hcju/isabelleps huggingface dataset:
surjective: fixes r :: "('e, 'f) Group_scheme" shows "r = \<lparr>carrier = carrier r, Group.top = (\<cdot>\<^bsub>r\<^esub>), iop = iop r, one = \<one>\<^bsub>r\<^esub>, \<dots> = Group.more r\<rparr>"
simps_1: fixes \<alpha> :: "game" shows "Dualo (Agame \<alpha>) = Agame \<alpha>^d"
pdouble_residue_eq: fixes a :: "int"   and p :: "int \<times> int \<times> int" shows "pdouble a p = mpdouble (int p) a p"
nnf_simps_1: fixes P :: "bool"   and Q :: "bool" shows "(\<not> (P \<and> Q)) = (\<not> P \<or> \<not> Q)"
LVE_LV2: shows "\<Union> (SS ` LVE) \<subseteq> \<Union> (funStar SS ` LVE)"
x_2: shows "(f \<circ> closest_point S) x = x"
simps_1: fixes iostep :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e" shows "mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e [] iostep \<sigma> = Some ([], \<sigma>)"
Separation_axioms: shows "Separation (remove_vertex v) v0 v1 S"
Rep_quot: fixes x :: "'b quot" shows "Rep_quot x \<in> quot"
Max_by_ge: fixes A :: "'a set"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "x \<in> A" shows "f x \<le> f (Max_by f A)"
closed_Inter: fixes "open" :: "'d set \<Rightarrow> bool"   and K :: "'d set set" assumes "class.topological_space open"   and "\<forall>S\<in>K. topological_space.closed open S" shows "topological_space.closed open (\<Inter> K)"
elims: fixes x :: "bool"   and y :: "ternaryvalue" assumes "bool_to_ternary x = y" obtains "x" and "y = TernaryTrue"   | "\<not> x" and "y = TernaryFalse"
inj_on_id2: fixes A :: "'c set" shows "inj_on (\<lambda>x. x) A"
Fun_in_terms_of_comp: fixes f :: "'s" assumes "arr f" shows "Fun f = restrict (img \<circ> (\<cdot>) f \<circ> mkPoint (local.dom f)) (Dom f)"
axioms_2: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" assumes "limit_cone J C D a \<chi>" shows "category J"
coprime_elems: fixes A :: "'a set"   and B :: "'a set" assumes "ideal R A"   and "ideal R B"   and "coprime_ideals R A B" shows "\<exists>a\<in>A. \<exists>b\<in>B. a \<plusminus> b = 1\<^sub>r"
compact_simple_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "simple_path g" shows "compact (path_image g)"
genby_add_closed: fixes g :: "'a"   and S :: "'a set"   and h :: "'a" assumes "g \<in> \<langle>S\<rangle>"   and "h \<in> \<langle>S\<rangle>" shows "g + h \<in> \<langle>S\<rangle>"
coeff: fixes x :: "'a poly" shows "poly.coeff x \<in> {f. \<forall>\<^sub>\<infinity>n. f n = (0::'a)}"
simps: fixes uu :: "('e \<times> 'f \<times> 'd) tree"   and uv :: "'e"   and uw :: "'f"   and a :: "'d"   and ux :: "('e \<times> 'f \<times> 'd) tree" shows "pst_getmin \<langle>uu, (uv, uw, a), ux\<rangle> = a"
real_sqrt_sum_squares_mult_ge_zero: fixes x :: "real"   and y :: "real"   and xa :: "real"   and ya :: "real" shows "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2))"
gf: fixes x :: "'a" assumes "x \<in> S" shows "g x = f x"
axioms_4: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" assumes "Limit.cone J C D a \<chi>" shows "constant_functor J C a"
connected_space_closedin_eq: fixes X :: "'c topology" shows "connected_space X = (\<nexists>E1 E2. closedin X E1 \<and> closedin X E2 \<and> E1 \<union> E2 = topspace X \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
characteristic_iso_1: fixes f :: "'a" assumes "Left.ide f" shows "Left.in_hom (local.inv \<a>[trg f, trg f, f]) (trg f \<star> trg f \<star> f) ((trg f \<star> trg f) \<star> f)"
cqt_further_4: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<not>(\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) in v]"
linearI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>R b) = r *\<^sub>R f b" shows "linear f"
ij_le1_1: shows "i < dim_row (matricize I (A + B))"
tllist_all2_tfinite1_terminalD: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist" assumes "tllist_all2 P Q xs ys"   and "tfinite xs" shows "Q (terminal xs) (terminal ys)"
path_decomp_2: fixes xs :: "'a list"   and ys :: "'a list" assumes "path (xs @ ys)" shows "path ys"
filter_a1_map_rel1: shows "filter_a1_map pos tp a1_map' = rel1"
t1_space_singleton_Inter_open: fixes X :: "'b topology" shows "t1_space X = (\<forall>x\<in>topspace X. \<Inter> {U. openin X U \<and> x \<in> U} = {x})"
simps: fixes base :: "nat"   and a :: "hmultiset" assumes "2 \<le> base" shows "well_base\<^sub>h base a = (\<exists>M. a = M \<and> (\<forall>N\<in>#hmsetmset M. well_base\<^sub>h base N) \<and> well_base base (hmsetmset M))"
strip_while_rev: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "strip_while P (rev xs) = rev (dropWhile P xs)"
rep_eq: fixes x :: "'a poly"   and xa :: "'a poly" shows "poly.coeff (x + xa) = (\<lambda>n. poly.coeff x n + poly.coeff xa n)"
connectedin_sing: fixes X :: "'c topology"   and a :: "'c" shows "connectedin X {a} = (a \<in> topspace X)"
ccw_uminus3: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "det3 a b c \<noteq> 0" shows "ccw (- a) (- b) (- c) = ccw a b c"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_def: fixes A :: "'a set" shows "DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<equiv> \<mu> x. (\<sqinter>xa\<in>A \<rightarrow>  x) |-| SKIP"
that: fixes ns' :: "'node list" assumes "g \<turnstile> n-ns @ ns'\<rightarrow>defNode g \<phi>\<^sub>r"   and "set (butlast (ns @ ns')) \<inter> set ms = {}" shows "thesis"
mat_gt_arc_compat: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<ge>\<^sub>m B"   and "mat_gt_arc B C"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "C \<in> carrier_mat nr nc" shows "mat_gt_arc A C"
def_hashmap_size_int_def: shows "def_hashmap_size = (\<lambda>_. 16)"
coeffs_eq_Nil: fixes p :: "'a poly" shows "(coeffs p = []) = (p = 0)"
step_3_c_dims_main: fixes x :: "'b"   and l :: "nat"   and k :: "nat"   and i :: "(nat \<times> nat) list"   and A :: "'b mat" shows "dim_row (step_3_c x l k i A) = dim_row A \<and> dim_col (step_3_c x l k i A) = dim_col A"
add_edge_def: fixes v :: "'c"   and e :: "'d"   and v' :: "'c"   and g :: "('c, 'd, 'e) graph_scheme" shows "add_edge v e v' g \<equiv> \<lparr>nodes = {v, v'} \<union> nodes g, edges = insert (v, e, v') (edges g)\<rparr>"
le_T_adm: fixes u :: "'a \<Rightarrow> 'b process"   and v :: "'a \<Rightarrow> 'b process" assumes "cont u"   and "monofun v" shows "adm (\<lambda>x. u x \<sqsubseteq>\<^sub>T v x)"
update_def: fixes a :: "'d array"   and i :: "nat"   and x :: "'d"   and h :: "heap" shows "Array.update a i x h = Array.set a ((Array.get h a)[i := x]) h"
vars\<^sub>s\<^sub>s\<^sub>t_is_fv\<^sub>s\<^sub>s\<^sub>t_bvars\<^sub>s\<^sub>s\<^sub>t: fixes S :: "('d, 'c) stateful_strand_step list" shows "vars\<^sub>s\<^sub>s\<^sub>t S = fv\<^sub>s\<^sub>s\<^sub>t S \<union> bvars\<^sub>s\<^sub>s\<^sub>t S"
GF_advice_af_letter: fixes x :: "'b set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set" assumes "(x ## w) \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu>" shows "w \<Turnstile>\<^sub>n af_letter \<phi> x[X]\<^sub>\<nu>"
wfp_on_tranclp_imp_wfp_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wfp_on P\<^sup>+\<^sup>+ A" shows "wfp_on P A"
select_convs_2: fixes Leg0 :: "'b"   and Leg1 :: "'b"   and more :: "'c" shows "Leg1 \<lparr>Leg0 = Leg0, Leg1 = Leg1, \<dots> = more\<rparr> = Leg1"
lt_1: shows "x \<otimes> z \<prec> y \<otimes> z"
edge_pred_succ: fixes a :: "'b" shows "edge_pred M (edge_succ M a) = a"
totient_divisor_sum: fixes n :: "nat" shows "(\<Sum>d | d dvd n. totient d) = n"
trancl_rel_chain: fixes x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set" shows "((x, y) \<in> R\<^sup>+) = (\<exists>l. l \<noteq> [] \<and> 1 < length l \<and> hd l = x \<and> last l = y \<and> rel_chain l R)"
fx_cover: shows "f x \<subseteq> (\<Union>a\<in>C. ball a (e / 2))"
ext: fixes g :: "'s"   and f :: "'s" assumes "g \<cdot>\<^sub>S f \<noteq> S.null" shows "S.seq g f"
reduction_signs_def: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "nat list list"   and matr :: "rat mat" shows "reduction_signs p qs signs subsets matr = take_indices signs (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets matr))"
tendstoD: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and l :: "'d"   and F :: "'e filter"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "filterlim f (topological_space.nhds open l) F"   and "0 < e" shows "\<forall>\<^sub>F x in F. dist (f x) l < e"
preserves_arr: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "arr (src f)"
independent_explicit_finite_subsets: fixes A :: "'b set" shows "independent A = (\<forall>S\<subseteq>A. finite S \<longrightarrow> (\<forall>u. (\<Sum>v\<in>S. u v *s v) = (0::'b) \<longrightarrow> (\<forall>v\<in>S. u v = (0::'a))))"
dim_subset_UNIV_cart: fixes S :: "(real, 'b) vec set" shows "dim S \<le> CARD('b)"
a_diff_kde: shows "a + complex_of_real kde \<in> closed_segment (a - complex_of_real d) (a + complex_of_real e)"
path_entry_prepend_conv: fixes p :: "nat list"   and n :: "nat" assumes "path_entry (g_E G) p n"   and "p \<noteq> []" shows "\<exists>v. path_entry (g_E G) (tl p) v \<and> (v, n) \<in> g_E G"
commutative_squareE: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.commutative_square f g h k" obtains "C.arr f" and "C.arr g" and "C.arr h" and "C.arr k" and "C.cod f = C.cod g" and "C.dom h = C.dom k" and "C.dom f = C.cod h" and "C.dom g = C.cod k" and "f \<cdot> h = g \<cdot> k"
D: shows "t = t1 @ [tick] \<and> u = u1 @ [tick] \<and> s setinterleaves ((t1, u1), insert tick (ev ` A))"
dominates_trans: fixes g :: "'a \<Rightarrow> real"   and F :: "'a filter"   and f :: "'a \<Rightarrow> real"   and h :: "'a \<Rightarrow> real" assumes "\<forall>\<^sub>F x in F. 0 < g x"   and "dominates F f g"   and "dominates F g h" shows "dominates F f h"
sym_Un_converse: fixes r :: "('a \<times> 'a) set" shows "sym (r \<union> r\<inverse>)"
valid_Exit_path_descending_path: fixes n :: "'node"   and as :: "'edge list" assumes "n -as\<rightarrow>\<^sub>\<surd>* (_Exit_)" obtains as' :: "'edge list" where "n -as'\<rightarrow>\<^sub>\<surd>* (_Exit_)" and "set (sourcenodes as') \<subseteq> set (sourcenodes as)" and "\<forall>a'\<in>set as'. intra_kind (kind a') \<or> (\<exists>Q f p. kind a' = Q\<hookleftarrow>\<^bsub>p\<^esub>f)"
h_f_g_eq: fixes z :: "'c" shows "h (f z, g z) = z"
mstfe_prefix_input_in_V: fixes V :: "'d list set"   and M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list" assumes "minimal_sequence_to_failure_extending V M1 M2 vs xs" shows "map fst vs \<in> V"
card_carrier_vec: fixes n :: "nat" shows "card (carrier_vec n) = CARD('b) ^ n"
T9: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "|~ \<box>[A]_v \<longrightarrow> _actrans A v"
isomorphic_def: fixes a :: "'c"   and a' :: "'c" shows "(a \<cong>\<^sub>C a') = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>C a'\<guillemotright> \<and> C.iso f)"
invert_side_of_triangle_2: fixes h :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VoV.arr h"   and "VoV.comp f g = h"   and "VoV.iso g" shows "VoV.seq h (VoV.inv g) \<and> f = VoV.comp h (VoV.inv g)"
lift_lift: fixes i :: "nat"   and k :: "nat"   and t :: "dB" assumes "i < k + 1" shows "lift (lift t i) (Suc k) = lift (lift t k) i"
composable_char\<^sub>P\<^sub>B: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<tau> :: "'b"   and \<sigma> :: "'b" assumes "prebicategory V H \<a>" shows "(H \<tau> \<sigma> \<noteq> partial_magma.null V) = (partial_magma.arr V \<sigma> \<and> partial_magma.arr V \<tau> \<and> prebicategory.some_src V H \<tau> = prebicategory.some_trg V H \<sigma>)"
sourcesI: fixes a :: "'a"   and \<mu> :: "'a" assumes "weak_unit a"   and "\<mu> \<star> a \<noteq> null" shows "a \<in> sources \<mu>"
pdouble_residue_eq: fixes m :: "int"   and a :: "int"   and p :: "int \<times> int \<times> int" assumes "residues m" shows "cring.pdouble (residue_ring m) a p = mpdouble m a p"
fps_lr_inverse_0_iff_1: fixes f :: "'a fps"   and x :: "'a" shows "(fps_nth (fps_left_inverse f x) 0 = (0::'a)) = (x = (0::'a))"
lookup_join: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t1 :: "('b, 'c) mapping_rbt"   and t2 :: "('b, 'c) mapping_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Mapping2.lookup (RBT_Mapping2.join f t1 t2) = (\<lambda>k. case RBT_Mapping2.lookup t1 k of None \<Rightarrow> RBT_Mapping2.lookup t2 k | Some v1 \<Rightarrow> Some (case RBT_Mapping2.lookup t2 k of None \<Rightarrow> v1 | Some v2 \<Rightarrow> f k v1 v2))"
a: shows "(a, b) \<in> BNF_Def.Gr (Domain R) (\<lambda>x. SOME y. (x, y) \<in> R)"
empty_Diff: fixes A :: "'a set" shows "{}\<setminus>A = {}"
asym_lex: fixes R :: "('g \<times> 'g) set" assumes "asym R" shows "asym (lex R)"
hyps_2: shows "Nom a at b in (ps, a) # branch"
cSup_singleton: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Sup (\<eta> x) = x"
valid_def: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> ('d \<times> 'c) set \<times> bool"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace> \<equiv> \<forall>s. P s \<longrightarrow> (\<forall>(r, s')\<in>fst (f s). Q r s')"
Let_def: fixes s :: "'c"   and f :: "'c \<Rightarrow> 'a" shows "Let s f \<equiv> f s"
tl_rev_take_Suc: fixes n :: "nat"   and l :: "'b list" assumes "n + 1 \<le> length l" shows "rev (take n l) = tl (rev (take (Suc n) l))"
filter_min_aux_supset: fixes ys :: "'a list"   and xs :: "'a list" shows "set ys \<subseteq> set (local.filter_min_aux xs ys)"
of_fun_def: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "IArray.of_fun f n = IArray (map f [0..<n])"
r_ifeq_def: shows "r_ifeq \<equiv> Cn 4 r_ifz [r_dummy 2 r_eq, recf.Id 4 2, recf.Id 4 3]"
that: fixes X :: "'a multiset"   and Z :: "'a multiset"   and Y :: "'a multiset" assumes "M = X + Z"   and "N = Y + Z"   and "\<forall>y\<in>#Y. \<exists>x\<in>#X. (x, y) \<in> R" shows "thesis"
v0_R'_path: shows "v0 \<leadsto>(v0 # R')\<leadsto> z"
map_int_transE_nv2T: 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"   and infTp :: "'c \<Rightarrow> bool"   and pol :: "'c \<Rightarrow> 'b \<Rightarrow> epol"   and grdOf :: "('a, 'b) lit list \<Rightarrow> ('a, 'b) lit \<Rightarrow> Preliminaries.var \<Rightarrow> ('a, 'b) lit"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and Tl :: "'a trm list"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "ModelIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf intT intF intP"   and "list_all (Signature.wt wtFsym arOf resOf) Tl"   and "Struct.wtE (ModelIkPolMcalc2C.I_intT infTp intT) \<xi>"   and "\<And>x. \<lbrakk>\<not> infTp (Signature.tpOfV x); \<exists>T\<in>set Tl. x \<in> ProblemIkPol.nv2T T\<rbrakk> \<Longrightarrow> \<exists>l. Signature.wtL wtFsym wtPsym arOf resOf parOf l \<and> \<not> Struct.satL (ModelIkPolMcalc2C.I_intF arOf intT intF) (ModelIkPolMcalc2C.I_intP parOf pol intT intP) \<xi> l \<and> ProblemIkPol.isGuard pol x l" shows "map (Struct.int intF (ModelIkPolMcalc2C.transE intT \<xi>)) Tl = map (Struct.int (ModelIkPolMcalc2C.I_intF arOf intT intF) \<xi>) Tl"
simps: fixes x :: "primfun"   and a :: "'a"   and y :: "'b"   and b :: "'a" shows "merge_primfun (x, a) (y, b) = (x, a + b)"
lifted_reflexive: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "f = g" shows "f \<le>\<le> g"
intros_1: fixes f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "\<forall>\<^sub>F x in at_top. f x = 0"   and "length basis = Suc (expansion_level TYPE('a))" shows "is_expansion_aux MSLNil f basis"
simps_1: fixes l :: "linear_poly"   and r :: "QDelta"   and v :: "nat" shows "qdelta_constraint_to_atom' (LEQ_ns l r) v = Leq v r"
drlex_pm_refl: fixes s :: "'a \<Rightarrow>\<^sub>0 'b" shows "drlex_pm s s"
rows_transpose: fixes A :: "'a Matrix.mat" shows "rows A\<^sup>t = cols A"
sorted_distinct_imp_sorted_wrt: fixes xs :: "'b list" assumes "sorted xs"   and "distinct xs" shows "sorted_wrt (<) xs"
at_within_nhd: fixes "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and S :: "'d set"   and T :: "'d set"   and U :: "'d set" assumes "class.topological_space open"   and "x \<in> S"   and "open S"   and "T \<inter> S - {x} = U \<inter> S - {x}" shows "topological_space.at_within open x T = topological_space.at_within open x U"
simps_1: fixes x :: "'a"   and ys :: "'a ppath" shows "pp_fusion (Node x) ys = ys"
confluentpE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y1 :: "'b"   and y2 :: "'b" assumes "confluentp r"   and "r\<^sup>*\<^sup>* x y1"   and "r\<^sup>*\<^sup>* x y2" obtains z :: "'b" where "r\<^sup>*\<^sup>* y1 z" and "r\<^sup>*\<^sup>* y2 z"
id_dom_cod_2: fixes A :: "'o" assumes "A \<in> Ob" shows "Cod (Id A) = A"
derived_eq_singleton: fixes H :: "(nat \<times> nat) set" assumes "H \<subseteq> carrier S'" shows "derived S' H = {\<one>\<^bsub>S'\<^esub>}"
mult_right_le_one_le: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and x :: "'e"   and y :: "'e" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x"   and "less_eq zero y"   and "less_eq y one" shows "less_eq (times x y) x"
measure_UN_bound: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and M :: "'d measure"   and B :: "real" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"   and "\<And>I'. \<lbrakk>I' \<subseteq> I; finite I'\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure M (\<Union> (A ` I')) \<le> B" shows "Sigma_Algebra.measure M (\<Union> (A ` I)) \<le> B"
rsquarefree_def: fixes p :: "'a list" shows "rsquarefree p = (poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1))"
inv_ins: shows "queue_invar (insertList (filter (\<lambda>t. rank t = 0) (children (getMinTree q))) (meld (rev (filter (\<lambda>t. 0 < rank t) (children (getMinTree q)))) (remove1 (getMinTree q) q)))"
closure_of_eq: fixes X :: "'d topology"   and S :: "'d set" shows "(X closure_of S = S) = closedin X S"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a"   and unit :: "unit" shows "P (case unit of () \<Rightarrow> f) = (unit = () \<longrightarrow> P f)"
blinfun_apply: fixes x :: "'d \<Rightarrow>\<^sub>L 'e" shows "blinfun_apply x \<in> {f. bounded_linear f}"
add_pos_pos: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "{#} \<subset># a"   and "{#} \<subset># b" shows "{#} \<subset># a + b"
dimQ: fixes x :: "'c['d, 'e]" shows "dimQ x = QD('d)"
gcdI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a divides b"   and "a divides c"   and "\<And>y. \<lbrakk>y \<in> carrier G; y divides b; y divides c\<rbrakk> \<Longrightarrow> y divides a"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a \<sim> somegcd G b c"
le_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less" shows "less_eq x y = (less x y \<or> x = y)"
countable_image: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'g" assumes "countable A" shows "countable (f ` A)"
affine_independent_1: fixes a :: "'b" shows "\<not> affine_dependent {a}"
eq_onpE: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "eq_onp P x y" obtains "x = y" and "P y"
Max_image_commute: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'b" assumes "finite S" shows "(MAX x\<in>S. Max (f x ` S)) = (MAX y\<in>S. MAX x\<in>S. f x y)"
hyps_2: assumes "coll 0 a (sum_list xs)"   and "\<And>x. x \<in> set xs \<Longrightarrow> coll 0 x y"   and "\<And>x. x \<in> set xs \<Longrightarrow> coll 0 x (sum_list xs)"   and "\<And>x. x \<in> set xs \<Longrightarrow> 0 < snd x"   and "a \<noteq> 0" shows "coll 0 a y"
remove_def: fixes x :: "'d"   and dxs :: "'d dlist" shows "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"
dg_Set_incl_Set_is_arr: fixes A :: "V"   and B :: "V" assumes "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>"   and "A \<subseteq>\<^sub>\<circ> B" shows "incl_Rel A B : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B"
rel_G_mono: fixes L1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L1' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and L2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co1' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co2' :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra1' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra1 :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool" assumes "L1 \<le> L1'"   and "L2 \<le> L2'"   and "Co1 \<le> Co1'"   and "Co2 \<le> Co2'"   and "Contra1' \<le> Contra1"   and "Contra2' \<le> Contra2" shows "rel_G L1 L2 Co1 Co2 Contra1 Contra2 \<le> rel_G L1' L2' Co1' Co2' Contra1' Contra2'"
image_smult_cbox: fixes m :: "real"   and a :: "'a"   and b :: "'a" shows "(*\<^sub>R) m ` cbox a b = (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
fg_rcv_msg_def: fixes p :: "'f"   and q :: "'f"   and HO :: "'f set"   and SHO :: "'f set"   and config :: "('g, 'f, 'h, 'i) config_scheme"   and config' :: "('g, 'f, 'h, 'i) config_scheme" shows "fg_rcv_msg p q HO SHO config config' \<equiv> \<exists>m m'. (q, round config p, p, m) \<in> network config \<and> q \<in> HO \<and> config' = config\<lparr>rcvd := (rcvd config)(p := (rcvd config p)(q := if q \<in> SHO then Some m else Some m')), network := network config - {(q, round config p, p, m)}\<rparr>"
right_neutral: fixes a :: "'a" shows "a \<sqinter> top = a"
escape_rate_pos: fixes x :: "'a" shows "0 < escape_rate x"