hcju/isabelleps huggingface dataset:
ConjI': fixes A0 :: "formula"   and Gamma :: "formula list"   and A :: "(formula list \<times> formula list set) set"   and A1 :: "formula" assumes "A0 # Gamma \<in> deductions A"   and "A1 # Gamma \<in> deductions A"   and "Contrs \<subseteq> A"   and "Conjs \<subseteq> A"   and "Perms \<subseteq> A" shows "FConj Pos A0 A1 # Gamma \<in> deductions A"
a_1: shows "lead_coeff f \<noteq> (0::'a)"
norm_ge_square: fixes a :: "real"   and x :: "'a" shows "(a \<le> norm x) = (a \<le> 0 \<or> a\<^sup>2 \<le> inner x x)"
isOK_error: fixes x :: "'a" shows "isOK (Inl x) = False"
inject_2: fixes x2 :: "('e, 'f, 'g) Tree"   and y2 :: "('e, 'f, 'g) Tree" shows "(tNot x2 = tNot y2) = (x2 = y2)"
l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" shows "l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs to_tree_order \<equiv> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes to_tree_order"
order_iff_strict: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry" shows "(b \<preceq> a) = (b \<prec> a \<or> a = b)"
ccw_hd_last_half_segments_dirvec: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "1 < length (half_segments_of_aform X)" shows "ccw' 0 (dirvec (hd (half_segments_of_aform X))) (dirvec (last (half_segments_of_aform X)))"
Outside_def: fixes R :: "('c \<times> 'd) set"   and X :: "'c set" shows "Outside' X R = R - X \<times> Range R"
distinct_names: shows "all_distinct (Node (Node Tip a False Tip) b False (Node Tip c False (Node Tip f False Tip)))"
frameChainAppend: fixes xvec :: "name list"   and yvec :: "name list"   and F :: "'d frame" shows "\<lparr>\<nu>*(xvec @ yvec)\<rparr>F = \<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>*yvec\<rparr>F"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 (src f) (src f)"
not_eq_order_implies_strict: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<noteq> b"   and "b \<subseteq># a" shows "b \<subset># a"
weakly_preserves_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "D.isomorphic (F (src\<^sub>C \<mu>)) (src\<^sub>D (F \<mu>))"
Fun_canonical: fixes v :: "trm"   and T\<^sub>1 :: "POPLmark.type"   and T\<^sub>2 :: "POPLmark.type" assumes "[] \<turnstile> v : T\<^sub>1 \<rightarrow> T\<^sub>2"   and "v \<in> value" shows "\<exists>t S. v = (\<lambda>:S. t)"
id_func_preserves_id: shows "preserves_id (id_func AA)"
hyps_3: obtains xs :: "(accent \<times> 'b) list" where "snd ` set xs \<subseteq> M" and "(x, v) \<in> local.lconv xs"
rebase_id: fixes x :: "'p mod_ring" shows "@@x = x"
induced_automorphism_chamber_map: fixes C :: "'a set" assumes "folding_g.chamber C" shows "folding_g.chamber (\
 ` C)"
valid_unSimpGraph_def: fixes G :: "('a, 'b) graph" shows "valid_unSimpGraph G \<equiv> valid_unMultigraph G \<and> valid_unSimpGraph_axioms G"
card_of_mono1: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "ordLeq3 (card_of A) (card_of B)"
map_def: fixes f :: "'a" shows "local.map f \<equiv> if A.arr f then g else B.null"
induces_meta_adjunction: shows "meta_adjunction hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta>\<epsilon>.\<phi> \<eta>\<epsilon>.\<psi>"
supp_singleton: fixes x :: "'b" shows "supp {x} = supp x"
poly_power_zero_iff: fixes p :: "'a poly"   and n :: "nat"   and x :: "'a" shows "(poly (p ^ n) x = (0::'a)) = (n \<noteq> 0 \<and> poly p x = (0::'a))"
xcpt_subcls_Throwable: fixes C :: "String.literal"   and wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "C \<in> sys_xcpts"   and "wf_prog wf_md P" shows "P \<turnstile> C \<preceq>\<^sup>* Throwable"
simps: fixes q :: "'d"   and l :: "'e"   and Qs :: "'d set list"   and qr :: "'d"   and lr :: "'e"   and qsr :: "'d list" shows "r_matchc q l Qs qr \<rightarrow> lr qsr = (q = qr \<and> l = lr \<and> r_match Qs qsr)"
add: fixes a :: "'a"   and b :: "'a" shows "a \<oplus>\<^bsub>law_restrict A\<^esub> b = a \<oplus>\<^bsub>law_restrict B\<^esub> b"
span_card_ge_dim: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "V \<subseteq> span B"   and "finite B" shows "dim V \<le> card B"
assign_annos_eval_elim: fixes x :: "'Var"   and e :: "'AExp"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>(x \<leftarrow> e) \<otimes> annos, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "c' = Stop \<and> mds' = mds \<oplus> annos"
affine_dependent_imp_collinear_3: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "affine_dependent {a, b, c}" shows "collinear {a, b, c}"
dom_comp: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "CCC.seq g f" shows "CCC.dom (CCC.comp g f) = CCC.dom f"
fundamental_theorem_of_linear_inequalities_A_imp_D: fixes A :: "'a vec set"   and b :: "'a vec" assumes "A \<subseteq> carrier_vec n"   and "finite A"   and "b \<in> cone A" shows "\<nexists>c. c \<in> carrier_vec n \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)"
sup_top_right: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and top :: "'c"   and x :: "'c" assumes "class.bounded_lattice_top inf less_eq less sup top" shows "sup x top = top"
smc_SemiCAT_Comp_vdomain: fixes \<alpha> :: "V" shows "\<D>\<^sub>\<circ> (smc_SemiCAT \<alpha>\<lparr>Comp\<rparr>) = composable_arrs (smc_SemiCAT \<alpha>)"
lookup_impl_of: fixes t :: "('a, 'b) RBT.rbt" shows "rbt_lookup (impl_of t) = lookup t"
div_Suc': fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "Suc m div n = (if m mod n < n - Suc 0 then m div n else Suc (m div n))"
lens_plus_right_sublens: fixes Y :: "'a \<Longrightarrow> 'b"   and Z :: "'c \<Longrightarrow> 'b"   and X :: "'d \<Longrightarrow> 'b" assumes "vwb_lens Y"   and "Y \<bowtie> Z"   and "X \<subseteq>\<^sub>L Z" shows "X \<subseteq>\<^sub>L Y +\<^sub>L Z"
RSD_set_rel_def: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" shows "RSD_set_rel S A B = (indiff_set S B \<and> (\<forall>x\<in>A. \<forall>y\<in>B. S x y))"
assms_1: shows "n \<equiv> LEAST n. A $ i $ n \<noteq> (0::'b)"
rev_subsetD: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
bounded_subset_cbox_symmetric: fixes S :: "'c set" assumes "bounded S" obtains a :: "'c" where "S \<subseteq> cbox (- a) a"
cset_count_inj_seq: fixes A :: "'c cset" shows "inj_on (cset_count A) (rcset A)"
assms_2: shows "tmf_ivl_cs prec ord I a f = Some cs"
comp_fun_commute: fixes y :: "'a"   and x :: "'a" shows "add_mset y \<circ> add_mset x = add_mset x \<circ> add_mset y"
coset_neq_imp_empty_inter: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "subgroup H G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "H #> a \<noteq> H #> b" shows "(H #> a) \<inter> (H #> b) = {}"
smult_normalize_content_ff: fixes p :: "'b fract poly" shows "smult (content_ff p) (normalize_content_ff p) = p"
resChainPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<leadsto>[Rel] Q"   and "eqvt Rel"   and "xvec \<sharp>* \<Psi>"   and "\<And>\<Psi>' R S y. \<lbrakk>(\<Psi>', R, S) \<in> Rel; y \<sharp> \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>S) \<in> Rel" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<leadsto>[Rel] \<lparr>\<nu>*xvec\<rparr>Q"
wellformed_eq: fixes p :: "('a, 'b) Combinators list" shows "DenyAll \<in> set p \<longrightarrow> wellformed_policy1 p = wellformed_policy1_strong p"
nths_only_one: fixes xs :: "'a list"   and I :: "nat set"   and j :: "nat" assumes "{i. i < length xs \<and> i \<in> I} = {j}" shows "nths xs I = [xs ! j]"
has_products_def: fixes I :: "'b set" shows "has_products I \<equiv> I \<noteq> UNIV \<and> (\<forall>J D. discrete_diagram J (\<cdot>) D \<and> Collect (partial_magma.arr J) = I \<longrightarrow> (\<exists>a. has_as_product J D a))"
logical_subst_monotonicity: fixes \<Gamma> :: "(name \<times> ty) list"   and \<Gamma>' :: "(name \<times> ty) list"   and \<theta> :: "(name \<times> trm) list"   and \<theta>' :: "(name \<times> trm) list"   and \<Gamma>'' :: "(name \<times> ty) list" assumes "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"   and " \<Gamma>' \<subseteq> \<Gamma>'' "   and "valid \<Gamma>''" shows "\<Gamma>'' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
return_measurable: fixes N :: "'a measure" shows "return N \<in> N \<rightarrow>\<^sub>M subprob_algebra N"
single_iteration_inv: fixes i' :: "nat"   and i :: "nat"   and j' :: "nat"   and j :: "nat"   and n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i' \<le> i"   and "j' \<le> j"   and "j \<le> n" shows "fwi m n k i j i' j' = fwi m n k i' j' i' j'"
assms_2: assumes "subgraph H G \<and> verts G = verts H" shows "P"
IH_2: shows "t \<prec> t2 \<longrightarrow> (t, t2) \<in> trm_ord"
B_def: shows "B = {v \<in> natpermute k (m + 1). k \<notin> set v}"
return_time_some: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.returns_to f X P x" shows "auto_ll_on_open.return_time f X P x = (SOME t. 0 < t \<and> t \<in> auto_ll_on_open.existence_ivl0 f X x \<and> auto_ll_on_open.flow0 f X x t \<in> P \<and> (\<forall>s\<in>{0<..<t}. auto_ll_on_open.flow0 f X x s \<notin> P))"
simps_2: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option" shows "not_MT C [] = False"
finite_vars_term: fixes t :: "('b, 'a) Term.term" shows "finite (fv t)"
OclValid_def: fixes \<tau> :: "'a state \<times> 'a state"   and P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "\<tau> \<Turnstile> P \<equiv> P \<tau> = true \<tau>"
surj_const_mult: fixes a :: "'c" assumes "a \<in> carrier V" shows "(\<oplus>\<^bsub>V\<^esub>) a ` carrier V = carrier V"
Rep_ratfps_inverse: fixes x :: "'b ratfps" shows "Abs_ratfps (Rep_ratfps x) = x"
naturality: fixes f :: "'b" assumes "Adj.D.arr f" shows "Adj.\<eta> (Adj.D.cod f) \<cdot>\<^sub>D Adj.D.map f = Adj.GF.map f \<cdot>\<^sub>D Adj.\<eta> (Adj.D.dom f)"
ide_cod: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.ide (V.cod f)"
rep_eq: fixes x :: "'e"   and xa :: "'e set['f]" shows "bmember x xa = (x \<in> set_bset xa)"
B: fixes s :: "trm"   and n :: "trm"   and l :: "stack" assumes "t = [s]"   and "k = [y]n\<ggreater>l"   and "r = n[y::=s] \<star> l" shows "P"
fls_times_conv_base_factor_times: fixes f :: "'a fls"   and g :: "'a fls" shows "f * g = fls_shift (- (fls_subdegree f + fls_subdegree g)) (fls_base_factor f * fls_base_factor g)"
of_real_neg_numeral: fixes w :: "num" shows "of_real (- numeral w) = - numeral w"
le_supI2: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x b" shows "less_eq x (sup a b)"
plus_except: fixes p :: "'d \<Rightarrow>\<^sub>0 'e"   and t :: "'d" shows "p = monomial (lookup p t) t + except p {t}"
monad_fail_alt_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and fail :: "'c"   and alt :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_fail_alt return bind fail alt" shows "monad_fail_alt (return_writer return) (bind_writer return bind) (fail_writer fail) (alt_writer alt)"
closed_sequentially: fixes s :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "closed s"   and "\<forall>n. f n \<in> s"   and "f \<longlonglongrightarrow> l" shows "l \<in> s"
comp_in_homI': fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "VV.arr f"   and "VV.arr g"   and "VV.dom f = a"   and "VV.cod g = c"   and "VV.dom g = VV.cod f" shows "VV.in_hom (VV.comp g f) a c"
leg0_composite: shows "f_src.leg0 = src.leg0 \<cdot> f_src.prj\<^sub>0"
Liminf_eqI: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'd"   and x :: "'d" assumes "\<And>P. eventually P F \<Longrightarrow> Inf (f ` Collect P) \<le> x"   and "\<And>y. (\<And>P. eventually P F \<Longrightarrow> Inf (f ` Collect P) \<le> y) \<Longrightarrow> x \<le> y" shows "Liminf F f = x"
simps_3: fixes e :: "expr" shows "radicals (Inverse e) = radicals e"
fully_faithful_functor_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c" shows "fully_faithful_functor A B F \<equiv> faithful_functor A B F \<and> full_functor A B F"
rotate_conv_mod: fixes n :: "nat"   and xs :: "'g list" shows "rotate n xs = rotate (n mod length xs) xs"
isomorphic_to_irr_right: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleIso G smult V smult' T W"   and "IrrFinGroupRepresentation G smult' W" shows "IrrFinGroupRepresentation G smult V"
diff_intersect_right_idem: fixes M :: "'k multiset"   and N :: "'k multiset" shows "M - N \<inter># M = M - N"
simps_1: fixes uu :: "('c \<times> 'c) set"   and uv :: "'d \<Rightarrow> 'c"   and uw :: "'c" shows "sinks uu uv uw [] = {}"
Assign: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> Suc 0 \<Down> s[a/x]"
gd_powerprod_def: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_strict :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "gd_powerprod ord ord_strict \<equiv> ordered_powerprod ord ord_strict"
cone_iff_finite_cone: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n"   and "finite A" shows "n.cone A = n.finite_cone A"
simps_2: fixes n :: "nat"   and p :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "repeat (Suc n) p = p ;; repeat n p"
gate_def: fixes n :: "nat"   and A :: "complex Matrix.mat" shows "gate n A \<equiv> dim_row A = 2 ^ n \<and> square_mat A \<and> unitary A"
map_1: fixes f1 :: "'f \<Rightarrow> 'b"   and f2 :: "'g \<Rightarrow> 'd"   and f3 :: "'h \<Rightarrow> 'e"   and x1 :: "'f" shows "map_generat f1 f2 f3 (Pure x1) = Pure (f1 x1)"
simps_2: fixes x :: "'e"   and xs :: "'e list" shows "stream_raw (x # xs) = Yield x xs"
inf_cancel_left2: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "- x \<sqinter> a \<sqinter> (x \<sqinter> b) = \<bottom>"
closed_UN: fixes "open" :: "'b set \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'b set" assumes "class.topological_space open"   and "finite A"   and "\<forall>x\<in>A. topological_space.closed open (B x)" shows "topological_space.closed open (\<Union> (B ` A))"
F2map_congL: fixes x :: "('e, 'd, 'f) F2"   and f :: "'d \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'f" assumes "\<forall>a\<in>F2set2 x. f a = a"   and "\<forall>a\<in>F2set3 x. g a = a" shows "F2map id f g x = x"
dfree_Plus: fixes \<theta>\<^sub>1 :: "('a, 'b) trm"   and \<theta>\<^sub>2 :: "('a, 'b) trm" assumes "dfree \<theta>\<^sub>1"   and "dfree \<theta>\<^sub>2" shows "dfree (Plus \<theta>\<^sub>1 \<theta>\<^sub>2)"
\<epsilon>_map_simp: fixes x :: "'c" assumes "C.ide x" shows "\<epsilon>.map x = \<psi> x (G x)"
distrib_left_NO_MATCH: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'b"   and y :: "'b"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring plus times"   and "NO_MATCH (x div y) a" shows "times a (plus b c) = plus (times a b) (times a c)"
T_VAR: fixes \<Gamma> :: "(W.var \<times> tyS) list"   and x :: "W.var"   and S :: "tyS"   and T :: "ty" assumes "valid \<Gamma>"   and "(x, S) \<in> set \<Gamma>"   and "T \<prec> S" shows " \<Gamma> \<turnstile> Var x : T "
obj_def: fixes a :: "'a" shows "S.obj a \<equiv> S.arr a \<and> S.src a = a"
homotopic_with_imp_property: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "homotopic_with P X Y f g" shows "P f \<and> P g"
f_not_zero: shows "monom (UP R) \<one> d \<ominus>\<^bsub>UP R\<^esub> monom (UP R) \<one> 0 \<noteq> \<zero>\<^bsub>UP R\<^esub>"
set_dropWhileD: fixes x :: "'g"   and P :: "'g \<Rightarrow> bool"   and xs :: "'g list" assumes "x \<in> set (dropWhile P xs)" shows "x \<in> set xs"
ray_to_rel_frontier: fixes S :: "'a set"   and a :: "'a"   and l :: "'a" assumes "bounded S"   and "a \<in> rel_interior S"   and "a + l \<in> affine hull S"   and "l \<noteq> (0::'a)" obtains d :: "real" where "0 < d" and "a + d *\<^sub>R l \<in> rel_frontier S" and "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S"
emeasure_pair_measure_Times: fixes A :: "'c set"   and N :: "'c measure"   and B :: "'a set" assumes "A \<in> sets N"   and "B \<in> sets M1" shows "emeasure (N \<Otimes>\<^sub>M M1) (A \<times> B) = emeasure N A * emeasure M1 B"
satPB_tPB: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "Ik.wtE \<xi>" shows "TE.satPB (eenv \<xi>) (tC ` \<Phi>) = Ik.satPB \<xi> \<Phi>"
fder: fixes x :: "real" assumes "x \<in> {a..b} - k" shows "(f has_derivative (\<lambda>h. 0::'a)) (at x within {a..b})"
Suc: fixes lc :: "'a"   and f :: "'a poly"   and g :: "'a poly"   and h :: "'a poly"   and i :: "nat" shows "map_poly hom (divide_poly_main lc f g h i j) = divide_poly_main (hom lc) (map_poly hom f) (map_poly hom g) (map_poly hom h) i j"
sfilter_eq: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b stream"   and x :: "'b"   and s' :: "'b stream" assumes "ev (holds P) s" shows "(sfilter P s = x ## s') = (P x \<and> ((\<lambda>xs. \<not> holds P xs) suntil (\<lambda>xs. HLD {x} xs \<and> nxt (\<lambda>s. sfilter P s = s') xs)) s)"
euler_maclaurin_strong_raw_nat: fixes a :: "nat"   and b :: "nat"   and N :: "nat"   and Y :: "real set"   and fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "0 < N"   and "finite Y"   and "fs 0 = f"   and "\<And>k. k \<le> N \<Longrightarrow> continuous_on {real a..real b} (fs k)"   and "\<And>k x. \<lbrakk>k < N; x \<in> {real a..real b} - Y\<rbrakk> \<Longrightarrow> (fs k has_vector_derivative fs (Suc k) x) (at x)" shows "(\<Sum>i\<in>{a<..b}. f (real i)) - integral {real a..real b} f = (\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R (fs k (real b) - fs k (real a))) + EM_remainder' N (fs N) (real a) (real b)"
ldrop_add: fixes t :: "'b llist"   and i :: "nat"   and k :: "nat" shows "t \<up> (i + k) = t \<up> i \<up> k"
assms_2: shows "sspace_DAG (prob_proj PROB vs) lss"
LPDs_safe: fixes g :: "safety"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex" assumes "safe_regex Futu g (from_mregex mr \<phi>s)"   and "ms \<in> LPDs mr" shows "safe_regex Futu g (from_mregex ms \<phi>s)"
not_iSince: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(\<not> (P t1. t1 \<S> t2 I. Q t2)) = (\<box> t I. Q t \<longrightarrow> (\<diamond> t' I \<down>> t. \<not> P t'))"
AnnoFAcc: fixes is_lub :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'b expr"   and e' :: "'b expr"   and U :: "ty"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" assumes "is_lub,P,E \<turnstile> e \<leadsto> e'"   and "is_lub,P,E \<turnstile> e' :: U"   and "class_type_of' U = \<lfloor>C\<rfloor>"   and "P \<turnstile> C sees F:T (fm) in D"   and "is_Array U \<longrightarrow> F \<noteq> length" shows "is_lub,P,E \<turnstile> e\<bullet>F \<leadsto> e'\<bullet>F{D}"
subgroupI: fixes H :: "'a set" assumes "H \<subseteq> carrier (G\<lparr>carrier := S\<rparr>)"   and "H \<noteq> {}"   and "\<And>a. a \<in> H \<Longrightarrow> inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> a \<in> H"   and "\<And>a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> b \<in> H" shows "subgroup H (G\<lparr>carrier := S\<rparr>)"
oracle_encrypt4_def: shows "oracle_encrypt4 = (\<lambda>(k_prf, k_upf) s (msg1, msg0). case length msg1 = prf_clen \<and> length msg0 = prf_clen of True \<Rightarrow> spmf_of_set prf_domain \<bind> (\<lambda>x. spmf_of_set (nlists UNIV prf_clen) \<bind> (\<lambda>P. let c = P; t = upf_fun k_upf (x @ c) in return_spmf (Some (x, c, t), s))) | False \<Rightarrow> return_spmf (None, s))"
prj1_simps_arr: fixes f :: "'a"   and g :: "'a" shows "arr \<p>\<^sub>1[f, g] = cospan f g"
lastPrecedesNoElement: fixes l :: "'a list" assumes "uniq l" shows "\<nexists>a. a \<noteq> last l \<and> precedes (last l) a l"
set_times_elim: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "x \<in> A * B" obtains a :: "'a" and b :: "'a" where "x = a * b" and "a \<in> A" and "b \<in> B"
ord_pos: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G"   and "x [^] n = \<one>"   and "0 < n" shows "0 < ord x"
reduced_GB_is_reduced_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "is_reduced_GB (reduced_GB F)"
id: shows "four_block_mat P1 (0\<^sub>m n m) (0\<^sub>m m n) P2 * four_block_mat B1 UR LL B2 * four_block_mat Q1 (0\<^sub>m n m) (0\<^sub>m m n) Q2 = four_block_mat (P1 * B1 * Q1) (P1 * UR * Q2) (P2 * LL * Q1) (P2 * B2 * Q2)"
bv_to_int_Cons0: fixes bs :: "bit list" shows "bv_to_int (\<zero> # bs) = int (bv_to_nat bs)"
simplicial_complex_def: fixes \<C> :: "'b set set" shows "simplicial_complex \<C> \<equiv> finite \<C> \<and> (\<forall>S\<in>\<C>. \<exists>n. n simplex S) \<and> (\<forall>F S. S \<in> \<C> \<and> F face_of S \<longrightarrow> F \<in> \<C>) \<and> (\<forall>S S'. S \<in> \<C> \<and> S' \<in> \<C> \<longrightarrow> S \<inter> S' face_of S)"
simps_51: fixes f1 :: "nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and f3 :: "nat set \<Rightarrow> nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and f6 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'd"   and f7 :: "'d"   and x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'e" shows "(case x31\<not>x32:arrive(x33) of x:*cast(xa) \<Rightarrow> f1 x xa | x:deliver(xa) \<Rightarrow> f2 x xa | x\<not>xa:arrive(xb) \<Rightarrow> f3 x xa xb | connect(x, xa) \<Rightarrow> f4 x xa | disconnect(x, xa) \<Rightarrow> f5 x xa | x:newpkt(xa, xb) \<Rightarrow> f6 x xa xb | \<tau>\<^sub>n \<Rightarrow> f7) = f3 x31 x32 x33"