hcju/isabelleps huggingface dataset:
cong: fixes K :: "'a \<Rightarrow> 'a pmf" shows "MC_syntax.F K \<equiv> MC_syntax.F K"
Dxz: shows "hausdorff_distance (f ` {x--z}) {f x--f z} \<le> D"
4: shows "\<forall>av\<in>ivs. \<exists>rs\<in>pendingSeqs s1. av = dstate s1 \<star> rs"
proj_in_set_iff_1: fixes i :: "'d"   and d :: "'e"   and D :: "('d strand_label \<times> 'e) list" shows "((ln i, d) \<in> set (proj i D)) = ((ln i, d) \<in> set D)"
times_assn_proper: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and Q :: "heap \<times> nat set \<Rightarrow> bool" assumes "proper P"   and "proper Q" shows "proper (times_assn_raw P Q)"
diff_eq_diff_eq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.group_add minus plus zero uminus"   and "minus a b = minus c d" shows "(a = b) = (c = d)"
"1_2": shows "x = (case x' of (s, as) \<Rightarrow> Inf (PLS s as))"
Subst_preserves_ardenable: fixes rhs :: "'b trm set"   and xrhs :: "'b trm set"   and X :: "'b list set" assumes "ardenable rhs"   and "ardenable xrhs" shows "ardenable (Subst rhs X xrhs)"
r_univuniv2_converg: fixes u :: "nat"   and x :: "nat"   and y :: "nat" assumes "eval (r_universal 2) [u, u, x] \<down>" shows "eval r_univuniv2 [u, x, y] = eval r_phi [the (eval (r_universal 2) [u, u, x]), y]"
same_diag_ev_blocks: fixes n :: "nat"   and A :: "'b mat"   and B :: "'b mat" assumes "??.Jordan_Normal_Form_Existence.same_diag n A B"   and "??.Jordan_Normal_Form_Existence.ev_blocks n A" shows "??.Jordan_Normal_Form_Existence.ev_blocks n B"
type0_def: fixes init :: "'b"   and x :: "'a"   and y :: "'a" shows "type0 init x y = bernoulli_pmf (5 / 10) \<bind> (\<lambda>a. bernoulli_pmf (5 / 10) \<bind> (\<lambda>b. return_pmf ([x, y], [a, b], init)))"
monoid_rng_of_frac: shows "Group.monoid rec_monoid_rng_of_frac"
single_addition_pm_mem: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "int" assumes "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "n\<Odot>a\<^bsub>f,i,z\<^esub> \<in> addition_set f (aug_pm_set z i {a})"
of_int_max_real: fixes c :: "int"   and d :: "int" shows "real_of_int (max c d) = max (real_of_int c) (real_of_int d)"
meet_le: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "dlat le"   and "le z x"   and "le z y" shows "le z (dlat.meet le x y)"
box_Int_box: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" shows "box a b \<inter> box c d = box (\<Sum>i\<in>Basis. max (a \<bullet> i) (c \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b \<bullet> i) (d \<bullet> i) *\<^sub>R i)"
get_all_matching_src_ips_executable_def: fixes iface :: "iface"   and m :: "'b common_primitive match_expr" shows "get_all_matching_src_ips_executable iface m \<equiv> let (i_matches, rest1) = primitive_extractor (is_Iiface, iiface_sel) m in if \<forall>is\<in>set i_matches. case is of Pos i \<Rightarrow> match_iface i (iface_sel iface) | Neg i \<Rightarrow> \<not> match_iface i (iface_sel iface) then let (ip_matches, rest2) = primitive_extractor (is_Src, src_sel) rest1 in if \<not> has_primitive rest2 \<and> matcheq_matchAny rest2 then if ip_matches = [] then wordinterval_UNIV else l2wi_negation_type_intersect (NegPos_map ipt_iprange_to_interval ip_matches) else Empty_WordInterval else Empty_WordInterval"
image_set_system_finite: fixes point_set :: "'c set"   and block_collection :: "'c set multiset"   and f :: "'c \<Rightarrow> 'd" assumes "finite_incidence_system point_set block_collection" shows "finite_incidence_system (f ` point_set) (blocks_image block_collection f)"
setdist_triangle: fixes S :: "'b set"   and T :: "'b set"   and a :: "'b" shows "setdist S T \<le> setdist S {a} + setdist {a} T"
INF_principal_finite: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "finite X" shows "(INF x\<in>X. principal (f x)) = principal (\<Inter> (f ` X))"
sig_red_wf_dgrad_max_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "dickson_grading d"   and "F \<subseteq> dgrad_max_set d" shows "wfP (sig_red sing_reg top_tail F)\<inverse>\<inverse>"
bit_eq_rec: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "(a = b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b \<and> divide a (numeral.numeral one plus (num.Bit0 num.One)) = divide b (numeral.numeral one plus (num.Bit0 num.One)))"
W0_def: shows "W0 = {(x, t). (x, t) \<in> U \<times> C \<and> psi (x, t) < e}"
prems_6: shows "\<forall>l1 l2. subseq l1 l2 \<and> PProbl l2 \<longrightarrow> PProbl l1"
moebius_similarity_inf_iff: fixes a :: "complex"   and b :: "complex"   and z :: "complex_homo" assumes "a \<noteq> 0" shows "(moebius_pt (moebius_similarity a b) z = \<infinity>\<^sub>h) = (z = \<infinity>\<^sub>h)"
field_def: fixes R :: "('c, 'd) ring_scheme" shows "field R \<equiv> domain R \<and> field_axioms R"
simps_4: fixes c :: "int"   and e :: "Cooper.num" shows "\<beta> (NEq (CN 0 c e)) = [Neg e]"
less_imp_not_eq2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less"   and "less x y" shows "(y = x) = False"
full_pmdlI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and K :: "'k set" assumes "\<And>v. v \<in> keys p \<Longrightarrow> component_of_term v \<in> K" shows "p \<in> full_pmdl K"
GrpI: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and y :: "'c"   and A :: "'d set" assumes "f x = y"   and "x \<in> A" shows "BNF_Def.Grp A f x y"
eq: shows "L2_set (quat_proj (x + y)) {..3} = L2_set (\<lambda>i. quat_proj x i + quat_proj y i) {..3}"
1: fixes V_of :: "'a \<Rightarrow> V" assumes "inj V_of" shows "inj (\<lambda>x. ZFC_in_HOL.set (V_of ` x))"
case_prod_comp: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'd"   and x :: "'b \<times> 'c" shows "(case x of (a, b) \<Rightarrow> (f \<circ> g) a b) = f (g (fst x)) (snd x)"
eq_acom_iff_strip_anno: fixes C1 :: "'a acom"   and C2 :: "'a acom" shows "(C1 = C2) = (strip C1 = strip C2 \<and> (\<forall>p<length (annos C1). anno C1 p = anno C2 p))"
residuated_conjl1: fixes y :: "'a" shows "residuated (\<lambda>x. x \<lhd> y)"
assms_1: fixes s :: "'s" assumes "guard s"   and "I s" shows "f s \<le> bound"
concat_eq: fixes v\<^sub>1 :: "'b list"   and v\<^sub>2 :: "'b list"   and u\<^sub>1 :: "nat \<Rightarrow> 'b"   and u\<^sub>2 :: "nat \<Rightarrow> 'b" assumes "length v\<^sub>1 = length v\<^sub>2" shows "(v\<^sub>1 \<frown> u\<^sub>1 = v\<^sub>2 \<frown> u\<^sub>2) = (v\<^sub>1 = v\<^sub>2 \<and> u\<^sub>1 = u\<^sub>2)"
d: shows "Reflected_Multivariate_Polynomial.degree p = Reflected_Multivariate_Polynomial.degree q"
case_4: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b" shows "(case Public of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f4"
total: shows "\<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
ASSERT_le_iff: fixes \<Phi> :: "bool"   and S :: "'ma"   and S' :: "'ma" shows "(bind (ASSERT \<Phi>) (\<lambda>_. S) \<le> S') = ((S' \<noteq> top \<longrightarrow> \<Phi>) \<and> S \<le> S')"
target: fixes T :: "'procT" shows "TargetTerm T \<sim>\<lbrakk>\<cdot>\<rbrakk> TargetTerm T"
extreme_def: fixes p :: "alt \<Rightarrow> real"   and b :: "alt" shows "extreme p b \<equiv> b \<cdot>< p \<or> p <\<cdot> b"
neq_Nil_conv: fixes xs :: "'a list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y \<Colon> ys)"
fun_upd_eqvt: fixes p :: "perm"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" shows "p \<bullet> f(x := y) = (p \<bullet> f)(p \<bullet> x := p \<bullet> y)"
halloc_type_sound: fixes G :: "prog"   and s1 :: "state"   and oi :: "obj_tag"   and a :: "loc"   and x :: "abopt"   and s :: "st"   and L :: "lenv"   and T :: "ty"   and fs :: "vn \<Rightarrow> val option" assumes "G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> (x, s)"   and "wf_prog G"   and "s1\<Colon>\<preceq>(G, L)"   and "T = obj_ty \<lparr>tag = oi, values = fs\<rparr>"   and "is_type G T" shows "(x, s)\<Colon>\<preceq>(G, L) \<and> (x = None \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>T)"
emeasure_point_measure: fixes X :: "'b set"   and f :: "'b \<Rightarrow> ennreal"   and A :: "'b set" assumes "finite {a \<in> X. 0 < f a}"   and "X \<subseteq> A" shows "emeasure (point_measure A f) X = (\<Sum>a | a \<in> X \<and> 0 < f a. f a)"
simps_1: fixes carrier :: "'d set"   and eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and le :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and more :: "'e" shows "(\<sqsubseteq>\<^bsub>\<lparr>carrier = carrier, eq = eq, le = le, \<dots> = more\<rparr>\<^esub>) = le"
abs_eq: fixes x :: "('a, 'b, 'c, 'd) Tree" shows "Not\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tNot x)"
simps: fixes v :: "nat"   and c :: "nat" shows "local.fast_dice_roll v c = (if n \<le> v then if c < n then return_spmf c else local.fast_dice_roll (v - n) (c - n) else coin_spmf \<bind> (\<lambda>b. local.fast_dice_roll (2 * v) (2 * c + (if b then 1 else 0))))"
simps_1: fixes pi :: "('e \<times> 'e) list" shows "pi \<bullet> [] = []"
index_vec_of_scal: fixes x :: "'b" shows "vec_of_scal x $ 0 = x"
ax2: fixes l :: "Lines"   and m :: "Lines" shows "\<exists>P. incid P l \<and> incid P m"
cong_ex: fixes f1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f1 x = f2 x"   and "\<forall>\<^sub>F x in F. g1 x = g2 x" shows "(f1 \<in> o[F](g1)) = (f2 \<in> o[F](g2))"
space_distr: fixes M :: "'b measure"   and N :: "'a measure"   and f :: "'b \<Rightarrow> 'a" shows "space (distr M N f) = space N"
simps_2: fixes p :: "fm"   and q :: "fm" shows "iszlfm (Or p q) = (iszlfm p \<and> iszlfm q)"
zip_eq_Nil_iff: fixes xs :: "'c list"   and ys :: "'d list" shows "(xs || ys = []) = (xs = [] \<or> ys = [])"
to_nat_less_card: shows "mod_type_class.to_nat i < CARD('n)"
csingleton_inject: fixes a :: "'c"   and b :: "'c" assumes "csingle a = csingle b" shows "a = b"
posl: fixes x :: "'a"   and y :: "'a" assumes "D x y"   and "x \<oplus> y \<in> E" shows "x \<in> E"
fold_bind_random_permutation_altdef: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'b pmf"   and x :: "'d"   and A :: "'c set" shows "fold_bind_random_permutation f g x A = fold_random_permutation f x A \<bind> g"
vimage_inter_cong: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and y :: "'e set" assumes "\<And>w. w \<in> S \<Longrightarrow> f w = g w" shows "f -` y \<inter> S = g -` y \<inter> S"
that: fixes Q' :: "'state" assumes "Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle>"   and "P' \<sim>\<cdot> Q'" shows "thesis"
Suc_mono: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "Suc m < Suc n"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> bankerberos"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> bankerberos; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evs1 A B. \<lbrakk>evs1 \<in> bankerberos; P evs1\<rbrakk> \<Longrightarrow> P (Says A Server \<lbrace>Agent A, Agent B\<rbrace> # evs1)"   and "\<And>evs2 K A' A B. \<lbrakk>evs2 \<in> bankerberos; P evs2; Key K \<notin> used evs2; K \<in> symKeys; Says A' Server \<lbrace>Agent A, Agent B\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says Server A (Crypt (shrK A) \<lbrace>Number (CT evs2), Agent B, Key K, Crypt (shrK B) \<lbrace>Number (CT evs2), Agent A, Key K\<rbrace>\<rbrace>) # evs2)"   and "\<And>evs3 S A Tk B K Ticket. \<lbrakk>evs3 \<in> bankerberos; P evs3; Says S A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evs3; Says A Server \<lbrace>Agent A, Agent B\<rbrace> \<in> set evs3; \<not> expiredK Tk evs3\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Ticket, Crypt K \<lbrace>Agent A, Number (CT evs3)\<rbrace>\<rbrace> # evs3)"   and "\<And>evs4 A' B Tk A K Ta. \<lbrakk>evs4 \<in> bankerberos; P evs4; Says A' B \<lbrace>Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs4; \<not> expiredK Tk evs4; \<not> expiredA Ta evs4\<rbrakk> \<Longrightarrow> P (Says B A (Crypt K (Number Ta)) # evs4)"   and "\<And>evso A Tk B K Ticket. \<lbrakk>evso \<in> bankerberos; P evso; Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evso; expiredK Tk evso\<rbrakk> \<Longrightarrow> P (Notes Spy \<lbrace>Number Tk, Key K\<rbrace> # evso)" shows "P x"
S1: shows "pr_gr (loc_upb n x) = g_step (pr_gr (c_pair (c_pair n x) 0)) (c_fst (c_pair (c_pair n x) 0))"
is_Var_iff: fixes v :: "hf" shows "is_Var v = (v = succ (ord_of (nat_of_name (decode_Var v))))"
mapCollect_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and m :: "'b \<Rightarrow> 'c option" shows "mapCollect f m = {f k v |k v. m k = Some v}"
label_append: fixes a :: "'b list"   and b :: "'b list" shows "label_list (a @ b) = State_Monad.ap (State_Monad.ap (State_Monad.return (@)) (label_list a)) (label_list b)"
sat_dualI: fixes y :: "'a vec"   and A :: "'a mat"   and c :: "'a vec" assumes "[y\<^sub>v*A]=c"   and "\<forall>i<dim_vec y. (0::'a) \<le> y $ i" shows "y \<in> sat_dual A c"
hyps_free_acyclic: fixes nodes :: "'a stream"   and inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and vertices :: "'d fset"   and nodeOf :: "'d \<Rightarrow> 'a"   and edges :: "(('d \<times> 'c) \<times> 'd \<times> 'b) set"   and hyps :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and v :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "Acyclic_Graph nodes inPorts outPorts vertices nodeOf edges hyps"   and "Pre_Port_Graph.path edges v v pth"   and "Scoped_Graph.hyps_free nodeOf hyps pth" shows "pth = []"
square_diff_square_factored: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.comm_ring plus zero minus uminus times" shows "minus (times x x) (times y y) = times (plus x y) (minus x y)"
comp_of_ords: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lt :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder le lt" shows "comparator (comp_of_ords le lt)"
scalar_prod_left_unit: fixes v :: "'b vec"   and n :: "nat"   and i :: "nat" assumes "v \<in> carrier_vec n"   and "i < n" shows "unit_vec n i \<bullet> v = v $ i"
MONO': fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "f x \<le> f y"
rel_spmf_bindI2: fixes q :: "'b spmf"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c spmf"   and f :: "'b \<Rightarrow> 'd spmf" assumes "\<And>x. x \<in> set_spmf q \<Longrightarrow> rel_spmf R p (f x)"   and "lossless_spmf q" shows "rel_spmf R p (q \<bind> f)"
le_inf_iff: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "less_eq x (inf y z) = (less_eq x y \<and> less_eq x z)"
iarray_length_of_fun: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "IArray.length (IArray.of_fun f n) = n"
rl: fixes ys :: "('a \<times> 'b) list" assumes "oalist_inv_raw ys"   and "fst ` set xs \<inter> fst ` set ys = {}" shows "set (foldr update_by_pair xs ys) = set xs \<union> set ys"
name_swap_bij: fixes a :: "name"   and b :: "name"   and x :: "'d" shows "[(a, b)] \<bullet> [(a, b)] \<bullet> x = x"
g'_nonneg: fixes y :: "nat" shows "0 \<le> g' y"
cycle_free_diag: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and i :: "nat" assumes "linordered_monoid.cycle_free m n"   and "i \<le> n" shows "Le (0::'d) \<preceq> m i i"
rep_eq: fixes x :: "int" shows "map_option base (restricted_of_simple x) = (if x \<in> {0, 2, 4, 6} then Some x else None)"
hyps_2: shows "m'' \<in> set (predecessors g m')"
cycle_Cons: fixes x :: "'a"   and xs :: "'a list" shows "cycle (x # xs) = x ## cycle (xs @ [x])"
add_le_cancel_left0: fixes x :: "V"   and z :: "V" shows "x \<le> x + z"
branch_inv_Branching_Right: fixes b :: "bool list"   and l :: "tree"   and r :: "tree" shows "branch (Right # b) (Branching l r) = branch b r"
cl_def: fixes L :: "'b set set"   and r :: "'b set" shows "cl L r \<equiv> \<Inter> {x \<in> L. r \<subseteq> x}"
inv_partition_def: fixes S :: "'a set"   and X :: "'a set"   and P :: "'a set set" shows "inv_partition S X P = (inv_iv S X \<and> \<Union> P = X \<and> (\<forall>p\<in>P. \<exists>s\<in>V. p = {s} \<union> neighbors s) \<and> card P = card S \<and> finite P)"
Mapping_Mapping_lookup: fixes m :: "('a, 'b) mapping" shows "Mapping (Mapping.lookup m) = m"
conj_commute: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "conj x y = conj y x"
Y0_carrier: shows "Y0 \<subseteq> carrier_vec (n + 1)"
distinct_27: shows "ToLowerCaseOp \<noteq> string_unop.ToIntegerOp"
diag_lub: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "class.cpo below"   and "\<And>j. po.chain below (\<lambda>i. Y i j)"   and "\<And>i. po.chain below (Y i)" shows "po.lub below (range (\<lambda>i. po.lub below (range (Y i)))) = po.lub below (range (\<lambda>i. Y i i))"
assms_1: shows "\<nexists>t. local.prefixes t j \<up>"
Suc_length_conv: fixes n :: "nat"   and xs :: "'c list" shows "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
finite_jumpFs_combine: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "finite_jumpFs f a b"   and "finite_jumpFs f b c" shows "finite_jumpFs f a c"
cprod_def: fixes r1 :: "('b \<times> 'b) set"   and r2 :: "('c \<times> 'c) set" shows "r1 *c r2 = |Field r1 \<times> Field r2|"
simps_9: fixes b :: "'b"   and x :: "'b"   and AB :: "'b tree"   and C :: "'b tree"   and D :: "'b tree" assumes "b < x" shows "splay x \<langle>AB, b, \<langle>C, x, D\<rangle>\<rangle> = \<langle>\<langle>AB, b, C\<rangle>, x, D\<rangle>"
intros_1: fixes u :: "'c \<Rightarrow> 'b"   and r :: "'c"   and d :: "'b" assumes "u r \<le> d" shows "dbm_entry_val u (Some r) None (Le d)"
contour_integral_neg: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" assumes "f contour_integrable_on g" shows "contour_integral g (\<lambda>x. - f x) = - contour_integral g f"
not_fin_subst1: fixes M :: "trm"   and x :: "name"   and c :: "coname"   and y :: "name"   and P :: "trm" assumes "\<not> fin M x" shows "\<not> fin (M{c:=y.P}) x"
R_lemma: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.B2_algebra plus times one zero less_eq less star"   and "times x x = x" shows "star x = plus one x"
numeral_3_eq_3: shows "3 = Suc (Suc (Suc 0))"
saturated_upto_complete_if: fixes N :: "'a literal multiset set" assumes "sr.saturated_upto N"   and "\<not> satisfiable N" shows "{#} \<in> N"
simps_2: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'d"   and l :: "'d list" shows "top_sorted_abs R (h # l) = (list_all (\<lambda>x. \<not> R x h) l \<and> top_sorted_abs R l)"
in_angle_asym: fixes D :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "D InAngle A B C"   and "C InAngle A B D" shows "A B C CongA A B D"
crypts_crypt_1: fixes K :: "key"   and X :: "msg" shows "crypts {Crypt K X} \<subseteq> Crypt K ` crypts {X}"
conjugate_inj: fixes x :: "complex_homo"   and y :: "complex_homo" assumes "conjugate x = conjugate y" shows "x = y"
H: shows "\<forall>m'<m. \<forall>n. prod a {0..m'} $ n = (\<Sum>v\<in>natpermute n (m' + 1). \<Prod>j = 0..m'. a j $ v ! j)"
standard_maintains_identity_rules: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph"   and x :: "('b Standard_Constant, nat) labeled_graph \<times> ('b Standard_Constant, nat) labeled_graph"   and L :: "'b Standard_Constant set" assumes "standard' C G"   and "x \<in> identity_rules L" shows "maintained x G"
simps_11: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x21 :: "'a"   and x22 :: "'a list"   and y21 :: "'b"   and y22 :: "'b list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
ccompare_enat_def: shows "ccompare = Some (\<lambda>x y. if x = y then order.Eq else if x < y then order.Lt else order.Gt)"
ring_hom_def: fixes R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" shows "ring_hom R S = {h \<in> carrier R \<rightarrow> carrier S. (\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y \<and> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) \<and> h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
ImageI: fixes a :: "'e"   and b :: "'f"   and r :: "('e \<times> 'f) set"   and A :: "'e set" assumes "(a, b) \<in> r"   and "a \<in> A" shows "b \<in> r `` A"
ivl_rel_def: fixes S :: "('c \<times> 'd) set" shows "\<langle>S\<rangle>ivl_rel = (S \<times>\<^sub>r S) O br set_of_ivl top"
functor_axioms: shows "functor A C (K \<circ> F)"
subst_mono_neq: fixes t :: "('a, 'b) Term.term"   and u :: "('a, 'b) Term.term"   and s :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "t \<sqsubset> u" shows "t \<cdot> s \<sqsubset> u \<cdot> s"
nat_not_ge_1D: fixes x :: "nat" assumes "\<not> Suc 0 \<le> x" shows "x = 0"
lead_coeff_rebase_poly: fixes p :: "'q mod_ring poly" shows "lead_coeff (#p) = @lead_coeff p"
poly_gcd_monic_factor: fixes p :: "'c poly"   and q :: "'c poly"   and r :: "'c poly" assumes "monic p" shows "gcd (p * q) (p * r) = p * gcd q r"
l0_f2_f2': shows "f2 / (f2 - f2') < real l0"
Postdomination_axioms: fixes prog :: "cmd" shows "Postdomination sourcenode targetnode kind (valid_edge prog) (_Entry_) (_Exit_)"
infsetsum_cmult_left': fixes f :: "'d \<Rightarrow> 'c"   and c :: "'c"   and A :: "'d set" shows "(\<Sum>\<^sub>ax\<in>A. f x * c) = infsetsum f A * c"
vec_elements_h_def: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec" shows "vec_elements_h v = range (($h) v)"
Lim_transform_bound: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and net :: "'a filter" assumes "\<forall>\<^sub>F n in net. norm (f n) \<le> norm (g n)"   and "(g \<longlongrightarrow> (0::'c)) net" shows "(f \<longlongrightarrow> (0::'b)) net"
simps_2: fixes uv :: "variable"   and T :: "typ" shows "tvs (Fv uv T) = tvsT T"