hcju/isabelleps huggingface dataset:
IH: fixes y :: "nat" assumes "y < m"   and "0 < y"   and "0 < k" shows "\<exists>a. vdw_lemma k y l a"
map_vec_def: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b vec" shows "map_vec f v \<equiv> Matrix.vec (dim_vec v) (\<lambda>i. f (v $ i))"
Gauss_Jordan_PA_eq: fixes A :: "(('a, 'b) vec, 'c) vec" shows "snd (Gauss_Jordan_PA A) = Gauss_Jordan A"
set_iterator_genord_def: fixes iti :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S0 :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "set_iterator_genord iti S0 R \<equiv> \<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0"
okamoto: fixes \<eta> :: "nat" shows "okamoto (\<G> \<eta>)"
Lcm_fin_least: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "\<And>b. b \<in> A \<Longrightarrow> b dvd a" shows "Lcm\<^sub>f\<^sub>i\<^sub>n A dvd a"
lower_on_term: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list"   and t :: "'a trm" assumes "lower_on \<sigma> \<eta> (vars_of t)" shows "lower_or_eq (t \<lhd> \<sigma>) (t \<lhd> \<eta>)"
noread_nowrite: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and c :: "'c"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "'c"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "sifum_security_init dma \<C>_vars eval"   and "(((c, mds), mem), (c', mds'), mem') \<in> eval"   and "\<And>v. (((c, mds), mem(x := v)), (c', mds'), mem'(x := v)) \<in> eval" shows "mem x = mem' x"
newton_seq_fact4: fixes l :: "nat" assumes "f \<bullet> ns l \<noteq> \<zero>" shows "val_Zp (f' \<bullet> ns l) \<le> val_Zp (f \<bullet> ns l)"
Abs_cltn2_inverse: fixes y :: "((real, 3) vec, 3) vec set" assumes "y \<in> Collect invertible // invertible_proportionality" shows "Rep_cltn2 (Abs_cltn2 y) = y"
open_prod_def: fixes S :: "('b \<times> 'c) set" shows "open S = (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
SMP_subset_union_eq: fixes M :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set" assumes "M \<subseteq> SMP N" shows "SMP N = SMP (M \<union> N)"
compact_Int_closed: fixes S :: "'a set"   and T :: "'a set" assumes "compact S"   and "closed T" shows "compact (S \<inter> T)"
invar_def: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> ('b \<times> nat) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list option"   and M_invar :: "'a \<Rightarrow> bool"   and g :: "'a" assumes "wgraph_by_map M_empty M_update M_delete M_lookup M_invar" shows "wgraph_by_map.invar M_lookup M_invar g \<equiv> M_invar g \<and> finite (dom (M_lookup g)) \<and> (\<forall>u xs. M_lookup g u = Some xs \<longrightarrow> distinct (map fst xs) \<and> u \<notin> set (map fst xs) \<and> (\<forall>(v, d)\<in>set xs. (u, d) \<in> set (the_default [] (M_lookup g v))))"
VSLuckiestBlocks: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval (VSLuckiestBlocks \<phi>) xs = eval \<phi> xs"
th0: shows "\<forall>xa\<in>set (remdups (uset (simpfm p))). \<forall>y\<in>set (remdups (uset (simpfm p))). Ifm vs (x # bs) (case (xa, y) of (x, xa) \<Rightarrow> (case x of (b, a) \<Rightarrow> \<lambda>(d, c). msubst2 (simpfm p) ((- 2)\<^sub>p *\<^sub>p b *\<^sub>p d) (tm.Add (tm.Mul d a) (tm.Mul b c))) xa) = Ifm vs (x # bs) (case (y, xa) of (x, xa) \<Rightarrow> (case x of (b, a) \<Rightarrow> \<lambda>(d, c). msubst2 (simpfm p) ((- 2)\<^sub>p *\<^sub>p b *\<^sub>p d) (tm.Add (tm.Mul d a) (tm.Mul b c))) xa)"
distinct_3: fixes x1 :: "'b"   and x31 :: "'b exp"   and x32 :: "'b exp" shows "Var x1 \<noteq> Plus x31 x32"
MtransC_Trans: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state"   and cf'' :: "('test, 'atom) com \<times> 'state" assumes "cf \<rightarrow>*c cf'"   and "cf' \<rightarrow>*c cf''" shows "cf \<rightarrow>*c cf''"
collapse_6: fixes expr :: "'a expr" assumes "is_StaticOperationCall expr" shows "StaticOperationCall (expr.type expr) (un_StaticOperationCall2 expr) (expr.args expr) = expr"
add_num_frac: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and y :: "'d"   and z :: "'d"   and x :: "'d" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero" shows "plus z (divide x y) = divide (plus x (times z y)) y"
exp_times_self_strict_antimono: fixes y :: "real"   and x :: "real" assumes "y \<le> - 1"   and "x < y" shows "y * exp y < x * exp x"
all_interval_nat_def: fixes P :: "nat \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" shows "List.all_interval_nat P i j = (\<forall>n\<in>{i..<j}. P n)"
elims: fixes x :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort"   and y :: "'v" assumes "edge_end x = y" obtains v1 :: "'v" and p1 :: "'outPort" and v2 :: "'v" and p2 :: "'inPort" where "x = ((v1, p1), v2, p2)" and "y = v2"
induct: fixes x :: "('fun, 'var) extstrand_step list"   and P :: "('fun, 'var) extstrand_step list \<Rightarrow> bool" assumes "well_analyzed x"   and "P []"   and "\<And>A x. \<lbrakk>well_analyzed A; P A\<rbrakk> \<Longrightarrow> P (A @ [Step x])"   and "\<And>A t. \<lbrakk>well_analyzed A; P A; t \<in> subterms\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) - range Var\<rbrakk> \<Longrightarrow> P (A @ [Decomp t])" shows "P x"
graph_from_list_def: fixes l :: "('a \<times> 'a) list" shows "graph_from_list l = foldr ins_edge l graph_empty"
min_dist_def: fixes v :: "nat"   and v' :: "nat" shows "min_dist v v' = (LEAST d. dist v d v')"
max_ipv6_addr_max: shows "\<forall>a. a \<le> max_ipv6_addr"
fls_eq_above_subdegreeI: fixes N :: "int"   and f :: "'a fls"   and g :: "'a fls" assumes "N \<le> fls_subdegree f"   and "N \<le> fls_subdegree g"   and "\<forall>k\<ge>N. fls_nth f k = fls_nth g k" shows "f = g"
induct: fixes P :: "'b negation_type list \<Rightarrow> bool"   and a0 :: "'b negation_type list" assumes "P []"   and "\<And>e es. P es \<Longrightarrow> P (Pos e # es)"   and "\<And>e es. P es \<Longrightarrow> P (Neg e # es)" shows "P a0"
lesspoll_trans1: fixes X :: "'a set"   and Y :: "'b set"   and Z :: "'c set" assumes "X \<lesssim> Y"   and "Y \<prec> Z" shows "X \<prec> Z"
ent_conjI: fixes A :: "assn"   and B :: "assn"   and C :: "assn" assumes "A \<Longrightarrow>\<^sub>A B"   and "A \<Longrightarrow>\<^sub>A C" shows "A \<Longrightarrow>\<^sub>A B \<and>\<^sub>A C"
span_finite: fixes S :: "('b, 'c) vec set" assumes "finite S" shows "vec.span S = range (\<lambda>u. \<Sum>v\<in>S. u v *s v)"
runit_hcomp_2: fixes f :: "'b"   and g :: "'b" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<r>\<^sub>D\<^sup>-\<^sup>1[f \<star>\<^sub>D g] = \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, src\<^sub>D g] \<cdot>\<^sub>D (f \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[g])"
iarray_to_vec_vec_to_iarray: fixes A :: "('d, 'e) vec" shows "iarray_to_vec (vec_to_iarray A) = A"
join_sub': fixes R :: "nat set"   and L :: "nat set"   and n :: "nat"   and t1 :: "'a option list set"   and t2 :: "'a option list set" assumes "R \<subseteq> L"   and "table n L t1"   and "table n R t2" shows "join t2 True t1 = {as \<in> t1. proj_tuple_in_join True (join_mask n R) as t2}"
dom_eqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.ide C a"   and "partial_magma.arr C (C f a)" shows "partial_magma.dom C f = a"
cast_document_ptr_not_node_ptr_1: fixes document_ptr :: "(_) document_ptr"   and node_ptr :: "(_) node_ptr" shows "cast document_ptr \<noteq> cast node_ptr"
monic_square_free_irreducible_factorization: fixes f :: "'b poly" assumes "monic f"   and "square_free f" shows "\<exists>P. finite P \<and> f = \<Prod>P \<and> P \<subseteq> {q. irreducible q \<and> monic q}"
intI: fixes A :: "'a \<Rightarrow> bool" assumes "\<And>w. A w" shows "\<turnstile> A"
totient_prime_power: fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "0 < n" shows "totient (p ^ n) = p ^ (n - 1) * (p - 1)"
inj_on_id2: fixes A :: "'d set" shows "inj_on (\<lambda>x. x) A"
image_mset_is_empty_iff: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" shows "(image_mset f M = \<Empt>) = (M = \<Empt>)"
sum_delta_notmem_2: fixes x :: "'c"   and s :: "'c set"   and P :: "'c \<Rightarrow> 'd"   and Q :: "'c \<Rightarrow> 'd" assumes "x \<notin> s" shows "(\<Sum>y\<in>s. if x = y then P x else Q y) = sum Q s"
inv_inv_f_eq: shows "inv (inv (inv (inv (inv hom)))) = inv (inv (inv hom))"
preserves_reflects_arr: fixes f :: "'c local.term set" shows "arr (\<ll> f) = arr f"
partial_correctness_direct: fixes V :: "nat set"   and I :: "nat set"   and J :: "nat set"   and Q_I_pos :: "(nat set \<times> 'a option list set) set"   and Q :: "(nat set \<times> 'a option list set) set"   and Q_J_pos :: "(nat set \<times> 'a option list set) set"   and Q_I_neg :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and Q_J_neg :: "(nat set \<times> 'a option list set) set"   and R_I :: "'a option list set"   and X :: "('a option list \<times> 'a option list set) set"   and R :: "'a option list set"   and R_NQ :: "'a option list set"   and NQ_pos :: "(nat set \<times> 'a option list set) set"   and NQ_neg :: "(nat set \<times> 'a option list set) set"   and n :: "nat"   and z :: "'a option list" assumes "V = I \<union> J"   and "Set.is_empty (I \<inter> J)"   and "1 \<le> card I"   and "1 \<le> card J"   and "Q_I_pos = projectQuery I (filterQuery I Q)"   and "Q_J_pos = filterQuery J Q"   and "Q_I_neg = filterQueryNeg I Qn"   and "Q_J_neg = filterQuery J (Qn - Q_I_neg)"   and "R_I = genericJoin I Q_I_pos Q_I_neg"   and "X = {(t, genericJoin J (newQuery J Q_J_pos (I, t)) (newQuery J Q_J_neg (I, t))) |t. t \<in> R_I}"   and "R = (\<Union>(t, x)\<in>X. {merge xx t |xx. xx \<in> x})"   and "R_NQ = genericJoin J NQ_pos NQ_neg"   and "\<forall>x. (x \<in> R_I) = (wf_tuple n I x \<and> (\<forall>(A, X)\<in>Q_I_pos. restrict A x \<in> X) \<and> (\<forall>(A, X)\<in>Q_I_neg. restrict A x \<notin> X))"   and "\<forall>t\<in>R_I. \<forall>y. (y \<in> genericJoin J (newQuery J Q_J_pos (I, t)) (newQuery J Q_J_neg (I, t))) = (wf_tuple n J y \<and> (\<forall>(A, X)\<in>newQuery J Q_J_pos (I, t). restrict A y \<in> X) \<and> (\<forall>(A, X)\<in>newQuery J Q_J_neg (I, t). restrict A y \<notin> X))"   and "wf_tuple n V z \<and> (\<forall>(A, X)\<in>Q. restrict A z \<in> X) \<and> (\<forall>(A, X)\<in>Qn. restrict A z \<notin> X)"   and "rwf_query n V Q Qn" shows "z \<in> R"
coprime_mod_left_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "algebraic_semidom.coprime times one (modulo a b) b = algebraic_semidom.coprime times one a b"
Field_Osum: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" shows "Field (r \<union>o r') = Field r \<union> Field r'"
sds_wf: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "lowered.is_pref_profile R" shows "lowered R \<in> lowered.lotteries"
tendsto_upperbound: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a"   and F :: "'b filter"   and a :: "'a" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq (f i) a"   and "F \<noteq> bot" shows "less_eq x a"
posr: fixes x :: "'a"   and y :: "'a" assumes "D x y"   and "x \<oplus> y \<in> E" shows "y \<in> E"
fst_comp_apfst: fixes f :: "'c \<Rightarrow> 'e" shows "fst \<circ> apfst f = f \<circ> fst"
elims: fixes x :: "'b kdt"   and y :: "nat" assumes "KD_Tree.size_kdt x = y" obtains uu_ :: "(real, 'b) vec" where "x = Leaf uu_" and "y = 1"   | uv_ :: "'b" and uw_ :: "real" and l :: "'b kdt" and r :: "'b kdt" where "x = Node uv_ uw_ l r" and "y = KD_Tree.size_kdt l + KD_Tree.size_kdt r"
Necessary_def: fixes x :: "e" shows "Necessary x \<equiv> \<^bold>\<box>E! x"
meet_3_col_3: fixes l :: "Lines"   and m :: "Lines"   and A :: "Points"   and B :: "Points"   and P :: "Points" assumes "meet_3_in l m (line A B) P" shows "col A B P"
sub: fixes a :: "'a" assumes "a \<in> N" shows "a \<in> M"
linear_order_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
Not\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Not\<^sub>\<alpha> t\<^sub>\<alpha>)"
space_T: fixes s :: "'a cfg" shows "space (MC.T s) = space (stream_space (count_space UNIV))"
isl_def: fixes sum :: "'g + 'h" shows "isl sum = (\<exists>x1. sum = Inl x1)"
ih_10: shows "InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
ord_embed_alt: fixes f :: "'b \<Rightarrow> 'c" shows "ord_embed f = (mono f \<and> (\<forall>x y. f x \<le> f y \<longrightarrow> x \<le> y))"
correct: fixes msgs :: "bool \<times> bool"   and b :: "bool" shows "protocol_OT12 msgs b = funct_OT_12 msgs b"
snd1: fixes P :: "'a trm"   and P' :: "'a trm" assumes "P >> P'" shows "Snd P >> Snd P'"
delete_edge_valid': fixes G :: "('c, 'd) graph"   and v :: "'c"   and e :: "'d"   and v' :: "'c" assumes "forest G" shows "valid_graph (delete_edge v e v' G)"
from_nat_to_nat: fixes x :: "'d" shows "from_nat (to_nat x) = x"
hyps_2: fixes B :: "Core.term" assumes "\<tau>s = strip_all_vars B"   and "\<Theta>,\<Gamma> \<turnstile> B" shows "\<Theta>,\<Gamma> \<turnstile> fold (\<lambda>(x, \<tau>). subst_bv (Fv x \<tau>) \<circ> strip_all_single_body) (zip xs (strip_all_vars B)) B"
set_cond_pmf: fixes p :: "'b pmf"   and s :: "'b set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "set_pmf (cond_pmf p s) = set_pmf p \<inter> s"
no_deadlock: fixes s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "A.reachable s" shows "\<exists>s'. A.step s s'"
kth_obs_le_iff: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and i :: "nat"   and l :: "nat"   and j :: "nat" assumes "is_kth_obs \<pi> k i"   and "is_kth_obs \<pi> l j" shows "(k < l) = (i < j)"
red_ta_satisfiable: fixes P :: "'addr J_prog"   and t :: "'thread_id"   and e :: "'addr expr"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'heap) J_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" assumes "extTA2J P,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "\<exists>s. red_mthr.actions_ok s t ta"
multiplicity_normalize_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one p (normalize x) = factorial_semiring.multiplicity times one p x"
assms_2: shows "M \<equiv> mult_row A i (ass (A $ i $ n) div A $ i $ n)"
SPEC_trans: fixes x :: "'a nres"   and y :: "'a nres"   and Postcond :: "'a \<Rightarrow> bool" assumes "x \<le> y"   and "y \<le> SPEC Postcond" shows "x \<le> SPEC Postcond"
matchC_M_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchC_M theta c d \<equiv> \<forall>s t c' s'. s \<approx> t \<and> (c, s) \<rightarrow>c (c', s') \<longrightarrow> (\<exists>d' t'. (d, t) \<rightarrow>*c (d', t') \<and> s' \<approx> t' \<and> (c', d') \<in> theta) \<or> (\<exists>t'. (d, t) \<rightarrow>*t t' \<and> s' \<approx> t' \<and> discr c')"
elementary_empty: obtains p :: "'a set set" where "p division_of {}"
fun_cong: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f"   and x :: "'e" assumes "f = g" shows "f x = g x"
SNF_first_divides_all: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "Smith_normal_form_mat A"   and "A \<in> carrier_mat m n"   and "i < m"   and "j < n" shows "A $$ (0, 0) dvd A $$ (i, j)"
CONST_INTF_def: fixes c :: "'c"   and I :: "interface" shows "c ::\<^sub>i I \<equiv> True"
is_expansion_aux_expansion_level: fixes G :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "is_expansion_aux G g basis" shows "length basis = Suc (expansion_level TYPE('a))"
eexp_ineq_iff_5: fixes x :: "ereal" shows "0 \<le> eexp x"
ccw_two_up_contra: fixes t :: "real \<times> real"   and p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real"   and s :: "real \<times> real" assumes "ccw' t p q"   and "ccw' t q r"   and "ccw t s p"   and "ccw t s q"   and "ccw t s r"   and "ccw t p q"   and "ccw t q r"   and "ccw t r p"   and "distinct5 t s p q r" shows "False"
span_minimal: fixes Sa :: "'a set"   and T :: "'a set" assumes "Sa \<subseteq> S"   and "T \<subseteq> S"   and "Sa \<subseteq> T"   and "??.Real_Vector_Spaces.subspace T" shows "??.Real_Vector_Spaces.span Sa \<subseteq> T"
Schwarz3: fixes f :: "complex \<Rightarrow> complex"   and r :: "real" assumes "f holomorphic_on ball 0 r"   and "f 0 = 0" obtains h :: "complex \<Rightarrow> complex" where "h holomorphic_on ball 0 r" and "\<And>z. cmod z < r \<Longrightarrow> f z = z * h z" and "deriv f 0 = h 0"
lgridI_child: fixes d :: "nat"   and ds :: "nat set"   and p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and dir :: "dir"   and lm :: "nat" assumes "d \<in> ds"   and "p \<in> lgrid (child b dir d) ds lm" shows "p \<in> lgrid b ds lm"
predI: fixes P :: "bool"   and cid :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and n :: "nat" assumes "P" shows "sb.eval cid t t' n (sb.pred P)"
A: shows "(case_nat v \<rho>)(Suc x := fst (extract_pair v), Suc y := snd (extract_pair v)) \<circ> Suc = \<rho>(x := fst (extract_pair v), y := snd (extract_pair v))"
eval_def: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and default_node_properties :: "'d"   and receiver_violation :: "bool"   and G :: "'c graph"   and P :: "('c, 'd) TopoS_Params" assumes "SecurityInvariant sinvar default_node_properties receiver_violation" shows "SecurityInvariant.eval sinvar default_node_properties G P \<equiv> wf_graph G \<and> sinvar G (SecurityInvariant.node_props default_node_properties P)"
connected_def: fixes S :: "'a set" shows "connected S = (\<nexists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})"
tag_name_eq2_h3: fixes ptr' :: "(_) element_ptr" shows "|h3 \<turnstile> get_tag_name ptr'|\<^sub>r = |h' \<turnstile> get_tag_name ptr'|\<^sub>r"
Notes_Crypt_parts_spies: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Notes A \<lbrace>Agent B, X\<rbrace> \<in> set evs"   and "evs \<in> tls" shows "Crypt (pubK B) X \<in> parts (knows Spy evs)"
select_convs_4: fixes verts :: "'d set"   and arcs :: "'c set"   and tail :: "'c \<Rightarrow> 'd"   and head :: "'c \<Rightarrow> 'd"   and more :: "'e" shows "head \<lparr>pre_digraph.verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = head"
subset: fixes U :: "'b set"   and V :: "'b set" assumes "Subspace.subspace U V" shows "U \<subseteq> V"
Sup_least: fixes A :: "('b \<times> 'c) set"   and z :: "'b \<times> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod x z" shows "less_eq_prod (Sup_prod A) z"
ext_inject: fixes orderA :: "('d, 'f) gorder_scheme"   and orderB :: "('e, 'g) gorder_scheme"   and lower :: "'d \<Rightarrow> 'e"   and upper :: "'e \<Rightarrow> 'd"   and more :: "'h"   and orderA' :: "('d, 'f) gorder_scheme"   and orderB' :: "('e, 'g) gorder_scheme"   and lower' :: "'d \<Rightarrow> 'e"   and upper' :: "'e \<Rightarrow> 'd"   and more' :: "'h" shows "(\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr> = \<lparr>orderA = orderA', orderB = orderB', lower = lower', upper = upper', \<dots> = more'\<rparr>) = (orderA = orderA' \<and> orderB = orderB' \<and> lower = lower' \<and> upper = upper' \<and> more = more')"
mult1_union: fixes B :: "'b multiset"   and D :: "'b multiset"   and r :: "('b \<times> 'b) set"   and C :: "'b multiset" assumes "(B, D) \<in> mult1 r" shows "(C + B, C + D) \<in> mult1 r"
g'_def: shows "g' = (\<lambda>s. if s \<in> {s. c < Re s} then eval_fds f s else g s)"
assms: shows "list.set \<alpha>s \<subseteq> ON"
iterates_above_Sup_aux: fixes M :: "'a set"   and M' :: "'a set" assumes "M \<in> Chains leq"   and "M \<noteq> {}"   and "M' \<in> Chains leq"   and "M' \<noteq> {}"   and "\<And>x. x \<in> M \<Longrightarrow> x \<in> iterates_above (lub M') \<or> lub M' \<in> iterates_above x" shows "(lub M, lub M') \<in> leq \<or> lub M \<in> iterates_above (lub M')"
rot_circle_right_top_edges_neq': shows "rot_circle_right_edge \<noteq> rot_circle_left_edge"
nth_solution_def: fixes n :: "nat" shows "nth_solution n = pell_power fund_sol n"
append_T_imp_tickFree: fixes t :: "'c event list"   and s :: "'c event list"   and P :: "'c process" assumes "t @ s \<in> T P"   and "s \<noteq> []" shows "tickFree t"
measure_map_spmf: fixes f :: "'i \<Rightarrow> 'h"   and p :: "'i spmf"   and A :: "'h set" shows "Sigma_Algebra.measure (measure_spmf (map_spmf f p)) A = Sigma_Algebra.measure (measure_spmf p) (f -` A)"
transform_simps_1: fixes a :: "Arity"   and e :: "exp"   and x :: "Vars.var" shows "\<T>\<^bsub>a\<^esub> (App e x) = App (\<T>\<^bsub>inc\<cdot>a\<^esub> e) x"
optionFreshChain_1: fixes xvec :: "name list"   and x :: "'d" shows "xvec \<sharp>* Some x = xvec \<sharp>* x"
s2: shows "s2 = atomize_poly (idx_pm_of_pm bs s2')"
scale_prod: fixes s1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'b"   and a :: "'c"   and b :: "'d" assumes "module_prod s1 s2" shows "module_prod.scale s1 s2 x (a, b) = (s1 x a, s2 x b)"
dirty_directly_subsumes_def: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and s2 :: "nat"   and t1 :: "transition"   and t2 :: "transition" shows "dirty_directly_subsumes e1 e2 s1 s2 t1 t2 = (if t1 = t2 then True else directly_subsumes e1 e2 s1 s2 t1 t2)"
scalar_vec_one: fixes v :: "'a Matrix.vec" shows "(1::'a) \<cdot>\<^sub>v v = v"
Bij_Nat_to_nat_neq: fixes a :: "'b"   and b :: "'b" assumes "to_nat a \<noteq> to_nat b" shows "Bij_Nat.to_nat a \<noteq> Bij_Nat.to_nat b"
assms_3: shows "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"