hcju/isabelleps huggingface dataset:
lmeasure_integral: fixes S :: "'a set" assumes "S \<in> lmeasurable" shows "Sigma_Algebra.measure lebesgue S = integral S (\<lambda>x. 1)"
elims: fixes x :: "'h + 'i + 'j"   and y :: "('h + 'i) + 'j" assumes "lsumr x = y" obtains a :: "'h" where "x = Inl a" and "y = Inl (Inl a)"   | b :: "'i" where "x = Inr (Inl b)" and "y = Inl (Inr b)"   | c :: "'j" where "x = Inr (Inr c)" and "y = Inr c"
is_interval_uminusI: fixes X :: "'b set" assumes "is_interval X" shows "is_interval (uminus ` X)"
normal: fixes i :: "nat" assumes "i + 1 < length \<GG>" shows "\<GG> ! i \<lhd> G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>"
intro: fixes x :: "'e \<Longrightarrow> 'd" assumes "\<And>\<sigma> \<rho>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<rho>) = \<rho>" shows "bij_lens_axioms x"
p_def: fixes x :: "nat"   and z :: "nat"   and u :: "nat" shows "p x z u = p_s x (c_fst u) (c_fst (c_snd u)) + p_r (c_fst u) z (c_snd (c_snd u))"
measurable_case_nat: fixes g :: "'a \<Rightarrow> 'b"   and R :: "'a measure"   and N :: "'b measure"   and h :: "'a \<Rightarrow> nat \<Rightarrow> 'b"   and V :: "nat set"   and M :: "nat \<Rightarrow> 'b measure" assumes "g \<in> R \<rightarrow>\<^sub>M N"   and "h \<in> R \<rightarrow>\<^sub>M Pi\<^sub>M V M" shows "(\<lambda>x. case_nat (g x) (h x)) \<in> R \<rightarrow>\<^sub>M Pi\<^sub>M (shift_var_set V) (case_nat N M)"
linorder_on_def: fixes A :: "'a set"   and R :: "('a \<times> 'a) set" shows "linorder_on A R = (refl_on A R \<and> antisym R \<and> trans R \<and> total_on A R)"
ferrack2: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list" assumes "qfree p" shows "qfree (ferrack2 p) \<and> Ifm vs bs (ferrack2 p) = Ifm vs bs (E p)"
subset: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "H \<le> G" shows "H \<subseteq> carrier G"
simps_2: fixes \<mu> :: "'c" shows "\<^bold>\<lfloor>\<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^bold>\<rfloor> = \<^bold>\<langle>\<mu>\<^bold>\<rangle>"
myss3_def: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a"   and v :: "char list \<Longrightarrow> 'a" shows "myss3 x y a z v \<equiv> myss2 x y a z \<and> myss3_axioms x y a z v"
Arden_preserves_ardenable: fixes rhs :: "'b trm set"   and X :: "'b list set" assumes "ardenable rhs" shows "ardenable (Arden X rhs)"
Sup_lexord_def: fixes k :: "'b \<Rightarrow> 'c"   and c :: "'b set \<Rightarrow> 'b"   and s :: "'b set \<Rightarrow> 'b"   and A :: "'b set" shows "Sup_lexord k c s A = (let U = Sup (k ` A) in if \<exists>a\<in>A. k a = U then c {a \<in> A. k a = U} else s A)"
Z_good_alt: fixes g :: "nat \<Rightarrow> nat" assumes "g \<in> local.F_strict" shows "local.Z_good g = {x \<in> local.F_inj. strict_from_inj n x = g} \<times> {\<pi>. \<pi> permutes {0..<n}}"
segment_1: fixes a :: "'b"   and b :: "'b" shows "{a<--<b} \<equiv> {a--b} - {a, b}"
l_distr: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> carrier S"   and "y \<in> carrier S"   and "z \<in> carrier S" shows "(x \<oplus>\<^bsub>S\<^esub> y) \<otimes>\<^bsub>S\<^esub> z = x \<otimes>\<^bsub>S\<^esub> z \<oplus>\<^bsub>S\<^esub> y \<otimes>\<^bsub>S\<^esub> z"
qSwapAll_ident: fixes zs :: "'c"   and x :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and A :: "('a, 'b, 'c, 'd, 'e) qAbs" shows "X #[[x \<and> x]]_zs = X \<and> A $[[x \<and> x]]_zs = A"
perm_swap_2: fixes a :: "NominalSequents.var"   and b :: "NominalSequents.var"   and x :: "'a" shows "[(a, b)] \<bullet> [(b, a)] \<bullet> x = x"
dom_delete_aux: fixes ps :: "('c \<times> 'd) list"   and k :: "'c" assumes "distinct (map fst ps)" shows "fst ` set (delete_aux k ps) = fst ` set ps - {k}"
k0: fixes F :: "nat \<Rightarrow> 'a poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "k \<noteq> 0"
allE: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x. P x" obtains "P x"
exec_d_def: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list" shows "exec_d P t \<sigma> \<equiv> if check P \<sigma> then Normal (exec P t \<sigma>) else TypeError"
aux: fixes f :: "'b \<Rightarrow>\<^sub>0 'c" shows "(lookup f = (\<lambda>_. 0::'c)) = (f = 0)"
simps_14: fixes v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "'a" shows "(IF v THEN va ELSE vb {vc} \<le> SKIP {vd}) = False"
simps_3: fixes x :: "'d"   and t1 :: "'d ifex"   and t2 :: "'d ifex"   and env :: "('d, bool) mapping" shows "taut_test_rec (IF x t1 t2) env = (case Mapping.lookup env x of None \<Rightarrow> taut_test_rec t1 (Mapping.update x True env) \<and> taut_test_rec t2 (Mapping.update x False env) | Some b \<Rightarrow> taut_test_rec (if b then t1 else t2) env)"
numeral_code_1: shows "Numeral1 = (1::'a)"
assms_1: shows "continuous_on {real_of_int a..} f"
simps_3: fixes r :: "'a rexp"   and v :: "'a rexp"   and va :: "'a rexp list" shows "verund (r # v # va) = Plus r (verund (v # va))"
path_connected_space_topspace_empty: fixes X :: "'a topology" assumes "topspace X = {}" shows "path_connected_space X"
dirichlet_prod_assoc: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> 'a" shows "dirichlet_prod (dirichlet_prod f g) h = dirichlet_prod f (dirichlet_prod g h)"
AI_Some_measure: fixes m :: "const \<Rightarrow> nat"   and c :: "com" assumes "(strict {(x, y). x \<sqsubseteq> y})\<inverse> \<subseteq> measure m"   and "\<forall>x y. x \<sqsubseteq> y \<and> y \<sqsubseteq> x \<longrightarrow> m x = m y" shows "\<exists>c'. AI_const c = Some c'"
assms: shows "rt \<noteq> update rt dip (osn, kno, val, hops, nhip)"
bij_vimage_eq_inv_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set" assumes "bij f" shows "f -` A = inv f ` A"
comp_null_2: fixes f :: "'a \<times> 'a" shows "VV.comp f VV.null = VV.null"
assms_1: shows "x \<in> (InString \<circ> I) ` set (substr w l u)"
cases: fixes x :: "nat \<times> nat" obtains p :: "nat" and n :: "nat" where "x = (p, n)"
IN: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "weight \<Gamma> x \<le> d_IN f x" shows "x \<in> SAT \<Gamma> f"
bind_return_distr: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "M \<bind> return N \<circ> f = distr M N f"
assms_1: fixes f :: "'a" assumes "f \<in> space\<^sub>N N1" shows "f \<in> space\<^sub>N N2"
A_3: shows "DBM_reset (FW M n) n (v c) d M'"
isProperPrefixPrefixToLevel: fixes level :: "nat"   and t :: "('b \<times> bool) list" assumes "level < currentLevel t" shows "\<exists>s. prefixToLevel level t @ s = t \<and> s \<noteq> [] \<and> isDecision (hd s)"
assms_1: shows "\<forall>t\<in>S. \<exists>c d. c < d \<and> {c..d} = t"
vector_smult_lid: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "(1::'a) *s x = x"
assms_1: shows "outs_\<I> \<I> \<turnstile>\<^sub>R res1 \<approx> res2"
emb_vars: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s" shows "vars s \<subseteq> vars t"
T_BIT_competitive: fixes n :: "nat" assumes "n \<le> length qs"   and "init \<noteq> []"   and "\<forall>i<n. qs ! i \<in> set init" shows "T_BIT n \<le> (7 / 4 - 3 / real (4 * length init)) * real_of_int (T_A' n)"
smap2_szip: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and s1 :: "'e stream"   and s2 :: "'f stream" shows "smap2 f s1 s2 = smap (\<lambda>(x, y). f x y) (szip s1 s2)"
integrable_const: fixes M :: "'c pmf"   and a :: "'d" shows "integrable (measure_pmf M) (\<lambda>x. a)"
assms: shows "u \<preceq>\<^sub>F\<^sub>I v"
Throw: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and A :: "'e set"   and Q :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> A Throw Q,A"
distinct_887: fixes x241 :: "t"   and x242 :: "unop_i"   and x311 :: "t"   and x312 :: "cvtop"   and x313 :: "t"   and x314 :: "sx option" shows "Unop_i x241 x242 \<noteq> Cvtop x311 x312 x313 x314"
obtain_qFresh_card_of: fixes V :: "'var set"   and XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" assumes "|V| <o |UNIV|"   and "|XS| <o |UNIV|"   and "Ball XS qGood" shows "\<exists>x. x \<notin> V \<and> (\<forall>X\<in>XS. qAFresh xs x X \<and> qFresh xs x X)"
is_duplicateEdge_def: fixes g :: "graph"   and f :: "face"   and a :: "nat"   and b :: "nat" shows "is_duplicateEdge g f a b \<equiv> (a, b) \<in> \<E> g \<and> (a, b) \<notin> \<E> f \<and> (b, a) \<notin> \<E> f \<or> (b, a) \<in> \<E> g \<and> (b, a) \<notin> \<E> f \<and> (a, b) \<notin> \<E> f"
equivalence_and_prime_implicates: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set" assumes "equivalent S1 S2" shows "prime_implicates S1 = prime_implicates S2"
To_def: fixes x3 :: "trm"   and x1 :: "name"   and x2 :: "trm" shows "x3 to x1 in x2 \<equiv> Abs_trm (To_Rep (Rep_trm x3) ([x1].Rep_trm x2))"
carrier_polynomial_shell: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set"   and p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "subring K L"   and "p \<in> carrier (K [X]\<^bsub>L\<^esub>)" shows "p \<in> carrier (poly_ring L)"
inf_sup_ord_2: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y \<le> y"
ir_runs2sigs_upd_init_some2_raw: fixes runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and Ta :: "nat"   and s :: "(key \<times> nat \<times> nat) signal" assumes "runz Ra = Some (Init, [A, B], [aKey Kab, aNum Ts, aNum Ta])"   and "finite (dom runz)" shows "ir_runs2sigs (runz(Ra \<mapsto> (Init, [A, B], [aKey Kab, aNum Ts, aNum Ta, END]))) s = ((ir_runs2sigs runz)(Commit [A, B] (Kab, Ts, Ta) := Suc (card (ir_commit runz A B Kab Ts Ta)))) s"
unat_sub_if': fixes x :: "'a word"   and y :: "'a word" shows "unat (x - y) = (if unat y \<le> unat x then unat x - unat y else unat x + 2 ^ LENGTH('a) - unat y)"
rep_eq: fixes x :: "'c monom" shows "monom_degree x = monom_list_degree (Rep_monom x)"
conv_radius_mult_power_right: fixes c :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "conv_radius (\<lambda>n. f n * c ^ n) = conv_radius f / ereal (norm c)"
lower_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'t" shows "lower p t = except p {s. t \<preceq>\<^sub>t s}"
inject_7: fixes x91 :: "'b ltln"   and x92 :: "'b ltln"   and y91 :: "'b ltln"   and y92 :: "'b ltln" shows "(x91 R\<^sub>n x92 = y91 R\<^sub>n y92) = (x91 = y91 \<and> x92 = y92)"
assms_1: shows "Bseq (\<lambda>n. fds_nth f n * nat_power n s)"
exec_Normal_elim_cases_13: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Throw,Normal s\<rangle> \<Rightarrow> t" obtains "t = Abrupt s"
FF_def: shows "FF \<equiv> \<lambda>\<mu>\<nu>. if C'.VV.arr \<mu>\<nu> then (local.map (fst \<mu>\<nu>), local.map (snd \<mu>\<nu>)) else D.VV.null"
abs_eq: fixes xa :: "(char list \<times> 'a) list"   and x :: "(char list \<times> 'a) list" shows "(St xa \<le> St x) = less_eq_st_rep xa x"
run_one_step_basic_set_global_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x18 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Set_global x18) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
hyps: fixes xs :: "('a \<times> 'a) list" shows "progressing (xs @ ys') = (progressing xs \<and> progressing ys' \<and> (xs \<noteq> [] \<and> ys' \<noteq> [] \<longrightarrow> fst (last xs) \<noteq> snd (hd ys') \<or> snd (last xs) \<noteq> fst (hd ys')))"
filternew_flows_state_alt: fixes \<T> :: "'a stateful_policy" shows "filternew_flows_state \<T> = flows_state \<T> - backflows (flows_fix \<T>)"
Abs_Int_inv_mono_axioms: shows "Abs_Int_inv_mono num_ivl (+) in_ivl inv_plus_ivl inv_less_ivl \<gamma>_ivl"
uminus_add_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "- (A + B) = - B + - A"
closedin_closure_of: fixes X :: "'c topology"   and S :: "'c set" shows "closedin X (X closure_of S)"
rep_eq: fixes x :: "'b fset"   and xa :: "'c fset" shows "fset (x |\<times>| xa) = fset x \<times> fset xa"
GSMP_subterm: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and t' :: "('fun, 'var) Term.term" assumes "t \<in> GSMP M"   and "t' \<sqsubseteq> t" shows "t' \<in> GSMP M"
assms_5: shows "h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r owner_document"
ceiling_eq: fixes n :: "int"   and x :: "'a" assumes "of_int n < x"   and "x \<le> of_int n + (1::'a)" shows "\<lceil>x\<rceil> = n + 1"
pw_Inf_inres: fixes r :: "'a" shows "inres (Inf (f ` C)) r = (\<forall>M\<in>f ` C. inres M r)"
extf_snoc: fixes f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and xs :: "('s, 'v) tm list"   and x :: "('s, 'v) tm" shows "extf f gt (xs @ [x]) xs"
simps_1: fixes n :: "n" shows "add2 Z n = n"
f1: shows "Deliver (i, Append i e) \<in> set pre \<or> (\<exists>mo2. Deliver (i, Store e mo2 i) \<in> set pre)"
lin_data_prev': shows "linearize data_prev' = linearize data_prev @ [(nt, X)]"
CMC_def: shows "CMC \<equiv> CM \<and> CMC_axioms"
img_fact: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f" shows "S (set_category_given_img.mkArr S (set_category.some_img S \<AA>) \<AA> (set_category_data.set S (set_category.some_img S \<AA>) (set_category.img S \<AA> f)) (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.cod S f)) (\<lambda>x. x)) (set_category.corestr S \<AA> f) = f"
factorize_rat_poly_1: fixes alg :: "int_poly_factorization_algorithm"   and f :: "rat poly"   and c :: "rat"   and fs :: "(rat poly \<times> nat) list" assumes "factorize_rat_poly_generic alg f = (c, fs)" shows "square_free_factorization f (c, fs)"
syzygy_module_closed_times_monomial: fixes s :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and c :: "'a \<Rightarrow>\<^sub>0 'c" assumes "s \<in> pmdl.syzygy_module B" shows "monomial c 0 * s \<in> pmdl.syzygy_module B"
induct: fixes x :: "State"   and P :: "State \<Rightarrow> bool" assumes "applyExplainUIP_dom x"   and "\<And>state. \<lbrakk>getCn state \<noteq> 1 \<Longrightarrow> applyExplainUIP_dom (applyExplain (getCl state) state); getCn state \<noteq> 1 \<Longrightarrow> P (applyExplain (getCl state) state)\<rbrakk> \<Longrightarrow> P state" shows "P x"
max_bag_card_upper_bound_V: assumes "V \<noteq> {}" shows "max_bag_card \<le> card V"
N_1: shows "2 * (cmod z + d) \<le> real N"
assms: shows "t \<in> set_pmf (random_bst A)"
equivp_equiv: fixes A :: "('b \<times> 'b) set" shows "equiv UNIV A = equivp (\<lambda>x y. (x, y) \<in> A)"
is_rat_complex_iff: fixes x :: "complex" shows "(x \<in> \<rat>) = (Re x \<in> \<rat> \<and> Im x = 0)"
rank_invar_cons_down: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_invar (t # bq)" shows "SkewBinomialHeapStruc.rank_invar bq"
sets_T: fixes K :: "'c \<Rightarrow> 'c pmf"   and s :: "'c" shows "sets (MC_syntax.T K s) = sets (stream_space (count_space UNIV))"
is_ub_insert: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and A :: "'b set"   and y :: "'b" assumes "class.po below" shows "po.is_ub below (insert x A) y = (below x y \<and> po.is_ub below A y)"
LtK_parts_impl: fixes X :: "msg"   and K :: "ltkey" assumes "X \<in> valid" shows "LtK K \<notin> parts {X}"
exec_drest_cons_proj_eq_succ: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and s :: "('d, 'e) fmap"   and vs :: "'d set"   and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" assumes "fst a \<subseteq>\<^sub>f s" shows "exec_plan (fmrestrict_set vs s) (action_proj a vs # as) = exec_plan (fmrestrict_set vs (state_succ s a)) as"
len_takeWhile_comp: fixes n1 :: "'b"   and xs :: "'b list"   and n2 :: "'b" assumes "n1 \<in> set xs"   and "n2 \<in> set xs"   and "n1 \<noteq> n2" shows "length (takeWhile ((\<noteq>) n1) xs) \<noteq> length (takeWhile ((\<noteq>) n2) xs)"
j_suffix: shows "suffix j w \<Turnstile>\<^sub>n \<psi>"
open_complex_def: fixes U :: "complex set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
IsCompFun_def: fixes F :: "('d, 'e, 'f) seqauto set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" shows "IsCompFun F G = (dom G = \<Union> (States ` F) \<and> \<Union> (ran G) = F - {Root F G} \<and> RootEx F G \<and> OneAncestor F G \<and> NoCycles F G)"
U_idem_p_prod: shows "Multirelations.U \<parallel> Multirelations.U = Multirelations.U"
uv: fixes u :: "'a"   and v :: "'c" assumes "(u, v) \<in> H ` ball (x, y) r" shows "f (u, snd (G (u, v))) = v"
strict_order_iff: fixes a :: "'a"   and b :: "'a" shows "(a < b) = (a = min a b \<and> a \<noteq> b)"
b': shows "\<forall>r'\<in>R'. leftPrincipal r' (Modal M Ms) R' \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"
inc: shows "{x \<in> space M. upper_asymptotic_density {n. \<forall>l\<in>{1..n}. - delta l * real l < u n x - u (n - l) ((T ^^ l) x)} < 1 - d} \<subseteq> \<Union> (range Bad)"
xo: fixes x :: "real" assumes "xo \<le> x" shows "x < e * exp (Re s * x)"
REC_unfold: fixes B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" assumes "trimono B" shows "REC B = B (REC B)"
dgrad_set_varnum: fixes X :: "'b set" shows "dgrad_set (varnum X) 0 = .[X]"
append_Nil: fixes ys :: "'b list" shows "\<epsilon> \<cdot> ys = ys"
free_return_not_match: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Return], Undecided\<rangle> \<Rightarrow> t" shows "\<not> matches \<gamma> m p"
size_gen_5: fixes x51 :: "expr"   and x52 :: "stmt"   and x53 :: "stmt" shows "size_stmt (Cond x51 x52 x53) = size_stmt x52 + size_stmt x53 + Suc 0"
pos: fixes h :: "'a \<Rightarrow> real" assumes "h \<in> H" shows "\<forall>\<^sub>F x in F. 0 < h x"
resCasesF: fixes x :: "name"   and P :: "pi"   and \<alpha> :: "freeRes"   and xP' :: "pi"   and F :: "pi \<Rightarrow> bool" assumes "<\<nu>x>P \<longmapsto> \<alpha> \<prec> xP'"   and "\<And>P'. \<lbrakk>P \<longmapsto> \<alpha> \<prec> P'; x \<sharp> \<alpha>\<rbrakk> \<Longrightarrow> F (<\<nu>x>P')" shows "F xP'"
frrestrictionE: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<bullet> A" obtains a :: "V" and b :: "V" where "x = [a, b]\<^sub>\<circ>" and "b \<in>\<^sub>\<circ> A" and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
q_def: shows "q \<equiv> \<Prod>{p. prime p \<and> p dvd n \<and> \<not> p dvd a}"
simps_15: fixes v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "char list"   and ve :: "aexp"   and vf :: "'a" shows "(IF v THEN va ELSE vb {vc} \<sqsubseteq> vd ::= ve {vf}) = False"
neq_Nil_conv: fixes xs :: "'a list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
bi_uniqueDl: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and z :: "'a" assumes "bi_unique A"   and "A x y"   and "A z y" shows "x = z"
nat_power_add: fixes n :: "nat"   and a :: "'a"   and b :: "'a" assumes "0 < n" shows "nat_power n (a + b) = nat_power n a * nat_power n b"
exec_def: fixes iofun :: "'c io"   and world :: "\<^url>" shows "exec iofun world = snd (Rep_io iofun world)"
sym_Id_on: fixes A :: "'a set" shows "sym (Id_on A)"
assms_3: shows "\<forall>j. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<longrightarrow> k \<le> j"
dim_eq_full: fixes S :: "'b set" shows "(dim S = dimension) = (span S = UNIV)"
transitionD_2: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "A \<turnstile> l \<longrightarrow>\<^bsup>g,\<mu>,X\<^esup> l'" shows "(X, l') \<in> set_pmf \<mu>"
ord_eq_le_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a = b"   and "b \<le>p c" shows "a \<le>p c"
assms: fixes k :: "'a"   and v :: "'b" assumes "(k, v) \<in> set (fst (list_of_oalistx xs))" shows "f k v = g k v"
differentiableI_vector: fixes f :: "real \<Rightarrow> 'b"   and y :: "'b"   and F :: "real filter" assumes "(f has_vector_derivative y) F" shows "f differentiable F"
CDPlayer_CTRL_Action2_def: shows "CDPlayer_CTRL_Action2 = ({}, CDPlayer_CTRL_Update2)"
mult_cong_l: fixes a :: "'a list"   and a' :: "'a list"   and b :: "'a list" assumes "a \<sim>\<^bsub>poly_ring R\<^esub> a'"   and "a \<in> carrier (poly_ring R)"   and "a' \<in> carrier (poly_ring R)"   and "b \<in> carrier (poly_ring R)" shows "a \<otimes>\<^bsub>poly_ring R\<^esub> b \<sim>\<^bsub>poly_ring R\<^esub> a' \<otimes>\<^bsub>poly_ring R\<^esub> b"
pos_divide_le_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq (divide b c) a = less_eq b (times a c)"
q: shows "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
M'_def: shows "M' \<equiv> \<lambda>i' j'. if i' = i \<and> j' = j then Le r else if i' = j \<and> j' = i then Le (- r) else M i' j'"
healthy: fixes x :: "'b"   and s :: "'a" assumes "x \<in> S s" shows "healthy (wp (prog x))"
semiring_simprules_6: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "semiring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
map_of_entries: fixes t :: "('a, 'c) rbt" assumes "rbt_sorted t" shows "map_of (RBT_Impl.entries t) = rbt_lookup t"
poly_mult_var'_2: fixes p :: "'a list" assumes "set p \<subseteq> carrier R" shows "poly_mult p X = local.normalize (p @ [\<zero>])"
abs_sum_upto_moebius_mu_over_n_le: fixes x :: "real" shows "\<bar>sum_upto (\<lambda>n. moebius_mu n / real n) x\<bar> \<le> 1"
Pyz: shows "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz"
scc_of_is_scc: fixes E :: "('b \<times> 'b) set"   and v :: "'b" shows "is_scc E (scc_of E v)"
bintr_bintr_i: fixes x :: "int"   and n :: "nat"   and y :: "int" assumes "x = take_bit n y" shows "take_bit n x = take_bit n y"
act_assocD: fixes x :: "'a"   and y :: "'a"   and p :: "'b" shows "(D x y \<and> Dla (x \<oplus> y) p) = (Dla y p \<and> Dla x (\<alpha> y p))"
set_update_subsetI: fixes xs :: "'c list"   and A :: "'c set"   and x :: "'c"   and i :: "nat" assumes "set xs \<subseteq> A"   and "x \<in> A" shows "set (xs[i := x]) \<subseteq> A"
countable_V: shows "countable \<^bold>V\<^bsub>\<Psi>\<^esub>"