hcju/isabelleps huggingface dataset:
l2_inv5I: fixes x :: "l2_state" assumes "\<And>A B M. Confid A B M \<in> chan x \<or> Secure A B M \<in> chan x \<Longrightarrow> M \<in> dy_fake_msg (bad x) (ik x) (chan x)" shows "x \<in> l2_inv5"
divide_less_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide b c) a = (if less zero c then less b (times a c) else if less c zero then less (times a c) b else less zero a)"
gfp_fixpoint: fixes f :: "'c \<Rightarrow> 'c" assumes "mono f" shows "f (gfp f) = gfp f"
1: shows "(\<forall>x\<in>state_set (state_list s as). last x \<in> valid_states PROB) \<noteq> False"
fresh_var_eqE: fixes e :: "exp"   and x :: "Vars.var" assumes "fresh_var e = x" obtains "x \<notin> fv e"
IH: shows "Var x \<lhd> \<sigma> \<noteq> Var x \<or> Var x \<lhd> \<sigma> \<noteq> Var x \<lhd> map_subst f \<sigma> \<longrightarrow> Var x \<lhd> map_subst f \<sigma> = f (Var x \<lhd> \<sigma>)"
poly_lift_hom_degree_eq: fixes S :: "('e, 'f) ring_scheme"   and h :: "'a \<Rightarrow> 'e"   and f :: "nat \<Rightarrow> 'a" assumes "cring S"   and "h \<in> ring_hom R S"   and "h (f (deg R f)) \<noteq> \<zero>\<^bsub>S\<^esub>"   and "f \<in> carrier (UP R)" shows "deg S (poly_lift_hom R S h f) = deg R f"
game_ext_eq: fixes G :: "game"   and H :: "game" shows "(G = H) = (left_options G = left_options H \<and> right_options G = right_options H)"
plus_emeasure: fixes a :: "'a set"   and M :: "'a measure"   and b :: "'a set" assumes "a \<in> sets M"   and "b \<in> sets M"   and "a \<inter> b = {}" shows "emeasure M a + emeasure M b = emeasure M (a \<union> b)"
VVV_arr_char: fixes \<mu>\<nu>\<tau> :: "'a \<times> 'a \<times> 'a" shows "VVV.arr \<mu>\<nu>\<tau> = B.VVV.arr (DN \<mu>\<nu>\<tau>)"
convex_on_ereal_mem: fixes s :: "'b set"   and f :: "'b \<Rightarrow> ereal"   and x :: "'b"   and y :: "'b"   and u :: "real"   and v :: "real" assumes "Lower_Semicontinuous.convex_on s f"   and "x \<in> s"   and "y \<in> s"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> ereal u * f x + ereal v * f y"
offset_poly_def: fixes p :: "'a poly"   and h :: "'a" shows "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
Envbusy: fixes p :: "PrIds" shows "\<turnstile> $Calling memCh p \<longrightarrow> \<not> ENext p"
exists_no_Fvars: shows "\<exists>\<phi>. \<phi> \<in> fmla \<and> bprv \<phi> \<and> Fvars \<phi> = {}"
invertible_iff_is_unit: fixes A :: "(('c, 'd) vec, 'd) vec" shows "invertible A = (det A dvd (1::'c))"
Red_I_of_subset: fixes N :: "'f set"   and N' :: "'f set" assumes "N \<subseteq> N'" shows "Red_I N \<subseteq> Red_I N'"
deg_n_of_X: shows "deg_n R S X X = Suc 0"
map_of_subtract_entries_raw: fixes ys :: "('b \<times> 'c) list"   and xs :: "('b \<times> 'c) list"   and x :: "'b" assumes "distinct (map fst ys)" shows "map_of (subtract_entries_raw xs ys) x = (case map_of xs x of None \<Rightarrow> None | Some v \<Rightarrow> case map_of ys x of None \<Rightarrow> Some v | Some v' \<Rightarrow> Some (v - v'))"
gfp_def: 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> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.gfp Sup less_eq f = Sup {u. less_eq u (f u)}"
vars_extract_var_subset: fixes p :: "'a mpoly"   and v :: "nat" shows "vars (extract_var p v) \<subseteq> vars p"
1: shows "Broadcast (i1, Insert e1 n1) \<in> set (history j)"
sem2: fixes I :: "('sf, 'sc, 'sz) interp" shows "prog_sem I (EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) (OSing vid2 (Plus (Times (f1 fid2 vid1) (Var vid2)) (f1 fid3 vid1)))) (p1 vid1 vid1)) = {(\<nu>, mk_v I (OProd (OSing vid1 (f1 fid1 vid1)) (OSing vid2 (Plus (Times (f1 fid2 vid1) (Var vid2)) (f1 fid3 vid1)))) \<nu> (sol t)) |\<nu> sol t. 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OProd (OSing vid1 (f1 fid1 vid1)) (OSing vid2 (Plus (Times (f1 fid2 vid1) (Var vid2)) (f1 fid3 vid1)))))) {0..t} {x. mk_v I (OProd (OSing vid1 (f1 fid1 vid1)) (OSing vid2 (Plus (Times (f1 fid2 vid1) (Var vid2)) (f1 fid3 vid1)))) \<nu> x \<in> fml_sem I (p1 vid1 vid1)} \<and> VSagree (sol 0) (fst \<nu>) {x |x. Inl x \<in> FVP (EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) (OSing vid2 (Plus (Times (f1 fid2 vid1) (Var vid2)) (f1 fid3 vid1)))) (p1 vid1 vid1))}}"
it_dfs_def: shows "it_dfs \<equiv> init \<bind> WHILE cond step"
rot: fixes \<alpha> :: "real" shows "P (moebius_rotation \<alpha>)"
indmhom_surjec1: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "surjec\<^bsub>M,N\<^esub> f"   and "f \<in> mHom R M N" shows "surjec\<^bsub>M /\<^sub>m (ker\<^bsub>M,N\<^esub> f),N\<^esub> f\<^sup>\<flat>\<^bsub>R M, N\<^esub>"
noneg: shows "\<not> has_disc_negated is_Dst_Ports False m"
True_True_steps_starI: fixes p :: "bool list"   and q :: "bool list"   and A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" assumes "(p, q) \<in> NAe.steps A w" shows "(True # p, True # q) \<in> NAe.steps (RegExp2NAe.star A) w"
xs'_in: shows "(map real_of_float xs)[0 := x] all_in I"
hom_induced_default: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and p :: "int" assumes "\<not> (continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> c \<in> carrier (relative_homology_group p X S))" shows "hom_induced p X S Y T f c = \<one>\<^bsub>relative_homology_group p Y T\<^esub>"
language_def: shows "N.language \<equiv> {xs. N.nextl (init N) xs \<in> final N}"
card_product_dependent: fixes S :: "'a set"   and T :: "'a \<Rightarrow> 'b set" assumes "finite S"   and "\<forall>x\<in>S. finite (T x)" shows "card {(x, y). x \<in> S \<and> y \<in> T x} = (\<Sum>x\<in>S. card (T x))"
pset_pnTimes_Times: fixes s :: "'a rexp"   and r :: "'a rexp"   and t :: "'a rexp" assumes "s \<in> pset r" shows "pnTimes s t = Times s t"
composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> I"   and "b \<in> I" shows "a + b \<in> I"
reflect_poly_at_0_eq_0_iff: fixes p :: "'b poly" shows "(poly (reflect_poly p) (0::'b) = (0::'b)) = (p = 0)"
A: shows "set (fst (snd x) # fst x) \<subseteq> dgrad_p_set d m"
N_instance_M: shows "has_all_wt_instances_of \<Gamma> N M"
"2_1": shows "u = u\<^sub>1 @ [a] @ u\<^sub>2"
vsubset_transitive: fixes A :: "V"   and B :: "V"   and C :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "B \<subseteq>\<^sub>\<circ> C" shows "A \<subseteq>\<^sub>\<circ> C"
simps_7: fixes x8 :: "'d \<Rightarrow> ('d, 'e, 'f) com"   and y8 :: "'d \<Rightarrow> ('d, 'e, 'f) com" shows "(DynCom x8 = DynCom y8) = (x8 = y8)"
NEW: fixes x :: "Var"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and C :: "Class" assumes "CONTEXT x = low" shows "G \<rhd> New x C : Sec (\<lambda>(s, t, \<beta>). \<exists>l r. l \<notin> Dom (snd r) \<and> r \<equiv>\<^sub>\<beta> t \<and> s = (update (fst r) x (RVal (Loc l)), (l, C, []) # snd r))"
pell_mul_cnj_right: fixes z :: "'a \<times> 'a" assumes "solution z" shows "pell_mul z (pell_cnj z) = (1::'a, 0::'a)"
ab_def: shows "ab = [a, b]\<^sub>\<circ>"
measurable_count_space: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" shows "f \<in> count_space A \<rightarrow>\<^sub>M count_space UNIV"
ex_un_null: shows "\<exists>!n. \<forall>f. C.VxVxV.comp n f = n \<and> C.VxVxV.comp f n = n"
balE: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and stack :: "'b \<Rightarrow> 'c list"   and c :: "'b"   and T :: "'b list"   and c' :: "'b" assumes "balance_trace step stack"   and "balance_trace.bal step stack c T c'" obtains "traces.trace step c T c'" and "\<forall>c'\<in>set T. stack c \<lesssim> stack c'" and "stack c' = stack c"
coeff_Poly: fixes xs :: "'a list"   and i :: "nat" shows "coeff (Poly xs) i = nth_default (0::'a) xs i"
lrev_body_final_lrev_body3_eq': fixes r :: "'b llist \<rightarrow> 'b llist \<rightarrow> 'b llist"   and xs :: "'b llist" shows "lrev_body_final\<cdot>r\<cdot>xs = lrev_body3\<cdot>r\<cdot>xs"
simps_13: fixes v :: "variable"   and va :: "typ" shows "eta_reducible (Fv v va) = False"
sg_lemma: fixes x :: "nat" shows "rec_exec rec_sg [x] = (if x = 0 then 0 else 1)"
cyclic_group_commute: fixes G :: "('a, 'b) cyclic_group_scheme"   and a :: "'a"   and b :: "'a" assumes "Cyclic_Group.cyclic_group G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<otimes>\<^bsub>G\<^esub> b = b \<otimes>\<^bsub>G\<^esub> a"
token_stays_in_final_states: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and x :: "nat"   and n :: "nat"   and m :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n \<in> F" shows "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (n + m) \<in> F"
f_shrink_last_length: fixes xs :: "'d list"   and k :: "nat" shows "length (xs \<div>\<^sub>l k) = length xs div k"
induct: fixes P :: "'b formula list \<Rightarrow> 'b list \<Rightarrow> 'b formula list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b formula list"   and a1 :: "'b list"   and a2 :: "'b formula list"   and a3 :: "'b list" assumes "\<And>\<Gamma> A \<Delta> B. P (\<bottom> # \<Gamma>) A \<Delta> B"   and "\<And>A B. P [] A [] B"   and "\<And>k \<Gamma> A \<Delta> B. P \<Gamma> (k # A) \<Delta> B \<Longrightarrow> P (Atom k # \<Gamma>) A \<Delta> B"   and "\<And>F \<Gamma> A \<Delta> B. P \<Gamma> A (F # \<Delta>) B \<Longrightarrow> P (\<^bold>\<not> F # \<Gamma>) A \<Delta> B"   and "\<And>F G \<Gamma> A \<Delta> B. P (F # G # \<Gamma>) A \<Delta> B \<Longrightarrow> P (F \<^bold>\<and> G # \<Gamma>) A \<Delta> B"   and "\<And>F G \<Gamma> A \<Delta> B. \<lbrakk>P (F # \<Gamma>) A \<Delta> B; P (G # \<Gamma>) A \<Delta> B\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G # \<Gamma>) A \<Delta> B"   and "\<And>F G \<Gamma> A \<Delta> B. \<lbrakk>P \<Gamma> A (F # \<Delta>) B; P (G # \<Gamma>) A \<Delta> B\<rbrakk> \<Longrightarrow> P (F \<^bold>\<rightarrow> G # \<Gamma>) A \<Delta> B"   and "\<And>A \<Delta> B. P [] A \<Delta> B \<Longrightarrow> P [] A (\<bottom> # \<Delta>) B"   and "\<And>A k \<Delta> B. P [] A \<Delta> (k # B) \<Longrightarrow> P [] A (Atom k # \<Delta>) B"   and "\<And>A F \<Delta> B. P [F] A \<Delta> B \<Longrightarrow> P [] A (\<^bold>\<not> F # \<Delta>) B"   and "\<And>A F G \<Delta> B. \<lbrakk>P [] A (F # \<Delta>) B; P [] A (G # \<Delta>) B\<rbrakk> \<Longrightarrow> P [] A (F \<^bold>\<and> G # \<Delta>) B"   and "\<And>A F G \<Delta> B. P [] A (F # G # \<Delta>) B \<Longrightarrow> P [] A (F \<^bold>\<or> G # \<Delta>) B"   and "\<And>A F G \<Delta> B. P [F] A (G # \<Delta>) B \<Longrightarrow> P [] A (F \<^bold>\<rightarrow> G # \<Delta>) B" shows "P a0 a1 a2 a3"
bv_leqs_refl: fixes xs :: "bool list" shows "xs \<preceq>\<^sub>b xs"
fold_insert2: fixes A :: "'d set"   and x :: "'d"   and z :: "'e" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold g z (insert x A) = Finite_Set.fold g (g x z) A"
Suc_mask_eq_exp: fixes n :: "nat" shows "Suc (mask n) = 2 ^ n"
rel_conversep: fixes R :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option R\<inverse>\<inverse> = (rel_option R)\<inverse>\<inverse>"
val_Ig: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I" shows "Ig K v I \<in> I \<and> v (Ig K v I) = LI K v I"
2: shows "x \<le> - z \<and> phi z \<le> y \<longrightarrow> y \<squnion> w = Filters.filter.Abs_filter UNIV"
to_conjugate_space_inject: fixes x :: "'d"   and y :: "'d" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(to_conjugate_space x = to_conjugate_space y) = (x = y)"
dom_char: fixes f :: "'d" shows "D'.dom f = (if D'.arr f then D.dom f else D.null)"
dgrad_set_leI: fixes S :: "'a set"   and T :: "'a set"   and d :: "'a \<Rightarrow> nat" assumes "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. d s \<le> d t" shows "dgrad_set_le d S T"
mkChannels_gState_progress_rel: fixes prog :: "program"   and g :: "'a gState_scheme"   and cs :: "channel list"   and p :: "pState" assumes "gState_inv prog g"   and "set cs \<subseteq> Collect channel_inv" shows "(g, fst (mkChannels g p cs)) \<in> gState_progress_rel prog"
is_cube_form_def: fixes a :: "int"   and b :: "int" shows "is_cube_form a b = (\<exists>p q. a = p ^ 3 - 9 * p * q\<^sup>2 \<and> b = 3 * p\<^sup>2 * q - 3 * q ^ 3)"
assms: shows "pre_digraph.gen_iapath (with_proj slim) (verts3 (with_proj G)) u p v"
except_zero: fixes S :: "'c set" shows "except 0 S = 0"
2: shows "min_height (Node k v l r) = nat \<lfloor>log 2 (real (KD_Tree.size_kdt r))\<rfloor> + 1"
joinable_eq_rtscl: fixes R :: "('a \<times> 'a) set" assumes "Joinable.confluent (R\<^sup>*)" shows "joinable (R\<^sup>*) = (R \<union> R\<inverse>)\<^sup>*"
execs_drop_suffix0: fixes f :: "nat \<Rightarrow> com list \<times> state"   and c :: "com"   and cs :: "com list"   and s :: "state"   and k :: "nat"   and p :: "nat \<Rightarrow> com list" assumes "f 0 = (c # cs, s)"   and "\<forall>i. f i \<rightarrow> f (Suc i)"   and "\<forall>i<k. p i \<noteq> [] \<and> fst (f i) = p i @ cs"   and "fst (f k) = cs"   and "p k = []" shows "([c], s) \<rightarrow>\<^sup>* ([], snd (f k))"
gamma4_as_euclid_space_fun: shows "gamma4 = (\<lambda>x. (a + (b - a) * x) *\<^sub>R j + (g2 (a + (b - a) * x), 0))"
le_lt_comparator_of_2: shows "lt_of_comp comparator_of = (<)"
R01iai_def: shows "R01iai \<equiv> {(s, s'). signals s = signalsInit s'}"
arcs_del_arc: fixes a :: "'b" shows "arcs (del_arc a) = arcs G - {a}"
divmod_digit_0_1: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and b :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step"   and "less zero b"   and "less (modulo a (times (numeral.numeral one plus (num.Bit0 num.One)) b)) b" shows "times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (times (numeral.numeral one plus (num.Bit0 num.One)) b)) = divide a b"
set_remdups: fixes xs :: "'c list" shows "set (remdups xs) = set xs"
linear_span_iOp_closed: fixes A :: "'b set"   and H :: "'a set"   and a :: "'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "a \<in> linear_span R M A H" shows "-\<^sub>a a \<in> linear_span R M A H"
shrK_keys_of_2: fixes B :: "agent"   and A :: "agent" shows "shrK B A \<in> keys_of A"
subd_uminus: fixes p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" shows "subd p (- c) = - subd p c"
ide_hcomp: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "ide \<nu>"   and "ide \<mu>"   and "src \<nu> = trg \<mu>" shows "ide (\<nu> \<star> \<mu>)"
prems: shows "t \<rightarrow>\<^sub>\<beta> r''"
pdom_inter: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" shows "pdom (f \<inter>\<^sub>p g) \<subseteq> pdom f \<inter> pdom g"
signed_def: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus \<equiv> map_fun Word.rep id (ring_1.of_int one plus zero minus \<circ> signed_take_bit (LENGTH('l) - Suc 0))"
stable_imp_awp_ident: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> stable A" shows "awp F A = A"
select_convs_1: fixes lens_get :: "'a \<Rightarrow> 'b"   and lens_put :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and more :: "'c" shows "get\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_get"
exec_move_FAssI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and F :: "String.literal"   and D :: "String.literal"   and e' :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e\<bullet>F{D} := e') h s ta h' s'"
emp_in: fixes G :: "'b list set" shows "\<epsilon> \<in> \<langle>G\<rangle>"
subset: shows "supp (\<lambda>n. f (n + r)) \<subseteq> (\<lambda>n. n + r) -` supp f"
cases: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and a :: "('h, 'i, 'j, 'k) converter" assumes "colossless_converter \<I> \<I>' a" obtains conv :: "('h, 'i, 'j, 'k) converter" where "a = conv" and "\<And>a. a \<in> outs_\<I> \<I> \<Longrightarrow> colossless_gpv \<I>' (run_converter conv a)" and "\<And>a b conv'. \<lbrakk>a \<in> outs_\<I> \<I>; (b, conv') \<in> results_gpv \<I>' (run_converter conv a)\<rbrakk> \<Longrightarrow> colossless_converter \<I> \<I>' conv'"
nyinitcls_subset_class: fixes G :: "prog"   and s :: "state" shows "nyinitcls G s \<subseteq> {C. is_class G C}"
pref_mod_power': fixes u :: "'b list"   and w :: "'b list"   and l :: "nat" assumes "u \<le>np w \<^sup>@ l" obtains k :: "nat" and z :: "'b list" where "k < l" and "z \<le>np w" and "w \<^sup>@ k \<cdot> z = u"
List_rec_NIL: fixes c :: "'b"   and h :: "('c, unit) Old_Datatype.node set \<Rightarrow> ('c, unit) Old_Datatype.node set \<Rightarrow> 'b \<Rightarrow> 'b" shows "List_rec NIL c h = c"
rep_fmap_base: fixes P :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool"   and F :: "'c \<Rightarrow> 'd option" shows "P F = pred_set_fmap P (set_fmap F)"
f_def: shows "f \<equiv> \<lambda>u. (- ln u) powr z / (1 - u) * (- ln u + u - 1)"
add_rng_of_frac_def: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" shows "add_rng_of_frac X Y \<equiv> let x' = SOME x. x \<in> X; y' = SOME y. y \<in> Y in snd y' \<otimes> fst x' \<oplus> snd x' \<otimes> fst y' |\<^bsub>rel\<^esub> snd x' \<otimes> snd y'"
sparse_row_vector_addmult_spvec: fixes y :: "'a"   and a :: "(nat \<times> 'a) list"   and b :: "(nat \<times> 'a) list" shows "sparse_row_vector (addmult_spvec y a b) = sparse_row_vector a + scalar_mult y (sparse_row_vector b)"
pls_cong_ax_def: shows "pls_cong_ax = Var X1 EQ Var X2 AND Var X3 EQ Var X4 IMP pls (Var X1) (Var X3) EQ pls (Var X2) (Var X4)"
get_ancestors_same_root_node: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and ptr'' :: "(_) object_ptr"   and root_ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors"   and "ptr' \<in> set ancestors"   and "ptr'' \<in> set ancestors" shows "h \<turnstile> Shadow_DOM.get_root_node ptr' \<rightarrow>\<^sub>r root_ptr = h \<turnstile> Shadow_DOM.get_root_node ptr'' \<rightarrow>\<^sub>r root_ptr"
sin_zero_lemma: fixes x :: "real" assumes "0 \<le> x"   and "sin x = 0" shows "\<exists>n. even n \<and> x = real n * (pi / 2)"
intros_1: shows "(\<zero>, {}) \<in> sumComposeSet"
zhmset_of_le: fixes M :: "hmultiset"   and N :: "hmultiset" shows "(zhmset_of M \<le> zhmset_of N) = (M \<le> N)"
finite_Fvars_fmla: fixes A :: "fmla" shows "finite (Fvars A)"
zero_column_k: shows "\<forall>m\<ge>mod_type_class.from_nat 0. A $ m $ mod_type_class.from_nat k = (0::'a)"
parent_child_rel_finite: fixes h :: "(_) heap" shows "finite (Shadow_DOM.parent_child_rel h)"
power_inductl: fixes z :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set"   and n :: "nat" assumes "z \<union> x O y \<subseteq> y" shows "rel_dioid.power x n O z \<subseteq> y"
w_eqI: fixes a :: "'a \<Rightarrow> real"   and b :: "'a \<Rightarrow> real" assumes "a \<in> diff_fun_space"   and "b \<in> diff_fun_space"   and "\<And>x. x \<in> B \<Longrightarrow> a x = b x" shows "w a = w b"
edom_esing_up: fixes x :: "'b"   and n :: "'c" shows "edom (esing x\<cdot>(up\<cdot>n)) = {x}"
exec_fdom_empty_prob: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" assumes "prob_dom PROB = {}"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "exec_plan s as = fmempty"
sel_2: fixes x2 :: "'d" shows "projr (Inr x2) = x2"
increasing_tendsto: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in F. less_eq (f n) l"   and "\<And>x. less x l \<Longrightarrow> \<forall>\<^sub>F n in F. less x (f n)" shows "filterlim f (topological_space.nhds open l) F"
n_eq_ladder_n: shows "n = ladder_n L (length L - 1)"
termMOD_iwlsFSw: shows "iwlsFSw termMOD"
UNIV_eq_I: fixes A :: "'c set" assumes "\<And>x. x \<in> A" shows "UNIV = A"
bind_refine': fixes M :: "'b nres"   and R' :: "('b \<times> 'c) set"   and M' :: "'c nres"   and f :: "'b \<Rightarrow> 'd nres"   and R :: "('d \<times> 'e) set"   and f' :: "'c \<Rightarrow> 'e nres" assumes "M \<le> \<Down> R' M'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; inres M x; inres M' x'; nofail M; nofail M'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "M \<bind> f \<le> \<Down> R (M' \<bind> f')"
length_filter_conv_size_filter_mset: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "length (filter P xs) = size (filter_mset P (mset xs))"
a1: shows "infsetsum f A = Complex (Re (infsetsum f A)) (Im (infsetsum f A))"
gen_rwof'_imp_rwof: fixes s :: "'s" assumes "nofail gen_dfs"   and "gen_rwof' s" shows "gen_rwof s"
rel_restrict_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b set" shows "rel_restrict R A \<equiv> {(v, w). (v, w) \<in> R \<and> v \<notin> A \<and> w \<notin> A}"
IDirProd_empty_right: fixes A :: "'a set" assumes "A \<subseteq> carrier G" shows "IDirProd G A {} = generate G A"
cases: fixes x :: "('a \<times> 'b) bound" obtains l0 :: "'a" and r0 :: "'b" and l1 :: "'a" and r1 :: "'b" where "x = Bound (l0, r0) (l1, r1)"