hcju/isabelleps huggingface dataset:
conditional_entropy_eq_entropy: fixes S :: "'d measure"   and T :: "'e measure"   and Y :: "'a \<Rightarrow> 'e"   and Py :: "'e \<Rightarrow> real"   and X :: "'a \<Rightarrow> 'd"   and Pxy :: "'d \<times> 'e \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>x. x \<in> space T \<Longrightarrow> 0 \<le> Py x"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"   and "\<And>x y. \<lbrakk>x \<in> space S; y \<in> space T\<rbrakk> \<Longrightarrow> 0 \<le> Pxy (x, y)"   and "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"   and "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))" shows "conditional_entropy b S T X Y = entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) - entropy b T Y"
semiring_simprules_7: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
snd_diag_id: fixes z :: "'d" shows "(snd \<circ> (\<lambda>x. (x, x))) z = id z"
singleton_in_FIN: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> \<R>" shows "{f} \<in> FIN"
no_trailing_coeffs: fixes p :: "'b poly" shows "no_trailing ((=) (0::'b)) (coeffs p)"
sum_split: fixes I :: "'b set"   and p :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite I" shows "(\<Sum>i\<in>I. if p i then f i else g i) = (\<Sum>i | i \<in> I \<and> p i. f i) + (\<Sum>i | i \<in> I \<and> \<not> p i. g i)"
inorder_balL: fixes ba :: "bal"   and l :: "('b \<times> bal) tree"   and a :: "'b"   and r :: "('b \<times> bal) tree" assumes "ba = Lh \<longrightarrow> l \<noteq> \<langle>\<rangle>"   and "avl l" shows "Tree2.inorder (balL l a ba r) = Tree2.inorder l @ a # Tree2.inorder r"
iIN_iprev_nth_if: fixes n :: "nat"   and d :: "nat"   and a :: "nat" shows "[n\<dots>,d] \<leftarrow> a = (if a \<le> d then n + d - a else n)"
3: shows "\<Union> {f ta xa |ta xa. (ta, xa) \<in> set ((t, x) # txs)} = f t x \<union> \<Union> {f ta xa |ta xa. (ta, xa) \<in> set txs}"
subst_atm_mset_list_empty_iff: fixes AAs :: "'a multiset list"   and \<eta> :: "'s" shows "(AAs \<cdot>aml \<eta> = []) = (AAs = [])"
up_antitone: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less" shows "less_eq x y = ({ya. less_eq y ya} \<subseteq> {y. less_eq x y})"
in_lists_conv_set_subset: fixes ws :: "'b list"   and G :: "'b set" shows "(set ws \<subseteq> G) = (ws \<in> lists G)"
prob_eq_0: fixes A :: "'b set"   and M :: "'b pmf" assumes "A \<in> measure_pmf.events M" shows "(measure_pmf.prob M A = 0) = (AE x in measure_pmf M. x \<notin> A)"
wf_prog_def: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "wf_prog wf_md P \<equiv> wf_syscls P \<and> (\<forall>c\<in>set P. wf_cdecl wf_md P c) \<and> distinct_fst P"
10: shows "e * (w \<sqinter> - prim_EP w v e)\<^sup>+ * e = bot"
ring_hom_finsum: fixes h :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c"   and S :: "('c, 'd) ring_scheme"   and I :: "'e set"   and F :: "'e \<Rightarrow> nat \<Rightarrow> 'a" assumes "h \<in> ring_hom P S"   and "ring S"   and "finite I"   and "F \<in> I \<rightarrow> carrier P" shows "h (finsum P F I) = finsum S (h \<circ> F) I"
not_None_eq: fixes x :: "'a option" shows "(x \<noteq> \<bottom>) = (\<exists>y. x = \<lfloor>y\<rfloor>)"
select_correct_factor_def: fixes init :: "'a"   and polys :: "(int poly \<times> root_info) list" shows "local.select_correct_factor init polys = (case bnd_get init of (l, r) \<Rightarrow> local.select_correct_factor_main init polys [] l r 0)"
characteristic_weak_formula_def: fixes P :: "'state" shows "characteristic_weak_formula P \<equiv> Conj (Abs_bset {distinguishing_weak_formula P Q |Q. \<not> P \<equiv>\<cdot> Q})"
decomposeLS_def: fixes s :: "('e, 'f, 'g, 'h) com list \<times> 'f option \<times> 'h" shows "decomposeLS s = (case cPGM s of [] \<Rightarrow> {} | c # x \<Rightarrow> decompose_com c)"
sst_vars_proj_subset_2: fixes n :: "'c"   and A :: "('c strand_label \<times> ('b, 'a) stateful_strand_step) list" shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj n A) \<subseteq> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
stab_typed_some_imp_cl_typed: fixes s :: "s"   and i :: "nat"   and c :: "nat"   and cl :: "cl"   and \<S> :: "s_context" assumes "stab s i c = Some cl"   and "store_typing s \<S>"   and "i < length (inst s)" shows "\<exists>tf. cl_typing \<S> cl tf"
one_F: fixes g :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "zero + aggregation.sum_0 g A = aggregation.sum_0 g A"
eventually_False: fixes F :: "'b filter" shows "(\<forall>\<^sub>F x in F. False) = (F = bot)"
thetaSeqWT_WretrT: shows "thetaSeqWT \<subseteq> WretrT (thetaSeqWT \<union> WbisT)"
FALSE_LabelFunSem: fixes ST :: "('d, 'e, 'f) status" shows "FALSE \<notin> the (LabelFunSem (HA ST) ST)"
\<omega>_sum_take_drop: fixes \<alpha>s :: "V list"   and k :: "nat" shows "\<omega>_sum \<alpha>s = \<omega>_sum (take k \<alpha>s) + \<omega>_sum (drop k \<alpha>s)"
opp_half_apartment_def: shows "folding_f.Y \<equiv> {x \<in> X. \<exists>C\<in>folding_g.\<C> - f \<turnstile> folding_g.\<C>. order.greater_eq C x}"
M_wf: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set M)"
uminus: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "\<Theta>[F](\<lambda>x. - g x) = \<Theta>[F](g)"
eval_no_abrupt_lemma: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and w :: "vals"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w, s')" shows "normal s' \<longrightarrow> normal s"
arith_simps_58: fixes m :: "num"   and n :: "num" shows "numeral m * numeral n = numeral (m * n)"
tab_in_vhom': shows "\<guillemotleft>(\<phi> \<star> f) \<cdot> \<rho> : g \<Rightarrow> r' \<star> f\<guillemotright>"
usubstappf_pred: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and p :: "char"   and r :: "fml"   and U :: "variable set"   and \<theta> :: "trm"   and \<sigma>\<theta> :: "trm" assumes "SPreds \<sigma> p = Afml r"   and "FVF r \<inter> U = {}"   and "usubstappt \<sigma> U \<theta> = Aterm \<sigma>\<theta>" shows "usubstappf \<sigma> U (Pred p \<theta>) = usubsta (dotsubstt \<sigma>\<theta>) r"
assms: shows "A \<inter>\<^sub>\<circ> B = []\<^sub>\<circ>"
feq_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "feq a b = (fcompare a b = Eq)"
B: assumes "(\<lambda>x. g x powr q) \<in> O[F](\<lambda>x. g x powr p)" shows "(\<lambda>x. g x powr p) \<notin> o[F](\<lambda>x. g x powr q)"
cases: fixes P :: "'a prog"   and a1 :: "String.literal"   and a2 :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "P \<turnstile> a1 has_fields a2" obtains C :: "String.literal" and D :: "String.literal" and fs :: "fdecl list" and ms :: "'a option mdecl list" and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" and FDTs' :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" where "a1 = C" and "a2 = FDTs'" and "class P C = \<lfloor>(D__, fs__, ms__)\<rfloor>" and "C \<noteq> Object" and "P \<turnstile> D__ has_fields FDTs__" and "FDTs' = map (\<lambda>(F, Tm). ((F, C), Tm)) fs__ @ FDTs__"   | D :: "String.literal" and fs :: "fdecl list" and ms :: "'a option mdecl list" and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" where "a1 = Object" and "a2 = FDTs" and "class P Object = \<lfloor>(D, fs, ms)\<rfloor>" and "FDTs = map (\<lambda>(F, T). ((F, Object), T)) fs"
summable_sums_iff: fixes f :: "nat \<Rightarrow> 'c" shows "summable f = f sums suminf f"
hom_into_subgroup: fixes h :: "'a \<Rightarrow> 'b"   and G :: "('a, 'c) monoid_scheme"   and G' :: "('b, 'd) monoid_scheme"   and H :: "'b set" assumes "h \<in> hom G G'"   and "h ` carrier G \<subseteq> H" shows "h \<in> hom G (subgroup_generated G' H)"
degree_prod_eq: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c poly" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<noteq> 0" shows "degree (prod f I) = (\<Sum>i\<in>I. degree (f i))"
rpr_le_trans: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set"   and z :: "'c"   and A :: "'c set" assumes "x \<^bsub>r\<^esub>\<preceq> y"   and "y \<^bsub>r\<^esub>\<preceq> z"   and "rpr A r" shows "x \<^bsub>r\<^esub>\<preceq> z"
word_less_imp_diff_less: fixes k :: "'b word"   and n :: "'b word"   and m :: "'b word" assumes "k \<le> n"   and "n < m" shows "n - k < m"
split_sels_2: fixes P :: "'b \<Rightarrow> bool"   and f :: "unit + 'c \<times> 'c list_R1 \<Rightarrow> 'b"   and list_R1 :: "'c list_R1" shows "P (case list_R1 of list_R1 x \<Rightarrow> f x) = (\<not> (list_R1 = list_R1 (unR list_R1) \<and> \<not> P (f (unR list_R1))))"
simps_1: fixes n :: "nat" shows "T (VNat n) = TNat n"
unreach_l_x: shows "\<not> s\<turnstile> l reachable_from x"
homogeneous_setD_hom_components: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and a :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and b :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "homogeneous_set A"   and "a \<in> A"   and "b \<in> hom_components a" shows "b \<in> A"
simps_2: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool"   and arOf :: "'b \<Rightarrow> 'a list"   and resOf :: "'b \<Rightarrow> 'a"   and f :: "'b"   and Tl :: "'b trm list" assumes "Signature TYPE('a) wtFsym wtPsym" shows "Signature.wt wtFsym arOf resOf (Fn f Tl) = (wtFsym f \<and> list_all (Signature.wt wtFsym arOf resOf) Tl \<and> arOf f = map (Signature.tpOf resOf) Tl)"
simps: fixes w1 :: "32 Word.word"   and w2 :: "32 Word.word" shows "pl w1 w2 = (if w1 = NEG_INF then NEG_INF else if w2 = NEG_INF then NEG_INF else if w1 = POS_INF then if w2 = POS_INF then POS_INF else let sum = SCAST(32 \<rightarrow> 64) w2 + SCAST(32 \<rightarrow> 64) POS_INF in if SCAST(32 \<rightarrow> 64) POS_INF \<le>s sum then POS_INF else SCAST(64 \<rightarrow> 32) sum else if w2 = POS_INF then let sum = SCAST(32 \<rightarrow> 64) w1 + SCAST(32 \<rightarrow> 64) POS_INF in if SCAST(32 \<rightarrow> 64) POS_INF \<le>s sum then POS_INF else SCAST(64 \<rightarrow> 32) sum else let sum = SCAST(32 \<rightarrow> 64) w1 + SCAST(32 \<rightarrow> 64) w2 in if SCAST(32 \<rightarrow> 64) POS_INF \<le>s sum then POS_INF else if sum \<le>s SCAST(32 \<rightarrow> 64) NEG_INF then NEG_INF else SCAST(64 \<rightarrow> 32) sum)"
assms_5: fixes exp :: "exp"   and var :: "Vars.var" assumes "P exp" shows "P (Lam [var]. exp)"
linord_of_listP_domain: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "linord_of_listP x y xs" shows "x \<in> set xs \<and> y \<in> set xs"
sgn_int_pow_if: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and p :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "x \<noteq> zero" shows "power.power one times (sgn x) p = (if even p then one else sgn x)"
sel: fixes x2 :: "'d" shows "the (\<lfloor>x2\<rfloor>) = x2"
signed_take_bit_of_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n zero = zero"
IH: shows "(induced_map A ^^ n) x = (T ^^ (\<Sum>i<n. return_time_function A ((induced_map A ^^ i) x))) x"
step: fixes X :: "'a fset"   and Y :: "'b fset"   and x :: "'a"   and y :: "'b" assumes "rel_fset A X Y"   and "P X Y"   and "A x y"   and "x |\<notin>| X \<or> y |\<notin>| Y" shows "P (finsert x X) (finsert y Y)"
digit_encode_decode: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" shows "digit_encode ds (digit_decode ds is) = is"
b_rel_subtyping_merge_tL: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and Am :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P' :: "'a \<Rightarrow> bool" assumes "hn_ctxt A x y \<or>\<^sub>A hn_ctxt A' x y \<Longrightarrow>\<^sub>t hn_ctxt Am x y" shows "hn_ctxt A x y \<or>\<^sub>A hn_ctxt (b_assn A' P') x y \<Longrightarrow>\<^sub>t hn_ctxt Am x y"
abs_eq: fixes x :: "j \<Rightarrow> i \<Rightarrow> bool" shows "Semantics.d\<^sub>0 (make\<o> x) = Some x"
length_list_of_eucl: fixes x :: "'b" shows "length (list_of_eucl x) = DIM('b)"
S: shows "S \<in> FGModuleHomSet G (\<cdot>) V (\<star>) W"
maxim_singleton: fixes x :: "'a" assumes "x \<in> Field r" shows "r.maxim {x} = x"
written_read_dd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and v :: "'var"   and k :: "nat"   and j :: "nat" assumes "is_path \<pi>"   and "v \<in> local.reads (\<pi> k)"   and "v \<in> local.writes (\<pi> j)"   and "j < k" obtains l :: "nat" where "k dd\<^bsup>\<pi>,v\<^esup>\<rightarrow> l"
least_prefixpoint_fixpoint: fixes f :: "'a \<Rightarrow> 'a" assumes "has_least_prefixpoint f"   and "isotone f" shows "is_least_fixpoint f (p\<mu> f)"
inner_prod_with_times_mat: fixes u :: "complex vec"   and v :: "complex vec" assumes "dim_vec u = dim_vec v" shows "\<langle>u|v\<rangle> = (\<langle>|u\<rangle>| * |v\<rangle>) $$ (0, 0)"
assms: shows "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"
min_dist_succ: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and v :: "nat"   and v' :: "nat"   and v'' :: "nat" assumes "Graph.connected c v v'"   and "(v', v'') \<in> Graph.E c" shows "Graph.min_dist c v v'' \<le> Suc (Graph.min_dist c v v')"
hyps_1: shows "prefix \<subseteq> ccNeighbors x G"
cyclic_group_commute: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier \<G>"   and "b \<in> carrier \<G>" shows "a \<otimes> b = b \<otimes> a"
assms_1: shows "a \<noteq> []\<^sub>\<circ>"
count_const_lt_components_def: fixes hs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> 'e) list" shows "count_const_lt_components hs = length (remdups (filter (\<lambda>x. x \<noteq> None) (map (const_lt_component \<circ> fst) hs)))"
suffix_length_le: fixes xs :: "'c list"   and ys :: "'c list" assumes "suffix xs ys" shows "length xs \<le> length ys"
pmf_mix_deeper: fixes a :: "real"   and p :: "'b pmf"   and q :: "'b pmf"   and x :: "'b" assumes "a \<in> {0..1}" shows "pmf (mix_pmf a p q) x = a * pmf p x + pmf q x - a * pmf q x"
it_paths_intros_1: fixes t :: "'a rose_tree" shows "[] \<in> it_paths t"
open_UN: fixes "open" :: "'a set \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'a set" assumes "class.topological_space open"   and "\<forall>x\<in>A. open (B x)" shows "open (\<Union> (B ` A))"
ipresCons_imp_ipresSwapAll: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" assumes "ipresCons h hA MOD"   and "igSwapCls MOD"   and "igConsIPresIGWls MOD" shows "ipresSwapAll h hA MOD"
measurable_on_imp_borel_measurable_lebesgue: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f measurable_on S"   and "S \<in> sets lebesgue" shows "f \<in> borel_measurable (lebesgue_on S)"
nonOverlapping: fixes R :: "('c \<times> 'd set) set" assumes "R \<in> partitionValuedUniverse" shows "is_non_overlapping (Range R)"
a: shows "emeasure M WA = emeasure M (induced_map A --` WA)"
measurable_N: fixes M' :: "'c measure" shows "N \<rightarrow>\<^sub>M M' = M \<rightarrow>\<^sub>M M'"
induct: fixes P :: "('c, 'd) rose_tree\<^sub>m \<Rightarrow> bool"   and rose_tree\<^sub>m :: "('c, 'd) rose_tree\<^sub>m" assumes "\<And>xa. (\<And>xaa xaaa xaaaa. \<lbrakk>xaa \<in> set1_blindable\<^sub>m xa; xaaa \<in> snds xaa; xaaaa \<in> set xaaa\<rbrakk> \<Longrightarrow> P xaaaa) \<Longrightarrow> P (Tree\<^sub>m xa)" shows "P rose_tree\<^sub>m"
preciseD': fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and a :: "'a"   and p :: "'b"   and F :: "assn"   and h :: "heap \<times> nat set"   and a' :: "'a"   and F' :: "assn" assumes "precise R"   and "h \<Turnstile> R a p * F"   and "h \<Turnstile> R a' p * F'" shows "a = a'"
assms_4: fixes x :: "constraint" assumes "x \<in> set cs" shows "\<nexists>a b. GTPP a b = x"
assms: shows "x \<in> \<real>\<^sub>\<le>\<^sub>0"
simps_5: fixes bs :: "'b list"   and a :: "'b poly"   and b :: "'b poly" shows "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
\<Psi>o_\<Phi>o: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c"   and a :: "'b" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"   and "partial_magma.ide S a" shows "two_set_categories_bij_betw_Univ.\<Phi>o S' S \<AA> (inv_into (Collect (category.terminal S)) \<phi>) (two_set_categories_bij_betw_Univ.\<Phi>o S S' \<AA> \<phi> a) = a"
finsum_reindex: fixes f :: "'g \<Rightarrow> 'c"   and h :: "'h \<Rightarrow> 'g"   and A :: "'h set" assumes "f \<in> h ` A \<rightarrow> carrier V"   and "inj_on h A" shows "finsum V f (h ` A) = (\<Oplus>\<^bsub>V\<^esub>x\<in>A. f (h x))"
exec_append: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and s :: "'e"   and rs :: "('f \<times> 'g) list"   and rs' :: "('f \<times> 'g) list" shows "pre_RDR.exec \<delta> (pre_RDR.exec \<delta> s rs) rs' = pre_RDR.exec \<delta> s (append rs' rs)"
\<pi>_regular: fixes k :: "nat"   and T :: "(symbol \<times> character list) set" shows "regular (\<pi> k T)"
da_elim_cases_27: fixes Env :: "env"   and B :: "lname set"   and T :: "ty"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Cast T e\<rangle>\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
that: fixes c :: "real" assumes "\<And>s. 0 < Re s \<Longrightarrow> cmod (stirling_integral n s) \<le> c / Re s ^ n" shows "thesis"
chain_subset_def: fixes C :: "'c set set" shows "chain\<^sub>\<subseteq> C = (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
length_remdups_leq: fixes xs :: "'a list" shows "length (remdups xs) \<le> length xs"
list_all2_old: shows "list_all2 (\<lambda>x y. triple_eq_pair x y (\<lambda>tp'. filter_a1_map pos tp' a1_map') (\<lambda>ts' tp'. filter_a2_map I ts' tp' a2_map')) (map (\<lambda>(t, a1, a2). (t, if pos then join a1 True rel1 else a1 \<union> rel1, if left I \<le> nt - t then a2 \<union> join rel2 pos a1 else a2)) (drop (length done) auxlist)) (zip (linearize tss) [tp - len..<tp])"
sorted_sorted_wrt: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq = sorted_wrt less_eq"
that_2: assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t"   and "\<not> no_matching_Goto \<gamma> p rs\<^sub>1" shows "thesis"
lookup_eq_SomeD: fixes M :: "('b, 'a) mapping"   and x :: "'b"   and y :: "'a" assumes "Mapping.lookup M x = Some y" shows "x \<in> Mapping.keys M"
get_create: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c" assumes "bij_lens x" shows "create\<^bsub>x\<^esub> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
simps_2: fixes f :: "'a"   and g :: "'a" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> \<^bold>\<langle>g\<^bold>\<rangle> = \<^bold>\<langle>f \<cdot> g\<^bold>\<rangle>"
intro: fixes f :: "nat \<Rightarrow> complex mat"   and dim :: "nat" assumes "\<And>n. f n \<in> carrier_mat dim dim"   and "\<And>n. partial_density_operator (f n)"   and "\<And>n. f n \<le>\<^sub>L f (Suc n)" shows "matrix_seq dim f"
finite_Sum_any_not_eq_zero_weakenI: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite {a. \<exists>b. f a b \<noteq> (0::'d)}" shows "finite {a. Sum_any (f a) \<noteq> (0::'d)}"
scale_minus_right: fixes a :: "'a"   and x :: "'a" shows "a * - x = - (a * x)"
preserves_reflects_arr: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "functor A B F" shows "partial_magma.arr B (F f) = partial_magma.arr A f"
signof_def: fixes p :: "nat \<Rightarrow> nat" shows "signof p = (if sign p = 1 then 1::'d else - (1::'d))"
extendEmpty: fixes C :: "('a, 'b) sequent" shows "extend ( \<Empt> \<Rightarrow>* \<Empt>) C = C"
flatten_times: fixes p :: "'b \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'c"   and q :: "'b \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'c" shows "flatten (p * q) = flatten p * flatten q"
in_idealE_homomorphism_finite: fixes B :: "'a set"   and p :: "'a" assumes "finite B"   and "B \<subseteq> range h"   and "p \<in> range h"   and "p \<in> ideal B" obtains q :: "'a \<Rightarrow> 'a" where "\<And>b. q b \<in> range h" and "p = (\<Sum>b\<in>B. q b * b)"
case_2: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and x2 :: "'g" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
coprime_iff_invertible'_nat: fixes m :: "nat"   and a :: "nat" assumes "0 < m" shows "coprime a m = (\<exists>x\<ge>0. x < m \<and> [a * x = Suc 0] (mod m))"
ergodic_qmpt_axioms_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "ergodic_qmpt_axioms M T \<equiv> \<forall>A. A \<in> sets (qmpt.Invariants M T) \<longrightarrow> A \<in> null_sets M \<or> space M - A \<in> null_sets M"
xle: shows "x < (1 + real_of_int \<lfloor>x / pi\<rfloor>) * pi"
id: shows "(\<Prod>a\<leftarrow>as. [:a, 1::'a:]) = [:b, 1::'a:] * p"
quotientE: fixes X :: "'b set"   and A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "X \<in> A // r" obtains x :: "'b" where "X = r `` {x}" and "x \<in> A"
distinct_7: fixes x2 :: "qtname"   and x3 :: "qtname" shows "IfaceT x2 \<noteq> ClassT x3"
abs_summable_on_nat_iff': fixes f :: "nat \<Rightarrow> 'b" shows "(f abs_summable_on UNIV) = summable (\<lambda>n. norm (f n))"
max_norm_def: fixes A :: "((real, 'b) vec, 'a) vec" shows "\<parallel>A\<parallel>\<^sub>m\<^sub>a\<^sub>x = Max {\<bar>A $ i $ j\<bar> |i j. i \<in> UNIV \<and> j \<in> UNIV}"
asymptotics: shows "(\<lambda>x. sum_upto (\<lambda>n. mangoldt n / real n) x - ln x) \<in> O(\<lambda>_. 1)"
eventually_nhds_in_open: fixes "open" :: "'c set \<Rightarrow> bool"   and s :: "'c set"   and x :: "'c" assumes "class.topological_space open"   and "open s"   and "x \<in> s" shows "\<forall>\<^sub>F y in topological_space.nhds open x. y \<in> s"