hcju/isabelleps huggingface dataset:
assms_3: shows "dgrad (pp_of_term u) \<le> dgrad_max dgrad"
nths_Cons: fixes x :: "'b"   and l :: "'b list"   and A :: "nat set" shows "nths (x # l) A = (if 0 \<in> A then [x] else []) @ nths l {j. Suc j \<in> A}"
S4: shows "c_fst (Suc (Suc 3)) = 2"
convex_epigraph: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" shows "convex (epigraph S f) = (convex_on S f \<and> convex S)"
case_eq_if: fixes f :: "le_rel \<Rightarrow> linear_poly \<Rightarrow> 'b \<Rightarrow> 'a"   and le_constraint :: "'b le_constraint" shows "(case le_constraint of Le_Constraint x xa xb \<Rightarrow> f x xa xb) = f (lec_rel le_constraint) (lec_poly le_constraint) (lec_const le_constraint)"
set_cases: fixes e :: "'a"   and a :: "'a behaviour" assumes "e \<in> set_behaviour a" obtains "a = Terminates e"   | "a = Goes_wrong e"
EmptyMap_NoCycles: fixes SA :: "('d, 'e, 'f) seqauto" shows "NoCycles {SA} (EmptyMap (States SA))"
numer_cf7_def: shows "numer_cf7 \<equiv> \<lambda>x. x ^ 7 + 56 * x ^ 6 + 1512 * x ^ 5 + 25200 * x ^ 4 + 277200 * x ^ 3 + 1995840 * x\<^sup>2 + 8648640 * x + 17297280"
leg_simps_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "span_in_category C S" shows "partial_magma.arr C (Leg1 S)"
io_targets_succ: fixes q2 :: "'d"   and M :: "('e, 'f, 'd) FSM"   and q1 :: "'d"   and xy :: "'e \<times> 'f" assumes "q2 \<in> io_targets M q1 [xy]" shows "q2 \<in> succ M xy q1"
D: shows "(Asset n, SesKey SK) \<in> s \<and> Crypt (SesK SK) (Pwd n) \<in> used s"
finite_UN: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "finite A" shows "finite (\<Union> (B ` A)) = (\<forall>x\<in>A. finite (B x))"
Diff_subset: fixes A :: "'d set"   and B :: "'d set" shows "A - B \<subseteq> A"
k': shows "spr_simAbs k' \<in> (\<Union>a. SPRdet.sim_equiv_class a ` SPR.jkbpC)"
p_unique: fixes p' :: "real" assumes "(\<Sum>i<k. as ! i * bs ! i powr p') = 1" shows "p = p'"
stut_always: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "STUTINV F" shows "STUTINV \<box>F"
DERIV_cmult: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b"   and s :: "'b set"   and c :: "'b" assumes "(f has_field_derivative D) (at x within s)" shows "((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
homeomorphic_maps_imp_homeomorphic_space: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "homeomorphic_maps X Y f g" shows "X homeomorphic_space Y"
after_secure: fixes xs :: "'e list"   and P :: "'e process"   and I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f" assumes "xs \<in> traces P"   and "secure P I D" shows "secure (P \<setminus> xs) I D"
map_ext: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> set xs \<longrightarrow> f x = g x" shows "map f xs = map g xs"
gcd_int_def: fixes x :: "int"   and y :: "int" shows "gcd x y = int (gcd (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
bigo_zero: fixes g :: "'c \<Rightarrow> 'd" shows "0 \<in> O(g)"
wf_preallocatedE: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and h :: "'heap"   and C :: "String.literal" assumes "wf_prog wf_md P"   and "preallocated h"   and "C \<in> sys_xcpts" obtains "typeof_addr h (addr_of_sys_xcpt C) = \<lfloor>Class_type C\<rfloor>" and "P \<turnstile> C \<preceq>\<^sup>* Throwable"
degree_m_prod_mset_le: fixes fs :: "int poly multiset" shows "q.degree_m (\<Prod>\<^sub># fs) \<le> \<Sum>\<^sub># (image_mset q.degree_m fs)"
landau_function_family_axioms: shows "landau_function_family F (G * H)"
card_remdups_def: shows "card_remdups = length \<circ> remdups"
pr_var_def: fixes x :: "'e \<Longrightarrow> 'f" shows "&x = x"
degree_pcompose_le: fixes p :: "'a poly"   and q :: "'a poly" shows "degree (p \<circ>\<^sub>p q) \<le> degree p * degree q"
Keys_to_list_def: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "Keys_to_list ps = fold (\<lambda>p. merge_wrt (\<succ>\<^sub>t) (keys_to_list p)) ps []"
auth_pubkey_asset_ii: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset m, \<lbrace>Num 2, Auth_PubKey n\<rbrace>) \<in> s" shows "False"
sp_one: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and e :: "('d \<times> nat) edge"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "SubRel.subpath g v1 [e] v2 subs = (sub_rel_of g subs \<and> (v1 = src e \<or> (v1, src e) \<in> subs\<^sup>+) \<and> e \<in> edges g \<and> (tgt e = v2 \<or> (tgt e, v2) \<in> subs\<^sup>+))"
RModuleHomI: fixes T :: "'m \<Rightarrow> 'b"   and smult' :: "'r \<Rightarrow> 'b \<Rightarrow> 'b" assumes "GroupHom M T"   and "\<And>r m. \<lbrakk>r \<in> R; m \<in> M\<rbrakk> \<Longrightarrow> T (r \<cdot> m) = smult' r (T m)" shows "RModuleHom R (\<cdot>) M smult' T"
q_def: fixes v :: "real"   and s0 :: "real"   and t :: "real" shows "movement.q v s0 t = s0 + v * t"
zet_image_mem: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "A \<in> zet" shows "g ` A \<in> zet"
b: shows "n \<le> Max (snd ` nonzero_positions (Rep_matrix A))"
fls_def: shows "fls \<equiv> Robinson_Arithmetic.neg (zer EQ zer)"
assms_2: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs"   and "P (dropWhile (\<lambda>y. y \<noteq> x) xs)" shows "P (x # xs)"
induced_ghom_ginjec: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "ginj\<^bsub>(F / gker\<^bsub>F,G\<^esub> f),G\<^esub> (f\<dieresis>\<^bsub>F,G\<^esub>)"
hyps_3: shows "sum S F = {sum s F |s. \<forall>i\<in>F. s i \<in> S i}"
induct: fixes x1 :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and x2 :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and P :: "('a, 'b, 'varSort, 'var, 'c) qTerm set \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set \<Rightarrow> bool" assumes "(x1, x2) \<in> swapped"   and "\<And>X. P X X"   and "\<And>X Y Z. \<lbrakk>(X, Y) \<in> swapped; P X Y; (Y, Z) \<in> swapped; P Y Z\<rbrakk> \<Longrightarrow> P X Z"   and "\<And>X Y x y zs. \<lbrakk>(X, Y) \<in> swapped; P X Y\<rbrakk> \<Longrightarrow> P X (Y #[x \<and> y]_zs)" shows "P x1 x2"
bsaxy: shows "set (bs @ [a] @ f) \<subseteq> {x, y}"
red1Return: fixes e' :: "'addr expr1"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and x' :: "'addr val list"   and e :: "'addr expr1"   and x :: "'addr val list"   and exs :: "('addr expr1 \<times> 'addr val list) list"   and h :: "'heap" assumes "final e'" shows "uf,P,t \<turnstile>1 \<langle>(e', x')/(e, x) # exs,h\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>(inline_call e' e, x)/exs,h\<rangle>"
sqrt: shows "sqrt (2 * pi * (\<sigma>' + \<tau>')) * sqrt (2 * pi * (\<sigma>' * \<tau>') / (\<sigma>' + \<tau>')) = sqrt (2 * pi * \<sigma>') * sqrt (2 * pi * \<tau>')"
cod_simp: fixes f :: "'a" assumes "S.arr f" shows "S.cod f = B.cod f"
match: fixes pat :: "term"   and t :: "term"   and env :: "(name, term) fmap"   and n :: "nat" assumes "match pat t = Some env"   and "wellformed' n t" shows "fmpred (\<lambda>_. wellformed' n) env"
of_int_le_1_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus z) one = (z \<le> 1)"
higherPoly\<^sub>0_def: shows "higherPoly\<^sub>0 \<equiv> map_fun id (map_fun id (map_fun lookup Abs_poly_mapping)) (\<lambda>b i mp mon. if \<exists>j\<in>{b..<b + i}. 0 < lookup mon j then 0::'a else mp (higherPowers b i mon))"
ImplE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<rightarrow> \<psi> in v]" shows "[\<phi> in v] \<longrightarrow> [\<psi> in v]"
distinct_7: fixes x1 :: "'b"   and x5 :: "('b, 'c) fm" shows "Pro x1 \<noteq> (\<^bold>\<diamond> x5)"
subprob_emeasure_le_1: fixes X :: "'a set" shows "emeasure M X \<le> 1"
cases: fixes x :: "'a literal" obtains p :: "char list" and ts :: "'a list" where "x = Pos p__ ts__"   | p :: "char list" and ts :: "'a list" where "x = Neg p ts"
mult_eq_1: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) \<le> a"   and "a \<le> (1::'b)"   and "b \<le> (1::'b)" shows "(a * b = (1::'b)) = (a = (1::'b) \<and> b = (1::'b))"
preserves_reflects_arr: fixes f :: "('a E.term, 'a) arr \<times> ('a E.term, 'a) arr" shows "B.arr (cmp\<^sub>D\<^sub>N f) = VV.arr f"
filterpositions2_def: fixes P :: "'c \<Rightarrow> bool"   and l :: "'c list" shows "filterpositions2 P l = concat (map (\<lambda>n. if P (l ! n) then [n] else []) [0..<length l])"
division_ring_divide_zero: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a zero = zero"
cases: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'b" assumes "transition_system.path execute enabled a1 a2" obtains p :: "'b" where "a1 = []" and "a2 = pa__"   | a :: "'a" and p :: "'b" and r :: "'a list" where "a1 = a # r" and "a2 = p" and "enabled a p" and "transition_system.path execute enabled r (execute a p)"
obtain_qFresh: fixes V :: "'var set"   and XS :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and xs :: "'varSort" assumes "finite V \<or> |V| <o |UNIV|"   and "finite XS \<or> |XS| <o |UNIV|"   and "Ball XS qGood" shows "\<exists>x. x \<notin> V \<and> (\<forall>X\<in>XS. qAFresh xs x X \<and> qFresh xs x X)"
closed_orbit_period_zero_fixed_point: fixes x :: "'a" assumes "closed_orbit x"   and "period x = 0" shows "f x = (0::'a)"
siso_Sbis: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis"   and "PL_Indis.siso aval tval cval indis c" shows "(c, c) \<in> PL_Indis.Sbis aval tval cval indis"
restrict_eq: fixes A :: "'e set" shows "restrict A = filter (\<lambda>(k, v). k \<in> A)"
assms: fixes x :: "'a"   and y :: "'b" assumes "x \<in> fmran' xm" shows "f x (g x y)"
hcomplex_of_complex_image: shows "range hcomplex_of_complex = SComplex"
assms_1: shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
normalize_replace_primitive_matchexpr_preserves_normalized_not_has_disc_negated: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and disc2 :: "'b \<Rightarrow> bool"   and neg :: "bool"   and replace_f :: "'c negation_type \<Rightarrow> 'b match_expr"   and m' :: "'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<not> has_disc_negated disc2 neg m"   and "\<And>a. \<not> has_disc_negated disc2 neg (replace_f a)"   and "m' \<in> set (normalize_match (replace_primitive_matchexpr (disc, sel) replace_f m))" shows "\<not> has_disc_negated disc2 neg m'"
has_derivative_local_min: fixes f :: "'a \<Rightarrow> real"   and f' :: "'a \<Rightarrow> real"   and x :: "'a" assumes "(f has_derivative f') (at x)"   and "\<forall>\<^sub>F y in at x. f x \<le> f y" shows "f' = (\<lambda>h. 0)"
lookup_map_default': fixes x :: "'c"   and d :: "'b"   and f :: "'b \<Rightarrow> 'b"   and m :: "('c, 'b) mapping"   and y :: "'c" shows "lookup (map_default x d f m) y = (if x = y then Some (f (lookup_default d m x)) else lookup m y)"
product_group_def: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" shows "product_group I G \<equiv> \<lparr>carrier = \<Pi>\<^sub>E i\<in>I. carrier (G i), mult = \<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i, one = \<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>\<rparr>"
that: fixes i :: "nat" assumes "i \<le> n"   and "\<pi> i = l" shows "thesis"
hcomp_simps_2: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "hseq \<nu> \<mu>" shows "trg (\<nu> \<star> \<mu>) = trg \<nu>"
sequent_def: shows "sequent = (\<lambda>(atoms, nforms). map snd nforms @ map (\<lambda>(z, p, vs). FAtom z p vs) atoms)"
STL4_eve: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> \<diamond>F \<longrightarrow> \<diamond>G"
Id_onE: fixes c :: "'e \<times> 'e"   and A :: "'e set" assumes "c \<in> Id_on A" obtains x :: "'e" where "x \<in> A" and "c = (x, x)"
hom_int_pow: fixes h :: "'a \<Rightarrow> 'b"   and G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme"   and x :: "'a"   and n :: "int" assumes "h \<in> hom G H"   and "x \<in> carrier G"   and "Group.group G"   and "Group.group H" shows "h (x [^]\<^bsub>G\<^esub> n) = h x [^]\<^bsub>H\<^esub> n"
simps_11: fixes va :: "atom" shows "Z.nnf (Atom va) = Atom va"
app_vimageI2: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"   and "a \<in>\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> r `\<^sub>\<circ> A"
that: fixes n :: "nat" assumes "n \<noteq> 0"   and "x \<in> elts (y * ord_of_nat n)"   and "\<And>m. m < n \<Longrightarrow> x \<notin> elts (y * ord_of_nat m)" shows "thesis"
is_rbt_rbtreeify: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and kvs :: "('c \<times> 'd) list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (list.map fst kvs)"   and "distinct (list.map fst kvs)" shows "ord.is_rbt less (rbtreeify kvs)"
realistic_execution_def: fixes aseq :: "'action_t list list" shows "realistic_execution aseq \<equiv> set aseq \<subseteq> AS_set"
map_poly_eval_poly: fixes h :: "'c \<Rightarrow> 'b"   and p :: "'c poly" assumes "h (0::'c) = (0::'b)" shows "map_poly h p = eval_poly (\<lambda>a. [:h a:]) p [:0::'b, 1::'b:]"
liftT0_1: fixes i :: "nat"   and T :: "POPLmarkRecord.type" shows "\<up>\<^sub>\<tau> 0 i T = T"
rdomp_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and a :: "'a" shows "rdomp R a \<equiv> \<exists>h c. h \<Turnstile> R a c"
u: fixes a :: "'a\<^sub>m" assumes "a \<in> A" shows "bo a u"
indis_iff: fixes a :: "'state"   and d :: "'state"   and b :: "'state"   and c :: "'state" assumes "a \<approx> d"   and "b \<approx> d" shows "(a \<approx> c) = (b \<approx> c)"
sup_Un_eq2: fixes R :: "('e \<times> 'f) set"   and S :: "('e \<times> 'f) set" shows "sup (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
\<tau>rtrancl3p_trans: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and tls :: "('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and tls' :: "('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list"   and s'' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "mthr.\<tau>rtrancl3p s tls s'"   and "mthr.\<tau>rtrancl3p s' tls' s''" shows "mthr.\<tau>rtrancl3p s (tls @ tls') s''"
gfun_mem: fixes v1 :: "val"   and v2 :: "val"   and f :: "(val \<times> val) fset"   and x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> bval) list" assumes "(v1, v2) \<in> fset f"   and "good_fun f x e \<rho>" shows "\<forall>v\<in>good v1. \<exists>v'. (x, v) # \<rho> \<turnstile> e \<Down> v' \<and> v' \<in> good v2"
Finca_get_compile: fixes e :: "'fenv_std"   and x :: "'fun" shows "Finca_get (compile_env e) x = map_option compile_fundef (Fstd_get e x)"
resultant_impl_main: fixes div_exp :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> 'b"   and G2 :: "'b poly"   and G1 :: "'b poly" assumes "\<And>x y n. to_fract x ^ n / to_fract y ^ (n - 1) \<in> range to_fract \<Longrightarrow> to_fract (div_exp x y n) = to_fract x ^ n / to_fract y ^ (n - 1)"   and "length (coeffs G2) \<le> length (coeffs G1)" shows "resultant_impl_main div_exp G1 G2 = resultant G1 G2"
abs_zero: fixes abs :: "'d \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs zero = zero"
t_find_closest_eq_time_find_closest_tm: fixes p :: "int \<times> int"   and \<delta> :: "real"   and ps :: "(int \<times> int) list" shows "t_find_closest p \<delta> ps = time (find_closest_tm p \<delta> ps)"
o_def: fixes f :: "'e \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'e" shows "f \<circ> g = (\<lambda>x. f (g x))"
eq': shows "\<forall>x. (THE f'. \<forall>x. ((\<lambda>x. 0::'e) has_derivative f' x) (at x)) x = (\<lambda>_. 0::'e)"
connected_component_of_subset_topspace: fixes X :: "'b topology"   and x :: "'b" shows "connected_component_of_set X x \<subseteq> topspace X"
Bonk_Schramm_extension_unfolded_wo_def: shows "Bonk_Schramm_extension_unfolded_wo = (SOME r. well_order r \<and> (\<forall>x\<in>range basepoint. \<forall>y\<in>- range basepoint. (x, y) \<in> r) \<and> (\<forall>a b. (a, middle a b) \<in> r) \<and> (\<forall>a b. (b, middle a b) \<in> r) \<and> (\<forall>u n. (u n, would_be_Cauchy u) \<in> r))"
allowed_insert_deliver: fixes i :: "'a"   and e :: "'a \<times> 'b \<times> bool"   and n :: "'a option"   and j :: "nat" assumes "Deliver (i, Insert e n) \<in> set (history j)" shows "n = None \<or> (\<exists>i' n' n'' v b. n = Some n' \<and> Deliver (i', Insert (n', v, b) n'') \<sqsubset>\<^sup>j Deliver (i, Insert e n))"
distinct_3: fixes a1 :: "'d"   and a2 :: "'d"   and psi :: "('d, 'e, 'f) psi"   and psi' :: "('d, 'e, 'f) psi" shows "RIn a1 a2 psi \<noteq> RTau psi'"
assms: shows "x \<in> set (vwalk_to_vpath xs)"
simps_5: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Pos x1 of Pos x \<Rightarrow> f1 x | Neg x \<Rightarrow> f2 x) = f1 x1"
mt: shows "mtype(CT,m,Da) = Ds \<rightarrow> D"
if_if_eq_conj: fixes P :: "bool"   and Q :: "bool"   and x :: "'a"   and y :: "'a" shows "(if P then if Q then x else y else y) = (if P \<and> Q then x else y)"
is_nat_string_simps_2: fixes c :: "char"   and s :: "char list" shows "is_nat_string (c # s) = (is_digit_char c \<and> is_nat_string s)"
hconfD: fixes P :: "'a prog"   and h :: "heap"   and a :: "nat"   and obj :: "obj" assumes "P \<turnstile> h \<surd>"   and "h a = \<lfloor>obj\<rfloor>" shows "P,h \<turnstile> obj \<surd>"
bipartite_web_of_def: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "bipartite_web_of \<Gamma> = \<lparr>edge = \<lambda>uv uv'. case (uv, uv') of (Inl u, Inl a) \<Rightarrow> False | (Inl u, Inr v) \<Rightarrow> edge \<Gamma> u v \<or> u = v \<and> u \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub> \<and> u \<notin> A \<Gamma> \<and> v \<notin> B \<Gamma> | (Inr bb, b) \<Rightarrow> False, weight = \<lambda>uv. case uv of Inl u \<Rightarrow> if u \<in> B \<Gamma> then 0 else weight \<Gamma> u | Inr u \<Rightarrow> if u \<in> A \<Gamma> then 0 else weight \<Gamma> u, A = Inl ` (\<^bold>V\<^bsub>\<Gamma>\<^esub> - B \<Gamma>), B = Inr ` (- A \<Gamma>), \<dots> = web.more \<Gamma>\<rparr>"
UPFDefs_25: fixes S :: "'b set"   and p :: "'b \<Rightarrow> 'c decision option" shows "S \<triangleleft> p \<equiv> \<lambda>x. if x \<in> S then p x else \<bottom>"
ref_morph: fixes us :: "'b list list"   and G :: "'b list set"   and vs :: "'b list list" assumes "us \<in> lists \<langle>G\<rangle>"   and "vs \<in> lists \<langle>G\<rangle>" shows "Ref G us \<cdot> vs = (Ref G us) \<cdot> Ref G vs"
extends_trans: fixes S :: "'b list"   and S' :: "'b list"   and S'' :: "'b list" assumes "S \<lesssim> S'"   and "S' \<lesssim> S''" shows "S \<lesssim> S''"
WhileCondThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e' :: "nat exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and c :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
WT_restD_rfunc_usr: fixes \<I>_adv :: "('p, 'q) \<I>"   and \<I>_usr :: "('r, 's) \<I>"   and I :: "'t \<Rightarrow> bool"   and rest :: "('t, 'u, 'p, 'r, 'q, 's, 't) rest_scheme"   and y :: "'s"   and es :: "'u list"   and s' :: "'t"   and s :: "'t"   and x :: "'r" assumes "WT_rest \<I>_adv \<I>_usr I rest"   and "((y, es), s') \<in> set_spmf (rfunc_usr rest s x)"   and "x \<in> outs_\<I> \<I>_usr"   and "I s" shows "y \<in> responses_\<I> \<I>_usr x \<and> I s'"
prim_rev_iff: fixes u :: "'b list" shows "primitive (rev u) = primitive u"
elims: fixes x :: "fm"   and y :: "fm" assumes "not x = y" obtains p :: "fm" where "x = NOT p" and "y = p"   | "x = T" and "y = F"   | "x = F" and "y = T"   | v :: "Cooper.num" where "x = Lt v__" and "y = NOT (Lt v__)"   | v :: "Cooper.num" where "x = Le va__" and "y = NOT (Le va__)"   | v :: "Cooper.num" where "x = Gt vb__" and "y = NOT (Gt vb__)"   | v :: "Cooper.num" where "x = Ge vc__" and "y = NOT (Ge vc__)"   | v :: "Cooper.num" where "x = Eq vd__" and "y = NOT (Eq vd__)"   | v :: "Cooper.num" where "x = NEq ve__" and "y = NOT (NEq ve__)"   | v :: "int" and va :: "Cooper.num" where "x = Dvd vf__ vaa__" and "y = NOT (Dvd vf__ vaa__)"   | v :: "int" and va :: "Cooper.num" where "x = NDvd vg__ vab__" and "y = NOT (NDvd vg__ vab__)"   | v :: "fm" and va :: "fm" where "x = And vh__ vac__" and "y = NOT (And vh__ vac__)"   | v :: "fm" and va :: "fm" where "x = Or vi__ vad__" and "y = NOT (Or vi__ vad__)"   | v :: "fm" and va :: "fm" where "x = Imp vj__ vae__" and "y = NOT (Imp vj__ vae__)"   | v :: "fm" and va :: "fm" where "x = Iff vk__ va" and "y = NOT (Iff vk__ va)"   | v :: "fm" where "x = E vl__" and "y = NOT (E vl__)"   | v :: "fm" where "x = A vm__" and "y = NOT (A vm__)"   | v :: "nat" where "x = Closed vn__" and "y = NOT (Closed vn__)"   | v :: "nat" where "x = NClosed v" and "y = NOT (NClosed v)"
add_spmat_Nil2: fixes as :: "(nat \<times> (nat \<times> 'a) list) list" shows "add_spmat as [] = as"
assms: shows "q \<rightarrow> l qs \<in> \<delta>"
keys_monom_multE: fixes v :: "'b"   and c :: "'c"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (MPoly_Type_Class.punit.monom_mult c t p)" obtains u :: "'b" where "u \<in> keys p" and "v = MPoly_Type_Class.punit.splus t u"
comp: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and H :: "'b set"   and S :: "'b \<Rightarrow> 'c" assumes "GroupHom G T"   and "GroupHom H S"   and "order.greater_eq H (T ` G)" shows "GroupHom G (S \<circ> T)"