap: fixes xs :: "'b list"   and ys :: "'b list"   and n :: "nat" assumes "dist_perm xs ys" shows "Inv xs (swap n ys) = (if Suc n < length xs then if ys ! n < ys ! Suc n in xs then Inv xs ys \<union> {(ys ! n, ys ! Suc n)} else Inv xs ys - {(ys ! Suc n, ys ! n)} else Inv xs ys)"
edka_wf_rel_def: shows "edka_wf_rel \<equiv> inv_image (less_than_bool <*lex*> measure (\<lambda>cf. ek_analysis_defs.ekMeasure cf s t)) (\<lambda>(f, brk). (\<not> brk, cf_of f))"
adapted_init: fixes x :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "borel_adapt_stoch_proc F x" shows "\<exists>c. \<forall>w\<in>space M. x 0 w = c"
n2m_dtor_unfold_coG_coF_def: fixes s1 :: "'g \<Rightarrow> ('f, 'e) coG0"   and s2 :: "'e \<Rightarrow> ('g, 'e) coF0" shows "n2m_dtor_unfold_coG_coF s1 s2 = dtor_unfold_coF (BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf \<circ> map_pre_coGcoF\<^sub>M id (n2m_dtor_unfold_coG s1 s2) id \<circ> s2)"
root_len_dvd: fixes u :: "'b list"   and t :: "'b list" assumes "u \<in> t*" shows "\<^bold>|t\<^bold>| dvd \<^bold>|u\<^bold>|"
hta_add_rule_def: fixes r :: "('a, 'b) ta_rule"   and H :: "('a, 'b) hashedTa" shows "hta_add_rule r H \<equiv> init_hta (hta_Qi H) (ls.ins r (hta_\<delta> H))"
dual_category_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "dual_category C \<equiv> category C"
strict_mono_on_imp_less_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "strict_mono_on f A"   and "x \<in> A"   and "y \<in> A" shows "(f x < f y) = (x < y)"
that: fixes f :: "'a \<Rightarrow> 'b" assumes "bij_betw f A B" shows "thesis"
ugraph_property_def: fixes C :: "(nat set \<times> nat set set) set" shows "ugraph_property C \<equiv> \<forall>G\<in>C. \<forall>G'. G \<simeq> G' \<longrightarrow> G' \<in> C"
abs_eq: fixes x :: "('c \<times> 'd) list" shows "AList_Mapping.Mapping x = mapping.Mapping (map_of x)"
\<phi>'_\<phi>: fixes x :: "char list" assumes "x \<in> V" shows "\<phi>' (\<phi> x) = x"
vrange_vconst_on_ne: fixes A :: "V"   and c :: "V" assumes "A \<noteq> 0" shows "\<R>\<^sub>\<circ> (vconst_on A c) = set {c}"
to_show: assumes "R" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
eq_le_not: fixes a :: "nat"   and b :: "nat" assumes "a \<le> b"   and "\<not> a < b" shows "a = b"
notci_def: fixes e :: "nat"   and s :: "bddi" shows "notci e s \<equiv> fci s \<bind> (\<lambda>(f, s). tci s \<bind> (\<lambda>(t, s). iteci_lu e f t s))"
brc_\<alpha>_def: shows "brc_\<alpha> \<equiv> \<lambda>(Q, W, rcm). (hs.\<alpha> Q, set W, hm.\<alpha> rcm)"
1: shows "(f, g) \<in> \<langle>Rk\<rangle>map2set_rel R"
nnfis_minus_nnfis_integral_2: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and b :: "real"   and g :: "'b \<Rightarrow> real" assumes "a \<in> nnfis f M"   and "b \<in> nnfis g M"   and "measure_space M" shows "\<integral> (\<lambda>t. f t - g t) \<partial>M = a - b"
cSup_singleton: 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 x :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Sup {x} = x"
emeasure_sup_measure'_le2: fixes B :: "'b measure"   and C :: "'b measure"   and A :: "'b measure"   and X :: "'b set" assumes "sets B = sets C"   and "sets A = sets C"   and "X \<in> sets C"   and "\<And>Y. \<lbrakk>Y \<subseteq> X; Y \<in> sets A\<rbrakk> \<Longrightarrow> emeasure A Y \<le> emeasure C Y"   and "\<And>Y. \<lbrakk>Y \<subseteq> X; Y \<in> sets A\<rbrakk> \<Longrightarrow> emeasure B Y \<le> emeasure C Y" shows "emeasure (sup_measure' A B) X \<le> emeasure C X"
intros_1: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> parts H"
dropWhile_eq_drop: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "dropWhile P xs = drop (length (takeWhile P xs)) xs"
pairwise_subset: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "pairwise P S"   and "T \<subseteq> S" shows "pairwise P T"
coeff_zero_simp: fixes v :: "nat" shows "coeff 0 v = 0"
norm_le_1: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and k :: "nat" assumes "dcharacter n \<chi>" shows "cmod (\<chi> k) \<le> 1"
fps_of_poly_numeral': fixes n :: "num" shows "fps_of_poly [:numeral n:] = numeral n"
well_def_def: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "well_def prog \<equiv> healthy (wp prog) \<and> nearly_healthy (wlp prog) \<and> wp_under_wlp prog \<and> sub_distrib_pconj prog \<and> sublinear (wp prog) \<and> bd_cts (wp prog)"
finiteIncLevel: fixes A :: "(nat \<times> 'b) set" shows "finite (incLevel ` A) = finite A"
let2: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e1 :: "exp0"   and s' :: "'b state"   and err :: "v error_result"   and n :: "char list option"   and e2 :: "exp0" assumes "evaluate ck env s e1 (s', Rerr err)" shows "evaluate ck env s (exp0.Let n e1 e2) (s', Rerr err)"
assms_2: shows "moebius_pt (moebius_similarity a b) z = \<infinity>\<^sub>h"
foundation16: fixes \<tau> :: "'b state \<times> 'b state"   and X :: "'b state \<times> 'b state \<Rightarrow> 'c" shows "(\<tau> \<Turnstile> \<delta> X) = (X \<tau> \<noteq> \<bottom> \<and> X \<tau> \<noteq> null)"
card_Un_disjoint: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "card (A \<union> B) = card A + card B"
exhaust: fixes y :: "'a rexp" obtains "y = Zero"   | "y = One"   | x3 :: "'a" where "y = Atom x3"   | x41 :: "'a rexp" and x42 :: "'a rexp" where "y = Plus x41 x42"   | x51 :: "'a rexp" and x52 :: "'a rexp" where "y = Times x51 x52"   | x6 :: "'a rexp" where "y = Star x6"
lookup_mapp: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and p :: "'b \<Rightarrow>\<^sub>0 'c"   and k :: "'b" shows "lookup (Poly_Mapping.mapp f p) k = (f k (lookup p k) when k \<in> keys p)"
P1: shows "B' Out C'' C' \<and> Cong B' C'' B C"
size_less_than_median_of_medians: fixes xs :: "'a list" shows "size {#y \<in># mset xs. y < median (map median (chop 5 xs))#} \<le> nat \<lceil>7 / 10 * real (length xs) + 3\<rceil>"
FieldI1: fixes i :: "'b"   and j :: "'b"   and R :: "('b \<times> 'b) set" assumes "(i, j) \<in> R" shows "i \<in> Field R"
continuous_on_Im: fixes s :: "'b set"   and g :: "'b \<Rightarrow> complex" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. Im (g x))"
max_Sup_absorb_right: fixes X :: "'b set"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "X \<noteq> {}"   and "bdd_above (f ` X)"   and "bdd_above (g ` X)"   and "Sup (f ` X) \<le> Sup (g ` X)" shows "Sup (pointwise_max f g ` X) = Sup (g ` X)"
finite_min_cf_outgoing: fixes u :: "nat" shows "finite {l v |v. (u, v) \<in> cf.E}"
hermitean_eye': shows "hermitean (1, 0, 0, 1)"
cInf_eq_minimum: 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 z :: "'c"   and X :: "'c set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq z x" shows "Inf X = z"
cases: fixes x :: "('fsym, 'tp) efsym trm" obtains xa :: "Preliminaries.var" where "x = Var xa"   | f :: "('fsym, 'tp) efsym" and Tl :: "('fsym, 'tp) efsym trm list" where "x = Fn f Tl"
pseudo_divmod_def: fixes p :: "'b poly"   and q :: "'b poly" shows "pseudo_divmod p q \<equiv> if q = 0 then (0, p) else pseudo_divmod_main (lead_coeff q) 0 p q (degree p) (1 + length (coeffs p) - length (coeffs q))"
mapping_of_pmf_def: shows "mapping_of_pmf \<equiv> map_fun id mapping.Mapping (\<lambda>p x. if pmf p x = 0 then None else Some (pmf p x))"
type_to_set_2: assumes "\<exists>Rep Abs. type_definition Rep Abs {0..<p}" shows "p = int CARD('a)"
simps_1: fixes p :: "linear_poly"   and c :: "rat" shows "lec_of_constraint (LEQ p c) = Le_Constraint Leq_Rel p c"
gpath_is_empty_code_def: shows "gpath_is_empty_code \<equiv> \<lambda>x. as_is_empty (case snd x of (xa, xb, xca, xd) \<Rightarrow> xa)"
coeff_C_mult: fixes x :: "'b"   and a :: "'b poly"   and n :: "nat" shows "coeff (C x * a) n = x * coeff a n"
hyps_3: assumes "a \<in> B"   and "z \<in> A"   and "B \<subseteq> A" shows "\<exists>y'. y = f a y' \<and> fold_graph f z (B - {a}) y' \<and> y' \<in> A"
right_def: shows "right \<equiv> (rep_msg ---> abs_msg) freeright"
abs_eq: fixes xa :: "'c"   and x :: "'c" assumes "eq_onp in_p_image xa xa"   and "eq_onp in_p_image x x" shows "Abs_regular xa \<sqinter> Abs_regular x = Abs_regular (xa \<sqinter> x)"
subprob_not_empty: fixes p :: "'a spmf" shows "space (measure_spmf p) \<noteq> {}"
simps_6: fixes \<phi> :: "'a ltln"   and \<xi> :: "'a ltln" shows "suspendable (\<phi> R\<^sub>n \<xi>) = (\<phi> = false\<^sub>n \<and> pure_eventual \<xi> \<or> suspendable \<xi>)"
unit_vec_nonzero: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "unit_vec n i \<noteq> 0\<^sub>v n"
horner_sum_simps_2: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and x :: "'b"   and xs :: "'b list" shows "horner_sum f a (x # xs) = f x + a * horner_sum f a xs"
ptrm_swp_transfer: fixes a :: "'b"   and b :: "'b"   and X :: "'b ptrm"   and Y :: "'b ptrm" shows "[a \<leftrightarrow> b] \<bullet> X \<approx> Y = X \<approx> [a \<leftrightarrow> b] \<bullet> Y"
HyperN_def: shows "HyperN \<equiv> {b \<in> carrier_vec n. \<nexists>c B. c \<in> carrier_vec n \<and> B \<subseteq> A \<and> Suc (card B) = dim_span (insert b A) \<and> lin_indpt B \<and> (\<forall>a\<in>B. c \<bullet> a = (0::'a)) \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)}"
share_union: fixes a :: "nat"   and sb :: "'b memref list"   and A :: "nat \<Rightarrow> bool option"   and z :: "nat set"   and B :: "nat set" assumes "a \<in> dom (share sb (A \<oplus>\<^bsub>z\<^esub> B))"   and "a \<notin> dom A" shows "a \<in> dom (share sb (Map.empty \<oplus>\<^bsub>z\<^esub> B))"
comp_arr_point: fixes f :: "'s"   and x :: "'s" assumes "arr f"   and "\<guillemotleft>x : unity \<rightarrow> local.dom f\<guillemotright>" shows "f \<cdot> x = mkPoint (cod f) (Fun f (img x))"
abs_nchotomy: fixes A :: "('b, 'c, 'varSort, 'var, 'd) qAbs set" assumes "goodAbs A" shows "\<exists>xs x X. good X \<and> A = Abs xs x X"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "R x y" shows "ord_spmf P (f x) (g y)"
simps_7: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and q22 :: "'a \<Rightarrow> bool"   and g221 :: "'a \<Rightarrow> ('b, 'c) tllist"   and g222 :: "'a \<Rightarrow> 'a" assumes "p a" shows "is_TNil (corec_tllist p g1 g21 q22 g221 g222 a)"
tighten_bounds_2_2: fixes x :: "real_alg_2" assumes "invariant_2 x" shows "invariant_2 (tighten_bounds_2 x)"
abs_eq: shows "0 = Abs_poly (\<lambda>_. 0::'a)"
eq: shows "{M. set_mset M = A \<and> size M = k} = {}"
exp1_bounds: shows "exp 1 \<in> {8 / 3..11 / 4}"
KL_div_pos2: fixes S :: "'b set"   and a :: "'b \<Rightarrow> real"   and d :: "'b \<Rightarrow> real" assumes "finite S"   and "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i"   and "\<And>i. i \<in> S \<Longrightarrow> 0 < d i"   and "sum a S = 1"   and "sum d S = 1" shows "0 \<le> KL_div S a d"
point_dist_def: fixes A :: "point"   and B :: "point" shows "dist A B = sqrt ((abscissa (A - B))\<^sup>2 + (ordinate (A - B))\<^sup>2)"
Constrains_Int: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> A Co A'"   and "F \<in> B Co B'" shows "F \<in> A \<inter> B Co A' \<inter> B'"
r_def: shows "r = Abs_poly (\<lambda>k. coeff p (k + n))"
balls_1: fixes x :: "'a" assumes "x \<in> carrier" shows "x \<in> balls x"
simps_6: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and b2 :: "bool" shows "local.less_gexp_aux (Eq e1 e2) (Bc b2) = False"
MsgCJ1: shows "msgs cJ1 = (\<lambda>m. if \<exists>p. m = <p, inM \<exists>i<Suc j. procList ! i = p> then 1 else 0)"
FAccNull: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>1\<rangle>"
ddvd_trans: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times x y \<and> dvd.dvd times y x"   and "dvd.dvd times y z \<and> dvd.dvd times z y" shows "dvd.dvd times x z \<and> dvd.dvd times z x"
beta2_def: shows "beta2 \<equiv> $pc2 = #b \<and> pc2$ = #g \<and> y$ = Suc<$y> \<and> unchanged (x, sem, pc1)"
no_messages_if_no_channel: fixes t :: "('a, 'b, 'c) event llist"   and cid :: "nat"   and i :: "nat" assumes "cotrace init t"   and "channel cid = None" shows "msgs (coS t i) cid = []"
bin_cat_zero: fixes n :: "nat"   and w :: "int" shows "concat_bit n w 0 = take_bit n w"
inductive_set_def: fixes D :: "('c, 'd) Order_scheme" shows "inductive_set D = (\<forall>C. Chain D C \<longrightarrow> (\<exists>b. ub\<^bsub>D\<^esub> C b))"
simps: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" shows "translate_signature (ExeSignature cto tao sa) = (map_of cto, map_of tao, translate_osig sa)"
arr_inv: fixes f :: "'a" assumes "L'.C.iso f" shows "L'.C.arr (L'.C.inv f)"
dom_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "span_vertical_category C" shows "partial_magma.dom (span_vertical_category.vcomp C) = (\<lambda>\<mu>. if partial_magma.arr (span_vertical_category.vcomp C) \<mu> then \<lparr>Chn = span_in_category.apex C (Dom \<mu>), Dom = Dom \<mu>, Cod = Dom \<mu>\<rparr> else partial_magma.null (span_vertical_category.vcomp C))"
tr_par_tfr: fixes A' :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list" assumes "A' \<in> set (tr\<^sub>p\<^sub>c A [])"   and "tfr\<^sub>s\<^sub>s\<^sub>t (unlabel A)"   and "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}" shows "tfr\<^sub>s\<^sub>t (unlabel A')"
empty: fixes a :: "'a"   and b :: "'a" assumes "a \<in> H"   and "b \<in> H"   and "a \<noteq> b" shows "(a <# J) \<inter> (b <# J) = {}"
Inv_CompDiag: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Can t"   and "Can u"   and "Diag t"   and "Diag u"   and "Dom t = Cod u" shows "Inv (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Inv u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> Inv t"
unique_connecting_path_rev: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V" shows "v \<leadsto> w = rev (w \<leadsto> v)"
steps_refl: fixes P :: "'b"   and Cal :: "'b processCalculus" shows "P \<longmapsto>Cal* P"
assms_3: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state"   and cf'' :: "('test, 'atom) com \<times> 'state" assumes "cf \<rightarrow>*c cf'"   and "phi cf cf'"   and "cf' \<rightarrow>c cf''" shows "phi cf cf''"
ft_dghm_op_dghm_is_ft_dghm: shows "op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
representation_add: fixes basis :: "'b set"   and v :: "'b"   and u :: "'b" assumes "independent basis"   and "v \<in> span basis"   and "u \<in> span basis" shows "representation basis (u + v) = (\<lambda>b. representation basis u b + representation basis v b)"
x2x1: shows "x2 - x1 = (v * (1 - u)) *\<^sub>R (b - a)"
NI_indirect_sources_total_def: shows "NI_indirect_sources_total \<equiv> \<forall>execs a n. realistic_executions execs \<longrightarrow> (let s_f = run_total n s0 execs in output_f (run_total n s0 (ipurge_l execs (current s_f))) a = output_f (run_total n s0 (ipurge_r execs (current s_f))) a)"
s_AppLam: fixes v :: "Syntax.term"   and x :: "Syntax.var"   and \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and uu :: "mode" assumes "value v"   and "atom x \<sharp> (v, \<pi>)" shows "\<lless>\<pi>, App (Lam x e) v\<ggreater> uu\<rightarrow> \<lless>\<pi>, e[v / x]\<ggreater>"
proots_def: fixes p :: "'a poly" shows "proots p = {x. poly p x = (0::'a)}"
fresh_pVs: fixes a :: "atom" assumes "a \<sharp> p" shows "a \<sharp> p \<bullet> V"
simps_2: fixes X :: "'b set"   and d :: "int" assumes "Beta_Regions X" shows "intv'_const (Const' d) = d"
axioms_2: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h" shows "Group.group H"
homogeneous_setD_hom_components: fixes A :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and a :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and b :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "homogeneous_set A"   and "a \<in> A"   and "b \<in> hom_components a" shows "b \<in> A"
group_left_one: fixes x :: "'a" shows "(1::'a) * x = x"
substs_preserve_subterms: fixes t :: "'b trm"   and p :: "indices list"   and s :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "subterm t p s" shows "subterm (t \<lhd> \<sigma>) p (s \<lhd> \<sigma>)"