hcju/isabelleps huggingface dataset:
is_glb_maximal: fixes S :: "'a set"   and x :: "'a" assumes "S >| x"   and "x \<in> S" shows "S >>| x"
\<section>: fixes xs :: "'a list"   and ys :: "'a list" assumes "(xs, ys) \<in> lexord {(u, v). u < v}"   and "(ys, xs) \<in> lexord {(u, v). u < v}" shows "False"
Inf_fold_inf: 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 bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A" shows "Inf A = Finite_Set.fold inf top A"
denote_while_n_add: fixes x1 :: "nat \<Rightarrow> complex mat"   and x2 :: "com"   and \<rho>\<^sub>1 :: "complex mat"   and \<rho>\<^sub>2 :: "complex mat"   and k :: "nat" assumes "x1 0 \<in> carrier_mat d d"   and "x1 1 \<in> carrier_mat d d"   and "well_com x2"   and "measurement d 2 x1"   and "\<And>\<rho>\<^sub>1 \<rho>\<^sub>2. \<lbrakk>\<rho>\<^sub>1 \<in> carrier_mat d d; \<rho>\<^sub>2 \<in> carrier_mat d d; partial_density_operator \<rho>\<^sub>1; partial_density_operator \<rho>\<^sub>2; trace (\<rho>\<^sub>1 + \<rho>\<^sub>2) \<le> 1\<rbrakk> \<Longrightarrow> denote x2 (\<rho>\<^sub>1 + \<rho>\<^sub>2) = denote x2 \<rho>\<^sub>1 + denote x2 \<rho>\<^sub>2"   and "\<rho>\<^sub>1 \<in> carrier_mat d d"   and "\<rho>\<^sub>2 \<in> carrier_mat d d"   and "partial_density_operator \<rho>\<^sub>1"   and "partial_density_operator \<rho>\<^sub>2"   and "trace (\<rho>\<^sub>1 + \<rho>\<^sub>2) \<le> 1" shows "denote_while_n (x1 0) (x1 1) (denote x2) k (\<rho>\<^sub>1 + \<rho>\<^sub>2) = denote_while_n (x1 0) (x1 1) (denote x2) k \<rho>\<^sub>1 + denote_while_n (x1 0) (x1 1) (denote x2) k \<rho>\<^sub>2"
VIsViewOnE: shows "isViewOn \<V> E\<^bsub>ES1 \<parallel> ES2\<^esub>"
triangle_in_hom_1: shows "\<guillemotleft>(\<epsilon> \<star> f) \<bullet> \<a>\<^sup>-\<^sup>1[f, g, f] \<bullet> (f \<star> \<eta>) : f \<star> src f \<Rightarrow> trg f \<star> f\<guillemotright>"
ws'_def: fixes n :: "nat" shows "ws' n = snd (SOME (E', ws'). E' \<in> \<E> \<and> P \<turnstile> (E', ws') \<surd> \<and> (enat n \<le> llength E \<longrightarrow> enat n \<le> llength E') \<and> ltake (enat (n - 1)) E = ltake (enat (n - 1)) E' \<and> (0 < n \<longrightarrow> action_tid E' (n - 1) = action_tid E (n - 1) \<and> (if n - 1 \<in> read_actions E then sim_action else (=)) (action_obs E' (n - 1)) (action_obs E (n - 1)) \<and> (\<forall>i<n - 1. i \<in> read_actions E \<longrightarrow> ws' i = ws i)) \<and> (\<forall>r\<in>read_actions E'. n - 1 \<le> r \<longrightarrow> P,E' \<turnstile> ws' r \<le>hb r))"
abc_mopup_measure_induct: fixes P :: "(nat \<times> cell list \<times> cell list) \<times> nat \<Rightarrow> bool"   and f :: "nat \<Rightarrow> (nat \<times> cell list \<times> cell list) \<times> nat" assumes "\<And>n. \<not> P (f n) \<Longrightarrow> (f (Suc n), f n) \<in> abc_mopup_measure" shows "\<exists>n. P (f n)"
nhds_metric: fixes x :: "'a" shows "nhds x = (INF e\<in>{0<..}. principal {y. dist y x < e})"
P: shows "coprime (fib (n + 1)) (fib (n + 1 + 1))"
type_substitution: fixes x :: "name"   and T' :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and e :: "lam"   and T :: "ty"   and e' :: "lam" assumes "(x, T') # \<Gamma> \<turnstile> e : T"   and "\<Gamma> \<turnstile> e' : T'" shows "\<Gamma> \<turnstile> e[x::=e'] : T"
comp_Rel_def: fixes S :: "V"   and T :: "V" shows "S \<circ>\<^sub>R\<^sub>e\<^sub>l T = [S\<lparr>ArrVal\<rparr> \<circ>\<^sub>\<circ> T\<lparr>ArrVal\<rparr>, T\<lparr>ArrDom\<rparr>, S\<lparr>ArrCod\<rparr>]\<^sub>\<circ>"
interv_prod_list_conv_prod_set_nat: fixes f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "prod_list (map f [m..<n]) = prod f (set [m..<n])"
mult_stalk_def: shows "stk2.mult_stalk \<equiv> stk2.mult_rel"
ag_mOp_closed: fixes A :: "('d, 'e) aGroup_scheme"   and x :: "'d" assumes "aGroup A"   and "x \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> x \<in> carrier A"
set_bits_False_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and set_bits :: "(nat \<Rightarrow> bool) \<Rightarrow> 'a" assumes "class.bit_comprehension plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not set_bits" shows "set_bits (\<lambda>_. False) = zero"
nth_the_index: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "xs ! the_index xs x = x"
assms: shows "is_zero_row_upt_k i (Suc k) A"
mat_mult: fixes m :: "(('b, 'd) vec, 'a) vec"   and m' :: "(('b, 'c) vec, 'd) vec" shows "($) (m ** m') = mat_mult_row m m'"
simps_2: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and c :: "color"   and lt :: "('a, 'c) rbt"   and x :: "'a"   and v :: "'c"   and rt :: "('a, 'c) rbt" shows "rbt_map_entry k f (Branch c lt x v rt) = (if k < x then Branch c (rbt_map_entry k f lt) x v rt else if x < k then Branch c lt x v (rbt_map_entry k f rt) else Branch c lt x (f v) rt)"
prems_6: shows "set (normFaces (Fss ! i)) \<inter> set (normFaces newFs) \<subseteq> {}"
qSwapInp_def: fixes xs :: "'h"   and x :: "'i"   and y :: "'i"   and qinp :: "'e \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qTerm option" shows "qSwapInp xs x y qinp \<equiv> lift (qSwap xs x y) qinp"
subgroup_img_is_subgroup: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g"   and I :: "'e set" assumes "group_hom G H h"   and "subgroup I G" shows "subgroup (h ` I) H"
neq_Nil_rev_conv: fixes l :: "'b list" shows "(l \<noteq> []) = (\<exists>xs x. l = xs @ [x])"
linkable_bipartite_web_ofD: assumes "linkable (bipartite_web_of (bipartite_web_of \<Gamma>))"   and "countable \<^bold>E\<^bsub>bipartite_web_of \<Gamma>\<^esub>" shows "linkable (bipartite_web_of \<Gamma>)"
isCont_arctan: fixes x :: "real" shows "isCont arctan x"
at_within_open_NO_MATCH: fixes "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and s :: "'a set" assumes "class.topological_space open"   and "a \<in> s"   and "open s"   and "NO_MATCH UNIV s" shows "topological_space.at_within open a s = topological_space.at_within open a UNIV"
restrict_homogenize_extend_indets: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "restrict_indets (homogenize None (extend_indets p)) = p"
isSubobj_isSubobj_isSubobj: 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 D :: "char list"   and Cs :: "char list list" assumes "is_subobj P (C, [D])"   and "is_subobj P (D, D # Cs)" shows "is_subobj P (C, D # Cs)"
minus_minus: fixes x :: "'a" assumes "x \<in> UNIV" shows "- (- x) = x"
Gromov_converging_at_boundary_converges: fixes u :: "nat \<Rightarrow> 'c" assumes "Gromov_converging_at_boundary u" shows "\<exists>x\<in>Gromov_boundary. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x"
subring_polynomial_pow_not_zero: fixes K :: "'a list set"   and p :: "'a list list"   and n :: "nat" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "p \<noteq> []" shows "p [^]\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> n \<noteq> []"
join_ir_def: fixes i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "join_ir i r \<equiv> \<lambda>x. case x of I n \<Rightarrow> input2state i $ n | R n \<Rightarrow> r $ n"
even_word_unfold: fixes k :: "int" shows "even k = (\<exists>l. take_bit LENGTH('a) k = take_bit LENGTH('a) (2 * l))"
Compl_eq: fixes A :: "'d set" shows "- A = {x. x \<notin> A}"
n2l_l2n_id: fixes x :: "'b list" assumes "x \<noteq> []" shows "n2l (l2n x) = x"
surjective_map_axioms: shows "surjective_map Class M Partition"
DynProcStaticSpec: fixes P :: "'d set"   and S :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'f, 'g) com"   and Q :: "'d set"   and p :: "'d \<Rightarrow> 'f" assumes "P \<subseteq> {s \<in> S. \<exists>Z. init s \<in> P' Z \<and> (\<forall>\<tau>. \<tau> \<in> Q' Z \<longrightarrow> return s \<tau> \<in> R s \<tau>) \<and> (\<forall>\<tau>. \<tau> \<in> A' Z \<longrightarrow> return s \<tau> \<in> A)}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (R s t) c s t Q,A"   and "\<forall>s\<in>S. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) Call (p s) (Q' Z),(A' Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
pow_eq_int_mod: fixes a :: "'a"   and n :: "int"   and m :: "int" assumes "a \<in> carrier G"   and "a [^] n = a [^] m" shows "n mod int (ord a) = m mod int (ord a)"
P1: shows "1 < p\<^sup>2 + N * q\<^sup>2"
adhCE: fixes x :: "'a llist"   and A :: "'a llist set"   and U :: "'a llist set" assumes "x adh A" obtains "U \<notin> nhds x"   | "U \<inter> A \<noteq> {}"
wt_ge_num_args: fixes s :: "('s, 'v) tm" assumes "wary s" shows "num_args s \<le> wt s"
col_mat: fixes j :: "nat"   and nc :: "nat"   and nr :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "j < nc" shows "col (mat nr nc f) j = vec nr (\<lambda>i. f (i, j))"
prodrg_one: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" assumes "\<forall>k\<in>I. Ring (A k)" shows "1\<^sub>r\<^bsub>r\<Pi>\<^bsub>I\<^esub> A\<^esub> = prod_one I A"
rel_Class_iff: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b" shows "stk1.rel x y = (y \<in> Sigma I \<FF>1 \<and> x \<in> e1.Class y)"
pbij_lens_mwb: fixes x :: "'c \<Longrightarrow> 'd" assumes "pbij_lens x" shows "mwb_lens x"
\<phi>_3: fixes i :: "nat"   and x :: "'a" assumes "i \<in> {0, 1}"   and "x \<in> carrier" shows "\<phi> i x \<le> 1"
interp_ins_subset: fixes op_list :: "('a \<times> 'a option) list" shows "set (interp_ins op_list) \<subseteq> set (map fst op_list)"
isomorphicE: fixes a :: "'b"   and a' :: "'b" assumes "F.B.isomorphic a a'" obtains f :: "'b" where "F.B.in_hom f a a' \<and> F.B.iso f"
V_decomp: shows "V = attractor p K \<union> V' \<union> W1"
assms: fixes y :: "'s" shows "measure_pmf.expectation N (\<lambda>x. pmf (K x) y) \<le> pmf N y"
sorted_hd_Min: fixes l :: "'b list" assumes "sorted l"   and "l \<noteq> []" shows "hd l = Min (set l)"
some_counit_def: fixes f :: "'a" shows "T0.some_counit f \<equiv> SOME \<epsilon>. adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f f\<^sup>* (T0.some_unit f) \<epsilon>"
varSortAsSort_inj_def: fixes Delta :: "('a, 'b, 'c, 'd, 'e, 'f) signature_scheme" shows "varSortAsSort_inj Delta \<equiv> inj (varSortAsSort Delta)"
residue_of_one_2: fixes k :: "nat" assumes "0 < k" shows "\<one> k = 1"
subtract_entries_def: shows "subtract_entries \<equiv> map_fun alist.impl_of (map_fun alist.impl_of Alist) subtract_entries_raw"
wf_digraphI_app_iso: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism" assumes "wf_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "wf_digraph (app_iso hom G)"
Basis_nonneg: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and i :: "'e" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "less_eq zero i"
is_arg_min_def: fixes f :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "is_arg_min f P x = (P x \<and> (\<nexists>y. P y \<and> f y < f x))"
gen_lossless_gpv_def: shows "local.gen_lossless_gpv = (if co' then gfp else lfp) F"
rv: fixes i :: "'i" shows "random_variable (N' i) (Y i)"
case_cong: fixes B1 :: "('a \<times> 'b) option"   and B2 :: "('a \<times> 'b) option"   and x1 :: "'c"   and x2 :: "'c"   and y1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and y2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "B1 = B2"   and "x1 = x2"   and "y1 = y2" shows "(case B1 of None \<Rightarrow> x1 | Some (x, P) \<Rightarrow> y1 x P) = (case B2 of None \<Rightarrow> x2 | Some (x, P) \<Rightarrow> y2 x P)"
eq_inf_reflp: fixes w :: "'action stream" shows "w =\<^sub>I w"
simps_2: fixes x :: "'a"   and t :: "'a bro" shows "ins x (N1 t) = n1 (ins x t)"
set_bset: fixes x :: "'c set['d]" shows "set_bset x \<in> {A. |A| <o natLeq +c |UNIV|}"
vfsequence_vdomain: fixes xs :: "V" assumes "vfsequence xs" shows "\<D>\<^sub>\<circ> xs = vcard xs"
Fpow_Pow_finite: fixes A :: "'b set" shows "Fpow A = Pow A \<inter> {A. finite A}"
distinct_1: fixes x1 :: "exp0"   and x2 :: "v" shows "Exp x1 \<noteq> Val x2"
lmap_eq_lmap_conv_llist_all2: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist"   and g :: "'c \<Rightarrow> 'a"   and ys :: "'c llist" shows "(lmap f xs = lmap g ys) = llist_all2 (\<lambda>x y. f x = g y) xs ys"
nths_dims_uminus_vars2: shows "nths dims (- vars2) = nths dims vars1"
iso_def: fixes f :: "'a" shows "A.iso f \<equiv> \<exists>g. A.inverse_arrows f g"
0: shows "q \<otimes>\<^bsub>P\<^esub> p = Cring_Poly.truncate R q \<otimes>\<^bsub>P\<^esub> (Cring_Poly.truncate R p \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) \<oplus>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q) \<otimes>\<^bsub>P\<^esub> (Cring_Poly.truncate R p \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p))"
lt2: shows "lt (- monom_mult (lookup p (t \<oplus> lt q) / lc q) t q) = t \<oplus> lt q"
ring_axioms: shows "ring R (+) (\<cdot>) \<zero> \<one>"
legs_form_cospan_2: shows "C.cospan f.cod.leg0 src.cod.leg1"
bisimSubstStructCong: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>s Q" shows "P \<sim>\<^sup>s\<^sub>l Q"
abelian_homology_group: fixes p :: "int"   and X :: "'b topology" shows "comm_group (homology_group p X)"
proper_interval_list_aux_correct: fixes xs :: "'a list"   and ys :: "'a list" shows "local.proper_interval_list_aux xs ys = (\<exists>zs>xs. zs < ys)"
lub_sum: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "class.idempotent_semiring one times sup less_eq less bot"   and "\<forall>k. less_eq (f k) x" shows "less_eq (comm_monoid_add.sum sup bot f {k. True}) x"
in_P2R: fixes xy :: "'c \<times> 'd"   and \<phi> :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(xy \<in> P2R \<phi>) = \<phi> (fst xy) (snd xy)"
HoHV_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "B.HoHV \<mu> \<equiv> if B.VVV.arr \<mu> then snd (snd \<mu>) \<star> fst (snd \<mu>) \<star> fst \<mu> else B.null"
start_addrs_NewHeapElem_start_heap_obsD: fixes a :: "'addr" assumes "a \<in> set start_addrs" shows "\<exists>CTn. NewHeapElem a CTn \<in> set start_heap_obs"
minus_fract: fixes a :: "'a"   and b :: "'a" shows "- Fract a b = Fract (- a) b"
mem_box_real_1: fixes x :: "real"   and a :: "real"   and b :: "real" shows "(x \<in> box a b) = (a < x \<and> x < b)"
assms_3: shows "c \<turnstile> ev \<mapsto> c'"
primitiveI: fixes f :: "'b poly" assumes "\<And>x. (\<And>y. y \<in> set (coeffs f) \<Longrightarrow> x dvd y) \<Longrightarrow> x dvd (1::'b)" shows "primitive f"
IH: assumes "i < j" shows "similar_mat (step_3_a i j B) B"
star_spartitions_elim2: fixes x :: "'a list"   and z :: "'a list"   and A :: "'a list set" assumes "x @ z \<in> A\<star>"   and "x \<noteq> []" shows "\<exists>(u, v)\<in>Partitions x. \<exists>(u', v')\<in>Partitions z. strict_prefix u x \<and> u \<in> A\<star> \<and> v @ u' \<in> A \<and> v' \<in> A\<star>"
in_Vd_conv: fixes v :: "nat"   and d :: "nat" shows "(v \<in> Vd d) = (local.connected src v \<and> ndist v = d)"
index_of_pr_def: fixes f :: "nat \<Rightarrow> nat" shows "index_of_pr f = (SOME n. f = nat_to_pr n)"
hyperb_quad_ineq: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and deltaG :: "'b itself \<Rightarrow> real"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and t :: "'b" assumes "class.Gromov_hyperbolic_space dist uniformity open deltaG" shows "dist x y + dist z t \<le> max (dist x z + dist y t) (dist x t + dist y z) + 2 * deltaG TYPE('b)"
linear_injective_isomorphism: fixes f :: "'b \<Rightarrow> 'c" assumes "linear (*a) (*b) f"   and "inj f"   and "vs2.dim UNIV = vs1.dim UNIV" shows "\<exists>f'. linear (*b) (*a) f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
face: fixes f :: "'a set"   and s :: "'a set" shows "face f s = (\<exists>a\<in>s. f = s - {a})"
set_bits_bit_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and set_bits :: "(nat \<Rightarrow> bool) \<Rightarrow> 'b"   and a :: "'b" assumes "class.bit_comprehension plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not set_bits" shows "set_bits (bit a) = a"
lists_IntI: fixes l :: "'a list"   and A :: "'a set"   and B :: "'a set" assumes "l \<in> lists A"   and "l \<in> lists B" shows "l \<in> lists (A \<inter> B)"
meet_join_distrib2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "(y \<squnion> z) \<sqinter> x = y \<sqinter> x \<squnion> z \<sqinter> x"
gbinomial_eq_0: fixes a :: "int"   and k :: "nat" assumes "0 \<le> a"   and "a < int k" shows "a gchoose k = 0"
P_f: fixes S :: "'a" assumes "P S" shows "P (f S)"
desargues_config_not_col_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col B B' C'"
assms: shows "FI_RESULT [(B, B), (A, A)] emp emp F"
exhaust: fixes y :: "sign" obtains "y = pos"   | "y = neg"
reach_foldl_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d" assumes "\<Sigma> \<noteq> {}" shows "reach \<Sigma> \<delta> q\<^sub>0 = {foldl \<delta> q\<^sub>0 w |w. set w \<subseteq> \<Sigma>}"
basis_in_indep_in: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set" assumes "indep_system carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.basis_in indep \<E> X" shows "indep_system.indep_in indep \<E> X"
sublist_append_leftI: fixes xs :: "'c list"   and ps :: "'c list" shows "sublist xs (ps @ xs)"
range_None: fixes A :: "'a set" assumes "MaxR_opt A = None" shows "A \<inter> Field r = {}"
of_int_le_iff: fixes w :: "int"   and z :: "int" shows "(of_int w \<le> of_int z) = (w \<le> z)"
Int_stableD: fixes M :: "'c set set"   and a :: "'c set"   and b :: "'c set" assumes "Int_stable M"   and "a \<in> M"   and "b \<in> M" shows "a \<inter> b \<in> M"
a4: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> F"   and "A \<subseteq> B"   and "B \<subseteq> carrier" shows "B \<in> F"
set_image_le_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> V"   and B :: "V" assumes "small A" shows "(set (f ` A) \<le> B) = (\<forall>x\<in>A. f x \<in> elts B)"
countable_UN: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and A :: "'d \<Rightarrow> 'c set"   and X :: "'d set" assumes "sigma_algebra \<Omega> M"   and "A ` X \<subseteq> M" shows "\<Union> (A ` X) \<in> M"
abc_unhalt_append_eq: fixes P :: "nat list \<Rightarrow> bool"   and A :: "abc_inst list"   and args :: "nat list"   and B :: "abc_inst list"   and stp :: "nat" assumes "{P} A \<up>"   and "P args" shows "abc_steps_l (0, args) (A [+] B) stp = abc_steps_l (0, args) A stp"
preserves_equivalence: fixes f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" assumes "C.internal_equivalence f g \<eta> \<epsilon>" shows "D.internal_equivalence (F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
o_apply: fixes f :: "'s \<Rightarrow> 'r"   and g :: "'t \<Rightarrow> 's"   and x :: "'t" shows "(f \<circ> g) x = f (g x)"
T_Lam: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list"   and T\<^sub>1 :: "ty"   and t :: "trm"   and T\<^sub>2 :: "ty" assumes "x \<sharp> \<Gamma>"   and " (x, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2 " shows " \<Gamma> \<turnstile> Lam [x].t : T\<^sub>1\<rightarrow>T\<^sub>2 "
hoare_patial_seq_assoc: fixes S1 :: "com"   and S2 :: "com"   and S3 :: "com"   and A :: "complex mat"   and B :: "complex mat" assumes "well_com S1"   and "well_com S2"   and "well_com S3"   and "is_quantum_predicate A"   and "is_quantum_predicate B" shows "(\<turnstile>\<^sub>p {A} S1;; S2;; S3 {B}) = (\<turnstile>\<^sub>p {A} S1;; (S2;; S3) {B})"
select_convs_1: fixes pset :: "'b set"   and order :: "('b \<times> 'b) set"   and more :: "'c" shows "pset \<lparr>pset = pset, order = order, \<dots> = more\<rparr> = pset"
funct_OT_14_def: fixes M :: "'a \<times> 'a \<times> 'a \<times> 'a"   and C :: "bool \<times> bool" shows "funct_OT_14 M C = (let (c0, c1) = C; (m00, m01, m10, m11) = M in return_spmf ((), if c0 then if c1 then m11 else m10 else if c1 then m01 else m00))"
subNEq_2: shows "s = 0 \<or> Re (g s) = Re z"
parent_upper_bound_alt: fixes l :: "int"   and c :: "int" assumes "l \<le> parent l c" shows "parent l c < c"
fmdom'_alt_def: fixes m :: "('d, 'e) fmap" shows "fmdom' m = fset (fmdom m)"
equiv_class_eq: fixes A :: "'e set"   and r :: "('e \<times> 'e) set"   and a :: "'e"   and b :: "'e" assumes "equiv A r"   and "(a, b) \<in> r" shows "r `` {a} = r `` {b}"