hcju/isabelleps huggingface dataset:
NewC: fixes A :: "triple set"   and P :: "vassn"   and C :: "cname" shows "A \<turnstile>\<^sub>e {\<lambda>s. \<forall>a. new_Addr s = Addr a \<longrightarrow> P (Addr a) (new_obj a C s)} new C {P}"
inv_image_codomain_eq: fixes c :: "('c, 'd) chart" shows "inv_chart c ` codomain c = domain c"
surj_permute: fixes p :: "perm" shows "surj (permute p)"
scaleR2_def: fixes l :: "ereal"   and u :: "ereal"   and X :: "('a \<times> 'b) set" shows "scaleR2 l u X = (\<lambda>(r, x, y). (x, r *\<^sub>R y)) ` (ereal -` {l..u} \<times> X)"
proj2_incident_right_abs: fixes v :: "(real, 3) vec"   and p :: "proj2" assumes "v \<noteq> 0" shows "proj2_incident p (proj2_line_abs v) = (proj2_rep p \<bullet> v = 0)"
bij_betw_f_the_inv_into_f: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and y :: "'c" assumes "bij_betw f A B"   and "y \<in> B" shows "f (the_inv_into A f y) = y"
decomp: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "P = (\<lambda>s. Safe P s \<and> Live P s)"
leg1_composite: shows "gf.leg1 = g.leg1 \<cdot> gf.prj\<^sub>1"
VA': shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT \<theta>. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"
llength_eq_enat_lfiniteD: fixes xs :: "'c llist"   and n :: "nat" assumes "llength xs = enat n" shows "lfinite xs"
SS_def: fixes D :: "('c, 'd) Order_scheme" shows "SS D = \<lparr>carrier = {X. \<exists>a\<in>carrier D. X = segment D a}, rel = {XX \<in> {X. \<exists>a\<in>carrier D. X = segment D a} \<times> {X. \<exists>a\<in>carrier D. X = segment D a}. fst XX \<subseteq> snd XX}\<rparr>"
\<U>\<^sub>c_def: fixes state :: "('a, 'b) state" shows "\<U>\<^sub>c state = (case state of State x1 x2 x3 x4 x5 x6 \<Rightarrow> x6)"
assms_3: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> X"   and "P xs" shows "P (x # xs)"
select_convs_7: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_dport \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = p_dport"
zip_append: fixes xs :: "'a list"   and us :: "'b list"   and ys :: "'a list"   and vs :: "'b list" assumes "length xs = length us" shows "zip (xs @ ys) (us @ vs) = zip xs us @ zip ys vs"
matrix_right_invertible_gen: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "(\<exists>B. matrix f ** B = mat (1::'a)) = (\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> f \<circ> g = id)"
Value_chainE: fixes Y :: "nat \<Rightarrow> Value" assumes "chain Y" obtains "Y = (\<lambda>_. \<bottom>)"   | n :: "nat" and b :: "bool discr" where "Y = (\<lambda>m. if m < n__ then \<bottom> else B\<cdot>b)"   | n :: "nat" and Y' :: "nat \<Rightarrow> Value \<rightarrow> Value" where "Y = (\<lambda>m. if m < n then \<bottom> else Fn\<cdot>(Y' (m - n)))" and "chain Y'"
simps: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" shows "diff t1 t2 = (if t1 = \<langle>\<rangle> then \<langle>\<rangle> else if t2 = \<langle>\<rangle> then t1 else case t2 of \<langle>l2, (a, x), r2\<rangle> \<Rightarrow> let (l1, uu_, r1) = split t1 a; l' = diff l1 l2 in Let (diff r1 r2) (join2 l'))"
fun_def: fixes h' :: "'b \<Rightarrow> 'b"   and b :: "'b"   and Q :: "'c \<Rightarrow> bool"   and g :: "'c \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "recursive h' b" shows "recursive.fun h' b Q g h x = (if tailrec.recfun_aux_dom Q h x then recursive.recfun_aux h' Q g h x else b)"
ANDaux3: fixes y :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and a :: "'b" assumes "y \<in> set xs"   and "a \<in> set (net_list_aux [y])" shows "a \<in> set (net_list_aux xs)"
SN_imp_WN: fixes A :: "('d \<times> 'd) set" assumes "SN A" shows "WN A"
co14_def: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> 'k \<Rightarrow> 'l \<Rightarrow> 'm \<Rightarrow> 'n \<Rightarrow> 'o \<Rightarrow> 'p \<Rightarrow> 'b"   and x1 :: "'c"   and x2 :: "'d"   and x3 :: "'e"   and x4 :: "'f"   and x5 :: "'g"   and x6 :: "'h"   and x7 :: "'i"   and x8 :: "'j"   and x9 :: "'k"   and x10 :: "'l"   and x11 :: "'m"   and x12 :: "'n"   and x13 :: "'o"   and x14 :: "'p" shows "co14 f g x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 = f (g x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14)"
before_in_setD1: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" assumes "x < y in xs" shows "x \<in> set xs"
simps_4: fixes f1 :: "'c"   and f2 :: "n \<Rightarrow> 'c" shows "(case Z of Z \<Rightarrow> f1 | S x \<Rightarrow> f2 x) = f1"
assms_2: shows "v \<cdot> u = u \<cdot> v"
properD2: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and h :: "heap"   and as :: "nat set"   and h' :: "heap" assumes "proper P"   and "P (h, as)"   and "relH as h h'"   and "in_range (h', as)" shows "P (h', as)"
partitioningIps_length: shows "length (partitioningIps (extract_IPSets rs) [wordinterval_UNIV]) \<le> 2 ^ length (extract_IPSets rs) * length [wordinterval_UNIV]"
split_If2: fixes P :: "'a \<Rightarrow> bool"   and Q :: "tr"   and x :: "'a"   and y :: "'a" shows "P (If2 Q x y) = ((Q = \<bottom> \<longrightarrow> P \<bottom>) \<and> (Q = TT \<longrightarrow> P x) \<and> (Q = FF \<longrightarrow> P y))"
SBWriteVolatile': fixes sb' :: "'c memref list"   and sb :: "'c memref list"   and a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and \<theta> :: "nat \<Rightarrow> nat option"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and m :: "nat \<Rightarrow> nat"   and ghst :: "'d \<times> 'e \<times> 'f \<times> 'g" assumes "sb' = sb @ [Write\<^sub>s\<^sub>b True a (D, f) (f \<theta>) A L R W]" shows "(Write True a (D, f) A L R W # is, \<theta>, sb, m, ghst) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>, sb', m, ghst)"
map_add_upt': fixes ofs :: "nat"   and a :: "nat"   and b :: "nat" shows "map (\<lambda>i. i + ofs) [a..<b] = [a + ofs..<b + ofs]"
rifnz: fixes \<rho> :: "(nat \<times> val) list"   and e1 :: "exp"   and n :: "nat"   and e2 :: "exp"   and v :: "val"   and e3 :: "exp" assumes "\<rho> \<turnstile> e1 \<Rightarrow> VNat n"   and "n \<noteq> 0"   and "\<rho> \<turnstile> e2 \<Rightarrow> v" shows "\<rho> \<turnstile> EIf e1 e2 e3 \<Rightarrow> v"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "Left.arr f"   and "Left.dom f = a" shows "Left.comp f a = f"
ide_dom: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.arr f" shows "Maps.ide (Maps.dom f)"
sameDom_def: fixes inp1 :: "'b \<Rightarrow> 'c option"   and inp2 :: "'b \<Rightarrow> 'd option" shows "sameDom inp1 inp2 \<equiv> \<forall>i. (inp1 i = None) = (inp2 i = None)"
represents_div: fixes p :: "int poly"   and x :: "'a"   and q :: "int poly"   and y :: "'a" assumes "p represents x"   and "q represents y"   and "poly q 0 \<noteq> 0" shows "poly_div p q represents x / y"
Abs_errorable_cases: fixes x :: "'b\<^sub>\<bottom>" obtains y :: "'b option" where "x = Abs_errorable y" and "y \<in> UNIV"
elims: fixes x :: "('a, 'b) Combinators list"   and y :: "('a, 'b) Combinators list" assumes "insertDenies x = y" obtains xa :: "('a, 'b) Combinators" and xs :: "('a, 'b) Combinators list" where "x = xa # xs" and "y = (case xa of DenyAll \<Rightarrow> DenyAll # insertDenies xs | _ \<Rightarrow> DenyAllFromTo (first_srcNet xa) (first_destNet xa) \<oplus> DenyAllFromTo (first_destNet xa) (first_srcNet xa) \<oplus> xa # insertDenies xs)"   | "x = []" and "y = []"
strict_implies_order: fixes b :: "'a list"   and a :: "'a list" assumes "strict_prefix b a" shows "prefix b a"
GB_imp_crit_pair_cbelow_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'e"   and q :: "'t \<Rightarrow>\<^sub>0 'e" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis F"   and "p \<in> F"   and "q \<in> F"   and "p \<noteq> 0"   and "q \<noteq> 0" shows "crit_pair_cbelow_on d m F p q"
res: shows "gauss_jordan A (1\<^sub>m n) = (BB, C)"
select_convs_1: fixes init :: "'a set"   and trans :: "('a \<times> 'b \<times> 'a) set"   and more :: "'c" shows "init \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = init"
of_nat_le_of_nat_power_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "nat"   and w :: "nat"   and x :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (power.power one times (semiring_1.of_nat one plus zero b) w) (semiring_1.of_nat one plus zero x) = (b ^ w \<le> x)"
inv_on_f_f: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a" assumes "inj_on f A"   and "x \<in> A" shows "inv_on f A (f x) = x"
transfer_arrive: fixes s :: "'s net_state"   and H :: "nat set"   and K :: "nat set"   and m :: "'m"   and s' :: "'s net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'g"   and \<zeta> :: "'l net_state" assumes "(s, H\<not>K:arrive(m), s') \<in> automaton.trans (pnet np n)"   and "s \<in> reachable (pnet np n) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "wf_net_tree n" obtains \<sigma>'' :: "nat \<Rightarrow> 'g" and \<zeta>'' :: "'l net_state" where "((\<sigma>, \<zeta>), H\<not>K:arrive(m), \<sigma>'', \<zeta>'') \<in> automaton.trans (opnet onp n)" and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>'' j = \<sigma> j" and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>'', \<zeta>'')"
lim: fixes y :: "'a Bonk_Schramm_extension_unfolded" assumes "y \<in> extended_distance_set \<inter> underS Bonk_Schramm_extension_unfolded_wo (would_be_Cauchy u)" shows "(\<lambda>n. extended_distance y (u n)) \<longlonglongrightarrow> extended_distance y (would_be_Cauchy u)"
vreal_mult_closed: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>" shows "x *\<^sub>\<real> y \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"
factorization_preserves_finiteness: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and P1 :: "'b eclause"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "finite (cl_ecl P1)"   and "basic_superposition.factorization trm_ord sel filter_trms P1 C \<sigma> k C'" shows "finite (cl_ecl C) \<and> finite C'"
items_parts_subset: fixes H :: "msg set" shows "items H \<subseteq> parts H"
\<psi>\<^sub>2_is_state: shows "state (n + 1) (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"
dense_run_def: fixes \<rho> :: "'b run" shows "dense_run \<rho> \<equiv> \<forall>n. \<exists>c. hamlet (Rep_run \<rho> n c)"
sin_exp_eq: fixes z :: "complex" shows "sin z = (exp (\<i> * z) - exp (- (\<i> * z))) / (2 * \<i>)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<times> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "nat list \<times> nat" assumes "\<And>a x xs s. (\<And>xa. \<lbrakk>xa = s + a * x; cond (x # xs) xa\<rbrakk> \<Longrightarrow> P a (Suc x) (xs, s)) \<Longrightarrow> P a x (xs, s)" shows "P a0 a1 a2"
axioms_2: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "bounded_linear_axioms f"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "Left_a.isomorphic a a' = (\<exists>f. Left_a.in_hom f a a' \<and> Left_a.iso f)"
sums_add: fixes a :: "'a"   and b :: "'a" assumes "f sums a"   and "g sums b" shows "(\<lambda>n. f n + g n) sums (a + b)"
HeronConf_interp_stepwise_kills_cases: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
univalent_def: fixes R :: "('a \<times> 'b) set" shows "univalent R = (\<forall>x y z. (x, y) \<in> R \<and> (x, z) \<in> R \<longrightarrow> z = y)"
linorder_lattice_sup_ge2: fixes y :: "'a"   and x :: "'a" shows "y \<le> sup x y"
adjoint_functors_respects_naturally_isomorphic: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and F' :: "'d \<Rightarrow> 'c"   and G' :: "'c \<Rightarrow> 'd" assumes "adjoint_functors C D F G"   and "naturally_isomorphic D C F' F"   and "naturally_isomorphic C D G G'" shows "adjoint_functors C D F' G'"
UNION_eq: fixes B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "\<Union> (B ` A) = rel_ext (\<lambda>y. \<exists>x\<in>A. y \<in> B x)"
seqE: fixes g :: "'a"   and f :: "'a" assumes "V.seq g f" obtains "V.arr f" and "V.arr g" and "V.dom g = V.cod f"
strict_trans1: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a \<le>s b"   and "b <s c" shows "a <s c"
Agii_def: fixes A :: "('c, 'e) carrier_scheme"   and f :: "'c \<Rightarrow> 'd" shows "Agii A f = restrict f (carrier A)"
const_poly_dvd_const_poly_iff: fixes a :: "'a"   and b :: "'a" shows "([:a:] dvd [:b:]) = (a dvd b)"
exec_as_proj_valid_2: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "a \<in> PROB" shows "action_dom (fst a) (snd a) \<subseteq> prob_dom PROB"
p2_gt_0: fixes n :: "nat" shows "(0 < 2 ^ n) = (n < LENGTH('b))"
assms_5: shows "polynomial (\<lambda>x. real (q x))"
loop_def: shows "loop \<equiv> while (dwa_cond WA) (dwa_step WA) (dwa_initial WA)"
entails_def: fixes P :: "assn"   and Q :: "assn" shows "P \<Longrightarrow>\<^sub>A Q \<equiv> \<forall>h. h \<Turnstile> P \<longrightarrow> h \<Turnstile> Q"
struct_specI: fixes sel :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "sel_spec sel"   and "punit.ap_spec ap"   and "ab_spec ab"   and "punit.compl_struct compl" shows "punit.struct_spec sel ap ab compl"
iteration_unfoldl_distr: fixes y :: "'a"   and x :: "'a" shows "y + x \<cdot> x\<^sup>\<infinity> \<cdot> y = x\<^sup>\<infinity> \<cdot> y"
mult: fixes a :: "nat"   and b :: "nat" shows "g (a * b) = g a * g b"
adj_in_verts_2: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow> v" shows "v \<in> verts G"
is_rcoset_replistD_set: fixes G :: "'a set"   and H :: "'a set"   and gs :: "'a list" assumes "Group G"   and "Group.is_rcoset_replist G H gs" shows "set gs \<subseteq> G"
fmdom_drop: fixes a :: "'a"   and m :: "('a, 'b) fmap" shows "fmdom (fmdrop a m) = fmdom m |-| {|a|}"
is_sig_GB_in_def: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t" shows "is_sig_GB_in d G u = (\<forall>r. lt r = u \<longrightarrow> r \<in> dgrad_sig_set' (length fs) d \<longrightarrow> sig_red_zero (\<preceq>\<^sub>t) G r)"
mult_minus_distrib_mat_vec: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b vec"   and w :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc"   and "w \<in> carrier_vec nc" shows "A *\<^sub>v (v - w) = A *\<^sub>v v - A *\<^sub>v w"
False: shows "liminf (\<lambda>n. eNorm (\<LL> p M) (f n)) \<noteq> \<infinity>"
ty: shows "[] \<turnstile> v : (\<forall><:T\<^sub>1. T\<^sub>2)"
quicksort_correct_case7: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and xs0 :: "'c list"   and lo :: "nat"   and hi :: "nat"   and xs :: "'c list"   and xs' :: "'c list"   and p :: "nat"   and xs'' :: "'c list"   and xs''' :: "'c list" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. x \<noteq> y \<Longrightarrow> R (h x) (h y) \<or> R (h y) (h x)"   and "quicksort_pre R h xs0 lo hi xs"   and "partition_spec R h xs lo hi xs' p"   and "\<not> p - Suc 0 \<le> lo"   and "\<not> hi \<le> Suc p"   and "quicksort_post R h lo (p - Suc 0) xs' xs''"   and "quicksort_post R h (Suc p) hi xs'' xs'''" shows "quicksort_post R h lo hi xs xs'''"
path_component_sym: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "path_component S x y" shows "path_component S y x"
demand_not_0: fixes e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "demand (\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>) \<noteq> \<bottom>"
tauStepChainCase: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P" shows "\<Psi> \<rhd> Cases Cs \<Longrightarrow>\<^sub>\<tau> P'"
collect_pseudominimals_below_in_eq: fixes x :: "'a"   and P :: "'a set"   and f :: "'a \<Rightarrow> 'b set"   and A :: "'b set"   and a :: "'a set" assumes "x \<in> P"   and "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater (P .\<^bold>\<le> x) f"   and "f ` (P .\<^bold>\<le> x) = Pow A"   and "order.greater_eq {y. pseudominimal_in (P .\<^bold>\<le> x) y} a" shows "a = {y. pseudominimal_in (P .\<^bold>\<le> the_inv_into (P .\<^bold>\<le> x) f (\<Union> (f ` a))) y}"
mat_kernel_mult_eq: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and C :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nr"   and "C \<in> carrier_mat nr nr"   and "C * B = 1\<^sub>m nr" shows "mat_kernel (B * A) = mat_kernel A"
le_approx2: fixes P :: "'b process"   and Q :: "'b process"   and s :: "'b event list"   and X :: "'b event set" assumes "P \<sqsubseteq> Q"   and "s \<notin> D P" shows "((s, X) \<in> F Q) = ((s, X) \<in> F P)"
multp_def: fixes P :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and N :: "'k multiset"   and M :: "'k multiset" shows "multp P N M = (let Z = M \<inter># N; X = M - Z in X \<noteq> {#} \<and> (let Y = N - Z in \<forall>y\<in>#Y. \<exists>x\<in>#X. P y x))"
simps_2: fixes x :: "'a"   and l :: "'a list" shows "perm2 (x # l) = (\<lambda>n. rotateLeft (length (x # rotateLeft (n mod (1 + length l)) (perm2 l (n div (1 + length l)))) - n mod (1 + length l) mod length (x # rotateLeft (n mod (1 + length l)) (perm2 l (n div (1 + length l))))) (x # rotateLeft (n mod (1 + length l)) (perm2 l (n div (1 + length l)))))"
mult_cong_l: fixes a :: "'a"   and a' :: "'a"   and b :: "'a" assumes "a \<sim> a'"   and "a \<in> carrier G"   and "a' \<in> carrier G"   and "b \<in> carrier G" shows "a \<otimes> b \<sim> a' \<otimes> b"
comp_cod_arr: fixes f :: "'a2"   and b :: "'a2" assumes "A2.arr f"   and "A2.cod f = b" shows "b \<cdot>\<^sub>A\<^sub>2 f = f"
simps_1: fixes x :: "'v" shows "min_passign (PWt x) = wt_sym (min_ground_head (Var x))"
snd_min_key_val: fixes xs :: "('b, 'c) oalist_ntm"   and ko :: "'b nat_term_order" assumes "fst (list_of_oalist_ntm xs) \<noteq> []" shows "snd (OAlist_min_key_val_ntm ko xs) = OAlist_lookup_ntm xs (fst (OAlist_min_key_val_ntm ko xs))"
\<epsilon>_in_terms_of_\<psi>: fixes x :: "'a" assumes "Adj.C.ide x" shows "Adj.\<epsilon> x = \<psi> x (G x)"
QP: shows "(\<mu> and\<^sub>r \<eta> \<in> old (fst (n, ns)) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst (n, ns)) \<union> new (fst (n, ns))) \<and> (\<forall>nd\<in>ns. \<mu> and\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd)"
enumerate_in_set: fixes S :: "'a set"   and n :: "nat" assumes "infinite S" shows "enum S n \<in> S"
assms: shows "F \<in> Pi I (S.set \<circ> D) \<inter> S.extensional' I"
list_neq_prefix_cases: fixes ls :: "'a list"   and ls' :: "'a list" assumes "ls \<noteq> ls'"   and "ls \<noteq> []"   and "ls' \<noteq> []" obtains xs :: "'a list" and x :: "'a" and x' :: "'a" and ys :: "'a list" and ys' :: "'a list" where "ls = xs__ @ [x] @ ys" and "ls' = xs__ @ [x'] @ ys'" and "x \<noteq> x'"   | xs :: "'a list" where "ls = ls' @ xsa__" and "xsa__ \<noteq> []"   | xs :: "'a list" where "ls @ xs = ls'" and "xs \<noteq> []"
has_path_def: fixes n :: "nat"   and S :: "(nat \<times> nat) set"   and i :: "nat"   and j :: "nat" shows "has_path n S i j = (\<exists>p. is_path n S p \<and> hd p = i \<and> last p = j)"
iso_Inf_subdistl: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "order.mono less_eq f" shows "f \<circ> Inf \<le> Inf \<circ> (`) f"