hcju/isabelleps huggingface dataset:
ideal_generated_singleton_dvd: fixes S :: "'a set"   and d :: "'a"   and x :: "'a" assumes "ideal_generated S = ideal_generated {d}"   and "x \<in> S" shows "d dvd x"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<^sub>A f = n \<and> f \<cdot>\<^sub>A n = n"
uniform_limit: fixes g :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and X :: "'e set"   and l :: "'e \<Rightarrow> 'a"   and F :: "'d filter" assumes "uniform_limit X g l F" shows "uniform_limit X (\<lambda>a b. f (g a b)) (\<lambda>a. f (l a)) F"
rep_eq: fixes x :: "'a pmf" shows "st_vec (st_vec_of_pmf x) = vec_lambda (pmf x)"
create_id_valid: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i1 :: "'a"   and i2 :: "'a"   and e :: "'b" assumes "xs prefix of j"   and "Deliver (i1, Create i2 e) \<in> set xs" shows "i1 = i2"
subset_implies_mult: fixes A :: "'b multiset"   and B :: "'b multiset"   and r :: "('b \<times> 'b) set" assumes "A \<subset># B" shows "(A, B) \<in> mult r"
rr: shows "real_of r = real_of_rat (ma_rat r)"
word_bw_lcs_2: fixes y :: "'b word"   and x :: "'b word"   and z :: "'b word" shows "y OR x OR z = x OR y OR z"
P_def: shows "P \<equiv> \<Sum>x = 0..p. smult (b x) (Bernstein_Poly_01 x p)"
induct: fixes P :: "nat \<times> nat \<Rightarrow> bool"   and a0 :: "nat \<times> nat" assumes "\<And>n a. n \<le> 2 \<Longrightarrow> P (a, n)"   and "\<And>n a. \<lbrakk>2 < n; P (a, nat \<lfloor>real n / 2\<rfloor>)\<rbrakk> \<Longrightarrow> P (a, n)" shows "P a0"
le_log2_of_power: fixes n :: "nat"   and m :: "nat" assumes "2 ^ n \<le> m" shows "real n \<le> Transcendental.log 2 (real m)"
b_assn_pure_conv: fixes R :: "('b \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" shows "b_assn (pure R) P = pure (b_rel R P)"
D_seq: fixes P :: "'a process"   and Q :: "'a process" shows "D (P `;` Q) = div_seq P Q"
take_nths_01_splice: fixes xs :: "'b list"   and ys :: "'b list" assumes "length xs = length ys \<or> length xs = length ys + 1" shows "take_nths 0 (Suc 0) (splice xs ys) = xs \<and> take_nths (Suc 0) (Suc 0) (splice xs ys) = ys"
ctr: fixes z :: "complex" shows "Complex.cnj z = Complex (complex.Re z) (- Im z)"
elims: fixes x :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and y :: "'a \<Rightarrow> ('b, 'a, 'c) val option" assumes "x\<^sub>\<sigma> = y" obtains \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" and e :: "('b, 'a, 'c) expr" where "x = (\<sigma>, \<tau>, e)" and "y = \<sigma>"
teq: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat n n"   and "partial_density_operator \<rho>" shows "trace (A * \<rho>) = trace (B * \<rho>)"
simps_2: fixes e :: "'env"   and x :: "'key \<times> 'val"   and xs :: "('key \<times> 'val) list" shows "add_list e (x # xs) = add (add_list e xs) (fst x) (snd x)"
fst_conv: fixes x1 :: "'a"   and x2 :: "'c" shows "fst (x1, x2) = x1"
successive_glue: fixes P :: "'b \<Rightarrow> bool"   and zs :: "'b list"   and z :: "'b"   and x :: "'b"   and xs :: "'b list" assumes "successive P (zs @ [z])"   and "successive P (x # xs)"   and "\<not> P z \<or> \<not> P x" shows "successive P (zs @ [z] @ x # xs)"
Un_left_absorb: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> (A \<union> B) = A \<union> B"
x_def: fixes n :: "nat" shows "x n = (\<Sum>i\<in>{..<n * 2 ^ n} - {0}. real i / 2 ^ n * Measure.measure M (A n i))"
h8: assumes "0 < Av"   and "Bv \<le> 0" shows "(Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
assms_1: shows "(P, Q \<parallel> !Q) \<in> bangRel Rel"
simps_1: fixes scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "'b" shows "scf_term scf (Var x) = Var x"
Red_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option" shows "Red P E = {((e, s), e', s'). P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>}"
prv_neg_zer_suc: fixes t :: "'trm" assumes "t \<in> atrm" shows "prv (neg (eql zer (suc t)))"
countably_compact_imp_compact_second_countable: fixes U :: "'a set" assumes "countably_compact U" shows "compact U"
polyfun_rootbound_finite: fixes n :: "nat"   and c :: "nat \<Rightarrow> 'b" assumes "\<exists>k\<le>n. c k \<noteq> (0::'b)" shows "finite {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'b)}"
max_list_Max: fixes ls :: "'a list" assumes "ls \<noteq> []" shows "Missing_Lemmas.max_list ls = Max (set ls)"
rep_eq: fixes x :: "'a fset" shows "F x = set.F (fset x)"
HMA_eigen_vector: shows "(HMA_M ===> HMA_V ===> (=)) eigenvector eigen_vector"
suminf_add: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "summable f"   and "summable g" shows "suminf f + suminf g = (\<Sum>n. f n + g n)"
negGorbit_list_nth: fixes i :: "nat"   and gs :: "'a list"   and smult :: "('c, 'a) aezfun \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'d \<Rightarrow> 'b"   and as :: "'d list" assumes "i < length gs" shows "aezfun_scalar_mult.negGorbit_list smult gs T as ! i = map (aezfun_scalar_mult.Gmult smult (- gs ! i) \<circ> T) as"
dom_char: fixes f :: "'a \<times> 'a" shows "B.VxV.dom f = (if B.arr (fst f) \<and> B.arr (snd f) then (B.dom (fst f), B.dom (snd f)) else (B.null, B.null))"
finsum_def: fixes G :: "('i, 'j) ring_scheme" shows "finsum G = finprod (add_monoid G)"
Compl_set_less_Compl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less" shows "ord.set_less less_eq (- A) (- B) = ord.set_less less_eq B A"
is_sum_cases: fixes z :: "V" shows "(\<exists>x. z = ZFC_Cardinals.Inl x \<or> z = ZFC_Cardinals.Inr x) \<or> \<not> is_sum z"
minor: fixes z :: "'a \<Rightarrow> 'b"   and i :: "'a" assumes "z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))"   and "P (product_topology X I)"   and "i \<in> I" shows "P (subtopology (product_topology X I) (\<Pi>\<^sub>E j\<in>I. if j = i then topspace (X i) else {z j}))"
tensor_case: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u" shows "F (\<F>C.mkarr (t \<^bold>\<otimes> u)) = F (\<F>C.mkarr t) \<otimes>\<^sub>D F (\<F>C.mkarr u)"
simps_3: fixes a :: "float interval option"   and f :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option" shows "lift_bin a None f = None"
lift_add: fixes lift :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" assumes "pfd_homomorphism lift" shows "lift (a + b) = lift a + lift b"
singl_def: fixes a :: "'a" shows "singl a = [a]"
components_are_iso: fixes a :: "'c" assumes "C.ide a" shows "C.iso (trg\<^sub>C a)"
bangSC: fixes P :: "pi" shows "!P \<sim>\<^sup>s P \<parallel> !P"
IH: fixes i :: "nat" assumes "\<And>xs. xs \<in> set xss \<Longrightarrow> length xs = d"   and "i < length xss" shows "fixed_length_sublist (concat xss) d i = xss ! i"
vec_vs: fixes n :: "nat" shows "vectorspace class_ring (module_vec TYPE('b) n)"
separating_weakening: fixes G :: "('k, 'l) graph_scheme"   and A :: "'k set"   and B :: "'k set"   and S :: "'k set"   and S' :: "'k set" assumes "separating_gen G A B S"   and "S \<subseteq> S'" shows "separating_gen G A B S'"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>\<epsilon>_inv.map f : C.map a \<rightarrow>\<^sub>C (F \<circ> G) b\<guillemotright>"
inj_on_imp_bij_betw: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "bij_betw f A (f ` A)"
that: fixes \<sigma>' :: "nat \<Rightarrow> 'g"   and \<zeta>' :: "'l net_state" assumes "((\<sigma>, \<zeta>), mR:*cast(m), \<sigma>', \<zeta>') \<in> automaton.trans (opnet onp n)"   and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>' j = \<sigma> j"   and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>', \<zeta>')" shows "thesis"
cong: fixes I :: "('c \<times> 'd) spmf"   and range :: "'c \<Rightarrow> 'b set"   and F :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "ETP_base.R1 I range F \<equiv> ETP_base.R1 I range F"
positive_def: fixes M :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" shows "positive M \<mu> = (\<mu> {} = 0)"
simps_2: fixes n :: "nat"   and ns :: "nat list"   and i :: "nat" shows "offs (n # ns) i = i # offs ns (i + n)"
poly_root_induct_alt: fixes Q :: "'b poly \<Rightarrow> bool"   and p :: "'b poly" assumes "Q 0"   and "\<And>p. (\<And>a. poly p a \<noteq> (0::'b)) \<Longrightarrow> Q p"   and "\<And>a p. Q p \<Longrightarrow> Q ([:- a, 1::'b:] * p)" shows "Q p"
bivar_func_eq: fixes A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<forall>a\<in>A. \<forall>b\<in>B. f a b = g a b" shows "(\<lambda>x\<in>A. restrict (f x) B) = (\<lambda>x\<in>A. restrict (g x) B)"
differing_vars_initially_empty: fixes i :: "nat"   and n :: "nat"   and x :: "'Var"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "i \<sqsubset> n" shows "x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 (zip (replicate n mem\<^sub>1) (replicate n mem\<^sub>2)) i"
cong: fixes K :: "'b \<Rightarrow> 'b measure" shows "discrete_Markov_process.K' K \<equiv> discrete_Markov_process.K' K"
norm_assertion_simps_32: fixes x :: "assn" shows "x \<or>\<^sub>A x = x"
normal_of_segment_def: shows "normal_of_segment = (\<lambda>((a0, a1), b0, b1). (b1 - a1, a0 - b0))"
set_cases: fixes e :: "'a"   and a :: "'a tree" assumes "e \<in> set_tree a" obtains z1 :: "nat" where "a = Leaf z1__ e"   | z1 :: "nat" and z2 :: "'a tree" and z3 :: "'a tree" where "a = Node z1a__ z2__ z3__" and "e \<in> set_tree z2__"   | z1 :: "nat" and z2 :: "'a tree" and z3 :: "'a tree" where "a = Node z1 z2 z3" and "e \<in> set_tree z3"
assms: shows "(xsi, xs) \<in> \<langle>appr_rell\<rangle>list_rel"
someinit_def: fixes i :: "nat" shows "opn.someinit i \<equiv> SOME x. x \<in> (fst \<circ> id) ` init (paodv i)"
n_o_narrow: fixes S1 :: "'av st option"   and X :: "char list set"   and S2 :: "'av st option" assumes "top_on\<^sub>o S1 (- X)"   and "top_on\<^sub>o S2 (- X)"   and "finite X"   and "S2 \<le> S1"   and "S1 \<triangle> S2 < S1" shows "n\<^sub>o (S1 \<triangle> S2) X < n\<^sub>o S1 X"
l_neg: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "\<ominus>\<^bsub>V\<^esub> x + x = 0\<^sub>v n"
vimage_def: fixes f :: "'d \<Rightarrow> 'e"   and B :: "'e set" shows "f -` B \<equiv> {x. f x \<in> B}"
intro: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set" assumes "M.Problem wtFsym wtPsym arOf resOf parOf \<Phi>" shows "M_Problem wtFsym wtPsym arOf resOf parOf \<Phi>"
prob_space_distr: fixes f :: "'b \<Rightarrow> 'd"   and M' :: "'d measure" assumes "X.random_variable M' f" shows "prob_space (distr (distr M S X) M' f)"
desargues_config_2D_rkABCabc: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and a :: "Points"   and R :: "Points"   and b :: "Points"   and c :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {Q, A', a} = 2"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {R, A, a} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2" shows "4 \<le> rk {A, B, C, a, b, c}"
SUP_is_Sup: fixes y :: "'d"   and F :: "('d \<Rightarrow> 'c) set" shows "(SUP f\<in>F. f y) = \<Squnion>{f y |f. f \<in> F}"
mono_Span_sublist: fixes Us :: "'a list"   and Vs :: "'a list" assumes "set Us \<subseteq> set Vs"   and "set Vs \<subseteq> carrier R" shows "Span K Us \<subseteq> Span K Vs"
transfer_arrive': fixes s :: "'s net_state"   and H :: "nat set"   and K :: "nat set"   and m :: "'m"   and s' :: "'s net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'g"   and \<zeta> :: "'l net_state"   and \<sigma>' :: "nat \<Rightarrow> 'g"   and \<zeta>' :: "'l net_state" assumes "(s, H\<not>K:arrive(m), s') \<in> automaton.trans (pnet np n)"   and "s \<in> reachable (pnet np n) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>', \<zeta>')"   and "wf_net_tree n" shows "((\<sigma>, \<zeta>), H\<not>K:arrive(m), \<sigma>', \<zeta>') \<in> automaton.trans (opnet onp n)"
Inter_greatest: fixes A :: "'a set set"   and C :: "'a set" assumes "\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X" shows "C \<subseteq> \<Inter> A"
sel_3: fixes x :: "quat" shows "Im2 (- x) = - Im2 x"
sum_add_split_nat_ivl: fixes m :: "nat"   and k :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> 'a" assumes "m \<le> k"   and "k \<le> n"   and "\<And>i. \<lbrakk>m \<le> i; i < k\<rbrakk> \<Longrightarrow> g i = f i"   and "\<And>i. \<lbrakk>k \<le> i; i < n\<rbrakk> \<Longrightarrow> h i = f i" shows "sum g {m..<k} + sum h {k..<n} = sum f {m..<n}"
mod_eq_self_iff_div_eq_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "(modulo a b = a) = (divide a b = zero)"
simps_2: fixes c :: "'b"   and c' :: "'b poly"   and n' :: "nat"   and p' :: "'b poly" shows "(c)\<^sub>p *\<^sub>p CN c' n' p' = (if c = (0::'b) then 0\<^sub>p else CN ((c)\<^sub>p *\<^sub>p c') n' ((c)\<^sub>p *\<^sub>p p'))"
take_bit_minus_one_eq_mask: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   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 "take_bit n (uminus one) = mask n"
Object_mdecls_def: shows "Object_mdecls \<equiv> []"
fine_division_exists: fixes g :: "'b \<Rightarrow> 'b set"   and a :: "'b"   and b :: "'b" assumes "gauge g" obtains p :: "('b \<times> 'b set) set" where "p tagged_division_of cbox a b" and "g fine p"
vsv_vimageI2': fixes b :: "V"   and a :: "V" assumes "b = \<GG>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<GG>\<lparr>ObjMap\<rparr>)"
distinct_15: fixes x51 :: "nat"   and x52 :: "recf"   and x53 :: "recf" shows "S \<noteq> Pr x51 x52 x53"
ex_ev: shows "S t i \<turnstile> ev \<mapsto> S t (Suc i)"
finite_measure_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "B \<in> PS.events" shows "PS.prob A \<le> PS.prob B"
CH_def: fixes X :: "'x set" shows "CH X = (\<Union>h. Ch h X)"
in_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> \<Theta>[F](h)) = (g \<in> \<Theta>[F](h))"
split2: shows "{y \<in> Pow (insert x F). card y = 1} = {y \<in> Pow F. card y = 1} \<union> {{x}}"
eq_divide_eq_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 b :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(one = divide b a) = (a \<noteq> zero \<and> a = b)"
psubst_subst_compose: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y"   and "goodEnv rho" shows "X #[Y / y]_ys #[rho] = X #[rho [y \<leftarrow> Y #[rho]]_ys]"
has_vector_derivative_sum: fixes I :: "'c set"   and f :: "'c \<Rightarrow> real \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and net :: "real filter" assumes "\<And>i. i \<in> I \<Longrightarrow> (f i has_vector_derivative f' i) net" shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_vector_derivative sum f' I) net"
simps: fixes v :: "nat \<Rightarrow> 'a"   and b :: "(nat \<Rightarrow> 'a option) \<times> (nat \<Rightarrow> 'a option)" shows "v \<Turnstile>\<^sub>b b = (\<forall>x. in_bounds x v b)"
List_hd_not_in_tl: fixes h :: "'b \<Rightarrow> 'b ref"   and a :: "'b"   and as :: "'b list" assumes "List h (h a) as" shows "a \<notin> set as"
inf1E: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "inf A B x" obtains "A x" and "B x"
assms_3: shows "r \<in> B \<rightarrow>\<^sub>M tree_sigma A"
convex_hull_insert_Int_eq: fixes z :: "'b"   and S :: "'b set"   and T :: "'b set"   and U :: "'b set" assumes "z \<in> rel_interior S"   and "T \<subseteq> rel_frontier S"   and "U \<subseteq> rel_frontier S"   and "convex S"   and "convex T"   and "convex U" shows "convex hull insert z T \<inter> convex hull insert z U = convex hull insert z (T \<inter> U)"
assms_11: shows "indexed_eval (P # Ps) i = indexed_eval (Q # Qs) i"
setVar_gState_progress_rel: fixes prog :: "program"   and g :: "'a gState_scheme"   and v :: "varRef"   and val :: "integer"   and p :: "pState" assumes "gState_inv prog g" shows "(g, fst (setVar v val g p)) \<in> gState_progress_rel prog"
monotone_map_prod: fixes ordc :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordd :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "monotone (rel_prod orda ordb) (rel_prod ordc ordd) (map_prod f g) = (monotone orda ordc f \<and> monotone ordb ordd g)"
fmrel_cases: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'g \<rightharpoonup>\<^sub>f 'e"   and n :: "'g \<rightharpoonup>\<^sub>f 'f"   and x :: "'g" assumes "fmrel P m n" obtains "fmlookup m x = None" and "fmlookup n x = None"   | a :: "'e" and b :: "'f" where "fmlookup m x = Some a" and "fmlookup n x = Some b" and "P a b"
extractable_red: fixes K :: "msg set"   and I :: "msg set" shows "extractable K I \<subseteq> I"
cases: fixes xs :: "'a list"   and a1 :: "nat"   and a2 :: "nat" assumes "inversionsp xs a1 a2" obtains i :: "nat" and j :: "nat" where "a1 = i" and "a2 = j" and "i < j" and "j < length xs" and "xs ! j < xs ! i"
condition_def: fixes P :: "'c \<Rightarrow> bool"   and L :: "'c \<Rightarrow> ('d \<times> 'c) set \<times> bool"   and R :: "'c \<Rightarrow> ('d \<times> 'c) set \<times> bool" shows "condition P   L   R \<equiv> \<lambda>s. if P s then L s else R s"
C: shows "steps' init qs Strat n = steps' init (qs @ a) Strat n"
standard_decomp_singleton: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" shows "standard_decomp (poly_deg h) [(h, U)]"
span_induct': fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and B :: "('a \<Rightarrow>\<^sub>0 'b) set"   and P :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool" assumes "p \<in> phull B"   and "P 0"   and "\<And>a q p. \<lbrakk>a \<in> phull B; P a; p \<in> B; q \<noteq> (0::'b)\<rbrakk> \<Longrightarrow> P (a + q \<cdot> p)" shows "P p"
dbproj_Cons_2: fixes i :: "'d"   and j :: "'d"   and d :: "'e"   and D :: "('d \<times> 'e) list" assumes "i \<noteq> j" shows "dbproj j ((i, d) # D) = dbproj j D"
qSwap_preserves_qAFresh_distinct: fixes ys :: "'a"   and zs :: "'a"   and y :: "'b"   and z1 :: "'b"   and z2 :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm" assumes "ys \<noteq> zs \<or> y \<notin> {z1, z2}" shows "qAFresh ys y (X #[[z1 \<and> z2]]_zs) = qAFresh ys y X"
inorder_sorted: fixes t :: "interval_tree" shows "tree_sorted t = Lists_Ex.strict_sorted (in_traverse t)"
simps_2: fixes b :: "bexp"   and res :: "bool"   and S :: "'av st option" shows "inv_bval' (bexp.Not b) res S = inv_bval' b (\<not> res) S"
simps_3: fixes jmps :: "jump set"   and j :: "jump"   and s :: "stmt" shows "jumpNestingOkS jmps (j\<bullet> s) = jumpNestingOkS ({j} \<union> jmps) s"
priSK_pubSK: fixes X :: "agent"   and X' :: "agent" shows "priSK X \<noteq> pubSK X'"
trd_aux_red_rtrancl: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" shows "(red (set fs))\<^sup>*\<^sup>* p (trd_aux fs p r - r)"
distinct_34: fixes x6 :: "'b ptrm"   and x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm" shows "PFst x6 \<noteq> PFn x41 x42 x43"
inf_sup_aci_5: fixes inf :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and less_eq :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and less :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and sup :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and x :: "'m"   and y :: "'m" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
inject_3: fixes x3 :: "val option"   and y3 :: "val option" shows "(Abs x3 = Abs y3) = (x3 = y3)"
simps_9: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and x1 :: "'c" shows "map_result f1 f2 (Rval x1) = Rval (f1 x1)"
assms_4: shows "e \<le> v * - v\<^sup>T"
hyps: fixes i :: "nat" shows "i < length zs * length y \<longrightarrow> vec_vec_Tensor zs y ! i = zs ! (i div length y) * y ! (i mod length y)"
spec\<^sub>2_def: fixes p :: "prog" shows "spec\<^sub>2 p \<equiv> vars p = vars\<^sub>0 \<and> (\<exists>s\<^sub>L s\<^sub>H. body_split p s\<^sub>L s\<^sub>H \<and> wfs \<Gamma>\<^sub>0 s\<^sub>L \<and> wfs \<Gamma>\<^sub>0 s\<^sub>H \<and> low_proc_non0\<^sub>2 s\<^sub>L \<and> low_proc_0\<^sub>2 s\<^sub>L \<and> low_proc_no_input_change s\<^sub>L \<and> high_proc\<^sub>2 s\<^sub>H \<and> high_proc_no_low_output_change s\<^sub>H)"
lowlink_more_cong: fixes s :: "('a, 'b) tarjan_state_scheme"   and s' :: "('a, 'b) tarjan_state_scheme" assumes "state.more s = state.more s'" shows "lowlink s = lowlink s'"
prems_1: shows "crdt_ops (xs @ [(oid, oper)] @ ys @ [y]) deps"
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 "SATp \<Gamma> f x"
no_overlapsI: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('a, 'c) flow_entry_match list" assumes "check_no_overlap \<gamma> t"   and "distinct t" shows "no_overlaps \<gamma> t"
integrable_component: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and y :: "'b" assumes "f integrable_on A" shows "(\<lambda>x. f x \<bullet> y) integrable_on A"