hcju/isabelleps huggingface dataset:
returns_to_outsideI: fixes t :: "real"   and x :: "'a"   and P :: "'a set" assumes "0 \<le> t"   and "t \<in> existence_ivl0 x"   and "flow0 x t \<in> P"   and "x \<notin> P"   and "closed P" shows "returns_to P x"
not_Stuck: shows "\<Gamma>|\<^bsub>M\<^esub>\<turnstile> \<langle>Catch c1 c2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
unique: fixes G2 :: "'a set" assumes "G2 \<in> \<G>"   and "G \<noteq> G2" shows "\<not> ps \<subseteq> G2"
Scheffe_lemma1: fixes M :: "'b measure"   and F :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>n. integrable M (F n)"   and "integrable M f"   and "AE x in M. (\<lambda>n. F n x) \<longlonglongrightarrow> f x"   and "limsup (\<lambda>n. \<integral>\<^sup>+ x. ennreal (norm (F n x)) \<partial>M) \<le> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M" shows "(\<lambda>n. \<integral>\<^sup>+ x. ennreal (norm (F n x - f x)) \<partial>M) \<longlonglongrightarrow> 0"
Q_ats_1: shows "Q ats (<) \<B>\<^sub>i\<^sub>u \<B>\<^sub>i\<^sub>l \<B>\<^sub>u \<B>\<^sub>l \<B>\<^sub>i\<^sub>u_update \<I>\<^sub>u \<I>\<^sub>l Leq Geq s"
simps_11: fixes prec :: "nat"   and x :: "nat"   and a :: "floatarith"   and vs :: "float interval option list" shows "isDERIV_approx prec x (Sqrt a) vs = (isDERIV_approx prec x a vs \<and> (case approx prec a vs of None \<Rightarrow> False | Some ivl \<Rightarrow> 0 < lower ivl))"
simps_11: fixes f :: "'f"   and c\<^sub>1 :: "('d, 'e, 'g) com"   and c\<^sub>2 :: "('d, 'e, 'g) com" shows "mark_guards f (Catch c\<^sub>1 c\<^sub>2) = Catch (mark_guards f c\<^sub>1) (mark_guards f c\<^sub>2)"
jumpF_IVT: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a < b"   and "continuous_on {a<..<b} f"   and "(0 < jumpF f (at_right a) \<or> continuous (at_right a) f \<and> 0 < f a) \<and> (jumpF f (at_left b) < 0 \<or> continuous (at_left b) f \<and> f b < 0) \<or> (jumpF f (at_right a) < 0 \<or> continuous (at_right a) f \<and> f a < 0) \<and> (0 < jumpF f (at_left b) \<or> continuous (at_left b) f \<and> 0 < f b)" shows "\<exists>x>a. x < b \<and> f x = 0"
step_def: fixes s :: "('v, 'es) state_scheme" shows "step s \<equiv> if is_empty_stack s then get_new_root s \<bind> (\<lambda>v0. do_new_root v0 s) else get_pending s \<bind> (\<lambda>(u, Vs, s). case Vs of None \<Rightarrow> do_finish u s | Some v \<Rightarrow> if is_discovered v s then do_known_edge u v s else do_discover u v s)"
minus_def: fixes x :: "'a"   and y :: "'a" shows "x - y = x \<sqinter> - y"
abs_zmultiset_def: shows "abs_zmultiset \<equiv> quot_type.abs equiv_zmset Abs_zmultiset"
run_sim: fixes ra :: "nat \<Rightarrow> 'a" assumes "a.is_run ra" obtains rb :: "nat \<Rightarrow> 'b" where "b.is_run rb" and "(ra, rb) \<in> run_rel R"
r_into_rtranclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "r x y" shows "r\<^sup>*\<^sup>* x y"
those_someD: fixes xs :: "'a option list"   and ys :: "'a list" assumes "those xs = Some ys" shows "xs = map Some ys"
sec_has_right_codom: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set"   and \<pp> :: "'a set" assumes "s \<in> \<O> U"   and "\<pp> \<in> U" shows "s \<pp> \<in> R \<^bsub>\<pp> (+) (\<cdot>) \<zero>\<^esub>"
LLL_factorization_def: fixes f :: "int poly" shows "LLL_factorization f = (let p = suitable_prime_bz f; (uu_, fs) = finite_field_factorization_int p f; n = degree f; no = \<parallel>f\<parallel>\<^sup>2; B = sqrt_int_ceiling (2 ^ (5 * (n - 1) * (n - 1)) * no ^ (2 * (n - 1))); l = find_exponent p B; us = hensel_lifting p l f fs; pl = p ^ l in LLL_implementation.LLL_reconstruction p pl f us)"
assms_2: shows "m1 f|` S' \<sqsubseteq> m2 f|` S'"
h1: shows "\<turnstile>\<^sub>p {adjoint (tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) * P * tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)} Utrans (tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) {P}"
setsum_add_nat_ivl2: fixes n :: "nat"   and p :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "n \<le> p" shows "sum f {..<n} + sum f {n..p} = sum f {..p}"
const_update_all: fixes b' :: "'b"   and b :: "'b" assumes "finite UNIV" shows "Finite_Set.fold (\<lambda>a. upd a b') (cnst b) UNIV = cnst b'"
nonzero_deriv_nonconstant: fixes f :: "'a \<Rightarrow> 'a"   and df :: "'a"   and \<xi> :: "'a"   and S :: "'a set" assumes "(f has_field_derivative df) (at \<xi>)"   and "open S"   and "\<xi> \<in> S"   and "df \<noteq> (0::'a)" shows "\<not> f constant_on S"
intros: fixes t :: "trm"   and s :: "trm" shows " App t s \<guillemotright> t"
unstream_gen_length: fixes g :: "('e, 'f) generator"   and s :: "'f" shows "gen_length_cons g 0 s = length (unstream g s)"
distributed_RN_deriv: fixes S :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and Px :: "'b \<Rightarrow> ennreal" assumes "distributed M S X Px" shows "AE x in S. RN_deriv S (density S Px) x = Px x"
AlphaImpl: fixes A :: "('a, 'b) form"   and B :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "\<turnstile> Neg A # B # G" shows "\<turnstile> Impl A B # G"
not_MOST: fixes P :: "'a \<Rightarrow> bool" shows "(\<not> (\<forall>\<^sub>\<infinity>x. P x)) = (\<exists>\<^sub>\<infinity>x. \<not> P x)"
rcosets: fixes H :: "'g set"   and g :: "'g" assumes "Subgroup H"   and "g \<in> G" shows "rcoset_rel H `` {g} = H + {g}"
comp_null_2: fixes f :: "('obj, 'obj) concrete_category.arr" shows "f \<cdot> null = null"
proper_ParT_properL: fixes cl :: "('d, 'e, 'f) cmd list" shows "proper (ParT cl) = properL cl"
less_linear: fixes x :: "'a"   and y :: "'a" shows "x < y \<or> x = y \<or> y < x"
transpose_of_prod: fixes M :: "complex mat"   and N :: "complex mat" assumes "dim_col M = dim_row N" shows "M * N\<^sup>t = N\<^sup>t * M\<^sup>t"
is_node_ptr_kind_none: fixes ptr :: "(_) object_ptr" assumes "\<not> is_node_ptr_kind ptr" shows "cast ptr = None"
nth_ucast: fixes w :: "'c word"   and n :: "nat" shows "UCAST('c \<rightarrow> 'b) w !! n = (w !! n \<and> n < LENGTH('b))"
that: fixes A :: "fm" assumes "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e"   and "atom (decode_Var v) \<sharp> A" shows "thesis"
Id_on_def: fixes A :: "'e set" shows "Id_on A = (\<Union>x\<in>A. {(x, x)})"
le_infI1: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<le> x" shows "inf a b \<le> x"
borel_measurable_nn_transfer_operator_iterates: fixes f :: "'a \<Rightarrow> ennreal"   and n :: "nat" assumes "f \<in> borel_measurable M" shows "(nn_transfer_operator ^^ n) f \<in> borel_measurable M"
bind_stateT_def: shows "bind = (\<Lambda> m k. StateT\<cdot>(\<Lambda> s. runStateT\<cdot>m\<cdot>s \<bind> (\<Lambda> (x, s'). runStateT\<cdot>(k\<cdot>x)\<cdot>s')))"
add_assocD: fixes y :: "'c \<times> 'd"   and z :: "'c \<times> 'd"   and x :: "'c \<times> 'd" shows "(ps_prod_D y z \<and> ps_prod_D x (ps_prod_times y z)) = (ps_prod_D x y \<and> ps_prod_D (ps_prod_times x y) z)"
Diff: fixes a :: "'d set"   and M :: "'d measure"   and b :: "'d set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a - b \<in> sets M"
simps_4: fixes \<phi> :: "('e, 'f, 'd) formula" shows "FVP (? \<phi>) = FVF \<phi>"
simps_2: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b stream" shows "stl (sfilter P s) = sfilter P (stl (sdrop_while (Not \<circ> P) s))"
eval\<^sub>1_preserves_len: fixes P :: "nat exp prog"   and e\<^sub>0 :: "nat exp"   and h\<^sub>0 :: "heap"   and ls\<^sub>0 :: "val list"   and sh\<^sub>0 :: "sheap"   and e\<^sub>1 :: "nat exp"   and h\<^sub>1 :: "heap"   and ls\<^sub>1 :: "val list"   and sh\<^sub>1 :: "sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0,(h\<^sub>0, ls\<^sub>0, sh\<^sub>0)\<rangle> \<Rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, ls\<^sub>1, sh\<^sub>1)\<rangle>" shows "length ls\<^sub>0 = length ls\<^sub>1"
sqmatrix_mult_1_right: fixes A :: "('c, 'd) sqmatrix" shows "A \<cdot> 1 = A"
islimpt_greaterThanLessThan1: fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "a islimpt {a<..<b}"
iso_forward: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "(x, y) \<in> r"   and "iso r r' f" shows "(f x, f y) \<in> r'"
prems_1: shows "C \<in> f \<turnstile> \<C>"
existence_ivl_maximal_interval: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "real \<Rightarrow> 'b"   and S :: "real set"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X"   and "(x solves_ode f) S X"   and "x t0 = x0"   and "t0 \<in> S"   and "is_interval S"   and "S \<subseteq> T" shows "S \<subseteq> ll_on_open.existence_ivl T f X t0 x0"
residualInject'': fixes \<alpha> :: "'d action"   and \<beta> :: "'d action"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "bn \<alpha> = bn \<beta>" shows "(\<alpha> \<prec> P = \<beta> \<prec> Q) = (\<alpha> = \<beta> \<and> P = Q)"
paste_def: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" shows "P +* Q = P outside Domain Q \<union> Q"
edge_step: fixes a :: "'form"   and b :: "nat list"   and ba :: "('form, 'var) out_port"   and aa :: "'form"   and bb :: "nat list"   and bc :: "('form, 'var) antecedent" assumes "(((a, b), ba), (aa, bb), bc) \<in> edges" obtains i :: "nat" where "a = aa" and "b = bb @ [i__]" and "bc = in_port_at (aa, bb) i__" and "hyps (nodeOf (a, b)) ba = None"   | i :: "nat" where "a = aa" and "prefix (b @ [i]) bb" and "hyps (nodeOf (a, b)) ba = Some (in_port_at (a, b) i)"
Borsukian_retraction_gen: fixes S :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and k :: "'b \<Rightarrow> 'a" assumes "Borsukian S"   and "continuous_on S h"   and "h ` S = T"   and "continuous_on T k"   and "k ` T \<subseteq> S"   and "\<And>y. y \<in> T \<Longrightarrow> h (k y) = y" shows "Borsukian T"
wto': shows "wset_thread_ok (wset S') (thr S')"
cmp_in_hom: fixes \<mu>\<nu> :: "'b \<times> 'b" assumes "B.VV.arr \<mu>\<nu>" shows "\<guillemotleft>cmp \<mu>\<nu> : H\<^sub>DoGF_GF.map (B.VV.dom \<mu>\<nu>) \<Rightarrow>\<^sub>D GFoH\<^sub>B.map (B.VV.cod \<mu>\<nu>)\<guillemotright>"
Gramian_matrix_def: fixes G :: "'b vec list"   and k :: "nat" shows "Gramian_matrix G k = (let M = mat k n (\<lambda>(i, j). G ! i $ j) in M * M\<^sup>T)"
less_pnat_def: fixes x :: "pnat"   and y :: "pnat" shows "x < y \<equiv> x \<le> y \<and> x \<noteq> y"
assms_2: shows "snd y \<noteq> (0::'a)"
homeomorphic_Borsukian_eq: fixes S :: "'c set"   and T :: "'d set" assumes "S homeomorphic T" shows "Borsukian S = Borsukian T"
interp_ext_def: fixes Functions :: "'a \<Rightarrow> (real, 'c) vec \<Rightarrow> real"   and Predicates :: "'c \<Rightarrow> (real, 'c) vec \<Rightarrow> bool"   and Contexts :: "'b \<Rightarrow> ((real, 'c) vec \<times> (real, 'c) vec) set \<Rightarrow> ((real, 'c) vec \<times> (real, 'c) vec) set"   and Programs :: "'c \<Rightarrow> (((real, 'c) vec \<times> (real, 'c) vec) \<times> (real, 'c) vec \<times> (real, 'c) vec) set"   and ODEs :: "'c \<Rightarrow> (real, 'c) vec \<Rightarrow> (real, 'c) vec"   and ODEBV :: "'c \<Rightarrow> 'c set"   and more :: "'d" shows "\<lparr>Functions = Functions, Predicates = Predicates, Contexts = Contexts, Programs = Programs, ODEs = ODEs, ODEBV = ODEBV, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons interp_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple (Record.iso_tuple_cons Record.tuple_iso_tuple Functions (Record.iso_tuple_cons Record.tuple_iso_tuple Predicates Contexts)) (Record.iso_tuple_cons Record.tuple_iso_tuple Programs (Record.iso_tuple_cons Record.tuple_iso_tuple ODEs ODEBV))) more"
integral_eq_cases: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and N :: "'e measure"   and g :: "'e \<Rightarrow> 'd" assumes "integrable M f = integrable N g"   and "\<lbrakk>integrable M f; integrable N g\<rbrakk> \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g" shows "integral\<^sup>L M f = integral\<^sup>L N g"
simps_2: fixes x :: "'c"   and ys :: "'c list"   and z :: "'c"   and zs :: "'c list" shows "replace x ys (z # zs) = (if z = x then ys @ zs else z # replace x ys zs)"
eq_iff: fixes a :: "'b list"   and b :: "'b list" shows "(a = b) = (a \<le>s b \<and> b \<le>s a)"
g_I_def: shows "g_I \<equiv> {(High, High), (Low, Low), (Low, High)}"
ideal_n_prod_prime: fixes P :: "'a set"   and n :: "nat"   and J :: "nat \<Rightarrow> 'a set" assumes "prime_ideal R P"   and "\<forall>k\<le>n. ideal R (J k)"   and "i\<Pi>\<^bsub>R,n\<^esub> J \<subseteq> P" shows "\<exists>i\<le>n. J i \<subseteq> P"
lset_conv_lnth: fixes xs :: "'b llist" shows "lset xs = {xs $ n |n. enat n < llength xs}"
assms: shows "gap i (Suc t) \<noteq> gap i t"
ch_equiv1: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" assumes "long_ch_by_ord f X"   and "finite X" shows "long_ch_by_ord2 f X"
zero_game_def: shows "zero_game \<equiv> Game zempty zempty"
cnf_of_operator_precondition_encoding_subset_encoding: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_all_operator_preconditions \<Pi> (\<Pi>\<^sub>\<O>) t) \<subseteq> cnf (\<Phi> \<Pi> t)"
less_le_trans: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ko.lt ko x y"   and "ko.le ko y z" shows "ko.lt ko x z"
fin_NotL_elim: fixes a :: "coname"   and M :: "trm"   and x :: "name"   and y :: "name" assumes "fin (NotL <a>.M x) y" shows "x = y \<and> x \<sharp> M"
pmf_positive_iff: fixes p :: "'e pmf"   and x :: "'e" shows "(0 < pmf p x) = (x \<in> set_pmf p)"
chain_map_0: fixes p :: "nat"   and g :: "'d \<Rightarrow> 'c" shows "chain_map p g 0 = 0"
cbm: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set" assumes "class.arelations meets \<I> e m b ov d s f" shows "b O m \<subseteq> b"
iso_is_arr: fixes f :: "'a" assumes "Right_a.iso f" shows "Right_a.arr f"
add_comm: fixes x :: "('a \<times> ('b \<times> 'a) list) set"   and y :: "('a \<times> ('b \<times> 'a) list) set" shows "x \<union> y = y \<union> x"
LessThan_induct: fixes F :: "'c program"   and A :: "'c set"   and f :: "'c \<Rightarrow> nat"   and B :: "'c set" assumes "\<And>m. F \<in> A \<inter> f -` {m} \<longmapsto>w A \<inter> f -` {..<m} \<union> B" shows "F \<in> A \<longmapsto>w B"
a: shows "x \<in> f \<or> x \<in> EX' (lfp (\<lambda>Z. f \<union> EX' Z))"
block_set_nempty_imp_block_ex: assumes "\<B> \<noteq> {#}" shows "\<exists>bl. bl \<in># \<B>"
sum_finite_imp_summands_finite_1: fixes a :: "eint"   and b :: "eint" assumes "a + b \<noteq> \<infinity>" shows "a \<noteq> \<infinity>"
fps_conv_radius_def: fixes f :: "'a fps" shows "fps_conv_radius f = conv_radius (fps_nth f)"
Arr_rep_ARR: fixes f :: "'c local.term set" assumes "ARR f" shows "Arr (rep f)"
hyps: fixes xs :: "'a list"   and ys :: "'a list" assumes "i < length xs"   and "i < length ys"   and "(xs ! i, ys ! i) \<in> R"   and "\<forall>j<i. xs ! j = ys ! j" shows "(xs, ys) \<in> lexord R"
distinct_mset_rempdups_union_mset: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "distinct_mset A"   and "distinct_mset B" shows "A \<union># B = remdups_mset (A + B)"
cons_post_rule: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and Q' :: "'a \<Rightarrow> assn" assumes "<P> c <Q>"   and "\<And>x. Q x \<Longrightarrow>\<^sub>A Q' x" shows "<P> c <Q'>"
poly_mult_const'_2: fixes p :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "a \<in> carrier R" shows "poly_mult p [a] = local.normalize (map ((\<otimes>) a) p)"
wfrec_fixpoint: fixes R :: "('a \<times> 'a) set"   and F :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" assumes "wf R"   and "adm_wf R F" shows "wfrec R F = F (wfrec R F)"
comp_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<sqinter> z \<le> y \<sqinter> z"
l_inv: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "field R"   and "x \<in> carrier R"   and "x \<noteq> \<zero>\<^bsub>R\<^esub>" shows "inv\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> x = \<one>\<^bsub>R\<^esub>"
trss_ret: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and p :: "'b"   and r :: "'a list"   and c :: "'a list multiset" shows "((return fg p # r, c), LRet, r, c) \<in> trss fg"
valid_edge_def: fixes prog :: "cmd"   and a :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node" shows "valid_edge prog a \<equiv> prog \<turnstile> sourcenode a -kind a\<rightarrow> targetnode a"
proj_monom_mult: fixes k :: "nat"   and c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (monom_mult c t p) = local.punit.monom_mult c t (proj_poly k p)"
HPhase1or2ReadElse_Inv2b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "Inv2b s"   and "HPhase1or2ReadElse s s' p d q" shows "Inv2b s'"
lookup_mult_scalar_monomial_right: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and c :: "'c"   and u :: "'t" shows "lookup (p \<odot> monomial c v) u = (lookup p (pp_of_term u - pp_of_term v) * c when v adds\<^sub>t u)"
swap_cols_rows_similar: fixes A :: "'b mat"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat n n"   and "k < n"   and "l < n" shows "similar_mat (swap_cols_rows k l A) A"
typ_of_def: fixes t :: "Core.term" shows "typ_of t \<equiv> typ_of1 [] t"
continuous_within_avoid: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "'c" assumes "continuous (at x within s) f"   and "f x \<noteq> a" shows "\<exists>e>0. \<forall>y\<in>s. dist x y < e \<longrightarrow> f y \<noteq> a"
ex: shows "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
lmult_0: fixes m :: "'c" assumes "m \<in> carrier M" shows "\<zero> \<odot>\<^bsub>M\<^esub> m = \<zero>\<^bsub>M\<^esub>"
fls_deriv_add: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_deriv (f + g) = fls_deriv f + fls_deriv g"
power_order_eq_one: fixes G :: "('b, 'c) monoid_scheme"   and a :: "'b" assumes "comm_group G"   and "finite (carrier G)"   and "a \<in> carrier G" shows "a [^]\<^bsub>G\<^esub> card (carrier G) = \<one>\<^bsub>G\<^esub>"
differentiableI: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and net :: "'b filter" assumes "(f has_derivative f') net" shows "f differentiable net"
trans_mult2_s: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "s O ns \<subseteq> s"   and "refl ns"   and "trans ns" shows "trans (mult2_s ns s)"
more_than_one_mset_mset_diff: fixes a :: "'a"   and M :: "'a multiset" assumes "a \<in># M - {#a#}" shows "set_mset (M - {#a#}) = set_mset M"
permutes_bij_inv_into: fixes p :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "p permutes A"   and "bij_betw f A B" shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B"
bidirected_digraph: fixes G :: "'a pair_pre_digraph" assumes "pair_bidirected_digraph G" shows "bidirected_digraph (with_proj G) (swap_in (parcs G))"
dom_b: shows "\<D>\<^sub>\<circ> b = J"
excess'_other: fixes x :: "nat" assumes "x \<noteq> u"   and "x \<noteq> v" shows "excess f' x = excess f x"
scene_indep_self_compl: fixes A :: "'a scene" shows "A \<bowtie>\<^sub>S - A"
symp: shows "symp (\<lambda>a b. cmp a b = Equiv)"
applyEdge_dRETURN: fixes prog :: "program"   and e :: "edge"   and p :: "pState"   and g :: "gState\<^sub>I" shows "nres_of (dRETURN (applyEdge_impl prog e p g)) \<le> applyEdge prog e p g"
axioms_4: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
phiDefs_def: fixes p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and n :: "'node" shows "uninst.phiDefs p g n \<equiv> {v. (n, v) \<in> dom (p g)}"
inv_unique': fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<otimes> y = \<one>"   and "y \<otimes> x = \<one>" shows "y = inv x"
fbd_surj: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>" shows "\<exists>f. bd\<^sub>\<F> f = \<phi>"
obj_simps_1: fixes a :: "'a" assumes "B.obj a" shows "B.arr a"
holf0: fixes n :: "nat" shows "\<F> n holomorphic_on ball z0 r"
h'_f: fixes x :: "'v" assumes "x \<in> X" shows "integral\<^sup>N (count_space (E' `` X)) (h' x) = f x"
HSem_restr: fixes h :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> 'Value" shows "\<lbrace>h\<rbrace>\<rho> f|` (- domA h) = \<lbrace>h\<rbrace>\<rho>"
const_ConstD: fixes u :: "'a \<Rightarrow> real"   and X :: "'a set"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and k :: "'a \<Rightarrow> nat"   and c :: "'a"   and d :: "nat" assumes "u \<in> Regions.region X I r"   and "Regions.valid_region X k I r"   and "c \<in> X"   and "u c = real d"   and "d \<le> k c" shows "I c = Regions.intv.Const d"
path_last_io_target: fixes M :: "('d, 'e, 'f) FSM"   and xs :: "('d \<times> 'e) list"   and tr :: "'f list"   and q :: "'f" assumes "path M (xs || tr) q"   and "length xs = length tr"   and "0 < length xs" shows "last tr \<in> io_targets M q xs"
assms_1: shows "0 < the (eval r_result1 [t, i, x])"
property_empty_interval: fixes P :: "'a set \<Rightarrow> bool" assumes "\<forall>a b. content (cbox a b) = 0 \<longrightarrow> P (cbox a b)" shows "P {}"
Evar: fixes x :: "nat"   and \<rho> :: "(nat \<times> val) list" shows "E (EVar x) \<rho> = {v. \<exists>v'. lookup \<rho> x = Some v' \<and> v \<sqsubseteq> v'}"
eq1: shows "fps_radical r k a ^ k $ 0 = (\<Prod>j = 0..h. fps_radical r k a $ replicate k 0 ! j)"
cong_r_def: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" shows "cong_r F a b \<equiv> d1 a b \<in> F"
card_of_Fpow_lists: fixes A :: "'b set" shows "|Fpow A| \<le>o |lists A|"
differentiable_on_eqI: fixes g :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "g differentiable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x"   and "open S" shows "f differentiable_on S"
op_ivl_rep_of_sets_def: fixes XS :: "'c set set" shows "op_ivl_rep_of_sets XS = FORWEAK XS (RETURN (0::'c, 0::'c)) op_ivl_rep_of_set (\<lambda>(i, s) (i', s'). RETURN (inf i i' ::: lv_rel, sup s s' ::: lv_rel))"
disjoint: shows "{Suc m} \<inter> {0..<Suc m} = {}"
assms_6: shows "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>"