hcju/isabelleps huggingface dataset:
comp_cong: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and x :: "'f"   and f' :: "'g \<Rightarrow> 'd"   and g' :: "'h \<Rightarrow> 'g"   and x' :: "'h" assumes "f (g x) = f' (g' x')" shows "(f \<circ> g) x = (f' \<circ> g') x'"
aux9: fixes x :: "'a"   and y :: "'a" shows "x + - x \<cdot> y = x + y"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Grave of Acute \<Rightarrow> f1 | Grave \<Rightarrow> f2 | Macron \<Rightarrow> f3) = f2"
cases: fixes a :: "'b list list list" assumes "map_head'_side a" obtains xss :: "'b list list list" where "a = xss" and "\<And>x xa. \<lbrakk>x \<in> set xss; xa \<in> set x\<rbrakk> \<Longrightarrow> xa \<noteq> []"
exec_Normal_elim_cases_11: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and p :: "'a"   and s :: "'b"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow> t" obtains bdy :: "('b, 'a, 'c) com" where "\<Gamma> p = Some bdy" and "\<Gamma>\<turnstile> \<langle>bdy,Normal s\<rangle> \<Rightarrow> t"   | "t = Stuck" and "\<Gamma> p = None"
getEnv_psubstEnv_Some: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "rho xs x = Some X" shows "rho &[rho'] xs x = Some X #[rho']"
fmlookup_dom_iff: fixes x :: "'a"   and m :: "('a, 'b) fmap" shows "(x |\<in>| fmdom m) = (\<exists>a. fmlookup m x = Some a)"
Pd_Xd: fixes x :: "'a"   and y :: "'a"   and d :: "'b" assumes "(x, y) \<in> Pd_singletons_for_ds X {Xd x} d" shows "Xd x = d \<and> Xd y = d"
real_sqrt_sum_squares_triangle_ineq: fixes a :: "real"   and c :: "real"   and b :: "real"   and d :: "real" shows "sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)"
sub_of_denot_equivt: fixes E :: "char list \<Rightarrow> hterm"   and t :: "fterm" shows "eval\<^sub>t E HFun (t \<cdot>\<^sub>t sub_of_denot E) = eval\<^sub>t E HFun t"
normalize_quot_proj_2: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" shows "snd (normalize_quot (a, b)) = normalize b div gcd a b"
compass_defs_1: shows "east = proj2_abs (vector [1, 0, 1])"
3: shows "smap (r \<circ> Suc) nats = smap snd wr"
map_cong: fixes x :: "'d \<Rightarrow> 'e"   and ya :: "'d \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "x = ya"   and "\<And>z. z \<in> range ya \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> ya"
unique_favorites: fixes i :: "'agent" assumes "i \<in> agents" shows "favorites R i = {favorite R i}"
arcs_elem_1: fixes a :: "nat"   and b :: "nat"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "(a, b) \<in> set (arcs i j xs)" shows "a \<in> set (i # xs)"
list_all2_append_singletonI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and x :: "'c"   and y :: "'d" assumes "list_all2 P xs ys"   and "P x y" shows "list_all2 P (xs @ [x]) (ys @ [y])"
alpha_tau_zero: fixes x :: "'a" shows "\<nu> (\<tau> x) = (0::'a)"
stone_phi_complement: fixes x :: "'c" shows "complement (stone_phi (Abs_regular (- x))) (stone_phi (Abs_regular (- - x)))"
length_prefix_ex: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "\<exists>ys zs. xs = ys @ zs \<and> length ys = n"
tbisim_NoneI: fixes w :: "bool"   and t :: "'t"   and m :: "'m1"   and m' :: "'m2" shows "tbisim w t None m None m'"
alt_group_is_sign_kernel: fixes n :: "nat" shows "carrier (alt_group n) = kernel (sym_group n) sign_img sign"
neval_S_n_6: shows "lcn' = (cmn', mem')"
altc_parametric: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique R" shows "rel_fun (rel_cset R) (rel_fun (rel_fun R (=)) (=)) altc altc"
inner_Basis_minus2: fixes b1 :: "'a"   and b2 :: "'a"   and v :: "'a" assumes "b1 \<in> S"   and "b2 \<in> S" shows "local.inner_Basis v (b1 - b2) = local.inner_Basis v b1 - local.inner_Basis v b2"
ltl_prop_implication_implies_ltl_implication: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n \<phi>"   and "\<phi> \<longrightarrow>\<^sub>P \<psi>" shows "w \<Turnstile>\<^sub>n \<psi>"
z: shows "z = Quad (a3, b3, c3, d3)"
I: shows "\<not> loose_bvar1 (eta_norm body) (Suc lev)"
AllN_1: fixes init :: "real list"   and i :: "nat" assumes "length init = i" shows "eval F (init @ xs) = eval (liftFm i amount F) (init @ I @ xs)"
intros: 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 C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "Shares D \<in> set Bs" shows "P \<turnstile> C \<prec>\<^sub>S D"
scalar_prod_right_unit: fixes i :: "nat"   and n :: "nat"   and v :: "'a Matrix.vec" assumes "i < n" shows "v \<bullet> unit_vec n i = v $ i"
FAssStatic: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and t :: "ty"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C has F,Static:t in D" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"
inv_comp_right: fixes f :: "'a \<Rightarrow> 'a" assumes "bijection f" shows "f \<circ> inv' f = id"
the_elem_def: fixes X :: "'c set" shows "the_elem X = (THE x. X = {x})"
length_int_to_bv_upper_limit_lem1: fixes i :: "int"   and k :: "nat" assumes "i < - 1"   and "- (2 ^ (k - 1)) \<le> i" shows "length (int_to_bv i) \<le> k"
take_bit_take_bit: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and m :: "nat"   and n :: "nat"   and a :: "'k" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit m (take_bit n a) = take_bit (min m n) a"
g'_nonneg: shows "\<forall>\<^sub>F x in at_top. 0 \<le> g' x"
strict_sorted_h: shows "strict_sorted (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0})"
matchrel_mono: fixes x :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)"   and y :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)"   and M :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset" assumes "(x, y) \<in> matchrel" shows "((fst x + M, snd x), fst y + M, snd y) \<in> matchrel"
r_A: shows "rank A = rank (Finite_Cartesian_Product.transpose A ** A)"
permutes_inverses_2: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" assumes "p permutes S" shows "inv p (p x) = x"
rel_imp_elem: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> r" shows "x \<in> A"
perm_MOST: fixes f :: "atom \<Rightarrow> atom" assumes "f \<in> perm" shows "\<forall>\<^sub>\<infinity>x. f x = x"
allBlocksRead_def: fixes s :: "state"   and p :: "Proc" shows "allBlocksRead s p = block ` allRdBlks s p"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs" shows "l_new_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
dfs_invariant': fixes S :: "bool list"   and xs :: "nat list" assumes "rquot_invariant A S"   and "list_all (dfa_is_node A) xs" shows "rquot_invariant A (dfs S xs)"
complete_tcsigs_def: fixes sub :: "('a \<times> 'a) set"   and tcs :: "'b \<Rightarrow> ('a \<Rightarrow> 'c option) option" shows "complete_tcsigs sub tcs \<equiv> \<forall>ars\<in>ran tcs. \<forall>(c\<^sub>1, c\<^sub>2)\<in>sub. c\<^sub>1 \<in> dom ars \<longrightarrow> c\<^sub>2 \<in> dom ars"
D2: shows "D2 = (v\<^sub>o - v\<^sub>e)\<^sup>2 - 4 * ((a\<^sub>o - a\<^sub>e) / 2) * (s\<^sub>o - s\<^sub>e)"
nonepd: shows "\<forall>i>0. \<not> \<pi> l pd\<rightarrow> \<pi>' i"
dvd_imp_mod_0: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and modulo :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and a :: "'m"   and b :: "'m" assumes "class.semidom_modulo divide plus minus zero times one modulo"   and "dvd.dvd times a b" shows "modulo b a = zero"
compatibleI: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "class.cpo below"   and "below x z"   and "below y z"   and "\<And>a. \<lbrakk>below x a; below y a\<rbrakk> \<Longrightarrow> below z a" shows "cpo.compatible below x y"
t_def: shows "t = take l s @ to_woot_char m # drop l s"
F_empty_iff: fixes x :: "'a" shows "(F x = {}) = (\<forall>i. i \<notin> F x)"
commute: fixes a :: "'e"   and b :: "'e" shows "a \<cdot> b = b \<cdot> a"
br_def: fixes \<alpha> :: "'d \<Rightarrow> 'e"   and I :: "'d \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
roots_def: fixes p :: "'a list" shows "roots p = Abs_multiset (alg_mult p)"
acquired_pending_write_mono_in: fixes x :: "nat"   and xs :: "'b memref list"   and A :: "nat set"   and B :: "nat set" assumes "x \<in> acquired False xs A" shows "x \<in> acquired True xs B"
is_rendered_commutative_by_cone: fixes a :: "'c"   and \<chi> :: "(bool, unit) J.arr \<Rightarrow> 'c" assumes "local.cone a \<chi>" shows "is_rendered_commutative_by (\<chi> (J.MkIde False)) (\<chi> (J.MkIde True))"
neg_less_eq_nonneg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) a = less_eq zero a"
is_up_down: fixes c :: "'b word \<Rightarrow> 'c word"   and d :: "'c word \<Rightarrow> 'b word" shows "is_up c = is_down d"
fv\<^sub>s\<^sub>e\<^sub>t_subst_img_eq: fixes X :: "'c set"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and Y :: "'c set" assumes "X \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` (Y - X)) = fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` Y) - X"
hom: shows "homeomorphic_map (subtopology (Euclidean_space m) S) (subtopology (Euclidean_space n) T) f"
assms: shows "\<Gamma> \<oplus> A = \<Delta> \<oplus> A"
inf_sup_aci_5: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and x :: "'g"   and y :: "'g" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
assms_2: fixes u :: "'a" assumes "u \<in> carrier L"   and "f u \<sqsubseteq> u" shows "x \<sqsubseteq> u"
ae_eq: shows "AE x in distr M MX X. log b (enn2real (RN_deriv MX (distr M MX X) x)) = log b (f x)"
midpoint_def: fixes a :: "'c"   and b :: "'c" shows "midpoint a b = (a + b) /\<^sub>R 2"
pp_def: shows "pp = smult (inverse (lead_coeff p)) p"
preserves_ide: fixes a :: "('a S.E.term, 'a) S.arr" assumes "S.ide a" shows "S.ide (S.src a)"
restrict_map_empty: fixes D :: "'a set" shows "Map.empty |` D = Map.empty"
cr_rename_gba_impl_def: shows "cr_rename_gba_impl \<equiv> \<lambda>x xa. let y' = gen_image (\<lambda>x. foldli (id x)) [] (glist_insert (=)) name_impl x; y'a = gen_image (\<lambda>x. foldli (id x)) [] (glist_insert (=)) name_impl (filter (\<lambda>xb. map2set_memb (\<lambda>k t. rbt_lookup t k) 0 (incoming_impl xb)) x) in build_succ_impl x \<bind> (\<lambda>xb. let y'b = \<lambda>xc. the_default [] (Impl_Array_Map.iam_lookup xc xb); y'c = build_F_impl x xa in pn_map_impl x \<bind> (\<lambda>xc. let y'd = \<lambda>xd xe. case Impl_Array_Map.iam_lookup xd xc of None \<Rightarrow> False | Some (xf, xg) \<Rightarrow> Gen_Set.gen_ball (\<lambda>x. foldli (id x)) xf xe \<and> Gen_Set.gen_ball (\<lambda>x. foldli (id x)) xg (\<lambda>xh. \<not> xe xh) in RETURN \<lparr>gi_V = y', gi_E = y'b, gi_V0 = y'a, gbgi_F = y'c, gbai_L = y'd\<rparr>))"
SeqHaddP_unique: fixes x :: "tm"   and s :: "tm"   and w :: "tm"   and y :: "tm"   and s' :: "tm"   and y' :: "tm" shows "{OrdP x, SeqHaddP s w x y, SeqHaddP s' w x y'} \<turnstile> y' EQ y"
af_G_prefix_lang: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and xs :: "'b set list"   and ys :: "'b set list" assumes "w \<Turnstile>\<^sub>n af (G\<^sub>n \<phi>) (xs @ ys)" shows "w \<Turnstile>\<^sub>n af (G\<^sub>n \<phi>) ys"
scan_eq_cons: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and a :: "'b"   and b :: "'b"   and w :: "'b list" shows "(scan f xs a = b # w) = (\<exists>y ys. xs = y # ys \<and> f y a = b \<and> scan f ys (f y a) = w)"
unit_eq_div2: 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "(a = divide c b) = (times a b = c)"
cblinfun_extension_exists_finite_dim: fixes S :: "'c set"   and \<phi> :: "'c \<Rightarrow> 'd" assumes "cindependent S"   and "cspan S = UNIV" shows "cblinfun_extension_exists S \<phi>"
ideal_incl_iff: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "(I \<subseteq> J) = (J = \<Union> ((+>) I ` J))"
qPsubstAbs_preserves_qGoodAbs: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodEnv rho"   and "qGoodAbs A" shows "qGoodAbs A $[[rho]]"
cos_periodic_pi: fixes x :: "real" shows "cos (x + pi) = - cos x"
x2_def: shows "x2 \<equiv> (- (v\<^sub>o - v\<^sub>e) - sqrt D2) / (2 * ((a\<^sub>o - a\<^sub>e) / 2))"
lift_id: fixes inp :: "'a \<Rightarrow> 'b option" shows "lift id inp = inp"
not_UNIF1_singleton_Var_right_2: fixes x :: "'c"   and y :: "'c" assumes "\<not> UNIF1 (subst x (Var y)) {#(Var x, Var y)#} {#}" shows "x = y"
Embranch_0: fixes rgs :: "recf list"   and k :: "nat"   and rcs :: "recf list"   and xs :: "nat list" assumes "length rgs = k"   and "length rcs = k"   and "0 < k"   and "\<forall>j<k. rec_exec (rcs ! j) xs = 0" shows "Embranch (zip (map rec_exec rgs) (map (\<lambda>r args. 0 < rec_exec r args) rcs)) xs = 0"
th3_7_2: fixes b :: "(real, 'a) vec"   and c :: "(real, 'a) vec"   and a :: "(real, 'a) vec"   and d :: "(real, 'a) vec" assumes "b \<noteq> c"   and "B\<^sub>\<real> a b c"   and "B\<^sub>\<real> b c d" shows "B\<^sub>\<real> a b d"
isoE: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C"   and "category.iso C f" obtains g :: "'a" where "category.inverse_arrows C f g"
lebesgue_integrable_real_affine_iff: fixes c :: "real"   and f :: "real \<Rightarrow> 'b"   and t :: "real" assumes "c \<noteq> 0" shows "integrable lebesgue (\<lambda>x. f (t + c * x)) = integrable lebesgue f"
OUT_h'_B: fixes x :: "'v" assumes "x \<in> B \<Gamma>" shows "d_OUT h' x = 0"
ring_hom_restrict: fixes f :: "('c multiset \<Rightarrow> 'a) \<Rightarrow> 'd"   and S :: "('d, 'e) ring_scheme"   and g :: "('c multiset \<Rightarrow> 'a) \<Rightarrow> 'd" assumes "f \<in> ring_hom L S"   and "\<And>r. r \<in> carrier L \<Longrightarrow> f r = g r" shows "g \<in> ring_hom L S"
lm019: fixes X :: "'a set"   and y :: "'a" assumes "1 \<le> card X"   and "\<forall>x\<in>X. x < y" shows "Max X < y"
support_adapt_def: fixes Mkt :: "(bool stream, 'b) discrete_market"   and pf :: "'b \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real" shows "support_adapt Mkt pf = (\<forall>asset\<in>support_set pf. borel_adapt_stoch_proc G (prices Mkt asset))"
a1: shows "low_equal s1 s2 \<and> fst (memory_read 10 addra s1) = fst (memory_read 10 addra s2)"
transfer: shows "rel_fun (=) (pcr_finfun (=) (=)) (\<lambda>b x. b) finfun_const"
Seed_max_final_ex: fixes p :: "nat" shows "\<exists>f\<in>set (finals Seed\<^bsub>p\<^esub>). |vertices f| = maxGon p"
sq_mtx_diag_simps_3: fixes f :: "'a \<Rightarrow> real"   and i :: "'a" shows "sq_mtx_diag f $$ i = axis i (f i)"
implies_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<leadsto> z = (x \<leadsto> z) \<sqinter> (y \<leadsto> z)"
quasisym_not_greater: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b" assumes "compare cmp a b \<noteq> Greater"   and "compare cmp b a \<noteq> Greater" shows "compare cmp a b = Equiv"
subgroup_incl: fixes I :: "'a set"   and J :: "'a set" assumes "subgroup I G"   and "subgroup J G"   and "I \<subseteq> J" shows "subgroup I (G\<lparr>carrier := J\<rparr>)"
freeliftid_trivial_P: fixes ps :: "('g \<times> bool) list" assumes "ps \<in> P" shows "freeword_funlift id (Abs_freeword ps) = (0::'g)"
outer_inv_def: shows "outer_inv = (\<lambda>it (blues, reds, cyc). case cyc of NO_CYC \<Rightarrow> add_inv blues reds {} \<and> gen_dfs_outer E reachable V0 it blues False | REACH v list \<Rightarrow> False | CIRC pr pl \<Rightarrow> \<exists>v0\<in>V0. \<exists>v. pl \<noteq> [] \<and> path E v pl v \<and> path E v0 pr v \<and> set pl \<inter> F \<noteq> {})"
lnth_list_of_stream: fixes xs :: "'a stream" shows "(?!) (llist_of_stream xs) = (!!) xs"
interaction_bounded_by: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat" assumes "interaction_bound consider gpv \<le> n" shows "interaction_bounded_by consider gpv n"
DERIV_subset: fixes f :: "'b \<Rightarrow> 'b"   and f' :: "'b"   and x :: "'b"   and s :: "'b set"   and t :: "'b set" assumes "(f has_field_derivative f') (at x within s)"   and "t \<subseteq> s" shows "(f has_field_derivative f') (at x within t)"
det3_nonneg_scaleR1_eq: fixes e :: "real"   and xr :: "real \<times> real"   and P :: "real \<times> real" assumes "0 < e" shows "(0 \<le> det3 0 (e *\<^sub>R xr) P) = (0 \<le> det3 0 xr P)"
is_inf_def: fixes x :: "'c"   and y :: "'c"   and inf :: "'c" shows "is_inf x y inf = (inf \<sqsubseteq> x \<and> inf \<sqsubseteq> y \<and> (\<forall>z. z \<sqsubseteq> x \<and> z \<sqsubseteq> y \<longrightarrow> z \<sqsubseteq> inf))"
B: shows "\<forall>n<Suc (length ws). P ((ws @ [w]) ! n) (drop (Suc n) ws @ drop (Suc n - length ws) [w])"
reachable_from_ArrLoc_impl_Arr: fixes s :: "Store"   and l :: "Location"   and T :: "Arraytype"   and a :: "nat"   and i :: "nat" assumes "s\<turnstile> l reachable_from (s@@arrLoc T a i)" shows "s\<turnstile> l reachable_from arrV T a"
option_rel_Some2: fixes A :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and x :: "'l option"   and y :: "'m" shows "rel_option A x (Some y) = (\<exists>x'. x = Some x' \<and> A x' y)"
embed_embedS_iso: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" shows "embed r r' f = (embedS r r' f \<or> iso r r' f)"
one_step_methodI: fixes m :: "'c set \<Rightarrow> real \<Rightarrow> real \<Rightarrow> 'c set \<Rightarrow> ('c set \<times> 'd) option nres" assumes "\<And>X0 CX hl hu. m X0 hl hu CX \<le> SPEC (case_option True (\<lambda>(res, err). nonempty res \<and> (\<forall>x0\<in>X0. \<forall>h\<in>{hl..hu}. x0 \<in> Csafe \<longrightarrow> 0 \<le> h \<longrightarrow> h \<in> existence_ivl0 x0 \<longrightarrow> (\<forall>h'\<in>{0..h}. flow0 x0 h' \<in> CX) \<longrightarrow> x0 + h *\<^sub>R ode x0 \<in> CX \<longrightarrow> flow0 x0 h \<in> res)))" shows "one_step_method m"
contains_propagates_fall: fixes n :: "nat"   and g :: "form" assumes "infinite (deriv s)"   and "contains f n (0, FAll g)" shows "\<exists>y. contains f (Suc (n + y)) (0, finst g (newvar (sfv (s_of_ns (snd (f (n + y)))))))"
assms: fixes u :: "nat \<Rightarrow> 'a"   and x :: "'a" shows "u \<longlonglongrightarrow> x = (\<lambda>n. f (u n)) \<longlonglongrightarrow> f x"
f_cd_w: fixes w :: "complex" assumes "w \<in> s" shows "f field_differentiable at w"
intro: fixes opset :: "('c \<times> 'd) set"   and deps :: "'d \<Rightarrow> 'c set" assumes "\<And>oid op1 op2. \<lbrakk>(oid, op1) \<in> opset; (oid, op2) \<in> opset\<rbrakk> \<Longrightarrow> op1 = op2"   and "\<And>oid oper ref. \<lbrakk>(oid, oper) \<in> opset; ref \<in> deps oper\<rbrakk> \<Longrightarrow> ref < oid"   and "finite opset" shows "opset opset deps"
SN_defs: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "SN_on r A = (\<nexists>f. f 0 \<in> A \<and> (\<forall>i. (f i, f (Suc i)) \<in> r))"
gt_same: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t = head s"   and "chkchop_same (>\<^sub>t) t s"   and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)" shows "t >\<^sub>t s"
edge_to_node: fixes g :: "'g"   and e :: "'node \<times> 'edgeD \<times> 'node" assumes "invar g"   and "e \<in> \<alpha>e g" obtains "getFrom e \<in> set (\<alpha>n g)" and "getTo e \<in> set (\<alpha>n g)"
comp_char: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "g \<cdot> f = (if seq g f then MkArr (Dom f) (Cod g) (vertical_composite.map (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Map f) (Map g)) else null)"
simps_2: fixes x :: "'g"   and xs :: "'g list"   and ys :: "'g list" shows "splice (x # xs) ys = x # splice ys xs"
is_distr_def: fixes p :: "'h \<Rightarrow> ('i \<Rightarrow> 'j option) \<Rightarrow> 'g \<Rightarrow> 'd decision option" shows "is_distr p = (\<lambda>g f. \<forall>N P1 P2. g o\<^sub>f (p N (P1 \<Oplus> P2) \<circ> f) = (g o\<^sub>f (p N P1 \<circ> f)) \<Oplus> (g o\<^sub>f (p N P2 \<circ> f)))"
rotate_is_Nil_conv: fixes n :: "nat"   and xs :: "'b list" shows "(rotate n xs = \<epsilon>) = (xs = \<epsilon>)"
inR: assumes "V \<noteq> []" shows "(c # V, c # V) \<in> R"
wellformed1_sortedQ: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "wellformed_policy1 p" shows "wellformed_policy1 (qsort p l)"
assms: shows "a \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"